%------------------------------------------------------------------------------
% File     : ITP213_3 : TPTP v8.2.0. Released v8.0.0.
% Domain   : Interactive Theorem Proving
% Problem  : Sledgehammer problem Hoare_Triple 00154_004690
% 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    : 0025_Hoare_Triple_00154_004690 [Des22]

% Status   : Theorem
% Rating   : 0.00 v8.2.0, 0.67 v8.1.0
% Syntax   : Number of formulae    : 14832 (3861 unt;4056 typ;   0 def)
%            Number of atoms       : 26140 (8178 equ)
%            Maximal formula atoms :   38 (   2 avg)
%            Number of connectives : 16860 (2071   ~; 428   |;1407   &)
%                                         (2741 <=>;10213  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   21 (   5 avg)
%            Maximal term depth    :   36 (   2 avg)
%            Number of FOOLs       : 1974 ( 593 fml;1381 var)
%            Number of X terms     :  499 (   0  []; 466 ite;  33 let)
%            Number of types       :  931 ( 930 usr)
%            Number of type conns  : 4046 (2587   >;1459   *;   0   +;   0  <<)
%            Number of predicates  :  217 ( 214 usr;   3 prp; 0-4 aty)
%            Number of functors    : 2922 (2922 usr; 548 con; 0-4 aty)
%            Number of variables   : 28628 (28086   !; 542   ?;28628   :)
% 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 15:20:39.129
%------------------------------------------------------------------------------
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tff(ty_n_t__fun_It__Rat__Orat_Mt__Num__Onum_J,type,
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tff(ty_n_t__fun_It__Num__Onum_Mt__Rat__Orat_J,type,
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tff(ty_n_t__fun_It__Num__Onum_Mt__Num__Onum_J,type,
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tff(ty_n_t__fun_It__Num__Onum_Mt__Nat__Onat_J,type,
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tff(ty_n_t__fun_It__Num__Onum_Mt__Int__Oint_J,type,
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tff(ty_n_t__fun_It__Nat__Onat_Mt__Rat__Orat_J,type,
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tff(ty_n_t__fun_It__Nat__Onat_Mt__Num__Onum_J,type,
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tff(ty_n_t__fun_It__Nat__Onat_Mt__Nat__Onat_J,type,
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tff(ty_n_t__fun_It__Nat__Onat_Mt__Int__Oint_J,type,
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tff(ty_n_t__fun_It__List__Olist_I_Eo_J_M_Eo_J,type,
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tff(ty_n_t__fun_It__Int__Oint_Mt__Rat__Orat_J,type,
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    aTP_Lamm_aat: fun_Code_integer_rat > fun_se8983711159220854466er_rat ).

tff(sy_c_ATP_058Lamm__aau____,type,
    aTP_Lamm_aau: fun_o_nat > fun_set_o_fun_o_nat ).

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

tff(sy_c_ATP_058Lamm__aaw____,type,
    aTP_Lamm_aaw: fun_Co8499238673658658103nteger > fun_Code_integer_o ).

tff(sy_c_ATP_058Lamm__aax____,type,
    aTP_Lamm_aax: fun_int_assn > fun_int_o ).

tff(sy_c_ATP_058Lamm__aay____,type,
    aTP_Lamm_aay: fun_Co4490034186337852034r_assn > fun_Code_integer_o ).

tff(sy_c_ATP_058Lamm__aaz____,type,
    aTP_Lamm_aaz: fun_int_rat > fun_int_o ).

tff(sy_c_ATP_058Lamm__ab____,type,
    aTP_Lamm_ab: set_Pr958786334691620121nt_int > fun_Pr3238374608701176162_int_o ).

tff(sy_c_ATP_058Lamm__aba____,type,
    aTP_Lamm_aba: fun_Code_integer_rat > fun_Code_integer_o ).

tff(sy_c_ATP_058Lamm__abb____,type,
    aTP_Lamm_abb: fun_int_nat > fun_int_o ).

tff(sy_c_ATP_058Lamm__abc____,type,
    aTP_Lamm_abc: fun_Code_integer_nat > fun_Code_integer_o ).

tff(sy_c_ATP_058Lamm__abd____,type,
    aTP_Lamm_abd: fun_Code_integer_int > fun_Code_integer_o ).

tff(sy_c_ATP_058Lamm__abe____,type,
    aTP_Lamm_abe: fun_nat_Code_integer > fun_nat_o ).

tff(sy_c_ATP_058Lamm__abf____,type,
    aTP_Lamm_abf: fun_rat_fun_rat_o ).

tff(sy_c_ATP_058Lamm__abg____,type,
    aTP_Lamm_abg: nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__abh____,type,
    aTP_Lamm_abh: rat > fun_rat_fun_nat_rat ).

tff(sy_c_ATP_058Lamm__abi____,type,
    aTP_Lamm_abi: code_natural > fun_Co740390889742897050atural ).

tff(sy_c_ATP_058Lamm__abj____,type,
    aTP_Lamm_abj: int > fun_int_fun_nat_int ).

tff(sy_c_ATP_058Lamm__abk____,type,
    aTP_Lamm_abk: code_integer > fun_Co6226381647958292740nteger ).

tff(sy_c_ATP_058Lamm__abl____,type,
    aTP_Lamm_abl: nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__abm____,type,
    aTP_Lamm_abm: nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__abn____,type,
    aTP_Lamm_abn: nat > fun_na3244735409550470327at_nat ).

tff(sy_c_ATP_058Lamm__abo____,type,
    aTP_Lamm_abo: fun_nat_nat > fun_na3244735409550470327at_nat ).

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

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

tff(sy_c_ATP_058Lamm__abr____,type,
    aTP_Lamm_abr: fun_nat_int > fun_na7654559027617327379at_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_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_nat_fun_nat_int > fun_nat_fun_nat_int ).

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

tff(sy_c_ATP_058Lamm__abx____,type,
    aTP_Lamm_abx: fun_int_o > fun_fu1617438229336688381nt_rat ).

tff(sy_c_ATP_058Lamm__aby____,type,
    aTP_Lamm_aby: fun_Code_integer_o > fun_fu8767785434844327952er_rat ).

tff(sy_c_ATP_058Lamm__abz____,type,
    aTP_Lamm_abz: fun_Code_integer_o > fun_fu2498091903632033272er_nat ).

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

tff(sy_c_ATP_058Lamm__aca____,type,
    aTP_Lamm_aca: fun_Code_integer_o > fun_fu1007793439646807308er_int ).

tff(sy_c_ATP_058Lamm__acb____,type,
    aTP_Lamm_acb: fun_nat_o > fun_fu6579618355643354601at_rat ).

tff(sy_c_ATP_058Lamm__acc____,type,
    aTP_Lamm_acc: fun_list_nat_o > fun_fu1597678101255499801at_rat ).

tff(sy_c_ATP_058Lamm__acd____,type,
    aTP_Lamm_acd: fun_int_o > fun_fu3890538715107687308nteger ).

tff(sy_c_ATP_058Lamm__ace____,type,
    aTP_Lamm_ace: fun_Code_integer_o > fun_fu2790023284655001759nteger ).

tff(sy_c_ATP_058Lamm__acf____,type,
    aTP_Lamm_acf: fun_nat_o > fun_fu6327154928193479288nteger ).

tff(sy_c_ATP_058Lamm__acg____,type,
    aTP_Lamm_acg: fun_int_o > fun_fu5832215286117076717t_assn ).

tff(sy_c_ATP_058Lamm__ach____,type,
    aTP_Lamm_ach: fun_Code_integer_o > fun_fu4755565544374692864r_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: nat > fun_na3244735409550470327at_nat ).

tff(sy_c_ATP_058Lamm__ack____,type,
    aTP_Lamm_ack: rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__acl____,type,
    aTP_Lamm_acl: rat > fun_ra5513060666784058567at_rat ).

tff(sy_c_ATP_058Lamm__acm____,type,
    aTP_Lamm_acm: int > fun_in6793278964266839919at_int ).

tff(sy_c_ATP_058Lamm__acn____,type,
    aTP_Lamm_acn: code_integer > fun_Co5080488584548667985nteger ).

tff(sy_c_ATP_058Lamm__aco____,type,
    aTP_Lamm_aco: rat > fun_ra5513060666784058567at_rat ).

tff(sy_c_ATP_058Lamm__acp____,type,
    aTP_Lamm_acp: int > fun_in6793278964266839919at_int ).

tff(sy_c_ATP_058Lamm__acq____,type,
    aTP_Lamm_acq: code_integer > fun_Co5080488584548667985nteger ).

tff(sy_c_ATP_058Lamm__acr____,type,
    aTP_Lamm_acr: rat > fun_nat_fun_nat_rat ).

tff(sy_c_ATP_058Lamm__acs____,type,
    aTP_Lamm_acs: fun_Code_integer_o > fun_fu8094397175230428611eger_o ).

tff(sy_c_ATP_058Lamm__act____,type,
    aTP_Lamm_act: fun_Code_integer_o > fun_fu8094397175230428611eger_o ).

tff(sy_c_ATP_058Lamm__acu____,type,
    aTP_Lamm_acu: fun_Pr4796943543331071514_nat_o > fun_fu3275390434952296369_nat_o ).

tff(sy_c_ATP_058Lamm__acv____,type,
    aTP_Lamm_acv: int > fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__acw____,type,
    aTP_Lamm_acw: int > fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__acx____,type,
    aTP_Lamm_acx: int > fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__acy____,type,
    aTP_Lamm_acy: int > fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__acz____,type,
    aTP_Lamm_acz: set_Code_integer > fun_se7289222521248229150eger_o ).

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

tff(sy_c_ATP_058Lamm__ada____,type,
    aTP_Lamm_ada: set_o > fun_fu5257111247350578329un_o_o ).

tff(sy_c_ATP_058Lamm__adb____,type,
    aTP_Lamm_adb: set_int > fun_fu138107234227827011_int_o ).

tff(sy_c_ATP_058Lamm__adc____,type,
    aTP_Lamm_adc: set_int > fun_fu7864316720324808067_int_o ).

tff(sy_c_ATP_058Lamm__add____,type,
    aTP_Lamm_add: set_o > fun_fu7138923643431704310un_o_o ).

tff(sy_c_ATP_058Lamm__ade____,type,
    aTP_Lamm_ade: set_o > fun_fu1426755399112167834un_o_o ).

tff(sy_c_ATP_058Lamm__adf____,type,
    aTP_Lamm_adf: set_o > fun_fu7893852151532352809un_o_o ).

tff(sy_c_ATP_058Lamm__adg____,type,
    aTP_Lamm_adg: set_nat > fun_fu7243318134412071974_nat_o ).

tff(sy_c_ATP_058Lamm__adh____,type,
    aTP_Lamm_adh: set_nat > fun_fu4988404976222497994_nat_o ).

tff(sy_c_ATP_058Lamm__adi____,type,
    aTP_Lamm_adi: set_nat > fun_fu1730513324920958169_nat_o ).

tff(sy_c_ATP_058Lamm__adj____,type,
    aTP_Lamm_adj: set_int > fun_fu4838637399144961554_int_o ).

tff(sy_c_ATP_058Lamm__adk____,type,
    aTP_Lamm_adk: set_int > fun_fu2583724240955387574_int_o ).

tff(sy_c_ATP_058Lamm__adl____,type,
    aTP_Lamm_adl: set_int > fun_fu1985264055766458053_int_o ).

tff(sy_c_ATP_058Lamm__adm____,type,
    aTP_Lamm_adm: set_Code_integer > fun_fu8693872822272901183eger_o ).

tff(sy_c_ATP_058Lamm__adn____,type,
    aTP_Lamm_adn: nat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__ado____,type,
    aTP_Lamm_ado: fun_nat_fun_nat_int ).

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

tff(sy_c_ATP_058Lamm__adq____,type,
    aTP_Lamm_adq: nat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__adr____,type,
    aTP_Lamm_adr: fun_nat_fun_nat_int ).

tff(sy_c_ATP_058Lamm__ads____,type,
    aTP_Lamm_ads: nat > fun_nat_Code_integer ).

tff(sy_c_ATP_058Lamm__adt____,type,
    aTP_Lamm_adt: fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__adu____,type,
    aTP_Lamm_adu: fun_Co2081214437782444048eger_o ).

tff(sy_c_ATP_058Lamm__adv____,type,
    aTP_Lamm_adv: fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__adw____,type,
    aTP_Lamm_adw: fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__adx____,type,
    aTP_Lamm_adx: fun_int_nat > fun_nat_fun_int_nat ).

tff(sy_c_ATP_058Lamm__ady____,type,
    aTP_Lamm_ady: fun_int_nat > fun_nat_fun_int_o ).

tff(sy_c_ATP_058Lamm__adz____,type,
    aTP_Lamm_adz: fun_int_nat > fun_nat_fun_int_o ).

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

tff(sy_c_ATP_058Lamm__aea____,type,
    aTP_Lamm_aea: fun_Code_integer_nat > fun_na5492297256294505719er_nat ).

tff(sy_c_ATP_058Lamm__aeb____,type,
    aTP_Lamm_aeb: fun_Code_integer_nat > fun_na6318815019539281885eger_o ).

tff(sy_c_ATP_058Lamm__aec____,type,
    aTP_Lamm_aec: fun_Code_integer_nat > fun_na6318815019539281885eger_o ).

tff(sy_c_ATP_058Lamm__aed____,type,
    aTP_Lamm_aed: fun_Code_integer_int > fun_in2718163036992669103er_int ).

tff(sy_c_ATP_058Lamm__aee____,type,
    aTP_Lamm_aee: fun_Code_integer_int > fun_in4891666191172872833eger_o ).

tff(sy_c_ATP_058Lamm__aef____,type,
    aTP_Lamm_aef: fun_Code_integer_int > fun_in4891666191172872833eger_o ).

tff(sy_c_ATP_058Lamm__aeg____,type,
    aTP_Lamm_aeg: fun_nat_int > fun_int_fun_nat_int ).

tff(sy_c_ATP_058Lamm__aeh____,type,
    aTP_Lamm_aeh: fun_nat_int > fun_int_fun_nat_o ).

tff(sy_c_ATP_058Lamm__aei____,type,
    aTP_Lamm_aei: fun_nat_int > fun_int_fun_nat_o ).

tff(sy_c_ATP_058Lamm__aej____,type,
    aTP_Lamm_aej: fun_int_Code_integer > fun_Co3011819262099091808nteger ).

tff(sy_c_ATP_058Lamm__aek____,type,
    aTP_Lamm_aek: fun_int_Code_integer > fun_Co8353275306710069249_int_o ).

tff(sy_c_ATP_058Lamm__ael____,type,
    aTP_Lamm_ael: fun_int_Code_integer > fun_Co8353275306710069249_int_o ).

tff(sy_c_ATP_058Lamm__aem____,type,
    aTP_Lamm_aem: fun_Co8499238673658658103nteger > fun_Co1981900195452921617nteger ).

tff(sy_c_ATP_058Lamm__aen____,type,
    aTP_Lamm_aen: fun_Co8499238673658658103nteger > fun_Co2081214437782444048eger_o ).

tff(sy_c_ATP_058Lamm__aeo____,type,
    aTP_Lamm_aeo: fun_Co8499238673658658103nteger > fun_Co2081214437782444048eger_o ).

tff(sy_c_ATP_058Lamm__aep____,type,
    aTP_Lamm_aep: fun_nat_Code_integer > fun_Co6226381647958292740nteger ).

tff(sy_c_ATP_058Lamm__aeq____,type,
    aTP_Lamm_aeq: fun_nat_Code_integer > fun_Co1375048237525739357_nat_o ).

tff(sy_c_ATP_058Lamm__aer____,type,
    aTP_Lamm_aer: fun_nat_Code_integer > fun_Co1375048237525739357_nat_o ).

tff(sy_c_ATP_058Lamm__aes____,type,
    aTP_Lamm_aes: fun_int_Code_natural > fun_Co6749200540738471926atural ).

tff(sy_c_ATP_058Lamm__aet____,type,
    aTP_Lamm_aet: fun_int_Code_natural > fun_Co7174943067365291894_int_o ).

tff(sy_c_ATP_058Lamm__aeu____,type,
    aTP_Lamm_aeu: fun_int_Code_natural > fun_Co7174943067365291894_int_o ).

tff(sy_c_ATP_058Lamm__aev____,type,
    aTP_Lamm_aev: fun_Co3734316457315071362atural > fun_Co8514937155073724839atural ).

tff(sy_c_ATP_058Lamm__aew____,type,
    aTP_Lamm_aew: fun_Co3734316457315071362atural > fun_Co2157892728433036165eger_o ).

tff(sy_c_ATP_058Lamm__aex____,type,
    aTP_Lamm_aex: fun_Co3734316457315071362atural > fun_Co2157892728433036165eger_o ).

tff(sy_c_ATP_058Lamm__aey____,type,
    aTP_Lamm_aey: fun_nat_Code_natural > fun_Co740390889742897050atural ).

tff(sy_c_ATP_058Lamm__aez____,type,
    aTP_Lamm_aez: fun_nat_Code_natural > fun_Co196715998180962002_nat_o ).

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

tff(sy_c_ATP_058Lamm__afa____,type,
    aTP_Lamm_afa: fun_nat_Code_natural > fun_Co196715998180962002_nat_o ).

tff(sy_c_ATP_058Lamm__afb____,type,
    aTP_Lamm_afb: fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__afc____,type,
    aTP_Lamm_afc: num > fun_na4138050610385705875at_nat ).

tff(sy_c_ATP_058Lamm__afd____,type,
    aTP_Lamm_afd: num > fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__afe____,type,
    aTP_Lamm_afe: num > fun_Co2470855966376689427nteger ).

tff(sy_c_ATP_058Lamm__aff____,type,
    aTP_Lamm_aff: fun_int_Code_integer > fun_fu6361890901329191134nteger ).

tff(sy_c_ATP_058Lamm__afg____,type,
    aTP_Lamm_afg: fun_Co8499238673658658103nteger > fun_fu2729256655946442782nteger ).

tff(sy_c_ATP_058Lamm__afh____,type,
    aTP_Lamm_afh: fun_nat_Code_integer > fun_fu7898940055406642654nteger ).

tff(sy_c_ATP_058Lamm__afi____,type,
    aTP_Lamm_afi: fun_int_rat > fun_fu3922030663826099033nt_rat ).

tff(sy_c_ATP_058Lamm__afj____,type,
    aTP_Lamm_afj: fun_Code_integer_rat > fun_fu991479512444375705er_rat ).

tff(sy_c_ATP_058Lamm__afk____,type,
    aTP_Lamm_afk: fun_nat_rat > fun_fu3162257917499486297at_rat ).

tff(sy_c_ATP_058Lamm__afl____,type,
    aTP_Lamm_afl: fun_int_nat > fun_fu1078644964449000785nt_nat ).

tff(sy_c_ATP_058Lamm__afm____,type,
    aTP_Lamm_afm: fun_Code_integer_nat > fun_fu7371465849922053265er_nat ).

tff(sy_c_ATP_058Lamm__afn____,type,
    aTP_Lamm_afn: fun_Code_integer_int > fun_fu4881382704990326509er_int ).

tff(sy_c_ATP_058Lamm__afo____,type,
    aTP_Lamm_afo: fun_nat_int > fun_fu7052161110045437101at_int ).

tff(sy_c_ATP_058Lamm__afp____,type,
    aTP_Lamm_afp: fun_int_o > fun_fu3628599891403802633nteger ).

tff(sy_c_ATP_058Lamm__afq____,type,
    aTP_Lamm_afq: fun_Code_integer_o > fun_fu1763652423599918571nteger ).

tff(sy_c_ATP_058Lamm__afr____,type,
    aTP_Lamm_afr: fun_nat_o > fun_fu2499394402704830801nteger ).

tff(sy_c_ATP_058Lamm__afs____,type,
    aTP_Lamm_afs: fun_int_o > fun_fu1008360527747082367nt_rat ).

tff(sy_c_ATP_058Lamm__aft____,type,
    aTP_Lamm_aft: fun_Code_integer_o > fun_fu722387130092441185er_rat ).

tff(sy_c_ATP_058Lamm__afu____,type,
    aTP_Lamm_afu: fun_nat_o > fun_fu8328862064278946247at_rat ).

tff(sy_c_ATP_058Lamm__afv____,type,
    aTP_Lamm_afv: fun_int_o > fun_fu8905809029841901679nt_nat ).

tff(sy_c_ATP_058Lamm__afw____,type,
    aTP_Lamm_afw: fun_Code_integer_o > fun_fu7724909810345500241er_nat ).

tff(sy_c_ATP_058Lamm__afx____,type,
    aTP_Lamm_afx: fun_Code_integer_o > fun_fu403917627589892361er_int ).

tff(sy_c_ATP_058Lamm__afy____,type,
    aTP_Lamm_afy: fun_nat_o > fun_fu8464300013002972271at_int ).

tff(sy_c_ATP_058Lamm__afz____,type,
    aTP_Lamm_afz: fun_na1668100129766719005at_nat > fun_Pr557098546508347797at_nat ).

tff(sy_c_ATP_058Lamm__ag____,type,
    aTP_Lamm_ag: fun_nat_o ).

tff(sy_c_ATP_058Lamm__aga____,type,
    aTP_Lamm_aga: fun_na5914473681949881652_nat_o > fun_Pr3455331679158536364_nat_o ).

tff(sy_c_ATP_058Lamm__agb____,type,
    aTP_Lamm_agb: fun_o_o > fun_fu138107234227827011_int_o ).

tff(sy_c_ATP_058Lamm__agc____,type,
    aTP_Lamm_agc: fun_o_int > fun_fu6453395667394816515nt_int ).

tff(sy_c_ATP_058Lamm__agd____,type,
    aTP_Lamm_agd: fun_int_o > fun_fu1481857817119020493_int_o ).

tff(sy_c_ATP_058Lamm__age____,type,
    aTP_Lamm_age: fun_int_int > fun_fu6092417073193843705nt_int ).

tff(sy_c_ATP_058Lamm__agf____,type,
    aTP_Lamm_agf: fun_Pr3238374608701176162_int_o > fun_fu4592178885959468770_int_o ).

tff(sy_c_ATP_058Lamm__agg____,type,
    aTP_Lamm_agg: fun_Pr982841511892225912nt_int > fun_fu717549361983437348nt_int ).

tff(sy_c_ATP_058Lamm__agh____,type,
    aTP_Lamm_agh: fun_o_8988054265482332260nt_int > fun_fu8427683764667544036nt_int ).

tff(sy_c_ATP_058Lamm__agi____,type,
    aTP_Lamm_agi: fun_in4211310926729683942nt_int > fun_fu918106826131887214nt_int ).

tff(sy_c_ATP_058Lamm__agj____,type,
    aTP_Lamm_agj: fun_Pr2478283820680522243nt_int > fun_fu3898734532474705027nt_int ).

tff(sy_c_ATP_058Lamm__agk____,type,
    aTP_Lamm_agk: fun_fu3275390434952296369_nat_o > fun_fu6384561554008368101_nat_o ).

tff(sy_c_ATP_058Lamm__agl____,type,
    aTP_Lamm_agl: fun_Pr2477239776880934807at_nat > fun_na1668100129766719005at_nat ).

tff(sy_c_ATP_058Lamm__agm____,type,
    aTP_Lamm_agm: fun_Pr2028404973800837934_nat_o > fun_na5914473681949881652_nat_o ).

tff(sy_c_ATP_058Lamm__agn____,type,
    aTP_Lamm_agn: fun_Pr2478283820680522243nt_int > fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__ago____,type,
    aTP_Lamm_ago: fun_Pr3238374608701176162_int_o > fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__agp____,type,
    aTP_Lamm_agp: fun_Pr982841511892225912nt_int > fun_int_fun_int_int ).

tff(sy_c_ATP_058Lamm__agq____,type,
    aTP_Lamm_agq: fun_na4138050610385705875at_nat ).

tff(sy_c_ATP_058Lamm__agr____,type,
    aTP_Lamm_agr: num > fun_num_int ).

tff(sy_c_ATP_058Lamm__ags____,type,
    aTP_Lamm_ags: fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__agt____,type,
    aTP_Lamm_agt: fun_na4138050610385705875at_nat ).

tff(sy_c_ATP_058Lamm__agu____,type,
    aTP_Lamm_agu: fun_Co2470855966376689427nteger ).

tff(sy_c_ATP_058Lamm__agv____,type,
    aTP_Lamm_agv: fun_num_option_num ).

tff(sy_c_ATP_058Lamm__agw____,type,
    aTP_Lamm_agw: fun_int_fun_int_int ).

tff(sy_c_ATP_058Lamm__agx____,type,
    aTP_Lamm_agx: nat > fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__agy____,type,
    aTP_Lamm_agy: fun_nat_fun_nat_nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__agz____,type,
    aTP_Lamm_agz: fun_nat_fun_nat_nat > fun_nat_nat ).

tff(sy_c_ATP_058Lamm__ah____,type,
    aTP_Lamm_ah: fun_nat_o ).

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

tff(sy_c_ATP_058Lamm__ahb____,type,
    aTP_Lamm_ahb: fun_nat_fun_nat_int > fun_nat_fun_nat_int ).

tff(sy_c_ATP_058Lamm__ahc____,type,
    aTP_Lamm_ahc: fun_nat_fun_nat_int > fun_nat_int ).

tff(sy_c_ATP_058Lamm__ahd____,type,
    aTP_Lamm_ahd: fun_a_6093291656547882009Heap_a > fun_a_3171520138701809467it_nat ).

tff(sy_c_ATP_058Lamm__ahe____,type,
    aTP_Lamm_ahe: fun_a_6093291656547882009Heap_a > fun_a_3426355041879316131it_nat ).

tff(sy_c_ATP_058Lamm__ahf____,type,
    aTP_Lamm_ahf: fun_int_option_int > fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__ahg____,type,
    aTP_Lamm_ahg: heap_Time_Heap_a > fun_fu4855298819597708819it_nat ).

tff(sy_c_ATP_058Lamm__ahh____,type,
    aTP_Lamm_ahh: fun_a_2990225006182978344t_unit > fun_a_6373226321756256586it_nat ).

tff(sy_c_ATP_058Lamm__ahi____,type,
    aTP_Lamm_ahi: fun_a_2990225006182978344t_unit > fun_a_4942204877778174258it_nat ).

tff(sy_c_ATP_058Lamm__ahj____,type,
    aTP_Lamm_ahj: heap_Time_Heap_a > fun_fu6488529940522614227it_nat ).

tff(sy_c_ATP_058Lamm__ahk____,type,
    aTP_Lamm_ahk: fun_se2170385732985038782_nat_o ).

tff(sy_c_ATP_058Lamm__ahl____,type,
    aTP_Lamm_ahl: fun_se2051630309975736070_int_o ).

tff(sy_c_ATP_058Lamm__ahm____,type,
    aTP_Lamm_ahm: fun_se5529282185252607396eger_o ).

tff(sy_c_ATP_058Lamm__ahn____,type,
    aTP_Lamm_ahn: fun_se5639961202347202350_nat_o ).

tff(sy_c_ATP_058Lamm__aho____,type,
    aTP_Lamm_aho: fun_fu2352890940381120596et_nat > fun_fu7582738562687708775_nat_o ).

tff(sy_c_ATP_058Lamm__ahp____,type,
    aTP_Lamm_ahp: fun_fu1905361174868762994et_nat > fun_fu4927615759029954091_nat_o ).

tff(sy_c_ATP_058Lamm__ahq____,type,
    aTP_Lamm_ahq: fun_fu4253467608928286509nt_int > fun_fu4695829907076200203_int_o ).

tff(sy_c_ATP_058Lamm__ahr____,type,
    aTP_Lamm_ahr: fun_fu5631754973857388633nt_int > fun_fu6286765928754279603_int_o ).

tff(sy_c_ATP_058Lamm__ahs____,type,
    aTP_Lamm_ahs: fun_a_1313742400928293293it_nat > fun_fu4836394952439539509_nat_o ).

tff(sy_c_ATP_058Lamm__aht____,type,
    aTP_Lamm_aht: fun_int_option_int > fun_fu7409062071265970153_int_o ).

tff(sy_c_ATP_058Lamm__ahu____,type,
    aTP_Lamm_ahu: fun_nat_nat > fun_se4014881416909572940_nat_o ).

tff(sy_c_ATP_058Lamm__ahv____,type,
    aTP_Lamm_ahv: fun_int_nat > fun_se5614109888078709212_int_o ).

tff(sy_c_ATP_058Lamm__ahw____,type,
    aTP_Lamm_ahw: fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__ahx____,type,
    aTP_Lamm_ahx: fun_na4138050610385705875at_nat ).

tff(sy_c_ATP_058Lamm__ahy____,type,
    aTP_Lamm_ahy: fun_Co2470855966376689427nteger ).

tff(sy_c_ATP_058Lamm__ahz____,type,
    aTP_Lamm_ahz: int > fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__ai____,type,
    aTP_Lamm_ai: set_Pr8693737435421807431at_nat > fun_Pr363642936170290612_nat_o ).

tff(sy_c_ATP_058Lamm__aia____,type,
    aTP_Lamm_aia: int > fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__aib____,type,
    aTP_Lamm_aib: $o > fun_fu138107234227827011_int_o ).

tff(sy_c_ATP_058Lamm__aic____,type,
    aTP_Lamm_aic: nat > fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__aid____,type,
    aTP_Lamm_aid: fun_nat_nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__aie____,type,
    aTP_Lamm_aie: fun_nat_int > fun_nat_fun_nat_int ).

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

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

tff(sy_c_ATP_058Lamm__aih____,type,
    aTP_Lamm_aih: fun_nat_int > fun_nat_fun_nat_int ).

tff(sy_c_ATP_058Lamm__aii____,type,
    aTP_Lamm_aii: fun_nat_fun_nat_nat > fun_nat_nat ).

tff(sy_c_ATP_058Lamm__aij____,type,
    aTP_Lamm_aij: fun_nat_fun_nat_nat > fun_nat_nat ).

tff(sy_c_ATP_058Lamm__aik____,type,
    aTP_Lamm_aik: fun_nat_fun_nat_int > fun_nat_int ).

tff(sy_c_ATP_058Lamm__ail____,type,
    aTP_Lamm_ail: code_integer > fun_na8856381646659805649nteger ).

tff(sy_c_ATP_058Lamm__aim____,type,
    aTP_Lamm_aim: rat > fun_na1148497095348402375at_rat ).

tff(sy_c_ATP_058Lamm__ain____,type,
    aTP_Lamm_ain: int > fun_na2382631619178260847at_int ).

tff(sy_c_ATP_058Lamm__aio____,type,
    aTP_Lamm_aio: code_integer > fun_na8856381646659805649nteger ).

tff(sy_c_ATP_058Lamm__aip____,type,
    aTP_Lamm_aip: rat > fun_na1148497095348402375at_rat ).

tff(sy_c_ATP_058Lamm__aiq____,type,
    aTP_Lamm_aiq: int > fun_na2382631619178260847at_int ).

tff(sy_c_ATP_058Lamm__air____,type,
    aTP_Lamm_air: code_integer > fun_na8135612474337863287nteger ).

tff(sy_c_ATP_058Lamm__ais____,type,
    aTP_Lamm_ais: rat > fun_nat_fun_nat_rat ).

tff(sy_c_ATP_058Lamm__ait____,type,
    aTP_Lamm_ait: int > fun_nat_fun_nat_int ).

tff(sy_c_ATP_058Lamm__aiu____,type,
    aTP_Lamm_aiu: fun_int_Code_integer > fun_Co968421022063998397nteger ).

tff(sy_c_ATP_058Lamm__aiv____,type,
    aTP_Lamm_aiv: fun_nat_Code_integer > fun_Co1741750543637222241nteger ).

tff(sy_c_ATP_058Lamm__aiw____,type,
    aTP_Lamm_aiw: fun_o_Code_integer > fun_Co2498591179490635159nteger ).

tff(sy_c_ATP_058Lamm__aix____,type,
    aTP_Lamm_aix: fun_int_rat > fun_ra1107035566227760819at_rat ).

tff(sy_c_ATP_058Lamm__aiy____,type,
    aTP_Lamm_aiy: fun_nat_rat > fun_ra7317100687414072407at_rat ).

tff(sy_c_ATP_058Lamm__aiz____,type,
    aTP_Lamm_aiz: fun_o_rat > fun_ra154387494674143703at_rat ).

tff(sy_c_ATP_058Lamm__aj____,type,
    aTP_Lamm_aj: set_Pr8551490117392284871at_nat > fun_Pr1099658227000185396_nat_o ).

tff(sy_c_ATP_058Lamm__aja____,type,
    aTP_Lamm_aja: fun_int_int > fun_in5653523039088217947at_int ).

tff(sy_c_ATP_058Lamm__ajb____,type,
    aTP_Lamm_ajb: fun_nat_int > fun_in2640216123419753727at_int ).

tff(sy_c_ATP_058Lamm__ajc____,type,
    aTP_Lamm_ajc: fun_int_nat > fun_na34515594046663331at_nat ).

tff(sy_c_ATP_058Lamm__ajd____,type,
    aTP_Lamm_ajd: fun_nat_nat > fun_na6244580715232974919at_nat ).

tff(sy_c_ATP_058Lamm__aje____,type,
    aTP_Lamm_aje: nat > fun_num_option_num ).

tff(sy_c_ATP_058Lamm__ajf____,type,
    aTP_Lamm_ajf: nat > fun_num_option_num ).

tff(sy_c_ATP_058Lamm__ajg____,type,
    aTP_Lamm_ajg: num > fun_nat_option_num ).

tff(sy_c_ATP_058Lamm__ajh____,type,
    aTP_Lamm_ajh: fun_na7595719129879163854on_num ).

tff(sy_c_ATP_058Lamm__aji____,type,
    aTP_Lamm_aji: set_o > fun_fu6781074197733401937un_o_o ).

tff(sy_c_ATP_058Lamm__ajj____,type,
    aTP_Lamm_ajj: set_int > fun_fu1805223966450871909_int_o ).

tff(sy_c_ATP_058Lamm__ajk____,type,
    aTP_Lamm_ajk: set_Code_integer > fun_fu2182969643484777475eger_o ).

tff(sy_c_ATP_058Lamm__ajl____,type,
    aTP_Lamm_ajl: set_o > fun_fu7756609662939579309un_o_o ).

tff(sy_c_ATP_058Lamm__ajm____,type,
    aTP_Lamm_ajm: set_nat > fun_fu8207122332086543297_nat_o ).

tff(sy_c_ATP_058Lamm__ajn____,type,
    aTP_Lamm_ajn: set_Code_integer > fun_fu1992263728252513447eger_o ).

tff(sy_c_ATP_058Lamm__ajo____,type,
    aTP_Lamm_ajo: set_o > fun_fu3819205551299168222un_o_o ).

tff(sy_c_ATP_058Lamm__ajp____,type,
    aTP_Lamm_ajp: set_nat > fun_fu5838931486547983824_nat_o ).

tff(sy_c_ATP_058Lamm__ajq____,type,
    aTP_Lamm_ajq: set_int > fun_fu5271299750659287320_int_o ).

tff(sy_c_ATP_058Lamm__ajr____,type,
    aTP_Lamm_ajr: set_Code_integer > fun_fu855086506144988854eger_o ).

tff(sy_c_ATP_058Lamm__ajs____,type,
    aTP_Lamm_ajs: nat > fun_set_nat_nat ).

tff(sy_c_ATP_058Lamm__ajt____,type,
    aTP_Lamm_ajt: nat > fun_list_nat_nat ).

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

tff(sy_c_ATP_058Lamm__ajv____,type,
    aTP_Lamm_ajv: int > fun_list_nat_int ).

tff(sy_c_ATP_058Lamm__ajw____,type,
    aTP_Lamm_ajw: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__ajx____,type,
    aTP_Lamm_ajx: fun_int_fun_nat_int ).

tff(sy_c_ATP_058Lamm__ajy____,type,
    aTP_Lamm_ajy: fun_int_fun_int_int ).

tff(sy_c_ATP_058Lamm__ajz____,type,
    aTP_Lamm_ajz: rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__ak____,type,
    aTP_Lamm_ak: set_Pr4329608150637261639at_nat > fun_Pr3059359313727237556_nat_o ).

tff(sy_c_ATP_058Lamm__aka____,type,
    aTP_Lamm_aka: code_integer > fun_nat_Code_integer ).

tff(sy_c_ATP_058Lamm__akb____,type,
    aTP_Lamm_akb: rat > fun_set_nat_rat ).

tff(sy_c_ATP_058Lamm__akc____,type,
    aTP_Lamm_akc: rat > fun_list_nat_rat ).

tff(sy_c_ATP_058Lamm__akd____,type,
    aTP_Lamm_akd: fun_int_o > fun_int_Code_integer ).

tff(sy_c_ATP_058Lamm__ake____,type,
    aTP_Lamm_ake: fun_Code_integer_o > fun_Co8499238673658658103nteger ).

tff(sy_c_ATP_058Lamm__akf____,type,
    aTP_Lamm_akf: fun_nat_o > fun_nat_Code_integer ).

tff(sy_c_ATP_058Lamm__akg____,type,
    aTP_Lamm_akg: fun_int_o > fun_int_nat ).

tff(sy_c_ATP_058Lamm__akh____,type,
    aTP_Lamm_akh: fun_Code_integer_o > fun_Code_integer_nat ).

tff(sy_c_ATP_058Lamm__aki____,type,
    aTP_Lamm_aki: fun_Code_integer_o > fun_Code_integer_int ).

tff(sy_c_ATP_058Lamm__akj____,type,
    aTP_Lamm_akj: fun_nat_o > fun_nat_int ).

tff(sy_c_ATP_058Lamm__akk____,type,
    aTP_Lamm_akk: fun_nat_o > fun_nat_nat ).

tff(sy_c_ATP_058Lamm__akl____,type,
    aTP_Lamm_akl: fun_int_o > fun_int_int ).

tff(sy_c_ATP_058Lamm__akm____,type,
    aTP_Lamm_akm: fun_set_nat_o > fun_se7586849273513198688nteger ).

tff(sy_c_ATP_058Lamm__akn____,type,
    aTP_Lamm_akn: fun_op8989173174992673105on_num > fun_fu1574286537503582443on_num ).

tff(sy_c_ATP_058Lamm__ako____,type,
    aTP_Lamm_ako: fun_option_num_o > fun_fu2897328509465618819_nat_o ).

tff(sy_c_ATP_058Lamm__akp____,type,
    aTP_Lamm_akp: fun_option_num_nat > fun_fu1502755112258691281at_nat ).

tff(sy_c_ATP_058Lamm__akq____,type,
    aTP_Lamm_akq: fun_o_option_num > fun_fu2797749360310127211on_num ).

tff(sy_c_ATP_058Lamm__akr____,type,
    aTP_Lamm_akr: fun_o_o > fun_fu4001317426178549763_nat_o ).

tff(sy_c_ATP_058Lamm__aks____,type,
    aTP_Lamm_aks: fun_o_nat > fun_fu318872218122388049at_nat ).

tff(sy_c_ATP_058Lamm__akt____,type,
    aTP_Lamm_akt: fun_nat_option_num > fun_fu5130958774822480081on_num ).

tff(sy_c_ATP_058Lamm__aku____,type,
    aTP_Lamm_aku: fun_nat_o > fun_fu5477517762487726877_nat_o ).

tff(sy_c_ATP_058Lamm__akv____,type,
    aTP_Lamm_akv: set_set_nat > fun_na3711450577076729278_nat_o ).

tff(sy_c_ATP_058Lamm__akw____,type,
    aTP_Lamm_akw: set_list_nat > fun_na1108963418173300452_nat_o ).

tff(sy_c_ATP_058Lamm__akx____,type,
    aTP_Lamm_akx: set_nat > fun_na8563630686737515764_nat_o ).

tff(sy_c_ATP_058Lamm__aky____,type,
    aTP_Lamm_aky: set_Code_integer > fun_na7849980549906154535eger_o ).

tff(sy_c_ATP_058Lamm__akz____,type,
    aTP_Lamm_akz: set_Pr1261947904930325089at_nat > fun_na4518284863240432007_nat_o ).

tff(sy_c_ATP_058Lamm__al____,type,
    aTP_Lamm_al: set_Pr1261947904930325089at_nat > fun_fu3275390434952296369_nat_o ).

tff(sy_c_ATP_058Lamm__ala____,type,
    aTP_Lamm_ala: set_int > fun_na8635581178960477080_int_o ).

tff(sy_c_ATP_058Lamm__alb____,type,
    aTP_Lamm_alb: set_o > fun_fu4605530428521801667un_o_o ).

tff(sy_c_ATP_058Lamm__alc____,type,
    aTP_Lamm_alc: set_o > fun_fu4605530428521801667un_o_o ).

tff(sy_c_ATP_058Lamm__ald____,type,
    aTP_Lamm_ald: set_o > fun_fu4625457934454704336_o_nat ).

tff(sy_c_ATP_058Lamm__ale____,type,
    aTP_Lamm_ale: set_int > fun_fu2630858903579709668_o_nat ).

tff(sy_c_ATP_058Lamm__alf____,type,
    aTP_Lamm_alf: set_Code_integer > fun_fu5820097178109300859eger_o ).

tff(sy_c_ATP_058Lamm__alg____,type,
    aTP_Lamm_alg: set_Code_integer > fun_fu9082890457842154773_o_nat ).

tff(sy_c_ATP_058Lamm__alh____,type,
    aTP_Lamm_alh: set_int > fun_fu7616189625787487449_int_o ).

tff(sy_c_ATP_058Lamm__ali____,type,
    aTP_Lamm_ali: set_o > fun_fu6500062642585991509un_o_o ).

tff(sy_c_ATP_058Lamm__alj____,type,
    aTP_Lamm_alj: set_o > fun_fu2395600473883792738nt_nat ).

tff(sy_c_ATP_058Lamm__alk____,type,
    aTP_Lamm_alk: set_int > fun_fu138107234227827011_int_o ).

tff(sy_c_ATP_058Lamm__all____,type,
    aTP_Lamm_all: set_int > fun_fu2566930710778725758nt_nat ).

tff(sy_c_ATP_058Lamm__alm____,type,
    aTP_Lamm_alm: set_Code_integer > fun_fu5813846785102284803eger_o ).

tff(sy_c_ATP_058Lamm__aln____,type,
    aTP_Lamm_aln: set_Code_integer > fun_fu716243638958221197nt_nat ).

tff(sy_c_ATP_058Lamm__alo____,type,
    aTP_Lamm_alo: set_Code_integer > fun_fu5848294987682527897eger_o ).

tff(sy_c_ATP_058Lamm__alp____,type,
    aTP_Lamm_alp: set_o > fun_fu5285309056923805367un_o_o ).

tff(sy_c_ATP_058Lamm__alq____,type,
    aTP_Lamm_alq: set_o > fun_fu2439999276549649860er_nat ).

tff(sy_c_ATP_058Lamm__alr____,type,
    aTP_Lamm_alr: set_int > fun_fu3779984648686517950er_nat ).

tff(sy_c_ATP_058Lamm__als____,type,
    aTP_Lamm_als: set_Code_integer > fun_fu8958601512183576259eger_o ).

tff(sy_c_ATP_058Lamm__alt____,type,
    aTP_Lamm_alt: set_Code_integer > fun_fu8958601512183576259eger_o ).

tff(sy_c_ATP_058Lamm__alu____,type,
    aTP_Lamm_alu: set_Code_integer > fun_fu6385016345819965645er_nat ).

tff(sy_c_ATP_058Lamm__alv____,type,
    aTP_Lamm_alv: set_nat > fun_fu6443148758731022984_o_nat ).

tff(sy_c_ATP_058Lamm__alw____,type,
    aTP_Lamm_alw: fun_set_nat_nat ).

tff(sy_c_ATP_058Lamm__alx____,type,
    aTP_Lamm_alx: fun_list_nat_nat ).

tff(sy_c_ATP_058Lamm__aly____,type,
    aTP_Lamm_aly: set_nat > fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__alz____,type,
    aTP_Lamm_alz: set_nat > fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__am____,type,
    aTP_Lamm_am: set_o > fun_fun_o_o_fun_o_o ).

tff(sy_c_ATP_058Lamm__ama____,type,
    aTP_Lamm_ama: fun_set_nat_nat > fun_set_nat_nat ).

tff(sy_c_ATP_058Lamm__amb____,type,
    aTP_Lamm_amb: fun_list_nat_nat > fun_list_nat_nat ).

tff(sy_c_ATP_058Lamm__amc____,type,
    aTP_Lamm_amc: fun_nat_nat > fun_nat_nat ).

tff(sy_c_ATP_058Lamm__amd____,type,
    aTP_Lamm_amd: set_nat > fun_fu2757636626010989786nt_nat ).

tff(sy_c_ATP_058Lamm__ame____,type,
    aTP_Lamm_ame: set_nat > fun_fu513762604956806682er_nat ).

tff(sy_c_ATP_058Lamm__amf____,type,
    aTP_Lamm_amf: fun_Co8521658056209057742er_num ).

tff(sy_c_ATP_058Lamm__amg____,type,
    aTP_Lamm_amg: fun_Co1092983284983208416er_int ).

tff(sy_c_ATP_058Lamm__amh____,type,
    aTP_Lamm_amh: fun_nat_o ).

tff(sy_c_ATP_058Lamm__ami____,type,
    aTP_Lamm_ami: code_integer > fun_Co3544761417380347684eger_o ).

tff(sy_c_ATP_058Lamm__amj____,type,
    aTP_Lamm_amj: fun_Co3583066429914935172er_nat ).

tff(sy_c_ATP_058Lamm__amk____,type,
    aTP_Lamm_amk: code_integer > fun_Co2470855966376689427nteger ).

tff(sy_c_ATP_058Lamm__aml____,type,
    aTP_Lamm_aml: code_integer > fun_Co2470855966376689427nteger ).

tff(sy_c_ATP_058Lamm__amm____,type,
    aTP_Lamm_amm: code_integer > fun_Co2470855966376689427nteger ).

tff(sy_c_ATP_058Lamm__amn____,type,
    aTP_Lamm_amn: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__amo____,type,
    aTP_Lamm_amo: fun_num_option_num ).

tff(sy_c_ATP_058Lamm__amp____,type,
    aTP_Lamm_amp: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__amq____,type,
    aTP_Lamm_amq: nat > fun_na4143055713577223196at_nat ).

tff(sy_c_ATP_058Lamm__amr____,type,
    aTP_Lamm_amr: fun_na1668100129766719005at_nat ).

tff(sy_c_ATP_058Lamm__ams____,type,
    aTP_Lamm_ams: fun_na4138050610385705875at_nat ).

tff(sy_c_ATP_058Lamm__amt____,type,
    aTP_Lamm_amt: nat > fun_na7928631014926962227_nat_o ).

tff(sy_c_ATP_058Lamm__amu____,type,
    aTP_Lamm_amu: fun_na5914473681949881652_nat_o ).

tff(sy_c_ATP_058Lamm__amv____,type,
    aTP_Lamm_amv: nat > fun_na7928631014926962227_nat_o ).

tff(sy_c_ATP_058Lamm__amw____,type,
    aTP_Lamm_amw: fun_na5914473681949881652_nat_o ).

tff(sy_c_ATP_058Lamm__amx____,type,
    aTP_Lamm_amx: nat > fun_na4143055713577223196at_nat ).

tff(sy_c_ATP_058Lamm__amy____,type,
    aTP_Lamm_amy: fun_na1668100129766719005at_nat ).

tff(sy_c_ATP_058Lamm__amz____,type,
    aTP_Lamm_amz: nat > fun_na4143055713577223196at_nat ).

tff(sy_c_ATP_058Lamm__an____,type,
    aTP_Lamm_an: set_Pr958786334691620121nt_int > fun_fu3068557634114553409_int_o ).

tff(sy_c_ATP_058Lamm__ana____,type,
    aTP_Lamm_ana: fun_na1668100129766719005at_nat ).

tff(sy_c_ATP_058Lamm__anb____,type,
    aTP_Lamm_anb: fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__anc____,type,
    aTP_Lamm_anc: fun_nat_set_nat ).

tff(sy_c_ATP_058Lamm__and____,type,
    aTP_Lamm_and: fun_int_nat ).

tff(sy_c_ATP_058Lamm__ane____,type,
    aTP_Lamm_ane: nat > fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__anf____,type,
    aTP_Lamm_anf: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__ang____,type,
    aTP_Lamm_ang: nat > fun_na8563630686737515764_nat_o ).

tff(sy_c_ATP_058Lamm__anh____,type,
    aTP_Lamm_anh: nat > fun_na7234455742697417626_nat_o ).

tff(sy_c_ATP_058Lamm__ani____,type,
    aTP_Lamm_ani: nat > fun_na7234455742697417626_nat_o ).

tff(sy_c_ATP_058Lamm__anj____,type,
    aTP_Lamm_anj: nat > fun_na7234455742697417626_nat_o ).

tff(sy_c_ATP_058Lamm__ank____,type,
    aTP_Lamm_ank: fun_nat_nat ).

tff(sy_c_ATP_058Lamm__anl____,type,
    aTP_Lamm_anl: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__anm____,type,
    aTP_Lamm_anm: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__ann____,type,
    aTP_Lamm_ann: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__ano____,type,
    aTP_Lamm_ano: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__anp____,type,
    aTP_Lamm_anp: rat > fun_int_o ).

tff(sy_c_ATP_058Lamm__anq____,type,
    aTP_Lamm_anq: int > fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__anr____,type,
    aTP_Lamm_anr: fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__ans____,type,
    aTP_Lamm_ans: int > fun_in2204242111582549184nt_int ).

tff(sy_c_ATP_058Lamm__ant____,type,
    aTP_Lamm_ant: rat > fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__anu____,type,
    aTP_Lamm_anu: int > fun_in2204242111582549184nt_int ).

tff(sy_c_ATP_058Lamm__anv____,type,
    aTP_Lamm_anv: rat > fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__anw____,type,
    aTP_Lamm_anw: int > fun_in5570076352459579423_int_o ).

tff(sy_c_ATP_058Lamm__anx____,type,
    aTP_Lamm_anx: rat > fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__any____,type,
    aTP_Lamm_any: int > fun_in5570076352459579423_int_o ).

tff(sy_c_ATP_058Lamm__anz____,type,
    aTP_Lamm_anz: rat > fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__ao____,type,
    aTP_Lamm_ao: set_list_nat > fun_fu1994797156054922755_nat_o ).

tff(sy_c_ATP_058Lamm__aoa____,type,
    aTP_Lamm_aoa: fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__aob____,type,
    aTP_Lamm_aob: int > fun_in2204242111582549184nt_int ).

tff(sy_c_ATP_058Lamm__aoc____,type,
    aTP_Lamm_aoc: rat > fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__aod____,type,
    aTP_Lamm_aod: int > fun_in2204242111582549184nt_int ).

tff(sy_c_ATP_058Lamm__aoe____,type,
    aTP_Lamm_aoe: rat > fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__aof____,type,
    aTP_Lamm_aof: fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__aog____,type,
    aTP_Lamm_aog: nat > fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__aoh____,type,
    aTP_Lamm_aoh: fun_Pr3238374608701176162_int_o ).

tff(sy_c_ATP_058Lamm__aoi____,type,
    aTP_Lamm_aoi: fun_Pr1308109350235310159nt_int ).

tff(sy_c_ATP_058Lamm__aoj____,type,
    aTP_Lamm_aoj: fun_Pr2478283820680522243nt_int ).

tff(sy_c_ATP_058Lamm__aok____,type,
    aTP_Lamm_aok: fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__aol____,type,
    aTP_Lamm_aol: fun_Pr2478283820680522243nt_int ).

tff(sy_c_ATP_058Lamm__aom____,type,
    aTP_Lamm_aom: fun_Pr1308109350235310159nt_int ).

tff(sy_c_ATP_058Lamm__aon____,type,
    aTP_Lamm_aon: fun_int_int ).

tff(sy_c_ATP_058Lamm__aoo____,type,
    aTP_Lamm_aoo: rat > fun_Pr3238374608701176162_int_o ).

tff(sy_c_ATP_058Lamm__aop____,type,
    aTP_Lamm_aop: fun_num_fun_num_int ).

tff(sy_c_ATP_058Lamm__aoq____,type,
    aTP_Lamm_aoq: fun_na1476630412504255114at_nat ).

tff(sy_c_ATP_058Lamm__aor____,type,
    aTP_Lamm_aor: set_int > fun_int_o ).

tff(sy_c_ATP_058Lamm__aos____,type,
    aTP_Lamm_aos: nat > fun_na7928631014926962227_nat_o ).

tff(sy_c_ATP_058Lamm__aot____,type,
    aTP_Lamm_aot: fun_na5914473681949881652_nat_o ).

tff(sy_c_ATP_058Lamm__aou____,type,
    aTP_Lamm_aou: nat > fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__aov____,type,
    aTP_Lamm_aov: fun_nat_o > fun_nat_o ).

tff(sy_c_ATP_058Lamm__aow____,type,
    aTP_Lamm_aow: fun_nat_o > fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__aox____,type,
    aTP_Lamm_aox: nat > fun_he5234484606919320760it_nat ).

tff(sy_c_ATP_058Lamm__aoy____,type,
    aTP_Lamm_aoy: code_natural > fun_Co9168753396221475450atural ).

tff(sy_c_ATP_058Lamm__aoz____,type,
    aTP_Lamm_aoz: produc7822875418678951345atural > fun_Co1890133979025130057atural ).

tff(sy_c_ATP_058Lamm__ap____,type,
    aTP_Lamm_ap: set_set_nat > fun_fu2408719596384163971_nat_o ).

tff(sy_c_ATP_058Lamm__apa____,type,
    aTP_Lamm_apa: code_natural > fun_Co1105491363547889651atural ).

tff(sy_c_ATP_058Lamm__apb____,type,
    aTP_Lamm_apb: fun_Co1105491363547889651atural ).

tff(sy_c_ATP_058Lamm__apc____,type,
    aTP_Lamm_apc: code_natural > fun_Co1105491363547889651atural ).

tff(sy_c_ATP_058Lamm__apd____,type,
    aTP_Lamm_apd: fun_na6128946301132549280et_nat ).

tff(sy_c_ATP_058Lamm__aq____,type,
    aTP_Lamm_aq: set_nat > fun_fu4001317426178549763_nat_o ).

tff(sy_c_ATP_058Lamm__ar____,type,
    aTP_Lamm_ar: set_int > fun_fu3883539188510166275_int_o ).

tff(sy_c_ATP_058Lamm__as____,type,
    aTP_Lamm_as: set_Pr1261947904930325089at_nat > fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__at____,type,
    aTP_Lamm_at: set_Pr1312406793078187350et_nat > fun_fu8419763957888848259_nat_o ).

tff(sy_c_ATP_058Lamm__au____,type,
    aTP_Lamm_au: set_Pr4040394263813967220et_nat > fun_fu2611741329380808165_nat_o ).

tff(sy_c_ATP_058Lamm__av____,type,
    aTP_Lamm_av: set_Pr6700895561065273913nt_int > fun_fu6865642474282033866_int_o ).

tff(sy_c_ATP_058Lamm__aw____,type,
    aTP_Lamm_aw: set_Pr4166680992137330533nt_int > fun_fu2978539155105838238_int_o ).

tff(sy_c_ATP_058Lamm__ax____,type,
    aTP_Lamm_ax: set_Pr7098220151150636591it_nat > fun_a_2467521563176467050_nat_o ).

tff(sy_c_ATP_058Lamm__ay____,type,
    aTP_Lamm_ay: set_Pr8693737435421807431at_nat > fun_Pr2028404973800837934_nat_o ).

tff(sy_c_ATP_058Lamm__az____,type,
    aTP_Lamm_az: set_Pr8551490117392284871at_nat > fun_mu7838234450475261998_nat_o ).

tff(sy_c_ATP_058Lamm__ba____,type,
    aTP_Lamm_ba: set_Pr4329608150637261639at_nat > fun_se5639961202347202350_nat_o ).

tff(sy_c_ATP_058Lamm__bb____,type,
    aTP_Lamm_bb: fun_Pr3238374608701176162_int_o ).

tff(sy_c_ATP_058Lamm__bc____,type,
    aTP_Lamm_bc: fun_list_nat_o ).

tff(sy_c_ATP_058Lamm__bd____,type,
    aTP_Lamm_bd: fun_set_nat_o ).

tff(sy_c_ATP_058Lamm__be____,type,
    aTP_Lamm_be: fun_Pr4796943543331071514_nat_o ).

tff(sy_c_ATP_058Lamm__bf____,type,
    aTP_Lamm_bf: fun_o_o ).

tff(sy_c_ATP_058Lamm__bg____,type,
    aTP_Lamm_bg: fun_nat_o ).

tff(sy_c_ATP_058Lamm__bh____,type,
    aTP_Lamm_bh: fun_int_o ).

tff(sy_c_ATP_058Lamm__bi____,type,
    aTP_Lamm_bi: set_o > fun_set_o_fun_o_o ).

tff(sy_c_ATP_058Lamm__bj____,type,
    aTP_Lamm_bj: set_Pr958786334691620121nt_int > fun_se4767462807232413326_int_o ).

tff(sy_c_ATP_058Lamm__bk____,type,
    aTP_Lamm_bk: set_list_nat > fun_se1865396762483378772_nat_o ).

tff(sy_c_ATP_058Lamm__bl____,type,
    aTP_Lamm_bl: set_set_nat > fun_se2389560805064864136_nat_o ).

tff(sy_c_ATP_058Lamm__bm____,type,
    aTP_Lamm_bm: set_int > fun_se867122708194164540_int_o ).

tff(sy_c_ATP_058Lamm__bn____,type,
    aTP_Lamm_bn: set_nat > fun_se382663106265794036_nat_o ).

tff(sy_c_ATP_058Lamm__bo____,type,
    aTP_Lamm_bo: set_Pr1261947904930325089at_nat > fun_se374060482380105870_nat_o ).

tff(sy_c_ATP_058Lamm__bp____,type,
    aTP_Lamm_bp: fun_Pr3238374608701176162_int_o > fun_fu3068557634114553409_int_o ).

tff(sy_c_ATP_058Lamm__bq____,type,
    aTP_Lamm_bq: fun_list_nat_o > fun_fu1994797156054922755_nat_o ).

tff(sy_c_ATP_058Lamm__br____,type,
    aTP_Lamm_br: fun_set_nat_o > fun_fu2408719596384163971_nat_o ).

tff(sy_c_ATP_058Lamm__bs____,type,
    aTP_Lamm_bs: fun_int_o > fun_fu3883539188510166275_int_o ).

tff(sy_c_ATP_058Lamm__bt____,type,
    aTP_Lamm_bt: fun_nat_o > fun_fu4001317426178549763_nat_o ).

tff(sy_c_ATP_058Lamm__bu____,type,
    aTP_Lamm_bu: fun_Pr4796943543331071514_nat_o > fun_fu3275390434952296369_nat_o ).

tff(sy_c_ATP_058Lamm__bv____,type,
    aTP_Lamm_bv: set_Pr1261947904930325089at_nat > fun_se374060482380105870_nat_o ).

tff(sy_c_ATP_058Lamm__bw____,type,
    aTP_Lamm_bw: set_o > fun_set_o_fun_o_o ).

tff(sy_c_ATP_058Lamm__bx____,type,
    aTP_Lamm_bx: set_Pr958786334691620121nt_int > fun_se4767462807232413326_int_o ).

tff(sy_c_ATP_058Lamm__by____,type,
    aTP_Lamm_by: set_list_nat > fun_se1865396762483378772_nat_o ).

tff(sy_c_ATP_058Lamm__bz____,type,
    aTP_Lamm_bz: set_set_nat > fun_se2389560805064864136_nat_o ).

tff(sy_c_ATP_058Lamm__ca____,type,
    aTP_Lamm_ca: set_int > fun_se867122708194164540_int_o ).

tff(sy_c_ATP_058Lamm__cb____,type,
    aTP_Lamm_cb: set_Pr8693737435421807431at_nat > fun_se2369204539228100110_nat_o ).

tff(sy_c_ATP_058Lamm__cc____,type,
    aTP_Lamm_cc: set_Pr8551490117392284871at_nat > fun_se3981709065971081742_nat_o ).

tff(sy_c_ATP_058Lamm__cd____,type,
    aTP_Lamm_cd: set_Pr4329608150637261639at_nat > fun_se5282722391638144526_nat_o ).

tff(sy_c_ATP_058Lamm__ce____,type,
    aTP_Lamm_ce: set_nat > fun_se382663106265794036_nat_o ).

tff(sy_c_ATP_058Lamm__cf____,type,
    aTP_Lamm_cf: fun_Pr3238374608701176162_int_o > fun_fu3068557634114553409_int_o ).

tff(sy_c_ATP_058Lamm__cg____,type,
    aTP_Lamm_cg: fun_list_nat_o > fun_fu1994797156054922755_nat_o ).

tff(sy_c_ATP_058Lamm__ch____,type,
    aTP_Lamm_ch: fun_set_nat_o > fun_fu2408719596384163971_nat_o ).

tff(sy_c_ATP_058Lamm__ci____,type,
    aTP_Lamm_ci: fun_int_o > fun_fu3883539188510166275_int_o ).

tff(sy_c_ATP_058Lamm__cj____,type,
    aTP_Lamm_cj: fun_Pr363642936170290612_nat_o > fun_fu2988548197490738277_nat_o ).

tff(sy_c_ATP_058Lamm__ck____,type,
    aTP_Lamm_ck: fun_Pr1099658227000185396_nat_o > fun_fu714798943701216869_nat_o ).

tff(sy_c_ATP_058Lamm__cl____,type,
    aTP_Lamm_cl: fun_Pr3059359313727237556_nat_o > fun_fu1322491050725084261_nat_o ).

tff(sy_c_ATP_058Lamm__cm____,type,
    aTP_Lamm_cm: fun_nat_o > fun_fu4001317426178549763_nat_o ).

tff(sy_c_ATP_058Lamm__cn____,type,
    aTP_Lamm_cn: heap_e7401611519738050253t_unit > fun_he4135464443191253079_nat_o ).

tff(sy_c_ATP_058Lamm__co____,type,
    aTP_Lamm_co: heap_e7401611519738050253t_unit > fun_se382663106265794036_nat_o ).

tff(sy_c_ATP_058Lamm__cp____,type,
    aTP_Lamm_cp: fun_se2896342228535362350_nat_o ).

tff(sy_c_ATP_058Lamm__cq____,type,
    aTP_Lamm_cq: fun_se2896342228535362350_nat_o ).

tff(sy_c_ATP_058Lamm__cr____,type,
    aTP_Lamm_cr: fun_se5750412846314922030_nat_o ).

tff(sy_c_ATP_058Lamm__cs____,type,
    aTP_Lamm_cs: fun_se5750412846314922030_nat_o ).

tff(sy_c_ATP_058Lamm__ct____,type,
    aTP_Lamm_ct: fun_se5672428230568430382_nat_o ).

tff(sy_c_ATP_058Lamm__cu____,type,
    aTP_Lamm_cu: fun_se5672428230568430382_nat_o ).

tff(sy_c_ATP_058Lamm__cv____,type,
    aTP_Lamm_cv: fun_se5639961202347202350_nat_o ).

tff(sy_c_ATP_058Lamm__cw____,type,
    aTP_Lamm_cw: fun_se5639961202347202350_nat_o ).

tff(sy_c_ATP_058Lamm__cx____,type,
    aTP_Lamm_cx: fun_se5038631536132265326et_o_o ).

tff(sy_c_ATP_058Lamm__cy____,type,
    aTP_Lamm_cy: fun_se5038631536132265326et_o_o ).

tff(sy_c_ATP_058Lamm__cz____,type,
    aTP_Lamm_cz: fun_se2170385732985038782_nat_o ).

tff(sy_c_ATP_058Lamm__da____,type,
    aTP_Lamm_da: fun_se2170385732985038782_nat_o ).

tff(sy_c_ATP_058Lamm__db____,type,
    aTP_Lamm_db: fun_assn_fun_assn_o ).

tff(sy_c_ATP_058Lamm__dc____,type,
    aTP_Lamm_dc: fun_assn_fun_assn_o ).

tff(sy_c_ATP_058Lamm__dd____,type,
    aTP_Lamm_dd: fun_se2051630309975736070_int_o ).

tff(sy_c_ATP_058Lamm__de____,type,
    aTP_Lamm_de: fun_se2051630309975736070_int_o ).

tff(sy_c_ATP_058Lamm__df____,type,
    aTP_Lamm_df: fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__dg____,type,
    aTP_Lamm_dg: fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__dh____,type,
    aTP_Lamm_dh: fun_rat_fun_rat_o ).

tff(sy_c_ATP_058Lamm__di____,type,
    aTP_Lamm_di: fun_rat_fun_rat_o ).

tff(sy_c_ATP_058Lamm__dj____,type,
    aTP_Lamm_dj: fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__dk____,type,
    aTP_Lamm_dk: fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__dl____,type,
    aTP_Lamm_dl: set_Pr8693737435421807431at_nat > fun_se8505878878902388681at_nat ).

tff(sy_c_ATP_058Lamm__dm____,type,
    aTP_Lamm_dm: option2158460348403737357at_nat > fun_op4346303324771211497at_nat ).

tff(sy_c_ATP_058Lamm__dn____,type,
    aTP_Lamm_dn: set_Pr8551490117392284871at_nat > fun_se2186744213282142921at_nat ).

tff(sy_c_ATP_058Lamm__do____,type,
    aTP_Lamm_do: option2498585697089621389at_nat > fun_op3311566623873074537at_nat ).

tff(sy_c_ATP_058Lamm__dp____,type,
    aTP_Lamm_dp: set_Pr4329608150637261639at_nat > fun_se7901873761543946185at_nat ).

tff(sy_c_ATP_058Lamm__dq____,type,
    aTP_Lamm_dq: option1583680563626158861at_nat > fun_op291006984297869545at_nat ).

tff(sy_c_ATP_058Lamm__dr____,type,
    aTP_Lamm_dr: set_nat > fun_se7316093745529328985et_nat ).

tff(sy_c_ATP_058Lamm__ds____,type,
    aTP_Lamm_ds: option_set_nat > fun_op386870758422531628et_nat ).

tff(sy_c_ATP_058Lamm__dt____,type,
    aTP_Lamm_dt: nat > fun_nat_option_nat ).

tff(sy_c_ATP_058Lamm__du____,type,
    aTP_Lamm_du: option_nat > fun_op7680796716376270858on_nat ).

tff(sy_c_ATP_058Lamm__dv____,type,
    aTP_Lamm_dv: fun_int_int > fun_fu7117095973476296707um_int ).

tff(sy_c_ATP_058Lamm__dw____,type,
    aTP_Lamm_dw: fun_int_option_num > fun_fu7964671872712412353on_num ).

tff(sy_c_ATP_058Lamm__dx____,type,
    aTP_Lamm_dx: fun_int_num > fun_fu5322398707847370225um_num ).

tff(sy_c_ATP_058Lamm__dy____,type,
    aTP_Lamm_dy: fun_option_num_int > fun_fu1947565687807238081um_int ).

tff(sy_c_ATP_058Lamm__dz____,type,
    aTP_Lamm_dz: fun_op8989173174992673105on_num > fun_fu2987389051837124479on_num ).

tff(sy_c_ATP_058Lamm__ea____,type,
    aTP_Lamm_ea: fun_option_num_num > fun_fu152868422178311599um_num ).

tff(sy_c_ATP_058Lamm__eb____,type,
    aTP_Lamm_eb: fun_num_int > fun_fu726849872193282289um_int ).

tff(sy_c_ATP_058Lamm__ec____,type,
    aTP_Lamm_ec: fun_num_option_num > fun_fu4719417816141531311on_num ).

tff(sy_c_ATP_058Lamm__ed____,type,
    aTP_Lamm_ed: fun_num_num > fun_fu8155524643419131615um_num ).

tff(sy_c_ATP_058Lamm__ee____,type,
    aTP_Lamm_ee: set_nat > fun_se7316093745529328985et_nat ).

tff(sy_c_ATP_058Lamm__ef____,type,
    aTP_Lamm_ef: option_set_nat > fun_se7316093745529328985et_nat ).

tff(sy_c_ATP_058Lamm__eg____,type,
    aTP_Lamm_eg: assn > fun_assn_option_assn ).

tff(sy_c_ATP_058Lamm__eh____,type,
    aTP_Lamm_eh: option_assn > fun_assn_option_assn ).

tff(sy_c_ATP_058Lamm__ei____,type,
    aTP_Lamm_ei: nat > fun_nat_option_nat ).

tff(sy_c_ATP_058Lamm__ej____,type,
    aTP_Lamm_ej: option_nat > fun_nat_option_nat ).

tff(sy_c_ATP_058Lamm__ek____,type,
    aTP_Lamm_ek: product_unit > fun_Pr8303579546892975945t_unit ).

tff(sy_c_ATP_058Lamm__el____,type,
    aTP_Lamm_el: option_Product_unit > fun_Pr8303579546892975945t_unit ).

tff(sy_c_ATP_058Lamm__em____,type,
    aTP_Lamm_em: set_Pr1261947904930325089at_nat > fun_se5244420045238757833at_nat ).

tff(sy_c_ATP_058Lamm__en____,type,
    aTP_Lamm_en: option8963830502488799655at_nat > fun_se5244420045238757833at_nat ).

tff(sy_c_ATP_058Lamm__eo____,type,
    aTP_Lamm_eo: fun_Pr6460944699146269935_nat_o ).

tff(sy_c_ATP_058Lamm__ep____,type,
    aTP_Lamm_ep: assn > fun_as3747249873083764768_nat_o ).

tff(sy_c_ATP_058Lamm__eq____,type,
    aTP_Lamm_eq: assn > fun_as3747249873083764768_nat_o ).

tff(sy_c_ATP_058Lamm__er____,type,
    aTP_Lamm_er: fun_nat_option_nat ).

tff(sy_c_ATP_058Lamm__es____,type,
    aTP_Lamm_es: fun_num_o ).

tff(sy_c_ATP_058Lamm__et____,type,
    aTP_Lamm_et: fun_num_o ).

tff(sy_c_ATP_058Lamm__eu____,type,
    aTP_Lamm_eu: option_set_int > fun_set_int_o ).

tff(sy_c_ATP_058Lamm__ev____,type,
    aTP_Lamm_ev: option_rat > fun_rat_o ).

tff(sy_c_ATP_058Lamm__ew____,type,
    aTP_Lamm_ew: option_num > fun_num_o ).

tff(sy_c_ATP_058Lamm__ex____,type,
    aTP_Lamm_ex: option_nat > fun_nat_o ).

tff(sy_c_ATP_058Lamm__ey____,type,
    aTP_Lamm_ey: option_int > fun_int_o ).

tff(sy_c_ATP_058Lamm__ez____,type,
    aTP_Lamm_ez: option_assn > fun_assn_o ).

tff(sy_c_ATP_058Lamm__fa____,type,
    aTP_Lamm_fa: option_rat > fun_rat_o ).

tff(sy_c_ATP_058Lamm__fb____,type,
    aTP_Lamm_fb: fun_num_fun_num_o ).

tff(sy_c_ATP_058Lamm__fc____,type,
    aTP_Lamm_fc: option_num > fun_num_o ).

tff(sy_c_ATP_058Lamm__fd____,type,
    aTP_Lamm_fd: option_nat > fun_nat_o ).

tff(sy_c_ATP_058Lamm__fe____,type,
    aTP_Lamm_fe: option_int > fun_int_o ).

tff(sy_c_ATP_058Lamm__ff____,type,
    aTP_Lamm_ff: produc3260487557148687353it_nat > fun_Pr5746006151868627512_nat_o ).

tff(sy_c_ATP_058Lamm__fg____,type,
    aTP_Lamm_fg: fun_num_fun_num_o ).

tff(sy_c_ATP_058Lamm__fh____,type,
    aTP_Lamm_fh: produc8664842809031399944it_nat > fun_Pr8032424337255632745_nat_o ).

tff(sy_c_ATP_058Lamm__fi____,type,
    aTP_Lamm_fi: fun_he8413527710729349838unit_o > fun_fu9056938625493858259it_nat ).

tff(sy_c_ATP_058Lamm__fj____,type,
    aTP_Lamm_fj: fun_he8413527710729349838unit_o > fun_fu2492676192234506515it_nat ).

tff(sy_c_ATP_058Lamm__fk____,type,
    aTP_Lamm_fk: fun_he5234484606919320760it_nat ).

tff(sy_c_ATP_058Lamm__fl____,type,
    aTP_Lamm_fl: fun_Pr7750636411365408485it_nat > fun_Pr7916406164169562037it_nat ).

tff(sy_c_ATP_058Lamm__fm____,type,
    aTP_Lamm_fm: fun_Pr7750636411365408485it_nat > fun_op5883308099655130629it_nat ).

tff(sy_c_ATP_058Lamm__fn____,type,
    aTP_Lamm_fn: fun_Pr1003294667014919570it_nat > fun_Pr3631287881653221602it_nat ).

tff(sy_c_ATP_058Lamm__fo____,type,
    aTP_Lamm_fo: fun_Pr1003294667014919570it_nat > fun_op4112804339236761266it_nat ).

tff(sy_c_ATP_058Lamm__fp____,type,
    aTP_Lamm_fp: fun_num_fun_num_num > fun_nu1049378303802735384on_num ).

tff(sy_c_ATP_058Lamm__fq____,type,
    aTP_Lamm_fq: fun_num_fun_num_num > fun_op3323321805596408808on_num ).

tff(sy_c_ATP_058Lamm__fr____,type,
    aTP_Lamm_fr: fun_Co8499238673658658103nteger ).

tff(sy_c_ATP_058Lamm__fs____,type,
    aTP_Lamm_fs: fun_assn_assn ).

tff(sy_c_ATP_058Lamm__ft____,type,
    aTP_Lamm_ft: fun_rat_rat ).

tff(sy_c_ATP_058Lamm__fu____,type,
    aTP_Lamm_fu: fun_nat_nat ).

tff(sy_c_ATP_058Lamm__fv____,type,
    aTP_Lamm_fv: fun_int_int ).

tff(sy_c_ATP_058Lamm__fw____,type,
    aTP_Lamm_fw: fun_Co8499238673658658103nteger ).

tff(sy_c_ATP_058Lamm__fx____,type,
    aTP_Lamm_fx: fun_rat_rat ).

tff(sy_c_ATP_058Lamm__fy____,type,
    aTP_Lamm_fy: fun_nat_nat ).

tff(sy_c_ATP_058Lamm__fz____,type,
    aTP_Lamm_fz: fun_int_int ).

tff(sy_c_ATP_058Lamm__ga____,type,
    aTP_Lamm_ga: fun_he6342020308413958580unit_a > fun_he1921015881043195817it_nat ).

tff(sy_c_ATP_058Lamm__gb____,type,
    aTP_Lamm_gb: fun_he5123621388056232515t_unit > fun_he5234484606919320760it_nat ).

tff(sy_c_ATP_058Lamm__gc____,type,
    aTP_Lamm_gc: fun_rat_rat ).

tff(sy_c_ATP_058Lamm__gd____,type,
    aTP_Lamm_gd: fun_int_int ).

tff(sy_c_ATP_058Lamm__ge____,type,
    aTP_Lamm_ge: fun_nat_nat ).

tff(sy_c_ATP_058Lamm__gf____,type,
    aTP_Lamm_gf: fun_Co8499238673658658103nteger ).

tff(sy_c_ATP_058Lamm__gg____,type,
    aTP_Lamm_gg: rat > fun_nat_fun_rat_rat ).

tff(sy_c_ATP_058Lamm__gh____,type,
    aTP_Lamm_gh: int > fun_nat_fun_int_int ).

tff(sy_c_ATP_058Lamm__gi____,type,
    aTP_Lamm_gi: code_integer > fun_na7078387940644931716nteger ).

tff(sy_c_ATP_058Lamm__gj____,type,
    aTP_Lamm_gj: nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__gk____,type,
    aTP_Lamm_gk: set_Pr1261947904930325089at_nat > fun_se374060482380105870_nat_o ).

tff(sy_c_ATP_058Lamm__gl____,type,
    aTP_Lamm_gl: set_o > fun_set_o_fun_o_o ).

tff(sy_c_ATP_058Lamm__gm____,type,
    aTP_Lamm_gm: set_Pr958786334691620121nt_int > fun_se4767462807232413326_int_o ).

tff(sy_c_ATP_058Lamm__gn____,type,
    aTP_Lamm_gn: set_list_nat > fun_se1865396762483378772_nat_o ).

tff(sy_c_ATP_058Lamm__go____,type,
    aTP_Lamm_go: set_set_nat > fun_se2389560805064864136_nat_o ).

tff(sy_c_ATP_058Lamm__gp____,type,
    aTP_Lamm_gp: set_int > fun_se867122708194164540_int_o ).

tff(sy_c_ATP_058Lamm__gq____,type,
    aTP_Lamm_gq: set_nat > fun_se382663106265794036_nat_o ).

tff(sy_c_ATP_058Lamm__gr____,type,
    aTP_Lamm_gr: fun_Pr3238374608701176162_int_o > fun_fu3068557634114553409_int_o ).

tff(sy_c_ATP_058Lamm__gs____,type,
    aTP_Lamm_gs: fun_list_nat_o > fun_fu1994797156054922755_nat_o ).

tff(sy_c_ATP_058Lamm__gt____,type,
    aTP_Lamm_gt: fun_set_nat_o > fun_fu2408719596384163971_nat_o ).

tff(sy_c_ATP_058Lamm__gu____,type,
    aTP_Lamm_gu: fun_int_o > fun_fu3883539188510166275_int_o ).

tff(sy_c_ATP_058Lamm__gv____,type,
    aTP_Lamm_gv: fun_Pr363642936170290612_nat_o > fun_fu2988548197490738277_nat_o ).

tff(sy_c_ATP_058Lamm__gw____,type,
    aTP_Lamm_gw: fun_Pr1099658227000185396_nat_o > fun_fu714798943701216869_nat_o ).

tff(sy_c_ATP_058Lamm__gx____,type,
    aTP_Lamm_gx: fun_Pr3059359313727237556_nat_o > fun_fu1322491050725084261_nat_o ).

tff(sy_c_ATP_058Lamm__gy____,type,
    aTP_Lamm_gy: fun_nat_o > fun_fu4001317426178549763_nat_o ).

tff(sy_c_ATP_058Lamm__gz____,type,
    aTP_Lamm_gz: assn > fun_Pr6460944699146269935_nat_o ).

tff(sy_c_ATP_058Lamm__ha____,type,
    aTP_Lamm_ha: fun_Pr3238374608701176162_int_o > fun_Pr3238374608701176162_int_o ).

tff(sy_c_ATP_058Lamm__hb____,type,
    aTP_Lamm_hb: fun_list_nat_o > fun_list_nat_o ).

tff(sy_c_ATP_058Lamm__hc____,type,
    aTP_Lamm_hc: fun_set_nat_o > fun_set_nat_o ).

tff(sy_c_ATP_058Lamm__hd____,type,
    aTP_Lamm_hd: fun_nat_o > fun_nat_o ).

tff(sy_c_ATP_058Lamm__he____,type,
    aTP_Lamm_he: fun_int_o > fun_int_o ).

tff(sy_c_ATP_058Lamm__hf____,type,
    aTP_Lamm_hf: set_Pr1261947904930325089at_nat > fun_Pr4796943543331071514_nat_o ).

tff(sy_c_ATP_058Lamm__hg____,type,
    aTP_Lamm_hg: set_o > fun_o_o ).

tff(sy_c_ATP_058Lamm__hh____,type,
    aTP_Lamm_hh: set_Pr958786334691620121nt_int > fun_Pr3238374608701176162_int_o ).

tff(sy_c_ATP_058Lamm__hi____,type,
    aTP_Lamm_hi: set_list_nat > fun_list_nat_o ).

tff(sy_c_ATP_058Lamm__hj____,type,
    aTP_Lamm_hj: set_set_nat > fun_set_nat_o ).

tff(sy_c_ATP_058Lamm__hk____,type,
    aTP_Lamm_hk: set_nat > fun_nat_o ).

tff(sy_c_ATP_058Lamm__hl____,type,
    aTP_Lamm_hl: set_int > fun_int_o ).

tff(sy_c_ATP_058Lamm__hm____,type,
    aTP_Lamm_hm: rat > fun_nat_fun_rat_rat ).

tff(sy_c_ATP_058Lamm__hn____,type,
    aTP_Lamm_hn: rat > fun_rat_fun_rat_rat ).

tff(sy_c_ATP_058Lamm__ho____,type,
    aTP_Lamm_ho: rat > fun_rat_fun_rat_rat ).

tff(sy_c_ATP_058Lamm__hp____,type,
    aTP_Lamm_hp: rat > fun_rat_fun_rat_rat ).

tff(sy_c_ATP_058Lamm__hq____,type,
    aTP_Lamm_hq: rat > fun_rat_fun_rat_rat ).

tff(sy_c_ATP_058Lamm__hr____,type,
    aTP_Lamm_hr: fun_rat_fun_rat_rat ).

tff(sy_c_ATP_058Lamm__hs____,type,
    aTP_Lamm_hs: fun_rat_fun_rat_rat ).

tff(sy_c_ATP_058Lamm__ht____,type,
    aTP_Lamm_ht: fun_int_fun_int_int ).

tff(sy_c_ATP_058Lamm__hu____,type,
    aTP_Lamm_hu: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__hv____,type,
    aTP_Lamm_hv: code_integer > fun_Co8499238673658658103nteger ).

tff(sy_c_ATP_058Lamm__hw____,type,
    aTP_Lamm_hw: fun_rat_fun_rat_rat ).

tff(sy_c_ATP_058Lamm__hx____,type,
    aTP_Lamm_hx: fun_int_fun_int_int ).

tff(sy_c_ATP_058Lamm__hy____,type,
    aTP_Lamm_hy: code_integer > fun_Co8499238673658658103nteger ).

tff(sy_c_ATP_058Lamm__hz____,type,
    aTP_Lamm_hz: fun_int_o > fun_fu5221552698205305849nt_int ).

tff(sy_c_ATP_058Lamm__ia____,type,
    aTP_Lamm_ia: fun_nat_o > fun_fu960360787657196261at_int ).

tff(sy_c_ATP_058Lamm__ib____,type,
    aTP_Lamm_ib: fun_int_o > fun_fu973771059875708133nt_nat ).

tff(sy_c_ATP_058Lamm__ic____,type,
    aTP_Lamm_ic: fun_nat_o > fun_fu5935951186182374353at_nat ).

tff(sy_c_ATP_058Lamm__id____,type,
    aTP_Lamm_id: fun_nat_o > fun_fu108291945127110899et_nat ).

tff(sy_c_ATP_058Lamm__ie____,type,
    aTP_Lamm_ie: fun_list_nat_o > fun_fu9217243180581622273at_nat ).

tff(sy_c_ATP_058Lamm__if____,type,
    aTP_Lamm_if: fun_set_nat_o > fun_fu3253069483457113587at_nat ).

tff(sy_c_ATP_058Lamm__ig____,type,
    aTP_Lamm_ig: fun_Pr3238374608701176162_int_o > fun_fu7573985428940033010nt_nat ).

tff(sy_c_ATP_058Lamm__ih____,type,
    aTP_Lamm_ih: fun_Pr4796943543331071514_nat_o > fun_fu6048389457699795770at_nat ).

tff(sy_c_ATP_058Lamm__ii____,type,
    aTP_Lamm_ii: fun_fu3275390434952296369_nat_o ).

tff(sy_c_ATP_058Lamm__ij____,type,
    aTP_Lamm_ij: fun_int_o > fun_fu1754067819247394666at_nat ).

tff(sy_c_ATP_058Lamm__ik____,type,
    aTP_Lamm_ik: fun_rat_fun_rat_rat ).

tff(sy_c_ATP_058Lamm__il____,type,
    aTP_Lamm_il: fun_nat_nat > fun_fu7002938529518989751at_nat ).

tff(sy_c_ATP_058Lamm__im____,type,
    aTP_Lamm_im: fun_nat_nat > fun_fu8905809029841901679nt_nat ).

tff(sy_c_ATP_058Lamm__in____,type,
    aTP_Lamm_in: fun_nat_int > fun_fu4512855384587262995at_int ).

tff(sy_c_ATP_058Lamm__io____,type,
    aTP_Lamm_io: fun_nat_int > fun_fu6415725884910174923nt_int ).

tff(sy_c_ATP_058Lamm__ip____,type,
    aTP_Lamm_ip: fun_int_nat > fun_fu1731011121079923219at_nat ).

tff(sy_c_ATP_058Lamm__iq____,type,
    aTP_Lamm_iq: fun_int_nat > fun_fu3633881621402835147nt_nat ).

tff(sy_c_ATP_058Lamm__ir____,type,
    aTP_Lamm_ir: fun_int_int > fun_fu8464300013002972271at_int ).

tff(sy_c_ATP_058Lamm__is____,type,
    aTP_Lamm_is: fun_int_int > fun_fu1143798476471108391nt_int ).

tff(sy_c_ATP_058Lamm__it____,type,
    aTP_Lamm_it: fun_set_nat_nat > fun_fu1470602711428196973at_nat ).

tff(sy_c_ATP_058Lamm__iu____,type,
    aTP_Lamm_iu: fun_set_nat_int > fun_fu8203891603351246025at_int ).

tff(sy_c_ATP_058Lamm__iv____,type,
    aTP_Lamm_iv: fun_o_o > fun_se7410564775176163214un_o_o ).

tff(sy_c_ATP_058Lamm__iw____,type,
    aTP_Lamm_iw: fun_o_o > fun_se7410564775176163214un_o_o ).

tff(sy_c_ATP_058Lamm__ix____,type,
    aTP_Lamm_ix: fun_nat_o > fun_se6580191135780181254un_o_o ).

tff(sy_c_ATP_058Lamm__iy____,type,
    aTP_Lamm_iy: fun_nat_o > fun_se5710877183863993418_nat_o ).

tff(sy_c_ATP_058Lamm__iz____,type,
    aTP_Lamm_iz: fun_int_o > fun_se7788308268708561250un_o_o ).

tff(sy_c_ATP_058Lamm__ja____,type,
    aTP_Lamm_ja: fun_int_o > fun_se7460340907868704146_int_o ).

tff(sy_c_ATP_058Lamm__jb____,type,
    aTP_Lamm_jb: fun_o_nat > fun_se659592270355342_nat_o ).

tff(sy_c_ATP_058Lamm__jc____,type,
    aTP_Lamm_jc: fun_o_nat > fun_se8896292107381019842un_o_o ).

tff(sy_c_ATP_058Lamm__jd____,type,
    aTP_Lamm_jd: fun_nat_nat > fun_se2874302864653534470_nat_o ).

tff(sy_c_ATP_058Lamm__je____,type,
    aTP_Lamm_je: fun_nat_nat > fun_se2874302864653534470_nat_o ).

tff(sy_c_ATP_058Lamm__jf____,type,
    aTP_Lamm_jf: fun_int_nat > fun_se2032030289318007650_nat_o ).

tff(sy_c_ATP_058Lamm__jg____,type,
    aTP_Lamm_jg: fun_int_nat > fun_se2103980781540968966_int_o ).

tff(sy_c_ATP_058Lamm__jh____,type,
    aTP_Lamm_jh: fun_o_int > fun_se5483692266196779150_int_o ).

tff(sy_c_ATP_058Lamm__ji____,type,
    aTP_Lamm_ji: fun_o_int > fun_se1227706266003314534un_o_o ).

tff(sy_c_ATP_058Lamm__jj____,type,
    aTP_Lamm_jj: fun_nat_int > fun_se8357335538579958278_int_o ).

tff(sy_c_ATP_058Lamm__jk____,type,
    aTP_Lamm_jk: fun_nat_int > fun_se8285385046356996962_nat_o ).

tff(sy_c_ATP_058Lamm__jl____,type,
    aTP_Lamm_jl: fun_int_int > fun_se7515062963244431458_int_o ).

tff(sy_c_ATP_058Lamm__jm____,type,
    aTP_Lamm_jm: fun_int_int > fun_se7515062963244431458_int_o ).

tff(sy_c_ATP_058Lamm__jn____,type,
    aTP_Lamm_jn: fun_list_nat_o > fun_se6300720826873464854un_o_o ).

tff(sy_c_ATP_058Lamm__jo____,type,
    aTP_Lamm_jo: fun_list_nat_o > fun_se7651465595257595690_nat_o ).

tff(sy_c_ATP_058Lamm__jp____,type,
    aTP_Lamm_jp: fun_int_nat > fun_int_int ).

tff(sy_c_ATP_058Lamm__jq____,type,
    aTP_Lamm_jq: fun_nat_nat > fun_nat_int ).

tff(sy_c_ATP_058Lamm__jr____,type,
    aTP_Lamm_jr: fun_nat_nat > fun_nat_Code_integer ).

tff(sy_c_ATP_058Lamm__js____,type,
    aTP_Lamm_js: fun_nat_nat > fun_nat_nat ).

tff(sy_c_ATP_058Lamm__jt____,type,
    aTP_Lamm_jt: fun_int_int > fun_int_rat ).

tff(sy_c_ATP_058Lamm__ju____,type,
    aTP_Lamm_ju: fun_int_int > fun_int_int ).

tff(sy_c_ATP_058Lamm__jv____,type,
    aTP_Lamm_jv: fun_nat_nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__jw____,type,
    aTP_Lamm_jw: fun_int_int > fun_int_fun_int_int ).

tff(sy_c_ATP_058Lamm__jx____,type,
    aTP_Lamm_jx: fun_nat_nat > fun_nat_nat ).

tff(sy_c_ATP_058Lamm__jy____,type,
    aTP_Lamm_jy: fun_nat_nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__jz____,type,
    aTP_Lamm_jz: fun_nat_nat > fun_na3244735409550470327at_nat ).

tff(sy_c_ATP_058Lamm__ka____,type,
    aTP_Lamm_ka: fun_na2338080690716861136at_nat > fun_na2338080690716861136at_nat ).

tff(sy_c_ATP_058Lamm__kb____,type,
    aTP_Lamm_kb: fun_nat_rat > fun_nat_rat ).

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

tff(sy_c_ATP_058Lamm__kd____,type,
    aTP_Lamm_kd: fun_nat_rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__ke____,type,
    aTP_Lamm_ke: fun_nat_int > fun_nat_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_nat_Code_integer > fun_na7078387940644931716nteger ).

tff(sy_c_ATP_058Lamm__kj____,type,
    aTP_Lamm_kj: fun_na2338080690716861136at_nat > fun_na1945018436707187408at_nat ).

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

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

tff(sy_c_ATP_058Lamm__km____,type,
    aTP_Lamm_km: fun_nat_nat > fun_nat_fun_nat_nat ).

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

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

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

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

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

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

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

tff(sy_c_ATP_058Lamm__ku____,type,
    aTP_Lamm_ku: fun_o_Code_integer > fun_fu2786480252100984515nteger ).

tff(sy_c_ATP_058Lamm__kv____,type,
    aTP_Lamm_kv: fun_nat_Code_integer > fun_fu2499394402704830801nteger ).

tff(sy_c_ATP_058Lamm__kw____,type,
    aTP_Lamm_kw: fun_int_Code_integer > fun_fu3628599891403802633nteger ).

tff(sy_c_ATP_058Lamm__kx____,type,
    aTP_Lamm_kx: fun_o_rat > fun_fu1989461620956114563_o_rat ).

tff(sy_c_ATP_058Lamm__ky____,type,
    aTP_Lamm_ky: fun_nat_rat > fun_fu8328862064278946247at_rat ).

tff(sy_c_ATP_058Lamm__kz____,type,
    aTP_Lamm_kz: fun_int_rat > fun_fu1008360527747082367nt_rat ).

tff(sy_c_ATP_058Lamm__la____,type,
    aTP_Lamm_la: fun_o_int > fun_fu6940589326549028739_o_int ).

tff(sy_c_ATP_058Lamm__lb____,type,
    aTP_Lamm_lb: fun_nat_int > fun_fu8464300013002972271at_int ).

tff(sy_c_ATP_058Lamm__lc____,type,
    aTP_Lamm_lc: fun_int_int > fun_fu1143798476471108391nt_int ).

tff(sy_c_ATP_058Lamm__ld____,type,
    aTP_Lamm_ld: fun_se7586849273513198688nteger > fun_fu5510336631606277565nteger ).

tff(sy_c_ATP_058Lamm__le____,type,
    aTP_Lamm_le: fun_Pr2602117396471556196at_nat > fun_fu7732904369943091395at_nat ).

tff(sy_c_ATP_058Lamm__lf____,type,
    aTP_Lamm_lf: fun_o_nat > fun_fu6701067901881939075_o_nat ).

tff(sy_c_ATP_058Lamm__lg____,type,
    aTP_Lamm_lg: fun_set_nat_nat > fun_fu1590586450949517859at_nat ).

tff(sy_c_ATP_058Lamm__lh____,type,
    aTP_Lamm_lh: fun_int_nat > fun_fu8905809029841901679nt_nat ).

tff(sy_c_ATP_058Lamm__li____,type,
    aTP_Lamm_li: fun_nat_nat > fun_fu7002938529518989751at_nat ).

tff(sy_c_ATP_058Lamm__lj____,type,
    aTP_Lamm_lj: fun_nat_fun_nat_nat > fun_se1179353343636554400at_nat ).

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

tff(sy_c_ATP_058Lamm__ll____,type,
    aTP_Lamm_ll: fun_nat_fun_nat_nat > fun_se1179353343636554400at_nat ).

tff(sy_c_ATP_058Lamm__lm____,type,
    aTP_Lamm_lm: fun_int_fun_int_int > fun_se4827636401813558708nt_int ).

tff(sy_c_ATP_058Lamm__ln____,type,
    aTP_Lamm_ln: fun_int_fun_int_int > fun_int_fun_int_int ).

tff(sy_c_ATP_058Lamm__lo____,type,
    aTP_Lamm_lo: fun_int_fun_int_int > fun_se4827636401813558708nt_int ).

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: fun_int_int > fun_fu1143798476471108391nt_int ).

tff(sy_c_ATP_058Lamm__lr____,type,
    aTP_Lamm_lr: fun_nat_nat > fun_fu4413992040155564420at_nat ).

tff(sy_c_ATP_058Lamm__ls____,type,
    aTP_Lamm_ls: fun_nat_nat > fun_fu2454420574033104442at_nat ).

tff(sy_c_ATP_058Lamm__lt____,type,
    aTP_Lamm_lt: fun_int_int > fun_fu923337204662071120nt_int ).

tff(sy_c_ATP_058Lamm__lu____,type,
    aTP_Lamm_lu: fun_int_int > fun_fu3739269683040691718nt_int ).

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

tff(sy_c_ATP_058Lamm__lw____,type,
    aTP_Lamm_lw: fun_int_int > fun_int_fun_int_int ).

tff(sy_c_ATP_058Lamm__lx____,type,
    aTP_Lamm_lx: nat > fun_fu7002938529518989751at_nat ).

tff(sy_c_ATP_058Lamm__ly____,type,
    aTP_Lamm_ly: int > fun_fu1143798476471108391nt_int ).

tff(sy_c_ATP_058Lamm__lz____,type,
    aTP_Lamm_lz: fun_int_int > fun_fu1143798476471108391nt_int ).

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

tff(sy_c_ATP_058Lamm__mb____,type,
    aTP_Lamm_mb: nat > fun_ra8505079968215247908at_rat ).

tff(sy_c_ATP_058Lamm__mc____,type,
    aTP_Lamm_mc: nat > fun_ra8505079968215247908at_rat ).

tff(sy_c_ATP_058Lamm__md____,type,
    aTP_Lamm_md: rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__me____,type,
    aTP_Lamm_me: code_natural > fun_nat_Code_natural ).

tff(sy_c_ATP_058Lamm__mf____,type,
    aTP_Lamm_mf: code_integer > fun_nat_Code_integer ).

tff(sy_c_ATP_058Lamm__mg____,type,
    aTP_Lamm_mg: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__mh____,type,
    aTP_Lamm_mh: fun_int_fun_nat_int ).

tff(sy_c_ATP_058Lamm__mi____,type,
    aTP_Lamm_mi: nat > fun_ra8505079968215247908at_rat ).

tff(sy_c_ATP_058Lamm__mj____,type,
    aTP_Lamm_mj: fun_nat_nat ).

tff(sy_c_ATP_058Lamm__mk____,type,
    aTP_Lamm_mk: fun_nat_int ).

tff(sy_c_ATP_058Lamm__ml____,type,
    aTP_Lamm_ml: fun_int_int ).

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

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

tff(sy_c_ATP_058Lamm__mo____,type,
    aTP_Lamm_mo: fun_nat_fun_nat_nat > fun_se1179353343636554400at_nat ).

tff(sy_c_ATP_058Lamm__mp____,type,
    aTP_Lamm_mp: fun_nat_fun_nat_nat > fun_se1179353343636554400at_nat ).

tff(sy_c_ATP_058Lamm__mq____,type,
    aTP_Lamm_mq: fun_nat_fun_nat_int > fun_se7912642235559603452at_int ).

tff(sy_c_ATP_058Lamm__mr____,type,
    aTP_Lamm_mr: fun_nat_fun_nat_int > fun_nat_fun_nat_int ).

tff(sy_c_ATP_058Lamm__ms____,type,
    aTP_Lamm_ms: fun_nat_fun_nat_int > fun_se7912642235559603452at_int ).

tff(sy_c_ATP_058Lamm__mt____,type,
    aTP_Lamm_mt: fun_nat_fun_int_int > fun_se2640714827120536920at_int ).

tff(sy_c_ATP_058Lamm__mu____,type,
    aTP_Lamm_mu: fun_nat_fun_int_int > fun_int_fun_nat_int ).

tff(sy_c_ATP_058Lamm__mv____,type,
    aTP_Lamm_mv: fun_nat_fun_int_int > fun_se876191773397849432nt_int ).

tff(sy_c_ATP_058Lamm__mw____,type,
    aTP_Lamm_mw: fun_int_fun_nat_int > fun_se876191773397849432nt_int ).

tff(sy_c_ATP_058Lamm__mx____,type,
    aTP_Lamm_mx: fun_int_fun_nat_int > fun_nat_fun_int_int ).

tff(sy_c_ATP_058Lamm__my____,type,
    aTP_Lamm_my: fun_int_fun_nat_int > fun_se2640714827120536920at_int ).

tff(sy_c_ATP_058Lamm__mz____,type,
    aTP_Lamm_mz: fun_int_fun_int_int > fun_se4827636401813558708nt_int ).

tff(sy_c_ATP_058Lamm__na____,type,
    aTP_Lamm_na: fun_int_fun_int_int > fun_se4827636401813558708nt_int ).

tff(sy_c_ATP_058Lamm__nb____,type,
    aTP_Lamm_nb: fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__nc____,type,
    aTP_Lamm_nc: fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__nd____,type,
    aTP_Lamm_nd: fun_nat_nat > fun_fu7002938529518989751at_nat ).

tff(sy_c_ATP_058Lamm__ne____,type,
    aTP_Lamm_ne: fun_nat_int > fun_fu8464300013002972271at_int ).

tff(sy_c_ATP_058Lamm__nf____,type,
    aTP_Lamm_nf: fun_int_int > fun_fu1143798476471108391nt_int ).

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

tff(sy_c_ATP_058Lamm__nh____,type,
    aTP_Lamm_nh: fun_nat_int > fun_nat_fun_nat_int ).

tff(sy_c_ATP_058Lamm__ni____,type,
    aTP_Lamm_ni: fun_int_int > fun_nat_fun_int_int ).

tff(sy_c_ATP_058Lamm__nj____,type,
    aTP_Lamm_nj: fun_nat_int > fun_int_fun_nat_int ).

tff(sy_c_ATP_058Lamm__nk____,type,
    aTP_Lamm_nk: fun_nat_Code_integer > fun_nat_Code_integer ).

tff(sy_c_ATP_058Lamm__nl____,type,
    aTP_Lamm_nl: fun_nat_assn > fun_nat_assn ).

tff(sy_c_ATP_058Lamm__nm____,type,
    aTP_Lamm_nm: fun_num_fun_num_o ).

tff(sy_c_ATP_058Lamm__nn____,type,
    aTP_Lamm_nn: nat > fun_fu7002938529518989751at_nat ).

tff(sy_c_ATP_058Lamm__no____,type,
    aTP_Lamm_no: int > fun_fu4512855384587262995at_int ).

tff(sy_c_ATP_058Lamm__np____,type,
    aTP_Lamm_np: int > fun_fu6415725884910174923nt_int ).

tff(sy_c_ATP_058Lamm__nq____,type,
    aTP_Lamm_nq: fun_nat_int > fun_nat_fun_nat_int ).

tff(sy_c_ATP_058Lamm__nr____,type,
    aTP_Lamm_nr: fun_nat_int > fun_na7654559027617327379at_int ).

tff(sy_c_ATP_058Lamm__ns____,type,
    aTP_Lamm_ns: nat > fun_fu6327154928193479288nteger ).

tff(sy_c_ATP_058Lamm__nt____,type,
    aTP_Lamm_nt: nat > fun_fu6327154928193479288nteger ).

tff(sy_c_ATP_058Lamm__nu____,type,
    aTP_Lamm_nu: nat > fun_fu3959123160307459545t_assn ).

tff(sy_c_ATP_058Lamm__nv____,type,
    aTP_Lamm_nv: nat > fun_fu3959123160307459545t_assn ).

tff(sy_c_ATP_058Lamm__nw____,type,
    aTP_Lamm_nw: nat > fun_fu6579618355643354601at_rat ).

tff(sy_c_ATP_058Lamm__nx____,type,
    aTP_Lamm_nx: nat > fun_fu6579618355643354601at_rat ).

tff(sy_c_ATP_058Lamm__ny____,type,
    aTP_Lamm_ny: nat > fun_fu5935951186182374353at_nat ).

tff(sy_c_ATP_058Lamm__nz____,type,
    aTP_Lamm_nz: nat > fun_fu5935951186182374353at_nat ).

tff(sy_c_ATP_058Lamm__oa____,type,
    aTP_Lamm_oa: nat > fun_fu960360787657196261at_int ).

tff(sy_c_ATP_058Lamm__ob____,type,
    aTP_Lamm_ob: nat > fun_fu960360787657196261at_int ).

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

tff(sy_c_ATP_058Lamm__od____,type,
    aTP_Lamm_od: fun_nat_int > fun_nat_int ).

tff(sy_c_ATP_058Lamm__oe____,type,
    aTP_Lamm_oe: fun_nat_Code_integer > fun_na7078387940644931716nteger ).

tff(sy_c_ATP_058Lamm__of____,type,
    aTP_Lamm_of: fun_nat_assn > fun_na2454419004871237530n_assn ).

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

tff(sy_c_ATP_058Lamm__oh____,type,
    aTP_Lamm_oh: fun_nat_nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__oi____,type,
    aTP_Lamm_oi: fun_nat_int > fun_nat_fun_int_int ).

tff(sy_c_ATP_058Lamm__oj____,type,
    aTP_Lamm_oj: rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__ok____,type,
    aTP_Lamm_ok: rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__ol____,type,
    aTP_Lamm_ol: code_integer > fun_nat_Code_integer ).

tff(sy_c_ATP_058Lamm__om____,type,
    aTP_Lamm_om: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__on____,type,
    aTP_Lamm_on: fun_int_fun_nat_int ).

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

tff(sy_c_ATP_058Lamm__op____,type,
    aTP_Lamm_op: code_integer > fun_na8135612474337863287nteger ).

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

tff(sy_c_ATP_058Lamm__or____,type,
    aTP_Lamm_or: int > fun_nat_fun_nat_int ).

tff(sy_c_ATP_058Lamm__os____,type,
    aTP_Lamm_os: rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__ot____,type,
    aTP_Lamm_ot: nat > fun_fu6327154928193479288nteger ).

tff(sy_c_ATP_058Lamm__ou____,type,
    aTP_Lamm_ou: nat > fun_fu695563712076452518at_nat ).

tff(sy_c_ATP_058Lamm__ov____,type,
    aTP_Lamm_ov: nat > fun_fu695563712076452518at_nat ).

tff(sy_c_ATP_058Lamm__ow____,type,
    aTP_Lamm_ow: nat > fun_fu6579618355643354601at_rat ).

tff(sy_c_ATP_058Lamm__ox____,type,
    aTP_Lamm_ox: nat > fun_fu960360787657196261at_int ).

tff(sy_c_ATP_058Lamm__oy____,type,
    aTP_Lamm_oy: nat > fun_fu5935951186182374353at_nat ).

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

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_Code_integer ).

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

tff(sy_c_ATP_058Lamm__pd____,type,
    aTP_Lamm_pd: fun_nat_fun_nat_int ).

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

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

tff(sy_c_ATP_058Lamm__pg____,type,
    aTP_Lamm_pg: fun_nat_rat > fun_fu8328862064278946247at_rat ).

tff(sy_c_ATP_058Lamm__ph____,type,
    aTP_Lamm_ph: $o > fun_fu2786480252100984515nteger ).

tff(sy_c_ATP_058Lamm__pi____,type,
    aTP_Lamm_pi: nat > fun_fu2499394402704830801nteger ).

tff(sy_c_ATP_058Lamm__pj____,type,
    aTP_Lamm_pj: int > fun_fu3628599891403802633nteger ).

tff(sy_c_ATP_058Lamm__pk____,type,
    aTP_Lamm_pk: code_integer > fun_fu1763652423599918571nteger ).

tff(sy_c_ATP_058Lamm__pl____,type,
    aTP_Lamm_pl: $o > fun_fu1989461620956114563_o_rat ).

tff(sy_c_ATP_058Lamm__pm____,type,
    aTP_Lamm_pm: nat > fun_fu8328862064278946247at_rat ).

tff(sy_c_ATP_058Lamm__pn____,type,
    aTP_Lamm_pn: int > fun_fu1008360527747082367nt_rat ).

tff(sy_c_ATP_058Lamm__po____,type,
    aTP_Lamm_po: code_integer > fun_fu722387130092441185er_rat ).

tff(sy_c_ATP_058Lamm__pp____,type,
    aTP_Lamm_pp: $o > fun_fu6701067901881939075_o_nat ).

tff(sy_c_ATP_058Lamm__pq____,type,
    aTP_Lamm_pq: int > fun_fu8905809029841901679nt_nat ).

tff(sy_c_ATP_058Lamm__pr____,type,
    aTP_Lamm_pr: $o > fun_fu2786480252100984515nteger ).

tff(sy_c_ATP_058Lamm__ps____,type,
    aTP_Lamm_ps: nat > fun_fu2499394402704830801nteger ).

tff(sy_c_ATP_058Lamm__pt____,type,
    aTP_Lamm_pt: int > fun_fu3628599891403802633nteger ).

tff(sy_c_ATP_058Lamm__pu____,type,
    aTP_Lamm_pu: code_integer > fun_fu1763652423599918571nteger ).

tff(sy_c_ATP_058Lamm__pv____,type,
    aTP_Lamm_pv: $o > fun_fu1989461620956114563_o_rat ).

tff(sy_c_ATP_058Lamm__pw____,type,
    aTP_Lamm_pw: nat > fun_fu8328862064278946247at_rat ).

tff(sy_c_ATP_058Lamm__px____,type,
    aTP_Lamm_px: int > fun_fu1008360527747082367nt_rat ).

tff(sy_c_ATP_058Lamm__py____,type,
    aTP_Lamm_py: code_integer > fun_fu722387130092441185er_rat ).

tff(sy_c_ATP_058Lamm__pz____,type,
    aTP_Lamm_pz: $o > fun_fu6701067901881939075_o_nat ).

tff(sy_c_ATP_058Lamm__qa____,type,
    aTP_Lamm_qa: int > fun_fu8905809029841901679nt_nat ).

tff(sy_c_ATP_058Lamm__qb____,type,
    aTP_Lamm_qb: $o > fun_fu2786480252100984515nteger ).

tff(sy_c_ATP_058Lamm__qc____,type,
    aTP_Lamm_qc: nat > fun_fu2499394402704830801nteger ).

tff(sy_c_ATP_058Lamm__qd____,type,
    aTP_Lamm_qd: int > fun_fu3628599891403802633nteger ).

tff(sy_c_ATP_058Lamm__qe____,type,
    aTP_Lamm_qe: code_integer > fun_fu1763652423599918571nteger ).

tff(sy_c_ATP_058Lamm__qf____,type,
    aTP_Lamm_qf: $o > fun_fu704708528062463619o_assn ).

tff(sy_c_ATP_058Lamm__qg____,type,
    aTP_Lamm_qg: nat > fun_fu2995346897264222311t_assn ).

tff(sy_c_ATP_058Lamm__qh____,type,
    aTP_Lamm_qh: int > fun_fu6568868615275847455t_assn ).

tff(sy_c_ATP_058Lamm__qi____,type,
    aTP_Lamm_qi: code_integer > fun_fu5082522747898945281r_assn ).

tff(sy_c_ATP_058Lamm__qj____,type,
    aTP_Lamm_qj: $o > fun_fu1989461620956114563_o_rat ).

tff(sy_c_ATP_058Lamm__qk____,type,
    aTP_Lamm_qk: nat > fun_fu8328862064278946247at_rat ).

tff(sy_c_ATP_058Lamm__ql____,type,
    aTP_Lamm_ql: $o > fun_fu2786480252100984515nteger ).

tff(sy_c_ATP_058Lamm__qm____,type,
    aTP_Lamm_qm: nat > fun_fu2499394402704830801nteger ).

tff(sy_c_ATP_058Lamm__qn____,type,
    aTP_Lamm_qn: int > fun_fu3628599891403802633nteger ).

tff(sy_c_ATP_058Lamm__qo____,type,
    aTP_Lamm_qo: code_integer > fun_fu1763652423599918571nteger ).

tff(sy_c_ATP_058Lamm__qp____,type,
    aTP_Lamm_qp: $o > fun_fu704708528062463619o_assn ).

tff(sy_c_ATP_058Lamm__qq____,type,
    aTP_Lamm_qq: nat > fun_fu2995346897264222311t_assn ).

tff(sy_c_ATP_058Lamm__qr____,type,
    aTP_Lamm_qr: int > fun_fu6568868615275847455t_assn ).

tff(sy_c_ATP_058Lamm__qs____,type,
    aTP_Lamm_qs: code_integer > fun_fu5082522747898945281r_assn ).

tff(sy_c_ATP_058Lamm__qt____,type,
    aTP_Lamm_qt: $o > fun_fu1989461620956114563_o_rat ).

tff(sy_c_ATP_058Lamm__qu____,type,
    aTP_Lamm_qu: nat > fun_fu8328862064278946247at_rat ).

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

tff(sy_c_ATP_058Lamm__qw____,type,
    aTP_Lamm_qw: set_int > fun_se2051630309975736070_int_o ).

tff(sy_c_ATP_058Lamm__qx____,type,
    aTP_Lamm_qx: set_Code_integer > fun_se5529282185252607396eger_o ).

tff(sy_c_ATP_058Lamm__qy____,type,
    aTP_Lamm_qy: set_Pr1261947904930325089at_nat > fun_se5639961202347202350_nat_o ).

tff(sy_c_ATP_058Lamm__qz____,type,
    aTP_Lamm_qz: set_nat > fun_se2170385732985038782_nat_o ).

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: fun_nat_o > fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__rc____,type,
    aTP_Lamm_rc: fun_nat_nat > fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__rd____,type,
    aTP_Lamm_rd: set_nat > fun_fu9085782028513977109_nat_o ).

tff(sy_c_ATP_058Lamm__re____,type,
    aTP_Lamm_re: set_nat > fun_fu5572043292636673283_o_nat ).

tff(sy_c_ATP_058Lamm__rf____,type,
    aTP_Lamm_rf: fun_o_fun_nat_nat > fun_nat_fun_o_nat ).

tff(sy_c_ATP_058Lamm__rg____,type,
    aTP_Lamm_rg: set_o > fun_fu7066070072329265113un_o_o ).

tff(sy_c_ATP_058Lamm__rh____,type,
    aTP_Lamm_rh: set_o > fun_fu4422690180167752641at_nat ).

tff(sy_c_ATP_058Lamm__ri____,type,
    aTP_Lamm_ri: set_nat > fun_fu927436805355353155_nat_o ).

tff(sy_c_ATP_058Lamm__rj____,type,
    aTP_Lamm_rj: set_nat > fun_fu6398909621167755501nt_nat ).

tff(sy_c_ATP_058Lamm__rk____,type,
    aTP_Lamm_rk: fun_int_fun_nat_nat > fun_nat_fun_int_nat ).

tff(sy_c_ATP_058Lamm__rl____,type,
    aTP_Lamm_rl: set_int > fun_fu7493154764834273859_int_o ).

tff(sy_c_ATP_058Lamm__rm____,type,
    aTP_Lamm_rm: set_int > fun_fu3180206027003385489at_nat ).

tff(sy_c_ATP_058Lamm__rn____,type,
    aTP_Lamm_rn: set_nat > fun_fu8096999557949053699_nat_o ).

tff(sy_c_ATP_058Lamm__ro____,type,
    aTP_Lamm_ro: set_nat > fun_fu4176442768802536380er_nat ).

tff(sy_c_ATP_058Lamm__rp____,type,
    aTP_Lamm_rp: fun_Co3312552843764780407at_nat > fun_na5492297256294505719er_nat ).

tff(sy_c_ATP_058Lamm__rq____,type,
    aTP_Lamm_rq: set_Code_integer > fun_fu6309186030181755779eger_o ).

tff(sy_c_ATP_058Lamm__rr____,type,
    aTP_Lamm_rr: set_Code_integer > fun_fu1095011443884336175at_nat ).

tff(sy_c_ATP_058Lamm__rs____,type,
    aTP_Lamm_rs: set_int > fun_fu3997622796447777885_int_o ).

tff(sy_c_ATP_058Lamm__rt____,type,
    aTP_Lamm_rt: set_int > fun_fu5188028542856013059_o_int ).

tff(sy_c_ATP_058Lamm__ru____,type,
    aTP_Lamm_ru: fun_o_fun_int_int > fun_int_fun_o_int ).

tff(sy_c_ATP_058Lamm__rv____,type,
    aTP_Lamm_rv: set_o > fun_fu2881495813246281945un_o_o ).

tff(sy_c_ATP_058Lamm__rw____,type,
    aTP_Lamm_rw: set_o > fun_fu8188496484073410517nt_int ).

tff(sy_c_ATP_058Lamm__rx____,type,
    aTP_Lamm_rx: set_int > fun_fu5646853623633916995_int_o ).

tff(sy_c_ATP_058Lamm__ry____,type,
    aTP_Lamm_ry: set_int > fun_fu4584101141160326545at_int ).

tff(sy_c_ATP_058Lamm__rz____,type,
    aTP_Lamm_rz: set_nat > fun_fu8304507701009772099_nat_o ).

tff(sy_c_ATP_058Lamm__sa____,type,
    aTP_Lamm_sa: set_nat > fun_fu7802804735324696557nt_int ).

tff(sy_c_ATP_058Lamm__sb____,type,
    aTP_Lamm_sb: set_int > fun_fu354264921998911491_int_o ).

tff(sy_c_ATP_058Lamm__sc____,type,
    aTP_Lamm_sc: set_int > fun_fu8759299743271765508er_int ).

tff(sy_c_ATP_058Lamm__sd____,type,
    aTP_Lamm_sd: fun_Co3009391273526075439nt_int > fun_in2718163036992669103er_int ).

tff(sy_c_ATP_058Lamm__se____,type,
    aTP_Lamm_se: set_Code_integer > fun_fu7527167023631164035eger_o ).

tff(sy_c_ATP_058Lamm__sf____,type,
    aTP_Lamm_sf: set_Code_integer > fun_fu1493779525457017043nt_int ).

tff(sy_c_ATP_058Lamm__sg____,type,
    aTP_Lamm_sg: set_o > fun_fu865690600344432797un_o_o ).

tff(sy_c_ATP_058Lamm__sh____,type,
    aTP_Lamm_sh: set_o > fun_fu4640364191220304065at_nat ).

tff(sy_c_ATP_058Lamm__si____,type,
    aTP_Lamm_si: fun_nat_fun_o_nat > fun_o_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__sj____,type,
    aTP_Lamm_sj: set_nat > fun_fu2885402556529144793_nat_o ).

tff(sy_c_ATP_058Lamm__sk____,type,
    aTP_Lamm_sk: set_nat > fun_fu4618758991028193283_o_nat ).

tff(sy_c_ATP_058Lamm__sl____,type,
    aTP_Lamm_sl: set_int > fun_fu9148247276982919825at_nat ).

tff(sy_c_ATP_058Lamm__sm____,type,
    aTP_Lamm_sm: fun_nat_fun_int_nat > fun_int_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__sn____,type,
    aTP_Lamm_sn: set_nat > fun_fu3143578834292514029nt_nat ).

tff(sy_c_ATP_058Lamm__so____,type,
    aTP_Lamm_so: set_Code_integer > fun_fu7601192970096278275eger_o ).

tff(sy_c_ATP_058Lamm__sp____,type,
    aTP_Lamm_sp: set_Code_integer > fun_fu8336550301028287791at_nat ).

tff(sy_c_ATP_058Lamm__sq____,type,
    aTP_Lamm_sq: fun_na5492297256294505719er_nat > fun_Co3312552843764780407at_nat ).

tff(sy_c_ATP_058Lamm__sr____,type,
    aTP_Lamm_sr: set_nat > fun_fu165634461008800387_nat_o ).

tff(sy_c_ATP_058Lamm__ss____,type,
    aTP_Lamm_ss: set_nat > fun_fu792289584686185660er_nat ).

tff(sy_c_ATP_058Lamm__st____,type,
    aTP_Lamm_st: set_nat > fun_fu6436183194761443139_nat_o ).

tff(sy_c_ATP_058Lamm__su____,type,
    aTP_Lamm_su: set_nat > fun_fu4073184632390761289at_nat ).

tff(sy_c_ATP_058Lamm__sv____,type,
    aTP_Lamm_sv: set_nat > fun_fu6436183194761443139_nat_o ).

tff(sy_c_ATP_058Lamm__sw____,type,
    aTP_Lamm_sw: set_nat > fun_fu4073184632390761289at_nat ).

tff(sy_c_ATP_058Lamm__sx____,type,
    aTP_Lamm_sx: set_nat > fun_fu5572043292636673283_o_nat ).

tff(sy_c_ATP_058Lamm__sy____,type,
    aTP_Lamm_sy: set_o > fun_fu4422690180167752641at_nat ).

tff(sy_c_ATP_058Lamm__sz____,type,
    aTP_Lamm_sz: set_nat > fun_fu6398909621167755501nt_nat ).

tff(sy_c_ATP_058Lamm__ta____,type,
    aTP_Lamm_ta: set_int > fun_fu3180206027003385489at_nat ).

tff(sy_c_ATP_058Lamm__tb____,type,
    aTP_Lamm_tb: set_nat > fun_fu4176442768802536380er_nat ).

tff(sy_c_ATP_058Lamm__tc____,type,
    aTP_Lamm_tc: set_Code_integer > fun_fu1095011443884336175at_nat ).

tff(sy_c_ATP_058Lamm__td____,type,
    aTP_Lamm_td: set_nat > fun_fu731591423980087811_o_int ).

tff(sy_c_ATP_058Lamm__te____,type,
    aTP_Lamm_te: fun_o_fun_nat_int > fun_nat_fun_o_int ).

tff(sy_c_ATP_058Lamm__tf____,type,
    aTP_Lamm_tf: set_o > fun_fu1432006813369012345at_int ).

tff(sy_c_ATP_058Lamm__tg____,type,
    aTP_Lamm_tg: set_nat > fun_fu202463024092729020er_int ).

tff(sy_c_ATP_058Lamm__th____,type,
    aTP_Lamm_th: fun_Co822469698833053651at_int > fun_na3002214111362778963er_int ).

tff(sy_c_ATP_058Lamm__ti____,type,
    aTP_Lamm_ti: set_Code_integer > fun_fu4854240044958656303at_int ).

tff(sy_c_ATP_058Lamm__tj____,type,
    aTP_Lamm_tj: set_int > fun_fu5188028542856013059_o_int ).

tff(sy_c_ATP_058Lamm__tk____,type,
    aTP_Lamm_tk: set_o > fun_fu8188496484073410517nt_int ).

tff(sy_c_ATP_058Lamm__tl____,type,
    aTP_Lamm_tl: set_int > fun_fu8759299743271765508er_int ).

tff(sy_c_ATP_058Lamm__tm____,type,
    aTP_Lamm_tm: set_Code_integer > fun_fu1493779525457017043nt_int ).

tff(sy_c_ATP_058Lamm__tn____,type,
    aTP_Lamm_tn: set_o > fun_fu4640364191220304065at_nat ).

tff(sy_c_ATP_058Lamm__to____,type,
    aTP_Lamm_to: set_nat > fun_fu4618758991028193283_o_nat ).

tff(sy_c_ATP_058Lamm__tp____,type,
    aTP_Lamm_tp: set_int > fun_fu9148247276982919825at_nat ).

tff(sy_c_ATP_058Lamm__tq____,type,
    aTP_Lamm_tq: set_nat > fun_fu3143578834292514029nt_nat ).

tff(sy_c_ATP_058Lamm__tr____,type,
    aTP_Lamm_tr: set_Code_integer > fun_fu8336550301028287791at_nat ).

tff(sy_c_ATP_058Lamm__ts____,type,
    aTP_Lamm_ts: set_nat > fun_fu792289584686185660er_nat ).

tff(sy_c_ATP_058Lamm__tt____,type,
    aTP_Lamm_tt: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__tu____,type,
    aTP_Lamm_tu: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__tv____,type,
    aTP_Lamm_tv: set_o > fun_fu3819205551299168222un_o_o ).

tff(sy_c_ATP_058Lamm__tw____,type,
    aTP_Lamm_tw: set_o > fun_fu8716660765737269145un_o_o ).

tff(sy_c_ATP_058Lamm__tx____,type,
    aTP_Lamm_tx: set_nat > fun_fu5838931486547983824_nat_o ).

tff(sy_c_ATP_058Lamm__ty____,type,
    aTP_Lamm_ty: set_nat > fun_fu8532066432991188095_nat_o ).

tff(sy_c_ATP_058Lamm__tz____,type,
    aTP_Lamm_tz: set_int > fun_fu5271299750659287320_int_o ).

tff(sy_c_ATP_058Lamm__ua____,type,
    aTP_Lamm_ua: set_int > fun_fu63825462011493995_int_o ).

tff(sy_c_ATP_058Lamm__ub____,type,
    aTP_Lamm_ub: set_Code_integer > fun_fu855086506144988854eger_o ).

tff(sy_c_ATP_058Lamm__uc____,type,
    aTP_Lamm_uc: set_Code_integer > fun_fu4083202260958266008eger_o ).

tff(sy_c_ATP_058Lamm__ud____,type,
    aTP_Lamm_ud: set_o > fun_fu6337961365400742745un_o_o ).

tff(sy_c_ATP_058Lamm__ue____,type,
    aTP_Lamm_ue: set_o > fun_fu2641566564518291801un_o_o ).

tff(sy_c_ATP_058Lamm__uf____,type,
    aTP_Lamm_uf: set_nat > fun_fu7145897819039579925_nat_o ).

tff(sy_c_ATP_058Lamm__ug____,type,
    aTP_Lamm_ug: set_nat > fun_fu1806770557662746761_nat_o ).

tff(sy_c_ATP_058Lamm__uh____,type,
    aTP_Lamm_uh: set_int > fun_fu3473604023002724957_int_o ).

tff(sy_c_ATP_058Lamm__ui____,type,
    aTP_Lamm_ui: set_int > fun_fu7714913403879666293_int_o ).

tff(sy_c_ATP_058Lamm__uj____,type,
    aTP_Lamm_uj: set_Code_integer > fun_fu4523568072766896123eger_o ).

tff(sy_c_ATP_058Lamm__uk____,type,
    aTP_Lamm_uk: set_Code_integer > fun_fu7141499545569947298eger_o ).

tff(sy_c_ATP_058Lamm__ul____,type,
    aTP_Lamm_ul: set_o > fun_fu6781074197733401937un_o_o ).

tff(sy_c_ATP_058Lamm__um____,type,
    aTP_Lamm_um: set_o > fun_fu5915781709770235737un_o_o ).

tff(sy_c_ATP_058Lamm__un____,type,
    aTP_Lamm_un: set_nat > fun_fu5477517762487726877_nat_o ).

tff(sy_c_ATP_058Lamm__uo____,type,
    aTP_Lamm_uo: set_nat > fun_fu6075382464776002713_nat_o ).

tff(sy_c_ATP_058Lamm__up____,type,
    aTP_Lamm_up: set_o > fun_fu3819205551299168222un_o_o ).

tff(sy_c_ATP_058Lamm__uq____,type,
    aTP_Lamm_uq: set_o > fun_fu8716660765737269145un_o_o ).

tff(sy_c_ATP_058Lamm__ur____,type,
    aTP_Lamm_ur: set_nat > fun_fu5838931486547983824_nat_o ).

tff(sy_c_ATP_058Lamm__us____,type,
    aTP_Lamm_us: set_nat > fun_fu8532066432991188095_nat_o ).

tff(sy_c_ATP_058Lamm__ut____,type,
    aTP_Lamm_ut: set_int > fun_fu5271299750659287320_int_o ).

tff(sy_c_ATP_058Lamm__uu____,type,
    aTP_Lamm_uu: set_int > fun_fu63825462011493995_int_o ).

tff(sy_c_ATP_058Lamm__uv____,type,
    aTP_Lamm_uv: set_Code_integer > fun_fu855086506144988854eger_o ).

tff(sy_c_ATP_058Lamm__uw____,type,
    aTP_Lamm_uw: set_Code_integer > fun_fu4083202260958266008eger_o ).

tff(sy_c_ATP_058Lamm__ux____,type,
    aTP_Lamm_ux: set_o > fun_fu3143923321121214633un_o_o ).

tff(sy_c_ATP_058Lamm__uy____,type,
    aTP_Lamm_uy: set_o > fun_fu5861359551626625881un_o_o ).

tff(sy_c_ATP_058Lamm__uz____,type,
    aTP_Lamm_uz: set_nat > fun_fu1116345184836134341_nat_o ).

tff(sy_c_ATP_058Lamm__va____,type,
    aTP_Lamm_va: set_nat > fun_fu3109912920687205353_nat_o ).

tff(sy_c_ATP_058Lamm__vb____,type,
    aTP_Lamm_vb: set_int > fun_fu2934462529731303693_int_o ).

tff(sy_c_ATP_058Lamm__vc____,type,
    aTP_Lamm_vc: set_int > fun_fu607692679349591509_int_o ).

tff(sy_c_ATP_058Lamm__vd____,type,
    aTP_Lamm_vd: set_Code_integer > fun_fu4844139638962010283eger_o ).

tff(sy_c_ATP_058Lamm__ve____,type,
    aTP_Lamm_ve: set_Code_integer > fun_fu4735791719682850818eger_o ).

tff(sy_c_ATP_058Lamm__vf____,type,
    aTP_Lamm_vf: set_o > fun_fu6337961365400742745un_o_o ).

tff(sy_c_ATP_058Lamm__vg____,type,
    aTP_Lamm_vg: set_o > fun_fu2641566564518291801un_o_o ).

tff(sy_c_ATP_058Lamm__vh____,type,
    aTP_Lamm_vh: set_nat > fun_fu7145897819039579925_nat_o ).

tff(sy_c_ATP_058Lamm__vi____,type,
    aTP_Lamm_vi: set_nat > fun_fu1806770557662746761_nat_o ).

tff(sy_c_ATP_058Lamm__vj____,type,
    aTP_Lamm_vj: fun_o_Code_integer > fun_fu7181763470137098718nteger ).

tff(sy_c_ATP_058Lamm__vk____,type,
    aTP_Lamm_vk: fun_nat_Code_integer > fun_fu7898940055406642654nteger ).

tff(sy_c_ATP_058Lamm__vl____,type,
    aTP_Lamm_vl: fun_int_Code_integer > fun_fu6361890901329191134nteger ).

tff(sy_c_ATP_058Lamm__vm____,type,
    aTP_Lamm_vm: set_Code_integer > fun_fu8094397175230428611eger_o ).

tff(sy_c_ATP_058Lamm__vn____,type,
    aTP_Lamm_vn: fun_Co8499238673658658103nteger > fun_fu2729256655946442782nteger ).

tff(sy_c_ATP_058Lamm__vo____,type,
    aTP_Lamm_vo: fun_o_rat > fun_fu830079035465870425_o_rat ).

tff(sy_c_ATP_058Lamm__vp____,type,
    aTP_Lamm_vp: fun_nat_rat > fun_fu3162257917499486297at_rat ).

tff(sy_c_ATP_058Lamm__vq____,type,
    aTP_Lamm_vq: fun_int_rat > fun_fu3922030663826099033nt_rat ).

tff(sy_c_ATP_058Lamm__vr____,type,
    aTP_Lamm_vr: fun_Code_integer_rat > fun_fu991479512444375705er_rat ).

tff(sy_c_ATP_058Lamm__vs____,type,
    aTP_Lamm_vs: fun_o_nat > fun_fu7210065372943547985_o_nat ).

tff(sy_c_ATP_058Lamm__vt____,type,
    aTP_Lamm_vt: fun_int_nat > fun_fu1078644964449000785nt_nat ).

tff(sy_c_ATP_058Lamm__vu____,type,
    aTP_Lamm_vu: fun_Pr5160692531401422620nt_nat > fun_Pr3238374608701176162_int_o ).

tff(sy_c_ATP_058Lamm__vv____,type,
    aTP_Lamm_vv: fun_Pr5160692531401422620nt_nat > fun_fu3068557634114553409_int_o ).

tff(sy_c_ATP_058Lamm__vw____,type,
    aTP_Lamm_vw: fun_list_nat_nat > fun_list_nat_o ).

tff(sy_c_ATP_058Lamm__vx____,type,
    aTP_Lamm_vx: fun_list_nat_nat > fun_fu1994797156054922755_nat_o ).

tff(sy_c_ATP_058Lamm__vy____,type,
    aTP_Lamm_vy: fun_set_nat_nat > fun_set_nat_o ).

tff(sy_c_ATP_058Lamm__vz____,type,
    aTP_Lamm_vz: fun_set_nat_nat > fun_fu2408719596384163971_nat_o ).

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

tff(sy_c_ATP_058Lamm__wb____,type,
    aTP_Lamm_wb: fun_nat_nat > fun_fu4001317426178549763_nat_o ).

tff(sy_c_ATP_058Lamm__wc____,type,
    aTP_Lamm_wc: fun_int_nat > fun_int_o ).

tff(sy_c_ATP_058Lamm__wd____,type,
    aTP_Lamm_wd: fun_int_nat > fun_fu3883539188510166275_int_o ).

tff(sy_c_ATP_058Lamm__we____,type,
    aTP_Lamm_we: fun_Code_integer_nat > fun_Code_integer_o ).

tff(sy_c_ATP_058Lamm__wf____,type,
    aTP_Lamm_wf: fun_Code_integer_nat > fun_fu8094397175230428611eger_o ).

tff(sy_c_ATP_058Lamm__wg____,type,
    aTP_Lamm_wg: fun_Pr2602117396471556196at_nat > fun_Pr4796943543331071514_nat_o ).

tff(sy_c_ATP_058Lamm__wh____,type,
    aTP_Lamm_wh: fun_Pr2602117396471556196at_nat > fun_fu3275390434952296369_nat_o ).

tff(sy_c_ATP_058Lamm__wi____,type,
    aTP_Lamm_wi: set_o > fun_fu904678272606472990un_o_o ).

tff(sy_c_ATP_058Lamm__wj____,type,
    aTP_Lamm_wj: set_o > fun_fu167450956818022195at_nat ).

tff(sy_c_ATP_058Lamm__wk____,type,
    aTP_Lamm_wk: set_int > fun_fu2109032829546145134_int_o ).

tff(sy_c_ATP_058Lamm__wl____,type,
    aTP_Lamm_wl: set_int > fun_fu6978439502566113929at_nat ).

tff(sy_c_ATP_058Lamm__wm____,type,
    aTP_Lamm_wm: set_Code_integer > fun_fu5459709170960931212eger_o ).

tff(sy_c_ATP_058Lamm__wn____,type,
    aTP_Lamm_wn: set_Code_integer > fun_fu4917398172150038123at_nat ).

tff(sy_c_ATP_058Lamm__wo____,type,
    aTP_Lamm_wo: set_o > fun_fu8793767842574475222un_o_o ).

tff(sy_c_ATP_058Lamm__wp____,type,
    aTP_Lamm_wp: set_o > fun_fu1073998408884275435nt_int ).

tff(sy_c_ATP_058Lamm__wq____,type,
    aTP_Lamm_wq: set_nat > fun_fu7718009545821314414_nat_o ).

tff(sy_c_ATP_058Lamm__wr____,type,
    aTP_Lamm_wr: set_nat > fun_fu4179064381821566273nt_int ).

tff(sy_c_ATP_058Lamm__ws____,type,
    aTP_Lamm_ws: set_Code_integer > fun_fu1028834580611891924eger_o ).

tff(sy_c_ATP_058Lamm__wt____,type,
    aTP_Lamm_wt: set_Code_integer > fun_fu6780308020184860635nt_int ).

tff(sy_c_ATP_058Lamm__wu____,type,
    aTP_Lamm_wu: set_nat > fun_fu2944589659160432777nt_nat ).

tff(sy_c_ATP_058Lamm__wv____,type,
    aTP_Lamm_wv: set_nat > fun_fu7243318134412071974_nat_o ).

tff(sy_c_ATP_058Lamm__ww____,type,
    aTP_Lamm_ww: set_nat > fun_fu5935951186182374353at_nat ).

tff(sy_c_ATP_058Lamm__wx____,type,
    aTP_Lamm_wx: set_int > fun_fu8212914225227247425at_int ).

tff(sy_c_ATP_058Lamm__wy____,type,
    aTP_Lamm_wy: set_int > fun_fu2583724240955387574_int_o ).

tff(sy_c_ATP_058Lamm__wz____,type,
    aTP_Lamm_wz: set_int > fun_fu5221552698205305849nt_int ).

tff(sy_c_ATP_058Lamm__xa____,type,
    aTP_Lamm_xa: fun_o_Code_integer > fun_fu7181763470137098718nteger ).

tff(sy_c_ATP_058Lamm__xb____,type,
    aTP_Lamm_xb: fun_nat_Code_integer > fun_fu7898940055406642654nteger ).

tff(sy_c_ATP_058Lamm__xc____,type,
    aTP_Lamm_xc: fun_int_Code_integer > fun_fu6361890901329191134nteger ).

tff(sy_c_ATP_058Lamm__xd____,type,
    aTP_Lamm_xd: fun_Co8499238673658658103nteger > fun_fu2729256655946442782nteger ).

tff(sy_c_ATP_058Lamm__xe____,type,
    aTP_Lamm_xe: fun_o_assn > fun_fu7714003655882793385o_assn ).

tff(sy_c_ATP_058Lamm__xf____,type,
    aTP_Lamm_xf: fun_nat_assn > fun_fu2792963925203051945t_assn ).

tff(sy_c_ATP_058Lamm__xg____,type,
    aTP_Lamm_xg: fun_int_assn > fun_fu2048648494261043369t_assn ).

tff(sy_c_ATP_058Lamm__xh____,type,
    aTP_Lamm_xh: fun_Co4490034186337852034r_assn > fun_fu1898041513128675305r_assn ).

tff(sy_c_ATP_058Lamm__xi____,type,
    aTP_Lamm_xi: fun_o_rat > fun_fu830079035465870425_o_rat ).

tff(sy_c_ATP_058Lamm__xj____,type,
    aTP_Lamm_xj: fun_nat_rat > fun_fu3162257917499486297at_rat ).

tff(sy_c_ATP_058Lamm__xk____,type,
    aTP_Lamm_xk: set_o > fun_fu167450956818022195at_nat ).

tff(sy_c_ATP_058Lamm__xl____,type,
    aTP_Lamm_xl: set_int > fun_fu6978439502566113929at_nat ).

tff(sy_c_ATP_058Lamm__xm____,type,
    aTP_Lamm_xm: set_Code_integer > fun_fu4917398172150038123at_nat ).

tff(sy_c_ATP_058Lamm__xn____,type,
    aTP_Lamm_xn: set_o > fun_fu4065359935906217011at_int ).

tff(sy_c_ATP_058Lamm__xo____,type,
    aTP_Lamm_xo: set_Code_integer > fun_fu548297510352026403at_int ).

tff(sy_c_ATP_058Lamm__xp____,type,
    aTP_Lamm_xp: set_o > fun_fu1073998408884275435nt_int ).

tff(sy_c_ATP_058Lamm__xq____,type,
    aTP_Lamm_xq: set_Code_integer > fun_fu6780308020184860635nt_int ).

tff(sy_c_ATP_058Lamm__xr____,type,
    aTP_Lamm_xr: set_nat > fun_fu2944589659160432777nt_nat ).

tff(sy_c_ATP_058Lamm__xs____,type,
    aTP_Lamm_xs: set_nat > fun_fu5935951186182374353at_nat ).

tff(sy_c_ATP_058Lamm__xt____,type,
    aTP_Lamm_xt: set_nat > fun_fu7170425908843507849at_int ).

tff(sy_c_ATP_058Lamm__xu____,type,
    aTP_Lamm_xu: rat > fun_rat_fun_rat_o ).

tff(sy_c_ATP_058Lamm__xv____,type,
    aTP_Lamm_xv: nat > fun_nat_fun_nat_rat ).

tff(sy_c_ATP_058Lamm__xw____,type,
    aTP_Lamm_xw: fun_na2338080690716861136at_nat > fun_na2338080690716861136at_nat ).

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

tff(sy_c_ATP_058Lamm__xy____,type,
    aTP_Lamm_xy: fun_nat_int > fun_nat_int ).

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

tff(sy_c_ATP_058Lamm__ya____,type,
    aTP_Lamm_ya: fun_nat_Code_integer > fun_nat_Code_integer ).

tff(sy_c_ATP_058Lamm__yb____,type,
    aTP_Lamm_yb: fun_nat_assn > fun_nat_assn ).

tff(sy_c_ATP_058Lamm__yc____,type,
    aTP_Lamm_yc: fun_nat_rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__yd____,type,
    aTP_Lamm_yd: fun_nat_nat > fun_nat_nat ).

tff(sy_c_ATP_058Lamm__ye____,type,
    aTP_Lamm_ye: fun_nat_int > fun_nat_int ).

tff(sy_c_ATP_058Lamm__yf____,type,
    aTP_Lamm_yf: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__yg____,type,
    aTP_Lamm_yg: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__yh____,type,
    aTP_Lamm_yh: fun_Pr5160692531401422620nt_nat > fun_Pr9018117161326028078_int_o ).

tff(sy_c_ATP_058Lamm__yi____,type,
    aTP_Lamm_yi: fun_list_nat_nat > fun_li1025265955262268170_nat_o ).

tff(sy_c_ATP_058Lamm__yj____,type,
    aTP_Lamm_yj: fun_set_nat_nat > fun_se2170385732985038782_nat_o ).

tff(sy_c_ATP_058Lamm__yk____,type,
    aTP_Lamm_yk: fun_nat_nat > fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__yl____,type,
    aTP_Lamm_yl: fun_int_nat > fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__ym____,type,
    aTP_Lamm_ym: fun_Code_integer_nat > fun_Co2081214437782444048eger_o ).

tff(sy_c_ATP_058Lamm__yn____,type,
    aTP_Lamm_yn: fun_Pr2602117396471556196at_nat > fun_Pr2028404973800837934_nat_o ).

tff(sy_c_ATP_058Lamm__yo____,type,
    aTP_Lamm_yo: fun_o_Code_integer > fun_se6637447080540490323nteger ).

tff(sy_c_ATP_058Lamm__yp____,type,
    aTP_Lamm_yp: fun_int_Code_integer > fun_se6579715966475752165nteger ).

tff(sy_c_ATP_058Lamm__yq____,type,
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tff(sy_c_ATP_058Lamm__yr____,type,
    aTP_Lamm_yr: fun_o_rat > fun_set_o_fun_o_rat ).

tff(sy_c_ATP_058Lamm__ys____,type,
    aTP_Lamm_ys: fun_int_rat > fun_se937733209267607904nt_rat ).

tff(sy_c_ATP_058Lamm__yt____,type,
    aTP_Lamm_yt: fun_Code_integer_rat > fun_se8983711159220854466er_rat ).

tff(sy_c_ATP_058Lamm__yu____,type,
    aTP_Lamm_yu: fun_o_nat > fun_set_o_fun_o_nat ).

tff(sy_c_ATP_058Lamm__yv____,type,
    aTP_Lamm_yv: fun_int_nat > fun_se7317719546745285464nt_nat ).

tff(sy_c_ATP_058Lamm__yw____,type,
    aTP_Lamm_yw: fun_Code_integer_nat > fun_se6140325459843756218er_nat ).

tff(sy_c_ATP_058Lamm__yx____,type,
    aTP_Lamm_yx: fun_o_int > fun_set_o_fun_o_int ).

tff(sy_c_ATP_058Lamm__yy____,type,
    aTP_Lamm_yy: set_nat > fun_fu8277437853836697484_nat_o ).

tff(sy_c_ATP_058Lamm__yz____,type,
    aTP_Lamm_yz: set_nat > fun_fu2612476005019691883er_nat ).

tff(sy_c_ATP_058Lamm__za____,type,
    aTP_Lamm_za: set_nat > fun_fu5979305884941020809nt_nat ).

tff(sy_c_ATP_058Lamm__zb____,type,
    aTP_Lamm_zb: set_int > fun_fu5178197999446659393at_int ).

tff(sy_c_ATP_058Lamm__zc____,type,
    aTP_Lamm_zc: set_int > fun_fu8854380836519874772_int_o ).

tff(sy_c_ATP_058Lamm__zd____,type,
    aTP_Lamm_zd: set_int > fun_fu2840411409734105307er_int ).

tff(sy_c_ATP_058Lamm__ze____,type,
    aTP_Lamm_ze: set_o > fun_fu167450956818022195at_nat ).

tff(sy_c_ATP_058Lamm__zf____,type,
    aTP_Lamm_zf: set_int > fun_fu6978439502566113929at_nat ).

tff(sy_c_ATP_058Lamm__zg____,type,
    aTP_Lamm_zg: set_Code_integer > fun_fu4917398172150038123at_nat ).

tff(sy_c_ATP_058Lamm__zh____,type,
    aTP_Lamm_zh: set_nat > fun_fu5935951186182374353at_nat ).

tff(sy_c_ATP_058Lamm__zi____,type,
    aTP_Lamm_zi: set_o > fun_fu1073998408884275435nt_int ).

tff(sy_c_ATP_058Lamm__zj____,type,
    aTP_Lamm_zj: set_nat > fun_fu4179064381821566273nt_int ).

tff(sy_c_ATP_058Lamm__zk____,type,
    aTP_Lamm_zk: fun_int_Code_integer > fun_int_o ).

tff(sy_c_ATP_058Lamm__zl____,type,
    aTP_Lamm_zl: fun_Co8499238673658658103nteger > fun_Code_integer_o ).

tff(sy_c_ATP_058Lamm__zm____,type,
    aTP_Lamm_zm: fun_int_rat > fun_int_o ).

tff(sy_c_ATP_058Lamm__zn____,type,
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tff(sy_c_ATP_058Lamm__zo____,type,
    aTP_Lamm_zo: fun_int_nat > fun_int_o ).

tff(sy_c_ATP_058Lamm__zp____,type,
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tff(sy_c_ATP_058Lamm__zq____,type,
    aTP_Lamm_zq: fun_Code_integer_int > fun_Code_integer_o ).

tff(sy_c_ATP_058Lamm__zr____,type,
    aTP_Lamm_zr: fun_nat_Code_integer > fun_nat_o ).

tff(sy_c_ATP_058Lamm__zs____,type,
    aTP_Lamm_zs: fun_nat_rat > fun_nat_o ).

tff(sy_c_ATP_058Lamm__zt____,type,
    aTP_Lamm_zt: fun_nat_int > fun_nat_o ).

tff(sy_c_ATP_058Lamm__zu____,type,
    aTP_Lamm_zu: fun_Pr5160692531401422620nt_nat > fun_fu7601549123499533073_int_o ).

tff(sy_c_ATP_058Lamm__zv____,type,
    aTP_Lamm_zv: fun_list_nat_nat > fun_fu4472246792138249213_nat_o ).

tff(sy_c_ATP_058Lamm__zw____,type,
    aTP_Lamm_zw: fun_set_nat_nat > fun_fu1237770272497958833_nat_o ).

tff(sy_c_ATP_058Lamm__zx____,type,
    aTP_Lamm_zx: fun_nat_nat > fun_fu5477517762487726877_nat_o ).

tff(sy_c_ATP_058Lamm__zy____,type,
    aTP_Lamm_zy: fun_int_nat > fun_fu1805223966450871909_int_o ).

tff(sy_c_ATP_058Lamm__zz____,type,
    aTP_Lamm_zz: fun_Code_integer_nat > fun_fu2182969643484777475eger_o ).

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_Oentails,type,
    entails: fun_assn_fun_assn_o ).

tff(sy_c_Assertions_Oin__range,type,
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tff(sy_c_Assertions_Oin__range__rel,type,
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tff(sy_c_Assertions_Oone__assn__raw,type,
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tff(sy_c_Assertions_Oone__assn__raw__rel,type,
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tff(sy_c_Assertions_Opure__assn,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,
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tff(sy_c_BNF__Cardinal__Arithmetic_Oczero_001_Eo,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_OnatLeq,type,
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tff(sy_c_BNF__Cardinal__Order__Relation_OnatLess,type,
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tff(sy_c_BNF__Def_Orel__fun_001t__Int__Oint_001t__Int__Oint_001_Eo_001_Eo,type,
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tff(sy_c_BNF__Def_Orel__fun_001t__Int__Oint_001t__Int__Oint_001t__fun_It__Int__Oint_M_Eo_J_001t__fun_It__Int__Oint_M_Eo_J,type,
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tff(sy_c_Lattices_Oinf__class_Oinf_001t__Rat__Orat,type,
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tff(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_I_Eo_J,type,
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tff(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Code____Numeral__Ointeger_J,type,
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tff(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Int__Oint_J,type,
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tff(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
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tff(sy_c_Lattices_Osemilattice__neutr__order_001t__Assertions__Oassn,type,
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tff(sy_c_Lattices_Osup__class_Osup_001t__Int__Oint,type,
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tff(sy_c_List_Oappend_001t__Int__Oint,type,
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tff(sy_c_List_Oproduct_001t__Nat__Onat_001t__Nat__Onat,type,
    product_nat_nat: ( list_nat * list_nat ) > list_P6011104703257516679at_nat ).

tff(sy_c_List_Oproduct_001tf__a_001t__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J,type,
    produc6966004487978991557it_nat: ( list_a * list_P131111800688179804it_nat ) > list_P6935614879863011209it_nat ).

tff(sy_c_List_Orotate1_001_Eo,type,
    rotate1_o: list_o > list_o ).

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

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

tff(sy_c_List_Osorted__wrt_001_Eo,type,
    sorted_wrt_o: fun_o_fun_o_o > fun_list_o_o ).

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

tff(sy_c_List_Osorted__wrt_001t__Nat__Onat,type,
    sorted_wrt_nat: fun_nat_fun_nat_o > fun_list_nat_o ).

tff(sy_c_List_Osorted__wrt_001t__Num__Onum,type,
    sorted_wrt_num: ( fun_num_fun_num_o * list_num ) > $o ).

tff(sy_c_List_Osorted__wrt_001t__Rat__Orat,type,
    sorted_wrt_rat: ( fun_rat_fun_rat_o * list_rat ) > $o ).

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

tff(sy_c_List_Oupt,type,
    upt: nat > fun_nat_list_nat ).

tff(sy_c_List_Oupto,type,
    upto: ( int * int ) > list_int ).

tff(sy_c_List_Oupto__aux,type,
    upto_aux: ( int * int * list_int ) > list_int ).

tff(sy_c_List_Oupto__rel,type,
    upto_rel: fun_Pr9018117161326028078_int_o ).

tff(sy_c_Misc_OEps__Opt_001t__Nat__Onat,type,
    eps_Opt_nat: fun_nat_o > option_nat ).

tff(sy_c_Misc_OEps__Opt_001t__Num__Onum,type,
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tff(sy_c_Misc_OEps__Opt_001t__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J,type,
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tff(sy_c_Misc_OEps__Opt_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
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tff(sy_c_Misc_Obijective_001t__Nat__Onat_001t__Nat__Onat,type,
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tff(sy_c_Misc_Obijective_001t__fun_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
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tff(sy_c_Misc_Odflt__None__set_001_Eo,type,
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tff(sy_c_Misc_Odflt__None__set_001t__Int__Oint,type,
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tff(sy_c_Misc_Odflt__None__set_001t__Nat__Onat,type,
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tff(sy_c_Misc_Omap__to__set_001t__Int__Oint_001t__Int__Oint,type,
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tff(sy_c_Misc_Omap__to__set_001t__Nat__Onat_001t__Nat__Onat,type,
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tff(sy_c_Misc_Orel__of_001t__Int__Oint_001t__Int__Oint,type,
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tff(sy_c_Misc_Orel__of_001t__Nat__Onat_001t__Nat__Onat,type,
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tff(sy_c_Misc_Oset__to__map_001t__Nat__Onat_001t__Nat__Onat,type,
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tff(sy_c_Misc_Oslice_001_Eo,type,
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tff(sy_c_Misc_Oslice_001t__Int__Oint,type,
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tff(sy_c_Misc_Oslice_001t__Nat__Onat,type,
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tff(sy_c_Misc_Othe__default_001t__Num__Onum,type,
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tff(sy_c_Misc_Othe__default_001t__Set__Oset_I_Eo_J,type,
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tff(sy_c_Misc_Othe__default_001t__Set__Oset_It__Int__Oint_J,type,
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tff(sy_c_Misc_Othe__default_001t__Set__Oset_It__Nat__Onat_J,type,
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tff(sy_c_Misc_Ouncurry_001t__Int__Oint_001t__Int__Oint_001_Eo,type,
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    size_s341701280123345136it_nat: list_P131111800688179804it_nat > nat ).

tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Rat__Orat_J,type,
    size_size_list_rat: list_rat > nat ).

tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_Itf__a_J,type,
    size_size_list_a: list_a > nat ).

tff(sy_c_Nat_Osize__class_Osize_001t__Multiset__Omultiset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    size_s8510653225128441779at_nat: multis2468970476368604999at_nat > nat ).

tff(sy_c_Nat_Osize__class_Osize_001t__Option__Ooption_It__Num__Onum_J,type,
    size_size_option_num: fun_option_num_nat ).

tff(sy_c_Nat_Osize__class_Osize_001t__Option__Ooption_It__Product____Type__Oprod_It__Product____Type__Ounit_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J_J,type,
    size_s8766407808098229740it_nat: option8956607266484857688it_nat > nat ).

tff(sy_c_Nat_Osize__class_Osize_001t__Option__Ooption_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J_J,type,
    size_s2363601347547812957it_nat: option3562590408128118217it_nat > nat ).

tff(sy_c_Nat__Bijection_Oprod__decode__aux,type,
    nat_prod_decode_aux: fun_na4138050610385705875at_nat ).

tff(sy_c_Nat__Bijection_Oprod__decode__aux__rel,type,
    nat_pr5047031295181774490ux_rel: fun_Pr2028404973800837934_nat_o ).

tff(sy_c_Nat__Bijection_Oset__decode,type,
    nat_set_decode: fun_nat_set_nat ).

tff(sy_c_Num_OBitM,type,
    bitM: fun_num_num ).

tff(sy_c_Num_Oinc,type,
    inc: fun_num_num ).

tff(sy_c_Num_Onum_OBit0,type,
    bit0: fun_num_num ).

tff(sy_c_Num_Onum_OBit1,type,
    bit1: fun_num_num ).

tff(sy_c_Num_Onum_OOne,type,
    one: num ).

tff(sy_c_Num_Onum_Ocase__num_001t__Option__Ooption_It__Num__Onum_J,type,
    case_num_option_num: ( option_num * fun_num_option_num * fun_num_option_num ) > fun_num_option_num ).

tff(sy_c_Num_Onum__of__nat,type,
    num_of_nat: fun_nat_num ).

tff(sy_c_Num_Onumeral__class_Onumeral_001t__Code____Numeral__Ointeger,type,
    numera6620942414471956472nteger: fun_num_Code_integer ).

tff(sy_c_Num_Onumeral__class_Onumeral_001t__Code____Numeral__Onatural,type,
    numera5444537566228673987atural: fun_num_Code_natural ).

tff(sy_c_Num_Onumeral__class_Onumeral_001t__Int__Oint,type,
    numeral_numeral_int: fun_num_int ).

tff(sy_c_Num_Onumeral__class_Onumeral_001t__Nat__Onat,type,
    numeral_numeral_nat: fun_num_nat ).

tff(sy_c_Num_Onumeral__class_Onumeral_001t__Rat__Orat,type,
    numeral_numeral_rat: fun_num_rat ).

tff(sy_c_Num_Opred__numeral,type,
    pred_numeral: fun_num_nat ).

tff(sy_c_Option_Ocombine__options_001t__Num__Onum,type,
    combine_options_num: ( fun_num_fun_num_num * option_num ) > fun_op8989173174992673105on_num ).

tff(sy_c_Option_Ocombine__options_001t__Product____Type__Oprod_It__Product____Type__Ounit_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J,type,
    combin4318129983670048329it_nat: ( fun_Pr1003294667014919570it_nat * option8956607266484857688it_nat * option8956607266484857688it_nat ) > option8956607266484857688it_nat ).

tff(sy_c_Option_Ocombine__options_001t__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J,type,
    combin5819755802923621690it_nat: ( fun_Pr7750636411365408485it_nat * option3562590408128118217it_nat * option3562590408128118217it_nat ) > option3562590408128118217it_nat ).

tff(sy_c_Option_Ooption_ONone_001t__Assertions__Oassn,type,
    none_assn: option_assn ).

tff(sy_c_Option_Ooption_ONone_001t__Nat__Onat,type,
    none_nat: option_nat ).

tff(sy_c_Option_Ooption_ONone_001t__Num__Onum,type,
    none_num: option_num ).

tff(sy_c_Option_Ooption_ONone_001t__Product____Type__Oprod_It__Product____Type__Ounit_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J,type,
    none_P9117596204409417319it_nat: option8956607266484857688it_nat ).

tff(sy_c_Option_Ooption_ONone_001t__Product____Type__Oprod_It__fun_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
    none_P1716269551890668430nt_int: option6186025235080601855nt_int ).

tff(sy_c_Option_Ooption_ONone_001t__Product____Type__Oprod_It__fun_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_J,type,
    none_P7891021755743879167et_nat: option4955102875562834416et_nat ).

tff(sy_c_Option_Ooption_ONone_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,
    none_P2758821087663475997et_nat: option5899113203645146382et_nat ).

tff(sy_c_Option_Ooption_ONone_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,
    none_P3112835510759997538nt_int: option1544258867535432659nt_int ).

tff(sy_c_Option_Ooption_ONone_001t__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J,type,
    none_P2651198173097904984it_nat: option3562590408128118217it_nat ).

tff(sy_c_Option_Ooption_ONone_001t__Product____Type__Ounit,type,
    none_Product_unit: option_Product_unit ).

tff(sy_c_Option_Ooption_ONone_001t__Set__Oset_I_Eo_J,type,
    none_set_o: option_set_o ).

tff(sy_c_Option_Ooption_ONone_001t__Set__Oset_It__Int__Oint_J,type,
    none_set_int: option_set_int ).

tff(sy_c_Option_Ooption_ONone_001t__Set__Oset_It__Nat__Onat_J,type,
    none_set_nat: option_set_nat ).

tff(sy_c_Option_Ooption_ONone_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    none_s625347054029921090at_nat: option8963830502488799655at_nat ).

tff(sy_c_Option_Ooption_OSome_001t__Assertions__Oassn,type,
    some_assn: assn > option_assn ).

tff(sy_c_Option_Ooption_OSome_001t__Int__Oint,type,
    some_int: fun_int_option_int ).

tff(sy_c_Option_Ooption_OSome_001t__Nat__Onat,type,
    some_nat: fun_nat_option_nat ).

tff(sy_c_Option_Ooption_OSome_001t__Num__Onum,type,
    some_num: fun_num_option_num ).

tff(sy_c_Option_Ooption_OSome_001t__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J,type,
    some_P1054239925786823975it_nat: produc6653097349344004940it_nat > option233860712434008220it_nat ).

tff(sy_c_Option_Ooption_OSome_001t__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J,type,
    some_P624177172695371229et_nat: produc3658429121746597890et_nat > option936205604648967762et_nat ).

tff(sy_c_Option_Ooption_OSome_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    some_P4184893108420464158nt_int: product_prod_int_int > option4624381673175914239nt_int ).

tff(sy_c_Option_Ooption_OSome_001t__Product____Type__Oprod_It__Product____Type__Ounit_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J,type,
    some_P1914260805536162275it_nat: fun_Pr7644814098969791837it_nat ).

tff(sy_c_Option_Ooption_OSome_001t__Product____Type__Oprod_It__fun_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
    some_P8798507657793832714nt_int: produc4788131091073775919nt_int > option6186025235080601855nt_int ).

tff(sy_c_Option_Ooption_OSome_001t__Product____Type__Oprod_It__fun_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_J,type,
    some_P994283844982041979et_nat: produc8799533983703829792et_nat > option4955102875562834416et_nat ).

tff(sy_c_Option_Ooption_OSome_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,
    some_P3006888012174695577et_nat: produc3112576180687729086et_nat > option5899113203645146382et_nat ).

tff(sy_c_Option_Ooption_OSome_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,
    some_P9145269820779039710nt_int: produc3043004577601930755nt_int > option1544258867535432659nt_int ).

tff(sy_c_Option_Ooption_OSome_001t__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J,type,
    some_P7913643980934408916it_nat: fun_Pr2403869735082759103it_nat ).

tff(sy_c_Option_Ooption_OSome_001t__Product____Type__Ounit,type,
    some_Product_unit: product_unit > option_Product_unit ).

tff(sy_c_Option_Ooption_OSome_001t__Rat__Orat,type,
    some_rat: rat > option_rat ).

tff(sy_c_Option_Ooption_OSome_001t__Set__Oset_I_Eo_J,type,
    some_set_o: set_o > option_set_o ).

tff(sy_c_Option_Ooption_OSome_001t__Set__Oset_It__Int__Oint_J,type,
    some_set_int: set_int > option_set_int ).

tff(sy_c_Option_Ooption_OSome_001t__Set__Oset_It__Nat__Onat_J,type,
    some_set_nat: set_nat > option_set_nat ).

tff(sy_c_Option_Ooption_OSome_001t__Set__Oset_It__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_J,type,
    some_s287724117700012716at_nat: set_Pr8551490117392284871at_nat > option2498585697089621389at_nat ).

tff(sy_c_Option_Ooption_OSome_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    some_s147305329494351046at_nat: set_Pr1261947904930325089at_nat > option8963830502488799655at_nat ).

tff(sy_c_Option_Ooption_OSome_001t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    some_s579751585967276588at_nat: set_Pr8693737435421807431at_nat > option2158460348403737357at_nat ).

tff(sy_c_Option_Ooption_OSome_001t__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J,type,
    some_s5890477192898017836at_nat: set_Pr4329608150637261639at_nat > option1583680563626158861at_nat ).

tff(sy_c_Option_Ooption_Ocase__option_001_Eo_001t__Assertions__Oassn,type,
    case_option_o_assn: ( $o * fun_assn_o * option_assn ) > $o ).

tff(sy_c_Option_Ooption_Ocase__option_001_Eo_001t__Int__Oint,type,
    case_option_o_int: ( $o * fun_int_o * option_int ) > $o ).

tff(sy_c_Option_Ooption_Ocase__option_001_Eo_001t__Nat__Onat,type,
    case_option_o_nat: ( $o * fun_nat_o * option_nat ) > $o ).

tff(sy_c_Option_Ooption_Ocase__option_001_Eo_001t__Num__Onum,type,
    case_option_o_num: ( $o * fun_num_o ) > fun_option_num_o ).

tff(sy_c_Option_Ooption_Ocase__option_001_Eo_001t__Product____Type__Oprod_It__Product____Type__Ounit_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J,type,
    case_o2686588417244861013it_nat: ( $o * fun_Pr8032424337255632745_nat_o ) > fun_op479301348043730329_nat_o ).

tff(sy_c_Option_Ooption_Ocase__option_001_Eo_001t__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J,type,
    case_o3267218206291580230it_nat: ( $o * fun_Pr5746006151868627512_nat_o ) > fun_op9133723461703387112_nat_o ).

tff(sy_c_Option_Ooption_Ocase__option_001_Eo_001t__Rat__Orat,type,
    case_option_o_rat: ( $o * fun_rat_o * option_rat ) > $o ).

tff(sy_c_Option_Ooption_Ocase__option_001_Eo_001t__Set__Oset_It__Int__Oint_J,type,
    case_o223999843215110191et_int: ( $o * fun_set_int_o * option_set_int ) > $o ).

tff(sy_c_Option_Ooption_Ocase__option_001t__Code____Numeral__Ointeger_001t__Num__Onum,type,
    case_o356765784539232260er_num: ( code_integer * fun_num_Code_integer * option_num ) > code_integer ).

tff(sy_c_Option_Ooption_Ocase__option_001t__Code____Numeral__Onatural_001t__Num__Onum,type,
    case_o5621594795226839503al_num: ( code_natural * fun_num_Code_natural * option_num ) > code_natural ).

tff(sy_c_Option_Ooption_Ocase__option_001t__Int__Oint_001t__Num__Onum,type,
    case_option_int_num: ( int * fun_num_int ) > fun_option_num_int ).

tff(sy_c_Option_Ooption_Ocase__option_001t__Nat__Onat_001t__Num__Onum,type,
    case_option_nat_num: ( nat * fun_num_nat ) > fun_option_num_nat ).

tff(sy_c_Option_Ooption_Ocase__option_001t__Num__Onum_001t__Num__Onum,type,
    case_option_num_num: ( num * fun_num_num ) > fun_option_num_num ).

tff(sy_c_Option_Ooption_Ocase__option_001t__Option__Ooption_It__Assertions__Oassn_J_001t__Assertions__Oassn,type,
    case_o4484465799723439917n_assn: ( option_assn * fun_assn_option_assn * option_assn ) > option_assn ).

tff(sy_c_Option_Ooption_Ocase__option_001t__Option__Ooption_It__Nat__Onat_J_001t__Nat__Onat,type,
    case_o7429725398727453821at_nat: ( option_nat * fun_nat_option_nat * option_nat ) > option_nat ).

tff(sy_c_Option_Ooption_Ocase__option_001t__Option__Ooption_It__Num__Onum_J_001t__Num__Onum,type,
    case_o6005452278849405969um_num: ( option_num * fun_num_option_num ) > fun_op8989173174992673105on_num ).

tff(sy_c_Option_Ooption_Ocase__option_001t__Option__Ooption_It__Product____Type__Oprod_It__Product____Type__Ounit_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J_J_001t__Product____Type__Oprod_It__Product____Type__Ounit_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J,type,
    case_o2963978774867076333it_nat: ( option8956607266484857688it_nat * fun_Pr7644814098969791837it_nat * option8956607266484857688it_nat ) > option8956607266484857688it_nat ).

tff(sy_c_Option_Ooption_Ocase__option_001t__Option__Ooption_It__Product____Type__Oprod_It__Product____Type__Ounit_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J_J_001t__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J,type,
    case_o7727800614283616222it_nat: ( option8956607266484857688it_nat * fun_Pr8191437870481986254it_nat * option3562590408128118217it_nat ) > option8956607266484857688it_nat ).

tff(sy_c_Option_Ooption_Ocase__option_001t__Option__Ooption_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J_J_001t__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J,type,
    case_o6061097939050036047it_nat: ( option3562590408128118217it_nat * fun_Pr2403869735082759103it_nat * option3562590408128118217it_nat ) > option3562590408128118217it_nat ).

tff(sy_c_Option_Ooption_Ocase__option_001t__Option__Ooption_It__Product____Type__Ounit_J_001t__Product____Type__Ounit,type,
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tff(sy_c_Option_Ooption_Ocase__option_001t__Option__Ooption_It__Set__Oset_It__Nat__Onat_J_J_001t__Set__Oset_It__Nat__Onat_J,type,
    case_o4054078431260844265et_nat: ( option_set_nat * fun_se7316093745529328985et_nat * option_set_nat ) > option_set_nat ).

tff(sy_c_Option_Ooption_Ocase__option_001t__Option__Ooption_It__Set__Oset_It__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_J_J_001t__Set__Oset_It__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_J,type,
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    aa_fun6119220767042692876it_nat: ( fun_fu4855298819597708819it_nat * fun_a_6093291656547882009Heap_a ) > fun_he1921015881043195817it_nat ).

tff(sy_c_aa_001tf__a_001_Eo,type,
    aa_a_o: ( fun_a_o * a ) > $o ).

tff(sy_c_aa_001tf__a_001t__Assertions__Oassn,type,
    aa_a_assn: ( fun_a_assn * a ) > assn ).

tff(sy_c_aa_001tf__a_001t__Heap____Time____Monad__OHeap_It__Product____Type__Ounit_J,type,
    aa_a_H4862568901524997081t_unit: ( fun_a_2990225006182978344t_unit * a ) > heap_T5738788834812785303t_unit ).

tff(sy_c_aa_001tf__a_001t__Heap____Time____Monad__OHeap_Itf__a_J,type,
    aa_a_H8829438773920479562Heap_a: ( fun_a_6093291656547882009Heap_a * a ) > heap_Time_Heap_a ).

tff(sy_c_aa_001tf__a_001t__Option__Ooption_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J,type,
    aa_a_o5231022909227610846it_nat: ( fun_a_1313742400928293293it_nat * a ) > option233860712434008220it_nat ).

tff(sy_c_aa_001tf__a_001t__fun_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__fun_It__Nat__Onat_Mt__Option__Ooption_It__Product____Type__Oprod_It__Product____Type__Ounit_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J_J_J_J,type,
    aa_a_f112692477493148739it_nat: ( fun_a_6373226321756256586it_nat * a ) > fun_he8369595706569872389it_nat ).

tff(sy_c_aa_001tf__a_001t__fun_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__fun_It__Nat__Onat_Mt__Option__Ooption_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J_J_J_J,type,
    aa_a_f1761598956578211508it_nat: ( fun_a_3171520138701809467it_nat * a ) > fun_he8849654208132081782it_nat ).

tff(sy_c_aa_001tf__a_001t__fun_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_M_Eo_J,type,
    aa_a_f337466037417904483_nat_o: ( fun_a_2467521563176467050_nat_o * a ) > fun_Pr7971309923680890661_nat_o ).

tff(sy_c_aa_001tf__a_001t__fun_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_Mt__Nat__Onat_J,type,
    aa_a_f8203466477562882395at_nat: ( fun_a_5416464254913058730at_nat * a ) > fun_Pr4343998633453697049at_nat ).

tff(sy_c_aa_001tf__a_001t__fun_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_Mt__Option__Ooption_It__Product____Type__Oprod_It__Product____Type__Ounit_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J_J_J,type,
    aa_a_f2724909083865345251it_nat: ( fun_a_4942204877778174258it_nat * a ) > fun_Pr5651933991979716257it_nat ).

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    aa_a_f2042851359257671380it_nat: ( fun_a_3426355041879316131it_nat * a ) > fun_Pr7273243183861550738it_nat ).

tff(sy_c_aa_001tf__a_001t__fun_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J_J,type,
    aa_a_f2234720956197727492it_nat: ( fun_a_576316402698418899it_nat * a ) > fun_Pr2713655712603142594it_nat ).

tff(sy_c_aa_001tf__a_001t__fun_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_Mt__Set__Oset_I_Eo_J_J,type,
    aa_a_f7646208301878272067_set_o: ( fun_a_6228866161775505738_set_o * a ) > fun_Pr8691148245588442629_set_o ).

tff(sy_c_aa_001tf__a_001t__fun_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_Mt__Set__Oset_It__Int__Oint_J_J,type,
    aa_a_f4666765590483879021et_int: ( fun_a_3360290013704583612et_int * a ) > fun_Pr1844946900262394667et_int ).

tff(sy_c_aa_001tf__a_001t__fun_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_Mt__Set__Oset_It__Nat__Onat_J_J,type,
    aa_a_f7156848735415605777et_nat: ( fun_a_8173838432492502368et_nat * a ) > fun_Pr4335030045194121423et_nat ).

tff(sy_c_aa_001tf__a_001t__fun_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    aa_a_f1049478142650162092at_nat: ( fun_a_8304141829328133171at_nat * a ) > fun_Pr4340503698666692462at_nat ).

tff(sy_c_aa_001tf__a_001t__fun_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_Mt__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
    aa_a_f1497801039697847239et_nat: ( fun_a_5925824989894720534et_nat * a ) > fun_Pr435983442313151109et_nat ).

tff(sy_c_fequal_001_Eo,type,
    fequal_o: fun_o_fun_o_o ).

tff(sy_c_fequal_001t__Int__Oint,type,
    fequal_int: fun_int_fun_int_o ).

tff(sy_c_fequal_001t__Nat__Onat,type,
    fequal_nat: fun_nat_fun_nat_o ).

tff(sy_c_fequal_001t__Num__Onum,type,
    fequal_num: fun_num_fun_num_o ).

tff(sy_c_fequal_001t__Product____Type__Oprod_It__Product____Type__Ounit_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J,type,
    fequal1407654770548522611it_nat: produc8664842809031399944it_nat > fun_Pr8032424337255632745_nat_o ).

tff(sy_c_fequal_001t__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J,type,
    fequal7405809337025681252it_nat: produc3260487557148687353it_nat > fun_Pr5746006151868627512_nat_o ).

tff(sy_c_member_001_Eo,type,
    member_o: ( $o * set_o ) > $o ).

tff(sy_c_member_001t__Assertions__Oassn,type,
    member_assn: ( assn * set_assn ) > $o ).

tff(sy_c_member_001t__Code____Numeral__Ointeger,type,
    member_Code_integer: code_integer > fun_se7289222521248229150eger_o ).

tff(sy_c_member_001t__Int__Oint,type,
    member_int: int > fun_set_int_o ).

tff(sy_c_member_001t__List__Olist_I_Eo_J,type,
    member_list_o: ( list_o * set_list_o ) > $o ).

tff(sy_c_member_001t__List__Olist_It__Nat__Onat_J,type,
    member_list_nat: ( list_nat * set_list_nat ) > $o ).

tff(sy_c_member_001t__Nat__Onat,type,
    member_nat: nat > fun_set_nat_o ).

tff(sy_c_member_001t__Num__Onum,type,
    member_num: ( num * set_num ) > $o ).

tff(sy_c_member_001t__Option__Ooption_It__Num__Onum_J,type,
    member_option_num: ( option_num * set_option_num ) > $o ).

tff(sy_c_member_001t__Option__Ooption_It__Product____Type__Oprod_It__Product____Type__Ounit_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J_J,type,
    member5596548051065438575it_nat: ( option8956607266484857688it_nat * set_op239864471688321678it_nat ) > $o ).

tff(sy_c_member_001t__Option__Ooption_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J_J,type,
    member1310632352865511136it_nat: ( option3562590408128118217it_nat * set_op4617111635268443007it_nat ) > $o ).

tff(sy_c_member_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    member5262025264175285858nt_int: ( product_prod_int_int * set_Pr958786334691620121nt_int ) > $o ).

tff(sy_c_member_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,
    member6689249552917799696at_nat: ( produc4166570645942440679at_nat * set_Pr8551490117392284871at_nat ) > $o ).

tff(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    member8440522571783428010at_nat: product_prod_nat_nat > fun_se4782206882984618170_nat_o ).

tff(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    member8206827879206165904at_nat: ( produc859450856879609959at_nat * set_Pr8693737435421807431at_nat ) > $o ).

tff(sy_c_member_001t__Product____Type__Oprod_It__Set__Oset_It__Code____Numeral__Ointeger_J_Mt__Set__Oset_It__Code____Numeral__Ointeger_J_J,type,
    member4307123515891402160nteger: ( produc6491284506569428743nteger * set_Pr7577011563204128103nteger ) > $o ).

tff(sy_c_member_001t__Product____Type__Oprod_It__Set__Oset_It__Int__Oint_J_Mt__Set__Oset_It__Int__Oint_J_J,type,
    member2572552093476627150et_int: ( produc2115011035271226405et_int * set_Pr2522554150109002629et_int ) > $o ).

tff(sy_c_member_001t__Product____Type__Oprod_It__Set__Oset_It__Nat__Onat_J_Mt__Set__Oset_It__Nat__Onat_J_J,type,
    member8277197624267554838et_nat: ( produc7819656566062154093et_nat * set_Pr5488025237498180813et_nat ) > $o ).

tff(sy_c_member_001t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_I_Eo_M_Eo_J_J_Mt__Set__Oset_It__Product____Type__Oprod_I_Eo_M_Eo_J_J_J,type,
    member444158400953824016od_o_o: ( produc2934264451710624999od_o_o * set_Pr1932065953672099015od_o_o ) > $o ).

tff(sy_c_member_001t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_I_Eo_M_Eo_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    member4095101504841534314at_nat: ( produc732395585841259969at_nat * set_Pr457366540195662369at_nat ) > $o ).

tff(sy_c_member_001t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    member8757157785044589968at_nat: ( produc3843707927480180839at_nat * set_Pr4329608150637261639at_nat ) > $o ).

tff(sy_c_member_001t__Product____Type__Oprod_It__fun_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
    member5890090881338784710nt_int: ( produc4788131091073775919nt_int * set_Pr4166680992137330533nt_int ) > $o ).

tff(sy_c_member_001t__Product____Type__Oprod_It__fun_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_J,type,
    member4513763875858039863et_nat: ( produc8799533983703829792et_nat * set_Pr1312406793078187350et_nat ) > $o ).

tff(sy_c_member_001t__Product____Type__Oprod_It__fun_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_Mt__Product____Type__Oprod_It__fun_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_J_J,type,
    member9174161442226468181et_nat: ( produc3112576180687729086et_nat * set_Pr4040394263813967220et_nat ) > $o ).

tff(sy_c_member_001t__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__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J,type,
    member4554412811331277712it_nat: ( produc3260487557148687353it_nat * set_Pr7098220151150636591it_nat ) > $o ).

tff(sy_c_member_001t__Rat__Orat,type,
    member_rat: ( rat * set_rat ) > $o ).

tff(sy_c_member_001t__Set__Oset_It__Int__Oint_J,type,
    member_set_int: ( set_int * set_set_int ) > $o ).

tff(sy_c_member_001t__Set__Oset_It__Nat__Onat_J,type,
    member_set_nat: set_nat > fun_set_set_nat_o ).

tff(sy_v_P,type,
    p: assn ).

tff(sy_v_Q,type,
    q: fun_a_assn ).

tff(sy_v_R,type,
    r: assn ).

tff(sy_v_as1____,type,
    as1: set_nat ).

tff(sy_v_as2____,type,
    as2: set_nat ).

tff(sy_v_as____,type,
    as: set_nat ).

tff(sy_v_c,type,
    c: heap_Time_Heap_a ).

tff(sy_v_h_H____,type,
    h: heap_e7401611519738050253t_unit ).

tff(sy_v_h____,type,
    h2: heap_e7401611519738050253t_unit ).

tff(sy_v_r____,type,
    r2: a ).

tff(sy_v_t____,type,
    t: nat ).

% Relevant facts (10763)
tff(fact_0_DJ,axiom,
    aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,as1),as2) = bot_bot_set_nat ).

% DJ
tff(fact_1__092_060open_062_Ih_H_M_Aas2_J_A_092_060Turnstile_062_AR_092_060close_062,axiom,
    aa_Pro653082453696054616_nat_o(rep_assn(r),produc7507926704131184380et_nat(h,as2)) ).

% \<open>(h', as2) \<Turnstile> R\<close>
tff(fact_2_M2,axiom,
    aa_Pro653082453696054616_nat_o(rep_assn(r),produc7507926704131184380et_nat(h2,as2)) ).

% M2
tff(fact_3_DJN,axiom,
    aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,hoare_new_addrs(h2,as1,h)),as2) = bot_bot_set_nat ).

% DJN
tff(fact_4__092_060open_062new__addrs_Ah_Aas_Ah_H_A_061_Anew__addrs_Ah_Aas1_Ah_H_A_092_060union_062_Aas2_092_060close_062,axiom,
    hoare_new_addrs(h2,as,h) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,hoare_new_addrs(h2,as1,h)),as2) ).

% \<open>new_addrs h as h' = new_addrs h as1 h' \<union> as2\<close>
tff(fact_5_new__addr__refl,axiom,
    ! [Ha: heap_e7401611519738050253t_unit,Asa: set_nat] : hoare_new_addrs(Ha,Asa,Ha) = Asa ).

% new_addr_refl
tff(fact_6_M1,axiom,
    aa_Pro653082453696054616_nat_o(rep_assn(p),produc7507926704131184380et_nat(h2,as1)) ).

% M1
tff(fact_7_MDL,axiom,
    aa_Pro653082453696054616_nat_o(rep_assn(aa_a_assn(q,r2)),produc7507926704131184380et_nat(h,hoare_new_addrs(h2,as1,h))) ).

% MDL
tff(fact_8__092_060open_062_Ih_M_Aas_J_A_092_060Turnstile_062_AP_A_K_AR_092_060close_062,axiom,
    aa_Pro653082453696054616_nat_o(rep_assn(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,p),r)),produc7507926704131184380et_nat(h2,as)) ).

% \<open>(h, as) \<Turnstile> P * R\<close>
tff(fact_9__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062as1_Aas2_O_A_092_060lbrakk_062as_A_061_Aas1_A_092_060union_062_Aas2_059_Aas1_A_092_060inter_062_Aas2_A_061_A_123_125_059_A_Ih_M_Aas1_J_A_092_060Turnstile_062_AP_059_A_Ih_M_Aas2_J_A_092_060Turnstile_062_AR_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
    ~ ! [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(rep_assn(p),produc7507926704131184380et_nat(h2,As1))
           => ~ aa_Pro653082453696054616_nat_o(rep_assn(r),produc7507926704131184380et_nat(h2,As2)) ) ) ) ).

% \<open>\<And>thesis. (\<And>as1 as2. \<lbrakk>as = as1 \<union> as2; as1 \<inter> as2 = {}; (h, as1) \<Turnstile> P; (h, as2) \<Turnstile> R\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
tff(fact_10_Rep__assn__inject,axiom,
    ! [X: assn,Y: assn] :
      ( ( rep_assn(X) = rep_assn(Y) )
    <=> ( X = Y ) ) ).

% Rep_assn_inject
tff(fact_11_mod__h__bot__iff_I5_J,axiom,
    ! [Pa: assn,Qa: assn,Ha: heap_e7401611519738050253t_unit] :
      ( aa_Pro653082453696054616_nat_o(rep_assn(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Pa),Qa)),produc7507926704131184380et_nat(Ha,bot_bot_set_nat))
    <=> ( aa_Pro653082453696054616_nat_o(rep_assn(Pa),produc7507926704131184380et_nat(Ha,bot_bot_set_nat))
        & aa_Pro653082453696054616_nat_o(rep_assn(Qa),produc7507926704131184380et_nat(Ha,bot_bot_set_nat)) ) ) ).

% mod_h_bot_iff(5)
tff(fact_12__092_060open_062relH_Aas2_Ah_Ah_H_092_060close_062,axiom,
    aa_hea7994211679605476167unit_o(relH(as2,h2),h) ).

% \<open>relH as2 h h'\<close>
tff(fact_13_prod_Oinject,axiom,
    ! [X1: fun_Pr6460944699146269935_nat_o,X2: produc3658429121746597890et_nat,Y1: fun_Pr6460944699146269935_nat_o,Y2: produc3658429121746597890et_nat] :
      ( ( aa_Pro434340453207536480et_nat(aa_fun3926903090414666066et_nat(produc6017135503889078682et_nat,X1),X2) = aa_Pro434340453207536480et_nat(aa_fun3926903090414666066et_nat(produc6017135503889078682et_nat,Y1),Y2) )
    <=> ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
tff(fact_14_prod_Oinject,axiom,
    ! [X1: fun_Pr6460944699146269935_nat_o,X2: produc8799533983703829792et_nat,Y1: fun_Pr6460944699146269935_nat_o,Y2: produc8799533983703829792et_nat] :
      ( ( aa_Pro1979504083478021276et_nat(aa_fun615918787490101902et_nat(produc7291855603632349880et_nat,X1),X2) = aa_Pro1979504083478021276et_nat(aa_fun615918787490101902et_nat(produc7291855603632349880et_nat,Y1),Y2) )
    <=> ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
tff(fact_15_prod_Oinject,axiom,
    ! [X1: fun_Pr7763680713840462443e_term,X2: product_prod_int_int,Y1: fun_Pr7763680713840462443e_term,Y2: product_prod_int_int] :
      ( ( aa_Pro5748361610964203648nt_int(aa_fun3265472288063244902nt_int(produc7320969046608631221nt_int,X1),X2) = aa_Pro5748361610964203648nt_int(aa_fun3265472288063244902nt_int(produc7320969046608631221nt_int,Y1),Y2) )
    <=> ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
tff(fact_16_prod_Oinject,axiom,
    ! [X1: fun_in5730379987311621695e_term,X2: product_prod_int_int,Y1: fun_in5730379987311621695e_term,Y2: product_prod_int_int] :
      ( ( aa_Pro8800328608558162860nt_int(aa_fun260840925927807678nt_int(produc4729913460834754785nt_int,X1),X2) = aa_Pro8800328608558162860nt_int(aa_fun260840925927807678nt_int(produc4729913460834754785nt_int,Y1),Y2) )
    <=> ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
tff(fact_17_prod_Oinject,axiom,
    ! [X1: a,X2: produc6653097349344004940it_nat,Y1: a,Y2: produc6653097349344004940it_nat] :
      ( ( aa_Pro8078911840264235907it_nat(aa_a_f2234720956197727492it_nat(produc9178034014595674355it_nat,X1),X2) = aa_Pro8078911840264235907it_nat(aa_a_f2234720956197727492it_nat(produc9178034014595674355it_nat,Y1),Y2) )
    <=> ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
tff(fact_18_old_Oprod_Oinject,axiom,
    ! [A: fun_Pr6460944699146269935_nat_o,B: produc3658429121746597890et_nat,A2: fun_Pr6460944699146269935_nat_o,B2: produc3658429121746597890et_nat] :
      ( ( aa_Pro434340453207536480et_nat(aa_fun3926903090414666066et_nat(produc6017135503889078682et_nat,A),B) = aa_Pro434340453207536480et_nat(aa_fun3926903090414666066et_nat(produc6017135503889078682et_nat,A2),B2) )
    <=> ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
tff(fact_19_old_Oprod_Oinject,axiom,
    ! [A: fun_Pr6460944699146269935_nat_o,B: produc8799533983703829792et_nat,A2: fun_Pr6460944699146269935_nat_o,B2: produc8799533983703829792et_nat] :
      ( ( aa_Pro1979504083478021276et_nat(aa_fun615918787490101902et_nat(produc7291855603632349880et_nat,A),B) = aa_Pro1979504083478021276et_nat(aa_fun615918787490101902et_nat(produc7291855603632349880et_nat,A2),B2) )
    <=> ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
tff(fact_20_old_Oprod_Oinject,axiom,
    ! [A: fun_Pr7763680713840462443e_term,B: product_prod_int_int,A2: fun_Pr7763680713840462443e_term,B2: product_prod_int_int] :
      ( ( aa_Pro5748361610964203648nt_int(aa_fun3265472288063244902nt_int(produc7320969046608631221nt_int,A),B) = aa_Pro5748361610964203648nt_int(aa_fun3265472288063244902nt_int(produc7320969046608631221nt_int,A2),B2) )
    <=> ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
tff(fact_21_old_Oprod_Oinject,axiom,
    ! [A: fun_in5730379987311621695e_term,B: product_prod_int_int,A2: fun_in5730379987311621695e_term,B2: product_prod_int_int] :
      ( ( aa_Pro8800328608558162860nt_int(aa_fun260840925927807678nt_int(produc4729913460834754785nt_int,A),B) = aa_Pro8800328608558162860nt_int(aa_fun260840925927807678nt_int(produc4729913460834754785nt_int,A2),B2) )
    <=> ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
tff(fact_22_old_Oprod_Oinject,axiom,
    ! [A: a,B: produc6653097349344004940it_nat,A2: a,B2: produc6653097349344004940it_nat] :
      ( ( aa_Pro8078911840264235907it_nat(aa_a_f2234720956197727492it_nat(produc9178034014595674355it_nat,A),B) = aa_Pro8078911840264235907it_nat(aa_a_f2234720956197727492it_nat(produc9178034014595674355it_nat,A2),B2) )
    <=> ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
tff(fact_23_mod__starD,axiom,
    ! [A3: assn,B3: assn,Ha: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(rep_assn(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A3),B3)),Ha)
     => ? [H1: produc3658429121746597890et_nat,H2: produc3658429121746597890et_nat] :
          ( aa_Pro653082453696054616_nat_o(rep_assn(A3),H1)
          & aa_Pro653082453696054616_nat_o(rep_assn(B3),H2) ) ) ).

% mod_starD
tff(fact_24_mod__starE,axiom,
    ! [A: assn,B: assn,Ha: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(rep_assn(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A),B)),Ha)
     => ~ ( ? [X_1: produc3658429121746597890et_nat] : aa_Pro653082453696054616_nat_o(rep_assn(A),X_1)
         => ! [H_2: produc3658429121746597890et_nat] : ~ aa_Pro653082453696054616_nat_o(rep_assn(B),H_2) ) ) ).

% mod_starE
tff(fact_25_assms,axiom,
    hoare_hoare_triple_a(p,c,q) ).

% assms
tff(fact_26__092_060open_062lim_Ah_A_092_060le_062_Alim_Ah_H_092_060close_062,axiom,
    aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,lim_Product_unit(h2)),lim_Product_unit(h)) ).

% \<open>lim h \<le> lim h'\<close>
tff(fact_27__092_060open_062as_A_061_Aas1_A_092_060union_062_Aas2_092_060close_062,axiom,
    as = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,as1),as2) ).

% \<open>as = as1 \<union> as2\<close>
tff(fact_28_mod__h__bot__indep,axiom,
    ! [Pa: assn,Ha: heap_e7401611519738050253t_unit,H_a: heap_e7401611519738050253t_unit] :
      ( aa_Pro653082453696054616_nat_o(rep_assn(Pa),produc7507926704131184380et_nat(Ha,bot_bot_set_nat))
    <=> aa_Pro653082453696054616_nat_o(rep_assn(Pa),produc7507926704131184380et_nat(H_a,bot_bot_set_nat)) ) ).

% mod_h_bot_indep
tff(fact_29_relH__dist__union,axiom,
    ! [Asa: set_nat,As: set_nat,Ha: heap_e7401611519738050253t_unit,H_a: heap_e7401611519738050253t_unit] :
      ( aa_hea7994211679605476167unit_o(relH(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Asa),As),Ha),H_a)
    <=> ( aa_hea7994211679605476167unit_o(relH(Asa,Ha),H_a)
        & aa_hea7994211679605476167unit_o(relH(As,Ha),H_a) ) ) ).

% relH_dist_union
tff(fact_30_times__assn__raw_Ocases,axiom,
    ! [X: produc3112576180687729086et_nat] :
      ~ ! [P: fun_Pr6460944699146269935_nat_o,Q: fun_Pr6460944699146269935_nat_o,H: heap_e7401611519738050253t_unit,As3: set_nat] : X != aa_Pro1979504083478021276et_nat(aa_fun615918787490101902et_nat(produc7291855603632349880et_nat,P),aa_Pro434340453207536480et_nat(aa_fun3926903090414666066et_nat(produc6017135503889078682et_nat,Q),produc7507926704131184380et_nat(H,As3))) ).

% times_assn_raw.cases
tff(fact_31_relH__sym,axiom,
    ! [Asa: set_nat,Ha: heap_e7401611519738050253t_unit,H_a: heap_e7401611519738050253t_unit] :
      ( aa_hea7994211679605476167unit_o(relH(Asa,Ha),H_a)
     => aa_hea7994211679605476167unit_o(relH(Asa,H_a),Ha) ) ).

% relH_sym
tff(fact_32_relH__trans,axiom,
    ! [Asa: set_nat,H12: heap_e7401611519738050253t_unit,H22: heap_e7401611519738050253t_unit,H3: heap_e7401611519738050253t_unit] :
      ( aa_hea7994211679605476167unit_o(relH(Asa,H12),H22)
     => ( aa_hea7994211679605476167unit_o(relH(Asa,H22),H3)
       => aa_hea7994211679605476167unit_o(relH(Asa,H12),H3) ) ) ).

% relH_trans
tff(fact_33_mod__relH,axiom,
    ! [Asa: set_nat,Ha: heap_e7401611519738050253t_unit,H_a: heap_e7401611519738050253t_unit,Pa: assn] :
      ( aa_hea7994211679605476167unit_o(relH(Asa,Ha),H_a)
     => ( aa_Pro653082453696054616_nat_o(rep_assn(Pa),produc7507926704131184380et_nat(Ha,Asa))
      <=> aa_Pro653082453696054616_nat_o(rep_assn(Pa),produc7507926704131184380et_nat(H_a,Asa)) ) ) ).

% mod_relH
tff(fact_34_hoare__triple__preI,axiom,
    ! [Pa: assn,Ca: heap_Time_Heap_a,Qa: fun_a_assn] :
      ( ! [H: produc3658429121746597890et_nat] :
          ( aa_Pro653082453696054616_nat_o(rep_assn(Pa),H)
         => hoare_hoare_triple_a(Pa,Ca,Qa) )
     => hoare_hoare_triple_a(Pa,Ca,Qa) ) ).

% hoare_triple_preI
tff(fact_35_mod__star__conv,axiom,
    ! [A3: assn,B3: assn,Ha: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(rep_assn(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A3),B3)),Ha)
    <=> ? [Hr: heap_e7401611519738050253t_unit,As12: set_nat,As22: set_nat] :
          ( ( Ha = produc7507926704131184380et_nat(Hr,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(rep_assn(A3),produc7507926704131184380et_nat(Hr,As12))
          & aa_Pro653082453696054616_nat_o(rep_assn(B3),produc7507926704131184380et_nat(Hr,As22)) ) ) ).

% mod_star_conv
tff(fact_36_star__assnI,axiom,
    ! [Pa: assn,Ha: heap_e7401611519738050253t_unit,Asa: set_nat,Qa: assn,As: set_nat] :
      ( aa_Pro653082453696054616_nat_o(rep_assn(Pa),produc7507926704131184380et_nat(Ha,Asa))
     => ( aa_Pro653082453696054616_nat_o(rep_assn(Qa),produc7507926704131184380et_nat(Ha,As))
       => ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Asa),As) = bot_bot_set_nat )
         => aa_Pro653082453696054616_nat_o(rep_assn(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Pa),Qa)),produc7507926704131184380et_nat(Ha,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Asa),As))) ) ) ) ).

% star_assnI
tff(fact_37_prod__induct4,axiom,
    ! [Pa: fun_Pr1227177680593602867_nat_o,X: produc3112576180687729086et_nat] :
      ( ! [A4: fun_Pr6460944699146269935_nat_o,B4: fun_Pr6460944699146269935_nat_o,C: heap_e7401611519738050253t_unit,D: set_nat] : aa_Pro2671238023309943196_nat_o(Pa,aa_Pro1979504083478021276et_nat(aa_fun615918787490101902et_nat(produc7291855603632349880et_nat,A4),aa_Pro434340453207536480et_nat(aa_fun3926903090414666066et_nat(produc6017135503889078682et_nat,B4),produc7507926704131184380et_nat(C,D))))
     => aa_Pro2671238023309943196_nat_o(Pa,X) ) ).

% prod_induct4
tff(fact_38_prod__induct3,axiom,
    ! [Pa: fun_Pr2620521676128474833_nat_o,X: produc8799533983703829792et_nat] :
      ( ! [A4: fun_Pr6460944699146269935_nat_o,B4: heap_e7401611519738050253t_unit,C: set_nat] : aa_Pro5643009357358181178_nat_o(Pa,aa_Pro434340453207536480et_nat(aa_fun3926903090414666066et_nat(produc6017135503889078682et_nat,A4),produc7507926704131184380et_nat(B4,C)))
     => aa_Pro5643009357358181178_nat_o(Pa,X) ) ).

% prod_induct3
tff(fact_39_prod__induct3,axiom,
    ! [Pa: fun_Pr1227177680593602867_nat_o,X: produc3112576180687729086et_nat] :
      ( ! [A4: fun_Pr6460944699146269935_nat_o,B4: fun_Pr6460944699146269935_nat_o,C: produc3658429121746597890et_nat] : aa_Pro2671238023309943196_nat_o(Pa,aa_Pro1979504083478021276et_nat(aa_fun615918787490101902et_nat(produc7291855603632349880et_nat,A4),aa_Pro434340453207536480et_nat(aa_fun3926903090414666066et_nat(produc6017135503889078682et_nat,B4),C)))
     => aa_Pro2671238023309943196_nat_o(Pa,X) ) ).

% prod_induct3
tff(fact_40_prod__induct3,axiom,
    ! [Pa: fun_Pr3540825087398332590_int_o,X: produc3043004577601930755nt_int] :
      ( ! [A4: fun_Pr7763680713840462443e_term,B4: int,C: int] : aa_Pro2601676399929874711_int_o(Pa,aa_Pro5748361610964203648nt_int(aa_fun3265472288063244902nt_int(produc7320969046608631221nt_int,A4),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,B4),C)))
     => aa_Pro2601676399929874711_int_o(Pa,X) ) ).

% prod_induct3
tff(fact_41_prod__induct3,axiom,
    ! [Pa: fun_Pr1196100989263358594_int_o,X: produc4788131091073775919nt_int] :
      ( ! [A4: fun_in5730379987311621695e_term,B4: int,C: int] : aa_Pro5451197003417746155_int_o(Pa,aa_Pro8800328608558162860nt_int(aa_fun260840925927807678nt_int(produc4729913460834754785nt_int,A4),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,B4),C)))
     => aa_Pro5451197003417746155_int_o(Pa,X) ) ).

% prod_induct3
tff(fact_42_prod__induct3,axiom,
    ! [Pa: fun_Pr5746006151868627512_nat_o,X: produc3260487557148687353it_nat] :
      ( ! [A4: a,B4: heap_e7401611519738050253t_unit,C: nat] : aa_Pro6426884464393220769_nat_o(Pa,aa_Pro8078911840264235907it_nat(aa_a_f2234720956197727492it_nat(produc9178034014595674355it_nat,A4),produc584006145561248582it_nat(B4,C)))
     => aa_Pro6426884464393220769_nat_o(Pa,X) ) ).

% prod_induct3
tff(fact_43_prod__cases4,axiom,
    ! [Y: produc3112576180687729086et_nat] :
      ~ ! [A4: fun_Pr6460944699146269935_nat_o,B4: fun_Pr6460944699146269935_nat_o,C: 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(C,D))) ).

% prod_cases4
tff(fact_44_prod__cases3,axiom,
    ! [Y: produc8799533983703829792et_nat] :
      ~ ! [A4: fun_Pr6460944699146269935_nat_o,B4: heap_e7401611519738050253t_unit,C: set_nat] : Y != aa_Pro434340453207536480et_nat(aa_fun3926903090414666066et_nat(produc6017135503889078682et_nat,A4),produc7507926704131184380et_nat(B4,C)) ).

% prod_cases3
tff(fact_45_prod__cases3,axiom,
    ! [Y: produc3112576180687729086et_nat] :
      ~ ! [A4: fun_Pr6460944699146269935_nat_o,B4: fun_Pr6460944699146269935_nat_o,C: produc3658429121746597890et_nat] : Y != aa_Pro1979504083478021276et_nat(aa_fun615918787490101902et_nat(produc7291855603632349880et_nat,A4),aa_Pro434340453207536480et_nat(aa_fun3926903090414666066et_nat(produc6017135503889078682et_nat,B4),C)) ).

% prod_cases3
tff(fact_46_prod__cases3,axiom,
    ! [Y: produc3043004577601930755nt_int] :
      ~ ! [A4: fun_Pr7763680713840462443e_term,B4: int,C: int] : Y != aa_Pro5748361610964203648nt_int(aa_fun3265472288063244902nt_int(produc7320969046608631221nt_int,A4),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,B4),C)) ).

% prod_cases3
tff(fact_47_prod__cases3,axiom,
    ! [Y: produc4788131091073775919nt_int] :
      ~ ! [A4: fun_in5730379987311621695e_term,B4: int,C: int] : Y != aa_Pro8800328608558162860nt_int(aa_fun260840925927807678nt_int(produc4729913460834754785nt_int,A4),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,B4),C)) ).

% prod_cases3
tff(fact_48_prod__cases3,axiom,
    ! [Y: produc3260487557148687353it_nat] :
      ~ ! [A4: a,B4: heap_e7401611519738050253t_unit,C: nat] : Y != aa_Pro8078911840264235907it_nat(aa_a_f2234720956197727492it_nat(produc9178034014595674355it_nat,A4),produc584006145561248582it_nat(B4,C)) ).

% prod_cases3
tff(fact_49_Pair__inject,axiom,
    ! [A: fun_Pr6460944699146269935_nat_o,B: produc3658429121746597890et_nat,A2: fun_Pr6460944699146269935_nat_o,B2: produc3658429121746597890et_nat] :
      ( ( aa_Pro434340453207536480et_nat(aa_fun3926903090414666066et_nat(produc6017135503889078682et_nat,A),B) = aa_Pro434340453207536480et_nat(aa_fun3926903090414666066et_nat(produc6017135503889078682et_nat,A2),B2) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
tff(fact_50_Pair__inject,axiom,
    ! [A: fun_Pr6460944699146269935_nat_o,B: produc8799533983703829792et_nat,A2: fun_Pr6460944699146269935_nat_o,B2: produc8799533983703829792et_nat] :
      ( ( aa_Pro1979504083478021276et_nat(aa_fun615918787490101902et_nat(produc7291855603632349880et_nat,A),B) = aa_Pro1979504083478021276et_nat(aa_fun615918787490101902et_nat(produc7291855603632349880et_nat,A2),B2) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
tff(fact_51_Pair__inject,axiom,
    ! [A: fun_Pr7763680713840462443e_term,B: product_prod_int_int,A2: fun_Pr7763680713840462443e_term,B2: product_prod_int_int] :
      ( ( aa_Pro5748361610964203648nt_int(aa_fun3265472288063244902nt_int(produc7320969046608631221nt_int,A),B) = aa_Pro5748361610964203648nt_int(aa_fun3265472288063244902nt_int(produc7320969046608631221nt_int,A2),B2) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
tff(fact_52_Pair__inject,axiom,
    ! [A: fun_in5730379987311621695e_term,B: product_prod_int_int,A2: fun_in5730379987311621695e_term,B2: product_prod_int_int] :
      ( ( aa_Pro8800328608558162860nt_int(aa_fun260840925927807678nt_int(produc4729913460834754785nt_int,A),B) = aa_Pro8800328608558162860nt_int(aa_fun260840925927807678nt_int(produc4729913460834754785nt_int,A2),B2) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
tff(fact_53_Pair__inject,axiom,
    ! [A: a,B: produc6653097349344004940it_nat,A2: a,B2: produc6653097349344004940it_nat] :
      ( ( aa_Pro8078911840264235907it_nat(aa_a_f2234720956197727492it_nat(produc9178034014595674355it_nat,A),B) = aa_Pro8078911840264235907it_nat(aa_a_f2234720956197727492it_nat(produc9178034014595674355it_nat,A2),B2) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
tff(fact_54_prod__cases,axiom,
    ! [Pa: fun_Pr2620521676128474833_nat_o,P2: produc8799533983703829792et_nat] :
      ( ! [A4: fun_Pr6460944699146269935_nat_o,B4: produc3658429121746597890et_nat] : aa_Pro5643009357358181178_nat_o(Pa,aa_Pro434340453207536480et_nat(aa_fun3926903090414666066et_nat(produc6017135503889078682et_nat,A4),B4))
     => aa_Pro5643009357358181178_nat_o(Pa,P2) ) ).

% prod_cases
tff(fact_55_prod__cases,axiom,
    ! [Pa: fun_Pr1227177680593602867_nat_o,P2: produc3112576180687729086et_nat] :
      ( ! [A4: fun_Pr6460944699146269935_nat_o,B4: produc8799533983703829792et_nat] : aa_Pro2671238023309943196_nat_o(Pa,aa_Pro1979504083478021276et_nat(aa_fun615918787490101902et_nat(produc7291855603632349880et_nat,A4),B4))
     => aa_Pro2671238023309943196_nat_o(Pa,P2) ) ).

% prod_cases
tff(fact_56_prod__cases,axiom,
    ! [Pa: fun_Pr3540825087398332590_int_o,P2: produc3043004577601930755nt_int] :
      ( ! [A4: fun_Pr7763680713840462443e_term,B4: product_prod_int_int] : aa_Pro2601676399929874711_int_o(Pa,aa_Pro5748361610964203648nt_int(aa_fun3265472288063244902nt_int(produc7320969046608631221nt_int,A4),B4))
     => aa_Pro2601676399929874711_int_o(Pa,P2) ) ).

% prod_cases
tff(fact_57_prod__cases,axiom,
    ! [Pa: fun_Pr1196100989263358594_int_o,P2: produc4788131091073775919nt_int] :
      ( ! [A4: fun_in5730379987311621695e_term,B4: product_prod_int_int] : aa_Pro5451197003417746155_int_o(Pa,aa_Pro8800328608558162860nt_int(aa_fun260840925927807678nt_int(produc4729913460834754785nt_int,A4),B4))
     => aa_Pro5451197003417746155_int_o(Pa,P2) ) ).

% prod_cases
tff(fact_58_prod__cases,axiom,
    ! [Pa: fun_Pr5746006151868627512_nat_o,P2: produc3260487557148687353it_nat] :
      ( ! [A4: a,B4: produc6653097349344004940it_nat] : aa_Pro6426884464393220769_nat_o(Pa,aa_Pro8078911840264235907it_nat(aa_a_f2234720956197727492it_nat(produc9178034014595674355it_nat,A4),B4))
     => aa_Pro6426884464393220769_nat_o(Pa,P2) ) ).

% prod_cases
tff(fact_59_surj__pair,axiom,
    ! [P2: produc8799533983703829792et_nat] :
    ? [X3: fun_Pr6460944699146269935_nat_o,Y3: produc3658429121746597890et_nat] : P2 = aa_Pro434340453207536480et_nat(aa_fun3926903090414666066et_nat(produc6017135503889078682et_nat,X3),Y3) ).

% surj_pair
tff(fact_60_surj__pair,axiom,
    ! [P2: produc3112576180687729086et_nat] :
    ? [X3: fun_Pr6460944699146269935_nat_o,Y3: produc8799533983703829792et_nat] : P2 = aa_Pro1979504083478021276et_nat(aa_fun615918787490101902et_nat(produc7291855603632349880et_nat,X3),Y3) ).

% surj_pair
tff(fact_61_surj__pair,axiom,
    ! [P2: produc3043004577601930755nt_int] :
    ? [X3: fun_Pr7763680713840462443e_term,Y3: product_prod_int_int] : P2 = aa_Pro5748361610964203648nt_int(aa_fun3265472288063244902nt_int(produc7320969046608631221nt_int,X3),Y3) ).

% surj_pair
tff(fact_62_surj__pair,axiom,
    ! [P2: produc4788131091073775919nt_int] :
    ? [X3: fun_in5730379987311621695e_term,Y3: product_prod_int_int] : P2 = aa_Pro8800328608558162860nt_int(aa_fun260840925927807678nt_int(produc4729913460834754785nt_int,X3),Y3) ).

% surj_pair
tff(fact_63_surj__pair,axiom,
    ! [P2: produc3260487557148687353it_nat] :
    ? [X3: a,Y3: produc6653097349344004940it_nat] : P2 = aa_Pro8078911840264235907it_nat(aa_a_f2234720956197727492it_nat(produc9178034014595674355it_nat,X3),Y3) ).

% surj_pair
tff(fact_64_mem__Collect__eq,axiom,
    ! [A: product_prod_nat_nat,Pa: fun_Pr4796943543331071514_nat_o] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(A),collec3392354462482085612at_nat(Pa))
    <=> aa_Pro6150637604445964947_nat_o(Pa,A) ) ).

% mem_Collect_eq
tff(fact_65_mem__Collect__eq,axiom,
    ! [A: $o,Pa: fun_o_o] :
      ( member_o((A),collect_o(Pa))
    <=> aa_o_o(Pa,(A)) ) ).

% mem_Collect_eq
tff(fact_66_mem__Collect__eq,axiom,
    ! [A: product_prod_int_int,Pa: fun_Pr3238374608701176162_int_o] :
      ( member5262025264175285858nt_int(A,collec213857154873943460nt_int(Pa))
    <=> aa_Pro4592068669816069595_int_o(Pa,A) ) ).

% mem_Collect_eq
tff(fact_67_mem__Collect__eq,axiom,
    ! [A: list_nat,Pa: fun_list_nat_o] :
      ( member_list_nat(A,collect_list_nat(Pa))
    <=> aa_list_nat_o(Pa,A) ) ).

% mem_Collect_eq
tff(fact_68_mem__Collect__eq,axiom,
    ! [A: set_nat,Pa: fun_set_nat_o] :
      ( aa_set_set_nat_o(member_set_nat(A),collect_set_nat(Pa))
    <=> aa_set_nat_o(Pa,A) ) ).

% mem_Collect_eq
tff(fact_69_mem__Collect__eq,axiom,
    ! [A: nat,Pa: fun_nat_o] :
      ( aa_set_nat_o(member_nat(A),collect_nat(Pa))
    <=> aa_nat_o(Pa,A) ) ).

% mem_Collect_eq
tff(fact_70_mem__Collect__eq,axiom,
    ! [A: int,Pa: fun_int_o] :
      ( aa_set_int_o(member_int(A),collect_int(Pa))
    <=> aa_int_o(Pa,A) ) ).

% mem_Collect_eq
tff(fact_71_Collect__mem__eq,axiom,
    ! [A3: set_Pr1261947904930325089at_nat] : collec3392354462482085612at_nat(aTP_Lamm_a(A3)) = A3 ).

% Collect_mem_eq
tff(fact_72_Collect__mem__eq,axiom,
    ! [A3: set_o] : collect_o(aTP_Lamm_aa(A3)) = A3 ).

% Collect_mem_eq
tff(fact_73_Collect__mem__eq,axiom,
    ! [A3: set_Pr958786334691620121nt_int] : collec213857154873943460nt_int(aTP_Lamm_ab(A3)) = A3 ).

% Collect_mem_eq
tff(fact_74_Collect__mem__eq,axiom,
    ! [A3: set_list_nat] : collect_list_nat(aTP_Lamm_ac(A3)) = A3 ).

% Collect_mem_eq
tff(fact_75_Collect__mem__eq,axiom,
    ! [A3: set_set_nat] : collect_set_nat(aTP_Lamm_ad(A3)) = A3 ).

% Collect_mem_eq
tff(fact_76_Collect__mem__eq,axiom,
    ! [A3: set_nat] : collect_nat(aTP_Lamm_ae(A3)) = A3 ).

% Collect_mem_eq
tff(fact_77_Collect__mem__eq,axiom,
    ! [A3: set_int] : collect_int(aTP_Lamm_af(A3)) = A3 ).

% Collect_mem_eq
tff(fact_78_Collect__cong,axiom,
    ! [Pa: fun_Pr3238374608701176162_int_o,Qa: fun_Pr3238374608701176162_int_o] :
      ( ! [X3: product_prod_int_int] :
          ( aa_Pro4592068669816069595_int_o(Pa,X3)
        <=> aa_Pro4592068669816069595_int_o(Qa,X3) )
     => ( collec213857154873943460nt_int(Pa) = collec213857154873943460nt_int(Qa) ) ) ).

% Collect_cong
tff(fact_79_Collect__cong,axiom,
    ! [Pa: fun_list_nat_o,Qa: fun_list_nat_o] :
      ( ! [X3: list_nat] :
          ( aa_list_nat_o(Pa,X3)
        <=> aa_list_nat_o(Qa,X3) )
     => ( collect_list_nat(Pa) = collect_list_nat(Qa) ) ) ).

% Collect_cong
tff(fact_80_Collect__cong,axiom,
    ! [Pa: fun_set_nat_o,Qa: fun_set_nat_o] :
      ( ! [X3: set_nat] :
          ( aa_set_nat_o(Pa,X3)
        <=> aa_set_nat_o(Qa,X3) )
     => ( collect_set_nat(Pa) = collect_set_nat(Qa) ) ) ).

% Collect_cong
tff(fact_81_Collect__cong,axiom,
    ! [Pa: fun_nat_o,Qa: fun_nat_o] :
      ( ! [X3: nat] :
          ( aa_nat_o(Pa,X3)
        <=> aa_nat_o(Qa,X3) )
     => ( collect_nat(Pa) = collect_nat(Qa) ) ) ).

% Collect_cong
tff(fact_82_Collect__cong,axiom,
    ! [Pa: fun_int_o,Qa: fun_int_o] :
      ( ! [X3: int] :
          ( aa_int_o(Pa,X3)
        <=> aa_int_o(Qa,X3) )
     => ( collect_int(Pa) = collect_int(Qa) ) ) ).

% Collect_cong
tff(fact_83_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_84_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_85_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_86_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_87_old_Oprod_Oexhaust,axiom,
    ! [Y: produc3260487557148687353it_nat] :
      ~ ! [A4: a,B4: produc6653097349344004940it_nat] : Y != aa_Pro8078911840264235907it_nat(aa_a_f2234720956197727492it_nat(produc9178034014595674355it_nat,A4),B4) ).

% old.prod.exhaust
tff(fact_88_one__assn__raw_Ocases,axiom,
    ! [X: produc3658429121746597890et_nat] :
      ~ ! [H: heap_e7401611519738050253t_unit,As3: set_nat] : X != produc7507926704131184380et_nat(H,As3) ).

% one_assn_raw.cases
tff(fact_89_assn__times__assoc,axiom,
    ! [Pa: assn,Qa: assn,Ra: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Pa),Qa)),Ra) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Pa),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Qa),Ra)) ).

% assn_times_assoc
tff(fact_90_assn__times__comm,axiom,
    ! [Pa: assn,Qa: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Pa),Qa) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Qa),Pa) ).

% assn_times_comm
tff(fact_91_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_92_Int__Un__eq_I4_J,axiom,
    ! [T: set_Pr8693737435421807431at_nat,S: set_Pr8693737435421807431at_nat] : aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,T),aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(inf_in4302113700860409141at_nat,S),T)) = T ).

% Int_Un_eq(4)
tff(fact_93_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_94_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_95_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_96_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_97_Int__Un__eq_I3_J,axiom,
    ! [S: set_Pr8693737435421807431at_nat,T: set_Pr8693737435421807431at_nat] : aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,S),aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(inf_in4302113700860409141at_nat,S),T)) = S ).

% Int_Un_eq(3)
tff(fact_98_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_99_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_100_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_101_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_102_Int__Un__eq_I2_J,axiom,
    ! [S: set_Pr8693737435421807431at_nat,T: set_Pr8693737435421807431at_nat] : aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(inf_in4302113700860409141at_nat,S),T)),T) = T ).

% Int_Un_eq(2)
tff(fact_103_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_104_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_105_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_106_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_107_Int__Un__eq_I1_J,axiom,
    ! [S: set_Pr8693737435421807431at_nat,T: set_Pr8693737435421807431at_nat] : aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(inf_in4302113700860409141at_nat,S),T)),S) = S ).

% Int_Un_eq(1)
tff(fact_108_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_109_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_110_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_111_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_112_Un__Int__eq_I4_J,axiom,
    ! [T: set_Pr8693737435421807431at_nat,S: set_Pr8693737435421807431at_nat] : aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(inf_in4302113700860409141at_nat,T),aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,S),T)) = T ).

% Un_Int_eq(4)
tff(fact_113_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_114_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_115_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_116_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_117_Un__Int__eq_I3_J,axiom,
    ! [S: set_Pr8693737435421807431at_nat,T: set_Pr8693737435421807431at_nat] : aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(inf_in4302113700860409141at_nat,S),aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,S),T)) = S ).

% Un_Int_eq(3)
tff(fact_118_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_119_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_120_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_121_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_122_Un__Int__eq_I2_J,axiom,
    ! [S: set_Pr8693737435421807431at_nat,T: set_Pr8693737435421807431at_nat] : aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(inf_in4302113700860409141at_nat,aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,S),T)),T) = T ).

% Un_Int_eq(2)
tff(fact_123_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_124_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_125_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_126_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_127_Un__Int__eq_I1_J,axiom,
    ! [S: set_Pr8693737435421807431at_nat,T: set_Pr8693737435421807431at_nat] : aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(inf_in4302113700860409141at_nat,aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,S),T)),S) = S ).

% Un_Int_eq(1)
tff(fact_128_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_129_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_130_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_131_Un__empty,axiom,
    ! [A3: set_Pr8693737435421807431at_nat,B3: set_Pr8693737435421807431at_nat] :
      ( ( aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,A3),B3) = bot_bo5327735625951526323at_nat )
    <=> ( ( A3 = bot_bo5327735625951526323at_nat )
        & ( B3 = bot_bo5327735625951526323at_nat ) ) ) ).

% Un_empty
tff(fact_132_Un__empty,axiom,
    ! [A3: set_Pr8551490117392284871at_nat,B3: set_Pr8551490117392284871at_nat] :
      ( ( aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A3),B3) = bot_bo8422036546324065075at_nat )
    <=> ( ( A3 = bot_bo8422036546324065075at_nat )
        & ( B3 = bot_bo8422036546324065075at_nat ) ) ) ).

% Un_empty
tff(fact_133_Un__empty,axiom,
    ! [A3: set_Pr4329608150637261639at_nat,B3: set_Pr4329608150637261639at_nat] :
      ( ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A3),B3) = bot_bo228742789529271731at_nat )
    <=> ( ( A3 = bot_bo228742789529271731at_nat )
        & ( B3 = bot_bo228742789529271731at_nat ) ) ) ).

% Un_empty
tff(fact_134_Un__empty,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
      ( ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,A3),B3) = bot_bo2099793752762293965at_nat )
    <=> ( ( A3 = bot_bo2099793752762293965at_nat )
        & ( B3 = bot_bo2099793752762293965at_nat ) ) ) ).

% Un_empty
tff(fact_135_Un__empty,axiom,
    ! [A3: set_o,B3: set_o] :
      ( ( aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,A3),B3) = bot_bot_set_o )
    <=> ( ( A3 = bot_bot_set_o )
        & ( B3 = bot_bot_set_o ) ) ) ).

% Un_empty
tff(fact_136_Un__empty,axiom,
    ! [A3: set_nat,B3: set_nat] :
      ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A3),B3) = bot_bot_set_nat )
    <=> ( ( A3 = bot_bot_set_nat )
        & ( B3 = bot_bot_set_nat ) ) ) ).

% Un_empty
tff(fact_137_Un__empty,axiom,
    ! [A3: set_int,B3: set_int] :
      ( ( aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A3),B3) = bot_bot_set_int )
    <=> ( ( A3 = bot_bot_set_int )
        & ( B3 = bot_bot_set_int ) ) ) ).

% Un_empty
tff(fact_138_inf__sup__absorb,axiom,
    ! [X: assn,Y: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),Y)) = X ).

% inf_sup_absorb
tff(fact_139_inf__sup__absorb,axiom,
    ! [X: product_unit,Y: product_unit] : aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,X),aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(sup_sup_Product_unit,X),Y)) = X ).

% inf_sup_absorb
tff(fact_140_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_141_inf__sup__absorb,axiom,
    ! [X: set_Pr8693737435421807431at_nat,Y: set_Pr8693737435421807431at_nat] : aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(inf_in4302113700860409141at_nat,X),aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,X),Y)) = X ).

% inf_sup_absorb
tff(fact_142_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_143_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_144_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_145_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_146_sup__inf__absorb,axiom,
    ! [X: assn,Y: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),Y)) = X ).

% sup_inf_absorb
tff(fact_147_sup__inf__absorb,axiom,
    ! [X: product_unit,Y: product_unit] : aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(sup_sup_Product_unit,X),aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,X),Y)) = X ).

% sup_inf_absorb
tff(fact_148_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_149_sup__inf__absorb,axiom,
    ! [X: set_Pr8693737435421807431at_nat,Y: set_Pr8693737435421807431at_nat] : aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,X),aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(inf_in4302113700860409141at_nat,X),Y)) = X ).

% sup_inf_absorb
tff(fact_150_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_151_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_152_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_153_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_154_sup__bot__left,axiom,
    ! [X: set_Pr8693737435421807431at_nat] : aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,bot_bo5327735625951526323at_nat),X) = X ).

% sup_bot_left
tff(fact_155_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_156_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_157_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_158_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_159_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_160_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_161_empty__Collect__eq,axiom,
    ! [Pa: fun_Pr3238374608701176162_int_o] :
      ( ( bot_bo1796632182523588997nt_int = collec213857154873943460nt_int(Pa) )
    <=> ! [X4: product_prod_int_int] : ~ aa_Pro4592068669816069595_int_o(Pa,X4) ) ).

% empty_Collect_eq
tff(fact_162_empty__Collect__eq,axiom,
    ! [Pa: fun_list_nat_o] :
      ( ( bot_bot_set_list_nat = collect_list_nat(Pa) )
    <=> ! [X4: list_nat] : ~ aa_list_nat_o(Pa,X4) ) ).

% empty_Collect_eq
tff(fact_163_empty__Collect__eq,axiom,
    ! [Pa: fun_set_nat_o] :
      ( ( bot_bot_set_set_nat = collect_set_nat(Pa) )
    <=> ! [X4: set_nat] : ~ aa_set_nat_o(Pa,X4) ) ).

% empty_Collect_eq
tff(fact_164_empty__Collect__eq,axiom,
    ! [Pa: fun_Pr4796943543331071514_nat_o] :
      ( ( bot_bo2099793752762293965at_nat = collec3392354462482085612at_nat(Pa) )
    <=> ! [X4: product_prod_nat_nat] : ~ aa_Pro6150637604445964947_nat_o(Pa,X4) ) ).

% empty_Collect_eq
tff(fact_165_empty__Collect__eq,axiom,
    ! [Pa: fun_o_o] :
      ( ( bot_bot_set_o = collect_o(Pa) )
    <=> ! [X4: $o] : ~ aa_o_o(Pa,(X4)) ) ).

% empty_Collect_eq
tff(fact_166_empty__Collect__eq,axiom,
    ! [Pa: fun_nat_o] :
      ( ( bot_bot_set_nat = collect_nat(Pa) )
    <=> ! [X4: nat] : ~ aa_nat_o(Pa,X4) ) ).

% empty_Collect_eq
tff(fact_167_empty__Collect__eq,axiom,
    ! [Pa: fun_int_o] :
      ( ( bot_bot_set_int = collect_int(Pa) )
    <=> ! [X4: int] : ~ aa_int_o(Pa,X4) ) ).

% empty_Collect_eq
tff(fact_168_Collect__empty__eq,axiom,
    ! [Pa: fun_Pr3238374608701176162_int_o] :
      ( ( collec213857154873943460nt_int(Pa) = bot_bo1796632182523588997nt_int )
    <=> ! [X4: product_prod_int_int] : ~ aa_Pro4592068669816069595_int_o(Pa,X4) ) ).

% Collect_empty_eq
tff(fact_169_Collect__empty__eq,axiom,
    ! [Pa: fun_list_nat_o] :
      ( ( collect_list_nat(Pa) = bot_bot_set_list_nat )
    <=> ! [X4: list_nat] : ~ aa_list_nat_o(Pa,X4) ) ).

% Collect_empty_eq
tff(fact_170_Collect__empty__eq,axiom,
    ! [Pa: fun_set_nat_o] :
      ( ( collect_set_nat(Pa) = bot_bot_set_set_nat )
    <=> ! [X4: set_nat] : ~ aa_set_nat_o(Pa,X4) ) ).

% Collect_empty_eq
tff(fact_171_Collect__empty__eq,axiom,
    ! [Pa: fun_Pr4796943543331071514_nat_o] :
      ( ( collec3392354462482085612at_nat(Pa) = bot_bo2099793752762293965at_nat )
    <=> ! [X4: product_prod_nat_nat] : ~ aa_Pro6150637604445964947_nat_o(Pa,X4) ) ).

% Collect_empty_eq
tff(fact_172_Collect__empty__eq,axiom,
    ! [Pa: fun_o_o] :
      ( ( collect_o(Pa) = bot_bot_set_o )
    <=> ! [X4: $o] : ~ aa_o_o(Pa,(X4)) ) ).

% Collect_empty_eq
tff(fact_173_Collect__empty__eq,axiom,
    ! [Pa: fun_nat_o] :
      ( ( collect_nat(Pa) = bot_bot_set_nat )
    <=> ! [X4: nat] : ~ aa_nat_o(Pa,X4) ) ).

% Collect_empty_eq
tff(fact_174_Collect__empty__eq,axiom,
    ! [Pa: fun_int_o] :
      ( ( collect_int(Pa) = bot_bot_set_int )
    <=> ! [X4: int] : ~ aa_int_o(Pa,X4) ) ).

% Collect_empty_eq
tff(fact_175_all__not__in__conv,axiom,
    ! [A3: set_set_nat] :
      ( ! [X4: set_nat] : ~ aa_set_set_nat_o(member_set_nat(X4),A3)
    <=> ( A3 = bot_bot_set_set_nat ) ) ).

% all_not_in_conv
tff(fact_176_all__not__in__conv,axiom,
    ! [A3: set_Pr1261947904930325089at_nat] :
      ( ! [X4: product_prod_nat_nat] : ~ aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X4),A3)
    <=> ( A3 = bot_bo2099793752762293965at_nat ) ) ).

% all_not_in_conv
tff(fact_177_all__not__in__conv,axiom,
    ! [A3: set_o] :
      ( ! [X4: $o] : ~ member_o((X4),A3)
    <=> ( A3 = bot_bot_set_o ) ) ).

% all_not_in_conv
tff(fact_178_all__not__in__conv,axiom,
    ! [A3: set_nat] :
      ( ! [X4: nat] : ~ aa_set_nat_o(member_nat(X4),A3)
    <=> ( A3 = bot_bot_set_nat ) ) ).

% all_not_in_conv
tff(fact_179_all__not__in__conv,axiom,
    ! [A3: set_int] :
      ( ! [X4: int] : ~ aa_set_int_o(member_int(X4),A3)
    <=> ( A3 = bot_bot_set_int ) ) ).

% all_not_in_conv
tff(fact_180_empty__iff,axiom,
    ! [Ca: set_nat] : ~ aa_set_set_nat_o(member_set_nat(Ca),bot_bot_set_set_nat) ).

% empty_iff
tff(fact_181_empty__iff,axiom,
    ! [Ca: product_prod_nat_nat] : ~ aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Ca),bot_bo2099793752762293965at_nat) ).

% empty_iff
tff(fact_182_empty__iff,axiom,
    ! [Ca: $o] : ~ member_o((Ca),bot_bot_set_o) ).

% empty_iff
tff(fact_183_empty__iff,axiom,
    ! [Ca: nat] : ~ aa_set_nat_o(member_nat(Ca),bot_bot_set_nat) ).

% empty_iff
tff(fact_184_empty__iff,axiom,
    ! [Ca: int] : ~ aa_set_int_o(member_int(Ca),bot_bot_set_int) ).

% empty_iff
tff(fact_185_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_186_inf__right__idem,axiom,
    ! [X: assn,Y: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),Y)),Y) = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),Y) ).

% inf_right_idem
tff(fact_187_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_188_inf__right__idem,axiom,
    ! [X: product_unit,Y: product_unit] : aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,X),Y)),Y) = aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,X),Y) ).

% inf_right_idem
tff(fact_189_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_190_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_191_inf_Oright__idem,axiom,
    ! [A: assn,B: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),B)),B) = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),B) ).

% inf.right_idem
tff(fact_192_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_193_inf_Oright__idem,axiom,
    ! [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,A),B)),B) = aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,A),B) ).

% inf.right_idem
tff(fact_194_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_195_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_196_inf__left__idem,axiom,
    ! [X: assn,Y: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),Y)) = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),Y) ).

% inf_left_idem
tff(fact_197_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_198_inf__left__idem,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,X),Y)) = aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,X),Y) ).

% inf_left_idem
tff(fact_199_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_200_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_201_inf_Oleft__idem,axiom,
    ! [A: assn,B: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),B)) = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),B) ).

% inf.left_idem
tff(fact_202_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_203_inf_Oleft__idem,axiom,
    ! [A: product_unit,B: product_unit] : aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,A),aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,A),B)) = aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,A),B) ).

% inf.left_idem
tff(fact_204_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_205_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_206_inf__idem,axiom,
    ! [X: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),X) = X ).

% inf_idem
tff(fact_207_inf__idem,axiom,
    ! [X: nat] : aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X),X) = X ).

% inf_idem
tff(fact_208_inf__idem,axiom,
    ! [X: product_unit] : aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,X),X) = X ).

% inf_idem
tff(fact_209_inf__idem,axiom,
    ! [X: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),X) = X ).

% inf_idem
tff(fact_210_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_211_inf_Oidem,axiom,
    ! [A: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),A) = A ).

% inf.idem
tff(fact_212_inf_Oidem,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),A) = A ).

% inf.idem
tff(fact_213_inf_Oidem,axiom,
    ! [A: product_unit] : aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,A),A) = A ).

% inf.idem
tff(fact_214_inf_Oidem,axiom,
    ! [A: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),A) = A ).

% inf.idem
tff(fact_215_sup_Oright__idem,axiom,
    ! [A: set_Pr8693737435421807431at_nat,B: set_Pr8693737435421807431at_nat] : aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,A),B)),B) = aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,A),B) ).

% sup.right_idem
tff(fact_216_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_217_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_218_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_219_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_220_sup__left__idem,axiom,
    ! [X: set_Pr8693737435421807431at_nat,Y: set_Pr8693737435421807431at_nat] : aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,X),aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,X),Y)) = aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,X),Y) ).

% sup_left_idem
tff(fact_221_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_222_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_223_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_224_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_225_sup_Oleft__idem,axiom,
    ! [A: set_Pr8693737435421807431at_nat,B: set_Pr8693737435421807431at_nat] : aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,A),aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,A),B)) = aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,A),B) ).

% sup.left_idem
tff(fact_226_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_227_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_228_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_229_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_230_sup__idem,axiom,
    ! [X: set_Pr8693737435421807431at_nat] : aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,X),X) = X ).

% sup_idem
tff(fact_231_sup__idem,axiom,
    ! [X: set_Pr8551490117392284871at_nat] : aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,X),X) = X ).

% sup_idem
tff(fact_232_sup__idem,axiom,
    ! [X: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),X) = X ).

% sup_idem
tff(fact_233_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_234_sup__idem,axiom,
    ! [X: nat] : aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X),X) = X ).

% sup_idem
tff(fact_235_sup_Oidem,axiom,
    ! [A: set_Pr8693737435421807431at_nat] : aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,A),A) = A ).

% sup.idem
tff(fact_236_sup_Oidem,axiom,
    ! [A: set_Pr8551490117392284871at_nat] : aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A),A) = A ).

% sup.idem
tff(fact_237_sup_Oidem,axiom,
    ! [A: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),A) = A ).

% sup.idem
tff(fact_238_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_239_sup_Oidem,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),A) = A ).

% sup.idem
tff(fact_240_Int__iff,axiom,
    ! [Ca: $o,A3: set_o,B3: set_o] :
      ( member_o((Ca),inf_inf_set_o(A3,B3))
    <=> ( member_o((Ca),A3)
        & member_o((Ca),B3) ) ) ).

% Int_iff
tff(fact_241_Int__iff,axiom,
    ! [Ca: set_nat,A3: set_set_nat,B3: set_set_nat] :
      ( aa_set_set_nat_o(member_set_nat(Ca),inf_inf_set_set_nat(A3,B3))
    <=> ( aa_set_set_nat_o(member_set_nat(Ca),A3)
        & aa_set_set_nat_o(member_set_nat(Ca),B3) ) ) ).

% Int_iff
tff(fact_242_Int__iff,axiom,
    ! [Ca: int,A3: set_int,B3: set_int] :
      ( aa_set_int_o(member_int(Ca),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A3),B3))
    <=> ( aa_set_int_o(member_int(Ca),A3)
        & aa_set_int_o(member_int(Ca),B3) ) ) ).

% Int_iff
tff(fact_243_Int__iff,axiom,
    ! [Ca: nat,A3: set_nat,B3: set_nat] :
      ( aa_set_nat_o(member_nat(Ca),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A3),B3))
    <=> ( aa_set_nat_o(member_nat(Ca),A3)
        & aa_set_nat_o(member_nat(Ca),B3) ) ) ).

% Int_iff
tff(fact_244_Int__iff,axiom,
    ! [Ca: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Ca),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A3),B3))
    <=> ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Ca),A3)
        & aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Ca),B3) ) ) ).

% Int_iff
tff(fact_245_IntI,axiom,
    ! [Ca: $o,A3: set_o,B3: set_o] :
      ( member_o((Ca),A3)
     => ( member_o((Ca),B3)
       => member_o((Ca),inf_inf_set_o(A3,B3)) ) ) ).

% IntI
tff(fact_246_IntI,axiom,
    ! [Ca: set_nat,A3: set_set_nat,B3: set_set_nat] :
      ( aa_set_set_nat_o(member_set_nat(Ca),A3)
     => ( aa_set_set_nat_o(member_set_nat(Ca),B3)
       => aa_set_set_nat_o(member_set_nat(Ca),inf_inf_set_set_nat(A3,B3)) ) ) ).

% IntI
tff(fact_247_IntI,axiom,
    ! [Ca: int,A3: set_int,B3: set_int] :
      ( aa_set_int_o(member_int(Ca),A3)
     => ( aa_set_int_o(member_int(Ca),B3)
       => aa_set_int_o(member_int(Ca),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A3),B3)) ) ) ).

% IntI
tff(fact_248_IntI,axiom,
    ! [Ca: nat,A3: set_nat,B3: set_nat] :
      ( aa_set_nat_o(member_nat(Ca),A3)
     => ( aa_set_nat_o(member_nat(Ca),B3)
       => aa_set_nat_o(member_nat(Ca),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A3),B3)) ) ) ).

% IntI
tff(fact_249_IntI,axiom,
    ! [Ca: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Ca),A3)
     => ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Ca),B3)
       => aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Ca),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A3),B3)) ) ) ).

% IntI
tff(fact_250_Un__iff,axiom,
    ! [Ca: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Ca),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,A3),B3))
    <=> ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Ca),A3)
        | aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Ca),B3) ) ) ).

% Un_iff
tff(fact_251_Un__iff,axiom,
    ! [Ca: $o,A3: set_o,B3: set_o] :
      ( member_o((Ca),aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,A3),B3))
    <=> ( member_o((Ca),A3)
        | member_o((Ca),B3) ) ) ).

% Un_iff
tff(fact_252_Un__iff,axiom,
    ! [Ca: set_nat,A3: set_set_nat,B3: set_set_nat] :
      ( aa_set_set_nat_o(member_set_nat(Ca),sup_sup_set_set_nat(A3,B3))
    <=> ( aa_set_set_nat_o(member_set_nat(Ca),A3)
        | aa_set_set_nat_o(member_set_nat(Ca),B3) ) ) ).

% Un_iff
tff(fact_253_Un__iff,axiom,
    ! [Ca: int,A3: set_int,B3: set_int] :
      ( aa_set_int_o(member_int(Ca),aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A3),B3))
    <=> ( aa_set_int_o(member_int(Ca),A3)
        | aa_set_int_o(member_int(Ca),B3) ) ) ).

% Un_iff
tff(fact_254_Un__iff,axiom,
    ! [Ca: produc859450856879609959at_nat,A3: set_Pr8693737435421807431at_nat,B3: set_Pr8693737435421807431at_nat] :
      ( member8206827879206165904at_nat(Ca,aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,A3),B3))
    <=> ( member8206827879206165904at_nat(Ca,A3)
        | member8206827879206165904at_nat(Ca,B3) ) ) ).

% Un_iff
tff(fact_255_Un__iff,axiom,
    ! [Ca: produc4166570645942440679at_nat,A3: set_Pr8551490117392284871at_nat,B3: set_Pr8551490117392284871at_nat] :
      ( member6689249552917799696at_nat(Ca,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A3),B3))
    <=> ( member6689249552917799696at_nat(Ca,A3)
        | member6689249552917799696at_nat(Ca,B3) ) ) ).

% Un_iff
tff(fact_256_Un__iff,axiom,
    ! [Ca: produc3843707927480180839at_nat,A3: set_Pr4329608150637261639at_nat,B3: set_Pr4329608150637261639at_nat] :
      ( member8757157785044589968at_nat(Ca,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A3),B3))
    <=> ( member8757157785044589968at_nat(Ca,A3)
        | member8757157785044589968at_nat(Ca,B3) ) ) ).

% Un_iff
tff(fact_257_Un__iff,axiom,
    ! [Ca: nat,A3: set_nat,B3: set_nat] :
      ( aa_set_nat_o(member_nat(Ca),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A3),B3))
    <=> ( aa_set_nat_o(member_nat(Ca),A3)
        | aa_set_nat_o(member_nat(Ca),B3) ) ) ).

% Un_iff
tff(fact_258_UnCI,axiom,
    ! [Ca: product_prod_nat_nat,B3: set_Pr1261947904930325089at_nat,A3: set_Pr1261947904930325089at_nat] :
      ( ( ~ aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Ca),B3)
       => aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Ca),A3) )
     => aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Ca),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,A3),B3)) ) ).

% UnCI
tff(fact_259_UnCI,axiom,
    ! [Ca: $o,B3: set_o,A3: set_o] :
      ( ( ~ member_o((Ca),B3)
       => member_o((Ca),A3) )
     => member_o((Ca),aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,A3),B3)) ) ).

% UnCI
tff(fact_260_UnCI,axiom,
    ! [Ca: set_nat,B3: set_set_nat,A3: set_set_nat] :
      ( ( ~ aa_set_set_nat_o(member_set_nat(Ca),B3)
       => aa_set_set_nat_o(member_set_nat(Ca),A3) )
     => aa_set_set_nat_o(member_set_nat(Ca),sup_sup_set_set_nat(A3,B3)) ) ).

% UnCI
tff(fact_261_UnCI,axiom,
    ! [Ca: int,B3: set_int,A3: set_int] :
      ( ( ~ aa_set_int_o(member_int(Ca),B3)
       => aa_set_int_o(member_int(Ca),A3) )
     => aa_set_int_o(member_int(Ca),aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A3),B3)) ) ).

% UnCI
tff(fact_262_UnCI,axiom,
    ! [Ca: produc859450856879609959at_nat,B3: set_Pr8693737435421807431at_nat,A3: set_Pr8693737435421807431at_nat] :
      ( ( ~ member8206827879206165904at_nat(Ca,B3)
       => member8206827879206165904at_nat(Ca,A3) )
     => member8206827879206165904at_nat(Ca,aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,A3),B3)) ) ).

% UnCI
tff(fact_263_UnCI,axiom,
    ! [Ca: produc4166570645942440679at_nat,B3: set_Pr8551490117392284871at_nat,A3: set_Pr8551490117392284871at_nat] :
      ( ( ~ member6689249552917799696at_nat(Ca,B3)
       => member6689249552917799696at_nat(Ca,A3) )
     => member6689249552917799696at_nat(Ca,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A3),B3)) ) ).

% UnCI
tff(fact_264_UnCI,axiom,
    ! [Ca: produc3843707927480180839at_nat,B3: set_Pr4329608150637261639at_nat,A3: set_Pr4329608150637261639at_nat] :
      ( ( ~ member8757157785044589968at_nat(Ca,B3)
       => member8757157785044589968at_nat(Ca,A3) )
     => member8757157785044589968at_nat(Ca,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A3),B3)) ) ).

% UnCI
tff(fact_265_UnCI,axiom,
    ! [Ca: nat,B3: set_nat,A3: set_nat] :
      ( ( ~ aa_set_nat_o(member_nat(Ca),B3)
       => aa_set_nat_o(member_nat(Ca),A3) )
     => aa_set_nat_o(member_nat(Ca),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A3),B3)) ) ).

% UnCI
tff(fact_266_mod__or__dist,axiom,
    ! [Pa: assn,Qa: assn,Ha: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(rep_assn(aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Pa),Qa)),Ha)
    <=> ( aa_Pro653082453696054616_nat_o(rep_assn(Pa),Ha)
        | aa_Pro653082453696054616_nat_o(rep_assn(Qa),Ha) ) ) ).

% mod_or_dist
tff(fact_267_star__false__left,axiom,
    ! [Pa: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,bot_bot_assn),Pa) = bot_bot_assn ).

% star_false_left
tff(fact_268_star__false__right,axiom,
    ! [Pa: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Pa),bot_bot_assn) = bot_bot_assn ).

% star_false_right
tff(fact_269_inf_Obounded__iff,axiom,
    ! [A: set_nat,B: set_nat,Ca: 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),Ca))
    <=> ( 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),Ca) ) ) ).

% inf.bounded_iff
tff(fact_270_inf_Obounded__iff,axiom,
    ! [A: assn,B: assn,Ca: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,B),Ca))
    <=> ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),B)
        & aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),Ca) ) ) ).

% inf.bounded_iff
tff(fact_271_inf_Obounded__iff,axiom,
    ! [A: product_unit,B: product_unit,Ca: product_unit] :
      ( aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,A),aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,B),Ca))
    <=> ( aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,A),B)
        & aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,A),Ca) ) ) ).

% inf.bounded_iff
tff(fact_272_inf_Obounded__iff,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,Ca: 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),Ca))
    <=> ( 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),Ca) ) ) ).

% inf.bounded_iff
tff(fact_273_inf_Obounded__iff,axiom,
    ! [A: set_int,B: set_int,Ca: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,B),Ca))
    <=> ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A),B)
        & aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A),Ca) ) ) ).

% inf.bounded_iff
tff(fact_274_inf_Obounded__iff,axiom,
    ! [A: rat,B: rat,Ca: 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),Ca))
    <=> ( 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),Ca) ) ) ).

% inf.bounded_iff
tff(fact_275_inf_Obounded__iff,axiom,
    ! [A: nat,B: nat,Ca: 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),Ca))
    <=> ( 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),Ca) ) ) ).

% inf.bounded_iff
tff(fact_276_inf_Obounded__iff,axiom,
    ! [A: int,B: int,Ca: 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),Ca))
    <=> ( 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),Ca) ) ) ).

% inf.bounded_iff
tff(fact_277_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_278_le__inf__iff,axiom,
    ! [X: assn,Y: assn,Z: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Y),Z))
    <=> ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X),Y)
        & aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X),Z) ) ) ).

% le_inf_iff
tff(fact_279_le__inf__iff,axiom,
    ! [X: product_unit,Y: product_unit,Z: product_unit] :
      ( aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,X),aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,Y),Z))
    <=> ( aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,X),Y)
        & aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,X),Z) ) ) ).

% le_inf_iff
tff(fact_280_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_281_le__inf__iff,axiom,
    ! [X: set_int,Y: set_int,Z: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,X),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,Y),Z))
    <=> ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,X),Y)
        & aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,X),Z) ) ) ).

% le_inf_iff
tff(fact_282_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_283_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_284_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_285_sup_Obounded__iff,axiom,
    ! [B: set_Pr8693737435421807431at_nat,Ca: set_Pr8693737435421807431at_nat,A: set_Pr8693737435421807431at_nat] :
      ( ord_le3000389064537975527at_nat(aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,B),Ca),A)
    <=> ( ord_le3000389064537975527at_nat(B,A)
        & ord_le3000389064537975527at_nat(Ca,A) ) ) ).

% sup.bounded_iff
tff(fact_286_sup_Obounded__iff,axiom,
    ! [B: set_Pr8551490117392284871at_nat,Ca: set_Pr8551490117392284871at_nat,A: set_Pr8551490117392284871at_nat] :
      ( ord_le8081472938463900775at_nat(aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,B),Ca),A)
    <=> ( ord_le8081472938463900775at_nat(B,A)
        & ord_le8081472938463900775at_nat(Ca,A) ) ) ).

% sup.bounded_iff
tff(fact_287_sup_Obounded__iff,axiom,
    ! [B: set_Pr4329608150637261639at_nat,Ca: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat] :
      ( ord_le1268244103169919719at_nat(aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,B),Ca),A)
    <=> ( ord_le1268244103169919719at_nat(B,A)
        & ord_le1268244103169919719at_nat(Ca,A) ) ) ).

% sup.bounded_iff
tff(fact_288_sup_Obounded__iff,axiom,
    ! [B: set_nat,Ca: 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),Ca)),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,Ca),A) ) ) ).

% sup.bounded_iff
tff(fact_289_sup_Obounded__iff,axiom,
    ! [B: set_int,Ca: set_int,A: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,B),Ca)),A)
    <=> ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,B),A)
        & aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,Ca),A) ) ) ).

% sup.bounded_iff
tff(fact_290_sup_Obounded__iff,axiom,
    ! [B: rat,Ca: 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),Ca)),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,Ca),A) ) ) ).

% sup.bounded_iff
tff(fact_291_sup_Obounded__iff,axiom,
    ! [B: nat,Ca: 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),Ca)),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,Ca),A) ) ) ).

% sup.bounded_iff
tff(fact_292_sup_Obounded__iff,axiom,
    ! [B: int,Ca: 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),Ca)),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,Ca),A) ) ) ).

% sup.bounded_iff
tff(fact_293_le__sup__iff,axiom,
    ! [X: set_Pr8693737435421807431at_nat,Y: set_Pr8693737435421807431at_nat,Z: set_Pr8693737435421807431at_nat] :
      ( ord_le3000389064537975527at_nat(aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,X),Y),Z)
    <=> ( ord_le3000389064537975527at_nat(X,Z)
        & ord_le3000389064537975527at_nat(Y,Z) ) ) ).

% le_sup_iff
tff(fact_294_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_295_le__sup__iff,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat,Z: set_Pr4329608150637261639at_nat] :
      ( ord_le1268244103169919719at_nat(aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),Y),Z)
    <=> ( ord_le1268244103169919719at_nat(X,Z)
        & ord_le1268244103169919719at_nat(Y,Z) ) ) ).

% le_sup_iff
tff(fact_296_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_297_le__sup__iff,axiom,
    ! [X: set_int,Y: set_int,Z: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,X),Y)),Z)
    <=> ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,X),Z)
        & aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,Y),Z) ) ) ).

% le_sup_iff
tff(fact_298_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_299_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_300_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_301_inf__bot__right,axiom,
    ! [X: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),bot_bot_assn) = bot_bot_assn ).

% inf_bot_right
tff(fact_302_inf__bot__right,axiom,
    ! [X: product_unit] : aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,X),bot_bot_Product_unit) = bot_bot_Product_unit ).

% inf_bot_right
tff(fact_303_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_304_inf__bot__right,axiom,
    ! [X: set_o] : inf_inf_set_o(X,bot_bot_set_o) = bot_bot_set_o ).

% inf_bot_right
tff(fact_305_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_306_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_307_inf__bot__left,axiom,
    ! [X: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,bot_bot_assn),X) = bot_bot_assn ).

% inf_bot_left
tff(fact_308_inf__bot__left,axiom,
    ! [X: product_unit] : aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,bot_bot_Product_unit),X) = bot_bot_Product_unit ).

% inf_bot_left
tff(fact_309_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_310_inf__bot__left,axiom,
    ! [X: set_o] : inf_inf_set_o(bot_bot_set_o,X) = bot_bot_set_o ).

% inf_bot_left
tff(fact_311_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_312_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_313_sup__bot_Oright__neutral,axiom,
    ! [A: set_Pr8693737435421807431at_nat] : aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,A),bot_bo5327735625951526323at_nat) = A ).

% sup_bot.right_neutral
tff(fact_314_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_315_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_316_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_317_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_318_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_319_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_320_sup__bot_Oneutr__eq__iff,axiom,
    ! [A: set_Pr8693737435421807431at_nat,B: set_Pr8693737435421807431at_nat] :
      ( ( bot_bo5327735625951526323at_nat = aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,A),B) )
    <=> ( ( A = bot_bo5327735625951526323at_nat )
        & ( B = bot_bo5327735625951526323at_nat ) ) ) ).

% sup_bot.neutr_eq_iff
tff(fact_321_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_322_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_323_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_324_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_325_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_326_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_327_sup__bot_Oleft__neutral,axiom,
    ! [A: set_Pr8693737435421807431at_nat] : aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,bot_bo5327735625951526323at_nat),A) = A ).

% sup_bot.left_neutral
tff(fact_328_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_329_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_330_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_331_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_332_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_333_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_334_sup__bot_Oeq__neutr__iff,axiom,
    ! [A: set_Pr8693737435421807431at_nat,B: set_Pr8693737435421807431at_nat] :
      ( ( aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,A),B) = bot_bo5327735625951526323at_nat )
    <=> ( ( A = bot_bo5327735625951526323at_nat )
        & ( B = bot_bo5327735625951526323at_nat ) ) ) ).

% sup_bot.eq_neutr_iff
tff(fact_335_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_336_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_337_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_338_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_339_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_340_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_341_sup__eq__bot__iff,axiom,
    ! [X: set_Pr8693737435421807431at_nat,Y: set_Pr8693737435421807431at_nat] :
      ( ( aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,X),Y) = bot_bo5327735625951526323at_nat )
    <=> ( ( X = bot_bo5327735625951526323at_nat )
        & ( Y = bot_bo5327735625951526323at_nat ) ) ) ).

% sup_eq_bot_iff
tff(fact_342_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_343_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_344_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_345_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_346_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_347_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_348_bot__eq__sup__iff,axiom,
    ! [X: set_Pr8693737435421807431at_nat,Y: set_Pr8693737435421807431at_nat] :
      ( ( bot_bo5327735625951526323at_nat = aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,X),Y) )
    <=> ( ( X = bot_bo5327735625951526323at_nat )
        & ( Y = bot_bo5327735625951526323at_nat ) ) ) ).

% bot_eq_sup_iff
tff(fact_349_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_350_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_351_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_352_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_353_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_354_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_355_sup__bot__right,axiom,
    ! [X: set_Pr8693737435421807431at_nat] : aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,X),bot_bo5327735625951526323at_nat) = X ).

% sup_bot_right
tff(fact_356_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_357_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_358_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_359_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_360_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_361_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_362_mod__h__bot__iff_I6_J,axiom,
    ! [Pa: assn,Qa: assn,Ha: heap_e7401611519738050253t_unit] :
      ( aa_Pro653082453696054616_nat_o(rep_assn(aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Pa),Qa)),produc7507926704131184380et_nat(Ha,bot_bot_set_nat))
    <=> ( aa_Pro653082453696054616_nat_o(rep_assn(Pa),produc7507926704131184380et_nat(Ha,bot_bot_set_nat))
        & aa_Pro653082453696054616_nat_o(rep_assn(Qa),produc7507926704131184380et_nat(Ha,bot_bot_set_nat)) ) ) ).

% mod_h_bot_iff(6)
tff(fact_363_mod__h__bot__iff_I7_J,axiom,
    ! [Pa: assn,Qa: assn,Ha: heap_e7401611519738050253t_unit] :
      ( aa_Pro653082453696054616_nat_o(rep_assn(aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Pa),Qa)),produc7507926704131184380et_nat(Ha,bot_bot_set_nat))
    <=> ( aa_Pro653082453696054616_nat_o(rep_assn(Pa),produc7507926704131184380et_nat(Ha,bot_bot_set_nat))
        | aa_Pro653082453696054616_nat_o(rep_assn(Qa),produc7507926704131184380et_nat(Ha,bot_bot_set_nat)) ) ) ).

% mod_h_bot_iff(7)
tff(fact_364_bot__set__def,axiom,
    bot_bo1796632182523588997nt_int = collec213857154873943460nt_int(bot_bo3516217578726220278_int_o) ).

% bot_set_def
tff(fact_365_bot__set__def,axiom,
    bot_bot_set_list_nat = collect_list_nat(bot_bo8828579575001852285_nat_o) ).

% bot_set_def
tff(fact_366_bot__set__def,axiom,
    bot_bot_set_set_nat = collect_set_nat(bot_bo3967814874310145751_nat_o) ).

% bot_set_def
tff(fact_367_bot__set__def,axiom,
    bot_bo2099793752762293965at_nat = collec3392354462482085612at_nat(bot_bo5074786513356115630_nat_o) ).

% bot_set_def
tff(fact_368_bot__set__def,axiom,
    bot_bot_set_o = collect_o(bot_bot_fun_o_o) ).

% bot_set_def
tff(fact_369_bot__set__def,axiom,
    bot_bot_set_nat = collect_nat(bot_bot_fun_nat_o) ).

% bot_set_def
tff(fact_370_bot__set__def,axiom,
    bot_bot_set_int = collect_int(bot_bot_fun_int_o) ).

% bot_set_def
tff(fact_371_mod__and__dist,axiom,
    ! [Pa: assn,Qa: assn,Ha: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(rep_assn(aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Pa),Qa)),Ha)
    <=> ( aa_Pro653082453696054616_nat_o(rep_assn(Pa),Ha)
        & aa_Pro653082453696054616_nat_o(rep_assn(Qa),Ha) ) ) ).

% mod_and_dist
tff(fact_372_mod__false,axiom,
    ! [Ha: produc3658429121746597890et_nat] : ~ aa_Pro653082453696054616_nat_o(rep_assn(bot_bot_assn),Ha) ).

% mod_false
tff(fact_373_star__or__dist1,axiom,
    ! [A3: assn,B3: assn,C2: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A3),B3)),C2) = aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A3),C2)),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,B3),C2)) ).

% star_or_dist1
tff(fact_374_star__or__dist2,axiom,
    ! [C2: assn,A3: assn,B3: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,C2),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A3),B3)) = aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,C2),A3)),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,C2),B3)) ).

% star_or_dist2
tff(fact_375_ex__in__conv,axiom,
    ! [A3: set_set_nat] :
      ( ? [X4: set_nat] : aa_set_set_nat_o(member_set_nat(X4),A3)
    <=> ( A3 != bot_bot_set_set_nat ) ) ).

% ex_in_conv
tff(fact_376_ex__in__conv,axiom,
    ! [A3: set_Pr1261947904930325089at_nat] :
      ( ? [X4: product_prod_nat_nat] : aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X4),A3)
    <=> ( A3 != bot_bo2099793752762293965at_nat ) ) ).

% ex_in_conv
tff(fact_377_ex__in__conv,axiom,
    ! [A3: set_o] :
      ( ? [X4: $o] : member_o((X4),A3)
    <=> ( A3 != bot_bot_set_o ) ) ).

% ex_in_conv
tff(fact_378_ex__in__conv,axiom,
    ! [A3: set_nat] :
      ( ? [X4: nat] : aa_set_nat_o(member_nat(X4),A3)
    <=> ( A3 != bot_bot_set_nat ) ) ).

% ex_in_conv
tff(fact_379_ex__in__conv,axiom,
    ! [A3: set_int] :
      ( ? [X4: int] : aa_set_int_o(member_int(X4),A3)
    <=> ( A3 != bot_bot_set_int ) ) ).

% ex_in_conv
tff(fact_380_equals0I,axiom,
    ! [A3: set_set_nat] :
      ( ! [Y3: set_nat] : ~ aa_set_set_nat_o(member_set_nat(Y3),A3)
     => ( A3 = bot_bot_set_set_nat ) ) ).

% equals0I
tff(fact_381_equals0I,axiom,
    ! [A3: set_Pr1261947904930325089at_nat] :
      ( ! [Y3: product_prod_nat_nat] : ~ aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Y3),A3)
     => ( A3 = bot_bo2099793752762293965at_nat ) ) ).

% equals0I
tff(fact_382_equals0I,axiom,
    ! [A3: set_o] :
      ( ! [Y3: $o] : ~ member_o((Y3),A3)
     => ( A3 = bot_bot_set_o ) ) ).

% equals0I
tff(fact_383_equals0I,axiom,
    ! [A3: set_nat] :
      ( ! [Y3: nat] : ~ aa_set_nat_o(member_nat(Y3),A3)
     => ( A3 = bot_bot_set_nat ) ) ).

% equals0I
tff(fact_384_equals0I,axiom,
    ! [A3: set_int] :
      ( ! [Y3: int] : ~ aa_set_int_o(member_int(Y3),A3)
     => ( A3 = bot_bot_set_int ) ) ).

% equals0I
tff(fact_385_equals0D,axiom,
    ! [A3: set_set_nat,A: set_nat] :
      ( ( A3 = bot_bot_set_set_nat )
     => ~ aa_set_set_nat_o(member_set_nat(A),A3) ) ).

% equals0D
tff(fact_386_equals0D,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat] :
      ( ( A3 = bot_bo2099793752762293965at_nat )
     => ~ aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(A),A3) ) ).

% equals0D
tff(fact_387_equals0D,axiom,
    ! [A3: set_o,A: $o] :
      ( ( A3 = bot_bot_set_o )
     => ~ member_o((A),A3) ) ).

% equals0D
tff(fact_388_equals0D,axiom,
    ! [A3: set_nat,A: nat] :
      ( ( A3 = bot_bot_set_nat )
     => ~ aa_set_nat_o(member_nat(A),A3) ) ).

% equals0D
tff(fact_389_equals0D,axiom,
    ! [A3: set_int,A: int] :
      ( ( A3 = bot_bot_set_int )
     => ~ aa_set_int_o(member_int(A),A3) ) ).

% equals0D
tff(fact_390_emptyE,axiom,
    ! [A: set_nat] : ~ aa_set_set_nat_o(member_set_nat(A),bot_bot_set_set_nat) ).

% emptyE
tff(fact_391_emptyE,axiom,
    ! [A: product_prod_nat_nat] : ~ aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(A),bot_bo2099793752762293965at_nat) ).

% emptyE
tff(fact_392_emptyE,axiom,
    ! [A: $o] : ~ member_o((A),bot_bot_set_o) ).

% emptyE
tff(fact_393_emptyE,axiom,
    ! [A: nat] : ~ aa_set_nat_o(member_nat(A),bot_bot_set_nat) ).

% emptyE
tff(fact_394_emptyE,axiom,
    ! [A: int] : ~ aa_set_int_o(member_int(A),bot_bot_set_int) ).

% emptyE
tff(fact_395_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_396_inf__left__commute,axiom,
    ! [X: assn,Y: assn,Z: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Y),Z)) = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Y),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),Z)) ).

% inf_left_commute
tff(fact_397_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_398_inf__left__commute,axiom,
    ! [X: product_unit,Y: product_unit,Z: product_unit] : aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,X),aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,Y),Z)) = aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,Y),aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,X),Z)) ).

% inf_left_commute
tff(fact_399_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_400_inf_Oleft__commute,axiom,
    ! [B: set_nat,A: set_nat,Ca: 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),Ca)) = 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),Ca)) ).

% inf.left_commute
tff(fact_401_inf_Oleft__commute,axiom,
    ! [B: assn,A: assn,Ca: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,B),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),Ca)) = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,B),Ca)) ).

% inf.left_commute
tff(fact_402_inf_Oleft__commute,axiom,
    ! [B: nat,A: nat,Ca: 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),Ca)) = aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,B),Ca)) ).

% inf.left_commute
tff(fact_403_inf_Oleft__commute,axiom,
    ! [B: product_unit,A: product_unit,Ca: product_unit] : aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,B),aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,A),Ca)) = aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,A),aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,B),Ca)) ).

% inf.left_commute
tff(fact_404_inf_Oleft__commute,axiom,
    ! [B: set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat,Ca: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),Ca)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B),Ca)) ).

% inf.left_commute
tff(fact_405_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_406_inf__commute,axiom,
    ! [X: assn,Y: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),Y) = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Y),X) ).

% inf_commute
tff(fact_407_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_408_inf__commute,axiom,
    ! [X: product_unit,Y: product_unit] : aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,X),Y) = aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,Y),X) ).

% inf_commute
tff(fact_409_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_410_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_411_inf_Ocommute,axiom,
    ! [A: assn,B: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),B) = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,B),A) ).

% inf.commute
tff(fact_412_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_413_inf_Ocommute,axiom,
    ! [A: product_unit,B: product_unit] : aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,A),B) = aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,B),A) ).

% inf.commute
tff(fact_414_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_415_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_416_inf__assoc,axiom,
    ! [X: assn,Y: assn,Z: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),Y)),Z) = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Y),Z)) ).

% inf_assoc
tff(fact_417_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_418_inf__assoc,axiom,
    ! [X: product_unit,Y: product_unit,Z: product_unit] : aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,X),Y)),Z) = aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,X),aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,Y),Z)) ).

% inf_assoc
tff(fact_419_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_420_inf_Oassoc,axiom,
    ! [A: set_nat,B: set_nat,Ca: 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)),Ca) = 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),Ca)) ).

% inf.assoc
tff(fact_421_inf_Oassoc,axiom,
    ! [A: assn,B: assn,Ca: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),B)),Ca) = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,B),Ca)) ).

% inf.assoc
tff(fact_422_inf_Oassoc,axiom,
    ! [A: nat,B: nat,Ca: 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)),Ca) = aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,B),Ca)) ).

% inf.assoc
tff(fact_423_inf_Oassoc,axiom,
    ! [A: product_unit,B: product_unit,Ca: product_unit] : aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,A),B)),Ca) = aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,A),aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,B),Ca)) ).

% inf.assoc
tff(fact_424_inf_Oassoc,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,Ca: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B)),Ca) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B),Ca)) ).

% inf.assoc
tff(fact_425_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_426_inf__sup__aci_I1_J,axiom,
    ! [X: assn,Y: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),Y) = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Y),X) ).

% inf_sup_aci(1)
tff(fact_427_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_428_inf__sup__aci_I1_J,axiom,
    ! [X: product_unit,Y: product_unit] : aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,X),Y) = aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,Y),X) ).

% inf_sup_aci(1)
tff(fact_429_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_430_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_431_inf__sup__aci_I2_J,axiom,
    ! [X: assn,Y: assn,Z: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),Y)),Z) = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Y),Z)) ).

% inf_sup_aci(2)
tff(fact_432_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_433_inf__sup__aci_I2_J,axiom,
    ! [X: product_unit,Y: product_unit,Z: product_unit] : aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,X),Y)),Z) = aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,X),aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,Y),Z)) ).

% inf_sup_aci(2)
tff(fact_434_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_435_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_436_inf__sup__aci_I3_J,axiom,
    ! [X: assn,Y: assn,Z: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Y),Z)) = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Y),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),Z)) ).

% inf_sup_aci(3)
tff(fact_437_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_438_inf__sup__aci_I3_J,axiom,
    ! [X: product_unit,Y: product_unit,Z: product_unit] : aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,X),aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,Y),Z)) = aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,Y),aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,X),Z)) ).

% inf_sup_aci(3)
tff(fact_439_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_440_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_441_inf__sup__aci_I4_J,axiom,
    ! [X: assn,Y: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),Y)) = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),Y) ).

% inf_sup_aci(4)
tff(fact_442_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_443_inf__sup__aci_I4_J,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,X),Y)) = aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,X),Y) ).

% inf_sup_aci(4)
tff(fact_444_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_445_sup__left__commute,axiom,
    ! [X: set_Pr8693737435421807431at_nat,Y: set_Pr8693737435421807431at_nat,Z: set_Pr8693737435421807431at_nat] : aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,X),aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,Y),Z)) = aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,Y),aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,X),Z)) ).

% sup_left_commute
tff(fact_446_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_447_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_448_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_449_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_450_sup_Oleft__commute,axiom,
    ! [B: set_Pr8693737435421807431at_nat,A: set_Pr8693737435421807431at_nat,Ca: set_Pr8693737435421807431at_nat] : aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,B),aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,A),Ca)) = aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,A),aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,B),Ca)) ).

% sup.left_commute
tff(fact_451_sup_Oleft__commute,axiom,
    ! [B: set_Pr8551490117392284871at_nat,A: set_Pr8551490117392284871at_nat,Ca: set_Pr8551490117392284871at_nat] : aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,B),aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A),Ca)) = aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A),aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,B),Ca)) ).

% sup.left_commute
tff(fact_452_sup_Oleft__commute,axiom,
    ! [B: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat,Ca: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,B),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),Ca)) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,B),Ca)) ).

% sup.left_commute
tff(fact_453_sup_Oleft__commute,axiom,
    ! [B: set_nat,A: set_nat,Ca: 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),Ca)) = 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),Ca)) ).

% sup.left_commute
tff(fact_454_sup_Oleft__commute,axiom,
    ! [B: nat,A: nat,Ca: 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),Ca)) = aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,B),Ca)) ).

% sup.left_commute
tff(fact_455_sup__commute,axiom,
    ! [X: set_Pr8693737435421807431at_nat,Y: set_Pr8693737435421807431at_nat] : aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,X),Y) = aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,Y),X) ).

% sup_commute
tff(fact_456_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_457_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_458_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_459_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_460_sup_Ocommute,axiom,
    ! [A: set_Pr8693737435421807431at_nat,B: set_Pr8693737435421807431at_nat] : aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,A),B) = aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,B),A) ).

% sup.commute
tff(fact_461_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_462_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_463_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_464_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_465_sup__assoc,axiom,
    ! [X: set_Pr8693737435421807431at_nat,Y: set_Pr8693737435421807431at_nat,Z: set_Pr8693737435421807431at_nat] : aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,X),Y)),Z) = aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,X),aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,Y),Z)) ).

% sup_assoc
tff(fact_466_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_467_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_468_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_469_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_470_sup_Oassoc,axiom,
    ! [A: set_Pr8693737435421807431at_nat,B: set_Pr8693737435421807431at_nat,Ca: set_Pr8693737435421807431at_nat] : aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,A),B)),Ca) = aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,A),aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,B),Ca)) ).

% sup.assoc
tff(fact_471_sup_Oassoc,axiom,
    ! [A: set_Pr8551490117392284871at_nat,B: set_Pr8551490117392284871at_nat,Ca: set_Pr8551490117392284871at_nat] : aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A),B)),Ca) = aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A),aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,B),Ca)) ).

% sup.assoc
tff(fact_472_sup_Oassoc,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat,Ca: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),B)),Ca) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,B),Ca)) ).

% sup.assoc
tff(fact_473_sup_Oassoc,axiom,
    ! [A: set_nat,B: set_nat,Ca: 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)),Ca) = 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),Ca)) ).

% sup.assoc
tff(fact_474_sup_Oassoc,axiom,
    ! [A: nat,B: nat,Ca: 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)),Ca) = aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,B),Ca)) ).

% sup.assoc
tff(fact_475_inf__sup__aci_I5_J,axiom,
    ! [X: set_Pr8693737435421807431at_nat,Y: set_Pr8693737435421807431at_nat] : aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,X),Y) = aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,Y),X) ).

% inf_sup_aci(5)
tff(fact_476_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_477_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_478_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_479_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_480_inf__sup__aci_I6_J,axiom,
    ! [X: set_Pr8693737435421807431at_nat,Y: set_Pr8693737435421807431at_nat,Z: set_Pr8693737435421807431at_nat] : aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,X),Y)),Z) = aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,X),aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,Y),Z)) ).

% inf_sup_aci(6)
tff(fact_481_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_482_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_483_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_484_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_485_inf__sup__aci_I7_J,axiom,
    ! [X: set_Pr8693737435421807431at_nat,Y: set_Pr8693737435421807431at_nat,Z: set_Pr8693737435421807431at_nat] : aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,X),aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,Y),Z)) = aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,Y),aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,X),Z)) ).

% inf_sup_aci(7)
tff(fact_486_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_487_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_488_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_489_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_490_inf__sup__aci_I8_J,axiom,
    ! [X: set_Pr8693737435421807431at_nat,Y: set_Pr8693737435421807431at_nat] : aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,X),aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,X),Y)) = aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,X),Y) ).

% inf_sup_aci(8)
tff(fact_491_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_492_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_493_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_494_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_495_Int__left__commute,axiom,
    ! [A3: set_nat,B3: set_nat,C2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A3),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B3),C2)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B3),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A3),C2)) ).

% Int_left_commute
tff(fact_496_Int__left__commute,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A3),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B3),C2)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B3),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A3),C2)) ).

% Int_left_commute
tff(fact_497_Int__left__absorb,axiom,
    ! [A3: set_nat,B3: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A3),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A3),B3)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A3),B3) ).

% Int_left_absorb
tff(fact_498_Int__left__absorb,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A3),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A3),B3)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A3),B3) ).

% Int_left_absorb
tff(fact_499_Int__commute,axiom,
    ! [A3: set_nat,B3: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A3),B3) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B3),A3) ).

% Int_commute
tff(fact_500_Int__commute,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A3),B3) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B3),A3) ).

% Int_commute
tff(fact_501_Int__absorb,axiom,
    ! [A3: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A3),A3) = A3 ).

% Int_absorb
tff(fact_502_Int__absorb,axiom,
    ! [A3: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A3),A3) = A3 ).

% Int_absorb
tff(fact_503_Int__assoc,axiom,
    ! [A3: set_nat,B3: set_nat,C2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A3),B3)),C2) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A3),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B3),C2)) ).

% Int_assoc
tff(fact_504_Int__assoc,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A3),B3)),C2) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A3),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B3),C2)) ).

% Int_assoc
tff(fact_505_IntD2,axiom,
    ! [Ca: $o,A3: set_o,B3: set_o] :
      ( member_o((Ca),inf_inf_set_o(A3,B3))
     => member_o((Ca),B3) ) ).

% IntD2
tff(fact_506_IntD2,axiom,
    ! [Ca: set_nat,A3: set_set_nat,B3: set_set_nat] :
      ( aa_set_set_nat_o(member_set_nat(Ca),inf_inf_set_set_nat(A3,B3))
     => aa_set_set_nat_o(member_set_nat(Ca),B3) ) ).

% IntD2
tff(fact_507_IntD2,axiom,
    ! [Ca: int,A3: set_int,B3: set_int] :
      ( aa_set_int_o(member_int(Ca),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A3),B3))
     => aa_set_int_o(member_int(Ca),B3) ) ).

% IntD2
tff(fact_508_IntD2,axiom,
    ! [Ca: nat,A3: set_nat,B3: set_nat] :
      ( aa_set_nat_o(member_nat(Ca),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A3),B3))
     => aa_set_nat_o(member_nat(Ca),B3) ) ).

% IntD2
tff(fact_509_IntD2,axiom,
    ! [Ca: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Ca),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A3),B3))
     => aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Ca),B3) ) ).

% IntD2
tff(fact_510_IntD1,axiom,
    ! [Ca: $o,A3: set_o,B3: set_o] :
      ( member_o((Ca),inf_inf_set_o(A3,B3))
     => member_o((Ca),A3) ) ).

% IntD1
tff(fact_511_IntD1,axiom,
    ! [Ca: set_nat,A3: set_set_nat,B3: set_set_nat] :
      ( aa_set_set_nat_o(member_set_nat(Ca),inf_inf_set_set_nat(A3,B3))
     => aa_set_set_nat_o(member_set_nat(Ca),A3) ) ).

% IntD1
tff(fact_512_IntD1,axiom,
    ! [Ca: int,A3: set_int,B3: set_int] :
      ( aa_set_int_o(member_int(Ca),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A3),B3))
     => aa_set_int_o(member_int(Ca),A3) ) ).

% IntD1
tff(fact_513_IntD1,axiom,
    ! [Ca: nat,A3: set_nat,B3: set_nat] :
      ( aa_set_nat_o(member_nat(Ca),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A3),B3))
     => aa_set_nat_o(member_nat(Ca),A3) ) ).

% IntD1
tff(fact_514_IntD1,axiom,
    ! [Ca: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Ca),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A3),B3))
     => aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Ca),A3) ) ).

% IntD1
tff(fact_515_IntE,axiom,
    ! [Ca: $o,A3: set_o,B3: set_o] :
      ( member_o((Ca),inf_inf_set_o(A3,B3))
     => ~ ( member_o((Ca),A3)
         => ~ member_o((Ca),B3) ) ) ).

% IntE
tff(fact_516_IntE,axiom,
    ! [Ca: set_nat,A3: set_set_nat,B3: set_set_nat] :
      ( aa_set_set_nat_o(member_set_nat(Ca),inf_inf_set_set_nat(A3,B3))
     => ~ ( aa_set_set_nat_o(member_set_nat(Ca),A3)
         => ~ aa_set_set_nat_o(member_set_nat(Ca),B3) ) ) ).

% IntE
tff(fact_517_IntE,axiom,
    ! [Ca: int,A3: set_int,B3: set_int] :
      ( aa_set_int_o(member_int(Ca),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A3),B3))
     => ~ ( aa_set_int_o(member_int(Ca),A3)
         => ~ aa_set_int_o(member_int(Ca),B3) ) ) ).

% IntE
tff(fact_518_IntE,axiom,
    ! [Ca: nat,A3: set_nat,B3: set_nat] :
      ( aa_set_nat_o(member_nat(Ca),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A3),B3))
     => ~ ( aa_set_nat_o(member_nat(Ca),A3)
         => ~ aa_set_nat_o(member_nat(Ca),B3) ) ) ).

% IntE
tff(fact_519_IntE,axiom,
    ! [Ca: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Ca),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A3),B3))
     => ~ ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Ca),A3)
         => ~ aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Ca),B3) ) ) ).

% IntE
tff(fact_520_Un__left__commute,axiom,
    ! [A3: set_Pr8693737435421807431at_nat,B3: set_Pr8693737435421807431at_nat,C2: set_Pr8693737435421807431at_nat] : aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,A3),aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,B3),C2)) = aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,B3),aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,A3),C2)) ).

% Un_left_commute
tff(fact_521_Un__left__commute,axiom,
    ! [A3: set_Pr8551490117392284871at_nat,B3: set_Pr8551490117392284871at_nat,C2: set_Pr8551490117392284871at_nat] : aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A3),aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,B3),C2)) = aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,B3),aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A3),C2)) ).

% Un_left_commute
tff(fact_522_Un__left__commute,axiom,
    ! [A3: set_Pr4329608150637261639at_nat,B3: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A3),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,B3),C2)) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,B3),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A3),C2)) ).

% Un_left_commute
tff(fact_523_Un__left__commute,axiom,
    ! [A3: set_nat,B3: set_nat,C2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A3),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,B3),C2)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,B3),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A3),C2)) ).

% Un_left_commute
tff(fact_524_Un__left__absorb,axiom,
    ! [A3: set_Pr8693737435421807431at_nat,B3: set_Pr8693737435421807431at_nat] : aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,A3),aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,A3),B3)) = aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,A3),B3) ).

% Un_left_absorb
tff(fact_525_Un__left__absorb,axiom,
    ! [A3: set_Pr8551490117392284871at_nat,B3: set_Pr8551490117392284871at_nat] : aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A3),aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A3),B3)) = aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A3),B3) ).

% Un_left_absorb
tff(fact_526_Un__left__absorb,axiom,
    ! [A3: set_Pr4329608150637261639at_nat,B3: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A3),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A3),B3)) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A3),B3) ).

% Un_left_absorb
tff(fact_527_Un__left__absorb,axiom,
    ! [A3: set_nat,B3: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A3),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A3),B3)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A3),B3) ).

% Un_left_absorb
tff(fact_528_Un__commute,axiom,
    ! [A3: set_Pr8693737435421807431at_nat,B3: set_Pr8693737435421807431at_nat] : aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,A3),B3) = aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,B3),A3) ).

% Un_commute
tff(fact_529_Un__commute,axiom,
    ! [A3: set_Pr8551490117392284871at_nat,B3: set_Pr8551490117392284871at_nat] : aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A3),B3) = aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,B3),A3) ).

% Un_commute
tff(fact_530_Un__commute,axiom,
    ! [A3: set_Pr4329608150637261639at_nat,B3: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A3),B3) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,B3),A3) ).

% Un_commute
tff(fact_531_Un__commute,axiom,
    ! [A3: set_nat,B3: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A3),B3) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,B3),A3) ).

% Un_commute
tff(fact_532_Un__absorb,axiom,
    ! [A3: set_Pr8693737435421807431at_nat] : aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,A3),A3) = A3 ).

% Un_absorb
tff(fact_533_Un__absorb,axiom,
    ! [A3: set_Pr8551490117392284871at_nat] : aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A3),A3) = A3 ).

% Un_absorb
tff(fact_534_Un__absorb,axiom,
    ! [A3: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A3),A3) = A3 ).

% Un_absorb
tff(fact_535_Un__absorb,axiom,
    ! [A3: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A3),A3) = A3 ).

% Un_absorb
tff(fact_536_Un__assoc,axiom,
    ! [A3: set_Pr8693737435421807431at_nat,B3: set_Pr8693737435421807431at_nat,C2: set_Pr8693737435421807431at_nat] : aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,A3),B3)),C2) = aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,A3),aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,B3),C2)) ).

% Un_assoc
tff(fact_537_Un__assoc,axiom,
    ! [A3: set_Pr8551490117392284871at_nat,B3: 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,A3),B3)),C2) = aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A3),aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,B3),C2)) ).

% Un_assoc
tff(fact_538_Un__assoc,axiom,
    ! [A3: set_Pr4329608150637261639at_nat,B3: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A3),B3)),C2) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A3),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,B3),C2)) ).

% Un_assoc
tff(fact_539_Un__assoc,axiom,
    ! [A3: set_nat,B3: set_nat,C2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A3),B3)),C2) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A3),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,B3),C2)) ).

% Un_assoc
tff(fact_540_ball__Un,axiom,
    ! [A3: set_Pr8693737435421807431at_nat,B3: set_Pr8693737435421807431at_nat,Pa: fun_Pr363642936170290612_nat_o] :
      ( ! [X4: produc859450856879609959at_nat] :
          ( member8206827879206165904at_nat(X4,aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,A3),B3))
         => aa_Pro602569247023830573_nat_o(Pa,X4) )
    <=> ( ! [X4: produc859450856879609959at_nat] :
            ( member8206827879206165904at_nat(X4,A3)
           => aa_Pro602569247023830573_nat_o(Pa,X4) )
        & ! [X4: produc859450856879609959at_nat] :
            ( member8206827879206165904at_nat(X4,B3)
           => aa_Pro602569247023830573_nat_o(Pa,X4) ) ) ) ).

% ball_Un
tff(fact_541_ball__Un,axiom,
    ! [A3: set_Pr8551490117392284871at_nat,B3: set_Pr8551490117392284871at_nat,Pa: fun_Pr1099658227000185396_nat_o] :
      ( ! [X4: produc4166570645942440679at_nat] :
          ( member6689249552917799696at_nat(X4,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A3),B3))
         => aa_Pro8143006071014924973_nat_o(Pa,X4) )
    <=> ( ! [X4: produc4166570645942440679at_nat] :
            ( member6689249552917799696at_nat(X4,A3)
           => aa_Pro8143006071014924973_nat_o(Pa,X4) )
        & ! [X4: produc4166570645942440679at_nat] :
            ( member6689249552917799696at_nat(X4,B3)
           => aa_Pro8143006071014924973_nat_o(Pa,X4) ) ) ) ).

% ball_Un
tff(fact_542_ball__Un,axiom,
    ! [A3: set_Pr4329608150637261639at_nat,B3: set_Pr4329608150637261639at_nat,Pa: fun_Pr3059359313727237556_nat_o] :
      ( ! [X4: produc3843707927480180839at_nat] :
          ( member8757157785044589968at_nat(X4,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A3),B3))
         => aa_Pro5947980084102686253_nat_o(Pa,X4) )
    <=> ( ! [X4: produc3843707927480180839at_nat] :
            ( member8757157785044589968at_nat(X4,A3)
           => aa_Pro5947980084102686253_nat_o(Pa,X4) )
        & ! [X4: produc3843707927480180839at_nat] :
            ( member8757157785044589968at_nat(X4,B3)
           => aa_Pro5947980084102686253_nat_o(Pa,X4) ) ) ) ).

% ball_Un
tff(fact_543_ball__Un,axiom,
    ! [A3: set_nat,B3: set_nat,Pa: fun_nat_o] :
      ( ! [X4: nat] :
          ( aa_set_nat_o(member_nat(X4),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A3),B3))
         => aa_nat_o(Pa,X4) )
    <=> ( ! [X4: nat] :
            ( aa_set_nat_o(member_nat(X4),A3)
           => aa_nat_o(Pa,X4) )
        & ! [X4: nat] :
            ( aa_set_nat_o(member_nat(X4),B3)
           => aa_nat_o(Pa,X4) ) ) ) ).

% ball_Un
tff(fact_544_bex__Un,axiom,
    ! [A3: set_Pr8693737435421807431at_nat,B3: set_Pr8693737435421807431at_nat,Pa: fun_Pr363642936170290612_nat_o] :
      ( ? [X4: produc859450856879609959at_nat] :
          ( member8206827879206165904at_nat(X4,aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,A3),B3))
          & aa_Pro602569247023830573_nat_o(Pa,X4) )
    <=> ( ? [X4: produc859450856879609959at_nat] :
            ( member8206827879206165904at_nat(X4,A3)
            & aa_Pro602569247023830573_nat_o(Pa,X4) )
        | ? [X4: produc859450856879609959at_nat] :
            ( member8206827879206165904at_nat(X4,B3)
            & aa_Pro602569247023830573_nat_o(Pa,X4) ) ) ) ).

% bex_Un
tff(fact_545_bex__Un,axiom,
    ! [A3: set_Pr8551490117392284871at_nat,B3: set_Pr8551490117392284871at_nat,Pa: fun_Pr1099658227000185396_nat_o] :
      ( ? [X4: produc4166570645942440679at_nat] :
          ( member6689249552917799696at_nat(X4,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A3),B3))
          & aa_Pro8143006071014924973_nat_o(Pa,X4) )
    <=> ( ? [X4: produc4166570645942440679at_nat] :
            ( member6689249552917799696at_nat(X4,A3)
            & aa_Pro8143006071014924973_nat_o(Pa,X4) )
        | ? [X4: produc4166570645942440679at_nat] :
            ( member6689249552917799696at_nat(X4,B3)
            & aa_Pro8143006071014924973_nat_o(Pa,X4) ) ) ) ).

% bex_Un
tff(fact_546_bex__Un,axiom,
    ! [A3: set_Pr4329608150637261639at_nat,B3: set_Pr4329608150637261639at_nat,Pa: fun_Pr3059359313727237556_nat_o] :
      ( ? [X4: produc3843707927480180839at_nat] :
          ( member8757157785044589968at_nat(X4,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A3),B3))
          & aa_Pro5947980084102686253_nat_o(Pa,X4) )
    <=> ( ? [X4: produc3843707927480180839at_nat] :
            ( member8757157785044589968at_nat(X4,A3)
            & aa_Pro5947980084102686253_nat_o(Pa,X4) )
        | ? [X4: produc3843707927480180839at_nat] :
            ( member8757157785044589968at_nat(X4,B3)
            & aa_Pro5947980084102686253_nat_o(Pa,X4) ) ) ) ).

% bex_Un
tff(fact_547_bex__Un,axiom,
    ! [A3: set_nat,B3: set_nat,Pa: fun_nat_o] :
      ( ? [X4: nat] :
          ( aa_set_nat_o(member_nat(X4),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A3),B3))
          & aa_nat_o(Pa,X4) )
    <=> ( ? [X4: nat] :
            ( aa_set_nat_o(member_nat(X4),A3)
            & aa_nat_o(Pa,X4) )
        | ? [X4: nat] :
            ( aa_set_nat_o(member_nat(X4),B3)
            & aa_nat_o(Pa,X4) ) ) ) ).

% bex_Un
tff(fact_548_UnI2,axiom,
    ! [Ca: product_prod_nat_nat,B3: set_Pr1261947904930325089at_nat,A3: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Ca),B3)
     => aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Ca),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,A3),B3)) ) ).

% UnI2
tff(fact_549_UnI2,axiom,
    ! [Ca: $o,B3: set_o,A3: set_o] :
      ( member_o((Ca),B3)
     => member_o((Ca),aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,A3),B3)) ) ).

% UnI2
tff(fact_550_UnI2,axiom,
    ! [Ca: set_nat,B3: set_set_nat,A3: set_set_nat] :
      ( aa_set_set_nat_o(member_set_nat(Ca),B3)
     => aa_set_set_nat_o(member_set_nat(Ca),sup_sup_set_set_nat(A3,B3)) ) ).

% UnI2
tff(fact_551_UnI2,axiom,
    ! [Ca: int,B3: set_int,A3: set_int] :
      ( aa_set_int_o(member_int(Ca),B3)
     => aa_set_int_o(member_int(Ca),aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A3),B3)) ) ).

% UnI2
tff(fact_552_UnI2,axiom,
    ! [Ca: produc859450856879609959at_nat,B3: set_Pr8693737435421807431at_nat,A3: set_Pr8693737435421807431at_nat] :
      ( member8206827879206165904at_nat(Ca,B3)
     => member8206827879206165904at_nat(Ca,aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,A3),B3)) ) ).

% UnI2
tff(fact_553_UnI2,axiom,
    ! [Ca: produc4166570645942440679at_nat,B3: set_Pr8551490117392284871at_nat,A3: set_Pr8551490117392284871at_nat] :
      ( member6689249552917799696at_nat(Ca,B3)
     => member6689249552917799696at_nat(Ca,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A3),B3)) ) ).

% UnI2
tff(fact_554_UnI2,axiom,
    ! [Ca: produc3843707927480180839at_nat,B3: set_Pr4329608150637261639at_nat,A3: set_Pr4329608150637261639at_nat] :
      ( member8757157785044589968at_nat(Ca,B3)
     => member8757157785044589968at_nat(Ca,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A3),B3)) ) ).

% UnI2
tff(fact_555_UnI2,axiom,
    ! [Ca: nat,B3: set_nat,A3: set_nat] :
      ( aa_set_nat_o(member_nat(Ca),B3)
     => aa_set_nat_o(member_nat(Ca),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A3),B3)) ) ).

% UnI2
tff(fact_556_UnI1,axiom,
    ! [Ca: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Ca),A3)
     => aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Ca),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,A3),B3)) ) ).

% UnI1
tff(fact_557_UnI1,axiom,
    ! [Ca: $o,A3: set_o,B3: set_o] :
      ( member_o((Ca),A3)
     => member_o((Ca),aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,A3),B3)) ) ).

% UnI1
tff(fact_558_UnI1,axiom,
    ! [Ca: set_nat,A3: set_set_nat,B3: set_set_nat] :
      ( aa_set_set_nat_o(member_set_nat(Ca),A3)
     => aa_set_set_nat_o(member_set_nat(Ca),sup_sup_set_set_nat(A3,B3)) ) ).

% UnI1
tff(fact_559_UnI1,axiom,
    ! [Ca: int,A3: set_int,B3: set_int] :
      ( aa_set_int_o(member_int(Ca),A3)
     => aa_set_int_o(member_int(Ca),aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A3),B3)) ) ).

% UnI1
tff(fact_560_UnI1,axiom,
    ! [Ca: produc859450856879609959at_nat,A3: set_Pr8693737435421807431at_nat,B3: set_Pr8693737435421807431at_nat] :
      ( member8206827879206165904at_nat(Ca,A3)
     => member8206827879206165904at_nat(Ca,aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,A3),B3)) ) ).

% UnI1
tff(fact_561_UnI1,axiom,
    ! [Ca: produc4166570645942440679at_nat,A3: set_Pr8551490117392284871at_nat,B3: set_Pr8551490117392284871at_nat] :
      ( member6689249552917799696at_nat(Ca,A3)
     => member6689249552917799696at_nat(Ca,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A3),B3)) ) ).

% UnI1
tff(fact_562_UnI1,axiom,
    ! [Ca: produc3843707927480180839at_nat,A3: set_Pr4329608150637261639at_nat,B3: set_Pr4329608150637261639at_nat] :
      ( member8757157785044589968at_nat(Ca,A3)
     => member8757157785044589968at_nat(Ca,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A3),B3)) ) ).

% UnI1
tff(fact_563_UnI1,axiom,
    ! [Ca: nat,A3: set_nat,B3: set_nat] :
      ( aa_set_nat_o(member_nat(Ca),A3)
     => aa_set_nat_o(member_nat(Ca),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A3),B3)) ) ).

% UnI1
tff(fact_564_UnE,axiom,
    ! [Ca: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Ca),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,A3),B3))
     => ( ~ aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Ca),A3)
       => aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Ca),B3) ) ) ).

% UnE
tff(fact_565_UnE,axiom,
    ! [Ca: $o,A3: set_o,B3: set_o] :
      ( member_o((Ca),aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,A3),B3))
     => ( ~ member_o((Ca),A3)
       => member_o((Ca),B3) ) ) ).

% UnE
tff(fact_566_UnE,axiom,
    ! [Ca: set_nat,A3: set_set_nat,B3: set_set_nat] :
      ( aa_set_set_nat_o(member_set_nat(Ca),sup_sup_set_set_nat(A3,B3))
     => ( ~ aa_set_set_nat_o(member_set_nat(Ca),A3)
       => aa_set_set_nat_o(member_set_nat(Ca),B3) ) ) ).

% UnE
tff(fact_567_UnE,axiom,
    ! [Ca: int,A3: set_int,B3: set_int] :
      ( aa_set_int_o(member_int(Ca),aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A3),B3))
     => ( ~ aa_set_int_o(member_int(Ca),A3)
       => aa_set_int_o(member_int(Ca),B3) ) ) ).

% UnE
tff(fact_568_UnE,axiom,
    ! [Ca: produc859450856879609959at_nat,A3: set_Pr8693737435421807431at_nat,B3: set_Pr8693737435421807431at_nat] :
      ( member8206827879206165904at_nat(Ca,aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,A3),B3))
     => ( ~ member8206827879206165904at_nat(Ca,A3)
       => member8206827879206165904at_nat(Ca,B3) ) ) ).

% UnE
tff(fact_569_UnE,axiom,
    ! [Ca: produc4166570645942440679at_nat,A3: set_Pr8551490117392284871at_nat,B3: set_Pr8551490117392284871at_nat] :
      ( member6689249552917799696at_nat(Ca,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A3),B3))
     => ( ~ member6689249552917799696at_nat(Ca,A3)
       => member6689249552917799696at_nat(Ca,B3) ) ) ).

% UnE
tff(fact_570_UnE,axiom,
    ! [Ca: produc3843707927480180839at_nat,A3: set_Pr4329608150637261639at_nat,B3: set_Pr4329608150637261639at_nat] :
      ( member8757157785044589968at_nat(Ca,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A3),B3))
     => ( ~ member8757157785044589968at_nat(Ca,A3)
       => member8757157785044589968at_nat(Ca,B3) ) ) ).

% UnE
tff(fact_571_UnE,axiom,
    ! [Ca: nat,A3: set_nat,B3: set_nat] :
      ( aa_set_nat_o(member_nat(Ca),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A3),B3))
     => ( ~ aa_set_nat_o(member_nat(Ca),A3)
       => aa_set_nat_o(member_nat(Ca),B3) ) ) ).

% UnE
tff(fact_572_inf_OcoboundedI2,axiom,
    ! [B: set_nat,Ca: set_nat,A: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B),Ca)
     => 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)),Ca) ) ).

% inf.coboundedI2
tff(fact_573_inf_OcoboundedI2,axiom,
    ! [B: assn,Ca: assn,A: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,B),Ca)
     => aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),B)),Ca) ) ).

% inf.coboundedI2
tff(fact_574_inf_OcoboundedI2,axiom,
    ! [B: product_unit,Ca: product_unit,A: product_unit] :
      ( aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,B),Ca)
     => aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,A),B)),Ca) ) ).

% inf.coboundedI2
tff(fact_575_inf_OcoboundedI2,axiom,
    ! [B: set_Pr1261947904930325089at_nat,Ca: set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,B),Ca)
     => aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B)),Ca) ) ).

% inf.coboundedI2
tff(fact_576_inf_OcoboundedI2,axiom,
    ! [B: set_int,Ca: set_int,A: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,B),Ca)
     => 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,A),B)),Ca) ) ).

% inf.coboundedI2
tff(fact_577_inf_OcoboundedI2,axiom,
    ! [B: rat,Ca: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),Ca)
     => 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)),Ca) ) ).

% inf.coboundedI2
tff(fact_578_inf_OcoboundedI2,axiom,
    ! [B: nat,Ca: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),Ca)
     => 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)),Ca) ) ).

% inf.coboundedI2
tff(fact_579_inf_OcoboundedI2,axiom,
    ! [B: int,Ca: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),Ca)
     => 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)),Ca) ) ).

% inf.coboundedI2
tff(fact_580_inf_OcoboundedI1,axiom,
    ! [A: set_nat,Ca: set_nat,B: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),Ca)
     => 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)),Ca) ) ).

% inf.coboundedI1
tff(fact_581_inf_OcoboundedI1,axiom,
    ! [A: assn,Ca: assn,B: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),Ca)
     => aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),B)),Ca) ) ).

% inf.coboundedI1
tff(fact_582_inf_OcoboundedI1,axiom,
    ! [A: product_unit,Ca: product_unit,B: product_unit] :
      ( aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,A),Ca)
     => aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,A),B)),Ca) ) ).

% inf.coboundedI1
tff(fact_583_inf_OcoboundedI1,axiom,
    ! [A: set_Pr1261947904930325089at_nat,Ca: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A),Ca)
     => aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B)),Ca) ) ).

% inf.coboundedI1
tff(fact_584_inf_OcoboundedI1,axiom,
    ! [A: set_int,Ca: set_int,B: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A),Ca)
     => 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,A),B)),Ca) ) ).

% inf.coboundedI1
tff(fact_585_inf_OcoboundedI1,axiom,
    ! [A: rat,Ca: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),Ca)
     => 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)),Ca) ) ).

% inf.coboundedI1
tff(fact_586_inf_OcoboundedI1,axiom,
    ! [A: nat,Ca: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),Ca)
     => 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)),Ca) ) ).

% inf.coboundedI1
tff(fact_587_inf_OcoboundedI1,axiom,
    ! [A: int,Ca: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),Ca)
     => 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)),Ca) ) ).

% inf.coboundedI1
tff(fact_588_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_589_inf_Oabsorb__iff2,axiom,
    ! [B: assn,A: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,B),A)
    <=> ( aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),B) = B ) ) ).

% inf.absorb_iff2
tff(fact_590_inf_Oabsorb__iff2,axiom,
    ! [B: product_unit,A: product_unit] :
      ( aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,B),A)
    <=> ( aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,A),B) = B ) ) ).

% inf.absorb_iff2
tff(fact_591_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_592_inf_Oabsorb__iff2,axiom,
    ! [B: set_int,A: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,B),A)
    <=> ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A),B) = B ) ) ).

% inf.absorb_iff2
tff(fact_593_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_594_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_595_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_596_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_597_inf_Oabsorb__iff1,axiom,
    ! [A: assn,B: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),B)
    <=> ( aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),B) = A ) ) ).

% inf.absorb_iff1
tff(fact_598_inf_Oabsorb__iff1,axiom,
    ! [A: product_unit,B: product_unit] :
      ( aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,A),B)
    <=> ( aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,A),B) = A ) ) ).

% inf.absorb_iff1
tff(fact_599_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_600_inf_Oabsorb__iff1,axiom,
    ! [A: set_int,B: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A),B)
    <=> ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A),B) = A ) ) ).

% inf.absorb_iff1
tff(fact_601_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_602_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_603_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_604_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_605_inf_Ocobounded2,axiom,
    ! [A: assn,B: assn] : aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),B)),B) ).

% inf.cobounded2
tff(fact_606_inf_Ocobounded2,axiom,
    ! [A: product_unit,B: product_unit] : aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,A),B)),B) ).

% inf.cobounded2
tff(fact_607_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_608_inf_Ocobounded2,axiom,
    ! [A: set_int,B: set_int] : 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,A),B)),B) ).

% inf.cobounded2
tff(fact_609_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_610_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_611_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_612_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_613_inf_Ocobounded1,axiom,
    ! [A: assn,B: assn] : aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),B)),A) ).

% inf.cobounded1
tff(fact_614_inf_Ocobounded1,axiom,
    ! [A: product_unit,B: product_unit] : aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,A),B)),A) ).

% inf.cobounded1
tff(fact_615_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_616_inf_Ocobounded1,axiom,
    ! [A: set_int,B: set_int] : 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,A),B)),A) ).

% inf.cobounded1
tff(fact_617_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_618_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_619_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_620_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_621_inf_Oorder__iff,axiom,
    ! [A: assn,B: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),B)
    <=> ( A = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),B) ) ) ).

% inf.order_iff
tff(fact_622_inf_Oorder__iff,axiom,
    ! [A: product_unit,B: product_unit] :
      ( aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,A),B)
    <=> ( A = aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,A),B) ) ) ).

% inf.order_iff
tff(fact_623_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_624_inf_Oorder__iff,axiom,
    ! [A: set_int,B: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A),B)
    <=> ( A = aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A),B) ) ) ).

% inf.order_iff
tff(fact_625_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_626_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_627_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_628_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_629_inf__greatest,axiom,
    ! [X: assn,Y: assn,Z: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X),Y)
     => ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X),Z)
       => aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Y),Z)) ) ) ).

% inf_greatest
tff(fact_630_inf__greatest,axiom,
    ! [X: product_unit,Y: product_unit,Z: product_unit] :
      ( aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,X),Y)
     => ( aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,X),Z)
       => aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,X),aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,Y),Z)) ) ) ).

% inf_greatest
tff(fact_631_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_632_inf__greatest,axiom,
    ! [X: set_int,Y: set_int,Z: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,X),Y)
     => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,X),Z)
       => aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,X),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,Y),Z)) ) ) ).

% inf_greatest
tff(fact_633_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_634_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_635_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_636_inf_OboundedI,axiom,
    ! [A: set_nat,B: set_nat,Ca: 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),Ca)
       => 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),Ca)) ) ) ).

% inf.boundedI
tff(fact_637_inf_OboundedI,axiom,
    ! [A: assn,B: assn,Ca: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),B)
     => ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),Ca)
       => aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,B),Ca)) ) ) ).

% inf.boundedI
tff(fact_638_inf_OboundedI,axiom,
    ! [A: product_unit,B: product_unit,Ca: product_unit] :
      ( aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,A),B)
     => ( aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,A),Ca)
       => aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,A),aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,B),Ca)) ) ) ).

% inf.boundedI
tff(fact_639_inf_OboundedI,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,Ca: 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),Ca)
       => aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B),Ca)) ) ) ).

% inf.boundedI
tff(fact_640_inf_OboundedI,axiom,
    ! [A: set_int,B: set_int,Ca: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A),B)
     => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A),Ca)
       => aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,B),Ca)) ) ) ).

% inf.boundedI
tff(fact_641_inf_OboundedI,axiom,
    ! [A: rat,B: rat,Ca: 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),Ca)
       => 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),Ca)) ) ) ).

% inf.boundedI
tff(fact_642_inf_OboundedI,axiom,
    ! [A: nat,B: nat,Ca: 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),Ca)
       => 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),Ca)) ) ) ).

% inf.boundedI
tff(fact_643_inf_OboundedI,axiom,
    ! [A: int,B: int,Ca: 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),Ca)
       => 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),Ca)) ) ) ).

% inf.boundedI
tff(fact_644_inf_OboundedE,axiom,
    ! [A: set_nat,B: set_nat,Ca: 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),Ca))
     => ~ ( 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),Ca) ) ) ).

% inf.boundedE
tff(fact_645_inf_OboundedE,axiom,
    ! [A: assn,B: assn,Ca: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,B),Ca))
     => ~ ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),B)
         => ~ aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),Ca) ) ) ).

% inf.boundedE
tff(fact_646_inf_OboundedE,axiom,
    ! [A: product_unit,B: product_unit,Ca: product_unit] :
      ( aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,A),aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,B),Ca))
     => ~ ( aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,A),B)
         => ~ aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,A),Ca) ) ) ).

% inf.boundedE
tff(fact_647_inf_OboundedE,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,Ca: 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),Ca))
     => ~ ( 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),Ca) ) ) ).

% inf.boundedE
tff(fact_648_inf_OboundedE,axiom,
    ! [A: set_int,B: set_int,Ca: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,B),Ca))
     => ~ ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A),B)
         => ~ aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A),Ca) ) ) ).

% inf.boundedE
tff(fact_649_inf_OboundedE,axiom,
    ! [A: rat,B: rat,Ca: 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),Ca))
     => ~ ( 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),Ca) ) ) ).

% inf.boundedE
tff(fact_650_inf_OboundedE,axiom,
    ! [A: nat,B: nat,Ca: 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),Ca))
     => ~ ( 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),Ca) ) ) ).

% inf.boundedE
tff(fact_651_inf_OboundedE,axiom,
    ! [A: int,B: int,Ca: 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),Ca))
     => ~ ( 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),Ca) ) ) ).

% inf.boundedE
tff(fact_652_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_653_inf__absorb2,axiom,
    ! [Y: assn,X: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,Y),X)
     => ( aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),Y) = Y ) ) ).

% inf_absorb2
tff(fact_654_inf__absorb2,axiom,
    ! [Y: product_unit,X: product_unit] :
      ( aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,Y),X)
     => ( aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,X),Y) = Y ) ) ).

% inf_absorb2
tff(fact_655_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_656_inf__absorb2,axiom,
    ! [Y: set_int,X: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,Y),X)
     => ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,X),Y) = Y ) ) ).

% inf_absorb2
tff(fact_657_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_658_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_659_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_660_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_661_inf__absorb1,axiom,
    ! [X: assn,Y: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X),Y)
     => ( aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),Y) = X ) ) ).

% inf_absorb1
tff(fact_662_inf__absorb1,axiom,
    ! [X: product_unit,Y: product_unit] :
      ( aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,X),Y)
     => ( aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,X),Y) = X ) ) ).

% inf_absorb1
tff(fact_663_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_664_inf__absorb1,axiom,
    ! [X: set_int,Y: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,X),Y)
     => ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,X),Y) = X ) ) ).

% inf_absorb1
tff(fact_665_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_666_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_667_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_668_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_669_inf_Oabsorb2,axiom,
    ! [B: assn,A: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,B),A)
     => ( aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),B) = B ) ) ).

% inf.absorb2
tff(fact_670_inf_Oabsorb2,axiom,
    ! [B: product_unit,A: product_unit] :
      ( aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,B),A)
     => ( aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,A),B) = B ) ) ).

% inf.absorb2
tff(fact_671_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_672_inf_Oabsorb2,axiom,
    ! [B: set_int,A: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,B),A)
     => ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A),B) = B ) ) ).

% inf.absorb2
tff(fact_673_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_674_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_675_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_676_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_677_inf_Oabsorb1,axiom,
    ! [A: assn,B: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),B)
     => ( aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),B) = A ) ) ).

% inf.absorb1
tff(fact_678_inf_Oabsorb1,axiom,
    ! [A: product_unit,B: product_unit] :
      ( aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,A),B)
     => ( aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,A),B) = A ) ) ).

% inf.absorb1
tff(fact_679_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_680_inf_Oabsorb1,axiom,
    ! [A: set_int,B: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A),B)
     => ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A),B) = A ) ) ).

% inf.absorb1
tff(fact_681_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_682_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_683_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_684_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_685_le__iff__inf,axiom,
    ! [X: assn,Y: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X),Y)
    <=> ( aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),Y) = X ) ) ).

% le_iff_inf
tff(fact_686_le__iff__inf,axiom,
    ! [X: product_unit,Y: product_unit] :
      ( aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,X),Y)
    <=> ( aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,X),Y) = X ) ) ).

% le_iff_inf
tff(fact_687_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_688_le__iff__inf,axiom,
    ! [X: set_int,Y: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,X),Y)
    <=> ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,X),Y) = X ) ) ).

% le_iff_inf
tff(fact_689_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_690_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_691_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_692_inf__unique,axiom,
    ! [F: fun_se1909708154429037836et_nat,X: set_nat,Y: set_nat] :
      ( ! [X3: set_nat,Y3: set_nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(F,X3),Y3)),X3)
     => ( ! [X3: set_nat,Y3: set_nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(F,X3),Y3)),Y3)
       => ( ! [X3: set_nat,Y3: set_nat,Z2: set_nat] :
              ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X3),Y3)
             => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X3),Z2)
               => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X3),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(F,Y3),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_693_inf__unique,axiom,
    ! [F: fun_as3983628580203722482n_assn,X: assn,Y: assn] :
      ( ! [X3: assn,Y3: assn] : aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_assn_assn(aa_ass954072293579886259n_assn(F,X3),Y3)),X3)
     => ( ! [X3: assn,Y3: assn] : aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_assn_assn(aa_ass954072293579886259n_assn(F,X3),Y3)),Y3)
       => ( ! [X3: assn,Y3: assn,Z2: assn] :
              ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X3),Y3)
             => ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X3),Z2)
               => aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X3),aa_assn_assn(aa_ass954072293579886259n_assn(F,Y3),Z2)) ) )
         => ( aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),Y) = aa_assn_assn(aa_ass954072293579886259n_assn(F,X),Y) ) ) ) ) ).

% inf_unique
tff(fact_694_inf__unique,axiom,
    ! [F: fun_Pr3553901264588531491t_unit,X: product_unit,Y: product_unit] :
      ( ! [X3: product_unit,Y3: product_unit] : aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(F,X3),Y3)),X3)
     => ( ! [X3: product_unit,Y3: product_unit] : aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(F,X3),Y3)),Y3)
       => ( ! [X3: product_unit,Y3: product_unit,Z2: product_unit] :
              ( aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,X3),Y3)
             => ( aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,X3),Z2)
               => aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,X3),aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(F,Y3),Z2)) ) )
         => ( aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,X),Y) = aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(F,X),Y) ) ) ) ) ).

% inf_unique
tff(fact_695_inf__unique,axiom,
    ! [F: fun_se3560235952666129655at_nat,X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
      ( ! [X3: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat] : aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(F,X3),Y3)),X3)
     => ( ! [X3: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat] : aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(F,X3),Y3)),Y3)
       => ( ! [X3: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat,Z2: set_Pr1261947904930325089at_nat] :
              ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,X3),Y3)
             => ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,X3),Z2)
               => aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,X3),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(F,Y3),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_696_inf__unique,axiom,
    ! [F: fun_se6573205548028396576et_int,X: set_int,Y: set_int] :
      ( ! [X3: set_int,Y3: set_int] : aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,aa_set_int_set_int(aa_set4234024024286713313et_int(F,X3),Y3)),X3)
     => ( ! [X3: set_int,Y3: set_int] : aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,aa_set_int_set_int(aa_set4234024024286713313et_int(F,X3),Y3)),Y3)
       => ( ! [X3: set_int,Y3: set_int,Z2: set_int] :
              ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,X3),Y3)
             => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,X3),Z2)
               => aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,X3),aa_set_int_set_int(aa_set4234024024286713313et_int(F,Y3),Z2)) ) )
         => ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,X),Y) = aa_set_int_set_int(aa_set4234024024286713313et_int(F,X),Y) ) ) ) ) ).

% inf_unique
tff(fact_697_inf__unique,axiom,
    ! [F: fun_rat_fun_rat_rat,X: rat,Y: rat] :
      ( ! [X3: rat,Y3: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(F,X3),Y3)),X3)
     => ( ! [X3: rat,Y3: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(F,X3),Y3)),Y3)
       => ( ! [X3: rat,Y3: rat,Z2: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y3)
             => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Z2)
               => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),aa_rat_rat(aa_rat_fun_rat_rat(F,Y3),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_698_inf__unique,axiom,
    ! [F: fun_nat_fun_nat_nat,X: nat,Y: nat] :
      ( ! [X3: nat,Y3: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(F,X3),Y3)),X3)
     => ( ! [X3: nat,Y3: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(F,X3),Y3)),Y3)
       => ( ! [X3: nat,Y3: nat,Z2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),Y3)
             => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),Z2)
               => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),aa_nat_nat(aa_nat_fun_nat_nat(F,Y3),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_699_inf__unique,axiom,
    ! [F: fun_int_fun_int_int,X: int,Y: int] :
      ( ! [X3: int,Y3: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(F,X3),Y3)),X3)
     => ( ! [X3: int,Y3: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(F,X3),Y3)),Y3)
       => ( ! [X3: int,Y3: int,Z2: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X3),Y3)
             => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X3),Z2)
               => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X3),aa_int_int(aa_int_fun_int_int(F,Y3),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_700_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_701_inf_OorderI,axiom,
    ! [A: assn,B: assn] :
      ( ( A = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),B) )
     => aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),B) ) ).

% inf.orderI
tff(fact_702_inf_OorderI,axiom,
    ! [A: product_unit,B: product_unit] :
      ( ( A = aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,A),B) )
     => aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,A),B) ) ).

% inf.orderI
tff(fact_703_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_704_inf_OorderI,axiom,
    ! [A: set_int,B: set_int] :
      ( ( A = aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A),B) )
     => aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A),B) ) ).

% inf.orderI
tff(fact_705_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_706_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_707_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_708_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_709_inf_OorderE,axiom,
    ! [A: assn,B: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),B)
     => ( A = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),B) ) ) ).

% inf.orderE
tff(fact_710_inf_OorderE,axiom,
    ! [A: product_unit,B: product_unit] :
      ( aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,A),B)
     => ( A = aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,A),B) ) ) ).

% inf.orderE
tff(fact_711_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_712_inf_OorderE,axiom,
    ! [A: set_int,B: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A),B)
     => ( A = aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A),B) ) ) ).

% inf.orderE
tff(fact_713_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_714_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_715_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_716_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_717_le__infI2,axiom,
    ! [B: assn,X: assn,A: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,B),X)
     => aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),B)),X) ) ).

% le_infI2
tff(fact_718_le__infI2,axiom,
    ! [B: product_unit,X: product_unit,A: product_unit] :
      ( aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,B),X)
     => aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,A),B)),X) ) ).

% le_infI2
tff(fact_719_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_720_le__infI2,axiom,
    ! [B: set_int,X: set_int,A: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,B),X)
     => 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,A),B)),X) ) ).

% le_infI2
tff(fact_721_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_722_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_723_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_724_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_725_le__infI1,axiom,
    ! [A: assn,X: assn,B: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),X)
     => aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),B)),X) ) ).

% le_infI1
tff(fact_726_le__infI1,axiom,
    ! [A: product_unit,X: product_unit,B: product_unit] :
      ( aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,A),X)
     => aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,A),B)),X) ) ).

% le_infI1
tff(fact_727_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_728_le__infI1,axiom,
    ! [A: set_int,X: set_int,B: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A),X)
     => 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,A),B)),X) ) ).

% le_infI1
tff(fact_729_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_730_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_731_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_732_inf__mono,axiom,
    ! [A: set_nat,Ca: set_nat,B: set_nat,D2: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),Ca)
     => ( 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,Ca),D2)) ) ) ).

% inf_mono
tff(fact_733_inf__mono,axiom,
    ! [A: assn,Ca: assn,B: assn,D2: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),Ca)
     => ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,B),D2)
       => aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),B)),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Ca),D2)) ) ) ).

% inf_mono
tff(fact_734_inf__mono,axiom,
    ! [A: product_unit,Ca: product_unit,B: product_unit,D2: product_unit] :
      ( aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,A),Ca)
     => ( aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,B),D2)
       => aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,A),B)),aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,Ca),D2)) ) ) ).

% inf_mono
tff(fact_735_inf__mono,axiom,
    ! [A: set_Pr1261947904930325089at_nat,Ca: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,D2: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A),Ca)
     => ( 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,Ca),D2)) ) ) ).

% inf_mono
tff(fact_736_inf__mono,axiom,
    ! [A: set_int,Ca: set_int,B: set_int,D2: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A),Ca)
     => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,B),D2)
       => 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,A),B)),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,Ca),D2)) ) ) ).

% inf_mono
tff(fact_737_inf__mono,axiom,
    ! [A: rat,Ca: rat,B: rat,D2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),Ca)
     => ( 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,Ca),D2)) ) ) ).

% inf_mono
tff(fact_738_inf__mono,axiom,
    ! [A: nat,Ca: nat,B: nat,D2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),Ca)
     => ( 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,Ca),D2)) ) ) ).

% inf_mono
tff(fact_739_inf__mono,axiom,
    ! [A: int,Ca: int,B: int,D2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),Ca)
     => ( 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,Ca),D2)) ) ) ).

% inf_mono
tff(fact_740_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_741_le__infI,axiom,
    ! [X: assn,A: assn,B: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X),A)
     => ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X),B)
       => aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),B)) ) ) ).

% le_infI
tff(fact_742_le__infI,axiom,
    ! [X: product_unit,A: product_unit,B: product_unit] :
      ( aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,X),A)
     => ( aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,X),B)
       => aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,X),aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,A),B)) ) ) ).

% le_infI
tff(fact_743_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_744_le__infI,axiom,
    ! [X: set_int,A: set_int,B: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,X),A)
     => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,X),B)
       => aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,X),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A),B)) ) ) ).

% le_infI
tff(fact_745_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_746_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_747_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_748_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_749_le__infE,axiom,
    ! [X: assn,A: assn,B: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),B))
     => ~ ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X),A)
         => ~ aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X),B) ) ) ).

% le_infE
tff(fact_750_le__infE,axiom,
    ! [X: product_unit,A: product_unit,B: product_unit] :
      ( aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,X),aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,A),B))
     => ~ ( aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,X),A)
         => ~ aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,X),B) ) ) ).

% le_infE
tff(fact_751_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_752_le__infE,axiom,
    ! [X: set_int,A: set_int,B: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,X),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A),B))
     => ~ ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,X),A)
         => ~ aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,X),B) ) ) ).

% le_infE
tff(fact_753_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_754_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_755_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_756_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_757_inf__le2,axiom,
    ! [X: assn,Y: assn] : aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),Y)),Y) ).

% inf_le2
tff(fact_758_inf__le2,axiom,
    ! [X: product_unit,Y: product_unit] : aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,X),Y)),Y) ).

% inf_le2
tff(fact_759_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_760_inf__le2,axiom,
    ! [X: set_int,Y: set_int] : 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,X),Y)),Y) ).

% inf_le2
tff(fact_761_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_762_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_763_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_764_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_765_inf__le1,axiom,
    ! [X: assn,Y: assn] : aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),Y)),X) ).

% inf_le1
tff(fact_766_inf__le1,axiom,
    ! [X: product_unit,Y: product_unit] : aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,X),Y)),X) ).

% inf_le1
tff(fact_767_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_768_inf__le1,axiom,
    ! [X: set_int,Y: set_int] : 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,X),Y)),X) ).

% inf_le1
tff(fact_769_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_770_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_771_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_772_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_773_inf__sup__ord_I1_J,axiom,
    ! [X: assn,Y: assn] : aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),Y)),X) ).

% inf_sup_ord(1)
tff(fact_774_inf__sup__ord_I1_J,axiom,
    ! [X: product_unit,Y: product_unit] : aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,X),Y)),X) ).

% inf_sup_ord(1)
tff(fact_775_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_776_inf__sup__ord_I1_J,axiom,
    ! [X: set_int,Y: set_int] : 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,X),Y)),X) ).

% inf_sup_ord(1)
tff(fact_777_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_778_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_779_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_780_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_781_inf__sup__ord_I2_J,axiom,
    ! [X: assn,Y: assn] : aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),Y)),Y) ).

% inf_sup_ord(2)
tff(fact_782_inf__sup__ord_I2_J,axiom,
    ! [X: product_unit,Y: product_unit] : aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,X),Y)),Y) ).

% inf_sup_ord(2)
tff(fact_783_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_784_inf__sup__ord_I2_J,axiom,
    ! [X: set_int,Y: set_int] : 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,X),Y)),Y) ).

% inf_sup_ord(2)
tff(fact_785_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_786_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_787_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_788_sup_OcoboundedI2,axiom,
    ! [Ca: set_Pr8693737435421807431at_nat,B: set_Pr8693737435421807431at_nat,A: set_Pr8693737435421807431at_nat] :
      ( ord_le3000389064537975527at_nat(Ca,B)
     => ord_le3000389064537975527at_nat(Ca,aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,A),B)) ) ).

% sup.coboundedI2
tff(fact_789_sup_OcoboundedI2,axiom,
    ! [Ca: set_Pr8551490117392284871at_nat,B: set_Pr8551490117392284871at_nat,A: set_Pr8551490117392284871at_nat] :
      ( ord_le8081472938463900775at_nat(Ca,B)
     => ord_le8081472938463900775at_nat(Ca,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A),B)) ) ).

% sup.coboundedI2
tff(fact_790_sup_OcoboundedI2,axiom,
    ! [Ca: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat] :
      ( ord_le1268244103169919719at_nat(Ca,B)
     => ord_le1268244103169919719at_nat(Ca,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),B)) ) ).

% sup.coboundedI2
tff(fact_791_sup_OcoboundedI2,axiom,
    ! [Ca: set_nat,B: set_nat,A: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Ca),B)
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Ca),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B)) ) ).

% sup.coboundedI2
tff(fact_792_sup_OcoboundedI2,axiom,
    ! [Ca: set_int,B: set_int,A: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,Ca),B)
     => aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,Ca),aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A),B)) ) ).

% sup.coboundedI2
tff(fact_793_sup_OcoboundedI2,axiom,
    ! [Ca: rat,B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Ca),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Ca),aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,A),B)) ) ).

% sup.coboundedI2
tff(fact_794_sup_OcoboundedI2,axiom,
    ! [Ca: nat,B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ca),B)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ca),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B)) ) ).

% sup.coboundedI2
tff(fact_795_sup_OcoboundedI2,axiom,
    ! [Ca: int,B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ca),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ca),aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B)) ) ).

% sup.coboundedI2
tff(fact_796_sup_OcoboundedI1,axiom,
    ! [Ca: set_Pr8693737435421807431at_nat,A: set_Pr8693737435421807431at_nat,B: set_Pr8693737435421807431at_nat] :
      ( ord_le3000389064537975527at_nat(Ca,A)
     => ord_le3000389064537975527at_nat(Ca,aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,A),B)) ) ).

% sup.coboundedI1
tff(fact_797_sup_OcoboundedI1,axiom,
    ! [Ca: set_Pr8551490117392284871at_nat,A: set_Pr8551490117392284871at_nat,B: set_Pr8551490117392284871at_nat] :
      ( ord_le8081472938463900775at_nat(Ca,A)
     => ord_le8081472938463900775at_nat(Ca,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A),B)) ) ).

% sup.coboundedI1
tff(fact_798_sup_OcoboundedI1,axiom,
    ! [Ca: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
      ( ord_le1268244103169919719at_nat(Ca,A)
     => ord_le1268244103169919719at_nat(Ca,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),B)) ) ).

% sup.coboundedI1
tff(fact_799_sup_OcoboundedI1,axiom,
    ! [Ca: set_nat,A: set_nat,B: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Ca),A)
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Ca),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B)) ) ).

% sup.coboundedI1
tff(fact_800_sup_OcoboundedI1,axiom,
    ! [Ca: set_int,A: set_int,B: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,Ca),A)
     => aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,Ca),aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A),B)) ) ).

% sup.coboundedI1
tff(fact_801_sup_OcoboundedI1,axiom,
    ! [Ca: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Ca),A)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Ca),aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,A),B)) ) ).

% sup.coboundedI1
tff(fact_802_sup_OcoboundedI1,axiom,
    ! [Ca: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ca),A)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ca),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B)) ) ).

% sup.coboundedI1
tff(fact_803_sup_OcoboundedI1,axiom,
    ! [Ca: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ca),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ca),aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B)) ) ).

% sup.coboundedI1
tff(fact_804_sup_Oabsorb__iff2,axiom,
    ! [A: set_Pr8693737435421807431at_nat,B: set_Pr8693737435421807431at_nat] :
      ( ord_le3000389064537975527at_nat(A,B)
    <=> ( aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,A),B) = B ) ) ).

% sup.absorb_iff2
tff(fact_805_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_806_sup_Oabsorb__iff2,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
      ( ord_le1268244103169919719at_nat(A,B)
    <=> ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),B) = B ) ) ).

% sup.absorb_iff2
tff(fact_807_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_808_sup_Oabsorb__iff2,axiom,
    ! [A: set_int,B: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A),B)
    <=> ( aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A),B) = B ) ) ).

% sup.absorb_iff2
tff(fact_809_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_810_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_811_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_812_sup_Oabsorb__iff1,axiom,
    ! [B: set_Pr8693737435421807431at_nat,A: set_Pr8693737435421807431at_nat] :
      ( ord_le3000389064537975527at_nat(B,A)
    <=> ( aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,A),B) = A ) ) ).

% sup.absorb_iff1
tff(fact_813_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_814_sup_Oabsorb__iff1,axiom,
    ! [B: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat] :
      ( ord_le1268244103169919719at_nat(B,A)
    <=> ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),B) = A ) ) ).

% sup.absorb_iff1
tff(fact_815_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_816_sup_Oabsorb__iff1,axiom,
    ! [B: set_int,A: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,B),A)
    <=> ( aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A),B) = A ) ) ).

% sup.absorb_iff1
tff(fact_817_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_818_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_819_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_820_sup_Ocobounded2,axiom,
    ! [B: set_Pr8693737435421807431at_nat,A: set_Pr8693737435421807431at_nat] : ord_le3000389064537975527at_nat(B,aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,A),B)) ).

% sup.cobounded2
tff(fact_821_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_822_sup_Ocobounded2,axiom,
    ! [B: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat] : ord_le1268244103169919719at_nat(B,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),B)) ).

% sup.cobounded2
tff(fact_823_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_824_sup_Ocobounded2,axiom,
    ! [B: set_int,A: set_int] : aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,B),aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A),B)) ).

% sup.cobounded2
tff(fact_825_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_826_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_827_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_828_sup_Ocobounded1,axiom,
    ! [A: set_Pr8693737435421807431at_nat,B: set_Pr8693737435421807431at_nat] : ord_le3000389064537975527at_nat(A,aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,A),B)) ).

% sup.cobounded1
tff(fact_829_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_830_sup_Ocobounded1,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] : ord_le1268244103169919719at_nat(A,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),B)) ).

% sup.cobounded1
tff(fact_831_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_832_sup_Ocobounded1,axiom,
    ! [A: set_int,B: set_int] : aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A),aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A),B)) ).

% sup.cobounded1
tff(fact_833_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_834_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_835_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_836_sup_Oorder__iff,axiom,
    ! [B: set_Pr8693737435421807431at_nat,A: set_Pr8693737435421807431at_nat] :
      ( ord_le3000389064537975527at_nat(B,A)
    <=> ( A = aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,A),B) ) ) ).

% sup.order_iff
tff(fact_837_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_838_sup_Oorder__iff,axiom,
    ! [B: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat] :
      ( ord_le1268244103169919719at_nat(B,A)
    <=> ( A = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),B) ) ) ).

% sup.order_iff
tff(fact_839_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_840_sup_Oorder__iff,axiom,
    ! [B: set_int,A: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,B),A)
    <=> ( A = aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A),B) ) ) ).

% sup.order_iff
tff(fact_841_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_842_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_843_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_844_sup_OboundedI,axiom,
    ! [B: set_Pr8693737435421807431at_nat,A: set_Pr8693737435421807431at_nat,Ca: set_Pr8693737435421807431at_nat] :
      ( ord_le3000389064537975527at_nat(B,A)
     => ( ord_le3000389064537975527at_nat(Ca,A)
       => ord_le3000389064537975527at_nat(aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,B),Ca),A) ) ) ).

% sup.boundedI
tff(fact_845_sup_OboundedI,axiom,
    ! [B: set_Pr8551490117392284871at_nat,A: set_Pr8551490117392284871at_nat,Ca: set_Pr8551490117392284871at_nat] :
      ( ord_le8081472938463900775at_nat(B,A)
     => ( ord_le8081472938463900775at_nat(Ca,A)
       => ord_le8081472938463900775at_nat(aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,B),Ca),A) ) ) ).

% sup.boundedI
tff(fact_846_sup_OboundedI,axiom,
    ! [B: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat,Ca: set_Pr4329608150637261639at_nat] :
      ( ord_le1268244103169919719at_nat(B,A)
     => ( ord_le1268244103169919719at_nat(Ca,A)
       => ord_le1268244103169919719at_nat(aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,B),Ca),A) ) ) ).

% sup.boundedI
tff(fact_847_sup_OboundedI,axiom,
    ! [B: set_nat,A: set_nat,Ca: 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,Ca),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),Ca)),A) ) ) ).

% sup.boundedI
tff(fact_848_sup_OboundedI,axiom,
    ! [B: set_int,A: set_int,Ca: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,B),A)
     => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,Ca),A)
       => aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,B),Ca)),A) ) ) ).

% sup.boundedI
tff(fact_849_sup_OboundedI,axiom,
    ! [B: rat,A: rat,Ca: 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,Ca),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),Ca)),A) ) ) ).

% sup.boundedI
tff(fact_850_sup_OboundedI,axiom,
    ! [B: nat,A: nat,Ca: 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,Ca),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),Ca)),A) ) ) ).

% sup.boundedI
tff(fact_851_sup_OboundedI,axiom,
    ! [B: int,A: int,Ca: 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,Ca),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),Ca)),A) ) ) ).

% sup.boundedI
tff(fact_852_sup_OboundedE,axiom,
    ! [B: set_Pr8693737435421807431at_nat,Ca: set_Pr8693737435421807431at_nat,A: set_Pr8693737435421807431at_nat] :
      ( ord_le3000389064537975527at_nat(aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,B),Ca),A)
     => ~ ( ord_le3000389064537975527at_nat(B,A)
         => ~ ord_le3000389064537975527at_nat(Ca,A) ) ) ).

% sup.boundedE
tff(fact_853_sup_OboundedE,axiom,
    ! [B: set_Pr8551490117392284871at_nat,Ca: set_Pr8551490117392284871at_nat,A: set_Pr8551490117392284871at_nat] :
      ( ord_le8081472938463900775at_nat(aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,B),Ca),A)
     => ~ ( ord_le8081472938463900775at_nat(B,A)
         => ~ ord_le8081472938463900775at_nat(Ca,A) ) ) ).

% sup.boundedE
tff(fact_854_sup_OboundedE,axiom,
    ! [B: set_Pr4329608150637261639at_nat,Ca: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat] :
      ( ord_le1268244103169919719at_nat(aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,B),Ca),A)
     => ~ ( ord_le1268244103169919719at_nat(B,A)
         => ~ ord_le1268244103169919719at_nat(Ca,A) ) ) ).

% sup.boundedE
tff(fact_855_sup_OboundedE,axiom,
    ! [B: set_nat,Ca: 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),Ca)),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,Ca),A) ) ) ).

% sup.boundedE
tff(fact_856_sup_OboundedE,axiom,
    ! [B: set_int,Ca: set_int,A: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,B),Ca)),A)
     => ~ ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,B),A)
         => ~ aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,Ca),A) ) ) ).

% sup.boundedE
tff(fact_857_sup_OboundedE,axiom,
    ! [B: rat,Ca: 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),Ca)),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,Ca),A) ) ) ).

% sup.boundedE
tff(fact_858_sup_OboundedE,axiom,
    ! [B: nat,Ca: 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),Ca)),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,Ca),A) ) ) ).

% sup.boundedE
tff(fact_859_sup_OboundedE,axiom,
    ! [B: int,Ca: 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),Ca)),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,Ca),A) ) ) ).

% sup.boundedE
tff(fact_860_sup__absorb2,axiom,
    ! [X: set_Pr8693737435421807431at_nat,Y: set_Pr8693737435421807431at_nat] :
      ( ord_le3000389064537975527at_nat(X,Y)
     => ( aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,X),Y) = Y ) ) ).

% sup_absorb2
tff(fact_861_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_862_sup__absorb2,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] :
      ( ord_le1268244103169919719at_nat(X,Y)
     => ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),Y) = Y ) ) ).

% sup_absorb2
tff(fact_863_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_864_sup__absorb2,axiom,
    ! [X: set_int,Y: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,X),Y)
     => ( aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,X),Y) = Y ) ) ).

% sup_absorb2
tff(fact_865_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_866_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_867_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_868_sup__absorb1,axiom,
    ! [Y: set_Pr8693737435421807431at_nat,X: set_Pr8693737435421807431at_nat] :
      ( ord_le3000389064537975527at_nat(Y,X)
     => ( aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,X),Y) = X ) ) ).

% sup_absorb1
tff(fact_869_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_870_sup__absorb1,axiom,
    ! [Y: set_Pr4329608150637261639at_nat,X: set_Pr4329608150637261639at_nat] :
      ( ord_le1268244103169919719at_nat(Y,X)
     => ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),Y) = X ) ) ).

% sup_absorb1
tff(fact_871_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_872_sup__absorb1,axiom,
    ! [Y: set_int,X: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,Y),X)
     => ( aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,X),Y) = X ) ) ).

% sup_absorb1
tff(fact_873_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_874_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_875_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_876_sup_Oabsorb2,axiom,
    ! [A: set_Pr8693737435421807431at_nat,B: set_Pr8693737435421807431at_nat] :
      ( ord_le3000389064537975527at_nat(A,B)
     => ( aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,A),B) = B ) ) ).

% sup.absorb2
tff(fact_877_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_878_sup_Oabsorb2,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
      ( ord_le1268244103169919719at_nat(A,B)
     => ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),B) = B ) ) ).

% sup.absorb2
tff(fact_879_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_880_sup_Oabsorb2,axiom,
    ! [A: set_int,B: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A),B)
     => ( aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A),B) = B ) ) ).

% sup.absorb2
tff(fact_881_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_882_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_883_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_884_sup_Oabsorb1,axiom,
    ! [B: set_Pr8693737435421807431at_nat,A: set_Pr8693737435421807431at_nat] :
      ( ord_le3000389064537975527at_nat(B,A)
     => ( aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,A),B) = A ) ) ).

% sup.absorb1
tff(fact_885_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_886_sup_Oabsorb1,axiom,
    ! [B: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat] :
      ( ord_le1268244103169919719at_nat(B,A)
     => ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),B) = A ) ) ).

% sup.absorb1
tff(fact_887_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_888_sup_Oabsorb1,axiom,
    ! [B: set_int,A: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,B),A)
     => ( aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A),B) = A ) ) ).

% sup.absorb1
tff(fact_889_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_890_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_891_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_892_sup__unique,axiom,
    ! [F: fun_se8464659365716347869at_nat,X: set_Pr8693737435421807431at_nat,Y: set_Pr8693737435421807431at_nat] :
      ( ! [X3: set_Pr8693737435421807431at_nat,Y3: set_Pr8693737435421807431at_nat] : ord_le3000389064537975527at_nat(X3,aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(F,X3),Y3))
     => ( ! [X3: set_Pr8693737435421807431at_nat,Y3: set_Pr8693737435421807431at_nat] : ord_le3000389064537975527at_nat(Y3,aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(F,X3),Y3))
       => ( ! [X3: set_Pr8693737435421807431at_nat,Y3: set_Pr8693737435421807431at_nat,Z2: set_Pr8693737435421807431at_nat] :
              ( ord_le3000389064537975527at_nat(Y3,X3)
             => ( ord_le3000389064537975527at_nat(Z2,X3)
               => ord_le3000389064537975527at_nat(aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(F,Y3),Z2),X3) ) )
         => ( aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,X),Y) = aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(F,X),Y) ) ) ) ) ).

% sup_unique
tff(fact_893_sup__unique,axiom,
    ! [F: fun_se6682931024248017501at_nat,X: set_Pr8551490117392284871at_nat,Y: set_Pr8551490117392284871at_nat] :
      ( ! [X3: set_Pr8551490117392284871at_nat,Y3: set_Pr8551490117392284871at_nat] : ord_le8081472938463900775at_nat(X3,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(F,X3),Y3))
     => ( ! [X3: set_Pr8551490117392284871at_nat,Y3: set_Pr8551490117392284871at_nat] : ord_le8081472938463900775at_nat(Y3,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(F,X3),Y3))
       => ( ! [X3: set_Pr8551490117392284871at_nat,Y3: set_Pr8551490117392284871at_nat,Z2: set_Pr8551490117392284871at_nat] :
              ( ord_le8081472938463900775at_nat(Y3,X3)
             => ( ord_le8081472938463900775at_nat(Z2,X3)
               => ord_le8081472938463900775at_nat(aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(F,Y3),Z2),X3) ) )
         => ( aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,X),Y) = aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(F,X),Y) ) ) ) ) ).

% sup_unique
tff(fact_894_sup__unique,axiom,
    ! [F: fun_se177437144654672861at_nat,X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] :
      ( ! [X3: set_Pr4329608150637261639at_nat,Y3: set_Pr4329608150637261639at_nat] : ord_le1268244103169919719at_nat(X3,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(F,X3),Y3))
     => ( ! [X3: set_Pr4329608150637261639at_nat,Y3: set_Pr4329608150637261639at_nat] : ord_le1268244103169919719at_nat(Y3,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(F,X3),Y3))
       => ( ! [X3: set_Pr4329608150637261639at_nat,Y3: set_Pr4329608150637261639at_nat,Z2: set_Pr4329608150637261639at_nat] :
              ( ord_le1268244103169919719at_nat(Y3,X3)
             => ( ord_le1268244103169919719at_nat(Z2,X3)
               => ord_le1268244103169919719at_nat(aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(F,Y3),Z2),X3) ) )
         => ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),Y) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(F,X),Y) ) ) ) ) ).

% sup_unique
tff(fact_895_sup__unique,axiom,
    ! [F: fun_se1909708154429037836et_nat,X: set_nat,Y: set_nat] :
      ( ! [X3: set_nat,Y3: set_nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X3),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(F,X3),Y3))
     => ( ! [X3: set_nat,Y3: set_nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Y3),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(F,X3),Y3))
       => ( ! [X3: set_nat,Y3: set_nat,Z2: set_nat] :
              ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Y3),X3)
             => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Z2),X3)
               => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(F,Y3),Z2)),X3) ) )
         => ( 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_896_sup__unique,axiom,
    ! [F: fun_se6573205548028396576et_int,X: set_int,Y: set_int] :
      ( ! [X3: set_int,Y3: set_int] : aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,X3),aa_set_int_set_int(aa_set4234024024286713313et_int(F,X3),Y3))
     => ( ! [X3: set_int,Y3: set_int] : aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,Y3),aa_set_int_set_int(aa_set4234024024286713313et_int(F,X3),Y3))
       => ( ! [X3: set_int,Y3: set_int,Z2: set_int] :
              ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,Y3),X3)
             => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,Z2),X3)
               => aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,aa_set_int_set_int(aa_set4234024024286713313et_int(F,Y3),Z2)),X3) ) )
         => ( aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,X),Y) = aa_set_int_set_int(aa_set4234024024286713313et_int(F,X),Y) ) ) ) ) ).

% sup_unique
tff(fact_897_sup__unique,axiom,
    ! [F: fun_rat_fun_rat_rat,X: rat,Y: rat] :
      ( ! [X3: rat,Y3: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),aa_rat_rat(aa_rat_fun_rat_rat(F,X3),Y3))
     => ( ! [X3: rat,Y3: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y3),aa_rat_rat(aa_rat_fun_rat_rat(F,X3),Y3))
       => ( ! [X3: rat,Y3: rat,Z2: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y3),X3)
             => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Z2),X3)
               => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(F,Y3),Z2)),X3) ) )
         => ( 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_898_sup__unique,axiom,
    ! [F: fun_nat_fun_nat_nat,X: nat,Y: nat] :
      ( ! [X3: nat,Y3: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),aa_nat_nat(aa_nat_fun_nat_nat(F,X3),Y3))
     => ( ! [X3: nat,Y3: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y3),aa_nat_nat(aa_nat_fun_nat_nat(F,X3),Y3))
       => ( ! [X3: nat,Y3: nat,Z2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y3),X3)
             => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Z2),X3)
               => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(F,Y3),Z2)),X3) ) )
         => ( 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_899_sup__unique,axiom,
    ! [F: fun_int_fun_int_int,X: int,Y: int] :
      ( ! [X3: int,Y3: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X3),aa_int_int(aa_int_fun_int_int(F,X3),Y3))
     => ( ! [X3: int,Y3: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y3),aa_int_int(aa_int_fun_int_int(F,X3),Y3))
       => ( ! [X3: int,Y3: int,Z2: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y3),X3)
             => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z2),X3)
               => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(F,Y3),Z2)),X3) ) )
         => ( 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_900_sup_OorderI,axiom,
    ! [A: set_Pr8693737435421807431at_nat,B: set_Pr8693737435421807431at_nat] :
      ( ( A = aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,A),B) )
     => ord_le3000389064537975527at_nat(B,A) ) ).

% sup.orderI
tff(fact_901_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_902_sup_OorderI,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
      ( ( A = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),B) )
     => ord_le1268244103169919719at_nat(B,A) ) ).

% sup.orderI
tff(fact_903_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_904_sup_OorderI,axiom,
    ! [A: set_int,B: set_int] :
      ( ( A = aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A),B) )
     => aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,B),A) ) ).

% sup.orderI
tff(fact_905_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_906_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_907_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_908_sup_OorderE,axiom,
    ! [B: set_Pr8693737435421807431at_nat,A: set_Pr8693737435421807431at_nat] :
      ( ord_le3000389064537975527at_nat(B,A)
     => ( A = aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,A),B) ) ) ).

% sup.orderE
tff(fact_909_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_910_sup_OorderE,axiom,
    ! [B: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat] :
      ( ord_le1268244103169919719at_nat(B,A)
     => ( A = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),B) ) ) ).

% sup.orderE
tff(fact_911_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_912_sup_OorderE,axiom,
    ! [B: set_int,A: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,B),A)
     => ( A = aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A),B) ) ) ).

% sup.orderE
tff(fact_913_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_914_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_915_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_916_le__iff__sup,axiom,
    ! [X: set_Pr8693737435421807431at_nat,Y: set_Pr8693737435421807431at_nat] :
      ( ord_le3000389064537975527at_nat(X,Y)
    <=> ( aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,X),Y) = Y ) ) ).

% le_iff_sup
tff(fact_917_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_918_le__iff__sup,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] :
      ( ord_le1268244103169919719at_nat(X,Y)
    <=> ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),Y) = Y ) ) ).

% le_iff_sup
tff(fact_919_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_920_le__iff__sup,axiom,
    ! [X: set_int,Y: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,X),Y)
    <=> ( aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,X),Y) = Y ) ) ).

% le_iff_sup
tff(fact_921_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_922_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_923_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_924_sup__least,axiom,
    ! [Y: set_Pr8693737435421807431at_nat,X: set_Pr8693737435421807431at_nat,Z: set_Pr8693737435421807431at_nat] :
      ( ord_le3000389064537975527at_nat(Y,X)
     => ( ord_le3000389064537975527at_nat(Z,X)
       => ord_le3000389064537975527at_nat(aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,Y),Z),X) ) ) ).

% sup_least
tff(fact_925_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_926_sup__least,axiom,
    ! [Y: set_Pr4329608150637261639at_nat,X: set_Pr4329608150637261639at_nat,Z: set_Pr4329608150637261639at_nat] :
      ( ord_le1268244103169919719at_nat(Y,X)
     => ( ord_le1268244103169919719at_nat(Z,X)
       => ord_le1268244103169919719at_nat(aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,Y),Z),X) ) ) ).

% sup_least
tff(fact_927_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_928_sup__least,axiom,
    ! [Y: set_int,X: set_int,Z: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,Y),X)
     => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,Z),X)
       => aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,Y),Z)),X) ) ) ).

% sup_least
tff(fact_929_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_930_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_931_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_932_sup__mono,axiom,
    ! [A: set_Pr8693737435421807431at_nat,Ca: set_Pr8693737435421807431at_nat,B: set_Pr8693737435421807431at_nat,D2: set_Pr8693737435421807431at_nat] :
      ( ord_le3000389064537975527at_nat(A,Ca)
     => ( ord_le3000389064537975527at_nat(B,D2)
       => ord_le3000389064537975527at_nat(aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,A),B),aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,Ca),D2)) ) ) ).

% sup_mono
tff(fact_933_sup__mono,axiom,
    ! [A: set_Pr8551490117392284871at_nat,Ca: set_Pr8551490117392284871at_nat,B: set_Pr8551490117392284871at_nat,D2: set_Pr8551490117392284871at_nat] :
      ( ord_le8081472938463900775at_nat(A,Ca)
     => ( 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,Ca),D2)) ) ) ).

% sup_mono
tff(fact_934_sup__mono,axiom,
    ! [A: set_Pr4329608150637261639at_nat,Ca: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat,D2: set_Pr4329608150637261639at_nat] :
      ( ord_le1268244103169919719at_nat(A,Ca)
     => ( ord_le1268244103169919719at_nat(B,D2)
       => ord_le1268244103169919719at_nat(aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),B),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,Ca),D2)) ) ) ).

% sup_mono
tff(fact_935_sup__mono,axiom,
    ! [A: set_nat,Ca: set_nat,B: set_nat,D2: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),Ca)
     => ( 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,Ca),D2)) ) ) ).

% sup_mono
tff(fact_936_sup__mono,axiom,
    ! [A: set_int,Ca: set_int,B: set_int,D2: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A),Ca)
     => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,B),D2)
       => aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A),B)),aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,Ca),D2)) ) ) ).

% sup_mono
tff(fact_937_sup__mono,axiom,
    ! [A: rat,Ca: rat,B: rat,D2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),Ca)
     => ( 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,Ca),D2)) ) ) ).

% sup_mono
tff(fact_938_sup__mono,axiom,
    ! [A: nat,Ca: nat,B: nat,D2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),Ca)
     => ( 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,Ca),D2)) ) ) ).

% sup_mono
tff(fact_939_sup__mono,axiom,
    ! [A: int,Ca: int,B: int,D2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),Ca)
     => ( 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,Ca),D2)) ) ) ).

% sup_mono
tff(fact_940_sup_Omono,axiom,
    ! [Ca: set_Pr8693737435421807431at_nat,A: set_Pr8693737435421807431at_nat,D2: set_Pr8693737435421807431at_nat,B: set_Pr8693737435421807431at_nat] :
      ( ord_le3000389064537975527at_nat(Ca,A)
     => ( ord_le3000389064537975527at_nat(D2,B)
       => ord_le3000389064537975527at_nat(aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,Ca),D2),aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,A),B)) ) ) ).

% sup.mono
tff(fact_941_sup_Omono,axiom,
    ! [Ca: set_Pr8551490117392284871at_nat,A: set_Pr8551490117392284871at_nat,D2: set_Pr8551490117392284871at_nat,B: set_Pr8551490117392284871at_nat] :
      ( ord_le8081472938463900775at_nat(Ca,A)
     => ( ord_le8081472938463900775at_nat(D2,B)
       => ord_le8081472938463900775at_nat(aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,Ca),D2),aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A),B)) ) ) ).

% sup.mono
tff(fact_942_sup_Omono,axiom,
    ! [Ca: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat,D2: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
      ( ord_le1268244103169919719at_nat(Ca,A)
     => ( ord_le1268244103169919719at_nat(D2,B)
       => ord_le1268244103169919719at_nat(aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,Ca),D2),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),B)) ) ) ).

% sup.mono
tff(fact_943_sup_Omono,axiom,
    ! [Ca: set_nat,A: set_nat,D2: set_nat,B: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Ca),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,Ca),D2)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B)) ) ) ).

% sup.mono
tff(fact_944_sup_Omono,axiom,
    ! [Ca: set_int,A: set_int,D2: set_int,B: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,Ca),A)
     => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,D2),B)
       => aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,Ca),D2)),aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A),B)) ) ) ).

% sup.mono
tff(fact_945_sup_Omono,axiom,
    ! [Ca: rat,A: rat,D2: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Ca),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,Ca),D2)),aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,A),B)) ) ) ).

% sup.mono
tff(fact_946_sup_Omono,axiom,
    ! [Ca: nat,A: nat,D2: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ca),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,Ca),D2)),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B)) ) ) ).

% sup.mono
tff(fact_947_sup_Omono,axiom,
    ! [Ca: int,A: int,D2: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ca),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,Ca),D2)),aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B)) ) ) ).

% sup.mono
tff(fact_948_le__supI2,axiom,
    ! [X: set_Pr8693737435421807431at_nat,B: set_Pr8693737435421807431at_nat,A: set_Pr8693737435421807431at_nat] :
      ( ord_le3000389064537975527at_nat(X,B)
     => ord_le3000389064537975527at_nat(X,aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,A),B)) ) ).

% le_supI2
tff(fact_949_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_950_le__supI2,axiom,
    ! [X: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat] :
      ( ord_le1268244103169919719at_nat(X,B)
     => ord_le1268244103169919719at_nat(X,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),B)) ) ).

% le_supI2
tff(fact_951_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_952_le__supI2,axiom,
    ! [X: set_int,B: set_int,A: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,X),B)
     => aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,X),aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A),B)) ) ).

% le_supI2
tff(fact_953_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_954_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_955_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_956_le__supI1,axiom,
    ! [X: set_Pr8693737435421807431at_nat,A: set_Pr8693737435421807431at_nat,B: set_Pr8693737435421807431at_nat] :
      ( ord_le3000389064537975527at_nat(X,A)
     => ord_le3000389064537975527at_nat(X,aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,A),B)) ) ).

% le_supI1
tff(fact_957_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_958_le__supI1,axiom,
    ! [X: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
      ( ord_le1268244103169919719at_nat(X,A)
     => ord_le1268244103169919719at_nat(X,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),B)) ) ).

% le_supI1
tff(fact_959_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_960_le__supI1,axiom,
    ! [X: set_int,A: set_int,B: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,X),A)
     => aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,X),aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A),B)) ) ).

% le_supI1
tff(fact_961_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_962_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_963_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_964_sup__ge2,axiom,
    ! [Y: set_Pr8693737435421807431at_nat,X: set_Pr8693737435421807431at_nat] : ord_le3000389064537975527at_nat(Y,aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,X),Y)) ).

% sup_ge2
tff(fact_965_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_966_sup__ge2,axiom,
    ! [Y: set_Pr4329608150637261639at_nat,X: set_Pr4329608150637261639at_nat] : ord_le1268244103169919719at_nat(Y,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),Y)) ).

% sup_ge2
tff(fact_967_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_968_sup__ge2,axiom,
    ! [Y: set_int,X: set_int] : aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,Y),aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,X),Y)) ).

% sup_ge2
tff(fact_969_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_970_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_971_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_972_sup__ge1,axiom,
    ! [X: set_Pr8693737435421807431at_nat,Y: set_Pr8693737435421807431at_nat] : ord_le3000389064537975527at_nat(X,aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,X),Y)) ).

% sup_ge1
tff(fact_973_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_974_sup__ge1,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] : ord_le1268244103169919719at_nat(X,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),Y)) ).

% sup_ge1
tff(fact_975_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_976_sup__ge1,axiom,
    ! [X: set_int,Y: set_int] : aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,X),aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,X),Y)) ).

% sup_ge1
tff(fact_977_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_978_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_979_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_980_le__supI,axiom,
    ! [A: set_Pr8693737435421807431at_nat,X: set_Pr8693737435421807431at_nat,B: set_Pr8693737435421807431at_nat] :
      ( ord_le3000389064537975527at_nat(A,X)
     => ( ord_le3000389064537975527at_nat(B,X)
       => ord_le3000389064537975527at_nat(aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,A),B),X) ) ) ).

% le_supI
tff(fact_981_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_982_le__supI,axiom,
    ! [A: set_Pr4329608150637261639at_nat,X: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
      ( ord_le1268244103169919719at_nat(A,X)
     => ( ord_le1268244103169919719at_nat(B,X)
       => ord_le1268244103169919719at_nat(aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),B),X) ) ) ).

% le_supI
tff(fact_983_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_984_le__supI,axiom,
    ! [A: set_int,X: set_int,B: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A),X)
     => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,B),X)
       => aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A),B)),X) ) ) ).

% le_supI
tff(fact_985_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_986_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_987_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_988_le__supE,axiom,
    ! [A: set_Pr8693737435421807431at_nat,B: set_Pr8693737435421807431at_nat,X: set_Pr8693737435421807431at_nat] :
      ( ord_le3000389064537975527at_nat(aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,A),B),X)
     => ~ ( ord_le3000389064537975527at_nat(A,X)
         => ~ ord_le3000389064537975527at_nat(B,X) ) ) ).

% le_supE
tff(fact_989_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_990_le__supE,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat,X: set_Pr4329608150637261639at_nat] :
      ( ord_le1268244103169919719at_nat(aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),B),X)
     => ~ ( ord_le1268244103169919719at_nat(A,X)
         => ~ ord_le1268244103169919719at_nat(B,X) ) ) ).

% le_supE
tff(fact_991_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_992_le__supE,axiom,
    ! [A: set_int,B: set_int,X: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A),B)),X)
     => ~ ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A),X)
         => ~ aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,B),X) ) ) ).

% le_supE
tff(fact_993_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_994_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_995_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_996_inf__sup__ord_I3_J,axiom,
    ! [X: set_Pr8693737435421807431at_nat,Y: set_Pr8693737435421807431at_nat] : ord_le3000389064537975527at_nat(X,aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,X),Y)) ).

% inf_sup_ord(3)
tff(fact_997_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_998_inf__sup__ord_I3_J,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] : ord_le1268244103169919719at_nat(X,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),Y)) ).

% inf_sup_ord(3)
tff(fact_999_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_1000_inf__sup__ord_I3_J,axiom,
    ! [X: set_int,Y: set_int] : aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,X),aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,X),Y)) ).

% inf_sup_ord(3)
tff(fact_1001_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_1002_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_1003_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_1004_inf__sup__ord_I4_J,axiom,
    ! [Y: set_Pr8693737435421807431at_nat,X: set_Pr8693737435421807431at_nat] : ord_le3000389064537975527at_nat(Y,aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,X),Y)) ).

% inf_sup_ord(4)
tff(fact_1005_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_1006_inf__sup__ord_I4_J,axiom,
    ! [Y: set_Pr4329608150637261639at_nat,X: set_Pr4329608150637261639at_nat] : ord_le1268244103169919719at_nat(Y,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),Y)) ).

% inf_sup_ord(4)
tff(fact_1007_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_1008_inf__sup__ord_I4_J,axiom,
    ! [Y: set_int,X: set_int] : aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,Y),aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,X),Y)) ).

% inf_sup_ord(4)
tff(fact_1009_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_1010_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_1011_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_1012_disjoint__iff__not__equal,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
      ( ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A3),B3) = bot_bo2099793752762293965at_nat )
    <=> ! [X4: product_prod_nat_nat] :
          ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X4),A3)
         => ! [Xa: product_prod_nat_nat] :
              ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Xa),B3)
             => ( X4 != Xa ) ) ) ) ).

% disjoint_iff_not_equal
tff(fact_1013_disjoint__iff__not__equal,axiom,
    ! [A3: set_o,B3: set_o] :
      ( ( inf_inf_set_o(A3,B3) = bot_bot_set_o )
    <=> ! [X4: $o] :
          ( member_o((X4),A3)
         => ! [Xa: $o] :
              ( member_o((Xa),B3)
             => ( (X4)
              <=> ~ (Xa) ) ) ) ) ).

% disjoint_iff_not_equal
tff(fact_1014_disjoint__iff__not__equal,axiom,
    ! [A3: set_nat,B3: set_nat] :
      ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A3),B3) = bot_bot_set_nat )
    <=> ! [X4: nat] :
          ( aa_set_nat_o(member_nat(X4),A3)
         => ! [Xa: nat] :
              ( aa_set_nat_o(member_nat(Xa),B3)
             => ( X4 != Xa ) ) ) ) ).

% disjoint_iff_not_equal
tff(fact_1015_disjoint__iff__not__equal,axiom,
    ! [A3: set_int,B3: set_int] :
      ( ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A3),B3) = bot_bot_set_int )
    <=> ! [X4: int] :
          ( aa_set_int_o(member_int(X4),A3)
         => ! [Xa: int] :
              ( aa_set_int_o(member_int(Xa),B3)
             => ( X4 != Xa ) ) ) ) ).

% disjoint_iff_not_equal
tff(fact_1016_Int__empty__right,axiom,
    ! [A3: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A3),bot_bo2099793752762293965at_nat) = bot_bo2099793752762293965at_nat ).

% Int_empty_right
tff(fact_1017_Int__empty__right,axiom,
    ! [A3: set_o] : inf_inf_set_o(A3,bot_bot_set_o) = bot_bot_set_o ).

% Int_empty_right
tff(fact_1018_Int__empty__right,axiom,
    ! [A3: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A3),bot_bot_set_nat) = bot_bot_set_nat ).

% Int_empty_right
tff(fact_1019_Int__empty__right,axiom,
    ! [A3: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A3),bot_bot_set_int) = bot_bot_set_int ).

% Int_empty_right
tff(fact_1020_Int__empty__left,axiom,
    ! [B3: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,bot_bo2099793752762293965at_nat),B3) = bot_bo2099793752762293965at_nat ).

% Int_empty_left
tff(fact_1021_Int__empty__left,axiom,
    ! [B3: set_o] : inf_inf_set_o(bot_bot_set_o,B3) = bot_bot_set_o ).

% Int_empty_left
tff(fact_1022_Int__empty__left,axiom,
    ! [B3: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,bot_bot_set_nat),B3) = bot_bot_set_nat ).

% Int_empty_left
tff(fact_1023_Int__empty__left,axiom,
    ! [B3: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,bot_bot_set_int),B3) = bot_bot_set_int ).

% Int_empty_left
tff(fact_1024_disjoint__iff,axiom,
    ! [A3: set_set_nat,B3: set_set_nat] :
      ( ( inf_inf_set_set_nat(A3,B3) = bot_bot_set_set_nat )
    <=> ! [X4: set_nat] :
          ( aa_set_set_nat_o(member_set_nat(X4),A3)
         => ~ aa_set_set_nat_o(member_set_nat(X4),B3) ) ) ).

% disjoint_iff
tff(fact_1025_disjoint__iff,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
      ( ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A3),B3) = bot_bo2099793752762293965at_nat )
    <=> ! [X4: product_prod_nat_nat] :
          ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X4),A3)
         => ~ aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X4),B3) ) ) ).

% disjoint_iff
tff(fact_1026_disjoint__iff,axiom,
    ! [A3: set_o,B3: set_o] :
      ( ( inf_inf_set_o(A3,B3) = bot_bot_set_o )
    <=> ! [X4: $o] :
          ( member_o((X4),A3)
         => ~ member_o((X4),B3) ) ) ).

% disjoint_iff
tff(fact_1027_disjoint__iff,axiom,
    ! [A3: set_nat,B3: set_nat] :
      ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A3),B3) = bot_bot_set_nat )
    <=> ! [X4: nat] :
          ( aa_set_nat_o(member_nat(X4),A3)
         => ~ aa_set_nat_o(member_nat(X4),B3) ) ) ).

% disjoint_iff
tff(fact_1028_disjoint__iff,axiom,
    ! [A3: set_int,B3: set_int] :
      ( ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A3),B3) = bot_bot_set_int )
    <=> ! [X4: int] :
          ( aa_set_int_o(member_int(X4),A3)
         => ~ aa_set_int_o(member_int(X4),B3) ) ) ).

% disjoint_iff
tff(fact_1029_Int__emptyI,axiom,
    ! [A3: set_set_nat,B3: set_set_nat] :
      ( ! [X3: set_nat] :
          ( aa_set_set_nat_o(member_set_nat(X3),A3)
         => ~ aa_set_set_nat_o(member_set_nat(X3),B3) )
     => ( inf_inf_set_set_nat(A3,B3) = bot_bot_set_set_nat ) ) ).

% Int_emptyI
tff(fact_1030_Int__emptyI,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
      ( ! [X3: product_prod_nat_nat] :
          ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X3),A3)
         => ~ aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X3),B3) )
     => ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A3),B3) = bot_bo2099793752762293965at_nat ) ) ).

% Int_emptyI
tff(fact_1031_Int__emptyI,axiom,
    ! [A3: set_o,B3: set_o] :
      ( ! [X3: $o] :
          ( member_o((X3),A3)
         => ~ member_o((X3),B3) )
     => ( inf_inf_set_o(A3,B3) = bot_bot_set_o ) ) ).

% Int_emptyI
tff(fact_1032_Int__emptyI,axiom,
    ! [A3: set_nat,B3: set_nat] :
      ( ! [X3: nat] :
          ( aa_set_nat_o(member_nat(X3),A3)
         => ~ aa_set_nat_o(member_nat(X3),B3) )
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A3),B3) = bot_bot_set_nat ) ) ).

% Int_emptyI
tff(fact_1033_Int__emptyI,axiom,
    ! [A3: set_int,B3: set_int] :
      ( ! [X3: int] :
          ( aa_set_int_o(member_int(X3),A3)
         => ~ aa_set_int_o(member_int(X3),B3) )
     => ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A3),B3) = bot_bot_set_int ) ) ).

% Int_emptyI
tff(fact_1034_sup__inf__distrib2,axiom,
    ! [Y: assn,Z: assn,X: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Y),Z)),X) = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Y),X)),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Z),X)) ).

% sup_inf_distrib2
tff(fact_1035_sup__inf__distrib2,axiom,
    ! [Y: product_unit,Z: product_unit,X: product_unit] : aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(sup_sup_Product_unit,aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,Y),Z)),X) = aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(sup_sup_Product_unit,Y),X)),aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(sup_sup_Product_unit,Z),X)) ).

% sup_inf_distrib2
tff(fact_1036_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_1037_sup__inf__distrib2,axiom,
    ! [Y: set_Pr8693737435421807431at_nat,Z: set_Pr8693737435421807431at_nat,X: set_Pr8693737435421807431at_nat] : aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(inf_in4302113700860409141at_nat,Y),Z)),X) = aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(inf_in4302113700860409141at_nat,aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,Y),X)),aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,Z),X)) ).

% sup_inf_distrib2
tff(fact_1038_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_1039_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_1040_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_1041_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_1042_sup__inf__distrib1,axiom,
    ! [X: assn,Y: assn,Z: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Y),Z)) = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),Y)),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),Z)) ).

% sup_inf_distrib1
tff(fact_1043_sup__inf__distrib1,axiom,
    ! [X: product_unit,Y: product_unit,Z: product_unit] : aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(sup_sup_Product_unit,X),aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,Y),Z)) = aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(sup_sup_Product_unit,X),Y)),aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(sup_sup_Product_unit,X),Z)) ).

% sup_inf_distrib1
tff(fact_1044_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_1045_sup__inf__distrib1,axiom,
    ! [X: set_Pr8693737435421807431at_nat,Y: set_Pr8693737435421807431at_nat,Z: set_Pr8693737435421807431at_nat] : aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,X),aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(inf_in4302113700860409141at_nat,Y),Z)) = aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(inf_in4302113700860409141at_nat,aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,X),Y)),aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,X),Z)) ).

% sup_inf_distrib1
tff(fact_1046_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_1047_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_1048_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_1049_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_1050_inf__sup__distrib2,axiom,
    ! [Y: assn,Z: assn,X: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Y),Z)),X) = aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Y),X)),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Z),X)) ).

% inf_sup_distrib2
tff(fact_1051_inf__sup__distrib2,axiom,
    ! [Y: product_unit,Z: product_unit,X: product_unit] : aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(sup_sup_Product_unit,Y),Z)),X) = aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(sup_sup_Product_unit,aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,Y),X)),aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,Z),X)) ).

% inf_sup_distrib2
tff(fact_1052_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_1053_inf__sup__distrib2,axiom,
    ! [Y: set_Pr8693737435421807431at_nat,Z: set_Pr8693737435421807431at_nat,X: set_Pr8693737435421807431at_nat] : aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(inf_in4302113700860409141at_nat,aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,Y),Z)),X) = aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(inf_in4302113700860409141at_nat,Y),X)),aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(inf_in4302113700860409141at_nat,Z),X)) ).

% inf_sup_distrib2
tff(fact_1054_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_1055_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_1056_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_1057_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_1058_inf__sup__distrib1,axiom,
    ! [X: assn,Y: assn,Z: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Y),Z)) = aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),Y)),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),Z)) ).

% inf_sup_distrib1
tff(fact_1059_inf__sup__distrib1,axiom,
    ! [X: product_unit,Y: product_unit,Z: product_unit] : aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,X),aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(sup_sup_Product_unit,Y),Z)) = aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(sup_sup_Product_unit,aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,X),Y)),aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,X),Z)) ).

% inf_sup_distrib1
tff(fact_1060_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_1061_inf__sup__distrib1,axiom,
    ! [X: set_Pr8693737435421807431at_nat,Y: set_Pr8693737435421807431at_nat,Z: set_Pr8693737435421807431at_nat] : aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(inf_in4302113700860409141at_nat,X),aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,Y),Z)) = aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(inf_in4302113700860409141at_nat,X),Y)),aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(inf_in4302113700860409141at_nat,X),Z)) ).

% inf_sup_distrib1
tff(fact_1062_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_1063_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_1064_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_1065_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_1066_distrib__imp2,axiom,
    ! [X: assn,Y: assn,Z: assn] :
      ( ! [X3: assn,Y3: assn,Z2: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X3),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Y3),Z2)) = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X3),Y3)),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X3),Z2))
     => ( aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Y),Z)) = aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),Y)),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),Z)) ) ) ).

% distrib_imp2
tff(fact_1067_distrib__imp2,axiom,
    ! [X: product_unit,Y: product_unit,Z: product_unit] :
      ( ! [X3: product_unit,Y3: product_unit,Z2: product_unit] : aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(sup_sup_Product_unit,X3),aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,Y3),Z2)) = aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(sup_sup_Product_unit,X3),Y3)),aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(sup_sup_Product_unit,X3),Z2))
     => ( aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,X),aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(sup_sup_Product_unit,Y),Z)) = aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(sup_sup_Product_unit,aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,X),Y)),aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,X),Z)) ) ) ).

% distrib_imp2
tff(fact_1068_distrib__imp2,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat] :
      ( ! [X3: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat,Z2: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,X3),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Y3),Z2)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,X3),Y3)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,X3),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_1069_distrib__imp2,axiom,
    ! [X: set_Pr8693737435421807431at_nat,Y: set_Pr8693737435421807431at_nat,Z: set_Pr8693737435421807431at_nat] :
      ( ! [X3: set_Pr8693737435421807431at_nat,Y3: set_Pr8693737435421807431at_nat,Z2: set_Pr8693737435421807431at_nat] : aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,X3),aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(inf_in4302113700860409141at_nat,Y3),Z2)) = aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(inf_in4302113700860409141at_nat,aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,X3),Y3)),aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,X3),Z2))
     => ( aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(inf_in4302113700860409141at_nat,X),aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,Y),Z)) = aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(inf_in4302113700860409141at_nat,X),Y)),aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(inf_in4302113700860409141at_nat,X),Z)) ) ) ).

% distrib_imp2
tff(fact_1070_distrib__imp2,axiom,
    ! [X: set_Pr8551490117392284871at_nat,Y: set_Pr8551490117392284871at_nat,Z: set_Pr8551490117392284871at_nat] :
      ( ! [X3: set_Pr8551490117392284871at_nat,Y3: set_Pr8551490117392284871at_nat,Z2: set_Pr8551490117392284871at_nat] : aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,X3),aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(inf_in1697001100524423349at_nat,Y3),Z2)) = aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(inf_in1697001100524423349at_nat,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,X3),Y3)),aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,X3),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_1071_distrib__imp2,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat,Z: set_Pr4329608150637261639at_nat] :
      ( ! [X3: set_Pr4329608150637261639at_nat,Y3: set_Pr4329608150637261639at_nat,Z2: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X3),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,Y3),Z2)) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X3),Y3)),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X3),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_1072_distrib__imp2,axiom,
    ! [X: set_nat,Y: set_nat,Z: set_nat] :
      ( ! [X3: set_nat,Y3: set_nat,Z2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X3),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Y3),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,X3),Y3)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X3),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_1073_distrib__imp2,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ! [X3: nat,Y3: nat,Z2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X3),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,Y3),Z2)) = aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X3),Y3)),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X3),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_1074_distrib__imp1,axiom,
    ! [X: assn,Y: assn,Z: assn] :
      ( ! [X3: assn,Y3: assn,Z2: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X3),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Y3),Z2)) = aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X3),Y3)),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X3),Z2))
     => ( aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Y),Z)) = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),Y)),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),Z)) ) ) ).

% distrib_imp1
tff(fact_1075_distrib__imp1,axiom,
    ! [X: product_unit,Y: product_unit,Z: product_unit] :
      ( ! [X3: product_unit,Y3: product_unit,Z2: product_unit] : aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,X3),aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(sup_sup_Product_unit,Y3),Z2)) = aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(sup_sup_Product_unit,aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,X3),Y3)),aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,X3),Z2))
     => ( aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(sup_sup_Product_unit,X),aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,Y),Z)) = aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(sup_sup_Product_unit,X),Y)),aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(sup_sup_Product_unit,X),Z)) ) ) ).

% distrib_imp1
tff(fact_1076_distrib__imp1,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat] :
      ( ! [X3: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat,Z2: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X3),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,Y3),Z2)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X3),Y3)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X3),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_1077_distrib__imp1,axiom,
    ! [X: set_Pr8693737435421807431at_nat,Y: set_Pr8693737435421807431at_nat,Z: set_Pr8693737435421807431at_nat] :
      ( ! [X3: set_Pr8693737435421807431at_nat,Y3: set_Pr8693737435421807431at_nat,Z2: set_Pr8693737435421807431at_nat] : aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(inf_in4302113700860409141at_nat,X3),aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,Y3),Z2)) = aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(inf_in4302113700860409141at_nat,X3),Y3)),aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(inf_in4302113700860409141at_nat,X3),Z2))
     => ( aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,X),aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(inf_in4302113700860409141at_nat,Y),Z)) = aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(inf_in4302113700860409141at_nat,aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,X),Y)),aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,X),Z)) ) ) ).

% distrib_imp1
tff(fact_1078_distrib__imp1,axiom,
    ! [X: set_Pr8551490117392284871at_nat,Y: set_Pr8551490117392284871at_nat,Z: set_Pr8551490117392284871at_nat] :
      ( ! [X3: set_Pr8551490117392284871at_nat,Y3: set_Pr8551490117392284871at_nat,Z2: set_Pr8551490117392284871at_nat] : aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(inf_in1697001100524423349at_nat,X3),aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,Y3),Z2)) = aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(inf_in1697001100524423349at_nat,X3),Y3)),aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(inf_in1697001100524423349at_nat,X3),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_1079_distrib__imp1,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat,Z: set_Pr4329608150637261639at_nat] :
      ( ! [X3: set_Pr4329608150637261639at_nat,Y3: set_Pr4329608150637261639at_nat,Z2: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,X3),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,Y3),Z2)) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,X3),Y3)),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,X3),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_1080_distrib__imp1,axiom,
    ! [X: set_nat,Y: set_nat,Z: set_nat] :
      ( ! [X3: set_nat,Y3: set_nat,Z2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X3),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Y3),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,X3),Y3)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X3),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_1081_distrib__imp1,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ! [X3: nat,Y3: nat,Z2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X3),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,Y3),Z2)) = aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X3),Y3)),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X3),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_1082_Un__empty__right,axiom,
    ! [A3: set_Pr8693737435421807431at_nat] : aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,A3),bot_bo5327735625951526323at_nat) = A3 ).

% Un_empty_right
tff(fact_1083_Un__empty__right,axiom,
    ! [A3: set_Pr8551490117392284871at_nat] : aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A3),bot_bo8422036546324065075at_nat) = A3 ).

% Un_empty_right
tff(fact_1084_Un__empty__right,axiom,
    ! [A3: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A3),bot_bo228742789529271731at_nat) = A3 ).

% Un_empty_right
tff(fact_1085_Un__empty__right,axiom,
    ! [A3: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,A3),bot_bo2099793752762293965at_nat) = A3 ).

% Un_empty_right
tff(fact_1086_Un__empty__right,axiom,
    ! [A3: set_o] : aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,A3),bot_bot_set_o) = A3 ).

% Un_empty_right
tff(fact_1087_Un__empty__right,axiom,
    ! [A3: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A3),bot_bot_set_nat) = A3 ).

% Un_empty_right
tff(fact_1088_Un__empty__right,axiom,
    ! [A3: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A3),bot_bot_set_int) = A3 ).

% Un_empty_right
tff(fact_1089_Un__empty__left,axiom,
    ! [B3: set_Pr8693737435421807431at_nat] : aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,bot_bo5327735625951526323at_nat),B3) = B3 ).

% Un_empty_left
tff(fact_1090_Un__empty__left,axiom,
    ! [B3: set_Pr8551490117392284871at_nat] : aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,bot_bo8422036546324065075at_nat),B3) = B3 ).

% Un_empty_left
tff(fact_1091_Un__empty__left,axiom,
    ! [B3: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,bot_bo228742789529271731at_nat),B3) = B3 ).

% Un_empty_left
tff(fact_1092_Un__empty__left,axiom,
    ! [B3: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,bot_bo2099793752762293965at_nat),B3) = B3 ).

% Un_empty_left
tff(fact_1093_Un__empty__left,axiom,
    ! [B3: set_o] : aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,bot_bot_set_o),B3) = B3 ).

% Un_empty_left
tff(fact_1094_Un__empty__left,axiom,
    ! [B3: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,bot_bot_set_nat),B3) = B3 ).

% Un_empty_left
tff(fact_1095_Un__empty__left,axiom,
    ! [B3: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,bot_bot_set_int),B3) = B3 ).

% Un_empty_left
tff(fact_1096_Un__Int__distrib2,axiom,
    ! [B3: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat,A3: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B3),C2)),A3) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,B3),A3)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,C2),A3)) ).

% Un_Int_distrib2
tff(fact_1097_Un__Int__distrib2,axiom,
    ! [B3: set_Pr8693737435421807431at_nat,C2: set_Pr8693737435421807431at_nat,A3: set_Pr8693737435421807431at_nat] : aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(inf_in4302113700860409141at_nat,B3),C2)),A3) = aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(inf_in4302113700860409141at_nat,aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,B3),A3)),aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,C2),A3)) ).

% Un_Int_distrib2
tff(fact_1098_Un__Int__distrib2,axiom,
    ! [B3: set_Pr8551490117392284871at_nat,C2: set_Pr8551490117392284871at_nat,A3: set_Pr8551490117392284871at_nat] : aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(inf_in1697001100524423349at_nat,B3),C2)),A3) = aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(inf_in1697001100524423349at_nat,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,B3),A3)),aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,C2),A3)) ).

% Un_Int_distrib2
tff(fact_1099_Un__Int__distrib2,axiom,
    ! [B3: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat,A3: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,B3),C2)),A3) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,B3),A3)),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,C2),A3)) ).

% Un_Int_distrib2
tff(fact_1100_Un__Int__distrib2,axiom,
    ! [B3: set_nat,C2: set_nat,A3: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B3),C2)),A3) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,B3),A3)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,C2),A3)) ).

% Un_Int_distrib2
tff(fact_1101_Int__Un__distrib2,axiom,
    ! [B3: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat,A3: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,B3),C2)),A3) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B3),A3)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,C2),A3)) ).

% Int_Un_distrib2
tff(fact_1102_Int__Un__distrib2,axiom,
    ! [B3: set_Pr8693737435421807431at_nat,C2: set_Pr8693737435421807431at_nat,A3: set_Pr8693737435421807431at_nat] : aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(inf_in4302113700860409141at_nat,aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,B3),C2)),A3) = aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(inf_in4302113700860409141at_nat,B3),A3)),aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(inf_in4302113700860409141at_nat,C2),A3)) ).

% Int_Un_distrib2
tff(fact_1103_Int__Un__distrib2,axiom,
    ! [B3: set_Pr8551490117392284871at_nat,C2: set_Pr8551490117392284871at_nat,A3: set_Pr8551490117392284871at_nat] : aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(inf_in1697001100524423349at_nat,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,B3),C2)),A3) = aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(inf_in1697001100524423349at_nat,B3),A3)),aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(inf_in1697001100524423349at_nat,C2),A3)) ).

% Int_Un_distrib2
tff(fact_1104_Int__Un__distrib2,axiom,
    ! [B3: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat,A3: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,B3),C2)),A3) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,B3),A3)),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,C2),A3)) ).

% Int_Un_distrib2
tff(fact_1105_Int__Un__distrib2,axiom,
    ! [B3: set_nat,C2: set_nat,A3: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,B3),C2)),A3) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B3),A3)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,C2),A3)) ).

% Int_Un_distrib2
tff(fact_1106_Un__Int__distrib,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,A3),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B3),C2)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,A3),B3)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,A3),C2)) ).

% Un_Int_distrib
tff(fact_1107_Un__Int__distrib,axiom,
    ! [A3: set_Pr8693737435421807431at_nat,B3: set_Pr8693737435421807431at_nat,C2: set_Pr8693737435421807431at_nat] : aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,A3),aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(inf_in4302113700860409141at_nat,B3),C2)) = aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(inf_in4302113700860409141at_nat,aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,A3),B3)),aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,A3),C2)) ).

% Un_Int_distrib
tff(fact_1108_Un__Int__distrib,axiom,
    ! [A3: set_Pr8551490117392284871at_nat,B3: set_Pr8551490117392284871at_nat,C2: set_Pr8551490117392284871at_nat] : aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A3),aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(inf_in1697001100524423349at_nat,B3),C2)) = aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(inf_in1697001100524423349at_nat,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A3),B3)),aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A3),C2)) ).

% Un_Int_distrib
tff(fact_1109_Un__Int__distrib,axiom,
    ! [A3: set_Pr4329608150637261639at_nat,B3: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A3),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,B3),C2)) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A3),B3)),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A3),C2)) ).

% Un_Int_distrib
tff(fact_1110_Un__Int__distrib,axiom,
    ! [A3: set_nat,B3: set_nat,C2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A3),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B3),C2)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A3),B3)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A3),C2)) ).

% Un_Int_distrib
tff(fact_1111_Int__Un__distrib,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A3),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,B3),C2)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A3),B3)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A3),C2)) ).

% Int_Un_distrib
tff(fact_1112_Int__Un__distrib,axiom,
    ! [A3: set_Pr8693737435421807431at_nat,B3: set_Pr8693737435421807431at_nat,C2: set_Pr8693737435421807431at_nat] : aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(inf_in4302113700860409141at_nat,A3),aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,B3),C2)) = aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(inf_in4302113700860409141at_nat,A3),B3)),aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(inf_in4302113700860409141at_nat,A3),C2)) ).

% Int_Un_distrib
tff(fact_1113_Int__Un__distrib,axiom,
    ! [A3: set_Pr8551490117392284871at_nat,B3: set_Pr8551490117392284871at_nat,C2: set_Pr8551490117392284871at_nat] : aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(inf_in1697001100524423349at_nat,A3),aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,B3),C2)) = aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(inf_in1697001100524423349at_nat,A3),B3)),aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(inf_in1697001100524423349at_nat,A3),C2)) ).

% Int_Un_distrib
tff(fact_1114_Int__Un__distrib,axiom,
    ! [A3: set_Pr4329608150637261639at_nat,B3: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,A3),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,B3),C2)) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,A3),B3)),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,A3),C2)) ).

% Int_Un_distrib
tff(fact_1115_Int__Un__distrib,axiom,
    ! [A3: set_nat,B3: set_nat,C2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A3),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,B3),C2)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A3),B3)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A3),C2)) ).

% Int_Un_distrib
tff(fact_1116_Un__Int__crazy,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A3),B3)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B3),C2))),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,C2),A3)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,A3),B3)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,B3),C2))),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,C2),A3)) ).

% Un_Int_crazy
tff(fact_1117_Un__Int__crazy,axiom,
    ! [A3: set_Pr8693737435421807431at_nat,B3: set_Pr8693737435421807431at_nat,C2: set_Pr8693737435421807431at_nat] : aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(inf_in4302113700860409141at_nat,A3),B3)),aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(inf_in4302113700860409141at_nat,B3),C2))),aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(inf_in4302113700860409141at_nat,C2),A3)) = aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(inf_in4302113700860409141at_nat,aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(inf_in4302113700860409141at_nat,aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,A3),B3)),aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,B3),C2))),aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,C2),A3)) ).

% Un_Int_crazy
tff(fact_1118_Un__Int__crazy,axiom,
    ! [A3: set_Pr8551490117392284871at_nat,B3: 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,A3),B3)),aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(inf_in1697001100524423349at_nat,B3),C2))),aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(inf_in1697001100524423349at_nat,C2),A3)) = 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,A3),B3)),aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,B3),C2))),aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,C2),A3)) ).

% Un_Int_crazy
tff(fact_1119_Un__Int__crazy,axiom,
    ! [A3: set_Pr4329608150637261639at_nat,B3: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,A3),B3)),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,B3),C2))),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,C2),A3)) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A3),B3)),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,B3),C2))),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,C2),A3)) ).

% Un_Int_crazy
tff(fact_1120_Un__Int__crazy,axiom,
    ! [A3: set_nat,B3: set_nat,C2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A3),B3)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B3),C2))),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,C2),A3)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A3),B3)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,B3),C2))),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,C2),A3)) ).

% Un_Int_crazy
tff(fact_1121_distrib__sup__le,axiom,
    ! [X: assn,Y: assn,Z: assn] : aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Y),Z))),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),Y)),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),Z))) ).

% distrib_sup_le
tff(fact_1122_distrib__sup__le,axiom,
    ! [X: product_unit,Y: product_unit,Z: product_unit] : aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(sup_sup_Product_unit,X),aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,Y),Z))),aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(sup_sup_Product_unit,X),Y)),aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(sup_sup_Product_unit,X),Z))) ).

% distrib_sup_le
tff(fact_1123_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_1124_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_1125_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_1126_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_1127_distrib__sup__le,axiom,
    ! [X: set_int,Y: set_int,Z: set_int] : aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,X),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,Y),Z))),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,X),Y)),aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,X),Z))) ).

% distrib_sup_le
tff(fact_1128_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_1129_distrib__sup__le,axiom,
    ! [X: set_Pr8693737435421807431at_nat,Y: set_Pr8693737435421807431at_nat,Z: set_Pr8693737435421807431at_nat] : ord_le3000389064537975527at_nat(aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,X),aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(inf_in4302113700860409141at_nat,Y),Z)),aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(inf_in4302113700860409141at_nat,aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,X),Y)),aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,X),Z))) ).

% distrib_sup_le
tff(fact_1130_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_1131_distrib__inf__le,axiom,
    ! [X: assn,Y: assn,Z: assn] : aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),Y)),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),Z))),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Y),Z))) ).

% distrib_inf_le
tff(fact_1132_distrib__inf__le,axiom,
    ! [X: product_unit,Y: product_unit,Z: product_unit] : aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(sup_sup_Product_unit,aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,X),Y)),aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,X),Z))),aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,X),aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(sup_sup_Product_unit,Y),Z))) ).

% distrib_inf_le
tff(fact_1133_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_1134_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_1135_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_1136_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_1137_distrib__inf__le,axiom,
    ! [X: set_int,Y: set_int,Z: set_int] : aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,X),Y)),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,X),Z))),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,X),aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,Y),Z))) ).

% distrib_inf_le
tff(fact_1138_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_1139_distrib__inf__le,axiom,
    ! [X: set_Pr8693737435421807431at_nat,Y: set_Pr8693737435421807431at_nat,Z: set_Pr8693737435421807431at_nat] : ord_le3000389064537975527at_nat(aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(inf_in4302113700860409141at_nat,X),Y)),aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(inf_in4302113700860409141at_nat,X),Z)),aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(inf_in4302113700860409141at_nat,X),aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,Y),Z))) ).

% distrib_inf_le
tff(fact_1140_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_1141_boolean__algebra_Oconj__zero__right,axiom,
    ! [X: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),bot_bot_assn) = bot_bot_assn ).

% boolean_algebra.conj_zero_right
tff(fact_1142_boolean__algebra_Oconj__zero__right,axiom,
    ! [X: product_unit] : aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,X),bot_bot_Product_unit) = bot_bot_Product_unit ).

% boolean_algebra.conj_zero_right
tff(fact_1143_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_1144_boolean__algebra_Oconj__zero__right,axiom,
    ! [X: set_o] : inf_inf_set_o(X,bot_bot_set_o) = bot_bot_set_o ).

% boolean_algebra.conj_zero_right
tff(fact_1145_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_1146_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_1147_boolean__algebra_Oconj__zero__left,axiom,
    ! [X: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,bot_bot_assn),X) = bot_bot_assn ).

% boolean_algebra.conj_zero_left
tff(fact_1148_boolean__algebra_Oconj__zero__left,axiom,
    ! [X: product_unit] : aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,bot_bot_Product_unit),X) = bot_bot_Product_unit ).

% boolean_algebra.conj_zero_left
tff(fact_1149_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_1150_boolean__algebra_Oconj__zero__left,axiom,
    ! [X: set_o] : inf_inf_set_o(bot_bot_set_o,X) = bot_bot_set_o ).

% boolean_algebra.conj_zero_left
tff(fact_1151_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_1152_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_1153_EX,axiom,
    aa_hea4724376117875161946it_nat(heap_Time_execute_a(c),h2) = aa_Pro5302094286594872192it_nat(some_P7913643980934408916it_nat,aa_Pro8078911840264235907it_nat(aa_a_f2234720956197727492it_nat(produc9178034014595674355it_nat,r2),produc584006145561248582it_nat(h,t))) ).

% EX
tff(fact_1154_times__assn__raw_Oelims_I3_J,axiom,
    ! [X: fun_Pr6460944699146269935_nat_o,Xa2: fun_Pr6460944699146269935_nat_o,Xb: produc3658429121746597890et_nat] :
      ( ~ aa_Pro653082453696054616_nat_o(aa_fun2638059832795814268_nat_o(times_assn_raw(X),Xa2),Xb)
     => ~ ! [H: heap_e7401611519738050253t_unit,As3: set_nat] :
            ( ( Xb = produc7507926704131184380et_nat(H,As3) )
           => ? [As13: set_nat,As23: set_nat] :
                ( ( As3 = 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(Xa2,produc7507926704131184380et_nat(H,As23)) ) ) ) ).

% times_assn_raw.elims(3)
tff(fact_1155_times__assn__raw_Oelims_I2_J,axiom,
    ! [X: fun_Pr6460944699146269935_nat_o,Xa2: fun_Pr6460944699146269935_nat_o,Xb: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(aa_fun2638059832795814268_nat_o(times_assn_raw(X),Xa2),Xb)
     => ~ ! [H: heap_e7401611519738050253t_unit,As3: set_nat] :
            ( ( Xb = produc7507926704131184380et_nat(H,As3) )
           => ~ ? [As1: set_nat,As2: set_nat] :
                  ( ( As3 = 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(Xa2,produc7507926704131184380et_nat(H,As2)) ) ) ) ).

% times_assn_raw.elims(2)
tff(fact_1156_times__assn__raw_Oelims_I1_J,axiom,
    ! [X: fun_Pr6460944699146269935_nat_o,Xa2: fun_Pr6460944699146269935_nat_o,Xb: produc3658429121746597890et_nat,Y: $o] :
      ( ( aa_Pro653082453696054616_nat_o(aa_fun2638059832795814268_nat_o(times_assn_raw(X),Xa2),Xb)
      <=> (Y) )
     => ~ ! [H: heap_e7401611519738050253t_unit,As3: set_nat] :
            ( ( Xb = produc7507926704131184380et_nat(H,As3) )
           => ( (Y)
            <=> ~ ? [As12: set_nat,As22: set_nat] :
                    ( ( As3 = 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(Xa2,produc7507926704131184380et_nat(H,As22)) ) ) ) ) ).

% times_assn_raw.elims(1)
tff(fact_1157_times__assn__raw_Osimps,axiom,
    ! [Pa: fun_Pr6460944699146269935_nat_o,Qa: fun_Pr6460944699146269935_nat_o,Ha: heap_e7401611519738050253t_unit,Asa: set_nat] :
      ( aa_Pro653082453696054616_nat_o(aa_fun2638059832795814268_nat_o(times_assn_raw(Pa),Qa),produc7507926704131184380et_nat(Ha,Asa))
    <=> ? [As12: set_nat,As22: set_nat] :
          ( ( Asa = 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(Pa,produc7507926704131184380et_nat(Ha,As12))
          & aa_Pro653082453696054616_nat_o(Qa,produc7507926704131184380et_nat(Ha,As22)) ) ) ).

% times_assn_raw.simps
tff(fact_1158_dual__order_Orefl,axiom,
    ! [A: set_int] : aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A),A) ).

% dual_order.refl
tff(fact_1159_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_1160_dual__order_Orefl,axiom,
    ! [A: num] : aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),A) ).

% dual_order.refl
tff(fact_1161_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_1162_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_1163_order__refl,axiom,
    ! [X: set_int] : aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,X),X) ).

% order_refl
tff(fact_1164_order__refl,axiom,
    ! [X: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),X) ).

% order_refl
tff(fact_1165_order__refl,axiom,
    ! [X: num] : aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X),X) ).

% order_refl
tff(fact_1166_order__refl,axiom,
    ! [X: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),X) ).

% order_refl
tff(fact_1167_order__refl,axiom,
    ! [X: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),X) ).

% order_refl
tff(fact_1168_subset__empty,axiom,
    ! [A3: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A3),bot_bo2099793752762293965at_nat)
    <=> ( A3 = bot_bo2099793752762293965at_nat ) ) ).

% subset_empty
tff(fact_1169_subset__empty,axiom,
    ! [A3: set_o] :
      ( ord_less_eq_set_o(A3,bot_bot_set_o)
    <=> ( A3 = bot_bot_set_o ) ) ).

% subset_empty
tff(fact_1170_subset__empty,axiom,
    ! [A3: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A3),bot_bot_set_nat)
    <=> ( A3 = bot_bot_set_nat ) ) ).

% subset_empty
tff(fact_1171_subset__empty,axiom,
    ! [A3: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A3),bot_bot_set_int)
    <=> ( A3 = bot_bot_set_int ) ) ).

% subset_empty
tff(fact_1172_empty__subsetI,axiom,
    ! [A3: set_Pr1261947904930325089at_nat] : aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,bot_bo2099793752762293965at_nat),A3) ).

% empty_subsetI
tff(fact_1173_empty__subsetI,axiom,
    ! [A3: set_o] : ord_less_eq_set_o(bot_bot_set_o,A3) ).

% empty_subsetI
tff(fact_1174_empty__subsetI,axiom,
    ! [A3: set_nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,bot_bot_set_nat),A3) ).

% empty_subsetI
tff(fact_1175_empty__subsetI,axiom,
    ! [A3: set_int] : aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,bot_bot_set_int),A3) ).

% empty_subsetI
tff(fact_1176_Int__subset__iff,axiom,
    ! [C2: set_nat,A3: set_nat,B3: 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,A3),B3))
    <=> ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,C2),A3)
        & aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,C2),B3) ) ) ).

% Int_subset_iff
tff(fact_1177_Int__subset__iff,axiom,
    ! [C2: set_Pr1261947904930325089at_nat,A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,C2),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A3),B3))
    <=> ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,C2),A3)
        & aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,C2),B3) ) ) ).

% Int_subset_iff
tff(fact_1178_Int__subset__iff,axiom,
    ! [C2: set_int,A3: set_int,B3: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,C2),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A3),B3))
    <=> ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,C2),A3)
        & aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,C2),B3) ) ) ).

% Int_subset_iff
tff(fact_1179_Un__subset__iff,axiom,
    ! [A3: set_Pr8693737435421807431at_nat,B3: set_Pr8693737435421807431at_nat,C2: set_Pr8693737435421807431at_nat] :
      ( ord_le3000389064537975527at_nat(aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,A3),B3),C2)
    <=> ( ord_le3000389064537975527at_nat(A3,C2)
        & ord_le3000389064537975527at_nat(B3,C2) ) ) ).

% Un_subset_iff
tff(fact_1180_Un__subset__iff,axiom,
    ! [A3: set_Pr8551490117392284871at_nat,B3: set_Pr8551490117392284871at_nat,C2: set_Pr8551490117392284871at_nat] :
      ( ord_le8081472938463900775at_nat(aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A3),B3),C2)
    <=> ( ord_le8081472938463900775at_nat(A3,C2)
        & ord_le8081472938463900775at_nat(B3,C2) ) ) ).

% Un_subset_iff
tff(fact_1181_Un__subset__iff,axiom,
    ! [A3: set_Pr4329608150637261639at_nat,B3: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat] :
      ( ord_le1268244103169919719at_nat(aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A3),B3),C2)
    <=> ( ord_le1268244103169919719at_nat(A3,C2)
        & ord_le1268244103169919719at_nat(B3,C2) ) ) ).

% Un_subset_iff
tff(fact_1182_Un__subset__iff,axiom,
    ! [A3: set_nat,B3: 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,A3),B3)),C2)
    <=> ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A3),C2)
        & aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B3),C2) ) ) ).

% Un_subset_iff
tff(fact_1183_Un__subset__iff,axiom,
    ! [A3: set_int,B3: set_int,C2: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A3),B3)),C2)
    <=> ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A3),C2)
        & aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,B3),C2) ) ) ).

% Un_subset_iff
tff(fact_1184_less__eq__assn__def,axiom,
    ! [A: assn,B: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),B)
    <=> ( A = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),B) ) ) ).

% less_eq_assn_def
tff(fact_1185_Int__mono,axiom,
    ! [A3: set_nat,C2: set_nat,B3: set_nat,D3: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A3),C2)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B3),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,A3),B3)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,C2),D3)) ) ) ).

% Int_mono
tff(fact_1186_Int__mono,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat,D3: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A3),C2)
     => ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,B3),D3)
       => aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A3),B3)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,C2),D3)) ) ) ).

% Int_mono
tff(fact_1187_Int__mono,axiom,
    ! [A3: set_int,C2: set_int,B3: set_int,D3: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A3),C2)
     => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,B3),D3)
       => 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,A3),B3)),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,C2),D3)) ) ) ).

% Int_mono
tff(fact_1188_Int__lower1,axiom,
    ! [A3: set_nat,B3: 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,A3),B3)),A3) ).

% Int_lower1
tff(fact_1189_Int__lower1,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] : aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A3),B3)),A3) ).

% Int_lower1
tff(fact_1190_Int__lower1,axiom,
    ! [A3: set_int,B3: set_int] : 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,A3),B3)),A3) ).

% Int_lower1
tff(fact_1191_Int__lower2,axiom,
    ! [A3: set_nat,B3: 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,A3),B3)),B3) ).

% Int_lower2
tff(fact_1192_Int__lower2,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] : aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A3),B3)),B3) ).

% Int_lower2
tff(fact_1193_Int__lower2,axiom,
    ! [A3: set_int,B3: set_int] : 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,A3),B3)),B3) ).

% Int_lower2
tff(fact_1194_Int__absorb1,axiom,
    ! [B3: set_nat,A3: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B3),A3)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A3),B3) = B3 ) ) ).

% Int_absorb1
tff(fact_1195_Int__absorb1,axiom,
    ! [B3: set_Pr1261947904930325089at_nat,A3: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,B3),A3)
     => ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A3),B3) = B3 ) ) ).

% Int_absorb1
tff(fact_1196_Int__absorb1,axiom,
    ! [B3: set_int,A3: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,B3),A3)
     => ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A3),B3) = B3 ) ) ).

% Int_absorb1
tff(fact_1197_Int__absorb2,axiom,
    ! [A3: set_nat,B3: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A3),B3)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A3),B3) = A3 ) ) ).

% Int_absorb2
tff(fact_1198_Int__absorb2,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A3),B3)
     => ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A3),B3) = A3 ) ) ).

% Int_absorb2
tff(fact_1199_Int__absorb2,axiom,
    ! [A3: set_int,B3: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A3),B3)
     => ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A3),B3) = A3 ) ) ).

% Int_absorb2
tff(fact_1200_Int__greatest,axiom,
    ! [C2: set_nat,A3: set_nat,B3: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,C2),A3)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,C2),B3)
       => 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,A3),B3)) ) ) ).

% Int_greatest
tff(fact_1201_Int__greatest,axiom,
    ! [C2: set_Pr1261947904930325089at_nat,A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,C2),A3)
     => ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,C2),B3)
       => aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,C2),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A3),B3)) ) ) ).

% Int_greatest
tff(fact_1202_Int__greatest,axiom,
    ! [C2: set_int,A3: set_int,B3: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,C2),A3)
     => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,C2),B3)
       => aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,C2),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A3),B3)) ) ) ).

% Int_greatest
tff(fact_1203_Int__Collect__mono,axiom,
    ! [A3: set_o,B3: set_o,Pa: fun_o_o,Qa: fun_o_o] :
      ( ord_less_eq_set_o(A3,B3)
     => ( ! [X3: $o] :
            ( member_o((X3),A3)
           => ( aa_o_o(Pa,(X3))
             => aa_o_o(Qa,(X3)) ) )
       => ord_less_eq_set_o(inf_inf_set_o(A3,collect_o(Pa)),inf_inf_set_o(B3,collect_o(Qa))) ) ) ).

% Int_Collect_mono
tff(fact_1204_Int__Collect__mono,axiom,
    ! [A3: set_Pr958786334691620121nt_int,B3: set_Pr958786334691620121nt_int,Pa: fun_Pr3238374608701176162_int_o,Qa: fun_Pr3238374608701176162_int_o] :
      ( ord_le2843351958646193337nt_int(A3,B3)
     => ( ! [X3: product_prod_int_int] :
            ( member5262025264175285858nt_int(X3,A3)
           => ( aa_Pro4592068669816069595_int_o(Pa,X3)
             => aa_Pro4592068669816069595_int_o(Qa,X3) ) )
       => ord_le2843351958646193337nt_int(inf_in2269163501485487111nt_int(A3,collec213857154873943460nt_int(Pa)),inf_in2269163501485487111nt_int(B3,collec213857154873943460nt_int(Qa))) ) ) ).

% Int_Collect_mono
tff(fact_1205_Int__Collect__mono,axiom,
    ! [A3: set_list_nat,B3: set_list_nat,Pa: fun_list_nat_o,Qa: fun_list_nat_o] :
      ( ord_le6045566169113846134st_nat(A3,B3)
     => ( ! [X3: list_nat] :
            ( member_list_nat(X3,A3)
           => ( aa_list_nat_o(Pa,X3)
             => aa_list_nat_o(Qa,X3) ) )
       => ord_le6045566169113846134st_nat(inf_inf_set_list_nat(A3,collect_list_nat(Pa)),inf_inf_set_list_nat(B3,collect_list_nat(Qa))) ) ) ).

% Int_Collect_mono
tff(fact_1206_Int__Collect__mono,axiom,
    ! [A3: set_set_nat,B3: set_set_nat,Pa: fun_set_nat_o,Qa: fun_set_nat_o] :
      ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(A3),B3)
     => ( ! [X3: set_nat] :
            ( aa_set_set_nat_o(member_set_nat(X3),A3)
           => ( aa_set_nat_o(Pa,X3)
             => aa_set_nat_o(Qa,X3) ) )
       => aa_set_set_nat_o(ord_le6893508408891458716et_nat(inf_inf_set_set_nat(A3,collect_set_nat(Pa))),inf_inf_set_set_nat(B3,collect_set_nat(Qa))) ) ) ).

% Int_Collect_mono
tff(fact_1207_Int__Collect__mono,axiom,
    ! [A3: set_nat,B3: set_nat,Pa: fun_nat_o,Qa: fun_nat_o] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A3),B3)
     => ( ! [X3: nat] :
            ( aa_set_nat_o(member_nat(X3),A3)
           => ( aa_nat_o(Pa,X3)
             => aa_nat_o(Qa,X3) ) )
       => 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,A3),collect_nat(Pa))),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B3),collect_nat(Qa))) ) ) ).

% Int_Collect_mono
tff(fact_1208_Int__Collect__mono,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat,Pa: fun_Pr4796943543331071514_nat_o,Qa: fun_Pr4796943543331071514_nat_o] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A3),B3)
     => ( ! [X3: product_prod_nat_nat] :
            ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X3),A3)
           => ( aa_Pro6150637604445964947_nat_o(Pa,X3)
             => aa_Pro6150637604445964947_nat_o(Qa,X3) ) )
       => aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A3),collec3392354462482085612at_nat(Pa))),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B3),collec3392354462482085612at_nat(Qa))) ) ) ).

% Int_Collect_mono
tff(fact_1209_Int__Collect__mono,axiom,
    ! [A3: set_int,B3: set_int,Pa: fun_int_o,Qa: fun_int_o] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A3),B3)
     => ( ! [X3: int] :
            ( aa_set_int_o(member_int(X3),A3)
           => ( aa_int_o(Pa,X3)
             => aa_int_o(Qa,X3) ) )
       => 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,A3),collect_int(Pa))),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,B3),collect_int(Qa))) ) ) ).

% Int_Collect_mono
tff(fact_1210_Un__mono,axiom,
    ! [A3: set_Pr8693737435421807431at_nat,C2: set_Pr8693737435421807431at_nat,B3: set_Pr8693737435421807431at_nat,D3: set_Pr8693737435421807431at_nat] :
      ( ord_le3000389064537975527at_nat(A3,C2)
     => ( ord_le3000389064537975527at_nat(B3,D3)
       => ord_le3000389064537975527at_nat(aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,A3),B3),aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,C2),D3)) ) ) ).

% Un_mono
tff(fact_1211_Un__mono,axiom,
    ! [A3: set_Pr8551490117392284871at_nat,C2: set_Pr8551490117392284871at_nat,B3: set_Pr8551490117392284871at_nat,D3: set_Pr8551490117392284871at_nat] :
      ( ord_le8081472938463900775at_nat(A3,C2)
     => ( ord_le8081472938463900775at_nat(B3,D3)
       => ord_le8081472938463900775at_nat(aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A3),B3),aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,C2),D3)) ) ) ).

% Un_mono
tff(fact_1212_Un__mono,axiom,
    ! [A3: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat,B3: set_Pr4329608150637261639at_nat,D3: set_Pr4329608150637261639at_nat] :
      ( ord_le1268244103169919719at_nat(A3,C2)
     => ( ord_le1268244103169919719at_nat(B3,D3)
       => ord_le1268244103169919719at_nat(aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A3),B3),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,C2),D3)) ) ) ).

% Un_mono
tff(fact_1213_Un__mono,axiom,
    ! [A3: set_nat,C2: set_nat,B3: set_nat,D3: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A3),C2)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B3),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,A3),B3)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,C2),D3)) ) ) ).

% Un_mono
tff(fact_1214_Un__mono,axiom,
    ! [A3: set_int,C2: set_int,B3: set_int,D3: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A3),C2)
     => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,B3),D3)
       => aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A3),B3)),aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,C2),D3)) ) ) ).

% Un_mono
tff(fact_1215_Un__least,axiom,
    ! [A3: set_Pr8693737435421807431at_nat,C2: set_Pr8693737435421807431at_nat,B3: set_Pr8693737435421807431at_nat] :
      ( ord_le3000389064537975527at_nat(A3,C2)
     => ( ord_le3000389064537975527at_nat(B3,C2)
       => ord_le3000389064537975527at_nat(aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,A3),B3),C2) ) ) ).

% Un_least
tff(fact_1216_Un__least,axiom,
    ! [A3: set_Pr8551490117392284871at_nat,C2: set_Pr8551490117392284871at_nat,B3: set_Pr8551490117392284871at_nat] :
      ( ord_le8081472938463900775at_nat(A3,C2)
     => ( ord_le8081472938463900775at_nat(B3,C2)
       => ord_le8081472938463900775at_nat(aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A3),B3),C2) ) ) ).

% Un_least
tff(fact_1217_Un__least,axiom,
    ! [A3: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat,B3: set_Pr4329608150637261639at_nat] :
      ( ord_le1268244103169919719at_nat(A3,C2)
     => ( ord_le1268244103169919719at_nat(B3,C2)
       => ord_le1268244103169919719at_nat(aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A3),B3),C2) ) ) ).

% Un_least
tff(fact_1218_Un__least,axiom,
    ! [A3: set_nat,C2: set_nat,B3: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A3),C2)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B3),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,A3),B3)),C2) ) ) ).

% Un_least
tff(fact_1219_Un__least,axiom,
    ! [A3: set_int,C2: set_int,B3: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A3),C2)
     => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,B3),C2)
       => aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A3),B3)),C2) ) ) ).

% Un_least
tff(fact_1220_Un__upper1,axiom,
    ! [A3: set_Pr8693737435421807431at_nat,B3: set_Pr8693737435421807431at_nat] : ord_le3000389064537975527at_nat(A3,aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,A3),B3)) ).

% Un_upper1
tff(fact_1221_Un__upper1,axiom,
    ! [A3: set_Pr8551490117392284871at_nat,B3: set_Pr8551490117392284871at_nat] : ord_le8081472938463900775at_nat(A3,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A3),B3)) ).

% Un_upper1
tff(fact_1222_Un__upper1,axiom,
    ! [A3: set_Pr4329608150637261639at_nat,B3: set_Pr4329608150637261639at_nat] : ord_le1268244103169919719at_nat(A3,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A3),B3)) ).

% Un_upper1
tff(fact_1223_Un__upper1,axiom,
    ! [A3: set_nat,B3: set_nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A3),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A3),B3)) ).

% Un_upper1
tff(fact_1224_Un__upper1,axiom,
    ! [A3: set_int,B3: set_int] : aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A3),aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A3),B3)) ).

% Un_upper1
tff(fact_1225_Un__upper2,axiom,
    ! [B3: set_Pr8693737435421807431at_nat,A3: set_Pr8693737435421807431at_nat] : ord_le3000389064537975527at_nat(B3,aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,A3),B3)) ).

% Un_upper2
tff(fact_1226_Un__upper2,axiom,
    ! [B3: set_Pr8551490117392284871at_nat,A3: set_Pr8551490117392284871at_nat] : ord_le8081472938463900775at_nat(B3,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A3),B3)) ).

% Un_upper2
tff(fact_1227_Un__upper2,axiom,
    ! [B3: set_Pr4329608150637261639at_nat,A3: set_Pr4329608150637261639at_nat] : ord_le1268244103169919719at_nat(B3,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A3),B3)) ).

% Un_upper2
tff(fact_1228_Un__upper2,axiom,
    ! [B3: set_nat,A3: set_nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B3),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A3),B3)) ).

% Un_upper2
tff(fact_1229_Un__upper2,axiom,
    ! [B3: set_int,A3: set_int] : aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,B3),aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A3),B3)) ).

% Un_upper2
tff(fact_1230_Un__absorb1,axiom,
    ! [A3: set_Pr8693737435421807431at_nat,B3: set_Pr8693737435421807431at_nat] :
      ( ord_le3000389064537975527at_nat(A3,B3)
     => ( aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,A3),B3) = B3 ) ) ).

% Un_absorb1
tff(fact_1231_Un__absorb1,axiom,
    ! [A3: set_Pr8551490117392284871at_nat,B3: set_Pr8551490117392284871at_nat] :
      ( ord_le8081472938463900775at_nat(A3,B3)
     => ( aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A3),B3) = B3 ) ) ).

% Un_absorb1
tff(fact_1232_Un__absorb1,axiom,
    ! [A3: set_Pr4329608150637261639at_nat,B3: set_Pr4329608150637261639at_nat] :
      ( ord_le1268244103169919719at_nat(A3,B3)
     => ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A3),B3) = B3 ) ) ).

% Un_absorb1
tff(fact_1233_Un__absorb1,axiom,
    ! [A3: set_nat,B3: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A3),B3)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A3),B3) = B3 ) ) ).

% Un_absorb1
tff(fact_1234_Un__absorb1,axiom,
    ! [A3: set_int,B3: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A3),B3)
     => ( aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A3),B3) = B3 ) ) ).

% Un_absorb1
tff(fact_1235_Un__absorb2,axiom,
    ! [B3: set_Pr8693737435421807431at_nat,A3: set_Pr8693737435421807431at_nat] :
      ( ord_le3000389064537975527at_nat(B3,A3)
     => ( aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,A3),B3) = A3 ) ) ).

% Un_absorb2
tff(fact_1236_Un__absorb2,axiom,
    ! [B3: set_Pr8551490117392284871at_nat,A3: set_Pr8551490117392284871at_nat] :
      ( ord_le8081472938463900775at_nat(B3,A3)
     => ( aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A3),B3) = A3 ) ) ).

% Un_absorb2
tff(fact_1237_Un__absorb2,axiom,
    ! [B3: set_Pr4329608150637261639at_nat,A3: set_Pr4329608150637261639at_nat] :
      ( ord_le1268244103169919719at_nat(B3,A3)
     => ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A3),B3) = A3 ) ) ).

% Un_absorb2
tff(fact_1238_Un__absorb2,axiom,
    ! [B3: set_nat,A3: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B3),A3)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A3),B3) = A3 ) ) ).

% Un_absorb2
tff(fact_1239_Un__absorb2,axiom,
    ! [B3: set_int,A3: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,B3),A3)
     => ( aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A3),B3) = A3 ) ) ).

% Un_absorb2
tff(fact_1240_subset__UnE,axiom,
    ! [C2: set_Pr8693737435421807431at_nat,A3: set_Pr8693737435421807431at_nat,B3: set_Pr8693737435421807431at_nat] :
      ( ord_le3000389064537975527at_nat(C2,aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,A3),B3))
     => ~ ! [A5: set_Pr8693737435421807431at_nat] :
            ( ord_le3000389064537975527at_nat(A5,A3)
           => ! [B5: set_Pr8693737435421807431at_nat] :
                ( ord_le3000389064537975527at_nat(B5,B3)
               => ( C2 != aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,A5),B5) ) ) ) ) ).

% subset_UnE
tff(fact_1241_subset__UnE,axiom,
    ! [C2: set_Pr8551490117392284871at_nat,A3: set_Pr8551490117392284871at_nat,B3: set_Pr8551490117392284871at_nat] :
      ( ord_le8081472938463900775at_nat(C2,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A3),B3))
     => ~ ! [A5: set_Pr8551490117392284871at_nat] :
            ( ord_le8081472938463900775at_nat(A5,A3)
           => ! [B5: set_Pr8551490117392284871at_nat] :
                ( ord_le8081472938463900775at_nat(B5,B3)
               => ( C2 != aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A5),B5) ) ) ) ) ).

% subset_UnE
tff(fact_1242_subset__UnE,axiom,
    ! [C2: set_Pr4329608150637261639at_nat,A3: set_Pr4329608150637261639at_nat,B3: set_Pr4329608150637261639at_nat] :
      ( ord_le1268244103169919719at_nat(C2,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A3),B3))
     => ~ ! [A5: set_Pr4329608150637261639at_nat] :
            ( ord_le1268244103169919719at_nat(A5,A3)
           => ! [B5: set_Pr4329608150637261639at_nat] :
                ( ord_le1268244103169919719at_nat(B5,B3)
               => ( C2 != aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A5),B5) ) ) ) ) ).

% subset_UnE
tff(fact_1243_subset__UnE,axiom,
    ! [C2: set_nat,A3: set_nat,B3: 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,A3),B3))
     => ~ ! [A5: set_nat] :
            ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A5),A3)
           => ! [B5: set_nat] :
                ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B5),B3)
               => ( C2 != aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A5),B5) ) ) ) ) ).

% subset_UnE
tff(fact_1244_subset__UnE,axiom,
    ! [C2: set_int,A3: set_int,B3: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,C2),aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A3),B3))
     => ~ ! [A5: set_int] :
            ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A5),A3)
           => ! [B5: set_int] :
                ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,B5),B3)
               => ( C2 != aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A5),B5) ) ) ) ) ).

% subset_UnE
tff(fact_1245_subset__Un__eq,axiom,
    ! [A3: set_Pr8693737435421807431at_nat,B3: set_Pr8693737435421807431at_nat] :
      ( ord_le3000389064537975527at_nat(A3,B3)
    <=> ( aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,A3),B3) = B3 ) ) ).

% subset_Un_eq
tff(fact_1246_subset__Un__eq,axiom,
    ! [A3: set_Pr8551490117392284871at_nat,B3: set_Pr8551490117392284871at_nat] :
      ( ord_le8081472938463900775at_nat(A3,B3)
    <=> ( aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A3),B3) = B3 ) ) ).

% subset_Un_eq
tff(fact_1247_subset__Un__eq,axiom,
    ! [A3: set_Pr4329608150637261639at_nat,B3: set_Pr4329608150637261639at_nat] :
      ( ord_le1268244103169919719at_nat(A3,B3)
    <=> ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A3),B3) = B3 ) ) ).

% subset_Un_eq
tff(fact_1248_subset__Un__eq,axiom,
    ! [A3: set_nat,B3: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A3),B3)
    <=> ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A3),B3) = B3 ) ) ).

% subset_Un_eq
tff(fact_1249_subset__Un__eq,axiom,
    ! [A3: set_int,B3: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A3),B3)
    <=> ( aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A3),B3) = B3 ) ) ).

% subset_Un_eq
tff(fact_1250_relH__subset,axiom,
    ! [Bs: set_nat,Ha: heap_e7401611519738050253t_unit,H_a: heap_e7401611519738050253t_unit,Asa: set_nat] :
      ( aa_hea7994211679605476167unit_o(relH(Bs,Ha),H_a)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Asa),Bs)
       => aa_hea7994211679605476167unit_o(relH(Asa,Ha),H_a) ) ) ).

% relH_subset
tff(fact_1251_Un__Int__assoc__eq,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat] :
      ( ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A3),B3)),C2) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A3),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,B3),C2)) )
    <=> aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,C2),A3) ) ).

% Un_Int_assoc_eq
tff(fact_1252_Un__Int__assoc__eq,axiom,
    ! [A3: set_Pr8693737435421807431at_nat,B3: set_Pr8693737435421807431at_nat,C2: set_Pr8693737435421807431at_nat] :
      ( ( aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(inf_in4302113700860409141at_nat,A3),B3)),C2) = aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(inf_in4302113700860409141at_nat,A3),aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,B3),C2)) )
    <=> ord_le3000389064537975527at_nat(C2,A3) ) ).

% Un_Int_assoc_eq
tff(fact_1253_Un__Int__assoc__eq,axiom,
    ! [A3: set_Pr8551490117392284871at_nat,B3: 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,A3),B3)),C2) = aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(inf_in1697001100524423349at_nat,A3),aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,B3),C2)) )
    <=> ord_le8081472938463900775at_nat(C2,A3) ) ).

% Un_Int_assoc_eq
tff(fact_1254_Un__Int__assoc__eq,axiom,
    ! [A3: set_Pr4329608150637261639at_nat,B3: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat] :
      ( ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,A3),B3)),C2) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,A3),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,B3),C2)) )
    <=> ord_le1268244103169919719at_nat(C2,A3) ) ).

% Un_Int_assoc_eq
tff(fact_1255_Un__Int__assoc__eq,axiom,
    ! [A3: set_nat,B3: set_nat,C2: set_nat] :
      ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A3),B3)),C2) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A3),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,B3),C2)) )
    <=> aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,C2),A3) ) ).

% Un_Int_assoc_eq
tff(fact_1256_Un__Int__assoc__eq,axiom,
    ! [A3: set_int,B3: set_int,C2: set_int] :
      ( ( aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A3),B3)),C2) = aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A3),aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,B3),C2)) )
    <=> aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,C2),A3) ) ).

% Un_Int_assoc_eq
tff(fact_1257_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_1258_nle__le,axiom,
    ! [A: num,B: num] :
      ( ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),A)
        & ( B != A ) ) ) ).

% nle_le
tff(fact_1259_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_1260_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_1261_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_1262_le__cases3,axiom,
    ! [X: num,Y: num,Z: num] :
      ( ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X),Y)
       => ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Y),Z) )
     => ( ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Y),X)
         => ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X),Z) )
       => ( ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X),Z)
           => ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Z),Y) )
         => ( ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Z),Y)
             => ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Y),X) )
           => ( ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Y),Z)
               => ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Z),X) )
             => ~ ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Z),X)
                 => ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X),Y) ) ) ) ) ) ) ).

% le_cases3
tff(fact_1263_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_1264_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_1265_order__class_Oorder__eq__iff,axiom,
    ! [X: set_int,Y: set_int] :
      ( ( X = Y )
    <=> ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,X),Y)
        & aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,Y),X) ) ) ).

% order_class.order_eq_iff
tff(fact_1266_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_1267_order__class_Oorder__eq__iff,axiom,
    ! [X: num,Y: num] :
      ( ( X = Y )
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X),Y)
        & aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Y),X) ) ) ).

% order_class.order_eq_iff
tff(fact_1268_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_1269_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_1270_ord__eq__le__trans,axiom,
    ! [A: set_int,B: set_int,Ca: set_int] :
      ( ( A = B )
     => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,B),Ca)
       => aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A),Ca) ) ) ).

% ord_eq_le_trans
tff(fact_1271_ord__eq__le__trans,axiom,
    ! [A: rat,B: rat,Ca: rat] :
      ( ( A = B )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),Ca)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),Ca) ) ) ).

% ord_eq_le_trans
tff(fact_1272_ord__eq__le__trans,axiom,
    ! [A: num,B: num,Ca: num] :
      ( ( A = B )
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),Ca)
       => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),Ca) ) ) ).

% ord_eq_le_trans
tff(fact_1273_ord__eq__le__trans,axiom,
    ! [A: nat,B: nat,Ca: nat] :
      ( ( A = B )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),Ca)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),Ca) ) ) ).

% ord_eq_le_trans
tff(fact_1274_ord__eq__le__trans,axiom,
    ! [A: int,B: int,Ca: int] :
      ( ( A = B )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),Ca)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),Ca) ) ) ).

% ord_eq_le_trans
tff(fact_1275_ord__le__eq__trans,axiom,
    ! [A: set_int,B: set_int,Ca: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A),B)
     => ( ( B = Ca )
       => aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A),Ca) ) ) ).

% ord_le_eq_trans
tff(fact_1276_ord__le__eq__trans,axiom,
    ! [A: rat,B: rat,Ca: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( ( B = Ca )
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),Ca) ) ) ).

% ord_le_eq_trans
tff(fact_1277_ord__le__eq__trans,axiom,
    ! [A: num,B: num,Ca: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
     => ( ( B = Ca )
       => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),Ca) ) ) ).

% ord_le_eq_trans
tff(fact_1278_ord__le__eq__trans,axiom,
    ! [A: nat,B: nat,Ca: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( ( B = Ca )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),Ca) ) ) ).

% ord_le_eq_trans
tff(fact_1279_ord__le__eq__trans,axiom,
    ! [A: int,B: int,Ca: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( ( B = Ca )
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),Ca) ) ) ).

% ord_le_eq_trans
tff(fact_1280_order__antisym,axiom,
    ! [X: set_int,Y: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,X),Y)
     => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,Y),X)
       => ( X = Y ) ) ) ).

% order_antisym
tff(fact_1281_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_1282_order__antisym,axiom,
    ! [X: num,Y: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X),Y)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Y),X)
       => ( X = Y ) ) ) ).

% order_antisym
tff(fact_1283_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_1284_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_1285_order_Otrans,axiom,
    ! [A: set_int,B: set_int,Ca: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A),B)
     => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,B),Ca)
       => aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A),Ca) ) ) ).

% order.trans
tff(fact_1286_order_Otrans,axiom,
    ! [A: rat,B: rat,Ca: 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),Ca)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),Ca) ) ) ).

% order.trans
tff(fact_1287_order_Otrans,axiom,
    ! [A: num,B: num,Ca: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),Ca)
       => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),Ca) ) ) ).

% order.trans
tff(fact_1288_order_Otrans,axiom,
    ! [A: nat,B: nat,Ca: 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),Ca)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),Ca) ) ) ).

% order.trans
tff(fact_1289_order_Otrans,axiom,
    ! [A: int,B: int,Ca: 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),Ca)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),Ca) ) ) ).

% order.trans
tff(fact_1290_order__trans,axiom,
    ! [X: set_int,Y: set_int,Z: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,X),Y)
     => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,Y),Z)
       => aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,X),Z) ) ) ).

% order_trans
tff(fact_1291_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_1292_order__trans,axiom,
    ! [X: num,Y: num,Z: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X),Y)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Y),Z)
       => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X),Z) ) ) ).

% order_trans
tff(fact_1293_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_1294_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_1295_linorder__wlog,axiom,
    ! [Pa: 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(Pa,A4),B4) )
     => ( ! [A4: rat,B4: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(Pa,B4),A4)
           => aa_rat_o(aa_rat_fun_rat_o(Pa,A4),B4) )
       => aa_rat_o(aa_rat_fun_rat_o(Pa,A),B) ) ) ).

% linorder_wlog
tff(fact_1296_linorder__wlog,axiom,
    ! [Pa: fun_num_fun_num_o,A: num,B: num] :
      ( ! [A4: num,B4: num] :
          ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A4),B4)
         => aa_num_o(aa_num_fun_num_o(Pa,A4),B4) )
     => ( ! [A4: num,B4: num] :
            ( aa_num_o(aa_num_fun_num_o(Pa,B4),A4)
           => aa_num_o(aa_num_fun_num_o(Pa,A4),B4) )
       => aa_num_o(aa_num_fun_num_o(Pa,A),B) ) ) ).

% linorder_wlog
tff(fact_1297_linorder__wlog,axiom,
    ! [Pa: 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(Pa,A4),B4) )
     => ( ! [A4: nat,B4: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(Pa,B4),A4)
           => aa_nat_o(aa_nat_fun_nat_o(Pa,A4),B4) )
       => aa_nat_o(aa_nat_fun_nat_o(Pa,A),B) ) ) ).

% linorder_wlog
tff(fact_1298_linorder__wlog,axiom,
    ! [Pa: 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(Pa,A4),B4) )
     => ( ! [A4: int,B4: int] :
            ( aa_int_o(aa_int_fun_int_o(Pa,B4),A4)
           => aa_int_o(aa_int_fun_int_o(Pa,A4),B4) )
       => aa_int_o(aa_int_fun_int_o(Pa,A),B) ) ) ).

% linorder_wlog
tff(fact_1299_dual__order_Oeq__iff,axiom,
    ! [A: set_int,B: set_int] :
      ( ( A = B )
    <=> ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,B),A)
        & aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A),B) ) ) ).

% dual_order.eq_iff
tff(fact_1300_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_1301_dual__order_Oeq__iff,axiom,
    ! [A: num,B: num] :
      ( ( A = B )
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),A)
        & aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B) ) ) ).

% dual_order.eq_iff
tff(fact_1302_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_1303_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_1304_dual__order_Oantisym,axiom,
    ! [B: set_int,A: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,B),A)
     => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A),B)
       => ( A = B ) ) ) ).

% dual_order.antisym
tff(fact_1305_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_1306_dual__order_Oantisym,axiom,
    ! [B: num,A: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),A)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
       => ( A = B ) ) ) ).

% dual_order.antisym
tff(fact_1307_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_1308_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_1309_dual__order_Otrans,axiom,
    ! [B: set_int,A: set_int,Ca: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,B),A)
     => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,Ca),B)
       => aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,Ca),A) ) ) ).

% dual_order.trans
tff(fact_1310_dual__order_Otrans,axiom,
    ! [B: rat,A: rat,Ca: 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,Ca),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Ca),A) ) ) ).

% dual_order.trans
tff(fact_1311_dual__order_Otrans,axiom,
    ! [B: num,A: num,Ca: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),A)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Ca),B)
       => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Ca),A) ) ) ).

% dual_order.trans
tff(fact_1312_dual__order_Otrans,axiom,
    ! [B: nat,A: nat,Ca: 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,Ca),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ca),A) ) ) ).

% dual_order.trans
tff(fact_1313_dual__order_Otrans,axiom,
    ! [B: int,A: int,Ca: 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,Ca),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ca),A) ) ) ).

% dual_order.trans
tff(fact_1314_antisym,axiom,
    ! [A: set_int,B: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A),B)
     => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,B),A)
       => ( A = B ) ) ) ).

% antisym
tff(fact_1315_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_1316_antisym,axiom,
    ! [A: num,B: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),A)
       => ( A = B ) ) ) ).

% antisym
tff(fact_1317_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_1318_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_1319_Orderings_Oorder__eq__iff,axiom,
    ! [A: set_int,B: set_int] :
      ( ( A = B )
    <=> ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A),B)
        & aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,B),A) ) ) ).

% Orderings.order_eq_iff
tff(fact_1320_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_1321_Orderings_Oorder__eq__iff,axiom,
    ! [A: num,B: num] :
      ( ( A = B )
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
        & aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),A) ) ) ).

% Orderings.order_eq_iff
tff(fact_1322_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_1323_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_1324_order__subst1,axiom,
    ! [A: rat,F: fun_rat_rat,B: rat,Ca: 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),Ca)
       => ( ! [X3: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(F,X3)),aa_rat_rat(F,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(F,Ca)) ) ) ) ).

% order_subst1
tff(fact_1325_order__subst1,axiom,
    ! [A: rat,F: fun_num_rat,B: num,Ca: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_num_rat(F,B))
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),Ca)
       => ( ! [X3: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(F,X3)),aa_num_rat(F,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_num_rat(F,Ca)) ) ) ) ).

% order_subst1
tff(fact_1326_order__subst1,axiom,
    ! [A: rat,F: fun_nat_rat,B: nat,Ca: 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),Ca)
       => ( ! [X3: nat,Y3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,X3)),aa_nat_rat(F,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_nat_rat(F,Ca)) ) ) ) ).

% order_subst1
tff(fact_1327_order__subst1,axiom,
    ! [A: rat,F: fun_int_rat,B: int,Ca: 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),Ca)
       => ( ! [X3: int,Y3: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X3),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(F,X3)),aa_int_rat(F,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_int_rat(F,Ca)) ) ) ) ).

% order_subst1
tff(fact_1328_order__subst1,axiom,
    ! [A: num,F: fun_rat_num,B: rat,Ca: rat] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),aa_rat_num(F,B))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),Ca)
       => ( ! [X3: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y3)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_rat_num(F,X3)),aa_rat_num(F,Y3)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),aa_rat_num(F,Ca)) ) ) ) ).

% order_subst1
tff(fact_1329_order__subst1,axiom,
    ! [A: num,F: fun_num_num,B: num,Ca: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),aa_num_num(F,B))
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),Ca)
       => ( ! [X3: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Y3)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(F,X3)),aa_num_num(F,Y3)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),aa_num_num(F,Ca)) ) ) ) ).

% order_subst1
tff(fact_1330_order__subst1,axiom,
    ! [A: num,F: fun_nat_num,B: nat,Ca: nat] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),aa_nat_num(F,B))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),Ca)
       => ( ! [X3: nat,Y3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),Y3)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(F,X3)),aa_nat_num(F,Y3)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),aa_nat_num(F,Ca)) ) ) ) ).

% order_subst1
tff(fact_1331_order__subst1,axiom,
    ! [A: num,F: fun_int_num,B: int,Ca: int] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),aa_int_num(F,B))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),Ca)
       => ( ! [X3: int,Y3: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X3),Y3)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_int_num(F,X3)),aa_int_num(F,Y3)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),aa_int_num(F,Ca)) ) ) ) ).

% order_subst1
tff(fact_1332_order__subst1,axiom,
    ! [A: nat,F: fun_rat_nat,B: rat,Ca: 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),Ca)
       => ( ! [X3: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y3)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_rat_nat(F,X3)),aa_rat_nat(F,Y3)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),aa_rat_nat(F,Ca)) ) ) ) ).

% order_subst1
tff(fact_1333_order__subst1,axiom,
    ! [A: nat,F: fun_num_nat,B: num,Ca: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),aa_num_nat(F,B))
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),Ca)
       => ( ! [X3: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Y3)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(F,X3)),aa_num_nat(F,Y3)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),aa_num_nat(F,Ca)) ) ) ) ).

% order_subst1
tff(fact_1334_order__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_rat,Ca: 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)),Ca)
       => ( ! [X3: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(F,X3)),aa_rat_rat(F,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(F,A)),Ca) ) ) ) ).

% order_subst2
tff(fact_1335_order__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_num,Ca: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_rat_num(F,B)),Ca)
       => ( ! [X3: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y3)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_rat_num(F,X3)),aa_rat_num(F,Y3)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_rat_num(F,A)),Ca) ) ) ) ).

% order_subst2
tff(fact_1336_order__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_nat,Ca: 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)),Ca)
       => ( ! [X3: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y3)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_rat_nat(F,X3)),aa_rat_nat(F,Y3)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_rat_nat(F,A)),Ca) ) ) ) ).

% order_subst2
tff(fact_1337_order__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_int,Ca: 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)),Ca)
       => ( ! [X3: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y3)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(F,X3)),aa_rat_int(F,Y3)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(F,A)),Ca) ) ) ) ).

% order_subst2
tff(fact_1338_order__subst2,axiom,
    ! [A: num,B: num,F: fun_num_rat,Ca: rat] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(F,B)),Ca)
       => ( ! [X3: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(F,X3)),aa_num_rat(F,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(F,A)),Ca) ) ) ) ).

% order_subst2
tff(fact_1339_order__subst2,axiom,
    ! [A: num,B: num,F: fun_num_num,Ca: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(F,B)),Ca)
       => ( ! [X3: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Y3)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(F,X3)),aa_num_num(F,Y3)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(F,A)),Ca) ) ) ) ).

% order_subst2
tff(fact_1340_order__subst2,axiom,
    ! [A: num,B: num,F: fun_num_nat,Ca: nat] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(F,B)),Ca)
       => ( ! [X3: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Y3)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(F,X3)),aa_num_nat(F,Y3)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(F,A)),Ca) ) ) ) ).

% order_subst2
tff(fact_1341_order__subst2,axiom,
    ! [A: num,B: num,F: fun_num_int,Ca: int] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(F,B)),Ca)
       => ( ! [X3: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Y3)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(F,X3)),aa_num_int(F,Y3)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(F,A)),Ca) ) ) ) ).

% order_subst2
tff(fact_1342_order__subst2,axiom,
    ! [A: nat,B: nat,F: fun_nat_rat,Ca: 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)),Ca)
       => ( ! [X3: nat,Y3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,X3)),aa_nat_rat(F,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,A)),Ca) ) ) ) ).

% order_subst2
tff(fact_1343_order__subst2,axiom,
    ! [A: nat,B: nat,F: fun_nat_num,Ca: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(F,B)),Ca)
       => ( ! [X3: nat,Y3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),Y3)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(F,X3)),aa_nat_num(F,Y3)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(F,A)),Ca) ) ) ) ).

% order_subst2
tff(fact_1344_order__eq__refl,axiom,
    ! [X: set_int,Y: set_int] :
      ( ( X = Y )
     => aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,X),Y) ) ).

% order_eq_refl
tff(fact_1345_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_1346_order__eq__refl,axiom,
    ! [X: num,Y: num] :
      ( ( X = Y )
     => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X),Y) ) ).

% order_eq_refl
tff(fact_1347_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_1348_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_1349_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_1350_linorder__linear,axiom,
    ! [X: num,Y: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X),Y)
      | aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Y),X) ) ).

% linorder_linear
tff(fact_1351_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_1352_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_1353_ord__eq__le__subst,axiom,
    ! [A: rat,F: fun_rat_rat,B: rat,Ca: rat] :
      ( ( A = aa_rat_rat(F,B) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),Ca)
       => ( ! [X3: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(F,X3)),aa_rat_rat(F,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(F,Ca)) ) ) ) ).

% ord_eq_le_subst
tff(fact_1354_ord__eq__le__subst,axiom,
    ! [A: num,F: fun_rat_num,B: rat,Ca: rat] :
      ( ( A = aa_rat_num(F,B) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),Ca)
       => ( ! [X3: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y3)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_rat_num(F,X3)),aa_rat_num(F,Y3)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),aa_rat_num(F,Ca)) ) ) ) ).

% ord_eq_le_subst
tff(fact_1355_ord__eq__le__subst,axiom,
    ! [A: nat,F: fun_rat_nat,B: rat,Ca: rat] :
      ( ( A = aa_rat_nat(F,B) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),Ca)
       => ( ! [X3: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y3)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_rat_nat(F,X3)),aa_rat_nat(F,Y3)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),aa_rat_nat(F,Ca)) ) ) ) ).

% ord_eq_le_subst
tff(fact_1356_ord__eq__le__subst,axiom,
    ! [A: int,F: fun_rat_int,B: rat,Ca: rat] :
      ( ( A = aa_rat_int(F,B) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),Ca)
       => ( ! [X3: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y3)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(F,X3)),aa_rat_int(F,Y3)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_rat_int(F,Ca)) ) ) ) ).

% ord_eq_le_subst
tff(fact_1357_ord__eq__le__subst,axiom,
    ! [A: rat,F: fun_num_rat,B: num,Ca: num] :
      ( ( A = aa_num_rat(F,B) )
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),Ca)
       => ( ! [X3: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(F,X3)),aa_num_rat(F,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_num_rat(F,Ca)) ) ) ) ).

% ord_eq_le_subst
tff(fact_1358_ord__eq__le__subst,axiom,
    ! [A: num,F: fun_num_num,B: num,Ca: num] :
      ( ( A = aa_num_num(F,B) )
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),Ca)
       => ( ! [X3: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Y3)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(F,X3)),aa_num_num(F,Y3)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),aa_num_num(F,Ca)) ) ) ) ).

% ord_eq_le_subst
tff(fact_1359_ord__eq__le__subst,axiom,
    ! [A: nat,F: fun_num_nat,B: num,Ca: num] :
      ( ( A = aa_num_nat(F,B) )
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),Ca)
       => ( ! [X3: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Y3)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(F,X3)),aa_num_nat(F,Y3)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),aa_num_nat(F,Ca)) ) ) ) ).

% ord_eq_le_subst
tff(fact_1360_ord__eq__le__subst,axiom,
    ! [A: int,F: fun_num_int,B: num,Ca: num] :
      ( ( A = aa_num_int(F,B) )
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),Ca)
       => ( ! [X3: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Y3)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(F,X3)),aa_num_int(F,Y3)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_num_int(F,Ca)) ) ) ) ).

% ord_eq_le_subst
tff(fact_1361_ord__eq__le__subst,axiom,
    ! [A: rat,F: fun_nat_rat,B: nat,Ca: nat] :
      ( ( A = aa_nat_rat(F,B) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),Ca)
       => ( ! [X3: nat,Y3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,X3)),aa_nat_rat(F,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_nat_rat(F,Ca)) ) ) ) ).

% ord_eq_le_subst
tff(fact_1362_ord__eq__le__subst,axiom,
    ! [A: num,F: fun_nat_num,B: nat,Ca: nat] :
      ( ( A = aa_nat_num(F,B) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),Ca)
       => ( ! [X3: nat,Y3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),Y3)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(F,X3)),aa_nat_num(F,Y3)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),aa_nat_num(F,Ca)) ) ) ) ).

% ord_eq_le_subst
tff(fact_1363_ord__le__eq__subst,axiom,
    ! [A: rat,B: rat,F: fun_rat_rat,Ca: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( ( aa_rat_rat(F,B) = Ca )
       => ( ! [X3: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(F,X3)),aa_rat_rat(F,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(F,A)),Ca) ) ) ) ).

% ord_le_eq_subst
tff(fact_1364_ord__le__eq__subst,axiom,
    ! [A: rat,B: rat,F: fun_rat_num,Ca: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( ( aa_rat_num(F,B) = Ca )
       => ( ! [X3: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y3)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_rat_num(F,X3)),aa_rat_num(F,Y3)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_rat_num(F,A)),Ca) ) ) ) ).

% ord_le_eq_subst
tff(fact_1365_ord__le__eq__subst,axiom,
    ! [A: rat,B: rat,F: fun_rat_nat,Ca: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( ( aa_rat_nat(F,B) = Ca )
       => ( ! [X3: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y3)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_rat_nat(F,X3)),aa_rat_nat(F,Y3)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_rat_nat(F,A)),Ca) ) ) ) ).

% ord_le_eq_subst
tff(fact_1366_ord__le__eq__subst,axiom,
    ! [A: rat,B: rat,F: fun_rat_int,Ca: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( ( aa_rat_int(F,B) = Ca )
       => ( ! [X3: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y3)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(F,X3)),aa_rat_int(F,Y3)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(F,A)),Ca) ) ) ) ).

% ord_le_eq_subst
tff(fact_1367_ord__le__eq__subst,axiom,
    ! [A: num,B: num,F: fun_num_rat,Ca: rat] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
     => ( ( aa_num_rat(F,B) = Ca )
       => ( ! [X3: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(F,X3)),aa_num_rat(F,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(F,A)),Ca) ) ) ) ).

% ord_le_eq_subst
tff(fact_1368_ord__le__eq__subst,axiom,
    ! [A: num,B: num,F: fun_num_num,Ca: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
     => ( ( aa_num_num(F,B) = Ca )
       => ( ! [X3: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Y3)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(F,X3)),aa_num_num(F,Y3)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(F,A)),Ca) ) ) ) ).

% ord_le_eq_subst
tff(fact_1369_ord__le__eq__subst,axiom,
    ! [A: num,B: num,F: fun_num_nat,Ca: nat] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
     => ( ( aa_num_nat(F,B) = Ca )
       => ( ! [X3: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Y3)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(F,X3)),aa_num_nat(F,Y3)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(F,A)),Ca) ) ) ) ).

% ord_le_eq_subst
tff(fact_1370_ord__le__eq__subst,axiom,
    ! [A: num,B: num,F: fun_num_int,Ca: int] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
     => ( ( aa_num_int(F,B) = Ca )
       => ( ! [X3: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Y3)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(F,X3)),aa_num_int(F,Y3)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(F,A)),Ca) ) ) ) ).

% ord_le_eq_subst
tff(fact_1371_ord__le__eq__subst,axiom,
    ! [A: nat,B: nat,F: fun_nat_rat,Ca: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( ( aa_nat_rat(F,B) = Ca )
       => ( ! [X3: nat,Y3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,X3)),aa_nat_rat(F,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,A)),Ca) ) ) ) ).

% ord_le_eq_subst
tff(fact_1372_ord__le__eq__subst,axiom,
    ! [A: nat,B: nat,F: fun_nat_num,Ca: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( ( aa_nat_num(F,B) = Ca )
       => ( ! [X3: nat,Y3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),Y3)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(F,X3)),aa_nat_num(F,Y3)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(F,A)),Ca) ) ) ) ).

% ord_le_eq_subst
tff(fact_1373_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_1374_linorder__le__cases,axiom,
    ! [X: num,Y: num] :
      ( ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X),Y)
     => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Y),X) ) ).

% linorder_le_cases
tff(fact_1375_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_1376_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_1377_order__antisym__conv,axiom,
    ! [Y: set_int,X: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,Y),X)
     => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,X),Y)
      <=> ( X = Y ) ) ) ).

% order_antisym_conv
tff(fact_1378_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_1379_order__antisym__conv,axiom,
    ! [Y: num,X: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Y),X)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X),Y)
      <=> ( X = Y ) ) ) ).

% order_antisym_conv
tff(fact_1380_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_1381_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_1382_boolean__algebra__cancel_Oinf1,axiom,
    ! [A3: set_nat,K: set_nat,A: set_nat,B: set_nat] :
      ( ( A3 = 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,A3),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_1383_boolean__algebra__cancel_Oinf1,axiom,
    ! [A3: assn,K: assn,A: assn,B: assn] :
      ( ( A3 = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,K),A) )
     => ( aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A3),B) = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,K),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),B)) ) ) ).

% boolean_algebra_cancel.inf1
tff(fact_1384_boolean__algebra__cancel_Oinf1,axiom,
    ! [A3: nat,K: nat,A: nat,B: nat] :
      ( ( A3 = aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,K),A) )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A3),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_1385_boolean__algebra__cancel_Oinf1,axiom,
    ! [A3: product_unit,K: product_unit,A: product_unit,B: product_unit] :
      ( ( A3 = aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,K),A) )
     => ( aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,A3),B) = aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,K),aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,A),B)) ) ) ).

% boolean_algebra_cancel.inf1
tff(fact_1386_boolean__algebra__cancel_Oinf1,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,K: set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( A3 = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,K),A) )
     => ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A3),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_1387_boolean__algebra__cancel_Oinf2,axiom,
    ! [B3: set_nat,K: set_nat,B: set_nat,A: set_nat] :
      ( ( B3 = 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),B3) = 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_1388_boolean__algebra__cancel_Oinf2,axiom,
    ! [B3: assn,K: assn,B: assn,A: assn] :
      ( ( B3 = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,K),B) )
     => ( aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),B3) = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,K),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),B)) ) ) ).

% boolean_algebra_cancel.inf2
tff(fact_1389_boolean__algebra__cancel_Oinf2,axiom,
    ! [B3: nat,K: nat,B: nat,A: nat] :
      ( ( B3 = 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),B3) = 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_1390_boolean__algebra__cancel_Oinf2,axiom,
    ! [B3: product_unit,K: product_unit,B: product_unit,A: product_unit] :
      ( ( B3 = aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,K),B) )
     => ( aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,A),B3) = aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,K),aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,A),B)) ) ) ).

% boolean_algebra_cancel.inf2
tff(fact_1391_boolean__algebra__cancel_Oinf2,axiom,
    ! [B3: set_Pr1261947904930325089at_nat,K: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat] :
      ( ( B3 = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,K),B) )
     => ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B3) = 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_1392_boolean__algebra__cancel_Osup1,axiom,
    ! [A3: set_Pr8693737435421807431at_nat,K: set_Pr8693737435421807431at_nat,A: set_Pr8693737435421807431at_nat,B: set_Pr8693737435421807431at_nat] :
      ( ( A3 = aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,K),A) )
     => ( aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,A3),B) = aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,K),aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,A),B)) ) ) ).

% boolean_algebra_cancel.sup1
tff(fact_1393_boolean__algebra__cancel_Osup1,axiom,
    ! [A3: set_Pr8551490117392284871at_nat,K: set_Pr8551490117392284871at_nat,A: set_Pr8551490117392284871at_nat,B: set_Pr8551490117392284871at_nat] :
      ( ( A3 = aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,K),A) )
     => ( aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A3),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_1394_boolean__algebra__cancel_Osup1,axiom,
    ! [A3: set_Pr4329608150637261639at_nat,K: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
      ( ( A3 = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,K),A) )
     => ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A3),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_1395_boolean__algebra__cancel_Osup1,axiom,
    ! [A3: set_nat,K: set_nat,A: set_nat,B: set_nat] :
      ( ( A3 = 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,A3),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_1396_boolean__algebra__cancel_Osup1,axiom,
    ! [A3: nat,K: nat,A: nat,B: nat] :
      ( ( A3 = aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,K),A) )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A3),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_1397_boolean__algebra__cancel_Osup2,axiom,
    ! [B3: set_Pr8693737435421807431at_nat,K: set_Pr8693737435421807431at_nat,B: set_Pr8693737435421807431at_nat,A: set_Pr8693737435421807431at_nat] :
      ( ( B3 = aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,K),B) )
     => ( aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,A),B3) = aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,K),aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,A),B)) ) ) ).

% boolean_algebra_cancel.sup2
tff(fact_1398_boolean__algebra__cancel_Osup2,axiom,
    ! [B3: set_Pr8551490117392284871at_nat,K: set_Pr8551490117392284871at_nat,B: set_Pr8551490117392284871at_nat,A: set_Pr8551490117392284871at_nat] :
      ( ( B3 = aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,K),B) )
     => ( aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A),B3) = 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_1399_boolean__algebra__cancel_Osup2,axiom,
    ! [B3: set_Pr4329608150637261639at_nat,K: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat] :
      ( ( B3 = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,K),B) )
     => ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),B3) = 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_1400_boolean__algebra__cancel_Osup2,axiom,
    ! [B3: set_nat,K: set_nat,B: set_nat,A: set_nat] :
      ( ( B3 = 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),B3) = 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_1401_boolean__algebra__cancel_Osup2,axiom,
    ! [B3: nat,K: nat,B: nat,A: nat] :
      ( ( B3 = 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),B3) = 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_1402_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_1403_bot_Oextremum,axiom,
    ! [A: set_o] : ord_less_eq_set_o(bot_bot_set_o,A) ).

% bot.extremum
tff(fact_1404_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_1405_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_1406_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_1407_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_1408_bot_Oextremum__unique,axiom,
    ! [A: set_o] :
      ( ord_less_eq_set_o(A,bot_bot_set_o)
    <=> ( A = bot_bot_set_o ) ) ).

% bot.extremum_unique
tff(fact_1409_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_1410_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_1411_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_1412_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_1413_bot_Oextremum__uniqueI,axiom,
    ! [A: set_o] :
      ( ord_less_eq_set_o(A,bot_bot_set_o)
     => ( A = bot_bot_set_o ) ) ).

% bot.extremum_uniqueI
tff(fact_1414_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_1415_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_1416_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_1417_boolean__algebra_Odisj__zero__right,axiom,
    ! [X: set_Pr8693737435421807431at_nat] : aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,X),bot_bo5327735625951526323at_nat) = X ).

% boolean_algebra.disj_zero_right
tff(fact_1418_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_1419_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_1420_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_1421_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_1422_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_1423_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_1424_boolean__algebra_Oconj__disj__distrib,axiom,
    ! [X: assn,Y: assn,Z: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Y),Z)) = aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),Y)),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),Z)) ).

% boolean_algebra.conj_disj_distrib
tff(fact_1425_boolean__algebra_Oconj__disj__distrib,axiom,
    ! [X: product_unit,Y: product_unit,Z: product_unit] : aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,X),aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(sup_sup_Product_unit,Y),Z)) = aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(sup_sup_Product_unit,aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,X),Y)),aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,X),Z)) ).

% boolean_algebra.conj_disj_distrib
tff(fact_1426_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_1427_boolean__algebra_Oconj__disj__distrib,axiom,
    ! [X: set_Pr8693737435421807431at_nat,Y: set_Pr8693737435421807431at_nat,Z: set_Pr8693737435421807431at_nat] : aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(inf_in4302113700860409141at_nat,X),aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,Y),Z)) = aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(inf_in4302113700860409141at_nat,X),Y)),aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(inf_in4302113700860409141at_nat,X),Z)) ).

% boolean_algebra.conj_disj_distrib
tff(fact_1428_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_1429_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_1430_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_1431_boolean__algebra_Odisj__conj__distrib,axiom,
    ! [X: assn,Y: assn,Z: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Y),Z)) = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),Y)),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),Z)) ).

% boolean_algebra.disj_conj_distrib
tff(fact_1432_boolean__algebra_Odisj__conj__distrib,axiom,
    ! [X: product_unit,Y: product_unit,Z: product_unit] : aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(sup_sup_Product_unit,X),aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,Y),Z)) = aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(sup_sup_Product_unit,X),Y)),aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(sup_sup_Product_unit,X),Z)) ).

% boolean_algebra.disj_conj_distrib
tff(fact_1433_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_1434_boolean__algebra_Odisj__conj__distrib,axiom,
    ! [X: set_Pr8693737435421807431at_nat,Y: set_Pr8693737435421807431at_nat,Z: set_Pr8693737435421807431at_nat] : aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,X),aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(inf_in4302113700860409141at_nat,Y),Z)) = aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(inf_in4302113700860409141at_nat,aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,X),Y)),aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,X),Z)) ).

% boolean_algebra.disj_conj_distrib
tff(fact_1435_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_1436_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_1437_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_1438_boolean__algebra_Oconj__disj__distrib2,axiom,
    ! [Y: assn,Z: assn,X: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Y),Z)),X) = aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Y),X)),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Z),X)) ).

% boolean_algebra.conj_disj_distrib2
tff(fact_1439_boolean__algebra_Oconj__disj__distrib2,axiom,
    ! [Y: product_unit,Z: product_unit,X: product_unit] : aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(sup_sup_Product_unit,Y),Z)),X) = aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(sup_sup_Product_unit,aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,Y),X)),aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,Z),X)) ).

% boolean_algebra.conj_disj_distrib2
tff(fact_1440_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_1441_boolean__algebra_Oconj__disj__distrib2,axiom,
    ! [Y: set_Pr8693737435421807431at_nat,Z: set_Pr8693737435421807431at_nat,X: set_Pr8693737435421807431at_nat] : aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(inf_in4302113700860409141at_nat,aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,Y),Z)),X) = aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(inf_in4302113700860409141at_nat,Y),X)),aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(inf_in4302113700860409141at_nat,Z),X)) ).

% boolean_algebra.conj_disj_distrib2
tff(fact_1442_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_1443_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_1444_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_1445_boolean__algebra_Odisj__conj__distrib2,axiom,
    ! [Y: assn,Z: assn,X: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Y),Z)),X) = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Y),X)),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Z),X)) ).

% boolean_algebra.disj_conj_distrib2
tff(fact_1446_boolean__algebra_Odisj__conj__distrib2,axiom,
    ! [Y: product_unit,Z: product_unit,X: product_unit] : aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(sup_sup_Product_unit,aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,Y),Z)),X) = aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(sup_sup_Product_unit,Y),X)),aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(sup_sup_Product_unit,Z),X)) ).

% boolean_algebra.disj_conj_distrib2
tff(fact_1447_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_1448_boolean__algebra_Odisj__conj__distrib2,axiom,
    ! [Y: set_Pr8693737435421807431at_nat,Z: set_Pr8693737435421807431at_nat,X: set_Pr8693737435421807431at_nat] : aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(inf_in4302113700860409141at_nat,Y),Z)),X) = aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(inf_in4302113700860409141at_nat,aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,Y),X)),aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,Z),X)) ).

% boolean_algebra.disj_conj_distrib2
tff(fact_1449_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_1450_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_1451_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_1452_sup__Some,axiom,
    ! [X: set_Pr8693737435421807431at_nat,Y: set_Pr8693737435421807431at_nat] : sup_su1419176701018045153at_nat(some_s579751585967276588at_nat(X),some_s579751585967276588at_nat(Y)) = some_s579751585967276588at_nat(aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,X),Y)) ).

% sup_Some
tff(fact_1453_sup__Some,axiom,
    ! [X: set_Pr8551490117392284871at_nat,Y: set_Pr8551490117392284871at_nat] : sup_su5873096292857799009at_nat(some_s287724117700012716at_nat(X),some_s287724117700012716at_nat(Y)) = some_s287724117700012716at_nat(aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,X),Y)) ).

% sup_Some
tff(fact_1454_sup__Some,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] : sup_su2273273666271716065at_nat(some_s5890477192898017836at_nat(X),some_s5890477192898017836at_nat(Y)) = some_s5890477192898017836at_nat(aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),Y)) ).

% sup_Some
tff(fact_1455_sup__Some,axiom,
    ! [X: set_nat,Y: set_nat] : sup_su3598758113090618626et_nat(some_set_nat(X),some_set_nat(Y)) = some_set_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Y)) ).

% sup_Some
tff(fact_1456_sup__Some,axiom,
    ! [X: nat,Y: nat] : sup_sup_option_nat(aa_nat_option_nat(some_nat,X),aa_nat_option_nat(some_nat,Y)) = aa_nat_option_nat(some_nat,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X),Y)) ).

% sup_Some
tff(fact_1457_inf__Some,axiom,
    ! [X: set_nat,Y: set_nat] : inf_in7812914138253463912et_nat(some_set_nat(X),some_set_nat(Y)) = some_set_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),Y)) ).

% inf_Some
tff(fact_1458_inf__Some,axiom,
    ! [X: assn,Y: assn] : inf_inf_option_assn(some_assn(X),some_assn(Y)) = some_assn(aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),Y)) ).

% inf_Some
tff(fact_1459_inf__Some,axiom,
    ! [X: nat,Y: nat] : inf_inf_option_nat(aa_nat_option_nat(some_nat,X),aa_nat_option_nat(some_nat,Y)) = aa_nat_option_nat(some_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X),Y)) ).

% inf_Some
tff(fact_1460_inf__Some,axiom,
    ! [X: product_unit,Y: product_unit] : inf_in5800606384722953409t_unit(some_Product_unit(X),some_Product_unit(Y)) = some_Product_unit(aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,X),Y)) ).

% inf_Some
tff(fact_1461_inf__Some,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] : inf_in777885744494410645at_nat(some_s147305329494351046at_nat(X),some_s147305329494351046at_nat(Y)) = some_s147305329494351046at_nat(aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),Y)) ).

% inf_Some
tff(fact_1462_less__eq__option__Some,axiom,
    ! [X: set_int,Y: set_int] :
      ( ord_le353528952715127954et_int(some_set_int(X),some_set_int(Y))
    <=> aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,X),Y) ) ).

% less_eq_option_Some
tff(fact_1463_less__eq__option__Some,axiom,
    ! [X: rat,Y: rat] :
      ( ord_le2406147912482264968on_rat(some_rat(X),some_rat(Y))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y) ) ).

% less_eq_option_Some
tff(fact_1464_less__eq__option__Some,axiom,
    ! [X: num,Y: num] :
      ( aa_option_num_o(aa_opt8674010985164783839_num_o(ord_le6622620407824499402on_num,aa_num_option_num(some_num,X)),aa_num_option_num(some_num,Y))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X),Y) ) ).

% less_eq_option_Some
tff(fact_1465_less__eq__option__Some,axiom,
    ! [X: nat,Y: nat] :
      ( ord_le5914376470875661696on_nat(aa_nat_option_nat(some_nat,X),aa_nat_option_nat(some_nat,Y))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y) ) ).

% less_eq_option_Some
tff(fact_1466_less__eq__option__Some,axiom,
    ! [X: int,Y: int] :
      ( ord_le1736525451366464988on_int(aa_int_option_int(some_int,X),aa_int_option_int(some_int,Y))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Y) ) ).

% less_eq_option_Some
tff(fact_1467__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062r_Ah_H_At_O_A_092_060lbrakk_062execute_Ac_Ah_A_061_ASome_A_Ir_M_Ah_H_M_At_J_059_A_Ih_H_M_Anew__addrs_Ah_Aas1_Ah_H_J_A_092_060Turnstile_062_AQ_Ar_059_ArelH_A_123a_O_Aa_A_060_Alim_Ah_A_092_060and_062_Aa_A_092_060notin_062_Aas1_125_Ah_Ah_H_059_Alim_Ah_A_092_060le_062_Alim_Ah_H_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
    ~ ! [R: a,H4: heap_e7401611519738050253t_unit] :
        ( ? [T2: nat] : aa_hea4724376117875161946it_nat(heap_Time_execute_a(c),h2) = aa_Pro5302094286594872192it_nat(some_P7913643980934408916it_nat,aa_Pro8078911840264235907it_nat(aa_a_f2234720956197727492it_nat(produc9178034014595674355it_nat,R),produc584006145561248582it_nat(H4,T2)))
       => ( aa_Pro653082453696054616_nat_o(rep_assn(aa_a_assn(q,R)),produc7507926704131184380et_nat(H4,hoare_new_addrs(h2,as1,H4)))
         => ( aa_hea7994211679605476167unit_o(relH(collect_nat(aTP_Lamm_ag),h2),H4)
           => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,lim_Product_unit(h2)),lim_Product_unit(H4)) ) ) ) ).

% \<open>\<And>thesis. (\<And>r h' t. \<lbrakk>execute c h = Some (r, h', t); (h', new_addrs h as1 h') \<Turnstile> Q r; relH {a. a < lim h \<and> a \<notin> as1} h h'; lim h \<le> lim h'\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
tff(fact_1468_option_Oinject,axiom,
    ! [X2: produc3260487557148687353it_nat,Y2: produc3260487557148687353it_nat] :
      ( ( aa_Pro5302094286594872192it_nat(some_P7913643980934408916it_nat,X2) = aa_Pro5302094286594872192it_nat(some_P7913643980934408916it_nat,Y2) )
    <=> ( X2 = Y2 ) ) ).

% option.inject
tff(fact_1469_option_Oinject,axiom,
    ! [X2: num,Y2: num] :
      ( ( aa_num_option_num(some_num,X2) = aa_num_option_num(some_num,Y2) )
    <=> ( X2 = Y2 ) ) ).

% option.inject
tff(fact_1470_option_Oinject,axiom,
    ! [X2: produc8664842809031399944it_nat,Y2: produc8664842809031399944it_nat] :
      ( ( aa_Pro8996696869555642526it_nat(some_P1914260805536162275it_nat,X2) = aa_Pro8996696869555642526it_nat(some_P1914260805536162275it_nat,Y2) )
    <=> ( X2 = Y2 ) ) ).

% option.inject
tff(fact_1471_bot__empty__eq,axiom,
    ! [X5: set_nat] :
      ( aa_set_nat_o(bot_bo3967814874310145751_nat_o,X5)
    <=> aa_set_set_nat_o(member_set_nat(X5),bot_bot_set_set_nat) ) ).

% bot_empty_eq
tff(fact_1472_bot__empty__eq,axiom,
    ! [X5: product_prod_nat_nat] :
      ( aa_Pro6150637604445964947_nat_o(bot_bo5074786513356115630_nat_o,X5)
    <=> aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X5),bot_bo2099793752762293965at_nat) ) ).

% bot_empty_eq
tff(fact_1473_bot__empty__eq,axiom,
    ! [X5: $o] :
      ( aa_o_o(bot_bot_fun_o_o,(X5))
    <=> member_o((X5),bot_bot_set_o) ) ).

% bot_empty_eq
tff(fact_1474_bot__empty__eq,axiom,
    ! [X5: nat] :
      ( aa_nat_o(bot_bot_fun_nat_o,X5)
    <=> aa_set_nat_o(member_nat(X5),bot_bot_set_nat) ) ).

% bot_empty_eq
tff(fact_1475_bot__empty__eq,axiom,
    ! [X5: int] :
      ( aa_int_o(bot_bot_fun_int_o,X5)
    <=> aa_set_int_o(member_int(X5),bot_bot_set_int) ) ).

% bot_empty_eq
tff(fact_1476_Collect__empty__eq__bot,axiom,
    ! [Pa: fun_Pr3238374608701176162_int_o] :
      ( ( collec213857154873943460nt_int(Pa) = bot_bo1796632182523588997nt_int )
    <=> ( Pa = bot_bo3516217578726220278_int_o ) ) ).

% Collect_empty_eq_bot
tff(fact_1477_Collect__empty__eq__bot,axiom,
    ! [Pa: fun_list_nat_o] :
      ( ( collect_list_nat(Pa) = bot_bot_set_list_nat )
    <=> ( Pa = bot_bo8828579575001852285_nat_o ) ) ).

% Collect_empty_eq_bot
tff(fact_1478_Collect__empty__eq__bot,axiom,
    ! [Pa: fun_set_nat_o] :
      ( ( collect_set_nat(Pa) = bot_bot_set_set_nat )
    <=> ( Pa = bot_bo3967814874310145751_nat_o ) ) ).

% Collect_empty_eq_bot
tff(fact_1479_Collect__empty__eq__bot,axiom,
    ! [Pa: fun_Pr4796943543331071514_nat_o] :
      ( ( collec3392354462482085612at_nat(Pa) = bot_bo2099793752762293965at_nat )
    <=> ( Pa = bot_bo5074786513356115630_nat_o ) ) ).

% Collect_empty_eq_bot
tff(fact_1480_Collect__empty__eq__bot,axiom,
    ! [Pa: fun_o_o] :
      ( ( collect_o(Pa) = bot_bot_set_o )
    <=> ( Pa = bot_bot_fun_o_o ) ) ).

% Collect_empty_eq_bot
tff(fact_1481_Collect__empty__eq__bot,axiom,
    ! [Pa: fun_nat_o] :
      ( ( collect_nat(Pa) = bot_bot_set_nat )
    <=> ( Pa = bot_bot_fun_nat_o ) ) ).

% Collect_empty_eq_bot
tff(fact_1482_Collect__empty__eq__bot,axiom,
    ! [Pa: fun_int_o] :
      ( ( collect_int(Pa) = bot_bot_set_int )
    <=> ( Pa = bot_bot_fun_int_o ) ) ).

% Collect_empty_eq_bot
tff(fact_1483_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_1484_disjoint__mono,axiom,
    ! [A: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A),A2)
     => ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,B),B2)
       => ( ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B2) = bot_bo2099793752762293965at_nat )
         => ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B) = bot_bo2099793752762293965at_nat ) ) ) ) ).

% disjoint_mono
tff(fact_1485_disjoint__mono,axiom,
    ! [A: set_o,A2: set_o,B: set_o,B2: set_o] :
      ( ord_less_eq_set_o(A,A2)
     => ( ord_less_eq_set_o(B,B2)
       => ( ( inf_inf_set_o(A2,B2) = bot_bot_set_o )
         => ( inf_inf_set_o(A,B) = bot_bot_set_o ) ) ) ) ).

% disjoint_mono
tff(fact_1486_disjoint__mono,axiom,
    ! [A: set_nat,A2: set_nat,B: set_nat,B2: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),A2)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B),B2)
       => ( ( 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(inf_inf_set_nat,A),B) = bot_bot_set_nat ) ) ) ) ).

% disjoint_mono
tff(fact_1487_disjoint__mono,axiom,
    ! [A: set_int,A2: set_int,B: set_int,B2: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A),A2)
     => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,B),B2)
       => ( ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),B2) = 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_1488_disjointI,axiom,
    ! [A: set_set_nat,B: set_set_nat] :
      ( ! [X3: set_nat] :
          ( aa_set_set_nat_o(member_set_nat(X3),A)
         => ~ aa_set_set_nat_o(member_set_nat(X3),B) )
     => ( inf_inf_set_set_nat(A,B) = bot_bot_set_set_nat ) ) ).

% disjointI
tff(fact_1489_disjointI,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ! [X3: product_prod_nat_nat] :
          ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X3),A)
         => ~ aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X3),B) )
     => ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B) = bot_bo2099793752762293965at_nat ) ) ).

% disjointI
tff(fact_1490_disjointI,axiom,
    ! [A: set_o,B: set_o] :
      ( ! [X3: $o] :
          ( member_o((X3),A)
         => ~ member_o((X3),B) )
     => ( inf_inf_set_o(A,B) = bot_bot_set_o ) ) ).

% disjointI
tff(fact_1491_disjointI,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ! [X3: nat] :
          ( aa_set_nat_o(member_nat(X3),A)
         => ~ aa_set_nat_o(member_nat(X3),B) )
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B) = bot_bot_set_nat ) ) ).

% disjointI
tff(fact_1492_disjointI,axiom,
    ! [A: set_int,B: set_int] :
      ( ! [X3: int] :
          ( aa_set_int_o(member_int(X3),A)
         => ~ aa_set_int_o(member_int(X3),B) )
     => ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A),B) = bot_bot_set_int ) ) ).

% disjointI
tff(fact_1493_le__some__optE,axiom,
    ! [M: set_int,X: option_set_int] :
      ( ord_le353528952715127954et_int(some_set_int(M),X)
     => ~ ! [M2: set_int] :
            ( ( X = some_set_int(M2) )
           => ~ aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,M),M2) ) ) ).

% le_some_optE
tff(fact_1494_le__some__optE,axiom,
    ! [M: rat,X: option_rat] :
      ( ord_le2406147912482264968on_rat(some_rat(M),X)
     => ~ ! [M2: rat] :
            ( ( X = some_rat(M2) )
           => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,M),M2) ) ) ).

% le_some_optE
tff(fact_1495_le__some__optE,axiom,
    ! [M: num,X: option_num] :
      ( aa_option_num_o(aa_opt8674010985164783839_num_o(ord_le6622620407824499402on_num,aa_num_option_num(some_num,M)),X)
     => ~ ! [M2: num] :
            ( ( X = aa_num_option_num(some_num,M2) )
           => ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,M),M2) ) ) ).

% le_some_optE
tff(fact_1496_le__some__optE,axiom,
    ! [M: nat,X: option_nat] :
      ( ord_le5914376470875661696on_nat(aa_nat_option_nat(some_nat,M),X)
     => ~ ! [M2: nat] :
            ( ( X = aa_nat_option_nat(some_nat,M2) )
           => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),M2) ) ) ).

% le_some_optE
tff(fact_1497_le__some__optE,axiom,
    ! [M: int,X: option_int] :
      ( ord_le1736525451366464988on_int(aa_int_option_int(some_int,M),X)
     => ~ ! [M2: int] :
            ( ( X = aa_int_option_int(some_int,M2) )
           => ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,M),M2) ) ) ).

% le_some_optE
tff(fact_1498_subsetI,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
      ( ! [X3: product_prod_nat_nat] :
          ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X3),A3)
         => aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X3),B3) )
     => aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A3),B3) ) ).

% subsetI
tff(fact_1499_subsetI,axiom,
    ! [A3: set_o,B3: set_o] :
      ( ! [X3: $o] :
          ( member_o((X3),A3)
         => member_o((X3),B3) )
     => ord_less_eq_set_o(A3,B3) ) ).

% subsetI
tff(fact_1500_subsetI,axiom,
    ! [A3: set_set_nat,B3: set_set_nat] :
      ( ! [X3: set_nat] :
          ( aa_set_set_nat_o(member_set_nat(X3),A3)
         => aa_set_set_nat_o(member_set_nat(X3),B3) )
     => aa_set_set_nat_o(ord_le6893508408891458716et_nat(A3),B3) ) ).

% subsetI
tff(fact_1501_subsetI,axiom,
    ! [A3: set_nat,B3: set_nat] :
      ( ! [X3: nat] :
          ( aa_set_nat_o(member_nat(X3),A3)
         => aa_set_nat_o(member_nat(X3),B3) )
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A3),B3) ) ).

% subsetI
tff(fact_1502_subsetI,axiom,
    ! [A3: set_int,B3: set_int] :
      ( ! [X3: int] :
          ( aa_set_int_o(member_int(X3),A3)
         => aa_set_int_o(member_int(X3),B3) )
     => aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A3),B3) ) ).

% subsetI
tff(fact_1503_subset__antisym,axiom,
    ! [A3: set_int,B3: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A3),B3)
     => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,B3),A3)
       => ( A3 = B3 ) ) ) ).

% subset_antisym
tff(fact_1504_RH1,axiom,
    aa_hea7994211679605476167unit_o(relH(collect_nat(aTP_Lamm_ag),h2),h) ).

% RH1
tff(fact_1505_less__option__Some,axiom,
    ! [X: assn,Y: assn] :
      ( ord_less_option_assn(some_assn(X),some_assn(Y))
    <=> aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X),Y) ) ).

% less_option_Some
tff(fact_1506_less__option__Some,axiom,
    ! [X: rat,Y: rat] :
      ( ord_less_option_rat(some_rat(X),some_rat(Y))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y) ) ).

% less_option_Some
tff(fact_1507_less__option__Some,axiom,
    ! [X: num,Y: num] :
      ( aa_option_num_o(aa_opt8674010985164783839_num_o(ord_less_option_num,aa_num_option_num(some_num,X)),aa_num_option_num(some_num,Y))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,X),Y) ) ).

% less_option_Some
tff(fact_1508_less__option__Some,axiom,
    ! [X: nat,Y: nat] :
      ( ord_less_option_nat(aa_nat_option_nat(some_nat,X),aa_nat_option_nat(some_nat,Y))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y) ) ).

% less_option_Some
tff(fact_1509_less__option__Some,axiom,
    ! [X: int,Y: int] :
      ( ord_less_option_int(aa_int_option_int(some_int,X),aa_int_option_int(some_int,Y))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,X),Y) ) ).

% less_option_Some
tff(fact_1510__092_060open_062relH_A_123a_O_Aa_A_060_Alim_Ah_A_092_060and_062_Aa_A_092_060notin_062_Aas_125_Ah_Ah_H_092_060close_062,axiom,
    aa_hea7994211679605476167unit_o(relH(collect_nat(aTP_Lamm_ah),h2),h) ).

% \<open>relH {a. a < lim h \<and> a \<notin> as} h h'\<close>
tff(fact_1511__092_060open_062as2_A_092_060subseteq_062_A_123a_O_Aa_A_060_Alim_Ah_A_092_060and_062_Aa_A_092_060notin_062_Aas1_125_092_060close_062,axiom,
    aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,as2),collect_nat(aTP_Lamm_ag)) ).

% \<open>as2 \<subseteq> {a. a < lim h \<and> a \<notin> as1}\<close>
tff(fact_1512__092_060open_062_123a_O_Aa_A_060_Alim_Ah_A_092_060and_062_Aa_A_092_060notin_062_Aas_125_A_092_060subseteq_062_A_123a_O_Aa_A_060_Alim_Ah_A_092_060and_062_Aa_A_092_060notin_062_Aas1_125_092_060close_062,axiom,
    aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,collect_nat(aTP_Lamm_ah)),collect_nat(aTP_Lamm_ag)) ).

% \<open>{a. a < lim h \<and> a \<notin> as} \<subseteq> {a. a < lim h \<and> a \<notin> as1}\<close>
tff(fact_1513_in__mono,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A3),B3)
     => ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X),A3)
       => aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X),B3) ) ) ).

% in_mono
tff(fact_1514_in__mono,axiom,
    ! [A3: set_o,B3: set_o,X: $o] :
      ( ord_less_eq_set_o(A3,B3)
     => ( member_o((X),A3)
       => member_o((X),B3) ) ) ).

% in_mono
tff(fact_1515_in__mono,axiom,
    ! [A3: set_set_nat,B3: set_set_nat,X: set_nat] :
      ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(A3),B3)
     => ( aa_set_set_nat_o(member_set_nat(X),A3)
       => aa_set_set_nat_o(member_set_nat(X),B3) ) ) ).

% in_mono
tff(fact_1516_in__mono,axiom,
    ! [A3: set_nat,B3: set_nat,X: nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A3),B3)
     => ( aa_set_nat_o(member_nat(X),A3)
       => aa_set_nat_o(member_nat(X),B3) ) ) ).

% in_mono
tff(fact_1517_in__mono,axiom,
    ! [A3: set_int,B3: set_int,X: int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A3),B3)
     => ( aa_set_int_o(member_int(X),A3)
       => aa_set_int_o(member_int(X),B3) ) ) ).

% in_mono
tff(fact_1518_subsetD,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat,Ca: product_prod_nat_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A3),B3)
     => ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Ca),A3)
       => aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Ca),B3) ) ) ).

% subsetD
tff(fact_1519_subsetD,axiom,
    ! [A3: set_o,B3: set_o,Ca: $o] :
      ( ord_less_eq_set_o(A3,B3)
     => ( member_o((Ca),A3)
       => member_o((Ca),B3) ) ) ).

% subsetD
tff(fact_1520_subsetD,axiom,
    ! [A3: set_set_nat,B3: set_set_nat,Ca: set_nat] :
      ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(A3),B3)
     => ( aa_set_set_nat_o(member_set_nat(Ca),A3)
       => aa_set_set_nat_o(member_set_nat(Ca),B3) ) ) ).

% subsetD
tff(fact_1521_subsetD,axiom,
    ! [A3: set_nat,B3: set_nat,Ca: nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A3),B3)
     => ( aa_set_nat_o(member_nat(Ca),A3)
       => aa_set_nat_o(member_nat(Ca),B3) ) ) ).

% subsetD
tff(fact_1522_subsetD,axiom,
    ! [A3: set_int,B3: set_int,Ca: int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A3),B3)
     => ( aa_set_int_o(member_int(Ca),A3)
       => aa_set_int_o(member_int(Ca),B3) ) ) ).

% subsetD
tff(fact_1523_equalityE,axiom,
    ! [A3: set_int,B3: set_int] :
      ( ( A3 = B3 )
     => ~ ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A3),B3)
         => ~ aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,B3),A3) ) ) ).

% equalityE
tff(fact_1524_subset__eq,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A3),B3)
    <=> ! [X4: product_prod_nat_nat] :
          ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X4),A3)
         => aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X4),B3) ) ) ).

% subset_eq
tff(fact_1525_subset__eq,axiom,
    ! [A3: set_o,B3: set_o] :
      ( ord_less_eq_set_o(A3,B3)
    <=> ! [X4: $o] :
          ( member_o((X4),A3)
         => member_o((X4),B3) ) ) ).

% subset_eq
tff(fact_1526_subset__eq,axiom,
    ! [A3: set_set_nat,B3: set_set_nat] :
      ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(A3),B3)
    <=> ! [X4: set_nat] :
          ( aa_set_set_nat_o(member_set_nat(X4),A3)
         => aa_set_set_nat_o(member_set_nat(X4),B3) ) ) ).

% subset_eq
tff(fact_1527_subset__eq,axiom,
    ! [A3: set_nat,B3: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A3),B3)
    <=> ! [X4: nat] :
          ( aa_set_nat_o(member_nat(X4),A3)
         => aa_set_nat_o(member_nat(X4),B3) ) ) ).

% subset_eq
tff(fact_1528_subset__eq,axiom,
    ! [A3: set_int,B3: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A3),B3)
    <=> ! [X4: int] :
          ( aa_set_int_o(member_int(X4),A3)
         => aa_set_int_o(member_int(X4),B3) ) ) ).

% subset_eq
tff(fact_1529_equalityD1,axiom,
    ! [A3: set_int,B3: set_int] :
      ( ( A3 = B3 )
     => aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A3),B3) ) ).

% equalityD1
tff(fact_1530_equalityD2,axiom,
    ! [A3: set_int,B3: set_int] :
      ( ( A3 = B3 )
     => aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,B3),A3) ) ).

% equalityD2
tff(fact_1531_subset__iff,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A3),B3)
    <=> ! [T3: product_prod_nat_nat] :
          ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(T3),A3)
         => aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(T3),B3) ) ) ).

% subset_iff
tff(fact_1532_subset__iff,axiom,
    ! [A3: set_o,B3: set_o] :
      ( ord_less_eq_set_o(A3,B3)
    <=> ! [T3: $o] :
          ( member_o((T3),A3)
         => member_o((T3),B3) ) ) ).

% subset_iff
tff(fact_1533_subset__iff,axiom,
    ! [A3: set_set_nat,B3: set_set_nat] :
      ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(A3),B3)
    <=> ! [T3: set_nat] :
          ( aa_set_set_nat_o(member_set_nat(T3),A3)
         => aa_set_set_nat_o(member_set_nat(T3),B3) ) ) ).

% subset_iff
tff(fact_1534_subset__iff,axiom,
    ! [A3: set_nat,B3: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A3),B3)
    <=> ! [T3: nat] :
          ( aa_set_nat_o(member_nat(T3),A3)
         => aa_set_nat_o(member_nat(T3),B3) ) ) ).

% subset_iff
tff(fact_1535_subset__iff,axiom,
    ! [A3: set_int,B3: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A3),B3)
    <=> ! [T3: int] :
          ( aa_set_int_o(member_int(T3),A3)
         => aa_set_int_o(member_int(T3),B3) ) ) ).

% subset_iff
tff(fact_1536_subset__refl,axiom,
    ! [A3: set_int] : aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A3),A3) ).

% subset_refl
tff(fact_1537_Collect__mono,axiom,
    ! [Pa: fun_Pr3238374608701176162_int_o,Qa: fun_Pr3238374608701176162_int_o] :
      ( ! [X3: product_prod_int_int] :
          ( aa_Pro4592068669816069595_int_o(Pa,X3)
         => aa_Pro4592068669816069595_int_o(Qa,X3) )
     => ord_le2843351958646193337nt_int(collec213857154873943460nt_int(Pa),collec213857154873943460nt_int(Qa)) ) ).

% Collect_mono
tff(fact_1538_Collect__mono,axiom,
    ! [Pa: fun_list_nat_o,Qa: fun_list_nat_o] :
      ( ! [X3: list_nat] :
          ( aa_list_nat_o(Pa,X3)
         => aa_list_nat_o(Qa,X3) )
     => ord_le6045566169113846134st_nat(collect_list_nat(Pa),collect_list_nat(Qa)) ) ).

% Collect_mono
tff(fact_1539_Collect__mono,axiom,
    ! [Pa: fun_set_nat_o,Qa: fun_set_nat_o] :
      ( ! [X3: set_nat] :
          ( aa_set_nat_o(Pa,X3)
         => aa_set_nat_o(Qa,X3) )
     => aa_set_set_nat_o(ord_le6893508408891458716et_nat(collect_set_nat(Pa)),collect_set_nat(Qa)) ) ).

% Collect_mono
tff(fact_1540_Collect__mono,axiom,
    ! [Pa: fun_nat_o,Qa: fun_nat_o] :
      ( ! [X3: nat] :
          ( aa_nat_o(Pa,X3)
         => aa_nat_o(Qa,X3) )
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,collect_nat(Pa)),collect_nat(Qa)) ) ).

% Collect_mono
tff(fact_1541_Collect__mono,axiom,
    ! [Pa: fun_int_o,Qa: fun_int_o] :
      ( ! [X3: int] :
          ( aa_int_o(Pa,X3)
         => aa_int_o(Qa,X3) )
     => aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,collect_int(Pa)),collect_int(Qa)) ) ).

% Collect_mono
tff(fact_1542_subset__trans,axiom,
    ! [A3: set_int,B3: set_int,C2: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A3),B3)
     => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,B3),C2)
       => aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A3),C2) ) ) ).

% subset_trans
tff(fact_1543_set__eq__subset,axiom,
    ! [A3: set_int,B3: set_int] :
      ( ( A3 = B3 )
    <=> ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A3),B3)
        & aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,B3),A3) ) ) ).

% set_eq_subset
tff(fact_1544_sup__Un__eq,axiom,
    ! [Ra: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat,X5: product_prod_nat_nat] :
      ( aa_Pro6150637604445964947_nat_o(aa_fun359341089595923442_nat_o(sup_su5606187154157927750_nat_o(aTP_Lamm_a(Ra)),aTP_Lamm_a(S)),X5)
    <=> aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X5),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,Ra),S)) ) ).

% sup_Un_eq
tff(fact_1545_sup__Un__eq,axiom,
    ! [Ra: set_o,S: set_o,X5: $o] :
      ( aa_o_o(sup_sup_fun_o_o(aTP_Lamm_aa(Ra),aTP_Lamm_aa(S)),(X5))
    <=> member_o((X5),aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,Ra),S)) ) ).

% sup_Un_eq
tff(fact_1546_sup__Un__eq,axiom,
    ! [Ra: set_set_nat,S: set_set_nat,X5: set_nat] :
      ( aa_set_nat_o(sup_su7503774895437956671_nat_o(aTP_Lamm_ad(Ra),aTP_Lamm_ad(S)),X5)
    <=> aa_set_set_nat_o(member_set_nat(X5),sup_sup_set_set_nat(Ra,S)) ) ).

% sup_Un_eq
tff(fact_1547_sup__Un__eq,axiom,
    ! [Ra: set_int,S: set_int,X5: int] :
      ( aa_int_o(sup_sup_fun_int_o(aTP_Lamm_af(Ra),aTP_Lamm_af(S)),X5)
    <=> aa_set_int_o(member_int(X5),aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,Ra),S)) ) ).

% sup_Un_eq
tff(fact_1548_sup__Un__eq,axiom,
    ! [Ra: set_Pr8693737435421807431at_nat,S: set_Pr8693737435421807431at_nat,X5: produc859450856879609959at_nat] :
      ( aa_Pro602569247023830573_nat_o(sup_su5487351348554007264_nat_o(aTP_Lamm_ai(Ra),aTP_Lamm_ai(S)),X5)
    <=> member8206827879206165904at_nat(X5,aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,Ra),S)) ) ).

% sup_Un_eq
tff(fact_1549_sup__Un__eq,axiom,
    ! [Ra: set_Pr8551490117392284871at_nat,S: set_Pr8551490117392284871at_nat,X5: produc4166570645942440679at_nat] :
      ( aa_Pro8143006071014924973_nat_o(sup_su5351388498073598816_nat_o(aTP_Lamm_aj(Ra),aTP_Lamm_aj(S)),X5)
    <=> member6689249552917799696at_nat(X5,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,Ra),S)) ) ).

% sup_Un_eq
tff(fact_1550_sup__Un__eq,axiom,
    ! [Ra: set_Pr4329608150637261639at_nat,S: set_Pr4329608150637261639at_nat,X5: produc3843707927480180839at_nat] :
      ( aa_Pro5947980084102686253_nat_o(sup_su2398265404452057824_nat_o(aTP_Lamm_ak(Ra),aTP_Lamm_ak(S)),X5)
    <=> member8757157785044589968at_nat(X5,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,Ra),S)) ) ).

% sup_Un_eq
tff(fact_1551_sup__Un__eq,axiom,
    ! [Ra: set_nat,S: set_nat,X5: nat] :
      ( aa_nat_o(sup_sup_fun_nat_o(aTP_Lamm_ae(Ra),aTP_Lamm_ae(S)),X5)
    <=> aa_set_nat_o(member_nat(X5),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Ra),S)) ) ).

% sup_Un_eq
tff(fact_1552_Collect__subset,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,Pa: fun_Pr4796943543331071514_nat_o] : aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,collec3392354462482085612at_nat(aa_fun359341089595923442_nat_o(aTP_Lamm_al(A3),Pa))),A3) ).

% Collect_subset
tff(fact_1553_Collect__subset,axiom,
    ! [A3: set_o,Pa: fun_o_o] : ord_less_eq_set_o(collect_o(aa_fun_o_o_fun_o_o(aTP_Lamm_am(A3),Pa)),A3) ).

% Collect_subset
tff(fact_1554_Collect__subset,axiom,
    ! [A3: set_Pr958786334691620121nt_int,Pa: fun_Pr3238374608701176162_int_o] : ord_le2843351958646193337nt_int(collec213857154873943460nt_int(aa_fun5349570537165229698_int_o(aTP_Lamm_an(A3),Pa)),A3) ).

% Collect_subset
tff(fact_1555_Collect__subset,axiom,
    ! [A3: set_list_nat,Pa: fun_list_nat_o] : ord_le6045566169113846134st_nat(collect_list_nat(aa_fun6155608546934989052_nat_o(aTP_Lamm_ao(A3),Pa)),A3) ).

% Collect_subset
tff(fact_1556_Collect__subset,axiom,
    ! [A3: set_set_nat,Pa: fun_set_nat_o] : aa_set_set_nat_o(ord_le6893508408891458716et_nat(collect_set_nat(aa_fun751667557118977660_nat_o(aTP_Lamm_ap(A3),Pa))),A3) ).

% Collect_subset
tff(fact_1557_Collect__subset,axiom,
    ! [A3: set_nat,Pa: fun_nat_o] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,collect_nat(aa_fun5937193402907943164_nat_o(aTP_Lamm_aq(A3),Pa))),A3) ).

% Collect_subset
tff(fact_1558_Collect__subset,axiom,
    ! [A3: set_int,Pa: fun_int_o] : aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_ar(A3),Pa))),A3) ).

% Collect_subset
tff(fact_1559_inf__Int__eq,axiom,
    ! [Ra: set_o,S: set_o,X5: $o] :
      ( aa_o_o(inf_inf_fun_o_o(aTP_Lamm_aa(Ra),aTP_Lamm_aa(S)),(X5))
    <=> member_o((X5),inf_inf_set_o(Ra,S)) ) ).

% inf_Int_eq
tff(fact_1560_inf__Int__eq,axiom,
    ! [Ra: set_set_nat,S: set_set_nat,X5: set_nat] :
      ( aa_set_nat_o(inf_in7602907407060649817_nat_o(aTP_Lamm_ad(Ra),aTP_Lamm_ad(S)),X5)
    <=> aa_set_set_nat_o(member_set_nat(X5),inf_inf_set_set_nat(Ra,S)) ) ).

% inf_Int_eq
tff(fact_1561_inf__Int__eq,axiom,
    ! [Ra: set_int,S: set_int,X5: int] :
      ( aa_int_o(inf_inf_fun_int_o(aTP_Lamm_af(Ra),aTP_Lamm_af(S)),X5)
    <=> aa_set_int_o(member_int(X5),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,Ra),S)) ) ).

% inf_Int_eq
tff(fact_1562_inf__Int__eq,axiom,
    ! [Ra: set_nat,S: set_nat,X5: nat] :
      ( aa_nat_o(inf_inf_fun_nat_o(aTP_Lamm_ae(Ra),aTP_Lamm_ae(S)),X5)
    <=> aa_set_nat_o(member_nat(X5),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Ra),S)) ) ).

% inf_Int_eq
tff(fact_1563_inf__Int__eq,axiom,
    ! [Ra: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat,X5: product_prod_nat_nat] :
      ( aa_Pro6150637604445964947_nat_o(aa_fun359341089595923442_nat_o(inf_in3341167027036195756_nat_o(aTP_Lamm_a(Ra)),aTP_Lamm_a(S)),X5)
    <=> aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X5),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Ra),S)) ) ).

% inf_Int_eq
tff(fact_1564_sup__Un__eq2,axiom,
    ! [Ra: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat,X5: nat,Xa3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(sup_su8083968640029086462_nat_o(aTP_Lamm_as(Ra),aTP_Lamm_as(S)),X5),Xa3)
    <=> aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,X5),Xa3)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,Ra),S)) ) ).

% sup_Un_eq2
tff(fact_1565_sup__Un__eq2,axiom,
    ! [Ra: set_Pr1312406793078187350et_nat,S: set_Pr1312406793078187350et_nat,X5: fun_Pr6460944699146269935_nat_o,Xa3: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(aa_fun2638059832795814268_nat_o(sup_su4970993554803434671_nat_o(aTP_Lamm_at(Ra),aTP_Lamm_at(S)),X5),Xa3)
    <=> member4513763875858039863et_nat(aa_Pro434340453207536480et_nat(aa_fun3926903090414666066et_nat(produc6017135503889078682et_nat,X5),Xa3),sup_su2387659076810923138et_nat(Ra,S)) ) ).

% sup_Un_eq2
tff(fact_1566_sup__Un__eq2,axiom,
    ! [Ra: set_Pr4040394263813967220et_nat,S: set_Pr4040394263813967220et_nat,X5: fun_Pr6460944699146269935_nat_o,Xa3: produc8799533983703829792et_nat] :
      ( aa_Pro5643009357358181178_nat_o(aa_fun912680268133416158_nat_o(sup_su6780828561269846801_nat_o(aTP_Lamm_au(Ra),aTP_Lamm_au(S)),X5),Xa3)
    <=> member9174161442226468181et_nat(aa_Pro1979504083478021276et_nat(aa_fun615918787490101902et_nat(produc7291855603632349880et_nat,X5),Xa3),sup_su5858552540197408416et_nat(Ra,S)) ) ).

% sup_Un_eq2
tff(fact_1567_sup__Un__eq2,axiom,
    ! [Ra: set_Pr6700895561065273913nt_int,S: set_Pr6700895561065273913nt_int,X5: fun_Pr7763680713840462443e_term,Xa3: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aa_fun3538162476766722827_int_o(sup_su2175524065394371574_int_o(aTP_Lamm_av(Ra),aTP_Lamm_av(S)),X5),Xa3)
    <=> member753276645541486746nt_int(aa_Pro5748361610964203648nt_int(aa_fun3265472288063244902nt_int(produc7320969046608631221nt_int,X5),Xa3),sup_su2017857570560100197nt_int(Ra,S)) ) ).

% sup_Un_eq2
tff(fact_1568_sup__Un__eq2,axiom,
    ! [Ra: set_Pr4166680992137330533nt_int,S: set_Pr4166680992137330533nt_int,X5: fun_in5730379987311621695e_term,Xa3: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aa_fun586359091965693151_int_o(sup_su7230269426179251658_int_o(aTP_Lamm_aw(Ra),aTP_Lamm_aw(S)),X5),Xa3)
    <=> member5890090881338784710nt_int(aa_Pro8800328608558162860nt_int(aa_fun260840925927807678nt_int(produc4729913460834754785nt_int,X5),Xa3),sup_su7513280395484302481nt_int(Ra,S)) ) ).

% sup_Un_eq2
tff(fact_1569_sup__Un__eq2,axiom,
    ! [Ra: set_Pr7098220151150636591it_nat,S: set_Pr7098220151150636591it_nat,X5: a,Xa3: produc6653097349344004940it_nat] :
      ( aa_Pro1202363709425262478_nat_o(aa_a_f337466037417904483_nat_o(sup_su5814904253042911126_nat_o(aTP_Lamm_ax(Ra),aTP_Lamm_ax(S)),X5),Xa3)
    <=> member4554412811331277712it_nat(aa_Pro8078911840264235907it_nat(aa_a_f2234720956197727492it_nat(produc9178034014595674355it_nat,X5),Xa3),sup_su3211442154794319707it_nat(Ra,S)) ) ).

% sup_Un_eq2
tff(fact_1570_sup__Un__eq2,axiom,
    ! [Ra: set_Pr8693737435421807431at_nat,S: set_Pr8693737435421807431at_nat,X5: product_prod_nat_nat,Xa3: product_prod_nat_nat] :
      ( aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(sup_su52541223548345602_nat_o(aTP_Lamm_ay(Ra),aTP_Lamm_ay(S)),X5),Xa3)
    <=> member8206827879206165904at_nat(produc6161850002892822231at_nat(X5,Xa3),aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,Ra),S)) ) ).

% sup_Un_eq2
tff(fact_1571_sup__Un__eq2,axiom,
    ! [Ra: set_Pr8551490117392284871at_nat,S: set_Pr8551490117392284871at_nat,X5: multis2468970476368604999at_nat,Xa3: multis2468970476368604999at_nat] :
      ( aa_mul869302825175964237_nat_o(aa_mul3916465897682844511_nat_o(sup_su9034197199115871746_nat_o(aTP_Lamm_az(Ra),aTP_Lamm_az(S)),X5),Xa3)
    <=> member6689249552917799696at_nat(produc4348348721325984599at_nat(X5,Xa3),aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,Ra),S)) ) ).

% sup_Un_eq2
tff(fact_1572_sup__Un__eq2,axiom,
    ! [Ra: set_Pr4329608150637261639at_nat,S: set_Pr4329608150637261639at_nat,X5: set_Pr1261947904930325089at_nat,Xa3: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(sup_su5424986631308460290_nat_o(aTP_Lamm_ba(Ra),aTP_Lamm_ba(S)),X5),Xa3)
    <=> member8757157785044589968at_nat(produc2922128104949294807at_nat(X5,Xa3),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,Ra),S)) ) ).

% sup_Un_eq2
tff(fact_1573_less__eq__set__def,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A3),B3)
    <=> aa_fun8408151826566693824at_o_o(ord_le1572679713340831354_nat_o(aTP_Lamm_a(A3)),aTP_Lamm_a(B3)) ) ).

% less_eq_set_def
tff(fact_1574_less__eq__set__def,axiom,
    ! [A3: set_o,B3: set_o] :
      ( ord_less_eq_set_o(A3,B3)
    <=> ord_less_eq_fun_o_o(aTP_Lamm_aa(A3),aTP_Lamm_aa(B3)) ) ).

% less_eq_set_def
tff(fact_1575_less__eq__set__def,axiom,
    ! [A3: set_set_nat,B3: set_set_nat] :
      ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(A3),B3)
    <=> ord_le6468474429079055883_nat_o(aTP_Lamm_ad(A3),aTP_Lamm_ad(B3)) ) ).

% less_eq_set_def
tff(fact_1576_less__eq__set__def,axiom,
    ! [A3: set_nat,B3: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A3),B3)
    <=> ord_le7240017405068453697_nat_o(aTP_Lamm_ae(A3),aTP_Lamm_ae(B3)) ) ).

% less_eq_set_def
tff(fact_1577_less__eq__set__def,axiom,
    ! [A3: set_int,B3: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A3),B3)
    <=> ord_le5418975059635765733_int_o(aTP_Lamm_af(A3),aTP_Lamm_af(B3)) ) ).

% less_eq_set_def
tff(fact_1578_inf__Int__eq2,axiom,
    ! [Ra: set_Pr1312406793078187350et_nat,S: set_Pr1312406793078187350et_nat,X5: fun_Pr6460944699146269935_nat_o,Xa3: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(aa_fun2638059832795814268_nat_o(inf_in3476382694143140117_nat_o(aTP_Lamm_at(Ra),aTP_Lamm_at(S)),X5),Xa3)
    <=> member4513763875858039863et_nat(aa_Pro434340453207536480et_nat(aa_fun3926903090414666066et_nat(produc6017135503889078682et_nat,X5),Xa3),inf_in4791168559842171624et_nat(Ra,S)) ) ).

% inf_Int_eq2
tff(fact_1579_inf__Int__eq2,axiom,
    ! [Ra: set_Pr4040394263813967220et_nat,S: set_Pr4040394263813967220et_nat,X5: fun_Pr6460944699146269935_nat_o,Xa3: produc8799533983703829792et_nat] :
      ( aa_Pro5643009357358181178_nat_o(aa_fun912680268133416158_nat_o(inf_in6333540121576287351_nat_o(aTP_Lamm_au(Ra),aTP_Lamm_au(S)),X5),Xa3)
    <=> member9174161442226468181et_nat(aa_Pro1979504083478021276et_nat(aa_fun615918787490101902et_nat(produc7291855603632349880et_nat,X5),Xa3),inf_in6821596156192049670et_nat(Ra,S)) ) ).

% inf_Int_eq2
tff(fact_1580_inf__Int__eq2,axiom,
    ! [Ra: set_Pr6700895561065273913nt_int,S: set_Pr6700895561065273913nt_int,X5: fun_Pr7763680713840462443e_term,Xa3: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aa_fun3538162476766722827_int_o(inf_in132845422080707420_int_o(aTP_Lamm_av(Ra),aTP_Lamm_av(S)),X5),Xa3)
    <=> member753276645541486746nt_int(aa_Pro5748361610964203648nt_int(aa_fun3265472288063244902nt_int(produc7320969046608631221nt_int,X5),Xa3),inf_in3653535833467300043nt_int(Ra,S)) ) ).

% inf_Int_eq2
tff(fact_1581_inf__Int__eq2,axiom,
    ! [Ra: set_Pr4166680992137330533nt_int,S: set_Pr4166680992137330533nt_int,X5: fun_in5730379987311621695e_term,Xa3: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aa_fun586359091965693151_int_o(inf_in4973581681881435440_int_o(aTP_Lamm_aw(Ra),aTP_Lamm_aw(S)),X5),Xa3)
    <=> member5890090881338784710nt_int(aa_Pro8800328608558162860nt_int(aa_fun260840925927807678nt_int(produc4729913460834754785nt_int,X5),Xa3),inf_in8549327947507203575nt_int(Ra,S)) ) ).

% inf_Int_eq2
tff(fact_1582_inf__Int__eq2,axiom,
    ! [Ra: set_Pr7098220151150636591it_nat,S: set_Pr7098220151150636591it_nat,X5: a,Xa3: produc6653097349344004940it_nat] :
      ( aa_Pro1202363709425262478_nat_o(aa_a_f337466037417904483_nat_o(inf_in2619332623043422460_nat_o(aTP_Lamm_ax(Ra),aTP_Lamm_ax(S)),X5),Xa3)
    <=> member4554412811331277712it_nat(aa_Pro8078911840264235907it_nat(aa_a_f2234720956197727492it_nat(produc9178034014595674355it_nat,X5),Xa3),inf_in3493107508024165057it_nat(Ra,S)) ) ).

% inf_Int_eq2
tff(fact_1583_inf__Int__eq2,axiom,
    ! [Ra: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat,X5: nat,Xa3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(inf_in375002649321709592_nat_o(aTP_Lamm_as(Ra),aTP_Lamm_as(S)),X5),Xa3)
    <=> aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,X5),Xa3)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Ra),S)) ) ).

% inf_Int_eq2
tff(fact_1584_Collect__mono__iff,axiom,
    ! [Pa: fun_Pr3238374608701176162_int_o,Qa: fun_Pr3238374608701176162_int_o] :
      ( ord_le2843351958646193337nt_int(collec213857154873943460nt_int(Pa),collec213857154873943460nt_int(Qa))
    <=> ! [X4: product_prod_int_int] :
          ( aa_Pro4592068669816069595_int_o(Pa,X4)
         => aa_Pro4592068669816069595_int_o(Qa,X4) ) ) ).

% Collect_mono_iff
tff(fact_1585_Collect__mono__iff,axiom,
    ! [Pa: fun_list_nat_o,Qa: fun_list_nat_o] :
      ( ord_le6045566169113846134st_nat(collect_list_nat(Pa),collect_list_nat(Qa))
    <=> ! [X4: list_nat] :
          ( aa_list_nat_o(Pa,X4)
         => aa_list_nat_o(Qa,X4) ) ) ).

% Collect_mono_iff
tff(fact_1586_Collect__mono__iff,axiom,
    ! [Pa: fun_set_nat_o,Qa: fun_set_nat_o] :
      ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(collect_set_nat(Pa)),collect_set_nat(Qa))
    <=> ! [X4: set_nat] :
          ( aa_set_nat_o(Pa,X4)
         => aa_set_nat_o(Qa,X4) ) ) ).

% Collect_mono_iff
tff(fact_1587_Collect__mono__iff,axiom,
    ! [Pa: fun_nat_o,Qa: fun_nat_o] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,collect_nat(Pa)),collect_nat(Qa))
    <=> ! [X4: nat] :
          ( aa_nat_o(Pa,X4)
         => aa_nat_o(Qa,X4) ) ) ).

% Collect_mono_iff
tff(fact_1588_Collect__mono__iff,axiom,
    ! [Pa: fun_int_o,Qa: fun_int_o] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,collect_int(Pa)),collect_int(Qa))
    <=> ! [X4: int] :
          ( aa_int_o(Pa,X4)
         => aa_int_o(Qa,X4) ) ) ).

% Collect_mono_iff
tff(fact_1589_bot__empty__eq2,axiom,
    ! [X5: fun_Pr6460944699146269935_nat_o,Xa3: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(aa_fun2638059832795814268_nat_o(bot_bo3509460380234399255_nat_o,X5),Xa3)
    <=> member4513763875858039863et_nat(aa_Pro434340453207536480et_nat(aa_fun3926903090414666066et_nat(produc6017135503889078682et_nat,X5),Xa3),bot_bo1493351289776401898et_nat) ) ).

% bot_empty_eq2
tff(fact_1590_bot__empty__eq2,axiom,
    ! [X5: fun_Pr6460944699146269935_nat_o,Xa3: produc8799533983703829792et_nat] :
      ( aa_Pro5643009357358181178_nat_o(aa_fun912680268133416158_nat_o(bot_bo5031374944739956857_nat_o,X5),Xa3)
    <=> member9174161442226468181et_nat(aa_Pro1979504083478021276et_nat(aa_fun615918787490101902et_nat(produc7291855603632349880et_nat,X5),Xa3),bot_bo6121529163330120712et_nat) ) ).

% bot_empty_eq2
tff(fact_1591_bot__empty__eq2,axiom,
    ! [X5: fun_Pr7763680713840462443e_term,Xa3: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aa_fun3538162476766722827_int_o(bot_bo1117279750869432670_int_o,X5),Xa3)
    <=> member753276645541486746nt_int(aa_Pro5748361610964203648nt_int(aa_fun3265472288063244902nt_int(produc7320969046608631221nt_int,X5),Xa3),bot_bo2501116328598523085nt_int) ) ).

% bot_empty_eq2
tff(fact_1592_bot__empty__eq2,axiom,
    ! [X5: fun_in5730379987311621695e_term,Xa3: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aa_fun586359091965693151_int_o(bot_bo6524246659364023090_int_o,X5),Xa3)
    <=> member5890090881338784710nt_int(aa_Pro8800328608558162860nt_int(aa_fun260840925927807678nt_int(produc4729913460834754785nt_int,X5),Xa3),bot_bo5189594446080201209nt_int) ) ).

% bot_empty_eq2
tff(fact_1593_bot__empty__eq2,axiom,
    ! [X5: a,Xa3: produc6653097349344004940it_nat] :
      ( aa_Pro1202363709425262478_nat_o(aa_a_f337466037417904483_nat_o(bot_bo2291057277214747902_nat_o,X5),Xa3)
    <=> member4554412811331277712it_nat(aa_Pro8078911840264235907it_nat(aa_a_f2234720956197727492it_nat(produc9178034014595674355it_nat,X5),Xa3),bot_bo8480459777671986371it_nat) ) ).

% bot_empty_eq2
tff(fact_1594_bot__empty__eq2,axiom,
    ! [X5: nat,Xa3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(bot_bo1118997454112379798_nat_o,X5),Xa3)
    <=> aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,X5),Xa3)),bot_bo2099793752762293965at_nat) ) ).

% bot_empty_eq2
tff(fact_1595_pred__equals__eq2,axiom,
    ! [Ra: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
      ( ! [X4: nat,Xa: nat] :
          ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,X4),Xa)),Ra)
        <=> aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,X4),Xa)),S) )
    <=> ( Ra = S ) ) ).

% pred_equals_eq2
tff(fact_1596_pred__equals__eq2,axiom,
    ! [Ra: set_Pr1312406793078187350et_nat,S: set_Pr1312406793078187350et_nat] :
      ( ! [X4: fun_Pr6460944699146269935_nat_o,Xa: produc3658429121746597890et_nat] :
          ( member4513763875858039863et_nat(aa_Pro434340453207536480et_nat(aa_fun3926903090414666066et_nat(produc6017135503889078682et_nat,X4),Xa),Ra)
        <=> member4513763875858039863et_nat(aa_Pro434340453207536480et_nat(aa_fun3926903090414666066et_nat(produc6017135503889078682et_nat,X4),Xa),S) )
    <=> ( Ra = S ) ) ).

% pred_equals_eq2
tff(fact_1597_pred__equals__eq2,axiom,
    ! [Ra: set_Pr4040394263813967220et_nat,S: set_Pr4040394263813967220et_nat] :
      ( ! [X4: fun_Pr6460944699146269935_nat_o,Xa: produc8799533983703829792et_nat] :
          ( member9174161442226468181et_nat(aa_Pro1979504083478021276et_nat(aa_fun615918787490101902et_nat(produc7291855603632349880et_nat,X4),Xa),Ra)
        <=> member9174161442226468181et_nat(aa_Pro1979504083478021276et_nat(aa_fun615918787490101902et_nat(produc7291855603632349880et_nat,X4),Xa),S) )
    <=> ( Ra = S ) ) ).

% pred_equals_eq2
tff(fact_1598_pred__equals__eq2,axiom,
    ! [Ra: set_Pr6700895561065273913nt_int,S: set_Pr6700895561065273913nt_int] :
      ( ! [X4: fun_Pr7763680713840462443e_term,Xa: product_prod_int_int] :
          ( member753276645541486746nt_int(aa_Pro5748361610964203648nt_int(aa_fun3265472288063244902nt_int(produc7320969046608631221nt_int,X4),Xa),Ra)
        <=> member753276645541486746nt_int(aa_Pro5748361610964203648nt_int(aa_fun3265472288063244902nt_int(produc7320969046608631221nt_int,X4),Xa),S) )
    <=> ( Ra = S ) ) ).

% pred_equals_eq2
tff(fact_1599_pred__equals__eq2,axiom,
    ! [Ra: set_Pr4166680992137330533nt_int,S: set_Pr4166680992137330533nt_int] :
      ( ! [X4: fun_in5730379987311621695e_term,Xa: product_prod_int_int] :
          ( member5890090881338784710nt_int(aa_Pro8800328608558162860nt_int(aa_fun260840925927807678nt_int(produc4729913460834754785nt_int,X4),Xa),Ra)
        <=> member5890090881338784710nt_int(aa_Pro8800328608558162860nt_int(aa_fun260840925927807678nt_int(produc4729913460834754785nt_int,X4),Xa),S) )
    <=> ( Ra = S ) ) ).

% pred_equals_eq2
tff(fact_1600_pred__equals__eq2,axiom,
    ! [Ra: set_Pr7098220151150636591it_nat,S: set_Pr7098220151150636591it_nat] :
      ( ! [X4: a,Xa: produc6653097349344004940it_nat] :
          ( member4554412811331277712it_nat(aa_Pro8078911840264235907it_nat(aa_a_f2234720956197727492it_nat(produc9178034014595674355it_nat,X4),Xa),Ra)
        <=> member4554412811331277712it_nat(aa_Pro8078911840264235907it_nat(aa_a_f2234720956197727492it_nat(produc9178034014595674355it_nat,X4),Xa),S) )
    <=> ( Ra = S ) ) ).

% pred_equals_eq2
tff(fact_1601_pred__subset__eq2,axiom,
    ! [Ra: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
      ( ord_le768339454706049738_nat_o(aTP_Lamm_as(Ra),aTP_Lamm_as(S))
    <=> aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,Ra),S) ) ).

% pred_subset_eq2
tff(fact_1602_pred__subset__eq2,axiom,
    ! [Ra: set_Pr1312406793078187350et_nat,S: set_Pr1312406793078187350et_nat] :
      ( ord_le3580378406483493347_nat_o(aTP_Lamm_at(Ra),aTP_Lamm_at(S))
    <=> ord_le3919382008627836342et_nat(Ra,S) ) ).

% pred_subset_eq2
tff(fact_1603_pred__subset__eq2,axiom,
    ! [Ra: set_Pr4040394263813967220et_nat,S: set_Pr4040394263813967220et_nat] :
      ( ord_le2834457679758710341_nat_o(aTP_Lamm_au(Ra),aTP_Lamm_au(S))
    <=> ord_le5189643068899110356et_nat(Ra,S) ) ).

% pred_subset_eq2
tff(fact_1604_pred__subset__eq2,axiom,
    ! [Ra: set_Pr6700895561065273913nt_int,S: set_Pr6700895561065273913nt_int] :
      ( ord_le4806730579060245290_int_o(aTP_Lamm_av(Ra),aTP_Lamm_av(S))
    <=> ord_le3990897393281686169nt_int(Ra,S) ) ).

% pred_subset_eq2
tff(fact_1605_pred__subset__eq2,axiom,
    ! [Ra: set_Pr4166680992137330533nt_int,S: set_Pr4166680992137330533nt_int] :
      ( ord_le2876595303158420734_int_o(aTP_Lamm_aw(Ra),aTP_Lamm_aw(S))
    <=> ord_le3197784583289274309nt_int(Ra,S) ) ).

% pred_subset_eq2
tff(fact_1606_pred__subset__eq2,axiom,
    ! [Ra: set_Pr7098220151150636591it_nat,S: set_Pr7098220151150636591it_nat] :
      ( ord_le6609387309041105610_nat_o(aTP_Lamm_ax(Ra),aTP_Lamm_ax(S))
    <=> ord_le4478181985103859343it_nat(Ra,S) ) ).

% pred_subset_eq2
tff(fact_1607_lt__ex,axiom,
    ! [X: rat] :
    ? [Y3: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y3),X) ).

% lt_ex
tff(fact_1608_lt__ex,axiom,
    ! [X: int] :
    ? [Y3: int] : aa_int_o(aa_int_fun_int_o(ord_less_int,Y3),X) ).

% lt_ex
tff(fact_1609_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_1610_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_1611_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_1612_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_1613_less__imp__neq,axiom,
    ! [X: assn,Y: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X),Y)
     => ( X != Y ) ) ).

% less_imp_neq
tff(fact_1614_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_1615_less__imp__neq,axiom,
    ! [X: num,Y: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,X),Y)
     => ( X != Y ) ) ).

% less_imp_neq
tff(fact_1616_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_1617_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_1618_order_Oasym,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_assn,B),A) ) ).

% order.asym
tff(fact_1619_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_1620_order_Oasym,axiom,
    ! [A: num,B: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,A),B)
     => ~ aa_num_o(aa_num_fun_num_o(ord_less_num,B),A) ) ).

% order.asym
tff(fact_1621_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_1622_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_1623_ord__eq__less__trans,axiom,
    ! [A: assn,B: assn,Ca: assn] :
      ( ( A = B )
     => ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,B),Ca)
       => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),Ca) ) ) ).

% ord_eq_less_trans
tff(fact_1624_ord__eq__less__trans,axiom,
    ! [A: rat,B: rat,Ca: rat] :
      ( ( A = B )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),Ca)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),Ca) ) ) ).

% ord_eq_less_trans
tff(fact_1625_ord__eq__less__trans,axiom,
    ! [A: num,B: num,Ca: num] :
      ( ( A = B )
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,B),Ca)
       => aa_num_o(aa_num_fun_num_o(ord_less_num,A),Ca) ) ) ).

% ord_eq_less_trans
tff(fact_1626_ord__eq__less__trans,axiom,
    ! [A: nat,B: nat,Ca: nat] :
      ( ( A = B )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),Ca)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),Ca) ) ) ).

% ord_eq_less_trans
tff(fact_1627_ord__eq__less__trans,axiom,
    ! [A: int,B: int,Ca: int] :
      ( ( A = B )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),Ca)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,A),Ca) ) ) ).

% ord_eq_less_trans
tff(fact_1628_ord__less__eq__trans,axiom,
    ! [A: assn,B: assn,Ca: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),B)
     => ( ( B = Ca )
       => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),Ca) ) ) ).

% ord_less_eq_trans
tff(fact_1629_ord__less__eq__trans,axiom,
    ! [A: rat,B: rat,Ca: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( ( B = Ca )
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),Ca) ) ) ).

% ord_less_eq_trans
tff(fact_1630_ord__less__eq__trans,axiom,
    ! [A: num,B: num,Ca: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,A),B)
     => ( ( B = Ca )
       => aa_num_o(aa_num_fun_num_o(ord_less_num,A),Ca) ) ) ).

% ord_less_eq_trans
tff(fact_1631_ord__less__eq__trans,axiom,
    ! [A: nat,B: nat,Ca: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( ( B = Ca )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),Ca) ) ) ).

% ord_less_eq_trans
tff(fact_1632_ord__less__eq__trans,axiom,
    ! [A: int,B: int,Ca: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( ( B = Ca )
       => aa_int_o(aa_int_fun_int_o(ord_less_int,A),Ca) ) ) ).

% ord_less_eq_trans
tff(fact_1633_less__induct,axiom,
    ! [Pa: fun_nat_o,A: nat] :
      ( ! [X3: nat] :
          ( ! [Y4: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y4),X3)
             => aa_nat_o(Pa,Y4) )
         => aa_nat_o(Pa,X3) )
     => aa_nat_o(Pa,A) ) ).

% less_induct
tff(fact_1634_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_1635_antisym__conv3,axiom,
    ! [Y: num,X: num] :
      ( ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Y),X)
     => ( ~ aa_num_o(aa_num_fun_num_o(ord_less_num,X),Y)
      <=> ( X = Y ) ) ) ).

% antisym_conv3
tff(fact_1636_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_1637_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_1638_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_1639_linorder__cases,axiom,
    ! [X: num,Y: num] :
      ( ~ aa_num_o(aa_num_fun_num_o(ord_less_num,X),Y)
     => ( ( X != Y )
       => aa_num_o(aa_num_fun_num_o(ord_less_num,Y),X) ) ) ).

% linorder_cases
tff(fact_1640_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_1641_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_1642_dual__order_Oasym,axiom,
    ! [B: assn,A: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,B),A)
     => ~ aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),B) ) ).

% dual_order.asym
tff(fact_1643_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_1644_dual__order_Oasym,axiom,
    ! [B: num,A: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,B),A)
     => ~ aa_num_o(aa_num_fun_num_o(ord_less_num,A),B) ) ).

% dual_order.asym
tff(fact_1645_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_1646_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_1647_dual__order_Oirrefl,axiom,
    ! [A: assn] : ~ aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),A) ).

% dual_order.irrefl
tff(fact_1648_dual__order_Oirrefl,axiom,
    ! [A: rat] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),A) ).

% dual_order.irrefl
tff(fact_1649_dual__order_Oirrefl,axiom,
    ! [A: num] : ~ aa_num_o(aa_num_fun_num_o(ord_less_num,A),A) ).

% dual_order.irrefl
tff(fact_1650_dual__order_Oirrefl,axiom,
    ! [A: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),A) ).

% dual_order.irrefl
tff(fact_1651_dual__order_Oirrefl,axiom,
    ! [A: int] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,A),A) ).

% dual_order.irrefl
tff(fact_1652_exists__least__iff,axiom,
    ! [Pa: fun_nat_o] :
      ( ? [X_12: nat] : aa_nat_o(Pa,X_12)
    <=> ? [N: nat] :
          ( aa_nat_o(Pa,N)
          & ! [M3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M3),N)
             => ~ aa_nat_o(Pa,M3) ) ) ) ).

% exists_least_iff
tff(fact_1653_linorder__less__wlog,axiom,
    ! [Pa: 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(Pa,A4),B4) )
     => ( ! [A4: rat] : aa_rat_o(aa_rat_fun_rat_o(Pa,A4),A4)
       => ( ! [A4: rat,B4: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(Pa,B4),A4)
             => aa_rat_o(aa_rat_fun_rat_o(Pa,A4),B4) )
         => aa_rat_o(aa_rat_fun_rat_o(Pa,A),B) ) ) ) ).

% linorder_less_wlog
tff(fact_1654_linorder__less__wlog,axiom,
    ! [Pa: fun_num_fun_num_o,A: num,B: num] :
      ( ! [A4: num,B4: num] :
          ( aa_num_o(aa_num_fun_num_o(ord_less_num,A4),B4)
         => aa_num_o(aa_num_fun_num_o(Pa,A4),B4) )
     => ( ! [A4: num] : aa_num_o(aa_num_fun_num_o(Pa,A4),A4)
       => ( ! [A4: num,B4: num] :
              ( aa_num_o(aa_num_fun_num_o(Pa,B4),A4)
             => aa_num_o(aa_num_fun_num_o(Pa,A4),B4) )
         => aa_num_o(aa_num_fun_num_o(Pa,A),B) ) ) ) ).

% linorder_less_wlog
tff(fact_1655_linorder__less__wlog,axiom,
    ! [Pa: 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(Pa,A4),B4) )
     => ( ! [A4: nat] : aa_nat_o(aa_nat_fun_nat_o(Pa,A4),A4)
       => ( ! [A4: nat,B4: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(Pa,B4),A4)
             => aa_nat_o(aa_nat_fun_nat_o(Pa,A4),B4) )
         => aa_nat_o(aa_nat_fun_nat_o(Pa,A),B) ) ) ) ).

% linorder_less_wlog
tff(fact_1656_linorder__less__wlog,axiom,
    ! [Pa: 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(Pa,A4),B4) )
     => ( ! [A4: int] : aa_int_o(aa_int_fun_int_o(Pa,A4),A4)
       => ( ! [A4: int,B4: int] :
              ( aa_int_o(aa_int_fun_int_o(Pa,B4),A4)
             => aa_int_o(aa_int_fun_int_o(Pa,A4),B4) )
         => aa_int_o(aa_int_fun_int_o(Pa,A),B) ) ) ) ).

% linorder_less_wlog
tff(fact_1657_order_Ostrict__trans,axiom,
    ! [A: assn,B: assn,Ca: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),B)
     => ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,B),Ca)
       => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),Ca) ) ) ).

% order.strict_trans
tff(fact_1658_order_Ostrict__trans,axiom,
    ! [A: rat,B: rat,Ca: 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),Ca)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),Ca) ) ) ).

% order.strict_trans
tff(fact_1659_order_Ostrict__trans,axiom,
    ! [A: num,B: num,Ca: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,A),B)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,B),Ca)
       => aa_num_o(aa_num_fun_num_o(ord_less_num,A),Ca) ) ) ).

% order.strict_trans
tff(fact_1660_order_Ostrict__trans,axiom,
    ! [A: nat,B: nat,Ca: 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),Ca)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),Ca) ) ) ).

% order.strict_trans
tff(fact_1661_order_Ostrict__trans,axiom,
    ! [A: int,B: int,Ca: 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),Ca)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,A),Ca) ) ) ).

% order.strict_trans
tff(fact_1662_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_1663_not__less__iff__gr__or__eq,axiom,
    ! [X: num,Y: num] :
      ( ~ aa_num_o(aa_num_fun_num_o(ord_less_num,X),Y)
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_num,Y),X)
        | ( X = Y ) ) ) ).

% not_less_iff_gr_or_eq
tff(fact_1664_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_1665_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_1666_dual__order_Ostrict__trans,axiom,
    ! [B: assn,A: assn,Ca: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,B),A)
     => ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,Ca),B)
       => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,Ca),A) ) ) ).

% dual_order.strict_trans
tff(fact_1667_dual__order_Ostrict__trans,axiom,
    ! [B: rat,A: rat,Ca: 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,Ca),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Ca),A) ) ) ).

% dual_order.strict_trans
tff(fact_1668_dual__order_Ostrict__trans,axiom,
    ! [B: num,A: num,Ca: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,B),A)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,Ca),B)
       => aa_num_o(aa_num_fun_num_o(ord_less_num,Ca),A) ) ) ).

% dual_order.strict_trans
tff(fact_1669_dual__order_Ostrict__trans,axiom,
    ! [B: nat,A: nat,Ca: 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,Ca),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ca),A) ) ) ).

% dual_order.strict_trans
tff(fact_1670_dual__order_Ostrict__trans,axiom,
    ! [B: int,A: int,Ca: 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,Ca),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,Ca),A) ) ) ).

% dual_order.strict_trans
tff(fact_1671_order_Ostrict__implies__not__eq,axiom,
    ! [A: assn,B: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),B)
     => ( A != B ) ) ).

% order.strict_implies_not_eq
tff(fact_1672_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_1673_order_Ostrict__implies__not__eq,axiom,
    ! [A: num,B: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,A),B)
     => ( A != B ) ) ).

% order.strict_implies_not_eq
tff(fact_1674_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_1675_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_1676_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B: assn,A: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,B),A)
     => ( A != B ) ) ).

% dual_order.strict_implies_not_eq
tff(fact_1677_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_1678_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B: num,A: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,B),A)
     => ( A != B ) ) ).

% dual_order.strict_implies_not_eq
tff(fact_1679_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_1680_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_1681_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_1682_linorder__neqE,axiom,
    ! [X: num,Y: num] :
      ( ( X != Y )
     => ( ~ aa_num_o(aa_num_fun_num_o(ord_less_num,X),Y)
       => aa_num_o(aa_num_fun_num_o(ord_less_num,Y),X) ) ) ).

% linorder_neqE
tff(fact_1683_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_1684_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_1685_order__less__asym,axiom,
    ! [X: assn,Y: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X),Y)
     => ~ aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,Y),X) ) ).

% order_less_asym
tff(fact_1686_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_1687_order__less__asym,axiom,
    ! [X: num,Y: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,X),Y)
     => ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Y),X) ) ).

% order_less_asym
tff(fact_1688_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_1689_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_1690_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_1691_linorder__neq__iff,axiom,
    ! [X: num,Y: num] :
      ( ( X != Y )
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_num,X),Y)
        | aa_num_o(aa_num_fun_num_o(ord_less_num,Y),X) ) ) ).

% linorder_neq_iff
tff(fact_1692_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_1693_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_1694_order__less__asym_H,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_assn,B),A) ) ).

% order_less_asym'
tff(fact_1695_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_1696_order__less__asym_H,axiom,
    ! [A: num,B: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,A),B)
     => ~ aa_num_o(aa_num_fun_num_o(ord_less_num,B),A) ) ).

% order_less_asym'
tff(fact_1697_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_1698_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_1699_order__less__trans,axiom,
    ! [X: assn,Y: assn,Z: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X),Y)
     => ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,Y),Z)
       => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X),Z) ) ) ).

% order_less_trans
tff(fact_1700_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_1701_order__less__trans,axiom,
    ! [X: num,Y: num,Z: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,X),Y)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,Y),Z)
       => aa_num_o(aa_num_fun_num_o(ord_less_num,X),Z) ) ) ).

% order_less_trans
tff(fact_1702_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_1703_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_1704_ord__eq__less__subst,axiom,
    ! [A: assn,F: fun_assn_assn,B: assn,Ca: assn] :
      ( ( A = aa_assn_assn(F,B) )
     => ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,B),Ca)
       => ( ! [X3: assn,Y3: assn] :
              ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X3),Y3)
             => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_assn_assn(F,X3)),aa_assn_assn(F,Y3)) )
         => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),aa_assn_assn(F,Ca)) ) ) ) ).

% ord_eq_less_subst
tff(fact_1705_ord__eq__less__subst,axiom,
    ! [A: rat,F: fun_assn_rat,B: assn,Ca: assn] :
      ( ( A = aa_assn_rat(F,B) )
     => ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,B),Ca)
       => ( ! [X3: assn,Y3: assn] :
              ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X3),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_assn_rat(F,X3)),aa_assn_rat(F,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_assn_rat(F,Ca)) ) ) ) ).

% ord_eq_less_subst
tff(fact_1706_ord__eq__less__subst,axiom,
    ! [A: num,F: fun_assn_num,B: assn,Ca: assn] :
      ( ( A = aa_assn_num(F,B) )
     => ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,B),Ca)
       => ( ! [X3: assn,Y3: assn] :
              ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X3),Y3)
             => aa_num_o(aa_num_fun_num_o(ord_less_num,aa_assn_num(F,X3)),aa_assn_num(F,Y3)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_num,A),aa_assn_num(F,Ca)) ) ) ) ).

% ord_eq_less_subst
tff(fact_1707_ord__eq__less__subst,axiom,
    ! [A: nat,F: fun_assn_nat,B: assn,Ca: assn] :
      ( ( A = aa_assn_nat(F,B) )
     => ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,B),Ca)
       => ( ! [X3: assn,Y3: assn] :
              ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X3),Y3)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_assn_nat(F,X3)),aa_assn_nat(F,Y3)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),aa_assn_nat(F,Ca)) ) ) ) ).

% ord_eq_less_subst
tff(fact_1708_ord__eq__less__subst,axiom,
    ! [A: int,F: fun_assn_int,B: assn,Ca: assn] :
      ( ( A = aa_assn_int(F,B) )
     => ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,B),Ca)
       => ( ! [X3: assn,Y3: assn] :
              ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X3),Y3)
             => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_assn_int(F,X3)),aa_assn_int(F,Y3)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_assn_int(F,Ca)) ) ) ) ).

% ord_eq_less_subst
tff(fact_1709_ord__eq__less__subst,axiom,
    ! [A: assn,F: fun_rat_assn,B: rat,Ca: rat] :
      ( ( A = aa_rat_assn(F,B) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),Ca)
       => ( ! [X3: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Y3)
             => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_rat_assn(F,X3)),aa_rat_assn(F,Y3)) )
         => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),aa_rat_assn(F,Ca)) ) ) ) ).

% ord_eq_less_subst
tff(fact_1710_ord__eq__less__subst,axiom,
    ! [A: rat,F: fun_rat_rat,B: rat,Ca: rat] :
      ( ( A = aa_rat_rat(F,B) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),Ca)
       => ( ! [X3: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(F,X3)),aa_rat_rat(F,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(F,Ca)) ) ) ) ).

% ord_eq_less_subst
tff(fact_1711_ord__eq__less__subst,axiom,
    ! [A: num,F: fun_rat_num,B: rat,Ca: rat] :
      ( ( A = aa_rat_num(F,B) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),Ca)
       => ( ! [X3: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Y3)
             => aa_num_o(aa_num_fun_num_o(ord_less_num,aa_rat_num(F,X3)),aa_rat_num(F,Y3)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_num,A),aa_rat_num(F,Ca)) ) ) ) ).

% ord_eq_less_subst
tff(fact_1712_ord__eq__less__subst,axiom,
    ! [A: nat,F: fun_rat_nat,B: rat,Ca: rat] :
      ( ( A = aa_rat_nat(F,B) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),Ca)
       => ( ! [X3: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Y3)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_rat_nat(F,X3)),aa_rat_nat(F,Y3)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),aa_rat_nat(F,Ca)) ) ) ) ).

% ord_eq_less_subst
tff(fact_1713_ord__eq__less__subst,axiom,
    ! [A: int,F: fun_rat_int,B: rat,Ca: rat] :
      ( ( A = aa_rat_int(F,B) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),Ca)
       => ( ! [X3: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Y3)
             => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(F,X3)),aa_rat_int(F,Y3)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_rat_int(F,Ca)) ) ) ) ).

% ord_eq_less_subst
tff(fact_1714_ord__less__eq__subst,axiom,
    ! [A: assn,B: assn,F: fun_assn_assn,Ca: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),B)
     => ( ( aa_assn_assn(F,B) = Ca )
       => ( ! [X3: assn,Y3: assn] :
              ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X3),Y3)
             => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_assn_assn(F,X3)),aa_assn_assn(F,Y3)) )
         => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_assn_assn(F,A)),Ca) ) ) ) ).

% ord_less_eq_subst
tff(fact_1715_ord__less__eq__subst,axiom,
    ! [A: assn,B: assn,F: fun_assn_rat,Ca: rat] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),B)
     => ( ( aa_assn_rat(F,B) = Ca )
       => ( ! [X3: assn,Y3: assn] :
              ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X3),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_assn_rat(F,X3)),aa_assn_rat(F,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_assn_rat(F,A)),Ca) ) ) ) ).

% ord_less_eq_subst
tff(fact_1716_ord__less__eq__subst,axiom,
    ! [A: assn,B: assn,F: fun_assn_num,Ca: num] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),B)
     => ( ( aa_assn_num(F,B) = Ca )
       => ( ! [X3: assn,Y3: assn] :
              ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X3),Y3)
             => aa_num_o(aa_num_fun_num_o(ord_less_num,aa_assn_num(F,X3)),aa_assn_num(F,Y3)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_num,aa_assn_num(F,A)),Ca) ) ) ) ).

% ord_less_eq_subst
tff(fact_1717_ord__less__eq__subst,axiom,
    ! [A: assn,B: assn,F: fun_assn_nat,Ca: nat] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),B)
     => ( ( aa_assn_nat(F,B) = Ca )
       => ( ! [X3: assn,Y3: assn] :
              ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X3),Y3)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_assn_nat(F,X3)),aa_assn_nat(F,Y3)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_assn_nat(F,A)),Ca) ) ) ) ).

% ord_less_eq_subst
tff(fact_1718_ord__less__eq__subst,axiom,
    ! [A: assn,B: assn,F: fun_assn_int,Ca: int] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),B)
     => ( ( aa_assn_int(F,B) = Ca )
       => ( ! [X3: assn,Y3: assn] :
              ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X3),Y3)
             => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_assn_int(F,X3)),aa_assn_int(F,Y3)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_assn_int(F,A)),Ca) ) ) ) ).

% ord_less_eq_subst
tff(fact_1719_ord__less__eq__subst,axiom,
    ! [A: rat,B: rat,F: fun_rat_assn,Ca: assn] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( ( aa_rat_assn(F,B) = Ca )
       => ( ! [X3: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Y3)
             => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_rat_assn(F,X3)),aa_rat_assn(F,Y3)) )
         => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_rat_assn(F,A)),Ca) ) ) ) ).

% ord_less_eq_subst
tff(fact_1720_ord__less__eq__subst,axiom,
    ! [A: rat,B: rat,F: fun_rat_rat,Ca: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( ( aa_rat_rat(F,B) = Ca )
       => ( ! [X3: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(F,X3)),aa_rat_rat(F,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(F,A)),Ca) ) ) ) ).

% ord_less_eq_subst
tff(fact_1721_ord__less__eq__subst,axiom,
    ! [A: rat,B: rat,F: fun_rat_num,Ca: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( ( aa_rat_num(F,B) = Ca )
       => ( ! [X3: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Y3)
             => aa_num_o(aa_num_fun_num_o(ord_less_num,aa_rat_num(F,X3)),aa_rat_num(F,Y3)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_num,aa_rat_num(F,A)),Ca) ) ) ) ).

% ord_less_eq_subst
tff(fact_1722_ord__less__eq__subst,axiom,
    ! [A: rat,B: rat,F: fun_rat_nat,Ca: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( ( aa_rat_nat(F,B) = Ca )
       => ( ! [X3: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Y3)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_rat_nat(F,X3)),aa_rat_nat(F,Y3)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_rat_nat(F,A)),Ca) ) ) ) ).

% ord_less_eq_subst
tff(fact_1723_ord__less__eq__subst,axiom,
    ! [A: rat,B: rat,F: fun_rat_int,Ca: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( ( aa_rat_int(F,B) = Ca )
       => ( ! [X3: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Y3)
             => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(F,X3)),aa_rat_int(F,Y3)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(F,A)),Ca) ) ) ) ).

% ord_less_eq_subst
tff(fact_1724_order__less__irrefl,axiom,
    ! [X: assn] : ~ aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X),X) ).

% order_less_irrefl
tff(fact_1725_order__less__irrefl,axiom,
    ! [X: rat] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),X) ).

% order_less_irrefl
tff(fact_1726_order__less__irrefl,axiom,
    ! [X: num] : ~ aa_num_o(aa_num_fun_num_o(ord_less_num,X),X) ).

% order_less_irrefl
tff(fact_1727_order__less__irrefl,axiom,
    ! [X: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),X) ).

% order_less_irrefl
tff(fact_1728_order__less__irrefl,axiom,
    ! [X: int] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,X),X) ).

% order_less_irrefl
tff(fact_1729_order__less__subst1,axiom,
    ! [A: assn,F: fun_assn_assn,B: assn,Ca: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),aa_assn_assn(F,B))
     => ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,B),Ca)
       => ( ! [X3: assn,Y3: assn] :
              ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X3),Y3)
             => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_assn_assn(F,X3)),aa_assn_assn(F,Y3)) )
         => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),aa_assn_assn(F,Ca)) ) ) ) ).

% order_less_subst1
tff(fact_1730_order__less__subst1,axiom,
    ! [A: assn,F: fun_rat_assn,B: rat,Ca: rat] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),aa_rat_assn(F,B))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),Ca)
       => ( ! [X3: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Y3)
             => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_rat_assn(F,X3)),aa_rat_assn(F,Y3)) )
         => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),aa_rat_assn(F,Ca)) ) ) ) ).

% order_less_subst1
tff(fact_1731_order__less__subst1,axiom,
    ! [A: assn,F: fun_num_assn,B: num,Ca: num] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),aa_num_assn(F,B))
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,B),Ca)
       => ( ! [X3: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_num,X3),Y3)
             => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_num_assn(F,X3)),aa_num_assn(F,Y3)) )
         => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),aa_num_assn(F,Ca)) ) ) ) ).

% order_less_subst1
tff(fact_1732_order__less__subst1,axiom,
    ! [A: assn,F: fun_nat_assn,B: nat,Ca: nat] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),aa_nat_assn(F,B))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),Ca)
       => ( ! [X3: nat,Y3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Y3)
             => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_nat_assn(F,X3)),aa_nat_assn(F,Y3)) )
         => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),aa_nat_assn(F,Ca)) ) ) ) ).

% order_less_subst1
tff(fact_1733_order__less__subst1,axiom,
    ! [A: assn,F: fun_int_assn,B: int,Ca: int] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),aa_int_assn(F,B))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),Ca)
       => ( ! [X3: int,Y3: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Y3)
             => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_int_assn(F,X3)),aa_int_assn(F,Y3)) )
         => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),aa_int_assn(F,Ca)) ) ) ) ).

% order_less_subst1
tff(fact_1734_order__less__subst1,axiom,
    ! [A: rat,F: fun_assn_rat,B: assn,Ca: assn] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_assn_rat(F,B))
     => ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,B),Ca)
       => ( ! [X3: assn,Y3: assn] :
              ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X3),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_assn_rat(F,X3)),aa_assn_rat(F,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_assn_rat(F,Ca)) ) ) ) ).

% order_less_subst1
tff(fact_1735_order__less__subst1,axiom,
    ! [A: rat,F: fun_rat_rat,B: rat,Ca: 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),Ca)
       => ( ! [X3: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(F,X3)),aa_rat_rat(F,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(F,Ca)) ) ) ) ).

% order_less_subst1
tff(fact_1736_order__less__subst1,axiom,
    ! [A: rat,F: fun_num_rat,B: num,Ca: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_num_rat(F,B))
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,B),Ca)
       => ( ! [X3: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_num,X3),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(F,X3)),aa_num_rat(F,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_num_rat(F,Ca)) ) ) ) ).

% order_less_subst1
tff(fact_1737_order__less__subst1,axiom,
    ! [A: rat,F: fun_nat_rat,B: nat,Ca: 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),Ca)
       => ( ! [X3: nat,Y3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(F,X3)),aa_nat_rat(F,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_nat_rat(F,Ca)) ) ) ) ).

% order_less_subst1
tff(fact_1738_order__less__subst1,axiom,
    ! [A: rat,F: fun_int_rat,B: int,Ca: 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),Ca)
       => ( ! [X3: int,Y3: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(F,X3)),aa_int_rat(F,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_int_rat(F,Ca)) ) ) ) ).

% order_less_subst1
tff(fact_1739_order__less__subst2,axiom,
    ! [A: assn,B: assn,F: fun_assn_assn,Ca: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),B)
     => ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_assn_assn(F,B)),Ca)
       => ( ! [X3: assn,Y3: assn] :
              ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X3),Y3)
             => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_assn_assn(F,X3)),aa_assn_assn(F,Y3)) )
         => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_assn_assn(F,A)),Ca) ) ) ) ).

% order_less_subst2
tff(fact_1740_order__less__subst2,axiom,
    ! [A: assn,B: assn,F: fun_assn_rat,Ca: rat] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_assn_rat(F,B)),Ca)
       => ( ! [X3: assn,Y3: assn] :
              ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X3),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_assn_rat(F,X3)),aa_assn_rat(F,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_assn_rat(F,A)),Ca) ) ) ) ).

% order_less_subst2
tff(fact_1741_order__less__subst2,axiom,
    ! [A: assn,B: assn,F: fun_assn_num,Ca: num] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),B)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,aa_assn_num(F,B)),Ca)
       => ( ! [X3: assn,Y3: assn] :
              ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X3),Y3)
             => aa_num_o(aa_num_fun_num_o(ord_less_num,aa_assn_num(F,X3)),aa_assn_num(F,Y3)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_num,aa_assn_num(F,A)),Ca) ) ) ) ).

% order_less_subst2
tff(fact_1742_order__less__subst2,axiom,
    ! [A: assn,B: assn,F: fun_assn_nat,Ca: nat] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_assn_nat(F,B)),Ca)
       => ( ! [X3: assn,Y3: assn] :
              ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X3),Y3)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_assn_nat(F,X3)),aa_assn_nat(F,Y3)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_assn_nat(F,A)),Ca) ) ) ) ).

% order_less_subst2
tff(fact_1743_order__less__subst2,axiom,
    ! [A: assn,B: assn,F: fun_assn_int,Ca: int] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_assn_int(F,B)),Ca)
       => ( ! [X3: assn,Y3: assn] :
              ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X3),Y3)
             => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_assn_int(F,X3)),aa_assn_int(F,Y3)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_assn_int(F,A)),Ca) ) ) ) ).

% order_less_subst2
tff(fact_1744_order__less__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_assn,Ca: assn] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_rat_assn(F,B)),Ca)
       => ( ! [X3: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Y3)
             => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_rat_assn(F,X3)),aa_rat_assn(F,Y3)) )
         => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_rat_assn(F,A)),Ca) ) ) ) ).

% order_less_subst2
tff(fact_1745_order__less__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_rat,Ca: 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)),Ca)
       => ( ! [X3: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(F,X3)),aa_rat_rat(F,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(F,A)),Ca) ) ) ) ).

% order_less_subst2
tff(fact_1746_order__less__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_num,Ca: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,aa_rat_num(F,B)),Ca)
       => ( ! [X3: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Y3)
             => aa_num_o(aa_num_fun_num_o(ord_less_num,aa_rat_num(F,X3)),aa_rat_num(F,Y3)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_num,aa_rat_num(F,A)),Ca) ) ) ) ).

% order_less_subst2
tff(fact_1747_order__less__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_nat,Ca: 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)),Ca)
       => ( ! [X3: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Y3)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_rat_nat(F,X3)),aa_rat_nat(F,Y3)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_rat_nat(F,A)),Ca) ) ) ) ).

% order_less_subst2
tff(fact_1748_order__less__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_int,Ca: 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)),Ca)
       => ( ! [X3: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Y3)
             => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(F,X3)),aa_rat_int(F,Y3)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(F,A)),Ca) ) ) ) ).

% order_less_subst2
tff(fact_1749_order__less__not__sym,axiom,
    ! [X: assn,Y: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X),Y)
     => ~ aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,Y),X) ) ).

% order_less_not_sym
tff(fact_1750_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_1751_order__less__not__sym,axiom,
    ! [X: num,Y: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,X),Y)
     => ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Y),X) ) ).

% order_less_not_sym
tff(fact_1752_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_1753_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_1754_order__less__imp__triv,axiom,
    ! [X: assn,Y: assn,Pa: $o] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X),Y)
     => ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,Y),X)
       => (Pa) ) ) ).

% order_less_imp_triv
tff(fact_1755_order__less__imp__triv,axiom,
    ! [X: rat,Y: rat,Pa: $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)
       => (Pa) ) ) ).

% order_less_imp_triv
tff(fact_1756_order__less__imp__triv,axiom,
    ! [X: num,Y: num,Pa: $o] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,X),Y)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,Y),X)
       => (Pa) ) ) ).

% order_less_imp_triv
tff(fact_1757_order__less__imp__triv,axiom,
    ! [X: nat,Y: nat,Pa: $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)
       => (Pa) ) ) ).

% order_less_imp_triv
tff(fact_1758_order__less__imp__triv,axiom,
    ! [X: int,Y: int,Pa: $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)
       => (Pa) ) ) ).

% order_less_imp_triv
tff(fact_1759_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_1760_linorder__less__linear,axiom,
    ! [X: num,Y: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,X),Y)
      | ( X = Y )
      | aa_num_o(aa_num_fun_num_o(ord_less_num,Y),X) ) ).

% linorder_less_linear
tff(fact_1761_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_1762_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_1763_order__less__imp__not__eq,axiom,
    ! [X: assn,Y: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X),Y)
     => ( X != Y ) ) ).

% order_less_imp_not_eq
tff(fact_1764_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_1765_order__less__imp__not__eq,axiom,
    ! [X: num,Y: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,X),Y)
     => ( X != Y ) ) ).

% order_less_imp_not_eq
tff(fact_1766_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_1767_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_1768_order__less__imp__not__eq2,axiom,
    ! [X: assn,Y: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X),Y)
     => ( Y != X ) ) ).

% order_less_imp_not_eq2
tff(fact_1769_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_1770_order__less__imp__not__eq2,axiom,
    ! [X: num,Y: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,X),Y)
     => ( Y != X ) ) ).

% order_less_imp_not_eq2
tff(fact_1771_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_1772_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_1773_order__less__imp__not__less,axiom,
    ! [X: assn,Y: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X),Y)
     => ~ aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,Y),X) ) ).

% order_less_imp_not_less
tff(fact_1774_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_1775_order__less__imp__not__less,axiom,
    ! [X: num,Y: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,X),Y)
     => ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Y),X) ) ).

% order_less_imp_not_less
tff(fact_1776_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_1777_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_1778_subset__Collect__conv,axiom,
    ! [S: set_Pr958786334691620121nt_int,Pa: fun_Pr3238374608701176162_int_o] :
      ( ord_le2843351958646193337nt_int(S,collec213857154873943460nt_int(Pa))
    <=> ! [X4: product_prod_int_int] :
          ( member5262025264175285858nt_int(X4,S)
         => aa_Pro4592068669816069595_int_o(Pa,X4) ) ) ).

% subset_Collect_conv
tff(fact_1779_subset__Collect__conv,axiom,
    ! [S: set_list_nat,Pa: fun_list_nat_o] :
      ( ord_le6045566169113846134st_nat(S,collect_list_nat(Pa))
    <=> ! [X4: list_nat] :
          ( member_list_nat(X4,S)
         => aa_list_nat_o(Pa,X4) ) ) ).

% subset_Collect_conv
tff(fact_1780_subset__Collect__conv,axiom,
    ! [S: set_set_nat,Pa: fun_set_nat_o] :
      ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(S),collect_set_nat(Pa))
    <=> ! [X4: set_nat] :
          ( aa_set_set_nat_o(member_set_nat(X4),S)
         => aa_set_nat_o(Pa,X4) ) ) ).

% subset_Collect_conv
tff(fact_1781_subset__Collect__conv,axiom,
    ! [S: set_nat,Pa: fun_nat_o] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,S),collect_nat(Pa))
    <=> ! [X4: nat] :
          ( aa_set_nat_o(member_nat(X4),S)
         => aa_nat_o(Pa,X4) ) ) ).

% subset_Collect_conv
tff(fact_1782_subset__Collect__conv,axiom,
    ! [S: set_int,Pa: fun_int_o] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,S),collect_int(Pa))
    <=> ! [X4: int] :
          ( aa_set_int_o(member_int(X4),S)
         => aa_int_o(Pa,X4) ) ) ).

% subset_Collect_conv
tff(fact_1783_pred__subset__eq,axiom,
    ! [Ra: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
      ( aa_fun8408151826566693824at_o_o(ord_le1572679713340831354_nat_o(aTP_Lamm_a(Ra)),aTP_Lamm_a(S))
    <=> aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,Ra),S) ) ).

% pred_subset_eq
tff(fact_1784_pred__subset__eq,axiom,
    ! [Ra: set_o,S: set_o] :
      ( ord_less_eq_fun_o_o(aTP_Lamm_aa(Ra),aTP_Lamm_aa(S))
    <=> ord_less_eq_set_o(Ra,S) ) ).

% pred_subset_eq
tff(fact_1785_pred__subset__eq,axiom,
    ! [Ra: set_set_nat,S: set_set_nat] :
      ( ord_le6468474429079055883_nat_o(aTP_Lamm_ad(Ra),aTP_Lamm_ad(S))
    <=> aa_set_set_nat_o(ord_le6893508408891458716et_nat(Ra),S) ) ).

% pred_subset_eq
tff(fact_1786_pred__subset__eq,axiom,
    ! [Ra: set_nat,S: set_nat] :
      ( ord_le7240017405068453697_nat_o(aTP_Lamm_ae(Ra),aTP_Lamm_ae(S))
    <=> aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Ra),S) ) ).

% pred_subset_eq
tff(fact_1787_pred__subset__eq,axiom,
    ! [Ra: set_int,S: set_int] :
      ( ord_le5418975059635765733_int_o(aTP_Lamm_af(Ra),aTP_Lamm_af(S))
    <=> aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,Ra),S) ) ).

% pred_subset_eq
tff(fact_1788_Set_Oempty__def,axiom,
    bot_bo1796632182523588997nt_int = collec213857154873943460nt_int(aTP_Lamm_bb) ).

% Set.empty_def
tff(fact_1789_Set_Oempty__def,axiom,
    bot_bot_set_list_nat = collect_list_nat(aTP_Lamm_bc) ).

% Set.empty_def
tff(fact_1790_Set_Oempty__def,axiom,
    bot_bot_set_set_nat = collect_set_nat(aTP_Lamm_bd) ).

% Set.empty_def
tff(fact_1791_Set_Oempty__def,axiom,
    bot_bo2099793752762293965at_nat = collec3392354462482085612at_nat(aTP_Lamm_be) ).

% Set.empty_def
tff(fact_1792_Set_Oempty__def,axiom,
    bot_bot_set_o = collect_o(aTP_Lamm_bf) ).

% Set.empty_def
tff(fact_1793_Set_Oempty__def,axiom,
    bot_bot_set_nat = collect_nat(aTP_Lamm_bg) ).

% Set.empty_def
tff(fact_1794_Set_Oempty__def,axiom,
    bot_bot_set_int = collect_int(aTP_Lamm_bh) ).

% Set.empty_def
tff(fact_1795_Int__def,axiom,
    ! [A3: set_o,B3: set_o] : inf_inf_set_o(A3,B3) = collect_o(aa_set_o_fun_o_o(aTP_Lamm_bi(A3),B3)) ).

% Int_def
tff(fact_1796_Int__def,axiom,
    ! [A3: set_Pr958786334691620121nt_int,B3: set_Pr958786334691620121nt_int] : inf_in2269163501485487111nt_int(A3,B3) = collec213857154873943460nt_int(aa_set2224860389584585407_int_o(aTP_Lamm_bj(A3),B3)) ).

% Int_def
tff(fact_1797_Int__def,axiom,
    ! [A3: set_list_nat,B3: set_list_nat] : inf_inf_set_list_nat(A3,B3) = collect_list_nat(aa_set6675653781276780861_nat_o(aTP_Lamm_bk(A3),B3)) ).

% Int_def
tff(fact_1798_Int__def,axiom,
    ! [A3: set_set_nat,B3: set_set_nat] : inf_inf_set_set_nat(A3,B3) = collect_set_nat(aa_set4404905041378927473_nat_o(aTP_Lamm_bl(A3),B3)) ).

% Int_def
tff(fact_1799_Int__def,axiom,
    ! [A3: set_int,B3: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A3),B3) = collect_int(aa_set_int_fun_int_o(aTP_Lamm_bm(A3),B3)) ).

% Int_def
tff(fact_1800_Int__def,axiom,
    ! [A3: set_nat,B3: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A3),B3) = collect_nat(aa_set_nat_fun_nat_o(aTP_Lamm_bn(A3),B3)) ).

% Int_def
tff(fact_1801_Int__def,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A3),B3) = collec3392354462482085612at_nat(aa_set1857767853180004543_nat_o(aTP_Lamm_bo(A3),B3)) ).

% Int_def
tff(fact_1802_Int__Collect,axiom,
    ! [X: $o,A3: set_o,Pa: fun_o_o] :
      ( member_o((X),inf_inf_set_o(A3,collect_o(Pa)))
    <=> ( member_o((X),A3)
        & aa_o_o(Pa,(X)) ) ) ).

% Int_Collect
tff(fact_1803_Int__Collect,axiom,
    ! [X: product_prod_int_int,A3: set_Pr958786334691620121nt_int,Pa: fun_Pr3238374608701176162_int_o] :
      ( member5262025264175285858nt_int(X,inf_in2269163501485487111nt_int(A3,collec213857154873943460nt_int(Pa)))
    <=> ( member5262025264175285858nt_int(X,A3)
        & aa_Pro4592068669816069595_int_o(Pa,X) ) ) ).

% Int_Collect
tff(fact_1804_Int__Collect,axiom,
    ! [X: list_nat,A3: set_list_nat,Pa: fun_list_nat_o] :
      ( member_list_nat(X,inf_inf_set_list_nat(A3,collect_list_nat(Pa)))
    <=> ( member_list_nat(X,A3)
        & aa_list_nat_o(Pa,X) ) ) ).

% Int_Collect
tff(fact_1805_Int__Collect,axiom,
    ! [X: set_nat,A3: set_set_nat,Pa: fun_set_nat_o] :
      ( aa_set_set_nat_o(member_set_nat(X),inf_inf_set_set_nat(A3,collect_set_nat(Pa)))
    <=> ( aa_set_set_nat_o(member_set_nat(X),A3)
        & aa_set_nat_o(Pa,X) ) ) ).

% Int_Collect
tff(fact_1806_Int__Collect,axiom,
    ! [X: int,A3: set_int,Pa: fun_int_o] :
      ( aa_set_int_o(member_int(X),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A3),collect_int(Pa)))
    <=> ( aa_set_int_o(member_int(X),A3)
        & aa_int_o(Pa,X) ) ) ).

% Int_Collect
tff(fact_1807_Int__Collect,axiom,
    ! [X: nat,A3: set_nat,Pa: fun_nat_o] :
      ( aa_set_nat_o(member_nat(X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A3),collect_nat(Pa)))
    <=> ( aa_set_nat_o(member_nat(X),A3)
        & aa_nat_o(Pa,X) ) ) ).

% Int_Collect
tff(fact_1808_Int__Collect,axiom,
    ! [X: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,Pa: fun_Pr4796943543331071514_nat_o] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A3),collec3392354462482085612at_nat(Pa)))
    <=> ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X),A3)
        & aa_Pro6150637604445964947_nat_o(Pa,X) ) ) ).

% Int_Collect
tff(fact_1809_inf__set__def,axiom,
    ! [A3: set_o,B3: set_o] : inf_inf_set_o(A3,B3) = collect_o(inf_inf_fun_o_o(aTP_Lamm_aa(A3),aTP_Lamm_aa(B3))) ).

% inf_set_def
tff(fact_1810_inf__set__def,axiom,
    ! [A3: set_Pr958786334691620121nt_int,B3: set_Pr958786334691620121nt_int] : inf_in2269163501485487111nt_int(A3,B3) = collec213857154873943460nt_int(inf_in1782598092406300404_int_o(aTP_Lamm_ab(A3),aTP_Lamm_ab(B3))) ).

% inf_set_def
tff(fact_1811_inf__set__def,axiom,
    ! [A3: set_list_nat,B3: set_list_nat] : inf_inf_set_list_nat(A3,B3) = collect_list_nat(inf_in1107944134336960767_nat_o(aTP_Lamm_ac(A3),aTP_Lamm_ac(B3))) ).

% inf_set_def
tff(fact_1812_inf__set__def,axiom,
    ! [A3: set_set_nat,B3: set_set_nat] : inf_inf_set_set_nat(A3,B3) = collect_set_nat(inf_in7602907407060649817_nat_o(aTP_Lamm_ad(A3),aTP_Lamm_ad(B3))) ).

% inf_set_def
tff(fact_1813_inf__set__def,axiom,
    ! [A3: set_int,B3: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A3),B3) = collect_int(inf_inf_fun_int_o(aTP_Lamm_af(A3),aTP_Lamm_af(B3))) ).

% inf_set_def
tff(fact_1814_inf__set__def,axiom,
    ! [A3: set_nat,B3: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A3),B3) = collect_nat(inf_inf_fun_nat_o(aTP_Lamm_ae(A3),aTP_Lamm_ae(B3))) ).

% inf_set_def
tff(fact_1815_inf__set__def,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A3),B3) = collec3392354462482085612at_nat(aa_fun359341089595923442_nat_o(inf_in3341167027036195756_nat_o(aTP_Lamm_a(A3)),aTP_Lamm_a(B3))) ).

% inf_set_def
tff(fact_1816_Collect__conj__eq,axiom,
    ! [Pa: fun_Pr3238374608701176162_int_o,Qa: fun_Pr3238374608701176162_int_o] : collec213857154873943460nt_int(aa_fun5349570537165229698_int_o(aTP_Lamm_bp(Pa),Qa)) = inf_in2269163501485487111nt_int(collec213857154873943460nt_int(Pa),collec213857154873943460nt_int(Qa)) ).

% Collect_conj_eq
tff(fact_1817_Collect__conj__eq,axiom,
    ! [Pa: fun_list_nat_o,Qa: fun_list_nat_o] : collect_list_nat(aa_fun6155608546934989052_nat_o(aTP_Lamm_bq(Pa),Qa)) = inf_inf_set_list_nat(collect_list_nat(Pa),collect_list_nat(Qa)) ).

% Collect_conj_eq
tff(fact_1818_Collect__conj__eq,axiom,
    ! [Pa: fun_set_nat_o,Qa: fun_set_nat_o] : collect_set_nat(aa_fun751667557118977660_nat_o(aTP_Lamm_br(Pa),Qa)) = inf_inf_set_set_nat(collect_set_nat(Pa),collect_set_nat(Qa)) ).

% Collect_conj_eq
tff(fact_1819_Collect__conj__eq,axiom,
    ! [Pa: fun_int_o,Qa: fun_int_o] : collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_bs(Pa),Qa)) = aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,collect_int(Pa)),collect_int(Qa)) ).

% Collect_conj_eq
tff(fact_1820_Collect__conj__eq,axiom,
    ! [Pa: fun_nat_o,Qa: fun_nat_o] : collect_nat(aa_fun5937193402907943164_nat_o(aTP_Lamm_bt(Pa),Qa)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,collect_nat(Pa)),collect_nat(Qa)) ).

% Collect_conj_eq
tff(fact_1821_Collect__conj__eq,axiom,
    ! [Pa: fun_Pr4796943543331071514_nat_o,Qa: fun_Pr4796943543331071514_nat_o] : collec3392354462482085612at_nat(aa_fun359341089595923442_nat_o(aTP_Lamm_bu(Pa),Qa)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,collec3392354462482085612at_nat(Pa)),collec3392354462482085612at_nat(Qa)) ).

% Collect_conj_eq
tff(fact_1822_Un__def,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,A3),B3) = collec3392354462482085612at_nat(aa_set1857767853180004543_nat_o(aTP_Lamm_bv(A3),B3)) ).

% Un_def
tff(fact_1823_Un__def,axiom,
    ! [A3: set_o,B3: set_o] : aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,A3),B3) = collect_o(aa_set_o_fun_o_o(aTP_Lamm_bw(A3),B3)) ).

% Un_def
tff(fact_1824_Un__def,axiom,
    ! [A3: set_Pr958786334691620121nt_int,B3: set_Pr958786334691620121nt_int] : sup_su6024340866399070445nt_int(A3,B3) = collec213857154873943460nt_int(aa_set2224860389584585407_int_o(aTP_Lamm_bx(A3),B3)) ).

% Un_def
tff(fact_1825_Un__def,axiom,
    ! [A3: set_list_nat,B3: set_list_nat] : sup_sup_set_list_nat(A3,B3) = collect_list_nat(aa_set6675653781276780861_nat_o(aTP_Lamm_by(A3),B3)) ).

% Un_def
tff(fact_1826_Un__def,axiom,
    ! [A3: set_set_nat,B3: set_set_nat] : sup_sup_set_set_nat(A3,B3) = collect_set_nat(aa_set4404905041378927473_nat_o(aTP_Lamm_bz(A3),B3)) ).

% Un_def
tff(fact_1827_Un__def,axiom,
    ! [A3: set_int,B3: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A3),B3) = collect_int(aa_set_int_fun_int_o(aTP_Lamm_ca(A3),B3)) ).

% Un_def
tff(fact_1828_Un__def,axiom,
    ! [A3: set_Pr8693737435421807431at_nat,B3: set_Pr8693737435421807431at_nat] : aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,A3),B3) = collec7088162979684241874at_nat(aa_set6735692237302977855_nat_o(aTP_Lamm_cb(A3),B3)) ).

% Un_def
tff(fact_1829_Un__def,axiom,
    ! [A3: set_Pr8551490117392284871at_nat,B3: set_Pr8551490117392284871at_nat] : aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A3),B3) = collec5204685387357076818at_nat(aa_set1219845518824332607_nat_o(aTP_Lamm_cc(A3),B3)) ).

% Un_def
tff(fact_1830_Un__def,axiom,
    ! [A3: set_Pr4329608150637261639at_nat,B3: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A3),B3) = collec6321179662152712658at_nat(aa_set6353399069419443519_nat_o(aTP_Lamm_cd(A3),B3)) ).

% Un_def
tff(fact_1831_Un__def,axiom,
    ! [A3: set_nat,B3: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A3),B3) = collect_nat(aa_set_nat_fun_nat_o(aTP_Lamm_ce(A3),B3)) ).

% Un_def
tff(fact_1832_sup__set__def,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,A3),B3) = collec3392354462482085612at_nat(aa_fun359341089595923442_nat_o(sup_su5606187154157927750_nat_o(aTP_Lamm_a(A3)),aTP_Lamm_a(B3))) ).

% sup_set_def
tff(fact_1833_sup__set__def,axiom,
    ! [A3: set_o,B3: set_o] : aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,A3),B3) = collect_o(sup_sup_fun_o_o(aTP_Lamm_aa(A3),aTP_Lamm_aa(B3))) ).

% sup_set_def
tff(fact_1834_sup__set__def,axiom,
    ! [A3: set_Pr958786334691620121nt_int,B3: set_Pr958786334691620121nt_int] : sup_su6024340866399070445nt_int(A3,B3) = collec213857154873943460nt_int(sup_su4047618219528032398_int_o(aTP_Lamm_ab(A3),aTP_Lamm_ab(B3))) ).

% sup_set_def
tff(fact_1835_sup__set__def,axiom,
    ! [A3: set_list_nat,B3: set_list_nat] : sup_sup_set_list_nat(A3,B3) = collect_list_nat(sup_su6906368799827444965_nat_o(aTP_Lamm_ac(A3),aTP_Lamm_ac(B3))) ).

% sup_set_def
tff(fact_1836_sup__set__def,axiom,
    ! [A3: set_set_nat,B3: set_set_nat] : sup_sup_set_set_nat(A3,B3) = collect_set_nat(sup_su7503774895437956671_nat_o(aTP_Lamm_ad(A3),aTP_Lamm_ad(B3))) ).

% sup_set_def
tff(fact_1837_sup__set__def,axiom,
    ! [A3: set_int,B3: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A3),B3) = collect_int(sup_sup_fun_int_o(aTP_Lamm_af(A3),aTP_Lamm_af(B3))) ).

% sup_set_def
tff(fact_1838_sup__set__def,axiom,
    ! [A3: set_Pr8693737435421807431at_nat,B3: set_Pr8693737435421807431at_nat] : aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,A3),B3) = collec7088162979684241874at_nat(sup_su5487351348554007264_nat_o(aTP_Lamm_ai(A3),aTP_Lamm_ai(B3))) ).

% sup_set_def
tff(fact_1839_sup__set__def,axiom,
    ! [A3: set_Pr8551490117392284871at_nat,B3: set_Pr8551490117392284871at_nat] : aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A3),B3) = collec5204685387357076818at_nat(sup_su5351388498073598816_nat_o(aTP_Lamm_aj(A3),aTP_Lamm_aj(B3))) ).

% sup_set_def
tff(fact_1840_sup__set__def,axiom,
    ! [A3: set_Pr4329608150637261639at_nat,B3: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A3),B3) = collec6321179662152712658at_nat(sup_su2398265404452057824_nat_o(aTP_Lamm_ak(A3),aTP_Lamm_ak(B3))) ).

% sup_set_def
tff(fact_1841_sup__set__def,axiom,
    ! [A3: set_nat,B3: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A3),B3) = collect_nat(sup_sup_fun_nat_o(aTP_Lamm_ae(A3),aTP_Lamm_ae(B3))) ).

% sup_set_def
tff(fact_1842_Collect__disj__eq,axiom,
    ! [Pa: fun_Pr3238374608701176162_int_o,Qa: fun_Pr3238374608701176162_int_o] : collec213857154873943460nt_int(aa_fun5349570537165229698_int_o(aTP_Lamm_cf(Pa),Qa)) = sup_su6024340866399070445nt_int(collec213857154873943460nt_int(Pa),collec213857154873943460nt_int(Qa)) ).

% Collect_disj_eq
tff(fact_1843_Collect__disj__eq,axiom,
    ! [Pa: fun_list_nat_o,Qa: fun_list_nat_o] : collect_list_nat(aa_fun6155608546934989052_nat_o(aTP_Lamm_cg(Pa),Qa)) = sup_sup_set_list_nat(collect_list_nat(Pa),collect_list_nat(Qa)) ).

% Collect_disj_eq
tff(fact_1844_Collect__disj__eq,axiom,
    ! [Pa: fun_set_nat_o,Qa: fun_set_nat_o] : collect_set_nat(aa_fun751667557118977660_nat_o(aTP_Lamm_ch(Pa),Qa)) = sup_sup_set_set_nat(collect_set_nat(Pa),collect_set_nat(Qa)) ).

% Collect_disj_eq
tff(fact_1845_Collect__disj__eq,axiom,
    ! [Pa: fun_int_o,Qa: fun_int_o] : collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_ci(Pa),Qa)) = aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,collect_int(Pa)),collect_int(Qa)) ).

% Collect_disj_eq
tff(fact_1846_Collect__disj__eq,axiom,
    ! [Pa: fun_Pr363642936170290612_nat_o,Qa: fun_Pr363642936170290612_nat_o] : collec7088162979684241874at_nat(aa_fun6414004499577345958_nat_o(aTP_Lamm_cj(Pa),Qa)) = aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,collec7088162979684241874at_nat(Pa)),collec7088162979684241874at_nat(Qa)) ).

% Collect_disj_eq
tff(fact_1847_Collect__disj__eq,axiom,
    ! [Pa: fun_Pr1099658227000185396_nat_o,Qa: fun_Pr1099658227000185396_nat_o] : collec5204685387357076818at_nat(aa_fun9125434449681309094_nat_o(aTP_Lamm_ck(Pa),Qa)) = aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,collec5204685387357076818at_nat(Pa)),collec5204685387357076818at_nat(Qa)) ).

% Collect_disj_eq
tff(fact_1848_Collect__disj__eq,axiom,
    ! [Pa: fun_Pr3059359313727237556_nat_o,Qa: fun_Pr3059359313727237556_nat_o] : collec6321179662152712658at_nat(aa_fun2566793329686429606_nat_o(aTP_Lamm_cl(Pa),Qa)) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,collec6321179662152712658at_nat(Pa)),collec6321179662152712658at_nat(Qa)) ).

% Collect_disj_eq
tff(fact_1849_Collect__disj__eq,axiom,
    ! [Pa: fun_nat_o,Qa: fun_nat_o] : collect_nat(aa_fun5937193402907943164_nat_o(aTP_Lamm_cm(Pa),Qa)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,collect_nat(Pa)),collect_nat(Qa)) ).

% Collect_disj_eq
tff(fact_1850_exists__leI,axiom,
    ! [N2: nat,Pa: fun_nat_o] :
      ( ( ! [N3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N3),N2)
           => ~ aa_nat_o(Pa,N3) )
       => aa_nat_o(Pa,N2) )
     => ? [N4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N4),N2)
          & aa_nat_o(Pa,N4) ) ) ).

% exists_leI
tff(fact_1851_order__le__imp__less__or__eq,axiom,
    ! [X: assn,Y: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X),Y)
     => ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X),Y)
        | ( X = Y ) ) ) ).

% order_le_imp_less_or_eq
tff(fact_1852_order__le__imp__less__or__eq,axiom,
    ! [X: set_int,Y: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,X),Y)
     => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_set_int,X),Y)
        | ( X = Y ) ) ) ).

% order_le_imp_less_or_eq
tff(fact_1853_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_1854_order__le__imp__less__or__eq,axiom,
    ! [X: num,Y: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X),Y)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,X),Y)
        | ( X = Y ) ) ) ).

% order_le_imp_less_or_eq
tff(fact_1855_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_1856_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_1857_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_1858_linorder__le__less__linear,axiom,
    ! [X: num,Y: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X),Y)
      | aa_num_o(aa_num_fun_num_o(ord_less_num,Y),X) ) ).

% linorder_le_less_linear
tff(fact_1859_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_1860_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_1861_order__less__le__subst2,axiom,
    ! [A: assn,B: assn,F: fun_assn_assn,Ca: 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,aa_assn_assn(F,B)),Ca)
       => ( ! [X3: assn,Y3: assn] :
              ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X3),Y3)
             => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_assn_assn(F,X3)),aa_assn_assn(F,Y3)) )
         => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_assn_assn(F,A)),Ca) ) ) ) ).

% order_less_le_subst2
tff(fact_1862_order__less__le__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_assn,Ca: assn] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_rat_assn(F,B)),Ca)
       => ( ! [X3: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Y3)
             => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_rat_assn(F,X3)),aa_rat_assn(F,Y3)) )
         => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_rat_assn(F,A)),Ca) ) ) ) ).

% order_less_le_subst2
tff(fact_1863_order__less__le__subst2,axiom,
    ! [A: num,B: num,F: fun_num_assn,Ca: assn] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,A),B)
     => ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_num_assn(F,B)),Ca)
       => ( ! [X3: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_num,X3),Y3)
             => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_num_assn(F,X3)),aa_num_assn(F,Y3)) )
         => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_num_assn(F,A)),Ca) ) ) ) ).

% order_less_le_subst2
tff(fact_1864_order__less__le__subst2,axiom,
    ! [A: nat,B: nat,F: fun_nat_assn,Ca: assn] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_nat_assn(F,B)),Ca)
       => ( ! [X3: nat,Y3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Y3)
             => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_nat_assn(F,X3)),aa_nat_assn(F,Y3)) )
         => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_nat_assn(F,A)),Ca) ) ) ) ).

% order_less_le_subst2
tff(fact_1865_order__less__le__subst2,axiom,
    ! [A: int,B: int,F: fun_int_assn,Ca: assn] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_int_assn(F,B)),Ca)
       => ( ! [X3: int,Y3: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Y3)
             => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_int_assn(F,X3)),aa_int_assn(F,Y3)) )
         => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_int_assn(F,A)),Ca) ) ) ) ).

% order_less_le_subst2
tff(fact_1866_order__less__le__subst2,axiom,
    ! [A: assn,B: assn,F: fun_assn_rat,Ca: rat] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_assn_rat(F,B)),Ca)
       => ( ! [X3: assn,Y3: assn] :
              ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X3),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_assn_rat(F,X3)),aa_assn_rat(F,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_assn_rat(F,A)),Ca) ) ) ) ).

% order_less_le_subst2
tff(fact_1867_order__less__le__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_rat,Ca: 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)),Ca)
       => ( ! [X3: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(F,X3)),aa_rat_rat(F,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(F,A)),Ca) ) ) ) ).

% order_less_le_subst2
tff(fact_1868_order__less__le__subst2,axiom,
    ! [A: num,B: num,F: fun_num_rat,Ca: rat] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(F,B)),Ca)
       => ( ! [X3: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_num,X3),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(F,X3)),aa_num_rat(F,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(F,A)),Ca) ) ) ) ).

% order_less_le_subst2
tff(fact_1869_order__less__le__subst2,axiom,
    ! [A: nat,B: nat,F: fun_nat_rat,Ca: 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)),Ca)
       => ( ! [X3: nat,Y3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(F,X3)),aa_nat_rat(F,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(F,A)),Ca) ) ) ) ).

% order_less_le_subst2
tff(fact_1870_order__less__le__subst2,axiom,
    ! [A: int,B: int,F: fun_int_rat,Ca: 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)),Ca)
       => ( ! [X3: int,Y3: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(F,X3)),aa_int_rat(F,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(F,A)),Ca) ) ) ) ).

% order_less_le_subst2
tff(fact_1871_order__less__le__subst1,axiom,
    ! [A: assn,F: fun_rat_assn,B: rat,Ca: rat] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),aa_rat_assn(F,B))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),Ca)
       => ( ! [X3: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y3)
             => aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_rat_assn(F,X3)),aa_rat_assn(F,Y3)) )
         => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),aa_rat_assn(F,Ca)) ) ) ) ).

% order_less_le_subst1
tff(fact_1872_order__less__le__subst1,axiom,
    ! [A: rat,F: fun_rat_rat,B: rat,Ca: 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),Ca)
       => ( ! [X3: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(F,X3)),aa_rat_rat(F,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(F,Ca)) ) ) ) ).

% order_less_le_subst1
tff(fact_1873_order__less__le__subst1,axiom,
    ! [A: num,F: fun_rat_num,B: rat,Ca: rat] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,A),aa_rat_num(F,B))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),Ca)
       => ( ! [X3: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y3)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_rat_num(F,X3)),aa_rat_num(F,Y3)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_num,A),aa_rat_num(F,Ca)) ) ) ) ).

% order_less_le_subst1
tff(fact_1874_order__less__le__subst1,axiom,
    ! [A: nat,F: fun_rat_nat,B: rat,Ca: 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),Ca)
       => ( ! [X3: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y3)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_rat_nat(F,X3)),aa_rat_nat(F,Y3)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),aa_rat_nat(F,Ca)) ) ) ) ).

% order_less_le_subst1
tff(fact_1875_order__less__le__subst1,axiom,
    ! [A: int,F: fun_rat_int,B: rat,Ca: 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),Ca)
       => ( ! [X3: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y3)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(F,X3)),aa_rat_int(F,Y3)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_rat_int(F,Ca)) ) ) ) ).

% order_less_le_subst1
tff(fact_1876_order__less__le__subst1,axiom,
    ! [A: assn,F: fun_num_assn,B: num,Ca: num] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),aa_num_assn(F,B))
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),Ca)
       => ( ! [X3: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Y3)
             => aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_num_assn(F,X3)),aa_num_assn(F,Y3)) )
         => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),aa_num_assn(F,Ca)) ) ) ) ).

% order_less_le_subst1
tff(fact_1877_order__less__le__subst1,axiom,
    ! [A: rat,F: fun_num_rat,B: num,Ca: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_num_rat(F,B))
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),Ca)
       => ( ! [X3: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(F,X3)),aa_num_rat(F,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_num_rat(F,Ca)) ) ) ) ).

% order_less_le_subst1
tff(fact_1878_order__less__le__subst1,axiom,
    ! [A: num,F: fun_num_num,B: num,Ca: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,A),aa_num_num(F,B))
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),Ca)
       => ( ! [X3: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Y3)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(F,X3)),aa_num_num(F,Y3)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_num,A),aa_num_num(F,Ca)) ) ) ) ).

% order_less_le_subst1
tff(fact_1879_order__less__le__subst1,axiom,
    ! [A: nat,F: fun_num_nat,B: num,Ca: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),aa_num_nat(F,B))
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),Ca)
       => ( ! [X3: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Y3)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(F,X3)),aa_num_nat(F,Y3)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),aa_num_nat(F,Ca)) ) ) ) ).

% order_less_le_subst1
tff(fact_1880_order__less__le__subst1,axiom,
    ! [A: int,F: fun_num_int,B: num,Ca: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_num_int(F,B))
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),Ca)
       => ( ! [X3: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Y3)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(F,X3)),aa_num_int(F,Y3)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_num_int(F,Ca)) ) ) ) ).

% order_less_le_subst1
tff(fact_1881_order__le__less__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_assn,Ca: assn] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_rat_assn(F,B)),Ca)
       => ( ! [X3: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y3)
             => aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_rat_assn(F,X3)),aa_rat_assn(F,Y3)) )
         => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_rat_assn(F,A)),Ca) ) ) ) ).

% order_le_less_subst2
tff(fact_1882_order__le__less__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_rat,Ca: 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)),Ca)
       => ( ! [X3: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(F,X3)),aa_rat_rat(F,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(F,A)),Ca) ) ) ) ).

% order_le_less_subst2
tff(fact_1883_order__le__less__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_num,Ca: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,aa_rat_num(F,B)),Ca)
       => ( ! [X3: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y3)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_rat_num(F,X3)),aa_rat_num(F,Y3)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_num,aa_rat_num(F,A)),Ca) ) ) ) ).

% order_le_less_subst2
tff(fact_1884_order__le__less__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_nat,Ca: 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)),Ca)
       => ( ! [X3: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y3)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_rat_nat(F,X3)),aa_rat_nat(F,Y3)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_rat_nat(F,A)),Ca) ) ) ) ).

% order_le_less_subst2
tff(fact_1885_order__le__less__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_int,Ca: 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)),Ca)
       => ( ! [X3: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y3)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(F,X3)),aa_rat_int(F,Y3)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(F,A)),Ca) ) ) ) ).

% order_le_less_subst2
tff(fact_1886_order__le__less__subst2,axiom,
    ! [A: num,B: num,F: fun_num_assn,Ca: assn] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
     => ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_num_assn(F,B)),Ca)
       => ( ! [X3: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Y3)
             => aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_num_assn(F,X3)),aa_num_assn(F,Y3)) )
         => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_num_assn(F,A)),Ca) ) ) ) ).

% order_le_less_subst2
tff(fact_1887_order__le__less__subst2,axiom,
    ! [A: num,B: num,F: fun_num_rat,Ca: rat] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(F,B)),Ca)
       => ( ! [X3: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(F,X3)),aa_num_rat(F,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(F,A)),Ca) ) ) ) ).

% order_le_less_subst2
tff(fact_1888_order__le__less__subst2,axiom,
    ! [A: num,B: num,F: fun_num_num,Ca: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,aa_num_num(F,B)),Ca)
       => ( ! [X3: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Y3)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(F,X3)),aa_num_num(F,Y3)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_num,aa_num_num(F,A)),Ca) ) ) ) ).

% order_le_less_subst2
tff(fact_1889_order__le__less__subst2,axiom,
    ! [A: num,B: num,F: fun_num_nat,Ca: nat] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_num_nat(F,B)),Ca)
       => ( ! [X3: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Y3)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(F,X3)),aa_num_nat(F,Y3)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_num_nat(F,A)),Ca) ) ) ) ).

% order_le_less_subst2
tff(fact_1890_order__le__less__subst2,axiom,
    ! [A: num,B: num,F: fun_num_int,Ca: int] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(F,B)),Ca)
       => ( ! [X3: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Y3)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(F,X3)),aa_num_int(F,Y3)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(F,A)),Ca) ) ) ) ).

% order_le_less_subst2
tff(fact_1891_order__le__less__subst1,axiom,
    ! [A: assn,F: fun_assn_assn,B: assn,Ca: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),aa_assn_assn(F,B))
     => ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,B),Ca)
       => ( ! [X3: assn,Y3: assn] :
              ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X3),Y3)
             => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_assn_assn(F,X3)),aa_assn_assn(F,Y3)) )
         => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),aa_assn_assn(F,Ca)) ) ) ) ).

% order_le_less_subst1
tff(fact_1892_order__le__less__subst1,axiom,
    ! [A: assn,F: fun_rat_assn,B: rat,Ca: rat] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),aa_rat_assn(F,B))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),Ca)
       => ( ! [X3: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Y3)
             => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_rat_assn(F,X3)),aa_rat_assn(F,Y3)) )
         => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),aa_rat_assn(F,Ca)) ) ) ) ).

% order_le_less_subst1
tff(fact_1893_order__le__less__subst1,axiom,
    ! [A: assn,F: fun_num_assn,B: num,Ca: num] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),aa_num_assn(F,B))
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,B),Ca)
       => ( ! [X3: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_num,X3),Y3)
             => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_num_assn(F,X3)),aa_num_assn(F,Y3)) )
         => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),aa_num_assn(F,Ca)) ) ) ) ).

% order_le_less_subst1
tff(fact_1894_order__le__less__subst1,axiom,
    ! [A: assn,F: fun_nat_assn,B: nat,Ca: nat] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),aa_nat_assn(F,B))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),Ca)
       => ( ! [X3: nat,Y3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Y3)
             => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_nat_assn(F,X3)),aa_nat_assn(F,Y3)) )
         => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),aa_nat_assn(F,Ca)) ) ) ) ).

% order_le_less_subst1
tff(fact_1895_order__le__less__subst1,axiom,
    ! [A: assn,F: fun_int_assn,B: int,Ca: int] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),aa_int_assn(F,B))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),Ca)
       => ( ! [X3: int,Y3: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Y3)
             => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_int_assn(F,X3)),aa_int_assn(F,Y3)) )
         => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),aa_int_assn(F,Ca)) ) ) ) ).

% order_le_less_subst1
tff(fact_1896_order__le__less__subst1,axiom,
    ! [A: rat,F: fun_assn_rat,B: assn,Ca: assn] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_assn_rat(F,B))
     => ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,B),Ca)
       => ( ! [X3: assn,Y3: assn] :
              ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X3),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_assn_rat(F,X3)),aa_assn_rat(F,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_assn_rat(F,Ca)) ) ) ) ).

% order_le_less_subst1
tff(fact_1897_order__le__less__subst1,axiom,
    ! [A: rat,F: fun_rat_rat,B: rat,Ca: 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),Ca)
       => ( ! [X3: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(F,X3)),aa_rat_rat(F,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(F,Ca)) ) ) ) ).

% order_le_less_subst1
tff(fact_1898_order__le__less__subst1,axiom,
    ! [A: rat,F: fun_num_rat,B: num,Ca: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_num_rat(F,B))
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,B),Ca)
       => ( ! [X3: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_num,X3),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(F,X3)),aa_num_rat(F,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_num_rat(F,Ca)) ) ) ) ).

% order_le_less_subst1
tff(fact_1899_order__le__less__subst1,axiom,
    ! [A: rat,F: fun_nat_rat,B: nat,Ca: 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),Ca)
       => ( ! [X3: nat,Y3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(F,X3)),aa_nat_rat(F,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_nat_rat(F,Ca)) ) ) ) ).

% order_le_less_subst1
tff(fact_1900_order__le__less__subst1,axiom,
    ! [A: rat,F: fun_int_rat,B: int,Ca: 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),Ca)
       => ( ! [X3: int,Y3: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(F,X3)),aa_int_rat(F,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_int_rat(F,Ca)) ) ) ) ).

% order_le_less_subst1
tff(fact_1901_order__less__le__trans,axiom,
    ! [X: assn,Y: assn,Z: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X),Y)
     => ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,Y),Z)
       => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X),Z) ) ) ).

% order_less_le_trans
tff(fact_1902_order__less__le__trans,axiom,
    ! [X: set_int,Y: set_int,Z: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_set_int,X),Y)
     => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,Y),Z)
       => aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_set_int,X),Z) ) ) ).

% order_less_le_trans
tff(fact_1903_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_1904_order__less__le__trans,axiom,
    ! [X: num,Y: num,Z: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,X),Y)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Y),Z)
       => aa_num_o(aa_num_fun_num_o(ord_less_num,X),Z) ) ) ).

% order_less_le_trans
tff(fact_1905_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_1906_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_1907_order__le__less__trans,axiom,
    ! [X: assn,Y: assn,Z: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X),Y)
     => ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,Y),Z)
       => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X),Z) ) ) ).

% order_le_less_trans
tff(fact_1908_order__le__less__trans,axiom,
    ! [X: set_int,Y: set_int,Z: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,X),Y)
     => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_set_int,Y),Z)
       => aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_set_int,X),Z) ) ) ).

% order_le_less_trans
tff(fact_1909_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_1910_order__le__less__trans,axiom,
    ! [X: num,Y: num,Z: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X),Y)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,Y),Z)
       => aa_num_o(aa_num_fun_num_o(ord_less_num,X),Z) ) ) ).

% order_le_less_trans
tff(fact_1911_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_1912_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_1913_order__neq__le__trans,axiom,
    ! [A: assn,B: assn] :
      ( ( A != B )
     => ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),B)
       => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),B) ) ) ).

% order_neq_le_trans
tff(fact_1914_order__neq__le__trans,axiom,
    ! [A: set_int,B: set_int] :
      ( ( A != B )
     => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A),B)
       => aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_set_int,A),B) ) ) ).

% order_neq_le_trans
tff(fact_1915_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_1916_order__neq__le__trans,axiom,
    ! [A: num,B: num] :
      ( ( A != B )
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
       => aa_num_o(aa_num_fun_num_o(ord_less_num,A),B) ) ) ).

% order_neq_le_trans
tff(fact_1917_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_1918_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_1919_order__le__neq__trans,axiom,
    ! [A: assn,B: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),B)
     => ( ( A != B )
       => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),B) ) ) ).

% order_le_neq_trans
tff(fact_1920_order__le__neq__trans,axiom,
    ! [A: set_int,B: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A),B)
     => ( ( A != B )
       => aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_set_int,A),B) ) ) ).

% order_le_neq_trans
tff(fact_1921_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_1922_order__le__neq__trans,axiom,
    ! [A: num,B: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
     => ( ( A != B )
       => aa_num_o(aa_num_fun_num_o(ord_less_num,A),B) ) ) ).

% order_le_neq_trans
tff(fact_1923_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_1924_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_1925_order__less__imp__le,axiom,
    ! [X: assn,Y: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X),Y)
     => aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X),Y) ) ).

% order_less_imp_le
tff(fact_1926_order__less__imp__le,axiom,
    ! [X: set_int,Y: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_set_int,X),Y)
     => aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,X),Y) ) ).

% order_less_imp_le
tff(fact_1927_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_1928_order__less__imp__le,axiom,
    ! [X: num,Y: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,X),Y)
     => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X),Y) ) ).

% order_less_imp_le
tff(fact_1929_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_1930_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_1931_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_1932_linorder__not__less,axiom,
    ! [X: num,Y: num] :
      ( ~ aa_num_o(aa_num_fun_num_o(ord_less_num,X),Y)
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Y),X) ) ).

% linorder_not_less
tff(fact_1933_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_1934_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_1935_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_1936_linorder__not__le,axiom,
    ! [X: num,Y: num] :
      ( ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X),Y)
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,Y),X) ) ).

% linorder_not_le
tff(fact_1937_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_1938_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_1939_order__less__le,axiom,
    ! [X: assn,Y: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X),Y)
    <=> ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X),Y)
        & ( X != Y ) ) ) ).

% order_less_le
tff(fact_1940_order__less__le,axiom,
    ! [X: set_int,Y: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_set_int,X),Y)
    <=> ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,X),Y)
        & ( X != Y ) ) ) ).

% order_less_le
tff(fact_1941_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_1942_order__less__le,axiom,
    ! [X: num,Y: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,X),Y)
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X),Y)
        & ( X != Y ) ) ) ).

% order_less_le
tff(fact_1943_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_1944_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_1945_order__le__less,axiom,
    ! [X: assn,Y: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X),Y)
    <=> ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X),Y)
        | ( X = Y ) ) ) ).

% order_le_less
tff(fact_1946_order__le__less,axiom,
    ! [X: set_int,Y: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,X),Y)
    <=> ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_set_int,X),Y)
        | ( X = Y ) ) ) ).

% order_le_less
tff(fact_1947_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_1948_order__le__less,axiom,
    ! [X: num,Y: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X),Y)
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_num,X),Y)
        | ( X = Y ) ) ) ).

% order_le_less
tff(fact_1949_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_1950_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_1951_dual__order_Ostrict__implies__order,axiom,
    ! [B: assn,A: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,B),A)
     => aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,B),A) ) ).

% dual_order.strict_implies_order
tff(fact_1952_dual__order_Ostrict__implies__order,axiom,
    ! [B: set_int,A: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_set_int,B),A)
     => aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,B),A) ) ).

% dual_order.strict_implies_order
tff(fact_1953_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_1954_dual__order_Ostrict__implies__order,axiom,
    ! [B: num,A: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,B),A)
     => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),A) ) ).

% dual_order.strict_implies_order
tff(fact_1955_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_1956_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_1957_order_Ostrict__implies__order,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) ) ).

% order.strict_implies_order
tff(fact_1958_order_Ostrict__implies__order,axiom,
    ! [A: set_int,B: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_set_int,A),B)
     => aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A),B) ) ).

% order.strict_implies_order
tff(fact_1959_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_1960_order_Ostrict__implies__order,axiom,
    ! [A: num,B: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,A),B)
     => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B) ) ).

% order.strict_implies_order
tff(fact_1961_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_1962_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_1963_dual__order_Ostrict__iff__not,axiom,
    ! [B: assn,A: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,B),A)
    <=> ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,B),A)
        & ~ aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),B) ) ) ).

% dual_order.strict_iff_not
tff(fact_1964_dual__order_Ostrict__iff__not,axiom,
    ! [B: set_int,A: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_set_int,B),A)
    <=> ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,B),A)
        & ~ aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A),B) ) ) ).

% dual_order.strict_iff_not
tff(fact_1965_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_1966_dual__order_Ostrict__iff__not,axiom,
    ! [B: num,A: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,B),A)
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),A)
        & ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B) ) ) ).

% dual_order.strict_iff_not
tff(fact_1967_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_1968_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_1969_dual__order_Ostrict__trans2,axiom,
    ! [B: assn,A: assn,Ca: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,B),A)
     => ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,Ca),B)
       => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,Ca),A) ) ) ).

% dual_order.strict_trans2
tff(fact_1970_dual__order_Ostrict__trans2,axiom,
    ! [B: set_int,A: set_int,Ca: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_set_int,B),A)
     => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,Ca),B)
       => aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_set_int,Ca),A) ) ) ).

% dual_order.strict_trans2
tff(fact_1971_dual__order_Ostrict__trans2,axiom,
    ! [B: rat,A: rat,Ca: 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,Ca),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Ca),A) ) ) ).

% dual_order.strict_trans2
tff(fact_1972_dual__order_Ostrict__trans2,axiom,
    ! [B: num,A: num,Ca: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,B),A)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Ca),B)
       => aa_num_o(aa_num_fun_num_o(ord_less_num,Ca),A) ) ) ).

% dual_order.strict_trans2
tff(fact_1973_dual__order_Ostrict__trans2,axiom,
    ! [B: nat,A: nat,Ca: 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,Ca),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ca),A) ) ) ).

% dual_order.strict_trans2
tff(fact_1974_dual__order_Ostrict__trans2,axiom,
    ! [B: int,A: int,Ca: 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,Ca),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,Ca),A) ) ) ).

% dual_order.strict_trans2
tff(fact_1975_dual__order_Ostrict__trans1,axiom,
    ! [B: assn,A: assn,Ca: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,B),A)
     => ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,Ca),B)
       => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,Ca),A) ) ) ).

% dual_order.strict_trans1
tff(fact_1976_dual__order_Ostrict__trans1,axiom,
    ! [B: set_int,A: set_int,Ca: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,B),A)
     => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_set_int,Ca),B)
       => aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_set_int,Ca),A) ) ) ).

% dual_order.strict_trans1
tff(fact_1977_dual__order_Ostrict__trans1,axiom,
    ! [B: rat,A: rat,Ca: 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,Ca),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Ca),A) ) ) ).

% dual_order.strict_trans1
tff(fact_1978_dual__order_Ostrict__trans1,axiom,
    ! [B: num,A: num,Ca: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),A)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,Ca),B)
       => aa_num_o(aa_num_fun_num_o(ord_less_num,Ca),A) ) ) ).

% dual_order.strict_trans1
tff(fact_1979_dual__order_Ostrict__trans1,axiom,
    ! [B: nat,A: nat,Ca: 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,Ca),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ca),A) ) ) ).

% dual_order.strict_trans1
tff(fact_1980_dual__order_Ostrict__trans1,axiom,
    ! [B: int,A: int,Ca: 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,Ca),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,Ca),A) ) ) ).

% dual_order.strict_trans1
tff(fact_1981_dual__order_Ostrict__iff__order,axiom,
    ! [B: assn,A: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,B),A)
    <=> ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,B),A)
        & ( A != B ) ) ) ).

% dual_order.strict_iff_order
tff(fact_1982_dual__order_Ostrict__iff__order,axiom,
    ! [B: set_int,A: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_set_int,B),A)
    <=> ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,B),A)
        & ( A != B ) ) ) ).

% dual_order.strict_iff_order
tff(fact_1983_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_1984_dual__order_Ostrict__iff__order,axiom,
    ! [B: num,A: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,B),A)
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),A)
        & ( A != B ) ) ) ).

% dual_order.strict_iff_order
tff(fact_1985_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_1986_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_1987_dual__order_Oorder__iff__strict,axiom,
    ! [B: assn,A: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,B),A)
    <=> ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,B),A)
        | ( A = B ) ) ) ).

% dual_order.order_iff_strict
tff(fact_1988_dual__order_Oorder__iff__strict,axiom,
    ! [B: set_int,A: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,B),A)
    <=> ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_set_int,B),A)
        | ( A = B ) ) ) ).

% dual_order.order_iff_strict
tff(fact_1989_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_1990_dual__order_Oorder__iff__strict,axiom,
    ! [B: num,A: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),A)
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_num,B),A)
        | ( A = B ) ) ) ).

% dual_order.order_iff_strict
tff(fact_1991_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_1992_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_1993_dense__le__bounded,axiom,
    ! [X: rat,Y: rat,Z: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y)
     => ( ! [W: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),W)
           => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,W),Y)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,W),Z) ) )
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),Z) ) ) ).

% dense_le_bounded
tff(fact_1994_dense__ge__bounded,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z),X)
     => ( ! [W: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z),W)
           => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,W),X)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),W) ) )
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),Z) ) ) ).

% dense_ge_bounded
tff(fact_1995_order_Ostrict__iff__not,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)
        & ~ aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,B),A) ) ) ).

% order.strict_iff_not
tff(fact_1996_order_Ostrict__iff__not,axiom,
    ! [A: set_int,B: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_set_int,A),B)
    <=> ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A),B)
        & ~ aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,B),A) ) ) ).

% order.strict_iff_not
tff(fact_1997_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_1998_order_Ostrict__iff__not,axiom,
    ! [A: num,B: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,A),B)
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
        & ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),A) ) ) ).

% order.strict_iff_not
tff(fact_1999_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_2000_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_2001_order_Ostrict__trans2,axiom,
    ! [A: assn,B: assn,Ca: 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,B),Ca)
       => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),Ca) ) ) ).

% order.strict_trans2
tff(fact_2002_order_Ostrict__trans2,axiom,
    ! [A: set_int,B: set_int,Ca: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_set_int,A),B)
     => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,B),Ca)
       => aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_set_int,A),Ca) ) ) ).

% order.strict_trans2
tff(fact_2003_order_Ostrict__trans2,axiom,
    ! [A: rat,B: rat,Ca: 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),Ca)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),Ca) ) ) ).

% order.strict_trans2
tff(fact_2004_order_Ostrict__trans2,axiom,
    ! [A: num,B: num,Ca: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,A),B)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),Ca)
       => aa_num_o(aa_num_fun_num_o(ord_less_num,A),Ca) ) ) ).

% order.strict_trans2
tff(fact_2005_order_Ostrict__trans2,axiom,
    ! [A: nat,B: nat,Ca: 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),Ca)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),Ca) ) ) ).

% order.strict_trans2
tff(fact_2006_order_Ostrict__trans2,axiom,
    ! [A: int,B: int,Ca: 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),Ca)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,A),Ca) ) ) ).

% order.strict_trans2
tff(fact_2007_order_Ostrict__trans1,axiom,
    ! [A: assn,B: assn,Ca: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),B)
     => ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,B),Ca)
       => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),Ca) ) ) ).

% order.strict_trans1
tff(fact_2008_order_Ostrict__trans1,axiom,
    ! [A: set_int,B: set_int,Ca: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A),B)
     => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_set_int,B),Ca)
       => aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_set_int,A),Ca) ) ) ).

% order.strict_trans1
tff(fact_2009_order_Ostrict__trans1,axiom,
    ! [A: rat,B: rat,Ca: 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),Ca)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),Ca) ) ) ).

% order.strict_trans1
tff(fact_2010_order_Ostrict__trans1,axiom,
    ! [A: num,B: num,Ca: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,B),Ca)
       => aa_num_o(aa_num_fun_num_o(ord_less_num,A),Ca) ) ) ).

% order.strict_trans1
tff(fact_2011_order_Ostrict__trans1,axiom,
    ! [A: nat,B: nat,Ca: 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),Ca)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),Ca) ) ) ).

% order.strict_trans1
tff(fact_2012_order_Ostrict__trans1,axiom,
    ! [A: int,B: int,Ca: 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),Ca)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,A),Ca) ) ) ).

% order.strict_trans1
tff(fact_2013_order_Ostrict__iff__order,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 ) ) ) ).

% order.strict_iff_order
tff(fact_2014_order_Ostrict__iff__order,axiom,
    ! [A: set_int,B: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_set_int,A),B)
    <=> ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A),B)
        & ( A != B ) ) ) ).

% order.strict_iff_order
tff(fact_2015_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_2016_order_Ostrict__iff__order,axiom,
    ! [A: num,B: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,A),B)
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
        & ( A != B ) ) ) ).

% order.strict_iff_order
tff(fact_2017_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_2018_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_2019_order_Oorder__iff__strict,axiom,
    ! [A: assn,B: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),B)
    <=> ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),B)
        | ( A = B ) ) ) ).

% order.order_iff_strict
tff(fact_2020_order_Oorder__iff__strict,axiom,
    ! [A: set_int,B: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A),B)
    <=> ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_set_int,A),B)
        | ( A = B ) ) ) ).

% order.order_iff_strict
tff(fact_2021_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_2022_order_Oorder__iff__strict,axiom,
    ! [A: num,B: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_num,A),B)
        | ( A = B ) ) ) ).

% order.order_iff_strict
tff(fact_2023_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_2024_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_2025_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_2026_not__le__imp__less,axiom,
    ! [Y: num,X: num] :
      ( ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Y),X)
     => aa_num_o(aa_num_fun_num_o(ord_less_num,X),Y) ) ).

% not_le_imp_less
tff(fact_2027_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_2028_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_2029_less__le__not__le,axiom,
    ! [X: assn,Y: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X),Y)
    <=> ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X),Y)
        & ~ aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,Y),X) ) ) ).

% less_le_not_le
tff(fact_2030_less__le__not__le,axiom,
    ! [X: set_int,Y: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_set_int,X),Y)
    <=> ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,X),Y)
        & ~ aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,Y),X) ) ) ).

% less_le_not_le
tff(fact_2031_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_2032_less__le__not__le,axiom,
    ! [X: num,Y: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,X),Y)
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X),Y)
        & ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Y),X) ) ) ).

% less_le_not_le
tff(fact_2033_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_2034_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_2035_dense__le,axiom,
    ! [Y: rat,Z: rat] :
      ( ! [X3: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Y)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Z) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),Z) ) ).

% dense_le
tff(fact_2036_dense__ge,axiom,
    ! [Z: rat,Y: rat] :
      ( ! [X3: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z),X3)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),X3) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),Z) ) ).

% dense_ge
tff(fact_2037_antisym__conv2,axiom,
    ! [X: assn,Y: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X),Y)
     => ( ~ aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X),Y)
      <=> ( X = Y ) ) ) ).

% antisym_conv2
tff(fact_2038_antisym__conv2,axiom,
    ! [X: set_int,Y: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,X),Y)
     => ( ~ aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_set_int,X),Y)
      <=> ( X = Y ) ) ) ).

% antisym_conv2
tff(fact_2039_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_2040_antisym__conv2,axiom,
    ! [X: num,Y: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X),Y)
     => ( ~ aa_num_o(aa_num_fun_num_o(ord_less_num,X),Y)
      <=> ( X = Y ) ) ) ).

% antisym_conv2
tff(fact_2041_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_2042_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_2043_antisym__conv1,axiom,
    ! [X: assn,Y: assn] :
      ( ~ aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X),Y)
     => ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X),Y)
      <=> ( X = Y ) ) ) ).

% antisym_conv1
tff(fact_2044_antisym__conv1,axiom,
    ! [X: set_int,Y: set_int] :
      ( ~ aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_set_int,X),Y)
     => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,X),Y)
      <=> ( X = Y ) ) ) ).

% antisym_conv1
tff(fact_2045_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_2046_antisym__conv1,axiom,
    ! [X: num,Y: num] :
      ( ~ aa_num_o(aa_num_fun_num_o(ord_less_num,X),Y)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X),Y)
      <=> ( X = Y ) ) ) ).

% antisym_conv1
tff(fact_2047_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_2048_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_2049_nless__le,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 ) ) ) ).

% nless_le
tff(fact_2050_nless__le,axiom,
    ! [A: set_int,B: set_int] :
      ( ~ aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_set_int,A),B)
    <=> ( ~ aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A),B)
        | ( A = B ) ) ) ).

% nless_le
tff(fact_2051_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_2052_nless__le,axiom,
    ! [A: num,B: num] :
      ( ~ aa_num_o(aa_num_fun_num_o(ord_less_num,A),B)
    <=> ( ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
        | ( A = B ) ) ) ).

% nless_le
tff(fact_2053_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_2054_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_2055_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_2056_leI,axiom,
    ! [X: num,Y: num] :
      ( ~ aa_num_o(aa_num_fun_num_o(ord_less_num,X),Y)
     => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Y),X) ) ).

% leI
tff(fact_2057_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_2058_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_2059_leD,axiom,
    ! [Y: assn,X: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,Y),X)
     => ~ aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X),Y) ) ).

% leD
tff(fact_2060_leD,axiom,
    ! [Y: set_int,X: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,Y),X)
     => ~ aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_set_int,X),Y) ) ).

% leD
tff(fact_2061_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_2062_leD,axiom,
    ! [Y: num,X: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Y),X)
     => ~ aa_num_o(aa_num_fun_num_o(ord_less_num,X),Y) ) ).

% leD
tff(fact_2063_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_2064_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_2065_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_2066_bot_Onot__eq__extremum,axiom,
    ! [A: set_o] :
      ( ( A != bot_bot_set_o )
    <=> ord_less_set_o(bot_bot_set_o,A) ) ).

% bot.not_eq_extremum
tff(fact_2067_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_2068_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_2069_bot_Onot__eq__extremum,axiom,
    ! [A: assn] :
      ( ( A != bot_bot_assn )
    <=> aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,bot_bot_assn),A) ) ).

% bot.not_eq_extremum
tff(fact_2070_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_2071_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_2072_bot_Oextremum__strict,axiom,
    ! [A: set_o] : ~ ord_less_set_o(A,bot_bot_set_o) ).

% bot.extremum_strict
tff(fact_2073_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_2074_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_2075_bot_Oextremum__strict,axiom,
    ! [A: assn] : ~ aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),bot_bot_assn) ).

% bot.extremum_strict
tff(fact_2076_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_2077_inf_Ostrict__coboundedI2,axiom,
    ! [B: set_nat,Ca: set_nat,A: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,B),Ca)
     => 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)),Ca) ) ).

% inf.strict_coboundedI2
tff(fact_2078_inf_Ostrict__coboundedI2,axiom,
    ! [B: product_unit,Ca: product_unit,A: product_unit] :
      ( aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le361264281704409273t_unit,B),Ca)
     => aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le361264281704409273t_unit,aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,A),B)),Ca) ) ).

% inf.strict_coboundedI2
tff(fact_2079_inf_Ostrict__coboundedI2,axiom,
    ! [B: set_Pr1261947904930325089at_nat,Ca: set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le7866589430770878221at_nat,B),Ca)
     => aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le7866589430770878221at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B)),Ca) ) ).

% inf.strict_coboundedI2
tff(fact_2080_inf_Ostrict__coboundedI2,axiom,
    ! [B: assn,Ca: assn,A: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,B),Ca)
     => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),B)),Ca) ) ).

% inf.strict_coboundedI2
tff(fact_2081_inf_Ostrict__coboundedI2,axiom,
    ! [B: rat,Ca: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),Ca)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,A),B)),Ca) ) ).

% inf.strict_coboundedI2
tff(fact_2082_inf_Ostrict__coboundedI2,axiom,
    ! [B: nat,Ca: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),Ca)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B)),Ca) ) ).

% inf.strict_coboundedI2
tff(fact_2083_inf_Ostrict__coboundedI2,axiom,
    ! [B: int,Ca: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),Ca)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(inf_inf_int,A),B)),Ca) ) ).

% inf.strict_coboundedI2
tff(fact_2084_inf_Ostrict__coboundedI1,axiom,
    ! [A: set_nat,Ca: set_nat,B: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,A),Ca)
     => 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)),Ca) ) ).

% inf.strict_coboundedI1
tff(fact_2085_inf_Ostrict__coboundedI1,axiom,
    ! [A: product_unit,Ca: product_unit,B: product_unit] :
      ( aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le361264281704409273t_unit,A),Ca)
     => aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le361264281704409273t_unit,aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,A),B)),Ca) ) ).

% inf.strict_coboundedI1
tff(fact_2086_inf_Ostrict__coboundedI1,axiom,
    ! [A: set_Pr1261947904930325089at_nat,Ca: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le7866589430770878221at_nat,A),Ca)
     => aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le7866589430770878221at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B)),Ca) ) ).

% inf.strict_coboundedI1
tff(fact_2087_inf_Ostrict__coboundedI1,axiom,
    ! [A: assn,Ca: assn,B: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),Ca)
     => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),B)),Ca) ) ).

% inf.strict_coboundedI1
tff(fact_2088_inf_Ostrict__coboundedI1,axiom,
    ! [A: rat,Ca: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),Ca)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,A),B)),Ca) ) ).

% inf.strict_coboundedI1
tff(fact_2089_inf_Ostrict__coboundedI1,axiom,
    ! [A: nat,Ca: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),Ca)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B)),Ca) ) ).

% inf.strict_coboundedI1
tff(fact_2090_inf_Ostrict__coboundedI1,axiom,
    ! [A: int,Ca: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),Ca)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(inf_inf_int,A),B)),Ca) ) ).

% inf.strict_coboundedI1
tff(fact_2091_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_2092_inf_Ostrict__order__iff,axiom,
    ! [A: product_unit,B: product_unit] :
      ( aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le361264281704409273t_unit,A),B)
    <=> ( ( A = aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,A),B) )
        & ( A != B ) ) ) ).

% inf.strict_order_iff
tff(fact_2093_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_2094_inf_Ostrict__order__iff,axiom,
    ! [A: assn,B: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),B)
    <=> ( ( A = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),B) )
        & ( A != B ) ) ) ).

% inf.strict_order_iff
tff(fact_2095_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_2096_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_2097_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_2098_inf_Ostrict__boundedE,axiom,
    ! [A: set_nat,B: set_nat,Ca: 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),Ca))
     => ~ ( 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),Ca) ) ) ).

% inf.strict_boundedE
tff(fact_2099_inf_Ostrict__boundedE,axiom,
    ! [A: product_unit,B: product_unit,Ca: product_unit] :
      ( aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le361264281704409273t_unit,A),aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,B),Ca))
     => ~ ( aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le361264281704409273t_unit,A),B)
         => ~ aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le361264281704409273t_unit,A),Ca) ) ) ).

% inf.strict_boundedE
tff(fact_2100_inf_Ostrict__boundedE,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,Ca: 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),Ca))
     => ~ ( 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),Ca) ) ) ).

% inf.strict_boundedE
tff(fact_2101_inf_Ostrict__boundedE,axiom,
    ! [A: assn,B: assn,Ca: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,B),Ca))
     => ~ ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),B)
         => ~ aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),Ca) ) ) ).

% inf.strict_boundedE
tff(fact_2102_inf_Ostrict__boundedE,axiom,
    ! [A: rat,B: rat,Ca: 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),Ca))
     => ~ ( 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),Ca) ) ) ).

% inf.strict_boundedE
tff(fact_2103_inf_Ostrict__boundedE,axiom,
    ! [A: nat,B: nat,Ca: 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),Ca))
     => ~ ( 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),Ca) ) ) ).

% inf.strict_boundedE
tff(fact_2104_inf_Ostrict__boundedE,axiom,
    ! [A: int,B: int,Ca: 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),Ca))
     => ~ ( 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),Ca) ) ) ).

% inf.strict_boundedE
tff(fact_2105_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_2106_inf_Oabsorb4,axiom,
    ! [B: product_unit,A: product_unit] :
      ( aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le361264281704409273t_unit,B),A)
     => ( aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,A),B) = B ) ) ).

% inf.absorb4
tff(fact_2107_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_2108_inf_Oabsorb4,axiom,
    ! [B: assn,A: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,B),A)
     => ( aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),B) = B ) ) ).

% inf.absorb4
tff(fact_2109_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_2110_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_2111_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_2112_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_2113_inf_Oabsorb3,axiom,
    ! [A: product_unit,B: product_unit] :
      ( aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le361264281704409273t_unit,A),B)
     => ( aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,A),B) = A ) ) ).

% inf.absorb3
tff(fact_2114_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_2115_inf_Oabsorb3,axiom,
    ! [A: assn,B: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),B)
     => ( aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),B) = A ) ) ).

% inf.absorb3
tff(fact_2116_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_2117_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_2118_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_2119_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_2120_less__infI2,axiom,
    ! [B: product_unit,X: product_unit,A: product_unit] :
      ( aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le361264281704409273t_unit,B),X)
     => aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le361264281704409273t_unit,aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,A),B)),X) ) ).

% less_infI2
tff(fact_2121_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_2122_less__infI2,axiom,
    ! [B: assn,X: assn,A: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,B),X)
     => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),B)),X) ) ).

% less_infI2
tff(fact_2123_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_2124_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_2125_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_2126_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_2127_less__infI1,axiom,
    ! [A: product_unit,X: product_unit,B: product_unit] :
      ( aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le361264281704409273t_unit,A),X)
     => aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le361264281704409273t_unit,aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,A),B)),X) ) ).

% less_infI1
tff(fact_2128_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_2129_less__infI1,axiom,
    ! [A: assn,X: assn,B: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),X)
     => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),B)),X) ) ).

% less_infI1
tff(fact_2130_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_2131_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_2132_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_2133_sup_Ostrict__coboundedI2,axiom,
    ! [Ca: set_Pr8693737435421807431at_nat,B: set_Pr8693737435421807431at_nat,A: set_Pr8693737435421807431at_nat] :
      ( ord_le6428140832669894131at_nat(Ca,B)
     => ord_le6428140832669894131at_nat(Ca,aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,A),B)) ) ).

% sup.strict_coboundedI2
tff(fact_2134_sup_Ostrict__coboundedI2,axiom,
    ! [Ca: set_Pr8551490117392284871at_nat,B: set_Pr8551490117392284871at_nat,A: set_Pr8551490117392284871at_nat] :
      ( ord_le7642048601412989811at_nat(Ca,B)
     => ord_le7642048601412989811at_nat(Ca,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A),B)) ) ).

% sup.strict_coboundedI2
tff(fact_2135_sup_Ostrict__coboundedI2,axiom,
    ! [Ca: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat] :
      ( ord_le2604355607129572851at_nat(Ca,B)
     => ord_le2604355607129572851at_nat(Ca,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),B)) ) ).

% sup.strict_coboundedI2
tff(fact_2136_sup_Ostrict__coboundedI2,axiom,
    ! [Ca: set_nat,B: set_nat,A: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,Ca),B)
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,Ca),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B)) ) ).

% sup.strict_coboundedI2
tff(fact_2137_sup_Ostrict__coboundedI2,axiom,
    ! [Ca: assn,B: assn,A: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,Ca),B)
     => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,Ca),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),B)) ) ).

% sup.strict_coboundedI2
tff(fact_2138_sup_Ostrict__coboundedI2,axiom,
    ! [Ca: rat,B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Ca),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Ca),aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,A),B)) ) ).

% sup.strict_coboundedI2
tff(fact_2139_sup_Ostrict__coboundedI2,axiom,
    ! [Ca: nat,B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ca),B)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ca),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B)) ) ).

% sup.strict_coboundedI2
tff(fact_2140_sup_Ostrict__coboundedI2,axiom,
    ! [Ca: int,B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Ca),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,Ca),aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B)) ) ).

% sup.strict_coboundedI2
tff(fact_2141_sup_Ostrict__coboundedI1,axiom,
    ! [Ca: set_Pr8693737435421807431at_nat,A: set_Pr8693737435421807431at_nat,B: set_Pr8693737435421807431at_nat] :
      ( ord_le6428140832669894131at_nat(Ca,A)
     => ord_le6428140832669894131at_nat(Ca,aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,A),B)) ) ).

% sup.strict_coboundedI1
tff(fact_2142_sup_Ostrict__coboundedI1,axiom,
    ! [Ca: set_Pr8551490117392284871at_nat,A: set_Pr8551490117392284871at_nat,B: set_Pr8551490117392284871at_nat] :
      ( ord_le7642048601412989811at_nat(Ca,A)
     => ord_le7642048601412989811at_nat(Ca,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A),B)) ) ).

% sup.strict_coboundedI1
tff(fact_2143_sup_Ostrict__coboundedI1,axiom,
    ! [Ca: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
      ( ord_le2604355607129572851at_nat(Ca,A)
     => ord_le2604355607129572851at_nat(Ca,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),B)) ) ).

% sup.strict_coboundedI1
tff(fact_2144_sup_Ostrict__coboundedI1,axiom,
    ! [Ca: set_nat,A: set_nat,B: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,Ca),A)
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,Ca),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B)) ) ).

% sup.strict_coboundedI1
tff(fact_2145_sup_Ostrict__coboundedI1,axiom,
    ! [Ca: assn,A: assn,B: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,Ca),A)
     => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,Ca),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),B)) ) ).

% sup.strict_coboundedI1
tff(fact_2146_sup_Ostrict__coboundedI1,axiom,
    ! [Ca: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Ca),A)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Ca),aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,A),B)) ) ).

% sup.strict_coboundedI1
tff(fact_2147_sup_Ostrict__coboundedI1,axiom,
    ! [Ca: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ca),A)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ca),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B)) ) ).

% sup.strict_coboundedI1
tff(fact_2148_sup_Ostrict__coboundedI1,axiom,
    ! [Ca: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Ca),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,Ca),aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B)) ) ).

% sup.strict_coboundedI1
tff(fact_2149_sup_Ostrict__order__iff,axiom,
    ! [B: set_Pr8693737435421807431at_nat,A: set_Pr8693737435421807431at_nat] :
      ( ord_le6428140832669894131at_nat(B,A)
    <=> ( ( A = aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,A),B) )
        & ( A != B ) ) ) ).

% sup.strict_order_iff
tff(fact_2150_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_2151_sup_Ostrict__order__iff,axiom,
    ! [B: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat] :
      ( 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_2152_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_2153_sup_Ostrict__order__iff,axiom,
    ! [B: assn,A: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,B),A)
    <=> ( ( A = aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),B) )
        & ( A != B ) ) ) ).

% sup.strict_order_iff
tff(fact_2154_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_2155_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_2156_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_2157_sup_Ostrict__boundedE,axiom,
    ! [B: set_Pr8693737435421807431at_nat,Ca: set_Pr8693737435421807431at_nat,A: set_Pr8693737435421807431at_nat] :
      ( ord_le6428140832669894131at_nat(aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,B),Ca),A)
     => ~ ( ord_le6428140832669894131at_nat(B,A)
         => ~ ord_le6428140832669894131at_nat(Ca,A) ) ) ).

% sup.strict_boundedE
tff(fact_2158_sup_Ostrict__boundedE,axiom,
    ! [B: set_Pr8551490117392284871at_nat,Ca: set_Pr8551490117392284871at_nat,A: set_Pr8551490117392284871at_nat] :
      ( ord_le7642048601412989811at_nat(aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,B),Ca),A)
     => ~ ( ord_le7642048601412989811at_nat(B,A)
         => ~ ord_le7642048601412989811at_nat(Ca,A) ) ) ).

% sup.strict_boundedE
tff(fact_2159_sup_Ostrict__boundedE,axiom,
    ! [B: set_Pr4329608150637261639at_nat,Ca: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat] :
      ( ord_le2604355607129572851at_nat(aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,B),Ca),A)
     => ~ ( ord_le2604355607129572851at_nat(B,A)
         => ~ ord_le2604355607129572851at_nat(Ca,A) ) ) ).

% sup.strict_boundedE
tff(fact_2160_sup_Ostrict__boundedE,axiom,
    ! [B: set_nat,Ca: 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),Ca)),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,Ca),A) ) ) ).

% sup.strict_boundedE
tff(fact_2161_sup_Ostrict__boundedE,axiom,
    ! [B: assn,Ca: assn,A: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,B),Ca)),A)
     => ~ ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,B),A)
         => ~ aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,Ca),A) ) ) ).

% sup.strict_boundedE
tff(fact_2162_sup_Ostrict__boundedE,axiom,
    ! [B: rat,Ca: 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),Ca)),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,Ca),A) ) ) ).

% sup.strict_boundedE
tff(fact_2163_sup_Ostrict__boundedE,axiom,
    ! [B: nat,Ca: 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),Ca)),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,Ca),A) ) ) ).

% sup.strict_boundedE
tff(fact_2164_sup_Ostrict__boundedE,axiom,
    ! [B: int,Ca: 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),Ca)),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,Ca),A) ) ) ).

% sup.strict_boundedE
tff(fact_2165_sup_Oabsorb4,axiom,
    ! [A: set_Pr8693737435421807431at_nat,B: set_Pr8693737435421807431at_nat] :
      ( ord_le6428140832669894131at_nat(A,B)
     => ( aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,A),B) = B ) ) ).

% sup.absorb4
tff(fact_2166_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_2167_sup_Oabsorb4,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
      ( ord_le2604355607129572851at_nat(A,B)
     => ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),B) = B ) ) ).

% sup.absorb4
tff(fact_2168_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_2169_sup_Oabsorb4,axiom,
    ! [A: assn,B: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),B)
     => ( aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),B) = B ) ) ).

% sup.absorb4
tff(fact_2170_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_2171_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_2172_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_2173_sup_Oabsorb3,axiom,
    ! [B: set_Pr8693737435421807431at_nat,A: set_Pr8693737435421807431at_nat] :
      ( ord_le6428140832669894131at_nat(B,A)
     => ( aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,A),B) = A ) ) ).

% sup.absorb3
tff(fact_2174_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_2175_sup_Oabsorb3,axiom,
    ! [B: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat] :
      ( ord_le2604355607129572851at_nat(B,A)
     => ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),B) = A ) ) ).

% sup.absorb3
tff(fact_2176_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_2177_sup_Oabsorb3,axiom,
    ! [B: assn,A: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,B),A)
     => ( aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),B) = A ) ) ).

% sup.absorb3
tff(fact_2178_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_2179_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_2180_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_2181_less__supI2,axiom,
    ! [X: set_Pr8693737435421807431at_nat,B: set_Pr8693737435421807431at_nat,A: set_Pr8693737435421807431at_nat] :
      ( ord_le6428140832669894131at_nat(X,B)
     => ord_le6428140832669894131at_nat(X,aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,A),B)) ) ).

% less_supI2
tff(fact_2182_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_2183_less__supI2,axiom,
    ! [X: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat] :
      ( ord_le2604355607129572851at_nat(X,B)
     => ord_le2604355607129572851at_nat(X,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),B)) ) ).

% less_supI2
tff(fact_2184_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_2185_less__supI2,axiom,
    ! [X: assn,B: assn,A: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X),B)
     => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),B)) ) ).

% less_supI2
tff(fact_2186_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_2187_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_2188_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_2189_less__supI1,axiom,
    ! [X: set_Pr8693737435421807431at_nat,A: set_Pr8693737435421807431at_nat,B: set_Pr8693737435421807431at_nat] :
      ( ord_le6428140832669894131at_nat(X,A)
     => ord_le6428140832669894131at_nat(X,aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,A),B)) ) ).

% less_supI1
tff(fact_2190_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_2191_less__supI1,axiom,
    ! [X: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
      ( ord_le2604355607129572851at_nat(X,A)
     => ord_le2604355607129572851at_nat(X,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),B)) ) ).

% less_supI1
tff(fact_2192_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_2193_less__supI1,axiom,
    ! [X: assn,A: assn,B: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X),A)
     => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),B)) ) ).

% less_supI1
tff(fact_2194_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_2195_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_2196_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_2197_new__addrs__def,axiom,
    ! [Ha: heap_e7401611519738050253t_unit,Asa: set_nat,H_a: heap_e7401611519738050253t_unit] : hoare_new_addrs(Ha,Asa,H_a) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Asa),collect_nat(aa_hea9023553427033036112_nat_o(aTP_Lamm_cn(Ha),H_a))) ).

% new_addrs_def
tff(fact_2198_ord__eq__le__eq__trans,axiom,
    ! [A: set_int,B: set_int,Ca: set_int,D2: set_int] :
      ( ( A = B )
     => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,B),Ca)
       => ( ( Ca = D2 )
         => aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A),D2) ) ) ) ).

% ord_eq_le_eq_trans
tff(fact_2199_ord__eq__le__eq__trans,axiom,
    ! [A: rat,B: rat,Ca: rat,D2: rat] :
      ( ( A = B )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),Ca)
       => ( ( Ca = D2 )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),D2) ) ) ) ).

% ord_eq_le_eq_trans
tff(fact_2200_ord__eq__le__eq__trans,axiom,
    ! [A: num,B: num,Ca: num,D2: num] :
      ( ( A = B )
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),Ca)
       => ( ( Ca = D2 )
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),D2) ) ) ) ).

% ord_eq_le_eq_trans
tff(fact_2201_ord__eq__le__eq__trans,axiom,
    ! [A: nat,B: nat,Ca: nat,D2: nat] :
      ( ( A = B )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),Ca)
       => ( ( Ca = D2 )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),D2) ) ) ) ).

% ord_eq_le_eq_trans
tff(fact_2202_ord__eq__le__eq__trans,axiom,
    ! [A: int,B: int,Ca: int,D2: int] :
      ( ( A = B )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),Ca)
       => ( ( Ca = D2 )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),D2) ) ) ) ).

% ord_eq_le_eq_trans
tff(fact_2203_bex2I,axiom,
    ! [A: nat,B: nat,S: set_Pr1261947904930325089at_nat,Pa: fun_nat_fun_nat_o] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,A),B)),S)
     => ( ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,A),B)),S)
         => aa_nat_o(aa_nat_fun_nat_o(Pa,A),B) )
       => ? [A4: nat,B4: nat] :
            ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,A4),B4)),S)
            & aa_nat_o(aa_nat_fun_nat_o(Pa,A4),B4) ) ) ) ).

% bex2I
tff(fact_2204_bex2I,axiom,
    ! [A: fun_Pr6460944699146269935_nat_o,B: produc3658429121746597890et_nat,S: set_Pr1312406793078187350et_nat,Pa: 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(Pa,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(Pa,A4),B4) ) ) ) ).

% bex2I
tff(fact_2205_bex2I,axiom,
    ! [A: fun_Pr6460944699146269935_nat_o,B: produc8799533983703829792et_nat,S: set_Pr4040394263813967220et_nat,Pa: 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(Pa,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(Pa,A4),B4) ) ) ) ).

% bex2I
tff(fact_2206_bex2I,axiom,
    ! [A: fun_Pr7763680713840462443e_term,B: product_prod_int_int,S: set_Pr6700895561065273913nt_int,Pa: 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(Pa,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(Pa,A4),B4) ) ) ) ).

% bex2I
tff(fact_2207_bex2I,axiom,
    ! [A: fun_in5730379987311621695e_term,B: product_prod_int_int,S: set_Pr4166680992137330533nt_int,Pa: 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(Pa,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(Pa,A4),B4) ) ) ) ).

% bex2I
tff(fact_2208_bex2I,axiom,
    ! [A: a,B: produc6653097349344004940it_nat,S: set_Pr7098220151150636591it_nat,Pa: fun_a_2467521563176467050_nat_o] :
      ( member4554412811331277712it_nat(aa_Pro8078911840264235907it_nat(aa_a_f2234720956197727492it_nat(produc9178034014595674355it_nat,A),B),S)
     => ( ( member4554412811331277712it_nat(aa_Pro8078911840264235907it_nat(aa_a_f2234720956197727492it_nat(produc9178034014595674355it_nat,A),B),S)
         => aa_Pro1202363709425262478_nat_o(aa_a_f337466037417904483_nat_o(Pa,A),B) )
       => ? [A4: a,B4: produc6653097349344004940it_nat] :
            ( member4554412811331277712it_nat(aa_Pro8078911840264235907it_nat(aa_a_f2234720956197727492it_nat(produc9178034014595674355it_nat,A4),B4),S)
            & aa_Pro1202363709425262478_nat_o(aa_a_f337466037417904483_nat_o(Pa,A4),B4) ) ) ) ).

% bex2I
tff(fact_2209_subrelI,axiom,
    ! [Ra2: set_Pr1261947904930325089at_nat,S2: set_Pr1261947904930325089at_nat] :
      ( ! [X3: nat,Y3: nat] :
          ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,X3),Y3)),Ra2)
         => aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,X3),Y3)),S2) )
     => aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,Ra2),S2) ) ).

% subrelI
tff(fact_2210_subrelI,axiom,
    ! [Ra2: set_Pr1312406793078187350et_nat,S2: set_Pr1312406793078187350et_nat] :
      ( ! [X3: fun_Pr6460944699146269935_nat_o,Y3: produc3658429121746597890et_nat] :
          ( member4513763875858039863et_nat(aa_Pro434340453207536480et_nat(aa_fun3926903090414666066et_nat(produc6017135503889078682et_nat,X3),Y3),Ra2)
         => member4513763875858039863et_nat(aa_Pro434340453207536480et_nat(aa_fun3926903090414666066et_nat(produc6017135503889078682et_nat,X3),Y3),S2) )
     => ord_le3919382008627836342et_nat(Ra2,S2) ) ).

% subrelI
tff(fact_2211_subrelI,axiom,
    ! [Ra2: set_Pr4040394263813967220et_nat,S2: set_Pr4040394263813967220et_nat] :
      ( ! [X3: fun_Pr6460944699146269935_nat_o,Y3: produc8799533983703829792et_nat] :
          ( member9174161442226468181et_nat(aa_Pro1979504083478021276et_nat(aa_fun615918787490101902et_nat(produc7291855603632349880et_nat,X3),Y3),Ra2)
         => member9174161442226468181et_nat(aa_Pro1979504083478021276et_nat(aa_fun615918787490101902et_nat(produc7291855603632349880et_nat,X3),Y3),S2) )
     => ord_le5189643068899110356et_nat(Ra2,S2) ) ).

% subrelI
tff(fact_2212_subrelI,axiom,
    ! [Ra2: set_Pr6700895561065273913nt_int,S2: set_Pr6700895561065273913nt_int] :
      ( ! [X3: fun_Pr7763680713840462443e_term,Y3: product_prod_int_int] :
          ( member753276645541486746nt_int(aa_Pro5748361610964203648nt_int(aa_fun3265472288063244902nt_int(produc7320969046608631221nt_int,X3),Y3),Ra2)
         => member753276645541486746nt_int(aa_Pro5748361610964203648nt_int(aa_fun3265472288063244902nt_int(produc7320969046608631221nt_int,X3),Y3),S2) )
     => ord_le3990897393281686169nt_int(Ra2,S2) ) ).

% subrelI
tff(fact_2213_subrelI,axiom,
    ! [Ra2: set_Pr4166680992137330533nt_int,S2: set_Pr4166680992137330533nt_int] :
      ( ! [X3: fun_in5730379987311621695e_term,Y3: product_prod_int_int] :
          ( member5890090881338784710nt_int(aa_Pro8800328608558162860nt_int(aa_fun260840925927807678nt_int(produc4729913460834754785nt_int,X3),Y3),Ra2)
         => member5890090881338784710nt_int(aa_Pro8800328608558162860nt_int(aa_fun260840925927807678nt_int(produc4729913460834754785nt_int,X3),Y3),S2) )
     => ord_le3197784583289274309nt_int(Ra2,S2) ) ).

% subrelI
tff(fact_2214_subrelI,axiom,
    ! [Ra2: set_Pr7098220151150636591it_nat,S2: set_Pr7098220151150636591it_nat] :
      ( ! [X3: a,Y3: produc6653097349344004940it_nat] :
          ( member4554412811331277712it_nat(aa_Pro8078911840264235907it_nat(aa_a_f2234720956197727492it_nat(produc9178034014595674355it_nat,X3),Y3),Ra2)
         => member4554412811331277712it_nat(aa_Pro8078911840264235907it_nat(aa_a_f2234720956197727492it_nat(produc9178034014595674355it_nat,X3),Y3),S2) )
     => ord_le4478181985103859343it_nat(Ra2,S2) ) ).

% subrelI
tff(fact_2215_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_2216_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_2217_memb__imp__not__empty,axiom,
    ! [X: $o,S: set_o] :
      ( member_o((X),S)
     => ( S != bot_bot_set_o ) ) ).

% memb_imp_not_empty
tff(fact_2218_memb__imp__not__empty,axiom,
    ! [X: nat,S: set_nat] :
      ( aa_set_nat_o(member_nat(X),S)
     => ( S != bot_bot_set_nat ) ) ).

% memb_imp_not_empty
tff(fact_2219_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_2220_set__notEmptyE,axiom,
    ! [S: set_set_nat] :
      ( ( S != bot_bot_set_set_nat )
     => ~ ! [X3: set_nat] : ~ aa_set_set_nat_o(member_set_nat(X3),S) ) ).

% set_notEmptyE
tff(fact_2221_set__notEmptyE,axiom,
    ! [S: set_Pr1261947904930325089at_nat] :
      ( ( S != bot_bo2099793752762293965at_nat )
     => ~ ! [X3: product_prod_nat_nat] : ~ aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X3),S) ) ).

% set_notEmptyE
tff(fact_2222_set__notEmptyE,axiom,
    ! [S: set_o] :
      ( ( S != bot_bot_set_o )
     => ~ ! [X3: $o] : ~ member_o((X3),S) ) ).

% set_notEmptyE
tff(fact_2223_set__notEmptyE,axiom,
    ! [S: set_nat] :
      ( ( S != bot_bot_set_nat )
     => ~ ! [X3: nat] : ~ aa_set_nat_o(member_nat(X3),S) ) ).

% set_notEmptyE
tff(fact_2224_set__notEmptyE,axiom,
    ! [S: set_int] :
      ( ( S != bot_bot_set_int )
     => ~ ! [X3: int] : ~ aa_set_int_o(member_int(X3),S) ) ).

% set_notEmptyE
tff(fact_2225_inter__eq__subsetI,axiom,
    ! [S: set_nat,S3: set_nat,A3: set_nat,B3: 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,A3),S3) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B3),S3) )
       => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A3),S) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B3),S) ) ) ) ).

% inter_eq_subsetI
tff(fact_2226_inter__eq__subsetI,axiom,
    ! [S: set_Pr1261947904930325089at_nat,S3: set_Pr1261947904930325089at_nat,A3: set_Pr1261947904930325089at_nat,B3: 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,A3),S3) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B3),S3) )
       => ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A3),S) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B3),S) ) ) ) ).

% inter_eq_subsetI
tff(fact_2227_inter__eq__subsetI,axiom,
    ! [S: set_int,S3: set_int,A3: set_int,B3: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,S),S3)
     => ( ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A3),S3) = aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,B3),S3) )
       => ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A3),S) = aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,B3),S) ) ) ) ).

% inter_eq_subsetI
tff(fact_2228_hoare__triple__def,axiom,
    ! [Pa: assn,Ca: heap_T5738788834812785303t_unit,Qa: fun_Pr6626665806141890376t_assn] :
      ( hoare_8945653483474564448t_unit(Pa,Ca,Qa)
    <=> ! [H5: heap_e7401611519738050253t_unit,As4: set_nat] :
          ( aa_Pro653082453696054616_nat_o(rep_assn(Pa),produc7507926704131184380et_nat(H5,As4))
         => ? [R2: product_unit,H6: heap_e7401611519738050253t_unit] :
              ( ? [T3: nat] : aa_hea684056619475940329it_nat(heap_T875086893843062177t_unit(Ca),H5) = aa_Pro8996696869555642526it_nat(some_P1914260805536162275it_nat,produc7133225469290080770it_nat(R2,produc584006145561248582it_nat(H6,T3)))
              & aa_Pro653082453696054616_nat_o(rep_assn(aa_Product_unit_assn(Qa,R2)),produc7507926704131184380et_nat(H6,hoare_new_addrs(H5,As4,H6)))
              & aa_hea7994211679605476167unit_o(relH(collect_nat(aa_set_nat_fun_nat_o(aTP_Lamm_co(H5),As4)),H5),H6)
              & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,lim_Product_unit(H5)),lim_Product_unit(H6)) ) ) ) ).

% hoare_triple_def
tff(fact_2229_hoare__triple__def,axiom,
    ! [Pa: assn,Ca: heap_Time_Heap_a,Qa: fun_a_assn] :
      ( hoare_hoare_triple_a(Pa,Ca,Qa)
    <=> ! [H5: heap_e7401611519738050253t_unit,As4: set_nat] :
          ( aa_Pro653082453696054616_nat_o(rep_assn(Pa),produc7507926704131184380et_nat(H5,As4))
         => ? [R2: a,H6: heap_e7401611519738050253t_unit] :
              ( ? [T3: nat] : aa_hea4724376117875161946it_nat(heap_Time_execute_a(Ca),H5) = aa_Pro5302094286594872192it_nat(some_P7913643980934408916it_nat,aa_Pro8078911840264235907it_nat(aa_a_f2234720956197727492it_nat(produc9178034014595674355it_nat,R2),produc584006145561248582it_nat(H6,T3)))
              & aa_Pro653082453696054616_nat_o(rep_assn(aa_a_assn(Qa,R2)),produc7507926704131184380et_nat(H6,hoare_new_addrs(H5,As4,H6)))
              & aa_hea7994211679605476167unit_o(relH(collect_nat(aa_set_nat_fun_nat_o(aTP_Lamm_co(H5),As4)),H5),H6)
              & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,lim_Product_unit(H5)),lim_Product_unit(H6)) ) ) ) ).

% hoare_triple_def
tff(fact_2230_hoare__tripleI,axiom,
    ! [Pa: assn,Ca: heap_T5738788834812785303t_unit,Qa: fun_Pr6626665806141890376t_assn] :
      ( ! [H: heap_e7401611519738050253t_unit,As3: set_nat] :
          ( aa_Pro653082453696054616_nat_o(rep_assn(Pa),produc7507926704131184380et_nat(H,As3))
         => ? [R3: product_unit,H7: heap_e7401611519738050253t_unit,T4: nat] :
              ( ( aa_hea684056619475940329it_nat(heap_T875086893843062177t_unit(Ca),H) = aa_Pro8996696869555642526it_nat(some_P1914260805536162275it_nat,produc7133225469290080770it_nat(R3,produc584006145561248582it_nat(H7,T4))) )
              & aa_Pro653082453696054616_nat_o(rep_assn(aa_Product_unit_assn(Qa,R3)),produc7507926704131184380et_nat(H7,hoare_new_addrs(H,As3,H7)))
              & aa_hea7994211679605476167unit_o(relH(collect_nat(aa_set_nat_fun_nat_o(aTP_Lamm_co(H),As3)),H),H7)
              & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,lim_Product_unit(H)),lim_Product_unit(H7)) ) )
     => hoare_8945653483474564448t_unit(Pa,Ca,Qa) ) ).

% hoare_tripleI
tff(fact_2231_hoare__tripleI,axiom,
    ! [Pa: assn,Ca: heap_Time_Heap_a,Qa: fun_a_assn] :
      ( ! [H: heap_e7401611519738050253t_unit,As3: set_nat] :
          ( aa_Pro653082453696054616_nat_o(rep_assn(Pa),produc7507926704131184380et_nat(H,As3))
         => ? [R3: a,H7: heap_e7401611519738050253t_unit,T4: nat] :
              ( ( aa_hea4724376117875161946it_nat(heap_Time_execute_a(Ca),H) = aa_Pro5302094286594872192it_nat(some_P7913643980934408916it_nat,aa_Pro8078911840264235907it_nat(aa_a_f2234720956197727492it_nat(produc9178034014595674355it_nat,R3),produc584006145561248582it_nat(H7,T4))) )
              & aa_Pro653082453696054616_nat_o(rep_assn(aa_a_assn(Qa,R3)),produc7507926704131184380et_nat(H7,hoare_new_addrs(H,As3,H7)))
              & aa_hea7994211679605476167unit_o(relH(collect_nat(aa_set_nat_fun_nat_o(aTP_Lamm_co(H),As3)),H),H7)
              & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,lim_Product_unit(H)),lim_Product_unit(H7)) ) )
     => hoare_hoare_triple_a(Pa,Ca,Qa) ) ).

% hoare_tripleI
tff(fact_2232_hoare__tripleE,axiom,
    ! [Pa: assn,Ca: heap_T5738788834812785303t_unit,Qa: fun_Pr6626665806141890376t_assn,Ha: heap_e7401611519738050253t_unit,Asa: set_nat] :
      ( hoare_8945653483474564448t_unit(Pa,Ca,Qa)
     => ( aa_Pro653082453696054616_nat_o(rep_assn(Pa),produc7507926704131184380et_nat(Ha,Asa))
       => ~ ! [R: product_unit,H4: heap_e7401611519738050253t_unit] :
              ( ? [T2: nat] : aa_hea684056619475940329it_nat(heap_T875086893843062177t_unit(Ca),Ha) = aa_Pro8996696869555642526it_nat(some_P1914260805536162275it_nat,produc7133225469290080770it_nat(R,produc584006145561248582it_nat(H4,T2)))
             => ( aa_Pro653082453696054616_nat_o(rep_assn(aa_Product_unit_assn(Qa,R)),produc7507926704131184380et_nat(H4,hoare_new_addrs(Ha,Asa,H4)))
               => ( aa_hea7994211679605476167unit_o(relH(collect_nat(aa_set_nat_fun_nat_o(aTP_Lamm_co(Ha),Asa)),Ha),H4)
                 => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,lim_Product_unit(Ha)),lim_Product_unit(H4)) ) ) ) ) ) ).

% hoare_tripleE
tff(fact_2233_hoare__tripleE,axiom,
    ! [Pa: assn,Ca: heap_Time_Heap_a,Qa: fun_a_assn,Ha: heap_e7401611519738050253t_unit,Asa: set_nat] :
      ( hoare_hoare_triple_a(Pa,Ca,Qa)
     => ( aa_Pro653082453696054616_nat_o(rep_assn(Pa),produc7507926704131184380et_nat(Ha,Asa))
       => ~ ! [R: a,H4: heap_e7401611519738050253t_unit] :
              ( ? [T2: nat] : aa_hea4724376117875161946it_nat(heap_Time_execute_a(Ca),Ha) = aa_Pro5302094286594872192it_nat(some_P7913643980934408916it_nat,aa_Pro8078911840264235907it_nat(aa_a_f2234720956197727492it_nat(produc9178034014595674355it_nat,R),produc584006145561248582it_nat(H4,T2)))
             => ( aa_Pro653082453696054616_nat_o(rep_assn(aa_a_assn(Qa,R)),produc7507926704131184380et_nat(H4,hoare_new_addrs(Ha,Asa,H4)))
               => ( aa_hea7994211679605476167unit_o(relH(collect_nat(aa_set_nat_fun_nat_o(aTP_Lamm_co(Ha),Asa)),Ha),H4)
                 => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,lim_Product_unit(Ha)),lim_Product_unit(H4)) ) ) ) ) ) ).

% hoare_tripleE
tff(fact_2234_sup__bot_Osemilattice__neutr__order__axioms,axiom,
    semila1307241728879085301at_nat(sup_su718114333110466843at_nat,bot_bo5327735625951526323at_nat,aTP_Lamm_cp,aTP_Lamm_cq) ).

% sup_bot.semilattice_neutr_order_axioms
tff(fact_2235_sup__bot_Osemilattice__neutr__order__axioms,axiom,
    semila5207306265035627125at_nat(sup_su3035147773818789531at_nat,bot_bo8422036546324065075at_nat,aTP_Lamm_cr,aTP_Lamm_cs) ).

% sup_bot.semilattice_neutr_order_axioms
tff(fact_2236_sup__bot_Osemilattice__neutr__order__axioms,axiom,
    semila6534579987270727413at_nat(sup_su5525570899277871387at_nat,bot_bo228742789529271731at_nat,aTP_Lamm_ct,aTP_Lamm_cu) ).

% sup_bot.semilattice_neutr_order_axioms
tff(fact_2237_sup__bot_Osemilattice__neutr__order__axioms,axiom,
    semila4918476307565957903at_nat(sup_su6327502436637775413at_nat,bot_bo2099793752762293965at_nat,aTP_Lamm_cv,aTP_Lamm_cw) ).

% sup_bot.semilattice_neutr_order_axioms
tff(fact_2238_sup__bot_Osemilattice__neutr__order__axioms,axiom,
    semila2554085542299052326_set_o(sup_sup_set_o,bot_bot_set_o,aTP_Lamm_cx,aTP_Lamm_cy) ).

% sup_bot.semilattice_neutr_order_axioms
tff(fact_2239_sup__bot_Osemilattice__neutr__order__axioms,axiom,
    semila1667268886620078168et_nat(sup_sup_set_nat,bot_bot_set_nat,aTP_Lamm_cz,aTP_Lamm_da) ).

% sup_bot.semilattice_neutr_order_axioms
tff(fact_2240_sup__bot_Osemilattice__neutr__order__axioms,axiom,
    semila8603258263270017530r_assn(sup_sup_assn,bot_bot_assn,aTP_Lamm_db,aTP_Lamm_dc) ).

% sup_bot.semilattice_neutr_order_axioms
tff(fact_2241_sup__bot_Osemilattice__neutr__order__axioms,axiom,
    semila6712789903965657268et_int(sup_sup_set_int,bot_bot_set_int,aTP_Lamm_dd,aTP_Lamm_de) ).

% sup_bot.semilattice_neutr_order_axioms
tff(fact_2242_nat__descend__induct,axiom,
    ! [N2: nat,Pa: fun_nat_o,M: nat] :
      ( ! [K2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),K2)
         => aa_nat_o(Pa,K2) )
     => ( ! [K2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),N2)
           => ( ! [I: nat] :
                  ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K2),I)
                 => aa_nat_o(Pa,I) )
             => aa_nat_o(Pa,K2) ) )
       => aa_nat_o(Pa,M) ) ) ).

% nat_descend_induct
tff(fact_2243_less__mono__imp__le__mono,axiom,
    ! [F: fun_nat_nat,I2: nat,J: nat] :
      ( ! [I3: nat,J2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),J2)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(F,I3)),aa_nat_nat(F,J2)) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),J)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F,I2)),aa_nat_nat(F,J)) ) ) ).

% less_mono_imp_le_mono
tff(fact_2244_le__neq__implies__less,axiom,
    ! [M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
     => ( ( M != N2 )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N2) ) ) ).

% le_neq_implies_less
tff(fact_2245_less__or__eq__imp__le,axiom,
    ! [M: nat,N2: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N2)
        | ( M = N2 ) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2) ) ).

% less_or_eq_imp_le
tff(fact_2246_le__eq__less__or__eq,axiom,
    ! [M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N2)
        | ( M = N2 ) ) ) ).

% le_eq_less_or_eq
tff(fact_2247_less__imp__le__nat,axiom,
    ! [M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2) ) ).

% less_imp_le_nat
tff(fact_2248_nat__less__le,axiom,
    ! [M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N2)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
        & ( M != N2 ) ) ) ).

% nat_less_le
tff(fact_2249_subset__emptyI,axiom,
    ! [A3: set_set_nat] :
      ( ! [X3: set_nat] : ~ aa_set_set_nat_o(member_set_nat(X3),A3)
     => aa_set_set_nat_o(ord_le6893508408891458716et_nat(A3),bot_bot_set_set_nat) ) ).

% subset_emptyI
tff(fact_2250_subset__emptyI,axiom,
    ! [A3: set_Pr1261947904930325089at_nat] :
      ( ! [X3: product_prod_nat_nat] : ~ aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X3),A3)
     => aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A3),bot_bo2099793752762293965at_nat) ) ).

% subset_emptyI
tff(fact_2251_subset__emptyI,axiom,
    ! [A3: set_o] :
      ( ! [X3: $o] : ~ member_o((X3),A3)
     => ord_less_eq_set_o(A3,bot_bot_set_o) ) ).

% subset_emptyI
tff(fact_2252_subset__emptyI,axiom,
    ! [A3: set_nat] :
      ( ! [X3: nat] : ~ aa_set_nat_o(member_nat(X3),A3)
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A3),bot_bot_set_nat) ) ).

% subset_emptyI
tff(fact_2253_subset__emptyI,axiom,
    ! [A3: set_int] :
      ( ! [X3: int] : ~ aa_set_int_o(member_int(X3),A3)
     => aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A3),bot_bot_set_int) ) ).

% subset_emptyI
tff(fact_2254_psubsetI,axiom,
    ! [A3: set_int,B3: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A3),B3)
     => ( ( A3 != B3 )
       => aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_set_int,A3),B3) ) ) ).

% psubsetI
tff(fact_2255_less__by__empty,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
      ( ( A3 = bot_bo2099793752762293965at_nat )
     => aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A3),B3) ) ).

% less_by_empty
tff(fact_2256_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_2257_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_2258_not__psubset__empty,axiom,
    ! [A3: set_Pr1261947904930325089at_nat] : ~ aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le7866589430770878221at_nat,A3),bot_bo2099793752762293965at_nat) ).

% not_psubset_empty
tff(fact_2259_not__psubset__empty,axiom,
    ! [A3: set_o] : ~ ord_less_set_o(A3,bot_bot_set_o) ).

% not_psubset_empty
tff(fact_2260_not__psubset__empty,axiom,
    ! [A3: set_nat] : ~ aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,A3),bot_bot_set_nat) ).

% not_psubset_empty
tff(fact_2261_not__psubset__empty,axiom,
    ! [A3: set_int] : ~ aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_set_int,A3),bot_bot_set_int) ).

% not_psubset_empty
tff(fact_2262_subset__iff__psubset__eq,axiom,
    ! [A3: set_int,B3: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A3),B3)
    <=> ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_set_int,A3),B3)
        | ( A3 = B3 ) ) ) ).

% subset_iff_psubset_eq
tff(fact_2263_subset__psubset__trans,axiom,
    ! [A3: set_int,B3: set_int,C2: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A3),B3)
     => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_set_int,B3),C2)
       => aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_set_int,A3),C2) ) ) ).

% subset_psubset_trans
tff(fact_2264_subset__not__subset__eq,axiom,
    ! [A3: set_int,B3: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_set_int,A3),B3)
    <=> ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A3),B3)
        & ~ aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,B3),A3) ) ) ).

% subset_not_subset_eq
tff(fact_2265_psubset__subset__trans,axiom,
    ! [A3: set_int,B3: set_int,C2: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_set_int,A3),B3)
     => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,B3),C2)
       => aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_set_int,A3),C2) ) ) ).

% psubset_subset_trans
tff(fact_2266_psubset__imp__subset,axiom,
    ! [A3: set_int,B3: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_set_int,A3),B3)
     => aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A3),B3) ) ).

% psubset_imp_subset
tff(fact_2267_psubset__eq,axiom,
    ! [A3: set_int,B3: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_set_int,A3),B3)
    <=> ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A3),B3)
        & ( A3 != B3 ) ) ) ).

% psubset_eq
tff(fact_2268_psubsetE,axiom,
    ! [A3: set_int,B3: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_set_int,A3),B3)
     => ~ ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A3),B3)
         => aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,B3),A3) ) ) ).

% psubsetE
tff(fact_2269_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_2270_ssubst__Pair__rhs,axiom,
    ! [Ra2: nat,S2: nat,Ra: set_Pr1261947904930325089at_nat,S4: nat] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Ra2),S2)),Ra)
     => ( ( S4 = S2 )
       => aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Ra2),S4)),Ra) ) ) ).

% ssubst_Pair_rhs
tff(fact_2271_ssubst__Pair__rhs,axiom,
    ! [Ra2: fun_Pr6460944699146269935_nat_o,S2: produc3658429121746597890et_nat,Ra: set_Pr1312406793078187350et_nat,S4: produc3658429121746597890et_nat] :
      ( member4513763875858039863et_nat(aa_Pro434340453207536480et_nat(aa_fun3926903090414666066et_nat(produc6017135503889078682et_nat,Ra2),S2),Ra)
     => ( ( S4 = S2 )
       => member4513763875858039863et_nat(aa_Pro434340453207536480et_nat(aa_fun3926903090414666066et_nat(produc6017135503889078682et_nat,Ra2),S4),Ra) ) ) ).

% ssubst_Pair_rhs
tff(fact_2272_ssubst__Pair__rhs,axiom,
    ! [Ra2: fun_Pr6460944699146269935_nat_o,S2: produc8799533983703829792et_nat,Ra: set_Pr4040394263813967220et_nat,S4: produc8799533983703829792et_nat] :
      ( member9174161442226468181et_nat(aa_Pro1979504083478021276et_nat(aa_fun615918787490101902et_nat(produc7291855603632349880et_nat,Ra2),S2),Ra)
     => ( ( S4 = S2 )
       => member9174161442226468181et_nat(aa_Pro1979504083478021276et_nat(aa_fun615918787490101902et_nat(produc7291855603632349880et_nat,Ra2),S4),Ra) ) ) ).

% ssubst_Pair_rhs
tff(fact_2273_ssubst__Pair__rhs,axiom,
    ! [Ra2: fun_Pr7763680713840462443e_term,S2: product_prod_int_int,Ra: set_Pr6700895561065273913nt_int,S4: product_prod_int_int] :
      ( member753276645541486746nt_int(aa_Pro5748361610964203648nt_int(aa_fun3265472288063244902nt_int(produc7320969046608631221nt_int,Ra2),S2),Ra)
     => ( ( S4 = S2 )
       => member753276645541486746nt_int(aa_Pro5748361610964203648nt_int(aa_fun3265472288063244902nt_int(produc7320969046608631221nt_int,Ra2),S4),Ra) ) ) ).

% ssubst_Pair_rhs
tff(fact_2274_ssubst__Pair__rhs,axiom,
    ! [Ra2: fun_in5730379987311621695e_term,S2: product_prod_int_int,Ra: set_Pr4166680992137330533nt_int,S4: product_prod_int_int] :
      ( member5890090881338784710nt_int(aa_Pro8800328608558162860nt_int(aa_fun260840925927807678nt_int(produc4729913460834754785nt_int,Ra2),S2),Ra)
     => ( ( S4 = S2 )
       => member5890090881338784710nt_int(aa_Pro8800328608558162860nt_int(aa_fun260840925927807678nt_int(produc4729913460834754785nt_int,Ra2),S4),Ra) ) ) ).

% ssubst_Pair_rhs
tff(fact_2275_ssubst__Pair__rhs,axiom,
    ! [Ra2: a,S2: produc6653097349344004940it_nat,Ra: set_Pr7098220151150636591it_nat,S4: produc6653097349344004940it_nat] :
      ( member4554412811331277712it_nat(aa_Pro8078911840264235907it_nat(aa_a_f2234720956197727492it_nat(produc9178034014595674355it_nat,Ra2),S2),Ra)
     => ( ( S4 = S2 )
       => member4554412811331277712it_nat(aa_Pro8078911840264235907it_nat(aa_a_f2234720956197727492it_nat(produc9178034014595674355it_nat,Ra2),S4),Ra) ) ) ).

% ssubst_Pair_rhs
tff(fact_2276_nat__neq__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( M != N2 )
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N2)
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),M) ) ) ).

% nat_neq_iff
tff(fact_2277_less__not__refl,axiom,
    ! [N2: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),N2) ).

% less_not_refl
tff(fact_2278_less__not__refl2,axiom,
    ! [N2: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),M)
     => ( M != N2 ) ) ).

% less_not_refl2
tff(fact_2279_less__not__refl3,axiom,
    ! [S2: nat,Ta: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,S2),Ta)
     => ( S2 != Ta ) ) ).

% less_not_refl3
tff(fact_2280_less__irrefl__nat,axiom,
    ! [N2: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),N2) ).

% less_irrefl_nat
tff(fact_2281_nat__less__induct,axiom,
    ! [Pa: fun_nat_o,N2: nat] :
      ( ! [N5: nat] :
          ( ! [M4: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M4),N5)
             => aa_nat_o(Pa,M4) )
         => aa_nat_o(Pa,N5) )
     => aa_nat_o(Pa,N2) ) ).

% nat_less_induct
tff(fact_2282_infinite__descent,axiom,
    ! [Pa: fun_nat_o,N2: nat] :
      ( ! [N5: nat] :
          ( ~ aa_nat_o(Pa,N5)
         => ? [M4: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M4),N5)
              & ~ aa_nat_o(Pa,M4) ) )
     => aa_nat_o(Pa,N2) ) ).

% infinite_descent
tff(fact_2283_linorder__neqE__nat,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_nat
tff(fact_2284_le__refl,axiom,
    ! [N2: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),N2) ).

% le_refl
tff(fact_2285_le__trans,axiom,
    ! [I2: nat,J: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),J)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,J),K)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),K) ) ) ).

% le_trans
tff(fact_2286_eq__imp__le,axiom,
    ! [M: nat,N2: nat] :
      ( ( M = N2 )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2) ) ).

% eq_imp_le
tff(fact_2287_le__antisym,axiom,
    ! [M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),M)
       => ( M = N2 ) ) ) ).

% le_antisym
tff(fact_2288_nat__le__linear,axiom,
    ! [M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
      | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),M) ) ).

% nat_le_linear
tff(fact_2289_Nat_Oex__has__greatest__nat,axiom,
    ! [Pa: fun_nat_o,K: nat,B: nat] :
      ( aa_nat_o(Pa,K)
     => ( ! [Y3: nat] :
            ( aa_nat_o(Pa,Y3)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y3),B) )
       => ? [X3: nat] :
            ( aa_nat_o(Pa,X3)
            & ! [Y4: nat] :
                ( aa_nat_o(Pa,Y4)
               => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y4),X3) ) ) ) ) ).

% Nat.ex_has_greatest_nat
tff(fact_2290_prop__restrict,axiom,
    ! [X: product_prod_nat_nat,Z3: set_Pr1261947904930325089at_nat,X6: set_Pr1261947904930325089at_nat,Pa: fun_Pr4796943543331071514_nat_o] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X),Z3)
     => ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,Z3),collec3392354462482085612at_nat(aa_fun359341089595923442_nat_o(aTP_Lamm_al(X6),Pa)))
       => aa_Pro6150637604445964947_nat_o(Pa,X) ) ) ).

% prop_restrict
tff(fact_2291_prop__restrict,axiom,
    ! [X: $o,Z3: set_o,X6: set_o,Pa: fun_o_o] :
      ( member_o((X),Z3)
     => ( ord_less_eq_set_o(Z3,collect_o(aa_fun_o_o_fun_o_o(aTP_Lamm_am(X6),Pa)))
       => aa_o_o(Pa,(X)) ) ) ).

% prop_restrict
tff(fact_2292_prop__restrict,axiom,
    ! [X: product_prod_int_int,Z3: set_Pr958786334691620121nt_int,X6: set_Pr958786334691620121nt_int,Pa: fun_Pr3238374608701176162_int_o] :
      ( member5262025264175285858nt_int(X,Z3)
     => ( ord_le2843351958646193337nt_int(Z3,collec213857154873943460nt_int(aa_fun5349570537165229698_int_o(aTP_Lamm_an(X6),Pa)))
       => aa_Pro4592068669816069595_int_o(Pa,X) ) ) ).

% prop_restrict
tff(fact_2293_prop__restrict,axiom,
    ! [X: list_nat,Z3: set_list_nat,X6: set_list_nat,Pa: fun_list_nat_o] :
      ( member_list_nat(X,Z3)
     => ( ord_le6045566169113846134st_nat(Z3,collect_list_nat(aa_fun6155608546934989052_nat_o(aTP_Lamm_ao(X6),Pa)))
       => aa_list_nat_o(Pa,X) ) ) ).

% prop_restrict
tff(fact_2294_prop__restrict,axiom,
    ! [X: set_nat,Z3: set_set_nat,X6: set_set_nat,Pa: fun_set_nat_o] :
      ( aa_set_set_nat_o(member_set_nat(X),Z3)
     => ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(Z3),collect_set_nat(aa_fun751667557118977660_nat_o(aTP_Lamm_ap(X6),Pa)))
       => aa_set_nat_o(Pa,X) ) ) ).

% prop_restrict
tff(fact_2295_prop__restrict,axiom,
    ! [X: nat,Z3: set_nat,X6: set_nat,Pa: fun_nat_o] :
      ( aa_set_nat_o(member_nat(X),Z3)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Z3),collect_nat(aa_fun5937193402907943164_nat_o(aTP_Lamm_aq(X6),Pa)))
       => aa_nat_o(Pa,X) ) ) ).

% prop_restrict
tff(fact_2296_prop__restrict,axiom,
    ! [X: int,Z3: set_int,X6: set_int,Pa: fun_int_o] :
      ( aa_set_int_o(member_int(X),Z3)
     => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,Z3),collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_ar(X6),Pa)))
       => aa_int_o(Pa,X) ) ) ).

% prop_restrict
tff(fact_2297_Collect__restrict,axiom,
    ! [X6: set_Pr1261947904930325089at_nat,Pa: fun_Pr4796943543331071514_nat_o] : aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,collec3392354462482085612at_nat(aa_fun359341089595923442_nat_o(aTP_Lamm_al(X6),Pa))),X6) ).

% Collect_restrict
tff(fact_2298_Collect__restrict,axiom,
    ! [X6: set_o,Pa: fun_o_o] : ord_less_eq_set_o(collect_o(aa_fun_o_o_fun_o_o(aTP_Lamm_am(X6),Pa)),X6) ).

% Collect_restrict
tff(fact_2299_Collect__restrict,axiom,
    ! [X6: set_Pr958786334691620121nt_int,Pa: fun_Pr3238374608701176162_int_o] : ord_le2843351958646193337nt_int(collec213857154873943460nt_int(aa_fun5349570537165229698_int_o(aTP_Lamm_an(X6),Pa)),X6) ).

% Collect_restrict
tff(fact_2300_Collect__restrict,axiom,
    ! [X6: set_list_nat,Pa: fun_list_nat_o] : ord_le6045566169113846134st_nat(collect_list_nat(aa_fun6155608546934989052_nat_o(aTP_Lamm_ao(X6),Pa)),X6) ).

% Collect_restrict
tff(fact_2301_Collect__restrict,axiom,
    ! [X6: set_set_nat,Pa: fun_set_nat_o] : aa_set_set_nat_o(ord_le6893508408891458716et_nat(collect_set_nat(aa_fun751667557118977660_nat_o(aTP_Lamm_ap(X6),Pa))),X6) ).

% Collect_restrict
tff(fact_2302_Collect__restrict,axiom,
    ! [X6: set_nat,Pa: fun_nat_o] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,collect_nat(aa_fun5937193402907943164_nat_o(aTP_Lamm_aq(X6),Pa))),X6) ).

% Collect_restrict
tff(fact_2303_Collect__restrict,axiom,
    ! [X6: set_int,Pa: fun_int_o] : aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_ar(X6),Pa))),X6) ).

% Collect_restrict
tff(fact_2304_minf_I8_J,axiom,
    ! [Ta: rat] :
    ? [Z2: rat] :
    ! [X5: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X5),Z2)
     => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Ta),X5) ) ).

% minf(8)
tff(fact_2305_minf_I8_J,axiom,
    ! [Ta: num] :
    ? [Z2: num] :
    ! [X5: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,X5),Z2)
     => ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Ta),X5) ) ).

% minf(8)
tff(fact_2306_minf_I8_J,axiom,
    ! [Ta: nat] :
    ? [Z2: nat] :
    ! [X5: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X5),Z2)
     => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ta),X5) ) ).

% minf(8)
tff(fact_2307_minf_I8_J,axiom,
    ! [Ta: int] :
    ? [Z2: int] :
    ! [X5: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X5),Z2)
     => ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ta),X5) ) ).

% minf(8)
tff(fact_2308_minf_I6_J,axiom,
    ! [Ta: rat] :
    ? [Z2: rat] :
    ! [X5: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X5),Z2)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X5),Ta) ) ).

% minf(6)
tff(fact_2309_minf_I6_J,axiom,
    ! [Ta: num] :
    ? [Z2: num] :
    ! [X5: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,X5),Z2)
     => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X5),Ta) ) ).

% minf(6)
tff(fact_2310_minf_I6_J,axiom,
    ! [Ta: nat] :
    ? [Z2: nat] :
    ! [X5: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X5),Z2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X5),Ta) ) ).

% minf(6)
tff(fact_2311_minf_I6_J,axiom,
    ! [Ta: int] :
    ? [Z2: int] :
    ! [X5: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X5),Z2)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X5),Ta) ) ).

% minf(6)
tff(fact_2312_pinf_I8_J,axiom,
    ! [Ta: rat] :
    ? [Z2: rat] :
    ! [X5: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z2),X5)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Ta),X5) ) ).

% pinf(8)
tff(fact_2313_pinf_I8_J,axiom,
    ! [Ta: num] :
    ? [Z2: num] :
    ! [X5: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Z2),X5)
     => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Ta),X5) ) ).

% pinf(8)
tff(fact_2314_pinf_I8_J,axiom,
    ! [Ta: nat] :
    ? [Z2: nat] :
    ! [X5: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z2),X5)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ta),X5) ) ).

% pinf(8)
tff(fact_2315_pinf_I8_J,axiom,
    ! [Ta: int] :
    ? [Z2: int] :
    ! [X5: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z2),X5)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ta),X5) ) ).

% pinf(8)
tff(fact_2316_pinf_I6_J,axiom,
    ! [Ta: rat] :
    ? [Z2: rat] :
    ! [X5: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z2),X5)
     => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X5),Ta) ) ).

% pinf(6)
tff(fact_2317_pinf_I6_J,axiom,
    ! [Ta: num] :
    ? [Z2: num] :
    ! [X5: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Z2),X5)
     => ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X5),Ta) ) ).

% pinf(6)
tff(fact_2318_pinf_I6_J,axiom,
    ! [Ta: nat] :
    ? [Z2: nat] :
    ! [X5: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z2),X5)
     => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X5),Ta) ) ).

% pinf(6)
tff(fact_2319_pinf_I6_J,axiom,
    ! [Ta: int] :
    ? [Z2: int] :
    ! [X5: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z2),X5)
     => ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X5),Ta) ) ).

% pinf(6)
tff(fact_2320_verit__comp__simplify1_I3_J,axiom,
    ! [B2: rat,A2: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),A2)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),B2) ) ).

% verit_comp_simplify1(3)
tff(fact_2321_verit__comp__simplify1_I3_J,axiom,
    ! [B2: num,A2: num] :
      ( ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B2),A2)
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,A2),B2) ) ).

% verit_comp_simplify1(3)
tff(fact_2322_verit__comp__simplify1_I3_J,axiom,
    ! [B2: nat,A2: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B2),A2)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),B2) ) ).

% verit_comp_simplify1(3)
tff(fact_2323_verit__comp__simplify1_I3_J,axiom,
    ! [B2: int,A2: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),A2)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A2),B2) ) ).

% verit_comp_simplify1(3)
tff(fact_2324_complete__interval,axiom,
    ! [A: nat,B: nat,Pa: fun_nat_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_o(Pa,A)
       => ( ~ aa_nat_o(Pa,B)
         => ? [C: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),C)
              & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),B)
              & ! [X5: nat] :
                  ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),X5)
                    & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X5),C) )
                 => aa_nat_o(Pa,X5) )
              & ! [D4: nat] :
                  ( ! [X3: nat] :
                      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),X3)
                        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),D4) )
                     => aa_nat_o(Pa,X3) )
                 => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,D4),C) ) ) ) ) ) ).

% complete_interval
tff(fact_2325_complete__interval,axiom,
    ! [A: int,B: int,Pa: fun_int_o] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_int_o(Pa,A)
       => ( ~ aa_int_o(Pa,B)
         => ? [C: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),C)
              & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),B)
              & ! [X5: int] :
                  ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),X5)
                    & aa_int_o(aa_int_fun_int_o(ord_less_int,X5),C) )
                 => aa_int_o(Pa,X5) )
              & ! [D4: int] :
                  ( ! [X3: int] :
                      ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),X3)
                        & aa_int_o(aa_int_fun_int_o(ord_less_int,X3),D4) )
                     => aa_int_o(Pa,X3) )
                 => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,D4),C) ) ) ) ) ) ).

% complete_interval
tff(fact_2326_bijective__Empty,axiom,
    aa_set3702741264012733747_nat_o(bijective_nat_nat,bot_bo2099793752762293965at_nat) ).

% bijective_Empty
tff(fact_2327_the__dflt__None__nonempty,axiom,
    ! [S: set_Pr1261947904930325089at_nat] :
      ( ( S != bot_bo2099793752762293965at_nat )
     => ( dflt_N6592383573632408824at_nat(S) = some_s147305329494351046at_nat(S) ) ) ).

% the_dflt_None_nonempty
tff(fact_2328_the__dflt__None__nonempty,axiom,
    ! [S: set_o] :
      ( ( S != bot_bot_set_o )
     => ( dflt_None_set_o(S) = some_set_o(S) ) ) ).

% the_dflt_None_nonempty
tff(fact_2329_the__dflt__None__nonempty,axiom,
    ! [S: set_nat] :
      ( ( S != bot_bot_set_nat )
     => ( dflt_None_set_nat(S) = some_set_nat(S) ) ) ).

% the_dflt_None_nonempty
tff(fact_2330_the__dflt__None__nonempty,axiom,
    ! [S: set_int] :
      ( ( S != bot_bot_set_int )
     => ( dflt_None_set_int(S) = some_set_int(S) ) ) ).

% the_dflt_None_nonempty
tff(fact_2331_times__assn__raw_Opelims_I3_J,axiom,
    ! [X: fun_Pr6460944699146269935_nat_o,Xa2: fun_Pr6460944699146269935_nat_o,Xb: produc3658429121746597890et_nat] :
      ( ~ aa_Pro653082453696054616_nat_o(aa_fun2638059832795814268_nat_o(times_assn_raw(X),Xa2),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,Xa2),Xb)))
       => ~ ! [H: heap_e7401611519738050253t_unit,As3: set_nat] :
              ( ( Xb = produc7507926704131184380et_nat(H,As3) )
             => ( 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,Xa2),produc7507926704131184380et_nat(H,As3))))
               => ? [As13: set_nat,As23: set_nat] :
                    ( ( As3 = 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(Xa2,produc7507926704131184380et_nat(H,As23)) ) ) ) ) ) ).

% times_assn_raw.pelims(3)
tff(fact_2332_psubsetD,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat,Ca: product_prod_nat_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le7866589430770878221at_nat,A3),B3)
     => ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Ca),A3)
       => aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Ca),B3) ) ) ).

% psubsetD
tff(fact_2333_psubsetD,axiom,
    ! [A3: set_o,B3: set_o,Ca: $o] :
      ( ord_less_set_o(A3,B3)
     => ( member_o((Ca),A3)
       => member_o((Ca),B3) ) ) ).

% psubsetD
tff(fact_2334_psubsetD,axiom,
    ! [A3: set_set_nat,B3: set_set_nat,Ca: set_nat] :
      ( aa_set_set_nat_o(ord_less_set_set_nat(A3),B3)
     => ( aa_set_set_nat_o(member_set_nat(Ca),A3)
       => aa_set_set_nat_o(member_set_nat(Ca),B3) ) ) ).

% psubsetD
tff(fact_2335_psubsetD,axiom,
    ! [A3: set_nat,B3: set_nat,Ca: nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,A3),B3)
     => ( aa_set_nat_o(member_nat(Ca),A3)
       => aa_set_nat_o(member_nat(Ca),B3) ) ) ).

% psubsetD
tff(fact_2336_psubsetD,axiom,
    ! [A3: set_int,B3: set_int,Ca: int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_set_int,A3),B3)
     => ( aa_set_int_o(member_int(Ca),A3)
       => aa_set_int_o(member_int(Ca),B3) ) ) ).

% psubsetD
tff(fact_2337_less__set__def,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le7866589430770878221at_nat,A3),B3)
    <=> aa_fun8408151826566693824at_o_o(ord_le9073604597836486254_nat_o(aTP_Lamm_a(A3)),aTP_Lamm_a(B3)) ) ).

% less_set_def
tff(fact_2338_less__set__def,axiom,
    ! [A3: set_o,B3: set_o] :
      ( ord_less_set_o(A3,B3)
    <=> ord_less_fun_o_o(aTP_Lamm_aa(A3),aTP_Lamm_aa(B3)) ) ).

% less_set_def
tff(fact_2339_less__set__def,axiom,
    ! [A3: set_set_nat,B3: set_set_nat] :
      ( aa_set_set_nat_o(ord_less_set_set_nat(A3),B3)
    <=> ord_le921202756137246487_nat_o(aTP_Lamm_ad(A3),aTP_Lamm_ad(B3)) ) ).

% less_set_def
tff(fact_2340_less__set__def,axiom,
    ! [A3: set_nat,B3: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,A3),B3)
    <=> ord_less_fun_nat_o(aTP_Lamm_ae(A3),aTP_Lamm_ae(B3)) ) ).

% less_set_def
tff(fact_2341_less__set__def,axiom,
    ! [A3: set_int,B3: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_set_int,A3),B3)
    <=> ord_less_fun_int_o(aTP_Lamm_af(A3),aTP_Lamm_af(B3)) ) ).

% less_set_def
tff(fact_2342_verit__la__disequality,axiom,
    ! [A: rat,B: rat] :
      ( ( A = B )
      | ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
      | ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A) ) ).

% verit_la_disequality
tff(fact_2343_verit__la__disequality,axiom,
    ! [A: num,B: num] :
      ( ( A = B )
      | ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
      | ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),A) ) ).

% verit_la_disequality
tff(fact_2344_verit__la__disequality,axiom,
    ! [A: nat,B: nat] :
      ( ( A = B )
      | ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
      | ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A) ) ).

% verit_la_disequality
tff(fact_2345_verit__la__disequality,axiom,
    ! [A: int,B: int] :
      ( ( A = B )
      | ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
      | ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A) ) ).

% verit_la_disequality
tff(fact_2346_verit__comp__simplify1_I2_J,axiom,
    ! [A: set_int] : aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A),A) ).

% verit_comp_simplify1(2)
tff(fact_2347_verit__comp__simplify1_I2_J,axiom,
    ! [A: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),A) ).

% verit_comp_simplify1(2)
tff(fact_2348_verit__comp__simplify1_I2_J,axiom,
    ! [A: num] : aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),A) ).

% verit_comp_simplify1(2)
tff(fact_2349_verit__comp__simplify1_I2_J,axiom,
    ! [A: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),A) ).

% verit_comp_simplify1(2)
tff(fact_2350_verit__comp__simplify1_I2_J,axiom,
    ! [A: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),A) ).

% verit_comp_simplify1(2)
tff(fact_2351_verit__comp__simplify1_I1_J,axiom,
    ! [A: assn] : ~ aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),A) ).

% verit_comp_simplify1(1)
tff(fact_2352_verit__comp__simplify1_I1_J,axiom,
    ! [A: rat] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),A) ).

% verit_comp_simplify1(1)
tff(fact_2353_verit__comp__simplify1_I1_J,axiom,
    ! [A: num] : ~ aa_num_o(aa_num_fun_num_o(ord_less_num,A),A) ).

% verit_comp_simplify1(1)
tff(fact_2354_verit__comp__simplify1_I1_J,axiom,
    ! [A: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),A) ).

% verit_comp_simplify1(1)
tff(fact_2355_verit__comp__simplify1_I1_J,axiom,
    ! [A: int] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,A),A) ).

% verit_comp_simplify1(1)
tff(fact_2356_pinf_I1_J,axiom,
    ! [Pa: fun_rat_o,P3: fun_rat_o,Qa: fun_rat_o,Q2: fun_rat_o] :
      ( ? [Z4: rat] :
        ! [X3: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z4),X3)
         => ( aa_rat_o(Pa,X3)
          <=> aa_rat_o(P3,X3) ) )
     => ( ? [Z4: rat] :
          ! [X3: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z4),X3)
           => ( aa_rat_o(Qa,X3)
            <=> aa_rat_o(Q2,X3) ) )
       => ? [Z2: rat] :
          ! [X5: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z2),X5)
           => ( ( aa_rat_o(Pa,X5)
                & aa_rat_o(Qa,X5) )
            <=> ( aa_rat_o(P3,X5)
                & aa_rat_o(Q2,X5) ) ) ) ) ) ).

% pinf(1)
tff(fact_2357_pinf_I1_J,axiom,
    ! [Pa: fun_num_o,P3: fun_num_o,Qa: fun_num_o,Q2: fun_num_o] :
      ( ? [Z4: num] :
        ! [X3: num] :
          ( aa_num_o(aa_num_fun_num_o(ord_less_num,Z4),X3)
         => ( aa_num_o(Pa,X3)
          <=> aa_num_o(P3,X3) ) )
     => ( ? [Z4: num] :
          ! [X3: num] :
            ( aa_num_o(aa_num_fun_num_o(ord_less_num,Z4),X3)
           => ( aa_num_o(Qa,X3)
            <=> aa_num_o(Q2,X3) ) )
       => ? [Z2: num] :
          ! [X5: num] :
            ( aa_num_o(aa_num_fun_num_o(ord_less_num,Z2),X5)
           => ( ( aa_num_o(Pa,X5)
                & aa_num_o(Qa,X5) )
            <=> ( aa_num_o(P3,X5)
                & aa_num_o(Q2,X5) ) ) ) ) ) ).

% pinf(1)
tff(fact_2358_pinf_I1_J,axiom,
    ! [Pa: fun_nat_o,P3: fun_nat_o,Qa: fun_nat_o,Q2: fun_nat_o] :
      ( ? [Z4: nat] :
        ! [X3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z4),X3)
         => ( aa_nat_o(Pa,X3)
          <=> aa_nat_o(P3,X3) ) )
     => ( ? [Z4: nat] :
          ! [X3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z4),X3)
           => ( aa_nat_o(Qa,X3)
            <=> aa_nat_o(Q2,X3) ) )
       => ? [Z2: nat] :
          ! [X5: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z2),X5)
           => ( ( aa_nat_o(Pa,X5)
                & aa_nat_o(Qa,X5) )
            <=> ( aa_nat_o(P3,X5)
                & aa_nat_o(Q2,X5) ) ) ) ) ) ).

% pinf(1)
tff(fact_2359_pinf_I1_J,axiom,
    ! [Pa: fun_int_o,P3: fun_int_o,Qa: fun_int_o,Q2: fun_int_o] :
      ( ? [Z4: int] :
        ! [X3: int] :
          ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z4),X3)
         => ( aa_int_o(Pa,X3)
          <=> aa_int_o(P3,X3) ) )
     => ( ? [Z4: int] :
          ! [X3: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z4),X3)
           => ( aa_int_o(Qa,X3)
            <=> aa_int_o(Q2,X3) ) )
       => ? [Z2: int] :
          ! [X5: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z2),X5)
           => ( ( aa_int_o(Pa,X5)
                & aa_int_o(Qa,X5) )
            <=> ( aa_int_o(P3,X5)
                & aa_int_o(Q2,X5) ) ) ) ) ) ).

% pinf(1)
tff(fact_2360_pinf_I2_J,axiom,
    ! [Pa: fun_rat_o,P3: fun_rat_o,Qa: fun_rat_o,Q2: fun_rat_o] :
      ( ? [Z4: rat] :
        ! [X3: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z4),X3)
         => ( aa_rat_o(Pa,X3)
          <=> aa_rat_o(P3,X3) ) )
     => ( ? [Z4: rat] :
          ! [X3: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z4),X3)
           => ( aa_rat_o(Qa,X3)
            <=> aa_rat_o(Q2,X3) ) )
       => ? [Z2: rat] :
          ! [X5: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z2),X5)
           => ( ( aa_rat_o(Pa,X5)
                | aa_rat_o(Qa,X5) )
            <=> ( aa_rat_o(P3,X5)
                | aa_rat_o(Q2,X5) ) ) ) ) ) ).

% pinf(2)
tff(fact_2361_pinf_I2_J,axiom,
    ! [Pa: fun_num_o,P3: fun_num_o,Qa: fun_num_o,Q2: fun_num_o] :
      ( ? [Z4: num] :
        ! [X3: num] :
          ( aa_num_o(aa_num_fun_num_o(ord_less_num,Z4),X3)
         => ( aa_num_o(Pa,X3)
          <=> aa_num_o(P3,X3) ) )
     => ( ? [Z4: num] :
          ! [X3: num] :
            ( aa_num_o(aa_num_fun_num_o(ord_less_num,Z4),X3)
           => ( aa_num_o(Qa,X3)
            <=> aa_num_o(Q2,X3) ) )
       => ? [Z2: num] :
          ! [X5: num] :
            ( aa_num_o(aa_num_fun_num_o(ord_less_num,Z2),X5)
           => ( ( aa_num_o(Pa,X5)
                | aa_num_o(Qa,X5) )
            <=> ( aa_num_o(P3,X5)
                | aa_num_o(Q2,X5) ) ) ) ) ) ).

% pinf(2)
tff(fact_2362_pinf_I2_J,axiom,
    ! [Pa: fun_nat_o,P3: fun_nat_o,Qa: fun_nat_o,Q2: fun_nat_o] :
      ( ? [Z4: nat] :
        ! [X3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z4),X3)
         => ( aa_nat_o(Pa,X3)
          <=> aa_nat_o(P3,X3) ) )
     => ( ? [Z4: nat] :
          ! [X3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z4),X3)
           => ( aa_nat_o(Qa,X3)
            <=> aa_nat_o(Q2,X3) ) )
       => ? [Z2: nat] :
          ! [X5: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z2),X5)
           => ( ( aa_nat_o(Pa,X5)
                | aa_nat_o(Qa,X5) )
            <=> ( aa_nat_o(P3,X5)
                | aa_nat_o(Q2,X5) ) ) ) ) ) ).

% pinf(2)
tff(fact_2363_pinf_I2_J,axiom,
    ! [Pa: fun_int_o,P3: fun_int_o,Qa: fun_int_o,Q2: fun_int_o] :
      ( ? [Z4: int] :
        ! [X3: int] :
          ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z4),X3)
         => ( aa_int_o(Pa,X3)
          <=> aa_int_o(P3,X3) ) )
     => ( ? [Z4: int] :
          ! [X3: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z4),X3)
           => ( aa_int_o(Qa,X3)
            <=> aa_int_o(Q2,X3) ) )
       => ? [Z2: int] :
          ! [X5: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z2),X5)
           => ( ( aa_int_o(Pa,X5)
                | aa_int_o(Qa,X5) )
            <=> ( aa_int_o(P3,X5)
                | aa_int_o(Q2,X5) ) ) ) ) ) ).

% pinf(2)
tff(fact_2364_pinf_I3_J,axiom,
    ! [Ta: rat] :
    ? [Z2: rat] :
    ! [X5: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z2),X5)
     => ( X5 != Ta ) ) ).

% pinf(3)
tff(fact_2365_pinf_I3_J,axiom,
    ! [Ta: num] :
    ? [Z2: num] :
    ! [X5: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Z2),X5)
     => ( X5 != Ta ) ) ).

% pinf(3)
tff(fact_2366_pinf_I3_J,axiom,
    ! [Ta: nat] :
    ? [Z2: nat] :
    ! [X5: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z2),X5)
     => ( X5 != Ta ) ) ).

% pinf(3)
tff(fact_2367_pinf_I3_J,axiom,
    ! [Ta: int] :
    ? [Z2: int] :
    ! [X5: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z2),X5)
     => ( X5 != Ta ) ) ).

% pinf(3)
tff(fact_2368_pinf_I4_J,axiom,
    ! [Ta: rat] :
    ? [Z2: rat] :
    ! [X5: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z2),X5)
     => ( X5 != Ta ) ) ).

% pinf(4)
tff(fact_2369_pinf_I4_J,axiom,
    ! [Ta: num] :
    ? [Z2: num] :
    ! [X5: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Z2),X5)
     => ( X5 != Ta ) ) ).

% pinf(4)
tff(fact_2370_pinf_I4_J,axiom,
    ! [Ta: nat] :
    ? [Z2: nat] :
    ! [X5: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z2),X5)
     => ( X5 != Ta ) ) ).

% pinf(4)
tff(fact_2371_pinf_I4_J,axiom,
    ! [Ta: int] :
    ? [Z2: int] :
    ! [X5: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z2),X5)
     => ( X5 != Ta ) ) ).

% pinf(4)
tff(fact_2372_pinf_I5_J,axiom,
    ! [Ta: rat] :
    ? [Z2: rat] :
    ! [X5: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z2),X5)
     => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X5),Ta) ) ).

% pinf(5)
tff(fact_2373_pinf_I5_J,axiom,
    ! [Ta: num] :
    ? [Z2: num] :
    ! [X5: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Z2),X5)
     => ~ aa_num_o(aa_num_fun_num_o(ord_less_num,X5),Ta) ) ).

% pinf(5)
tff(fact_2374_pinf_I5_J,axiom,
    ! [Ta: nat] :
    ? [Z2: nat] :
    ! [X5: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z2),X5)
     => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X5),Ta) ) ).

% pinf(5)
tff(fact_2375_pinf_I5_J,axiom,
    ! [Ta: int] :
    ? [Z2: int] :
    ! [X5: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z2),X5)
     => ~ aa_int_o(aa_int_fun_int_o(ord_less_int,X5),Ta) ) ).

% pinf(5)
tff(fact_2376_pinf_I7_J,axiom,
    ! [Ta: rat] :
    ? [Z2: rat] :
    ! [X5: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z2),X5)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Ta),X5) ) ).

% pinf(7)
tff(fact_2377_pinf_I7_J,axiom,
    ! [Ta: num] :
    ? [Z2: num] :
    ! [X5: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Z2),X5)
     => aa_num_o(aa_num_fun_num_o(ord_less_num,Ta),X5) ) ).

% pinf(7)
tff(fact_2378_pinf_I7_J,axiom,
    ! [Ta: nat] :
    ? [Z2: nat] :
    ! [X5: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z2),X5)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ta),X5) ) ).

% pinf(7)
tff(fact_2379_pinf_I7_J,axiom,
    ! [Ta: int] :
    ? [Z2: int] :
    ! [X5: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z2),X5)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,Ta),X5) ) ).

% pinf(7)
tff(fact_2380_minf_I1_J,axiom,
    ! [Pa: fun_rat_o,P3: fun_rat_o,Qa: fun_rat_o,Q2: fun_rat_o] :
      ( ? [Z4: rat] :
        ! [X3: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Z4)
         => ( aa_rat_o(Pa,X3)
          <=> aa_rat_o(P3,X3) ) )
     => ( ? [Z4: rat] :
          ! [X3: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Z4)
           => ( aa_rat_o(Qa,X3)
            <=> aa_rat_o(Q2,X3) ) )
       => ? [Z2: rat] :
          ! [X5: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X5),Z2)
           => ( ( aa_rat_o(Pa,X5)
                & aa_rat_o(Qa,X5) )
            <=> ( aa_rat_o(P3,X5)
                & aa_rat_o(Q2,X5) ) ) ) ) ) ).

% minf(1)
tff(fact_2381_minf_I1_J,axiom,
    ! [Pa: fun_num_o,P3: fun_num_o,Qa: fun_num_o,Q2: fun_num_o] :
      ( ? [Z4: num] :
        ! [X3: num] :
          ( aa_num_o(aa_num_fun_num_o(ord_less_num,X3),Z4)
         => ( aa_num_o(Pa,X3)
          <=> aa_num_o(P3,X3) ) )
     => ( ? [Z4: num] :
          ! [X3: num] :
            ( aa_num_o(aa_num_fun_num_o(ord_less_num,X3),Z4)
           => ( aa_num_o(Qa,X3)
            <=> aa_num_o(Q2,X3) ) )
       => ? [Z2: num] :
          ! [X5: num] :
            ( aa_num_o(aa_num_fun_num_o(ord_less_num,X5),Z2)
           => ( ( aa_num_o(Pa,X5)
                & aa_num_o(Qa,X5) )
            <=> ( aa_num_o(P3,X5)
                & aa_num_o(Q2,X5) ) ) ) ) ) ).

% minf(1)
tff(fact_2382_minf_I1_J,axiom,
    ! [Pa: fun_nat_o,P3: fun_nat_o,Qa: fun_nat_o,Q2: fun_nat_o] :
      ( ? [Z4: nat] :
        ! [X3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Z4)
         => ( aa_nat_o(Pa,X3)
          <=> aa_nat_o(P3,X3) ) )
     => ( ? [Z4: nat] :
          ! [X3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Z4)
           => ( aa_nat_o(Qa,X3)
            <=> aa_nat_o(Q2,X3) ) )
       => ? [Z2: nat] :
          ! [X5: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X5),Z2)
           => ( ( aa_nat_o(Pa,X5)
                & aa_nat_o(Qa,X5) )
            <=> ( aa_nat_o(P3,X5)
                & aa_nat_o(Q2,X5) ) ) ) ) ) ).

% minf(1)
tff(fact_2383_minf_I1_J,axiom,
    ! [Pa: fun_int_o,P3: fun_int_o,Qa: fun_int_o,Q2: fun_int_o] :
      ( ? [Z4: int] :
        ! [X3: int] :
          ( aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Z4)
         => ( aa_int_o(Pa,X3)
          <=> aa_int_o(P3,X3) ) )
     => ( ? [Z4: int] :
          ! [X3: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Z4)
           => ( aa_int_o(Qa,X3)
            <=> aa_int_o(Q2,X3) ) )
       => ? [Z2: int] :
          ! [X5: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_int,X5),Z2)
           => ( ( aa_int_o(Pa,X5)
                & aa_int_o(Qa,X5) )
            <=> ( aa_int_o(P3,X5)
                & aa_int_o(Q2,X5) ) ) ) ) ) ).

% minf(1)
tff(fact_2384_minf_I2_J,axiom,
    ! [Pa: fun_rat_o,P3: fun_rat_o,Qa: fun_rat_o,Q2: fun_rat_o] :
      ( ? [Z4: rat] :
        ! [X3: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Z4)
         => ( aa_rat_o(Pa,X3)
          <=> aa_rat_o(P3,X3) ) )
     => ( ? [Z4: rat] :
          ! [X3: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Z4)
           => ( aa_rat_o(Qa,X3)
            <=> aa_rat_o(Q2,X3) ) )
       => ? [Z2: rat] :
          ! [X5: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X5),Z2)
           => ( ( aa_rat_o(Pa,X5)
                | aa_rat_o(Qa,X5) )
            <=> ( aa_rat_o(P3,X5)
                | aa_rat_o(Q2,X5) ) ) ) ) ) ).

% minf(2)
tff(fact_2385_minf_I2_J,axiom,
    ! [Pa: fun_num_o,P3: fun_num_o,Qa: fun_num_o,Q2: fun_num_o] :
      ( ? [Z4: num] :
        ! [X3: num] :
          ( aa_num_o(aa_num_fun_num_o(ord_less_num,X3),Z4)
         => ( aa_num_o(Pa,X3)
          <=> aa_num_o(P3,X3) ) )
     => ( ? [Z4: num] :
          ! [X3: num] :
            ( aa_num_o(aa_num_fun_num_o(ord_less_num,X3),Z4)
           => ( aa_num_o(Qa,X3)
            <=> aa_num_o(Q2,X3) ) )
       => ? [Z2: num] :
          ! [X5: num] :
            ( aa_num_o(aa_num_fun_num_o(ord_less_num,X5),Z2)
           => ( ( aa_num_o(Pa,X5)
                | aa_num_o(Qa,X5) )
            <=> ( aa_num_o(P3,X5)
                | aa_num_o(Q2,X5) ) ) ) ) ) ).

% minf(2)
tff(fact_2386_minf_I2_J,axiom,
    ! [Pa: fun_nat_o,P3: fun_nat_o,Qa: fun_nat_o,Q2: fun_nat_o] :
      ( ? [Z4: nat] :
        ! [X3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Z4)
         => ( aa_nat_o(Pa,X3)
          <=> aa_nat_o(P3,X3) ) )
     => ( ? [Z4: nat] :
          ! [X3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Z4)
           => ( aa_nat_o(Qa,X3)
            <=> aa_nat_o(Q2,X3) ) )
       => ? [Z2: nat] :
          ! [X5: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X5),Z2)
           => ( ( aa_nat_o(Pa,X5)
                | aa_nat_o(Qa,X5) )
            <=> ( aa_nat_o(P3,X5)
                | aa_nat_o(Q2,X5) ) ) ) ) ) ).

% minf(2)
tff(fact_2387_minf_I2_J,axiom,
    ! [Pa: fun_int_o,P3: fun_int_o,Qa: fun_int_o,Q2: fun_int_o] :
      ( ? [Z4: int] :
        ! [X3: int] :
          ( aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Z4)
         => ( aa_int_o(Pa,X3)
          <=> aa_int_o(P3,X3) ) )
     => ( ? [Z4: int] :
          ! [X3: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Z4)
           => ( aa_int_o(Qa,X3)
            <=> aa_int_o(Q2,X3) ) )
       => ? [Z2: int] :
          ! [X5: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_int,X5),Z2)
           => ( ( aa_int_o(Pa,X5)
                | aa_int_o(Qa,X5) )
            <=> ( aa_int_o(P3,X5)
                | aa_int_o(Q2,X5) ) ) ) ) ) ).

% minf(2)
tff(fact_2388_minf_I3_J,axiom,
    ! [Ta: rat] :
    ? [Z2: rat] :
    ! [X5: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X5),Z2)
     => ( X5 != Ta ) ) ).

% minf(3)
tff(fact_2389_minf_I3_J,axiom,
    ! [Ta: num] :
    ? [Z2: num] :
    ! [X5: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,X5),Z2)
     => ( X5 != Ta ) ) ).

% minf(3)
tff(fact_2390_minf_I3_J,axiom,
    ! [Ta: nat] :
    ? [Z2: nat] :
    ! [X5: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X5),Z2)
     => ( X5 != Ta ) ) ).

% minf(3)
tff(fact_2391_minf_I3_J,axiom,
    ! [Ta: int] :
    ? [Z2: int] :
    ! [X5: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X5),Z2)
     => ( X5 != Ta ) ) ).

% minf(3)
tff(fact_2392_minf_I4_J,axiom,
    ! [Ta: rat] :
    ? [Z2: rat] :
    ! [X5: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X5),Z2)
     => ( X5 != Ta ) ) ).

% minf(4)
tff(fact_2393_minf_I4_J,axiom,
    ! [Ta: num] :
    ? [Z2: num] :
    ! [X5: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,X5),Z2)
     => ( X5 != Ta ) ) ).

% minf(4)
tff(fact_2394_minf_I4_J,axiom,
    ! [Ta: nat] :
    ? [Z2: nat] :
    ! [X5: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X5),Z2)
     => ( X5 != Ta ) ) ).

% minf(4)
tff(fact_2395_minf_I4_J,axiom,
    ! [Ta: int] :
    ? [Z2: int] :
    ! [X5: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X5),Z2)
     => ( X5 != Ta ) ) ).

% minf(4)
tff(fact_2396_minf_I5_J,axiom,
    ! [Ta: rat] :
    ? [Z2: rat] :
    ! [X5: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X5),Z2)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X5),Ta) ) ).

% minf(5)
tff(fact_2397_minf_I5_J,axiom,
    ! [Ta: num] :
    ? [Z2: num] :
    ! [X5: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,X5),Z2)
     => aa_num_o(aa_num_fun_num_o(ord_less_num,X5),Ta) ) ).

% minf(5)
tff(fact_2398_minf_I5_J,axiom,
    ! [Ta: nat] :
    ? [Z2: nat] :
    ! [X5: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X5),Z2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X5),Ta) ) ).

% minf(5)
tff(fact_2399_minf_I5_J,axiom,
    ! [Ta: int] :
    ? [Z2: int] :
    ! [X5: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X5),Z2)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,X5),Ta) ) ).

% minf(5)
tff(fact_2400_minf_I7_J,axiom,
    ! [Ta: rat] :
    ? [Z2: rat] :
    ! [X5: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X5),Z2)
     => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Ta),X5) ) ).

% minf(7)
tff(fact_2401_minf_I7_J,axiom,
    ! [Ta: num] :
    ? [Z2: num] :
    ! [X5: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,X5),Z2)
     => ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Ta),X5) ) ).

% minf(7)
tff(fact_2402_minf_I7_J,axiom,
    ! [Ta: nat] :
    ? [Z2: nat] :
    ! [X5: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X5),Z2)
     => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ta),X5) ) ).

% minf(7)
tff(fact_2403_minf_I7_J,axiom,
    ! [Ta: int] :
    ? [Z2: int] :
    ! [X5: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X5),Z2)
     => ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Ta),X5) ) ).

% minf(7)
tff(fact_2404_bijective__def,axiom,
    ! [Ra: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(bijective_nat_nat,Ra)
    <=> ( ! [X4: nat,Y5: nat,Z5: nat] :
            ( ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,X4),Y5)),Ra)
              & aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,X4),Z5)),Ra) )
           => ( Y5 = Z5 ) )
        & ! [X4: nat,Y5: nat,Z5: nat] :
            ( ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,X4),Z5)),Ra)
              & aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Y5),Z5)),Ra) )
           => ( X4 = Y5 ) ) ) ) ).

% bijective_def
tff(fact_2405_bijective__def,axiom,
    ! [Ra: set_Pr1312406793078187350et_nat] :
      ( biject6574446902887564708et_nat(Ra)
    <=> ( ! [X4: fun_Pr6460944699146269935_nat_o,Y5: produc3658429121746597890et_nat,Z5: produc3658429121746597890et_nat] :
            ( ( member4513763875858039863et_nat(aa_Pro434340453207536480et_nat(aa_fun3926903090414666066et_nat(produc6017135503889078682et_nat,X4),Y5),Ra)
              & member4513763875858039863et_nat(aa_Pro434340453207536480et_nat(aa_fun3926903090414666066et_nat(produc6017135503889078682et_nat,X4),Z5),Ra) )
           => ( Y5 = Z5 ) )
        & ! [X4: fun_Pr6460944699146269935_nat_o,Y5: fun_Pr6460944699146269935_nat_o,Z5: produc3658429121746597890et_nat] :
            ( ( member4513763875858039863et_nat(aa_Pro434340453207536480et_nat(aa_fun3926903090414666066et_nat(produc6017135503889078682et_nat,X4),Z5),Ra)
              & member4513763875858039863et_nat(aa_Pro434340453207536480et_nat(aa_fun3926903090414666066et_nat(produc6017135503889078682et_nat,Y5),Z5),Ra) )
           => ( X4 = Y5 ) ) ) ) ).

% bijective_def
tff(fact_2406_bijective__def,axiom,
    ! [Ra: set_Pr4040394263813967220et_nat] :
      ( biject4066639966817638850et_nat(Ra)
    <=> ( ! [X4: fun_Pr6460944699146269935_nat_o,Y5: produc8799533983703829792et_nat,Z5: produc8799533983703829792et_nat] :
            ( ( member9174161442226468181et_nat(aa_Pro1979504083478021276et_nat(aa_fun615918787490101902et_nat(produc7291855603632349880et_nat,X4),Y5),Ra)
              & member9174161442226468181et_nat(aa_Pro1979504083478021276et_nat(aa_fun615918787490101902et_nat(produc7291855603632349880et_nat,X4),Z5),Ra) )
           => ( Y5 = Z5 ) )
        & ! [X4: fun_Pr6460944699146269935_nat_o,Y5: fun_Pr6460944699146269935_nat_o,Z5: produc8799533983703829792et_nat] :
            ( ( member9174161442226468181et_nat(aa_Pro1979504083478021276et_nat(aa_fun615918787490101902et_nat(produc7291855603632349880et_nat,X4),Z5),Ra)
              & member9174161442226468181et_nat(aa_Pro1979504083478021276et_nat(aa_fun615918787490101902et_nat(produc7291855603632349880et_nat,Y5),Z5),Ra) )
           => ( X4 = Y5 ) ) ) ) ).

% bijective_def
tff(fact_2407_bijective__def,axiom,
    ! [Ra: set_Pr6700895561065273913nt_int] :
      ( biject1579208791189821119nt_int(Ra)
    <=> ( ! [X4: fun_Pr7763680713840462443e_term,Y5: product_prod_int_int,Z5: product_prod_int_int] :
            ( ( member753276645541486746nt_int(aa_Pro5748361610964203648nt_int(aa_fun3265472288063244902nt_int(produc7320969046608631221nt_int,X4),Y5),Ra)
              & member753276645541486746nt_int(aa_Pro5748361610964203648nt_int(aa_fun3265472288063244902nt_int(produc7320969046608631221nt_int,X4),Z5),Ra) )
           => ( Y5 = Z5 ) )
        & ! [X4: fun_Pr7763680713840462443e_term,Y5: fun_Pr7763680713840462443e_term,Z5: product_prod_int_int] :
            ( ( member753276645541486746nt_int(aa_Pro5748361610964203648nt_int(aa_fun3265472288063244902nt_int(produc7320969046608631221nt_int,X4),Z5),Ra)
              & member753276645541486746nt_int(aa_Pro5748361610964203648nt_int(aa_fun3265472288063244902nt_int(produc7320969046608631221nt_int,Y5),Z5),Ra) )
           => ( X4 = Y5 ) ) ) ) ).

% bijective_def
tff(fact_2408_bijective__def,axiom,
    ! [Ra: set_Pr4166680992137330533nt_int] :
      ( biject9165800892491502059nt_int(Ra)
    <=> ( ! [X4: fun_in5730379987311621695e_term,Y5: product_prod_int_int,Z5: product_prod_int_int] :
            ( ( member5890090881338784710nt_int(aa_Pro8800328608558162860nt_int(aa_fun260840925927807678nt_int(produc4729913460834754785nt_int,X4),Y5),Ra)
              & member5890090881338784710nt_int(aa_Pro8800328608558162860nt_int(aa_fun260840925927807678nt_int(produc4729913460834754785nt_int,X4),Z5),Ra) )
           => ( Y5 = Z5 ) )
        & ! [X4: fun_in5730379987311621695e_term,Y5: fun_in5730379987311621695e_term,Z5: product_prod_int_int] :
            ( ( member5890090881338784710nt_int(aa_Pro8800328608558162860nt_int(aa_fun260840925927807678nt_int(produc4729913460834754785nt_int,X4),Z5),Ra)
              & member5890090881338784710nt_int(aa_Pro8800328608558162860nt_int(aa_fun260840925927807678nt_int(produc4729913460834754785nt_int,Y5),Z5),Ra) )
           => ( X4 = Y5 ) ) ) ) ).

% bijective_def
tff(fact_2409_bijective__def,axiom,
    ! [Ra: set_Pr7098220151150636591it_nat] :
      ( biject8592306251674886141it_nat(Ra)
    <=> ( ! [X4: a,Y5: produc6653097349344004940it_nat,Z5: produc6653097349344004940it_nat] :
            ( ( member4554412811331277712it_nat(aa_Pro8078911840264235907it_nat(aa_a_f2234720956197727492it_nat(produc9178034014595674355it_nat,X4),Y5),Ra)
              & member4554412811331277712it_nat(aa_Pro8078911840264235907it_nat(aa_a_f2234720956197727492it_nat(produc9178034014595674355it_nat,X4),Z5),Ra) )
           => ( Y5 = Z5 ) )
        & ! [X4: a,Y5: a,Z5: produc6653097349344004940it_nat] :
            ( ( member4554412811331277712it_nat(aa_Pro8078911840264235907it_nat(aa_a_f2234720956197727492it_nat(produc9178034014595674355it_nat,X4),Z5),Ra)
              & member4554412811331277712it_nat(aa_Pro8078911840264235907it_nat(aa_a_f2234720956197727492it_nat(produc9178034014595674355it_nat,Y5),Z5),Ra) )
           => ( X4 = Y5 ) ) ) ) ).

% bijective_def
tff(fact_2410_times__assn__raw_Opelims_I1_J,axiom,
    ! [X: fun_Pr6460944699146269935_nat_o,Xa2: fun_Pr6460944699146269935_nat_o,Xb: produc3658429121746597890et_nat,Y: $o] :
      ( ( aa_Pro653082453696054616_nat_o(aa_fun2638059832795814268_nat_o(times_assn_raw(X),Xa2),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,Xa2),Xb)))
       => ~ ! [H: heap_e7401611519738050253t_unit,As3: set_nat] :
              ( ( Xb = produc7507926704131184380et_nat(H,As3) )
             => ( ( (Y)
                <=> ? [As12: set_nat,As22: set_nat] :
                      ( ( As3 = 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(Xa2,produc7507926704131184380et_nat(H,As22)) ) )
               => ~ 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,Xa2),produc7507926704131184380et_nat(H,As3)))) ) ) ) ) ).

% times_assn_raw.pelims(1)
tff(fact_2411_times__assn__raw_Opelims_I2_J,axiom,
    ! [X: fun_Pr6460944699146269935_nat_o,Xa2: fun_Pr6460944699146269935_nat_o,Xb: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(aa_fun2638059832795814268_nat_o(times_assn_raw(X),Xa2),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,Xa2),Xb)))
       => ~ ! [H: heap_e7401611519738050253t_unit,As3: set_nat] :
              ( ( Xb = produc7507926704131184380et_nat(H,As3) )
             => ( 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,Xa2),produc7507926704131184380et_nat(H,As3))))
               => ~ ? [As1: set_nat,As2: set_nat] :
                      ( ( As3 = 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(Xa2,produc7507926704131184380et_nat(H,As2)) ) ) ) ) ) ).

% times_assn_raw.pelims(2)
tff(fact_2412_the__dflt__None__set,axiom,
    ! [X: set_Pr1261947904930325089at_nat] : the_de3668382347305825784at_nat(bot_bo2099793752762293965at_nat,dflt_N6592383573632408824at_nat(X)) = X ).

% the_dflt_None_set
tff(fact_2413_the__dflt__None__set,axiom,
    ! [X: set_o] : the_default_set_o(bot_bot_set_o,dflt_None_set_o(X)) = X ).

% the_dflt_None_set
tff(fact_2414_the__dflt__None__set,axiom,
    ! [X: set_nat] : the_default_set_nat(bot_bot_set_nat,dflt_None_set_nat(X)) = X ).

% the_dflt_None_set
tff(fact_2415_the__dflt__None__set,axiom,
    ! [X: set_int] : the_default_set_int(bot_bot_set_int,dflt_None_set_int(X)) = X ).

% the_dflt_None_set
tff(fact_2416_accp__subset,axiom,
    ! [R1: fun_Pr7920380315054183110_nat_o,R22: fun_Pr7920380315054183110_nat_o] :
      ( ord_le5132100987546516070_nat_o(R1,R22)
     => ord_le5018287362838994063_nat_o(accp_P5801069581201407417et_nat(R22),accp_P5801069581201407417et_nat(R1)) ) ).

% accp_subset
tff(fact_2417_accp__subset,axiom,
    ! [R1: fun_Pr4799464687642144302_num_o,R22: fun_Pr4799464687642144302_num_o] :
      ( ord_le6421466325430216142_num_o(R1,R22)
     => ord_le3107050023622517222_num_o(accp_P3113834385874906142um_num(R22),accp_P3113834385874906142um_num(R1)) ) ).

% accp_subset
tff(fact_2418_accp__subset,axiom,
    ! [R1: fun_Pr2028404973800837934_nat_o,R22: fun_Pr2028404973800837934_nat_o] :
      ( ord_le3650406611588909774_nat_o(R1,R22)
     => aa_fun8408151826566693824at_o_o(ord_le1572679713340831354_nat_o(accp_P4275260045618599050at_nat(R22)),accp_P4275260045618599050at_nat(R1)) ) ).

% accp_subset
tff(fact_2419_accp__subset,axiom,
    ! [R1: fun_Pr9018117161326028078_int_o,R22: fun_Pr9018117161326028078_int_o] :
      ( ord_le1416746762259324110_int_o(R1,R22)
     => ord_le14110778710936002_int_o(accp_P1096762738010456898nt_int(R22),accp_P1096762738010456898nt_int(R1)) ) ).

% accp_subset
tff(fact_2420_accp__subset,axiom,
    ! [R1: fun_Pr3358449090565925198_nat_o,R22: fun_Pr3358449090565925198_nat_o] :
      ( ord_le6446978100474826990_nat_o(R1,R22)
     => ord_le3896530374514135251_nat_o(accp_P1394777952347926645et_nat(R22),accp_P1394777952347926645et_nat(R1)) ) ).

% accp_subset
tff(fact_2421_accp__subset__induct,axiom,
    ! [D3: fun_Pr6460944699146269935_nat_o,Ra: fun_Pr7920380315054183110_nat_o,X: produc3658429121746597890et_nat,Pa: fun_Pr6460944699146269935_nat_o] :
      ( ord_le5018287362838994063_nat_o(D3,accp_P5801069581201407417et_nat(Ra))
     => ( ! [X3: produc3658429121746597890et_nat,Z2: produc3658429121746597890et_nat] :
            ( aa_Pro653082453696054616_nat_o(D3,X3)
           => ( aa_Pro653082453696054616_nat_o(aa_Pro7278237734719282863_nat_o(Ra,Z2),X3)
             => aa_Pro653082453696054616_nat_o(D3,Z2) ) )
       => ( aa_Pro653082453696054616_nat_o(D3,X)
         => ( ! [X3: produc3658429121746597890et_nat] :
                ( aa_Pro653082453696054616_nat_o(D3,X3)
               => ( ! [Z4: produc3658429121746597890et_nat] :
                      ( aa_Pro653082453696054616_nat_o(aa_Pro7278237734719282863_nat_o(Ra,Z4),X3)
                     => aa_Pro653082453696054616_nat_o(Pa,Z4) )
                 => aa_Pro653082453696054616_nat_o(Pa,X3) ) )
           => aa_Pro653082453696054616_nat_o(Pa,X) ) ) ) ) ).

% accp_subset_induct
tff(fact_2422_accp__subset__induct,axiom,
    ! [D3: fun_Pr6331313853612757382_num_o,Ra: fun_Pr4799464687642144302_num_o,X: product_prod_num_num,Pa: fun_Pr6331313853612757382_num_o] :
      ( ord_le3107050023622517222_num_o(D3,accp_P3113834385874906142um_num(Ra))
     => ( ! [X3: product_prod_num_num,Z2: product_prod_num_num] :
            ( aa_Pro7685007914727650815_num_o(D3,X3)
           => ( aa_Pro7685007914727650815_num_o(aa_Pro4218333719626769503_num_o(Ra,Z2),X3)
             => aa_Pro7685007914727650815_num_o(D3,Z2) ) )
       => ( aa_Pro7685007914727650815_num_o(D3,X)
         => ( ! [X3: product_prod_num_num] :
                ( aa_Pro7685007914727650815_num_o(D3,X3)
               => ( ! [Z4: product_prod_num_num] :
                      ( aa_Pro7685007914727650815_num_o(aa_Pro4218333719626769503_num_o(Ra,Z4),X3)
                     => aa_Pro7685007914727650815_num_o(Pa,Z4) )
                 => aa_Pro7685007914727650815_num_o(Pa,X3) ) )
           => aa_Pro7685007914727650815_num_o(Pa,X) ) ) ) ) ).

% accp_subset_induct
tff(fact_2423_accp__subset__induct,axiom,
    ! [D3: fun_Pr4796943543331071514_nat_o,Ra: fun_Pr2028404973800837934_nat_o,X: product_prod_nat_nat,Pa: fun_Pr4796943543331071514_nat_o] :
      ( aa_fun8408151826566693824at_o_o(ord_le1572679713340831354_nat_o(D3),accp_P4275260045618599050at_nat(Ra))
     => ( ! [X3: product_prod_nat_nat,Z2: product_prod_nat_nat] :
            ( aa_Pro6150637604445964947_nat_o(D3,X3)
           => ( aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(Ra,Z2),X3)
             => aa_Pro6150637604445964947_nat_o(D3,Z2) ) )
       => ( aa_Pro6150637604445964947_nat_o(D3,X)
         => ( ! [X3: product_prod_nat_nat] :
                ( aa_Pro6150637604445964947_nat_o(D3,X3)
               => ( ! [Z4: product_prod_nat_nat] :
                      ( aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(Ra,Z4),X3)
                     => aa_Pro6150637604445964947_nat_o(Pa,Z4) )
                 => aa_Pro6150637604445964947_nat_o(Pa,X3) ) )
           => aa_Pro6150637604445964947_nat_o(Pa,X) ) ) ) ) ).

% accp_subset_induct
tff(fact_2424_accp__subset__induct,axiom,
    ! [D3: fun_Pr3238374608701176162_int_o,Ra: fun_Pr9018117161326028078_int_o,X: product_prod_int_int,Pa: fun_Pr3238374608701176162_int_o] :
      ( ord_le14110778710936002_int_o(D3,accp_P1096762738010456898nt_int(Ra))
     => ( ! [X3: product_prod_int_int,Z2: product_prod_int_int] :
            ( aa_Pro4592068669816069595_int_o(D3,X3)
           => ( aa_Pro4592068669816069595_int_o(aa_Pro3952998155787061599_int_o(Ra,Z2),X3)
             => aa_Pro4592068669816069595_int_o(D3,Z2) ) )
       => ( aa_Pro4592068669816069595_int_o(D3,X)
         => ( ! [X3: product_prod_int_int] :
                ( aa_Pro4592068669816069595_int_o(D3,X3)
               => ( ! [Z4: product_prod_int_int] :
                      ( aa_Pro4592068669816069595_int_o(aa_Pro3952998155787061599_int_o(Ra,Z4),X3)
                     => aa_Pro4592068669816069595_int_o(Pa,Z4) )
                 => aa_Pro4592068669816069595_int_o(Pa,X3) ) )
           => aa_Pro4592068669816069595_int_o(Pa,X) ) ) ) ) ).

% accp_subset_induct
tff(fact_2425_accp__subset__induct,axiom,
    ! [D3: fun_Pr1227177680593602867_nat_o,Ra: fun_Pr3358449090565925198_nat_o,X: produc3112576180687729086et_nat,Pa: fun_Pr1227177680593602867_nat_o] :
      ( ord_le3896530374514135251_nat_o(D3,accp_P1394777952347926645et_nat(Ra))
     => ( ! [X3: produc3112576180687729086et_nat,Z2: produc3112576180687729086et_nat] :
            ( aa_Pro2671238023309943196_nat_o(D3,X3)
           => ( aa_Pro2671238023309943196_nat_o(aa_Pro4671852207856306999_nat_o(Ra,Z2),X3)
             => aa_Pro2671238023309943196_nat_o(D3,Z2) ) )
       => ( aa_Pro2671238023309943196_nat_o(D3,X)
         => ( ! [X3: produc3112576180687729086et_nat] :
                ( aa_Pro2671238023309943196_nat_o(D3,X3)
               => ( ! [Z4: produc3112576180687729086et_nat] :
                      ( aa_Pro2671238023309943196_nat_o(aa_Pro4671852207856306999_nat_o(Ra,Z4),X3)
                     => aa_Pro2671238023309943196_nat_o(Pa,Z4) )
                 => aa_Pro2671238023309943196_nat_o(Pa,X3) ) )
           => aa_Pro2671238023309943196_nat_o(Pa,X) ) ) ) ) ).

% accp_subset_induct
tff(fact_2426_bot_Oordering__top__axioms,axiom,
    aa_set3702741264012733747_nat_o(orderi2172309028950807442at_nat(aTP_Lamm_cv,aTP_Lamm_cw),bot_bo2099793752762293965at_nat) ).

% bot.ordering_top_axioms
tff(fact_2427_bot_Oordering__top__axioms,axiom,
    ordering_top_set_o(aTP_Lamm_cx,aTP_Lamm_cy,bot_bot_set_o) ).

% bot.ordering_top_axioms
tff(fact_2428_bot_Oordering__top__axioms,axiom,
    aa_set_nat_o(ordering_top_set_nat(aTP_Lamm_cz,aTP_Lamm_da),bot_bot_set_nat) ).

% bot.ordering_top_axioms
tff(fact_2429_bot_Oordering__top__axioms,axiom,
    ordering_top_assn(aTP_Lamm_db,aTP_Lamm_dc,bot_bot_assn) ).

% bot.ordering_top_axioms
tff(fact_2430_bot_Oordering__top__axioms,axiom,
    aa_set_int_o(ordering_top_set_int(aTP_Lamm_dd,aTP_Lamm_de),bot_bot_set_int) ).

% bot.ordering_top_axioms
tff(fact_2431_bot_Oordering__top__axioms,axiom,
    aa_nat_o(ordering_top_nat(aTP_Lamm_df,aTP_Lamm_dg),bot_bot_nat) ).

% bot.ordering_top_axioms
tff(fact_2432_mod__h__bot__iff_I1_J,axiom,
    ! [B: $o,Ha: heap_e7401611519738050253t_unit] :
      ( aa_Pro653082453696054616_nat_o(rep_assn(aa_o_assn(pure_assn,(B))),produc7507926704131184380et_nat(Ha,bot_bot_set_nat))
    <=> (B) ) ).

% mod_h_bot_iff(1)
tff(fact_2433_hoare__triple__effect,axiom,
    ! [Pa: assn,Ca: heap_Time_Heap_a,Qa: fun_a_assn,Ha: heap_e7401611519738050253t_unit,Asa: set_nat] :
      ( hoare_hoare_triple_a(Pa,Ca,Qa)
     => ( aa_Pro653082453696054616_nat_o(rep_assn(Pa),produc7507926704131184380et_nat(Ha,Asa))
       => ? [H4: heap_e7401611519738050253t_unit,R: a,T2: nat] :
            ( aa_nat_o(heap_Time_effect_a(Ca,Ha,H4,R),T2)
            & aa_Pro653082453696054616_nat_o(rep_assn(aa_a_assn(Qa,R)),produc7507926704131184380et_nat(H4,hoare_new_addrs(Ha,Asa,H4))) ) ) ) ).

% hoare_triple_effect
tff(fact_2434_Sup__fin_Osemilattice__order__set__axioms,axiom,
    lattic5672175589177086241at_nat(sup_su718114333110466843at_nat,aTP_Lamm_cp,aTP_Lamm_cq) ).

% Sup_fin.semilattice_order_set_axioms
tff(fact_2435_Sup__fin_Osemilattice__order__set__axioms,axiom,
    lattic5174734335190411425at_nat(sup_su3035147773818789531at_nat,aTP_Lamm_cr,aTP_Lamm_cs) ).

% Sup_fin.semilattice_order_set_axioms
tff(fact_2436_Sup__fin_Osemilattice__order__set__axioms,axiom,
    lattic6976360593897529633at_nat(sup_su5525570899277871387at_nat,aTP_Lamm_ct,aTP_Lamm_cu) ).

% Sup_fin.semilattice_order_set_axioms
tff(fact_2437_Sup__fin_Osemilattice__order__set__axioms,axiom,
    lattic3109210760196336428et_nat(sup_sup_set_nat,aTP_Lamm_cz,aTP_Lamm_da) ).

% Sup_fin.semilattice_order_set_axioms
tff(fact_2438_Sup__fin_Osemilattice__order__set__axioms,axiom,
    lattic5623431474481994958t_assn(sup_sup_assn,aTP_Lamm_db,aTP_Lamm_dc) ).

% Sup_fin.semilattice_order_set_axioms
tff(fact_2439_Sup__fin_Osemilattice__order__set__axioms,axiom,
    lattic8154731777541915528et_int(sup_sup_set_int,aTP_Lamm_dd,aTP_Lamm_de) ).

% Sup_fin.semilattice_order_set_axioms
tff(fact_2440_Sup__fin_Osemilattice__order__set__axioms,axiom,
    lattic5374021519246970238et_rat(sup_sup_rat,aTP_Lamm_dh,aTP_Lamm_di) ).

% Sup_fin.semilattice_order_set_axioms
tff(fact_2441_Sup__fin_Osemilattice__order__set__axioms,axiom,
    lattic6009151579333465974et_nat(sup_sup_nat,aTP_Lamm_df,aTP_Lamm_dg) ).

% Sup_fin.semilattice_order_set_axioms
tff(fact_2442_Sup__fin_Osemilattice__order__set__axioms,axiom,
    lattic6006661108824415698et_int(sup_sup_int,aTP_Lamm_dj,aTP_Lamm_dk) ).

% Sup_fin.semilattice_order_set_axioms
tff(fact_2443_wand__assnI,axiom,
    ! [Ha: heap_e7401611519738050253t_unit,Asa: set_nat,Qa: assn,Ra: assn] :
      ( aa_Pro653082453696054616_nat_o(in_range,produc7507926704131184380et_nat(Ha,Asa))
     => ( ! [H4: heap_e7401611519738050253t_unit,As5: set_nat] :
            ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Asa),As5) = bot_bot_set_nat )
           => ( aa_hea7994211679605476167unit_o(relH(Asa,Ha),H4)
             => ( aa_Pro653082453696054616_nat_o(in_range,produc7507926704131184380et_nat(H4,Asa))
               => ( aa_Pro653082453696054616_nat_o(rep_assn(Qa),produc7507926704131184380et_nat(H4,As5))
                 => aa_Pro653082453696054616_nat_o(rep_assn(Ra),produc7507926704131184380et_nat(H4,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Asa),As5))) ) ) ) )
       => aa_Pro653082453696054616_nat_o(rep_assn(aa_assn_assn(aa_ass954072293579886259n_assn(wand_assn,Qa),Ra)),produc7507926704131184380et_nat(Ha,Asa)) ) ) ).

% wand_assnI
tff(fact_2444_pure__assn__eq__conv,axiom,
    ! [Pa: $o,Qa: $o] :
      ( ( aa_o_assn(pure_assn,(Pa)) = aa_o_assn(pure_assn,(Qa)) )
    <=> ( (Pa)
      <=> (Qa) ) ) ).

% pure_assn_eq_conv
tff(fact_2445_merge__pure__star,axiom,
    ! [A: $o,B: $o] :
      aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_o_assn(pure_assn,(A))),aa_o_assn(pure_assn,(B))) = aa_o_assn(pure_assn,
        ( (A)
        & (B) )) ).

% merge_pure_star
tff(fact_2446_merge__pure__or,axiom,
    ! [A: $o,B: $o] :
      aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,aa_o_assn(pure_assn,(A))),aa_o_assn(pure_assn,(B))) = aa_o_assn(pure_assn,
        ( (A)
        | (B) )) ).

% merge_pure_or
tff(fact_2447_pure__assn__eq__false__iff,axiom,
    ! [Pa: $o] :
      ( ( aa_o_assn(pure_assn,(Pa)) = bot_bot_assn )
    <=> ~ (Pa) ) ).

% pure_assn_eq_false_iff
tff(fact_2448_pure__false,axiom,
    aa_o_assn(pure_assn,$false) = bot_bot_assn ).

% pure_false
tff(fact_2449_merge__pure__and,axiom,
    ! [A: $o,B: $o] :
      aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,aa_o_assn(pure_assn,(A))),aa_o_assn(pure_assn,(B))) = aa_o_assn(pure_assn,
        ( (A)
        & (B) )) ).

% merge_pure_and
tff(fact_2450_in__range__empty,axiom,
    ! [Ha: heap_e7401611519738050253t_unit] : aa_Pro653082453696054616_nat_o(in_range,produc7507926704131184380et_nat(Ha,bot_bot_set_nat)) ).

% in_range_empty
tff(fact_2451_mod__pure__star__dist,axiom,
    ! [Pa: assn,B: $o,Ha: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(rep_assn(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Pa),aa_o_assn(pure_assn,(B)))),Ha)
    <=> ( aa_Pro653082453696054616_nat_o(rep_assn(Pa),Ha)
        & (B) ) ) ).

% mod_pure_star_dist
tff(fact_2452_in__range__dist__union,axiom,
    ! [Ha: heap_e7401611519738050253t_unit,Asa: set_nat,As: set_nat] :
      ( aa_Pro653082453696054616_nat_o(in_range,produc7507926704131184380et_nat(Ha,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Asa),As)))
    <=> ( aa_Pro653082453696054616_nat_o(in_range,produc7507926704131184380et_nat(Ha,Asa))
        & aa_Pro653082453696054616_nat_o(in_range,produc7507926704131184380et_nat(Ha,As)) ) ) ).

% in_range_dist_union
tff(fact_2453_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_2454_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_2455_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_2456_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_2457_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_2458_models__in__range,axiom,
    ! [Pa: assn,Ha: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(rep_assn(Pa),Ha)
     => aa_Pro653082453696054616_nat_o(in_range,Ha) ) ).

% models_in_range
tff(fact_2459_relH__in__rangeI_I2_J,axiom,
    ! [Asa: set_nat,Ha: heap_e7401611519738050253t_unit,H_a: heap_e7401611519738050253t_unit] :
      ( aa_hea7994211679605476167unit_o(relH(Asa,Ha),H_a)
     => aa_Pro653082453696054616_nat_o(in_range,produc7507926704131184380et_nat(H_a,Asa)) ) ).

% relH_in_rangeI(2)
tff(fact_2460_relH__in__rangeI_I1_J,axiom,
    ! [Asa: set_nat,Ha: heap_e7401611519738050253t_unit,H_a: heap_e7401611519738050253t_unit] :
      ( aa_hea7994211679605476167unit_o(relH(Asa,Ha),H_a)
     => aa_Pro653082453696054616_nat_o(in_range,produc7507926704131184380et_nat(Ha,Asa)) ) ).

% relH_in_rangeI(1)
tff(fact_2461_relH__refl,axiom,
    ! [Ha: heap_e7401611519738050253t_unit,Asa: set_nat] :
      ( aa_Pro653082453696054616_nat_o(in_range,produc7507926704131184380et_nat(Ha,Asa))
     => aa_hea7994211679605476167unit_o(relH(Asa,Ha),Ha) ) ).

% relH_refl
tff(fact_2462_in__range__subset,axiom,
    ! [Asa: set_nat,As: set_nat,Ha: heap_e7401611519738050253t_unit] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Asa),As)
     => ( aa_Pro653082453696054616_nat_o(in_range,produc7507926704131184380et_nat(Ha,As))
       => aa_Pro653082453696054616_nat_o(in_range,produc7507926704131184380et_nat(Ha,Asa)) ) ) ).

% in_range_subset
tff(fact_2463_effectI,axiom,
    ! [Ca: heap_T5738788834812785303t_unit,Ha: heap_e7401611519738050253t_unit,Ra2: product_unit,H_a: heap_e7401611519738050253t_unit,N2: nat] :
      ( ( aa_hea684056619475940329it_nat(heap_T875086893843062177t_unit(Ca),Ha) = aa_Pro8996696869555642526it_nat(some_P1914260805536162275it_nat,produc7133225469290080770it_nat(Ra2,produc584006145561248582it_nat(H_a,N2))) )
     => aa_nat_o(heap_T6553295506729943825t_unit(Ca,Ha,H_a,Ra2),N2) ) ).

% effectI
tff(fact_2464_effectI,axiom,
    ! [Ca: heap_Time_Heap_a,Ha: heap_e7401611519738050253t_unit,Ra2: a,H_a: heap_e7401611519738050253t_unit,N2: nat] :
      ( ( aa_hea4724376117875161946it_nat(heap_Time_execute_a(Ca),Ha) = aa_Pro5302094286594872192it_nat(some_P7913643980934408916it_nat,aa_Pro8078911840264235907it_nat(aa_a_f2234720956197727492it_nat(produc9178034014595674355it_nat,Ra2),produc584006145561248582it_nat(H_a,N2))) )
     => aa_nat_o(heap_Time_effect_a(Ca,Ha,H_a,Ra2),N2) ) ).

% effectI
tff(fact_2465_effect__def,axiom,
    ! [Ca: heap_T5738788834812785303t_unit,Ha: heap_e7401611519738050253t_unit,H_a: heap_e7401611519738050253t_unit,Ra2: product_unit,N2: nat] :
      ( aa_nat_o(heap_T6553295506729943825t_unit(Ca,Ha,H_a,Ra2),N2)
    <=> ( aa_hea684056619475940329it_nat(heap_T875086893843062177t_unit(Ca),Ha) = aa_Pro8996696869555642526it_nat(some_P1914260805536162275it_nat,produc7133225469290080770it_nat(Ra2,produc584006145561248582it_nat(H_a,N2))) ) ) ).

% effect_def
tff(fact_2466_effect__def,axiom,
    ! [Ca: heap_Time_Heap_a,Ha: heap_e7401611519738050253t_unit,H_a: heap_e7401611519738050253t_unit,Ra2: a,N2: nat] :
      ( aa_nat_o(heap_Time_effect_a(Ca,Ha,H_a,Ra2),N2)
    <=> ( aa_hea4724376117875161946it_nat(heap_Time_execute_a(Ca),Ha) = aa_Pro5302094286594872192it_nat(some_P7913643980934408916it_nat,aa_Pro8078911840264235907it_nat(aa_a_f2234720956197727492it_nat(produc9178034014595674355it_nat,Ra2),produc584006145561248582it_nat(H_a,N2))) ) ) ).

% effect_def
tff(fact_2467_in__range_Oelims_I3_J,axiom,
    ! [X: produc3658429121746597890et_nat] :
      ( ~ aa_Pro653082453696054616_nat_o(in_range,X)
     => ~ ! [H: heap_e7401611519738050253t_unit,As3: set_nat] :
            ( ( X = produc7507926704131184380et_nat(H,As3) )
           => ! [X3: nat] :
                ( aa_set_nat_o(member_nat(X3),As3)
               => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),lim_Product_unit(H)) ) ) ) ).

% in_range.elims(3)
tff(fact_2468_in__range_Oelims_I2_J,axiom,
    ! [X: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(in_range,X)
     => ~ ! [H: heap_e7401611519738050253t_unit,As3: set_nat] :
            ( ( X = produc7507926704131184380et_nat(H,As3) )
           => ~ ! [X5: nat] :
                  ( aa_set_nat_o(member_nat(X5),As3)
                 => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X5),lim_Product_unit(H)) ) ) ) ).

% in_range.elims(2)
tff(fact_2469_in__range_Oelims_I1_J,axiom,
    ! [X: produc3658429121746597890et_nat,Y: $o] :
      ( ( aa_Pro653082453696054616_nat_o(in_range,X)
      <=> (Y) )
     => ~ ! [H: heap_e7401611519738050253t_unit,As3: set_nat] :
            ( ( X = produc7507926704131184380et_nat(H,As3) )
           => ( (Y)
            <=> ~ ! [X4: nat] :
                    ( aa_set_nat_o(member_nat(X4),As3)
                   => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X4),lim_Product_unit(H)) ) ) ) ) ).

% in_range.elims(1)
tff(fact_2470_in__range_Osimps,axiom,
    ! [Ha: heap_e7401611519738050253t_unit,Asa: set_nat] :
      ( aa_Pro653082453696054616_nat_o(in_range,produc7507926704131184380et_nat(Ha,Asa))
    <=> ! [X4: nat] :
          ( aa_set_nat_o(member_nat(X4),Asa)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X4),lim_Product_unit(Ha)) ) ) ).

% in_range.simps
tff(fact_2471_Heap__eqI,axiom,
    ! [F: heap_Time_Heap_a,G: heap_Time_Heap_a] :
      ( ! [H: heap_e7401611519738050253t_unit] : aa_hea4724376117875161946it_nat(heap_Time_execute_a(F),H) = aa_hea4724376117875161946it_nat(heap_Time_execute_a(G),H)
     => ( F = G ) ) ).

% Heap_eqI
tff(fact_2472_Inf__fin_Osemilattice__order__set__axioms,axiom,
    lattic3109210760196336428et_nat(inf_inf_set_nat,ord_less_eq_set_nat,ord_less_set_nat) ).

% Inf_fin.semilattice_order_set_axioms
tff(fact_2473_Inf__fin_Osemilattice__order__set__axioms,axiom,
    lattic5111740090977526247t_unit(inf_inf_Product_unit,ord_le3221252021190050221t_unit,ord_le361264281704409273t_unit) ).

% Inf_fin.semilattice_order_set_axioms
tff(fact_2474_Inf__fin_Osemilattice__order__set__axioms,axiom,
    lattic6529551498584149819at_nat(inf_in2572325071724192079at_nat,ord_le3146513528884898305at_nat,ord_le7866589430770878221at_nat) ).

% Inf_fin.semilattice_order_set_axioms
tff(fact_2475_Inf__fin_Osemilattice__order__set__axioms,axiom,
    lattic5623431474481994958t_assn(inf_inf_assn,ord_less_eq_assn,ord_less_assn) ).

% Inf_fin.semilattice_order_set_axioms
tff(fact_2476_Inf__fin_Osemilattice__order__set__axioms,axiom,
    lattic8154731777541915528et_int(inf_inf_set_int,ord_less_eq_set_int,ord_less_set_int) ).

% Inf_fin.semilattice_order_set_axioms
tff(fact_2477_Inf__fin_Osemilattice__order__set__axioms,axiom,
    lattic5374021519246970238et_rat(inf_inf_rat,ord_less_eq_rat,ord_less_rat) ).

% Inf_fin.semilattice_order_set_axioms
tff(fact_2478_Inf__fin_Osemilattice__order__set__axioms,axiom,
    lattic6009151579333465974et_nat(inf_inf_nat,ord_less_eq_nat,ord_less_nat) ).

% Inf_fin.semilattice_order_set_axioms
tff(fact_2479_Inf__fin_Osemilattice__order__set__axioms,axiom,
    lattic6006661108824415698et_int(inf_inf_int,ord_less_eq_int,ord_less_int) ).

% Inf_fin.semilattice_order_set_axioms
tff(fact_2480_the__default_Osimps_I1_J,axiom,
    ! [Uu: produc3260487557148687353it_nat,X: produc3260487557148687353it_nat] : the_de542837501814275618it_nat(Uu,aa_Pro5302094286594872192it_nat(some_P7913643980934408916it_nat,X)) = X ).

% the_default.simps(1)
tff(fact_2481_the__default_Osimps_I1_J,axiom,
    ! [Uu: num,X: num] : aa_option_num_num(the_default_num(Uu),aa_num_option_num(some_num,X)) = X ).

% the_default.simps(1)
tff(fact_2482_the__default_Osimps_I1_J,axiom,
    ! [Uu: produc8664842809031399944it_nat,X: produc8664842809031399944it_nat] : the_de2487931475039285041it_nat(Uu,aa_Pro8996696869555642526it_nat(some_P1914260805536162275it_nat,X)) = X ).

% the_default.simps(1)
tff(fact_2483_wand__raw_Oelims_I3_J,axiom,
    ! [X: fun_Pr6460944699146269935_nat_o,Xa2: fun_Pr6460944699146269935_nat_o,Xb: produc3658429121746597890et_nat] :
      ( ~ aa_Pro653082453696054616_nat_o(aa_fun2638059832795814268_nat_o(wand_raw(X),Xa2),Xb)
     => ~ ! [H: heap_e7401611519738050253t_unit,As3: set_nat] :
            ( ( Xb = produc7507926704131184380et_nat(H,As3) )
           => ( aa_Pro653082453696054616_nat_o(in_range,produc7507926704131184380et_nat(H,As3))
              & ! [H4: heap_e7401611519738050253t_unit,As5: set_nat] :
                  ( ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,As3),As5) = bot_bot_set_nat )
                    & aa_hea7994211679605476167unit_o(relH(As3,H),H4)
                    & aa_Pro653082453696054616_nat_o(in_range,produc7507926704131184380et_nat(H4,As3))
                    & aa_Pro653082453696054616_nat_o(X,produc7507926704131184380et_nat(H4,As5)) )
                 => aa_Pro653082453696054616_nat_o(Xa2,produc7507926704131184380et_nat(H4,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,As3),As5))) ) ) ) ) ).

% wand_raw.elims(3)
tff(fact_2484_wand__raw_Oelims_I2_J,axiom,
    ! [X: fun_Pr6460944699146269935_nat_o,Xa2: fun_Pr6460944699146269935_nat_o,Xb: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(aa_fun2638059832795814268_nat_o(wand_raw(X),Xa2),Xb)
     => ~ ! [H: heap_e7401611519738050253t_unit,As3: set_nat] :
            ( ( Xb = produc7507926704131184380et_nat(H,As3) )
           => ~ ( aa_Pro653082453696054616_nat_o(in_range,produc7507926704131184380et_nat(H,As3))
                & ! [H7: heap_e7401611519738050253t_unit,As6: set_nat] :
                    ( ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,As3),As6) = bot_bot_set_nat )
                      & aa_hea7994211679605476167unit_o(relH(As3,H),H7)
                      & aa_Pro653082453696054616_nat_o(in_range,produc7507926704131184380et_nat(H7,As3))
                      & aa_Pro653082453696054616_nat_o(X,produc7507926704131184380et_nat(H7,As6)) )
                   => aa_Pro653082453696054616_nat_o(Xa2,produc7507926704131184380et_nat(H7,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,As3),As6))) ) ) ) ) ).

% wand_raw.elims(2)
tff(fact_2485_wand__raw_Oelims_I1_J,axiom,
    ! [X: fun_Pr6460944699146269935_nat_o,Xa2: fun_Pr6460944699146269935_nat_o,Xb: produc3658429121746597890et_nat,Y: $o] :
      ( ( aa_Pro653082453696054616_nat_o(aa_fun2638059832795814268_nat_o(wand_raw(X),Xa2),Xb)
      <=> (Y) )
     => ~ ! [H: heap_e7401611519738050253t_unit,As3: set_nat] :
            ( ( Xb = produc7507926704131184380et_nat(H,As3) )
           => ( (Y)
            <=> ~ ( aa_Pro653082453696054616_nat_o(in_range,produc7507926704131184380et_nat(H,As3))
                  & ! [H6: heap_e7401611519738050253t_unit,As7: set_nat] :
                      ( ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,As3),As7) = bot_bot_set_nat )
                        & aa_hea7994211679605476167unit_o(relH(As3,H),H6)
                        & aa_Pro653082453696054616_nat_o(in_range,produc7507926704131184380et_nat(H6,As3))
                        & aa_Pro653082453696054616_nat_o(X,produc7507926704131184380et_nat(H6,As7)) )
                     => aa_Pro653082453696054616_nat_o(Xa2,produc7507926704131184380et_nat(H6,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,As3),As7))) ) ) ) ) ) ).

% wand_raw.elims(1)
tff(fact_2486_wand__raw_Osimps,axiom,
    ! [Pa: fun_Pr6460944699146269935_nat_o,Qa: fun_Pr6460944699146269935_nat_o,Ha: heap_e7401611519738050253t_unit,Asa: set_nat] :
      ( aa_Pro653082453696054616_nat_o(aa_fun2638059832795814268_nat_o(wand_raw(Pa),Qa),produc7507926704131184380et_nat(Ha,Asa))
    <=> ( aa_Pro653082453696054616_nat_o(in_range,produc7507926704131184380et_nat(Ha,Asa))
        & ! [H6: heap_e7401611519738050253t_unit,As7: set_nat] :
            ( ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Asa),As7) = bot_bot_set_nat )
              & aa_hea7994211679605476167unit_o(relH(Asa,Ha),H6)
              & aa_Pro653082453696054616_nat_o(in_range,produc7507926704131184380et_nat(H6,Asa))
              & aa_Pro653082453696054616_nat_o(Pa,produc7507926704131184380et_nat(H6,As7)) )
           => aa_Pro653082453696054616_nat_o(Qa,produc7507926704131184380et_nat(H6,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Asa),As7))) ) ) ) ).

% wand_raw.simps
tff(fact_2487_in__measure,axiom,
    ! [X: nat,Y: nat,F: fun_nat_nat] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,X),Y)),measure_nat(F))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(F,X)),aa_nat_nat(F,Y)) ) ).

% in_measure
tff(fact_2488_wand__raw_Opelims_I3_J,axiom,
    ! [X: fun_Pr6460944699146269935_nat_o,Xa2: fun_Pr6460944699146269935_nat_o,Xb: produc3658429121746597890et_nat] :
      ( ~ aa_Pro653082453696054616_nat_o(aa_fun2638059832795814268_nat_o(wand_raw(X),Xa2),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,Xa2),Xb)))
       => ~ ! [H: heap_e7401611519738050253t_unit,As3: set_nat] :
              ( ( Xb = produc7507926704131184380et_nat(H,As3) )
             => ( 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,Xa2),produc7507926704131184380et_nat(H,As3))))
               => ( aa_Pro653082453696054616_nat_o(in_range,produc7507926704131184380et_nat(H,As3))
                  & ! [H4: heap_e7401611519738050253t_unit,As5: set_nat] :
                      ( ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,As3),As5) = bot_bot_set_nat )
                        & aa_hea7994211679605476167unit_o(relH(As3,H),H4)
                        & aa_Pro653082453696054616_nat_o(in_range,produc7507926704131184380et_nat(H4,As3))
                        & aa_Pro653082453696054616_nat_o(X,produc7507926704131184380et_nat(H4,As5)) )
                     => aa_Pro653082453696054616_nat_o(Xa2,produc7507926704131184380et_nat(H4,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,As3),As5))) ) ) ) ) ) ) ).

% wand_raw.pelims(3)
tff(fact_2489_wand__raw_Opelims_I2_J,axiom,
    ! [X: fun_Pr6460944699146269935_nat_o,Xa2: fun_Pr6460944699146269935_nat_o,Xb: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(aa_fun2638059832795814268_nat_o(wand_raw(X),Xa2),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,Xa2),Xb)))
       => ~ ! [H: heap_e7401611519738050253t_unit,As3: set_nat] :
              ( ( Xb = produc7507926704131184380et_nat(H,As3) )
             => ( 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,Xa2),produc7507926704131184380et_nat(H,As3))))
               => ~ ( aa_Pro653082453696054616_nat_o(in_range,produc7507926704131184380et_nat(H,As3))
                    & ! [H7: heap_e7401611519738050253t_unit,As6: set_nat] :
                        ( ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,As3),As6) = bot_bot_set_nat )
                          & aa_hea7994211679605476167unit_o(relH(As3,H),H7)
                          & aa_Pro653082453696054616_nat_o(in_range,produc7507926704131184380et_nat(H7,As3))
                          & aa_Pro653082453696054616_nat_o(X,produc7507926704131184380et_nat(H7,As6)) )
                       => aa_Pro653082453696054616_nat_o(Xa2,produc7507926704131184380et_nat(H7,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,As3),As6))) ) ) ) ) ) ) ).

% wand_raw.pelims(2)
tff(fact_2490_wand__raw_Opelims_I1_J,axiom,
    ! [X: fun_Pr6460944699146269935_nat_o,Xa2: fun_Pr6460944699146269935_nat_o,Xb: produc3658429121746597890et_nat,Y: $o] :
      ( ( aa_Pro653082453696054616_nat_o(aa_fun2638059832795814268_nat_o(wand_raw(X),Xa2),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,Xa2),Xb)))
       => ~ ! [H: heap_e7401611519738050253t_unit,As3: set_nat] :
              ( ( Xb = produc7507926704131184380et_nat(H,As3) )
             => ( ( (Y)
                <=> ( aa_Pro653082453696054616_nat_o(in_range,produc7507926704131184380et_nat(H,As3))
                    & ! [H6: heap_e7401611519738050253t_unit,As7: set_nat] :
                        ( ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,As3),As7) = bot_bot_set_nat )
                          & aa_hea7994211679605476167unit_o(relH(As3,H),H6)
                          & aa_Pro653082453696054616_nat_o(in_range,produc7507926704131184380et_nat(H6,As3))
                          & aa_Pro653082453696054616_nat_o(X,produc7507926704131184380et_nat(H6,As7)) )
                       => aa_Pro653082453696054616_nat_o(Xa2,produc7507926704131184380et_nat(H6,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,As3),As7))) ) ) )
               => ~ 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,Xa2),produc7507926704131184380et_nat(H,As3)))) ) ) ) ) ).

% wand_raw.pelims(1)
tff(fact_2491_in__range_Opelims_I3_J,axiom,
    ! [X: produc3658429121746597890et_nat] :
      ( ~ aa_Pro653082453696054616_nat_o(in_range,X)
     => ( aa_Pro653082453696054616_nat_o(accp_P5801069581201407417et_nat(in_range_rel),X)
       => ~ ! [H: heap_e7401611519738050253t_unit,As3: set_nat] :
              ( ( X = produc7507926704131184380et_nat(H,As3) )
             => ( aa_Pro653082453696054616_nat_o(accp_P5801069581201407417et_nat(in_range_rel),produc7507926704131184380et_nat(H,As3))
               => ! [X3: nat] :
                    ( aa_set_nat_o(member_nat(X3),As3)
                   => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),lim_Product_unit(H)) ) ) ) ) ) ).

% in_range.pelims(3)
tff(fact_2492_in__range_Opelims_I2_J,axiom,
    ! [X: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(in_range,X)
     => ( aa_Pro653082453696054616_nat_o(accp_P5801069581201407417et_nat(in_range_rel),X)
       => ~ ! [H: heap_e7401611519738050253t_unit,As3: set_nat] :
              ( ( X = produc7507926704131184380et_nat(H,As3) )
             => ( aa_Pro653082453696054616_nat_o(accp_P5801069581201407417et_nat(in_range_rel),produc7507926704131184380et_nat(H,As3))
               => ~ ! [X5: nat] :
                      ( aa_set_nat_o(member_nat(X5),As3)
                     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X5),lim_Product_unit(H)) ) ) ) ) ) ).

% in_range.pelims(2)
tff(fact_2493_in__range_Opelims_I1_J,axiom,
    ! [X: produc3658429121746597890et_nat,Y: $o] :
      ( ( aa_Pro653082453696054616_nat_o(in_range,X)
      <=> (Y) )
     => ( aa_Pro653082453696054616_nat_o(accp_P5801069581201407417et_nat(in_range_rel),X)
       => ~ ! [H: heap_e7401611519738050253t_unit,As3: set_nat] :
              ( ( X = produc7507926704131184380et_nat(H,As3) )
             => ( ( (Y)
                <=> ! [X4: nat] :
                      ( aa_set_nat_o(member_nat(X4),As3)
                     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X4),lim_Product_unit(H)) ) )
               => ~ aa_Pro653082453696054616_nat_o(accp_P5801069581201407417et_nat(in_range_rel),produc7507926704131184380et_nat(H,As3)) ) ) ) ) ).

% in_range.pelims(1)
tff(fact_2494_wand__assn__def,axiom,
    ! [Pa: assn,Qa: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(wand_assn,Pa),Qa) = abs_assn(aa_fun2638059832795814268_nat_o(wand_raw(rep_assn(Pa)),rep_assn(Qa))) ).

% wand_assn_def
tff(fact_2495_sup__option__def,axiom,
    ! [X: option2158460348403737357at_nat,Y: option2158460348403737357at_nat] : sup_su1419176701018045153at_nat(X,Y) = case_o7467001542679133189at_nat(Y,aa_opt2198001859980869274at_nat(aTP_Lamm_dm(X),Y),X) ).

% sup_option_def
tff(fact_2496_sup__option__def,axiom,
    ! [X: option2498585697089621389at_nat,Y: option2498585697089621389at_nat] : sup_su5873096292857799009at_nat(X,Y) = case_o8757249484648823045at_nat(Y,aa_opt1741456117285354266at_nat(aTP_Lamm_do(X),Y),X) ).

% sup_option_def
tff(fact_2497_sup__option__def,axiom,
    ! [X: option1583680563626158861at_nat,Y: option1583680563626158861at_nat] : sup_su2273273666271716065at_nat(X,Y) = case_o9080059180780454917at_nat(Y,aa_opt1027706798495913626at_nat(aTP_Lamm_dq(X),Y),X) ).

% sup_option_def
tff(fact_2498_sup__option__def,axiom,
    ! [X: option_set_nat,Y: option_set_nat] : sup_su3598758113090618626et_nat(X,Y) = case_o4054078431260844265et_nat(Y,aa_opt5648106117674462701et_nat(aTP_Lamm_ds(X),Y),X) ).

% sup_option_def
tff(fact_2499_sup__option__def,axiom,
    ! [X: option_nat,Y: option_nat] : sup_sup_option_nat(X,Y) = case_o7429725398727453821at_nat(Y,aa_opt1558076126722380235on_nat(aTP_Lamm_du(X),Y),X) ).

% sup_option_def
tff(fact_2500_dflt__None__set__def,axiom,
    ! [S: set_Pr1261947904930325089at_nat] :
      dflt_N6592383573632408824at_nat(S) = $ite(S = bot_bo2099793752762293965at_nat,none_s625347054029921090at_nat,some_s147305329494351046at_nat(S)) ).

% dflt_None_set_def
tff(fact_2501_dflt__None__set__def,axiom,
    ! [S: set_o] :
      dflt_None_set_o(S) = $ite(S = bot_bot_set_o,none_set_o,some_set_o(S)) ).

% dflt_None_set_def
tff(fact_2502_dflt__None__set__def,axiom,
    ! [S: set_nat] :
      dflt_None_set_nat(S) = $ite(S = bot_bot_set_nat,none_set_nat,some_set_nat(S)) ).

% dflt_None_set_def
tff(fact_2503_dflt__None__set__def,axiom,
    ! [S: set_int] :
      dflt_None_set_int(S) = $ite(S = bot_bot_set_int,none_set_int,some_set_int(S)) ).

% dflt_None_set_def
tff(fact_2504_times__assn__def,axiom,
    ! [Pa: assn,Qa: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Pa),Qa) = abs_assn(aa_fun2638059832795814268_nat_o(times_assn_raw(rep_assn(Pa)),rep_assn(Qa))) ).

% times_assn_def
tff(fact_2505_Set_Ois__empty__def,axiom,
    ! [A3: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(is_emp1662574758705540307at_nat,A3)
    <=> ( A3 = bot_bo2099793752762293965at_nat ) ) ).

% Set.is_empty_def
tff(fact_2506_Set_Ois__empty__def,axiom,
    ! [A3: set_o] :
      ( is_empty_o(A3)
    <=> ( A3 = bot_bot_set_o ) ) ).

% Set.is_empty_def
tff(fact_2507_Set_Ois__empty__def,axiom,
    ! [A3: set_nat] :
      ( aa_set_nat_o(is_empty_nat,A3)
    <=> ( A3 = bot_bot_set_nat ) ) ).

% Set.is_empty_def
tff(fact_2508_Set_Ois__empty__def,axiom,
    ! [A3: set_int] :
      ( aa_set_int_o(is_empty_int,A3)
    <=> ( A3 = bot_bot_set_int ) ) ).

% Set.is_empty_def
tff(fact_2509_hoare__triple__success,axiom,
    ! [Pa: assn,Ca: heap_Time_Heap_a,Qa: fun_a_assn,Ha: heap_e7401611519738050253t_unit,Asa: set_nat] :
      ( hoare_hoare_triple_a(Pa,Ca,Qa)
     => ( aa_Pro653082453696054616_nat_o(rep_assn(Pa),produc7507926704131184380et_nat(Ha,Asa))
       => aa_hea7994211679605476167unit_o(heap_Time_success_a(Ca),Ha) ) ) ).

% hoare_triple_success
tff(fact_2510_subset__Collect__iff,axiom,
    ! [B3: set_Pr1261947904930325089at_nat,A3: set_Pr1261947904930325089at_nat,Pa: fun_Pr4796943543331071514_nat_o] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,B3),A3)
     => ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,B3),collec3392354462482085612at_nat(aa_fun359341089595923442_nat_o(aTP_Lamm_al(A3),Pa)))
      <=> ! [X4: product_prod_nat_nat] :
            ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X4),B3)
           => aa_Pro6150637604445964947_nat_o(Pa,X4) ) ) ) ).

% subset_Collect_iff
tff(fact_2511_subset__Collect__iff,axiom,
    ! [B3: set_o,A3: set_o,Pa: fun_o_o] :
      ( ord_less_eq_set_o(B3,A3)
     => ( ord_less_eq_set_o(B3,collect_o(aa_fun_o_o_fun_o_o(aTP_Lamm_am(A3),Pa)))
      <=> ! [X4: $o] :
            ( member_o((X4),B3)
           => aa_o_o(Pa,(X4)) ) ) ) ).

% subset_Collect_iff
tff(fact_2512_subset__Collect__iff,axiom,
    ! [B3: set_Pr958786334691620121nt_int,A3: set_Pr958786334691620121nt_int,Pa: fun_Pr3238374608701176162_int_o] :
      ( ord_le2843351958646193337nt_int(B3,A3)
     => ( ord_le2843351958646193337nt_int(B3,collec213857154873943460nt_int(aa_fun5349570537165229698_int_o(aTP_Lamm_an(A3),Pa)))
      <=> ! [X4: product_prod_int_int] :
            ( member5262025264175285858nt_int(X4,B3)
           => aa_Pro4592068669816069595_int_o(Pa,X4) ) ) ) ).

% subset_Collect_iff
tff(fact_2513_subset__Collect__iff,axiom,
    ! [B3: set_list_nat,A3: set_list_nat,Pa: fun_list_nat_o] :
      ( ord_le6045566169113846134st_nat(B3,A3)
     => ( ord_le6045566169113846134st_nat(B3,collect_list_nat(aa_fun6155608546934989052_nat_o(aTP_Lamm_ao(A3),Pa)))
      <=> ! [X4: list_nat] :
            ( member_list_nat(X4,B3)
           => aa_list_nat_o(Pa,X4) ) ) ) ).

% subset_Collect_iff
tff(fact_2514_subset__Collect__iff,axiom,
    ! [B3: set_set_nat,A3: set_set_nat,Pa: fun_set_nat_o] :
      ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(B3),A3)
     => ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(B3),collect_set_nat(aa_fun751667557118977660_nat_o(aTP_Lamm_ap(A3),Pa)))
      <=> ! [X4: set_nat] :
            ( aa_set_set_nat_o(member_set_nat(X4),B3)
           => aa_set_nat_o(Pa,X4) ) ) ) ).

% subset_Collect_iff
tff(fact_2515_subset__Collect__iff,axiom,
    ! [B3: set_nat,A3: set_nat,Pa: fun_nat_o] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B3),A3)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B3),collect_nat(aa_fun5937193402907943164_nat_o(aTP_Lamm_aq(A3),Pa)))
      <=> ! [X4: nat] :
            ( aa_set_nat_o(member_nat(X4),B3)
           => aa_nat_o(Pa,X4) ) ) ) ).

% subset_Collect_iff
tff(fact_2516_subset__Collect__iff,axiom,
    ! [B3: set_int,A3: set_int,Pa: fun_int_o] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,B3),A3)
     => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,B3),collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_ar(A3),Pa)))
      <=> ! [X4: int] :
            ( aa_set_int_o(member_int(X4),B3)
           => aa_int_o(Pa,X4) ) ) ) ).

% subset_Collect_iff
tff(fact_2517_subset__CollectI,axiom,
    ! [B3: set_Pr1261947904930325089at_nat,A3: set_Pr1261947904930325089at_nat,Qa: fun_Pr4796943543331071514_nat_o,Pa: fun_Pr4796943543331071514_nat_o] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,B3),A3)
     => ( ! [X3: product_prod_nat_nat] :
            ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X3),B3)
           => ( aa_Pro6150637604445964947_nat_o(Qa,X3)
             => aa_Pro6150637604445964947_nat_o(Pa,X3) ) )
       => aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,collec3392354462482085612at_nat(aa_fun359341089595923442_nat_o(aTP_Lamm_al(B3),Qa))),collec3392354462482085612at_nat(aa_fun359341089595923442_nat_o(aTP_Lamm_al(A3),Pa))) ) ) ).

% subset_CollectI
tff(fact_2518_subset__CollectI,axiom,
    ! [B3: set_o,A3: set_o,Qa: fun_o_o,Pa: fun_o_o] :
      ( ord_less_eq_set_o(B3,A3)
     => ( ! [X3: $o] :
            ( member_o((X3),B3)
           => ( aa_o_o(Qa,(X3))
             => aa_o_o(Pa,(X3)) ) )
       => ord_less_eq_set_o(collect_o(aa_fun_o_o_fun_o_o(aTP_Lamm_am(B3),Qa)),collect_o(aa_fun_o_o_fun_o_o(aTP_Lamm_am(A3),Pa))) ) ) ).

% subset_CollectI
tff(fact_2519_subset__CollectI,axiom,
    ! [B3: set_Pr958786334691620121nt_int,A3: set_Pr958786334691620121nt_int,Qa: fun_Pr3238374608701176162_int_o,Pa: fun_Pr3238374608701176162_int_o] :
      ( ord_le2843351958646193337nt_int(B3,A3)
     => ( ! [X3: product_prod_int_int] :
            ( member5262025264175285858nt_int(X3,B3)
           => ( aa_Pro4592068669816069595_int_o(Qa,X3)
             => aa_Pro4592068669816069595_int_o(Pa,X3) ) )
       => ord_le2843351958646193337nt_int(collec213857154873943460nt_int(aa_fun5349570537165229698_int_o(aTP_Lamm_an(B3),Qa)),collec213857154873943460nt_int(aa_fun5349570537165229698_int_o(aTP_Lamm_an(A3),Pa))) ) ) ).

% subset_CollectI
tff(fact_2520_subset__CollectI,axiom,
    ! [B3: set_list_nat,A3: set_list_nat,Qa: fun_list_nat_o,Pa: fun_list_nat_o] :
      ( ord_le6045566169113846134st_nat(B3,A3)
     => ( ! [X3: list_nat] :
            ( member_list_nat(X3,B3)
           => ( aa_list_nat_o(Qa,X3)
             => aa_list_nat_o(Pa,X3) ) )
       => ord_le6045566169113846134st_nat(collect_list_nat(aa_fun6155608546934989052_nat_o(aTP_Lamm_ao(B3),Qa)),collect_list_nat(aa_fun6155608546934989052_nat_o(aTP_Lamm_ao(A3),Pa))) ) ) ).

% subset_CollectI
tff(fact_2521_subset__CollectI,axiom,
    ! [B3: set_set_nat,A3: set_set_nat,Qa: fun_set_nat_o,Pa: fun_set_nat_o] :
      ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(B3),A3)
     => ( ! [X3: set_nat] :
            ( aa_set_set_nat_o(member_set_nat(X3),B3)
           => ( aa_set_nat_o(Qa,X3)
             => aa_set_nat_o(Pa,X3) ) )
       => aa_set_set_nat_o(ord_le6893508408891458716et_nat(collect_set_nat(aa_fun751667557118977660_nat_o(aTP_Lamm_ap(B3),Qa))),collect_set_nat(aa_fun751667557118977660_nat_o(aTP_Lamm_ap(A3),Pa))) ) ) ).

% subset_CollectI
tff(fact_2522_subset__CollectI,axiom,
    ! [B3: set_nat,A3: set_nat,Qa: fun_nat_o,Pa: fun_nat_o] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B3),A3)
     => ( ! [X3: nat] :
            ( aa_set_nat_o(member_nat(X3),B3)
           => ( aa_nat_o(Qa,X3)
             => aa_nat_o(Pa,X3) ) )
       => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,collect_nat(aa_fun5937193402907943164_nat_o(aTP_Lamm_aq(B3),Qa))),collect_nat(aa_fun5937193402907943164_nat_o(aTP_Lamm_aq(A3),Pa))) ) ) ).

% subset_CollectI
tff(fact_2523_subset__CollectI,axiom,
    ! [B3: set_int,A3: set_int,Qa: fun_int_o,Pa: fun_int_o] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,B3),A3)
     => ( ! [X3: int] :
            ( aa_set_int_o(member_int(X3),B3)
           => ( aa_int_o(Qa,X3)
             => aa_int_o(Pa,X3) ) )
       => aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_ar(B3),Qa))),collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_ar(A3),Pa))) ) ) ).

% subset_CollectI
tff(fact_2524_conj__subset__def,axiom,
    ! [A3: set_Pr958786334691620121nt_int,Pa: fun_Pr3238374608701176162_int_o,Qa: fun_Pr3238374608701176162_int_o] :
      ( ord_le2843351958646193337nt_int(A3,collec213857154873943460nt_int(aa_fun5349570537165229698_int_o(aTP_Lamm_bp(Pa),Qa)))
    <=> ( ord_le2843351958646193337nt_int(A3,collec213857154873943460nt_int(Pa))
        & ord_le2843351958646193337nt_int(A3,collec213857154873943460nt_int(Qa)) ) ) ).

% conj_subset_def
tff(fact_2525_conj__subset__def,axiom,
    ! [A3: set_list_nat,Pa: fun_list_nat_o,Qa: fun_list_nat_o] :
      ( ord_le6045566169113846134st_nat(A3,collect_list_nat(aa_fun6155608546934989052_nat_o(aTP_Lamm_bq(Pa),Qa)))
    <=> ( ord_le6045566169113846134st_nat(A3,collect_list_nat(Pa))
        & ord_le6045566169113846134st_nat(A3,collect_list_nat(Qa)) ) ) ).

% conj_subset_def
tff(fact_2526_conj__subset__def,axiom,
    ! [A3: set_set_nat,Pa: fun_set_nat_o,Qa: fun_set_nat_o] :
      ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(A3),collect_set_nat(aa_fun751667557118977660_nat_o(aTP_Lamm_br(Pa),Qa)))
    <=> ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(A3),collect_set_nat(Pa))
        & aa_set_set_nat_o(ord_le6893508408891458716et_nat(A3),collect_set_nat(Qa)) ) ) ).

% conj_subset_def
tff(fact_2527_conj__subset__def,axiom,
    ! [A3: set_nat,Pa: fun_nat_o,Qa: fun_nat_o] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A3),collect_nat(aa_fun5937193402907943164_nat_o(aTP_Lamm_bt(Pa),Qa)))
    <=> ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A3),collect_nat(Pa))
        & aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A3),collect_nat(Qa)) ) ) ).

% conj_subset_def
tff(fact_2528_conj__subset__def,axiom,
    ! [A3: set_int,Pa: fun_int_o,Qa: fun_int_o] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A3),collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_bs(Pa),Qa)))
    <=> ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A3),collect_int(Pa))
        & aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A3),collect_int(Qa)) ) ) ).

% conj_subset_def
tff(fact_2529_not__None__eq,axiom,
    ! [X: option3562590408128118217it_nat] :
      ( ( X != none_P2651198173097904984it_nat )
    <=> ? [Y5: produc3260487557148687353it_nat] : X = aa_Pro5302094286594872192it_nat(some_P7913643980934408916it_nat,Y5) ) ).

% not_None_eq
tff(fact_2530_not__None__eq,axiom,
    ! [X: option_num] :
      ( ( X != none_num )
    <=> ? [Y5: num] : X = aa_num_option_num(some_num,Y5) ) ).

% not_None_eq
tff(fact_2531_not__None__eq,axiom,
    ! [X: option8956607266484857688it_nat] :
      ( ( X != none_P9117596204409417319it_nat )
    <=> ? [Y5: produc8664842809031399944it_nat] : X = aa_Pro8996696869555642526it_nat(some_P1914260805536162275it_nat,Y5) ) ).

% not_None_eq
tff(fact_2532_not__Some__eq,axiom,
    ! [X: option3562590408128118217it_nat] :
      ( ! [Y5: produc3260487557148687353it_nat] : X != aa_Pro5302094286594872192it_nat(some_P7913643980934408916it_nat,Y5)
    <=> ( X = none_P2651198173097904984it_nat ) ) ).

% not_Some_eq
tff(fact_2533_not__Some__eq,axiom,
    ! [X: option_num] :
      ( ! [Y5: num] : X != aa_num_option_num(some_num,Y5)
    <=> ( X = none_num ) ) ).

% not_Some_eq
tff(fact_2534_not__Some__eq,axiom,
    ! [X: option8956607266484857688it_nat] :
      ( ! [Y5: produc8664842809031399944it_nat] : X != aa_Pro8996696869555642526it_nat(some_P1914260805536162275it_nat,Y5)
    <=> ( X = none_P9117596204409417319it_nat ) ) ).

% not_Some_eq
tff(fact_2535_Rep__assn__inverse,axiom,
    ! [X: assn] : abs_assn(rep_assn(X)) = X ).

% Rep_assn_inverse
tff(fact_2536_less__eq__option__None__code,axiom,
    ! [X: option_num] : aa_option_num_o(aa_opt8674010985164783839_num_o(ord_le6622620407824499402on_num,none_num),X) ).

% less_eq_option_None_code
tff(fact_2537_less__option__None,axiom,
    ! [X: option_num] : ~ aa_option_num_o(aa_opt8674010985164783839_num_o(ord_less_option_num,X),none_num) ).

% less_option_None
tff(fact_2538_less__eq__option__Some__None,axiom,
    ! [X: num] : ~ aa_option_num_o(aa_opt8674010985164783839_num_o(ord_le6622620407824499402on_num,aa_num_option_num(some_num,X)),none_num) ).

% less_eq_option_Some_None
tff(fact_2539_less__option__None__Some__code,axiom,
    ! [X: num] : aa_option_num_o(aa_opt8674010985164783839_num_o(ord_less_option_num,none_num),aa_num_option_num(some_num,X)) ).

% less_option_None_Some_code
tff(fact_2540_the__dflt__None__empty,axiom,
    dflt_N6592383573632408824at_nat(bot_bo2099793752762293965at_nat) = none_s625347054029921090at_nat ).

% the_dflt_None_empty
tff(fact_2541_the__dflt__None__empty,axiom,
    dflt_None_set_o(bot_bot_set_o) = none_set_o ).

% the_dflt_None_empty
tff(fact_2542_the__dflt__None__empty,axiom,
    dflt_None_set_nat(bot_bot_set_nat) = none_set_nat ).

% the_dflt_None_empty
tff(fact_2543_the__dflt__None__empty,axiom,
    dflt_None_set_int(bot_bot_set_int) = none_set_int ).

% the_dflt_None_empty
tff(fact_2544_option_Osimps_I4_J,axiom,
    ! [F1: int,F22: fun_num_int] : aa_option_num_int(case_option_int_num(F1,F22),none_num) = F1 ).

% option.simps(4)
tff(fact_2545_option_Osimps_I4_J,axiom,
    ! [F1: option_num,F22: fun_num_option_num] : aa_opt5369159329878751634on_num(case_o6005452278849405969um_num(F1,F22),none_num) = F1 ).

% option.simps(4)
tff(fact_2546_option_Osimps_I4_J,axiom,
    ! [F1: num,F22: fun_num_num] : aa_option_num_num(case_option_num_num(F1,F22),none_num) = F1 ).

% option.simps(4)
tff(fact_2547_bot__option__def,axiom,
    bot_bot_option_num = none_num ).

% bot_option_def
tff(fact_2548_option_Ocase__distrib,axiom,
    ! [Ha: fun_int_int,F1: int,F22: fun_num_int,Option: option_num] : aa_int_int(Ha,aa_option_num_int(case_option_int_num(F1,F22),Option)) = aa_option_num_int(case_option_int_num(aa_int_int(Ha,F1),aa_fun329774391934724804um_int(aTP_Lamm_dv(Ha),F22)),Option) ).

% option.case_distrib
tff(fact_2549_option_Ocase__distrib,axiom,
    ! [Ha: fun_int_option_num,F1: int,F22: fun_num_int,Option: option_num] : aa_int_option_num(Ha,aa_option_num_int(case_option_int_num(F1,F22),Option)) = aa_opt5369159329878751634on_num(case_o6005452278849405969um_num(aa_int_option_num(Ha,F1),aa_fun3858197967482829954on_num(aTP_Lamm_dw(Ha),F22)),Option) ).

% option.case_distrib
tff(fact_2550_option_Ocase__distrib,axiom,
    ! [Ha: fun_int_num,F1: int,F22: fun_num_int,Option: option_num] : aa_int_num(Ha,aa_option_num_int(case_option_int_num(F1,F22),Option)) = aa_option_num_num(case_option_num_num(aa_int_num(Ha,F1),aa_fun5215869348392759218um_num(aTP_Lamm_dx(Ha),F22)),Option) ).

% option.case_distrib
tff(fact_2551_option_Ocase__distrib,axiom,
    ! [Ha: fun_option_num_int,F1: option_num,F22: fun_num_option_num,Option: option_num] : aa_option_num_int(Ha,aa_opt5369159329878751634on_num(case_o6005452278849405969um_num(F1,F22),Option)) = aa_option_num_int(case_option_int_num(aa_option_num_int(Ha,F1),aa_fun2256179333612926594um_int(aTP_Lamm_dy(Ha),F22)),Option) ).

% option.case_distrib
tff(fact_2552_option_Ocase__distrib,axiom,
    ! [Ha: fun_op8989173174992673105on_num,F1: option_num,F22: fun_num_option_num,Option: option_num] : aa_opt5369159329878751634on_num(Ha,aa_opt5369159329878751634on_num(case_o6005452278849405969um_num(F1,F22),Option)) = aa_opt5369159329878751634on_num(case_o6005452278849405969um_num(aa_opt5369159329878751634on_num(Ha,F1),aa_fun3636977833914307392on_num(aTP_Lamm_dz(Ha),F22)),Option) ).

% option.case_distrib
tff(fact_2553_option_Ocase__distrib,axiom,
    ! [Ha: fun_option_num_num,F1: option_num,F22: fun_num_option_num,Option: option_num] : aa_option_num_num(Ha,aa_opt5369159329878751634on_num(case_o6005452278849405969um_num(F1,F22),Option)) = aa_option_num_num(case_option_num_num(aa_option_num_num(Ha,F1),aa_fun7142274290070961008um_num(aTP_Lamm_ea(Ha),F22)),Option) ).

% option.case_distrib
tff(fact_2554_option_Ocase__distrib,axiom,
    ! [Ha: fun_num_int,F1: num,F22: fun_num_num,Option: option_num] : aa_num_int(Ha,aa_option_num_num(case_option_num_num(F1,F22),Option)) = aa_option_num_int(case_option_int_num(aa_num_int(Ha,F1),aa_fun3162900327506486194um_int(aTP_Lamm_eb(Ha),F22)),Option) ).

% option.case_distrib
tff(fact_2555_option_Ocase__distrib,axiom,
    ! [Ha: fun_num_option_num,F1: num,F22: fun_num_num,Option: option_num] : aa_num_option_num(Ha,aa_option_num_num(case_option_num_num(F1,F22),Option)) = aa_opt5369159329878751634on_num(case_o6005452278849405969um_num(aa_num_option_num(Ha,F1),aa_fun612943910911948912on_num(aTP_Lamm_ec(Ha),F22)),Option) ).

% option.case_distrib
tff(fact_2556_option_Ocase__distrib,axiom,
    ! [Ha: fun_num_num,F1: num,F22: fun_num_num,Option: option_num] : aa_num_num(Ha,aa_option_num_num(case_option_num_num(F1,F22),Option)) = aa_option_num_num(case_option_num_num(aa_num_num(Ha,F1),aa_fun8048995283964520608um_num(aTP_Lamm_ed(Ha),F22)),Option) ).

% option.case_distrib
tff(fact_2557_option_Odistinct_I1_J,axiom,
    ! [X2: produc3260487557148687353it_nat] : none_P2651198173097904984it_nat != aa_Pro5302094286594872192it_nat(some_P7913643980934408916it_nat,X2) ).

% option.distinct(1)
tff(fact_2558_option_Odistinct_I1_J,axiom,
    ! [X2: num] : none_num != aa_num_option_num(some_num,X2) ).

% option.distinct(1)
tff(fact_2559_option_Odistinct_I1_J,axiom,
    ! [X2: produc8664842809031399944it_nat] : none_P9117596204409417319it_nat != aa_Pro8996696869555642526it_nat(some_P1914260805536162275it_nat,X2) ).

% option.distinct(1)
tff(fact_2560_option_OdiscI,axiom,
    ! [Option: option3562590408128118217it_nat,X2: produc3260487557148687353it_nat] :
      ( ( Option = aa_Pro5302094286594872192it_nat(some_P7913643980934408916it_nat,X2) )
     => ( Option != none_P2651198173097904984it_nat ) ) ).

% option.discI
tff(fact_2561_option_OdiscI,axiom,
    ! [Option: option_num,X2: num] :
      ( ( Option = aa_num_option_num(some_num,X2) )
     => ( Option != none_num ) ) ).

% option.discI
tff(fact_2562_option_OdiscI,axiom,
    ! [Option: option8956607266484857688it_nat,X2: produc8664842809031399944it_nat] :
      ( ( Option = aa_Pro8996696869555642526it_nat(some_P1914260805536162275it_nat,X2) )
     => ( Option != none_P9117596204409417319it_nat ) ) ).

% option.discI
tff(fact_2563_option_Oexhaust,axiom,
    ! [Y: option3562590408128118217it_nat] :
      ( ( Y != none_P2651198173097904984it_nat )
     => ~ ! [X22: produc3260487557148687353it_nat] : Y != aa_Pro5302094286594872192it_nat(some_P7913643980934408916it_nat,X22) ) ).

% option.exhaust
tff(fact_2564_option_Oexhaust,axiom,
    ! [Y: option_num] :
      ( ( Y != none_num )
     => ~ ! [X22: num] : Y != aa_num_option_num(some_num,X22) ) ).

% option.exhaust
tff(fact_2565_option_Oexhaust,axiom,
    ! [Y: option8956607266484857688it_nat] :
      ( ( Y != none_P9117596204409417319it_nat )
     => ~ ! [X22: produc8664842809031399944it_nat] : Y != aa_Pro8996696869555642526it_nat(some_P1914260805536162275it_nat,X22) ) ).

% option.exhaust
tff(fact_2566_split__option__ex,axiom,
    ! [Pa: fun_op9133723461703387112_nat_o] :
      ( ? [X_12: option3562590408128118217it_nat] : aa_opt9064029888611036241_nat_o(Pa,X_12)
    <=> ( aa_opt9064029888611036241_nat_o(Pa,none_P2651198173097904984it_nat)
        | ? [X4: produc3260487557148687353it_nat] : aa_opt9064029888611036241_nat_o(Pa,aa_Pro5302094286594872192it_nat(some_P7913643980934408916it_nat,X4)) ) ) ).

% split_option_ex
tff(fact_2567_split__option__ex,axiom,
    ! [Pa: fun_option_num_o] :
      ( ? [X_12: option_num] : aa_option_num_o(Pa,X_12)
    <=> ( aa_option_num_o(Pa,none_num)
        | ? [X4: num] : aa_option_num_o(Pa,aa_num_option_num(some_num,X4)) ) ) ).

% split_option_ex
tff(fact_2568_split__option__ex,axiom,
    ! [Pa: fun_op479301348043730329_nat_o] :
      ( ? [X_12: option8956607266484857688it_nat] : aa_opt8360315729766332930_nat_o(Pa,X_12)
    <=> ( aa_opt8360315729766332930_nat_o(Pa,none_P9117596204409417319it_nat)
        | ? [X4: produc8664842809031399944it_nat] : aa_opt8360315729766332930_nat_o(Pa,aa_Pro8996696869555642526it_nat(some_P1914260805536162275it_nat,X4)) ) ) ).

% split_option_ex
tff(fact_2569_split__option__all,axiom,
    ! [Pa: fun_op9133723461703387112_nat_o] :
      ( ! [X_12: option3562590408128118217it_nat] : aa_opt9064029888611036241_nat_o(Pa,X_12)
    <=> ( aa_opt9064029888611036241_nat_o(Pa,none_P2651198173097904984it_nat)
        & ! [X4: produc3260487557148687353it_nat] : aa_opt9064029888611036241_nat_o(Pa,aa_Pro5302094286594872192it_nat(some_P7913643980934408916it_nat,X4)) ) ) ).

% split_option_all
tff(fact_2570_split__option__all,axiom,
    ! [Pa: fun_option_num_o] :
      ( ! [X_12: option_num] : aa_option_num_o(Pa,X_12)
    <=> ( aa_option_num_o(Pa,none_num)
        & ! [X4: num] : aa_option_num_o(Pa,aa_num_option_num(some_num,X4)) ) ) ).

% split_option_all
tff(fact_2571_split__option__all,axiom,
    ! [Pa: fun_op479301348043730329_nat_o] :
      ( ! [X_12: option8956607266484857688it_nat] : aa_opt8360315729766332930_nat_o(Pa,X_12)
    <=> ( aa_opt8360315729766332930_nat_o(Pa,none_P9117596204409417319it_nat)
        & ! [X4: produc8664842809031399944it_nat] : aa_opt8360315729766332930_nat_o(Pa,aa_Pro8996696869555642526it_nat(some_P1914260805536162275it_nat,X4)) ) ) ).

% split_option_all
tff(fact_2572_combine__options__cases,axiom,
    ! [X: option3562590408128118217it_nat,Pa: fun_op5061235789045561272_nat_o,Y: option3562590408128118217it_nat] :
      ( ( ( X = none_P2651198173097904984it_nat )
       => aa_opt9064029888611036241_nat_o(aa_opt9017216189638898209_nat_o(Pa,X),Y) )
     => ( ( ( Y = none_P2651198173097904984it_nat )
         => aa_opt9064029888611036241_nat_o(aa_opt9017216189638898209_nat_o(Pa,X),Y) )
       => ( ! [A4: produc3260487557148687353it_nat,B4: produc3260487557148687353it_nat] :
              ( ( X = aa_Pro5302094286594872192it_nat(some_P7913643980934408916it_nat,A4) )
             => ( ( Y = aa_Pro5302094286594872192it_nat(some_P7913643980934408916it_nat,B4) )
               => aa_opt9064029888611036241_nat_o(aa_opt9017216189638898209_nat_o(Pa,X),Y) ) )
         => aa_opt9064029888611036241_nat_o(aa_opt9017216189638898209_nat_o(Pa,X),Y) ) ) ) ).

% combine_options_cases
tff(fact_2573_combine__options__cases,axiom,
    ! [X: option3562590408128118217it_nat,Pa: fun_op8016730479996091735_num_o,Y: option_num] :
      ( ( ( X = none_P2651198173097904984it_nat )
       => aa_option_num_o(aa_opt1040307220371850048_num_o(Pa,X),Y) )
     => ( ( ( Y = none_num )
         => aa_option_num_o(aa_opt1040307220371850048_num_o(Pa,X),Y) )
       => ( ! [A4: produc3260487557148687353it_nat,B4: num] :
              ( ( X = aa_Pro5302094286594872192it_nat(some_P7913643980934408916it_nat,A4) )
             => ( ( Y = aa_num_option_num(some_num,B4) )
               => aa_option_num_o(aa_opt1040307220371850048_num_o(Pa,X),Y) ) )
         => aa_option_num_o(aa_opt1040307220371850048_num_o(Pa,X),Y) ) ) ) ).

% combine_options_cases
tff(fact_2574_combine__options__cases,axiom,
    ! [X: option3562590408128118217it_nat,Pa: fun_op636363337204153321_nat_o,Y: option8956607266484857688it_nat] :
      ( ( ( X = none_P2651198173097904984it_nat )
       => aa_opt8360315729766332930_nat_o(aa_opt7300124240507411154_nat_o(Pa,X),Y) )
     => ( ( ( Y = none_P9117596204409417319it_nat )
         => aa_opt8360315729766332930_nat_o(aa_opt7300124240507411154_nat_o(Pa,X),Y) )
       => ( ! [A4: produc3260487557148687353it_nat,B4: produc8664842809031399944it_nat] :
              ( ( X = aa_Pro5302094286594872192it_nat(some_P7913643980934408916it_nat,A4) )
             => ( ( Y = aa_Pro8996696869555642526it_nat(some_P1914260805536162275it_nat,B4) )
               => aa_opt8360315729766332930_nat_o(aa_opt7300124240507411154_nat_o(Pa,X),Y) ) )
         => aa_opt8360315729766332930_nat_o(aa_opt7300124240507411154_nat_o(Pa,X),Y) ) ) ) ).

% combine_options_cases
tff(fact_2575_combine__options__cases,axiom,
    ! [X: option_num,Pa: fun_op7041624247167716439_nat_o,Y: option3562590408128118217it_nat] :
      ( ( ( X = none_num )
       => aa_opt9064029888611036241_nat_o(aa_opt3534265616007204544_nat_o(Pa,X),Y) )
     => ( ( ( Y = none_P2651198173097904984it_nat )
         => aa_opt9064029888611036241_nat_o(aa_opt3534265616007204544_nat_o(Pa,X),Y) )
       => ( ! [A4: num,B4: produc3260487557148687353it_nat] :
              ( ( X = aa_num_option_num(some_num,A4) )
             => ( ( Y = aa_Pro5302094286594872192it_nat(some_P7913643980934408916it_nat,B4) )
               => aa_opt9064029888611036241_nat_o(aa_opt3534265616007204544_nat_o(Pa,X),Y) ) )
         => aa_opt9064029888611036241_nat_o(aa_opt3534265616007204544_nat_o(Pa,X),Y) ) ) ) ).

% combine_options_cases
tff(fact_2576_combine__options__cases,axiom,
    ! [X: option_num,Pa: fun_op493924057549976310_num_o,Y: option_num] :
      ( ( ( X = none_num )
       => aa_option_num_o(aa_opt8674010985164783839_num_o(Pa,X),Y) )
     => ( ( ( Y = none_num )
         => aa_option_num_o(aa_opt8674010985164783839_num_o(Pa,X),Y) )
       => ( ! [A4: num,B4: num] :
              ( ( X = aa_num_option_num(some_num,A4) )
             => ( ( Y = aa_num_option_num(some_num,B4) )
               => aa_option_num_o(aa_opt8674010985164783839_num_o(Pa,X),Y) ) )
         => aa_option_num_o(aa_opt8674010985164783839_num_o(Pa,X),Y) ) ) ) ).

% combine_options_cases
tff(fact_2577_combine__options__cases,axiom,
    ! [X: option_num,Pa: fun_op298641349230941576_nat_o,Y: option8956607266484857688it_nat] :
      ( ( ( X = none_num )
       => aa_opt8360315729766332930_nat_o(aa_opt6324340006354107505_nat_o(Pa,X),Y) )
     => ( ( ( Y = none_P9117596204409417319it_nat )
         => aa_opt8360315729766332930_nat_o(aa_opt6324340006354107505_nat_o(Pa,X),Y) )
       => ( ! [A4: num,B4: produc8664842809031399944it_nat] :
              ( ( X = aa_num_option_num(some_num,A4) )
             => ( ( Y = aa_Pro8996696869555642526it_nat(some_P1914260805536162275it_nat,B4) )
               => aa_opt8360315729766332930_nat_o(aa_opt6324340006354107505_nat_o(Pa,X),Y) ) )
         => aa_opt8360315729766332930_nat_o(aa_opt6324340006354107505_nat_o(Pa,X),Y) ) ) ) ).

% combine_options_cases
tff(fact_2578_combine__options__cases,axiom,
    ! [X: option8956607266484857688it_nat,Pa: fun_op2843162390950497001_nat_o,Y: option3562590408128118217it_nat] :
      ( ( ( X = none_P9117596204409417319it_nat )
       => aa_opt9064029888611036241_nat_o(aa_opt4896439242367626578_nat_o(Pa,X),Y) )
     => ( ( ( Y = none_P2651198173097904984it_nat )
         => aa_opt9064029888611036241_nat_o(aa_opt4896439242367626578_nat_o(Pa,X),Y) )
       => ( ! [A4: produc8664842809031399944it_nat,B4: produc3260487557148687353it_nat] :
              ( ( X = aa_Pro8996696869555642526it_nat(some_P1914260805536162275it_nat,A4) )
             => ( ( Y = aa_Pro5302094286594872192it_nat(some_P7913643980934408916it_nat,B4) )
               => aa_opt9064029888611036241_nat_o(aa_opt4896439242367626578_nat_o(Pa,X),Y) ) )
         => aa_opt9064029888611036241_nat_o(aa_opt4896439242367626578_nat_o(Pa,X),Y) ) ) ) ).

% combine_options_cases
tff(fact_2579_combine__options__cases,axiom,
    ! [X: option8956607266484857688it_nat,Pa: fun_op6018817604785332104_num_o,Y: option_num] :
      ( ( ( X = none_P9117596204409417319it_nat )
       => aa_option_num_o(aa_opt3964967581558639985_num_o(Pa,X),Y) )
     => ( ( ( Y = none_num )
         => aa_option_num_o(aa_opt3964967581558639985_num_o(Pa,X),Y) )
       => ( ! [A4: produc8664842809031399944it_nat,B4: num] :
              ( ( X = aa_Pro8996696869555642526it_nat(some_P1914260805536162275it_nat,A4) )
             => ( ( Y = aa_num_option_num(some_num,B4) )
               => aa_option_num_o(aa_opt3964967581558639985_num_o(Pa,X),Y) ) )
         => aa_option_num_o(aa_opt3964967581558639985_num_o(Pa,X),Y) ) ) ) ).

% combine_options_cases
tff(fact_2580_combine__options__cases,axiom,
    ! [X: option8956607266484857688it_nat,Pa: fun_op1633239139951409690_nat_o,Y: option8956607266484857688it_nat] :
      ( ( ( X = none_P9117596204409417319it_nat )
       => aa_opt8360315729766332930_nat_o(aa_opt6948326687743049987_nat_o(Pa,X),Y) )
     => ( ( ( Y = none_P9117596204409417319it_nat )
         => aa_opt8360315729766332930_nat_o(aa_opt6948326687743049987_nat_o(Pa,X),Y) )
       => ( ! [A4: produc8664842809031399944it_nat,B4: produc8664842809031399944it_nat] :
              ( ( X = aa_Pro8996696869555642526it_nat(some_P1914260805536162275it_nat,A4) )
             => ( ( Y = aa_Pro8996696869555642526it_nat(some_P1914260805536162275it_nat,B4) )
               => aa_opt8360315729766332930_nat_o(aa_opt6948326687743049987_nat_o(Pa,X),Y) ) )
         => aa_opt8360315729766332930_nat_o(aa_opt6948326687743049987_nat_o(Pa,X),Y) ) ) ) ).

% combine_options_cases
tff(fact_2581_less__eq__option__None,axiom,
    ! [X: option_num] : aa_option_num_o(aa_opt8674010985164783839_num_o(ord_le6622620407824499402on_num,none_num),X) ).

% less_eq_option_None
tff(fact_2582_less__eq__option__None__is__None,axiom,
    ! [X: option_num] :
      ( aa_option_num_o(aa_opt8674010985164783839_num_o(ord_le6622620407824499402on_num,X),none_num)
     => ( X = none_num ) ) ).

% less_eq_option_None_is_None
tff(fact_2583_Abs__assn__eqI_I1_J,axiom,
    ! [Pa: fun_Pr6460944699146269935_nat_o,Pr: assn] :
      ( ! [H: produc3658429121746597890et_nat] :
          ( aa_Pro653082453696054616_nat_o(Pa,H)
        <=> aa_Pro653082453696054616_nat_o(rep_assn(Pr),H) )
     => ( abs_assn(Pa) = Pr ) ) ).

% Abs_assn_eqI(1)
tff(fact_2584_Abs__assn__eqI_I2_J,axiom,
    ! [Pa: fun_Pr6460944699146269935_nat_o,Pr: assn] :
      ( ! [H: produc3658429121746597890et_nat] :
          ( aa_Pro653082453696054616_nat_o(Pa,H)
        <=> aa_Pro653082453696054616_nat_o(rep_assn(Pr),H) )
     => ( Pr = abs_assn(Pa) ) ) ).

% Abs_assn_eqI(2)
tff(fact_2585_option_Osimps_I5_J,axiom,
    ! [F1: int,F22: fun_num_int,X2: num] : aa_option_num_int(case_option_int_num(F1,F22),aa_num_option_num(some_num,X2)) = aa_num_int(F22,X2) ).

% option.simps(5)
tff(fact_2586_option_Osimps_I5_J,axiom,
    ! [F1: option_num,F22: fun_num_option_num,X2: num] : aa_opt5369159329878751634on_num(case_o6005452278849405969um_num(F1,F22),aa_num_option_num(some_num,X2)) = aa_num_option_num(F22,X2) ).

% option.simps(5)
tff(fact_2587_option_Osimps_I5_J,axiom,
    ! [F1: num,F22: fun_num_num,X2: num] : aa_option_num_num(case_option_num_num(F1,F22),aa_num_option_num(some_num,X2)) = aa_num_num(F22,X2) ).

% option.simps(5)
tff(fact_2588_the__default_Osimps_I2_J,axiom,
    ! [X: num] : aa_option_num_num(the_default_num(X),none_num) = X ).

% the_default.simps(2)
tff(fact_2589_inf__option__def,axiom,
    ! [X: option_set_nat,Y: option_set_nat] : inf_in7812914138253463912et_nat(X,Y) = case_o4054078431260844265et_nat(none_set_nat,aTP_Lamm_ef(Y),X) ).

% inf_option_def
tff(fact_2590_inf__option__def,axiom,
    ! [X: option_assn,Y: option_assn] : inf_inf_option_assn(X,Y) = case_o4484465799723439917n_assn(none_assn,aTP_Lamm_eh(Y),X) ).

% inf_option_def
tff(fact_2591_inf__option__def,axiom,
    ! [X: option_nat,Y: option_nat] : inf_inf_option_nat(X,Y) = case_o7429725398727453821at_nat(none_nat,aTP_Lamm_ej(Y),X) ).

% inf_option_def
tff(fact_2592_inf__option__def,axiom,
    ! [X: option_Product_unit,Y: option_Product_unit] : inf_in5800606384722953409t_unit(X,Y) = case_o2985186191577459077t_unit(none_Product_unit,aTP_Lamm_el(Y),X) ).

% inf_option_def
tff(fact_2593_inf__option__def,axiom,
    ! [X: option8963830502488799655at_nat,Y: option8963830502488799655at_nat] : inf_in777885744494410645at_nat(X,Y) = case_o311359030874850053at_nat(none_s625347054029921090at_nat,aTP_Lamm_en(Y),X) ).

% inf_option_def
tff(fact_2594_bot__assn__def,axiom,
    bot_bot_assn = abs_assn(aTP_Lamm_eo) ).

% bot_assn_def
tff(fact_2595_less__option__None__is__Some,axiom,
    ! [X: option_num] :
      ( aa_option_num_o(aa_opt8674010985164783839_num_o(ord_less_option_num,none_num),X)
     => ? [Z2: num] : X = aa_num_option_num(some_num,Z2) ) ).

% less_option_None_is_Some
tff(fact_2596_less__option__None__Some,axiom,
    ! [X: num] : aa_option_num_o(aa_opt8674010985164783839_num_o(ord_less_option_num,none_num),aa_num_option_num(some_num,X)) ).

% less_option_None_Some
tff(fact_2597_sup__assn__def,axiom,
    ! [Pa: assn,Qa: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Pa),Qa) = abs_assn(aa_ass4243947775541419017_nat_o(aTP_Lamm_ep(Pa),Qa)) ).

% sup_assn_def
tff(fact_2598_inf__assn__def,axiom,
    ! [Pa: assn,Qa: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Pa),Qa) = abs_assn(aa_ass4243947775541419017_nat_o(aTP_Lamm_eq(Pa),Qa)) ).

% inf_assn_def
tff(fact_2599_successE,axiom,
    ! [F: heap_T5738788834812785303t_unit,Ha: heap_e7401611519738050253t_unit] :
      ( aa_hea7994211679605476167unit_o(heap_T584514906347983379t_unit(F),Ha)
     => ~ ! [R: product_unit,H4: produc6653097349344004940it_nat] : aa_hea684056619475940329it_nat(heap_T875086893843062177t_unit(F),Ha) != aa_Pro8996696869555642526it_nat(some_P1914260805536162275it_nat,produc7133225469290080770it_nat(R,H4)) ) ).

% successE
tff(fact_2600_successE,axiom,
    ! [F: heap_Time_Heap_a,Ha: heap_e7401611519738050253t_unit] :
      ( aa_hea7994211679605476167unit_o(heap_Time_success_a(F),Ha)
     => ~ ! [R: a,H4: produc6653097349344004940it_nat] : aa_hea4724376117875161946it_nat(heap_Time_execute_a(F),Ha) != aa_Pro5302094286594872192it_nat(some_P7913643980934408916it_nat,aa_Pro8078911840264235907it_nat(aa_a_f2234720956197727492it_nat(produc9178034014595674355it_nat,R),H4)) ) ).

% successE
tff(fact_2601_rel__of__empty,axiom,
    ! [Pa: fun_Pr4796943543331071514_nat_o] : rel_of_nat_nat(aTP_Lamm_er,Pa) = bot_bo2099793752762293965at_nat ).

% rel_of_empty
tff(fact_2602_set__to__map__empty,axiom,
    ! [X5: nat] : aa_nat_option_nat(set_to_map_nat_nat(bot_bo2099793752762293965at_nat),X5) = none_nat ).

% set_to_map_empty
tff(fact_2603_map__to__set__empty,axiom,
    map_to_set_nat_nat(aTP_Lamm_er) = bot_bo2099793752762293965at_nat ).

% map_to_set_empty
tff(fact_2604_disjE__realizer2,axiom,
    ! [Pa: $o,Qa: fun_num_o,X: option_num,Ra: fun_int_o,F: int,G: fun_num_int] :
      ( aa_option_num_o(case_option_o_num((Pa),Qa),X)
     => ( ( (Pa)
         => aa_int_o(Ra,F) )
       => ( ! [Q3: num] :
              ( aa_num_o(Qa,Q3)
             => aa_int_o(Ra,aa_num_int(G,Q3)) )
         => aa_int_o(Ra,aa_option_num_int(case_option_int_num(F,G),X)) ) ) ) ).

% disjE_realizer2
tff(fact_2605_disjE__realizer2,axiom,
    ! [Pa: $o,Qa: fun_num_o,X: option_num,Ra: fun_option_num_o,F: option_num,G: fun_num_option_num] :
      ( aa_option_num_o(case_option_o_num((Pa),Qa),X)
     => ( ( (Pa)
         => aa_option_num_o(Ra,F) )
       => ( ! [Q3: num] :
              ( aa_num_o(Qa,Q3)
             => aa_option_num_o(Ra,aa_num_option_num(G,Q3)) )
         => aa_option_num_o(Ra,aa_opt5369159329878751634on_num(case_o6005452278849405969um_num(F,G),X)) ) ) ) ).

% disjE_realizer2
tff(fact_2606_disjE__realizer2,axiom,
    ! [Pa: $o,Qa: fun_num_o,X: option_num,Ra: fun_num_o,F: num,G: fun_num_num] :
      ( aa_option_num_o(case_option_o_num((Pa),Qa),X)
     => ( ( (Pa)
         => aa_num_o(Ra,F) )
       => ( ! [Q3: num] :
              ( aa_num_o(Qa,Q3)
             => aa_num_o(Ra,aa_num_num(G,Q3)) )
         => aa_num_o(Ra,aa_option_num_num(case_option_num_num(F,G),X)) ) ) ) ).

% disjE_realizer2
tff(fact_2607_pure__assn__def,axiom,
    ! [B: $o] : aa_o_assn(pure_assn,(B)) = abs_assn(pure_a825153325127701367it_nat((B))) ).

% pure_assn_def
tff(fact_2608_set__to__map__empty__iff_I2_J,axiom,
    ! [S: set_Pr1261947904930325089at_nat] :
      ( ( aTP_Lamm_er = set_to_map_nat_nat(S) )
    <=> ( S = bot_bo2099793752762293965at_nat ) ) ).

% set_to_map_empty_iff(2)
tff(fact_2609_not__Some__eq2,axiom,
    ! [V: option4955102875562834416et_nat] :
      ( ! [X4: fun_Pr6460944699146269935_nat_o,Y5: produc3658429121746597890et_nat] : V != some_P994283844982041979et_nat(aa_Pro434340453207536480et_nat(aa_fun3926903090414666066et_nat(produc6017135503889078682et_nat,X4),Y5))
    <=> ( V = none_P7891021755743879167et_nat ) ) ).

% not_Some_eq2
tff(fact_2610_not__Some__eq2,axiom,
    ! [V: option5899113203645146382et_nat] :
      ( ! [X4: fun_Pr6460944699146269935_nat_o,Y5: produc8799533983703829792et_nat] : V != some_P3006888012174695577et_nat(aa_Pro1979504083478021276et_nat(aa_fun615918787490101902et_nat(produc7291855603632349880et_nat,X4),Y5))
    <=> ( V = none_P2758821087663475997et_nat ) ) ).

% not_Some_eq2
tff(fact_2611_not__Some__eq2,axiom,
    ! [V: option1544258867535432659nt_int] :
      ( ! [X4: fun_Pr7763680713840462443e_term,Y5: product_prod_int_int] : V != some_P9145269820779039710nt_int(aa_Pro5748361610964203648nt_int(aa_fun3265472288063244902nt_int(produc7320969046608631221nt_int,X4),Y5))
    <=> ( V = none_P3112835510759997538nt_int ) ) ).

% not_Some_eq2
tff(fact_2612_not__Some__eq2,axiom,
    ! [V: option6186025235080601855nt_int] :
      ( ! [X4: fun_in5730379987311621695e_term,Y5: product_prod_int_int] : V != some_P8798507657793832714nt_int(aa_Pro8800328608558162860nt_int(aa_fun260840925927807678nt_int(produc4729913460834754785nt_int,X4),Y5))
    <=> ( V = none_P1716269551890668430nt_int ) ) ).

% not_Some_eq2
tff(fact_2613_not__Some__eq2,axiom,
    ! [V: option3562590408128118217it_nat] :
      ( ! [X4: a,Y5: produc6653097349344004940it_nat] : V != aa_Pro5302094286594872192it_nat(some_P7913643980934408916it_nat,aa_Pro8078911840264235907it_nat(aa_a_f2234720956197727492it_nat(produc9178034014595674355it_nat,X4),Y5))
    <=> ( V = none_P2651198173097904984it_nat ) ) ).

% not_Some_eq2
tff(fact_2614_not__Some__eq2,axiom,
    ! [V: option8956607266484857688it_nat] :
      ( ! [X4: product_unit,Y5: produc6653097349344004940it_nat] : V != aa_Pro8996696869555642526it_nat(some_P1914260805536162275it_nat,produc7133225469290080770it_nat(X4,Y5))
    <=> ( V = none_P9117596204409417319it_nat ) ) ).

% not_Some_eq2
tff(fact_2615_execute__bind_I2_J,axiom,
    ! [F: heap_Time_Heap_a,Ha: heap_e7401611519738050253t_unit,G: fun_a_6093291656547882009Heap_a] :
      ( ( aa_hea4724376117875161946it_nat(heap_Time_execute_a(F),Ha) = none_P2651198173097904984it_nat )
     => ( aa_hea4724376117875161946it_nat(heap_Time_execute_a(heap_Time_bind_a_a(F,G)),Ha) = none_P2651198173097904984it_nat ) ) ).

% execute_bind(2)
tff(fact_2616_option_Odisc__eq__case_I2_J,axiom,
    ! [Option: option_num] :
      ( ( Option != none_num )
    <=> aa_option_num_o(case_option_o_num($false,aTP_Lamm_es),Option) ) ).

% option.disc_eq_case(2)
tff(fact_2617_option_Odisc__eq__case_I1_J,axiom,
    ! [Option: option_num] :
      ( ( Option = none_num )
    <=> aa_option_num_o(case_option_o_num($true,aTP_Lamm_et),Option) ) ).

% option.disc_eq_case(1)
tff(fact_2618_success__def,axiom,
    ! [F: heap_Time_Heap_a,Ha: heap_e7401611519738050253t_unit] :
      ( aa_hea7994211679605476167unit_o(heap_Time_success_a(F),Ha)
    <=> ( aa_hea4724376117875161946it_nat(heap_Time_execute_a(F),Ha) != none_P2651198173097904984it_nat ) ) ).

% success_def
tff(fact_2619_successI,axiom,
    ! [F: heap_Time_Heap_a,Ha: heap_e7401611519738050253t_unit] :
      ( ( aa_hea4724376117875161946it_nat(heap_Time_execute_a(F),Ha) != none_P2651198173097904984it_nat )
     => aa_hea7994211679605476167unit_o(heap_Time_success_a(F),Ha) ) ).

% successI
tff(fact_2620_case__optionE,axiom,
    ! [Pa: $o,Qa: fun_Pr5746006151868627512_nat_o,X: option3562590408128118217it_nat] :
      ( aa_opt9064029888611036241_nat_o(case_o3267218206291580230it_nat((Pa),Qa),X)
     => ( ( ( X = none_P2651198173097904984it_nat )
         => ~ (Pa) )
       => ~ ! [Y3: produc3260487557148687353it_nat] :
              ( ( X = aa_Pro5302094286594872192it_nat(some_P7913643980934408916it_nat,Y3) )
             => ~ aa_Pro6426884464393220769_nat_o(Qa,Y3) ) ) ) ).

% case_optionE
tff(fact_2621_case__optionE,axiom,
    ! [Pa: $o,Qa: fun_num_o,X: option_num] :
      ( aa_option_num_o(case_option_o_num((Pa),Qa),X)
     => ( ( ( X = none_num )
         => ~ (Pa) )
       => ~ ! [Y3: num] :
              ( ( X = aa_num_option_num(some_num,Y3) )
             => ~ aa_num_o(Qa,Y3) ) ) ) ).

% case_optionE
tff(fact_2622_case__optionE,axiom,
    ! [Pa: $o,Qa: fun_Pr8032424337255632745_nat_o,X: option8956607266484857688it_nat] :
      ( aa_opt8360315729766332930_nat_o(case_o2686588417244861013it_nat((Pa),Qa),X)
     => ( ( ( X = none_P9117596204409417319it_nat )
         => ~ (Pa) )
       => ~ ! [Y3: produc8664842809031399944it_nat] :
              ( ( X = aa_Pro8996696869555642526it_nat(some_P1914260805536162275it_nat,Y3) )
             => ~ aa_Pro4744234731172327378_nat_o(Qa,Y3) ) ) ) ).

% case_optionE
tff(fact_2623_map__to__set__empty__iff_I1_J,axiom,
    ! [M: fun_nat_option_nat] :
      ( ( map_to_set_nat_nat(M) = bot_bo2099793752762293965at_nat )
    <=> ! [X4: nat] : aa_nat_option_nat(M,X4) = none_nat ) ).

% map_to_set_empty_iff(1)
tff(fact_2624_map__to__set__empty__iff_I2_J,axiom,
    ! [M: fun_nat_option_nat] :
      ( ( bot_bo2099793752762293965at_nat = map_to_set_nat_nat(M) )
    <=> ! [X4: nat] : aa_nat_option_nat(M,X4) = none_nat ) ).

% map_to_set_empty_iff(2)
tff(fact_2625_less__eq__option__def,axiom,
    ! [X: option_set_int,Y: option_set_int] :
      ( ord_le353528952715127954et_int(X,Y)
    <=> case_o223999843215110191et_int($true,aTP_Lamm_eu(Y),X) ) ).

% less_eq_option_def
tff(fact_2626_less__eq__option__def,axiom,
    ! [X: option_rat,Y: option_rat] :
      ( ord_le2406147912482264968on_rat(X,Y)
    <=> case_option_o_rat($true,aTP_Lamm_ev(Y),X) ) ).

% less_eq_option_def
tff(fact_2627_less__eq__option__def,axiom,
    ! [X: option_num,Y: option_num] :
      ( aa_option_num_o(aa_opt8674010985164783839_num_o(ord_le6622620407824499402on_num,X),Y)
    <=> aa_option_num_o(case_option_o_num($true,aTP_Lamm_ew(Y)),X) ) ).

% less_eq_option_def
tff(fact_2628_less__eq__option__def,axiom,
    ! [X: option_nat,Y: option_nat] :
      ( ord_le5914376470875661696on_nat(X,Y)
    <=> case_option_o_nat($true,aTP_Lamm_ex(Y),X) ) ).

% less_eq_option_def
tff(fact_2629_less__eq__option__def,axiom,
    ! [X: option_int,Y: option_int] :
      ( ord_le1736525451366464988on_int(X,Y)
    <=> case_option_o_int($true,aTP_Lamm_ey(Y),X) ) ).

% less_eq_option_def
tff(fact_2630_less__option__def,axiom,
    ! [X: option_assn,Y: option_assn] :
      ( ord_less_option_assn(X,Y)
    <=> case_option_o_assn($false,aTP_Lamm_ez(X),Y) ) ).

% less_option_def
tff(fact_2631_less__option__def,axiom,
    ! [X: option_rat,Y: option_rat] :
      ( ord_less_option_rat(X,Y)
    <=> case_option_o_rat($false,aTP_Lamm_fa(X),Y) ) ).

% less_option_def
tff(fact_2632_less__option__def,axiom,
    ! [X: option_num,Y: option_num] :
      ( aa_option_num_o(aa_opt8674010985164783839_num_o(ord_less_option_num,X),Y)
    <=> aa_option_num_o(case_option_o_num($false,aTP_Lamm_fc(X)),Y) ) ).

% less_option_def
tff(fact_2633_less__option__def,axiom,
    ! [X: option_nat,Y: option_nat] :
      ( ord_less_option_nat(X,Y)
    <=> case_option_o_nat($false,aTP_Lamm_fd(X),Y) ) ).

% less_option_def
tff(fact_2634_less__option__def,axiom,
    ! [X: option_int,Y: option_int] :
      ( ord_less_option_int(X,Y)
    <=> case_option_o_int($false,aTP_Lamm_fe(X),Y) ) ).

% less_option_def
tff(fact_2635_Heap__cases,axiom,
    ! [F: heap_T5738788834812785303t_unit,Ha: heap_e7401611519738050253t_unit] :
      ( ! [X3: product_unit,H4: produc6653097349344004940it_nat] : aa_hea684056619475940329it_nat(heap_T875086893843062177t_unit(F),Ha) != aa_Pro8996696869555642526it_nat(some_P1914260805536162275it_nat,produc7133225469290080770it_nat(X3,H4))
     => ( aa_hea684056619475940329it_nat(heap_T875086893843062177t_unit(F),Ha) = none_P9117596204409417319it_nat ) ) ).

% Heap_cases
tff(fact_2636_Heap__cases,axiom,
    ! [F: heap_Time_Heap_a,Ha: heap_e7401611519738050253t_unit] :
      ( ! [X3: a,H4: produc6653097349344004940it_nat] : aa_hea4724376117875161946it_nat(heap_Time_execute_a(F),Ha) != aa_Pro5302094286594872192it_nat(some_P7913643980934408916it_nat,aa_Pro8078911840264235907it_nat(aa_a_f2234720956197727492it_nat(produc9178034014595674355it_nat,X3),H4))
     => ( aa_hea4724376117875161946it_nat(heap_Time_execute_a(F),Ha) = none_P2651198173097904984it_nat ) ) ).

% Heap_cases
tff(fact_2637_timeFrame_Ocases,axiom,
    ! [X: produc4453839368661128058it_nat] :
      ( ! [N5: nat,R: a,H: heap_e7401611519738050253t_unit,N4: nat] : X != produc4111269673004989362it_nat(N5,aa_Pro5302094286594872192it_nat(some_P7913643980934408916it_nat,aa_Pro8078911840264235907it_nat(aa_a_f2234720956197727492it_nat(produc9178034014595674355it_nat,R),produc584006145561248582it_nat(H,N4))))
     => ~ ! [N5: nat] : X != produc4111269673004989362it_nat(N5,none_P2651198173097904984it_nat) ) ).

% timeFrame.cases
tff(fact_2638_timeFrame_Ocases,axiom,
    ! [X: produc3911288613690379145it_nat] :
      ( ! [N5: nat,R: product_unit,H: heap_e7401611519738050253t_unit,N4: nat] : X != produc638857205735767105it_nat(N5,aa_Pro8996696869555642526it_nat(some_P1914260805536162275it_nat,produc7133225469290080770it_nat(R,produc584006145561248582it_nat(H,N4))))
     => ~ ! [N5: nat] : X != produc638857205735767105it_nat(N5,none_P9117596204409417319it_nat) ) ).

% timeFrame.cases
tff(fact_2639_set__to__map__empty__iff_I1_J,axiom,
    ! [S: set_Pr1261947904930325089at_nat] :
      ( ! [X4: nat] : aa_nat_option_nat(set_to_map_nat_nat(S),X4) = none_nat
    <=> ( S = bot_bo2099793752762293965at_nat ) ) ).

% set_to_map_empty_iff(1)
tff(fact_2640_pure__assn__raw_Oelims_I3_J,axiom,
    ! [X: $o,Xa2: produc3658429121746597890et_nat] :
      ( ~ aa_Pro653082453696054616_nat_o(pure_a825153325127701367it_nat((X)),Xa2)
     => ~ ! [H: heap_e7401611519738050253t_unit,As3: set_nat] :
            ( ( Xa2 = produc7507926704131184380et_nat(H,As3) )
           => ( ( As3 = bot_bot_set_nat )
              & (X) ) ) ) ).

% pure_assn_raw.elims(3)
tff(fact_2641_pure__assn__raw_Oelims_I2_J,axiom,
    ! [X: $o,Xa2: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(pure_a825153325127701367it_nat((X)),Xa2)
     => ~ ! [H: heap_e7401611519738050253t_unit,As3: set_nat] :
            ( ( Xa2 = produc7507926704131184380et_nat(H,As3) )
           => ~ ( ( As3 = bot_bot_set_nat )
                & (X) ) ) ) ).

% pure_assn_raw.elims(2)
tff(fact_2642_pure__assn__raw_Oelims_I1_J,axiom,
    ! [X: $o,Xa2: produc3658429121746597890et_nat,Y: $o] :
      ( ( aa_Pro653082453696054616_nat_o(pure_a825153325127701367it_nat((X)),Xa2)
      <=> (Y) )
     => ~ ! [H: heap_e7401611519738050253t_unit,As3: set_nat] :
            ( ( Xa2 = produc7507926704131184380et_nat(H,As3) )
           => ( (Y)
            <=> ~ ( ( As3 = bot_bot_set_nat )
                  & (X) ) ) ) ) ).

% pure_assn_raw.elims(1)
tff(fact_2643_pure__assn__raw_Osimps,axiom,
    ! [B: $o,Ha: heap_e7401611519738050253t_unit,Asa: set_nat] :
      ( aa_Pro653082453696054616_nat_o(pure_a825153325127701367it_nat((B)),produc7507926704131184380et_nat(Ha,Asa))
    <=> ( ( Asa = bot_bot_set_nat )
        & (B) ) ) ).

% pure_assn_raw.simps
tff(fact_2644_pure__assn__raw_Opelims_I1_J,axiom,
    ! [X: $o,Xa2: produc3658429121746597890et_nat,Y: $o] :
      ( ( aa_Pro653082453696054616_nat_o(pure_a825153325127701367it_nat((X)),Xa2)
      <=> (Y) )
     => ( accp_P8458817951426537472et_nat(pure_a6022498039421069578it_nat,produc3762353314782720579et_nat((X),Xa2))
       => ~ ! [H: heap_e7401611519738050253t_unit,As3: set_nat] :
              ( ( Xa2 = produc7507926704131184380et_nat(H,As3) )
             => ( ( (Y)
                <=> ( ( As3 = bot_bot_set_nat )
                    & (X) ) )
               => ~ accp_P8458817951426537472et_nat(pure_a6022498039421069578it_nat,produc3762353314782720579et_nat((X),produc7507926704131184380et_nat(H,As3))) ) ) ) ) ).

% pure_assn_raw.pelims(1)
tff(fact_2645_pure__assn__raw_Opelims_I2_J,axiom,
    ! [X: $o,Xa2: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(pure_a825153325127701367it_nat((X)),Xa2)
     => ( accp_P8458817951426537472et_nat(pure_a6022498039421069578it_nat,produc3762353314782720579et_nat((X),Xa2))
       => ~ ! [H: heap_e7401611519738050253t_unit,As3: set_nat] :
              ( ( Xa2 = produc7507926704131184380et_nat(H,As3) )
             => ( accp_P8458817951426537472et_nat(pure_a6022498039421069578it_nat,produc3762353314782720579et_nat((X),produc7507926704131184380et_nat(H,As3)))
               => ~ ( ( As3 = bot_bot_set_nat )
                    & (X) ) ) ) ) ) ).

% pure_assn_raw.pelims(2)
tff(fact_2646_pure__assn__raw_Opelims_I3_J,axiom,
    ! [X: $o,Xa2: produc3658429121746597890et_nat] :
      ( ~ aa_Pro653082453696054616_nat_o(pure_a825153325127701367it_nat((X)),Xa2)
     => ( accp_P8458817951426537472et_nat(pure_a6022498039421069578it_nat,produc3762353314782720579et_nat((X),Xa2))
       => ~ ! [H: heap_e7401611519738050253t_unit,As3: set_nat] :
              ( ( Xa2 = produc7507926704131184380et_nat(H,As3) )
             => ( accp_P8458817951426537472et_nat(pure_a6022498039421069578it_nat,produc3762353314782720579et_nat((X),produc7507926704131184380et_nat(H,As3)))
               => ( ( As3 = bot_bot_set_nat )
                  & (X) ) ) ) ) ) ).

% pure_assn_raw.pelims(3)
tff(fact_2647_execute__bind_I1_J,axiom,
    ! [F: heap_T5738788834812785303t_unit,Ha: heap_e7401611519738050253t_unit,X: product_unit,H_a: heap_e7401611519738050253t_unit,N2: nat,G: fun_Pr5215307506330550056Heap_a] :
      ( ( aa_hea684056619475940329it_nat(heap_T875086893843062177t_unit(F),Ha) = aa_Pro8996696869555642526it_nat(some_P1914260805536162275it_nat,produc7133225469290080770it_nat(X,produc584006145561248582it_nat(H_a,N2))) )
     => ( aa_hea4724376117875161946it_nat(heap_Time_execute_a(heap_T757603679106148408unit_a(F,G)),Ha) = heap_T7616092557645711335rame_a(N2,aa_hea4724376117875161946it_nat(heap_Time_execute_a(aa_Pro7886978635576167513Heap_a(G,X)),H_a)) ) ) ).

% execute_bind(1)
tff(fact_2648_execute__bind_I1_J,axiom,
    ! [F: heap_Time_Heap_a,Ha: heap_e7401611519738050253t_unit,X: a,H_a: heap_e7401611519738050253t_unit,N2: nat,G: fun_a_6093291656547882009Heap_a] :
      ( ( aa_hea4724376117875161946it_nat(heap_Time_execute_a(F),Ha) = aa_Pro5302094286594872192it_nat(some_P7913643980934408916it_nat,aa_Pro8078911840264235907it_nat(aa_a_f2234720956197727492it_nat(produc9178034014595674355it_nat,X),produc584006145561248582it_nat(H_a,N2))) )
     => ( aa_hea4724376117875161946it_nat(heap_Time_execute_a(heap_Time_bind_a_a(F,G)),Ha) = heap_T7616092557645711335rame_a(N2,aa_hea4724376117875161946it_nat(heap_Time_execute_a(aa_a_H8829438773920479562Heap_a(G,X)),H_a)) ) ) ).

% execute_bind(1)
tff(fact_2649_execute__bind__eq__SomeI,axiom,
    ! [F: heap_T5738788834812785303t_unit,Ha: heap_e7401611519738050253t_unit,X: product_unit,H_a: heap_e7401611519738050253t_unit,N2: nat,G: fun_Pr292408699144966967t_unit,Y: product_unit,H8: heap_e7401611519738050253t_unit,N6: nat] :
      ( ( aa_hea684056619475940329it_nat(heap_T875086893843062177t_unit(F),Ha) = aa_Pro8996696869555642526it_nat(some_P1914260805536162275it_nat,produc7133225469290080770it_nat(X,produc584006145561248582it_nat(H_a,N2))) )
     => ( ( aa_hea684056619475940329it_nat(heap_T875086893843062177t_unit(aa_Pro1108561329362217960t_unit(G,X)),H_a) = aa_Pro8996696869555642526it_nat(some_P1914260805536162275it_nat,produc7133225469290080770it_nat(Y,produc584006145561248582it_nat(H8,N6))) )
       => ( aa_hea684056619475940329it_nat(heap_T875086893843062177t_unit(heap_T2633723481742716231t_unit(F,G)),Ha) = aa_Pro8996696869555642526it_nat(some_P1914260805536162275it_nat,produc7133225469290080770it_nat(Y,produc584006145561248582it_nat(H8,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),N6)))) ) ) ) ).

% execute_bind_eq_SomeI
tff(fact_2650_execute__bind__eq__SomeI,axiom,
    ! [F: heap_T5738788834812785303t_unit,Ha: heap_e7401611519738050253t_unit,X: product_unit,H_a: heap_e7401611519738050253t_unit,N2: nat,G: fun_Pr5215307506330550056Heap_a,Y: a,H8: heap_e7401611519738050253t_unit,N6: nat] :
      ( ( aa_hea684056619475940329it_nat(heap_T875086893843062177t_unit(F),Ha) = aa_Pro8996696869555642526it_nat(some_P1914260805536162275it_nat,produc7133225469290080770it_nat(X,produc584006145561248582it_nat(H_a,N2))) )
     => ( ( aa_hea4724376117875161946it_nat(heap_Time_execute_a(aa_Pro7886978635576167513Heap_a(G,X)),H_a) = aa_Pro5302094286594872192it_nat(some_P7913643980934408916it_nat,aa_Pro8078911840264235907it_nat(aa_a_f2234720956197727492it_nat(produc9178034014595674355it_nat,Y),produc584006145561248582it_nat(H8,N6))) )
       => ( aa_hea4724376117875161946it_nat(heap_Time_execute_a(heap_T757603679106148408unit_a(F,G)),Ha) = aa_Pro5302094286594872192it_nat(some_P7913643980934408916it_nat,aa_Pro8078911840264235907it_nat(aa_a_f2234720956197727492it_nat(produc9178034014595674355it_nat,Y),produc584006145561248582it_nat(H8,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),N6)))) ) ) ) ).

% execute_bind_eq_SomeI
tff(fact_2651_execute__bind__eq__SomeI,axiom,
    ! [F: heap_Time_Heap_a,Ha: heap_e7401611519738050253t_unit,X: a,H_a: heap_e7401611519738050253t_unit,N2: nat,G: fun_a_2990225006182978344t_unit,Y: product_unit,H8: heap_e7401611519738050253t_unit,N6: nat] :
      ( ( aa_hea4724376117875161946it_nat(heap_Time_execute_a(F),Ha) = aa_Pro5302094286594872192it_nat(some_P7913643980934408916it_nat,aa_Pro8078911840264235907it_nat(aa_a_f2234720956197727492it_nat(produc9178034014595674355it_nat,X),produc584006145561248582it_nat(H_a,N2))) )
     => ( ( aa_hea684056619475940329it_nat(heap_T875086893843062177t_unit(aa_a_H4862568901524997081t_unit(G,X)),H_a) = aa_Pro8996696869555642526it_nat(some_P1914260805536162275it_nat,produc7133225469290080770it_nat(Y,produc584006145561248582it_nat(H8,N6))) )
       => ( aa_hea684056619475940329it_nat(heap_T875086893843062177t_unit(heap_T8685611227969916822t_unit(F,G)),Ha) = aa_Pro8996696869555642526it_nat(some_P1914260805536162275it_nat,produc7133225469290080770it_nat(Y,produc584006145561248582it_nat(H8,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),N6)))) ) ) ) ).

% execute_bind_eq_SomeI
tff(fact_2652_execute__bind__eq__SomeI,axiom,
    ! [F: heap_Time_Heap_a,Ha: heap_e7401611519738050253t_unit,X: a,H_a: heap_e7401611519738050253t_unit,N2: nat,G: fun_a_6093291656547882009Heap_a,Y: a,H8: heap_e7401611519738050253t_unit,N6: nat] :
      ( ( aa_hea4724376117875161946it_nat(heap_Time_execute_a(F),Ha) = aa_Pro5302094286594872192it_nat(some_P7913643980934408916it_nat,aa_Pro8078911840264235907it_nat(aa_a_f2234720956197727492it_nat(produc9178034014595674355it_nat,X),produc584006145561248582it_nat(H_a,N2))) )
     => ( ( aa_hea4724376117875161946it_nat(heap_Time_execute_a(aa_a_H8829438773920479562Heap_a(G,X)),H_a) = aa_Pro5302094286594872192it_nat(some_P7913643980934408916it_nat,aa_Pro8078911840264235907it_nat(aa_a_f2234720956197727492it_nat(produc9178034014595674355it_nat,Y),produc584006145561248582it_nat(H8,N6))) )
       => ( aa_hea4724376117875161946it_nat(heap_Time_execute_a(heap_Time_bind_a_a(F,G)),Ha) = aa_Pro5302094286594872192it_nat(some_P7913643980934408916it_nat,aa_Pro8078911840264235907it_nat(aa_a_f2234720956197727492it_nat(produc9178034014595674355it_nat,Y),produc584006145561248582it_nat(H8,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),N6)))) ) ) ) ).

% execute_bind_eq_SomeI
tff(fact_2653_set__to__map__def,axiom,
    ! [S: set_Pr1261947904930325089at_nat,K: nat] : aa_nat_option_nat(set_to_map_nat_nat(S),K) = eps_Opt_nat(aa_nat_fun_nat_o(aTP_Lamm_as(S),K)) ).

% set_to_map_def
tff(fact_2654_set__to__map__def,axiom,
    ! [S: set_Pr1312406793078187350et_nat,K: fun_Pr6460944699146269935_nat_o] : aa_fun5824947343344787589et_nat(set_to2037224588991580346et_nat(S),K) = aa_fun5824947343344787589et_nat(eps_Op2013419657081471078et_nat,aa_fun2638059832795814268_nat_o(aTP_Lamm_at(S),K)) ).

% set_to_map_def
tff(fact_2655_set__to__map__def,axiom,
    ! [S: set_Pr4040394263813967220et_nat,K: fun_Pr6460944699146269935_nat_o] : aa_fun1716447855799474851et_nat(set_to3162555076294590936et_nat(S),K) = eps_Op5936278548196508548et_nat(aa_fun912680268133416158_nat_o(aTP_Lamm_au(S),K)) ).

% set_to_map_def
tff(fact_2656_set__to__map__def,axiom,
    ! [S: set_Pr6700895561065273913nt_int,K: fun_Pr7763680713840462443e_term] : aa_fun251074914522852630nt_int(set_to8251664086025573589nt_int(S),K) = eps_Op2446201859369042517nt_int(aa_fun3538162476766722827_int_o(aTP_Lamm_av(S),K)) ).

% set_to_map_def
tff(fact_2657_set__to__map__def,axiom,
    ! [S: set_Pr4166680992137330533nt_int,K: fun_in5730379987311621695e_term] : aa_fun3970441718214872642nt_int(set_to2016340311434469377nt_int(S),K) = eps_Op2446201859369042517nt_int(aa_fun586359091965693151_int_o(aTP_Lamm_aw(S),K)) ).

% set_to_map_def
tff(fact_2658_set__to__map__def,axiom,
    ! [S: set_Pr7098220151150636591it_nat,K: a] : aa_a_o5231022909227610846it_nat(set_to2008915276733485075it_nat(S),K) = eps_Op8301357815426737072it_nat(aa_a_f337466037417904483_nat_o(aTP_Lamm_ax(S),K)) ).

% set_to_map_def
tff(fact_2659_execute__guard_I2_J,axiom,
    ! [Pa: fun_he8413527710729349838unit_o,Ha: heap_e7401611519738050253t_unit,F: fun_he4944555133437423336it_nat] :
      ( aa_hea7994211679605476167unit_o(Pa,Ha)
     => ( aa_hea684056619475940329it_nat(heap_T875086893843062177t_unit(heap_T8440541562793052209t_unit(Pa,F)),Ha) = aa_Pro8996696869555642526it_nat(some_P1914260805536162275it_nat,aa_hea256619718444519449it_nat(F,Ha)) ) ) ).

% execute_guard(2)
tff(fact_2660_execute__guard_I2_J,axiom,
    ! [Pa: fun_he8413527710729349838unit_o,Ha: heap_e7401611519738050253t_unit,F: fun_he8889679731534893401it_nat] :
      ( aa_hea7994211679605476167unit_o(Pa,Ha)
     => ( aa_hea4724376117875161946it_nat(heap_Time_execute_a(heap_Time_guard_a(Pa,F)),Ha) = aa_Pro5302094286594872192it_nat(some_P7913643980934408916it_nat,aa_hea5645563107516622602it_nat(F,Ha)) ) ) ).

% execute_guard(2)
tff(fact_2661_mlex__leq,axiom,
    ! [F: fun_nat_nat,X: nat,Y: nat,Ra: set_Pr1261947904930325089at_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F,X)),aa_nat_nat(F,Y))
     => ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,X),Y)),Ra)
       => aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,X),Y)),mlex_prod_nat(F,Ra)) ) ) ).

% mlex_leq
tff(fact_2662_curryI,axiom,
    ! [F: fun_Pr2620521676128474833_nat_o,A: fun_Pr6460944699146269935_nat_o,B: produc3658429121746597890et_nat] :
      ( aa_Pro5643009357358181178_nat_o(F,aa_Pro434340453207536480et_nat(aa_fun3926903090414666066et_nat(produc6017135503889078682et_nat,A),B))
     => aa_Pro653082453696054616_nat_o(aa_fun2638059832795814268_nat_o(produc4701769847872139056_nat_o(F),A),B) ) ).

% curryI
tff(fact_2663_curryI,axiom,
    ! [F: fun_Pr1227177680593602867_nat_o,A: fun_Pr6460944699146269935_nat_o,B: produc8799533983703829792et_nat] :
      ( aa_Pro2671238023309943196_nat_o(F,aa_Pro1979504083478021276et_nat(aa_fun615918787490101902et_nat(produc7291855603632349880et_nat,A),B))
     => aa_Pro5643009357358181178_nat_o(aa_fun912680268133416158_nat_o(produc3333590387107866898_nat_o(F),A),B) ) ).

% curryI
tff(fact_2664_curryI,axiom,
    ! [F: fun_Pr3540825087398332590_int_o,A: fun_Pr7763680713840462443e_term,B: product_prod_int_int] :
      ( aa_Pro2601676399929874711_int_o(F,aa_Pro5748361610964203648nt_int(aa_fun3265472288063244902nt_int(produc7320969046608631221nt_int,A),B))
     => aa_Pro4592068669816069595_int_o(aa_fun3538162476766722827_int_o(produc443495763101479893_int_o(F),A),B) ) ).

% curryI
tff(fact_2665_curryI,axiom,
    ! [F: fun_Pr1196100989263358594_int_o,A: fun_in5730379987311621695e_term,B: product_prod_int_int] :
      ( aa_Pro5451197003417746155_int_o(F,aa_Pro8800328608558162860nt_int(aa_fun260840925927807678nt_int(produc4729913460834754785nt_int,A),B))
     => aa_Pro4592068669816069595_int_o(aa_fun586359091965693151_int_o(produc2168430993391648169_int_o(F),A),B) ) ).

% curryI
tff(fact_2666_curryI,axiom,
    ! [F: fun_Pr5746006151868627512_nat_o,A: a,B: produc6653097349344004940it_nat] :
      ( aa_Pro6426884464393220769_nat_o(F,aa_Pro8078911840264235907it_nat(aa_a_f2234720956197727492it_nat(produc9178034014595674355it_nat,A),B))
     => aa_Pro1202363709425262478_nat_o(aa_a_f337466037417904483_nat_o(produc3709033845735585303_nat_o(F),A),B) ) ).

% curryI
tff(fact_2667_nat__add__left__cancel__less,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),N2))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N2) ) ).

% nat_add_left_cancel_less
tff(fact_2668_nat__add__left__cancel__le,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),N2))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2) ) ).

% nat_add_left_cancel_le
tff(fact_2669_some__opt__sym__eq__trivial,axiom,
    ! [X: produc3260487557148687353it_nat] : eps_Op1741419982095519837it_nat(fequal7405809337025681252it_nat(X)) = aa_Pro5302094286594872192it_nat(some_P7913643980934408916it_nat,X) ).

% some_opt_sym_eq_trivial
tff(fact_2670_some__opt__sym__eq__trivial,axiom,
    ! [X: num] : eps_Opt_num(aa_num_fun_num_o(fequal_num,X)) = aa_num_option_num(some_num,X) ).

% some_opt_sym_eq_trivial
tff(fact_2671_some__opt__sym__eq__trivial,axiom,
    ! [X: produc8664842809031399944it_nat] : eps_Op3393321821070424684it_nat(fequal1407654770548522611it_nat(X)) = aa_Pro8996696869555642526it_nat(some_P1914260805536162275it_nat,X) ).

% some_opt_sym_eq_trivial
tff(fact_2672_some__opt__eq__trivial,axiom,
    ! [X: produc3260487557148687353it_nat] : eps_Op1741419982095519837it_nat(aTP_Lamm_ff(X)) = aa_Pro5302094286594872192it_nat(some_P7913643980934408916it_nat,X) ).

% some_opt_eq_trivial
tff(fact_2673_some__opt__eq__trivial,axiom,
    ! [X: num] : eps_Opt_num(aa_num_fun_num_o(aTP_Lamm_fg,X)) = aa_num_option_num(some_num,X) ).

% some_opt_eq_trivial
tff(fact_2674_some__opt__eq__trivial,axiom,
    ! [X: produc8664842809031399944it_nat] : eps_Op3393321821070424684it_nat(aTP_Lamm_fh(X)) = aa_Pro8996696869555642526it_nat(some_P1914260805536162275it_nat,X) ).

% some_opt_eq_trivial
tff(fact_2675_some__opt__false__trivial,axiom,
    eps_Opt_num(aTP_Lamm_et) = none_num ).

% some_opt_false_trivial
tff(fact_2676_add__lessD1,axiom,
    ! [I2: nat,J: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),J)),K)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),K) ) ).

% add_lessD1
tff(fact_2677_add__less__mono,axiom,
    ! [I2: nat,J: nat,K: nat,L: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),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,I2),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L)) ) ) ).

% add_less_mono
tff(fact_2678_not__add__less1,axiom,
    ! [I2: nat,J: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),J)),I2) ).

% not_add_less1
tff(fact_2679_not__add__less2,axiom,
    ! [J: nat,I2: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),I2)),I2) ).

% not_add_less2
tff(fact_2680_add__less__mono1,axiom,
    ! [I2: nat,J: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),J)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K)) ) ).

% add_less_mono1
tff(fact_2681_trans__less__add1,axiom,
    ! [I2: nat,J: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),J)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),M)) ) ).

% trans_less_add1
tff(fact_2682_trans__less__add2,axiom,
    ! [I2: nat,J: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),J)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),J)) ) ).

% trans_less_add2
tff(fact_2683_less__add__eq__less,axiom,
    ! [K: nat,L: nat,M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),L)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),L) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),N2) )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N2) ) ) ).

% less_add_eq_less
tff(fact_2684_add__leE,axiom,
    ! [M: nat,K: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K)),N2)
     => ~ ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
         => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),N2) ) ) ).

% add_leE
tff(fact_2685_le__add1,axiom,
    ! [N2: nat,M: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),M)) ).

% le_add1
tff(fact_2686_le__add2,axiom,
    ! [N2: nat,M: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N2)) ).

% le_add2
tff(fact_2687_add__leD1,axiom,
    ! [M: nat,K: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K)),N2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2) ) ).

% add_leD1
tff(fact_2688_add__leD2,axiom,
    ! [M: nat,K: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K)),N2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),N2) ) ).

% add_leD2
tff(fact_2689_le__Suc__ex,axiom,
    ! [K: nat,L: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),L)
     => ? [N5: nat] : L = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),N5) ) ).

% le_Suc_ex
tff(fact_2690_add__le__mono,axiom,
    ! [I2: nat,J: nat,K: nat,L: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),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,I2),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L)) ) ) ).

% add_le_mono
tff(fact_2691_add__le__mono1,axiom,
    ! [I2: nat,J: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),J)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K)) ) ).

% add_le_mono1
tff(fact_2692_trans__le__add1,axiom,
    ! [I2: nat,J: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),J)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),M)) ) ).

% trans_le_add1
tff(fact_2693_trans__le__add2,axiom,
    ! [I2: nat,J: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),J)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),J)) ) ).

% trans_le_add2
tff(fact_2694_nat__le__iff__add,axiom,
    ! [M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
    <=> ? [K3: nat] : N2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K3) ) ).

% nat_le_iff_add
tff(fact_2695_curryE,axiom,
    ! [F: fun_Pr2620521676128474833_nat_o,A: fun_Pr6460944699146269935_nat_o,B: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(aa_fun2638059832795814268_nat_o(produc4701769847872139056_nat_o(F),A),B)
     => aa_Pro5643009357358181178_nat_o(F,aa_Pro434340453207536480et_nat(aa_fun3926903090414666066et_nat(produc6017135503889078682et_nat,A),B)) ) ).

% curryE
tff(fact_2696_curryE,axiom,
    ! [F: fun_Pr1227177680593602867_nat_o,A: fun_Pr6460944699146269935_nat_o,B: produc8799533983703829792et_nat] :
      ( aa_Pro5643009357358181178_nat_o(aa_fun912680268133416158_nat_o(produc3333590387107866898_nat_o(F),A),B)
     => aa_Pro2671238023309943196_nat_o(F,aa_Pro1979504083478021276et_nat(aa_fun615918787490101902et_nat(produc7291855603632349880et_nat,A),B)) ) ).

% curryE
tff(fact_2697_curryE,axiom,
    ! [F: fun_Pr3540825087398332590_int_o,A: fun_Pr7763680713840462443e_term,B: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aa_fun3538162476766722827_int_o(produc443495763101479893_int_o(F),A),B)
     => aa_Pro2601676399929874711_int_o(F,aa_Pro5748361610964203648nt_int(aa_fun3265472288063244902nt_int(produc7320969046608631221nt_int,A),B)) ) ).

% curryE
tff(fact_2698_curryE,axiom,
    ! [F: fun_Pr1196100989263358594_int_o,A: fun_in5730379987311621695e_term,B: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aa_fun586359091965693151_int_o(produc2168430993391648169_int_o(F),A),B)
     => aa_Pro5451197003417746155_int_o(F,aa_Pro8800328608558162860nt_int(aa_fun260840925927807678nt_int(produc4729913460834754785nt_int,A),B)) ) ).

% curryE
tff(fact_2699_curryE,axiom,
    ! [F: fun_Pr5746006151868627512_nat_o,A: a,B: produc6653097349344004940it_nat] :
      ( aa_Pro1202363709425262478_nat_o(aa_a_f337466037417904483_nat_o(produc3709033845735585303_nat_o(F),A),B)
     => aa_Pro6426884464393220769_nat_o(F,aa_Pro8078911840264235907it_nat(aa_a_f2234720956197727492it_nat(produc9178034014595674355it_nat,A),B)) ) ).

% curryE
tff(fact_2700_curryD,axiom,
    ! [F: fun_Pr2620521676128474833_nat_o,A: fun_Pr6460944699146269935_nat_o,B: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(aa_fun2638059832795814268_nat_o(produc4701769847872139056_nat_o(F),A),B)
     => aa_Pro5643009357358181178_nat_o(F,aa_Pro434340453207536480et_nat(aa_fun3926903090414666066et_nat(produc6017135503889078682et_nat,A),B)) ) ).

% curryD
tff(fact_2701_curryD,axiom,
    ! [F: fun_Pr1227177680593602867_nat_o,A: fun_Pr6460944699146269935_nat_o,B: produc8799533983703829792et_nat] :
      ( aa_Pro5643009357358181178_nat_o(aa_fun912680268133416158_nat_o(produc3333590387107866898_nat_o(F),A),B)
     => aa_Pro2671238023309943196_nat_o(F,aa_Pro1979504083478021276et_nat(aa_fun615918787490101902et_nat(produc7291855603632349880et_nat,A),B)) ) ).

% curryD
tff(fact_2702_curryD,axiom,
    ! [F: fun_Pr3540825087398332590_int_o,A: fun_Pr7763680713840462443e_term,B: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aa_fun3538162476766722827_int_o(produc443495763101479893_int_o(F),A),B)
     => aa_Pro2601676399929874711_int_o(F,aa_Pro5748361610964203648nt_int(aa_fun3265472288063244902nt_int(produc7320969046608631221nt_int,A),B)) ) ).

% curryD
tff(fact_2703_curryD,axiom,
    ! [F: fun_Pr1196100989263358594_int_o,A: fun_in5730379987311621695e_term,B: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aa_fun586359091965693151_int_o(produc2168430993391648169_int_o(F),A),B)
     => aa_Pro5451197003417746155_int_o(F,aa_Pro8800328608558162860nt_int(aa_fun260840925927807678nt_int(produc4729913460834754785nt_int,A),B)) ) ).

% curryD
tff(fact_2704_curryD,axiom,
    ! [F: fun_Pr5746006151868627512_nat_o,A: a,B: produc6653097349344004940it_nat] :
      ( aa_Pro1202363709425262478_nat_o(aa_a_f337466037417904483_nat_o(produc3709033845735585303_nat_o(F),A),B)
     => aa_Pro6426884464393220769_nat_o(F,aa_Pro8078911840264235907it_nat(aa_a_f2234720956197727492it_nat(produc9178034014595674355it_nat,A),B)) ) ).

% curryD
tff(fact_2705_timeFrame_Osimps_I1_J,axiom,
    ! [N2: nat,Ra2: a,Ha: heap_e7401611519738050253t_unit,N6: nat] : heap_T7616092557645711335rame_a(N2,aa_Pro5302094286594872192it_nat(some_P7913643980934408916it_nat,aa_Pro8078911840264235907it_nat(aa_a_f2234720956197727492it_nat(produc9178034014595674355it_nat,Ra2),produc584006145561248582it_nat(Ha,N6)))) = aa_Pro5302094286594872192it_nat(some_P7913643980934408916it_nat,aa_Pro8078911840264235907it_nat(aa_a_f2234720956197727492it_nat(produc9178034014595674355it_nat,Ra2),produc584006145561248582it_nat(Ha,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),N6)))) ).

% timeFrame.simps(1)
tff(fact_2706_timeFrame_Osimps_I1_J,axiom,
    ! [N2: nat,Ra2: product_unit,Ha: heap_e7401611519738050253t_unit,N6: nat] : heap_T3616969660504097270t_unit(N2,aa_Pro8996696869555642526it_nat(some_P1914260805536162275it_nat,produc7133225469290080770it_nat(Ra2,produc584006145561248582it_nat(Ha,N6)))) = aa_Pro8996696869555642526it_nat(some_P1914260805536162275it_nat,produc7133225469290080770it_nat(Ra2,produc584006145561248582it_nat(Ha,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),N6)))) ).

% timeFrame.simps(1)
tff(fact_2707_timeFrame_Oelims,axiom,
    ! [X: nat,Xa2: option3562590408128118217it_nat,Y: option3562590408128118217it_nat] :
      ( ( heap_T7616092557645711335rame_a(X,Xa2) = Y )
     => ( ! [R: a,H: heap_e7401611519738050253t_unit,N4: nat] :
            ( ( Xa2 = aa_Pro5302094286594872192it_nat(some_P7913643980934408916it_nat,aa_Pro8078911840264235907it_nat(aa_a_f2234720956197727492it_nat(produc9178034014595674355it_nat,R),produc584006145561248582it_nat(H,N4))) )
           => ( Y != aa_Pro5302094286594872192it_nat(some_P7913643980934408916it_nat,aa_Pro8078911840264235907it_nat(aa_a_f2234720956197727492it_nat(produc9178034014595674355it_nat,R),produc584006145561248582it_nat(H,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),N4)))) ) )
       => ~ ( ( Xa2 = none_P2651198173097904984it_nat )
           => ( Y != none_P2651198173097904984it_nat ) ) ) ) ).

% timeFrame.elims
tff(fact_2708_timeFrame_Oelims,axiom,
    ! [X: nat,Xa2: option8956607266484857688it_nat,Y: option8956607266484857688it_nat] :
      ( ( heap_T3616969660504097270t_unit(X,Xa2) = Y )
     => ( ! [R: product_unit,H: heap_e7401611519738050253t_unit,N4: nat] :
            ( ( Xa2 = aa_Pro8996696869555642526it_nat(some_P1914260805536162275it_nat,produc7133225469290080770it_nat(R,produc584006145561248582it_nat(H,N4))) )
           => ( Y != aa_Pro8996696869555642526it_nat(some_P1914260805536162275it_nat,produc7133225469290080770it_nat(R,produc584006145561248582it_nat(H,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),N4)))) ) )
       => ~ ( ( Xa2 = none_P9117596204409417319it_nat )
           => ( Y != none_P9117596204409417319it_nat ) ) ) ) ).

% timeFrame.elims
tff(fact_2709_mono__nat__linear__lb,axiom,
    ! [F: fun_nat_nat,M: nat,K: nat] :
      ( ! [M5: nat,N5: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M5),N5)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(F,M5)),aa_nat_nat(F,N5)) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(F,M)),K)),aa_nat_nat(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K))) ) ).

% mono_nat_linear_lb
tff(fact_2710_Eps__Opt__eq__Some,axiom,
    ! [Pa: fun_Pr5746006151868627512_nat_o,X: produc3260487557148687353it_nat] :
      ( ! [X7: produc3260487557148687353it_nat] :
          ( aa_Pro6426884464393220769_nat_o(Pa,X)
         => ( aa_Pro6426884464393220769_nat_o(Pa,X7)
           => ( X7 = X ) ) )
     => ( ( eps_Op1741419982095519837it_nat(Pa) = aa_Pro5302094286594872192it_nat(some_P7913643980934408916it_nat,X) )
      <=> aa_Pro6426884464393220769_nat_o(Pa,X) ) ) ).

% Eps_Opt_eq_Some
tff(fact_2711_Eps__Opt__eq__Some,axiom,
    ! [Pa: fun_num_o,X: num] :
      ( ! [X7: num] :
          ( aa_num_o(Pa,X)
         => ( aa_num_o(Pa,X7)
           => ( X7 = X ) ) )
     => ( ( eps_Opt_num(Pa) = aa_num_option_num(some_num,X) )
      <=> aa_num_o(Pa,X) ) ) ).

% Eps_Opt_eq_Some
tff(fact_2712_Eps__Opt__eq__Some,axiom,
    ! [Pa: fun_Pr8032424337255632745_nat_o,X: produc8664842809031399944it_nat] :
      ( ! [X7: produc8664842809031399944it_nat] :
          ( aa_Pro4744234731172327378_nat_o(Pa,X)
         => ( aa_Pro4744234731172327378_nat_o(Pa,X7)
           => ( X7 = X ) ) )
     => ( ( eps_Op3393321821070424684it_nat(Pa) = aa_Pro8996696869555642526it_nat(some_P1914260805536162275it_nat,X) )
      <=> aa_Pro4744234731172327378_nat_o(Pa,X) ) ) ).

% Eps_Opt_eq_Some
tff(fact_2713_Eps__Opt__eq__Some__implies,axiom,
    ! [Pa: fun_Pr5746006151868627512_nat_o,X: produc3260487557148687353it_nat] :
      ( ( eps_Op1741419982095519837it_nat(Pa) = aa_Pro5302094286594872192it_nat(some_P7913643980934408916it_nat,X) )
     => aa_Pro6426884464393220769_nat_o(Pa,X) ) ).

% Eps_Opt_eq_Some_implies
tff(fact_2714_Eps__Opt__eq__Some__implies,axiom,
    ! [Pa: fun_num_o,X: num] :
      ( ( eps_Opt_num(Pa) = aa_num_option_num(some_num,X) )
     => aa_num_o(Pa,X) ) ).

% Eps_Opt_eq_Some_implies
tff(fact_2715_Eps__Opt__eq__Some__implies,axiom,
    ! [Pa: fun_Pr8032424337255632745_nat_o,X: produc8664842809031399944it_nat] :
      ( ( eps_Op3393321821070424684it_nat(Pa) = aa_Pro8996696869555642526it_nat(some_P1914260805536162275it_nat,X) )
     => aa_Pro4744234731172327378_nat_o(Pa,X) ) ).

% Eps_Opt_eq_Some_implies
tff(fact_2716_execute__guard_I1_J,axiom,
    ! [Pa: fun_he8413527710729349838unit_o,Ha: heap_e7401611519738050253t_unit,F: fun_he8889679731534893401it_nat] :
      ( ~ aa_hea7994211679605476167unit_o(Pa,Ha)
     => ( aa_hea4724376117875161946it_nat(heap_Time_execute_a(heap_Time_guard_a(Pa,F)),Ha) = none_P2651198173097904984it_nat ) ) ).

% execute_guard(1)
tff(fact_2717_mlex__less,axiom,
    ! [F: fun_nat_nat,X: nat,Y: nat,Ra: set_Pr1261947904930325089at_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(F,X)),aa_nat_nat(F,Y))
     => aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,X),Y)),mlex_prod_nat(F,Ra)) ) ).

% mlex_less
tff(fact_2718_mlex__iff,axiom,
    ! [X: nat,Y: nat,F: fun_nat_nat,Ra: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,X),Y)),mlex_prod_nat(F,Ra))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(F,X)),aa_nat_nat(F,Y))
        | ( ( aa_nat_nat(F,X) = aa_nat_nat(F,Y) )
          & aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,X),Y)),Ra) ) ) ) ).

% mlex_iff
tff(fact_2719_timeFrame_Opelims,axiom,
    ! [X: nat,Xa2: option3562590408128118217it_nat,Y: option3562590408128118217it_nat] :
      ( ( heap_T7616092557645711335rame_a(X,Xa2) = Y )
     => ( accp_P6263092265436569219it_nat(heap_T8132184524487034138_rel_a,produc4111269673004989362it_nat(X,Xa2))
       => ( ! [R: a,H: heap_e7401611519738050253t_unit,N4: nat] :
              ( ( Xa2 = aa_Pro5302094286594872192it_nat(some_P7913643980934408916it_nat,aa_Pro8078911840264235907it_nat(aa_a_f2234720956197727492it_nat(produc9178034014595674355it_nat,R),produc584006145561248582it_nat(H,N4))) )
             => ( ( Y = aa_Pro5302094286594872192it_nat(some_P7913643980934408916it_nat,aa_Pro8078911840264235907it_nat(aa_a_f2234720956197727492it_nat(produc9178034014595674355it_nat,R),produc584006145561248582it_nat(H,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),N4)))) )
               => ~ accp_P6263092265436569219it_nat(heap_T8132184524487034138_rel_a,produc4111269673004989362it_nat(X,aa_Pro5302094286594872192it_nat(some_P7913643980934408916it_nat,aa_Pro8078911840264235907it_nat(aa_a_f2234720956197727492it_nat(produc9178034014595674355it_nat,R),produc584006145561248582it_nat(H,N4))))) ) )
         => ~ ( ( Xa2 = none_P2651198173097904984it_nat )
             => ( ( Y = none_P2651198173097904984it_nat )
               => ~ accp_P6263092265436569219it_nat(heap_T8132184524487034138_rel_a,produc4111269673004989362it_nat(X,none_P2651198173097904984it_nat)) ) ) ) ) ) ).

% timeFrame.pelims
tff(fact_2720_timeFrame_Opelims,axiom,
    ! [X: nat,Xa2: option8956607266484857688it_nat,Y: option8956607266484857688it_nat] :
      ( ( heap_T3616969660504097270t_unit(X,Xa2) = Y )
     => ( accp_P414730952086964626it_nat(heap_T996182799752388649t_unit,produc638857205735767105it_nat(X,Xa2))
       => ( ! [R: product_unit,H: heap_e7401611519738050253t_unit,N4: nat] :
              ( ( Xa2 = aa_Pro8996696869555642526it_nat(some_P1914260805536162275it_nat,produc7133225469290080770it_nat(R,produc584006145561248582it_nat(H,N4))) )
             => ( ( Y = aa_Pro8996696869555642526it_nat(some_P1914260805536162275it_nat,produc7133225469290080770it_nat(R,produc584006145561248582it_nat(H,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),N4)))) )
               => ~ accp_P414730952086964626it_nat(heap_T996182799752388649t_unit,produc638857205735767105it_nat(X,aa_Pro8996696869555642526it_nat(some_P1914260805536162275it_nat,produc7133225469290080770it_nat(R,produc584006145561248582it_nat(H,N4))))) ) )
         => ~ ( ( Xa2 = none_P9117596204409417319it_nat )
             => ( ( Y = none_P9117596204409417319it_nat )
               => ~ accp_P414730952086964626it_nat(heap_T996182799752388649t_unit,produc638857205735767105it_nat(X,none_P9117596204409417319it_nat)) ) ) ) ) ) ).

% timeFrame.pelims
tff(fact_2721_add__less__cancel__left,axiom,
    ! [Ca: 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,Ca),A)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Ca),B))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ).

% add_less_cancel_left
tff(fact_2722_add__less__cancel__left,axiom,
    ! [Ca: 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,Ca),A)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ca),B))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B) ) ).

% add_less_cancel_left
tff(fact_2723_add__less__cancel__left,axiom,
    ! [Ca: 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,Ca),A)),aa_int_int(aa_int_fun_int_int(plus_plus_int,Ca),B))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ).

% add_less_cancel_left
tff(fact_2724_add__less__cancel__right,axiom,
    ! [A: rat,Ca: 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),Ca)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),Ca))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ).

% add_less_cancel_right
tff(fact_2725_add__less__cancel__right,axiom,
    ! [A: nat,Ca: 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),Ca)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),Ca))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B) ) ).

% add_less_cancel_right
tff(fact_2726_add__less__cancel__right,axiom,
    ! [A: int,Ca: 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),Ca)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),Ca))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ).

% add_less_cancel_right
tff(fact_2727_add__le__cancel__left,axiom,
    ! [Ca: 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,Ca),A)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Ca),B))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ).

% add_le_cancel_left
tff(fact_2728_add__le__cancel__left,axiom,
    ! [Ca: 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,Ca),A)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ca),B))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B) ) ).

% add_le_cancel_left
tff(fact_2729_add__le__cancel__left,axiom,
    ! [Ca: 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,Ca),A)),aa_int_int(aa_int_fun_int_int(plus_plus_int,Ca),B))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ).

% add_le_cancel_left
tff(fact_2730_add__le__cancel__right,axiom,
    ! [A: rat,Ca: 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),Ca)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),Ca))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ).

% add_le_cancel_right
tff(fact_2731_add__le__cancel__right,axiom,
    ! [A: nat,Ca: 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),Ca)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),Ca))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B) ) ).

% add_le_cancel_right
tff(fact_2732_add__le__cancel__right,axiom,
    ! [A: int,Ca: 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),Ca)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),Ca))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ).

% add_le_cancel_right
tff(fact_2733_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I2: rat,J: rat,K: rat,L: rat] :
      ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,I2),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,I2),K)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,J),L)) ) ).

% add_mono_thms_linordered_field(4)
tff(fact_2734_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I2: nat,J: nat,K: nat,L: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),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,I2),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L)) ) ).

% add_mono_thms_linordered_field(4)
tff(fact_2735_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I2: int,J: int,K: int,L: int] :
      ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I2),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,I2),K)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L)) ) ).

% add_mono_thms_linordered_field(4)
tff(fact_2736_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I2: rat,J: rat,K: rat,L: rat] :
      ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,I2),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,I2),K)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,J),L)) ) ).

% add_mono_thms_linordered_field(3)
tff(fact_2737_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I2: nat,J: nat,K: nat,L: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),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,I2),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L)) ) ).

% add_mono_thms_linordered_field(3)
tff(fact_2738_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I2: int,J: int,K: int,L: int] :
      ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,I2),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,I2),K)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L)) ) ).

% add_mono_thms_linordered_field(3)
tff(fact_2739_add__le__less__mono,axiom,
    ! [A: rat,B: rat,Ca: 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,Ca),D2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),Ca)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),D2)) ) ) ).

% add_le_less_mono
tff(fact_2740_add__le__less__mono,axiom,
    ! [A: nat,B: nat,Ca: 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,Ca),D2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),Ca)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),D2)) ) ) ).

% add_le_less_mono
tff(fact_2741_add__le__less__mono,axiom,
    ! [A: int,B: int,Ca: 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,Ca),D2)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),Ca)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),D2)) ) ) ).

% add_le_less_mono
tff(fact_2742_add__less__le__mono,axiom,
    ! [A: rat,B: rat,Ca: 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,Ca),D2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),Ca)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),D2)) ) ) ).

% add_less_le_mono
tff(fact_2743_add__less__le__mono,axiom,
    ! [A: nat,B: nat,Ca: 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,Ca),D2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),Ca)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),D2)) ) ) ).

% add_less_le_mono
tff(fact_2744_add__less__le__mono,axiom,
    ! [A: int,B: int,Ca: 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,Ca),D2)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),Ca)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),D2)) ) ) ).

% add_less_le_mono
tff(fact_2745_crossproduct__noteq,axiom,
    ! [A: code_integer,B: code_integer,Ca: code_integer,D2: code_integer] :
      ( ( ( A != B )
        & ( Ca != D2 ) )
    <=> ( aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),Ca)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),D2)) != aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),D2)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),Ca)) ) ) ).

% crossproduct_noteq
tff(fact_2746_crossproduct__noteq,axiom,
    ! [A: rat,B: rat,Ca: rat,D2: rat] :
      ( ( ( A != B )
        & ( Ca != D2 ) )
    <=> ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),Ca)),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),Ca)) ) ) ).

% crossproduct_noteq
tff(fact_2747_crossproduct__noteq,axiom,
    ! [A: nat,B: nat,Ca: nat,D2: nat] :
      ( ( ( A != B )
        & ( Ca != D2 ) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),Ca)),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),Ca)) ) ) ).

% crossproduct_noteq
tff(fact_2748_crossproduct__noteq,axiom,
    ! [A: int,B: int,Ca: int,D2: int] :
      ( ( ( A != B )
        & ( Ca != D2 ) )
    <=> ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),Ca)),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),Ca)) ) ) ).

% crossproduct_noteq
tff(fact_2749_mult__le__mono2,axiom,
    ! [I2: nat,J: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),J)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),I2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),J)) ) ).

% mult_le_mono2
tff(fact_2750_mult__le__mono1,axiom,
    ! [I2: nat,J: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),J)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I2),K)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),K)) ) ).

% mult_le_mono1
tff(fact_2751_mult__le__mono,axiom,
    ! [I2: nat,J: nat,K: nat,L: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),J)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),L)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I2),K)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),L)) ) ) ).

% mult_le_mono
tff(fact_2752_le__square,axiom,
    ! [M: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),M)) ).

% le_square
tff(fact_2753_le__cube,axiom,
    ! [M: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),M))) ).

% le_cube
tff(fact_2754_add__mult__distrib,axiom,
    ! [M: nat,N2: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N2)),K) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),K)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N2),K)) ).

% add_mult_distrib
tff(fact_2755_add__mult__distrib2,axiom,
    ! [K: nat,M: nat,N2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),N2)) ).

% add_mult_distrib2
tff(fact_2756_mlex__bound,axiom,
    ! [A: nat,A3: nat,B: nat,N7: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),A3)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),N7)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),N7)),B)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A3),N7)) ) ) ).

% mlex_bound
tff(fact_2757_mlex__fst__decrI,axiom,
    ! [A: nat,A2: nat,B: nat,N7: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),A2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),N7)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B2),N7)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),N7)),B)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),N7)),B2)) ) ) ) ).

% mlex_fst_decrI
tff(fact_2758_mlex__snd__decrI,axiom,
    ! [A: nat,A2: nat,B: nat,B2: nat,N7: nat] :
      ( ( A = A2 )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),B2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),N7)),B)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),N7)),B2)) ) ) ).

% mlex_snd_decrI
tff(fact_2759_mlex__leI,axiom,
    ! [A: nat,A2: nat,B: nat,B2: nat,N7: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),A2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),B2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),N7)),B)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),N7)),B2)) ) ) ).

% mlex_leI
tff(fact_2760_mult_Oassoc,axiom,
    ! [A: code_integer,B: code_integer,Ca: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),Ca) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),Ca)) ).

% mult.assoc
tff(fact_2761_mult_Oassoc,axiom,
    ! [A: assn,B: assn,Ca: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A),B)),Ca) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,B),Ca)) ).

% mult.assoc
tff(fact_2762_mult_Oassoc,axiom,
    ! [A: rat,B: rat,Ca: 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)),Ca) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),Ca)) ).

% mult.assoc
tff(fact_2763_mult_Oassoc,axiom,
    ! [A: nat,B: nat,Ca: 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)),Ca) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),Ca)) ).

% mult.assoc
tff(fact_2764_mult_Oassoc,axiom,
    ! [A: int,B: int,Ca: 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)),Ca) = aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),Ca)) ).

% mult.assoc
tff(fact_2765_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_2766_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_2767_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_2768_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_2769_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_2770_ab__semigroup__mult__class_Omult_Oleft__commute,axiom,
    ! [B: code_integer,A: code_integer,Ca: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),Ca)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),Ca)) ).

% ab_semigroup_mult_class.mult.left_commute
tff(fact_2771_ab__semigroup__mult__class_Omult_Oleft__commute,axiom,
    ! [B: assn,A: assn,Ca: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,B),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A),Ca)) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,B),Ca)) ).

% ab_semigroup_mult_class.mult.left_commute
tff(fact_2772_ab__semigroup__mult__class_Omult_Oleft__commute,axiom,
    ! [B: rat,A: rat,Ca: 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),Ca)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),Ca)) ).

% ab_semigroup_mult_class.mult.left_commute
tff(fact_2773_ab__semigroup__mult__class_Omult_Oleft__commute,axiom,
    ! [B: nat,A: nat,Ca: 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),Ca)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),Ca)) ).

% ab_semigroup_mult_class.mult.left_commute
tff(fact_2774_ab__semigroup__mult__class_Omult_Oleft__commute,axiom,
    ! [B: int,A: int,Ca: 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),Ca)) = aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),Ca)) ).

% ab_semigroup_mult_class.mult.left_commute
tff(fact_2775_add__le__imp__le__right,axiom,
    ! [A: rat,Ca: 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),Ca)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),Ca))
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ).

% add_le_imp_le_right
tff(fact_2776_add__le__imp__le__right,axiom,
    ! [A: nat,Ca: 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),Ca)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),Ca))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B) ) ).

% add_le_imp_le_right
tff(fact_2777_add__le__imp__le__right,axiom,
    ! [A: int,Ca: 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),Ca)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),Ca))
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ).

% add_le_imp_le_right
tff(fact_2778_add__le__imp__le__left,axiom,
    ! [Ca: 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,Ca),A)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Ca),B))
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ).

% add_le_imp_le_left
tff(fact_2779_add__le__imp__le__left,axiom,
    ! [Ca: 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,Ca),A)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ca),B))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B) ) ).

% add_le_imp_le_left
tff(fact_2780_add__le__imp__le__left,axiom,
    ! [Ca: 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,Ca),A)),aa_int_int(aa_int_fun_int_int(plus_plus_int,Ca),B))
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ).

% add_le_imp_le_left
tff(fact_2781_le__iff__add,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
    <=> ? [C3: nat] : B = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C3) ) ).

% le_iff_add
tff(fact_2782_add__right__mono,axiom,
    ! [A: rat,B: rat,Ca: 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),Ca)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),Ca)) ) ).

% add_right_mono
tff(fact_2783_add__right__mono,axiom,
    ! [A: nat,B: nat,Ca: 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),Ca)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),Ca)) ) ).

% add_right_mono
tff(fact_2784_add__right__mono,axiom,
    ! [A: int,B: int,Ca: 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),Ca)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),Ca)) ) ).

% add_right_mono
tff(fact_2785_less__eqE,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ~ ! [C: nat] : B != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C) ) ).

% less_eqE
tff(fact_2786_add__left__mono,axiom,
    ! [A: rat,B: rat,Ca: 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,Ca),A)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Ca),B)) ) ).

% add_left_mono
tff(fact_2787_add__left__mono,axiom,
    ! [A: nat,B: nat,Ca: 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,Ca),A)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ca),B)) ) ).

% add_left_mono
tff(fact_2788_add__left__mono,axiom,
    ! [A: int,B: int,Ca: 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,Ca),A)),aa_int_int(aa_int_fun_int_int(plus_plus_int,Ca),B)) ) ).

% add_left_mono
tff(fact_2789_add__mono,axiom,
    ! [A: rat,B: rat,Ca: 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,Ca),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),Ca)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),D2)) ) ) ).

% add_mono
tff(fact_2790_add__mono,axiom,
    ! [A: nat,B: nat,Ca: 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,Ca),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),Ca)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),D2)) ) ) ).

% add_mono
tff(fact_2791_add__mono,axiom,
    ! [A: int,B: int,Ca: 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,Ca),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),Ca)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),D2)) ) ) ).

% add_mono
tff(fact_2792_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I2: rat,J: rat,K: rat,L: rat] :
      ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,I2),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,I2),K)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,J),L)) ) ).

% add_mono_thms_linordered_semiring(1)
tff(fact_2793_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I2: nat,J: nat,K: nat,L: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),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,I2),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L)) ) ).

% add_mono_thms_linordered_semiring(1)
tff(fact_2794_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I2: int,J: int,K: int,L: int] :
      ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I2),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,I2),K)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L)) ) ).

% add_mono_thms_linordered_semiring(1)
tff(fact_2795_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I2: rat,J: rat,K: rat,L: rat] :
      ( ( ( I2 = 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,I2),K)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,J),L)) ) ).

% add_mono_thms_linordered_semiring(2)
tff(fact_2796_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I2: nat,J: nat,K: nat,L: nat] :
      ( ( ( I2 = 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,I2),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L)) ) ).

% add_mono_thms_linordered_semiring(2)
tff(fact_2797_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I2: int,J: int,K: int,L: int] :
      ( ( ( I2 = 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,I2),K)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L)) ) ).

% add_mono_thms_linordered_semiring(2)
tff(fact_2798_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I2: rat,J: rat,K: rat,L: rat] :
      ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,I2),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,I2),K)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,J),L)) ) ).

% add_mono_thms_linordered_semiring(3)
tff(fact_2799_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I2: nat,J: nat,K: nat,L: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),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,I2),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L)) ) ).

% add_mono_thms_linordered_semiring(3)
tff(fact_2800_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I2: int,J: int,K: int,L: int] :
      ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I2),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,I2),K)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L)) ) ).

% add_mono_thms_linordered_semiring(3)
tff(fact_2801_add__less__imp__less__right,axiom,
    ! [A: rat,Ca: 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),Ca)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),Ca))
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ).

% add_less_imp_less_right
tff(fact_2802_add__less__imp__less__right,axiom,
    ! [A: nat,Ca: 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),Ca)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),Ca))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B) ) ).

% add_less_imp_less_right
tff(fact_2803_add__less__imp__less__right,axiom,
    ! [A: int,Ca: 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),Ca)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),Ca))
     => aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ).

% add_less_imp_less_right
tff(fact_2804_add__less__imp__less__left,axiom,
    ! [Ca: 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,Ca),A)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Ca),B))
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ).

% add_less_imp_less_left
tff(fact_2805_add__less__imp__less__left,axiom,
    ! [Ca: 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,Ca),A)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ca),B))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B) ) ).

% add_less_imp_less_left
tff(fact_2806_add__less__imp__less__left,axiom,
    ! [Ca: 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,Ca),A)),aa_int_int(aa_int_fun_int_int(plus_plus_int,Ca),B))
     => aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ).

% add_less_imp_less_left
tff(fact_2807_add__strict__right__mono,axiom,
    ! [A: rat,B: rat,Ca: 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),Ca)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),Ca)) ) ).

% add_strict_right_mono
tff(fact_2808_add__strict__right__mono,axiom,
    ! [A: nat,B: nat,Ca: 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),Ca)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),Ca)) ) ).

% add_strict_right_mono
tff(fact_2809_add__strict__right__mono,axiom,
    ! [A: int,B: int,Ca: 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),Ca)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),Ca)) ) ).

% add_strict_right_mono
tff(fact_2810_add__strict__left__mono,axiom,
    ! [A: rat,B: rat,Ca: 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,Ca),A)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Ca),B)) ) ).

% add_strict_left_mono
tff(fact_2811_add__strict__left__mono,axiom,
    ! [A: nat,B: nat,Ca: 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,Ca),A)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ca),B)) ) ).

% add_strict_left_mono
tff(fact_2812_add__strict__left__mono,axiom,
    ! [A: int,B: int,Ca: 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,Ca),A)),aa_int_int(aa_int_fun_int_int(plus_plus_int,Ca),B)) ) ).

% add_strict_left_mono
tff(fact_2813_add__strict__mono,axiom,
    ! [A: rat,B: rat,Ca: 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,Ca),D2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),Ca)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),D2)) ) ) ).

% add_strict_mono
tff(fact_2814_add__strict__mono,axiom,
    ! [A: nat,B: nat,Ca: 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,Ca),D2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),Ca)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),D2)) ) ) ).

% add_strict_mono
tff(fact_2815_add__strict__mono,axiom,
    ! [A: int,B: int,Ca: 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,Ca),D2)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),Ca)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),D2)) ) ) ).

% add_strict_mono
tff(fact_2816_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I2: rat,J: rat,K: rat,L: rat] :
      ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,I2),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,I2),K)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,J),L)) ) ).

% add_mono_thms_linordered_field(1)
tff(fact_2817_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I2: nat,J: nat,K: nat,L: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),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,I2),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L)) ) ).

% add_mono_thms_linordered_field(1)
tff(fact_2818_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I2: int,J: int,K: int,L: int] :
      ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,I2),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,I2),K)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L)) ) ).

% add_mono_thms_linordered_field(1)
tff(fact_2819_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I2: rat,J: rat,K: rat,L: rat] :
      ( ( ( I2 = 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,I2),K)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,J),L)) ) ).

% add_mono_thms_linordered_field(2)
tff(fact_2820_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I2: nat,J: nat,K: nat,L: nat] :
      ( ( ( I2 = 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,I2),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L)) ) ).

% add_mono_thms_linordered_field(2)
tff(fact_2821_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I2: int,J: int,K: int,L: int] :
      ( ( ( I2 = 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,I2),K)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L)) ) ).

% add_mono_thms_linordered_field(2)
tff(fact_2822_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I2: rat,J: rat,K: rat,L: rat] :
      ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,I2),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,I2),K)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,J),L)) ) ).

% add_mono_thms_linordered_field(5)
tff(fact_2823_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I2: nat,J: nat,K: nat,L: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),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,I2),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L)) ) ).

% add_mono_thms_linordered_field(5)
tff(fact_2824_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I2: int,J: int,K: int,L: int] :
      ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,I2),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,I2),K)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L)) ) ).

% add_mono_thms_linordered_field(5)
tff(fact_2825_crossproduct__eq,axiom,
    ! [W2: code_integer,Y: code_integer,X: code_integer,Z: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,W2),Y)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,X),Z)) = aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,W2),Z)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,X),Y)) )
    <=> ( ( W2 = X )
        | ( Y = Z ) ) ) ).

% crossproduct_eq
tff(fact_2826_crossproduct__eq,axiom,
    ! [W2: 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,W2),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,W2),Z)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X),Y)) )
    <=> ( ( W2 = X )
        | ( Y = Z ) ) ) ).

% crossproduct_eq
tff(fact_2827_crossproduct__eq,axiom,
    ! [W2: 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,W2),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,W2),Z)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,X),Y)) )
    <=> ( ( W2 = X )
        | ( Y = Z ) ) ) ).

% crossproduct_eq
tff(fact_2828_crossproduct__eq,axiom,
    ! [W2: 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,W2),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,W2),Z)),aa_int_int(aa_int_fun_int_int(times_times_int,X),Y)) )
    <=> ( ( W2 = X )
        | ( Y = Z ) ) ) ).

% crossproduct_eq
tff(fact_2829_ring__class_Oring__distribs_I2_J,axiom,
    ! [A: code_integer,B: code_integer,Ca: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(A),B)),Ca) = aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),Ca)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),Ca)) ).

% ring_class.ring_distribs(2)
tff(fact_2830_ring__class_Oring__distribs_I2_J,axiom,
    ! [A: rat,B: rat,Ca: 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)),Ca) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),Ca)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),Ca)) ).

% ring_class.ring_distribs(2)
tff(fact_2831_ring__class_Oring__distribs_I2_J,axiom,
    ! [A: int,B: int,Ca: 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)),Ca) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),Ca)),aa_int_int(aa_int_fun_int_int(times_times_int,B),Ca)) ).

% ring_class.ring_distribs(2)
tff(fact_2832_ring__class_Oring__distribs_I1_J,axiom,
    ! [A: code_integer,B: code_integer,Ca: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(B),Ca)) = aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),Ca)) ).

% ring_class.ring_distribs(1)
tff(fact_2833_ring__class_Oring__distribs_I1_J,axiom,
    ! [A: rat,B: rat,Ca: 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),Ca)) = 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),Ca)) ).

% ring_class.ring_distribs(1)
tff(fact_2834_ring__class_Oring__distribs_I1_J,axiom,
    ! [A: int,B: int,Ca: 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),Ca)) = 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),Ca)) ).

% ring_class.ring_distribs(1)
tff(fact_2835_comm__semiring__class_Odistrib,axiom,
    ! [A: code_integer,B: code_integer,Ca: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(A),B)),Ca) = aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),Ca)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),Ca)) ).

% comm_semiring_class.distrib
tff(fact_2836_comm__semiring__class_Odistrib,axiom,
    ! [A: rat,B: rat,Ca: 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)),Ca) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),Ca)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),Ca)) ).

% comm_semiring_class.distrib
tff(fact_2837_comm__semiring__class_Odistrib,axiom,
    ! [A: nat,B: nat,Ca: 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)),Ca) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),Ca)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),Ca)) ).

% comm_semiring_class.distrib
tff(fact_2838_comm__semiring__class_Odistrib,axiom,
    ! [A: int,B: int,Ca: 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)),Ca) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),Ca)),aa_int_int(aa_int_fun_int_int(times_times_int,B),Ca)) ).

% comm_semiring_class.distrib
tff(fact_2839_distrib__left,axiom,
    ! [A: code_integer,B: code_integer,Ca: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(B),Ca)) = aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),Ca)) ).

% distrib_left
tff(fact_2840_distrib__left,axiom,
    ! [A: rat,B: rat,Ca: 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),Ca)) = 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),Ca)) ).

% distrib_left
tff(fact_2841_distrib__left,axiom,
    ! [A: nat,B: nat,Ca: 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),Ca)) = 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),Ca)) ).

% distrib_left
tff(fact_2842_distrib__left,axiom,
    ! [A: int,B: int,Ca: 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),Ca)) = 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),Ca)) ).

% distrib_left
tff(fact_2843_distrib__right,axiom,
    ! [A: code_integer,B: code_integer,Ca: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(A),B)),Ca) = aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),Ca)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),Ca)) ).

% distrib_right
tff(fact_2844_distrib__right,axiom,
    ! [A: rat,B: rat,Ca: 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)),Ca) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),Ca)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),Ca)) ).

% distrib_right
tff(fact_2845_distrib__right,axiom,
    ! [A: nat,B: nat,Ca: 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)),Ca) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),Ca)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),Ca)) ).

% distrib_right
tff(fact_2846_distrib__right,axiom,
    ! [A: int,B: int,Ca: 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)),Ca) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),Ca)),aa_int_int(aa_int_fun_int_int(times_times_int,B),Ca)) ).

% distrib_right
tff(fact_2847_combine__common__factor,axiom,
    ! [A: code_integer,E: code_integer,B: code_integer,Ca: code_integer] : aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),E)),aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),E)),Ca)) = aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(A),B)),E)),Ca) ).

% combine_common_factor
tff(fact_2848_combine__common__factor,axiom,
    ! [A: rat,E: rat,B: rat,Ca: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),E)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),E)),Ca)) = 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)),E)),Ca) ).

% combine_common_factor
tff(fact_2849_combine__common__factor,axiom,
    ! [A: nat,E: nat,B: nat,Ca: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),E)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),E)),Ca)) = 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)),E)),Ca) ).

% combine_common_factor
tff(fact_2850_combine__common__factor,axiom,
    ! [A: int,E: int,B: int,Ca: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),E)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),E)),Ca)) = 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)),E)),Ca) ).

% combine_common_factor
tff(fact_2851_execute__raise,axiom,
    ! [S2: list_char,X5: heap_e7401611519738050253t_unit] : aa_hea4724376117875161946it_nat(heap_Time_execute_a(heap_Time_raise_a(S2)),X5) = none_P2651198173097904984it_nat ).

% execute_raise
tff(fact_2852_guard__def,axiom,
    ! [Pa: fun_he8413527710729349838unit_o,F: fun_he8889679731534893401it_nat] : heap_Time_guard_a(Pa,F) = heap_Time_Heap_a2(aa_fun3370411987892275404it_nat(aTP_Lamm_fi(Pa),F)) ).

% guard_def
tff(fact_2853_guard__def,axiom,
    ! [Pa: fun_he8413527710729349838unit_o,F: fun_he4944555133437423336it_nat] : heap_T8440541562793052209t_unit(Pa,F) = heap_T6183433275982383450t_unit(aa_fun2986507860694160780it_nat(aTP_Lamm_fj(Pa),F)) ).

% guard_def
tff(fact_2854_execute__assert_I2_J,axiom,
    ! [Pa: fun_a_o,X: a,Ha: heap_e7401611519738050253t_unit] :
      ( ~ aa_a_o(Pa,X)
     => ( aa_hea4724376117875161946it_nat(heap_Time_execute_a(aa_a_H8829438773920479562Heap_a(heap_Time_assert_a(Pa),X)),Ha) = none_P2651198173097904984it_nat ) ) ).

% execute_assert(2)
tff(fact_2855_ent__pure__post__iff,axiom,
    ! [Pa: assn,Qa: assn,B: $o] :
      ( aa_assn_o(aa_assn_fun_assn_o(entails,Pa),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Qa),aa_o_assn(pure_assn,(B))))
    <=> ( ! [H5: produc3658429121746597890et_nat] :
            ( aa_Pro653082453696054616_nat_o(rep_assn(Pa),H5)
           => (B) )
        & aa_assn_o(aa_assn_fun_assn_o(entails,Pa),Qa) ) ) ).

% ent_pure_post_iff
tff(fact_2856_Heap__execute,axiom,
    ! [F: heap_Time_Heap_a] : heap_Time_Heap_a2(heap_Time_execute_a(F)) = F ).

% Heap_execute
tff(fact_2857_Heap__execute,axiom,
    ! [F: heap_T5738788834812785303t_unit] : heap_T6183433275982383450t_unit(heap_T875086893843062177t_unit(F)) = F ).

% Heap_execute
tff(fact_2858_ent__pure__pre__iff,axiom,
    ! [Pa: assn,B: $o,Qa: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(entails,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Pa),aa_o_assn(pure_assn,(B)))),Qa)
    <=> ( (B)
       => aa_assn_o(aa_assn_fun_assn_o(entails,Pa),Qa) ) ) ).

% ent_pure_pre_iff
tff(fact_2859_ent__false__iff,axiom,
    ! [Pa: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(entails,Pa),bot_bot_assn)
    <=> ! [H5: produc3658429121746597890et_nat] : ~ aa_Pro653082453696054616_nat_o(rep_assn(Pa),H5) ) ).

% ent_false_iff
tff(fact_2860_ent__trans,axiom,
    ! [Pa: assn,Qa: assn,Ra: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(entails,Pa),Qa)
     => ( aa_assn_o(aa_assn_fun_assn_o(entails,Qa),Ra)
       => aa_assn_o(aa_assn_fun_assn_o(entails,Pa),Ra) ) ) ).

% ent_trans
tff(fact_2861_ent__refl,axiom,
    ! [Pa: assn] : aa_assn_o(aa_assn_fun_assn_o(entails,Pa),Pa) ).

% ent_refl
tff(fact_2862_ent__iffI,axiom,
    ! [A3: assn,B3: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(entails,A3),B3)
     => ( aa_assn_o(aa_assn_fun_assn_o(entails,B3),A3)
       => ( A3 = B3 ) ) ) ).

% ent_iffI
tff(fact_2863_raise__def,axiom,
    ! [S2: list_char] : heap_T2927564422264180874t_unit(S2) = heap_T6183433275982383450t_unit(aTP_Lamm_fk) ).

% raise_def
tff(fact_2864_entails__def,axiom,
    ! [Pa: assn,Qa: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(entails,Pa),Qa)
    <=> ! [H5: produc3658429121746597890et_nat] :
          ( aa_Pro653082453696054616_nat_o(rep_assn(Pa),H5)
         => aa_Pro653082453696054616_nat_o(rep_assn(Qa),H5) ) ) ).

% entails_def
tff(fact_2865_entailsI,axiom,
    ! [Pa: assn,Qa: assn] :
      ( ! [H: produc3658429121746597890et_nat] :
          ( aa_Pro653082453696054616_nat_o(rep_assn(Pa),H)
         => aa_Pro653082453696054616_nat_o(rep_assn(Qa),H) )
     => aa_assn_o(aa_assn_fun_assn_o(entails,Pa),Qa) ) ).

% entailsI
tff(fact_2866_entailsD,axiom,
    ! [Pa: assn,Qa: assn,Ha: produc3658429121746597890et_nat] :
      ( aa_assn_o(aa_assn_fun_assn_o(entails,Pa),Qa)
     => ( aa_Pro653082453696054616_nat_o(rep_assn(Pa),Ha)
       => aa_Pro653082453696054616_nat_o(rep_assn(Qa),Ha) ) ) ).

% entailsD
tff(fact_2867_ent__fwd,axiom,
    ! [Pa: assn,Ha: produc3658429121746597890et_nat,Qa: assn] :
      ( aa_Pro653082453696054616_nat_o(rep_assn(Pa),Ha)
     => ( aa_assn_o(aa_assn_fun_assn_o(entails,Pa),Qa)
       => aa_Pro653082453696054616_nat_o(rep_assn(Qa),Ha) ) ) ).

% ent_fwd
tff(fact_2868_ent__star__mono,axiom,
    ! [Pa: assn,P3: assn,Qa: assn,Q2: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(entails,Pa),P3)
     => ( aa_assn_o(aa_assn_fun_assn_o(entails,Qa),Q2)
       => aa_assn_o(aa_assn_fun_assn_o(entails,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Pa),Qa)),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,P3),Q2)) ) ) ).

% ent_star_mono
tff(fact_2869_ent__disjE,axiom,
    ! [A3: assn,C2: assn,B3: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(entails,A3),C2)
     => ( aa_assn_o(aa_assn_fun_assn_o(entails,B3),C2)
       => aa_assn_o(aa_assn_fun_assn_o(entails,aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A3),B3)),C2) ) ) ).

% ent_disjE
tff(fact_2870_ent__disjI1,axiom,
    ! [Pa: assn,Qa: assn,Ra: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(entails,aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Pa),Qa)),Ra)
     => aa_assn_o(aa_assn_fun_assn_o(entails,Pa),Ra) ) ).

% ent_disjI1
tff(fact_2871_ent__disjI2,axiom,
    ! [Pa: assn,Qa: assn,Ra: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(entails,aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Pa),Qa)),Ra)
     => aa_assn_o(aa_assn_fun_assn_o(entails,Qa),Ra) ) ).

% ent_disjI2
tff(fact_2872_ent__disjI1_H,axiom,
    ! [A3: assn,B3: assn,C2: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(entails,A3),B3)
     => aa_assn_o(aa_assn_fun_assn_o(entails,A3),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,B3),C2)) ) ).

% ent_disjI1'
tff(fact_2873_ent__disjI2_H,axiom,
    ! [A3: assn,C2: assn,B3: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(entails,A3),C2)
     => aa_assn_o(aa_assn_fun_assn_o(entails,A3),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,B3),C2)) ) ).

% ent_disjI2'
tff(fact_2874_ent__disjI1__direct,axiom,
    ! [A3: assn,B3: assn] : aa_assn_o(aa_assn_fun_assn_o(entails,A3),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A3),B3)) ).

% ent_disjI1_direct
tff(fact_2875_ent__disjI2__direct,axiom,
    ! [B3: assn,A3: assn] : aa_assn_o(aa_assn_fun_assn_o(entails,B3),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A3),B3)) ).

% ent_disjI2_direct
tff(fact_2876_execute_Osimps,axiom,
    ! [F: fun_he1921015881043195817it_nat] : heap_Time_execute_a(heap_Time_Heap_a2(F)) = F ).

% execute.simps
tff(fact_2877_execute_Osimps,axiom,
    ! [F: fun_he5234484606919320760it_nat] : heap_T875086893843062177t_unit(heap_T6183433275982383450t_unit(F)) = F ).

% execute.simps
tff(fact_2878_ent__false,axiom,
    ! [Pa: assn] : aa_assn_o(aa_assn_fun_assn_o(entails,bot_bot_assn),Pa) ).

% ent_false
tff(fact_2879_ent__conjI,axiom,
    ! [A3: assn,B3: assn,C2: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(entails,A3),B3)
     => ( aa_assn_o(aa_assn_fun_assn_o(entails,A3),C2)
       => aa_assn_o(aa_assn_fun_assn_o(entails,A3),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,B3),C2)) ) ) ).

% ent_conjI
tff(fact_2880_ent__conjE1,axiom,
    ! [A3: assn,C2: assn,B3: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(entails,A3),C2)
     => aa_assn_o(aa_assn_fun_assn_o(entails,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A3),B3)),C2) ) ).

% ent_conjE1
tff(fact_2881_ent__conjE2,axiom,
    ! [B3: assn,C2: assn,A3: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(entails,B3),C2)
     => aa_assn_o(aa_assn_fun_assn_o(entails,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A3),B3)),C2) ) ).

% ent_conjE2
tff(fact_2882_ent__wandI,axiom,
    ! [Qa: assn,Pa: assn,Ra: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(entails,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Qa),Pa)),Ra)
     => aa_assn_o(aa_assn_fun_assn_o(entails,Pa),aa_assn_assn(aa_ass954072293579886259n_assn(wand_assn,Qa),Ra)) ) ).

% ent_wandI
tff(fact_2883_ent__mp,axiom,
    ! [Pa: assn,Qa: assn] : aa_assn_o(aa_assn_fun_assn_o(entails,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Pa),aa_assn_assn(aa_ass954072293579886259n_assn(wand_assn,Pa),Qa))),Qa) ).

% ent_mp
tff(fact_2884_linorder__neqE__linordered__idom,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_linordered_idom
tff(fact_2885_linorder__neqE__linordered__idom,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_linordered_idom
tff(fact_2886_execute__assert_I1_J,axiom,
    ! [Pa: fun_Product_unit_o,X: product_unit,Ha: heap_e7401611519738050253t_unit] :
      ( aa_Product_unit_o(Pa,X)
     => ( aa_hea684056619475940329it_nat(heap_T875086893843062177t_unit(aa_Pro1108561329362217960t_unit(heap_T4208721593536448476t_unit(Pa),X)),Ha) = aa_Pro8996696869555642526it_nat(some_P1914260805536162275it_nat,produc7133225469290080770it_nat(X,produc584006145561248582it_nat(Ha,one_one_nat))) ) ) ).

% execute_assert(1)
tff(fact_2887_execute__assert_I1_J,axiom,
    ! [Pa: fun_a_o,X: a,Ha: heap_e7401611519738050253t_unit] :
      ( aa_a_o(Pa,X)
     => ( aa_hea4724376117875161946it_nat(heap_Time_execute_a(aa_a_H8829438773920479562Heap_a(heap_Time_assert_a(Pa),X)),Ha) = aa_Pro5302094286594872192it_nat(some_P7913643980934408916it_nat,aa_Pro8078911840264235907it_nat(aa_a_f2234720956197727492it_nat(produc9178034014595674355it_nat,X),produc584006145561248582it_nat(Ha,one_one_nat))) ) ) ).

% execute_assert(1)
tff(fact_2888_Heap__lub__empty,axiom,
    heap_T4436805183663264329t_unit(bot_bo8105976889876132193t_unit) = heap_T6183433275982383450t_unit(aTP_Lamm_fk) ).

% Heap_lub_empty
tff(fact_2889_ent__pure__post__iff__sng,axiom,
    ! [Pa: assn,B: $o] :
      ( aa_assn_o(aa_assn_fun_assn_o(entails,Pa),aa_o_assn(pure_assn,(B)))
    <=> ( ! [H5: produc3658429121746597890et_nat] :
            ( aa_Pro653082453696054616_nat_o(rep_assn(Pa),H5)
           => (B) )
        & aa_assn_o(aa_assn_fun_assn_o(entails,Pa),one_one_assn) ) ) ).

% ent_pure_post_iff_sng
tff(fact_2890_pairself_Opelims,axiom,
    ! [X: fun_in5730379987311621695e_term,Xa2: product_prod_int_int,Y: produc6576344331059438605e_term] :
      ( ( pairse3534876335208825186e_term(X,Xa2) = Y )
     => ( aa_Pro5451197003417746155_int_o(accp_P3253464921323968230nt_int(pairse6848479906795794847e_term),aa_Pro8800328608558162860nt_int(aa_fun260840925927807678nt_int(produc4729913460834754785nt_int,X),Xa2))
       => ~ ! [A4: int,B4: int] :
              ( ( Xa2 = 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_2891_one__assn__raw_Osimps,axiom,
    ! [Ha: heap_e7401611519738050253t_unit,Asa: set_nat] :
      ( aa_Pro653082453696054616_nat_o(one_assn_raw,produc7507926704131184380et_nat(Ha,Asa))
    <=> ( Asa = bot_bot_set_nat ) ) ).

% one_assn_raw.simps
tff(fact_2892_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,As3: set_nat] :
            ( ( X = produc7507926704131184380et_nat(H,As3) )
           => ( (Y)
            <=> ( As3 != bot_bot_set_nat ) ) ) ) ).

% one_assn_raw.elims(1)
tff(fact_2893_one__assn__raw_Oelims_I2_J,axiom,
    ! [X: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(one_assn_raw,X)
     => ~ ! [H: heap_e7401611519738050253t_unit,As3: set_nat] :
            ( ( X = produc7507926704131184380et_nat(H,As3) )
           => ( As3 != bot_bot_set_nat ) ) ) ).

% one_assn_raw.elims(2)
tff(fact_2894_one__assn__raw_Oelims_I3_J,axiom,
    ! [X: produc3658429121746597890et_nat] :
      ( ~ aa_Pro653082453696054616_nat_o(one_assn_raw,X)
     => ~ ! [H: heap_e7401611519738050253t_unit,As3: set_nat] :
            ( ( X = produc7507926704131184380et_nat(H,As3) )
           => ( As3 = bot_bot_set_nat ) ) ) ).

% one_assn_raw.elims(3)
tff(fact_2895_mult__le__cancel2,axiom,
    ! [M: nat,K: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),K)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N2),K))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2) ) ) ).

% mult_le_cancel2
tff(fact_2896_combine__options__def,axiom,
    ! [F: fun_Pr7750636411365408485it_nat,X: option3562590408128118217it_nat,Y: option3562590408128118217it_nat] : combin5819755802923621690it_nat(F,X,Y) = case_o6061097939050036047it_nat(Y,aa_opt996909609682481350it_nat(aTP_Lamm_fm(F),Y),X) ).

% combine_options_def
tff(fact_2897_combine__options__def,axiom,
    ! [F: fun_Pr1003294667014919570it_nat,X: option8956607266484857688it_nat,Y: option8956607266484857688it_nat] : combin4318129983670048329it_nat(F,X,Y) = case_o2963978774867076333it_nat(Y,aa_opt7728438819757478003it_nat(aTP_Lamm_fo(F),Y),X) ).

% combine_options_def
tff(fact_2898_combine__options__def,axiom,
    ! [F: fun_num_fun_num_num,X: option_num,Y: option_num] : aa_opt5369159329878751634on_num(combine_options_num(F,X),Y) = aa_opt5369159329878751634on_num(case_o6005452278849405969um_num(Y,aa_opt8899475237019873193on_num(aTP_Lamm_fq(F),Y)),X) ).

% combine_options_def
tff(fact_2899_le__zero__eq,axiom,
    ! [N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),zero_zero_nat)
    <=> ( N2 = zero_zero_nat ) ) ).

% le_zero_eq
tff(fact_2900_not__gr__zero,axiom,
    ! [N2: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)
    <=> ( N2 = zero_zero_nat ) ) ).

% not_gr_zero
tff(fact_2901_mult__zero__left,axiom,
    ! [A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,zero_z3403309356797280102nteger),A) = zero_z3403309356797280102nteger ).

% mult_zero_left
tff(fact_2902_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_2903_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_2904_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_2905_mult__zero__right,axiom,
    ! [A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),zero_z3403309356797280102nteger) = zero_z3403309356797280102nteger ).

% mult_zero_right
tff(fact_2906_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_2907_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_2908_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_2909_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_2910_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_2911_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_2912_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_2913_mult__cancel__left,axiom,
    ! [Ca: code_integer,A: code_integer,B: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Ca),A) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Ca),B) )
    <=> ( ( Ca = zero_z3403309356797280102nteger )
        | ( A = B ) ) ) ).

% mult_cancel_left
tff(fact_2914_mult__cancel__left,axiom,
    ! [Ca: rat,A: rat,B: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Ca),A) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Ca),B) )
    <=> ( ( Ca = zero_zero_rat )
        | ( A = B ) ) ) ).

% mult_cancel_left
tff(fact_2915_mult__cancel__left,axiom,
    ! [Ca: nat,A: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ca),A) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ca),B) )
    <=> ( ( Ca = zero_zero_nat )
        | ( A = B ) ) ) ).

% mult_cancel_left
tff(fact_2916_mult__cancel__left,axiom,
    ! [Ca: int,A: int,B: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,Ca),A) = aa_int_int(aa_int_fun_int_int(times_times_int,Ca),B) )
    <=> ( ( Ca = zero_zero_int )
        | ( A = B ) ) ) ).

% mult_cancel_left
tff(fact_2917_mult__cancel__right,axiom,
    ! [A: code_integer,Ca: code_integer,B: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),Ca) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),Ca) )
    <=> ( ( Ca = zero_z3403309356797280102nteger )
        | ( A = B ) ) ) ).

% mult_cancel_right
tff(fact_2918_mult__cancel__right,axiom,
    ! [A: rat,Ca: rat,B: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),Ca) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),Ca) )
    <=> ( ( Ca = zero_zero_rat )
        | ( A = B ) ) ) ).

% mult_cancel_right
tff(fact_2919_mult__cancel__right,axiom,
    ! [A: nat,Ca: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),Ca) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),Ca) )
    <=> ( ( Ca = zero_zero_nat )
        | ( A = B ) ) ) ).

% mult_cancel_right
tff(fact_2920_mult__cancel__right,axiom,
    ! [A: int,Ca: int,B: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,A),Ca) = aa_int_int(aa_int_fun_int_int(times_times_int,B),Ca) )
    <=> ( ( Ca = zero_zero_int )
        | ( A = B ) ) ) ).

% mult_cancel_right
tff(fact_2921_mult_Oright__neutral,axiom,
    ! [A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),one_one_Code_integer) = A ).

% mult.right_neutral
tff(fact_2922_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_2923_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_2924_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_2925_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_2926_mult__1,axiom,
    ! [A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,one_one_Code_integer),A) = A ).

% mult_1
tff(fact_2927_mult__1,axiom,
    ! [A: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,one_one_assn),A) = A ).

% mult_1
tff(fact_2928_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_2929_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_2930_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_2931_less__nat__zero__code,axiom,
    ! [N2: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),zero_zero_nat) ).

% less_nat_zero_code
tff(fact_2932_neq0__conv,axiom,
    ! [N2: nat] :
      ( ( N2 != zero_zero_nat )
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2) ) ).

% neq0_conv
tff(fact_2933_bot__nat__0_Onot__eq__extremum,axiom,
    ! [A: nat] :
      ( ( A != zero_zero_nat )
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A) ) ).

% bot_nat_0.not_eq_extremum
tff(fact_2934_le0,axiom,
    ! [N2: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),N2) ).

% le0
tff(fact_2935_bot__nat__0_Oextremum,axiom,
    ! [A: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A) ).

% bot_nat_0.extremum
tff(fact_2936_add__is__0,axiom,
    ! [M: nat,N2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N2) = zero_zero_nat )
    <=> ( ( M = zero_zero_nat )
        & ( N2 = zero_zero_nat ) ) ) ).

% add_is_0
tff(fact_2937_Nat_Oadd__0__right,axiom,
    ! [M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),zero_zero_nat) = M ).

% Nat.add_0_right
tff(fact_2938_mult__cancel2,axiom,
    ! [M: nat,K: nat,N2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),K) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N2),K) )
    <=> ( ( M = N2 )
        | ( K = zero_zero_nat ) ) ) ).

% mult_cancel2
tff(fact_2939_mult__cancel1,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),N2) )
    <=> ( ( M = N2 )
        | ( K = zero_zero_nat ) ) ) ).

% mult_cancel1
tff(fact_2940_mult__0__right,axiom,
    ! [M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),zero_zero_nat) = zero_zero_nat ).

% mult_0_right
tff(fact_2941_mult__is__0,axiom,
    ! [M: nat,N2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N2) = zero_zero_nat )
    <=> ( ( M = zero_zero_nat )
        | ( N2 = zero_zero_nat ) ) ) ).

% mult_is_0
tff(fact_2942_nat__mult__eq__1__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N2) = one_one_nat )
    <=> ( ( M = one_one_nat )
        & ( N2 = one_one_nat ) ) ) ).

% nat_mult_eq_1_iff
tff(fact_2943_nat__1__eq__mult__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( one_one_nat = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N2) )
    <=> ( ( M = one_one_nat )
        & ( N2 = one_one_nat ) ) ) ).

% nat_1_eq_mult_iff
tff(fact_2944_combine__options__simps_I3_J,axiom,
    ! [F: fun_Pr7750636411365408485it_nat,A: produc3260487557148687353it_nat,B: produc3260487557148687353it_nat] : combin5819755802923621690it_nat(F,aa_Pro5302094286594872192it_nat(some_P7913643980934408916it_nat,A),aa_Pro5302094286594872192it_nat(some_P7913643980934408916it_nat,B)) = aa_Pro5302094286594872192it_nat(some_P7913643980934408916it_nat,aa_Pro4202338132779708464it_nat(aa_Pro2263720180353132454it_nat(F,A),B)) ).

% combine_options_simps(3)
tff(fact_2945_combine__options__simps_I3_J,axiom,
    ! [F: fun_num_fun_num_num,A: num,B: num] : aa_opt5369159329878751634on_num(combine_options_num(F,aa_num_option_num(some_num,A)),aa_num_option_num(some_num,B)) = aa_num_option_num(some_num,aa_num_num(aa_num_fun_num_num(F,A),B)) ).

% combine_options_simps(3)
tff(fact_2946_combine__options__simps_I3_J,axiom,
    ! [F: fun_Pr1003294667014919570it_nat,A: produc8664842809031399944it_nat,B: produc8664842809031399944it_nat] : combin4318129983670048329it_nat(F,aa_Pro8996696869555642526it_nat(some_P1914260805536162275it_nat,A),aa_Pro8996696869555642526it_nat(some_P1914260805536162275it_nat,B)) = aa_Pro8996696869555642526it_nat(some_P1914260805536162275it_nat,aa_Pro550487673543749710it_nat(aa_Pro4096753284154920275it_nat(F,A),B)) ).

% combine_options_simps(3)
tff(fact_2947_combine__options__simps_I2_J,axiom,
    ! [F: fun_num_fun_num_num,X: option_num] : aa_opt5369159329878751634on_num(combine_options_num(F,X),none_num) = X ).

% combine_options_simps(2)
tff(fact_2948_combine__options__simps_I1_J,axiom,
    ! [F: fun_num_fun_num_num,Y: option_num] : aa_opt5369159329878751634on_num(combine_options_num(F,none_num),Y) = Y ).

% combine_options_simps(1)
tff(fact_2949_pure__true,axiom,
    aa_o_assn(pure_assn,$true) = one_one_assn ).

% pure_true
tff(fact_2950_pure__assn__eq__emp__iff,axiom,
    ! [Pa: $o] :
      ( ( aa_o_assn(pure_assn,(Pa)) = one_one_assn )
    <=> (Pa) ) ).

% pure_assn_eq_emp_iff
tff(fact_2951_assn__basic__inequalities_I3_J,axiom,
    bot_bot_assn != one_one_assn ).

% assn_basic_inequalities(3)
tff(fact_2952_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(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_2953_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_2954_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_2955_double__add__le__zero__iff__single__add__le__zero,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(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_2956_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_2957_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_2958_le__add__same__cancel2,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(B),A))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),B) ) ).

% le_add_same_cancel2
tff(fact_2959_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_2960_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_2961_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_2962_le__add__same__cancel1,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(A),B))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),B) ) ).

% le_add_same_cancel1
tff(fact_2963_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_2964_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_2965_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_2966_add__le__same__cancel2,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(A),B)),B)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),zero_z3403309356797280102nteger) ) ).

% add_le_same_cancel2
tff(fact_2967_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_2968_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_2969_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_2970_add__le__same__cancel1,axiom,
    ! [B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(B),A)),B)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),zero_z3403309356797280102nteger) ) ).

% add_le_same_cancel1
tff(fact_2971_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_2972_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_2973_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_2974_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(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_2975_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_2976_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_2977_double__add__less__zero__iff__single__add__less__zero,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(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_2978_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_2979_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_2980_less__add__same__cancel2,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(B),A))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B) ) ).

% less_add_same_cancel2
tff(fact_2981_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_2982_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_2983_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_2984_less__add__same__cancel1,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(A),B))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B) ) ).

% less_add_same_cancel1
tff(fact_2985_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_2986_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_2987_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_2988_add__less__same__cancel2,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(A),B)),B)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),zero_z3403309356797280102nteger) ) ).

% add_less_same_cancel2
tff(fact_2989_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_2990_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_2991_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_2992_add__less__same__cancel1,axiom,
    ! [B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(B),A)),B)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),zero_z3403309356797280102nteger) ) ).

% add_less_same_cancel1
tff(fact_2993_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_2994_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_2995_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_2996_mult__cancel__left1,axiom,
    ! [Ca: code_integer,B: code_integer] :
      ( ( Ca = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Ca),B) )
    <=> ( ( Ca = zero_z3403309356797280102nteger )
        | ( B = one_one_Code_integer ) ) ) ).

% mult_cancel_left1
tff(fact_2997_mult__cancel__left1,axiom,
    ! [Ca: rat,B: rat] :
      ( ( Ca = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Ca),B) )
    <=> ( ( Ca = zero_zero_rat )
        | ( B = one_one_rat ) ) ) ).

% mult_cancel_left1
tff(fact_2998_mult__cancel__left1,axiom,
    ! [Ca: int,B: int] :
      ( ( Ca = aa_int_int(aa_int_fun_int_int(times_times_int,Ca),B) )
    <=> ( ( Ca = zero_zero_int )
        | ( B = one_one_int ) ) ) ).

% mult_cancel_left1
tff(fact_2999_mult__cancel__left2,axiom,
    ! [Ca: code_integer,A: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Ca),A) = Ca )
    <=> ( ( Ca = zero_z3403309356797280102nteger )
        | ( A = one_one_Code_integer ) ) ) ).

% mult_cancel_left2
tff(fact_3000_mult__cancel__left2,axiom,
    ! [Ca: rat,A: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Ca),A) = Ca )
    <=> ( ( Ca = zero_zero_rat )
        | ( A = one_one_rat ) ) ) ).

% mult_cancel_left2
tff(fact_3001_mult__cancel__left2,axiom,
    ! [Ca: int,A: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,Ca),A) = Ca )
    <=> ( ( Ca = zero_zero_int )
        | ( A = one_one_int ) ) ) ).

% mult_cancel_left2
tff(fact_3002_mult__cancel__right1,axiom,
    ! [Ca: code_integer,B: code_integer] :
      ( ( Ca = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),Ca) )
    <=> ( ( Ca = zero_z3403309356797280102nteger )
        | ( B = one_one_Code_integer ) ) ) ).

% mult_cancel_right1
tff(fact_3003_mult__cancel__right1,axiom,
    ! [Ca: rat,B: rat] :
      ( ( Ca = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),Ca) )
    <=> ( ( Ca = zero_zero_rat )
        | ( B = one_one_rat ) ) ) ).

% mult_cancel_right1
tff(fact_3004_mult__cancel__right1,axiom,
    ! [Ca: int,B: int] :
      ( ( Ca = aa_int_int(aa_int_fun_int_int(times_times_int,B),Ca) )
    <=> ( ( Ca = zero_zero_int )
        | ( B = one_one_int ) ) ) ).

% mult_cancel_right1
tff(fact_3005_mult__cancel__right2,axiom,
    ! [A: code_integer,Ca: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),Ca) = Ca )
    <=> ( ( Ca = zero_z3403309356797280102nteger )
        | ( A = one_one_Code_integer ) ) ) ).

% mult_cancel_right2
tff(fact_3006_mult__cancel__right2,axiom,
    ! [A: rat,Ca: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),Ca) = Ca )
    <=> ( ( Ca = zero_zero_rat )
        | ( A = one_one_rat ) ) ) ).

% mult_cancel_right2
tff(fact_3007_mult__cancel__right2,axiom,
    ! [A: int,Ca: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,A),Ca) = Ca )
    <=> ( ( Ca = zero_zero_int )
        | ( A = one_one_int ) ) ) ).

% mult_cancel_right2
tff(fact_3008_add__gr__0,axiom,
    ! [M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N2))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M)
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2) ) ) ).

% add_gr_0
tff(fact_3009_mult__less__cancel2,axiom,
    ! [M: nat,K: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),K)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N2),K))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N2) ) ) ).

% mult_less_cancel2
tff(fact_3010_nat__0__less__mult__iff,axiom,
    ! [M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N2))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2) ) ) ).

% nat_0_less_mult_iff
tff(fact_3011_less__one,axiom,
    ! [N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),one_one_nat)
    <=> ( N2 = zero_zero_nat ) ) ).

% less_one
tff(fact_3012_ent__pure__pre__iff__sng,axiom,
    ! [B: $o,Qa: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(entails,aa_o_assn(pure_assn,(B))),Qa)
    <=> ( (B)
       => aa_assn_o(aa_assn_fun_assn_o(entails,one_one_assn),Qa) ) ) ).

% ent_pure_pre_iff_sng
tff(fact_3013_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_3014_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_3015_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_3016_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_3017_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_3018_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_3019_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_3020_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_3021_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_3022_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_3023_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_3024_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_3025_zero__less__one,axiom,
    aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),one_one_Code_integer) ).

% zero_less_one
tff(fact_3026_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_3027_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_3028_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_3029_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_3030_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_3031_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_3032_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_3033_bot__nat__def,axiom,
    bot_bot_nat = zero_zero_nat ).

% bot_nat_def
tff(fact_3034_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_3035_mult__eq__self__implies__10,axiom,
    ! [M: nat,N2: nat] :
      ( ( M = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N2) )
     => ( ( N2 = one_one_nat )
        | ( M = zero_zero_nat ) ) ) ).

% mult_eq_self_implies_10
tff(fact_3036_one__assn__def,axiom,
    one_one_assn = abs_assn(one_assn_raw) ).

% one_assn_def
tff(fact_3037_mult__left__le,axiom,
    ! [Ca: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Ca),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),Ca)),A) ) ) ).

% mult_left_le
tff(fact_3038_mult__left__le,axiom,
    ! [Ca: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Ca),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),Ca)),A) ) ) ).

% mult_left_le
tff(fact_3039_mult__left__le,axiom,
    ! [Ca: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ca),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),Ca)),A) ) ) ).

% mult_left_le
tff(fact_3040_mult__left__le,axiom,
    ! [Ca: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ca),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),Ca)),A) ) ) ).

% mult_left_le
tff(fact_3041_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_3042_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_3043_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_3044_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_3045_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_3046_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_3047_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_3048_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_3049_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_3050_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_3051_zero__less__two,axiom,
    aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(one_one_Code_integer),one_one_Code_integer)) ).

% zero_less_two
tff(fact_3052_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_3053_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_3054_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_3055_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_3056_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_3057_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_3058_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_3059_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_3060_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_3061_mult_Ocomm__neutral,axiom,
    ! [A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),one_one_Code_integer) = A ).

% mult.comm_neutral
tff(fact_3062_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_3063_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_3064_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_3065_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_3066_zero__less__iff__neq__zero,axiom,
    ! [N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)
    <=> ( N2 != zero_zero_nat ) ) ).

% zero_less_iff_neq_zero
tff(fact_3067_gr__implies__not__zero,axiom,
    ! [M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N2)
     => ( N2 != zero_zero_nat ) ) ).

% gr_implies_not_zero
tff(fact_3068_not__less__zero,axiom,
    ! [N2: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),zero_zero_nat) ).

% not_less_zero
tff(fact_3069_gr__zeroI,axiom,
    ! [N2: nat] :
      ( ( N2 != zero_zero_nat )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2) ) ).

% gr_zeroI
tff(fact_3070_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_3071_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_3072_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_3073_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_3074_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_3075_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_3076_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_3077_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_3078_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_3079_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_3080_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_3081_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_3082_mult__left__cancel,axiom,
    ! [Ca: code_integer,A: code_integer,B: code_integer] :
      ( ( Ca != zero_z3403309356797280102nteger )
     => ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Ca),A) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Ca),B) )
      <=> ( A = B ) ) ) ).

% mult_left_cancel
tff(fact_3083_mult__left__cancel,axiom,
    ! [Ca: rat,A: rat,B: rat] :
      ( ( Ca != zero_zero_rat )
     => ( ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Ca),A) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Ca),B) )
      <=> ( A = B ) ) ) ).

% mult_left_cancel
tff(fact_3084_mult__left__cancel,axiom,
    ! [Ca: nat,A: nat,B: nat] :
      ( ( Ca != zero_zero_nat )
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ca),A) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ca),B) )
      <=> ( A = B ) ) ) ).

% mult_left_cancel
tff(fact_3085_mult__left__cancel,axiom,
    ! [Ca: int,A: int,B: int] :
      ( ( Ca != zero_zero_int )
     => ( ( aa_int_int(aa_int_fun_int_int(times_times_int,Ca),A) = aa_int_int(aa_int_fun_int_int(times_times_int,Ca),B) )
      <=> ( A = B ) ) ) ).

% mult_left_cancel
tff(fact_3086_mult__right__cancel,axiom,
    ! [Ca: code_integer,A: code_integer,B: code_integer] :
      ( ( Ca != zero_z3403309356797280102nteger )
     => ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),Ca) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),Ca) )
      <=> ( A = B ) ) ) ).

% mult_right_cancel
tff(fact_3087_mult__right__cancel,axiom,
    ! [Ca: rat,A: rat,B: rat] :
      ( ( Ca != zero_zero_rat )
     => ( ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),Ca) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),Ca) )
      <=> ( A = B ) ) ) ).

% mult_right_cancel
tff(fact_3088_mult__right__cancel,axiom,
    ! [Ca: nat,A: nat,B: nat] :
      ( ( Ca != zero_zero_nat )
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),Ca) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),Ca) )
      <=> ( A = B ) ) ) ).

% mult_right_cancel
tff(fact_3089_mult__right__cancel,axiom,
    ! [Ca: int,A: int,B: int] :
      ( ( Ca != zero_zero_int )
     => ( ( aa_int_int(aa_int_fun_int_int(times_times_int,A),Ca) = aa_int_int(aa_int_fun_int_int(times_times_int,B),Ca) )
      <=> ( A = B ) ) ) ).

% mult_right_cancel
tff(fact_3090_infinite__descent0,axiom,
    ! [Pa: fun_nat_o,N2: nat] :
      ( aa_nat_o(Pa,zero_zero_nat)
     => ( ! [N5: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N5)
           => ( ~ aa_nat_o(Pa,N5)
             => ? [M4: nat] :
                  ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M4),N5)
                  & ~ aa_nat_o(Pa,M4) ) ) )
       => aa_nat_o(Pa,N2) ) ) ).

% infinite_descent0
tff(fact_3091_gr__implies__not0,axiom,
    ! [M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N2)
     => ( N2 != zero_zero_nat ) ) ).

% gr_implies_not0
tff(fact_3092_less__zeroE,axiom,
    ! [N2: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),zero_zero_nat) ).

% less_zeroE
tff(fact_3093_not__less0,axiom,
    ! [N2: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),zero_zero_nat) ).

% not_less0
tff(fact_3094_not__gr0,axiom,
    ! [N2: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)
    <=> ( N2 = zero_zero_nat ) ) ).

% not_gr0
tff(fact_3095_gr0I,axiom,
    ! [N2: nat] :
      ( ( N2 != zero_zero_nat )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2) ) ).

% gr0I
tff(fact_3096_bot__nat__0_Oextremum__strict,axiom,
    ! [A: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),zero_zero_nat) ).

% bot_nat_0.extremum_strict
tff(fact_3097_nat__mult__1__right,axiom,
    ! [N2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N2),one_one_nat) = N2 ).

% nat_mult_1_right
tff(fact_3098_nat__mult__1,axiom,
    ! [N2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,one_one_nat),N2) = N2 ).

% nat_mult_1
tff(fact_3099_le__0__eq,axiom,
    ! [N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),zero_zero_nat)
    <=> ( N2 = zero_zero_nat ) ) ).

% le_0_eq
tff(fact_3100_bot__nat__0_Oextremum__uniqueI,axiom,
    ! [A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),zero_zero_nat)
     => ( A = zero_zero_nat ) ) ).

% bot_nat_0.extremum_uniqueI
tff(fact_3101_bot__nat__0_Oextremum__unique,axiom,
    ! [A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),zero_zero_nat)
    <=> ( A = zero_zero_nat ) ) ).

% bot_nat_0.extremum_unique
tff(fact_3102_less__eq__nat_Osimps_I1_J,axiom,
    ! [N2: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),N2) ).

% less_eq_nat.simps(1)
tff(fact_3103_plus__nat_Oadd__0,axiom,
    ! [N2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,zero_zero_nat),N2) = N2 ).

% plus_nat.add_0
tff(fact_3104_add__eq__self__zero,axiom,
    ! [M: nat,N2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N2) = M )
     => ( N2 = zero_zero_nat ) ) ).

% add_eq_self_zero
tff(fact_3105_mult__0,axiom,
    ! [N2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,zero_zero_nat),N2) = zero_zero_nat ).

% mult_0
tff(fact_3106_assn__one__left,axiom,
    ! [Pa: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,one_one_assn),Pa) = Pa ).

% assn_one_left
tff(fact_3107_mult__le__cancel__left1,axiom,
    ! [Ca: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Ca),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Ca),B))
    <=> ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),Ca)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),B) )
        & ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Ca),zero_z3403309356797280102nteger)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),one_one_Code_integer) ) ) ) ).

% mult_le_cancel_left1
tff(fact_3108_mult__le__cancel__left1,axiom,
    ! [Ca: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Ca),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Ca),B))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Ca)
         => 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,Ca),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_3109_mult__le__cancel__left1,axiom,
    ! [Ca: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ca),aa_int_int(aa_int_fun_int_int(times_times_int,Ca),B))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Ca)
         => 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,Ca),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_3110_mult__le__cancel__left2,axiom,
    ! [Ca: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Ca),A)),Ca)
    <=> ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),Ca)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),one_one_Code_integer) )
        & ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Ca),zero_z3403309356797280102nteger)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),A) ) ) ) ).

% mult_le_cancel_left2
tff(fact_3111_mult__le__cancel__left2,axiom,
    ! [Ca: 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,Ca),A)),Ca)
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Ca)
         => 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,Ca),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_3112_mult__le__cancel__left2,axiom,
    ! [Ca: 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,Ca),A)),Ca)
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Ca)
         => 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,Ca),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_3113_mult__le__cancel__right1,axiom,
    ! [Ca: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Ca),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),Ca))
    <=> ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),Ca)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),B) )
        & ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Ca),zero_z3403309356797280102nteger)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),one_one_Code_integer) ) ) ) ).

% mult_le_cancel_right1
tff(fact_3114_mult__le__cancel__right1,axiom,
    ! [Ca: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Ca),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),Ca))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Ca)
         => 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,Ca),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_3115_mult__le__cancel__right1,axiom,
    ! [Ca: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ca),aa_int_int(aa_int_fun_int_int(times_times_int,B),Ca))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Ca)
         => 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,Ca),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_3116_mult__le__cancel__right2,axiom,
    ! [A: code_integer,Ca: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),Ca)),Ca)
    <=> ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),Ca)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),one_one_Code_integer) )
        & ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Ca),zero_z3403309356797280102nteger)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),A) ) ) ) ).

% mult_le_cancel_right2
tff(fact_3117_mult__le__cancel__right2,axiom,
    ! [A: rat,Ca: 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),Ca)),Ca)
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Ca)
         => 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,Ca),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_3118_mult__le__cancel__right2,axiom,
    ! [A: int,Ca: 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),Ca)),Ca)
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Ca)
         => 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,Ca),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_3119_mult__less__cancel__left1,axiom,
    ! [Ca: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Ca),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Ca),B))
    <=> ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),Ca)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),B) )
        & ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Ca),zero_z3403309356797280102nteger)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),one_one_Code_integer) ) ) ) ).

% mult_less_cancel_left1
tff(fact_3120_mult__less__cancel__left1,axiom,
    ! [Ca: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Ca),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Ca),B))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Ca)
         => 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,Ca),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),one_one_rat) ) ) ) ).

% mult_less_cancel_left1
tff(fact_3121_mult__less__cancel__left1,axiom,
    ! [Ca: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Ca),aa_int_int(aa_int_fun_int_int(times_times_int,Ca),B))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Ca)
         => 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,Ca),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,B),one_one_int) ) ) ) ).

% mult_less_cancel_left1
tff(fact_3122_mult__less__cancel__left2,axiom,
    ! [Ca: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Ca),A)),Ca)
    <=> ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),Ca)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),one_one_Code_integer) )
        & ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Ca),zero_z3403309356797280102nteger)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),A) ) ) ) ).

% mult_less_cancel_left2
tff(fact_3123_mult__less__cancel__left2,axiom,
    ! [Ca: 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,Ca),A)),Ca)
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Ca)
         => 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,Ca),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),A) ) ) ) ).

% mult_less_cancel_left2
tff(fact_3124_mult__less__cancel__left2,axiom,
    ! [Ca: 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,Ca),A)),Ca)
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Ca)
         => 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,Ca),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A) ) ) ) ).

% mult_less_cancel_left2
tff(fact_3125_mult__less__cancel__right1,axiom,
    ! [Ca: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Ca),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),Ca))
    <=> ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),Ca)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),B) )
        & ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Ca),zero_z3403309356797280102nteger)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),one_one_Code_integer) ) ) ) ).

% mult_less_cancel_right1
tff(fact_3126_mult__less__cancel__right1,axiom,
    ! [Ca: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Ca),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),Ca))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Ca)
         => 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,Ca),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),one_one_rat) ) ) ) ).

% mult_less_cancel_right1
tff(fact_3127_mult__less__cancel__right1,axiom,
    ! [Ca: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Ca),aa_int_int(aa_int_fun_int_int(times_times_int,B),Ca))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Ca)
         => 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,Ca),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,B),one_one_int) ) ) ) ).

% mult_less_cancel_right1
tff(fact_3128_mult__less__cancel__right2,axiom,
    ! [A: code_integer,Ca: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),Ca)),Ca)
    <=> ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),Ca)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),one_one_Code_integer) )
        & ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Ca),zero_z3403309356797280102nteger)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),A) ) ) ) ).

% mult_less_cancel_right2
tff(fact_3129_mult__less__cancel__right2,axiom,
    ! [A: rat,Ca: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),Ca)),Ca)
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Ca)
         => 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,Ca),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),A) ) ) ) ).

% mult_less_cancel_right2
tff(fact_3130_mult__less__cancel__right2,axiom,
    ! [A: int,Ca: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),Ca)),Ca)
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Ca)
         => 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,Ca),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A) ) ) ) ).

% mult_less_cancel_right2
tff(fact_3131_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(plus_p5714425477246183910nteger(U),V) = one_one_Code_integer )
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,U),X)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,V),Y))),A) ) ) ) ) ) ).

% convex_bound_le
tff(fact_3132_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_3133_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_3134_lambda__one,axiom,
    aTP_Lamm_fr = aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,one_one_Code_integer) ).

% lambda_one
tff(fact_3135_lambda__one,axiom,
    aTP_Lamm_fs = aa_ass954072293579886259n_assn(times_times_assn,one_one_assn) ).

% lambda_one
tff(fact_3136_lambda__one,axiom,
    aTP_Lamm_ft = aa_rat_fun_rat_rat(times_times_rat,one_one_rat) ).

% lambda_one
tff(fact_3137_lambda__one,axiom,
    aTP_Lamm_fu = aa_nat_fun_nat_nat(times_times_nat,one_one_nat) ).

% lambda_one
tff(fact_3138_lambda__one,axiom,
    aTP_Lamm_fv = aa_int_fun_int_int(times_times_int,one_one_int) ).

% lambda_one
tff(fact_3139_lambda__zero,axiom,
    aTP_Lamm_fw = aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,zero_z3403309356797280102nteger) ).

% lambda_zero
tff(fact_3140_lambda__zero,axiom,
    aTP_Lamm_fx = aa_rat_fun_rat_rat(times_times_rat,zero_zero_rat) ).

% lambda_zero
tff(fact_3141_lambda__zero,axiom,
    aTP_Lamm_fy = aa_nat_fun_nat_nat(times_times_nat,zero_zero_nat) ).

% lambda_zero
tff(fact_3142_lambda__zero,axiom,
    aTP_Lamm_fz = aa_int_fun_int_int(times_times_int,zero_zero_int) ).

% lambda_zero
tff(fact_3143_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(plus_p5714425477246183910nteger(U),V) = one_one_Code_integer )
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,U),X)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,V),Y))),A) ) ) ) ) ) ).

% convex_bound_lt
tff(fact_3144_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_3145_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_3146_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(plus_p5714425477246183910nteger(A),one_one_Code_integer)),aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(B),one_one_Code_integer)) ) ).

% add_mono1
tff(fact_3147_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_3148_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_3149_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_3150_less__add__one,axiom,
    ! [A: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(A),one_one_Code_integer)) ).

% less_add_one
tff(fact_3151_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_3152_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_3153_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_3154_less__1__mult,axiom,
    ! [M: code_integer,N2: 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),N2)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,M),N2)) ) ) ).

% less_1_mult
tff(fact_3155_less__1__mult,axiom,
    ! [M: rat,N2: 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),N2)
       => 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),N2)) ) ) ).

% less_1_mult
tff(fact_3156_less__1__mult,axiom,
    ! [M: nat,N2: 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),N2)
       => 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),N2)) ) ) ).

% less_1_mult
tff(fact_3157_less__1__mult,axiom,
    ! [M: int,N2: 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),N2)
       => 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),N2)) ) ) ).

% less_1_mult
tff(fact_3158_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(plus_p5714425477246183910nteger(X),Y) = zero_z3403309356797280102nteger )
        <=> ( ( X = zero_z3403309356797280102nteger )
            & ( Y = zero_z3403309356797280102nteger ) ) ) ) ) ).

% add_nonpos_eq_0_iff
tff(fact_3159_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_3160_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_3161_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_3162_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(plus_p5714425477246183910nteger(X),Y) = zero_z3403309356797280102nteger )
        <=> ( ( X = zero_z3403309356797280102nteger )
            & ( Y = zero_z3403309356797280102nteger ) ) ) ) ) ).

% add_nonneg_eq_0_iff
tff(fact_3163_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_3164_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_3165_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_3166_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(plus_p5714425477246183910nteger(A),B)),zero_z3403309356797280102nteger) ) ) ).

% add_nonpos_nonpos
tff(fact_3167_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_3168_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_3169_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_3170_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(plus_p5714425477246183910nteger(A),B)) ) ) ).

% add_nonneg_nonneg
tff(fact_3171_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_3172_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_3173_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_3174_add__increasing2,axiom,
    ! [Ca: code_integer,B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),Ca)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),A)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(A),Ca)) ) ) ).

% add_increasing2
tff(fact_3175_add__increasing2,axiom,
    ! [Ca: rat,B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Ca)
     => ( 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),Ca)) ) ) ).

% add_increasing2
tff(fact_3176_add__increasing2,axiom,
    ! [Ca: nat,B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Ca)
     => ( 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),Ca)) ) ) ).

% add_increasing2
tff(fact_3177_add__increasing2,axiom,
    ! [Ca: int,B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Ca)
     => ( 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),Ca)) ) ) ).

% add_increasing2
tff(fact_3178_add__decreasing2,axiom,
    ! [Ca: code_integer,A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Ca),zero_z3403309356797280102nteger)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(A),Ca)),B) ) ) ).

% add_decreasing2
tff(fact_3179_add__decreasing2,axiom,
    ! [Ca: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Ca),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),Ca)),B) ) ) ).

% add_decreasing2
tff(fact_3180_add__decreasing2,axiom,
    ! [Ca: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ca),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),Ca)),B) ) ) ).

% add_decreasing2
tff(fact_3181_add__decreasing2,axiom,
    ! [Ca: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ca),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),Ca)),B) ) ) ).

% add_decreasing2
tff(fact_3182_add__increasing,axiom,
    ! [A: code_integer,B: code_integer,Ca: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),Ca)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(A),Ca)) ) ) ).

% add_increasing
tff(fact_3183_add__increasing,axiom,
    ! [A: rat,B: rat,Ca: 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),Ca)
       => 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),Ca)) ) ) ).

% add_increasing
tff(fact_3184_add__increasing,axiom,
    ! [A: nat,B: nat,Ca: 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),Ca)
       => 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),Ca)) ) ) ).

% add_increasing
tff(fact_3185_add__increasing,axiom,
    ! [A: int,B: int,Ca: 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),Ca)
       => 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),Ca)) ) ) ).

% add_increasing
tff(fact_3186_add__decreasing,axiom,
    ! [A: code_integer,Ca: 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,Ca),B)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(A),Ca)),B) ) ) ).

% add_decreasing
tff(fact_3187_add__decreasing,axiom,
    ! [A: rat,Ca: 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,Ca),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),Ca)),B) ) ) ).

% add_decreasing
tff(fact_3188_add__decreasing,axiom,
    ! [A: nat,Ca: 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,Ca),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),Ca)),B) ) ) ).

% add_decreasing
tff(fact_3189_add__decreasing,axiom,
    ! [A: int,Ca: 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,Ca),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),Ca)),B) ) ) ).

% add_decreasing
tff(fact_3190_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: code_integer,B: code_integer,Ca: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),Ca)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Ca),A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Ca),B)) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
tff(fact_3191_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: rat,B: rat,Ca: 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),Ca)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Ca),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Ca),B)) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
tff(fact_3192_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: nat,B: nat,Ca: 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),Ca)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ca),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ca),B)) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
tff(fact_3193_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: int,B: int,Ca: 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),Ca)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,Ca),A)),aa_int_int(aa_int_fun_int_int(times_times_int,Ca),B)) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
tff(fact_3194_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_3195_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_3196_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_3197_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_3198_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_3199_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_3200_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_3201_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_3202_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_3203_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_3204_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_3205_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_3206_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_3207_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_3208_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_3209_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_3210_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_3211_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_3212_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_3213_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_3214_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_3215_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_3216_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_3217_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_3218_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_3219_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_3220_mult__right__mono,axiom,
    ! [A: code_integer,B: code_integer,Ca: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),Ca)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),Ca)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),Ca)) ) ) ).

% mult_right_mono
tff(fact_3221_mult__right__mono,axiom,
    ! [A: rat,B: rat,Ca: 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),Ca)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),Ca)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),Ca)) ) ) ).

% mult_right_mono
tff(fact_3222_mult__right__mono,axiom,
    ! [A: nat,B: nat,Ca: 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),Ca)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),Ca)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),Ca)) ) ) ).

% mult_right_mono
tff(fact_3223_mult__right__mono,axiom,
    ! [A: int,B: int,Ca: 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),Ca)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),Ca)),aa_int_int(aa_int_fun_int_int(times_times_int,B),Ca)) ) ) ).

% mult_right_mono
tff(fact_3224_mult__right__mono__neg,axiom,
    ! [B: code_integer,A: code_integer,Ca: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),A)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Ca),zero_z3403309356797280102nteger)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),Ca)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),Ca)) ) ) ).

% mult_right_mono_neg
tff(fact_3225_mult__right__mono__neg,axiom,
    ! [B: rat,A: rat,Ca: 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,Ca),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),Ca)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),Ca)) ) ) ).

% mult_right_mono_neg
tff(fact_3226_mult__right__mono__neg,axiom,
    ! [B: int,A: int,Ca: 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,Ca),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),Ca)),aa_int_int(aa_int_fun_int_int(times_times_int,B),Ca)) ) ) ).

% mult_right_mono_neg
tff(fact_3227_mult__left__mono,axiom,
    ! [A: code_integer,B: code_integer,Ca: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),Ca)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Ca),A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Ca),B)) ) ) ).

% mult_left_mono
tff(fact_3228_mult__left__mono,axiom,
    ! [A: rat,B: rat,Ca: 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),Ca)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Ca),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Ca),B)) ) ) ).

% mult_left_mono
tff(fact_3229_mult__left__mono,axiom,
    ! [A: nat,B: nat,Ca: 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),Ca)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ca),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ca),B)) ) ) ).

% mult_left_mono
tff(fact_3230_mult__left__mono,axiom,
    ! [A: int,B: int,Ca: 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),Ca)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,Ca),A)),aa_int_int(aa_int_fun_int_int(times_times_int,Ca),B)) ) ) ).

% mult_left_mono
tff(fact_3231_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_3232_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_3233_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_3234_mult__left__mono__neg,axiom,
    ! [B: code_integer,A: code_integer,Ca: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),A)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Ca),zero_z3403309356797280102nteger)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Ca),A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Ca),B)) ) ) ).

% mult_left_mono_neg
tff(fact_3235_mult__left__mono__neg,axiom,
    ! [B: rat,A: rat,Ca: 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,Ca),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,Ca),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Ca),B)) ) ) ).

% mult_left_mono_neg
tff(fact_3236_mult__left__mono__neg,axiom,
    ! [B: int,A: int,Ca: 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,Ca),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,Ca),A)),aa_int_int(aa_int_fun_int_int(times_times_int,Ca),B)) ) ) ).

% mult_left_mono_neg
tff(fact_3237_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_3238_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_3239_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_3240_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_3241_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_3242_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_3243_mult__mono_H,axiom,
    ! [A: code_integer,B: code_integer,Ca: 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,Ca),D2)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)
         => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),Ca)
           => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),Ca)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),D2)) ) ) ) ) ).

% mult_mono'
tff(fact_3244_mult__mono_H,axiom,
    ! [A: rat,B: rat,Ca: 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,Ca),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),Ca)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),Ca)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),D2)) ) ) ) ) ).

% mult_mono'
tff(fact_3245_mult__mono_H,axiom,
    ! [A: nat,B: nat,Ca: 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,Ca),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),Ca)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),Ca)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D2)) ) ) ) ) ).

% mult_mono'
tff(fact_3246_mult__mono_H,axiom,
    ! [A: int,B: int,Ca: 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,Ca),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),Ca)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),Ca)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D2)) ) ) ) ) ).

% mult_mono'
tff(fact_3247_mult__mono,axiom,
    ! [A: code_integer,B: code_integer,Ca: 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,Ca),D2)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),B)
         => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),Ca)
           => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),Ca)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),D2)) ) ) ) ) ).

% mult_mono
tff(fact_3248_mult__mono,axiom,
    ! [A: rat,B: rat,Ca: 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,Ca),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),Ca)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),Ca)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),D2)) ) ) ) ) ).

% mult_mono
tff(fact_3249_mult__mono,axiom,
    ! [A: nat,B: nat,Ca: 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,Ca),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),Ca)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),Ca)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D2)) ) ) ) ) ).

% mult_mono
tff(fact_3250_mult__mono,axiom,
    ! [A: int,B: int,Ca: 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,Ca),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),Ca)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),Ca)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D2)) ) ) ) ) ).

% mult_mono
tff(fact_3251_pos__add__strict,axiom,
    ! [A: code_integer,B: code_integer,Ca: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),Ca)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(A),Ca)) ) ) ).

% pos_add_strict
tff(fact_3252_pos__add__strict,axiom,
    ! [A: rat,B: rat,Ca: 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),Ca)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),Ca)) ) ) ).

% pos_add_strict
tff(fact_3253_pos__add__strict,axiom,
    ! [A: nat,B: nat,Ca: 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),Ca)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),Ca)) ) ) ).

% pos_add_strict
tff(fact_3254_pos__add__strict,axiom,
    ! [A: int,B: int,Ca: 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),Ca)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,B),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),Ca)) ) ) ).

% pos_add_strict
tff(fact_3255_canonically__ordered__monoid__add__class_OlessE,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ~ ! [C: nat] :
            ( ( B = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C) )
           => ( C = zero_zero_nat ) ) ) ).

% canonically_ordered_monoid_add_class.lessE
tff(fact_3256_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(plus_p5714425477246183910nteger(A),B)) ) ) ).

% add_pos_pos
tff(fact_3257_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_3258_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_3259_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_3260_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(plus_p5714425477246183910nteger(A),B)),zero_z3403309356797280102nteger) ) ) ).

% add_neg_neg
tff(fact_3261_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_3262_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_3263_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_3264_add__less__zeroD,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(X),Y)),zero_z3403309356797280102nteger)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,X),zero_z3403309356797280102nteger)
        | aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Y),zero_z3403309356797280102nteger) ) ) ).

% add_less_zeroD
tff(fact_3265_add__less__zeroD,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,X),Y)),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),zero_zero_rat)
        | aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),zero_zero_rat) ) ) ).

% add_less_zeroD
tff(fact_3266_add__less__zeroD,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,X),Y)),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),zero_zero_int)
        | aa_int_o(aa_int_fun_int_o(ord_less_int,Y),zero_zero_int) ) ) ).

% add_less_zeroD
tff(fact_3267_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A: code_integer,B: code_integer,Ca: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),Ca)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Ca),A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Ca),B)) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
tff(fact_3268_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A: rat,B: rat,Ca: 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),Ca)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Ca),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Ca),B)) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
tff(fact_3269_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A: nat,B: nat,Ca: 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),Ca)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ca),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ca),B)) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
tff(fact_3270_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A: int,B: int,Ca: 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),Ca)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,Ca),A)),aa_int_int(aa_int_fun_int_int(times_times_int,Ca),B)) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
tff(fact_3271_mult__less__cancel__right__disj,axiom,
    ! [A: code_integer,Ca: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),Ca)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),Ca))
    <=> ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),Ca)
          & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B) )
        | ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Ca),zero_z3403309356797280102nteger)
          & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),A) ) ) ) ).

% mult_less_cancel_right_disj
tff(fact_3272_mult__less__cancel__right__disj,axiom,
    ! [A: rat,Ca: 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),Ca)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),Ca))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Ca)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Ca),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A) ) ) ) ).

% mult_less_cancel_right_disj
tff(fact_3273_mult__less__cancel__right__disj,axiom,
    ! [A: int,Ca: 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),Ca)),aa_int_int(aa_int_fun_int_int(times_times_int,B),Ca))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Ca)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_int,Ca),zero_zero_int)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,B),A) ) ) ) ).

% mult_less_cancel_right_disj
tff(fact_3274_mult__strict__right__mono,axiom,
    ! [A: code_integer,B: code_integer,Ca: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),Ca)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),Ca)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),Ca)) ) ) ).

% mult_strict_right_mono
tff(fact_3275_mult__strict__right__mono,axiom,
    ! [A: rat,B: rat,Ca: 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),Ca)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),Ca)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),Ca)) ) ) ).

% mult_strict_right_mono
tff(fact_3276_mult__strict__right__mono,axiom,
    ! [A: nat,B: nat,Ca: 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),Ca)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),Ca)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),Ca)) ) ) ).

% mult_strict_right_mono
tff(fact_3277_mult__strict__right__mono,axiom,
    ! [A: int,B: int,Ca: 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),Ca)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),Ca)),aa_int_int(aa_int_fun_int_int(times_times_int,B),Ca)) ) ) ).

% mult_strict_right_mono
tff(fact_3278_mult__strict__right__mono__neg,axiom,
    ! [B: code_integer,A: code_integer,Ca: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),A)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Ca),zero_z3403309356797280102nteger)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),Ca)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),Ca)) ) ) ).

% mult_strict_right_mono_neg
tff(fact_3279_mult__strict__right__mono__neg,axiom,
    ! [B: rat,A: rat,Ca: 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,Ca),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),Ca)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),Ca)) ) ) ).

% mult_strict_right_mono_neg
tff(fact_3280_mult__strict__right__mono__neg,axiom,
    ! [B: int,A: int,Ca: 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,Ca),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),Ca)),aa_int_int(aa_int_fun_int_int(times_times_int,B),Ca)) ) ) ).

% mult_strict_right_mono_neg
tff(fact_3281_mult__less__cancel__left__disj,axiom,
    ! [Ca: code_integer,A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Ca),A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Ca),B))
    <=> ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),Ca)
          & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B) )
        | ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Ca),zero_z3403309356797280102nteger)
          & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),A) ) ) ) ).

% mult_less_cancel_left_disj
tff(fact_3282_mult__less__cancel__left__disj,axiom,
    ! [Ca: 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,Ca),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Ca),B))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Ca)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Ca),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A) ) ) ) ).

% mult_less_cancel_left_disj
tff(fact_3283_mult__less__cancel__left__disj,axiom,
    ! [Ca: 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,Ca),A)),aa_int_int(aa_int_fun_int_int(times_times_int,Ca),B))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Ca)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_int,Ca),zero_zero_int)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,B),A) ) ) ) ).

% mult_less_cancel_left_disj
tff(fact_3284_mult__strict__left__mono,axiom,
    ! [A: code_integer,B: code_integer,Ca: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),Ca)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Ca),A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Ca),B)) ) ) ).

% mult_strict_left_mono
tff(fact_3285_mult__strict__left__mono,axiom,
    ! [A: rat,B: rat,Ca: 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),Ca)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Ca),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Ca),B)) ) ) ).

% mult_strict_left_mono
tff(fact_3286_mult__strict__left__mono,axiom,
    ! [A: nat,B: nat,Ca: 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),Ca)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ca),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ca),B)) ) ) ).

% mult_strict_left_mono
tff(fact_3287_mult__strict__left__mono,axiom,
    ! [A: int,B: int,Ca: 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),Ca)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,Ca),A)),aa_int_int(aa_int_fun_int_int(times_times_int,Ca),B)) ) ) ).

% mult_strict_left_mono
tff(fact_3288_mult__strict__left__mono__neg,axiom,
    ! [B: code_integer,A: code_integer,Ca: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),A)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Ca),zero_z3403309356797280102nteger)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Ca),A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Ca),B)) ) ) ).

% mult_strict_left_mono_neg
tff(fact_3289_mult__strict__left__mono__neg,axiom,
    ! [B: rat,A: rat,Ca: 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,Ca),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,Ca),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Ca),B)) ) ) ).

% mult_strict_left_mono_neg
tff(fact_3290_mult__strict__left__mono__neg,axiom,
    ! [B: int,A: int,Ca: 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,Ca),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,Ca),A)),aa_int_int(aa_int_fun_int_int(times_times_int,Ca),B)) ) ) ).

% mult_strict_left_mono_neg
tff(fact_3291_mult__less__cancel__left__pos,axiom,
    ! [Ca: code_integer,A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),Ca)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Ca),A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Ca),B))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B) ) ) ).

% mult_less_cancel_left_pos
tff(fact_3292_mult__less__cancel__left__pos,axiom,
    ! [Ca: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Ca)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Ca),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Ca),B))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ) ).

% mult_less_cancel_left_pos
tff(fact_3293_mult__less__cancel__left__pos,axiom,
    ! [Ca: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Ca)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,Ca),A)),aa_int_int(aa_int_fun_int_int(times_times_int,Ca),B))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ) ).

% mult_less_cancel_left_pos
tff(fact_3294_mult__less__cancel__left__neg,axiom,
    ! [Ca: code_integer,A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Ca),zero_z3403309356797280102nteger)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Ca),A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Ca),B))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),A) ) ) ).

% mult_less_cancel_left_neg
tff(fact_3295_mult__less__cancel__left__neg,axiom,
    ! [Ca: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Ca),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,Ca),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Ca),B))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A) ) ) ).

% mult_less_cancel_left_neg
tff(fact_3296_mult__less__cancel__left__neg,axiom,
    ! [Ca: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Ca),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,Ca),A)),aa_int_int(aa_int_fun_int_int(times_times_int,Ca),B))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,B),A) ) ) ).

% mult_less_cancel_left_neg
tff(fact_3297_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_3298_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_3299_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_3300_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_3301_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_3302_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_3303_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_3304_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_3305_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_3306_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_3307_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_3308_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_3309_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_3310_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_3311_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_3312_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_3313_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_3314_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_3315_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_3316_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_3317_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_3318_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_3319_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_3320_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_3321_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_3322_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_3323_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_3324_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_3325_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_3326_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_3327_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_3328_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_3329_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_3330_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_3331_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_3332_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_3333_add__scale__eq__noteq,axiom,
    ! [Ra2: code_integer,A: code_integer,B: code_integer,Ca: code_integer,D2: code_integer] :
      ( ( Ra2 != zero_z3403309356797280102nteger )
     => ( ( ( A = B )
          & ( Ca != D2 ) )
       => ( aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Ra2),Ca)) != aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(B),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Ra2),D2)) ) ) ) ).

% add_scale_eq_noteq
tff(fact_3334_add__scale__eq__noteq,axiom,
    ! [Ra2: rat,A: rat,B: rat,Ca: rat,D2: rat] :
      ( ( Ra2 != zero_zero_rat )
     => ( ( ( A = B )
          & ( Ca != D2 ) )
       => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Ra2),Ca)) != aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Ra2),D2)) ) ) ) ).

% add_scale_eq_noteq
tff(fact_3335_add__scale__eq__noteq,axiom,
    ! [Ra2: nat,A: nat,B: nat,Ca: nat,D2: nat] :
      ( ( Ra2 != zero_zero_nat )
     => ( ( ( A = B )
          & ( Ca != D2 ) )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ra2),Ca)) != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ra2),D2)) ) ) ) ).

% add_scale_eq_noteq
tff(fact_3336_add__scale__eq__noteq,axiom,
    ! [Ra2: int,A: int,B: int,Ca: int,D2: int] :
      ( ( Ra2 != zero_zero_int )
     => ( ( ( A = B )
          & ( Ca != D2 ) )
       => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,Ra2),Ca)) != aa_int_int(aa_int_fun_int_int(plus_plus_int,B),aa_int_int(aa_int_fun_int_int(times_times_int,Ra2),D2)) ) ) ) ).

% add_scale_eq_noteq
tff(fact_3337_ex__least__nat__le,axiom,
    ! [Pa: fun_nat_o,N2: nat] :
      ( aa_nat_o(Pa,N2)
     => ( ~ aa_nat_o(Pa,zero_zero_nat)
       => ? [K2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),N2)
            & ! [I: nat] :
                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),K2)
               => ~ aa_nat_o(Pa,I) )
            & aa_nat_o(Pa,K2) ) ) ) ).

% ex_least_nat_le
tff(fact_3338_less__imp__add__positive,axiom,
    ! [I2: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),J)
     => ? [K2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K2)
          & ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),K2) = J ) ) ) ).

% less_imp_add_positive
tff(fact_3339_mult__less__mono1,axiom,
    ! [I2: nat,J: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),J)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I2),K)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),K)) ) ) ).

% mult_less_mono1
tff(fact_3340_mult__less__mono2,axiom,
    ! [I2: nat,J: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),J)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),I2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),J)) ) ) ).

% mult_less_mono2
tff(fact_3341_bot__nat__0_Oordering__top__axioms,axiom,
    aa_nat_o(ordering_top_nat(aTP_Lamm_df,aTP_Lamm_dg),zero_zero_nat) ).

% bot_nat_0.ordering_top_axioms
tff(fact_3342_add__strict__increasing2,axiom,
    ! [A: code_integer,B: code_integer,Ca: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),Ca)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(A),Ca)) ) ) ).

% add_strict_increasing2
tff(fact_3343_add__strict__increasing2,axiom,
    ! [A: rat,B: rat,Ca: 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),Ca)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),Ca)) ) ) ).

% add_strict_increasing2
tff(fact_3344_add__strict__increasing2,axiom,
    ! [A: nat,B: nat,Ca: 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),Ca)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),Ca)) ) ) ).

% add_strict_increasing2
tff(fact_3345_add__strict__increasing2,axiom,
    ! [A: int,B: int,Ca: 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),Ca)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,B),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),Ca)) ) ) ).

% add_strict_increasing2
tff(fact_3346_add__strict__increasing,axiom,
    ! [A: code_integer,B: code_integer,Ca: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),Ca)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(A),Ca)) ) ) ).

% add_strict_increasing
tff(fact_3347_add__strict__increasing,axiom,
    ! [A: rat,B: rat,Ca: 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),Ca)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),Ca)) ) ) ).

% add_strict_increasing
tff(fact_3348_add__strict__increasing,axiom,
    ! [A: nat,B: nat,Ca: 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),Ca)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),Ca)) ) ) ).

% add_strict_increasing
tff(fact_3349_add__strict__increasing,axiom,
    ! [A: int,B: int,Ca: 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),Ca)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,B),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),Ca)) ) ) ).

% add_strict_increasing
tff(fact_3350_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(plus_p5714425477246183910nteger(A),B)) ) ) ).

% add_pos_nonneg
tff(fact_3351_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_3352_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_3353_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_3354_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(plus_p5714425477246183910nteger(A),B)),zero_z3403309356797280102nteger) ) ) ).

% add_nonpos_neg
tff(fact_3355_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_3356_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_3357_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_3358_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(plus_p5714425477246183910nteger(A),B)) ) ) ).

% add_nonneg_pos
tff(fact_3359_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_3360_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_3361_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_3362_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(plus_p5714425477246183910nteger(A),B)),zero_z3403309356797280102nteger) ) ) ).

% add_neg_nonpos
tff(fact_3363_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_3364_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_3365_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_3366_mult__less__le__imp__less,axiom,
    ! [A: code_integer,B: code_integer,Ca: 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,Ca),D2)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)
         => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),Ca)
           => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),Ca)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),D2)) ) ) ) ) ).

% mult_less_le_imp_less
tff(fact_3367_mult__less__le__imp__less,axiom,
    ! [A: rat,B: rat,Ca: 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,Ca),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),Ca)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),Ca)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),D2)) ) ) ) ) ).

% mult_less_le_imp_less
tff(fact_3368_mult__less__le__imp__less,axiom,
    ! [A: nat,B: nat,Ca: 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,Ca),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),Ca)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),Ca)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D2)) ) ) ) ) ).

% mult_less_le_imp_less
tff(fact_3369_mult__less__le__imp__less,axiom,
    ! [A: int,B: int,Ca: 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,Ca),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),Ca)
           => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),Ca)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D2)) ) ) ) ) ).

% mult_less_le_imp_less
tff(fact_3370_mult__le__less__imp__less,axiom,
    ! [A: code_integer,B: code_integer,Ca: 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,Ca),D2)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A)
         => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),Ca)
           => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),Ca)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),D2)) ) ) ) ) ).

% mult_le_less_imp_less
tff(fact_3371_mult__le__less__imp__less,axiom,
    ! [A: rat,B: rat,Ca: 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,Ca),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),Ca)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),Ca)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),D2)) ) ) ) ) ).

% mult_le_less_imp_less
tff(fact_3372_mult__le__less__imp__less,axiom,
    ! [A: nat,B: nat,Ca: 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,Ca),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),Ca)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),Ca)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D2)) ) ) ) ) ).

% mult_le_less_imp_less
tff(fact_3373_mult__le__less__imp__less,axiom,
    ! [A: int,B: int,Ca: 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,Ca),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),Ca)
           => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),Ca)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D2)) ) ) ) ) ).

% mult_le_less_imp_less
tff(fact_3374_mult__right__le__imp__le,axiom,
    ! [A: code_integer,Ca: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),Ca)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),Ca))
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),Ca)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B) ) ) ).

% mult_right_le_imp_le
tff(fact_3375_mult__right__le__imp__le,axiom,
    ! [A: rat,Ca: 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),Ca)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),Ca))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Ca)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ) ).

% mult_right_le_imp_le
tff(fact_3376_mult__right__le__imp__le,axiom,
    ! [A: nat,Ca: 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),Ca)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),Ca))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Ca)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B) ) ) ).

% mult_right_le_imp_le
tff(fact_3377_mult__right__le__imp__le,axiom,
    ! [A: int,Ca: 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),Ca)),aa_int_int(aa_int_fun_int_int(times_times_int,B),Ca))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Ca)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ) ).

% mult_right_le_imp_le
tff(fact_3378_mult__left__le__imp__le,axiom,
    ! [Ca: code_integer,A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Ca),A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Ca),B))
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),Ca)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B) ) ) ).

% mult_left_le_imp_le
tff(fact_3379_mult__left__le__imp__le,axiom,
    ! [Ca: 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,Ca),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Ca),B))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Ca)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ) ).

% mult_left_le_imp_le
tff(fact_3380_mult__left__le__imp__le,axiom,
    ! [Ca: 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,Ca),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ca),B))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Ca)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B) ) ) ).

% mult_left_le_imp_le
tff(fact_3381_mult__left__le__imp__le,axiom,
    ! [Ca: 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,Ca),A)),aa_int_int(aa_int_fun_int_int(times_times_int,Ca),B))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Ca)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ) ).

% mult_left_le_imp_le
tff(fact_3382_mult__le__cancel__left__pos,axiom,
    ! [Ca: code_integer,A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),Ca)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Ca),A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Ca),B))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B) ) ) ).

% mult_le_cancel_left_pos
tff(fact_3383_mult__le__cancel__left__pos,axiom,
    ! [Ca: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Ca)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Ca),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Ca),B))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ) ).

% mult_le_cancel_left_pos
tff(fact_3384_mult__le__cancel__left__pos,axiom,
    ! [Ca: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Ca)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,Ca),A)),aa_int_int(aa_int_fun_int_int(times_times_int,Ca),B))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ) ).

% mult_le_cancel_left_pos
tff(fact_3385_mult__le__cancel__left__neg,axiom,
    ! [Ca: code_integer,A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Ca),zero_z3403309356797280102nteger)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Ca),A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Ca),B))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),A) ) ) ).

% mult_le_cancel_left_neg
tff(fact_3386_mult__le__cancel__left__neg,axiom,
    ! [Ca: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Ca),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,Ca),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Ca),B))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A) ) ) ).

% mult_le_cancel_left_neg
tff(fact_3387_mult__le__cancel__left__neg,axiom,
    ! [Ca: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Ca),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,Ca),A)),aa_int_int(aa_int_fun_int_int(times_times_int,Ca),B))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A) ) ) ).

% mult_le_cancel_left_neg
tff(fact_3388_mult__less__cancel__right,axiom,
    ! [A: code_integer,Ca: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),Ca)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),Ca))
    <=> ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),Ca)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B) )
        & ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Ca),zero_z3403309356797280102nteger)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),A) ) ) ) ).

% mult_less_cancel_right
tff(fact_3389_mult__less__cancel__right,axiom,
    ! [A: rat,Ca: 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),Ca)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),Ca))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Ca)
         => 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,Ca),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A) ) ) ) ).

% mult_less_cancel_right
tff(fact_3390_mult__less__cancel__right,axiom,
    ! [A: int,Ca: 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),Ca)),aa_int_int(aa_int_fun_int_int(times_times_int,B),Ca))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Ca)
         => 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,Ca),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,B),A) ) ) ) ).

% mult_less_cancel_right
tff(fact_3391_mult__strict__mono_H,axiom,
    ! [A: code_integer,B: code_integer,Ca: 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,Ca),D2)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)
         => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),Ca)
           => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),Ca)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),D2)) ) ) ) ) ).

% mult_strict_mono'
tff(fact_3392_mult__strict__mono_H,axiom,
    ! [A: rat,B: rat,Ca: 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,Ca),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),Ca)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),Ca)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),D2)) ) ) ) ) ).

% mult_strict_mono'
tff(fact_3393_mult__strict__mono_H,axiom,
    ! [A: nat,B: nat,Ca: 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,Ca),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),Ca)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),Ca)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D2)) ) ) ) ) ).

% mult_strict_mono'
tff(fact_3394_mult__strict__mono_H,axiom,
    ! [A: int,B: int,Ca: 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,Ca),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),Ca)
           => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),Ca)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D2)) ) ) ) ) ).

% mult_strict_mono'
tff(fact_3395_mult__right__less__imp__less,axiom,
    ! [A: code_integer,Ca: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),Ca)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),Ca))
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),Ca)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B) ) ) ).

% mult_right_less_imp_less
tff(fact_3396_mult__right__less__imp__less,axiom,
    ! [A: rat,Ca: 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),Ca)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),Ca))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Ca)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ) ).

% mult_right_less_imp_less
tff(fact_3397_mult__right__less__imp__less,axiom,
    ! [A: nat,Ca: 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),Ca)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),Ca))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Ca)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B) ) ) ).

% mult_right_less_imp_less
tff(fact_3398_mult__right__less__imp__less,axiom,
    ! [A: int,Ca: 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),Ca)),aa_int_int(aa_int_fun_int_int(times_times_int,B),Ca))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Ca)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ) ).

% mult_right_less_imp_less
tff(fact_3399_mult__less__cancel__left,axiom,
    ! [Ca: code_integer,A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Ca),A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Ca),B))
    <=> ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),Ca)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B) )
        & ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Ca),zero_z3403309356797280102nteger)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),A) ) ) ) ).

% mult_less_cancel_left
tff(fact_3400_mult__less__cancel__left,axiom,
    ! [Ca: 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,Ca),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Ca),B))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Ca)
         => 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,Ca),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A) ) ) ) ).

% mult_less_cancel_left
tff(fact_3401_mult__less__cancel__left,axiom,
    ! [Ca: 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,Ca),A)),aa_int_int(aa_int_fun_int_int(times_times_int,Ca),B))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Ca)
         => 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,Ca),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,B),A) ) ) ) ).

% mult_less_cancel_left
tff(fact_3402_mult__strict__mono,axiom,
    ! [A: code_integer,B: code_integer,Ca: 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,Ca),D2)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B)
         => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),Ca)
           => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),Ca)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),D2)) ) ) ) ) ).

% mult_strict_mono
tff(fact_3403_mult__strict__mono,axiom,
    ! [A: rat,B: rat,Ca: 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,Ca),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),Ca)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),Ca)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),D2)) ) ) ) ) ).

% mult_strict_mono
tff(fact_3404_mult__strict__mono,axiom,
    ! [A: nat,B: nat,Ca: 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,Ca),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),Ca)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),Ca)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D2)) ) ) ) ) ).

% mult_strict_mono
tff(fact_3405_mult__strict__mono,axiom,
    ! [A: int,B: int,Ca: 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,Ca),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),Ca)
           => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),Ca)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D2)) ) ) ) ) ).

% mult_strict_mono
tff(fact_3406_mult__left__less__imp__less,axiom,
    ! [Ca: code_integer,A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Ca),A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Ca),B))
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),Ca)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B) ) ) ).

% mult_left_less_imp_less
tff(fact_3407_mult__left__less__imp__less,axiom,
    ! [Ca: 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,Ca),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Ca),B))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Ca)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ) ).

% mult_left_less_imp_less
tff(fact_3408_mult__left__less__imp__less,axiom,
    ! [Ca: 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,Ca),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ca),B))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Ca)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B) ) ) ).

% mult_left_less_imp_less
tff(fact_3409_mult__left__less__imp__less,axiom,
    ! [Ca: 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,Ca),A)),aa_int_int(aa_int_fun_int_int(times_times_int,Ca),B))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Ca)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ) ).

% mult_left_less_imp_less
tff(fact_3410_mult__le__cancel__right,axiom,
    ! [A: code_integer,Ca: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),Ca)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),Ca))
    <=> ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),Ca)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B) )
        & ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Ca),zero_z3403309356797280102nteger)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),A) ) ) ) ).

% mult_le_cancel_right
tff(fact_3411_mult__le__cancel__right,axiom,
    ! [A: rat,Ca: 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),Ca)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),Ca))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Ca)
         => 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,Ca),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A) ) ) ) ).

% mult_le_cancel_right
tff(fact_3412_mult__le__cancel__right,axiom,
    ! [A: int,Ca: 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),Ca)),aa_int_int(aa_int_fun_int_int(times_times_int,B),Ca))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Ca)
         => 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,Ca),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A) ) ) ) ).

% mult_le_cancel_right
tff(fact_3413_mult__le__cancel__left,axiom,
    ! [Ca: code_integer,A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Ca),A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Ca),B))
    <=> ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),Ca)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B) )
        & ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Ca),zero_z3403309356797280102nteger)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),A) ) ) ) ).

% mult_le_cancel_left
tff(fact_3414_mult__le__cancel__left,axiom,
    ! [Ca: 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,Ca),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Ca),B))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Ca)
         => 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,Ca),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A) ) ) ) ).

% mult_le_cancel_left
tff(fact_3415_mult__le__cancel__left,axiom,
    ! [Ca: 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,Ca),A)),aa_int_int(aa_int_fun_int_int(times_times_int,Ca),B))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Ca)
         => 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,Ca),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A) ) ) ) ).

% mult_le_cancel_left
tff(fact_3416_sum__squares__ge__zero,axiom,
    ! [X: code_integer,Y: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,X),X)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Y),Y))) ).

% sum_squares_ge_zero
tff(fact_3417_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_3418_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_3419_not__sum__squares__lt__zero,axiom,
    ! [X: code_integer,Y: code_integer] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(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_3420_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_3421_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_3422_mod__emp__simp,axiom,
    ! [Ha: heap_e7401611519738050253t_unit] : aa_Pro653082453696054616_nat_o(rep_assn(one_one_assn),produc7507926704131184380et_nat(Ha,bot_bot_set_nat)) ).

% mod_emp_simp
tff(fact_3423_nat__mult__le__cancel__disj,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),N2))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2) ) ) ).

% nat_mult_le_cancel_disj
tff(fact_3424_nat__mult__less__cancel__disj,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),N2))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N2) ) ) ).

% nat_mult_less_cancel_disj
tff(fact_3425_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_3426_nat__mult__le__cancel1,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),N2))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2) ) ) ).

% nat_mult_le_cancel1
tff(fact_3427_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(plus_p5714425477246183910nteger(A),one_one_Code_integer)),B) ) ).

% discrete
tff(fact_3428_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_3429_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_3430_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(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_3431_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_3432_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_3433_sum__squares__le__zero__iff,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(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_3434_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_3435_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_3436_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_3437_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_3438_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_3439_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_3440_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_3441_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_3442_linordered__field__no__lb,axiom,
    ! [X5: rat] :
    ? [Y3: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y3),X5) ).

% linordered_field_no_lb
tff(fact_3443_linordered__field__no__ub,axiom,
    ! [X5: rat] :
    ? [X_1: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X5),X_1) ).

% linordered_field_no_ub
tff(fact_3444_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_3445_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_3446_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_3447_sum__squares__eq__zero__iff,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(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_3448_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_3449_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_3450_nat__mult__eq__cancel1,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),N2) )
      <=> ( M = N2 ) ) ) ).

% nat_mult_eq_cancel1
tff(fact_3451_nat__mult__less__cancel1,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),N2))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N2) ) ) ).

% nat_mult_less_cancel1
tff(fact_3452_field__le__epsilon,axiom,
    ! [X: rat,Y: rat] :
      ( ! [E2: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),E2)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Y),E2)) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y) ) ).

% field_le_epsilon
tff(fact_3453_divides__aux__eq,axiom,
    ! [Q4: code_integer,Ra2: code_integer] :
      ( unique5706413561485394159nteger(produc1086072967326762835nteger(Q4,Ra2))
    <=> ( Ra2 = zero_z3403309356797280102nteger ) ) ).

% divides_aux_eq
tff(fact_3454_divides__aux__eq,axiom,
    ! [Q4: nat,Ra2: nat] :
      ( aa_Pro6150637604445964947_nat_o(unique6322359934112328802ux_nat,aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Q4),Ra2))
    <=> ( Ra2 = zero_zero_nat ) ) ).

% divides_aux_eq
tff(fact_3455_divides__aux__eq,axiom,
    ! [Q4: int,Ra2: int] :
      ( aa_Pro4592068669816069595_int_o(unique6319869463603278526ux_int,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q4),Ra2))
    <=> ( Ra2 = zero_zero_int ) ) ).

% divides_aux_eq
tff(fact_3456_tap__def,axiom,
    ! [F: fun_he6342020308413958580unit_a] : heap_Time_tap_a(F) = heap_Time_Heap_a2(aTP_Lamm_ga(F)) ).

% tap_def
tff(fact_3457_tap__def,axiom,
    ! [F: fun_he5123621388056232515t_unit] : heap_T560649228465745139t_unit(F) = heap_T6183433275982383450t_unit(aTP_Lamm_gb(F)) ).

% tap_def
tff(fact_3458_execute__tap,axiom,
    ! [F: fun_he5123621388056232515t_unit,Ha: heap_e7401611519738050253t_unit] : aa_hea684056619475940329it_nat(heap_T875086893843062177t_unit(heap_T560649228465745139t_unit(F)),Ha) = aa_Pro8996696869555642526it_nat(some_P1914260805536162275it_nat,produc7133225469290080770it_nat(aa_hea4146273921685161788t_unit(F,Ha),produc584006145561248582it_nat(Ha,one_one_nat))) ).

% execute_tap
tff(fact_3459_execute__tap,axiom,
    ! [F: fun_he6342020308413958580unit_a,Ha: heap_e7401611519738050253t_unit] : aa_hea4724376117875161946it_nat(heap_Time_execute_a(heap_Time_tap_a(F)),Ha) = aa_Pro5302094286594872192it_nat(some_P7913643980934408916it_nat,aa_Pro8078911840264235907it_nat(aa_a_f2234720956197727492it_nat(produc9178034014595674355it_nat,aa_hea5100639427662539565unit_a(F,Ha)),produc584006145561248582it_nat(Ha,one_one_nat))) ).

% execute_tap
tff(fact_3460_one__assn__raw_Opelims_I3_J,axiom,
    ! [X: produc3658429121746597890et_nat] :
      ( ~ aa_Pro653082453696054616_nat_o(one_assn_raw,X)
     => ( aa_Pro653082453696054616_nat_o(accp_P5801069581201407417et_nat(one_assn_raw_rel),X)
       => ~ ! [H: heap_e7401611519738050253t_unit,As3: set_nat] :
              ( ( X = produc7507926704131184380et_nat(H,As3) )
             => ( aa_Pro653082453696054616_nat_o(accp_P5801069581201407417et_nat(one_assn_raw_rel),produc7507926704131184380et_nat(H,As3))
               => ( As3 = bot_bot_set_nat ) ) ) ) ) ).

% one_assn_raw.pelims(3)
tff(fact_3461_one__assn__raw_Opelims_I2_J,axiom,
    ! [X: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(one_assn_raw,X)
     => ( aa_Pro653082453696054616_nat_o(accp_P5801069581201407417et_nat(one_assn_raw_rel),X)
       => ~ ! [H: heap_e7401611519738050253t_unit,As3: set_nat] :
              ( ( X = produc7507926704131184380et_nat(H,As3) )
             => ( aa_Pro653082453696054616_nat_o(accp_P5801069581201407417et_nat(one_assn_raw_rel),produc7507926704131184380et_nat(H,As3))
               => ( As3 != bot_bot_set_nat ) ) ) ) ) ).

% one_assn_raw.pelims(2)
tff(fact_3462_one__assn__raw_Opelims_I1_J,axiom,
    ! [X: produc3658429121746597890et_nat,Y: $o] :
      ( ( aa_Pro653082453696054616_nat_o(one_assn_raw,X)
      <=> (Y) )
     => ( aa_Pro653082453696054616_nat_o(accp_P5801069581201407417et_nat(one_assn_raw_rel),X)
       => ~ ! [H: heap_e7401611519738050253t_unit,As3: set_nat] :
              ( ( X = produc7507926704131184380et_nat(H,As3) )
             => ( ( (Y)
                <=> ( As3 = bot_bot_set_nat ) )
               => ~ aa_Pro653082453696054616_nat_o(accp_P5801069581201407417et_nat(one_assn_raw_rel),produc7507926704131184380et_nat(H,As3)) ) ) ) ) ).

% one_assn_raw.pelims(1)
tff(fact_3463_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_3464_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_3465_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_3466_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_3467_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_3468_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_3469_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_3470_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_3471_bounded__Max__nat,axiom,
    ! [Pa: fun_nat_o,X: nat,M6: nat] :
      ( aa_nat_o(Pa,X)
     => ( ! [X3: nat] :
            ( aa_nat_o(Pa,X3)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),M6) )
       => ~ ! [M5: nat] :
              ( aa_nat_o(Pa,M5)
             => ~ ! [X5: nat] :
                    ( aa_nat_o(Pa,X5)
                   => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X5),M5) ) ) ) ) ).

% bounded_Max_nat
tff(fact_3472_le__numeral__extra_I3_J,axiom,
    aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),zero_z3403309356797280102nteger) ).

% le_numeral_extra(3)
tff(fact_3473_le__numeral__extra_I3_J,axiom,
    aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),zero_zero_rat) ).

% le_numeral_extra(3)
tff(fact_3474_le__numeral__extra_I3_J,axiom,
    aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),zero_zero_nat) ).

% le_numeral_extra(3)
tff(fact_3475_le__numeral__extra_I3_J,axiom,
    aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),zero_zero_int) ).

% le_numeral_extra(3)
tff(fact_3476_less__numeral__extra_I3_J,axiom,
    ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),zero_z3403309356797280102nteger) ).

% less_numeral_extra(3)
tff(fact_3477_less__numeral__extra_I3_J,axiom,
    ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),zero_zero_rat) ).

% less_numeral_extra(3)
tff(fact_3478_less__numeral__extra_I3_J,axiom,
    ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),zero_zero_nat) ).

% less_numeral_extra(3)
tff(fact_3479_less__numeral__extra_I3_J,axiom,
    ~ aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),zero_zero_int) ).

% less_numeral_extra(3)
tff(fact_3480_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_3481_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_3482_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_3483_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_3484_power__decreasing__iff,axiom,
    ! [B: code_integer,M: nat,N2: 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),N2))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),M) ) ) ) ).

% power_decreasing_iff
tff(fact_3485_power__decreasing__iff,axiom,
    ! [B: rat,M: nat,N2: 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),N2))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),M) ) ) ) ).

% power_decreasing_iff
tff(fact_3486_power__decreasing__iff,axiom,
    ! [B: nat,M: nat,N2: 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),N2))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),M) ) ) ) ).

% power_decreasing_iff
tff(fact_3487_power__decreasing__iff,axiom,
    ! [B: int,M: nat,N2: 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),N2))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),M) ) ) ) ).

% power_decreasing_iff
tff(fact_3488_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_3489_fold__atLeastAtMost__nat_Oelims,axiom,
    ! [X: fun_nat_fun_nat_nat,Xa2: nat,Xb: nat,Xc: nat,Y: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(X,Xa2),Xb),Xc) = Y )
     => ( Y = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Xa2),Xc,aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(X,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xa2),one_one_nat)),Xb),aa_nat_nat(aa_nat_fun_nat_nat(X,Xa2),Xc))) ) ) ).

% fold_atLeastAtMost_nat.elims
tff(fact_3490_power__strict__mono,axiom,
    ! [A: code_integer,B: code_integer,N2: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),N2)),aa_nat_Code_integer(power_8256067586552552935nteger(B),N2)) ) ) ) ).

% power_strict_mono
tff(fact_3491_power__strict__mono,axiom,
    ! [A: rat,B: rat,N2: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(A),N2)),aa_nat_rat(power_power_rat(B),N2)) ) ) ) ).

% power_strict_mono
tff(fact_3492_power__strict__mono,axiom,
    ! [A: nat,B: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N2)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),N2)) ) ) ) ).

% power_strict_mono
tff(fact_3493_power__strict__mono,axiom,
    ! [A: int,B: int,N2: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A),N2)),aa_nat_int(aa_int_fun_nat_int(power_power_int,B),N2)) ) ) ) ).

% power_strict_mono
tff(fact_3494_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_3495_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_3496_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_3497_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_3498_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_3499_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_3500_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_3501_times__divide__eq__right,axiom,
    ! [A: rat,B: rat,Ca: 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),Ca)) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)),Ca) ).

% times_divide_eq_right
tff(fact_3502_divide__divide__eq__right,axiom,
    ! [A: rat,B: rat,Ca: 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),Ca)) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),Ca)),B) ).

% divide_divide_eq_right
tff(fact_3503_divide__divide__eq__left,axiom,
    ! [A: rat,B: rat,Ca: 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)),Ca) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),Ca)) ).

% divide_divide_eq_left
tff(fact_3504_times__divide__eq__left,axiom,
    ! [B: rat,Ca: 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),Ca)),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)),Ca) ).

% times_divide_eq_left
tff(fact_3505_nat__zero__less__power__iff,axiom,
    ! [X: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X),N2))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),X)
        | ( N2 = zero_zero_nat ) ) ) ).

% nat_zero_less_power_iff
tff(fact_3506_mult__divide__mult__cancel__left__if,axiom,
    ! [Ca: 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,Ca),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Ca),B)) = $ite(Ca = 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_3507_nonzero__mult__divide__mult__cancel__left,axiom,
    ! [Ca: rat,A: rat,B: rat] :
      ( ( Ca != 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,Ca),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Ca),B)) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),B) ) ) ).

% nonzero_mult_divide_mult_cancel_left
tff(fact_3508_nonzero__mult__divide__mult__cancel__left2,axiom,
    ! [Ca: rat,A: rat,B: rat] :
      ( ( Ca != 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,Ca),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),Ca)) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),B) ) ) ).

% nonzero_mult_divide_mult_cancel_left2
tff(fact_3509_nonzero__mult__divide__mult__cancel__right,axiom,
    ! [Ca: rat,A: rat,B: rat] :
      ( ( Ca != 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),Ca)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),Ca)) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),B) ) ) ).

% nonzero_mult_divide_mult_cancel_right
tff(fact_3510_nonzero__mult__divide__mult__cancel__right2,axiom,
    ! [Ca: rat,A: rat,B: rat] :
      ( ( Ca != 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),Ca)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Ca),B)) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),B) ) ) ).

% nonzero_mult_divide_mult_cancel_right2
tff(fact_3511_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_3512_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_3513_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_3514_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_3515_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_3516_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_3517_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_3518_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_3519_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_3520_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_3521_power__inject__exp,axiom,
    ! [A: code_integer,M: nat,N2: 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),N2) )
      <=> ( M = N2 ) ) ) ).

% power_inject_exp
tff(fact_3522_power__inject__exp,axiom,
    ! [A: rat,M: nat,N2: 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),N2) )
      <=> ( M = N2 ) ) ) ).

% power_inject_exp
tff(fact_3523_power__inject__exp,axiom,
    ! [A: nat,M: nat,N2: 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),N2) )
      <=> ( M = N2 ) ) ) ).

% power_inject_exp
tff(fact_3524_power__inject__exp,axiom,
    ! [A: int,M: nat,N2: 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),N2) )
      <=> ( M = N2 ) ) ) ).

% power_inject_exp
tff(fact_3525_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_3526_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_3527_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_3528_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_3529_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_3530_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_3531_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_3532_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_3533_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_3534_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_3535_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_3536_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_3537_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_3538_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_3539_power__eq__0__iff,axiom,
    ! [A: code_integer,N2: nat] :
      ( ( aa_nat_Code_integer(power_8256067586552552935nteger(A),N2) = zero_z3403309356797280102nteger )
    <=> ( ( A = zero_z3403309356797280102nteger )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2) ) ) ).

% power_eq_0_iff
tff(fact_3540_power__eq__0__iff,axiom,
    ! [A: rat,N2: nat] :
      ( ( aa_nat_rat(power_power_rat(A),N2) = zero_zero_rat )
    <=> ( ( A = zero_zero_rat )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2) ) ) ).

% power_eq_0_iff
tff(fact_3541_power__eq__0__iff,axiom,
    ! [A: nat,N2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N2) = zero_zero_nat )
    <=> ( ( A = zero_zero_nat )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2) ) ) ).

% power_eq_0_iff
tff(fact_3542_power__eq__0__iff,axiom,
    ! [A: int,N2: nat] :
      ( ( aa_nat_int(aa_int_fun_nat_int(power_power_int,A),N2) = zero_zero_int )
    <=> ( ( A = zero_zero_int )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2) ) ) ).

% power_eq_0_iff
tff(fact_3543_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_3544_power__strict__decreasing__iff,axiom,
    ! [B: code_integer,M: nat,N2: 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),N2))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),M) ) ) ) ).

% power_strict_decreasing_iff
tff(fact_3545_power__strict__decreasing__iff,axiom,
    ! [B: rat,M: nat,N2: 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),N2))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),M) ) ) ) ).

% power_strict_decreasing_iff
tff(fact_3546_power__strict__decreasing__iff,axiom,
    ! [B: nat,M: nat,N2: 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),N2))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),M) ) ) ) ).

% power_strict_decreasing_iff
tff(fact_3547_power__strict__decreasing__iff,axiom,
    ! [B: int,M: nat,N2: 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),N2))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),M) ) ) ) ).

% power_strict_decreasing_iff
tff(fact_3548_power__mono__iff,axiom,
    ! [A: code_integer,B: code_integer,N2: nat] :
      ( 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_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)
         => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),N2)),aa_nat_Code_integer(power_8256067586552552935nteger(B),N2))
          <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B) ) ) ) ) ).

% power_mono_iff
tff(fact_3549_power__mono__iff,axiom,
    ! [A: rat,B: rat,N2: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),N2)),aa_nat_rat(power_power_rat(B),N2))
          <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ) ) ) ).

% power_mono_iff
tff(fact_3550_power__mono__iff,axiom,
    ! [A: nat,B: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N2)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),N2))
          <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B) ) ) ) ) ).

% power_mono_iff
tff(fact_3551_power__mono__iff,axiom,
    ! [A: int,B: int,N2: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A),N2)),aa_nat_int(aa_int_fun_nat_int(power_power_int,B),N2))
          <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ) ) ) ).

% power_mono_iff
tff(fact_3552_power__commuting__commutes,axiom,
    ! [X: code_integer,Y: code_integer,N2: 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),N2)),Y) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Y),aa_nat_Code_integer(power_8256067586552552935nteger(X),N2)) ) ) ).

% power_commuting_commutes
tff(fact_3553_power__commuting__commutes,axiom,
    ! [X: assn,Y: assn,N2: 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),N2)),Y) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Y),aa_nat_assn(power_power_assn(X),N2)) ) ) ).

% power_commuting_commutes
tff(fact_3554_power__commuting__commutes,axiom,
    ! [X: rat,Y: rat,N2: 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),N2)),Y) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),aa_nat_rat(power_power_rat(X),N2)) ) ) ).

% power_commuting_commutes
tff(fact_3555_power__commuting__commutes,axiom,
    ! [X: nat,Y: nat,N2: 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),N2)),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),N2)) ) ) ).

% power_commuting_commutes
tff(fact_3556_power__commuting__commutes,axiom,
    ! [X: int,Y: int,N2: 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),N2)),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),N2)) ) ) ).

% power_commuting_commutes
tff(fact_3557_power__mult__distrib,axiom,
    ! [A: code_integer,B: code_integer,N2: nat] : aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),N2) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),N2)),aa_nat_Code_integer(power_8256067586552552935nteger(B),N2)) ).

% power_mult_distrib
tff(fact_3558_power__mult__distrib,axiom,
    ! [A: assn,B: assn,N2: nat] : aa_nat_assn(power_power_assn(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A),B)),N2) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_nat_assn(power_power_assn(A),N2)),aa_nat_assn(power_power_assn(B),N2)) ).

% power_mult_distrib
tff(fact_3559_power__mult__distrib,axiom,
    ! [A: rat,B: rat,N2: nat] : aa_nat_rat(power_power_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)),N2) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(A),N2)),aa_nat_rat(power_power_rat(B),N2)) ).

% power_mult_distrib
tff(fact_3560_power__mult__distrib,axiom,
    ! [A: nat,B: nat,N2: 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)),N2) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N2)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),N2)) ).

% power_mult_distrib
tff(fact_3561_power__mult__distrib,axiom,
    ! [A: int,B: int,N2: 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)),N2) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A),N2)),aa_nat_int(aa_int_fun_nat_int(power_power_int,B),N2)) ).

% power_mult_distrib
tff(fact_3562_power__commutes,axiom,
    ! [A: code_integer,N2: nat] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),N2)),A) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),aa_nat_Code_integer(power_8256067586552552935nteger(A),N2)) ).

% power_commutes
tff(fact_3563_power__commutes,axiom,
    ! [A: assn,N2: nat] : aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_nat_assn(power_power_assn(A),N2)),A) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A),aa_nat_assn(power_power_assn(A),N2)) ).

% power_commutes
tff(fact_3564_power__commutes,axiom,
    ! [A: rat,N2: nat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(A),N2)),A) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_nat_rat(power_power_rat(A),N2)) ).

% power_commutes
tff(fact_3565_power__commutes,axiom,
    ! [A: nat,N2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N2)),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),N2)) ).

% power_commutes
tff(fact_3566_power__commutes,axiom,
    ! [A: int,N2: nat] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A),N2)),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),N2)) ).

% power_commutes
tff(fact_3567_divide__divide__eq__left_H,axiom,
    ! [A: rat,B: rat,Ca: 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)),Ca) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Ca),B)) ).

% divide_divide_eq_left'
tff(fact_3568_divide__divide__times__eq,axiom,
    ! [X: rat,Y: rat,Z: rat,W2: 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),W2)) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X),W2)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Z)) ).

% divide_divide_times_eq
tff(fact_3569_times__divide__times__eq,axiom,
    ! [X: rat,Y: rat,Z: rat,W2: 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),W2)) = 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),W2)) ).

% times_divide_times_eq
tff(fact_3570_power__mono,axiom,
    ! [A: code_integer,B: code_integer,N2: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),N2)),aa_nat_Code_integer(power_8256067586552552935nteger(B),N2)) ) ) ).

% power_mono
tff(fact_3571_power__mono,axiom,
    ! [A: rat,B: rat,N2: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),N2)),aa_nat_rat(power_power_rat(B),N2)) ) ) ).

% power_mono
tff(fact_3572_power__mono,axiom,
    ! [A: nat,B: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N2)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),N2)) ) ) ).

% power_mono
tff(fact_3573_power__mono,axiom,
    ! [A: int,B: int,N2: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A),N2)),aa_nat_int(aa_int_fun_nat_int(power_power_int,B),N2)) ) ) ).

% power_mono
tff(fact_3574_zero__le__power,axiom,
    ! [A: code_integer,N2: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(power_8256067586552552935nteger(A),N2)) ) ).

% zero_le_power
tff(fact_3575_zero__le__power,axiom,
    ! [A: rat,N2: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),N2)) ) ).

% zero_le_power
tff(fact_3576_zero__le__power,axiom,
    ! [A: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N2)) ) ).

% zero_le_power
tff(fact_3577_zero__le__power,axiom,
    ! [A: int,N2: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(aa_int_fun_nat_int(power_power_int,A),N2)) ) ).

% zero_le_power
tff(fact_3578_zero__less__power,axiom,
    ! [A: code_integer,N2: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(power_8256067586552552935nteger(A),N2)) ) ).

% zero_less_power
tff(fact_3579_zero__less__power,axiom,
    ! [A: rat,N2: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),N2)) ) ).

% zero_less_power
tff(fact_3580_zero__less__power,axiom,
    ! [A: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N2)) ) ).

% zero_less_power
tff(fact_3581_zero__less__power,axiom,
    ! [A: int,N2: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(aa_int_fun_nat_int(power_power_int,A),N2)) ) ).

% zero_less_power
tff(fact_3582_one__le__power,axiom,
    ! [A: code_integer,N2: 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),N2)) ) ).

% one_le_power
tff(fact_3583_one__le__power,axiom,
    ! [A: rat,N2: 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),N2)) ) ).

% one_le_power
tff(fact_3584_one__le__power,axiom,
    ! [A: nat,N2: 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),N2)) ) ).

% one_le_power
tff(fact_3585_one__le__power,axiom,
    ! [A: int,N2: 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),N2)) ) ).

% one_le_power
tff(fact_3586_divide__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(divide_divide_rat,A),B)),zero_zero_rat)
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),zero_zero_rat) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B) ) ) ) ).

% divide_le_0_iff
tff(fact_3587_divide__right__mono,axiom,
    ! [A: rat,B: rat,Ca: 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),Ca)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),Ca)),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),Ca)) ) ) ).

% divide_right_mono
tff(fact_3588_zero__le__divide__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),B))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),zero_zero_rat) ) ) ) ).

% zero_le_divide_iff
tff(fact_3589_divide__nonneg__nonneg,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,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,X),Y)) ) ) ).

% divide_nonneg_nonneg
tff(fact_3590_divide__nonneg__nonpos,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,Y),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)),zero_zero_rat) ) ) ).

% divide_nonneg_nonpos
tff(fact_3591_divide__nonpos__nonneg,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,zero_zero_rat),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)),zero_zero_rat) ) ) ).

% divide_nonpos_nonneg
tff(fact_3592_divide__nonpos__nonpos,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_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,X),Y)) ) ) ).

% divide_nonpos_nonpos
tff(fact_3593_divide__right__mono__neg,axiom,
    ! [A: rat,B: rat,Ca: 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,Ca),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),Ca)),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),Ca)) ) ) ).

% divide_right_mono_neg
tff(fact_3594_divide__neg__neg,axiom,
    ! [X: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),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(divide_divide_rat,X),Y)) ) ) ).

% divide_neg_neg
tff(fact_3595_divide__neg__pos,axiom,
    ! [X: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),zero_zero_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(divide_divide_rat,X),Y)),zero_zero_rat) ) ) ).

% divide_neg_pos
tff(fact_3596_divide__pos__neg,axiom,
    ! [X: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),X)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),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)),zero_zero_rat) ) ) ).

% divide_pos_neg
tff(fact_3597_divide__pos__pos,axiom,
    ! [X: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),X)
     => ( 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,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,X),Y)) ) ) ).

% divide_pos_pos
tff(fact_3598_divide__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(divide_divide_rat,A),B)),zero_zero_rat)
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),zero_zero_rat) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B) ) ) ) ).

% divide_less_0_iff
tff(fact_3599_divide__less__cancel,axiom,
    ! [A: rat,Ca: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),Ca)),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),Ca))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Ca)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Ca),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A) )
        & ( Ca != zero_zero_rat ) ) ) ).

% divide_less_cancel
tff(fact_3600_zero__less__divide__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),B))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),zero_zero_rat) ) ) ) ).

% zero_less_divide_iff
tff(fact_3601_divide__strict__right__mono,axiom,
    ! [A: rat,B: rat,Ca: 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),Ca)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),Ca)),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),Ca)) ) ) ).

% divide_strict_right_mono
tff(fact_3602_divide__strict__right__mono__neg,axiom,
    ! [B: rat,A: rat,Ca: 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,Ca),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,A),Ca)),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),Ca)) ) ) ).

% divide_strict_right_mono_neg
tff(fact_3603_left__right__inverse__power,axiom,
    ! [X: code_integer,Y: code_integer,N2: 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),N2)),aa_nat_Code_integer(power_8256067586552552935nteger(Y),N2)) = one_one_Code_integer ) ) ).

% left_right_inverse_power
tff(fact_3604_left__right__inverse__power,axiom,
    ! [X: assn,Y: assn,N2: 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),N2)),aa_nat_assn(power_power_assn(Y),N2)) = one_one_assn ) ) ).

% left_right_inverse_power
tff(fact_3605_left__right__inverse__power,axiom,
    ! [X: rat,Y: rat,N2: 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),N2)),aa_nat_rat(power_power_rat(Y),N2)) = one_one_rat ) ) ).

% left_right_inverse_power
tff(fact_3606_left__right__inverse__power,axiom,
    ! [X: nat,Y: nat,N2: 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),N2)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Y),N2)) = one_one_nat ) ) ).

% left_right_inverse_power
tff(fact_3607_left__right__inverse__power,axiom,
    ! [X: int,Y: int,N2: 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),N2)),aa_nat_int(aa_int_fun_nat_int(power_power_int,Y),N2)) = one_one_int ) ) ).

% left_right_inverse_power
tff(fact_3608_frac__eq__eq,axiom,
    ! [Y: rat,Z: rat,X: rat,W2: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( Z != zero_zero_rat )
       => ( ( aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,X),Y) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,W2),Z) )
        <=> ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X),Z) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,W2),Y) ) ) ) ) ).

% frac_eq_eq
tff(fact_3609_divide__eq__eq,axiom,
    ! [B: rat,Ca: rat,A: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),Ca) = A )
    <=> $ite(Ca != zero_zero_rat,B = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),Ca),A = zero_zero_rat) ) ).

% divide_eq_eq
tff(fact_3610_eq__divide__eq,axiom,
    ! [A: rat,B: rat,Ca: rat] :
      ( ( A = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),Ca) )
    <=> $ite(Ca != zero_zero_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),Ca) = B,A = zero_zero_rat) ) ).

% eq_divide_eq
tff(fact_3611_divide__eq__imp,axiom,
    ! [Ca: rat,B: rat,A: rat] :
      ( ( Ca != zero_zero_rat )
     => ( ( B = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),Ca) )
       => ( aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),Ca) = A ) ) ) ).

% divide_eq_imp
tff(fact_3612_eq__divide__imp,axiom,
    ! [Ca: rat,A: rat,B: rat] :
      ( ( Ca != zero_zero_rat )
     => ( ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),Ca) = B )
       => ( A = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),Ca) ) ) ) ).

% eq_divide_imp
tff(fact_3613_nonzero__divide__eq__eq,axiom,
    ! [Ca: rat,B: rat,A: rat] :
      ( ( Ca != zero_zero_rat )
     => ( ( aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),Ca) = A )
      <=> ( B = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),Ca) ) ) ) ).

% nonzero_divide_eq_eq
tff(fact_3614_nonzero__eq__divide__eq,axiom,
    ! [Ca: rat,A: rat,B: rat] :
      ( ( Ca != zero_zero_rat )
     => ( ( A = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),Ca) )
      <=> ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),Ca) = B ) ) ) ).

% nonzero_eq_divide_eq
tff(fact_3615_power__add,axiom,
    ! [A: code_integer,M: nat,N2: nat] : aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N2)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),M)),aa_nat_Code_integer(power_8256067586552552935nteger(A),N2)) ).

% power_add
tff(fact_3616_power__add,axiom,
    ! [A: assn,M: nat,N2: nat] : aa_nat_assn(power_power_assn(A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N2)) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_nat_assn(power_power_assn(A),M)),aa_nat_assn(power_power_assn(A),N2)) ).

% power_add
tff(fact_3617_power__add,axiom,
    ! [A: rat,M: nat,N2: nat] : aa_nat_rat(power_power_rat(A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N2)) = 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),N2)) ).

% power_add
tff(fact_3618_power__add,axiom,
    ! [A: nat,M: nat,N2: 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),N2)) = 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),N2)) ).

% power_add
tff(fact_3619_power__add,axiom,
    ! [A: int,M: nat,N2: 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),N2)) = 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),N2)) ).

% power_add
tff(fact_3620_nat__power__less__imp__less,axiom,
    ! [I2: nat,M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),I2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,I2),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,I2),N2))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N2) ) ) ).

% nat_power_less_imp_less
tff(fact_3621_power__less__imp__less__base,axiom,
    ! [A: code_integer,N2: nat,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),N2)),aa_nat_Code_integer(power_8256067586552552935nteger(B),N2))
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),B)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B) ) ) ).

% power_less_imp_less_base
tff(fact_3622_power__less__imp__less__base,axiom,
    ! [A: rat,N2: nat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(A),N2)),aa_nat_rat(power_power_rat(B),N2))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ) ).

% power_less_imp_less_base
tff(fact_3623_power__less__imp__less__base,axiom,
    ! [A: nat,N2: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N2)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),N2))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B) ) ) ).

% power_less_imp_less_base
tff(fact_3624_power__less__imp__less__base,axiom,
    ! [A: int,N2: nat,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A),N2)),aa_nat_int(aa_int_fun_nat_int(power_power_int,B),N2))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ) ).

% power_less_imp_less_base
tff(fact_3625_power__le__one,axiom,
    ! [A: code_integer,N2: 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),N2)),one_one_Code_integer) ) ) ).

% power_le_one
tff(fact_3626_power__le__one,axiom,
    ! [A: rat,N2: 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),N2)),one_one_rat) ) ) ).

% power_le_one
tff(fact_3627_power__le__one,axiom,
    ! [A: nat,N2: 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),N2)),one_one_nat) ) ) ).

% power_le_one
tff(fact_3628_power__le__one,axiom,
    ! [A: int,N2: 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),N2)),one_one_int) ) ) ).

% power_le_one
tff(fact_3629_frac__le,axiom,
    ! [Y: rat,X: rat,W2: rat,Z: rat] :
      ( 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,X),Y)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),W2)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,W2),Z)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,X),Z)),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,Y),W2)) ) ) ) ) ).

% frac_le
tff(fact_3630_frac__less,axiom,
    ! [X: rat,Y: rat,W2: rat,Z: 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_rat,X),Y)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),W2)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,W2),Z)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,X),Z)),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,Y),W2)) ) ) ) ) ).

% frac_less
tff(fact_3631_frac__less2,axiom,
    ! [X: rat,Y: rat,W2: rat,Z: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),X)
     => ( 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,zero_zero_rat),W2)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,W2),Z)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,X),Z)),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,Y),W2)) ) ) ) ) ).

% frac_less2
tff(fact_3632_divide__le__cancel,axiom,
    ! [A: rat,Ca: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),Ca)),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),Ca))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Ca)
         => 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,Ca),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A) ) ) ) ).

% divide_le_cancel
tff(fact_3633_divide__nonneg__neg,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_rat,Y),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)),zero_zero_rat) ) ) ).

% divide_nonneg_neg
tff(fact_3634_divide__nonneg__pos,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_rat,zero_zero_rat),Y)
       => 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,X),Y)) ) ) ).

% divide_nonneg_pos
tff(fact_3635_divide__nonpos__neg,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_rat,Y),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(divide_divide_rat,X),Y)) ) ) ).

% divide_nonpos_neg
tff(fact_3636_divide__nonpos__pos,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_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(divide_divide_rat,X),Y)),zero_zero_rat) ) ) ).

% divide_nonpos_pos
tff(fact_3637_unique__euclidean__semiring__numeral__class_Odiv__less,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B)
       => ( aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),B) = zero_z3403309356797280102nteger ) ) ) ).

% unique_euclidean_semiring_numeral_class.div_less
tff(fact_3638_unique__euclidean__semiring__numeral__class_Odiv__less,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B) = zero_zero_nat ) ) ) ).

% unique_euclidean_semiring_numeral_class.div_less
tff(fact_3639_unique__euclidean__semiring__numeral__class_Odiv__less,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B) = zero_zero_int ) ) ) ).

% unique_euclidean_semiring_numeral_class.div_less
tff(fact_3640_div__positive,axiom,
    ! [B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),A)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),B)) ) ) ).

% div_positive
tff(fact_3641_div__positive,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)) ) ) ).

% div_positive
tff(fact_3642_div__positive,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)) ) ) ).

% div_positive
tff(fact_3643_power__gt1__lemma,axiom,
    ! [A: code_integer,N2: 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),N2))) ) ).

% power_gt1_lemma
tff(fact_3644_power__gt1__lemma,axiom,
    ! [A: rat,N2: 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),N2))) ) ).

% power_gt1_lemma
tff(fact_3645_power__gt1__lemma,axiom,
    ! [A: nat,N2: 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),N2))) ) ).

% power_gt1_lemma
tff(fact_3646_power__gt1__lemma,axiom,
    ! [A: int,N2: 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),N2))) ) ).

% power_gt1_lemma
tff(fact_3647_power__less__power__Suc,axiom,
    ! [A: code_integer,N2: 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),N2)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),aa_nat_Code_integer(power_8256067586552552935nteger(A),N2))) ) ).

% power_less_power_Suc
tff(fact_3648_power__less__power__Suc,axiom,
    ! [A: rat,N2: 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),N2)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_nat_rat(power_power_rat(A),N2))) ) ).

% power_less_power_Suc
tff(fact_3649_power__less__power__Suc,axiom,
    ! [A: nat,N2: 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),N2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N2))) ) ).

% power_less_power_Suc
tff(fact_3650_power__less__power__Suc,axiom,
    ! [A: int,N2: 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),N2)),aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_nat_int(aa_int_fun_nat_int(power_power_int,A),N2))) ) ).

% power_less_power_Suc
tff(fact_3651_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
    ! [Ca: code_integer,A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),Ca)
     => ( aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),Ca)) = aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),B)),Ca) ) ) ).

% unique_euclidean_semiring_numeral_class.div_mult2_eq
tff(fact_3652_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
    ! [Ca: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Ca)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),Ca)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),Ca) ) ) ).

% unique_euclidean_semiring_numeral_class.div_mult2_eq
tff(fact_3653_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
    ! [Ca: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Ca)
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),Ca)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),Ca) ) ) ).

% unique_euclidean_semiring_numeral_class.div_mult2_eq
tff(fact_3654_divide__less__eq,axiom,
    ! [B: rat,Ca: 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),Ca)),A)
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Ca),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),Ca)),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Ca),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),Ca)),B),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)) ) ) ).

% divide_less_eq
tff(fact_3655_less__divide__eq,axiom,
    ! [A: rat,B: rat,Ca: 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),Ca))
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Ca),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),Ca)),B),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Ca),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),Ca)),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)) ) ) ).

% less_divide_eq
tff(fact_3656_neg__divide__less__eq,axiom,
    ! [Ca: rat,B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Ca),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),Ca)),A)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),Ca)),B) ) ) ).

% neg_divide_less_eq
tff(fact_3657_neg__less__divide__eq,axiom,
    ! [Ca: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Ca),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),Ca))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),Ca)) ) ) ).

% neg_less_divide_eq
tff(fact_3658_pos__divide__less__eq,axiom,
    ! [Ca: rat,B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Ca)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),Ca)),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),Ca)) ) ) ).

% pos_divide_less_eq
tff(fact_3659_pos__less__divide__eq,axiom,
    ! [Ca: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Ca)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),Ca))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),Ca)),B) ) ) ).

% pos_less_divide_eq
tff(fact_3660_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_3661_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_3662_divide__strict__left__mono,axiom,
    ! [B: rat,A: rat,Ca: 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),Ca)
       => ( 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,Ca),A)),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,Ca),B)) ) ) ) ).

% divide_strict_left_mono
tff(fact_3663_divide__strict__left__mono__neg,axiom,
    ! [A: rat,B: rat,Ca: 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,Ca),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,Ca),A)),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,Ca),B)) ) ) ) ).

% divide_strict_left_mono_neg
tff(fact_3664_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_3665_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_3666_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_3667_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_3668_add__frac__eq,axiom,
    ! [Y: rat,Z: rat,X: rat,W2: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( Z != zero_zero_rat )
       => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,X),Y)),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,W2),Z)) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X),Z)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,W2),Y))),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Z)) ) ) ) ).

% add_frac_eq
tff(fact_3669_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_3670_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_3671_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_3672_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_3673_power__less__imp__less__exp,axiom,
    ! [A: code_integer,M: nat,N2: 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),N2))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N2) ) ) ).

% power_less_imp_less_exp
tff(fact_3674_power__less__imp__less__exp,axiom,
    ! [A: rat,M: nat,N2: 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),N2))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N2) ) ) ).

% power_less_imp_less_exp
tff(fact_3675_power__less__imp__less__exp,axiom,
    ! [A: nat,M: nat,N2: 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),N2))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N2) ) ) ).

% power_less_imp_less_exp
tff(fact_3676_power__less__imp__less__exp,axiom,
    ! [A: int,M: nat,N2: 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),N2))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N2) ) ) ).

% power_less_imp_less_exp
tff(fact_3677_power__strict__increasing,axiom,
    ! [N2: nat,N7: nat,A: code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),N7)
     => ( 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),N2)),aa_nat_Code_integer(power_8256067586552552935nteger(A),N7)) ) ) ).

% power_strict_increasing
tff(fact_3678_power__strict__increasing,axiom,
    ! [N2: nat,N7: nat,A: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),N7)
     => ( 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),N2)),aa_nat_rat(power_power_rat(A),N7)) ) ) ).

% power_strict_increasing
tff(fact_3679_power__strict__increasing,axiom,
    ! [N2: nat,N7: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),N7)
     => ( 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),N2)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N7)) ) ) ).

% power_strict_increasing
tff(fact_3680_power__strict__increasing,axiom,
    ! [N2: nat,N7: nat,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),N7)
     => ( 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),N2)),aa_nat_int(aa_int_fun_nat_int(power_power_int,A),N7)) ) ) ).

% power_strict_increasing
tff(fact_3681_power__increasing,axiom,
    ! [N2: nat,N7: nat,A: code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),N7)
     => ( 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),N2)),aa_nat_Code_integer(power_8256067586552552935nteger(A),N7)) ) ) ).

% power_increasing
tff(fact_3682_power__increasing,axiom,
    ! [N2: nat,N7: nat,A: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),N7)
     => ( 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),N2)),aa_nat_rat(power_power_rat(A),N7)) ) ) ).

% power_increasing
tff(fact_3683_power__increasing,axiom,
    ! [N2: nat,N7: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),N7)
     => ( 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),N2)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N7)) ) ) ).

% power_increasing
tff(fact_3684_power__increasing,axiom,
    ! [N2: nat,N7: nat,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),N7)
     => ( 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),N2)),aa_nat_int(aa_int_fun_nat_int(power_power_int,A),N7)) ) ) ).

% power_increasing
tff(fact_3685_zero__power,axiom,
    ! [N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)
     => ( aa_nat_Code_integer(power_8256067586552552935nteger(zero_z3403309356797280102nteger),N2) = zero_z3403309356797280102nteger ) ) ).

% zero_power
tff(fact_3686_zero__power,axiom,
    ! [N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)
     => ( aa_nat_rat(power_power_rat(zero_zero_rat),N2) = zero_zero_rat ) ) ).

% zero_power
tff(fact_3687_zero__power,axiom,
    ! [N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,zero_zero_nat),N2) = zero_zero_nat ) ) ).

% zero_power
tff(fact_3688_zero__power,axiom,
    ! [N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)
     => ( aa_nat_int(aa_int_fun_nat_int(power_power_int,zero_zero_int),N2) = zero_zero_int ) ) ).

% zero_power
tff(fact_3689_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_3690_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_3691_nat__mult__div__cancel1,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),N2)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N2) ) ) ).

% nat_mult_div_cancel1
tff(fact_3692_fold__atLeastAtMost__nat_Opelims,axiom,
    ! [X: fun_nat_fun_nat_nat,Xa2: nat,Xb: nat,Xc: nat,Y: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(X,Xa2),Xb),Xc) = Y )
     => ( accp_P4694664897847628410at_nat(set_fo3699595496184130361el_nat,produc2818902965508115497at_nat(X,produc487386426758144856at_nat(Xa2,aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Xb),Xc))))
       => ~ ( ( Y = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Xa2),Xc,aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(X,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xa2),one_one_nat)),Xb),aa_nat_nat(aa_nat_fun_nat_nat(X,Xa2),Xc))) )
           => ~ accp_P4694664897847628410at_nat(set_fo3699595496184130361el_nat,produc2818902965508115497at_nat(X,produc487386426758144856at_nat(Xa2,aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Xb),Xc)))) ) ) ) ).

% fold_atLeastAtMost_nat.pelims
tff(fact_3693_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(aa_nat8803788312118491388at_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_3694_power__Suc__less,axiom,
    ! [A: code_integer,N2: 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),N2))),aa_nat_Code_integer(power_8256067586552552935nteger(A),N2)) ) ) ).

% power_Suc_less
tff(fact_3695_power__Suc__less,axiom,
    ! [A: rat,N2: 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),N2))),aa_nat_rat(power_power_rat(A),N2)) ) ) ).

% power_Suc_less
tff(fact_3696_power__Suc__less,axiom,
    ! [A: nat,N2: 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),N2))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N2)) ) ) ).

% power_Suc_less
tff(fact_3697_power__Suc__less,axiom,
    ! [A: int,N2: 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),N2))),aa_nat_int(aa_int_fun_nat_int(power_power_int,A),N2)) ) ) ).

% power_Suc_less
tff(fact_3698_divide__left__mono__neg,axiom,
    ! [A: rat,B: rat,Ca: 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,Ca),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,Ca),A)),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,Ca),B)) ) ) ) ).

% divide_left_mono_neg
tff(fact_3699_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_3700_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_3701_pos__le__divide__eq,axiom,
    ! [Ca: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Ca)
     => ( 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),Ca))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),Ca)),B) ) ) ).

% pos_le_divide_eq
tff(fact_3702_pos__divide__le__eq,axiom,
    ! [Ca: rat,B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Ca)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),Ca)),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),Ca)) ) ) ).

% pos_divide_le_eq
tff(fact_3703_neg__le__divide__eq,axiom,
    ! [Ca: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Ca),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),Ca))
      <=> 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),Ca)) ) ) ).

% neg_le_divide_eq
tff(fact_3704_neg__divide__le__eq,axiom,
    ! [Ca: rat,B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Ca),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),Ca)),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),Ca)),B) ) ) ).

% neg_divide_le_eq
tff(fact_3705_divide__left__mono,axiom,
    ! [B: rat,A: rat,Ca: 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),Ca)
       => ( 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,Ca),A)),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,Ca),B)) ) ) ) ).

% divide_left_mono
tff(fact_3706_le__divide__eq,axiom,
    ! [A: rat,B: rat,Ca: 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),Ca))
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Ca),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),Ca)),B),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Ca),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),Ca)),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)) ) ) ).

% le_divide_eq
tff(fact_3707_divide__le__eq,axiom,
    ! [B: rat,Ca: 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),Ca)),A)
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Ca),
          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),Ca)),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Ca),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),Ca)),B),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)) ) ) ).

% divide_le_eq
tff(fact_3708_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_3709_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_3710_power__strict__decreasing,axiom,
    ! [N2: nat,N7: nat,A: code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),N7)
     => ( 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),N7)),aa_nat_Code_integer(power_8256067586552552935nteger(A),N2)) ) ) ) ).

% power_strict_decreasing
tff(fact_3711_power__strict__decreasing,axiom,
    ! [N2: nat,N7: nat,A: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),N7)
     => ( 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),N7)),aa_nat_rat(power_power_rat(A),N2)) ) ) ) ).

% power_strict_decreasing
tff(fact_3712_power__strict__decreasing,axiom,
    ! [N2: nat,N7: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),N7)
     => ( 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),N7)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N2)) ) ) ) ).

% power_strict_decreasing
tff(fact_3713_power__strict__decreasing,axiom,
    ! [N2: nat,N7: nat,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),N7)
     => ( 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),N7)),aa_nat_int(aa_int_fun_nat_int(power_power_int,A),N2)) ) ) ) ).

% power_strict_decreasing
tff(fact_3714_power__decreasing,axiom,
    ! [N2: nat,N7: nat,A: code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),N7)
     => ( 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),N7)),aa_nat_Code_integer(power_8256067586552552935nteger(A),N2)) ) ) ) ).

% power_decreasing
tff(fact_3715_power__decreasing,axiom,
    ! [N2: nat,N7: nat,A: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),N7)
     => ( 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),N7)),aa_nat_rat(power_power_rat(A),N2)) ) ) ) ).

% power_decreasing
tff(fact_3716_power__decreasing,axiom,
    ! [N2: nat,N7: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),N7)
     => ( 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),N7)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N2)) ) ) ) ).

% power_decreasing
tff(fact_3717_power__decreasing,axiom,
    ! [N2: nat,N7: nat,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),N7)
     => ( 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),N7)),aa_nat_int(aa_int_fun_nat_int(power_power_int,A),N2)) ) ) ) ).

% power_decreasing
tff(fact_3718_power__eq__iff__eq__base,axiom,
    ! [N2: nat,A: code_integer,B: code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)
     => ( 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_nat_Code_integer(power_8256067586552552935nteger(A),N2) = aa_nat_Code_integer(power_8256067586552552935nteger(B),N2) )
          <=> ( A = B ) ) ) ) ) ).

% power_eq_iff_eq_base
tff(fact_3719_power__eq__iff__eq__base,axiom,
    ! [N2: nat,A: rat,B: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
         => ( ( aa_nat_rat(power_power_rat(A),N2) = aa_nat_rat(power_power_rat(B),N2) )
          <=> ( A = B ) ) ) ) ) ).

% power_eq_iff_eq_base
tff(fact_3720_power__eq__iff__eq__base,axiom,
    ! [N2: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B)
         => ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N2) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),N2) )
          <=> ( A = B ) ) ) ) ) ).

% power_eq_iff_eq_base
tff(fact_3721_power__eq__iff__eq__base,axiom,
    ! [N2: nat,A: int,B: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
         => ( ( aa_nat_int(aa_int_fun_nat_int(power_power_int,A),N2) = aa_nat_int(aa_int_fun_nat_int(power_power_int,B),N2) )
          <=> ( A = B ) ) ) ) ) ).

% power_eq_iff_eq_base
tff(fact_3722_power__eq__imp__eq__base,axiom,
    ! [A: code_integer,N2: nat,B: code_integer] :
      ( ( aa_nat_Code_integer(power_8256067586552552935nteger(A),N2) = aa_nat_Code_integer(power_8256067586552552935nteger(B),N2) )
     => ( 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_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)
           => ( A = B ) ) ) ) ) ).

% power_eq_imp_eq_base
tff(fact_3723_power__eq__imp__eq__base,axiom,
    ! [A: rat,N2: nat,B: rat] :
      ( ( aa_nat_rat(power_power_rat(A),N2) = aa_nat_rat(power_power_rat(B),N2) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)
           => ( A = B ) ) ) ) ) ).

% power_eq_imp_eq_base
tff(fact_3724_power__eq__imp__eq__base,axiom,
    ! [A: nat,N2: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N2) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),N2) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)
           => ( A = B ) ) ) ) ) ).

% power_eq_imp_eq_base
tff(fact_3725_power__eq__imp__eq__base,axiom,
    ! [A: int,N2: nat,B: int] :
      ( ( aa_nat_int(aa_int_fun_nat_int(power_power_int,A),N2) = aa_nat_int(aa_int_fun_nat_int(power_power_int,B),N2) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)
           => ( A = B ) ) ) ) ) ).

% power_eq_imp_eq_base
tff(fact_3726_power__le__imp__le__exp,axiom,
    ! [A: code_integer,M: nat,N2: 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),N2))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2) ) ) ).

% power_le_imp_le_exp
tff(fact_3727_power__le__imp__le__exp,axiom,
    ! [A: rat,M: nat,N2: 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),N2))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2) ) ) ).

% power_le_imp_le_exp
tff(fact_3728_power__le__imp__le__exp,axiom,
    ! [A: nat,M: nat,N2: 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),N2))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2) ) ) ).

% power_le_imp_le_exp
tff(fact_3729_power__le__imp__le__exp,axiom,
    ! [A: int,M: nat,N2: 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),N2))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2) ) ) ).

% power_le_imp_le_exp
tff(fact_3730_self__le__power,axiom,
    ! [A: code_integer,N2: 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),N2)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),aa_nat_Code_integer(power_8256067586552552935nteger(A),N2)) ) ) ).

% self_le_power
tff(fact_3731_self__le__power,axiom,
    ! [A: rat,N2: 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),N2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_nat_rat(power_power_rat(A),N2)) ) ) ).

% self_le_power
tff(fact_3732_self__le__power,axiom,
    ! [A: nat,N2: 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),N2)
       => 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),N2)) ) ) ).

% self_le_power
tff(fact_3733_self__le__power,axiom,
    ! [A: int,N2: 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),N2)
       => 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),N2)) ) ) ).

% self_le_power
tff(fact_3734_one__less__power,axiom,
    ! [A: code_integer,N2: 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),N2)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),aa_nat_Code_integer(power_8256067586552552935nteger(A),N2)) ) ) ).

% one_less_power
tff(fact_3735_one__less__power,axiom,
    ! [A: rat,N2: 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),N2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_nat_rat(power_power_rat(A),N2)) ) ) ).

% one_less_power
tff(fact_3736_one__less__power,axiom,
    ! [A: nat,N2: 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),N2)
       => 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),N2)) ) ) ).

% one_less_power
tff(fact_3737_one__less__power,axiom,
    ! [A: int,N2: 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),N2)
       => 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),N2)) ) ) ).

% one_less_power
tff(fact_3738_div__mult__self1,axiom,
    ! [B: nat,A: nat,Ca: 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,Ca),B))),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ca),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)) ) ) ).

% div_mult_self1
tff(fact_3739_div__mult__self1,axiom,
    ! [B: int,A: int,Ca: 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,Ca),B))),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Ca),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)) ) ) ).

% div_mult_self1
tff(fact_3740_div__mult__self1,axiom,
    ! [B: code_integer,A: code_integer,Ca: code_integer] :
      ( ( B != zero_z3403309356797280102nteger )
     => ( aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Ca),B))),B) = aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(Ca),aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),B)) ) ) ).

% div_mult_self1
tff(fact_3741_div__mult__self1,axiom,
    ! [B: code_natural,A: code_natural,Ca: code_natural] :
      ( ( B != zero_z2226904508553997617atural )
     => ( divide5121882707175180666atural(plus_p4538020629002901425atural(A,times_2397367101498566445atural(Ca,B)),B) = plus_p4538020629002901425atural(Ca,divide5121882707175180666atural(A,B)) ) ) ).

% div_mult_self1
tff(fact_3742_div__mult__self2,axiom,
    ! [B: nat,A: nat,Ca: 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),Ca))),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ca),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)) ) ) ).

% div_mult_self2
tff(fact_3743_div__mult__self2,axiom,
    ! [B: int,A: int,Ca: 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),Ca))),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Ca),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)) ) ) ).

% div_mult_self2
tff(fact_3744_div__mult__self2,axiom,
    ! [B: code_integer,A: code_integer,Ca: code_integer] :
      ( ( B != zero_z3403309356797280102nteger )
     => ( aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),Ca))),B) = aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(Ca),aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),B)) ) ) ).

% div_mult_self2
tff(fact_3745_div__mult__self2,axiom,
    ! [B: code_natural,A: code_natural,Ca: code_natural] :
      ( ( B != zero_z2226904508553997617atural )
     => ( divide5121882707175180666atural(plus_p4538020629002901425atural(A,times_2397367101498566445atural(B,Ca)),B) = plus_p4538020629002901425atural(Ca,divide5121882707175180666atural(A,B)) ) ) ).

% div_mult_self2
tff(fact_3746_div__mult__self3,axiom,
    ! [B: nat,Ca: 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,Ca),B)),A)),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ca),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)) ) ) ).

% div_mult_self3
tff(fact_3747_div__mult__self3,axiom,
    ! [B: int,Ca: 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,Ca),B)),A)),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Ca),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)) ) ) ).

% div_mult_self3
tff(fact_3748_div__mult__self3,axiom,
    ! [B: code_integer,Ca: code_integer,A: code_integer] :
      ( ( B != zero_z3403309356797280102nteger )
     => ( aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Ca),B)),A)),B) = aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(Ca),aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),B)) ) ) ).

% div_mult_self3
tff(fact_3749_div__mult__self3,axiom,
    ! [B: code_natural,Ca: code_natural,A: code_natural] :
      ( ( B != zero_z2226904508553997617atural )
     => ( divide5121882707175180666atural(plus_p4538020629002901425atural(times_2397367101498566445atural(Ca,B),A),B) = plus_p4538020629002901425atural(Ca,divide5121882707175180666atural(A,B)) ) ) ).

% div_mult_self3
tff(fact_3750_div__mult__self4,axiom,
    ! [B: nat,Ca: 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),Ca)),A)),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ca),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)) ) ) ).

% div_mult_self4
tff(fact_3751_div__mult__self4,axiom,
    ! [B: int,Ca: 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),Ca)),A)),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Ca),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)) ) ) ).

% div_mult_self4
tff(fact_3752_div__mult__self4,axiom,
    ! [B: code_integer,Ca: code_integer,A: code_integer] :
      ( ( B != zero_z3403309356797280102nteger )
     => ( aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),Ca)),A)),B) = aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(Ca),aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),B)) ) ) ).

% div_mult_self4
tff(fact_3753_div__mult__self4,axiom,
    ! [B: code_natural,Ca: code_natural,A: code_natural] :
      ( ( B != zero_z2226904508553997617atural )
     => ( divide5121882707175180666atural(plus_p4538020629002901425atural(times_2397367101498566445atural(B,Ca),A),B) = plus_p4538020629002901425atural(Ca,divide5121882707175180666atural(A,B)) ) ) ).

% div_mult_self4
tff(fact_3754_div__mult__self1__is__m,axiom,
    ! [N2: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N2),M)),N2) = M ) ) ).

% div_mult_self1_is_m
tff(fact_3755_div__mult__self__is__m,axiom,
    ! [N2: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N2)),N2) = M ) ) ).

% div_mult_self_is_m
tff(fact_3756_div__mult__mult1,axiom,
    ! [Ca: nat,A: nat,B: nat] :
      ( ( Ca != 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,Ca),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ca),B)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B) ) ) ).

% div_mult_mult1
tff(fact_3757_div__mult__mult1,axiom,
    ! [Ca: int,A: int,B: int] :
      ( ( Ca != 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,Ca),A)),aa_int_int(aa_int_fun_int_int(times_times_int,Ca),B)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B) ) ) ).

% div_mult_mult1
tff(fact_3758_div__mult__mult1,axiom,
    ! [Ca: code_integer,A: code_integer,B: code_integer] :
      ( ( Ca != zero_z3403309356797280102nteger )
     => ( aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Ca),A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Ca),B)) = aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),B) ) ) ).

% div_mult_mult1
tff(fact_3759_div__mult__mult1,axiom,
    ! [Ca: code_natural,A: code_natural,B: code_natural] :
      ( ( Ca != zero_z2226904508553997617atural )
     => ( divide5121882707175180666atural(times_2397367101498566445atural(Ca,A),times_2397367101498566445atural(Ca,B)) = divide5121882707175180666atural(A,B) ) ) ).

% div_mult_mult1
tff(fact_3760_div__mult__mult2,axiom,
    ! [Ca: nat,A: nat,B: nat] :
      ( ( Ca != 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),Ca)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),Ca)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B) ) ) ).

% div_mult_mult2
tff(fact_3761_div__mult__mult2,axiom,
    ! [Ca: int,A: int,B: int] :
      ( ( Ca != 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),Ca)),aa_int_int(aa_int_fun_int_int(times_times_int,B),Ca)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B) ) ) ).

% div_mult_mult2
tff(fact_3762_div__mult__mult2,axiom,
    ! [Ca: code_integer,A: code_integer,B: code_integer] :
      ( ( Ca != zero_z3403309356797280102nteger )
     => ( aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),Ca)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),Ca)) = aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),B) ) ) ).

% div_mult_mult2
tff(fact_3763_div__mult__mult2,axiom,
    ! [Ca: code_natural,A: code_natural,B: code_natural] :
      ( ( Ca != zero_z2226904508553997617atural )
     => ( divide5121882707175180666atural(times_2397367101498566445atural(A,Ca),times_2397367101498566445atural(B,Ca)) = divide5121882707175180666atural(A,B) ) ) ).

% div_mult_mult2
tff(fact_3764_div__mult__mult1__if,axiom,
    ! [Ca: 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,Ca),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ca),B)) = $ite(Ca = 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_3765_div__mult__mult1__if,axiom,
    ! [Ca: 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,Ca),A)),aa_int_int(aa_int_fun_int_int(times_times_int,Ca),B)) = $ite(Ca = 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_3766_div__mult__mult1__if,axiom,
    ! [Ca: code_integer,A: code_integer,B: code_integer] :
      aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Ca),A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Ca),B)) = $ite(Ca = zero_z3403309356797280102nteger,zero_z3403309356797280102nteger,aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),B)) ).

% div_mult_mult1_if
tff(fact_3767_div__mult__mult1__if,axiom,
    ! [Ca: code_natural,A: code_natural,B: code_natural] :
      divide5121882707175180666atural(times_2397367101498566445atural(Ca,A),times_2397367101498566445atural(Ca,B)) = $ite(Ca = zero_z2226904508553997617atural,zero_z2226904508553997617atural,divide5121882707175180666atural(A,B)) ).

% div_mult_mult1_if
tff(fact_3768_div__less,axiom,
    ! [M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N2)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N2) = zero_zero_nat ) ) ).

% div_less
tff(fact_3769_div__le__mono,axiom,
    ! [M: nat,N2: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),K)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N2),K)) ) ).

% div_le_mono
tff(fact_3770_div__le__dividend,axiom,
    ! [M: nat,N2: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N2)),M) ).

% div_le_dividend
tff(fact_3771_Euclidean__Division_Odiv__eq__0__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N2) = zero_zero_nat )
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N2)
        | ( N2 = zero_zero_nat ) ) ) ).

% Euclidean_Division.div_eq_0_iff
tff(fact_3772_less__mult__imp__div__less,axiom,
    ! [M: nat,I2: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I2),N2))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N2)),I2) ) ).

% less_mult_imp_div_less
tff(fact_3773_times__div__less__eq__dividend,axiom,
    ! [N2: nat,M: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N2))),M) ).

% times_div_less_eq_dividend
tff(fact_3774_div__times__less__eq__dividend,axiom,
    ! [M: nat,N2: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N2)),N2)),M) ).

% div_times_less_eq_dividend
tff(fact_3775_div__greater__zero__iff,axiom,
    ! [M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N2))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),M)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2) ) ) ).

% div_greater_zero_iff
tff(fact_3776_div__le__mono2,axiom,
    ! [M: nat,N2: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,K),N2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,K),M)) ) ) ).

% div_le_mono2
tff(fact_3777_div__less__iff__less__mult,axiom,
    ! [Q4: nat,M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Q4)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Q4)),N2)
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N2),Q4)) ) ) ).

% div_less_iff_less_mult
tff(fact_3778_div__eq__dividend__iff,axiom,
    ! [M: nat,N2: 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),N2) = M )
      <=> ( N2 = one_one_nat ) ) ) ).

% div_eq_dividend_iff
tff(fact_3779_div__less__dividend,axiom,
    ! [N2: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),N2)
     => ( 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),N2)),M) ) ) ).

% div_less_dividend
tff(fact_3780_less__eq__div__iff__mult__less__eq,axiom,
    ! [Q4: nat,M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Q4)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N2),Q4))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Q4)),N2) ) ) ).

% less_eq_div_iff_mult_less_eq
tff(fact_3781_split__div,axiom,
    ! [Pa: fun_nat_o,M: nat,N2: nat] :
      ( aa_nat_o(Pa,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N2))
    <=> ( ( ( N2 = zero_zero_nat )
         => aa_nat_o(Pa,zero_zero_nat) )
        & ( ( N2 != zero_zero_nat )
         => ! [I4: nat,J3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J3),N2)
             => ( ( M = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N2),I4)),J3) )
               => aa_nat_o(Pa,I4) ) ) ) ) ) ).

% split_div
tff(fact_3782_dividend__less__div__times,axiom,
    ! [N2: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N2)),N2))) ) ).

% dividend_less_div_times
tff(fact_3783_dividend__less__times__div,axiom,
    ! [N2: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N2)))) ) ).

% dividend_less_times_div
tff(fact_3784_verit__le__mono__div,axiom,
    ! [A3: nat,B3: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A3),B3)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,
              aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A3),N2)),
                $ite(aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B3),N2) = zero_zero_nat,one_one_nat,zero_zero_nat))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B3),N2)) ) ) ).

% verit_le_mono_div
tff(fact_3785_of__nat__zero__less__power__iff,axiom,
    ! [X: nat,N2: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_nat_rat(power_power_rat(aa_nat_rat(semiri681578069525770553at_rat,X)),N2))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),X)
        | ( N2 = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
tff(fact_3786_of__nat__zero__less__power__iff,axiom,
    ! [X: nat,N2: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_nat_int(semiri1314217659103216013at_int,X)),N2))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),X)
        | ( N2 = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
tff(fact_3787_of__nat__zero__less__power__iff,axiom,
    ! [X: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(semiri1316708129612266289at_nat,X)),N2))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),X)
        | ( N2 = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
tff(fact_3788_of__nat__zero__less__power__iff,axiom,
    ! [X: nat,N2: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(power_8256067586552552935nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,X)),N2))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),X)
        | ( N2 = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
tff(fact_3789_distrib__right__NO__MATCH,axiom,
    ! [X: nat,Y: nat,Ca: code_integer,A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_nat7420681472272698950eger_o(nO_MAT8694959213317031834nteger,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X),Y)),Ca)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(A),B)),Ca) = aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),Ca)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),Ca)) ) ) ).

% distrib_right_NO_MATCH
tff(fact_3790_distrib__right__NO__MATCH,axiom,
    ! [X: nat,Y: nat,Ca: 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)),Ca)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),Ca) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),Ca)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),Ca)) ) ) ).

% distrib_right_NO_MATCH
tff(fact_3791_distrib__right__NO__MATCH,axiom,
    ! [X: nat,Y: nat,Ca: 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)),Ca)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),Ca) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),Ca)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),Ca)) ) ) ).

% distrib_right_NO_MATCH
tff(fact_3792_distrib__right__NO__MATCH,axiom,
    ! [X: nat,Y: nat,Ca: 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)),Ca)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),Ca) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),Ca)),aa_int_int(aa_int_fun_int_int(times_times_int,B),Ca)) ) ) ).

% distrib_right_NO_MATCH
tff(fact_3793_distrib__right__NO__MATCH,axiom,
    ! [X: int,Y: int,Ca: code_integer,A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_int5993532643906289898eger_o(nO_MAT9066612773553876470nteger,aa_int_int(aa_int_fun_int_int(divide_divide_int,X),Y)),Ca)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(A),B)),Ca) = aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),Ca)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),Ca)) ) ) ).

% distrib_right_NO_MATCH
tff(fact_3794_distrib__right__NO__MATCH,axiom,
    ! [X: int,Y: int,Ca: 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)),Ca)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),Ca) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),Ca)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),Ca)) ) ) ).

% distrib_right_NO_MATCH
tff(fact_3795_distrib__right__NO__MATCH,axiom,
    ! [X: int,Y: int,Ca: 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)),Ca)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),Ca) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),Ca)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),Ca)) ) ) ).

% distrib_right_NO_MATCH
tff(fact_3796_distrib__right__NO__MATCH,axiom,
    ! [X: int,Y: int,Ca: 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)),Ca)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),Ca) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),Ca)),aa_int_int(aa_int_fun_int_int(times_times_int,B),Ca)) ) ) ).

% distrib_right_NO_MATCH
tff(fact_3797_distrib__right__NO__MATCH,axiom,
    ! [X: code_integer,Y: code_integer,Ca: code_integer,A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(nO_MAT8252062027627875367nteger,aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(X),Y)),Ca)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(A),B)),Ca) = aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),Ca)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),Ca)) ) ) ).

% distrib_right_NO_MATCH
tff(fact_3798_distrib__right__NO__MATCH,axiom,
    ! [X: code_integer,Y: code_integer,Ca: rat,A: rat,B: rat] :
      ( aa_rat_o(nO_MAT7795273704451493282er_rat(aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(X),Y)),Ca)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),Ca) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),Ca)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),Ca)) ) ) ).

% distrib_right_NO_MATCH
tff(fact_3799_distrib__left__NO__MATCH,axiom,
    ! [X: nat,Y: nat,A: code_integer,B: code_integer,Ca: code_integer] :
      ( aa_Code_integer_o(aa_nat7420681472272698950eger_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(plus_p5714425477246183910nteger(B),Ca)) = aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),Ca)) ) ) ).

% distrib_left_NO_MATCH
tff(fact_3800_distrib__left__NO__MATCH,axiom,
    ! [X: nat,Y: nat,A: rat,B: rat,Ca: 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),Ca)) = 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),Ca)) ) ) ).

% distrib_left_NO_MATCH
tff(fact_3801_distrib__left__NO__MATCH,axiom,
    ! [X: nat,Y: nat,A: nat,B: nat,Ca: 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),Ca)) = 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),Ca)) ) ) ).

% distrib_left_NO_MATCH
tff(fact_3802_distrib__left__NO__MATCH,axiom,
    ! [X: nat,Y: nat,A: int,B: int,Ca: 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),Ca)) = 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),Ca)) ) ) ).

% distrib_left_NO_MATCH
tff(fact_3803_distrib__left__NO__MATCH,axiom,
    ! [X: int,Y: int,A: code_integer,B: code_integer,Ca: code_integer] :
      ( aa_Code_integer_o(aa_int5993532643906289898eger_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(plus_p5714425477246183910nteger(B),Ca)) = aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),Ca)) ) ) ).

% distrib_left_NO_MATCH
tff(fact_3804_distrib__left__NO__MATCH,axiom,
    ! [X: int,Y: int,A: rat,B: rat,Ca: 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),Ca)) = 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),Ca)) ) ) ).

% distrib_left_NO_MATCH
tff(fact_3805_distrib__left__NO__MATCH,axiom,
    ! [X: int,Y: int,A: nat,B: nat,Ca: 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),Ca)) = 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),Ca)) ) ) ).

% distrib_left_NO_MATCH
tff(fact_3806_distrib__left__NO__MATCH,axiom,
    ! [X: int,Y: int,A: int,B: int,Ca: 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),Ca)) = 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),Ca)) ) ) ).

% distrib_left_NO_MATCH
tff(fact_3807_distrib__left__NO__MATCH,axiom,
    ! [X: code_integer,Y: code_integer,A: code_integer,B: code_integer,Ca: 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(plus_p5714425477246183910nteger(B),Ca)) = aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),Ca)) ) ) ).

% distrib_left_NO_MATCH
tff(fact_3808_distrib__left__NO__MATCH,axiom,
    ! [X: code_integer,Y: code_integer,A: rat,B: rat,Ca: 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),Ca)) = 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),Ca)) ) ) ).

% distrib_left_NO_MATCH
tff(fact_3809_split__div_H,axiom,
    ! [Pa: fun_nat_o,M: nat,N2: nat] :
      ( aa_nat_o(Pa,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N2))
    <=> ( ( ( N2 = zero_zero_nat )
          & aa_nat_o(Pa,zero_zero_nat) )
        | ? [Q5: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N2),Q5)),M)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N2),aa_nat_nat(suc,Q5)))
            & aa_nat_o(Pa,Q5) ) ) ) ).

% split_div'
tff(fact_3810_power__minus__mult,axiom,
    ! [N2: nat,A: code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),one_one_nat))),A) = aa_nat_Code_integer(power_8256067586552552935nteger(A),N2) ) ) ).

% power_minus_mult
tff(fact_3811_power__minus__mult,axiom,
    ! [N2: nat,A: assn] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)
     => ( 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,N2),one_one_nat))),A) = aa_nat_assn(power_power_assn(A),N2) ) ) ).

% power_minus_mult
tff(fact_3812_power__minus__mult,axiom,
    ! [N2: nat,A: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)
     => ( 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,N2),one_one_nat))),A) = aa_nat_rat(power_power_rat(A),N2) ) ) ).

% power_minus_mult
tff(fact_3813_power__minus__mult,axiom,
    ! [N2: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)
     => ( 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,N2),one_one_nat))),A) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N2) ) ) ).

% power_minus_mult
tff(fact_3814_power__minus__mult,axiom,
    ! [N2: nat,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)
     => ( 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,N2),one_one_nat))),A) = aa_nat_int(aa_int_fun_nat_int(power_power_int,A),N2) ) ) ).

% power_minus_mult
tff(fact_3815_power__eq__if,axiom,
    ! [P2: code_integer,M: nat] :
      aa_nat_Code_integer(power_8256067586552552935nteger(P2),M) = $ite(M = zero_zero_nat,one_one_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,P2),aa_nat_Code_integer(power_8256067586552552935nteger(P2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat)))) ).

% power_eq_if
tff(fact_3816_power__eq__if,axiom,
    ! [P2: assn,M: nat] :
      aa_nat_assn(power_power_assn(P2),M) = $ite(M = zero_zero_nat,one_one_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,P2),aa_nat_assn(power_power_assn(P2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat)))) ).

% power_eq_if
tff(fact_3817_power__eq__if,axiom,
    ! [P2: rat,M: nat] :
      aa_nat_rat(power_power_rat(P2),M) = $ite(M = zero_zero_nat,one_one_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,P2),aa_nat_rat(power_power_rat(P2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat)))) ).

% power_eq_if
tff(fact_3818_power__eq__if,axiom,
    ! [P2: nat,M: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,P2),M) = $ite(M = zero_zero_nat,one_one_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,P2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,P2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat)))) ).

% power_eq_if
tff(fact_3819_power__eq__if,axiom,
    ! [P2: int,M: nat] :
      aa_nat_int(aa_int_fun_nat_int(power_power_int,P2),M) = $ite(M = zero_zero_nat,one_one_int,aa_int_int(aa_int_fun_int_int(times_times_int,P2),aa_nat_int(aa_int_fun_nat_int(power_power_int,P2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat)))) ).

% power_eq_if
tff(fact_3820_power__diff__power__eq,axiom,
    ! [A: nat,M: nat,N2: 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),N2)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),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),N2)),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,N2),M)))) ) ) ).

% power_diff_power_eq
tff(fact_3821_power__diff__power__eq,axiom,
    ! [A: int,M: nat,N2: 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),N2)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),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),N2)),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,N2),M)))) ) ) ).

% power_diff_power_eq
tff(fact_3822_power__diff__power__eq,axiom,
    ! [A: code_integer,M: nat,N2: nat] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_nat_Code_integer(power_8256067586552552935nteger(A),M)),aa_nat_Code_integer(power_8256067586552552935nteger(A),N2)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),M),aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N2)),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,N2),M)))) ) ) ).

% power_diff_power_eq
tff(fact_3823_power__diff__power__eq,axiom,
    ! [A: code_natural,M: nat,N2: nat] :
      ( ( A != zero_z2226904508553997617atural )
     => ( divide5121882707175180666atural(aa_nat_Code_natural(power_7079662738309270450atural(A),M),aa_nat_Code_natural(power_7079662738309270450atural(A),N2)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),M),aa_nat_Code_natural(power_7079662738309270450atural(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N2)),divide5121882707175180666atural(one_one_Code_natural,aa_nat_Code_natural(power_7079662738309270450atural(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),M)))) ) ) ).

% power_diff_power_eq
tff(fact_3824_le__divide__eq__numeral_I1_J,axiom,
    ! [W2: num,B: rat,Ca: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,W2)),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),Ca))
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Ca),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,W2)),Ca)),B),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Ca),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_num_rat(numeral_numeral_rat,W2)),Ca)),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,W2)),zero_zero_rat)) ) ) ).

% le_divide_eq_numeral(1)
tff(fact_3825_old_Onat_Oinject,axiom,
    ! [Nat: nat,Nat2: nat] :
      ( ( aa_nat_nat(suc,Nat) = aa_nat_nat(suc,Nat2) )
    <=> ( Nat = Nat2 ) ) ).

% old.nat.inject
tff(fact_3826_nat_Oinject,axiom,
    ! [X2: nat,Y2: nat] :
      ( ( aa_nat_nat(suc,X2) = aa_nat_nat(suc,Y2) )
    <=> ( X2 = Y2 ) ) ).

% nat.inject
tff(fact_3827_of__nat__eq__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( aa_nat_int(semiri1314217659103216013at_int,M) = aa_nat_int(semiri1314217659103216013at_int,N2) )
    <=> ( M = N2 ) ) ).

% of_nat_eq_iff
tff(fact_3828_of__nat__eq__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( aa_nat_nat(semiri1316708129612266289at_nat,M) = aa_nat_nat(semiri1316708129612266289at_nat,N2) )
    <=> ( M = N2 ) ) ).

% of_nat_eq_iff
tff(fact_3829_of__nat__eq__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( aa_nat_Code_integer(semiri4939895301339042750nteger,M) = aa_nat_Code_integer(semiri4939895301339042750nteger,N2) )
    <=> ( M = N2 ) ) ).

% of_nat_eq_iff
tff(fact_3830_numeral__le__iff,axiom,
    ! [M: num,N2: num] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_num_Code_integer(numera6620942414471956472nteger,M)),aa_num_Code_integer(numera6620942414471956472nteger,N2))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,M),N2) ) ).

% numeral_le_iff
tff(fact_3831_numeral__le__iff,axiom,
    ! [M: num,N2: num] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,aa_num_Code_natural(numera5444537566228673987atural,M)),aa_num_Code_natural(numera5444537566228673987atural,N2))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,M),N2) ) ).

% numeral_le_iff
tff(fact_3832_numeral__le__iff,axiom,
    ! [M: num,N2: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,M)),aa_num_rat(numeral_numeral_rat,N2))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,M),N2) ) ).

% numeral_le_iff
tff(fact_3833_numeral__le__iff,axiom,
    ! [M: num,N2: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,N2))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,M),N2) ) ).

% numeral_le_iff
tff(fact_3834_numeral__le__iff,axiom,
    ! [M: num,N2: num] :
      ( 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,N2))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,M),N2) ) ).

% numeral_le_iff
tff(fact_3835_numeral__less__iff,axiom,
    ! [M: num,N2: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,N2))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,M),N2) ) ).

% numeral_less_iff
tff(fact_3836_numeral__less__iff,axiom,
    ! [M: num,N2: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,N2))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,M),N2) ) ).

% numeral_less_iff
tff(fact_3837_numeral__less__iff,axiom,
    ! [M: num,N2: num] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_num_Code_integer(numera6620942414471956472nteger,M)),aa_num_Code_integer(numera6620942414471956472nteger,N2))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,M),N2) ) ).

% numeral_less_iff
tff(fact_3838_numeral__less__iff,axiom,
    ! [M: num,N2: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(numeral_numeral_rat,M)),aa_num_rat(numeral_numeral_rat,N2))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,M),N2) ) ).

% numeral_less_iff
tff(fact_3839_numeral__less__iff,axiom,
    ! [M: num,N2: num] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,aa_num_Code_natural(numera5444537566228673987atural,M)),aa_num_Code_natural(numera5444537566228673987atural,N2))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,M),N2) ) ).

% numeral_less_iff
tff(fact_3840_mult__numeral__left__semiring__numeral,axiom,
    ! [V: num,W2: num,Z: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,V)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,W2)),Z)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(times_times_num,V),W2))),Z) ).

% mult_numeral_left_semiring_numeral
tff(fact_3841_mult__numeral__left__semiring__numeral,axiom,
    ! [V: num,W2: 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,W2)),Z)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(times_times_num,V),W2))),Z) ).

% mult_numeral_left_semiring_numeral
tff(fact_3842_mult__numeral__left__semiring__numeral,axiom,
    ! [V: num,W2: 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,W2)),Z)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(aa_num_fun_num_num(times_times_num,V),W2))),Z) ).

% mult_numeral_left_semiring_numeral
tff(fact_3843_mult__numeral__left__semiring__numeral,axiom,
    ! [V: num,W2: num,Z: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,V)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,W2)),Z)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(aa_num_fun_num_num(times_times_num,V),W2))),Z) ).

% mult_numeral_left_semiring_numeral
tff(fact_3844_mult__numeral__left__semiring__numeral,axiom,
    ! [V: num,W2: num,Z: code_natural] : times_2397367101498566445atural(aa_num_Code_natural(numera5444537566228673987atural,V),times_2397367101498566445atural(aa_num_Code_natural(numera5444537566228673987atural,W2),Z)) = times_2397367101498566445atural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(aa_num_fun_num_num(times_times_num,V),W2)),Z) ).

% mult_numeral_left_semiring_numeral
tff(fact_3845_numeral__times__numeral,axiom,
    ! [M: num,N2: num] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,N2)) = aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(times_times_num,M),N2)) ).

% numeral_times_numeral
tff(fact_3846_numeral__times__numeral,axiom,
    ! [M: num,N2: 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,N2)) = aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(times_times_num,M),N2)) ).

% numeral_times_numeral
tff(fact_3847_numeral__times__numeral,axiom,
    ! [M: num,N2: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,M)),aa_num_Code_integer(numera6620942414471956472nteger,N2)) = aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(aa_num_fun_num_num(times_times_num,M),N2)) ).

% numeral_times_numeral
tff(fact_3848_numeral__times__numeral,axiom,
    ! [M: num,N2: num] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,M)),aa_num_rat(numeral_numeral_rat,N2)) = aa_num_rat(numeral_numeral_rat,aa_num_num(aa_num_fun_num_num(times_times_num,M),N2)) ).

% numeral_times_numeral
tff(fact_3849_numeral__times__numeral,axiom,
    ! [M: num,N2: num] : times_2397367101498566445atural(aa_num_Code_natural(numera5444537566228673987atural,M),aa_num_Code_natural(numera5444537566228673987atural,N2)) = aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(aa_num_fun_num_num(times_times_num,M),N2)) ).

% numeral_times_numeral
tff(fact_3850_power__add__numeral,axiom,
    ! [A: code_integer,M: num,N2: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_num_nat(numeral_numeral_nat,M))),aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_num_nat(numeral_numeral_nat,N2))) = aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,M),N2))) ).

% power_add_numeral
tff(fact_3851_power__add__numeral,axiom,
    ! [A: assn,M: num,N2: num] : aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_nat_assn(power_power_assn(A),aa_num_nat(numeral_numeral_nat,M))),aa_nat_assn(power_power_assn(A),aa_num_nat(numeral_numeral_nat,N2))) = aa_nat_assn(power_power_assn(A),aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,M),N2))) ).

% power_add_numeral
tff(fact_3852_power__add__numeral,axiom,
    ! [A: rat,M: num,N2: num] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,M))),aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,N2))) = aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,M),N2))) ).

% power_add_numeral
tff(fact_3853_power__add__numeral,axiom,
    ! [A: nat,M: num,N2: num] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_num_nat(numeral_numeral_nat,M))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_num_nat(numeral_numeral_nat,N2))) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,M),N2))) ).

% power_add_numeral
tff(fact_3854_power__add__numeral,axiom,
    ! [A: int,M: num,N2: num] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A),aa_num_nat(numeral_numeral_nat,M))),aa_nat_int(aa_int_fun_nat_int(power_power_int,A),aa_num_nat(numeral_numeral_nat,N2))) = aa_nat_int(aa_int_fun_nat_int(power_power_int,A),aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,M),N2))) ).

% power_add_numeral
tff(fact_3855_power__add__numeral2,axiom,
    ! [A: code_integer,M: num,N2: num,B: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_num_nat(numeral_numeral_nat,M))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_num_nat(numeral_numeral_nat,N2))),B)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,M),N2)))),B) ).

% power_add_numeral2
tff(fact_3856_power__add__numeral2,axiom,
    ! [A: assn,M: num,N2: num,B: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_nat_assn(power_power_assn(A),aa_num_nat(numeral_numeral_nat,M))),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_nat_assn(power_power_assn(A),aa_num_nat(numeral_numeral_nat,N2))),B)) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_nat_assn(power_power_assn(A),aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,M),N2)))),B) ).

% power_add_numeral2
tff(fact_3857_power__add__numeral2,axiom,
    ! [A: rat,M: num,N2: num,B: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,M))),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,N2))),B)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,M),N2)))),B) ).

% power_add_numeral2
tff(fact_3858_power__add__numeral2,axiom,
    ! [A: nat,M: num,N2: num,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_num_nat(numeral_numeral_nat,M))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_num_nat(numeral_numeral_nat,N2))),B)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,M),N2)))),B) ).

% power_add_numeral2
tff(fact_3859_power__add__numeral2,axiom,
    ! [A: int,M: num,N2: 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),aa_num_nat(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),aa_num_nat(numeral_numeral_nat,N2))),B)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A),aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,M),N2)))),B) ).

% power_add_numeral2
tff(fact_3860_lessI,axiom,
    ! [N2: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),aa_nat_nat(suc,N2)) ).

% lessI
tff(fact_3861_Suc__mono,axiom,
    ! [M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,M)),aa_nat_nat(suc,N2)) ) ).

% Suc_mono
tff(fact_3862_Suc__less__eq,axiom,
    ! [M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,M)),aa_nat_nat(suc,N2))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N2) ) ).

% Suc_less_eq
tff(fact_3863_Suc__le__mono,axiom,
    ! [N2: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,N2)),aa_nat_nat(suc,M))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),M) ) ).

% Suc_le_mono
tff(fact_3864_add__Suc__right,axiom,
    ! [M: nat,N2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),aa_nat_nat(suc,N2)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N2)) ).

% add_Suc_right
tff(fact_3865_diff__self__eq__0,axiom,
    ! [M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),M) = zero_zero_nat ).

% diff_self_eq_0
tff(fact_3866_diff__0__eq__0,axiom,
    ! [N2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,zero_zero_nat),N2) = zero_zero_nat ).

% diff_0_eq_0
tff(fact_3867_Suc__diff__diff,axiom,
    ! [M: nat,N2: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,M)),N2)),aa_nat_nat(suc,K)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N2)),K) ).

% Suc_diff_diff
tff(fact_3868_diff__Suc__Suc,axiom,
    ! [M: nat,N2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,M)),aa_nat_nat(suc,N2)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N2) ).

% diff_Suc_Suc
tff(fact_3869_diff__diff__cancel,axiom,
    ! [I2: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),N2)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),I2)) = I2 ) ) ).

% diff_diff_cancel
tff(fact_3870_diff__diff__left,axiom,
    ! [I2: nat,J: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I2),J)),K) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K)) ).

% diff_diff_left
tff(fact_3871_mod__less,axiom,
    ! [M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N2)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N2) = M ) ) ).

% mod_less
tff(fact_3872_zero__comp__diff__simps_I1_J,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) ) ).

% zero_comp_diff_simps(1)
tff(fact_3873_zero__comp__diff__simps_I1_J,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A) ) ).

% zero_comp_diff_simps(1)
tff(fact_3874_zero__comp__diff__simps_I1_J,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A) ) ).

% zero_comp_diff_simps(1)
tff(fact_3875_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_3876_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_3877_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_3878_zero__comp__diff__simps_I2_J,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) ) ).

% zero_comp_diff_simps(2)
tff(fact_3879_zero__comp__diff__simps_I2_J,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A) ) ).

% zero_comp_diff_simps(2)
tff(fact_3880_zero__comp__diff__simps_I2_J,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,B),A) ) ).

% zero_comp_diff_simps(2)
tff(fact_3881_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_3882_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_3883_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_3884_le__add__diff__inverse,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B)) = A ) ) ).

% le_add_diff_inverse
tff(fact_3885_le__add__diff__inverse,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B)) = A ) ) ).

% le_add_diff_inverse
tff(fact_3886_le__add__diff__inverse,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,B),aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)) = A ) ) ).

% le_add_diff_inverse
tff(fact_3887_le__add__diff__inverse2,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B)),B) = A ) ) ).

% le_add_diff_inverse2
tff(fact_3888_le__add__diff__inverse2,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B)),B) = A ) ) ).

% le_add_diff_inverse2
tff(fact_3889_le__add__diff__inverse2,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),B) = A ) ) ).

% le_add_diff_inverse2
tff(fact_3890_distrib__left__numeral,axiom,
    ! [V: num,B: nat,Ca: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,V)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),Ca)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,V)),B)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,V)),Ca)) ).

% distrib_left_numeral
tff(fact_3891_distrib__left__numeral,axiom,
    ! [V: num,B: int,Ca: 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),Ca)) = 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)),Ca)) ).

% distrib_left_numeral
tff(fact_3892_distrib__left__numeral,axiom,
    ! [V: num,B: code_integer,Ca: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,V)),aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(B),Ca)) = aa_Cod1783974652533396728nteger(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)),Ca)) ).

% distrib_left_numeral
tff(fact_3893_distrib__left__numeral,axiom,
    ! [V: num,B: rat,Ca: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,V)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),Ca)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,V)),B)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,V)),Ca)) ).

% distrib_left_numeral
tff(fact_3894_distrib__left__numeral,axiom,
    ! [V: num,B: code_natural,Ca: code_natural] : times_2397367101498566445atural(aa_num_Code_natural(numera5444537566228673987atural,V),plus_p4538020629002901425atural(B,Ca)) = plus_p4538020629002901425atural(times_2397367101498566445atural(aa_num_Code_natural(numera5444537566228673987atural,V),B),times_2397367101498566445atural(aa_num_Code_natural(numera5444537566228673987atural,V),Ca)) ).

% distrib_left_numeral
tff(fact_3895_distrib__right__numeral,axiom,
    ! [A: nat,B: nat,V: num] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),aa_num_nat(numeral_numeral_nat,V)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_num_nat(numeral_numeral_nat,V))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),aa_num_nat(numeral_numeral_nat,V))) ).

% distrib_right_numeral
tff(fact_3896_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_3897_distrib__right__numeral,axiom,
    ! [A: code_integer,B: code_integer,V: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(A),B)),aa_num_Code_integer(numera6620942414471956472nteger,V)) = aa_Cod1783974652533396728nteger(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_3898_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)),aa_num_rat(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),aa_num_rat(numeral_numeral_rat,V))),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),aa_num_rat(numeral_numeral_rat,V))) ).

% distrib_right_numeral
tff(fact_3899_distrib__right__numeral,axiom,
    ! [A: code_natural,B: code_natural,V: num] : times_2397367101498566445atural(plus_p4538020629002901425atural(A,B),aa_num_Code_natural(numera5444537566228673987atural,V)) = plus_p4538020629002901425atural(times_2397367101498566445atural(A,aa_num_Code_natural(numera5444537566228673987atural,V)),times_2397367101498566445atural(B,aa_num_Code_natural(numera5444537566228673987atural,V))) ).

% distrib_right_numeral
tff(fact_3900_right__diff__distrib__numeral,axiom,
    ! [V: num,B: int,Ca: 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),Ca)) = 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)),Ca)) ).

% right_diff_distrib_numeral
tff(fact_3901_right__diff__distrib__numeral,axiom,
    ! [V: num,B: code_integer,Ca: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,V)),aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(B),Ca)) = 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)),Ca)) ).

% right_diff_distrib_numeral
tff(fact_3902_right__diff__distrib__numeral,axiom,
    ! [V: num,B: rat,Ca: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,V)),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,B),Ca)) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,V)),B)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,V)),Ca)) ).

% right_diff_distrib_numeral
tff(fact_3903_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_3904_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_3905_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)),aa_num_rat(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),aa_num_rat(numeral_numeral_rat,V))),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),aa_num_rat(numeral_numeral_rat,V))) ).

% left_diff_distrib_numeral
tff(fact_3906_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_3907_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_3908_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_3909_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_3910_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_3911_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_3912_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_3913_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_3914_of__nat__eq__0__iff,axiom,
    ! [M: nat] :
      ( ( aa_nat_rat(semiri681578069525770553at_rat,M) = zero_zero_rat )
    <=> ( M = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
tff(fact_3915_of__nat__eq__0__iff,axiom,
    ! [M: nat] :
      ( ( aa_nat_int(semiri1314217659103216013at_int,M) = zero_zero_int )
    <=> ( M = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
tff(fact_3916_of__nat__eq__0__iff,axiom,
    ! [M: nat] :
      ( ( aa_nat_nat(semiri1316708129612266289at_nat,M) = zero_zero_nat )
    <=> ( M = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
tff(fact_3917_of__nat__eq__0__iff,axiom,
    ! [M: nat] :
      ( ( aa_nat_Code_integer(semiri4939895301339042750nteger,M) = zero_z3403309356797280102nteger )
    <=> ( M = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
tff(fact_3918_of__nat__0__eq__iff,axiom,
    ! [N2: nat] :
      ( ( zero_zero_rat = aa_nat_rat(semiri681578069525770553at_rat,N2) )
    <=> ( zero_zero_nat = N2 ) ) ).

% of_nat_0_eq_iff
tff(fact_3919_of__nat__0__eq__iff,axiom,
    ! [N2: nat] :
      ( ( zero_zero_int = aa_nat_int(semiri1314217659103216013at_int,N2) )
    <=> ( zero_zero_nat = N2 ) ) ).

% of_nat_0_eq_iff
tff(fact_3920_of__nat__0__eq__iff,axiom,
    ! [N2: nat] :
      ( ( zero_zero_nat = aa_nat_nat(semiri1316708129612266289at_nat,N2) )
    <=> ( zero_zero_nat = N2 ) ) ).

% of_nat_0_eq_iff
tff(fact_3921_of__nat__0__eq__iff,axiom,
    ! [N2: nat] :
      ( ( zero_z3403309356797280102nteger = aa_nat_Code_integer(semiri4939895301339042750nteger,N2) )
    <=> ( zero_zero_nat = N2 ) ) ).

% of_nat_0_eq_iff
tff(fact_3922_of__nat__0,axiom,
    aa_nat_rat(semiri681578069525770553at_rat,zero_zero_nat) = zero_zero_rat ).

% of_nat_0
tff(fact_3923_of__nat__0,axiom,
    aa_nat_int(semiri1314217659103216013at_int,zero_zero_nat) = zero_zero_int ).

% of_nat_0
tff(fact_3924_of__nat__0,axiom,
    aa_nat_nat(semiri1316708129612266289at_nat,zero_zero_nat) = zero_zero_nat ).

% of_nat_0
tff(fact_3925_of__nat__0,axiom,
    aa_nat_Code_integer(semiri4939895301339042750nteger,zero_zero_nat) = zero_z3403309356797280102nteger ).

% of_nat_0
tff(fact_3926_mod__mult__self1,axiom,
    ! [A: nat,Ca: 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,Ca),B))),B) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) ).

% mod_mult_self1
tff(fact_3927_mod__mult__self1,axiom,
    ! [A: int,Ca: 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,Ca),B))),B) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) ).

% mod_mult_self1
tff(fact_3928_mod__mult__self1,axiom,
    ! [A: code_integer,Ca: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Ca),B))),B) = aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A),B) ).

% mod_mult_self1
tff(fact_3929_mod__mult__self1,axiom,
    ! [A: code_natural,Ca: code_natural,B: code_natural] : modulo8411746178871703098atural(plus_p4538020629002901425atural(A,times_2397367101498566445atural(Ca,B)),B) = modulo8411746178871703098atural(A,B) ).

% mod_mult_self1
tff(fact_3930_mod__mult__self2,axiom,
    ! [A: nat,B: nat,Ca: 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),Ca))),B) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) ).

% mod_mult_self2
tff(fact_3931_mod__mult__self2,axiom,
    ! [A: int,B: int,Ca: 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),Ca))),B) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) ).

% mod_mult_self2
tff(fact_3932_mod__mult__self2,axiom,
    ! [A: code_integer,B: code_integer,Ca: code_integer] : aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),Ca))),B) = aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A),B) ).

% mod_mult_self2
tff(fact_3933_mod__mult__self2,axiom,
    ! [A: code_natural,B: code_natural,Ca: code_natural] : modulo8411746178871703098atural(plus_p4538020629002901425atural(A,times_2397367101498566445atural(B,Ca)),B) = modulo8411746178871703098atural(A,B) ).

% mod_mult_self2
tff(fact_3934_mod__mult__self3,axiom,
    ! [Ca: 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,Ca),B)),A)),B) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) ).

% mod_mult_self3
tff(fact_3935_mod__mult__self3,axiom,
    ! [Ca: 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,Ca),B)),A)),B) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) ).

% mod_mult_self3
tff(fact_3936_mod__mult__self3,axiom,
    ! [Ca: code_integer,B: code_integer,A: code_integer] : aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Ca),B)),A)),B) = aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A),B) ).

% mod_mult_self3
tff(fact_3937_mod__mult__self3,axiom,
    ! [Ca: code_natural,B: code_natural,A: code_natural] : modulo8411746178871703098atural(plus_p4538020629002901425atural(times_2397367101498566445atural(Ca,B),A),B) = modulo8411746178871703098atural(A,B) ).

% mod_mult_self3
tff(fact_3938_mod__mult__self4,axiom,
    ! [B: nat,Ca: 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),Ca)),A)),B) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) ).

% mod_mult_self4
tff(fact_3939_mod__mult__self4,axiom,
    ! [B: int,Ca: 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),Ca)),A)),B) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) ).

% mod_mult_self4
tff(fact_3940_mod__mult__self4,axiom,
    ! [B: code_integer,Ca: code_integer,A: code_integer] : aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),Ca)),A)),B) = aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A),B) ).

% mod_mult_self4
tff(fact_3941_mod__mult__self4,axiom,
    ! [B: code_natural,Ca: code_natural,A: code_natural] : modulo8411746178871703098atural(plus_p4538020629002901425atural(times_2397367101498566445atural(B,Ca),A),B) = modulo8411746178871703098atural(A,B) ).

% mod_mult_self4
tff(fact_3942_of__nat__less__iff,axiom,
    ! [M: nat,N2: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,M)),aa_nat_rat(semiri681578069525770553at_rat,N2))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N2) ) ).

% of_nat_less_iff
tff(fact_3943_of__nat__less__iff,axiom,
    ! [M: nat,N2: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,N2))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N2) ) ).

% of_nat_less_iff
tff(fact_3944_of__nat__less__iff,axiom,
    ! [M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,N2))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N2) ) ).

% of_nat_less_iff
tff(fact_3945_of__nat__less__iff,axiom,
    ! [M: nat,N2: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,M)),aa_nat_Code_integer(semiri4939895301339042750nteger,N2))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N2) ) ).

% of_nat_less_iff
tff(fact_3946_of__nat__le__iff,axiom,
    ! [M: nat,N2: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,M)),aa_nat_Code_integer(semiri4939895301339042750nteger,N2))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2) ) ).

% of_nat_le_iff
tff(fact_3947_of__nat__le__iff,axiom,
    ! [M: nat,N2: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(semiri681578069525770553at_rat,M)),aa_nat_rat(semiri681578069525770553at_rat,N2))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2) ) ).

% of_nat_le_iff
tff(fact_3948_of__nat__le__iff,axiom,
    ! [M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,N2))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2) ) ).

% of_nat_le_iff
tff(fact_3949_of__nat__le__iff,axiom,
    ! [M: nat,N2: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,N2))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2) ) ).

% of_nat_le_iff
tff(fact_3950_zero__less__Suc,axiom,
    ! [N2: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(suc,N2)) ).

% zero_less_Suc
tff(fact_3951_less__Suc0,axiom,
    ! [N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),aa_nat_nat(suc,zero_zero_nat))
    <=> ( N2 = zero_zero_nat ) ) ).

% less_Suc0
tff(fact_3952_of__nat__add,axiom,
    ! [M: nat,N2: nat] : aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N2)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(semiri681578069525770553at_rat,M)),aa_nat_rat(semiri681578069525770553at_rat,N2)) ).

% of_nat_add
tff(fact_3953_of__nat__add,axiom,
    ! [M: nat,N2: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N2)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,N2)) ).

% of_nat_add
tff(fact_3954_of__nat__add,axiom,
    ! [M: nat,N2: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,N2)) ).

% of_nat_add
tff(fact_3955_of__nat__add,axiom,
    ! [M: nat,N2: nat] : aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N2)) = aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,M)),aa_nat_Code_integer(semiri4939895301339042750nteger,N2)) ).

% of_nat_add
tff(fact_3956_zero__less__diff,axiom,
    ! [N2: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),M))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N2) ) ).

% zero_less_diff
tff(fact_3957_mult__eq__1__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N2) = aa_nat_nat(suc,zero_zero_nat) )
    <=> ( ( M = aa_nat_nat(suc,zero_zero_nat) )
        & ( N2 = aa_nat_nat(suc,zero_zero_nat) ) ) ) ).

% mult_eq_1_iff
tff(fact_3958_one__eq__mult__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( aa_nat_nat(suc,zero_zero_nat) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N2) )
    <=> ( ( M = aa_nat_nat(suc,zero_zero_nat) )
        & ( N2 = aa_nat_nat(suc,zero_zero_nat) ) ) ) ).

% one_eq_mult_iff
tff(fact_3959_of__nat__mult,axiom,
    ! [M: nat,N2: nat] : aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N2)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,M)),aa_nat_rat(semiri681578069525770553at_rat,N2)) ).

% of_nat_mult
tff(fact_3960_of__nat__mult,axiom,
    ! [M: nat,N2: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N2)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,N2)) ).

% of_nat_mult
tff(fact_3961_of__nat__mult,axiom,
    ! [M: nat,N2: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N2)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,N2)) ).

% of_nat_mult
tff(fact_3962_of__nat__mult,axiom,
    ! [M: nat,N2: nat] : aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N2)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,M)),aa_nat_Code_integer(semiri4939895301339042750nteger,N2)) ).

% of_nat_mult
tff(fact_3963_diff__is__0__eq_H,axiom,
    ! [M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N2) = zero_zero_nat ) ) ).

% diff_is_0_eq'
tff(fact_3964_diff__is__0__eq,axiom,
    ! [M: nat,N2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N2) = zero_zero_nat )
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2) ) ).

% diff_is_0_eq
tff(fact_3965_of__nat__eq__1__iff,axiom,
    ! [N2: nat] :
      ( ( aa_nat_rat(semiri681578069525770553at_rat,N2) = one_one_rat )
    <=> ( N2 = one_one_nat ) ) ).

% of_nat_eq_1_iff
tff(fact_3966_of__nat__eq__1__iff,axiom,
    ! [N2: nat] :
      ( ( aa_nat_int(semiri1314217659103216013at_int,N2) = one_one_int )
    <=> ( N2 = one_one_nat ) ) ).

% of_nat_eq_1_iff
tff(fact_3967_of__nat__eq__1__iff,axiom,
    ! [N2: nat] :
      ( ( aa_nat_nat(semiri1316708129612266289at_nat,N2) = one_one_nat )
    <=> ( N2 = one_one_nat ) ) ).

% of_nat_eq_1_iff
tff(fact_3968_of__nat__eq__1__iff,axiom,
    ! [N2: nat] :
      ( ( aa_nat_Code_integer(semiri4939895301339042750nteger,N2) = one_one_Code_integer )
    <=> ( N2 = one_one_nat ) ) ).

% of_nat_eq_1_iff
tff(fact_3969_of__nat__1__eq__iff,axiom,
    ! [N2: nat] :
      ( ( one_one_rat = aa_nat_rat(semiri681578069525770553at_rat,N2) )
    <=> ( N2 = one_one_nat ) ) ).

% of_nat_1_eq_iff
tff(fact_3970_of__nat__1__eq__iff,axiom,
    ! [N2: nat] :
      ( ( one_one_int = aa_nat_int(semiri1314217659103216013at_int,N2) )
    <=> ( N2 = one_one_nat ) ) ).

% of_nat_1_eq_iff
tff(fact_3971_of__nat__1__eq__iff,axiom,
    ! [N2: nat] :
      ( ( one_one_nat = aa_nat_nat(semiri1316708129612266289at_nat,N2) )
    <=> ( N2 = one_one_nat ) ) ).

% of_nat_1_eq_iff
tff(fact_3972_of__nat__1__eq__iff,axiom,
    ! [N2: nat] :
      ( ( one_one_Code_integer = aa_nat_Code_integer(semiri4939895301339042750nteger,N2) )
    <=> ( N2 = one_one_nat ) ) ).

% of_nat_1_eq_iff
tff(fact_3973_of__nat__1,axiom,
    aa_nat_rat(semiri681578069525770553at_rat,one_one_nat) = one_one_rat ).

% of_nat_1
tff(fact_3974_of__nat__1,axiom,
    aa_nat_int(semiri1314217659103216013at_int,one_one_nat) = one_one_int ).

% of_nat_1
tff(fact_3975_of__nat__1,axiom,
    aa_nat_nat(semiri1316708129612266289at_nat,one_one_nat) = one_one_nat ).

% of_nat_1
tff(fact_3976_of__nat__1,axiom,
    aa_nat_Code_integer(semiri4939895301339042750nteger,one_one_nat) = one_one_Code_integer ).

% of_nat_1
tff(fact_3977_mult__Suc__right,axiom,
    ! [M: nat,N2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),aa_nat_nat(suc,N2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N2)) ).

% mult_Suc_right
tff(fact_3978_Nat_Odiff__diff__right,axiom,
    ! [K: nat,J: nat,I2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),J)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),K)),J) ) ) ).

% Nat.diff_diff_right
tff(fact_3979_Nat_Oadd__diff__assoc2,axiom,
    ! [K: nat,J: nat,I2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),J)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K)),I2) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),I2)),K) ) ) ).

% Nat.add_diff_assoc2
tff(fact_3980_Nat_Oadd__diff__assoc,axiom,
    ! [K: nat,J: nat,I2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),J)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),J)),K) ) ) ).

% Nat.add_diff_assoc
tff(fact_3981_diff__Suc__1,axiom,
    ! [N2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,N2)),one_one_nat) = N2 ).

% diff_Suc_1
tff(fact_3982_divide__le__eq__numeral1_I1_J,axiom,
    ! [B: rat,W2: num,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),aa_num_rat(numeral_numeral_rat,W2))),A)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_num_rat(numeral_numeral_rat,W2))) ) ).

% divide_le_eq_numeral1(1)
tff(fact_3983_le__divide__eq__numeral1_I1_J,axiom,
    ! [A: rat,B: rat,W2: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),aa_num_rat(numeral_numeral_rat,W2)))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_num_rat(numeral_numeral_rat,W2))),B) ) ).

% le_divide_eq_numeral1(1)
tff(fact_3984_divide__eq__eq__numeral1_I1_J,axiom,
    ! [B: rat,W2: num,A: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),aa_num_rat(numeral_numeral_rat,W2)) = A )
    <=> $ite(aa_num_rat(numeral_numeral_rat,W2) != zero_zero_rat,B = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_num_rat(numeral_numeral_rat,W2)),A = zero_zero_rat) ) ).

% divide_eq_eq_numeral1(1)
tff(fact_3985_eq__divide__eq__numeral1_I1_J,axiom,
    ! [A: rat,B: rat,W2: num] :
      ( ( A = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),aa_num_rat(numeral_numeral_rat,W2)) )
    <=> $ite(aa_num_rat(numeral_numeral_rat,W2) != zero_zero_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_num_rat(numeral_numeral_rat,W2)) = B,A = zero_zero_rat) ) ).

% eq_divide_eq_numeral1(1)
tff(fact_3986_divide__less__eq__numeral1_I1_J,axiom,
    ! [B: rat,W2: 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_num_rat(numeral_numeral_rat,W2))),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),aa_num_rat(numeral_numeral_rat,W2))) ) ).

% divide_less_eq_numeral1(1)
tff(fact_3987_less__divide__eq__numeral1_I1_J,axiom,
    ! [A: rat,B: rat,W2: 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_num_rat(numeral_numeral_rat,W2)))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_num_rat(numeral_numeral_rat,W2))),B) ) ).

% less_divide_eq_numeral1(1)
tff(fact_3988_of__nat__le__0__iff,axiom,
    ! [M: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,M)),zero_z3403309356797280102nteger)
    <=> ( M = zero_zero_nat ) ) ).

% of_nat_le_0_iff
tff(fact_3989_of__nat__le__0__iff,axiom,
    ! [M: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(semiri681578069525770553at_rat,M)),zero_zero_rat)
    <=> ( M = zero_zero_nat ) ) ).

% of_nat_le_0_iff
tff(fact_3990_of__nat__le__0__iff,axiom,
    ! [M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),zero_zero_nat)
    <=> ( M = zero_zero_nat ) ) ).

% of_nat_le_0_iff
tff(fact_3991_of__nat__le__0__iff,axiom,
    ! [M: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,M)),zero_zero_int)
    <=> ( M = zero_zero_nat ) ) ).

% of_nat_le_0_iff
tff(fact_3992_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_3993_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_3994_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_3995_of__nat__Suc,axiom,
    ! [M: nat] : aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(suc,M)) = aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(one_one_Code_integer),aa_nat_Code_integer(semiri4939895301339042750nteger,M)) ).

% of_nat_Suc
tff(fact_3996_Suc__pred,axiom,
    ! [N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)
     => ( aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),aa_nat_nat(suc,zero_zero_nat))) = N2 ) ) ).

% Suc_pred
tff(fact_3997_one__le__mult__iff,axiom,
    ! [M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N2))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),M)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),N2) ) ) ).

% one_le_mult_iff
tff(fact_3998_diff__Suc__diff__eq1,axiom,
    ! [K: nat,J: nat,I2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),J)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I2),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),K)),aa_nat_nat(suc,J)) ) ) ).

% diff_Suc_diff_eq1
tff(fact_3999_diff__Suc__diff__eq2,axiom,
    ! [K: nat,J: nat,I2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),J)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K))),I2) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,J)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),I2)) ) ) ).

% diff_Suc_diff_eq2
tff(fact_4000_Suc__diff,axiom,
    ! [M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),M)
       => ( aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),M)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat)) ) ) ) ).

% Suc_diff
tff(fact_4001_of__nat__0__less__iff,axiom,
    ! [N2: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_nat_rat(semiri681578069525770553at_rat,N2))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2) ) ).

% of_nat_0_less_iff
tff(fact_4002_of__nat__0__less__iff,axiom,
    ! [N2: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(semiri1314217659103216013at_int,N2))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2) ) ).

% of_nat_0_less_iff
tff(fact_4003_of__nat__0__less__iff,axiom,
    ! [N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(semiri1316708129612266289at_nat,N2))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2) ) ).

% of_nat_0_less_iff
tff(fact_4004_of__nat__0__less__iff,axiom,
    ! [N2: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(semiri4939895301339042750nteger,N2))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2) ) ).

% of_nat_0_less_iff
tff(fact_4005_Suc__diff__1,axiom,
    ! [N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)
     => ( aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),one_one_nat)) = N2 ) ) ).

% Suc_diff_1
tff(fact_4006_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W2: nat,X: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(aa_nat_rat(semiri681578069525770553at_rat,B)),W2)),aa_nat_rat(semiri681578069525770553at_rat,X))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W2)),X) ) ).

% of_nat_less_of_nat_power_cancel_iff
tff(fact_4007_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W2: nat,X: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_nat_int(semiri1314217659103216013at_int,B)),W2)),aa_nat_int(semiri1314217659103216013at_int,X))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W2)),X) ) ).

% of_nat_less_of_nat_power_cancel_iff
tff(fact_4008_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W2: nat,X: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(semiri1316708129612266289at_nat,B)),W2)),aa_nat_nat(semiri1316708129612266289at_nat,X))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W2)),X) ) ).

% of_nat_less_of_nat_power_cancel_iff
tff(fact_4009_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W2: nat,X: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,B)),W2)),aa_nat_Code_integer(semiri4939895301339042750nteger,X))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W2)),X) ) ).

% of_nat_less_of_nat_power_cancel_iff
tff(fact_4010_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W2: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,X)),aa_nat_rat(power_power_rat(aa_nat_rat(semiri681578069525770553at_rat,B)),W2))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W2)) ) ).

% of_nat_power_less_of_nat_cancel_iff
tff(fact_4011_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W2: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,X)),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_nat_int(semiri1314217659103216013at_int,B)),W2))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W2)) ) ).

% of_nat_power_less_of_nat_cancel_iff
tff(fact_4012_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(semiri1316708129612266289at_nat,X)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(semiri1316708129612266289at_nat,B)),W2))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W2)) ) ).

% of_nat_power_less_of_nat_cancel_iff
tff(fact_4013_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W2: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,X)),aa_nat_Code_integer(power_8256067586552552935nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,B)),W2))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W2)) ) ).

% of_nat_power_less_of_nat_cancel_iff
tff(fact_4014_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W2: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,X)),aa_nat_Code_integer(power_8256067586552552935nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,B)),W2))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W2)) ) ).

% of_nat_power_le_of_nat_cancel_iff
tff(fact_4015_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W2: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(semiri681578069525770553at_rat,X)),aa_nat_rat(power_power_rat(aa_nat_rat(semiri681578069525770553at_rat,B)),W2))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W2)) ) ).

% of_nat_power_le_of_nat_cancel_iff
tff(fact_4016_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(semiri1316708129612266289at_nat,X)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(semiri1316708129612266289at_nat,B)),W2))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W2)) ) ).

% of_nat_power_le_of_nat_cancel_iff
tff(fact_4017_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W2: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,X)),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_nat_int(semiri1314217659103216013at_int,B)),W2))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W2)) ) ).

% of_nat_power_le_of_nat_cancel_iff
tff(fact_4018_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W2: nat,X: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,B)),W2)),aa_nat_Code_integer(semiri4939895301339042750nteger,X))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W2)),X) ) ).

% of_nat_le_of_nat_power_cancel_iff
tff(fact_4019_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W2: nat,X: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(aa_nat_rat(semiri681578069525770553at_rat,B)),W2)),aa_nat_rat(semiri681578069525770553at_rat,X))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W2)),X) ) ).

% of_nat_le_of_nat_power_cancel_iff
tff(fact_4020_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W2: nat,X: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(semiri1316708129612266289at_nat,B)),W2)),aa_nat_nat(semiri1316708129612266289at_nat,X))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W2)),X) ) ).

% of_nat_le_of_nat_power_cancel_iff
tff(fact_4021_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W2: nat,X: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_nat_int(semiri1314217659103216013at_int,B)),W2)),aa_nat_int(semiri1314217659103216013at_int,X))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W2)),X) ) ).

% of_nat_le_of_nat_power_cancel_iff
tff(fact_4022_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [X: nat,I2: num,N2: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,X)),aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,I2)),N2))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I2)),N2)) ) ).

% of_nat_less_numeral_power_cancel_iff
tff(fact_4023_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [X: nat,I2: num,N2: nat] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,aa_nat_Code_natural(semiri3763490453095760265atural,X)),aa_nat_Code_natural(power_7079662738309270450atural(aa_num_Code_natural(numera5444537566228673987atural,I2)),N2))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I2)),N2)) ) ).

% of_nat_less_numeral_power_cancel_iff
tff(fact_4024_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [X: nat,I2: num,N2: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,X)),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,I2)),N2))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I2)),N2)) ) ).

% of_nat_less_numeral_power_cancel_iff
tff(fact_4025_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [X: nat,I2: num,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(semiri1316708129612266289at_nat,X)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I2)),N2))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I2)),N2)) ) ).

% of_nat_less_numeral_power_cancel_iff
tff(fact_4026_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [X: nat,I2: num,N2: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,X)),aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,I2)),N2))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I2)),N2)) ) ).

% of_nat_less_numeral_power_cancel_iff
tff(fact_4027_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [I2: num,N2: nat,X: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,I2)),N2)),aa_nat_rat(semiri681578069525770553at_rat,X))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I2)),N2)),X) ) ).

% numeral_power_less_of_nat_cancel_iff
tff(fact_4028_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [I2: num,N2: nat,X: nat] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,aa_nat_Code_natural(power_7079662738309270450atural(aa_num_Code_natural(numera5444537566228673987atural,I2)),N2)),aa_nat_Code_natural(semiri3763490453095760265atural,X))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I2)),N2)),X) ) ).

% numeral_power_less_of_nat_cancel_iff
tff(fact_4029_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [I2: num,N2: nat,X: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,I2)),N2)),aa_nat_int(semiri1314217659103216013at_int,X))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I2)),N2)),X) ) ).

% numeral_power_less_of_nat_cancel_iff
tff(fact_4030_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [I2: num,N2: nat,X: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I2)),N2)),aa_nat_nat(semiri1316708129612266289at_nat,X))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I2)),N2)),X) ) ).

% numeral_power_less_of_nat_cancel_iff
tff(fact_4031_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [I2: num,N2: nat,X: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,I2)),N2)),aa_nat_Code_integer(semiri4939895301339042750nteger,X))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I2)),N2)),X) ) ).

% numeral_power_less_of_nat_cancel_iff
tff(fact_4032_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [X: nat,I2: num,N2: nat] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,aa_nat_Code_natural(semiri3763490453095760265atural,X)),aa_nat_Code_natural(power_7079662738309270450atural(aa_num_Code_natural(numera5444537566228673987atural,I2)),N2))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I2)),N2)) ) ).

% of_nat_le_numeral_power_cancel_iff
tff(fact_4033_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [X: nat,I2: num,N2: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,X)),aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,I2)),N2))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I2)),N2)) ) ).

% of_nat_le_numeral_power_cancel_iff
tff(fact_4034_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [X: nat,I2: num,N2: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(semiri681578069525770553at_rat,X)),aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,I2)),N2))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I2)),N2)) ) ).

% of_nat_le_numeral_power_cancel_iff
tff(fact_4035_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [X: nat,I2: num,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(semiri1316708129612266289at_nat,X)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I2)),N2))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I2)),N2)) ) ).

% of_nat_le_numeral_power_cancel_iff
tff(fact_4036_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [X: nat,I2: num,N2: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,X)),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,I2)),N2))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I2)),N2)) ) ).

% of_nat_le_numeral_power_cancel_iff
tff(fact_4037_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I2: num,N2: nat,X: nat] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,aa_nat_Code_natural(power_7079662738309270450atural(aa_num_Code_natural(numera5444537566228673987atural,I2)),N2)),aa_nat_Code_natural(semiri3763490453095760265atural,X))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I2)),N2)),X) ) ).

% numeral_power_le_of_nat_cancel_iff
tff(fact_4038_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I2: num,N2: nat,X: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,I2)),N2)),aa_nat_Code_integer(semiri4939895301339042750nteger,X))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I2)),N2)),X) ) ).

% numeral_power_le_of_nat_cancel_iff
tff(fact_4039_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I2: num,N2: nat,X: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,I2)),N2)),aa_nat_rat(semiri681578069525770553at_rat,X))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I2)),N2)),X) ) ).

% numeral_power_le_of_nat_cancel_iff
tff(fact_4040_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I2: num,N2: nat,X: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I2)),N2)),aa_nat_nat(semiri1316708129612266289at_nat,X))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I2)),N2)),X) ) ).

% numeral_power_le_of_nat_cancel_iff
tff(fact_4041_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I2: num,N2: nat,X: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,I2)),N2)),aa_nat_int(semiri1314217659103216013at_int,X))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I2)),N2)),X) ) ).

% numeral_power_le_of_nat_cancel_iff
tff(fact_4042_mod__induct,axiom,
    ! [Pa: fun_nat_o,N2: nat,P2: nat,M: nat] :
      ( aa_nat_o(Pa,N2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),P2)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),P2)
         => ( ! [N5: nat] :
                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N5),P2)
               => ( aa_nat_o(Pa,N5)
                 => aa_nat_o(Pa,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,N5)),P2)) ) )
           => aa_nat_o(Pa,M) ) ) ) ) ).

% mod_induct
tff(fact_4043_mod__if,axiom,
    ! [M: nat,N2: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N2) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N2),M,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N2)),N2)) ).

% mod_if
tff(fact_4044_mod__Suc__le__divisor,axiom,
    ! [M: nat,N2: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_nat_nat(suc,N2))),N2) ).

% mod_Suc_le_divisor
tff(fact_4045_le__mod__geq,axiom,
    ! [N2: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),M)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N2) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N2)),N2) ) ) ).

% le_mod_geq
tff(fact_4046_Suc__diff__Suc,axiom,
    ! [N2: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),M)
     => ( aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),aa_nat_nat(suc,N2))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N2) ) ) ).

% Suc_diff_Suc
tff(fact_4047_diff__less__Suc,axiom,
    ! [M: nat,N2: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N2)),aa_nat_nat(suc,M)) ).

% diff_less_Suc
tff(fact_4048_Suc__diff__le,axiom,
    ! [N2: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),M)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,M)),N2) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N2)) ) ) ).

% Suc_diff_le
tff(fact_4049_zero__induct__lemma,axiom,
    ! [Pa: fun_nat_o,K: nat,I2: nat] :
      ( aa_nat_o(Pa,K)
     => ( ! [N5: nat] :
            ( aa_nat_o(Pa,aa_nat_nat(suc,N5))
           => aa_nat_o(Pa,N5) )
       => aa_nat_o(Pa,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),I2)) ) ) ).

% zero_induct_lemma
tff(fact_4050_diff__commute,axiom,
    ! [I2: nat,J: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I2),J)),K) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I2),K)),J) ).

% diff_commute
tff(fact_4051_n__not__Suc__n,axiom,
    ! [N2: nat] : N2 != aa_nat_nat(suc,N2) ).

% n_not_Suc_n
tff(fact_4052_Suc__inject,axiom,
    ! [X: nat,Y: nat] :
      ( ( aa_nat_nat(suc,X) = aa_nat_nat(suc,Y) )
     => ( X = Y ) ) ).

% Suc_inject
tff(fact_4053_of__nat__diff,axiom,
    ! [N2: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),M)
     => ( aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N2)) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_nat_rat(semiri681578069525770553at_rat,M)),aa_nat_rat(semiri681578069525770553at_rat,N2)) ) ) ).

% of_nat_diff
tff(fact_4054_of__nat__diff,axiom,
    ! [N2: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),M)
     => ( aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N2)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,N2)) ) ) ).

% of_nat_diff
tff(fact_4055_of__nat__diff,axiom,
    ! [N2: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),M)
     => ( aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N2)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,N2)) ) ) ).

% of_nat_diff
tff(fact_4056_of__nat__diff,axiom,
    ! [N2: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),M)
     => ( aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N2)) = aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,M)),aa_nat_Code_integer(semiri4939895301339042750nteger,N2)) ) ) ).

% of_nat_diff
tff(fact_4057_mod__geq,axiom,
    ! [M: nat,N2: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N2)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N2) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N2)),N2) ) ) ).

% mod_geq
tff(fact_4058_Suc__to__right,axiom,
    ! [N2: nat,M: nat] :
      ( ( aa_nat_nat(suc,N2) = M )
     => ( N2 = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),aa_nat_nat(suc,zero_zero_nat)) ) ) ).

% Suc_to_right
tff(fact_4059_diff__Suc__eq__diff__pred,axiom,
    ! [M: nat,N2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),aa_nat_nat(suc,N2)) = 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)),N2) ).

% diff_Suc_eq_diff_pred
tff(fact_4060_of__nat__neq__0,axiom,
    ! [N2: nat] : aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,N2)) != zero_zero_rat ).

% of_nat_neq_0
tff(fact_4061_of__nat__neq__0,axiom,
    ! [N2: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,N2)) != zero_zero_int ).

% of_nat_neq_0
tff(fact_4062_of__nat__neq__0,axiom,
    ! [N2: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(suc,N2)) != zero_zero_nat ).

% of_nat_neq_0
tff(fact_4063_of__nat__neq__0,axiom,
    ! [N2: nat] : aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(suc,N2)) != zero_z3403309356797280102nteger ).

% of_nat_neq_0
tff(fact_4064_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_4065_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_4066_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_4067_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_4068_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_4069_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_4070_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_4071_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_4072_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_4073_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_4074_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_4075_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_4076_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_4077_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_4078_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_4079_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_4080_int__power__div__base,axiom,
    ! [M: nat,K: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),K)
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,K),M)),K) = aa_nat_int(aa_int_fun_nat_int(power_power_int,K),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),aa_nat_nat(suc,zero_zero_nat))) ) ) ) ).

% int_power_div_base
tff(fact_4081_diff__Suc__less,axiom,
    ! [N2: nat,I2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),aa_nat_nat(suc,I2))),N2) ) ).

% diff_Suc_less
tff(fact_4082_mod__mult__eq,axiom,
    ! [A: nat,Ca: 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),Ca)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B),Ca))),Ca) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),Ca) ).

% mod_mult_eq
tff(fact_4083_mod__mult__eq,axiom,
    ! [A: int,Ca: 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),Ca)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B),Ca))),Ca) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),Ca) ).

% mod_mult_eq
tff(fact_4084_mod__mult__eq,axiom,
    ! [A: code_integer,Ca: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A),Ca)),aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(B),Ca))),Ca) = aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),Ca) ).

% mod_mult_eq
tff(fact_4085_mod__mult__eq,axiom,
    ! [A: code_natural,Ca: code_natural,B: code_natural] : modulo8411746178871703098atural(times_2397367101498566445atural(modulo8411746178871703098atural(A,Ca),modulo8411746178871703098atural(B,Ca)),Ca) = modulo8411746178871703098atural(times_2397367101498566445atural(A,B),Ca) ).

% mod_mult_eq
tff(fact_4086_mod__mult__cong,axiom,
    ! [A: nat,Ca: nat,A2: nat,B: nat,B2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),Ca) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),Ca) )
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B),Ca) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B2),Ca) )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),Ca) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),B2)),Ca) ) ) ) ).

% mod_mult_cong
tff(fact_4087_mod__mult__cong,axiom,
    ! [A: int,Ca: int,A2: int,B: int,B2: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),Ca) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),Ca) )
     => ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B),Ca) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B2),Ca) )
       => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),Ca) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),B2)),Ca) ) ) ) ).

% mod_mult_cong
tff(fact_4088_mod__mult__cong,axiom,
    ! [A: code_integer,Ca: code_integer,A2: code_integer,B: code_integer,B2: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A),Ca) = aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A2),Ca) )
     => ( ( aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(B),Ca) = aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(B2),Ca) )
       => ( aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),Ca) = aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A2),B2)),Ca) ) ) ) ).

% mod_mult_cong
tff(fact_4089_mod__mult__cong,axiom,
    ! [A: code_natural,Ca: code_natural,A2: code_natural,B: code_natural,B2: code_natural] :
      ( ( modulo8411746178871703098atural(A,Ca) = modulo8411746178871703098atural(A2,Ca) )
     => ( ( modulo8411746178871703098atural(B,Ca) = modulo8411746178871703098atural(B2,Ca) )
       => ( modulo8411746178871703098atural(times_2397367101498566445atural(A,B),Ca) = modulo8411746178871703098atural(times_2397367101498566445atural(A2,B2),Ca) ) ) ) ).

% mod_mult_cong
tff(fact_4090_mod__mult__mult2,axiom,
    ! [A: nat,Ca: 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),Ca)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),Ca)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)),Ca) ).

% mod_mult_mult2
tff(fact_4091_mod__mult__mult2,axiom,
    ! [A: int,Ca: 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),Ca)),aa_int_int(aa_int_fun_int_int(times_times_int,B),Ca)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)),Ca) ).

% mod_mult_mult2
tff(fact_4092_mod__mult__mult2,axiom,
    ! [A: code_integer,Ca: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),Ca)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),Ca)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A),B)),Ca) ).

% mod_mult_mult2
tff(fact_4093_mod__mult__mult2,axiom,
    ! [A: code_natural,Ca: code_natural,B: code_natural] : modulo8411746178871703098atural(times_2397367101498566445atural(A,Ca),times_2397367101498566445atural(B,Ca)) = times_2397367101498566445atural(modulo8411746178871703098atural(A,B),Ca) ).

% mod_mult_mult2
tff(fact_4094_mult__mod__right,axiom,
    ! [Ca: nat,A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ca),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,Ca),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ca),B)) ).

% mult_mod_right
tff(fact_4095_mult__mod__right,axiom,
    ! [Ca: int,A: int,B: int] : aa_int_int(aa_int_fun_int_int(times_times_int,Ca),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,Ca),A)),aa_int_int(aa_int_fun_int_int(times_times_int,Ca),B)) ).

% mult_mod_right
tff(fact_4096_mult__mod__right,axiom,
    ! [Ca: code_integer,A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Ca),aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A),B)) = aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Ca),A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Ca),B)) ).

% mult_mod_right
tff(fact_4097_mult__mod__right,axiom,
    ! [Ca: code_natural,A: code_natural,B: code_natural] : times_2397367101498566445atural(Ca,modulo8411746178871703098atural(A,B)) = modulo8411746178871703098atural(times_2397367101498566445atural(Ca,A),times_2397367101498566445atural(Ca,B)) ).

% mult_mod_right
tff(fact_4098_mod__mult__left__eq,axiom,
    ! [A: nat,Ca: 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),Ca)),B)),Ca) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),Ca) ).

% mod_mult_left_eq
tff(fact_4099_mod__mult__left__eq,axiom,
    ! [A: int,Ca: 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),Ca)),B)),Ca) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),Ca) ).

% mod_mult_left_eq
tff(fact_4100_mod__mult__left__eq,axiom,
    ! [A: code_integer,Ca: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A),Ca)),B)),Ca) = aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),Ca) ).

% mod_mult_left_eq
tff(fact_4101_mod__mult__left__eq,axiom,
    ! [A: code_natural,Ca: code_natural,B: code_natural] : modulo8411746178871703098atural(times_2397367101498566445atural(modulo8411746178871703098atural(A,Ca),B),Ca) = modulo8411746178871703098atural(times_2397367101498566445atural(A,B),Ca) ).

% mod_mult_left_eq
tff(fact_4102_mod__mult__right__eq,axiom,
    ! [A: nat,B: nat,Ca: 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),Ca))),Ca) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),Ca) ).

% mod_mult_right_eq
tff(fact_4103_mod__mult__right__eq,axiom,
    ! [A: int,B: int,Ca: 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),Ca))),Ca) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),Ca) ).

% mod_mult_right_eq
tff(fact_4104_mod__mult__right__eq,axiom,
    ! [A: code_integer,B: code_integer,Ca: code_integer] : aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(B),Ca))),Ca) = aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),Ca) ).

% mod_mult_right_eq
tff(fact_4105_mod__mult__right__eq,axiom,
    ! [A: code_natural,B: code_natural,Ca: code_natural] : modulo8411746178871703098atural(times_2397367101498566445atural(A,modulo8411746178871703098atural(B,Ca)),Ca) = modulo8411746178871703098atural(times_2397367101498566445atural(A,B),Ca) ).

% mod_mult_right_eq
tff(fact_4106_diff__mono,axiom,
    ! [A: rat,B: rat,D2: rat,Ca: 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),Ca)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),Ca)),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,B),D2)) ) ) ).

% diff_mono
tff(fact_4107_diff__mono,axiom,
    ! [A: int,B: int,D2: int,Ca: 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),Ca)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),Ca)),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),D2)) ) ) ).

% diff_mono
tff(fact_4108_diff__left__mono,axiom,
    ! [B: rat,A: rat,Ca: 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,Ca),A)),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,Ca),B)) ) ).

% diff_left_mono
tff(fact_4109_diff__left__mono,axiom,
    ! [B: int,A: int,Ca: 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,Ca),A)),aa_int_int(aa_int_fun_int_int(minus_minus_int,Ca),B)) ) ).

% diff_left_mono
tff(fact_4110_diff__right__mono,axiom,
    ! [A: rat,B: rat,Ca: 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),Ca)),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,B),Ca)) ) ).

% diff_right_mono
tff(fact_4111_diff__right__mono,axiom,
    ! [A: int,B: int,Ca: 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),Ca)),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),Ca)) ) ).

% diff_right_mono
tff(fact_4112_diff__eq__diff__less__eq,axiom,
    ! [A: rat,B: rat,Ca: 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,Ca),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,Ca),D2) ) ) ).

% diff_eq_diff_less_eq
tff(fact_4113_diff__eq__diff__less__eq,axiom,
    ! [A: int,B: int,Ca: 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,Ca),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,Ca),D2) ) ) ).

% diff_eq_diff_less_eq
tff(fact_4114_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_4115_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_4116_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_4117_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_4118_diff__strict__mono,axiom,
    ! [A: rat,B: rat,D2: rat,Ca: 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),Ca)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),Ca)),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,B),D2)) ) ) ).

% diff_strict_mono
tff(fact_4119_diff__strict__mono,axiom,
    ! [A: int,B: int,D2: int,Ca: 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),Ca)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),Ca)),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),D2)) ) ) ).

% diff_strict_mono
tff(fact_4120_diff__eq__diff__less,axiom,
    ! [A: rat,B: rat,Ca: 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,Ca),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,Ca),D2) ) ) ).

% diff_eq_diff_less
tff(fact_4121_diff__eq__diff__less,axiom,
    ! [A: int,B: int,Ca: 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,Ca),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,Ca),D2) ) ) ).

% diff_eq_diff_less
tff(fact_4122_diff__strict__left__mono,axiom,
    ! [B: rat,A: rat,Ca: 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,Ca),A)),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,Ca),B)) ) ).

% diff_strict_left_mono
tff(fact_4123_diff__strict__left__mono,axiom,
    ! [B: int,A: int,Ca: 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,Ca),A)),aa_int_int(aa_int_fun_int_int(minus_minus_int,Ca),B)) ) ).

% diff_strict_left_mono
tff(fact_4124_diff__strict__right__mono,axiom,
    ! [A: rat,B: rat,Ca: 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),Ca)),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,B),Ca)) ) ).

% diff_strict_right_mono
tff(fact_4125_diff__strict__right__mono,axiom,
    ! [A: int,B: int,Ca: 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),Ca)),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),Ca)) ) ).

% diff_strict_right_mono
tff(fact_4126_right__diff__distrib_H,axiom,
    ! [A: code_integer,B: code_integer,Ca: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(B),Ca)) = aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),Ca)) ).

% right_diff_distrib'
tff(fact_4127_right__diff__distrib_H,axiom,
    ! [A: rat,B: rat,Ca: 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),Ca)) = 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),Ca)) ).

% right_diff_distrib'
tff(fact_4128_right__diff__distrib_H,axiom,
    ! [A: nat,B: nat,Ca: 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),Ca)) = 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),Ca)) ).

% right_diff_distrib'
tff(fact_4129_right__diff__distrib_H,axiom,
    ! [A: int,B: int,Ca: 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),Ca)) = 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),Ca)) ).

% right_diff_distrib'
tff(fact_4130_left__diff__distrib_H,axiom,
    ! [B: code_integer,Ca: code_integer,A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(B),Ca)),A) = aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Ca),A)) ).

% left_diff_distrib'
tff(fact_4131_left__diff__distrib_H,axiom,
    ! [B: rat,Ca: 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),Ca)),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,Ca),A)) ).

% left_diff_distrib'
tff(fact_4132_left__diff__distrib_H,axiom,
    ! [B: nat,Ca: 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),Ca)),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,Ca),A)) ).

% left_diff_distrib'
tff(fact_4133_left__diff__distrib_H,axiom,
    ! [B: int,Ca: 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),Ca)),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,Ca),A)) ).

% left_diff_distrib'
tff(fact_4134_right__diff__distrib,axiom,
    ! [A: code_integer,B: code_integer,Ca: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(B),Ca)) = aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),Ca)) ).

% right_diff_distrib
tff(fact_4135_right__diff__distrib,axiom,
    ! [A: rat,B: rat,Ca: 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),Ca)) = 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),Ca)) ).

% right_diff_distrib
tff(fact_4136_right__diff__distrib,axiom,
    ! [A: int,B: int,Ca: 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),Ca)) = 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),Ca)) ).

% right_diff_distrib
tff(fact_4137_left__diff__distrib,axiom,
    ! [A: code_integer,B: code_integer,Ca: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(A),B)),Ca) = aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),Ca)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),Ca)) ).

% left_diff_distrib
tff(fact_4138_left__diff__distrib,axiom,
    ! [A: rat,B: rat,Ca: 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)),Ca) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),Ca)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),Ca)) ).

% left_diff_distrib
tff(fact_4139_left__diff__distrib,axiom,
    ! [A: int,B: int,Ca: 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)),Ca) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),Ca)),aa_int_int(aa_int_fun_int_int(times_times_int,B),Ca)) ).

% left_diff_distrib
tff(fact_4140_inf__period_I1_J,axiom,
    ! [Pa: fun_Code_integer_o,D3: code_integer,Qa: fun_Code_integer_o] :
      ( ! [X3: code_integer,K2: code_integer] :
          ( aa_Code_integer_o(Pa,X3)
        <=> aa_Code_integer_o(Pa,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(X3),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,K2),D3))) )
     => ( ! [X3: code_integer,K2: code_integer] :
            ( aa_Code_integer_o(Qa,X3)
          <=> aa_Code_integer_o(Qa,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(X3),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,K2),D3))) )
       => ! [X5: code_integer,K4: code_integer] :
            ( ( aa_Code_integer_o(Pa,X5)
              & aa_Code_integer_o(Qa,X5) )
          <=> ( aa_Code_integer_o(Pa,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(X5),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,K4),D3)))
              & aa_Code_integer_o(Qa,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(X5),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,K4),D3))) ) ) ) ) ).

% inf_period(1)
tff(fact_4141_inf__period_I1_J,axiom,
    ! [Pa: fun_rat_o,D3: rat,Qa: fun_rat_o] :
      ( ! [X3: rat,K2: rat] :
          ( aa_rat_o(Pa,X3)
        <=> aa_rat_o(Pa,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,X3),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,K2),D3))) )
     => ( ! [X3: rat,K2: rat] :
            ( aa_rat_o(Qa,X3)
          <=> aa_rat_o(Qa,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,X3),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,K2),D3))) )
       => ! [X5: rat,K4: rat] :
            ( ( aa_rat_o(Pa,X5)
              & aa_rat_o(Qa,X5) )
          <=> ( aa_rat_o(Pa,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,X5),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,K4),D3)))
              & aa_rat_o(Qa,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,X5),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,K4),D3))) ) ) ) ) ).

% inf_period(1)
tff(fact_4142_inf__period_I1_J,axiom,
    ! [Pa: fun_int_o,D3: int,Qa: fun_int_o] :
      ( ! [X3: int,K2: int] :
          ( aa_int_o(Pa,X3)
        <=> aa_int_o(Pa,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),aa_int_int(aa_int_fun_int_int(times_times_int,K2),D3))) )
     => ( ! [X3: int,K2: int] :
            ( aa_int_o(Qa,X3)
          <=> aa_int_o(Qa,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),aa_int_int(aa_int_fun_int_int(times_times_int,K2),D3))) )
       => ! [X5: int,K4: int] :
            ( ( aa_int_o(Pa,X5)
              & aa_int_o(Qa,X5) )
          <=> ( aa_int_o(Pa,aa_int_int(aa_int_fun_int_int(minus_minus_int,X5),aa_int_int(aa_int_fun_int_int(times_times_int,K4),D3)))
              & aa_int_o(Qa,aa_int_int(aa_int_fun_int_int(minus_minus_int,X5),aa_int_int(aa_int_fun_int_int(times_times_int,K4),D3))) ) ) ) ) ).

% inf_period(1)
tff(fact_4143_inf__period_I2_J,axiom,
    ! [Pa: fun_Code_integer_o,D3: code_integer,Qa: fun_Code_integer_o] :
      ( ! [X3: code_integer,K2: code_integer] :
          ( aa_Code_integer_o(Pa,X3)
        <=> aa_Code_integer_o(Pa,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(X3),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,K2),D3))) )
     => ( ! [X3: code_integer,K2: code_integer] :
            ( aa_Code_integer_o(Qa,X3)
          <=> aa_Code_integer_o(Qa,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(X3),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,K2),D3))) )
       => ! [X5: code_integer,K4: code_integer] :
            ( ( aa_Code_integer_o(Pa,X5)
              | aa_Code_integer_o(Qa,X5) )
          <=> ( aa_Code_integer_o(Pa,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(X5),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,K4),D3)))
              | aa_Code_integer_o(Qa,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(X5),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,K4),D3))) ) ) ) ) ).

% inf_period(2)
tff(fact_4144_inf__period_I2_J,axiom,
    ! [Pa: fun_rat_o,D3: rat,Qa: fun_rat_o] :
      ( ! [X3: rat,K2: rat] :
          ( aa_rat_o(Pa,X3)
        <=> aa_rat_o(Pa,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,X3),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,K2),D3))) )
     => ( ! [X3: rat,K2: rat] :
            ( aa_rat_o(Qa,X3)
          <=> aa_rat_o(Qa,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,X3),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,K2),D3))) )
       => ! [X5: rat,K4: rat] :
            ( ( aa_rat_o(Pa,X5)
              | aa_rat_o(Qa,X5) )
          <=> ( aa_rat_o(Pa,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,X5),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,K4),D3)))
              | aa_rat_o(Qa,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,X5),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,K4),D3))) ) ) ) ) ).

% inf_period(2)
tff(fact_4145_inf__period_I2_J,axiom,
    ! [Pa: fun_int_o,D3: int,Qa: fun_int_o] :
      ( ! [X3: int,K2: int] :
          ( aa_int_o(Pa,X3)
        <=> aa_int_o(Pa,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),aa_int_int(aa_int_fun_int_int(times_times_int,K2),D3))) )
     => ( ! [X3: int,K2: int] :
            ( aa_int_o(Qa,X3)
          <=> aa_int_o(Qa,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),aa_int_int(aa_int_fun_int_int(times_times_int,K2),D3))) )
       => ! [X5: int,K4: int] :
            ( ( aa_int_o(Pa,X5)
              | aa_int_o(Qa,X5) )
          <=> ( aa_int_o(Pa,aa_int_int(aa_int_fun_int_int(minus_minus_int,X5),aa_int_int(aa_int_fun_int_int(times_times_int,K4),D3)))
              | aa_int_o(Qa,aa_int_int(aa_int_fun_int_int(minus_minus_int,X5),aa_int_int(aa_int_fun_int_int(times_times_int,K4),D3))) ) ) ) ) ).

% inf_period(2)
tff(fact_4146_mod__less__eq__dividend,axiom,
    ! [M: nat,N2: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N2)),M) ).

% mod_less_eq_dividend
tff(fact_4147_nat__int__comparison_I2_J,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,A)),aa_nat_int(semiri1314217659103216013at_int,B)) ) ).

% nat_int_comparison(2)
tff(fact_4148_nat__int__comparison_I3_J,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,A)),aa_nat_int(semiri1314217659103216013at_int,B)) ) ).

% nat_int_comparison(3)
tff(fact_4149_diffs0__imp__equal,axiom,
    ! [M: nat,N2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N2) = zero_zero_nat )
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),M) = zero_zero_nat )
       => ( M = N2 ) ) ) ).

% diffs0_imp_equal
tff(fact_4150_minus__nat_Odiff__0,axiom,
    ! [M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),zero_zero_nat) = M ).

% minus_nat.diff_0
tff(fact_4151_diff__less__mono2,axiom,
    ! [M: nat,N2: nat,L: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),L)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,L),N2)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,L),M)) ) ) ).

% diff_less_mono2
tff(fact_4152_less__imp__diff__less,axiom,
    ! [J: nat,K: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J),K)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),N2)),K) ) ).

% less_imp_diff_less
tff(fact_4153_not0__implies__Suc,axiom,
    ! [N2: nat] :
      ( ( N2 != zero_zero_nat )
     => ? [M5: nat] : N2 = aa_nat_nat(suc,M5) ) ).

% not0_implies_Suc
tff(fact_4154_Zero__not__Suc,axiom,
    ! [M: nat] : zero_zero_nat != aa_nat_nat(suc,M) ).

% Zero_not_Suc
tff(fact_4155_Zero__neq__Suc,axiom,
    ! [M: nat] : zero_zero_nat != aa_nat_nat(suc,M) ).

% Zero_neq_Suc
tff(fact_4156_Suc__neq__Zero,axiom,
    ! [M: nat] : aa_nat_nat(suc,M) != zero_zero_nat ).

% Suc_neq_Zero
tff(fact_4157_zero__induct,axiom,
    ! [Pa: fun_nat_o,K: nat] :
      ( aa_nat_o(Pa,K)
     => ( ! [N5: nat] :
            ( aa_nat_o(Pa,aa_nat_nat(suc,N5))
           => aa_nat_o(Pa,N5) )
       => aa_nat_o(Pa,zero_zero_nat) ) ) ).

% zero_induct
tff(fact_4158_diff__induct,axiom,
    ! [Pa: fun_nat_fun_nat_o,M: nat,N2: nat] :
      ( ! [X3: nat] : aa_nat_o(aa_nat_fun_nat_o(Pa,X3),zero_zero_nat)
     => ( ! [Y3: nat] : aa_nat_o(aa_nat_fun_nat_o(Pa,zero_zero_nat),aa_nat_nat(suc,Y3))
       => ( ! [X3: nat,Y3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(Pa,X3),Y3)
             => aa_nat_o(aa_nat_fun_nat_o(Pa,aa_nat_nat(suc,X3)),aa_nat_nat(suc,Y3)) )
         => aa_nat_o(aa_nat_fun_nat_o(Pa,M),N2) ) ) ) ).

% diff_induct
tff(fact_4159_nat__induct,axiom,
    ! [Pa: fun_nat_o,N2: nat] :
      ( aa_nat_o(Pa,zero_zero_nat)
     => ( ! [N5: nat] :
            ( aa_nat_o(Pa,N5)
           => aa_nat_o(Pa,aa_nat_nat(suc,N5)) )
       => aa_nat_o(Pa,N2) ) ) ).

% nat_induct
tff(fact_4160_old_Onat_Oexhaust,axiom,
    ! [Y: nat] :
      ( ( Y != zero_zero_nat )
     => ~ ! [Nat3: nat] : Y != aa_nat_nat(suc,Nat3) ) ).

% old.nat.exhaust
tff(fact_4161_nat_OdiscI,axiom,
    ! [Nat: nat,X2: nat] :
      ( ( Nat = aa_nat_nat(suc,X2) )
     => ( Nat != zero_zero_nat ) ) ).

% nat.discI
tff(fact_4162_old_Onat_Odistinct_I1_J,axiom,
    ! [Nat2: nat] : zero_zero_nat != aa_nat_nat(suc,Nat2) ).

% old.nat.distinct(1)
tff(fact_4163_old_Onat_Odistinct_I2_J,axiom,
    ! [Nat2: nat] : aa_nat_nat(suc,Nat2) != zero_zero_nat ).

% old.nat.distinct(2)
tff(fact_4164_nat_Odistinct_I1_J,axiom,
    ! [X2: nat] : zero_zero_nat != aa_nat_nat(suc,X2) ).

% nat.distinct(1)
tff(fact_4165_Nat_OlessE,axiom,
    ! [I2: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),K)
     => ( ( K != aa_nat_nat(suc,I2) )
       => ~ ! [J2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),J2)
             => ( K != aa_nat_nat(suc,J2) ) ) ) ) ).

% Nat.lessE
tff(fact_4166_Suc__lessD,axiom,
    ! [M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,M)),N2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N2) ) ).

% Suc_lessD
tff(fact_4167_Suc__lessE,axiom,
    ! [I2: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,I2)),K)
     => ~ ! [J2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),J2)
           => ( K != aa_nat_nat(suc,J2) ) ) ) ).

% Suc_lessE
tff(fact_4168_Suc__lessI,axiom,
    ! [M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N2)
     => ( ( aa_nat_nat(suc,M) != N2 )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,M)),N2) ) ) ).

% Suc_lessI
tff(fact_4169_less__SucE,axiom,
    ! [M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),aa_nat_nat(suc,N2))
     => ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N2)
       => ( M = N2 ) ) ) ).

% less_SucE
tff(fact_4170_less__SucI,axiom,
    ! [M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),aa_nat_nat(suc,N2)) ) ).

% less_SucI
tff(fact_4171_Ex__less__Suc,axiom,
    ! [N2: nat,Pa: fun_nat_o] :
      ( ? [I4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),aa_nat_nat(suc,N2))
          & aa_nat_o(Pa,I4) )
    <=> ( aa_nat_o(Pa,N2)
        | ? [I4: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),N2)
            & aa_nat_o(Pa,I4) ) ) ) ).

% Ex_less_Suc
tff(fact_4172_less__Suc__eq,axiom,
    ! [M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),aa_nat_nat(suc,N2))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N2)
        | ( M = N2 ) ) ) ).

% less_Suc_eq
tff(fact_4173_not__less__eq,axiom,
    ! [M: nat,N2: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N2)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),aa_nat_nat(suc,M)) ) ).

% not_less_eq
tff(fact_4174_All__less__Suc,axiom,
    ! [N2: nat,Pa: fun_nat_o] :
      ( ! [I4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),aa_nat_nat(suc,N2))
         => aa_nat_o(Pa,I4) )
    <=> ( aa_nat_o(Pa,N2)
        & ! [I4: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),N2)
           => aa_nat_o(Pa,I4) ) ) ) ).

% All_less_Suc
tff(fact_4175_Suc__less__eq2,axiom,
    ! [N2: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,N2)),M)
    <=> ? [M7: nat] :
          ( ( M = aa_nat_nat(suc,M7) )
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),M7) ) ) ).

% Suc_less_eq2
tff(fact_4176_less__antisym,axiom,
    ! [N2: nat,M: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),aa_nat_nat(suc,M))
       => ( M = N2 ) ) ) ).

% less_antisym
tff(fact_4177_Suc__less__SucD,axiom,
    ! [M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,M)),aa_nat_nat(suc,N2))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N2) ) ).

% Suc_less_SucD
tff(fact_4178_less__trans__Suc,axiom,
    ! [I2: nat,J: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),J)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J),K)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,I2)),K) ) ) ).

% less_trans_Suc
tff(fact_4179_less__Suc__induct,axiom,
    ! [I2: nat,J: nat,Pa: fun_nat_fun_nat_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),J)
     => ( ! [I3: nat] : aa_nat_o(aa_nat_fun_nat_o(Pa,I3),aa_nat_nat(suc,I3))
       => ( ! [I3: nat,J2: nat,K2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),J2)
             => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J2),K2)
               => ( aa_nat_o(aa_nat_fun_nat_o(Pa,I3),J2)
                 => ( aa_nat_o(aa_nat_fun_nat_o(Pa,J2),K2)
                   => aa_nat_o(aa_nat_fun_nat_o(Pa,I3),K2) ) ) ) )
         => aa_nat_o(aa_nat_fun_nat_o(Pa,I2),J) ) ) ) ).

% less_Suc_induct
tff(fact_4180_strict__inc__induct,axiom,
    ! [I2: nat,J: nat,Pa: fun_nat_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),J)
     => ( ! [I3: nat] :
            ( ( J = aa_nat_nat(suc,I3) )
           => aa_nat_o(Pa,I3) )
       => ( ! [I3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),J)
             => ( aa_nat_o(Pa,aa_nat_nat(suc,I3))
               => aa_nat_o(Pa,I3) ) )
         => aa_nat_o(Pa,I2) ) ) ) ).

% strict_inc_induct
tff(fact_4181_not__less__less__Suc__eq,axiom,
    ! [N2: nat,M: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),aa_nat_nat(suc,M))
      <=> ( N2 = M ) ) ) ).

% not_less_less_Suc_eq
tff(fact_4182_eq__diff__iff,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),N2)
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),K) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),K) )
        <=> ( M = N2 ) ) ) ) ).

% eq_diff_iff
tff(fact_4183_le__diff__iff,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),N2)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),K)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),K))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2) ) ) ) ).

% le_diff_iff
tff(fact_4184_Nat_Odiff__diff__eq,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),N2)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),K)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),K)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N2) ) ) ) ).

% Nat.diff_diff_eq
tff(fact_4185_diff__le__mono,axiom,
    ! [M: nat,N2: nat,L: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),L)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),L)) ) ).

% diff_le_mono
tff(fact_4186_diff__le__self,axiom,
    ! [M: nat,N2: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N2)),M) ).

% diff_le_self
tff(fact_4187_le__diff__iff_H,axiom,
    ! [A: nat,Ca: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),Ca)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),Ca)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Ca),A)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Ca),B))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A) ) ) ) ).

% le_diff_iff'
tff(fact_4188_diff__le__mono2,axiom,
    ! [M: nat,N2: nat,L: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,L),N2)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,L),M)) ) ).

% diff_le_mono2
tff(fact_4189_diff__add__inverse2,axiom,
    ! [M: nat,N2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N2)),N2) = M ).

% diff_add_inverse2
tff(fact_4190_diff__add__inverse,axiom,
    ! [N2: nat,M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),M)),N2) = M ).

% diff_add_inverse
tff(fact_4191_diff__cancel2,axiom,
    ! [M: nat,K: nat,N2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),K)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N2) ).

% diff_cancel2
tff(fact_4192_Nat_Odiff__cancel,axiom,
    ! [K: nat,M: nat,N2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),N2)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N2) ).

% Nat.diff_cancel
tff(fact_4193_Suc__leD,axiom,
    ! [M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,M)),N2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2) ) ).

% Suc_leD
tff(fact_4194_le__SucE,axiom,
    ! [M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(suc,N2))
     => ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
       => ( M = aa_nat_nat(suc,N2) ) ) ) ).

% le_SucE
tff(fact_4195_le__SucI,axiom,
    ! [M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(suc,N2)) ) ).

% le_SucI
tff(fact_4196_Suc__le__D,axiom,
    ! [N2: nat,M8: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,N2)),M8)
     => ? [M5: nat] : M8 = aa_nat_nat(suc,M5) ) ).

% Suc_le_D
tff(fact_4197_le__Suc__eq,axiom,
    ! [M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(suc,N2))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
        | ( M = aa_nat_nat(suc,N2) ) ) ) ).

% le_Suc_eq
tff(fact_4198_Suc__n__not__le__n,axiom,
    ! [N2: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,N2)),N2) ).

% Suc_n_not_le_n
tff(fact_4199_not__less__eq__eq,axiom,
    ! [M: nat,N2: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,N2)),M) ) ).

% not_less_eq_eq
tff(fact_4200_full__nat__induct,axiom,
    ! [Pa: fun_nat_o,N2: nat] :
      ( ! [N5: nat] :
          ( ! [M4: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,M4)),N5)
             => aa_nat_o(Pa,M4) )
         => aa_nat_o(Pa,N5) )
     => aa_nat_o(Pa,N2) ) ).

% full_nat_induct
tff(fact_4201_nat__induct__at__least,axiom,
    ! [M: nat,N2: nat,Pa: fun_nat_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
     => ( aa_nat_o(Pa,M)
       => ( ! [N5: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N5)
             => ( aa_nat_o(Pa,N5)
               => aa_nat_o(Pa,aa_nat_nat(suc,N5)) ) )
         => aa_nat_o(Pa,N2) ) ) ) ).

% nat_induct_at_least
tff(fact_4202_transitive__stepwise__le,axiom,
    ! [M: nat,N2: nat,Ra: fun_nat_fun_nat_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
     => ( ! [X3: nat] : aa_nat_o(aa_nat_fun_nat_o(Ra,X3),X3)
       => ( ! [X3: nat,Y3: nat,Z2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(Ra,X3),Y3)
             => ( aa_nat_o(aa_nat_fun_nat_o(Ra,Y3),Z2)
               => aa_nat_o(aa_nat_fun_nat_o(Ra,X3),Z2) ) )
         => ( ! [N5: nat] : aa_nat_o(aa_nat_fun_nat_o(Ra,N5),aa_nat_nat(suc,N5))
           => aa_nat_o(aa_nat_fun_nat_o(Ra,M),N2) ) ) ) ) ).

% transitive_stepwise_le
tff(fact_4203_add__Suc__shift,axiom,
    ! [M: nat,N2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(suc,M)),N2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),aa_nat_nat(suc,N2)) ).

% add_Suc_shift
tff(fact_4204_add__Suc,axiom,
    ! [M: nat,N2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(suc,M)),N2) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N2)) ).

% add_Suc
tff(fact_4205_nat__arith_Osuc1,axiom,
    ! [A3: nat,K: nat,A: nat] :
      ( ( A3 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),A) )
     => ( aa_nat_nat(suc,A3) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),aa_nat_nat(suc,A)) ) ) ).

% nat_arith.suc1
tff(fact_4206_diff__mult__distrib,axiom,
    ! [M: nat,N2: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N2)),K) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),K)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N2),K)) ).

% diff_mult_distrib
tff(fact_4207_diff__mult__distrib2,axiom,
    ! [K: nat,M: nat,N2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N2)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),N2)) ).

% diff_mult_distrib2
tff(fact_4208_Suc__mult__cancel1,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K)),M) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K)),N2) )
    <=> ( M = N2 ) ) ).

% Suc_mult_cancel1
tff(fact_4209_left__diff__distrib__NO__MATCH,axiom,
    ! [X: nat,Y: nat,Ca: code_integer,A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_nat7420681472272698950eger_o(nO_MAT8694959213317031834nteger,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X),Y)),Ca)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(A),B)),Ca) = aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),Ca)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),Ca)) ) ) ).

% left_diff_distrib_NO_MATCH
tff(fact_4210_left__diff__distrib__NO__MATCH,axiom,
    ! [X: nat,Y: nat,Ca: 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)),Ca)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B)),Ca) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),Ca)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),Ca)) ) ) ).

% left_diff_distrib_NO_MATCH
tff(fact_4211_left__diff__distrib__NO__MATCH,axiom,
    ! [X: nat,Y: nat,Ca: 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)),Ca)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),Ca) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),Ca)),aa_int_int(aa_int_fun_int_int(times_times_int,B),Ca)) ) ) ).

% left_diff_distrib_NO_MATCH
tff(fact_4212_left__diff__distrib__NO__MATCH,axiom,
    ! [X: int,Y: int,Ca: code_integer,A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_int5993532643906289898eger_o(nO_MAT9066612773553876470nteger,aa_int_int(aa_int_fun_int_int(divide_divide_int,X),Y)),Ca)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(A),B)),Ca) = aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),Ca)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),Ca)) ) ) ).

% left_diff_distrib_NO_MATCH
tff(fact_4213_left__diff__distrib__NO__MATCH,axiom,
    ! [X: int,Y: int,Ca: 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)),Ca)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B)),Ca) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),Ca)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),Ca)) ) ) ).

% left_diff_distrib_NO_MATCH
tff(fact_4214_left__diff__distrib__NO__MATCH,axiom,
    ! [X: int,Y: int,Ca: 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)),Ca)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),Ca) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),Ca)),aa_int_int(aa_int_fun_int_int(times_times_int,B),Ca)) ) ) ).

% left_diff_distrib_NO_MATCH
tff(fact_4215_left__diff__distrib__NO__MATCH,axiom,
    ! [X: code_integer,Y: code_integer,Ca: code_integer,A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(nO_MAT8252062027627875367nteger,aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(X),Y)),Ca)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(A),B)),Ca) = aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),Ca)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),Ca)) ) ) ).

% left_diff_distrib_NO_MATCH
tff(fact_4216_left__diff__distrib__NO__MATCH,axiom,
    ! [X: code_integer,Y: code_integer,Ca: rat,A: rat,B: rat] :
      ( aa_rat_o(nO_MAT7795273704451493282er_rat(aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(X),Y)),Ca)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B)),Ca) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),Ca)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),Ca)) ) ) ).

% left_diff_distrib_NO_MATCH
tff(fact_4217_left__diff__distrib__NO__MATCH,axiom,
    ! [X: code_integer,Y: code_integer,Ca: int,A: int,B: int] :
      ( aa_int_o(aa_Cod8477617969808203498_int_o(nO_MAT8427913294028938742er_int,aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(X),Y)),Ca)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),Ca) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),Ca)),aa_int_int(aa_int_fun_int_int(times_times_int,B),Ca)) ) ) ).

% left_diff_distrib_NO_MATCH
tff(fact_4218_left__diff__distrib__NO__MATCH,axiom,
    ! [X: rat,Y: rat,Ca: 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)),Ca)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(A),B)),Ca) = aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),Ca)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),Ca)) ) ) ).

% left_diff_distrib_NO_MATCH
tff(fact_4219_right__diff__distrib__NO__MATCH,axiom,
    ! [X: nat,Y: nat,A: code_integer,B: code_integer,Ca: code_integer] :
      ( aa_Code_integer_o(aa_nat7420681472272698950eger_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),Ca)) = aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),Ca)) ) ) ).

% right_diff_distrib_NO_MATCH
tff(fact_4220_right__diff__distrib__NO__MATCH,axiom,
    ! [X: nat,Y: nat,A: rat,B: rat,Ca: 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),Ca)) = 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),Ca)) ) ) ).

% right_diff_distrib_NO_MATCH
tff(fact_4221_right__diff__distrib__NO__MATCH,axiom,
    ! [X: nat,Y: nat,A: int,B: int,Ca: 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),Ca)) = 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),Ca)) ) ) ).

% right_diff_distrib_NO_MATCH
tff(fact_4222_right__diff__distrib__NO__MATCH,axiom,
    ! [X: int,Y: int,A: code_integer,B: code_integer,Ca: code_integer] :
      ( aa_Code_integer_o(aa_int5993532643906289898eger_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),Ca)) = aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),Ca)) ) ) ).

% right_diff_distrib_NO_MATCH
tff(fact_4223_right__diff__distrib__NO__MATCH,axiom,
    ! [X: int,Y: int,A: rat,B: rat,Ca: 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),Ca)) = 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),Ca)) ) ) ).

% right_diff_distrib_NO_MATCH
tff(fact_4224_right__diff__distrib__NO__MATCH,axiom,
    ! [X: int,Y: int,A: int,B: int,Ca: 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),Ca)) = 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),Ca)) ) ) ).

% right_diff_distrib_NO_MATCH
tff(fact_4225_right__diff__distrib__NO__MATCH,axiom,
    ! [X: code_integer,Y: code_integer,A: code_integer,B: code_integer,Ca: 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),Ca)) = aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),Ca)) ) ) ).

% right_diff_distrib_NO_MATCH
tff(fact_4226_right__diff__distrib__NO__MATCH,axiom,
    ! [X: code_integer,Y: code_integer,A: rat,B: rat,Ca: 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),Ca)) = 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),Ca)) ) ) ).

% right_diff_distrib_NO_MATCH
tff(fact_4227_right__diff__distrib__NO__MATCH,axiom,
    ! [X: code_integer,Y: code_integer,A: int,B: int,Ca: int] :
      ( aa_int_o(aa_Cod8477617969808203498_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),Ca)) = 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),Ca)) ) ) ).

% right_diff_distrib_NO_MATCH
tff(fact_4228_right__diff__distrib__NO__MATCH,axiom,
    ! [X: rat,Y: rat,A: code_integer,B: code_integer,Ca: 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),Ca)) = aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),Ca)) ) ) ).

% right_diff_distrib_NO_MATCH
tff(fact_4229_mod__mult2__eq_H,axiom,
    ! [A: code_natural,M: nat,N2: nat] : modulo8411746178871703098atural(A,times_2397367101498566445atural(aa_nat_Code_natural(semiri3763490453095760265atural,M),aa_nat_Code_natural(semiri3763490453095760265atural,N2))) = plus_p4538020629002901425atural(times_2397367101498566445atural(aa_nat_Code_natural(semiri3763490453095760265atural,M),modulo8411746178871703098atural(divide5121882707175180666atural(A,aa_nat_Code_natural(semiri3763490453095760265atural,M)),aa_nat_Code_natural(semiri3763490453095760265atural,N2))),modulo8411746178871703098atural(A,aa_nat_Code_natural(semiri3763490453095760265atural,M))) ).

% mod_mult2_eq'
tff(fact_4230_mod__mult2__eq_H,axiom,
    ! [A: int,M: nat,N2: 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,N2))) = 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,N2)))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_nat_int(semiri1314217659103216013at_int,M))) ).

% mod_mult2_eq'
tff(fact_4231_mod__mult2__eq_H,axiom,
    ! [A: nat,M: nat,N2: 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,N2))) = 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,N2)))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(semiri1316708129612266289at_nat,M))) ).

% mod_mult2_eq'
tff(fact_4232_mod__mult2__eq_H,axiom,
    ! [A: code_integer,M: nat,N2: nat] : aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,M)),aa_nat_Code_integer(semiri4939895301339042750nteger,N2))) = aa_Cod1783974652533396728nteger(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,N2)))),aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A),aa_nat_Code_integer(semiri4939895301339042750nteger,M))) ).

% mod_mult2_eq'
tff(fact_4233_nz__le__conv__less,axiom,
    ! [K: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),M)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),aa_nat_nat(suc,zero_zero_nat))),M) ) ) ).

% nz_le_conv_less
tff(fact_4234_Suc__pred_H,axiom,
    ! [N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)
     => ( N2 = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),one_one_nat)) ) ) ).

% Suc_pred'
tff(fact_4235_Suc__diff__eq__diff__pred,axiom,
    ! [N2: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,M)),N2) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),one_one_nat)) ) ) ).

% Suc_diff_eq_diff_pred
tff(fact_4236_add__eq__if,axiom,
    ! [M: nat,N2: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N2) = $ite(M = zero_zero_nat,N2,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)),N2))) ).

% add_eq_if
tff(fact_4237_Suc__n__minus__m__eq,axiom,
    ! [M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),M)
       => ( aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),M)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat)) ) ) ) ).

% Suc_n_minus_m_eq
tff(fact_4238_div__geq,axiom,
    ! [N2: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)
     => ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N2)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N2) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N2)),N2)) ) ) ) ).

% div_geq
tff(fact_4239_div__if,axiom,
    ! [M: nat,N2: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N2) = $ite(
        ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N2)
        | ( N2 = zero_zero_nat ) ),
        zero_zero_nat,
        aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N2)),N2)) ) ).

% div_if
tff(fact_4240_nat__less__as__int,axiom,
    ! [X5: nat,Xa3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X5),Xa3)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,X5)),aa_nat_int(semiri1314217659103216013at_int,Xa3)) ) ).

% nat_less_as_int
tff(fact_4241_nat__leq__as__int,axiom,
    ! [X5: nat,Xa3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X5),Xa3)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,X5)),aa_nat_int(semiri1314217659103216013at_int,Xa3)) ) ).

% nat_leq_as_int
tff(fact_4242_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A),B)),A) ) ).

% unique_euclidean_semiring_numeral_class.mod_less_eq_dividend
tff(fact_4243_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)),A) ) ).

% unique_euclidean_semiring_numeral_class.mod_less_eq_dividend
tff(fact_4244_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)),A) ) ).

% unique_euclidean_semiring_numeral_class.mod_less_eq_dividend
tff(fact_4245_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)),B) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_bound
tff(fact_4246_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)),B) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_bound
tff(fact_4247_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
    ! [B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A),B)),B) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_bound
tff(fact_4248_Suc__times__mod__eq,axiom,
    ! [M: nat,N2: 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),N2))),M) = one_one_nat ) ) ).

% Suc_times_mod_eq
tff(fact_4249_of__nat__0__le__iff,axiom,
    ! [N2: nat] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(semiri4939895301339042750nteger,N2)) ).

% of_nat_0_le_iff
tff(fact_4250_of__nat__0__le__iff,axiom,
    ! [N2: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(semiri681578069525770553at_rat,N2)) ).

% of_nat_0_le_iff
tff(fact_4251_of__nat__0__le__iff,axiom,
    ! [N2: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(semiri1316708129612266289at_nat,N2)) ).

% of_nat_0_le_iff
tff(fact_4252_of__nat__0__le__iff,axiom,
    ! [N2: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(semiri1314217659103216013at_int,N2)) ).

% of_nat_0_le_iff
tff(fact_4253_of__nat__less__0__iff,axiom,
    ! [M: nat] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,M)),zero_zero_rat) ).

% of_nat_less_0_iff
tff(fact_4254_of__nat__less__0__iff,axiom,
    ! [M: nat] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,M)),zero_zero_int) ).

% of_nat_less_0_iff
tff(fact_4255_of__nat__less__0__iff,axiom,
    ! [M: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),zero_zero_nat) ).

% of_nat_less_0_iff
tff(fact_4256_of__nat__less__0__iff,axiom,
    ! [M: nat] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,M)),zero_z3403309356797280102nteger) ).

% of_nat_less_0_iff
tff(fact_4257_mod__eqE,axiom,
    ! [A: int,Ca: int,B: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),Ca) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B),Ca) )
     => ~ ! [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,Ca),D)) ) ).

% mod_eqE
tff(fact_4258_mod__eqE,axiom,
    ! [A: code_integer,Ca: code_integer,B: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A),Ca) = aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(B),Ca) )
     => ~ ! [D: code_integer] : B != aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Ca),D)) ) ).

% mod_eqE
tff(fact_4259_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_4260_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_4261_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_4262_le__div__geq,axiom,
    ! [N2: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),M)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N2) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N2)),N2)) ) ) ) ).

% le_div_geq
tff(fact_4263_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_4264_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_4265_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_4266_add__le__imp__le__diff,axiom,
    ! [I2: rat,K: rat,N2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,I2),K)),N2)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,I2),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,N2),K)) ) ).

% add_le_imp_le_diff
tff(fact_4267_add__le__imp__le__diff,axiom,
    ! [I2: nat,K: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),K)),N2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),K)) ) ).

% add_le_imp_le_diff
tff(fact_4268_add__le__imp__le__diff,axiom,
    ! [I2: int,K: int,N2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I2),K)),N2)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I2),aa_int_int(aa_int_fun_int_int(minus_minus_int,N2),K)) ) ).

% add_le_imp_le_diff
tff(fact_4269_add__le__add__imp__diff__le,axiom,
    ! [I2: rat,K: rat,N2: rat,J: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,I2),K)),N2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,N2),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,J),K))
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,I2),K)),N2)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,N2),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,J),K))
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,N2),K)),J) ) ) ) ) ).

% add_le_add_imp_diff_le
tff(fact_4270_add__le__add__imp__diff__le,axiom,
    ! [I2: nat,K: nat,N2: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),K)),N2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K))
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),K)),N2)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K))
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),K)),J) ) ) ) ) ).

% add_le_add_imp_diff_le
tff(fact_4271_add__le__add__imp__diff__le,axiom,
    ! [I2: int,K: int,N2: int,J: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I2),K)),N2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,N2),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),K))
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I2),K)),N2)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,N2),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),K))
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,N2),K)),J) ) ) ) ) ).

% add_le_add_imp_diff_le
tff(fact_4272_diff__le__eq,axiom,
    ! [A: rat,B: rat,Ca: 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)),Ca)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Ca),B)) ) ).

% diff_le_eq
tff(fact_4273_diff__le__eq,axiom,
    ! [A: int,B: int,Ca: 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)),Ca)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,Ca),B)) ) ).

% diff_le_eq
tff(fact_4274_le__diff__eq,axiom,
    ! [A: rat,Ca: 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,Ca),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)),Ca) ) ).

% le_diff_eq
tff(fact_4275_le__diff__eq,axiom,
    ! [A: int,Ca: 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,Ca),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)),Ca) ) ).

% le_diff_eq
tff(fact_4276_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_4277_le__add__diff,axiom,
    ! [A: nat,B: nat,Ca: 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,Ca),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),Ca)),A)) ) ).

% le_add_diff
tff(fact_4278_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
    ! [A: nat,B: nat,Ca: 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,Ca),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,Ca),A)),B) ) ) ).

% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
tff(fact_4279_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
    ! [A: nat,B: nat,Ca: 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,Ca),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,Ca),B)),A) ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
tff(fact_4280_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
    ! [A: nat,B: nat,Ca: 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,Ca),B)),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ca),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A)) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
tff(fact_4281_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
    ! [A: nat,B: nat,Ca: 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)),Ca) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),Ca)),A) ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
tff(fact_4282_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
    ! [A: nat,B: nat,Ca: 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),Ca)),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)),Ca) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
tff(fact_4283_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
    ! [A: nat,B: nat,Ca: 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,Ca),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,Ca),A)),B) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_diff_right
tff(fact_4284_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_4285_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
    ! [A: nat,B: nat,Ca: 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) = Ca )
        <=> ( B = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ca),A) ) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
tff(fact_4286_less__imp__of__nat__less,axiom,
    ! [M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N2)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,M)),aa_nat_rat(semiri681578069525770553at_rat,N2)) ) ).

% less_imp_of_nat_less
tff(fact_4287_less__imp__of__nat__less,axiom,
    ! [M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N2)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,N2)) ) ).

% less_imp_of_nat_less
tff(fact_4288_less__imp__of__nat__less,axiom,
    ! [M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,N2)) ) ).

% less_imp_of_nat_less
tff(fact_4289_less__imp__of__nat__less,axiom,
    ! [M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N2)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,M)),aa_nat_Code_integer(semiri4939895301339042750nteger,N2)) ) ).

% less_imp_of_nat_less
tff(fact_4290_of__nat__less__imp__less,axiom,
    ! [M: nat,N2: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,M)),aa_nat_rat(semiri681578069525770553at_rat,N2))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N2) ) ).

% of_nat_less_imp_less
tff(fact_4291_of__nat__less__imp__less,axiom,
    ! [M: nat,N2: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,N2))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N2) ) ).

% of_nat_less_imp_less
tff(fact_4292_of__nat__less__imp__less,axiom,
    ! [M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,N2))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N2) ) ).

% of_nat_less_imp_less
tff(fact_4293_of__nat__less__imp__less,axiom,
    ! [M: nat,N2: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,M)),aa_nat_Code_integer(semiri4939895301339042750nteger,N2))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N2) ) ).

% of_nat_less_imp_less
tff(fact_4294_of__nat__mono,axiom,
    ! [I2: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),J)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,I2)),aa_nat_Code_integer(semiri4939895301339042750nteger,J)) ) ).

% of_nat_mono
tff(fact_4295_of__nat__mono,axiom,
    ! [I2: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),J)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(semiri681578069525770553at_rat,I2)),aa_nat_rat(semiri681578069525770553at_rat,J)) ) ).

% of_nat_mono
tff(fact_4296_of__nat__mono,axiom,
    ! [I2: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),J)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(semiri1316708129612266289at_nat,I2)),aa_nat_nat(semiri1316708129612266289at_nat,J)) ) ).

% of_nat_mono
tff(fact_4297_of__nat__mono,axiom,
    ! [I2: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),J)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,I2)),aa_nat_int(semiri1314217659103216013at_int,J)) ) ).

% of_nat_mono
tff(fact_4298_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_4299_diff__shunt__var,axiom,
    ! [X: set_o,Y: set_o] :
      ( ( minus_minus_set_o(X,Y) = bot_bot_set_o )
    <=> ord_less_eq_set_o(X,Y) ) ).

% diff_shunt_var
tff(fact_4300_diff__shunt__var,axiom,
    ! [X: assn,Y: assn] :
      ( ( aa_assn_assn(aa_ass954072293579886259n_assn(minus_minus_assn,X),Y) = bot_bot_assn )
    <=> aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X),Y) ) ).

% diff_shunt_var
tff(fact_4301_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_4302_diff__shunt__var,axiom,
    ! [X: set_int,Y: set_int] :
      ( ( aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,X),Y) = bot_bot_set_int )
    <=> aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,X),Y) ) ).

% diff_shunt_var
tff(fact_4303_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [A: rat,B: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B)) = A ) ) ).

% linordered_semidom_class.add_diff_inverse
tff(fact_4304_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [A: nat,B: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B)) = A ) ) ).

% linordered_semidom_class.add_diff_inverse
tff(fact_4305_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [A: int,B: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,B),aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)) = A ) ) ).

% linordered_semidom_class.add_diff_inverse
tff(fact_4306_diff__less__eq,axiom,
    ! [A: rat,B: rat,Ca: 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)),Ca)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Ca),B)) ) ).

% diff_less_eq
tff(fact_4307_diff__less__eq,axiom,
    ! [A: int,B: int,Ca: 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)),Ca)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,Ca),B)) ) ).

% diff_less_eq
tff(fact_4308_less__diff__eq,axiom,
    ! [A: rat,Ca: 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,Ca),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)),Ca) ) ).

% less_diff_eq
tff(fact_4309_less__diff__eq,axiom,
    ! [A: int,Ca: 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,Ca),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)),Ca) ) ).

% less_diff_eq
tff(fact_4310_div__mult2__eq_H,axiom,
    ! [A: code_natural,M: nat,N2: nat] : divide5121882707175180666atural(A,times_2397367101498566445atural(aa_nat_Code_natural(semiri3763490453095760265atural,M),aa_nat_Code_natural(semiri3763490453095760265atural,N2))) = divide5121882707175180666atural(divide5121882707175180666atural(A,aa_nat_Code_natural(semiri3763490453095760265atural,M)),aa_nat_Code_natural(semiri3763490453095760265atural,N2)) ).

% div_mult2_eq'
tff(fact_4311_div__mult2__eq_H,axiom,
    ! [A: int,M: nat,N2: 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,N2))) = 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,N2)) ).

% div_mult2_eq'
tff(fact_4312_div__mult2__eq_H,axiom,
    ! [A: nat,M: nat,N2: 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,N2))) = 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,N2)) ).

% div_mult2_eq'
tff(fact_4313_div__mult2__eq_H,axiom,
    ! [A: code_integer,M: nat,N2: nat] : aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,M)),aa_nat_Code_integer(semiri4939895301339042750nteger,N2))) = aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),aa_nat_Code_integer(semiri4939895301339042750nteger,M))),aa_nat_Code_integer(semiri4939895301339042750nteger,N2)) ).

% div_mult2_eq'
tff(fact_4314_eq__add__iff1,axiom,
    ! [A: code_integer,E: code_integer,Ca: code_integer,B: code_integer,D2: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),E)),Ca) = aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),E)),D2) )
    <=> ( aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(A),B)),E)),Ca) = D2 ) ) ).

% eq_add_iff1
tff(fact_4315_eq__add__iff1,axiom,
    ! [A: rat,E: rat,Ca: 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),E)),Ca) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),E)),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)),E)),Ca) = D2 ) ) ).

% eq_add_iff1
tff(fact_4316_eq__add__iff1,axiom,
    ! [A: int,E: int,Ca: 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),E)),Ca) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),E)),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)),E)),Ca) = D2 ) ) ).

% eq_add_iff1
tff(fact_4317_eq__add__iff2,axiom,
    ! [A: code_integer,E: code_integer,Ca: code_integer,B: code_integer,D2: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),E)),Ca) = aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),E)),D2) )
    <=> ( Ca = aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(B),A)),E)),D2) ) ) ).

% eq_add_iff2
tff(fact_4318_eq__add__iff2,axiom,
    ! [A: rat,E: rat,Ca: 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),E)),Ca) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),E)),D2) )
    <=> ( Ca = 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)),E)),D2) ) ) ).

% eq_add_iff2
tff(fact_4319_eq__add__iff2,axiom,
    ! [A: int,E: int,Ca: 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),E)),Ca) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),E)),D2) )
    <=> ( Ca = 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)),E)),D2) ) ) ).

% eq_add_iff2
tff(fact_4320_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(plus_p5714425477246183910nteger(X),Y)),aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(X),Y)) ).

% square_diff_square_factored
tff(fact_4321_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_4322_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_4323_mod__less__divisor,axiom,
    ! [N2: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N2)),N2) ) ).

% mod_less_divisor
tff(fact_4324_zero__le__numeral,axiom,
    ! [N2: num] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_num_Code_integer(numera6620942414471956472nteger,N2)) ).

% zero_le_numeral
tff(fact_4325_zero__le__numeral,axiom,
    ! [N2: num] : aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,zero_z2226904508553997617atural),aa_num_Code_natural(numera5444537566228673987atural,N2)) ).

% zero_le_numeral
tff(fact_4326_zero__le__numeral,axiom,
    ! [N2: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_num_rat(numeral_numeral_rat,N2)) ).

% zero_le_numeral
tff(fact_4327_zero__le__numeral,axiom,
    ! [N2: num] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,N2)) ).

% zero_le_numeral
tff(fact_4328_zero__le__numeral,axiom,
    ! [N2: num] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_num_int(numeral_numeral_int,N2)) ).

% zero_le_numeral
tff(fact_4329_not__numeral__le__zero,axiom,
    ! [N2: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_num_Code_integer(numera6620942414471956472nteger,N2)),zero_z3403309356797280102nteger) ).

% not_numeral_le_zero
tff(fact_4330_not__numeral__le__zero,axiom,
    ! [N2: num] : ~ aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,aa_num_Code_natural(numera5444537566228673987atural,N2)),zero_z2226904508553997617atural) ).

% not_numeral_le_zero
tff(fact_4331_not__numeral__le__zero,axiom,
    ! [N2: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,N2)),zero_zero_rat) ).

% not_numeral_le_zero
tff(fact_4332_not__numeral__le__zero,axiom,
    ! [N2: num] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,N2)),zero_zero_nat) ).

% not_numeral_le_zero
tff(fact_4333_not__numeral__le__zero,axiom,
    ! [N2: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,N2)),zero_zero_int) ).

% not_numeral_le_zero
tff(fact_4334_zero__less__numeral,axiom,
    ! [N2: num] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,N2)) ).

% zero_less_numeral
tff(fact_4335_zero__less__numeral,axiom,
    ! [N2: num] : aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_num_int(numeral_numeral_int,N2)) ).

% zero_less_numeral
tff(fact_4336_zero__less__numeral,axiom,
    ! [N2: num] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_num_Code_integer(numera6620942414471956472nteger,N2)) ).

% zero_less_numeral
tff(fact_4337_zero__less__numeral,axiom,
    ! [N2: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_num_rat(numeral_numeral_rat,N2)) ).

% zero_less_numeral
tff(fact_4338_zero__less__numeral,axiom,
    ! [N2: num] : aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,zero_z2226904508553997617atural),aa_num_Code_natural(numera5444537566228673987atural,N2)) ).

% zero_less_numeral
tff(fact_4339_not__numeral__less__zero,axiom,
    ! [N2: num] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_num_nat(numeral_numeral_nat,N2)),zero_zero_nat) ).

% not_numeral_less_zero
tff(fact_4340_not__numeral__less__zero,axiom,
    ! [N2: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,N2)),zero_zero_int) ).

% not_numeral_less_zero
tff(fact_4341_not__numeral__less__zero,axiom,
    ! [N2: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_num_Code_integer(numera6620942414471956472nteger,N2)),zero_z3403309356797280102nteger) ).

% not_numeral_less_zero
tff(fact_4342_not__numeral__less__zero,axiom,
    ! [N2: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(numeral_numeral_rat,N2)),zero_zero_rat) ).

% not_numeral_less_zero
tff(fact_4343_not__numeral__less__zero,axiom,
    ! [N2: num] : ~ aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,aa_num_Code_natural(numera5444537566228673987atural,N2)),zero_z2226904508553997617atural) ).

% not_numeral_less_zero
tff(fact_4344_one__le__numeral,axiom,
    ! [N2: num] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),aa_num_Code_integer(numera6620942414471956472nteger,N2)) ).

% one_le_numeral
tff(fact_4345_one__le__numeral,axiom,
    ! [N2: num] : aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,one_one_Code_natural),aa_num_Code_natural(numera5444537566228673987atural,N2)) ).

% one_le_numeral
tff(fact_4346_one__le__numeral,axiom,
    ! [N2: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_num_rat(numeral_numeral_rat,N2)) ).

% one_le_numeral
tff(fact_4347_one__le__numeral,axiom,
    ! [N2: num] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,N2)) ).

% one_le_numeral
tff(fact_4348_one__le__numeral,axiom,
    ! [N2: num] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_num_int(numeral_numeral_int,N2)) ).

% one_le_numeral
tff(fact_4349_not__numeral__less__one,axiom,
    ! [N2: num] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_num_nat(numeral_numeral_nat,N2)),one_one_nat) ).

% not_numeral_less_one
tff(fact_4350_not__numeral__less__one,axiom,
    ! [N2: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,N2)),one_one_int) ).

% not_numeral_less_one
tff(fact_4351_not__numeral__less__one,axiom,
    ! [N2: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_num_Code_integer(numera6620942414471956472nteger,N2)),one_one_Code_integer) ).

% not_numeral_less_one
tff(fact_4352_not__numeral__less__one,axiom,
    ! [N2: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(numeral_numeral_rat,N2)),one_one_rat) ).

% not_numeral_less_one
tff(fact_4353_not__numeral__less__one,axiom,
    ! [N2: num] : ~ aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,aa_num_Code_natural(numera5444537566228673987atural,N2)),one_one_Code_natural) ).

% not_numeral_less_one
tff(fact_4354_lift__Suc__mono__less,axiom,
    ! [F: fun_nat_assn,N2: nat,N6: nat] :
      ( ! [N5: nat] : aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_nat_assn(F,N5)),aa_nat_assn(F,aa_nat_nat(suc,N5)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),N6)
       => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_nat_assn(F,N2)),aa_nat_assn(F,N6)) ) ) ).

% lift_Suc_mono_less
tff(fact_4355_lift__Suc__mono__less,axiom,
    ! [F: fun_nat_rat,N2: nat,N6: nat] :
      ( ! [N5: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(F,N5)),aa_nat_rat(F,aa_nat_nat(suc,N5)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),N6)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(F,N2)),aa_nat_rat(F,N6)) ) ) ).

% lift_Suc_mono_less
tff(fact_4356_lift__Suc__mono__less,axiom,
    ! [F: fun_nat_num,N2: nat,N6: nat] :
      ( ! [N5: nat] : aa_num_o(aa_num_fun_num_o(ord_less_num,aa_nat_num(F,N5)),aa_nat_num(F,aa_nat_nat(suc,N5)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),N6)
       => aa_num_o(aa_num_fun_num_o(ord_less_num,aa_nat_num(F,N2)),aa_nat_num(F,N6)) ) ) ).

% lift_Suc_mono_less
tff(fact_4357_lift__Suc__mono__less,axiom,
    ! [F: fun_nat_nat,N2: nat,N6: nat] :
      ( ! [N5: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(F,N5)),aa_nat_nat(F,aa_nat_nat(suc,N5)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),N6)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(F,N2)),aa_nat_nat(F,N6)) ) ) ).

% lift_Suc_mono_less
tff(fact_4358_lift__Suc__mono__less,axiom,
    ! [F: fun_nat_int,N2: nat,N6: nat] :
      ( ! [N5: nat] : aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(F,N5)),aa_nat_int(F,aa_nat_nat(suc,N5)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),N6)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(F,N2)),aa_nat_int(F,N6)) ) ) ).

% lift_Suc_mono_less
tff(fact_4359_lift__Suc__mono__less__iff,axiom,
    ! [F: fun_nat_assn,N2: nat,M: nat] :
      ( ! [N5: nat] : aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_nat_assn(F,N5)),aa_nat_assn(F,aa_nat_nat(suc,N5)))
     => ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_nat_assn(F,N2)),aa_nat_assn(F,M))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),M) ) ) ).

% lift_Suc_mono_less_iff
tff(fact_4360_lift__Suc__mono__less__iff,axiom,
    ! [F: fun_nat_rat,N2: nat,M: nat] :
      ( ! [N5: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(F,N5)),aa_nat_rat(F,aa_nat_nat(suc,N5)))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(F,N2)),aa_nat_rat(F,M))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),M) ) ) ).

% lift_Suc_mono_less_iff
tff(fact_4361_lift__Suc__mono__less__iff,axiom,
    ! [F: fun_nat_num,N2: nat,M: nat] :
      ( ! [N5: nat] : aa_num_o(aa_num_fun_num_o(ord_less_num,aa_nat_num(F,N5)),aa_nat_num(F,aa_nat_nat(suc,N5)))
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,aa_nat_num(F,N2)),aa_nat_num(F,M))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),M) ) ) ).

% lift_Suc_mono_less_iff
tff(fact_4362_lift__Suc__mono__less__iff,axiom,
    ! [F: fun_nat_nat,N2: nat,M: nat] :
      ( ! [N5: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(F,N5)),aa_nat_nat(F,aa_nat_nat(suc,N5)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(F,N2)),aa_nat_nat(F,M))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),M) ) ) ).

% lift_Suc_mono_less_iff
tff(fact_4363_lift__Suc__mono__less__iff,axiom,
    ! [F: fun_nat_int,N2: nat,M: nat] :
      ( ! [N5: nat] : aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(F,N5)),aa_nat_int(F,aa_nat_nat(suc,N5)))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(F,N2)),aa_nat_int(F,M))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),M) ) ) ).

% lift_Suc_mono_less_iff
tff(fact_4364_lift__Suc__mono__le,axiom,
    ! [F: fun_nat_set_int,N2: nat,N6: nat] :
      ( ! [N5: nat] : aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,aa_nat_set_int(F,N5)),aa_nat_set_int(F,aa_nat_nat(suc,N5)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),N6)
       => aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,aa_nat_set_int(F,N2)),aa_nat_set_int(F,N6)) ) ) ).

% lift_Suc_mono_le
tff(fact_4365_lift__Suc__mono__le,axiom,
    ! [F: fun_nat_rat,N2: nat,N6: nat] :
      ( ! [N5: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,N5)),aa_nat_rat(F,aa_nat_nat(suc,N5)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),N6)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,N2)),aa_nat_rat(F,N6)) ) ) ).

% lift_Suc_mono_le
tff(fact_4366_lift__Suc__mono__le,axiom,
    ! [F: fun_nat_num,N2: nat,N6: nat] :
      ( ! [N5: nat] : aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(F,N5)),aa_nat_num(F,aa_nat_nat(suc,N5)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),N6)
       => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(F,N2)),aa_nat_num(F,N6)) ) ) ).

% lift_Suc_mono_le
tff(fact_4367_lift__Suc__mono__le,axiom,
    ! [F: fun_nat_nat,N2: nat,N6: nat] :
      ( ! [N5: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F,N5)),aa_nat_nat(F,aa_nat_nat(suc,N5)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),N6)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F,N2)),aa_nat_nat(F,N6)) ) ) ).

% lift_Suc_mono_le
tff(fact_4368_lift__Suc__mono__le,axiom,
    ! [F: fun_nat_int,N2: nat,N6: nat] :
      ( ! [N5: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,N5)),aa_nat_int(F,aa_nat_nat(suc,N5)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),N6)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,N2)),aa_nat_int(F,N6)) ) ) ).

% lift_Suc_mono_le
tff(fact_4369_lift__Suc__antimono__le,axiom,
    ! [F: fun_nat_set_int,N2: nat,N6: nat] :
      ( ! [N5: nat] : aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,aa_nat_set_int(F,aa_nat_nat(suc,N5))),aa_nat_set_int(F,N5))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),N6)
       => aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,aa_nat_set_int(F,N6)),aa_nat_set_int(F,N2)) ) ) ).

% lift_Suc_antimono_le
tff(fact_4370_lift__Suc__antimono__le,axiom,
    ! [F: fun_nat_rat,N2: nat,N6: nat] :
      ( ! [N5: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,aa_nat_nat(suc,N5))),aa_nat_rat(F,N5))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),N6)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,N6)),aa_nat_rat(F,N2)) ) ) ).

% lift_Suc_antimono_le
tff(fact_4371_lift__Suc__antimono__le,axiom,
    ! [F: fun_nat_num,N2: nat,N6: nat] :
      ( ! [N5: nat] : aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(F,aa_nat_nat(suc,N5))),aa_nat_num(F,N5))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),N6)
       => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(F,N6)),aa_nat_num(F,N2)) ) ) ).

% lift_Suc_antimono_le
tff(fact_4372_lift__Suc__antimono__le,axiom,
    ! [F: fun_nat_nat,N2: nat,N6: nat] :
      ( ! [N5: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F,aa_nat_nat(suc,N5))),aa_nat_nat(F,N5))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),N6)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F,N6)),aa_nat_nat(F,N2)) ) ) ).

% lift_Suc_antimono_le
tff(fact_4373_lift__Suc__antimono__le,axiom,
    ! [F: fun_nat_int,N2: nat,N6: nat] :
      ( ! [N5: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,aa_nat_nat(suc,N5))),aa_nat_int(F,N5))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),N6)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,N6)),aa_nat_int(F,N2)) ) ) ).

% lift_Suc_antimono_le
tff(fact_4374_diff__less,axiom,
    ! [N2: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N2)),M) ) ) ).

% diff_less
tff(fact_4375_power__Suc2,axiom,
    ! [A: code_integer,N2: nat] : aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_nat_nat(suc,N2)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),N2)),A) ).

% power_Suc2
tff(fact_4376_power__Suc2,axiom,
    ! [A: assn,N2: nat] : aa_nat_assn(power_power_assn(A),aa_nat_nat(suc,N2)) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_nat_assn(power_power_assn(A),N2)),A) ).

% power_Suc2
tff(fact_4377_power__Suc2,axiom,
    ! [A: rat,N2: nat] : aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,N2)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(A),N2)),A) ).

% power_Suc2
tff(fact_4378_power__Suc2,axiom,
    ! [A: nat,N2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(suc,N2)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N2)),A) ).

% power_Suc2
tff(fact_4379_power__Suc2,axiom,
    ! [A: int,N2: nat] : aa_nat_int(aa_int_fun_nat_int(power_power_int,A),aa_nat_nat(suc,N2)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A),N2)),A) ).

% power_Suc2
tff(fact_4380_power__Suc,axiom,
    ! [A: code_integer,N2: nat] : aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_nat_nat(suc,N2)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),aa_nat_Code_integer(power_8256067586552552935nteger(A),N2)) ).

% power_Suc
tff(fact_4381_power__Suc,axiom,
    ! [A: assn,N2: nat] : aa_nat_assn(power_power_assn(A),aa_nat_nat(suc,N2)) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A),aa_nat_assn(power_power_assn(A),N2)) ).

% power_Suc
tff(fact_4382_power__Suc,axiom,
    ! [A: rat,N2: nat] : aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,N2)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_nat_rat(power_power_rat(A),N2)) ).

% power_Suc
tff(fact_4383_power__Suc,axiom,
    ! [A: nat,N2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(suc,N2)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N2)) ).

% power_Suc
tff(fact_4384_power__Suc,axiom,
    ! [A: int,N2: nat] : aa_nat_int(aa_int_fun_nat_int(power_power_int,A),aa_nat_nat(suc,N2)) = aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_nat_int(aa_int_fun_nat_int(power_power_int,A),N2)) ).

% power_Suc
tff(fact_4385_Ex__less__Suc2,axiom,
    ! [N2: nat,Pa: fun_nat_o] :
      ( ? [I4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),aa_nat_nat(suc,N2))
          & aa_nat_o(Pa,I4) )
    <=> ( aa_nat_o(Pa,zero_zero_nat)
        | ? [I4: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),N2)
            & aa_nat_o(Pa,aa_nat_nat(suc,I4)) ) ) ) ).

% Ex_less_Suc2
tff(fact_4386_gr0__conv__Suc,axiom,
    ! [N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)
    <=> ? [M3: nat] : N2 = aa_nat_nat(suc,M3) ) ).

% gr0_conv_Suc
tff(fact_4387_All__less__Suc2,axiom,
    ! [N2: nat,Pa: fun_nat_o] :
      ( ! [I4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),aa_nat_nat(suc,N2))
         => aa_nat_o(Pa,I4) )
    <=> ( aa_nat_o(Pa,zero_zero_nat)
        & ! [I4: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),N2)
           => aa_nat_o(Pa,aa_nat_nat(suc,I4)) ) ) ) ).

% All_less_Suc2
tff(fact_4388_gr0__implies__Suc,axiom,
    ! [N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)
     => ? [M5: nat] : N2 = aa_nat_nat(suc,M5) ) ).

% gr0_implies_Suc
tff(fact_4389_less__Suc__eq__0__disj,axiom,
    ! [M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),aa_nat_nat(suc,N2))
    <=> ( ( M = zero_zero_nat )
        | ? [J3: nat] :
            ( ( M = aa_nat_nat(suc,J3) )
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J3),N2) ) ) ) ).

% less_Suc_eq_0_disj
tff(fact_4390_diff__add__0,axiom,
    ! [N2: nat,M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),M)) = zero_zero_nat ).

% diff_add_0
tff(fact_4391_less__diff__iff,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),N2)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),K)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),K))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N2) ) ) ) ).

% less_diff_iff
tff(fact_4392_diff__less__mono,axiom,
    ! [A: nat,B: nat,Ca: 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,Ca),A)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),Ca)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),Ca)) ) ) ).

% diff_less_mono
tff(fact_4393_less__diff__conv,axiom,
    ! [I2: nat,J: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),K)),J) ) ).

% less_diff_conv
tff(fact_4394_add__diff__inverse__nat,axiom,
    ! [M: nat,N2: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N2)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N2)) = M ) ) ).

% add_diff_inverse_nat
tff(fact_4395_nat__compl__induct_H,axiom,
    ! [Pa: fun_nat_o,N2: nat] :
      ( aa_nat_o(Pa,zero_zero_nat)
     => ( ! [N5: nat] :
            ( ! [Nn: nat] :
                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nn),N5)
               => aa_nat_o(Pa,Nn) )
           => aa_nat_o(Pa,aa_nat_nat(suc,N5)) )
       => aa_nat_o(Pa,N2) ) ) ).

% nat_compl_induct'
tff(fact_4396_nat__compl__induct,axiom,
    ! [Pa: fun_nat_o,N2: nat] :
      ( aa_nat_o(Pa,zero_zero_nat)
     => ( ! [N5: nat] :
            ( ! [Nn: nat] :
                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nn),N5)
               => aa_nat_o(Pa,Nn) )
           => aa_nat_o(Pa,aa_nat_nat(suc,N5)) )
       => aa_nat_o(Pa,N2) ) ) ).

% nat_compl_induct
tff(fact_4397_add__is__1,axiom,
    ! [M: nat,N2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N2) = aa_nat_nat(suc,zero_zero_nat) )
    <=> ( ( ( M = aa_nat_nat(suc,zero_zero_nat) )
          & ( N2 = zero_zero_nat ) )
        | ( ( M = zero_zero_nat )
          & ( N2 = aa_nat_nat(suc,zero_zero_nat) ) ) ) ) ).

% add_is_1
tff(fact_4398_one__is__add,axiom,
    ! [M: nat,N2: nat] :
      ( ( aa_nat_nat(suc,zero_zero_nat) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N2) )
    <=> ( ( ( M = aa_nat_nat(suc,zero_zero_nat) )
          & ( N2 = zero_zero_nat ) )
        | ( ( M = zero_zero_nat )
          & ( N2 = aa_nat_nat(suc,zero_zero_nat) ) ) ) ) ).

% one_is_add
tff(fact_4399_le__diff__conv,axiom,
    ! [J: nat,K: nat,I2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K)),I2)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,J),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),K)) ) ).

% le_diff_conv
tff(fact_4400_Nat_Ole__diff__conv2,axiom,
    ! [K: nat,J: nat,I2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),J)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),K)),J) ) ) ).

% Nat.le_diff_conv2
tff(fact_4401_Nat_Odiff__add__assoc,axiom,
    ! [K: nat,J: nat,I2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),J)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),J)),K) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K)) ) ) ).

% Nat.diff_add_assoc
tff(fact_4402_Nat_Odiff__add__assoc2,axiom,
    ! [K: nat,J: nat,I2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),J)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),I2)),K) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K)),I2) ) ) ).

% Nat.diff_add_assoc2
tff(fact_4403_Nat_Ole__imp__diff__is__add,axiom,
    ! [I2: nat,J: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),J)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),I2) = K )
      <=> ( J = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),I2) ) ) ) ).

% Nat.le_imp_diff_is_add
tff(fact_4404_Suc__leI,axiom,
    ! [M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,M)),N2) ) ).

% Suc_leI
tff(fact_4405_Suc__le__eq,axiom,
    ! [M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,M)),N2)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N2) ) ).

% Suc_le_eq
tff(fact_4406_dec__induct,axiom,
    ! [I2: nat,J: nat,Pa: fun_nat_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),J)
     => ( aa_nat_o(Pa,I2)
       => ( ! [N5: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),N5)
             => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N5),J)
               => ( aa_nat_o(Pa,N5)
                 => aa_nat_o(Pa,aa_nat_nat(suc,N5)) ) ) )
         => aa_nat_o(Pa,J) ) ) ) ).

% dec_induct
tff(fact_4407_inc__induct,axiom,
    ! [I2: nat,J: nat,Pa: fun_nat_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),J)
     => ( aa_nat_o(Pa,J)
       => ( ! [N5: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),N5)
             => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N5),J)
               => ( aa_nat_o(Pa,aa_nat_nat(suc,N5))
                 => aa_nat_o(Pa,N5) ) ) )
         => aa_nat_o(Pa,I2) ) ) ) ).

% inc_induct
tff(fact_4408_Suc__le__lessD,axiom,
    ! [M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,M)),N2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N2) ) ).

% Suc_le_lessD
tff(fact_4409_le__less__Suc__eq,axiom,
    ! [M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),aa_nat_nat(suc,M))
      <=> ( N2 = M ) ) ) ).

% le_less_Suc_eq
tff(fact_4410_less__Suc__eq__le,axiom,
    ! [M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),aa_nat_nat(suc,N2))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2) ) ).

% less_Suc_eq_le
tff(fact_4411_less__eq__Suc__le,axiom,
    ! [N2: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),M)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,N2)),M) ) ).

% less_eq_Suc_le
tff(fact_4412_le__imp__less__Suc,axiom,
    ! [M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),aa_nat_nat(suc,N2)) ) ).

% le_imp_less_Suc
tff(fact_4413_nat__in__between__eq_I2_J,axiom,
    ! [A: nat,B: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),aa_nat_nat(suc,A)) )
    <=> ( B = A ) ) ).

% nat_in_between_eq(2)
tff(fact_4414_nat__in__between__eq_I1_J,axiom,
    ! [A: nat,B: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),aa_nat_nat(suc,A)) )
    <=> ( B = aa_nat_nat(suc,A) ) ) ).

% nat_in_between_eq(1)
tff(fact_4415_less__natE,axiom,
    ! [M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N2)
     => ~ ! [Q3: nat] : N2 != aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Q3)) ) ).

% less_natE
tff(fact_4416_less__add__Suc1,axiom,
    ! [I2: nat,M: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),M))) ).

% less_add_Suc1
tff(fact_4417_less__add__Suc2,axiom,
    ! [I2: nat,M: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),I2))) ).

% less_add_Suc2
tff(fact_4418_less__iff__Suc__add,axiom,
    ! [M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N2)
    <=> ? [K3: nat] : N2 = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K3)) ) ).

% less_iff_Suc_add
tff(fact_4419_less__imp__Suc__add,axiom,
    ! [M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N2)
     => ? [K2: nat] : N2 = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K2)) ) ).

% less_imp_Suc_add
tff(fact_4420_Suc__mult__less__cancel1,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K)),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K)),N2))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N2) ) ).

% Suc_mult_less_cancel1
tff(fact_4421_One__nat__def,axiom,
    one_one_nat = aa_nat_nat(suc,zero_zero_nat) ).

% One_nat_def
tff(fact_4422_Suc__mult__le__cancel1,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K)),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K)),N2))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2) ) ).

% Suc_mult_le_cancel1
tff(fact_4423_mult__Suc,axiom,
    ! [M: nat,N2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,M)),N2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N2)) ).

% mult_Suc
tff(fact_4424_Suc__eq__plus1,axiom,
    ! [N2: nat] : aa_nat_nat(suc,N2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),one_one_nat) ).

% Suc_eq_plus1
tff(fact_4425_plus__1__eq__Suc,axiom,
    aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat) = suc ).

% plus_1_eq_Suc
tff(fact_4426_Suc__eq__plus1__left,axiom,
    ! [N2: nat] : aa_nat_nat(suc,N2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),N2) ).

% Suc_eq_plus1_left
tff(fact_4427_Suc__div__le__mono,axiom,
    ! [M: nat,N2: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,M)),N2)) ).

% Suc_div_le_mono
tff(fact_4428_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B)
       => ( aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A),B) = A ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_less
tff(fact_4429_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) = A ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_less
tff(fact_4430_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
       => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) = A ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_less
tff(fact_4431_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
    ! [B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A),B)) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_sign
tff(fact_4432_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_sign
tff(fact_4433_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_sign
tff(fact_4434_cancel__div__mod__rules_I2_J,axiom,
    ! [B: nat,A: nat,Ca: 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))),Ca) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),Ca) ).

% cancel_div_mod_rules(2)
tff(fact_4435_cancel__div__mod__rules_I2_J,axiom,
    ! [B: int,A: int,Ca: 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))),Ca) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),Ca) ).

% cancel_div_mod_rules(2)
tff(fact_4436_cancel__div__mod__rules_I2_J,axiom,
    ! [B: code_integer,A: code_integer,Ca: code_integer] : aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),B))),aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A),B))),Ca) = aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(A),Ca) ).

% cancel_div_mod_rules(2)
tff(fact_4437_cancel__div__mod__rules_I2_J,axiom,
    ! [B: code_natural,A: code_natural,Ca: code_natural] : plus_p4538020629002901425atural(plus_p4538020629002901425atural(times_2397367101498566445atural(B,divide5121882707175180666atural(A,B)),modulo8411746178871703098atural(A,B)),Ca) = plus_p4538020629002901425atural(A,Ca) ).

% cancel_div_mod_rules(2)
tff(fact_4438_cancel__div__mod__rules_I1_J,axiom,
    ! [A: nat,B: nat,Ca: 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))),Ca) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),Ca) ).

% cancel_div_mod_rules(1)
tff(fact_4439_cancel__div__mod__rules_I1_J,axiom,
    ! [A: int,B: int,Ca: 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))),Ca) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),Ca) ).

% cancel_div_mod_rules(1)
tff(fact_4440_cancel__div__mod__rules_I1_J,axiom,
    ! [A: code_integer,B: code_integer,Ca: code_integer] : aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),B)),B)),aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A),B))),Ca) = aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(A),Ca) ).

% cancel_div_mod_rules(1)
tff(fact_4441_cancel__div__mod__rules_I1_J,axiom,
    ! [A: code_natural,B: code_natural,Ca: code_natural] : plus_p4538020629002901425atural(plus_p4538020629002901425atural(times_2397367101498566445atural(divide5121882707175180666atural(A,B),B),modulo8411746178871703098atural(A,B)),Ca) = plus_p4538020629002901425atural(A,Ca) ).

% cancel_div_mod_rules(1)
tff(fact_4442_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_4443_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_4444_mod__div__decomp,axiom,
    ! [A: code_integer,B: code_integer] : A = aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),B)),B)),aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A),B)) ).

% mod_div_decomp
tff(fact_4445_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_4446_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_4447_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_4448_div__mult__mod__eq,axiom,
    ! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(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_4449_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_4450_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_4451_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_4452_mod__div__mult__eq,axiom,
    ! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(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_4453_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_4454_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_4455_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_4456_mod__mult__div__eq,axiom,
    ! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(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_4457_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_4458_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_4459_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_4460_mult__div__mod__eq,axiom,
    ! [B: code_integer,A: code_integer] : aa_Cod1783974652533396728nteger(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_4461_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_4462_div__mult1__eq,axiom,
    ! [A: nat,B: nat,Ca: 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)),Ca) = 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),Ca))),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),Ca))),Ca)) ).

% div_mult1_eq
tff(fact_4463_div__mult1__eq,axiom,
    ! [A: int,B: int,Ca: 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)),Ca) = 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),Ca))),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),Ca))),Ca)) ).

% div_mult1_eq
tff(fact_4464_div__mult1__eq,axiom,
    ! [A: code_integer,B: code_integer,Ca: code_integer] : aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),Ca) = aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(B),Ca))),aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(B),Ca))),Ca)) ).

% div_mult1_eq
tff(fact_4465_div__mult1__eq,axiom,
    ! [A: code_natural,B: code_natural,Ca: code_natural] : divide5121882707175180666atural(times_2397367101498566445atural(A,B),Ca) = plus_p4538020629002901425atural(times_2397367101498566445atural(A,divide5121882707175180666atural(B,Ca)),divide5121882707175180666atural(times_2397367101498566445atural(A,modulo8411746178871703098atural(B,Ca)),Ca)) ).

% div_mult1_eq
tff(fact_4466_le__add__iff1,axiom,
    ! [A: code_integer,E: code_integer,Ca: code_integer,B: code_integer,D2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),E)),Ca)),aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),E)),D2))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(A),B)),E)),Ca)),D2) ) ).

% le_add_iff1
tff(fact_4467_le__add__iff1,axiom,
    ! [A: rat,E: rat,Ca: 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),E)),Ca)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),E)),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)),E)),Ca)),D2) ) ).

% le_add_iff1
tff(fact_4468_le__add__iff1,axiom,
    ! [A: int,E: int,Ca: 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),E)),Ca)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),E)),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)),E)),Ca)),D2) ) ).

% le_add_iff1
tff(fact_4469_le__add__iff2,axiom,
    ! [A: code_integer,E: code_integer,Ca: code_integer,B: code_integer,D2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),E)),Ca)),aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),E)),D2))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Ca),aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(B),A)),E)),D2)) ) ).

% le_add_iff2
tff(fact_4470_le__add__iff2,axiom,
    ! [A: rat,E: rat,Ca: 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),E)),Ca)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),E)),D2))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Ca),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)),E)),D2)) ) ).

% le_add_iff2
tff(fact_4471_le__add__iff2,axiom,
    ! [A: int,E: int,Ca: 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),E)),Ca)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),E)),D2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ca),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)),E)),D2)) ) ).

% le_add_iff2
tff(fact_4472_less__add__iff1,axiom,
    ! [A: code_integer,E: code_integer,Ca: code_integer,B: code_integer,D2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),E)),Ca)),aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),E)),D2))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(A),B)),E)),Ca)),D2) ) ).

% less_add_iff1
tff(fact_4473_less__add__iff1,axiom,
    ! [A: rat,E: rat,Ca: 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),E)),Ca)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),E)),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)),E)),Ca)),D2) ) ).

% less_add_iff1
tff(fact_4474_less__add__iff1,axiom,
    ! [A: int,E: int,Ca: 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),E)),Ca)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),E)),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)),E)),Ca)),D2) ) ).

% less_add_iff1
tff(fact_4475_less__add__iff2,axiom,
    ! [A: code_integer,E: code_integer,Ca: code_integer,B: code_integer,D2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),E)),Ca)),aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),E)),D2))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Ca),aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(B),A)),E)),D2)) ) ).

% less_add_iff2
tff(fact_4476_less__add__iff2,axiom,
    ! [A: rat,E: rat,Ca: 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),E)),Ca)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),E)),D2))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Ca),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)),E)),D2)) ) ).

% less_add_iff2
tff(fact_4477_less__add__iff2,axiom,
    ! [A: int,E: int,Ca: 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),E)),Ca)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),E)),D2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Ca),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)),E)),D2)) ) ).

% less_add_iff2
tff(fact_4478_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(plus_p5714425477246183910nteger(X),one_one_Code_integer)),aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(X),one_one_Code_integer)) ).

% square_diff_one_factored
tff(fact_4479_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_4480_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_4481_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_4482_diff__frac__eq,axiom,
    ! [Y: rat,Z: rat,X: rat,W2: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( Z != zero_zero_rat )
       => ( aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,X),Y)),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,W2),Z)) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X),Z)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,W2),Y))),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Z)) ) ) ) ).

% diff_frac_eq
tff(fact_4483_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_4484_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_4485_mod__le__divisor,axiom,
    ! [N2: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N2)),N2) ) ).

% mod_le_divisor
tff(fact_4486_power__inject__base,axiom,
    ! [A: code_integer,N2: nat,B: code_integer] :
      ( ( aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_nat_nat(suc,N2)) = aa_nat_Code_integer(power_8256067586552552935nteger(B),aa_nat_nat(suc,N2)) )
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),B)
         => ( A = B ) ) ) ) ).

% power_inject_base
tff(fact_4487_power__inject__base,axiom,
    ! [A: rat,N2: nat,B: rat] :
      ( ( aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,N2)) = aa_nat_rat(power_power_rat(B),aa_nat_nat(suc,N2)) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
         => ( A = B ) ) ) ) ).

% power_inject_base
tff(fact_4488_power__inject__base,axiom,
    ! [A: nat,N2: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(suc,N2)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),aa_nat_nat(suc,N2)) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B)
         => ( A = B ) ) ) ) ).

% power_inject_base
tff(fact_4489_power__inject__base,axiom,
    ! [A: int,N2: nat,B: int] :
      ( ( aa_nat_int(aa_int_fun_nat_int(power_power_int,A),aa_nat_nat(suc,N2)) = aa_nat_int(aa_int_fun_nat_int(power_power_int,B),aa_nat_nat(suc,N2)) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
         => ( A = B ) ) ) ) ).

% power_inject_base
tff(fact_4490_power__le__imp__le__base,axiom,
    ! [A: code_integer,N2: nat,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_nat_nat(suc,N2))),aa_nat_Code_integer(power_8256067586552552935nteger(B),aa_nat_nat(suc,N2)))
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),B)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B) ) ) ).

% power_le_imp_le_base
tff(fact_4491_power__le__imp__le__base,axiom,
    ! [A: rat,N2: nat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,N2))),aa_nat_rat(power_power_rat(B),aa_nat_nat(suc,N2)))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ) ).

% power_le_imp_le_base
tff(fact_4492_power__le__imp__le__base,axiom,
    ! [A: nat,N2: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(suc,N2))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),aa_nat_nat(suc,N2)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B) ) ) ).

% power_le_imp_le_base
tff(fact_4493_power__le__imp__le__base,axiom,
    ! [A: int,N2: nat,B: 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,N2))),aa_nat_int(aa_int_fun_nat_int(power_power_int,B),aa_nat_nat(suc,N2)))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ) ).

% power_le_imp_le_base
tff(fact_4494_divide__eq__eq__numeral_I1_J,axiom,
    ! [B: rat,Ca: rat,W2: num] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),Ca) = aa_num_rat(numeral_numeral_rat,W2) )
    <=> $ite(Ca != zero_zero_rat,B = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,W2)),Ca),aa_num_rat(numeral_numeral_rat,W2) = zero_zero_rat) ) ).

% divide_eq_eq_numeral(1)
tff(fact_4495_eq__divide__eq__numeral_I1_J,axiom,
    ! [W2: num,B: rat,Ca: rat] :
      ( ( aa_num_rat(numeral_numeral_rat,W2) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),Ca) )
    <=> $ite(Ca != zero_zero_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,W2)),Ca) = B,aa_num_rat(numeral_numeral_rat,W2) = zero_zero_rat) ) ).

% eq_divide_eq_numeral(1)
tff(fact_4496_mod__eq__nat1E,axiom,
    ! [M: nat,Q4: nat,N2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Q4) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N2),Q4) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),M)
       => ~ ! [S5: nat] : M != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Q4),S5)) ) ) ).

% mod_eq_nat1E
tff(fact_4497_mod__eq__nat2E,axiom,
    ! [M: nat,Q4: nat,N2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Q4) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N2),Q4) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
       => ~ ! [S5: nat] : N2 != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Q4),S5)) ) ) ).

% mod_eq_nat2E
tff(fact_4498_nat__mod__eq__lemma,axiom,
    ! [X: nat,N2: nat,Y: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),N2) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Y),N2) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),X)
       => ? [Q3: nat] : X = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N2),Q3)) ) ) ).

% nat_mod_eq_lemma
tff(fact_4499_div__less__mono,axiom,
    ! [A3: nat,B3: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A3),B3)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A3),N2) = zero_zero_nat )
         => ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B3),N2) = zero_zero_nat )
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A3),N2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B3),N2)) ) ) ) ) ).

% div_less_mono
tff(fact_4500_power__gt1,axiom,
    ! [A: code_integer,N2: 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,N2))) ) ).

% power_gt1
tff(fact_4501_power__gt1,axiom,
    ! [A: rat,N2: 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,N2))) ) ).

% power_gt1
tff(fact_4502_power__gt1,axiom,
    ! [A: nat,N2: 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,N2))) ) ).

% power_gt1
tff(fact_4503_power__gt1,axiom,
    ! [A: int,N2: 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,N2))) ) ).

% power_gt1
tff(fact_4504_nat__diff__split,axiom,
    ! [Pa: fun_nat_o,A: nat,B: nat] :
      ( aa_nat_o(Pa,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B))
    <=> ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
         => aa_nat_o(Pa,zero_zero_nat) )
        & ! [D5: nat] :
            ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),D5) )
           => aa_nat_o(Pa,D5) ) ) ) ).

% nat_diff_split
tff(fact_4505_nat__diff__split__asm,axiom,
    ! [Pa: fun_nat_o,A: nat,B: nat] :
      ( aa_nat_o(Pa,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B))
    <=> ~ ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
            & ~ aa_nat_o(Pa,zero_zero_nat) )
          | ? [D5: nat] :
              ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),D5) )
              & ~ aa_nat_o(Pa,D5) ) ) ) ).

% nat_diff_split_asm
tff(fact_4506_less__diff__conv2,axiom,
    ! [K: nat,J: nat,I2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),J)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K)),I2)
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),K)) ) ) ).

% less_diff_conv2
tff(fact_4507_ex__least__nat__less,axiom,
    ! [Pa: fun_nat_o,N2: nat] :
      ( aa_nat_o(Pa,N2)
     => ( ~ aa_nat_o(Pa,zero_zero_nat)
       => ? [K2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K2),N2)
            & ! [I: nat] :
                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),K2)
               => ~ aa_nat_o(Pa,I) )
            & aa_nat_o(Pa,aa_nat_nat(suc,K2)) ) ) ) ).

% ex_least_nat_less
tff(fact_4508_n__less__n__mult__m,axiom,
    ! [N2: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),M)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N2),M)) ) ) ).

% n_less_n_mult_m
tff(fact_4509_n__less__m__mult__n,axiom,
    ! [N2: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),M)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N2)) ) ) ).

% n_less_m_mult_n
tff(fact_4510_one__less__mult,axiom,
    ! [N2: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),N2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),M)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N2)) ) ) ).

% one_less_mult
tff(fact_4511_nat__induct__non__zero,axiom,
    ! [N2: nat,Pa: fun_nat_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)
     => ( aa_nat_o(Pa,one_one_nat)
       => ( ! [N5: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N5)
             => ( aa_nat_o(Pa,N5)
               => aa_nat_o(Pa,aa_nat_nat(suc,N5)) ) )
         => aa_nat_o(Pa,N2) ) ) ) ).

% nat_induct_non_zero
tff(fact_4512_nat__eq__add__iff1,axiom,
    ! [J: nat,I2: nat,U: nat,M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,J),I2)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I2),U)),M) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),N2) )
      <=> ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I2),J)),U)),M) = N2 ) ) ) ).

% nat_eq_add_iff1
tff(fact_4513_nat__eq__add__iff2,axiom,
    ! [I2: nat,J: nat,U: nat,M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),J)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I2),U)),M) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),N2) )
      <=> ( M = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),I2)),U)),N2) ) ) ) ).

% nat_eq_add_iff2
tff(fact_4514_nat__le__add__iff1,axiom,
    ! [J: nat,I2: nat,U: nat,M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,J),I2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I2),U)),M)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),N2))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I2),J)),U)),M)),N2) ) ) ).

% nat_le_add_iff1
tff(fact_4515_nat__le__add__iff2,axiom,
    ! [I2: nat,J: nat,U: nat,M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),J)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I2),U)),M)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),N2))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),I2)),U)),N2)) ) ) ).

% nat_le_add_iff2
tff(fact_4516_nat__diff__add__eq1,axiom,
    ! [J: nat,I2: nat,U: nat,M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,J),I2)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I2),U)),M)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),N2)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I2),J)),U)),M)),N2) ) ) ).

% nat_diff_add_eq1
tff(fact_4517_nat__diff__add__eq2,axiom,
    ! [I2: nat,J: nat,U: nat,M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),J)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I2),U)),M)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),N2)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),I2)),U)),N2)) ) ) ).

% nat_diff_add_eq2
tff(fact_4518_power__gt__expt,axiom,
    ! [N2: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),N2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,N2),K)) ) ).

% power_gt_expt
tff(fact_4519_nat__one__le__power,axiom,
    ! [I2: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),I2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,I2),N2)) ) ).

% nat_one_le_power
tff(fact_4520_frac__le__eq,axiom,
    ! [Y: rat,Z: rat,X: rat,W2: 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,W2),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,W2),Y))),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Z))),zero_zero_rat) ) ) ) ).

% frac_le_eq
tff(fact_4521_frac__less__eq,axiom,
    ! [Y: rat,Z: rat,X: rat,W2: 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,W2),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,W2),Y))),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Z))),zero_zero_rat) ) ) ) ).

% frac_less_eq
tff(fact_4522_split__mod,axiom,
    ! [Pa: fun_nat_o,M: nat,N2: nat] :
      ( aa_nat_o(Pa,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N2))
    <=> ( ( ( N2 = zero_zero_nat )
         => aa_nat_o(Pa,M) )
        & ( ( N2 != zero_zero_nat )
         => ! [I4: nat,J3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J3),N2)
             => ( ( M = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N2),I4)),J3) )
               => aa_nat_o(Pa,J3) ) ) ) ) ) ).

% split_mod
tff(fact_4523_divide__less__eq__numeral_I1_J,axiom,
    ! [B: rat,Ca: rat,W2: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),Ca)),aa_num_rat(numeral_numeral_rat,W2))
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Ca),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,W2)),Ca)),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Ca),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_num_rat(numeral_numeral_rat,W2)),Ca)),B),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_num_rat(numeral_numeral_rat,W2))) ) ) ).

% divide_less_eq_numeral(1)
tff(fact_4524_less__divide__eq__numeral_I1_J,axiom,
    ! [W2: num,B: rat,Ca: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(numeral_numeral_rat,W2)),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),Ca))
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Ca),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,W2)),Ca)),B),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Ca),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_num_rat(numeral_numeral_rat,W2)),Ca)),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(numeral_numeral_rat,W2)),zero_zero_rat)) ) ) ).

% less_divide_eq_numeral(1)
tff(fact_4525_power__Suc__le__self,axiom,
    ! [A: code_integer,N2: 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,N2))),A) ) ) ).

% power_Suc_le_self
tff(fact_4526_power__Suc__le__self,axiom,
    ! [A: rat,N2: 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,N2))),A) ) ) ).

% power_Suc_le_self
tff(fact_4527_power__Suc__le__self,axiom,
    ! [A: nat,N2: 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,N2))),A) ) ) ).

% power_Suc_le_self
tff(fact_4528_power__Suc__le__self,axiom,
    ! [A: int,N2: 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,N2))),A) ) ) ).

% power_Suc_le_self
tff(fact_4529_power__Suc__less__one,axiom,
    ! [A: code_integer,N2: 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,N2))),one_one_Code_integer) ) ) ).

% power_Suc_less_one
tff(fact_4530_power__Suc__less__one,axiom,
    ! [A: rat,N2: 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,N2))),one_one_rat) ) ) ).

% power_Suc_less_one
tff(fact_4531_power__Suc__less__one,axiom,
    ! [A: nat,N2: 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,N2))),one_one_nat) ) ) ).

% power_Suc_less_one
tff(fact_4532_power__Suc__less__one,axiom,
    ! [A: int,N2: 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,N2))),one_one_int) ) ) ).

% power_Suc_less_one
tff(fact_4533_power__diff,axiom,
    ! [A: nat,N2: nat,M: nat] :
      ( ( A != zero_zero_nat )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),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),N2)) = 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),N2)) ) ) ) ).

% power_diff
tff(fact_4534_power__diff,axiom,
    ! [A: int,N2: nat,M: nat] :
      ( ( A != zero_zero_int )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),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),N2)) = 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),N2)) ) ) ) ).

% power_diff
tff(fact_4535_power__diff,axiom,
    ! [A: code_integer,N2: nat,M: nat] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),M)
       => ( aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N2)) = aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_nat_Code_integer(power_8256067586552552935nteger(A),M)),aa_nat_Code_integer(power_8256067586552552935nteger(A),N2)) ) ) ) ).

% power_diff
tff(fact_4536_power__diff,axiom,
    ! [A: rat,N2: nat,M: nat] :
      ( ( A != zero_zero_rat )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),M)
       => ( aa_nat_rat(power_power_rat(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N2)) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_nat_rat(power_power_rat(A),M)),aa_nat_rat(power_power_rat(A),N2)) ) ) ) ).

% power_diff
tff(fact_4537_power__diff,axiom,
    ! [A: code_natural,N2: nat,M: nat] :
      ( ( A != zero_z2226904508553997617atural )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),M)
       => ( aa_nat_Code_natural(power_7079662738309270450atural(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N2)) = divide5121882707175180666atural(aa_nat_Code_natural(power_7079662738309270450atural(A),M),aa_nat_Code_natural(power_7079662738309270450atural(A),N2)) ) ) ) ).

% power_diff
tff(fact_4538_nat__less__add__iff1,axiom,
    ! [J: nat,I2: nat,U: nat,M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,J),I2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I2),U)),M)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),N2))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I2),J)),U)),M)),N2) ) ) ).

% nat_less_add_iff1
tff(fact_4539_nat__less__add__iff2,axiom,
    ! [I2: nat,J: nat,U: nat,M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),J)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I2),U)),M)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),N2))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),I2)),U)),N2)) ) ) ).

% nat_less_add_iff2
tff(fact_4540_mult__eq__if,axiom,
    ! [M: nat,N2: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N2) = $ite(M = zero_zero_nat,zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),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)),N2))) ).

% mult_eq_if
tff(fact_4541_div__nat__eqI,axiom,
    ! [N2: nat,Q4: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N2),Q4)),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N2),aa_nat_nat(suc,Q4)))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N2) = Q4 ) ) ) ).

% div_nat_eqI
tff(fact_4542_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
    ! [Ca: code_integer,A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),Ca)
     => ( aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),Ca)) = aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),B)),Ca))),aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A),B)) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_mult2_eq
tff(fact_4543_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
    ! [Ca: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Ca)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),Ca)) = 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)),Ca))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_mult2_eq
tff(fact_4544_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
    ! [Ca: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Ca)
     => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),Ca)) = 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)),Ca))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_mult2_eq
tff(fact_4545_scaling__mono,axiom,
    ! [U: rat,V: rat,Ra2: 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),Ra2)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Ra2),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,Ra2),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,V),U))),S2))),V) ) ) ) ).

% scaling_mono
tff(fact_4546_divide__le__eq__numeral_I1_J,axiom,
    ! [B: rat,Ca: rat,W2: 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),Ca)),aa_num_rat(numeral_numeral_rat,W2))
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Ca),
          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_num_rat(numeral_numeral_rat,W2)),Ca)),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Ca),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_num_rat(numeral_numeral_rat,W2)),Ca)),B),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_num_rat(numeral_numeral_rat,W2))) ) ) ).

% divide_le_eq_numeral(1)
tff(fact_4547_nat__approx__posE,axiom,
    ! [E: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),E)
     => ~ ! [N5: 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,N5)))),E) ) ).

% nat_approx_posE
tff(fact_4548_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)
     => ? [N5: 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,N5)),X)) ) ).

% ex_less_of_nat_mult
tff(fact_4549_of__nat__code,axiom,
    ! [N2: nat] : aa_nat_rat(semiri681578069525770553at_rat,N2) = aa_rat_rat(semiri7787848453975740701ux_rat(aTP_Lamm_gc,N2),zero_zero_rat) ).

% of_nat_code
tff(fact_4550_of__nat__code,axiom,
    ! [N2: nat] : aa_nat_int(semiri1314217659103216013at_int,N2) = aa_int_int(aa_nat_fun_int_int(semiri8420488043553186161ux_int(aTP_Lamm_gd),N2),zero_zero_int) ).

% of_nat_code
tff(fact_4551_of__nat__code,axiom,
    ! [N2: nat] : aa_nat_nat(semiri1316708129612266289at_nat,N2) = aa_nat_nat(aa_nat_fun_nat_nat(semiri8422978514062236437ux_nat(aTP_Lamm_ge),N2),zero_zero_nat) ).

% of_nat_code
tff(fact_4552_of__nat__code,axiom,
    ! [N2: nat] : aa_nat_Code_integer(semiri4939895301339042750nteger,N2) = aa_Cod1783974652533396728nteger(semiri4055485073559036834nteger(aTP_Lamm_gf,N2),zero_z3403309356797280102nteger) ).

% of_nat_code
tff(fact_4553_gcd__nat__induct,axiom,
    ! [Pa: fun_nat_fun_nat_o,M: nat,N2: nat] :
      ( ! [M5: nat] : aa_nat_o(aa_nat_fun_nat_o(Pa,M5),zero_zero_nat)
     => ( ! [M5: nat,N5: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N5)
           => ( aa_nat_o(aa_nat_fun_nat_o(Pa,N5),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M5),N5))
             => aa_nat_o(aa_nat_fun_nat_o(Pa,M5),N5) ) )
       => aa_nat_o(aa_nat_fun_nat_o(Pa,M),N2) ) ) ).

% gcd_nat_induct
tff(fact_4554_option_Osize__gen_I2_J,axiom,
    ! [X: fun_Pr4403862266400233414at_nat,X2: produc3260487557148687353it_nat] : size_o1731029420616209814it_nat(X,aa_Pro5302094286594872192it_nat(some_P7913643980934408916it_nat,X2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_Pro545002926299402503at_nat(X,X2)),aa_nat_nat(suc,zero_zero_nat)) ).

% option.size_gen(2)
tff(fact_4555_option_Osize__gen_I2_J,axiom,
    ! [X: fun_num_nat,X2: num] : aa_option_num_nat(size_option_num(X),aa_num_option_num(some_num,X2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(X,X2)),aa_nat_nat(suc,zero_zero_nat)) ).

% option.size_gen(2)
tff(fact_4556_option_Osize__gen_I2_J,axiom,
    ! [X: fun_Pr367382672041210069at_nat,X2: produc8664842809031399944it_nat] : size_o29782932136985253it_nat(X,aa_Pro8996696869555642526it_nat(some_P1914260805536162275it_nat,X2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_Pro5228051289423594006at_nat(X,X2)),aa_nat_nat(suc,zero_zero_nat)) ).

% option.size_gen(2)
tff(fact_4557_pochhammer__code,axiom,
    ! [A: rat,N2: nat] :
      aa_nat_rat(comm_s4028243227959126397er_rat(A),N2) = $ite(N2 = zero_zero_nat,one_one_rat,aa_rat_rat(set_fo1949268297981939178at_rat(aTP_Lamm_gg(A),zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),one_one_nat)),one_one_rat)) ).

% pochhammer_code
tff(fact_4558_pochhammer__code,axiom,
    ! [A: int,N2: nat] :
      aa_nat_int(aa_int_fun_nat_int(comm_s4660882817536571857er_int,A),N2) = $ite(N2 = zero_zero_nat,one_one_int,aa_int_int(aa_nat_fun_int_int(set_fo2581907887559384638at_int(aTP_Lamm_gh(A),zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),one_one_nat)),one_one_int)) ).

% pochhammer_code
tff(fact_4559_pochhammer__code,axiom,
    ! [A: code_integer,N2: nat] :
      aa_nat_Code_integer(comm_s8582702949713902594nteger(A),N2) = $ite(N2 = zero_zero_nat,one_one_Code_integer,aa_Cod1783974652533396728nteger(set_fo1084959871951514735nteger(aTP_Lamm_gi(A),zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),one_one_nat)),one_one_Code_integer)) ).

% pochhammer_code
tff(fact_4560_pochhammer__code,axiom,
    ! [A: nat,N2: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,A),N2) = $ite(N2 = zero_zero_nat,one_one_nat,aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(aTP_Lamm_gj(A),zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),one_one_nat)),one_one_nat)) ).

% pochhammer_code
tff(fact_4561_option_Osize__gen_I1_J,axiom,
    ! [X: fun_num_nat] : aa_option_num_nat(size_option_num(X),none_num) = aa_nat_nat(suc,zero_zero_nat) ).

% option.size_gen(1)
tff(fact_4562_Diff__cancel,axiom,
    ! [A3: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A3),A3) = bot_bo2099793752762293965at_nat ).

% Diff_cancel
tff(fact_4563_Diff__cancel,axiom,
    ! [A3: set_o] : minus_minus_set_o(A3,A3) = bot_bot_set_o ).

% Diff_cancel
tff(fact_4564_Diff__cancel,axiom,
    ! [A3: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A3),A3) = bot_bot_set_int ).

% Diff_cancel
tff(fact_4565_Diff__cancel,axiom,
    ! [A3: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A3),A3) = bot_bot_set_nat ).

% Diff_cancel
tff(fact_4566_empty__Diff,axiom,
    ! [A3: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,bot_bo2099793752762293965at_nat),A3) = bot_bo2099793752762293965at_nat ).

% empty_Diff
tff(fact_4567_empty__Diff,axiom,
    ! [A3: set_o] : minus_minus_set_o(bot_bot_set_o,A3) = bot_bot_set_o ).

% empty_Diff
tff(fact_4568_empty__Diff,axiom,
    ! [A3: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,bot_bot_set_int),A3) = bot_bot_set_int ).

% empty_Diff
tff(fact_4569_empty__Diff,axiom,
    ! [A3: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,bot_bot_set_nat),A3) = bot_bot_set_nat ).

% empty_Diff
tff(fact_4570_Diff__empty,axiom,
    ! [A3: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A3),bot_bo2099793752762293965at_nat) = A3 ).

% Diff_empty
tff(fact_4571_Diff__empty,axiom,
    ! [A3: set_o] : minus_minus_set_o(A3,bot_bot_set_o) = A3 ).

% Diff_empty
tff(fact_4572_Diff__empty,axiom,
    ! [A3: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A3),bot_bot_set_int) = A3 ).

% Diff_empty
tff(fact_4573_Diff__empty,axiom,
    ! [A3: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A3),bot_bot_set_nat) = A3 ).

% Diff_empty
tff(fact_4574_Un__Diff__cancel2,axiom,
    ! [B3: set_Pr8693737435421807431at_nat,A3: set_Pr8693737435421807431at_nat] : aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(minus_8321449233255521966at_nat,B3),A3)),A3) = aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,B3),A3) ).

% Un_Diff_cancel2
tff(fact_4575_Un__Diff__cancel2,axiom,
    ! [B3: set_Pr8551490117392284871at_nat,A3: set_Pr8551490117392284871at_nat] : aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(minus_5060654252129873198at_nat,B3),A3)),A3) = aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,B3),A3) ).

% Un_Diff_cancel2
tff(fact_4576_Un__Diff__cancel2,axiom,
    ! [B3: set_Pr4329608150637261639at_nat,A3: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,B3),A3)),A3) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,B3),A3) ).

% Un_Diff_cancel2
tff(fact_4577_Un__Diff__cancel2,axiom,
    ! [B3: set_nat,A3: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,B3),A3)),A3) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,B3),A3) ).

% Un_Diff_cancel2
tff(fact_4578_Un__Diff__cancel,axiom,
    ! [A3: set_Pr8693737435421807431at_nat,B3: set_Pr8693737435421807431at_nat] : aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,A3),aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(minus_8321449233255521966at_nat,B3),A3)) = aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,A3),B3) ).

% Un_Diff_cancel
tff(fact_4579_Un__Diff__cancel,axiom,
    ! [A3: set_Pr8551490117392284871at_nat,B3: set_Pr8551490117392284871at_nat] : aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A3),aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(minus_5060654252129873198at_nat,B3),A3)) = aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A3),B3) ).

% Un_Diff_cancel
tff(fact_4580_Un__Diff__cancel,axiom,
    ! [A3: set_Pr4329608150637261639at_nat,B3: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A3),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,B3),A3)) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A3),B3) ).

% Un_Diff_cancel
tff(fact_4581_Un__Diff__cancel,axiom,
    ! [A3: set_nat,B3: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A3),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,B3),A3)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A3),B3) ).

% Un_Diff_cancel
tff(fact_4582_Diff__eq__empty__iff,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
      ( ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A3),B3) = bot_bo2099793752762293965at_nat )
    <=> aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A3),B3) ) ).

% Diff_eq_empty_iff
tff(fact_4583_Diff__eq__empty__iff,axiom,
    ! [A3: set_o,B3: set_o] :
      ( ( minus_minus_set_o(A3,B3) = bot_bot_set_o )
    <=> ord_less_eq_set_o(A3,B3) ) ).

% Diff_eq_empty_iff
tff(fact_4584_Diff__eq__empty__iff,axiom,
    ! [A3: set_nat,B3: set_nat] :
      ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A3),B3) = bot_bot_set_nat )
    <=> aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A3),B3) ) ).

% Diff_eq_empty_iff
tff(fact_4585_Diff__eq__empty__iff,axiom,
    ! [A3: set_int,B3: set_int] :
      ( ( aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A3),B3) = bot_bot_set_int )
    <=> aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A3),B3) ) ).

% Diff_eq_empty_iff
tff(fact_4586_Diff__disjoint,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A3),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,B3),A3)) = bot_bo2099793752762293965at_nat ).

% Diff_disjoint
tff(fact_4587_Diff__disjoint,axiom,
    ! [A3: set_o,B3: set_o] : inf_inf_set_o(A3,minus_minus_set_o(B3,A3)) = bot_bot_set_o ).

% Diff_disjoint
tff(fact_4588_Diff__disjoint,axiom,
    ! [A3: set_int,B3: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A3),aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,B3),A3)) = bot_bot_set_int ).

% Diff_disjoint
tff(fact_4589_Diff__disjoint,axiom,
    ! [A3: set_nat,B3: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A3),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,B3),A3)) = bot_bot_set_nat ).

% Diff_disjoint
tff(fact_4590_mod__pos__pos__trivial,axiom,
    ! [K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),L)
       => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),L) = K ) ) ) ).

% mod_pos_pos_trivial
tff(fact_4591_mod__neg__neg__trivial,axiom,
    ! [K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,L),K)
       => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),L) = K ) ) ) ).

% mod_neg_neg_trivial
tff(fact_4592_div__pos__pos__trivial,axiom,
    ! [K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),L)
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,K),L) = zero_zero_int ) ) ) ).

% div_pos_pos_trivial
tff(fact_4593_div__neg__neg__trivial,axiom,
    ! [K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,L),K)
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,K),L) = zero_zero_int ) ) ) ).

% div_neg_neg_trivial
tff(fact_4594_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_4595_verit__le__mono__div__int,axiom,
    ! [A3: int,B3: int,N2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A3),B3)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),N2)
       => 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,A3),N2)),
                $ite(aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B3),N2) = zero_zero_int,one_one_int,zero_zero_int))),aa_int_int(aa_int_fun_int_int(divide_divide_int,B3),N2)) ) ) ).

% verit_le_mono_div_int
tff(fact_4596_zdiv__mono__strict,axiom,
    ! [A3: int,B3: int,N2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A3),B3)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),N2)
       => ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A3),N2) = zero_zero_int )
         => ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B3),N2) = zero_zero_int )
           => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A3),N2)),aa_int_int(aa_int_fun_int_int(divide_divide_int,B3),N2)) ) ) ) ) ).

% zdiv_mono_strict
tff(fact_4597_split__pos__lemma,axiom,
    ! [K: int,Pa: fun_int_fun_int_o,N2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),K)
     => ( aa_int_o(aa_int_fun_int_o(Pa,aa_int_int(aa_int_fun_int_int(divide_divide_int,N2),K)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,N2),K))
      <=> ! [I4: int,J3: int] :
            ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),J3)
              & aa_int_o(aa_int_fun_int_o(ord_less_int,J3),K)
              & ( N2 = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,K),I4)),J3) ) )
           => aa_int_o(aa_int_fun_int_o(Pa,I4),J3) ) ) ) ).

% split_pos_lemma
tff(fact_4598_split__neg__lemma,axiom,
    ! [K: int,Pa: fun_int_fun_int_o,N2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(Pa,aa_int_int(aa_int_fun_int_int(divide_divide_int,N2),K)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,N2),K))
      <=> ! [I4: int,J3: int] :
            ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),J3)
              & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,J3),zero_zero_int)
              & ( N2 = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,K),I4)),J3) ) )
           => aa_int_o(aa_int_fun_int_o(Pa,I4),J3) ) ) ) ).

% split_neg_lemma
tff(fact_4599_zmod__zmult2__eq,axiom,
    ! [Ca: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Ca)
     => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),Ca)) = 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)),Ca))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)) ) ) ).

% zmod_zmult2_eq
tff(fact_4600_Euclidean__Division_Opos__mod__sign,axiom,
    ! [L: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),L)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),L)) ) ).

% Euclidean_Division.pos_mod_sign
tff(fact_4601_neg__mod__sign,axiom,
    ! [L: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,L),zero_zero_int)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),L)),zero_zero_int) ) ).

% neg_mod_sign
tff(fact_4602_Euclidean__Division_Opos__mod__bound,axiom,
    ! [L: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),L)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),L)),L) ) ).

% Euclidean_Division.pos_mod_bound
tff(fact_4603_neg__mod__bound,axiom,
    ! [L: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,L),zero_zero_int)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,L),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),L)) ) ).

% neg_mod_bound
tff(fact_4604_mod__pos__neg__trivial,axiom,
    ! [K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),K)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,K),L)),zero_zero_int)
       => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),L) = aa_int_int(aa_int_fun_int_int(plus_plus_int,K),L) ) ) ) ).

% mod_pos_neg_trivial
tff(fact_4605_zmod__le__nonneg__dividend,axiom,
    ! [M: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),M)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,M),K)),M) ) ).

% zmod_le_nonneg_dividend
tff(fact_4606_mod__pos__geq,axiom,
    ! [L: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),L)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,L),K)
       => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),L) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,K),L)),L) ) ) ) ).

% mod_pos_geq
tff(fact_4607_neg__mod__conj,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)),zero_zero_int)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,B),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)) ) ) ).

% neg_mod_conj
tff(fact_4608_pos__mod__conj,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B))
        & aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)),B) ) ) ).

% pos_mod_conj
tff(fact_4609_zmod__trivial__iff,axiom,
    ! [I2: int,K: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,I2),K) = I2 )
    <=> ( ( K = zero_zero_int )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),I2)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,I2),K) )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I2),zero_zero_int)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,K),I2) ) ) ) ).

% zmod_trivial_iff
tff(fact_4610_plusinfinity,axiom,
    ! [D2: int,P3: fun_int_o,Pa: fun_int_o] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D2)
     => ( ! [X3: int,K2: int] :
            ( aa_int_o(P3,X3)
          <=> aa_int_o(P3,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),aa_int_int(aa_int_fun_int_int(times_times_int,K2),D2))) )
       => ( ? [Z4: int] :
            ! [X3: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z4),X3)
             => ( aa_int_o(Pa,X3)
              <=> aa_int_o(P3,X3) ) )
         => ( ? [X_13: int] : aa_int_o(P3,X_13)
           => ? [X_1: int] : aa_int_o(Pa,X_1) ) ) ) ) ).

% plusinfinity
tff(fact_4611_minusinfinity,axiom,
    ! [D2: int,P1: fun_int_o,Pa: fun_int_o] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D2)
     => ( ! [X3: int,K2: int] :
            ( aa_int_o(P1,X3)
          <=> aa_int_o(P1,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),aa_int_int(aa_int_fun_int_int(times_times_int,K2),D2))) )
       => ( ? [Z4: int] :
            ! [X3: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Z4)
             => ( aa_int_o(Pa,X3)
              <=> aa_int_o(P1,X3) ) )
         => ( ? [X_13: int] : aa_int_o(P1,X_13)
           => ? [X_1: int] : aa_int_o(Pa,X_1) ) ) ) ) ).

% minusinfinity
tff(fact_4612_decr__mult__lemma,axiom,
    ! [D2: int,Pa: fun_int_o,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D2)
     => ( ! [X3: int] :
            ( aa_int_o(Pa,X3)
           => aa_int_o(Pa,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),D2)) )
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K)
         => ! [X5: int] :
              ( aa_int_o(Pa,X5)
             => aa_int_o(Pa,aa_int_int(aa_int_fun_int_int(minus_minus_int,X5),aa_int_int(aa_int_fun_int_int(times_times_int,K),D2))) ) ) ) ) ).

% decr_mult_lemma
tff(fact_4613_int__mod__pos__eq,axiom,
    ! [A: int,B: int,Q4: int,Ra2: int] :
      ( ( A = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),Q4)),Ra2) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Ra2)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Ra2),B)
         => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) = Ra2 ) ) ) ) ).

% int_mod_pos_eq
tff(fact_4614_int__mod__neg__eq,axiom,
    ! [A: int,B: int,Q4: int,Ra2: int] :
      ( ( A = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),Q4)),Ra2) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ra2),zero_zero_int)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),Ra2)
         => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) = Ra2 ) ) ) ) ).

% int_mod_neg_eq
tff(fact_4615_split__zmod,axiom,
    ! [Pa: fun_int_o,N2: int,K: int] :
      ( aa_int_o(Pa,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,N2),K))
    <=> ( ( ( K = zero_zero_int )
         => aa_int_o(Pa,N2) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),K)
         => ! [I4: int,J3: int] :
              ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),J3)
                & aa_int_o(aa_int_fun_int_o(ord_less_int,J3),K)
                & ( N2 = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,K),I4)),J3) ) )
             => aa_int_o(Pa,J3) ) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int)
         => ! [I4: int,J3: int] :
              ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),J3)
                & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,J3),zero_zero_int)
                & ( N2 = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,K),I4)),J3) ) )
             => aa_int_o(Pa,J3) ) ) ) ) ).

% split_zmod
tff(fact_4616_q__pos__lemma,axiom,
    ! [B2: int,Q6: int,R4: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B2),Q6)),R4))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,R4),B2)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B2)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Q6) ) ) ) ).

% q_pos_lemma
tff(fact_4617_zdiv__mono2__lemma,axiom,
    ! [B: int,Q4: int,Ra2: int,B2: int,Q6: int,R4: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),Q4)),Ra2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B2),Q6)),R4) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B2),Q6)),R4))
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,R4),B2)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Ra2)
           => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B2)
             => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),B)
               => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Q4),Q6) ) ) ) ) ) ) ).

% zdiv_mono2_lemma
tff(fact_4618_incr__mult__lemma,axiom,
    ! [D2: int,Pa: fun_int_o,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D2)
     => ( ! [X3: int] :
            ( aa_int_o(Pa,X3)
           => aa_int_o(Pa,aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),D2)) )
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K)
         => ! [X5: int] :
              ( aa_int_o(Pa,X5)
             => aa_int_o(Pa,aa_int_int(aa_int_fun_int_int(plus_plus_int,X5),aa_int_int(aa_int_fun_int_int(times_times_int,K),D2))) ) ) ) ) ).

% incr_mult_lemma
tff(fact_4619_zdiv__mono2__neg__lemma,axiom,
    ! [B: int,Q4: int,Ra2: int,B2: int,Q6: int,R4: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),Q4)),Ra2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B2),Q6)),R4) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B2),Q6)),R4)),zero_zero_int)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Ra2),B)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),R4)
           => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B2)
             => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),B)
               => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Q6),Q4) ) ) ) ) ) ) ).

% zdiv_mono2_neg_lemma
tff(fact_4620_unique__quotient__lemma,axiom,
    ! [B: int,Q6: int,R4: int,Q4: int,Ra2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),Q6)),R4)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),Q4)),Ra2))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),R4)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,R4),B)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Ra2),B)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Q6),Q4) ) ) ) ) ).

% unique_quotient_lemma
tff(fact_4621_unique__quotient__lemma__neg,axiom,
    ! [B: int,Q6: int,R4: int,Q4: int,Ra2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),Q6)),R4)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),Q4)),Ra2))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ra2),zero_zero_int)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),Ra2)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),R4)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Q4),Q6) ) ) ) ) ).

% unique_quotient_lemma_neg
tff(fact_4622_imp__le__cong,axiom,
    ! [X: int,X8: int,Pa: $o,P3: $o] :
      ( ( X = X8 )
     => ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X8)
         => ( (Pa)
          <=> (P3) ) )
       => ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X)
           => (Pa) )
        <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X8)
           => (P3) ) ) ) ) ).

% imp_le_cong
tff(fact_4623_conj__le__cong,axiom,
    ! [X: int,X8: int,Pa: $o,P3: $o] :
      ( ( X = X8 )
     => ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X8)
         => ( (Pa)
          <=> (P3) ) )
       => ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X)
            & (Pa) )
        <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X8)
            & (P3) ) ) ) ) ).

% conj_le_cong
tff(fact_4624_verit__la__generic,axiom,
    ! [A: int,X: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),X)
      | ( A = X )
      | aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),A) ) ).

% verit_la_generic
tff(fact_4625_split__zdiv,axiom,
    ! [Pa: fun_int_o,N2: int,K: int] :
      ( aa_int_o(Pa,aa_int_int(aa_int_fun_int_int(divide_divide_int,N2),K))
    <=> ( ( ( K = zero_zero_int )
         => aa_int_o(Pa,zero_zero_int) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),K)
         => ! [I4: int,J3: int] :
              ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),J3)
                & aa_int_o(aa_int_fun_int_o(ord_less_int,J3),K)
                & ( N2 = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,K),I4)),J3) ) )
             => aa_int_o(Pa,I4) ) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int)
         => ! [I4: int,J3: int] :
              ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),J3)
                & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,J3),zero_zero_int)
                & ( N2 = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,K),I4)),J3) ) )
             => aa_int_o(Pa,I4) ) ) ) ) ).

% split_zdiv
tff(fact_4626_int__div__neg__eq,axiom,
    ! [A: int,B: int,Q4: int,Ra2: int] :
      ( ( A = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),Q4)),Ra2) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ra2),zero_zero_int)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),Ra2)
         => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B) = Q4 ) ) ) ) ).

% int_div_neg_eq
tff(fact_4627_int__div__pos__eq,axiom,
    ! [A: int,B: int,Q4: int,Ra2: int] :
      ( ( A = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),Q4)),Ra2) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Ra2)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Ra2),B)
         => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B) = Q4 ) ) ) ) ).

% int_div_pos_eq
tff(fact_4628_zdiv__zmult2__eq,axiom,
    ! [Ca: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Ca)
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),Ca)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),Ca) ) ) ).

% zdiv_zmult2_eq
tff(fact_4629_nonneg1__imp__zdiv__pos__iff,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B))
      <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B) ) ) ) ).

% nonneg1_imp_zdiv_pos_iff
tff(fact_4630_pos__imp__zdiv__nonneg__iff,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A) ) ) ).

% pos_imp_zdiv_nonneg_iff
tff(fact_4631_neg__imp__zdiv__nonneg__iff,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int) ) ) ).

% neg_imp_zdiv_nonneg_iff
tff(fact_4632_pos__imp__zdiv__pos__iff,axiom,
    ! [K: int,I2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),K)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(divide_divide_int,I2),K))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),I2) ) ) ).

% pos_imp_zdiv_pos_iff
tff(fact_4633_div__nonpos__pos__le0,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),zero_zero_int) ) ) ).

% div_nonpos_pos_le0
tff(fact_4634_div__nonneg__neg__le0,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),zero_zero_int) ) ) ).

% div_nonneg_neg_le0
tff(fact_4635_div__positive__int,axiom,
    ! [L: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,L),K)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),L)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(divide_divide_int,K),L)) ) ) ).

% div_positive_int
tff(fact_4636_div__int__pos__iff,axiom,
    ! [K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(divide_divide_int,K),L))
    <=> ( ( K = zero_zero_int )
        | ( L = zero_zero_int )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K)
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),L) )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,L),zero_zero_int) ) ) ) ).

% div_int_pos_iff
tff(fact_4637_zdiv__mono2__neg,axiom,
    ! [A: int,B2: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B2)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),B)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B2)),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)) ) ) ) ).

% zdiv_mono2_neg
tff(fact_4638_zdiv__mono1__neg,axiom,
    ! [A: int,A2: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),A2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B)),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)) ) ) ).

% zdiv_mono1_neg
tff(fact_4639_zdiv__eq__0__iff,axiom,
    ! [I2: int,K: int] :
      ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,I2),K) = zero_zero_int )
    <=> ( ( K = zero_zero_int )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),I2)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,I2),K) )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I2),zero_zero_int)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,K),I2) ) ) ) ).

% zdiv_eq_0_iff
tff(fact_4640_zdiv__mono2,axiom,
    ! [A: int,B2: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B2)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),B)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B2)) ) ) ) ).

% zdiv_mono2
tff(fact_4641_zdiv__mono1,axiom,
    ! [A: int,A2: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),A2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B)) ) ) ).

% zdiv_mono1
tff(fact_4642_pos__imp__zdiv__neg__iff,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),zero_zero_int)
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int) ) ) ).

% pos_imp_zdiv_neg_iff
tff(fact_4643_neg__imp__zdiv__neg__iff,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),zero_zero_int)
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A) ) ) ).

% neg_imp_zdiv_neg_iff
tff(fact_4644_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_4645_div__neg__pos__less0,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),zero_zero_int) ) ) ).

% div_neg_pos_less0
tff(fact_4646_Diff__mono,axiom,
    ! [A3: set_nat,C2: set_nat,D3: set_nat,B3: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A3),C2)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,D3),B3)
       => 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,A3),B3)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,C2),D3)) ) ) ).

% Diff_mono
tff(fact_4647_Diff__mono,axiom,
    ! [A3: set_int,C2: set_int,D3: set_int,B3: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A3),C2)
     => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,D3),B3)
       => aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A3),B3)),aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,C2),D3)) ) ) ).

% Diff_mono
tff(fact_4648_Diff__subset,axiom,
    ! [A3: set_nat,B3: 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,A3),B3)),A3) ).

% Diff_subset
tff(fact_4649_Diff__subset,axiom,
    ! [A3: set_int,B3: set_int] : aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A3),B3)),A3) ).

% Diff_subset
tff(fact_4650_double__diff,axiom,
    ! [A3: set_nat,B3: set_nat,C2: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A3),B3)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B3),C2)
       => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,B3),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,C2),A3)) = A3 ) ) ) ).

% double_diff
tff(fact_4651_double__diff,axiom,
    ! [A3: set_int,B3: set_int,C2: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A3),B3)
     => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,B3),C2)
       => ( aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,B3),aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,C2),A3)) = A3 ) ) ) ).

% double_diff
tff(fact_4652_int__ops_I6_J,axiom,
    ! [A: nat,B: nat] :
      aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B)) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,A)),aa_nat_int(semiri1314217659103216013at_int,B)),zero_zero_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(semiri1314217659103216013at_int,A)),aa_nat_int(semiri1314217659103216013at_int,B))) ).

% int_ops(6)
tff(fact_4653_Diff__Int__distrib2,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A3),B3)),C2) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A3),C2)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B3),C2)) ).

% Diff_Int_distrib2
tff(fact_4654_Diff__Int__distrib2,axiom,
    ! [A3: set_nat,B3: set_nat,C2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A3),B3)),C2) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A3),C2)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B3),C2)) ).

% Diff_Int_distrib2
tff(fact_4655_Diff__Int__distrib,axiom,
    ! [C2: set_Pr1261947904930325089at_nat,A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,C2),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A3),B3)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,C2),A3)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,C2),B3)) ).

% Diff_Int_distrib
tff(fact_4656_Diff__Int__distrib,axiom,
    ! [C2: set_nat,A3: set_nat,B3: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,C2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A3),B3)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,C2),A3)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,C2),B3)) ).

% Diff_Int_distrib
tff(fact_4657_Diff__Diff__Int,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A3),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A3),B3)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A3),B3) ).

% Diff_Diff_Int
tff(fact_4658_Diff__Diff__Int,axiom,
    ! [A3: set_nat,B3: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A3),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A3),B3)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A3),B3) ).

% Diff_Diff_Int
tff(fact_4659_Diff__Int2,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A3),C2)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B3),C2)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A3),C2)),B3) ).

% Diff_Int2
tff(fact_4660_Diff__Int2,axiom,
    ! [A3: set_nat,C2: set_nat,B3: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A3),C2)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B3),C2)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A3),C2)),B3) ).

% Diff_Int2
tff(fact_4661_Int__Diff,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A3),B3)),C2) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A3),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,B3),C2)) ).

% Int_Diff
tff(fact_4662_Int__Diff,axiom,
    ! [A3: set_nat,B3: set_nat,C2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A3),B3)),C2) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A3),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,B3),C2)) ).

% Int_Diff
tff(fact_4663_set__diff__diff__left,axiom,
    ! [A3: set_Pr8693737435421807431at_nat,B3: set_Pr8693737435421807431at_nat,C2: set_Pr8693737435421807431at_nat] : aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(minus_8321449233255521966at_nat,aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(minus_8321449233255521966at_nat,A3),B3)),C2) = aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(minus_8321449233255521966at_nat,A3),aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,B3),C2)) ).

% set_diff_diff_left
tff(fact_4664_set__diff__diff__left,axiom,
    ! [A3: set_Pr8551490117392284871at_nat,B3: 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,A3),B3)),C2) = aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(minus_5060654252129873198at_nat,A3),aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,B3),C2)) ).

% set_diff_diff_left
tff(fact_4665_set__diff__diff__left,axiom,
    ! [A3: set_Pr4329608150637261639at_nat,B3: 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,A3),B3)),C2) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,A3),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,B3),C2)) ).

% set_diff_diff_left
tff(fact_4666_set__diff__diff__left,axiom,
    ! [A3: set_nat,B3: 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,A3),B3)),C2) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A3),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,B3),C2)) ).

% set_diff_diff_left
tff(fact_4667_Un__Diff,axiom,
    ! [A3: set_Pr8693737435421807431at_nat,B3: set_Pr8693737435421807431at_nat,C2: set_Pr8693737435421807431at_nat] : aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(minus_8321449233255521966at_nat,aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,A3),B3)),C2) = aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(minus_8321449233255521966at_nat,A3),C2)),aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(minus_8321449233255521966at_nat,B3),C2)) ).

% Un_Diff
tff(fact_4668_Un__Diff,axiom,
    ! [A3: set_Pr8551490117392284871at_nat,B3: 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,A3),B3)),C2) = aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(minus_5060654252129873198at_nat,A3),C2)),aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(minus_5060654252129873198at_nat,B3),C2)) ).

% Un_Diff
tff(fact_4669_Un__Diff,axiom,
    ! [A3: set_Pr4329608150637261639at_nat,B3: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A3),B3)),C2) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,A3),C2)),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,B3),C2)) ).

% Un_Diff
tff(fact_4670_Un__Diff,axiom,
    ! [A3: set_nat,B3: set_nat,C2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A3),B3)),C2) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A3),C2)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,B3),C2)) ).

% Un_Diff
tff(fact_4671_psubset__imp__ex__mem,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le7866589430770878221at_nat,A3),B3)
     => ? [B4: product_prod_nat_nat] : aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(B4),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,B3),A3)) ) ).

% psubset_imp_ex_mem
tff(fact_4672_psubset__imp__ex__mem,axiom,
    ! [A3: set_o,B3: set_o] :
      ( ord_less_set_o(A3,B3)
     => ? [B4: $o] : member_o((B4),minus_minus_set_o(B3,A3)) ) ).

% psubset_imp_ex_mem
tff(fact_4673_psubset__imp__ex__mem,axiom,
    ! [A3: set_set_nat,B3: set_set_nat] :
      ( aa_set_set_nat_o(ord_less_set_set_nat(A3),B3)
     => ? [B4: set_nat] : aa_set_set_nat_o(member_set_nat(B4),minus_2163939370556025621et_nat(B3,A3)) ) ).

% psubset_imp_ex_mem
tff(fact_4674_psubset__imp__ex__mem,axiom,
    ! [A3: set_int,B3: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_set_int,A3),B3)
     => ? [B4: int] : aa_set_int_o(member_int(B4),aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,B3),A3)) ) ).

% psubset_imp_ex_mem
tff(fact_4675_psubset__imp__ex__mem,axiom,
    ! [A3: set_nat,B3: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,A3),B3)
     => ? [B4: nat] : aa_set_nat_o(member_nat(B4),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,B3),A3)) ) ).

% psubset_imp_ex_mem
tff(fact_4676_subset__minus__empty,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A3),B3)
     => ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A3),B3) = bot_bo2099793752762293965at_nat ) ) ).

% subset_minus_empty
tff(fact_4677_subset__minus__empty,axiom,
    ! [A3: set_o,B3: set_o] :
      ( ord_less_eq_set_o(A3,B3)
     => ( minus_minus_set_o(A3,B3) = bot_bot_set_o ) ) ).

% subset_minus_empty
tff(fact_4678_subset__minus__empty,axiom,
    ! [A3: set_nat,B3: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A3),B3)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A3),B3) = bot_bot_set_nat ) ) ).

% subset_minus_empty
tff(fact_4679_subset__minus__empty,axiom,
    ! [A3: set_int,B3: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A3),B3)
     => ( aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A3),B3) = bot_bot_set_int ) ) ).

% subset_minus_empty
tff(fact_4680_disjoint__alt__simp2,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
      ( ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A3),B3) != A3 )
    <=> ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A3),B3) != bot_bo2099793752762293965at_nat ) ) ).

% disjoint_alt_simp2
tff(fact_4681_disjoint__alt__simp2,axiom,
    ! [A3: set_o,B3: set_o] :
      ( ( minus_minus_set_o(A3,B3) != A3 )
    <=> ( inf_inf_set_o(A3,B3) != bot_bot_set_o ) ) ).

% disjoint_alt_simp2
tff(fact_4682_disjoint__alt__simp2,axiom,
    ! [A3: set_int,B3: set_int] :
      ( ( aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A3),B3) != A3 )
    <=> ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A3),B3) != bot_bot_set_int ) ) ).

% disjoint_alt_simp2
tff(fact_4683_disjoint__alt__simp2,axiom,
    ! [A3: set_nat,B3: set_nat] :
      ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A3),B3) != A3 )
    <=> ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A3),B3) != bot_bot_set_nat ) ) ).

% disjoint_alt_simp2
tff(fact_4684_disjoint__alt__simp1,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
      ( ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A3),B3) = A3 )
    <=> ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A3),B3) = bot_bo2099793752762293965at_nat ) ) ).

% disjoint_alt_simp1
tff(fact_4685_disjoint__alt__simp1,axiom,
    ! [A3: set_o,B3: set_o] :
      ( ( minus_minus_set_o(A3,B3) = A3 )
    <=> ( inf_inf_set_o(A3,B3) = bot_bot_set_o ) ) ).

% disjoint_alt_simp1
tff(fact_4686_disjoint__alt__simp1,axiom,
    ! [A3: set_int,B3: set_int] :
      ( ( aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A3),B3) = A3 )
    <=> ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A3),B3) = bot_bot_set_int ) ) ).

% disjoint_alt_simp1
tff(fact_4687_disjoint__alt__simp1,axiom,
    ! [A3: set_nat,B3: set_nat] :
      ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A3),B3) = A3 )
    <=> ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A3),B3) = bot_bot_set_nat ) ) ).

% disjoint_alt_simp1
tff(fact_4688_Int__Diff__disjoint,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A3),B3)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A3),B3)) = bot_bo2099793752762293965at_nat ).

% Int_Diff_disjoint
tff(fact_4689_Int__Diff__disjoint,axiom,
    ! [A3: set_o,B3: set_o] : inf_inf_set_o(inf_inf_set_o(A3,B3),minus_minus_set_o(A3,B3)) = bot_bot_set_o ).

% Int_Diff_disjoint
tff(fact_4690_Int__Diff__disjoint,axiom,
    ! [A3: set_int,B3: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A3),B3)),aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A3),B3)) = bot_bot_set_int ).

% Int_Diff_disjoint
tff(fact_4691_Int__Diff__disjoint,axiom,
    ! [A3: set_nat,B3: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A3),B3)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A3),B3)) = bot_bot_set_nat ).

% Int_Diff_disjoint
tff(fact_4692_Diff__triv,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
      ( ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A3),B3) = bot_bo2099793752762293965at_nat )
     => ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A3),B3) = A3 ) ) ).

% Diff_triv
tff(fact_4693_Diff__triv,axiom,
    ! [A3: set_o,B3: set_o] :
      ( ( inf_inf_set_o(A3,B3) = bot_bot_set_o )
     => ( minus_minus_set_o(A3,B3) = A3 ) ) ).

% Diff_triv
tff(fact_4694_Diff__triv,axiom,
    ! [A3: set_int,B3: set_int] :
      ( ( 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(minus_minus_set_int,A3),B3) = A3 ) ) ).

% Diff_triv
tff(fact_4695_Diff__triv,axiom,
    ! [A3: set_nat,B3: set_nat] :
      ( ( 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(minus_minus_set_nat,A3),B3) = A3 ) ) ).

% Diff_triv
tff(fact_4696_Diff__partition,axiom,
    ! [A3: set_Pr8693737435421807431at_nat,B3: set_Pr8693737435421807431at_nat] :
      ( ord_le3000389064537975527at_nat(A3,B3)
     => ( aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,A3),aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(minus_8321449233255521966at_nat,B3),A3)) = B3 ) ) ).

% Diff_partition
tff(fact_4697_Diff__partition,axiom,
    ! [A3: set_Pr8551490117392284871at_nat,B3: set_Pr8551490117392284871at_nat] :
      ( ord_le8081472938463900775at_nat(A3,B3)
     => ( aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A3),aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(minus_5060654252129873198at_nat,B3),A3)) = B3 ) ) ).

% Diff_partition
tff(fact_4698_Diff__partition,axiom,
    ! [A3: set_Pr4329608150637261639at_nat,B3: set_Pr4329608150637261639at_nat] :
      ( ord_le1268244103169919719at_nat(A3,B3)
     => ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A3),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,B3),A3)) = B3 ) ) ).

% Diff_partition
tff(fact_4699_Diff__partition,axiom,
    ! [A3: set_nat,B3: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A3),B3)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A3),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,B3),A3)) = B3 ) ) ).

% Diff_partition
tff(fact_4700_Diff__partition,axiom,
    ! [A3: set_int,B3: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A3),B3)
     => ( aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A3),aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,B3),A3)) = B3 ) ) ).

% Diff_partition
tff(fact_4701_Diff__subset__conv,axiom,
    ! [A3: set_Pr8693737435421807431at_nat,B3: set_Pr8693737435421807431at_nat,C2: set_Pr8693737435421807431at_nat] :
      ( ord_le3000389064537975527at_nat(aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(minus_8321449233255521966at_nat,A3),B3),C2)
    <=> ord_le3000389064537975527at_nat(A3,aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,B3),C2)) ) ).

% Diff_subset_conv
tff(fact_4702_Diff__subset__conv,axiom,
    ! [A3: set_Pr8551490117392284871at_nat,B3: set_Pr8551490117392284871at_nat,C2: set_Pr8551490117392284871at_nat] :
      ( ord_le8081472938463900775at_nat(aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(minus_5060654252129873198at_nat,A3),B3),C2)
    <=> ord_le8081472938463900775at_nat(A3,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,B3),C2)) ) ).

% Diff_subset_conv
tff(fact_4703_Diff__subset__conv,axiom,
    ! [A3: set_Pr4329608150637261639at_nat,B3: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat] :
      ( ord_le1268244103169919719at_nat(aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,A3),B3),C2)
    <=> ord_le1268244103169919719at_nat(A3,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,B3),C2)) ) ).

% Diff_subset_conv
tff(fact_4704_Diff__subset__conv,axiom,
    ! [A3: set_nat,B3: 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,A3),B3)),C2)
    <=> aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A3),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,B3),C2)) ) ).

% Diff_subset_conv
tff(fact_4705_Diff__subset__conv,axiom,
    ! [A3: set_int,B3: set_int,C2: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A3),B3)),C2)
    <=> aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A3),aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,B3),C2)) ) ).

% Diff_subset_conv
tff(fact_4706_pochhammer__pos,axiom,
    ! [X: code_integer,N2: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),X)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(comm_s8582702949713902594nteger(X),N2)) ) ).

% pochhammer_pos
tff(fact_4707_pochhammer__pos,axiom,
    ! [X: rat,N2: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),X)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_nat_rat(comm_s4028243227959126397er_rat(X),N2)) ) ).

% pochhammer_pos
tff(fact_4708_pochhammer__pos,axiom,
    ! [X: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),X)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,X),N2)) ) ).

% pochhammer_pos
tff(fact_4709_pochhammer__pos,axiom,
    ! [X: int,N2: nat] :
      ( 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,zero_zero_int),aa_nat_int(aa_int_fun_nat_int(comm_s4660882817536571857er_int,X),N2)) ) ).

% pochhammer_pos
tff(fact_4710_Un__Diff__Int,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A3),B3)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A3),B3)) = A3 ).

% Un_Diff_Int
tff(fact_4711_Un__Diff__Int,axiom,
    ! [A3: set_Pr8693737435421807431at_nat,B3: set_Pr8693737435421807431at_nat] : aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(minus_8321449233255521966at_nat,A3),B3)),aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(inf_in4302113700860409141at_nat,A3),B3)) = A3 ).

% Un_Diff_Int
tff(fact_4712_Un__Diff__Int,axiom,
    ! [A3: set_Pr8551490117392284871at_nat,B3: set_Pr8551490117392284871at_nat] : aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(minus_5060654252129873198at_nat,A3),B3)),aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(inf_in1697001100524423349at_nat,A3),B3)) = A3 ).

% Un_Diff_Int
tff(fact_4713_Un__Diff__Int,axiom,
    ! [A3: set_Pr4329608150637261639at_nat,B3: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,A3),B3)),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,A3),B3)) = A3 ).

% Un_Diff_Int
tff(fact_4714_Un__Diff__Int,axiom,
    ! [A3: set_nat,B3: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A3),B3)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A3),B3)) = A3 ).

% Un_Diff_Int
tff(fact_4715_Int__Diff__Un,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A3),B3)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A3),B3)) = A3 ).

% Int_Diff_Un
tff(fact_4716_Int__Diff__Un,axiom,
    ! [A3: set_Pr8693737435421807431at_nat,B3: set_Pr8693737435421807431at_nat] : aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(inf_in4302113700860409141at_nat,A3),B3)),aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(minus_8321449233255521966at_nat,A3),B3)) = A3 ).

% Int_Diff_Un
tff(fact_4717_Int__Diff__Un,axiom,
    ! [A3: set_Pr8551490117392284871at_nat,B3: set_Pr8551490117392284871at_nat] : aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(inf_in1697001100524423349at_nat,A3),B3)),aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(minus_5060654252129873198at_nat,A3),B3)) = A3 ).

% Int_Diff_Un
tff(fact_4718_Int__Diff__Un,axiom,
    ! [A3: set_Pr4329608150637261639at_nat,B3: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,A3),B3)),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,A3),B3)) = A3 ).

% Int_Diff_Un
tff(fact_4719_Int__Diff__Un,axiom,
    ! [A3: set_nat,B3: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A3),B3)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A3),B3)) = A3 ).

% Int_Diff_Un
tff(fact_4720_Diff__Int,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A3),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B3),C2)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A3),B3)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A3),C2)) ).

% Diff_Int
tff(fact_4721_Diff__Int,axiom,
    ! [A3: set_Pr8693737435421807431at_nat,B3: set_Pr8693737435421807431at_nat,C2: set_Pr8693737435421807431at_nat] : aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(minus_8321449233255521966at_nat,A3),aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(inf_in4302113700860409141at_nat,B3),C2)) = aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(minus_8321449233255521966at_nat,A3),B3)),aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(minus_8321449233255521966at_nat,A3),C2)) ).

% Diff_Int
tff(fact_4722_Diff__Int,axiom,
    ! [A3: set_Pr8551490117392284871at_nat,B3: set_Pr8551490117392284871at_nat,C2: set_Pr8551490117392284871at_nat] : aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(minus_5060654252129873198at_nat,A3),aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(inf_in1697001100524423349at_nat,B3),C2)) = aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(minus_5060654252129873198at_nat,A3),B3)),aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(minus_5060654252129873198at_nat,A3),C2)) ).

% Diff_Int
tff(fact_4723_Diff__Int,axiom,
    ! [A3: set_Pr4329608150637261639at_nat,B3: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,A3),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,B3),C2)) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,A3),B3)),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,A3),C2)) ).

% Diff_Int
tff(fact_4724_Diff__Int,axiom,
    ! [A3: set_nat,B3: set_nat,C2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A3),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B3),C2)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A3),B3)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A3),C2)) ).

% Diff_Int
tff(fact_4725_Diff__Un,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A3),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,B3),C2)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A3),B3)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A3),C2)) ).

% Diff_Un
tff(fact_4726_Diff__Un,axiom,
    ! [A3: set_Pr8693737435421807431at_nat,B3: set_Pr8693737435421807431at_nat,C2: set_Pr8693737435421807431at_nat] : aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(minus_8321449233255521966at_nat,A3),aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,B3),C2)) = aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(inf_in4302113700860409141at_nat,aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(minus_8321449233255521966at_nat,A3),B3)),aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(minus_8321449233255521966at_nat,A3),C2)) ).

% Diff_Un
tff(fact_4727_Diff__Un,axiom,
    ! [A3: set_Pr8551490117392284871at_nat,B3: set_Pr8551490117392284871at_nat,C2: set_Pr8551490117392284871at_nat] : aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(minus_5060654252129873198at_nat,A3),aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,B3),C2)) = aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(inf_in1697001100524423349at_nat,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(minus_5060654252129873198at_nat,A3),B3)),aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(minus_5060654252129873198at_nat,A3),C2)) ).

% Diff_Un
tff(fact_4728_Diff__Un,axiom,
    ! [A3: set_Pr4329608150637261639at_nat,B3: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,A3),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,B3),C2)) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,A3),B3)),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,A3),C2)) ).

% Diff_Un
tff(fact_4729_Diff__Un,axiom,
    ! [A3: set_nat,B3: set_nat,C2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A3),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,B3),C2)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A3),B3)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A3),C2)) ).

% Diff_Un
tff(fact_4730_pochhammer__neq__0__mono,axiom,
    ! [A: rat,M: nat,N2: nat] :
      ( ( aa_nat_rat(comm_s4028243227959126397er_rat(A),M) != zero_zero_rat )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),M)
       => ( aa_nat_rat(comm_s4028243227959126397er_rat(A),N2) != zero_zero_rat ) ) ) ).

% pochhammer_neq_0_mono
tff(fact_4731_pochhammer__eq__0__mono,axiom,
    ! [A: rat,N2: nat,M: nat] :
      ( ( aa_nat_rat(comm_s4028243227959126397er_rat(A),N2) = zero_zero_rat )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),M)
       => ( aa_nat_rat(comm_s4028243227959126397er_rat(A),M) = zero_zero_rat ) ) ) ).

% pochhammer_eq_0_mono
tff(fact_4732_zdiff__int__split,axiom,
    ! [Pa: fun_int_o,X: nat,Y: nat] :
      ( aa_int_o(Pa,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,X),Y)))
    <=> ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),X)
         => aa_int_o(Pa,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(semiri1314217659103216013at_int,X)),aa_nat_int(semiri1314217659103216013at_int,Y))) )
        & ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y)
         => aa_int_o(Pa,zero_zero_int) ) ) ) ).

% zdiff_int_split
tff(fact_4733_pochhammer__nonneg,axiom,
    ! [X: code_integer,N2: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),X)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(comm_s8582702949713902594nteger(X),N2)) ) ).

% pochhammer_nonneg
tff(fact_4734_pochhammer__nonneg,axiom,
    ! [X: rat,N2: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),X)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(comm_s4028243227959126397er_rat(X),N2)) ) ).

% pochhammer_nonneg
tff(fact_4735_pochhammer__nonneg,axiom,
    ! [X: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),X)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,X),N2)) ) ).

% pochhammer_nonneg
tff(fact_4736_pochhammer__nonneg,axiom,
    ! [X: int,N2: nat] :
      ( 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_eq_int,zero_zero_int),aa_nat_int(aa_int_fun_nat_int(comm_s4660882817536571857er_int,X),N2)) ) ).

% pochhammer_nonneg
tff(fact_4737_disjoint__alt__simp3,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le7866589430770878221at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A3),B3)),A3)
    <=> ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A3),B3) != bot_bo2099793752762293965at_nat ) ) ).

% disjoint_alt_simp3
tff(fact_4738_disjoint__alt__simp3,axiom,
    ! [A3: set_o,B3: set_o] :
      ( ord_less_set_o(minus_minus_set_o(A3,B3),A3)
    <=> ( inf_inf_set_o(A3,B3) != bot_bot_set_o ) ) ).

% disjoint_alt_simp3
tff(fact_4739_disjoint__alt__simp3,axiom,
    ! [A3: set_int,B3: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_set_int,aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A3),B3)),A3)
    <=> ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A3),B3) != bot_bot_set_int ) ) ).

% disjoint_alt_simp3
tff(fact_4740_disjoint__alt__simp3,axiom,
    ! [A3: set_nat,B3: 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,A3),B3)),A3)
    <=> ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A3),B3) != bot_bot_set_nat ) ) ).

% disjoint_alt_simp3
tff(fact_4741_pochhammer__rec,axiom,
    ! [A: code_integer,N2: nat] : aa_nat_Code_integer(comm_s8582702949713902594nteger(A),aa_nat_nat(suc,N2)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),aa_nat_Code_integer(comm_s8582702949713902594nteger(aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(A),one_one_Code_integer)),N2)) ).

% pochhammer_rec
tff(fact_4742_pochhammer__rec,axiom,
    ! [A: rat,N2: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(A),aa_nat_nat(suc,N2)) = 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)),N2)) ).

% pochhammer_rec
tff(fact_4743_pochhammer__rec,axiom,
    ! [A: nat,N2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,A),aa_nat_nat(suc,N2)) = 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)),N2)) ).

% pochhammer_rec
tff(fact_4744_pochhammer__rec,axiom,
    ! [A: int,N2: nat] : aa_nat_int(aa_int_fun_nat_int(comm_s4660882817536571857er_int,A),aa_nat_nat(suc,N2)) = 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)),N2)) ).

% pochhammer_rec
tff(fact_4745_pochhammer__rec_H,axiom,
    ! [Z: rat,N2: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(Z),aa_nat_nat(suc,N2)) = 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,N2))),aa_nat_rat(comm_s4028243227959126397er_rat(Z),N2)) ).

% pochhammer_rec'
tff(fact_4746_pochhammer__rec_H,axiom,
    ! [Z: int,N2: nat] : aa_nat_int(aa_int_fun_nat_int(comm_s4660882817536571857er_int,Z),aa_nat_nat(suc,N2)) = 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,N2))),aa_nat_int(aa_int_fun_nat_int(comm_s4660882817536571857er_int,Z),N2)) ).

% pochhammer_rec'
tff(fact_4747_pochhammer__rec_H,axiom,
    ! [Z: nat,N2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,Z),aa_nat_nat(suc,N2)) = 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,N2))),aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,Z),N2)) ).

% pochhammer_rec'
tff(fact_4748_pochhammer__rec_H,axiom,
    ! [Z: code_integer,N2: nat] : aa_nat_Code_integer(comm_s8582702949713902594nteger(Z),aa_nat_nat(suc,N2)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(Z),aa_nat_Code_integer(semiri4939895301339042750nteger,N2))),aa_nat_Code_integer(comm_s8582702949713902594nteger(Z),N2)) ).

% pochhammer_rec'
tff(fact_4749_pochhammer__Suc,axiom,
    ! [A: rat,N2: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(A),aa_nat_nat(suc,N2)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(comm_s4028243227959126397er_rat(A),N2)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),aa_nat_rat(semiri681578069525770553at_rat,N2))) ).

% pochhammer_Suc
tff(fact_4750_pochhammer__Suc,axiom,
    ! [A: int,N2: nat] : aa_nat_int(aa_int_fun_nat_int(comm_s4660882817536571857er_int,A),aa_nat_nat(suc,N2)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(aa_int_fun_nat_int(comm_s4660882817536571857er_int,A),N2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_nat_int(semiri1314217659103216013at_int,N2))) ).

% pochhammer_Suc
tff(fact_4751_pochhammer__Suc,axiom,
    ! [A: nat,N2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,A),aa_nat_nat(suc,N2)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,A),N2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(semiri1316708129612266289at_nat,N2))) ).

% pochhammer_Suc
tff(fact_4752_pochhammer__Suc,axiom,
    ! [A: code_integer,N2: nat] : aa_nat_Code_integer(comm_s8582702949713902594nteger(A),aa_nat_nat(suc,N2)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(comm_s8582702949713902594nteger(A),N2)),aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(A),aa_nat_Code_integer(semiri4939895301339042750nteger,N2))) ).

% pochhammer_Suc
tff(fact_4753_pochhammer__product_H,axiom,
    ! [Z: rat,N2: nat,M: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(Z),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),M)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(comm_s4028243227959126397er_rat(Z),N2)),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Z),aa_nat_rat(semiri681578069525770553at_rat,N2))),M)) ).

% pochhammer_product'
tff(fact_4754_pochhammer__product_H,axiom,
    ! [Z: int,N2: 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,N2),M)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(aa_int_fun_nat_int(comm_s4660882817536571857er_int,Z),N2)),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,N2))),M)) ).

% pochhammer_product'
tff(fact_4755_pochhammer__product_H,axiom,
    ! [Z: nat,N2: 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,N2),M)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,Z),N2)),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,N2))),M)) ).

% pochhammer_product'
tff(fact_4756_pochhammer__product_H,axiom,
    ! [Z: code_integer,N2: nat,M: nat] : aa_nat_Code_integer(comm_s8582702949713902594nteger(Z),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),M)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(comm_s8582702949713902594nteger(Z),N2)),aa_nat_Code_integer(comm_s8582702949713902594nteger(aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(Z),aa_nat_Code_integer(semiri4939895301339042750nteger,N2))),M)) ).

% pochhammer_product'
tff(fact_4757_pochhammer__product,axiom,
    ! [M: nat,N2: nat,Z: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
     => ( aa_nat_rat(comm_s4028243227959126397er_rat(Z),N2) = 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,N2),M))) ) ) ).

% pochhammer_product
tff(fact_4758_pochhammer__product,axiom,
    ! [M: nat,N2: nat,Z: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
     => ( aa_nat_int(aa_int_fun_nat_int(comm_s4660882817536571857er_int,Z),N2) = 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,N2),M))) ) ) ).

% pochhammer_product
tff(fact_4759_pochhammer__product,axiom,
    ! [M: nat,N2: nat,Z: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,Z),N2) = 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,N2),M))) ) ) ).

% pochhammer_product
tff(fact_4760_pochhammer__product,axiom,
    ! [M: nat,N2: nat,Z: code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
     => ( aa_nat_Code_integer(comm_s8582702949713902594nteger(Z),N2) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(comm_s8582702949713902594nteger(Z),M)),aa_nat_Code_integer(comm_s8582702949713902594nteger(aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(Z),aa_nat_Code_integer(semiri4939895301339042750nteger,M))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),M))) ) ) ).

% pochhammer_product
tff(fact_4761_real__arch__simple,axiom,
    ! [X: rat] :
    ? [N5: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),aa_nat_rat(semiri681578069525770553at_rat,N5)) ).

% real_arch_simple
tff(fact_4762_reals__Archimedean2,axiom,
    ! [X: rat] :
    ? [N5: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),aa_nat_rat(semiri681578069525770553at_rat,N5)) ).

% reals_Archimedean2
tff(fact_4763_zle__diff1__eq,axiom,
    ! [W2: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,W2),aa_int_int(aa_int_fun_int_int(minus_minus_int,Z),one_one_int))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,W2),Z) ) ).

% zle_diff1_eq
tff(fact_4764_zle__add1__eq__le,axiom,
    ! [W2: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,W2),aa_int_int(aa_int_fun_int_int(plus_plus_int,Z),one_one_int))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,W2),Z) ) ).

% zle_add1_eq_le
tff(fact_4765_zero__less__imp__eq__int,axiom,
    ! [K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),K)
     => ? [N5: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N5)
          & ( K = aa_nat_int(semiri1314217659103216013at_int,N5) ) ) ) ).

% zero_less_imp_eq_int
tff(fact_4766_pos__int__cases,axiom,
    ! [K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),K)
     => ~ ! [N5: nat] :
            ( ( K = aa_nat_int(semiri1314217659103216013at_int,N5) )
           => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N5) ) ) ).

% pos_int_cases
tff(fact_4767_zmult__zless__mono2__lemma,axiom,
    ! [I2: int,J: int,K: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,I2),J)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,K)),I2)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,K)),J)) ) ) ).

% zmult_zless_mono2_lemma
tff(fact_4768_nonneg__int__cases,axiom,
    ! [K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K)
     => ~ ! [N5: nat] : K != aa_nat_int(semiri1314217659103216013at_int,N5) ) ).

% nonneg_int_cases
tff(fact_4769_zero__le__imp__eq__int,axiom,
    ! [K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K)
     => ? [N5: nat] : K = aa_nat_int(semiri1314217659103216013at_int,N5) ) ).

% zero_le_imp_eq_int
tff(fact_4770_zle__int,axiom,
    ! [M: nat,N2: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,N2))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2) ) ).

% zle_int
tff(fact_4771_Diff__idemp,axiom,
    ! [A3: set_nat,B3: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A3),B3)),B3) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A3),B3) ).

% Diff_idemp
tff(fact_4772_Diff__iff,axiom,
    ! [Ca: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Ca),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A3),B3))
    <=> ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Ca),A3)
        & ~ aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Ca),B3) ) ) ).

% Diff_iff
tff(fact_4773_Diff__iff,axiom,
    ! [Ca: $o,A3: set_o,B3: set_o] :
      ( member_o((Ca),minus_minus_set_o(A3,B3))
    <=> ( member_o((Ca),A3)
        & ~ member_o((Ca),B3) ) ) ).

% Diff_iff
tff(fact_4774_Diff__iff,axiom,
    ! [Ca: set_nat,A3: set_set_nat,B3: set_set_nat] :
      ( aa_set_set_nat_o(member_set_nat(Ca),minus_2163939370556025621et_nat(A3,B3))
    <=> ( aa_set_set_nat_o(member_set_nat(Ca),A3)
        & ~ aa_set_set_nat_o(member_set_nat(Ca),B3) ) ) ).

% Diff_iff
tff(fact_4775_Diff__iff,axiom,
    ! [Ca: int,A3: set_int,B3: set_int] :
      ( aa_set_int_o(member_int(Ca),aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A3),B3))
    <=> ( aa_set_int_o(member_int(Ca),A3)
        & ~ aa_set_int_o(member_int(Ca),B3) ) ) ).

% Diff_iff
tff(fact_4776_Diff__iff,axiom,
    ! [Ca: nat,A3: set_nat,B3: set_nat] :
      ( aa_set_nat_o(member_nat(Ca),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A3),B3))
    <=> ( aa_set_nat_o(member_nat(Ca),A3)
        & ~ aa_set_nat_o(member_nat(Ca),B3) ) ) ).

% Diff_iff
tff(fact_4777_DiffI,axiom,
    ! [Ca: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Ca),A3)
     => ( ~ aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Ca),B3)
       => aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Ca),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A3),B3)) ) ) ).

% DiffI
tff(fact_4778_DiffI,axiom,
    ! [Ca: $o,A3: set_o,B3: set_o] :
      ( member_o((Ca),A3)
     => ( ~ member_o((Ca),B3)
       => member_o((Ca),minus_minus_set_o(A3,B3)) ) ) ).

% DiffI
tff(fact_4779_DiffI,axiom,
    ! [Ca: set_nat,A3: set_set_nat,B3: set_set_nat] :
      ( aa_set_set_nat_o(member_set_nat(Ca),A3)
     => ( ~ aa_set_set_nat_o(member_set_nat(Ca),B3)
       => aa_set_set_nat_o(member_set_nat(Ca),minus_2163939370556025621et_nat(A3,B3)) ) ) ).

% DiffI
tff(fact_4780_DiffI,axiom,
    ! [Ca: int,A3: set_int,B3: set_int] :
      ( aa_set_int_o(member_int(Ca),A3)
     => ( ~ aa_set_int_o(member_int(Ca),B3)
       => aa_set_int_o(member_int(Ca),aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A3),B3)) ) ) ).

% DiffI
tff(fact_4781_DiffI,axiom,
    ! [Ca: nat,A3: set_nat,B3: set_nat] :
      ( aa_set_nat_o(member_nat(Ca),A3)
     => ( ~ aa_set_nat_o(member_nat(Ca),B3)
       => aa_set_nat_o(member_nat(Ca),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A3),B3)) ) ) ).

% DiffI
tff(fact_4782_minus__set__def,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A3),B3) = collec3392354462482085612at_nat(aa_fun359341089595923442_nat_o(minus_2386458686304230003_nat_o(aTP_Lamm_a(A3)),aTP_Lamm_a(B3))) ).

% minus_set_def
tff(fact_4783_minus__set__def,axiom,
    ! [A3: set_o,B3: set_o] : minus_minus_set_o(A3,B3) = collect_o(minus_minus_fun_o_o(aTP_Lamm_aa(A3),aTP_Lamm_aa(B3))) ).

% minus_set_def
tff(fact_4784_minus__set__def,axiom,
    ! [A3: set_Pr958786334691620121nt_int,B3: set_Pr958786334691620121nt_int] : minus_1052850069191792384nt_int(A3,B3) = collec213857154873943460nt_int(minus_827889751674334651_int_o(aTP_Lamm_ab(A3),aTP_Lamm_ab(B3))) ).

% minus_set_def
tff(fact_4785_minus__set__def,axiom,
    ! [A3: set_list_nat,B3: set_list_nat] : minus_7954133019191499631st_nat(A3,B3) = collect_list_nat(minus_5761417318614613624_nat_o(aTP_Lamm_ac(A3),aTP_Lamm_ac(B3))) ).

% minus_set_def
tff(fact_4786_minus__set__def,axiom,
    ! [A3: set_set_nat,B3: set_set_nat] : minus_2163939370556025621et_nat(A3,B3) = collect_set_nat(minus_6087510627544308050_nat_o(aTP_Lamm_ad(A3),aTP_Lamm_ad(B3))) ).

% minus_set_def
tff(fact_4787_minus__set__def,axiom,
    ! [A3: set_int,B3: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A3),B3) = collect_int(minus_7151557278095657260_int_o(aTP_Lamm_af(A3),aTP_Lamm_af(B3))) ).

% minus_set_def
tff(fact_4788_minus__set__def,axiom,
    ! [A3: set_nat,B3: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A3),B3) = collect_nat(minus_8972599623528345224_nat_o(aTP_Lamm_ae(A3),aTP_Lamm_ae(B3))) ).

% minus_set_def
tff(fact_4789_DiffD2,axiom,
    ! [Ca: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Ca),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A3),B3))
     => ~ aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Ca),B3) ) ).

% DiffD2
tff(fact_4790_DiffD2,axiom,
    ! [Ca: $o,A3: set_o,B3: set_o] :
      ( member_o((Ca),minus_minus_set_o(A3,B3))
     => ~ member_o((Ca),B3) ) ).

% DiffD2
tff(fact_4791_DiffD2,axiom,
    ! [Ca: set_nat,A3: set_set_nat,B3: set_set_nat] :
      ( aa_set_set_nat_o(member_set_nat(Ca),minus_2163939370556025621et_nat(A3,B3))
     => ~ aa_set_set_nat_o(member_set_nat(Ca),B3) ) ).

% DiffD2
tff(fact_4792_DiffD2,axiom,
    ! [Ca: int,A3: set_int,B3: set_int] :
      ( aa_set_int_o(member_int(Ca),aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A3),B3))
     => ~ aa_set_int_o(member_int(Ca),B3) ) ).

% DiffD2
tff(fact_4793_DiffD2,axiom,
    ! [Ca: nat,A3: set_nat,B3: set_nat] :
      ( aa_set_nat_o(member_nat(Ca),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A3),B3))
     => ~ aa_set_nat_o(member_nat(Ca),B3) ) ).

% DiffD2
tff(fact_4794_DiffD1,axiom,
    ! [Ca: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Ca),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A3),B3))
     => aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Ca),A3) ) ).

% DiffD1
tff(fact_4795_DiffD1,axiom,
    ! [Ca: $o,A3: set_o,B3: set_o] :
      ( member_o((Ca),minus_minus_set_o(A3,B3))
     => member_o((Ca),A3) ) ).

% DiffD1
tff(fact_4796_DiffD1,axiom,
    ! [Ca: set_nat,A3: set_set_nat,B3: set_set_nat] :
      ( aa_set_set_nat_o(member_set_nat(Ca),minus_2163939370556025621et_nat(A3,B3))
     => aa_set_set_nat_o(member_set_nat(Ca),A3) ) ).

% DiffD1
tff(fact_4797_DiffD1,axiom,
    ! [Ca: int,A3: set_int,B3: set_int] :
      ( aa_set_int_o(member_int(Ca),aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A3),B3))
     => aa_set_int_o(member_int(Ca),A3) ) ).

% DiffD1
tff(fact_4798_DiffD1,axiom,
    ! [Ca: nat,A3: set_nat,B3: set_nat] :
      ( aa_set_nat_o(member_nat(Ca),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A3),B3))
     => aa_set_nat_o(member_nat(Ca),A3) ) ).

% DiffD1
tff(fact_4799_DiffE,axiom,
    ! [Ca: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Ca),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A3),B3))
     => ~ ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Ca),A3)
         => aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Ca),B3) ) ) ).

% DiffE
tff(fact_4800_DiffE,axiom,
    ! [Ca: $o,A3: set_o,B3: set_o] :
      ( member_o((Ca),minus_minus_set_o(A3,B3))
     => ~ ( member_o((Ca),A3)
         => member_o((Ca),B3) ) ) ).

% DiffE
tff(fact_4801_DiffE,axiom,
    ! [Ca: set_nat,A3: set_set_nat,B3: set_set_nat] :
      ( aa_set_set_nat_o(member_set_nat(Ca),minus_2163939370556025621et_nat(A3,B3))
     => ~ ( aa_set_set_nat_o(member_set_nat(Ca),A3)
         => aa_set_set_nat_o(member_set_nat(Ca),B3) ) ) ).

% DiffE
tff(fact_4802_DiffE,axiom,
    ! [Ca: int,A3: set_int,B3: set_int] :
      ( aa_set_int_o(member_int(Ca),aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A3),B3))
     => ~ ( aa_set_int_o(member_int(Ca),A3)
         => aa_set_int_o(member_int(Ca),B3) ) ) ).

% DiffE
tff(fact_4803_DiffE,axiom,
    ! [Ca: nat,A3: set_nat,B3: set_nat] :
      ( aa_set_nat_o(member_nat(Ca),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A3),B3))
     => ~ ( aa_set_nat_o(member_nat(Ca),A3)
         => aa_set_nat_o(member_nat(Ca),B3) ) ) ).

% DiffE
tff(fact_4804_set__diff__eq,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A3),B3) = collec3392354462482085612at_nat(aa_set1857767853180004543_nat_o(aTP_Lamm_gk(A3),B3)) ).

% set_diff_eq
tff(fact_4805_set__diff__eq,axiom,
    ! [A3: set_o,B3: set_o] : minus_minus_set_o(A3,B3) = collect_o(aa_set_o_fun_o_o(aTP_Lamm_gl(A3),B3)) ).

% set_diff_eq
tff(fact_4806_set__diff__eq,axiom,
    ! [A3: set_Pr958786334691620121nt_int,B3: set_Pr958786334691620121nt_int] : minus_1052850069191792384nt_int(A3,B3) = collec213857154873943460nt_int(aa_set2224860389584585407_int_o(aTP_Lamm_gm(A3),B3)) ).

% set_diff_eq
tff(fact_4807_set__diff__eq,axiom,
    ! [A3: set_list_nat,B3: set_list_nat] : minus_7954133019191499631st_nat(A3,B3) = collect_list_nat(aa_set6675653781276780861_nat_o(aTP_Lamm_gn(A3),B3)) ).

% set_diff_eq
tff(fact_4808_set__diff__eq,axiom,
    ! [A3: set_set_nat,B3: set_set_nat] : minus_2163939370556025621et_nat(A3,B3) = collect_set_nat(aa_set4404905041378927473_nat_o(aTP_Lamm_go(A3),B3)) ).

% set_diff_eq
tff(fact_4809_set__diff__eq,axiom,
    ! [A3: set_int,B3: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A3),B3) = collect_int(aa_set_int_fun_int_o(aTP_Lamm_gp(A3),B3)) ).

% set_diff_eq
tff(fact_4810_set__diff__eq,axiom,
    ! [A3: set_nat,B3: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A3),B3) = collect_nat(aa_set_nat_fun_nat_o(aTP_Lamm_gq(A3),B3)) ).

% set_diff_eq
tff(fact_4811_int__ge__induct,axiom,
    ! [K: int,I2: int,Pa: fun_int_o] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),I2)
     => ( aa_int_o(Pa,K)
       => ( ! [I3: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),I3)
             => ( aa_int_o(Pa,I3)
               => aa_int_o(Pa,aa_int_int(aa_int_fun_int_int(plus_plus_int,I3),one_one_int)) ) )
         => aa_int_o(Pa,I2) ) ) ) ).

% int_ge_induct
tff(fact_4812_int__le__induct,axiom,
    ! [I2: int,K: int,Pa: fun_int_o] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I2),K)
     => ( aa_int_o(Pa,K)
       => ( ! [I3: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I3),K)
             => ( aa_int_o(Pa,I3)
               => aa_int_o(Pa,aa_int_int(aa_int_fun_int_int(minus_minus_int,I3),one_one_int)) ) )
         => aa_int_o(Pa,I2) ) ) ) ).

% int_le_induct
tff(fact_4813_int__induct,axiom,
    ! [Pa: fun_int_o,K: int,I2: int] :
      ( aa_int_o(Pa,K)
     => ( ! [I3: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),I3)
           => ( aa_int_o(Pa,I3)
             => aa_int_o(Pa,aa_int_int(aa_int_fun_int_int(plus_plus_int,I3),one_one_int)) ) )
       => ( ! [I3: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I3),K)
             => ( aa_int_o(Pa,I3)
               => aa_int_o(Pa,aa_int_int(aa_int_fun_int_int(minus_minus_int,I3),one_one_int)) ) )
         => aa_int_o(Pa,I2) ) ) ) ).

% int_induct
tff(fact_4814_int__gr__induct,axiom,
    ! [K: int,I2: int,Pa: fun_int_o] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),I2)
     => ( aa_int_o(Pa,aa_int_int(aa_int_fun_int_int(plus_plus_int,K),one_one_int))
       => ( ! [I3: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),I3)
             => ( aa_int_o(Pa,I3)
               => aa_int_o(Pa,aa_int_int(aa_int_fun_int_int(plus_plus_int,I3),one_one_int)) ) )
         => aa_int_o(Pa,I2) ) ) ) ).

% int_gr_induct
tff(fact_4815_zless__add1__eq,axiom,
    ! [W2: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,W2),aa_int_int(aa_int_fun_int_int(plus_plus_int,Z),one_one_int))
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,W2),Z)
        | ( W2 = Z ) ) ) ).

% zless_add1_eq
tff(fact_4816_int__less__induct,axiom,
    ! [I2: int,K: int,Pa: fun_int_o] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,I2),K)
     => ( aa_int_o(Pa,aa_int_int(aa_int_fun_int_int(minus_minus_int,K),one_one_int))
       => ( ! [I3: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,I3),K)
             => ( aa_int_o(Pa,I3)
               => aa_int_o(Pa,aa_int_int(aa_int_fun_int_int(minus_minus_int,I3),one_one_int)) ) )
         => aa_int_o(Pa,I2) ) ) ) ).

% int_less_induct
tff(fact_4817_less__eq__int__code_I1_J,axiom,
    aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),zero_zero_int) ).

% less_eq_int_code(1)
tff(fact_4818_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_4819_less__int__code_I1_J,axiom,
    ~ aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),zero_zero_int) ).

% less_int_code(1)
tff(fact_4820_zmult__zless__mono2,axiom,
    ! [I2: int,J: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,I2),J)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),K)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,K),I2)),aa_int_int(aa_int_fun_int_int(times_times_int,K),J)) ) ) ).

% zmult_zless_mono2
tff(fact_4821_pos__zmult__eq__1__iff,axiom,
    ! [M: int,N2: 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),N2) = one_one_int )
      <=> ( ( M = one_one_int )
          & ( N2 = one_one_int ) ) ) ) ).

% pos_zmult_eq_1_iff
tff(fact_4822_add1__zle__eq,axiom,
    ! [W2: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,W2),one_one_int)),Z)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,W2),Z) ) ).

% add1_zle_eq
tff(fact_4823_zless__imp__add1__zle,axiom,
    ! [W2: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,W2),Z)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,W2),one_one_int)),Z) ) ).

% zless_imp_add1_zle
tff(fact_4824_zle__iff__zadd,axiom,
    ! [W2: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,W2),Z)
    <=> ? [N: nat] : Z = aa_int_int(aa_int_fun_int_int(plus_plus_int,W2),aa_nat_int(semiri1314217659103216013at_int,N)) ) ).

% zle_iff_zadd
tff(fact_4825_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_4826_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_4827_zless__iff__Suc__zadd,axiom,
    ! [W2: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,W2),Z)
    <=> ? [N: nat] : Z = aa_int_int(aa_int_fun_int_int(plus_plus_int,W2),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,N))) ) ).

% zless_iff_Suc_zadd
tff(fact_4828_nat0__intermed__int__val,axiom,
    ! [N2: nat,F: fun_nat_int,K: int] :
      ( ! [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),N2)
         => 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,I3),one_one_nat))),aa_nat_int(F,I3)))),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,N2))
         => ? [I3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I3),N2)
              & ( aa_nat_int(F,I3) = K ) ) ) ) ) ).

% nat0_intermed_int_val
tff(fact_4829_eucl__rel__int__iff,axiom,
    ! [K: int,L: int,Q4: int,Ra2: int] :
      ( aa_Pro4592068669816069595_int_o(eucl_rel_int(K,L),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q4),Ra2))
    <=> ( ( K = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,L),Q4)),Ra2) )
        & $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),Ra2)
            & aa_int_o(aa_int_fun_int_o(ord_less_int,Ra2),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),Ra2)
              & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ra2),zero_zero_int) ),
              Q4 = zero_zero_int ) ) ) ) ).

% eucl_rel_int_iff
tff(fact_4830_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_4831_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_4832_pochhammer__minus_H,axiom,
    ! [B: code_integer,K: nat] : aa_nat_Code_integer(comm_s8582702949713902594nteger(aa_Cod1783974652533396728nteger(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_4833_boolean__algebra__class_Oboolean__algebra_Odouble__compl,axiom,
    ! [X: assn] : aa_assn_assn(uminus_uminus_assn,aa_assn_assn(uminus_uminus_assn,X)) = X ).

% boolean_algebra_class.boolean_algebra.double_compl
tff(fact_4834_boolean__algebra__class_Oboolean__algebra_Odouble__compl,axiom,
    ! [X: product_unit] : uminus2952777764628376836t_unit(uminus2952777764628376836t_unit(X)) = X ).

% boolean_algebra_class.boolean_algebra.double_compl
tff(fact_4835_boolean__algebra__class_Oboolean__algebra_Ocompl__eq__compl__iff,axiom,
    ! [X: assn,Y: assn] :
      ( ( aa_assn_assn(uminus_uminus_assn,X) = aa_assn_assn(uminus_uminus_assn,Y) )
    <=> ( X = Y ) ) ).

% boolean_algebra_class.boolean_algebra.compl_eq_compl_iff
tff(fact_4836_boolean__algebra__class_Oboolean__algebra_Ocompl__eq__compl__iff,axiom,
    ! [X: product_unit,Y: product_unit] :
      ( ( uminus2952777764628376836t_unit(X) = uminus2952777764628376836t_unit(Y) )
    <=> ( X = Y ) ) ).

% boolean_algebra_class.boolean_algebra.compl_eq_compl_iff
tff(fact_4837_Compl__anti__mono,axiom,
    ! [A3: set_int,B3: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A3),B3)
     => aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,uminus1532241313380277803et_int(B3)),uminus1532241313380277803et_int(A3)) ) ).

% Compl_anti_mono
tff(fact_4838_Compl__subset__Compl__iff,axiom,
    ! [A3: set_int,B3: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,uminus1532241313380277803et_int(A3)),uminus1532241313380277803et_int(B3))
    <=> aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,B3),A3) ) ).

% Compl_subset_Compl_iff
tff(fact_4839_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_4840_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_4841_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_4842_compl__le__compl__iff,axiom,
    ! [X: assn,Y: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_assn_assn(uminus_uminus_assn,X)),aa_assn_assn(uminus_uminus_assn,Y))
    <=> aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,Y),X) ) ).

% compl_le_compl_iff
tff(fact_4843_compl__le__compl__iff,axiom,
    ! [X: product_unit,Y: product_unit] :
      ( aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,uminus2952777764628376836t_unit(X)),uminus2952777764628376836t_unit(Y))
    <=> aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,Y),X) ) ).

% compl_le_compl_iff
tff(fact_4844_compl__le__compl__iff,axiom,
    ! [X: set_int,Y: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,uminus1532241313380277803et_int(X)),uminus1532241313380277803et_int(Y))
    <=> aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,Y),X) ) ).

% compl_le_compl_iff
tff(fact_4845_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_4846_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_4847_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_4848_compl__less__compl__iff,axiom,
    ! [X: assn,Y: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_assn_assn(uminus_uminus_assn,X)),aa_assn_assn(uminus_uminus_assn,Y))
    <=> aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,Y),X) ) ).

% compl_less_compl_iff
tff(fact_4849_compl__less__compl__iff,axiom,
    ! [X: product_unit,Y: product_unit] :
      ( aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le361264281704409273t_unit,uminus2952777764628376836t_unit(X)),uminus2952777764628376836t_unit(Y))
    <=> aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le361264281704409273t_unit,Y),X) ) ).

% compl_less_compl_iff
tff(fact_4850_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_4851_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_4852_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_4853_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_4854_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_4855_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_4856_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_4857_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_4858_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_4859_Compl__disjoint2,axiom,
    ! [A3: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,uminus6524753893492686040at_nat(A3)),A3) = bot_bo2099793752762293965at_nat ).

% Compl_disjoint2
tff(fact_4860_Compl__disjoint2,axiom,
    ! [A3: set_o] : inf_inf_set_o(uminus_uminus_set_o(A3),A3) = bot_bot_set_o ).

% Compl_disjoint2
tff(fact_4861_Compl__disjoint2,axiom,
    ! [A3: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,uminus5710092332889474511et_nat(A3)),A3) = bot_bot_set_nat ).

% Compl_disjoint2
tff(fact_4862_Compl__disjoint2,axiom,
    ! [A3: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,uminus1532241313380277803et_int(A3)),A3) = bot_bot_set_int ).

% Compl_disjoint2
tff(fact_4863_Compl__disjoint,axiom,
    ! [A3: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A3),uminus6524753893492686040at_nat(A3)) = bot_bo2099793752762293965at_nat ).

% Compl_disjoint
tff(fact_4864_Compl__disjoint,axiom,
    ! [A3: set_o] : inf_inf_set_o(A3,uminus_uminus_set_o(A3)) = bot_bot_set_o ).

% Compl_disjoint
tff(fact_4865_Compl__disjoint,axiom,
    ! [A3: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A3),uminus5710092332889474511et_nat(A3)) = bot_bot_set_nat ).

% Compl_disjoint
tff(fact_4866_Compl__disjoint,axiom,
    ! [A3: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A3),uminus1532241313380277803et_int(A3)) = bot_bot_set_int ).

% Compl_disjoint
tff(fact_4867_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_4868_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_4869_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_4870_inter__compl__diff__conv,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A3),uminus6524753893492686040at_nat(B3)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A3),B3) ).

% inter_compl_diff_conv
tff(fact_4871_inter__compl__diff__conv,axiom,
    ! [A3: set_nat,B3: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A3),uminus5710092332889474511et_nat(B3)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A3),B3) ).

% inter_compl_diff_conv
tff(fact_4872_Diff__Compl,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A3),uminus6524753893492686040at_nat(B3)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A3),B3) ).

% Diff_Compl
tff(fact_4873_Diff__Compl,axiom,
    ! [A3: set_nat,B3: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A3),uminus5710092332889474511et_nat(B3)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A3),B3) ).

% Diff_Compl
tff(fact_4874_abs__of__nat,axiom,
    ! [N2: nat] : aa_rat_rat(abs_abs_rat,aa_nat_rat(semiri681578069525770553at_rat,N2)) = aa_nat_rat(semiri681578069525770553at_rat,N2) ).

% abs_of_nat
tff(fact_4875_abs__of__nat,axiom,
    ! [N2: nat] : aa_int_int(abs_abs_int,aa_nat_int(semiri1314217659103216013at_int,N2)) = aa_nat_int(semiri1314217659103216013at_int,N2) ).

% abs_of_nat
tff(fact_4876_abs__of__nat,axiom,
    ! [N2: nat] : aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_nat_Code_integer(semiri4939895301339042750nteger,N2)) = aa_nat_Code_integer(semiri4939895301339042750nteger,N2) ).

% abs_of_nat
tff(fact_4877_Compl__Diff__eq,axiom,
    ! [A3: set_Pr8693737435421807431at_nat,B3: set_Pr8693737435421807431at_nat] : uminus4384627049435823934at_nat(aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(minus_8321449233255521966at_nat,A3),B3)) = aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,uminus4384627049435823934at_nat(A3)),B3) ).

% Compl_Diff_eq
tff(fact_4878_Compl__Diff__eq,axiom,
    ! [A3: set_Pr8551490117392284871at_nat,B3: set_Pr8551490117392284871at_nat] : uminus2330091110623919550at_nat(aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(minus_5060654252129873198at_nat,A3),B3)) = aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,uminus2330091110623919550at_nat(A3)),B3) ).

% Compl_Diff_eq
tff(fact_4879_Compl__Diff__eq,axiom,
    ! [A3: set_Pr4329608150637261639at_nat,B3: set_Pr4329608150637261639at_nat] : uminus935396558254630718at_nat(aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,A3),B3)) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,uminus935396558254630718at_nat(A3)),B3) ).

% Compl_Diff_eq
tff(fact_4880_Compl__Diff__eq,axiom,
    ! [A3: set_nat,B3: set_nat] : uminus5710092332889474511et_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A3),B3)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,uminus5710092332889474511et_nat(A3)),B3) ).

% Compl_Diff_eq
tff(fact_4881_negative__zle,axiom,
    ! [N2: 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,N2))),aa_nat_int(semiri1314217659103216013at_int,M)) ).

% negative_zle
tff(fact_4882_mod__not__dist,axiom,
    ! [Pa: assn,Ha: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(rep_assn(aa_assn_assn(uminus_uminus_assn,Pa)),Ha)
    <=> ( aa_Pro653082453696054616_nat_o(in_range,Ha)
        & ~ aa_Pro653082453696054616_nat_o(rep_assn(Pa),Ha) ) ) ).

% mod_not_dist
tff(fact_4883_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_4884_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_4885_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_4886_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_4887_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_4888_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_4889_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_4890_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_4891_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_4892_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_4893_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_4894_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_4895_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_4896_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_4897_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_4898_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_4899_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_4900_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_4901_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_4902_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_4903_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_4904_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_4905_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_4906_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_4907_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_4908_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_4909_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_4910_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_4911_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_4912_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_4913_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_4914_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_4915_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_4916_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_4917_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_4918_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_4919_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_4920_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_4921_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_4922_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_4923_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_4924_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_4925_boolean__algebra_Oconj__cancel__right,axiom,
    ! [X: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),uminus6524753893492686040at_nat(X)) = bot_bo2099793752762293965at_nat ).

% boolean_algebra.conj_cancel_right
tff(fact_4926_boolean__algebra_Oconj__cancel__right,axiom,
    ! [X: set_o] : inf_inf_set_o(X,uminus_uminus_set_o(X)) = bot_bot_set_o ).

% boolean_algebra.conj_cancel_right
tff(fact_4927_boolean__algebra_Oconj__cancel__right,axiom,
    ! [X: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),uminus5710092332889474511et_nat(X)) = bot_bot_set_nat ).

% boolean_algebra.conj_cancel_right
tff(fact_4928_boolean__algebra_Oconj__cancel__right,axiom,
    ! [X: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,X),uminus1532241313380277803et_int(X)) = bot_bot_set_int ).

% boolean_algebra.conj_cancel_right
tff(fact_4929_boolean__algebra_Oconj__cancel__right,axiom,
    ! [X: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),aa_assn_assn(uminus_uminus_assn,X)) = bot_bot_assn ).

% boolean_algebra.conj_cancel_right
tff(fact_4930_boolean__algebra_Oconj__cancel__right,axiom,
    ! [X: product_unit] : aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,X),uminus2952777764628376836t_unit(X)) = bot_bot_Product_unit ).

% boolean_algebra.conj_cancel_right
tff(fact_4931_boolean__algebra_Oconj__cancel__left,axiom,
    ! [X: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,uminus6524753893492686040at_nat(X)),X) = bot_bo2099793752762293965at_nat ).

% boolean_algebra.conj_cancel_left
tff(fact_4932_boolean__algebra_Oconj__cancel__left,axiom,
    ! [X: set_o] : inf_inf_set_o(uminus_uminus_set_o(X),X) = bot_bot_set_o ).

% boolean_algebra.conj_cancel_left
tff(fact_4933_boolean__algebra_Oconj__cancel__left,axiom,
    ! [X: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,uminus5710092332889474511et_nat(X)),X) = bot_bot_set_nat ).

% boolean_algebra.conj_cancel_left
tff(fact_4934_boolean__algebra_Oconj__cancel__left,axiom,
    ! [X: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,uminus1532241313380277803et_int(X)),X) = bot_bot_set_int ).

% boolean_algebra.conj_cancel_left
tff(fact_4935_boolean__algebra_Oconj__cancel__left,axiom,
    ! [X: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,aa_assn_assn(uminus_uminus_assn,X)),X) = bot_bot_assn ).

% boolean_algebra.conj_cancel_left
tff(fact_4936_boolean__algebra_Oconj__cancel__left,axiom,
    ! [X: product_unit] : aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,uminus2952777764628376836t_unit(X)),X) = bot_bot_Product_unit ).

% boolean_algebra.conj_cancel_left
tff(fact_4937_inf__compl__bot__right,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Y),uminus6524753893492686040at_nat(X))) = bot_bo2099793752762293965at_nat ).

% inf_compl_bot_right
tff(fact_4938_inf__compl__bot__right,axiom,
    ! [X: set_o,Y: set_o] : inf_inf_set_o(X,inf_inf_set_o(Y,uminus_uminus_set_o(X))) = bot_bot_set_o ).

% inf_compl_bot_right
tff(fact_4939_inf__compl__bot__right,axiom,
    ! [X: set_nat,Y: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Y),uminus5710092332889474511et_nat(X))) = bot_bot_set_nat ).

% inf_compl_bot_right
tff(fact_4940_inf__compl__bot__right,axiom,
    ! [X: set_int,Y: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,X),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,Y),uminus1532241313380277803et_int(X))) = bot_bot_set_int ).

% inf_compl_bot_right
tff(fact_4941_inf__compl__bot__right,axiom,
    ! [X: assn,Y: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Y),aa_assn_assn(uminus_uminus_assn,X))) = bot_bot_assn ).

% inf_compl_bot_right
tff(fact_4942_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),uminus2952777764628376836t_unit(X))) = bot_bot_Product_unit ).

% inf_compl_bot_right
tff(fact_4943_inf__compl__bot__left2,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,uminus6524753893492686040at_nat(X)),Y)) = bot_bo2099793752762293965at_nat ).

% inf_compl_bot_left2
tff(fact_4944_inf__compl__bot__left2,axiom,
    ! [X: set_o,Y: set_o] : inf_inf_set_o(X,inf_inf_set_o(uminus_uminus_set_o(X),Y)) = bot_bot_set_o ).

% inf_compl_bot_left2
tff(fact_4945_inf__compl__bot__left2,axiom,
    ! [X: set_nat,Y: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,uminus5710092332889474511et_nat(X)),Y)) = bot_bot_set_nat ).

% inf_compl_bot_left2
tff(fact_4946_inf__compl__bot__left2,axiom,
    ! [X: set_int,Y: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,X),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,uminus1532241313380277803et_int(X)),Y)) = bot_bot_set_int ).

% inf_compl_bot_left2
tff(fact_4947_inf__compl__bot__left2,axiom,
    ! [X: assn,Y: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,aa_assn_assn(uminus_uminus_assn,X)),Y)) = bot_bot_assn ).

% inf_compl_bot_left2
tff(fact_4948_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,uminus2952777764628376836t_unit(X)),Y)) = bot_bot_Product_unit ).

% inf_compl_bot_left2
tff(fact_4949_inf__compl__bot__left1,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,uminus6524753893492686040at_nat(X)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),Y)) = bot_bo2099793752762293965at_nat ).

% inf_compl_bot_left1
tff(fact_4950_inf__compl__bot__left1,axiom,
    ! [X: set_o,Y: set_o] : inf_inf_set_o(uminus_uminus_set_o(X),inf_inf_set_o(X,Y)) = bot_bot_set_o ).

% inf_compl_bot_left1
tff(fact_4951_inf__compl__bot__left1,axiom,
    ! [X: set_nat,Y: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,uminus5710092332889474511et_nat(X)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),Y)) = bot_bot_set_nat ).

% inf_compl_bot_left1
tff(fact_4952_inf__compl__bot__left1,axiom,
    ! [X: set_int,Y: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,uminus1532241313380277803et_int(X)),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,X),Y)) = bot_bot_set_int ).

% inf_compl_bot_left1
tff(fact_4953_inf__compl__bot__left1,axiom,
    ! [X: assn,Y: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,aa_assn_assn(uminus_uminus_assn,X)),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),Y)) = bot_bot_assn ).

% inf_compl_bot_left1
tff(fact_4954_inf__compl__bot__left1,axiom,
    ! [X: product_unit,Y: product_unit] : aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_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_4955_boolean__algebra_Ode__Morgan__disj,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] : uminus6524753893492686040at_nat(aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,X),Y)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,uminus6524753893492686040at_nat(X)),uminus6524753893492686040at_nat(Y)) ).

% boolean_algebra.de_Morgan_disj
tff(fact_4956_boolean__algebra_Ode__Morgan__disj,axiom,
    ! [X: set_Pr8693737435421807431at_nat,Y: set_Pr8693737435421807431at_nat] : uminus4384627049435823934at_nat(aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,X),Y)) = aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(inf_in4302113700860409141at_nat,uminus4384627049435823934at_nat(X)),uminus4384627049435823934at_nat(Y)) ).

% boolean_algebra.de_Morgan_disj
tff(fact_4957_boolean__algebra_Ode__Morgan__disj,axiom,
    ! [X: set_Pr8551490117392284871at_nat,Y: set_Pr8551490117392284871at_nat] : uminus2330091110623919550at_nat(aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,X),Y)) = aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(inf_in1697001100524423349at_nat,uminus2330091110623919550at_nat(X)),uminus2330091110623919550at_nat(Y)) ).

% boolean_algebra.de_Morgan_disj
tff(fact_4958_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_4959_boolean__algebra_Ode__Morgan__disj,axiom,
    ! [X: set_nat,Y: 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,uminus5710092332889474511et_nat(X)),uminus5710092332889474511et_nat(Y)) ).

% boolean_algebra.de_Morgan_disj
tff(fact_4960_boolean__algebra_Ode__Morgan__disj,axiom,
    ! [X: assn,Y: assn] : aa_assn_assn(uminus_uminus_assn,aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),Y)) = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,aa_assn_assn(uminus_uminus_assn,X)),aa_assn_assn(uminus_uminus_assn,Y)) ).

% boolean_algebra.de_Morgan_disj
tff(fact_4961_boolean__algebra_Ode__Morgan__disj,axiom,
    ! [X: product_unit,Y: product_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,uminus2952777764628376836t_unit(X)),uminus2952777764628376836t_unit(Y)) ).

% boolean_algebra.de_Morgan_disj
tff(fact_4962_boolean__algebra_Ode__Morgan__conj,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] : uminus6524753893492686040at_nat(aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),Y)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,uminus6524753893492686040at_nat(X)),uminus6524753893492686040at_nat(Y)) ).

% boolean_algebra.de_Morgan_conj
tff(fact_4963_boolean__algebra_Ode__Morgan__conj,axiom,
    ! [X: set_Pr8693737435421807431at_nat,Y: set_Pr8693737435421807431at_nat] : uminus4384627049435823934at_nat(aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(inf_in4302113700860409141at_nat,X),Y)) = aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,uminus4384627049435823934at_nat(X)),uminus4384627049435823934at_nat(Y)) ).

% boolean_algebra.de_Morgan_conj
tff(fact_4964_boolean__algebra_Ode__Morgan__conj,axiom,
    ! [X: set_Pr8551490117392284871at_nat,Y: set_Pr8551490117392284871at_nat] : uminus2330091110623919550at_nat(aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(inf_in1697001100524423349at_nat,X),Y)) = aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,uminus2330091110623919550at_nat(X)),uminus2330091110623919550at_nat(Y)) ).

% boolean_algebra.de_Morgan_conj
tff(fact_4965_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_4966_boolean__algebra_Ode__Morgan__conj,axiom,
    ! [X: set_nat,Y: 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,uminus5710092332889474511et_nat(X)),uminus5710092332889474511et_nat(Y)) ).

% boolean_algebra.de_Morgan_conj
tff(fact_4967_boolean__algebra_Ode__Morgan__conj,axiom,
    ! [X: assn,Y: assn] : aa_assn_assn(uminus_uminus_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),Y)) = aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,aa_assn_assn(uminus_uminus_assn,X)),aa_assn_assn(uminus_uminus_assn,Y)) ).

% boolean_algebra.de_Morgan_conj
tff(fact_4968_boolean__algebra_Ode__Morgan__conj,axiom,
    ! [X: product_unit,Y: product_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,uminus2952777764628376836t_unit(X)),uminus2952777764628376836t_unit(Y)) ).

% boolean_algebra.de_Morgan_conj
tff(fact_4969_negative__zless,axiom,
    ! [N2: 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,N2)))),aa_nat_int(semiri1314217659103216013at_int,M)) ).

% negative_zless
tff(fact_4970_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_4971_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_4972_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_4973_zero__le__divide__abs__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),aa_rat_rat(abs_abs_rat,B)))
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
        | ( B = zero_zero_rat ) ) ) ).

% zero_le_divide_abs_iff
tff(fact_4974_divide__le__0__abs__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),aa_rat_rat(abs_abs_rat,B))),zero_zero_rat)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)
        | ( B = zero_zero_rat ) ) ) ).

% divide_le_0_abs_iff
tff(fact_4975_left__minus__one__mult__self,axiom,
    ! [N2: 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)),N2)),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)),N2)),A)) = A ).

% left_minus_one_mult_self
tff(fact_4976_left__minus__one__mult__self,axiom,
    ! [N2: nat,A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),N2)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),N2)),A)) = A ).

% left_minus_one_mult_self
tff(fact_4977_left__minus__one__mult__self,axiom,
    ! [N2: 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)),N2)),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)),N2)),A)) = A ).

% left_minus_one_mult_self
tff(fact_4978_minus__one__mult__self,axiom,
    ! [N2: 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)),N2)),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,one_one_int)),N2)) = one_one_int ).

% minus_one_mult_self
tff(fact_4979_minus__one__mult__self,axiom,
    ! [N2: nat] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),N2)),aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),N2)) = one_one_Code_integer ).

% minus_one_mult_self
tff(fact_4980_minus__one__mult__self,axiom,
    ! [N2: 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)),N2)),aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),N2)) = one_one_rat ).

% minus_one_mult_self
tff(fact_4981_semiring__norm_I172_J,axiom,
    ! [V: num,W2: 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,W2))),Y)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(times_times_num,V),W2))),Y) ).

% semiring_norm(172)
tff(fact_4982_semiring__norm_I172_J,axiom,
    ! [V: num,W2: 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,W2))),Y)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(aa_num_fun_num_num(times_times_num,V),W2))),Y) ).

% semiring_norm(172)
tff(fact_4983_semiring__norm_I172_J,axiom,
    ! [V: num,W2: num,Y: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2))),Y)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(aa_num_fun_num_num(times_times_num,V),W2))),Y) ).

% semiring_norm(172)
tff(fact_4984_semiring__norm_I171_J,axiom,
    ! [V: num,W2: 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,W2))),Y)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(times_times_num,V),W2)))),Y) ).

% semiring_norm(171)
tff(fact_4985_semiring__norm_I171_J,axiom,
    ! [V: num,W2: 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,W2))),Y)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(aa_num_fun_num_num(times_times_num,V),W2)))),Y) ).

% semiring_norm(171)
tff(fact_4986_semiring__norm_I171_J,axiom,
    ! [V: num,W2: num,Y: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,V)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2))),Y)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(aa_num_fun_num_num(times_times_num,V),W2)))),Y) ).

% semiring_norm(171)
tff(fact_4987_semiring__norm_I170_J,axiom,
    ! [V: num,W2: 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,W2)),Y)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(times_times_num,V),W2)))),Y) ).

% semiring_norm(170)
tff(fact_4988_semiring__norm_I170_J,axiom,
    ! [V: num,W2: 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,W2)),Y)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(aa_num_fun_num_num(times_times_num,V),W2)))),Y) ).

% semiring_norm(170)
tff(fact_4989_semiring__norm_I170_J,axiom,
    ! [V: num,W2: num,Y: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,W2)),Y)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(aa_num_fun_num_num(times_times_num,V),W2)))),Y) ).

% semiring_norm(170)
tff(fact_4990_mult__neg__numeral__simps_I3_J,axiom,
    ! [M: num,N2: 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,N2))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(times_times_num,M),N2))) ).

% mult_neg_numeral_simps(3)
tff(fact_4991_mult__neg__numeral__simps_I3_J,axiom,
    ! [M: num,N2: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,M)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,N2))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(aa_num_fun_num_num(times_times_num,M),N2))) ).

% mult_neg_numeral_simps(3)
tff(fact_4992_mult__neg__numeral__simps_I3_J,axiom,
    ! [M: num,N2: num] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,M)),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,N2))) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(aa_num_fun_num_num(times_times_num,M),N2))) ).

% mult_neg_numeral_simps(3)
tff(fact_4993_mult__neg__numeral__simps_I2_J,axiom,
    ! [M: num,N2: 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,N2)) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(times_times_num,M),N2))) ).

% mult_neg_numeral_simps(2)
tff(fact_4994_mult__neg__numeral__simps_I2_J,axiom,
    ! [M: num,N2: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,M))),aa_num_Code_integer(numera6620942414471956472nteger,N2)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(aa_num_fun_num_num(times_times_num,M),N2))) ).

% mult_neg_numeral_simps(2)
tff(fact_4995_mult__neg__numeral__simps_I2_J,axiom,
    ! [M: num,N2: num] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,M))),aa_num_rat(numeral_numeral_rat,N2)) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(aa_num_fun_num_num(times_times_num,M),N2))) ).

% mult_neg_numeral_simps(2)
tff(fact_4996_mult__neg__numeral__simps_I1_J,axiom,
    ! [M: num,N2: 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,N2))) = aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(times_times_num,M),N2)) ).

% mult_neg_numeral_simps(1)
tff(fact_4997_mult__neg__numeral__simps_I1_J,axiom,
    ! [M: num,N2: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,M))),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,N2))) = aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(aa_num_fun_num_num(times_times_num,M),N2)) ).

% mult_neg_numeral_simps(1)
tff(fact_4998_mult__neg__numeral__simps_I1_J,axiom,
    ! [M: num,N2: num] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,M))),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,N2))) = aa_num_rat(numeral_numeral_rat,aa_num_num(aa_num_fun_num_num(times_times_num,M),N2)) ).

% mult_neg_numeral_simps(1)
tff(fact_4999_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_5000_neg__numeral__le__iff,axiom,
    ! [M: num,N2: 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,N2)))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,N2),M) ) ).

% neg_numeral_le_iff
tff(fact_5001_neg__numeral__le__iff,axiom,
    ! [M: num,N2: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,M))),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,N2)))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,N2),M) ) ).

% neg_numeral_le_iff
tff(fact_5002_neg__numeral__le__iff,axiom,
    ! [M: num,N2: 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,N2)))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,N2),M) ) ).

% neg_numeral_le_iff
tff(fact_5003_neg__numeral__less__iff,axiom,
    ! [M: num,N2: 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,N2)))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,N2),M) ) ).

% neg_numeral_less_iff
tff(fact_5004_neg__numeral__less__iff,axiom,
    ! [M: num,N2: 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,N2)))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,N2),M) ) ).

% neg_numeral_less_iff
tff(fact_5005_neg__numeral__less__iff,axiom,
    ! [M: num,N2: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,M))),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,N2)))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,N2),M) ) ).

% neg_numeral_less_iff
tff(fact_5006_divide__le__eq__numeral1_I2_J,axiom,
    ! [B: rat,W2: num,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2)))),A)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2)))),B) ) ).

% divide_le_eq_numeral1(2)
tff(fact_5007_le__divide__eq__numeral1_I2_J,axiom,
    ! [A: rat,B: rat,W2: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2))))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2)))) ) ).

% le_divide_eq_numeral1(2)
tff(fact_5008_eq__divide__eq__numeral1_I2_J,axiom,
    ! [A: rat,B: rat,W2: num] :
      ( ( A = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2))) )
    <=> $ite(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2)) != zero_zero_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2))) = B,A = zero_zero_rat) ) ).

% eq_divide_eq_numeral1(2)
tff(fact_5009_divide__eq__eq__numeral1_I2_J,axiom,
    ! [B: rat,W2: num,A: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2))) = A )
    <=> $ite(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2)) != zero_zero_rat,B = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2))),A = zero_zero_rat) ) ).

% divide_eq_eq_numeral1(2)
tff(fact_5010_less__divide__eq__numeral1_I2_J,axiom,
    ! [A: rat,B: rat,W2: 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,aa_num_rat(numeral_numeral_rat,W2))))
    <=> 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,aa_num_rat(numeral_numeral_rat,W2)))) ) ).

% less_divide_eq_numeral1(2)
tff(fact_5011_divide__less__eq__numeral1_I2_J,axiom,
    ! [B: rat,W2: 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,aa_num_rat(numeral_numeral_rat,W2)))),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,aa_num_rat(numeral_numeral_rat,W2)))),B) ) ).

% divide_less_eq_numeral1(2)
tff(fact_5012_zero__less__power__abs__iff,axiom,
    ! [A: code_integer,N2: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A)),N2))
    <=> ( ( A != zero_z3403309356797280102nteger )
        | ( N2 = zero_zero_nat ) ) ) ).

% zero_less_power_abs_iff
tff(fact_5013_zero__less__power__abs__iff,axiom,
    ! [A: rat,N2: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_nat_rat(power_power_rat(aa_rat_rat(abs_abs_rat,A)),N2))
    <=> ( ( A != zero_zero_rat )
        | ( N2 = zero_zero_nat ) ) ) ).

% zero_less_power_abs_iff
tff(fact_5014_zero__less__power__abs__iff,axiom,
    ! [A: int,N2: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(abs_abs_int,A)),N2))
    <=> ( ( A != zero_zero_int )
        | ( N2 = zero_zero_nat ) ) ) ).

% zero_less_power_abs_iff
tff(fact_5015_small__lazy_H_Ocases,axiom,
    ! [X: product_prod_int_int] :
      ~ ! [D: int,I3: int] : X != aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,D),I3) ).

% small_lazy'.cases
tff(fact_5016_exhaustive__int_H_Ocases,axiom,
    ! [X: produc4788131091073775919nt_int] :
      ~ ! [F2: fun_in5730379987311621695e_term,D: int,I3: int] : X != aa_Pro8800328608558162860nt_int(aa_fun260840925927807678nt_int(produc4729913460834754785nt_int,F2),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,D),I3)) ).

% exhaustive_int'.cases
tff(fact_5017_full__exhaustive__int_H_Ocases,axiom,
    ! [X: produc3043004577601930755nt_int] :
      ~ ! [F2: fun_Pr7763680713840462443e_term,D: int,I3: int] : X != aa_Pro5748361610964203648nt_int(aa_fun3265472288063244902nt_int(produc7320969046608631221nt_int,F2),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,D),I3)) ).

% full_exhaustive_int'.cases
tff(fact_5018_unique__remainder,axiom,
    ! [A: int,B: int,Q4: int,Ra2: int,Q6: int,R4: int] :
      ( aa_Pro4592068669816069595_int_o(eucl_rel_int(A,B),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q4),Ra2))
     => ( aa_Pro4592068669816069595_int_o(eucl_rel_int(A,B),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q6),R4))
       => ( Ra2 = R4 ) ) ) ).

% unique_remainder
tff(fact_5019_unique__quotient,axiom,
    ! [A: int,B: int,Q4: int,Ra2: int,Q6: int,R4: int] :
      ( aa_Pro4592068669816069595_int_o(eucl_rel_int(A,B),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q4),Ra2))
     => ( aa_Pro4592068669816069595_int_o(eucl_rel_int(A,B),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q6),R4))
       => ( Q4 = Q6 ) ) ) ).

% unique_quotient
tff(fact_5020_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_5021_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_5022_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_5023_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_5024_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_5025_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_5026_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_5027_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_5028_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_5029_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_5030_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_5031_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_5032_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_5033_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_5034_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_5035_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_5036_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_5037_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_5038_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_5039_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_5040_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_5041_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_5042_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_5043_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_5044_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_5045_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_5046_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_5047_abs__if__raw,axiom,
    ! [X5: int] :
      aa_int_int(abs_abs_int,X5) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_int,X5),zero_zero_int),aa_int_int(uminus_uminus_int,X5),X5) ).

% abs_if_raw
tff(fact_5048_abs__if__raw,axiom,
    ! [X5: code_integer] :
      aa_Cod1783974652533396728nteger(abs_abs_Code_integer,X5) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,X5),zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,X5),X5) ).

% abs_if_raw
tff(fact_5049_abs__if__raw,axiom,
    ! [X5: rat] :
      aa_rat_rat(abs_abs_rat,X5) = $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X5),zero_zero_rat),aa_rat_rat(uminus_uminus_rat,X5),X5) ).

% abs_if_raw
tff(fact_5050_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_5051_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_5052_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_5053_zabs__def,axiom,
    ! [I2: int] :
      aa_int_int(abs_abs_int,I2) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_int,I2),zero_zero_int),aa_int_int(uminus_uminus_int,I2),I2) ).

% zabs_def
tff(fact_5054_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_5055_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_5056_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_5057_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_5058_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_5059_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_5060_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_5061_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_5062_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_5063_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_5064_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_5065_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_5066_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_5067_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_5068_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_5069_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_5070_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_5071_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_5072_compl__le__swap2,axiom,
    ! [Y: assn,X: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_assn_assn(uminus_uminus_assn,Y)),X)
     => aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_assn_assn(uminus_uminus_assn,X)),Y) ) ).

% compl_le_swap2
tff(fact_5073_compl__le__swap2,axiom,
    ! [Y: product_unit,X: product_unit] :
      ( aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,uminus2952777764628376836t_unit(Y)),X)
     => aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,uminus2952777764628376836t_unit(X)),Y) ) ).

% compl_le_swap2
tff(fact_5074_compl__le__swap2,axiom,
    ! [Y: set_int,X: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,uminus1532241313380277803et_int(Y)),X)
     => aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,uminus1532241313380277803et_int(X)),Y) ) ).

% compl_le_swap2
tff(fact_5075_compl__le__swap1,axiom,
    ! [Y: assn,X: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,Y),aa_assn_assn(uminus_uminus_assn,X))
     => aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X),aa_assn_assn(uminus_uminus_assn,Y)) ) ).

% compl_le_swap1
tff(fact_5076_compl__le__swap1,axiom,
    ! [Y: product_unit,X: product_unit] :
      ( aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,Y),uminus2952777764628376836t_unit(X))
     => aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,X),uminus2952777764628376836t_unit(Y)) ) ).

% compl_le_swap1
tff(fact_5077_compl__le__swap1,axiom,
    ! [Y: set_int,X: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,Y),uminus1532241313380277803et_int(X))
     => aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,X),uminus1532241313380277803et_int(Y)) ) ).

% compl_le_swap1
tff(fact_5078_compl__mono,axiom,
    ! [X: assn,Y: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X),Y)
     => aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_assn_assn(uminus_uminus_assn,Y)),aa_assn_assn(uminus_uminus_assn,X)) ) ).

% compl_mono
tff(fact_5079_compl__mono,axiom,
    ! [X: product_unit,Y: product_unit] :
      ( aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,X),Y)
     => aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,uminus2952777764628376836t_unit(Y)),uminus2952777764628376836t_unit(X)) ) ).

% compl_mono
tff(fact_5080_compl__mono,axiom,
    ! [X: set_int,Y: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,X),Y)
     => aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,uminus1532241313380277803et_int(Y)),uminus1532241313380277803et_int(X)) ) ).

% compl_mono
tff(fact_5081_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_5082_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_5083_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_5084_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_5085_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_5086_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_5087_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_5088_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_5089_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_5090_compl__less__swap1,axiom,
    ! [Y: assn,X: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,Y),aa_assn_assn(uminus_uminus_assn,X))
     => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X),aa_assn_assn(uminus_uminus_assn,Y)) ) ).

% compl_less_swap1
tff(fact_5091_compl__less__swap1,axiom,
    ! [Y: product_unit,X: product_unit] :
      ( aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le361264281704409273t_unit,Y),uminus2952777764628376836t_unit(X))
     => aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le361264281704409273t_unit,X),uminus2952777764628376836t_unit(Y)) ) ).

% compl_less_swap1
tff(fact_5092_compl__less__swap2,axiom,
    ! [Y: assn,X: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_assn_assn(uminus_uminus_assn,Y)),X)
     => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_assn_assn(uminus_uminus_assn,X)),Y) ) ).

% compl_less_swap2
tff(fact_5093_compl__less__swap2,axiom,
    ! [Y: product_unit,X: product_unit] :
      ( aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le361264281704409273t_unit,uminus2952777764628376836t_unit(Y)),X)
     => aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le361264281704409273t_unit,uminus2952777764628376836t_unit(X)),Y) ) ).

% compl_less_swap2
tff(fact_5094_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_5095_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_5096_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_5097_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_5098_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_5099_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_5100_minus__assn__def,axiom,
    ! [A: assn,B: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(minus_minus_assn,A),B) = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),aa_assn_assn(uminus_uminus_assn,B)) ).

% minus_assn_def
tff(fact_5101_Collect__imp__eq,axiom,
    ! [Pa: fun_Pr3238374608701176162_int_o,Qa: fun_Pr3238374608701176162_int_o] : collec213857154873943460nt_int(aa_fun5349570537165229698_int_o(aTP_Lamm_gr(Pa),Qa)) = sup_su6024340866399070445nt_int(uminus6221592323253981072nt_int(collec213857154873943460nt_int(Pa)),collec213857154873943460nt_int(Qa)) ).

% Collect_imp_eq
tff(fact_5102_Collect__imp__eq,axiom,
    ! [Pa: fun_list_nat_o,Qa: fun_list_nat_o] : collect_list_nat(aa_fun6155608546934989052_nat_o(aTP_Lamm_gs(Pa),Qa)) = sup_sup_set_list_nat(uminus3195874150345416415st_nat(collect_list_nat(Pa)),collect_list_nat(Qa)) ).

% Collect_imp_eq
tff(fact_5103_Collect__imp__eq,axiom,
    ! [Pa: fun_set_nat_o,Qa: fun_set_nat_o] : collect_set_nat(aa_fun751667557118977660_nat_o(aTP_Lamm_gt(Pa),Qa)) = sup_sup_set_set_nat(uminus613421341184616069et_nat(collect_set_nat(Pa)),collect_set_nat(Qa)) ).

% Collect_imp_eq
tff(fact_5104_Collect__imp__eq,axiom,
    ! [Pa: fun_int_o,Qa: fun_int_o] : collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_gu(Pa),Qa)) = aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,uminus1532241313380277803et_int(collect_int(Pa))),collect_int(Qa)) ).

% Collect_imp_eq
tff(fact_5105_Collect__imp__eq,axiom,
    ! [Pa: fun_Pr363642936170290612_nat_o,Qa: fun_Pr363642936170290612_nat_o] : collec7088162979684241874at_nat(aa_fun6414004499577345958_nat_o(aTP_Lamm_gv(Pa),Qa)) = aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,uminus4384627049435823934at_nat(collec7088162979684241874at_nat(Pa))),collec7088162979684241874at_nat(Qa)) ).

% Collect_imp_eq
tff(fact_5106_Collect__imp__eq,axiom,
    ! [Pa: fun_Pr1099658227000185396_nat_o,Qa: fun_Pr1099658227000185396_nat_o] : collec5204685387357076818at_nat(aa_fun9125434449681309094_nat_o(aTP_Lamm_gw(Pa),Qa)) = aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,uminus2330091110623919550at_nat(collec5204685387357076818at_nat(Pa))),collec5204685387357076818at_nat(Qa)) ).

% Collect_imp_eq
tff(fact_5107_Collect__imp__eq,axiom,
    ! [Pa: fun_Pr3059359313727237556_nat_o,Qa: fun_Pr3059359313727237556_nat_o] : collec6321179662152712658at_nat(aa_fun2566793329686429606_nat_o(aTP_Lamm_gx(Pa),Qa)) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,uminus935396558254630718at_nat(collec6321179662152712658at_nat(Pa))),collec6321179662152712658at_nat(Qa)) ).

% Collect_imp_eq
tff(fact_5108_Collect__imp__eq,axiom,
    ! [Pa: fun_nat_o,Qa: fun_nat_o] : collect_nat(aa_fun5937193402907943164_nat_o(aTP_Lamm_gy(Pa),Qa)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,uminus5710092332889474511et_nat(collect_nat(Pa))),collect_nat(Qa)) ).

% Collect_imp_eq
tff(fact_5109_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_5110_div__int__unique,axiom,
    ! [K: int,L: int,Q4: int,Ra2: int] :
      ( aa_Pro4592068669816069595_int_o(eucl_rel_int(K,L),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q4),Ra2))
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,K),L) = Q4 ) ) ).

% div_int_unique
tff(fact_5111_zminus1__lemma,axiom,
    ! [A: int,B: int,Q4: int,Ra2: int] :
      ( aa_Pro4592068669816069595_int_o(eucl_rel_int(A,B),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q4),Ra2))
     => ( ( 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(Ra2 = zero_zero_int,aa_int_int(uminus_uminus_int,Q4),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,Q4)),one_one_int))),
              $ite(Ra2 = zero_zero_int,zero_zero_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B),Ra2)))) ) ) ).

% zminus1_lemma
tff(fact_5112_mod__int__unique,axiom,
    ! [K: int,L: int,Q4: int,Ra2: int] :
      ( aa_Pro4592068669816069595_int_o(eucl_rel_int(K,L),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q4),Ra2))
     => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),L) = Ra2 ) ) ).

% mod_int_unique
tff(fact_5113_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_5114_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_5115_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_5116_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_5117_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_5118_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_5119_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_5120_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_5121_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_5122_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(plus_p5714425477246183910nteger(A),B))),aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A)),aa_Cod1783974652533396728nteger(abs_abs_Code_integer,B))) ).

% abs_triangle_ineq
tff(fact_5123_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_5124_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_5125_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_5126_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_5127_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_5128_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_5129_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_5130_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_5131_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_5132_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_5133_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_5134_abs__mult__less,axiom,
    ! [A: code_integer,Ca: code_integer,B: code_integer,D2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A)),Ca)
     => ( 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,Ca),D2)) ) ) ).

% abs_mult_less
tff(fact_5135_abs__mult__less,axiom,
    ! [A: rat,Ca: rat,B: rat,D2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(abs_abs_rat,A)),Ca)
     => ( 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,Ca),D2)) ) ) ).

% abs_mult_less
tff(fact_5136_abs__mult__less,axiom,
    ! [A: int,Ca: int,B: int,D2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,A)),Ca)
     => ( 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,Ca),D2)) ) ) ).

% abs_mult_less
tff(fact_5137_neg__numeral__le__numeral,axiom,
    ! [M: num,N2: num] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,M))),aa_num_Code_integer(numera6620942414471956472nteger,N2)) ).

% neg_numeral_le_numeral
tff(fact_5138_neg__numeral__le__numeral,axiom,
    ! [M: num,N2: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,M))),aa_num_rat(numeral_numeral_rat,N2)) ).

% neg_numeral_le_numeral
tff(fact_5139_neg__numeral__le__numeral,axiom,
    ! [M: num,N2: 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,N2)) ).

% neg_numeral_le_numeral
tff(fact_5140_not__numeral__le__neg__numeral,axiom,
    ! [M: num,N2: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_num_Code_integer(numera6620942414471956472nteger,M)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,N2))) ).

% not_numeral_le_neg_numeral
tff(fact_5141_not__numeral__le__neg__numeral,axiom,
    ! [M: num,N2: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,M)),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,N2))) ).

% not_numeral_le_neg_numeral
tff(fact_5142_not__numeral__le__neg__numeral,axiom,
    ! [M: num,N2: 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,N2))) ).

% not_numeral_le_neg_numeral
tff(fact_5143_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_5144_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_5145_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_5146_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_5147_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_5148_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_5149_neg__numeral__less__numeral,axiom,
    ! [M: num,N2: 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,N2)) ).

% neg_numeral_less_numeral
tff(fact_5150_neg__numeral__less__numeral,axiom,
    ! [M: num,N2: num] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,M))),aa_num_Code_integer(numera6620942414471956472nteger,N2)) ).

% neg_numeral_less_numeral
tff(fact_5151_neg__numeral__less__numeral,axiom,
    ! [M: num,N2: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,M))),aa_num_rat(numeral_numeral_rat,N2)) ).

% neg_numeral_less_numeral
tff(fact_5152_not__numeral__less__neg__numeral,axiom,
    ! [M: num,N2: 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,N2))) ).

% not_numeral_less_neg_numeral
tff(fact_5153_not__numeral__less__neg__numeral,axiom,
    ! [M: num,N2: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_num_Code_integer(numera6620942414471956472nteger,M)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,N2))) ).

% not_numeral_less_neg_numeral
tff(fact_5154_not__numeral__less__neg__numeral,axiom,
    ! [M: num,N2: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(numeral_numeral_rat,M)),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,N2))) ).

% not_numeral_less_neg_numeral
tff(fact_5155_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_5156_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_5157_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_5158_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_5159_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_5160_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_5161_numeral__times__minus__swap,axiom,
    ! [W2: num,X: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,W2)),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,W2))) ).

% numeral_times_minus_swap
tff(fact_5162_numeral__times__minus__swap,axiom,
    ! [W2: num,X: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,W2)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,X)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,X),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,W2))) ).

% numeral_times_minus_swap
tff(fact_5163_numeral__times__minus__swap,axiom,
    ! [W2: num,X: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,W2)),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,aa_num_rat(numeral_numeral_rat,W2))) ).

% numeral_times_minus_swap
tff(fact_5164_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_5165_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_5166_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_5167_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),uminus6524753893492686040at_nat(Y)) ).

% diff_eq
tff(fact_5168_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),uminus5710092332889474511et_nat(Y)) ).

% diff_eq
tff(fact_5169_diff__eq,axiom,
    ! [X: assn,Y: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(minus_minus_assn,X),Y) = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),aa_assn_assn(uminus_uminus_assn,Y)) ).

% diff_eq
tff(fact_5170_diff__eq,axiom,
    ! [X: product_unit,Y: product_unit] : aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(minus_3524152463667985524t_unit,X),Y) = aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,X),uminus2952777764628376836t_unit(Y)) ).

% diff_eq
tff(fact_5171_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,uminus6524753893492686040at_nat(X)),A)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),B)) = bot_bo2099793752762293965at_nat ).

% inf_cancel_left2
tff(fact_5172_inf__cancel__left2,axiom,
    ! [X: set_o,A: set_o,B: set_o] : inf_inf_set_o(inf_inf_set_o(uminus_uminus_set_o(X),A),inf_inf_set_o(X,B)) = bot_bot_set_o ).

% inf_cancel_left2
tff(fact_5173_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,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_5174_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,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_5175_inf__cancel__left2,axiom,
    ! [X: assn,A: assn,B: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,aa_assn_assn(uminus_uminus_assn,X)),A)),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),B)) = bot_bot_assn ).

% inf_cancel_left2
tff(fact_5176_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,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_5177_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,uminus6524753893492686040at_nat(X)),B)) = bot_bo2099793752762293965at_nat ).

% inf_cancel_left1
tff(fact_5178_inf__cancel__left1,axiom,
    ! [X: set_o,A: set_o,B: set_o] : inf_inf_set_o(inf_inf_set_o(X,A),inf_inf_set_o(uminus_uminus_set_o(X),B)) = bot_bot_set_o ).

% inf_cancel_left1
tff(fact_5179_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,uminus5710092332889474511et_nat(X)),B)) = bot_bot_set_nat ).

% inf_cancel_left1
tff(fact_5180_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,uminus1532241313380277803et_int(X)),B)) = bot_bot_set_int ).

% inf_cancel_left1
tff(fact_5181_inf__cancel__left1,axiom,
    ! [X: assn,A: assn,B: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),A)),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,aa_assn_assn(uminus_uminus_assn,X)),B)) = bot_bot_assn ).

% inf_cancel_left1
tff(fact_5182_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,uminus2952777764628376836t_unit(X)),B)) = bot_bot_Product_unit ).

% inf_cancel_left1
tff(fact_5183_subset__Compl__self__eq,axiom,
    ! [A3: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A3),uminus6524753893492686040at_nat(A3))
    <=> ( A3 = bot_bo2099793752762293965at_nat ) ) ).

% subset_Compl_self_eq
tff(fact_5184_subset__Compl__self__eq,axiom,
    ! [A3: set_o] :
      ( ord_less_eq_set_o(A3,uminus_uminus_set_o(A3))
    <=> ( A3 = bot_bot_set_o ) ) ).

% subset_Compl_self_eq
tff(fact_5185_subset__Compl__self__eq,axiom,
    ! [A3: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A3),uminus5710092332889474511et_nat(A3))
    <=> ( A3 = bot_bot_set_nat ) ) ).

% subset_Compl_self_eq
tff(fact_5186_subset__Compl__self__eq,axiom,
    ! [A3: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A3),uminus1532241313380277803et_int(A3))
    <=> ( A3 = bot_bot_set_int ) ) ).

% subset_Compl_self_eq
tff(fact_5187_Compl__Int,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] : uminus6524753893492686040at_nat(aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A3),B3)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,uminus6524753893492686040at_nat(A3)),uminus6524753893492686040at_nat(B3)) ).

% Compl_Int
tff(fact_5188_Compl__Int,axiom,
    ! [A3: set_Pr8693737435421807431at_nat,B3: set_Pr8693737435421807431at_nat] : uminus4384627049435823934at_nat(aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(inf_in4302113700860409141at_nat,A3),B3)) = aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,uminus4384627049435823934at_nat(A3)),uminus4384627049435823934at_nat(B3)) ).

% Compl_Int
tff(fact_5189_Compl__Int,axiom,
    ! [A3: set_Pr8551490117392284871at_nat,B3: set_Pr8551490117392284871at_nat] : uminus2330091110623919550at_nat(aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(inf_in1697001100524423349at_nat,A3),B3)) = aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,uminus2330091110623919550at_nat(A3)),uminus2330091110623919550at_nat(B3)) ).

% Compl_Int
tff(fact_5190_Compl__Int,axiom,
    ! [A3: set_Pr4329608150637261639at_nat,B3: set_Pr4329608150637261639at_nat] : uminus935396558254630718at_nat(aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,A3),B3)) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,uminus935396558254630718at_nat(A3)),uminus935396558254630718at_nat(B3)) ).

% Compl_Int
tff(fact_5191_Compl__Int,axiom,
    ! [A3: set_nat,B3: set_nat] : uminus5710092332889474511et_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A3),B3)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,uminus5710092332889474511et_nat(A3)),uminus5710092332889474511et_nat(B3)) ).

% Compl_Int
tff(fact_5192_Compl__Un,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] : uminus6524753893492686040at_nat(aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,A3),B3)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,uminus6524753893492686040at_nat(A3)),uminus6524753893492686040at_nat(B3)) ).

% Compl_Un
tff(fact_5193_Compl__Un,axiom,
    ! [A3: set_Pr8693737435421807431at_nat,B3: set_Pr8693737435421807431at_nat] : uminus4384627049435823934at_nat(aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,A3),B3)) = aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(inf_in4302113700860409141at_nat,uminus4384627049435823934at_nat(A3)),uminus4384627049435823934at_nat(B3)) ).

% Compl_Un
tff(fact_5194_Compl__Un,axiom,
    ! [A3: set_Pr8551490117392284871at_nat,B3: set_Pr8551490117392284871at_nat] : uminus2330091110623919550at_nat(aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A3),B3)) = aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(inf_in1697001100524423349at_nat,uminus2330091110623919550at_nat(A3)),uminus2330091110623919550at_nat(B3)) ).

% Compl_Un
tff(fact_5195_Compl__Un,axiom,
    ! [A3: set_Pr4329608150637261639at_nat,B3: set_Pr4329608150637261639at_nat] : uminus935396558254630718at_nat(aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A3),B3)) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,uminus935396558254630718at_nat(A3)),uminus935396558254630718at_nat(B3)) ).

% Compl_Un
tff(fact_5196_Compl__Un,axiom,
    ! [A3: set_nat,B3: set_nat] : uminus5710092332889474511et_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A3),B3)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,uminus5710092332889474511et_nat(A3)),uminus5710092332889474511et_nat(B3)) ).

% Compl_Un
tff(fact_5197_not__int__zless__negative,axiom,
    ! [N2: nat,M: nat] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,N2)),aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,M))) ).

% not_int_zless_negative
tff(fact_5198_dense__eq0__I,axiom,
    ! [X: rat] :
      ( ! [E2: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),E2)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,X)),E2) )
     => ( X = zero_zero_rat ) ) ).

% dense_eq0_I
tff(fact_5199_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_5200_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_5201_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_5202_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_5203_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_5204_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_5205_abs__diff__triangle__ineq,axiom,
    ! [A: code_integer,B: code_integer,Ca: 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(plus_p5714425477246183910nteger(A),B)),aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(Ca),D2)))),aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(A),Ca))),aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(B),D2)))) ).

% abs_diff_triangle_ineq
tff(fact_5206_abs__diff__triangle__ineq,axiom,
    ! [A: rat,B: rat,Ca: 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,Ca),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),Ca))),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_5207_abs__diff__triangle__ineq,axiom,
    ! [A: int,B: int,Ca: 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,Ca),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),Ca))),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_5208_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(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A)),aa_Cod1783974652533396728nteger(abs_abs_Code_integer,B))) ).

% abs_triangle_ineq4
tff(fact_5209_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_5210_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_5211_abs__diff__le__iff,axiom,
    ! [X: code_integer,A: code_integer,Ra2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(X),A))),Ra2)
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(A),Ra2)),X)
        & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,X),aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(A),Ra2)) ) ) ).

% abs_diff_le_iff
tff(fact_5212_abs__diff__le__iff,axiom,
    ! [X: rat,A: rat,Ra2: 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,X),A))),Ra2)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),Ra2)),X)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),Ra2)) ) ) ).

% abs_diff_le_iff
tff(fact_5213_abs__diff__le__iff,axiom,
    ! [X: int,A: int,Ra2: 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,X),A))),Ra2)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),Ra2)),X)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),Ra2)) ) ) ).

% abs_diff_le_iff
tff(fact_5214_abs__diff__less__iff,axiom,
    ! [X: code_integer,A: code_integer,Ra2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(X),A))),Ra2)
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(A),Ra2)),X)
        & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,X),aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(A),Ra2)) ) ) ).

% abs_diff_less_iff
tff(fact_5215_abs__diff__less__iff,axiom,
    ! [X: rat,A: rat,Ra2: rat] :
      ( 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),A))),Ra2)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),Ra2)),X)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),Ra2)) ) ) ).

% abs_diff_less_iff
tff(fact_5216_abs__diff__less__iff,axiom,
    ! [X: int,A: int,Ra2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),A))),Ra2)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),Ra2)),X)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,X),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),Ra2)) ) ) ).

% abs_diff_less_iff
tff(fact_5217_abs__div__pos,axiom,
    ! [Y: rat,X: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Y)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(abs_abs_rat,X)),Y) = aa_rat_rat(abs_abs_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,X),Y)) ) ) ).

% abs_div_pos
tff(fact_5218_zero__le__power__abs,axiom,
    ! [A: code_integer,N2: nat] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A)),N2)) ).

% zero_le_power_abs
tff(fact_5219_zero__le__power__abs,axiom,
    ! [A: rat,N2: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(aa_rat_rat(abs_abs_rat,A)),N2)) ).

% zero_le_power_abs
tff(fact_5220_zero__le__power__abs,axiom,
    ! [A: int,N2: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(abs_abs_int,A)),N2)) ).

% zero_le_power_abs
tff(fact_5221_not__zero__le__neg__numeral,axiom,
    ! [N2: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,N2))) ).

% not_zero_le_neg_numeral
tff(fact_5222_not__zero__le__neg__numeral,axiom,
    ! [N2: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,N2))) ).

% not_zero_le_neg_numeral
tff(fact_5223_not__zero__le__neg__numeral,axiom,
    ! [N2: 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,N2))) ).

% not_zero_le_neg_numeral
tff(fact_5224_neg__numeral__le__zero,axiom,
    ! [N2: num] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,N2))),zero_z3403309356797280102nteger) ).

% neg_numeral_le_zero
tff(fact_5225_neg__numeral__le__zero,axiom,
    ! [N2: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,N2))),zero_zero_rat) ).

% neg_numeral_le_zero
tff(fact_5226_neg__numeral__le__zero,axiom,
    ! [N2: 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,N2))),zero_zero_int) ).

% neg_numeral_le_zero
tff(fact_5227_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_5228_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_5229_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_5230_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_5231_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_5232_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_5233_not__zero__less__neg__numeral,axiom,
    ! [N2: 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,N2))) ).

% not_zero_less_neg_numeral
tff(fact_5234_not__zero__less__neg__numeral,axiom,
    ! [N2: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,N2))) ).

% not_zero_less_neg_numeral
tff(fact_5235_not__zero__less__neg__numeral,axiom,
    ! [N2: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,N2))) ).

% not_zero_less_neg_numeral
tff(fact_5236_neg__numeral__less__zero,axiom,
    ! [N2: num] : aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N2))),zero_zero_int) ).

% neg_numeral_less_zero
tff(fact_5237_neg__numeral__less__zero,axiom,
    ! [N2: num] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,N2))),zero_z3403309356797280102nteger) ).

% neg_numeral_less_zero
tff(fact_5238_neg__numeral__less__zero,axiom,
    ! [N2: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,N2))),zero_zero_rat) ).

% neg_numeral_less_zero
tff(fact_5239_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_5240_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_5241_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_5242_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_5243_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_5244_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_5245_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_5246_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,aa_num_rat(numeral_numeral_rat,M))) ).

% not_one_le_neg_numeral
tff(fact_5247_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_5248_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_5249_not__numeral__le__neg__one,axiom,
    ! [M: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,M)),aa_rat_rat(uminus_uminus_rat,one_one_rat)) ).

% not_numeral_le_neg_one
tff(fact_5250_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_5251_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_5252_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,aa_num_rat(numeral_numeral_rat,M))),aa_rat_rat(uminus_uminus_rat,one_one_rat)) ).

% neg_numeral_le_neg_one
tff(fact_5253_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_5254_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_5255_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)),aa_num_rat(numeral_numeral_rat,M)) ).

% neg_one_le_numeral
tff(fact_5256_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_5257_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_5258_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,aa_num_rat(numeral_numeral_rat,M))),one_one_rat) ).

% neg_numeral_le_one
tff(fact_5259_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_5260_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_5261_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_5262_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,aa_num_rat(numeral_numeral_rat,M))) ).

% not_neg_one_less_neg_numeral
tff(fact_5263_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_5264_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_5265_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,aa_num_rat(numeral_numeral_rat,M))) ).

% not_one_less_neg_numeral
tff(fact_5266_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_5267_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_5268_not__numeral__less__neg__one,axiom,
    ! [M: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(numeral_numeral_rat,M)),aa_rat_rat(uminus_uminus_rat,one_one_rat)) ).

% not_numeral_less_neg_one
tff(fact_5269_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_5270_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_5271_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)),aa_num_rat(numeral_numeral_rat,M)) ).

% neg_one_less_numeral
tff(fact_5272_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_5273_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_5274_neg__numeral__less__one,axiom,
    ! [M: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,M))),one_one_rat) ).

% neg_numeral_less_one
tff(fact_5275_nonzero__neg__divide__eq__eq2,axiom,
    ! [B: rat,Ca: rat,A: rat] :
      ( ( B != zero_zero_rat )
     => ( ( Ca = 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,Ca),B) = aa_rat_rat(uminus_uminus_rat,A) ) ) ) ).

% nonzero_neg_divide_eq_eq2
tff(fact_5276_nonzero__neg__divide__eq__eq,axiom,
    ! [B: rat,A: rat,Ca: rat] :
      ( ( B != zero_zero_rat )
     => ( ( aa_rat_rat(uminus_uminus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),B)) = Ca )
      <=> ( aa_rat_rat(uminus_uminus_rat,A) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Ca),B) ) ) ) ).

% nonzero_neg_divide_eq_eq
tff(fact_5277_minus__divide__eq__eq,axiom,
    ! [B: rat,Ca: rat,A: rat] :
      ( ( aa_rat_rat(uminus_uminus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),Ca)) = A )
    <=> $ite(Ca != zero_zero_rat,aa_rat_rat(uminus_uminus_rat,B) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),Ca),A = zero_zero_rat) ) ).

% minus_divide_eq_eq
tff(fact_5278_eq__minus__divide__eq,axiom,
    ! [A: rat,B: rat,Ca: rat] :
      ( ( A = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),Ca)) )
    <=> $ite(Ca != zero_zero_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),Ca) = aa_rat_rat(uminus_uminus_rat,B),A = zero_zero_rat) ) ).

% eq_minus_divide_eq
tff(fact_5279_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),uminus6524753893492686040at_nat(Y)) ) ).

% inf_shunt
tff(fact_5280_inf__shunt,axiom,
    ! [X: set_o,Y: set_o] :
      ( ( inf_inf_set_o(X,Y) = bot_bot_set_o )
    <=> ord_less_eq_set_o(X,uminus_uminus_set_o(Y)) ) ).

% inf_shunt
tff(fact_5281_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),uminus5710092332889474511et_nat(Y)) ) ).

% inf_shunt
tff(fact_5282_inf__shunt,axiom,
    ! [X: assn,Y: assn] :
      ( ( aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),Y) = bot_bot_assn )
    <=> aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X),aa_assn_assn(uminus_uminus_assn,Y)) ) ).

% inf_shunt
tff(fact_5283_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(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,X),uminus2952777764628376836t_unit(Y)) ) ).

% inf_shunt
tff(fact_5284_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),uminus1532241313380277803et_int(Y)) ) ).

% inf_shunt
tff(fact_5285_sup__neg__inf,axiom,
    ! [P2: set_Pr1261947904930325089at_nat,Q4: set_Pr1261947904930325089at_nat,Ra2: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,P2),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,Q4),Ra2))
    <=> aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,P2),uminus6524753893492686040at_nat(Q4))),Ra2) ) ).

% sup_neg_inf
tff(fact_5286_sup__neg__inf,axiom,
    ! [P2: set_Pr8693737435421807431at_nat,Q4: set_Pr8693737435421807431at_nat,Ra2: set_Pr8693737435421807431at_nat] :
      ( ord_le3000389064537975527at_nat(P2,aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,Q4),Ra2))
    <=> ord_le3000389064537975527at_nat(aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(inf_in4302113700860409141at_nat,P2),uminus4384627049435823934at_nat(Q4)),Ra2) ) ).

% sup_neg_inf
tff(fact_5287_sup__neg__inf,axiom,
    ! [P2: set_Pr8551490117392284871at_nat,Q4: set_Pr8551490117392284871at_nat,Ra2: set_Pr8551490117392284871at_nat] :
      ( ord_le8081472938463900775at_nat(P2,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,Q4),Ra2))
    <=> ord_le8081472938463900775at_nat(aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(inf_in1697001100524423349at_nat,P2),uminus2330091110623919550at_nat(Q4)),Ra2) ) ).

% sup_neg_inf
tff(fact_5288_sup__neg__inf,axiom,
    ! [P2: set_Pr4329608150637261639at_nat,Q4: set_Pr4329608150637261639at_nat,Ra2: set_Pr4329608150637261639at_nat] :
      ( ord_le1268244103169919719at_nat(P2,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,Q4),Ra2))
    <=> ord_le1268244103169919719at_nat(aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,P2),uminus935396558254630718at_nat(Q4)),Ra2) ) ).

% sup_neg_inf
tff(fact_5289_sup__neg__inf,axiom,
    ! [P2: set_nat,Q4: set_nat,Ra2: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,P2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Q4),Ra2))
    <=> 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,P2),uminus5710092332889474511et_nat(Q4))),Ra2) ) ).

% sup_neg_inf
tff(fact_5290_sup__neg__inf,axiom,
    ! [P2: assn,Q4: assn,Ra2: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,P2),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Q4),Ra2))
    <=> aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,P2),aa_assn_assn(uminus_uminus_assn,Q4))),Ra2) ) ).

% sup_neg_inf
tff(fact_5291_sup__neg__inf,axiom,
    ! [P2: product_unit,Q4: product_unit,Ra2: product_unit] :
      ( aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,P2),aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(sup_sup_Product_unit,Q4),Ra2))
    <=> aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,P2),uminus2952777764628376836t_unit(Q4))),Ra2) ) ).

% sup_neg_inf
tff(fact_5292_sup__neg__inf,axiom,
    ! [P2: set_int,Q4: set_int,Ra2: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,P2),aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,Q4),Ra2))
    <=> 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,P2),uminus1532241313380277803et_int(Q4))),Ra2) ) ).

% sup_neg_inf
tff(fact_5293_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),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_5294_shunt2,axiom,
    ! [X: set_Pr8693737435421807431at_nat,Y: set_Pr8693737435421807431at_nat,Z: set_Pr8693737435421807431at_nat] :
      ( ord_le3000389064537975527at_nat(aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(inf_in4302113700860409141at_nat,X),uminus4384627049435823934at_nat(Y)),Z)
    <=> ord_le3000389064537975527at_nat(X,aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,Y),Z)) ) ).

% shunt2
tff(fact_5295_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),uminus2330091110623919550at_nat(Y)),Z)
    <=> ord_le8081472938463900775at_nat(X,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,Y),Z)) ) ).

% shunt2
tff(fact_5296_shunt2,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat,Z: set_Pr4329608150637261639at_nat] :
      ( ord_le1268244103169919719at_nat(aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,X),uminus935396558254630718at_nat(Y)),Z)
    <=> ord_le1268244103169919719at_nat(X,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,Y),Z)) ) ).

% shunt2
tff(fact_5297_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),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_5298_shunt2,axiom,
    ! [X: assn,Y: assn,Z: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),aa_assn_assn(uminus_uminus_assn,Y))),Z)
    <=> aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Y),Z)) ) ).

% shunt2
tff(fact_5299_shunt2,axiom,
    ! [X: product_unit,Y: product_unit,Z: product_unit] :
      ( aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,X),uminus2952777764628376836t_unit(Y))),Z)
    <=> aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,X),aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(sup_sup_Product_unit,Y),Z)) ) ).

% shunt2
tff(fact_5300_shunt2,axiom,
    ! [X: set_int,Y: set_int,Z: set_int] :
      ( 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,X),uminus1532241313380277803et_int(Y))),Z)
    <=> aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,X),aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,Y),Z)) ) ).

% shunt2
tff(fact_5301_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,uminus6524753893492686040at_nat(Y)),Z)) ) ).

% shunt1
tff(fact_5302_shunt1,axiom,
    ! [X: set_Pr8693737435421807431at_nat,Y: set_Pr8693737435421807431at_nat,Z: set_Pr8693737435421807431at_nat] :
      ( ord_le3000389064537975527at_nat(aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(inf_in4302113700860409141at_nat,X),Y),Z)
    <=> ord_le3000389064537975527at_nat(X,aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,uminus4384627049435823934at_nat(Y)),Z)) ) ).

% shunt1
tff(fact_5303_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,uminus2330091110623919550at_nat(Y)),Z)) ) ).

% shunt1
tff(fact_5304_shunt1,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat,Z: set_Pr4329608150637261639at_nat] :
      ( ord_le1268244103169919719at_nat(aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,X),Y),Z)
    <=> ord_le1268244103169919719at_nat(X,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,uminus935396558254630718at_nat(Y)),Z)) ) ).

% shunt1
tff(fact_5305_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,uminus5710092332889474511et_nat(Y)),Z)) ) ).

% shunt1
tff(fact_5306_shunt1,axiom,
    ! [X: assn,Y: assn,Z: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),Y)),Z)
    <=> aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,aa_assn_assn(uminus_uminus_assn,Y)),Z)) ) ).

% shunt1
tff(fact_5307_shunt1,axiom,
    ! [X: product_unit,Y: product_unit,Z: product_unit] :
      ( aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,X),Y)),Z)
    <=> aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,X),aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(sup_sup_Product_unit,uminus2952777764628376836t_unit(Y)),Z)) ) ).

% shunt1
tff(fact_5308_shunt1,axiom,
    ! [X: set_int,Y: set_int,Z: set_int] :
      ( 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,X),Y)),Z)
    <=> aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,X),aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,uminus1532241313380277803et_int(Y)),Z)) ) ).

% shunt1
tff(fact_5309_power__minus,axiom,
    ! [A: int,N2: nat] : aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,A)),N2) = 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)),N2)),aa_nat_int(aa_int_fun_nat_int(power_power_int,A),N2)) ).

% power_minus
tff(fact_5310_power__minus,axiom,
    ! [A: code_integer,N2: nat] : aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),N2) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),N2)),aa_nat_Code_integer(power_8256067586552552935nteger(A),N2)) ).

% power_minus
tff(fact_5311_power__minus,axiom,
    ! [A: rat,N2: nat] : aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,A)),N2) = 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)),N2)),aa_nat_rat(power_power_rat(A),N2)) ).

% power_minus
tff(fact_5312_eucl__rel__int__dividesI,axiom,
    ! [L: int,K: int,Q4: int] :
      ( ( L != zero_zero_int )
     => ( ( K = aa_int_int(aa_int_fun_int_int(times_times_int,Q4),L) )
       => aa_Pro4592068669816069595_int_o(eucl_rel_int(K,L),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q4),zero_zero_int)) ) ) ).

% eucl_rel_int_dividesI
tff(fact_5313_disjoint__eq__subset__Compl,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
      ( ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A3),B3) = bot_bo2099793752762293965at_nat )
    <=> aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A3),uminus6524753893492686040at_nat(B3)) ) ).

% disjoint_eq_subset_Compl
tff(fact_5314_disjoint__eq__subset__Compl,axiom,
    ! [A3: set_o,B3: set_o] :
      ( ( inf_inf_set_o(A3,B3) = bot_bot_set_o )
    <=> ord_less_eq_set_o(A3,uminus_uminus_set_o(B3)) ) ).

% disjoint_eq_subset_Compl
tff(fact_5315_disjoint__eq__subset__Compl,axiom,
    ! [A3: set_nat,B3: set_nat] :
      ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A3),B3) = bot_bot_set_nat )
    <=> aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A3),uminus5710092332889474511et_nat(B3)) ) ).

% disjoint_eq_subset_Compl
tff(fact_5316_disjoint__eq__subset__Compl,axiom,
    ! [A3: set_int,B3: set_int] :
      ( ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A3),B3) = bot_bot_set_int )
    <=> aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A3),uminus1532241313380277803et_int(B3)) ) ).

% disjoint_eq_subset_Compl
tff(fact_5317_abs__mod__less,axiom,
    ! [L: int,K: int] :
      ( ( L != zero_zero_int )
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),L))),aa_int_int(abs_abs_int,L)) ) ).

% abs_mod_less
tff(fact_5318_int__cases4,axiom,
    ! [M: int] :
      ( ! [N5: nat] : M != aa_nat_int(semiri1314217659103216013at_int,N5)
     => ~ ! [N5: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N5)
           => ( M != aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N5)) ) ) ) ).

% int_cases4
tff(fact_5319_int__zle__neg,axiom,
    ! [N2: nat,M: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,N2)),aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,M)))
    <=> ( ( N2 = zero_zero_nat )
        & ( M = zero_zero_nat ) ) ) ).

% int_zle_neg
tff(fact_5320_negative__zle__0,axiom,
    ! [N2: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N2))),zero_zero_int) ).

% negative_zle_0
tff(fact_5321_nonpos__int__cases,axiom,
    ! [K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),zero_zero_int)
     => ~ ! [N5: nat] : K != aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N5)) ) ).

% nonpos_int_cases
tff(fact_5322_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_5323_abs__add__one__gt__zero,axiom,
    ! [X: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(one_one_Code_integer),aa_Cod1783974652533396728nteger(abs_abs_Code_integer,X))) ).

% abs_add_one_gt_zero
tff(fact_5324_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_5325_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_5326_pos__minus__divide__less__eq,axiom,
    ! [Ca: rat,B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Ca)
     => ( 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),Ca))),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),Ca)) ) ) ).

% pos_minus_divide_less_eq
tff(fact_5327_pos__less__minus__divide__eq,axiom,
    ! [Ca: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Ca)
     => ( 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),Ca)))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),Ca)),aa_rat_rat(uminus_uminus_rat,B)) ) ) ).

% pos_less_minus_divide_eq
tff(fact_5328_neg__minus__divide__less__eq,axiom,
    ! [Ca: rat,B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Ca),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),Ca))),A)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),Ca)),aa_rat_rat(uminus_uminus_rat,B)) ) ) ).

% neg_minus_divide_less_eq
tff(fact_5329_neg__less__minus__divide__eq,axiom,
    ! [Ca: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Ca),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),Ca)))
      <=> 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),Ca)) ) ) ).

% neg_less_minus_divide_eq
tff(fact_5330_minus__divide__less__eq,axiom,
    ! [B: rat,Ca: 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),Ca))),A)
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Ca),
          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),Ca)),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Ca),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),Ca)),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_5331_less__minus__divide__eq,axiom,
    ! [A: rat,B: rat,Ca: 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),Ca)))
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Ca),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),Ca)),aa_rat_rat(uminus_uminus_rat,B)),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Ca),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),Ca)),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)) ) ) ).

% less_minus_divide_eq
tff(fact_5332_divide__eq__eq__numeral_I2_J,axiom,
    ! [B: rat,Ca: rat,W2: num] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),Ca) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2)) )
    <=> $ite(Ca != zero_zero_rat,B = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2))),Ca),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2)) = zero_zero_rat) ) ).

% divide_eq_eq_numeral(2)
tff(fact_5333_eq__divide__eq__numeral_I2_J,axiom,
    ! [W2: num,B: rat,Ca: rat] :
      ( ( aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2)) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),Ca) )
    <=> $ite(Ca != zero_zero_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2))),Ca) = B,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2)) = zero_zero_rat) ) ).

% eq_divide_eq_numeral(2)
tff(fact_5334_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_5335_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_5336_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_5337_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_5338_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_5339_uminus__assn__def,axiom,
    ! [Pa: assn] : aa_assn_assn(uminus_uminus_assn,Pa) = abs_assn(aTP_Lamm_gz(Pa)) ).

% uminus_assn_def
tff(fact_5340_int__cases3,axiom,
    ! [K: int] :
      ( ( K != zero_zero_int )
     => ( ! [N5: nat] :
            ( ( K = aa_nat_int(semiri1314217659103216013at_int,N5) )
           => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N5) )
       => ~ ! [N5: nat] :
              ( ( K = aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N5)) )
             => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N5) ) ) ) ).

% int_cases3
tff(fact_5341_pochhammer__of__nat__eq__0__lemma,axiom,
    ! [N2: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),K)
     => ( aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(uminus_uminus_rat,aa_nat_rat(semiri681578069525770553at_rat,N2))),K) = zero_zero_rat ) ) ).

% pochhammer_of_nat_eq_0_lemma
tff(fact_5342_pochhammer__of__nat__eq__0__lemma,axiom,
    ! [N2: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),K)
     => ( aa_nat_int(aa_int_fun_nat_int(comm_s4660882817536571857er_int,aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N2))),K) = zero_zero_int ) ) ).

% pochhammer_of_nat_eq_0_lemma
tff(fact_5343_pochhammer__of__nat__eq__0__lemma,axiom,
    ! [N2: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),K)
     => ( aa_nat_Code_integer(comm_s8582702949713902594nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,N2))),K) = zero_z3403309356797280102nteger ) ) ).

% pochhammer_of_nat_eq_0_lemma
tff(fact_5344_pochhammer__of__nat__eq__0__iff,axiom,
    ! [N2: nat,K: nat] :
      ( ( aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(uminus_uminus_rat,aa_nat_rat(semiri681578069525770553at_rat,N2))),K) = zero_zero_rat )
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),K) ) ).

% pochhammer_of_nat_eq_0_iff
tff(fact_5345_pochhammer__of__nat__eq__0__iff,axiom,
    ! [N2: 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,N2))),K) = zero_zero_int )
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),K) ) ).

% pochhammer_of_nat_eq_0_iff
tff(fact_5346_pochhammer__of__nat__eq__0__iff,axiom,
    ! [N2: nat,K: nat] :
      ( ( aa_nat_Code_integer(comm_s8582702949713902594nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,N2))),K) = zero_z3403309356797280102nteger )
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),K) ) ).

% pochhammer_of_nat_eq_0_iff
tff(fact_5347_pochhammer__eq__0__iff,axiom,
    ! [A: rat,N2: nat] :
      ( ( aa_nat_rat(comm_s4028243227959126397er_rat(A),N2) = zero_zero_rat )
    <=> ? [K3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K3),N2)
          & ( A = aa_rat_rat(uminus_uminus_rat,aa_nat_rat(semiri681578069525770553at_rat,K3)) ) ) ) ).

% pochhammer_eq_0_iff
tff(fact_5348_not__zle__0__negative,axiom,
    ! [N2: 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,N2)))) ).

% not_zle_0_negative
tff(fact_5349_pochhammer__of__nat__eq__0__lemma_H,axiom,
    ! [K: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),N2)
     => ( aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(uminus_uminus_rat,aa_nat_rat(semiri681578069525770553at_rat,N2))),K) != zero_zero_rat ) ) ).

% pochhammer_of_nat_eq_0_lemma'
tff(fact_5350_pochhammer__of__nat__eq__0__lemma_H,axiom,
    ! [K: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),N2)
     => ( aa_nat_int(aa_int_fun_nat_int(comm_s4660882817536571857er_int,aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N2))),K) != zero_zero_int ) ) ).

% pochhammer_of_nat_eq_0_lemma'
tff(fact_5351_pochhammer__of__nat__eq__0__lemma_H,axiom,
    ! [K: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),N2)
     => ( aa_nat_Code_integer(comm_s8582702949713902594nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,N2))),K) != zero_z3403309356797280102nteger ) ) ).

% pochhammer_of_nat_eq_0_lemma'
tff(fact_5352_negD,axiom,
    ! [X: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),zero_zero_int)
     => ? [N5: nat] : X = aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,N5))) ) ).

% negD
tff(fact_5353_negative__zless__0,axiom,
    ! [N2: 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,N2)))),zero_zero_int) ).

% negative_zless_0
tff(fact_5354_verit__less__mono__div__int2,axiom,
    ! [A3: int,B3: int,N2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A3),B3)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(uminus_uminus_int,N2))
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,B3),N2)),aa_int_int(aa_int_fun_int_int(divide_divide_int,A3),N2)) ) ) ).

% verit_less_mono_div_int2
tff(fact_5355_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_5356_power__minus_H,axiom,
    ! [X: int,N2: 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)),N2) = 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)),N2)),aa_nat_int(aa_int_fun_nat_int(power_power_int,X),N2)) ) ) ).

% power_minus'
tff(fact_5357_power__minus_H,axiom,
    ! [X: code_integer,N2: 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)),N2) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),N2)),aa_nat_Code_integer(power_8256067586552552935nteger(X),N2)) ) ) ).

% power_minus'
tff(fact_5358_power__minus_H,axiom,
    ! [X: rat,N2: 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)),N2) = 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)),N2)),aa_nat_rat(power_power_rat(X),N2)) ) ) ).

% power_minus'
tff(fact_5359_pos__minus__divide__le__eq,axiom,
    ! [Ca: rat,B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Ca)
     => ( 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),Ca))),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),Ca)) ) ) ).

% pos_minus_divide_le_eq
tff(fact_5360_pos__le__minus__divide__eq,axiom,
    ! [Ca: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Ca)
     => ( 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),Ca)))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),Ca)),aa_rat_rat(uminus_uminus_rat,B)) ) ) ).

% pos_le_minus_divide_eq
tff(fact_5361_neg__minus__divide__le__eq,axiom,
    ! [Ca: rat,B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Ca),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),Ca))),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),Ca)),aa_rat_rat(uminus_uminus_rat,B)) ) ) ).

% neg_minus_divide_le_eq
tff(fact_5362_neg__le__minus__divide__eq,axiom,
    ! [Ca: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Ca),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),Ca)))
      <=> 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),Ca)) ) ) ).

% neg_le_minus_divide_eq
tff(fact_5363_minus__divide__le__eq,axiom,
    ! [B: rat,Ca: 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),Ca))),A)
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Ca),
          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),Ca)),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Ca),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),Ca)),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_5364_le__minus__divide__eq,axiom,
    ! [A: rat,B: rat,Ca: 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),Ca)))
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Ca),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),Ca)),aa_rat_rat(uminus_uminus_rat,B)),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Ca),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),Ca)),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)) ) ) ).

% le_minus_divide_eq
tff(fact_5365_less__divide__eq__numeral_I2_J,axiom,
    ! [W2: num,B: rat,Ca: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2))),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),Ca))
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Ca),
          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,aa_num_rat(numeral_numeral_rat,W2))),Ca)),B),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Ca),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,aa_num_rat(numeral_numeral_rat,W2))),Ca)),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2))),zero_zero_rat)) ) ) ).

% less_divide_eq_numeral(2)
tff(fact_5366_divide__less__eq__numeral_I2_J,axiom,
    ! [B: rat,Ca: rat,W2: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),Ca)),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2)))
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Ca),
          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,aa_num_rat(numeral_numeral_rat,W2))),Ca)),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Ca),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,aa_num_rat(numeral_numeral_rat,W2))),Ca)),B),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2)))) ) ) ).

% divide_less_eq_numeral(2)
tff(fact_5367_neg__int__cases,axiom,
    ! [K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int)
     => ~ ! [N5: nat] :
            ( ( K = aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N5)) )
           => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N5) ) ) ).

% neg_int_cases
tff(fact_5368_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_5369_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_5370_divide__le__eq__numeral_I2_J,axiom,
    ! [B: rat,Ca: rat,W2: 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),Ca)),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2)))
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Ca),
          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,aa_num_rat(numeral_numeral_rat,W2))),Ca)),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Ca),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,aa_num_rat(numeral_numeral_rat,W2))),Ca)),B),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2)))) ) ) ).

% divide_le_eq_numeral(2)
tff(fact_5371_le__divide__eq__numeral_I2_J,axiom,
    ! [W2: num,B: rat,Ca: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2))),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),Ca))
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Ca),
          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,aa_num_rat(numeral_numeral_rat,W2))),Ca)),B),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Ca),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,aa_num_rat(numeral_numeral_rat,W2))),Ca)),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2))),zero_zero_rat)) ) ) ).

% le_divide_eq_numeral(2)
tff(fact_5372_nat__intermed__int__val,axiom,
    ! [M: nat,N2: nat,F: fun_nat_int,K: int] :
      ( ! [I3: nat] :
          ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),I3)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),N2) )
         => 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,I3))),aa_nat_int(F,I3)))),one_one_int) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,M)),K)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),aa_nat_int(F,N2))
           => ? [I3: nat] :
                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),I3)
                & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I3),N2)
                & ( aa_nat_int(F,I3) = K ) ) ) ) ) ) ).

% nat_intermed_int_val
tff(fact_5373_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_5374_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_5375_pochhammer__absorb__comp,axiom,
    ! [Ra2: 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,Ra2),aa_nat_rat(semiri681578069525770553at_rat,K))),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(uminus_uminus_rat,Ra2)),K)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Ra2),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,Ra2)),one_one_rat)),K)) ).

% pochhammer_absorb_comp
tff(fact_5376_pochhammer__absorb__comp,axiom,
    ! [Ra2: 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,Ra2),aa_nat_int(semiri1314217659103216013at_int,K))),aa_nat_int(aa_int_fun_nat_int(comm_s4660882817536571857er_int,aa_int_int(uminus_uminus_int,Ra2)),K)) = aa_int_int(aa_int_fun_int_int(times_times_int,Ra2),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,Ra2)),one_one_int)),K)) ).

% pochhammer_absorb_comp
tff(fact_5377_pochhammer__absorb__comp,axiom,
    ! [Ra2: code_integer,K: nat] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(Ra2),aa_nat_Code_integer(semiri4939895301339042750nteger,K))),aa_nat_Code_integer(comm_s8582702949713902594nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Ra2)),K)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Ra2),aa_nat_Code_integer(comm_s8582702949713902594nteger(aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Ra2)),one_one_Code_integer)),K)) ).

% pochhammer_absorb_comp
tff(fact_5378_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_5379_nat__ivt__aux,axiom,
    ! [N2: nat,F: fun_nat_int,K: int] :
      ( ! [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),N2)
         => 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,I3))),aa_nat_int(F,I3)))),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,N2))
         => ? [I3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I3),N2)
              & ( aa_nat_int(F,I3) = K ) ) ) ) ) ).

% nat_ivt_aux
tff(fact_5380_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_5381_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_5382_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(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(B),aa_nat_Code_integer(semiri4939895301339042750nteger,K))),one_one_Code_integer)),K)) ).

% pochhammer_minus
tff(fact_5383_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),N2)
     => ( 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,N2),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,N2),K)) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
tff(fact_5384_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),N2)
     => ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),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,N2),K)) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
tff(fact_5385_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),N2)
     => ( 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,N2),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,N2),K)) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
tff(fact_5386_upto_Opinduct,axiom,
    ! [A0: int,A1: int,Pa: 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))
     => ( ! [I3: int,J2: int] :
            ( aa_Pro4592068669816069595_int_o(accp_P1096762738010456898nt_int(upto_rel),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,I3),J2))
           => ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I3),J2)
               => aa_int_o(aa_int_fun_int_o(Pa,aa_int_int(aa_int_fun_int_int(plus_plus_int,I3),one_one_int)),J2) )
             => aa_int_o(aa_int_fun_int_o(Pa,I3),J2) ) )
       => aa_int_o(aa_int_fun_int_o(Pa,A0),A1) ) ) ).

% upto.pinduct
tff(fact_5387_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_5388_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,aa_num_rat(numeral_numeral_rat,V))),one_one_rat)),X) ) ).

% neg_numeral_le_ceiling
tff(fact_5389_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,aa_num_rat(numeral_numeral_rat,V))),one_one_rat)) ) ).

% ceiling_less_neg_numeral
tff(fact_5390_option_Osize_I3_J,axiom,
    aa_option_num_nat(size_size_option_num,none_num) = aa_nat_nat(suc,zero_zero_nat) ).

% option.size(3)
tff(fact_5391_ComplI,axiom,
    ! [Ca: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat] :
      ( ~ aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Ca),A3)
     => aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Ca),uminus6524753893492686040at_nat(A3)) ) ).

% ComplI
tff(fact_5392_ComplI,axiom,
    ! [Ca: $o,A3: set_o] :
      ( ~ member_o((Ca),A3)
     => member_o((Ca),uminus_uminus_set_o(A3)) ) ).

% ComplI
tff(fact_5393_ComplI,axiom,
    ! [Ca: set_nat,A3: set_set_nat] :
      ( ~ aa_set_set_nat_o(member_set_nat(Ca),A3)
     => aa_set_set_nat_o(member_set_nat(Ca),uminus613421341184616069et_nat(A3)) ) ).

% ComplI
tff(fact_5394_ComplI,axiom,
    ! [Ca: nat,A3: set_nat] :
      ( ~ aa_set_nat_o(member_nat(Ca),A3)
     => aa_set_nat_o(member_nat(Ca),uminus5710092332889474511et_nat(A3)) ) ).

% ComplI
tff(fact_5395_ComplI,axiom,
    ! [Ca: int,A3: set_int] :
      ( ~ aa_set_int_o(member_int(Ca),A3)
     => aa_set_int_o(member_int(Ca),uminus1532241313380277803et_int(A3)) ) ).

% ComplI
tff(fact_5396_Compl__iff,axiom,
    ! [Ca: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Ca),uminus6524753893492686040at_nat(A3))
    <=> ~ aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Ca),A3) ) ).

% Compl_iff
tff(fact_5397_Compl__iff,axiom,
    ! [Ca: $o,A3: set_o] :
      ( member_o((Ca),uminus_uminus_set_o(A3))
    <=> ~ member_o((Ca),A3) ) ).

% Compl_iff
tff(fact_5398_Compl__iff,axiom,
    ! [Ca: set_nat,A3: set_set_nat] :
      ( aa_set_set_nat_o(member_set_nat(Ca),uminus613421341184616069et_nat(A3))
    <=> ~ aa_set_set_nat_o(member_set_nat(Ca),A3) ) ).

% Compl_iff
tff(fact_5399_Compl__iff,axiom,
    ! [Ca: nat,A3: set_nat] :
      ( aa_set_nat_o(member_nat(Ca),uminus5710092332889474511et_nat(A3))
    <=> ~ aa_set_nat_o(member_nat(Ca),A3) ) ).

% Compl_iff
tff(fact_5400_Compl__iff,axiom,
    ! [Ca: int,A3: set_int] :
      ( aa_set_int_o(member_int(Ca),uminus1532241313380277803et_int(A3))
    <=> ~ aa_set_int_o(member_int(Ca),A3) ) ).

% Compl_iff
tff(fact_5401_ceiling__le__zero,axiom,
    ! [X: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(archim2889992004027027881ng_rat,X)),zero_zero_int)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),zero_zero_rat) ) ).

% ceiling_le_zero
tff(fact_5402_zero__less__ceiling,axiom,
    ! [X: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_rat_int(archim2889992004027027881ng_rat,X))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),X) ) ).

% zero_less_ceiling
tff(fact_5403_ceiling__le__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_num_int(numeral_numeral_int,V))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),aa_num_rat(numeral_numeral_rat,V)) ) ).

% ceiling_le_numeral
tff(fact_5404_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_5405_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_5406_numeral__less__ceiling,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(archim2889992004027027881ng_rat,X))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(numeral_numeral_rat,V)),X) ) ).

% numeral_less_ceiling
tff(fact_5407_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_5408_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_5409_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_5410_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_5411_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,aa_num_rat(numeral_numeral_rat,V)),one_one_rat)) ) ).

% ceiling_less_numeral
tff(fact_5412_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,aa_num_rat(numeral_numeral_rat,V)),one_one_rat)),X) ) ).

% numeral_le_ceiling
tff(fact_5413_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,aa_num_rat(numeral_numeral_rat,V))) ) ).

% ceiling_le_neg_numeral
tff(fact_5414_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,aa_num_rat(numeral_numeral_rat,V))),X) ) ).

% neg_numeral_less_ceiling
tff(fact_5415_Collect__neg__eq,axiom,
    ! [Pa: fun_Pr3238374608701176162_int_o] : collec213857154873943460nt_int(aTP_Lamm_ha(Pa)) = uminus6221592323253981072nt_int(collec213857154873943460nt_int(Pa)) ).

% Collect_neg_eq
tff(fact_5416_Collect__neg__eq,axiom,
    ! [Pa: fun_list_nat_o] : collect_list_nat(aTP_Lamm_hb(Pa)) = uminus3195874150345416415st_nat(collect_list_nat(Pa)) ).

% Collect_neg_eq
tff(fact_5417_Collect__neg__eq,axiom,
    ! [Pa: fun_set_nat_o] : collect_set_nat(aTP_Lamm_hc(Pa)) = uminus613421341184616069et_nat(collect_set_nat(Pa)) ).

% Collect_neg_eq
tff(fact_5418_Collect__neg__eq,axiom,
    ! [Pa: fun_nat_o] : collect_nat(aTP_Lamm_hd(Pa)) = uminus5710092332889474511et_nat(collect_nat(Pa)) ).

% Collect_neg_eq
tff(fact_5419_Collect__neg__eq,axiom,
    ! [Pa: fun_int_o] : collect_int(aTP_Lamm_he(Pa)) = uminus1532241313380277803et_int(collect_int(Pa)) ).

% Collect_neg_eq
tff(fact_5420_Compl__eq,axiom,
    ! [A3: set_Pr1261947904930325089at_nat] : uminus6524753893492686040at_nat(A3) = collec3392354462482085612at_nat(aTP_Lamm_hf(A3)) ).

% Compl_eq
tff(fact_5421_Compl__eq,axiom,
    ! [A3: set_o] : uminus_uminus_set_o(A3) = collect_o(aTP_Lamm_hg(A3)) ).

% Compl_eq
tff(fact_5422_Compl__eq,axiom,
    ! [A3: set_Pr958786334691620121nt_int] : uminus6221592323253981072nt_int(A3) = collec213857154873943460nt_int(aTP_Lamm_hh(A3)) ).

% Compl_eq
tff(fact_5423_Compl__eq,axiom,
    ! [A3: set_list_nat] : uminus3195874150345416415st_nat(A3) = collect_list_nat(aTP_Lamm_hi(A3)) ).

% Compl_eq
tff(fact_5424_Compl__eq,axiom,
    ! [A3: set_set_nat] : uminus613421341184616069et_nat(A3) = collect_set_nat(aTP_Lamm_hj(A3)) ).

% Compl_eq
tff(fact_5425_Compl__eq,axiom,
    ! [A3: set_nat] : uminus5710092332889474511et_nat(A3) = collect_nat(aTP_Lamm_hk(A3)) ).

% Compl_eq
tff(fact_5426_Compl__eq,axiom,
    ! [A3: set_int] : uminus1532241313380277803et_int(A3) = collect_int(aTP_Lamm_hl(A3)) ).

% Compl_eq
tff(fact_5427_ComplD,axiom,
    ! [Ca: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Ca),uminus6524753893492686040at_nat(A3))
     => ~ aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Ca),A3) ) ).

% ComplD
tff(fact_5428_ComplD,axiom,
    ! [Ca: $o,A3: set_o] :
      ( member_o((Ca),uminus_uminus_set_o(A3))
     => ~ member_o((Ca),A3) ) ).

% ComplD
tff(fact_5429_ComplD,axiom,
    ! [Ca: set_nat,A3: set_set_nat] :
      ( aa_set_set_nat_o(member_set_nat(Ca),uminus613421341184616069et_nat(A3))
     => ~ aa_set_set_nat_o(member_set_nat(Ca),A3) ) ).

% ComplD
tff(fact_5430_ComplD,axiom,
    ! [Ca: nat,A3: set_nat] :
      ( aa_set_nat_o(member_nat(Ca),uminus5710092332889474511et_nat(A3))
     => ~ aa_set_nat_o(member_nat(Ca),A3) ) ).

% ComplD
tff(fact_5431_ComplD,axiom,
    ! [Ca: int,A3: set_int] :
      ( aa_set_int_o(member_int(Ca),uminus1532241313380277803et_int(A3))
     => ~ aa_set_int_o(member_int(Ca),A3) ) ).

% ComplD
tff(fact_5432_uminus__set__def,axiom,
    ! [A3: set_Pr1261947904930325089at_nat] : uminus6524753893492686040at_nat(A3) = collec3392354462482085612at_nat(aa_fun359341089595923442_nat_o(uminus1662154291547880931_nat_o,aTP_Lamm_a(A3))) ).

% uminus_set_def
tff(fact_5433_uminus__set__def,axiom,
    ! [A3: set_o] : uminus_uminus_set_o(A3) = collect_o(uminus6910356288409981324un_o_o(aTP_Lamm_aa(A3))) ).

% uminus_set_def
tff(fact_5434_uminus__set__def,axiom,
    ! [A3: set_Pr958786334691620121nt_int] : uminus6221592323253981072nt_int(A3) = collec213857154873943460nt_int(uminus103585356917985579_int_o(aTP_Lamm_ab(A3))) ).

% uminus_set_def
tff(fact_5435_uminus__set__def,axiom,
    ! [A3: set_list_nat] : uminus3195874150345416415st_nat(A3) = collect_list_nat(uminus8676031017987022600_nat_o(aTP_Lamm_ac(A3))) ).

% uminus_set_def
tff(fact_5436_uminus__set__def,axiom,
    ! [A3: set_set_nat] : uminus613421341184616069et_nat(A3) = collect_set_nat(uminus1495274395075166690_nat_o(aTP_Lamm_ad(A3))) ).

% uminus_set_def
tff(fact_5437_uminus__set__def,axiom,
    ! [A3: set_nat] : uminus5710092332889474511et_nat(A3) = collect_nat(uminus6183504819137795864_nat_o(aTP_Lamm_ae(A3))) ).

% uminus_set_def
tff(fact_5438_uminus__set__def,axiom,
    ! [A3: set_int] : uminus1532241313380277803et_int(A3) = collect_int(uminus4362462473705107900_int_o(aTP_Lamm_af(A3))) ).

% uminus_set_def
tff(fact_5439_size__neq__size__imp__neq,axiom,
    ! [X: list_nat,Y: list_nat] :
      ( ( aa_list_nat_nat(size_size_list_nat,X) != aa_list_nat_nat(size_size_list_nat,Y) )
     => ( X != Y ) ) ).

% size_neq_size_imp_neq
tff(fact_5440_size__neq__size__imp__neq,axiom,
    ! [X: list_o,Y: list_o] :
      ( ( size_size_list_o(X) != size_size_list_o(Y) )
     => ( X != Y ) ) ).

% size_neq_size_imp_neq
tff(fact_5441_ceiling__mono,axiom,
    ! [Y: rat,X: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),X)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(archim2889992004027027881ng_rat,Y)),aa_rat_int(archim2889992004027027881ng_rat,X)) ) ).

% ceiling_mono
tff(fact_5442_ceiling__less__cancel,axiom,
    ! [X: rat,Y: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(archim2889992004027027881ng_rat,X)),aa_rat_int(archim2889992004027027881ng_rat,Y))
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y) ) ).

% ceiling_less_cancel
tff(fact_5443_ceiling__add__le,axiom,
    ! [X: rat,Y: rat] : 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(plus_plus_rat,X),Y))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_rat_int(archim2889992004027027881ng_rat,X)),aa_rat_int(archim2889992004027027881ng_rat,Y))) ).

% ceiling_add_le
tff(fact_5444_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_5445_option_Osize_I4_J,axiom,
    ! [X2: produc3260487557148687353it_nat] : size_s2363601347547812957it_nat(aa_Pro5302094286594872192it_nat(some_P7913643980934408916it_nat,X2)) = aa_nat_nat(suc,zero_zero_nat) ).

% option.size(4)
tff(fact_5446_option_Osize_I4_J,axiom,
    ! [X2: num] : aa_option_num_nat(size_size_option_num,aa_num_option_num(some_num,X2)) = aa_nat_nat(suc,zero_zero_nat) ).

% option.size(4)
tff(fact_5447_option_Osize_I4_J,axiom,
    ! [X2: produc8664842809031399944it_nat] : size_s8766407808098229740it_nat(aa_Pro8996696869555642526it_nat(some_P1914260805536162275it_nat,X2)) = aa_nat_nat(suc,zero_zero_nat) ).

% option.size(4)
tff(fact_5448_neg__numeral__power__less__of__int__cancel__iff,axiom,
    ! [X: num,N2: 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))),N2)),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))),N2)),A) ) ).

% neg_numeral_power_less_of_int_cancel_iff
tff(fact_5449_neg__numeral__power__less__of__int__cancel__iff,axiom,
    ! [X: num,N2: 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))),N2)),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))),N2)),A) ) ).

% neg_numeral_power_less_of_int_cancel_iff
tff(fact_5450_neg__numeral__power__less__of__int__cancel__iff,axiom,
    ! [X: num,N2: nat,A: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,X))),N2)),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))),N2)),A) ) ).

% neg_numeral_power_less_of_int_cancel_iff
tff(fact_5451_of__int__less__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N2: 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))),N2))
    <=> 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))),N2)) ) ).

% of_int_less_neg_numeral_power_cancel_iff
tff(fact_5452_of__int__less__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N2: 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))),N2))
    <=> 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))),N2)) ) ).

% of_int_less_neg_numeral_power_cancel_iff
tff(fact_5453_of__int__less__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N2: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(ring_1_of_int_rat,A)),aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,X))),N2))
    <=> 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))),N2)) ) ).

% of_int_less_neg_numeral_power_cancel_iff
tff(fact_5454_neg__numeral__power__le__of__int__cancel__iff,axiom,
    ! [X: num,N2: 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))),N2)),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))),N2)),A) ) ).

% neg_numeral_power_le_of_int_cancel_iff
tff(fact_5455_neg__numeral__power__le__of__int__cancel__iff,axiom,
    ! [X: num,N2: nat,A: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,X))),N2)),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))),N2)),A) ) ).

% neg_numeral_power_le_of_int_cancel_iff
tff(fact_5456_neg__numeral__power__le__of__int__cancel__iff,axiom,
    ! [X: num,N2: 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))),N2)),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))),N2)),A) ) ).

% neg_numeral_power_le_of_int_cancel_iff
tff(fact_5457_of__int__le__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N2: 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))),N2))
    <=> 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))),N2)) ) ).

% of_int_le_neg_numeral_power_cancel_iff
tff(fact_5458_of__int__le__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N2: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,A)),aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,X))),N2))
    <=> 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))),N2)) ) ).

% of_int_le_neg_numeral_power_cancel_iff
tff(fact_5459_of__int__le__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N2: 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))),N2))
    <=> 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))),N2)) ) ).

% of_int_le_neg_numeral_power_cancel_iff
tff(fact_5460_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,aa_num_rat(numeral_numeral_rat,V))),one_one_rat)) ) ).

% floor_le_neg_numeral
tff(fact_5461_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,aa_num_rat(numeral_numeral_rat,V))),one_one_rat)),X) ) ).

% neg_numeral_less_floor
tff(fact_5462_nat__le__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(nat2,A)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,X)),N2))
    <=> 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_num_int(numeral_numeral_int,X)),N2)) ) ).

% nat_le_numeral_power_cancel_iff
tff(fact_5463_of__int__le__iff,axiom,
    ! [W2: int,Z: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,W2)),aa_int_rat(ring_1_of_int_rat,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,W2),Z) ) ).

% of_int_le_iff
tff(fact_5464_of__int__le__iff,axiom,
    ! [W2: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(ring_1_of_int_int,W2)),aa_int_int(ring_1_of_int_int,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,W2),Z) ) ).

% of_int_le_iff
tff(fact_5465_of__int__less__iff,axiom,
    ! [W2: int,Z: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(ring_1_of_int_rat,W2)),aa_int_rat(ring_1_of_int_rat,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,W2),Z) ) ).

% of_int_less_iff
tff(fact_5466_of__int__less__iff,axiom,
    ! [W2: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(ring_1_of_int_int,W2)),aa_int_int(ring_1_of_int_int,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,W2),Z) ) ).

% of_int_less_iff
tff(fact_5467_of__int__mult,axiom,
    ! [W2: int,Z: int] : aa_int_Code_integer(ring_18347121197199848620nteger,aa_int_int(aa_int_fun_int_int(times_times_int,W2),Z)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_int_Code_integer(ring_18347121197199848620nteger,W2)),aa_int_Code_integer(ring_18347121197199848620nteger,Z)) ).

% of_int_mult
tff(fact_5468_of__int__mult,axiom,
    ! [W2: int,Z: int] : aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(times_times_int,W2),Z)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_int_rat(ring_1_of_int_rat,W2)),aa_int_rat(ring_1_of_int_rat,Z)) ).

% of_int_mult
tff(fact_5469_of__int__mult,axiom,
    ! [W2: int,Z: int] : aa_int_int(ring_1_of_int_int,aa_int_int(aa_int_fun_int_int(times_times_int,W2),Z)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(ring_1_of_int_int,W2)),aa_int_int(ring_1_of_int_int,Z)) ).

% of_int_mult
tff(fact_5470_nat__le__0,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),zero_zero_int)
     => ( aa_int_nat(nat2,Z) = zero_zero_nat ) ) ).

% nat_le_0
tff(fact_5471_nat__0__iff,axiom,
    ! [I2: int] :
      ( ( aa_int_nat(nat2,I2) = zero_zero_nat )
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I2),zero_zero_int) ) ).

% nat_0_iff
tff(fact_5472_zless__nat__conj,axiom,
    ! [W2: int,Z: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_int_nat(nat2,W2)),aa_int_nat(nat2,Z))
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Z)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,W2),Z) ) ) ).

% zless_nat_conj
tff(fact_5473_int__nat__eq,axiom,
    ! [Z: int] :
      aa_nat_int(semiri1314217659103216013at_int,aa_int_nat(nat2,Z)) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z),Z,zero_zero_int) ).

% int_nat_eq
tff(fact_5474_of__int__le__0__iff,axiom,
    ! [Z: int] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_int_Code_integer(ring_18347121197199848620nteger,Z)),zero_z3403309356797280102nteger)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),zero_zero_int) ) ).

% of_int_le_0_iff
tff(fact_5475_of__int__le__0__iff,axiom,
    ! [Z: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Z)),zero_zero_rat)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),zero_zero_int) ) ).

% of_int_le_0_iff
tff(fact_5476_of__int__le__0__iff,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(ring_1_of_int_int,Z)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),zero_zero_int) ) ).

% of_int_le_0_iff
tff(fact_5477_of__int__0__le__iff,axiom,
    ! [Z: int] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_int_Code_integer(ring_18347121197199848620nteger,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z) ) ).

% of_int_0_le_iff
tff(fact_5478_of__int__0__le__iff,axiom,
    ! [Z: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_int_rat(ring_1_of_int_rat,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z) ) ).

% of_int_0_le_iff
tff(fact_5479_of__int__0__le__iff,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(ring_1_of_int_int,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z) ) ).

% of_int_0_le_iff
tff(fact_5480_of__int__less__0__iff,axiom,
    ! [Z: int] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_int_Code_integer(ring_18347121197199848620nteger,Z)),zero_z3403309356797280102nteger)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Z),zero_zero_int) ) ).

% of_int_less_0_iff
tff(fact_5481_of__int__less__0__iff,axiom,
    ! [Z: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(ring_1_of_int_rat,Z)),zero_zero_rat)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Z),zero_zero_int) ) ).

% of_int_less_0_iff
tff(fact_5482_of__int__less__0__iff,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(ring_1_of_int_int,Z)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Z),zero_zero_int) ) ).

% of_int_less_0_iff
tff(fact_5483_of__int__0__less__iff,axiom,
    ! [Z: int] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_int_Code_integer(ring_18347121197199848620nteger,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Z) ) ).

% of_int_0_less_iff
tff(fact_5484_of__int__0__less__iff,axiom,
    ! [Z: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_int_rat(ring_1_of_int_rat,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Z) ) ).

% of_int_0_less_iff
tff(fact_5485_of__int__0__less__iff,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(ring_1_of_int_int,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Z) ) ).

% of_int_0_less_iff
tff(fact_5486_of__int__numeral__le__iff,axiom,
    ! [N2: num,Z: int] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_num_Code_integer(numera6620942414471956472nteger,N2)),aa_int_Code_integer(ring_18347121197199848620nteger,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,N2)),Z) ) ).

% of_int_numeral_le_iff
tff(fact_5487_of__int__numeral__le__iff,axiom,
    ! [N2: num,Z: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,N2)),aa_int_rat(ring_1_of_int_rat,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,N2)),Z) ) ).

% of_int_numeral_le_iff
tff(fact_5488_of__int__numeral__le__iff,axiom,
    ! [N2: num,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,N2)),aa_int_int(ring_1_of_int_int,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,N2)),Z) ) ).

% of_int_numeral_le_iff
tff(fact_5489_of__int__le__numeral__iff,axiom,
    ! [Z: int,N2: num] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_int_Code_integer(ring_18347121197199848620nteger,Z)),aa_num_Code_integer(numera6620942414471956472nteger,N2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),aa_num_int(numeral_numeral_int,N2)) ) ).

% of_int_le_numeral_iff
tff(fact_5490_of__int__le__numeral__iff,axiom,
    ! [Z: int,N2: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Z)),aa_num_rat(numeral_numeral_rat,N2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),aa_num_int(numeral_numeral_int,N2)) ) ).

% of_int_le_numeral_iff
tff(fact_5491_of__int__le__numeral__iff,axiom,
    ! [Z: int,N2: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(ring_1_of_int_int,Z)),aa_num_int(numeral_numeral_int,N2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),aa_num_int(numeral_numeral_int,N2)) ) ).

% of_int_le_numeral_iff
tff(fact_5492_of__int__numeral__less__iff,axiom,
    ! [N2: num,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,N2)),aa_int_int(ring_1_of_int_int,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,N2)),Z) ) ).

% of_int_numeral_less_iff
tff(fact_5493_of__int__numeral__less__iff,axiom,
    ! [N2: num,Z: int] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_num_Code_integer(numera6620942414471956472nteger,N2)),aa_int_Code_integer(ring_18347121197199848620nteger,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,N2)),Z) ) ).

% of_int_numeral_less_iff
tff(fact_5494_of__int__numeral__less__iff,axiom,
    ! [N2: num,Z: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(numeral_numeral_rat,N2)),aa_int_rat(ring_1_of_int_rat,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,N2)),Z) ) ).

% of_int_numeral_less_iff
tff(fact_5495_of__int__less__numeral__iff,axiom,
    ! [Z: int,N2: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(ring_1_of_int_int,Z)),aa_num_int(numeral_numeral_int,N2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Z),aa_num_int(numeral_numeral_int,N2)) ) ).

% of_int_less_numeral_iff
tff(fact_5496_of__int__less__numeral__iff,axiom,
    ! [Z: int,N2: num] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_int_Code_integer(ring_18347121197199848620nteger,Z)),aa_num_Code_integer(numera6620942414471956472nteger,N2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Z),aa_num_int(numeral_numeral_int,N2)) ) ).

% of_int_less_numeral_iff
tff(fact_5497_of__int__less__numeral__iff,axiom,
    ! [Z: int,N2: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(ring_1_of_int_rat,Z)),aa_num_rat(numeral_numeral_rat,N2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Z),aa_num_int(numeral_numeral_int,N2)) ) ).

% of_int_less_numeral_iff
tff(fact_5498_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_5499_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_5500_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_5501_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_5502_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_5503_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_5504_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_5505_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_5506_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_5507_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_5508_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_5509_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_5510_zero__le__floor,axiom,
    ! [X: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_rat_int(archim3151403230148437115or_rat,X))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),X) ) ).

% zero_le_floor
tff(fact_5511_floor__less__zero,axiom,
    ! [X: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(archim3151403230148437115or_rat,X)),zero_zero_int)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),zero_zero_rat) ) ).

% floor_less_zero
tff(fact_5512_numeral__le__floor,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(archim3151403230148437115or_rat,X))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,V)),X) ) ).

% numeral_le_floor
tff(fact_5513_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_5514_zero__less__nat__eq,axiom,
    ! [Z: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_int_nat(nat2,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Z) ) ).

% zero_less_nat_eq
tff(fact_5515_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_5516_floor__less__numeral,axiom,
    ! [X: rat,V: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_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_num_rat(numeral_numeral_rat,V)) ) ).

% floor_less_numeral
tff(fact_5517_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_5518_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_5519_of__nat__nat,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z)
     => ( aa_nat_rat(semiri681578069525770553at_rat,aa_int_nat(nat2,Z)) = aa_int_rat(ring_1_of_int_rat,Z) ) ) ).

% of_nat_nat
tff(fact_5520_of__nat__nat,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z)
     => ( aa_nat_int(semiri1314217659103216013at_int,aa_int_nat(nat2,Z)) = aa_int_int(ring_1_of_int_int,Z) ) ) ).

% of_nat_nat
tff(fact_5521_of__nat__nat,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z)
     => ( aa_nat_Code_integer(semiri4939895301339042750nteger,aa_int_nat(nat2,Z)) = aa_int_Code_integer(ring_18347121197199848620nteger,Z) ) ) ).

% of_nat_nat
tff(fact_5522_of__int__power__le__of__int__cancel__iff,axiom,
    ! [X: int,B: int,W2: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,X)),aa_nat_rat(power_power_rat(aa_int_rat(ring_1_of_int_rat,B)),W2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),aa_nat_int(aa_int_fun_nat_int(power_power_int,B),W2)) ) ).

% of_int_power_le_of_int_cancel_iff
tff(fact_5523_of__int__power__le__of__int__cancel__iff,axiom,
    ! [X: int,B: int,W2: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(ring_1_of_int_int,X)),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(ring_1_of_int_int,B)),W2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),aa_nat_int(aa_int_fun_nat_int(power_power_int,B),W2)) ) ).

% of_int_power_le_of_int_cancel_iff
tff(fact_5524_of__int__le__of__int__power__cancel__iff,axiom,
    ! [B: int,W2: nat,X: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(aa_int_rat(ring_1_of_int_rat,B)),W2)),aa_int_rat(ring_1_of_int_rat,X))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,B),W2)),X) ) ).

% of_int_le_of_int_power_cancel_iff
tff(fact_5525_of__int__le__of__int__power__cancel__iff,axiom,
    ! [B: int,W2: nat,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,aa_int_int(ring_1_of_int_int,B)),W2)),aa_int_int(ring_1_of_int_int,X))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,B),W2)),X) ) ).

% of_int_le_of_int_power_cancel_iff
tff(fact_5526_of__int__power__less__of__int__cancel__iff,axiom,
    ! [X: int,B: int,W2: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(ring_1_of_int_rat,X)),aa_nat_rat(power_power_rat(aa_int_rat(ring_1_of_int_rat,B)),W2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,X),aa_nat_int(aa_int_fun_nat_int(power_power_int,B),W2)) ) ).

% of_int_power_less_of_int_cancel_iff
tff(fact_5527_of__int__power__less__of__int__cancel__iff,axiom,
    ! [X: int,B: int,W2: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(ring_1_of_int_int,X)),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(ring_1_of_int_int,B)),W2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,X),aa_nat_int(aa_int_fun_nat_int(power_power_int,B),W2)) ) ).

% of_int_power_less_of_int_cancel_iff
tff(fact_5528_of__int__less__of__int__power__cancel__iff,axiom,
    ! [B: int,W2: nat,X: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(aa_int_rat(ring_1_of_int_rat,B)),W2)),aa_int_rat(ring_1_of_int_rat,X))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,B),W2)),X) ) ).

% of_int_less_of_int_power_cancel_iff
tff(fact_5529_of__int__less__of__int__power__cancel__iff,axiom,
    ! [B: int,W2: nat,X: 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(ring_1_of_int_int,B)),W2)),aa_int_int(ring_1_of_int_int,X))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,B),W2)),X) ) ).

% of_int_less_of_int_power_cancel_iff
tff(fact_5530_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_5531_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,aa_num_rat(numeral_numeral_rat,V)),one_one_rat)),X) ) ).

% numeral_less_floor
tff(fact_5532_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,aa_num_rat(numeral_numeral_rat,V)),one_one_rat)) ) ).

% floor_le_numeral
tff(fact_5533_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,aa_num_rat(numeral_numeral_rat,V))),X) ) ).

% neg_numeral_le_floor
tff(fact_5534_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,aa_num_rat(numeral_numeral_rat,V))) ) ).

% floor_less_neg_numeral
tff(fact_5535_of__int__le__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N2: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_int_Code_integer(ring_18347121197199848620nteger,A)),aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,X)),N2))
    <=> 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_num_int(numeral_numeral_int,X)),N2)) ) ).

% of_int_le_numeral_power_cancel_iff
tff(fact_5536_of__int__le__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N2: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,A)),aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,X)),N2))
    <=> 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_num_int(numeral_numeral_int,X)),N2)) ) ).

% of_int_le_numeral_power_cancel_iff
tff(fact_5537_of__int__le__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N2: 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_num_int(numeral_numeral_int,X)),N2))
    <=> 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_num_int(numeral_numeral_int,X)),N2)) ) ).

% of_int_le_numeral_power_cancel_iff
tff(fact_5538_numeral__power__le__of__int__cancel__iff,axiom,
    ! [X: num,N2: nat,A: int] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,X)),N2)),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_num_int(numeral_numeral_int,X)),N2)),A) ) ).

% numeral_power_le_of_int_cancel_iff
tff(fact_5539_numeral__power__le__of__int__cancel__iff,axiom,
    ! [X: num,N2: nat,A: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,X)),N2)),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_num_int(numeral_numeral_int,X)),N2)),A) ) ).

% numeral_power_le_of_int_cancel_iff
tff(fact_5540_numeral__power__le__of__int__cancel__iff,axiom,
    ! [X: num,N2: 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_num_int(numeral_numeral_int,X)),N2)),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_num_int(numeral_numeral_int,X)),N2)),A) ) ).

% numeral_power_le_of_int_cancel_iff
tff(fact_5541_of__int__less__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N2: 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_num_int(numeral_numeral_int,X)),N2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,X)),N2)) ) ).

% of_int_less_numeral_power_cancel_iff
tff(fact_5542_of__int__less__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N2: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_int_Code_integer(ring_18347121197199848620nteger,A)),aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,X)),N2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,X)),N2)) ) ).

% of_int_less_numeral_power_cancel_iff
tff(fact_5543_of__int__less__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N2: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(ring_1_of_int_rat,A)),aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,X)),N2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,X)),N2)) ) ).

% of_int_less_numeral_power_cancel_iff
tff(fact_5544_numeral__power__less__of__int__cancel__iff,axiom,
    ! [X: num,N2: 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_num_int(numeral_numeral_int,X)),N2)),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_num_int(numeral_numeral_int,X)),N2)),A) ) ).

% numeral_power_less_of_int_cancel_iff
tff(fact_5545_numeral__power__less__of__int__cancel__iff,axiom,
    ! [X: num,N2: nat,A: int] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,X)),N2)),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_num_int(numeral_numeral_int,X)),N2)),A) ) ).

% numeral_power_less_of_int_cancel_iff
tff(fact_5546_numeral__power__less__of__int__cancel__iff,axiom,
    ! [X: num,N2: nat,A: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,X)),N2)),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_num_int(numeral_numeral_int,X)),N2)),A) ) ).

% numeral_power_less_of_int_cancel_iff
tff(fact_5547_numeral__power__less__nat__cancel__iff,axiom,
    ! [X: num,N2: nat,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,X)),N2)),aa_int_nat(nat2,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,X)),N2)),A) ) ).

% numeral_power_less_nat_cancel_iff
tff(fact_5548_nat__less__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_int_nat(nat2,A)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,X)),N2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,X)),N2)) ) ).

% nat_less_numeral_power_cancel_iff
tff(fact_5549_numeral__power__le__nat__cancel__iff,axiom,
    ! [X: num,N2: nat,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,X)),N2)),aa_int_nat(nat2,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_num_int(numeral_numeral_int,X)),N2)),A) ) ).

% numeral_power_le_nat_cancel_iff
tff(fact_5550_of__int__floor__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(archim3151403230148437115or_rat,X))),X) ).

% of_int_floor_le
tff(fact_5551_le__floor__iff,axiom,
    ! [Z: int,X: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),aa_rat_int(archim3151403230148437115or_rat,X))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Z)),X) ) ).

% le_floor_iff
tff(fact_5552_floor__less__iff,axiom,
    ! [X: rat,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(archim3151403230148437115or_rat,X)),Z)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),aa_int_rat(ring_1_of_int_rat,Z)) ) ).

% floor_less_iff
tff(fact_5553_ex__le__of__int,axiom,
    ! [X: rat] :
    ? [Z2: int] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),aa_int_rat(ring_1_of_int_rat,Z2)) ).

% ex_le_of_int
tff(fact_5554_ex__less__of__int,axiom,
    ! [X: rat] :
    ? [Z2: int] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),aa_int_rat(ring_1_of_int_rat,Z2)) ).

% ex_less_of_int
tff(fact_5555_ex__of__int__less,axiom,
    ! [X: rat] :
    ? [Z2: int] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(ring_1_of_int_rat,Z2)),X) ).

% ex_of_int_less
tff(fact_5556_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_5557_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_5558_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_5559_length__induct,axiom,
    ! [Pa: fun_list_nat_o,Xs: list_nat] :
      ( ! [Xs2: list_nat] :
          ( ! [Ys: list_nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_list_nat_nat(size_size_list_nat,Ys)),aa_list_nat_nat(size_size_list_nat,Xs2))
             => aa_list_nat_o(Pa,Ys) )
         => aa_list_nat_o(Pa,Xs2) )
     => aa_list_nat_o(Pa,Xs) ) ).

% length_induct
tff(fact_5560_length__induct,axiom,
    ! [Pa: fun_list_o_o,Xs: list_o] :
      ( ! [Xs2: list_o] :
          ( ! [Ys: list_o] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,size_size_list_o(Ys)),size_size_list_o(Xs2))
             => aa_list_o_o(Pa,Ys) )
         => aa_list_o_o(Pa,Xs2) )
     => aa_list_o_o(Pa,Xs) ) ).

% length_induct
tff(fact_5561_of__nat__floor,axiom,
    ! [Ra2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Ra2)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(semiri681578069525770553at_rat,aa_int_nat(nat2,aa_rat_int(archim3151403230148437115or_rat,Ra2)))),Ra2) ) ).

% of_nat_floor
tff(fact_5562_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_5563_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_5564_floor__split,axiom,
    ! [Pa: fun_int_o,Ta: rat] :
      ( aa_int_o(Pa,aa_rat_int(archim3151403230148437115or_rat,Ta))
    <=> ! [I4: int] :
          ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,I4)),Ta)
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Ta),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_int_rat(ring_1_of_int_rat,I4)),one_one_rat)) )
         => aa_int_o(Pa,I4) ) ) ).

% floor_split
tff(fact_5565_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_5566_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_5567_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_5568_floor__mono,axiom,
    ! [X: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(archim3151403230148437115or_rat,X)),aa_rat_int(archim3151403230148437115or_rat,Y)) ) ).

% floor_mono
tff(fact_5569_floor__less__cancel,axiom,
    ! [X: rat,Y: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(archim3151403230148437115or_rat,X)),aa_rat_int(archim3151403230148437115or_rat,Y))
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y) ) ).

% floor_less_cancel
tff(fact_5570_nat__mono,axiom,
    ! [X: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Y)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(nat2,X)),aa_int_nat(nat2,Y)) ) ).

% nat_mono
tff(fact_5571_eq__nat__nat__iff,axiom,
    ! [Z: int,Z6: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z6)
       => ( ( aa_int_nat(nat2,Z) = aa_int_nat(nat2,Z6) )
        <=> ( Z = Z6 ) ) ) ) ).

% eq_nat_nat_iff
tff(fact_5572_all__nat,axiom,
    ! [Pa: fun_nat_o] :
      ( ! [X_12: nat] : aa_nat_o(Pa,X_12)
    <=> ! [X4: int] :
          ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X4)
         => aa_nat_o(Pa,aa_int_nat(nat2,X4)) ) ) ).

% all_nat
tff(fact_5573_ex__nat,axiom,
    ! [Pa: fun_nat_o] :
      ( ? [X_12: nat] : aa_nat_o(Pa,X_12)
    <=> ? [X4: int] :
          ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X4)
          & aa_nat_o(Pa,aa_int_nat(nat2,X4)) ) ) ).

% ex_nat
tff(fact_5574_le__of__int__ceiling,axiom,
    ! [X: rat] : 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))) ).

% le_of_int_ceiling
tff(fact_5575_floor__le__ceiling,axiom,
    ! [X: rat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(archim3151403230148437115or_rat,X)),aa_rat_int(archim2889992004027027881ng_rat,X)) ).

% floor_le_ceiling
tff(fact_5576_floor__divide__lower,axiom,
    ! [Q4: rat,P2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Q4)
     => 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,P2),Q4)))),Q4)),P2) ) ).

% floor_divide_lower
tff(fact_5577_of__int__of__nat,axiom,
    ! [K: int] :
      aa_int_rat(ring_1_of_int_rat,K) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int),aa_rat_rat(uminus_uminus_rat,aa_nat_rat(semiri681578069525770553at_rat,aa_int_nat(nat2,aa_int_int(uminus_uminus_int,K)))),aa_nat_rat(semiri681578069525770553at_rat,aa_int_nat(nat2,K))) ).

% of_int_of_nat
tff(fact_5578_of__int__of__nat,axiom,
    ! [K: int] :
      aa_int_int(ring_1_of_int_int,K) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int),aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_int_nat(nat2,aa_int_int(uminus_uminus_int,K)))),aa_nat_int(semiri1314217659103216013at_int,aa_int_nat(nat2,K))) ).

% of_int_of_nat
tff(fact_5579_of__int__of__nat,axiom,
    ! [K: int] :
      aa_int_Code_integer(ring_18347121197199848620nteger,K) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,aa_int_nat(nat2,aa_int_int(uminus_uminus_int,K)))),aa_nat_Code_integer(semiri4939895301339042750nteger,aa_int_nat(nat2,K))) ).

% of_int_of_nat
tff(fact_5580_floor__divide__upper,axiom,
    ! [Q4: rat,P2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Q4)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,P2),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,P2),Q4)))),one_one_rat)),Q4)) ) ).

% floor_divide_upper
tff(fact_5581_nat__mono__iff,axiom,
    ! [Z: int,W2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Z)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_int_nat(nat2,W2)),aa_int_nat(nat2,Z))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,W2),Z) ) ) ).

% nat_mono_iff
tff(fact_5582_le__floor__add,axiom,
    ! [X: rat,Y: rat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_rat_int(archim3151403230148437115or_rat,X)),aa_rat_int(archim3151403230148437115or_rat,Y))),aa_rat_int(archim3151403230148437115or_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),Y))) ).

% le_floor_add
tff(fact_5583_zless__nat__eq__int__zless,axiom,
    ! [M: nat,Z: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),aa_int_nat(nat2,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,M)),Z) ) ).

% zless_nat_eq_int_zless
tff(fact_5584_nat__le__iff,axiom,
    ! [X: int,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(nat2,X)),N2)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),aa_nat_int(semiri1314217659103216013at_int,N2)) ) ).

% nat_le_iff
tff(fact_5585_nat__0__le,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z)
     => ( aa_nat_int(semiri1314217659103216013at_int,aa_int_nat(nat2,Z)) = Z ) ) ).

% nat_0_le
tff(fact_5586_int__eq__iff,axiom,
    ! [M: nat,Z: int] :
      ( ( aa_nat_int(semiri1314217659103216013at_int,M) = Z )
    <=> ( ( M = aa_int_nat(nat2,Z) )
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z) ) ) ).

% int_eq_iff
tff(fact_5587_of__nat__ceiling,axiom,
    ! [Ra2: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Ra2),aa_nat_rat(semiri681578069525770553at_rat,aa_int_nat(nat2,aa_rat_int(archim2889992004027027881ng_rat,Ra2)))) ).

% of_nat_ceiling
tff(fact_5588_ceiling__le__iff,axiom,
    ! [X: rat,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(archim2889992004027027881ng_rat,X)),Z)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),aa_int_rat(ring_1_of_int_rat,Z)) ) ).

% ceiling_le_iff
tff(fact_5589_less__ceiling__iff,axiom,
    ! [Z: int,X: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z),aa_rat_int(archim2889992004027027881ng_rat,X))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(ring_1_of_int_rat,Z)),X) ) ).

% less_ceiling_iff
tff(fact_5590_nat__plus__as__int,axiom,
    ! [X5: nat,Xa3: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X5),Xa3) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,X5)),aa_nat_int(semiri1314217659103216013at_int,Xa3))) ).

% nat_plus_as_int
tff(fact_5591_nat__times__as__int,axiom,
    ! [X5: nat,Xa3: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,X5),Xa3) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,X5)),aa_nat_int(semiri1314217659103216013at_int,Xa3))) ).

% nat_times_as_int
tff(fact_5592_nat__minus__as__int,axiom,
    ! [X5: nat,Xa3: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,X5),Xa3) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(semiri1314217659103216013at_int,X5)),aa_nat_int(semiri1314217659103216013at_int,Xa3))) ).

% nat_minus_as_int
tff(fact_5593_nat__div__as__int,axiom,
    ! [X5: nat,Xa3: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X5),Xa3) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_nat_int(semiri1314217659103216013at_int,X5)),aa_nat_int(semiri1314217659103216013at_int,Xa3))) ).

% nat_div_as_int
tff(fact_5594_nat__mod__as__int,axiom,
    ! [X5: nat,Xa3: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X5),Xa3) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_nat_int(semiri1314217659103216013at_int,X5)),aa_nat_int(semiri1314217659103216013at_int,Xa3))) ).

% nat_mod_as_int
tff(fact_5595_of__int__nonneg,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_int_Code_integer(ring_18347121197199848620nteger,Z)) ) ).

% of_int_nonneg
tff(fact_5596_of__int__nonneg,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_int_rat(ring_1_of_int_rat,Z)) ) ).

% of_int_nonneg
tff(fact_5597_of__int__nonneg,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(ring_1_of_int_int,Z)) ) ).

% of_int_nonneg
tff(fact_5598_of__int__pos,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Z)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_int_Code_integer(ring_18347121197199848620nteger,Z)) ) ).

% of_int_pos
tff(fact_5599_of__int__pos,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Z)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_int_rat(ring_1_of_int_rat,Z)) ) ).

% of_int_pos
tff(fact_5600_of__int__pos,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Z)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(ring_1_of_int_int,Z)) ) ).

% of_int_pos
tff(fact_5601_of__int__leD,axiom,
    ! [N2: 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,N2))),X)
     => ( ( N2 = zero_zero_int )
        | aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),X) ) ) ).

% of_int_leD
tff(fact_5602_of__int__leD,axiom,
    ! [N2: 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,N2))),X)
     => ( ( N2 = zero_zero_int )
        | aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),X) ) ) ).

% of_int_leD
tff(fact_5603_of__int__leD,axiom,
    ! [N2: 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,N2))),X)
     => ( ( N2 = zero_zero_int )
        | aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),X) ) ) ).

% of_int_leD
tff(fact_5604_of__int__lessD,axiom,
    ! [N2: 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,N2))),X)
     => ( ( N2 = zero_zero_int )
        | aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),X) ) ) ).

% of_int_lessD
tff(fact_5605_of__int__lessD,axiom,
    ! [N2: 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,N2))),X)
     => ( ( N2 = zero_zero_int )
        | aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),X) ) ) ).

% of_int_lessD
tff(fact_5606_of__int__lessD,axiom,
    ! [N2: 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,N2))),X)
     => ( ( N2 = zero_zero_int )
        | aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),X) ) ) ).

% of_int_lessD
tff(fact_5607_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_5608_floor__exists1,axiom,
    ! [X: rat] :
    ? [X3: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,X3)),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,X3),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 = X3 ) ) ) ).

% floor_exists1
tff(fact_5609_nat__less__eq__zless,axiom,
    ! [W2: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),W2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_int_nat(nat2,W2)),aa_int_nat(nat2,Z))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,W2),Z) ) ) ).

% nat_less_eq_zless
tff(fact_5610_nat__le__eq__zle,axiom,
    ! [W2: int,Z: int] :
      ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),W2)
        | aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(nat2,W2)),aa_int_nat(nat2,Z))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,W2),Z) ) ) ).

% nat_le_eq_zle
tff(fact_5611_nat__eq__iff,axiom,
    ! [W2: int,M: nat] :
      ( ( aa_int_nat(nat2,W2) = M )
    <=> $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),W2),W2 = aa_nat_int(semiri1314217659103216013at_int,M),M = zero_zero_nat) ) ).

% nat_eq_iff
tff(fact_5612_nat__eq__iff2,axiom,
    ! [M: nat,W2: int] :
      ( ( M = aa_int_nat(nat2,W2) )
    <=> $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),W2),W2 = aa_nat_int(semiri1314217659103216013at_int,M),M = zero_zero_nat) ) ).

% nat_eq_iff2
tff(fact_5613_of__nat__less__of__int__iff,axiom,
    ! [N2: nat,X: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,N2)),aa_int_rat(ring_1_of_int_rat,X))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,N2)),X) ) ).

% of_nat_less_of_int_iff
tff(fact_5614_of__nat__less__of__int__iff,axiom,
    ! [N2: nat,X: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,N2)),aa_int_int(ring_1_of_int_int,X))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,N2)),X) ) ).

% of_nat_less_of_int_iff
tff(fact_5615_of__nat__less__of__int__iff,axiom,
    ! [N2: nat,X: int] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,N2)),aa_int_Code_integer(ring_18347121197199848620nteger,X))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,N2)),X) ) ).

% of_nat_less_of_int_iff
tff(fact_5616_split__nat,axiom,
    ! [Pa: fun_nat_o,I2: int] :
      ( aa_nat_o(Pa,aa_int_nat(nat2,I2))
    <=> ( ! [N: nat] :
            ( ( I2 = aa_nat_int(semiri1314217659103216013at_int,N) )
           => aa_nat_o(Pa,N) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_int,I2),zero_zero_int)
         => aa_nat_o(Pa,zero_zero_nat) ) ) ) ).

% split_nat
tff(fact_5617_le__nat__iff,axiom,
    ! [K: int,N2: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),aa_int_nat(nat2,K))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,N2)),K) ) ) ).

% le_nat_iff
tff(fact_5618_nat__add__distrib,axiom,
    ! [Z: int,Z6: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z6)
       => ( aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(plus_plus_int,Z),Z6)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_int_nat(nat2,Z)),aa_int_nat(nat2,Z6)) ) ) ) ).

% nat_add_distrib
tff(fact_5619_nat__mult__distrib,axiom,
    ! [Z: int,Z6: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z)
     => ( aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(times_times_int,Z),Z6)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_int_nat(nat2,Z)),aa_int_nat(nat2,Z6)) ) ) ).

% nat_mult_distrib
tff(fact_5620_nat__diff__distrib,axiom,
    ! [Z6: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z6)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z6),Z)
       => ( aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(minus_minus_int,Z),Z6)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_int_nat(nat2,Z)),aa_int_nat(nat2,Z6)) ) ) ) ).

% nat_diff_distrib
tff(fact_5621_nat__diff__distrib_H,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_nat(nat2,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),Y)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_int_nat(nat2,X)),aa_int_nat(nat2,Y)) ) ) ) ).

% nat_diff_distrib'
tff(fact_5622_nat__abs__triangle__ineq,axiom,
    ! [K: int,L: int] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,K),L)))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,K))),aa_int_nat(nat2,aa_int_int(abs_abs_int,L)))) ).

% nat_abs_triangle_ineq
tff(fact_5623_nat__div__distrib,axiom,
    ! [X: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X)
     => ( aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(divide_divide_int,X),Y)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_int_nat(nat2,X)),aa_int_nat(nat2,Y)) ) ) ).

% nat_div_distrib
tff(fact_5624_nat__div__distrib_H,axiom,
    ! [Y: int,X: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
     => ( aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(divide_divide_int,X),Y)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_int_nat(nat2,X)),aa_int_nat(nat2,Y)) ) ) ).

% nat_div_distrib'
tff(fact_5625_nat__mod__distrib,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_nat(nat2,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,X),Y)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_int_nat(nat2,X)),aa_int_nat(nat2,Y)) ) ) ) ).

% nat_mod_distrib
tff(fact_5626_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_5627_nat__power__eq,axiom,
    ! [Z: int,N2: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z)
     => ( aa_int_nat(nat2,aa_nat_int(aa_int_fun_nat_int(power_power_int,Z),N2)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_int_nat(nat2,Z)),N2) ) ) ).

% nat_power_eq
tff(fact_5628_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_5629_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_5630_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_5631_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_5632_ceiling__split,axiom,
    ! [Pa: fun_int_o,Ta: rat] :
      ( aa_int_o(Pa,aa_rat_int(archim2889992004027027881ng_rat,Ta))
    <=> ! [I4: 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,I4)),one_one_rat)),Ta)
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Ta),aa_int_rat(ring_1_of_int_rat,I4)) )
         => aa_int_o(Pa,I4) ) ) ).

% ceiling_split
tff(fact_5633_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_5634_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_5635_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_5636_nat__less__iff,axiom,
    ! [W2: int,M: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),W2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_int_nat(nat2,W2)),M)
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,W2),aa_nat_int(semiri1314217659103216013at_int,M)) ) ) ).

% nat_less_iff
tff(fact_5637_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_5638_nat__abs__int__diff,axiom,
    ! [A: nat,B: nat] :
      aa_int_nat(nat2,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(semiri1314217659103216013at_int,A)),aa_nat_int(semiri1314217659103216013at_int,B)))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B)) ).

% nat_abs_int_diff
tff(fact_5639_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_5640_ceiling__divide__upper,axiom,
    ! [Q4: rat,P2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Q4)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,P2),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,P2),Q4)))),Q4)) ) ).

% ceiling_divide_upper
tff(fact_5641_ceiling__divide__lower,axiom,
    ! [Q4: rat,P2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Q4)
     => 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,P2),Q4)))),one_one_rat)),Q4)),P2) ) ).

% ceiling_divide_lower
tff(fact_5642_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_5643_eucl__rel__int_Osimps,axiom,
    ! [A1: int,A22: int,A32: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(eucl_rel_int(A1,A22),A32)
    <=> ( ? [K3: int] :
            ( ( A1 = K3 )
            & ( A22 = zero_zero_int )
            & ( A32 = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,zero_zero_int),K3) ) )
        | ? [L2: int,K3: int,Q5: int] :
            ( ( A1 = K3 )
            & ( A22 = L2 )
            & ( A32 = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q5),zero_zero_int) )
            & ( L2 != zero_zero_int )
            & ( K3 = aa_int_int(aa_int_fun_int_int(times_times_int,Q5),L2) ) )
        | ? [R2: int,L2: int,K3: int,Q5: int] :
            ( ( A1 = K3 )
            & ( A22 = L2 )
            & ( A32 = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q5),R2) )
            & ( aa_int_int(sgn_sgn_int,R2) = aa_int_int(sgn_sgn_int,L2) )
            & aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,R2)),aa_int_int(abs_abs_int,L2))
            & ( K3 = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Q5),L2)),R2) ) ) ) ) ).

% eucl_rel_int.simps
tff(fact_5644_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) ) )
       => ( ! [Q3: int] :
              ( ( A32 = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q3),zero_zero_int) )
             => ( ( A22 != zero_zero_int )
               => ( A1 != aa_int_int(aa_int_fun_int_int(times_times_int,Q3),A22) ) ) )
         => ~ ! [R: int,Q3: int] :
                ( ( A32 = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q3),R) )
               => ( ( aa_int_int(sgn_sgn_int,R) = aa_int_int(sgn_sgn_int,A22) )
                 => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,R)),aa_int_int(abs_abs_int,A22))
                   => ( A1 != aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Q3),A22)),R) ) ) ) ) ) ) ) ).

% eucl_rel_int.cases
tff(fact_5645_nonempty__has__size,axiom,
    ! [S: multis2468970476368604999at_nat] :
      ( ( S != zero_z1048942125864253310at_nat )
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),size_s8510653225128441779at_nat(S)) ) ).

% nonempty_has_size
tff(fact_5646_mult__ceiling__le__Ints,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
     => ( member_rat(A,ring_1_Ints_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim2889992004027027881ng_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)))),aa_int_rat(ring_1_of_int_rat,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_Ints
tff(fact_5647_mult__ceiling__le__Ints,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
     => ( member_rat(A,ring_1_Ints_rat)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(ring_1_of_int_int,aa_rat_int(archim2889992004027027881ng_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)))),aa_int_int(ring_1_of_int_int,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_Ints
tff(fact_5648_sgn__less,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,A)),zero_z3403309356797280102nteger)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),zero_z3403309356797280102nteger) ) ).

% sgn_less
tff(fact_5649_sgn__less,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(sgn_sgn_rat,A)),zero_zero_rat)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat) ) ).

% sgn_less
tff(fact_5650_sgn__less,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(sgn_sgn_int,A)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int) ) ).

% sgn_less
tff(fact_5651_sgn__greater,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,A))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A) ) ).

% sgn_greater
tff(fact_5652_sgn__greater,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(sgn_sgn_rat,A))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A) ) ).

% sgn_greater
tff(fact_5653_sgn__greater,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(sgn_sgn_int,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A) ) ).

% sgn_greater
tff(fact_5654_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_5655_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_5656_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_5657_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_5658_frac__gt__0__iff,axiom,
    ! [X: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(archimedean_frac_rat,X))
    <=> ~ member_rat(X,ring_1_Ints_rat) ) ).

% frac_gt_0_iff
tff(fact_5659_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_5660_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_5661_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_5662_union__diff__assoc,axiom,
    ! [C2: multis2468970476368604999at_nat,B3: multis2468970476368604999at_nat,A3: multis2468970476368604999at_nat] :
      ( ( minus_4286766774447292334at_nat(C2,B3) = zero_z1048942125864253310at_nat )
     => ( minus_4286766774447292334at_nat(aa_mul5754855110707352700at_nat(plus_p7104986032573967614at_nat(A3),B3),C2) = aa_mul5754855110707352700at_nat(plus_p7104986032573967614at_nat(A3),minus_4286766774447292334at_nat(B3,C2)) ) ) ).

% union_diff_assoc
tff(fact_5663_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_5664_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_5665_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_5666_Ints__mult,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_set5643387332196575111eger_o(member_Code_integer(A),ring_11222124179247155820nteger)
     => ( aa_set5643387332196575111eger_o(member_Code_integer(B),ring_11222124179247155820nteger)
       => aa_set5643387332196575111eger_o(member_Code_integer(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),ring_11222124179247155820nteger) ) ) ).

% Ints_mult
tff(fact_5667_Ints__mult,axiom,
    ! [A: rat,B: rat] :
      ( member_rat(A,ring_1_Ints_rat)
     => ( member_rat(B,ring_1_Ints_rat)
       => member_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B),ring_1_Ints_rat) ) ) ).

% Ints_mult
tff(fact_5668_Ints__mult,axiom,
    ! [A: int,B: int] :
      ( aa_set_int_o(member_int(A),ring_1_Ints_int)
     => ( aa_set_int_o(member_int(B),ring_1_Ints_int)
       => aa_set_int_o(member_int(aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),ring_1_Ints_int) ) ) ).

% Ints_mult
tff(fact_5669_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_5670_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_5671_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_5672_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_5673_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_5674_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_5675_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_5676_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_5677_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_5678_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_5679_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_5680_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_5681_frac__ge__0,axiom,
    ! [X: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(archimedean_frac_rat,X)) ).

% frac_ge_0
tff(fact_5682_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_5683_frac__unique__iff,axiom,
    ! [X: rat,A: rat] :
      ( ( aa_rat_rat(archimedean_frac_rat,X) = A )
    <=> ( member_rat(aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,X),A),ring_1_Ints_rat)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),one_one_rat) ) ) ).

% frac_unique_iff
tff(fact_5684_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_5685_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_5686_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_5687_zsgn__def,axiom,
    ! [I2: int] :
      aa_int_int(sgn_sgn_int,I2) = $ite(
        I2 = zero_zero_int,
        zero_zero_int,
        $ite(aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),I2),one_one_int,aa_int_int(uminus_uminus_int,one_one_int)) ) ).

% zsgn_def
tff(fact_5688_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_5689_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_5690_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_5691_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_5692_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_5693_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_5694_Ints__odd__less__0,axiom,
    ! [A: code_integer] :
      ( aa_set5643387332196575111eger_o(member_Code_integer(A),ring_11222124179247155820nteger)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(one_one_Code_integer),A)),A)),zero_z3403309356797280102nteger)
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),zero_z3403309356797280102nteger) ) ) ).

% Ints_odd_less_0
tff(fact_5695_Ints__odd__less__0,axiom,
    ! [A: rat] :
      ( member_rat(A,ring_1_Ints_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(plus_plus_rat,one_one_rat),A)),A)),zero_zero_rat)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat) ) ) ).

% Ints_odd_less_0
tff(fact_5696_Ints__odd__less__0,axiom,
    ! [A: int] :
      ( aa_set_int_o(member_int(A),ring_1_Ints_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),A)),A)),zero_zero_int)
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int) ) ) ).

% Ints_odd_less_0
tff(fact_5697_Ints__nonzero__abs__ge1,axiom,
    ! [X: code_integer] :
      ( aa_set5643387332196575111eger_o(member_Code_integer(X),ring_11222124179247155820nteger)
     => ( ( X != zero_z3403309356797280102nteger )
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(abs_abs_Code_integer,X)) ) ) ).

% Ints_nonzero_abs_ge1
tff(fact_5698_Ints__nonzero__abs__ge1,axiom,
    ! [X: rat] :
      ( member_rat(X,ring_1_Ints_rat)
     => ( ( X != zero_zero_rat )
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_rat_rat(abs_abs_rat,X)) ) ) ).

% Ints_nonzero_abs_ge1
tff(fact_5699_Ints__nonzero__abs__ge1,axiom,
    ! [X: int] :
      ( aa_set_int_o(member_int(X),ring_1_Ints_int)
     => ( ( X != zero_zero_int )
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_int_int(abs_abs_int,X)) ) ) ).

% Ints_nonzero_abs_ge1
tff(fact_5700_Ints__nonzero__abs__less1,axiom,
    ! [X: code_integer] :
      ( aa_set5643387332196575111eger_o(member_Code_integer(X),ring_11222124179247155820nteger)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,X)),one_one_Code_integer)
       => ( X = zero_z3403309356797280102nteger ) ) ) ).

% Ints_nonzero_abs_less1
tff(fact_5701_Ints__nonzero__abs__less1,axiom,
    ! [X: rat] :
      ( member_rat(X,ring_1_Ints_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(abs_abs_rat,X)),one_one_rat)
       => ( X = zero_zero_rat ) ) ) ).

% Ints_nonzero_abs_less1
tff(fact_5702_Ints__nonzero__abs__less1,axiom,
    ! [X: int] :
      ( aa_set_int_o(member_int(X),ring_1_Ints_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,X)),one_one_int)
       => ( X = zero_zero_int ) ) ) ).

% Ints_nonzero_abs_less1
tff(fact_5703_Ints__eq__abs__less1,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( aa_set5643387332196575111eger_o(member_Code_integer(X),ring_11222124179247155820nteger)
     => ( aa_set5643387332196575111eger_o(member_Code_integer(Y),ring_11222124179247155820nteger)
       => ( ( X = Y )
        <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(X),Y))),one_one_Code_integer) ) ) ) ).

% Ints_eq_abs_less1
tff(fact_5704_Ints__eq__abs__less1,axiom,
    ! [X: rat,Y: rat] :
      ( member_rat(X,ring_1_Ints_rat)
     => ( member_rat(Y,ring_1_Ints_rat)
       => ( ( X = Y )
        <=> 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),Y))),one_one_rat) ) ) ) ).

% Ints_eq_abs_less1
tff(fact_5705_Ints__eq__abs__less1,axiom,
    ! [X: int,Y: int] :
      ( aa_set_int_o(member_int(X),ring_1_Ints_int)
     => ( aa_set_int_o(member_int(Y),ring_1_Ints_int)
       => ( ( X = Y )
        <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),Y))),one_one_int) ) ) ) ).

% Ints_eq_abs_less1
tff(fact_5706_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_5707_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_5708_eucl__rel__int__remainderI,axiom,
    ! [Ra2: int,L: int,K: int,Q4: int] :
      ( ( aa_int_int(sgn_sgn_int,Ra2) = aa_int_int(sgn_sgn_int,L) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,Ra2)),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,Q4),L)),Ra2) )
         => aa_Pro4592068669816069595_int_o(eucl_rel_int(K,L),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q4),Ra2)) ) ) ) ).

% eucl_rel_int_remainderI
tff(fact_5709_le__mult__floor__Ints,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
     => ( member_rat(A,ring_1_Ints_rat)
       => 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(times_times_int,aa_rat_int(archim3151403230148437115or_rat,A)),aa_rat_int(archim3151403230148437115or_rat,B)))),aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim3151403230148437115or_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)))) ) ) ).

% le_mult_floor_Ints
tff(fact_5710_le__mult__floor__Ints,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
     => ( member_rat(A,ring_1_Ints_rat)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(ring_1_of_int_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_int_int(ring_1_of_int_int,aa_rat_int(archim3151403230148437115or_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)))) ) ) ).

% le_mult_floor_Ints
tff(fact_5711_slice__len,axiom,
    ! [From: nat,To: nat,Xs: list_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,From),To)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,To),aa_list_nat_nat(size_size_list_nat,Xs))
       => ( aa_list_nat_nat(size_size_list_nat,slice_nat(From,To,Xs)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,To),From) ) ) ) ).

% slice_len
tff(fact_5712_slice__len,axiom,
    ! [From: nat,To: nat,Xs: list_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,From),To)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,To),size_size_list_o(Xs))
       => ( size_size_list_o(slice_o(From,To,Xs)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,To),From) ) ) ) ).

% slice_len
tff(fact_5713_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_5714_pochhammer__same,axiom,
    ! [N2: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(uminus_uminus_rat,aa_nat_rat(semiri681578069525770553at_rat,N2))),N2) = 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)),N2)),aa_nat_rat(semiri773545260158071498ct_rat,N2)) ).

% pochhammer_same
tff(fact_5715_pochhammer__same,axiom,
    ! [N2: nat] : aa_nat_int(aa_int_fun_nat_int(comm_s4660882817536571857er_int,aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N2))),N2) = 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)),N2)),aa_nat_int(semiri1406184849735516958ct_int,N2)) ).

% pochhammer_same
tff(fact_5716_pochhammer__same,axiom,
    ! [N2: nat] : aa_nat_Code_integer(comm_s8582702949713902594nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,N2))),N2) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),N2)),aa_nat_Code_integer(semiri3624122377584611663nteger,N2)) ).

% pochhammer_same
tff(fact_5717_fact__reduce,axiom,
    ! [N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)
     => ( aa_nat_rat(semiri773545260158071498ct_rat,N2) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,N2)),aa_nat_rat(semiri773545260158071498ct_rat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),one_one_nat))) ) ) ).

% fact_reduce
tff(fact_5718_fact__reduce,axiom,
    ! [N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)
     => ( aa_nat_int(semiri1406184849735516958ct_int,N2) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,N2)),aa_nat_int(semiri1406184849735516958ct_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),one_one_nat))) ) ) ).

% fact_reduce
tff(fact_5719_fact__reduce,axiom,
    ! [N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)
     => ( aa_nat_Code_integer(semiri3624122377584611663nteger,N2) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,N2)),aa_nat_Code_integer(semiri3624122377584611663nteger,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),one_one_nat))) ) ) ).

% fact_reduce
tff(fact_5720_fact__reduce,axiom,
    ! [N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)
     => ( aa_nat_nat(semiri1408675320244567234ct_nat,N2) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,N2)),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),one_one_nat))) ) ) ).

% fact_reduce
tff(fact_5721_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_5722_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_5723_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_5724_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_5725_rotate1__length01,axiom,
    ! [Xs: list_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_list_nat_nat(size_size_list_nat,Xs)),one_one_nat)
     => ( rotate1_nat(Xs) = Xs ) ) ).

% rotate1_length01
tff(fact_5726_rotate1__length01,axiom,
    ! [Xs: list_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_size_list_o(Xs)),one_one_nat)
     => ( rotate1_o(Xs) = Xs ) ) ).

% rotate1_length01
tff(fact_5727_cpmi,axiom,
    ! [D3: int,Pa: fun_int_o,P3: fun_int_o,B3: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D3)
     => ( ? [Z4: int] :
          ! [X3: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Z4)
           => ( aa_int_o(Pa,X3)
            <=> aa_int_o(P3,X3) ) )
       => ( ! [X3: 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),B3)
                     => ( X3 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb2),Xa3) ) ) )
             => ( aa_int_o(Pa,X3)
               => aa_int_o(Pa,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),D3)) ) )
         => ( ! [X3: int,K2: int] :
                ( aa_int_o(P3,X3)
              <=> aa_int_o(P3,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),aa_int_int(aa_int_fun_int_int(times_times_int,K2),D3))) )
           => ( ? [X_12: int] : aa_int_o(Pa,X_12)
            <=> ( ? [X4: int] :
                    ( aa_set_int_o(member_int(X4),set_or1266510415728281911st_int(one_one_int,D3))
                    & aa_int_o(P3,X4) )
                | ? [X4: int] :
                    ( aa_set_int_o(member_int(X4),set_or1266510415728281911st_int(one_one_int,D3))
                    & ? [Xa: int] :
                        ( aa_set_int_o(member_int(Xa),B3)
                        & aa_int_o(Pa,aa_int_int(aa_int_fun_int_int(plus_plus_int,Xa),X4)) ) ) ) ) ) ) ) ) ).

% cpmi
tff(fact_5728_Icc__eq__Icc,axiom,
    ! [L: set_int,Ha: set_int,L3: set_int,H_a: set_int] :
      ( ( set_or370866239135849197et_int(L,Ha) = set_or370866239135849197et_int(L3,H_a) )
    <=> ( ( ( L = L3 )
          & ( Ha = H_a ) )
        | ( ~ aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,L),Ha)
          & ~ aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,L3),H_a) ) ) ) ).

% Icc_eq_Icc
tff(fact_5729_Icc__eq__Icc,axiom,
    ! [L: rat,Ha: rat,L3: rat,H_a: rat] :
      ( ( set_or633870826150836451st_rat(L,Ha) = set_or633870826150836451st_rat(L3,H_a) )
    <=> ( ( ( L = L3 )
          & ( Ha = H_a ) )
        | ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,L),Ha)
          & ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,L3),H_a) ) ) ) ).

% Icc_eq_Icc
tff(fact_5730_Icc__eq__Icc,axiom,
    ! [L: num,Ha: num,L3: num,H_a: num] :
      ( ( set_or7049704709247886629st_num(L,Ha) = set_or7049704709247886629st_num(L3,H_a) )
    <=> ( ( ( L = L3 )
          & ( Ha = H_a ) )
        | ( ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,L),Ha)
          & ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,L3),H_a) ) ) ) ).

% Icc_eq_Icc
tff(fact_5731_Icc__eq__Icc,axiom,
    ! [L: int,Ha: int,L3: int,H_a: int] :
      ( ( set_or1266510415728281911st_int(L,Ha) = set_or1266510415728281911st_int(L3,H_a) )
    <=> ( ( ( L = L3 )
          & ( Ha = H_a ) )
        | ( ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,L),Ha)
          & ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,L3),H_a) ) ) ) ).

% Icc_eq_Icc
tff(fact_5732_Icc__eq__Icc,axiom,
    ! [L: nat,Ha: nat,L3: nat,H_a: nat] :
      ( ( aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,L),Ha) = aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,L3),H_a) )
    <=> ( ( ( L = L3 )
          & ( Ha = H_a ) )
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,L),Ha)
          & ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,L3),H_a) ) ) ) ).

% Icc_eq_Icc
tff(fact_5733_Icc__eq__Icc,axiom,
    ! [L: code_integer,Ha: code_integer,L3: code_integer,H_a: code_integer] :
      ( ( set_or189985376899183464nteger(L,Ha) = set_or189985376899183464nteger(L3,H_a) )
    <=> ( ( ( L = L3 )
          & ( Ha = H_a ) )
        | ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,L),Ha)
          & ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,L3),H_a) ) ) ) ).

% Icc_eq_Icc
tff(fact_5734_atLeastAtMost__iff,axiom,
    ! [I2: $o,L: $o,U: $o] :
      ( member_o((I2),set_or8904488021354931149Most_o((L),(U)))
    <=> ( aa_o_o(aa_o_fun_o_o(ord_less_eq_o,(L)),(I2))
        & aa_o_o(aa_o_fun_o_o(ord_less_eq_o,(I2)),(U)) ) ) ).

% atLeastAtMost_iff
tff(fact_5735_atLeastAtMost__iff,axiom,
    ! [I2: set_nat,L: set_nat,U: set_nat] :
      ( aa_set_set_nat_o(member_set_nat(I2),set_or4548717258645045905et_nat(L,U))
    <=> ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,L),I2)
        & aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,I2),U) ) ) ).

% atLeastAtMost_iff
tff(fact_5736_atLeastAtMost__iff,axiom,
    ! [I2: set_int,L: set_int,U: set_int] :
      ( member_set_int(I2,set_or370866239135849197et_int(L,U))
    <=> ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,L),I2)
        & aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,I2),U) ) ) ).

% atLeastAtMost_iff
tff(fact_5737_atLeastAtMost__iff,axiom,
    ! [I2: rat,L: rat,U: rat] :
      ( member_rat(I2,set_or633870826150836451st_rat(L,U))
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,L),I2)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,I2),U) ) ) ).

% atLeastAtMost_iff
tff(fact_5738_atLeastAtMost__iff,axiom,
    ! [I2: num,L: num,U: num] :
      ( member_num(I2,set_or7049704709247886629st_num(L,U))
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,L),I2)
        & aa_num_o(aa_num_fun_num_o(ord_less_eq_num,I2),U) ) ) ).

% atLeastAtMost_iff
tff(fact_5739_atLeastAtMost__iff,axiom,
    ! [I2: int,L: int,U: int] :
      ( aa_set_int_o(member_int(I2),set_or1266510415728281911st_int(L,U))
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,L),I2)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I2),U) ) ) ).

% atLeastAtMost_iff
tff(fact_5740_atLeastAtMost__iff,axiom,
    ! [I2: nat,L: nat,U: nat] :
      ( aa_set_nat_o(member_nat(I2),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,L),U))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,L),I2)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),U) ) ) ).

% atLeastAtMost_iff
tff(fact_5741_atLeastAtMost__iff,axiom,
    ! [I2: code_integer,L: code_integer,U: code_integer] :
      ( aa_set5643387332196575111eger_o(member_Code_integer(I2),set_or189985376899183464nteger(L,U))
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,L),I2)
        & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,I2),U) ) ) ).

% atLeastAtMost_iff
tff(fact_5742_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_5743_atLeastatMost__empty__iff2,axiom,
    ! [A: set_int,B: set_int] :
      ( ( bot_bot_set_set_int = set_or370866239135849197et_int(A,B) )
    <=> ~ aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A),B) ) ).

% atLeastatMost_empty_iff2
tff(fact_5744_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_5745_atLeastatMost__empty__iff2,axiom,
    ! [A: num,B: num] :
      ( ( bot_bot_set_num = set_or7049704709247886629st_num(A,B) )
    <=> ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B) ) ).

% atLeastatMost_empty_iff2
tff(fact_5746_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_5747_atLeastatMost__empty__iff2,axiom,
    ! [A: nat,B: nat] :
      ( ( bot_bot_set_nat = aa_nat_set_nat(aa_nat1818925681583257441et_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_5748_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_5749_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_5750_atLeastatMost__empty__iff,axiom,
    ! [A: set_int,B: set_int] :
      ( ( set_or370866239135849197et_int(A,B) = bot_bot_set_set_int )
    <=> ~ aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A),B) ) ).

% atLeastatMost_empty_iff
tff(fact_5751_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_5752_atLeastatMost__empty__iff,axiom,
    ! [A: num,B: num] :
      ( ( set_or7049704709247886629st_num(A,B) = bot_bot_set_num )
    <=> ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B) ) ).

% atLeastatMost_empty_iff
tff(fact_5753_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_5754_atLeastatMost__empty__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( aa_nat_set_nat(aa_nat1818925681583257441et_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_5755_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_5756_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_5757_atLeastatMost__empty,axiom,
    ! [B: assn,A: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,B),A)
     => ( set_or7959216805967363635t_assn(A,B) = bot_bot_set_assn ) ) ).

% atLeastatMost_empty
tff(fact_5758_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_5759_atLeastatMost__empty,axiom,
    ! [B: num,A: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,B),A)
     => ( set_or7049704709247886629st_num(A,B) = bot_bot_set_num ) ) ).

% atLeastatMost_empty
tff(fact_5760_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_5761_atLeastatMost__empty,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A)
     => ( aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,A),B) = bot_bot_set_nat ) ) ).

% atLeastatMost_empty
tff(fact_5762_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_5763_atLeastatMost__subset__iff,axiom,
    ! [A: set_int,B: set_int,Ca: set_int,D2: set_int] :
      ( ord_le4403425263959731960et_int(set_or370866239135849197et_int(A,B),set_or370866239135849197et_int(Ca,D2))
    <=> ( ~ aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A),B)
        | ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,Ca),A)
          & aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,B),D2) ) ) ) ).

% atLeastatMost_subset_iff
tff(fact_5764_atLeastatMost__subset__iff,axiom,
    ! [A: rat,B: rat,Ca: rat,D2: rat] :
      ( ord_less_eq_set_rat(set_or633870826150836451st_rat(A,B),set_or633870826150836451st_rat(Ca,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,Ca),A)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),D2) ) ) ) ).

% atLeastatMost_subset_iff
tff(fact_5765_atLeastatMost__subset__iff,axiom,
    ! [A: num,B: num,Ca: num,D2: num] :
      ( ord_less_eq_set_num(set_or7049704709247886629st_num(A,B),set_or7049704709247886629st_num(Ca,D2))
    <=> ( ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
        | ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Ca),A)
          & aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),D2) ) ) ) ).

% atLeastatMost_subset_iff
tff(fact_5766_atLeastatMost__subset__iff,axiom,
    ! [A: int,B: int,Ca: int,D2: int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,set_or1266510415728281911st_int(A,B)),set_or1266510415728281911st_int(Ca,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,Ca),A)
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),D2) ) ) ) ).

% atLeastatMost_subset_iff
tff(fact_5767_atLeastatMost__subset__iff,axiom,
    ! [A: nat,B: nat,Ca: nat,D2: nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,A),B)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,Ca),D2))
    <=> ( ~ 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,Ca),A)
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),D2) ) ) ) ).

% atLeastatMost_subset_iff
tff(fact_5768_atLeastatMost__subset__iff,axiom,
    ! [A: code_integer,B: code_integer,Ca: code_integer,D2: code_integer] :
      ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(set_or189985376899183464nteger(A,B)),set_or189985376899183464nteger(Ca,D2))
    <=> ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B)
        | ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Ca),A)
          & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),D2) ) ) ) ).

% atLeastatMost_subset_iff
tff(fact_5769_slice__complete,axiom,
    ! [Xs: list_nat] : slice_nat(zero_zero_nat,aa_list_nat_nat(size_size_list_nat,Xs),Xs) = Xs ).

% slice_complete
tff(fact_5770_slice__complete,axiom,
    ! [Xs: list_o] : slice_o(zero_zero_nat,size_size_list_o(Xs),Xs) = Xs ).

% slice_complete
tff(fact_5771_fact__Suc,axiom,
    ! [N2: nat] : aa_nat_rat(semiri773545260158071498ct_rat,aa_nat_nat(suc,N2)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,N2))),aa_nat_rat(semiri773545260158071498ct_rat,N2)) ).

% fact_Suc
tff(fact_5772_fact__Suc,axiom,
    ! [N2: nat] : aa_nat_int(semiri1406184849735516958ct_int,aa_nat_nat(suc,N2)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,N2))),aa_nat_int(semiri1406184849735516958ct_int,N2)) ).

% fact_Suc
tff(fact_5773_fact__Suc,axiom,
    ! [N2: nat] : aa_nat_Code_integer(semiri3624122377584611663nteger,aa_nat_nat(suc,N2)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(suc,N2))),aa_nat_Code_integer(semiri3624122377584611663nteger,N2)) ).

% fact_Suc
tff(fact_5774_fact__Suc,axiom,
    ! [N2: nat] : aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(suc,N2)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(suc,N2))),aa_nat_nat(semiri1408675320244567234ct_nat,N2)) ).

% fact_Suc
tff(fact_5775_mset__distrib,axiom,
    ! [A3: multis2468970476368604999at_nat,B3: multis2468970476368604999at_nat,M6: multis2468970476368604999at_nat,N7: multis2468970476368604999at_nat] :
      ( ( aa_mul5754855110707352700at_nat(plus_p7104986032573967614at_nat(A3),B3) = aa_mul5754855110707352700at_nat(plus_p7104986032573967614at_nat(M6),N7) )
     => ~ ! [Am: multis2468970476368604999at_nat,An: multis2468970476368604999at_nat] :
            ( ( A3 = aa_mul5754855110707352700at_nat(plus_p7104986032573967614at_nat(Am),An) )
           => ! [Bm: multis2468970476368604999at_nat,Bn: multis2468970476368604999at_nat] :
                ( ( B3 = aa_mul5754855110707352700at_nat(plus_p7104986032573967614at_nat(Bm),Bn) )
               => ( ( M6 = aa_mul5754855110707352700at_nat(plus_p7104986032573967614at_nat(Am),Bm) )
                 => ( N7 != aa_mul5754855110707352700at_nat(plus_p7104986032573967614at_nat(An),Bn) ) ) ) ) ) ).

% mset_distrib
tff(fact_5776_fact__mono__nat,axiom,
    ! [M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(semiri1408675320244567234ct_nat,M)),aa_nat_nat(semiri1408675320244567234ct_nat,N2)) ) ).

% fact_mono_nat
tff(fact_5777_fact__ge__self,axiom,
    ! [N2: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),aa_nat_nat(semiri1408675320244567234ct_nat,N2)) ).

% fact_ge_self
tff(fact_5778_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)
       => ( 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_5779_ivl__disj__un__two__touch_I4_J,axiom,
    ! [L: num,M: num,U: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,L),M)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,M),U)
       => ( sup_sup_set_num(set_or7049704709247886629st_num(L,M),set_or7049704709247886629st_num(M,U)) = set_or7049704709247886629st_num(L,U) ) ) ) ).

% ivl_disj_un_two_touch(4)
tff(fact_5780_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_5781_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(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,L),M)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M),U)) = aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,L),U) ) ) ) ).

% ivl_disj_un_two_touch(4)
tff(fact_5782_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_5783_fact__less__mono__nat,axiom,
    ! [M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(semiri1408675320244567234ct_nat,M)),aa_nat_nat(semiri1408675320244567234ct_nat,N2)) ) ) ).

% fact_less_mono_nat
tff(fact_5784_fact__ge__zero,axiom,
    ! [N2: nat] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(semiri3624122377584611663nteger,N2)) ).

% fact_ge_zero
tff(fact_5785_fact__ge__zero,axiom,
    ! [N2: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(semiri773545260158071498ct_rat,N2)) ).

% fact_ge_zero
tff(fact_5786_fact__ge__zero,axiom,
    ! [N2: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(semiri1406184849735516958ct_int,N2)) ).

% fact_ge_zero
tff(fact_5787_fact__ge__zero,axiom,
    ! [N2: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(semiri1408675320244567234ct_nat,N2)) ).

% fact_ge_zero
tff(fact_5788_fact__not__neg,axiom,
    ! [N2: nat] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(semiri3624122377584611663nteger,N2)),zero_z3403309356797280102nteger) ).

% fact_not_neg
tff(fact_5789_fact__not__neg,axiom,
    ! [N2: nat] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(semiri773545260158071498ct_rat,N2)),zero_zero_rat) ).

% fact_not_neg
tff(fact_5790_fact__not__neg,axiom,
    ! [N2: nat] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1406184849735516958ct_int,N2)),zero_zero_int) ).

% fact_not_neg
tff(fact_5791_fact__not__neg,axiom,
    ! [N2: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(semiri1408675320244567234ct_nat,N2)),zero_zero_nat) ).

% fact_not_neg
tff(fact_5792_fact__gt__zero,axiom,
    ! [N2: nat] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(semiri3624122377584611663nteger,N2)) ).

% fact_gt_zero
tff(fact_5793_fact__gt__zero,axiom,
    ! [N2: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_nat_rat(semiri773545260158071498ct_rat,N2)) ).

% fact_gt_zero
tff(fact_5794_fact__gt__zero,axiom,
    ! [N2: nat] : aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(semiri1406184849735516958ct_int,N2)) ).

% fact_gt_zero
tff(fact_5795_fact__gt__zero,axiom,
    ! [N2: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(semiri1408675320244567234ct_nat,N2)) ).

% fact_gt_zero
tff(fact_5796_fact__ge__1,axiom,
    ! [N2: nat] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),aa_nat_Code_integer(semiri3624122377584611663nteger,N2)) ).

% fact_ge_1
tff(fact_5797_fact__ge__1,axiom,
    ! [N2: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_nat_rat(semiri773545260158071498ct_rat,N2)) ).

% fact_ge_1
tff(fact_5798_fact__ge__1,axiom,
    ! [N2: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_nat_int(semiri1406184849735516958ct_int,N2)) ).

% fact_ge_1
tff(fact_5799_fact__ge__1,axiom,
    ! [N2: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),aa_nat_nat(semiri1408675320244567234ct_nat,N2)) ).

% fact_ge_1
tff(fact_5800_fact__mono,axiom,
    ! [M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(semiri773545260158071498ct_rat,M)),aa_nat_rat(semiri773545260158071498ct_rat,N2)) ) ).

% fact_mono
tff(fact_5801_fact__mono,axiom,
    ! [M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1406184849735516958ct_int,M)),aa_nat_int(semiri1406184849735516958ct_int,N2)) ) ).

% fact_mono
tff(fact_5802_fact__mono,axiom,
    ! [M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(semiri1408675320244567234ct_nat,M)),aa_nat_nat(semiri1408675320244567234ct_nat,N2)) ) ).

% fact_mono
tff(fact_5803_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_5804_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_5805_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_5806_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_5807_gbinomial__ge__n__over__k__pow__k,axiom,
    ! [K: nat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(semiri681578069525770553at_rat,K)),A)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),aa_nat_rat(semiri681578069525770553at_rat,K))),K)),aa_nat_rat(gbinomial_rat(A),K)) ) ).

% gbinomial_ge_n_over_k_pow_k
tff(fact_5808_atLeastatMost__psubset__iff,axiom,
    ! [A: assn,B: assn,Ca: assn,D2: assn] :
      ( ord_less_set_assn(set_or7959216805967363635t_assn(A,B),set_or7959216805967363635t_assn(Ca,D2))
    <=> ( ( ~ aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),B)
          | ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,Ca),A)
            & aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,B),D2)
            & ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,Ca),A)
              | aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,B),D2) ) ) )
        & aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,Ca),D2) ) ) ).

% atLeastatMost_psubset_iff
tff(fact_5809_atLeastatMost__psubset__iff,axiom,
    ! [A: set_int,B: set_int,Ca: set_int,D2: set_int] :
      ( ord_less_set_set_int(set_or370866239135849197et_int(A,B),set_or370866239135849197et_int(Ca,D2))
    <=> ( ( ~ aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A),B)
          | ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,Ca),A)
            & aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,B),D2)
            & ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_set_int,Ca),A)
              | aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_set_int,B),D2) ) ) )
        & aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,Ca),D2) ) ) ).

% atLeastatMost_psubset_iff
tff(fact_5810_atLeastatMost__psubset__iff,axiom,
    ! [A: rat,B: rat,Ca: rat,D2: rat] :
      ( ord_less_set_rat(set_or633870826150836451st_rat(A,B),set_or633870826150836451st_rat(Ca,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,Ca),A)
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),D2)
            & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Ca),A)
              | aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),D2) ) ) )
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Ca),D2) ) ) ).

% atLeastatMost_psubset_iff
tff(fact_5811_atLeastatMost__psubset__iff,axiom,
    ! [A: num,B: num,Ca: num,D2: num] :
      ( ord_less_set_num(set_or7049704709247886629st_num(A,B),set_or7049704709247886629st_num(Ca,D2))
    <=> ( ( ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
          | ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Ca),A)
            & aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),D2)
            & ( aa_num_o(aa_num_fun_num_o(ord_less_num,Ca),A)
              | aa_num_o(aa_num_fun_num_o(ord_less_num,B),D2) ) ) )
        & aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Ca),D2) ) ) ).

% atLeastatMost_psubset_iff
tff(fact_5812_atLeastatMost__psubset__iff,axiom,
    ! [A: int,B: int,Ca: int,D2: int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_set_int,set_or1266510415728281911st_int(A,B)),set_or1266510415728281911st_int(Ca,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,Ca),A)
            & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),D2)
            & ( aa_int_o(aa_int_fun_int_o(ord_less_int,Ca),A)
              | aa_int_o(aa_int_fun_int_o(ord_less_int,B),D2) ) ) )
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ca),D2) ) ) ).

% atLeastatMost_psubset_iff
tff(fact_5813_atLeastatMost__psubset__iff,axiom,
    ! [A: nat,B: nat,Ca: nat,D2: nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,A),B)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,Ca),D2))
    <=> ( ( ~ 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,Ca),A)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),D2)
            & ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ca),A)
              | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),D2) ) ) )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ca),D2) ) ) ).

% atLeastatMost_psubset_iff
tff(fact_5814_atLeastatMost__psubset__iff,axiom,
    ! [A: code_integer,B: code_integer,Ca: code_integer,D2: code_integer] :
      ( aa_set5643387332196575111eger_o(ord_le1307284697595431911nteger(set_or189985376899183464nteger(A,B)),set_or189985376899183464nteger(Ca,D2))
    <=> ( ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B)
          | ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Ca),A)
            & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),D2)
            & ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Ca),A)
              | aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),D2) ) ) )
        & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Ca),D2) ) ) ).

% atLeastatMost_psubset_iff
tff(fact_5815_fact__ge__Suc__0__nat,axiom,
    ! [N2: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),aa_nat_nat(semiri1408675320244567234ct_nat,N2)) ).

% fact_ge_Suc_0_nat
tff(fact_5816_fact__less__mono,axiom,
    ! [M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(semiri773545260158071498ct_rat,M)),aa_nat_rat(semiri773545260158071498ct_rat,N2)) ) ) ).

% fact_less_mono
tff(fact_5817_fact__less__mono,axiom,
    ! [M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N2)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1406184849735516958ct_int,M)),aa_nat_int(semiri1406184849735516958ct_int,N2)) ) ) ).

% fact_less_mono
tff(fact_5818_fact__less__mono,axiom,
    ! [M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(semiri1408675320244567234ct_nat,M)),aa_nat_nat(semiri1408675320244567234ct_nat,N2)) ) ) ).

% fact_less_mono
tff(fact_5819_fact__mod,axiom,
    ! [M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
     => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_nat_int(semiri1406184849735516958ct_int,N2)),aa_nat_int(semiri1406184849735516958ct_int,M)) = zero_zero_int ) ) ).

% fact_mod
tff(fact_5820_fact__mod,axiom,
    ! [M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
     => ( aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_nat_Code_integer(semiri3624122377584611663nteger,N2)),aa_nat_Code_integer(semiri3624122377584611663nteger,M)) = zero_z3403309356797280102nteger ) ) ).

% fact_mod
tff(fact_5821_fact__mod,axiom,
    ! [M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
     => ( modulo8411746178871703098atural(aa_nat_Code_natural(semiri2447717529341329178atural,N2),aa_nat_Code_natural(semiri2447717529341329178atural,M)) = zero_z2226904508553997617atural ) ) ).

% fact_mod
tff(fact_5822_fact__mod,axiom,
    ! [M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(semiri1408675320244567234ct_nat,N2)),aa_nat_nat(semiri1408675320244567234ct_nat,M)) = zero_zero_nat ) ) ).

% fact_mod
tff(fact_5823_fact__le__power,axiom,
    ! [N2: nat] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(semiri3624122377584611663nteger,N2)),aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,N2),N2))) ).

% fact_le_power
tff(fact_5824_fact__le__power,axiom,
    ! [N2: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(semiri773545260158071498ct_rat,N2)),aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,N2),N2))) ).

% fact_le_power
tff(fact_5825_fact__le__power,axiom,
    ! [N2: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1406184849735516958ct_int,N2)),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,N2),N2))) ).

% fact_le_power
tff(fact_5826_fact__le__power,axiom,
    ! [N2: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(semiri1408675320244567234ct_nat,N2)),aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,N2),N2))) ).

% fact_le_power
tff(fact_5827_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_5828_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_5829_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_5830_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_hm(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_5831_fact__div__fact__le__pow,axiom,
    ! [Ra2: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ra2),N2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(semiri1408675320244567234ct_nat,N2)),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),Ra2)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,N2),Ra2)) ) ).

% fact_div_fact_le_pow
tff(fact_5832_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_5833_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_5834_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_5835_gbinomial__index__swap,axiom,
    ! [K: nat,N2: 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,N2))),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)),N2)),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)),N2)) ).

% gbinomial_index_swap
tff(fact_5836_periodic__finite__ex,axiom,
    ! [D2: int,Pa: fun_int_o] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D2)
     => ( ! [X3: int,K2: int] :
            ( aa_int_o(Pa,X3)
          <=> aa_int_o(Pa,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),aa_int_int(aa_int_fun_int_int(times_times_int,K2),D2))) )
       => ( ? [X_12: int] : aa_int_o(Pa,X_12)
        <=> ? [X4: int] :
              ( aa_set_int_o(member_int(X4),set_or1266510415728281911st_int(one_one_int,D2))
              & aa_int_o(Pa,X4) ) ) ) ) ).

% periodic_finite_ex
tff(fact_5837_aset_I7_J,axiom,
    ! [D3: int,A3: set_int,Ta: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D3)
     => ! [X5: 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),A3)
                 => ( X5 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb3),Xa4) ) ) )
         => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Ta),X5)
           => aa_int_o(aa_int_fun_int_o(ord_less_int,Ta),aa_int_int(aa_int_fun_int_int(plus_plus_int,X5),D3)) ) ) ) ).

% aset(7)
tff(fact_5838_aset_I5_J,axiom,
    ! [D3: int,Ta: int,A3: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D3)
     => ( aa_set_int_o(member_int(Ta),A3)
       => ! [X5: 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),A3)
                   => ( X5 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb3),Xa4) ) ) )
           => ( aa_int_o(aa_int_fun_int_o(ord_less_int,X5),Ta)
             => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X5),D3)),Ta) ) ) ) ) ).

% aset(5)
tff(fact_5839_aset_I4_J,axiom,
    ! [D3: int,Ta: int,A3: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D3)
     => ( aa_set_int_o(member_int(Ta),A3)
       => ! [X5: 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),A3)
                   => ( X5 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb3),Xa4) ) ) )
           => ( ( X5 != Ta )
             => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,X5),D3) != Ta ) ) ) ) ) ).

% aset(4)
tff(fact_5840_aset_I3_J,axiom,
    ! [D3: int,Ta: int,A3: 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,Ta),one_one_int)),A3)
       => ! [X5: 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),A3)
                   => ( X5 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb3),Xa4) ) ) )
           => ( ( X5 = Ta )
             => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,X5),D3) = Ta ) ) ) ) ) ).

% aset(3)
tff(fact_5841_bset_I7_J,axiom,
    ! [D3: int,Ta: int,B3: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D3)
     => ( aa_set_int_o(member_int(Ta),B3)
       => ! [X5: 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),B3)
                   => ( X5 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb3),Xa4) ) ) )
           => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Ta),X5)
             => aa_int_o(aa_int_fun_int_o(ord_less_int,Ta),aa_int_int(aa_int_fun_int_int(minus_minus_int,X5),D3)) ) ) ) ) ).

% bset(7)
tff(fact_5842_bset_I5_J,axiom,
    ! [D3: int,B3: set_int,Ta: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D3)
     => ! [X5: 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),B3)
                 => ( X5 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb3),Xa4) ) ) )
         => ( aa_int_o(aa_int_fun_int_o(ord_less_int,X5),Ta)
           => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X5),D3)),Ta) ) ) ) ).

% bset(5)
tff(fact_5843_bset_I4_J,axiom,
    ! [D3: int,Ta: int,B3: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D3)
     => ( aa_set_int_o(member_int(Ta),B3)
       => ! [X5: 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),B3)
                   => ( X5 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb3),Xa4) ) ) )
           => ( ( X5 != Ta )
             => ( aa_int_int(aa_int_fun_int_int(minus_minus_int,X5),D3) != Ta ) ) ) ) ) ).

% bset(4)
tff(fact_5844_bset_I3_J,axiom,
    ! [D3: int,Ta: int,B3: 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,Ta),one_one_int)),B3)
       => ! [X5: 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),B3)
                   => ( X5 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb3),Xa4) ) ) )
           => ( ( X5 = Ta )
             => ( aa_int_int(aa_int_fun_int_int(minus_minus_int,X5),D3) = Ta ) ) ) ) ) ).

% bset(3)
tff(fact_5845_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_5846_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_5847_aset_I8_J,axiom,
    ! [D3: int,A3: set_int,Ta: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D3)
     => ! [X5: 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),A3)
                 => ( X5 != 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,Ta),X5)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ta),aa_int_int(aa_int_fun_int_int(plus_plus_int,X5),D3)) ) ) ) ).

% aset(8)
tff(fact_5848_aset_I6_J,axiom,
    ! [D3: int,Ta: int,A3: 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,Ta),one_one_int)),A3)
       => ! [X5: 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),A3)
                   => ( X5 != 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,X5),Ta)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X5),D3)),Ta) ) ) ) ) ).

% aset(6)
tff(fact_5849_bset_I8_J,axiom,
    ! [D3: int,Ta: int,B3: 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,Ta),one_one_int)),B3)
       => ! [X5: 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),B3)
                   => ( X5 != 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,Ta),X5)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ta),aa_int_int(aa_int_fun_int_int(minus_minus_int,X5),D3)) ) ) ) ) ).

% bset(8)
tff(fact_5850_bset_I6_J,axiom,
    ! [D3: int,B3: set_int,Ta: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D3)
     => ! [X5: 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),B3)
                 => ( X5 != 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,X5),Ta)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X5),D3)),Ta) ) ) ) ).

% bset(6)
tff(fact_5851_cppi,axiom,
    ! [D3: int,Pa: fun_int_o,P3: fun_int_o,A3: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D3)
     => ( ? [Z4: int] :
          ! [X3: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z4),X3)
           => ( aa_int_o(Pa,X3)
            <=> aa_int_o(P3,X3) ) )
       => ( ! [X3: 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),A3)
                     => ( X3 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb2),Xa3) ) ) )
             => ( aa_int_o(Pa,X3)
               => aa_int_o(Pa,aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),D3)) ) )
         => ( ! [X3: int,K2: int] :
                ( aa_int_o(P3,X3)
              <=> aa_int_o(P3,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),aa_int_int(aa_int_fun_int_int(times_times_int,K2),D3))) )
           => ( ? [X_12: int] : aa_int_o(Pa,X_12)
            <=> ( ? [X4: int] :
                    ( aa_set_int_o(member_int(X4),set_or1266510415728281911st_int(one_one_int,D3))
                    & aa_int_o(P3,X4) )
                | ? [X4: int] :
                    ( aa_set_int_o(member_int(X4),set_or1266510415728281911st_int(one_one_int,D3))
                    & ? [Xa: int] :
                        ( aa_set_int_o(member_int(Xa),A3)
                        & aa_int_o(Pa,aa_int_int(aa_int_fun_int_int(minus_minus_int,Xa),X4)) ) ) ) ) ) ) ) ) ).

% cppi
tff(fact_5852_slice__nth,axiom,
    ! [From: nat,To: nat,Xs: list_int,I2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,From),To)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,To),size_size_list_int(Xs))
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,To),From))
         => ( aa_nat_int(nth_int(slice_int(From,To,Xs)),I2) = aa_nat_int(nth_int(Xs),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,From),I2)) ) ) ) ) ).

% slice_nth
tff(fact_5853_slice__nth,axiom,
    ! [From: nat,To: nat,Xs: list_nat,I2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,From),To)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,To),aa_list_nat_nat(size_size_list_nat,Xs))
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,To),From))
         => ( aa_nat_nat(nth_nat(slice_nat(From,To,Xs)),I2) = aa_nat_nat(nth_nat(Xs),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,From),I2)) ) ) ) ) ).

% slice_nth
tff(fact_5854_slice__nth,axiom,
    ! [From: nat,To: nat,Xs: list_o,I2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,From),To)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,To),size_size_list_o(Xs))
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,To),From))
         => ( aa_nat_o(nth_o(slice_o(From,To,Xs)),I2)
          <=> aa_nat_o(nth_o(Xs),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,From),I2)) ) ) ) ) ).

% slice_nth
tff(fact_5855_image__affinity__atLeastAtMost__div__diff,axiom,
    ! [M: rat,Ca: rat,A: rat,B: rat] :
      image_rat_rat(aa_rat_fun_rat_rat(aTP_Lamm_hn(M),Ca),set_or633870826150836451st_rat(A,B)) = $ite(
        set_or633870826150836451st_rat(A,B) = bot_bot_set_rat,
        bot_bot_set_rat,
        $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),M),set_or633870826150836451st_rat(aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),M)),Ca),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),M)),Ca)),set_or633870826150836451st_rat(aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),M)),Ca),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),M)),Ca))) ) ).

% image_affinity_atLeastAtMost_div_diff
tff(fact_5856_image__affinity__atLeastAtMost__div,axiom,
    ! [M: rat,Ca: rat,A: rat,B: rat] :
      image_rat_rat(aa_rat_fun_rat_rat(aTP_Lamm_ho(M),Ca),set_or633870826150836451st_rat(A,B)) = $ite(
        set_or633870826150836451st_rat(A,B) = bot_bot_set_rat,
        bot_bot_set_rat,
        $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),M),set_or633870826150836451st_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),M)),Ca),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),M)),Ca)),set_or633870826150836451st_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),M)),Ca),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),M)),Ca))) ) ).

% image_affinity_atLeastAtMost_div
tff(fact_5857_image__affinity__atLeastAtMost__diff,axiom,
    ! [M: rat,Ca: rat,A: rat,B: rat] :
      image_rat_rat(aa_rat_fun_rat_rat(aTP_Lamm_hp(M),Ca),set_or633870826150836451st_rat(A,B)) = $ite(
        set_or633870826150836451st_rat(A,B) = bot_bot_set_rat,
        bot_bot_set_rat,
        $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),M),set_or633870826150836451st_rat(aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,M),A)),Ca),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,M),B)),Ca)),set_or633870826150836451st_rat(aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,M),B)),Ca),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,M),A)),Ca))) ) ).

% image_affinity_atLeastAtMost_diff
tff(fact_5858_image__affinity__atLeastAtMost,axiom,
    ! [M: rat,Ca: rat,A: rat,B: rat] :
      image_rat_rat(aa_rat_fun_rat_rat(aTP_Lamm_hq(M),Ca),set_or633870826150836451st_rat(A,B)) = $ite(
        set_or633870826150836451st_rat(A,B) = bot_bot_set_rat,
        bot_bot_set_rat,
        $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),M),set_or633870826150836451st_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,M),A)),Ca),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,M),B)),Ca)),set_or633870826150836451st_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,M),B)),Ca),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,M),A)),Ca))) ) ).

% image_affinity_atLeastAtMost
tff(fact_5859_image__mult__atLeastAtMost__if_H,axiom,
    ! [Ca: rat,X: rat,Y: rat] :
      image_rat_rat(aa_rat_fun_rat_rat(aTP_Lamm_hr,Ca),set_or633870826150836451st_rat(X,Y)) = $ite(
        aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y),
        $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Ca),set_or633870826150836451st_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X),Ca),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Ca)),set_or633870826150836451st_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Ca),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X),Ca))),
        bot_bot_set_rat ) ).

% image_mult_atLeastAtMost_if'
tff(fact_5860_prod__decode__aux_Oelims,axiom,
    ! [X: nat,Xa2: nat,Y: product_prod_nat_nat] :
      ( ( aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(nat_prod_decode_aux,X),Xa2) = Y )
     => ( Y = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa2),X),aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,X),Xa2)),aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(nat_prod_decode_aux,aa_nat_nat(suc,X)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Xa2),aa_nat_nat(suc,X)))) ) ) ).

% prod_decode_aux.elims
tff(fact_5861_image__eqI,axiom,
    ! [B: $o,F: fun_o_o,X: $o,A3: set_o] :
      ( ( (B)
      <=> aa_o_o(F,(X)) )
     => ( member_o((X),A3)
       => member_o((B),image_o_o(F,A3)) ) ) ).

% image_eqI
tff(fact_5862_image__eqI,axiom,
    ! [B: nat,F: fun_o_nat,X: $o,A3: set_o] :
      ( ( B = aa_o_nat(F,(X)) )
     => ( member_o((X),A3)
       => aa_set_nat_o(member_nat(B),image_o_nat(F,A3)) ) ) ).

% image_eqI
tff(fact_5863_image__eqI,axiom,
    ! [B: int,F: fun_o_int,X: $o,A3: set_o] :
      ( ( B = aa_o_int(F,(X)) )
     => ( member_o((X),A3)
       => aa_set_int_o(member_int(B),image_o_int(F,A3)) ) ) ).

% image_eqI
tff(fact_5864_image__eqI,axiom,
    ! [B: $o,F: fun_nat_o,X: nat,A3: set_nat] :
      ( ( (B)
      <=> aa_nat_o(F,X) )
     => ( aa_set_nat_o(member_nat(X),A3)
       => member_o((B),image_nat_o(F,A3)) ) ) ).

% image_eqI
tff(fact_5865_image__eqI,axiom,
    ! [B: nat,F: fun_nat_nat,X: nat,A3: set_nat] :
      ( ( B = aa_nat_nat(F,X) )
     => ( aa_set_nat_o(member_nat(X),A3)
       => aa_set_nat_o(member_nat(B),image_nat_nat(F,A3)) ) ) ).

% image_eqI
tff(fact_5866_image__eqI,axiom,
    ! [B: int,F: fun_nat_int,X: nat,A3: set_nat] :
      ( ( B = aa_nat_int(F,X) )
     => ( aa_set_nat_o(member_nat(X),A3)
       => aa_set_int_o(member_int(B),image_nat_int(F,A3)) ) ) ).

% image_eqI
tff(fact_5867_image__eqI,axiom,
    ! [B: $o,F: fun_int_o,X: int,A3: set_int] :
      ( ( (B)
      <=> aa_int_o(F,X) )
     => ( aa_set_int_o(member_int(X),A3)
       => member_o((B),image_int_o(F,A3)) ) ) ).

% image_eqI
tff(fact_5868_image__eqI,axiom,
    ! [B: nat,F: fun_int_nat,X: int,A3: set_int] :
      ( ( B = aa_int_nat(F,X) )
     => ( aa_set_int_o(member_int(X),A3)
       => aa_set_nat_o(member_nat(B),image_int_nat(F,A3)) ) ) ).

% image_eqI
tff(fact_5869_image__eqI,axiom,
    ! [B: int,F: fun_int_int,X: int,A3: set_int] :
      ( ( B = aa_int_int(F,X) )
     => ( aa_set_int_o(member_int(X),A3)
       => aa_set_int_o(member_int(B),image_int_int(F,A3)) ) ) ).

% image_eqI
tff(fact_5870_image__eqI,axiom,
    ! [B: set_nat,F: fun_o_set_nat,X: $o,A3: set_o] :
      ( ( B = aa_o_set_nat(F,(X)) )
     => ( member_o((X),A3)
       => aa_set_set_nat_o(member_set_nat(B),image_o_set_nat(F,A3)) ) ) ).

% image_eqI
tff(fact_5871_image__ident,axiom,
    ! [Y6: set_nat] : image_nat_nat(aTP_Lamm_fu,Y6) = Y6 ).

% image_ident
tff(fact_5872_image__ident,axiom,
    ! [Y6: set_int] : image_int_int(aTP_Lamm_fv,Y6) = Y6 ).

% image_ident
tff(fact_5873_image__empty,axiom,
    ! [F: fun_o_o] : image_o_o(F,bot_bot_set_o) = bot_bot_set_o ).

% image_empty
tff(fact_5874_image__empty,axiom,
    ! [F: fun_o_nat] : image_o_nat(F,bot_bot_set_o) = bot_bot_set_nat ).

% image_empty
tff(fact_5875_image__empty,axiom,
    ! [F: fun_o_int] : image_o_int(F,bot_bot_set_o) = bot_bot_set_int ).

% image_empty
tff(fact_5876_image__empty,axiom,
    ! [F: fun_nat_o] : image_nat_o(F,bot_bot_set_nat) = bot_bot_set_o ).

% image_empty
tff(fact_5877_image__empty,axiom,
    ! [F: fun_nat_nat] : image_nat_nat(F,bot_bot_set_nat) = bot_bot_set_nat ).

% image_empty
tff(fact_5878_image__empty,axiom,
    ! [F: fun_nat_int] : image_nat_int(F,bot_bot_set_nat) = bot_bot_set_int ).

% image_empty
tff(fact_5879_image__empty,axiom,
    ! [F: fun_int_o] : image_int_o(F,bot_bot_set_int) = bot_bot_set_o ).

% image_empty
tff(fact_5880_image__empty,axiom,
    ! [F: fun_int_nat] : image_int_nat(F,bot_bot_set_int) = bot_bot_set_nat ).

% image_empty
tff(fact_5881_image__empty,axiom,
    ! [F: fun_int_int] : image_int_int(F,bot_bot_set_int) = bot_bot_set_int ).

% image_empty
tff(fact_5882_image__empty,axiom,
    ! [F: fun_nat_set_nat] : image_nat_set_nat(F,bot_bot_set_nat) = bot_bot_set_set_nat ).

% image_empty
tff(fact_5883_empty__is__image,axiom,
    ! [F: fun_o_o,A3: set_o] :
      ( ( bot_bot_set_o = image_o_o(F,A3) )
    <=> ( A3 = bot_bot_set_o ) ) ).

% empty_is_image
tff(fact_5884_empty__is__image,axiom,
    ! [F: fun_nat_o,A3: set_nat] :
      ( ( bot_bot_set_o = image_nat_o(F,A3) )
    <=> ( A3 = bot_bot_set_nat ) ) ).

% empty_is_image
tff(fact_5885_empty__is__image,axiom,
    ! [F: fun_int_o,A3: set_int] :
      ( ( bot_bot_set_o = image_int_o(F,A3) )
    <=> ( A3 = bot_bot_set_int ) ) ).

% empty_is_image
tff(fact_5886_empty__is__image,axiom,
    ! [F: fun_o_nat,A3: set_o] :
      ( ( bot_bot_set_nat = image_o_nat(F,A3) )
    <=> ( A3 = bot_bot_set_o ) ) ).

% empty_is_image
tff(fact_5887_empty__is__image,axiom,
    ! [F: fun_nat_nat,A3: set_nat] :
      ( ( bot_bot_set_nat = image_nat_nat(F,A3) )
    <=> ( A3 = bot_bot_set_nat ) ) ).

% empty_is_image
tff(fact_5888_empty__is__image,axiom,
    ! [F: fun_int_nat,A3: set_int] :
      ( ( bot_bot_set_nat = image_int_nat(F,A3) )
    <=> ( A3 = bot_bot_set_int ) ) ).

% empty_is_image
tff(fact_5889_empty__is__image,axiom,
    ! [F: fun_o_int,A3: set_o] :
      ( ( bot_bot_set_int = image_o_int(F,A3) )
    <=> ( A3 = bot_bot_set_o ) ) ).

% empty_is_image
tff(fact_5890_empty__is__image,axiom,
    ! [F: fun_nat_int,A3: set_nat] :
      ( ( bot_bot_set_int = image_nat_int(F,A3) )
    <=> ( A3 = bot_bot_set_nat ) ) ).

% empty_is_image
tff(fact_5891_empty__is__image,axiom,
    ! [F: fun_int_int,A3: set_int] :
      ( ( bot_bot_set_int = image_int_int(F,A3) )
    <=> ( A3 = bot_bot_set_int ) ) ).

% empty_is_image
tff(fact_5892_empty__is__image,axiom,
    ! [F: fun_nat_set_nat,A3: set_nat] :
      ( ( bot_bot_set_set_nat = image_nat_set_nat(F,A3) )
    <=> ( A3 = bot_bot_set_nat ) ) ).

% empty_is_image
tff(fact_5893_image__is__empty,axiom,
    ! [F: fun_o_o,A3: set_o] :
      ( ( image_o_o(F,A3) = bot_bot_set_o )
    <=> ( A3 = bot_bot_set_o ) ) ).

% image_is_empty
tff(fact_5894_image__is__empty,axiom,
    ! [F: fun_nat_o,A3: set_nat] :
      ( ( image_nat_o(F,A3) = bot_bot_set_o )
    <=> ( A3 = bot_bot_set_nat ) ) ).

% image_is_empty
tff(fact_5895_image__is__empty,axiom,
    ! [F: fun_int_o,A3: set_int] :
      ( ( image_int_o(F,A3) = bot_bot_set_o )
    <=> ( A3 = bot_bot_set_int ) ) ).

% image_is_empty
tff(fact_5896_image__is__empty,axiom,
    ! [F: fun_o_nat,A3: set_o] :
      ( ( image_o_nat(F,A3) = bot_bot_set_nat )
    <=> ( A3 = bot_bot_set_o ) ) ).

% image_is_empty
tff(fact_5897_image__is__empty,axiom,
    ! [F: fun_nat_nat,A3: set_nat] :
      ( ( image_nat_nat(F,A3) = bot_bot_set_nat )
    <=> ( A3 = bot_bot_set_nat ) ) ).

% image_is_empty
tff(fact_5898_image__is__empty,axiom,
    ! [F: fun_int_nat,A3: set_int] :
      ( ( image_int_nat(F,A3) = bot_bot_set_nat )
    <=> ( A3 = bot_bot_set_int ) ) ).

% image_is_empty
tff(fact_5899_image__is__empty,axiom,
    ! [F: fun_o_int,A3: set_o] :
      ( ( image_o_int(F,A3) = bot_bot_set_int )
    <=> ( A3 = bot_bot_set_o ) ) ).

% image_is_empty
tff(fact_5900_image__is__empty,axiom,
    ! [F: fun_nat_int,A3: set_nat] :
      ( ( image_nat_int(F,A3) = bot_bot_set_int )
    <=> ( A3 = bot_bot_set_nat ) ) ).

% image_is_empty
tff(fact_5901_image__is__empty,axiom,
    ! [F: fun_int_int,A3: set_int] :
      ( ( image_int_int(F,A3) = bot_bot_set_int )
    <=> ( A3 = bot_bot_set_int ) ) ).

% image_is_empty
tff(fact_5902_image__is__empty,axiom,
    ! [F: fun_nat_set_nat,A3: set_nat] :
      ( ( image_nat_set_nat(F,A3) = bot_bot_set_set_nat )
    <=> ( A3 = bot_bot_set_nat ) ) ).

% image_is_empty
tff(fact_5903_image__add__0,axiom,
    ! [S: set_Code_integer] : image_4470545334726330049nteger(plus_p5714425477246183910nteger(zero_z3403309356797280102nteger),S) = S ).

% image_add_0
tff(fact_5904_image__add__0,axiom,
    ! [S: set_mu2057375006010111271at_nat] : image_2516384619306764229at_nat(plus_p7104986032573967614at_nat(zero_z1048942125864253310at_nat),S) = S ).

% image_add_0
tff(fact_5905_image__add__0,axiom,
    ! [S: set_rat] : image_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,zero_zero_rat),S) = S ).

% image_add_0
tff(fact_5906_image__add__0,axiom,
    ! [S: set_nat] : image_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,zero_zero_nat),S) = S ).

% image_add_0
tff(fact_5907_image__add__0,axiom,
    ! [S: set_int] : image_int_int(aa_int_fun_int_int(plus_plus_int,zero_zero_int),S) = S ).

% image_add_0
tff(fact_5908_image__add__atLeastAtMost_H,axiom,
    ! [K: rat,I2: rat,J: rat] : image_rat_rat(aa_rat_fun_rat_rat(aTP_Lamm_hs,K),set_or633870826150836451st_rat(I2,J)) = set_or633870826150836451st_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,I2),K),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,J),K)) ).

% image_add_atLeastAtMost'
tff(fact_5909_image__add__atLeastAtMost_H,axiom,
    ! [K: int,I2: int,J: int] : image_int_int(aa_int_fun_int_int(aTP_Lamm_ht,K),set_or1266510415728281911st_int(I2,J)) = set_or1266510415728281911st_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,I2),K),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),K)) ).

% image_add_atLeastAtMost'
tff(fact_5910_image__add__atLeastAtMost_H,axiom,
    ! [K: nat,I2: nat,J: nat] : image_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_hu,K),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,I2),J)) = aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K)) ).

% image_add_atLeastAtMost'
tff(fact_5911_image__add__atLeastAtMost_H,axiom,
    ! [K: code_integer,I2: code_integer,J: code_integer] : image_4470545334726330049nteger(aTP_Lamm_hv(K),set_or189985376899183464nteger(I2,J)) = set_or189985376899183464nteger(aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(I2),K),aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(J),K)) ).

% image_add_atLeastAtMost'
tff(fact_5912_image__minus__const__atLeastAtMost_H,axiom,
    ! [D2: rat,A: rat,B: rat] : image_rat_rat(aa_rat_fun_rat_rat(aTP_Lamm_hw,D2),set_or633870826150836451st_rat(A,B)) = set_or633870826150836451st_rat(aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),D2),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,B),D2)) ).

% image_minus_const_atLeastAtMost'
tff(fact_5913_image__minus__const__atLeastAtMost_H,axiom,
    ! [D2: int,A: int,B: int] : image_int_int(aa_int_fun_int_int(aTP_Lamm_hx,D2),set_or1266510415728281911st_int(A,B)) = set_or1266510415728281911st_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,A),D2),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),D2)) ).

% image_minus_const_atLeastAtMost'
tff(fact_5914_image__minus__const__atLeastAtMost_H,axiom,
    ! [D2: code_integer,A: code_integer,B: code_integer] : image_4470545334726330049nteger(aTP_Lamm_hy(D2),set_or189985376899183464nteger(A,B)) = set_or189985376899183464nteger(aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(A),D2),aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(B),D2)) ).

% image_minus_const_atLeastAtMost'
tff(fact_5915_if__image__distrib,axiom,
    ! [Pa: fun_int_o,F: fun_int_int,G: fun_int_int,S: set_int] : image_int_int(aa_fun6280345759651049192nt_int(aa_fun3693603986580398266nt_int(aTP_Lamm_hz(Pa),F),G),S) = aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,image_int_int(F,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,S),collect_int(Pa)))),image_int_int(G,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,S),collect_int(aTP_Lamm_he(Pa))))) ).

% if_image_distrib
tff(fact_5916_if__image__distrib,axiom,
    ! [Pa: fun_nat_o,F: fun_nat_int,G: fun_nat_int,S: set_nat] : image_nat_int(aa_fun5565043122120104496at_int(aa_fun5140708168904641446at_int(aTP_Lamm_ia(Pa),F),G),S) = aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,image_nat_int(F,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,S),collect_nat(Pa)))),image_nat_int(G,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,S),collect_nat(aTP_Lamm_hd(Pa))))) ).

% if_image_distrib
tff(fact_5917_if__image__distrib,axiom,
    ! [Pa: fun_int_o,F: fun_int_nat,G: fun_int_nat,S: set_int] : image_int_nat(aa_fun6506752150744536624nt_nat(aa_fun5407591398502139302nt_nat(aTP_Lamm_ib(Pa),F),G),S) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,image_int_nat(F,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,S),collect_int(Pa)))),image_int_nat(G,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,S),collect_int(aTP_Lamm_he(Pa))))) ).

% if_image_distrib
tff(fact_5918_if__image__distrib,axiom,
    ! [Pa: fun_nat_o,F: fun_nat_nat,G: fun_nat_nat,S: set_nat] : image_nat_nat(aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(aTP_Lamm_ic(Pa),F),G),S) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,image_nat_nat(F,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,S),collect_nat(Pa)))),image_nat_nat(G,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,S),collect_nat(aTP_Lamm_hd(Pa))))) ).

% if_image_distrib
tff(fact_5919_if__image__distrib,axiom,
    ! [Pa: fun_nat_o,F: fun_nat_set_nat,G: fun_nat_set_nat,S: set_nat] : image_nat_set_nat(aa_fun850356524293783268et_nat(aa_fun2558789851993377716et_nat(aTP_Lamm_id(Pa),F),G),S) = sup_sup_set_set_nat(image_nat_set_nat(F,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,S),collect_nat(Pa))),image_nat_set_nat(G,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,S),collect_nat(aTP_Lamm_hd(Pa))))) ).

% if_image_distrib
tff(fact_5920_if__image__distrib,axiom,
    ! [Pa: fun_list_nat_o,F: fun_list_nat_nat,G: fun_list_nat_nat,S: set_list_nat] : image_list_nat_nat(aa_fun8775324257704122008at_nat(aa_fun7930739032552464066at_nat(aTP_Lamm_ie(Pa),F),G),S) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,image_list_nat_nat(F,inf_inf_set_list_nat(S,collect_list_nat(Pa)))),image_list_nat_nat(G,inf_inf_set_list_nat(S,collect_list_nat(aTP_Lamm_hb(Pa))))) ).

% if_image_distrib
tff(fact_5921_if__image__distrib,axiom,
    ! [Pa: fun_set_nat_o,F: fun_set_nat_nat,G: fun_set_nat_nat,S: set_set_nat] : image_set_nat_nat(aa_fun3386010016858254564at_nat(aa_fun8118849421014274228at_nat(aTP_Lamm_if(Pa),F),G),S) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,image_set_nat_nat(F,inf_inf_set_set_nat(S,collect_set_nat(Pa)))),image_set_nat_nat(G,inf_inf_set_set_nat(S,collect_set_nat(aTP_Lamm_hc(Pa))))) ).

% if_image_distrib
tff(fact_5922_if__image__distrib,axiom,
    ! [Pa: fun_Pr3238374608701176162_int_o,F: fun_Pr5160692531401422620nt_nat,G: fun_Pr5160692531401422620nt_nat,S: set_Pr958786334691620121nt_int] : image_5044651549707136836nt_nat(aa_fun6016190284160528700nt_nat(aa_fun1083184158259774627nt_nat(aTP_Lamm_ig(Pa),F),G),S) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,image_5044651549707136836nt_nat(F,inf_in2269163501485487111nt_int(S,collec213857154873943460nt_int(Pa)))),image_5044651549707136836nt_nat(G,inf_in2269163501485487111nt_int(S,collec213857154873943460nt_int(aTP_Lamm_ha(Pa))))) ).

% if_image_distrib
tff(fact_5923_if__image__distrib,axiom,
    ! [Pa: fun_Pr4796943543331071514_nat_o,F: fun_Pr2602117396471556196at_nat,G: fun_Pr2602117396471556196at_nat,S: set_Pr1261947904930325089at_nat] : image_2486076414777270412at_nat(aa_fun2351638177499875132at_nat(aa_fun1768432373507419115at_nat(aTP_Lamm_ih(Pa),F),G),S) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,image_2486076414777270412at_nat(F,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,S),collec3392354462482085612at_nat(Pa)))),image_2486076414777270412at_nat(G,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,S),collec3392354462482085612at_nat(aa_fun359341089595923442_nat_o(aTP_Lamm_ii,Pa))))) ).

% if_image_distrib
tff(fact_5924_if__image__distrib,axiom,
    ! [Pa: fun_int_o,F: fun_in5957238474111007380at_nat,G: fun_in5957238474111007380at_nat,S: set_int] : image_6031046145702598900at_nat(aa_fun6444483342604292028at_nat(aa_fun5672742635520905243at_nat(aTP_Lamm_ij(Pa),F),G),S) = aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,image_6031046145702598900at_nat(F,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,S),collect_int(Pa)))),image_6031046145702598900at_nat(G,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,S),collect_int(aTP_Lamm_he(Pa))))) ).

% if_image_distrib
tff(fact_5925_image__mult__atLeastAtMost,axiom,
    ! [D2: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),D2)
     => ( image_rat_rat(aa_rat_fun_rat_rat(times_times_rat,D2),set_or633870826150836451st_rat(A,B)) = set_or633870826150836451st_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,D2),A),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,D2),B)) ) ) ).

% image_mult_atLeastAtMost
tff(fact_5926_image__divide__atLeastAtMost,axiom,
    ! [D2: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),D2)
     => ( image_rat_rat(aa_rat_fun_rat_rat(aTP_Lamm_ik,D2),set_or633870826150836451st_rat(A,B)) = set_or633870826150836451st_rat(aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),D2),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),D2)) ) ) ).

% image_divide_atLeastAtMost
tff(fact_5927_rev__image__eqI,axiom,
    ! [X: $o,A3: set_o,B: $o,F: fun_o_o] :
      ( member_o((X),A3)
     => ( ( (B)
        <=> aa_o_o(F,(X)) )
       => member_o((B),image_o_o(F,A3)) ) ) ).

% rev_image_eqI
tff(fact_5928_rev__image__eqI,axiom,
    ! [X: $o,A3: set_o,B: nat,F: fun_o_nat] :
      ( member_o((X),A3)
     => ( ( B = aa_o_nat(F,(X)) )
       => aa_set_nat_o(member_nat(B),image_o_nat(F,A3)) ) ) ).

% rev_image_eqI
tff(fact_5929_rev__image__eqI,axiom,
    ! [X: $o,A3: set_o,B: int,F: fun_o_int] :
      ( member_o((X),A3)
     => ( ( B = aa_o_int(F,(X)) )
       => aa_set_int_o(member_int(B),image_o_int(F,A3)) ) ) ).

% rev_image_eqI
tff(fact_5930_rev__image__eqI,axiom,
    ! [X: nat,A3: set_nat,B: $o,F: fun_nat_o] :
      ( aa_set_nat_o(member_nat(X),A3)
     => ( ( (B)
        <=> aa_nat_o(F,X) )
       => member_o((B),image_nat_o(F,A3)) ) ) ).

% rev_image_eqI
tff(fact_5931_rev__image__eqI,axiom,
    ! [X: nat,A3: set_nat,B: nat,F: fun_nat_nat] :
      ( aa_set_nat_o(member_nat(X),A3)
     => ( ( B = aa_nat_nat(F,X) )
       => aa_set_nat_o(member_nat(B),image_nat_nat(F,A3)) ) ) ).

% rev_image_eqI
tff(fact_5932_rev__image__eqI,axiom,
    ! [X: nat,A3: set_nat,B: int,F: fun_nat_int] :
      ( aa_set_nat_o(member_nat(X),A3)
     => ( ( B = aa_nat_int(F,X) )
       => aa_set_int_o(member_int(B),image_nat_int(F,A3)) ) ) ).

% rev_image_eqI
tff(fact_5933_rev__image__eqI,axiom,
    ! [X: int,A3: set_int,B: $o,F: fun_int_o] :
      ( aa_set_int_o(member_int(X),A3)
     => ( ( (B)
        <=> aa_int_o(F,X) )
       => member_o((B),image_int_o(F,A3)) ) ) ).

% rev_image_eqI
tff(fact_5934_rev__image__eqI,axiom,
    ! [X: int,A3: set_int,B: nat,F: fun_int_nat] :
      ( aa_set_int_o(member_int(X),A3)
     => ( ( B = aa_int_nat(F,X) )
       => aa_set_nat_o(member_nat(B),image_int_nat(F,A3)) ) ) ).

% rev_image_eqI
tff(fact_5935_rev__image__eqI,axiom,
    ! [X: int,A3: set_int,B: int,F: fun_int_int] :
      ( aa_set_int_o(member_int(X),A3)
     => ( ( B = aa_int_int(F,X) )
       => aa_set_int_o(member_int(B),image_int_int(F,A3)) ) ) ).

% rev_image_eqI
tff(fact_5936_rev__image__eqI,axiom,
    ! [X: $o,A3: set_o,B: set_nat,F: fun_o_set_nat] :
      ( member_o((X),A3)
     => ( ( B = aa_o_set_nat(F,(X)) )
       => aa_set_set_nat_o(member_set_nat(B),image_o_set_nat(F,A3)) ) ) ).

% rev_image_eqI
tff(fact_5937_ball__imageD,axiom,
    ! [F: fun_nat_set_nat,A3: set_nat,Pa: fun_set_nat_o] :
      ( ! [X3: set_nat] :
          ( aa_set_set_nat_o(member_set_nat(X3),image_nat_set_nat(F,A3))
         => aa_set_nat_o(Pa,X3) )
     => ! [X5: nat] :
          ( aa_set_nat_o(member_nat(X5),A3)
         => aa_set_nat_o(Pa,aa_nat_set_nat(F,X5)) ) ) ).

% ball_imageD
tff(fact_5938_ball__imageD,axiom,
    ! [F: fun_nat_nat,A3: set_nat,Pa: fun_nat_o] :
      ( ! [X3: nat] :
          ( aa_set_nat_o(member_nat(X3),image_nat_nat(F,A3))
         => aa_nat_o(Pa,X3) )
     => ! [X5: nat] :
          ( aa_set_nat_o(member_nat(X5),A3)
         => aa_nat_o(Pa,aa_nat_nat(F,X5)) ) ) ).

% ball_imageD
tff(fact_5939_ball__imageD,axiom,
    ! [F: fun_nat_int,A3: set_nat,Pa: fun_int_o] :
      ( ! [X3: int] :
          ( aa_set_int_o(member_int(X3),image_nat_int(F,A3))
         => aa_int_o(Pa,X3) )
     => ! [X5: nat] :
          ( aa_set_nat_o(member_nat(X5),A3)
         => aa_int_o(Pa,aa_nat_int(F,X5)) ) ) ).

% ball_imageD
tff(fact_5940_ball__imageD,axiom,
    ! [F: fun_int_nat,A3: set_int,Pa: fun_nat_o] :
      ( ! [X3: nat] :
          ( aa_set_nat_o(member_nat(X3),image_int_nat(F,A3))
         => aa_nat_o(Pa,X3) )
     => ! [X5: int] :
          ( aa_set_int_o(member_int(X5),A3)
         => aa_nat_o(Pa,aa_int_nat(F,X5)) ) ) ).

% ball_imageD
tff(fact_5941_ball__imageD,axiom,
    ! [F: fun_int_int,A3: set_int,Pa: fun_int_o] :
      ( ! [X3: int] :
          ( aa_set_int_o(member_int(X3),image_int_int(F,A3))
         => aa_int_o(Pa,X3) )
     => ! [X5: int] :
          ( aa_set_int_o(member_int(X5),A3)
         => aa_int_o(Pa,aa_int_int(F,X5)) ) ) ).

% ball_imageD
tff(fact_5942_image__cong,axiom,
    ! [M6: set_nat,N7: set_nat,F: fun_nat_set_nat,G: fun_nat_set_nat] :
      ( ( M6 = N7 )
     => ( ! [X3: nat] :
            ( aa_set_nat_o(member_nat(X3),N7)
           => ( aa_nat_set_nat(F,X3) = aa_nat_set_nat(G,X3) ) )
       => ( image_nat_set_nat(F,M6) = image_nat_set_nat(G,N7) ) ) ) ).

% image_cong
tff(fact_5943_image__cong,axiom,
    ! [M6: set_nat,N7: set_nat,F: fun_nat_nat,G: fun_nat_nat] :
      ( ( M6 = N7 )
     => ( ! [X3: nat] :
            ( aa_set_nat_o(member_nat(X3),N7)
           => ( aa_nat_nat(F,X3) = aa_nat_nat(G,X3) ) )
       => ( image_nat_nat(F,M6) = image_nat_nat(G,N7) ) ) ) ).

% image_cong
tff(fact_5944_image__cong,axiom,
    ! [M6: set_nat,N7: set_nat,F: fun_nat_int,G: fun_nat_int] :
      ( ( M6 = N7 )
     => ( ! [X3: nat] :
            ( aa_set_nat_o(member_nat(X3),N7)
           => ( aa_nat_int(F,X3) = aa_nat_int(G,X3) ) )
       => ( image_nat_int(F,M6) = image_nat_int(G,N7) ) ) ) ).

% image_cong
tff(fact_5945_image__cong,axiom,
    ! [M6: set_int,N7: set_int,F: fun_int_nat,G: fun_int_nat] :
      ( ( M6 = N7 )
     => ( ! [X3: int] :
            ( aa_set_int_o(member_int(X3),N7)
           => ( aa_int_nat(F,X3) = aa_int_nat(G,X3) ) )
       => ( image_int_nat(F,M6) = image_int_nat(G,N7) ) ) ) ).

% image_cong
tff(fact_5946_image__cong,axiom,
    ! [M6: set_int,N7: set_int,F: fun_int_int,G: fun_int_int] :
      ( ( M6 = N7 )
     => ( ! [X3: int] :
            ( aa_set_int_o(member_int(X3),N7)
           => ( aa_int_int(F,X3) = aa_int_int(G,X3) ) )
       => ( image_int_int(F,M6) = image_int_int(G,N7) ) ) ) ).

% image_cong
tff(fact_5947_bex__imageD,axiom,
    ! [F: fun_nat_set_nat,A3: set_nat,Pa: fun_set_nat_o] :
      ( ? [X5: set_nat] :
          ( aa_set_set_nat_o(member_set_nat(X5),image_nat_set_nat(F,A3))
          & aa_set_nat_o(Pa,X5) )
     => ? [X3: nat] :
          ( aa_set_nat_o(member_nat(X3),A3)
          & aa_set_nat_o(Pa,aa_nat_set_nat(F,X3)) ) ) ).

% bex_imageD
tff(fact_5948_bex__imageD,axiom,
    ! [F: fun_nat_nat,A3: set_nat,Pa: fun_nat_o] :
      ( ? [X5: nat] :
          ( aa_set_nat_o(member_nat(X5),image_nat_nat(F,A3))
          & aa_nat_o(Pa,X5) )
     => ? [X3: nat] :
          ( aa_set_nat_o(member_nat(X3),A3)
          & aa_nat_o(Pa,aa_nat_nat(F,X3)) ) ) ).

% bex_imageD
tff(fact_5949_bex__imageD,axiom,
    ! [F: fun_nat_int,A3: set_nat,Pa: fun_int_o] :
      ( ? [X5: int] :
          ( aa_set_int_o(member_int(X5),image_nat_int(F,A3))
          & aa_int_o(Pa,X5) )
     => ? [X3: nat] :
          ( aa_set_nat_o(member_nat(X3),A3)
          & aa_int_o(Pa,aa_nat_int(F,X3)) ) ) ).

% bex_imageD
tff(fact_5950_bex__imageD,axiom,
    ! [F: fun_int_nat,A3: set_int,Pa: fun_nat_o] :
      ( ? [X5: nat] :
          ( aa_set_nat_o(member_nat(X5),image_int_nat(F,A3))
          & aa_nat_o(Pa,X5) )
     => ? [X3: int] :
          ( aa_set_int_o(member_int(X3),A3)
          & aa_nat_o(Pa,aa_int_nat(F,X3)) ) ) ).

% bex_imageD
tff(fact_5951_bex__imageD,axiom,
    ! [F: fun_int_int,A3: set_int,Pa: fun_int_o] :
      ( ? [X5: int] :
          ( aa_set_int_o(member_int(X5),image_int_int(F,A3))
          & aa_int_o(Pa,X5) )
     => ? [X3: int] :
          ( aa_set_int_o(member_int(X3),A3)
          & aa_int_o(Pa,aa_int_int(F,X3)) ) ) ).

% bex_imageD
tff(fact_5952_image__iff,axiom,
    ! [Z: set_nat,F: fun_nat_set_nat,A3: set_nat] :
      ( aa_set_set_nat_o(member_set_nat(Z),image_nat_set_nat(F,A3))
    <=> ? [X4: nat] :
          ( aa_set_nat_o(member_nat(X4),A3)
          & ( Z = aa_nat_set_nat(F,X4) ) ) ) ).

% image_iff
tff(fact_5953_image__iff,axiom,
    ! [Z: nat,F: fun_nat_nat,A3: set_nat] :
      ( aa_set_nat_o(member_nat(Z),image_nat_nat(F,A3))
    <=> ? [X4: nat] :
          ( aa_set_nat_o(member_nat(X4),A3)
          & ( Z = aa_nat_nat(F,X4) ) ) ) ).

% image_iff
tff(fact_5954_image__iff,axiom,
    ! [Z: nat,F: fun_int_nat,A3: set_int] :
      ( aa_set_nat_o(member_nat(Z),image_int_nat(F,A3))
    <=> ? [X4: int] :
          ( aa_set_int_o(member_int(X4),A3)
          & ( Z = aa_int_nat(F,X4) ) ) ) ).

% image_iff
tff(fact_5955_image__iff,axiom,
    ! [Z: int,F: fun_nat_int,A3: set_nat] :
      ( aa_set_int_o(member_int(Z),image_nat_int(F,A3))
    <=> ? [X4: nat] :
          ( aa_set_nat_o(member_nat(X4),A3)
          & ( Z = aa_nat_int(F,X4) ) ) ) ).

% image_iff
tff(fact_5956_image__iff,axiom,
    ! [Z: int,F: fun_int_int,A3: set_int] :
      ( aa_set_int_o(member_int(Z),image_int_int(F,A3))
    <=> ? [X4: int] :
          ( aa_set_int_o(member_int(X4),A3)
          & ( Z = aa_int_int(F,X4) ) ) ) ).

% image_iff
tff(fact_5957_imageI,axiom,
    ! [X: $o,A3: set_o,F: fun_o_o] :
      ( member_o((X),A3)
     => member_o(aa_o_o(F,(X)),image_o_o(F,A3)) ) ).

% imageI
tff(fact_5958_imageI,axiom,
    ! [X: $o,A3: set_o,F: fun_o_nat] :
      ( member_o((X),A3)
     => aa_set_nat_o(member_nat(aa_o_nat(F,(X))),image_o_nat(F,A3)) ) ).

% imageI
tff(fact_5959_imageI,axiom,
    ! [X: $o,A3: set_o,F: fun_o_int] :
      ( member_o((X),A3)
     => aa_set_int_o(member_int(aa_o_int(F,(X))),image_o_int(F,A3)) ) ).

% imageI
tff(fact_5960_imageI,axiom,
    ! [X: nat,A3: set_nat,F: fun_nat_o] :
      ( aa_set_nat_o(member_nat(X),A3)
     => member_o(aa_nat_o(F,X),image_nat_o(F,A3)) ) ).

% imageI
tff(fact_5961_imageI,axiom,
    ! [X: nat,A3: set_nat,F: fun_nat_nat] :
      ( aa_set_nat_o(member_nat(X),A3)
     => aa_set_nat_o(member_nat(aa_nat_nat(F,X)),image_nat_nat(F,A3)) ) ).

% imageI
tff(fact_5962_imageI,axiom,
    ! [X: nat,A3: set_nat,F: fun_nat_int] :
      ( aa_set_nat_o(member_nat(X),A3)
     => aa_set_int_o(member_int(aa_nat_int(F,X)),image_nat_int(F,A3)) ) ).

% imageI
tff(fact_5963_imageI,axiom,
    ! [X: int,A3: set_int,F: fun_int_o] :
      ( aa_set_int_o(member_int(X),A3)
     => member_o(aa_int_o(F,X),image_int_o(F,A3)) ) ).

% imageI
tff(fact_5964_imageI,axiom,
    ! [X: int,A3: set_int,F: fun_int_nat] :
      ( aa_set_int_o(member_int(X),A3)
     => aa_set_nat_o(member_nat(aa_int_nat(F,X)),image_int_nat(F,A3)) ) ).

% imageI
tff(fact_5965_imageI,axiom,
    ! [X: int,A3: set_int,F: fun_int_int] :
      ( aa_set_int_o(member_int(X),A3)
     => aa_set_int_o(member_int(aa_int_int(F,X)),image_int_int(F,A3)) ) ).

% imageI
tff(fact_5966_imageI,axiom,
    ! [X: $o,A3: set_o,F: fun_o_set_nat] :
      ( member_o((X),A3)
     => aa_set_set_nat_o(member_set_nat(aa_o_set_nat(F,(X))),image_o_set_nat(F,A3)) ) ).

% imageI
tff(fact_5967_imageE,axiom,
    ! [B: $o,F: fun_o_o,A3: set_o] :
      ( member_o((B),image_o_o(F,A3))
     => ~ ! [X3: $o] :
            ( ( (B)
            <=> aa_o_o(F,(X3)) )
           => ~ member_o((X3),A3) ) ) ).

% imageE
tff(fact_5968_imageE,axiom,
    ! [B: $o,F: fun_nat_o,A3: set_nat] :
      ( member_o((B),image_nat_o(F,A3))
     => ~ ! [X3: nat] :
            ( ( (B)
            <=> aa_nat_o(F,X3) )
           => ~ aa_set_nat_o(member_nat(X3),A3) ) ) ).

% imageE
tff(fact_5969_imageE,axiom,
    ! [B: $o,F: fun_int_o,A3: set_int] :
      ( member_o((B),image_int_o(F,A3))
     => ~ ! [X3: int] :
            ( ( (B)
            <=> aa_int_o(F,X3) )
           => ~ aa_set_int_o(member_int(X3),A3) ) ) ).

% imageE
tff(fact_5970_imageE,axiom,
    ! [B: nat,F: fun_o_nat,A3: set_o] :
      ( aa_set_nat_o(member_nat(B),image_o_nat(F,A3))
     => ~ ! [X3: $o] :
            ( ( B = aa_o_nat(F,(X3)) )
           => ~ member_o((X3),A3) ) ) ).

% imageE
tff(fact_5971_imageE,axiom,
    ! [B: nat,F: fun_nat_nat,A3: set_nat] :
      ( aa_set_nat_o(member_nat(B),image_nat_nat(F,A3))
     => ~ ! [X3: nat] :
            ( ( B = aa_nat_nat(F,X3) )
           => ~ aa_set_nat_o(member_nat(X3),A3) ) ) ).

% imageE
tff(fact_5972_imageE,axiom,
    ! [B: nat,F: fun_int_nat,A3: set_int] :
      ( aa_set_nat_o(member_nat(B),image_int_nat(F,A3))
     => ~ ! [X3: int] :
            ( ( B = aa_int_nat(F,X3) )
           => ~ aa_set_int_o(member_int(X3),A3) ) ) ).

% imageE
tff(fact_5973_imageE,axiom,
    ! [B: int,F: fun_o_int,A3: set_o] :
      ( aa_set_int_o(member_int(B),image_o_int(F,A3))
     => ~ ! [X3: $o] :
            ( ( B = aa_o_int(F,(X3)) )
           => ~ member_o((X3),A3) ) ) ).

% imageE
tff(fact_5974_imageE,axiom,
    ! [B: int,F: fun_nat_int,A3: set_nat] :
      ( aa_set_int_o(member_int(B),image_nat_int(F,A3))
     => ~ ! [X3: nat] :
            ( ( B = aa_nat_int(F,X3) )
           => ~ aa_set_nat_o(member_nat(X3),A3) ) ) ).

% imageE
tff(fact_5975_imageE,axiom,
    ! [B: int,F: fun_int_int,A3: set_int] :
      ( aa_set_int_o(member_int(B),image_int_int(F,A3))
     => ~ ! [X3: int] :
            ( ( B = aa_int_int(F,X3) )
           => ~ aa_set_int_o(member_int(X3),A3) ) ) ).

% imageE
tff(fact_5976_imageE,axiom,
    ! [B: $o,F: fun_set_nat_o,A3: set_set_nat] :
      ( member_o((B),image_set_nat_o(F,A3))
     => ~ ! [X3: set_nat] :
            ( ( (B)
            <=> aa_set_nat_o(F,X3) )
           => ~ aa_set_set_nat_o(member_set_nat(X3),A3) ) ) ).

% imageE
tff(fact_5977_image__image,axiom,
    ! [F: fun_nat_nat,G: fun_nat_nat,A3: set_nat] : image_nat_nat(F,image_nat_nat(G,A3)) = image_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_il(F),G),A3) ).

% image_image
tff(fact_5978_image__image,axiom,
    ! [F: fun_nat_nat,G: fun_int_nat,A3: set_int] : image_nat_nat(F,image_int_nat(G,A3)) = image_int_nat(aa_fun6506752150744536624nt_nat(aTP_Lamm_im(F),G),A3) ).

% image_image
tff(fact_5979_image__image,axiom,
    ! [F: fun_nat_int,G: fun_nat_nat,A3: set_nat] : image_nat_int(F,image_nat_nat(G,A3)) = image_nat_int(aa_fun1613598493704395220at_int(aTP_Lamm_in(F),G),A3) ).

% image_image
tff(fact_5980_image__image,axiom,
    ! [F: fun_nat_int,G: fun_int_nat,A3: set_int] : image_nat_int(F,image_int_nat(G,A3)) = image_int_int(aa_fun2328901131235339916nt_int(aTP_Lamm_io(F),G),A3) ).

% image_image
tff(fact_5981_image__image,axiom,
    ! [F: fun_int_nat,G: fun_nat_int,A3: set_nat] : image_int_nat(F,image_nat_int(G,A3)) = image_nat_nat(aa_fun519522104774525396at_nat(aTP_Lamm_ip(F),G),A3) ).

% image_image
tff(fact_5982_image__image,axiom,
    ! [F: fun_int_nat,G: fun_int_int,A3: set_int] : image_int_nat(F,image_int_int(G,A3)) = image_int_nat(aa_fun1234824742305470092nt_nat(aTP_Lamm_iq(F),G),A3) ).

% image_image
tff(fact_5983_image__image,axiom,
    ! [F: fun_int_int,G: fun_nat_int,A3: set_nat] : image_int_int(F,image_nat_int(G,A3)) = image_nat_int(aa_fun5565043122120104496at_int(aTP_Lamm_ir(F),G),A3) ).

% image_image
tff(fact_5984_image__image,axiom,
    ! [F: fun_int_int,G: fun_int_int,A3: set_int] : image_int_int(F,image_int_int(G,A3)) = image_int_int(aa_fun6280345759651049192nt_int(aTP_Lamm_is(F),G),A3) ).

% image_image
tff(fact_5985_image__image,axiom,
    ! [F: fun_set_nat_nat,G: fun_nat_set_nat,A3: set_nat] : image_set_nat_nat(F,image_nat_set_nat(G,A3)) = image_nat_nat(aa_fun4820382421437094446at_nat(aTP_Lamm_it(F),G),A3) ).

% image_image
tff(fact_5986_image__image,axiom,
    ! [F: fun_set_nat_int,G: fun_nat_set_nat,A3: set_nat] : image_set_nat_int(F,image_nat_set_nat(G,A3)) = image_nat_int(aa_fun642531401927897738at_int(aTP_Lamm_iu(F),G),A3) ).

% image_image
tff(fact_5987_Compr__image__eq,axiom,
    ! [F: fun_o_o,A3: set_o,Pa: fun_o_o] : collect_o(aa_fun_o_o_fun_o_o(aa_set8294393945046410815un_o_o(aTP_Lamm_iv(F),A3),Pa)) = image_o_o(F,collect_o(aa_fun_o_o_fun_o_o(aa_set8294393945046410815un_o_o(aTP_Lamm_iw(F),A3),Pa))) ).

% Compr_image_eq
tff(fact_5988_Compr__image__eq,axiom,
    ! [F: fun_nat_o,A3: set_nat,Pa: fun_o_o] : collect_o(aa_fun_o_o_fun_o_o(aa_set1656415104011699143un_o_o(aTP_Lamm_ix(F),A3),Pa)) = image_nat_o(F,collect_nat(aa_fun_o_o_fun_nat_o(aa_set2987053916987590195_nat_o(aTP_Lamm_iy(F),A3),Pa))) ).

% Compr_image_eq
tff(fact_5989_Compr__image__eq,axiom,
    ! [F: fun_int_o,A3: set_int,Pa: fun_o_o] : collect_o(aa_fun_o_o_fun_o_o(aa_set2864532236940079139un_o_o(aTP_Lamm_iz(F),A3),Pa)) = image_int_o(F,collect_int(aa_fun_o_o_fun_int_o(aa_set2419422181098893691_int_o(aTP_Lamm_ja(F),A3),Pa))) ).

% Compr_image_eq
tff(fact_5990_Compr__image__eq,axiom,
    ! [F: fun_o_nat,A3: set_o,Pa: fun_nat_o] : collect_nat(aa_fun5937193402907943164_nat_o(aa_set1339306056044713023_nat_o(aTP_Lamm_jb(F),A3),Pa)) = image_o_nat(F,collect_o(aa_fun_nat_o_fun_o_o(aa_set26024703028173499un_o_o(aTP_Lamm_jc(F),A3),Pa))) ).

% Compr_image_eq
tff(fact_5991_Compr__image__eq,axiom,
    ! [F: fun_nat_nat,A3: set_nat,Pa: fun_nat_o] : collect_nat(aa_fun5937193402907943164_nat_o(aa_set1405850572773641671_nat_o(aTP_Lamm_jd(F),A3),Pa)) = image_nat_nat(F,collect_nat(aa_fun5937193402907943164_nat_o(aa_set1405850572773641671_nat_o(aTP_Lamm_je(F),A3),Pa))) ).

% Compr_image_eq
tff(fact_5992_Compr__image__eq,axiom,
    ! [F: fun_int_nat,A3: set_int,Pa: fun_nat_o] : collect_nat(aa_fun5937193402907943164_nat_o(aa_set563577997438114851_nat_o(aTP_Lamm_jf(F),A3),Pa)) = image_int_nat(F,collect_int(aa_fun4116151057475255200_int_o(aa_set7541805066622444743_int_o(aTP_Lamm_jg(F),A3),Pa))) ).

% Compr_image_eq
tff(fact_5993_Compr__image__eq,axiom,
    ! [F: fun_o_int,A3: set_o,Pa: fun_int_o] : collect_int(aa_fun6243517787477317628_int_o(aa_set1221527818376329535_int_o(aTP_Lamm_jh(F),A3),Pa)) = image_o_int(F,collect_o(aa_fun_int_o_fun_o_o(aa_set6365818687085129567un_o_o(aTP_Lamm_ji(F),A3),Pa))) ).

% Compr_image_eq
tff(fact_5994_Compr__image__eq,axiom,
    ! [F: fun_nat_int,A3: set_nat,Pa: fun_int_o] : collect_int(aa_fun6243517787477317628_int_o(aa_set1288072335105258183_int_o(aTP_Lamm_jj(F),A3),Pa)) = image_nat_int(F,collect_nat(aa_fun8064560132910005592_nat_o(aa_set3533217302775704099_nat_o(aTP_Lamm_jk(F),A3),Pa))) ).

% Compr_image_eq
tff(fact_5995_Compr__image__eq,axiom,
    ! [F: fun_int_int,A3: set_int,Pa: fun_int_o] : collect_int(aa_fun6243517787477317628_int_o(aa_set445799759769731363_int_o(aTP_Lamm_jl(F),A3),Pa)) = image_int_int(F,collect_int(aa_fun6243517787477317628_int_o(aa_set445799759769731363_int_o(aTP_Lamm_jm(F),A3),Pa))) ).

% Compr_image_eq
tff(fact_5996_Compr__image__eq,axiom,
    ! [F: fun_list_nat_o,A3: set_list_nat,Pa: fun_o_o] : collect_o(aa_fun_o_o_fun_o_o(aa_set4082382446602818263un_o_o(aTP_Lamm_jn(F),A3),Pa)) = image_list_nat_o(F,collect_list_nat(aa_fun3550071730982498768_nat_o(aa_set7739190917479332627_nat_o(aTP_Lamm_jo(F),A3),Pa))) ).

% Compr_image_eq
tff(fact_5997_prod__decode__aux_Ocases,axiom,
    ! [X: product_prod_nat_nat] :
      ~ ! [K2: nat,M5: nat] : X != aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,K2),M5) ).

% prod_decode_aux.cases
tff(fact_5998_subset__image__iff,axiom,
    ! [B3: set_set_nat,F: fun_nat_set_nat,A3: set_nat] :
      ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(B3),image_nat_set_nat(F,A3))
    <=> ? [AA: set_nat] :
          ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,AA),A3)
          & ( B3 = image_nat_set_nat(F,AA) ) ) ) ).

% subset_image_iff
tff(fact_5999_subset__image__iff,axiom,
    ! [B3: set_nat,F: fun_nat_nat,A3: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B3),image_nat_nat(F,A3))
    <=> ? [AA: set_nat] :
          ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,AA),A3)
          & ( B3 = image_nat_nat(F,AA) ) ) ) ).

% subset_image_iff
tff(fact_6000_subset__image__iff,axiom,
    ! [B3: set_nat,F: fun_int_nat,A3: set_int] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B3),image_int_nat(F,A3))
    <=> ? [AA: set_int] :
          ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,AA),A3)
          & ( B3 = image_int_nat(F,AA) ) ) ) ).

% subset_image_iff
tff(fact_6001_subset__image__iff,axiom,
    ! [B3: set_int,F: fun_nat_int,A3: set_nat] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,B3),image_nat_int(F,A3))
    <=> ? [AA: set_nat] :
          ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,AA),A3)
          & ( B3 = image_nat_int(F,AA) ) ) ) ).

% subset_image_iff
tff(fact_6002_subset__image__iff,axiom,
    ! [B3: set_int,F: fun_int_int,A3: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,B3),image_int_int(F,A3))
    <=> ? [AA: set_int] :
          ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,AA),A3)
          & ( B3 = image_int_int(F,AA) ) ) ) ).

% subset_image_iff
tff(fact_6003_image__subset__iff,axiom,
    ! [F: fun_nat_set_nat,A3: set_nat,B3: set_set_nat] :
      ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(image_nat_set_nat(F,A3)),B3)
    <=> ! [X4: nat] :
          ( aa_set_nat_o(member_nat(X4),A3)
         => aa_set_set_nat_o(member_set_nat(aa_nat_set_nat(F,X4)),B3) ) ) ).

% image_subset_iff
tff(fact_6004_image__subset__iff,axiom,
    ! [F: fun_nat_nat,A3: set_nat,B3: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,image_nat_nat(F,A3)),B3)
    <=> ! [X4: nat] :
          ( aa_set_nat_o(member_nat(X4),A3)
         => aa_set_nat_o(member_nat(aa_nat_nat(F,X4)),B3) ) ) ).

% image_subset_iff
tff(fact_6005_image__subset__iff,axiom,
    ! [F: fun_int_nat,A3: set_int,B3: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,image_int_nat(F,A3)),B3)
    <=> ! [X4: int] :
          ( aa_set_int_o(member_int(X4),A3)
         => aa_set_nat_o(member_nat(aa_int_nat(F,X4)),B3) ) ) ).

% image_subset_iff
tff(fact_6006_image__subset__iff,axiom,
    ! [F: fun_nat_int,A3: set_nat,B3: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,image_nat_int(F,A3)),B3)
    <=> ! [X4: nat] :
          ( aa_set_nat_o(member_nat(X4),A3)
         => aa_set_int_o(member_int(aa_nat_int(F,X4)),B3) ) ) ).

% image_subset_iff
tff(fact_6007_image__subset__iff,axiom,
    ! [F: fun_int_int,A3: set_int,B3: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,image_int_int(F,A3)),B3)
    <=> ! [X4: int] :
          ( aa_set_int_o(member_int(X4),A3)
         => aa_set_int_o(member_int(aa_int_int(F,X4)),B3) ) ) ).

% image_subset_iff
tff(fact_6008_subset__imageE,axiom,
    ! [B3: set_set_nat,F: fun_nat_set_nat,A3: set_nat] :
      ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(B3),image_nat_set_nat(F,A3))
     => ~ ! [C4: set_nat] :
            ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,C4),A3)
           => ( B3 != image_nat_set_nat(F,C4) ) ) ) ).

% subset_imageE
tff(fact_6009_subset__imageE,axiom,
    ! [B3: set_nat,F: fun_nat_nat,A3: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B3),image_nat_nat(F,A3))
     => ~ ! [C4: set_nat] :
            ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,C4),A3)
           => ( B3 != image_nat_nat(F,C4) ) ) ) ).

% subset_imageE
tff(fact_6010_subset__imageE,axiom,
    ! [B3: set_nat,F: fun_int_nat,A3: set_int] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B3),image_int_nat(F,A3))
     => ~ ! [C4: set_int] :
            ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,C4),A3)
           => ( B3 != image_int_nat(F,C4) ) ) ) ).

% subset_imageE
tff(fact_6011_subset__imageE,axiom,
    ! [B3: set_int,F: fun_nat_int,A3: set_nat] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,B3),image_nat_int(F,A3))
     => ~ ! [C4: set_nat] :
            ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,C4),A3)
           => ( B3 != image_nat_int(F,C4) ) ) ) ).

% subset_imageE
tff(fact_6012_subset__imageE,axiom,
    ! [B3: set_int,F: fun_int_int,A3: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,B3),image_int_int(F,A3))
     => ~ ! [C4: set_int] :
            ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,C4),A3)
           => ( B3 != image_int_int(F,C4) ) ) ) ).

% subset_imageE
tff(fact_6013_image__subsetI,axiom,
    ! [A3: set_o,F: fun_o_o,B3: set_o] :
      ( ! [X3: $o] :
          ( member_o((X3),A3)
         => member_o(aa_o_o(F,(X3)),B3) )
     => ord_less_eq_set_o(image_o_o(F,A3),B3) ) ).

% image_subsetI
tff(fact_6014_image__subsetI,axiom,
    ! [A3: set_o,F: fun_o_nat,B3: set_nat] :
      ( ! [X3: $o] :
          ( member_o((X3),A3)
         => aa_set_nat_o(member_nat(aa_o_nat(F,(X3))),B3) )
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,image_o_nat(F,A3)),B3) ) ).

% image_subsetI
tff(fact_6015_image__subsetI,axiom,
    ! [A3: set_nat,F: fun_nat_o,B3: set_o] :
      ( ! [X3: nat] :
          ( aa_set_nat_o(member_nat(X3),A3)
         => member_o(aa_nat_o(F,X3),B3) )
     => ord_less_eq_set_o(image_nat_o(F,A3),B3) ) ).

% image_subsetI
tff(fact_6016_image__subsetI,axiom,
    ! [A3: set_nat,F: fun_nat_nat,B3: set_nat] :
      ( ! [X3: nat] :
          ( aa_set_nat_o(member_nat(X3),A3)
         => aa_set_nat_o(member_nat(aa_nat_nat(F,X3)),B3) )
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,image_nat_nat(F,A3)),B3) ) ).

% image_subsetI
tff(fact_6017_image__subsetI,axiom,
    ! [A3: set_int,F: fun_int_o,B3: set_o] :
      ( ! [X3: int] :
          ( aa_set_int_o(member_int(X3),A3)
         => member_o(aa_int_o(F,X3),B3) )
     => ord_less_eq_set_o(image_int_o(F,A3),B3) ) ).

% image_subsetI
tff(fact_6018_image__subsetI,axiom,
    ! [A3: set_int,F: fun_int_nat,B3: set_nat] :
      ( ! [X3: int] :
          ( aa_set_int_o(member_int(X3),A3)
         => aa_set_nat_o(member_nat(aa_int_nat(F,X3)),B3) )
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,image_int_nat(F,A3)),B3) ) ).

% image_subsetI
tff(fact_6019_image__subsetI,axiom,
    ! [A3: set_o,F: fun_o_int,B3: set_int] :
      ( ! [X3: $o] :
          ( member_o((X3),A3)
         => aa_set_int_o(member_int(aa_o_int(F,(X3))),B3) )
     => aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,image_o_int(F,A3)),B3) ) ).

% image_subsetI
tff(fact_6020_image__subsetI,axiom,
    ! [A3: set_nat,F: fun_nat_int,B3: set_int] :
      ( ! [X3: nat] :
          ( aa_set_nat_o(member_nat(X3),A3)
         => aa_set_int_o(member_int(aa_nat_int(F,X3)),B3) )
     => aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,image_nat_int(F,A3)),B3) ) ).

% image_subsetI
tff(fact_6021_image__subsetI,axiom,
    ! [A3: set_int,F: fun_int_int,B3: set_int] :
      ( ! [X3: int] :
          ( aa_set_int_o(member_int(X3),A3)
         => aa_set_int_o(member_int(aa_int_int(F,X3)),B3) )
     => aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,image_int_int(F,A3)),B3) ) ).

% image_subsetI
tff(fact_6022_image__subsetI,axiom,
    ! [A3: set_o,F: fun_o_set_nat,B3: set_set_nat] :
      ( ! [X3: $o] :
          ( member_o((X3),A3)
         => aa_set_set_nat_o(member_set_nat(aa_o_set_nat(F,(X3))),B3) )
     => aa_set_set_nat_o(ord_le6893508408891458716et_nat(image_o_set_nat(F,A3)),B3) ) ).

% image_subsetI
tff(fact_6023_image__mono,axiom,
    ! [A3: set_nat,B3: set_nat,F: fun_nat_set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A3),B3)
     => aa_set_set_nat_o(ord_le6893508408891458716et_nat(image_nat_set_nat(F,A3)),image_nat_set_nat(F,B3)) ) ).

% image_mono
tff(fact_6024_image__mono,axiom,
    ! [A3: set_nat,B3: set_nat,F: fun_nat_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A3),B3)
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,image_nat_nat(F,A3)),image_nat_nat(F,B3)) ) ).

% image_mono
tff(fact_6025_image__mono,axiom,
    ! [A3: set_nat,B3: set_nat,F: fun_nat_int] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A3),B3)
     => aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,image_nat_int(F,A3)),image_nat_int(F,B3)) ) ).

% image_mono
tff(fact_6026_image__mono,axiom,
    ! [A3: set_int,B3: set_int,F: fun_int_nat] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A3),B3)
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,image_int_nat(F,A3)),image_int_nat(F,B3)) ) ).

% image_mono
tff(fact_6027_image__mono,axiom,
    ! [A3: set_int,B3: set_int,F: fun_int_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A3),B3)
     => aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,image_int_int(F,A3)),image_int_int(F,B3)) ) ).

% image_mono
tff(fact_6028_image__Un,axiom,
    ! [F: fun_int_int,A3: set_int,B3: set_int] : image_int_int(F,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A3),B3)) = aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,image_int_int(F,A3)),image_int_int(F,B3)) ).

% image_Un
tff(fact_6029_image__Un,axiom,
    ! [F: fun_int_nat,A3: set_int,B3: set_int] : image_int_nat(F,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A3),B3)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,image_int_nat(F,A3)),image_int_nat(F,B3)) ).

% image_Un
tff(fact_6030_image__Un,axiom,
    ! [F: fun_nat_int,A3: set_nat,B3: set_nat] : image_nat_int(F,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A3),B3)) = aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,image_nat_int(F,A3)),image_nat_int(F,B3)) ).

% image_Un
tff(fact_6031_image__Un,axiom,
    ! [F: fun_nat_nat,A3: set_nat,B3: set_nat] : image_nat_nat(F,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A3),B3)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,image_nat_nat(F,A3)),image_nat_nat(F,B3)) ).

% image_Un
tff(fact_6032_image__Un,axiom,
    ! [F: fun_nat_set_nat,A3: set_nat,B3: set_nat] : image_nat_set_nat(F,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A3),B3)) = sup_sup_set_set_nat(image_nat_set_nat(F,A3),image_nat_set_nat(F,B3)) ).

% image_Un
tff(fact_6033_image__Un,axiom,
    ! [F: fun_Pr2871540613828922186at_nat,A3: set_Pr8693737435421807431at_nat,B3: set_Pr8693737435421807431at_nat] : image_6160980552072806642at_nat(F,aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,A3),B3)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,image_6160980552072806642at_nat(F,A3)),image_6160980552072806642at_nat(F,B3)) ).

% image_Un
tff(fact_6034_image__Un,axiom,
    ! [F: fun_na3202795955365919216at_nat,A3: set_nat,B3: set_nat] : image_3276603626957510736at_nat(F,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A3),B3)) = aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,image_3276603626957510736at_nat(F,A3)),image_3276603626957510736at_nat(F,B3)) ).

% image_Un
tff(fact_6035_image__Un,axiom,
    ! [F: fun_Pr7574179848310716106at_nat,A3: set_Pr8551490117392284871at_nat,B3: set_Pr8551490117392284871at_nat] : image_6009382790362589554at_nat(F,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A3),B3)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,image_6009382790362589554at_nat(F,A3)),image_6009382790362589554at_nat(F,B3)) ).

% image_Un
tff(fact_6036_image__Un,axiom,
    ! [F: fun_Pr1253225528153422666at_nat,A3: set_Pr4329608150637261639at_nat,B3: set_Pr4329608150637261639at_nat] : image_3198525501578754290at_nat(F,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A3),B3)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,image_3198525501578754290at_nat(F,A3)),image_3198525501578754290at_nat(F,B3)) ).

% image_Un
tff(fact_6037_image__Un,axiom,
    ! [F: fun_na8407688532849143408at_nat,A3: set_nat,B3: set_nat] : image_5319605298250079696at_nat(F,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A3),B3)) = aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,image_5319605298250079696at_nat(F,A3)),image_5319605298250079696at_nat(F,B3)) ).

% image_Un
tff(fact_6038_image__Collect__subsetI,axiom,
    ! [Pa: fun_nat_o,F: fun_nat_o,B3: set_o] :
      ( ! [X3: nat] :
          ( aa_nat_o(Pa,X3)
         => member_o(aa_nat_o(F,X3),B3) )
     => ord_less_eq_set_o(image_nat_o(F,collect_nat(Pa)),B3) ) ).

% image_Collect_subsetI
tff(fact_6039_image__Collect__subsetI,axiom,
    ! [Pa: fun_nat_o,F: fun_nat_nat,B3: set_nat] :
      ( ! [X3: nat] :
          ( aa_nat_o(Pa,X3)
         => aa_set_nat_o(member_nat(aa_nat_nat(F,X3)),B3) )
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,image_nat_nat(F,collect_nat(Pa))),B3) ) ).

% image_Collect_subsetI
tff(fact_6040_image__Collect__subsetI,axiom,
    ! [Pa: fun_int_o,F: fun_int_o,B3: set_o] :
      ( ! [X3: int] :
          ( aa_int_o(Pa,X3)
         => member_o(aa_int_o(F,X3),B3) )
     => ord_less_eq_set_o(image_int_o(F,collect_int(Pa)),B3) ) ).

% image_Collect_subsetI
tff(fact_6041_image__Collect__subsetI,axiom,
    ! [Pa: fun_int_o,F: fun_int_nat,B3: set_nat] :
      ( ! [X3: int] :
          ( aa_int_o(Pa,X3)
         => aa_set_nat_o(member_nat(aa_int_nat(F,X3)),B3) )
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,image_int_nat(F,collect_int(Pa))),B3) ) ).

% image_Collect_subsetI
tff(fact_6042_image__Collect__subsetI,axiom,
    ! [Pa: fun_nat_o,F: fun_nat_int,B3: set_int] :
      ( ! [X3: nat] :
          ( aa_nat_o(Pa,X3)
         => aa_set_int_o(member_int(aa_nat_int(F,X3)),B3) )
     => aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,image_nat_int(F,collect_nat(Pa))),B3) ) ).

% image_Collect_subsetI
tff(fact_6043_image__Collect__subsetI,axiom,
    ! [Pa: fun_int_o,F: fun_int_int,B3: set_int] :
      ( ! [X3: int] :
          ( aa_int_o(Pa,X3)
         => aa_set_int_o(member_int(aa_int_int(F,X3)),B3) )
     => aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,image_int_int(F,collect_int(Pa))),B3) ) ).

% image_Collect_subsetI
tff(fact_6044_image__Collect__subsetI,axiom,
    ! [Pa: fun_list_nat_o,F: fun_list_nat_o,B3: set_o] :
      ( ! [X3: list_nat] :
          ( aa_list_nat_o(Pa,X3)
         => member_o(aa_list_nat_o(F,X3),B3) )
     => ord_less_eq_set_o(image_list_nat_o(F,collect_list_nat(Pa)),B3) ) ).

% image_Collect_subsetI
tff(fact_6045_image__Collect__subsetI,axiom,
    ! [Pa: fun_list_nat_o,F: fun_list_nat_nat,B3: set_nat] :
      ( ! [X3: list_nat] :
          ( aa_list_nat_o(Pa,X3)
         => aa_set_nat_o(member_nat(aa_list_nat_nat(F,X3)),B3) )
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,image_list_nat_nat(F,collect_list_nat(Pa))),B3) ) ).

% image_Collect_subsetI
tff(fact_6046_image__Collect__subsetI,axiom,
    ! [Pa: fun_set_nat_o,F: fun_set_nat_o,B3: set_o] :
      ( ! [X3: set_nat] :
          ( aa_set_nat_o(Pa,X3)
         => member_o(aa_set_nat_o(F,X3),B3) )
     => ord_less_eq_set_o(image_set_nat_o(F,collect_set_nat(Pa)),B3) ) ).

% image_Collect_subsetI
tff(fact_6047_image__Collect__subsetI,axiom,
    ! [Pa: fun_set_nat_o,F: fun_set_nat_nat,B3: set_nat] :
      ( ! [X3: set_nat] :
          ( aa_set_nat_o(Pa,X3)
         => aa_set_nat_o(member_nat(aa_set_nat_nat(F,X3)),B3) )
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,image_set_nat_nat(F,collect_set_nat(Pa))),B3) ) ).

% image_Collect_subsetI
tff(fact_6048_obtain__list__from__elements,axiom,
    ! [N2: nat,Pa: fun_int_fun_nat_o] :
      ( ! [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),N2)
         => ? [Li: int] : aa_nat_o(aa_int_fun_nat_o(Pa,Li),I3) )
     => ~ ! [L4: list_int] :
            ( ( size_size_list_int(L4) = N2 )
           => ~ ! [I: nat] :
                  ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),N2)
                 => aa_nat_o(aa_int_fun_nat_o(Pa,aa_nat_int(nth_int(L4),I)),I) ) ) ) ).

% obtain_list_from_elements
tff(fact_6049_obtain__list__from__elements,axiom,
    ! [N2: nat,Pa: fun_nat_fun_nat_o] :
      ( ! [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),N2)
         => ? [Li: nat] : aa_nat_o(aa_nat_fun_nat_o(Pa,Li),I3) )
     => ~ ! [L4: list_nat] :
            ( ( aa_list_nat_nat(size_size_list_nat,L4) = N2 )
           => ~ ! [I: nat] :
                  ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),N2)
                 => aa_nat_o(aa_nat_fun_nat_o(Pa,aa_nat_nat(nth_nat(L4),I)),I) ) ) ) ).

% obtain_list_from_elements
tff(fact_6050_obtain__list__from__elements,axiom,
    ! [N2: nat,Pa: fun_o_fun_nat_o] :
      ( ! [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),N2)
         => ? [Li: $o] : aa_nat_o(aa_o_fun_nat_o(Pa,(Li)),I3) )
     => ~ ! [L4: list_o] :
            ( ( size_size_list_o(L4) = N2 )
           => ~ ! [I: nat] :
                  ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),N2)
                 => aa_nat_o(aa_o_fun_nat_o(Pa,aa_nat_o(nth_o(L4),I)),I) ) ) ) ).

% obtain_list_from_elements
tff(fact_6051_list__eq__iff__nth__eq,axiom,
    ! [Xs: list_int,Ys2: list_int] :
      ( ( Xs = Ys2 )
    <=> ( ( size_size_list_int(Xs) = size_size_list_int(Ys2) )
        & ! [I4: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),size_size_list_int(Xs))
           => ( aa_nat_int(nth_int(Xs),I4) = aa_nat_int(nth_int(Ys2),I4) ) ) ) ) ).

% list_eq_iff_nth_eq
tff(fact_6052_list__eq__iff__nth__eq,axiom,
    ! [Xs: list_nat,Ys2: list_nat] :
      ( ( Xs = Ys2 )
    <=> ( ( aa_list_nat_nat(size_size_list_nat,Xs) = aa_list_nat_nat(size_size_list_nat,Ys2) )
        & ! [I4: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),aa_list_nat_nat(size_size_list_nat,Xs))
           => ( aa_nat_nat(nth_nat(Xs),I4) = aa_nat_nat(nth_nat(Ys2),I4) ) ) ) ) ).

% list_eq_iff_nth_eq
tff(fact_6053_list__eq__iff__nth__eq,axiom,
    ! [Xs: list_o,Ys2: list_o] :
      ( ( Xs = Ys2 )
    <=> ( ( size_size_list_o(Xs) = size_size_list_o(Ys2) )
        & ! [I4: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),size_size_list_o(Xs))
           => ( aa_nat_o(nth_o(Xs),I4)
            <=> aa_nat_o(nth_o(Ys2),I4) ) ) ) ) ).

% list_eq_iff_nth_eq
tff(fact_6054_Skolem__list__nth,axiom,
    ! [K: nat,Pa: fun_nat_fun_int_o] :
      ( ! [I4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),K)
         => ? [X_12: int] : aa_int_o(aa_nat_fun_int_o(Pa,I4),X_12) )
    <=> ? [Xs3: list_int] :
          ( ( size_size_list_int(Xs3) = K )
          & ! [I4: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),K)
             => aa_int_o(aa_nat_fun_int_o(Pa,I4),aa_nat_int(nth_int(Xs3),I4)) ) ) ) ).

% Skolem_list_nth
tff(fact_6055_Skolem__list__nth,axiom,
    ! [K: nat,Pa: fun_nat_fun_nat_o] :
      ( ! [I4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),K)
         => ? [X_12: nat] : aa_nat_o(aa_nat_fun_nat_o(Pa,I4),X_12) )
    <=> ? [Xs3: list_nat] :
          ( ( aa_list_nat_nat(size_size_list_nat,Xs3) = K )
          & ! [I4: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),K)
             => aa_nat_o(aa_nat_fun_nat_o(Pa,I4),aa_nat_nat(nth_nat(Xs3),I4)) ) ) ) ).

% Skolem_list_nth
tff(fact_6056_Skolem__list__nth,axiom,
    ! [K: nat,Pa: fun_nat_fun_o_o] :
      ( ! [I4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),K)
         => ? [X_12: $o] : aa_o_o(aa_nat_fun_o_o(Pa,I4),(X_12)) )
    <=> ? [Xs3: list_o] :
          ( ( size_size_list_o(Xs3) = K )
          & ! [I4: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),K)
             => aa_o_o(aa_nat_fun_o_o(Pa,I4),aa_nat_o(nth_o(Xs3),I4)) ) ) ) ).

% Skolem_list_nth
tff(fact_6057_nth__equalityI,axiom,
    ! [Xs: list_int,Ys2: list_int] :
      ( ( size_size_list_int(Xs) = size_size_list_int(Ys2) )
     => ( ! [I3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),size_size_list_int(Xs))
           => ( aa_nat_int(nth_int(Xs),I3) = aa_nat_int(nth_int(Ys2),I3) ) )
       => ( Xs = Ys2 ) ) ) ).

% nth_equalityI
tff(fact_6058_nth__equalityI,axiom,
    ! [Xs: list_nat,Ys2: list_nat] :
      ( ( aa_list_nat_nat(size_size_list_nat,Xs) = aa_list_nat_nat(size_size_list_nat,Ys2) )
     => ( ! [I3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),aa_list_nat_nat(size_size_list_nat,Xs))
           => ( aa_nat_nat(nth_nat(Xs),I3) = aa_nat_nat(nth_nat(Ys2),I3) ) )
       => ( Xs = Ys2 ) ) ) ).

% nth_equalityI
tff(fact_6059_nth__equalityI,axiom,
    ! [Xs: list_o,Ys2: list_o] :
      ( ( size_size_list_o(Xs) = size_size_list_o(Ys2) )
     => ( ! [I3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),size_size_list_o(Xs))
           => ( aa_nat_o(nth_o(Xs),I3)
            <=> aa_nat_o(nth_o(Ys2),I3) ) )
       => ( Xs = Ys2 ) ) ) ).

% nth_equalityI
tff(fact_6060_image__Int__subset,axiom,
    ! [F: fun_int_nat,A3: set_int,B3: set_int] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,image_int_nat(F,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A3),B3))),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,image_int_nat(F,A3)),image_int_nat(F,B3))) ).

% image_Int_subset
tff(fact_6061_image__Int__subset,axiom,
    ! [F: fun_nat_set_nat,A3: set_nat,B3: set_nat] : aa_set_set_nat_o(ord_le6893508408891458716et_nat(image_nat_set_nat(F,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A3),B3))),inf_inf_set_set_nat(image_nat_set_nat(F,A3),image_nat_set_nat(F,B3))) ).

% image_Int_subset
tff(fact_6062_image__Int__subset,axiom,
    ! [F: fun_nat_nat,A3: set_nat,B3: set_nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,image_nat_nat(F,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A3),B3))),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,image_nat_nat(F,A3)),image_nat_nat(F,B3))) ).

% image_Int_subset
tff(fact_6063_image__Int__subset,axiom,
    ! [F: fun_na1476630412504255114at_nat,A3: set_nat,B3: set_nat] : aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,image_5846123807819985514at_nat(F,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A3),B3))),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,image_5846123807819985514at_nat(F,A3)),image_5846123807819985514at_nat(F,B3))) ).

% image_Int_subset
tff(fact_6064_image__Int__subset,axiom,
    ! [F: fun_Pr2602117396471556196at_nat,A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,image_2486076414777270412at_nat(F,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A3),B3))),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,image_2486076414777270412at_nat(F,A3)),image_2486076414777270412at_nat(F,B3))) ).

% image_Int_subset
tff(fact_6065_image__Int__subset,axiom,
    ! [F: fun_Pr2118491707810114563at_nat,A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] : aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,image_5168914502847457605at_nat(F,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A3),B3))),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,image_5168914502847457605at_nat(F,A3)),image_5168914502847457605at_nat(F,B3))) ).

% image_Int_subset
tff(fact_6066_image__Int__subset,axiom,
    ! [F: fun_int_int,A3: set_int,B3: set_int] : aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,image_int_int(F,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A3),B3))),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,image_int_int(F,A3)),image_int_int(F,B3))) ).

% image_Int_subset
tff(fact_6067_image__Int__subset,axiom,
    ! [F: fun_nat_int,A3: set_nat,B3: set_nat] : aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,image_nat_int(F,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A3),B3))),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,image_nat_int(F,A3)),image_nat_int(F,B3))) ).

% image_Int_subset
tff(fact_6068_image__Int__subset,axiom,
    ! [F: fun_Pr7647638413817135296at_int,A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] : aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,image_2483585944268220136at_int(F,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A3),B3))),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,image_2483585944268220136at_int(F,A3)),image_2483585944268220136at_int(F,B3))) ).

% image_Int_subset
tff(fact_6069_image__diff__subset,axiom,
    ! [F: fun_nat_set_nat,A3: set_nat,B3: set_nat] : aa_set_set_nat_o(ord_le6893508408891458716et_nat(minus_2163939370556025621et_nat(image_nat_set_nat(F,A3),image_nat_set_nat(F,B3))),image_nat_set_nat(F,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A3),B3))) ).

% image_diff_subset
tff(fact_6070_image__diff__subset,axiom,
    ! [F: fun_int_nat,A3: set_int,B3: set_int] : 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,image_int_nat(F,A3)),image_int_nat(F,B3))),image_int_nat(F,aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A3),B3))) ).

% image_diff_subset
tff(fact_6071_image__diff__subset,axiom,
    ! [F: fun_nat_nat,A3: set_nat,B3: 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,image_nat_nat(F,A3)),image_nat_nat(F,B3))),image_nat_nat(F,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A3),B3))) ).

% image_diff_subset
tff(fact_6072_image__diff__subset,axiom,
    ! [F: fun_int_int,A3: set_int,B3: set_int] : aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,image_int_int(F,A3)),image_int_int(F,B3))),image_int_int(F,aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A3),B3))) ).

% image_diff_subset
tff(fact_6073_image__diff__subset,axiom,
    ! [F: fun_nat_int,A3: set_nat,B3: set_nat] : aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,image_nat_int(F,A3)),image_nat_int(F,B3))),image_nat_int(F,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A3),B3))) ).

% image_diff_subset
tff(fact_6074_ex__nat__less,axiom,
    ! [N2: nat,Pa: fun_nat_o] :
      ( ? [M3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M3),N2)
          & aa_nat_o(Pa,M3) )
    <=> ? [X4: nat] :
          ( aa_set_nat_o(member_nat(X4),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,zero_zero_nat),N2))
          & aa_nat_o(Pa,X4) ) ) ).

% ex_nat_less
tff(fact_6075_all__nat__less,axiom,
    ! [N2: nat,Pa: fun_nat_o] :
      ( ! [M3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M3),N2)
         => aa_nat_o(Pa,M3) )
    <=> ! [X4: nat] :
          ( aa_set_nat_o(member_nat(X4),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,zero_zero_nat),N2))
         => aa_nat_o(Pa,X4) ) ) ).

% all_nat_less
tff(fact_6076_None__notin__image__Some,axiom,
    ! [A3: set_Pr7098220151150636591it_nat] : ~ member1310632352865511136it_nat(none_P2651198173097904984it_nat,image_2003495143432273353it_nat(some_P7913643980934408916it_nat,A3)) ).

% None_notin_image_Some
tff(fact_6077_None__notin__image__Some,axiom,
    ! [A3: set_num] : ~ member_option_num(none_num,image_num_option_num(some_num,A3)) ).

% None_notin_image_Some
tff(fact_6078_None__notin__image__Some,axiom,
    ! [A3: set_Pr1354866905816374718it_nat] : ~ member5596548051065438575it_nat(none_P9117596204409417319it_nat,image_3455408117458022631it_nat(some_P1914260805536162275it_nat,A3)) ).

% None_notin_image_Some
tff(fact_6079_nth__rotate1,axiom,
    ! [N2: nat,Xs: list_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),size_size_list_int(Xs))
     => ( aa_nat_int(nth_int(rotate1_int(Xs)),N2) = aa_nat_int(nth_int(Xs),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,N2)),size_size_list_int(Xs))) ) ) ).

% nth_rotate1
tff(fact_6080_nth__rotate1,axiom,
    ! [N2: nat,Xs: list_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),aa_list_nat_nat(size_size_list_nat,Xs))
     => ( aa_nat_nat(nth_nat(rotate1_nat(Xs)),N2) = aa_nat_nat(nth_nat(Xs),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,N2)),aa_list_nat_nat(size_size_list_nat,Xs))) ) ) ).

% nth_rotate1
tff(fact_6081_nth__rotate1,axiom,
    ! [N2: nat,Xs: list_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),size_size_list_o(Xs))
     => ( aa_nat_o(nth_o(rotate1_o(Xs)),N2)
      <=> aa_nat_o(nth_o(Xs),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,N2)),size_size_list_o(Xs))) ) ) ).

% nth_rotate1
tff(fact_6082_image__mult__atLeastAtMost__if,axiom,
    ! [Ca: rat,X: rat,Y: rat] :
      image_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Ca),set_or633870826150836451st_rat(X,Y)) = $ite(
        aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Ca),
        set_or633870826150836451st_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Ca),X),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Ca),Y)),
        $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y),set_or633870826150836451st_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Ca),Y),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Ca),X)),bot_bot_set_rat) ) ).

% image_mult_atLeastAtMost_if
tff(fact_6083_prod__decode__aux_Osimps,axiom,
    ! [K: nat,M: nat] :
      aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_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(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),M)),aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_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_6084_prod__decode__aux_Opelims,axiom,
    ! [X: nat,Xa2: nat,Y: product_prod_nat_nat] :
      ( ( aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(nat_prod_decode_aux,X),Xa2) = Y )
     => ( aa_Pro6150637604445964947_nat_o(accp_P4275260045618599050at_nat(nat_pr5047031295181774490ux_rel),aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,X),Xa2))
       => ~ ( ( Y = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa2),X),aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,X),Xa2)),aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(nat_prod_decode_aux,aa_nat_nat(suc,X)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Xa2),aa_nat_nat(suc,X)))) )
           => ~ aa_Pro6150637604445964947_nat_o(accp_P4275260045618599050at_nat(nat_pr5047031295181774490ux_rel),aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,X),Xa2)) ) ) ) ).

% prod_decode_aux.pelims
tff(fact_6085_product__nth,axiom,
    ! [N2: nat,Xs: list_int,Ys2: list_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_int(Xs)),size_size_list_int(Ys2)))
     => ( nth_Pr4439495888332055232nt_int(product_int_int(Xs,Ys2),N2) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_nat_int(nth_int(Xs),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N2),size_size_list_int(Ys2)))),aa_nat_int(nth_int(Ys2),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N2),size_size_list_int(Ys2)))) ) ) ).

% product_nth
tff(fact_6086_product__nth,axiom,
    ! [N2: nat,Xs: list_int,Ys2: list_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_int(Xs)),aa_list_nat_nat(size_size_list_nat,Ys2)))
     => ( nth_Pr8617346907841251940nt_nat(product_int_nat(Xs,Ys2),N2) = product_Pair_int_nat(aa_nat_int(nth_int(Xs),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N2),aa_list_nat_nat(size_size_list_nat,Ys2))),aa_nat_nat(nth_nat(Ys2),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N2),aa_list_nat_nat(size_size_list_nat,Ys2)))) ) ) ).

% product_nth
tff(fact_6087_product__nth,axiom,
    ! [N2: nat,Xs: list_int,Ys2: list_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_int(Xs)),size_size_list_o(Ys2)))
     => ( nth_Pr7514405829937366042_int_o(product_int_o(Xs,Ys2),N2) = product_Pair_int_o(aa_nat_int(nth_int(Xs),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N2),size_size_list_o(Ys2))),aa_nat_o(nth_o(Ys2),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N2),size_size_list_o(Ys2)))) ) ) ).

% product_nth
tff(fact_6088_product__nth,axiom,
    ! [N2: nat,Xs: list_nat,Ys2: list_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_list_nat_nat(size_size_list_nat,Xs)),size_size_list_int(Ys2)))
     => ( nth_Pr3440142176431000676at_int(product_nat_int(Xs,Ys2),N2) = product_Pair_nat_int(aa_nat_nat(nth_nat(Xs),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N2),size_size_list_int(Ys2))),aa_nat_int(nth_int(Ys2),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N2),size_size_list_int(Ys2)))) ) ) ).

% product_nth
tff(fact_6089_product__nth,axiom,
    ! [N2: nat,Xs: list_nat,Ys2: list_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_list_nat_nat(size_size_list_nat,Xs)),aa_list_nat_nat(size_size_list_nat,Ys2)))
     => ( aa_nat7603221163087113075at_nat(nth_Pr7617993195940197384at_nat(product_nat_nat(Xs,Ys2)),N2) = aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,aa_nat_nat(nth_nat(Xs),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N2),aa_list_nat_nat(size_size_list_nat,Ys2)))),aa_nat_nat(nth_nat(Ys2),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N2),aa_list_nat_nat(size_size_list_nat,Ys2)))) ) ) ).

% product_nth
tff(fact_6090_product__nth,axiom,
    ! [N2: nat,Xs: list_nat,Ys2: list_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_list_nat_nat(size_size_list_nat,Xs)),size_size_list_o(Ys2)))
     => ( nth_Pr112076138515278198_nat_o(product_nat_o(Xs,Ys2),N2) = product_Pair_nat_o(aa_nat_nat(nth_nat(Xs),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N2),size_size_list_o(Ys2))),aa_nat_o(nth_o(Ys2),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N2),size_size_list_o(Ys2)))) ) ) ).

% product_nth
tff(fact_6091_product__nth,axiom,
    ! [N2: nat,Xs: list_o,Ys2: list_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_o(Xs)),size_size_list_int(Ys2)))
     => ( nth_Pr1649062631805364268_o_int(product_o_int(Xs,Ys2),N2) = product_Pair_o_int(aa_nat_o(nth_o(Xs),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N2),size_size_list_int(Ys2))),aa_nat_int(nth_int(Ys2),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N2),size_size_list_int(Ys2)))) ) ) ).

% product_nth
tff(fact_6092_product__nth,axiom,
    ! [N2: nat,Xs: list_o,Ys2: list_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_o(Xs)),aa_list_nat_nat(size_size_list_nat,Ys2)))
     => ( nth_Pr5826913651314560976_o_nat(product_o_nat(Xs,Ys2),N2) = product_Pair_o_nat(aa_nat_o(nth_o(Xs),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N2),aa_list_nat_nat(size_size_list_nat,Ys2))),aa_nat_nat(nth_nat(Ys2),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N2),aa_list_nat_nat(size_size_list_nat,Ys2)))) ) ) ).

% product_nth
tff(fact_6093_product__nth,axiom,
    ! [N2: nat,Xs: list_o,Ys2: list_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_o(Xs)),size_size_list_o(Ys2)))
     => ( nth_Product_prod_o_o(product_o_o(Xs,Ys2),N2) = product_Pair_o_o(aa_nat_o(nth_o(Xs),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N2),size_size_list_o(Ys2))),aa_nat_o(nth_o(Ys2),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N2),size_size_list_o(Ys2)))) ) ) ).

% product_nth
tff(fact_6094_product__nth,axiom,
    ! [N2: nat,Xs: list_a,Ys2: list_P131111800688179804it_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_a(Xs)),size_s341701280123345136it_nat(Ys2)))
     => ( nth_Pr8687047199098054322it_nat(produc6966004487978991557it_nat(Xs,Ys2),N2) = aa_Pro8078911840264235907it_nat(aa_a_f2234720956197727492it_nat(produc9178034014595674355it_nat,nth_a(Xs,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N2),size_s341701280123345136it_nat(Ys2)))),nth_Pr4561283279332054789it_nat(Ys2,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N2),size_s341701280123345136it_nat(Ys2)))) ) ) ).

% product_nth
tff(fact_6095_translation__subtract__Compl,axiom,
    ! [A: rat,Ta: set_rat] : image_rat_rat(aa_rat_fun_rat_rat(aTP_Lamm_hw,A),uminus2201863774496077783et_rat(Ta)) = uminus2201863774496077783et_rat(image_rat_rat(aa_rat_fun_rat_rat(aTP_Lamm_hw,A),Ta)) ).

% translation_subtract_Compl
tff(fact_6096_translation__subtract__Compl,axiom,
    ! [A: int,Ta: set_int] : image_int_int(aa_int_fun_int_int(aTP_Lamm_hx,A),uminus1532241313380277803et_int(Ta)) = uminus1532241313380277803et_int(image_int_int(aa_int_fun_int_int(aTP_Lamm_hx,A),Ta)) ).

% translation_subtract_Compl
tff(fact_6097_translation__subtract__diff,axiom,
    ! [A: rat,S2: set_rat,Ta: set_rat] : image_rat_rat(aa_rat_fun_rat_rat(aTP_Lamm_hw,A),minus_minus_set_rat(S2,Ta)) = minus_minus_set_rat(image_rat_rat(aa_rat_fun_rat_rat(aTP_Lamm_hw,A),S2),image_rat_rat(aa_rat_fun_rat_rat(aTP_Lamm_hw,A),Ta)) ).

% translation_subtract_diff
tff(fact_6098_translation__subtract__diff,axiom,
    ! [A: int,S2: set_int,Ta: set_int] : image_int_int(aa_int_fun_int_int(aTP_Lamm_hx,A),aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,S2),Ta)) = aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,image_int_int(aa_int_fun_int_int(aTP_Lamm_hx,A),S2)),image_int_int(aa_int_fun_int_int(aTP_Lamm_hx,A),Ta)) ).

% translation_subtract_diff
tff(fact_6099_translation__subtract__Int,axiom,
    ! [A: rat,S2: set_rat,Ta: set_rat] : image_rat_rat(aa_rat_fun_rat_rat(aTP_Lamm_hw,A),inf_inf_set_rat(S2,Ta)) = inf_inf_set_rat(image_rat_rat(aa_rat_fun_rat_rat(aTP_Lamm_hw,A),S2),image_rat_rat(aa_rat_fun_rat_rat(aTP_Lamm_hw,A),Ta)) ).

% translation_subtract_Int
tff(fact_6100_translation__subtract__Int,axiom,
    ! [A: int,S2: set_int,Ta: set_int] : image_int_int(aa_int_fun_int_int(aTP_Lamm_hx,A),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,S2),Ta)) = aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,image_int_int(aa_int_fun_int_int(aTP_Lamm_hx,A),S2)),image_int_int(aa_int_fun_int_int(aTP_Lamm_hx,A),Ta)) ).

% translation_subtract_Int
tff(fact_6101_sum__gp,axiom,
    ! [X: rat,M: nat,N2: nat] :
      aa_set_nat_rat(groups2906978787729119204at_rat(power_power_rat(X)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M),N2)) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),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,N2),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,N2)))),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,one_one_rat),X))) ) ).

% sum_gp
tff(fact_6102_of__nat__sum,axiom,
    ! [F: fun_int_nat,A3: set_int] : aa_nat_int(semiri1314217659103216013at_int,groups4541462559716669496nt_nat(F,A3)) = groups4538972089207619220nt_int(aTP_Lamm_jp(F),A3) ).

% of_nat_sum
tff(fact_6103_of__nat__sum,axiom,
    ! [F: fun_nat_nat,A3: set_nat] : aa_nat_int(semiri1314217659103216013at_int,aa_set_nat_nat(groups3542108847815614940at_nat(F),A3)) = aa_set_nat_int(groups3539618377306564664at_int(aTP_Lamm_jq(F)),A3) ).

% of_nat_sum
tff(fact_6104_of__nat__sum,axiom,
    ! [F: fun_nat_nat,A3: set_nat] : aa_nat_Code_integer(semiri4939895301339042750nteger,aa_set_nat_nat(groups3542108847815614940at_nat(F),A3)) = aa_set871585252387937313nteger(groups7501900531339628137nteger(aTP_Lamm_jr(F)),A3) ).

% of_nat_sum
tff(fact_6105_of__nat__sum,axiom,
    ! [F: fun_nat_nat,A3: set_nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),A3)) = aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_js(F)),A3) ).

% of_nat_sum
tff(fact_6106_of__int__sum,axiom,
    ! [F: fun_int_int,A3: set_int] : aa_int_rat(ring_1_of_int_rat,groups4538972089207619220nt_int(F,A3)) = groups3906332499630173760nt_rat(aTP_Lamm_jt(F),A3) ).

% of_int_sum
tff(fact_6107_of__int__sum,axiom,
    ! [F: fun_int_int,A3: set_int] : aa_int_int(ring_1_of_int_int,groups4538972089207619220nt_int(F,A3)) = groups4538972089207619220nt_int(aTP_Lamm_ju(F),A3) ).

% of_int_sum
tff(fact_6108_sum_Ocl__ivl__Suc,axiom,
    ! [G: fun_nat_Code_integer,M: nat,N2: nat] :
      aa_set871585252387937313nteger(groups7501900531339628137nteger(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M),aa_nat_nat(suc,N2))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,N2)),M),zero_z3403309356797280102nteger,aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_set871585252387937313nteger(groups7501900531339628137nteger(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M),N2))),aa_nat_Code_integer(G,aa_nat_nat(suc,N2)))) ).

% sum.cl_ivl_Suc
tff(fact_6109_sum_Ocl__ivl__Suc,axiom,
    ! [G: fun_na2338080690716861136at_nat,M: nat,N2: nat] :
      groups6857163185585827899at_nat(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M),aa_nat_nat(suc,N2))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,N2)),M),zero_z1048942125864253310at_nat,aa_mul5754855110707352700at_nat(plus_p7104986032573967614at_nat(groups6857163185585827899at_nat(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M),N2))),aa_nat2240542744625857721at_nat(G,aa_nat_nat(suc,N2)))) ).

% sum.cl_ivl_Suc
tff(fact_6110_sum_Ocl__ivl__Suc,axiom,
    ! [G: fun_nat_rat,M: nat,N2: nat] :
      aa_set_nat_rat(groups2906978787729119204at_rat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M),aa_nat_nat(suc,N2))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,N2)),M),zero_zero_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_set_nat_rat(groups2906978787729119204at_rat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M),N2))),aa_nat_rat(G,aa_nat_nat(suc,N2)))) ).

% sum.cl_ivl_Suc
tff(fact_6111_sum_Ocl__ivl__Suc,axiom,
    ! [G: fun_nat_int,M: nat,N2: nat] :
      aa_set_nat_int(groups3539618377306564664at_int(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M),aa_nat_nat(suc,N2))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,N2)),M),zero_zero_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_set_nat_int(groups3539618377306564664at_int(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M),N2))),aa_nat_int(G,aa_nat_nat(suc,N2)))) ).

% sum.cl_ivl_Suc
tff(fact_6112_sum_Ocl__ivl__Suc,axiom,
    ! [G: fun_nat_nat,M: nat,N2: nat] :
      aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M),aa_nat_nat(suc,N2))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,N2)),M),zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M),N2))),aa_nat_nat(G,aa_nat_nat(suc,N2)))) ).

% sum.cl_ivl_Suc
tff(fact_6113_mod__sum__eq,axiom,
    ! [F: fun_nat_nat,A: nat,A3: 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_jv(F),A)),A3)),A) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),A3)),A) ).

% mod_sum_eq
tff(fact_6114_mod__sum__eq,axiom,
    ! [F: fun_int_int,A: int,A3: set_int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,groups4538972089207619220nt_int(aa_int_fun_int_int(aTP_Lamm_jw(F),A),A3)),A) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,groups4538972089207619220nt_int(F,A3)),A) ).

% mod_sum_eq
tff(fact_6115_sum_Oshift__bounds__cl__Suc__ivl,axiom,
    ! [G: fun_nat_nat,M: nat,N2: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,aa_nat_nat(suc,M)),aa_nat_nat(suc,N2))) = aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_jx(G)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M),N2)) ).

% sum.shift_bounds_cl_Suc_ivl
tff(fact_6116_sum_Oshift__bounds__cl__nat__ivl,axiom,
    ! [G: fun_nat_nat,M: nat,K: nat,N2: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(aa_nat1818925681583257441et_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,N2),K))) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_jy(G),K)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M),N2)) ).

% sum.shift_bounds_cl_nat_ivl
tff(fact_6117_sum_OatLeastAtMost__rev,axiom,
    ! [G: fun_nat_nat,N2: nat,M: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,N2),M)) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_jz(G),N2),M)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,N2),M)) ).

% sum.atLeastAtMost_rev
tff(fact_6118_sum_Onat__ivl__Suc_H,axiom,
    ! [M: nat,N2: nat,G: fun_na2338080690716861136at_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(suc,N2))
     => ( groups6857163185585827899at_nat(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M),aa_nat_nat(suc,N2))) = aa_mul5754855110707352700at_nat(plus_p7104986032573967614at_nat(aa_nat2240542744625857721at_nat(G,aa_nat_nat(suc,N2))),groups6857163185585827899at_nat(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M),N2))) ) ) ).

% sum.nat_ivl_Suc'
tff(fact_6119_sum_Onat__ivl__Suc_H,axiom,
    ! [M: nat,N2: nat,G: fun_nat_rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(suc,N2))
     => ( aa_set_nat_rat(groups2906978787729119204at_rat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M),aa_nat_nat(suc,N2))) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(G,aa_nat_nat(suc,N2))),aa_set_nat_rat(groups2906978787729119204at_rat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M),N2))) ) ) ).

% sum.nat_ivl_Suc'
tff(fact_6120_sum_Onat__ivl__Suc_H,axiom,
    ! [M: nat,N2: nat,G: fun_nat_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(suc,N2))
     => ( aa_set_nat_int(groups3539618377306564664at_int(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M),aa_nat_nat(suc,N2))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(G,aa_nat_nat(suc,N2))),aa_set_nat_int(groups3539618377306564664at_int(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M),N2))) ) ) ).

% sum.nat_ivl_Suc'
tff(fact_6121_sum_Onat__ivl__Suc_H,axiom,
    ! [M: nat,N2: nat,G: fun_nat_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(suc,N2))
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M),aa_nat_nat(suc,N2))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(G,aa_nat_nat(suc,N2))),aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M),N2))) ) ) ).

% sum.nat_ivl_Suc'
tff(fact_6122_sum_OatLeast__Suc__atMost,axiom,
    ! [M: nat,N2: nat,G: fun_na2338080690716861136at_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
     => ( groups6857163185585827899at_nat(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M),N2)) = aa_mul5754855110707352700at_nat(plus_p7104986032573967614at_nat(aa_nat2240542744625857721at_nat(G,M)),groups6857163185585827899at_nat(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,aa_nat_nat(suc,M)),N2))) ) ) ).

% sum.atLeast_Suc_atMost
tff(fact_6123_sum_OatLeast__Suc__atMost,axiom,
    ! [M: nat,N2: nat,G: fun_nat_rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
     => ( aa_set_nat_rat(groups2906978787729119204at_rat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M),N2)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(G,M)),aa_set_nat_rat(groups2906978787729119204at_rat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,aa_nat_nat(suc,M)),N2))) ) ) ).

% sum.atLeast_Suc_atMost
tff(fact_6124_sum_OatLeast__Suc__atMost,axiom,
    ! [M: nat,N2: nat,G: fun_nat_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
     => ( aa_set_nat_int(groups3539618377306564664at_int(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M),N2)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(G,M)),aa_set_nat_int(groups3539618377306564664at_int(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,aa_nat_nat(suc,M)),N2))) ) ) ).

% sum.atLeast_Suc_atMost
tff(fact_6125_sum_OatLeast__Suc__atMost,axiom,
    ! [M: nat,N2: nat,G: fun_nat_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M),N2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(G,M)),aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,aa_nat_nat(suc,M)),N2))) ) ) ).

% sum.atLeast_Suc_atMost
tff(fact_6126_sum_OSuc__reindex__ivl,axiom,
    ! [M: nat,N2: nat,G: fun_na2338080690716861136at_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
     => ( aa_mul5754855110707352700at_nat(plus_p7104986032573967614at_nat(groups6857163185585827899at_nat(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M),N2))),aa_nat2240542744625857721at_nat(G,aa_nat_nat(suc,N2))) = aa_mul5754855110707352700at_nat(plus_p7104986032573967614at_nat(aa_nat2240542744625857721at_nat(G,M)),groups6857163185585827899at_nat(aTP_Lamm_ka(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M),N2))) ) ) ).

% sum.Suc_reindex_ivl
tff(fact_6127_sum_OSuc__reindex__ivl,axiom,
    ! [M: nat,N2: nat,G: fun_nat_rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_set_nat_rat(groups2906978787729119204at_rat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M),N2))),aa_nat_rat(G,aa_nat_nat(suc,N2))) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(G,M)),aa_set_nat_rat(groups2906978787729119204at_rat(aTP_Lamm_kb(G)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M),N2))) ) ) ).

% sum.Suc_reindex_ivl
tff(fact_6128_sum_OSuc__reindex__ivl,axiom,
    ! [M: nat,N2: nat,G: fun_nat_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_set_nat_int(groups3539618377306564664at_int(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M),N2))),aa_nat_int(G,aa_nat_nat(suc,N2))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(G,M)),aa_set_nat_int(groups3539618377306564664at_int(aTP_Lamm_kc(G)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M),N2))) ) ) ).

% sum.Suc_reindex_ivl
tff(fact_6129_sum_OSuc__reindex__ivl,axiom,
    ! [M: nat,N2: nat,G: fun_nat_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M),N2))),aa_nat_nat(G,aa_nat_nat(suc,N2))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(G,M)),aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_jx(G)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M),N2))) ) ) ).

% sum.Suc_reindex_ivl
tff(fact_6130_sum__Suc__diff,axiom,
    ! [M: nat,N2: nat,F: fun_nat_rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(suc,N2))
     => ( aa_set_nat_rat(groups2906978787729119204at_rat(aTP_Lamm_kd(F)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M),N2)) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_nat_rat(F,aa_nat_nat(suc,N2))),aa_nat_rat(F,M)) ) ) ).

% sum_Suc_diff
tff(fact_6131_sum__Suc__diff,axiom,
    ! [M: nat,N2: nat,F: fun_nat_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(suc,N2))
     => ( aa_set_nat_int(groups3539618377306564664at_int(aTP_Lamm_ke(F)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M),N2)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(F,aa_nat_nat(suc,N2))),aa_nat_int(F,M)) ) ) ).

% sum_Suc_diff
tff(fact_6132_sum__power__add,axiom,
    ! [X: code_integer,M: nat,I5: set_nat] : aa_set871585252387937313nteger(groups7501900531339628137nteger(aa_nat2010205248571845816nteger(aTP_Lamm_kf(X),M)),I5) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(X),M)),aa_set871585252387937313nteger(groups7501900531339628137nteger(power_8256067586552552935nteger(X)),I5)) ).

% sum_power_add
tff(fact_6133_sum__power__add,axiom,
    ! [X: rat,M: nat,I5: set_nat] : aa_set_nat_rat(groups2906978787729119204at_rat(aa_nat_fun_nat_rat(aTP_Lamm_kg(X),M)),I5) = 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)),I5)) ).

% sum_power_add
tff(fact_6134_sum__power__add,axiom,
    ! [X: int,M: nat,I5: set_nat] : aa_set_nat_int(groups3539618377306564664at_int(aa_nat_fun_nat_int(aTP_Lamm_kh(X),M)),I5) = 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)),I5)) ).

% sum_power_add
tff(fact_6135_sum__atLeastAtMost__code,axiom,
    ! [F: fun_nat_Code_integer,A: nat,B: nat] : aa_set871585252387937313nteger(groups7501900531339628137nteger(F),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,A),B)) = aa_Cod1783974652533396728nteger(set_fo1084959871951514735nteger(aTP_Lamm_ki(F),A,B),zero_z3403309356797280102nteger) ).

% sum_atLeastAtMost_code
tff(fact_6136_sum__atLeastAtMost__code,axiom,
    ! [F: fun_na2338080690716861136at_nat,A: nat,B: nat] : groups6857163185585827899at_nat(F,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,A),B)) = set_fo4497565046347964853at_nat(aTP_Lamm_kj(F),A,B,zero_z1048942125864253310at_nat) ).

% sum_atLeastAtMost_code
tff(fact_6137_sum__atLeastAtMost__code,axiom,
    ! [F: fun_nat_rat,A: nat,B: nat] : aa_set_nat_rat(groups2906978787729119204at_rat(F),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,A),B)) = aa_rat_rat(set_fo1949268297981939178at_rat(aTP_Lamm_kk(F),A,B),zero_zero_rat) ).

% sum_atLeastAtMost_code
tff(fact_6138_sum__atLeastAtMost__code,axiom,
    ! [F: fun_nat_int,A: nat,B: nat] : aa_set_nat_int(groups3539618377306564664at_int(F),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,A),B)) = aa_int_int(aa_nat_fun_int_int(set_fo2581907887559384638at_int(aTP_Lamm_kl(F),A),B),zero_zero_int) ).

% sum_atLeastAtMost_code
tff(fact_6139_sum__atLeastAtMost__code,axiom,
    ! [F: fun_nat_nat,A: nat,B: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(F),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,A),B)) = aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(aTP_Lamm_km(F),A),B),zero_zero_nat) ).

% sum_atLeastAtMost_code
tff(fact_6140_sum_Oub__add__nat,axiom,
    ! [M: nat,N2: nat,G: fun_na2338080690716861136at_nat,P2: 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,N2),one_one_nat))
     => ( groups6857163185585827899at_nat(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),P2))) = aa_mul5754855110707352700at_nat(plus_p7104986032573967614at_nat(groups6857163185585827899at_nat(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M),N2))),groups6857163185585827899at_nat(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),P2)))) ) ) ).

% sum.ub_add_nat
tff(fact_6141_sum_Oub__add__nat,axiom,
    ! [M: nat,N2: nat,G: fun_nat_rat,P2: 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,N2),one_one_nat))
     => ( aa_set_nat_rat(groups2906978787729119204at_rat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),P2))) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_set_nat_rat(groups2906978787729119204at_rat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M),N2))),aa_set_nat_rat(groups2906978787729119204at_rat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),P2)))) ) ) ).

% sum.ub_add_nat
tff(fact_6142_sum_Oub__add__nat,axiom,
    ! [M: nat,N2: nat,G: fun_nat_int,P2: 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,N2),one_one_nat))
     => ( aa_set_nat_int(groups3539618377306564664at_int(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),P2))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_set_nat_int(groups3539618377306564664at_int(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M),N2))),aa_set_nat_int(groups3539618377306564664at_int(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),P2)))) ) ) ).

% sum.ub_add_nat
tff(fact_6143_sum_Oub__add__nat,axiom,
    ! [M: nat,N2: nat,G: fun_nat_nat,P2: 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,N2),one_one_nat))
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),P2))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M),N2))),aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),P2)))) ) ) ).

% sum.ub_add_nat
tff(fact_6144_sum__natinterval__diff,axiom,
    ! [F: fun_nat_Code_integer,M: nat,N2: nat] :
      aa_set871585252387937313nteger(groups7501900531339628137nteger(aTP_Lamm_kn(F)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M),N2)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2),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,N2),one_one_nat))),zero_z3403309356797280102nteger) ).

% sum_natinterval_diff
tff(fact_6145_sum__natinterval__diff,axiom,
    ! [F: fun_nat_rat,M: nat,N2: nat] :
      aa_set_nat_rat(groups2906978787729119204at_rat(aTP_Lamm_ko(F)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M),N2)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2),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,N2),one_one_nat))),zero_zero_rat) ).

% sum_natinterval_diff
tff(fact_6146_sum__natinterval__diff,axiom,
    ! [F: fun_nat_int,M: nat,N2: nat] :
      aa_set_nat_int(groups3539618377306564664at_int(aTP_Lamm_kp(F)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M),N2)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2),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,N2),one_one_nat))),zero_zero_int) ).

% sum_natinterval_diff
tff(fact_6147_sum__telescope_H_H,axiom,
    ! [M: nat,N2: nat,F: fun_nat_rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
     => ( aa_set_nat_rat(groups2906978787729119204at_rat(aTP_Lamm_kq(F)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,aa_nat_nat(suc,M)),N2)) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_nat_rat(F,N2)),aa_nat_rat(F,M)) ) ) ).

% sum_telescope''
tff(fact_6148_sum__telescope_H_H,axiom,
    ! [M: nat,N2: nat,F: fun_nat_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
     => ( aa_set_nat_int(groups3539618377306564664at_int(aTP_Lamm_kr(F)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,aa_nat_nat(suc,M)),N2)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(F,N2)),aa_nat_int(F,M)) ) ) ).

% sum_telescope''
tff(fact_6149_sum__gp__multiplied,axiom,
    ! [M: nat,N2: nat,X: code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
     => ( 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(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M),N2))) = aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_nat_Code_integer(power_8256067586552552935nteger(X),M)),aa_nat_Code_integer(power_8256067586552552935nteger(X),aa_nat_nat(suc,N2))) ) ) ).

% sum_gp_multiplied
tff(fact_6150_sum__gp__multiplied,axiom,
    ! [M: nat,N2: nat,X: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
     => ( 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(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M),N2))) = 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,N2))) ) ) ).

% sum_gp_multiplied
tff(fact_6151_sum__gp__multiplied,axiom,
    ! [M: nat,N2: nat,X: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
     => ( 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(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M),N2))) = 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,N2))) ) ) ).

% sum_gp_multiplied
tff(fact_6152_gbinomial__sum__up__index,axiom,
    ! [K: nat,N2: nat] : aa_set_nat_rat(groups2906978787729119204at_rat(aTP_Lamm_ks(K)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,zero_zero_nat),N2)) = aa_nat_rat(gbinomial_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(semiri681578069525770553at_rat,N2)),one_one_rat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),one_one_nat)) ).

% gbinomial_sum_up_index
tff(fact_6153_sum__gp__offset,axiom,
    ! [X: rat,M: nat,N2: nat] :
      aa_set_nat_rat(groups2906978787729119204at_rat(power_power_rat(X)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N2))) = $ite(X = one_one_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(semiri681578069525770553at_rat,N2)),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,N2))))),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,one_one_rat),X))) ).

% sum_gp_offset
tff(fact_6154_translation__Int,axiom,
    ! [A: rat,S2: set_rat,Ta: set_rat] : image_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),inf_inf_set_rat(S2,Ta)) = inf_inf_set_rat(image_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),S2),image_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),Ta)) ).

% translation_Int
tff(fact_6155_translation__Int,axiom,
    ! [A: int,S2: set_int,Ta: set_int] : image_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,S2),Ta)) = aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,image_int_int(aa_int_fun_int_int(plus_plus_int,A),S2)),image_int_int(aa_int_fun_int_int(plus_plus_int,A),Ta)) ).

% translation_Int
tff(fact_6156_sum__abs__ge__zero,axiom,
    ! [F: fun_int_int,A3: set_int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),groups4538972089207619220nt_int(aTP_Lamm_kt(F),A3)) ).

% sum_abs_ge_zero
tff(fact_6157_sum__abs,axiom,
    ! [F: fun_int_int,A3: set_int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,groups4538972089207619220nt_int(F,A3))),groups4538972089207619220nt_int(aTP_Lamm_kt(F),A3)) ).

% sum_abs
tff(fact_6158_sum_Oempty,axiom,
    ! [G: fun_o_Code_integer] : groups4406642042086082107nteger(G,bot_bot_set_o) = zero_z3403309356797280102nteger ).

% sum.empty
tff(fact_6159_sum_Oempty,axiom,
    ! [G: fun_o_rat] : groups7872700643590313910_o_rat(G,bot_bot_set_o) = zero_zero_rat ).

% sum.empty
tff(fact_6160_sum_Oempty,axiom,
    ! [G: fun_o_nat] : groups8507830703676809646_o_nat(G,bot_bot_set_o) = zero_zero_nat ).

% sum.empty
tff(fact_6161_sum_Oempty,axiom,
    ! [G: fun_o_int] : groups8505340233167759370_o_int(G,bot_bot_set_o) = zero_zero_int ).

% sum.empty
tff(fact_6162_sum_Oempty,axiom,
    ! [G: fun_nat_Code_integer] : aa_set871585252387937313nteger(groups7501900531339628137nteger(G),bot_bot_set_nat) = zero_z3403309356797280102nteger ).

% sum.empty
tff(fact_6163_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_6164_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_6165_sum_Oempty,axiom,
    ! [G: fun_int_Code_integer] : groups7873554091576472773nteger(G,bot_bot_set_int) = zero_z3403309356797280102nteger ).

% sum.empty
tff(fact_6166_sum_Oempty,axiom,
    ! [G: fun_int_rat] : groups3906332499630173760nt_rat(G,bot_bot_set_int) = zero_zero_rat ).

% sum.empty
tff(fact_6167_sum_Oempty,axiom,
    ! [G: fun_int_nat] : groups4541462559716669496nt_nat(G,bot_bot_set_int) = zero_zero_nat ).

% sum.empty
tff(fact_6168_convex__sum__bound__le,axiom,
    ! [I5: set_o,X: fun_o_Code_integer,A: fun_o_Code_integer,B: code_integer,Delta: code_integer] :
      ( ! [I3: $o] :
          ( member_o((I3),I5)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_o_Code_integer(X,(I3))) )
     => ( ( groups4406642042086082107nteger(X,I5) = one_one_Code_integer )
       => ( ! [I3: $o] :
              ( member_o((I3),I5)
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_o_Code_integer(A,(I3))),B))),Delta) )
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(groups4406642042086082107nteger(aa_fun8520278569782659900nteger(aTP_Lamm_ku(X),A),I5)),B))),Delta) ) ) ) ).

% convex_sum_bound_le
tff(fact_6169_convex__sum__bound__le,axiom,
    ! [I5: set_nat,X: fun_nat_Code_integer,A: fun_nat_Code_integer,B: code_integer,Delta: code_integer] :
      ( ! [I3: nat] :
          ( aa_set_nat_o(member_nat(I3),I5)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(X,I3)) )
     => ( ( aa_set871585252387937313nteger(groups7501900531339628137nteger(X),I5) = one_one_Code_integer )
       => ( ! [I3: nat] :
              ( aa_set_nat_o(member_nat(I3),I5)
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_nat_Code_integer(A,I3)),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_kv(X),A)),I5)),B))),Delta) ) ) ) ).

% convex_sum_bound_le
tff(fact_6170_convex__sum__bound__le,axiom,
    ! [I5: set_int,X: fun_int_Code_integer,A: fun_int_Code_integer,B: code_integer,Delta: code_integer] :
      ( ! [I3: int] :
          ( aa_set_int_o(member_int(I3),I5)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_int_Code_integer(X,I3)) )
     => ( ( groups7873554091576472773nteger(X,I5) = one_one_Code_integer )
       => ( ! [I3: int] :
              ( aa_set_int_o(member_int(I3),I5)
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_int_Code_integer(A,I3)),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_kw(X),A),I5)),B))),Delta) ) ) ) ).

% convex_sum_bound_le
tff(fact_6171_convex__sum__bound__le,axiom,
    ! [I5: set_o,X: fun_o_rat,A: fun_o_rat,B: rat,Delta: rat] :
      ( ! [I3: $o] :
          ( member_o((I3),I5)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_o_rat(X,(I3))) )
     => ( ( groups7872700643590313910_o_rat(X,I5) = one_one_rat )
       => ( ! [I3: $o] :
              ( member_o((I3),I5)
             => 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_o_rat(A,(I3))),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,groups7872700643590313910_o_rat(aa_fun4336163974892599420_o_rat(aTP_Lamm_kx(X),A),I5)),B))),Delta) ) ) ) ).

% convex_sum_bound_le
tff(fact_6172_convex__sum__bound__le,axiom,
    ! [I5: set_nat,X: fun_nat_rat,A: fun_nat_rat,B: rat,Delta: rat] :
      ( ! [I3: nat] :
          ( aa_set_nat_o(member_nat(I3),I5)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(X,I3)) )
     => ( ( aa_set_nat_rat(groups2906978787729119204at_rat(X),I5) = one_one_rat )
       => ( ! [I3: nat] :
              ( aa_set_nat_o(member_nat(I3),I5)
             => 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,I3)),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_ky(X),A)),I5)),B))),Delta) ) ) ) ).

% convex_sum_bound_le
tff(fact_6173_convex__sum__bound__le,axiom,
    ! [I5: set_int,X: fun_int_rat,A: fun_int_rat,B: rat,Delta: rat] :
      ( ! [I3: int] :
          ( aa_set_int_o(member_int(I3),I5)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_int_rat(X,I3)) )
     => ( ( groups3906332499630173760nt_rat(X,I5) = one_one_rat )
       => ( ! [I3: int] :
              ( aa_set_int_o(member_int(I3),I5)
             => 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,I3)),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_kz(X),A),I5)),B))),Delta) ) ) ) ).

% convex_sum_bound_le
tff(fact_6174_convex__sum__bound__le,axiom,
    ! [I5: set_o,X: fun_o_int,A: fun_o_int,B: int,Delta: int] :
      ( ! [I3: $o] :
          ( member_o((I3),I5)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_o_int(X,(I3))) )
     => ( ( groups8505340233167759370_o_int(X,I5) = one_one_int )
       => ( ! [I3: $o] :
              ( member_o((I3),I5)
             => 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_o_int(A,(I3))),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,groups8505340233167759370_o_int(aa_fun4727766026823762812_o_int(aTP_Lamm_la(X),A),I5)),B))),Delta) ) ) ) ).

% convex_sum_bound_le
tff(fact_6175_convex__sum__bound__le,axiom,
    ! [I5: set_nat,X: fun_nat_int,A: fun_nat_int,B: int,Delta: int] :
      ( ! [I3: nat] :
          ( aa_set_nat_o(member_nat(I3),I5)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(X,I3)) )
     => ( ( aa_set_nat_int(groups3539618377306564664at_int(X),I5) = one_one_int )
       => ( ! [I3: nat] :
              ( aa_set_nat_o(member_nat(I3),I5)
             => 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,I3)),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_lb(X),A)),I5)),B))),Delta) ) ) ) ).

% convex_sum_bound_le
tff(fact_6176_convex__sum__bound__le,axiom,
    ! [I5: set_int,X: fun_int_int,A: fun_int_int,B: int,Delta: int] :
      ( ! [I3: int] :
          ( aa_set_int_o(member_int(I3),I5)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(X,I3)) )
     => ( ( groups4538972089207619220nt_int(X,I5) = one_one_int )
       => ( ! [I3: int] :
              ( aa_set_int_o(member_int(I3),I5)
             => 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,I3)),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_lc(X),A),I5)),B))),Delta) ) ) ) ).

% convex_sum_bound_le
tff(fact_6177_convex__sum__bound__le,axiom,
    ! [I5: set_set_nat,X: fun_se7586849273513198688nteger,A: fun_se7586849273513198688nteger,B: code_integer,Delta: code_integer] :
      ( ! [I3: set_nat] :
          ( aa_set_set_nat_o(member_set_nat(I3),I5)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_set871585252387937313nteger(X,I3)) )
     => ( ( groups9190459664516455967nteger(X,I5) = one_one_Code_integer )
       => ( ! [I3: set_nat] :
              ( aa_set_set_nat_o(member_set_nat(I3),I5)
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_set871585252387937313nteger(A,I3)),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_ld(X),A),I5)),B))),Delta) ) ) ) ).

% convex_sum_bound_le
tff(fact_6178_abs__sum__abs,axiom,
    ! [F: fun_int_int,A3: set_int] : aa_int_int(abs_abs_int,groups4538972089207619220nt_int(aTP_Lamm_kt(F),A3)) = groups4538972089207619220nt_int(aTP_Lamm_kt(F),A3) ).

% abs_sum_abs
tff(fact_6179_sum_Oneutral__const,axiom,
    ! [A3: set_nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_fy),A3) = zero_zero_nat ).

% sum.neutral_const
tff(fact_6180_sum_Oneutral__const,axiom,
    ! [A3: set_int] : groups4538972089207619220nt_int(aTP_Lamm_fz,A3) = zero_zero_int ).

% sum.neutral_const
tff(fact_6181_int__sum,axiom,
    ! [F: fun_int_nat,A3: set_int] : aa_nat_int(semiri1314217659103216013at_int,groups4541462559716669496nt_nat(F,A3)) = groups4538972089207619220nt_int(aTP_Lamm_jp(F),A3) ).

% int_sum
tff(fact_6182_int__sum,axiom,
    ! [F: fun_nat_nat,A3: set_nat] : aa_nat_int(semiri1314217659103216013at_int,aa_set_nat_nat(groups3542108847815614940at_nat(F),A3)) = aa_set_nat_int(groups3539618377306564664at_int(aTP_Lamm_jq(F)),A3) ).

% int_sum
tff(fact_6183_sum__subtractf__nat,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,G: fun_Pr2602117396471556196at_nat,F: fun_Pr2602117396471556196at_nat] :
      ( ! [X3: product_prod_nat_nat] :
          ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X3),A3)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_Pro4243173770044397973at_nat(G,X3)),aa_Pro4243173770044397973at_nat(F,X3)) )
     => ( groups977919841031483927at_nat(aa_fun2351638177499875132at_nat(aTP_Lamm_le(G),F),A3) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups977919841031483927at_nat(F,A3)),groups977919841031483927at_nat(G,A3)) ) ) ).

% sum_subtractf_nat
tff(fact_6184_sum__subtractf__nat,axiom,
    ! [A3: set_o,G: fun_o_nat,F: fun_o_nat] :
      ( ! [X3: $o] :
          ( member_o((X3),A3)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_o_nat(G,(X3))),aa_o_nat(F,(X3))) )
     => ( groups8507830703676809646_o_nat(aa_fun6176012476734143100_o_nat(aTP_Lamm_lf(G),F),A3) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups8507830703676809646_o_nat(F,A3)),groups8507830703676809646_o_nat(G,A3)) ) ) ).

% sum_subtractf_nat
tff(fact_6185_sum__subtractf__nat,axiom,
    ! [A3: set_set_nat,G: fun_set_nat_nat,F: fun_set_nat_nat] :
      ( ! [X3: set_nat] :
          ( aa_set_set_nat_o(member_set_nat(X3),A3)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(G,X3)),aa_set_nat_nat(F,X3)) )
     => ( groups8294997508430121362at_nat(aa_fun3386010016858254564at_nat(aTP_Lamm_lg(G),F),A3) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups8294997508430121362at_nat(F,A3)),groups8294997508430121362at_nat(G,A3)) ) ) ).

% sum_subtractf_nat
tff(fact_6186_sum__subtractf__nat,axiom,
    ! [A3: set_int,G: fun_int_nat,F: fun_int_nat] :
      ( ! [X3: int] :
          ( aa_set_int_o(member_int(X3),A3)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(G,X3)),aa_int_nat(F,X3)) )
     => ( groups4541462559716669496nt_nat(aa_fun6506752150744536624nt_nat(aTP_Lamm_lh(G),F),A3) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups4541462559716669496nt_nat(F,A3)),groups4541462559716669496nt_nat(G,A3)) ) ) ).

% sum_subtractf_nat
tff(fact_6187_sum__subtractf__nat,axiom,
    ! [A3: set_nat,G: fun_nat_nat,F: fun_nat_nat] :
      ( ! [X3: nat] :
          ( aa_set_nat_o(member_nat(X3),A3)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(G,X3)),aa_nat_nat(F,X3)) )
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_li(G),F)),A3) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),A3)),aa_set_nat_nat(groups3542108847815614940at_nat(G),A3)) ) ) ).

% sum_subtractf_nat
tff(fact_6188_sum__SucD,axiom,
    ! [F: fun_nat_nat,A3: set_nat,N2: nat] :
      ( ( aa_set_nat_nat(groups3542108847815614940at_nat(F),A3) = aa_nat_nat(suc,N2) )
     => ? [X3: nat] :
          ( aa_set_nat_o(member_nat(X3),A3)
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(F,X3)) ) ) ).

% sum_SucD
tff(fact_6189_sum_Oswap,axiom,
    ! [G: fun_nat_fun_nat_nat,B3: set_nat,A3: set_nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_set6537093552120046177at_nat(aTP_Lamm_lj(G),B3)),A3) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_set6537093552120046177at_nat(aTP_Lamm_ll(G),A3)),B3) ).

% sum.swap
tff(fact_6190_sum_Oswap,axiom,
    ! [G: fun_int_fun_int_int,B3: set_int,A3: set_int] : groups4538972089207619220nt_int(aa_set7310040872927613301nt_int(aTP_Lamm_lm(G),B3),A3) = groups4538972089207619220nt_int(aa_set7310040872927613301nt_int(aTP_Lamm_lo(G),A3),B3) ).

% sum.swap
tff(fact_6191_sum__mono,axiom,
    ! [K5: set_o,F: fun_o_rat,G: fun_o_rat] :
      ( ! [I3: $o] :
          ( member_o((I3),K5)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_o_rat(F,(I3))),aa_o_rat(G,(I3))) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups7872700643590313910_o_rat(F,K5)),groups7872700643590313910_o_rat(G,K5)) ) ).

% sum_mono
tff(fact_6192_sum__mono,axiom,
    ! [K5: set_nat,F: fun_nat_rat,G: fun_nat_rat] :
      ( ! [I3: nat] :
          ( aa_set_nat_o(member_nat(I3),K5)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,I3)),aa_nat_rat(G,I3)) )
     => 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_6193_sum__mono,axiom,
    ! [K5: set_int,F: fun_int_rat,G: fun_int_rat] :
      ( ! [I3: int] :
          ( aa_set_int_o(member_int(I3),K5)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(F,I3)),aa_int_rat(G,I3)) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups3906332499630173760nt_rat(F,K5)),groups3906332499630173760nt_rat(G,K5)) ) ).

% sum_mono
tff(fact_6194_sum__mono,axiom,
    ! [K5: set_o,F: fun_o_nat,G: fun_o_nat] :
      ( ! [I3: $o] :
          ( member_o((I3),K5)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_o_nat(F,(I3))),aa_o_nat(G,(I3))) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,groups8507830703676809646_o_nat(F,K5)),groups8507830703676809646_o_nat(G,K5)) ) ).

% sum_mono
tff(fact_6195_sum__mono,axiom,
    ! [K5: set_int,F: fun_int_nat,G: fun_int_nat] :
      ( ! [I3: int] :
          ( aa_set_int_o(member_int(I3),K5)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(F,I3)),aa_int_nat(G,I3)) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,groups4541462559716669496nt_nat(F,K5)),groups4541462559716669496nt_nat(G,K5)) ) ).

% sum_mono
tff(fact_6196_sum__mono,axiom,
    ! [K5: set_o,F: fun_o_int,G: fun_o_int] :
      ( ! [I3: $o] :
          ( member_o((I3),K5)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_o_int(F,(I3))),aa_o_int(G,(I3))) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,groups8505340233167759370_o_int(F,K5)),groups8505340233167759370_o_int(G,K5)) ) ).

% sum_mono
tff(fact_6197_sum__mono,axiom,
    ! [K5: set_nat,F: fun_nat_int,G: fun_nat_int] :
      ( ! [I3: nat] :
          ( aa_set_nat_o(member_nat(I3),K5)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,I3)),aa_nat_int(G,I3)) )
     => 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_6198_sum__mono,axiom,
    ! [K5: set_nat,F: fun_nat_nat,G: fun_nat_nat] :
      ( ! [I3: nat] :
          ( aa_set_nat_o(member_nat(I3),K5)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F,I3)),aa_nat_nat(G,I3)) )
     => 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_6199_sum__mono,axiom,
    ! [K5: set_int,F: fun_int_int,G: fun_int_int] :
      ( ! [I3: int] :
          ( aa_set_int_o(member_int(I3),K5)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(F,I3)),aa_int_int(G,I3)) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,groups4538972089207619220nt_int(F,K5)),groups4538972089207619220nt_int(G,K5)) ) ).

% sum_mono
tff(fact_6200_sum__mono,axiom,
    ! [K5: set_set_nat,F: fun_set_nat_rat,G: fun_set_nat_rat] :
      ( ! [I3: set_nat] :
          ( aa_set_set_nat_o(member_set_nat(I3),K5)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_set_nat_rat(F,I3)),aa_set_nat_rat(G,I3)) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups7659867448343625626at_rat(F,K5)),groups7659867448343625626at_rat(G,K5)) ) ).

% sum_mono
tff(fact_6201_sum_Odistrib,axiom,
    ! [G: fun_nat_nat,Ha: fun_nat_nat,A3: set_nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_lp(G),Ha)),A3) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(G),A3)),aa_set_nat_nat(groups3542108847815614940at_nat(Ha),A3)) ).

% sum.distrib
tff(fact_6202_sum_Odistrib,axiom,
    ! [G: fun_int_int,Ha: fun_int_int,A3: set_int] : groups4538972089207619220nt_int(aa_fun6280345759651049192nt_int(aTP_Lamm_lq(G),Ha),A3) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups4538972089207619220nt_int(G,A3)),groups4538972089207619220nt_int(Ha,A3)) ).

% sum.distrib
tff(fact_6203_sum__product,axiom,
    ! [F: fun_nat_nat,A3: set_nat,G: fun_nat_nat,B3: set_nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),A3)),aa_set_nat_nat(groups3542108847815614940at_nat(G),B3)) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_set6537093552120046177at_nat(aa_fun4561837934398617467at_nat(aTP_Lamm_ls(F),G),B3)),A3) ).

% sum_product
tff(fact_6204_sum__product,axiom,
    ! [F: fun_int_int,A3: set_int,G: fun_int_int,B3: set_int] : aa_int_int(aa_int_fun_int_int(times_times_int,groups4538972089207619220nt_int(F,A3)),groups4538972089207619220nt_int(G,B3)) = groups4538972089207619220nt_int(aa_set7310040872927613301nt_int(aa_fun8076508010538293575nt_int(aTP_Lamm_lu(F),G),B3),A3) ).

% sum_product
tff(fact_6205_sum__distrib__right,axiom,
    ! [F: fun_nat_nat,A3: set_nat,Ra2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),A3)),Ra2) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_lv(F),Ra2)),A3) ).

% sum_distrib_right
tff(fact_6206_sum__distrib__right,axiom,
    ! [F: fun_int_int,A3: set_int,Ra2: int] : aa_int_int(aa_int_fun_int_int(times_times_int,groups4538972089207619220nt_int(F,A3)),Ra2) = groups4538972089207619220nt_int(aa_int_fun_int_int(aTP_Lamm_lw(F),Ra2),A3) ).

% sum_distrib_right
tff(fact_6207_sum__distrib__left,axiom,
    ! [Ra2: nat,F: fun_nat_nat,A3: set_nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ra2),aa_set_nat_nat(groups3542108847815614940at_nat(F),A3)) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_lx(Ra2),F)),A3) ).

% sum_distrib_left
tff(fact_6208_sum__distrib__left,axiom,
    ! [Ra2: int,F: fun_int_int,A3: set_int] : aa_int_int(aa_int_fun_int_int(times_times_int,Ra2),groups4538972089207619220nt_int(F,A3)) = groups4538972089207619220nt_int(aa_fun6280345759651049192nt_int(aTP_Lamm_ly(Ra2),F),A3) ).

% sum_distrib_left
tff(fact_6209_sum__subtractf,axiom,
    ! [F: fun_int_int,G: fun_int_int,A3: set_int] : groups4538972089207619220nt_int(aa_fun6280345759651049192nt_int(aTP_Lamm_lz(F),G),A3) = aa_int_int(aa_int_fun_int_int(minus_minus_int,groups4538972089207619220nt_int(F,A3)),groups4538972089207619220nt_int(G,A3)) ).

% sum_subtractf
tff(fact_6210_sum__negf,axiom,
    ! [F: fun_int_int,A3: set_int] : groups4538972089207619220nt_int(aTP_Lamm_ma(F),A3) = aa_int_int(uminus_uminus_int,groups4538972089207619220nt_int(F,A3)) ).

% sum_negf
tff(fact_6211_sum__nonneg,axiom,
    ! [A3: set_o,F: fun_o_Code_integer] :
      ( ! [X3: $o] :
          ( member_o((X3),A3)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_o_Code_integer(F,(X3))) )
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),groups4406642042086082107nteger(F,A3)) ) ).

% sum_nonneg
tff(fact_6212_sum__nonneg,axiom,
    ! [A3: set_nat,F: fun_nat_Code_integer] :
      ( ! [X3: nat] :
          ( aa_set_nat_o(member_nat(X3),A3)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(F,X3)) )
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_set871585252387937313nteger(groups7501900531339628137nteger(F),A3)) ) ).

% sum_nonneg
tff(fact_6213_sum__nonneg,axiom,
    ! [A3: set_int,F: fun_int_Code_integer] :
      ( ! [X3: int] :
          ( aa_set_int_o(member_int(X3),A3)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_int_Code_integer(F,X3)) )
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),groups7873554091576472773nteger(F,A3)) ) ).

% sum_nonneg
tff(fact_6214_sum__nonneg,axiom,
    ! [A3: set_o,F: fun_o_rat] :
      ( ! [X3: $o] :
          ( member_o((X3),A3)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_o_rat(F,(X3))) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),groups7872700643590313910_o_rat(F,A3)) ) ).

% sum_nonneg
tff(fact_6215_sum__nonneg,axiom,
    ! [A3: set_nat,F: fun_nat_rat] :
      ( ! [X3: nat] :
          ( aa_set_nat_o(member_nat(X3),A3)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(F,X3)) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_set_nat_rat(groups2906978787729119204at_rat(F),A3)) ) ).

% sum_nonneg
tff(fact_6216_sum__nonneg,axiom,
    ! [A3: set_int,F: fun_int_rat] :
      ( ! [X3: int] :
          ( aa_set_int_o(member_int(X3),A3)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_int_rat(F,X3)) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),groups3906332499630173760nt_rat(F,A3)) ) ).

% sum_nonneg
tff(fact_6217_sum__nonneg,axiom,
    ! [A3: set_o,F: fun_o_nat] :
      ( ! [X3: $o] :
          ( member_o((X3),A3)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_o_nat(F,(X3))) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),groups8507830703676809646_o_nat(F,A3)) ) ).

% sum_nonneg
tff(fact_6218_sum__nonneg,axiom,
    ! [A3: set_int,F: fun_int_nat] :
      ( ! [X3: int] :
          ( aa_set_int_o(member_int(X3),A3)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_int_nat(F,X3)) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),groups4541462559716669496nt_nat(F,A3)) ) ).

% sum_nonneg
tff(fact_6219_sum__nonneg,axiom,
    ! [A3: set_o,F: fun_o_int] :
      ( ! [X3: $o] :
          ( member_o((X3),A3)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_o_int(F,(X3))) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),groups8505340233167759370_o_int(F,A3)) ) ).

% sum_nonneg
tff(fact_6220_sum__nonneg,axiom,
    ! [A3: set_nat,F: fun_nat_int] :
      ( ! [X3: nat] :
          ( aa_set_nat_o(member_nat(X3),A3)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(F,X3)) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_set_nat_int(groups3539618377306564664at_int(F),A3)) ) ).

% sum_nonneg
tff(fact_6221_sum__nonpos,axiom,
    ! [A3: set_o,F: fun_o_Code_integer] :
      ( ! [X3: $o] :
          ( member_o((X3),A3)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_o_Code_integer(F,(X3))),zero_z3403309356797280102nteger) )
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,groups4406642042086082107nteger(F,A3)),zero_z3403309356797280102nteger) ) ).

% sum_nonpos
tff(fact_6222_sum__nonpos,axiom,
    ! [A3: set_nat,F: fun_nat_Code_integer] :
      ( ! [X3: nat] :
          ( aa_set_nat_o(member_nat(X3),A3)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(F,X3)),zero_z3403309356797280102nteger) )
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_set871585252387937313nteger(groups7501900531339628137nteger(F),A3)),zero_z3403309356797280102nteger) ) ).

% sum_nonpos
tff(fact_6223_sum__nonpos,axiom,
    ! [A3: set_int,F: fun_int_Code_integer] :
      ( ! [X3: int] :
          ( aa_set_int_o(member_int(X3),A3)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_int_Code_integer(F,X3)),zero_z3403309356797280102nteger) )
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,groups7873554091576472773nteger(F,A3)),zero_z3403309356797280102nteger) ) ).

% sum_nonpos
tff(fact_6224_sum__nonpos,axiom,
    ! [A3: set_o,F: fun_o_rat] :
      ( ! [X3: $o] :
          ( member_o((X3),A3)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_o_rat(F,(X3))),zero_zero_rat) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups7872700643590313910_o_rat(F,A3)),zero_zero_rat) ) ).

% sum_nonpos
tff(fact_6225_sum__nonpos,axiom,
    ! [A3: set_nat,F: fun_nat_rat] :
      ( ! [X3: nat] :
          ( aa_set_nat_o(member_nat(X3),A3)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,X3)),zero_zero_rat) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_set_nat_rat(groups2906978787729119204at_rat(F),A3)),zero_zero_rat) ) ).

% sum_nonpos
tff(fact_6226_sum__nonpos,axiom,
    ! [A3: set_int,F: fun_int_rat] :
      ( ! [X3: int] :
          ( aa_set_int_o(member_int(X3),A3)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(F,X3)),zero_zero_rat) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups3906332499630173760nt_rat(F,A3)),zero_zero_rat) ) ).

% sum_nonpos
tff(fact_6227_sum__nonpos,axiom,
    ! [A3: set_o,F: fun_o_nat] :
      ( ! [X3: $o] :
          ( member_o((X3),A3)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_o_nat(F,(X3))),zero_zero_nat) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,groups8507830703676809646_o_nat(F,A3)),zero_zero_nat) ) ).

% sum_nonpos
tff(fact_6228_sum__nonpos,axiom,
    ! [A3: set_int,F: fun_int_nat] :
      ( ! [X3: int] :
          ( aa_set_int_o(member_int(X3),A3)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(F,X3)),zero_zero_nat) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,groups4541462559716669496nt_nat(F,A3)),zero_zero_nat) ) ).

% sum_nonpos
tff(fact_6229_sum__nonpos,axiom,
    ! [A3: set_o,F: fun_o_int] :
      ( ! [X3: $o] :
          ( member_o((X3),A3)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_o_int(F,(X3))),zero_zero_int) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,groups8505340233167759370_o_int(F,A3)),zero_zero_int) ) ).

% sum_nonpos
tff(fact_6230_sum__nonpos,axiom,
    ! [A3: set_nat,F: fun_nat_int] :
      ( ! [X3: nat] :
          ( aa_set_nat_o(member_nat(X3),A3)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,X3)),zero_zero_int) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_set_nat_int(groups3539618377306564664at_int(F),A3)),zero_zero_int) ) ).

% sum_nonpos
tff(fact_6231_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_mb(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_mc(M),A),X),Y)),aa_nat_set_nat(set_ord_atMost_nat,M)) ).

% gbinomial_partial_sum_poly_xpos
tff(fact_6232_sum__gp0,axiom,
    ! [X: rat,N2: nat] :
      aa_set_nat_rat(groups2906978787729119204at_rat(power_power_rat(X)),aa_nat_set_nat(set_ord_atMost_nat,N2)) = $ite(X = one_one_rat,aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),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,N2)))),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,one_one_rat),X))) ).

% sum_gp0
tff(fact_6233_divmod__nat__def,axiom,
    ! [M: nat,N2: nat] : aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(divmod_nat,M),N2) = aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N2)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N2)) ).

% divmod_nat_def
tff(fact_6234_sorted__in__between,axiom,
    ! [I2: nat,J: nat,L: list_o,X: $o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),I2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),J)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J),size_size_list_o(L))
         => ( aa_list_o_o(sorted_wrt_o(ord_less_eq_o),L)
           => ( aa_o_o(aa_o_fun_o_o(ord_less_eq_o,aa_nat_o(nth_o(L),I2)),(X))
             => ( aa_o_o(aa_o_fun_o_o(ord_less_o,(X)),aa_nat_o(nth_o(L),J))
               => ~ ! [K2: nat] :
                      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),K2)
                     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K2),J)
                       => ( aa_o_o(aa_o_fun_o_o(ord_less_eq_o,aa_nat_o(nth_o(L),K2)),(X))
                         => ~ aa_o_o(aa_o_fun_o_o(ord_less_o,(X)),aa_nat_o(nth_o(L),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K2),one_one_nat))) ) ) ) ) ) ) ) ) ) ).

% sorted_in_between
tff(fact_6235_sorted__in__between,axiom,
    ! [I2: nat,J: nat,L: list_rat,X: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),I2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),J)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J),size_size_list_rat(L))
         => ( sorted_wrt_rat(ord_less_eq_rat,L)
           => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(nth_rat(L),I2)),X)
             => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),aa_nat_rat(nth_rat(L),J))
               => ~ ! [K2: nat] :
                      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),K2)
                     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K2),J)
                       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(nth_rat(L),K2)),X)
                         => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),aa_nat_rat(nth_rat(L),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K2),one_one_nat))) ) ) ) ) ) ) ) ) ) ).

% sorted_in_between
tff(fact_6236_sorted__in__between,axiom,
    ! [I2: nat,J: nat,L: list_num,X: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),I2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),J)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J),size_size_list_num(L))
         => ( sorted_wrt_num(ord_less_eq_num,L)
           => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(nth_num(L),I2)),X)
             => ( aa_num_o(aa_num_fun_num_o(ord_less_num,X),aa_nat_num(nth_num(L),J))
               => ~ ! [K2: nat] :
                      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),K2)
                     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K2),J)
                       => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(nth_num(L),K2)),X)
                         => ~ aa_num_o(aa_num_fun_num_o(ord_less_num,X),aa_nat_num(nth_num(L),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K2),one_one_nat))) ) ) ) ) ) ) ) ) ) ).

% sorted_in_between
tff(fact_6237_sorted__in__between,axiom,
    ! [I2: nat,J: nat,L: list_nat,X: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),I2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),J)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J),aa_list_nat_nat(size_size_list_nat,L))
         => ( aa_list_nat_o(sorted_wrt_nat(ord_less_eq_nat),L)
           => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(nth_nat(L),I2)),X)
             => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),aa_nat_nat(nth_nat(L),J))
               => ~ ! [K2: nat] :
                      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),K2)
                     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K2),J)
                       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(nth_nat(L),K2)),X)
                         => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),aa_nat_nat(nth_nat(L),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K2),one_one_nat))) ) ) ) ) ) ) ) ) ) ).

% sorted_in_between
tff(fact_6238_sorted__in__between,axiom,
    ! [I2: nat,J: nat,L: list_int,X: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),I2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),J)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J),size_size_list_int(L))
         => ( sorted_wrt_int(ord_less_eq_int,L)
           => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(nth_int(L),I2)),X)
             => ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),aa_nat_int(nth_int(L),J))
               => ~ ! [K2: nat] :
                      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),K2)
                     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K2),J)
                       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(nth_int(L),K2)),X)
                         => ~ aa_int_o(aa_int_fun_int_o(ord_less_int,X),aa_nat_int(nth_int(L),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K2),one_one_nat))) ) ) ) ) ) ) ) ) ) ).

% sorted_in_between
tff(fact_6239_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_md(A)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,zero_zero_nat),K))),aa_nat_rat(semiri773545260158071498ct_rat,aa_nat_nat(suc,K))) ).

% gbinomial_Suc
tff(fact_6240_gbinomial__Suc,axiom,
    ! [A: code_natural,K: nat] : aa_nat_Code_natural(gbinom7368847122466276068atural(A),aa_nat_nat(suc,K)) = divide5121882707175180666atural(groups2279045934846249631atural(aTP_Lamm_me(A),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,zero_zero_nat),K)),aa_nat_Code_natural(semiri2447717529341329178atural,aa_nat_nat(suc,K))) ).

% gbinomial_Suc
tff(fact_6241_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_mf(A)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,zero_zero_nat),K))),aa_nat_Code_integer(semiri3624122377584611663nteger,aa_nat_nat(suc,K))) ).

% gbinomial_Suc
tff(fact_6242_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_mg,A)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,zero_zero_nat),K))),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(suc,K))) ).

% gbinomial_Suc
tff(fact_6243_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_mh,A)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,zero_zero_nat),K))),aa_nat_int(semiri1406184849735516958ct_int,aa_nat_nat(suc,K))) ).

% gbinomial_Suc
tff(fact_6244_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_mb(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_mi(M),A),X),Y)),aa_nat_set_nat(set_ord_atMost_nat,M)) ).

% gbinomial_partial_sum_poly
tff(fact_6245_atMost__iff,axiom,
    ! [I2: $o,K: $o] :
      ( member_o((I2),set_ord_atMost_o((K)))
    <=> aa_o_o(aa_o_fun_o_o(ord_less_eq_o,(I2)),(K)) ) ).

% atMost_iff
tff(fact_6246_atMost__iff,axiom,
    ! [I2: set_nat,K: set_nat] :
      ( aa_set_set_nat_o(member_set_nat(I2),set_or4236626031148496127et_nat(K))
    <=> aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,I2),K) ) ).

% atMost_iff
tff(fact_6247_atMost__iff,axiom,
    ! [I2: set_int,K: set_int] :
      ( member_set_int(I2,set_or58775011639299419et_int(K))
    <=> aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,I2),K) ) ).

% atMost_iff
tff(fact_6248_atMost__iff,axiom,
    ! [I2: rat,K: rat] :
      ( member_rat(I2,set_ord_atMost_rat(K))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,I2),K) ) ).

% atMost_iff
tff(fact_6249_atMost__iff,axiom,
    ! [I2: num,K: num] :
      ( member_num(I2,set_ord_atMost_num(K))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,I2),K) ) ).

% atMost_iff
tff(fact_6250_atMost__iff,axiom,
    ! [I2: nat,K: nat] :
      ( aa_set_nat_o(member_nat(I2),aa_nat_set_nat(set_ord_atMost_nat,K))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),K) ) ).

% atMost_iff
tff(fact_6251_atMost__iff,axiom,
    ! [I2: int,K: int] :
      ( aa_set_int_o(member_int(I2),set_ord_atMost_int(K))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I2),K) ) ).

% atMost_iff
tff(fact_6252_prod_Oneutral__const,axiom,
    ! [A3: set_nat] : aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_mj),A3) = one_one_nat ).

% prod.neutral_const
tff(fact_6253_prod_Oneutral__const,axiom,
    ! [A3: set_nat] : aa_set_nat_int(groups705719431365010083at_int(aTP_Lamm_mk),A3) = one_one_int ).

% prod.neutral_const
tff(fact_6254_prod_Oneutral__const,axiom,
    ! [A3: set_int] : groups1705073143266064639nt_int(aTP_Lamm_ml,A3) = one_one_int ).

% prod.neutral_const
tff(fact_6255_of__nat__prod,axiom,
    ! [F: fun_int_nat,A3: set_int] : aa_nat_int(semiri1314217659103216013at_int,groups1707563613775114915nt_nat(F,A3)) = groups1705073143266064639nt_int(aTP_Lamm_jp(F),A3) ).

% of_nat_prod
tff(fact_6256_of__nat__prod,axiom,
    ! [F: fun_nat_nat,A3: set_nat] : aa_nat_Code_integer(semiri4939895301339042750nteger,aa_set_nat_nat(groups708209901874060359at_nat(F),A3)) = aa_set871585252387937313nteger(groups3455450783089532116nteger(aTP_Lamm_jr(F)),A3) ).

% of_nat_prod
tff(fact_6257_of__nat__prod,axiom,
    ! [F: fun_nat_nat,A3: set_nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_set_nat_nat(groups708209901874060359at_nat(F),A3)) = aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_js(F)),A3) ).

% of_nat_prod
tff(fact_6258_of__nat__prod,axiom,
    ! [F: fun_nat_nat,A3: set_nat] : aa_nat_int(semiri1314217659103216013at_int,aa_set_nat_nat(groups708209901874060359at_nat(F),A3)) = aa_set_nat_int(groups705719431365010083at_int(aTP_Lamm_jq(F)),A3) ).

% of_nat_prod
tff(fact_6259_of__int__prod,axiom,
    ! [F: fun_nat_int,A3: set_nat] : aa_int_rat(ring_1_of_int_rat,aa_set_nat_int(groups705719431365010083at_int(F),A3)) = aa_set_nat_rat(groups73079841787564623at_rat(aTP_Lamm_mm(F)),A3) ).

% of_int_prod
tff(fact_6260_of__int__prod,axiom,
    ! [F: fun_nat_int,A3: set_nat] : aa_int_int(ring_1_of_int_int,aa_set_nat_int(groups705719431365010083at_int(F),A3)) = aa_set_nat_int(groups705719431365010083at_int(aTP_Lamm_mn(F)),A3) ).

% of_int_prod
tff(fact_6261_of__int__prod,axiom,
    ! [F: fun_int_int,A3: set_int] : aa_int_rat(ring_1_of_int_rat,groups1705073143266064639nt_int(F,A3)) = groups1072433553688619179nt_rat(aTP_Lamm_jt(F),A3) ).

% of_int_prod
tff(fact_6262_of__int__prod,axiom,
    ! [F: fun_int_int,A3: set_int] : aa_int_int(ring_1_of_int_int,groups1705073143266064639nt_int(F,A3)) = groups1705073143266064639nt_int(aTP_Lamm_ju(F),A3) ).

% of_int_prod
tff(fact_6263_prod_Oempty,axiom,
    ! [G: fun_o_Code_integer] : groups7694694392188491536nteger(G,bot_bot_set_o) = one_one_Code_integer ).

% prod.empty
tff(fact_6264_prod_Oempty,axiom,
    ! [G: fun_o_assn] : groups5301882518646026715o_assn(G,bot_bot_set_o) = one_one_assn ).

% prod.empty
tff(fact_6265_prod_Oempty,axiom,
    ! [G: fun_o_rat] : groups2869687844427037835_o_rat(G,bot_bot_set_o) = one_one_rat ).

% prod.empty
tff(fact_6266_prod_Oempty,axiom,
    ! [G: fun_o_nat] : groups3504817904513533571_o_nat(G,bot_bot_set_o) = one_one_nat ).

% prod.empty
tff(fact_6267_prod_Oempty,axiom,
    ! [G: fun_o_int] : groups3502327434004483295_o_int(G,bot_bot_set_o) = one_one_int ).

% prod.empty
tff(fact_6268_prod_Oempty,axiom,
    ! [G: fun_nat_Code_integer] : aa_set871585252387937313nteger(groups3455450783089532116nteger(G),bot_bot_set_nat) = one_one_Code_integer ).

% prod.empty
tff(fact_6269_prod_Oempty,axiom,
    ! [G: fun_nat_assn] : groups6906906614972039071t_assn(G,bot_bot_set_nat) = one_one_assn ).

% prod.empty
tff(fact_6270_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_6271_prod_Oempty,axiom,
    ! [G: fun_int_Code_integer] : groups3827104343326376752nteger(G,bot_bot_set_int) = one_one_Code_integer ).

% prod.empty
tff(fact_6272_prod_Oempty,axiom,
    ! [G: fun_int_assn] : groups7882442080178216443t_assn(G,bot_bot_set_int) = one_one_assn ).

% prod.empty
tff(fact_6273_atMost__subset__iff,axiom,
    ! [X: set_int,Y: set_int] :
      ( ord_le4403425263959731960et_int(set_or58775011639299419et_int(X),set_or58775011639299419et_int(Y))
    <=> aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,X),Y) ) ).

% atMost_subset_iff
tff(fact_6274_atMost__subset__iff,axiom,
    ! [X: rat,Y: rat] :
      ( ord_less_eq_set_rat(set_ord_atMost_rat(X),set_ord_atMost_rat(Y))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y) ) ).

% atMost_subset_iff
tff(fact_6275_atMost__subset__iff,axiom,
    ! [X: num,Y: num] :
      ( ord_less_eq_set_num(set_ord_atMost_num(X),set_ord_atMost_num(Y))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X),Y) ) ).

% atMost_subset_iff
tff(fact_6276_atMost__subset__iff,axiom,
    ! [X: nat,Y: nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_nat_set_nat(set_ord_atMost_nat,X)),aa_nat_set_nat(set_ord_atMost_nat,Y))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y) ) ).

% atMost_subset_iff
tff(fact_6277_atMost__subset__iff,axiom,
    ! [X: int,Y: int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,set_ord_atMost_int(X)),set_ord_atMost_int(Y))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Y) ) ).

% atMost_subset_iff
tff(fact_6278_Icc__subset__Iic__iff,axiom,
    ! [L: set_int,Ha: set_int,H_a: set_int] :
      ( ord_le4403425263959731960et_int(set_or370866239135849197et_int(L,Ha),set_or58775011639299419et_int(H_a))
    <=> ( ~ aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,L),Ha)
        | aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,Ha),H_a) ) ) ).

% Icc_subset_Iic_iff
tff(fact_6279_Icc__subset__Iic__iff,axiom,
    ! [L: rat,Ha: rat,H_a: rat] :
      ( ord_less_eq_set_rat(set_or633870826150836451st_rat(L,Ha),set_ord_atMost_rat(H_a))
    <=> ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,L),Ha)
        | aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Ha),H_a) ) ) ).

% Icc_subset_Iic_iff
tff(fact_6280_Icc__subset__Iic__iff,axiom,
    ! [L: num,Ha: num,H_a: num] :
      ( ord_less_eq_set_num(set_or7049704709247886629st_num(L,Ha),set_ord_atMost_num(H_a))
    <=> ( ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,L),Ha)
        | aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Ha),H_a) ) ) ).

% Icc_subset_Iic_iff
tff(fact_6281_Icc__subset__Iic__iff,axiom,
    ! [L: int,Ha: int,H_a: int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,set_or1266510415728281911st_int(L,Ha)),set_ord_atMost_int(H_a))
    <=> ( ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,L),Ha)
        | aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ha),H_a) ) ) ).

% Icc_subset_Iic_iff
tff(fact_6282_Icc__subset__Iic__iff,axiom,
    ! [L: nat,Ha: nat,H_a: nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,L),Ha)),aa_nat_set_nat(set_ord_atMost_nat,H_a))
    <=> ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,L),Ha)
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ha),H_a) ) ) ).

% Icc_subset_Iic_iff
tff(fact_6283_Icc__subset__Iic__iff,axiom,
    ! [L: code_integer,Ha: code_integer,H_a: code_integer] :
      ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(set_or189985376899183464nteger(L,Ha)),set_or9101266186257409494nteger(H_a))
    <=> ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,L),Ha)
        | aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Ha),H_a) ) ) ).

% Icc_subset_Iic_iff
tff(fact_6284_prod_OatMost__Suc,axiom,
    ! [G: fun_nat_Code_integer,N2: nat] : aa_set871585252387937313nteger(groups3455450783089532116nteger(G),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,N2))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_set871585252387937313nteger(groups3455450783089532116nteger(G),aa_nat_set_nat(set_ord_atMost_nat,N2))),aa_nat_Code_integer(G,aa_nat_nat(suc,N2))) ).

% prod.atMost_Suc
tff(fact_6285_prod_OatMost__Suc,axiom,
    ! [G: fun_nat_assn,N2: nat] : groups6906906614972039071t_assn(G,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,N2))) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,groups6906906614972039071t_assn(G,aa_nat_set_nat(set_ord_atMost_nat,N2))),aa_nat_assn(G,aa_nat_nat(suc,N2))) ).

% prod.atMost_Suc
tff(fact_6286_prod_OatMost__Suc,axiom,
    ! [G: fun_nat_rat,N2: nat] : aa_set_nat_rat(groups73079841787564623at_rat(G),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,N2))) = 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,N2))),aa_nat_rat(G,aa_nat_nat(suc,N2))) ).

% prod.atMost_Suc
tff(fact_6287_prod_OatMost__Suc,axiom,
    ! [G: fun_nat_nat,N2: nat] : aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,N2))) = 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,N2))),aa_nat_nat(G,aa_nat_nat(suc,N2))) ).

% prod.atMost_Suc
tff(fact_6288_prod_OatMost__Suc,axiom,
    ! [G: fun_nat_int,N2: nat] : aa_set_nat_int(groups705719431365010083at_int(G),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,N2))) = 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,N2))),aa_nat_int(G,aa_nat_nat(suc,N2))) ).

% prod.atMost_Suc
tff(fact_6289_prod_Ocl__ivl__Suc,axiom,
    ! [G: fun_nat_Code_integer,M: nat,N2: nat] :
      aa_set871585252387937313nteger(groups3455450783089532116nteger(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M),aa_nat_nat(suc,N2))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,N2)),M),one_one_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_set871585252387937313nteger(groups3455450783089532116nteger(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M),N2))),aa_nat_Code_integer(G,aa_nat_nat(suc,N2)))) ).

% prod.cl_ivl_Suc
tff(fact_6290_prod_Ocl__ivl__Suc,axiom,
    ! [G: fun_nat_assn,M: nat,N2: nat] :
      groups6906906614972039071t_assn(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M),aa_nat_nat(suc,N2))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,N2)),M),one_one_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,groups6906906614972039071t_assn(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M),N2))),aa_nat_assn(G,aa_nat_nat(suc,N2)))) ).

% prod.cl_ivl_Suc
tff(fact_6291_prod_Ocl__ivl__Suc,axiom,
    ! [G: fun_nat_rat,M: nat,N2: nat] :
      aa_set_nat_rat(groups73079841787564623at_rat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M),aa_nat_nat(suc,N2))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,N2)),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(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M),N2))),aa_nat_rat(G,aa_nat_nat(suc,N2)))) ).

% prod.cl_ivl_Suc
tff(fact_6292_prod_Ocl__ivl__Suc,axiom,
    ! [G: fun_nat_nat,M: nat,N2: nat] :
      aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M),aa_nat_nat(suc,N2))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,N2)),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(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M),N2))),aa_nat_nat(G,aa_nat_nat(suc,N2)))) ).

% prod.cl_ivl_Suc
tff(fact_6293_prod_Ocl__ivl__Suc,axiom,
    ! [G: fun_nat_int,M: nat,N2: nat] :
      aa_set_nat_int(groups705719431365010083at_int(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M),aa_nat_nat(suc,N2))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,N2)),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(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M),N2))),aa_nat_int(G,aa_nat_nat(suc,N2)))) ).

% prod.cl_ivl_Suc
tff(fact_6294_prod_Oswap,axiom,
    ! [G: fun_nat_fun_nat_nat,B3: set_nat,A3: set_nat] : aa_set_nat_nat(groups708209901874060359at_nat(aa_set6537093552120046177at_nat(aTP_Lamm_mo(G),B3)),A3) = aa_set_nat_nat(groups708209901874060359at_nat(aa_set6537093552120046177at_nat(aTP_Lamm_mp(G),A3)),B3) ).

% prod.swap
tff(fact_6295_prod_Oswap,axiom,
    ! [G: fun_nat_fun_nat_int,B3: set_nat,A3: set_nat] : aa_set_nat_int(groups705719431365010083at_int(aa_set2359242532610849469at_int(aTP_Lamm_mq(G),B3)),A3) = aa_set_nat_int(groups705719431365010083at_int(aa_set2359242532610849469at_int(aTP_Lamm_ms(G),A3)),B3) ).

% prod.swap
tff(fact_6296_prod_Oswap,axiom,
    ! [G: fun_nat_fun_int_int,B3: set_int,A3: set_nat] : aa_set_nat_int(groups705719431365010083at_int(aa_set6310687161026558745at_int(aTP_Lamm_mt(G),B3)),A3) = groups1705073143266064639nt_int(aa_set3358596244511904025nt_int(aTP_Lamm_mv(G),A3),B3) ).

% prod.swap
tff(fact_6297_prod_Oswap,axiom,
    ! [G: fun_int_fun_nat_int,B3: set_nat,A3: set_int] : groups1705073143266064639nt_int(aa_set3358596244511904025nt_int(aTP_Lamm_mw(G),B3),A3) = aa_set_nat_int(groups705719431365010083at_int(aa_set6310687161026558745at_int(aTP_Lamm_my(G),A3)),B3) ).

% prod.swap
tff(fact_6298_prod_Oswap,axiom,
    ! [G: fun_int_fun_int_int,B3: set_int,A3: set_int] : groups1705073143266064639nt_int(aa_set7310040872927613301nt_int(aTP_Lamm_mz(G),B3),A3) = groups1705073143266064639nt_int(aa_set7310040872927613301nt_int(aTP_Lamm_na(G),A3),B3) ).

% prod.swap
tff(fact_6299_sorted__wrt__true,axiom,
    ! [Xs: list_nat] : aa_list_nat_o(sorted_wrt_nat(aTP_Lamm_nb),Xs) ).

% sorted_wrt_true
tff(fact_6300_sorted__wrt__true,axiom,
    ! [Xs: list_int] : sorted_wrt_int(aTP_Lamm_nc,Xs) ).

% sorted_wrt_true
tff(fact_6301_not__empty__eq__Iic__eq__empty,axiom,
    ! [Ha: $o] : bot_bot_set_o != set_ord_atMost_o((Ha)) ).

% not_empty_eq_Iic_eq_empty
tff(fact_6302_not__empty__eq__Iic__eq__empty,axiom,
    ! [Ha: nat] : bot_bot_set_nat != aa_nat_set_nat(set_ord_atMost_nat,Ha) ).

% not_empty_eq_Iic_eq_empty
tff(fact_6303_not__empty__eq__Iic__eq__empty,axiom,
    ! [Ha: int] : bot_bot_set_int != set_ord_atMost_int(Ha) ).

% not_empty_eq_Iic_eq_empty
tff(fact_6304_prod_Odistrib,axiom,
    ! [G: fun_nat_nat,Ha: fun_nat_nat,A3: set_nat] : aa_set_nat_nat(groups708209901874060359at_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_nd(G),Ha)),A3) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(groups708209901874060359at_nat(G),A3)),aa_set_nat_nat(groups708209901874060359at_nat(Ha),A3)) ).

% prod.distrib
tff(fact_6305_prod_Odistrib,axiom,
    ! [G: fun_nat_int,Ha: fun_nat_int,A3: set_nat] : aa_set_nat_int(groups705719431365010083at_int(aa_fun5565043122120104496at_int(aTP_Lamm_ne(G),Ha)),A3) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_set_nat_int(groups705719431365010083at_int(G),A3)),aa_set_nat_int(groups705719431365010083at_int(Ha),A3)) ).

% prod.distrib
tff(fact_6306_prod_Odistrib,axiom,
    ! [G: fun_int_int,Ha: fun_int_int,A3: set_int] : groups1705073143266064639nt_int(aa_fun6280345759651049192nt_int(aTP_Lamm_nf(G),Ha),A3) = aa_int_int(aa_int_fun_int_int(times_times_int,groups1705073143266064639nt_int(G,A3)),groups1705073143266064639nt_int(Ha,A3)) ).

% prod.distrib
tff(fact_6307_prod__power__distrib,axiom,
    ! [F: fun_nat_nat,A3: set_nat,N2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_set_nat_nat(groups708209901874060359at_nat(F),A3)),N2) = aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_ng(F),N2)),A3) ).

% prod_power_distrib
tff(fact_6308_prod__power__distrib,axiom,
    ! [F: fun_nat_int,A3: set_nat,N2: nat] : aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_set_nat_int(groups705719431365010083at_int(F),A3)),N2) = aa_set_nat_int(groups705719431365010083at_int(aa_nat_fun_nat_int(aTP_Lamm_nh(F),N2)),A3) ).

% prod_power_distrib
tff(fact_6309_prod__power__distrib,axiom,
    ! [F: fun_int_int,A3: set_int,N2: nat] : aa_nat_int(aa_int_fun_nat_int(power_power_int,groups1705073143266064639nt_int(F,A3)),N2) = groups1705073143266064639nt_int(aa_nat_fun_int_int(aTP_Lamm_ni(F),N2),A3) ).

% prod_power_distrib
tff(fact_6310_mod__prod__eq,axiom,
    ! [F: fun_nat_nat,A: nat,A3: 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_jv(F),A)),A3)),A) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_set_nat_nat(groups708209901874060359at_nat(F),A3)),A) ).

% mod_prod_eq
tff(fact_6311_mod__prod__eq,axiom,
    ! [F: fun_nat_int,A: int,A3: 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_nj(F),A)),A3)),A) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_set_nat_int(groups705719431365010083at_int(F),A3)),A) ).

% mod_prod_eq
tff(fact_6312_mod__prod__eq,axiom,
    ! [F: fun_int_int,A: int,A3: set_int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,groups1705073143266064639nt_int(aa_int_fun_int_int(aTP_Lamm_jw(F),A),A3)),A) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,groups1705073143266064639nt_int(F,A3)),A) ).

% mod_prod_eq
tff(fact_6313_prod_OatMost__Suc__shift,axiom,
    ! [G: fun_nat_Code_integer,N2: nat] : aa_set871585252387937313nteger(groups3455450783089532116nteger(G),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,N2))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(G,zero_zero_nat)),aa_set871585252387937313nteger(groups3455450783089532116nteger(aTP_Lamm_nk(G)),aa_nat_set_nat(set_ord_atMost_nat,N2))) ).

% prod.atMost_Suc_shift
tff(fact_6314_prod_OatMost__Suc__shift,axiom,
    ! [G: fun_nat_assn,N2: nat] : groups6906906614972039071t_assn(G,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,N2))) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_nat_assn(G,zero_zero_nat)),groups6906906614972039071t_assn(aTP_Lamm_nl(G),aa_nat_set_nat(set_ord_atMost_nat,N2))) ).

% prod.atMost_Suc_shift
tff(fact_6315_prod_OatMost__Suc__shift,axiom,
    ! [G: fun_nat_rat,N2: nat] : aa_set_nat_rat(groups73079841787564623at_rat(G),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,N2))) = 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_kb(G)),aa_nat_set_nat(set_ord_atMost_nat,N2))) ).

% prod.atMost_Suc_shift
tff(fact_6316_prod_OatMost__Suc__shift,axiom,
    ! [G: fun_nat_nat,N2: nat] : aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,N2))) = 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_jx(G)),aa_nat_set_nat(set_ord_atMost_nat,N2))) ).

% prod.atMost_Suc_shift
tff(fact_6317_prod_OatMost__Suc__shift,axiom,
    ! [G: fun_nat_int,N2: nat] : aa_set_nat_int(groups705719431365010083at_int(G),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,N2))) = 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_kc(G)),aa_nat_set_nat(set_ord_atMost_nat,N2))) ).

% prod.atMost_Suc_shift
tff(fact_6318_atMost__def,axiom,
    ! [U: set_nat] : set_or4236626031148496127et_nat(U) = collect_set_nat(aa_set7861427066675100071_nat_o(aTP_Lamm_cz,U)) ).

% atMost_def
tff(fact_6319_atMost__def,axiom,
    ! [U: set_int] : set_or58775011639299419et_int(U) = collect_set_int(aa_set5425576183772390127_int_o(aTP_Lamm_dd,U)) ).

% atMost_def
tff(fact_6320_atMost__def,axiom,
    ! [U: rat] : set_ord_atMost_rat(U) = collect_rat(aa_rat_fun_rat_o(aTP_Lamm_dh,U)) ).

% atMost_def
tff(fact_6321_atMost__def,axiom,
    ! [U: num] : set_ord_atMost_num(U) = collect_num(aa_num_fun_num_o(aTP_Lamm_nm,U)) ).

% atMost_def
tff(fact_6322_atMost__def,axiom,
    ! [U: nat] : aa_nat_set_nat(set_ord_atMost_nat,U) = collect_nat(aa_nat_fun_nat_o(aTP_Lamm_df,U)) ).

% atMost_def
tff(fact_6323_atMost__def,axiom,
    ! [U: int] : set_ord_atMost_int(U) = collect_int(aa_int_fun_int_o(aTP_Lamm_dj,U)) ).

% atMost_def
tff(fact_6324_strict__sorted__imp__sorted,axiom,
    ! [Xs: list_rat] :
      ( sorted_wrt_rat(ord_less_rat,Xs)
     => sorted_wrt_rat(ord_less_eq_rat,Xs) ) ).

% strict_sorted_imp_sorted
tff(fact_6325_strict__sorted__imp__sorted,axiom,
    ! [Xs: list_num] :
      ( sorted_wrt_num(ord_less_num,Xs)
     => sorted_wrt_num(ord_less_eq_num,Xs) ) ).

% strict_sorted_imp_sorted
tff(fact_6326_strict__sorted__imp__sorted,axiom,
    ! [Xs: list_nat] :
      ( aa_list_nat_o(sorted_wrt_nat(ord_less_nat),Xs)
     => aa_list_nat_o(sorted_wrt_nat(ord_less_eq_nat),Xs) ) ).

% strict_sorted_imp_sorted
tff(fact_6327_strict__sorted__imp__sorted,axiom,
    ! [Xs: list_int] :
      ( sorted_wrt_int(ord_less_int,Xs)
     => sorted_wrt_int(ord_less_eq_int,Xs) ) ).

% strict_sorted_imp_sorted
tff(fact_6328_prod__nonneg,axiom,
    ! [A3: set_nat,F: fun_nat_nat] :
      ( ! [X3: nat] :
          ( aa_set_nat_o(member_nat(X3),A3)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(F,X3)) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_set_nat_nat(groups708209901874060359at_nat(F),A3)) ) ).

% prod_nonneg
tff(fact_6329_prod__nonneg,axiom,
    ! [A3: set_nat,F: fun_nat_int] :
      ( ! [X3: nat] :
          ( aa_set_nat_o(member_nat(X3),A3)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(F,X3)) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_set_nat_int(groups705719431365010083at_int(F),A3)) ) ).

% prod_nonneg
tff(fact_6330_prod__nonneg,axiom,
    ! [A3: set_int,F: fun_int_int] :
      ( ! [X3: int] :
          ( aa_set_int_o(member_int(X3),A3)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(F,X3)) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),groups1705073143266064639nt_int(F,A3)) ) ).

% prod_nonneg
tff(fact_6331_prod__mono,axiom,
    ! [A3: set_o,F: fun_o_Code_integer,G: fun_o_Code_integer] :
      ( ! [I3: $o] :
          ( member_o((I3),A3)
         => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_o_Code_integer(F,(I3)))
            & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_o_Code_integer(F,(I3))),aa_o_Code_integer(G,(I3))) ) )
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,groups7694694392188491536nteger(F,A3)),groups7694694392188491536nteger(G,A3)) ) ).

% prod_mono
tff(fact_6332_prod__mono,axiom,
    ! [A3: set_nat,F: fun_nat_Code_integer,G: fun_nat_Code_integer] :
      ( ! [I3: nat] :
          ( aa_set_nat_o(member_nat(I3),A3)
         => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(F,I3))
            & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(F,I3)),aa_nat_Code_integer(G,I3)) ) )
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_set871585252387937313nteger(groups3455450783089532116nteger(F),A3)),aa_set871585252387937313nteger(groups3455450783089532116nteger(G),A3)) ) ).

% prod_mono
tff(fact_6333_prod__mono,axiom,
    ! [A3: set_int,F: fun_int_Code_integer,G: fun_int_Code_integer] :
      ( ! [I3: int] :
          ( aa_set_int_o(member_int(I3),A3)
         => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_int_Code_integer(F,I3))
            & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_int_Code_integer(F,I3)),aa_int_Code_integer(G,I3)) ) )
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,groups3827104343326376752nteger(F,A3)),groups3827104343326376752nteger(G,A3)) ) ).

% prod_mono
tff(fact_6334_prod__mono,axiom,
    ! [A3: set_o,F: fun_o_rat,G: fun_o_rat] :
      ( ! [I3: $o] :
          ( member_o((I3),A3)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_o_rat(F,(I3)))
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_o_rat(F,(I3))),aa_o_rat(G,(I3))) ) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups2869687844427037835_o_rat(F,A3)),groups2869687844427037835_o_rat(G,A3)) ) ).

% prod_mono
tff(fact_6335_prod__mono,axiom,
    ! [A3: set_nat,F: fun_nat_rat,G: fun_nat_rat] :
      ( ! [I3: nat] :
          ( aa_set_nat_o(member_nat(I3),A3)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(F,I3))
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,I3)),aa_nat_rat(G,I3)) ) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_set_nat_rat(groups73079841787564623at_rat(F),A3)),aa_set_nat_rat(groups73079841787564623at_rat(G),A3)) ) ).

% prod_mono
tff(fact_6336_prod__mono,axiom,
    ! [A3: set_int,F: fun_int_rat,G: fun_int_rat] :
      ( ! [I3: int] :
          ( aa_set_int_o(member_int(I3),A3)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_int_rat(F,I3))
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(F,I3)),aa_int_rat(G,I3)) ) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups1072433553688619179nt_rat(F,A3)),groups1072433553688619179nt_rat(G,A3)) ) ).

% prod_mono
tff(fact_6337_prod__mono,axiom,
    ! [A3: set_o,F: fun_o_nat,G: fun_o_nat] :
      ( ! [I3: $o] :
          ( member_o((I3),A3)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_o_nat(F,(I3)))
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_o_nat(F,(I3))),aa_o_nat(G,(I3))) ) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,groups3504817904513533571_o_nat(F,A3)),groups3504817904513533571_o_nat(G,A3)) ) ).

% prod_mono
tff(fact_6338_prod__mono,axiom,
    ! [A3: set_int,F: fun_int_nat,G: fun_int_nat] :
      ( ! [I3: int] :
          ( aa_set_int_o(member_int(I3),A3)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_int_nat(F,I3))
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(F,I3)),aa_int_nat(G,I3)) ) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,groups1707563613775114915nt_nat(F,A3)),groups1707563613775114915nt_nat(G,A3)) ) ).

% prod_mono
tff(fact_6339_prod__mono,axiom,
    ! [A3: set_o,F: fun_o_int,G: fun_o_int] :
      ( ! [I3: $o] :
          ( member_o((I3),A3)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_o_int(F,(I3)))
            & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_o_int(F,(I3))),aa_o_int(G,(I3))) ) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,groups3502327434004483295_o_int(F,A3)),groups3502327434004483295_o_int(G,A3)) ) ).

% prod_mono
tff(fact_6340_prod__mono,axiom,
    ! [A3: set_nat,F: fun_nat_nat,G: fun_nat_nat] :
      ( ! [I3: nat] :
          ( aa_set_nat_o(member_nat(I3),A3)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(F,I3))
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F,I3)),aa_nat_nat(G,I3)) ) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(groups708209901874060359at_nat(F),A3)),aa_set_nat_nat(groups708209901874060359at_nat(G),A3)) ) ).

% prod_mono
tff(fact_6341_prod__pos,axiom,
    ! [A3: set_nat,F: fun_nat_nat] :
      ( ! [X3: nat] :
          ( aa_set_nat_o(member_nat(X3),A3)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(F,X3)) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_set_nat_nat(groups708209901874060359at_nat(F),A3)) ) ).

% prod_pos
tff(fact_6342_prod__pos,axiom,
    ! [A3: set_nat,F: fun_nat_int] :
      ( ! [X3: nat] :
          ( aa_set_nat_o(member_nat(X3),A3)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(F,X3)) )
     => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_set_nat_int(groups705719431365010083at_int(F),A3)) ) ).

% prod_pos
tff(fact_6343_prod__pos,axiom,
    ! [A3: set_int,F: fun_int_int] :
      ( ! [X3: int] :
          ( aa_set_int_o(member_int(X3),A3)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(F,X3)) )
     => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),groups1705073143266064639nt_int(F,A3)) ) ).

% prod_pos
tff(fact_6344_prod__ge__1,axiom,
    ! [A3: set_o,F: fun_o_Code_integer] :
      ( ! [X3: $o] :
          ( member_o((X3),A3)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),aa_o_Code_integer(F,(X3))) )
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),groups7694694392188491536nteger(F,A3)) ) ).

% prod_ge_1
tff(fact_6345_prod__ge__1,axiom,
    ! [A3: set_nat,F: fun_nat_Code_integer] :
      ( ! [X3: nat] :
          ( aa_set_nat_o(member_nat(X3),A3)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),aa_nat_Code_integer(F,X3)) )
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),aa_set871585252387937313nteger(groups3455450783089532116nteger(F),A3)) ) ).

% prod_ge_1
tff(fact_6346_prod__ge__1,axiom,
    ! [A3: set_int,F: fun_int_Code_integer] :
      ( ! [X3: int] :
          ( aa_set_int_o(member_int(X3),A3)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),aa_int_Code_integer(F,X3)) )
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),groups3827104343326376752nteger(F,A3)) ) ).

% prod_ge_1
tff(fact_6347_prod__ge__1,axiom,
    ! [A3: set_o,F: fun_o_rat] :
      ( ! [X3: $o] :
          ( member_o((X3),A3)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_o_rat(F,(X3))) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),groups2869687844427037835_o_rat(F,A3)) ) ).

% prod_ge_1
tff(fact_6348_prod__ge__1,axiom,
    ! [A3: set_nat,F: fun_nat_rat] :
      ( ! [X3: nat] :
          ( aa_set_nat_o(member_nat(X3),A3)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_nat_rat(F,X3)) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_set_nat_rat(groups73079841787564623at_rat(F),A3)) ) ).

% prod_ge_1
tff(fact_6349_prod__ge__1,axiom,
    ! [A3: set_int,F: fun_int_rat] :
      ( ! [X3: int] :
          ( aa_set_int_o(member_int(X3),A3)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_int_rat(F,X3)) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),groups1072433553688619179nt_rat(F,A3)) ) ).

% prod_ge_1
tff(fact_6350_prod__ge__1,axiom,
    ! [A3: set_o,F: fun_o_nat] :
      ( ! [X3: $o] :
          ( member_o((X3),A3)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),aa_o_nat(F,(X3))) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),groups3504817904513533571_o_nat(F,A3)) ) ).

% prod_ge_1
tff(fact_6351_prod__ge__1,axiom,
    ! [A3: set_int,F: fun_int_nat] :
      ( ! [X3: int] :
          ( aa_set_int_o(member_int(X3),A3)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),aa_int_nat(F,X3)) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),groups1707563613775114915nt_nat(F,A3)) ) ).

% prod_ge_1
tff(fact_6352_prod__ge__1,axiom,
    ! [A3: set_o,F: fun_o_int] :
      ( ! [X3: $o] :
          ( member_o((X3),A3)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_o_int(F,(X3))) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),groups3502327434004483295_o_int(F,A3)) ) ).

% prod_ge_1
tff(fact_6353_prod__ge__1,axiom,
    ! [A3: set_nat,F: fun_nat_nat] :
      ( ! [X3: nat] :
          ( aa_set_nat_o(member_nat(X3),A3)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),aa_nat_nat(F,X3)) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),aa_set_nat_nat(groups708209901874060359at_nat(F),A3)) ) ).

% prod_ge_1
tff(fact_6354_sorted__wrt__less__idx,axiom,
    ! [Ns: list_nat,I2: nat] :
      ( aa_list_nat_o(sorted_wrt_nat(ord_less_nat),Ns)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),aa_list_nat_nat(size_size_list_nat,Ns))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),aa_nat_nat(nth_nat(Ns),I2)) ) ) ).

% sorted_wrt_less_idx
tff(fact_6355_not__Iic__le__Icc,axiom,
    ! [Ha: int,L3: int,H_a: int] : ~ aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,set_ord_atMost_int(Ha)),set_or1266510415728281911st_int(L3,H_a)) ).

% not_Iic_le_Icc
tff(fact_6356_prod_Oshift__bounds__cl__Suc__ivl,axiom,
    ! [G: fun_nat_nat,M: nat,N2: nat] : aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,aa_nat_nat(suc,M)),aa_nat_nat(suc,N2))) = aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_jx(G)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M),N2)) ).

% prod.shift_bounds_cl_Suc_ivl
tff(fact_6357_prod_Oshift__bounds__cl__Suc__ivl,axiom,
    ! [G: fun_nat_int,M: nat,N2: nat] : aa_set_nat_int(groups705719431365010083at_int(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,aa_nat_nat(suc,M)),aa_nat_nat(suc,N2))) = aa_set_nat_int(groups705719431365010083at_int(aTP_Lamm_kc(G)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M),N2)) ).

% prod.shift_bounds_cl_Suc_ivl
tff(fact_6358_power__sum,axiom,
    ! [Ca: nat,F: fun_nat_nat,A3: set_nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Ca),aa_set_nat_nat(groups3542108847815614940at_nat(F),A3)) = aa_set_nat_nat(groups708209901874060359at_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_nn(Ca),F)),A3) ).

% power_sum
tff(fact_6359_power__sum,axiom,
    ! [Ca: int,F: fun_nat_nat,A3: set_nat] : aa_nat_int(aa_int_fun_nat_int(power_power_int,Ca),aa_set_nat_nat(groups3542108847815614940at_nat(F),A3)) = aa_set_nat_int(groups705719431365010083at_int(aa_fun1613598493704395220at_int(aTP_Lamm_no(Ca),F)),A3) ).

% power_sum
tff(fact_6360_power__sum,axiom,
    ! [Ca: int,F: fun_int_nat,A3: set_int] : aa_nat_int(aa_int_fun_nat_int(power_power_int,Ca),groups4541462559716669496nt_nat(F,A3)) = groups1705073143266064639nt_int(aa_fun2328901131235339916nt_int(aTP_Lamm_np(Ca),F),A3) ).

% power_sum
tff(fact_6361_prod_Oshift__bounds__cl__nat__ivl,axiom,
    ! [G: fun_nat_nat,M: nat,K: nat,N2: nat] : aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(aa_nat1818925681583257441et_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,N2),K))) = aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_jy(G),K)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M),N2)) ).

% prod.shift_bounds_cl_nat_ivl
tff(fact_6362_prod_Oshift__bounds__cl__nat__ivl,axiom,
    ! [G: fun_nat_int,M: nat,K: nat,N2: nat] : aa_set_nat_int(groups705719431365010083at_int(G),aa_nat_set_nat(aa_nat1818925681583257441et_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,N2),K))) = aa_set_nat_int(groups705719431365010083at_int(aa_nat_fun_nat_int(aTP_Lamm_nq(G),K)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M),N2)) ).

% prod.shift_bounds_cl_nat_ivl
tff(fact_6363_prod__le__1,axiom,
    ! [A3: set_o,F: fun_o_Code_integer] :
      ( ! [X3: $o] :
          ( member_o((X3),A3)
         => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_o_Code_integer(F,(X3)))
            & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_o_Code_integer(F,(X3))),one_one_Code_integer) ) )
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,groups7694694392188491536nteger(F,A3)),one_one_Code_integer) ) ).

% prod_le_1
tff(fact_6364_prod__le__1,axiom,
    ! [A3: set_nat,F: fun_nat_Code_integer] :
      ( ! [X3: nat] :
          ( aa_set_nat_o(member_nat(X3),A3)
         => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(F,X3))
            & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(F,X3)),one_one_Code_integer) ) )
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_set871585252387937313nteger(groups3455450783089532116nteger(F),A3)),one_one_Code_integer) ) ).

% prod_le_1
tff(fact_6365_prod__le__1,axiom,
    ! [A3: set_int,F: fun_int_Code_integer] :
      ( ! [X3: int] :
          ( aa_set_int_o(member_int(X3),A3)
         => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_int_Code_integer(F,X3))
            & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_int_Code_integer(F,X3)),one_one_Code_integer) ) )
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,groups3827104343326376752nteger(F,A3)),one_one_Code_integer) ) ).

% prod_le_1
tff(fact_6366_prod__le__1,axiom,
    ! [A3: set_o,F: fun_o_rat] :
      ( ! [X3: $o] :
          ( member_o((X3),A3)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_o_rat(F,(X3)))
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_o_rat(F,(X3))),one_one_rat) ) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups2869687844427037835_o_rat(F,A3)),one_one_rat) ) ).

% prod_le_1
tff(fact_6367_prod__le__1,axiom,
    ! [A3: set_nat,F: fun_nat_rat] :
      ( ! [X3: nat] :
          ( aa_set_nat_o(member_nat(X3),A3)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(F,X3))
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,X3)),one_one_rat) ) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_set_nat_rat(groups73079841787564623at_rat(F),A3)),one_one_rat) ) ).

% prod_le_1
tff(fact_6368_prod__le__1,axiom,
    ! [A3: set_int,F: fun_int_rat] :
      ( ! [X3: int] :
          ( aa_set_int_o(member_int(X3),A3)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_int_rat(F,X3))
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(F,X3)),one_one_rat) ) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups1072433553688619179nt_rat(F,A3)),one_one_rat) ) ).

% prod_le_1
tff(fact_6369_prod__le__1,axiom,
    ! [A3: set_o,F: fun_o_nat] :
      ( ! [X3: $o] :
          ( member_o((X3),A3)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_o_nat(F,(X3)))
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_o_nat(F,(X3))),one_one_nat) ) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,groups3504817904513533571_o_nat(F,A3)),one_one_nat) ) ).

% prod_le_1
tff(fact_6370_prod__le__1,axiom,
    ! [A3: set_int,F: fun_int_nat] :
      ( ! [X3: int] :
          ( aa_set_int_o(member_int(X3),A3)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_int_nat(F,X3))
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(F,X3)),one_one_nat) ) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,groups1707563613775114915nt_nat(F,A3)),one_one_nat) ) ).

% prod_le_1
tff(fact_6371_prod__le__1,axiom,
    ! [A3: set_o,F: fun_o_int] :
      ( ! [X3: $o] :
          ( member_o((X3),A3)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_o_int(F,(X3)))
            & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_o_int(F,(X3))),one_one_int) ) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,groups3502327434004483295_o_int(F,A3)),one_one_int) ) ).

% prod_le_1
tff(fact_6372_prod__le__1,axiom,
    ! [A3: set_nat,F: fun_nat_nat] :
      ( ! [X3: nat] :
          ( aa_set_nat_o(member_nat(X3),A3)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(F,X3))
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F,X3)),one_one_nat) ) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(groups708209901874060359at_nat(F),A3)),one_one_nat) ) ).

% prod_le_1
tff(fact_6373_sorted__wrt__iff__nth__less,axiom,
    ! [Pa: fun_int_fun_int_o,Xs: list_int] :
      ( sorted_wrt_int(Pa,Xs)
    <=> ! [I4: nat,J3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),J3)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J3),size_size_list_int(Xs))
           => aa_int_o(aa_int_fun_int_o(Pa,aa_nat_int(nth_int(Xs),I4)),aa_nat_int(nth_int(Xs),J3)) ) ) ) ).

% sorted_wrt_iff_nth_less
tff(fact_6374_sorted__wrt__iff__nth__less,axiom,
    ! [Pa: fun_nat_fun_nat_o,Xs: list_nat] :
      ( aa_list_nat_o(sorted_wrt_nat(Pa),Xs)
    <=> ! [I4: nat,J3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),J3)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J3),aa_list_nat_nat(size_size_list_nat,Xs))
           => aa_nat_o(aa_nat_fun_nat_o(Pa,aa_nat_nat(nth_nat(Xs),I4)),aa_nat_nat(nth_nat(Xs),J3)) ) ) ) ).

% sorted_wrt_iff_nth_less
tff(fact_6375_sorted__wrt__iff__nth__less,axiom,
    ! [Pa: fun_o_fun_o_o,Xs: list_o] :
      ( aa_list_o_o(sorted_wrt_o(Pa),Xs)
    <=> ! [I4: nat,J3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),J3)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J3),size_size_list_o(Xs))
           => aa_o_o(aa_o_fun_o_o(Pa,aa_nat_o(nth_o(Xs),I4)),aa_nat_o(nth_o(Xs),J3)) ) ) ) ).

% sorted_wrt_iff_nth_less
tff(fact_6376_sorted__wrt__nth__less,axiom,
    ! [Pa: fun_int_fun_int_o,Xs: list_int,I2: nat,J: nat] :
      ( sorted_wrt_int(Pa,Xs)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),J)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J),size_size_list_int(Xs))
         => aa_int_o(aa_int_fun_int_o(Pa,aa_nat_int(nth_int(Xs),I2)),aa_nat_int(nth_int(Xs),J)) ) ) ) ).

% sorted_wrt_nth_less
tff(fact_6377_sorted__wrt__nth__less,axiom,
    ! [Pa: fun_nat_fun_nat_o,Xs: list_nat,I2: nat,J: nat] :
      ( aa_list_nat_o(sorted_wrt_nat(Pa),Xs)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),J)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J),aa_list_nat_nat(size_size_list_nat,Xs))
         => aa_nat_o(aa_nat_fun_nat_o(Pa,aa_nat_nat(nth_nat(Xs),I2)),aa_nat_nat(nth_nat(Xs),J)) ) ) ) ).

% sorted_wrt_nth_less
tff(fact_6378_sorted__wrt__nth__less,axiom,
    ! [Pa: fun_o_fun_o_o,Xs: list_o,I2: nat,J: nat] :
      ( aa_list_o_o(sorted_wrt_o(Pa),Xs)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),J)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J),size_size_list_o(Xs))
         => aa_o_o(aa_o_fun_o_o(Pa,aa_nat_o(nth_o(Xs),I2)),aa_nat_o(nth_o(Xs),J)) ) ) ) ).

% sorted_wrt_nth_less
tff(fact_6379_sorted__wrt01,axiom,
    ! [Xs: list_int,Pa: fun_int_fun_int_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_size_list_int(Xs)),one_one_nat)
     => sorted_wrt_int(Pa,Xs) ) ).

% sorted_wrt01
tff(fact_6380_sorted__wrt01,axiom,
    ! [Xs: list_nat,Pa: fun_nat_fun_nat_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_list_nat_nat(size_size_list_nat,Xs)),one_one_nat)
     => aa_list_nat_o(sorted_wrt_nat(Pa),Xs) ) ).

% sorted_wrt01
tff(fact_6381_sorted__wrt01,axiom,
    ! [Xs: list_o,Pa: fun_o_fun_o_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_size_list_o(Xs)),one_one_nat)
     => aa_list_o_o(sorted_wrt_o(Pa),Xs) ) ).

% sorted_wrt01
tff(fact_6382_prod_OatLeastAtMost__rev,axiom,
    ! [G: fun_nat_nat,N2: nat,M: nat] : aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,N2),M)) = aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_jz(G),N2),M)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,N2),M)) ).

% prod.atLeastAtMost_rev
tff(fact_6383_prod_OatLeastAtMost__rev,axiom,
    ! [G: fun_nat_int,N2: nat,M: nat] : aa_set_nat_int(groups705719431365010083at_int(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,N2),M)) = aa_set_nat_int(groups705719431365010083at_int(aa_nat_fun_nat_int(aa_nat1293065384339088724at_int(aTP_Lamm_nr(G),N2),M)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,N2),M)) ).

% prod.atLeastAtMost_rev
tff(fact_6384_prod_Ozero__middle,axiom,
    ! [P2: nat,K: nat,G: fun_nat_Code_integer,Ha: fun_nat_Code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),P2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),P2)
       => ( aa_set871585252387937313nteger(groups3455450783089532116nteger(aa_fun589076786913273746nteger(aa_fun3918145391673965625nteger(aTP_Lamm_ns(K),G),Ha)),aa_nat_set_nat(set_ord_atMost_nat,P2)) = aa_set871585252387937313nteger(groups3455450783089532116nteger(aa_fun589076786913273746nteger(aa_fun3918145391673965625nteger(aTP_Lamm_nt(K),G),Ha)),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,P2),aa_nat_nat(suc,zero_zero_nat)))) ) ) ) ).

% prod.zero_middle
tff(fact_6385_prod_Ozero__middle,axiom,
    ! [P2: nat,K: nat,G: fun_nat_assn,Ha: fun_nat_assn] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),P2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),P2)
       => ( groups6906906614972039071t_assn(aa_fun785744391571715112t_assn(aa_fun6591345337174305434t_assn(aTP_Lamm_nu(K),G),Ha),aa_nat_set_nat(set_ord_atMost_nat,P2)) = groups6906906614972039071t_assn(aa_fun785744391571715112t_assn(aa_fun6591345337174305434t_assn(aTP_Lamm_nv(K),G),Ha),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,P2),aa_nat_nat(suc,zero_zero_nat)))) ) ) ) ).

% prod.zero_middle
tff(fact_6386_prod_Ozero__middle,axiom,
    ! [P2: nat,K: nat,G: fun_nat_rat,Ha: fun_nat_rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),P2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),P2)
       => ( aa_set_nat_rat(groups73079841787564623at_rat(aa_fun765758790203053448at_rat(aa_fun3638354034882855594at_rat(aTP_Lamm_nw(K),G),Ha)),aa_nat_set_nat(set_ord_atMost_nat,P2)) = aa_set_nat_rat(groups73079841787564623at_rat(aa_fun765758790203053448at_rat(aa_fun3638354034882855594at_rat(aTP_Lamm_nx(K),G),Ha)),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,P2),aa_nat_nat(suc,zero_zero_nat)))) ) ) ) ).

% prod.zero_middle
tff(fact_6387_prod_Ozero__middle,axiom,
    ! [P2: nat,K: nat,G: fun_nat_nat,Ha: fun_nat_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),P2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),P2)
       => ( aa_set_nat_nat(groups708209901874060359at_nat(aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(aTP_Lamm_ny(K),G),Ha)),aa_nat_set_nat(set_ord_atMost_nat,P2)) = aa_set_nat_nat(groups708209901874060359at_nat(aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(aTP_Lamm_nz(K),G),Ha)),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,P2),aa_nat_nat(suc,zero_zero_nat)))) ) ) ) ).

% prod.zero_middle
tff(fact_6388_prod_Ozero__middle,axiom,
    ! [P2: nat,K: nat,G: fun_nat_int,Ha: fun_nat_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),P2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),P2)
       => ( aa_set_nat_int(groups705719431365010083at_int(aa_fun5565043122120104496at_int(aa_fun5140708168904641446at_int(aTP_Lamm_oa(K),G),Ha)),aa_nat_set_nat(set_ord_atMost_nat,P2)) = aa_set_nat_int(groups705719431365010083at_int(aa_fun5565043122120104496at_int(aa_fun5140708168904641446at_int(aTP_Lamm_ob(K),G),Ha)),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,P2),aa_nat_nat(suc,zero_zero_nat)))) ) ) ) ).

% prod.zero_middle
tff(fact_6389_sorted__iff__nth__mono__less,axiom,
    ! [Xs: list_o] :
      ( aa_list_o_o(sorted_wrt_o(ord_less_eq_o),Xs)
    <=> ! [I4: nat,J3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),J3)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J3),size_size_list_o(Xs))
           => aa_o_o(aa_o_fun_o_o(ord_less_eq_o,aa_nat_o(nth_o(Xs),I4)),aa_nat_o(nth_o(Xs),J3)) ) ) ) ).

% sorted_iff_nth_mono_less
tff(fact_6390_sorted__iff__nth__mono__less,axiom,
    ! [Xs: list_rat] :
      ( sorted_wrt_rat(ord_less_eq_rat,Xs)
    <=> ! [I4: nat,J3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),J3)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J3),size_size_list_rat(Xs))
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(nth_rat(Xs),I4)),aa_nat_rat(nth_rat(Xs),J3)) ) ) ) ).

% sorted_iff_nth_mono_less
tff(fact_6391_sorted__iff__nth__mono__less,axiom,
    ! [Xs: list_num] :
      ( sorted_wrt_num(ord_less_eq_num,Xs)
    <=> ! [I4: nat,J3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),J3)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J3),size_size_list_num(Xs))
           => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(nth_num(Xs),I4)),aa_nat_num(nth_num(Xs),J3)) ) ) ) ).

% sorted_iff_nth_mono_less
tff(fact_6392_sorted__iff__nth__mono__less,axiom,
    ! [Xs: list_nat] :
      ( aa_list_nat_o(sorted_wrt_nat(ord_less_eq_nat),Xs)
    <=> ! [I4: nat,J3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),J3)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J3),aa_list_nat_nat(size_size_list_nat,Xs))
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(nth_nat(Xs),I4)),aa_nat_nat(nth_nat(Xs),J3)) ) ) ) ).

% sorted_iff_nth_mono_less
tff(fact_6393_sorted__iff__nth__mono__less,axiom,
    ! [Xs: list_int] :
      ( sorted_wrt_int(ord_less_eq_int,Xs)
    <=> ! [I4: nat,J3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),J3)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J3),size_size_list_int(Xs))
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(nth_int(Xs),I4)),aa_nat_int(nth_int(Xs),J3)) ) ) ) ).

% sorted_iff_nth_mono_less
tff(fact_6394_sorted01,axiom,
    ! [Xs: list_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_size_list_o(Xs)),one_one_nat)
     => aa_list_o_o(sorted_wrt_o(ord_less_eq_o),Xs) ) ).

% sorted01
tff(fact_6395_sorted01,axiom,
    ! [Xs: list_rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_size_list_rat(Xs)),one_one_nat)
     => sorted_wrt_rat(ord_less_eq_rat,Xs) ) ).

% sorted01
tff(fact_6396_sorted01,axiom,
    ! [Xs: list_num] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_size_list_num(Xs)),one_one_nat)
     => sorted_wrt_num(ord_less_eq_num,Xs) ) ).

% sorted01
tff(fact_6397_sorted01,axiom,
    ! [Xs: list_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_list_nat_nat(size_size_list_nat,Xs)),one_one_nat)
     => aa_list_nat_o(sorted_wrt_nat(ord_less_eq_nat),Xs) ) ).

% sorted01
tff(fact_6398_sorted01,axiom,
    ! [Xs: list_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_size_list_int(Xs)),one_one_nat)
     => sorted_wrt_int(ord_less_eq_int,Xs) ) ).

% sorted01
tff(fact_6399_prod_OatLeast0__atMost__Suc,axiom,
    ! [G: fun_nat_Code_integer,N2: nat] : aa_set871585252387937313nteger(groups3455450783089532116nteger(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,zero_zero_nat),aa_nat_nat(suc,N2))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_set871585252387937313nteger(groups3455450783089532116nteger(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,zero_zero_nat),N2))),aa_nat_Code_integer(G,aa_nat_nat(suc,N2))) ).

% prod.atLeast0_atMost_Suc
tff(fact_6400_prod_OatLeast0__atMost__Suc,axiom,
    ! [G: fun_nat_assn,N2: nat] : groups6906906614972039071t_assn(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,zero_zero_nat),aa_nat_nat(suc,N2))) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,groups6906906614972039071t_assn(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,zero_zero_nat),N2))),aa_nat_assn(G,aa_nat_nat(suc,N2))) ).

% prod.atLeast0_atMost_Suc
tff(fact_6401_prod_OatLeast0__atMost__Suc,axiom,
    ! [G: fun_nat_rat,N2: nat] : aa_set_nat_rat(groups73079841787564623at_rat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,zero_zero_nat),aa_nat_nat(suc,N2))) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_set_nat_rat(groups73079841787564623at_rat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,zero_zero_nat),N2))),aa_nat_rat(G,aa_nat_nat(suc,N2))) ).

% prod.atLeast0_atMost_Suc
tff(fact_6402_prod_OatLeast0__atMost__Suc,axiom,
    ! [G: fun_nat_nat,N2: nat] : aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,zero_zero_nat),aa_nat_nat(suc,N2))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,zero_zero_nat),N2))),aa_nat_nat(G,aa_nat_nat(suc,N2))) ).

% prod.atLeast0_atMost_Suc
tff(fact_6403_prod_OatLeast0__atMost__Suc,axiom,
    ! [G: fun_nat_int,N2: nat] : aa_set_nat_int(groups705719431365010083at_int(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,zero_zero_nat),aa_nat_nat(suc,N2))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_set_nat_int(groups705719431365010083at_int(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,zero_zero_nat),N2))),aa_nat_int(G,aa_nat_nat(suc,N2))) ).

% prod.atLeast0_atMost_Suc
tff(fact_6404_prod_Onat__ivl__Suc_H,axiom,
    ! [M: nat,N2: nat,G: fun_nat_Code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(suc,N2))
     => ( aa_set871585252387937313nteger(groups3455450783089532116nteger(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M),aa_nat_nat(suc,N2))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(G,aa_nat_nat(suc,N2))),aa_set871585252387937313nteger(groups3455450783089532116nteger(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M),N2))) ) ) ).

% prod.nat_ivl_Suc'
tff(fact_6405_prod_Onat__ivl__Suc_H,axiom,
    ! [M: nat,N2: nat,G: fun_nat_assn] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(suc,N2))
     => ( groups6906906614972039071t_assn(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M),aa_nat_nat(suc,N2))) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_nat_assn(G,aa_nat_nat(suc,N2))),groups6906906614972039071t_assn(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M),N2))) ) ) ).

% prod.nat_ivl_Suc'
tff(fact_6406_prod_Onat__ivl__Suc_H,axiom,
    ! [M: nat,N2: nat,G: fun_nat_rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(suc,N2))
     => ( aa_set_nat_rat(groups73079841787564623at_rat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M),aa_nat_nat(suc,N2))) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(G,aa_nat_nat(suc,N2))),aa_set_nat_rat(groups73079841787564623at_rat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M),N2))) ) ) ).

% prod.nat_ivl_Suc'
tff(fact_6407_prod_Onat__ivl__Suc_H,axiom,
    ! [M: nat,N2: nat,G: fun_nat_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(suc,N2))
     => ( aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M),aa_nat_nat(suc,N2))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(G,aa_nat_nat(suc,N2))),aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M),N2))) ) ) ).

% prod.nat_ivl_Suc'
tff(fact_6408_prod_Onat__ivl__Suc_H,axiom,
    ! [M: nat,N2: nat,G: fun_nat_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(suc,N2))
     => ( aa_set_nat_int(groups705719431365010083at_int(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M),aa_nat_nat(suc,N2))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(G,aa_nat_nat(suc,N2))),aa_set_nat_int(groups705719431365010083at_int(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M),N2))) ) ) ).

% prod.nat_ivl_Suc'
tff(fact_6409_prod_OatLeast__Suc__atMost,axiom,
    ! [M: nat,N2: nat,G: fun_nat_Code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
     => ( aa_set871585252387937313nteger(groups3455450783089532116nteger(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M),N2)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(G,M)),aa_set871585252387937313nteger(groups3455450783089532116nteger(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,aa_nat_nat(suc,M)),N2))) ) ) ).

% prod.atLeast_Suc_atMost
tff(fact_6410_prod_OatLeast__Suc__atMost,axiom,
    ! [M: nat,N2: nat,G: fun_nat_assn] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
     => ( groups6906906614972039071t_assn(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M),N2)) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_nat_assn(G,M)),groups6906906614972039071t_assn(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,aa_nat_nat(suc,M)),N2))) ) ) ).

% prod.atLeast_Suc_atMost
tff(fact_6411_prod_OatLeast__Suc__atMost,axiom,
    ! [M: nat,N2: nat,G: fun_nat_rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
     => ( aa_set_nat_rat(groups73079841787564623at_rat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M),N2)) = 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(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,aa_nat_nat(suc,M)),N2))) ) ) ).

% prod.atLeast_Suc_atMost
tff(fact_6412_prod_OatLeast__Suc__atMost,axiom,
    ! [M: nat,N2: nat,G: fun_nat_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
     => ( aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M),N2)) = 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(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,aa_nat_nat(suc,M)),N2))) ) ) ).

% prod.atLeast_Suc_atMost
tff(fact_6413_prod_OatLeast__Suc__atMost,axiom,
    ! [M: nat,N2: nat,G: fun_nat_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
     => ( aa_set_nat_int(groups705719431365010083at_int(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M),N2)) = 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(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,aa_nat_nat(suc,M)),N2))) ) ) ).

% prod.atLeast_Suc_atMost
tff(fact_6414_prod_OSuc__reindex__ivl,axiom,
    ! [M: nat,N2: nat,G: fun_nat_Code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_set871585252387937313nteger(groups3455450783089532116nteger(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M),N2))),aa_nat_Code_integer(G,aa_nat_nat(suc,N2))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(G,M)),aa_set871585252387937313nteger(groups3455450783089532116nteger(aTP_Lamm_nk(G)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M),N2))) ) ) ).

% prod.Suc_reindex_ivl
tff(fact_6415_prod_OSuc__reindex__ivl,axiom,
    ! [M: nat,N2: nat,G: fun_nat_assn] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
     => ( aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,groups6906906614972039071t_assn(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M),N2))),aa_nat_assn(G,aa_nat_nat(suc,N2))) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_nat_assn(G,M)),groups6906906614972039071t_assn(aTP_Lamm_nl(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M),N2))) ) ) ).

% prod.Suc_reindex_ivl
tff(fact_6416_prod_OSuc__reindex__ivl,axiom,
    ! [M: nat,N2: nat,G: fun_nat_rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_set_nat_rat(groups73079841787564623at_rat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M),N2))),aa_nat_rat(G,aa_nat_nat(suc,N2))) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(G,M)),aa_set_nat_rat(groups73079841787564623at_rat(aTP_Lamm_kb(G)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M),N2))) ) ) ).

% prod.Suc_reindex_ivl
tff(fact_6417_prod_OSuc__reindex__ivl,axiom,
    ! [M: nat,N2: nat,G: fun_nat_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M),N2))),aa_nat_nat(G,aa_nat_nat(suc,N2))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(G,M)),aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_jx(G)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M),N2))) ) ) ).

% prod.Suc_reindex_ivl
tff(fact_6418_prod_OSuc__reindex__ivl,axiom,
    ! [M: nat,N2: nat,G: fun_nat_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_set_nat_int(groups705719431365010083at_int(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M),N2))),aa_nat_int(G,aa_nat_nat(suc,N2))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(G,M)),aa_set_nat_int(groups705719431365010083at_int(aTP_Lamm_kc(G)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M),N2))) ) ) ).

% prod.Suc_reindex_ivl
tff(fact_6419_sum_OatMost__Suc__shift,axiom,
    ! [G: fun_na2338080690716861136at_nat,N2: nat] : groups6857163185585827899at_nat(G,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,N2))) = aa_mul5754855110707352700at_nat(plus_p7104986032573967614at_nat(aa_nat2240542744625857721at_nat(G,zero_zero_nat)),groups6857163185585827899at_nat(aTP_Lamm_ka(G),aa_nat_set_nat(set_ord_atMost_nat,N2))) ).

% sum.atMost_Suc_shift
tff(fact_6420_sum_OatMost__Suc__shift,axiom,
    ! [G: fun_nat_rat,N2: nat] : aa_set_nat_rat(groups2906978787729119204at_rat(G),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,N2))) = 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_kb(G)),aa_nat_set_nat(set_ord_atMost_nat,N2))) ).

% sum.atMost_Suc_shift
tff(fact_6421_sum_OatMost__Suc__shift,axiom,
    ! [G: fun_nat_int,N2: nat] : aa_set_nat_int(groups3539618377306564664at_int(G),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,N2))) = 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_kc(G)),aa_nat_set_nat(set_ord_atMost_nat,N2))) ).

% sum.atMost_Suc_shift
tff(fact_6422_sum_OatMost__Suc__shift,axiom,
    ! [G: fun_nat_nat,N2: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,N2))) = 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_jx(G)),aa_nat_set_nat(set_ord_atMost_nat,N2))) ).

% sum.atMost_Suc_shift
tff(fact_6423_sum__telescope,axiom,
    ! [F: fun_nat_rat,I2: nat] : aa_set_nat_rat(groups2906978787729119204at_rat(aTP_Lamm_oc(F)),aa_nat_set_nat(set_ord_atMost_nat,I2)) = 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,I2))) ).

% sum_telescope
tff(fact_6424_sum__telescope,axiom,
    ! [F: fun_nat_int,I2: nat] : aa_set_nat_int(groups3539618377306564664at_int(aTP_Lamm_od(F)),aa_nat_set_nat(set_ord_atMost_nat,I2)) = 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,I2))) ).

% sum_telescope
tff(fact_6425_fact__prod,axiom,
    ! [N2: nat] : aa_nat_int(semiri1406184849735516958ct_int,N2) = aa_nat_int(semiri1314217659103216013at_int,aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_fu),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,one_one_nat),N2))) ).

% fact_prod
tff(fact_6426_fact__prod,axiom,
    ! [N2: nat] : aa_nat_Code_integer(semiri3624122377584611663nteger,N2) = aa_nat_Code_integer(semiri4939895301339042750nteger,aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_fu),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,one_one_nat),N2))) ).

% fact_prod
tff(fact_6427_fact__prod,axiom,
    ! [N2: nat] : aa_nat_nat(semiri1408675320244567234ct_nat,N2) = aa_nat_nat(semiri1316708129612266289at_nat,aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_fu),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,one_one_nat),N2))) ).

% fact_prod
tff(fact_6428_prod__atLeastAtMost__code,axiom,
    ! [F: fun_nat_Code_integer,A: nat,B: nat] : aa_set871585252387937313nteger(groups3455450783089532116nteger(F),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,A),B)) = aa_Cod1783974652533396728nteger(set_fo1084959871951514735nteger(aTP_Lamm_oe(F),A,B),one_one_Code_integer) ).

% prod_atLeastAtMost_code
tff(fact_6429_prod__atLeastAtMost__code,axiom,
    ! [F: fun_nat_assn,A: nat,B: nat] : groups6906906614972039071t_assn(F,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,A),B)) = aa_assn_assn(set_fo1959793692361082170t_assn(aTP_Lamm_of(F),A,B),one_one_assn) ).

% prod_atLeastAtMost_code
tff(fact_6430_prod__atLeastAtMost__code,axiom,
    ! [F: fun_nat_rat,A: nat,B: nat] : aa_set_nat_rat(groups73079841787564623at_rat(F),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,A),B)) = aa_rat_rat(set_fo1949268297981939178at_rat(aTP_Lamm_og(F),A,B),one_one_rat) ).

% prod_atLeastAtMost_code
tff(fact_6431_prod__atLeastAtMost__code,axiom,
    ! [F: fun_nat_nat,A: nat,B: nat] : aa_set_nat_nat(groups708209901874060359at_nat(F),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,A),B)) = aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(aTP_Lamm_oh(F),A),B),one_one_nat) ).

% prod_atLeastAtMost_code
tff(fact_6432_prod__atLeastAtMost__code,axiom,
    ! [F: fun_nat_int,A: nat,B: nat] : aa_set_nat_int(groups705719431365010083at_int(F),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,A),B)) = aa_int_int(aa_nat_fun_int_int(set_fo2581907887559384638at_int(aTP_Lamm_oi(F),A),B),one_one_int) ).

% prod_atLeastAtMost_code
tff(fact_6433_sorted__iff__nth__Suc,axiom,
    ! [Xs: list_o] :
      ( aa_list_o_o(sorted_wrt_o(ord_less_eq_o),Xs)
    <=> ! [I4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,I4)),size_size_list_o(Xs))
         => aa_o_o(aa_o_fun_o_o(ord_less_eq_o,aa_nat_o(nth_o(Xs),I4)),aa_nat_o(nth_o(Xs),aa_nat_nat(suc,I4))) ) ) ).

% sorted_iff_nth_Suc
tff(fact_6434_sorted__iff__nth__Suc,axiom,
    ! [Xs: list_rat] :
      ( sorted_wrt_rat(ord_less_eq_rat,Xs)
    <=> ! [I4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,I4)),size_size_list_rat(Xs))
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(nth_rat(Xs),I4)),aa_nat_rat(nth_rat(Xs),aa_nat_nat(suc,I4))) ) ) ).

% sorted_iff_nth_Suc
tff(fact_6435_sorted__iff__nth__Suc,axiom,
    ! [Xs: list_num] :
      ( sorted_wrt_num(ord_less_eq_num,Xs)
    <=> ! [I4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,I4)),size_size_list_num(Xs))
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(nth_num(Xs),I4)),aa_nat_num(nth_num(Xs),aa_nat_nat(suc,I4))) ) ) ).

% sorted_iff_nth_Suc
tff(fact_6436_sorted__iff__nth__Suc,axiom,
    ! [Xs: list_nat] :
      ( aa_list_nat_o(sorted_wrt_nat(ord_less_eq_nat),Xs)
    <=> ! [I4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,I4)),aa_list_nat_nat(size_size_list_nat,Xs))
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(nth_nat(Xs),I4)),aa_nat_nat(nth_nat(Xs),aa_nat_nat(suc,I4))) ) ) ).

% sorted_iff_nth_Suc
tff(fact_6437_sorted__iff__nth__Suc,axiom,
    ! [Xs: list_int] :
      ( sorted_wrt_int(ord_less_eq_int,Xs)
    <=> ! [I4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,I4)),size_size_list_int(Xs))
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(nth_int(Xs),I4)),aa_nat_int(nth_int(Xs),aa_nat_nat(suc,I4))) ) ) ).

% sorted_iff_nth_Suc
tff(fact_6438_sorted__nth__mono,axiom,
    ! [Xs: list_o,I2: nat,J: nat] :
      ( aa_list_o_o(sorted_wrt_o(ord_less_eq_o),Xs)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),J)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J),size_size_list_o(Xs))
         => aa_o_o(aa_o_fun_o_o(ord_less_eq_o,aa_nat_o(nth_o(Xs),I2)),aa_nat_o(nth_o(Xs),J)) ) ) ) ).

% sorted_nth_mono
tff(fact_6439_sorted__nth__mono,axiom,
    ! [Xs: list_rat,I2: nat,J: nat] :
      ( sorted_wrt_rat(ord_less_eq_rat,Xs)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),J)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J),size_size_list_rat(Xs))
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(nth_rat(Xs),I2)),aa_nat_rat(nth_rat(Xs),J)) ) ) ) ).

% sorted_nth_mono
tff(fact_6440_sorted__nth__mono,axiom,
    ! [Xs: list_num,I2: nat,J: nat] :
      ( sorted_wrt_num(ord_less_eq_num,Xs)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),J)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J),size_size_list_num(Xs))
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(nth_num(Xs),I2)),aa_nat_num(nth_num(Xs),J)) ) ) ) ).

% sorted_nth_mono
tff(fact_6441_sorted__nth__mono,axiom,
    ! [Xs: list_nat,I2: nat,J: nat] :
      ( aa_list_nat_o(sorted_wrt_nat(ord_less_eq_nat),Xs)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),J)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J),aa_list_nat_nat(size_size_list_nat,Xs))
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(nth_nat(Xs),I2)),aa_nat_nat(nth_nat(Xs),J)) ) ) ) ).

% sorted_nth_mono
tff(fact_6442_sorted__nth__mono,axiom,
    ! [Xs: list_int,I2: nat,J: nat] :
      ( sorted_wrt_int(ord_less_eq_int,Xs)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),J)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J),size_size_list_int(Xs))
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(nth_int(Xs),I2)),aa_nat_int(nth_int(Xs),J)) ) ) ) ).

% sorted_nth_mono
tff(fact_6443_sorted__iff__nth__mono,axiom,
    ! [Xs: list_o] :
      ( aa_list_o_o(sorted_wrt_o(ord_less_eq_o),Xs)
    <=> ! [I4: nat,J3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I4),J3)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J3),size_size_list_o(Xs))
           => aa_o_o(aa_o_fun_o_o(ord_less_eq_o,aa_nat_o(nth_o(Xs),I4)),aa_nat_o(nth_o(Xs),J3)) ) ) ) ).

% sorted_iff_nth_mono
tff(fact_6444_sorted__iff__nth__mono,axiom,
    ! [Xs: list_rat] :
      ( sorted_wrt_rat(ord_less_eq_rat,Xs)
    <=> ! [I4: nat,J3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I4),J3)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J3),size_size_list_rat(Xs))
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(nth_rat(Xs),I4)),aa_nat_rat(nth_rat(Xs),J3)) ) ) ) ).

% sorted_iff_nth_mono
tff(fact_6445_sorted__iff__nth__mono,axiom,
    ! [Xs: list_num] :
      ( sorted_wrt_num(ord_less_eq_num,Xs)
    <=> ! [I4: nat,J3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I4),J3)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J3),size_size_list_num(Xs))
           => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(nth_num(Xs),I4)),aa_nat_num(nth_num(Xs),J3)) ) ) ) ).

% sorted_iff_nth_mono
tff(fact_6446_sorted__iff__nth__mono,axiom,
    ! [Xs: list_nat] :
      ( aa_list_nat_o(sorted_wrt_nat(ord_less_eq_nat),Xs)
    <=> ! [I4: nat,J3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I4),J3)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J3),aa_list_nat_nat(size_size_list_nat,Xs))
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(nth_nat(Xs),I4)),aa_nat_nat(nth_nat(Xs),J3)) ) ) ) ).

% sorted_iff_nth_mono
tff(fact_6447_sorted__iff__nth__mono,axiom,
    ! [Xs: list_int] :
      ( sorted_wrt_int(ord_less_eq_int,Xs)
    <=> ! [I4: nat,J3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I4),J3)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J3),size_size_list_int(Xs))
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(nth_int(Xs),I4)),aa_nat_int(nth_int(Xs),J3)) ) ) ) ).

% sorted_iff_nth_mono
tff(fact_6448_prod_Oub__add__nat,axiom,
    ! [M: nat,N2: nat,G: fun_nat_Code_integer,P2: 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,N2),one_one_nat))
     => ( aa_set871585252387937313nteger(groups3455450783089532116nteger(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),P2))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_set871585252387937313nteger(groups3455450783089532116nteger(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M),N2))),aa_set871585252387937313nteger(groups3455450783089532116nteger(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),P2)))) ) ) ).

% prod.ub_add_nat
tff(fact_6449_prod_Oub__add__nat,axiom,
    ! [M: nat,N2: nat,G: fun_nat_assn,P2: 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,N2),one_one_nat))
     => ( groups6906906614972039071t_assn(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),P2))) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,groups6906906614972039071t_assn(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M),N2))),groups6906906614972039071t_assn(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),P2)))) ) ) ).

% prod.ub_add_nat
tff(fact_6450_prod_Oub__add__nat,axiom,
    ! [M: nat,N2: nat,G: fun_nat_rat,P2: 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,N2),one_one_nat))
     => ( aa_set_nat_rat(groups73079841787564623at_rat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),P2))) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_set_nat_rat(groups73079841787564623at_rat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M),N2))),aa_set_nat_rat(groups73079841787564623at_rat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),P2)))) ) ) ).

% prod.ub_add_nat
tff(fact_6451_prod_Oub__add__nat,axiom,
    ! [M: nat,N2: nat,G: fun_nat_nat,P2: 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,N2),one_one_nat))
     => ( aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),P2))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M),N2))),aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),P2)))) ) ) ).

% prod.ub_add_nat
tff(fact_6452_prod_Oub__add__nat,axiom,
    ! [M: nat,N2: nat,G: fun_nat_int,P2: 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,N2),one_one_nat))
     => ( aa_set_nat_int(groups705719431365010083at_int(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),P2))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_set_nat_int(groups705719431365010083at_int(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M),N2))),aa_set_nat_int(groups705719431365010083at_int(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),P2)))) ) ) ).

% prod.ub_add_nat
tff(fact_6453_gbinomial__parallel__sum,axiom,
    ! [A: rat,N2: nat] : aa_set_nat_rat(groups2906978787729119204at_rat(aTP_Lamm_oj(A)),aa_nat_set_nat(set_ord_atMost_nat,N2)) = 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,N2))),one_one_rat)),N2) ).

% gbinomial_parallel_sum
tff(fact_6454_fact__eq__fact__times,axiom,
    ! [N2: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),M)
     => ( aa_nat_nat(semiri1408675320244567234ct_nat,M) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1408675320244567234ct_nat,N2)),aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_fu),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,aa_nat_nat(suc,N2)),M))) ) ) ).

% fact_eq_fact_times
tff(fact_6455_sum__gp__basic,axiom,
    ! [X: code_integer,N2: 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,N2))) = aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(one_one_Code_integer),aa_nat_Code_integer(power_8256067586552552935nteger(X),aa_nat_nat(suc,N2))) ).

% sum_gp_basic
tff(fact_6456_sum__gp__basic,axiom,
    ! [X: rat,N2: 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,N2))) = 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,N2))) ).

% sum_gp_basic
tff(fact_6457_sum__gp__basic,axiom,
    ! [X: int,N2: 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,N2))) = 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,N2))) ).

% sum_gp_basic
tff(fact_6458_pochhammer__Suc__prod,axiom,
    ! [A: rat,N2: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(A),aa_nat_nat(suc,N2)) = aa_set_nat_rat(groups73079841787564623at_rat(aTP_Lamm_ok(A)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,zero_zero_nat),N2)) ).

% pochhammer_Suc_prod
tff(fact_6459_pochhammer__Suc__prod,axiom,
    ! [A: code_integer,N2: nat] : aa_nat_Code_integer(comm_s8582702949713902594nteger(A),aa_nat_nat(suc,N2)) = aa_set871585252387937313nteger(groups3455450783089532116nteger(aTP_Lamm_ol(A)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,zero_zero_nat),N2)) ).

% pochhammer_Suc_prod
tff(fact_6460_pochhammer__Suc__prod,axiom,
    ! [A: nat,N2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,A),aa_nat_nat(suc,N2)) = aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_om,A)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,zero_zero_nat),N2)) ).

% pochhammer_Suc_prod
tff(fact_6461_pochhammer__Suc__prod,axiom,
    ! [A: int,N2: nat] : aa_nat_int(aa_int_fun_nat_int(comm_s4660882817536571857er_int,A),aa_nat_nat(suc,N2)) = aa_set_nat_int(groups705719431365010083at_int(aa_int_fun_nat_int(aTP_Lamm_on,A)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,zero_zero_nat),N2)) ).

% pochhammer_Suc_prod
tff(fact_6462_sum__power__shift,axiom,
    ! [M: nat,N2: nat,X: code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
     => ( aa_set871585252387937313nteger(groups7501900531339628137nteger(power_8256067586552552935nteger(X)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M),N2)) = 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,N2),M)))) ) ) ).

% sum_power_shift
tff(fact_6463_sum__power__shift,axiom,
    ! [M: nat,N2: nat,X: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
     => ( aa_set_nat_rat(groups2906978787729119204at_rat(power_power_rat(X)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M),N2)) = 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,N2),M)))) ) ) ).

% sum_power_shift
tff(fact_6464_sum__power__shift,axiom,
    ! [M: nat,N2: nat,X: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
     => ( aa_set_nat_int(groups3539618377306564664at_int(aa_int_fun_nat_int(power_power_int,X)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M),N2)) = 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,N2),M)))) ) ) ).

% sum_power_shift
tff(fact_6465_pochhammer__prod__rev,axiom,
    ! [A: rat,N2: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(A),N2) = aa_set_nat_rat(groups73079841787564623at_rat(aa_nat_fun_nat_rat(aTP_Lamm_oo(A),N2)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,one_one_nat),N2)) ).

% pochhammer_prod_rev
tff(fact_6466_pochhammer__prod__rev,axiom,
    ! [A: code_integer,N2: nat] : aa_nat_Code_integer(comm_s8582702949713902594nteger(A),N2) = aa_set871585252387937313nteger(groups3455450783089532116nteger(aa_nat2010205248571845816nteger(aTP_Lamm_op(A),N2)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,one_one_nat),N2)) ).

% pochhammer_prod_rev
tff(fact_6467_pochhammer__prod__rev,axiom,
    ! [A: nat,N2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,A),N2) = aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_oq(A),N2)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,one_one_nat),N2)) ).

% pochhammer_prod_rev
tff(fact_6468_pochhammer__prod__rev,axiom,
    ! [A: int,N2: nat] : aa_nat_int(aa_int_fun_nat_int(comm_s4660882817536571857er_int,A),N2) = aa_set_nat_int(groups705719431365010083at_int(aa_nat_fun_nat_int(aTP_Lamm_or(A),N2)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,one_one_nat),N2)) ).

% pochhammer_prod_rev
tff(fact_6469_fact__div__fact,axiom,
    ! [N2: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),M)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(semiri1408675320244567234ct_nat,M)),aa_nat_nat(semiri1408675320244567234ct_nat,N2)) = aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_fu),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),one_one_nat)),M)) ) ) ).

% fact_div_fact
tff(fact_6470_gbinomial__sum__lower__neg,axiom,
    ! [A: rat,M: nat] : aa_set_nat_rat(groups2906978787729119204at_rat(aTP_Lamm_os(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_6471_pochhammer__Suc__prod__rev,axiom,
    ! [A: rat,N2: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(A),aa_nat_nat(suc,N2)) = aa_set_nat_rat(groups73079841787564623at_rat(aa_nat_fun_nat_rat(aTP_Lamm_oo(A),N2)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,zero_zero_nat),N2)) ).

% pochhammer_Suc_prod_rev
tff(fact_6472_pochhammer__Suc__prod__rev,axiom,
    ! [A: code_integer,N2: nat] : aa_nat_Code_integer(comm_s8582702949713902594nteger(A),aa_nat_nat(suc,N2)) = aa_set871585252387937313nteger(groups3455450783089532116nteger(aa_nat2010205248571845816nteger(aTP_Lamm_op(A),N2)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,zero_zero_nat),N2)) ).

% pochhammer_Suc_prod_rev
tff(fact_6473_pochhammer__Suc__prod__rev,axiom,
    ! [A: nat,N2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,A),aa_nat_nat(suc,N2)) = aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_oq(A),N2)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,zero_zero_nat),N2)) ).

% pochhammer_Suc_prod_rev
tff(fact_6474_pochhammer__Suc__prod__rev,axiom,
    ! [A: int,N2: nat] : aa_nat_int(aa_int_fun_nat_int(comm_s4660882817536571857er_int,A),aa_nat_nat(suc,N2)) = aa_set_nat_int(groups705719431365010083at_int(aa_nat_fun_nat_int(aTP_Lamm_or(A),N2)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,zero_zero_nat),N2)) ).

% pochhammer_Suc_prod_rev
tff(fact_6475_sum_Ozero__middle,axiom,
    ! [P2: nat,K: nat,G: fun_nat_Code_integer,Ha: fun_nat_Code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),P2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),P2)
       => ( aa_set871585252387937313nteger(groups7501900531339628137nteger(aa_fun589076786913273746nteger(aa_fun3918145391673965625nteger(aTP_Lamm_ot(K),G),Ha)),aa_nat_set_nat(set_ord_atMost_nat,P2)) = aa_set871585252387937313nteger(groups7501900531339628137nteger(aa_fun589076786913273746nteger(aa_fun3918145391673965625nteger(aTP_Lamm_nt(K),G),Ha)),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,P2),aa_nat_nat(suc,zero_zero_nat)))) ) ) ) ).

% sum.zero_middle
tff(fact_6476_sum_Ozero__middle,axiom,
    ! [P2: nat,K: nat,G: fun_na2338080690716861136at_nat,Ha: fun_na2338080690716861136at_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),P2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),P2)
       => ( groups6857163185585827899at_nat(aa_fun4141531684813184444at_nat(aa_fun8230936523094545239at_nat(aTP_Lamm_ou(K),G),Ha),aa_nat_set_nat(set_ord_atMost_nat,P2)) = groups6857163185585827899at_nat(aa_fun4141531684813184444at_nat(aa_fun8230936523094545239at_nat(aTP_Lamm_ov(K),G),Ha),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,P2),aa_nat_nat(suc,zero_zero_nat)))) ) ) ) ).

% sum.zero_middle
tff(fact_6477_sum_Ozero__middle,axiom,
    ! [P2: nat,K: nat,G: fun_nat_rat,Ha: fun_nat_rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),P2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),P2)
       => ( aa_set_nat_rat(groups2906978787729119204at_rat(aa_fun765758790203053448at_rat(aa_fun3638354034882855594at_rat(aTP_Lamm_ow(K),G),Ha)),aa_nat_set_nat(set_ord_atMost_nat,P2)) = aa_set_nat_rat(groups2906978787729119204at_rat(aa_fun765758790203053448at_rat(aa_fun3638354034882855594at_rat(aTP_Lamm_nx(K),G),Ha)),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,P2),aa_nat_nat(suc,zero_zero_nat)))) ) ) ) ).

% sum.zero_middle
tff(fact_6478_sum_Ozero__middle,axiom,
    ! [P2: nat,K: nat,G: fun_nat_int,Ha: fun_nat_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),P2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),P2)
       => ( aa_set_nat_int(groups3539618377306564664at_int(aa_fun5565043122120104496at_int(aa_fun5140708168904641446at_int(aTP_Lamm_ox(K),G),Ha)),aa_nat_set_nat(set_ord_atMost_nat,P2)) = aa_set_nat_int(groups3539618377306564664at_int(aa_fun5565043122120104496at_int(aa_fun5140708168904641446at_int(aTP_Lamm_ob(K),G),Ha)),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,P2),aa_nat_nat(suc,zero_zero_nat)))) ) ) ) ).

% sum.zero_middle
tff(fact_6479_sum_Ozero__middle,axiom,
    ! [P2: nat,K: nat,G: fun_nat_nat,Ha: fun_nat_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),P2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),P2)
       => ( aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(aTP_Lamm_oy(K),G),Ha)),aa_nat_set_nat(set_ord_atMost_nat,P2)) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(aTP_Lamm_nz(K),G),Ha)),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,P2),aa_nat_nat(suc,zero_zero_nat)))) ) ) ) ).

% sum.zero_middle
tff(fact_6480_choose__alternating__sum,axiom,
    ! [N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)
     => ( aa_set_nat_rat(groups2906978787729119204at_rat(aTP_Lamm_oz(N2)),aa_nat_set_nat(set_ord_atMost_nat,N2)) = zero_zero_rat ) ) ).

% choose_alternating_sum
tff(fact_6481_choose__alternating__sum,axiom,
    ! [N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)
     => ( aa_set_nat_int(groups3539618377306564664at_int(aa_nat_fun_nat_int(aTP_Lamm_pa,N2)),aa_nat_set_nat(set_ord_atMost_nat,N2)) = zero_zero_int ) ) ).

% choose_alternating_sum
tff(fact_6482_choose__alternating__sum,axiom,
    ! [N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)
     => ( aa_set871585252387937313nteger(groups7501900531339628137nteger(aTP_Lamm_pb(N2)),aa_nat_set_nat(set_ord_atMost_nat,N2)) = zero_z3403309356797280102nteger ) ) ).

% choose_alternating_sum
tff(fact_6483_choose__alternating__linear__sum,axiom,
    ! [N2: nat] :
      ( ( N2 != one_one_nat )
     => ( aa_set_nat_rat(groups2906978787729119204at_rat(aTP_Lamm_pc(N2)),aa_nat_set_nat(set_ord_atMost_nat,N2)) = zero_zero_rat ) ) ).

% choose_alternating_linear_sum
tff(fact_6484_choose__alternating__linear__sum,axiom,
    ! [N2: nat] :
      ( ( N2 != one_one_nat )
     => ( aa_set_nat_int(groups3539618377306564664at_int(aa_nat_fun_nat_int(aTP_Lamm_pd,N2)),aa_nat_set_nat(set_ord_atMost_nat,N2)) = zero_zero_int ) ) ).

% choose_alternating_linear_sum
tff(fact_6485_choose__alternating__linear__sum,axiom,
    ! [N2: nat] :
      ( ( N2 != one_one_nat )
     => ( aa_set871585252387937313nteger(groups7501900531339628137nteger(aTP_Lamm_pe(N2)),aa_nat_set_nat(set_ord_atMost_nat,N2)) = zero_z3403309356797280102nteger ) ) ).

% choose_alternating_linear_sum
tff(fact_6486_pochhammer__times__pochhammer__half,axiom,
    ! [Z: rat,N2: nat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(comm_s4028243227959126397er_rat(Z),aa_nat_nat(suc,N2))),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),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))))),aa_nat_nat(suc,N2))) = aa_set_nat_rat(groups73079841787564623at_rat(aTP_Lamm_pf(Z)),aa_nat_set_nat(aa_nat1818925681583257441et_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,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2)),one_one_nat))) ).

% pochhammer_times_pochhammer_half
tff(fact_6487_sum__zero__power_H,axiom,
    ! [Ca: fun_nat_rat,D2: fun_nat_rat,A3: set_nat] :
      aa_set_nat_rat(groups2906978787729119204at_rat(aa_fun765758790203053448at_rat(aTP_Lamm_pg(Ca),D2)),A3) = $ite(
        ( aa_set_nat_o(finite_finite_nat,A3)
        & aa_set_nat_o(member_nat(zero_zero_nat),A3) ),
        aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_nat_rat(Ca,zero_zero_nat)),aa_nat_rat(D2,zero_zero_nat)),
        zero_zero_rat ) ).

% sum_zero_power'
tff(fact_6488_nth__enumerate__eq,axiom,
    ! [M: nat,Xs: list_int,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),size_size_list_int(Xs))
     => ( nth_Pr3440142176431000676at_int(enumerate_int(N2,Xs),M) = product_Pair_nat_int(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),M),aa_nat_int(nth_int(Xs),M)) ) ) ).

% nth_enumerate_eq
tff(fact_6489_nth__enumerate__eq,axiom,
    ! [M: nat,Xs: list_nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),aa_list_nat_nat(size_size_list_nat,Xs))
     => ( aa_nat7603221163087113075at_nat(nth_Pr7617993195940197384at_nat(enumerate_nat(N2,Xs)),M) = aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),M)),aa_nat_nat(nth_nat(Xs),M)) ) ) ).

% nth_enumerate_eq
tff(fact_6490_nth__enumerate__eq,axiom,
    ! [M: nat,Xs: list_o,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),size_size_list_o(Xs))
     => ( nth_Pr112076138515278198_nat_o(enumerate_o(N2,Xs),M) = product_Pair_nat_o(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),M),aa_nat_o(nth_o(Xs),M)) ) ) ).

% nth_enumerate_eq
tff(fact_6491_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_md(A)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),K))),aa_nat_rat(semiri773545260158071498ct_rat,K)) ).

% gbinomial_prod_rev
tff(fact_6492_gbinomial__prod__rev,axiom,
    ! [A: code_natural,K: nat] : aa_nat_Code_natural(gbinom7368847122466276068atural(A),K) = divide5121882707175180666atural(groups2279045934846249631atural(aTP_Lamm_me(A),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),K)),aa_nat_Code_natural(semiri2447717529341329178atural,K)) ).

% gbinomial_prod_rev
tff(fact_6493_gbinomial__prod__rev,axiom,
    ! [A: code_integer,K: nat] : aa_nat_Code_integer(gbinom8545251970709558553nteger(A),K) = aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_set871585252387937313nteger(groups3455450783089532116nteger(aTP_Lamm_mf(A)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),K))),aa_nat_Code_integer(semiri3624122377584611663nteger,K)) ).

% gbinomial_prod_rev
tff(fact_6494_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_mg,A)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),K))),aa_nat_nat(semiri1408675320244567234ct_nat,K)) ).

% gbinomial_prod_rev
tff(fact_6495_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_mh,A)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),K))),aa_nat_int(semiri1406184849735516958ct_int,K)) ).

% gbinomial_prod_rev
tff(fact_6496_semiring__norm_I78_J,axiom,
    ! [M: num,N2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,aa_num_num(bit0,M)),aa_num_num(bit0,N2))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,M),N2) ) ).

% semiring_norm(78)
tff(fact_6497_semiring__norm_I71_J,axiom,
    ! [M: num,N2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(bit0,M)),aa_num_num(bit0,N2))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,M),N2) ) ).

% semiring_norm(71)
tff(fact_6498_semiring__norm_I75_J,axiom,
    ! [M: num] : ~ aa_num_o(aa_num_fun_num_o(ord_less_num,M),one) ).

% semiring_norm(75)
tff(fact_6499_semiring__norm_I68_J,axiom,
    ! [N2: num] : aa_num_o(aa_num_fun_num_o(ord_less_eq_num,one),N2) ).

% semiring_norm(68)
tff(fact_6500_atLeastLessThan__iff,axiom,
    ! [I2: $o,L: $o,U: $o] :
      ( member_o((I2),set_or7139685690850216873Than_o((L),(U)))
    <=> ( aa_o_o(aa_o_fun_o_o(ord_less_eq_o,(L)),(I2))
        & aa_o_o(aa_o_fun_o_o(ord_less_o,(I2)),(U)) ) ) ).

% atLeastLessThan_iff
tff(fact_6501_atLeastLessThan__iff,axiom,
    ! [I2: set_nat,L: set_nat,U: set_nat] :
      ( aa_set_set_nat_o(member_set_nat(I2),set_or3540276404033026485et_nat(L,U))
    <=> ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,L),I2)
        & aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,I2),U) ) ) ).

% atLeastLessThan_iff
tff(fact_6502_atLeastLessThan__iff,axiom,
    ! [I2: assn,L: assn,U: assn] :
      ( member_assn(I2,set_or5523502641901747543n_assn(L,U))
    <=> ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,L),I2)
        & aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,I2),U) ) ) ).

% atLeastLessThan_iff
tff(fact_6503_atLeastLessThan__iff,axiom,
    ! [I2: set_int,L: set_int,U: set_int] :
      ( member_set_int(I2,set_or8585797421378605585et_int(L,U))
    <=> ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,L),I2)
        & aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_set_int,I2),U) ) ) ).

% atLeastLessThan_iff
tff(fact_6504_atLeastLessThan__iff,axiom,
    ! [I2: rat,L: rat,U: rat] :
      ( member_rat(I2,set_or4029947393144176647an_rat(L,U))
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,L),I2)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,I2),U) ) ) ).

% atLeastLessThan_iff
tff(fact_6505_atLeastLessThan__iff,axiom,
    ! [I2: num,L: num,U: num] :
      ( member_num(I2,set_or1222409239386451017an_num(L,U))
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,L),I2)
        & aa_num_o(aa_num_fun_num_o(ord_less_num,I2),U) ) ) ).

% atLeastLessThan_iff
tff(fact_6506_atLeastLessThan__iff,axiom,
    ! [I2: nat,L: nat,U: nat] :
      ( aa_set_nat_o(member_nat(I2),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,L),U))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,L),I2)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),U) ) ) ).

% atLeastLessThan_iff
tff(fact_6507_atLeastLessThan__iff,axiom,
    ! [I2: int,L: int,U: int] :
      ( aa_set_int_o(member_int(I2),set_or4662586982721622107an_int(L,U))
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,L),I2)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,I2),U) ) ) ).

% atLeastLessThan_iff
tff(fact_6508_atLeastLessThan__iff,axiom,
    ! [I2: code_integer,L: code_integer,U: code_integer] :
      ( aa_set5643387332196575111eger_o(member_Code_integer(I2),set_or8404916559141939852nteger(L,U))
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,L),I2)
        & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,I2),U) ) ) ).

% atLeastLessThan_iff
tff(fact_6509_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_6510_atLeastLessThan__empty,axiom,
    ! [B: set_int,A: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,B),A)
     => ( set_or8585797421378605585et_int(A,B) = bot_bot_set_set_int ) ) ).

% atLeastLessThan_empty
tff(fact_6511_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_6512_atLeastLessThan__empty,axiom,
    ! [B: num,A: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),A)
     => ( set_or1222409239386451017an_num(A,B) = bot_bot_set_num ) ) ).

% atLeastLessThan_empty
tff(fact_6513_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(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,A),B) = bot_bot_set_nat ) ) ).

% atLeastLessThan_empty
tff(fact_6514_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_6515_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_6516_infinite__Icc__iff,axiom,
    ! [A: rat,B: rat] :
      ( ~ finite_finite_rat(set_or633870826150836451st_rat(A,B))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ).

% infinite_Icc_iff
tff(fact_6517_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_6518_atLeastLessThan__empty__iff2,axiom,
    ! [A: assn,B: assn] :
      ( ( bot_bot_set_assn = set_or5523502641901747543n_assn(A,B) )
    <=> ~ aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),B) ) ).

% atLeastLessThan_empty_iff2
tff(fact_6519_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_6520_atLeastLessThan__empty__iff2,axiom,
    ! [A: num,B: num] :
      ( ( bot_bot_set_num = set_or1222409239386451017an_num(A,B) )
    <=> ~ aa_num_o(aa_num_fun_num_o(ord_less_num,A),B) ) ).

% atLeastLessThan_empty_iff2
tff(fact_6521_atLeastLessThan__empty__iff2,axiom,
    ! [A: nat,B: nat] :
      ( ( bot_bot_set_nat = aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,A),B) )
    <=> ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B) ) ).

% atLeastLessThan_empty_iff2
tff(fact_6522_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_6523_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_6524_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_6525_atLeastLessThan__empty__iff,axiom,
    ! [A: assn,B: assn] :
      ( ( set_or5523502641901747543n_assn(A,B) = bot_bot_set_assn )
    <=> ~ aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),B) ) ).

% atLeastLessThan_empty_iff
tff(fact_6526_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_6527_atLeastLessThan__empty__iff,axiom,
    ! [A: num,B: num] :
      ( ( set_or1222409239386451017an_num(A,B) = bot_bot_set_num )
    <=> ~ aa_num_o(aa_num_fun_num_o(ord_less_num,A),B) ) ).

% atLeastLessThan_empty_iff
tff(fact_6528_atLeastLessThan__empty__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( aa_nat_set_nat(aa_nat1818925681583257441et_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_6529_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_6530_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_6531_infinite__Ico__iff,axiom,
    ! [A: rat,B: rat] :
      ( ~ finite_finite_rat(set_or4029947393144176647an_rat(A,B))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ).

% infinite_Ico_iff
tff(fact_6532_ivl__subset,axiom,
    ! [I2: rat,J: rat,M: rat,N2: rat] :
      ( ord_less_eq_set_rat(set_or4029947393144176647an_rat(I2,J),set_or4029947393144176647an_rat(M,N2))
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,J),I2)
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,M),I2)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,J),N2) ) ) ) ).

% ivl_subset
tff(fact_6533_ivl__subset,axiom,
    ! [I2: num,J: num,M: num,N2: num] :
      ( ord_less_eq_set_num(set_or1222409239386451017an_num(I2,J),set_or1222409239386451017an_num(M,N2))
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,J),I2)
        | ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,M),I2)
          & aa_num_o(aa_num_fun_num_o(ord_less_eq_num,J),N2) ) ) ) ).

% ivl_subset
tff(fact_6534_ivl__subset,axiom,
    ! [I2: nat,J: nat,M: nat,N2: nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,I2),J)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,M),N2))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,J),I2)
        | ( 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,J),N2) ) ) ) ).

% ivl_subset
tff(fact_6535_ivl__subset,axiom,
    ! [I2: int,J: int,M: int,N2: int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,set_or4662586982721622107an_int(I2,J)),set_or4662586982721622107an_int(M,N2))
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,J),I2)
        | ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,M),I2)
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,J),N2) ) ) ) ).

% ivl_subset
tff(fact_6536_ivl__subset,axiom,
    ! [I2: code_integer,J: code_integer,M: code_integer,N2: code_integer] :
      ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(set_or8404916559141939852nteger(I2,J)),set_or8404916559141939852nteger(M,N2))
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,J),I2)
        | ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,M),I2)
          & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,J),N2) ) ) ) ).

% ivl_subset
tff(fact_6537_ivl__diff,axiom,
    ! [I2: rat,N2: rat,M: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,I2),N2)
     => ( minus_minus_set_rat(set_or4029947393144176647an_rat(I2,M),set_or4029947393144176647an_rat(I2,N2)) = set_or4029947393144176647an_rat(N2,M) ) ) ).

% ivl_diff
tff(fact_6538_ivl__diff,axiom,
    ! [I2: num,N2: num,M: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,I2),N2)
     => ( minus_minus_set_num(set_or1222409239386451017an_num(I2,M),set_or1222409239386451017an_num(I2,N2)) = set_or1222409239386451017an_num(N2,M) ) ) ).

% ivl_diff
tff(fact_6539_ivl__diff,axiom,
    ! [I2: nat,N2: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),N2)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,I2),M)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,I2),N2)) = aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,N2),M) ) ) ).

% ivl_diff
tff(fact_6540_ivl__diff,axiom,
    ! [I2: int,N2: int,M: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I2),N2)
     => ( aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,set_or4662586982721622107an_int(I2,M)),set_or4662586982721622107an_int(I2,N2)) = set_or4662586982721622107an_int(N2,M) ) ) ).

% ivl_diff
tff(fact_6541_ivl__diff,axiom,
    ! [I2: code_integer,N2: code_integer,M: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,I2),N2)
     => ( minus_2355218937544613996nteger(set_or8404916559141939852nteger(I2,M),set_or8404916559141939852nteger(I2,N2)) = set_or8404916559141939852nteger(N2,M) ) ) ).

% ivl_diff
tff(fact_6542_binomial__eq__0__iff,axiom,
    ! [N2: nat,K: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,N2),K) = zero_zero_nat )
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),K) ) ).

% binomial_eq_0_iff
tff(fact_6543_semiring__norm_I69_J,axiom,
    ! [M: num] : ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(bit0,M)),one) ).

% semiring_norm(69)
tff(fact_6544_semiring__norm_I76_J,axiom,
    ! [N2: num] : aa_num_o(aa_num_fun_num_o(ord_less_num,one),aa_num_num(bit0,N2)) ).

% semiring_norm(76)
tff(fact_6545_sum_Odelta_H,axiom,
    ! [S: set_o,A: $o,B: fun_o_Code_integer] :
      ( finite_finite_o(S)
     => ( groups4406642042086082107nteger(aa_fun8520278569782659900nteger(aTP_Lamm_ph((A)),B),S) = $ite(member_o((A),S),aa_o_Code_integer(B,(A)),zero_z3403309356797280102nteger) ) ) ).

% sum.delta'
tff(fact_6546_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_pi(A),B)),S) = $ite(aa_set_nat_o(member_nat(A),S),aa_nat_Code_integer(B,A),zero_z3403309356797280102nteger) ) ) ).

% sum.delta'
tff(fact_6547_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_pj(A),B),S) = $ite(aa_set_int_o(member_int(A),S),aa_int_Code_integer(B,A),zero_z3403309356797280102nteger) ) ) ).

% sum.delta'
tff(fact_6548_sum_Odelta_H,axiom,
    ! [S: set_Code_integer,A: code_integer,B: fun_Co8499238673658658103nteger] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,S)
     => ( groups879477027807139574nteger(aa_fun4435975922079263404nteger(aTP_Lamm_pk(A),B),S) = $ite(aa_set5643387332196575111eger_o(member_Code_integer(A),S),aa_Cod1783974652533396728nteger(B,A),zero_z3403309356797280102nteger) ) ) ).

% sum.delta'
tff(fact_6549_sum_Odelta_H,axiom,
    ! [S: set_o,A: $o,B: fun_o_rat] :
      ( finite_finite_o(S)
     => ( groups7872700643590313910_o_rat(aa_fun4336163974892599420_o_rat(aTP_Lamm_pl((A)),B),S) = $ite(member_o((A),S),aa_o_rat(B,(A)),zero_zero_rat) ) ) ).

% sum.delta'
tff(fact_6550_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_pm(A),B)),S) = $ite(aa_set_nat_o(member_nat(A),S),aa_nat_rat(B,A),zero_zero_rat) ) ) ).

% sum.delta'
tff(fact_6551_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_pn(A),B),S) = $ite(aa_set_int_o(member_int(A),S),aa_int_rat(B,A),zero_zero_rat) ) ) ).

% sum.delta'
tff(fact_6552_sum_Odelta_H,axiom,
    ! [S: set_Code_integer,A: code_integer,B: fun_Code_integer_rat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,S)
     => ( groups6602215022474089585er_rat(aa_fun8169474305930283170er_rat(aTP_Lamm_po(A),B),S) = $ite(aa_set5643387332196575111eger_o(member_Code_integer(A),S),aa_Code_integer_rat(B,A),zero_zero_rat) ) ) ).

% sum.delta'
tff(fact_6553_sum_Odelta_H,axiom,
    ! [S: set_o,A: $o,B: fun_o_nat] :
      ( finite_finite_o(S)
     => ( groups8507830703676809646_o_nat(aa_fun6176012476734143100_o_nat(aTP_Lamm_pp((A)),B),S) = $ite(member_o((A),S),aa_o_nat(B,(A)),zero_zero_nat) ) ) ).

% sum.delta'
tff(fact_6554_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_pq(A),B),S) = $ite(aa_set_int_o(member_int(A),S),aa_int_nat(B,A),zero_zero_nat) ) ) ).

% sum.delta'
tff(fact_6555_sum_Odelta,axiom,
    ! [S: set_o,A: $o,B: fun_o_Code_integer] :
      ( finite_finite_o(S)
     => ( groups4406642042086082107nteger(aa_fun8520278569782659900nteger(aTP_Lamm_pr((A)),B),S) = $ite(member_o((A),S),aa_o_Code_integer(B,(A)),zero_z3403309356797280102nteger) ) ) ).

% sum.delta
tff(fact_6556_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_ps(A),B)),S) = $ite(aa_set_nat_o(member_nat(A),S),aa_nat_Code_integer(B,A),zero_z3403309356797280102nteger) ) ) ).

% sum.delta
tff(fact_6557_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_pt(A),B),S) = $ite(aa_set_int_o(member_int(A),S),aa_int_Code_integer(B,A),zero_z3403309356797280102nteger) ) ) ).

% sum.delta
tff(fact_6558_sum_Odelta,axiom,
    ! [S: set_Code_integer,A: code_integer,B: fun_Co8499238673658658103nteger] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,S)
     => ( groups879477027807139574nteger(aa_fun4435975922079263404nteger(aTP_Lamm_pu(A),B),S) = $ite(aa_set5643387332196575111eger_o(member_Code_integer(A),S),aa_Cod1783974652533396728nteger(B,A),zero_z3403309356797280102nteger) ) ) ).

% sum.delta
tff(fact_6559_sum_Odelta,axiom,
    ! [S: set_o,A: $o,B: fun_o_rat] :
      ( finite_finite_o(S)
     => ( groups7872700643590313910_o_rat(aa_fun4336163974892599420_o_rat(aTP_Lamm_pv((A)),B),S) = $ite(member_o((A),S),aa_o_rat(B,(A)),zero_zero_rat) ) ) ).

% sum.delta
tff(fact_6560_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_pw(A),B)),S) = $ite(aa_set_nat_o(member_nat(A),S),aa_nat_rat(B,A),zero_zero_rat) ) ) ).

% sum.delta
tff(fact_6561_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_px(A),B),S) = $ite(aa_set_int_o(member_int(A),S),aa_int_rat(B,A),zero_zero_rat) ) ) ).

% sum.delta
tff(fact_6562_sum_Odelta,axiom,
    ! [S: set_Code_integer,A: code_integer,B: fun_Code_integer_rat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,S)
     => ( groups6602215022474089585er_rat(aa_fun8169474305930283170er_rat(aTP_Lamm_py(A),B),S) = $ite(aa_set5643387332196575111eger_o(member_Code_integer(A),S),aa_Code_integer_rat(B,A),zero_zero_rat) ) ) ).

% sum.delta
tff(fact_6563_sum_Odelta,axiom,
    ! [S: set_o,A: $o,B: fun_o_nat] :
      ( finite_finite_o(S)
     => ( groups8507830703676809646_o_nat(aa_fun6176012476734143100_o_nat(aTP_Lamm_pz((A)),B),S) = $ite(member_o((A),S),aa_o_nat(B,(A)),zero_zero_nat) ) ) ).

% sum.delta
tff(fact_6564_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_qa(A),B),S) = $ite(aa_set_int_o(member_int(A),S),aa_int_nat(B,A),zero_zero_nat) ) ) ).

% sum.delta
tff(fact_6565_image__add__atLeastLessThan_H,axiom,
    ! [K: rat,I2: rat,J: rat] : image_rat_rat(aa_rat_fun_rat_rat(aTP_Lamm_hs,K),set_or4029947393144176647an_rat(I2,J)) = set_or4029947393144176647an_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,I2),K),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,J),K)) ).

% image_add_atLeastLessThan'
tff(fact_6566_image__add__atLeastLessThan_H,axiom,
    ! [K: nat,I2: nat,J: nat] : image_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_hu,K),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,I2),J)) = aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K)) ).

% image_add_atLeastLessThan'
tff(fact_6567_image__add__atLeastLessThan_H,axiom,
    ! [K: int,I2: int,J: int] : image_int_int(aa_int_fun_int_int(aTP_Lamm_ht,K),set_or4662586982721622107an_int(I2,J)) = set_or4662586982721622107an_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,I2),K),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),K)) ).

% image_add_atLeastLessThan'
tff(fact_6568_image__add__atLeastLessThan_H,axiom,
    ! [K: code_integer,I2: code_integer,J: code_integer] : image_4470545334726330049nteger(aTP_Lamm_hv(K),set_or8404916559141939852nteger(I2,J)) = set_or8404916559141939852nteger(aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(I2),K),aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(J),K)) ).

% image_add_atLeastLessThan'
tff(fact_6569_prod_Odelta_H,axiom,
    ! [S: set_o,A: $o,B: fun_o_Code_integer] :
      ( finite_finite_o(S)
     => ( groups7694694392188491536nteger(aa_fun8520278569782659900nteger(aTP_Lamm_qb((A)),B),S) = $ite(member_o((A),S),aa_o_Code_integer(B,(A)),one_one_Code_integer) ) ) ).

% prod.delta'
tff(fact_6570_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_qc(A),B)),S) = $ite(aa_set_nat_o(member_nat(A),S),aa_nat_Code_integer(B,A),one_one_Code_integer) ) ) ).

% prod.delta'
tff(fact_6571_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_qd(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_6572_prod_Odelta_H,axiom,
    ! [S: set_Code_integer,A: code_integer,B: fun_Co8499238673658658103nteger] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,S)
     => ( groups3674199335183972705nteger(aa_fun4435975922079263404nteger(aTP_Lamm_qe(A),B),S) = $ite(aa_set5643387332196575111eger_o(member_Code_integer(A),S),aa_Cod1783974652533396728nteger(B,A),one_one_Code_integer) ) ) ).

% prod.delta'
tff(fact_6573_prod_Odelta_H,axiom,
    ! [S: set_o,A: $o,B: fun_o_assn] :
      ( finite_finite_o(S)
     => ( groups5301882518646026715o_assn(aa_fun4536517337817438332o_assn(aTP_Lamm_qf((A)),B),S) = $ite(member_o((A),S),aa_o_assn(B,(A)),one_one_assn) ) ) ).

% prod.delta'
tff(fact_6574_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_qg(A),B),S) = $ite(aa_set_nat_o(member_nat(A),S),aa_nat_assn(B,A),one_one_assn) ) ) ).

% prod.delta'
tff(fact_6575_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_qh(A),B),S) = $ite(aa_set_int_o(member_int(A),S),aa_int_assn(B,A),one_one_assn) ) ) ).

% prod.delta'
tff(fact_6576_prod_Odelta_H,axiom,
    ! [S: set_Code_integer,A: code_integer,B: fun_Co4490034186337852034r_assn] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,S)
     => ( groups1304777262505850412r_assn(aa_fun4119713415300323650r_assn(aTP_Lamm_qi(A),B),S) = $ite(aa_set5643387332196575111eger_o(member_Code_integer(A),S),aa_Code_integer_assn(B,A),one_one_assn) ) ) ).

% prod.delta'
tff(fact_6577_prod_Odelta_H,axiom,
    ! [S: set_o,A: $o,B: fun_o_rat] :
      ( finite_finite_o(S)
     => ( groups2869687844427037835_o_rat(aa_fun4336163974892599420_o_rat(aTP_Lamm_qj((A)),B),S) = $ite(member_o((A),S),aa_o_rat(B,(A)),one_one_rat) ) ) ).

% prod.delta'
tff(fact_6578_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_qk(A),B)),S) = $ite(aa_set_nat_o(member_nat(A),S),aa_nat_rat(B,A),one_one_rat) ) ) ).

% prod.delta'
tff(fact_6579_prod_Odelta,axiom,
    ! [S: set_o,A: $o,B: fun_o_Code_integer] :
      ( finite_finite_o(S)
     => ( groups7694694392188491536nteger(aa_fun8520278569782659900nteger(aTP_Lamm_ql((A)),B),S) = $ite(member_o((A),S),aa_o_Code_integer(B,(A)),one_one_Code_integer) ) ) ).

% prod.delta
tff(fact_6580_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_qm(A),B)),S) = $ite(aa_set_nat_o(member_nat(A),S),aa_nat_Code_integer(B,A),one_one_Code_integer) ) ) ).

% prod.delta
tff(fact_6581_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_qn(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_6582_prod_Odelta,axiom,
    ! [S: set_Code_integer,A: code_integer,B: fun_Co8499238673658658103nteger] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,S)
     => ( groups3674199335183972705nteger(aa_fun4435975922079263404nteger(aTP_Lamm_qo(A),B),S) = $ite(aa_set5643387332196575111eger_o(member_Code_integer(A),S),aa_Cod1783974652533396728nteger(B,A),one_one_Code_integer) ) ) ).

% prod.delta
tff(fact_6583_prod_Odelta,axiom,
    ! [S: set_o,A: $o,B: fun_o_assn] :
      ( finite_finite_o(S)
     => ( groups5301882518646026715o_assn(aa_fun4536517337817438332o_assn(aTP_Lamm_qp((A)),B),S) = $ite(member_o((A),S),aa_o_assn(B,(A)),one_one_assn) ) ) ).

% prod.delta
tff(fact_6584_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_qq(A),B),S) = $ite(aa_set_nat_o(member_nat(A),S),aa_nat_assn(B,A),one_one_assn) ) ) ).

% prod.delta
tff(fact_6585_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_qr(A),B),S) = $ite(aa_set_int_o(member_int(A),S),aa_int_assn(B,A),one_one_assn) ) ) ).

% prod.delta
tff(fact_6586_prod_Odelta,axiom,
    ! [S: set_Code_integer,A: code_integer,B: fun_Co4490034186337852034r_assn] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,S)
     => ( groups1304777262505850412r_assn(aa_fun4119713415300323650r_assn(aTP_Lamm_qs(A),B),S) = $ite(aa_set5643387332196575111eger_o(member_Code_integer(A),S),aa_Code_integer_assn(B,A),one_one_assn) ) ) ).

% prod.delta
tff(fact_6587_prod_Odelta,axiom,
    ! [S: set_o,A: $o,B: fun_o_rat] :
      ( finite_finite_o(S)
     => ( groups2869687844427037835_o_rat(aa_fun4336163974892599420_o_rat(aTP_Lamm_qt((A)),B),S) = $ite(member_o((A),S),aa_o_rat(B,(A)),one_one_rat) ) ) ).

% prod.delta
tff(fact_6588_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_qu(A),B)),S) = $ite(aa_set_nat_o(member_nat(A),S),aa_nat_rat(B,A),one_one_rat) ) ) ).

% prod.delta
tff(fact_6589_zero__less__binomial__iff,axiom,
    ! [N2: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(binomial,N2),K))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),N2) ) ).

% zero_less_binomial_iff
tff(fact_6590_prod__pos__nat__iff,axiom,
    ! [A3: set_int,F: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,A3)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),groups1707563613775114915nt_nat(F,A3))
      <=> ! [X4: int] :
            ( aa_set_int_o(member_int(X4),A3)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_int_nat(F,X4)) ) ) ) ).

% prod_pos_nat_iff
tff(fact_6591_prod__pos__nat__iff,axiom,
    ! [A3: set_Code_integer,F: fun_Code_integer_nat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A3)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),groups3190895334310489300er_nat(F,A3))
      <=> ! [X4: code_integer] :
            ( aa_set5643387332196575111eger_o(member_Code_integer(X4),A3)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_Code_integer_nat(F,X4)) ) ) ) ).

% prod_pos_nat_iff
tff(fact_6592_prod__pos__nat__iff,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,F: fun_Pr2602117396471556196at_nat] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A3)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),groups4077766827762148844at_nat(F,A3))
      <=> ! [X4: product_prod_nat_nat] :
            ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X4),A3)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_Pro4243173770044397973at_nat(F,X4)) ) ) ) ).

% prod_pos_nat_iff
tff(fact_6593_prod__pos__nat__iff,axiom,
    ! [A3: set_nat,F: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_set_nat_nat(groups708209901874060359at_nat(F),A3))
      <=> ! [X4: nat] :
            ( aa_set_nat_o(member_nat(X4),A3)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(F,X4)) ) ) ) ).

% prod_pos_nat_iff
tff(fact_6594_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_6595_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,aa_num_rat(numeral_numeral_rat,M)))
    <=> ( M != one ) ) ).

% not_neg_one_le_neg_numeral_iff
tff(fact_6596_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_6597_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_6598_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_6599_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,aa_num_rat(numeral_numeral_rat,M))),aa_rat_rat(uminus_uminus_rat,one_one_rat))
    <=> ( M != one ) ) ).

% neg_numeral_less_neg_one_iff
tff(fact_6600_numeral__le__one__iff,axiom,
    ! [N2: num] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_num_Code_integer(numera6620942414471956472nteger,N2)),one_one_Code_integer)
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,N2),one) ) ).

% numeral_le_one_iff
tff(fact_6601_numeral__le__one__iff,axiom,
    ! [N2: num] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,aa_num_Code_natural(numera5444537566228673987atural,N2)),one_one_Code_natural)
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,N2),one) ) ).

% numeral_le_one_iff
tff(fact_6602_numeral__le__one__iff,axiom,
    ! [N2: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,N2)),one_one_rat)
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,N2),one) ) ).

% numeral_le_one_iff
tff(fact_6603_numeral__le__one__iff,axiom,
    ! [N2: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,N2)),one_one_nat)
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,N2),one) ) ).

% numeral_le_one_iff
tff(fact_6604_numeral__le__one__iff,axiom,
    ! [N2: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,N2)),one_one_int)
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,N2),one) ) ).

% numeral_le_one_iff
tff(fact_6605_one__less__numeral__iff,axiom,
    ! [N2: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,N2))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,one),N2) ) ).

% one_less_numeral_iff
tff(fact_6606_one__less__numeral__iff,axiom,
    ! [N2: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_num_int(numeral_numeral_int,N2))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,one),N2) ) ).

% one_less_numeral_iff
tff(fact_6607_one__less__numeral__iff,axiom,
    ! [N2: num] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),aa_num_Code_integer(numera6620942414471956472nteger,N2))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,one),N2) ) ).

% one_less_numeral_iff
tff(fact_6608_one__less__numeral__iff,axiom,
    ! [N2: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_num_rat(numeral_numeral_rat,N2))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,one),N2) ) ).

% one_less_numeral_iff
tff(fact_6609_one__less__numeral__iff,axiom,
    ! [N2: num] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,one_one_Code_natural),aa_num_Code_natural(numera5444537566228673987atural,N2))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,one),N2) ) ).

% one_less_numeral_iff
tff(fact_6610_power2__less__eq__zero__iff,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),zero_z3403309356797280102nteger)
    <=> ( A = zero_z3403309356797280102nteger ) ) ).

% power2_less_eq_zero_iff
tff(fact_6611_power2__less__eq__zero__iff,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),zero_zero_rat)
    <=> ( A = zero_zero_rat ) ) ).

% power2_less_eq_zero_iff
tff(fact_6612_power2__less__eq__zero__iff,axiom,
    ! [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,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),zero_zero_int)
    <=> ( A = zero_zero_int ) ) ).

% power2_less_eq_zero_iff
tff(fact_6613_power2__eq__iff__nonneg,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_nat_Code_integer(power_8256067586552552935nteger(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_Code_integer(power_8256067586552552935nteger(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
        <=> ( X = Y ) ) ) ) ).

% power2_eq_iff_nonneg
tff(fact_6614_power2__eq__iff__nonneg,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_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
        <=> ( X = Y ) ) ) ) ).

% power2_eq_iff_nonneg
tff(fact_6615_power2__eq__iff__nonneg,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(power_power_nat,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
        <=> ( X = Y ) ) ) ) ).

% power2_eq_iff_nonneg
tff(fact_6616_power2__eq__iff__nonneg,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_nat_int(aa_int_fun_nat_int(power_power_int,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_int(aa_int_fun_nat_int(power_power_int,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
        <=> ( X = Y ) ) ) ) ).

% power2_eq_iff_nonneg
tff(fact_6617_zero__less__power2,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
    <=> ( A != zero_z3403309356797280102nteger ) ) ).

% zero_less_power2
tff(fact_6618_zero__less__power2,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
    <=> ( A != zero_zero_rat ) ) ).

% zero_less_power2
tff(fact_6619_zero__less__power2,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(aa_int_fun_nat_int(power_power_int,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
    <=> ( A != zero_zero_int ) ) ).

% zero_less_power2
tff(fact_6620_sum_Oop__ivl__Suc,axiom,
    ! [G: fun_nat_Code_integer,M: nat,N2: nat] :
      aa_set871585252387937313nteger(groups7501900531339628137nteger(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,M),aa_nat_nat(suc,N2))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),M),zero_z3403309356797280102nteger,aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_set871585252387937313nteger(groups7501900531339628137nteger(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,M),N2))),aa_nat_Code_integer(G,N2))) ).

% sum.op_ivl_Suc
tff(fact_6621_sum_Oop__ivl__Suc,axiom,
    ! [G: fun_na2338080690716861136at_nat,M: nat,N2: nat] :
      groups6857163185585827899at_nat(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,M),aa_nat_nat(suc,N2))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),M),zero_z1048942125864253310at_nat,aa_mul5754855110707352700at_nat(plus_p7104986032573967614at_nat(groups6857163185585827899at_nat(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,M),N2))),aa_nat2240542744625857721at_nat(G,N2))) ).

% sum.op_ivl_Suc
tff(fact_6622_sum_Oop__ivl__Suc,axiom,
    ! [G: fun_nat_rat,M: nat,N2: nat] :
      aa_set_nat_rat(groups2906978787729119204at_rat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,M),aa_nat_nat(suc,N2))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),M),zero_zero_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_set_nat_rat(groups2906978787729119204at_rat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,M),N2))),aa_nat_rat(G,N2))) ).

% sum.op_ivl_Suc
tff(fact_6623_sum_Oop__ivl__Suc,axiom,
    ! [G: fun_nat_int,M: nat,N2: nat] :
      aa_set_nat_int(groups3539618377306564664at_int(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,M),aa_nat_nat(suc,N2))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),M),zero_zero_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_set_nat_int(groups3539618377306564664at_int(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,M),N2))),aa_nat_int(G,N2))) ).

% sum.op_ivl_Suc
tff(fact_6624_sum_Oop__ivl__Suc,axiom,
    ! [G: fun_nat_nat,M: nat,N2: nat] :
      aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,M),aa_nat_nat(suc,N2))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),M),zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,M),N2))),aa_nat_nat(G,N2))) ).

% sum.op_ivl_Suc
tff(fact_6625_prod_Oop__ivl__Suc,axiom,
    ! [G: fun_nat_Code_integer,M: nat,N2: nat] :
      aa_set871585252387937313nteger(groups3455450783089532116nteger(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,M),aa_nat_nat(suc,N2))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),M),one_one_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_set871585252387937313nteger(groups3455450783089532116nteger(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,M),N2))),aa_nat_Code_integer(G,N2))) ).

% prod.op_ivl_Suc
tff(fact_6626_prod_Oop__ivl__Suc,axiom,
    ! [G: fun_nat_assn,M: nat,N2: nat] :
      groups6906906614972039071t_assn(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,M),aa_nat_nat(suc,N2))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),M),one_one_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,groups6906906614972039071t_assn(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,M),N2))),aa_nat_assn(G,N2))) ).

% prod.op_ivl_Suc
tff(fact_6627_prod_Oop__ivl__Suc,axiom,
    ! [G: fun_nat_rat,M: nat,N2: nat] :
      aa_set_nat_rat(groups73079841787564623at_rat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,M),aa_nat_nat(suc,N2))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),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(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,M),N2))),aa_nat_rat(G,N2))) ).

% prod.op_ivl_Suc
tff(fact_6628_prod_Oop__ivl__Suc,axiom,
    ! [G: fun_nat_nat,M: nat,N2: nat] :
      aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,M),aa_nat_nat(suc,N2))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),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(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,M),N2))),aa_nat_nat(G,N2))) ).

% prod.op_ivl_Suc
tff(fact_6629_prod_Oop__ivl__Suc,axiom,
    ! [G: fun_nat_int,M: nat,N2: nat] :
      aa_set_nat_int(groups705719431365010083at_int(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,M),aa_nat_nat(suc,N2))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),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(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,M),N2))),aa_nat_int(G,N2))) ).

% prod.op_ivl_Suc
tff(fact_6630_sum__zero__power,axiom,
    ! [Ca: fun_nat_rat,A3: set_nat] :
      aa_set_nat_rat(groups2906978787729119204at_rat(aTP_Lamm_qv(Ca)),A3) = $ite(
        ( aa_set_nat_o(finite_finite_nat,A3)
        & aa_set_nat_o(member_nat(zero_zero_nat),A3) ),
        aa_nat_rat(Ca,zero_zero_nat),
        zero_zero_rat ) ).

% sum_zero_power
tff(fact_6631_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,aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),X) ) ).

% one_less_floor
tff(fact_6632_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),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))) ) ).

% floor_le_one
tff(fact_6633_mod2__gr__0,axiom,
    ! [M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_nat ) ) ).

% mod2_gr_0
tff(fact_6634_int__prod,axiom,
    ! [F: fun_int_nat,A3: set_int] : aa_nat_int(semiri1314217659103216013at_int,groups1707563613775114915nt_nat(F,A3)) = groups1705073143266064639nt_int(aTP_Lamm_jp(F),A3) ).

% int_prod
tff(fact_6635_int__prod,axiom,
    ! [F: fun_nat_nat,A3: set_nat] : aa_nat_int(semiri1314217659103216013at_int,aa_set_nat_nat(groups708209901874060359at_nat(F),A3)) = aa_set_nat_int(groups705719431365010083at_int(aTP_Lamm_jq(F)),A3) ).

% int_prod
tff(fact_6636_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_qw(S),S4))) ) ).

% finite_if_eq_beyond_finite
tff(fact_6637_finite__if__eq__beyond__finite,axiom,
    ! [S: set_Code_integer,S4: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,S)
     => finite6931041176100689706nteger(collec574505750873337192nteger(aa_set4044230130517860621eger_o(aTP_Lamm_qx(S),S4))) ) ).

% finite_if_eq_beyond_finite
tff(fact_6638_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_qy(S),S4))) ) ).

% finite_if_eq_beyond_finite
tff(fact_6639_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_qz(S),S4))) ) ).

% finite_if_eq_beyond_finite
tff(fact_6640_infinite__Ico,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ~ finite_finite_rat(set_or4029947393144176647an_rat(A,B)) ) ).

% infinite_Ico
tff(fact_6641_binomial__le__pow2,axiom,
    ! [N2: nat,K: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,N2),K)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2)) ).

% binomial_le_pow2
tff(fact_6642_atLeastLessThan__eq__iff,axiom,
    ! [A: rat,B: rat,Ca: 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,Ca),D2)
       => ( ( set_or4029947393144176647an_rat(A,B) = set_or4029947393144176647an_rat(Ca,D2) )
        <=> ( ( A = Ca )
            & ( B = D2 ) ) ) ) ) ).

% atLeastLessThan_eq_iff
tff(fact_6643_atLeastLessThan__eq__iff,axiom,
    ! [A: num,B: num,Ca: num,D2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,A),B)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,Ca),D2)
       => ( ( set_or1222409239386451017an_num(A,B) = set_or1222409239386451017an_num(Ca,D2) )
        <=> ( ( A = Ca )
            & ( B = D2 ) ) ) ) ) ).

% atLeastLessThan_eq_iff
tff(fact_6644_atLeastLessThan__eq__iff,axiom,
    ! [A: nat,B: nat,Ca: 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,Ca),D2)
       => ( ( aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,A),B) = aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,Ca),D2) )
        <=> ( ( A = Ca )
            & ( B = D2 ) ) ) ) ) ).

% atLeastLessThan_eq_iff
tff(fact_6645_atLeastLessThan__eq__iff,axiom,
    ! [A: int,B: int,Ca: 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,Ca),D2)
       => ( ( set_or4662586982721622107an_int(A,B) = set_or4662586982721622107an_int(Ca,D2) )
        <=> ( ( A = Ca )
            & ( B = D2 ) ) ) ) ) ).

% atLeastLessThan_eq_iff
tff(fact_6646_atLeastLessThan__eq__iff,axiom,
    ! [A: code_integer,B: code_integer,Ca: 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,Ca),D2)
       => ( ( set_or8404916559141939852nteger(A,B) = set_or8404916559141939852nteger(Ca,D2) )
        <=> ( ( A = Ca )
            & ( B = D2 ) ) ) ) ) ).

% atLeastLessThan_eq_iff
tff(fact_6647_atLeastLessThan__inj_I1_J,axiom,
    ! [A: rat,B: rat,Ca: rat,D2: rat] :
      ( ( set_or4029947393144176647an_rat(A,B) = set_or4029947393144176647an_rat(Ca,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,Ca),D2)
         => ( A = Ca ) ) ) ) ).

% atLeastLessThan_inj(1)
tff(fact_6648_atLeastLessThan__inj_I1_J,axiom,
    ! [A: num,B: num,Ca: num,D2: num] :
      ( ( set_or1222409239386451017an_num(A,B) = set_or1222409239386451017an_num(Ca,D2) )
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,A),B)
       => ( aa_num_o(aa_num_fun_num_o(ord_less_num,Ca),D2)
         => ( A = Ca ) ) ) ) ).

% atLeastLessThan_inj(1)
tff(fact_6649_atLeastLessThan__inj_I1_J,axiom,
    ! [A: nat,B: nat,Ca: nat,D2: nat] :
      ( ( aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,A),B) = aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,Ca),D2) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ca),D2)
         => ( A = Ca ) ) ) ) ).

% atLeastLessThan_inj(1)
tff(fact_6650_atLeastLessThan__inj_I1_J,axiom,
    ! [A: int,B: int,Ca: int,D2: int] :
      ( ( set_or4662586982721622107an_int(A,B) = set_or4662586982721622107an_int(Ca,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,Ca),D2)
         => ( A = Ca ) ) ) ) ).

% atLeastLessThan_inj(1)
tff(fact_6651_atLeastLessThan__inj_I1_J,axiom,
    ! [A: code_integer,B: code_integer,Ca: code_integer,D2: code_integer] :
      ( ( set_or8404916559141939852nteger(A,B) = set_or8404916559141939852nteger(Ca,D2) )
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Ca),D2)
         => ( A = Ca ) ) ) ) ).

% atLeastLessThan_inj(1)
tff(fact_6652_atLeastLessThan__inj_I2_J,axiom,
    ! [A: rat,B: rat,Ca: rat,D2: rat] :
      ( ( set_or4029947393144176647an_rat(A,B) = set_or4029947393144176647an_rat(Ca,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,Ca),D2)
         => ( B = D2 ) ) ) ) ).

% atLeastLessThan_inj(2)
tff(fact_6653_atLeastLessThan__inj_I2_J,axiom,
    ! [A: num,B: num,Ca: num,D2: num] :
      ( ( set_or1222409239386451017an_num(A,B) = set_or1222409239386451017an_num(Ca,D2) )
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,A),B)
       => ( aa_num_o(aa_num_fun_num_o(ord_less_num,Ca),D2)
         => ( B = D2 ) ) ) ) ).

% atLeastLessThan_inj(2)
tff(fact_6654_atLeastLessThan__inj_I2_J,axiom,
    ! [A: nat,B: nat,Ca: nat,D2: nat] :
      ( ( aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,A),B) = aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,Ca),D2) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ca),D2)
         => ( B = D2 ) ) ) ) ).

% atLeastLessThan_inj(2)
tff(fact_6655_atLeastLessThan__inj_I2_J,axiom,
    ! [A: int,B: int,Ca: int,D2: int] :
      ( ( set_or4662586982721622107an_int(A,B) = set_or4662586982721622107an_int(Ca,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,Ca),D2)
         => ( B = D2 ) ) ) ) ).

% atLeastLessThan_inj(2)
tff(fact_6656_atLeastLessThan__inj_I2_J,axiom,
    ! [A: code_integer,B: code_integer,Ca: code_integer,D2: code_integer] :
      ( ( set_or8404916559141939852nteger(A,B) = set_or8404916559141939852nteger(Ca,D2) )
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Ca),D2)
         => ( B = D2 ) ) ) ) ).

% atLeastLessThan_inj(2)
tff(fact_6657_subset__eq__atLeast0__lessThan__finite,axiom,
    ! [N7: set_nat,N2: nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,N7),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),N2))
     => aa_set_nat_o(finite_finite_nat,N7) ) ).

% subset_eq_atLeast0_lessThan_finite
tff(fact_6658_choose__row__sum,axiom,
    ! [N2: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(binomial,N2)),aa_nat_set_nat(set_ord_atMost_nat,N2)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2) ).

% choose_row_sum
tff(fact_6659_finite__nat__set__iff__bounded,axiom,
    ! [N7: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,N7)
    <=> ? [M3: nat] :
        ! [X4: nat] :
          ( aa_set_nat_o(member_nat(X4),N7)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X4),M3) ) ) ).

% finite_nat_set_iff_bounded
tff(fact_6660_bounded__nat__set__is__finite,axiom,
    ! [N7: set_nat,N2: nat] :
      ( ! [X3: nat] :
          ( aa_set_nat_o(member_nat(X3),N7)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),N2) )
     => aa_set_nat_o(finite_finite_nat,N7) ) ).

% bounded_nat_set_is_finite
tff(fact_6661_finite__nat__set__iff__bounded__le,axiom,
    ! [N7: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,N7)
    <=> ? [M3: nat] :
        ! [X4: nat] :
          ( aa_set_nat_o(member_nat(X4),N7)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X4),M3) ) ) ).

% finite_nat_set_iff_bounded_le
tff(fact_6662_choose__square__sum,axiom,
    ! [N2: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_ra,N2)),aa_nat_set_nat(set_ord_atMost_nat,N2)) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2)),N2) ).

% choose_square_sum
tff(fact_6663_mult__2,axiom,
    ! [Z: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Z) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Z),Z) ).

% mult_2
tff(fact_6664_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_6665_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(plus_p5714425477246183910nteger(Z),Z) ).

% mult_2
tff(fact_6666_mult__2,axiom,
    ! [Z: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_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_6667_mult__2,axiom,
    ! [Z: code_natural] : times_2397367101498566445atural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one)),Z) = plus_p4538020629002901425atural(Z,Z) ).

% mult_2
tff(fact_6668_mult__2__right,axiom,
    ! [Z: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Z),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Z),Z) ).

% mult_2_right
tff(fact_6669_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_6670_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(plus_p5714425477246183910nteger(Z),Z) ).

% mult_2_right
tff(fact_6671_mult__2__right,axiom,
    ! [Z: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Z),aa_num_rat(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_6672_mult__2__right,axiom,
    ! [Z: code_natural] : times_2397367101498566445atural(Z,aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))) = plus_p4538020629002901425atural(Z,Z) ).

% mult_2_right
tff(fact_6673_left__add__twice,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)),B) ).

% left_add_twice
tff(fact_6674_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_6675_left__add__twice,axiom,
    ! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(A),aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(A),B)) = aa_Cod1783974652533396728nteger(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_6676_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,aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),A)),B) ).

% left_add_twice
tff(fact_6677_left__add__twice,axiom,
    ! [A: code_natural,B: code_natural] : plus_p4538020629002901425atural(A,plus_p4538020629002901425atural(A,B)) = plus_p4538020629002901425atural(times_2397367101498566445atural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one)),A),B) ).

% left_add_twice
tff(fact_6678_le__num__One__iff,axiom,
    ! [X: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X),one)
    <=> ( X = one ) ) ).

% le_num_One_iff
tff(fact_6679_power4__eq__xxxx,axiom,
    ! [X: code_integer] : aa_nat_Code_integer(power_8256067586552552935nteger(X),aa_num_nat(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_6680_power4__eq__xxxx,axiom,
    ! [X: assn] : aa_nat_assn(power_power_assn(X),aa_num_nat(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_6681_power4__eq__xxxx,axiom,
    ! [X: rat] : aa_nat_rat(power_power_rat(X),aa_num_nat(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_6682_power4__eq__xxxx,axiom,
    ! [X: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(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_6683_power4__eq__xxxx,axiom,
    ! [X: int] : aa_nat_int(aa_int_fun_nat_int(power_power_int,X),aa_num_nat(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_6684_power2__eq__square,axiom,
    ! [A: code_integer] : aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),A) ).

% power2_eq_square
tff(fact_6685_power2__eq__square,axiom,
    ! [A: assn] : aa_nat_assn(power_power_assn(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A),A) ).

% power2_eq_square
tff(fact_6686_power2__eq__square,axiom,
    ! [A: rat] : aa_nat_rat(power_power_rat(A),aa_num_nat(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_6687_power2__eq__square,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),A) ).

% power2_eq_square
tff(fact_6688_power2__eq__square,axiom,
    ! [A: int] : aa_nat_int(aa_int_fun_nat_int(power_power_int,A),aa_num_nat(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_6689_finite__M__bounded__by__nat,axiom,
    ! [Pa: fun_nat_o,I2: nat] : aa_set_nat_o(finite_finite_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_rb(Pa),I2))) ).

% finite_M_bounded_by_nat
tff(fact_6690_finite__less__ub,axiom,
    ! [F: fun_nat_nat,U: nat] :
      ( ! [N5: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N5),aa_nat_nat(F,N5))
     => aa_set_nat_o(finite_finite_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_rc(F),U))) ) ).

% finite_less_ub
tff(fact_6691_sum_Oswap__restrict,axiom,
    ! [A3: set_o,B3: set_nat,G: fun_o_fun_nat_nat,Ra: fun_o_fun_nat_o] :
      ( finite_finite_o(A3)
     => ( aa_set_nat_o(finite_finite_nat,B3)
       => ( groups8507830703676809646_o_nat(aa_fun6229755271245457521_o_nat(aa_fun802073206315457404_o_nat(aTP_Lamm_re(B3),G),Ra),A3) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun1827338972888073639at_nat(aa_fun390978558358138354at_nat(aTP_Lamm_rh(A3),G),Ra)),B3) ) ) ) ).

% sum.swap_restrict
tff(fact_6692_sum_Oswap__restrict,axiom,
    ! [A3: set_int,B3: set_nat,G: fun_int_fun_nat_nat,Ra: fun_int_fun_nat_o] :
      ( aa_set_int_o(finite_finite_int,A3)
     => ( aa_set_nat_o(finite_finite_nat,B3)
       => ( groups4541462559716669496nt_nat(aa_fun4650797359919590027nt_nat(aa_fun8303507722728657494nt_nat(aTP_Lamm_rj(B3),G),Ra),A3) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun3651443648018535471at_nat(aa_fun6116586148035635706at_nat(aTP_Lamm_rm(A3),G),Ra)),B3) ) ) ) ).

% sum.swap_restrict
tff(fact_6693_sum_Oswap__restrict,axiom,
    ! [A3: set_Code_integer,B3: set_nat,G: fun_Co3312552843764780407at_nat,Ra: fun_Co1375048237525739357_nat_o] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A3)
     => ( aa_set_nat_o(finite_finite_nat,B3)
       => ( groups7237345082560585321er_nat(aa_fun6683192525214198859er_nat(aa_fun2143651634668453157er_nat(aTP_Lamm_ro(B3),G),Ra),A3) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun7456160887942273982at_nat(aa_fun4417447297828429592at_nat(aTP_Lamm_rr(A3),G),Ra)),B3) ) ) ) ).

% sum.swap_restrict
tff(fact_6694_sum_Oswap__restrict,axiom,
    ! [A3: set_o,B3: set_int,G: fun_o_fun_int_int,Ra: fun_o_fun_int_o] :
      ( finite_finite_o(A3)
     => ( aa_set_int_o(finite_finite_int,B3)
       => ( groups8505340233167759370_o_int(aa_fun7462986433439723305_o_int(aa_fun4201161283678742396_o_int(aTP_Lamm_rt(B3),G),Ra),A3) = groups4538972089207619220nt_int(aa_fun7681507786902443267nt_int(aa_fun5020185059765800966nt_int(aTP_Lamm_rw(A3),G),Ra),B3) ) ) ) ).

% sum.swap_restrict
tff(fact_6695_sum_Oswap__restrict,axiom,
    ! [A3: set_nat,B3: set_int,G: fun_nat_fun_int_int,Ra: fun_nat_fun_int_o] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( aa_set_int_o(finite_finite_int,B3)
       => ( aa_set_nat_int(groups3539618377306564664at_int(aa_fun4489122510365962891at_int(aa_fun1158358564283266810at_int(aTP_Lamm_ry(B3),G),Ra)),A3) = groups4538972089207619220nt_int(aa_fun5488476222267017447nt_int(aa_fun3345280138976288598nt_int(aTP_Lamm_sa(A3),G),Ra),B3) ) ) ) ).

% sum.swap_restrict
tff(fact_6696_sum_Oswap__restrict,axiom,
    ! [A3: set_Code_integer,B3: set_int,G: fun_Co3009391273526075439nt_int,Ra: fun_Co8353275306710069249_int_o] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A3)
     => ( aa_set_int_o(finite_finite_int,B3)
       => ( groups7234854612051535045er_int(aa_fun5771563549434713419er_int(aa_fun4265095070934345581er_int(aTP_Lamm_sc(B3),G),Ra),A3) = groups4538972089207619220nt_int(aa_fun4086957665101867802nt_int(aa_fun5679615576903115196nt_int(aTP_Lamm_sf(A3),G),Ra),B3) ) ) ) ).

% sum.swap_restrict
tff(fact_6697_sum_Oswap__restrict,axiom,
    ! [A3: set_nat,B3: set_o,G: fun_nat_fun_o_nat,Ra: fun_nat_fun_o_o] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( finite_finite_o(B3)
       => ( aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun4178193321845537899at_nat(aa_fun3391070971708407026at_nat(aTP_Lamm_sh(B3),G),Ra)),A3) = groups8507830703676809646_o_nat(aa_fun5720900557306247341_o_nat(aa_fun4151889263603125372_o_nat(aTP_Lamm_sk(A3),G),Ra),B3) ) ) ) ).

% sum.swap_restrict
tff(fact_6698_sum_Oswap__restrict,axiom,
    ! [A3: set_nat,B3: set_int,G: fun_nat_fun_int_nat,Ra: fun_nat_fun_int_o] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( aa_set_int_o(finite_finite_int,B3)
       => ( aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun8666973529875159599at_nat(aa_fun3399039426249668090at_nat(aTP_Lamm_sl(B3),G),Ra)),A3) = groups4541462559716669496nt_nat(aa_fun442955204921438347nt_nat(aa_fun5585961000942689878nt_nat(aTP_Lamm_sn(A3),G),Ra),B3) ) ) ) ).

% sum.swap_restrict
tff(fact_6699_sum_Oswap__restrict,axiom,
    ! [A3: set_nat,B3: set_Code_integer,G: fun_na5492297256294505719er_nat,Ra: fun_na6318815019539281885eger_o] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B3)
       => ( aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun2058600731543071806at_nat(aa_fun4864976072519958552at_nat(aTP_Lamm_sp(B3),G),Ra)),A3) = groups7237345082560585321er_nat(aa_fun7264493923017702603er_nat(aa_fun7529132916787797541er_nat(aTP_Lamm_ss(A3),G),Ra),B3) ) ) ) ).

% sum.swap_restrict
tff(fact_6700_sum_Oswap__restrict,axiom,
    ! [A3: set_nat,B3: set_nat,G: fun_nat_fun_nat_nat,Ra: fun_nat_fun_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( aa_set_nat_o(finite_finite_nat,B3)
       => ( aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun8857679445107423627at_nat(aa_fun7975631875395124402at_nat(aTP_Lamm_su(B3),G),Ra)),A3) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun8857679445107423627at_nat(aa_fun7975631875395124402at_nat(aTP_Lamm_sw(A3),G),Ra)),B3) ) ) ) ).

% sum.swap_restrict
tff(fact_6701_prod_Oswap__restrict,axiom,
    ! [A3: set_o,B3: set_nat,G: fun_o_fun_nat_nat,Ra: fun_o_fun_nat_o] :
      ( finite_finite_o(A3)
     => ( aa_set_nat_o(finite_finite_nat,B3)
       => ( groups3504817904513533571_o_nat(aa_fun6229755271245457521_o_nat(aa_fun802073206315457404_o_nat(aTP_Lamm_sx(B3),G),Ra),A3) = aa_set_nat_nat(groups708209901874060359at_nat(aa_fun1827338972888073639at_nat(aa_fun390978558358138354at_nat(aTP_Lamm_sy(A3),G),Ra)),B3) ) ) ) ).

% prod.swap_restrict
tff(fact_6702_prod_Oswap__restrict,axiom,
    ! [A3: set_int,B3: set_nat,G: fun_int_fun_nat_nat,Ra: fun_int_fun_nat_o] :
      ( aa_set_int_o(finite_finite_int,A3)
     => ( aa_set_nat_o(finite_finite_nat,B3)
       => ( groups1707563613775114915nt_nat(aa_fun4650797359919590027nt_nat(aa_fun8303507722728657494nt_nat(aTP_Lamm_sz(B3),G),Ra),A3) = aa_set_nat_nat(groups708209901874060359at_nat(aa_fun3651443648018535471at_nat(aa_fun6116586148035635706at_nat(aTP_Lamm_ta(A3),G),Ra)),B3) ) ) ) ).

% prod.swap_restrict
tff(fact_6703_prod_Oswap__restrict,axiom,
    ! [A3: set_Code_integer,B3: set_nat,G: fun_Co3312552843764780407at_nat,Ra: fun_Co1375048237525739357_nat_o] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A3)
     => ( aa_set_nat_o(finite_finite_nat,B3)
       => ( groups3190895334310489300er_nat(aa_fun6683192525214198859er_nat(aa_fun2143651634668453157er_nat(aTP_Lamm_tb(B3),G),Ra),A3) = aa_set_nat_nat(groups708209901874060359at_nat(aa_fun7456160887942273982at_nat(aa_fun4417447297828429592at_nat(aTP_Lamm_tc(A3),G),Ra)),B3) ) ) ) ).

% prod.swap_restrict
tff(fact_6704_prod_Oswap__restrict,axiom,
    ! [A3: set_o,B3: set_nat,G: fun_o_fun_nat_int,Ra: fun_o_fun_nat_o] :
      ( finite_finite_o(A3)
     => ( aa_set_nat_o(finite_finite_nat,B3)
       => ( groups3502327434004483295_o_int(aa_fun2051904251736260813_o_int(aa_fun7508458648369839740_o_int(aTP_Lamm_td(B3),G),Ra),A3) = aa_set_nat_int(groups705719431365010083at_int(aa_fun6872859990233652739at_int(aa_fun8947132502270365866at_int(aTP_Lamm_tf(A3),G),Ra)),B3) ) ) ) ).

% prod.swap_restrict
tff(fact_6705_prod_Oswap__restrict,axiom,
    ! [A3: set_Code_integer,B3: set_nat,G: fun_Co822469698833053651at_int,Ra: fun_Co1375048237525739357_nat_o] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A3)
     => ( aa_set_nat_o(finite_finite_nat,B3)
       => ( groups3188404863801439024er_int(aa_fun2505341505705002151er_int(aa_fun493137163814837797er_int(aTP_Lamm_tg(B3),G),Ra),A3) = aa_set_nat_int(groups705719431365010083at_int(aa_fun3278309868433077274at_int(aa_fun1276769135904165912at_int(aTP_Lamm_ti(A3),G),Ra)),B3) ) ) ) ).

% prod.swap_restrict
tff(fact_6706_prod_Oswap__restrict,axiom,
    ! [A3: set_o,B3: set_int,G: fun_o_fun_int_int,Ra: fun_o_fun_int_o] :
      ( finite_finite_o(A3)
     => ( aa_set_int_o(finite_finite_int,B3)
       => ( groups3502327434004483295_o_int(aa_fun7462986433439723305_o_int(aa_fun4201161283678742396_o_int(aTP_Lamm_tj(B3),G),Ra),A3) = groups1705073143266064639nt_int(aa_fun7681507786902443267nt_int(aa_fun5020185059765800966nt_int(aTP_Lamm_tk(A3),G),Ra),B3) ) ) ) ).

% prod.swap_restrict
tff(fact_6707_prod_Oswap__restrict,axiom,
    ! [A3: set_Code_integer,B3: set_int,G: fun_Co3009391273526075439nt_int,Ra: fun_Co8353275306710069249_int_o] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A3)
     => ( aa_set_int_o(finite_finite_int,B3)
       => ( groups3188404863801439024er_int(aa_fun5771563549434713419er_int(aa_fun4265095070934345581er_int(aTP_Lamm_tl(B3),G),Ra),A3) = groups1705073143266064639nt_int(aa_fun4086957665101867802nt_int(aa_fun5679615576903115196nt_int(aTP_Lamm_tm(A3),G),Ra),B3) ) ) ) ).

% prod.swap_restrict
tff(fact_6708_prod_Oswap__restrict,axiom,
    ! [A3: set_nat,B3: set_o,G: fun_nat_fun_o_nat,Ra: fun_nat_fun_o_o] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( finite_finite_o(B3)
       => ( aa_set_nat_nat(groups708209901874060359at_nat(aa_fun4178193321845537899at_nat(aa_fun3391070971708407026at_nat(aTP_Lamm_tn(B3),G),Ra)),A3) = groups3504817904513533571_o_nat(aa_fun5720900557306247341_o_nat(aa_fun4151889263603125372_o_nat(aTP_Lamm_to(A3),G),Ra),B3) ) ) ) ).

% prod.swap_restrict
tff(fact_6709_prod_Oswap__restrict,axiom,
    ! [A3: set_nat,B3: set_int,G: fun_nat_fun_int_nat,Ra: fun_nat_fun_int_o] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( aa_set_int_o(finite_finite_int,B3)
       => ( aa_set_nat_nat(groups708209901874060359at_nat(aa_fun8666973529875159599at_nat(aa_fun3399039426249668090at_nat(aTP_Lamm_tp(B3),G),Ra)),A3) = groups1707563613775114915nt_nat(aa_fun442955204921438347nt_nat(aa_fun5585961000942689878nt_nat(aTP_Lamm_tq(A3),G),Ra),B3) ) ) ) ).

% prod.swap_restrict
tff(fact_6710_prod_Oswap__restrict,axiom,
    ! [A3: set_nat,B3: set_Code_integer,G: fun_na5492297256294505719er_nat,Ra: fun_na6318815019539281885eger_o] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B3)
       => ( aa_set_nat_nat(groups708209901874060359at_nat(aa_fun2058600731543071806at_nat(aa_fun4864976072519958552at_nat(aTP_Lamm_tr(B3),G),Ra)),A3) = groups3190895334310489300er_nat(aa_fun7264493923017702603er_nat(aa_fun7529132916787797541er_nat(aTP_Lamm_ts(A3),G),Ra),B3) ) ) ) ).

% prod.swap_restrict
tff(fact_6711_less__exp,axiom,
    ! [N2: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2)) ).

% less_exp
tff(fact_6712_self__le__ge2__pow,axiom,
    ! [K: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,K),M)) ) ).

% self_le_ge2_pow
tff(fact_6713_power2__nat__le__eq__le,axiom,
    ! [M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,N2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2) ) ).

% power2_nat_le_eq_le
tff(fact_6714_power2__nat__le__imp__le,axiom,
    ! [M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),N2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2) ) ).

% power2_nat_le_imp_le
tff(fact_6715_sum__power2,axiom,
    ! [K: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),K)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K)),one_one_nat) ).

% sum_power2
tff(fact_6716_Sum__Ico__nat,axiom,
    ! [M: nat,N2: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_fu),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,M),N2)) = 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,N2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),one_one_nat))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat)))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% Sum_Ico_nat
tff(fact_6717_binomial__eq__0,axiom,
    ! [N2: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),K)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,N2),K) = zero_zero_nat ) ) ).

% binomial_eq_0
tff(fact_6718_half__gt__zero__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,A),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A) ) ).

% half_gt_zero_iff
tff(fact_6719_half__gt__zero,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)))) ) ).

% half_gt_zero
tff(fact_6720_atLeastLessThan__subset__iff,axiom,
    ! [A: rat,B: rat,Ca: rat,D2: rat] :
      ( ord_less_eq_set_rat(set_or4029947393144176647an_rat(A,B),set_or4029947393144176647an_rat(Ca,D2))
     => ( 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,Ca),A)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),D2) ) ) ) ).

% atLeastLessThan_subset_iff
tff(fact_6721_atLeastLessThan__subset__iff,axiom,
    ! [A: num,B: num,Ca: num,D2: num] :
      ( ord_less_eq_set_num(set_or1222409239386451017an_num(A,B),set_or1222409239386451017an_num(Ca,D2))
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),A)
        | ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Ca),A)
          & aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),D2) ) ) ) ).

% atLeastLessThan_subset_iff
tff(fact_6722_atLeastLessThan__subset__iff,axiom,
    ! [A: nat,B: nat,Ca: nat,D2: nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,A),B)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,Ca),D2))
     => ( 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,Ca),A)
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),D2) ) ) ) ).

% atLeastLessThan_subset_iff
tff(fact_6723_atLeastLessThan__subset__iff,axiom,
    ! [A: int,B: int,Ca: int,D2: int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,set_or4662586982721622107an_int(A,B)),set_or4662586982721622107an_int(Ca,D2))
     => ( 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,Ca),A)
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),D2) ) ) ) ).

% atLeastLessThan_subset_iff
tff(fact_6724_atLeastLessThan__subset__iff,axiom,
    ! [A: code_integer,B: code_integer,Ca: code_integer,D2: code_integer] :
      ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(set_or8404916559141939852nteger(A,B)),set_or8404916559141939852nteger(Ca,D2))
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),A)
        | ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Ca),A)
          & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),D2) ) ) ) ).

% atLeastLessThan_subset_iff
tff(fact_6725_zero__le__power2,axiom,
    ! [A: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% zero_le_power2
tff(fact_6726_zero__le__power2,axiom,
    ! [A: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% zero_le_power2
tff(fact_6727_zero__le__power2,axiom,
    ! [A: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(aa_int_fun_nat_int(power_power_int,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% zero_le_power2
tff(fact_6728_power2__eq__imp__eq,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( aa_nat_Code_integer(power_8256067586552552935nteger(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_Code_integer(power_8256067586552552935nteger(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),X)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),Y)
         => ( X = Y ) ) ) ) ).

% power2_eq_imp_eq
tff(fact_6729_power2__eq__imp__eq,axiom,
    ! [X: rat,Y: rat] :
      ( ( aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
     => ( 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)
         => ( X = Y ) ) ) ) ).

% power2_eq_imp_eq
tff(fact_6730_power2__eq__imp__eq,axiom,
    ! [X: nat,Y: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
     => ( 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)
         => ( X = Y ) ) ) ) ).

% power2_eq_imp_eq
tff(fact_6731_power2__eq__imp__eq,axiom,
    ! [X: int,Y: int] :
      ( ( aa_nat_int(aa_int_fun_nat_int(power_power_int,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_int(aa_int_fun_nat_int(power_power_int,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
         => ( X = Y ) ) ) ) ).

% power2_eq_imp_eq
tff(fact_6732_power2__le__imp__le,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_Code_integer(power_8256067586552552935nteger(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),Y)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,X),Y) ) ) ).

% power2_le_imp_le
tff(fact_6733_power2__le__imp__le,axiom,
    ! [X: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
     => ( 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,X),Y) ) ) ).

% power2_le_imp_le
tff(fact_6734_power2__le__imp__le,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(power_power_nat,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Y)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y) ) ) ).

% power2_le_imp_le
tff(fact_6735_power2__le__imp__le,axiom,
    ! [X: int,Y: 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),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(aa_int_fun_nat_int(power_power_int,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Y) ) ) ).

% power2_le_imp_le
tff(fact_6736_power2__less__0,axiom,
    ! [A: code_integer] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),zero_z3403309356797280102nteger) ).

% power2_less_0
tff(fact_6737_power2__less__0,axiom,
    ! [A: rat] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),zero_zero_rat) ).

% power2_less_0
tff(fact_6738_power2__less__0,axiom,
    ! [A: 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_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),zero_zero_int) ).

% power2_less_0
tff(fact_6739_binomial__symmetric,axiom,
    ! [K: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),N2)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,N2),K) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,N2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),K)) ) ) ).

% binomial_symmetric
tff(fact_6740_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)
       => ( 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_6741_ivl__disj__un__two_I3_J,axiom,
    ! [L: num,M: num,U: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,L),M)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,M),U)
       => ( sup_sup_set_num(set_or1222409239386451017an_num(L,M),set_or1222409239386451017an_num(M,U)) = set_or1222409239386451017an_num(L,U) ) ) ) ).

% ivl_disj_un_two(3)
tff(fact_6742_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(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,L),M)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,M),U)) = aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,L),U) ) ) ) ).

% ivl_disj_un_two(3)
tff(fact_6743_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_6744_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_6745_ex__nat__less__eq,axiom,
    ! [N2: nat,Pa: fun_nat_o] :
      ( ? [M3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M3),N2)
          & aa_nat_o(Pa,M3) )
    <=> ? [X4: nat] :
          ( aa_set_nat_o(member_nat(X4),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),N2))
          & aa_nat_o(Pa,X4) ) ) ).

% ex_nat_less_eq
tff(fact_6746_all__nat__less__eq,axiom,
    ! [N2: nat,Pa: fun_nat_o] :
      ( ! [M3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M3),N2)
         => aa_nat_o(Pa,M3) )
    <=> ! [X4: nat] :
          ( aa_set_nat_o(member_nat(X4),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),N2))
         => aa_nat_o(Pa,X4) ) ) ).

% all_nat_less_eq
tff(fact_6747_less__2__cases__iff,axiom,
    ! [N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))
    <=> ( ( N2 = zero_zero_nat )
        | ( N2 = aa_nat_nat(suc,zero_zero_nat) ) ) ) ).

% less_2_cases_iff
tff(fact_6748_less__2__cases,axiom,
    ! [N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))
     => ( ( N2 = zero_zero_nat )
        | ( N2 = aa_nat_nat(suc,zero_zero_nat) ) ) ) ).

% less_2_cases
tff(fact_6749_ivl__disj__int__two_I3_J,axiom,
    ! [L: $o,M: $o,U: $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_6750_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(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,L),M)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,M),U)) = bot_bot_set_nat ).

% ivl_disj_int_two(3)
tff(fact_6751_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_6752_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_6753_abs__le__square__iff,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,X)),aa_Cod1783974652533396728nteger(abs_abs_Code_integer,Y))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_Code_integer(power_8256067586552552935nteger(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ).

% abs_le_square_iff
tff(fact_6754_abs__le__square__iff,axiom,
    ! [X: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,X)),aa_rat_rat(abs_abs_rat,Y))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ).

% abs_le_square_iff
tff(fact_6755_abs__le__square__iff,axiom,
    ! [X: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,X)),aa_int_int(abs_abs_int,Y))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(aa_int_fun_nat_int(power_power_int,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ).

% abs_le_square_iff
tff(fact_6756_ex__min__if__finite,axiom,
    ! [S: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,S)
     => ( ( S != bot_bo3990330152332043303nteger )
       => ? [X3: code_integer] :
            ( aa_set5643387332196575111eger_o(member_Code_integer(X3),S)
            & ~ ? [Xa3: code_integer] :
                  ( aa_set5643387332196575111eger_o(member_Code_integer(Xa3),S)
                  & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Xa3),X3) ) ) ) ) ).

% ex_min_if_finite
tff(fact_6757_ex__min__if__finite,axiom,
    ! [S: set_o] :
      ( finite_finite_o(S)
     => ( ( S != bot_bot_set_o )
       => ? [X3: $o] :
            ( member_o((X3),S)
            & ~ ? [Xa3: $o] :
                  ( member_o((Xa3),S)
                  & aa_o_o(aa_o_fun_o_o(ord_less_o,(Xa3)),(X3)) ) ) ) ) ).

% ex_min_if_finite
tff(fact_6758_ex__min__if__finite,axiom,
    ! [S: set_assn] :
      ( finite_finite_assn(S)
     => ( ( S != bot_bot_set_assn )
       => ? [X3: assn] :
            ( member_assn(X3,S)
            & ~ ? [Xa3: assn] :
                  ( member_assn(Xa3,S)
                  & aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,Xa3),X3) ) ) ) ) ).

% ex_min_if_finite
tff(fact_6759_ex__min__if__finite,axiom,
    ! [S: set_rat] :
      ( finite_finite_rat(S)
     => ( ( S != bot_bot_set_rat )
       => ? [X3: rat] :
            ( member_rat(X3,S)
            & ~ ? [Xa3: rat] :
                  ( member_rat(Xa3,S)
                  & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa3),X3) ) ) ) ) ).

% ex_min_if_finite
tff(fact_6760_ex__min__if__finite,axiom,
    ! [S: set_num] :
      ( finite_finite_num(S)
     => ( ( S != bot_bot_set_num )
       => ? [X3: num] :
            ( member_num(X3,S)
            & ~ ? [Xa3: num] :
                  ( member_num(Xa3,S)
                  & aa_num_o(aa_num_fun_num_o(ord_less_num,Xa3),X3) ) ) ) ) ).

% ex_min_if_finite
tff(fact_6761_ex__min__if__finite,axiom,
    ! [S: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,S)
     => ( ( S != bot_bot_set_nat )
       => ? [X3: nat] :
            ( aa_set_nat_o(member_nat(X3),S)
            & ~ ? [Xa3: nat] :
                  ( aa_set_nat_o(member_nat(Xa3),S)
                  & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa3),X3) ) ) ) ) ).

% ex_min_if_finite
tff(fact_6762_ex__min__if__finite,axiom,
    ! [S: set_int] :
      ( aa_set_int_o(finite_finite_int,S)
     => ( ( S != bot_bot_set_int )
       => ? [X3: int] :
            ( aa_set_int_o(member_int(X3),S)
            & ~ ? [Xa3: int] :
                  ( aa_set_int_o(member_int(Xa3),S)
                  & aa_int_o(aa_int_fun_int_o(ord_less_int,Xa3),X3) ) ) ) ) ).

% ex_min_if_finite
tff(fact_6763_infinite__growing,axiom,
    ! [X6: set_Code_integer] :
      ( ( X6 != bot_bo3990330152332043303nteger )
     => ( ! [X3: code_integer] :
            ( aa_set5643387332196575111eger_o(member_Code_integer(X3),X6)
           => ? [Xa3: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(Xa3),X6)
                & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,X3),Xa3) ) )
       => ~ aa_set5643387332196575111eger_o(finite6017078050557962740nteger,X6) ) ) ).

% infinite_growing
tff(fact_6764_infinite__growing,axiom,
    ! [X6: set_o] :
      ( ( X6 != bot_bot_set_o )
     => ( ! [X3: $o] :
            ( member_o((X3),X6)
           => ? [Xa3: $o] :
                ( member_o((Xa3),X6)
                & aa_o_o(aa_o_fun_o_o(ord_less_o,(X3)),(Xa3)) ) )
       => ~ finite_finite_o(X6) ) ) ).

% infinite_growing
tff(fact_6765_infinite__growing,axiom,
    ! [X6: set_rat] :
      ( ( X6 != bot_bot_set_rat )
     => ( ! [X3: rat] :
            ( member_rat(X3,X6)
           => ? [Xa3: rat] :
                ( member_rat(Xa3,X6)
                & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Xa3) ) )
       => ~ finite_finite_rat(X6) ) ) ).

% infinite_growing
tff(fact_6766_infinite__growing,axiom,
    ! [X6: set_num] :
      ( ( X6 != bot_bot_set_num )
     => ( ! [X3: num] :
            ( member_num(X3,X6)
           => ? [Xa3: num] :
                ( member_num(Xa3,X6)
                & aa_num_o(aa_num_fun_num_o(ord_less_num,X3),Xa3) ) )
       => ~ finite_finite_num(X6) ) ) ).

% infinite_growing
tff(fact_6767_infinite__growing,axiom,
    ! [X6: set_nat] :
      ( ( X6 != bot_bot_set_nat )
     => ( ! [X3: nat] :
            ( aa_set_nat_o(member_nat(X3),X6)
           => ? [Xa3: nat] :
                ( aa_set_nat_o(member_nat(Xa3),X6)
                & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Xa3) ) )
       => ~ aa_set_nat_o(finite_finite_nat,X6) ) ) ).

% infinite_growing
tff(fact_6768_infinite__growing,axiom,
    ! [X6: set_int] :
      ( ( X6 != bot_bot_set_int )
     => ( ! [X3: int] :
            ( aa_set_int_o(member_int(X3),X6)
           => ? [Xa3: int] :
                ( aa_set_int_o(member_int(Xa3),X6)
                & aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Xa3) ) )
       => ~ aa_set_int_o(finite_finite_int,X6) ) ) ).

% infinite_growing
tff(fact_6769_binomial__le__pow,axiom,
    ! [Ra2: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ra2),N2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,N2),Ra2)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,N2),Ra2)) ) ).

% binomial_le_pow
tff(fact_6770_diff__le__diff__pow,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N2)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,K),N2))) ) ).

% diff_le_diff_pow
tff(fact_6771_binomial__r__part__sum,axiom,
    ! [M: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)),one_one_nat))),aa_nat_set_nat(set_ord_atMost_nat,M)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)) ).

% binomial_r_part_sum
tff(fact_6772_atLeastLessThan0,axiom,
    ! [M: nat] : aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,M),zero_zero_nat) = bot_bot_set_nat ).

% atLeastLessThan0
tff(fact_6773_choose__linear__sum,axiom,
    ! [N2: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_tt,N2)),aa_nat_set_nat(set_ord_atMost_nat,N2)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),one_one_nat))) ).

% choose_linear_sum
tff(fact_6774_sum__mono__inv,axiom,
    ! [F: fun_o_rat,I5: set_o,G: fun_o_rat,I2: $o] :
      ( ( groups7872700643590313910_o_rat(F,I5) = groups7872700643590313910_o_rat(G,I5) )
     => ( ! [I3: $o] :
            ( member_o((I3),I5)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_o_rat(F,(I3))),aa_o_rat(G,(I3))) )
       => ( member_o((I2),I5)
         => ( finite_finite_o(I5)
           => ( aa_o_rat(F,(I2)) = aa_o_rat(G,(I2)) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_6775_sum__mono__inv,axiom,
    ! [F: fun_nat_rat,I5: set_nat,G: fun_nat_rat,I2: nat] :
      ( ( aa_set_nat_rat(groups2906978787729119204at_rat(F),I5) = aa_set_nat_rat(groups2906978787729119204at_rat(G),I5) )
     => ( ! [I3: nat] :
            ( aa_set_nat_o(member_nat(I3),I5)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,I3)),aa_nat_rat(G,I3)) )
       => ( aa_set_nat_o(member_nat(I2),I5)
         => ( aa_set_nat_o(finite_finite_nat,I5)
           => ( aa_nat_rat(F,I2) = aa_nat_rat(G,I2) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_6776_sum__mono__inv,axiom,
    ! [F: fun_int_rat,I5: set_int,G: fun_int_rat,I2: int] :
      ( ( groups3906332499630173760nt_rat(F,I5) = groups3906332499630173760nt_rat(G,I5) )
     => ( ! [I3: int] :
            ( aa_set_int_o(member_int(I3),I5)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(F,I3)),aa_int_rat(G,I3)) )
       => ( aa_set_int_o(member_int(I2),I5)
         => ( aa_set_int_o(finite_finite_int,I5)
           => ( aa_int_rat(F,I2) = aa_int_rat(G,I2) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_6777_sum__mono__inv,axiom,
    ! [F: fun_Code_integer_rat,I5: set_Code_integer,G: fun_Code_integer_rat,I2: code_integer] :
      ( ( groups6602215022474089585er_rat(F,I5) = groups6602215022474089585er_rat(G,I5) )
     => ( ! [I3: code_integer] :
            ( aa_set5643387332196575111eger_o(member_Code_integer(I3),I5)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_Code_integer_rat(F,I3)),aa_Code_integer_rat(G,I3)) )
       => ( aa_set5643387332196575111eger_o(member_Code_integer(I2),I5)
         => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,I5)
           => ( aa_Code_integer_rat(F,I2) = aa_Code_integer_rat(G,I2) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_6778_sum__mono__inv,axiom,
    ! [F: fun_o_nat,I5: set_o,G: fun_o_nat,I2: $o] :
      ( ( groups8507830703676809646_o_nat(F,I5) = groups8507830703676809646_o_nat(G,I5) )
     => ( ! [I3: $o] :
            ( member_o((I3),I5)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_o_nat(F,(I3))),aa_o_nat(G,(I3))) )
       => ( member_o((I2),I5)
         => ( finite_finite_o(I5)
           => ( aa_o_nat(F,(I2)) = aa_o_nat(G,(I2)) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_6779_sum__mono__inv,axiom,
    ! [F: fun_int_nat,I5: set_int,G: fun_int_nat,I2: int] :
      ( ( groups4541462559716669496nt_nat(F,I5) = groups4541462559716669496nt_nat(G,I5) )
     => ( ! [I3: int] :
            ( aa_set_int_o(member_int(I3),I5)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(F,I3)),aa_int_nat(G,I3)) )
       => ( aa_set_int_o(member_int(I2),I5)
         => ( aa_set_int_o(finite_finite_int,I5)
           => ( aa_int_nat(F,I2) = aa_int_nat(G,I2) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_6780_sum__mono__inv,axiom,
    ! [F: fun_Code_integer_nat,I5: set_Code_integer,G: fun_Code_integer_nat,I2: code_integer] :
      ( ( groups7237345082560585321er_nat(F,I5) = groups7237345082560585321er_nat(G,I5) )
     => ( ! [I3: code_integer] :
            ( aa_set5643387332196575111eger_o(member_Code_integer(I3),I5)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_Code_integer_nat(F,I3)),aa_Code_integer_nat(G,I3)) )
       => ( aa_set5643387332196575111eger_o(member_Code_integer(I2),I5)
         => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,I5)
           => ( aa_Code_integer_nat(F,I2) = aa_Code_integer_nat(G,I2) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_6781_sum__mono__inv,axiom,
    ! [F: fun_o_int,I5: set_o,G: fun_o_int,I2: $o] :
      ( ( groups8505340233167759370_o_int(F,I5) = groups8505340233167759370_o_int(G,I5) )
     => ( ! [I3: $o] :
            ( member_o((I3),I5)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_o_int(F,(I3))),aa_o_int(G,(I3))) )
       => ( member_o((I2),I5)
         => ( finite_finite_o(I5)
           => ( aa_o_int(F,(I2)) = aa_o_int(G,(I2)) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_6782_sum__mono__inv,axiom,
    ! [F: fun_nat_int,I5: set_nat,G: fun_nat_int,I2: nat] :
      ( ( aa_set_nat_int(groups3539618377306564664at_int(F),I5) = aa_set_nat_int(groups3539618377306564664at_int(G),I5) )
     => ( ! [I3: nat] :
            ( aa_set_nat_o(member_nat(I3),I5)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,I3)),aa_nat_int(G,I3)) )
       => ( aa_set_nat_o(member_nat(I2),I5)
         => ( aa_set_nat_o(finite_finite_nat,I5)
           => ( aa_nat_int(F,I2) = aa_nat_int(G,I2) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_6783_sum__mono__inv,axiom,
    ! [F: fun_Code_integer_int,I5: set_Code_integer,G: fun_Code_integer_int,I2: code_integer] :
      ( ( groups7234854612051535045er_int(F,I5) = groups7234854612051535045er_int(G,I5) )
     => ( ! [I3: code_integer] :
            ( aa_set5643387332196575111eger_o(member_Code_integer(I3),I5)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_Code_integer_int(F,I3)),aa_Code_integer_int(G,I3)) )
       => ( aa_set5643387332196575111eger_o(member_Code_integer(I2),I5)
         => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,I5)
           => ( aa_Code_integer_int(F,I2) = aa_Code_integer_int(G,I2) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_6784_infinite__Icc,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ~ finite_finite_rat(set_or633870826150836451st_rat(A,B)) ) ).

% infinite_Icc
tff(fact_6785_mult__numeral__1__right,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_num_nat(numeral_numeral_nat,one)) = A ).

% mult_numeral_1_right
tff(fact_6786_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_6787_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_6788_mult__numeral__1__right,axiom,
    ! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_num_rat(numeral_numeral_rat,one)) = A ).

% mult_numeral_1_right
tff(fact_6789_mult__numeral__1__right,axiom,
    ! [A: code_natural] : times_2397367101498566445atural(A,aa_num_Code_natural(numera5444537566228673987atural,one)) = A ).

% mult_numeral_1_right
tff(fact_6790_mult__numeral__1,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,one)),A) = A ).

% mult_numeral_1
tff(fact_6791_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_6792_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_6793_mult__numeral__1,axiom,
    ! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,one)),A) = A ).

% mult_numeral_1
tff(fact_6794_mult__numeral__1,axiom,
    ! [A: code_natural] : times_2397367101498566445atural(aa_num_Code_natural(numera5444537566228673987atural,one),A) = A ).

% mult_numeral_1
tff(fact_6795_sum__choose__upper,axiom,
    ! [M: nat,N2: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_tu,M)),aa_nat_set_nat(set_ord_atMost_nat,N2)) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,N2)),aa_nat_nat(suc,M)) ).

% sum_choose_upper
tff(fact_6796_sum_Oshift__bounds__Suc__ivl,axiom,
    ! [G: fun_nat_nat,M: nat,N2: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,aa_nat_nat(suc,M)),aa_nat_nat(suc,N2))) = aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_jx(G)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,M),N2)) ).

% sum.shift_bounds_Suc_ivl
tff(fact_6797_sum_Oshift__bounds__nat__ivl,axiom,
    ! [G: fun_nat_nat,M: nat,K: nat,N2: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(aa_nat1818925681583257441et_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,N2),K))) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_jy(G),K)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,M),N2)) ).

% sum.shift_bounds_nat_ivl
tff(fact_6798_prod_Oshift__bounds__Suc__ivl,axiom,
    ! [G: fun_nat_nat,M: nat,N2: nat] : aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,aa_nat_nat(suc,M)),aa_nat_nat(suc,N2))) = aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_jx(G)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,M),N2)) ).

% prod.shift_bounds_Suc_ivl
tff(fact_6799_prod_Oshift__bounds__Suc__ivl,axiom,
    ! [G: fun_nat_int,M: nat,N2: nat] : aa_set_nat_int(groups705719431365010083at_int(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,aa_nat_nat(suc,M)),aa_nat_nat(suc,N2))) = aa_set_nat_int(groups705719431365010083at_int(aTP_Lamm_kc(G)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,M),N2)) ).

% prod.shift_bounds_Suc_ivl
tff(fact_6800_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_o,X: fun_o_Code_integer,Y: fun_o_Code_integer] :
      ( finite_finite_o(collect_o(aa_fun8192762245619933327un_o_o(aTP_Lamm_tv(I5),X)))
     => ( finite_finite_o(collect_o(aa_fun8192762245619933327un_o_o(aTP_Lamm_tv(I5),Y)))
       => finite_finite_o(collect_o(aa_fun8192762245619933327un_o_o(aa_fun1865401534164435218un_o_o(aTP_Lamm_tw(I5),X),Y))) ) ) ).

% sum.finite_Collect_op
tff(fact_6801_sum_Ofinite__Collect__op,axiom,
    ! [I5: 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_tx(I5),X)))
     => ( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun3480067038916266681_nat_o(aTP_Lamm_tx(I5),Y)))
       => aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun3480067038916266681_nat_o(aa_fun2745573882733218024_nat_o(aTP_Lamm_ty(I5),X),Y))) ) ) ).

% sum.finite_Collect_op
tff(fact_6802_sum_Ofinite__Collect__op,axiom,
    ! [I5: 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_tz(I5),X)))
     => ( aa_set_int_o(finite_finite_int,collect_int(aa_fun3336537925265328129_int_o(aTP_Lamm_tz(I5),Y)))
       => aa_set_int_o(finite_finite_int,collect_int(aa_fun3336537925265328129_int_o(aa_fun8818272713621706452_int_o(aTP_Lamm_ua(I5),X),Y))) ) ) ).

% sum.finite_Collect_op
tff(fact_6803_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_Code_integer,X: fun_Co8499238673658658103nteger,Y: fun_Co8499238673658658103nteger] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,collect_Code_integer(aa_fun7159535342743012639eger_o(aTP_Lamm_ub(I5),X)))
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,collect_Code_integer(aa_fun7159535342743012639eger_o(aTP_Lamm_ub(I5),Y)))
       => aa_set5643387332196575111eger_o(finite6017078050557962740nteger,collect_Code_integer(aa_fun7159535342743012639eger_o(aa_fun5296778454019918337eger_o(aTP_Lamm_uc(I5),X),Y))) ) ) ).

% sum.finite_Collect_op
tff(fact_6804_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_o,X: fun_o_rat,Y: fun_o_rat] :
      ( finite_finite_o(collect_o(aa_fun_o_rat_fun_o_o(aTP_Lamm_ud(I5),X)))
     => ( finite_finite_o(collect_o(aa_fun_o_rat_fun_o_o(aTP_Lamm_ud(I5),Y)))
       => finite_finite_o(collect_o(aa_fun_o_rat_fun_o_o(aa_fun5510981055783399506un_o_o(aTP_Lamm_ue(I5),X),Y))) ) ) ).

% sum.finite_Collect_op
tff(fact_6805_sum_Ofinite__Collect__op,axiom,
    ! [I5: 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_uf(I5),X)))
     => ( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun1118977351124560894_nat_o(aTP_Lamm_uf(I5),Y)))
       => aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun1118977351124560894_nat_o(aa_fun8314661095172591474_nat_o(aTP_Lamm_ug(I5),X),Y))) ) ) ).

% sum.finite_Collect_op
tff(fact_6806_sum_Ofinite__Collect__op,axiom,
    ! [I5: 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_uh(I5),X)))
     => ( aa_set_int_o(finite_finite_int,collect_int(aa_fun7094158214180239686_int_o(aTP_Lamm_uh(I5),Y)))
       => aa_set_int_o(finite_finite_int,collect_int(aa_fun7094158214180239686_int_o(aa_fun1445893531507182942_int_o(aTP_Lamm_ui(I5),X),Y))) ) ) ).

% sum.finite_Collect_op
tff(fact_6807_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_Code_integer,X: fun_Code_integer_rat,Y: fun_Code_integer_rat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,collect_Code_integer(aa_fun6395433285121891940eger_o(aTP_Lamm_uj(I5),X)))
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,collect_Code_integer(aa_fun6395433285121891940eger_o(aTP_Lamm_uj(I5),Y)))
       => aa_set5643387332196575111eger_o(finite6017078050557962740nteger,collect_Code_integer(aa_fun6395433285121891940eger_o(aa_fun4095556841599367307eger_o(aTP_Lamm_uk(I5),X),Y))) ) ) ).

% sum.finite_Collect_op
tff(fact_6808_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_o,X: fun_o_nat,Y: fun_o_nat] :
      ( finite_finite_o(collect_o(aa_fun_o_nat_fun_o_o(aTP_Lamm_ul(I5),X)))
     => ( finite_finite_o(collect_o(aa_fun_o_nat_fun_o_o(aTP_Lamm_ul(I5),Y)))
       => finite_finite_o(collect_o(aa_fun_o_nat_fun_o_o(aa_fun5833554292436378194un_o_o(aTP_Lamm_um(I5),X),Y))) ) ) ).

% sum.finite_Collect_op
tff(fact_6809_sum_Ofinite__Collect__op,axiom,
    ! [I5: 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_un(I5),X)))
     => ( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun8673969331427483654_nat_o(aTP_Lamm_un(I5),Y)))
       => aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun8673969331427483654_nat_o(aa_fun6348600740168882050_nat_o(aTP_Lamm_uo(I5),X),Y))) ) ) ).

% sum.finite_Collect_op
tff(fact_6810_prod_Oshift__bounds__nat__ivl,axiom,
    ! [G: fun_nat_nat,M: nat,K: nat,N2: nat] : aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(aa_nat1818925681583257441et_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,N2),K))) = aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_jy(G),K)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,M),N2)) ).

% prod.shift_bounds_nat_ivl
tff(fact_6811_prod_Oshift__bounds__nat__ivl,axiom,
    ! [G: fun_nat_int,M: nat,K: nat,N2: nat] : aa_set_nat_int(groups705719431365010083at_int(G),aa_nat_set_nat(aa_nat1818925681583257441et_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,N2),K))) = aa_set_nat_int(groups705719431365010083at_int(aa_nat_fun_nat_int(aTP_Lamm_nq(G),K)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,M),N2)) ).

% prod.shift_bounds_nat_ivl
tff(fact_6812_divmod__digit__0_I2_J,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B))),B)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) ) ) ) ).

% divmod_digit_0(2)
tff(fact_6813_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_6814_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_6815_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_o,X: fun_o_Code_integer,Y: fun_o_Code_integer] :
      ( finite_finite_o(collect_o(aa_fun8192762245619933327un_o_o(aTP_Lamm_up(I5),X)))
     => ( finite_finite_o(collect_o(aa_fun8192762245619933327un_o_o(aTP_Lamm_up(I5),Y)))
       => finite_finite_o(collect_o(aa_fun8192762245619933327un_o_o(aa_fun1865401534164435218un_o_o(aTP_Lamm_uq(I5),X),Y))) ) ) ).

% prod.finite_Collect_op
tff(fact_6816_prod_Ofinite__Collect__op,axiom,
    ! [I5: 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_ur(I5),X)))
     => ( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun3480067038916266681_nat_o(aTP_Lamm_ur(I5),Y)))
       => aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun3480067038916266681_nat_o(aa_fun2745573882733218024_nat_o(aTP_Lamm_us(I5),X),Y))) ) ) ).

% prod.finite_Collect_op
tff(fact_6817_prod_Ofinite__Collect__op,axiom,
    ! [I5: 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_ut(I5),X)))
     => ( aa_set_int_o(finite_finite_int,collect_int(aa_fun3336537925265328129_int_o(aTP_Lamm_ut(I5),Y)))
       => aa_set_int_o(finite_finite_int,collect_int(aa_fun3336537925265328129_int_o(aa_fun8818272713621706452_int_o(aTP_Lamm_uu(I5),X),Y))) ) ) ).

% prod.finite_Collect_op
tff(fact_6818_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_Code_integer,X: fun_Co8499238673658658103nteger,Y: fun_Co8499238673658658103nteger] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,collect_Code_integer(aa_fun7159535342743012639eger_o(aTP_Lamm_uv(I5),X)))
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,collect_Code_integer(aa_fun7159535342743012639eger_o(aTP_Lamm_uv(I5),Y)))
       => aa_set5643387332196575111eger_o(finite6017078050557962740nteger,collect_Code_integer(aa_fun7159535342743012639eger_o(aa_fun5296778454019918337eger_o(aTP_Lamm_uw(I5),X),Y))) ) ) ).

% prod.finite_Collect_op
tff(fact_6819_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_o,X: fun_o_assn,Y: fun_o_assn] :
      ( finite_finite_o(collect_o(aa_fun8382565201914911066un_o_o(aTP_Lamm_ux(I5),X)))
     => ( finite_finite_o(collect_o(aa_fun8382565201914911066un_o_o(aTP_Lamm_ux(I5),Y)))
       => finite_finite_o(collect_o(aa_fun8382565201914911066un_o_o(aa_fun5794833117214906962un_o_o(aTP_Lamm_uy(I5),X),Y))) ) ) ).

% prod.finite_Collect_op
tff(fact_6820_prod_Ofinite__Collect__op,axiom,
    ! [I5: 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_uz(I5),X)))
     => ( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun7183233715851567790_nat_o(aTP_Lamm_uz(I5),Y)))
       => aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun7183233715851567790_nat_o(aa_fun8595640264602358482_nat_o(aTP_Lamm_va(I5),X),Y))) ) ) ).

% prod.finite_Collect_op
tff(fact_6821_prod_Ofinite__Collect__op,axiom,
    ! [I5: 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_vb(I5),X)))
     => ( aa_set_int_o(finite_finite_int,collect_int(aa_fun202081646129719286_int_o(aTP_Lamm_vb(I5),Y)))
       => aa_set_int_o(finite_finite_int,collect_int(aa_fun202081646129719286_int_o(aa_fun871127997271750846_int_o(aTP_Lamm_vc(I5),X),Y))) ) ) ).

% prod.finite_Collect_op
tff(fact_6822_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_Code_integer,X: fun_Co4490034186337852034r_assn,Y: fun_Co4490034186337852034r_assn] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,collect_Code_integer(aa_fun6538703224075513620eger_o(aTP_Lamm_vd(I5),X)))
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,collect_Code_integer(aa_fun6538703224075513620eger_o(aTP_Lamm_vd(I5),Y)))
       => aa_set5643387332196575111eger_o(finite6017078050557962740nteger,collect_Code_integer(aa_fun6538703224075513620eger_o(aa_fun4422309413433313771eger_o(aTP_Lamm_ve(I5),X),Y))) ) ) ).

% prod.finite_Collect_op
tff(fact_6823_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_o,X: fun_o_rat,Y: fun_o_rat] :
      ( finite_finite_o(collect_o(aa_fun_o_rat_fun_o_o(aTP_Lamm_vf(I5),X)))
     => ( finite_finite_o(collect_o(aa_fun_o_rat_fun_o_o(aTP_Lamm_vf(I5),Y)))
       => finite_finite_o(collect_o(aa_fun_o_rat_fun_o_o(aa_fun5510981055783399506un_o_o(aTP_Lamm_vg(I5),X),Y))) ) ) ).

% prod.finite_Collect_op
tff(fact_6824_prod_Ofinite__Collect__op,axiom,
    ! [I5: 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_vh(I5),X)))
     => ( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun1118977351124560894_nat_o(aTP_Lamm_vh(I5),Y)))
       => aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun1118977351124560894_nat_o(aa_fun8314661095172591474_nat_o(aTP_Lamm_vi(I5),X),Y))) ) ) ).

% prod.finite_Collect_op
tff(fact_6825_power2__less__imp__less,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_Code_integer(power_8256067586552552935nteger(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),Y)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,X),Y) ) ) ).

% power2_less_imp_less
tff(fact_6826_power2__less__imp__less,axiom,
    ! [X: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
     => ( 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_rat,X),Y) ) ) ).

% power2_less_imp_less
tff(fact_6827_power2__less__imp__less,axiom,
    ! [X: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Y)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y) ) ) ).

% power2_less_imp_less
tff(fact_6828_power2__less__imp__less,axiom,
    ! [X: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(aa_int_fun_nat_int(power_power_int,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,X),Y) ) ) ).

% power2_less_imp_less
tff(fact_6829_sum__power2__le__zero__iff,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_nat_Code_integer(power_8256067586552552935nteger(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_Code_integer(power_8256067586552552935nteger(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),zero_z3403309356797280102nteger)
    <=> ( ( X = zero_z3403309356797280102nteger )
        & ( Y = zero_z3403309356797280102nteger ) ) ) ).

% sum_power2_le_zero_iff
tff(fact_6830_sum__power2__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_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),zero_zero_rat)
    <=> ( ( X = zero_zero_rat )
        & ( Y = zero_zero_rat ) ) ) ).

% sum_power2_le_zero_iff
tff(fact_6831_sum__power2__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_nat_int(aa_int_fun_nat_int(power_power_int,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(aa_int_fun_nat_int(power_power_int,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),zero_zero_int)
    <=> ( ( X = zero_zero_int )
        & ( Y = zero_zero_int ) ) ) ).

% sum_power2_le_zero_iff
tff(fact_6832_sum__power2__ge__zero,axiom,
    ! [X: code_integer,Y: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_nat_Code_integer(power_8256067586552552935nteger(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_Code_integer(power_8256067586552552935nteger(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% sum_power2_ge_zero
tff(fact_6833_sum__power2__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_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% sum_power2_ge_zero
tff(fact_6834_sum__power2__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_nat_int(aa_int_fun_nat_int(power_power_int,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(aa_int_fun_nat_int(power_power_int,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% sum_power2_ge_zero
tff(fact_6835_sum__power2__gt__zero__iff,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_nat_Code_integer(power_8256067586552552935nteger(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_Code_integer(power_8256067586552552935nteger(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
    <=> ( ( X != zero_z3403309356797280102nteger )
        | ( Y != zero_z3403309356797280102nteger ) ) ) ).

% sum_power2_gt_zero_iff
tff(fact_6836_sum__power2__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_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
    <=> ( ( X != zero_zero_rat )
        | ( Y != zero_zero_rat ) ) ) ).

% sum_power2_gt_zero_iff
tff(fact_6837_sum__power2__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_nat_int(aa_int_fun_nat_int(power_power_int,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(aa_int_fun_nat_int(power_power_int,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
    <=> ( ( X != zero_zero_int )
        | ( Y != zero_zero_int ) ) ) ).

% sum_power2_gt_zero_iff
tff(fact_6838_not__sum__power2__lt__zero,axiom,
    ! [X: code_integer,Y: code_integer] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_nat_Code_integer(power_8256067586552552935nteger(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_Code_integer(power_8256067586552552935nteger(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),zero_z3403309356797280102nteger) ).

% not_sum_power2_lt_zero
tff(fact_6839_not__sum__power2__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_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),zero_zero_rat) ).

% not_sum_power2_lt_zero
tff(fact_6840_not__sum__power2__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_nat_int(aa_int_fun_nat_int(power_power_int,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(aa_int_fun_nat_int(power_power_int,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),zero_zero_int) ).

% not_sum_power2_lt_zero
tff(fact_6841_numeral__code_I2_J,axiom,
    ! [N2: num] :
      aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,N2)) = $let(
        m: nat,
        m:= aa_num_nat(numeral_numeral_nat,N2),
        aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,m),m) ) ).

% numeral_code(2)
tff(fact_6842_numeral__code_I2_J,axiom,
    ! [N2: num] :
      aa_num_int(numeral_numeral_int,aa_num_num(bit0,N2)) = $let(
        m: int,
        m:= aa_num_int(numeral_numeral_int,N2),
        aa_int_int(aa_int_fun_int_int(plus_plus_int,m),m) ) ).

% numeral_code(2)
tff(fact_6843_numeral__code_I2_J,axiom,
    ! [N2: num] :
      aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,N2)) = $let(
        m: code_integer,
        m:= aa_num_Code_integer(numera6620942414471956472nteger,N2),
        aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(m),m) ) ).

% numeral_code(2)
tff(fact_6844_numeral__code_I2_J,axiom,
    ! [N2: num] :
      aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,N2)) = $let(
        m: rat,
        m:= aa_num_rat(numeral_numeral_rat,N2),
        aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,m),m) ) ).

% numeral_code(2)
tff(fact_6845_numeral__code_I2_J,axiom,
    ! [N2: num] :
      aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,N2)) = $let(
        m: code_natural,
        m:= aa_num_Code_natural(numera5444537566228673987atural,N2),
        plus_p4538020629002901425atural(m,m) ) ).

% numeral_code(2)
tff(fact_6846_sum_Ointer__filter,axiom,
    ! [A3: set_o,G: fun_o_Code_integer,Pa: fun_o_o] :
      ( finite_finite_o(A3)
     => ( groups4406642042086082107nteger(G,collect_o(aa_fun_o_o_fun_o_o(aTP_Lamm_am(A3),Pa))) = groups4406642042086082107nteger(aa_fun6542514109734600263nteger(aTP_Lamm_vj(G),Pa),A3) ) ) ).

% sum.inter_filter
tff(fact_6847_sum_Ointer__filter,axiom,
    ! [A3: set_nat,G: fun_nat_Code_integer,Pa: fun_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( aa_set871585252387937313nteger(groups7501900531339628137nteger(G),collect_nat(aa_fun5937193402907943164_nat_o(aTP_Lamm_aq(A3),Pa))) = aa_set871585252387937313nteger(groups7501900531339628137nteger(aa_fun3969837373441324047nteger(aTP_Lamm_vk(G),Pa)),A3) ) ) ).

% sum.inter_filter
tff(fact_6848_sum_Ointer__filter,axiom,
    ! [A3: set_int,G: fun_int_Code_integer,Pa: fun_int_o] :
      ( aa_set_int_o(finite_finite_int,A3)
     => ( groups7873554091576472773nteger(G,collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_ar(A3),Pa))) = groups7873554091576472773nteger(aa_fun6019004165459918095nteger(aTP_Lamm_vl(G),Pa),A3) ) ) ).

% sum.inter_filter
tff(fact_6849_sum_Ointer__filter,axiom,
    ! [A3: set_Code_integer,G: fun_Co8499238673658658103nteger,Pa: fun_Code_integer_o] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A3)
     => ( groups879477027807139574nteger(G,collect_Code_integer(aa_fun8214818745163655228eger_o(aTP_Lamm_vm(A3),Pa))) = groups879477027807139574nteger(aa_fun791458909507501263nteger(aTP_Lamm_vn(G),Pa),A3) ) ) ).

% sum.inter_filter
tff(fact_6850_sum_Ointer__filter,axiom,
    ! [A3: set_o,G: fun_o_rat,Pa: fun_o_o] :
      ( finite_finite_o(A3)
     => ( groups7872700643590313910_o_rat(G,collect_o(aa_fun_o_o_fun_o_o(aTP_Lamm_am(A3),Pa))) = groups7872700643590313910_o_rat(aa_fun_o_o_fun_o_rat(aTP_Lamm_vo(G),Pa),A3) ) ) ).

% sum.inter_filter
tff(fact_6851_sum_Ointer__filter,axiom,
    ! [A3: set_nat,G: fun_nat_rat,Pa: fun_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( aa_set_nat_rat(groups2906978787729119204at_rat(G),collect_nat(aa_fun5937193402907943164_nat_o(aTP_Lamm_aq(A3),Pa))) = aa_set_nat_rat(groups2906978787729119204at_rat(aa_fun696632391729880074at_rat(aTP_Lamm_vp(G),Pa)),A3) ) ) ).

% sum.inter_filter
tff(fact_6852_sum_Ointer__filter,axiom,
    ! [A3: set_int,G: fun_int_rat,Pa: fun_int_o] :
      ( aa_set_int_o(finite_finite_int,A3)
     => ( groups3906332499630173760nt_rat(G,collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_ar(A3),Pa))) = groups3906332499630173760nt_rat(aa_fun268837275264525578nt_rat(aTP_Lamm_vq(G),Pa),A3) ) ) ).

% sum.inter_filter
tff(fact_6853_sum_Ointer__filter,axiom,
    ! [A3: set_Code_integer,G: fun_Code_integer_rat,Pa: fun_Code_integer_o] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A3)
     => ( groups6602215022474089585er_rat(G,collect_Code_integer(aa_fun8214818745163655228eger_o(aTP_Lamm_vm(A3),Pa))) = groups6602215022474089585er_rat(aa_fun25512833367352522er_rat(aTP_Lamm_vr(G),Pa),A3) ) ) ).

% sum.inter_filter
tff(fact_6854_sum_Ointer__filter,axiom,
    ! [A3: set_o,G: fun_o_nat,Pa: fun_o_o] :
      ( finite_finite_o(A3)
     => ( groups8507830703676809646_o_nat(G,collect_o(aa_fun_o_o_fun_o_o(aTP_Lamm_am(A3),Pa))) = groups8507830703676809646_o_nat(aa_fun_o_o_fun_o_nat(aTP_Lamm_vs(G),Pa),A3) ) ) ).

% sum.inter_filter
tff(fact_6855_sum_Ointer__filter,axiom,
    ! [A3: set_int,G: fun_int_nat,Pa: fun_int_o] :
      ( aa_set_int_o(finite_finite_int,A3)
     => ( groups4541462559716669496nt_nat(G,collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_ar(A3),Pa))) = groups4541462559716669496nt_nat(aa_fun3777065833657922306nt_nat(aTP_Lamm_vt(G),Pa),A3) ) ) ).

% sum.inter_filter
tff(fact_6856_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)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),X)),Y)) ).

% power2_sum
tff(fact_6857_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)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(aa_int_fun_nat_int(power_power_int,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),X)),Y)) ).

% power2_sum
tff(fact_6858_power2__sum,axiom,
    ! [X: code_integer,Y: code_integer] : aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(X),Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_nat_Code_integer(power_8256067586552552935nteger(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_Code_integer(power_8256067586552552935nteger(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),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_6859_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)),aa_num_nat(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),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),X)),Y)) ).

% power2_sum
tff(fact_6860_power2__sum,axiom,
    ! [X: code_natural,Y: code_natural] : aa_nat_Code_natural(power_7079662738309270450atural(plus_p4538020629002901425atural(X,Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = plus_p4538020629002901425atural(plus_p4538020629002901425atural(aa_nat_Code_natural(power_7079662738309270450atural(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_Code_natural(power_7079662738309270450atural(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),times_2397367101498566445atural(times_2397367101498566445atural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one)),X),Y)) ).

% power2_sum
tff(fact_6861_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),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_Code_integer) ) ) ).

% square_le_1
tff(fact_6862_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),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_rat) ) ) ).

% square_le_1
tff(fact_6863_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),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_int) ) ) ).

% square_le_1
tff(fact_6864_filter__preserves__multiset,axiom,
    ! [M6: fun_Pr5160692531401422620nt_nat,Pa: fun_Pr3238374608701176162_int_o] :
      ( finite2998713641127702882nt_int(collec213857154873943460nt_int(aTP_Lamm_vu(M6)))
     => finite2998713641127702882nt_int(collec213857154873943460nt_int(aa_fun5349570537165229698_int_o(aTP_Lamm_vv(M6),Pa))) ) ).

% filter_preserves_multiset
tff(fact_6865_filter__preserves__multiset,axiom,
    ! [M6: fun_list_nat_nat,Pa: fun_list_nat_o] :
      ( finite8100373058378681591st_nat(collect_list_nat(aTP_Lamm_vw(M6)))
     => finite8100373058378681591st_nat(collect_list_nat(aa_fun6155608546934989052_nat_o(aTP_Lamm_vx(M6),Pa))) ) ).

% filter_preserves_multiset
tff(fact_6866_filter__preserves__multiset,axiom,
    ! [M6: fun_set_nat_nat,Pa: fun_set_nat_o] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,collect_set_nat(aTP_Lamm_vy(M6)))
     => aa_set_set_nat_o(finite1152437895449049373et_nat,collect_set_nat(aa_fun751667557118977660_nat_o(aTP_Lamm_vz(M6),Pa))) ) ).

% filter_preserves_multiset
tff(fact_6867_filter__preserves__multiset,axiom,
    ! [M6: fun_nat_nat,Pa: fun_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,collect_nat(aTP_Lamm_wa(M6)))
     => aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun5937193402907943164_nat_o(aTP_Lamm_wb(M6),Pa))) ) ).

% filter_preserves_multiset
tff(fact_6868_filter__preserves__multiset,axiom,
    ! [M6: fun_int_nat,Pa: fun_int_o] :
      ( aa_set_int_o(finite_finite_int,collect_int(aTP_Lamm_wc(M6)))
     => aa_set_int_o(finite_finite_int,collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_wd(M6),Pa))) ) ).

% filter_preserves_multiset
tff(fact_6869_filter__preserves__multiset,axiom,
    ! [M6: fun_Code_integer_nat,Pa: fun_Code_integer_o] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,collect_Code_integer(aTP_Lamm_we(M6)))
     => aa_set5643387332196575111eger_o(finite6017078050557962740nteger,collect_Code_integer(aa_fun8214818745163655228eger_o(aTP_Lamm_wf(M6),Pa))) ) ).

% filter_preserves_multiset
tff(fact_6870_filter__preserves__multiset,axiom,
    ! [M6: fun_Pr2602117396471556196at_nat,Pa: fun_Pr4796943543331071514_nat_o] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,collec3392354462482085612at_nat(aTP_Lamm_wg(M6)))
     => aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,collec3392354462482085612at_nat(aa_fun359341089595923442_nat_o(aTP_Lamm_wh(M6),Pa))) ) ).

% filter_preserves_multiset
tff(fact_6871_zero__le__even__power_H,axiom,
    ! [A: code_integer,N2: nat] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2))) ).

% zero_le_even_power'
tff(fact_6872_zero__le__even__power_H,axiom,
    ! [A: rat,N2: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2))) ).

% zero_le_even_power'
tff(fact_6873_zero__le__even__power_H,axiom,
    ! [A: int,N2: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(aa_int_fun_nat_int(power_power_int,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2))) ).

% zero_le_even_power'
tff(fact_6874_power2__le__iff__abs__le,axiom,
    ! [Y: code_integer,X: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),Y)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_Code_integer(power_8256067586552552935nteger(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,X)),Y) ) ) ).

% power2_le_iff_abs_le
tff(fact_6875_power2__le__iff__abs__le,axiom,
    ! [Y: rat,X: rat] :
      ( 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,aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,X)),Y) ) ) ).

% power2_le_iff_abs_le
tff(fact_6876_power2__le__iff__abs__le,axiom,
    ! [Y: int,X: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(aa_int_fun_nat_int(power_power_int,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,X)),Y) ) ) ).

% power2_le_iff_abs_le
tff(fact_6877_sum_Oimage__gen,axiom,
    ! [S: set_o,Ha: fun_o_nat,G: fun_o_nat] :
      ( finite_finite_o(S)
     => ( groups8507830703676809646_o_nat(Ha,S) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun1225607396849667164at_nat(aa_fun2442258407354871084at_nat(aTP_Lamm_wj(S),Ha),G)),image_o_nat(G,S)) ) ) ).

% sum.image_gen
tff(fact_6878_sum_Oimage__gen,axiom,
    ! [S: set_int,Ha: fun_int_nat,G: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,S)
     => ( groups4541462559716669496nt_nat(Ha,S) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun5507398438843482068at_nat(aa_fun3220669823809117514at_nat(aTP_Lamm_wl(S),Ha),G)),image_int_nat(G,S)) ) ) ).

% sum.image_gen
tff(fact_6879_sum_Oimage__gen,axiom,
    ! [S: set_Code_integer,Ha: fun_Code_integer_nat,G: fun_Code_integer_nat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,S)
     => ( groups7237345082560585321er_nat(Ha,S) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun1707803128846032901at_nat(aa_fun2075405564639334828at_nat(aTP_Lamm_wn(S),Ha),G)),image_951025933927791156er_nat(G,S)) ) ) ).

% sum.image_gen
tff(fact_6880_sum_Oimage__gen,axiom,
    ! [S: set_o,Ha: fun_o_int,G: fun_o_int] :
      ( finite_finite_o(S)
     => ( groups8505340233167759370_o_int(Ha,S) = groups4538972089207619220nt_int(aa_fun1998554717657234288nt_int(aa_fun5578626826553213156nt_int(aTP_Lamm_wp(S),Ha),G),image_o_int(G,S)) ) ) ).

% sum.image_gen
tff(fact_6881_sum_Oimage__gen,axiom,
    ! [S: set_nat,Ha: fun_nat_int,G: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,S)
     => ( aa_set_nat_int(groups3539618377306564664at_int(Ha),S) = groups4538972089207619220nt_int(aa_fun6564396834021159052nt_int(aa_fun7327629743597663234nt_int(aTP_Lamm_wr(S),Ha),G),image_nat_int(G,S)) ) ) ).

% sum.image_gen
tff(fact_6882_sum_Oimage__gen,axiom,
    ! [S: set_Code_integer,Ha: fun_Code_integer_int,G: fun_Code_integer_int] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,S)
     => ( groups7234854612051535045er_int(Ha,S) = groups4538972089207619220nt_int(aa_fun2480750449653600025nt_int(aa_fun6168136379806246172nt_int(aTP_Lamm_wt(S),Ha),G),image_948535463418740880er_int(G,S)) ) ) ).

% sum.image_gen
tff(fact_6883_sum_Oimage__gen,axiom,
    ! [S: set_nat,Ha: fun_nat_nat,G: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,S)
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(Ha),S) = groups4541462559716669496nt_nat(aa_fun1518875816675579952nt_nat(aa_fun3769689747080337738nt_nat(aTP_Lamm_wu(S),Ha),G),image_nat_int(G,S)) ) ) ).

% sum.image_gen
tff(fact_6884_sum_Oimage__gen,axiom,
    ! [S: set_nat,Ha: fun_nat_nat,G: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,S)
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(Ha),S) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(aTP_Lamm_ww(S),Ha),G)),image_nat_nat(G,S)) ) ) ).

% sum.image_gen
tff(fact_6885_sum_Oimage__gen,axiom,
    ! [S: set_int,Ha: fun_int_int,G: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,S)
     => ( groups4538972089207619220nt_int(Ha,S) = aa_set_nat_int(groups3539618377306564664at_int(aa_fun1329547419334285360at_int(aa_fun6778609820326443010at_int(aTP_Lamm_wx(S),Ha),G)),image_int_nat(G,S)) ) ) ).

% sum.image_gen
tff(fact_6886_sum_Oimage__gen,axiom,
    ! [S: set_int,Ha: fun_int_int,G: fun_int_int] :
      ( aa_set_int_o(finite_finite_int,S)
     => ( groups4538972089207619220nt_int(Ha,S) = groups4538972089207619220nt_int(aa_fun6280345759651049192nt_int(aa_fun3693603986580398266nt_int(aTP_Lamm_wz(S),Ha),G),image_int_int(G,S)) ) ) ).

% sum.image_gen
tff(fact_6887_abs__square__le__1,axiom,
    ! [X: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(X),aa_num_nat(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_6888_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),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_rat)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,X)),one_one_rat) ) ).

% abs_square_le_1
tff(fact_6889_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),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,X)),one_one_int) ) ).

% abs_square_le_1
tff(fact_6890_abs__square__less__1,axiom,
    ! [X: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(X),aa_num_nat(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_6891_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),aa_num_nat(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_6892_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),aa_num_nat(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_6893_prod_Ointer__filter,axiom,
    ! [A3: set_o,G: fun_o_Code_integer,Pa: fun_o_o] :
      ( finite_finite_o(A3)
     => ( groups7694694392188491536nteger(G,collect_o(aa_fun_o_o_fun_o_o(aTP_Lamm_am(A3),Pa))) = groups7694694392188491536nteger(aa_fun6542514109734600263nteger(aTP_Lamm_xa(G),Pa),A3) ) ) ).

% prod.inter_filter
tff(fact_6894_prod_Ointer__filter,axiom,
    ! [A3: set_nat,G: fun_nat_Code_integer,Pa: fun_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( aa_set871585252387937313nteger(groups3455450783089532116nteger(G),collect_nat(aa_fun5937193402907943164_nat_o(aTP_Lamm_aq(A3),Pa))) = aa_set871585252387937313nteger(groups3455450783089532116nteger(aa_fun3969837373441324047nteger(aTP_Lamm_xb(G),Pa)),A3) ) ) ).

% prod.inter_filter
tff(fact_6895_prod_Ointer__filter,axiom,
    ! [A3: set_int,G: fun_int_Code_integer,Pa: fun_int_o] :
      ( aa_set_int_o(finite_finite_int,A3)
     => ( groups3827104343326376752nteger(G,collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_ar(A3),Pa))) = groups3827104343326376752nteger(aa_fun6019004165459918095nteger(aTP_Lamm_xc(G),Pa),A3) ) ) ).

% prod.inter_filter
tff(fact_6896_prod_Ointer__filter,axiom,
    ! [A3: set_Code_integer,G: fun_Co8499238673658658103nteger,Pa: fun_Code_integer_o] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A3)
     => ( groups3674199335183972705nteger(G,collect_Code_integer(aa_fun8214818745163655228eger_o(aTP_Lamm_vm(A3),Pa))) = groups3674199335183972705nteger(aa_fun791458909507501263nteger(aTP_Lamm_xd(G),Pa),A3) ) ) ).

% prod.inter_filter
tff(fact_6897_prod_Ointer__filter,axiom,
    ! [A3: set_o,G: fun_o_assn,Pa: fun_o_o] :
      ( finite_finite_o(A3)
     => ( groups5301882518646026715o_assn(G,collect_o(aa_fun_o_o_fun_o_o(aTP_Lamm_am(A3),Pa))) = groups5301882518646026715o_assn(aa_fun8560762093604944786o_assn(aTP_Lamm_xe(G),Pa),A3) ) ) ).

% prod.inter_filter
tff(fact_6898_prod_Ointer__filter,axiom,
    ! [A3: set_nat,G: fun_nat_assn,Pa: fun_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( groups6906906614972039071t_assn(G,collect_nat(aa_fun5937193402907943164_nat_o(aTP_Lamm_aq(A3),Pa))) = groups6906906614972039071t_assn(aa_fun3990007403578212698t_assn(aTP_Lamm_xf(G),Pa),A3) ) ) ).

% prod.inter_filter
tff(fact_6899_prod_Ointer__filter,axiom,
    ! [A3: set_int,G: fun_int_assn,Pa: fun_int_o] :
      ( aa_set_int_o(finite_finite_int,A3)
     => ( groups7882442080178216443t_assn(G,collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_ar(A3),Pa))) = groups7882442080178216443t_assn(aa_fun9028805181350005338t_assn(aTP_Lamm_xg(G),Pa),A3) ) ) ).

% prod.inter_filter
tff(fact_6900_prod_Ointer__filter,axiom,
    ! [A3: set_Code_integer,G: fun_Co4490034186337852034r_assn,Pa: fun_Code_integer_o] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A3)
     => ( groups1304777262505850412r_assn(G,collect_Code_integer(aa_fun8214818745163655228eger_o(aTP_Lamm_vm(A3),Pa))) = groups1304777262505850412r_assn(aa_fun6392223008197429786r_assn(aTP_Lamm_xh(G),Pa),A3) ) ) ).

% prod.inter_filter
tff(fact_6901_prod_Ointer__filter,axiom,
    ! [A3: set_o,G: fun_o_rat,Pa: fun_o_o] :
      ( finite_finite_o(A3)
     => ( groups2869687844427037835_o_rat(G,collect_o(aa_fun_o_o_fun_o_o(aTP_Lamm_am(A3),Pa))) = groups2869687844427037835_o_rat(aa_fun_o_o_fun_o_rat(aTP_Lamm_xi(G),Pa),A3) ) ) ).

% prod.inter_filter
tff(fact_6902_prod_Ointer__filter,axiom,
    ! [A3: set_nat,G: fun_nat_rat,Pa: fun_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( aa_set_nat_rat(groups73079841787564623at_rat(G),collect_nat(aa_fun5937193402907943164_nat_o(aTP_Lamm_aq(A3),Pa))) = aa_set_nat_rat(groups73079841787564623at_rat(aa_fun696632391729880074at_rat(aTP_Lamm_xj(G),Pa)),A3) ) ) ).

% prod.inter_filter
tff(fact_6903_prod_Oimage__gen,axiom,
    ! [S: set_o,Ha: fun_o_nat,G: fun_o_nat] :
      ( finite_finite_o(S)
     => ( groups3504817904513533571_o_nat(Ha,S) = aa_set_nat_nat(groups708209901874060359at_nat(aa_fun1225607396849667164at_nat(aa_fun2442258407354871084at_nat(aTP_Lamm_xk(S),Ha),G)),image_o_nat(G,S)) ) ) ).

% prod.image_gen
tff(fact_6904_prod_Oimage__gen,axiom,
    ! [S: set_int,Ha: fun_int_nat,G: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,S)
     => ( groups1707563613775114915nt_nat(Ha,S) = aa_set_nat_nat(groups708209901874060359at_nat(aa_fun5507398438843482068at_nat(aa_fun3220669823809117514at_nat(aTP_Lamm_xl(S),Ha),G)),image_int_nat(G,S)) ) ) ).

% prod.image_gen
tff(fact_6905_prod_Oimage__gen,axiom,
    ! [S: set_Code_integer,Ha: fun_Code_integer_nat,G: fun_Code_integer_nat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,S)
     => ( groups3190895334310489300er_nat(Ha,S) = aa_set_nat_nat(groups708209901874060359at_nat(aa_fun1707803128846032901at_nat(aa_fun2075405564639334828at_nat(aTP_Lamm_xm(S),Ha),G)),image_951025933927791156er_nat(G,S)) ) ) ).

% prod.image_gen
tff(fact_6906_prod_Oimage__gen,axiom,
    ! [S: set_o,Ha: fun_o_int,G: fun_o_nat] :
      ( finite_finite_o(S)
     => ( groups3502327434004483295_o_int(Ha,S) = aa_set_nat_int(groups705719431365010083at_int(aa_fun6271128414195246264at_int(aa_fun8663632660299257900at_int(aTP_Lamm_xn(S),Ha),G)),image_o_nat(G,S)) ) ) ).

% prod.image_gen
tff(fact_6907_prod_Oimage__gen,axiom,
    ! [S: set_Code_integer,Ha: fun_Code_integer_int,G: fun_Code_integer_nat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,S)
     => ( groups3188404863801439024er_int(Ha,S) = aa_set_nat_int(groups705719431365010083at_int(aa_fun6753324146191612001at_int(aa_fun29770176697515108at_int(aTP_Lamm_xo(S),Ha),G)),image_951025933927791156er_nat(G,S)) ) ) ).

% prod.image_gen
tff(fact_6908_prod_Oimage__gen,axiom,
    ! [S: set_o,Ha: fun_o_int,G: fun_o_int] :
      ( finite_finite_o(S)
     => ( groups3502327434004483295_o_int(Ha,S) = groups1705073143266064639nt_int(aa_fun1998554717657234288nt_int(aa_fun5578626826553213156nt_int(aTP_Lamm_xp(S),Ha),G),image_o_int(G,S)) ) ) ).

% prod.image_gen
tff(fact_6909_prod_Oimage__gen,axiom,
    ! [S: set_Code_integer,Ha: fun_Code_integer_int,G: fun_Code_integer_int] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,S)
     => ( groups3188404863801439024er_int(Ha,S) = groups1705073143266064639nt_int(aa_fun2480750449653600025nt_int(aa_fun6168136379806246172nt_int(aTP_Lamm_xq(S),Ha),G),image_948535463418740880er_int(G,S)) ) ) ).

% prod.image_gen
tff(fact_6910_prod_Oimage__gen,axiom,
    ! [S: set_nat,Ha: fun_nat_nat,G: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,S)
     => ( aa_set_nat_nat(groups708209901874060359at_nat(Ha),S) = groups1707563613775114915nt_nat(aa_fun1518875816675579952nt_nat(aa_fun3769689747080337738nt_nat(aTP_Lamm_xr(S),Ha),G),image_nat_int(G,S)) ) ) ).

% prod.image_gen
tff(fact_6911_prod_Oimage__gen,axiom,
    ! [S: set_nat,Ha: fun_nat_nat,G: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,S)
     => ( aa_set_nat_nat(groups708209901874060359at_nat(Ha),S) = aa_set_nat_nat(groups708209901874060359at_nat(aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(aTP_Lamm_xs(S),Ha),G)),image_nat_nat(G,S)) ) ) ).

% prod.image_gen
tff(fact_6912_prod_Oimage__gen,axiom,
    ! [S: set_nat,Ha: fun_nat_int,G: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,S)
     => ( aa_set_nat_int(groups705719431365010083at_int(Ha),S) = aa_set_nat_int(groups705719431365010083at_int(aa_fun1613598493704395220at_int(aa_fun1189263540488932170at_int(aTP_Lamm_xt(S),Ha),G)),image_nat_nat(G,S)) ) ) ).

% prod.image_gen
tff(fact_6913_minus__power__mult__self,axiom,
    ! [A: int,N2: 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)),N2)),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,A)),N2)) = aa_nat_int(aa_int_fun_nat_int(power_power_int,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2)) ).

% minus_power_mult_self
tff(fact_6914_minus__power__mult__self,axiom,
    ! [A: code_integer,N2: nat] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),N2)),aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),N2)) = aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2)) ).

% minus_power_mult_self
tff(fact_6915_minus__power__mult__self,axiom,
    ! [A: rat,N2: 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)),N2)),aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,A)),N2)) = aa_nat_rat(power_power_rat(A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2)) ).

% minus_power_mult_self
tff(fact_6916_power__odd__eq,axiom,
    ! [A: code_integer,N2: nat] : aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),aa_nat_Code_integer(power_8256067586552552935nteger(aa_nat_Code_integer(power_8256067586552552935nteger(A),N2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% power_odd_eq
tff(fact_6917_power__odd__eq,axiom,
    ! [A: assn,N2: nat] : aa_nat_assn(power_power_assn(A),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2))) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A),aa_nat_assn(power_power_assn(aa_nat_assn(power_power_assn(A),N2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% power_odd_eq
tff(fact_6918_power__odd__eq,axiom,
    ! [A: rat,N2: nat] : aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2))) = 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),N2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% power_odd_eq
tff(fact_6919_power__odd__eq,axiom,
    ! [A: nat,N2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2))) = 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),N2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% power_odd_eq
tff(fact_6920_power__odd__eq,axiom,
    ! [A: int,N2: 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,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2))) = 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),N2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% power_odd_eq
tff(fact_6921_Suc__n__div__2__gt__zero,axiom,
    ! [N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,N2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ).

% Suc_n_div_2_gt_zero
tff(fact_6922_div__2__gt__zero,axiom,
    ! [N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),N2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ).

% div_2_gt_zero
tff(fact_6923_finite__int__segment,axiom,
    ! [A: rat,B: rat] : finite_finite_rat(collect_rat(aa_rat_fun_rat_o(aTP_Lamm_xu(A),B))) ).

% finite_int_segment
tff(fact_6924_zero__less__binomial,axiom,
    ! [K: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),N2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(binomial,N2),K)) ) ).

% zero_less_binomial
tff(fact_6925_sum_Oivl__cong,axiom,
    ! [A: nat,Ca: nat,B: nat,D2: nat,G: fun_nat_nat,Ha: fun_nat_nat] :
      ( ( A = Ca )
     => ( ( B = D2 )
       => ( ! [X3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ca),X3)
             => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),D2)
               => ( aa_nat_nat(G,X3) = aa_nat_nat(Ha,X3) ) ) )
         => ( aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,A),B)) = aa_set_nat_nat(groups3542108847815614940at_nat(Ha),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,Ca),D2)) ) ) ) ) ).

% sum.ivl_cong
tff(fact_6926_sum_Oivl__cong,axiom,
    ! [A: int,Ca: int,B: int,D2: int,G: fun_int_int,Ha: fun_int_int] :
      ( ( A = Ca )
     => ( ( B = D2 )
       => ( ! [X3: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ca),X3)
             => ( aa_int_o(aa_int_fun_int_o(ord_less_int,X3),D2)
               => ( aa_int_int(G,X3) = aa_int_int(Ha,X3) ) ) )
         => ( groups4538972089207619220nt_int(G,set_or4662586982721622107an_int(A,B)) = groups4538972089207619220nt_int(Ha,set_or4662586982721622107an_int(Ca,D2)) ) ) ) ) ).

% sum.ivl_cong
tff(fact_6927_prod_Oivl__cong,axiom,
    ! [A: nat,Ca: nat,B: nat,D2: nat,G: fun_nat_nat,Ha: fun_nat_nat] :
      ( ( A = Ca )
     => ( ( B = D2 )
       => ( ! [X3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ca),X3)
             => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),D2)
               => ( aa_nat_nat(G,X3) = aa_nat_nat(Ha,X3) ) ) )
         => ( aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,A),B)) = aa_set_nat_nat(groups708209901874060359at_nat(Ha),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,Ca),D2)) ) ) ) ) ).

% prod.ivl_cong
tff(fact_6928_prod_Oivl__cong,axiom,
    ! [A: nat,Ca: nat,B: nat,D2: nat,G: fun_nat_int,Ha: fun_nat_int] :
      ( ( A = Ca )
     => ( ( B = D2 )
       => ( ! [X3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ca),X3)
             => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),D2)
               => ( aa_nat_int(G,X3) = aa_nat_int(Ha,X3) ) ) )
         => ( aa_set_nat_int(groups705719431365010083at_int(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,A),B)) = aa_set_nat_int(groups705719431365010083at_int(Ha),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,Ca),D2)) ) ) ) ) ).

% prod.ivl_cong
tff(fact_6929_prod_Oivl__cong,axiom,
    ! [A: int,Ca: int,B: int,D2: int,G: fun_int_int,Ha: fun_int_int] :
      ( ( A = Ca )
     => ( ( B = D2 )
       => ( ! [X3: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ca),X3)
             => ( aa_int_o(aa_int_fun_int_o(ord_less_int,X3),D2)
               => ( aa_int_int(G,X3) = aa_int_int(Ha,X3) ) ) )
         => ( groups1705073143266064639nt_int(G,set_or4662586982721622107an_int(A,B)) = groups1705073143266064639nt_int(Ha,set_or4662586982721622107an_int(Ca,D2)) ) ) ) ) ).

% prod.ivl_cong
tff(fact_6930_choose__mult,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,N2),M)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,M),K)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,N2),K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),K)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),K))) ) ) ) ).

% choose_mult
tff(fact_6931_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)
       => ( 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_6932_ivl__disj__un__two_I7_J,axiom,
    ! [L: num,M: num,U: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,L),M)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,M),U)
       => ( sup_sup_set_num(set_or1222409239386451017an_num(L,M),set_or7049704709247886629st_num(M,U)) = set_or7049704709247886629st_num(L,U) ) ) ) ).

% ivl_disj_un_two(7)
tff(fact_6933_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(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,L),M)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M),U)) = aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,L),U) ) ) ) ).

% ivl_disj_un_two(7)
tff(fact_6934_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_6935_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_6936_binomial__altdef__of__nat,axiom,
    ! [K: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),N2)
     => ( aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,N2),K)) = aa_set_nat_rat(groups73079841787564623at_rat(aa_nat_fun_nat_rat(aTP_Lamm_xv(K),N2)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),K)) ) ) ).

% binomial_altdef_of_nat
tff(fact_6937_divmod__digit__0_I1_J,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B))),B)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B))) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B) ) ) ) ).

% divmod_digit_0(1)
tff(fact_6938_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_6939_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_6940_ivl__disj__int__two_I7_J,axiom,
    ! [L: $o,M: $o,U: $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_6941_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(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,L),M)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M),U)) = bot_bot_set_nat ).

% ivl_disj_int_two(7)
tff(fact_6942_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_6943_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_6944_sum_OatLeastLessThan__concat,axiom,
    ! [M: nat,N2: nat,P2: nat,G: fun_na2338080690716861136at_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),P2)
       => ( aa_mul5754855110707352700at_nat(plus_p7104986032573967614at_nat(groups6857163185585827899at_nat(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,M),N2))),groups6857163185585827899at_nat(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,N2),P2))) = groups6857163185585827899at_nat(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,M),P2)) ) ) ) ).

% sum.atLeastLessThan_concat
tff(fact_6945_sum_OatLeastLessThan__concat,axiom,
    ! [M: nat,N2: nat,P2: nat,G: fun_nat_rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),P2)
       => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_set_nat_rat(groups2906978787729119204at_rat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,M),N2))),aa_set_nat_rat(groups2906978787729119204at_rat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,N2),P2))) = aa_set_nat_rat(groups2906978787729119204at_rat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,M),P2)) ) ) ) ).

% sum.atLeastLessThan_concat
tff(fact_6946_sum_OatLeastLessThan__concat,axiom,
    ! [M: nat,N2: nat,P2: nat,G: fun_nat_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),P2)
       => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_set_nat_int(groups3539618377306564664at_int(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,M),N2))),aa_set_nat_int(groups3539618377306564664at_int(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,N2),P2))) = aa_set_nat_int(groups3539618377306564664at_int(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,M),P2)) ) ) ) ).

% sum.atLeastLessThan_concat
tff(fact_6947_sum_OatLeastLessThan__concat,axiom,
    ! [M: nat,N2: nat,P2: nat,G: fun_nat_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),P2)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,M),N2))),aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,N2),P2))) = aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,M),P2)) ) ) ) ).

% sum.atLeastLessThan_concat
tff(fact_6948_sum__diff__nat__ivl,axiom,
    ! [M: nat,N2: nat,P2: nat,F: fun_nat_rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),P2)
       => ( aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_set_nat_rat(groups2906978787729119204at_rat(F),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,M),P2))),aa_set_nat_rat(groups2906978787729119204at_rat(F),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,M),N2))) = aa_set_nat_rat(groups2906978787729119204at_rat(F),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,N2),P2)) ) ) ) ).

% sum_diff_nat_ivl
tff(fact_6949_sum__diff__nat__ivl,axiom,
    ! [M: nat,N2: nat,P2: nat,F: fun_nat_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),P2)
       => ( aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_set_nat_int(groups3539618377306564664at_int(F),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,M),P2))),aa_set_nat_int(groups3539618377306564664at_int(F),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,M),N2))) = aa_set_nat_int(groups3539618377306564664at_int(F),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,N2),P2)) ) ) ) ).

% sum_diff_nat_ivl
tff(fact_6950_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)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(aa_int_fun_nat_int(power_power_int,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),X)),Y)) ).

% power2_diff
tff(fact_6951_power2__diff,axiom,
    ! [X: code_integer,Y: code_integer] : aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(X),Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_nat_Code_integer(power_8256067586552552935nteger(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_Code_integer(power_8256067586552552935nteger(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),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_6952_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)),aa_num_nat(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),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),X)),Y)) ).

% power2_diff
tff(fact_6953_binomial__code,axiom,
    ! [N2: nat,K: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(binomial,N2),K) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),K),
        zero_zero_nat,
        $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,N2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),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,N2),K)),one_one_nat)),N2),one_one_nat)),aa_nat_nat(semiri1408675320244567234ct_nat,K))) ) ).

% binomial_code
tff(fact_6954_prod_OatLeastLessThan__concat,axiom,
    ! [M: nat,N2: nat,P2: nat,G: fun_nat_Code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),P2)
       => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_set871585252387937313nteger(groups3455450783089532116nteger(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,M),N2))),aa_set871585252387937313nteger(groups3455450783089532116nteger(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,N2),P2))) = aa_set871585252387937313nteger(groups3455450783089532116nteger(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,M),P2)) ) ) ) ).

% prod.atLeastLessThan_concat
tff(fact_6955_prod_OatLeastLessThan__concat,axiom,
    ! [M: nat,N2: nat,P2: nat,G: fun_nat_assn] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),P2)
       => ( aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,groups6906906614972039071t_assn(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,M),N2))),groups6906906614972039071t_assn(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,N2),P2))) = groups6906906614972039071t_assn(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,M),P2)) ) ) ) ).

% prod.atLeastLessThan_concat
tff(fact_6956_prod_OatLeastLessThan__concat,axiom,
    ! [M: nat,N2: nat,P2: nat,G: fun_nat_rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),P2)
       => ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_set_nat_rat(groups73079841787564623at_rat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,M),N2))),aa_set_nat_rat(groups73079841787564623at_rat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,N2),P2))) = aa_set_nat_rat(groups73079841787564623at_rat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,M),P2)) ) ) ) ).

% prod.atLeastLessThan_concat
tff(fact_6957_prod_OatLeastLessThan__concat,axiom,
    ! [M: nat,N2: nat,P2: nat,G: fun_nat_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),P2)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,M),N2))),aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,N2),P2))) = aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,M),P2)) ) ) ) ).

% prod.atLeastLessThan_concat
tff(fact_6958_prod_OatLeastLessThan__concat,axiom,
    ! [M: nat,N2: nat,P2: nat,G: fun_nat_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),P2)
       => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_set_nat_int(groups705719431365010083at_int(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,M),N2))),aa_set_nat_int(groups705719431365010083at_int(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,N2),P2))) = aa_set_nat_int(groups705719431365010083at_int(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,M),P2)) ) ) ) ).

% prod.atLeastLessThan_concat
tff(fact_6959_odd__0__le__power__imp__0__le,axiom,
    ! [A: code_integer,N2: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2))))
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A) ) ).

% odd_0_le_power_imp_0_le
tff(fact_6960_odd__0__le__power__imp__0__le,axiom,
    ! [A: rat,N2: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2))))
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A) ) ).

% odd_0_le_power_imp_0_le
tff(fact_6961_odd__0__le__power__imp__0__le,axiom,
    ! [A: int,N2: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),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,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2))))
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A) ) ).

% odd_0_le_power_imp_0_le
tff(fact_6962_odd__power__less__zero,axiom,
    ! [A: code_integer,N2: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),zero_z3403309356797280102nteger)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2)))),zero_z3403309356797280102nteger) ) ).

% odd_power_less_zero
tff(fact_6963_odd__power__less__zero,axiom,
    ! [A: rat,N2: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2)))),zero_zero_rat) ) ).

% odd_power_less_zero
tff(fact_6964_odd__power__less__zero,axiom,
    ! [A: int,N2: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2)))),zero_zero_int) ) ).

% odd_power_less_zero
tff(fact_6965_sum__nonneg__eq__0__iff,axiom,
    ! [A3: set_o,F: fun_o_Code_integer] :
      ( finite_finite_o(A3)
     => ( ! [X3: $o] :
            ( member_o((X3),A3)
           => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_o_Code_integer(F,(X3))) )
       => ( ( groups4406642042086082107nteger(F,A3) = zero_z3403309356797280102nteger )
        <=> ! [X4: $o] :
              ( member_o((X4),A3)
             => ( aa_o_Code_integer(F,(X4)) = zero_z3403309356797280102nteger ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_6966_sum__nonneg__eq__0__iff,axiom,
    ! [A3: set_nat,F: fun_nat_Code_integer] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( ! [X3: nat] :
            ( aa_set_nat_o(member_nat(X3),A3)
           => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(F,X3)) )
       => ( ( aa_set871585252387937313nteger(groups7501900531339628137nteger(F),A3) = zero_z3403309356797280102nteger )
        <=> ! [X4: nat] :
              ( aa_set_nat_o(member_nat(X4),A3)
             => ( aa_nat_Code_integer(F,X4) = zero_z3403309356797280102nteger ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_6967_sum__nonneg__eq__0__iff,axiom,
    ! [A3: set_int,F: fun_int_Code_integer] :
      ( aa_set_int_o(finite_finite_int,A3)
     => ( ! [X3: int] :
            ( aa_set_int_o(member_int(X3),A3)
           => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_int_Code_integer(F,X3)) )
       => ( ( groups7873554091576472773nteger(F,A3) = zero_z3403309356797280102nteger )
        <=> ! [X4: int] :
              ( aa_set_int_o(member_int(X4),A3)
             => ( aa_int_Code_integer(F,X4) = zero_z3403309356797280102nteger ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_6968_sum__nonneg__eq__0__iff,axiom,
    ! [A3: set_Code_integer,F: fun_Co8499238673658658103nteger] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A3)
     => ( ! [X3: code_integer] :
            ( aa_set5643387332196575111eger_o(member_Code_integer(X3),A3)
           => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(F,X3)) )
       => ( ( groups879477027807139574nteger(F,A3) = zero_z3403309356797280102nteger )
        <=> ! [X4: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X4),A3)
             => ( aa_Cod1783974652533396728nteger(F,X4) = zero_z3403309356797280102nteger ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_6969_sum__nonneg__eq__0__iff,axiom,
    ! [A3: set_o,F: fun_o_rat] :
      ( finite_finite_o(A3)
     => ( ! [X3: $o] :
            ( member_o((X3),A3)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_o_rat(F,(X3))) )
       => ( ( groups7872700643590313910_o_rat(F,A3) = zero_zero_rat )
        <=> ! [X4: $o] :
              ( member_o((X4),A3)
             => ( aa_o_rat(F,(X4)) = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_6970_sum__nonneg__eq__0__iff,axiom,
    ! [A3: set_nat,F: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( ! [X3: nat] :
            ( aa_set_nat_o(member_nat(X3),A3)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(F,X3)) )
       => ( ( aa_set_nat_rat(groups2906978787729119204at_rat(F),A3) = zero_zero_rat )
        <=> ! [X4: nat] :
              ( aa_set_nat_o(member_nat(X4),A3)
             => ( aa_nat_rat(F,X4) = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_6971_sum__nonneg__eq__0__iff,axiom,
    ! [A3: set_int,F: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,A3)
     => ( ! [X3: int] :
            ( aa_set_int_o(member_int(X3),A3)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_int_rat(F,X3)) )
       => ( ( groups3906332499630173760nt_rat(F,A3) = zero_zero_rat )
        <=> ! [X4: int] :
              ( aa_set_int_o(member_int(X4),A3)
             => ( aa_int_rat(F,X4) = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_6972_sum__nonneg__eq__0__iff,axiom,
    ! [A3: set_Code_integer,F: fun_Code_integer_rat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A3)
     => ( ! [X3: code_integer] :
            ( aa_set5643387332196575111eger_o(member_Code_integer(X3),A3)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_Code_integer_rat(F,X3)) )
       => ( ( groups6602215022474089585er_rat(F,A3) = zero_zero_rat )
        <=> ! [X4: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X4),A3)
             => ( aa_Code_integer_rat(F,X4) = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_6973_sum__nonneg__eq__0__iff,axiom,
    ! [A3: set_o,F: fun_o_nat] :
      ( finite_finite_o(A3)
     => ( ! [X3: $o] :
            ( member_o((X3),A3)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_o_nat(F,(X3))) )
       => ( ( groups8507830703676809646_o_nat(F,A3) = zero_zero_nat )
        <=> ! [X4: $o] :
              ( member_o((X4),A3)
             => ( aa_o_nat(F,(X4)) = zero_zero_nat ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_6974_sum__nonneg__eq__0__iff,axiom,
    ! [A3: set_int,F: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,A3)
     => ( ! [X3: int] :
            ( aa_set_int_o(member_int(X3),A3)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_int_nat(F,X3)) )
       => ( ( groups4541462559716669496nt_nat(F,A3) = zero_zero_nat )
        <=> ! [X4: int] :
              ( aa_set_int_o(member_int(X4),A3)
             => ( aa_int_nat(F,X4) = zero_zero_nat ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_6975_sum__le__included,axiom,
    ! [S2: set_nat,Ta: set_nat,G: fun_nat_Code_integer,I2: fun_nat_nat,F: fun_nat_Code_integer] :
      ( aa_set_nat_o(finite_finite_nat,S2)
     => ( aa_set_nat_o(finite_finite_nat,Ta)
       => ( ! [X3: nat] :
              ( aa_set_nat_o(member_nat(X3),Ta)
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(G,X3)) )
         => ( ! [X3: nat] :
                ( aa_set_nat_o(member_nat(X3),S2)
               => ? [Xa3: nat] :
                    ( aa_set_nat_o(member_nat(Xa3),Ta)
                    & ( aa_nat_nat(I2,Xa3) = X3 )
                    & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(F,X3)),aa_nat_Code_integer(G,Xa3)) ) )
           => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_set871585252387937313nteger(groups7501900531339628137nteger(F),S2)),aa_set871585252387937313nteger(groups7501900531339628137nteger(G),Ta)) ) ) ) ) ).

% sum_le_included
tff(fact_6976_sum__le__included,axiom,
    ! [S2: set_nat,Ta: set_int,G: fun_int_Code_integer,I2: fun_int_nat,F: fun_nat_Code_integer] :
      ( aa_set_nat_o(finite_finite_nat,S2)
     => ( aa_set_int_o(finite_finite_int,Ta)
       => ( ! [X3: int] :
              ( aa_set_int_o(member_int(X3),Ta)
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_int_Code_integer(G,X3)) )
         => ( ! [X3: nat] :
                ( aa_set_nat_o(member_nat(X3),S2)
               => ? [Xa3: int] :
                    ( aa_set_int_o(member_int(Xa3),Ta)
                    & ( aa_int_nat(I2,Xa3) = X3 )
                    & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(F,X3)),aa_int_Code_integer(G,Xa3)) ) )
           => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_set871585252387937313nteger(groups7501900531339628137nteger(F),S2)),groups7873554091576472773nteger(G,Ta)) ) ) ) ) ).

% sum_le_included
tff(fact_6977_sum__le__included,axiom,
    ! [S2: set_nat,Ta: set_Code_integer,G: fun_Co8499238673658658103nteger,I2: fun_Code_integer_nat,F: fun_nat_Code_integer] :
      ( aa_set_nat_o(finite_finite_nat,S2)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,Ta)
       => ( ! [X3: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X3),Ta)
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(G,X3)) )
         => ( ! [X3: nat] :
                ( aa_set_nat_o(member_nat(X3),S2)
               => ? [Xa3: code_integer] :
                    ( aa_set5643387332196575111eger_o(member_Code_integer(Xa3),Ta)
                    & ( aa_Code_integer_nat(I2,Xa3) = X3 )
                    & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(F,X3)),aa_Cod1783974652533396728nteger(G,Xa3)) ) )
           => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_set871585252387937313nteger(groups7501900531339628137nteger(F),S2)),groups879477027807139574nteger(G,Ta)) ) ) ) ) ).

% sum_le_included
tff(fact_6978_sum__le__included,axiom,
    ! [S2: set_int,Ta: set_nat,G: fun_nat_Code_integer,I2: fun_nat_int,F: fun_int_Code_integer] :
      ( aa_set_int_o(finite_finite_int,S2)
     => ( aa_set_nat_o(finite_finite_nat,Ta)
       => ( ! [X3: nat] :
              ( aa_set_nat_o(member_nat(X3),Ta)
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(G,X3)) )
         => ( ! [X3: int] :
                ( aa_set_int_o(member_int(X3),S2)
               => ? [Xa3: nat] :
                    ( aa_set_nat_o(member_nat(Xa3),Ta)
                    & ( aa_nat_int(I2,Xa3) = X3 )
                    & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_int_Code_integer(F,X3)),aa_nat_Code_integer(G,Xa3)) ) )
           => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,groups7873554091576472773nteger(F,S2)),aa_set871585252387937313nteger(groups7501900531339628137nteger(G),Ta)) ) ) ) ) ).

% sum_le_included
tff(fact_6979_sum__le__included,axiom,
    ! [S2: set_int,Ta: set_int,G: fun_int_Code_integer,I2: fun_int_int,F: fun_int_Code_integer] :
      ( aa_set_int_o(finite_finite_int,S2)
     => ( aa_set_int_o(finite_finite_int,Ta)
       => ( ! [X3: int] :
              ( aa_set_int_o(member_int(X3),Ta)
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_int_Code_integer(G,X3)) )
         => ( ! [X3: int] :
                ( aa_set_int_o(member_int(X3),S2)
               => ? [Xa3: int] :
                    ( aa_set_int_o(member_int(Xa3),Ta)
                    & ( aa_int_int(I2,Xa3) = X3 )
                    & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_int_Code_integer(F,X3)),aa_int_Code_integer(G,Xa3)) ) )
           => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,groups7873554091576472773nteger(F,S2)),groups7873554091576472773nteger(G,Ta)) ) ) ) ) ).

% sum_le_included
tff(fact_6980_sum__le__included,axiom,
    ! [S2: set_int,Ta: set_Code_integer,G: fun_Co8499238673658658103nteger,I2: fun_Code_integer_int,F: fun_int_Code_integer] :
      ( aa_set_int_o(finite_finite_int,S2)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,Ta)
       => ( ! [X3: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X3),Ta)
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(G,X3)) )
         => ( ! [X3: int] :
                ( aa_set_int_o(member_int(X3),S2)
               => ? [Xa3: code_integer] :
                    ( aa_set5643387332196575111eger_o(member_Code_integer(Xa3),Ta)
                    & ( aa_Code_integer_int(I2,Xa3) = X3 )
                    & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_int_Code_integer(F,X3)),aa_Cod1783974652533396728nteger(G,Xa3)) ) )
           => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,groups7873554091576472773nteger(F,S2)),groups879477027807139574nteger(G,Ta)) ) ) ) ) ).

% sum_le_included
tff(fact_6981_sum__le__included,axiom,
    ! [S2: set_Code_integer,Ta: set_nat,G: fun_nat_Code_integer,I2: fun_nat_Code_integer,F: fun_Co8499238673658658103nteger] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,S2)
     => ( aa_set_nat_o(finite_finite_nat,Ta)
       => ( ! [X3: nat] :
              ( aa_set_nat_o(member_nat(X3),Ta)
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(G,X3)) )
         => ( ! [X3: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(X3),S2)
               => ? [Xa3: nat] :
                    ( aa_set_nat_o(member_nat(Xa3),Ta)
                    & ( aa_nat_Code_integer(I2,Xa3) = X3 )
                    & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(F,X3)),aa_nat_Code_integer(G,Xa3)) ) )
           => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,groups879477027807139574nteger(F,S2)),aa_set871585252387937313nteger(groups7501900531339628137nteger(G),Ta)) ) ) ) ) ).

% sum_le_included
tff(fact_6982_sum__le__included,axiom,
    ! [S2: set_Code_integer,Ta: set_int,G: fun_int_Code_integer,I2: fun_int_Code_integer,F: fun_Co8499238673658658103nteger] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,S2)
     => ( aa_set_int_o(finite_finite_int,Ta)
       => ( ! [X3: int] :
              ( aa_set_int_o(member_int(X3),Ta)
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_int_Code_integer(G,X3)) )
         => ( ! [X3: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(X3),S2)
               => ? [Xa3: int] :
                    ( aa_set_int_o(member_int(Xa3),Ta)
                    & ( aa_int_Code_integer(I2,Xa3) = X3 )
                    & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(F,X3)),aa_int_Code_integer(G,Xa3)) ) )
           => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,groups879477027807139574nteger(F,S2)),groups7873554091576472773nteger(G,Ta)) ) ) ) ) ).

% sum_le_included
tff(fact_6983_sum__le__included,axiom,
    ! [S2: set_Code_integer,Ta: set_Code_integer,G: fun_Co8499238673658658103nteger,I2: fun_Co8499238673658658103nteger,F: fun_Co8499238673658658103nteger] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,S2)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,Ta)
       => ( ! [X3: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X3),Ta)
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(G,X3)) )
         => ( ! [X3: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(X3),S2)
               => ? [Xa3: code_integer] :
                    ( aa_set5643387332196575111eger_o(member_Code_integer(Xa3),Ta)
                    & ( aa_Cod1783974652533396728nteger(I2,Xa3) = X3 )
                    & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(F,X3)),aa_Cod1783974652533396728nteger(G,Xa3)) ) )
           => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,groups879477027807139574nteger(F,S2)),groups879477027807139574nteger(G,Ta)) ) ) ) ) ).

% sum_le_included
tff(fact_6984_sum__le__included,axiom,
    ! [S2: set_nat,Ta: set_nat,G: fun_nat_rat,I2: fun_nat_nat,F: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,S2)
     => ( aa_set_nat_o(finite_finite_nat,Ta)
       => ( ! [X3: nat] :
              ( aa_set_nat_o(member_nat(X3),Ta)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(G,X3)) )
         => ( ! [X3: nat] :
                ( aa_set_nat_o(member_nat(X3),S2)
               => ? [Xa3: nat] :
                    ( aa_set_nat_o(member_nat(Xa3),Ta)
                    & ( aa_nat_nat(I2,Xa3) = X3 )
                    & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,X3)),aa_nat_rat(G,Xa3)) ) )
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_set_nat_rat(groups2906978787729119204at_rat(F),S2)),aa_set_nat_rat(groups2906978787729119204at_rat(G),Ta)) ) ) ) ) ).

% sum_le_included
tff(fact_6985_sum__strict__mono__ex1,axiom,
    ! [A3: set_nat,F: fun_nat_rat,G: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( ! [X3: nat] :
            ( aa_set_nat_o(member_nat(X3),A3)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,X3)),aa_nat_rat(G,X3)) )
       => ( ? [X5: nat] :
              ( aa_set_nat_o(member_nat(X5),A3)
              & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(F,X5)),aa_nat_rat(G,X5)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_set_nat_rat(groups2906978787729119204at_rat(F),A3)),aa_set_nat_rat(groups2906978787729119204at_rat(G),A3)) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_6986_sum__strict__mono__ex1,axiom,
    ! [A3: set_int,F: fun_int_rat,G: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,A3)
     => ( ! [X3: int] :
            ( aa_set_int_o(member_int(X3),A3)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(F,X3)),aa_int_rat(G,X3)) )
       => ( ? [X5: int] :
              ( aa_set_int_o(member_int(X5),A3)
              & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(F,X5)),aa_int_rat(G,X5)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,groups3906332499630173760nt_rat(F,A3)),groups3906332499630173760nt_rat(G,A3)) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_6987_sum__strict__mono__ex1,axiom,
    ! [A3: set_Code_integer,F: fun_Code_integer_rat,G: fun_Code_integer_rat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A3)
     => ( ! [X3: code_integer] :
            ( aa_set5643387332196575111eger_o(member_Code_integer(X3),A3)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_Code_integer_rat(F,X3)),aa_Code_integer_rat(G,X3)) )
       => ( ? [X5: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X5),A3)
              & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_Code_integer_rat(F,X5)),aa_Code_integer_rat(G,X5)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,groups6602215022474089585er_rat(F,A3)),groups6602215022474089585er_rat(G,A3)) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_6988_sum__strict__mono__ex1,axiom,
    ! [A3: set_int,F: fun_int_nat,G: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,A3)
     => ( ! [X3: int] :
            ( aa_set_int_o(member_int(X3),A3)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(F,X3)),aa_int_nat(G,X3)) )
       => ( ? [X5: int] :
              ( aa_set_int_o(member_int(X5),A3)
              & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_int_nat(F,X5)),aa_int_nat(G,X5)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,groups4541462559716669496nt_nat(F,A3)),groups4541462559716669496nt_nat(G,A3)) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_6989_sum__strict__mono__ex1,axiom,
    ! [A3: set_Code_integer,F: fun_Code_integer_nat,G: fun_Code_integer_nat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A3)
     => ( ! [X3: code_integer] :
            ( aa_set5643387332196575111eger_o(member_Code_integer(X3),A3)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_Code_integer_nat(F,X3)),aa_Code_integer_nat(G,X3)) )
       => ( ? [X5: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X5),A3)
              & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_Code_integer_nat(F,X5)),aa_Code_integer_nat(G,X5)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,groups7237345082560585321er_nat(F,A3)),groups7237345082560585321er_nat(G,A3)) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_6990_sum__strict__mono__ex1,axiom,
    ! [A3: set_nat,F: fun_nat_int,G: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( ! [X3: nat] :
            ( aa_set_nat_o(member_nat(X3),A3)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,X3)),aa_nat_int(G,X3)) )
       => ( ? [X5: nat] :
              ( aa_set_nat_o(member_nat(X5),A3)
              & aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(F,X5)),aa_nat_int(G,X5)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_set_nat_int(groups3539618377306564664at_int(F),A3)),aa_set_nat_int(groups3539618377306564664at_int(G),A3)) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_6991_sum__strict__mono__ex1,axiom,
    ! [A3: set_Code_integer,F: fun_Code_integer_int,G: fun_Code_integer_int] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A3)
     => ( ! [X3: code_integer] :
            ( aa_set5643387332196575111eger_o(member_Code_integer(X3),A3)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_Code_integer_int(F,X3)),aa_Code_integer_int(G,X3)) )
       => ( ? [X5: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X5),A3)
              & aa_int_o(aa_int_fun_int_o(ord_less_int,aa_Code_integer_int(F,X5)),aa_Code_integer_int(G,X5)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,groups7234854612051535045er_int(F,A3)),groups7234854612051535045er_int(G,A3)) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_6992_sum__strict__mono__ex1,axiom,
    ! [A3: set_nat,F: fun_nat_nat,G: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( ! [X3: nat] :
            ( aa_set_nat_o(member_nat(X3),A3)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F,X3)),aa_nat_nat(G,X3)) )
       => ( ? [X5: nat] :
              ( aa_set_nat_o(member_nat(X5),A3)
              & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(F,X5)),aa_nat_nat(G,X5)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),A3)),aa_set_nat_nat(groups3542108847815614940at_nat(G),A3)) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_6993_sum__strict__mono__ex1,axiom,
    ! [A3: set_int,F: fun_int_int,G: fun_int_int] :
      ( aa_set_int_o(finite_finite_int,A3)
     => ( ! [X3: int] :
            ( aa_set_int_o(member_int(X3),A3)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(F,X3)),aa_int_int(G,X3)) )
       => ( ? [X5: int] :
              ( aa_set_int_o(member_int(X5),A3)
              & aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(F,X5)),aa_int_int(G,X5)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,groups4538972089207619220nt_int(F,A3)),groups4538972089207619220nt_int(G,A3)) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_6994_sum__strict__mono__ex1,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,F: fun_Pr8317260874932935276at_rat,G: fun_Pr8317260874932935276at_rat] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A3)
     => ( ! [X3: product_prod_nat_nat] :
            ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X3),A3)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_Pro3608043709957902237at_rat(F,X3)),aa_Pro3608043709957902237at_rat(G,X3)) )
       => ( ? [X5: product_prod_nat_nat] :
              ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X5),A3)
              & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_Pro3608043709957902237at_rat(F,X5)),aa_Pro3608043709957902237at_rat(G,X5)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,groups342789780944988191at_rat(F,A3)),groups342789780944988191at_rat(G,A3)) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_6995_sum__strict__mono,axiom,
    ! [A3: set_Code_integer,F: fun_Code_integer_rat,G: fun_Code_integer_rat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A3)
     => ( ( A3 != bot_bo3990330152332043303nteger )
       => ( ! [X3: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X3),A3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_Code_integer_rat(F,X3)),aa_Code_integer_rat(G,X3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,groups6602215022474089585er_rat(F,A3)),groups6602215022474089585er_rat(G,A3)) ) ) ) ).

% sum_strict_mono
tff(fact_6996_sum__strict__mono,axiom,
    ! [A3: set_o,F: fun_o_rat,G: fun_o_rat] :
      ( finite_finite_o(A3)
     => ( ( A3 != bot_bot_set_o )
       => ( ! [X3: $o] :
              ( member_o((X3),A3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_o_rat(F,(X3))),aa_o_rat(G,(X3))) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,groups7872700643590313910_o_rat(F,A3)),groups7872700643590313910_o_rat(G,A3)) ) ) ) ).

% sum_strict_mono
tff(fact_6997_sum__strict__mono,axiom,
    ! [A3: set_nat,F: fun_nat_rat,G: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( ( A3 != bot_bot_set_nat )
       => ( ! [X3: nat] :
              ( aa_set_nat_o(member_nat(X3),A3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(F,X3)),aa_nat_rat(G,X3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_set_nat_rat(groups2906978787729119204at_rat(F),A3)),aa_set_nat_rat(groups2906978787729119204at_rat(G),A3)) ) ) ) ).

% sum_strict_mono
tff(fact_6998_sum__strict__mono,axiom,
    ! [A3: set_int,F: fun_int_rat,G: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,A3)
     => ( ( A3 != bot_bot_set_int )
       => ( ! [X3: int] :
              ( aa_set_int_o(member_int(X3),A3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(F,X3)),aa_int_rat(G,X3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,groups3906332499630173760nt_rat(F,A3)),groups3906332499630173760nt_rat(G,A3)) ) ) ) ).

% sum_strict_mono
tff(fact_6999_sum__strict__mono,axiom,
    ! [A3: set_Code_integer,F: fun_Code_integer_nat,G: fun_Code_integer_nat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A3)
     => ( ( A3 != bot_bo3990330152332043303nteger )
       => ( ! [X3: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X3),A3)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_Code_integer_nat(F,X3)),aa_Code_integer_nat(G,X3)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,groups7237345082560585321er_nat(F,A3)),groups7237345082560585321er_nat(G,A3)) ) ) ) ).

% sum_strict_mono
tff(fact_7000_sum__strict__mono,axiom,
    ! [A3: set_o,F: fun_o_nat,G: fun_o_nat] :
      ( finite_finite_o(A3)
     => ( ( A3 != bot_bot_set_o )
       => ( ! [X3: $o] :
              ( member_o((X3),A3)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_o_nat(F,(X3))),aa_o_nat(G,(X3))) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,groups8507830703676809646_o_nat(F,A3)),groups8507830703676809646_o_nat(G,A3)) ) ) ) ).

% sum_strict_mono
tff(fact_7001_sum__strict__mono,axiom,
    ! [A3: set_int,F: fun_int_nat,G: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,A3)
     => ( ( A3 != bot_bot_set_int )
       => ( ! [X3: int] :
              ( aa_set_int_o(member_int(X3),A3)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_int_nat(F,X3)),aa_int_nat(G,X3)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,groups4541462559716669496nt_nat(F,A3)),groups4541462559716669496nt_nat(G,A3)) ) ) ) ).

% sum_strict_mono
tff(fact_7002_sum__strict__mono,axiom,
    ! [A3: set_Code_integer,F: fun_Code_integer_int,G: fun_Code_integer_int] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A3)
     => ( ( A3 != bot_bo3990330152332043303nteger )
       => ( ! [X3: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X3),A3)
             => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_Code_integer_int(F,X3)),aa_Code_integer_int(G,X3)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,groups7234854612051535045er_int(F,A3)),groups7234854612051535045er_int(G,A3)) ) ) ) ).

% sum_strict_mono
tff(fact_7003_sum__strict__mono,axiom,
    ! [A3: set_o,F: fun_o_int,G: fun_o_int] :
      ( finite_finite_o(A3)
     => ( ( A3 != bot_bot_set_o )
       => ( ! [X3: $o] :
              ( member_o((X3),A3)
             => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_o_int(F,(X3))),aa_o_int(G,(X3))) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,groups8505340233167759370_o_int(F,A3)),groups8505340233167759370_o_int(G,A3)) ) ) ) ).

% sum_strict_mono
tff(fact_7004_sum__strict__mono,axiom,
    ! [A3: set_nat,F: fun_nat_int,G: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( ( A3 != bot_bot_set_nat )
       => ( ! [X3: nat] :
              ( aa_set_nat_o(member_nat(X3),A3)
             => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(F,X3)),aa_nat_int(G,X3)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_set_nat_int(groups3539618377306564664at_int(F),A3)),aa_set_nat_int(groups3539618377306564664at_int(G),A3)) ) ) ) ).

% sum_strict_mono
tff(fact_7005_prod_Orelated,axiom,
    ! [Ra: fun_Co2081214437782444048eger_o,S: set_nat,Ha: fun_nat_Code_integer,G: fun_nat_Code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(Ra,one_one_Code_integer),one_one_Code_integer)
     => ( ! [X12: code_integer,Y12: code_integer,X22: code_integer,Y22: code_integer] :
            ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(Ra,X12),X22)
              & aa_Code_integer_o(aa_Cod326357343816142457eger_o(Ra,Y12),Y22) )
           => aa_Code_integer_o(aa_Cod326357343816142457eger_o(Ra,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,X12),Y12)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,X22),Y22)) )
       => ( aa_set_nat_o(finite_finite_nat,S)
         => ( ! [X3: nat] :
                ( aa_set_nat_o(member_nat(X3),S)
               => aa_Code_integer_o(aa_Cod326357343816142457eger_o(Ra,aa_nat_Code_integer(Ha,X3)),aa_nat_Code_integer(G,X3)) )
           => aa_Code_integer_o(aa_Cod326357343816142457eger_o(Ra,aa_set871585252387937313nteger(groups3455450783089532116nteger(Ha),S)),aa_set871585252387937313nteger(groups3455450783089532116nteger(G),S)) ) ) ) ) ).

% prod.related
tff(fact_7006_prod_Orelated,axiom,
    ! [Ra: fun_Co2081214437782444048eger_o,S: set_int,Ha: fun_int_Code_integer,G: fun_int_Code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(Ra,one_one_Code_integer),one_one_Code_integer)
     => ( ! [X12: code_integer,Y12: code_integer,X22: code_integer,Y22: code_integer] :
            ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(Ra,X12),X22)
              & aa_Code_integer_o(aa_Cod326357343816142457eger_o(Ra,Y12),Y22) )
           => aa_Code_integer_o(aa_Cod326357343816142457eger_o(Ra,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,X12),Y12)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,X22),Y22)) )
       => ( aa_set_int_o(finite_finite_int,S)
         => ( ! [X3: int] :
                ( aa_set_int_o(member_int(X3),S)
               => aa_Code_integer_o(aa_Cod326357343816142457eger_o(Ra,aa_int_Code_integer(Ha,X3)),aa_int_Code_integer(G,X3)) )
           => aa_Code_integer_o(aa_Cod326357343816142457eger_o(Ra,groups3827104343326376752nteger(Ha,S)),groups3827104343326376752nteger(G,S)) ) ) ) ) ).

% prod.related
tff(fact_7007_prod_Orelated,axiom,
    ! [Ra: fun_Co2081214437782444048eger_o,S: set_Code_integer,Ha: fun_Co8499238673658658103nteger,G: fun_Co8499238673658658103nteger] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(Ra,one_one_Code_integer),one_one_Code_integer)
     => ( ! [X12: code_integer,Y12: code_integer,X22: code_integer,Y22: code_integer] :
            ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(Ra,X12),X22)
              & aa_Code_integer_o(aa_Cod326357343816142457eger_o(Ra,Y12),Y22) )
           => aa_Code_integer_o(aa_Cod326357343816142457eger_o(Ra,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,X12),Y12)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,X22),Y22)) )
       => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,S)
         => ( ! [X3: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(X3),S)
               => aa_Code_integer_o(aa_Cod326357343816142457eger_o(Ra,aa_Cod1783974652533396728nteger(Ha,X3)),aa_Cod1783974652533396728nteger(G,X3)) )
           => aa_Code_integer_o(aa_Cod326357343816142457eger_o(Ra,groups3674199335183972705nteger(Ha,S)),groups3674199335183972705nteger(G,S)) ) ) ) ) ).

% prod.related
tff(fact_7008_prod_Orelated,axiom,
    ! [Ra: fun_assn_fun_assn_o,S: set_nat,Ha: fun_nat_assn,G: fun_nat_assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(Ra,one_one_assn),one_one_assn)
     => ( ! [X12: assn,Y12: assn,X22: assn,Y22: assn] :
            ( ( aa_assn_o(aa_assn_fun_assn_o(Ra,X12),X22)
              & aa_assn_o(aa_assn_fun_assn_o(Ra,Y12),Y22) )
           => aa_assn_o(aa_assn_fun_assn_o(Ra,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,X12),Y12)),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,X22),Y22)) )
       => ( aa_set_nat_o(finite_finite_nat,S)
         => ( ! [X3: nat] :
                ( aa_set_nat_o(member_nat(X3),S)
               => aa_assn_o(aa_assn_fun_assn_o(Ra,aa_nat_assn(Ha,X3)),aa_nat_assn(G,X3)) )
           => aa_assn_o(aa_assn_fun_assn_o(Ra,groups6906906614972039071t_assn(Ha,S)),groups6906906614972039071t_assn(G,S)) ) ) ) ) ).

% prod.related
tff(fact_7009_prod_Orelated,axiom,
    ! [Ra: fun_assn_fun_assn_o,S: set_int,Ha: fun_int_assn,G: fun_int_assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(Ra,one_one_assn),one_one_assn)
     => ( ! [X12: assn,Y12: assn,X22: assn,Y22: assn] :
            ( ( aa_assn_o(aa_assn_fun_assn_o(Ra,X12),X22)
              & aa_assn_o(aa_assn_fun_assn_o(Ra,Y12),Y22) )
           => aa_assn_o(aa_assn_fun_assn_o(Ra,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,X12),Y12)),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,X22),Y22)) )
       => ( aa_set_int_o(finite_finite_int,S)
         => ( ! [X3: int] :
                ( aa_set_int_o(member_int(X3),S)
               => aa_assn_o(aa_assn_fun_assn_o(Ra,aa_int_assn(Ha,X3)),aa_int_assn(G,X3)) )
           => aa_assn_o(aa_assn_fun_assn_o(Ra,groups7882442080178216443t_assn(Ha,S)),groups7882442080178216443t_assn(G,S)) ) ) ) ) ).

% prod.related
tff(fact_7010_prod_Orelated,axiom,
    ! [Ra: fun_assn_fun_assn_o,S: set_Code_integer,Ha: fun_Co4490034186337852034r_assn,G: fun_Co4490034186337852034r_assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(Ra,one_one_assn),one_one_assn)
     => ( ! [X12: assn,Y12: assn,X22: assn,Y22: assn] :
            ( ( aa_assn_o(aa_assn_fun_assn_o(Ra,X12),X22)
              & aa_assn_o(aa_assn_fun_assn_o(Ra,Y12),Y22) )
           => aa_assn_o(aa_assn_fun_assn_o(Ra,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,X12),Y12)),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,X22),Y22)) )
       => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,S)
         => ( ! [X3: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(X3),S)
               => aa_assn_o(aa_assn_fun_assn_o(Ra,aa_Code_integer_assn(Ha,X3)),aa_Code_integer_assn(G,X3)) )
           => aa_assn_o(aa_assn_fun_assn_o(Ra,groups1304777262505850412r_assn(Ha,S)),groups1304777262505850412r_assn(G,S)) ) ) ) ) ).

% prod.related
tff(fact_7011_prod_Orelated,axiom,
    ! [Ra: fun_rat_fun_rat_o,S: set_nat,Ha: fun_nat_rat,G: fun_nat_rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(Ra,one_one_rat),one_one_rat)
     => ( ! [X12: rat,Y12: rat,X22: rat,Y22: rat] :
            ( ( aa_rat_o(aa_rat_fun_rat_o(Ra,X12),X22)
              & aa_rat_o(aa_rat_fun_rat_o(Ra,Y12),Y22) )
           => aa_rat_o(aa_rat_fun_rat_o(Ra,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X12),Y12)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X22),Y22)) )
       => ( aa_set_nat_o(finite_finite_nat,S)
         => ( ! [X3: nat] :
                ( aa_set_nat_o(member_nat(X3),S)
               => aa_rat_o(aa_rat_fun_rat_o(Ra,aa_nat_rat(Ha,X3)),aa_nat_rat(G,X3)) )
           => aa_rat_o(aa_rat_fun_rat_o(Ra,aa_set_nat_rat(groups73079841787564623at_rat(Ha),S)),aa_set_nat_rat(groups73079841787564623at_rat(G),S)) ) ) ) ) ).

% prod.related
tff(fact_7012_prod_Orelated,axiom,
    ! [Ra: fun_rat_fun_rat_o,S: set_int,Ha: fun_int_rat,G: fun_int_rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(Ra,one_one_rat),one_one_rat)
     => ( ! [X12: rat,Y12: rat,X22: rat,Y22: rat] :
            ( ( aa_rat_o(aa_rat_fun_rat_o(Ra,X12),X22)
              & aa_rat_o(aa_rat_fun_rat_o(Ra,Y12),Y22) )
           => aa_rat_o(aa_rat_fun_rat_o(Ra,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X12),Y12)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X22),Y22)) )
       => ( aa_set_int_o(finite_finite_int,S)
         => ( ! [X3: int] :
                ( aa_set_int_o(member_int(X3),S)
               => aa_rat_o(aa_rat_fun_rat_o(Ra,aa_int_rat(Ha,X3)),aa_int_rat(G,X3)) )
           => aa_rat_o(aa_rat_fun_rat_o(Ra,groups1072433553688619179nt_rat(Ha,S)),groups1072433553688619179nt_rat(G,S)) ) ) ) ) ).

% prod.related
tff(fact_7013_prod_Orelated,axiom,
    ! [Ra: fun_rat_fun_rat_o,S: set_Code_integer,Ha: fun_Code_integer_rat,G: fun_Code_integer_rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(Ra,one_one_rat),one_one_rat)
     => ( ! [X12: rat,Y12: rat,X22: rat,Y22: rat] :
            ( ( aa_rat_o(aa_rat_fun_rat_o(Ra,X12),X22)
              & aa_rat_o(aa_rat_fun_rat_o(Ra,Y12),Y22) )
           => aa_rat_o(aa_rat_fun_rat_o(Ra,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X12),Y12)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X22),Y22)) )
       => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,S)
         => ( ! [X3: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(X3),S)
               => aa_rat_o(aa_rat_fun_rat_o(Ra,aa_Code_integer_rat(Ha,X3)),aa_Code_integer_rat(G,X3)) )
           => aa_rat_o(aa_rat_fun_rat_o(Ra,groups2555765274223993564er_rat(Ha,S)),groups2555765274223993564er_rat(G,S)) ) ) ) ) ).

% prod.related
tff(fact_7014_prod_Orelated,axiom,
    ! [Ra: fun_nat_fun_nat_o,S: set_int,Ha: fun_int_nat,G: fun_int_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(Ra,one_one_nat),one_one_nat)
     => ( ! [X12: nat,Y12: nat,X22: nat,Y22: nat] :
            ( ( aa_nat_o(aa_nat_fun_nat_o(Ra,X12),X22)
              & aa_nat_o(aa_nat_fun_nat_o(Ra,Y12),Y22) )
           => aa_nat_o(aa_nat_fun_nat_o(Ra,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,X12),Y12)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,X22),Y22)) )
       => ( aa_set_int_o(finite_finite_int,S)
         => ( ! [X3: int] :
                ( aa_set_int_o(member_int(X3),S)
               => aa_nat_o(aa_nat_fun_nat_o(Ra,aa_int_nat(Ha,X3)),aa_int_nat(G,X3)) )
           => aa_nat_o(aa_nat_fun_nat_o(Ra,groups1707563613775114915nt_nat(Ha,S)),groups1707563613775114915nt_nat(G,S)) ) ) ) ) ).

% prod.related
tff(fact_7015_ex__power__ivl1,axiom,
    ! [B: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),K)
       => ? [N5: 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),N5)),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,N5),one_one_nat))) ) ) ) ).

% ex_power_ivl1
tff(fact_7016_ex__power__ivl2,axiom,
    ! [B: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K)
       => ? [N5: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),N5)),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,N5),one_one_nat))) ) ) ) ).

% ex_power_ivl2
tff(fact_7017_atLeastLessThan__add__Un,axiom,
    ! [I2: nat,J: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),J)
     => ( aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,I2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,I2),J)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,J),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K))) ) ) ).

% atLeastLessThan_add_Un
tff(fact_7018_mask__eq__sum__exp,axiom,
    ! [N2: 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))),N2)),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_dg,N2))) ).

% mask_eq_sum_exp
tff(fact_7019_mask__eq__sum__exp,axiom,
    ! [N2: nat] : aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),N2)),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_dg,N2))) ).

% mask_eq_sum_exp
tff(fact_7020_mask__eq__sum__exp,axiom,
    ! [N2: nat] : minus_7197305767214868737atural(aa_nat_Code_natural(power_7079662738309270450atural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))),N2),one_one_Code_natural) = groups6325495683096345652atural(power_7079662738309270450atural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))),collect_nat(aa_nat_fun_nat_o(aTP_Lamm_dg,N2))) ).

% mask_eq_sum_exp
tff(fact_7021_mask__eq__sum__exp,axiom,
    ! [N2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2)),one_one_nat) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),collect_nat(aa_nat_fun_nat_o(aTP_Lamm_dg,N2))) ).

% mask_eq_sum_exp
tff(fact_7022_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_7023_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_7024_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,aa_num_rat(numeral_numeral_rat,one))),B) = aa_rat_rat(uminus_uminus_rat,B) ).

% mult_1s_ring_1(1)
tff(fact_7025_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_7026_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_7027_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,aa_num_rat(numeral_numeral_rat,one))) = aa_rat_rat(uminus_uminus_rat,B) ).

% mult_1s_ring_1(2)
tff(fact_7028_sum_Oin__pairs,axiom,
    ! [G: fun_na2338080690716861136at_nat,M: nat,N2: nat] : groups6857163185585827899at_nat(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2)))) = groups6857163185585827899at_nat(aTP_Lamm_xw(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M),N2)) ).

% sum.in_pairs
tff(fact_7029_sum_Oin__pairs,axiom,
    ! [G: fun_nat_rat,M: nat,N2: nat] : aa_set_nat_rat(groups2906978787729119204at_rat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2)))) = aa_set_nat_rat(groups2906978787729119204at_rat(aTP_Lamm_xx(G)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M),N2)) ).

% sum.in_pairs
tff(fact_7030_sum_Oin__pairs,axiom,
    ! [G: fun_nat_int,M: nat,N2: nat] : aa_set_nat_int(groups3539618377306564664at_int(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2)))) = aa_set_nat_int(groups3539618377306564664at_int(aTP_Lamm_xy(G)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M),N2)) ).

% sum.in_pairs
tff(fact_7031_sum_Oin__pairs,axiom,
    ! [G: fun_nat_nat,M: nat,N2: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2)))) = aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_xz(G)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M),N2)) ).

% sum.in_pairs
tff(fact_7032_mask__eq__sum__exp__nat,axiom,
    ! [N2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2)),aa_nat_nat(suc,zero_zero_nat)) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),collect_nat(aa_nat_fun_nat_o(aTP_Lamm_dg,N2))) ).

% mask_eq_sum_exp_nat
tff(fact_7033_prod_Oin__pairs,axiom,
    ! [G: fun_nat_Code_integer,M: nat,N2: nat] : aa_set871585252387937313nteger(groups3455450783089532116nteger(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2)))) = aa_set871585252387937313nteger(groups3455450783089532116nteger(aTP_Lamm_ya(G)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M),N2)) ).

% prod.in_pairs
tff(fact_7034_prod_Oin__pairs,axiom,
    ! [G: fun_nat_assn,M: nat,N2: nat] : groups6906906614972039071t_assn(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2)))) = groups6906906614972039071t_assn(aTP_Lamm_yb(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M),N2)) ).

% prod.in_pairs
tff(fact_7035_prod_Oin__pairs,axiom,
    ! [G: fun_nat_rat,M: nat,N2: nat] : aa_set_nat_rat(groups73079841787564623at_rat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2)))) = aa_set_nat_rat(groups73079841787564623at_rat(aTP_Lamm_yc(G)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M),N2)) ).

% prod.in_pairs
tff(fact_7036_prod_Oin__pairs,axiom,
    ! [G: fun_nat_nat,M: nat,N2: nat] : aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2)))) = aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_yd(G)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M),N2)) ).

% prod.in_pairs
tff(fact_7037_prod_Oin__pairs,axiom,
    ! [G: fun_nat_int,M: nat,N2: nat] : aa_set_nat_int(groups705719431365010083at_int(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2)))) = aa_set_nat_int(groups705719431365010083at_int(aTP_Lamm_ye(G)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M),N2)) ).

% prod.in_pairs
tff(fact_7038_gauss__sum__nat,axiom,
    ! [N2: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_fu),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,zero_zero_nat),N2)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N2),aa_nat_nat(suc,N2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% gauss_sum_nat
tff(fact_7039_prod__int__eq,axiom,
    ! [I2: nat,J: nat] : aa_set_nat_int(groups705719431365010083at_int(semiri1314217659103216013at_int),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,I2),J)) = groups1705073143266064639nt_int(aTP_Lamm_fv,set_or1266510415728281911st_int(aa_nat_int(semiri1314217659103216013at_int,I2),aa_nat_int(semiri1314217659103216013at_int,J))) ).

% prod_int_eq
tff(fact_7040_sum_Oin__pairs__0,axiom,
    ! [G: fun_na2338080690716861136at_nat,N2: nat] : groups6857163185585827899at_nat(G,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2)))) = groups6857163185585827899at_nat(aTP_Lamm_xw(G),aa_nat_set_nat(set_ord_atMost_nat,N2)) ).

% sum.in_pairs_0
tff(fact_7041_sum_Oin__pairs__0,axiom,
    ! [G: fun_nat_rat,N2: 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,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2)))) = aa_set_nat_rat(groups2906978787729119204at_rat(aTP_Lamm_xx(G)),aa_nat_set_nat(set_ord_atMost_nat,N2)) ).

% sum.in_pairs_0
tff(fact_7042_sum_Oin__pairs__0,axiom,
    ! [G: fun_nat_int,N2: 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,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2)))) = aa_set_nat_int(groups3539618377306564664at_int(aTP_Lamm_xy(G)),aa_nat_set_nat(set_ord_atMost_nat,N2)) ).

% sum.in_pairs_0
tff(fact_7043_sum_Oin__pairs__0,axiom,
    ! [G: fun_nat_nat,N2: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2)))) = aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_xz(G)),aa_nat_set_nat(set_ord_atMost_nat,N2)) ).

% sum.in_pairs_0
tff(fact_7044_prod_Oin__pairs__0,axiom,
    ! [G: fun_nat_Code_integer,N2: 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,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2)))) = aa_set871585252387937313nteger(groups3455450783089532116nteger(aTP_Lamm_ya(G)),aa_nat_set_nat(set_ord_atMost_nat,N2)) ).

% prod.in_pairs_0
tff(fact_7045_prod_Oin__pairs__0,axiom,
    ! [G: fun_nat_assn,N2: 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,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2)))) = groups6906906614972039071t_assn(aTP_Lamm_yb(G),aa_nat_set_nat(set_ord_atMost_nat,N2)) ).

% prod.in_pairs_0
tff(fact_7046_prod_Oin__pairs__0,axiom,
    ! [G: fun_nat_rat,N2: 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,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2)))) = aa_set_nat_rat(groups73079841787564623at_rat(aTP_Lamm_yc(G)),aa_nat_set_nat(set_ord_atMost_nat,N2)) ).

% prod.in_pairs_0
tff(fact_7047_prod_Oin__pairs__0,axiom,
    ! [G: fun_nat_nat,N2: nat] : aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2)))) = aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_yd(G)),aa_nat_set_nat(set_ord_atMost_nat,N2)) ).

% prod.in_pairs_0
tff(fact_7048_prod_Oin__pairs__0,axiom,
    ! [G: fun_nat_int,N2: 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,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2)))) = aa_set_nat_int(groups705719431365010083at_int(aTP_Lamm_ye(G)),aa_nat_set_nat(set_ord_atMost_nat,N2)) ).

% prod.in_pairs_0
tff(fact_7049_sum__choose__lower,axiom,
    ! [Ra2: nat,N2: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_yf,Ra2)),aa_nat_set_nat(set_ord_atMost_nat,N2)) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ra2),N2))),N2) ).

% sum_choose_lower
tff(fact_7050_choose__rising__sum_I2_J,axiom,
    ! [N2: nat,M: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_yg,N2)),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,N2),M)),one_one_nat)),M) ).

% choose_rising_sum(2)
tff(fact_7051_choose__rising__sum_I1_J,axiom,
    ! [N2: nat,M: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_yg,N2)),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,N2),M)),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),one_one_nat)) ).

% choose_rising_sum(1)
tff(fact_7052_sum__nonneg__0,axiom,
    ! [S2: set_o,F: fun_o_Code_integer,I2: $o] :
      ( finite_finite_o(S2)
     => ( ! [I3: $o] :
            ( member_o((I3),S2)
           => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_o_Code_integer(F,(I3))) )
       => ( ( groups4406642042086082107nteger(F,S2) = zero_z3403309356797280102nteger )
         => ( member_o((I2),S2)
           => ( aa_o_Code_integer(F,(I2)) = zero_z3403309356797280102nteger ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_7053_sum__nonneg__0,axiom,
    ! [S2: set_nat,F: fun_nat_Code_integer,I2: nat] :
      ( aa_set_nat_o(finite_finite_nat,S2)
     => ( ! [I3: nat] :
            ( aa_set_nat_o(member_nat(I3),S2)
           => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(F,I3)) )
       => ( ( aa_set871585252387937313nteger(groups7501900531339628137nteger(F),S2) = zero_z3403309356797280102nteger )
         => ( aa_set_nat_o(member_nat(I2),S2)
           => ( aa_nat_Code_integer(F,I2) = zero_z3403309356797280102nteger ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_7054_sum__nonneg__0,axiom,
    ! [S2: set_int,F: fun_int_Code_integer,I2: int] :
      ( aa_set_int_o(finite_finite_int,S2)
     => ( ! [I3: int] :
            ( aa_set_int_o(member_int(I3),S2)
           => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_int_Code_integer(F,I3)) )
       => ( ( groups7873554091576472773nteger(F,S2) = zero_z3403309356797280102nteger )
         => ( aa_set_int_o(member_int(I2),S2)
           => ( aa_int_Code_integer(F,I2) = zero_z3403309356797280102nteger ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_7055_sum__nonneg__0,axiom,
    ! [S2: set_Code_integer,F: fun_Co8499238673658658103nteger,I2: code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,S2)
     => ( ! [I3: code_integer] :
            ( aa_set5643387332196575111eger_o(member_Code_integer(I3),S2)
           => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(F,I3)) )
       => ( ( groups879477027807139574nteger(F,S2) = zero_z3403309356797280102nteger )
         => ( aa_set5643387332196575111eger_o(member_Code_integer(I2),S2)
           => ( aa_Cod1783974652533396728nteger(F,I2) = zero_z3403309356797280102nteger ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_7056_sum__nonneg__0,axiom,
    ! [S2: set_o,F: fun_o_rat,I2: $o] :
      ( finite_finite_o(S2)
     => ( ! [I3: $o] :
            ( member_o((I3),S2)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_o_rat(F,(I3))) )
       => ( ( groups7872700643590313910_o_rat(F,S2) = zero_zero_rat )
         => ( member_o((I2),S2)
           => ( aa_o_rat(F,(I2)) = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_7057_sum__nonneg__0,axiom,
    ! [S2: set_nat,F: fun_nat_rat,I2: nat] :
      ( aa_set_nat_o(finite_finite_nat,S2)
     => ( ! [I3: nat] :
            ( aa_set_nat_o(member_nat(I3),S2)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(F,I3)) )
       => ( ( aa_set_nat_rat(groups2906978787729119204at_rat(F),S2) = zero_zero_rat )
         => ( aa_set_nat_o(member_nat(I2),S2)
           => ( aa_nat_rat(F,I2) = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_7058_sum__nonneg__0,axiom,
    ! [S2: set_int,F: fun_int_rat,I2: int] :
      ( aa_set_int_o(finite_finite_int,S2)
     => ( ! [I3: int] :
            ( aa_set_int_o(member_int(I3),S2)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_int_rat(F,I3)) )
       => ( ( groups3906332499630173760nt_rat(F,S2) = zero_zero_rat )
         => ( aa_set_int_o(member_int(I2),S2)
           => ( aa_int_rat(F,I2) = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_7059_sum__nonneg__0,axiom,
    ! [S2: set_Code_integer,F: fun_Code_integer_rat,I2: code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,S2)
     => ( ! [I3: code_integer] :
            ( aa_set5643387332196575111eger_o(member_Code_integer(I3),S2)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_Code_integer_rat(F,I3)) )
       => ( ( groups6602215022474089585er_rat(F,S2) = zero_zero_rat )
         => ( aa_set5643387332196575111eger_o(member_Code_integer(I2),S2)
           => ( aa_Code_integer_rat(F,I2) = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_7060_sum__nonneg__0,axiom,
    ! [S2: set_o,F: fun_o_nat,I2: $o] :
      ( finite_finite_o(S2)
     => ( ! [I3: $o] :
            ( member_o((I3),S2)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_o_nat(F,(I3))) )
       => ( ( groups8507830703676809646_o_nat(F,S2) = zero_zero_nat )
         => ( member_o((I2),S2)
           => ( aa_o_nat(F,(I2)) = zero_zero_nat ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_7061_sum__nonneg__0,axiom,
    ! [S2: set_int,F: fun_int_nat,I2: int] :
      ( aa_set_int_o(finite_finite_int,S2)
     => ( ! [I3: int] :
            ( aa_set_int_o(member_int(I3),S2)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_int_nat(F,I3)) )
       => ( ( groups4541462559716669496nt_nat(F,S2) = zero_zero_nat )
         => ( aa_set_int_o(member_int(I2),S2)
           => ( aa_int_nat(F,I2) = zero_zero_nat ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_7062_sum__nonneg__leq__bound,axiom,
    ! [S2: set_o,F: fun_o_Code_integer,B3: code_integer,I2: $o] :
      ( finite_finite_o(S2)
     => ( ! [I3: $o] :
            ( member_o((I3),S2)
           => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_o_Code_integer(F,(I3))) )
       => ( ( groups4406642042086082107nteger(F,S2) = B3 )
         => ( member_o((I2),S2)
           => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_o_Code_integer(F,(I2))),B3) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_7063_sum__nonneg__leq__bound,axiom,
    ! [S2: set_nat,F: fun_nat_Code_integer,B3: code_integer,I2: nat] :
      ( aa_set_nat_o(finite_finite_nat,S2)
     => ( ! [I3: nat] :
            ( aa_set_nat_o(member_nat(I3),S2)
           => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(F,I3)) )
       => ( ( aa_set871585252387937313nteger(groups7501900531339628137nteger(F),S2) = B3 )
         => ( aa_set_nat_o(member_nat(I2),S2)
           => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(F,I2)),B3) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_7064_sum__nonneg__leq__bound,axiom,
    ! [S2: set_int,F: fun_int_Code_integer,B3: code_integer,I2: int] :
      ( aa_set_int_o(finite_finite_int,S2)
     => ( ! [I3: int] :
            ( aa_set_int_o(member_int(I3),S2)
           => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_int_Code_integer(F,I3)) )
       => ( ( groups7873554091576472773nteger(F,S2) = B3 )
         => ( aa_set_int_o(member_int(I2),S2)
           => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_int_Code_integer(F,I2)),B3) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_7065_sum__nonneg__leq__bound,axiom,
    ! [S2: set_Code_integer,F: fun_Co8499238673658658103nteger,B3: code_integer,I2: code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,S2)
     => ( ! [I3: code_integer] :
            ( aa_set5643387332196575111eger_o(member_Code_integer(I3),S2)
           => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(F,I3)) )
       => ( ( groups879477027807139574nteger(F,S2) = B3 )
         => ( aa_set5643387332196575111eger_o(member_Code_integer(I2),S2)
           => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(F,I2)),B3) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_7066_sum__nonneg__leq__bound,axiom,
    ! [S2: set_o,F: fun_o_rat,B3: rat,I2: $o] :
      ( finite_finite_o(S2)
     => ( ! [I3: $o] :
            ( member_o((I3),S2)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_o_rat(F,(I3))) )
       => ( ( groups7872700643590313910_o_rat(F,S2) = B3 )
         => ( member_o((I2),S2)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_o_rat(F,(I2))),B3) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_7067_sum__nonneg__leq__bound,axiom,
    ! [S2: set_nat,F: fun_nat_rat,B3: rat,I2: nat] :
      ( aa_set_nat_o(finite_finite_nat,S2)
     => ( ! [I3: nat] :
            ( aa_set_nat_o(member_nat(I3),S2)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(F,I3)) )
       => ( ( aa_set_nat_rat(groups2906978787729119204at_rat(F),S2) = B3 )
         => ( aa_set_nat_o(member_nat(I2),S2)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,I2)),B3) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_7068_sum__nonneg__leq__bound,axiom,
    ! [S2: set_int,F: fun_int_rat,B3: rat,I2: int] :
      ( aa_set_int_o(finite_finite_int,S2)
     => ( ! [I3: int] :
            ( aa_set_int_o(member_int(I3),S2)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_int_rat(F,I3)) )
       => ( ( groups3906332499630173760nt_rat(F,S2) = B3 )
         => ( aa_set_int_o(member_int(I2),S2)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(F,I2)),B3) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_7069_sum__nonneg__leq__bound,axiom,
    ! [S2: set_Code_integer,F: fun_Code_integer_rat,B3: rat,I2: code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,S2)
     => ( ! [I3: code_integer] :
            ( aa_set5643387332196575111eger_o(member_Code_integer(I3),S2)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_Code_integer_rat(F,I3)) )
       => ( ( groups6602215022474089585er_rat(F,S2) = B3 )
         => ( aa_set5643387332196575111eger_o(member_Code_integer(I2),S2)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_Code_integer_rat(F,I2)),B3) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_7070_sum__nonneg__leq__bound,axiom,
    ! [S2: set_o,F: fun_o_nat,B3: nat,I2: $o] :
      ( finite_finite_o(S2)
     => ( ! [I3: $o] :
            ( member_o((I3),S2)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_o_nat(F,(I3))) )
       => ( ( groups8507830703676809646_o_nat(F,S2) = B3 )
         => ( member_o((I2),S2)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_o_nat(F,(I2))),B3) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_7071_sum__nonneg__leq__bound,axiom,
    ! [S2: set_int,F: fun_int_nat,B3: nat,I2: int] :
      ( aa_set_int_o(finite_finite_int,S2)
     => ( ! [I3: int] :
            ( aa_set_int_o(member_int(I3),S2)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_int_nat(F,I3)) )
       => ( ( groups4541462559716669496nt_nat(F,S2) = B3 )
         => ( aa_set_int_o(member_int(I2),S2)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(F,I2)),B3) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_7072_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(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(X),M)),M) ) ) ) ) ).

% mod_double_modulus
tff(fact_7073_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,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),M) )
          | ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),M)),M) ) ) ) ) ).

% mod_double_modulus
tff(fact_7074_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_7075_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_7076_divmod__digit__1_I2_J,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B)))
         => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B))),B) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) ) ) ) ) ).

% divmod_digit_1(2)
tff(fact_7077_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_7078_add__mset__in__multiset,axiom,
    ! [M6: fun_Pr5160692531401422620nt_nat,A: product_prod_int_int] :
      ( finite2998713641127702882nt_int(collec213857154873943460nt_int(aTP_Lamm_vu(M6)))
     => finite2998713641127702882nt_int(collec213857154873943460nt_int(aa_Pro3952998155787061599_int_o(aTP_Lamm_yh(M6),A))) ) ).

% add_mset_in_multiset
tff(fact_7079_add__mset__in__multiset,axiom,
    ! [M6: fun_list_nat_nat,A: list_nat] :
      ( finite8100373058378681591st_nat(collect_list_nat(aTP_Lamm_vw(M6)))
     => finite8100373058378681591st_nat(collect_list_nat(aa_lis557685189327716851_nat_o(aTP_Lamm_yi(M6),A))) ) ).

% add_mset_in_multiset
tff(fact_7080_add__mset__in__multiset,axiom,
    ! [M6: fun_set_nat_nat,A: set_nat] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,collect_set_nat(aTP_Lamm_vy(M6)))
     => aa_set_set_nat_o(finite1152437895449049373et_nat,collect_set_nat(aa_set7861427066675100071_nat_o(aTP_Lamm_yj(M6),A))) ) ).

% add_mset_in_multiset
tff(fact_7081_add__mset__in__multiset,axiom,
    ! [M6: fun_nat_nat,A: nat] :
      ( aa_set_nat_o(finite_finite_nat,collect_nat(aTP_Lamm_wa(M6)))
     => aa_set_nat_o(finite_finite_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_yk(M6),A))) ) ).

% add_mset_in_multiset
tff(fact_7082_add__mset__in__multiset,axiom,
    ! [M6: fun_int_nat,A: int] :
      ( aa_set_int_o(finite_finite_int,collect_int(aTP_Lamm_wc(M6)))
     => aa_set_int_o(finite_finite_int,collect_int(aa_int_fun_int_o(aTP_Lamm_yl(M6),A))) ) ).

% add_mset_in_multiset
tff(fact_7083_add__mset__in__multiset,axiom,
    ! [M6: fun_Code_integer_nat,A: code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,collect_Code_integer(aTP_Lamm_we(M6)))
     => aa_set5643387332196575111eger_o(finite6017078050557962740nteger,collect_Code_integer(aa_Cod326357343816142457eger_o(aTP_Lamm_ym(M6),A))) ) ).

% add_mset_in_multiset
tff(fact_7084_add__mset__in__multiset,axiom,
    ! [M6: fun_Pr2602117396471556196at_nat,A: product_prod_nat_nat] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,collec3392354462482085612at_nat(aTP_Lamm_wg(M6)))
     => aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,collec3392354462482085612at_nat(aa_Pro989595756709598047_nat_o(aTP_Lamm_yn(M6),A))) ) ).

% add_mset_in_multiset
tff(fact_7085_sum_Ointer__restrict,axiom,
    ! [A3: set_o,G: fun_o_Code_integer,B3: set_o] :
      ( finite_finite_o(A3)
     => ( groups4406642042086082107nteger(G,inf_inf_set_o(A3,B3)) = groups4406642042086082107nteger(aa_set9049949509649958604nteger(aTP_Lamm_yo(G),B3),A3) ) ) ).

% sum.inter_restrict
tff(fact_7086_sum_Ointer__restrict,axiom,
    ! [A3: set_int,G: fun_int_Code_integer,B3: set_int] :
      ( aa_set_int_o(finite_finite_int,A3)
     => ( groups7873554091576472773nteger(G,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A3),B3)) = groups7873554091576472773nteger(aa_set5517701783735052070nteger(aTP_Lamm_yp(G),B3),A3) ) ) ).

% sum.inter_restrict
tff(fact_7087_sum_Ointer__restrict,axiom,
    ! [A3: set_Code_integer,G: fun_Co8499238673658658103nteger,B3: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A3)
     => ( groups879477027807139574nteger(G,inf_in1364745209274528805nteger(A3,B3)) = groups879477027807139574nteger(aa_set4294055438810586376nteger(aTP_Lamm_yq(G),B3),A3) ) ) ).

% sum.inter_restrict
tff(fact_7088_sum_Ointer__restrict,axiom,
    ! [A3: set_o,G: fun_o_rat,B3: set_o] :
      ( finite_finite_o(A3)
     => ( groups7872700643590313910_o_rat(G,inf_inf_set_o(A3,B3)) = groups7872700643590313910_o_rat(aa_set_o_fun_o_rat(aTP_Lamm_yr(G),B3),A3) ) ) ).

% sum.inter_restrict
tff(fact_7089_sum_Ointer__restrict,axiom,
    ! [A3: set_int,G: fun_int_rat,B3: set_int] :
      ( aa_set_int_o(finite_finite_int,A3)
     => ( groups3906332499630173760nt_rat(G,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A3),B3)) = groups3906332499630173760nt_rat(aa_set7979663334043413281nt_rat(aTP_Lamm_ys(G),B3),A3) ) ) ).

% sum.inter_restrict
tff(fact_7090_sum_Ointer__restrict,axiom,
    ! [A3: set_Code_integer,G: fun_Code_integer_rat,B3: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A3)
     => ( groups6602215022474089585er_rat(G,inf_in1364745209274528805nteger(A3,B3)) = groups6602215022474089585er_rat(aa_set7344733238996822275er_rat(aTP_Lamm_yt(G),B3),A3) ) ) ).

% sum.inter_restrict
tff(fact_7091_sum_Ointer__restrict,axiom,
    ! [A3: set_o,G: fun_o_nat,B3: set_o] :
      ( finite_finite_o(A3)
     => ( groups8507830703676809646_o_nat(G,inf_inf_set_o(A3,B3)) = groups8507830703676809646_o_nat(aa_set_o_fun_o_nat(aTP_Lamm_yu(G),B3),A3) ) ) ).

% sum.inter_restrict
tff(fact_7092_sum_Ointer__restrict,axiom,
    ! [A3: set_int,G: fun_int_nat,B3: set_int] :
      ( aa_set_int_o(finite_finite_int,A3)
     => ( groups4541462559716669496nt_nat(G,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A3),B3)) = groups4541462559716669496nt_nat(aa_set2264519855582034201nt_nat(aTP_Lamm_yv(G),B3),A3) ) ) ).

% sum.inter_restrict
tff(fact_7093_sum_Ointer__restrict,axiom,
    ! [A3: set_Code_integer,G: fun_Code_integer_nat,B3: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A3)
     => ( groups7237345082560585321er_nat(G,inf_in1364745209274528805nteger(A3,B3)) = groups7237345082560585321er_nat(aa_set1629589760535443195er_nat(aTP_Lamm_yw(G),B3),A3) ) ) ).

% sum.inter_restrict
tff(fact_7094_sum_Ointer__restrict,axiom,
    ! [A3: set_o,G: fun_o_int,B3: set_o] :
      ( finite_finite_o(A3)
     => ( groups8505340233167759370_o_int(G,inf_inf_set_o(A3,B3)) = groups8505340233167759370_o_int(aa_set_o_fun_o_int(aTP_Lamm_yx(G),B3),A3) ) ) ).

% sum.inter_restrict
tff(fact_7095_sum_Ogroup,axiom,
    ! [S: set_nat,T: set_Code_integer,G: fun_nat_Code_integer,Ha: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,S)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,T)
       => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(image_1215581382706833972nteger(G,S)),T)
         => ( groups7237345082560585321er_nat(aa_fun2720235278490593797er_nat(aa_fun676797813428960940er_nat(aTP_Lamm_yz(S),G),Ha),T) = aa_set_nat_nat(groups3542108847815614940at_nat(Ha),S) ) ) ) ) ).

% sum.group
tff(fact_7096_sum_Ogroup,axiom,
    ! [S: set_nat,T: set_int,G: fun_nat_int,Ha: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,S)
     => ( aa_set_int_o(finite_finite_int,T)
       => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,image_nat_int(G,S)),T)
         => ( groups4541462559716669496nt_nat(aa_fun6790803225114646484nt_nat(aa_fun5866268260113680714nt_nat(aTP_Lamm_za(S),G),Ha),T) = aa_set_nat_nat(groups3542108847815614940at_nat(Ha),S) ) ) ) ) ).

% sum.group
tff(fact_7097_sum_Ogroup,axiom,
    ! [S: set_int,T: set_nat,G: fun_int_nat,Ha: fun_int_int] :
      ( aa_set_int_o(finite_finite_int,S)
     => ( aa_set_nat_o(finite_finite_nat,T)
       => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,image_int_nat(G,S)),T)
         => ( aa_set_nat_int(groups3539618377306564664at_int(aa_fun5280992047749994636at_int(aa_fun4682031307293100034at_int(aTP_Lamm_zb(S),G),Ha)),T) = groups4538972089207619220nt_int(Ha,S) ) ) ) ) ).

% sum.group
tff(fact_7098_sum_Ogroup,axiom,
    ! [S: set_int,T: set_Code_integer,G: fun_int_Code_integer,Ha: fun_int_int] :
      ( aa_set_int_o(finite_finite_int,S)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,T)
       => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(image_1587234942943678608nteger(G,S)),T)
         => ( groups7234854612051535045er_int(aa_fun2092574682676764953er_int(aa_fun9146121993134396956er_int(aTP_Lamm_zd(S),G),Ha),T) = groups4538972089207619220nt_int(Ha,S) ) ) ) ) ).

% sum.group
tff(fact_7099_sum_Ogroup,axiom,
    ! [S: set_o,T: set_nat,G: fun_o_nat,Ha: fun_o_nat] :
      ( finite_finite_o(S)
     => ( aa_set_nat_o(finite_finite_nat,T)
       => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,image_o_nat(G,S)),T)
         => ( aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun1225607396849667164at_nat(aa_fun2442258407354871084at_nat(aTP_Lamm_ze(S),G),Ha)),T) = groups8507830703676809646_o_nat(Ha,S) ) ) ) ) ).

% sum.group
tff(fact_7100_sum_Ogroup,axiom,
    ! [S: set_int,T: set_nat,G: fun_int_nat,Ha: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,S)
     => ( aa_set_nat_o(finite_finite_nat,T)
       => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,image_int_nat(G,S)),T)
         => ( aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun5507398438843482068at_nat(aa_fun3220669823809117514at_nat(aTP_Lamm_zf(S),G),Ha)),T) = groups4541462559716669496nt_nat(Ha,S) ) ) ) ) ).

% sum.group
tff(fact_7101_sum_Ogroup,axiom,
    ! [S: set_Code_integer,T: set_nat,G: fun_Code_integer_nat,Ha: fun_Code_integer_nat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,S)
     => ( aa_set_nat_o(finite_finite_nat,T)
       => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,image_951025933927791156er_nat(G,S)),T)
         => ( aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun1707803128846032901at_nat(aa_fun2075405564639334828at_nat(aTP_Lamm_zg(S),G),Ha)),T) = groups7237345082560585321er_nat(Ha,S) ) ) ) ) ).

% sum.group
tff(fact_7102_sum_Ogroup,axiom,
    ! [S: set_nat,T: set_nat,G: fun_nat_nat,Ha: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,S)
     => ( aa_set_nat_o(finite_finite_nat,T)
       => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,image_nat_nat(G,S)),T)
         => ( aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(aTP_Lamm_zh(S),G),Ha)),T) = aa_set_nat_nat(groups3542108847815614940at_nat(Ha),S) ) ) ) ) ).

% sum.group
tff(fact_7103_sum_Ogroup,axiom,
    ! [S: set_o,T: set_int,G: fun_o_int,Ha: fun_o_int] :
      ( finite_finite_o(S)
     => ( aa_set_int_o(finite_finite_int,T)
       => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,image_o_int(G,S)),T)
         => ( groups4538972089207619220nt_int(aa_fun1998554717657234288nt_int(aa_fun5578626826553213156nt_int(aTP_Lamm_zi(S),G),Ha),T) = groups8505340233167759370_o_int(Ha,S) ) ) ) ) ).

% sum.group
tff(fact_7104_sum_Ogroup,axiom,
    ! [S: set_nat,T: set_int,G: fun_nat_int,Ha: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,S)
     => ( aa_set_int_o(finite_finite_int,T)
       => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,image_nat_int(G,S)),T)
         => ( groups4538972089207619220nt_int(aa_fun6564396834021159052nt_int(aa_fun7327629743597663234nt_int(aTP_Lamm_zj(S),G),Ha),T) = aa_set_nat_int(groups3539618377306564664at_int(Ha),S) ) ) ) ) ).

% sum.group
tff(fact_7105_sum_Osetdiff__irrelevant,axiom,
    ! [A3: set_int,G: fun_int_Code_integer] :
      ( aa_set_int_o(finite_finite_int,A3)
     => ( groups7873554091576472773nteger(G,aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A3),collect_int(aTP_Lamm_zk(G)))) = groups7873554091576472773nteger(G,A3) ) ) ).

% sum.setdiff_irrelevant
tff(fact_7106_sum_Osetdiff__irrelevant,axiom,
    ! [A3: set_Code_integer,G: fun_Co8499238673658658103nteger] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A3)
     => ( groups879477027807139574nteger(G,minus_2355218937544613996nteger(A3,collect_Code_integer(aTP_Lamm_zl(G)))) = groups879477027807139574nteger(G,A3) ) ) ).

% sum.setdiff_irrelevant
tff(fact_7107_sum_Osetdiff__irrelevant,axiom,
    ! [A3: set_int,G: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,A3)
     => ( groups3906332499630173760nt_rat(G,aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A3),collect_int(aTP_Lamm_zm(G)))) = groups3906332499630173760nt_rat(G,A3) ) ) ).

% sum.setdiff_irrelevant
tff(fact_7108_sum_Osetdiff__irrelevant,axiom,
    ! [A3: set_Code_integer,G: fun_Code_integer_rat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A3)
     => ( groups6602215022474089585er_rat(G,minus_2355218937544613996nteger(A3,collect_Code_integer(aTP_Lamm_zn(G)))) = groups6602215022474089585er_rat(G,A3) ) ) ).

% sum.setdiff_irrelevant
tff(fact_7109_sum_Osetdiff__irrelevant,axiom,
    ! [A3: set_int,G: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,A3)
     => ( groups4541462559716669496nt_nat(G,aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A3),collect_int(aTP_Lamm_zo(G)))) = groups4541462559716669496nt_nat(G,A3) ) ) ).

% sum.setdiff_irrelevant
tff(fact_7110_sum_Osetdiff__irrelevant,axiom,
    ! [A3: set_Code_integer,G: fun_Code_integer_nat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A3)
     => ( groups7237345082560585321er_nat(G,minus_2355218937544613996nteger(A3,collect_Code_integer(aTP_Lamm_zp(G)))) = groups7237345082560585321er_nat(G,A3) ) ) ).

% sum.setdiff_irrelevant
tff(fact_7111_sum_Osetdiff__irrelevant,axiom,
    ! [A3: set_Code_integer,G: fun_Code_integer_int] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A3)
     => ( groups7234854612051535045er_int(G,minus_2355218937544613996nteger(A3,collect_Code_integer(aTP_Lamm_zq(G)))) = groups7234854612051535045er_int(G,A3) ) ) ).

% sum.setdiff_irrelevant
tff(fact_7112_sum_Osetdiff__irrelevant,axiom,
    ! [A3: set_nat,G: fun_nat_Code_integer] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( aa_set871585252387937313nteger(groups7501900531339628137nteger(G),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A3),collect_nat(aTP_Lamm_zr(G)))) = aa_set871585252387937313nteger(groups7501900531339628137nteger(G),A3) ) ) ).

% sum.setdiff_irrelevant
tff(fact_7113_sum_Osetdiff__irrelevant,axiom,
    ! [A3: set_nat,G: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( aa_set_nat_rat(groups2906978787729119204at_rat(G),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A3),collect_nat(aTP_Lamm_zs(G)))) = aa_set_nat_rat(groups2906978787729119204at_rat(G),A3) ) ) ).

% sum.setdiff_irrelevant
tff(fact_7114_sum_Osetdiff__irrelevant,axiom,
    ! [A3: set_nat,G: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( aa_set_nat_int(groups3539618377306564664at_int(G),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A3),collect_nat(aTP_Lamm_zt(G)))) = aa_set_nat_int(groups3539618377306564664at_int(G),A3) ) ) ).

% sum.setdiff_irrelevant
tff(fact_7115_diff__preserves__multiset,axiom,
    ! [M6: fun_Pr5160692531401422620nt_nat,N7: fun_Pr5160692531401422620nt_nat] :
      ( finite2998713641127702882nt_int(collec213857154873943460nt_int(aTP_Lamm_vu(M6)))
     => finite2998713641127702882nt_int(collec213857154873943460nt_int(aa_fun6302685600795359042_int_o(aTP_Lamm_zu(M6),N7))) ) ).

% diff_preserves_multiset
tff(fact_7116_diff__preserves__multiset,axiom,
    ! [M6: fun_list_nat_nat,N7: fun_list_nat_nat] :
      ( finite8100373058378681591st_nat(collect_list_nat(aTP_Lamm_vw(M6)))
     => finite8100373058378681591st_nat(collect_list_nat(aa_fun7915899554530022_nat_o(aTP_Lamm_zv(M6),N7))) ) ).

% diff_preserves_multiset
tff(fact_7117_diff__preserves__multiset,axiom,
    ! [M6: fun_set_nat_nat,N7: fun_set_nat_nat] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,collect_set_nat(aTP_Lamm_vy(M6)))
     => aa_set_set_nat_o(finite1152437895449049373et_nat,collect_set_nat(aa_fun1332161875654541722_nat_o(aTP_Lamm_zw(M6),N7))) ) ).

% diff_preserves_multiset
tff(fact_7118_diff__preserves__multiset,axiom,
    ! [M6: fun_nat_nat,N7: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,collect_nat(aTP_Lamm_wa(M6)))
     => aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun8673969331427483654_nat_o(aTP_Lamm_zx(M6),N7))) ) ).

% diff_preserves_multiset
tff(fact_7119_diff__preserves__multiset,axiom,
    ! [M6: fun_int_nat,N7: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,collect_int(aTP_Lamm_wc(M6)))
     => aa_set_int_o(finite_finite_int,collect_int(aa_fun5425778157628386638_int_o(aTP_Lamm_zy(M6),N7))) ) ).

% diff_preserves_multiset
tff(fact_7120_diff__preserves__multiset,axiom,
    ! [M6: fun_Code_integer_nat,N7: fun_Code_integer_nat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,collect_Code_integer(aTP_Lamm_we(M6)))
     => aa_set5643387332196575111eger_o(finite6017078050557962740nteger,collect_Code_integer(aa_fun4054834855839773292eger_o(aTP_Lamm_zz(M6),N7))) ) ).

% diff_preserves_multiset
tff(fact_7121_diff__preserves__multiset,axiom,
    ! [M6: fun_Pr2602117396471556196at_nat,N7: fun_Pr2602117396471556196at_nat] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,collec3392354462482085612at_nat(aTP_Lamm_wg(M6)))
     => aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,collec3392354462482085612at_nat(aa_fun741873263805325634_nat_o(aTP_Lamm_aaa(M6),N7))) ) ).

% diff_preserves_multiset
tff(fact_7122_prod_Ogroup,axiom,
    ! [S: set_nat,T: set_Code_integer,G: fun_nat_Code_integer,Ha: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,S)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,T)
       => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(image_1215581382706833972nteger(G,S)),T)
         => ( groups3190895334310489300er_nat(aa_fun2720235278490593797er_nat(aa_fun676797813428960940er_nat(aTP_Lamm_aab(S),G),Ha),T) = aa_set_nat_nat(groups708209901874060359at_nat(Ha),S) ) ) ) ) ).

% prod.group
tff(fact_7123_prod_Ogroup,axiom,
    ! [S: set_nat,T: set_int,G: fun_nat_int,Ha: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,S)
     => ( aa_set_int_o(finite_finite_int,T)
       => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,image_nat_int(G,S)),T)
         => ( groups1707563613775114915nt_nat(aa_fun6790803225114646484nt_nat(aa_fun5866268260113680714nt_nat(aTP_Lamm_aac(S),G),Ha),T) = aa_set_nat_nat(groups708209901874060359at_nat(Ha),S) ) ) ) ) ).

% prod.group
tff(fact_7124_prod_Ogroup,axiom,
    ! [S: set_nat,T: set_Code_integer,G: fun_nat_Code_integer,Ha: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,S)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,T)
       => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(image_1215581382706833972nteger(G,S)),T)
         => ( groups3188404863801439024er_int(aa_fun2934847258012291773er_int(aa_fun2579177667528128868er_int(aTP_Lamm_aad(S),G),Ha),T) = aa_set_nat_int(groups705719431365010083at_int(Ha),S) ) ) ) ) ).

% prod.group
tff(fact_7125_prod_Ogroup,axiom,
    ! [S: set_int,T: set_Code_integer,G: fun_int_Code_integer,Ha: fun_int_int] :
      ( aa_set_int_o(finite_finite_int,S)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,T)
       => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(image_1587234942943678608nteger(G,S)),T)
         => ( groups3188404863801439024er_int(aa_fun2092574682676764953er_int(aa_fun9146121993134396956er_int(aTP_Lamm_aae(S),G),Ha),T) = groups1705073143266064639nt_int(Ha,S) ) ) ) ) ).

% prod.group
tff(fact_7126_prod_Ogroup,axiom,
    ! [S: set_o,T: set_nat,G: fun_o_nat,Ha: fun_o_nat] :
      ( finite_finite_o(S)
     => ( aa_set_nat_o(finite_finite_nat,T)
       => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,image_o_nat(G,S)),T)
         => ( aa_set_nat_nat(groups708209901874060359at_nat(aa_fun1225607396849667164at_nat(aa_fun2442258407354871084at_nat(aTP_Lamm_aaf(S),G),Ha)),T) = groups3504817904513533571_o_nat(Ha,S) ) ) ) ) ).

% prod.group
tff(fact_7127_prod_Ogroup,axiom,
    ! [S: set_int,T: set_nat,G: fun_int_nat,Ha: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,S)
     => ( aa_set_nat_o(finite_finite_nat,T)
       => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,image_int_nat(G,S)),T)
         => ( aa_set_nat_nat(groups708209901874060359at_nat(aa_fun5507398438843482068at_nat(aa_fun3220669823809117514at_nat(aTP_Lamm_aag(S),G),Ha)),T) = groups1707563613775114915nt_nat(Ha,S) ) ) ) ) ).

% prod.group
tff(fact_7128_prod_Ogroup,axiom,
    ! [S: set_Code_integer,T: set_nat,G: fun_Code_integer_nat,Ha: fun_Code_integer_nat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,S)
     => ( aa_set_nat_o(finite_finite_nat,T)
       => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,image_951025933927791156er_nat(G,S)),T)
         => ( aa_set_nat_nat(groups708209901874060359at_nat(aa_fun1707803128846032901at_nat(aa_fun2075405564639334828at_nat(aTP_Lamm_aah(S),G),Ha)),T) = groups3190895334310489300er_nat(Ha,S) ) ) ) ) ).

% prod.group
tff(fact_7129_prod_Ogroup,axiom,
    ! [S: set_nat,T: set_nat,G: fun_nat_nat,Ha: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,S)
     => ( aa_set_nat_o(finite_finite_nat,T)
       => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,image_nat_nat(G,S)),T)
         => ( aa_set_nat_nat(groups708209901874060359at_nat(aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(aTP_Lamm_aai(S),G),Ha)),T) = aa_set_nat_nat(groups708209901874060359at_nat(Ha),S) ) ) ) ) ).

% prod.group
tff(fact_7130_prod_Ogroup,axiom,
    ! [S: set_o,T: set_nat,G: fun_o_nat,Ha: fun_o_int] :
      ( finite_finite_o(S)
     => ( aa_set_nat_o(finite_finite_nat,T)
       => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,image_o_nat(G,S)),T)
         => ( aa_set_nat_int(groups705719431365010083at_int(aa_fun999201005756179732at_int(aa_fun3903619890838853604at_int(aTP_Lamm_aaj(S),G),Ha)),T) = groups3502327434004483295_o_int(Ha,S) ) ) ) ) ).

% prod.group
tff(fact_7131_prod_Ogroup,axiom,
    ! [S: set_Code_integer,T: set_nat,G: fun_Code_integer_nat,Ha: fun_Code_integer_int] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,S)
     => ( aa_set_nat_o(finite_finite_nat,T)
       => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,image_951025933927791156er_nat(G,S)),T)
         => ( aa_set_nat_int(groups705719431365010083at_int(aa_fun1481396737752545469at_int(aa_fun3536767048123317348at_int(aTP_Lamm_aak(S),G),Ha)),T) = groups3188404863801439024er_int(Ha,S) ) ) ) ) ).

% prod.group
tff(fact_7132_subset__eq__atLeast0__atMost__finite,axiom,
    ! [N7: set_nat,N2: nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,N7),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,zero_zero_nat),N2))
     => aa_set_nat_o(finite_finite_nat,N7) ) ).

% subset_eq_atLeast0_atMost_finite
tff(fact_7133_prod_Ointer__restrict,axiom,
    ! [A3: set_o,G: fun_o_Code_integer,B3: set_o] :
      ( finite_finite_o(A3)
     => ( groups7694694392188491536nteger(G,inf_inf_set_o(A3,B3)) = groups7694694392188491536nteger(aa_set9049949509649958604nteger(aTP_Lamm_aal(G),B3),A3) ) ) ).

% prod.inter_restrict
tff(fact_7134_prod_Ointer__restrict,axiom,
    ! [A3: set_int,G: fun_int_Code_integer,B3: set_int] :
      ( aa_set_int_o(finite_finite_int,A3)
     => ( groups3827104343326376752nteger(G,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A3),B3)) = groups3827104343326376752nteger(aa_set5517701783735052070nteger(aTP_Lamm_aam(G),B3),A3) ) ) ).

% prod.inter_restrict
tff(fact_7135_prod_Ointer__restrict,axiom,
    ! [A3: set_Code_integer,G: fun_Co8499238673658658103nteger,B3: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A3)
     => ( groups3674199335183972705nteger(G,inf_in1364745209274528805nteger(A3,B3)) = groups3674199335183972705nteger(aa_set4294055438810586376nteger(aTP_Lamm_aan(G),B3),A3) ) ) ).

% prod.inter_restrict
tff(fact_7136_prod_Ointer__restrict,axiom,
    ! [A3: set_o,G: fun_o_assn,B3: set_o] :
      ( finite_finite_o(A3)
     => ( groups5301882518646026715o_assn(G,inf_inf_set_o(A3,B3)) = groups5301882518646026715o_assn(aa_set_o_fun_o_assn(aTP_Lamm_aao(G),B3),A3) ) ) ).

% prod.inter_restrict
tff(fact_7137_prod_Ointer__restrict,axiom,
    ! [A3: set_int,G: fun_int_assn,B3: set_int] :
      ( aa_set_int_o(finite_finite_int,A3)
     => ( groups7882442080178216443t_assn(G,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A3),B3)) = groups7882442080178216443t_assn(aa_set5310648174735401585t_assn(aTP_Lamm_aap(G),B3),A3) ) ) ).

% prod.inter_restrict
tff(fact_7138_prod_Ointer__restrict,axiom,
    ! [A3: set_Code_integer,G: fun_Co4490034186337852034r_assn,B3: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A3)
     => ( groups1304777262505850412r_assn(G,inf_in1364745209274528805nteger(A3,B3)) = groups1304777262505850412r_assn(aa_set2437433236263590995r_assn(aTP_Lamm_aaq(G),B3),A3) ) ) ).

% prod.inter_restrict
tff(fact_7139_prod_Ointer__restrict,axiom,
    ! [A3: set_o,G: fun_o_rat,B3: set_o] :
      ( finite_finite_o(A3)
     => ( groups2869687844427037835_o_rat(G,inf_inf_set_o(A3,B3)) = groups2869687844427037835_o_rat(aa_set_o_fun_o_rat(aTP_Lamm_aar(G),B3),A3) ) ) ).

% prod.inter_restrict
tff(fact_7140_prod_Ointer__restrict,axiom,
    ! [A3: set_int,G: fun_int_rat,B3: set_int] :
      ( aa_set_int_o(finite_finite_int,A3)
     => ( groups1072433553688619179nt_rat(G,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A3),B3)) = groups1072433553688619179nt_rat(aa_set7979663334043413281nt_rat(aTP_Lamm_aas(G),B3),A3) ) ) ).

% prod.inter_restrict
tff(fact_7141_prod_Ointer__restrict,axiom,
    ! [A3: set_Code_integer,G: fun_Code_integer_rat,B3: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A3)
     => ( groups2555765274223993564er_rat(G,inf_in1364745209274528805nteger(A3,B3)) = groups2555765274223993564er_rat(aa_set7344733238996822275er_rat(aTP_Lamm_aat(G),B3),A3) ) ) ).

% prod.inter_restrict
tff(fact_7142_prod_Ointer__restrict,axiom,
    ! [A3: set_o,G: fun_o_nat,B3: set_o] :
      ( finite_finite_o(A3)
     => ( groups3504817904513533571_o_nat(G,inf_inf_set_o(A3,B3)) = groups3504817904513533571_o_nat(aa_set_o_fun_o_nat(aTP_Lamm_aau(G),B3),A3) ) ) ).

% prod.inter_restrict
tff(fact_7143_power__numeral__even,axiom,
    ! [Z: code_integer,W2: num] :
      aa_nat_Code_integer(power_8256067586552552935nteger(Z),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,W2))) = $let(
        w: code_integer,
        w:= aa_nat_Code_integer(power_8256067586552552935nteger(Z),aa_num_nat(numeral_numeral_nat,W2)),
        aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,w),w) ) ).

% power_numeral_even
tff(fact_7144_power__numeral__even,axiom,
    ! [Z: assn,W2: num] :
      aa_nat_assn(power_power_assn(Z),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,W2))) = $let(
        w: assn,
        w:= aa_nat_assn(power_power_assn(Z),aa_num_nat(numeral_numeral_nat,W2)),
        aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,w),w) ) ).

% power_numeral_even
tff(fact_7145_power__numeral__even,axiom,
    ! [Z: rat,W2: num] :
      aa_nat_rat(power_power_rat(Z),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,W2))) = $let(
        w: rat,
        w:= aa_nat_rat(power_power_rat(Z),aa_num_nat(numeral_numeral_nat,W2)),
        aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,w),w) ) ).

% power_numeral_even
tff(fact_7146_power__numeral__even,axiom,
    ! [Z: nat,W2: num] :
      aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Z),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,W2))) = $let(
        w: nat,
        w:= aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Z),aa_num_nat(numeral_numeral_nat,W2)),
        aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,w),w) ) ).

% power_numeral_even
tff(fact_7147_power__numeral__even,axiom,
    ! [Z: int,W2: num] :
      aa_nat_int(aa_int_fun_nat_int(power_power_int,Z),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,W2))) = $let(
        w: int,
        w:= aa_nat_int(aa_int_fun_nat_int(power_power_int,Z),aa_num_nat(numeral_numeral_nat,W2)),
        aa_int_int(aa_int_fun_int_int(times_times_int,w),w) ) ).

% power_numeral_even
tff(fact_7148_prod_Osetdiff__irrelevant,axiom,
    ! [A3: set_int,G: fun_int_Code_integer] :
      ( aa_set_int_o(finite_finite_int,A3)
     => ( groups3827104343326376752nteger(G,aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A3),collect_int(aTP_Lamm_aav(G)))) = groups3827104343326376752nteger(G,A3) ) ) ).

% prod.setdiff_irrelevant
tff(fact_7149_prod_Osetdiff__irrelevant,axiom,
    ! [A3: set_Code_integer,G: fun_Co8499238673658658103nteger] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A3)
     => ( groups3674199335183972705nteger(G,minus_2355218937544613996nteger(A3,collect_Code_integer(aTP_Lamm_aaw(G)))) = groups3674199335183972705nteger(G,A3) ) ) ).

% prod.setdiff_irrelevant
tff(fact_7150_prod_Osetdiff__irrelevant,axiom,
    ! [A3: set_int,G: fun_int_assn] :
      ( aa_set_int_o(finite_finite_int,A3)
     => ( groups7882442080178216443t_assn(G,aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A3),collect_int(aTP_Lamm_aax(G)))) = groups7882442080178216443t_assn(G,A3) ) ) ).

% prod.setdiff_irrelevant
tff(fact_7151_prod_Osetdiff__irrelevant,axiom,
    ! [A3: set_Code_integer,G: fun_Co4490034186337852034r_assn] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A3)
     => ( groups1304777262505850412r_assn(G,minus_2355218937544613996nteger(A3,collect_Code_integer(aTP_Lamm_aay(G)))) = groups1304777262505850412r_assn(G,A3) ) ) ).

% prod.setdiff_irrelevant
tff(fact_7152_prod_Osetdiff__irrelevant,axiom,
    ! [A3: set_int,G: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,A3)
     => ( groups1072433553688619179nt_rat(G,aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A3),collect_int(aTP_Lamm_aaz(G)))) = groups1072433553688619179nt_rat(G,A3) ) ) ).

% prod.setdiff_irrelevant
tff(fact_7153_prod_Osetdiff__irrelevant,axiom,
    ! [A3: set_Code_integer,G: fun_Code_integer_rat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A3)
     => ( groups2555765274223993564er_rat(G,minus_2355218937544613996nteger(A3,collect_Code_integer(aTP_Lamm_aba(G)))) = groups2555765274223993564er_rat(G,A3) ) ) ).

% prod.setdiff_irrelevant
tff(fact_7154_prod_Osetdiff__irrelevant,axiom,
    ! [A3: set_int,G: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,A3)
     => ( groups1707563613775114915nt_nat(G,aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A3),collect_int(aTP_Lamm_abb(G)))) = groups1707563613775114915nt_nat(G,A3) ) ) ).

% prod.setdiff_irrelevant
tff(fact_7155_prod_Osetdiff__irrelevant,axiom,
    ! [A3: set_Code_integer,G: fun_Code_integer_nat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A3)
     => ( groups3190895334310489300er_nat(G,minus_2355218937544613996nteger(A3,collect_Code_integer(aTP_Lamm_abc(G)))) = groups3190895334310489300er_nat(G,A3) ) ) ).

% prod.setdiff_irrelevant
tff(fact_7156_prod_Osetdiff__irrelevant,axiom,
    ! [A3: set_Code_integer,G: fun_Code_integer_int] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A3)
     => ( groups3188404863801439024er_int(G,minus_2355218937544613996nteger(A3,collect_Code_integer(aTP_Lamm_abd(G)))) = groups3188404863801439024er_int(G,A3) ) ) ).

% prod.setdiff_irrelevant
tff(fact_7157_prod_Osetdiff__irrelevant,axiom,
    ! [A3: set_nat,G: fun_nat_Code_integer] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( aa_set871585252387937313nteger(groups3455450783089532116nteger(G),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A3),collect_nat(aTP_Lamm_abe(G)))) = aa_set871585252387937313nteger(groups3455450783089532116nteger(G),A3) ) ) ).

% prod.setdiff_irrelevant
tff(fact_7158_finite__abs__int__segment,axiom,
    ! [A: rat] : finite_finite_rat(collect_rat(aa_rat_fun_rat_o(aTP_Lamm_abf,A))) ).

% finite_abs_int_segment
tff(fact_7159_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_7160_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_7161_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_7162_arith__series__nat,axiom,
    ! [A: nat,D2: nat,N2: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_abg(A),D2)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,zero_zero_nat),N2)) = 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,N2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N2),D2)))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% arith_series_nat
tff(fact_7163_atLeastAtMost__subseteq__atLeastLessThan__iff,axiom,
    ! [A: assn,B: assn,Ca: assn,D2: assn] :
      ( ord_less_eq_set_assn(set_or7959216805967363635t_assn(A,B),set_or5523502641901747543n_assn(Ca,D2))
    <=> ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),B)
       => ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,Ca),A)
          & aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,B),D2) ) ) ) ).

% atLeastAtMost_subseteq_atLeastLessThan_iff
tff(fact_7164_atLeastAtMost__subseteq__atLeastLessThan__iff,axiom,
    ! [A: set_int,B: set_int,Ca: set_int,D2: set_int] :
      ( ord_le4403425263959731960et_int(set_or370866239135849197et_int(A,B),set_or8585797421378605585et_int(Ca,D2))
    <=> ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A),B)
       => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,Ca),A)
          & aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_set_int,B),D2) ) ) ) ).

% atLeastAtMost_subseteq_atLeastLessThan_iff
tff(fact_7165_atLeastAtMost__subseteq__atLeastLessThan__iff,axiom,
    ! [A: rat,B: rat,Ca: rat,D2: rat] :
      ( ord_less_eq_set_rat(set_or633870826150836451st_rat(A,B),set_or4029947393144176647an_rat(Ca,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,Ca),A)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),D2) ) ) ) ).

% atLeastAtMost_subseteq_atLeastLessThan_iff
tff(fact_7166_atLeastAtMost__subseteq__atLeastLessThan__iff,axiom,
    ! [A: num,B: num,Ca: num,D2: num] :
      ( ord_less_eq_set_num(set_or7049704709247886629st_num(A,B),set_or1222409239386451017an_num(Ca,D2))
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
       => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Ca),A)
          & aa_num_o(aa_num_fun_num_o(ord_less_num,B),D2) ) ) ) ).

% atLeastAtMost_subseteq_atLeastLessThan_iff
tff(fact_7167_atLeastAtMost__subseteq__atLeastLessThan__iff,axiom,
    ! [A: nat,B: nat,Ca: nat,D2: nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,A),B)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,Ca),D2))
    <=> ( 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,Ca),A)
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),D2) ) ) ) ).

% atLeastAtMost_subseteq_atLeastLessThan_iff
tff(fact_7168_atLeastAtMost__subseteq__atLeastLessThan__iff,axiom,
    ! [A: int,B: int,Ca: int,D2: int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,set_or1266510415728281911st_int(A,B)),set_or4662586982721622107an_int(Ca,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,Ca),A)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,B),D2) ) ) ) ).

% atLeastAtMost_subseteq_atLeastLessThan_iff
tff(fact_7169_atLeastAtMost__subseteq__atLeastLessThan__iff,axiom,
    ! [A: code_integer,B: code_integer,Ca: code_integer,D2: code_integer] :
      ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(set_or189985376899183464nteger(A,B)),set_or8404916559141939852nteger(Ca,D2))
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Ca),A)
          & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),D2) ) ) ) ).

% atLeastAtMost_subseteq_atLeastLessThan_iff
tff(fact_7170_atLeastLessThan__subseteq__atLeastAtMost__iff,axiom,
    ! [A: rat,B: rat,Ca: rat,D2: rat] :
      ( ord_less_eq_set_rat(set_or4029947393144176647an_rat(A,B),set_or633870826150836451st_rat(Ca,D2))
    <=> ( 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,Ca),A)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),D2) ) ) ) ).

% atLeastLessThan_subseteq_atLeastAtMost_iff
tff(fact_7171_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)
       => ( 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_7172_ivl__disj__un__two__touch_I2_J,axiom,
    ! [L: num,M: num,U: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,L),M)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,M),U)
       => ( sup_sup_set_num(set_or7049704709247886629st_num(L,M),set_or1222409239386451017an_num(M,U)) = set_or1222409239386451017an_num(L,U) ) ) ) ).

% ivl_disj_un_two_touch(2)
tff(fact_7173_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(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,L),M)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,M),U)) = aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,L),U) ) ) ) ).

% ivl_disj_un_two_touch(2)
tff(fact_7174_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_7175_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_7176_Sum__Icc__nat,axiom,
    ! [M: nat,N2: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_fu),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M),N2)) = 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,N2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),one_one_nat))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat)))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% Sum_Icc_nat
tff(fact_7177_sum_OatLeast__Suc__lessThan,axiom,
    ! [M: nat,N2: nat,G: fun_na2338080690716861136at_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N2)
     => ( groups6857163185585827899at_nat(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,M),N2)) = aa_mul5754855110707352700at_nat(plus_p7104986032573967614at_nat(aa_nat2240542744625857721at_nat(G,M)),groups6857163185585827899at_nat(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,aa_nat_nat(suc,M)),N2))) ) ) ).

% sum.atLeast_Suc_lessThan
tff(fact_7178_sum_OatLeast__Suc__lessThan,axiom,
    ! [M: nat,N2: nat,G: fun_nat_rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N2)
     => ( aa_set_nat_rat(groups2906978787729119204at_rat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,M),N2)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(G,M)),aa_set_nat_rat(groups2906978787729119204at_rat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,aa_nat_nat(suc,M)),N2))) ) ) ).

% sum.atLeast_Suc_lessThan
tff(fact_7179_sum_OatLeast__Suc__lessThan,axiom,
    ! [M: nat,N2: nat,G: fun_nat_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N2)
     => ( aa_set_nat_int(groups3539618377306564664at_int(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,M),N2)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(G,M)),aa_set_nat_int(groups3539618377306564664at_int(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,aa_nat_nat(suc,M)),N2))) ) ) ).

% sum.atLeast_Suc_lessThan
tff(fact_7180_sum_OatLeast__Suc__lessThan,axiom,
    ! [M: nat,N2: nat,G: fun_nat_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N2)
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,M),N2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(G,M)),aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,aa_nat_nat(suc,M)),N2))) ) ) ).

% sum.atLeast_Suc_lessThan
tff(fact_7181_sum_OatLeastLessThan__Suc,axiom,
    ! [A: nat,B: nat,G: fun_na2338080690716861136at_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( groups6857163185585827899at_nat(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,A),aa_nat_nat(suc,B))) = aa_mul5754855110707352700at_nat(plus_p7104986032573967614at_nat(groups6857163185585827899at_nat(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,A),B))),aa_nat2240542744625857721at_nat(G,B)) ) ) ).

% sum.atLeastLessThan_Suc
tff(fact_7182_sum_OatLeastLessThan__Suc,axiom,
    ! [A: nat,B: nat,G: fun_nat_rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_set_nat_rat(groups2906978787729119204at_rat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,A),aa_nat_nat(suc,B))) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_set_nat_rat(groups2906978787729119204at_rat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,A),B))),aa_nat_rat(G,B)) ) ) ).

% sum.atLeastLessThan_Suc
tff(fact_7183_sum_OatLeastLessThan__Suc,axiom,
    ! [A: nat,B: nat,G: fun_nat_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_set_nat_int(groups3539618377306564664at_int(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,A),aa_nat_nat(suc,B))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_set_nat_int(groups3539618377306564664at_int(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,A),B))),aa_nat_int(G,B)) ) ) ).

% sum.atLeastLessThan_Suc
tff(fact_7184_sum_OatLeastLessThan__Suc,axiom,
    ! [A: nat,B: nat,G: fun_nat_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,A),aa_nat_nat(suc,B))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,A),B))),aa_nat_nat(G,B)) ) ) ).

% sum.atLeastLessThan_Suc
tff(fact_7185_prod_OatLeast0__lessThan__Suc,axiom,
    ! [G: fun_nat_Code_integer,N2: nat] : aa_set871585252387937313nteger(groups3455450783089532116nteger(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),aa_nat_nat(suc,N2))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_set871585252387937313nteger(groups3455450783089532116nteger(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),N2))),aa_nat_Code_integer(G,N2)) ).

% prod.atLeast0_lessThan_Suc
tff(fact_7186_prod_OatLeast0__lessThan__Suc,axiom,
    ! [G: fun_nat_assn,N2: nat] : groups6906906614972039071t_assn(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),aa_nat_nat(suc,N2))) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,groups6906906614972039071t_assn(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),N2))),aa_nat_assn(G,N2)) ).

% prod.atLeast0_lessThan_Suc
tff(fact_7187_prod_OatLeast0__lessThan__Suc,axiom,
    ! [G: fun_nat_rat,N2: nat] : aa_set_nat_rat(groups73079841787564623at_rat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),aa_nat_nat(suc,N2))) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_set_nat_rat(groups73079841787564623at_rat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),N2))),aa_nat_rat(G,N2)) ).

% prod.atLeast0_lessThan_Suc
tff(fact_7188_prod_OatLeast0__lessThan__Suc,axiom,
    ! [G: fun_nat_nat,N2: nat] : aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),aa_nat_nat(suc,N2))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),N2))),aa_nat_nat(G,N2)) ).

% prod.atLeast0_lessThan_Suc
tff(fact_7189_prod_OatLeast0__lessThan__Suc,axiom,
    ! [G: fun_nat_int,N2: nat] : aa_set_nat_int(groups705719431365010083at_int(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),aa_nat_nat(suc,N2))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_set_nat_int(groups705719431365010083at_int(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),N2))),aa_nat_int(G,N2)) ).

% prod.atLeast0_lessThan_Suc
tff(fact_7190_prod_OatLeast__Suc__lessThan,axiom,
    ! [M: nat,N2: nat,G: fun_nat_Code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N2)
     => ( aa_set871585252387937313nteger(groups3455450783089532116nteger(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,M),N2)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(G,M)),aa_set871585252387937313nteger(groups3455450783089532116nteger(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,aa_nat_nat(suc,M)),N2))) ) ) ).

% prod.atLeast_Suc_lessThan
tff(fact_7191_prod_OatLeast__Suc__lessThan,axiom,
    ! [M: nat,N2: nat,G: fun_nat_assn] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N2)
     => ( groups6906906614972039071t_assn(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,M),N2)) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_nat_assn(G,M)),groups6906906614972039071t_assn(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,aa_nat_nat(suc,M)),N2))) ) ) ).

% prod.atLeast_Suc_lessThan
tff(fact_7192_prod_OatLeast__Suc__lessThan,axiom,
    ! [M: nat,N2: nat,G: fun_nat_rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N2)
     => ( aa_set_nat_rat(groups73079841787564623at_rat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,M),N2)) = 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(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,aa_nat_nat(suc,M)),N2))) ) ) ).

% prod.atLeast_Suc_lessThan
tff(fact_7193_prod_OatLeast__Suc__lessThan,axiom,
    ! [M: nat,N2: nat,G: fun_nat_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N2)
     => ( aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,M),N2)) = 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(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,aa_nat_nat(suc,M)),N2))) ) ) ).

% prod.atLeast_Suc_lessThan
tff(fact_7194_prod_OatLeast__Suc__lessThan,axiom,
    ! [M: nat,N2: nat,G: fun_nat_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N2)
     => ( aa_set_nat_int(groups705719431365010083at_int(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,M),N2)) = 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(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,aa_nat_nat(suc,M)),N2))) ) ) ).

% prod.atLeast_Suc_lessThan
tff(fact_7195_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(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,A),aa_nat_nat(suc,B))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_set871585252387937313nteger(groups3455450783089532116nteger(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,A),B))),aa_nat_Code_integer(G,B)) ) ) ).

% prod.atLeastLessThan_Suc
tff(fact_7196_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(aa_nat1818925681583257441et_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(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,A),B))),aa_nat_assn(G,B)) ) ) ).

% prod.atLeastLessThan_Suc
tff(fact_7197_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(aa_nat1818925681583257441et_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(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,A),B))),aa_nat_rat(G,B)) ) ) ).

% prod.atLeastLessThan_Suc
tff(fact_7198_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(aa_nat1818925681583257441et_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(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,A),B))),aa_nat_nat(G,B)) ) ) ).

% prod.atLeastLessThan_Suc
tff(fact_7199_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(aa_nat1818925681583257441et_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(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,A),B))),aa_nat_int(G,B)) ) ) ).

% prod.atLeastLessThan_Suc
tff(fact_7200_sum_Olast__plus,axiom,
    ! [M: nat,N2: nat,G: fun_na2338080690716861136at_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
     => ( groups6857163185585827899at_nat(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M),N2)) = aa_mul5754855110707352700at_nat(plus_p7104986032573967614at_nat(aa_nat2240542744625857721at_nat(G,N2)),groups6857163185585827899at_nat(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,M),N2))) ) ) ).

% sum.last_plus
tff(fact_7201_sum_Olast__plus,axiom,
    ! [M: nat,N2: nat,G: fun_nat_rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
     => ( aa_set_nat_rat(groups2906978787729119204at_rat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M),N2)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(G,N2)),aa_set_nat_rat(groups2906978787729119204at_rat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,M),N2))) ) ) ).

% sum.last_plus
tff(fact_7202_sum_Olast__plus,axiom,
    ! [M: nat,N2: nat,G: fun_nat_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
     => ( aa_set_nat_int(groups3539618377306564664at_int(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M),N2)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(G,N2)),aa_set_nat_int(groups3539618377306564664at_int(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,M),N2))) ) ) ).

% sum.last_plus
tff(fact_7203_sum_Olast__plus,axiom,
    ! [M: nat,N2: nat,G: fun_nat_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M),N2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(G,N2)),aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,M),N2))) ) ) ).

% sum.last_plus
tff(fact_7204_prod_Olast__plus,axiom,
    ! [M: nat,N2: nat,G: fun_nat_Code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
     => ( aa_set871585252387937313nteger(groups3455450783089532116nteger(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M),N2)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(G,N2)),aa_set871585252387937313nteger(groups3455450783089532116nteger(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,M),N2))) ) ) ).

% prod.last_plus
tff(fact_7205_prod_Olast__plus,axiom,
    ! [M: nat,N2: nat,G: fun_nat_assn] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
     => ( groups6906906614972039071t_assn(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M),N2)) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_nat_assn(G,N2)),groups6906906614972039071t_assn(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,M),N2))) ) ) ).

% prod.last_plus
tff(fact_7206_prod_Olast__plus,axiom,
    ! [M: nat,N2: nat,G: fun_nat_rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
     => ( aa_set_nat_rat(groups73079841787564623at_rat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M),N2)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(G,N2)),aa_set_nat_rat(groups73079841787564623at_rat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,M),N2))) ) ) ).

% prod.last_plus
tff(fact_7207_prod_Olast__plus,axiom,
    ! [M: nat,N2: nat,G: fun_nat_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
     => ( aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M),N2)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(G,N2)),aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,M),N2))) ) ) ).

% prod.last_plus
tff(fact_7208_prod_Olast__plus,axiom,
    ! [M: nat,N2: nat,G: fun_nat_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
     => ( aa_set_nat_int(groups705719431365010083at_int(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M),N2)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(G,N2)),aa_set_nat_int(groups705719431365010083at_int(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,M),N2))) ) ) ).

% prod.last_plus
tff(fact_7209_sum__pos2,axiom,
    ! [I5: set_o,I2: $o,F: fun_o_Code_integer] :
      ( finite_finite_o(I5)
     => ( member_o((I2),I5)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_o_Code_integer(F,(I2)))
         => ( ! [I3: $o] :
                ( member_o((I3),I5)
               => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_o_Code_integer(F,(I3))) )
           => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),groups4406642042086082107nteger(F,I5)) ) ) ) ) ).

% sum_pos2
tff(fact_7210_sum__pos2,axiom,
    ! [I5: set_nat,I2: nat,F: fun_nat_Code_integer] :
      ( aa_set_nat_o(finite_finite_nat,I5)
     => ( aa_set_nat_o(member_nat(I2),I5)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(F,I2))
         => ( ! [I3: nat] :
                ( aa_set_nat_o(member_nat(I3),I5)
               => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(F,I3)) )
           => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_set871585252387937313nteger(groups7501900531339628137nteger(F),I5)) ) ) ) ) ).

% sum_pos2
tff(fact_7211_sum__pos2,axiom,
    ! [I5: set_int,I2: int,F: fun_int_Code_integer] :
      ( aa_set_int_o(finite_finite_int,I5)
     => ( aa_set_int_o(member_int(I2),I5)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_int_Code_integer(F,I2))
         => ( ! [I3: int] :
                ( aa_set_int_o(member_int(I3),I5)
               => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_int_Code_integer(F,I3)) )
           => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),groups7873554091576472773nteger(F,I5)) ) ) ) ) ).

% sum_pos2
tff(fact_7212_sum__pos2,axiom,
    ! [I5: set_Code_integer,I2: code_integer,F: fun_Co8499238673658658103nteger] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,I5)
     => ( aa_set5643387332196575111eger_o(member_Code_integer(I2),I5)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(F,I2))
         => ( ! [I3: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(I3),I5)
               => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(F,I3)) )
           => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),groups879477027807139574nteger(F,I5)) ) ) ) ) ).

% sum_pos2
tff(fact_7213_sum__pos2,axiom,
    ! [I5: set_o,I2: $o,F: fun_o_rat] :
      ( finite_finite_o(I5)
     => ( member_o((I2),I5)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_o_rat(F,(I2)))
         => ( ! [I3: $o] :
                ( member_o((I3),I5)
               => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_o_rat(F,(I3))) )
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),groups7872700643590313910_o_rat(F,I5)) ) ) ) ) ).

% sum_pos2
tff(fact_7214_sum__pos2,axiom,
    ! [I5: set_nat,I2: nat,F: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,I5)
     => ( aa_set_nat_o(member_nat(I2),I5)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_nat_rat(F,I2))
         => ( ! [I3: nat] :
                ( aa_set_nat_o(member_nat(I3),I5)
               => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(F,I3)) )
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_set_nat_rat(groups2906978787729119204at_rat(F),I5)) ) ) ) ) ).

% sum_pos2
tff(fact_7215_sum__pos2,axiom,
    ! [I5: set_int,I2: int,F: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,I5)
     => ( aa_set_int_o(member_int(I2),I5)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_int_rat(F,I2))
         => ( ! [I3: int] :
                ( aa_set_int_o(member_int(I3),I5)
               => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_int_rat(F,I3)) )
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),groups3906332499630173760nt_rat(F,I5)) ) ) ) ) ).

% sum_pos2
tff(fact_7216_sum__pos2,axiom,
    ! [I5: set_Code_integer,I2: code_integer,F: fun_Code_integer_rat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,I5)
     => ( aa_set5643387332196575111eger_o(member_Code_integer(I2),I5)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_Code_integer_rat(F,I2))
         => ( ! [I3: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(I3),I5)
               => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_Code_integer_rat(F,I3)) )
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),groups6602215022474089585er_rat(F,I5)) ) ) ) ) ).

% sum_pos2
tff(fact_7217_sum__pos2,axiom,
    ! [I5: set_o,I2: $o,F: fun_o_nat] :
      ( finite_finite_o(I5)
     => ( member_o((I2),I5)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_o_nat(F,(I2)))
         => ( ! [I3: $o] :
                ( member_o((I3),I5)
               => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_o_nat(F,(I3))) )
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),groups8507830703676809646_o_nat(F,I5)) ) ) ) ) ).

% sum_pos2
tff(fact_7218_sum__pos2,axiom,
    ! [I5: set_int,I2: int,F: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,I5)
     => ( aa_set_int_o(member_int(I2),I5)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_int_nat(F,I2))
         => ( ! [I3: int] :
                ( aa_set_int_o(member_int(I3),I5)
               => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_int_nat(F,I3)) )
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),groups4541462559716669496nt_nat(F,I5)) ) ) ) ) ).

% sum_pos2
tff(fact_7219_sum__pos,axiom,
    ! [I5: set_Code_integer,F: fun_Co8499238673658658103nteger] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,I5)
     => ( ( I5 != bot_bo3990330152332043303nteger )
       => ( ! [I3: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(I3),I5)
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(F,I3)) )
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),groups879477027807139574nteger(F,I5)) ) ) ) ).

% sum_pos
tff(fact_7220_sum__pos,axiom,
    ! [I5: set_o,F: fun_o_Code_integer] :
      ( finite_finite_o(I5)
     => ( ( I5 != bot_bot_set_o )
       => ( ! [I3: $o] :
              ( member_o((I3),I5)
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_o_Code_integer(F,(I3))) )
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),groups4406642042086082107nteger(F,I5)) ) ) ) ).

% sum_pos
tff(fact_7221_sum__pos,axiom,
    ! [I5: set_nat,F: fun_nat_Code_integer] :
      ( aa_set_nat_o(finite_finite_nat,I5)
     => ( ( I5 != bot_bot_set_nat )
       => ( ! [I3: nat] :
              ( aa_set_nat_o(member_nat(I3),I5)
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(F,I3)) )
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_set871585252387937313nteger(groups7501900531339628137nteger(F),I5)) ) ) ) ).

% sum_pos
tff(fact_7222_sum__pos,axiom,
    ! [I5: set_int,F: fun_int_Code_integer] :
      ( aa_set_int_o(finite_finite_int,I5)
     => ( ( I5 != bot_bot_set_int )
       => ( ! [I3: int] :
              ( aa_set_int_o(member_int(I3),I5)
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_int_Code_integer(F,I3)) )
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),groups7873554091576472773nteger(F,I5)) ) ) ) ).

% sum_pos
tff(fact_7223_sum__pos,axiom,
    ! [I5: set_Code_integer,F: fun_Code_integer_rat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,I5)
     => ( ( I5 != bot_bo3990330152332043303nteger )
       => ( ! [I3: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(I3),I5)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_Code_integer_rat(F,I3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),groups6602215022474089585er_rat(F,I5)) ) ) ) ).

% sum_pos
tff(fact_7224_sum__pos,axiom,
    ! [I5: set_o,F: fun_o_rat] :
      ( finite_finite_o(I5)
     => ( ( I5 != bot_bot_set_o )
       => ( ! [I3: $o] :
              ( member_o((I3),I5)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_o_rat(F,(I3))) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),groups7872700643590313910_o_rat(F,I5)) ) ) ) ).

% sum_pos
tff(fact_7225_sum__pos,axiom,
    ! [I5: set_nat,F: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,I5)
     => ( ( I5 != bot_bot_set_nat )
       => ( ! [I3: nat] :
              ( aa_set_nat_o(member_nat(I3),I5)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_nat_rat(F,I3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_set_nat_rat(groups2906978787729119204at_rat(F),I5)) ) ) ) ).

% sum_pos
tff(fact_7226_sum__pos,axiom,
    ! [I5: set_int,F: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,I5)
     => ( ( I5 != bot_bot_set_int )
       => ( ! [I3: int] :
              ( aa_set_int_o(member_int(I3),I5)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_int_rat(F,I3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),groups3906332499630173760nt_rat(F,I5)) ) ) ) ).

% sum_pos
tff(fact_7227_sum__pos,axiom,
    ! [I5: set_Code_integer,F: fun_Code_integer_nat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,I5)
     => ( ( I5 != bot_bo3990330152332043303nteger )
       => ( ! [I3: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(I3),I5)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_Code_integer_nat(F,I3)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),groups7237345082560585321er_nat(F,I5)) ) ) ) ).

% sum_pos
tff(fact_7228_sum__pos,axiom,
    ! [I5: set_o,F: fun_o_nat] :
      ( finite_finite_o(I5)
     => ( ( I5 != bot_bot_set_o )
       => ( ! [I3: $o] :
              ( member_o((I3),I5)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_o_nat(F,(I3))) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),groups8507830703676809646_o_nat(F,I5)) ) ) ) ).

% sum_pos
tff(fact_7229_less__1__prod2,axiom,
    ! [I5: set_o,I2: $o,F: fun_o_Code_integer] :
      ( finite_finite_o(I5)
     => ( member_o((I2),I5)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),aa_o_Code_integer(F,(I2)))
         => ( ! [I3: $o] :
                ( member_o((I3),I5)
               => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),aa_o_Code_integer(F,(I3))) )
           => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),groups7694694392188491536nteger(F,I5)) ) ) ) ) ).

% less_1_prod2
tff(fact_7230_less__1__prod2,axiom,
    ! [I5: set_nat,I2: nat,F: fun_nat_Code_integer] :
      ( aa_set_nat_o(finite_finite_nat,I5)
     => ( aa_set_nat_o(member_nat(I2),I5)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),aa_nat_Code_integer(F,I2))
         => ( ! [I3: nat] :
                ( aa_set_nat_o(member_nat(I3),I5)
               => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),aa_nat_Code_integer(F,I3)) )
           => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),aa_set871585252387937313nteger(groups3455450783089532116nteger(F),I5)) ) ) ) ) ).

% less_1_prod2
tff(fact_7231_less__1__prod2,axiom,
    ! [I5: set_int,I2: int,F: fun_int_Code_integer] :
      ( aa_set_int_o(finite_finite_int,I5)
     => ( aa_set_int_o(member_int(I2),I5)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),aa_int_Code_integer(F,I2))
         => ( ! [I3: int] :
                ( aa_set_int_o(member_int(I3),I5)
               => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),aa_int_Code_integer(F,I3)) )
           => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),groups3827104343326376752nteger(F,I5)) ) ) ) ) ).

% less_1_prod2
tff(fact_7232_less__1__prod2,axiom,
    ! [I5: set_Code_integer,I2: code_integer,F: fun_Co8499238673658658103nteger] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,I5)
     => ( aa_set5643387332196575111eger_o(member_Code_integer(I2),I5)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(F,I2))
         => ( ! [I3: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(I3),I5)
               => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(F,I3)) )
           => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),groups3674199335183972705nteger(F,I5)) ) ) ) ) ).

% less_1_prod2
tff(fact_7233_less__1__prod2,axiom,
    ! [I5: set_o,I2: $o,F: fun_o_rat] :
      ( finite_finite_o(I5)
     => ( member_o((I2),I5)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_o_rat(F,(I2)))
         => ( ! [I3: $o] :
                ( member_o((I3),I5)
               => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_o_rat(F,(I3))) )
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),groups2869687844427037835_o_rat(F,I5)) ) ) ) ) ).

% less_1_prod2
tff(fact_7234_less__1__prod2,axiom,
    ! [I5: set_nat,I2: nat,F: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,I5)
     => ( aa_set_nat_o(member_nat(I2),I5)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_nat_rat(F,I2))
         => ( ! [I3: nat] :
                ( aa_set_nat_o(member_nat(I3),I5)
               => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_nat_rat(F,I3)) )
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_set_nat_rat(groups73079841787564623at_rat(F),I5)) ) ) ) ) ).

% less_1_prod2
tff(fact_7235_less__1__prod2,axiom,
    ! [I5: set_int,I2: int,F: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,I5)
     => ( aa_set_int_o(member_int(I2),I5)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_int_rat(F,I2))
         => ( ! [I3: int] :
                ( aa_set_int_o(member_int(I3),I5)
               => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_int_rat(F,I3)) )
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),groups1072433553688619179nt_rat(F,I5)) ) ) ) ) ).

% less_1_prod2
tff(fact_7236_less__1__prod2,axiom,
    ! [I5: set_Code_integer,I2: code_integer,F: fun_Code_integer_rat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,I5)
     => ( aa_set5643387332196575111eger_o(member_Code_integer(I2),I5)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_Code_integer_rat(F,I2))
         => ( ! [I3: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(I3),I5)
               => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_Code_integer_rat(F,I3)) )
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),groups2555765274223993564er_rat(F,I5)) ) ) ) ) ).

% less_1_prod2
tff(fact_7237_less__1__prod2,axiom,
    ! [I5: set_o,I2: $o,F: fun_o_int] :
      ( finite_finite_o(I5)
     => ( member_o((I2),I5)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_o_int(F,(I2)))
         => ( ! [I3: $o] :
                ( member_o((I3),I5)
               => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_o_int(F,(I3))) )
           => aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),groups3502327434004483295_o_int(F,I5)) ) ) ) ) ).

% less_1_prod2
tff(fact_7238_less__1__prod2,axiom,
    ! [I5: set_Code_integer,I2: code_integer,F: fun_Code_integer_int] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,I5)
     => ( aa_set5643387332196575111eger_o(member_Code_integer(I2),I5)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_Code_integer_int(F,I2))
         => ( ! [I3: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(I3),I5)
               => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_Code_integer_int(F,I3)) )
           => aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),groups3188404863801439024er_int(F,I5)) ) ) ) ) ).

% less_1_prod2
tff(fact_7239_less__1__prod,axiom,
    ! [I5: set_Code_integer,F: fun_Co8499238673658658103nteger] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,I5)
     => ( ( I5 != bot_bo3990330152332043303nteger )
       => ( ! [I3: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(I3),I5)
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(F,I3)) )
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),groups3674199335183972705nteger(F,I5)) ) ) ) ).

% less_1_prod
tff(fact_7240_less__1__prod,axiom,
    ! [I5: set_o,F: fun_o_Code_integer] :
      ( finite_finite_o(I5)
     => ( ( I5 != bot_bot_set_o )
       => ( ! [I3: $o] :
              ( member_o((I3),I5)
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),aa_o_Code_integer(F,(I3))) )
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),groups7694694392188491536nteger(F,I5)) ) ) ) ).

% less_1_prod
tff(fact_7241_less__1__prod,axiom,
    ! [I5: set_nat,F: fun_nat_Code_integer] :
      ( aa_set_nat_o(finite_finite_nat,I5)
     => ( ( I5 != bot_bot_set_nat )
       => ( ! [I3: nat] :
              ( aa_set_nat_o(member_nat(I3),I5)
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),aa_nat_Code_integer(F,I3)) )
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),aa_set871585252387937313nteger(groups3455450783089532116nteger(F),I5)) ) ) ) ).

% less_1_prod
tff(fact_7242_less__1__prod,axiom,
    ! [I5: set_int,F: fun_int_Code_integer] :
      ( aa_set_int_o(finite_finite_int,I5)
     => ( ( I5 != bot_bot_set_int )
       => ( ! [I3: int] :
              ( aa_set_int_o(member_int(I3),I5)
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),aa_int_Code_integer(F,I3)) )
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),groups3827104343326376752nteger(F,I5)) ) ) ) ).

% less_1_prod
tff(fact_7243_less__1__prod,axiom,
    ! [I5: set_Code_integer,F: fun_Code_integer_rat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,I5)
     => ( ( I5 != bot_bo3990330152332043303nteger )
       => ( ! [I3: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(I3),I5)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_Code_integer_rat(F,I3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),groups2555765274223993564er_rat(F,I5)) ) ) ) ).

% less_1_prod
tff(fact_7244_less__1__prod,axiom,
    ! [I5: set_o,F: fun_o_rat] :
      ( finite_finite_o(I5)
     => ( ( I5 != bot_bot_set_o )
       => ( ! [I3: $o] :
              ( member_o((I3),I5)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_o_rat(F,(I3))) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),groups2869687844427037835_o_rat(F,I5)) ) ) ) ).

% less_1_prod
tff(fact_7245_less__1__prod,axiom,
    ! [I5: set_nat,F: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,I5)
     => ( ( I5 != bot_bot_set_nat )
       => ( ! [I3: nat] :
              ( aa_set_nat_o(member_nat(I3),I5)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_nat_rat(F,I3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_set_nat_rat(groups73079841787564623at_rat(F),I5)) ) ) ) ).

% less_1_prod
tff(fact_7246_less__1__prod,axiom,
    ! [I5: set_int,F: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,I5)
     => ( ( I5 != bot_bot_set_int )
       => ( ! [I3: int] :
              ( aa_set_int_o(member_int(I3),I5)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_int_rat(F,I3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),groups1072433553688619179nt_rat(F,I5)) ) ) ) ).

% less_1_prod
tff(fact_7247_less__1__prod,axiom,
    ! [I5: set_Code_integer,F: fun_Code_integer_int] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,I5)
     => ( ( I5 != bot_bo3990330152332043303nteger )
       => ( ! [I3: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(I3),I5)
             => aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_Code_integer_int(F,I3)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),groups3188404863801439024er_int(F,I5)) ) ) ) ).

% less_1_prod
tff(fact_7248_less__1__prod,axiom,
    ! [I5: set_o,F: fun_o_int] :
      ( finite_finite_o(I5)
     => ( ( I5 != bot_bot_set_o )
       => ( ! [I3: $o] :
              ( member_o((I3),I5)
             => aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_o_int(F,(I3))) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),groups3502327434004483295_o_int(F,I5)) ) ) ) ).

% less_1_prod
tff(fact_7249_binomial__fact__lemma,axiom,
    ! [K: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),N2)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1408675320244567234ct_nat,K)),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),K)))),aa_nat_nat(aa_nat_fun_nat_nat(binomial,N2),K)) = aa_nat_nat(semiri1408675320244567234ct_nat,N2) ) ) ).

% binomial_fact_lemma
tff(fact_7250_sum_Osame__carrier,axiom,
    ! [C2: set_o,A3: set_o,B3: set_o,G: fun_o_Code_integer,Ha: fun_o_Code_integer] :
      ( finite_finite_o(C2)
     => ( ord_less_eq_set_o(A3,C2)
       => ( ord_less_eq_set_o(B3,C2)
         => ( ! [A4: $o] :
                ( member_o((A4),minus_minus_set_o(C2,A3))
               => ( aa_o_Code_integer(G,(A4)) = zero_z3403309356797280102nteger ) )
           => ( ! [B4: $o] :
                  ( member_o((B4),minus_minus_set_o(C2,B3))
                 => ( aa_o_Code_integer(Ha,(B4)) = zero_z3403309356797280102nteger ) )
             => ( ( groups4406642042086082107nteger(G,A3) = groups4406642042086082107nteger(Ha,B3) )
              <=> ( groups4406642042086082107nteger(G,C2) = groups4406642042086082107nteger(Ha,C2) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_7251_sum_Osame__carrier,axiom,
    ! [C2: set_Code_integer,A3: set_Code_integer,B3: set_Code_integer,G: fun_Co8499238673658658103nteger,Ha: fun_Co8499238673658658103nteger] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,C2)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(A3),C2)
       => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(B3),C2)
         => ( ! [A4: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(A4),minus_2355218937544613996nteger(C2,A3))
               => ( aa_Cod1783974652533396728nteger(G,A4) = zero_z3403309356797280102nteger ) )
           => ( ! [B4: code_integer] :
                  ( aa_set5643387332196575111eger_o(member_Code_integer(B4),minus_2355218937544613996nteger(C2,B3))
                 => ( aa_Cod1783974652533396728nteger(Ha,B4) = zero_z3403309356797280102nteger ) )
             => ( ( groups879477027807139574nteger(G,A3) = groups879477027807139574nteger(Ha,B3) )
              <=> ( groups879477027807139574nteger(G,C2) = groups879477027807139574nteger(Ha,C2) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_7252_sum_Osame__carrier,axiom,
    ! [C2: set_o,A3: set_o,B3: set_o,G: fun_o_rat,Ha: fun_o_rat] :
      ( finite_finite_o(C2)
     => ( ord_less_eq_set_o(A3,C2)
       => ( ord_less_eq_set_o(B3,C2)
         => ( ! [A4: $o] :
                ( member_o((A4),minus_minus_set_o(C2,A3))
               => ( aa_o_rat(G,(A4)) = zero_zero_rat ) )
           => ( ! [B4: $o] :
                  ( member_o((B4),minus_minus_set_o(C2,B3))
                 => ( aa_o_rat(Ha,(B4)) = zero_zero_rat ) )
             => ( ( groups7872700643590313910_o_rat(G,A3) = groups7872700643590313910_o_rat(Ha,B3) )
              <=> ( groups7872700643590313910_o_rat(G,C2) = groups7872700643590313910_o_rat(Ha,C2) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_7253_sum_Osame__carrier,axiom,
    ! [C2: set_Code_integer,A3: set_Code_integer,B3: set_Code_integer,G: fun_Code_integer_rat,Ha: fun_Code_integer_rat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,C2)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(A3),C2)
       => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(B3),C2)
         => ( ! [A4: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(A4),minus_2355218937544613996nteger(C2,A3))
               => ( aa_Code_integer_rat(G,A4) = zero_zero_rat ) )
           => ( ! [B4: code_integer] :
                  ( aa_set5643387332196575111eger_o(member_Code_integer(B4),minus_2355218937544613996nteger(C2,B3))
                 => ( aa_Code_integer_rat(Ha,B4) = zero_zero_rat ) )
             => ( ( groups6602215022474089585er_rat(G,A3) = groups6602215022474089585er_rat(Ha,B3) )
              <=> ( groups6602215022474089585er_rat(G,C2) = groups6602215022474089585er_rat(Ha,C2) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_7254_sum_Osame__carrier,axiom,
    ! [C2: set_o,A3: set_o,B3: set_o,G: fun_o_nat,Ha: fun_o_nat] :
      ( finite_finite_o(C2)
     => ( ord_less_eq_set_o(A3,C2)
       => ( ord_less_eq_set_o(B3,C2)
         => ( ! [A4: $o] :
                ( member_o((A4),minus_minus_set_o(C2,A3))
               => ( aa_o_nat(G,(A4)) = zero_zero_nat ) )
           => ( ! [B4: $o] :
                  ( member_o((B4),minus_minus_set_o(C2,B3))
                 => ( aa_o_nat(Ha,(B4)) = zero_zero_nat ) )
             => ( ( groups8507830703676809646_o_nat(G,A3) = groups8507830703676809646_o_nat(Ha,B3) )
              <=> ( groups8507830703676809646_o_nat(G,C2) = groups8507830703676809646_o_nat(Ha,C2) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_7255_sum_Osame__carrier,axiom,
    ! [C2: set_Code_integer,A3: set_Code_integer,B3: set_Code_integer,G: fun_Code_integer_nat,Ha: fun_Code_integer_nat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,C2)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(A3),C2)
       => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(B3),C2)
         => ( ! [A4: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(A4),minus_2355218937544613996nteger(C2,A3))
               => ( aa_Code_integer_nat(G,A4) = zero_zero_nat ) )
           => ( ! [B4: code_integer] :
                  ( aa_set5643387332196575111eger_o(member_Code_integer(B4),minus_2355218937544613996nteger(C2,B3))
                 => ( aa_Code_integer_nat(Ha,B4) = zero_zero_nat ) )
             => ( ( groups7237345082560585321er_nat(G,A3) = groups7237345082560585321er_nat(Ha,B3) )
              <=> ( groups7237345082560585321er_nat(G,C2) = groups7237345082560585321er_nat(Ha,C2) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_7256_sum_Osame__carrier,axiom,
    ! [C2: set_o,A3: set_o,B3: set_o,G: fun_o_int,Ha: fun_o_int] :
      ( finite_finite_o(C2)
     => ( ord_less_eq_set_o(A3,C2)
       => ( ord_less_eq_set_o(B3,C2)
         => ( ! [A4: $o] :
                ( member_o((A4),minus_minus_set_o(C2,A3))
               => ( aa_o_int(G,(A4)) = zero_zero_int ) )
           => ( ! [B4: $o] :
                  ( member_o((B4),minus_minus_set_o(C2,B3))
                 => ( aa_o_int(Ha,(B4)) = zero_zero_int ) )
             => ( ( groups8505340233167759370_o_int(G,A3) = groups8505340233167759370_o_int(Ha,B3) )
              <=> ( groups8505340233167759370_o_int(G,C2) = groups8505340233167759370_o_int(Ha,C2) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_7257_sum_Osame__carrier,axiom,
    ! [C2: set_Code_integer,A3: set_Code_integer,B3: set_Code_integer,G: fun_Code_integer_int,Ha: fun_Code_integer_int] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,C2)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(A3),C2)
       => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(B3),C2)
         => ( ! [A4: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(A4),minus_2355218937544613996nteger(C2,A3))
               => ( aa_Code_integer_int(G,A4) = zero_zero_int ) )
           => ( ! [B4: code_integer] :
                  ( aa_set5643387332196575111eger_o(member_Code_integer(B4),minus_2355218937544613996nteger(C2,B3))
                 => ( aa_Code_integer_int(Ha,B4) = zero_zero_int ) )
             => ( ( groups7234854612051535045er_int(G,A3) = groups7234854612051535045er_int(Ha,B3) )
              <=> ( groups7234854612051535045er_int(G,C2) = groups7234854612051535045er_int(Ha,C2) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_7258_sum_Osame__carrier,axiom,
    ! [C2: set_nat,A3: set_nat,B3: set_nat,G: fun_nat_Code_integer,Ha: 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,A3),C2)
       => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B3),C2)
         => ( ! [A4: nat] :
                ( aa_set_nat_o(member_nat(A4),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,C2),A3))
               => ( aa_nat_Code_integer(G,A4) = zero_z3403309356797280102nteger ) )
           => ( ! [B4: nat] :
                  ( aa_set_nat_o(member_nat(B4),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,C2),B3))
                 => ( aa_nat_Code_integer(Ha,B4) = zero_z3403309356797280102nteger ) )
             => ( ( aa_set871585252387937313nteger(groups7501900531339628137nteger(G),A3) = aa_set871585252387937313nteger(groups7501900531339628137nteger(Ha),B3) )
              <=> ( aa_set871585252387937313nteger(groups7501900531339628137nteger(G),C2) = aa_set871585252387937313nteger(groups7501900531339628137nteger(Ha),C2) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_7259_sum_Osame__carrier,axiom,
    ! [C2: set_nat,A3: set_nat,B3: set_nat,G: fun_nat_rat,Ha: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,C2)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A3),C2)
       => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B3),C2)
         => ( ! [A4: nat] :
                ( aa_set_nat_o(member_nat(A4),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,C2),A3))
               => ( aa_nat_rat(G,A4) = zero_zero_rat ) )
           => ( ! [B4: nat] :
                  ( aa_set_nat_o(member_nat(B4),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,C2),B3))
                 => ( aa_nat_rat(Ha,B4) = zero_zero_rat ) )
             => ( ( aa_set_nat_rat(groups2906978787729119204at_rat(G),A3) = aa_set_nat_rat(groups2906978787729119204at_rat(Ha),B3) )
              <=> ( aa_set_nat_rat(groups2906978787729119204at_rat(G),C2) = aa_set_nat_rat(groups2906978787729119204at_rat(Ha),C2) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_7260_sum_Osame__carrierI,axiom,
    ! [C2: set_o,A3: set_o,B3: set_o,G: fun_o_Code_integer,Ha: fun_o_Code_integer] :
      ( finite_finite_o(C2)
     => ( ord_less_eq_set_o(A3,C2)
       => ( ord_less_eq_set_o(B3,C2)
         => ( ! [A4: $o] :
                ( member_o((A4),minus_minus_set_o(C2,A3))
               => ( aa_o_Code_integer(G,(A4)) = zero_z3403309356797280102nteger ) )
           => ( ! [B4: $o] :
                  ( member_o((B4),minus_minus_set_o(C2,B3))
                 => ( aa_o_Code_integer(Ha,(B4)) = zero_z3403309356797280102nteger ) )
             => ( ( groups4406642042086082107nteger(G,C2) = groups4406642042086082107nteger(Ha,C2) )
               => ( groups4406642042086082107nteger(G,A3) = groups4406642042086082107nteger(Ha,B3) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_7261_sum_Osame__carrierI,axiom,
    ! [C2: set_Code_integer,A3: set_Code_integer,B3: set_Code_integer,G: fun_Co8499238673658658103nteger,Ha: fun_Co8499238673658658103nteger] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,C2)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(A3),C2)
       => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(B3),C2)
         => ( ! [A4: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(A4),minus_2355218937544613996nteger(C2,A3))
               => ( aa_Cod1783974652533396728nteger(G,A4) = zero_z3403309356797280102nteger ) )
           => ( ! [B4: code_integer] :
                  ( aa_set5643387332196575111eger_o(member_Code_integer(B4),minus_2355218937544613996nteger(C2,B3))
                 => ( aa_Cod1783974652533396728nteger(Ha,B4) = zero_z3403309356797280102nteger ) )
             => ( ( groups879477027807139574nteger(G,C2) = groups879477027807139574nteger(Ha,C2) )
               => ( groups879477027807139574nteger(G,A3) = groups879477027807139574nteger(Ha,B3) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_7262_sum_Osame__carrierI,axiom,
    ! [C2: set_o,A3: set_o,B3: set_o,G: fun_o_rat,Ha: fun_o_rat] :
      ( finite_finite_o(C2)
     => ( ord_less_eq_set_o(A3,C2)
       => ( ord_less_eq_set_o(B3,C2)
         => ( ! [A4: $o] :
                ( member_o((A4),minus_minus_set_o(C2,A3))
               => ( aa_o_rat(G,(A4)) = zero_zero_rat ) )
           => ( ! [B4: $o] :
                  ( member_o((B4),minus_minus_set_o(C2,B3))
                 => ( aa_o_rat(Ha,(B4)) = zero_zero_rat ) )
             => ( ( groups7872700643590313910_o_rat(G,C2) = groups7872700643590313910_o_rat(Ha,C2) )
               => ( groups7872700643590313910_o_rat(G,A3) = groups7872700643590313910_o_rat(Ha,B3) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_7263_sum_Osame__carrierI,axiom,
    ! [C2: set_Code_integer,A3: set_Code_integer,B3: set_Code_integer,G: fun_Code_integer_rat,Ha: fun_Code_integer_rat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,C2)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(A3),C2)
       => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(B3),C2)
         => ( ! [A4: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(A4),minus_2355218937544613996nteger(C2,A3))
               => ( aa_Code_integer_rat(G,A4) = zero_zero_rat ) )
           => ( ! [B4: code_integer] :
                  ( aa_set5643387332196575111eger_o(member_Code_integer(B4),minus_2355218937544613996nteger(C2,B3))
                 => ( aa_Code_integer_rat(Ha,B4) = zero_zero_rat ) )
             => ( ( groups6602215022474089585er_rat(G,C2) = groups6602215022474089585er_rat(Ha,C2) )
               => ( groups6602215022474089585er_rat(G,A3) = groups6602215022474089585er_rat(Ha,B3) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_7264_sum_Osame__carrierI,axiom,
    ! [C2: set_o,A3: set_o,B3: set_o,G: fun_o_nat,Ha: fun_o_nat] :
      ( finite_finite_o(C2)
     => ( ord_less_eq_set_o(A3,C2)
       => ( ord_less_eq_set_o(B3,C2)
         => ( ! [A4: $o] :
                ( member_o((A4),minus_minus_set_o(C2,A3))
               => ( aa_o_nat(G,(A4)) = zero_zero_nat ) )
           => ( ! [B4: $o] :
                  ( member_o((B4),minus_minus_set_o(C2,B3))
                 => ( aa_o_nat(Ha,(B4)) = zero_zero_nat ) )
             => ( ( groups8507830703676809646_o_nat(G,C2) = groups8507830703676809646_o_nat(Ha,C2) )
               => ( groups8507830703676809646_o_nat(G,A3) = groups8507830703676809646_o_nat(Ha,B3) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_7265_sum_Osame__carrierI,axiom,
    ! [C2: set_Code_integer,A3: set_Code_integer,B3: set_Code_integer,G: fun_Code_integer_nat,Ha: fun_Code_integer_nat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,C2)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(A3),C2)
       => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(B3),C2)
         => ( ! [A4: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(A4),minus_2355218937544613996nteger(C2,A3))
               => ( aa_Code_integer_nat(G,A4) = zero_zero_nat ) )
           => ( ! [B4: code_integer] :
                  ( aa_set5643387332196575111eger_o(member_Code_integer(B4),minus_2355218937544613996nteger(C2,B3))
                 => ( aa_Code_integer_nat(Ha,B4) = zero_zero_nat ) )
             => ( ( groups7237345082560585321er_nat(G,C2) = groups7237345082560585321er_nat(Ha,C2) )
               => ( groups7237345082560585321er_nat(G,A3) = groups7237345082560585321er_nat(Ha,B3) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_7266_sum_Osame__carrierI,axiom,
    ! [C2: set_o,A3: set_o,B3: set_o,G: fun_o_int,Ha: fun_o_int] :
      ( finite_finite_o(C2)
     => ( ord_less_eq_set_o(A3,C2)
       => ( ord_less_eq_set_o(B3,C2)
         => ( ! [A4: $o] :
                ( member_o((A4),minus_minus_set_o(C2,A3))
               => ( aa_o_int(G,(A4)) = zero_zero_int ) )
           => ( ! [B4: $o] :
                  ( member_o((B4),minus_minus_set_o(C2,B3))
                 => ( aa_o_int(Ha,(B4)) = zero_zero_int ) )
             => ( ( groups8505340233167759370_o_int(G,C2) = groups8505340233167759370_o_int(Ha,C2) )
               => ( groups8505340233167759370_o_int(G,A3) = groups8505340233167759370_o_int(Ha,B3) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_7267_sum_Osame__carrierI,axiom,
    ! [C2: set_Code_integer,A3: set_Code_integer,B3: set_Code_integer,G: fun_Code_integer_int,Ha: fun_Code_integer_int] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,C2)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(A3),C2)
       => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(B3),C2)
         => ( ! [A4: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(A4),minus_2355218937544613996nteger(C2,A3))
               => ( aa_Code_integer_int(G,A4) = zero_zero_int ) )
           => ( ! [B4: code_integer] :
                  ( aa_set5643387332196575111eger_o(member_Code_integer(B4),minus_2355218937544613996nteger(C2,B3))
                 => ( aa_Code_integer_int(Ha,B4) = zero_zero_int ) )
             => ( ( groups7234854612051535045er_int(G,C2) = groups7234854612051535045er_int(Ha,C2) )
               => ( groups7234854612051535045er_int(G,A3) = groups7234854612051535045er_int(Ha,B3) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_7268_sum_Osame__carrierI,axiom,
    ! [C2: set_nat,A3: set_nat,B3: set_nat,G: fun_nat_Code_integer,Ha: 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,A3),C2)
       => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B3),C2)
         => ( ! [A4: nat] :
                ( aa_set_nat_o(member_nat(A4),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,C2),A3))
               => ( aa_nat_Code_integer(G,A4) = zero_z3403309356797280102nteger ) )
           => ( ! [B4: nat] :
                  ( aa_set_nat_o(member_nat(B4),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,C2),B3))
                 => ( aa_nat_Code_integer(Ha,B4) = zero_z3403309356797280102nteger ) )
             => ( ( aa_set871585252387937313nteger(groups7501900531339628137nteger(G),C2) = aa_set871585252387937313nteger(groups7501900531339628137nteger(Ha),C2) )
               => ( aa_set871585252387937313nteger(groups7501900531339628137nteger(G),A3) = aa_set871585252387937313nteger(groups7501900531339628137nteger(Ha),B3) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_7269_sum_Osame__carrierI,axiom,
    ! [C2: set_nat,A3: set_nat,B3: set_nat,G: fun_nat_rat,Ha: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,C2)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A3),C2)
       => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B3),C2)
         => ( ! [A4: nat] :
                ( aa_set_nat_o(member_nat(A4),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,C2),A3))
               => ( aa_nat_rat(G,A4) = zero_zero_rat ) )
           => ( ! [B4: nat] :
                  ( aa_set_nat_o(member_nat(B4),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,C2),B3))
                 => ( aa_nat_rat(Ha,B4) = zero_zero_rat ) )
             => ( ( aa_set_nat_rat(groups2906978787729119204at_rat(G),C2) = aa_set_nat_rat(groups2906978787729119204at_rat(Ha),C2) )
               => ( aa_set_nat_rat(groups2906978787729119204at_rat(G),A3) = aa_set_nat_rat(groups2906978787729119204at_rat(Ha),B3) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_7270_sum_Omono__neutral__left,axiom,
    ! [T: set_Code_integer,S: set_Code_integer,G: fun_Co8499238673658658103nteger] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,T)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(S),T)
       => ( ! [X3: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X3),minus_2355218937544613996nteger(T,S))
             => ( aa_Cod1783974652533396728nteger(G,X3) = zero_z3403309356797280102nteger ) )
         => ( groups879477027807139574nteger(G,S) = groups879477027807139574nteger(G,T) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_7271_sum_Omono__neutral__left,axiom,
    ! [T: set_Code_integer,S: set_Code_integer,G: fun_Code_integer_rat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,T)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(S),T)
       => ( ! [X3: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X3),minus_2355218937544613996nteger(T,S))
             => ( aa_Code_integer_rat(G,X3) = zero_zero_rat ) )
         => ( groups6602215022474089585er_rat(G,S) = groups6602215022474089585er_rat(G,T) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_7272_sum_Omono__neutral__left,axiom,
    ! [T: set_Code_integer,S: set_Code_integer,G: fun_Code_integer_nat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,T)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(S),T)
       => ( ! [X3: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X3),minus_2355218937544613996nteger(T,S))
             => ( aa_Code_integer_nat(G,X3) = zero_zero_nat ) )
         => ( groups7237345082560585321er_nat(G,S) = groups7237345082560585321er_nat(G,T) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_7273_sum_Omono__neutral__left,axiom,
    ! [T: set_Code_integer,S: set_Code_integer,G: fun_Code_integer_int] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,T)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(S),T)
       => ( ! [X3: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X3),minus_2355218937544613996nteger(T,S))
             => ( aa_Code_integer_int(G,X3) = zero_zero_int ) )
         => ( groups7234854612051535045er_int(G,S) = groups7234854612051535045er_int(G,T) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_7274_sum_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)
       => ( ! [X3: nat] :
              ( aa_set_nat_o(member_nat(X3),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,T),S))
             => ( aa_nat_Code_integer(G,X3) = zero_z3403309356797280102nteger ) )
         => ( aa_set871585252387937313nteger(groups7501900531339628137nteger(G),S) = aa_set871585252387937313nteger(groups7501900531339628137nteger(G),T) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_7275_sum_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)
       => ( ! [X3: nat] :
              ( aa_set_nat_o(member_nat(X3),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,T),S))
             => ( aa_nat_rat(G,X3) = zero_zero_rat ) )
         => ( aa_set_nat_rat(groups2906978787729119204at_rat(G),S) = aa_set_nat_rat(groups2906978787729119204at_rat(G),T) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_7276_sum_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)
       => ( ! [X3: nat] :
              ( aa_set_nat_o(member_nat(X3),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,T),S))
             => ( aa_nat_int(G,X3) = zero_zero_int ) )
         => ( aa_set_nat_int(groups3539618377306564664at_int(G),S) = aa_set_nat_int(groups3539618377306564664at_int(G),T) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_7277_sum_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)
       => ( ! [X3: int] :
              ( aa_set_int_o(member_int(X3),aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,T),S))
             => ( aa_int_Code_integer(G,X3) = zero_z3403309356797280102nteger ) )
         => ( groups7873554091576472773nteger(G,S) = groups7873554091576472773nteger(G,T) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_7278_sum_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)
       => ( ! [X3: int] :
              ( aa_set_int_o(member_int(X3),aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,T),S))
             => ( aa_int_rat(G,X3) = zero_zero_rat ) )
         => ( groups3906332499630173760nt_rat(G,S) = groups3906332499630173760nt_rat(G,T) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_7279_sum_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)
       => ( ! [X3: int] :
              ( aa_set_int_o(member_int(X3),aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,T),S))
             => ( aa_int_nat(G,X3) = zero_zero_nat ) )
         => ( groups4541462559716669496nt_nat(G,S) = groups4541462559716669496nt_nat(G,T) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_7280_sum_Omono__neutral__right,axiom,
    ! [T: set_Code_integer,S: set_Code_integer,G: fun_Co8499238673658658103nteger] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,T)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(S),T)
       => ( ! [X3: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X3),minus_2355218937544613996nteger(T,S))
             => ( aa_Cod1783974652533396728nteger(G,X3) = zero_z3403309356797280102nteger ) )
         => ( groups879477027807139574nteger(G,T) = groups879477027807139574nteger(G,S) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_7281_sum_Omono__neutral__right,axiom,
    ! [T: set_Code_integer,S: set_Code_integer,G: fun_Code_integer_rat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,T)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(S),T)
       => ( ! [X3: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X3),minus_2355218937544613996nteger(T,S))
             => ( aa_Code_integer_rat(G,X3) = zero_zero_rat ) )
         => ( groups6602215022474089585er_rat(G,T) = groups6602215022474089585er_rat(G,S) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_7282_sum_Omono__neutral__right,axiom,
    ! [T: set_Code_integer,S: set_Code_integer,G: fun_Code_integer_nat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,T)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(S),T)
       => ( ! [X3: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X3),minus_2355218937544613996nteger(T,S))
             => ( aa_Code_integer_nat(G,X3) = zero_zero_nat ) )
         => ( groups7237345082560585321er_nat(G,T) = groups7237345082560585321er_nat(G,S) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_7283_sum_Omono__neutral__right,axiom,
    ! [T: set_Code_integer,S: set_Code_integer,G: fun_Code_integer_int] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,T)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(S),T)
       => ( ! [X3: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X3),minus_2355218937544613996nteger(T,S))
             => ( aa_Code_integer_int(G,X3) = zero_zero_int ) )
         => ( groups7234854612051535045er_int(G,T) = groups7234854612051535045er_int(G,S) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_7284_sum_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)
       => ( ! [X3: nat] :
              ( aa_set_nat_o(member_nat(X3),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,T),S))
             => ( aa_nat_Code_integer(G,X3) = zero_z3403309356797280102nteger ) )
         => ( aa_set871585252387937313nteger(groups7501900531339628137nteger(G),T) = aa_set871585252387937313nteger(groups7501900531339628137nteger(G),S) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_7285_sum_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)
       => ( ! [X3: nat] :
              ( aa_set_nat_o(member_nat(X3),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,T),S))
             => ( aa_nat_rat(G,X3) = zero_zero_rat ) )
         => ( aa_set_nat_rat(groups2906978787729119204at_rat(G),T) = aa_set_nat_rat(groups2906978787729119204at_rat(G),S) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_7286_sum_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)
       => ( ! [X3: nat] :
              ( aa_set_nat_o(member_nat(X3),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,T),S))
             => ( aa_nat_int(G,X3) = zero_zero_int ) )
         => ( aa_set_nat_int(groups3539618377306564664at_int(G),T) = aa_set_nat_int(groups3539618377306564664at_int(G),S) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_7287_sum_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)
       => ( ! [X3: int] :
              ( aa_set_int_o(member_int(X3),aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,T),S))
             => ( aa_int_Code_integer(G,X3) = zero_z3403309356797280102nteger ) )
         => ( groups7873554091576472773nteger(G,T) = groups7873554091576472773nteger(G,S) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_7288_sum_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)
       => ( ! [X3: int] :
              ( aa_set_int_o(member_int(X3),aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,T),S))
             => ( aa_int_rat(G,X3) = zero_zero_rat ) )
         => ( groups3906332499630173760nt_rat(G,T) = groups3906332499630173760nt_rat(G,S) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_7289_sum_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)
       => ( ! [X3: int] :
              ( aa_set_int_o(member_int(X3),aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,T),S))
             => ( aa_int_nat(G,X3) = zero_zero_nat ) )
         => ( groups4541462559716669496nt_nat(G,T) = groups4541462559716669496nt_nat(G,S) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_7290_sum_Omono__neutral__cong__left,axiom,
    ! [T: set_o,S: set_o,Ha: fun_o_Code_integer,G: fun_o_Code_integer] :
      ( finite_finite_o(T)
     => ( ord_less_eq_set_o(S,T)
       => ( ! [X3: $o] :
              ( member_o((X3),minus_minus_set_o(T,S))
             => ( aa_o_Code_integer(Ha,(X3)) = zero_z3403309356797280102nteger ) )
         => ( ! [X3: $o] :
                ( member_o((X3),S)
               => ( aa_o_Code_integer(G,(X3)) = aa_o_Code_integer(Ha,(X3)) ) )
           => ( groups4406642042086082107nteger(G,S) = groups4406642042086082107nteger(Ha,T) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_7291_sum_Omono__neutral__cong__left,axiom,
    ! [T: set_Code_integer,S: set_Code_integer,Ha: fun_Co8499238673658658103nteger,G: fun_Co8499238673658658103nteger] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,T)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(S),T)
       => ( ! [X3: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X3),minus_2355218937544613996nteger(T,S))
             => ( aa_Cod1783974652533396728nteger(Ha,X3) = zero_z3403309356797280102nteger ) )
         => ( ! [X3: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(X3),S)
               => ( aa_Cod1783974652533396728nteger(G,X3) = aa_Cod1783974652533396728nteger(Ha,X3) ) )
           => ( groups879477027807139574nteger(G,S) = groups879477027807139574nteger(Ha,T) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_7292_sum_Omono__neutral__cong__left,axiom,
    ! [T: set_o,S: set_o,Ha: fun_o_rat,G: fun_o_rat] :
      ( finite_finite_o(T)
     => ( ord_less_eq_set_o(S,T)
       => ( ! [X3: $o] :
              ( member_o((X3),minus_minus_set_o(T,S))
             => ( aa_o_rat(Ha,(X3)) = zero_zero_rat ) )
         => ( ! [X3: $o] :
                ( member_o((X3),S)
               => ( aa_o_rat(G,(X3)) = aa_o_rat(Ha,(X3)) ) )
           => ( groups7872700643590313910_o_rat(G,S) = groups7872700643590313910_o_rat(Ha,T) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_7293_sum_Omono__neutral__cong__left,axiom,
    ! [T: set_Code_integer,S: set_Code_integer,Ha: fun_Code_integer_rat,G: fun_Code_integer_rat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,T)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(S),T)
       => ( ! [X3: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X3),minus_2355218937544613996nteger(T,S))
             => ( aa_Code_integer_rat(Ha,X3) = zero_zero_rat ) )
         => ( ! [X3: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(X3),S)
               => ( aa_Code_integer_rat(G,X3) = aa_Code_integer_rat(Ha,X3) ) )
           => ( groups6602215022474089585er_rat(G,S) = groups6602215022474089585er_rat(Ha,T) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_7294_sum_Omono__neutral__cong__left,axiom,
    ! [T: set_o,S: set_o,Ha: fun_o_nat,G: fun_o_nat] :
      ( finite_finite_o(T)
     => ( ord_less_eq_set_o(S,T)
       => ( ! [X3: $o] :
              ( member_o((X3),minus_minus_set_o(T,S))
             => ( aa_o_nat(Ha,(X3)) = zero_zero_nat ) )
         => ( ! [X3: $o] :
                ( member_o((X3),S)
               => ( aa_o_nat(G,(X3)) = aa_o_nat(Ha,(X3)) ) )
           => ( groups8507830703676809646_o_nat(G,S) = groups8507830703676809646_o_nat(Ha,T) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_7295_sum_Omono__neutral__cong__left,axiom,
    ! [T: set_Code_integer,S: set_Code_integer,Ha: fun_Code_integer_nat,G: fun_Code_integer_nat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,T)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(S),T)
       => ( ! [X3: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X3),minus_2355218937544613996nteger(T,S))
             => ( aa_Code_integer_nat(Ha,X3) = zero_zero_nat ) )
         => ( ! [X3: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(X3),S)
               => ( aa_Code_integer_nat(G,X3) = aa_Code_integer_nat(Ha,X3) ) )
           => ( groups7237345082560585321er_nat(G,S) = groups7237345082560585321er_nat(Ha,T) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_7296_sum_Omono__neutral__cong__left,axiom,
    ! [T: set_o,S: set_o,Ha: fun_o_int,G: fun_o_int] :
      ( finite_finite_o(T)
     => ( ord_less_eq_set_o(S,T)
       => ( ! [X3: $o] :
              ( member_o((X3),minus_minus_set_o(T,S))
             => ( aa_o_int(Ha,(X3)) = zero_zero_int ) )
         => ( ! [X3: $o] :
                ( member_o((X3),S)
               => ( aa_o_int(G,(X3)) = aa_o_int(Ha,(X3)) ) )
           => ( groups8505340233167759370_o_int(G,S) = groups8505340233167759370_o_int(Ha,T) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_7297_sum_Omono__neutral__cong__left,axiom,
    ! [T: set_Code_integer,S: set_Code_integer,Ha: fun_Code_integer_int,G: fun_Code_integer_int] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,T)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(S),T)
       => ( ! [X3: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X3),minus_2355218937544613996nteger(T,S))
             => ( aa_Code_integer_int(Ha,X3) = zero_zero_int ) )
         => ( ! [X3: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(X3),S)
               => ( aa_Code_integer_int(G,X3) = aa_Code_integer_int(Ha,X3) ) )
           => ( groups7234854612051535045er_int(G,S) = groups7234854612051535045er_int(Ha,T) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_7298_sum_Omono__neutral__cong__left,axiom,
    ! [T: set_nat,S: set_nat,Ha: 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)
       => ( ! [X3: nat] :
              ( aa_set_nat_o(member_nat(X3),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,T),S))
             => ( aa_nat_Code_integer(Ha,X3) = zero_z3403309356797280102nteger ) )
         => ( ! [X3: nat] :
                ( aa_set_nat_o(member_nat(X3),S)
               => ( aa_nat_Code_integer(G,X3) = aa_nat_Code_integer(Ha,X3) ) )
           => ( aa_set871585252387937313nteger(groups7501900531339628137nteger(G),S) = aa_set871585252387937313nteger(groups7501900531339628137nteger(Ha),T) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_7299_sum_Omono__neutral__cong__left,axiom,
    ! [T: set_nat,S: set_nat,Ha: 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)
       => ( ! [X3: nat] :
              ( aa_set_nat_o(member_nat(X3),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,T),S))
             => ( aa_nat_rat(Ha,X3) = zero_zero_rat ) )
         => ( ! [X3: nat] :
                ( aa_set_nat_o(member_nat(X3),S)
               => ( aa_nat_rat(G,X3) = aa_nat_rat(Ha,X3) ) )
           => ( aa_set_nat_rat(groups2906978787729119204at_rat(G),S) = aa_set_nat_rat(groups2906978787729119204at_rat(Ha),T) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_7300_sum_Omono__neutral__cong__right,axiom,
    ! [T: set_o,S: set_o,G: fun_o_Code_integer,Ha: fun_o_Code_integer] :
      ( finite_finite_o(T)
     => ( ord_less_eq_set_o(S,T)
       => ( ! [X3: $o] :
              ( member_o((X3),minus_minus_set_o(T,S))
             => ( aa_o_Code_integer(G,(X3)) = zero_z3403309356797280102nteger ) )
         => ( ! [X3: $o] :
                ( member_o((X3),S)
               => ( aa_o_Code_integer(G,(X3)) = aa_o_Code_integer(Ha,(X3)) ) )
           => ( groups4406642042086082107nteger(G,T) = groups4406642042086082107nteger(Ha,S) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_7301_sum_Omono__neutral__cong__right,axiom,
    ! [T: set_Code_integer,S: set_Code_integer,G: fun_Co8499238673658658103nteger,Ha: fun_Co8499238673658658103nteger] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,T)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(S),T)
       => ( ! [X3: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X3),minus_2355218937544613996nteger(T,S))
             => ( aa_Cod1783974652533396728nteger(G,X3) = zero_z3403309356797280102nteger ) )
         => ( ! [X3: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(X3),S)
               => ( aa_Cod1783974652533396728nteger(G,X3) = aa_Cod1783974652533396728nteger(Ha,X3) ) )
           => ( groups879477027807139574nteger(G,T) = groups879477027807139574nteger(Ha,S) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_7302_sum_Omono__neutral__cong__right,axiom,
    ! [T: set_o,S: set_o,G: fun_o_rat,Ha: fun_o_rat] :
      ( finite_finite_o(T)
     => ( ord_less_eq_set_o(S,T)
       => ( ! [X3: $o] :
              ( member_o((X3),minus_minus_set_o(T,S))
             => ( aa_o_rat(G,(X3)) = zero_zero_rat ) )
         => ( ! [X3: $o] :
                ( member_o((X3),S)
               => ( aa_o_rat(G,(X3)) = aa_o_rat(Ha,(X3)) ) )
           => ( groups7872700643590313910_o_rat(G,T) = groups7872700643590313910_o_rat(Ha,S) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_7303_sum_Omono__neutral__cong__right,axiom,
    ! [T: set_Code_integer,S: set_Code_integer,G: fun_Code_integer_rat,Ha: fun_Code_integer_rat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,T)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(S),T)
       => ( ! [X3: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X3),minus_2355218937544613996nteger(T,S))
             => ( aa_Code_integer_rat(G,X3) = zero_zero_rat ) )
         => ( ! [X3: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(X3),S)
               => ( aa_Code_integer_rat(G,X3) = aa_Code_integer_rat(Ha,X3) ) )
           => ( groups6602215022474089585er_rat(G,T) = groups6602215022474089585er_rat(Ha,S) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_7304_sum_Omono__neutral__cong__right,axiom,
    ! [T: set_o,S: set_o,G: fun_o_nat,Ha: fun_o_nat] :
      ( finite_finite_o(T)
     => ( ord_less_eq_set_o(S,T)
       => ( ! [X3: $o] :
              ( member_o((X3),minus_minus_set_o(T,S))
             => ( aa_o_nat(G,(X3)) = zero_zero_nat ) )
         => ( ! [X3: $o] :
                ( member_o((X3),S)
               => ( aa_o_nat(G,(X3)) = aa_o_nat(Ha,(X3)) ) )
           => ( groups8507830703676809646_o_nat(G,T) = groups8507830703676809646_o_nat(Ha,S) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_7305_sum_Omono__neutral__cong__right,axiom,
    ! [T: set_Code_integer,S: set_Code_integer,G: fun_Code_integer_nat,Ha: fun_Code_integer_nat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,T)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(S),T)
       => ( ! [X3: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X3),minus_2355218937544613996nteger(T,S))
             => ( aa_Code_integer_nat(G,X3) = zero_zero_nat ) )
         => ( ! [X3: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(X3),S)
               => ( aa_Code_integer_nat(G,X3) = aa_Code_integer_nat(Ha,X3) ) )
           => ( groups7237345082560585321er_nat(G,T) = groups7237345082560585321er_nat(Ha,S) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_7306_sum_Omono__neutral__cong__right,axiom,
    ! [T: set_o,S: set_o,G: fun_o_int,Ha: fun_o_int] :
      ( finite_finite_o(T)
     => ( ord_less_eq_set_o(S,T)
       => ( ! [X3: $o] :
              ( member_o((X3),minus_minus_set_o(T,S))
             => ( aa_o_int(G,(X3)) = zero_zero_int ) )
         => ( ! [X3: $o] :
                ( member_o((X3),S)
               => ( aa_o_int(G,(X3)) = aa_o_int(Ha,(X3)) ) )
           => ( groups8505340233167759370_o_int(G,T) = groups8505340233167759370_o_int(Ha,S) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_7307_sum_Omono__neutral__cong__right,axiom,
    ! [T: set_Code_integer,S: set_Code_integer,G: fun_Code_integer_int,Ha: fun_Code_integer_int] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,T)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(S),T)
       => ( ! [X3: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X3),minus_2355218937544613996nteger(T,S))
             => ( aa_Code_integer_int(G,X3) = zero_zero_int ) )
         => ( ! [X3: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(X3),S)
               => ( aa_Code_integer_int(G,X3) = aa_Code_integer_int(Ha,X3) ) )
           => ( groups7234854612051535045er_int(G,T) = groups7234854612051535045er_int(Ha,S) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_7308_sum_Omono__neutral__cong__right,axiom,
    ! [T: set_nat,S: set_nat,G: fun_nat_Code_integer,Ha: 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)
       => ( ! [X3: nat] :
              ( aa_set_nat_o(member_nat(X3),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,T),S))
             => ( aa_nat_Code_integer(G,X3) = zero_z3403309356797280102nteger ) )
         => ( ! [X3: nat] :
                ( aa_set_nat_o(member_nat(X3),S)
               => ( aa_nat_Code_integer(G,X3) = aa_nat_Code_integer(Ha,X3) ) )
           => ( aa_set871585252387937313nteger(groups7501900531339628137nteger(G),T) = aa_set871585252387937313nteger(groups7501900531339628137nteger(Ha),S) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_7309_sum_Omono__neutral__cong__right,axiom,
    ! [T: set_nat,S: set_nat,G: fun_nat_rat,Ha: 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)
       => ( ! [X3: nat] :
              ( aa_set_nat_o(member_nat(X3),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,T),S))
             => ( aa_nat_rat(G,X3) = zero_zero_rat ) )
         => ( ! [X3: nat] :
                ( aa_set_nat_o(member_nat(X3),S)
               => ( aa_nat_rat(G,X3) = aa_nat_rat(Ha,X3) ) )
           => ( aa_set_nat_rat(groups2906978787729119204at_rat(G),T) = aa_set_nat_rat(groups2906978787729119204at_rat(Ha),S) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_7310_sum_Osubset__diff,axiom,
    ! [B3: set_Code_integer,A3: set_Code_integer,G: fun_Code_integer_rat] :
      ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(B3),A3)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A3)
       => ( groups6602215022474089585er_rat(G,A3) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,groups6602215022474089585er_rat(G,minus_2355218937544613996nteger(A3,B3))),groups6602215022474089585er_rat(G,B3)) ) ) ) ).

% sum.subset_diff
tff(fact_7311_sum_Osubset__diff,axiom,
    ! [B3: set_Code_integer,A3: set_Code_integer,G: fun_Code_integer_nat] :
      ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(B3),A3)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A3)
       => ( groups7237345082560585321er_nat(G,A3) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups7237345082560585321er_nat(G,minus_2355218937544613996nteger(A3,B3))),groups7237345082560585321er_nat(G,B3)) ) ) ) ).

% sum.subset_diff
tff(fact_7312_sum_Osubset__diff,axiom,
    ! [B3: set_Code_integer,A3: set_Code_integer,G: fun_Code_integer_int] :
      ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(B3),A3)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A3)
       => ( groups7234854612051535045er_int(G,A3) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups7234854612051535045er_int(G,minus_2355218937544613996nteger(A3,B3))),groups7234854612051535045er_int(G,B3)) ) ) ) ).

% sum.subset_diff
tff(fact_7313_sum_Osubset__diff,axiom,
    ! [B3: set_nat,A3: set_nat,G: fun_nat_rat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B3),A3)
     => ( aa_set_nat_o(finite_finite_nat,A3)
       => ( aa_set_nat_rat(groups2906978787729119204at_rat(G),A3) = 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,A3),B3))),aa_set_nat_rat(groups2906978787729119204at_rat(G),B3)) ) ) ) ).

% sum.subset_diff
tff(fact_7314_sum_Osubset__diff,axiom,
    ! [B3: set_nat,A3: set_nat,G: fun_nat_int] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B3),A3)
     => ( aa_set_nat_o(finite_finite_nat,A3)
       => ( aa_set_nat_int(groups3539618377306564664at_int(G),A3) = 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,A3),B3))),aa_set_nat_int(groups3539618377306564664at_int(G),B3)) ) ) ) ).

% sum.subset_diff
tff(fact_7315_sum_Osubset__diff,axiom,
    ! [B3: set_int,A3: set_int,G: fun_int_rat] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,B3),A3)
     => ( aa_set_int_o(finite_finite_int,A3)
       => ( groups3906332499630173760nt_rat(G,A3) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,groups3906332499630173760nt_rat(G,aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A3),B3))),groups3906332499630173760nt_rat(G,B3)) ) ) ) ).

% sum.subset_diff
tff(fact_7316_sum_Osubset__diff,axiom,
    ! [B3: set_int,A3: set_int,G: fun_int_nat] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,B3),A3)
     => ( aa_set_int_o(finite_finite_int,A3)
       => ( groups4541462559716669496nt_nat(G,A3) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups4541462559716669496nt_nat(G,aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A3),B3))),groups4541462559716669496nt_nat(G,B3)) ) ) ) ).

% sum.subset_diff
tff(fact_7317_sum_Osubset__diff,axiom,
    ! [B3: set_nat,A3: set_nat,G: fun_nat_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B3),A3)
     => ( aa_set_nat_o(finite_finite_nat,A3)
       => ( aa_set_nat_nat(groups3542108847815614940at_nat(G),A3) = 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,A3),B3))),aa_set_nat_nat(groups3542108847815614940at_nat(G),B3)) ) ) ) ).

% sum.subset_diff
tff(fact_7318_sum_Osubset__diff,axiom,
    ! [B3: set_int,A3: set_int,G: fun_int_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,B3),A3)
     => ( aa_set_int_o(finite_finite_int,A3)
       => ( groups4538972089207619220nt_int(G,A3) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups4538972089207619220nt_int(G,aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A3),B3))),groups4538972089207619220nt_int(G,B3)) ) ) ) ).

% sum.subset_diff
tff(fact_7319_sum_Osubset__diff,axiom,
    ! [B3: set_Pr1261947904930325089at_nat,A3: set_Pr1261947904930325089at_nat,G: fun_Pr8317260874932935276at_rat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,B3),A3)
     => ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A3)
       => ( groups342789780944988191at_rat(G,A3) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,groups342789780944988191at_rat(G,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A3),B3))),groups342789780944988191at_rat(G,B3)) ) ) ) ).

% sum.subset_diff
tff(fact_7320_sum__diff,axiom,
    ! [A3: set_Code_integer,B3: set_Code_integer,F: fun_Code_integer_rat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A3)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(B3),A3)
       => ( groups6602215022474089585er_rat(F,minus_2355218937544613996nteger(A3,B3)) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,groups6602215022474089585er_rat(F,A3)),groups6602215022474089585er_rat(F,B3)) ) ) ) ).

% sum_diff
tff(fact_7321_sum__diff,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat,F: fun_Pr8317260874932935276at_rat] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A3)
     => ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,B3),A3)
       => ( groups342789780944988191at_rat(F,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A3),B3)) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,groups342789780944988191at_rat(F,A3)),groups342789780944988191at_rat(F,B3)) ) ) ) ).

% sum_diff
tff(fact_7322_sum__diff,axiom,
    ! [A3: set_Code_integer,B3: set_Code_integer,F: fun_Code_integer_int] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A3)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(B3),A3)
       => ( groups7234854612051535045er_int(F,minus_2355218937544613996nteger(A3,B3)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,groups7234854612051535045er_int(F,A3)),groups7234854612051535045er_int(F,B3)) ) ) ) ).

% sum_diff
tff(fact_7323_sum__diff,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat,F: fun_Pr7647638413817135296at_int] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A3)
     => ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,B3),A3)
       => ( groups975429370522433651at_int(F,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A3),B3)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,groups975429370522433651at_int(F,A3)),groups975429370522433651at_int(F,B3)) ) ) ) ).

% sum_diff
tff(fact_7324_sum__diff,axiom,
    ! [A3: set_nat,B3: set_nat,F: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B3),A3)
       => ( aa_set_nat_rat(groups2906978787729119204at_rat(F),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A3),B3)) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_set_nat_rat(groups2906978787729119204at_rat(F),A3)),aa_set_nat_rat(groups2906978787729119204at_rat(F),B3)) ) ) ) ).

% sum_diff
tff(fact_7325_sum__diff,axiom,
    ! [A3: set_nat,B3: set_nat,F: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B3),A3)
       => ( aa_set_nat_int(groups3539618377306564664at_int(F),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A3),B3)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_set_nat_int(groups3539618377306564664at_int(F),A3)),aa_set_nat_int(groups3539618377306564664at_int(F),B3)) ) ) ) ).

% sum_diff
tff(fact_7326_sum__diff,axiom,
    ! [A3: set_int,B3: set_int,F: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,A3)
     => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,B3),A3)
       => ( groups3906332499630173760nt_rat(F,aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A3),B3)) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,groups3906332499630173760nt_rat(F,A3)),groups3906332499630173760nt_rat(F,B3)) ) ) ) ).

% sum_diff
tff(fact_7327_sum__diff,axiom,
    ! [A3: set_int,B3: set_int,F: fun_int_int] :
      ( aa_set_int_o(finite_finite_int,A3)
     => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,B3),A3)
       => ( groups4538972089207619220nt_int(F,aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A3),B3)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,groups4538972089207619220nt_int(F,A3)),groups4538972089207619220nt_int(F,B3)) ) ) ) ).

% sum_diff
tff(fact_7328_sum_Omono__neutral__cong,axiom,
    ! [T: set_o,S: set_o,Ha: fun_o_Code_integer,G: fun_o_Code_integer] :
      ( finite_finite_o(T)
     => ( finite_finite_o(S)
       => ( ! [I3: $o] :
              ( member_o((I3),minus_minus_set_o(T,S))
             => ( aa_o_Code_integer(Ha,(I3)) = zero_z3403309356797280102nteger ) )
         => ( ! [I3: $o] :
                ( member_o((I3),minus_minus_set_o(S,T))
               => ( aa_o_Code_integer(G,(I3)) = zero_z3403309356797280102nteger ) )
           => ( ! [X3: $o] :
                  ( member_o((X3),inf_inf_set_o(S,T))
                 => ( aa_o_Code_integer(G,(X3)) = aa_o_Code_integer(Ha,(X3)) ) )
             => ( groups4406642042086082107nteger(G,S) = groups4406642042086082107nteger(Ha,T) ) ) ) ) ) ) ).

% sum.mono_neutral_cong
tff(fact_7329_sum_Omono__neutral__cong,axiom,
    ! [T: set_int,S: set_int,Ha: 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)
       => ( ! [I3: int] :
              ( aa_set_int_o(member_int(I3),aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,T),S))
             => ( aa_int_Code_integer(Ha,I3) = zero_z3403309356797280102nteger ) )
         => ( ! [I3: int] :
                ( aa_set_int_o(member_int(I3),aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,S),T))
               => ( aa_int_Code_integer(G,I3) = zero_z3403309356797280102nteger ) )
           => ( ! [X3: int] :
                  ( aa_set_int_o(member_int(X3),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,S),T))
                 => ( aa_int_Code_integer(G,X3) = aa_int_Code_integer(Ha,X3) ) )
             => ( groups7873554091576472773nteger(G,S) = groups7873554091576472773nteger(Ha,T) ) ) ) ) ) ) ).

% sum.mono_neutral_cong
tff(fact_7330_sum_Omono__neutral__cong,axiom,
    ! [T: set_Code_integer,S: set_Code_integer,Ha: fun_Co8499238673658658103nteger,G: fun_Co8499238673658658103nteger] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,T)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,S)
       => ( ! [I3: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(I3),minus_2355218937544613996nteger(T,S))
             => ( aa_Cod1783974652533396728nteger(Ha,I3) = zero_z3403309356797280102nteger ) )
         => ( ! [I3: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(I3),minus_2355218937544613996nteger(S,T))
               => ( aa_Cod1783974652533396728nteger(G,I3) = zero_z3403309356797280102nteger ) )
           => ( ! [X3: code_integer] :
                  ( aa_set5643387332196575111eger_o(member_Code_integer(X3),inf_in1364745209274528805nteger(S,T))
                 => ( aa_Cod1783974652533396728nteger(G,X3) = aa_Cod1783974652533396728nteger(Ha,X3) ) )
             => ( groups879477027807139574nteger(G,S) = groups879477027807139574nteger(Ha,T) ) ) ) ) ) ) ).

% sum.mono_neutral_cong
tff(fact_7331_sum_Omono__neutral__cong,axiom,
    ! [T: set_o,S: set_o,Ha: fun_o_rat,G: fun_o_rat] :
      ( finite_finite_o(T)
     => ( finite_finite_o(S)
       => ( ! [I3: $o] :
              ( member_o((I3),minus_minus_set_o(T,S))
             => ( aa_o_rat(Ha,(I3)) = zero_zero_rat ) )
         => ( ! [I3: $o] :
                ( member_o((I3),minus_minus_set_o(S,T))
               => ( aa_o_rat(G,(I3)) = zero_zero_rat ) )
           => ( ! [X3: $o] :
                  ( member_o((X3),inf_inf_set_o(S,T))
                 => ( aa_o_rat(G,(X3)) = aa_o_rat(Ha,(X3)) ) )
             => ( groups7872700643590313910_o_rat(G,S) = groups7872700643590313910_o_rat(Ha,T) ) ) ) ) ) ) ).

% sum.mono_neutral_cong
tff(fact_7332_sum_Omono__neutral__cong,axiom,
    ! [T: set_int,S: set_int,Ha: fun_int_rat,G: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,T)
     => ( aa_set_int_o(finite_finite_int,S)
       => ( ! [I3: int] :
              ( aa_set_int_o(member_int(I3),aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,T),S))
             => ( aa_int_rat(Ha,I3) = zero_zero_rat ) )
         => ( ! [I3: int] :
                ( aa_set_int_o(member_int(I3),aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,S),T))
               => ( aa_int_rat(G,I3) = zero_zero_rat ) )
           => ( ! [X3: int] :
                  ( aa_set_int_o(member_int(X3),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,S),T))
                 => ( aa_int_rat(G,X3) = aa_int_rat(Ha,X3) ) )
             => ( groups3906332499630173760nt_rat(G,S) = groups3906332499630173760nt_rat(Ha,T) ) ) ) ) ) ) ).

% sum.mono_neutral_cong
tff(fact_7333_sum_Omono__neutral__cong,axiom,
    ! [T: set_Code_integer,S: set_Code_integer,Ha: fun_Code_integer_rat,G: fun_Code_integer_rat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,T)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,S)
       => ( ! [I3: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(I3),minus_2355218937544613996nteger(T,S))
             => ( aa_Code_integer_rat(Ha,I3) = zero_zero_rat ) )
         => ( ! [I3: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(I3),minus_2355218937544613996nteger(S,T))
               => ( aa_Code_integer_rat(G,I3) = zero_zero_rat ) )
           => ( ! [X3: code_integer] :
                  ( aa_set5643387332196575111eger_o(member_Code_integer(X3),inf_in1364745209274528805nteger(S,T))
                 => ( aa_Code_integer_rat(G,X3) = aa_Code_integer_rat(Ha,X3) ) )
             => ( groups6602215022474089585er_rat(G,S) = groups6602215022474089585er_rat(Ha,T) ) ) ) ) ) ) ).

% sum.mono_neutral_cong
tff(fact_7334_sum_Omono__neutral__cong,axiom,
    ! [T: set_o,S: set_o,Ha: fun_o_nat,G: fun_o_nat] :
      ( finite_finite_o(T)
     => ( finite_finite_o(S)
       => ( ! [I3: $o] :
              ( member_o((I3),minus_minus_set_o(T,S))
             => ( aa_o_nat(Ha,(I3)) = zero_zero_nat ) )
         => ( ! [I3: $o] :
                ( member_o((I3),minus_minus_set_o(S,T))
               => ( aa_o_nat(G,(I3)) = zero_zero_nat ) )
           => ( ! [X3: $o] :
                  ( member_o((X3),inf_inf_set_o(S,T))
                 => ( aa_o_nat(G,(X3)) = aa_o_nat(Ha,(X3)) ) )
             => ( groups8507830703676809646_o_nat(G,S) = groups8507830703676809646_o_nat(Ha,T) ) ) ) ) ) ) ).

% sum.mono_neutral_cong
tff(fact_7335_sum_Omono__neutral__cong,axiom,
    ! [T: set_int,S: set_int,Ha: fun_int_nat,G: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,T)
     => ( aa_set_int_o(finite_finite_int,S)
       => ( ! [I3: int] :
              ( aa_set_int_o(member_int(I3),aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,T),S))
             => ( aa_int_nat(Ha,I3) = zero_zero_nat ) )
         => ( ! [I3: int] :
                ( aa_set_int_o(member_int(I3),aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,S),T))
               => ( aa_int_nat(G,I3) = zero_zero_nat ) )
           => ( ! [X3: int] :
                  ( aa_set_int_o(member_int(X3),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,S),T))
                 => ( aa_int_nat(G,X3) = aa_int_nat(Ha,X3) ) )
             => ( groups4541462559716669496nt_nat(G,S) = groups4541462559716669496nt_nat(Ha,T) ) ) ) ) ) ) ).

% sum.mono_neutral_cong
tff(fact_7336_sum_Omono__neutral__cong,axiom,
    ! [T: set_Code_integer,S: set_Code_integer,Ha: fun_Code_integer_nat,G: fun_Code_integer_nat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,T)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,S)
       => ( ! [I3: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(I3),minus_2355218937544613996nteger(T,S))
             => ( aa_Code_integer_nat(Ha,I3) = zero_zero_nat ) )
         => ( ! [I3: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(I3),minus_2355218937544613996nteger(S,T))
               => ( aa_Code_integer_nat(G,I3) = zero_zero_nat ) )
           => ( ! [X3: code_integer] :
                  ( aa_set5643387332196575111eger_o(member_Code_integer(X3),inf_in1364745209274528805nteger(S,T))
                 => ( aa_Code_integer_nat(G,X3) = aa_Code_integer_nat(Ha,X3) ) )
             => ( groups7237345082560585321er_nat(G,S) = groups7237345082560585321er_nat(Ha,T) ) ) ) ) ) ) ).

% sum.mono_neutral_cong
tff(fact_7337_sum_Omono__neutral__cong,axiom,
    ! [T: set_o,S: set_o,Ha: fun_o_int,G: fun_o_int] :
      ( finite_finite_o(T)
     => ( finite_finite_o(S)
       => ( ! [I3: $o] :
              ( member_o((I3),minus_minus_set_o(T,S))
             => ( aa_o_int(Ha,(I3)) = zero_zero_int ) )
         => ( ! [I3: $o] :
                ( member_o((I3),minus_minus_set_o(S,T))
               => ( aa_o_int(G,(I3)) = zero_zero_int ) )
           => ( ! [X3: $o] :
                  ( member_o((X3),inf_inf_set_o(S,T))
                 => ( aa_o_int(G,(X3)) = aa_o_int(Ha,(X3)) ) )
             => ( groups8505340233167759370_o_int(G,S) = groups8505340233167759370_o_int(Ha,T) ) ) ) ) ) ) ).

% sum.mono_neutral_cong
tff(fact_7338_prod_Osubset__diff,axiom,
    ! [B3: set_Code_integer,A3: set_Code_integer,G: fun_Co8499238673658658103nteger] :
      ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(B3),A3)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A3)
       => ( groups3674199335183972705nteger(G,A3) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,groups3674199335183972705nteger(G,minus_2355218937544613996nteger(A3,B3))),groups3674199335183972705nteger(G,B3)) ) ) ) ).

% prod.subset_diff
tff(fact_7339_prod_Osubset__diff,axiom,
    ! [B3: set_nat,A3: set_nat,G: fun_nat_Code_integer] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B3),A3)
     => ( aa_set_nat_o(finite_finite_nat,A3)
       => ( aa_set871585252387937313nteger(groups3455450783089532116nteger(G),A3) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_set871585252387937313nteger(groups3455450783089532116nteger(G),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A3),B3))),aa_set871585252387937313nteger(groups3455450783089532116nteger(G),B3)) ) ) ) ).

% prod.subset_diff
tff(fact_7340_prod_Osubset__diff,axiom,
    ! [B3: set_Code_integer,A3: set_Code_integer,G: fun_Co4490034186337852034r_assn] :
      ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(B3),A3)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A3)
       => ( groups1304777262505850412r_assn(G,A3) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,groups1304777262505850412r_assn(G,minus_2355218937544613996nteger(A3,B3))),groups1304777262505850412r_assn(G,B3)) ) ) ) ).

% prod.subset_diff
tff(fact_7341_prod_Osubset__diff,axiom,
    ! [B3: set_nat,A3: set_nat,G: fun_nat_assn] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B3),A3)
     => ( aa_set_nat_o(finite_finite_nat,A3)
       => ( groups6906906614972039071t_assn(G,A3) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,groups6906906614972039071t_assn(G,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A3),B3))),groups6906906614972039071t_assn(G,B3)) ) ) ) ).

% prod.subset_diff
tff(fact_7342_prod_Osubset__diff,axiom,
    ! [B3: set_Code_integer,A3: set_Code_integer,G: fun_Code_integer_rat] :
      ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(B3),A3)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A3)
       => ( groups2555765274223993564er_rat(G,A3) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,groups2555765274223993564er_rat(G,minus_2355218937544613996nteger(A3,B3))),groups2555765274223993564er_rat(G,B3)) ) ) ) ).

% prod.subset_diff
tff(fact_7343_prod_Osubset__diff,axiom,
    ! [B3: set_nat,A3: set_nat,G: fun_nat_rat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B3),A3)
     => ( aa_set_nat_o(finite_finite_nat,A3)
       => ( aa_set_nat_rat(groups73079841787564623at_rat(G),A3) = 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,A3),B3))),aa_set_nat_rat(groups73079841787564623at_rat(G),B3)) ) ) ) ).

% prod.subset_diff
tff(fact_7344_prod_Osubset__diff,axiom,
    ! [B3: set_Code_integer,A3: set_Code_integer,G: fun_Code_integer_nat] :
      ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(B3),A3)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A3)
       => ( groups3190895334310489300er_nat(G,A3) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,groups3190895334310489300er_nat(G,minus_2355218937544613996nteger(A3,B3))),groups3190895334310489300er_nat(G,B3)) ) ) ) ).

% prod.subset_diff
tff(fact_7345_prod_Osubset__diff,axiom,
    ! [B3: set_Code_integer,A3: set_Code_integer,G: fun_Code_integer_int] :
      ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(B3),A3)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A3)
       => ( groups3188404863801439024er_int(G,A3) = aa_int_int(aa_int_fun_int_int(times_times_int,groups3188404863801439024er_int(G,minus_2355218937544613996nteger(A3,B3))),groups3188404863801439024er_int(G,B3)) ) ) ) ).

% prod.subset_diff
tff(fact_7346_prod_Osubset__diff,axiom,
    ! [B3: set_int,A3: set_int,G: fun_int_Code_integer] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,B3),A3)
     => ( aa_set_int_o(finite_finite_int,A3)
       => ( groups3827104343326376752nteger(G,A3) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,groups3827104343326376752nteger(G,aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A3),B3))),groups3827104343326376752nteger(G,B3)) ) ) ) ).

% prod.subset_diff
tff(fact_7347_prod_Osubset__diff,axiom,
    ! [B3: set_int,A3: set_int,G: fun_int_assn] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,B3),A3)
     => ( aa_set_int_o(finite_finite_int,A3)
       => ( groups7882442080178216443t_assn(G,A3) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,groups7882442080178216443t_assn(G,aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A3),B3))),groups7882442080178216443t_assn(G,B3)) ) ) ) ).

% prod.subset_diff
tff(fact_7348_prod_Osame__carrier,axiom,
    ! [C2: set_o,A3: set_o,B3: set_o,G: fun_o_Code_integer,Ha: fun_o_Code_integer] :
      ( finite_finite_o(C2)
     => ( ord_less_eq_set_o(A3,C2)
       => ( ord_less_eq_set_o(B3,C2)
         => ( ! [A4: $o] :
                ( member_o((A4),minus_minus_set_o(C2,A3))
               => ( aa_o_Code_integer(G,(A4)) = one_one_Code_integer ) )
           => ( ! [B4: $o] :
                  ( member_o((B4),minus_minus_set_o(C2,B3))
                 => ( aa_o_Code_integer(Ha,(B4)) = one_one_Code_integer ) )
             => ( ( groups7694694392188491536nteger(G,A3) = groups7694694392188491536nteger(Ha,B3) )
              <=> ( groups7694694392188491536nteger(G,C2) = groups7694694392188491536nteger(Ha,C2) ) ) ) ) ) ) ) ).

% prod.same_carrier
tff(fact_7349_prod_Osame__carrier,axiom,
    ! [C2: set_Code_integer,A3: set_Code_integer,B3: set_Code_integer,G: fun_Co8499238673658658103nteger,Ha: fun_Co8499238673658658103nteger] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,C2)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(A3),C2)
       => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(B3),C2)
         => ( ! [A4: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(A4),minus_2355218937544613996nteger(C2,A3))
               => ( aa_Cod1783974652533396728nteger(G,A4) = one_one_Code_integer ) )
           => ( ! [B4: code_integer] :
                  ( aa_set5643387332196575111eger_o(member_Code_integer(B4),minus_2355218937544613996nteger(C2,B3))
                 => ( aa_Cod1783974652533396728nteger(Ha,B4) = one_one_Code_integer ) )
             => ( ( groups3674199335183972705nteger(G,A3) = groups3674199335183972705nteger(Ha,B3) )
              <=> ( groups3674199335183972705nteger(G,C2) = groups3674199335183972705nteger(Ha,C2) ) ) ) ) ) ) ) ).

% prod.same_carrier
tff(fact_7350_prod_Osame__carrier,axiom,
    ! [C2: set_o,A3: set_o,B3: set_o,G: fun_o_assn,Ha: fun_o_assn] :
      ( finite_finite_o(C2)
     => ( ord_less_eq_set_o(A3,C2)
       => ( ord_less_eq_set_o(B3,C2)
         => ( ! [A4: $o] :
                ( member_o((A4),minus_minus_set_o(C2,A3))
               => ( aa_o_assn(G,(A4)) = one_one_assn ) )
           => ( ! [B4: $o] :
                  ( member_o((B4),minus_minus_set_o(C2,B3))
                 => ( aa_o_assn(Ha,(B4)) = one_one_assn ) )
             => ( ( groups5301882518646026715o_assn(G,A3) = groups5301882518646026715o_assn(Ha,B3) )
              <=> ( groups5301882518646026715o_assn(G,C2) = groups5301882518646026715o_assn(Ha,C2) ) ) ) ) ) ) ) ).

% prod.same_carrier
tff(fact_7351_prod_Osame__carrier,axiom,
    ! [C2: set_Code_integer,A3: set_Code_integer,B3: set_Code_integer,G: fun_Co4490034186337852034r_assn,Ha: fun_Co4490034186337852034r_assn] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,C2)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(A3),C2)
       => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(B3),C2)
         => ( ! [A4: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(A4),minus_2355218937544613996nteger(C2,A3))
               => ( aa_Code_integer_assn(G,A4) = one_one_assn ) )
           => ( ! [B4: code_integer] :
                  ( aa_set5643387332196575111eger_o(member_Code_integer(B4),minus_2355218937544613996nteger(C2,B3))
                 => ( aa_Code_integer_assn(Ha,B4) = one_one_assn ) )
             => ( ( groups1304777262505850412r_assn(G,A3) = groups1304777262505850412r_assn(Ha,B3) )
              <=> ( groups1304777262505850412r_assn(G,C2) = groups1304777262505850412r_assn(Ha,C2) ) ) ) ) ) ) ) ).

% prod.same_carrier
tff(fact_7352_prod_Osame__carrier,axiom,
    ! [C2: set_o,A3: set_o,B3: set_o,G: fun_o_rat,Ha: fun_o_rat] :
      ( finite_finite_o(C2)
     => ( ord_less_eq_set_o(A3,C2)
       => ( ord_less_eq_set_o(B3,C2)
         => ( ! [A4: $o] :
                ( member_o((A4),minus_minus_set_o(C2,A3))
               => ( aa_o_rat(G,(A4)) = one_one_rat ) )
           => ( ! [B4: $o] :
                  ( member_o((B4),minus_minus_set_o(C2,B3))
                 => ( aa_o_rat(Ha,(B4)) = one_one_rat ) )
             => ( ( groups2869687844427037835_o_rat(G,A3) = groups2869687844427037835_o_rat(Ha,B3) )
              <=> ( groups2869687844427037835_o_rat(G,C2) = groups2869687844427037835_o_rat(Ha,C2) ) ) ) ) ) ) ) ).

% prod.same_carrier
tff(fact_7353_prod_Osame__carrier,axiom,
    ! [C2: set_Code_integer,A3: set_Code_integer,B3: set_Code_integer,G: fun_Code_integer_rat,Ha: fun_Code_integer_rat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,C2)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(A3),C2)
       => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(B3),C2)
         => ( ! [A4: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(A4),minus_2355218937544613996nteger(C2,A3))
               => ( aa_Code_integer_rat(G,A4) = one_one_rat ) )
           => ( ! [B4: code_integer] :
                  ( aa_set5643387332196575111eger_o(member_Code_integer(B4),minus_2355218937544613996nteger(C2,B3))
                 => ( aa_Code_integer_rat(Ha,B4) = one_one_rat ) )
             => ( ( groups2555765274223993564er_rat(G,A3) = groups2555765274223993564er_rat(Ha,B3) )
              <=> ( groups2555765274223993564er_rat(G,C2) = groups2555765274223993564er_rat(Ha,C2) ) ) ) ) ) ) ) ).

% prod.same_carrier
tff(fact_7354_prod_Osame__carrier,axiom,
    ! [C2: set_o,A3: set_o,B3: set_o,G: fun_o_nat,Ha: fun_o_nat] :
      ( finite_finite_o(C2)
     => ( ord_less_eq_set_o(A3,C2)
       => ( ord_less_eq_set_o(B3,C2)
         => ( ! [A4: $o] :
                ( member_o((A4),minus_minus_set_o(C2,A3))
               => ( aa_o_nat(G,(A4)) = one_one_nat ) )
           => ( ! [B4: $o] :
                  ( member_o((B4),minus_minus_set_o(C2,B3))
                 => ( aa_o_nat(Ha,(B4)) = one_one_nat ) )
             => ( ( groups3504817904513533571_o_nat(G,A3) = groups3504817904513533571_o_nat(Ha,B3) )
              <=> ( groups3504817904513533571_o_nat(G,C2) = groups3504817904513533571_o_nat(Ha,C2) ) ) ) ) ) ) ) ).

% prod.same_carrier
tff(fact_7355_prod_Osame__carrier,axiom,
    ! [C2: set_Code_integer,A3: set_Code_integer,B3: set_Code_integer,G: fun_Code_integer_nat,Ha: fun_Code_integer_nat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,C2)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(A3),C2)
       => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(B3),C2)
         => ( ! [A4: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(A4),minus_2355218937544613996nteger(C2,A3))
               => ( aa_Code_integer_nat(G,A4) = one_one_nat ) )
           => ( ! [B4: code_integer] :
                  ( aa_set5643387332196575111eger_o(member_Code_integer(B4),minus_2355218937544613996nteger(C2,B3))
                 => ( aa_Code_integer_nat(Ha,B4) = one_one_nat ) )
             => ( ( groups3190895334310489300er_nat(G,A3) = groups3190895334310489300er_nat(Ha,B3) )
              <=> ( groups3190895334310489300er_nat(G,C2) = groups3190895334310489300er_nat(Ha,C2) ) ) ) ) ) ) ) ).

% prod.same_carrier
tff(fact_7356_prod_Osame__carrier,axiom,
    ! [C2: set_o,A3: set_o,B3: set_o,G: fun_o_int,Ha: fun_o_int] :
      ( finite_finite_o(C2)
     => ( ord_less_eq_set_o(A3,C2)
       => ( ord_less_eq_set_o(B3,C2)
         => ( ! [A4: $o] :
                ( member_o((A4),minus_minus_set_o(C2,A3))
               => ( aa_o_int(G,(A4)) = one_one_int ) )
           => ( ! [B4: $o] :
                  ( member_o((B4),minus_minus_set_o(C2,B3))
                 => ( aa_o_int(Ha,(B4)) = one_one_int ) )
             => ( ( groups3502327434004483295_o_int(G,A3) = groups3502327434004483295_o_int(Ha,B3) )
              <=> ( groups3502327434004483295_o_int(G,C2) = groups3502327434004483295_o_int(Ha,C2) ) ) ) ) ) ) ) ).

% prod.same_carrier
tff(fact_7357_prod_Osame__carrier,axiom,
    ! [C2: set_Code_integer,A3: set_Code_integer,B3: set_Code_integer,G: fun_Code_integer_int,Ha: fun_Code_integer_int] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,C2)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(A3),C2)
       => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(B3),C2)
         => ( ! [A4: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(A4),minus_2355218937544613996nteger(C2,A3))
               => ( aa_Code_integer_int(G,A4) = one_one_int ) )
           => ( ! [B4: code_integer] :
                  ( aa_set5643387332196575111eger_o(member_Code_integer(B4),minus_2355218937544613996nteger(C2,B3))
                 => ( aa_Code_integer_int(Ha,B4) = one_one_int ) )
             => ( ( groups3188404863801439024er_int(G,A3) = groups3188404863801439024er_int(Ha,B3) )
              <=> ( groups3188404863801439024er_int(G,C2) = groups3188404863801439024er_int(Ha,C2) ) ) ) ) ) ) ) ).

% prod.same_carrier
tff(fact_7358_prod_Osame__carrierI,axiom,
    ! [C2: set_o,A3: set_o,B3: set_o,G: fun_o_Code_integer,Ha: fun_o_Code_integer] :
      ( finite_finite_o(C2)
     => ( ord_less_eq_set_o(A3,C2)
       => ( ord_less_eq_set_o(B3,C2)
         => ( ! [A4: $o] :
                ( member_o((A4),minus_minus_set_o(C2,A3))
               => ( aa_o_Code_integer(G,(A4)) = one_one_Code_integer ) )
           => ( ! [B4: $o] :
                  ( member_o((B4),minus_minus_set_o(C2,B3))
                 => ( aa_o_Code_integer(Ha,(B4)) = one_one_Code_integer ) )
             => ( ( groups7694694392188491536nteger(G,C2) = groups7694694392188491536nteger(Ha,C2) )
               => ( groups7694694392188491536nteger(G,A3) = groups7694694392188491536nteger(Ha,B3) ) ) ) ) ) ) ) ).

% prod.same_carrierI
tff(fact_7359_prod_Osame__carrierI,axiom,
    ! [C2: set_Code_integer,A3: set_Code_integer,B3: set_Code_integer,G: fun_Co8499238673658658103nteger,Ha: fun_Co8499238673658658103nteger] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,C2)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(A3),C2)
       => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(B3),C2)
         => ( ! [A4: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(A4),minus_2355218937544613996nteger(C2,A3))
               => ( aa_Cod1783974652533396728nteger(G,A4) = one_one_Code_integer ) )
           => ( ! [B4: code_integer] :
                  ( aa_set5643387332196575111eger_o(member_Code_integer(B4),minus_2355218937544613996nteger(C2,B3))
                 => ( aa_Cod1783974652533396728nteger(Ha,B4) = one_one_Code_integer ) )
             => ( ( groups3674199335183972705nteger(G,C2) = groups3674199335183972705nteger(Ha,C2) )
               => ( groups3674199335183972705nteger(G,A3) = groups3674199335183972705nteger(Ha,B3) ) ) ) ) ) ) ) ).

% prod.same_carrierI
tff(fact_7360_prod_Osame__carrierI,axiom,
    ! [C2: set_o,A3: set_o,B3: set_o,G: fun_o_assn,Ha: fun_o_assn] :
      ( finite_finite_o(C2)
     => ( ord_less_eq_set_o(A3,C2)
       => ( ord_less_eq_set_o(B3,C2)
         => ( ! [A4: $o] :
                ( member_o((A4),minus_minus_set_o(C2,A3))
               => ( aa_o_assn(G,(A4)) = one_one_assn ) )
           => ( ! [B4: $o] :
                  ( member_o((B4),minus_minus_set_o(C2,B3))
                 => ( aa_o_assn(Ha,(B4)) = one_one_assn ) )
             => ( ( groups5301882518646026715o_assn(G,C2) = groups5301882518646026715o_assn(Ha,C2) )
               => ( groups5301882518646026715o_assn(G,A3) = groups5301882518646026715o_assn(Ha,B3) ) ) ) ) ) ) ) ).

% prod.same_carrierI
tff(fact_7361_prod_Osame__carrierI,axiom,
    ! [C2: set_Code_integer,A3: set_Code_integer,B3: set_Code_integer,G: fun_Co4490034186337852034r_assn,Ha: fun_Co4490034186337852034r_assn] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,C2)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(A3),C2)
       => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(B3),C2)
         => ( ! [A4: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(A4),minus_2355218937544613996nteger(C2,A3))
               => ( aa_Code_integer_assn(G,A4) = one_one_assn ) )
           => ( ! [B4: code_integer] :
                  ( aa_set5643387332196575111eger_o(member_Code_integer(B4),minus_2355218937544613996nteger(C2,B3))
                 => ( aa_Code_integer_assn(Ha,B4) = one_one_assn ) )
             => ( ( groups1304777262505850412r_assn(G,C2) = groups1304777262505850412r_assn(Ha,C2) )
               => ( groups1304777262505850412r_assn(G,A3) = groups1304777262505850412r_assn(Ha,B3) ) ) ) ) ) ) ) ).

% prod.same_carrierI
tff(fact_7362_prod_Osame__carrierI,axiom,
    ! [C2: set_o,A3: set_o,B3: set_o,G: fun_o_rat,Ha: fun_o_rat] :
      ( finite_finite_o(C2)
     => ( ord_less_eq_set_o(A3,C2)
       => ( ord_less_eq_set_o(B3,C2)
         => ( ! [A4: $o] :
                ( member_o((A4),minus_minus_set_o(C2,A3))
               => ( aa_o_rat(G,(A4)) = one_one_rat ) )
           => ( ! [B4: $o] :
                  ( member_o((B4),minus_minus_set_o(C2,B3))
                 => ( aa_o_rat(Ha,(B4)) = one_one_rat ) )
             => ( ( groups2869687844427037835_o_rat(G,C2) = groups2869687844427037835_o_rat(Ha,C2) )
               => ( groups2869687844427037835_o_rat(G,A3) = groups2869687844427037835_o_rat(Ha,B3) ) ) ) ) ) ) ) ).

% prod.same_carrierI
tff(fact_7363_prod_Osame__carrierI,axiom,
    ! [C2: set_Code_integer,A3: set_Code_integer,B3: set_Code_integer,G: fun_Code_integer_rat,Ha: fun_Code_integer_rat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,C2)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(A3),C2)
       => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(B3),C2)
         => ( ! [A4: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(A4),minus_2355218937544613996nteger(C2,A3))
               => ( aa_Code_integer_rat(G,A4) = one_one_rat ) )
           => ( ! [B4: code_integer] :
                  ( aa_set5643387332196575111eger_o(member_Code_integer(B4),minus_2355218937544613996nteger(C2,B3))
                 => ( aa_Code_integer_rat(Ha,B4) = one_one_rat ) )
             => ( ( groups2555765274223993564er_rat(G,C2) = groups2555765274223993564er_rat(Ha,C2) )
               => ( groups2555765274223993564er_rat(G,A3) = groups2555765274223993564er_rat(Ha,B3) ) ) ) ) ) ) ) ).

% prod.same_carrierI
tff(fact_7364_prod_Osame__carrierI,axiom,
    ! [C2: set_o,A3: set_o,B3: set_o,G: fun_o_nat,Ha: fun_o_nat] :
      ( finite_finite_o(C2)
     => ( ord_less_eq_set_o(A3,C2)
       => ( ord_less_eq_set_o(B3,C2)
         => ( ! [A4: $o] :
                ( member_o((A4),minus_minus_set_o(C2,A3))
               => ( aa_o_nat(G,(A4)) = one_one_nat ) )
           => ( ! [B4: $o] :
                  ( member_o((B4),minus_minus_set_o(C2,B3))
                 => ( aa_o_nat(Ha,(B4)) = one_one_nat ) )
             => ( ( groups3504817904513533571_o_nat(G,C2) = groups3504817904513533571_o_nat(Ha,C2) )
               => ( groups3504817904513533571_o_nat(G,A3) = groups3504817904513533571_o_nat(Ha,B3) ) ) ) ) ) ) ) ).

% prod.same_carrierI
tff(fact_7365_prod_Osame__carrierI,axiom,
    ! [C2: set_Code_integer,A3: set_Code_integer,B3: set_Code_integer,G: fun_Code_integer_nat,Ha: fun_Code_integer_nat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,C2)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(A3),C2)
       => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(B3),C2)
         => ( ! [A4: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(A4),minus_2355218937544613996nteger(C2,A3))
               => ( aa_Code_integer_nat(G,A4) = one_one_nat ) )
           => ( ! [B4: code_integer] :
                  ( aa_set5643387332196575111eger_o(member_Code_integer(B4),minus_2355218937544613996nteger(C2,B3))
                 => ( aa_Code_integer_nat(Ha,B4) = one_one_nat ) )
             => ( ( groups3190895334310489300er_nat(G,C2) = groups3190895334310489300er_nat(Ha,C2) )
               => ( groups3190895334310489300er_nat(G,A3) = groups3190895334310489300er_nat(Ha,B3) ) ) ) ) ) ) ) ).

% prod.same_carrierI
tff(fact_7366_prod_Osame__carrierI,axiom,
    ! [C2: set_o,A3: set_o,B3: set_o,G: fun_o_int,Ha: fun_o_int] :
      ( finite_finite_o(C2)
     => ( ord_less_eq_set_o(A3,C2)
       => ( ord_less_eq_set_o(B3,C2)
         => ( ! [A4: $o] :
                ( member_o((A4),minus_minus_set_o(C2,A3))
               => ( aa_o_int(G,(A4)) = one_one_int ) )
           => ( ! [B4: $o] :
                  ( member_o((B4),minus_minus_set_o(C2,B3))
                 => ( aa_o_int(Ha,(B4)) = one_one_int ) )
             => ( ( groups3502327434004483295_o_int(G,C2) = groups3502327434004483295_o_int(Ha,C2) )
               => ( groups3502327434004483295_o_int(G,A3) = groups3502327434004483295_o_int(Ha,B3) ) ) ) ) ) ) ) ).

% prod.same_carrierI
tff(fact_7367_prod_Osame__carrierI,axiom,
    ! [C2: set_Code_integer,A3: set_Code_integer,B3: set_Code_integer,G: fun_Code_integer_int,Ha: fun_Code_integer_int] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,C2)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(A3),C2)
       => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(B3),C2)
         => ( ! [A4: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(A4),minus_2355218937544613996nteger(C2,A3))
               => ( aa_Code_integer_int(G,A4) = one_one_int ) )
           => ( ! [B4: code_integer] :
                  ( aa_set5643387332196575111eger_o(member_Code_integer(B4),minus_2355218937544613996nteger(C2,B3))
                 => ( aa_Code_integer_int(Ha,B4) = one_one_int ) )
             => ( ( groups3188404863801439024er_int(G,C2) = groups3188404863801439024er_int(Ha,C2) )
               => ( groups3188404863801439024er_int(G,A3) = groups3188404863801439024er_int(Ha,B3) ) ) ) ) ) ) ) ).

% prod.same_carrierI
tff(fact_7368_prod_Omono__neutral__left,axiom,
    ! [T: set_Code_integer,S: set_Code_integer,G: fun_Co8499238673658658103nteger] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,T)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(S),T)
       => ( ! [X3: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X3),minus_2355218937544613996nteger(T,S))
             => ( aa_Cod1783974652533396728nteger(G,X3) = one_one_Code_integer ) )
         => ( groups3674199335183972705nteger(G,S) = groups3674199335183972705nteger(G,T) ) ) ) ) ).

% prod.mono_neutral_left
tff(fact_7369_prod_Omono__neutral__left,axiom,
    ! [T: set_Code_integer,S: set_Code_integer,G: fun_Co4490034186337852034r_assn] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,T)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(S),T)
       => ( ! [X3: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X3),minus_2355218937544613996nteger(T,S))
             => ( aa_Code_integer_assn(G,X3) = one_one_assn ) )
         => ( groups1304777262505850412r_assn(G,S) = groups1304777262505850412r_assn(G,T) ) ) ) ) ).

% prod.mono_neutral_left
tff(fact_7370_prod_Omono__neutral__left,axiom,
    ! [T: set_Code_integer,S: set_Code_integer,G: fun_Code_integer_rat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,T)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(S),T)
       => ( ! [X3: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X3),minus_2355218937544613996nteger(T,S))
             => ( aa_Code_integer_rat(G,X3) = one_one_rat ) )
         => ( groups2555765274223993564er_rat(G,S) = groups2555765274223993564er_rat(G,T) ) ) ) ) ).

% prod.mono_neutral_left
tff(fact_7371_prod_Omono__neutral__left,axiom,
    ! [T: set_Code_integer,S: set_Code_integer,G: fun_Code_integer_nat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,T)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(S),T)
       => ( ! [X3: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X3),minus_2355218937544613996nteger(T,S))
             => ( aa_Code_integer_nat(G,X3) = one_one_nat ) )
         => ( groups3190895334310489300er_nat(G,S) = groups3190895334310489300er_nat(G,T) ) ) ) ) ).

% prod.mono_neutral_left
tff(fact_7372_prod_Omono__neutral__left,axiom,
    ! [T: set_Code_integer,S: set_Code_integer,G: fun_Code_integer_int] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,T)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(S),T)
       => ( ! [X3: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X3),minus_2355218937544613996nteger(T,S))
             => ( aa_Code_integer_int(G,X3) = one_one_int ) )
         => ( groups3188404863801439024er_int(G,S) = groups3188404863801439024er_int(G,T) ) ) ) ) ).

% prod.mono_neutral_left
tff(fact_7373_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)
       => ( ! [X3: nat] :
              ( aa_set_nat_o(member_nat(X3),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,T),S))
             => ( aa_nat_Code_integer(G,X3) = one_one_Code_integer ) )
         => ( aa_set871585252387937313nteger(groups3455450783089532116nteger(G),S) = aa_set871585252387937313nteger(groups3455450783089532116nteger(G),T) ) ) ) ) ).

% prod.mono_neutral_left
tff(fact_7374_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)
       => ( ! [X3: nat] :
              ( aa_set_nat_o(member_nat(X3),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,T),S))
             => ( aa_nat_assn(G,X3) = one_one_assn ) )
         => ( groups6906906614972039071t_assn(G,S) = groups6906906614972039071t_assn(G,T) ) ) ) ) ).

% prod.mono_neutral_left
tff(fact_7375_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)
       => ( ! [X3: nat] :
              ( aa_set_nat_o(member_nat(X3),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,T),S))
             => ( aa_nat_rat(G,X3) = 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_7376_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)
       => ( ! [X3: int] :
              ( aa_set_int_o(member_int(X3),aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,T),S))
             => ( aa_int_Code_integer(G,X3) = one_one_Code_integer ) )
         => ( groups3827104343326376752nteger(G,S) = groups3827104343326376752nteger(G,T) ) ) ) ) ).

% prod.mono_neutral_left
tff(fact_7377_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)
       => ( ! [X3: int] :
              ( aa_set_int_o(member_int(X3),aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,T),S))
             => ( aa_int_assn(G,X3) = one_one_assn ) )
         => ( groups7882442080178216443t_assn(G,S) = groups7882442080178216443t_assn(G,T) ) ) ) ) ).

% prod.mono_neutral_left
tff(fact_7378_prod_Omono__neutral__right,axiom,
    ! [T: set_Code_integer,S: set_Code_integer,G: fun_Co8499238673658658103nteger] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,T)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(S),T)
       => ( ! [X3: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X3),minus_2355218937544613996nteger(T,S))
             => ( aa_Cod1783974652533396728nteger(G,X3) = one_one_Code_integer ) )
         => ( groups3674199335183972705nteger(G,T) = groups3674199335183972705nteger(G,S) ) ) ) ) ).

% prod.mono_neutral_right
tff(fact_7379_prod_Omono__neutral__right,axiom,
    ! [T: set_Code_integer,S: set_Code_integer,G: fun_Co4490034186337852034r_assn] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,T)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(S),T)
       => ( ! [X3: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X3),minus_2355218937544613996nteger(T,S))
             => ( aa_Code_integer_assn(G,X3) = one_one_assn ) )
         => ( groups1304777262505850412r_assn(G,T) = groups1304777262505850412r_assn(G,S) ) ) ) ) ).

% prod.mono_neutral_right
tff(fact_7380_prod_Omono__neutral__right,axiom,
    ! [T: set_Code_integer,S: set_Code_integer,G: fun_Code_integer_rat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,T)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(S),T)
       => ( ! [X3: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X3),minus_2355218937544613996nteger(T,S))
             => ( aa_Code_integer_rat(G,X3) = one_one_rat ) )
         => ( groups2555765274223993564er_rat(G,T) = groups2555765274223993564er_rat(G,S) ) ) ) ) ).

% prod.mono_neutral_right
tff(fact_7381_prod_Omono__neutral__right,axiom,
    ! [T: set_Code_integer,S: set_Code_integer,G: fun_Code_integer_nat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,T)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(S),T)
       => ( ! [X3: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X3),minus_2355218937544613996nteger(T,S))
             => ( aa_Code_integer_nat(G,X3) = one_one_nat ) )
         => ( groups3190895334310489300er_nat(G,T) = groups3190895334310489300er_nat(G,S) ) ) ) ) ).

% prod.mono_neutral_right
tff(fact_7382_prod_Omono__neutral__right,axiom,
    ! [T: set_Code_integer,S: set_Code_integer,G: fun_Code_integer_int] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,T)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(S),T)
       => ( ! [X3: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X3),minus_2355218937544613996nteger(T,S))
             => ( aa_Code_integer_int(G,X3) = one_one_int ) )
         => ( groups3188404863801439024er_int(G,T) = groups3188404863801439024er_int(G,S) ) ) ) ) ).

% prod.mono_neutral_right
tff(fact_7383_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)
       => ( ! [X3: nat] :
              ( aa_set_nat_o(member_nat(X3),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,T),S))
             => ( aa_nat_Code_integer(G,X3) = one_one_Code_integer ) )
         => ( aa_set871585252387937313nteger(groups3455450783089532116nteger(G),T) = aa_set871585252387937313nteger(groups3455450783089532116nteger(G),S) ) ) ) ) ).

% prod.mono_neutral_right
tff(fact_7384_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)
       => ( ! [X3: nat] :
              ( aa_set_nat_o(member_nat(X3),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,T),S))
             => ( aa_nat_assn(G,X3) = one_one_assn ) )
         => ( groups6906906614972039071t_assn(G,T) = groups6906906614972039071t_assn(G,S) ) ) ) ) ).

% prod.mono_neutral_right
tff(fact_7385_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)
       => ( ! [X3: nat] :
              ( aa_set_nat_o(member_nat(X3),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,T),S))
             => ( aa_nat_rat(G,X3) = 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_7386_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)
       => ( ! [X3: int] :
              ( aa_set_int_o(member_int(X3),aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,T),S))
             => ( aa_int_Code_integer(G,X3) = one_one_Code_integer ) )
         => ( groups3827104343326376752nteger(G,T) = groups3827104343326376752nteger(G,S) ) ) ) ) ).

% prod.mono_neutral_right
tff(fact_7387_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)
       => ( ! [X3: int] :
              ( aa_set_int_o(member_int(X3),aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,T),S))
             => ( aa_int_assn(G,X3) = one_one_assn ) )
         => ( groups7882442080178216443t_assn(G,T) = groups7882442080178216443t_assn(G,S) ) ) ) ) ).

% prod.mono_neutral_right
tff(fact_7388_prod_Omono__neutral__cong__left,axiom,
    ! [T: set_o,S: set_o,Ha: fun_o_Code_integer,G: fun_o_Code_integer] :
      ( finite_finite_o(T)
     => ( ord_less_eq_set_o(S,T)
       => ( ! [X3: $o] :
              ( member_o((X3),minus_minus_set_o(T,S))
             => ( aa_o_Code_integer(Ha,(X3)) = one_one_Code_integer ) )
         => ( ! [X3: $o] :
                ( member_o((X3),S)
               => ( aa_o_Code_integer(G,(X3)) = aa_o_Code_integer(Ha,(X3)) ) )
           => ( groups7694694392188491536nteger(G,S) = groups7694694392188491536nteger(Ha,T) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
tff(fact_7389_prod_Omono__neutral__cong__left,axiom,
    ! [T: set_Code_integer,S: set_Code_integer,Ha: fun_Co8499238673658658103nteger,G: fun_Co8499238673658658103nteger] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,T)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(S),T)
       => ( ! [X3: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X3),minus_2355218937544613996nteger(T,S))
             => ( aa_Cod1783974652533396728nteger(Ha,X3) = one_one_Code_integer ) )
         => ( ! [X3: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(X3),S)
               => ( aa_Cod1783974652533396728nteger(G,X3) = aa_Cod1783974652533396728nteger(Ha,X3) ) )
           => ( groups3674199335183972705nteger(G,S) = groups3674199335183972705nteger(Ha,T) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
tff(fact_7390_prod_Omono__neutral__cong__left,axiom,
    ! [T: set_o,S: set_o,Ha: fun_o_assn,G: fun_o_assn] :
      ( finite_finite_o(T)
     => ( ord_less_eq_set_o(S,T)
       => ( ! [X3: $o] :
              ( member_o((X3),minus_minus_set_o(T,S))
             => ( aa_o_assn(Ha,(X3)) = one_one_assn ) )
         => ( ! [X3: $o] :
                ( member_o((X3),S)
               => ( aa_o_assn(G,(X3)) = aa_o_assn(Ha,(X3)) ) )
           => ( groups5301882518646026715o_assn(G,S) = groups5301882518646026715o_assn(Ha,T) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
tff(fact_7391_prod_Omono__neutral__cong__left,axiom,
    ! [T: set_Code_integer,S: set_Code_integer,Ha: fun_Co4490034186337852034r_assn,G: fun_Co4490034186337852034r_assn] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,T)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(S),T)
       => ( ! [X3: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X3),minus_2355218937544613996nteger(T,S))
             => ( aa_Code_integer_assn(Ha,X3) = one_one_assn ) )
         => ( ! [X3: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(X3),S)
               => ( aa_Code_integer_assn(G,X3) = aa_Code_integer_assn(Ha,X3) ) )
           => ( groups1304777262505850412r_assn(G,S) = groups1304777262505850412r_assn(Ha,T) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
tff(fact_7392_prod_Omono__neutral__cong__left,axiom,
    ! [T: set_o,S: set_o,Ha: fun_o_rat,G: fun_o_rat] :
      ( finite_finite_o(T)
     => ( ord_less_eq_set_o(S,T)
       => ( ! [X3: $o] :
              ( member_o((X3),minus_minus_set_o(T,S))
             => ( aa_o_rat(Ha,(X3)) = one_one_rat ) )
         => ( ! [X3: $o] :
                ( member_o((X3),S)
               => ( aa_o_rat(G,(X3)) = aa_o_rat(Ha,(X3)) ) )
           => ( groups2869687844427037835_o_rat(G,S) = groups2869687844427037835_o_rat(Ha,T) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
tff(fact_7393_prod_Omono__neutral__cong__left,axiom,
    ! [T: set_Code_integer,S: set_Code_integer,Ha: fun_Code_integer_rat,G: fun_Code_integer_rat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,T)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(S),T)
       => ( ! [X3: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X3),minus_2355218937544613996nteger(T,S))
             => ( aa_Code_integer_rat(Ha,X3) = one_one_rat ) )
         => ( ! [X3: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(X3),S)
               => ( aa_Code_integer_rat(G,X3) = aa_Code_integer_rat(Ha,X3) ) )
           => ( groups2555765274223993564er_rat(G,S) = groups2555765274223993564er_rat(Ha,T) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
tff(fact_7394_prod_Omono__neutral__cong__left,axiom,
    ! [T: set_o,S: set_o,Ha: fun_o_nat,G: fun_o_nat] :
      ( finite_finite_o(T)
     => ( ord_less_eq_set_o(S,T)
       => ( ! [X3: $o] :
              ( member_o((X3),minus_minus_set_o(T,S))
             => ( aa_o_nat(Ha,(X3)) = one_one_nat ) )
         => ( ! [X3: $o] :
                ( member_o((X3),S)
               => ( aa_o_nat(G,(X3)) = aa_o_nat(Ha,(X3)) ) )
           => ( groups3504817904513533571_o_nat(G,S) = groups3504817904513533571_o_nat(Ha,T) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
tff(fact_7395_prod_Omono__neutral__cong__left,axiom,
    ! [T: set_Code_integer,S: set_Code_integer,Ha: fun_Code_integer_nat,G: fun_Code_integer_nat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,T)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(S),T)
       => ( ! [X3: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X3),minus_2355218937544613996nteger(T,S))
             => ( aa_Code_integer_nat(Ha,X3) = one_one_nat ) )
         => ( ! [X3: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(X3),S)
               => ( aa_Code_integer_nat(G,X3) = aa_Code_integer_nat(Ha,X3) ) )
           => ( groups3190895334310489300er_nat(G,S) = groups3190895334310489300er_nat(Ha,T) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
tff(fact_7396_prod_Omono__neutral__cong__left,axiom,
    ! [T: set_o,S: set_o,Ha: fun_o_int,G: fun_o_int] :
      ( finite_finite_o(T)
     => ( ord_less_eq_set_o(S,T)
       => ( ! [X3: $o] :
              ( member_o((X3),minus_minus_set_o(T,S))
             => ( aa_o_int(Ha,(X3)) = one_one_int ) )
         => ( ! [X3: $o] :
                ( member_o((X3),S)
               => ( aa_o_int(G,(X3)) = aa_o_int(Ha,(X3)) ) )
           => ( groups3502327434004483295_o_int(G,S) = groups3502327434004483295_o_int(Ha,T) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
tff(fact_7397_prod_Omono__neutral__cong__left,axiom,
    ! [T: set_Code_integer,S: set_Code_integer,Ha: fun_Code_integer_int,G: fun_Code_integer_int] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,T)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(S),T)
       => ( ! [X3: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X3),minus_2355218937544613996nteger(T,S))
             => ( aa_Code_integer_int(Ha,X3) = one_one_int ) )
         => ( ! [X3: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(X3),S)
               => ( aa_Code_integer_int(G,X3) = aa_Code_integer_int(Ha,X3) ) )
           => ( groups3188404863801439024er_int(G,S) = groups3188404863801439024er_int(Ha,T) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
tff(fact_7398_prod_Omono__neutral__cong__right,axiom,
    ! [T: set_o,S: set_o,G: fun_o_Code_integer,Ha: fun_o_Code_integer] :
      ( finite_finite_o(T)
     => ( ord_less_eq_set_o(S,T)
       => ( ! [X3: $o] :
              ( member_o((X3),minus_minus_set_o(T,S))
             => ( aa_o_Code_integer(G,(X3)) = one_one_Code_integer ) )
         => ( ! [X3: $o] :
                ( member_o((X3),S)
               => ( aa_o_Code_integer(G,(X3)) = aa_o_Code_integer(Ha,(X3)) ) )
           => ( groups7694694392188491536nteger(G,T) = groups7694694392188491536nteger(Ha,S) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
tff(fact_7399_prod_Omono__neutral__cong__right,axiom,
    ! [T: set_Code_integer,S: set_Code_integer,G: fun_Co8499238673658658103nteger,Ha: fun_Co8499238673658658103nteger] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,T)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(S),T)
       => ( ! [X3: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X3),minus_2355218937544613996nteger(T,S))
             => ( aa_Cod1783974652533396728nteger(G,X3) = one_one_Code_integer ) )
         => ( ! [X3: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(X3),S)
               => ( aa_Cod1783974652533396728nteger(G,X3) = aa_Cod1783974652533396728nteger(Ha,X3) ) )
           => ( groups3674199335183972705nteger(G,T) = groups3674199335183972705nteger(Ha,S) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
tff(fact_7400_prod_Omono__neutral__cong__right,axiom,
    ! [T: set_o,S: set_o,G: fun_o_assn,Ha: fun_o_assn] :
      ( finite_finite_o(T)
     => ( ord_less_eq_set_o(S,T)
       => ( ! [X3: $o] :
              ( member_o((X3),minus_minus_set_o(T,S))
             => ( aa_o_assn(G,(X3)) = one_one_assn ) )
         => ( ! [X3: $o] :
                ( member_o((X3),S)
               => ( aa_o_assn(G,(X3)) = aa_o_assn(Ha,(X3)) ) )
           => ( groups5301882518646026715o_assn(G,T) = groups5301882518646026715o_assn(Ha,S) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
tff(fact_7401_prod_Omono__neutral__cong__right,axiom,
    ! [T: set_Code_integer,S: set_Code_integer,G: fun_Co4490034186337852034r_assn,Ha: fun_Co4490034186337852034r_assn] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,T)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(S),T)
       => ( ! [X3: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X3),minus_2355218937544613996nteger(T,S))
             => ( aa_Code_integer_assn(G,X3) = one_one_assn ) )
         => ( ! [X3: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(X3),S)
               => ( aa_Code_integer_assn(G,X3) = aa_Code_integer_assn(Ha,X3) ) )
           => ( groups1304777262505850412r_assn(G,T) = groups1304777262505850412r_assn(Ha,S) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
tff(fact_7402_prod_Omono__neutral__cong__right,axiom,
    ! [T: set_o,S: set_o,G: fun_o_rat,Ha: fun_o_rat] :
      ( finite_finite_o(T)
     => ( ord_less_eq_set_o(S,T)
       => ( ! [X3: $o] :
              ( member_o((X3),minus_minus_set_o(T,S))
             => ( aa_o_rat(G,(X3)) = one_one_rat ) )
         => ( ! [X3: $o] :
                ( member_o((X3),S)
               => ( aa_o_rat(G,(X3)) = aa_o_rat(Ha,(X3)) ) )
           => ( groups2869687844427037835_o_rat(G,T) = groups2869687844427037835_o_rat(Ha,S) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
tff(fact_7403_prod_Omono__neutral__cong__right,axiom,
    ! [T: set_Code_integer,S: set_Code_integer,G: fun_Code_integer_rat,Ha: fun_Code_integer_rat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,T)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(S),T)
       => ( ! [X3: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X3),minus_2355218937544613996nteger(T,S))
             => ( aa_Code_integer_rat(G,X3) = one_one_rat ) )
         => ( ! [X3: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(X3),S)
               => ( aa_Code_integer_rat(G,X3) = aa_Code_integer_rat(Ha,X3) ) )
           => ( groups2555765274223993564er_rat(G,T) = groups2555765274223993564er_rat(Ha,S) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
tff(fact_7404_prod_Omono__neutral__cong__right,axiom,
    ! [T: set_o,S: set_o,G: fun_o_nat,Ha: fun_o_nat] :
      ( finite_finite_o(T)
     => ( ord_less_eq_set_o(S,T)
       => ( ! [X3: $o] :
              ( member_o((X3),minus_minus_set_o(T,S))
             => ( aa_o_nat(G,(X3)) = one_one_nat ) )
         => ( ! [X3: $o] :
                ( member_o((X3),S)
               => ( aa_o_nat(G,(X3)) = aa_o_nat(Ha,(X3)) ) )
           => ( groups3504817904513533571_o_nat(G,T) = groups3504817904513533571_o_nat(Ha,S) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
tff(fact_7405_prod_Omono__neutral__cong__right,axiom,
    ! [T: set_Code_integer,S: set_Code_integer,G: fun_Code_integer_nat,Ha: fun_Code_integer_nat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,T)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(S),T)
       => ( ! [X3: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X3),minus_2355218937544613996nteger(T,S))
             => ( aa_Code_integer_nat(G,X3) = one_one_nat ) )
         => ( ! [X3: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(X3),S)
               => ( aa_Code_integer_nat(G,X3) = aa_Code_integer_nat(Ha,X3) ) )
           => ( groups3190895334310489300er_nat(G,T) = groups3190895334310489300er_nat(Ha,S) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
tff(fact_7406_prod_Omono__neutral__cong__right,axiom,
    ! [T: set_o,S: set_o,G: fun_o_int,Ha: fun_o_int] :
      ( finite_finite_o(T)
     => ( ord_less_eq_set_o(S,T)
       => ( ! [X3: $o] :
              ( member_o((X3),minus_minus_set_o(T,S))
             => ( aa_o_int(G,(X3)) = one_one_int ) )
         => ( ! [X3: $o] :
                ( member_o((X3),S)
               => ( aa_o_int(G,(X3)) = aa_o_int(Ha,(X3)) ) )
           => ( groups3502327434004483295_o_int(G,T) = groups3502327434004483295_o_int(Ha,S) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
tff(fact_7407_prod_Omono__neutral__cong__right,axiom,
    ! [T: set_Code_integer,S: set_Code_integer,G: fun_Code_integer_int,Ha: fun_Code_integer_int] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,T)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(S),T)
       => ( ! [X3: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X3),minus_2355218937544613996nteger(T,S))
             => ( aa_Code_integer_int(G,X3) = one_one_int ) )
         => ( ! [X3: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(X3),S)
               => ( aa_Code_integer_int(G,X3) = aa_Code_integer_int(Ha,X3) ) )
           => ( groups3188404863801439024er_int(G,T) = groups3188404863801439024er_int(Ha,S) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
tff(fact_7408_sum_Ounion__inter,axiom,
    ! [A3: set_int,B3: set_int,G: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,A3)
     => ( aa_set_int_o(finite_finite_int,B3)
       => ( 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,A3),B3))),groups3906332499630173760nt_rat(G,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A3),B3))) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,groups3906332499630173760nt_rat(G,A3)),groups3906332499630173760nt_rat(G,B3)) ) ) ) ).

% sum.union_inter
tff(fact_7409_sum_Ounion__inter,axiom,
    ! [A3: set_Code_integer,B3: set_Code_integer,G: fun_Code_integer_rat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A3)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B3)
       => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,groups6602215022474089585er_rat(G,sup_su848401254843788991nteger(A3,B3))),groups6602215022474089585er_rat(G,inf_in1364745209274528805nteger(A3,B3))) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,groups6602215022474089585er_rat(G,A3)),groups6602215022474089585er_rat(G,B3)) ) ) ) ).

% sum.union_inter
tff(fact_7410_sum_Ounion__inter,axiom,
    ! [A3: set_int,B3: set_int,G: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,A3)
     => ( aa_set_int_o(finite_finite_int,B3)
       => ( 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,A3),B3))),groups4541462559716669496nt_nat(G,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A3),B3))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups4541462559716669496nt_nat(G,A3)),groups4541462559716669496nt_nat(G,B3)) ) ) ) ).

% sum.union_inter
tff(fact_7411_sum_Ounion__inter,axiom,
    ! [A3: set_Code_integer,B3: set_Code_integer,G: fun_Code_integer_nat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A3)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B3)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups7237345082560585321er_nat(G,sup_su848401254843788991nteger(A3,B3))),groups7237345082560585321er_nat(G,inf_in1364745209274528805nteger(A3,B3))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups7237345082560585321er_nat(G,A3)),groups7237345082560585321er_nat(G,B3)) ) ) ) ).

% sum.union_inter
tff(fact_7412_sum_Ounion__inter,axiom,
    ! [A3: set_Code_integer,B3: set_Code_integer,G: fun_Code_integer_int] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A3)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B3)
       => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,groups7234854612051535045er_int(G,sup_su848401254843788991nteger(A3,B3))),groups7234854612051535045er_int(G,inf_in1364745209274528805nteger(A3,B3))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups7234854612051535045er_int(G,A3)),groups7234854612051535045er_int(G,B3)) ) ) ) ).

% sum.union_inter
tff(fact_7413_sum_Ounion__inter,axiom,
    ! [A3: set_nat,B3: set_nat,G: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( aa_set_nat_o(finite_finite_nat,B3)
       => ( 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,A3),B3))),aa_set_nat_rat(groups2906978787729119204at_rat(G),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A3),B3))) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_set_nat_rat(groups2906978787729119204at_rat(G),A3)),aa_set_nat_rat(groups2906978787729119204at_rat(G),B3)) ) ) ) ).

% sum.union_inter
tff(fact_7414_sum_Ounion__inter,axiom,
    ! [A3: set_nat,B3: set_nat,G: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( aa_set_nat_o(finite_finite_nat,B3)
       => ( 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,A3),B3))),aa_set_nat_int(groups3539618377306564664at_int(G),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A3),B3))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_set_nat_int(groups3539618377306564664at_int(G),A3)),aa_set_nat_int(groups3539618377306564664at_int(G),B3)) ) ) ) ).

% sum.union_inter
tff(fact_7415_sum_Ounion__inter,axiom,
    ! [A3: set_nat,B3: set_nat,G: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( aa_set_nat_o(finite_finite_nat,B3)
       => ( 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,A3),B3))),aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A3),B3))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(G),A3)),aa_set_nat_nat(groups3542108847815614940at_nat(G),B3)) ) ) ) ).

% sum.union_inter
tff(fact_7416_sum_Ounion__inter,axiom,
    ! [A3: set_int,B3: set_int,G: fun_int_int] :
      ( aa_set_int_o(finite_finite_int,A3)
     => ( aa_set_int_o(finite_finite_int,B3)
       => ( 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,A3),B3))),groups4538972089207619220nt_int(G,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A3),B3))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups4538972089207619220nt_int(G,A3)),groups4538972089207619220nt_int(G,B3)) ) ) ) ).

% sum.union_inter
tff(fact_7417_sum_Ounion__inter,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat,G: fun_Pr8317260874932935276at_rat] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A3)
     => ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,B3)
       => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,groups342789780944988191at_rat(G,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,A3),B3))),groups342789780944988191at_rat(G,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A3),B3))) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,groups342789780944988191at_rat(G,A3)),groups342789780944988191at_rat(G,B3)) ) ) ) ).

% sum.union_inter
tff(fact_7418_sum_OInt__Diff,axiom,
    ! [A3: set_int,G: fun_int_rat,B3: set_int] :
      ( aa_set_int_o(finite_finite_int,A3)
     => ( groups3906332499630173760nt_rat(G,A3) = 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,A3),B3))),groups3906332499630173760nt_rat(G,aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A3),B3))) ) ) ).

% sum.Int_Diff
tff(fact_7419_sum_OInt__Diff,axiom,
    ! [A3: set_Code_integer,G: fun_Code_integer_rat,B3: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A3)
     => ( groups6602215022474089585er_rat(G,A3) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,groups6602215022474089585er_rat(G,inf_in1364745209274528805nteger(A3,B3))),groups6602215022474089585er_rat(G,minus_2355218937544613996nteger(A3,B3))) ) ) ).

% sum.Int_Diff
tff(fact_7420_sum_OInt__Diff,axiom,
    ! [A3: set_int,G: fun_int_nat,B3: set_int] :
      ( aa_set_int_o(finite_finite_int,A3)
     => ( groups4541462559716669496nt_nat(G,A3) = 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,A3),B3))),groups4541462559716669496nt_nat(G,aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A3),B3))) ) ) ).

% sum.Int_Diff
tff(fact_7421_sum_OInt__Diff,axiom,
    ! [A3: set_Code_integer,G: fun_Code_integer_nat,B3: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A3)
     => ( groups7237345082560585321er_nat(G,A3) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups7237345082560585321er_nat(G,inf_in1364745209274528805nteger(A3,B3))),groups7237345082560585321er_nat(G,minus_2355218937544613996nteger(A3,B3))) ) ) ).

% sum.Int_Diff
tff(fact_7422_sum_OInt__Diff,axiom,
    ! [A3: set_Code_integer,G: fun_Code_integer_int,B3: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A3)
     => ( groups7234854612051535045er_int(G,A3) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups7234854612051535045er_int(G,inf_in1364745209274528805nteger(A3,B3))),groups7234854612051535045er_int(G,minus_2355218937544613996nteger(A3,B3))) ) ) ).

% sum.Int_Diff
tff(fact_7423_sum_OInt__Diff,axiom,
    ! [A3: set_nat,G: fun_nat_rat,B3: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( aa_set_nat_rat(groups2906978787729119204at_rat(G),A3) = 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,A3),B3))),aa_set_nat_rat(groups2906978787729119204at_rat(G),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A3),B3))) ) ) ).

% sum.Int_Diff
tff(fact_7424_sum_OInt__Diff,axiom,
    ! [A3: set_nat,G: fun_nat_int,B3: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( aa_set_nat_int(groups3539618377306564664at_int(G),A3) = 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,A3),B3))),aa_set_nat_int(groups3539618377306564664at_int(G),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A3),B3))) ) ) ).

% sum.Int_Diff
tff(fact_7425_sum_OInt__Diff,axiom,
    ! [A3: set_nat,G: fun_nat_nat,B3: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(G),A3) = 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,A3),B3))),aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A3),B3))) ) ) ).

% sum.Int_Diff
tff(fact_7426_sum_OInt__Diff,axiom,
    ! [A3: set_int,G: fun_int_int,B3: set_int] :
      ( aa_set_int_o(finite_finite_int,A3)
     => ( groups4538972089207619220nt_int(G,A3) = 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,A3),B3))),groups4538972089207619220nt_int(G,aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A3),B3))) ) ) ).

% sum.Int_Diff
tff(fact_7427_sum_OInt__Diff,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,G: fun_Pr8317260874932935276at_rat,B3: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A3)
     => ( groups342789780944988191at_rat(G,A3) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,groups342789780944988191at_rat(G,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A3),B3))),groups342789780944988191at_rat(G,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A3),B3))) ) ) ).

% sum.Int_Diff
tff(fact_7428_prod_Ounion__inter,axiom,
    ! [A3: set_int,B3: set_int,G: fun_int_Code_integer] :
      ( aa_set_int_o(finite_finite_int,A3)
     => ( aa_set_int_o(finite_finite_int,B3)
       => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,groups3827104343326376752nteger(G,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A3),B3))),groups3827104343326376752nteger(G,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A3),B3))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,groups3827104343326376752nteger(G,A3)),groups3827104343326376752nteger(G,B3)) ) ) ) ).

% prod.union_inter
tff(fact_7429_prod_Ounion__inter,axiom,
    ! [A3: set_Code_integer,B3: set_Code_integer,G: fun_Co8499238673658658103nteger] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A3)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B3)
       => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,groups3674199335183972705nteger(G,sup_su848401254843788991nteger(A3,B3))),groups3674199335183972705nteger(G,inf_in1364745209274528805nteger(A3,B3))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,groups3674199335183972705nteger(G,A3)),groups3674199335183972705nteger(G,B3)) ) ) ) ).

% prod.union_inter
tff(fact_7430_prod_Ounion__inter,axiom,
    ! [A3: set_nat,B3: set_nat,G: fun_nat_Code_integer] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( aa_set_nat_o(finite_finite_nat,B3)
       => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_set871585252387937313nteger(groups3455450783089532116nteger(G),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A3),B3))),aa_set871585252387937313nteger(groups3455450783089532116nteger(G),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A3),B3))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_set871585252387937313nteger(groups3455450783089532116nteger(G),A3)),aa_set871585252387937313nteger(groups3455450783089532116nteger(G),B3)) ) ) ) ).

% prod.union_inter
tff(fact_7431_prod_Ounion__inter,axiom,
    ! [A3: set_int,B3: set_int,G: fun_int_assn] :
      ( aa_set_int_o(finite_finite_int,A3)
     => ( aa_set_int_o(finite_finite_int,B3)
       => ( aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,groups7882442080178216443t_assn(G,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A3),B3))),groups7882442080178216443t_assn(G,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A3),B3))) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,groups7882442080178216443t_assn(G,A3)),groups7882442080178216443t_assn(G,B3)) ) ) ) ).

% prod.union_inter
tff(fact_7432_prod_Ounion__inter,axiom,
    ! [A3: set_Code_integer,B3: set_Code_integer,G: fun_Co4490034186337852034r_assn] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A3)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B3)
       => ( aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,groups1304777262505850412r_assn(G,sup_su848401254843788991nteger(A3,B3))),groups1304777262505850412r_assn(G,inf_in1364745209274528805nteger(A3,B3))) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,groups1304777262505850412r_assn(G,A3)),groups1304777262505850412r_assn(G,B3)) ) ) ) ).

% prod.union_inter
tff(fact_7433_prod_Ounion__inter,axiom,
    ! [A3: set_nat,B3: set_nat,G: fun_nat_assn] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( aa_set_nat_o(finite_finite_nat,B3)
       => ( aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,groups6906906614972039071t_assn(G,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A3),B3))),groups6906906614972039071t_assn(G,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A3),B3))) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,groups6906906614972039071t_assn(G,A3)),groups6906906614972039071t_assn(G,B3)) ) ) ) ).

% prod.union_inter
tff(fact_7434_prod_Ounion__inter,axiom,
    ! [A3: set_int,B3: set_int,G: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,A3)
     => ( aa_set_int_o(finite_finite_int,B3)
       => ( 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,A3),B3))),groups1072433553688619179nt_rat(G,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A3),B3))) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,groups1072433553688619179nt_rat(G,A3)),groups1072433553688619179nt_rat(G,B3)) ) ) ) ).

% prod.union_inter
tff(fact_7435_prod_Ounion__inter,axiom,
    ! [A3: set_Code_integer,B3: set_Code_integer,G: fun_Code_integer_rat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A3)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B3)
       => ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,groups2555765274223993564er_rat(G,sup_su848401254843788991nteger(A3,B3))),groups2555765274223993564er_rat(G,inf_in1364745209274528805nteger(A3,B3))) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,groups2555765274223993564er_rat(G,A3)),groups2555765274223993564er_rat(G,B3)) ) ) ) ).

% prod.union_inter
tff(fact_7436_prod_Ounion__inter,axiom,
    ! [A3: set_nat,B3: set_nat,G: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( aa_set_nat_o(finite_finite_nat,B3)
       => ( 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,A3),B3))),aa_set_nat_rat(groups73079841787564623at_rat(G),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A3),B3))) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_set_nat_rat(groups73079841787564623at_rat(G),A3)),aa_set_nat_rat(groups73079841787564623at_rat(G),B3)) ) ) ) ).

% prod.union_inter
tff(fact_7437_prod_Ounion__inter,axiom,
    ! [A3: set_int,B3: set_int,G: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,A3)
     => ( aa_set_int_o(finite_finite_int,B3)
       => ( 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,A3),B3))),groups1707563613775114915nt_nat(G,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A3),B3))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,groups1707563613775114915nt_nat(G,A3)),groups1707563613775114915nt_nat(G,B3)) ) ) ) ).

% prod.union_inter
tff(fact_7438_prod_OInt__Diff,axiom,
    ! [A3: set_int,G: fun_int_Code_integer,B3: set_int] :
      ( aa_set_int_o(finite_finite_int,A3)
     => ( groups3827104343326376752nteger(G,A3) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,groups3827104343326376752nteger(G,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A3),B3))),groups3827104343326376752nteger(G,aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A3),B3))) ) ) ).

% prod.Int_Diff
tff(fact_7439_prod_OInt__Diff,axiom,
    ! [A3: set_Code_integer,G: fun_Co8499238673658658103nteger,B3: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A3)
     => ( groups3674199335183972705nteger(G,A3) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,groups3674199335183972705nteger(G,inf_in1364745209274528805nteger(A3,B3))),groups3674199335183972705nteger(G,minus_2355218937544613996nteger(A3,B3))) ) ) ).

% prod.Int_Diff
tff(fact_7440_prod_OInt__Diff,axiom,
    ! [A3: set_nat,G: fun_nat_Code_integer,B3: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( aa_set871585252387937313nteger(groups3455450783089532116nteger(G),A3) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_set871585252387937313nteger(groups3455450783089532116nteger(G),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A3),B3))),aa_set871585252387937313nteger(groups3455450783089532116nteger(G),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A3),B3))) ) ) ).

% prod.Int_Diff
tff(fact_7441_prod_OInt__Diff,axiom,
    ! [A3: set_int,G: fun_int_assn,B3: set_int] :
      ( aa_set_int_o(finite_finite_int,A3)
     => ( groups7882442080178216443t_assn(G,A3) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,groups7882442080178216443t_assn(G,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A3),B3))),groups7882442080178216443t_assn(G,aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A3),B3))) ) ) ).

% prod.Int_Diff
tff(fact_7442_prod_OInt__Diff,axiom,
    ! [A3: set_Code_integer,G: fun_Co4490034186337852034r_assn,B3: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A3)
     => ( groups1304777262505850412r_assn(G,A3) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,groups1304777262505850412r_assn(G,inf_in1364745209274528805nteger(A3,B3))),groups1304777262505850412r_assn(G,minus_2355218937544613996nteger(A3,B3))) ) ) ).

% prod.Int_Diff
tff(fact_7443_prod_OInt__Diff,axiom,
    ! [A3: set_nat,G: fun_nat_assn,B3: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( groups6906906614972039071t_assn(G,A3) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,groups6906906614972039071t_assn(G,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A3),B3))),groups6906906614972039071t_assn(G,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A3),B3))) ) ) ).

% prod.Int_Diff
tff(fact_7444_prod_OInt__Diff,axiom,
    ! [A3: set_int,G: fun_int_rat,B3: set_int] :
      ( aa_set_int_o(finite_finite_int,A3)
     => ( groups1072433553688619179nt_rat(G,A3) = 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,A3),B3))),groups1072433553688619179nt_rat(G,aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A3),B3))) ) ) ).

% prod.Int_Diff
tff(fact_7445_prod_OInt__Diff,axiom,
    ! [A3: set_Code_integer,G: fun_Code_integer_rat,B3: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A3)
     => ( groups2555765274223993564er_rat(G,A3) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,groups2555765274223993564er_rat(G,inf_in1364745209274528805nteger(A3,B3))),groups2555765274223993564er_rat(G,minus_2355218937544613996nteger(A3,B3))) ) ) ).

% prod.Int_Diff
tff(fact_7446_prod_OInt__Diff,axiom,
    ! [A3: set_nat,G: fun_nat_rat,B3: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( aa_set_nat_rat(groups73079841787564623at_rat(G),A3) = 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,A3),B3))),aa_set_nat_rat(groups73079841787564623at_rat(G),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A3),B3))) ) ) ).

% prod.Int_Diff
tff(fact_7447_prod_OInt__Diff,axiom,
    ! [A3: set_int,G: fun_int_nat,B3: set_int] :
      ( aa_set_int_o(finite_finite_int,A3)
     => ( groups1707563613775114915nt_nat(G,A3) = 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,A3),B3))),groups1707563613775114915nt_nat(G,aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A3),B3))) ) ) ).

% prod.Int_Diff
tff(fact_7448_prod_Omono__neutral__cong,axiom,
    ! [T: set_o,S: set_o,Ha: fun_o_Code_integer,G: fun_o_Code_integer] :
      ( finite_finite_o(T)
     => ( finite_finite_o(S)
       => ( ! [I3: $o] :
              ( member_o((I3),minus_minus_set_o(T,S))
             => ( aa_o_Code_integer(Ha,(I3)) = one_one_Code_integer ) )
         => ( ! [I3: $o] :
                ( member_o((I3),minus_minus_set_o(S,T))
               => ( aa_o_Code_integer(G,(I3)) = one_one_Code_integer ) )
           => ( ! [X3: $o] :
                  ( member_o((X3),inf_inf_set_o(S,T))
                 => ( aa_o_Code_integer(G,(X3)) = aa_o_Code_integer(Ha,(X3)) ) )
             => ( groups7694694392188491536nteger(G,S) = groups7694694392188491536nteger(Ha,T) ) ) ) ) ) ) ).

% prod.mono_neutral_cong
tff(fact_7449_prod_Omono__neutral__cong,axiom,
    ! [T: set_int,S: set_int,Ha: 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)
       => ( ! [I3: int] :
              ( aa_set_int_o(member_int(I3),aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,T),S))
             => ( aa_int_Code_integer(Ha,I3) = one_one_Code_integer ) )
         => ( ! [I3: int] :
                ( aa_set_int_o(member_int(I3),aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,S),T))
               => ( aa_int_Code_integer(G,I3) = one_one_Code_integer ) )
           => ( ! [X3: int] :
                  ( aa_set_int_o(member_int(X3),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,S),T))
                 => ( aa_int_Code_integer(G,X3) = aa_int_Code_integer(Ha,X3) ) )
             => ( groups3827104343326376752nteger(G,S) = groups3827104343326376752nteger(Ha,T) ) ) ) ) ) ) ).

% prod.mono_neutral_cong
tff(fact_7450_prod_Omono__neutral__cong,axiom,
    ! [T: set_Code_integer,S: set_Code_integer,Ha: fun_Co8499238673658658103nteger,G: fun_Co8499238673658658103nteger] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,T)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,S)
       => ( ! [I3: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(I3),minus_2355218937544613996nteger(T,S))
             => ( aa_Cod1783974652533396728nteger(Ha,I3) = one_one_Code_integer ) )
         => ( ! [I3: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(I3),minus_2355218937544613996nteger(S,T))
               => ( aa_Cod1783974652533396728nteger(G,I3) = one_one_Code_integer ) )
           => ( ! [X3: code_integer] :
                  ( aa_set5643387332196575111eger_o(member_Code_integer(X3),inf_in1364745209274528805nteger(S,T))
                 => ( aa_Cod1783974652533396728nteger(G,X3) = aa_Cod1783974652533396728nteger(Ha,X3) ) )
             => ( groups3674199335183972705nteger(G,S) = groups3674199335183972705nteger(Ha,T) ) ) ) ) ) ) ).

% prod.mono_neutral_cong
tff(fact_7451_prod_Omono__neutral__cong,axiom,
    ! [T: set_o,S: set_o,Ha: fun_o_assn,G: fun_o_assn] :
      ( finite_finite_o(T)
     => ( finite_finite_o(S)
       => ( ! [I3: $o] :
              ( member_o((I3),minus_minus_set_o(T,S))
             => ( aa_o_assn(Ha,(I3)) = one_one_assn ) )
         => ( ! [I3: $o] :
                ( member_o((I3),minus_minus_set_o(S,T))
               => ( aa_o_assn(G,(I3)) = one_one_assn ) )
           => ( ! [X3: $o] :
                  ( member_o((X3),inf_inf_set_o(S,T))
                 => ( aa_o_assn(G,(X3)) = aa_o_assn(Ha,(X3)) ) )
             => ( groups5301882518646026715o_assn(G,S) = groups5301882518646026715o_assn(Ha,T) ) ) ) ) ) ) ).

% prod.mono_neutral_cong
tff(fact_7452_prod_Omono__neutral__cong,axiom,
    ! [T: set_int,S: set_int,Ha: fun_int_assn,G: fun_int_assn] :
      ( aa_set_int_o(finite_finite_int,T)
     => ( aa_set_int_o(finite_finite_int,S)
       => ( ! [I3: int] :
              ( aa_set_int_o(member_int(I3),aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,T),S))
             => ( aa_int_assn(Ha,I3) = one_one_assn ) )
         => ( ! [I3: int] :
                ( aa_set_int_o(member_int(I3),aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,S),T))
               => ( aa_int_assn(G,I3) = one_one_assn ) )
           => ( ! [X3: int] :
                  ( aa_set_int_o(member_int(X3),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,S),T))
                 => ( aa_int_assn(G,X3) = aa_int_assn(Ha,X3) ) )
             => ( groups7882442080178216443t_assn(G,S) = groups7882442080178216443t_assn(Ha,T) ) ) ) ) ) ) ).

% prod.mono_neutral_cong
tff(fact_7453_prod_Omono__neutral__cong,axiom,
    ! [T: set_Code_integer,S: set_Code_integer,Ha: fun_Co4490034186337852034r_assn,G: fun_Co4490034186337852034r_assn] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,T)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,S)
       => ( ! [I3: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(I3),minus_2355218937544613996nteger(T,S))
             => ( aa_Code_integer_assn(Ha,I3) = one_one_assn ) )
         => ( ! [I3: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(I3),minus_2355218937544613996nteger(S,T))
               => ( aa_Code_integer_assn(G,I3) = one_one_assn ) )
           => ( ! [X3: code_integer] :
                  ( aa_set5643387332196575111eger_o(member_Code_integer(X3),inf_in1364745209274528805nteger(S,T))
                 => ( aa_Code_integer_assn(G,X3) = aa_Code_integer_assn(Ha,X3) ) )
             => ( groups1304777262505850412r_assn(G,S) = groups1304777262505850412r_assn(Ha,T) ) ) ) ) ) ) ).

% prod.mono_neutral_cong
tff(fact_7454_prod_Omono__neutral__cong,axiom,
    ! [T: set_o,S: set_o,Ha: fun_o_rat,G: fun_o_rat] :
      ( finite_finite_o(T)
     => ( finite_finite_o(S)
       => ( ! [I3: $o] :
              ( member_o((I3),minus_minus_set_o(T,S))
             => ( aa_o_rat(Ha,(I3)) = one_one_rat ) )
         => ( ! [I3: $o] :
                ( member_o((I3),minus_minus_set_o(S,T))
               => ( aa_o_rat(G,(I3)) = one_one_rat ) )
           => ( ! [X3: $o] :
                  ( member_o((X3),inf_inf_set_o(S,T))
                 => ( aa_o_rat(G,(X3)) = aa_o_rat(Ha,(X3)) ) )
             => ( groups2869687844427037835_o_rat(G,S) = groups2869687844427037835_o_rat(Ha,T) ) ) ) ) ) ) ).

% prod.mono_neutral_cong
tff(fact_7455_prod_Omono__neutral__cong,axiom,
    ! [T: set_int,S: set_int,Ha: fun_int_rat,G: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,T)
     => ( aa_set_int_o(finite_finite_int,S)
       => ( ! [I3: int] :
              ( aa_set_int_o(member_int(I3),aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,T),S))
             => ( aa_int_rat(Ha,I3) = one_one_rat ) )
         => ( ! [I3: int] :
                ( aa_set_int_o(member_int(I3),aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,S),T))
               => ( aa_int_rat(G,I3) = one_one_rat ) )
           => ( ! [X3: int] :
                  ( aa_set_int_o(member_int(X3),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,S),T))
                 => ( aa_int_rat(G,X3) = aa_int_rat(Ha,X3) ) )
             => ( groups1072433553688619179nt_rat(G,S) = groups1072433553688619179nt_rat(Ha,T) ) ) ) ) ) ) ).

% prod.mono_neutral_cong
tff(fact_7456_prod_Omono__neutral__cong,axiom,
    ! [T: set_Code_integer,S: set_Code_integer,Ha: fun_Code_integer_rat,G: fun_Code_integer_rat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,T)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,S)
       => ( ! [I3: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(I3),minus_2355218937544613996nteger(T,S))
             => ( aa_Code_integer_rat(Ha,I3) = one_one_rat ) )
         => ( ! [I3: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(I3),minus_2355218937544613996nteger(S,T))
               => ( aa_Code_integer_rat(G,I3) = one_one_rat ) )
           => ( ! [X3: code_integer] :
                  ( aa_set5643387332196575111eger_o(member_Code_integer(X3),inf_in1364745209274528805nteger(S,T))
                 => ( aa_Code_integer_rat(G,X3) = aa_Code_integer_rat(Ha,X3) ) )
             => ( groups2555765274223993564er_rat(G,S) = groups2555765274223993564er_rat(Ha,T) ) ) ) ) ) ) ).

% prod.mono_neutral_cong
tff(fact_7457_prod_Omono__neutral__cong,axiom,
    ! [T: set_o,S: set_o,Ha: fun_o_nat,G: fun_o_nat] :
      ( finite_finite_o(T)
     => ( finite_finite_o(S)
       => ( ! [I3: $o] :
              ( member_o((I3),minus_minus_set_o(T,S))
             => ( aa_o_nat(Ha,(I3)) = one_one_nat ) )
         => ( ! [I3: $o] :
                ( member_o((I3),minus_minus_set_o(S,T))
               => ( aa_o_nat(G,(I3)) = one_one_nat ) )
           => ( ! [X3: $o] :
                  ( member_o((X3),inf_inf_set_o(S,T))
                 => ( aa_o_nat(G,(X3)) = aa_o_nat(Ha,(X3)) ) )
             => ( groups3504817904513533571_o_nat(G,S) = groups3504817904513533571_o_nat(Ha,T) ) ) ) ) ) ) ).

% prod.mono_neutral_cong
tff(fact_7458_prod__int__plus__eq,axiom,
    ! [I2: nat,J: nat] : aa_set_nat_int(groups705719431365010083at_int(semiri1314217659103216013at_int),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,I2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),J))) = groups1705073143266064639nt_int(aTP_Lamm_fv,set_or1266510415728281911st_int(aa_nat_int(semiri1314217659103216013at_int,I2),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),J)))) ).

% prod_int_plus_eq
tff(fact_7459_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_7460_double__arith__series,axiom,
    ! [A: rat,D2: rat,N2: nat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),aa_set_nat_rat(groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aTP_Lamm_abh(A),D2)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,zero_zero_nat),N2))) = 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,N2)),one_one_rat)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,N2)),D2))) ).

% double_arith_series
tff(fact_7461_double__arith__series,axiom,
    ! [A: code_natural,D2: code_natural,N2: nat] : times_2397367101498566445atural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one)),groups6325495683096345652atural(aa_Cod5320533616963873627atural(aTP_Lamm_abi(A),D2),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,zero_zero_nat),N2))) = times_2397367101498566445atural(plus_p4538020629002901425atural(aa_nat_Code_natural(semiri3763490453095760265atural,N2),one_one_Code_natural),plus_p4538020629002901425atural(times_2397367101498566445atural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one)),A),times_2397367101498566445atural(aa_nat_Code_natural(semiri3763490453095760265atural,N2),D2))) ).

% double_arith_series
tff(fact_7462_double__arith__series,axiom,
    ! [A: int,D2: int,N2: 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_abj(A),D2)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,zero_zero_nat),N2))) = 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,N2)),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,N2)),D2))) ).

% double_arith_series
tff(fact_7463_double__arith__series,axiom,
    ! [A: code_integer,D2: code_integer,N2: nat] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_set871585252387937313nteger(groups7501900531339628137nteger(aa_Cod1578151519121547077nteger(aTP_Lamm_abk(A),D2)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,zero_zero_nat),N2))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,N2)),one_one_Code_integer)),aa_Cod1783974652533396728nteger(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,N2)),D2))) ).

% double_arith_series
tff(fact_7464_double__arith__series,axiom,
    ! [A: nat,D2: nat,N2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_abl(A),D2)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,zero_zero_nat),N2))) = 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,N2)),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,N2)),D2))) ).

% double_arith_series
tff(fact_7465_double__gauss__sum,axiom,
    ! [N2: nat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),aa_set_nat_rat(groups2906978787729119204at_rat(semiri681578069525770553at_rat),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,zero_zero_nat),N2))) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,N2)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(semiri681578069525770553at_rat,N2)),one_one_rat)) ).

% double_gauss_sum
tff(fact_7466_double__gauss__sum,axiom,
    ! [N2: nat] : times_2397367101498566445atural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one)),groups6325495683096345652atural(semiri3763490453095760265atural,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,zero_zero_nat),N2))) = times_2397367101498566445atural(aa_nat_Code_natural(semiri3763490453095760265atural,N2),plus_p4538020629002901425atural(aa_nat_Code_natural(semiri3763490453095760265atural,N2),one_one_Code_natural)) ).

% double_gauss_sum
tff(fact_7467_double__gauss__sum,axiom,
    ! [N2: 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(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,zero_zero_nat),N2))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,N2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,N2)),one_one_int)) ).

% double_gauss_sum
tff(fact_7468_double__gauss__sum,axiom,
    ! [N2: 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(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,zero_zero_nat),N2))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,N2)),aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,N2)),one_one_Code_integer)) ).

% double_gauss_sum
tff(fact_7469_double__gauss__sum,axiom,
    ! [N2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_set_nat_nat(groups3542108847815614940at_nat(semiri1316708129612266289at_nat),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,zero_zero_nat),N2))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,N2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(semiri1316708129612266289at_nat,N2)),one_one_nat)) ).

% double_gauss_sum
tff(fact_7470_sum__choose__diagonal,axiom,
    ! [M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_abm(M),N2)),aa_nat_set_nat(set_ord_atMost_nat,M)) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,N2)),M) ) ) ).

% sum_choose_diagonal
tff(fact_7471_sum__Suc__diff_H,axiom,
    ! [M: nat,N2: nat,F: fun_nat_rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
     => ( aa_set_nat_rat(groups2906978787729119204at_rat(aTP_Lamm_kd(F)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,M),N2)) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_nat_rat(F,N2)),aa_nat_rat(F,M)) ) ) ).

% sum_Suc_diff'
tff(fact_7472_sum__Suc__diff_H,axiom,
    ! [M: nat,N2: nat,F: fun_nat_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
     => ( aa_set_nat_int(groups3539618377306564664at_int(aTP_Lamm_ke(F)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,M),N2)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(F,N2)),aa_nat_int(F,M)) ) ) ).

% sum_Suc_diff'
tff(fact_7473_sum__diff__nat,axiom,
    ! [B3: set_Code_integer,A3: set_Code_integer,F: fun_Code_integer_nat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B3)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(B3),A3)
       => ( groups7237345082560585321er_nat(F,minus_2355218937544613996nteger(A3,B3)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups7237345082560585321er_nat(F,A3)),groups7237345082560585321er_nat(F,B3)) ) ) ) ).

% sum_diff_nat
tff(fact_7474_sum__diff__nat,axiom,
    ! [B3: set_Pr1261947904930325089at_nat,A3: set_Pr1261947904930325089at_nat,F: fun_Pr2602117396471556196at_nat] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,B3)
     => ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,B3),A3)
       => ( groups977919841031483927at_nat(F,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A3),B3)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups977919841031483927at_nat(F,A3)),groups977919841031483927at_nat(F,B3)) ) ) ) ).

% sum_diff_nat
tff(fact_7475_sum__diff__nat,axiom,
    ! [B3: set_int,A3: set_int,F: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,B3)
     => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,B3),A3)
       => ( groups4541462559716669496nt_nat(F,aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A3),B3)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups4541462559716669496nt_nat(F,A3)),groups4541462559716669496nt_nat(F,B3)) ) ) ) ).

% sum_diff_nat
tff(fact_7476_sum__diff__nat,axiom,
    ! [B3: set_nat,A3: set_nat,F: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,B3)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B3),A3)
       => ( aa_set_nat_nat(groups3542108847815614940at_nat(F),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A3),B3)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),A3)),aa_set_nat_nat(groups3542108847815614940at_nat(F),B3)) ) ) ) ).

% sum_diff_nat
tff(fact_7477_vandermonde,axiom,
    ! [M: nat,N2: nat,Ra2: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_abn(M),N2),Ra2)),aa_nat_set_nat(set_ord_atMost_nat,Ra2)) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N2)),Ra2) ).

% vandermonde
tff(fact_7478_sum_OatLeastLessThan__rev,axiom,
    ! [G: fun_nat_nat,N2: nat,M: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,N2),M)) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_abo(G),N2),M)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,N2),M)) ).

% sum.atLeastLessThan_rev
tff(fact_7479_pos__eucl__rel__int__mult__2,axiom,
    ! [B: int,A: int,Q4: int,Ra2: 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,Q4),Ra2))
       => 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,Q4),aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Ra2)))) ) ) ).

% pos_eucl_rel_int_mult_2
tff(fact_7480_sum_Onested__swap,axiom,
    ! [A: fun_nat_fun_nat_nat,N2: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_abp(A)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,zero_zero_nat),N2)) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_abq(A),N2)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),N2)) ).

% sum.nested_swap
tff(fact_7481_prod_OatLeastLessThan__rev,axiom,
    ! [G: fun_nat_nat,N2: nat,M: nat] : aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,N2),M)) = aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_abo(G),N2),M)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,N2),M)) ).

% prod.atLeastLessThan_rev
tff(fact_7482_prod_OatLeastLessThan__rev,axiom,
    ! [G: fun_nat_int,N2: nat,M: nat] : aa_set_nat_int(groups705719431365010083at_int(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,N2),M)) = aa_set_nat_int(groups705719431365010083at_int(aa_nat_fun_nat_int(aa_nat1293065384339088724at_int(aTP_Lamm_abr(G),N2),M)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,N2),M)) ).

% prod.atLeastLessThan_rev
tff(fact_7483_prod_Onested__swap,axiom,
    ! [A: fun_nat_fun_nat_nat,N2: nat] : aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_abs(A)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,zero_zero_nat),N2)) = aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_abt(A),N2)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),N2)) ).

% prod.nested_swap
tff(fact_7484_prod_Onested__swap,axiom,
    ! [A: fun_nat_fun_nat_int,N2: nat] : aa_set_nat_int(groups705719431365010083at_int(aTP_Lamm_abu(A)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,zero_zero_nat),N2)) = aa_set_nat_int(groups705719431365010083at_int(aa_nat_fun_nat_int(aTP_Lamm_abv(A),N2)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),N2)) ).

% prod.nested_swap
tff(fact_7485_fact__prod__rev,axiom,
    ! [N2: nat] : aa_nat_int(semiri1406184849735516958ct_int,N2) = aa_nat_int(semiri1314217659103216013at_int,aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),N2))) ).

% fact_prod_rev
tff(fact_7486_fact__prod__rev,axiom,
    ! [N2: nat] : aa_nat_Code_integer(semiri3624122377584611663nteger,N2) = aa_nat_Code_integer(semiri4939895301339042750nteger,aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),N2))) ).

% fact_prod_rev
tff(fact_7487_fact__prod__rev,axiom,
    ! [N2: nat] : aa_nat_nat(semiri1408675320244567234ct_nat,N2) = aa_nat_nat(semiri1316708129612266289at_nat,aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),N2))) ).

% fact_prod_rev
tff(fact_7488_gbinomial__sum__nat__pow2,axiom,
    ! [M: nat] : aa_set_nat_rat(groups2906978787729119204at_rat(aTP_Lamm_abw(M)),aa_nat_set_nat(set_ord_atMost_nat,M)) = aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),M) ).

% gbinomial_sum_nat_pow2
tff(fact_7489_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(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_7490_divmod__digit__1_I1_J,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B)))
         => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B)))),one_one_nat) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B) ) ) ) ) ).

% divmod_digit_1(1)
tff(fact_7491_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_7492_Sum__Icc__int,axiom,
    ! [M: int,N2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,M),N2)
     => ( groups4538972089207619220nt_int(aTP_Lamm_fv,set_or1266510415728281911st_int(M,N2)) = 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,N2),aa_int_int(aa_int_fun_int_int(plus_plus_int,N2),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_7493_sum_OIf__cases,axiom,
    ! [A3: set_int,Pa: fun_int_o,Ha: fun_int_rat,G: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,A3)
     => ( groups3906332499630173760nt_rat(aa_fun1481061427733998144nt_rat(aa_fun2191249852558612414nt_rat(aTP_Lamm_abx(Pa),Ha),G),A3) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,groups3906332499630173760nt_rat(Ha,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A3),collect_int(Pa)))),groups3906332499630173760nt_rat(G,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A3),uminus1532241313380277803et_int(collect_int(Pa))))) ) ) ).

% sum.If_cases
tff(fact_7494_sum_OIf__cases,axiom,
    ! [A3: set_Code_integer,Pa: fun_Code_integer_o,Ha: fun_Code_integer_rat,G: fun_Code_integer_rat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A3)
     => ( groups6602215022474089585er_rat(aa_fun8169474305930283170er_rat(aa_fun1894662059565830609er_rat(aTP_Lamm_aby(Pa),Ha),G),A3) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,groups6602215022474089585er_rat(Ha,inf_in1364745209274528805nteger(A3,collect_Code_integer(Pa)))),groups6602215022474089585er_rat(G,inf_in1364745209274528805nteger(A3,uminus804700908173204444nteger(collect_Code_integer(Pa))))) ) ) ).

% sum.If_cases
tff(fact_7495_sum_OIf__cases,axiom,
    ! [A3: set_int,Pa: fun_int_o,Ha: fun_int_nat,G: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,A3)
     => ( groups4541462559716669496nt_nat(aa_fun6506752150744536624nt_nat(aa_fun5407591398502139302nt_nat(aTP_Lamm_ib(Pa),Ha),G),A3) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups4541462559716669496nt_nat(Ha,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A3),collect_int(Pa)))),groups4541462559716669496nt_nat(G,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A3),uminus1532241313380277803et_int(collect_int(Pa))))) ) ) ).

% sum.If_cases
tff(fact_7496_sum_OIf__cases,axiom,
    ! [A3: set_Code_integer,Pa: fun_Code_integer_o,Ha: fun_Code_integer_nat,G: fun_Code_integer_nat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A3)
     => ( groups7237345082560585321er_nat(aa_fun3076867170244285586er_nat(aa_fun7635586485349294009er_nat(aTP_Lamm_abz(Pa),Ha),G),A3) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups7237345082560585321er_nat(Ha,inf_in1364745209274528805nteger(A3,collect_Code_integer(Pa)))),groups7237345082560585321er_nat(G,inf_in1364745209274528805nteger(A3,uminus804700908173204444nteger(collect_Code_integer(Pa))))) ) ) ).

% sum.If_cases
tff(fact_7497_sum_OIf__cases,axiom,
    ! [A3: set_Code_integer,Pa: fun_Code_integer_o,Ha: fun_Code_integer_int,G: fun_Code_integer_int] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A3)
     => ( groups7234854612051535045er_int(aa_fun3291479149765983562er_int(aa_fun4411173778969602765er_int(aTP_Lamm_aca(Pa),Ha),G),A3) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups7234854612051535045er_int(Ha,inf_in1364745209274528805nteger(A3,collect_Code_integer(Pa)))),groups7234854612051535045er_int(G,inf_in1364745209274528805nteger(A3,uminus804700908173204444nteger(collect_Code_integer(Pa))))) ) ) ).

% sum.If_cases
tff(fact_7498_sum_OIf__cases,axiom,
    ! [A3: set_nat,Pa: fun_nat_o,Ha: fun_nat_rat,G: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( aa_set_nat_rat(groups2906978787729119204at_rat(aa_fun765758790203053448at_rat(aa_fun3638354034882855594at_rat(aTP_Lamm_acb(Pa),Ha),G)),A3) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_set_nat_rat(groups2906978787729119204at_rat(Ha),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A3),collect_nat(Pa)))),aa_set_nat_rat(groups2906978787729119204at_rat(G),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A3),uminus5710092332889474511et_nat(collect_nat(Pa))))) ) ) ).

% sum.If_cases
tff(fact_7499_sum_OIf__cases,axiom,
    ! [A3: set_nat,Pa: fun_nat_o,Ha: fun_nat_int,G: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( aa_set_nat_int(groups3539618377306564664at_int(aa_fun5565043122120104496at_int(aa_fun5140708168904641446at_int(aTP_Lamm_ia(Pa),Ha),G)),A3) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_set_nat_int(groups3539618377306564664at_int(Ha),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A3),collect_nat(Pa)))),aa_set_nat_int(groups3539618377306564664at_int(G),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A3),uminus5710092332889474511et_nat(collect_nat(Pa))))) ) ) ).

% sum.If_cases
tff(fact_7500_sum_OIf__cases,axiom,
    ! [A3: set_nat,Pa: fun_nat_o,Ha: fun_nat_nat,G: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(aTP_Lamm_ic(Pa),Ha),G)),A3) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(Ha),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A3),collect_nat(Pa)))),aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A3),uminus5710092332889474511et_nat(collect_nat(Pa))))) ) ) ).

% sum.If_cases
tff(fact_7501_sum_OIf__cases,axiom,
    ! [A3: set_int,Pa: fun_int_o,Ha: fun_int_int,G: fun_int_int] :
      ( aa_set_int_o(finite_finite_int,A3)
     => ( groups4538972089207619220nt_int(aa_fun6280345759651049192nt_int(aa_fun3693603986580398266nt_int(aTP_Lamm_hz(Pa),Ha),G),A3) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups4538972089207619220nt_int(Ha,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A3),collect_int(Pa)))),groups4538972089207619220nt_int(G,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A3),uminus1532241313380277803et_int(collect_int(Pa))))) ) ) ).

% sum.If_cases
tff(fact_7502_sum_OIf__cases,axiom,
    ! [A3: set_list_nat,Pa: fun_list_nat_o,Ha: fun_list_nat_rat,G: fun_list_nat_rat] :
      ( finite8100373058378681591st_nat(A3)
     => ( groups3760926236672600436at_rat(aa_fun1630595088783574696at_rat(aa_fun6629217458047550682at_rat(aTP_Lamm_acc(Pa),Ha),G),A3) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,groups3760926236672600436at_rat(Ha,inf_inf_set_list_nat(A3,collect_list_nat(Pa)))),groups3760926236672600436at_rat(G,inf_inf_set_list_nat(A3,uminus3195874150345416415st_nat(collect_list_nat(Pa))))) ) ) ).

% sum.If_cases
tff(fact_7503_prod_OIf__cases,axiom,
    ! [A3: set_int,Pa: fun_int_o,Ha: fun_int_Code_integer,G: fun_int_Code_integer] :
      ( aa_set_int_o(finite_finite_int,A3)
     => ( groups3827104343326376752nteger(aa_fun5304498221708291146nteger(aa_fun3938006231294128461nteger(aTP_Lamm_acd(Pa),Ha),G),A3) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,groups3827104343326376752nteger(Ha,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A3),collect_int(Pa)))),groups3827104343326376752nteger(G,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A3),uminus1532241313380277803et_int(collect_int(Pa))))) ) ) ).

% prod.If_cases
tff(fact_7504_prod_OIf__cases,axiom,
    ! [A3: set_Code_integer,Pa: fun_Code_integer_o,Ha: fun_Co8499238673658658103nteger,G: fun_Co8499238673658658103nteger] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A3)
     => ( groups3674199335183972705nteger(aa_fun4435975922079263404nteger(aa_fun2957433868855875424nteger(aTP_Lamm_ace(Pa),Ha),G),A3) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,groups3674199335183972705nteger(Ha,inf_in1364745209274528805nteger(A3,collect_Code_integer(Pa)))),groups3674199335183972705nteger(G,inf_in1364745209274528805nteger(A3,uminus804700908173204444nteger(collect_Code_integer(Pa))))) ) ) ).

% prod.If_cases
tff(fact_7505_prod_OIf__cases,axiom,
    ! [A3: set_nat,Pa: fun_nat_o,Ha: fun_nat_Code_integer,G: fun_nat_Code_integer] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( aa_set871585252387937313nteger(groups3455450783089532116nteger(aa_fun589076786913273746nteger(aa_fun3918145391673965625nteger(aTP_Lamm_acf(Pa),Ha),G)),A3) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_set871585252387937313nteger(groups3455450783089532116nteger(Ha),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A3),collect_nat(Pa)))),aa_set871585252387937313nteger(groups3455450783089532116nteger(G),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A3),uminus5710092332889474511et_nat(collect_nat(Pa))))) ) ) ).

% prod.If_cases
tff(fact_7506_prod_OIf__cases,axiom,
    ! [A3: set_int,Pa: fun_int_o,Ha: fun_int_assn,G: fun_int_assn] :
      ( aa_set_int_o(finite_finite_int,A3)
     => ( groups7882442080178216443t_assn(aa_fun919007281442365664t_assn(aa_fun7795160494388424622t_assn(aTP_Lamm_acg(Pa),Ha),G),A3) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,groups7882442080178216443t_assn(Ha,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A3),collect_int(Pa)))),groups7882442080178216443t_assn(G,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A3),uminus1532241313380277803et_int(collect_int(Pa))))) ) ) ).

% prod.If_cases
tff(fact_7507_prod_OIf__cases,axiom,
    ! [A3: set_Code_integer,Pa: fun_Code_integer_o,Ha: fun_Co4490034186337852034r_assn,G: fun_Co4490034186337852034r_assn] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A3)
     => ( groups1304777262505850412r_assn(aa_fun4119713415300323650r_assn(aa_fun8067317676786998721r_assn(aTP_Lamm_ach(Pa),Ha),G),A3) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,groups1304777262505850412r_assn(Ha,inf_in1364745209274528805nteger(A3,collect_Code_integer(Pa)))),groups1304777262505850412r_assn(G,inf_in1364745209274528805nteger(A3,uminus804700908173204444nteger(collect_Code_integer(Pa))))) ) ) ).

% prod.If_cases
tff(fact_7508_prod_OIf__cases,axiom,
    ! [A3: set_nat,Pa: fun_nat_o,Ha: fun_nat_assn,G: fun_nat_assn] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( groups6906906614972039071t_assn(aa_fun785744391571715112t_assn(aa_fun6591345337174305434t_assn(aTP_Lamm_aci(Pa),Ha),G),A3) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,groups6906906614972039071t_assn(Ha,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A3),collect_nat(Pa)))),groups6906906614972039071t_assn(G,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A3),uminus5710092332889474511et_nat(collect_nat(Pa))))) ) ) ).

% prod.If_cases
tff(fact_7509_prod_OIf__cases,axiom,
    ! [A3: set_int,Pa: fun_int_o,Ha: fun_int_rat,G: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,A3)
     => ( groups1072433553688619179nt_rat(aa_fun1481061427733998144nt_rat(aa_fun2191249852558612414nt_rat(aTP_Lamm_abx(Pa),Ha),G),A3) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,groups1072433553688619179nt_rat(Ha,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A3),collect_int(Pa)))),groups1072433553688619179nt_rat(G,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A3),uminus1532241313380277803et_int(collect_int(Pa))))) ) ) ).

% prod.If_cases
tff(fact_7510_prod_OIf__cases,axiom,
    ! [A3: set_Code_integer,Pa: fun_Code_integer_o,Ha: fun_Code_integer_rat,G: fun_Code_integer_rat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A3)
     => ( groups2555765274223993564er_rat(aa_fun8169474305930283170er_rat(aa_fun1894662059565830609er_rat(aTP_Lamm_aby(Pa),Ha),G),A3) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,groups2555765274223993564er_rat(Ha,inf_in1364745209274528805nteger(A3,collect_Code_integer(Pa)))),groups2555765274223993564er_rat(G,inf_in1364745209274528805nteger(A3,uminus804700908173204444nteger(collect_Code_integer(Pa))))) ) ) ).

% prod.If_cases
tff(fact_7511_prod_OIf__cases,axiom,
    ! [A3: set_nat,Pa: fun_nat_o,Ha: fun_nat_rat,G: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( aa_set_nat_rat(groups73079841787564623at_rat(aa_fun765758790203053448at_rat(aa_fun3638354034882855594at_rat(aTP_Lamm_acb(Pa),Ha),G)),A3) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_set_nat_rat(groups73079841787564623at_rat(Ha),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A3),collect_nat(Pa)))),aa_set_nat_rat(groups73079841787564623at_rat(G),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A3),uminus5710092332889474511et_nat(collect_nat(Pa))))) ) ) ).

% prod.If_cases
tff(fact_7512_prod_OIf__cases,axiom,
    ! [A3: set_int,Pa: fun_int_o,Ha: fun_int_nat,G: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,A3)
     => ( groups1707563613775114915nt_nat(aa_fun6506752150744536624nt_nat(aa_fun5407591398502139302nt_nat(aTP_Lamm_ib(Pa),Ha),G),A3) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,groups1707563613775114915nt_nat(Ha,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A3),collect_int(Pa)))),groups1707563613775114915nt_nat(G,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A3),uminus1532241313380277803et_int(collect_int(Pa))))) ) ) ).

% prod.If_cases
tff(fact_7513_fact__double,axiom,
    ! [N2: nat] : aa_nat_rat(semiri773545260158071498ct_rat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2)) = 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_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2))),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,one_one_rat),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)))),N2))),aa_nat_rat(semiri773545260158071498ct_rat,N2)) ).

% fact_double
tff(fact_7514_binomial__ge__n__over__k__pow__k,axiom,
    ! [K: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),N2)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_nat_rat(semiri681578069525770553at_rat,N2)),aa_nat_rat(semiri681578069525770553at_rat,K))),K)),aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,N2),K))) ) ).

% binomial_ge_n_over_k_pow_k
tff(fact_7515_double__gauss__sum__from__Suc__0,axiom,
    ! [N2: nat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),aa_set_nat_rat(groups2906978787729119204at_rat(semiri681578069525770553at_rat),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,aa_nat_nat(suc,zero_zero_nat)),N2))) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,N2)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(semiri681578069525770553at_rat,N2)),one_one_rat)) ).

% double_gauss_sum_from_Suc_0
tff(fact_7516_double__gauss__sum__from__Suc__0,axiom,
    ! [N2: nat] : times_2397367101498566445atural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one)),groups6325495683096345652atural(semiri3763490453095760265atural,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,aa_nat_nat(suc,zero_zero_nat)),N2))) = times_2397367101498566445atural(aa_nat_Code_natural(semiri3763490453095760265atural,N2),plus_p4538020629002901425atural(aa_nat_Code_natural(semiri3763490453095760265atural,N2),one_one_Code_natural)) ).

% double_gauss_sum_from_Suc_0
tff(fact_7517_double__gauss__sum__from__Suc__0,axiom,
    ! [N2: 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(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,aa_nat_nat(suc,zero_zero_nat)),N2))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,N2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,N2)),one_one_int)) ).

% double_gauss_sum_from_Suc_0
tff(fact_7518_double__gauss__sum__from__Suc__0,axiom,
    ! [N2: 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(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,aa_nat_nat(suc,zero_zero_nat)),N2))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,N2)),aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,N2)),one_one_Code_integer)) ).

% double_gauss_sum_from_Suc_0
tff(fact_7519_double__gauss__sum__from__Suc__0,axiom,
    ! [N2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_set_nat_nat(groups3542108847815614940at_nat(semiri1316708129612266289at_nat),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,aa_nat_nat(suc,zero_zero_nat)),N2))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,N2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(semiri1316708129612266289at_nat,N2)),one_one_nat)) ).

% double_gauss_sum_from_Suc_0
tff(fact_7520_choose__reduce__nat,axiom,
    ! [N2: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,N2),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,N2),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,N2),one_one_nat)),K)) ) ) ) ).

% choose_reduce_nat
tff(fact_7521_arith__series,axiom,
    ! [A: code_natural,D2: code_natural,N2: nat] : groups6325495683096345652atural(aa_Cod5320533616963873627atural(aTP_Lamm_abi(A),D2),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,zero_zero_nat),N2)) = divide5121882707175180666atural(times_2397367101498566445atural(plus_p4538020629002901425atural(aa_nat_Code_natural(semiri3763490453095760265atural,N2),one_one_Code_natural),plus_p4538020629002901425atural(times_2397367101498566445atural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one)),A),times_2397367101498566445atural(aa_nat_Code_natural(semiri3763490453095760265atural,N2),D2))),aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))) ).

% arith_series
tff(fact_7522_arith__series,axiom,
    ! [A: int,D2: int,N2: nat] : aa_set_nat_int(groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_abj(A),D2)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,zero_zero_nat),N2)) = 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,N2)),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,N2)),D2)))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% arith_series
tff(fact_7523_arith__series,axiom,
    ! [A: code_integer,D2: code_integer,N2: nat] : aa_set871585252387937313nteger(groups7501900531339628137nteger(aa_Cod1578151519121547077nteger(aTP_Lamm_abk(A),D2)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,zero_zero_nat),N2)) = aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,N2)),one_one_Code_integer)),aa_Cod1783974652533396728nteger(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,N2)),D2)))),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) ).

% arith_series
tff(fact_7524_arith__series,axiom,
    ! [A: nat,D2: nat,N2: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_abl(A),D2)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,zero_zero_nat),N2)) = 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,N2)),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,N2)),D2)))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% arith_series
tff(fact_7525_gauss__sum,axiom,
    ! [N2: nat] : groups6325495683096345652atural(semiri3763490453095760265atural,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,zero_zero_nat),N2)) = divide5121882707175180666atural(times_2397367101498566445atural(aa_nat_Code_natural(semiri3763490453095760265atural,N2),plus_p4538020629002901425atural(aa_nat_Code_natural(semiri3763490453095760265atural,N2),one_one_Code_natural)),aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))) ).

% gauss_sum
tff(fact_7526_gauss__sum,axiom,
    ! [N2: nat] : aa_set_nat_int(groups3539618377306564664at_int(semiri1314217659103216013at_int),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,zero_zero_nat),N2)) = 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,N2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,N2)),one_one_int))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% gauss_sum
tff(fact_7527_gauss__sum,axiom,
    ! [N2: nat] : aa_set871585252387937313nteger(groups7501900531339628137nteger(semiri4939895301339042750nteger),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,zero_zero_nat),N2)) = aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,N2)),aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,N2)),one_one_Code_integer))),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) ).

% gauss_sum
tff(fact_7528_gauss__sum,axiom,
    ! [N2: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(semiri1316708129612266289at_nat),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,zero_zero_nat),N2)) = 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,N2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(semiri1316708129612266289at_nat,N2)),one_one_nat))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% gauss_sum
tff(fact_7529_times__binomial__minus1__eq,axiom,
    ! [K: nat,N2: 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,N2),K)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N2),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),one_one_nat))) ) ) ).

% times_binomial_minus1_eq
tff(fact_7530_sum_Ohead__if,axiom,
    ! [G: fun_nat_Code_integer,M: nat,N2: nat] :
      aa_set871585252387937313nteger(groups7501900531339628137nteger(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M),N2)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),M),zero_z3403309356797280102nteger,aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_set871585252387937313nteger(groups7501900531339628137nteger(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,M),N2))),aa_nat_Code_integer(G,N2))) ).

% sum.head_if
tff(fact_7531_sum_Ohead__if,axiom,
    ! [G: fun_na2338080690716861136at_nat,M: nat,N2: nat] :
      groups6857163185585827899at_nat(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M),N2)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),M),zero_z1048942125864253310at_nat,aa_mul5754855110707352700at_nat(plus_p7104986032573967614at_nat(groups6857163185585827899at_nat(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,M),N2))),aa_nat2240542744625857721at_nat(G,N2))) ).

% sum.head_if
tff(fact_7532_sum_Ohead__if,axiom,
    ! [G: fun_nat_rat,M: nat,N2: nat] :
      aa_set_nat_rat(groups2906978787729119204at_rat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M),N2)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),M),zero_zero_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_set_nat_rat(groups2906978787729119204at_rat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,M),N2))),aa_nat_rat(G,N2))) ).

% sum.head_if
tff(fact_7533_sum_Ohead__if,axiom,
    ! [G: fun_nat_int,M: nat,N2: nat] :
      aa_set_nat_int(groups3539618377306564664at_int(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M),N2)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),M),zero_zero_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_set_nat_int(groups3539618377306564664at_int(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,M),N2))),aa_nat_int(G,N2))) ).

% sum.head_if
tff(fact_7534_sum_Ohead__if,axiom,
    ! [G: fun_nat_nat,M: nat,N2: nat] :
      aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M),N2)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),M),zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,M),N2))),aa_nat_nat(G,N2))) ).

% sum.head_if
tff(fact_7535_neg__eucl__rel__int__mult__2,axiom,
    ! [B: int,A: int,Q4: int,Ra2: 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,Q4),Ra2))
       => 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,Q4),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Ra2)),one_one_int))) ) ) ).

% neg_eucl_rel_int_mult_2
tff(fact_7536_prod_Ohead__if,axiom,
    ! [G: fun_nat_Code_integer,M: nat,N2: nat] :
      aa_set871585252387937313nteger(groups3455450783089532116nteger(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M),N2)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),M),one_one_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_set871585252387937313nteger(groups3455450783089532116nteger(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,M),N2))),aa_nat_Code_integer(G,N2))) ).

% prod.head_if
tff(fact_7537_prod_Ohead__if,axiom,
    ! [G: fun_nat_assn,M: nat,N2: nat] :
      groups6906906614972039071t_assn(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M),N2)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),M),one_one_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,groups6906906614972039071t_assn(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,M),N2))),aa_nat_assn(G,N2))) ).

% prod.head_if
tff(fact_7538_prod_Ohead__if,axiom,
    ! [G: fun_nat_rat,M: nat,N2: nat] :
      aa_set_nat_rat(groups73079841787564623at_rat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M),N2)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),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(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,M),N2))),aa_nat_rat(G,N2))) ).

% prod.head_if
tff(fact_7539_prod_Ohead__if,axiom,
    ! [G: fun_nat_nat,M: nat,N2: nat] :
      aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M),N2)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),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(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,M),N2))),aa_nat_nat(G,N2))) ).

% prod.head_if
tff(fact_7540_prod_Ohead__if,axiom,
    ! [G: fun_nat_int,M: nat,N2: nat] :
      aa_set_nat_int(groups705719431365010083at_int(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M),N2)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),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(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,M),N2))),aa_nat_int(G,N2))) ).

% prod.head_if
tff(fact_7541_prod__mono__strict,axiom,
    ! [A3: set_Code_integer,F: fun_Co8499238673658658103nteger,G: fun_Co8499238673658658103nteger] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A3)
     => ( ! [I3: code_integer] :
            ( aa_set5643387332196575111eger_o(member_Code_integer(I3),A3)
           => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(F,I3))
              & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(F,I3)),aa_Cod1783974652533396728nteger(G,I3)) ) )
       => ( ( A3 != bot_bo3990330152332043303nteger )
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,groups3674199335183972705nteger(F,A3)),groups3674199335183972705nteger(G,A3)) ) ) ) ).

% prod_mono_strict
tff(fact_7542_prod__mono__strict,axiom,
    ! [A3: set_o,F: fun_o_Code_integer,G: fun_o_Code_integer] :
      ( finite_finite_o(A3)
     => ( ! [I3: $o] :
            ( member_o((I3),A3)
           => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_o_Code_integer(F,(I3)))
              & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_o_Code_integer(F,(I3))),aa_o_Code_integer(G,(I3))) ) )
       => ( ( A3 != bot_bot_set_o )
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,groups7694694392188491536nteger(F,A3)),groups7694694392188491536nteger(G,A3)) ) ) ) ).

% prod_mono_strict
tff(fact_7543_prod__mono__strict,axiom,
    ! [A3: set_nat,F: fun_nat_Code_integer,G: fun_nat_Code_integer] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( ! [I3: nat] :
            ( aa_set_nat_o(member_nat(I3),A3)
           => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(F,I3))
              & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(F,I3)),aa_nat_Code_integer(G,I3)) ) )
       => ( ( A3 != bot_bot_set_nat )
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_set871585252387937313nteger(groups3455450783089532116nteger(F),A3)),aa_set871585252387937313nteger(groups3455450783089532116nteger(G),A3)) ) ) ) ).

% prod_mono_strict
tff(fact_7544_prod__mono__strict,axiom,
    ! [A3: set_int,F: fun_int_Code_integer,G: fun_int_Code_integer] :
      ( aa_set_int_o(finite_finite_int,A3)
     => ( ! [I3: int] :
            ( aa_set_int_o(member_int(I3),A3)
           => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_int_Code_integer(F,I3))
              & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_int_Code_integer(F,I3)),aa_int_Code_integer(G,I3)) ) )
       => ( ( A3 != bot_bot_set_int )
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,groups3827104343326376752nteger(F,A3)),groups3827104343326376752nteger(G,A3)) ) ) ) ).

% prod_mono_strict
tff(fact_7545_prod__mono__strict,axiom,
    ! [A3: set_Code_integer,F: fun_Code_integer_rat,G: fun_Code_integer_rat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A3)
     => ( ! [I3: code_integer] :
            ( aa_set5643387332196575111eger_o(member_Code_integer(I3),A3)
           => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_Code_integer_rat(F,I3))
              & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_Code_integer_rat(F,I3)),aa_Code_integer_rat(G,I3)) ) )
       => ( ( A3 != bot_bo3990330152332043303nteger )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,groups2555765274223993564er_rat(F,A3)),groups2555765274223993564er_rat(G,A3)) ) ) ) ).

% prod_mono_strict
tff(fact_7546_prod__mono__strict,axiom,
    ! [A3: set_o,F: fun_o_rat,G: fun_o_rat] :
      ( finite_finite_o(A3)
     => ( ! [I3: $o] :
            ( member_o((I3),A3)
           => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_o_rat(F,(I3)))
              & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_o_rat(F,(I3))),aa_o_rat(G,(I3))) ) )
       => ( ( A3 != bot_bot_set_o )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,groups2869687844427037835_o_rat(F,A3)),groups2869687844427037835_o_rat(G,A3)) ) ) ) ).

% prod_mono_strict
tff(fact_7547_prod__mono__strict,axiom,
    ! [A3: set_nat,F: fun_nat_rat,G: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( ! [I3: nat] :
            ( aa_set_nat_o(member_nat(I3),A3)
           => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(F,I3))
              & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(F,I3)),aa_nat_rat(G,I3)) ) )
       => ( ( A3 != bot_bot_set_nat )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_set_nat_rat(groups73079841787564623at_rat(F),A3)),aa_set_nat_rat(groups73079841787564623at_rat(G),A3)) ) ) ) ).

% prod_mono_strict
tff(fact_7548_prod__mono__strict,axiom,
    ! [A3: set_int,F: fun_int_rat,G: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,A3)
     => ( ! [I3: int] :
            ( aa_set_int_o(member_int(I3),A3)
           => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_int_rat(F,I3))
              & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(F,I3)),aa_int_rat(G,I3)) ) )
       => ( ( A3 != bot_bot_set_int )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,groups1072433553688619179nt_rat(F,A3)),groups1072433553688619179nt_rat(G,A3)) ) ) ) ).

% prod_mono_strict
tff(fact_7549_prod__mono__strict,axiom,
    ! [A3: set_Code_integer,F: fun_Code_integer_nat,G: fun_Code_integer_nat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A3)
     => ( ! [I3: code_integer] :
            ( aa_set5643387332196575111eger_o(member_Code_integer(I3),A3)
           => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_Code_integer_nat(F,I3))
              & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_Code_integer_nat(F,I3)),aa_Code_integer_nat(G,I3)) ) )
       => ( ( A3 != bot_bo3990330152332043303nteger )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,groups3190895334310489300er_nat(F,A3)),groups3190895334310489300er_nat(G,A3)) ) ) ) ).

% prod_mono_strict
tff(fact_7550_prod__mono__strict,axiom,
    ! [A3: set_o,F: fun_o_nat,G: fun_o_nat] :
      ( finite_finite_o(A3)
     => ( ! [I3: $o] :
            ( member_o((I3),A3)
           => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_o_nat(F,(I3)))
              & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_o_nat(F,(I3))),aa_o_nat(G,(I3))) ) )
       => ( ( A3 != bot_bot_set_o )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,groups3504817904513533571_o_nat(F,A3)),groups3504817904513533571_o_nat(G,A3)) ) ) ) ).

% prod_mono_strict
tff(fact_7551_sum__mono2,axiom,
    ! [B3: set_o,A3: set_o,F: fun_o_Code_integer] :
      ( finite_finite_o(B3)
     => ( ord_less_eq_set_o(A3,B3)
       => ( ! [B4: $o] :
              ( member_o((B4),minus_minus_set_o(B3,A3))
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_o_Code_integer(F,(B4))) )
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,groups4406642042086082107nteger(F,A3)),groups4406642042086082107nteger(F,B3)) ) ) ) ).

% sum_mono2
tff(fact_7552_sum__mono2,axiom,
    ! [B3: set_Code_integer,A3: set_Code_integer,F: fun_Co8499238673658658103nteger] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B3)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(A3),B3)
       => ( ! [B4: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(B4),minus_2355218937544613996nteger(B3,A3))
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(F,B4)) )
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,groups879477027807139574nteger(F,A3)),groups879477027807139574nteger(F,B3)) ) ) ) ).

% sum_mono2
tff(fact_7553_sum__mono2,axiom,
    ! [B3: set_nat,A3: set_nat,F: fun_nat_Code_integer] :
      ( aa_set_nat_o(finite_finite_nat,B3)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A3),B3)
       => ( ! [B4: nat] :
              ( aa_set_nat_o(member_nat(B4),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,B3),A3))
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(F,B4)) )
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_set871585252387937313nteger(groups7501900531339628137nteger(F),A3)),aa_set871585252387937313nteger(groups7501900531339628137nteger(F),B3)) ) ) ) ).

% sum_mono2
tff(fact_7554_sum__mono2,axiom,
    ! [B3: set_o,A3: set_o,F: fun_o_rat] :
      ( finite_finite_o(B3)
     => ( ord_less_eq_set_o(A3,B3)
       => ( ! [B4: $o] :
              ( member_o((B4),minus_minus_set_o(B3,A3))
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_o_rat(F,(B4))) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups7872700643590313910_o_rat(F,A3)),groups7872700643590313910_o_rat(F,B3)) ) ) ) ).

% sum_mono2
tff(fact_7555_sum__mono2,axiom,
    ! [B3: set_Code_integer,A3: set_Code_integer,F: fun_Code_integer_rat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B3)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(A3),B3)
       => ( ! [B4: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(B4),minus_2355218937544613996nteger(B3,A3))
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_Code_integer_rat(F,B4)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups6602215022474089585er_rat(F,A3)),groups6602215022474089585er_rat(F,B3)) ) ) ) ).

% sum_mono2
tff(fact_7556_sum__mono2,axiom,
    ! [B3: set_nat,A3: set_nat,F: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,B3)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A3),B3)
       => ( ! [B4: nat] :
              ( aa_set_nat_o(member_nat(B4),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,B3),A3))
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(F,B4)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_set_nat_rat(groups2906978787729119204at_rat(F),A3)),aa_set_nat_rat(groups2906978787729119204at_rat(F),B3)) ) ) ) ).

% sum_mono2
tff(fact_7557_sum__mono2,axiom,
    ! [B3: set_o,A3: set_o,F: fun_o_nat] :
      ( finite_finite_o(B3)
     => ( ord_less_eq_set_o(A3,B3)
       => ( ! [B4: $o] :
              ( member_o((B4),minus_minus_set_o(B3,A3))
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_o_nat(F,(B4))) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,groups8507830703676809646_o_nat(F,A3)),groups8507830703676809646_o_nat(F,B3)) ) ) ) ).

% sum_mono2
tff(fact_7558_sum__mono2,axiom,
    ! [B3: set_Code_integer,A3: set_Code_integer,F: fun_Code_integer_nat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B3)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(A3),B3)
       => ( ! [B4: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(B4),minus_2355218937544613996nteger(B3,A3))
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_Code_integer_nat(F,B4)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,groups7237345082560585321er_nat(F,A3)),groups7237345082560585321er_nat(F,B3)) ) ) ) ).

% sum_mono2
tff(fact_7559_sum__mono2,axiom,
    ! [B3: set_o,A3: set_o,F: fun_o_int] :
      ( finite_finite_o(B3)
     => ( ord_less_eq_set_o(A3,B3)
       => ( ! [B4: $o] :
              ( member_o((B4),minus_minus_set_o(B3,A3))
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_o_int(F,(B4))) )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,groups8505340233167759370_o_int(F,A3)),groups8505340233167759370_o_int(F,B3)) ) ) ) ).

% sum_mono2
tff(fact_7560_sum__mono2,axiom,
    ! [B3: set_Code_integer,A3: set_Code_integer,F: fun_Code_integer_int] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B3)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(A3),B3)
       => ( ! [B4: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(B4),minus_2355218937544613996nteger(B3,A3))
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_Code_integer_int(F,B4)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,groups7234854612051535045er_int(F,A3)),groups7234854612051535045er_int(F,B3)) ) ) ) ).

% sum_mono2
tff(fact_7561_binomial__altdef__nat,axiom,
    ! [K: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),N2)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,N2),K) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(semiri1408675320244567234ct_nat,N2)),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,N2),K)))) ) ) ).

% binomial_altdef_nat
tff(fact_7562_sum_Ounion__inter__neutral,axiom,
    ! [A3: set_int,B3: set_int,G: fun_int_Code_integer] :
      ( aa_set_int_o(finite_finite_int,A3)
     => ( aa_set_int_o(finite_finite_int,B3)
       => ( ! [X3: int] :
              ( aa_set_int_o(member_int(X3),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A3),B3))
             => ( aa_int_Code_integer(G,X3) = zero_z3403309356797280102nteger ) )
         => ( groups7873554091576472773nteger(G,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A3),B3)) = aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(groups7873554091576472773nteger(G,A3)),groups7873554091576472773nteger(G,B3)) ) ) ) ) ).

% sum.union_inter_neutral
tff(fact_7563_sum_Ounion__inter__neutral,axiom,
    ! [A3: set_Code_integer,B3: set_Code_integer,G: fun_Co8499238673658658103nteger] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A3)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B3)
       => ( ! [X3: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X3),inf_in1364745209274528805nteger(A3,B3))
             => ( aa_Cod1783974652533396728nteger(G,X3) = zero_z3403309356797280102nteger ) )
         => ( groups879477027807139574nteger(G,sup_su848401254843788991nteger(A3,B3)) = aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(groups879477027807139574nteger(G,A3)),groups879477027807139574nteger(G,B3)) ) ) ) ) ).

% sum.union_inter_neutral
tff(fact_7564_sum_Ounion__inter__neutral,axiom,
    ! [A3: set_int,B3: set_int,G: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,A3)
     => ( aa_set_int_o(finite_finite_int,B3)
       => ( ! [X3: int] :
              ( aa_set_int_o(member_int(X3),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A3),B3))
             => ( aa_int_rat(G,X3) = zero_zero_rat ) )
         => ( groups3906332499630173760nt_rat(G,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A3),B3)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,groups3906332499630173760nt_rat(G,A3)),groups3906332499630173760nt_rat(G,B3)) ) ) ) ) ).

% sum.union_inter_neutral
tff(fact_7565_sum_Ounion__inter__neutral,axiom,
    ! [A3: set_Code_integer,B3: set_Code_integer,G: fun_Code_integer_rat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A3)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B3)
       => ( ! [X3: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X3),inf_in1364745209274528805nteger(A3,B3))
             => ( aa_Code_integer_rat(G,X3) = zero_zero_rat ) )
         => ( groups6602215022474089585er_rat(G,sup_su848401254843788991nteger(A3,B3)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,groups6602215022474089585er_rat(G,A3)),groups6602215022474089585er_rat(G,B3)) ) ) ) ) ).

% sum.union_inter_neutral
tff(fact_7566_sum_Ounion__inter__neutral,axiom,
    ! [A3: set_int,B3: set_int,G: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,A3)
     => ( aa_set_int_o(finite_finite_int,B3)
       => ( ! [X3: int] :
              ( aa_set_int_o(member_int(X3),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A3),B3))
             => ( aa_int_nat(G,X3) = zero_zero_nat ) )
         => ( groups4541462559716669496nt_nat(G,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A3),B3)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups4541462559716669496nt_nat(G,A3)),groups4541462559716669496nt_nat(G,B3)) ) ) ) ) ).

% sum.union_inter_neutral
tff(fact_7567_sum_Ounion__inter__neutral,axiom,
    ! [A3: set_Code_integer,B3: set_Code_integer,G: fun_Code_integer_nat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A3)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B3)
       => ( ! [X3: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X3),inf_in1364745209274528805nteger(A3,B3))
             => ( aa_Code_integer_nat(G,X3) = zero_zero_nat ) )
         => ( groups7237345082560585321er_nat(G,sup_su848401254843788991nteger(A3,B3)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups7237345082560585321er_nat(G,A3)),groups7237345082560585321er_nat(G,B3)) ) ) ) ) ).

% sum.union_inter_neutral
tff(fact_7568_sum_Ounion__inter__neutral,axiom,
    ! [A3: set_Code_integer,B3: set_Code_integer,G: fun_Code_integer_int] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A3)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B3)
       => ( ! [X3: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X3),inf_in1364745209274528805nteger(A3,B3))
             => ( aa_Code_integer_int(G,X3) = zero_zero_int ) )
         => ( groups7234854612051535045er_int(G,sup_su848401254843788991nteger(A3,B3)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups7234854612051535045er_int(G,A3)),groups7234854612051535045er_int(G,B3)) ) ) ) ) ).

% sum.union_inter_neutral
tff(fact_7569_sum_Ounion__inter__neutral,axiom,
    ! [A3: set_nat,B3: set_nat,G: fun_nat_Code_integer] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( aa_set_nat_o(finite_finite_nat,B3)
       => ( ! [X3: nat] :
              ( aa_set_nat_o(member_nat(X3),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A3),B3))
             => ( aa_nat_Code_integer(G,X3) = zero_z3403309356797280102nteger ) )
         => ( aa_set871585252387937313nteger(groups7501900531339628137nteger(G),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A3),B3)) = aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_set871585252387937313nteger(groups7501900531339628137nteger(G),A3)),aa_set871585252387937313nteger(groups7501900531339628137nteger(G),B3)) ) ) ) ) ).

% sum.union_inter_neutral
tff(fact_7570_sum_Ounion__inter__neutral,axiom,
    ! [A3: set_nat,B3: set_nat,G: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( aa_set_nat_o(finite_finite_nat,B3)
       => ( ! [X3: nat] :
              ( aa_set_nat_o(member_nat(X3),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A3),B3))
             => ( aa_nat_rat(G,X3) = zero_zero_rat ) )
         => ( aa_set_nat_rat(groups2906978787729119204at_rat(G),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A3),B3)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_set_nat_rat(groups2906978787729119204at_rat(G),A3)),aa_set_nat_rat(groups2906978787729119204at_rat(G),B3)) ) ) ) ) ).

% sum.union_inter_neutral
tff(fact_7571_sum_Ounion__inter__neutral,axiom,
    ! [A3: set_nat,B3: set_nat,G: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( aa_set_nat_o(finite_finite_nat,B3)
       => ( ! [X3: nat] :
              ( aa_set_nat_o(member_nat(X3),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A3),B3))
             => ( aa_nat_int(G,X3) = zero_zero_int ) )
         => ( aa_set_nat_int(groups3539618377306564664at_int(G),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A3),B3)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_set_nat_int(groups3539618377306564664at_int(G),A3)),aa_set_nat_int(groups3539618377306564664at_int(G),B3)) ) ) ) ) ).

% sum.union_inter_neutral
tff(fact_7572_sum__Un,axiom,
    ! [A3: set_int,B3: set_int,F: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,A3)
     => ( aa_set_int_o(finite_finite_int,B3)
       => ( groups3906332499630173760nt_rat(F,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A3),B3)) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,groups3906332499630173760nt_rat(F,A3)),groups3906332499630173760nt_rat(F,B3))),groups3906332499630173760nt_rat(F,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A3),B3))) ) ) ) ).

% sum_Un
tff(fact_7573_sum__Un,axiom,
    ! [A3: set_Code_integer,B3: set_Code_integer,F: fun_Code_integer_rat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A3)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B3)
       => ( groups6602215022474089585er_rat(F,sup_su848401254843788991nteger(A3,B3)) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,groups6602215022474089585er_rat(F,A3)),groups6602215022474089585er_rat(F,B3))),groups6602215022474089585er_rat(F,inf_in1364745209274528805nteger(A3,B3))) ) ) ) ).

% sum_Un
tff(fact_7574_sum__Un,axiom,
    ! [A3: set_nat,B3: set_nat,F: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( aa_set_nat_o(finite_finite_nat,B3)
       => ( aa_set_nat_rat(groups2906978787729119204at_rat(F),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A3),B3)) = 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),A3)),aa_set_nat_rat(groups2906978787729119204at_rat(F),B3))),aa_set_nat_rat(groups2906978787729119204at_rat(F),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A3),B3))) ) ) ) ).

% sum_Un
tff(fact_7575_sum__Un,axiom,
    ! [A3: set_Code_integer,B3: set_Code_integer,F: fun_Code_integer_int] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A3)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B3)
       => ( groups7234854612051535045er_int(F,sup_su848401254843788991nteger(A3,B3)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,groups7234854612051535045er_int(F,A3)),groups7234854612051535045er_int(F,B3))),groups7234854612051535045er_int(F,inf_in1364745209274528805nteger(A3,B3))) ) ) ) ).

% sum_Un
tff(fact_7576_sum__Un,axiom,
    ! [A3: set_nat,B3: set_nat,F: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( aa_set_nat_o(finite_finite_nat,B3)
       => ( aa_set_nat_int(groups3539618377306564664at_int(F),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A3),B3)) = 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),A3)),aa_set_nat_int(groups3539618377306564664at_int(F),B3))),aa_set_nat_int(groups3539618377306564664at_int(F),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A3),B3))) ) ) ) ).

% sum_Un
tff(fact_7577_sum__Un,axiom,
    ! [A3: set_int,B3: set_int,F: fun_int_int] :
      ( aa_set_int_o(finite_finite_int,A3)
     => ( aa_set_int_o(finite_finite_int,B3)
       => ( groups4538972089207619220nt_int(F,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A3),B3)) = 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,A3)),groups4538972089207619220nt_int(F,B3))),groups4538972089207619220nt_int(F,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A3),B3))) ) ) ) ).

% sum_Un
tff(fact_7578_sum__Un,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat,F: fun_Pr8317260874932935276at_rat] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A3)
     => ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,B3)
       => ( groups342789780944988191at_rat(F,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,A3),B3)) = 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,A3)),groups342789780944988191at_rat(F,B3))),groups342789780944988191at_rat(F,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A3),B3))) ) ) ) ).

% sum_Un
tff(fact_7579_sum__Un,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat,F: fun_Pr7647638413817135296at_int] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A3)
     => ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,B3)
       => ( groups975429370522433651at_int(F,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,A3),B3)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,groups975429370522433651at_int(F,A3)),groups975429370522433651at_int(F,B3))),groups975429370522433651at_int(F,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A3),B3))) ) ) ) ).

% sum_Un
tff(fact_7580_sum__Un,axiom,
    ! [A3: set_Pr8693737435421807431at_nat,B3: set_Pr8693737435421807431at_nat,F: fun_Pr8586684092290301266at_rat] :
      ( finite4392333629123659920at_nat(A3)
     => ( finite4392333629123659920at_nat(B3)
       => ( groups1265588324298845189at_rat(F,aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,A3),B3)) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,groups1265588324298845189at_rat(F,A3)),groups1265588324298845189at_rat(F,B3))),groups1265588324298845189at_rat(F,aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(inf_in4302113700860409141at_nat,A3),B3))) ) ) ) ).

% sum_Un
tff(fact_7581_sum__Un,axiom,
    ! [A3: set_Pr8693737435421807431at_nat,B3: set_Pr8693737435421807431at_nat,F: fun_Pr7917061631174501286at_int] :
      ( finite4392333629123659920at_nat(A3)
     => ( finite4392333629123659920at_nat(B3)
       => ( groups1898227913876290649at_int(F,aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,A3),B3)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,groups1898227913876290649at_int(F,A3)),groups1898227913876290649at_int(F,B3))),groups1898227913876290649at_int(F,aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(inf_in4302113700860409141at_nat,A3),B3))) ) ) ) ).

% sum_Un
tff(fact_7582_sum_Ounion__disjoint,axiom,
    ! [A3: set_Code_integer,B3: set_Code_integer,G: fun_Code_integer_rat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A3)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B3)
       => ( ( inf_in1364745209274528805nteger(A3,B3) = bot_bo3990330152332043303nteger )
         => ( groups6602215022474089585er_rat(G,sup_su848401254843788991nteger(A3,B3)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,groups6602215022474089585er_rat(G,A3)),groups6602215022474089585er_rat(G,B3)) ) ) ) ) ).

% sum.union_disjoint
tff(fact_7583_sum_Ounion__disjoint,axiom,
    ! [A3: set_Code_integer,B3: set_Code_integer,G: fun_Code_integer_nat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A3)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B3)
       => ( ( inf_in1364745209274528805nteger(A3,B3) = bot_bo3990330152332043303nteger )
         => ( groups7237345082560585321er_nat(G,sup_su848401254843788991nteger(A3,B3)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups7237345082560585321er_nat(G,A3)),groups7237345082560585321er_nat(G,B3)) ) ) ) ) ).

% sum.union_disjoint
tff(fact_7584_sum_Ounion__disjoint,axiom,
    ! [A3: set_Code_integer,B3: set_Code_integer,G: fun_Code_integer_int] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A3)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B3)
       => ( ( inf_in1364745209274528805nteger(A3,B3) = bot_bo3990330152332043303nteger )
         => ( groups7234854612051535045er_int(G,sup_su848401254843788991nteger(A3,B3)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups7234854612051535045er_int(G,A3)),groups7234854612051535045er_int(G,B3)) ) ) ) ) ).

% sum.union_disjoint
tff(fact_7585_sum_Ounion__disjoint,axiom,
    ! [A3: set_o,B3: set_o,G: fun_o_rat] :
      ( finite_finite_o(A3)
     => ( finite_finite_o(B3)
       => ( ( inf_inf_set_o(A3,B3) = bot_bot_set_o )
         => ( groups7872700643590313910_o_rat(G,aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,A3),B3)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,groups7872700643590313910_o_rat(G,A3)),groups7872700643590313910_o_rat(G,B3)) ) ) ) ) ).

% sum.union_disjoint
tff(fact_7586_sum_Ounion__disjoint,axiom,
    ! [A3: set_o,B3: set_o,G: fun_o_nat] :
      ( finite_finite_o(A3)
     => ( finite_finite_o(B3)
       => ( ( inf_inf_set_o(A3,B3) = bot_bot_set_o )
         => ( groups8507830703676809646_o_nat(G,aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,A3),B3)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups8507830703676809646_o_nat(G,A3)),groups8507830703676809646_o_nat(G,B3)) ) ) ) ) ).

% sum.union_disjoint
tff(fact_7587_sum_Ounion__disjoint,axiom,
    ! [A3: set_o,B3: set_o,G: fun_o_int] :
      ( finite_finite_o(A3)
     => ( finite_finite_o(B3)
       => ( ( inf_inf_set_o(A3,B3) = bot_bot_set_o )
         => ( groups8505340233167759370_o_int(G,aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,A3),B3)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups8505340233167759370_o_int(G,A3)),groups8505340233167759370_o_int(G,B3)) ) ) ) ) ).

% sum.union_disjoint
tff(fact_7588_sum_Ounion__disjoint,axiom,
    ! [A3: set_nat,B3: set_nat,G: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( aa_set_nat_o(finite_finite_nat,B3)
       => ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A3),B3) = bot_bot_set_nat )
         => ( aa_set_nat_rat(groups2906978787729119204at_rat(G),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A3),B3)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_set_nat_rat(groups2906978787729119204at_rat(G),A3)),aa_set_nat_rat(groups2906978787729119204at_rat(G),B3)) ) ) ) ) ).

% sum.union_disjoint
tff(fact_7589_sum_Ounion__disjoint,axiom,
    ! [A3: set_nat,B3: set_nat,G: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( aa_set_nat_o(finite_finite_nat,B3)
       => ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A3),B3) = bot_bot_set_nat )
         => ( aa_set_nat_int(groups3539618377306564664at_int(G),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A3),B3)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_set_nat_int(groups3539618377306564664at_int(G),A3)),aa_set_nat_int(groups3539618377306564664at_int(G),B3)) ) ) ) ) ).

% sum.union_disjoint
tff(fact_7590_sum_Ounion__disjoint,axiom,
    ! [A3: set_int,B3: set_int,G: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,A3)
     => ( aa_set_int_o(finite_finite_int,B3)
       => ( ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A3),B3) = bot_bot_set_int )
         => ( groups3906332499630173760nt_rat(G,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A3),B3)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,groups3906332499630173760nt_rat(G,A3)),groups3906332499630173760nt_rat(G,B3)) ) ) ) ) ).

% sum.union_disjoint
tff(fact_7591_sum_Ounion__disjoint,axiom,
    ! [A3: set_int,B3: set_int,G: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,A3)
     => ( aa_set_int_o(finite_finite_int,B3)
       => ( ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A3),B3) = bot_bot_set_int )
         => ( groups4541462559716669496nt_nat(G,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A3),B3)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups4541462559716669496nt_nat(G,A3)),groups4541462559716669496nt_nat(G,B3)) ) ) ) ) ).

% sum.union_disjoint
tff(fact_7592_prod_Ounion__inter__neutral,axiom,
    ! [A3: set_int,B3: set_int,G: fun_int_Code_integer] :
      ( aa_set_int_o(finite_finite_int,A3)
     => ( aa_set_int_o(finite_finite_int,B3)
       => ( ! [X3: int] :
              ( aa_set_int_o(member_int(X3),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A3),B3))
             => ( aa_int_Code_integer(G,X3) = one_one_Code_integer ) )
         => ( groups3827104343326376752nteger(G,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A3),B3)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,groups3827104343326376752nteger(G,A3)),groups3827104343326376752nteger(G,B3)) ) ) ) ) ).

% prod.union_inter_neutral
tff(fact_7593_prod_Ounion__inter__neutral,axiom,
    ! [A3: set_Code_integer,B3: set_Code_integer,G: fun_Co8499238673658658103nteger] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A3)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B3)
       => ( ! [X3: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X3),inf_in1364745209274528805nteger(A3,B3))
             => ( aa_Cod1783974652533396728nteger(G,X3) = one_one_Code_integer ) )
         => ( groups3674199335183972705nteger(G,sup_su848401254843788991nteger(A3,B3)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,groups3674199335183972705nteger(G,A3)),groups3674199335183972705nteger(G,B3)) ) ) ) ) ).

% prod.union_inter_neutral
tff(fact_7594_prod_Ounion__inter__neutral,axiom,
    ! [A3: set_nat,B3: set_nat,G: fun_nat_Code_integer] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( aa_set_nat_o(finite_finite_nat,B3)
       => ( ! [X3: nat] :
              ( aa_set_nat_o(member_nat(X3),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A3),B3))
             => ( aa_nat_Code_integer(G,X3) = one_one_Code_integer ) )
         => ( aa_set871585252387937313nteger(groups3455450783089532116nteger(G),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A3),B3)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_set871585252387937313nteger(groups3455450783089532116nteger(G),A3)),aa_set871585252387937313nteger(groups3455450783089532116nteger(G),B3)) ) ) ) ) ).

% prod.union_inter_neutral
tff(fact_7595_prod_Ounion__inter__neutral,axiom,
    ! [A3: set_int,B3: set_int,G: fun_int_assn] :
      ( aa_set_int_o(finite_finite_int,A3)
     => ( aa_set_int_o(finite_finite_int,B3)
       => ( ! [X3: int] :
              ( aa_set_int_o(member_int(X3),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A3),B3))
             => ( aa_int_assn(G,X3) = one_one_assn ) )
         => ( groups7882442080178216443t_assn(G,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A3),B3)) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,groups7882442080178216443t_assn(G,A3)),groups7882442080178216443t_assn(G,B3)) ) ) ) ) ).

% prod.union_inter_neutral
tff(fact_7596_prod_Ounion__inter__neutral,axiom,
    ! [A3: set_Code_integer,B3: set_Code_integer,G: fun_Co4490034186337852034r_assn] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A3)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B3)
       => ( ! [X3: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X3),inf_in1364745209274528805nteger(A3,B3))
             => ( aa_Code_integer_assn(G,X3) = one_one_assn ) )
         => ( groups1304777262505850412r_assn(G,sup_su848401254843788991nteger(A3,B3)) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,groups1304777262505850412r_assn(G,A3)),groups1304777262505850412r_assn(G,B3)) ) ) ) ) ).

% prod.union_inter_neutral
tff(fact_7597_prod_Ounion__inter__neutral,axiom,
    ! [A3: set_nat,B3: set_nat,G: fun_nat_assn] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( aa_set_nat_o(finite_finite_nat,B3)
       => ( ! [X3: nat] :
              ( aa_set_nat_o(member_nat(X3),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A3),B3))
             => ( aa_nat_assn(G,X3) = one_one_assn ) )
         => ( groups6906906614972039071t_assn(G,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A3),B3)) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,groups6906906614972039071t_assn(G,A3)),groups6906906614972039071t_assn(G,B3)) ) ) ) ) ).

% prod.union_inter_neutral
tff(fact_7598_prod_Ounion__inter__neutral,axiom,
    ! [A3: set_int,B3: set_int,G: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,A3)
     => ( aa_set_int_o(finite_finite_int,B3)
       => ( ! [X3: int] :
              ( aa_set_int_o(member_int(X3),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A3),B3))
             => ( aa_int_rat(G,X3) = one_one_rat ) )
         => ( groups1072433553688619179nt_rat(G,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A3),B3)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,groups1072433553688619179nt_rat(G,A3)),groups1072433553688619179nt_rat(G,B3)) ) ) ) ) ).

% prod.union_inter_neutral
tff(fact_7599_prod_Ounion__inter__neutral,axiom,
    ! [A3: set_Code_integer,B3: set_Code_integer,G: fun_Code_integer_rat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A3)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B3)
       => ( ! [X3: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X3),inf_in1364745209274528805nteger(A3,B3))
             => ( aa_Code_integer_rat(G,X3) = one_one_rat ) )
         => ( groups2555765274223993564er_rat(G,sup_su848401254843788991nteger(A3,B3)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,groups2555765274223993564er_rat(G,A3)),groups2555765274223993564er_rat(G,B3)) ) ) ) ) ).

% prod.union_inter_neutral
tff(fact_7600_prod_Ounion__inter__neutral,axiom,
    ! [A3: set_nat,B3: set_nat,G: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( aa_set_nat_o(finite_finite_nat,B3)
       => ( ! [X3: nat] :
              ( aa_set_nat_o(member_nat(X3),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A3),B3))
             => ( aa_nat_rat(G,X3) = one_one_rat ) )
         => ( aa_set_nat_rat(groups73079841787564623at_rat(G),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A3),B3)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_set_nat_rat(groups73079841787564623at_rat(G),A3)),aa_set_nat_rat(groups73079841787564623at_rat(G),B3)) ) ) ) ) ).

% prod.union_inter_neutral
tff(fact_7601_prod_Ounion__inter__neutral,axiom,
    ! [A3: set_int,B3: set_int,G: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,A3)
     => ( aa_set_int_o(finite_finite_int,B3)
       => ( ! [X3: int] :
              ( aa_set_int_o(member_int(X3),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A3),B3))
             => ( aa_int_nat(G,X3) = one_one_nat ) )
         => ( groups1707563613775114915nt_nat(G,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A3),B3)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,groups1707563613775114915nt_nat(G,A3)),groups1707563613775114915nt_nat(G,B3)) ) ) ) ) ).

% prod.union_inter_neutral
tff(fact_7602_prod_Ounion__disjoint,axiom,
    ! [A3: set_Code_integer,B3: set_Code_integer,G: fun_Co8499238673658658103nteger] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A3)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B3)
       => ( ( inf_in1364745209274528805nteger(A3,B3) = bot_bo3990330152332043303nteger )
         => ( groups3674199335183972705nteger(G,sup_su848401254843788991nteger(A3,B3)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,groups3674199335183972705nteger(G,A3)),groups3674199335183972705nteger(G,B3)) ) ) ) ) ).

% prod.union_disjoint
tff(fact_7603_prod_Ounion__disjoint,axiom,
    ! [A3: set_o,B3: set_o,G: fun_o_Code_integer] :
      ( finite_finite_o(A3)
     => ( finite_finite_o(B3)
       => ( ( inf_inf_set_o(A3,B3) = bot_bot_set_o )
         => ( groups7694694392188491536nteger(G,aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,A3),B3)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,groups7694694392188491536nteger(G,A3)),groups7694694392188491536nteger(G,B3)) ) ) ) ) ).

% prod.union_disjoint
tff(fact_7604_prod_Ounion__disjoint,axiom,
    ! [A3: set_nat,B3: set_nat,G: fun_nat_Code_integer] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( aa_set_nat_o(finite_finite_nat,B3)
       => ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A3),B3) = bot_bot_set_nat )
         => ( aa_set871585252387937313nteger(groups3455450783089532116nteger(G),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A3),B3)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_set871585252387937313nteger(groups3455450783089532116nteger(G),A3)),aa_set871585252387937313nteger(groups3455450783089532116nteger(G),B3)) ) ) ) ) ).

% prod.union_disjoint
tff(fact_7605_prod_Ounion__disjoint,axiom,
    ! [A3: set_int,B3: set_int,G: fun_int_Code_integer] :
      ( aa_set_int_o(finite_finite_int,A3)
     => ( aa_set_int_o(finite_finite_int,B3)
       => ( ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A3),B3) = bot_bot_set_int )
         => ( groups3827104343326376752nteger(G,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A3),B3)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,groups3827104343326376752nteger(G,A3)),groups3827104343326376752nteger(G,B3)) ) ) ) ) ).

% prod.union_disjoint
tff(fact_7606_prod_Ounion__disjoint,axiom,
    ! [A3: set_Code_integer,B3: set_Code_integer,G: fun_Co4490034186337852034r_assn] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A3)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B3)
       => ( ( inf_in1364745209274528805nteger(A3,B3) = bot_bo3990330152332043303nteger )
         => ( groups1304777262505850412r_assn(G,sup_su848401254843788991nteger(A3,B3)) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,groups1304777262505850412r_assn(G,A3)),groups1304777262505850412r_assn(G,B3)) ) ) ) ) ).

% prod.union_disjoint
tff(fact_7607_prod_Ounion__disjoint,axiom,
    ! [A3: set_o,B3: set_o,G: fun_o_assn] :
      ( finite_finite_o(A3)
     => ( finite_finite_o(B3)
       => ( ( inf_inf_set_o(A3,B3) = bot_bot_set_o )
         => ( groups5301882518646026715o_assn(G,aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,A3),B3)) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,groups5301882518646026715o_assn(G,A3)),groups5301882518646026715o_assn(G,B3)) ) ) ) ) ).

% prod.union_disjoint
tff(fact_7608_prod_Ounion__disjoint,axiom,
    ! [A3: set_nat,B3: set_nat,G: fun_nat_assn] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( aa_set_nat_o(finite_finite_nat,B3)
       => ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A3),B3) = bot_bot_set_nat )
         => ( groups6906906614972039071t_assn(G,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A3),B3)) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,groups6906906614972039071t_assn(G,A3)),groups6906906614972039071t_assn(G,B3)) ) ) ) ) ).

% prod.union_disjoint
tff(fact_7609_prod_Ounion__disjoint,axiom,
    ! [A3: set_int,B3: set_int,G: fun_int_assn] :
      ( aa_set_int_o(finite_finite_int,A3)
     => ( aa_set_int_o(finite_finite_int,B3)
       => ( ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A3),B3) = bot_bot_set_int )
         => ( groups7882442080178216443t_assn(G,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A3),B3)) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,groups7882442080178216443t_assn(G,A3)),groups7882442080178216443t_assn(G,B3)) ) ) ) ) ).

% prod.union_disjoint
tff(fact_7610_prod_Ounion__disjoint,axiom,
    ! [A3: set_Code_integer,B3: set_Code_integer,G: fun_Code_integer_rat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A3)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B3)
       => ( ( inf_in1364745209274528805nteger(A3,B3) = bot_bo3990330152332043303nteger )
         => ( groups2555765274223993564er_rat(G,sup_su848401254843788991nteger(A3,B3)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,groups2555765274223993564er_rat(G,A3)),groups2555765274223993564er_rat(G,B3)) ) ) ) ) ).

% prod.union_disjoint
tff(fact_7611_prod_Ounion__disjoint,axiom,
    ! [A3: set_o,B3: set_o,G: fun_o_rat] :
      ( finite_finite_o(A3)
     => ( finite_finite_o(B3)
       => ( ( inf_inf_set_o(A3,B3) = bot_bot_set_o )
         => ( groups2869687844427037835_o_rat(G,aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,A3),B3)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,groups2869687844427037835_o_rat(G,A3)),groups2869687844427037835_o_rat(G,B3)) ) ) ) ) ).

% prod.union_disjoint
tff(fact_7612_sum__Un2,axiom,
    ! [A3: set_int,B3: 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,A3),B3))
     => ( groups3906332499630173760nt_rat(F,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A3),B3)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,groups3906332499630173760nt_rat(F,aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A3),B3))),groups3906332499630173760nt_rat(F,aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,B3),A3)))),groups3906332499630173760nt_rat(F,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A3),B3))) ) ) ).

% sum_Un2
tff(fact_7613_sum__Un2,axiom,
    ! [A3: set_Code_integer,B3: set_Code_integer,F: fun_Code_integer_rat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,sup_su848401254843788991nteger(A3,B3))
     => ( groups6602215022474089585er_rat(F,sup_su848401254843788991nteger(A3,B3)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,groups6602215022474089585er_rat(F,minus_2355218937544613996nteger(A3,B3))),groups6602215022474089585er_rat(F,minus_2355218937544613996nteger(B3,A3)))),groups6602215022474089585er_rat(F,inf_in1364745209274528805nteger(A3,B3))) ) ) ).

% sum_Un2
tff(fact_7614_sum__Un2,axiom,
    ! [A3: set_int,B3: 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,A3),B3))
     => ( groups4541462559716669496nt_nat(F,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A3),B3)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups4541462559716669496nt_nat(F,aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A3),B3))),groups4541462559716669496nt_nat(F,aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,B3),A3)))),groups4541462559716669496nt_nat(F,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A3),B3))) ) ) ).

% sum_Un2
tff(fact_7615_sum__Un2,axiom,
    ! [A3: set_Code_integer,B3: set_Code_integer,F: fun_Code_integer_nat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,sup_su848401254843788991nteger(A3,B3))
     => ( groups7237345082560585321er_nat(F,sup_su848401254843788991nteger(A3,B3)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups7237345082560585321er_nat(F,minus_2355218937544613996nteger(A3,B3))),groups7237345082560585321er_nat(F,minus_2355218937544613996nteger(B3,A3)))),groups7237345082560585321er_nat(F,inf_in1364745209274528805nteger(A3,B3))) ) ) ).

% sum_Un2
tff(fact_7616_sum__Un2,axiom,
    ! [A3: set_Code_integer,B3: set_Code_integer,F: fun_Code_integer_int] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,sup_su848401254843788991nteger(A3,B3))
     => ( groups7234854612051535045er_int(F,sup_su848401254843788991nteger(A3,B3)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,groups7234854612051535045er_int(F,minus_2355218937544613996nteger(A3,B3))),groups7234854612051535045er_int(F,minus_2355218937544613996nteger(B3,A3)))),groups7234854612051535045er_int(F,inf_in1364745209274528805nteger(A3,B3))) ) ) ).

% sum_Un2
tff(fact_7617_sum__Un2,axiom,
    ! [A3: set_nat,B3: 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,A3),B3))
     => ( aa_set_nat_rat(groups2906978787729119204at_rat(F),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A3),B3)) = 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,A3),B3))),aa_set_nat_rat(groups2906978787729119204at_rat(F),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,B3),A3)))),aa_set_nat_rat(groups2906978787729119204at_rat(F),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A3),B3))) ) ) ).

% sum_Un2
tff(fact_7618_sum__Un2,axiom,
    ! [A3: set_nat,B3: 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,A3),B3))
     => ( aa_set_nat_int(groups3539618377306564664at_int(F),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A3),B3)) = 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,A3),B3))),aa_set_nat_int(groups3539618377306564664at_int(F),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,B3),A3)))),aa_set_nat_int(groups3539618377306564664at_int(F),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A3),B3))) ) ) ).

% sum_Un2
tff(fact_7619_sum__Un2,axiom,
    ! [A3: set_nat,B3: 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,A3),B3))
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(F),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A3),B3)) = 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,A3),B3))),aa_set_nat_nat(groups3542108847815614940at_nat(F),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,B3),A3)))),aa_set_nat_nat(groups3542108847815614940at_nat(F),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A3),B3))) ) ) ).

% sum_Un2
tff(fact_7620_sum__Un2,axiom,
    ! [A3: set_int,B3: 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,A3),B3))
     => ( groups4538972089207619220nt_int(F,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A3),B3)) = 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,aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A3),B3))),groups4538972089207619220nt_int(F,aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,B3),A3)))),groups4538972089207619220nt_int(F,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A3),B3))) ) ) ).

% sum_Un2
tff(fact_7621_sum__Un2,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat,F: fun_Pr8317260874932935276at_rat] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,A3),B3))
     => ( groups342789780944988191at_rat(F,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,A3),B3)) = 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,A3),B3))),groups342789780944988191at_rat(F,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,B3),A3)))),groups342789780944988191at_rat(F,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A3),B3))) ) ) ).

% sum_Un2
tff(fact_7622_sum_Ounion__diff2,axiom,
    ! [A3: set_int,B3: set_int,G: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,A3)
     => ( aa_set_int_o(finite_finite_int,B3)
       => ( groups3906332499630173760nt_rat(G,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A3),B3)) = 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,aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A3),B3))),groups3906332499630173760nt_rat(G,aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,B3),A3)))),groups3906332499630173760nt_rat(G,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A3),B3))) ) ) ) ).

% sum.union_diff2
tff(fact_7623_sum_Ounion__diff2,axiom,
    ! [A3: set_Code_integer,B3: set_Code_integer,G: fun_Code_integer_rat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A3)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B3)
       => ( groups6602215022474089585er_rat(G,sup_su848401254843788991nteger(A3,B3)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,groups6602215022474089585er_rat(G,minus_2355218937544613996nteger(A3,B3))),groups6602215022474089585er_rat(G,minus_2355218937544613996nteger(B3,A3)))),groups6602215022474089585er_rat(G,inf_in1364745209274528805nteger(A3,B3))) ) ) ) ).

% sum.union_diff2
tff(fact_7624_sum_Ounion__diff2,axiom,
    ! [A3: set_int,B3: set_int,G: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,A3)
     => ( aa_set_int_o(finite_finite_int,B3)
       => ( groups4541462559716669496nt_nat(G,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A3),B3)) = 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,aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A3),B3))),groups4541462559716669496nt_nat(G,aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,B3),A3)))),groups4541462559716669496nt_nat(G,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A3),B3))) ) ) ) ).

% sum.union_diff2
tff(fact_7625_sum_Ounion__diff2,axiom,
    ! [A3: set_Code_integer,B3: set_Code_integer,G: fun_Code_integer_nat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A3)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B3)
       => ( groups7237345082560585321er_nat(G,sup_su848401254843788991nteger(A3,B3)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups7237345082560585321er_nat(G,minus_2355218937544613996nteger(A3,B3))),groups7237345082560585321er_nat(G,minus_2355218937544613996nteger(B3,A3)))),groups7237345082560585321er_nat(G,inf_in1364745209274528805nteger(A3,B3))) ) ) ) ).

% sum.union_diff2
tff(fact_7626_sum_Ounion__diff2,axiom,
    ! [A3: set_Code_integer,B3: set_Code_integer,G: fun_Code_integer_int] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A3)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B3)
       => ( groups7234854612051535045er_int(G,sup_su848401254843788991nteger(A3,B3)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,groups7234854612051535045er_int(G,minus_2355218937544613996nteger(A3,B3))),groups7234854612051535045er_int(G,minus_2355218937544613996nteger(B3,A3)))),groups7234854612051535045er_int(G,inf_in1364745209274528805nteger(A3,B3))) ) ) ) ).

% sum.union_diff2
tff(fact_7627_sum_Ounion__diff2,axiom,
    ! [A3: set_nat,B3: set_nat,G: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( aa_set_nat_o(finite_finite_nat,B3)
       => ( aa_set_nat_rat(groups2906978787729119204at_rat(G),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A3),B3)) = 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,A3),B3))),aa_set_nat_rat(groups2906978787729119204at_rat(G),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,B3),A3)))),aa_set_nat_rat(groups2906978787729119204at_rat(G),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A3),B3))) ) ) ) ).

% sum.union_diff2
tff(fact_7628_sum_Ounion__diff2,axiom,
    ! [A3: set_nat,B3: set_nat,G: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( aa_set_nat_o(finite_finite_nat,B3)
       => ( aa_set_nat_int(groups3539618377306564664at_int(G),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A3),B3)) = 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,A3),B3))),aa_set_nat_int(groups3539618377306564664at_int(G),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,B3),A3)))),aa_set_nat_int(groups3539618377306564664at_int(G),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A3),B3))) ) ) ) ).

% sum.union_diff2
tff(fact_7629_sum_Ounion__diff2,axiom,
    ! [A3: set_nat,B3: set_nat,G: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( aa_set_nat_o(finite_finite_nat,B3)
       => ( aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A3),B3)) = 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,A3),B3))),aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,B3),A3)))),aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A3),B3))) ) ) ) ).

% sum.union_diff2
tff(fact_7630_sum_Ounion__diff2,axiom,
    ! [A3: set_int,B3: set_int,G: fun_int_int] :
      ( aa_set_int_o(finite_finite_int,A3)
     => ( aa_set_int_o(finite_finite_int,B3)
       => ( groups4538972089207619220nt_int(G,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A3),B3)) = 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,aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A3),B3))),groups4538972089207619220nt_int(G,aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,B3),A3)))),groups4538972089207619220nt_int(G,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A3),B3))) ) ) ) ).

% sum.union_diff2
tff(fact_7631_sum_Ounion__diff2,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat,G: fun_Pr8317260874932935276at_rat] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A3)
     => ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,B3)
       => ( groups342789780944988191at_rat(G,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,A3),B3)) = 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,A3),B3))),groups342789780944988191at_rat(G,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,B3),A3)))),groups342789780944988191at_rat(G,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A3),B3))) ) ) ) ).

% sum.union_diff2
tff(fact_7632_prod_Ounion__diff2,axiom,
    ! [A3: set_int,B3: set_int,G: fun_int_Code_integer] :
      ( aa_set_int_o(finite_finite_int,A3)
     => ( aa_set_int_o(finite_finite_int,B3)
       => ( groups3827104343326376752nteger(G,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A3),B3)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,groups3827104343326376752nteger(G,aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A3),B3))),groups3827104343326376752nteger(G,aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,B3),A3)))),groups3827104343326376752nteger(G,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A3),B3))) ) ) ) ).

% prod.union_diff2
tff(fact_7633_prod_Ounion__diff2,axiom,
    ! [A3: set_Code_integer,B3: set_Code_integer,G: fun_Co8499238673658658103nteger] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A3)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B3)
       => ( groups3674199335183972705nteger(G,sup_su848401254843788991nteger(A3,B3)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,groups3674199335183972705nteger(G,minus_2355218937544613996nteger(A3,B3))),groups3674199335183972705nteger(G,minus_2355218937544613996nteger(B3,A3)))),groups3674199335183972705nteger(G,inf_in1364745209274528805nteger(A3,B3))) ) ) ) ).

% prod.union_diff2
tff(fact_7634_prod_Ounion__diff2,axiom,
    ! [A3: set_nat,B3: set_nat,G: fun_nat_Code_integer] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( aa_set_nat_o(finite_finite_nat,B3)
       => ( aa_set871585252387937313nteger(groups3455450783089532116nteger(G),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A3),B3)) = 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,A3),B3))),aa_set871585252387937313nteger(groups3455450783089532116nteger(G),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,B3),A3)))),aa_set871585252387937313nteger(groups3455450783089532116nteger(G),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A3),B3))) ) ) ) ).

% prod.union_diff2
tff(fact_7635_prod_Ounion__diff2,axiom,
    ! [A3: set_int,B3: set_int,G: fun_int_assn] :
      ( aa_set_int_o(finite_finite_int,A3)
     => ( aa_set_int_o(finite_finite_int,B3)
       => ( groups7882442080178216443t_assn(G,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A3),B3)) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,groups7882442080178216443t_assn(G,aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A3),B3))),groups7882442080178216443t_assn(G,aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,B3),A3)))),groups7882442080178216443t_assn(G,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A3),B3))) ) ) ) ).

% prod.union_diff2
tff(fact_7636_prod_Ounion__diff2,axiom,
    ! [A3: set_Code_integer,B3: set_Code_integer,G: fun_Co4490034186337852034r_assn] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A3)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B3)
       => ( groups1304777262505850412r_assn(G,sup_su848401254843788991nteger(A3,B3)) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,groups1304777262505850412r_assn(G,minus_2355218937544613996nteger(A3,B3))),groups1304777262505850412r_assn(G,minus_2355218937544613996nteger(B3,A3)))),groups1304777262505850412r_assn(G,inf_in1364745209274528805nteger(A3,B3))) ) ) ) ).

% prod.union_diff2
tff(fact_7637_prod_Ounion__diff2,axiom,
    ! [A3: set_nat,B3: set_nat,G: fun_nat_assn] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( aa_set_nat_o(finite_finite_nat,B3)
       => ( groups6906906614972039071t_assn(G,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A3),B3)) = 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,A3),B3))),groups6906906614972039071t_assn(G,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,B3),A3)))),groups6906906614972039071t_assn(G,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A3),B3))) ) ) ) ).

% prod.union_diff2
tff(fact_7638_prod_Ounion__diff2,axiom,
    ! [A3: set_int,B3: set_int,G: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,A3)
     => ( aa_set_int_o(finite_finite_int,B3)
       => ( groups1072433553688619179nt_rat(G,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A3),B3)) = 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,aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A3),B3))),groups1072433553688619179nt_rat(G,aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,B3),A3)))),groups1072433553688619179nt_rat(G,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A3),B3))) ) ) ) ).

% prod.union_diff2
tff(fact_7639_prod_Ounion__diff2,axiom,
    ! [A3: set_Code_integer,B3: set_Code_integer,G: fun_Code_integer_rat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A3)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B3)
       => ( groups2555765274223993564er_rat(G,sup_su848401254843788991nteger(A3,B3)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,groups2555765274223993564er_rat(G,minus_2355218937544613996nteger(A3,B3))),groups2555765274223993564er_rat(G,minus_2355218937544613996nteger(B3,A3)))),groups2555765274223993564er_rat(G,inf_in1364745209274528805nteger(A3,B3))) ) ) ) ).

% prod.union_diff2
tff(fact_7640_prod_Ounion__diff2,axiom,
    ! [A3: set_nat,B3: set_nat,G: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( aa_set_nat_o(finite_finite_nat,B3)
       => ( aa_set_nat_rat(groups73079841787564623at_rat(G),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A3),B3)) = 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,A3),B3))),aa_set_nat_rat(groups73079841787564623at_rat(G),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,B3),A3)))),aa_set_nat_rat(groups73079841787564623at_rat(G),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A3),B3))) ) ) ) ).

% prod.union_diff2
tff(fact_7641_prod_Ounion__diff2,axiom,
    ! [A3: set_int,B3: set_int,G: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,A3)
     => ( aa_set_int_o(finite_finite_int,B3)
       => ( groups1707563613775114915nt_nat(G,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A3),B3)) = 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,aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A3),B3))),groups1707563613775114915nt_nat(G,aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,B3),A3)))),groups1707563613775114915nt_nat(G,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A3),B3))) ) ) ) ).

% prod.union_diff2
tff(fact_7642_pochhammer__double,axiom,
    ! [Z: rat,N2: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),Z)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2)) = 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,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2)))),aa_nat_rat(comm_s4028243227959126397er_rat(Z),N2))),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),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))))),N2)) ).

% pochhammer_double
tff(fact_7643_gauss__sum__from__Suc__0,axiom,
    ! [N2: nat] : groups6325495683096345652atural(semiri3763490453095760265atural,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,aa_nat_nat(suc,zero_zero_nat)),N2)) = divide5121882707175180666atural(times_2397367101498566445atural(aa_nat_Code_natural(semiri3763490453095760265atural,N2),plus_p4538020629002901425atural(aa_nat_Code_natural(semiri3763490453095760265atural,N2),one_one_Code_natural)),aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))) ).

% gauss_sum_from_Suc_0
tff(fact_7644_gauss__sum__from__Suc__0,axiom,
    ! [N2: nat] : aa_set_nat_int(groups3539618377306564664at_int(semiri1314217659103216013at_int),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,aa_nat_nat(suc,zero_zero_nat)),N2)) = 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,N2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,N2)),one_one_int))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% gauss_sum_from_Suc_0
tff(fact_7645_gauss__sum__from__Suc__0,axiom,
    ! [N2: nat] : aa_set871585252387937313nteger(groups7501900531339628137nteger(semiri4939895301339042750nteger),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,aa_nat_nat(suc,zero_zero_nat)),N2)) = aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,N2)),aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,N2)),one_one_Code_integer))),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) ).

% gauss_sum_from_Suc_0
tff(fact_7646_gauss__sum__from__Suc__0,axiom,
    ! [N2: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(semiri1316708129612266289at_nat),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,aa_nat_nat(suc,zero_zero_nat)),N2)) = 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,N2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(semiri1316708129612266289at_nat,N2)),one_one_nat))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% gauss_sum_from_Suc_0
tff(fact_7647_sum__Un__nat,axiom,
    ! [A3: set_int,B3: set_int,F: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,A3)
     => ( aa_set_int_o(finite_finite_int,B3)
       => ( groups4541462559716669496nt_nat(F,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A3),B3)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups4541462559716669496nt_nat(F,A3)),groups4541462559716669496nt_nat(F,B3))),groups4541462559716669496nt_nat(F,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A3),B3))) ) ) ) ).

% sum_Un_nat
tff(fact_7648_sum__Un__nat,axiom,
    ! [A3: set_Code_integer,B3: set_Code_integer,F: fun_Code_integer_nat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A3)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B3)
       => ( groups7237345082560585321er_nat(F,sup_su848401254843788991nteger(A3,B3)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups7237345082560585321er_nat(F,A3)),groups7237345082560585321er_nat(F,B3))),groups7237345082560585321er_nat(F,inf_in1364745209274528805nteger(A3,B3))) ) ) ) ).

% sum_Un_nat
tff(fact_7649_sum__Un__nat,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat,F: fun_Pr2602117396471556196at_nat] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A3)
     => ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,B3)
       => ( groups977919841031483927at_nat(F,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,A3),B3)) = 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,A3)),groups977919841031483927at_nat(F,B3))),groups977919841031483927at_nat(F,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A3),B3))) ) ) ) ).

% sum_Un_nat
tff(fact_7650_sum__Un__nat,axiom,
    ! [A3: set_Pr8693737435421807431at_nat,B3: set_Pr8693737435421807431at_nat,F: fun_Pr2871540613828922186at_nat] :
      ( finite4392333629123659920at_nat(A3)
     => ( finite4392333629123659920at_nat(B3)
       => ( groups1900718384385340925at_nat(F,aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,A3),B3)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups1900718384385340925at_nat(F,A3)),groups1900718384385340925at_nat(F,B3))),groups1900718384385340925at_nat(F,aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(inf_in4302113700860409141at_nat,A3),B3))) ) ) ) ).

% sum_Un_nat
tff(fact_7651_sum__Un__nat,axiom,
    ! [A3: set_Pr8551490117392284871at_nat,B3: set_Pr8551490117392284871at_nat,F: fun_Pr7574179848310716106at_nat] :
      ( finite1918287321285529104at_nat(A3)
     => ( finite1918287321285529104at_nat(B3)
       => ( groups7059897769266147197at_nat(F,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A3),B3)) = 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,A3)),groups7059897769266147197at_nat(F,B3))),groups7059897769266147197at_nat(F,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(inf_in1697001100524423349at_nat,A3),B3))) ) ) ) ).

% sum_Un_nat
tff(fact_7652_sum__Un__nat,axiom,
    ! [A3: set_Pr4329608150637261639at_nat,B3: set_Pr4329608150637261639at_nat,F: fun_Pr1253225528153422666at_nat] :
      ( finite4343798906461161616at_nat(A3)
     => ( finite4343798906461161616at_nat(B3)
       => ( groups3860910324918113789at_nat(F,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A3),B3)) = 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,A3)),groups3860910324918113789at_nat(F,B3))),groups3860910324918113789at_nat(F,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,A3),B3))) ) ) ) ).

% sum_Un_nat
tff(fact_7653_sum__Un__nat,axiom,
    ! [A3: set_nat,B3: set_nat,F: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( aa_set_nat_o(finite_finite_nat,B3)
       => ( aa_set_nat_nat(groups3542108847815614940at_nat(F),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A3),B3)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),A3)),aa_set_nat_nat(groups3542108847815614940at_nat(F),B3))),aa_set_nat_nat(groups3542108847815614940at_nat(F),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A3),B3))) ) ) ) ).

% sum_Un_nat
tff(fact_7654_sum_OatLeastLessThan__rev__at__least__Suc__atMost,axiom,
    ! [G: fun_nat_nat,N2: nat,M: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,N2),M)) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_jz(G),N2),M)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,aa_nat_nat(suc,N2)),M)) ).

% sum.atLeastLessThan_rev_at_least_Suc_atMost
tff(fact_7655_binomial,axiom,
    ! [A: nat,B: nat,N2: 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)),N2) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_acj(A),B),N2)),aa_nat_set_nat(set_ord_atMost_nat,N2)) ).

% binomial
tff(fact_7656_prod_OatLeastLessThan__rev__at__least__Suc__atMost,axiom,
    ! [G: fun_nat_nat,N2: nat,M: nat] : aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,N2),M)) = aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_jz(G),N2),M)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,aa_nat_nat(suc,N2)),M)) ).

% prod.atLeastLessThan_rev_at_least_Suc_atMost
tff(fact_7657_prod_OatLeastLessThan__rev__at__least__Suc__atMost,axiom,
    ! [G: fun_nat_int,N2: nat,M: nat] : aa_set_nat_int(groups705719431365010083at_int(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,N2),M)) = aa_set_nat_int(groups705719431365010083at_int(aa_nat_fun_nat_int(aa_nat1293065384339088724at_int(aTP_Lamm_nr(G),N2),M)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,aa_nat_nat(suc,N2)),M)) ).

% prod.atLeastLessThan_rev_at_least_Suc_atMost
tff(fact_7658_gchoose__row__sum__weighted,axiom,
    ! [Ra2: rat,M: nat] : aa_set_nat_rat(groups2906978787729119204at_rat(aTP_Lamm_ack(Ra2)),aa_nat_set_nat(aa_nat1818925681583257441et_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))),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)))),aa_nat_rat(gbinomial_rat(Ra2),aa_nat_nat(suc,M))) ).

% gchoose_row_sum_weighted
tff(fact_7659_pochhammer__prod,axiom,
    ! [A: rat,N2: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(A),N2) = aa_set_nat_rat(groups73079841787564623at_rat(aTP_Lamm_ok(A)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),N2)) ).

% pochhammer_prod
tff(fact_7660_pochhammer__prod,axiom,
    ! [A: code_integer,N2: nat] : aa_nat_Code_integer(comm_s8582702949713902594nteger(A),N2) = aa_set871585252387937313nteger(groups3455450783089532116nteger(aTP_Lamm_ol(A)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),N2)) ).

% pochhammer_prod
tff(fact_7661_pochhammer__prod,axiom,
    ! [A: nat,N2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,A),N2) = aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_om,A)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),N2)) ).

% pochhammer_prod
tff(fact_7662_pochhammer__prod,axiom,
    ! [A: int,N2: nat] : aa_nat_int(aa_int_fun_nat_int(comm_s4660882817536571857er_int,A),N2) = aa_set_nat_int(groups705719431365010083at_int(aa_int_fun_nat_int(aTP_Lamm_on,A)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),N2)) ).

% pochhammer_prod
tff(fact_7663_binomial__addition__formula,axiom,
    ! [N2: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,N2),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,N2),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,N2),one_one_nat)),K)) ) ) ).

% binomial_addition_formula
tff(fact_7664_binomial__fact,axiom,
    ! [K: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),N2)
     => ( aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,N2),K)) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_nat_rat(semiri773545260158071498ct_rat,N2)),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,N2),K)))) ) ) ).

% binomial_fact
tff(fact_7665_fact__binomial,axiom,
    ! [K: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),N2)
     => ( 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,N2),K))) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_nat_rat(semiri773545260158071498ct_rat,N2)),aa_nat_rat(semiri773545260158071498ct_rat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),K))) ) ) ).

% fact_binomial
tff(fact_7666_fact__split,axiom,
    ! [K: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),N2)
     => ( aa_nat_rat(semiri773545260158071498ct_rat,N2) = 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(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),K)),N2)))),aa_nat_rat(semiri773545260158071498ct_rat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),K))) ) ) ).

% fact_split
tff(fact_7667_fact__split,axiom,
    ! [K: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),N2)
     => ( aa_nat_int(semiri1406184849735516958ct_int,N2) = 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(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),K)),N2)))),aa_nat_int(semiri1406184849735516958ct_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),K))) ) ) ).

% fact_split
tff(fact_7668_fact__split,axiom,
    ! [K: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),N2)
     => ( aa_nat_Code_integer(semiri3624122377584611663nteger,N2) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,aa_set_nat_nat(groups708209901874060359at_nat(suc),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),K)),N2)))),aa_nat_Code_integer(semiri3624122377584611663nteger,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),K))) ) ) ).

% fact_split
tff(fact_7669_fact__split,axiom,
    ! [K: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),N2)
     => ( aa_nat_nat(semiri1408675320244567234ct_nat,N2) = 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(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),K)),N2)))),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),K))) ) ) ).

% fact_split
tff(fact_7670_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,aa_num_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(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)) ).

% gbinomial_r_part_sum
tff(fact_7671_sum__strict__mono2,axiom,
    ! [B3: set_o,A3: set_o,B: $o,F: fun_o_Code_integer] :
      ( finite_finite_o(B3)
     => ( ord_less_eq_set_o(A3,B3)
       => ( member_o((B),minus_minus_set_o(B3,A3))
         => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_o_Code_integer(F,(B)))
           => ( ! [X3: $o] :
                  ( member_o((X3),B3)
                 => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_o_Code_integer(F,(X3))) )
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,groups4406642042086082107nteger(F,A3)),groups4406642042086082107nteger(F,B3)) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_7672_sum__strict__mono2,axiom,
    ! [B3: set_Code_integer,A3: set_Code_integer,B: code_integer,F: fun_Co8499238673658658103nteger] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B3)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(A3),B3)
       => ( aa_set5643387332196575111eger_o(member_Code_integer(B),minus_2355218937544613996nteger(B3,A3))
         => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(F,B))
           => ( ! [X3: code_integer] :
                  ( aa_set5643387332196575111eger_o(member_Code_integer(X3),B3)
                 => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(F,X3)) )
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,groups879477027807139574nteger(F,A3)),groups879477027807139574nteger(F,B3)) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_7673_sum__strict__mono2,axiom,
    ! [B3: set_nat,A3: set_nat,B: nat,F: fun_nat_Code_integer] :
      ( aa_set_nat_o(finite_finite_nat,B3)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A3),B3)
       => ( aa_set_nat_o(member_nat(B),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,B3),A3))
         => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(F,B))
           => ( ! [X3: nat] :
                  ( aa_set_nat_o(member_nat(X3),B3)
                 => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(F,X3)) )
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_set871585252387937313nteger(groups7501900531339628137nteger(F),A3)),aa_set871585252387937313nteger(groups7501900531339628137nteger(F),B3)) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_7674_sum__strict__mono2,axiom,
    ! [B3: set_o,A3: set_o,B: $o,F: fun_o_rat] :
      ( finite_finite_o(B3)
     => ( ord_less_eq_set_o(A3,B3)
       => ( member_o((B),minus_minus_set_o(B3,A3))
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_o_rat(F,(B)))
           => ( ! [X3: $o] :
                  ( member_o((X3),B3)
                 => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_o_rat(F,(X3))) )
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,groups7872700643590313910_o_rat(F,A3)),groups7872700643590313910_o_rat(F,B3)) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_7675_sum__strict__mono2,axiom,
    ! [B3: set_Code_integer,A3: set_Code_integer,B: code_integer,F: fun_Code_integer_rat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B3)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(A3),B3)
       => ( aa_set5643387332196575111eger_o(member_Code_integer(B),minus_2355218937544613996nteger(B3,A3))
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_Code_integer_rat(F,B))
           => ( ! [X3: code_integer] :
                  ( aa_set5643387332196575111eger_o(member_Code_integer(X3),B3)
                 => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_Code_integer_rat(F,X3)) )
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,groups6602215022474089585er_rat(F,A3)),groups6602215022474089585er_rat(F,B3)) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_7676_sum__strict__mono2,axiom,
    ! [B3: set_nat,A3: set_nat,B: nat,F: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,B3)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A3),B3)
       => ( aa_set_nat_o(member_nat(B),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,B3),A3))
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_nat_rat(F,B))
           => ( ! [X3: nat] :
                  ( aa_set_nat_o(member_nat(X3),B3)
                 => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(F,X3)) )
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_set_nat_rat(groups2906978787729119204at_rat(F),A3)),aa_set_nat_rat(groups2906978787729119204at_rat(F),B3)) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_7677_sum__strict__mono2,axiom,
    ! [B3: set_o,A3: set_o,B: $o,F: fun_o_nat] :
      ( finite_finite_o(B3)
     => ( ord_less_eq_set_o(A3,B3)
       => ( member_o((B),minus_minus_set_o(B3,A3))
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_o_nat(F,(B)))
           => ( ! [X3: $o] :
                  ( member_o((X3),B3)
                 => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_o_nat(F,(X3))) )
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,groups8507830703676809646_o_nat(F,A3)),groups8507830703676809646_o_nat(F,B3)) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_7678_sum__strict__mono2,axiom,
    ! [B3: set_Code_integer,A3: set_Code_integer,B: code_integer,F: fun_Code_integer_nat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B3)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(A3),B3)
       => ( aa_set5643387332196575111eger_o(member_Code_integer(B),minus_2355218937544613996nteger(B3,A3))
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_Code_integer_nat(F,B))
           => ( ! [X3: code_integer] :
                  ( aa_set5643387332196575111eger_o(member_Code_integer(X3),B3)
                 => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_Code_integer_nat(F,X3)) )
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,groups7237345082560585321er_nat(F,A3)),groups7237345082560585321er_nat(F,B3)) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_7679_sum__strict__mono2,axiom,
    ! [B3: set_o,A3: set_o,B: $o,F: fun_o_int] :
      ( finite_finite_o(B3)
     => ( ord_less_eq_set_o(A3,B3)
       => ( member_o((B),minus_minus_set_o(B3,A3))
         => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_o_int(F,(B)))
           => ( ! [X3: $o] :
                  ( member_o((X3),B3)
                 => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_o_int(F,(X3))) )
             => aa_int_o(aa_int_fun_int_o(ord_less_int,groups8505340233167759370_o_int(F,A3)),groups8505340233167759370_o_int(F,B3)) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_7680_sum__strict__mono2,axiom,
    ! [B3: set_Code_integer,A3: set_Code_integer,B: code_integer,F: fun_Code_integer_int] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B3)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(A3),B3)
       => ( aa_set5643387332196575111eger_o(member_Code_integer(B),minus_2355218937544613996nteger(B3,A3))
         => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_Code_integer_int(F,B))
           => ( ! [X3: code_integer] :
                  ( aa_set5643387332196575111eger_o(member_Code_integer(X3),B3)
                 => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_Code_integer_int(F,X3)) )
             => aa_int_o(aa_int_fun_int_o(ord_less_int,groups7234854612051535045er_int(F,A3)),groups7234854612051535045er_int(F,B3)) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_7681_prod__mono2,axiom,
    ! [B3: set_o,A3: set_o,F: fun_o_Code_integer] :
      ( finite_finite_o(B3)
     => ( ord_less_eq_set_o(A3,B3)
       => ( ! [B4: $o] :
              ( member_o((B4),minus_minus_set_o(B3,A3))
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),aa_o_Code_integer(F,(B4))) )
         => ( ! [A4: $o] :
                ( member_o((A4),A3)
               => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_o_Code_integer(F,(A4))) )
           => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,groups7694694392188491536nteger(F,A3)),groups7694694392188491536nteger(F,B3)) ) ) ) ) ).

% prod_mono2
tff(fact_7682_prod__mono2,axiom,
    ! [B3: set_Code_integer,A3: set_Code_integer,F: fun_Co8499238673658658103nteger] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B3)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(A3),B3)
       => ( ! [B4: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(B4),minus_2355218937544613996nteger(B3,A3))
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(F,B4)) )
         => ( ! [A4: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(A4),A3)
               => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(F,A4)) )
           => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,groups3674199335183972705nteger(F,A3)),groups3674199335183972705nteger(F,B3)) ) ) ) ) ).

% prod_mono2
tff(fact_7683_prod__mono2,axiom,
    ! [B3: set_nat,A3: set_nat,F: fun_nat_Code_integer] :
      ( aa_set_nat_o(finite_finite_nat,B3)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A3),B3)
       => ( ! [B4: nat] :
              ( aa_set_nat_o(member_nat(B4),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,B3),A3))
             => 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(member_nat(A4),A3)
               => 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),A3)),aa_set871585252387937313nteger(groups3455450783089532116nteger(F),B3)) ) ) ) ) ).

% prod_mono2
tff(fact_7684_prod__mono2,axiom,
    ! [B3: set_o,A3: set_o,F: fun_o_rat] :
      ( finite_finite_o(B3)
     => ( ord_less_eq_set_o(A3,B3)
       => ( ! [B4: $o] :
              ( member_o((B4),minus_minus_set_o(B3,A3))
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_o_rat(F,(B4))) )
         => ( ! [A4: $o] :
                ( member_o((A4),A3)
               => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_o_rat(F,(A4))) )
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups2869687844427037835_o_rat(F,A3)),groups2869687844427037835_o_rat(F,B3)) ) ) ) ) ).

% prod_mono2
tff(fact_7685_prod__mono2,axiom,
    ! [B3: set_Code_integer,A3: set_Code_integer,F: fun_Code_integer_rat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B3)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(A3),B3)
       => ( ! [B4: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(B4),minus_2355218937544613996nteger(B3,A3))
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_Code_integer_rat(F,B4)) )
         => ( ! [A4: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(A4),A3)
               => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_Code_integer_rat(F,A4)) )
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups2555765274223993564er_rat(F,A3)),groups2555765274223993564er_rat(F,B3)) ) ) ) ) ).

% prod_mono2
tff(fact_7686_prod__mono2,axiom,
    ! [B3: set_nat,A3: set_nat,F: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,B3)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A3),B3)
       => ( ! [B4: nat] :
              ( aa_set_nat_o(member_nat(B4),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,B3),A3))
             => 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(member_nat(A4),A3)
               => 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),A3)),aa_set_nat_rat(groups73079841787564623at_rat(F),B3)) ) ) ) ) ).

% prod_mono2
tff(fact_7687_prod__mono2,axiom,
    ! [B3: set_o,A3: set_o,F: fun_o_int] :
      ( finite_finite_o(B3)
     => ( ord_less_eq_set_o(A3,B3)
       => ( ! [B4: $o] :
              ( member_o((B4),minus_minus_set_o(B3,A3))
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_o_int(F,(B4))) )
         => ( ! [A4: $o] :
                ( member_o((A4),A3)
               => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_o_int(F,(A4))) )
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,groups3502327434004483295_o_int(F,A3)),groups3502327434004483295_o_int(F,B3)) ) ) ) ) ).

% prod_mono2
tff(fact_7688_prod__mono2,axiom,
    ! [B3: set_Code_integer,A3: set_Code_integer,F: fun_Code_integer_int] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B3)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(A3),B3)
       => ( ! [B4: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(B4),minus_2355218937544613996nteger(B3,A3))
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_Code_integer_int(F,B4)) )
         => ( ! [A4: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(A4),A3)
               => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_Code_integer_int(F,A4)) )
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,groups3188404863801439024er_int(F,A3)),groups3188404863801439024er_int(F,B3)) ) ) ) ) ).

% prod_mono2
tff(fact_7689_prod__mono2,axiom,
    ! [B3: set_int,A3: set_int,F: fun_int_Code_integer] :
      ( aa_set_int_o(finite_finite_int,B3)
     => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A3),B3)
       => ( ! [B4: int] :
              ( aa_set_int_o(member_int(B4),aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,B3),A3))
             => 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),A3)
               => 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,A3)),groups3827104343326376752nteger(F,B3)) ) ) ) ) ).

% prod_mono2
tff(fact_7690_prod__mono2,axiom,
    ! [B3: set_int,A3: set_int,F: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,B3)
     => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A3),B3)
       => ( ! [B4: int] :
              ( aa_set_int_o(member_int(B4),aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,B3),A3))
             => 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),A3)
               => 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,A3)),groups1072433553688619179nt_rat(F,B3)) ) ) ) ) ).

% prod_mono2
tff(fact_7691_prod__Un,axiom,
    ! [A3: set_int,B3: set_int,F: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,A3)
     => ( aa_set_int_o(finite_finite_int,B3)
       => ( ! [X3: int] :
              ( aa_set_int_o(member_int(X3),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A3),B3))
             => ( aa_int_rat(F,X3) != zero_zero_rat ) )
         => ( groups1072433553688619179nt_rat(F,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A3),B3)) = 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,A3)),groups1072433553688619179nt_rat(F,B3))),groups1072433553688619179nt_rat(F,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A3),B3))) ) ) ) ) ).

% prod_Un
tff(fact_7692_prod__Un,axiom,
    ! [A3: set_Code_integer,B3: set_Code_integer,F: fun_Code_integer_rat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A3)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B3)
       => ( ! [X3: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X3),inf_in1364745209274528805nteger(A3,B3))
             => ( aa_Code_integer_rat(F,X3) != zero_zero_rat ) )
         => ( groups2555765274223993564er_rat(F,sup_su848401254843788991nteger(A3,B3)) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,groups2555765274223993564er_rat(F,A3)),groups2555765274223993564er_rat(F,B3))),groups2555765274223993564er_rat(F,inf_in1364745209274528805nteger(A3,B3))) ) ) ) ) ).

% prod_Un
tff(fact_7693_prod__Un,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat,F: fun_Pr8317260874932935276at_rat] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A3)
     => ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,B3)
       => ( ! [X3: product_prod_nat_nat] :
              ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X3),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A3),B3))
             => ( aa_Pro3608043709957902237at_rat(F,X3) != zero_zero_rat ) )
         => ( groups3442636767675653108at_rat(F,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,A3),B3)) = 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,A3)),groups3442636767675653108at_rat(F,B3))),groups3442636767675653108at_rat(F,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A3),B3))) ) ) ) ) ).

% prod_Un
tff(fact_7694_prod__Un,axiom,
    ! [A3: set_Pr8693737435421807431at_nat,B3: set_Pr8693737435421807431at_nat,F: fun_Pr8586684092290301266at_rat] :
      ( finite4392333629123659920at_nat(A3)
     => ( finite4392333629123659920at_nat(B3)
       => ( ! [X3: produc859450856879609959at_nat] :
              ( member8206827879206165904at_nat(X3,aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(inf_in4302113700860409141at_nat,A3),B3))
             => ( aa_Pro7042314473122848387at_rat(F,X3) != zero_zero_rat ) )
         => ( groups7136053025338371162at_rat(F,aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,A3),B3)) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,groups7136053025338371162at_rat(F,A3)),groups7136053025338371162at_rat(F,B3))),groups7136053025338371162at_rat(F,aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(inf_in4302113700860409141at_nat,A3),B3))) ) ) ) ) ).

% prod_Un
tff(fact_7695_prod__Un,axiom,
    ! [A3: set_Pr8551490117392284871at_nat,B3: set_Pr8551490117392284871at_nat,F: fun_Pr4065951289917319378at_rat] :
      ( finite1918287321285529104at_nat(A3)
     => ( finite1918287321285529104at_nat(B3)
       => ( ! [X3: produc4166570645942440679at_nat] :
              ( member6689249552917799696at_nat(X3,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(inf_in1697001100524423349at_nat,A3),B3))
             => ( aa_Pro6772482835110228483at_rat(F,X3) != zero_zero_rat ) )
         => ( groups5938585286922990810at_rat(F,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A3),B3)) = 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,A3)),groups5938585286922990810at_rat(F,B3))),groups5938585286922990810at_rat(F,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(inf_in1697001100524423349at_nat,A3),B3))) ) ) ) ) ).

% prod_Un
tff(fact_7696_prod__Un,axiom,
    ! [A3: set_Pr4329608150637261639at_nat,B3: set_Pr4329608150637261639at_nat,F: fun_Pr6968369006614801746at_rat] :
      ( finite4343798906461161616at_nat(A3)
     => ( finite4343798906461161616at_nat(B3)
       => ( ! [X3: produc3843707927480180839at_nat] :
              ( member8757157785044589968at_nat(X3,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,A3),B3))
             => ( aa_Pro5677984708781969027at_rat(F,X3) != zero_zero_rat ) )
         => ( groups8874911130973611098at_rat(F,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A3),B3)) = 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,A3)),groups8874911130973611098at_rat(F,B3))),groups8874911130973611098at_rat(F,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,A3),B3))) ) ) ) ) ).

% prod_Un
tff(fact_7697_prod__Un,axiom,
    ! [A3: set_nat,B3: set_nat,F: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( aa_set_nat_o(finite_finite_nat,B3)
       => ( ! [X3: nat] :
              ( aa_set_nat_o(member_nat(X3),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A3),B3))
             => ( aa_nat_rat(F,X3) != zero_zero_rat ) )
         => ( aa_set_nat_rat(groups73079841787564623at_rat(F),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A3),B3)) = 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),A3)),aa_set_nat_rat(groups73079841787564623at_rat(F),B3))),aa_set_nat_rat(groups73079841787564623at_rat(F),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A3),B3))) ) ) ) ) ).

% prod_Un
tff(fact_7698_gbinomial__partial__row__sum,axiom,
    ! [A: rat,M: nat] : aa_set_nat_rat(groups2906978787729119204at_rat(aTP_Lamm_ack(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)),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)))),aa_nat_rat(gbinomial_rat(A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),one_one_nat))) ).

% gbinomial_partial_row_sum
tff(fact_7699_binomial__ring,axiom,
    ! [A: rat,B: rat,N2: nat] : aa_nat_rat(power_power_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),N2) = aa_set_nat_rat(groups2906978787729119204at_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_acl(A),B),N2)),aa_nat_set_nat(set_ord_atMost_nat,N2)) ).

% binomial_ring
tff(fact_7700_binomial__ring,axiom,
    ! [A: int,B: int,N2: 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)),N2) = aa_set_nat_int(groups3539618377306564664at_int(aa_nat_fun_nat_int(aa_int431785320988601264at_int(aTP_Lamm_acm(A),B),N2)),aa_nat_set_nat(set_ord_atMost_nat,N2)) ).

% binomial_ring
tff(fact_7701_binomial__ring,axiom,
    ! [A: code_integer,B: code_integer,N2: nat] : aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(A),B)),N2) = aa_set871585252387937313nteger(groups7501900531339628137nteger(aa_nat2010205248571845816nteger(aa_Cod3140087502949435154nteger(aTP_Lamm_acn(A),B),N2)),aa_nat_set_nat(set_ord_atMost_nat,N2)) ).

% binomial_ring
tff(fact_7702_binomial__ring,axiom,
    ! [A: nat,B: nat,N2: 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)),N2) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_acj(A),B),N2)),aa_nat_set_nat(set_ord_atMost_nat,N2)) ).

% binomial_ring
tff(fact_7703_pochhammer__binomial__sum,axiom,
    ! [A: rat,B: rat,N2: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),N2) = aa_set_nat_rat(groups2906978787729119204at_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_aco(A),B),N2)),aa_nat_set_nat(set_ord_atMost_nat,N2)) ).

% pochhammer_binomial_sum
tff(fact_7704_pochhammer__binomial__sum,axiom,
    ! [A: int,B: int,N2: 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)),N2) = aa_set_nat_int(groups3539618377306564664at_int(aa_nat_fun_nat_int(aa_int431785320988601264at_int(aTP_Lamm_acp(A),B),N2)),aa_nat_set_nat(set_ord_atMost_nat,N2)) ).

% pochhammer_binomial_sum
tff(fact_7705_pochhammer__binomial__sum,axiom,
    ! [A: code_integer,B: code_integer,N2: nat] : aa_nat_Code_integer(comm_s8582702949713902594nteger(aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(A),B)),N2) = aa_set871585252387937313nteger(groups7501900531339628137nteger(aa_nat2010205248571845816nteger(aa_Cod3140087502949435154nteger(aTP_Lamm_acq(A),B),N2)),aa_nat_set_nat(set_ord_atMost_nat,N2)) ).

% pochhammer_binomial_sum
tff(fact_7706_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_acr(A),K)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),K)) ).

% gbinomial_altdef_of_nat
tff(fact_7707_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_md(A)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),K)) ).

% gbinomial_mult_fact'
tff(fact_7708_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_md(A)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),K)) ).

% gbinomial_mult_fact
tff(fact_7709_half__negative__int__iff,axiom,
    ! [K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int) ) ).

% half_negative_int_iff
tff(fact_7710_half__nonnegative__int__iff,axiom,
    ! [K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K) ) ).

% half_nonnegative_int_iff
tff(fact_7711_divmod__step__eq,axiom,
    ! [L: num,Q4: nat,Ra2: nat] :
      aa_Pro262675715343573244at_nat(unique5026877609467782581ep_nat(L),aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Q4),Ra2)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,L)),Ra2),aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Q4)),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Ra2),aa_num_nat(numeral_numeral_nat,L))),aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Q4)),Ra2)) ).

% divmod_step_eq
tff(fact_7712_divmod__step__eq,axiom,
    ! [L: num,Q4: int,Ra2: int] :
      aa_Pro6970504127699319548nt_int(unique5024387138958732305ep_int(L),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q4),Ra2)) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,L)),Ra2),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))),Q4)),one_one_int)),aa_int_int(aa_int_fun_int_int(minus_minus_int,Ra2),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))),Q4)),Ra2)) ).

% divmod_step_eq
tff(fact_7713_divmod__step__eq,axiom,
    ! [L: num,Q4: code_integer,Ra2: code_integer] :
      unique4921790084139445826nteger(L,produc1086072967326762835nteger(Q4,Ra2)) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_num_Code_integer(numera6620942414471956472nteger,L)),Ra2),produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Q4)),one_one_Code_integer),aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(Ra2),aa_num_Code_integer(numera6620942414471956472nteger,L))),produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Q4),Ra2)) ).

% divmod_step_eq
tff(fact_7714_mult__exp__mod__exp__eq,axiom,
    ! [M: nat,N2: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),M)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)) ) ) ).

% mult_exp_mod_exp_eq
tff(fact_7715_mult__exp__mod__exp__eq,axiom,
    ! [M: nat,N2: nat,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
     => ( 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))),N2)) = 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,N2),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_7716_mult__exp__mod__exp__eq,axiom,
    ! [M: nat,N2: nat,A: code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
     => ( 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))),N2)) = 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,N2),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_7717_mult__exp__mod__exp__eq,axiom,
    ! [M: nat,N2: nat,A: code_natural] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
     => ( modulo8411746178871703098atural(times_2397367101498566445atural(A,aa_nat_Code_natural(power_7079662738309270450atural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))),M)),aa_nat_Code_natural(power_7079662738309270450atural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))),N2)) = times_2397367101498566445atural(modulo8411746178871703098atural(A,aa_nat_Code_natural(power_7079662738309270450atural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),M))),aa_nat_Code_natural(power_7079662738309270450atural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))),M)) ) ) ).

% mult_exp_mod_exp_eq
tff(fact_7718_nat__bit__induct,axiom,
    ! [Pa: fun_nat_o,N2: nat] :
      ( aa_nat_o(Pa,zero_zero_nat)
     => ( ! [N5: nat] :
            ( aa_nat_o(Pa,N5)
           => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N5)
             => aa_nat_o(Pa,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N5)) ) )
       => ( ! [N5: nat] :
              ( aa_nat_o(Pa,N5)
             => aa_nat_o(Pa,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N5))) )
         => aa_nat_o(Pa,N2) ) ) ) ).

% nat_bit_induct
tff(fact_7719_finite__Collect__le__nat,axiom,
    ! [K: nat] : aa_set_nat_o(finite_finite_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_df,K))) ).

% finite_Collect_le_nat
tff(fact_7720_finite__Collect__conjI,axiom,
    ! [Pa: fun_Pr3238374608701176162_int_o,Qa: fun_Pr3238374608701176162_int_o] :
      ( ( finite2998713641127702882nt_int(collec213857154873943460nt_int(Pa))
        | finite2998713641127702882nt_int(collec213857154873943460nt_int(Qa)) )
     => finite2998713641127702882nt_int(collec213857154873943460nt_int(aa_fun5349570537165229698_int_o(aTP_Lamm_bp(Pa),Qa))) ) ).

% finite_Collect_conjI
tff(fact_7721_finite__Collect__conjI,axiom,
    ! [Pa: fun_list_nat_o,Qa: fun_list_nat_o] :
      ( ( finite8100373058378681591st_nat(collect_list_nat(Pa))
        | finite8100373058378681591st_nat(collect_list_nat(Qa)) )
     => finite8100373058378681591st_nat(collect_list_nat(aa_fun6155608546934989052_nat_o(aTP_Lamm_bq(Pa),Qa))) ) ).

% finite_Collect_conjI
tff(fact_7722_finite__Collect__conjI,axiom,
    ! [Pa: fun_set_nat_o,Qa: fun_set_nat_o] :
      ( ( aa_set_set_nat_o(finite1152437895449049373et_nat,collect_set_nat(Pa))
        | aa_set_set_nat_o(finite1152437895449049373et_nat,collect_set_nat(Qa)) )
     => aa_set_set_nat_o(finite1152437895449049373et_nat,collect_set_nat(aa_fun751667557118977660_nat_o(aTP_Lamm_br(Pa),Qa))) ) ).

% finite_Collect_conjI
tff(fact_7723_finite__Collect__conjI,axiom,
    ! [Pa: fun_nat_o,Qa: fun_nat_o] :
      ( ( aa_set_nat_o(finite_finite_nat,collect_nat(Pa))
        | aa_set_nat_o(finite_finite_nat,collect_nat(Qa)) )
     => aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun5937193402907943164_nat_o(aTP_Lamm_bt(Pa),Qa))) ) ).

% finite_Collect_conjI
tff(fact_7724_finite__Collect__conjI,axiom,
    ! [Pa: fun_int_o,Qa: fun_int_o] :
      ( ( aa_set_int_o(finite_finite_int,collect_int(Pa))
        | aa_set_int_o(finite_finite_int,collect_int(Qa)) )
     => aa_set_int_o(finite_finite_int,collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_bs(Pa),Qa))) ) ).

% finite_Collect_conjI
tff(fact_7725_finite__Collect__conjI,axiom,
    ! [Pa: fun_Code_integer_o,Qa: fun_Code_integer_o] :
      ( ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,collect_Code_integer(Pa))
        | aa_set5643387332196575111eger_o(finite6017078050557962740nteger,collect_Code_integer(Qa)) )
     => aa_set5643387332196575111eger_o(finite6017078050557962740nteger,collect_Code_integer(aa_fun8214818745163655228eger_o(aTP_Lamm_acs(Pa),Qa))) ) ).

% finite_Collect_conjI
tff(fact_7726_finite__Collect__conjI,axiom,
    ! [Pa: fun_Pr4796943543331071514_nat_o,Qa: fun_Pr4796943543331071514_nat_o] :
      ( ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,collec3392354462482085612at_nat(Pa))
        | aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,collec3392354462482085612at_nat(Qa)) )
     => aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,collec3392354462482085612at_nat(aa_fun359341089595923442_nat_o(aTP_Lamm_bu(Pa),Qa))) ) ).

% finite_Collect_conjI
tff(fact_7727_finite__Collect__disjI,axiom,
    ! [Pa: fun_Pr3238374608701176162_int_o,Qa: fun_Pr3238374608701176162_int_o] :
      ( finite2998713641127702882nt_int(collec213857154873943460nt_int(aa_fun5349570537165229698_int_o(aTP_Lamm_cf(Pa),Qa)))
    <=> ( finite2998713641127702882nt_int(collec213857154873943460nt_int(Pa))
        & finite2998713641127702882nt_int(collec213857154873943460nt_int(Qa)) ) ) ).

% finite_Collect_disjI
tff(fact_7728_finite__Collect__disjI,axiom,
    ! [Pa: fun_list_nat_o,Qa: fun_list_nat_o] :
      ( finite8100373058378681591st_nat(collect_list_nat(aa_fun6155608546934989052_nat_o(aTP_Lamm_cg(Pa),Qa)))
    <=> ( finite8100373058378681591st_nat(collect_list_nat(Pa))
        & finite8100373058378681591st_nat(collect_list_nat(Qa)) ) ) ).

% finite_Collect_disjI
tff(fact_7729_finite__Collect__disjI,axiom,
    ! [Pa: fun_set_nat_o,Qa: fun_set_nat_o] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,collect_set_nat(aa_fun751667557118977660_nat_o(aTP_Lamm_ch(Pa),Qa)))
    <=> ( aa_set_set_nat_o(finite1152437895449049373et_nat,collect_set_nat(Pa))
        & aa_set_set_nat_o(finite1152437895449049373et_nat,collect_set_nat(Qa)) ) ) ).

% finite_Collect_disjI
tff(fact_7730_finite__Collect__disjI,axiom,
    ! [Pa: fun_nat_o,Qa: fun_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun5937193402907943164_nat_o(aTP_Lamm_cm(Pa),Qa)))
    <=> ( aa_set_nat_o(finite_finite_nat,collect_nat(Pa))
        & aa_set_nat_o(finite_finite_nat,collect_nat(Qa)) ) ) ).

% finite_Collect_disjI
tff(fact_7731_finite__Collect__disjI,axiom,
    ! [Pa: fun_int_o,Qa: fun_int_o] :
      ( aa_set_int_o(finite_finite_int,collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_ci(Pa),Qa)))
    <=> ( aa_set_int_o(finite_finite_int,collect_int(Pa))
        & aa_set_int_o(finite_finite_int,collect_int(Qa)) ) ) ).

% finite_Collect_disjI
tff(fact_7732_finite__Collect__disjI,axiom,
    ! [Pa: fun_Code_integer_o,Qa: fun_Code_integer_o] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,collect_Code_integer(aa_fun8214818745163655228eger_o(aTP_Lamm_act(Pa),Qa)))
    <=> ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,collect_Code_integer(Pa))
        & aa_set5643387332196575111eger_o(finite6017078050557962740nteger,collect_Code_integer(Qa)) ) ) ).

% finite_Collect_disjI
tff(fact_7733_finite__Collect__disjI,axiom,
    ! [Pa: fun_Pr4796943543331071514_nat_o,Qa: fun_Pr4796943543331071514_nat_o] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,collec3392354462482085612at_nat(aa_fun359341089595923442_nat_o(aTP_Lamm_acu(Pa),Qa)))
    <=> ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,collec3392354462482085612at_nat(Pa))
        & aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,collec3392354462482085612at_nat(Qa)) ) ) ).

% finite_Collect_disjI
tff(fact_7734_finite__interval__int1,axiom,
    ! [A: int,B: int] : aa_set_int_o(finite_finite_int,collect_int(aa_int_fun_int_o(aTP_Lamm_acv(A),B))) ).

% finite_interval_int1
tff(fact_7735_finite__interval__int4,axiom,
    ! [A: int,B: int] : aa_set_int_o(finite_finite_int,collect_int(aa_int_fun_int_o(aTP_Lamm_acw(A),B))) ).

% finite_interval_int4
tff(fact_7736_of__nat__id,axiom,
    ! [N2: nat] : aa_nat_nat(semiri1316708129612266289at_nat,N2) = N2 ).

% of_nat_id
tff(fact_7737_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_7738_finite__Int,axiom,
    ! [F3: set_Code_integer,G2: set_Code_integer] :
      ( ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,F3)
        | aa_set5643387332196575111eger_o(finite6017078050557962740nteger,G2) )
     => aa_set5643387332196575111eger_o(finite6017078050557962740nteger,inf_in1364745209274528805nteger(F3,G2)) ) ).

% finite_Int
tff(fact_7739_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_7740_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_7741_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_7742_finite__Un,axiom,
    ! [F3: set_Code_integer,G2: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,sup_su848401254843788991nteger(F3,G2))
    <=> ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,F3)
        & aa_set5643387332196575111eger_o(finite6017078050557962740nteger,G2) ) ) ).

% finite_Un
tff(fact_7743_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_7744_finite__Un,axiom,
    ! [F3: set_Pr8693737435421807431at_nat,G2: set_Pr8693737435421807431at_nat] :
      ( finite4392333629123659920at_nat(aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,F3),G2))
    <=> ( finite4392333629123659920at_nat(F3)
        & finite4392333629123659920at_nat(G2) ) ) ).

% finite_Un
tff(fact_7745_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_7746_finite__Un,axiom,
    ! [F3: set_Pr4329608150637261639at_nat,G2: set_Pr4329608150637261639at_nat] :
      ( finite4343798906461161616at_nat(aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,F3),G2))
    <=> ( finite4343798906461161616at_nat(F3)
        & finite4343798906461161616at_nat(G2) ) ) ).

% finite_Un
tff(fact_7747_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_7748_finite__interval__int3,axiom,
    ! [A: int,B: int] : aa_set_int_o(finite_finite_int,collect_int(aa_int_fun_int_o(aTP_Lamm_acx(A),B))) ).

% finite_interval_int3
tff(fact_7749_finite__interval__int2,axiom,
    ! [A: int,B: int] : aa_set_int_o(finite_finite_int,collect_int(aa_int_fun_int_o(aTP_Lamm_acy(A),B))) ).

% finite_interval_int2
tff(fact_7750_finite__Collect__less__nat,axiom,
    ! [K: nat] : aa_set_nat_o(finite_finite_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_dg,K))) ).

% finite_Collect_less_nat
tff(fact_7751_finite__Collect__subsets,axiom,
    ! [A3: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => aa_set_set_nat_o(finite1152437895449049373et_nat,collect_set_nat(aa_set7861427066675100071_nat_o(aTP_Lamm_cz,A3))) ) ).

% finite_Collect_subsets
tff(fact_7752_finite__Collect__subsets,axiom,
    ! [A3: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A3)
     => finite6931041176100689706nteger(collec574505750873337192nteger(aTP_Lamm_acz(A3))) ) ).

% finite_Collect_subsets
tff(fact_7753_finite__Collect__subsets,axiom,
    ! [A3: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A3)
     => finite9047747110432174090at_nat(collec5514110066124741708at_nat(aa_set7675215768510388063_nat_o(aTP_Lamm_cv,A3))) ) ).

% finite_Collect_subsets
tff(fact_7754_finite__Collect__subsets,axiom,
    ! [A3: set_int] :
      ( aa_set_int_o(finite_finite_int,A3)
     => finite6197958912794628473et_int(collect_set_int(aa_set5425576183772390127_int_o(aTP_Lamm_dd,A3))) ) ).

% finite_Collect_subsets
tff(fact_7755_finite__maxlen,axiom,
    ! [M6: set_list_nat] :
      ( finite8100373058378681591st_nat(M6)
     => ? [N5: nat] :
        ! [X5: list_nat] :
          ( member_list_nat(X5,M6)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_list_nat_nat(size_size_list_nat,X5)),N5) ) ) ).

% finite_maxlen
tff(fact_7756_finite__maxlen,axiom,
    ! [M6: set_list_o] :
      ( finite_finite_list_o(M6)
     => ? [N5: nat] :
        ! [X5: list_o] :
          ( member_list_o(X5,M6)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,size_size_list_o(X5)),N5) ) ) ).

% finite_maxlen
tff(fact_7757_image__add__int__atLeastLessThan,axiom,
    ! [L: int,U: int] : image_int_int(aa_int_fun_int_int(aTP_Lamm_ht,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_7758_not__finite__existsD,axiom,
    ! [Pa: fun_Pr3238374608701176162_int_o] :
      ( ~ finite2998713641127702882nt_int(collec213857154873943460nt_int(Pa))
     => ? [X_1: product_prod_int_int] : aa_Pro4592068669816069595_int_o(Pa,X_1) ) ).

% not_finite_existsD
tff(fact_7759_not__finite__existsD,axiom,
    ! [Pa: fun_list_nat_o] :
      ( ~ finite8100373058378681591st_nat(collect_list_nat(Pa))
     => ? [X_1: list_nat] : aa_list_nat_o(Pa,X_1) ) ).

% not_finite_existsD
tff(fact_7760_not__finite__existsD,axiom,
    ! [Pa: fun_set_nat_o] :
      ( ~ aa_set_set_nat_o(finite1152437895449049373et_nat,collect_set_nat(Pa))
     => ? [X_1: set_nat] : aa_set_nat_o(Pa,X_1) ) ).

% not_finite_existsD
tff(fact_7761_not__finite__existsD,axiom,
    ! [Pa: fun_nat_o] :
      ( ~ aa_set_nat_o(finite_finite_nat,collect_nat(Pa))
     => ? [X_1: nat] : aa_nat_o(Pa,X_1) ) ).

% not_finite_existsD
tff(fact_7762_not__finite__existsD,axiom,
    ! [Pa: fun_int_o] :
      ( ~ aa_set_int_o(finite_finite_int,collect_int(Pa))
     => ? [X_1: int] : aa_int_o(Pa,X_1) ) ).

% not_finite_existsD
tff(fact_7763_not__finite__existsD,axiom,
    ! [Pa: fun_Code_integer_o] :
      ( ~ aa_set5643387332196575111eger_o(finite6017078050557962740nteger,collect_Code_integer(Pa))
     => ? [X_1: code_integer] : aa_Code_integer_o(Pa,X_1) ) ).

% not_finite_existsD
tff(fact_7764_not__finite__existsD,axiom,
    ! [Pa: fun_Pr4796943543331071514_nat_o] :
      ( ~ aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,collec3392354462482085612at_nat(Pa))
     => ? [X_1: product_prod_nat_nat] : aa_Pro6150637604445964947_nat_o(Pa,X_1) ) ).

% not_finite_existsD
tff(fact_7765_pigeonhole__infinite__rel,axiom,
    ! [A3: set_o,B3: set_nat,Ra: fun_o_fun_nat_o] :
      ( ~ finite_finite_o(A3)
     => ( aa_set_nat_o(finite_finite_nat,B3)
       => ( ! [X3: $o] :
              ( member_o((X3),A3)
             => ? [Xa3: nat] :
                  ( aa_set_nat_o(member_nat(Xa3),B3)
                  & aa_nat_o(aa_o_fun_nat_o(Ra,(X3)),Xa3) ) )
         => ? [X3: nat] :
              ( aa_set_nat_o(member_nat(X3),B3)
              & ~ finite_finite_o(collect_o(aa_nat_fun_o_o(aa_fun532071374735444762un_o_o(aTP_Lamm_rg(A3),Ra),X3))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_7766_pigeonhole__infinite__rel,axiom,
    ! [A3: set_o,B3: set_int,Ra: fun_o_fun_int_o] :
      ( ~ finite_finite_o(A3)
     => ( aa_set_int_o(finite_finite_int,B3)
       => ( ! [X3: $o] :
              ( member_o((X3),A3)
             => ? [Xa3: int] :
                  ( aa_set_int_o(member_int(Xa3),B3)
                  & aa_int_o(aa_o_fun_int_o(Ra,(X3)),Xa3) ) )
         => ? [X3: int] :
              ( aa_set_int_o(member_int(X3),B3)
              & ~ finite_finite_o(collect_o(aa_int_fun_o_o(aa_fun250872857246597658un_o_o(aTP_Lamm_rv(A3),Ra),X3))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_7767_pigeonhole__infinite__rel,axiom,
    ! [A3: set_o,B3: set_Code_integer,Ra: fun_o_4021313873839786239eger_o] :
      ( ~ finite_finite_o(A3)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B3)
       => ( ! [X3: $o] :
              ( member_o((X3),A3)
             => ? [Xa3: code_integer] :
                  ( aa_set5643387332196575111eger_o(member_Code_integer(Xa3),B3)
                  & aa_Code_integer_o(aa_o_f8517747772616379256eger_o(Ra,(X3)),Xa3) ) )
         => ? [X3: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X3),B3)
              & ~ finite_finite_o(collect_o(aa_Cod6379977994291289310un_o_o(aa_fun8950210611160600154un_o_o(aTP_Lamm_ada(A3),Ra),X3))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_7768_pigeonhole__infinite__rel,axiom,
    ! [A3: set_nat,B3: set_nat,Ra: fun_nat_fun_nat_o] :
      ( ~ aa_set_nat_o(finite_finite_nat,A3)
     => ( aa_set_nat_o(finite_finite_nat,B3)
       => ( ! [X3: nat] :
              ( aa_set_nat_o(member_nat(X3),A3)
             => ? [Xa3: nat] :
                  ( aa_set_nat_o(member_nat(Xa3),B3)
                  & aa_nat_o(aa_nat_fun_nat_o(Ra,X3),Xa3) ) )
         => ? [X3: nat] :
              ( aa_set_nat_o(member_nat(X3),B3)
              & ~ aa_set_nat_o(finite_finite_nat,collect_nat(aa_nat_fun_nat_o(aa_fun6699960423209655484_nat_o(aTP_Lamm_sv(A3),Ra),X3))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_7769_pigeonhole__infinite__rel,axiom,
    ! [A3: set_nat,B3: set_int,Ra: fun_nat_fun_int_o] :
      ( ~ aa_set_nat_o(finite_finite_nat,A3)
     => ( aa_set_int_o(finite_finite_int,B3)
       => ( ! [X3: nat] :
              ( aa_set_nat_o(member_nat(X3),A3)
             => ? [Xa3: int] :
                  ( aa_set_int_o(member_int(Xa3),B3)
                  & aa_int_o(aa_nat_fun_int_o(Ra,X3),Xa3) ) )
         => ? [X3: int] :
              ( aa_set_int_o(member_int(X3),B3)
              & ~ aa_set_nat_o(finite_finite_nat,collect_nat(aa_int_fun_nat_o(aa_fun7393466555336238524_nat_o(aTP_Lamm_rz(A3),Ra),X3))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_7770_pigeonhole__infinite__rel,axiom,
    ! [A3: set_nat,B3: set_Code_integer,Ra: fun_na6318815019539281885eger_o] :
      ( ~ aa_set_nat_o(finite_finite_nat,A3)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B3)
       => ( ! [X3: nat] :
              ( aa_set_nat_o(member_nat(X3),A3)
             => ? [Xa3: code_integer] :
                  ( aa_set5643387332196575111eger_o(member_Code_integer(Xa3),B3)
                  & aa_Code_integer_o(aa_nat7420681472272698950eger_o(Ra,X3),Xa3) ) )
         => ? [X3: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X3),B3)
              & ~ aa_set_nat_o(finite_finite_nat,collect_nat(aa_Cod1075288278386115654_nat_o(aa_fun3722190751602477436_nat_o(aTP_Lamm_sr(A3),Ra),X3))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_7771_pigeonhole__infinite__rel,axiom,
    ! [A3: set_int,B3: set_nat,Ra: fun_int_fun_nat_o] :
      ( ~ aa_set_int_o(finite_finite_int,A3)
     => ( aa_set_nat_o(finite_finite_nat,B3)
       => ( ! [X3: int] :
              ( aa_set_int_o(member_int(X3),A3)
             => ? [Xa3: nat] :
                  ( aa_set_nat_o(member_nat(Xa3),B3)
                  & aa_nat_o(aa_int_fun_nat_o(Ra,X3),Xa3) ) )
         => ? [X3: nat] :
              ( aa_set_nat_o(member_nat(X3),B3)
              & ~ aa_set_int_o(finite_finite_int,collect_int(aa_nat_fun_int_o(aa_fun5439836533389078972_int_o(aTP_Lamm_rl(A3),Ra),X3))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_7772_pigeonhole__infinite__rel,axiom,
    ! [A3: set_int,B3: set_int,Ra: fun_int_fun_int_o] :
      ( ~ aa_set_int_o(finite_finite_int,A3)
     => ( aa_set_int_o(finite_finite_int,B3)
       => ( ! [X3: int] :
              ( aa_set_int_o(member_int(X3),A3)
             => ? [Xa3: int] :
                  ( aa_set_int_o(member_int(Xa3),B3)
                  & aa_int_o(aa_int_fun_int_o(Ra,X3),Xa3) ) )
         => ? [X3: int] :
              ( aa_set_int_o(member_int(X3),B3)
              & ~ aa_set_int_o(finite_finite_int,collect_int(aa_int_fun_int_o(aa_fun6133342665515662012_int_o(aTP_Lamm_adb(A3),Ra),X3))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_7773_pigeonhole__infinite__rel,axiom,
    ! [A3: set_int,B3: set_Code_integer,Ra: fun_in4891666191172872833eger_o] :
      ( ~ aa_set_int_o(finite_finite_int,A3)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B3)
       => ( ! [X3: int] :
              ( aa_set_int_o(member_int(X3),A3)
             => ? [Xa3: code_integer] :
                  ( aa_set5643387332196575111eger_o(member_Code_integer(Xa3),B3)
                  & aa_Code_integer_o(aa_int5993532643906289898eger_o(Ra,X3),Xa3) ) )
         => ? [X3: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X3),B3)
              & ~ aa_set_int_o(finite_finite_int,collect_int(aa_Cod8477617969808203498_int_o(aa_fun9103777551025077884_int_o(aTP_Lamm_adc(A3),Ra),X3))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_7774_pigeonhole__infinite__rel,axiom,
    ! [A3: set_Code_integer,B3: set_nat,Ra: fun_Co1375048237525739357_nat_o] :
      ( ~ aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A3)
     => ( aa_set_nat_o(finite_finite_nat,B3)
       => ( ! [X3: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X3),A3)
             => ? [Xa3: nat] :
                  ( aa_set_nat_o(member_nat(Xa3),B3)
                  & aa_nat_o(aa_Cod1075288278386115654_nat_o(Ra,X3),Xa3) ) )
         => ? [X3: nat] :
              ( aa_set_nat_o(member_nat(X3),B3)
              & ~ aa_set5643387332196575111eger_o(finite6017078050557962740nteger,collect_Code_integer(aa_nat7420681472272698950eger_o(aa_fun7373950593701497468eger_o(aTP_Lamm_rq(A3),Ra),X3))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_7775_finite__has__maximal2,axiom,
    ! [A3: set_o,A: $o] :
      ( finite_finite_o(A3)
     => ( member_o((A),A3)
       => ? [X3: $o] :
            ( member_o((X3),A3)
            & aa_o_o(aa_o_fun_o_o(ord_less_eq_o,(A)),(X3))
            & ! [Xa3: $o] :
                ( member_o((Xa3),A3)
               => ( aa_o_o(aa_o_fun_o_o(ord_less_eq_o,(X3)),(Xa3))
                 => ( (X3)
                  <=> (Xa3) ) ) ) ) ) ) ).

% finite_has_maximal2
tff(fact_7776_finite__has__maximal2,axiom,
    ! [A3: set_set_nat,A: set_nat] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,A3)
     => ( aa_set_set_nat_o(member_set_nat(A),A3)
       => ? [X3: set_nat] :
            ( aa_set_set_nat_o(member_set_nat(X3),A3)
            & aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),X3)
            & ! [Xa3: set_nat] :
                ( aa_set_set_nat_o(member_set_nat(Xa3),A3)
               => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X3),Xa3)
                 => ( X3 = Xa3 ) ) ) ) ) ) ).

% finite_has_maximal2
tff(fact_7777_finite__has__maximal2,axiom,
    ! [A3: set_Code_integer,A: code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A3)
     => ( aa_set5643387332196575111eger_o(member_Code_integer(A),A3)
       => ? [X3: code_integer] :
            ( aa_set5643387332196575111eger_o(member_Code_integer(X3),A3)
            & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),X3)
            & ! [Xa3: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(Xa3),A3)
               => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,X3),Xa3)
                 => ( X3 = Xa3 ) ) ) ) ) ) ).

% finite_has_maximal2
tff(fact_7778_finite__has__maximal2,axiom,
    ! [A3: set_set_int,A: set_int] :
      ( finite6197958912794628473et_int(A3)
     => ( member_set_int(A,A3)
       => ? [X3: set_int] :
            ( member_set_int(X3,A3)
            & aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A),X3)
            & ! [Xa3: set_int] :
                ( member_set_int(Xa3,A3)
               => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,X3),Xa3)
                 => ( X3 = Xa3 ) ) ) ) ) ) ).

% finite_has_maximal2
tff(fact_7779_finite__has__maximal2,axiom,
    ! [A3: set_rat,A: rat] :
      ( finite_finite_rat(A3)
     => ( member_rat(A,A3)
       => ? [X3: rat] :
            ( member_rat(X3,A3)
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),X3)
            & ! [Xa3: rat] :
                ( member_rat(Xa3,A3)
               => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Xa3)
                 => ( X3 = Xa3 ) ) ) ) ) ) ).

% finite_has_maximal2
tff(fact_7780_finite__has__maximal2,axiom,
    ! [A3: set_num,A: num] :
      ( finite_finite_num(A3)
     => ( member_num(A,A3)
       => ? [X3: num] :
            ( member_num(X3,A3)
            & aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),X3)
            & ! [Xa3: num] :
                ( member_num(Xa3,A3)
               => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Xa3)
                 => ( X3 = Xa3 ) ) ) ) ) ) ).

% finite_has_maximal2
tff(fact_7781_finite__has__maximal2,axiom,
    ! [A3: set_nat,A: nat] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( aa_set_nat_o(member_nat(A),A3)
       => ? [X3: nat] :
            ( aa_set_nat_o(member_nat(X3),A3)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),X3)
            & ! [Xa3: nat] :
                ( aa_set_nat_o(member_nat(Xa3),A3)
               => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),Xa3)
                 => ( X3 = Xa3 ) ) ) ) ) ) ).

% finite_has_maximal2
tff(fact_7782_finite__has__maximal2,axiom,
    ! [A3: set_int,A: int] :
      ( aa_set_int_o(finite_finite_int,A3)
     => ( aa_set_int_o(member_int(A),A3)
       => ? [X3: int] :
            ( aa_set_int_o(member_int(X3),A3)
            & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),X3)
            & ! [Xa3: int] :
                ( aa_set_int_o(member_int(Xa3),A3)
               => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X3),Xa3)
                 => ( X3 = Xa3 ) ) ) ) ) ) ).

% finite_has_maximal2
tff(fact_7783_finite__has__minimal2,axiom,
    ! [A3: set_o,A: $o] :
      ( finite_finite_o(A3)
     => ( member_o((A),A3)
       => ? [X3: $o] :
            ( member_o((X3),A3)
            & aa_o_o(aa_o_fun_o_o(ord_less_eq_o,(X3)),(A))
            & ! [Xa3: $o] :
                ( member_o((Xa3),A3)
               => ( aa_o_o(aa_o_fun_o_o(ord_less_eq_o,(Xa3)),(X3))
                 => ( (X3)
                  <=> (Xa3) ) ) ) ) ) ) ).

% finite_has_minimal2
tff(fact_7784_finite__has__minimal2,axiom,
    ! [A3: set_set_nat,A: set_nat] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,A3)
     => ( aa_set_set_nat_o(member_set_nat(A),A3)
       => ? [X3: set_nat] :
            ( aa_set_set_nat_o(member_set_nat(X3),A3)
            & aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X3),A)
            & ! [Xa3: set_nat] :
                ( aa_set_set_nat_o(member_set_nat(Xa3),A3)
               => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Xa3),X3)
                 => ( X3 = Xa3 ) ) ) ) ) ) ).

% finite_has_minimal2
tff(fact_7785_finite__has__minimal2,axiom,
    ! [A3: set_Code_integer,A: code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A3)
     => ( aa_set5643387332196575111eger_o(member_Code_integer(A),A3)
       => ? [X3: code_integer] :
            ( aa_set5643387332196575111eger_o(member_Code_integer(X3),A3)
            & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,X3),A)
            & ! [Xa3: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(Xa3),A3)
               => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Xa3),X3)
                 => ( X3 = Xa3 ) ) ) ) ) ) ).

% finite_has_minimal2
tff(fact_7786_finite__has__minimal2,axiom,
    ! [A3: set_set_int,A: set_int] :
      ( finite6197958912794628473et_int(A3)
     => ( member_set_int(A,A3)
       => ? [X3: set_int] :
            ( member_set_int(X3,A3)
            & aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,X3),A)
            & ! [Xa3: set_int] :
                ( member_set_int(Xa3,A3)
               => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,Xa3),X3)
                 => ( X3 = Xa3 ) ) ) ) ) ) ).

% finite_has_minimal2
tff(fact_7787_finite__has__minimal2,axiom,
    ! [A3: set_rat,A: rat] :
      ( finite_finite_rat(A3)
     => ( member_rat(A,A3)
       => ? [X3: rat] :
            ( member_rat(X3,A3)
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),A)
            & ! [Xa3: rat] :
                ( member_rat(Xa3,A3)
               => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa3),X3)
                 => ( X3 = Xa3 ) ) ) ) ) ) ).

% finite_has_minimal2
tff(fact_7788_finite__has__minimal2,axiom,
    ! [A3: set_num,A: num] :
      ( finite_finite_num(A3)
     => ( member_num(A,A3)
       => ? [X3: num] :
            ( member_num(X3,A3)
            & aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),A)
            & ! [Xa3: num] :
                ( member_num(Xa3,A3)
               => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xa3),X3)
                 => ( X3 = Xa3 ) ) ) ) ) ) ).

% finite_has_minimal2
tff(fact_7789_finite__has__minimal2,axiom,
    ! [A3: set_nat,A: nat] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( aa_set_nat_o(member_nat(A),A3)
       => ? [X3: nat] :
            ( aa_set_nat_o(member_nat(X3),A3)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),A)
            & ! [Xa3: nat] :
                ( aa_set_nat_o(member_nat(Xa3),A3)
               => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa3),X3)
                 => ( X3 = Xa3 ) ) ) ) ) ) ).

% finite_has_minimal2
tff(fact_7790_finite__has__minimal2,axiom,
    ! [A3: set_int,A: int] :
      ( aa_set_int_o(finite_finite_int,A3)
     => ( aa_set_int_o(member_int(A),A3)
       => ? [X3: int] :
            ( aa_set_int_o(member_int(X3),A3)
            & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X3),A)
            & ! [Xa3: int] :
                ( aa_set_int_o(member_int(Xa3),A3)
               => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa3),X3)
                 => ( X3 = Xa3 ) ) ) ) ) ) ).

% finite_has_minimal2
tff(fact_7791_finite_OemptyI,axiom,
    aa_set5643387332196575111eger_o(finite6017078050557962740nteger,bot_bo3990330152332043303nteger) ).

% finite.emptyI
tff(fact_7792_finite_OemptyI,axiom,
    aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,bot_bo2099793752762293965at_nat) ).

% finite.emptyI
tff(fact_7793_finite_OemptyI,axiom,
    finite_finite_o(bot_bot_set_o) ).

% finite.emptyI
tff(fact_7794_finite_OemptyI,axiom,
    aa_set_nat_o(finite_finite_nat,bot_bot_set_nat) ).

% finite.emptyI
tff(fact_7795_finite_OemptyI,axiom,
    aa_set_int_o(finite_finite_int,bot_bot_set_int) ).

% finite.emptyI
tff(fact_7796_infinite__imp__nonempty,axiom,
    ! [S: set_Code_integer] :
      ( ~ aa_set5643387332196575111eger_o(finite6017078050557962740nteger,S)
     => ( S != bot_bo3990330152332043303nteger ) ) ).

% infinite_imp_nonempty
tff(fact_7797_infinite__imp__nonempty,axiom,
    ! [S: set_Pr1261947904930325089at_nat] :
      ( ~ aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,S)
     => ( S != bot_bo2099793752762293965at_nat ) ) ).

% infinite_imp_nonempty
tff(fact_7798_infinite__imp__nonempty,axiom,
    ! [S: set_o] :
      ( ~ finite_finite_o(S)
     => ( S != bot_bot_set_o ) ) ).

% infinite_imp_nonempty
tff(fact_7799_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_7800_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_7801_all__subset__image,axiom,
    ! [F: fun_nat_set_nat,A3: set_nat,Pa: fun_set_set_nat_o] :
      ( ! [B6: set_set_nat] :
          ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(B6),image_nat_set_nat(F,A3))
         => aa_set_set_nat_o(Pa,B6) )
    <=> ! [B6: set_nat] :
          ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B6),A3)
         => aa_set_set_nat_o(Pa,image_nat_set_nat(F,B6)) ) ) ).

% all_subset_image
tff(fact_7802_all__subset__image,axiom,
    ! [F: fun_nat_nat,A3: set_nat,Pa: fun_set_nat_o] :
      ( ! [B6: set_nat] :
          ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B6),image_nat_nat(F,A3))
         => aa_set_nat_o(Pa,B6) )
    <=> ! [B6: set_nat] :
          ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B6),A3)
         => aa_set_nat_o(Pa,image_nat_nat(F,B6)) ) ) ).

% all_subset_image
tff(fact_7803_all__subset__image,axiom,
    ! [F: fun_int_nat,A3: set_int,Pa: fun_set_nat_o] :
      ( ! [B6: set_nat] :
          ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B6),image_int_nat(F,A3))
         => aa_set_nat_o(Pa,B6) )
    <=> ! [B6: set_int] :
          ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,B6),A3)
         => aa_set_nat_o(Pa,image_int_nat(F,B6)) ) ) ).

% all_subset_image
tff(fact_7804_all__subset__image,axiom,
    ! [F: fun_nat_int,A3: set_nat,Pa: fun_set_int_o] :
      ( ! [B6: set_int] :
          ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,B6),image_nat_int(F,A3))
         => aa_set_int_o(Pa,B6) )
    <=> ! [B6: set_nat] :
          ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B6),A3)
         => aa_set_int_o(Pa,image_nat_int(F,B6)) ) ) ).

% all_subset_image
tff(fact_7805_all__subset__image,axiom,
    ! [F: fun_int_int,A3: set_int,Pa: fun_set_int_o] :
      ( ! [B6: set_int] :
          ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,B6),image_int_int(F,A3))
         => aa_set_int_o(Pa,B6) )
    <=> ! [B6: set_int] :
          ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,B6),A3)
         => aa_set_int_o(Pa,image_int_int(F,B6)) ) ) ).

% all_subset_image
tff(fact_7806_finite__subset,axiom,
    ! [A3: set_nat,B3: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A3),B3)
     => ( aa_set_nat_o(finite_finite_nat,B3)
       => aa_set_nat_o(finite_finite_nat,A3) ) ) ).

% finite_subset
tff(fact_7807_finite__subset,axiom,
    ! [A3: set_Code_integer,B3: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(A3),B3)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B3)
       => aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A3) ) ) ).

% finite_subset
tff(fact_7808_finite__subset,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A3),B3)
     => ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,B3)
       => aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A3) ) ) ).

% finite_subset
tff(fact_7809_finite__subset,axiom,
    ! [A3: set_int,B3: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A3),B3)
     => ( aa_set_int_o(finite_finite_int,B3)
       => aa_set_int_o(finite_finite_int,A3) ) ) ).

% finite_subset
tff(fact_7810_infinite__super,axiom,
    ! [S: set_nat,T: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,S),T)
     => ( ~ aa_set_nat_o(finite_finite_nat,S)
       => ~ aa_set_nat_o(finite_finite_nat,T) ) ) ).

% infinite_super
tff(fact_7811_infinite__super,axiom,
    ! [S: set_Code_integer,T: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(S),T)
     => ( ~ aa_set5643387332196575111eger_o(finite6017078050557962740nteger,S)
       => ~ aa_set5643387332196575111eger_o(finite6017078050557962740nteger,T) ) ) ).

% infinite_super
tff(fact_7812_infinite__super,axiom,
    ! [S: set_Pr1261947904930325089at_nat,T: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,S),T)
     => ( ~ aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,S)
       => ~ aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,T) ) ) ).

% infinite_super
tff(fact_7813_infinite__super,axiom,
    ! [S: set_int,T: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,S),T)
     => ( ~ aa_set_int_o(finite_finite_int,S)
       => ~ aa_set_int_o(finite_finite_int,T) ) ) ).

% infinite_super
tff(fact_7814_rev__finite__subset,axiom,
    ! [B3: set_nat,A3: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,B3)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A3),B3)
       => aa_set_nat_o(finite_finite_nat,A3) ) ) ).

% rev_finite_subset
tff(fact_7815_rev__finite__subset,axiom,
    ! [B3: set_Code_integer,A3: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B3)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(A3),B3)
       => aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A3) ) ) ).

% rev_finite_subset
tff(fact_7816_rev__finite__subset,axiom,
    ! [B3: set_Pr1261947904930325089at_nat,A3: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,B3)
     => ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A3),B3)
       => aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A3) ) ) ).

% rev_finite_subset
tff(fact_7817_rev__finite__subset,axiom,
    ! [B3: set_int,A3: set_int] :
      ( aa_set_int_o(finite_finite_int,B3)
     => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A3),B3)
       => aa_set_int_o(finite_finite_int,A3) ) ) ).

% rev_finite_subset
tff(fact_7818_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_7819_finite__UnI,axiom,
    ! [F3: set_Code_integer,G2: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,F3)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,G2)
       => aa_set5643387332196575111eger_o(finite6017078050557962740nteger,sup_su848401254843788991nteger(F3,G2)) ) ) ).

% finite_UnI
tff(fact_7820_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_7821_finite__UnI,axiom,
    ! [F3: set_Pr8693737435421807431at_nat,G2: set_Pr8693737435421807431at_nat] :
      ( finite4392333629123659920at_nat(F3)
     => ( finite4392333629123659920at_nat(G2)
       => finite4392333629123659920at_nat(aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,F3),G2)) ) ) ).

% finite_UnI
tff(fact_7822_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_7823_finite__UnI,axiom,
    ! [F3: set_Pr4329608150637261639at_nat,G2: set_Pr4329608150637261639at_nat] :
      ( finite4343798906461161616at_nat(F3)
     => ( finite4343798906461161616at_nat(G2)
       => finite4343798906461161616at_nat(aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,F3),G2)) ) ) ).

% finite_UnI
tff(fact_7824_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_7825_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_7826_Un__infinite,axiom,
    ! [S: set_Code_integer,T: set_Code_integer] :
      ( ~ aa_set5643387332196575111eger_o(finite6017078050557962740nteger,S)
     => ~ aa_set5643387332196575111eger_o(finite6017078050557962740nteger,sup_su848401254843788991nteger(S,T)) ) ).

% Un_infinite
tff(fact_7827_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_7828_Un__infinite,axiom,
    ! [S: set_Pr8693737435421807431at_nat,T: set_Pr8693737435421807431at_nat] :
      ( ~ finite4392333629123659920at_nat(S)
     => ~ finite4392333629123659920at_nat(aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,S),T)) ) ).

% Un_infinite
tff(fact_7829_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_7830_Un__infinite,axiom,
    ! [S: set_Pr4329608150637261639at_nat,T: set_Pr4329608150637261639at_nat] :
      ( ~ finite4343798906461161616at_nat(S)
     => ~ finite4343798906461161616at_nat(aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,S),T)) ) ).

% Un_infinite
tff(fact_7831_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_7832_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_7833_infinite__Un,axiom,
    ! [S: set_Code_integer,T: set_Code_integer] :
      ( ~ aa_set5643387332196575111eger_o(finite6017078050557962740nteger,sup_su848401254843788991nteger(S,T))
    <=> ( ~ aa_set5643387332196575111eger_o(finite6017078050557962740nteger,S)
        | ~ aa_set5643387332196575111eger_o(finite6017078050557962740nteger,T) ) ) ).

% infinite_Un
tff(fact_7834_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_7835_infinite__Un,axiom,
    ! [S: set_Pr8693737435421807431at_nat,T: set_Pr8693737435421807431at_nat] :
      ( ~ finite4392333629123659920at_nat(aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,S),T))
    <=> ( ~ finite4392333629123659920at_nat(S)
        | ~ finite4392333629123659920at_nat(T) ) ) ).

% infinite_Un
tff(fact_7836_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_7837_infinite__Un,axiom,
    ! [S: set_Pr4329608150637261639at_nat,T: set_Pr4329608150637261639at_nat] :
      ( ~ finite4343798906461161616at_nat(aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,S),T))
    <=> ( ~ finite4343798906461161616at_nat(S)
        | ~ finite4343798906461161616at_nat(T) ) ) ).

% infinite_Un
tff(fact_7838_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_7839_finite__psubset__induct,axiom,
    ! [A3: set_nat,Pa: fun_set_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( ! [A6: set_nat] :
            ( aa_set_nat_o(finite_finite_nat,A6)
           => ( ! [B7: set_nat] :
                  ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,B7),A6)
                 => aa_set_nat_o(Pa,B7) )
             => aa_set_nat_o(Pa,A6) ) )
       => aa_set_nat_o(Pa,A3) ) ) ).

% finite_psubset_induct
tff(fact_7840_finite__psubset__induct,axiom,
    ! [A3: set_int,Pa: fun_set_int_o] :
      ( aa_set_int_o(finite_finite_int,A3)
     => ( ! [A6: set_int] :
            ( aa_set_int_o(finite_finite_int,A6)
           => ( ! [B7: set_int] :
                  ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_set_int,B7),A6)
                 => aa_set_int_o(Pa,B7) )
             => aa_set_int_o(Pa,A6) ) )
       => aa_set_int_o(Pa,A3) ) ) ).

% finite_psubset_induct
tff(fact_7841_finite__psubset__induct,axiom,
    ! [A3: set_Code_integer,Pa: fun_se7289222521248229150eger_o] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A3)
     => ( ! [A6: set_Code_integer] :
            ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A6)
           => ( ! [B7: set_Code_integer] :
                  ( aa_set5643387332196575111eger_o(ord_le1307284697595431911nteger(B7),A6)
                 => aa_set5643387332196575111eger_o(Pa,B7) )
             => aa_set5643387332196575111eger_o(Pa,A6) ) )
       => aa_set5643387332196575111eger_o(Pa,A3) ) ) ).

% finite_psubset_induct
tff(fact_7842_finite__psubset__induct,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,Pa: fun_se4782206882984618170_nat_o] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A3)
     => ( ! [A6: set_Pr1261947904930325089at_nat] :
            ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A6)
           => ( ! [B7: set_Pr1261947904930325089at_nat] :
                  ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le7866589430770878221at_nat,B7),A6)
                 => aa_set3702741264012733747_nat_o(Pa,B7) )
             => aa_set3702741264012733747_nat_o(Pa,A6) ) )
       => aa_set3702741264012733747_nat_o(Pa,A3) ) ) ).

% finite_psubset_induct
tff(fact_7843_pigeonhole__infinite,axiom,
    ! [A3: set_o,F: fun_o_nat] :
      ( ~ finite_finite_o(A3)
     => ( aa_set_nat_o(finite_finite_nat,image_o_nat(F,A3))
       => ? [X3: $o] :
            ( member_o((X3),A3)
            & ~ finite_finite_o(collect_o(aa_o_fun_o_o(aa_fun8960405049393213551un_o_o(aTP_Lamm_add(A3),F),(X3)))) ) ) ) ).

% pigeonhole_infinite
tff(fact_7844_pigeonhole__infinite,axiom,
    ! [A3: set_o,F: fun_o_int] :
      ( ~ finite_finite_o(A3)
     => ( aa_set_int_o(finite_finite_int,image_o_int(F,A3))
       => ? [X3: $o] :
            ( member_o((X3),A3)
            & ~ finite_finite_o(collect_o(aa_o_fun_o_o(aa_fun3248236805073677075un_o_o(aTP_Lamm_ade(A3),F),(X3)))) ) ) ) ).

% pigeonhole_infinite
tff(fact_7845_pigeonhole__infinite,axiom,
    ! [A3: set_o,F: fun_o_Code_integer] :
      ( ~ finite_finite_o(A3)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,image_o_Code_integer(F,A3))
       => ? [X3: $o] :
            ( member_o((X3),A3)
            & ~ finite_finite_o(collect_o(aa_o_fun_o_o(aa_fun5562237597604540066un_o_o(aTP_Lamm_adf(A3),F),(X3)))) ) ) ) ).

% pigeonhole_infinite
tff(fact_7846_pigeonhole__infinite,axiom,
    ! [A3: set_nat,F: fun_nat_nat] :
      ( ~ aa_set_nat_o(finite_finite_nat,A3)
     => ( aa_set_nat_o(finite_finite_nat,image_nat_nat(F,A3))
       => ? [X3: nat] :
            ( aa_set_nat_o(member_nat(X3),A3)
            & ~ aa_set_nat_o(finite_finite_nat,collect_nat(aa_nat_fun_nat_o(aa_fun6824225949033397647_nat_o(aTP_Lamm_adg(A3),F),X3))) ) ) ) ).

% pigeonhole_infinite
tff(fact_7847_pigeonhole__infinite,axiom,
    ! [A3: set_nat,F: fun_nat_int] :
      ( ~ aa_set_nat_o(finite_finite_nat,A3)
     => ( aa_set_int_o(finite_finite_int,image_nat_int(F,A3))
       => ? [X3: nat] :
            ( aa_set_nat_o(member_nat(X3),A3)
            & ~ aa_set_nat_o(finite_finite_nat,collect_nat(aa_nat_fun_nat_o(aa_fun4569312790843823667_nat_o(aTP_Lamm_adh(A3),F),X3))) ) ) ) ).

% pigeonhole_infinite
tff(fact_7848_pigeonhole__infinite,axiom,
    ! [A3: set_nat,F: fun_nat_Code_integer] :
      ( ~ aa_set_nat_o(finite_finite_nat,A3)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,image_1215581382706833972nteger(F,A3))
       => ? [X3: nat] :
            ( aa_set_nat_o(member_nat(X3),A3)
            & ~ aa_set_nat_o(finite_finite_nat,collect_nat(aa_nat_fun_nat_o(aa_fun8436538915998086722_nat_o(aTP_Lamm_adi(A3),F),X3))) ) ) ) ).

% pigeonhole_infinite
tff(fact_7849_pigeonhole__infinite,axiom,
    ! [A3: set_int,F: fun_int_nat] :
      ( ~ aa_set_int_o(finite_finite_int,A3)
     => ( aa_set_nat_o(finite_finite_nat,image_int_nat(F,A3))
       => ? [X3: int] :
            ( aa_set_int_o(member_int(X3),A3)
            & ~ aa_set_int_o(finite_finite_int,collect_int(aa_int_fun_int_o(aa_fun927631379751134075_int_o(aTP_Lamm_adj(A3),F),X3))) ) ) ) ).

% pigeonhole_infinite
tff(fact_7850_pigeonhole__infinite,axiom,
    ! [A3: set_int,F: fun_int_int] :
      ( ~ aa_set_int_o(finite_finite_int,A3)
     => ( aa_set_int_o(finite_finite_int,image_int_int(F,A3))
       => ? [X3: int] :
            ( aa_set_int_o(member_int(X3),A3)
            & ~ aa_set_int_o(finite_finite_int,collect_int(aa_int_fun_int_o(aa_fun7896090258416335903_int_o(aTP_Lamm_adk(A3),F),X3))) ) ) ) ).

% pigeonhole_infinite
tff(fact_7851_pigeonhole__infinite,axiom,
    ! [A3: set_int,F: fun_int_Code_integer] :
      ( ~ aa_set_int_o(finite_finite_int,A3)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,image_1587234942943678608nteger(F,A3))
       => ? [X3: int] :
            ( aa_set_int_o(member_int(X3),A3)
            & ~ aa_set_int_o(finite_finite_int,collect_int(aa_int_fun_int_o(aa_fun5199375812828433454_int_o(aTP_Lamm_adl(A3),F),X3))) ) ) ) ).

% pigeonhole_infinite
tff(fact_7852_pigeonhole__infinite,axiom,
    ! [A3: set_Code_integer,F: fun_Code_integer_nat] :
      ( ~ aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A3)
     => ( aa_set_nat_o(finite_finite_nat,image_951025933927791156er_nat(F,A3))
       => ? [X3: code_integer] :
            ( aa_set5643387332196575111eger_o(member_Code_integer(X3),A3)
            & ~ aa_set5643387332196575111eger_o(finite6017078050557962740nteger,collect_Code_integer(aa_Cod326357343816142457eger_o(aa_fun7938718647838309544eger_o(aTP_Lamm_adm(A3),F),X3))) ) ) ) ).

% pigeonhole_infinite
tff(fact_7853_nat__seg__image__imp__finite,axiom,
    ! [A3: set_set_nat,F: fun_nat_set_nat,N2: nat] :
      ( ( A3 = image_nat_set_nat(F,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_dg,N2))) )
     => aa_set_set_nat_o(finite1152437895449049373et_nat,A3) ) ).

% nat_seg_image_imp_finite
tff(fact_7854_nat__seg__image__imp__finite,axiom,
    ! [A3: set_nat,F: fun_nat_nat,N2: nat] :
      ( ( A3 = image_nat_nat(F,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_dg,N2))) )
     => aa_set_nat_o(finite_finite_nat,A3) ) ).

% nat_seg_image_imp_finite
tff(fact_7855_nat__seg__image__imp__finite,axiom,
    ! [A3: set_int,F: fun_nat_int,N2: nat] :
      ( ( A3 = image_nat_int(F,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_dg,N2))) )
     => aa_set_int_o(finite_finite_int,A3) ) ).

% nat_seg_image_imp_finite
tff(fact_7856_nat__seg__image__imp__finite,axiom,
    ! [A3: set_Code_integer,F: fun_nat_Code_integer,N2: nat] :
      ( ( A3 = image_1215581382706833972nteger(F,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_dg,N2))) )
     => aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A3) ) ).

% nat_seg_image_imp_finite
tff(fact_7857_nat__seg__image__imp__finite,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,F: fun_na1476630412504255114at_nat,N2: nat] :
      ( ( A3 = image_5846123807819985514at_nat(F,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_dg,N2))) )
     => aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A3) ) ).

% nat_seg_image_imp_finite
tff(fact_7858_finite__conv__nat__seg__image,axiom,
    ! [A3: set_set_nat] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,A3)
    <=> ? [N: nat,F4: fun_nat_set_nat] : A3 = image_nat_set_nat(F4,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_dg,N))) ) ).

% finite_conv_nat_seg_image
tff(fact_7859_finite__conv__nat__seg__image,axiom,
    ! [A3: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A3)
    <=> ? [N: nat,F4: fun_nat_nat] : A3 = image_nat_nat(F4,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_dg,N))) ) ).

% finite_conv_nat_seg_image
tff(fact_7860_finite__conv__nat__seg__image,axiom,
    ! [A3: set_int] :
      ( aa_set_int_o(finite_finite_int,A3)
    <=> ? [N: nat,F4: fun_nat_int] : A3 = image_nat_int(F4,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_dg,N))) ) ).

% finite_conv_nat_seg_image
tff(fact_7861_finite__conv__nat__seg__image,axiom,
    ! [A3: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A3)
    <=> ? [N: nat,F4: fun_nat_Code_integer] : A3 = image_1215581382706833972nteger(F4,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_dg,N))) ) ).

% finite_conv_nat_seg_image
tff(fact_7862_finite__conv__nat__seg__image,axiom,
    ! [A3: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A3)
    <=> ? [N: nat,F4: fun_na1476630412504255114at_nat] : A3 = image_5846123807819985514at_nat(F4,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_dg,N))) ) ).

% finite_conv_nat_seg_image
tff(fact_7863_finite__has__minimal,axiom,
    ! [A3: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A3)
     => ( ( A3 != bot_bo3990330152332043303nteger )
       => ? [X3: code_integer] :
            ( aa_set5643387332196575111eger_o(member_Code_integer(X3),A3)
            & ! [Xa3: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(Xa3),A3)
               => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Xa3),X3)
                 => ( X3 = Xa3 ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_7864_finite__has__minimal,axiom,
    ! [A3: set_o] :
      ( finite_finite_o(A3)
     => ( ( A3 != bot_bot_set_o )
       => ? [X3: $o] :
            ( member_o((X3),A3)
            & ! [Xa3: $o] :
                ( member_o((Xa3),A3)
               => ( aa_o_o(aa_o_fun_o_o(ord_less_eq_o,(Xa3)),(X3))
                 => ( (X3)
                  <=> (Xa3) ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_7865_finite__has__minimal,axiom,
    ! [A3: set_set_int] :
      ( finite6197958912794628473et_int(A3)
     => ( ( A3 != bot_bot_set_set_int )
       => ? [X3: set_int] :
            ( member_set_int(X3,A3)
            & ! [Xa3: set_int] :
                ( member_set_int(Xa3,A3)
               => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,Xa3),X3)
                 => ( X3 = Xa3 ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_7866_finite__has__minimal,axiom,
    ! [A3: set_rat] :
      ( finite_finite_rat(A3)
     => ( ( A3 != bot_bot_set_rat )
       => ? [X3: rat] :
            ( member_rat(X3,A3)
            & ! [Xa3: rat] :
                ( member_rat(Xa3,A3)
               => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa3),X3)
                 => ( X3 = Xa3 ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_7867_finite__has__minimal,axiom,
    ! [A3: set_num] :
      ( finite_finite_num(A3)
     => ( ( A3 != bot_bot_set_num )
       => ? [X3: num] :
            ( member_num(X3,A3)
            & ! [Xa3: num] :
                ( member_num(Xa3,A3)
               => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xa3),X3)
                 => ( X3 = Xa3 ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_7868_finite__has__minimal,axiom,
    ! [A3: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( ( A3 != bot_bot_set_nat )
       => ? [X3: nat] :
            ( aa_set_nat_o(member_nat(X3),A3)
            & ! [Xa3: nat] :
                ( aa_set_nat_o(member_nat(Xa3),A3)
               => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa3),X3)
                 => ( X3 = Xa3 ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_7869_finite__has__minimal,axiom,
    ! [A3: set_int] :
      ( aa_set_int_o(finite_finite_int,A3)
     => ( ( A3 != bot_bot_set_int )
       => ? [X3: int] :
            ( aa_set_int_o(member_int(X3),A3)
            & ! [Xa3: int] :
                ( aa_set_int_o(member_int(Xa3),A3)
               => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa3),X3)
                 => ( X3 = Xa3 ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_7870_finite__has__maximal,axiom,
    ! [A3: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A3)
     => ( ( A3 != bot_bo3990330152332043303nteger )
       => ? [X3: code_integer] :
            ( aa_set5643387332196575111eger_o(member_Code_integer(X3),A3)
            & ! [Xa3: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(Xa3),A3)
               => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,X3),Xa3)
                 => ( X3 = Xa3 ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_7871_finite__has__maximal,axiom,
    ! [A3: set_o] :
      ( finite_finite_o(A3)
     => ( ( A3 != bot_bot_set_o )
       => ? [X3: $o] :
            ( member_o((X3),A3)
            & ! [Xa3: $o] :
                ( member_o((Xa3),A3)
               => ( aa_o_o(aa_o_fun_o_o(ord_less_eq_o,(X3)),(Xa3))
                 => ( (X3)
                  <=> (Xa3) ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_7872_finite__has__maximal,axiom,
    ! [A3: set_set_int] :
      ( finite6197958912794628473et_int(A3)
     => ( ( A3 != bot_bot_set_set_int )
       => ? [X3: set_int] :
            ( member_set_int(X3,A3)
            & ! [Xa3: set_int] :
                ( member_set_int(Xa3,A3)
               => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,X3),Xa3)
                 => ( X3 = Xa3 ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_7873_finite__has__maximal,axiom,
    ! [A3: set_rat] :
      ( finite_finite_rat(A3)
     => ( ( A3 != bot_bot_set_rat )
       => ? [X3: rat] :
            ( member_rat(X3,A3)
            & ! [Xa3: rat] :
                ( member_rat(Xa3,A3)
               => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Xa3)
                 => ( X3 = Xa3 ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_7874_finite__has__maximal,axiom,
    ! [A3: set_num] :
      ( finite_finite_num(A3)
     => ( ( A3 != bot_bot_set_num )
       => ? [X3: num] :
            ( member_num(X3,A3)
            & ! [Xa3: num] :
                ( member_num(Xa3,A3)
               => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Xa3)
                 => ( X3 = Xa3 ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_7875_finite__has__maximal,axiom,
    ! [A3: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( ( A3 != bot_bot_set_nat )
       => ? [X3: nat] :
            ( aa_set_nat_o(member_nat(X3),A3)
            & ! [Xa3: nat] :
                ( aa_set_nat_o(member_nat(Xa3),A3)
               => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),Xa3)
                 => ( X3 = Xa3 ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_7876_finite__has__maximal,axiom,
    ! [A3: set_int] :
      ( aa_set_int_o(finite_finite_int,A3)
     => ( ( A3 != bot_bot_set_int )
       => ? [X3: int] :
            ( aa_set_int_o(member_int(X3),A3)
            & ! [Xa3: int] :
                ( aa_set_int_o(member_int(Xa3),A3)
               => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X3),Xa3)
                 => ( X3 = Xa3 ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_7877_finite__surj,axiom,
    ! [A3: set_nat,B3: set_nat,F: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B3),image_nat_nat(F,A3))
       => aa_set_nat_o(finite_finite_nat,B3) ) ) ).

% finite_surj
tff(fact_7878_finite__surj,axiom,
    ! [A3: set_nat,B3: set_Code_integer,F: fun_nat_Code_integer] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(B3),image_1215581382706833972nteger(F,A3))
       => aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B3) ) ) ).

% finite_surj
tff(fact_7879_finite__surj,axiom,
    ! [A3: set_int,B3: set_nat,F: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,A3)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B3),image_int_nat(F,A3))
       => aa_set_nat_o(finite_finite_nat,B3) ) ) ).

% finite_surj
tff(fact_7880_finite__surj,axiom,
    ! [A3: set_int,B3: set_Code_integer,F: fun_int_Code_integer] :
      ( aa_set_int_o(finite_finite_int,A3)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(B3),image_1587234942943678608nteger(F,A3))
       => aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B3) ) ) ).

% finite_surj
tff(fact_7881_finite__surj,axiom,
    ! [A3: set_Code_integer,B3: set_nat,F: fun_Code_integer_nat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A3)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B3),image_951025933927791156er_nat(F,A3))
       => aa_set_nat_o(finite_finite_nat,B3) ) ) ).

% finite_surj
tff(fact_7882_finite__surj,axiom,
    ! [A3: set_Code_integer,B3: set_Code_integer,F: fun_Co8499238673658658103nteger] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A3)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(B3),image_4470545334726330049nteger(F,A3))
       => aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B3) ) ) ).

% finite_surj
tff(fact_7883_finite__surj,axiom,
    ! [A3: set_nat,B3: set_int,F: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,B3),image_nat_int(F,A3))
       => aa_set_int_o(finite_finite_int,B3) ) ) ).

% finite_surj
tff(fact_7884_finite__surj,axiom,
    ! [A3: set_int,B3: set_int,F: fun_int_int] :
      ( aa_set_int_o(finite_finite_int,A3)
     => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,B3),image_int_int(F,A3))
       => aa_set_int_o(finite_finite_int,B3) ) ) ).

% finite_surj
tff(fact_7885_finite__surj,axiom,
    ! [A3: set_Code_integer,B3: set_int,F: fun_Code_integer_int] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A3)
     => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,B3),image_948535463418740880er_int(F,A3))
       => aa_set_int_o(finite_finite_int,B3) ) ) ).

% finite_surj
tff(fact_7886_finite__surj,axiom,
    ! [A3: set_nat,B3: set_set_nat,F: fun_nat_set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(B3),image_nat_set_nat(F,A3))
       => aa_set_set_nat_o(finite1152437895449049373et_nat,B3) ) ) ).

% finite_surj
tff(fact_7887_finite__subset__image,axiom,
    ! [B3: set_nat,F: fun_nat_nat,A3: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,B3)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B3),image_nat_nat(F,A3))
       => ? [C4: set_nat] :
            ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,C4),A3)
            & aa_set_nat_o(finite_finite_nat,C4)
            & ( B3 = image_nat_nat(F,C4) ) ) ) ) ).

% finite_subset_image
tff(fact_7888_finite__subset__image,axiom,
    ! [B3: set_nat,F: fun_Code_integer_nat,A3: set_Code_integer] :
      ( aa_set_nat_o(finite_finite_nat,B3)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B3),image_951025933927791156er_nat(F,A3))
       => ? [C4: set_Code_integer] :
            ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(C4),A3)
            & aa_set5643387332196575111eger_o(finite6017078050557962740nteger,C4)
            & ( B3 = image_951025933927791156er_nat(F,C4) ) ) ) ) ).

% finite_subset_image
tff(fact_7889_finite__subset__image,axiom,
    ! [B3: set_Code_integer,F: fun_nat_Code_integer,A3: set_nat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B3)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(B3),image_1215581382706833972nteger(F,A3))
       => ? [C4: set_nat] :
            ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,C4),A3)
            & aa_set_nat_o(finite_finite_nat,C4)
            & ( B3 = image_1215581382706833972nteger(F,C4) ) ) ) ) ).

% finite_subset_image
tff(fact_7890_finite__subset__image,axiom,
    ! [B3: set_Code_integer,F: fun_Co8499238673658658103nteger,A3: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B3)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(B3),image_4470545334726330049nteger(F,A3))
       => ? [C4: set_Code_integer] :
            ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(C4),A3)
            & aa_set5643387332196575111eger_o(finite6017078050557962740nteger,C4)
            & ( B3 = image_4470545334726330049nteger(F,C4) ) ) ) ) ).

% finite_subset_image
tff(fact_7891_finite__subset__image,axiom,
    ! [B3: set_nat,F: fun_int_nat,A3: set_int] :
      ( aa_set_nat_o(finite_finite_nat,B3)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B3),image_int_nat(F,A3))
       => ? [C4: set_int] :
            ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,C4),A3)
            & aa_set_int_o(finite_finite_int,C4)
            & ( B3 = image_int_nat(F,C4) ) ) ) ) ).

% finite_subset_image
tff(fact_7892_finite__subset__image,axiom,
    ! [B3: set_Code_integer,F: fun_int_Code_integer,A3: set_int] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B3)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(B3),image_1587234942943678608nteger(F,A3))
       => ? [C4: set_int] :
            ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,C4),A3)
            & aa_set_int_o(finite_finite_int,C4)
            & ( B3 = image_1587234942943678608nteger(F,C4) ) ) ) ) ).

% finite_subset_image
tff(fact_7893_finite__subset__image,axiom,
    ! [B3: set_int,F: fun_nat_int,A3: set_nat] :
      ( aa_set_int_o(finite_finite_int,B3)
     => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,B3),image_nat_int(F,A3))
       => ? [C4: set_nat] :
            ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,C4),A3)
            & aa_set_nat_o(finite_finite_nat,C4)
            & ( B3 = image_nat_int(F,C4) ) ) ) ) ).

% finite_subset_image
tff(fact_7894_finite__subset__image,axiom,
    ! [B3: set_int,F: fun_Code_integer_int,A3: set_Code_integer] :
      ( aa_set_int_o(finite_finite_int,B3)
     => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,B3),image_948535463418740880er_int(F,A3))
       => ? [C4: set_Code_integer] :
            ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(C4),A3)
            & aa_set5643387332196575111eger_o(finite6017078050557962740nteger,C4)
            & ( B3 = image_948535463418740880er_int(F,C4) ) ) ) ) ).

% finite_subset_image
tff(fact_7895_finite__subset__image,axiom,
    ! [B3: set_int,F: fun_int_int,A3: set_int] :
      ( aa_set_int_o(finite_finite_int,B3)
     => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,B3),image_int_int(F,A3))
       => ? [C4: set_int] :
            ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,C4),A3)
            & aa_set_int_o(finite_finite_int,C4)
            & ( B3 = image_int_int(F,C4) ) ) ) ) ).

% finite_subset_image
tff(fact_7896_finite__subset__image,axiom,
    ! [B3: set_set_nat,F: fun_nat_set_nat,A3: set_nat] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,B3)
     => ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(B3),image_nat_set_nat(F,A3))
       => ? [C4: set_nat] :
            ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,C4),A3)
            & aa_set_nat_o(finite_finite_nat,C4)
            & ( B3 = image_nat_set_nat(F,C4) ) ) ) ) ).

% finite_subset_image
tff(fact_7897_ex__finite__subset__image,axiom,
    ! [F: fun_nat_nat,A3: set_nat,Pa: fun_set_nat_o] :
      ( ? [B6: set_nat] :
          ( aa_set_nat_o(finite_finite_nat,B6)
          & aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B6),image_nat_nat(F,A3))
          & aa_set_nat_o(Pa,B6) )
    <=> ? [B6: set_nat] :
          ( aa_set_nat_o(finite_finite_nat,B6)
          & aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B6),A3)
          & aa_set_nat_o(Pa,image_nat_nat(F,B6)) ) ) ).

% ex_finite_subset_image
tff(fact_7898_ex__finite__subset__image,axiom,
    ! [F: fun_Code_integer_nat,A3: set_Code_integer,Pa: fun_set_nat_o] :
      ( ? [B6: set_nat] :
          ( aa_set_nat_o(finite_finite_nat,B6)
          & aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B6),image_951025933927791156er_nat(F,A3))
          & aa_set_nat_o(Pa,B6) )
    <=> ? [B6: set_Code_integer] :
          ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B6)
          & aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(B6),A3)
          & aa_set_nat_o(Pa,image_951025933927791156er_nat(F,B6)) ) ) ).

% ex_finite_subset_image
tff(fact_7899_ex__finite__subset__image,axiom,
    ! [F: fun_nat_Code_integer,A3: set_nat,Pa: fun_se7289222521248229150eger_o] :
      ( ? [B6: set_Code_integer] :
          ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B6)
          & aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(B6),image_1215581382706833972nteger(F,A3))
          & aa_set5643387332196575111eger_o(Pa,B6) )
    <=> ? [B6: set_nat] :
          ( aa_set_nat_o(finite_finite_nat,B6)
          & aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B6),A3)
          & aa_set5643387332196575111eger_o(Pa,image_1215581382706833972nteger(F,B6)) ) ) ).

% ex_finite_subset_image
tff(fact_7900_ex__finite__subset__image,axiom,
    ! [F: fun_Co8499238673658658103nteger,A3: set_Code_integer,Pa: fun_se7289222521248229150eger_o] :
      ( ? [B6: set_Code_integer] :
          ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B6)
          & aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(B6),image_4470545334726330049nteger(F,A3))
          & aa_set5643387332196575111eger_o(Pa,B6) )
    <=> ? [B6: set_Code_integer] :
          ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B6)
          & aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(B6),A3)
          & aa_set5643387332196575111eger_o(Pa,image_4470545334726330049nteger(F,B6)) ) ) ).

% ex_finite_subset_image
tff(fact_7901_ex__finite__subset__image,axiom,
    ! [F: fun_int_nat,A3: set_int,Pa: fun_set_nat_o] :
      ( ? [B6: set_nat] :
          ( aa_set_nat_o(finite_finite_nat,B6)
          & aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B6),image_int_nat(F,A3))
          & aa_set_nat_o(Pa,B6) )
    <=> ? [B6: set_int] :
          ( aa_set_int_o(finite_finite_int,B6)
          & aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,B6),A3)
          & aa_set_nat_o(Pa,image_int_nat(F,B6)) ) ) ).

% ex_finite_subset_image
tff(fact_7902_ex__finite__subset__image,axiom,
    ! [F: fun_int_Code_integer,A3: set_int,Pa: fun_se7289222521248229150eger_o] :
      ( ? [B6: set_Code_integer] :
          ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B6)
          & aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(B6),image_1587234942943678608nteger(F,A3))
          & aa_set5643387332196575111eger_o(Pa,B6) )
    <=> ? [B6: set_int] :
          ( aa_set_int_o(finite_finite_int,B6)
          & aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,B6),A3)
          & aa_set5643387332196575111eger_o(Pa,image_1587234942943678608nteger(F,B6)) ) ) ).

% ex_finite_subset_image
tff(fact_7903_ex__finite__subset__image,axiom,
    ! [F: fun_nat_int,A3: set_nat,Pa: fun_set_int_o] :
      ( ? [B6: set_int] :
          ( aa_set_int_o(finite_finite_int,B6)
          & aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,B6),image_nat_int(F,A3))
          & aa_set_int_o(Pa,B6) )
    <=> ? [B6: set_nat] :
          ( aa_set_nat_o(finite_finite_nat,B6)
          & aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B6),A3)
          & aa_set_int_o(Pa,image_nat_int(F,B6)) ) ) ).

% ex_finite_subset_image
tff(fact_7904_ex__finite__subset__image,axiom,
    ! [F: fun_Code_integer_int,A3: set_Code_integer,Pa: fun_set_int_o] :
      ( ? [B6: set_int] :
          ( aa_set_int_o(finite_finite_int,B6)
          & aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,B6),image_948535463418740880er_int(F,A3))
          & aa_set_int_o(Pa,B6) )
    <=> ? [B6: set_Code_integer] :
          ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B6)
          & aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(B6),A3)
          & aa_set_int_o(Pa,image_948535463418740880er_int(F,B6)) ) ) ).

% ex_finite_subset_image
tff(fact_7905_ex__finite__subset__image,axiom,
    ! [F: fun_int_int,A3: set_int,Pa: fun_set_int_o] :
      ( ? [B6: set_int] :
          ( aa_set_int_o(finite_finite_int,B6)
          & aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,B6),image_int_int(F,A3))
          & aa_set_int_o(Pa,B6) )
    <=> ? [B6: set_int] :
          ( aa_set_int_o(finite_finite_int,B6)
          & aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,B6),A3)
          & aa_set_int_o(Pa,image_int_int(F,B6)) ) ) ).

% ex_finite_subset_image
tff(fact_7906_ex__finite__subset__image,axiom,
    ! [F: fun_nat_set_nat,A3: set_nat,Pa: fun_set_set_nat_o] :
      ( ? [B6: set_set_nat] :
          ( aa_set_set_nat_o(finite1152437895449049373et_nat,B6)
          & aa_set_set_nat_o(ord_le6893508408891458716et_nat(B6),image_nat_set_nat(F,A3))
          & aa_set_set_nat_o(Pa,B6) )
    <=> ? [B6: set_nat] :
          ( aa_set_nat_o(finite_finite_nat,B6)
          & aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B6),A3)
          & aa_set_set_nat_o(Pa,image_nat_set_nat(F,B6)) ) ) ).

% ex_finite_subset_image
tff(fact_7907_all__finite__subset__image,axiom,
    ! [F: fun_nat_nat,A3: set_nat,Pa: fun_set_nat_o] :
      ( ! [B6: set_nat] :
          ( ( aa_set_nat_o(finite_finite_nat,B6)
            & aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B6),image_nat_nat(F,A3)) )
         => aa_set_nat_o(Pa,B6) )
    <=> ! [B6: set_nat] :
          ( ( aa_set_nat_o(finite_finite_nat,B6)
            & aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B6),A3) )
         => aa_set_nat_o(Pa,image_nat_nat(F,B6)) ) ) ).

% all_finite_subset_image
tff(fact_7908_all__finite__subset__image,axiom,
    ! [F: fun_Code_integer_nat,A3: set_Code_integer,Pa: fun_set_nat_o] :
      ( ! [B6: set_nat] :
          ( ( aa_set_nat_o(finite_finite_nat,B6)
            & aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B6),image_951025933927791156er_nat(F,A3)) )
         => aa_set_nat_o(Pa,B6) )
    <=> ! [B6: set_Code_integer] :
          ( ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B6)
            & aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(B6),A3) )
         => aa_set_nat_o(Pa,image_951025933927791156er_nat(F,B6)) ) ) ).

% all_finite_subset_image
tff(fact_7909_all__finite__subset__image,axiom,
    ! [F: fun_nat_Code_integer,A3: set_nat,Pa: fun_se7289222521248229150eger_o] :
      ( ! [B6: set_Code_integer] :
          ( ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B6)
            & aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(B6),image_1215581382706833972nteger(F,A3)) )
         => aa_set5643387332196575111eger_o(Pa,B6) )
    <=> ! [B6: set_nat] :
          ( ( aa_set_nat_o(finite_finite_nat,B6)
            & aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B6),A3) )
         => aa_set5643387332196575111eger_o(Pa,image_1215581382706833972nteger(F,B6)) ) ) ).

% all_finite_subset_image
tff(fact_7910_all__finite__subset__image,axiom,
    ! [F: fun_Co8499238673658658103nteger,A3: set_Code_integer,Pa: fun_se7289222521248229150eger_o] :
      ( ! [B6: set_Code_integer] :
          ( ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B6)
            & aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(B6),image_4470545334726330049nteger(F,A3)) )
         => aa_set5643387332196575111eger_o(Pa,B6) )
    <=> ! [B6: set_Code_integer] :
          ( ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B6)
            & aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(B6),A3) )
         => aa_set5643387332196575111eger_o(Pa,image_4470545334726330049nteger(F,B6)) ) ) ).

% all_finite_subset_image
tff(fact_7911_all__finite__subset__image,axiom,
    ! [F: fun_int_nat,A3: set_int,Pa: fun_set_nat_o] :
      ( ! [B6: set_nat] :
          ( ( aa_set_nat_o(finite_finite_nat,B6)
            & aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B6),image_int_nat(F,A3)) )
         => aa_set_nat_o(Pa,B6) )
    <=> ! [B6: set_int] :
          ( ( aa_set_int_o(finite_finite_int,B6)
            & aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,B6),A3) )
         => aa_set_nat_o(Pa,image_int_nat(F,B6)) ) ) ).

% all_finite_subset_image
tff(fact_7912_all__finite__subset__image,axiom,
    ! [F: fun_int_Code_integer,A3: set_int,Pa: fun_se7289222521248229150eger_o] :
      ( ! [B6: set_Code_integer] :
          ( ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B6)
            & aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(B6),image_1587234942943678608nteger(F,A3)) )
         => aa_set5643387332196575111eger_o(Pa,B6) )
    <=> ! [B6: set_int] :
          ( ( aa_set_int_o(finite_finite_int,B6)
            & aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,B6),A3) )
         => aa_set5643387332196575111eger_o(Pa,image_1587234942943678608nteger(F,B6)) ) ) ).

% all_finite_subset_image
tff(fact_7913_all__finite__subset__image,axiom,
    ! [F: fun_nat_int,A3: set_nat,Pa: fun_set_int_o] :
      ( ! [B6: set_int] :
          ( ( aa_set_int_o(finite_finite_int,B6)
            & aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,B6),image_nat_int(F,A3)) )
         => aa_set_int_o(Pa,B6) )
    <=> ! [B6: set_nat] :
          ( ( aa_set_nat_o(finite_finite_nat,B6)
            & aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B6),A3) )
         => aa_set_int_o(Pa,image_nat_int(F,B6)) ) ) ).

% all_finite_subset_image
tff(fact_7914_all__finite__subset__image,axiom,
    ! [F: fun_Code_integer_int,A3: set_Code_integer,Pa: fun_set_int_o] :
      ( ! [B6: set_int] :
          ( ( aa_set_int_o(finite_finite_int,B6)
            & aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,B6),image_948535463418740880er_int(F,A3)) )
         => aa_set_int_o(Pa,B6) )
    <=> ! [B6: set_Code_integer] :
          ( ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B6)
            & aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(B6),A3) )
         => aa_set_int_o(Pa,image_948535463418740880er_int(F,B6)) ) ) ).

% all_finite_subset_image
tff(fact_7915_all__finite__subset__image,axiom,
    ! [F: fun_int_int,A3: set_int,Pa: fun_set_int_o] :
      ( ! [B6: set_int] :
          ( ( aa_set_int_o(finite_finite_int,B6)
            & aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,B6),image_int_int(F,A3)) )
         => aa_set_int_o(Pa,B6) )
    <=> ! [B6: set_int] :
          ( ( aa_set_int_o(finite_finite_int,B6)
            & aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,B6),A3) )
         => aa_set_int_o(Pa,image_int_int(F,B6)) ) ) ).

% all_finite_subset_image
tff(fact_7916_all__finite__subset__image,axiom,
    ! [F: fun_nat_set_nat,A3: set_nat,Pa: fun_set_set_nat_o] :
      ( ! [B6: set_set_nat] :
          ( ( aa_set_set_nat_o(finite1152437895449049373et_nat,B6)
            & aa_set_set_nat_o(ord_le6893508408891458716et_nat(B6),image_nat_set_nat(F,A3)) )
         => aa_set_set_nat_o(Pa,B6) )
    <=> ! [B6: set_nat] :
          ( ( aa_set_nat_o(finite_finite_nat,B6)
            & aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B6),A3) )
         => aa_set_set_nat_o(Pa,image_nat_set_nat(F,B6)) ) ) ).

% all_finite_subset_image
tff(fact_7917_not__exp__less__eq__0__int,axiom,
    ! [N2: nat] : ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N2)),zero_zero_int) ).

% not_exp_less_eq_0_int
tff(fact_7918_set__bit__0,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se7882103937844011126it_nat,zero_zero_nat),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% set_bit_0
tff(fact_7919_set__bit__0,axiom,
    ! [A: code_integer] : aa_Cod1783974652533396728nteger(bit_se2793503036327961859nteger(zero_zero_nat),A) = aa_Cod1783974652533396728nteger(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_7920_set__bit__0,axiom,
    ! [A: code_natural] : bit_se1617098188084679374atural(zero_zero_nat,A) = plus_p4538020629002901425atural(one_one_Code_natural,times_2397367101498566445atural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one)),divide5121882707175180666atural(A,aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))))) ).

% set_bit_0
tff(fact_7921_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_7922_unset__bit__0,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se4205575877204974255it_nat,zero_zero_nat),A) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% unset_bit_0
tff(fact_7923_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_7924_unset__bit__0,axiom,
    ! [A: code_natural] : bit_se7083795435491715335atural(zero_zero_nat,A) = times_2397367101498566445atural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one)),divide5121882707175180666atural(A,aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one)))) ).

% unset_bit_0
tff(fact_7925_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_7926_set__bit__Suc,axiom,
    ! [N2: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se7882103937844011126it_nat,aa_nat_nat(suc,N2)),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se7882103937844011126it_nat,N2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% set_bit_Suc
tff(fact_7927_set__bit__Suc,axiom,
    ! [N2: nat,A: code_integer] : aa_Cod1783974652533396728nteger(bit_se2793503036327961859nteger(aa_nat_nat(suc,N2)),A) = aa_Cod1783974652533396728nteger(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(N2),aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)))))) ).

% set_bit_Suc
tff(fact_7928_set__bit__Suc,axiom,
    ! [N2: nat,A: code_natural] : bit_se1617098188084679374atural(aa_nat_nat(suc,N2),A) = plus_p4538020629002901425atural(modulo8411746178871703098atural(A,aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))),times_2397367101498566445atural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one)),bit_se1617098188084679374atural(N2,divide5121882707175180666atural(A,aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one)))))) ).

% set_bit_Suc
tff(fact_7929_set__bit__Suc,axiom,
    ! [N2: nat,A: int] : aa_int_int(aa_nat_fun_int_int(bit_se7879613467334960850it_int,aa_nat_nat(suc,N2)),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,N2),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_7930_flip__bit__Suc,axiom,
    ! [N2: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2161824704523386999it_nat,aa_nat_nat(suc,N2)),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2161824704523386999it_nat,N2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% flip_bit_Suc
tff(fact_7931_flip__bit__Suc,axiom,
    ! [N2: nat,A: code_integer] : aa_Cod1783974652533396728nteger(bit_se1345352211410354436nteger(aa_nat_nat(suc,N2)),A) = aa_Cod1783974652533396728nteger(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(N2),aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)))))) ).

% flip_bit_Suc
tff(fact_7932_flip__bit__Suc,axiom,
    ! [N2: nat,A: code_natural] : bit_se168947363167071951atural(aa_nat_nat(suc,N2),A) = plus_p4538020629002901425atural(modulo8411746178871703098atural(A,aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))),times_2397367101498566445atural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one)),bit_se168947363167071951atural(N2,divide5121882707175180666atural(A,aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one)))))) ).

% flip_bit_Suc
tff(fact_7933_flip__bit__Suc,axiom,
    ! [N2: nat,A: int] : aa_int_int(aa_nat_fun_int_int(bit_se2159334234014336723it_int,aa_nat_nat(suc,N2)),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,N2),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_7934_unset__bit__Suc,axiom,
    ! [N2: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se4205575877204974255it_nat,aa_nat_nat(suc,N2)),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se4205575877204974255it_nat,N2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% unset_bit_Suc
tff(fact_7935_unset__bit__Suc,axiom,
    ! [N2: nat,A: code_integer] : aa_Cod1783974652533396728nteger(bit_se8260200283734997820nteger(aa_nat_nat(suc,N2)),A) = aa_Cod1783974652533396728nteger(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(N2),aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)))))) ).

% unset_bit_Suc
tff(fact_7936_unset__bit__Suc,axiom,
    ! [N2: nat,A: code_natural] : bit_se7083795435491715335atural(aa_nat_nat(suc,N2),A) = plus_p4538020629002901425atural(modulo8411746178871703098atural(A,aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))),times_2397367101498566445atural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one)),bit_se7083795435491715335atural(N2,divide5121882707175180666atural(A,aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one)))))) ).

% unset_bit_Suc
tff(fact_7937_unset__bit__Suc,axiom,
    ! [N2: nat,A: int] : aa_int_int(aa_nat_fun_int_int(bit_se4203085406695923979it_int,aa_nat_nat(suc,N2)),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,N2),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_7938_signed__take__bit__rec,axiom,
    ! [N2: nat,A: code_integer] :
      aa_Cod1783974652533396728nteger(bit_ri6519982836138164636nteger(N2),A) = $ite(N2 = zero_zero_nat,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)))),aa_Cod1783974652533396728nteger(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,N2),one_one_nat)),aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))))))) ).

% signed_take_bit_rec
tff(fact_7939_signed__take__bit__rec,axiom,
    ! [N2: nat,A: int] :
      aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,N2),A) = $ite(N2 = 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,N2),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_7940_unset__bit__nonnegative__int__iff,axiom,
    ! [N2: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_nat_fun_int_int(bit_se4203085406695923979it_int,N2),K))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K) ) ).

% unset_bit_nonnegative_int_iff
tff(fact_7941_set__bit__nonnegative__int__iff,axiom,
    ! [N2: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_nat_fun_int_int(bit_se7879613467334960850it_int,N2),K))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K) ) ).

% set_bit_nonnegative_int_iff
tff(fact_7942_flip__bit__nonnegative__int__iff,axiom,
    ! [N2: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_nat_fun_int_int(bit_se2159334234014336723it_int,N2),K))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K) ) ).

% flip_bit_nonnegative_int_iff
tff(fact_7943_unset__bit__negative__int__iff,axiom,
    ! [N2: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_nat_fun_int_int(bit_se4203085406695923979it_int,N2),K)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int) ) ).

% unset_bit_negative_int_iff
tff(fact_7944_set__bit__negative__int__iff,axiom,
    ! [N2: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_nat_fun_int_int(bit_se7879613467334960850it_int,N2),K)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int) ) ).

% set_bit_negative_int_iff
tff(fact_7945_flip__bit__negative__int__iff,axiom,
    ! [N2: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_nat_fun_int_int(bit_se2159334234014336723it_int,N2),K)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int) ) ).

% flip_bit_negative_int_iff
tff(fact_7946_unset__bit__less__eq,axiom,
    ! [N2: nat,K: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_nat_fun_int_int(bit_se4203085406695923979it_int,N2),K)),K) ).

% unset_bit_less_eq
tff(fact_7947_set__bit__greater__eq,axiom,
    ! [K: int,N2: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),aa_int_int(aa_nat_fun_int_int(bit_se7879613467334960850it_int,N2),K)) ).

% set_bit_greater_eq
tff(fact_7948_signed__take__bit__int__less__exp,axiom,
    ! [N2: nat,K: int] : aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,N2),K)),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N2)) ).

% signed_take_bit_int_less_exp
tff(fact_7949_signed__take__bit__int__greater__eq__self__iff,axiom,
    ! [K: int,N2: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,N2),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))),N2)) ) ).

% signed_take_bit_int_greater_eq_self_iff
tff(fact_7950_signed__take__bit__int__less__self__iff,axiom,
    ! [N2: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,N2),K)),K)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N2)),K) ) ).

% signed_take_bit_int_less_self_iff
tff(fact_7951_signed__take__bit__int__greater__eq__minus__exp,axiom,
    ! [N2: 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))),N2))),aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,N2),K)) ).

% signed_take_bit_int_greater_eq_minus_exp
tff(fact_7952_signed__take__bit__int__less__eq__self__iff,axiom,
    ! [N2: 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,N2),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))),N2))),K) ) ).

% signed_take_bit_int_less_eq_self_iff
tff(fact_7953_signed__take__bit__int__greater__self__iff,axiom,
    ! [K: int,N2: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,N2),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))),N2))) ) ).

% signed_take_bit_int_greater_self_iff
tff(fact_7954_signed__take__bit__int__less__eq,axiom,
    ! [N2: nat,K: 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_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N2)),K)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,N2),K)),aa_int_int(aa_int_fun_int_int(minus_minus_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,N2)))) ) ).

% signed_take_bit_int_less_eq
tff(fact_7955_signed__take__bit__int__eq__self,axiom,
    ! [N2: 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))),N2))),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))),N2))
       => ( aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,N2),K) = K ) ) ) ).

% signed_take_bit_int_eq_self
tff(fact_7956_signed__take__bit__int__eq__self__iff,axiom,
    ! [N2: nat,K: int] :
      ( ( aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,N2),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))),N2))),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))),N2)) ) ) ).

% signed_take_bit_int_eq_self_iff
tff(fact_7957_signed__take__bit__int__greater__eq,axiom,
    ! [K: int,N2: 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))),N2)))
     => 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,N2)))),aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,N2),K)) ) ).

% signed_take_bit_int_greater_eq
tff(fact_7958_signed__take__bit__Suc,axiom,
    ! [N2: nat,A: code_integer] : aa_Cod1783974652533396728nteger(bit_ri6519982836138164636nteger(aa_nat_nat(suc,N2)),A) = aa_Cod1783974652533396728nteger(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(N2),aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)))))) ).

% signed_take_bit_Suc
tff(fact_7959_signed__take__bit__Suc,axiom,
    ! [N2: nat,A: int] : aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,aa_nat_nat(suc,N2)),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,N2),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_7960_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),aa_num_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),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)))))
       => ( aa_rat_int(archim7778729529865785530nd_rat,X) = Y ) ) ) ).

% round_unique
tff(fact_7961_finite__int__iff__bounded__le,axiom,
    ! [S: set_int] :
      ( aa_set_int_o(finite_finite_int,S)
    <=> ? [K3: int] : aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,image_int_int(abs_abs_int,S)),set_ord_atMost_int(K3)) ) ).

% finite_int_iff_bounded_le
tff(fact_7962_choose__odd__sum,axiom,
    ! [N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),aa_set_nat_rat(groups2906978787729119204at_rat(aTP_Lamm_adn(N2)),aa_nat_set_nat(set_ord_atMost_nat,N2))) = aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),N2) ) ) ).

% choose_odd_sum
tff(fact_7963_choose__odd__sum,axiom,
    ! [N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)
     => ( 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_ado,N2)),aa_nat_set_nat(set_ord_atMost_nat,N2))) = aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N2) ) ) ).

% choose_odd_sum
tff(fact_7964_choose__odd__sum,axiom,
    ! [N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_set871585252387937313nteger(groups7501900531339628137nteger(aTP_Lamm_adp(N2)),aa_nat_set_nat(set_ord_atMost_nat,N2))) = aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),N2) ) ) ).

% choose_odd_sum
tff(fact_7965_choose__even__sum,axiom,
    ! [N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),aa_set_nat_rat(groups2906978787729119204at_rat(aTP_Lamm_adq(N2)),aa_nat_set_nat(set_ord_atMost_nat,N2))) = aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),N2) ) ) ).

% choose_even_sum
tff(fact_7966_choose__even__sum,axiom,
    ! [N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)
     => ( 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_adr,N2)),aa_nat_set_nat(set_ord_atMost_nat,N2))) = aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N2) ) ) ).

% choose_even_sum
tff(fact_7967_choose__even__sum,axiom,
    ! [N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_set871585252387937313nteger(groups7501900531339628137nteger(aTP_Lamm_ads(N2)),aa_nat_set_nat(set_ord_atMost_nat,N2))) = aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),N2) ) ) ).

% choose_even_sum
tff(fact_7968_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),aa_num_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_7969_finite__nat__iff__bounded__le,axiom,
    ! [S: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,S)
    <=> ? [K3: nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,S),aa_nat_set_nat(set_ord_atMost_nat,K3)) ) ).

% finite_nat_iff_bounded_le
tff(fact_7970_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_7971_dvd__mult__cancel__left,axiom,
    ! [Ca: 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,Ca),A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Ca),B))
    <=> ( ( Ca = zero_z3403309356797280102nteger )
        | aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B) ) ) ).

% dvd_mult_cancel_left
tff(fact_7972_dvd__mult__cancel__left,axiom,
    ! [Ca: 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,Ca),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Ca),B))
    <=> ( ( Ca = zero_zero_rat )
        | aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),B) ) ) ).

% dvd_mult_cancel_left
tff(fact_7973_dvd__mult__cancel__left,axiom,
    ! [Ca: 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,Ca),A)),aa_int_int(aa_int_fun_int_int(times_times_int,Ca),B))
    <=> ( ( Ca = zero_zero_int )
        | aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B) ) ) ).

% dvd_mult_cancel_left
tff(fact_7974_dvd__mult__cancel__right,axiom,
    ! [A: code_integer,Ca: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),Ca)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),Ca))
    <=> ( ( Ca = zero_z3403309356797280102nteger )
        | aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B) ) ) ).

% dvd_mult_cancel_right
tff(fact_7975_dvd__mult__cancel__right,axiom,
    ! [A: rat,Ca: 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),Ca)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),Ca))
    <=> ( ( Ca = zero_zero_rat )
        | aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),B) ) ) ).

% dvd_mult_cancel_right
tff(fact_7976_dvd__mult__cancel__right,axiom,
    ! [A: int,Ca: 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),Ca)),aa_int_int(aa_int_fun_int_int(times_times_int,B),Ca))
    <=> ( ( Ca = zero_zero_int )
        | aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B) ) ) ).

% dvd_mult_cancel_right
tff(fact_7977_dvd__times__left__cancel__iff,axiom,
    ! [A: code_integer,B: code_integer,Ca: 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),Ca))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),Ca) ) ) ).

% dvd_times_left_cancel_iff
tff(fact_7978_dvd__times__left__cancel__iff,axiom,
    ! [A: nat,B: nat,Ca: 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),Ca))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),Ca) ) ) ).

% dvd_times_left_cancel_iff
tff(fact_7979_dvd__times__left__cancel__iff,axiom,
    ! [A: int,B: int,Ca: 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),Ca))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),Ca) ) ) ).

% dvd_times_left_cancel_iff
tff(fact_7980_dvd__times__right__cancel__iff,axiom,
    ! [A: code_integer,B: code_integer,Ca: 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,Ca),A))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),Ca) ) ) ).

% dvd_times_right_cancel_iff
tff(fact_7981_dvd__times__right__cancel__iff,axiom,
    ! [A: nat,B: nat,Ca: 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,Ca),A))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),Ca) ) ) ).

% dvd_times_right_cancel_iff
tff(fact_7982_dvd__times__right__cancel__iff,axiom,
    ! [A: int,B: int,Ca: 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,Ca),A))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),Ca) ) ) ).

% dvd_times_right_cancel_iff
tff(fact_7983_dvd__add__times__triv__right__iff,axiom,
    ! [A: code_integer,B: code_integer,Ca: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(B),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Ca),A)))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B) ) ).

% dvd_add_times_triv_right_iff
tff(fact_7984_dvd__add__times__triv__right__iff,axiom,
    ! [A: rat,B: rat,Ca: 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,Ca),A)))
    <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),B) ) ).

% dvd_add_times_triv_right_iff
tff(fact_7985_dvd__add__times__triv__right__iff,axiom,
    ! [A: nat,B: nat,Ca: 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,Ca),A)))
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B) ) ).

% dvd_add_times_triv_right_iff
tff(fact_7986_dvd__add__times__triv__right__iff,axiom,
    ! [A: int,B: int,Ca: 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,Ca),A)))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B) ) ).

% dvd_add_times_triv_right_iff
tff(fact_7987_dvd__add__times__triv__left__iff,axiom,
    ! [A: code_integer,Ca: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Ca),A)),B))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B) ) ).

% dvd_add_times_triv_left_iff
tff(fact_7988_dvd__add__times__triv__left__iff,axiom,
    ! [A: rat,Ca: 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,Ca),A)),B))
    <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),B) ) ).

% dvd_add_times_triv_left_iff
tff(fact_7989_dvd__add__times__triv__left__iff,axiom,
    ! [A: nat,Ca: 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,Ca),A)),B))
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B) ) ).

% dvd_add_times_triv_left_iff
tff(fact_7990_dvd__add__times__triv__left__iff,axiom,
    ! [A: int,Ca: 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,Ca),A)),B))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B) ) ).

% dvd_add_times_triv_left_iff
tff(fact_7991_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_7992_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_7993_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_7994_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_7995_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_7996_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_7997_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_7998_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_7999_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_8000_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_8001_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_8002_dvd__1__iff__1,axiom,
    ! [M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,M),aa_nat_nat(suc,zero_zero_nat))
    <=> ( M = aa_nat_nat(suc,zero_zero_nat) ) ) ).

% dvd_1_iff_1
tff(fact_8003_dvd__1__left,axiom,
    ! [K: nat] : aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(suc,zero_zero_nat)),K) ).

% dvd_1_left
tff(fact_8004_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_8005_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_8006_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_8007_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_8008_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_8009_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_8010_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_8011_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_8012_pow__divides__pow__iff,axiom,
    ! [N2: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N2)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),N2))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B) ) ) ).

% pow_divides_pow_iff
tff(fact_8013_pow__divides__pow__iff,axiom,
    ! [N2: nat,A: int,B: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A),N2)),aa_nat_int(aa_int_fun_nat_int(power_power_int,B),N2))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B) ) ) ).

% pow_divides_pow_iff
tff(fact_8014_even__mult__iff,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
        | aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B) ) ) ).

% even_mult_iff
tff(fact_8015_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_8016_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_8017_even__mult__iff,axiom,
    ! [A: code_natural,B: code_natural] :
      ( aa_Code_natural_o(dvd_dvd_Code_natural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))),times_2397367101498566445atural(A,B))
    <=> ( aa_Code_natural_o(dvd_dvd_Code_natural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))),A)
        | aa_Code_natural_o(dvd_dvd_Code_natural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))),B) ) ) ).

% even_mult_iff
tff(fact_8018_even__power,axiom,
    ! [A: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N2))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2) ) ) ).

% even_power
tff(fact_8019_even__power,axiom,
    ! [A: int,N2: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_int(aa_int_fun_nat_int(power_power_int,A),N2))
    <=> ( 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),N2) ) ) ).

% even_power
tff(fact_8020_even__power,axiom,
    ! [A: code_integer,N2: nat] :
      ( 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(power_8256067586552552935nteger(A),N2))
    <=> ( 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),N2) ) ) ).

% even_power
tff(fact_8021_even__power,axiom,
    ! [A: code_natural,N2: nat] :
      ( aa_Code_natural_o(dvd_dvd_Code_natural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))),aa_nat_Code_natural(power_7079662738309270450atural(A),N2))
    <=> ( aa_Code_natural_o(dvd_dvd_Code_natural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))),A)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2) ) ) ).

% even_power
tff(fact_8022_zero__le__power__eq__numeral,axiom,
    ! [A: code_integer,W2: num] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_num_nat(numeral_numeral_nat,W2)))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W2))
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W2))
          & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A) ) ) ) ).

% zero_le_power_eq_numeral
tff(fact_8023_zero__le__power__eq__numeral,axiom,
    ! [A: rat,W2: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,W2)))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W2))
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W2))
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A) ) ) ) ).

% zero_le_power_eq_numeral
tff(fact_8024_zero__le__power__eq__numeral,axiom,
    ! [A: int,W2: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(aa_int_fun_nat_int(power_power_int,A),aa_num_nat(numeral_numeral_nat,W2)))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W2))
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W2))
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A) ) ) ) ).

% zero_le_power_eq_numeral
tff(fact_8025_power__less__zero__eq__numeral,axiom,
    ! [A: code_integer,W2: num] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_num_nat(numeral_numeral_nat,W2))),zero_z3403309356797280102nteger)
    <=> ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W2))
        & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),zero_z3403309356797280102nteger) ) ) ).

% power_less_zero_eq_numeral
tff(fact_8026_power__less__zero__eq__numeral,axiom,
    ! [A: rat,W2: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,W2))),zero_zero_rat)
    <=> ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W2))
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat) ) ) ).

% power_less_zero_eq_numeral
tff(fact_8027_power__less__zero__eq__numeral,axiom,
    ! [A: int,W2: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A),aa_num_nat(numeral_numeral_nat,W2))),zero_zero_int)
    <=> ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W2))
        & aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int) ) ) ).

% power_less_zero_eq_numeral
tff(fact_8028_power__less__zero__eq,axiom,
    ! [A: code_integer,N2: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),N2)),zero_z3403309356797280102nteger)
    <=> ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2)
        & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),zero_z3403309356797280102nteger) ) ) ).

% power_less_zero_eq
tff(fact_8029_power__less__zero__eq,axiom,
    ! [A: rat,N2: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(A),N2)),zero_zero_rat)
    <=> ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat) ) ) ).

% power_less_zero_eq
tff(fact_8030_power__less__zero__eq,axiom,
    ! [A: int,N2: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A),N2)),zero_zero_int)
    <=> ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int) ) ) ).

% power_less_zero_eq
tff(fact_8031_even__diff__nat,axiom,
    ! [M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N2))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N2)
        | aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N2)) ) ) ).

% even_diff_nat
tff(fact_8032_odd__two__times__div__two__succ,axiom,
    ! [A: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),one_one_nat) = A ) ) ).

% odd_two_times_div_two_succ
tff(fact_8033_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_8034_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(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_8035_odd__two__times__div__two__succ,axiom,
    ! [A: code_natural] :
      ( ~ aa_Code_natural_o(dvd_dvd_Code_natural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))),A)
     => ( plus_p4538020629002901425atural(times_2397367101498566445atural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one)),divide5121882707175180666atural(A,aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one)))),one_one_Code_natural) = A ) ) ).

% odd_two_times_div_two_succ
tff(fact_8036_zero__less__power__eq__numeral,axiom,
    ! [A: code_integer,W2: num] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_num_nat(numeral_numeral_nat,W2)))
    <=> ( ( aa_num_nat(numeral_numeral_nat,W2) = zero_zero_nat )
        | ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W2))
          & ( A != zero_z3403309356797280102nteger ) )
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W2))
          & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A) ) ) ) ).

% zero_less_power_eq_numeral
tff(fact_8037_zero__less__power__eq__numeral,axiom,
    ! [A: rat,W2: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,W2)))
    <=> ( ( aa_num_nat(numeral_numeral_nat,W2) = zero_zero_nat )
        | ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W2))
          & ( A != zero_zero_rat ) )
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W2))
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A) ) ) ) ).

% zero_less_power_eq_numeral
tff(fact_8038_zero__less__power__eq__numeral,axiom,
    ! [A: int,W2: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(aa_int_fun_nat_int(power_power_int,A),aa_num_nat(numeral_numeral_nat,W2)))
    <=> ( ( aa_num_nat(numeral_numeral_nat,W2) = zero_zero_nat )
        | ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W2))
          & ( A != zero_zero_int ) )
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W2))
          & aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A) ) ) ) ).

% zero_less_power_eq_numeral
tff(fact_8039_power__le__zero__eq__numeral,axiom,
    ! [A: code_integer,W2: num] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_num_nat(numeral_numeral_nat,W2))),zero_z3403309356797280102nteger)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,W2))
        & ( ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W2))
            & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),zero_z3403309356797280102nteger) )
          | ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W2))
            & ( A = zero_z3403309356797280102nteger ) ) ) ) ) ).

% power_le_zero_eq_numeral
tff(fact_8040_power__le__zero__eq__numeral,axiom,
    ! [A: rat,W2: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,W2))),zero_zero_rat)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,W2))
        & ( ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W2))
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat) )
          | ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W2))
            & ( A = zero_zero_rat ) ) ) ) ) ).

% power_le_zero_eq_numeral
tff(fact_8041_power__le__zero__eq__numeral,axiom,
    ! [A: int,W2: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A),aa_num_nat(numeral_numeral_nat,W2))),zero_zero_int)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,W2))
        & ( ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W2))
            & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int) )
          | ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W2))
            & ( A = zero_zero_int ) ) ) ) ) ).

% power_le_zero_eq_numeral
tff(fact_8042_even__succ__div__exp,axiom,
    ! [A: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),A)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2)) ) ) ) ).

% even_succ_div_exp
tff(fact_8043_even__succ__div__exp,axiom,
    ! [A: int,N2: 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),N2)
       => ( 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))),N2)) = 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))),N2)) ) ) ) ).

% even_succ_div_exp
tff(fact_8044_even__succ__div__exp,axiom,
    ! [A: code_integer,N2: 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),N2)
       => ( aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(one_one_Code_integer),A)),aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),N2)) = aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),N2)) ) ) ) ).

% even_succ_div_exp
tff(fact_8045_even__succ__div__exp,axiom,
    ! [A: code_natural,N2: nat] :
      ( aa_Code_natural_o(dvd_dvd_Code_natural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)
       => ( divide5121882707175180666atural(plus_p4538020629002901425atural(one_one_Code_natural,A),aa_nat_Code_natural(power_7079662738309270450atural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))),N2)) = divide5121882707175180666atural(A,aa_nat_Code_natural(power_7079662738309270450atural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))),N2)) ) ) ) ).

% even_succ_div_exp
tff(fact_8046_even__succ__mod__exp,axiom,
    ! [A: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),A)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2))) ) ) ) ).

% even_succ_mod_exp
tff(fact_8047_even__succ__mod__exp,axiom,
    ! [A: int,N2: 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),N2)
       => ( 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))),N2)) = 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))),N2))) ) ) ) ).

% even_succ_mod_exp
tff(fact_8048_even__succ__mod__exp,axiom,
    ! [A: code_integer,N2: 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),N2)
       => ( aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(one_one_Code_integer),A)),aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),N2)) = aa_Cod1783974652533396728nteger(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))),N2))) ) ) ) ).

% even_succ_mod_exp
tff(fact_8049_even__succ__mod__exp,axiom,
    ! [A: code_natural,N2: nat] :
      ( aa_Code_natural_o(dvd_dvd_Code_natural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)
       => ( modulo8411746178871703098atural(plus_p4538020629002901425atural(one_one_Code_natural,A),aa_nat_Code_natural(power_7079662738309270450atural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))),N2)) = plus_p4538020629002901425atural(one_one_Code_natural,modulo8411746178871703098atural(A,aa_nat_Code_natural(power_7079662738309270450atural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))),N2))) ) ) ) ).

% even_succ_mod_exp
tff(fact_8050_dvd__antisym,axiom,
    ! [M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,M),N2)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,N2),M)
       => ( M = N2 ) ) ) ).

% dvd_antisym
tff(fact_8051_division__decomp,axiom,
    ! [A: nat,B: nat,Ca: 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),Ca))
     => ? [B8: nat,C5: nat] :
          ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B8),C5) )
          & aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B8),B)
          & aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C5),Ca) ) ) ).

% division_decomp
tff(fact_8052_division__decomp,axiom,
    ! [A: int,B: int,Ca: 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),Ca))
     => ? [B8: int,C5: int] :
          ( ( A = aa_int_int(aa_int_fun_int_int(times_times_int,B8),C5) )
          & aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B8),B)
          & aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C5),Ca) ) ) ).

% division_decomp
tff(fact_8053_dvd__productE,axiom,
    ! [P2: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,P2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B))
     => ~ ! [X3: nat,Y3: nat] :
            ( ( P2 = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,X3),Y3) )
           => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,X3),A)
             => ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Y3),B) ) ) ) ).

% dvd_productE
tff(fact_8054_dvd__productE,axiom,
    ! [P2: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,P2),aa_int_int(aa_int_fun_int_int(times_times_int,A),B))
     => ~ ! [X3: int,Y3: int] :
            ( ( P2 = aa_int_int(aa_int_fun_int_int(times_times_int,X3),Y3) )
           => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,X3),A)
             => ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Y3),B) ) ) ) ).

% dvd_productE
tff(fact_8055_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_8056_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_8057_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_8058_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_8059_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_8060_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_8061_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_8062_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_8063_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_8064_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_8065_dvd__def,axiom,
    ! [B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),A)
    <=> ? [K3: code_integer] : A = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),K3) ) ).

% dvd_def
tff(fact_8066_dvd__def,axiom,
    ! [B: assn,A: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(dvd_dvd_assn,B),A)
    <=> ? [K3: assn] : A = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,B),K3) ) ).

% dvd_def
tff(fact_8067_dvd__def,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,B),A)
    <=> ? [K3: rat] : A = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),K3) ) ).

% dvd_def
tff(fact_8068_dvd__def,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),A)
    <=> ? [K3: nat] : A = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),K3) ) ).

% dvd_def
tff(fact_8069_dvd__def,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),A)
    <=> ? [K3: int] : A = aa_int_int(aa_int_fun_int_int(times_times_int,B),K3) ) ).

% dvd_def
tff(fact_8070_dvd__mult,axiom,
    ! [A: code_integer,Ca: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),Ca)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),Ca)) ) ).

% dvd_mult
tff(fact_8071_dvd__mult,axiom,
    ! [A: assn,Ca: assn,B: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(dvd_dvd_assn,A),Ca)
     => aa_assn_o(aa_assn_fun_assn_o(dvd_dvd_assn,A),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,B),Ca)) ) ).

% dvd_mult
tff(fact_8072_dvd__mult,axiom,
    ! [A: rat,Ca: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),Ca)
     => aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),Ca)) ) ).

% dvd_mult
tff(fact_8073_dvd__mult,axiom,
    ! [A: nat,Ca: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),Ca)
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),Ca)) ) ).

% dvd_mult
tff(fact_8074_dvd__mult,axiom,
    ! [A: int,Ca: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),Ca)
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),Ca)) ) ).

% dvd_mult
tff(fact_8075_dvd__mult2,axiom,
    ! [A: code_integer,B: code_integer,Ca: 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),Ca)) ) ).

% dvd_mult2
tff(fact_8076_dvd__mult2,axiom,
    ! [A: assn,B: assn,Ca: 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),Ca)) ) ).

% dvd_mult2
tff(fact_8077_dvd__mult2,axiom,
    ! [A: rat,B: rat,Ca: 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),Ca)) ) ).

% dvd_mult2
tff(fact_8078_dvd__mult2,axiom,
    ! [A: nat,B: nat,Ca: 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),Ca)) ) ).

% dvd_mult2
tff(fact_8079_dvd__mult2,axiom,
    ! [A: int,B: int,Ca: 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),Ca)) ) ).

% dvd_mult2
tff(fact_8080_dvd__mult__left,axiom,
    ! [A: code_integer,B: code_integer,Ca: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),Ca)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),Ca) ) ).

% dvd_mult_left
tff(fact_8081_dvd__mult__left,axiom,
    ! [A: assn,B: assn,Ca: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(dvd_dvd_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A),B)),Ca)
     => aa_assn_o(aa_assn_fun_assn_o(dvd_dvd_assn,A),Ca) ) ).

% dvd_mult_left
tff(fact_8082_dvd__mult__left,axiom,
    ! [A: rat,B: rat,Ca: 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)),Ca)
     => aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),Ca) ) ).

% dvd_mult_left
tff(fact_8083_dvd__mult__left,axiom,
    ! [A: nat,B: nat,Ca: 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)),Ca)
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),Ca) ) ).

% dvd_mult_left
tff(fact_8084_dvd__mult__left,axiom,
    ! [A: int,B: int,Ca: 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)),Ca)
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),Ca) ) ).

% dvd_mult_left
tff(fact_8085_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_8086_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_8087_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_8088_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_8089_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_8090_mult__dvd__mono,axiom,
    ! [A: code_integer,B: code_integer,Ca: 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,Ca),D2)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),Ca)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),D2)) ) ) ).

% mult_dvd_mono
tff(fact_8091_mult__dvd__mono,axiom,
    ! [A: assn,B: assn,Ca: 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,Ca),D2)
       => aa_assn_o(aa_assn_fun_assn_o(dvd_dvd_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A),Ca)),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,B),D2)) ) ) ).

% mult_dvd_mono
tff(fact_8092_mult__dvd__mono,axiom,
    ! [A: rat,B: rat,Ca: 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,Ca),D2)
       => aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),Ca)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),D2)) ) ) ).

% mult_dvd_mono
tff(fact_8093_mult__dvd__mono,axiom,
    ! [A: nat,B: nat,Ca: 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,Ca),D2)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),Ca)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D2)) ) ) ).

% mult_dvd_mono
tff(fact_8094_mult__dvd__mono,axiom,
    ! [A: int,B: int,Ca: 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,Ca),D2)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),Ca)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D2)) ) ) ).

% mult_dvd_mono
tff(fact_8095_dvd__mult__right,axiom,
    ! [A: code_integer,B: code_integer,Ca: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),Ca)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),Ca) ) ).

% dvd_mult_right
tff(fact_8096_dvd__mult__right,axiom,
    ! [A: assn,B: assn,Ca: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(dvd_dvd_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A),B)),Ca)
     => aa_assn_o(aa_assn_fun_assn_o(dvd_dvd_assn,B),Ca) ) ).

% dvd_mult_right
tff(fact_8097_dvd__mult__right,axiom,
    ! [A: rat,B: rat,Ca: 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)),Ca)
     => aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,B),Ca) ) ).

% dvd_mult_right
tff(fact_8098_dvd__mult__right,axiom,
    ! [A: nat,B: nat,Ca: 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)),Ca)
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),Ca) ) ).

% dvd_mult_right
tff(fact_8099_dvd__mult__right,axiom,
    ! [A: int,B: int,Ca: 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)),Ca)
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),Ca) ) ).

% dvd_mult_right
tff(fact_8100_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_8101_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_8102_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_8103_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_8104_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_8105_dvd__diff__nat,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K),N2)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N2)) ) ) ).

% dvd_diff_nat
tff(fact_8106_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_adt,A))),collect_nat(aa_nat_fun_nat_o(aTP_Lamm_adt,B)))
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B) ) ).

% subset_divisors_dvd
tff(fact_8107_subset__divisors__dvd,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(collect_Code_integer(aa_Cod326357343816142457eger_o(aTP_Lamm_adu,A))),collect_Code_integer(aa_Cod326357343816142457eger_o(aTP_Lamm_adu,B)))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B) ) ).

% subset_divisors_dvd
tff(fact_8108_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_adv,A))),collect_int(aa_int_fun_int_o(aTP_Lamm_adv,B)))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B) ) ).

% subset_divisors_dvd
tff(fact_8109_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_adt,A))),collect_nat(aa_nat_fun_nat_o(aTP_Lamm_adt,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_8110_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_adv,A))),collect_int(aa_int_fun_int_o(aTP_Lamm_adv,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_8111_strict__subset__divisors__dvd,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_set5643387332196575111eger_o(ord_le1307284697595431911nteger(collect_Code_integer(aa_Cod326357343816142457eger_o(aTP_Lamm_adu,A))),collect_Code_integer(aa_Cod326357343816142457eger_o(aTP_Lamm_adu,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_8112_pinf_I9_J,axiom,
    ! [D2: code_integer,S2: code_integer] :
    ? [Z2: code_integer] :
    ! [X5: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Z2),X5)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D2),aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(X5),S2))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D2),aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(X5),S2)) ) ) ).

% pinf(9)
tff(fact_8113_pinf_I9_J,axiom,
    ! [D2: rat,S2: rat] :
    ? [Z2: rat] :
    ! [X5: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z2),X5)
     => ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D2),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X5),S2))
      <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D2),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X5),S2)) ) ) ).

% pinf(9)
tff(fact_8114_pinf_I9_J,axiom,
    ! [D2: nat,S2: nat] :
    ? [Z2: nat] :
    ! [X5: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z2),X5)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X5),S2))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X5),S2)) ) ) ).

% pinf(9)
tff(fact_8115_pinf_I9_J,axiom,
    ! [D2: int,S2: int] :
    ? [Z2: int] :
    ! [X5: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z2),X5)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D2),aa_int_int(aa_int_fun_int_int(plus_plus_int,X5),S2))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D2),aa_int_int(aa_int_fun_int_int(plus_plus_int,X5),S2)) ) ) ).

% pinf(9)
tff(fact_8116_pinf_I10_J,axiom,
    ! [D2: code_integer,S2: code_integer] :
    ? [Z2: code_integer] :
    ! [X5: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Z2),X5)
     => ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D2),aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(X5),S2))
      <=> ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D2),aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(X5),S2)) ) ) ).

% pinf(10)
tff(fact_8117_pinf_I10_J,axiom,
    ! [D2: rat,S2: rat] :
    ? [Z2: rat] :
    ! [X5: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z2),X5)
     => ( ~ aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D2),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X5),S2))
      <=> ~ aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D2),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X5),S2)) ) ) ).

% pinf(10)
tff(fact_8118_pinf_I10_J,axiom,
    ! [D2: nat,S2: nat] :
    ? [Z2: nat] :
    ! [X5: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z2),X5)
     => ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X5),S2))
      <=> ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X5),S2)) ) ) ).

% pinf(10)
tff(fact_8119_pinf_I10_J,axiom,
    ! [D2: int,S2: int] :
    ? [Z2: int] :
    ! [X5: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z2),X5)
     => ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D2),aa_int_int(aa_int_fun_int_int(plus_plus_int,X5),S2))
      <=> ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D2),aa_int_int(aa_int_fun_int_int(plus_plus_int,X5),S2)) ) ) ).

% pinf(10)
tff(fact_8120_minf_I9_J,axiom,
    ! [D2: code_integer,S2: code_integer] :
    ? [Z2: code_integer] :
    ! [X5: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,X5),Z2)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D2),aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(X5),S2))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D2),aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(X5),S2)) ) ) ).

% minf(9)
tff(fact_8121_minf_I9_J,axiom,
    ! [D2: rat,S2: rat] :
    ? [Z2: rat] :
    ! [X5: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X5),Z2)
     => ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D2),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X5),S2))
      <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D2),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X5),S2)) ) ) ).

% minf(9)
tff(fact_8122_minf_I9_J,axiom,
    ! [D2: nat,S2: nat] :
    ? [Z2: nat] :
    ! [X5: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X5),Z2)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X5),S2))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X5),S2)) ) ) ).

% minf(9)
tff(fact_8123_minf_I9_J,axiom,
    ! [D2: int,S2: int] :
    ? [Z2: int] :
    ! [X5: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X5),Z2)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D2),aa_int_int(aa_int_fun_int_int(plus_plus_int,X5),S2))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D2),aa_int_int(aa_int_fun_int_int(plus_plus_int,X5),S2)) ) ) ).

% minf(9)
tff(fact_8124_minf_I10_J,axiom,
    ! [D2: code_integer,S2: code_integer] :
    ? [Z2: code_integer] :
    ! [X5: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,X5),Z2)
     => ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D2),aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(X5),S2))
      <=> ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D2),aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(X5),S2)) ) ) ).

% minf(10)
tff(fact_8125_minf_I10_J,axiom,
    ! [D2: rat,S2: rat] :
    ? [Z2: rat] :
    ! [X5: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X5),Z2)
     => ( ~ aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D2),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X5),S2))
      <=> ~ aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D2),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X5),S2)) ) ) ).

% minf(10)
tff(fact_8126_minf_I10_J,axiom,
    ! [D2: nat,S2: nat] :
    ? [Z2: nat] :
    ! [X5: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X5),Z2)
     => ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X5),S2))
      <=> ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X5),S2)) ) ) ).

% minf(10)
tff(fact_8127_minf_I10_J,axiom,
    ! [D2: int,S2: int] :
    ? [Z2: int] :
    ! [X5: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X5),Z2)
     => ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D2),aa_int_int(aa_int_fun_int_int(plus_plus_int,X5),S2))
      <=> ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D2),aa_int_int(aa_int_fun_int_int(plus_plus_int,X5),S2)) ) ) ).

% minf(10)
tff(fact_8128_unit__mult__right__cancel,axiom,
    ! [A: code_integer,B: code_integer,Ca: 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,Ca),A) )
      <=> ( B = Ca ) ) ) ).

% unit_mult_right_cancel
tff(fact_8129_unit__mult__right__cancel,axiom,
    ! [A: nat,B: nat,Ca: 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,Ca),A) )
      <=> ( B = Ca ) ) ) ).

% unit_mult_right_cancel
tff(fact_8130_unit__mult__right__cancel,axiom,
    ! [A: int,B: int,Ca: 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,Ca),A) )
      <=> ( B = Ca ) ) ) ).

% unit_mult_right_cancel
tff(fact_8131_unit__mult__left__cancel,axiom,
    ! [A: code_integer,B: code_integer,Ca: 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),Ca) )
      <=> ( B = Ca ) ) ) ).

% unit_mult_left_cancel
tff(fact_8132_unit__mult__left__cancel,axiom,
    ! [A: nat,B: nat,Ca: 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),Ca) )
      <=> ( B = Ca ) ) ) ).

% unit_mult_left_cancel
tff(fact_8133_unit__mult__left__cancel,axiom,
    ! [A: int,B: int,Ca: 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),Ca) )
      <=> ( B = Ca ) ) ) ).

% unit_mult_left_cancel
tff(fact_8134_mult__unit__dvd__iff_H,axiom,
    ! [A: code_integer,B: code_integer,Ca: 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)),Ca)
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),Ca) ) ) ).

% mult_unit_dvd_iff'
tff(fact_8135_mult__unit__dvd__iff_H,axiom,
    ! [A: nat,B: nat,Ca: 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)),Ca)
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),Ca) ) ) ).

% mult_unit_dvd_iff'
tff(fact_8136_mult__unit__dvd__iff_H,axiom,
    ! [A: int,B: int,Ca: 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)),Ca)
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),Ca) ) ) ).

% mult_unit_dvd_iff'
tff(fact_8137_dvd__mult__unit__iff_H,axiom,
    ! [B: code_integer,A: code_integer,Ca: 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),Ca))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),Ca) ) ) ).

% dvd_mult_unit_iff'
tff(fact_8138_dvd__mult__unit__iff_H,axiom,
    ! [B: nat,A: nat,Ca: 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),Ca))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),Ca) ) ) ).

% dvd_mult_unit_iff'
tff(fact_8139_dvd__mult__unit__iff_H,axiom,
    ! [B: int,A: int,Ca: 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),Ca))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),Ca) ) ) ).

% dvd_mult_unit_iff'
tff(fact_8140_mult__unit__dvd__iff,axiom,
    ! [B: code_integer,A: code_integer,Ca: 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)),Ca)
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),Ca) ) ) ).

% mult_unit_dvd_iff
tff(fact_8141_mult__unit__dvd__iff,axiom,
    ! [B: nat,A: nat,Ca: 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)),Ca)
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),Ca) ) ) ).

% mult_unit_dvd_iff
tff(fact_8142_mult__unit__dvd__iff,axiom,
    ! [B: int,A: int,Ca: 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)),Ca)
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),Ca) ) ) ).

% mult_unit_dvd_iff
tff(fact_8143_dvd__mult__unit__iff,axiom,
    ! [B: code_integer,A: code_integer,Ca: 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,Ca),B))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),Ca) ) ) ).

% dvd_mult_unit_iff
tff(fact_8144_dvd__mult__unit__iff,axiom,
    ! [B: nat,A: nat,Ca: 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,Ca),B))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),Ca) ) ) ).

% dvd_mult_unit_iff
tff(fact_8145_dvd__mult__unit__iff,axiom,
    ! [B: int,A: int,Ca: 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,Ca),B))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),Ca) ) ) ).

% dvd_mult_unit_iff
tff(fact_8146_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_8147_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_8148_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_8149_div__mult__div__if__dvd,axiom,
    ! [B: nat,A: nat,D2: nat,Ca: 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),Ca)
       => ( 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,Ca),D2)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),Ca)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D2)) ) ) ) ).

% div_mult_div_if_dvd
tff(fact_8150_div__mult__div__if__dvd,axiom,
    ! [B: int,A: int,D2: int,Ca: 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),Ca)
       => ( 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,Ca),D2)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),Ca)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D2)) ) ) ) ).

% div_mult_div_if_dvd
tff(fact_8151_div__mult__div__if__dvd,axiom,
    ! [B: code_integer,A: code_integer,D2: code_integer,Ca: 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),Ca)
       => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),B)),aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(Ca),D2)) = aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),Ca)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),D2)) ) ) ) ).

% div_mult_div_if_dvd
tff(fact_8152_div__mult__div__if__dvd,axiom,
    ! [B: code_natural,A: code_natural,D2: code_natural,Ca: code_natural] :
      ( aa_Code_natural_o(dvd_dvd_Code_natural(B),A)
     => ( aa_Code_natural_o(dvd_dvd_Code_natural(D2),Ca)
       => ( times_2397367101498566445atural(divide5121882707175180666atural(A,B),divide5121882707175180666atural(Ca,D2)) = divide5121882707175180666atural(times_2397367101498566445atural(A,Ca),times_2397367101498566445atural(B,D2)) ) ) ) ).

% div_mult_div_if_dvd
tff(fact_8153_dvd__mult__imp__div,axiom,
    ! [A: nat,Ca: 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),Ca)),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),Ca)) ) ).

% dvd_mult_imp_div
tff(fact_8154_dvd__mult__imp__div,axiom,
    ! [A: int,Ca: 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),Ca)),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),Ca)) ) ).

% dvd_mult_imp_div
tff(fact_8155_dvd__mult__imp__div,axiom,
    ! [A: code_integer,Ca: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),Ca)),B)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(B),Ca)) ) ).

% dvd_mult_imp_div
tff(fact_8156_dvd__mult__imp__div,axiom,
    ! [A: code_natural,Ca: code_natural,B: code_natural] :
      ( aa_Code_natural_o(dvd_dvd_Code_natural(times_2397367101498566445atural(A,Ca)),B)
     => aa_Code_natural_o(dvd_dvd_Code_natural(A),divide5121882707175180666atural(B,Ca)) ) ).

% dvd_mult_imp_div
tff(fact_8157_dvd__div__mult2__eq,axiom,
    ! [B: nat,Ca: 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),Ca)),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),Ca)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),Ca) ) ) ).

% dvd_div_mult2_eq
tff(fact_8158_dvd__div__mult2__eq,axiom,
    ! [B: int,Ca: 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),Ca)),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),Ca)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),Ca) ) ) ).

% dvd_div_mult2_eq
tff(fact_8159_dvd__div__mult2__eq,axiom,
    ! [B: code_integer,Ca: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),Ca)),A)
     => ( aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),Ca)) = aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),B)),Ca) ) ) ).

% dvd_div_mult2_eq
tff(fact_8160_dvd__div__mult2__eq,axiom,
    ! [B: code_natural,Ca: code_natural,A: code_natural] :
      ( aa_Code_natural_o(dvd_dvd_Code_natural(times_2397367101498566445atural(B,Ca)),A)
     => ( divide5121882707175180666atural(A,times_2397367101498566445atural(B,Ca)) = divide5121882707175180666atural(divide5121882707175180666atural(A,B),Ca) ) ) ).

% dvd_div_mult2_eq
tff(fact_8161_div__div__eq__right,axiom,
    ! [Ca: nat,B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Ca),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),Ca)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),Ca) ) ) ) ).

% div_div_eq_right
tff(fact_8162_div__div__eq__right,axiom,
    ! [Ca: int,B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Ca),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),Ca)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),Ca) ) ) ) ).

% div_div_eq_right
tff(fact_8163_div__div__eq__right,axiom,
    ! [Ca: code_integer,B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Ca),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),A)
       => ( aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(B),Ca)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),B)),Ca) ) ) ) ).

% div_div_eq_right
tff(fact_8164_div__div__eq__right,axiom,
    ! [Ca: code_natural,B: code_natural,A: code_natural] :
      ( aa_Code_natural_o(dvd_dvd_Code_natural(Ca),B)
     => ( aa_Code_natural_o(dvd_dvd_Code_natural(B),A)
       => ( divide5121882707175180666atural(A,divide5121882707175180666atural(B,Ca)) = times_2397367101498566445atural(divide5121882707175180666atural(A,B),Ca) ) ) ) ).

% div_div_eq_right
tff(fact_8165_div__mult__swap,axiom,
    ! [Ca: nat,B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Ca),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),Ca)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),Ca) ) ) ).

% div_mult_swap
tff(fact_8166_div__mult__swap,axiom,
    ! [Ca: int,B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Ca),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),Ca)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),Ca) ) ) ).

% div_mult_swap
tff(fact_8167_div__mult__swap,axiom,
    ! [Ca: code_integer,B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Ca),B)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(B),Ca)) = aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),Ca) ) ) ).

% div_mult_swap
tff(fact_8168_div__mult__swap,axiom,
    ! [Ca: code_natural,B: code_natural,A: code_natural] :
      ( aa_Code_natural_o(dvd_dvd_Code_natural(Ca),B)
     => ( times_2397367101498566445atural(A,divide5121882707175180666atural(B,Ca)) = divide5121882707175180666atural(times_2397367101498566445atural(A,B),Ca) ) ) ).

% div_mult_swap
tff(fact_8169_dvd__div__mult,axiom,
    ! [Ca: nat,B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Ca),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),Ca)),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)),Ca) ) ) ).

% dvd_div_mult
tff(fact_8170_dvd__div__mult,axiom,
    ! [Ca: int,B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Ca),B)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,B),Ca)),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)),Ca) ) ) ).

% dvd_div_mult
tff(fact_8171_dvd__div__mult,axiom,
    ! [Ca: code_integer,B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Ca),B)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(B),Ca)),A) = aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),A)),Ca) ) ) ).

% dvd_div_mult
tff(fact_8172_dvd__div__mult,axiom,
    ! [Ca: code_natural,B: code_natural,A: code_natural] :
      ( aa_Code_natural_o(dvd_dvd_Code_natural(Ca),B)
     => ( times_2397367101498566445atural(divide5121882707175180666atural(B,Ca),A) = divide5121882707175180666atural(times_2397367101498566445atural(B,A),Ca) ) ) ).

% dvd_div_mult
tff(fact_8173_dvd__pos__nat,axiom,
    ! [N2: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,M),N2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M) ) ) ).

% dvd_pos_nat
tff(fact_8174_nat__dvd__not__less,axiom,
    ! [M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N2)
       => ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,N2),M) ) ) ).

% nat_dvd_not_less
tff(fact_8175_dvd__power__le,axiom,
    ! [X: code_integer,Y: code_integer,N2: nat,M: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,X),Y)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),M)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(X),N2)),aa_nat_Code_integer(power_8256067586552552935nteger(Y),M)) ) ) ).

% dvd_power_le
tff(fact_8176_dvd__power__le,axiom,
    ! [X: nat,Y: nat,N2: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,X),Y)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),M)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X),N2)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Y),M)) ) ) ).

% dvd_power_le
tff(fact_8177_dvd__power__le,axiom,
    ! [X: int,Y: int,N2: nat,M: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,X),Y)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),M)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,X),N2)),aa_nat_int(aa_int_fun_nat_int(power_power_int,Y),M)) ) ) ).

% dvd_power_le
tff(fact_8178_power__le__dvd,axiom,
    ! [A: code_integer,N2: nat,B: code_integer,M: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(A),N2)),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(A),M)),B) ) ) ).

% power_le_dvd
tff(fact_8179_power__le__dvd,axiom,
    ! [A: nat,N2: nat,B: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N2)),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),M)),B) ) ) ).

% power_le_dvd
tff(fact_8180_power__le__dvd,axiom,
    ! [A: int,N2: nat,B: int,M: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A),N2)),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A),M)),B) ) ) ).

% power_le_dvd
tff(fact_8181_le__imp__power__dvd,axiom,
    ! [M: nat,N2: nat,A: code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(A),M)),aa_nat_Code_integer(power_8256067586552552935nteger(A),N2)) ) ).

% le_imp_power_dvd
tff(fact_8182_le__imp__power__dvd,axiom,
    ! [M: nat,N2: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_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),N2)) ) ).

% le_imp_power_dvd
tff(fact_8183_le__imp__power__dvd,axiom,
    ! [M: nat,N2: nat,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_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),N2)) ) ).

% le_imp_power_dvd
tff(fact_8184_dvd__minus__self,axiom,
    ! [M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),M))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),M)
        | aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,M),N2) ) ) ).

% dvd_minus_self
tff(fact_8185_less__eq__dvd__minus,axiom,
    ! [M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,M),N2)
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),M)) ) ) ).

% less_eq_dvd_minus
tff(fact_8186_dvd__diffD1,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N2))
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K),M)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),M)
         => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K),N2) ) ) ) ).

% dvd_diffD1
tff(fact_8187_dvd__diffD,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N2))
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K),N2)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),M)
         => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K),M) ) ) ) ).

% dvd_diffD
tff(fact_8188_fact__dvd,axiom,
    ! [N2: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),M)
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(semiri1406184849735516958ct_int,N2)),aa_nat_int(semiri1406184849735516958ct_int,M)) ) ).

% fact_dvd
tff(fact_8189_fact__dvd,axiom,
    ! [N2: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),M)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_nat_Code_integer(semiri3624122377584611663nteger,N2)),aa_nat_Code_integer(semiri3624122377584611663nteger,M)) ) ).

% fact_dvd
tff(fact_8190_fact__dvd,axiom,
    ! [N2: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),M)
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(semiri1408675320244567234ct_nat,N2)),aa_nat_nat(semiri1408675320244567234ct_nat,M)) ) ).

% fact_dvd
tff(fact_8191_unity__coeff__ex,axiom,
    ! [Pa: fun_Code_integer_o,L: code_integer] :
      ( ? [X4: code_integer] : aa_Code_integer_o(Pa,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(plus_p5714425477246183910nteger(X4),zero_z3403309356797280102nteger))
          & aa_Code_integer_o(Pa,X4) ) ) ).

% unity_coeff_ex
tff(fact_8192_unity__coeff__ex,axiom,
    ! [Pa: fun_rat_o,L: rat] :
      ( ? [X4: rat] : aa_rat_o(Pa,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(Pa,X4) ) ) ).

% unity_coeff_ex
tff(fact_8193_unity__coeff__ex,axiom,
    ! [Pa: fun_nat_o,L: nat] :
      ( ? [X4: nat] : aa_nat_o(Pa,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(Pa,X4) ) ) ).

% unity_coeff_ex
tff(fact_8194_unity__coeff__ex,axiom,
    ! [Pa: fun_int_o,L: int] :
      ( ? [X4: int] : aa_int_o(Pa,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(Pa,X4) ) ) ).

% unity_coeff_ex
tff(fact_8195_unit__dvdE,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),one_one_Code_integer)
     => ~ ( ( A != zero_z3403309356797280102nteger )
         => ! [C: code_integer] : B != aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C) ) ) ).

% unit_dvdE
tff(fact_8196_unit__dvdE,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),one_one_nat)
     => ~ ( ( A != zero_zero_nat )
         => ! [C: nat] : B != aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C) ) ) ).

% unit_dvdE
tff(fact_8197_unit__dvdE,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int)
     => ~ ( ( A != zero_zero_int )
         => ! [C: int] : B != aa_int_int(aa_int_fun_int_int(times_times_int,A),C) ) ) ).

% unit_dvdE
tff(fact_8198_inf__period_I4_J,axiom,
    ! [D2: code_integer,D3: code_integer,Ta: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D2),D3)
     => ! [X5: code_integer,K4: code_integer] :
          ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D2),aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(X5),Ta))
        <=> ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D2),aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(X5),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,K4),D3))),Ta)) ) ) ).

% inf_period(4)
tff(fact_8199_inf__period_I4_J,axiom,
    ! [D2: rat,D3: rat,Ta: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D2),D3)
     => ! [X5: rat,K4: rat] :
          ( ~ aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D2),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X5),Ta))
        <=> ~ 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,X5),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,K4),D3))),Ta)) ) ) ).

% inf_period(4)
tff(fact_8200_inf__period_I4_J,axiom,
    ! [D2: int,D3: int,Ta: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D2),D3)
     => ! [X5: int,K4: int] :
          ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D2),aa_int_int(aa_int_fun_int_int(plus_plus_int,X5),Ta))
        <=> ~ 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,X5),aa_int_int(aa_int_fun_int_int(times_times_int,K4),D3))),Ta)) ) ) ).

% inf_period(4)
tff(fact_8201_inf__period_I3_J,axiom,
    ! [D2: code_integer,D3: code_integer,Ta: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D2),D3)
     => ! [X5: code_integer,K4: code_integer] :
          ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D2),aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(X5),Ta))
        <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D2),aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(X5),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,K4),D3))),Ta)) ) ) ).

% inf_period(3)
tff(fact_8202_inf__period_I3_J,axiom,
    ! [D2: rat,D3: rat,Ta: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D2),D3)
     => ! [X5: rat,K4: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D2),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X5),Ta))
        <=> 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,X5),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,K4),D3))),Ta)) ) ) ).

% inf_period(3)
tff(fact_8203_inf__period_I3_J,axiom,
    ! [D2: int,D3: int,Ta: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D2),D3)
     => ! [X5: int,K4: int] :
          ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D2),aa_int_int(aa_int_fun_int_int(plus_plus_int,X5),Ta))
        <=> 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,X5),aa_int_int(aa_int_fun_int_int(times_times_int,K4),D3))),Ta)) ) ) ).

% inf_period(3)
tff(fact_8204_dvd__div__div__eq__mult,axiom,
    ! [A: nat,Ca: nat,B: nat,D2: nat] :
      ( ( A != zero_zero_nat )
     => ( ( Ca != 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,Ca),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),Ca) )
            <=> ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),Ca) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),D2) ) ) ) ) ) ) ).

% dvd_div_div_eq_mult
tff(fact_8205_dvd__div__div__eq__mult,axiom,
    ! [A: int,Ca: int,B: int,D2: int] :
      ( ( A != zero_zero_int )
     => ( ( Ca != 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,Ca),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),Ca) )
            <=> ( aa_int_int(aa_int_fun_int_int(times_times_int,B),Ca) = aa_int_int(aa_int_fun_int_int(times_times_int,A),D2) ) ) ) ) ) ) ).

% dvd_div_div_eq_mult
tff(fact_8206_dvd__div__div__eq__mult,axiom,
    ! [A: code_integer,Ca: code_integer,B: code_integer,D2: code_integer] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( ( Ca != 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,Ca),D2)
           => ( ( aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(B),A) = aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(D2),Ca) )
            <=> ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),Ca) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),D2) ) ) ) ) ) ) ).

% dvd_div_div_eq_mult
tff(fact_8207_dvd__div__div__eq__mult,axiom,
    ! [A: code_natural,Ca: code_natural,B: code_natural,D2: code_natural] :
      ( ( A != zero_z2226904508553997617atural )
     => ( ( Ca != zero_z2226904508553997617atural )
       => ( aa_Code_natural_o(dvd_dvd_Code_natural(A),B)
         => ( aa_Code_natural_o(dvd_dvd_Code_natural(Ca),D2)
           => ( ( divide5121882707175180666atural(B,A) = divide5121882707175180666atural(D2,Ca) )
            <=> ( times_2397367101498566445atural(B,Ca) = times_2397367101498566445atural(A,D2) ) ) ) ) ) ) ).

% dvd_div_div_eq_mult
tff(fact_8208_dvd__div__iff__mult,axiom,
    ! [Ca: nat,B: nat,A: nat] :
      ( ( Ca != zero_zero_nat )
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Ca),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),Ca))
        <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),Ca)),B) ) ) ) ).

% dvd_div_iff_mult
tff(fact_8209_dvd__div__iff__mult,axiom,
    ! [Ca: int,B: int,A: int] :
      ( ( Ca != zero_zero_int )
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Ca),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),Ca))
        <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),Ca)),B) ) ) ) ).

% dvd_div_iff_mult
tff(fact_8210_dvd__div__iff__mult,axiom,
    ! [Ca: code_integer,B: code_integer,A: code_integer] :
      ( ( Ca != zero_z3403309356797280102nteger )
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Ca),B)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(B),Ca))
        <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),Ca)),B) ) ) ) ).

% dvd_div_iff_mult
tff(fact_8211_dvd__div__iff__mult,axiom,
    ! [Ca: code_natural,B: code_natural,A: code_natural] :
      ( ( Ca != zero_z2226904508553997617atural )
     => ( aa_Code_natural_o(dvd_dvd_Code_natural(Ca),B)
       => ( aa_Code_natural_o(dvd_dvd_Code_natural(A),divide5121882707175180666atural(B,Ca))
        <=> aa_Code_natural_o(dvd_dvd_Code_natural(times_2397367101498566445atural(A,Ca)),B) ) ) ) ).

% dvd_div_iff_mult
tff(fact_8212_div__dvd__iff__mult,axiom,
    ! [B: nat,A: nat,Ca: 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)),Ca)
        <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ca),B)) ) ) ) ).

% div_dvd_iff_mult
tff(fact_8213_div__dvd__iff__mult,axiom,
    ! [B: int,A: int,Ca: 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)),Ca)
        <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,Ca),B)) ) ) ) ).

% div_dvd_iff_mult
tff(fact_8214_div__dvd__iff__mult,axiom,
    ! [B: code_integer,A: code_integer,Ca: 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)),Ca)
        <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Ca),B)) ) ) ) ).

% div_dvd_iff_mult
tff(fact_8215_div__dvd__iff__mult,axiom,
    ! [B: code_natural,A: code_natural,Ca: 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)),Ca)
        <=> aa_Code_natural_o(dvd_dvd_Code_natural(A),times_2397367101498566445atural(Ca,B)) ) ) ) ).

% div_dvd_iff_mult
tff(fact_8216_dvd__div__eq__mult,axiom,
    ! [A: nat,B: nat,Ca: 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) = Ca )
        <=> ( B = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ca),A) ) ) ) ) ).

% dvd_div_eq_mult
tff(fact_8217_dvd__div__eq__mult,axiom,
    ! [A: int,B: int,Ca: 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) = Ca )
        <=> ( B = aa_int_int(aa_int_fun_int_int(times_times_int,Ca),A) ) ) ) ) ).

% dvd_div_eq_mult
tff(fact_8218_dvd__div__eq__mult,axiom,
    ! [A: code_integer,B: code_integer,Ca: code_integer] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B)
       => ( ( aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(B),A) = Ca )
        <=> ( B = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Ca),A) ) ) ) ) ).

% dvd_div_eq_mult
tff(fact_8219_dvd__div__eq__mult,axiom,
    ! [A: code_natural,B: code_natural,Ca: code_natural] :
      ( ( A != zero_z2226904508553997617atural )
     => ( aa_Code_natural_o(dvd_dvd_Code_natural(A),B)
       => ( ( divide5121882707175180666atural(B,A) = Ca )
        <=> ( B = times_2397367101498566445atural(Ca,A) ) ) ) ) ).

% dvd_div_eq_mult
tff(fact_8220_is__unit__div__mult2__eq,axiom,
    ! [B: nat,Ca: 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,Ca),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),Ca)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),Ca) ) ) ) ).

% is_unit_div_mult2_eq
tff(fact_8221_is__unit__div__mult2__eq,axiom,
    ! [B: int,Ca: 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,Ca),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),Ca)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),Ca) ) ) ) ).

% is_unit_div_mult2_eq
tff(fact_8222_is__unit__div__mult2__eq,axiom,
    ! [B: code_integer,Ca: 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,Ca),one_one_Code_integer)
       => ( aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),Ca)) = aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),B)),Ca) ) ) ) ).

% is_unit_div_mult2_eq
tff(fact_8223_is__unit__div__mult2__eq,axiom,
    ! [B: code_natural,Ca: 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(Ca),one_one_Code_natural)
       => ( divide5121882707175180666atural(A,times_2397367101498566445atural(B,Ca)) = divide5121882707175180666atural(divide5121882707175180666atural(A,B),Ca) ) ) ) ).

% is_unit_div_mult2_eq
tff(fact_8224_unit__div__mult__swap,axiom,
    ! [Ca: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Ca),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),Ca)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),Ca) ) ) ).

% unit_div_mult_swap
tff(fact_8225_unit__div__mult__swap,axiom,
    ! [Ca: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Ca),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),Ca)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),Ca) ) ) ).

% unit_div_mult_swap
tff(fact_8226_unit__div__mult__swap,axiom,
    ! [Ca: code_integer,A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Ca),one_one_Code_integer)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(B),Ca)) = aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),Ca) ) ) ).

% unit_div_mult_swap
tff(fact_8227_unit__div__mult__swap,axiom,
    ! [Ca: code_natural,A: code_natural,B: code_natural] :
      ( aa_Code_natural_o(dvd_dvd_Code_natural(Ca),one_one_Code_natural)
     => ( times_2397367101498566445atural(A,divide5121882707175180666atural(B,Ca)) = divide5121882707175180666atural(times_2397367101498566445atural(A,B),Ca) ) ) ).

% unit_div_mult_swap
tff(fact_8228_unit__div__commute,axiom,
    ! [B: nat,A: nat,Ca: 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)),Ca) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),Ca)),B) ) ) ).

% unit_div_commute
tff(fact_8229_unit__div__commute,axiom,
    ! [B: int,A: int,Ca: 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)),Ca) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),Ca)),B) ) ) ).

% unit_div_commute
tff(fact_8230_unit__div__commute,axiom,
    ! [B: code_integer,A: code_integer,Ca: 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)),Ca) = aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),Ca)),B) ) ) ).

% unit_div_commute
tff(fact_8231_unit__div__commute,axiom,
    ! [B: code_natural,A: code_natural,Ca: code_natural] :
      ( aa_Code_natural_o(dvd_dvd_Code_natural(B),one_one_Code_natural)
     => ( times_2397367101498566445atural(divide5121882707175180666atural(A,B),Ca) = divide5121882707175180666atural(times_2397367101498566445atural(A,Ca),B) ) ) ).

% unit_div_commute
tff(fact_8232_div__mult__unit2,axiom,
    ! [Ca: nat,B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Ca),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),Ca)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),Ca) ) ) ) ).

% div_mult_unit2
tff(fact_8233_div__mult__unit2,axiom,
    ! [Ca: int,B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Ca),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),Ca)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),Ca) ) ) ) ).

% div_mult_unit2
tff(fact_8234_div__mult__unit2,axiom,
    ! [Ca: code_integer,B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Ca),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),Ca)) = aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),B)),Ca) ) ) ) ).

% div_mult_unit2
tff(fact_8235_div__mult__unit2,axiom,
    ! [Ca: code_natural,B: code_natural,A: code_natural] :
      ( aa_Code_natural_o(dvd_dvd_Code_natural(Ca),one_one_Code_natural)
     => ( aa_Code_natural_o(dvd_dvd_Code_natural(B),A)
       => ( divide5121882707175180666atural(A,times_2397367101498566445atural(B,Ca)) = divide5121882707175180666atural(divide5121882707175180666atural(A,B),Ca) ) ) ) ).

% div_mult_unit2
tff(fact_8236_unit__eq__div2,axiom,
    ! [B: nat,A: nat,Ca: 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,Ca),B) )
      <=> ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B) = Ca ) ) ) ).

% unit_eq_div2
tff(fact_8237_unit__eq__div2,axiom,
    ! [B: int,A: int,Ca: 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,Ca),B) )
      <=> ( aa_int_int(aa_int_fun_int_int(times_times_int,A),B) = Ca ) ) ) ).

% unit_eq_div2
tff(fact_8238_unit__eq__div2,axiom,
    ! [B: code_integer,A: code_integer,Ca: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer)
     => ( ( A = aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(Ca),B) )
      <=> ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B) = Ca ) ) ) ).

% unit_eq_div2
tff(fact_8239_unit__eq__div2,axiom,
    ! [B: code_natural,A: code_natural,Ca: code_natural] :
      ( aa_Code_natural_o(dvd_dvd_Code_natural(B),one_one_Code_natural)
     => ( ( A = divide5121882707175180666atural(Ca,B) )
      <=> ( times_2397367101498566445atural(A,B) = Ca ) ) ) ).

% unit_eq_div2
tff(fact_8240_unit__eq__div1,axiom,
    ! [B: nat,A: nat,Ca: 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) = Ca )
      <=> ( A = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ca),B) ) ) ) ).

% unit_eq_div1
tff(fact_8241_unit__eq__div1,axiom,
    ! [B: int,A: int,Ca: 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) = Ca )
      <=> ( A = aa_int_int(aa_int_fun_int_int(times_times_int,Ca),B) ) ) ) ).

% unit_eq_div1
tff(fact_8242_unit__eq__div1,axiom,
    ! [B: code_integer,A: code_integer,Ca: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer)
     => ( ( aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),B) = Ca )
      <=> ( A = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Ca),B) ) ) ) ).

% unit_eq_div1
tff(fact_8243_unit__eq__div1,axiom,
    ! [B: code_natural,A: code_natural,Ca: code_natural] :
      ( aa_Code_natural_o(dvd_dvd_Code_natural(B),one_one_Code_natural)
     => ( ( divide5121882707175180666atural(A,B) = Ca )
      <=> ( A = times_2397367101498566445atural(Ca,B) ) ) ) ).

% unit_eq_div1
tff(fact_8244_round__mono,axiom,
    ! [X: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(archim7778729529865785530nd_rat,X)),aa_rat_int(archim7778729529865785530nd_rat,Y)) ) ).

% round_mono
tff(fact_8245_prod__dvd__prod__subset,axiom,
    ! [B3: set_Code_integer,A3: set_Code_integer,F: fun_Code_integer_nat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B3)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(A3),B3)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,groups3190895334310489300er_nat(F,A3)),groups3190895334310489300er_nat(F,B3)) ) ) ).

% prod_dvd_prod_subset
tff(fact_8246_prod__dvd__prod__subset,axiom,
    ! [B3: set_Code_integer,A3: set_Code_integer,F: fun_Code_integer_int] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B3)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(A3),B3)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,groups3188404863801439024er_int(F,A3)),groups3188404863801439024er_int(F,B3)) ) ) ).

% prod_dvd_prod_subset
tff(fact_8247_prod__dvd__prod__subset,axiom,
    ! [B3: set_nat,A3: set_nat,F: fun_nat_Code_integer] :
      ( aa_set_nat_o(finite_finite_nat,B3)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A3),B3)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_set871585252387937313nteger(groups3455450783089532116nteger(F),A3)),aa_set871585252387937313nteger(groups3455450783089532116nteger(F),B3)) ) ) ).

% prod_dvd_prod_subset
tff(fact_8248_prod__dvd__prod__subset,axiom,
    ! [B3: set_Code_integer,A3: set_Code_integer,F: fun_Co8499238673658658103nteger] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B3)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(A3),B3)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,groups3674199335183972705nteger(F,A3)),groups3674199335183972705nteger(F,B3)) ) ) ).

% prod_dvd_prod_subset
tff(fact_8249_prod__dvd__prod__subset,axiom,
    ! [B3: set_int,A3: set_int,F: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,B3)
     => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A3),B3)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,groups1707563613775114915nt_nat(F,A3)),groups1707563613775114915nt_nat(F,B3)) ) ) ).

% prod_dvd_prod_subset
tff(fact_8250_prod__dvd__prod__subset,axiom,
    ! [B3: set_int,A3: set_int,F: fun_int_Code_integer] :
      ( aa_set_int_o(finite_finite_int,B3)
     => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A3),B3)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,groups3827104343326376752nteger(F,A3)),groups3827104343326376752nteger(F,B3)) ) ) ).

% prod_dvd_prod_subset
tff(fact_8251_prod__dvd__prod__subset,axiom,
    ! [B3: set_nat,A3: set_nat,F: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,B3)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A3),B3)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_set_nat_nat(groups708209901874060359at_nat(F),A3)),aa_set_nat_nat(groups708209901874060359at_nat(F),B3)) ) ) ).

% prod_dvd_prod_subset
tff(fact_8252_prod__dvd__prod__subset,axiom,
    ! [B3: set_nat,A3: set_nat,F: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,B3)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A3),B3)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_set_nat_int(groups705719431365010083at_int(F),A3)),aa_set_nat_int(groups705719431365010083at_int(F),B3)) ) ) ).

% prod_dvd_prod_subset
tff(fact_8253_prod__dvd__prod__subset,axiom,
    ! [B3: set_int,A3: set_int,F: fun_int_int] :
      ( aa_set_int_o(finite_finite_int,B3)
     => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A3),B3)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,groups1705073143266064639nt_int(F,A3)),groups1705073143266064639nt_int(F,B3)) ) ) ).

% prod_dvd_prod_subset
tff(fact_8254_prod__dvd__prod__subset,axiom,
    ! [B3: set_Pr1261947904930325089at_nat,A3: set_Pr1261947904930325089at_nat,F: fun_Pr2602117396471556196at_nat] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,B3)
     => ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A3),B3)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,groups4077766827762148844at_nat(F,A3)),groups4077766827762148844at_nat(F,B3)) ) ) ).

% prod_dvd_prod_subset
tff(fact_8255_prod__dvd__prod__subset2,axiom,
    ! [B3: set_o,A3: set_o,F: fun_o_nat,G: fun_o_nat] :
      ( finite_finite_o(B3)
     => ( ord_less_eq_set_o(A3,B3)
       => ( ! [A4: $o] :
              ( member_o((A4),A3)
             => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_o_nat(F,(A4))),aa_o_nat(G,(A4))) )
         => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,groups3504817904513533571_o_nat(F,A3)),groups3504817904513533571_o_nat(G,B3)) ) ) ) ).

% prod_dvd_prod_subset2
tff(fact_8256_prod__dvd__prod__subset2,axiom,
    ! [B3: set_Code_integer,A3: set_Code_integer,F: fun_Code_integer_nat,G: fun_Code_integer_nat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B3)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(A3),B3)
       => ( ! [A4: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(A4),A3)
             => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_Code_integer_nat(F,A4)),aa_Code_integer_nat(G,A4)) )
         => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,groups3190895334310489300er_nat(F,A3)),groups3190895334310489300er_nat(G,B3)) ) ) ) ).

% prod_dvd_prod_subset2
tff(fact_8257_prod__dvd__prod__subset2,axiom,
    ! [B3: set_o,A3: set_o,F: fun_o_int,G: fun_o_int] :
      ( finite_finite_o(B3)
     => ( ord_less_eq_set_o(A3,B3)
       => ( ! [A4: $o] :
              ( member_o((A4),A3)
             => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_o_int(F,(A4))),aa_o_int(G,(A4))) )
         => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,groups3502327434004483295_o_int(F,A3)),groups3502327434004483295_o_int(G,B3)) ) ) ) ).

% prod_dvd_prod_subset2
tff(fact_8258_prod__dvd__prod__subset2,axiom,
    ! [B3: set_Code_integer,A3: set_Code_integer,F: fun_Code_integer_int,G: fun_Code_integer_int] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B3)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(A3),B3)
       => ( ! [A4: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(A4),A3)
             => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_Code_integer_int(F,A4)),aa_Code_integer_int(G,A4)) )
         => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,groups3188404863801439024er_int(F,A3)),groups3188404863801439024er_int(G,B3)) ) ) ) ).

% prod_dvd_prod_subset2
tff(fact_8259_prod__dvd__prod__subset2,axiom,
    ! [B3: set_o,A3: set_o,F: fun_o_Code_integer,G: fun_o_Code_integer] :
      ( finite_finite_o(B3)
     => ( ord_less_eq_set_o(A3,B3)
       => ( ! [A4: $o] :
              ( member_o((A4),A3)
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_o_Code_integer(F,(A4))),aa_o_Code_integer(G,(A4))) )
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,groups7694694392188491536nteger(F,A3)),groups7694694392188491536nteger(G,B3)) ) ) ) ).

% prod_dvd_prod_subset2
tff(fact_8260_prod__dvd__prod__subset2,axiom,
    ! [B3: set_nat,A3: set_nat,F: fun_nat_Code_integer,G: fun_nat_Code_integer] :
      ( aa_set_nat_o(finite_finite_nat,B3)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A3),B3)
       => ( ! [A4: nat] :
              ( aa_set_nat_o(member_nat(A4),A3)
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_nat_Code_integer(F,A4)),aa_nat_Code_integer(G,A4)) )
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_set871585252387937313nteger(groups3455450783089532116nteger(F),A3)),aa_set871585252387937313nteger(groups3455450783089532116nteger(G),B3)) ) ) ) ).

% prod_dvd_prod_subset2
tff(fact_8261_prod__dvd__prod__subset2,axiom,
    ! [B3: set_Code_integer,A3: set_Code_integer,F: fun_Co8499238673658658103nteger,G: fun_Co8499238673658658103nteger] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B3)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(A3),B3)
       => ( ! [A4: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(A4),A3)
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(F,A4)),aa_Cod1783974652533396728nteger(G,A4)) )
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,groups3674199335183972705nteger(F,A3)),groups3674199335183972705nteger(G,B3)) ) ) ) ).

% prod_dvd_prod_subset2
tff(fact_8262_prod__dvd__prod__subset2,axiom,
    ! [B3: set_int,A3: set_int,F: fun_int_nat,G: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,B3)
     => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A3),B3)
       => ( ! [A4: int] :
              ( aa_set_int_o(member_int(A4),A3)
             => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_int_nat(F,A4)),aa_int_nat(G,A4)) )
         => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,groups1707563613775114915nt_nat(F,A3)),groups1707563613775114915nt_nat(G,B3)) ) ) ) ).

% prod_dvd_prod_subset2
tff(fact_8263_prod__dvd__prod__subset2,axiom,
    ! [B3: set_int,A3: set_int,F: fun_int_Code_integer,G: fun_int_Code_integer] :
      ( aa_set_int_o(finite_finite_int,B3)
     => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A3),B3)
       => ( ! [A4: int] :
              ( aa_set_int_o(member_int(A4),A3)
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_int_Code_integer(F,A4)),aa_int_Code_integer(G,A4)) )
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,groups3827104343326376752nteger(F,A3)),groups3827104343326376752nteger(G,B3)) ) ) ) ).

% prod_dvd_prod_subset2
tff(fact_8264_prod__dvd__prod__subset2,axiom,
    ! [B3: set_nat,A3: set_nat,F: fun_nat_nat,G: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,B3)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A3),B3)
       => ( ! [A4: nat] :
              ( aa_set_nat_o(member_nat(A4),A3)
             => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(F,A4)),aa_nat_nat(G,A4)) )
         => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_set_nat_nat(groups708209901874060359at_nat(F),A3)),aa_set_nat_nat(groups708209901874060359at_nat(G),B3)) ) ) ) ).

% prod_dvd_prod_subset2
tff(fact_8265_dvd__imp__le,axiom,
    ! [K: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K),N2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),N2) ) ) ).

% dvd_imp_le
tff(fact_8266_dvd__mult__cancel,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),N2))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,M),N2) ) ) ).

% dvd_mult_cancel
tff(fact_8267_nat__mult__dvd__cancel1,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),N2))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,M),N2) ) ) ).

% nat_mult_dvd_cancel1
tff(fact_8268_fact__fact__dvd__fact,axiom,
    ! [K: nat,N2: 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,N2))),aa_nat_Code_integer(semiri3624122377584611663nteger,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),N2))) ).

% fact_fact_dvd_fact
tff(fact_8269_fact__fact__dvd__fact,axiom,
    ! [K: nat,N2: 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,N2))),aa_nat_rat(semiri773545260158071498ct_rat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),N2))) ).

% fact_fact_dvd_fact
tff(fact_8270_fact__fact__dvd__fact,axiom,
    ! [K: nat,N2: 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,N2))),aa_nat_int(semiri1406184849735516958ct_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),N2))) ).

% fact_fact_dvd_fact
tff(fact_8271_fact__fact__dvd__fact,axiom,
    ! [K: nat,N2: 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,N2))),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),N2))) ).

% fact_fact_dvd_fact
tff(fact_8272_mod__greater__zero__iff__not__dvd,axiom,
    ! [M: nat,N2: 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),N2))
    <=> ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,N2),M) ) ).

% mod_greater_zero_iff_not_dvd
tff(fact_8273_mod__eq__dvd__iff__nat,axiom,
    ! [N2: nat,M: nat,Q4: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),M)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Q4) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N2),Q4) )
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Q4),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N2)) ) ) ).

% mod_eq_dvd_iff_nat
tff(fact_8274_floor__le__round,axiom,
    ! [X: rat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(archim3151403230148437115or_rat,X)),aa_rat_int(archim7778729529865785530nd_rat,X)) ).

% floor_le_round
tff(fact_8275_dvd__fact,axiom,
    ! [M: nat,N2: 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),N2)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,M),aa_nat_nat(semiri1408675320244567234ct_nat,N2)) ) ) ).

% dvd_fact
tff(fact_8276_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_adt,M))) ) ).

% finite_divisors_nat
tff(fact_8277_gcd__nat_Oordering__top__axioms,axiom,
    aa_nat_o(ordering_top_nat(dvd_dvd_nat,aTP_Lamm_adw),zero_zero_nat) ).

% gcd_nat.ordering_top_axioms
tff(fact_8278_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_8279_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_8280_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_8281_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_8282_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_8283_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_8284_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_8285_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_8286_is__unitE,axiom,
    ! [A: nat,Ca: 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,Ca),A) != aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ca),B4) ) ) ) ) ) ) ) ) ).

% is_unitE
tff(fact_8287_is__unitE,axiom,
    ! [A: int,Ca: 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,Ca),A) != aa_int_int(aa_int_fun_int_int(times_times_int,Ca),B4) ) ) ) ) ) ) ) ) ).

% is_unitE
tff(fact_8288_is__unitE,axiom,
    ! [A: code_integer,Ca: 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(Ca),A) != aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Ca),B4) ) ) ) ) ) ) ) ) ).

% is_unitE
tff(fact_8289_is__unitE,axiom,
    ! [A: code_natural,Ca: 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(Ca,A) != times_2397367101498566445atural(Ca,B4) ) ) ) ) ) ) ) ) ).

% is_unitE
tff(fact_8290_evenE,axiom,
    ! [A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
     => ~ ! [B4: nat] : A != aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B4) ) ).

% evenE
tff(fact_8291_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_8292_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_8293_evenE,axiom,
    ! [A: code_natural] :
      ( aa_Code_natural_o(dvd_dvd_Code_natural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))),A)
     => ~ ! [B4: code_natural] : A != times_2397367101498566445atural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one)),B4) ) ).

% evenE
tff(fact_8294_dvd__power__iff,axiom,
    ! [X: code_integer,M: nat,N2: 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),N2))
      <=> ( 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),N2) ) ) ) ).

% dvd_power_iff
tff(fact_8295_dvd__power__iff,axiom,
    ! [X: nat,M: nat,N2: 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),N2))
      <=> ( 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),N2) ) ) ) ).

% dvd_power_iff
tff(fact_8296_dvd__power__iff,axiom,
    ! [X: int,M: nat,N2: 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),N2))
      <=> ( 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),N2) ) ) ) ).

% dvd_power_iff
tff(fact_8297_dvd__power,axiom,
    ! [N2: nat,X: code_integer] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)
        | ( X = one_one_Code_integer ) )
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,X),aa_nat_Code_integer(power_8256067586552552935nteger(X),N2)) ) ).

% dvd_power
tff(fact_8298_dvd__power,axiom,
    ! [N2: nat,X: rat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)
        | ( X = one_one_rat ) )
     => aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,X),aa_nat_rat(power_power_rat(X),N2)) ) ).

% dvd_power
tff(fact_8299_dvd__power,axiom,
    ! [N2: nat,X: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)
        | ( 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),N2)) ) ).

% dvd_power
tff(fact_8300_dvd__power,axiom,
    ! [N2: nat,X: int] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)
        | ( 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),N2)) ) ).

% dvd_power
tff(fact_8301_dvd__mult__cancel1,axiom,
    ! [M: nat,N2: 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),N2)),M)
      <=> ( N2 = one_one_nat ) ) ) ).

% dvd_mult_cancel1
tff(fact_8302_dvd__mult__cancel2,axiom,
    ! [M: nat,N2: 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,N2),M)),M)
      <=> ( N2 = one_one_nat ) ) ) ).

% dvd_mult_cancel2
tff(fact_8303_choose__dvd,axiom,
    ! [K: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),N2)
     => 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,N2),K)))),aa_nat_Code_integer(semiri3624122377584611663nteger,N2)) ) ).

% choose_dvd
tff(fact_8304_choose__dvd,axiom,
    ! [K: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),N2)
     => 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,N2),K)))),aa_nat_rat(semiri773545260158071498ct_rat,N2)) ) ).

% choose_dvd
tff(fact_8305_choose__dvd,axiom,
    ! [K: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),N2)
     => 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,N2),K)))),aa_nat_int(semiri1406184849735516958ct_int,N2)) ) ).

% choose_dvd
tff(fact_8306_choose__dvd,axiom,
    ! [K: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),N2)
     => 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,N2),K)))),aa_nat_nat(semiri1408675320244567234ct_nat,N2)) ) ).

% choose_dvd
tff(fact_8307_dvd__minus__add,axiom,
    ! [Q4: nat,N2: nat,Ra2: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Q4),N2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Q4),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ra2),M))
       => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),Q4))
        <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ra2),M)),Q4))) ) ) ) ).

% dvd_minus_add
tff(fact_8308_mod__nat__eqI,axiom,
    ! [Ra2: nat,N2: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ra2),N2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ra2),M)
       => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,N2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Ra2))
         => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N2) = Ra2 ) ) ) ) ).

% mod_nat_eqI
tff(fact_8309_power__dvd__imp__le,axiom,
    ! [I2: nat,M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,I2),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,I2),N2))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),I2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2) ) ) ).

% power_dvd_imp_le
tff(fact_8310_even__two__times__div__two,axiom,
    ! [A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = A ) ) ).

% even_two_times_div_two
tff(fact_8311_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_8312_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_8313_even__two__times__div__two,axiom,
    ! [A: code_natural] :
      ( aa_Code_natural_o(dvd_dvd_Code_natural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))),A)
     => ( times_2397367101498566445atural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one)),divide5121882707175180666atural(A,aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one)))) = A ) ) ).

% even_two_times_div_two
tff(fact_8314_power__mono__odd,axiom,
    ! [N2: nat,A: rat,B: rat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),N2)),aa_nat_rat(power_power_rat(B),N2)) ) ) ).

% power_mono_odd
tff(fact_8315_power__mono__odd,axiom,
    ! [N2: nat,A: int,B: int] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A),N2)),aa_nat_int(aa_int_fun_nat_int(power_power_int,B),N2)) ) ) ).

% power_mono_odd
tff(fact_8316_odd__pos,axiom,
    ! [N2: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2) ) ).

% odd_pos
tff(fact_8317_dvd__power__iff__le,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,K),N2))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2) ) ) ).

% dvd_power_iff_le
tff(fact_8318_sum__div__partition,axiom,
    ! [A3: set_int,F: fun_int_nat,B: nat] :
      ( aa_set_int_o(finite_finite_int,A3)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,groups4541462559716669496nt_nat(F,A3)),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups4541462559716669496nt_nat(aa_nat_fun_int_nat(aTP_Lamm_adx(F),B),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A3),collect_int(aa_nat_fun_int_o(aTP_Lamm_ady(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,A3),collect_int(aa_nat_fun_int_o(aTP_Lamm_adz(F),B))))),B)) ) ) ).

% sum_div_partition
tff(fact_8319_sum__div__partition,axiom,
    ! [A3: set_Code_integer,F: fun_Code_integer_nat,B: nat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A3)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,groups7237345082560585321er_nat(F,A3)),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups7237345082560585321er_nat(aa_nat5852537043073606456er_nat(aTP_Lamm_aea(F),B),inf_in1364745209274528805nteger(A3,collect_Code_integer(aa_nat7420681472272698950eger_o(aTP_Lamm_aeb(F),B))))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,groups7237345082560585321er_nat(F,inf_in1364745209274528805nteger(A3,collect_Code_integer(aa_nat7420681472272698950eger_o(aTP_Lamm_aec(F),B))))),B)) ) ) ).

% sum_div_partition
tff(fact_8320_sum__div__partition,axiom,
    ! [A3: set_Code_integer,F: fun_Code_integer_int,B: int] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A3)
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,groups7234854612051535045er_int(F,A3)),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups7234854612051535045er_int(aa_int1390634949194299888er_int(aTP_Lamm_aed(F),B),inf_in1364745209274528805nteger(A3,collect_Code_integer(aa_int5993532643906289898eger_o(aTP_Lamm_aee(F),B))))),aa_int_int(aa_int_fun_int_int(divide_divide_int,groups7234854612051535045er_int(F,inf_in1364745209274528805nteger(A3,collect_Code_integer(aa_int5993532643906289898eger_o(aTP_Lamm_aef(F),B))))),B)) ) ) ).

% sum_div_partition
tff(fact_8321_sum__div__partition,axiom,
    ! [A3: set_nat,F: fun_nat_int,B: int] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_set_nat_int(groups3539618377306564664at_int(F),A3)),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_aeg(F),B)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A3),collect_nat(aa_int_fun_nat_o(aTP_Lamm_aeh(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,A3),collect_nat(aa_int_fun_nat_o(aTP_Lamm_aei(F),B))))),B)) ) ) ).

% sum_div_partition
tff(fact_8322_sum__div__partition,axiom,
    ! [A3: set_int,F: fun_int_Code_integer,B: code_integer] :
      ( aa_set_int_o(finite_finite_int,A3)
     => ( aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(groups7873554091576472773nteger(F,A3)),B) = aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(groups7873554091576472773nteger(aa_Cod1949805079358391713nteger(aTP_Lamm_aej(F),B),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A3),collect_int(aa_Cod8477617969808203498_int_o(aTP_Lamm_aek(F),B))))),aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(groups7873554091576472773nteger(F,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A3),collect_int(aa_Cod8477617969808203498_int_o(aTP_Lamm_ael(F),B))))),B)) ) ) ).

% sum_div_partition
tff(fact_8323_sum__div__partition,axiom,
    ! [A3: set_Code_integer,F: fun_Co8499238673658658103nteger,B: code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A3)
     => ( aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(groups879477027807139574nteger(F,A3)),B) = aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(groups879477027807139574nteger(aa_Cod4161879620478718930nteger(aTP_Lamm_aem(F),B),inf_in1364745209274528805nteger(A3,collect_Code_integer(aa_Cod326357343816142457eger_o(aTP_Lamm_aen(F),B))))),aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(groups879477027807139574nteger(F,inf_in1364745209274528805nteger(A3,collect_Code_integer(aa_Cod326357343816142457eger_o(aTP_Lamm_aeo(F),B))))),B)) ) ) ).

% sum_div_partition
tff(fact_8324_sum__div__partition,axiom,
    ! [A3: set_nat,F: fun_nat_Code_integer,B: code_integer] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_set871585252387937313nteger(groups7501900531339628137nteger(F),A3)),B) = aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_set871585252387937313nteger(groups7501900531339628137nteger(aa_Cod1578151519121547077nteger(aTP_Lamm_aep(F),B)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A3),collect_nat(aa_Cod1075288278386115654_nat_o(aTP_Lamm_aeq(F),B))))),aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_set871585252387937313nteger(groups7501900531339628137nteger(F),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A3),collect_nat(aa_Cod1075288278386115654_nat_o(aTP_Lamm_aer(F),B))))),B)) ) ) ).

% sum_div_partition
tff(fact_8325_sum__div__partition,axiom,
    ! [A3: set_int,F: fun_int_Code_natural,B: code_natural] :
      ( aa_set_int_o(finite_finite_int,A3)
     => ( divide5121882707175180666atural(groups6697149243333190288atural(F,A3),B) = plus_p4538020629002901425atural(groups6697149243333190288atural(aa_Cod5692187177200718263atural(aTP_Lamm_aes(F),B),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A3),collect_int(aa_Cod7299285730463426143_int_o(aTP_Lamm_aet(F),B)))),divide5121882707175180666atural(groups6697149243333190288atural(F,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A3),collect_int(aa_Cod7299285730463426143_int_o(aTP_Lamm_aeu(F),B)))),B)) ) ) ).

% sum_div_partition
tff(fact_8326_sum__div__partition,axiom,
    ! [A3: set_Code_integer,F: fun_Co3734316457315071362atural,B: code_natural] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A3)
     => ( divide5121882707175180666atural(groups8926444216418632897atural(F,A3),B) = plus_p4538020629002901425atural(groups8926444216418632897atural(aa_Cod1476545362447692776atural(aTP_Lamm_aev(F),B),inf_in1364745209274528805nteger(A3,collect_Code_integer(aa_Cod403035634466734574eger_o(aTP_Lamm_aew(F),B)))),divide5121882707175180666atural(groups8926444216418632897atural(F,inf_in1364745209274528805nteger(A3,collect_Code_integer(aa_Cod403035634466734574eger_o(aTP_Lamm_aex(F),B)))),B)) ) ) ).

% sum_div_partition
tff(fact_8327_sum__div__partition,axiom,
    ! [A3: set_nat,F: fun_nat_Code_natural,B: code_natural] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( divide5121882707175180666atural(groups6325495683096345652atural(F,A3),B) = plus_p4538020629002901425atural(groups6325495683096345652atural(aa_Cod5320533616963873627atural(aTP_Lamm_aey(F),B),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A3),collect_nat(aa_Cod9120328075896114107_nat_o(aTP_Lamm_aez(F),B)))),divide5121882707175180666atural(groups6325495683096345652atural(F,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A3),collect_nat(aa_Cod9120328075896114107_nat_o(aTP_Lamm_afa(F),B)))),B)) ) ) ).

% sum_div_partition
tff(fact_8328_round__diff__minimal,axiom,
    ! [Z: rat,M: int] : 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,Z),aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim7778729529865785530nd_rat,Z))))),aa_rat_rat(abs_abs_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,Z),aa_int_rat(ring_1_of_int_rat,M)))) ).

% round_diff_minimal
tff(fact_8329_oddE,axiom,
    ! [A: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
     => ~ ! [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,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B4)),one_one_nat) ) ).

% oddE
tff(fact_8330_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_8331_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(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_8332_oddE,axiom,
    ! [A: code_natural] :
      ( ~ aa_Code_natural_o(dvd_dvd_Code_natural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))),A)
     => ~ ! [B4: code_natural] : A != plus_p4538020629002901425atural(times_2397367101498566445atural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one)),B4),one_one_Code_natural) ) ).

% oddE
tff(fact_8333_zero__le__even__power,axiom,
    ! [N2: nat,A: code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(power_8256067586552552935nteger(A),N2)) ) ).

% zero_le_even_power
tff(fact_8334_zero__le__even__power,axiom,
    ! [N2: nat,A: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),N2)) ) ).

% zero_le_even_power
tff(fact_8335_zero__le__even__power,axiom,
    ! [N2: nat,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(aa_int_fun_nat_int(power_power_int,A),N2)) ) ).

% zero_le_even_power
tff(fact_8336_zero__le__odd__power,axiom,
    ! [N2: nat,A: code_integer] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(power_8256067586552552935nteger(A),N2))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A) ) ) ).

% zero_le_odd_power
tff(fact_8337_zero__le__odd__power,axiom,
    ! [N2: nat,A: rat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),N2))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A) ) ) ).

% zero_le_odd_power
tff(fact_8338_zero__le__odd__power,axiom,
    ! [N2: nat,A: int] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(aa_int_fun_nat_int(power_power_int,A),N2))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A) ) ) ).

% zero_le_odd_power
tff(fact_8339_zero__le__power__eq,axiom,
    ! [A: code_integer,N2: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(power_8256067586552552935nteger(A),N2))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2)
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2)
          & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A) ) ) ) ).

% zero_le_power_eq
tff(fact_8340_zero__le__power__eq,axiom,
    ! [A: rat,N2: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),N2))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2)
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A) ) ) ) ).

% zero_le_power_eq
tff(fact_8341_zero__le__power__eq,axiom,
    ! [A: int,N2: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(aa_int_fun_nat_int(power_power_int,A),N2))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2)
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2)
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A) ) ) ) ).

% zero_le_power_eq
tff(fact_8342_power__mono__even,axiom,
    ! [N2: nat,A: code_integer,B: code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A)),aa_Cod1783974652533396728nteger(abs_abs_Code_integer,B))
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),N2)),aa_nat_Code_integer(power_8256067586552552935nteger(B),N2)) ) ) ).

% power_mono_even
tff(fact_8343_power__mono__even,axiom,
    ! [N2: nat,A: rat,B: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B))
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),N2)),aa_nat_rat(power_power_rat(B),N2)) ) ) ).

% power_mono_even
tff(fact_8344_power__mono__even,axiom,
    ! [N2: nat,A: int,B: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B))
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A),N2)),aa_nat_int(aa_int_fun_nat_int(power_power_int,B),N2)) ) ) ).

% power_mono_even
tff(fact_8345_finite__transitivity__chain,axiom,
    ! [A3: set_set_nat,Ra: fun_se2170385732985038782_nat_o] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,A3)
     => ( ! [X3: set_nat] : ~ aa_set_nat_o(aa_set7861427066675100071_nat_o(Ra,X3),X3)
       => ( ! [X3: set_nat,Y3: set_nat,Z2: set_nat] :
              ( aa_set_nat_o(aa_set7861427066675100071_nat_o(Ra,X3),Y3)
             => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(Ra,Y3),Z2)
               => aa_set_nat_o(aa_set7861427066675100071_nat_o(Ra,X3),Z2) ) )
         => ( ! [X3: set_nat] :
                ( aa_set_set_nat_o(member_set_nat(X3),A3)
               => ? [Y4: set_nat] :
                    ( aa_set_set_nat_o(member_set_nat(Y4),A3)
                    & aa_set_nat_o(aa_set7861427066675100071_nat_o(Ra,X3),Y4) ) )
           => ( A3 = bot_bot_set_set_nat ) ) ) ) ) ).

% finite_transitivity_chain
tff(fact_8346_finite__transitivity__chain,axiom,
    ! [A3: set_Code_integer,Ra: fun_Co2081214437782444048eger_o] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A3)
     => ( ! [X3: code_integer] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(Ra,X3),X3)
       => ( ! [X3: code_integer,Y3: code_integer,Z2: code_integer] :
              ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(Ra,X3),Y3)
             => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(Ra,Y3),Z2)
               => aa_Code_integer_o(aa_Cod326357343816142457eger_o(Ra,X3),Z2) ) )
         => ( ! [X3: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(X3),A3)
               => ? [Y4: code_integer] :
                    ( aa_set5643387332196575111eger_o(member_Code_integer(Y4),A3)
                    & aa_Code_integer_o(aa_Cod326357343816142457eger_o(Ra,X3),Y4) ) )
           => ( A3 = bot_bo3990330152332043303nteger ) ) ) ) ) ).

% finite_transitivity_chain
tff(fact_8347_finite__transitivity__chain,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,Ra: fun_Pr2028404973800837934_nat_o] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A3)
     => ( ! [X3: product_prod_nat_nat] : ~ aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(Ra,X3),X3)
       => ( ! [X3: product_prod_nat_nat,Y3: product_prod_nat_nat,Z2: product_prod_nat_nat] :
              ( aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(Ra,X3),Y3)
             => ( aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(Ra,Y3),Z2)
               => aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(Ra,X3),Z2) ) )
         => ( ! [X3: product_prod_nat_nat] :
                ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X3),A3)
               => ? [Y4: product_prod_nat_nat] :
                    ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Y4),A3)
                    & aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(Ra,X3),Y4) ) )
           => ( A3 = bot_bo2099793752762293965at_nat ) ) ) ) ) ).

% finite_transitivity_chain
tff(fact_8348_finite__transitivity__chain,axiom,
    ! [A3: set_o,Ra: fun_o_fun_o_o] :
      ( finite_finite_o(A3)
     => ( ! [X3: $o] : ~ aa_o_o(aa_o_fun_o_o(Ra,(X3)),(X3))
       => ( ! [X3: $o,Y3: $o,Z2: $o] :
              ( aa_o_o(aa_o_fun_o_o(Ra,(X3)),(Y3))
             => ( aa_o_o(aa_o_fun_o_o(Ra,(Y3)),(Z2))
               => aa_o_o(aa_o_fun_o_o(Ra,(X3)),(Z2)) ) )
         => ( ! [X3: $o] :
                ( member_o((X3),A3)
               => ? [Y4: $o] :
                    ( member_o((Y4),A3)
                    & aa_o_o(aa_o_fun_o_o(Ra,(X3)),(Y4)) ) )
           => ( A3 = bot_bot_set_o ) ) ) ) ) ).

% finite_transitivity_chain
tff(fact_8349_finite__transitivity__chain,axiom,
    ! [A3: set_nat,Ra: fun_nat_fun_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( ! [X3: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(Ra,X3),X3)
       => ( ! [X3: nat,Y3: nat,Z2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(Ra,X3),Y3)
             => ( aa_nat_o(aa_nat_fun_nat_o(Ra,Y3),Z2)
               => aa_nat_o(aa_nat_fun_nat_o(Ra,X3),Z2) ) )
         => ( ! [X3: nat] :
                ( aa_set_nat_o(member_nat(X3),A3)
               => ? [Y4: nat] :
                    ( aa_set_nat_o(member_nat(Y4),A3)
                    & aa_nat_o(aa_nat_fun_nat_o(Ra,X3),Y4) ) )
           => ( A3 = bot_bot_set_nat ) ) ) ) ) ).

% finite_transitivity_chain
tff(fact_8350_finite__transitivity__chain,axiom,
    ! [A3: set_int,Ra: fun_int_fun_int_o] :
      ( aa_set_int_o(finite_finite_int,A3)
     => ( ! [X3: int] : ~ aa_int_o(aa_int_fun_int_o(Ra,X3),X3)
       => ( ! [X3: int,Y3: int,Z2: int] :
              ( aa_int_o(aa_int_fun_int_o(Ra,X3),Y3)
             => ( aa_int_o(aa_int_fun_int_o(Ra,Y3),Z2)
               => aa_int_o(aa_int_fun_int_o(Ra,X3),Z2) ) )
         => ( ! [X3: int] :
                ( aa_set_int_o(member_int(X3),A3)
               => ? [Y4: int] :
                    ( aa_set_int_o(member_int(Y4),A3)
                    & aa_int_o(aa_int_fun_int_o(Ra,X3),Y4) ) )
           => ( A3 = bot_bot_set_int ) ) ) ) ) ).

% finite_transitivity_chain
tff(fact_8351_zero__less__power__eq,axiom,
    ! [A: code_integer,N2: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(power_8256067586552552935nteger(A),N2))
    <=> ( ( N2 = zero_zero_nat )
        | ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2)
          & ( A != zero_z3403309356797280102nteger ) )
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2)
          & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A) ) ) ) ).

% zero_less_power_eq
tff(fact_8352_zero__less__power__eq,axiom,
    ! [A: rat,N2: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),N2))
    <=> ( ( N2 = zero_zero_nat )
        | ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2)
          & ( A != zero_zero_rat ) )
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A) ) ) ) ).

% zero_less_power_eq
tff(fact_8353_zero__less__power__eq,axiom,
    ! [A: int,N2: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(aa_int_fun_nat_int(power_power_int,A),N2))
    <=> ( ( N2 = zero_zero_nat )
        | ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2)
          & ( A != zero_zero_int ) )
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A) ) ) ) ).

% zero_less_power_eq
tff(fact_8354_unbounded__k__infinite,axiom,
    ! [K: nat,S: set_nat] :
      ( ! [M5: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),M5)
         => ? [N8: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M5),N8)
              & aa_set_nat_o(member_nat(N8),S) ) )
     => ~ aa_set_nat_o(finite_finite_nat,S) ) ).

% unbounded_k_infinite
tff(fact_8355_infinite__nat__iff__unbounded,axiom,
    ! [S: set_nat] :
      ( ~ aa_set_nat_o(finite_finite_nat,S)
    <=> ! [M3: nat] :
        ? [N: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M3),N)
          & aa_set_nat_o(member_nat(N),S) ) ) ).

% infinite_nat_iff_unbounded
tff(fact_8356_infinite__nat__iff__unbounded__le,axiom,
    ! [S: set_nat] :
      ( ~ aa_set_nat_o(finite_finite_nat,S)
    <=> ! [M3: nat] :
        ? [N: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M3),N)
          & aa_set_nat_o(member_nat(N),S) ) ) ).

% infinite_nat_iff_unbounded_le
tff(fact_8357_even__mask__div__iff_H,axiom,
    ! [M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2)))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2) ) ).

% even_mask_div_iff'
tff(fact_8358_even__mask__div__iff_H,axiom,
    ! [M: nat,N2: 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))),N2)))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2) ) ).

% even_mask_div_iff'
tff(fact_8359_even__mask__div__iff_H,axiom,
    ! [M: nat,N2: 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))),N2)))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2) ) ).

% even_mask_div_iff'
tff(fact_8360_even__mask__div__iff_H,axiom,
    ! [M: nat,N2: nat] :
      ( aa_Code_natural_o(dvd_dvd_Code_natural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))),divide5121882707175180666atural(minus_7197305767214868737atural(aa_nat_Code_natural(power_7079662738309270450atural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))),M),one_one_Code_natural),aa_nat_Code_natural(power_7079662738309270450atural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))),N2)))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2) ) ).

% even_mask_div_iff'
tff(fact_8361_power__le__zero__eq,axiom,
    ! [A: code_integer,N2: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),N2)),zero_z3403309356797280102nteger)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)
        & ( ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2)
            & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),zero_z3403309356797280102nteger) )
          | ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2)
            & ( A = zero_z3403309356797280102nteger ) ) ) ) ) ).

% power_le_zero_eq
tff(fact_8362_power__le__zero__eq,axiom,
    ! [A: rat,N2: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),N2)),zero_zero_rat)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)
        & ( ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2)
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat) )
          | ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2)
            & ( A = zero_zero_rat ) ) ) ) ) ).

% power_le_zero_eq
tff(fact_8363_power__le__zero__eq,axiom,
    ! [A: int,N2: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A),N2)),zero_zero_int)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)
        & ( ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2)
            & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int) )
          | ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2)
            & ( A = zero_zero_int ) ) ) ) ) ).

% power_le_zero_eq
tff(fact_8364_even__mask__div__iff,axiom,
    ! [M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2)))
    <=> ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2) = zero_zero_nat )
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2) ) ) ).

% even_mask_div_iff
tff(fact_8365_even__mask__div__iff,axiom,
    ! [M: nat,N2: 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))),N2)))
    <=> ( ( aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N2) = zero_zero_int )
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2) ) ) ).

% even_mask_div_iff
tff(fact_8366_even__mask__div__iff,axiom,
    ! [M: nat,N2: 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))),N2)))
    <=> ( ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),N2) = zero_z3403309356797280102nteger )
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2) ) ) ).

% even_mask_div_iff
tff(fact_8367_even__mask__div__iff,axiom,
    ! [M: nat,N2: nat] :
      ( aa_Code_natural_o(dvd_dvd_Code_natural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))),divide5121882707175180666atural(minus_7197305767214868737atural(aa_nat_Code_natural(power_7079662738309270450atural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))),M),one_one_Code_natural),aa_nat_Code_natural(power_7079662738309270450atural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))),N2)))
    <=> ( ( aa_nat_Code_natural(power_7079662738309270450atural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))),N2) = zero_z2226904508553997617atural )
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2) ) ) ).

% even_mask_div_iff
tff(fact_8368_even__mult__exp__div__exp__iff,axiom,
    ! [A: nat,M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2)))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),M)
        | ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2) = zero_zero_nat )
        | ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
          & aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),M)))) ) ) ) ).

% even_mult_exp_div_exp_iff
tff(fact_8369_even__mult__exp__div__exp__iff,axiom,
    ! [A: int,M: nat,N2: 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))),N2)))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),M)
        | ( aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N2) = zero_zero_int )
        | ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
          & 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,N2),M)))) ) ) ) ).

% even_mult_exp_div_exp_iff
tff(fact_8370_even__mult__exp__div__exp__iff,axiom,
    ! [A: code_integer,M: nat,N2: 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))),N2)))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),M)
        | ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),N2) = zero_z3403309356797280102nteger )
        | ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
          & 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,N2),M)))) ) ) ) ).

% even_mult_exp_div_exp_iff
tff(fact_8371_even__mult__exp__div__exp__iff,axiom,
    ! [A: code_natural,M: nat,N2: nat] :
      ( aa_Code_natural_o(dvd_dvd_Code_natural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))),divide5121882707175180666atural(times_2397367101498566445atural(A,aa_nat_Code_natural(power_7079662738309270450atural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))),M)),aa_nat_Code_natural(power_7079662738309270450atural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))),N2)))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),M)
        | ( aa_nat_Code_natural(power_7079662738309270450atural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))),N2) = zero_z2226904508553997617atural )
        | ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
          & aa_Code_natural_o(dvd_dvd_Code_natural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))),divide5121882707175180666atural(A,aa_nat_Code_natural(power_7079662738309270450atural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),M)))) ) ) ) ).

% even_mult_exp_div_exp_iff
tff(fact_8372_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),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))))) ).

% of_int_round_le
tff(fact_8373_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),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))))),aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim7778729529865785530nd_rat,X))) ).

% of_int_round_ge
tff(fact_8374_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),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))))),aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim7778729529865785530nd_rat,X))) ).

% of_int_round_gt
tff(fact_8375_infinite__int__iff__unbounded__le,axiom,
    ! [S: set_int] :
      ( ~ aa_set_int_o(finite_finite_int,S)
    <=> ! [M3: int] :
        ? [N: int] :
          ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,M3),aa_int_int(abs_abs_int,N))
          & aa_set_int_o(member_int(N),S) ) ) ).

% infinite_int_iff_unbounded_le
tff(fact_8376_infinite__int__iff__unbounded,axiom,
    ! [S: set_int] :
      ( ~ aa_set_int_o(finite_finite_int,S)
    <=> ! [M3: int] :
        ? [N: int] :
          ( aa_int_o(aa_int_fun_int_o(ord_less_int,M3),aa_int_int(abs_abs_int,N))
          & aa_set_int_o(member_int(N),S) ) ) ).

% infinite_int_iff_unbounded
tff(fact_8377_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),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)))) ).

% of_int_round_abs_le
tff(fact_8378_round__unique_H,axiom,
    ! [X: rat,N2: 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,N2)))),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,one_one_rat),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))))
     => ( aa_rat_int(archim7778729529865785530nd_rat,X) = N2 ) ) ).

% round_unique'
tff(fact_8379_in__finite__psubset,axiom,
    ! [A3: set_nat,B3: set_nat] :
      ( member8277197624267554838et_nat(produc4532415448927165861et_nat(A3,B3),finite_psubset_nat)
    <=> ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,A3),B3)
        & aa_set_nat_o(finite_finite_nat,B3) ) ) ).

% in_finite_psubset
tff(fact_8380_in__finite__psubset,axiom,
    ! [A3: set_int,B3: set_int] :
      ( member2572552093476627150et_int(produc6363374080413544029et_int(A3,B3),finite_psubset_int)
    <=> ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_set_int,A3),B3)
        & aa_set_int_o(finite_finite_int,B3) ) ) ).

% in_finite_psubset
tff(fact_8381_in__finite__psubset,axiom,
    ! [A3: set_Code_integer,B3: set_Code_integer] :
      ( member4307123515891402160nteger(produc7443773368509356479nteger(A3,B3),finite2416775604798480986nteger)
    <=> ( aa_set5643387332196575111eger_o(ord_le1307284697595431911nteger(A3),B3)
        & aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B3) ) ) ).

% in_finite_psubset
tff(fact_8382_in__finite__psubset,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
      ( member8757157785044589968at_nat(produc2922128104949294807at_nat(A3,B3),finite469560695537375940at_nat)
    <=> ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le7866589430770878221at_nat,A3),B3)
        & aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,B3) ) ) ).

% in_finite_psubset
tff(fact_8383_flip__bit__0,axiom,
    ! [A: code_integer] : aa_Cod1783974652533396728nteger(bit_se1345352211410354436nteger(zero_zero_nat),A) = aa_Cod1783974652533396728nteger(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_8384_flip__bit__0,axiom,
    ! [A: code_natural] : bit_se168947363167071951atural(zero_zero_nat,A) = plus_p4538020629002901425atural(zero_n8403883297036319079atural(aa_Code_natural_o(dvd_dvd_Code_natural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))),A)),times_2397367101498566445atural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one)),divide5121882707175180666atural(A,aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))))) ).

% flip_bit_0
tff(fact_8385_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,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% flip_bit_0
tff(fact_8386_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_8387_set__decode__def,axiom,
    ! [X: nat] : aa_nat_set_nat(nat_set_decode,X) = collect_nat(aa_nat_fun_nat_o(aTP_Lamm_afb,X)) ).

% set_decode_def
tff(fact_8388_take__bit__rec,axiom,
    ! [N2: nat,A: code_integer] :
      aa_Cod1783974652533396728nteger(bit_se1745604003318907178nteger(N2),A) = $ite(N2 = zero_zero_nat,zero_z3403309356797280102nteger,aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(bit_se1745604003318907178nteger(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),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_8389_take__bit__rec,axiom,
    ! [N2: nat,A: code_natural] :
      bit_se569199155075624693atural(N2,A) = $ite(N2 = zero_zero_nat,zero_z2226904508553997617atural,plus_p4538020629002901425atural(times_2397367101498566445atural(bit_se569199155075624693atural(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),one_one_nat),divide5121882707175180666atural(A,aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one)))),aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))),modulo8411746178871703098atural(A,aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))))) ).

% take_bit_rec
tff(fact_8390_take__bit__rec,axiom,
    ! [N2: nat,A: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N2),A) = $ite(N2 = 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,N2),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% take_bit_rec
tff(fact_8391_take__bit__rec,axiom,
    ! [N2: nat,A: int] :
      aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N2),A) = $ite(N2 = 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,N2),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_8392_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_afc(L)),Qr) ).

% divmod_step_def
tff(fact_8393_divmod__step__def,axiom,
    ! [L: num,Qr: product_prod_int_int] : aa_Pro6970504127699319548nt_int(unique5024387138958732305ep_int(L),Qr) = aa_Pro6970504127699319548nt_int(aa_fun8026974734089330650nt_int(produc4245557441103728435nt_int,aTP_Lamm_afd(L)),Qr) ).

% divmod_step_def
tff(fact_8394_divmod__step__def,axiom,
    ! [L: num,Qr: produc8923325533196201883nteger] : unique4921790084139445826nteger(L,Qr) = produc6916734918728496179nteger(aTP_Lamm_afe(L),Qr) ).

% divmod_step_def
tff(fact_8395_one__mod__2__pow__eq,axiom,
    ! [N2: nat] : aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(one_one_Code_integer),aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),N2)) = aa_o_Code_integer(zero_n356916108424825756nteger,aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)) ).

% one_mod_2_pow_eq
tff(fact_8396_one__mod__2__pow__eq,axiom,
    ! [N2: nat] : modulo8411746178871703098atural(one_one_Code_natural,aa_nat_Code_natural(power_7079662738309270450atural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))),N2)) = zero_n8403883297036319079atural(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)) ).

% one_mod_2_pow_eq
tff(fact_8397_one__mod__2__pow__eq,axiom,
    ! [N2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2)) = aa_o_nat(zero_n2687167440665602831ol_nat,aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)) ).

% one_mod_2_pow_eq
tff(fact_8398_one__mod__2__pow__eq,axiom,
    ! [N2: 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))),N2)) = aa_o_int(zero_n2684676970156552555ol_int,aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)) ).

% one_mod_2_pow_eq
tff(fact_8399_of__bool__less__eq__iff,axiom,
    ! [Pa: $o,Qa: $o] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_o_rat(zero_n2052037380579107095ol_rat,(Pa))),aa_o_rat(zero_n2052037380579107095ol_rat,(Qa)))
    <=> ( (Pa)
       => (Qa) ) ) ).

% of_bool_less_eq_iff
tff(fact_8400_of__bool__less__eq__iff,axiom,
    ! [Pa: $o,Qa: $o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_o_nat(zero_n2687167440665602831ol_nat,(Pa))),aa_o_nat(zero_n2687167440665602831ol_nat,(Qa)))
    <=> ( (Pa)
       => (Qa) ) ) ).

% of_bool_less_eq_iff
tff(fact_8401_of__bool__less__eq__iff,axiom,
    ! [Pa: $o,Qa: $o] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_o_int(zero_n2684676970156552555ol_int,(Pa))),aa_o_int(zero_n2684676970156552555ol_int,(Qa)))
    <=> ( (Pa)
       => (Qa) ) ) ).

% of_bool_less_eq_iff
tff(fact_8402_of__bool__less__iff,axiom,
    ! [Pa: $o,Qa: $o] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_o_rat(zero_n2052037380579107095ol_rat,(Pa))),aa_o_rat(zero_n2052037380579107095ol_rat,(Qa)))
    <=> ( ~ (Pa)
        & (Qa) ) ) ).

% of_bool_less_iff
tff(fact_8403_of__bool__less__iff,axiom,
    ! [Pa: $o,Qa: $o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_o_nat(zero_n2687167440665602831ol_nat,(Pa))),aa_o_nat(zero_n2687167440665602831ol_nat,(Qa)))
    <=> ( ~ (Pa)
        & (Qa) ) ) ).

% of_bool_less_iff
tff(fact_8404_of__bool__less__iff,axiom,
    ! [Pa: $o,Qa: $o] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_o_int(zero_n2684676970156552555ol_int,(Pa))),aa_o_int(zero_n2684676970156552555ol_int,(Qa)))
    <=> ( ~ (Pa)
        & (Qa) ) ) ).

% of_bool_less_iff
tff(fact_8405_of__nat__of__bool,axiom,
    ! [Pa: $o] : aa_nat_Code_integer(semiri4939895301339042750nteger,aa_o_nat(zero_n2687167440665602831ol_nat,(Pa))) = aa_o_Code_integer(zero_n356916108424825756nteger,(Pa)) ).

% of_nat_of_bool
tff(fact_8406_of__nat__of__bool,axiom,
    ! [Pa: $o] : aa_nat_nat(semiri1316708129612266289at_nat,aa_o_nat(zero_n2687167440665602831ol_nat,(Pa))) = aa_o_nat(zero_n2687167440665602831ol_nat,(Pa)) ).

% of_nat_of_bool
tff(fact_8407_of__nat__of__bool,axiom,
    ! [Pa: $o] : aa_nat_int(semiri1314217659103216013at_int,aa_o_nat(zero_n2687167440665602831ol_nat,(Pa))) = aa_o_int(zero_n2684676970156552555ol_int,(Pa)) ).

% of_nat_of_bool
tff(fact_8408_pair__imageI,axiom,
    ! [A: nat,B: nat,A3: set_Pr1261947904930325089at_nat,F: fun_nat_fun_nat_o] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,A),B)),A3)
     => member_o(aa_nat_o(aa_nat_fun_nat_o(F,A),B),image_3693632289388996572_nat_o(produc6081775807080527818_nat_o(F),A3)) ) ).

% pair_imageI
tff(fact_8409_pair__imageI,axiom,
    ! [A: nat,B: nat,A3: set_Pr1261947904930325089at_nat,F: fun_nat_fun_nat_nat] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,A),B)),A3)
     => aa_set_nat_o(member_nat(aa_nat_nat(aa_nat_fun_nat_nat(F,A),B)),image_2486076414777270412at_nat(produc6842872674320459806at_nat(F),A3)) ) ).

% pair_imageI
tff(fact_8410_pair__imageI,axiom,
    ! [A: nat,B: nat,A3: set_Pr1261947904930325089at_nat,F: fun_nat_fun_nat_int] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,A),B)),A3)
     => aa_set_int_o(member_int(aa_nat_int(aa_nat_fun_nat_int(F,A),B)),image_2483585944268220136at_int(produc6840382203811409530at_int(F),A3)) ) ).

% pair_imageI
tff(fact_8411_pair__imageI,axiom,
    ! [A: int,B: int,A3: set_Pr958786334691620121nt_int,F: fun_int_fun_int_o] :
      ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),B),A3)
     => member_o(aa_int_o(aa_int_fun_int_o(F,A),B),image_2135063354759101220_int_o(aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,F),A3)) ) ).

% pair_imageI
tff(fact_8412_pair__imageI,axiom,
    ! [A: int,B: int,A3: set_Pr958786334691620121nt_int,F: fun_int_fun_int_int] :
      ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),B),A3)
     => aa_set_int_o(member_int(aa_int_int(aa_int_fun_int_int(F,A),B)),image_5042161079198086560nt_int(aa_fun914008879417524284nt_int(produc8211389475949308722nt_int,F),A3)) ) ).

% pair_imageI
tff(fact_8413_pair__imageI,axiom,
    ! [A: nat,B: nat,A3: set_Pr1261947904930325089at_nat,F: fun_na6128946301132549280et_nat] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,A),B)),A3)
     => aa_set_set_nat_o(member_set_nat(aa_nat_set_nat(aa_nat1818925681583257441et_nat(F,A),B)),image_15824709712370754et_nat(produc6189476227299908564et_nat(F),A3)) ) ).

% pair_imageI
tff(fact_8414_pair__imageI,axiom,
    ! [A: nat,B: nat,A3: set_Pr1261947904930325089at_nat,F: fun_na4138050610385705875at_nat] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,A),B)),A3)
     => aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(F,A),B)),image_5168914502847457605at_nat(produc2626176000494625587at_nat(F),A3)) ) ).

% pair_imageI
tff(fact_8415_pair__imageI,axiom,
    ! [A: int,B: int,A3: set_Pr958786334691620121nt_int,F: fun_in1696471567231710611nt_int] :
      ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),B),A3)
     => member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(F,A),B),image_2653370878348428101nt_int(aa_fun8026974734089330650nt_int(produc4245557441103728435nt_int,F),A3)) ) ).

% pair_imageI
tff(fact_8416_pair__imageI,axiom,
    ! [A: a,B: produc6653097349344004940it_nat,A3: set_Pr7098220151150636591it_nat,F: fun_a_2467521563176467050_nat_o] :
      ( member4554412811331277712it_nat(aa_Pro8078911840264235907it_nat(aa_a_f2234720956197727492it_nat(produc9178034014595674355it_nat,A),B),A3)
     => member_o(aa_Pro1202363709425262478_nat_o(aa_a_f337466037417904483_nat_o(F,A),B),image_1991556864075471768_nat_o(produc8860486935419594360_nat_o(F),A3)) ) ).

% pair_imageI
tff(fact_8417_pair__imageI,axiom,
    ! [A: a,B: produc6653097349344004940it_nat,A3: set_Pr7098220151150636591it_nat,F: fun_a_5416464254913058730at_nat] :
      ( member4554412811331277712it_nat(aa_Pro8078911840264235907it_nat(aa_a_f2234720956197727492it_nat(produc9178034014595674355it_nat,A),B),A3)
     => aa_set_nat_o(member_nat(aa_Pro3734910839824185690at_nat(aa_a_f8203466477562882395at_nat(F,A),B)),image_1030127162643715024at_nat(produc7727125748877983984at_nat(F),A3)) ) ).

% pair_imageI
tff(fact_8418_case__prod__conv,axiom,
    ! [F: fun_na1668100129766719005at_nat,A: nat,B: nat] : aa_Pro8023792847356852296at_nat(aa_fun6875833325056119164at_nat(produc4528391901023682385at_nat,F),aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,A),B)) = aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(F,A),B) ).

% case_prod_conv
tff(fact_8419_case__prod__conv,axiom,
    ! [F: fun_na5914473681949881652_nat_o,A: nat,B: nat] : aa_Pro989595756709598047_nat_o(aa_fun5912894404062470332_nat_o(produc6395135316500328296_nat_o,F),aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,A),B)) = aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(F,A),B) ).

% case_prod_conv
tff(fact_8420_case__prod__conv,axiom,
    ! [F: fun_in1696471567231710611nt_int,A: int,B: int] : aa_Pro6970504127699319548nt_int(aa_fun8026974734089330650nt_int(produc4245557441103728435nt_int,F),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),B)) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(F,A),B) ).

% case_prod_conv
tff(fact_8421_case__prod__conv,axiom,
    ! [F: fun_int_fun_int_o,A: int,B: int] :
      ( aa_Pro4592068669816069595_int_o(aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,F),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),B))
    <=> aa_int_o(aa_int_fun_int_o(F,A),B) ) ).

% case_prod_conv
tff(fact_8422_case__prod__conv,axiom,
    ! [F: fun_int_fun_int_int,A: int,B: int] : aa_Pro6799258434465214121nt_int(aa_fun914008879417524284nt_int(produc8211389475949308722nt_int,F),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),B)) = aa_int_int(aa_int_fun_int_int(F,A),B) ).

% case_prod_conv
tff(fact_8423_curry__case__prod,axiom,
    ! [F: fun_na1668100129766719005at_nat] : produc4701859392678344816at_nat(aa_fun6875833325056119164at_nat(produc4528391901023682385at_nat,F)) = F ).

% curry_case_prod
tff(fact_8424_curry__case__prod,axiom,
    ! [F: fun_na5914473681949881652_nat_o] : produc2062876521443793799_nat_o(aa_fun5912894404062470332_nat_o(produc6395135316500328296_nat_o,F)) = F ).

% curry_case_prod
tff(fact_8425_curry__case__prod,axiom,
    ! [F: fun_in1696471567231710611nt_int] : produc8249235968001453780nt_int(aa_fun8026974734089330650nt_int(produc4245557441103728435nt_int,F)) = F ).

% curry_case_prod
tff(fact_8426_curry__case__prod,axiom,
    ! [F: fun_int_fun_int_o] : produc175634133007206835_int_o(aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,F)) = F ).

% curry_case_prod
tff(fact_8427_curry__case__prod,axiom,
    ! [F: fun_int_fun_int_int] : produc1016772743285680337nt_int(aa_fun914008879417524284nt_int(produc8211389475949308722nt_int,F)) = F ).

% curry_case_prod
tff(fact_8428_case__prod__curry,axiom,
    ! [F: fun_Pr2477239776880934807at_nat] : aa_fun6875833325056119164at_nat(produc4528391901023682385at_nat,produc4701859392678344816at_nat(F)) = F ).

% case_prod_curry
tff(fact_8429_case__prod__curry,axiom,
    ! [F: fun_Pr2028404973800837934_nat_o] : aa_fun5912894404062470332_nat_o(produc6395135316500328296_nat_o,produc2062876521443793799_nat_o(F)) = F ).

% case_prod_curry
tff(fact_8430_case__prod__curry,axiom,
    ! [F: fun_Pr2478283820680522243nt_int] : aa_fun8026974734089330650nt_int(produc4245557441103728435nt_int,produc8249235968001453780nt_int(F)) = F ).

% case_prod_curry
tff(fact_8431_case__prod__curry,axiom,
    ! [F: fun_Pr3238374608701176162_int_o] : aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,produc175634133007206835_int_o(F)) = F ).

% case_prod_curry
tff(fact_8432_case__prod__curry,axiom,
    ! [F: fun_Pr982841511892225912nt_int] : aa_fun914008879417524284nt_int(produc8211389475949308722nt_int,produc1016772743285680337nt_int(F)) = F ).

% case_prod_curry
tff(fact_8433_take__bit__of__Suc__0,axiom,
    ! [N2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N2),aa_nat_nat(suc,zero_zero_nat)) = aa_o_nat(zero_n2687167440665602831ol_nat,aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)) ).

% take_bit_of_Suc_0
tff(fact_8434_zero__less__of__bool__iff,axiom,
    ! [Pa: $o] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_o_Code_integer(zero_n356916108424825756nteger,(Pa)))
    <=> (Pa) ) ).

% zero_less_of_bool_iff
tff(fact_8435_zero__less__of__bool__iff,axiom,
    ! [Pa: $o] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_o_rat(zero_n2052037380579107095ol_rat,(Pa)))
    <=> (Pa) ) ).

% zero_less_of_bool_iff
tff(fact_8436_zero__less__of__bool__iff,axiom,
    ! [Pa: $o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_o_nat(zero_n2687167440665602831ol_nat,(Pa)))
    <=> (Pa) ) ).

% zero_less_of_bool_iff
tff(fact_8437_zero__less__of__bool__iff,axiom,
    ! [Pa: $o] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_o_int(zero_n2684676970156552555ol_int,(Pa)))
    <=> (Pa) ) ).

% zero_less_of_bool_iff
tff(fact_8438_of__bool__less__one__iff,axiom,
    ! [Pa: $o] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_o_Code_integer(zero_n356916108424825756nteger,(Pa))),one_one_Code_integer)
    <=> ~ (Pa) ) ).

% of_bool_less_one_iff
tff(fact_8439_of__bool__less__one__iff,axiom,
    ! [Pa: $o] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_o_rat(zero_n2052037380579107095ol_rat,(Pa))),one_one_rat)
    <=> ~ (Pa) ) ).

% of_bool_less_one_iff
tff(fact_8440_of__bool__less__one__iff,axiom,
    ! [Pa: $o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_o_nat(zero_n2687167440665602831ol_nat,(Pa))),one_one_nat)
    <=> ~ (Pa) ) ).

% of_bool_less_one_iff
tff(fact_8441_of__bool__less__one__iff,axiom,
    ! [Pa: $o] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_o_int(zero_n2684676970156552555ol_int,(Pa))),one_one_int)
    <=> ~ (Pa) ) ).

% of_bool_less_one_iff
tff(fact_8442_set__decode__zero,axiom,
    aa_nat_set_nat(nat_set_decode,zero_zero_nat) = bot_bot_set_nat ).

% set_decode_zero
tff(fact_8443_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_8444_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_8445_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_8446_take__bit__of__1,axiom,
    ! [N2: nat] : aa_Cod1783974652533396728nteger(bit_se1745604003318907178nteger(N2),one_one_Code_integer) = aa_o_Code_integer(zero_n356916108424825756nteger,aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)) ).

% take_bit_of_1
tff(fact_8447_take__bit__of__1,axiom,
    ! [N2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N2),one_one_nat) = aa_o_nat(zero_n2687167440665602831ol_nat,aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)) ).

% take_bit_of_1
tff(fact_8448_take__bit__of__1,axiom,
    ! [N2: nat] : aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N2),one_one_int) = aa_o_int(zero_n2684676970156552555ol_int,aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)) ).

% take_bit_of_1
tff(fact_8449_sgn__of__nat,axiom,
    ! [N2: nat] : aa_rat_rat(sgn_sgn_rat,aa_nat_rat(semiri681578069525770553at_rat,N2)) = aa_o_rat(zero_n2052037380579107095ol_rat,aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)) ).

% sgn_of_nat
tff(fact_8450_sgn__of__nat,axiom,
    ! [N2: nat] : aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,aa_nat_Code_integer(semiri4939895301339042750nteger,N2)) = aa_o_Code_integer(zero_n356916108424825756nteger,aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)) ).

% sgn_of_nat
tff(fact_8451_sgn__of__nat,axiom,
    ! [N2: nat] : aa_int_int(sgn_sgn_int,aa_nat_int(semiri1314217659103216013at_int,N2)) = aa_o_int(zero_n2684676970156552555ol_int,aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)) ).

% sgn_of_nat
tff(fact_8452_sum__mult__of__bool__eq,axiom,
    ! [A3: set_int,F: fun_int_Code_integer,Pa: fun_int_o] :
      ( aa_set_int_o(finite_finite_int,A3)
     => ( groups7873554091576472773nteger(aa_fun6019004165459918095nteger(aTP_Lamm_aff(F),Pa),A3) = groups7873554091576472773nteger(F,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A3),collect_int(Pa))) ) ) ).

% sum_mult_of_bool_eq
tff(fact_8453_sum__mult__of__bool__eq,axiom,
    ! [A3: set_Code_integer,F: fun_Co8499238673658658103nteger,Pa: fun_Code_integer_o] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A3)
     => ( groups879477027807139574nteger(aa_fun791458909507501263nteger(aTP_Lamm_afg(F),Pa),A3) = groups879477027807139574nteger(F,inf_in1364745209274528805nteger(A3,collect_Code_integer(Pa))) ) ) ).

% sum_mult_of_bool_eq
tff(fact_8454_sum__mult__of__bool__eq,axiom,
    ! [A3: set_nat,F: fun_nat_Code_integer,Pa: fun_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( aa_set871585252387937313nteger(groups7501900531339628137nteger(aa_fun3969837373441324047nteger(aTP_Lamm_afh(F),Pa)),A3) = aa_set871585252387937313nteger(groups7501900531339628137nteger(F),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A3),collect_nat(Pa))) ) ) ).

% sum_mult_of_bool_eq
tff(fact_8455_sum__mult__of__bool__eq,axiom,
    ! [A3: set_int,F: fun_int_rat,Pa: fun_int_o] :
      ( aa_set_int_o(finite_finite_int,A3)
     => ( groups3906332499630173760nt_rat(aa_fun268837275264525578nt_rat(aTP_Lamm_afi(F),Pa),A3) = groups3906332499630173760nt_rat(F,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A3),collect_int(Pa))) ) ) ).

% sum_mult_of_bool_eq
tff(fact_8456_sum__mult__of__bool__eq,axiom,
    ! [A3: set_Code_integer,F: fun_Code_integer_rat,Pa: fun_Code_integer_o] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A3)
     => ( groups6602215022474089585er_rat(aa_fun25512833367352522er_rat(aTP_Lamm_afj(F),Pa),A3) = groups6602215022474089585er_rat(F,inf_in1364745209274528805nteger(A3,collect_Code_integer(Pa))) ) ) ).

% sum_mult_of_bool_eq
tff(fact_8457_sum__mult__of__bool__eq,axiom,
    ! [A3: set_nat,F: fun_nat_rat,Pa: fun_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( aa_set_nat_rat(groups2906978787729119204at_rat(aa_fun696632391729880074at_rat(aTP_Lamm_afk(F),Pa)),A3) = aa_set_nat_rat(groups2906978787729119204at_rat(F),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A3),collect_nat(Pa))) ) ) ).

% sum_mult_of_bool_eq
tff(fact_8458_sum__mult__of__bool__eq,axiom,
    ! [A3: set_int,F: fun_int_nat,Pa: fun_int_o] :
      ( aa_set_int_o(finite_finite_int,A3)
     => ( groups4541462559716669496nt_nat(aa_fun3777065833657922306nt_nat(aTP_Lamm_afl(F),Pa),A3) = groups4541462559716669496nt_nat(F,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A3),collect_int(Pa))) ) ) ).

% sum_mult_of_bool_eq
tff(fact_8459_sum__mult__of__bool__eq,axiom,
    ! [A3: set_Code_integer,F: fun_Code_integer_nat,Pa: fun_Code_integer_o] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A3)
     => ( groups7237345082560585321er_nat(aa_fun3533741391760749250er_nat(aTP_Lamm_afm(F),Pa),A3) = groups7237345082560585321er_nat(F,inf_in1364745209274528805nteger(A3,collect_Code_integer(Pa))) ) ) ).

% sum_mult_of_bool_eq
tff(fact_8460_sum__mult__of__bool__eq,axiom,
    ! [A3: set_Code_integer,F: fun_Code_integer_int,Pa: fun_Code_integer_o] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A3)
     => ( groups7234854612051535045er_int(aa_fun8579262409106328350er_int(aTP_Lamm_afn(F),Pa),A3) = groups7234854612051535045er_int(F,inf_in1364745209274528805nteger(A3,collect_Code_integer(Pa))) ) ) ).

% sum_mult_of_bool_eq
tff(fact_8461_sum__mult__of__bool__eq,axiom,
    ! [A3: set_nat,F: fun_nat_int,Pa: fun_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( aa_set_nat_int(groups3539618377306564664at_int(aa_fun27009930614080094at_int(aTP_Lamm_afo(F),Pa)),A3) = aa_set_nat_int(groups3539618377306564664at_int(F),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A3),collect_nat(Pa))) ) ) ).

% sum_mult_of_bool_eq
tff(fact_8462_sum__of__bool__mult__eq,axiom,
    ! [A3: set_int,Pa: fun_int_o,F: fun_int_Code_integer] :
      ( aa_set_int_o(finite_finite_int,A3)
     => ( groups7873554091576472773nteger(aa_fun5304498221708291146nteger(aTP_Lamm_afp(Pa),F),A3) = groups7873554091576472773nteger(F,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A3),collect_int(Pa))) ) ) ).

% sum_of_bool_mult_eq
tff(fact_8463_sum__of__bool__mult__eq,axiom,
    ! [A3: set_Code_integer,Pa: fun_Code_integer_o,F: fun_Co8499238673658658103nteger] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A3)
     => ( groups879477027807139574nteger(aa_fun4435975922079263404nteger(aTP_Lamm_afq(Pa),F),A3) = groups879477027807139574nteger(F,inf_in1364745209274528805nteger(A3,collect_Code_integer(Pa))) ) ) ).

% sum_of_bool_mult_eq
tff(fact_8464_sum__of__bool__mult__eq,axiom,
    ! [A3: set_nat,Pa: fun_nat_o,F: fun_nat_Code_integer] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( aa_set871585252387937313nteger(groups7501900531339628137nteger(aa_fun589076786913273746nteger(aTP_Lamm_afr(Pa),F)),A3) = aa_set871585252387937313nteger(groups7501900531339628137nteger(F),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A3),collect_nat(Pa))) ) ) ).

% sum_of_bool_mult_eq
tff(fact_8465_sum__of__bool__mult__eq,axiom,
    ! [A3: set_int,Pa: fun_int_o,F: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,A3)
     => ( groups3906332499630173760nt_rat(aa_fun1481061427733998144nt_rat(aTP_Lamm_afs(Pa),F),A3) = groups3906332499630173760nt_rat(F,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A3),collect_int(Pa))) ) ) ).

% sum_of_bool_mult_eq
tff(fact_8466_sum__of__bool__mult__eq,axiom,
    ! [A3: set_Code_integer,Pa: fun_Code_integer_o,F: fun_Code_integer_rat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A3)
     => ( groups6602215022474089585er_rat(aa_fun8169474305930283170er_rat(aTP_Lamm_aft(Pa),F),A3) = groups6602215022474089585er_rat(F,inf_in1364745209274528805nteger(A3,collect_Code_integer(Pa))) ) ) ).

% sum_of_bool_mult_eq
tff(fact_8467_sum__of__bool__mult__eq,axiom,
    ! [A3: set_nat,Pa: fun_nat_o,F: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( aa_set_nat_rat(groups2906978787729119204at_rat(aa_fun765758790203053448at_rat(aTP_Lamm_afu(Pa),F)),A3) = aa_set_nat_rat(groups2906978787729119204at_rat(F),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A3),collect_nat(Pa))) ) ) ).

% sum_of_bool_mult_eq
tff(fact_8468_sum__of__bool__mult__eq,axiom,
    ! [A3: set_int,Pa: fun_int_o,F: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,A3)
     => ( groups4541462559716669496nt_nat(aa_fun6506752150744536624nt_nat(aTP_Lamm_afv(Pa),F),A3) = groups4541462559716669496nt_nat(F,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A3),collect_int(Pa))) ) ) ).

% sum_of_bool_mult_eq
tff(fact_8469_sum__of__bool__mult__eq,axiom,
    ! [A3: set_Code_integer,Pa: fun_Code_integer_o,F: fun_Code_integer_nat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A3)
     => ( groups7237345082560585321er_nat(aa_fun3076867170244285586er_nat(aTP_Lamm_afw(Pa),F),A3) = groups7237345082560585321er_nat(F,inf_in1364745209274528805nteger(A3,collect_Code_integer(Pa))) ) ) ).

% sum_of_bool_mult_eq
tff(fact_8470_sum__of__bool__mult__eq,axiom,
    ! [A3: set_Code_integer,Pa: fun_Code_integer_o,F: fun_Code_integer_int] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A3)
     => ( groups7234854612051535045er_int(aa_fun3291479149765983562er_int(aTP_Lamm_afx(Pa),F),A3) = groups7234854612051535045er_int(F,inf_in1364745209274528805nteger(A3,collect_Code_integer(Pa))) ) ) ).

% sum_of_bool_mult_eq
tff(fact_8471_sum__of__bool__mult__eq,axiom,
    ! [A3: set_nat,Pa: fun_nat_o,F: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( aa_set_nat_int(groups3539618377306564664at_int(aa_fun5565043122120104496at_int(aTP_Lamm_afy(Pa),F)),A3) = aa_set_nat_int(groups3539618377306564664at_int(F),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A3),collect_nat(Pa))) ) ) ).

% sum_of_bool_mult_eq
tff(fact_8472_take__bit__of__exp,axiom,
    ! [M: nat,N2: nat] : aa_Cod1783974652533396728nteger(bit_se1745604003318907178nteger(M),aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),N2)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_o_Code_integer(zero_n356916108424825756nteger,aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),M))),aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),N2)) ).

% take_bit_of_exp
tff(fact_8473_take__bit__of__exp,axiom,
    ! [M: nat,N2: nat] : bit_se569199155075624693atural(M,aa_nat_Code_natural(power_7079662738309270450atural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))),N2)) = times_2397367101498566445atural(zero_n8403883297036319079atural(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),M)),aa_nat_Code_natural(power_7079662738309270450atural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))),N2)) ).

% take_bit_of_exp
tff(fact_8474_take__bit__of__exp,axiom,
    ! [M: nat,N2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_o_nat(zero_n2687167440665602831ol_nat,aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),M))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2)) ).

% take_bit_of_exp
tff(fact_8475_take__bit__of__exp,axiom,
    ! [M: nat,N2: 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))),N2)) = 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,N2),M))),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N2)) ).

% take_bit_of_exp
tff(fact_8476_take__bit__of__2,axiom,
    ! [N2: nat] : aa_Cod1783974652533396728nteger(bit_se1745604003318907178nteger(N2),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,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2))),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) ).

% take_bit_of_2
tff(fact_8477_take__bit__of__2,axiom,
    ! [N2: nat] : bit_se569199155075624693atural(N2,aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))) = times_2397367101498566445atural(zero_n8403883297036319079atural(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2)),aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))) ).

% take_bit_of_2
tff(fact_8478_take__bit__of__2,axiom,
    ! [N2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_o_nat(zero_n2687167440665602831ol_nat,aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% take_bit_of_2
tff(fact_8479_take__bit__of__2,axiom,
    ! [N2: nat] : aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N2),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,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% take_bit_of_2
tff(fact_8480_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(aa_fun6875833325056119164at_nat(produc4528391901023682385at_nat,F),X),Y) = aa_Pro262675715343573244at_nat(produc2626176000494625587at_nat(aa_Pro6895555690143131534at_nat(aTP_Lamm_afz(F),Y)),X) ).

% nested_case_prod_simp
tff(fact_8481_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(aa_fun5912894404062470332_nat_o(produc6395135316500328296_nat_o,F),X),Y)
    <=> aa_Pro6150637604445964947_nat_o(produc6081775807080527818_nat_o(aa_Pro1957341097241355813_nat_o(aTP_Lamm_aga(F),Y)),X) ) ).

% nested_case_prod_simp
tff(fact_8482_case__prod__app,axiom,
    ! [F: fun_na1668100129766719005at_nat,X: product_prod_nat_nat,Y: product_prod_nat_nat] : aa_Pro262675715343573244at_nat(aa_Pro8023792847356852296at_nat(aa_fun6875833325056119164at_nat(produc4528391901023682385at_nat,F),X),Y) = aa_Pro262675715343573244at_nat(produc2626176000494625587at_nat(aa_Pro6895555690143131534at_nat(aTP_Lamm_afz(F),Y)),X) ).

% case_prod_app
tff(fact_8483_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(aa_fun5912894404062470332_nat_o(produc6395135316500328296_nat_o,F),X),Y)
    <=> aa_Pro6150637604445964947_nat_o(produc6081775807080527818_nat_o(aa_Pro1957341097241355813_nat_o(aTP_Lamm_aga(F),Y)),X) ) ).

% case_prod_app
tff(fact_8484_prod_Ocase__distrib,axiom,
    ! [Ha: fun_o_o,F: fun_int_fun_int_o,Prod: product_prod_int_int] :
      ( aa_o_o(Ha,aa_Pro4592068669816069595_int_o(aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,F),Prod))
    <=> aa_Pro4592068669816069595_int_o(aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,aa_fun6133342665515662012_int_o(aTP_Lamm_agb(Ha),F)),Prod) ) ).

% prod.case_distrib
tff(fact_8485_prod_Ocase__distrib,axiom,
    ! [Ha: fun_o_int,F: fun_int_fun_int_o,Prod: product_prod_int_int] : aa_o_int(Ha,aa_Pro4592068669816069595_int_o(aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,F),Prod)) = aa_Pro6799258434465214121nt_int(aa_fun914008879417524284nt_int(produc8211389475949308722nt_int,aa_fun7323606162024495668nt_int(aTP_Lamm_agc(Ha),F)),Prod) ).

% prod.case_distrib
tff(fact_8486_prod_Ocase__distrib,axiom,
    ! [Ha: fun_int_o,F: fun_int_fun_int_int,Prod: product_prod_int_int] :
      ( aa_int_o(Ha,aa_Pro6799258434465214121nt_int(aa_fun914008879417524284nt_int(produc8211389475949308722nt_int,F),Prod))
    <=> aa_Pro4592068669816069595_int_o(aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,aa_fun82482788810740534_int_o(aTP_Lamm_agd(Ha),F)),Prod) ) ).

% prod.case_distrib
tff(fact_8487_prod_Ocase__distrib,axiom,
    ! [Ha: fun_int_int,F: fun_int_fun_int_int,Prod: product_prod_int_int] : aa_int_int(Ha,aa_Pro6799258434465214121nt_int(aa_fun914008879417524284nt_int(produc8211389475949308722nt_int,F),Prod)) = aa_Pro6799258434465214121nt_int(aa_fun914008879417524284nt_int(produc8211389475949308722nt_int,aa_fun6119123525526402106nt_int(aTP_Lamm_age(Ha),F)),Prod) ).

% prod.case_distrib
tff(fact_8488_prod_Ocase__distrib,axiom,
    ! [Ha: fun_Pr3238374608701176162_int_o,F: fun_in1696471567231710611nt_int,Prod: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(Ha,aa_Pro6970504127699319548nt_int(aa_fun8026974734089330650nt_int(produc4245557441103728435nt_int,F),Prod))
    <=> aa_Pro4592068669816069595_int_o(aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,aa_fun4545072666309863515_int_o(aTP_Lamm_agf(Ha),F)),Prod) ) ).

% prod.case_distrib
tff(fact_8489_prod_Ocase__distrib,axiom,
    ! [Ha: fun_Pr982841511892225912nt_int,F: fun_in1696471567231710611nt_int,Prod: product_prod_int_int] : aa_Pro6799258434465214121nt_int(Ha,aa_Pro6970504127699319548nt_int(aa_fun8026974734089330650nt_int(produc4245557441103728435nt_int,F),Prod)) = aa_Pro6799258434465214121nt_int(aa_fun914008879417524284nt_int(produc8211389475949308722nt_int,aa_fun7461304428735181397nt_int(aTP_Lamm_agg(Ha),F)),Prod) ).

% prod.case_distrib
tff(fact_8490_prod_Ocase__distrib,axiom,
    ! [Ha: fun_o_8988054265482332260nt_int,F: fun_int_fun_int_o,Prod: product_prod_int_int] : aa_o_P3134438092330336605nt_int(Ha,aa_Pro4592068669816069595_int_o(aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,F),Prod)) = aa_Pro6970504127699319548nt_int(aa_fun8026974734089330650nt_int(produc4245557441103728435nt_int,aa_fun4023131440832735197nt_int(aTP_Lamm_agh(Ha),F)),Prod) ).

% prod.case_distrib
tff(fact_8491_prod_Ocase__distrib,axiom,
    ! [Ha: fun_in4211310926729683942nt_int,F: fun_int_fun_int_int,Prod: product_prod_int_int] : aa_int7462565939943104719nt_int(Ha,aa_Pro6799258434465214121nt_int(aa_fun914008879417524284nt_int(produc8211389475949308722nt_int,F),Prod)) = aa_Pro6970504127699319548nt_int(aa_fun8026974734089330650nt_int(produc4245557441103728435nt_int,aa_fun1034830265760468567nt_int(aTP_Lamm_agi(Ha),F)),Prod) ).

% prod.case_distrib
tff(fact_8492_prod_Ocase__distrib,axiom,
    ! [Ha: fun_Pr2478283820680522243nt_int,F: fun_in1696471567231710611nt_int,Prod: product_prod_int_int] : aa_Pro6970504127699319548nt_int(Ha,aa_Pro6970504127699319548nt_int(aa_fun8026974734089330650nt_int(produc4245557441103728435nt_int,F),Prod)) = aa_Pro6970504127699319548nt_int(aa_fun8026974734089330650nt_int(produc4245557441103728435nt_int,aa_fun4152872853588681340nt_int(aTP_Lamm_agj(Ha),F)),Prod) ).

% prod.case_distrib
tff(fact_8493_prod_Ocase__distrib,axiom,
    ! [Ha: fun_fu3275390434952296369_nat_o,F: fun_na5914473681949881652_nat_o,Prod: product_prod_nat_nat] : aa_fun359341089595923442_nat_o(Ha,aa_Pro989595756709598047_nat_o(aa_fun5912894404062470332_nat_o(produc6395135316500328296_nat_o,F),Prod)) = aa_Pro989595756709598047_nat_o(aa_fun5912894404062470332_nat_o(produc6395135316500328296_nat_o,aa_fun824319262034445606_nat_o(aTP_Lamm_agk(Ha),F)),Prod) ).

% prod.case_distrib
tff(fact_8494_take__bit__tightened,axiom,
    ! [N2: nat,A: nat,B: nat,M: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N2),A) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N2),B) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,M),A) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,M),B) ) ) ) ).

% take_bit_tightened
tff(fact_8495_take__bit__tightened,axiom,
    ! [N2: nat,A: int,B: int,M: nat] :
      ( ( aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N2),A) = aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N2),B) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
       => ( aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,M),A) = aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,M),B) ) ) ) ).

% take_bit_tightened
tff(fact_8496_take__bit__tightened__less__eq__nat,axiom,
    ! [M: nat,N2: nat,Q4: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,M),Q4)),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N2),Q4)) ) ).

% take_bit_tightened_less_eq_nat
tff(fact_8497_take__bit__nat__less__eq__self,axiom,
    ! [N2: nat,M: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N2),M)),M) ).

% take_bit_nat_less_eq_self
tff(fact_8498_of__bool__conj,axiom,
    ! [Pa: $o,Qa: $o] :
      aa_o_Code_integer(zero_n356916108424825756nteger,
        ( (Pa)
        & (Qa) )) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_o_Code_integer(zero_n356916108424825756nteger,(Pa))),aa_o_Code_integer(zero_n356916108424825756nteger,(Qa))) ).

% of_bool_conj
tff(fact_8499_of__bool__conj,axiom,
    ! [Pa: $o,Qa: $o] :
      aa_o_rat(zero_n2052037380579107095ol_rat,
        ( (Pa)
        & (Qa) )) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_o_rat(zero_n2052037380579107095ol_rat,(Pa))),aa_o_rat(zero_n2052037380579107095ol_rat,(Qa))) ).

% of_bool_conj
tff(fact_8500_of__bool__conj,axiom,
    ! [Pa: $o,Qa: $o] :
      aa_o_nat(zero_n2687167440665602831ol_nat,
        ( (Pa)
        & (Qa) )) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_o_nat(zero_n2687167440665602831ol_nat,(Pa))),aa_o_nat(zero_n2687167440665602831ol_nat,(Qa))) ).

% of_bool_conj
tff(fact_8501_of__bool__conj,axiom,
    ! [Pa: $o,Qa: $o] :
      aa_o_int(zero_n2684676970156552555ol_int,
        ( (Pa)
        & (Qa) )) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_o_int(zero_n2684676970156552555ol_int,(Pa))),aa_o_int(zero_n2684676970156552555ol_int,(Qa))) ).

% of_bool_conj
tff(fact_8502_split__cong,axiom,
    ! [Q4: product_prod_nat_nat,F: fun_na1668100129766719005at_nat,G: fun_na1668100129766719005at_nat,P2: product_prod_nat_nat] :
      ( ! [X3: nat,Y3: nat] :
          ( ( aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,X3),Y3) = Q4 )
         => ( aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(F,X3),Y3) = aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(G,X3),Y3) ) )
     => ( ( P2 = Q4 )
       => ( aa_Pro8023792847356852296at_nat(aa_fun6875833325056119164at_nat(produc4528391901023682385at_nat,F),P2) = aa_Pro8023792847356852296at_nat(aa_fun6875833325056119164at_nat(produc4528391901023682385at_nat,G),Q4) ) ) ) ).

% split_cong
tff(fact_8503_split__cong,axiom,
    ! [Q4: product_prod_nat_nat,F: fun_na5914473681949881652_nat_o,G: fun_na5914473681949881652_nat_o,P2: product_prod_nat_nat] :
      ( ! [X3: nat,Y3: nat] :
          ( ( aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,X3),Y3) = Q4 )
         => ( aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(F,X3),Y3) = aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(G,X3),Y3) ) )
     => ( ( P2 = Q4 )
       => ( aa_Pro989595756709598047_nat_o(aa_fun5912894404062470332_nat_o(produc6395135316500328296_nat_o,F),P2) = aa_Pro989595756709598047_nat_o(aa_fun5912894404062470332_nat_o(produc6395135316500328296_nat_o,G),Q4) ) ) ) ).

% split_cong
tff(fact_8504_split__cong,axiom,
    ! [Q4: product_prod_int_int,F: fun_in1696471567231710611nt_int,G: fun_in1696471567231710611nt_int,P2: product_prod_int_int] :
      ( ! [X3: int,Y3: int] :
          ( ( aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X3),Y3) = Q4 )
         => ( aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(F,X3),Y3) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(G,X3),Y3) ) )
     => ( ( P2 = Q4 )
       => ( aa_Pro6970504127699319548nt_int(aa_fun8026974734089330650nt_int(produc4245557441103728435nt_int,F),P2) = aa_Pro6970504127699319548nt_int(aa_fun8026974734089330650nt_int(produc4245557441103728435nt_int,G),Q4) ) ) ) ).

% split_cong
tff(fact_8505_split__cong,axiom,
    ! [Q4: product_prod_int_int,F: fun_int_fun_int_o,G: fun_int_fun_int_o,P2: product_prod_int_int] :
      ( ! [X3: int,Y3: int] :
          ( ( aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X3),Y3) = Q4 )
         => ( aa_int_o(aa_int_fun_int_o(F,X3),Y3)
          <=> aa_int_o(aa_int_fun_int_o(G,X3),Y3) ) )
     => ( ( P2 = Q4 )
       => ( aa_Pro4592068669816069595_int_o(aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,F),P2)
        <=> aa_Pro4592068669816069595_int_o(aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,G),Q4) ) ) ) ).

% split_cong
tff(fact_8506_split__cong,axiom,
    ! [Q4: product_prod_int_int,F: fun_int_fun_int_int,G: fun_int_fun_int_int,P2: product_prod_int_int] :
      ( ! [X3: int,Y3: int] :
          ( ( aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X3),Y3) = Q4 )
         => ( aa_int_int(aa_int_fun_int_int(F,X3),Y3) = aa_int_int(aa_int_fun_int_int(G,X3),Y3) ) )
     => ( ( P2 = Q4 )
       => ( aa_Pro6799258434465214121nt_int(aa_fun914008879417524284nt_int(produc8211389475949308722nt_int,F),P2) = aa_Pro6799258434465214121nt_int(aa_fun914008879417524284nt_int(produc8211389475949308722nt_int,G),Q4) ) ) ) ).

% split_cong
tff(fact_8507_old_Oprod_Ocase,axiom,
    ! [F: fun_na1668100129766719005at_nat,X1: nat,X2: nat] : aa_Pro8023792847356852296at_nat(aa_fun6875833325056119164at_nat(produc4528391901023682385at_nat,F),aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,X1),X2)) = aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(F,X1),X2) ).

% old.prod.case
tff(fact_8508_old_Oprod_Ocase,axiom,
    ! [F: fun_na5914473681949881652_nat_o,X1: nat,X2: nat] : aa_Pro989595756709598047_nat_o(aa_fun5912894404062470332_nat_o(produc6395135316500328296_nat_o,F),aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,X1),X2)) = aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(F,X1),X2) ).

% old.prod.case
tff(fact_8509_old_Oprod_Ocase,axiom,
    ! [F: fun_in1696471567231710611nt_int,X1: int,X2: int] : aa_Pro6970504127699319548nt_int(aa_fun8026974734089330650nt_int(produc4245557441103728435nt_int,F),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X1),X2)) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(F,X1),X2) ).

% old.prod.case
tff(fact_8510_old_Oprod_Ocase,axiom,
    ! [F: fun_int_fun_int_o,X1: int,X2: int] :
      ( aa_Pro4592068669816069595_int_o(aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,F),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X1),X2))
    <=> aa_int_o(aa_int_fun_int_o(F,X1),X2) ) ).

% old.prod.case
tff(fact_8511_old_Oprod_Ocase,axiom,
    ! [F: fun_int_fun_int_int,X1: int,X2: int] : aa_Pro6799258434465214121nt_int(aa_fun914008879417524284nt_int(produc8211389475949308722nt_int,F),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X1),X2)) = aa_int_int(aa_int_fun_int_int(F,X1),X2) ).

% old.prod.case
tff(fact_8512_case__prod__Pair__iden,axiom,
    ! [P2: produc8799533983703829792et_nat] : produc1949718844479134567et_nat(produc6017135503889078682et_nat,P2) = P2 ).

% case_prod_Pair_iden
tff(fact_8513_case__prod__Pair__iden,axiom,
    ! [P2: produc3112576180687729086et_nat] : produc2255458887868536227et_nat(produc7291855603632349880et_nat,P2) = P2 ).

% case_prod_Pair_iden
tff(fact_8514_case__prod__Pair__iden,axiom,
    ! [P2: produc3043004577601930755nt_int] : produc6823905543215200229nt_int(produc7320969046608631221nt_int,P2) = P2 ).

% case_prod_Pair_iden
tff(fact_8515_case__prod__Pair__iden,axiom,
    ! [P2: produc4788131091073775919nt_int] : produc7102243260366339645nt_int(produc4729913460834754785nt_int,P2) = P2 ).

% case_prod_Pair_iden
tff(fact_8516_case__prod__Pair__iden,axiom,
    ! [P2: produc3260487557148687353it_nat] : produc2183267410226956569it_nat(produc9178034014595674355it_nat,P2) = P2 ).

% case_prod_Pair_iden
tff(fact_8517_case__prod__Pair__iden,axiom,
    ! [P2: product_prod_int_int] : aa_Pro6970504127699319548nt_int(aa_fun8026974734089330650nt_int(produc4245557441103728435nt_int,product_Pair_int_int),P2) = P2 ).

% case_prod_Pair_iden
tff(fact_8518_take__bit__nat__eq,axiom,
    ! [K: int,N2: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N2),aa_int_nat(nat2,K)) = aa_int_nat(nat2,aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N2),K)) ) ) ).

% take_bit_nat_eq
tff(fact_8519_nat__take__bit__eq,axiom,
    ! [K: int,N2: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K)
     => ( aa_int_nat(nat2,aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N2),K)) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N2),aa_int_nat(nat2,K)) ) ) ).

% nat_take_bit_eq
tff(fact_8520_cond__case__prod__eta,axiom,
    ! [F: fun_na1668100129766719005at_nat,G: fun_Pr2477239776880934807at_nat] :
      ( ! [X3: nat,Y3: nat] : aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(F,X3),Y3) = aa_Pro8023792847356852296at_nat(G,aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,X3),Y3))
     => ( aa_fun6875833325056119164at_nat(produc4528391901023682385at_nat,F) = G ) ) ).

% cond_case_prod_eta
tff(fact_8521_cond__case__prod__eta,axiom,
    ! [F: fun_na5914473681949881652_nat_o,G: fun_Pr2028404973800837934_nat_o] :
      ( ! [X3: nat,Y3: nat] : aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(F,X3),Y3) = aa_Pro989595756709598047_nat_o(G,aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,X3),Y3))
     => ( aa_fun5912894404062470332_nat_o(produc6395135316500328296_nat_o,F) = G ) ) ).

% cond_case_prod_eta
tff(fact_8522_cond__case__prod__eta,axiom,
    ! [F: fun_in1696471567231710611nt_int,G: fun_Pr2478283820680522243nt_int] :
      ( ! [X3: int,Y3: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(F,X3),Y3) = aa_Pro6970504127699319548nt_int(G,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X3),Y3))
     => ( aa_fun8026974734089330650nt_int(produc4245557441103728435nt_int,F) = G ) ) ).

% cond_case_prod_eta
tff(fact_8523_cond__case__prod__eta,axiom,
    ! [F: fun_int_fun_int_o,G: fun_Pr3238374608701176162_int_o] :
      ( ! [X3: int,Y3: int] :
          ( aa_int_o(aa_int_fun_int_o(F,X3),Y3)
        <=> aa_Pro4592068669816069595_int_o(G,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X3),Y3)) )
     => ( aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,F) = G ) ) ).

% cond_case_prod_eta
tff(fact_8524_cond__case__prod__eta,axiom,
    ! [F: fun_int_fun_int_int,G: fun_Pr982841511892225912nt_int] :
      ( ! [X3: int,Y3: int] : aa_int_int(aa_int_fun_int_int(F,X3),Y3) = aa_Pro6799258434465214121nt_int(G,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X3),Y3))
     => ( aa_fun914008879417524284nt_int(produc8211389475949308722nt_int,F) = G ) ) ).

% cond_case_prod_eta
tff(fact_8525_case__prod__eta,axiom,
    ! [F: fun_Pr2477239776880934807at_nat] : aa_fun6875833325056119164at_nat(produc4528391901023682385at_nat,aTP_Lamm_agl(F)) = F ).

% case_prod_eta
tff(fact_8526_case__prod__eta,axiom,
    ! [F: fun_Pr2028404973800837934_nat_o] : aa_fun5912894404062470332_nat_o(produc6395135316500328296_nat_o,aTP_Lamm_agm(F)) = F ).

% case_prod_eta
tff(fact_8527_case__prod__eta,axiom,
    ! [F: fun_Pr2478283820680522243nt_int] : aa_fun8026974734089330650nt_int(produc4245557441103728435nt_int,aTP_Lamm_agn(F)) = F ).

% case_prod_eta
tff(fact_8528_case__prod__eta,axiom,
    ! [F: fun_Pr3238374608701176162_int_o] : aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,aTP_Lamm_ago(F)) = F ).

% case_prod_eta
tff(fact_8529_case__prod__eta,axiom,
    ! [F: fun_Pr982841511892225912nt_int] : aa_fun914008879417524284nt_int(produc8211389475949308722nt_int,aTP_Lamm_agp(F)) = F ).

% case_prod_eta
tff(fact_8530_case__prodE2,axiom,
    ! [Qa: fun_fu8391177546919232046_nat_o,Pa: fun_na1668100129766719005at_nat,Z: product_prod_nat_nat] :
      ( aa_fun3538214411680374423_nat_o(Qa,aa_Pro8023792847356852296at_nat(aa_fun6875833325056119164at_nat(produc4528391901023682385at_nat,Pa),Z))
     => ~ ! [X3: nat,Y3: nat] :
            ( ( Z = aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,X3),Y3) )
           => ~ aa_fun3538214411680374423_nat_o(Qa,aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(Pa,X3),Y3)) ) ) ).

% case_prodE2
tff(fact_8531_case__prodE2,axiom,
    ! [Qa: fun_fu6891447142836604759at_o_o,Pa: fun_na5914473681949881652_nat_o,Z: product_prod_nat_nat] :
      ( aa_fun8408151826566693824at_o_o(Qa,aa_Pro989595756709598047_nat_o(aa_fun5912894404062470332_nat_o(produc6395135316500328296_nat_o,Pa),Z))
     => ~ ! [X3: nat,Y3: nat] :
            ( ( Z = aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,X3),Y3) )
           => ~ aa_fun8408151826566693824at_o_o(Qa,aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(Pa,X3),Y3)) ) ) ).

% case_prodE2
tff(fact_8532_case__prodE2,axiom,
    ! [Qa: fun_Pr3238374608701176162_int_o,Pa: fun_in1696471567231710611nt_int,Z: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(Qa,aa_Pro6970504127699319548nt_int(aa_fun8026974734089330650nt_int(produc4245557441103728435nt_int,Pa),Z))
     => ~ ! [X3: int,Y3: int] :
            ( ( Z = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X3),Y3) )
           => ~ aa_Pro4592068669816069595_int_o(Qa,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(Pa,X3),Y3)) ) ) ).

% case_prodE2
tff(fact_8533_case__prodE2,axiom,
    ! [Qa: fun_o_o,Pa: fun_int_fun_int_o,Z: product_prod_int_int] :
      ( aa_o_o(Qa,aa_Pro4592068669816069595_int_o(aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,Pa),Z))
     => ~ ! [X3: int,Y3: int] :
            ( ( Z = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X3),Y3) )
           => ~ aa_o_o(Qa,aa_int_o(aa_int_fun_int_o(Pa,X3),Y3)) ) ) ).

% case_prodE2
tff(fact_8534_case__prodE2,axiom,
    ! [Qa: fun_int_o,Pa: fun_int_fun_int_int,Z: product_prod_int_int] :
      ( aa_int_o(Qa,aa_Pro6799258434465214121nt_int(aa_fun914008879417524284nt_int(produc8211389475949308722nt_int,Pa),Z))
     => ~ ! [X3: int,Y3: int] :
            ( ( Z = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X3),Y3) )
           => ~ aa_int_o(Qa,aa_int_int(aa_int_fun_int_int(Pa,X3),Y3)) ) ) ).

% case_prodE2
tff(fact_8535_zero__less__eq__of__bool,axiom,
    ! [Pa: $o] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_o_Code_integer(zero_n356916108424825756nteger,(Pa))) ).

% zero_less_eq_of_bool
tff(fact_8536_zero__less__eq__of__bool,axiom,
    ! [Pa: $o] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_o_rat(zero_n2052037380579107095ol_rat,(Pa))) ).

% zero_less_eq_of_bool
tff(fact_8537_zero__less__eq__of__bool,axiom,
    ! [Pa: $o] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_o_nat(zero_n2687167440665602831ol_nat,(Pa))) ).

% zero_less_eq_of_bool
tff(fact_8538_zero__less__eq__of__bool,axiom,
    ! [Pa: $o] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_o_int(zero_n2684676970156552555ol_int,(Pa))) ).

% zero_less_eq_of_bool
tff(fact_8539_of__bool__less__eq__one,axiom,
    ! [Pa: $o] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_o_Code_integer(zero_n356916108424825756nteger,(Pa))),one_one_Code_integer) ).

% of_bool_less_eq_one
tff(fact_8540_of__bool__less__eq__one,axiom,
    ! [Pa: $o] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_o_rat(zero_n2052037380579107095ol_rat,(Pa))),one_one_rat) ).

% of_bool_less_eq_one
tff(fact_8541_of__bool__less__eq__one,axiom,
    ! [Pa: $o] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_o_nat(zero_n2687167440665602831ol_nat,(Pa))),one_one_nat) ).

% of_bool_less_eq_one
tff(fact_8542_of__bool__less__eq__one,axiom,
    ! [Pa: $o] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_o_int(zero_n2684676970156552555ol_int,(Pa))),one_one_int) ).

% of_bool_less_eq_one
tff(fact_8543_take__bit__tightened__less__eq__int,axiom,
    ! [M: nat,N2: nat,K: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,M),K)),aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N2),K)) ) ).

% take_bit_tightened_less_eq_int
tff(fact_8544_take__bit__nonnegative,axiom,
    ! [N2: nat,K: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N2),K)) ).

% take_bit_nonnegative
tff(fact_8545_take__bit__int__less__eq__self__iff,axiom,
    ! [N2: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N2),K)),K)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K) ) ).

% take_bit_int_less_eq_self_iff
tff(fact_8546_not__take__bit__negative,axiom,
    ! [N2: nat,K: int] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N2),K)),zero_zero_int) ).

% not_take_bit_negative
tff(fact_8547_take__bit__int__greater__self__iff,axiom,
    ! [K: int,N2: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N2),K))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int) ) ).

% take_bit_int_greater_self_iff
tff(fact_8548_signed__take__bit__take__bit,axiom,
    ! [M: nat,N2: nat,A: int] :
      aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,M),aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N2),A)) = aa_int_int($ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),M),aa_nat_fun_int_int(bit_se2923211474154528505it_int,N2),aa_nat_fun_int_int(bit_ri631733984087533419it_int,M)),A) ).

% signed_take_bit_take_bit
tff(fact_8549_take__bit__unset__bit__eq,axiom,
    ! [N2: nat,M: nat,A: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N2),aa_nat_nat(aa_nat_fun_nat_nat(bit_se4205575877204974255it_nat,M),A)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),M),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N2),A),aa_nat_nat(aa_nat_fun_nat_nat(bit_se4205575877204974255it_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N2),A))) ).

% take_bit_unset_bit_eq
tff(fact_8550_take__bit__unset__bit__eq,axiom,
    ! [N2: nat,M: nat,A: int] :
      aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N2),aa_int_int(aa_nat_fun_int_int(bit_se4203085406695923979it_int,M),A)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),M),aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N2),A),aa_int_int(aa_nat_fun_int_int(bit_se4203085406695923979it_int,M),aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N2),A))) ).

% take_bit_unset_bit_eq
tff(fact_8551_take__bit__set__bit__eq,axiom,
    ! [N2: nat,M: nat,A: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N2),aa_nat_nat(aa_nat_fun_nat_nat(bit_se7882103937844011126it_nat,M),A)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),M),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N2),A),aa_nat_nat(aa_nat_fun_nat_nat(bit_se7882103937844011126it_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N2),A))) ).

% take_bit_set_bit_eq
tff(fact_8552_take__bit__set__bit__eq,axiom,
    ! [N2: nat,M: nat,A: int] :
      aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N2),aa_int_int(aa_nat_fun_int_int(bit_se7879613467334960850it_int,M),A)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),M),aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N2),A),aa_int_int(aa_nat_fun_int_int(bit_se7879613467334960850it_int,M),aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N2),A))) ).

% take_bit_set_bit_eq
tff(fact_8553_take__bit__flip__bit__eq,axiom,
    ! [N2: nat,M: nat,A: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N2),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2161824704523386999it_nat,M),A)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),M),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N2),A),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2161824704523386999it_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N2),A))) ).

% take_bit_flip_bit_eq
tff(fact_8554_take__bit__flip__bit__eq,axiom,
    ! [N2: nat,M: nat,A: int] :
      aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N2),aa_int_int(aa_nat_fun_int_int(bit_se2159334234014336723it_int,M),A)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),M),aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N2),A),aa_int_int(aa_nat_fun_int_int(bit_se2159334234014336723it_int,M),aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N2),A))) ).

% take_bit_flip_bit_eq
tff(fact_8555_zdvd__antisym__nonneg,axiom,
    ! [M: int,N2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),M)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),N2)
       => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,M),N2)
         => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,N2),M)
           => ( M = N2 ) ) ) ) ) ).

% zdvd_antisym_nonneg
tff(fact_8556_zdvd__not__zless,axiom,
    ! [M: int,N2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),M)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,M),N2)
       => ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,N2),M) ) ) ).

% zdvd_not_zless
tff(fact_8557_uncurry__def,axiom,
    ! [F: fun_na1668100129766719005at_nat] : uncurr9212869592684144700at_nat(F) = aa_fun6875833325056119164at_nat(produc4528391901023682385at_nat,F) ).

% uncurry_def
tff(fact_8558_uncurry__def,axiom,
    ! [F: fun_na5914473681949881652_nat_o] : uncurr8479020417594943315_nat_o(F) = aa_fun5912894404062470332_nat_o(produc6395135316500328296_nat_o,F) ).

% uncurry_def
tff(fact_8559_uncurry__def,axiom,
    ! [F: fun_in1696471567231710611nt_int] : uncurr7650761721940715016nt_int(F) = aa_fun8026974734089330650nt_int(produc4245557441103728435nt_int,F) ).

% uncurry_def
tff(fact_8560_uncurry__def,axiom,
    ! [F: fun_int_fun_int_o] : uncurry_int_int_o(F) = aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,F) ).

% uncurry_def
tff(fact_8561_uncurry__def,axiom,
    ! [F: fun_int_fun_int_int] : uncurry_int_int_int(F) = aa_fun914008879417524284nt_int(produc8211389475949308722nt_int,F) ).

% uncurry_def
tff(fact_8562_internal__case__prod__def,axiom,
    produc8104577556401574439at_nat = produc4528391901023682385at_nat ).

% internal_case_prod_def
tff(fact_8563_internal__case__prod__def,axiom,
    produc9189008374811844670_nat_o = produc6395135316500328296_nat_o ).

% internal_case_prod_def
tff(fact_8564_internal__case__prod__def,axiom,
    produc297006045350968285nt_int = produc4245557441103728435nt_int ).

% internal_case_prod_def
tff(fact_8565_internal__case__prod__def,axiom,
    produc8005341501107743676_int_o = produc4947309494688390418_int_o ).

% internal_case_prod_def
tff(fact_8566_internal__case__prod__def,axiom,
    produc7926200574084438792nt_int = produc8211389475949308722nt_int ).

% internal_case_prod_def
tff(fact_8567_finite__divisors__int,axiom,
    ! [I2: int] :
      ( ( I2 != zero_zero_int )
     => aa_set_int_o(finite_finite_int,collect_int(aa_int_fun_int_o(aTP_Lamm_adv,I2))) ) ).

% finite_divisors_int
tff(fact_8568_take__bit__signed__take__bit,axiom,
    ! [M: nat,N2: nat,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(suc,N2))
     => ( aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,M),aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,N2),A)) = aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,M),A) ) ) ).

% take_bit_signed_take_bit
tff(fact_8569_zdvd__imp__le,axiom,
    ! [Z: int,N2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Z),N2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),N2)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),N2) ) ) ).

% zdvd_imp_le
tff(fact_8570_dvd__imp__le__int,axiom,
    ! [I2: int,D2: int] :
      ( ( I2 != zero_zero_int )
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D2),I2)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,D2)),aa_int_int(abs_abs_int,I2)) ) ) ).

% dvd_imp_le_int
tff(fact_8571_subset__decode__imp__le,axiom,
    ! [M: nat,N2: nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_nat_set_nat(nat_set_decode,M)),aa_nat_set_nat(nat_set_decode,N2))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2) ) ).

% subset_decode_imp_le
tff(fact_8572_mod__int__pos__iff,axiom,
    ! [K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),L))
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,L),K)
        | ( ( L = zero_zero_int )
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K) )
        | aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),L) ) ) ).

% mod_int_pos_iff
tff(fact_8573_take__bit__Suc__bit0,axiom,
    ! [N2: nat,K: num] : aa_Cod1783974652533396728nteger(bit_se1745604003318907178nteger(aa_nat_nat(suc,N2)),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,K))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(bit_se1745604003318907178nteger(N2),aa_num_Code_integer(numera6620942414471956472nteger,K))),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) ).

% take_bit_Suc_bit0
tff(fact_8574_take__bit__Suc__bit0,axiom,
    ! [N2: nat,K: num] : bit_se569199155075624693atural(aa_nat_nat(suc,N2),aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,K))) = times_2397367101498566445atural(bit_se569199155075624693atural(N2,aa_num_Code_natural(numera5444537566228673987atural,K)),aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))) ).

% take_bit_Suc_bit0
tff(fact_8575_take__bit__Suc__bit0,axiom,
    ! [N2: nat,K: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,aa_nat_nat(suc,N2)),aa_num_nat(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,N2),aa_num_nat(numeral_numeral_nat,K))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% take_bit_Suc_bit0
tff(fact_8576_take__bit__Suc__bit0,axiom,
    ! [N2: nat,K: num] : aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,aa_nat_nat(suc,N2)),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,N2),aa_num_int(numeral_numeral_int,K))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% take_bit_Suc_bit0
tff(fact_8577_take__bit__nat__eq__self,axiom,
    ! [M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N2),M) = M ) ) ).

% take_bit_nat_eq_self
tff(fact_8578_take__bit__nat__less__exp,axiom,
    ! [N2: nat,M: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N2),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2)) ).

% take_bit_nat_less_exp
tff(fact_8579_take__bit__nat__eq__self__iff,axiom,
    ! [N2: nat,M: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N2),M) = M )
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2)) ) ).

% take_bit_nat_eq_self_iff
tff(fact_8580_take__bit__int__less__exp,axiom,
    ! [N2: nat,K: int] : aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N2),K)),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N2)) ).

% take_bit_int_less_exp
tff(fact_8581_take__bit__nat__less__self__iff,axiom,
    ! [N2: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N2),M)),M)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2)),M) ) ).

% take_bit_nat_less_self_iff
tff(fact_8582_bits__induct,axiom,
    ! [Pa: 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(Pa,A4) )
     => ( ! [A4: code_integer,B4: $o] :
            ( aa_Code_integer_o(Pa,A4)
           => ( ( aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(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(Pa,aa_Cod1783974652533396728nteger(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(Pa,A) ) ) ).

% bits_induct
tff(fact_8583_bits__induct,axiom,
    ! [Pa: fun_Code_natural_o,A: code_natural] :
      ( ! [A4: code_natural] :
          ( ( divide5121882707175180666atural(A4,aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))) = A4 )
         => aa_Code_natural_o(Pa,A4) )
     => ( ! [A4: code_natural,B4: $o] :
            ( aa_Code_natural_o(Pa,A4)
           => ( ( divide5121882707175180666atural(plus_p4538020629002901425atural(zero_n8403883297036319079atural((B4)),times_2397367101498566445atural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one)),A4)),aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))) = A4 )
             => aa_Code_natural_o(Pa,plus_p4538020629002901425atural(zero_n8403883297036319079atural((B4)),times_2397367101498566445atural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one)),A4))) ) )
       => aa_Code_natural_o(Pa,A) ) ) ).

% bits_induct
tff(fact_8584_bits__induct,axiom,
    ! [Pa: fun_nat_o,A: nat] :
      ( ! [A4: nat] :
          ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A4),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = A4 )
         => aa_nat_o(Pa,A4) )
     => ( ! [A4: nat,B4: $o] :
            ( aa_nat_o(Pa,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,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A4))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = A4 )
             => aa_nat_o(Pa,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,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A4))) ) )
       => aa_nat_o(Pa,A) ) ) ).

% bits_induct
tff(fact_8585_bits__induct,axiom,
    ! [Pa: 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(Pa,A4) )
     => ( ! [A4: int,B4: $o] :
            ( aa_int_o(Pa,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(Pa,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(Pa,A) ) ) ).

% bits_induct
tff(fact_8586_take__bit__int__greater__eq__self__iff,axiom,
    ! [K: int,N2: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N2),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))),N2)) ) ).

% take_bit_int_greater_eq_self_iff
tff(fact_8587_take__bit__int__less__self__iff,axiom,
    ! [N2: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N2),K)),K)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N2)),K) ) ).

% take_bit_int_less_self_iff
tff(fact_8588_divmod__nat__if,axiom,
    ! [M: nat,N2: nat] :
      aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(divmod_nat,M),N2) = $ite(
        ( ( N2 = zero_zero_nat )
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N2) ),
        aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,zero_zero_nat),M),
        aa_Pro262675715343573244at_nat(produc2626176000494625587at_nat(aTP_Lamm_agq),aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(divmod_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N2)),N2)) ) ).

% divmod_nat_if
tff(fact_8589_nat__dvd__iff,axiom,
    ! [Z: int,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_int_nat(nat2,Z)),M)
    <=> $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z),aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Z),aa_nat_int(semiri1314217659103216013at_int,M)),M = zero_zero_nat) ) ).

% nat_dvd_iff
tff(fact_8590_take__bit__int__eq__self__iff,axiom,
    ! [N2: nat,K: int] :
      ( ( aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N2),K) = K )
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,K),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N2)) ) ) ).

% take_bit_int_eq_self_iff
tff(fact_8591_take__bit__int__eq__self,axiom,
    ! [K: int,N2: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N2))
       => ( aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N2),K) = K ) ) ) ).

% take_bit_int_eq_self
tff(fact_8592_exp__mod__exp,axiom,
    ! [M: nat,N2: 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))),N2)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_o_Code_integer(zero_n356916108424825756nteger,aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N2))),aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),M)) ).

% exp_mod_exp
tff(fact_8593_exp__mod__exp,axiom,
    ! [M: nat,N2: nat] : modulo8411746178871703098atural(aa_nat_Code_natural(power_7079662738309270450atural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))),M),aa_nat_Code_natural(power_7079662738309270450atural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))),N2)) = times_2397367101498566445atural(zero_n8403883297036319079atural(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N2)),aa_nat_Code_natural(power_7079662738309270450atural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))),M)) ).

% exp_mod_exp
tff(fact_8594_exp__mod__exp,axiom,
    ! [M: nat,N2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2)) = 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),N2))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)) ).

% exp_mod_exp
tff(fact_8595_exp__mod__exp,axiom,
    ! [M: nat,N2: 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))),N2)) = 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),N2))),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_8596_take__bit__Suc,axiom,
    ! [N2: nat,A: code_integer] : aa_Cod1783974652533396728nteger(bit_se1745604003318907178nteger(aa_nat_nat(suc,N2)),A) = aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(bit_se1745604003318907178nteger(N2),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_8597_take__bit__Suc,axiom,
    ! [N2: nat,A: code_natural] : bit_se569199155075624693atural(aa_nat_nat(suc,N2),A) = plus_p4538020629002901425atural(times_2397367101498566445atural(bit_se569199155075624693atural(N2,divide5121882707175180666atural(A,aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one)))),aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))),modulo8411746178871703098atural(A,aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one)))) ).

% take_bit_Suc
tff(fact_8598_take__bit__Suc,axiom,
    ! [N2: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,aa_nat_nat(suc,N2)),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,N2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% take_bit_Suc
tff(fact_8599_take__bit__Suc,axiom,
    ! [N2: nat,A: int] : aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,aa_nat_nat(suc,N2)),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,N2),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_8600_take__bit__int__less__eq,axiom,
    ! [N2: nat,K: 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_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N2)),K)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N2),K)),aa_int_int(aa_int_fun_int_int(minus_minus_int,K),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N2))) ) ) ).

% take_bit_int_less_eq
tff(fact_8601_take__bit__int__greater__eq,axiom,
    ! [K: int,N2: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,K),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N2))),aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N2),K)) ) ).

% take_bit_int_greater_eq
tff(fact_8602_even__nat__iff,axiom,
    ! [K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_int_nat(nat2,K))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K) ) ) ).

% even_nat_iff
tff(fact_8603_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_afc(L)),Qr) ).

% divmod_step_nat_def
tff(fact_8604_exp__div__exp__eq,axiom,
    ! [M: nat,N2: 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))),N2)) = 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,N2),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),N2))) ).

% exp_div_exp_eq
tff(fact_8605_exp__div__exp__eq,axiom,
    ! [M: nat,N2: nat] :
      divide5121882707175180666atural(aa_nat_Code_natural(power_7079662738309270450atural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))),M),aa_nat_Code_natural(power_7079662738309270450atural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))),N2)) = times_2397367101498566445atural(zero_n8403883297036319079atural(( ( aa_nat_Code_natural(power_7079662738309270450atural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))),M) != zero_z2226904508553997617atural )
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),M) )),aa_nat_Code_natural(power_7079662738309270450atural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N2))) ).

% exp_div_exp_eq
tff(fact_8606_exp__div__exp__eq,axiom,
    ! [M: nat,N2: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,
          aa_o_nat(zero_n2687167440665602831ol_nat,
            ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M) != zero_zero_nat )
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),M) ))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N2))) ).

% exp_div_exp_eq
tff(fact_8607_exp__div__exp__eq,axiom,
    ! [M: nat,N2: 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))),N2)) = 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,N2),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),N2))) ).

% exp_div_exp_eq
tff(fact_8608_divmod__step__int__def,axiom,
    ! [L: num,Qr: product_prod_int_int] : aa_Pro6970504127699319548nt_int(unique5024387138958732305ep_int(L),Qr) = aa_Pro6970504127699319548nt_int(aa_fun8026974734089330650nt_int(produc4245557441103728435nt_int,aTP_Lamm_afd(L)),Qr) ).

% divmod_step_int_def
tff(fact_8609_take__bit__minus__small__eq,axiom,
    ! [K: int,N2: 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))),N2))
       => ( aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N2),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))),N2)),K) ) ) ) ).

% take_bit_minus_small_eq
tff(fact_8610_take__bit__numeral__minus__numeral__int,axiom,
    ! [M: num,N2: num] : aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,aa_num_nat(numeral_numeral_nat,M)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N2))) = aa_option_num_int(case_option_int_num(zero_zero_int,aTP_Lamm_agr(M)),aa_num_option_num(bit_take_bit_num(aa_num_nat(numeral_numeral_nat,M)),N2)) ).

% take_bit_numeral_minus_numeral_int
tff(fact_8611_divmod__algorithm__code_I5_J,axiom,
    ! [M: num,N2: num] : unique5052692396658037445od_int(aa_num_num(bit0,M),aa_num_num(bit0,N2)) = aa_Pro6970504127699319548nt_int(aa_fun8026974734089330650nt_int(produc4245557441103728435nt_int,aTP_Lamm_ags),unique5052692396658037445od_int(M,N2)) ).

% divmod_algorithm_code(5)
tff(fact_8612_divmod__algorithm__code_I5_J,axiom,
    ! [M: num,N2: num] : unique5055182867167087721od_nat(aa_num_num(bit0,M),aa_num_num(bit0,N2)) = aa_Pro262675715343573244at_nat(produc2626176000494625587at_nat(aTP_Lamm_agt),unique5055182867167087721od_nat(M,N2)) ).

% divmod_algorithm_code(5)
tff(fact_8613_divmod__algorithm__code_I5_J,axiom,
    ! [M: num,N2: num] : unique3479559517661332726nteger(aa_num_num(bit0,M),aa_num_num(bit0,N2)) = produc6916734918728496179nteger(aTP_Lamm_agu,unique3479559517661332726nteger(M,N2)) ).

% divmod_algorithm_code(5)
tff(fact_8614_Divides_Oadjust__div__eq,axiom,
    ! [Q4: int,Ra2: int] : aa_Pro6799258434465214121nt_int(adjust_div,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q4),Ra2)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Q4),aa_o_int(zero_n2684676970156552555ol_int,Ra2 != zero_zero_int)) ).

% Divides.adjust_div_eq
tff(fact_8615_divmod__divmod__step,axiom,
    ! [M: num,N2: num] :
      unique5055182867167087721od_nat(M,N2) = $ite(aa_num_o(aa_num_fun_num_o(ord_less_num,M),N2),aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,M)),aa_Pro262675715343573244at_nat(unique5026877609467782581ep_nat(N2),unique5055182867167087721od_nat(M,aa_num_num(bit0,N2)))) ).

% divmod_divmod_step
tff(fact_8616_divmod__divmod__step,axiom,
    ! [M: num,N2: num] :
      unique5052692396658037445od_int(M,N2) = $ite(aa_num_o(aa_num_fun_num_o(ord_less_num,M),N2),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(N2),unique5052692396658037445od_int(M,aa_num_num(bit0,N2)))) ).

% divmod_divmod_step
tff(fact_8617_divmod__divmod__step,axiom,
    ! [M: num,N2: num] :
      unique3479559517661332726nteger(M,N2) = $ite(aa_num_o(aa_num_fun_num_o(ord_less_num,M),N2),produc1086072967326762835nteger(zero_z3403309356797280102nteger,aa_num_Code_integer(numera6620942414471956472nteger,M)),unique4921790084139445826nteger(N2,unique3479559517661332726nteger(M,aa_num_num(bit0,N2)))) ).

% divmod_divmod_step
tff(fact_8618_divmod__algorithm__code_I3_J,axiom,
    ! [N2: num] : unique5052692396658037445od_int(one,aa_num_num(bit0,N2)) = 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_8619_divmod__algorithm__code_I3_J,axiom,
    ! [N2: num] : unique5055182867167087721od_nat(one,aa_num_num(bit0,N2)) = aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,one)) ).

% divmod_algorithm_code(3)
tff(fact_8620_divmod__algorithm__code_I3_J,axiom,
    ! [N2: num] : unique3479559517661332726nteger(one,aa_num_num(bit0,N2)) = produc1086072967326762835nteger(zero_z3403309356797280102nteger,aa_num_Code_integer(numera6620942414471956472nteger,one)) ).

% divmod_algorithm_code(3)
tff(fact_8621_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_8622_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_8623_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_8624_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_8625_case__prodI,axiom,
    ! [F: fun_a_2467521563176467050_nat_o,A: a,B: produc6653097349344004940it_nat] :
      ( aa_Pro1202363709425262478_nat_o(aa_a_f337466037417904483_nat_o(F,A),B)
     => aa_Pro6426884464393220769_nat_o(produc8860486935419594360_nat_o(F),aa_Pro8078911840264235907it_nat(aa_a_f2234720956197727492it_nat(produc9178034014595674355it_nat,A),B)) ) ).

% case_prodI
tff(fact_8626_case__prodI,axiom,
    ! [F: fun_int_fun_int_o,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(F,A),B)
     => aa_Pro4592068669816069595_int_o(aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,F),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),B)) ) ).

% case_prodI
tff(fact_8627_case__prodI2,axiom,
    ! [P2: produc8799533983703829792et_nat,Ca: fun_fu8419763957888848259_nat_o] :
      ( ! [A4: fun_Pr6460944699146269935_nat_o,B4: produc3658429121746597890et_nat] :
          ( ( P2 = aa_Pro434340453207536480et_nat(aa_fun3926903090414666066et_nat(produc6017135503889078682et_nat,A4),B4) )
         => aa_Pro653082453696054616_nat_o(aa_fun2638059832795814268_nat_o(Ca,A4),B4) )
     => aa_Pro5643009357358181178_nat_o(produc2915205735004773393_nat_o(Ca),P2) ) ).

% case_prodI2
tff(fact_8628_case__prodI2,axiom,
    ! [P2: produc3112576180687729086et_nat,Ca: fun_fu2611741329380808165_nat_o] :
      ( ! [A4: fun_Pr6460944699146269935_nat_o,B4: produc8799533983703829792et_nat] :
          ( ( P2 = aa_Pro1979504083478021276et_nat(aa_fun615918787490101902et_nat(produc7291855603632349880et_nat,A4),B4) )
         => aa_Pro5643009357358181178_nat_o(aa_fun912680268133416158_nat_o(Ca,A4),B4) )
     => aa_Pro2671238023309943196_nat_o(produc4465912260561112435_nat_o(Ca),P2) ) ).

% case_prodI2
tff(fact_8629_case__prodI2,axiom,
    ! [P2: produc3043004577601930755nt_int,Ca: fun_fu6865642474282033866_int_o] :
      ( ! [A4: fun_Pr7763680713840462443e_term,B4: product_prod_int_int] :
          ( ( P2 = aa_Pro5748361610964203648nt_int(aa_fun3265472288063244902nt_int(produc7320969046608631221nt_int,A4),B4) )
         => aa_Pro4592068669816069595_int_o(aa_fun3538162476766722827_int_o(Ca,A4),B4) )
     => aa_Pro2601676399929874711_int_o(produc2207054017359051062_int_o(Ca),P2) ) ).

% case_prodI2
tff(fact_8630_case__prodI2,axiom,
    ! [P2: produc4788131091073775919nt_int,Ca: fun_fu2978539155105838238_int_o] :
      ( ! [A4: fun_in5730379987311621695e_term,B4: product_prod_int_int] :
          ( ( P2 = aa_Pro8800328608558162860nt_int(aa_fun260840925927807678nt_int(produc4729913460834754785nt_int,A4),B4) )
         => aa_Pro4592068669816069595_int_o(aa_fun586359091965693151_int_o(Ca,A4),B4) )
     => aa_Pro5451197003417746155_int_o(produc7271678899836001034_int_o(Ca),P2) ) ).

% case_prodI2
tff(fact_8631_case__prodI2,axiom,
    ! [P2: produc3260487557148687353it_nat,Ca: fun_a_2467521563176467050_nat_o] :
      ( ! [A4: a,B4: produc6653097349344004940it_nat] :
          ( ( P2 = aa_Pro8078911840264235907it_nat(aa_a_f2234720956197727492it_nat(produc9178034014595674355it_nat,A4),B4) )
         => aa_Pro1202363709425262478_nat_o(aa_a_f337466037417904483_nat_o(Ca,A4),B4) )
     => aa_Pro6426884464393220769_nat_o(produc8860486935419594360_nat_o(Ca),P2) ) ).

% case_prodI2
tff(fact_8632_case__prodI2,axiom,
    ! [P2: product_prod_int_int,Ca: fun_int_fun_int_o] :
      ( ! [A4: int,B4: int] :
          ( ( P2 = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A4),B4) )
         => aa_int_o(aa_int_fun_int_o(Ca,A4),B4) )
     => aa_Pro4592068669816069595_int_o(aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,Ca),P2) ) ).

% case_prodI2
tff(fact_8633_mem__case__prodI,axiom,
    ! [Z: $o,Ca: fun_a_6228866161775505738_set_o,A: a,B: produc6653097349344004940it_nat] :
      ( member_o((Z),aa_Pro1465782317883297390_set_o(aa_a_f7646208301878272067_set_o(Ca,A),B))
     => member_o((Z),produc5350075035600711000_set_o(Ca,aa_Pro8078911840264235907it_nat(aa_a_f2234720956197727492it_nat(produc9178034014595674355it_nat,A),B))) ) ).

% mem_case_prodI
tff(fact_8634_mem__case__prodI,axiom,
    ! [Z: nat,Ca: fun_a_8173838432492502368et_nat,A: a,B: produc6653097349344004940it_nat] :
      ( aa_set_nat_o(member_nat(Z),aa_Pro6139939891222248720et_nat(aa_a_f7156848735415605777et_nat(Ca,A),B))
     => aa_set_nat_o(member_nat(Z),produc7315013382793309350et_nat(Ca,aa_Pro8078911840264235907it_nat(aa_a_f2234720956197727492it_nat(produc9178034014595674355it_nat,A),B))) ) ).

% mem_case_prodI
tff(fact_8635_mem__case__prodI,axiom,
    ! [Z: int,Ca: fun_a_3360290013704583612et_int,A: a,B: produc6653097349344004940it_nat] :
      ( aa_set_int_o(member_int(Z),aa_Pro1962088871713052012et_int(aa_a_f4666765590483879021et_int(Ca,A),B))
     => aa_set_int_o(member_int(Z),produc3137162363284112642et_int(Ca,aa_Pro8078911840264235907it_nat(aa_a_f2234720956197727492it_nat(produc9178034014595674355it_nat,A),B))) ) ).

% mem_case_prodI
tff(fact_8636_mem__case__prodI,axiom,
    ! [Z: set_nat,Ca: fun_a_5925824989894720534et_nat,A: a,B: produc6653097349344004940it_nat] :
      ( aa_set_set_nat_o(member_set_nat(Z),aa_Pro4298387719113726406et_nat(aa_a_f1497801039697847239et_nat(Ca,A),B))
     => aa_set_set_nat_o(member_set_nat(Z),produc8632796428668088668et_nat(Ca,aa_Pro8078911840264235907it_nat(aa_a_f2234720956197727492it_nat(produc9178034014595674355it_nat,A),B))) ) ).

% mem_case_prodI
tff(fact_8637_mem__case__prodI,axiom,
    ! [Z: product_prod_nat_nat,Ca: fun_a_8304141829328133171at_nat,A: a,B: produc6653097349344004940it_nat] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Z),aa_Pro5139103691568615511at_nat(aa_a_f1049478142650162092at_nat(Ca,A),B))
     => aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Z),produc8253449849199807809at_nat(Ca,aa_Pro8078911840264235907it_nat(aa_a_f2234720956197727492it_nat(produc9178034014595674355it_nat,A),B))) ) ).

% mem_case_prodI
tff(fact_8638_mem__case__prodI,axiom,
    ! [Z: $o,Ca: fun_fu4618385802829667198_set_o,A: fun_in5730379987311621695e_term,B: product_prod_int_int] :
      ( member_o((Z),aa_Pro2653658648327035067_set_o(aa_fun8934358842864497599_set_o(Ca,A),B))
     => member_o((Z),produc8509172437693241834_set_o(Ca,aa_Pro8800328608558162860nt_int(aa_fun260840925927807678nt_int(produc4729913460834754785nt_int,A),B))) ) ).

% mem_case_prodI
tff(fact_8639_mem__case__prodI,axiom,
    ! [Z: nat,Ca: fun_fu9079770784915425324et_nat,A: fun_in5730379987311621695e_term,B: product_prod_int_int] :
      ( aa_set_nat_o(member_nat(Z),aa_Pro3599192995643711491et_nat(aa_fun8409204002232819093et_nat(Ca,A),B))
     => aa_set_nat_o(member_nat(Z),produc5011043943518292116et_nat(Ca,aa_Pro8800328608558162860nt_int(aa_fun260840925927807678nt_int(produc4729913460834754785nt_int,A),B))) ) ).

% mem_case_prodI
tff(fact_8640_mem__case__prodI,axiom,
    ! [Z: int,Ca: 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(Ca,A),B))
     => aa_set_int_o(member_int(Z),produc833192924009095408et_int(Ca,aa_Pro8800328608558162860nt_int(aa_fun260840925927807678nt_int(produc4729913460834754785nt_int,A),B))) ) ).

% mem_case_prodI
tff(fact_8641_mem__case__prodI,axiom,
    ! [Z: set_nat,Ca: 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(Ca,A),B))
     => aa_set_set_nat_o(member_set_nat(Z),produc4945675248400274762et_nat(Ca,aa_Pro8800328608558162860nt_int(aa_fun260840925927807678nt_int(produc4729913460834754785nt_int,A),B))) ) ).

% mem_case_prodI
tff(fact_8642_mem__case__prodI,axiom,
    ! [Z: $o,Ca: fun_fu4934440599869564003_set_o,A: fun_Pr6460944699146269935_nat_o,B: produc3658429121746597890et_nat] :
      ( member_o((Z),aa_Pro3581020192872483896_set_o(aa_fun1634346356166085724_set_o(Ca,A),B))
     => member_o((Z),produc330442800291577585_set_o(Ca,aa_Pro434340453207536480et_nat(aa_fun3926903090414666066et_nat(produc6017135503889078682et_nat,A),B))) ) ).

% mem_case_prodI
tff(fact_8643_mem__case__prodI2,axiom,
    ! [P2: produc3260487557148687353it_nat,Z: $o,Ca: fun_a_6228866161775505738_set_o] :
      ( ! [A4: a,B4: produc6653097349344004940it_nat] :
          ( ( P2 = aa_Pro8078911840264235907it_nat(aa_a_f2234720956197727492it_nat(produc9178034014595674355it_nat,A4),B4) )
         => member_o((Z),aa_Pro1465782317883297390_set_o(aa_a_f7646208301878272067_set_o(Ca,A4),B4)) )
     => member_o((Z),produc5350075035600711000_set_o(Ca,P2)) ) ).

% mem_case_prodI2
tff(fact_8644_mem__case__prodI2,axiom,
    ! [P2: produc3260487557148687353it_nat,Z: nat,Ca: fun_a_8173838432492502368et_nat] :
      ( ! [A4: a,B4: produc6653097349344004940it_nat] :
          ( ( P2 = aa_Pro8078911840264235907it_nat(aa_a_f2234720956197727492it_nat(produc9178034014595674355it_nat,A4),B4) )
         => aa_set_nat_o(member_nat(Z),aa_Pro6139939891222248720et_nat(aa_a_f7156848735415605777et_nat(Ca,A4),B4)) )
     => aa_set_nat_o(member_nat(Z),produc7315013382793309350et_nat(Ca,P2)) ) ).

% mem_case_prodI2
tff(fact_8645_mem__case__prodI2,axiom,
    ! [P2: produc3260487557148687353it_nat,Z: int,Ca: fun_a_3360290013704583612et_int] :
      ( ! [A4: a,B4: produc6653097349344004940it_nat] :
          ( ( P2 = aa_Pro8078911840264235907it_nat(aa_a_f2234720956197727492it_nat(produc9178034014595674355it_nat,A4),B4) )
         => aa_set_int_o(member_int(Z),aa_Pro1962088871713052012et_int(aa_a_f4666765590483879021et_int(Ca,A4),B4)) )
     => aa_set_int_o(member_int(Z),produc3137162363284112642et_int(Ca,P2)) ) ).

% mem_case_prodI2
tff(fact_8646_mem__case__prodI2,axiom,
    ! [P2: produc3260487557148687353it_nat,Z: set_nat,Ca: fun_a_5925824989894720534et_nat] :
      ( ! [A4: a,B4: produc6653097349344004940it_nat] :
          ( ( P2 = aa_Pro8078911840264235907it_nat(aa_a_f2234720956197727492it_nat(produc9178034014595674355it_nat,A4),B4) )
         => aa_set_set_nat_o(member_set_nat(Z),aa_Pro4298387719113726406et_nat(aa_a_f1497801039697847239et_nat(Ca,A4),B4)) )
     => aa_set_set_nat_o(member_set_nat(Z),produc8632796428668088668et_nat(Ca,P2)) ) ).

% mem_case_prodI2
tff(fact_8647_mem__case__prodI2,axiom,
    ! [P2: produc3260487557148687353it_nat,Z: product_prod_nat_nat,Ca: fun_a_8304141829328133171at_nat] :
      ( ! [A4: a,B4: produc6653097349344004940it_nat] :
          ( ( P2 = aa_Pro8078911840264235907it_nat(aa_a_f2234720956197727492it_nat(produc9178034014595674355it_nat,A4),B4) )
         => aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Z),aa_Pro5139103691568615511at_nat(aa_a_f1049478142650162092at_nat(Ca,A4),B4)) )
     => aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Z),produc8253449849199807809at_nat(Ca,P2)) ) ).

% mem_case_prodI2
tff(fact_8648_mem__case__prodI2,axiom,
    ! [P2: produc4788131091073775919nt_int,Z: $o,Ca: fun_fu4618385802829667198_set_o] :
      ( ! [A4: fun_in5730379987311621695e_term,B4: product_prod_int_int] :
          ( ( P2 = aa_Pro8800328608558162860nt_int(aa_fun260840925927807678nt_int(produc4729913460834754785nt_int,A4),B4) )
         => member_o((Z),aa_Pro2653658648327035067_set_o(aa_fun8934358842864497599_set_o(Ca,A4),B4)) )
     => member_o((Z),produc8509172437693241834_set_o(Ca,P2)) ) ).

% mem_case_prodI2
tff(fact_8649_mem__case__prodI2,axiom,
    ! [P2: produc4788131091073775919nt_int,Z: nat,Ca: fun_fu9079770784915425324et_nat] :
      ( ! [A4: fun_in5730379987311621695e_term,B4: product_prod_int_int] :
          ( ( P2 = aa_Pro8800328608558162860nt_int(aa_fun260840925927807678nt_int(produc4729913460834754785nt_int,A4),B4) )
         => aa_set_nat_o(member_nat(Z),aa_Pro3599192995643711491et_nat(aa_fun8409204002232819093et_nat(Ca,A4),B4)) )
     => aa_set_nat_o(member_nat(Z),produc5011043943518292116et_nat(Ca,P2)) ) ).

% mem_case_prodI2
tff(fact_8650_mem__case__prodI2,axiom,
    ! [P2: produc4788131091073775919nt_int,Z: int,Ca: fun_fu4266222366127506568et_int] :
      ( ! [A4: fun_in5730379987311621695e_term,B4: product_prod_int_int] :
          ( ( P2 = aa_Pro8800328608558162860nt_int(aa_fun260840925927807678nt_int(produc4729913460834754785nt_int,A4),B4) )
         => aa_set_int_o(member_int(Z),aa_Pro8644714012989290591et_int(aa_fun5919120857301092337et_int(Ca,A4),B4)) )
     => aa_set_int_o(member_int(Z),produc833192924009095408et_int(Ca,P2)) ) ).

% mem_case_prodI2
tff(fact_8651_mem__case__prodI2,axiom,
    ! [P2: produc4788131091073775919nt_int,Z: set_nat,Ca: fun_fu5458086908000294626et_nat] :
      ( ! [A4: fun_in5730379987311621695e_term,B4: product_prod_int_int] :
          ( ( P2 = 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(Ca,A4),B4)) )
     => aa_set_set_nat_o(member_set_nat(Z),produc4945675248400274762et_nat(Ca,P2)) ) ).

% mem_case_prodI2
tff(fact_8652_mem__case__prodI2,axiom,
    ! [P2: produc8799533983703829792et_nat,Z: $o,Ca: fun_fu4934440599869564003_set_o] :
      ( ! [A4: fun_Pr6460944699146269935_nat_o,B4: produc3658429121746597890et_nat] :
          ( ( P2 = aa_Pro434340453207536480et_nat(aa_fun3926903090414666066et_nat(produc6017135503889078682et_nat,A4),B4) )
         => member_o((Z),aa_Pro3581020192872483896_set_o(aa_fun1634346356166085724_set_o(Ca,A4),B4)) )
     => member_o((Z),produc330442800291577585_set_o(Ca,P2)) ) ).

% mem_case_prodI2
tff(fact_8653_case__prodI2_H,axiom,
    ! [P2: product_prod_nat_nat,Ca: fun_na5914473681949881652_nat_o,X: product_prod_nat_nat] :
      ( ! [A4: nat,B4: nat] :
          ( ( aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,A4),B4) = P2 )
         => aa_Pro6150637604445964947_nat_o(aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(Ca,A4),B4),X) )
     => aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(aa_fun5912894404062470332_nat_o(produc6395135316500328296_nat_o,Ca),P2),X) ) ).

% case_prodI2'
tff(fact_8654_take__bit__num__simps_I2_J,axiom,
    ! [N2: nat] : aa_num_option_num(bit_take_bit_num(aa_nat_nat(suc,N2)),one) = aa_num_option_num(some_num,one) ).

% take_bit_num_simps(2)
tff(fact_8655_take__bit__num__simps_I5_J,axiom,
    ! [Ra2: num] : aa_num_option_num(bit_take_bit_num(aa_num_nat(numeral_numeral_nat,Ra2)),one) = aa_num_option_num(some_num,one) ).

% take_bit_num_simps(5)
tff(fact_8656_take__bit__num__simps_I3_J,axiom,
    ! [N2: nat,M: num] : aa_num_option_num(bit_take_bit_num(aa_nat_nat(suc,N2)),aa_num_num(bit0,M)) = aa_opt5369159329878751634on_num(case_o6005452278849405969um_num(none_num,aTP_Lamm_agv),aa_num_option_num(bit_take_bit_num(N2),M)) ).

% take_bit_num_simps(3)
tff(fact_8657_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_8658_divmod__algorithm__code_I2_J,axiom,
    ! [M: num] : unique5055182867167087721od_nat(M,one) = aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,aa_num_nat(numeral_numeral_nat,M)),zero_zero_nat) ).

% divmod_algorithm_code(2)
tff(fact_8659_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_8660_take__bit__numeral__numeral,axiom,
    ! [M: num,N2: num] : aa_Cod1783974652533396728nteger(bit_se1745604003318907178nteger(aa_num_nat(numeral_numeral_nat,M)),aa_num_Code_integer(numera6620942414471956472nteger,N2)) = case_o356765784539232260er_num(zero_z3403309356797280102nteger,numera6620942414471956472nteger,aa_num_option_num(bit_take_bit_num(aa_num_nat(numeral_numeral_nat,M)),N2)) ).

% take_bit_numeral_numeral
tff(fact_8661_take__bit__numeral__numeral,axiom,
    ! [M: num,N2: num] : bit_se569199155075624693atural(aa_num_nat(numeral_numeral_nat,M),aa_num_Code_natural(numera5444537566228673987atural,N2)) = case_o5621594795226839503al_num(zero_z2226904508553997617atural,numera5444537566228673987atural,aa_num_option_num(bit_take_bit_num(aa_num_nat(numeral_numeral_nat,M)),N2)) ).

% take_bit_numeral_numeral
tff(fact_8662_take__bit__numeral__numeral,axiom,
    ! [M: num,N2: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,N2)) = aa_option_num_nat(case_option_nat_num(zero_zero_nat,numeral_numeral_nat),aa_num_option_num(bit_take_bit_num(aa_num_nat(numeral_numeral_nat,M)),N2)) ).

% take_bit_numeral_numeral
tff(fact_8663_take__bit__numeral__numeral,axiom,
    ! [M: num,N2: num] : aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,aa_num_nat(numeral_numeral_nat,M)),aa_num_int(numeral_numeral_int,N2)) = aa_option_num_int(case_option_int_num(zero_zero_int,numeral_numeral_int),aa_num_option_num(bit_take_bit_num(aa_num_nat(numeral_numeral_nat,M)),N2)) ).

% take_bit_numeral_numeral
tff(fact_8664_mem__case__prodE,axiom,
    ! [Z: $o,Ca: fun_a_6228866161775505738_set_o,P2: produc3260487557148687353it_nat] :
      ( member_o((Z),produc5350075035600711000_set_o(Ca,P2))
     => ~ ! [X3: a,Y3: produc6653097349344004940it_nat] :
            ( ( P2 = aa_Pro8078911840264235907it_nat(aa_a_f2234720956197727492it_nat(produc9178034014595674355it_nat,X3),Y3) )
           => ~ member_o((Z),aa_Pro1465782317883297390_set_o(aa_a_f7646208301878272067_set_o(Ca,X3),Y3)) ) ) ).

% mem_case_prodE
tff(fact_8665_mem__case__prodE,axiom,
    ! [Z: nat,Ca: fun_a_8173838432492502368et_nat,P2: produc3260487557148687353it_nat] :
      ( aa_set_nat_o(member_nat(Z),produc7315013382793309350et_nat(Ca,P2))
     => ~ ! [X3: a,Y3: produc6653097349344004940it_nat] :
            ( ( P2 = aa_Pro8078911840264235907it_nat(aa_a_f2234720956197727492it_nat(produc9178034014595674355it_nat,X3),Y3) )
           => ~ aa_set_nat_o(member_nat(Z),aa_Pro6139939891222248720et_nat(aa_a_f7156848735415605777et_nat(Ca,X3),Y3)) ) ) ).

% mem_case_prodE
tff(fact_8666_mem__case__prodE,axiom,
    ! [Z: int,Ca: fun_a_3360290013704583612et_int,P2: produc3260487557148687353it_nat] :
      ( aa_set_int_o(member_int(Z),produc3137162363284112642et_int(Ca,P2))
     => ~ ! [X3: a,Y3: produc6653097349344004940it_nat] :
            ( ( P2 = aa_Pro8078911840264235907it_nat(aa_a_f2234720956197727492it_nat(produc9178034014595674355it_nat,X3),Y3) )
           => ~ aa_set_int_o(member_int(Z),aa_Pro1962088871713052012et_int(aa_a_f4666765590483879021et_int(Ca,X3),Y3)) ) ) ).

% mem_case_prodE
tff(fact_8667_mem__case__prodE,axiom,
    ! [Z: set_nat,Ca: fun_a_5925824989894720534et_nat,P2: produc3260487557148687353it_nat] :
      ( aa_set_set_nat_o(member_set_nat(Z),produc8632796428668088668et_nat(Ca,P2))
     => ~ ! [X3: a,Y3: produc6653097349344004940it_nat] :
            ( ( P2 = aa_Pro8078911840264235907it_nat(aa_a_f2234720956197727492it_nat(produc9178034014595674355it_nat,X3),Y3) )
           => ~ aa_set_set_nat_o(member_set_nat(Z),aa_Pro4298387719113726406et_nat(aa_a_f1497801039697847239et_nat(Ca,X3),Y3)) ) ) ).

% mem_case_prodE
tff(fact_8668_mem__case__prodE,axiom,
    ! [Z: product_prod_nat_nat,Ca: fun_a_8304141829328133171at_nat,P2: produc3260487557148687353it_nat] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Z),produc8253449849199807809at_nat(Ca,P2))
     => ~ ! [X3: a,Y3: produc6653097349344004940it_nat] :
            ( ( P2 = aa_Pro8078911840264235907it_nat(aa_a_f2234720956197727492it_nat(produc9178034014595674355it_nat,X3),Y3) )
           => ~ aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Z),aa_Pro5139103691568615511at_nat(aa_a_f1049478142650162092at_nat(Ca,X3),Y3)) ) ) ).

% mem_case_prodE
tff(fact_8669_mem__case__prodE,axiom,
    ! [Z: $o,Ca: fun_fu4618385802829667198_set_o,P2: produc4788131091073775919nt_int] :
      ( member_o((Z),produc8509172437693241834_set_o(Ca,P2))
     => ~ ! [X3: fun_in5730379987311621695e_term,Y3: product_prod_int_int] :
            ( ( P2 = aa_Pro8800328608558162860nt_int(aa_fun260840925927807678nt_int(produc4729913460834754785nt_int,X3),Y3) )
           => ~ member_o((Z),aa_Pro2653658648327035067_set_o(aa_fun8934358842864497599_set_o(Ca,X3),Y3)) ) ) ).

% mem_case_prodE
tff(fact_8670_mem__case__prodE,axiom,
    ! [Z: nat,Ca: fun_fu9079770784915425324et_nat,P2: produc4788131091073775919nt_int] :
      ( aa_set_nat_o(member_nat(Z),produc5011043943518292116et_nat(Ca,P2))
     => ~ ! [X3: fun_in5730379987311621695e_term,Y3: product_prod_int_int] :
            ( ( P2 = aa_Pro8800328608558162860nt_int(aa_fun260840925927807678nt_int(produc4729913460834754785nt_int,X3),Y3) )
           => ~ aa_set_nat_o(member_nat(Z),aa_Pro3599192995643711491et_nat(aa_fun8409204002232819093et_nat(Ca,X3),Y3)) ) ) ).

% mem_case_prodE
tff(fact_8671_mem__case__prodE,axiom,
    ! [Z: int,Ca: fun_fu4266222366127506568et_int,P2: produc4788131091073775919nt_int] :
      ( aa_set_int_o(member_int(Z),produc833192924009095408et_int(Ca,P2))
     => ~ ! [X3: fun_in5730379987311621695e_term,Y3: product_prod_int_int] :
            ( ( P2 = aa_Pro8800328608558162860nt_int(aa_fun260840925927807678nt_int(produc4729913460834754785nt_int,X3),Y3) )
           => ~ aa_set_int_o(member_int(Z),aa_Pro8644714012989290591et_int(aa_fun5919120857301092337et_int(Ca,X3),Y3)) ) ) ).

% mem_case_prodE
tff(fact_8672_mem__case__prodE,axiom,
    ! [Z: set_nat,Ca: fun_fu5458086908000294626et_nat,P2: produc4788131091073775919nt_int] :
      ( aa_set_set_nat_o(member_set_nat(Z),produc4945675248400274762et_nat(Ca,P2))
     => ~ ! [X3: fun_in5730379987311621695e_term,Y3: product_prod_int_int] :
            ( ( P2 = aa_Pro8800328608558162860nt_int(aa_fun260840925927807678nt_int(produc4729913460834754785nt_int,X3),Y3) )
           => ~ aa_set_set_nat_o(member_set_nat(Z),aa_Pro1963545105414287545et_nat(aa_fun8341152516354955083et_nat(Ca,X3),Y3)) ) ) ).

% mem_case_prodE
tff(fact_8673_mem__case__prodE,axiom,
    ! [Z: $o,Ca: fun_fu4934440599869564003_set_o,P2: produc8799533983703829792et_nat] :
      ( member_o((Z),produc330442800291577585_set_o(Ca,P2))
     => ~ ! [X3: fun_Pr6460944699146269935_nat_o,Y3: produc3658429121746597890et_nat] :
            ( ( P2 = aa_Pro434340453207536480et_nat(aa_fun3926903090414666066et_nat(produc6017135503889078682et_nat,X3),Y3) )
           => ~ member_o((Z),aa_Pro3581020192872483896_set_o(aa_fun1634346356166085724_set_o(Ca,X3),Y3)) ) ) ).

% mem_case_prodE
tff(fact_8674_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_8675_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_8676_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_8677_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_8678_case__prodD,axiom,
    ! [F: fun_a_2467521563176467050_nat_o,A: a,B: produc6653097349344004940it_nat] :
      ( aa_Pro6426884464393220769_nat_o(produc8860486935419594360_nat_o(F),aa_Pro8078911840264235907it_nat(aa_a_f2234720956197727492it_nat(produc9178034014595674355it_nat,A),B))
     => aa_Pro1202363709425262478_nat_o(aa_a_f337466037417904483_nat_o(F,A),B) ) ).

% case_prodD
tff(fact_8679_case__prodD,axiom,
    ! [F: fun_int_fun_int_o,A: int,B: int] :
      ( aa_Pro4592068669816069595_int_o(aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,F),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),B))
     => aa_int_o(aa_int_fun_int_o(F,A),B) ) ).

% case_prodD
tff(fact_8680_case__prodE,axiom,
    ! [Ca: fun_fu8419763957888848259_nat_o,P2: produc8799533983703829792et_nat] :
      ( aa_Pro5643009357358181178_nat_o(produc2915205735004773393_nat_o(Ca),P2)
     => ~ ! [X3: fun_Pr6460944699146269935_nat_o,Y3: produc3658429121746597890et_nat] :
            ( ( P2 = aa_Pro434340453207536480et_nat(aa_fun3926903090414666066et_nat(produc6017135503889078682et_nat,X3),Y3) )
           => ~ aa_Pro653082453696054616_nat_o(aa_fun2638059832795814268_nat_o(Ca,X3),Y3) ) ) ).

% case_prodE
tff(fact_8681_case__prodE,axiom,
    ! [Ca: fun_fu2611741329380808165_nat_o,P2: produc3112576180687729086et_nat] :
      ( aa_Pro2671238023309943196_nat_o(produc4465912260561112435_nat_o(Ca),P2)
     => ~ ! [X3: fun_Pr6460944699146269935_nat_o,Y3: produc8799533983703829792et_nat] :
            ( ( P2 = aa_Pro1979504083478021276et_nat(aa_fun615918787490101902et_nat(produc7291855603632349880et_nat,X3),Y3) )
           => ~ aa_Pro5643009357358181178_nat_o(aa_fun912680268133416158_nat_o(Ca,X3),Y3) ) ) ).

% case_prodE
tff(fact_8682_case__prodE,axiom,
    ! [Ca: fun_fu6865642474282033866_int_o,P2: produc3043004577601930755nt_int] :
      ( aa_Pro2601676399929874711_int_o(produc2207054017359051062_int_o(Ca),P2)
     => ~ ! [X3: fun_Pr7763680713840462443e_term,Y3: product_prod_int_int] :
            ( ( P2 = aa_Pro5748361610964203648nt_int(aa_fun3265472288063244902nt_int(produc7320969046608631221nt_int,X3),Y3) )
           => ~ aa_Pro4592068669816069595_int_o(aa_fun3538162476766722827_int_o(Ca,X3),Y3) ) ) ).

% case_prodE
tff(fact_8683_case__prodE,axiom,
    ! [Ca: fun_fu2978539155105838238_int_o,P2: produc4788131091073775919nt_int] :
      ( aa_Pro5451197003417746155_int_o(produc7271678899836001034_int_o(Ca),P2)
     => ~ ! [X3: fun_in5730379987311621695e_term,Y3: product_prod_int_int] :
            ( ( P2 = aa_Pro8800328608558162860nt_int(aa_fun260840925927807678nt_int(produc4729913460834754785nt_int,X3),Y3) )
           => ~ aa_Pro4592068669816069595_int_o(aa_fun586359091965693151_int_o(Ca,X3),Y3) ) ) ).

% case_prodE
tff(fact_8684_case__prodE,axiom,
    ! [Ca: fun_a_2467521563176467050_nat_o,P2: produc3260487557148687353it_nat] :
      ( aa_Pro6426884464393220769_nat_o(produc8860486935419594360_nat_o(Ca),P2)
     => ~ ! [X3: a,Y3: produc6653097349344004940it_nat] :
            ( ( P2 = aa_Pro8078911840264235907it_nat(aa_a_f2234720956197727492it_nat(produc9178034014595674355it_nat,X3),Y3) )
           => ~ aa_Pro1202363709425262478_nat_o(aa_a_f337466037417904483_nat_o(Ca,X3),Y3) ) ) ).

% case_prodE
tff(fact_8685_case__prodE,axiom,
    ! [Ca: fun_int_fun_int_o,P2: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,Ca),P2)
     => ~ ! [X3: int,Y3: int] :
            ( ( P2 = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X3),Y3) )
           => ~ aa_int_o(aa_int_fun_int_o(Ca,X3),Y3) ) ) ).

% case_prodE
tff(fact_8686_case__prodD_H,axiom,
    ! [Ra: fun_na5914473681949881652_nat_o,A: nat,B: nat,Ca: product_prod_nat_nat] :
      ( aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(aa_fun5912894404062470332_nat_o(produc6395135316500328296_nat_o,Ra),aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,A),B)),Ca)
     => aa_Pro6150637604445964947_nat_o(aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(Ra,A),B),Ca) ) ).

% case_prodD'
tff(fact_8687_case__prodE_H,axiom,
    ! [Ca: fun_na5914473681949881652_nat_o,P2: product_prod_nat_nat,Z: product_prod_nat_nat] :
      ( aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(aa_fun5912894404062470332_nat_o(produc6395135316500328296_nat_o,Ca),P2),Z)
     => ~ ! [X3: nat,Y3: nat] :
            ( ( P2 = aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,X3),Y3) )
           => ~ aa_Pro6150637604445964947_nat_o(aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(Ca,X3),Y3),Z) ) ) ).

% case_prodE'
tff(fact_8688_Divides_Oadjust__div__def,axiom,
    ! [Qr: product_prod_int_int] : aa_Pro6799258434465214121nt_int(adjust_div,Qr) = aa_Pro6799258434465214121nt_int(aa_fun914008879417524284nt_int(produc8211389475949308722nt_int,aTP_Lamm_agw),Qr) ).

% Divides.adjust_div_def
tff(fact_8689_Collect__case__prod__mono,axiom,
    ! [A3: fun_int_fun_int_o,B3: fun_int_fun_int_o] :
      ( ord_le77980682512674322_int_o(A3,B3)
     => ord_le2843351958646193337nt_int(collec213857154873943460nt_int(aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,A3)),collec213857154873943460nt_int(aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,B3))) ) ).

% Collect_case_prod_mono
tff(fact_8690_sum_Otriangle__reindex__eq,axiom,
    ! [G: fun_nat_fun_nat_nat,N2: nat] : groups977919841031483927at_nat(produc6842872674320459806at_nat(G),collec3392354462482085612at_nat(produc6081775807080527818_nat_o(aTP_Lamm_agx(N2)))) = aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_agz(G)),aa_nat_set_nat(set_ord_atMost_nat,N2)) ).

% sum.triangle_reindex_eq
tff(fact_8691_prod_Otriangle__reindex__eq,axiom,
    ! [G: fun_nat_fun_nat_nat,N2: nat] : groups4077766827762148844at_nat(produc6842872674320459806at_nat(G),collec3392354462482085612at_nat(produc6081775807080527818_nat_o(aTP_Lamm_agx(N2)))) = aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_aha(G)),aa_nat_set_nat(set_ord_atMost_nat,N2)) ).

% prod.triangle_reindex_eq
tff(fact_8692_prod_Otriangle__reindex__eq,axiom,
    ! [G: fun_nat_fun_nat_int,N2: nat] : groups4075276357253098568at_int(produc6840382203811409530at_int(G),collec3392354462482085612at_nat(produc6081775807080527818_nat_o(aTP_Lamm_agx(N2)))) = aa_set_nat_int(groups705719431365010083at_int(aTP_Lamm_ahc(G)),aa_nat_set_nat(set_ord_atMost_nat,N2)) ).

% prod.triangle_reindex_eq
tff(fact_8693_execute__bind__case,axiom,
    ! [F: heap_Time_Heap_a,G: fun_a_6093291656547882009Heap_a,Ha: heap_e7401611519738050253t_unit] : aa_hea4724376117875161946it_nat(heap_Time_execute_a(heap_Time_bind_a_a(F,G)),Ha) = case_o6061097939050036047it_nat(none_P2651198173097904984it_nat,produc7850007024774191849it_nat(aTP_Lamm_ahe(G)),aa_hea4724376117875161946it_nat(heap_Time_execute_a(F),Ha)) ).

% execute_bind_case
tff(fact_8694_map__to__set__def,axiom,
    ! [M: fun_int_option_int] : map_to_set_int_int(M) = collec213857154873943460nt_int(aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,aTP_Lamm_ahf(M))) ).

% map_to_set_def
tff(fact_8695_take__bit__num__eq__Some__imp,axiom,
    ! [M: nat,N2: num,Q4: num] :
      ( ( aa_num_option_num(bit_take_bit_num(M),N2) = aa_num_option_num(some_num,Q4) )
     => ( aa_Cod1783974652533396728nteger(bit_se1745604003318907178nteger(M),aa_num_Code_integer(numera6620942414471956472nteger,N2)) = aa_num_Code_integer(numera6620942414471956472nteger,Q4) ) ) ).

% take_bit_num_eq_Some_imp
tff(fact_8696_take__bit__num__eq__Some__imp,axiom,
    ! [M: nat,N2: num,Q4: num] :
      ( ( aa_num_option_num(bit_take_bit_num(M),N2) = aa_num_option_num(some_num,Q4) )
     => ( bit_se569199155075624693atural(M,aa_num_Code_natural(numera5444537566228673987atural,N2)) = aa_num_Code_natural(numera5444537566228673987atural,Q4) ) ) ).

% take_bit_num_eq_Some_imp
tff(fact_8697_take__bit__num__eq__Some__imp,axiom,
    ! [M: nat,N2: num,Q4: num] :
      ( ( aa_num_option_num(bit_take_bit_num(M),N2) = aa_num_option_num(some_num,Q4) )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,M),aa_num_nat(numeral_numeral_nat,N2)) = aa_num_nat(numeral_numeral_nat,Q4) ) ) ).

% take_bit_num_eq_Some_imp
tff(fact_8698_take__bit__num__eq__Some__imp,axiom,
    ! [M: nat,N2: num,Q4: num] :
      ( ( aa_num_option_num(bit_take_bit_num(M),N2) = aa_num_option_num(some_num,Q4) )
     => ( aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,M),aa_num_int(numeral_numeral_int,N2)) = aa_num_int(numeral_numeral_int,Q4) ) ) ).

% take_bit_num_eq_Some_imp
tff(fact_8699_Heap__Time__Monad_Obind__def,axiom,
    ! [F: heap_Time_Heap_a,G: fun_a_6093291656547882009Heap_a] : heap_Time_bind_a_a(F,G) = heap_Time_Heap_a2(aa_fun6119220767042692876it_nat(aTP_Lamm_ahg(F),G)) ).

% Heap_Time_Monad.bind_def
tff(fact_8700_Heap__Time__Monad_Obind__def,axiom,
    ! [F: heap_Time_Heap_a,G: fun_a_2990225006182978344t_unit] : heap_T8685611227969916822t_unit(F,G) = heap_T6183433275982383450t_unit(aa_fun372765568861188684it_nat(aTP_Lamm_ahj(F),G)) ).

% Heap_Time_Monad.bind_def
tff(fact_8701_finite__psubset__def,axiom,
    finite_psubset_nat = collec6662362479098859352et_nat(produc6247414631856714078_nat_o(aTP_Lamm_ahk)) ).

% finite_psubset_def
tff(fact_8702_finite__psubset__def,axiom,
    finite_psubset_int = collec957716948307931664et_int(produc4109468873575309990_int_o(aTP_Lamm_ahl)) ).

% finite_psubset_def
tff(fact_8703_finite__psubset__def,axiom,
    finite2416775604798480986nteger = collec2770208431294612722nteger(produc1658495936642081476eger_o(aTP_Lamm_ahm)) ).

% finite_psubset_def
tff(fact_8704_finite__psubset__def,axiom,
    finite469560695537375940at_nat = collec6321179662152712658at_nat(produc410239310623530412_nat_o(aTP_Lamm_ahn)) ).

% finite_psubset_def
tff(fact_8705_rel__of__def,axiom,
    ! [M: fun_fu2352890940381120596et_nat,Pa: fun_Pr2620521676128474833_nat_o] : rel_of7233497478102412281et_nat(M,Pa) = collec6193113176214168693et_nat(produc2915205735004773393_nat_o(aa_fun1045796833200626456_nat_o(aTP_Lamm_aho(M),Pa))) ).

% rel_of_def
tff(fact_8706_rel__of__def,axiom,
    ! [M: fun_fu1905361174868762994et_nat,Pa: fun_Pr1227177680593602867_nat_o] : rel_of4573248530116435351et_nat(M,Pa) = collec5916191906451531411et_nat(produc4465912260561112435_nat_o(aa_fun4422562838209026524_nat_o(aTP_Lamm_ahp(M),Pa))) ).

% rel_of_def
tff(fact_8707_rel__of__def,axiom,
    ! [M: fun_fu4253467608928286509nt_int,Pa: fun_Pr3540825087398332590_int_o] : rel_of9206080981073604500nt_int(M,Pa) = collec2586211367037964248nt_int(produc2207054017359051062_int_o(aa_fun1054583131507368252_int_o(aTP_Lamm_ahq(M),Pa))) ).

% rel_of_def
tff(fact_8708_rel__of__def,axiom,
    ! [M: fun_fu5631754973857388633nt_int,Pa: fun_Pr1196100989263358594_int_o] : rel_of6825086619541906112nt_int(M,Pa) = collec8113046454796679940nt_int(produc7271678899836001034_int_o(aa_fun8601777901039773924_int_o(aTP_Lamm_ahr(M),Pa))) ).

% rel_of_def
tff(fact_8709_rel__of__def,axiom,
    ! [M: fun_a_1313742400928293293it_nat,Pa: fun_Pr5746006151868627512_nat_o] : rel_of2761222537318653906it_nat(M,Pa) = collec4148678073425210574it_nat(produc8860486935419594360_nat_o(aa_fun7773039478287727974_nat_o(aTP_Lamm_ahs(M),Pa))) ).

% rel_of_def
tff(fact_8710_rel__of__def,axiom,
    ! [M: fun_int_option_int,Pa: fun_Pr3238374608701176162_int_o] : rel_of_int_int(M,Pa) = collec213857154873943460nt_int(aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,aa_fun5140975492127840594_int_o(aTP_Lamm_aht(M),Pa))) ).

% rel_of_def
tff(fact_8711_divmod__int__def,axiom,
    ! [M: num,N2: num] : unique5052692396658037445od_int(M,N2) = 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,N2))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,N2))) ).

% divmod_int_def
tff(fact_8712_divmod__def,axiom,
    ! [M: num,N2: num] : unique5052692396658037445od_int(M,N2) = 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,N2))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,N2))) ).

% divmod_def
tff(fact_8713_divmod__def,axiom,
    ! [M: num,N2: num] : unique5055182867167087721od_nat(M,N2) = aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,N2))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,N2))) ).

% divmod_def
tff(fact_8714_divmod__def,axiom,
    ! [M: num,N2: num] : unique3479559517661332726nteger(M,N2) = produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_num_Code_integer(numera6620942414471956472nteger,M)),aa_num_Code_integer(numera6620942414471956472nteger,N2)),aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_num_Code_integer(numera6620942414471956472nteger,M)),aa_num_Code_integer(numera6620942414471956472nteger,N2))) ).

% divmod_def
tff(fact_8715_divmod_H__nat__def,axiom,
    ! [M: num,N2: num] : unique5055182867167087721od_nat(M,N2) = aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,N2))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,N2))) ).

% divmod'_nat_def
tff(fact_8716_mlex__eq,axiom,
    ! [F: fun_nat_nat,Ra: set_Pr1261947904930325089at_nat] : mlex_prod_nat(F,Ra) = collec3392354462482085612at_nat(produc6081775807080527818_nat_o(aa_set2634951030311766213_nat_o(aTP_Lamm_ahu(F),Ra))) ).

% mlex_eq
tff(fact_8717_mlex__eq,axiom,
    ! [F: fun_int_nat,Ra: set_Pr958786334691620121nt_int] : mlex_prod_int(F,Ra) = collec213857154873943460nt_int(aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,aa_set742265667465749333_int_o(aTP_Lamm_ahv(F),Ra))) ).

% mlex_eq
tff(fact_8718_divmod__algorithm__code_I6_J,axiom,
    ! [M: num,N2: num] : unique5052692396658037445od_int(aa_num_num(bit1,M),aa_num_num(bit0,N2)) = aa_Pro6970504127699319548nt_int(aa_fun8026974734089330650nt_int(produc4245557441103728435nt_int,aTP_Lamm_ahw),unique5052692396658037445od_int(M,N2)) ).

% divmod_algorithm_code(6)
tff(fact_8719_divmod__algorithm__code_I6_J,axiom,
    ! [M: num,N2: num] : unique5055182867167087721od_nat(aa_num_num(bit1,M),aa_num_num(bit0,N2)) = aa_Pro262675715343573244at_nat(produc2626176000494625587at_nat(aTP_Lamm_ahx),unique5055182867167087721od_nat(M,N2)) ).

% divmod_algorithm_code(6)
tff(fact_8720_divmod__algorithm__code_I6_J,axiom,
    ! [M: num,N2: num] : unique3479559517661332726nteger(aa_num_num(bit1,M),aa_num_num(bit0,N2)) = produc6916734918728496179nteger(aTP_Lamm_ahy,unique3479559517661332726nteger(M,N2)) ).

% divmod_algorithm_code(6)
tff(fact_8721_int__ge__less__than2__def,axiom,
    ! [D2: int] : int_ge_less_than2(D2) = collec213857154873943460nt_int(aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,aTP_Lamm_ahz(D2))) ).

% int_ge_less_than2_def
tff(fact_8722_int__ge__less__than__def,axiom,
    ! [D2: int] : int_ge_less_than(D2) = collec213857154873943460nt_int(aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,aTP_Lamm_aia(D2))) ).

% int_ge_less_than_def
tff(fact_8723_divmod__algorithm__code_I8_J,axiom,
    ! [M: num,N2: num] :
      unique5055182867167087721od_nat(aa_num_num(bit1,M),aa_num_num(bit1,N2)) = $ite(aa_num_o(aa_num_fun_num_o(ord_less_num,M),N2),aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,M))),aa_Pro262675715343573244at_nat(unique5026877609467782581ep_nat(aa_num_num(bit1,N2)),unique5055182867167087721od_nat(aa_num_num(bit1,M),aa_num_num(bit0,aa_num_num(bit1,N2))))) ).

% divmod_algorithm_code(8)
tff(fact_8724_divmod__algorithm__code_I8_J,axiom,
    ! [M: num,N2: num] :
      unique5052692396658037445od_int(aa_num_num(bit1,M),aa_num_num(bit1,N2)) = $ite(aa_num_o(aa_num_fun_num_o(ord_less_num,M),N2),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,N2)),unique5052692396658037445od_int(aa_num_num(bit1,M),aa_num_num(bit0,aa_num_num(bit1,N2))))) ).

% divmod_algorithm_code(8)
tff(fact_8725_divmod__algorithm__code_I8_J,axiom,
    ! [M: num,N2: num] :
      unique3479559517661332726nteger(aa_num_num(bit1,M),aa_num_num(bit1,N2)) = $ite(aa_num_o(aa_num_fun_num_o(ord_less_num,M),N2),produc1086072967326762835nteger(zero_z3403309356797280102nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit1,M))),unique4921790084139445826nteger(aa_num_num(bit1,N2),unique3479559517661332726nteger(aa_num_num(bit1,M),aa_num_num(bit0,aa_num_num(bit1,N2))))) ).

% divmod_algorithm_code(8)
tff(fact_8726_split__part,axiom,
    ! [Pa: $o,Qa: fun_int_fun_int_o,X5: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,aa_fun6133342665515662012_int_o(aTP_Lamm_aib((Pa)),Qa)),X5)
    <=> ( (Pa)
        & aa_Pro4592068669816069595_int_o(aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,Qa),X5) ) ) ).

% split_part
tff(fact_8727_semiring__norm_I80_J,axiom,
    ! [M: num,N2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,aa_num_num(bit1,M)),aa_num_num(bit1,N2))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,M),N2) ) ).

% semiring_norm(80)
tff(fact_8728_semiring__norm_I73_J,axiom,
    ! [M: num,N2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(bit1,M)),aa_num_num(bit1,N2))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,M),N2) ) ).

% semiring_norm(73)
tff(fact_8729_semiring__norm_I72_J,axiom,
    ! [M: num,N2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(bit0,M)),aa_num_num(bit1,N2))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,M),N2) ) ).

% semiring_norm(72)
tff(fact_8730_semiring__norm_I81_J,axiom,
    ! [M: num,N2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,aa_num_num(bit1,M)),aa_num_num(bit0,N2))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,M),N2) ) ).

% semiring_norm(81)
tff(fact_8731_semiring__norm_I70_J,axiom,
    ! [M: num] : ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(bit1,M)),one) ).

% semiring_norm(70)
tff(fact_8732_semiring__norm_I77_J,axiom,
    ! [N2: num] : aa_num_o(aa_num_fun_num_o(ord_less_num,one),aa_num_num(bit1,N2)) ).

% semiring_norm(77)
tff(fact_8733_semiring__norm_I74_J,axiom,
    ! [M: num,N2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(bit1,M)),aa_num_num(bit0,N2))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,M),N2) ) ).

% semiring_norm(74)
tff(fact_8734_semiring__norm_I79_J,axiom,
    ! [M: num,N2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,aa_num_num(bit0,M)),aa_num_num(bit1,N2))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,M),N2) ) ).

% semiring_norm(79)
tff(fact_8735_take__bit__num__simps_I4_J,axiom,
    ! [N2: nat,M: num] : aa_num_option_num(bit_take_bit_num(aa_nat_nat(suc,N2)),aa_num_num(bit1,M)) = aa_num_option_num(some_num,aa_option_num_num(case_option_num_num(one,bit1),aa_num_option_num(bit_take_bit_num(N2),M))) ).

% take_bit_num_simps(4)
tff(fact_8736_divmod__algorithm__code_I4_J,axiom,
    ! [N2: num] : unique5052692396658037445od_int(one,aa_num_num(bit1,N2)) = 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_8737_divmod__algorithm__code_I4_J,axiom,
    ! [N2: num] : unique5055182867167087721od_nat(one,aa_num_num(bit1,N2)) = aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,one)) ).

% divmod_algorithm_code(4)
tff(fact_8738_divmod__algorithm__code_I4_J,axiom,
    ! [N2: num] : unique3479559517661332726nteger(one,aa_num_num(bit1,N2)) = produc1086072967326762835nteger(zero_z3403309356797280102nteger,aa_num_Code_integer(numera6620942414471956472nteger,one)) ).

% divmod_algorithm_code(4)
tff(fact_8739_divmod__algorithm__code_I7_J,axiom,
    ! [M: num,N2: num] :
      unique5055182867167087721od_nat(aa_num_num(bit0,M),aa_num_num(bit1,N2)) = $ite(aa_num_o(aa_num_fun_num_o(ord_less_eq_num,M),N2),aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,M))),aa_Pro262675715343573244at_nat(unique5026877609467782581ep_nat(aa_num_num(bit1,N2)),unique5055182867167087721od_nat(aa_num_num(bit0,M),aa_num_num(bit0,aa_num_num(bit1,N2))))) ).

% divmod_algorithm_code(7)
tff(fact_8740_divmod__algorithm__code_I7_J,axiom,
    ! [M: num,N2: num] :
      unique5052692396658037445od_int(aa_num_num(bit0,M),aa_num_num(bit1,N2)) = $ite(aa_num_o(aa_num_fun_num_o(ord_less_eq_num,M),N2),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,N2)),unique5052692396658037445od_int(aa_num_num(bit0,M),aa_num_num(bit0,aa_num_num(bit1,N2))))) ).

% divmod_algorithm_code(7)
tff(fact_8741_divmod__algorithm__code_I7_J,axiom,
    ! [M: num,N2: num] :
      unique3479559517661332726nteger(aa_num_num(bit0,M),aa_num_num(bit1,N2)) = $ite(aa_num_o(aa_num_fun_num_o(ord_less_eq_num,M),N2),produc1086072967326762835nteger(zero_z3403309356797280102nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,M))),unique4921790084139445826nteger(aa_num_num(bit1,N2),unique3479559517661332726nteger(aa_num_num(bit0,M),aa_num_num(bit0,aa_num_num(bit1,N2))))) ).

% divmod_algorithm_code(7)
tff(fact_8742_prod_Odisc__eq__case,axiom,
    ! [Prod: product_prod_int_int] : aa_Pro4592068669816069595_int_o(aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,aTP_Lamm_nc),Prod) ).

% prod.disc_eq_case
tff(fact_8743_xor__num_Ocases,axiom,
    ! [X: product_prod_num_num] :
      ( ( X != product_Pair_num_num(one,one) )
     => ( ! [N5: num] : X != product_Pair_num_num(one,aa_num_num(bit0,N5))
       => ( ! [N5: num] : X != product_Pair_num_num(one,aa_num_num(bit1,N5))
         => ( ! [M5: num] : X != product_Pair_num_num(aa_num_num(bit0,M5),one)
           => ( ! [M5: num,N5: num] : X != product_Pair_num_num(aa_num_num(bit0,M5),aa_num_num(bit0,N5))
             => ( ! [M5: num,N5: num] : X != product_Pair_num_num(aa_num_num(bit0,M5),aa_num_num(bit1,N5))
               => ( ! [M5: num] : X != product_Pair_num_num(aa_num_num(bit1,M5),one)
                 => ( ! [M5: num,N5: num] : X != product_Pair_num_num(aa_num_num(bit1,M5),aa_num_num(bit0,N5))
                   => ~ ! [M5: num,N5: num] : X != product_Pair_num_num(aa_num_num(bit1,M5),aa_num_num(bit1,N5)) ) ) ) ) ) ) ) ) ).

% xor_num.cases
tff(fact_8744_numeral__code_I3_J,axiom,
    ! [N2: num] :
      aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,N2)) = $let(
        m: nat,
        m:= aa_num_nat(numeral_numeral_nat,N2),
        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_8745_numeral__code_I3_J,axiom,
    ! [N2: num] :
      aa_num_int(numeral_numeral_int,aa_num_num(bit1,N2)) = $let(
        m: int,
        m:= aa_num_int(numeral_numeral_int,N2),
        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_8746_numeral__code_I3_J,axiom,
    ! [N2: num] :
      aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit1,N2)) = $let(
        m: code_integer,
        m:= aa_num_Code_integer(numera6620942414471956472nteger,N2),
        aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(m),m)),one_one_Code_integer) ) ).

% numeral_code(3)
tff(fact_8747_numeral__code_I3_J,axiom,
    ! [N2: num] :
      aa_num_rat(numeral_numeral_rat,aa_num_num(bit1,N2)) = $let(
        m: rat,
        m:= aa_num_rat(numeral_numeral_rat,N2),
        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_8748_numeral__code_I3_J,axiom,
    ! [N2: num] :
      aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit1,N2)) = $let(
        m: code_natural,
        m:= aa_num_Code_natural(numera5444537566228673987atural,N2),
        plus_p4538020629002901425atural(plus_p4538020629002901425atural(m,m),one_one_Code_natural) ) ).

% numeral_code(3)
tff(fact_8749_power__numeral__odd,axiom,
    ! [Z: code_integer,W2: num] :
      aa_nat_Code_integer(power_8256067586552552935nteger(Z),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,W2))) = $let(
        w: code_integer,
        w:= aa_nat_Code_integer(power_8256067586552552935nteger(Z),aa_num_nat(numeral_numeral_nat,W2)),
        aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Z),w)),w) ) ).

% power_numeral_odd
tff(fact_8750_power__numeral__odd,axiom,
    ! [Z: assn,W2: num] :
      aa_nat_assn(power_power_assn(Z),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,W2))) = $let(
        w: assn,
        w:= aa_nat_assn(power_power_assn(Z),aa_num_nat(numeral_numeral_nat,W2)),
        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_8751_power__numeral__odd,axiom,
    ! [Z: rat,W2: num] :
      aa_nat_rat(power_power_rat(Z),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,W2))) = $let(
        w: rat,
        w:= aa_nat_rat(power_power_rat(Z),aa_num_nat(numeral_numeral_nat,W2)),
        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_8752_power__numeral__odd,axiom,
    ! [Z: nat,W2: num] :
      aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Z),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,W2))) = $let(
        w: nat,
        w:= aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Z),aa_num_nat(numeral_numeral_nat,W2)),
        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_8753_power__numeral__odd,axiom,
    ! [Z: int,W2: num] :
      aa_nat_int(aa_int_fun_nat_int(power_power_int,Z),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,W2))) = $let(
        w: int,
        w:= aa_nat_int(aa_int_fun_nat_int(power_power_int,Z),aa_num_nat(numeral_numeral_nat,W2)),
        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_8754_power3__eq__cube,axiom,
    ! [A: code_integer] : aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_num_nat(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_8755_power3__eq__cube,axiom,
    ! [A: assn] : aa_nat_assn(power_power_assn(A),aa_num_nat(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_8756_power3__eq__cube,axiom,
    ! [A: rat] : aa_nat_rat(power_power_rat(A),aa_num_nat(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_8757_power3__eq__cube,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_num_nat(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_8758_power3__eq__cube,axiom,
    ! [A: int] : aa_nat_int(aa_int_fun_nat_int(power_power_int,A),aa_num_nat(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_8759_take__bit__Suc__bit1,axiom,
    ! [N2: nat,K: num] : aa_Cod1783974652533396728nteger(bit_se1745604003318907178nteger(aa_nat_nat(suc,N2)),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit1,K))) = aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(bit_se1745604003318907178nteger(N2),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_8760_take__bit__Suc__bit1,axiom,
    ! [N2: nat,K: num] : bit_se569199155075624693atural(aa_nat_nat(suc,N2),aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit1,K))) = plus_p4538020629002901425atural(times_2397367101498566445atural(bit_se569199155075624693atural(N2,aa_num_Code_natural(numera5444537566228673987atural,K)),aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))),one_one_Code_natural) ).

% take_bit_Suc_bit1
tff(fact_8761_take__bit__Suc__bit1,axiom,
    ! [N2: nat,K: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,aa_nat_nat(suc,N2)),aa_num_nat(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,N2),aa_num_nat(numeral_numeral_nat,K))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_nat) ).

% take_bit_Suc_bit1
tff(fact_8762_take__bit__Suc__bit1,axiom,
    ! [N2: nat,K: num] : aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,aa_nat_nat(suc,N2)),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,N2),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_8763_divmod__BitM__2__eq,axiom,
    ! [M: num] : unique5052692396658037445od_int(aa_num_num(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_8764_take__bit__num__simps_I6_J,axiom,
    ! [Ra2: num,M: num] : aa_num_option_num(bit_take_bit_num(aa_num_nat(numeral_numeral_nat,Ra2)),aa_num_num(bit0,M)) = aa_opt5369159329878751634on_num(case_o6005452278849405969um_num(none_num,aTP_Lamm_agv),aa_num_option_num(bit_take_bit_num(aa_num_nat(pred_numeral,Ra2)),M)) ).

% take_bit_num_simps(6)
tff(fact_8765_take__bit__numeral__bit1,axiom,
    ! [L: num,K: num] : aa_Cod1783974652533396728nteger(bit_se1745604003318907178nteger(aa_num_nat(numeral_numeral_nat,L)),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit1,K))) = aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(bit_se1745604003318907178nteger(aa_num_nat(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_8766_take__bit__numeral__bit1,axiom,
    ! [L: num,K: num] : bit_se569199155075624693atural(aa_num_nat(numeral_numeral_nat,L),aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit1,K))) = plus_p4538020629002901425atural(times_2397367101498566445atural(bit_se569199155075624693atural(aa_num_nat(pred_numeral,L),aa_num_Code_natural(numera5444537566228673987atural,K)),aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))),one_one_Code_natural) ).

% take_bit_numeral_bit1
tff(fact_8767_take__bit__numeral__bit1,axiom,
    ! [L: num,K: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,aa_num_nat(numeral_numeral_nat,L)),aa_num_nat(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,aa_num_nat(pred_numeral,L)),aa_num_nat(numeral_numeral_nat,K))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_nat) ).

% take_bit_numeral_bit1
tff(fact_8768_take__bit__numeral__bit1,axiom,
    ! [L: num,K: num] : aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,aa_num_nat(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,aa_num_nat(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_8769_prod_Otriangle__reindex,axiom,
    ! [G: fun_nat_fun_nat_nat,N2: nat] : groups4077766827762148844at_nat(produc6842872674320459806at_nat(G),collec3392354462482085612at_nat(produc6081775807080527818_nat_o(aTP_Lamm_aic(N2)))) = aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_aha(G)),aa_nat_set_nat(set_ord_lessThan_nat,N2)) ).

% prod.triangle_reindex
tff(fact_8770_prod_Otriangle__reindex,axiom,
    ! [G: fun_nat_fun_nat_int,N2: nat] : groups4075276357253098568at_int(produc6840382203811409530at_int(G),collec3392354462482085612at_nat(produc6081775807080527818_nat_o(aTP_Lamm_aic(N2)))) = aa_set_nat_int(groups705719431365010083at_int(aTP_Lamm_ahc(G)),aa_nat_set_nat(set_ord_lessThan_nat,N2)) ).

% prod.triangle_reindex
tff(fact_8771_take__bit__num__def,axiom,
    ! [N2: nat,M: num] :
      aa_num_option_num(bit_take_bit_num(N2),M) = $ite(aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N2),aa_num_nat(numeral_numeral_nat,M)) = zero_zero_nat,none_num,aa_num_option_num(some_num,aa_nat_num(num_of_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N2),aa_num_nat(numeral_numeral_nat,M))))) ).

% take_bit_num_def
tff(fact_8772_lessThan__iff,axiom,
    ! [I2: $o,K: $o] :
      ( member_o((I2),set_ord_lessThan_o((K)))
    <=> aa_o_o(aa_o_fun_o_o(ord_less_o,(I2)),(K)) ) ).

% lessThan_iff
tff(fact_8773_lessThan__iff,axiom,
    ! [I2: set_nat,K: set_nat] :
      ( aa_set_set_nat_o(member_set_nat(I2),set_or890127255671739683et_nat(K))
    <=> aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,I2),K) ) ).

% lessThan_iff
tff(fact_8774_lessThan__iff,axiom,
    ! [I2: assn,K: assn] :
      ( member_assn(I2,set_or7637083652282234053n_assn(K))
    <=> aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,I2),K) ) ).

% lessThan_iff
tff(fact_8775_lessThan__iff,axiom,
    ! [I2: rat,K: rat] :
      ( member_rat(I2,set_ord_lessThan_rat(K))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,I2),K) ) ).

% lessThan_iff
tff(fact_8776_lessThan__iff,axiom,
    ! [I2: num,K: num] :
      ( member_num(I2,set_ord_lessThan_num(K))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,I2),K) ) ).

% lessThan_iff
tff(fact_8777_lessThan__iff,axiom,
    ! [I2: nat,K: nat] :
      ( aa_set_nat_o(member_nat(I2),aa_nat_set_nat(set_ord_lessThan_nat,K))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),K) ) ).

% lessThan_iff
tff(fact_8778_lessThan__iff,axiom,
    ! [I2: int,K: int] :
      ( aa_set_int_o(member_int(I2),set_ord_lessThan_int(K))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,I2),K) ) ).

% lessThan_iff
tff(fact_8779_lessThan__subset__iff,axiom,
    ! [X: rat,Y: rat] :
      ( ord_less_eq_set_rat(set_ord_lessThan_rat(X),set_ord_lessThan_rat(Y))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y) ) ).

% lessThan_subset_iff
tff(fact_8780_lessThan__subset__iff,axiom,
    ! [X: num,Y: num] :
      ( ord_less_eq_set_num(set_ord_lessThan_num(X),set_ord_lessThan_num(Y))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X),Y) ) ).

% lessThan_subset_iff
tff(fact_8781_lessThan__subset__iff,axiom,
    ! [X: nat,Y: nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_nat_set_nat(set_ord_lessThan_nat,X)),aa_nat_set_nat(set_ord_lessThan_nat,Y))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y) ) ).

% lessThan_subset_iff
tff(fact_8782_lessThan__subset__iff,axiom,
    ! [X: int,Y: int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,set_ord_lessThan_int(X)),set_ord_lessThan_int(Y))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Y) ) ).

% lessThan_subset_iff
tff(fact_8783_lessThan__0,axiom,
    aa_nat_set_nat(set_ord_lessThan_nat,zero_zero_nat) = bot_bot_set_nat ).

% lessThan_0
tff(fact_8784_less__Suc__numeral,axiom,
    ! [N2: nat,K: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,N2)),aa_num_nat(numeral_numeral_nat,K))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),aa_num_nat(pred_numeral,K)) ) ).

% less_Suc_numeral
tff(fact_8785_less__numeral__Suc,axiom,
    ! [K: num,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_num_nat(numeral_numeral_nat,K)),aa_nat_nat(suc,N2))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_num_nat(pred_numeral,K)),N2) ) ).

% less_numeral_Suc
tff(fact_8786_prod_OlessThan__Suc,axiom,
    ! [G: fun_nat_Code_integer,N2: nat] : aa_set871585252387937313nteger(groups3455450783089532116nteger(G),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N2))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_set871585252387937313nteger(groups3455450783089532116nteger(G),aa_nat_set_nat(set_ord_lessThan_nat,N2))),aa_nat_Code_integer(G,N2)) ).

% prod.lessThan_Suc
tff(fact_8787_prod_OlessThan__Suc,axiom,
    ! [G: fun_nat_assn,N2: nat] : groups6906906614972039071t_assn(G,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N2))) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,groups6906906614972039071t_assn(G,aa_nat_set_nat(set_ord_lessThan_nat,N2))),aa_nat_assn(G,N2)) ).

% prod.lessThan_Suc
tff(fact_8788_prod_OlessThan__Suc,axiom,
    ! [G: fun_nat_rat,N2: nat] : aa_set_nat_rat(groups73079841787564623at_rat(G),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N2))) = 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,N2))),aa_nat_rat(G,N2)) ).

% prod.lessThan_Suc
tff(fact_8789_prod_OlessThan__Suc,axiom,
    ! [G: fun_nat_nat,N2: nat] : aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N2))) = 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,N2))),aa_nat_nat(G,N2)) ).

% prod.lessThan_Suc
tff(fact_8790_prod_OlessThan__Suc,axiom,
    ! [G: fun_nat_int,N2: nat] : aa_set_nat_int(groups705719431365010083at_int(G),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N2))) = 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,N2))),aa_nat_int(G,N2)) ).

% prod.lessThan_Suc
tff(fact_8791_le__Suc__numeral,axiom,
    ! [N2: nat,K: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,N2)),aa_num_nat(numeral_numeral_nat,K))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),aa_num_nat(pred_numeral,K)) ) ).

% le_Suc_numeral
tff(fact_8792_le__numeral__Suc,axiom,
    ! [K: num,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,K)),aa_nat_nat(suc,N2))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(pred_numeral,K)),N2) ) ).

% le_numeral_Suc
tff(fact_8793_take__bit__num__simps_I7_J,axiom,
    ! [Ra2: num,M: num] : aa_num_option_num(bit_take_bit_num(aa_num_nat(numeral_numeral_nat,Ra2)),aa_num_num(bit1,M)) = aa_num_option_num(some_num,aa_option_num_num(case_option_num_num(one,bit1),aa_num_option_num(bit_take_bit_num(aa_num_nat(pred_numeral,Ra2)),M))) ).

% take_bit_num_simps(7)
tff(fact_8794_lessThan__non__empty,axiom,
    ! [X: int] : set_ord_lessThan_int(X) != bot_bot_set_int ).

% lessThan_non_empty
tff(fact_8795_lessThan__def,axiom,
    ! [U: set_nat] : set_or890127255671739683et_nat(U) = collect_set_nat(aa_set7861427066675100071_nat_o(aTP_Lamm_da,U)) ).

% lessThan_def
tff(fact_8796_lessThan__def,axiom,
    ! [U: assn] : set_or7637083652282234053n_assn(U) = collect_assn(aa_assn_fun_assn_o(aTP_Lamm_dc,U)) ).

% lessThan_def
tff(fact_8797_lessThan__def,axiom,
    ! [U: rat] : set_ord_lessThan_rat(U) = collect_rat(aa_rat_fun_rat_o(aTP_Lamm_di,U)) ).

% lessThan_def
tff(fact_8798_lessThan__def,axiom,
    ! [U: num] : set_ord_lessThan_num(U) = collect_num(aa_num_fun_num_o(aTP_Lamm_fb,U)) ).

% lessThan_def
tff(fact_8799_lessThan__def,axiom,
    ! [U: nat] : aa_nat_set_nat(set_ord_lessThan_nat,U) = collect_nat(aa_nat_fun_nat_o(aTP_Lamm_dg,U)) ).

% lessThan_def
tff(fact_8800_lessThan__def,axiom,
    ! [U: int] : set_ord_lessThan_int(U) = collect_int(aa_int_fun_int_o(aTP_Lamm_dk,U)) ).

% lessThan_def
tff(fact_8801_lessThan__empty__iff,axiom,
    ! [N2: nat] :
      ( ( aa_nat_set_nat(set_ord_lessThan_nat,N2) = bot_bot_set_nat )
    <=> ( N2 = zero_zero_nat ) ) ).

% lessThan_empty_iff
tff(fact_8802_Iio__eq__empty__iff,axiom,
    ! [N2: $o] :
      ( ( set_ord_lessThan_o((N2)) = bot_bot_set_o )
    <=> ( (N2)
      <=> bot_bot_o ) ) ).

% Iio_eq_empty_iff
tff(fact_8803_Iio__eq__empty__iff,axiom,
    ! [N2: nat] :
      ( ( aa_nat_set_nat(set_ord_lessThan_nat,N2) = bot_bot_set_nat )
    <=> ( N2 = bot_bot_nat ) ) ).

% Iio_eq_empty_iff
tff(fact_8804_finite__nat__iff__bounded,axiom,
    ! [S: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,S)
    <=> ? [K3: nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,S),aa_nat_set_nat(set_ord_lessThan_nat,K3)) ) ).

% finite_nat_iff_bounded
tff(fact_8805_finite__nat__bounded,axiom,
    ! [S: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,S)
     => ? [K2: nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,S),aa_nat_set_nat(set_ord_lessThan_nat,K2)) ) ).

% finite_nat_bounded
tff(fact_8806_lessThan__strict__subset__iff,axiom,
    ! [M: rat,N2: rat] :
      ( ord_less_set_rat(set_ord_lessThan_rat(M),set_ord_lessThan_rat(N2))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,M),N2) ) ).

% lessThan_strict_subset_iff
tff(fact_8807_lessThan__strict__subset__iff,axiom,
    ! [M: num,N2: num] :
      ( ord_less_set_num(set_ord_lessThan_num(M),set_ord_lessThan_num(N2))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,M),N2) ) ).

% lessThan_strict_subset_iff
tff(fact_8808_lessThan__strict__subset__iff,axiom,
    ! [M: nat,N2: nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,aa_nat_set_nat(set_ord_lessThan_nat,M)),aa_nat_set_nat(set_ord_lessThan_nat,N2))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N2) ) ).

% lessThan_strict_subset_iff
tff(fact_8809_lessThan__strict__subset__iff,axiom,
    ! [M: int,N2: int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_set_int,set_ord_lessThan_int(M)),set_ord_lessThan_int(N2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,M),N2) ) ).

% lessThan_strict_subset_iff
tff(fact_8810_sum_Onat__diff__reindex,axiom,
    ! [G: fun_nat_nat,N2: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_aid(G),N2)),aa_nat_set_nat(set_ord_lessThan_nat,N2)) = aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_ord_lessThan_nat,N2)) ).

% sum.nat_diff_reindex
tff(fact_8811_prod_Onat__diff__reindex,axiom,
    ! [G: fun_nat_nat,N2: nat] : aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_aid(G),N2)),aa_nat_set_nat(set_ord_lessThan_nat,N2)) = aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_ord_lessThan_nat,N2)) ).

% prod.nat_diff_reindex
tff(fact_8812_prod_Onat__diff__reindex,axiom,
    ! [G: fun_nat_int,N2: nat] : aa_set_nat_int(groups705719431365010083at_int(aa_nat_fun_nat_int(aTP_Lamm_aie(G),N2)),aa_nat_set_nat(set_ord_lessThan_nat,N2)) = aa_set_nat_int(groups705719431365010083at_int(G),aa_nat_set_nat(set_ord_lessThan_nat,N2)) ).

% prod.nat_diff_reindex
tff(fact_8813_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)
     => ( 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_8814_ivl__disj__un__one_I2_J,axiom,
    ! [L: num,U: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,L),U)
     => ( sup_sup_set_num(set_ord_lessThan_num(L),set_or1222409239386451017an_num(L,U)) = set_ord_lessThan_num(U) ) ) ).

% ivl_disj_un_one(2)
tff(fact_8815_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(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,L),U)) = aa_nat_set_nat(set_ord_lessThan_nat,U) ) ) ).

% ivl_disj_un_one(2)
tff(fact_8816_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_8817_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_8818_ivl__disj__int__one_I4_J,axiom,
    ! [L: $o,U: $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_8819_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_8820_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(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,L),U)) = bot_bot_set_nat ).

% ivl_disj_int_one(4)
tff(fact_8821_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_8822_Iic__subset__Iio__iff,axiom,
    ! [A: rat,B: rat] :
      ( ord_less_eq_set_rat(set_ord_atMost_rat(A),set_ord_lessThan_rat(B))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ).

% Iic_subset_Iio_iff
tff(fact_8823_Iic__subset__Iio__iff,axiom,
    ! [A: num,B: num] :
      ( ord_less_eq_set_num(set_ord_atMost_num(A),set_ord_lessThan_num(B))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,A),B) ) ).

% Iic_subset_Iio_iff
tff(fact_8824_Iic__subset__Iio__iff,axiom,
    ! [A: nat,B: nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_nat_set_nat(set_ord_atMost_nat,A)),aa_nat_set_nat(set_ord_lessThan_nat,B))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B) ) ).

% Iic_subset_Iio_iff
tff(fact_8825_Iic__subset__Iio__iff,axiom,
    ! [A: int,B: int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,set_ord_atMost_int(A)),set_ord_lessThan_int(B))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ).

% Iic_subset_Iio_iff
tff(fact_8826_ivl__disj__int__one_I2_J,axiom,
    ! [L: $o,U: $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_8827_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(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,L),U)) = bot_bot_set_nat ).

% ivl_disj_int_one(2)
tff(fact_8828_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_8829_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_8830_sum__diff__distrib,axiom,
    ! [Qa: fun_int_nat,Pa: fun_int_nat,N2: int] :
      ( ! [X3: int] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(Qa,X3)),aa_int_nat(Pa,X3))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups4541462559716669496nt_nat(Pa,set_ord_lessThan_int(N2))),groups4541462559716669496nt_nat(Qa,set_ord_lessThan_int(N2))) = groups4541462559716669496nt_nat(aa_fun6506752150744536624nt_nat(aTP_Lamm_lh(Qa),Pa),set_ord_lessThan_int(N2)) ) ) ).

% sum_diff_distrib
tff(fact_8831_sum__diff__distrib,axiom,
    ! [Qa: fun_nat_nat,Pa: fun_nat_nat,N2: nat] :
      ( ! [X3: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(Qa,X3)),aa_nat_nat(Pa,X3))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(Pa),aa_nat_set_nat(set_ord_lessThan_nat,N2))),aa_set_nat_nat(groups3542108847815614940at_nat(Qa),aa_nat_set_nat(set_ord_lessThan_nat,N2))) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_li(Qa),Pa)),aa_nat_set_nat(set_ord_lessThan_nat,N2)) ) ) ).

% sum_diff_distrib
tff(fact_8832_sum_OlessThan__Suc__shift,axiom,
    ! [G: fun_na2338080690716861136at_nat,N2: nat] : groups6857163185585827899at_nat(G,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N2))) = aa_mul5754855110707352700at_nat(plus_p7104986032573967614at_nat(aa_nat2240542744625857721at_nat(G,zero_zero_nat)),groups6857163185585827899at_nat(aTP_Lamm_ka(G),aa_nat_set_nat(set_ord_lessThan_nat,N2))) ).

% sum.lessThan_Suc_shift
tff(fact_8833_sum_OlessThan__Suc__shift,axiom,
    ! [G: fun_nat_rat,N2: nat] : aa_set_nat_rat(groups2906978787729119204at_rat(G),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N2))) = 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_kb(G)),aa_nat_set_nat(set_ord_lessThan_nat,N2))) ).

% sum.lessThan_Suc_shift
tff(fact_8834_sum_OlessThan__Suc__shift,axiom,
    ! [G: fun_nat_int,N2: nat] : aa_set_nat_int(groups3539618377306564664at_int(G),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N2))) = 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_kc(G)),aa_nat_set_nat(set_ord_lessThan_nat,N2))) ).

% sum.lessThan_Suc_shift
tff(fact_8835_sum_OlessThan__Suc__shift,axiom,
    ! [G: fun_nat_nat,N2: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N2))) = 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_jx(G)),aa_nat_set_nat(set_ord_lessThan_nat,N2))) ).

% sum.lessThan_Suc_shift
tff(fact_8836_sum__lessThan__telescope_H,axiom,
    ! [F: fun_nat_rat,M: nat] : aa_set_nat_rat(groups2906978787729119204at_rat(aTP_Lamm_oc(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_8837_sum__lessThan__telescope_H,axiom,
    ! [F: fun_nat_int,M: nat] : aa_set_nat_int(groups3539618377306564664at_int(aTP_Lamm_od(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_8838_sum__lessThan__telescope,axiom,
    ! [F: fun_nat_rat,M: nat] : aa_set_nat_rat(groups2906978787729119204at_rat(aTP_Lamm_kd(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_8839_sum__lessThan__telescope,axiom,
    ! [F: fun_nat_int,M: nat] : aa_set_nat_int(groups3539618377306564664at_int(aTP_Lamm_ke(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_8840_prod_OlessThan__Suc__shift,axiom,
    ! [G: fun_nat_Code_integer,N2: nat] : aa_set871585252387937313nteger(groups3455450783089532116nteger(G),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N2))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(G,zero_zero_nat)),aa_set871585252387937313nteger(groups3455450783089532116nteger(aTP_Lamm_nk(G)),aa_nat_set_nat(set_ord_lessThan_nat,N2))) ).

% prod.lessThan_Suc_shift
tff(fact_8841_prod_OlessThan__Suc__shift,axiom,
    ! [G: fun_nat_assn,N2: nat] : groups6906906614972039071t_assn(G,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N2))) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_nat_assn(G,zero_zero_nat)),groups6906906614972039071t_assn(aTP_Lamm_nl(G),aa_nat_set_nat(set_ord_lessThan_nat,N2))) ).

% prod.lessThan_Suc_shift
tff(fact_8842_prod_OlessThan__Suc__shift,axiom,
    ! [G: fun_nat_rat,N2: nat] : aa_set_nat_rat(groups73079841787564623at_rat(G),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N2))) = 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_kb(G)),aa_nat_set_nat(set_ord_lessThan_nat,N2))) ).

% prod.lessThan_Suc_shift
tff(fact_8843_prod_OlessThan__Suc__shift,axiom,
    ! [G: fun_nat_nat,N2: nat] : aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N2))) = 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_jx(G)),aa_nat_set_nat(set_ord_lessThan_nat,N2))) ).

% prod.lessThan_Suc_shift
tff(fact_8844_prod_OlessThan__Suc__shift,axiom,
    ! [G: fun_nat_int,N2: nat] : aa_set_nat_int(groups705719431365010083at_int(G),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N2))) = 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_kc(G)),aa_nat_set_nat(set_ord_lessThan_nat,N2))) ).

% prod.lessThan_Suc_shift
tff(fact_8845_numeral__num__of__nat,axiom,
    ! [N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)
     => ( aa_num_nat(numeral_numeral_nat,aa_nat_num(num_of_nat,N2)) = N2 ) ) ).

% numeral_num_of_nat
tff(fact_8846_sum_OatLeast1__atMost__eq,axiom,
    ! [G: fun_nat_nat,N2: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,aa_nat_nat(suc,zero_zero_nat)),N2)) = aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_jx(G)),aa_nat_set_nat(set_ord_lessThan_nat,N2)) ).

% sum.atLeast1_atMost_eq
tff(fact_8847_num__of__nat__One,axiom,
    ! [N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),one_one_nat)
     => ( aa_nat_num(num_of_nat,N2) = one ) ) ).

% num_of_nat_One
tff(fact_8848_prod_OatLeast1__atMost__eq,axiom,
    ! [G: fun_nat_nat,N2: nat] : aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,aa_nat_nat(suc,zero_zero_nat)),N2)) = aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_jx(G)),aa_nat_set_nat(set_ord_lessThan_nat,N2)) ).

% prod.atLeast1_atMost_eq
tff(fact_8849_prod_OatLeast1__atMost__eq,axiom,
    ! [G: fun_nat_int,N2: nat] : aa_set_nat_int(groups705719431365010083at_int(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,aa_nat_nat(suc,zero_zero_nat)),N2)) = aa_set_nat_int(groups705719431365010083at_int(aTP_Lamm_kc(G)),aa_nat_set_nat(set_ord_lessThan_nat,N2)) ).

% prod.atLeast1_atMost_eq
tff(fact_8850_sum__bounds__lt__plus1,axiom,
    ! [F: fun_nat_nat,Mm: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_jx(F)),aa_nat_set_nat(set_ord_lessThan_nat,Mm)) = aa_set_nat_nat(groups3542108847815614940at_nat(F),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,one_one_nat),Mm)) ).

% sum_bounds_lt_plus1
tff(fact_8851_sum_Onat__group,axiom,
    ! [G: fun_nat_nat,K: nat,N2: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_aif(G),K)),aa_nat_set_nat(set_ord_lessThan_nat,N2)) = 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,N2),K))) ).

% sum.nat_group
tff(fact_8852_prod_Onat__group,axiom,
    ! [G: fun_nat_nat,K: nat,N2: nat] : aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_aig(G),K)),aa_nat_set_nat(set_ord_lessThan_nat,N2)) = 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,N2),K))) ).

% prod.nat_group
tff(fact_8853_prod_Onat__group,axiom,
    ! [G: fun_nat_int,K: nat,N2: nat] : aa_set_nat_int(groups705719431365010083at_int(aa_nat_fun_nat_int(aTP_Lamm_aih(G),K)),aa_nat_set_nat(set_ord_lessThan_nat,N2)) = 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,N2),K))) ).

% prod.nat_group
tff(fact_8854_sum_Onested__swap_H,axiom,
    ! [A: fun_nat_fun_nat_nat,N2: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_aii(A)),aa_nat_set_nat(set_ord_atMost_nat,N2)) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_abq(A),N2)),aa_nat_set_nat(set_ord_lessThan_nat,N2)) ).

% sum.nested_swap'
tff(fact_8855_prod_Onested__swap_H,axiom,
    ! [A: fun_nat_fun_nat_nat,N2: nat] : aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_aij(A)),aa_nat_set_nat(set_ord_atMost_nat,N2)) = aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_abt(A),N2)),aa_nat_set_nat(set_ord_lessThan_nat,N2)) ).

% prod.nested_swap'
tff(fact_8856_prod_Onested__swap_H,axiom,
    ! [A: fun_nat_fun_nat_int,N2: nat] : aa_set_nat_int(groups705719431365010083at_int(aTP_Lamm_aik(A)),aa_nat_set_nat(set_ord_atMost_nat,N2)) = aa_set_nat_int(groups705719431365010083at_int(aa_nat_fun_nat_int(aTP_Lamm_abv(A),N2)),aa_nat_set_nat(set_ord_lessThan_nat,N2)) ).

% prod.nested_swap'
tff(fact_8857_fact__numeral,axiom,
    ! [K: num] : aa_nat_int(semiri1406184849735516958ct_int,aa_num_nat(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,aa_num_nat(pred_numeral,K))) ).

% fact_numeral
tff(fact_8858_fact__numeral,axiom,
    ! [K: num] : aa_nat_Code_integer(semiri3624122377584611663nteger,aa_num_nat(numeral_numeral_nat,K)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,K)),aa_nat_Code_integer(semiri3624122377584611663nteger,aa_num_nat(pred_numeral,K))) ).

% fact_numeral
tff(fact_8859_fact__numeral,axiom,
    ! [K: num] : aa_nat_rat(semiri773545260158071498ct_rat,aa_num_nat(numeral_numeral_nat,K)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,K)),aa_nat_rat(semiri773545260158071498ct_rat,aa_num_nat(pred_numeral,K))) ).

% fact_numeral
tff(fact_8860_fact__numeral,axiom,
    ! [K: num] : aa_nat_Code_natural(semiri2447717529341329178atural,aa_num_nat(numeral_numeral_nat,K)) = times_2397367101498566445atural(aa_num_Code_natural(numera5444537566228673987atural,K),aa_nat_Code_natural(semiri2447717529341329178atural,aa_num_nat(pred_numeral,K))) ).

% fact_numeral
tff(fact_8861_fact__numeral,axiom,
    ! [K: num] : aa_nat_nat(semiri1408675320244567234ct_nat,aa_num_nat(numeral_numeral_nat,K)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,K)),aa_nat_nat(semiri1408675320244567234ct_nat,aa_num_nat(pred_numeral,K))) ).

% fact_numeral
tff(fact_8862_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)
     => ( 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_8863_ivl__disj__un__one_I4_J,axiom,
    ! [L: num,U: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,L),U)
     => ( sup_sup_set_num(set_ord_lessThan_num(L),set_or7049704709247886629st_num(L,U)) = set_ord_atMost_num(U) ) ) ).

% ivl_disj_un_one(4)
tff(fact_8864_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_8865_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(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,L),U)) = aa_nat_set_nat(set_ord_atMost_nat,U) ) ) ).

% ivl_disj_un_one(4)
tff(fact_8866_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_8867_one__diff__power__eq,axiom,
    ! [X: code_integer,N2: nat] : aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(one_one_Code_integer),aa_nat_Code_integer(power_8256067586552552935nteger(X),N2)) = 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,N2))) ).

% one_diff_power_eq
tff(fact_8868_one__diff__power__eq,axiom,
    ! [X: rat,N2: nat] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,one_one_rat),aa_nat_rat(power_power_rat(X),N2)) = 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,N2))) ).

% one_diff_power_eq
tff(fact_8869_one__diff__power__eq,axiom,
    ! [X: int,N2: 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),N2)) = 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,N2))) ).

% one_diff_power_eq
tff(fact_8870_power__diff__1__eq,axiom,
    ! [X: code_integer,N2: nat] : aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_nat_Code_integer(power_8256067586552552935nteger(X),N2)),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,N2))) ).

% power_diff_1_eq
tff(fact_8871_power__diff__1__eq,axiom,
    ! [X: rat,N2: nat] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_nat_rat(power_power_rat(X),N2)),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,N2))) ).

% power_diff_1_eq
tff(fact_8872_power__diff__1__eq,axiom,
    ! [X: int,N2: nat] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,X),N2)),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,N2))) ).

% power_diff_1_eq
tff(fact_8873_geometric__sum,axiom,
    ! [X: rat,N2: nat] :
      ( ( X != one_one_rat )
     => ( aa_set_nat_rat(groups2906978787729119204at_rat(power_power_rat(X)),aa_nat_set_nat(set_ord_lessThan_nat,N2)) = 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),N2)),one_one_rat)),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,X),one_one_rat)) ) ) ).

% geometric_sum
tff(fact_8874_image__atLeastZeroLessThan__int,axiom,
    ! [U: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),U)
     => ( set_or4662586982721622107an_int(zero_zero_int,U) = image_nat_int(semiri1314217659103216013at_int,aa_nat_set_nat(set_ord_lessThan_nat,aa_int_nat(nat2,U))) ) ) ).

% image_atLeastZeroLessThan_int
tff(fact_8875_sum_OatMost__shift,axiom,
    ! [G: fun_na2338080690716861136at_nat,N2: nat] : groups6857163185585827899at_nat(G,aa_nat_set_nat(set_ord_atMost_nat,N2)) = aa_mul5754855110707352700at_nat(plus_p7104986032573967614at_nat(aa_nat2240542744625857721at_nat(G,zero_zero_nat)),groups6857163185585827899at_nat(aTP_Lamm_ka(G),aa_nat_set_nat(set_ord_lessThan_nat,N2))) ).

% sum.atMost_shift
tff(fact_8876_sum_OatMost__shift,axiom,
    ! [G: fun_nat_rat,N2: nat] : aa_set_nat_rat(groups2906978787729119204at_rat(G),aa_nat_set_nat(set_ord_atMost_nat,N2)) = 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_kb(G)),aa_nat_set_nat(set_ord_lessThan_nat,N2))) ).

% sum.atMost_shift
tff(fact_8877_sum_OatMost__shift,axiom,
    ! [G: fun_nat_int,N2: nat] : aa_set_nat_int(groups3539618377306564664at_int(G),aa_nat_set_nat(set_ord_atMost_nat,N2)) = 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_kc(G)),aa_nat_set_nat(set_ord_lessThan_nat,N2))) ).

% sum.atMost_shift
tff(fact_8878_sum_OatMost__shift,axiom,
    ! [G: fun_nat_nat,N2: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_ord_atMost_nat,N2)) = 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_jx(G)),aa_nat_set_nat(set_ord_lessThan_nat,N2))) ).

% sum.atMost_shift
tff(fact_8879_prod_OatMost__shift,axiom,
    ! [G: fun_nat_Code_integer,N2: nat] : aa_set871585252387937313nteger(groups3455450783089532116nteger(G),aa_nat_set_nat(set_ord_atMost_nat,N2)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(G,zero_zero_nat)),aa_set871585252387937313nteger(groups3455450783089532116nteger(aTP_Lamm_nk(G)),aa_nat_set_nat(set_ord_lessThan_nat,N2))) ).

% prod.atMost_shift
tff(fact_8880_prod_OatMost__shift,axiom,
    ! [G: fun_nat_assn,N2: nat] : groups6906906614972039071t_assn(G,aa_nat_set_nat(set_ord_atMost_nat,N2)) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_nat_assn(G,zero_zero_nat)),groups6906906614972039071t_assn(aTP_Lamm_nl(G),aa_nat_set_nat(set_ord_lessThan_nat,N2))) ).

% prod.atMost_shift
tff(fact_8881_prod_OatMost__shift,axiom,
    ! [G: fun_nat_rat,N2: nat] : aa_set_nat_rat(groups73079841787564623at_rat(G),aa_nat_set_nat(set_ord_atMost_nat,N2)) = 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_kb(G)),aa_nat_set_nat(set_ord_lessThan_nat,N2))) ).

% prod.atMost_shift
tff(fact_8882_prod_OatMost__shift,axiom,
    ! [G: fun_nat_nat,N2: nat] : aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_ord_atMost_nat,N2)) = 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_jx(G)),aa_nat_set_nat(set_ord_lessThan_nat,N2))) ).

% prod.atMost_shift
tff(fact_8883_prod_OatMost__shift,axiom,
    ! [G: fun_nat_int,N2: nat] : aa_set_nat_int(groups705719431365010083at_int(G),aa_nat_set_nat(set_ord_atMost_nat,N2)) = 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_kc(G)),aa_nat_set_nat(set_ord_lessThan_nat,N2))) ).

% prod.atMost_shift
tff(fact_8884_num__of__nat__double,axiom,
    ! [N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)
     => ( aa_nat_num(num_of_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),N2)) = aa_num_num(bit0,aa_nat_num(num_of_nat,N2)) ) ) ).

% num_of_nat_double
tff(fact_8885_num__of__nat__plus__distrib,axiom,
    ! [M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)
       => ( aa_nat_num(num_of_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N2)) = aa_num_num(aa_num_fun_num_num(plus_plus_num,aa_nat_num(num_of_nat,M)),aa_nat_num(num_of_nat,N2)) ) ) ) ).

% num_of_nat_plus_distrib
tff(fact_8886_sum__gp__strict,axiom,
    ! [X: rat,N2: nat] :
      aa_set_nat_rat(groups2906978787729119204at_rat(power_power_rat(X)),aa_nat_set_nat(set_ord_lessThan_nat,N2)) = $ite(X = one_one_rat,aa_nat_rat(semiri681578069525770553at_rat,N2),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),N2))),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,one_one_rat),X))) ).

% sum_gp_strict
tff(fact_8887_diff__power__eq__sum,axiom,
    ! [X: code_integer,N2: nat,Y: code_integer] : aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_nat_Code_integer(power_8256067586552552935nteger(X),aa_nat_nat(suc,N2))),aa_nat_Code_integer(power_8256067586552552935nteger(Y),aa_nat_nat(suc,N2))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(X),Y)),aa_set871585252387937313nteger(groups7501900531339628137nteger(aa_Cod1578151519121547077nteger(aa_nat2351529209232473362nteger(aTP_Lamm_ail(X),N2),Y)),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N2)))) ).

% diff_power_eq_sum
tff(fact_8888_diff__power__eq__sum,axiom,
    ! [X: rat,N2: 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,N2))),aa_nat_rat(power_power_rat(Y),aa_nat_nat(suc,N2))) = 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_aim(X),N2),Y)),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N2)))) ).

% diff_power_eq_sum
tff(fact_8889_diff__power__eq__sum,axiom,
    ! [X: int,N2: 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,N2))),aa_nat_int(aa_int_fun_nat_int(power_power_int,Y),aa_nat_nat(suc,N2))) = 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_ain(X),N2),Y)),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N2)))) ).

% diff_power_eq_sum
tff(fact_8890_power__diff__sumr2,axiom,
    ! [X: code_integer,N2: nat,Y: code_integer] : aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_nat_Code_integer(power_8256067586552552935nteger(X),N2)),aa_nat_Code_integer(power_8256067586552552935nteger(Y),N2)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(X),Y)),aa_set871585252387937313nteger(groups7501900531339628137nteger(aa_Cod1578151519121547077nteger(aa_nat2351529209232473362nteger(aTP_Lamm_aio(X),N2),Y)),aa_nat_set_nat(set_ord_lessThan_nat,N2))) ).

% power_diff_sumr2
tff(fact_8891_power__diff__sumr2,axiom,
    ! [X: rat,N2: nat,Y: rat] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_nat_rat(power_power_rat(X),N2)),aa_nat_rat(power_power_rat(Y),N2)) = 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_aip(X),N2),Y)),aa_nat_set_nat(set_ord_lessThan_nat,N2))) ).

% power_diff_sumr2
tff(fact_8892_power__diff__sumr2,axiom,
    ! [X: int,N2: 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),N2)),aa_nat_int(aa_int_fun_nat_int(power_power_int,Y),N2)) = 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_aiq(X),N2),Y)),aa_nat_set_nat(set_ord_lessThan_nat,N2))) ).

% power_diff_sumr2
tff(fact_8893_take__bit__numeral__bit0,axiom,
    ! [L: num,K: num] : aa_Cod1783974652533396728nteger(bit_se1745604003318907178nteger(aa_num_nat(numeral_numeral_nat,L)),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,K))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(bit_se1745604003318907178nteger(aa_num_nat(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_8894_take__bit__numeral__bit0,axiom,
    ! [L: num,K: num] : bit_se569199155075624693atural(aa_num_nat(numeral_numeral_nat,L),aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,K))) = times_2397367101498566445atural(bit_se569199155075624693atural(aa_num_nat(pred_numeral,L),aa_num_Code_natural(numera5444537566228673987atural,K)),aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))) ).

% take_bit_numeral_bit0
tff(fact_8895_take__bit__numeral__bit0,axiom,
    ! [L: num,K: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,aa_num_nat(numeral_numeral_nat,L)),aa_num_nat(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,aa_num_nat(pred_numeral,L)),aa_num_nat(numeral_numeral_nat,K))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% take_bit_numeral_bit0
tff(fact_8896_take__bit__numeral__bit0,axiom,
    ! [L: num,K: num] : aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,aa_num_nat(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,aa_num_nat(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_8897_one__diff__power__eq_H,axiom,
    ! [X: code_integer,N2: nat] : aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(one_one_Code_integer),aa_nat_Code_integer(power_8256067586552552935nteger(X),N2)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(one_one_Code_integer),X)),aa_set871585252387937313nteger(groups7501900531339628137nteger(aa_nat2010205248571845816nteger(aTP_Lamm_air(X),N2)),aa_nat_set_nat(set_ord_lessThan_nat,N2))) ).

% one_diff_power_eq'
tff(fact_8898_one__diff__power__eq_H,axiom,
    ! [X: rat,N2: nat] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,one_one_rat),aa_nat_rat(power_power_rat(X),N2)) = 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_ais(X),N2)),aa_nat_set_nat(set_ord_lessThan_nat,N2))) ).

% one_diff_power_eq'
tff(fact_8899_one__diff__power__eq_H,axiom,
    ! [X: int,N2: 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),N2)) = 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_ait(X),N2)),aa_nat_set_nat(set_ord_lessThan_nat,N2))) ).

% one_diff_power_eq'
tff(fact_8900_sum_Otriangle__reindex,axiom,
    ! [G: fun_nat_fun_nat_nat,N2: nat] : groups977919841031483927at_nat(produc6842872674320459806at_nat(G),collec3392354462482085612at_nat(produc6081775807080527818_nat_o(aTP_Lamm_aic(N2)))) = aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_agz(G)),aa_nat_set_nat(set_ord_lessThan_nat,N2)) ).

% sum.triangle_reindex
tff(fact_8901_mask__numeral,axiom,
    ! [N2: num] : aa_nat_Code_integer(bit_se2119862282449309892nteger,aa_num_nat(numeral_numeral_nat,N2)) = aa_Cod1783974652533396728nteger(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,aa_num_nat(pred_numeral,N2)))) ).

% mask_numeral
tff(fact_8902_mask__numeral,axiom,
    ! [N2: num] : aa_nat_Code_natural(bit_se943457434206027407atural,aa_num_nat(numeral_numeral_nat,N2)) = plus_p4538020629002901425atural(one_one_Code_natural,times_2397367101498566445atural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one)),aa_nat_Code_natural(bit_se943457434206027407atural,aa_num_nat(pred_numeral,N2)))) ).

% mask_numeral
tff(fact_8903_mask__numeral,axiom,
    ! [N2: num] : aa_nat_nat(bit_se2002935070580805687sk_nat,aa_num_nat(numeral_numeral_nat,N2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(bit_se2002935070580805687sk_nat,aa_num_nat(pred_numeral,N2)))) ).

% mask_numeral
tff(fact_8904_mask__numeral,axiom,
    ! [N2: num] : aa_nat_int(bit_se2000444600071755411sk_int,aa_num_nat(numeral_numeral_nat,N2)) = 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,aa_num_nat(pred_numeral,N2)))) ).

% mask_numeral
tff(fact_8905_horner__sum__of__bool__2__less,axiom,
    ! [Bs: list_o] : aa_int_o(aa_int_fun_int_o(ord_less_int,groups9116527308978886569_o_int(zero_n2684676970156552555ol_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)),Bs)),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),size_size_list_o(Bs))) ).

% horner_sum_of_bool_2_less
tff(fact_8906_xor__numerals_I4_J,axiom,
    ! [X: num,Y: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,X))),aa_num_nat(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,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_num_nat(numeral_numeral_nat,X)),aa_num_nat(numeral_numeral_nat,Y)))) ).

% xor_numerals(4)
tff(fact_8907_xor__numerals_I4_J,axiom,
    ! [X: num,Y: num] : aa_Cod1783974652533396728nteger(bit_se3222712562003087583nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,X))),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit1,Y))) = aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(one_one_Code_integer),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(bit_se3222712562003087583nteger(aa_num_Code_integer(numera6620942414471956472nteger,X)),aa_num_Code_integer(numera6620942414471956472nteger,Y)))) ).

% xor_numerals(4)
tff(fact_8908_xor__numerals_I4_J,axiom,
    ! [X: num,Y: num] : bit_se2046307713759805098atural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,X)),aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit1,Y))) = plus_p4538020629002901425atural(one_one_Code_natural,times_2397367101498566445atural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one)),bit_se2046307713759805098atural(aa_num_Code_natural(numera5444537566228673987atural,X),aa_num_Code_natural(numera5444537566228673987atural,Y)))) ).

% xor_numerals(4)
tff(fact_8909_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_8910_xor__numerals_I6_J,axiom,
    ! [X: num,Y: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,X))),aa_num_nat(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,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_num_nat(numeral_numeral_nat,X)),aa_num_nat(numeral_numeral_nat,Y)))) ).

% xor_numerals(6)
tff(fact_8911_xor__numerals_I6_J,axiom,
    ! [X: num,Y: num] : aa_Cod1783974652533396728nteger(bit_se3222712562003087583nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit1,X))),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,Y))) = aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(one_one_Code_integer),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(bit_se3222712562003087583nteger(aa_num_Code_integer(numera6620942414471956472nteger,X)),aa_num_Code_integer(numera6620942414471956472nteger,Y)))) ).

% xor_numerals(6)
tff(fact_8912_xor__numerals_I6_J,axiom,
    ! [X: num,Y: num] : bit_se2046307713759805098atural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit1,X)),aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,Y))) = plus_p4538020629002901425atural(one_one_Code_natural,times_2397367101498566445atural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one)),bit_se2046307713759805098atural(aa_num_Code_natural(numera5444537566228673987atural,X),aa_num_Code_natural(numera5444537566228673987atural,Y)))) ).

% xor_numerals(6)
tff(fact_8913_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_8914_divmod__step__integer__def,axiom,
    ! [L: num,Qr: produc8923325533196201883nteger] : unique4921790084139445826nteger(L,Qr) = produc6916734918728496179nteger(aTP_Lamm_afe(L),Qr) ).

% divmod_step_integer_def
tff(fact_8915_mask__nat__positive__iff,axiom,
    ! [N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(bit_se2002935070580805687sk_nat,N2))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2) ) ).

% mask_nat_positive_iff
tff(fact_8916_xor__nonnegative__int__iff,axiom,
    ! [K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,K),L))
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K)
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),L) ) ) ).

% xor_nonnegative_int_iff
tff(fact_8917_xor__negative__int__iff,axiom,
    ! [K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,K),L)),zero_zero_int)
    <=> ~ ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int)
        <=> aa_int_o(aa_int_fun_int_o(ord_less_int,L),zero_zero_int) ) ) ).

% xor_negative_int_iff
tff(fact_8918_xor__numerals_I3_J,axiom,
    ! [X: num,Y: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,X))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Y))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_num_nat(numeral_numeral_nat,X)),aa_num_nat(numeral_numeral_nat,Y))) ).

% xor_numerals(3)
tff(fact_8919_xor__numerals_I3_J,axiom,
    ! [X: num,Y: num] : aa_Cod1783974652533396728nteger(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(bit_se3222712562003087583nteger(aa_num_Code_integer(numera6620942414471956472nteger,X)),aa_num_Code_integer(numera6620942414471956472nteger,Y))) ).

% xor_numerals(3)
tff(fact_8920_xor__numerals_I3_J,axiom,
    ! [X: num,Y: num] : bit_se2046307713759805098atural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,X)),aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,Y))) = times_2397367101498566445atural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one)),bit_se2046307713759805098atural(aa_num_Code_natural(numera5444537566228673987atural,X),aa_num_Code_natural(numera5444537566228673987atural,Y))) ).

% xor_numerals(3)
tff(fact_8921_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_8922_xor__numerals_I7_J,axiom,
    ! [X: num,Y: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,X))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Y))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_num_nat(numeral_numeral_nat,X)),aa_num_nat(numeral_numeral_nat,Y))) ).

% xor_numerals(7)
tff(fact_8923_xor__numerals_I7_J,axiom,
    ! [X: num,Y: num] : aa_Cod1783974652533396728nteger(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(bit_se3222712562003087583nteger(aa_num_Code_integer(numera6620942414471956472nteger,X)),aa_num_Code_integer(numera6620942414471956472nteger,Y))) ).

% xor_numerals(7)
tff(fact_8924_xor__numerals_I7_J,axiom,
    ! [X: num,Y: num] : bit_se2046307713759805098atural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit1,X)),aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit1,Y))) = times_2397367101498566445atural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one)),bit_se2046307713759805098atural(aa_num_Code_natural(numera5444537566228673987atural,X),aa_num_Code_natural(numera5444537566228673987atural,Y))) ).

% xor_numerals(7)
tff(fact_8925_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_8926_divmod__integer_H__def,axiom,
    ! [M: num,N2: num] : unique3479559517661332726nteger(M,N2) = produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_num_Code_integer(numera6620942414471956472nteger,M)),aa_num_Code_integer(numera6620942414471956472nteger,N2)),aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_num_Code_integer(numera6620942414471956472nteger,M)),aa_num_Code_integer(numera6620942414471956472nteger,N2))) ).

% divmod_integer'_def
tff(fact_8927_exhaustive__integer_H_Ocases,axiom,
    ! [X: produc7509114972854644512nteger] :
      ~ ! [F2: fun_Co3460408610320954864e_term,D: code_integer,I3: code_integer] : X != produc7264323995540650962nteger(F2,produc1086072967326762835nteger(D,I3)) ).

% exhaustive_integer'.cases
tff(fact_8928_full__exhaustive__integer_H_Ocases,axiom,
    ! [X: produc2453742332425399446nteger] :
      ~ ! [F2: fun_Pr5098669352780829690e_term,D: code_integer,I3: code_integer] : X != produc565448451787809096nteger(F2,produc1086072967326762835nteger(D,I3)) ).

% full_exhaustive_integer'.cases
tff(fact_8929_image__add__integer__atLeastLessThan,axiom,
    ! [L: code_integer,U: code_integer] : image_4470545334726330049nteger(aTP_Lamm_hv(L),set_or8404916559141939852nteger(zero_z3403309356797280102nteger,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(U),L))) = set_or8404916559141939852nteger(L,U) ).

% image_add_integer_atLeastLessThan
tff(fact_8930_less__eq__integer__code_I1_J,axiom,
    aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),zero_z3403309356797280102nteger) ).

% less_eq_integer_code(1)
tff(fact_8931_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_8932_atLeastLessThanPlusOne__atLeastAtMost__integer,axiom,
    ! [L: code_integer,U: code_integer] : set_or8404916559141939852nteger(L,aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(U),one_one_Code_integer)) = set_or189985376899183464nteger(L,U) ).

% atLeastLessThanPlusOne_atLeastAtMost_integer
tff(fact_8933_less__eq__mask,axiom,
    ! [N2: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),aa_nat_nat(bit_se2002935070580805687sk_nat,N2)) ).

% less_eq_mask
tff(fact_8934_XOR__lower,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,zero_zero_int),aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,X),Y)) ) ) ).

% XOR_lower
tff(fact_8935_mask__nonnegative__int,axiom,
    ! [N2: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(bit_se2000444600071755411sk_int,N2)) ).

% mask_nonnegative_int
tff(fact_8936_not__mask__negative__int,axiom,
    ! [N2: nat] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(bit_se2000444600071755411sk_int,N2)),zero_zero_int) ).

% not_mask_negative_int
tff(fact_8937_less__mask,axiom,
    ! [N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),N2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),aa_nat_nat(bit_se2002935070580805687sk_nat,N2)) ) ).

% less_mask
tff(fact_8938_mask__nat__less__exp,axiom,
    ! [N2: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(bit_se2002935070580805687sk_nat,N2)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2)) ).

% mask_nat_less_exp
tff(fact_8939_finite__int__iff__bounded,axiom,
    ! [S: set_int] :
      ( aa_set_int_o(finite_finite_int,S)
    <=> ? [K3: int] : aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,image_int_int(abs_abs_int,S)),set_ord_lessThan_int(K3)) ) ).

% finite_int_iff_bounded
tff(fact_8940_XOR__upper,axiom,
    ! [X: int,N2: nat,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_int,X),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N2))
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Y),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N2))
         => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,X),Y)),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N2)) ) ) ) ).

% XOR_upper
tff(fact_8941_horner__sum__eq__sum,axiom,
    ! [F: fun_int_Code_integer,A: code_integer,Xs: list_int] : groups8594936613892548326nteger(F,A,Xs) = aa_set871585252387937313nteger(groups7501900531339628137nteger(aa_lis2169558292809893284nteger(aa_Cod4217156690935544446nteger(aTP_Lamm_aiu(F),A),Xs)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),size_size_list_int(Xs))) ).

% horner_sum_eq_sum
tff(fact_8942_horner__sum__eq__sum,axiom,
    ! [F: fun_nat_Code_integer,A: code_integer,Xs: list_nat] : groups8223283053655703690nteger(F,A,Xs) = aa_set871585252387937313nteger(groups7501900531339628137nteger(aa_lis7000467330633774408nteger(aa_Cod9048065728759425570nteger(aTP_Lamm_aiv(F),A),Xs)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),aa_list_nat_nat(size_size_list_nat,Xs))) ).

% horner_sum_eq_sum
tff(fact_8943_horner__sum__eq__sum,axiom,
    ! [F: fun_o_Code_integer,A: code_integer,Xs: list_o] : groups3417619833198082522nteger(F,A,Xs) = aa_set871585252387937313nteger(groups7501900531339628137nteger(aa_lis2656634407596002188nteger(aa_Cod901939952664962432nteger(aTP_Lamm_aiw(F),A),Xs)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),size_size_list_o(Xs))) ).

% horner_sum_eq_sum
tff(fact_8944_horner__sum__eq__sum,axiom,
    ! [F: fun_int_rat,A: rat,Xs: list_int] : groups7852591826665563233nt_rat(F,A,Xs) = aa_set_nat_rat(groups2906978787729119204at_rat(aa_lis1742918661435654559at_rat(aa_rat8348276666120063220at_rat(aTP_Lamm_aix(F),A),Xs)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),size_size_list_int(Xs))) ).

% horner_sum_eq_sum
tff(fact_8945_horner__sum__eq__sum,axiom,
    ! [F: fun_nat_rat,A: rat,Xs: list_nat] : aa_list_nat_rat(groups6853238114764508677at_rat(F,A),Xs) = aa_set_nat_rat(groups2906978787729119204at_rat(aa_lis7014846069874721091at_rat(aa_rat4396832037704353944at_rat(aTP_Lamm_aiy(F),A),Xs)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),aa_list_nat_nat(size_size_list_nat,Xs))) ).

% horner_sum_eq_sum
tff(fact_8946_horner__sum__eq__sum,axiom,
    ! [F: fun_o_rat,A: rat,Xs: list_o] : groups8483887719401441109_o_rat(F,A,Xs) = aa_set_nat_rat(groups2906978787729119204at_rat(aa_lis2932070769283852167at_rat(aa_rat5396404679109611328at_rat(aTP_Lamm_aiz(F),A),Xs)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),size_size_list_o(Xs))) ).

% horner_sum_eq_sum
tff(fact_8947_horner__sum__eq__sum,axiom,
    ! [F: fun_int_int,A: int,Xs: list_int] : groups8485231416243008693nt_int(F,A,Xs) = aa_set_nat_int(groups3539618377306564664at_int(aa_lis1073296200319854579at_int(aa_int2572695527019912092at_int(aTP_Lamm_aja(F),A),Xs)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),size_size_list_int(Xs))) ).

% horner_sum_eq_sum
tff(fact_8948_horner__sum__eq__sum,axiom,
    ! [F: fun_nat_int,A: int,Xs: list_nat] : groups7485877704341954137at_int(F,A,Xs) = aa_set_nat_int(groups3539618377306564664at_int(aa_lis6345223608758921111at_int(aa_int7844622935458978624at_int(aTP_Lamm_ajb(F),A),Xs)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),aa_list_nat_nat(size_size_list_nat,Xs))) ).

% horner_sum_eq_sum
tff(fact_8949_horner__sum__eq__sum,axiom,
    ! [F: fun_int_nat,A: nat,Xs: list_int] : groups8487721886752058969nt_nat(F,A,Xs) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_lis5251147219829051287at_nat(aa_nat3853594844976941284at_nat(aTP_Lamm_ajc(F),A),Xs)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),size_size_list_int(Xs))) ).

% horner_sum_eq_sum
tff(fact_8950_horner__sum__eq__sum,axiom,
    ! [F: fun_nat_nat,A: nat,Xs: list_nat] : aa_list_nat_nat(groups7488368174851004413at_nat(F,A),Xs) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_lis1299702591413342011at_nat(aa_nat9125522253416007816at_nat(aTP_Lamm_ajd(F),A),Xs)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),aa_list_nat_nat(size_size_list_nat,Xs))) ).

% horner_sum_eq_sum
tff(fact_8951_integer__of__int__code,axiom,
    ! [K: int] :
      aa_int_Code_integer(code_integer_of_int,K) = $ite(
        aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int),
        aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_int_Code_integer(code_integer_of_int,aa_int_int(uminus_uminus_int,K))),
        $ite(
          K = zero_zero_int,
          zero_z3403309356797280102nteger,
          $let(
            l: code_integer,
            l:= aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_int_Code_integer(code_integer_of_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))),
            $ite(aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = zero_zero_int,l,aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(l),one_one_Code_integer)) ) ) ) ).

% integer_of_int_code
tff(fact_8952_integer__of__num_I3_J,axiom,
    ! [N2: num] :
      code_integer_of_num(aa_num_num(bit1,N2)) = $let(
        k: code_integer,
        k:= code_integer_of_num(N2),
        aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(k),k)),one_one_Code_integer) ) ).

% integer_of_num(3)
tff(fact_8953_take__bit__num__code,axiom,
    ! [N2: nat,M: num] : aa_num_option_num(bit_take_bit_num(N2),M) = produc478579273971653890on_num(aTP_Lamm_ajh,product_Pair_nat_num(N2,M)) ).

% take_bit_num_code
tff(fact_8954_even__sum__iff,axiom,
    ! [A3: set_o,F: fun_o_nat] :
      ( finite_finite_o(A3)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),groups8507830703676809646_o_nat(F,A3))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),finite_card_o(collect_o(aa_fun_o_nat_fun_o_o(aTP_Lamm_aji(A3),F)))) ) ) ).

% even_sum_iff
tff(fact_8955_even__sum__iff,axiom,
    ! [A3: set_int,F: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,A3)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),groups4541462559716669496nt_nat(F,A3))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),finite_card_int(collect_int(aa_fun5425778157628386638_int_o(aTP_Lamm_ajj(A3),F)))) ) ) ).

% even_sum_iff
tff(fact_8956_even__sum__iff,axiom,
    ! [A3: set_Code_integer,F: fun_Code_integer_nat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A3)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),groups7237345082560585321er_nat(F,A3))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),finite4902975817058060853nteger(collect_Code_integer(aa_fun4054834855839773292eger_o(aTP_Lamm_ajk(A3),F)))) ) ) ).

% even_sum_iff
tff(fact_8957_even__sum__iff,axiom,
    ! [A3: set_o,F: fun_o_int] :
      ( finite_finite_o(A3)
     => ( 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,A3))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),finite_card_o(collect_o(aa_fun_o_int_fun_o_o(aTP_Lamm_ajl(A3),F)))) ) ) ).

% even_sum_iff
tff(fact_8958_even__sum__iff,axiom,
    ! [A3: set_nat,F: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( 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),A3))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_set_nat_nat(finite_card_nat,collect_nat(aa_fun2180201864171524266_nat_o(aTP_Lamm_ajm(A3),F)))) ) ) ).

% even_sum_iff
tff(fact_8959_even__sum__iff,axiom,
    ! [A3: set_Code_integer,F: fun_Code_integer_int] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A3)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),groups7234854612051535045er_int(F,A3))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),finite4902975817058060853nteger(collect_Code_integer(aa_fun3864128940607509264eger_o(aTP_Lamm_ajn(A3),F)))) ) ) ).

% even_sum_iff
tff(fact_8960_even__sum__iff,axiom,
    ! [A3: set_o,F: fun_o_Code_integer] :
      ( finite_finite_o(A3)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),groups4406642042086082107nteger(F,A3))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),finite_card_o(collect_o(aa_fun8192762245619933327un_o_o(aTP_Lamm_ajo(A3),F)))) ) ) ).

% even_sum_iff
tff(fact_8961_even__sum__iff,axiom,
    ! [A3: set_nat,F: fun_nat_Code_integer] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( 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),A3))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_set_nat_nat(finite_card_nat,collect_nat(aa_fun3480067038916266681_nat_o(aTP_Lamm_ajp(A3),F)))) ) ) ).

% even_sum_iff
tff(fact_8962_even__sum__iff,axiom,
    ! [A3: set_int,F: fun_int_Code_integer] :
      ( aa_set_int_o(finite_finite_int,A3)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),groups7873554091576472773nteger(F,A3))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),finite_card_int(collect_int(aa_fun3336537925265328129_int_o(aTP_Lamm_ajq(A3),F)))) ) ) ).

% even_sum_iff
tff(fact_8963_even__sum__iff,axiom,
    ! [A3: set_Code_integer,F: fun_Co8499238673658658103nteger] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A3)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),groups879477027807139574nteger(F,A3))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),finite4902975817058060853nteger(collect_Code_integer(aa_fun7159535342743012639eger_o(aTP_Lamm_ajr(A3),F)))) ) ) ).

% even_sum_iff
tff(fact_8964_card__Collect__less__nat,axiom,
    ! [N2: nat] : aa_set_nat_nat(finite_card_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_dg,N2))) = N2 ).

% card_Collect_less_nat
tff(fact_8965_finite__atLeastLessThan__integer,axiom,
    ! [L: code_integer,U: code_integer] : aa_set5643387332196575111eger_o(finite6017078050557962740nteger,set_or8404916559141939852nteger(L,U)) ).

% finite_atLeastLessThan_integer
tff(fact_8966_finite__atLeastAtMost__integer,axiom,
    ! [L: code_integer,U: code_integer] : aa_set5643387332196575111eger_o(finite6017078050557962740nteger,set_or189985376899183464nteger(L,U)) ).

% finite_atLeastAtMost_integer
tff(fact_8967_card__Collect__le__nat,axiom,
    ! [N2: nat] : aa_set_nat_nat(finite_card_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_df,N2))) = aa_nat_nat(suc,N2) ).

% card_Collect_le_nat
tff(fact_8968_card_Oempty,axiom,
    finite_card_set_nat(bot_bot_set_set_nat) = zero_zero_nat ).

% card.empty
tff(fact_8969_card_Oempty,axiom,
    finite_card_list_nat(bot_bot_set_list_nat) = zero_zero_nat ).

% card.empty
tff(fact_8970_card_Oempty,axiom,
    finite711546835091564841at_nat(bot_bo2099793752762293965at_nat) = zero_zero_nat ).

% card.empty
tff(fact_8971_card_Oempty,axiom,
    finite_card_o(bot_bot_set_o) = zero_zero_nat ).

% card.empty
tff(fact_8972_card_Oempty,axiom,
    aa_set_nat_nat(finite_card_nat,bot_bot_set_nat) = zero_zero_nat ).

% card.empty
tff(fact_8973_card_Oempty,axiom,
    finite_card_int(bot_bot_set_int) = zero_zero_nat ).

% card.empty
tff(fact_8974_prod__constant,axiom,
    ! [Y: nat,A3: set_set_nat] : groups4248547760180025341at_nat(aTP_Lamm_ajs(Y),A3) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Y),finite_card_set_nat(A3)) ).

% prod_constant
tff(fact_8975_prod__constant,axiom,
    ! [Y: nat,A3: set_list_nat] : groups2907647131375434839at_nat(aTP_Lamm_ajt(Y),A3) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Y),finite_card_list_nat(A3)) ).

% prod_constant
tff(fact_8976_prod__constant,axiom,
    ! [Y: int,A3: set_set_nat] : groups4246057289670975065at_int(aTP_Lamm_aju(Y),A3) = aa_nat_int(aa_int_fun_nat_int(power_power_int,Y),finite_card_set_nat(A3)) ).

% prod_constant
tff(fact_8977_prod__constant,axiom,
    ! [Y: int,A3: set_list_nat] : groups2905156660866384563at_int(aTP_Lamm_ajv(Y),A3) = aa_nat_int(aa_int_fun_nat_int(power_power_int,Y),finite_card_list_nat(A3)) ).

% prod_constant
tff(fact_8978_prod__constant,axiom,
    ! [Y: nat,A3: set_nat] : aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_ajw,Y)),A3) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Y),aa_set_nat_nat(finite_card_nat,A3)) ).

% prod_constant
tff(fact_8979_prod__constant,axiom,
    ! [Y: int,A3: set_nat] : aa_set_nat_int(groups705719431365010083at_int(aa_int_fun_nat_int(aTP_Lamm_ajx,Y)),A3) = aa_nat_int(aa_int_fun_nat_int(power_power_int,Y),aa_set_nat_nat(finite_card_nat,A3)) ).

% prod_constant
tff(fact_8980_prod__constant,axiom,
    ! [Y: int,A3: set_int] : groups1705073143266064639nt_int(aa_int_fun_int_int(aTP_Lamm_ajy,Y),A3) = aa_nat_int(aa_int_fun_nat_int(power_power_int,Y),finite_card_int(A3)) ).

% prod_constant
tff(fact_8981_case__nat__numeral,axiom,
    ! [A: option_num,F: fun_nat_option_num,V: num] : aa_nat_option_num(case_nat_option_num(A,F),aa_num_nat(numeral_numeral_nat,V)) = aa_nat_option_num(F,aa_num_nat(pred_numeral,V)) ).

% case_nat_numeral
tff(fact_8982_case__nat__numeral,axiom,
    ! [A: $o,F: fun_nat_o,V: num] :
      ( aa_nat_o(case_nat_o((A),F),aa_num_nat(numeral_numeral_nat,V))
    <=> aa_nat_o(F,aa_num_nat(pred_numeral,V)) ) ).

% case_nat_numeral
tff(fact_8983_case__nat__numeral,axiom,
    ! [A: nat,F: fun_nat_nat,V: num] : aa_nat_nat(case_nat_nat(A,F),aa_num_nat(numeral_numeral_nat,V)) = aa_nat_nat(F,aa_num_nat(pred_numeral,V)) ).

% case_nat_numeral
tff(fact_8984_card__0__eq,axiom,
    ! [A3: set_set_nat] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,A3)
     => ( ( finite_card_set_nat(A3) = zero_zero_nat )
      <=> ( A3 = bot_bot_set_set_nat ) ) ) ).

% card_0_eq
tff(fact_8985_card__0__eq,axiom,
    ! [A3: set_list_nat] :
      ( finite8100373058378681591st_nat(A3)
     => ( ( finite_card_list_nat(A3) = zero_zero_nat )
      <=> ( A3 = bot_bot_set_list_nat ) ) ) ).

% card_0_eq
tff(fact_8986_card__0__eq,axiom,
    ! [A3: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A3)
     => ( ( finite4902975817058060853nteger(A3) = zero_zero_nat )
      <=> ( A3 = bot_bo3990330152332043303nteger ) ) ) ).

% card_0_eq
tff(fact_8987_card__0__eq,axiom,
    ! [A3: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A3)
     => ( ( finite711546835091564841at_nat(A3) = zero_zero_nat )
      <=> ( A3 = bot_bo2099793752762293965at_nat ) ) ) ).

% card_0_eq
tff(fact_8988_card__0__eq,axiom,
    ! [A3: set_o] :
      ( finite_finite_o(A3)
     => ( ( finite_card_o(A3) = zero_zero_nat )
      <=> ( A3 = bot_bot_set_o ) ) ) ).

% card_0_eq
tff(fact_8989_card__0__eq,axiom,
    ! [A3: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( ( aa_set_nat_nat(finite_card_nat,A3) = zero_zero_nat )
      <=> ( A3 = bot_bot_set_nat ) ) ) ).

% card_0_eq
tff(fact_8990_card__0__eq,axiom,
    ! [A3: set_int] :
      ( aa_set_int_o(finite_finite_int,A3)
     => ( ( finite_card_int(A3) = zero_zero_nat )
      <=> ( A3 = bot_bot_set_int ) ) ) ).

% card_0_eq
tff(fact_8991_sum__constant,axiom,
    ! [Y: rat,A3: set_nat] : aa_set_nat_rat(groups2906978787729119204at_rat(aTP_Lamm_ajz(Y)),A3) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,aa_set_nat_nat(finite_card_nat,A3))),Y) ).

% sum_constant
tff(fact_8992_sum__constant,axiom,
    ! [Y: int,A3: set_nat] : aa_set_nat_int(groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_ajx,Y)),A3) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,aa_set_nat_nat(finite_card_nat,A3))),Y) ).

% sum_constant
tff(fact_8993_sum__constant,axiom,
    ! [Y: code_integer,A3: set_nat] : aa_set871585252387937313nteger(groups7501900531339628137nteger(aTP_Lamm_aka(Y)),A3) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,aa_set_nat_nat(finite_card_nat,A3))),Y) ).

% sum_constant
tff(fact_8994_sum__constant,axiom,
    ! [Y: nat,A3: set_nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_ajw,Y)),A3) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,aa_set_nat_nat(finite_card_nat,A3))),Y) ).

% sum_constant
tff(fact_8995_sum__constant,axiom,
    ! [Y: int,A3: set_int] : groups4538972089207619220nt_int(aa_int_fun_int_int(aTP_Lamm_ajy,Y),A3) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,finite_card_int(A3))),Y) ).

% sum_constant
tff(fact_8996_sum__constant,axiom,
    ! [Y: rat,A3: set_set_nat] : groups7659867448343625626at_rat(aTP_Lamm_akb(Y),A3) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,finite_card_set_nat(A3))),Y) ).

% sum_constant
tff(fact_8997_sum__constant,axiom,
    ! [Y: rat,A3: set_list_nat] : groups3760926236672600436at_rat(aTP_Lamm_akc(Y),A3) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,finite_card_list_nat(A3))),Y) ).

% sum_constant
tff(fact_8998_sum__constant,axiom,
    ! [Y: int,A3: set_set_nat] : groups8292507037921071086at_int(aTP_Lamm_aju(Y),A3) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,finite_card_set_nat(A3))),Y) ).

% sum_constant
tff(fact_8999_sum__constant,axiom,
    ! [Y: int,A3: set_list_nat] : groups4393565826250045896at_int(aTP_Lamm_ajv(Y),A3) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,finite_card_list_nat(A3))),Y) ).

% sum_constant
tff(fact_9000_sum__constant,axiom,
    ! [Y: nat,A3: set_set_nat] : groups8294997508430121362at_nat(aTP_Lamm_ajs(Y),A3) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,finite_card_set_nat(A3))),Y) ).

% sum_constant
tff(fact_9001_case__nat__add__eq__if,axiom,
    ! [A: option_num,F: fun_nat_option_num,V: num,N2: nat] : aa_nat_option_num(case_nat_option_num(A,F),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,V)),N2)) = aa_nat_option_num(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(pred_numeral,V)),N2)) ).

% case_nat_add_eq_if
tff(fact_9002_case__nat__add__eq__if,axiom,
    ! [A: $o,F: fun_nat_o,V: num,N2: nat] :
      ( aa_nat_o(case_nat_o((A),F),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,V)),N2))
    <=> aa_nat_o(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(pred_numeral,V)),N2)) ) ).

% case_nat_add_eq_if
tff(fact_9003_case__nat__add__eq__if,axiom,
    ! [A: nat,F: fun_nat_nat,V: num,N2: nat] : aa_nat_nat(case_nat_nat(A,F),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,V)),N2)) = aa_nat_nat(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(pred_numeral,V)),N2)) ).

% case_nat_add_eq_if
tff(fact_9004_sum__of__bool__eq,axiom,
    ! [A3: set_int,Pa: fun_int_o] :
      ( aa_set_int_o(finite_finite_int,A3)
     => ( aa_set_int_o(finite_finite_int,A3)
       => ( groups7873554091576472773nteger(aTP_Lamm_akd(Pa),A3) = aa_nat_Code_integer(semiri4939895301339042750nteger,finite_card_int(aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A3),collect_int(Pa)))) ) ) ) ).

% sum_of_bool_eq
tff(fact_9005_sum__of__bool__eq,axiom,
    ! [A3: set_Code_integer,Pa: fun_Code_integer_o] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A3)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A3)
       => ( groups879477027807139574nteger(aTP_Lamm_ake(Pa),A3) = aa_nat_Code_integer(semiri4939895301339042750nteger,finite4902975817058060853nteger(inf_in1364745209274528805nteger(A3,collect_Code_integer(Pa)))) ) ) ) ).

% sum_of_bool_eq
tff(fact_9006_sum__of__bool__eq,axiom,
    ! [A3: set_nat,Pa: fun_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( aa_set_nat_o(finite_finite_nat,A3)
       => ( aa_set871585252387937313nteger(groups7501900531339628137nteger(aTP_Lamm_akf(Pa)),A3) = aa_nat_Code_integer(semiri4939895301339042750nteger,aa_set_nat_nat(finite_card_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A3),collect_nat(Pa)))) ) ) ) ).

% sum_of_bool_eq
tff(fact_9007_sum__of__bool__eq,axiom,
    ! [A3: set_int,Pa: fun_int_o] :
      ( aa_set_int_o(finite_finite_int,A3)
     => ( aa_set_int_o(finite_finite_int,A3)
       => ( groups4541462559716669496nt_nat(aTP_Lamm_akg(Pa),A3) = aa_nat_nat(semiri1316708129612266289at_nat,finite_card_int(aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A3),collect_int(Pa)))) ) ) ) ).

% sum_of_bool_eq
tff(fact_9008_sum__of__bool__eq,axiom,
    ! [A3: set_Code_integer,Pa: fun_Code_integer_o] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A3)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A3)
       => ( groups7237345082560585321er_nat(aTP_Lamm_akh(Pa),A3) = aa_nat_nat(semiri1316708129612266289at_nat,finite4902975817058060853nteger(inf_in1364745209274528805nteger(A3,collect_Code_integer(Pa)))) ) ) ) ).

% sum_of_bool_eq
tff(fact_9009_sum__of__bool__eq,axiom,
    ! [A3: set_Code_integer,Pa: fun_Code_integer_o] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A3)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A3)
       => ( groups7234854612051535045er_int(aTP_Lamm_aki(Pa),A3) = aa_nat_int(semiri1314217659103216013at_int,finite4902975817058060853nteger(inf_in1364745209274528805nteger(A3,collect_Code_integer(Pa)))) ) ) ) ).

% sum_of_bool_eq
tff(fact_9010_sum__of__bool__eq,axiom,
    ! [A3: set_nat,Pa: fun_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( aa_set_nat_o(finite_finite_nat,A3)
       => ( aa_set_nat_int(groups3539618377306564664at_int(aTP_Lamm_akj(Pa)),A3) = aa_nat_int(semiri1314217659103216013at_int,aa_set_nat_nat(finite_card_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A3),collect_nat(Pa)))) ) ) ) ).

% sum_of_bool_eq
tff(fact_9011_sum__of__bool__eq,axiom,
    ! [A3: set_nat,Pa: fun_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( aa_set_nat_o(finite_finite_nat,A3)
       => ( aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_akk(Pa)),A3) = aa_nat_nat(semiri1316708129612266289at_nat,aa_set_nat_nat(finite_card_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A3),collect_nat(Pa)))) ) ) ) ).

% sum_of_bool_eq
tff(fact_9012_sum__of__bool__eq,axiom,
    ! [A3: set_int,Pa: fun_int_o] :
      ( aa_set_int_o(finite_finite_int,A3)
     => ( aa_set_int_o(finite_finite_int,A3)
       => ( groups4538972089207619220nt_int(aTP_Lamm_akl(Pa),A3) = aa_nat_int(semiri1314217659103216013at_int,finite_card_int(aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A3),collect_int(Pa)))) ) ) ) ).

% sum_of_bool_eq
tff(fact_9013_sum__of__bool__eq,axiom,
    ! [A3: set_set_nat,Pa: fun_set_nat_o] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,A3)
     => ( aa_set_set_nat_o(finite1152437895449049373et_nat,A3)
       => ( groups9190459664516455967nteger(aTP_Lamm_akm(Pa),A3) = aa_nat_Code_integer(semiri4939895301339042750nteger,finite_card_set_nat(inf_inf_set_set_nat(A3,collect_set_nat(Pa)))) ) ) ) ).

% sum_of_bool_eq
tff(fact_9014_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_9015_less__integer__code_I1_J,axiom,
    ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),zero_z3403309356797280102nteger) ).

% less_integer_code(1)
tff(fact_9016_less__integer_Oabs__eq,axiom,
    ! [Xa2: int,X: int] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_int_Code_integer(code_integer_of_int,Xa2)),aa_int_Code_integer(code_integer_of_int,X))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Xa2),X) ) ).

% less_integer.abs_eq
tff(fact_9017_finite__atLeastZeroLessThan__integer,axiom,
    ! [U: code_integer] : aa_set5643387332196575111eger_o(finite6017078050557962740nteger,set_or8404916559141939852nteger(zero_z3403309356797280102nteger,U)) ).

% finite_atLeastZeroLessThan_integer
tff(fact_9018_nat_Ocase__distrib,axiom,
    ! [Ha: fun_op8989173174992673105on_num,F1: option_num,F22: fun_nat_option_num,Nat: nat] : aa_opt5369159329878751634on_num(Ha,aa_nat_option_num(case_nat_option_num(F1,F22),Nat)) = aa_nat_option_num(case_nat_option_num(aa_opt5369159329878751634on_num(Ha,F1),aa_fun7716400287237121964on_num(aTP_Lamm_akn(Ha),F22)),Nat) ).

% nat.case_distrib
tff(fact_9019_nat_Ocase__distrib,axiom,
    ! [Ha: fun_option_num_o,F1: option_num,F22: fun_nat_option_num,Nat: nat] :
      ( aa_option_num_o(Ha,aa_nat_option_num(case_nat_option_num(F1,F22),Nat))
    <=> aa_nat_o(case_nat_o(aa_option_num_o(Ha,F1),aa_fun1675964276245824108_nat_o(aTP_Lamm_ako(Ha),F22)),Nat) ) ).

% nat.case_distrib
tff(fact_9020_nat_Ocase__distrib,axiom,
    ! [Ha: fun_option_num_nat,F1: option_num,F22: fun_nat_option_num,Nat: nat] : aa_option_num_nat(Ha,aa_nat_option_num(case_nat_option_num(F1,F22),Nat)) = aa_nat_nat(case_nat_nat(aa_option_num_nat(Ha,F1),aa_fun7387201323300553874at_nat(aTP_Lamm_akp(Ha),F22)),Nat) ).

% nat.case_distrib
tff(fact_9021_nat_Ocase__distrib,axiom,
    ! [Ha: fun_o_option_num,F1: $o,F22: fun_nat_o,Nat: nat] : aa_o_option_num(Ha,aa_nat_o(case_nat_o((F1),F22),Nat)) = aa_nat_option_num(case_nat_option_num(aa_o_option_num(Ha,(F1)),aa_fun6990899178391151900on_num(aTP_Lamm_akq(Ha),F22)),Nat) ).

% nat.case_distrib
tff(fact_9022_nat_Ocase__distrib,axiom,
    ! [Ha: fun_o_o,F1: $o,F22: fun_nat_o,Nat: nat] :
      ( aa_o_o(Ha,aa_nat_o(case_nat_o((F1),F22),Nat))
    <=> aa_nat_o(case_nat_o(aa_o_o(Ha,(F1)),aa_fun5937193402907943164_nat_o(aTP_Lamm_akr(Ha),F22)),Nat) ) ).

% nat.case_distrib
tff(fact_9023_nat_Ocase__distrib,axiom,
    ! [Ha: fun_o_nat,F1: $o,F22: fun_nat_o,Nat: nat] : aa_o_nat(Ha,aa_nat_o(case_nat_o((F1),F22),Nat)) = aa_nat_nat(case_nat_nat(aa_o_nat(Ha,(F1)),aa_fun4204860950123276802at_nat(aTP_Lamm_aks(Ha),F22)),Nat) ).

% nat.case_distrib
tff(fact_9024_nat_Ocase__distrib,axiom,
    ! [Ha: fun_nat_option_num,F1: nat,F22: fun_nat_nat,Nat: nat] : aa_nat_option_num(Ha,aa_nat_nat(case_nat_nat(F1,F22),Nat)) = aa_nat_option_num(case_nat_option_num(aa_nat_option_num(Ha,F1),aa_fun6517009837249254290on_num(aTP_Lamm_akt(Ha),F22)),Nat) ).

% nat.case_distrib
tff(fact_9025_nat_Ocase__distrib,axiom,
    ! [Ha: fun_nat_o,F1: nat,F22: fun_nat_nat,Nat: nat] :
      ( aa_nat_o(Ha,aa_nat_nat(case_nat_nat(F1,F22),Nat))
    <=> aa_nat_o(case_nat_o(aa_nat_o(Ha,F1),aa_fun8673969331427483654_nat_o(aTP_Lamm_aku(Ha),F22)),Nat) ) ).

% nat.case_distrib
tff(fact_9026_nat_Ocase__distrib,axiom,
    ! [Ha: fun_nat_nat,F1: nat,F22: fun_nat_nat,Nat: nat] : aa_nat_nat(Ha,aa_nat_nat(case_nat_nat(F1,F22),Nat)) = aa_nat_nat(case_nat_nat(aa_nat_nat(Ha,F1),aa_fun5791449513213591928at_nat(aTP_Lamm_il(Ha),F22)),Nat) ).

% nat.case_distrib
tff(fact_9027_num_Ocase__distrib,axiom,
    ! [Ha: fun_op8989173174992673105on_num,F1: option_num,F22: fun_num_option_num,F32: fun_num_option_num,Num: num] : aa_opt5369159329878751634on_num(Ha,aa_num_option_num(case_num_option_num(F1,F22,F32),Num)) = aa_num_option_num(case_num_option_num(aa_opt5369159329878751634on_num(Ha,F1),aa_fun3636977833914307392on_num(aTP_Lamm_dz(Ha),F22),aa_fun3636977833914307392on_num(aTP_Lamm_dz(Ha),F32)),Num) ).

% num.case_distrib
tff(fact_9028_n__subsets,axiom,
    ! [A3: set_set_nat,K: nat] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,A3)
     => ( finite1149291290879098388et_nat(collect_set_set_nat(aa_nat5160323483289836711_nat_o(aTP_Lamm_akv(A3),K))) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,finite_card_set_nat(A3)),K) ) ) ).

% n_subsets
tff(fact_9029_n__subsets,axiom,
    ! [A3: set_list_nat,K: nat] :
      ( finite8100373058378681591st_nat(A3)
     => ( finite2364142230527598318st_nat(collect_set_list_nat(aa_nat7721107121239158477_nat_o(aTP_Lamm_akw(A3),K))) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,finite_card_list_nat(A3)),K) ) ) ).

% n_subsets
tff(fact_9030_n__subsets,axiom,
    ! [A3: set_nat,K: nat] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( finite_card_set_nat(collect_set_nat(aa_nat_fun_set_nat_o(aTP_Lamm_akx(A3),K))) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_set_nat_nat(finite_card_nat,A3)),K) ) ) ).

% n_subsets
tff(fact_9031_n__subsets,axiom,
    ! [A3: set_Code_integer,K: nat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A3)
     => ( finite1340570857867686763nteger(collec574505750873337192nteger(aa_nat2915507816091602832eger_o(aTP_Lamm_aky(A3),K))) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,finite4902975817058060853nteger(A3)),K) ) ) ).

% n_subsets
tff(fact_9032_n__subsets,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,K: nat] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A3)
     => ( finite4356350796350151305at_nat(collec5514110066124741708at_nat(aa_nat1727191234135941576_nat_o(aTP_Lamm_akz(A3),K))) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,finite711546835091564841at_nat(A3)),K) ) ) ).

% n_subsets
tff(fact_9033_n__subsets,axiom,
    ! [A3: set_int,K: nat] :
      ( aa_set_int_o(finite_finite_int,A3)
     => ( finite_card_set_int(collect_set_int(aa_nat_fun_set_int_o(aTP_Lamm_ala(A3),K))) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,finite_card_int(A3)),K) ) ) ).

% n_subsets
tff(fact_9034_old_Onat_Osimps_I4_J,axiom,
    ! [F1: option_num,F22: fun_nat_option_num] : aa_nat_option_num(case_nat_option_num(F1,F22),zero_zero_nat) = F1 ).

% old.nat.simps(4)
tff(fact_9035_old_Onat_Osimps_I4_J,axiom,
    ! [F1: $o,F22: fun_nat_o] :
      ( aa_nat_o(case_nat_o((F1),F22),zero_zero_nat)
    <=> (F1) ) ).

% old.nat.simps(4)
tff(fact_9036_old_Onat_Osimps_I4_J,axiom,
    ! [F1: nat,F22: fun_nat_nat] : aa_nat_nat(case_nat_nat(F1,F22),zero_zero_nat) = F1 ).

% old.nat.simps(4)
tff(fact_9037_old_Onat_Osimps_I5_J,axiom,
    ! [F1: option_num,F22: fun_nat_option_num,X2: nat] : aa_nat_option_num(case_nat_option_num(F1,F22),aa_nat_nat(suc,X2)) = aa_nat_option_num(F22,X2) ).

% old.nat.simps(5)
tff(fact_9038_old_Onat_Osimps_I5_J,axiom,
    ! [F1: $o,F22: fun_nat_o,X2: nat] :
      ( aa_nat_o(case_nat_o((F1),F22),aa_nat_nat(suc,X2))
    <=> aa_nat_o(F22,X2) ) ).

% old.nat.simps(5)
tff(fact_9039_old_Onat_Osimps_I5_J,axiom,
    ! [F1: nat,F22: fun_nat_nat,X2: nat] : aa_nat_nat(case_nat_nat(F1,F22),aa_nat_nat(suc,X2)) = aa_nat_nat(F22,X2) ).

% old.nat.simps(5)
tff(fact_9040_infinite__arbitrarily__large,axiom,
    ! [A3: set_set_nat,N2: nat] :
      ( ~ aa_set_set_nat_o(finite1152437895449049373et_nat,A3)
     => ? [B9: set_set_nat] :
          ( aa_set_set_nat_o(finite1152437895449049373et_nat,B9)
          & ( finite_card_set_nat(B9) = N2 )
          & aa_set_set_nat_o(ord_le6893508408891458716et_nat(B9),A3) ) ) ).

% infinite_arbitrarily_large
tff(fact_9041_infinite__arbitrarily__large,axiom,
    ! [A3: set_list_nat,N2: nat] :
      ( ~ finite8100373058378681591st_nat(A3)
     => ? [B9: set_list_nat] :
          ( finite8100373058378681591st_nat(B9)
          & ( finite_card_list_nat(B9) = N2 )
          & ord_le6045566169113846134st_nat(B9,A3) ) ) ).

% infinite_arbitrarily_large
tff(fact_9042_infinite__arbitrarily__large,axiom,
    ! [A3: set_nat,N2: nat] :
      ( ~ aa_set_nat_o(finite_finite_nat,A3)
     => ? [B9: set_nat] :
          ( aa_set_nat_o(finite_finite_nat,B9)
          & ( aa_set_nat_nat(finite_card_nat,B9) = N2 )
          & aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B9),A3) ) ) ).

% infinite_arbitrarily_large
tff(fact_9043_infinite__arbitrarily__large,axiom,
    ! [A3: set_Code_integer,N2: nat] :
      ( ~ aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A3)
     => ? [B9: set_Code_integer] :
          ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B9)
          & ( finite4902975817058060853nteger(B9) = N2 )
          & aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(B9),A3) ) ) ).

% infinite_arbitrarily_large
tff(fact_9044_infinite__arbitrarily__large,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,N2: nat] :
      ( ~ aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A3)
     => ? [B9: set_Pr1261947904930325089at_nat] :
          ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,B9)
          & ( finite711546835091564841at_nat(B9) = N2 )
          & aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,B9),A3) ) ) ).

% infinite_arbitrarily_large
tff(fact_9045_infinite__arbitrarily__large,axiom,
    ! [A3: set_int,N2: nat] :
      ( ~ aa_set_int_o(finite_finite_int,A3)
     => ? [B9: set_int] :
          ( aa_set_int_o(finite_finite_int,B9)
          & ( finite_card_int(B9) = N2 )
          & aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,B9),A3) ) ) ).

% infinite_arbitrarily_large
tff(fact_9046_card__subset__eq,axiom,
    ! [B3: set_set_nat,A3: set_set_nat] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,B3)
     => ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(A3),B3)
       => ( ( finite_card_set_nat(A3) = finite_card_set_nat(B3) )
         => ( A3 = B3 ) ) ) ) ).

% card_subset_eq
tff(fact_9047_card__subset__eq,axiom,
    ! [B3: set_list_nat,A3: set_list_nat] :
      ( finite8100373058378681591st_nat(B3)
     => ( ord_le6045566169113846134st_nat(A3,B3)
       => ( ( finite_card_list_nat(A3) = finite_card_list_nat(B3) )
         => ( A3 = B3 ) ) ) ) ).

% card_subset_eq
tff(fact_9048_card__subset__eq,axiom,
    ! [B3: set_nat,A3: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,B3)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A3),B3)
       => ( ( aa_set_nat_nat(finite_card_nat,A3) = aa_set_nat_nat(finite_card_nat,B3) )
         => ( A3 = B3 ) ) ) ) ).

% card_subset_eq
tff(fact_9049_card__subset__eq,axiom,
    ! [B3: set_Code_integer,A3: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B3)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(A3),B3)
       => ( ( finite4902975817058060853nteger(A3) = finite4902975817058060853nteger(B3) )
         => ( A3 = B3 ) ) ) ) ).

% card_subset_eq
tff(fact_9050_card__subset__eq,axiom,
    ! [B3: set_Pr1261947904930325089at_nat,A3: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,B3)
     => ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A3),B3)
       => ( ( finite711546835091564841at_nat(A3) = finite711546835091564841at_nat(B3) )
         => ( A3 = B3 ) ) ) ) ).

% card_subset_eq
tff(fact_9051_card__subset__eq,axiom,
    ! [B3: set_int,A3: set_int] :
      ( aa_set_int_o(finite_finite_int,B3)
     => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A3),B3)
       => ( ( finite_card_int(A3) = finite_card_int(B3) )
         => ( A3 = B3 ) ) ) ) ).

% card_subset_eq
tff(fact_9052_card__le__if__inj__on__rel,axiom,
    ! [B3: set_o,A3: set_o,Ra2: fun_o_fun_o_o] :
      ( finite_finite_o(B3)
     => ( ! [A4: $o] :
            ( member_o((A4),A3)
           => ? [B10: $o] :
                ( member_o((B10),B3)
                & aa_o_o(aa_o_fun_o_o(Ra2,(A4)),(B10)) ) )
       => ( ! [A12: $o,A23: $o,B4: $o] :
              ( member_o((A12),A3)
             => ( member_o((A23),A3)
               => ( member_o((B4),B3)
                 => ( aa_o_o(aa_o_fun_o_o(Ra2,(A12)),(B4))
                   => ( aa_o_o(aa_o_fun_o_o(Ra2,(A23)),(B4))
                     => ( (A12)
                      <=> (A23) ) ) ) ) ) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_o(A3)),finite_card_o(B3)) ) ) ) ).

% card_le_if_inj_on_rel
tff(fact_9053_card__le__if__inj__on__rel,axiom,
    ! [B3: set_o,A3: set_int,Ra2: fun_int_fun_o_o] :
      ( finite_finite_o(B3)
     => ( ! [A4: int] :
            ( aa_set_int_o(member_int(A4),A3)
           => ? [B10: $o] :
                ( member_o((B10),B3)
                & aa_o_o(aa_int_fun_o_o(Ra2,A4),(B10)) ) )
       => ( ! [A12: int,A23: int,B4: $o] :
              ( aa_set_int_o(member_int(A12),A3)
             => ( aa_set_int_o(member_int(A23),A3)
               => ( member_o((B4),B3)
                 => ( aa_o_o(aa_int_fun_o_o(Ra2,A12),(B4))
                   => ( aa_o_o(aa_int_fun_o_o(Ra2,A23),(B4))
                     => ( A12 = A23 ) ) ) ) ) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_int(A3)),finite_card_o(B3)) ) ) ) ).

% card_le_if_inj_on_rel
tff(fact_9054_card__le__if__inj__on__rel,axiom,
    ! [B3: set_o,A3: set_nat,Ra2: fun_nat_fun_o_o] :
      ( finite_finite_o(B3)
     => ( ! [A4: nat] :
            ( aa_set_nat_o(member_nat(A4),A3)
           => ? [B10: $o] :
                ( member_o((B10),B3)
                & aa_o_o(aa_nat_fun_o_o(Ra2,A4),(B10)) ) )
       => ( ! [A12: nat,A23: nat,B4: $o] :
              ( aa_set_nat_o(member_nat(A12),A3)
             => ( aa_set_nat_o(member_nat(A23),A3)
               => ( member_o((B4),B3)
                 => ( aa_o_o(aa_nat_fun_o_o(Ra2,A12),(B4))
                   => ( aa_o_o(aa_nat_fun_o_o(Ra2,A23),(B4))
                     => ( A12 = A23 ) ) ) ) ) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(finite_card_nat,A3)),finite_card_o(B3)) ) ) ) ).

% card_le_if_inj_on_rel
tff(fact_9055_card__le__if__inj__on__rel,axiom,
    ! [B3: set_nat,A3: set_o,Ra2: fun_o_fun_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,B3)
     => ( ! [A4: $o] :
            ( member_o((A4),A3)
           => ? [B10: nat] :
                ( aa_set_nat_o(member_nat(B10),B3)
                & aa_nat_o(aa_o_fun_nat_o(Ra2,(A4)),B10) ) )
       => ( ! [A12: $o,A23: $o,B4: nat] :
              ( member_o((A12),A3)
             => ( member_o((A23),A3)
               => ( aa_set_nat_o(member_nat(B4),B3)
                 => ( aa_nat_o(aa_o_fun_nat_o(Ra2,(A12)),B4)
                   => ( aa_nat_o(aa_o_fun_nat_o(Ra2,(A23)),B4)
                     => ( (A12)
                      <=> (A23) ) ) ) ) ) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_o(A3)),aa_set_nat_nat(finite_card_nat,B3)) ) ) ) ).

% card_le_if_inj_on_rel
tff(fact_9056_card__le__if__inj__on__rel,axiom,
    ! [B3: set_nat,A3: set_int,Ra2: fun_int_fun_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,B3)
     => ( ! [A4: int] :
            ( aa_set_int_o(member_int(A4),A3)
           => ? [B10: nat] :
                ( aa_set_nat_o(member_nat(B10),B3)
                & aa_nat_o(aa_int_fun_nat_o(Ra2,A4),B10) ) )
       => ( ! [A12: int,A23: int,B4: nat] :
              ( aa_set_int_o(member_int(A12),A3)
             => ( aa_set_int_o(member_int(A23),A3)
               => ( aa_set_nat_o(member_nat(B4),B3)
                 => ( aa_nat_o(aa_int_fun_nat_o(Ra2,A12),B4)
                   => ( aa_nat_o(aa_int_fun_nat_o(Ra2,A23),B4)
                     => ( A12 = A23 ) ) ) ) ) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_int(A3)),aa_set_nat_nat(finite_card_nat,B3)) ) ) ) ).

% card_le_if_inj_on_rel
tff(fact_9057_card__le__if__inj__on__rel,axiom,
    ! [B3: set_nat,A3: set_nat,Ra2: fun_nat_fun_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,B3)
     => ( ! [A4: nat] :
            ( aa_set_nat_o(member_nat(A4),A3)
           => ? [B10: nat] :
                ( aa_set_nat_o(member_nat(B10),B3)
                & aa_nat_o(aa_nat_fun_nat_o(Ra2,A4),B10) ) )
       => ( ! [A12: nat,A23: nat,B4: nat] :
              ( aa_set_nat_o(member_nat(A12),A3)
             => ( aa_set_nat_o(member_nat(A23),A3)
               => ( aa_set_nat_o(member_nat(B4),B3)
                 => ( aa_nat_o(aa_nat_fun_nat_o(Ra2,A12),B4)
                   => ( aa_nat_o(aa_nat_fun_nat_o(Ra2,A23),B4)
                     => ( A12 = A23 ) ) ) ) ) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(finite_card_nat,A3)),aa_set_nat_nat(finite_card_nat,B3)) ) ) ) ).

% card_le_if_inj_on_rel
tff(fact_9058_card__le__if__inj__on__rel,axiom,
    ! [B3: set_int,A3: set_o,Ra2: fun_o_fun_int_o] :
      ( aa_set_int_o(finite_finite_int,B3)
     => ( ! [A4: $o] :
            ( member_o((A4),A3)
           => ? [B10: int] :
                ( aa_set_int_o(member_int(B10),B3)
                & aa_int_o(aa_o_fun_int_o(Ra2,(A4)),B10) ) )
       => ( ! [A12: $o,A23: $o,B4: int] :
              ( member_o((A12),A3)
             => ( member_o((A23),A3)
               => ( aa_set_int_o(member_int(B4),B3)
                 => ( aa_int_o(aa_o_fun_int_o(Ra2,(A12)),B4)
                   => ( aa_int_o(aa_o_fun_int_o(Ra2,(A23)),B4)
                     => ( (A12)
                      <=> (A23) ) ) ) ) ) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_o(A3)),finite_card_int(B3)) ) ) ) ).

% card_le_if_inj_on_rel
tff(fact_9059_card__le__if__inj__on__rel,axiom,
    ! [B3: set_int,A3: set_int,Ra2: fun_int_fun_int_o] :
      ( aa_set_int_o(finite_finite_int,B3)
     => ( ! [A4: int] :
            ( aa_set_int_o(member_int(A4),A3)
           => ? [B10: int] :
                ( aa_set_int_o(member_int(B10),B3)
                & aa_int_o(aa_int_fun_int_o(Ra2,A4),B10) ) )
       => ( ! [A12: int,A23: int,B4: int] :
              ( aa_set_int_o(member_int(A12),A3)
             => ( aa_set_int_o(member_int(A23),A3)
               => ( aa_set_int_o(member_int(B4),B3)
                 => ( aa_int_o(aa_int_fun_int_o(Ra2,A12),B4)
                   => ( aa_int_o(aa_int_fun_int_o(Ra2,A23),B4)
                     => ( A12 = A23 ) ) ) ) ) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_int(A3)),finite_card_int(B3)) ) ) ) ).

% card_le_if_inj_on_rel
tff(fact_9060_card__le__if__inj__on__rel,axiom,
    ! [B3: set_int,A3: set_nat,Ra2: fun_nat_fun_int_o] :
      ( aa_set_int_o(finite_finite_int,B3)
     => ( ! [A4: nat] :
            ( aa_set_nat_o(member_nat(A4),A3)
           => ? [B10: int] :
                ( aa_set_int_o(member_int(B10),B3)
                & aa_int_o(aa_nat_fun_int_o(Ra2,A4),B10) ) )
       => ( ! [A12: nat,A23: nat,B4: int] :
              ( aa_set_nat_o(member_nat(A12),A3)
             => ( aa_set_nat_o(member_nat(A23),A3)
               => ( aa_set_int_o(member_int(B4),B3)
                 => ( aa_int_o(aa_nat_fun_int_o(Ra2,A12),B4)
                   => ( aa_int_o(aa_nat_fun_int_o(Ra2,A23),B4)
                     => ( A12 = A23 ) ) ) ) ) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(finite_card_nat,A3)),finite_card_int(B3)) ) ) ) ).

% card_le_if_inj_on_rel
tff(fact_9061_card__le__if__inj__on__rel,axiom,
    ! [B3: set_Code_integer,A3: set_o,Ra2: fun_o_4021313873839786239eger_o] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B3)
     => ( ! [A4: $o] :
            ( member_o((A4),A3)
           => ? [B10: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(B10),B3)
                & aa_Code_integer_o(aa_o_f8517747772616379256eger_o(Ra2,(A4)),B10) ) )
       => ( ! [A12: $o,A23: $o,B4: code_integer] :
              ( member_o((A12),A3)
             => ( member_o((A23),A3)
               => ( aa_set5643387332196575111eger_o(member_Code_integer(B4),B3)
                 => ( aa_Code_integer_o(aa_o_f8517747772616379256eger_o(Ra2,(A12)),B4)
                   => ( aa_Code_integer_o(aa_o_f8517747772616379256eger_o(Ra2,(A23)),B4)
                     => ( (A12)
                      <=> (A23) ) ) ) ) ) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_o(A3)),finite4902975817058060853nteger(B3)) ) ) ) ).

% card_le_if_inj_on_rel
tff(fact_9062_less__eq__integer_Oabs__eq,axiom,
    ! [Xa2: int,X: int] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_int_Code_integer(code_integer_of_int,Xa2)),aa_int_Code_integer(code_integer_of_int,X))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa2),X) ) ).

% less_eq_integer.abs_eq
tff(fact_9063_sum__multicount__gen,axiom,
    ! [S2: set_o,Ta: set_o,Ra: fun_o_fun_o_o,K: fun_o_nat] :
      ( finite_finite_o(S2)
     => ( finite_finite_o(Ta)
       => ( ! [X3: $o] :
              ( member_o((X3),Ta)
             => ( finite_card_o(collect_o(aa_o_fun_o_o(aa_fun6846671739589424444un_o_o(aTP_Lamm_alb(S2),Ra),(X3)))) = aa_o_nat(K,(X3)) ) )
         => ( groups8507830703676809646_o_nat(aa_fun8807093888949732041_o_nat(aTP_Lamm_ald(Ta),Ra),S2) = groups8507830703676809646_o_nat(K,Ta) ) ) ) ) ).

% sum_multicount_gen
tff(fact_9064_sum__multicount__gen,axiom,
    ! [S2: set_o,Ta: set_int,Ra: fun_o_fun_int_o,K: fun_int_nat] :
      ( finite_finite_o(S2)
     => ( aa_set_int_o(finite_finite_int,Ta)
       => ( ! [X3: int] :
              ( aa_set_int_o(member_int(X3),Ta)
             => ( finite_card_o(collect_o(aa_int_fun_o_o(aa_fun250872857246597658un_o_o(aTP_Lamm_rv(S2),Ra),X3))) = aa_int_nat(K,X3) ) )
         => ( groups8507830703676809646_o_nat(aa_fun2417465416094144205_o_nat(aTP_Lamm_ale(Ta),Ra),S2) = groups4541462559716669496nt_nat(K,Ta) ) ) ) ) ).

% sum_multicount_gen
tff(fact_9065_sum__multicount__gen,axiom,
    ! [S2: set_o,Ta: set_Code_integer,Ra: fun_o_4021313873839786239eger_o,K: fun_Code_integer_nat] :
      ( finite_finite_o(S2)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,Ta)
       => ( ! [X3: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X3),Ta)
             => ( finite_card_o(collect_o(aa_Cod6379977994291289310un_o_o(aa_fun8950210611160600154un_o_o(aTP_Lamm_ada(S2),Ra),X3))) = aa_Code_integer_nat(K,X3) ) )
         => ( groups8507830703676809646_o_nat(aa_fun5400594822745170686_o_nat(aTP_Lamm_alg(Ta),Ra),S2) = groups7237345082560585321er_nat(K,Ta) ) ) ) ) ).

% sum_multicount_gen
tff(fact_9066_sum__multicount__gen,axiom,
    ! [S2: set_int,Ta: set_o,Ra: fun_int_fun_o_o,K: fun_o_nat] :
      ( aa_set_int_o(finite_finite_int,S2)
     => ( finite_finite_o(Ta)
       => ( ! [X3: $o] :
              ( member_o((X3),Ta)
             => ( finite_card_int(collect_int(aa_o_fun_int_o(aa_fun5351653887499120666_int_o(aTP_Lamm_alh(S2),Ra),(X3)))) = aa_o_nat(K,(X3)) ) )
         => ( groups4541462559716669496nt_nat(aa_fun7603103345900947235nt_nat(aTP_Lamm_alj(Ta),Ra),S2) = groups8507830703676809646_o_nat(K,Ta) ) ) ) ) ).

% sum_multicount_gen
tff(fact_9067_sum__multicount__gen,axiom,
    ! [S2: set_int,Ta: set_int,Ra: fun_int_fun_int_o,K: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,S2)
     => ( aa_set_int_o(finite_finite_int,Ta)
       => ( ! [X3: int] :
              ( aa_set_int_o(member_int(X3),Ta)
             => ( finite_card_int(collect_int(aa_int_fun_int_o(aa_fun6133342665515662012_int_o(aTP_Lamm_adb(S2),Ra),X3))) = aa_int_nat(K,X3) ) )
         => ( groups4541462559716669496nt_nat(aa_fun4460091444687325999nt_nat(aTP_Lamm_all(Ta),Ra),S2) = groups4541462559716669496nt_nat(K,Ta) ) ) ) ) ).

% sum_multicount_gen
tff(fact_9068_sum__multicount__gen,axiom,
    ! [S2: set_int,Ta: set_Code_integer,Ra: fun_in4891666191172872833eger_o,K: fun_Code_integer_nat] :
      ( aa_set_int_o(finite_finite_int,S2)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,Ta)
       => ( ! [X3: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X3),Ta)
             => ( finite_card_int(collect_int(aa_Cod8477617969808203498_int_o(aa_fun9103777551025077884_int_o(aTP_Lamm_adc(S2),Ra),X3))) = aa_Code_integer_nat(K,X3) ) )
         => ( groups4541462559716669496nt_nat(aa_fun511150112467848510nt_nat(aTP_Lamm_aln(Ta),Ra),S2) = groups7237345082560585321er_nat(K,Ta) ) ) ) ) ).

% sum_multicount_gen
tff(fact_9069_sum__multicount__gen,axiom,
    ! [S2: set_Code_integer,Ta: set_o,Ra: fun_Co480733329897759773un_o_o,K: fun_o_nat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,S2)
     => ( finite_finite_o(Ta)
       => ( ! [X3: $o] :
              ( member_o((X3),Ta)
             => ( finite4902975817058060853nteger(collect_Code_integer(aa_o_f8517747772616379256eger_o(aa_fun3295616927821077850eger_o(aTP_Lamm_alo(S2),Ra),(X3)))) = aa_o_nat(K,(X3)) ) )
         => ( groups7237345082560585321er_nat(aa_fun7670077909833392133er_nat(aTP_Lamm_alq(Ta),Ra),S2) = groups8507830703676809646_o_nat(K,Ta) ) ) ) ) ).

% sum_multicount_gen
tff(fact_9070_sum__multicount__gen,axiom,
    ! [S2: set_Code_integer,Ta: set_int,Ra: fun_Co8353275306710069249_int_o,K: fun_int_nat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,S2)
     => ( aa_set_int_o(finite_finite_int,Ta)
       => ( ! [X3: int] :
              ( aa_set_int_o(member_int(X3),Ta)
             => ( finite4902975817058060853nteger(collect_Code_integer(aa_int5993532643906289898eger_o(aa_fun7355488674016760700eger_o(aTP_Lamm_se(S2),Ra),X3))) = aa_int_nat(K,X3) ) )
         => ( groups7237345082560585321er_nat(aa_fun726042532089134319er_nat(aTP_Lamm_alr(Ta),Ra),S2) = groups4541462559716669496nt_nat(K,Ta) ) ) ) ) ).

% sum_multicount_gen
tff(fact_9071_sum__multicount__gen,axiom,
    ! [S2: set_Code_integer,Ta: set_Code_integer,Ra: fun_Co2081214437782444048eger_o,K: fun_Code_integer_nat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,S2)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,Ta)
       => ( ! [X3: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X3),Ta)
             => ( finite4902975817058060853nteger(collect_Code_integer(aa_Cod326357343816142457eger_o(aa_fun7683620812777033532eger_o(aTP_Lamm_als(S2),Ra),X3))) = aa_Code_integer_nat(K,X3) ) )
         => ( groups7237345082560585321er_nat(aa_fun8504338775591962366er_nat(aTP_Lamm_alu(Ta),Ra),S2) = groups7237345082560585321er_nat(K,Ta) ) ) ) ) ).

% sum_multicount_gen
tff(fact_9072_sum__multicount__gen,axiom,
    ! [S2: set_o,Ta: set_nat,Ra: fun_o_fun_nat_o,K: fun_nat_nat] :
      ( finite_finite_o(S2)
     => ( aa_set_nat_o(finite_finite_nat,Ta)
       => ( ! [X3: nat] :
              ( aa_set_nat_o(member_nat(X3),Ta)
             => ( finite_card_o(collect_o(aa_nat_fun_o_o(aa_fun532071374735444762un_o_o(aTP_Lamm_rg(S2),Ra),X3))) = aa_nat_nat(K,X3) ) )
         => ( groups8507830703676809646_o_nat(aa_fun6229755271245457521_o_nat(aTP_Lamm_alv(Ta),Ra),S2) = aa_set_nat_nat(groups3542108847815614940at_nat(K),Ta) ) ) ) ) ).

% sum_multicount_gen
tff(fact_9073_card__eq__sum,axiom,
    ! [A3: set_set_nat] : finite_card_set_nat(A3) = groups8294997508430121362at_nat(aTP_Lamm_alw,A3) ).

% card_eq_sum
tff(fact_9074_card__eq__sum,axiom,
    ! [A3: set_list_nat] : finite_card_list_nat(A3) = groups4396056296759096172at_nat(aTP_Lamm_alx,A3) ).

% card_eq_sum
tff(fact_9075_card__eq__sum,axiom,
    ! [A3: set_nat] : aa_set_nat_nat(finite_card_nat,A3) = aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_mj),A3) ).

% card_eq_sum
tff(fact_9076_card__eq__0__iff,axiom,
    ! [A3: set_set_nat] :
      ( ( finite_card_set_nat(A3) = zero_zero_nat )
    <=> ( ( A3 = bot_bot_set_set_nat )
        | ~ aa_set_set_nat_o(finite1152437895449049373et_nat,A3) ) ) ).

% card_eq_0_iff
tff(fact_9077_card__eq__0__iff,axiom,
    ! [A3: set_list_nat] :
      ( ( finite_card_list_nat(A3) = zero_zero_nat )
    <=> ( ( A3 = bot_bot_set_list_nat )
        | ~ finite8100373058378681591st_nat(A3) ) ) ).

% card_eq_0_iff
tff(fact_9078_card__eq__0__iff,axiom,
    ! [A3: set_Code_integer] :
      ( ( finite4902975817058060853nteger(A3) = zero_zero_nat )
    <=> ( ( A3 = bot_bo3990330152332043303nteger )
        | ~ aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A3) ) ) ).

% card_eq_0_iff
tff(fact_9079_card__eq__0__iff,axiom,
    ! [A3: set_Pr1261947904930325089at_nat] :
      ( ( finite711546835091564841at_nat(A3) = zero_zero_nat )
    <=> ( ( A3 = bot_bo2099793752762293965at_nat )
        | ~ aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A3) ) ) ).

% card_eq_0_iff
tff(fact_9080_card__eq__0__iff,axiom,
    ! [A3: set_o] :
      ( ( finite_card_o(A3) = zero_zero_nat )
    <=> ( ( A3 = bot_bot_set_o )
        | ~ finite_finite_o(A3) ) ) ).

% card_eq_0_iff
tff(fact_9081_card__eq__0__iff,axiom,
    ! [A3: set_nat] :
      ( ( aa_set_nat_nat(finite_card_nat,A3) = zero_zero_nat )
    <=> ( ( A3 = bot_bot_set_nat )
        | ~ aa_set_nat_o(finite_finite_nat,A3) ) ) ).

% card_eq_0_iff
tff(fact_9082_card__eq__0__iff,axiom,
    ! [A3: set_int] :
      ( ( finite_card_int(A3) = zero_zero_nat )
    <=> ( ( A3 = bot_bot_set_int )
        | ~ aa_set_int_o(finite_finite_int,A3) ) ) ).

% card_eq_0_iff
tff(fact_9083_card__ge__0__finite,axiom,
    ! [A3: set_set_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),finite_card_set_nat(A3))
     => aa_set_set_nat_o(finite1152437895449049373et_nat,A3) ) ).

% card_ge_0_finite
tff(fact_9084_card__ge__0__finite,axiom,
    ! [A3: set_list_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),finite_card_list_nat(A3))
     => finite8100373058378681591st_nat(A3) ) ).

% card_ge_0_finite
tff(fact_9085_card__ge__0__finite,axiom,
    ! [A3: set_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_set_nat_nat(finite_card_nat,A3))
     => aa_set_nat_o(finite_finite_nat,A3) ) ).

% card_ge_0_finite
tff(fact_9086_card__ge__0__finite,axiom,
    ! [A3: set_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),finite_card_int(A3))
     => aa_set_int_o(finite_finite_int,A3) ) ).

% card_ge_0_finite
tff(fact_9087_card__ge__0__finite,axiom,
    ! [A3: set_Code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),finite4902975817058060853nteger(A3))
     => aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A3) ) ).

% card_ge_0_finite
tff(fact_9088_card__ge__0__finite,axiom,
    ! [A3: set_Pr1261947904930325089at_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),finite711546835091564841at_nat(A3))
     => aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A3) ) ).

% card_ge_0_finite
tff(fact_9089_card__image__le,axiom,
    ! [A3: set_nat,F: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_int(image_nat_int(F,A3))),aa_set_nat_nat(finite_card_nat,A3)) ) ).

% card_image_le
tff(fact_9090_card__image__le,axiom,
    ! [A3: set_nat,F: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(finite_card_nat,image_nat_nat(F,A3))),aa_set_nat_nat(finite_card_nat,A3)) ) ).

% card_image_le
tff(fact_9091_card__image__le,axiom,
    ! [A3: set_int,F: fun_int_int] :
      ( aa_set_int_o(finite_finite_int,A3)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_int(image_int_int(F,A3))),finite_card_int(A3)) ) ).

% card_image_le
tff(fact_9092_card__image__le,axiom,
    ! [A3: set_int,F: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,A3)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(finite_card_nat,image_int_nat(F,A3))),finite_card_int(A3)) ) ).

% card_image_le
tff(fact_9093_card__image__le,axiom,
    ! [A3: set_Code_integer,F: fun_Code_integer_nat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A3)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(finite_card_nat,image_951025933927791156er_nat(F,A3))),finite4902975817058060853nteger(A3)) ) ).

% card_image_le
tff(fact_9094_card__image__le,axiom,
    ! [A3: set_set_nat,F: fun_set_nat_nat] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,A3)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(finite_card_nat,image_set_nat_nat(F,A3))),finite_card_set_nat(A3)) ) ).

% card_image_le
tff(fact_9095_card__image__le,axiom,
    ! [A3: set_list_nat,F: fun_list_nat_nat] :
      ( finite8100373058378681591st_nat(A3)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(finite_card_nat,image_list_nat_nat(F,A3))),finite_card_list_nat(A3)) ) ).

% card_image_le
tff(fact_9096_card__image__le,axiom,
    ! [A3: set_nat,F: fun_nat_set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_set_nat(image_nat_set_nat(F,A3))),aa_set_nat_nat(finite_card_nat,A3)) ) ).

% card_image_le
tff(fact_9097_card__image__le,axiom,
    ! [A3: set_nat,F: fun_nat_list_nat] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_list_nat(image_nat_list_nat(F,A3))),aa_set_nat_nat(finite_card_nat,A3)) ) ).

% card_image_le
tff(fact_9098_card__image__le,axiom,
    ! [A3: set_int,F: fun_int_set_nat] :
      ( aa_set_int_o(finite_finite_int,A3)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_set_nat(image_int_set_nat(F,A3))),finite_card_int(A3)) ) ).

% card_image_le
tff(fact_9099_finite__if__finite__subsets__card__bdd,axiom,
    ! [F3: set_set_nat,C2: nat] :
      ( ! [G3: set_set_nat] :
          ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(G3),F3)
         => ( aa_set_set_nat_o(finite1152437895449049373et_nat,G3)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_set_nat(G3)),C2) ) )
     => ( aa_set_set_nat_o(finite1152437895449049373et_nat,F3)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_set_nat(F3)),C2) ) ) ).

% finite_if_finite_subsets_card_bdd
tff(fact_9100_finite__if__finite__subsets__card__bdd,axiom,
    ! [F3: set_list_nat,C2: nat] :
      ( ! [G3: set_list_nat] :
          ( ord_le6045566169113846134st_nat(G3,F3)
         => ( finite8100373058378681591st_nat(G3)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_list_nat(G3)),C2) ) )
     => ( finite8100373058378681591st_nat(F3)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_list_nat(F3)),C2) ) ) ).

% finite_if_finite_subsets_card_bdd
tff(fact_9101_finite__if__finite__subsets__card__bdd,axiom,
    ! [F3: set_nat,C2: nat] :
      ( ! [G3: set_nat] :
          ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,G3),F3)
         => ( aa_set_nat_o(finite_finite_nat,G3)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(finite_card_nat,G3)),C2) ) )
     => ( aa_set_nat_o(finite_finite_nat,F3)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(finite_card_nat,F3)),C2) ) ) ).

% finite_if_finite_subsets_card_bdd
tff(fact_9102_finite__if__finite__subsets__card__bdd,axiom,
    ! [F3: set_Code_integer,C2: nat] :
      ( ! [G3: set_Code_integer] :
          ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(G3),F3)
         => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,G3)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite4902975817058060853nteger(G3)),C2) ) )
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,F3)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite4902975817058060853nteger(F3)),C2) ) ) ).

% finite_if_finite_subsets_card_bdd
tff(fact_9103_finite__if__finite__subsets__card__bdd,axiom,
    ! [F3: set_Pr1261947904930325089at_nat,C2: nat] :
      ( ! [G3: set_Pr1261947904930325089at_nat] :
          ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,G3),F3)
         => ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,G3)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite711546835091564841at_nat(G3)),C2) ) )
     => ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,F3)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite711546835091564841at_nat(F3)),C2) ) ) ).

% finite_if_finite_subsets_card_bdd
tff(fact_9104_finite__if__finite__subsets__card__bdd,axiom,
    ! [F3: set_int,C2: nat] :
      ( ! [G3: set_int] :
          ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,G3),F3)
         => ( aa_set_int_o(finite_finite_int,G3)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_int(G3)),C2) ) )
     => ( aa_set_int_o(finite_finite_int,F3)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_int(F3)),C2) ) ) ).

% finite_if_finite_subsets_card_bdd
tff(fact_9105_card__seteq,axiom,
    ! [B3: set_set_nat,A3: set_set_nat] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,B3)
     => ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(A3),B3)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_set_nat(B3)),finite_card_set_nat(A3))
         => ( A3 = B3 ) ) ) ) ).

% card_seteq
tff(fact_9106_card__seteq,axiom,
    ! [B3: set_list_nat,A3: set_list_nat] :
      ( finite8100373058378681591st_nat(B3)
     => ( ord_le6045566169113846134st_nat(A3,B3)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_list_nat(B3)),finite_card_list_nat(A3))
         => ( A3 = B3 ) ) ) ) ).

% card_seteq
tff(fact_9107_card__seteq,axiom,
    ! [B3: set_nat,A3: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,B3)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A3),B3)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(finite_card_nat,B3)),aa_set_nat_nat(finite_card_nat,A3))
         => ( A3 = B3 ) ) ) ) ).

% card_seteq
tff(fact_9108_card__seteq,axiom,
    ! [B3: set_Code_integer,A3: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B3)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(A3),B3)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite4902975817058060853nteger(B3)),finite4902975817058060853nteger(A3))
         => ( A3 = B3 ) ) ) ) ).

% card_seteq
tff(fact_9109_card__seteq,axiom,
    ! [B3: set_Pr1261947904930325089at_nat,A3: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,B3)
     => ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A3),B3)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite711546835091564841at_nat(B3)),finite711546835091564841at_nat(A3))
         => ( A3 = B3 ) ) ) ) ).

% card_seteq
tff(fact_9110_card__seteq,axiom,
    ! [B3: set_int,A3: set_int] :
      ( aa_set_int_o(finite_finite_int,B3)
     => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A3),B3)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_int(B3)),finite_card_int(A3))
         => ( A3 = B3 ) ) ) ) ).

% card_seteq
tff(fact_9111_card__mono,axiom,
    ! [B3: set_set_nat,A3: set_set_nat] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,B3)
     => ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(A3),B3)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_set_nat(A3)),finite_card_set_nat(B3)) ) ) ).

% card_mono
tff(fact_9112_card__mono,axiom,
    ! [B3: set_list_nat,A3: set_list_nat] :
      ( finite8100373058378681591st_nat(B3)
     => ( ord_le6045566169113846134st_nat(A3,B3)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_list_nat(A3)),finite_card_list_nat(B3)) ) ) ).

% card_mono
tff(fact_9113_card__mono,axiom,
    ! [B3: set_nat,A3: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,B3)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A3),B3)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(finite_card_nat,A3)),aa_set_nat_nat(finite_card_nat,B3)) ) ) ).

% card_mono
tff(fact_9114_card__mono,axiom,
    ! [B3: set_Code_integer,A3: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B3)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(A3),B3)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite4902975817058060853nteger(A3)),finite4902975817058060853nteger(B3)) ) ) ).

% card_mono
tff(fact_9115_card__mono,axiom,
    ! [B3: set_Pr1261947904930325089at_nat,A3: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,B3)
     => ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A3),B3)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite711546835091564841at_nat(A3)),finite711546835091564841at_nat(B3)) ) ) ).

% card_mono
tff(fact_9116_card__mono,axiom,
    ! [B3: set_int,A3: set_int] :
      ( aa_set_int_o(finite_finite_int,B3)
     => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A3),B3)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_int(A3)),finite_card_int(B3)) ) ) ).

% card_mono
tff(fact_9117_obtain__subset__with__card__n,axiom,
    ! [N2: nat,S: set_set_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),finite_card_set_nat(S))
     => ~ ! [T5: set_set_nat] :
            ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(T5),S)
           => ( ( finite_card_set_nat(T5) = N2 )
             => ~ aa_set_set_nat_o(finite1152437895449049373et_nat,T5) ) ) ) ).

% obtain_subset_with_card_n
tff(fact_9118_obtain__subset__with__card__n,axiom,
    ! [N2: nat,S: set_list_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),finite_card_list_nat(S))
     => ~ ! [T5: set_list_nat] :
            ( ord_le6045566169113846134st_nat(T5,S)
           => ( ( finite_card_list_nat(T5) = N2 )
             => ~ finite8100373058378681591st_nat(T5) ) ) ) ).

% obtain_subset_with_card_n
tff(fact_9119_obtain__subset__with__card__n,axiom,
    ! [N2: nat,S: set_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),aa_set_nat_nat(finite_card_nat,S))
     => ~ ! [T5: set_nat] :
            ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,T5),S)
           => ( ( aa_set_nat_nat(finite_card_nat,T5) = N2 )
             => ~ aa_set_nat_o(finite_finite_nat,T5) ) ) ) ).

% obtain_subset_with_card_n
tff(fact_9120_obtain__subset__with__card__n,axiom,
    ! [N2: nat,S: set_Code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),finite4902975817058060853nteger(S))
     => ~ ! [T5: set_Code_integer] :
            ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(T5),S)
           => ( ( finite4902975817058060853nteger(T5) = N2 )
             => ~ aa_set5643387332196575111eger_o(finite6017078050557962740nteger,T5) ) ) ) ).

% obtain_subset_with_card_n
tff(fact_9121_obtain__subset__with__card__n,axiom,
    ! [N2: nat,S: set_Pr1261947904930325089at_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),finite711546835091564841at_nat(S))
     => ~ ! [T5: set_Pr1261947904930325089at_nat] :
            ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,T5),S)
           => ( ( finite711546835091564841at_nat(T5) = N2 )
             => ~ aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,T5) ) ) ) ).

% obtain_subset_with_card_n
tff(fact_9122_obtain__subset__with__card__n,axiom,
    ! [N2: nat,S: set_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),finite_card_int(S))
     => ~ ! [T5: set_int] :
            ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,T5),S)
           => ( ( finite_card_int(T5) = N2 )
             => ~ aa_set_int_o(finite_finite_int,T5) ) ) ) ).

% obtain_subset_with_card_n
tff(fact_9123_card__less__sym__Diff,axiom,
    ! [A3: set_set_nat,B3: set_set_nat] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,A3)
     => ( aa_set_set_nat_o(finite1152437895449049373et_nat,B3)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,finite_card_set_nat(A3)),finite_card_set_nat(B3))
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,finite_card_set_nat(minus_2163939370556025621et_nat(A3,B3))),finite_card_set_nat(minus_2163939370556025621et_nat(B3,A3))) ) ) ) ).

% card_less_sym_Diff
tff(fact_9124_card__less__sym__Diff,axiom,
    ! [A3: set_list_nat,B3: set_list_nat] :
      ( finite8100373058378681591st_nat(A3)
     => ( finite8100373058378681591st_nat(B3)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,finite_card_list_nat(A3)),finite_card_list_nat(B3))
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,finite_card_list_nat(minus_7954133019191499631st_nat(A3,B3))),finite_card_list_nat(minus_7954133019191499631st_nat(B3,A3))) ) ) ) ).

% card_less_sym_Diff
tff(fact_9125_card__less__sym__Diff,axiom,
    ! [A3: set_int,B3: set_int] :
      ( aa_set_int_o(finite_finite_int,A3)
     => ( aa_set_int_o(finite_finite_int,B3)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,finite_card_int(A3)),finite_card_int(B3))
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,finite_card_int(aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A3),B3))),finite_card_int(aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,B3),A3))) ) ) ) ).

% card_less_sym_Diff
tff(fact_9126_card__less__sym__Diff,axiom,
    ! [A3: set_Code_integer,B3: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A3)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B3)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,finite4902975817058060853nteger(A3)),finite4902975817058060853nteger(B3))
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,finite4902975817058060853nteger(minus_2355218937544613996nteger(A3,B3))),finite4902975817058060853nteger(minus_2355218937544613996nteger(B3,A3))) ) ) ) ).

% card_less_sym_Diff
tff(fact_9127_card__less__sym__Diff,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A3)
     => ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,B3)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,finite711546835091564841at_nat(A3)),finite711546835091564841at_nat(B3))
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,finite711546835091564841at_nat(aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A3),B3))),finite711546835091564841at_nat(aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,B3),A3))) ) ) ) ).

% card_less_sym_Diff
tff(fact_9128_card__less__sym__Diff,axiom,
    ! [A3: set_nat,B3: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( aa_set_nat_o(finite_finite_nat,B3)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_set_nat_nat(finite_card_nat,A3)),aa_set_nat_nat(finite_card_nat,B3))
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_set_nat_nat(finite_card_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A3),B3))),aa_set_nat_nat(finite_card_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,B3),A3))) ) ) ) ).

% card_less_sym_Diff
tff(fact_9129_card__le__sym__Diff,axiom,
    ! [A3: set_set_nat,B3: set_set_nat] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,A3)
     => ( aa_set_set_nat_o(finite1152437895449049373et_nat,B3)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_set_nat(A3)),finite_card_set_nat(B3))
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_set_nat(minus_2163939370556025621et_nat(A3,B3))),finite_card_set_nat(minus_2163939370556025621et_nat(B3,A3))) ) ) ) ).

% card_le_sym_Diff
tff(fact_9130_card__le__sym__Diff,axiom,
    ! [A3: set_list_nat,B3: set_list_nat] :
      ( finite8100373058378681591st_nat(A3)
     => ( finite8100373058378681591st_nat(B3)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_list_nat(A3)),finite_card_list_nat(B3))
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_list_nat(minus_7954133019191499631st_nat(A3,B3))),finite_card_list_nat(minus_7954133019191499631st_nat(B3,A3))) ) ) ) ).

% card_le_sym_Diff
tff(fact_9131_card__le__sym__Diff,axiom,
    ! [A3: set_int,B3: set_int] :
      ( aa_set_int_o(finite_finite_int,A3)
     => ( aa_set_int_o(finite_finite_int,B3)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_int(A3)),finite_card_int(B3))
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_int(aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A3),B3))),finite_card_int(aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,B3),A3))) ) ) ) ).

% card_le_sym_Diff
tff(fact_9132_card__le__sym__Diff,axiom,
    ! [A3: set_Code_integer,B3: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A3)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B3)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite4902975817058060853nteger(A3)),finite4902975817058060853nteger(B3))
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite4902975817058060853nteger(minus_2355218937544613996nteger(A3,B3))),finite4902975817058060853nteger(minus_2355218937544613996nteger(B3,A3))) ) ) ) ).

% card_le_sym_Diff
tff(fact_9133_card__le__sym__Diff,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A3)
     => ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,B3)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite711546835091564841at_nat(A3)),finite711546835091564841at_nat(B3))
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite711546835091564841at_nat(aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A3),B3))),finite711546835091564841at_nat(aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,B3),A3))) ) ) ) ).

% card_le_sym_Diff
tff(fact_9134_card__le__sym__Diff,axiom,
    ! [A3: set_nat,B3: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( aa_set_nat_o(finite_finite_nat,B3)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(finite_card_nat,A3)),aa_set_nat_nat(finite_card_nat,B3))
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(finite_card_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A3),B3))),aa_set_nat_nat(finite_card_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,B3),A3))) ) ) ) ).

% card_le_sym_Diff
tff(fact_9135_psubset__card__mono,axiom,
    ! [B3: set_set_nat,A3: set_set_nat] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,B3)
     => ( aa_set_set_nat_o(ord_less_set_set_nat(A3),B3)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,finite_card_set_nat(A3)),finite_card_set_nat(B3)) ) ) ).

% psubset_card_mono
tff(fact_9136_psubset__card__mono,axiom,
    ! [B3: set_list_nat,A3: set_list_nat] :
      ( finite8100373058378681591st_nat(B3)
     => ( ord_le1190675801316882794st_nat(A3,B3)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,finite_card_list_nat(A3)),finite_card_list_nat(B3)) ) ) ).

% psubset_card_mono
tff(fact_9137_psubset__card__mono,axiom,
    ! [B3: set_nat,A3: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,B3)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,A3),B3)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_set_nat_nat(finite_card_nat,A3)),aa_set_nat_nat(finite_card_nat,B3)) ) ) ).

% psubset_card_mono
tff(fact_9138_psubset__card__mono,axiom,
    ! [B3: set_int,A3: set_int] :
      ( aa_set_int_o(finite_finite_int,B3)
     => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_set_int,A3),B3)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,finite_card_int(A3)),finite_card_int(B3)) ) ) ).

% psubset_card_mono
tff(fact_9139_psubset__card__mono,axiom,
    ! [B3: set_Code_integer,A3: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B3)
     => ( aa_set5643387332196575111eger_o(ord_le1307284697595431911nteger(A3),B3)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,finite4902975817058060853nteger(A3)),finite4902975817058060853nteger(B3)) ) ) ).

% psubset_card_mono
tff(fact_9140_psubset__card__mono,axiom,
    ! [B3: set_Pr1261947904930325089at_nat,A3: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,B3)
     => ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le7866589430770878221at_nat,A3),B3)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,finite711546835091564841at_nat(A3)),finite711546835091564841at_nat(B3)) ) ) ).

% psubset_card_mono
tff(fact_9141_card__Un__le,axiom,
    ! [A3: set_set_nat,B3: set_set_nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_set_nat(sup_sup_set_set_nat(A3,B3))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,finite_card_set_nat(A3)),finite_card_set_nat(B3))) ).

% card_Un_le
tff(fact_9142_card__Un__le,axiom,
    ! [A3: set_list_nat,B3: set_list_nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_list_nat(sup_sup_set_list_nat(A3,B3))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,finite_card_list_nat(A3)),finite_card_list_nat(B3))) ).

% card_Un_le
tff(fact_9143_card__Un__le,axiom,
    ! [A3: set_Pr8693737435421807431at_nat,B3: set_Pr8693737435421807431at_nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite1207074278014112911at_nat(aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,A3),B3))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,finite1207074278014112911at_nat(A3)),finite1207074278014112911at_nat(B3))) ).

% card_Un_le
tff(fact_9144_card__Un__le,axiom,
    ! [A3: set_Pr8551490117392284871at_nat,B3: set_Pr8551490117392284871at_nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite7007076676225009423at_nat(aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A3),B3))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,finite7007076676225009423at_nat(A3)),finite7007076676225009423at_nat(B3))) ).

% card_Un_le
tff(fact_9145_card__Un__le,axiom,
    ! [A3: set_Pr4329608150637261639at_nat,B3: set_Pr4329608150637261639at_nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite3771342082235030671at_nat(aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A3),B3))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,finite3771342082235030671at_nat(A3)),finite3771342082235030671at_nat(B3))) ).

% card_Un_le
tff(fact_9146_card__Un__le,axiom,
    ! [A3: set_nat,B3: set_nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(finite_card_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A3),B3))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(finite_card_nat,A3)),aa_set_nat_nat(finite_card_nat,B3))) ).

% card_Un_le
tff(fact_9147_card__less__Suc2,axiom,
    ! [M6: set_nat,I2: nat] :
      ( ~ aa_set_nat_o(member_nat(zero_zero_nat),M6)
     => ( aa_set_nat_nat(finite_card_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_aly(M6),I2))) = aa_set_nat_nat(finite_card_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_alz(M6),I2))) ) ) ).

% card_less_Suc2
tff(fact_9148_card__less__Suc,axiom,
    ! [M6: set_nat,I2: nat] :
      ( aa_set_nat_o(member_nat(zero_zero_nat),M6)
     => ( aa_nat_nat(suc,aa_set_nat_nat(finite_card_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_aly(M6),I2)))) = aa_set_nat_nat(finite_card_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_alz(M6),I2))) ) ) ).

% card_less_Suc
tff(fact_9149_card__less,axiom,
    ! [M6: set_nat,I2: nat] :
      ( aa_set_nat_o(member_nat(zero_zero_nat),M6)
     => ( aa_set_nat_nat(finite_card_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_alz(M6),I2))) != zero_zero_nat ) ) ).

% card_less
tff(fact_9150_subset__card__intvl__is__intvl,axiom,
    ! [A3: set_nat,K: nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A3),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),aa_set_nat_nat(finite_card_nat,A3))))
     => ( A3 = aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),aa_set_nat_nat(finite_card_nat,A3))) ) ) ).

% subset_card_intvl_is_intvl
tff(fact_9151_sum__Suc,axiom,
    ! [F: fun_set_nat_nat,A3: set_set_nat] : groups8294997508430121362at_nat(aTP_Lamm_ama(F),A3) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups8294997508430121362at_nat(F,A3)),finite_card_set_nat(A3)) ).

% sum_Suc
tff(fact_9152_sum__Suc,axiom,
    ! [F: fun_list_nat_nat,A3: set_list_nat] : groups4396056296759096172at_nat(aTP_Lamm_amb(F),A3) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups4396056296759096172at_nat(F,A3)),finite_card_list_nat(A3)) ).

% sum_Suc
tff(fact_9153_sum__Suc,axiom,
    ! [F: fun_nat_nat,A3: set_nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_amc(F)),A3) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),A3)),aa_set_nat_nat(finite_card_nat,A3)) ).

% sum_Suc
tff(fact_9154_sum__multicount,axiom,
    ! [S: set_o,T: set_o,Ra: fun_o_fun_o_o,K: nat] :
      ( finite_finite_o(S)
     => ( finite_finite_o(T)
       => ( ! [X3: $o] :
              ( member_o((X3),T)
             => ( finite_card_o(collect_o(aa_o_fun_o_o(aa_fun6846671739589424444un_o_o(aTP_Lamm_alb(S),Ra),(X3)))) = K ) )
         => ( groups8507830703676809646_o_nat(aa_fun8807093888949732041_o_nat(aTP_Lamm_ald(T),Ra),S) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),finite_card_o(T)) ) ) ) ) ).

% sum_multicount
tff(fact_9155_sum__multicount,axiom,
    ! [S: set_o,T: set_nat,Ra: fun_o_fun_nat_o,K: nat] :
      ( finite_finite_o(S)
     => ( aa_set_nat_o(finite_finite_nat,T)
       => ( ! [X3: nat] :
              ( aa_set_nat_o(member_nat(X3),T)
             => ( finite_card_o(collect_o(aa_nat_fun_o_o(aa_fun532071374735444762un_o_o(aTP_Lamm_rg(S),Ra),X3))) = K ) )
         => ( groups8507830703676809646_o_nat(aa_fun6229755271245457521_o_nat(aTP_Lamm_alv(T),Ra),S) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),aa_set_nat_nat(finite_card_nat,T)) ) ) ) ) ).

% sum_multicount
tff(fact_9156_sum__multicount,axiom,
    ! [S: set_o,T: set_int,Ra: fun_o_fun_int_o,K: nat] :
      ( finite_finite_o(S)
     => ( aa_set_int_o(finite_finite_int,T)
       => ( ! [X3: int] :
              ( aa_set_int_o(member_int(X3),T)
             => ( finite_card_o(collect_o(aa_int_fun_o_o(aa_fun250872857246597658un_o_o(aTP_Lamm_rv(S),Ra),X3))) = K ) )
         => ( groups8507830703676809646_o_nat(aa_fun2417465416094144205_o_nat(aTP_Lamm_ale(T),Ra),S) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),finite_card_int(T)) ) ) ) ) ).

% sum_multicount
tff(fact_9157_sum__multicount,axiom,
    ! [S: set_o,T: set_Code_integer,Ra: fun_o_4021313873839786239eger_o,K: nat] :
      ( finite_finite_o(S)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,T)
       => ( ! [X3: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X3),T)
             => ( finite_card_o(collect_o(aa_Cod6379977994291289310un_o_o(aa_fun8950210611160600154un_o_o(aTP_Lamm_ada(S),Ra),X3))) = K ) )
         => ( groups8507830703676809646_o_nat(aa_fun5400594822745170686_o_nat(aTP_Lamm_alg(T),Ra),S) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),finite4902975817058060853nteger(T)) ) ) ) ) ).

% sum_multicount
tff(fact_9158_sum__multicount,axiom,
    ! [S: set_int,T: set_o,Ra: fun_int_fun_o_o,K: nat] :
      ( aa_set_int_o(finite_finite_int,S)
     => ( finite_finite_o(T)
       => ( ! [X3: $o] :
              ( member_o((X3),T)
             => ( finite_card_int(collect_int(aa_o_fun_int_o(aa_fun5351653887499120666_int_o(aTP_Lamm_alh(S),Ra),(X3)))) = K ) )
         => ( groups4541462559716669496nt_nat(aa_fun7603103345900947235nt_nat(aTP_Lamm_alj(T),Ra),S) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),finite_card_o(T)) ) ) ) ) ).

% sum_multicount
tff(fact_9159_sum__multicount,axiom,
    ! [S: set_int,T: set_nat,Ra: fun_int_fun_nat_o,K: nat] :
      ( aa_set_int_o(finite_finite_int,S)
     => ( aa_set_nat_o(finite_finite_nat,T)
       => ( ! [X3: nat] :
              ( aa_set_nat_o(member_nat(X3),T)
             => ( finite_card_int(collect_int(aa_nat_fun_int_o(aa_fun5439836533389078972_int_o(aTP_Lamm_rl(S),Ra),X3))) = K ) )
         => ( groups4541462559716669496nt_nat(aa_fun4650797359919590027nt_nat(aTP_Lamm_amd(T),Ra),S) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),aa_set_nat_nat(finite_card_nat,T)) ) ) ) ) ).

% sum_multicount
tff(fact_9160_sum__multicount,axiom,
    ! [S: set_int,T: set_int,Ra: fun_int_fun_int_o,K: nat] :
      ( aa_set_int_o(finite_finite_int,S)
     => ( aa_set_int_o(finite_finite_int,T)
       => ( ! [X3: int] :
              ( aa_set_int_o(member_int(X3),T)
             => ( finite_card_int(collect_int(aa_int_fun_int_o(aa_fun6133342665515662012_int_o(aTP_Lamm_adb(S),Ra),X3))) = K ) )
         => ( groups4541462559716669496nt_nat(aa_fun4460091444687325999nt_nat(aTP_Lamm_all(T),Ra),S) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),finite_card_int(T)) ) ) ) ) ).

% sum_multicount
tff(fact_9161_sum__multicount,axiom,
    ! [S: set_int,T: set_Code_integer,Ra: fun_in4891666191172872833eger_o,K: nat] :
      ( aa_set_int_o(finite_finite_int,S)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,T)
       => ( ! [X3: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X3),T)
             => ( finite_card_int(collect_int(aa_Cod8477617969808203498_int_o(aa_fun9103777551025077884_int_o(aTP_Lamm_adc(S),Ra),X3))) = K ) )
         => ( groups4541462559716669496nt_nat(aa_fun511150112467848510nt_nat(aTP_Lamm_aln(T),Ra),S) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),finite4902975817058060853nteger(T)) ) ) ) ) ).

% sum_multicount
tff(fact_9162_sum__multicount,axiom,
    ! [S: set_Code_integer,T: set_o,Ra: fun_Co480733329897759773un_o_o,K: nat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,S)
     => ( finite_finite_o(T)
       => ( ! [X3: $o] :
              ( member_o((X3),T)
             => ( finite4902975817058060853nteger(collect_Code_integer(aa_o_f8517747772616379256eger_o(aa_fun3295616927821077850eger_o(aTP_Lamm_alo(S),Ra),(X3)))) = K ) )
         => ( groups7237345082560585321er_nat(aa_fun7670077909833392133er_nat(aTP_Lamm_alq(T),Ra),S) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),finite_card_o(T)) ) ) ) ) ).

% sum_multicount
tff(fact_9163_sum__multicount,axiom,
    ! [S: set_Code_integer,T: set_nat,Ra: fun_Co1375048237525739357_nat_o,K: nat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,S)
     => ( aa_set_nat_o(finite_finite_nat,T)
       => ( ! [X3: nat] :
              ( aa_set_nat_o(member_nat(X3),T)
             => ( finite4902975817058060853nteger(collect_Code_integer(aa_nat7420681472272698950eger_o(aa_fun7373950593701497468eger_o(aTP_Lamm_rq(S),Ra),X3))) = K ) )
         => ( groups7237345082560585321er_nat(aa_fun6683192525214198859er_nat(aTP_Lamm_ame(T),Ra),S) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),aa_set_nat_nat(finite_card_nat,T)) ) ) ) ) ).

% sum_multicount
tff(fact_9164_card__gt__0__iff,axiom,
    ! [A3: set_set_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),finite_card_set_nat(A3))
    <=> ( ( A3 != bot_bot_set_set_nat )
        & aa_set_set_nat_o(finite1152437895449049373et_nat,A3) ) ) ).

% card_gt_0_iff
tff(fact_9165_card__gt__0__iff,axiom,
    ! [A3: set_list_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),finite_card_list_nat(A3))
    <=> ( ( A3 != bot_bot_set_list_nat )
        & finite8100373058378681591st_nat(A3) ) ) ).

% card_gt_0_iff
tff(fact_9166_card__gt__0__iff,axiom,
    ! [A3: set_Code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),finite4902975817058060853nteger(A3))
    <=> ( ( A3 != bot_bo3990330152332043303nteger )
        & aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A3) ) ) ).

% card_gt_0_iff
tff(fact_9167_card__gt__0__iff,axiom,
    ! [A3: set_Pr1261947904930325089at_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),finite711546835091564841at_nat(A3))
    <=> ( ( A3 != bot_bo2099793752762293965at_nat )
        & aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A3) ) ) ).

% card_gt_0_iff
tff(fact_9168_card__gt__0__iff,axiom,
    ! [A3: set_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),finite_card_o(A3))
    <=> ( ( A3 != bot_bot_set_o )
        & finite_finite_o(A3) ) ) ).

% card_gt_0_iff
tff(fact_9169_card__gt__0__iff,axiom,
    ! [A3: set_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_set_nat_nat(finite_card_nat,A3))
    <=> ( ( A3 != bot_bot_set_nat )
        & aa_set_nat_o(finite_finite_nat,A3) ) ) ).

% card_gt_0_iff
tff(fact_9170_card__gt__0__iff,axiom,
    ! [A3: set_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),finite_card_int(A3))
    <=> ( ( A3 != bot_bot_set_int )
        & aa_set_int_o(finite_finite_int,A3) ) ) ).

% card_gt_0_iff
tff(fact_9171_sum__bounded__below,axiom,
    ! [A3: set_nat,K5: code_integer,F: fun_nat_Code_integer] :
      ( ! [I3: nat] :
          ( aa_set_nat_o(member_nat(I3),A3)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,K5),aa_nat_Code_integer(F,I3)) )
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,aa_set_nat_nat(finite_card_nat,A3))),K5)),aa_set871585252387937313nteger(groups7501900531339628137nteger(F),A3)) ) ).

% sum_bounded_below
tff(fact_9172_sum__bounded__below,axiom,
    ! [A3: set_o,K5: rat,F: fun_o_rat] :
      ( ! [I3: $o] :
          ( member_o((I3),A3)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,K5),aa_o_rat(F,(I3))) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,finite_card_o(A3))),K5)),groups7872700643590313910_o_rat(F,A3)) ) ).

% sum_bounded_below
tff(fact_9173_sum__bounded__below,axiom,
    ! [A3: set_int,K5: rat,F: fun_int_rat] :
      ( ! [I3: int] :
          ( aa_set_int_o(member_int(I3),A3)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,K5),aa_int_rat(F,I3)) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,finite_card_int(A3))),K5)),groups3906332499630173760nt_rat(F,A3)) ) ).

% sum_bounded_below
tff(fact_9174_sum__bounded__below,axiom,
    ! [A3: set_nat,K5: rat,F: fun_nat_rat] :
      ( ! [I3: nat] :
          ( aa_set_nat_o(member_nat(I3),A3)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,K5),aa_nat_rat(F,I3)) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,aa_set_nat_nat(finite_card_nat,A3))),K5)),aa_set_nat_rat(groups2906978787729119204at_rat(F),A3)) ) ).

% sum_bounded_below
tff(fact_9175_sum__bounded__below,axiom,
    ! [A3: set_o,K5: nat,F: fun_o_nat] :
      ( ! [I3: $o] :
          ( member_o((I3),A3)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K5),aa_o_nat(F,(I3))) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,finite_card_o(A3))),K5)),groups8507830703676809646_o_nat(F,A3)) ) ).

% sum_bounded_below
tff(fact_9176_sum__bounded__below,axiom,
    ! [A3: set_int,K5: nat,F: fun_int_nat] :
      ( ! [I3: int] :
          ( aa_set_int_o(member_int(I3),A3)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K5),aa_int_nat(F,I3)) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,finite_card_int(A3))),K5)),groups4541462559716669496nt_nat(F,A3)) ) ).

% sum_bounded_below
tff(fact_9177_sum__bounded__below,axiom,
    ! [A3: set_o,K5: int,F: fun_o_int] :
      ( ! [I3: $o] :
          ( member_o((I3),A3)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K5),aa_o_int(F,(I3))) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,finite_card_o(A3))),K5)),groups8505340233167759370_o_int(F,A3)) ) ).

% sum_bounded_below
tff(fact_9178_sum__bounded__below,axiom,
    ! [A3: set_nat,K5: int,F: fun_nat_int] :
      ( ! [I3: nat] :
          ( aa_set_nat_o(member_nat(I3),A3)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K5),aa_nat_int(F,I3)) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,aa_set_nat_nat(finite_card_nat,A3))),K5)),aa_set_nat_int(groups3539618377306564664at_int(F),A3)) ) ).

% sum_bounded_below
tff(fact_9179_sum__bounded__below,axiom,
    ! [A3: set_nat,K5: nat,F: fun_nat_nat] :
      ( ! [I3: nat] :
          ( aa_set_nat_o(member_nat(I3),A3)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K5),aa_nat_nat(F,I3)) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,aa_set_nat_nat(finite_card_nat,A3))),K5)),aa_set_nat_nat(groups3542108847815614940at_nat(F),A3)) ) ).

% sum_bounded_below
tff(fact_9180_sum__bounded__below,axiom,
    ! [A3: set_int,K5: int,F: fun_int_int] :
      ( ! [I3: int] :
          ( aa_set_int_o(member_int(I3),A3)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K5),aa_int_int(F,I3)) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,finite_card_int(A3))),K5)),groups4538972089207619220nt_int(F,A3)) ) ).

% sum_bounded_below
tff(fact_9181_subset__eq__atLeast0__lessThan__card,axiom,
    ! [N7: set_nat,N2: nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,N7),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),N2))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(finite_card_nat,N7)),N2) ) ).

% subset_eq_atLeast0_lessThan_card
tff(fact_9182_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_fu),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),aa_set_nat_nat(finite_card_nat,S)))),aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_fu),S)) ).

% card_sum_le_nat_sum
tff(fact_9183_integer__of__num_I2_J,axiom,
    ! [N2: num] :
      code_integer_of_num(aa_num_num(bit0,N2)) = $let(
        k: code_integer,
        k:= code_integer_of_num(N2),
        aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(k),k) ) ).

% integer_of_num(2)
tff(fact_9184_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_amf,code_divmod_integer(K,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))))) ).

% num_of_integer_code
tff(fact_9185_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_9186_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_amg,code_divmod_integer(K,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))))) ) ).

% int_of_integer_code
tff(fact_9187_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_9188_nat_Odisc__eq__case_I2_J,axiom,
    ! [Nat: nat] :
      ( ( Nat != zero_zero_nat )
    <=> aa_nat_o(case_nat_o($false,aTP_Lamm_amh),Nat) ) ).

% nat.disc_eq_case(2)
tff(fact_9189_nat_Odisc__eq__case_I1_J,axiom,
    ! [Nat: nat] :
      ( ( Nat = zero_zero_nat )
    <=> aa_nat_o(case_nat_o($true,aTP_Lamm_bg),Nat) ) ).

% nat.disc_eq_case(1)
tff(fact_9190_int__of__integer__less__iff,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_Code_integer_int(code_int_of_integer,X)),aa_Code_integer_int(code_int_of_integer,Y))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,X),Y) ) ).

% int_of_integer_less_iff
tff(fact_9191_integer__less__iff,axiom,
    ! [K: code_integer,L: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,K),L)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_Code_integer_int(code_int_of_integer,K)),aa_Code_integer_int(code_int_of_integer,L)) ) ).

% integer_less_iff
tff(fact_9192_less__integer_Orep__eq,axiom,
    ! [X: code_integer,Xa2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,X),Xa2)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_Code_integer_int(code_int_of_integer,X)),aa_Code_integer_int(code_int_of_integer,Xa2)) ) ).

% less_integer.rep_eq
tff(fact_9193_integer__less__eq__iff,axiom,
    ! [K: code_integer,L: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,K),L)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_Code_integer_int(code_int_of_integer,K)),aa_Code_integer_int(code_int_of_integer,L)) ) ).

% integer_less_eq_iff
tff(fact_9194_less__eq__integer_Orep__eq,axiom,
    ! [X: code_integer,Xa2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,X),Xa2)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_Code_integer_int(code_int_of_integer,X)),aa_Code_integer_int(code_int_of_integer,Xa2)) ) ).

% less_eq_integer.rep_eq
tff(fact_9195_less__eq__nat_Osimps_I2_J,axiom,
    ! [M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,M)),N2)
    <=> aa_nat_o(case_nat_o($false,aa_nat_fun_nat_o(ord_less_eq_nat,M)),N2) ) ).

% less_eq_nat.simps(2)
tff(fact_9196_diff__Suc,axiom,
    ! [M: nat,N2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),aa_nat_nat(suc,N2)) = aa_nat_nat(case_nat_nat(zero_zero_nat,aTP_Lamm_fu),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N2)) ).

% diff_Suc
tff(fact_9197_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_ami(K),code_divmod_abs(K,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))))) ).

% bit_cut_integer_code
tff(fact_9198_nat__of__integer__code,axiom,
    ! [K: code_integer] :
      aa_Code_integer_nat(code_nat_of_integer,K) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,K),zero_z3403309356797280102nteger),zero_zero_nat,produc1555791787009142072er_nat(aTP_Lamm_amj,code_divmod_integer(K,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))))) ).

% nat_of_integer_code
tff(fact_9199_nat__of__integer__non__positive,axiom,
    ! [K: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,K),zero_z3403309356797280102nteger)
     => ( aa_Code_integer_nat(code_nat_of_integer,K) = zero_zero_nat ) ) ).

% nat_of_integer_non_positive
tff(fact_9200_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_9201_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_9202_nat__of__integer__less__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_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_Code_integer_nat(code_nat_of_integer,X)),aa_Code_integer_nat(code_nat_of_integer,Y))
        <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,X),Y) ) ) ) ).

% nat_of_integer_less_iff
tff(fact_9203_image__atLeastZeroLessThan__integer,axiom,
    ! [U: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),U)
     => ( set_or8404916559141939852nteger(zero_z3403309356797280102nteger,U) = image_1215581382706833972nteger(semiri4939895301339042750nteger,aa_nat_set_nat(set_ord_lessThan_nat,aa_Code_integer_nat(code_nat_of_integer,U))) ) ) ).

% image_atLeastZeroLessThan_integer
tff(fact_9204_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_9205_divmod__integer__code,axiom,
    ! [K: code_integer,L: code_integer] :
      code_divmod_integer(K,L) = $ite(
        K = zero_z3403309356797280102nteger,
        produc1086072967326762835nteger(zero_z3403309356797280102nteger,zero_z3403309356797280102nteger),
        $ite(
          aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),L),
          $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),K),code_divmod_abs(K,L),produc6916734918728496179nteger(aTP_Lamm_amk(L),code_divmod_abs(K,L))),
          $ite(
            L = zero_z3403309356797280102nteger,
            produc1086072967326762835nteger(zero_z3403309356797280102nteger,K),
            aa_Pro4693618923521437820nteger(aa_fun2509385170507305080nteger(produc6499014454317279255nteger,uminus1351360451143612070nteger),
              $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,K),zero_z3403309356797280102nteger),code_divmod_abs(K,L),produc6916734918728496179nteger(aTP_Lamm_aml(L),code_divmod_abs(K,L)))) ) ) ) ).

% divmod_integer_code
tff(fact_9206_signed__take__bit__nonnegative__iff,axiom,
    ! [N2: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,N2),K))
    <=> ~ aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,K),N2) ) ).

% signed_take_bit_nonnegative_iff
tff(fact_9207_signed__take__bit__negative__iff,axiom,
    ! [N2: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,N2),K)),zero_zero_int)
    <=> aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,K),N2) ) ).

% signed_take_bit_negative_iff
tff(fact_9208_bit__nat__iff,axiom,
    ! [K: int,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_int_nat(nat2,K)),N2)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K)
        & aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,K),N2) ) ) ).

% bit_nat_iff
tff(fact_9209_bit__imp__take__bit__positive,axiom,
    ! [N2: nat,M: nat,K: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),M)
     => ( aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,K),N2)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,M),K)) ) ) ).

% bit_imp_take_bit_positive
tff(fact_9210_int__bit__bound,axiom,
    ! [K: int] :
      ~ ! [N5: nat] :
          ( ! [M4: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N5),M4)
             => ( aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,K),M4)
              <=> aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,K),N5) ) )
         => ~ ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N5)
             => ( aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,K),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N5),one_one_nat))
              <=> ~ aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,K),N5) ) ) ) ).

% int_bit_bound
tff(fact_9211_pred__def,axiom,
    ! [Nat: nat] : aa_nat_nat(pred,Nat) = aa_nat_nat(case_nat_nat(zero_zero_nat,aTP_Lamm_fu),Nat) ).

% pred_def
tff(fact_9212_finite__enumerate,axiom,
    ! [S: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,S)
     => ? [R: fun_nat_nat] :
          ( aa_set_nat_o(strict1292158309912662752at_nat(R),aa_nat_set_nat(set_ord_lessThan_nat,aa_set_nat_nat(finite_card_nat,S)))
          & ! [N8: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N8),aa_set_nat_nat(finite_card_nat,S))
             => aa_set_nat_o(member_nat(aa_nat_nat(R,N8)),S) ) ) ) ).

% finite_enumerate
tff(fact_9213_or__nonnegative__int__iff,axiom,
    ! [K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,K),L))
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),L) ) ) ).

% or_nonnegative_int_iff
tff(fact_9214_push__bit__nonnegative__int__iff,axiom,
    ! [N2: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_nat_fun_int_int(bit_se545348938243370406it_int,N2),K))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K) ) ).

% push_bit_nonnegative_int_iff
tff(fact_9215_or__negative__int__iff,axiom,
    ! [K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,K),L)),zero_zero_int)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int)
        | aa_int_o(aa_int_fun_int_o(ord_less_int,L),zero_zero_int) ) ) ).

% or_negative_int_iff
tff(fact_9216_push__bit__negative__int__iff,axiom,
    ! [N2: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_nat_fun_int_int(bit_se545348938243370406it_int,N2),K)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int) ) ).

% push_bit_negative_int_iff
tff(fact_9217_OR__lower,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,zero_zero_int),aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,X),Y)) ) ) ).

% OR_lower
tff(fact_9218_or__greater__eq,axiom,
    ! [L: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),L)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,K),L)) ) ).

% or_greater_eq
tff(fact_9219_bit__push__bit__iff__int,axiom,
    ! [M: nat,K: int,N2: nat] :
      ( aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,aa_int_int(aa_nat_fun_int_int(bit_se545348938243370406it_int,M),K)),N2)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
        & aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,K),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),M)) ) ) ).

% bit_push_bit_iff_int
tff(fact_9220_bit__push__bit__iff__nat,axiom,
    ! [M: nat,Q4: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,M),Q4)),N2)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
        & aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,Q4),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),M)) ) ) ).

% bit_push_bit_iff_nat
tff(fact_9221_OR__upper,axiom,
    ! [X: int,N2: nat,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_int,X),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N2))
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Y),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N2))
         => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,X),Y)),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N2)) ) ) ) ).

% OR_upper
tff(fact_9222_Sup__nat__empty,axiom,
    aa_set_nat_nat(complete_Sup_Sup_nat,bot_bot_set_nat) = zero_zero_nat ).

% Sup_nat_empty
tff(fact_9223_divmod__integer__eq__cases,axiom,
    ! [K: code_integer,L: code_integer] :
      code_divmod_integer(K,L) = $ite(
        K = zero_z3403309356797280102nteger,
        produc1086072967326762835nteger(zero_z3403309356797280102nteger,zero_z3403309356797280102nteger),
        $ite(
          L = zero_z3403309356797280102nteger,
          produc1086072967326762835nteger(zero_z3403309356797280102nteger,K),
          aa_Pro4693618923521437820nteger(comp_C7848664941760151970nteger(comp_f6739083541385432444nteger(produc6499014454317279255nteger,times_3573771949741848930nteger),sgn_sgn_Code_integer,L),
            $ite(aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,K) = aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,L),code_divmod_abs(K,L),produc6916734918728496179nteger(aTP_Lamm_amm(L),code_divmod_abs(K,L)))) ) ) ).

% divmod_integer_eq_cases
tff(fact_9224_Inf__nat__def1,axiom,
    ! [K5: set_nat] :
      ( ( K5 != bot_bot_set_nat )
     => aa_set_nat_o(member_nat(aa_set_nat_nat(complete_Inf_Inf_nat,K5)),K5) ) ).

% Inf_nat_def1
tff(fact_9225_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_9226_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_9227_range__mod,axiom,
    ! [N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)
     => ( image_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_amn,N2),top_top_set_nat) = aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),N2) ) ) ).

% range_mod
tff(fact_9228_and__not__num_Osimps_I8_J,axiom,
    ! [M: num,N2: num] : aa_num_option_num(bit_and_not_num(aa_num_num(bit1,M)),aa_num_num(bit0,N2)) = aa_opt5369159329878751634on_num(case_o6005452278849405969um_num(aa_num_option_num(some_num,one),aTP_Lamm_amo),aa_num_option_num(bit_and_not_num(M),N2)) ).

% and_not_num.simps(8)
tff(fact_9229_merge__true__star,axiom,
    aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,top_top_assn),top_top_assn) = top_top_assn ).

% merge_true_star
tff(fact_9230_assn__basic__inequalities_I5_J,axiom,
    top_top_assn != bot_bot_assn ).

% assn_basic_inequalities(5)
tff(fact_9231_assn__basic__inequalities_I1_J,axiom,
    top_top_assn != one_one_assn ).

% assn_basic_inequalities(1)
tff(fact_9232_Suc__funpow,axiom,
    ! [N2: nat] : compow_fun_nat_nat(N2,suc) = aa_nat_fun_nat_nat(plus_plus_nat,N2) ).

% Suc_funpow
tff(fact_9233_and__nonnegative__int__iff,axiom,
    ! [K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,K),L))
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K)
        | aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),L) ) ) ).

% and_nonnegative_int_iff
tff(fact_9234_and__negative__int__iff,axiom,
    ! [K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,K),L)),zero_zero_int)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,L),zero_zero_int) ) ) ).

% and_negative_int_iff
tff(fact_9235_mod__true,axiom,
    ! [Ha: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(rep_assn(top_top_assn),Ha)
    <=> aa_Pro653082453696054616_nat_o(in_range,Ha) ) ).

% mod_true
tff(fact_9236_not__negative__int__iff,axiom,
    ! [K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(bit_ri7919022796975470100ot_int,K)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K) ) ).

% not_negative_int_iff
tff(fact_9237_not__nonnegative__int__iff,axiom,
    ! [K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(bit_ri7919022796975470100ot_int,K))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int) ) ).

% not_nonnegative_int_iff
tff(fact_9238_and__minus__numerals_I4_J,axiom,
    ! [M: num,N2: 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,N2)))) = aa_option_num_int(case_option_int_num(zero_zero_int,numeral_numeral_int),aa_num_option_num(bit_and_not_num(M),aa_num_num(bit0,N2))) ).

% and_minus_numerals(4)
tff(fact_9239_and__minus__numerals_I8_J,axiom,
    ! [N2: 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,N2)))),aa_num_int(numeral_numeral_int,M)) = aa_option_num_int(case_option_int_num(zero_zero_int,numeral_numeral_int),aa_num_option_num(bit_and_not_num(M),aa_num_num(bit0,N2))) ).

% and_minus_numerals(8)
tff(fact_9240_and__minus__numerals_I3_J,axiom,
    ! [M: num,N2: 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,N2)))) = aa_option_num_int(case_option_int_num(zero_zero_int,numeral_numeral_int),aa_num_option_num(bit_and_not_num(M),aa_num_num(bitM,N2))) ).

% and_minus_numerals(3)
tff(fact_9241_and__minus__numerals_I7_J,axiom,
    ! [N2: 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,N2)))),aa_num_int(numeral_numeral_int,M)) = aa_option_num_int(case_option_int_num(zero_zero_int,numeral_numeral_int),aa_num_option_num(bit_and_not_num(M),aa_num_num(bitM,N2))) ).

% and_minus_numerals(7)
tff(fact_9242_ent__true,axiom,
    ! [Pa: assn] : aa_assn_o(aa_assn_fun_assn_o(entails,Pa),top_top_assn) ).

% ent_true
tff(fact_9243_and__not__num__eq__Some__iff,axiom,
    ! [M: num,N2: num,Q4: num] :
      ( ( aa_num_option_num(bit_and_not_num(M),N2) = aa_num_option_num(some_num,Q4) )
    <=> ( 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,N2))) = aa_num_int(numeral_numeral_int,Q4) ) ) ).

% and_not_num_eq_Some_iff
tff(fact_9244_int__numeral__not__and__num,axiom,
    ! [M: num,N2: 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,N2)) = aa_option_num_int(case_option_int_num(zero_zero_int,numeral_numeral_int),aa_num_option_num(bit_and_not_num(N2),M)) ).

% int_numeral_not_and_num
tff(fact_9245_int__numeral__and__not__num,axiom,
    ! [M: num,N2: 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,N2))) = aa_option_num_int(case_option_int_num(zero_zero_int,numeral_numeral_int),aa_num_option_num(bit_and_not_num(M),N2)) ).

% int_numeral_and_not_num
tff(fact_9246_AND__lower,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),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,X),Y)) ) ).

% AND_lower
tff(fact_9247_AND__upper1,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,aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,X),Y)),X) ) ).

% AND_upper1
tff(fact_9248_AND__upper2,axiom,
    ! [Y: int,X: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,X),Y)),Y) ) ).

% AND_upper2
tff(fact_9249_AND__upper1_H,axiom,
    ! [Y: int,Z: int,Ya: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),Z)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,Y),Ya)),Z) ) ) ).

% AND_upper1'
tff(fact_9250_AND__upper2_H,axiom,
    ! [Y: int,Z: int,X: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),Z)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,X),Y)),Z) ) ) ).

% AND_upper2'
tff(fact_9251_mod__star__trueI,axiom,
    ! [Pa: assn,Ha: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(rep_assn(Pa),Ha)
     => aa_Pro653082453696054616_nat_o(rep_assn(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Pa),top_top_assn)),Ha) ) ).

% mod_star_trueI
tff(fact_9252_mod__star__trueE,axiom,
    ! [Pa: assn,Ha: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(rep_assn(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Pa),top_top_assn)),Ha)
     => ~ ! [H4: produc3658429121746597890et_nat] : ~ aa_Pro653082453696054616_nat_o(rep_assn(Pa),H4) ) ).

% mod_star_trueE
tff(fact_9253_top__assn__def,axiom,
    top_top_assn = abs_assn(in_range) ).

% top_assn_def
tff(fact_9254_and__less__eq,axiom,
    ! [L: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,L),zero_zero_int)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,K),L)),K) ) ).

% and_less_eq
tff(fact_9255_AND__upper1_H_H,axiom,
    ! [Y: int,Z: int,Ya: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Y),Z)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,Y),Ya)),Z) ) ) ).

% AND_upper1''
tff(fact_9256_AND__upper2_H_H,axiom,
    ! [Y: int,Z: int,X: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Y),Z)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,X),Y)),Z) ) ) ).

% AND_upper2''
tff(fact_9257_mod__h__bot__iff_I2_J,axiom,
    ! [Ha: heap_e7401611519738050253t_unit] : aa_Pro653082453696054616_nat_o(rep_assn(top_top_assn),produc7507926704131184380et_nat(Ha,bot_bot_set_nat)) ).

% mod_h_bot_iff(2)
tff(fact_9258_and__not__num_Osimps_I2_J,axiom,
    ! [N2: num] : aa_num_option_num(bit_and_not_num(one),aa_num_num(bit0,N2)) = aa_num_option_num(some_num,one) ).

% and_not_num.simps(2)
tff(fact_9259_and__not__num_Osimps_I4_J,axiom,
    ! [M: num] : aa_num_option_num(bit_and_not_num(aa_num_num(bit0,M)),one) = aa_num_option_num(some_num,aa_num_num(bit0,M)) ).

% and_not_num.simps(4)
tff(fact_9260_and__not__num_Osimps_I7_J,axiom,
    ! [M: num] : aa_num_option_num(bit_and_not_num(aa_num_num(bit1,M)),one) = aa_num_option_num(some_num,aa_num_num(bit0,M)) ).

% and_not_num.simps(7)
tff(fact_9261_and__int_Oelims,axiom,
    ! [X: int,Xa2: int,Y: int] :
      ( ( aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,X),Xa2) = 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(Xa2),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))),Xa2) ))),
            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))),Xa2) ))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,X),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(divide_divide_int,Xa2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ) ) ) ).

% and_int.elims
tff(fact_9262_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_9263_and__int_Opelims,axiom,
    ! [X: int,Xa2: int,Y: int] :
      ( ( aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,X),Xa2) = Y )
     => ( aa_Pro4592068669816069595_int_o(accp_P1096762738010456898nt_int(bit_and_int_rel),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X),Xa2))
       => ~ ( ( 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(Xa2),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))),Xa2) ))),
                  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))),Xa2) ))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,X),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(divide_divide_int,Xa2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ) )
           => ~ aa_Pro4592068669816069595_int_o(accp_P1096762738010456898nt_int(bit_and_int_rel),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X),Xa2)) ) ) ) ).

% and_int.pelims
tff(fact_9264_atLeastAtMostPlus1__int__conv,axiom,
    ! [M: int,N2: 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),N2))
     => ( set_or1266510415728281911st_int(M,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),N2)) = insert_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),N2),set_or1266510415728281911st_int(M,N2)) ) ) ).

% atLeastAtMostPlus1_int_conv
tff(fact_9265_simp__from__to,axiom,
    ! [I2: int,J: int] :
      set_or1266510415728281911st_int(I2,J) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_int,J),I2),bot_bot_set_int,insert_int(I2,set_or1266510415728281911st_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,I2),one_one_int),J))) ).

% simp_from_to
tff(fact_9266_and__int_Opinduct,axiom,
    ! [A0: int,A1: int,Pa: 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,L4: int] :
            ( aa_Pro4592068669816069595_int_o(accp_P1096762738010456898nt_int(bit_and_int_rel),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,K2),L4))
           => ( ( ~ ( 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(L4),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(Pa,aa_int_int(aa_int_fun_int_int(divide_divide_int,K2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(divide_divide_int,L4),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))) )
             => aa_int_o(aa_int_fun_int_o(Pa,K2),L4) ) )
       => aa_int_o(aa_int_fun_int_o(Pa,A0),A1) ) ) ).

% and_int.pinduct
tff(fact_9267_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_9268_atLeastLessThan__singleton,axiom,
    ! [M: nat] : aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,M),aa_nat_nat(suc,M)) = insert_nat(M,bot_bot_set_nat) ).

% atLeastLessThan_singleton
tff(fact_9269_atMost__0,axiom,
    aa_nat_set_nat(set_ord_atMost_nat,zero_zero_nat) = insert_nat(zero_zero_nat,bot_bot_set_nat) ).

% atMost_0
tff(fact_9270_UNIV__bool,axiom,
    top_top_set_o = insert_o($false,insert_o($true,bot_bot_set_o)) ).

% UNIV_bool
tff(fact_9271_atLeastAtMost__insertL,axiom,
    ! [M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
     => ( insert_nat(M,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,aa_nat_nat(suc,M)),N2)) = aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M),N2) ) ) ).

% atLeastAtMost_insertL
tff(fact_9272_atLeastAtMostSuc__conv,axiom,
    ! [M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(suc,N2))
     => ( aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M),aa_nat_nat(suc,N2)) = insert_nat(aa_nat_nat(suc,N2),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M),N2)) ) ) ).

% atLeastAtMostSuc_conv
tff(fact_9273_Icc__eq__insert__lb__nat,axiom,
    ! [M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
     => ( aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M),N2) = insert_nat(M,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,aa_nat_nat(suc,M)),N2)) ) ) ).

% Icc_eq_insert_lb_nat
tff(fact_9274_atLeastLessThanSuc,axiom,
    ! [M: nat,N2: nat] :
      aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,M),aa_nat_nat(suc,N2)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2),insert_nat(N2,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,M),N2)),bot_bot_set_nat) ).

% atLeastLessThanSuc
tff(fact_9275_atLeast1__atMost__eq__remove0,axiom,
    ! [N2: nat] : aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,aa_nat_nat(suc,zero_zero_nat)),N2) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,aa_nat_set_nat(set_ord_atMost_nat,N2)),insert_nat(zero_zero_nat,bot_bot_set_nat)) ).

% atLeast1_atMost_eq_remove0
tff(fact_9276_atLeast1__lessThan__eq__remove0,axiom,
    ! [N2: nat] : aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,aa_nat_nat(suc,zero_zero_nat)),N2) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,aa_nat_set_nat(set_ord_lessThan_nat,N2)),insert_nat(zero_zero_nat,bot_bot_set_nat)) ).

% atLeast1_lessThan_eq_remove0
tff(fact_9277_atLeastLessThan__nat__numeral,axiom,
    ! [M: nat,K: num] :
      aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,M),aa_num_nat(numeral_numeral_nat,K)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_num_nat(pred_numeral,K)),insert_nat(aa_num_nat(pred_numeral,K),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,M),aa_num_nat(pred_numeral,K))),bot_bot_set_nat) ).

% atLeastLessThan_nat_numeral
tff(fact_9278_image__minus__const__atLeastLessThan__nat,axiom,
    ! [Ca: nat,X: nat,Y: nat] :
      image_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_amp,Ca),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,X),Y)) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ca),Y),
        aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,X),Ca)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Y),Ca)),
        $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y),insert_nat(zero_zero_nat,bot_bot_set_nat),bot_bot_set_nat) ) ).

% image_minus_const_atLeastLessThan_nat
tff(fact_9279_and__not__num_Oelims,axiom,
    ! [X: num,Xa2: num,Y: option_num] :
      ( ( aa_num_option_num(bit_and_not_num(X),Xa2) = Y )
     => ( ( ( X = one )
         => ( ( Xa2 = one )
           => ( Y != none_num ) ) )
       => ( ( ( X = one )
           => ( ? [N5: num] : Xa2 = aa_num_num(bit0,N5)
             => ( Y != aa_num_option_num(some_num,one) ) ) )
         => ( ( ( X = one )
             => ( ? [N5: num] : Xa2 = aa_num_num(bit1,N5)
               => ( Y != none_num ) ) )
           => ( ! [M5: num] :
                  ( ( X = aa_num_num(bit0,M5) )
                 => ( ( Xa2 = one )
                   => ( Y != aa_num_option_num(some_num,aa_num_num(bit0,M5)) ) ) )
             => ( ! [M5: num] :
                    ( ( X = aa_num_num(bit0,M5) )
                   => ! [N5: num] :
                        ( ( Xa2 = aa_num_num(bit0,N5) )
                       => ( Y != aa_opt5369159329878751634on_num(map_option_num_num(bit0),aa_num_option_num(bit_and_not_num(M5),N5)) ) ) )
               => ( ! [M5: num] :
                      ( ( X = aa_num_num(bit0,M5) )
                     => ! [N5: num] :
                          ( ( Xa2 = aa_num_num(bit1,N5) )
                         => ( Y != aa_opt5369159329878751634on_num(map_option_num_num(bit0),aa_num_option_num(bit_and_not_num(M5),N5)) ) ) )
                 => ( ! [M5: num] :
                        ( ( X = aa_num_num(bit1,M5) )
                       => ( ( Xa2 = one )
                         => ( Y != aa_num_option_num(some_num,aa_num_num(bit0,M5)) ) ) )
                   => ( ! [M5: num] :
                          ( ( X = aa_num_num(bit1,M5) )
                         => ! [N5: num] :
                              ( ( Xa2 = aa_num_num(bit0,N5) )
                             => ( Y != aa_opt5369159329878751634on_num(case_o6005452278849405969um_num(aa_num_option_num(some_num,one),aTP_Lamm_amo),aa_num_option_num(bit_and_not_num(M5),N5)) ) ) )
                     => ~ ! [M5: num] :
                            ( ( X = aa_num_num(bit1,M5) )
                           => ! [N5: num] :
                                ( ( Xa2 = aa_num_num(bit1,N5) )
                               => ( Y != aa_opt5369159329878751634on_num(map_option_num_num(bit0),aa_num_option_num(bit_and_not_num(M5),N5)) ) ) ) ) ) ) ) ) ) ) ) ) ).

% and_not_num.elims
tff(fact_9280_and__not__num_Opelims,axiom,
    ! [X: num,Xa2: num,Y: option_num] :
      ( ( aa_num_option_num(bit_and_not_num(X),Xa2) = Y )
     => ( aa_Pro7685007914727650815_num_o(accp_P3113834385874906142um_num(bit_and_not_num_rel),product_Pair_num_num(X,Xa2))
       => ( ( ( X = one )
           => ( ( Xa2 = one )
             => ( ( Y = none_num )
               => ~ aa_Pro7685007914727650815_num_o(accp_P3113834385874906142um_num(bit_and_not_num_rel),product_Pair_num_num(one,one)) ) ) )
         => ( ( ( X = one )
             => ! [N5: num] :
                  ( ( Xa2 = aa_num_num(bit0,N5) )
                 => ( ( Y = aa_num_option_num(some_num,one) )
                   => ~ aa_Pro7685007914727650815_num_o(accp_P3113834385874906142um_num(bit_and_not_num_rel),product_Pair_num_num(one,aa_num_num(bit0,N5))) ) ) )
           => ( ( ( X = one )
               => ! [N5: num] :
                    ( ( Xa2 = aa_num_num(bit1,N5) )
                   => ( ( Y = none_num )
                     => ~ aa_Pro7685007914727650815_num_o(accp_P3113834385874906142um_num(bit_and_not_num_rel),product_Pair_num_num(one,aa_num_num(bit1,N5))) ) ) )
             => ( ! [M5: num] :
                    ( ( X = aa_num_num(bit0,M5) )
                   => ( ( Xa2 = one )
                     => ( ( Y = aa_num_option_num(some_num,aa_num_num(bit0,M5)) )
                       => ~ aa_Pro7685007914727650815_num_o(accp_P3113834385874906142um_num(bit_and_not_num_rel),product_Pair_num_num(aa_num_num(bit0,M5),one)) ) ) )
               => ( ! [M5: num] :
                      ( ( X = aa_num_num(bit0,M5) )
                     => ! [N5: num] :
                          ( ( Xa2 = aa_num_num(bit0,N5) )
                         => ( ( Y = aa_opt5369159329878751634on_num(map_option_num_num(bit0),aa_num_option_num(bit_and_not_num(M5),N5)) )
                           => ~ aa_Pro7685007914727650815_num_o(accp_P3113834385874906142um_num(bit_and_not_num_rel),product_Pair_num_num(aa_num_num(bit0,M5),aa_num_num(bit0,N5))) ) ) )
                 => ( ! [M5: num] :
                        ( ( X = aa_num_num(bit0,M5) )
                       => ! [N5: num] :
                            ( ( Xa2 = aa_num_num(bit1,N5) )
                           => ( ( Y = aa_opt5369159329878751634on_num(map_option_num_num(bit0),aa_num_option_num(bit_and_not_num(M5),N5)) )
                             => ~ aa_Pro7685007914727650815_num_o(accp_P3113834385874906142um_num(bit_and_not_num_rel),product_Pair_num_num(aa_num_num(bit0,M5),aa_num_num(bit1,N5))) ) ) )
                   => ( ! [M5: num] :
                          ( ( X = aa_num_num(bit1,M5) )
                         => ( ( Xa2 = one )
                           => ( ( Y = aa_num_option_num(some_num,aa_num_num(bit0,M5)) )
                             => ~ aa_Pro7685007914727650815_num_o(accp_P3113834385874906142um_num(bit_and_not_num_rel),product_Pair_num_num(aa_num_num(bit1,M5),one)) ) ) )
                     => ( ! [M5: num] :
                            ( ( X = aa_num_num(bit1,M5) )
                           => ! [N5: num] :
                                ( ( Xa2 = aa_num_num(bit0,N5) )
                               => ( ( Y = aa_opt5369159329878751634on_num(case_o6005452278849405969um_num(aa_num_option_num(some_num,one),aTP_Lamm_amo),aa_num_option_num(bit_and_not_num(M5),N5)) )
                                 => ~ aa_Pro7685007914727650815_num_o(accp_P3113834385874906142um_num(bit_and_not_num_rel),product_Pair_num_num(aa_num_num(bit1,M5),aa_num_num(bit0,N5))) ) ) )
                       => ~ ! [M5: num] :
                              ( ( X = aa_num_num(bit1,M5) )
                             => ! [N5: num] :
                                  ( ( Xa2 = aa_num_num(bit1,N5) )
                                 => ( ( Y = aa_opt5369159329878751634on_num(map_option_num_num(bit0),aa_num_option_num(bit_and_not_num(M5),N5)) )
                                   => ~ aa_Pro7685007914727650815_num_o(accp_P3113834385874906142um_num(bit_and_not_num_rel),product_Pair_num_num(aa_num_num(bit1,M5),aa_num_num(bit1,N5))) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% and_not_num.pelims
tff(fact_9281_and__num_Oelims,axiom,
    ! [X: num,Xa2: num,Y: option_num] :
      ( ( aa_num_option_num(bit_un7362597486090784418nd_num(X),Xa2) = Y )
     => ( ( ( X = one )
         => ( ( Xa2 = one )
           => ( Y != aa_num_option_num(some_num,one) ) ) )
       => ( ( ( X = one )
           => ( ? [N5: num] : Xa2 = aa_num_num(bit0,N5)
             => ( Y != none_num ) ) )
         => ( ( ( X = one )
             => ( ? [N5: num] : Xa2 = aa_num_num(bit1,N5)
               => ( Y != aa_num_option_num(some_num,one) ) ) )
           => ( ( ? [M5: num] : X = aa_num_num(bit0,M5)
               => ( ( Xa2 = one )
                 => ( Y != none_num ) ) )
             => ( ! [M5: num] :
                    ( ( X = aa_num_num(bit0,M5) )
                   => ! [N5: num] :
                        ( ( Xa2 = aa_num_num(bit0,N5) )
                       => ( Y != aa_opt5369159329878751634on_num(map_option_num_num(bit0),aa_num_option_num(bit_un7362597486090784418nd_num(M5),N5)) ) ) )
               => ( ! [M5: num] :
                      ( ( X = aa_num_num(bit0,M5) )
                     => ! [N5: num] :
                          ( ( Xa2 = aa_num_num(bit1,N5) )
                         => ( Y != aa_opt5369159329878751634on_num(map_option_num_num(bit0),aa_num_option_num(bit_un7362597486090784418nd_num(M5),N5)) ) ) )
                 => ( ( ? [M5: num] : X = aa_num_num(bit1,M5)
                     => ( ( Xa2 = one )
                       => ( Y != aa_num_option_num(some_num,one) ) ) )
                   => ( ! [M5: num] :
                          ( ( X = aa_num_num(bit1,M5) )
                         => ! [N5: num] :
                              ( ( Xa2 = aa_num_num(bit0,N5) )
                             => ( Y != aa_opt5369159329878751634on_num(map_option_num_num(bit0),aa_num_option_num(bit_un7362597486090784418nd_num(M5),N5)) ) ) )
                     => ~ ! [M5: num] :
                            ( ( X = aa_num_num(bit1,M5) )
                           => ! [N5: num] :
                                ( ( Xa2 = aa_num_num(bit1,N5) )
                               => ( Y != aa_opt5369159329878751634on_num(case_o6005452278849405969um_num(aa_num_option_num(some_num,one),aTP_Lamm_amo),aa_num_option_num(bit_un7362597486090784418nd_num(M5),N5)) ) ) ) ) ) ) ) ) ) ) ) ) ).

% and_num.elims
tff(fact_9282_xor__num_Oelims,axiom,
    ! [X: num,Xa2: num,Y: option_num] :
      ( ( aa_num_option_num(bit_un2480387367778600638or_num(X),Xa2) = Y )
     => ( ( ( X = one )
         => ( ( Xa2 = one )
           => ( Y != none_num ) ) )
       => ( ( ( X = one )
           => ! [N5: num] :
                ( ( Xa2 = aa_num_num(bit0,N5) )
               => ( Y != aa_num_option_num(some_num,aa_num_num(bit1,N5)) ) ) )
         => ( ( ( X = one )
             => ! [N5: num] :
                  ( ( Xa2 = aa_num_num(bit1,N5) )
                 => ( Y != aa_num_option_num(some_num,aa_num_num(bit0,N5)) ) ) )
           => ( ! [M5: num] :
                  ( ( X = aa_num_num(bit0,M5) )
                 => ( ( Xa2 = one )
                   => ( Y != aa_num_option_num(some_num,aa_num_num(bit1,M5)) ) ) )
             => ( ! [M5: num] :
                    ( ( X = aa_num_num(bit0,M5) )
                   => ! [N5: num] :
                        ( ( Xa2 = aa_num_num(bit0,N5) )
                       => ( Y != aa_opt5369159329878751634on_num(map_option_num_num(bit0),aa_num_option_num(bit_un2480387367778600638or_num(M5),N5)) ) ) )
               => ( ! [M5: num] :
                      ( ( X = aa_num_num(bit0,M5) )
                     => ! [N5: num] :
                          ( ( Xa2 = aa_num_num(bit1,N5) )
                         => ( Y != aa_num_option_num(some_num,aa_option_num_num(case_option_num_num(one,bit1),aa_num_option_num(bit_un2480387367778600638or_num(M5),N5))) ) ) )
                 => ( ! [M5: num] :
                        ( ( X = aa_num_num(bit1,M5) )
                       => ( ( Xa2 = one )
                         => ( Y != aa_num_option_num(some_num,aa_num_num(bit0,M5)) ) ) )
                   => ( ! [M5: num] :
                          ( ( X = aa_num_num(bit1,M5) )
                         => ! [N5: num] :
                              ( ( Xa2 = aa_num_num(bit0,N5) )
                             => ( Y != aa_num_option_num(some_num,aa_option_num_num(case_option_num_num(one,bit1),aa_num_option_num(bit_un2480387367778600638or_num(M5),N5))) ) ) )
                     => ~ ! [M5: num] :
                            ( ( X = aa_num_num(bit1,M5) )
                           => ! [N5: num] :
                                ( ( Xa2 = aa_num_num(bit1,N5) )
                               => ( Y != aa_opt5369159329878751634on_num(map_option_num_num(bit0),aa_num_option_num(bit_un2480387367778600638or_num(M5),N5)) ) ) ) ) ) ) ) ) ) ) ) ) ).

% xor_num.elims
tff(fact_9283_and__num_Osimps_I1_J,axiom,
    aa_num_option_num(bit_un7362597486090784418nd_num(one),one) = aa_num_option_num(some_num,one) ).

% and_num.simps(1)
tff(fact_9284_and__num_Osimps_I3_J,axiom,
    ! [N2: num] : aa_num_option_num(bit_un7362597486090784418nd_num(one),aa_num_num(bit1,N2)) = aa_num_option_num(some_num,one) ).

% and_num.simps(3)
tff(fact_9285_and__num_Osimps_I7_J,axiom,
    ! [M: num] : aa_num_option_num(bit_un7362597486090784418nd_num(aa_num_num(bit1,M)),one) = aa_num_option_num(some_num,one) ).

% and_num.simps(7)
tff(fact_9286_xor__num_Osimps_I2_J,axiom,
    ! [N2: num] : aa_num_option_num(bit_un2480387367778600638or_num(one),aa_num_num(bit0,N2)) = aa_num_option_num(some_num,aa_num_num(bit1,N2)) ).

% xor_num.simps(2)
tff(fact_9287_xor__num_Osimps_I3_J,axiom,
    ! [N2: num] : aa_num_option_num(bit_un2480387367778600638or_num(one),aa_num_num(bit1,N2)) = aa_num_option_num(some_num,aa_num_num(bit0,N2)) ).

% xor_num.simps(3)
tff(fact_9288_xor__num_Osimps_I4_J,axiom,
    ! [M: num] : aa_num_option_num(bit_un2480387367778600638or_num(aa_num_num(bit0,M)),one) = aa_num_option_num(some_num,aa_num_num(bit1,M)) ).

% xor_num.simps(4)
tff(fact_9289_xor__num_Osimps_I7_J,axiom,
    ! [M: num] : aa_num_option_num(bit_un2480387367778600638or_num(aa_num_num(bit1,M)),one) = aa_num_option_num(some_num,aa_num_num(bit0,M)) ).

% xor_num.simps(7)
tff(fact_9290_and__num_Osimps_I9_J,axiom,
    ! [M: num,N2: num] : aa_num_option_num(bit_un7362597486090784418nd_num(aa_num_num(bit1,M)),aa_num_num(bit1,N2)) = aa_opt5369159329878751634on_num(case_o6005452278849405969um_num(aa_num_option_num(some_num,one),aTP_Lamm_amo),aa_num_option_num(bit_un7362597486090784418nd_num(M),N2)) ).

% and_num.simps(9)
tff(fact_9291_xor__num_Osimps_I6_J,axiom,
    ! [M: num,N2: num] : aa_num_option_num(bit_un2480387367778600638or_num(aa_num_num(bit0,M)),aa_num_num(bit1,N2)) = aa_num_option_num(some_num,aa_option_num_num(case_option_num_num(one,bit1),aa_num_option_num(bit_un2480387367778600638or_num(M),N2))) ).

% xor_num.simps(6)
tff(fact_9292_xor__num_Osimps_I8_J,axiom,
    ! [M: num,N2: num] : aa_num_option_num(bit_un2480387367778600638or_num(aa_num_num(bit1,M)),aa_num_num(bit0,N2)) = aa_num_option_num(some_num,aa_option_num_num(case_option_num_num(one,bit1),aa_num_option_num(bit_un2480387367778600638or_num(M),N2))) ).

% xor_num.simps(8)
tff(fact_9293_and__num_Opelims,axiom,
    ! [X: num,Xa2: num,Y: option_num] :
      ( ( aa_num_option_num(bit_un7362597486090784418nd_num(X),Xa2) = Y )
     => ( aa_Pro7685007914727650815_num_o(accp_P3113834385874906142um_num(bit_un4731106466462545111um_rel),product_Pair_num_num(X,Xa2))
       => ( ( ( X = one )
           => ( ( Xa2 = one )
             => ( ( Y = aa_num_option_num(some_num,one) )
               => ~ aa_Pro7685007914727650815_num_o(accp_P3113834385874906142um_num(bit_un4731106466462545111um_rel),product_Pair_num_num(one,one)) ) ) )
         => ( ( ( X = one )
             => ! [N5: num] :
                  ( ( Xa2 = aa_num_num(bit0,N5) )
                 => ( ( Y = none_num )
                   => ~ aa_Pro7685007914727650815_num_o(accp_P3113834385874906142um_num(bit_un4731106466462545111um_rel),product_Pair_num_num(one,aa_num_num(bit0,N5))) ) ) )
           => ( ( ( X = one )
               => ! [N5: num] :
                    ( ( Xa2 = aa_num_num(bit1,N5) )
                   => ( ( Y = aa_num_option_num(some_num,one) )
                     => ~ aa_Pro7685007914727650815_num_o(accp_P3113834385874906142um_num(bit_un4731106466462545111um_rel),product_Pair_num_num(one,aa_num_num(bit1,N5))) ) ) )
             => ( ! [M5: num] :
                    ( ( X = aa_num_num(bit0,M5) )
                   => ( ( Xa2 = one )
                     => ( ( Y = none_num )
                       => ~ aa_Pro7685007914727650815_num_o(accp_P3113834385874906142um_num(bit_un4731106466462545111um_rel),product_Pair_num_num(aa_num_num(bit0,M5),one)) ) ) )
               => ( ! [M5: num] :
                      ( ( X = aa_num_num(bit0,M5) )
                     => ! [N5: num] :
                          ( ( Xa2 = aa_num_num(bit0,N5) )
                         => ( ( Y = aa_opt5369159329878751634on_num(map_option_num_num(bit0),aa_num_option_num(bit_un7362597486090784418nd_num(M5),N5)) )
                           => ~ aa_Pro7685007914727650815_num_o(accp_P3113834385874906142um_num(bit_un4731106466462545111um_rel),product_Pair_num_num(aa_num_num(bit0,M5),aa_num_num(bit0,N5))) ) ) )
                 => ( ! [M5: num] :
                        ( ( X = aa_num_num(bit0,M5) )
                       => ! [N5: num] :
                            ( ( Xa2 = aa_num_num(bit1,N5) )
                           => ( ( Y = aa_opt5369159329878751634on_num(map_option_num_num(bit0),aa_num_option_num(bit_un7362597486090784418nd_num(M5),N5)) )
                             => ~ aa_Pro7685007914727650815_num_o(accp_P3113834385874906142um_num(bit_un4731106466462545111um_rel),product_Pair_num_num(aa_num_num(bit0,M5),aa_num_num(bit1,N5))) ) ) )
                   => ( ! [M5: num] :
                          ( ( X = aa_num_num(bit1,M5) )
                         => ( ( Xa2 = one )
                           => ( ( Y = aa_num_option_num(some_num,one) )
                             => ~ aa_Pro7685007914727650815_num_o(accp_P3113834385874906142um_num(bit_un4731106466462545111um_rel),product_Pair_num_num(aa_num_num(bit1,M5),one)) ) ) )
                     => ( ! [M5: num] :
                            ( ( X = aa_num_num(bit1,M5) )
                           => ! [N5: num] :
                                ( ( Xa2 = aa_num_num(bit0,N5) )
                               => ( ( Y = aa_opt5369159329878751634on_num(map_option_num_num(bit0),aa_num_option_num(bit_un7362597486090784418nd_num(M5),N5)) )
                                 => ~ aa_Pro7685007914727650815_num_o(accp_P3113834385874906142um_num(bit_un4731106466462545111um_rel),product_Pair_num_num(aa_num_num(bit1,M5),aa_num_num(bit0,N5))) ) ) )
                       => ~ ! [M5: num] :
                              ( ( X = aa_num_num(bit1,M5) )
                             => ! [N5: num] :
                                  ( ( Xa2 = aa_num_num(bit1,N5) )
                                 => ( ( Y = aa_opt5369159329878751634on_num(case_o6005452278849405969um_num(aa_num_option_num(some_num,one),aTP_Lamm_amo),aa_num_option_num(bit_un7362597486090784418nd_num(M5),N5)) )
                                   => ~ aa_Pro7685007914727650815_num_o(accp_P3113834385874906142um_num(bit_un4731106466462545111um_rel),product_Pair_num_num(aa_num_num(bit1,M5),aa_num_num(bit1,N5))) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% and_num.pelims
tff(fact_9294_xor__num_Opelims,axiom,
    ! [X: num,Xa2: num,Y: option_num] :
      ( ( aa_num_option_num(bit_un2480387367778600638or_num(X),Xa2) = Y )
     => ( aa_Pro7685007914727650815_num_o(accp_P3113834385874906142um_num(bit_un2901131394128224187um_rel),product_Pair_num_num(X,Xa2))
       => ( ( ( X = one )
           => ( ( Xa2 = one )
             => ( ( Y = none_num )
               => ~ aa_Pro7685007914727650815_num_o(accp_P3113834385874906142um_num(bit_un2901131394128224187um_rel),product_Pair_num_num(one,one)) ) ) )
         => ( ( ( X = one )
             => ! [N5: num] :
                  ( ( Xa2 = aa_num_num(bit0,N5) )
                 => ( ( Y = aa_num_option_num(some_num,aa_num_num(bit1,N5)) )
                   => ~ aa_Pro7685007914727650815_num_o(accp_P3113834385874906142um_num(bit_un2901131394128224187um_rel),product_Pair_num_num(one,aa_num_num(bit0,N5))) ) ) )
           => ( ( ( X = one )
               => ! [N5: num] :
                    ( ( Xa2 = aa_num_num(bit1,N5) )
                   => ( ( Y = aa_num_option_num(some_num,aa_num_num(bit0,N5)) )
                     => ~ aa_Pro7685007914727650815_num_o(accp_P3113834385874906142um_num(bit_un2901131394128224187um_rel),product_Pair_num_num(one,aa_num_num(bit1,N5))) ) ) )
             => ( ! [M5: num] :
                    ( ( X = aa_num_num(bit0,M5) )
                   => ( ( Xa2 = one )
                     => ( ( Y = aa_num_option_num(some_num,aa_num_num(bit1,M5)) )
                       => ~ aa_Pro7685007914727650815_num_o(accp_P3113834385874906142um_num(bit_un2901131394128224187um_rel),product_Pair_num_num(aa_num_num(bit0,M5),one)) ) ) )
               => ( ! [M5: num] :
                      ( ( X = aa_num_num(bit0,M5) )
                     => ! [N5: num] :
                          ( ( Xa2 = aa_num_num(bit0,N5) )
                         => ( ( Y = aa_opt5369159329878751634on_num(map_option_num_num(bit0),aa_num_option_num(bit_un2480387367778600638or_num(M5),N5)) )
                           => ~ aa_Pro7685007914727650815_num_o(accp_P3113834385874906142um_num(bit_un2901131394128224187um_rel),product_Pair_num_num(aa_num_num(bit0,M5),aa_num_num(bit0,N5))) ) ) )
                 => ( ! [M5: num] :
                        ( ( X = aa_num_num(bit0,M5) )
                       => ! [N5: num] :
                            ( ( Xa2 = aa_num_num(bit1,N5) )
                           => ( ( Y = aa_num_option_num(some_num,aa_option_num_num(case_option_num_num(one,bit1),aa_num_option_num(bit_un2480387367778600638or_num(M5),N5))) )
                             => ~ aa_Pro7685007914727650815_num_o(accp_P3113834385874906142um_num(bit_un2901131394128224187um_rel),product_Pair_num_num(aa_num_num(bit0,M5),aa_num_num(bit1,N5))) ) ) )
                   => ( ! [M5: num] :
                          ( ( X = aa_num_num(bit1,M5) )
                         => ( ( Xa2 = one )
                           => ( ( Y = aa_num_option_num(some_num,aa_num_num(bit0,M5)) )
                             => ~ aa_Pro7685007914727650815_num_o(accp_P3113834385874906142um_num(bit_un2901131394128224187um_rel),product_Pair_num_num(aa_num_num(bit1,M5),one)) ) ) )
                     => ( ! [M5: num] :
                            ( ( X = aa_num_num(bit1,M5) )
                           => ! [N5: num] :
                                ( ( Xa2 = aa_num_num(bit0,N5) )
                               => ( ( Y = aa_num_option_num(some_num,aa_option_num_num(case_option_num_num(one,bit1),aa_num_option_num(bit_un2480387367778600638or_num(M5),N5))) )
                                 => ~ aa_Pro7685007914727650815_num_o(accp_P3113834385874906142um_num(bit_un2901131394128224187um_rel),product_Pair_num_num(aa_num_num(bit1,M5),aa_num_num(bit0,N5))) ) ) )
                       => ~ ! [M5: num] :
                              ( ( X = aa_num_num(bit1,M5) )
                             => ! [N5: num] :
                                  ( ( Xa2 = aa_num_num(bit1,N5) )
                                 => ( ( Y = aa_opt5369159329878751634on_num(map_option_num_num(bit0),aa_num_option_num(bit_un2480387367778600638or_num(M5),N5)) )
                                   => ~ aa_Pro7685007914727650815_num_o(accp_P3113834385874906142um_num(bit_un2901131394128224187um_rel),product_Pair_num_num(aa_num_num(bit1,M5),aa_num_num(bit1,N5))) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% xor_num.pelims
tff(fact_9295_times__int_Oabs__eq,axiom,
    ! [Xa2: 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,Xa2)),aa_Pro4240683299535347697at_int(abs_Integ,X)) = aa_Pro4240683299535347697at_int(abs_Integ,aa_Pro262675715343573244at_nat(aa_Pro8023792847356852296at_nat(aa_fun6875833325056119164at_nat(produc4528391901023682385at_nat,aTP_Lamm_amr),Xa2),X)) ).

% times_int.abs_eq
tff(fact_9296_eq__Abs__Integ,axiom,
    ! [Z: int] :
      ~ ! [X3: nat,Y3: nat] : Z != aa_Pro4240683299535347697at_int(abs_Integ,aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,X3),Y3)) ).

% eq_Abs_Integ
tff(fact_9297_zero__int__def,axiom,
    zero_zero_int = aa_Pro4240683299535347697at_int(abs_Integ,aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,zero_zero_nat),zero_zero_nat)) ).

% zero_int_def
tff(fact_9298_int__def,axiom,
    ! [N2: nat] : aa_nat_int(semiri1314217659103216013at_int,N2) = aa_Pro4240683299535347697at_int(abs_Integ,aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,N2),zero_zero_nat)) ).

% int_def
tff(fact_9299_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_ams),X)) ).

% uminus_int.abs_eq
tff(fact_9300_one__int__def,axiom,
    one_one_int = aa_Pro4240683299535347697at_int(abs_Integ,aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,one_one_nat),zero_zero_nat)) ).

% one_int_def
tff(fact_9301_less__int_Oabs__eq,axiom,
    ! [Xa2: 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,Xa2)),aa_Pro4240683299535347697at_int(abs_Integ,X))
    <=> aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(aa_fun5912894404062470332_nat_o(produc6395135316500328296_nat_o,aTP_Lamm_amu),Xa2),X) ) ).

% less_int.abs_eq
tff(fact_9302_less__eq__int_Oabs__eq,axiom,
    ! [Xa2: 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,Xa2)),aa_Pro4240683299535347697at_int(abs_Integ,X))
    <=> aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(aa_fun5912894404062470332_nat_o(produc6395135316500328296_nat_o,aTP_Lamm_amw),Xa2),X) ) ).

% less_eq_int.abs_eq
tff(fact_9303_plus__int_Oabs__eq,axiom,
    ! [Xa2: 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,Xa2)),aa_Pro4240683299535347697at_int(abs_Integ,X)) = aa_Pro4240683299535347697at_int(abs_Integ,aa_Pro262675715343573244at_nat(aa_Pro8023792847356852296at_nat(aa_fun6875833325056119164at_nat(produc4528391901023682385at_nat,aTP_Lamm_amy),Xa2),X)) ).

% plus_int.abs_eq
tff(fact_9304_minus__int_Oabs__eq,axiom,
    ! [Xa2: 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,Xa2)),aa_Pro4240683299535347697at_int(abs_Integ,X)) = aa_Pro4240683299535347697at_int(abs_Integ,aa_Pro262675715343573244at_nat(aa_Pro8023792847356852296at_nat(aa_fun6875833325056119164at_nat(produc4528391901023682385at_nat,aTP_Lamm_ana),Xa2),X)) ).

% minus_int.abs_eq
tff(fact_9305_less__eq__int_Orep__eq,axiom,
    ! [X: int,Xa2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Xa2)
    <=> aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(aa_fun5912894404062470332_nat_o(produc6395135316500328296_nat_o,aTP_Lamm_amw),aa_int1417691210696471063at_nat(rep_Integ,X)),aa_int1417691210696471063at_nat(rep_Integ,Xa2)) ) ).

% less_eq_int.rep_eq
tff(fact_9306_less__int_Orep__eq,axiom,
    ! [X: int,Xa2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Xa2)
    <=> aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(aa_fun5912894404062470332_nat_o(produc6395135316500328296_nat_o,aTP_Lamm_amu),aa_int1417691210696471063at_nat(rep_Integ,X)),aa_int1417691210696471063at_nat(rep_Integ,Xa2)) ) ).

% less_int.rep_eq
tff(fact_9307_uminus__int__def,axiom,
    uminus_uminus_int = aa_fun8399641135911395862nt_int(map_fu3667384564859982768at_int(rep_Integ,abs_Integ),produc2626176000494625587at_nat(aTP_Lamm_ams)) ).

% uminus_int_def
tff(fact_9308_pred__nat__def,axiom,
    pred_nat = collec3392354462482085612at_nat(produc6081775807080527818_nat_o(aTP_Lamm_anb)) ).

% pred_nat_def
tff(fact_9309_times__int__def,axiom,
    times_times_int = map_fu8294830438409148311nt_int(rep_Integ,map_fu3667384564859982768at_int(rep_Integ,abs_Integ),aa_fun6875833325056119164at_nat(produc4528391901023682385at_nat,aTP_Lamm_amr)) ).

% times_int_def
tff(fact_9310_minus__int__def,axiom,
    minus_minus_int = map_fu8294830438409148311nt_int(rep_Integ,map_fu3667384564859982768at_int(rep_Integ,abs_Integ),aa_fun6875833325056119164at_nat(produc4528391901023682385at_nat,aTP_Lamm_ana)) ).

% minus_int_def
tff(fact_9311_plus__int__def,axiom,
    plus_plus_int = map_fu8294830438409148311nt_int(rep_Integ,map_fu3667384564859982768at_int(rep_Integ,abs_Integ),aa_fun6875833325056119164at_nat(produc4528391901023682385at_nat,aTP_Lamm_amy)) ).

% plus_int_def
tff(fact_9312_Gcd__remove0__nat,axiom,
    ! [M6: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,M6)
     => ( aa_set_nat_nat(gcd_Gcd_nat,M6) = aa_set_nat_nat(gcd_Gcd_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,M6),insert_nat(zero_zero_nat,bot_bot_set_nat))) ) ) ).

% Gcd_remove0_nat
tff(fact_9313_num__of__nat_Osimps_I2_J,axiom,
    ! [N2: nat] :
      aa_nat_num(num_of_nat,aa_nat_nat(suc,N2)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2),aa_num_num(inc,aa_nat_num(num_of_nat,N2)),one) ).

% num_of_nat.simps(2)
tff(fact_9314_Gcd__eq__Max,axiom,
    ! [M6: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,M6)
     => ( ( M6 != bot_bot_set_nat )
       => ( ~ aa_set_nat_o(member_nat(zero_zero_nat),M6)
         => ( aa_set_nat_nat(gcd_Gcd_nat,M6) = aa_set_nat_nat(lattic8265883725875713057ax_nat,comple7806235888213564991et_nat(image_nat_set_nat(aTP_Lamm_anc,M6))) ) ) ) ) ).

% Gcd_eq_Max
tff(fact_9315_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_9316_Gcd__nat__abs__eq,axiom,
    ! [K5: set_int] : aa_set_nat_nat(gcd_Gcd_nat,image_int_nat(aTP_Lamm_and,K5)) = aa_int_nat(nat2,gcd_Gcd_int(K5)) ).

% Gcd_nat_abs_eq
tff(fact_9317_Max__divisors__self__nat,axiom,
    ! [N2: nat] :
      ( ( N2 != zero_zero_nat )
     => ( aa_set_nat_nat(lattic8265883725875713057ax_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_adt,N2))) = N2 ) ) ).

% Max_divisors_self_nat
tff(fact_9318_Gcd__int__eq,axiom,
    ! [N7: set_nat] : gcd_Gcd_int(image_nat_int(semiri1314217659103216013at_int,N7)) = aa_nat_int(semiri1314217659103216013at_int,aa_set_nat_nat(gcd_Gcd_nat,N7)) ).

% Gcd_int_eq
tff(fact_9319_Gcd__int__greater__eq__0,axiom,
    ! [K5: set_int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),gcd_Gcd_int(K5)) ).

% Gcd_int_greater_eq_0
tff(fact_9320_Sup__nat__def,axiom,
    ! [X6: set_nat] :
      aa_set_nat_nat(complete_Sup_Sup_nat,X6) = $ite(X6 = bot_bot_set_nat,zero_zero_nat,aa_set_nat_nat(lattic8265883725875713057ax_nat,X6)) ).

% Sup_nat_def
tff(fact_9321_card__le__Suc__Max,axiom,
    ! [S: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,S)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(finite_card_nat,S)),aa_nat_nat(suc,aa_set_nat_nat(lattic8265883725875713057ax_nat,S))) ) ).

% card_le_Suc_Max
tff(fact_9322_divide__nat__def,axiom,
    ! [M: nat,N2: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N2) = $ite(N2 = zero_zero_nat,zero_zero_nat,aa_set_nat_nat(lattic8265883725875713057ax_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_ane(M),N2)))) ).

% divide_nat_def
tff(fact_9323_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_9324_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_9325_Max__divisors__self__int,axiom,
    ! [N2: int] :
      ( ( N2 != zero_zero_int )
     => ( lattic8263393255366662781ax_int(collect_int(aa_int_fun_int_o(aTP_Lamm_adv,N2))) = aa_int_int(abs_abs_int,N2) ) ) ).

% Max_divisors_self_int
tff(fact_9326_greaterThan__Suc,axiom,
    ! [K: nat] : aa_nat_set_nat(set_or1210151606488870762an_nat,aa_nat_nat(suc,K)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,aa_nat_set_nat(set_or1210151606488870762an_nat,K)),insert_nat(aa_nat_nat(suc,K),bot_bot_set_nat)) ).

% greaterThan_Suc
tff(fact_9327_mono__Suc,axiom,
    order_mono_nat_nat(suc) ).

% mono_Suc
tff(fact_9328_mono__times__nat,axiom,
    ! [N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)
     => order_mono_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N2)) ) ).

% mono_times_nat
tff(fact_9329_mono__ge2__power__minus__self,axiom,
    ! [K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K)
     => order_mono_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_anf,K)) ) ).

% mono_ge2_power_minus_self
tff(fact_9330_binomial__def,axiom,
    ! [N2: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,N2),K) = finite_card_set_nat(collect_set_nat(aa_nat_fun_set_nat_o(aTP_Lamm_ang(N2),K))) ).

% binomial_def
tff(fact_9331_upto__aux__rec,axiom,
    ! [I2: int,J: int,Js: list_int] :
      upto_aux(I2,J,Js) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_int,J),I2),Js,upto_aux(I2,aa_int_int(aa_int_fun_int_int(minus_minus_int,J),one_one_int),cons_int(J,Js))) ).

% upto_aux_rec
tff(fact_9332_upto_Opsimps,axiom,
    ! [I2: int,J: int] :
      ( aa_Pro4592068669816069595_int_o(accp_P1096762738010456898nt_int(upto_rel),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,I2),J))
     => ( upto(I2,J) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I2),J),cons_int(I2,upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,I2),one_one_int),J)),nil_int) ) ) ).

% upto.psimps
tff(fact_9333_upto_Opelims,axiom,
    ! [X: int,Xa2: int,Y: list_int] :
      ( ( upto(X,Xa2) = Y )
     => ( aa_Pro4592068669816069595_int_o(accp_P1096762738010456898nt_int(upto_rel),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X),Xa2))
       => ~ ( ( Y = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Xa2),cons_int(X,upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,X),one_one_int),Xa2)),nil_int) )
           => ~ aa_Pro4592068669816069595_int_o(accp_P1096762738010456898nt_int(upto_rel),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X),Xa2)) ) ) ) ).

% upto.pelims
tff(fact_9334_sort__upto,axiom,
    ! [I2: int,J: int] : linord1735203802627413978nt_int(aTP_Lamm_fv,upto(I2,J)) = upto(I2,J) ).

% sort_upto
tff(fact_9335_upto__Nil,axiom,
    ! [I2: int,J: int] :
      ( ( upto(I2,J) = nil_int )
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,J),I2) ) ).

% upto_Nil
tff(fact_9336_upto__Nil2,axiom,
    ! [I2: int,J: int] :
      ( ( nil_int = upto(I2,J) )
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,J),I2) ) ).

% upto_Nil2
tff(fact_9337_upto__empty,axiom,
    ! [J: int,I2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,J),I2)
     => ( upto(I2,J) = nil_int ) ) ).

% upto_empty
tff(fact_9338_nth__upto,axiom,
    ! [I2: int,K: nat,J: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I2),aa_nat_int(semiri1314217659103216013at_int,K))),J)
     => ( aa_nat_int(nth_int(upto(I2,J)),K) = aa_int_int(aa_int_fun_int_int(plus_plus_int,I2),aa_nat_int(semiri1314217659103216013at_int,K)) ) ) ).

% nth_upto
tff(fact_9339_upto__rec__numeral_I1_J,axiom,
    ! [M: num,N2: num] :
      upto(aa_num_int(numeral_numeral_int,M),aa_num_int(numeral_numeral_int,N2)) = $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,N2)),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,N2))),nil_int) ).

% upto_rec_numeral(1)
tff(fact_9340_upto__rec__numeral_I4_J,axiom,
    ! [M: num,N2: 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,N2))) = $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,N2))),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,N2)))),nil_int) ).

% upto_rec_numeral(4)
tff(fact_9341_upto__rec__numeral_I3_J,axiom,
    ! [M: num,N2: num] :
      upto(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,N2)) = $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,N2)),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,N2))),nil_int) ).

% upto_rec_numeral(3)
tff(fact_9342_upto__rec__numeral_I2_J,axiom,
    ! [M: num,N2: num] :
      upto(aa_num_int(numeral_numeral_int,M),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N2))) = $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,N2))),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,N2)))),nil_int) ).

% upto_rec_numeral(2)
tff(fact_9343_sorted__upto,axiom,
    ! [M: int,N2: int] : sorted_wrt_int(ord_less_eq_int,upto(M,N2)) ).

% sorted_upto
tff(fact_9344_sorted__wrt__upto,axiom,
    ! [I2: int,J: int] : sorted_wrt_int(ord_less_int,upto(I2,J)) ).

% sorted_wrt_upto
tff(fact_9345_upto__split2,axiom,
    ! [I2: int,J: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I2),J)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,J),K)
       => ( upto(I2,K) = append_int(upto(I2,J),upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,J),one_one_int),K)) ) ) ) ).

% upto_split2
tff(fact_9346_upto__split1,axiom,
    ! [I2: int,J: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I2),J)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,J),K)
       => ( upto(I2,K) = append_int(upto(I2,aa_int_int(aa_int_fun_int_int(minus_minus_int,J),one_one_int)),upto(J,K)) ) ) ) ).

% upto_split1
tff(fact_9347_upto_Oelims,axiom,
    ! [X: int,Xa2: int,Y: list_int] :
      ( ( upto(X,Xa2) = Y )
     => ( Y = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Xa2),cons_int(X,upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,X),one_one_int),Xa2)),nil_int) ) ) ).

% upto.elims
tff(fact_9348_upto_Osimps,axiom,
    ! [I2: int,J: int] :
      upto(I2,J) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I2),J),cons_int(I2,upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,I2),one_one_int),J)),nil_int) ).

% upto.simps
tff(fact_9349_upto__rec1,axiom,
    ! [I2: int,J: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I2),J)
     => ( upto(I2,J) = cons_int(I2,upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,I2),one_one_int),J)) ) ) ).

% upto_rec1
tff(fact_9350_upto__rec2,axiom,
    ! [I2: int,J: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I2),J)
     => ( upto(I2,J) = append_int(upto(I2,aa_int_int(aa_int_fun_int_int(minus_minus_int,J),one_one_int)),cons_int(J,nil_int)) ) ) ).

% upto_rec2
tff(fact_9351_upto__split3,axiom,
    ! [I2: int,J: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I2),J)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,J),K)
       => ( upto(I2,K) = append_int(upto(I2,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_9352_card__length__sum__list__rec,axiom,
    ! [M: nat,N7: 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_anh(M),N7))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,finite_card_list_nat(collect_list_nat(aa_nat5883842100386203267_nat_o(aTP_Lamm_ani(M),N7)))),finite_card_list_nat(collect_list_nat(aa_nat5883842100386203267_nat_o(aTP_Lamm_anj(M),N7)))) ) ) ).

% card_length_sum_list_rec
tff(fact_9353_card__length__sum__list,axiom,
    ! [M: nat,N7: nat] : finite_card_list_nat(collect_list_nat(aa_nat5883842100386203267_nat_o(aTP_Lamm_anh(M),N7))) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N7),M)),one_one_nat)),N7) ).

% card_length_sum_list
tff(fact_9354_sort__upt,axiom,
    ! [M: nat,N2: nat] : linord738340561235409698at_nat(aTP_Lamm_fu,aa_nat_list_nat(upt(M),N2)) = aa_nat_list_nat(upt(M),N2) ).

% sort_upt
tff(fact_9355_upt__0__eq__Nil__conv,axiom,
    ! [J: nat] :
      ( ( aa_nat_list_nat(upt(zero_zero_nat),J) = nil_nat )
    <=> ( J = zero_zero_nat ) ) ).

% upt_0_eq_Nil_conv
tff(fact_9356_hd__upt,axiom,
    ! [I2: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),J)
     => ( aa_list_nat_nat(hd_nat,aa_nat_list_nat(upt(I2),J)) = I2 ) ) ).

% hd_upt
tff(fact_9357_upt__conv__Nil,axiom,
    ! [J: nat,I2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,J),I2)
     => ( aa_nat_list_nat(upt(I2),J) = nil_nat ) ) ).

% upt_conv_Nil
tff(fact_9358_upt__merge,axiom,
    ! [I2: nat,J: nat,K: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),J)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,J),K) )
     => ( append_nat(aa_nat_list_nat(upt(I2),J),aa_nat_list_nat(upt(J),K)) = aa_nat_list_nat(upt(I2),K) ) ) ).

% upt_merge
tff(fact_9359_upt__eq__Nil__conv,axiom,
    ! [I2: nat,J: nat] :
      ( ( aa_nat_list_nat(upt(I2),J) = nil_nat )
    <=> ( ( J = zero_zero_nat )
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,J),I2) ) ) ).

% upt_eq_Nil_conv
tff(fact_9360_take__upt,axiom,
    ! [I2: nat,M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),M)),N2)
     => ( take_nat(M,aa_nat_list_nat(upt(I2),N2)) = aa_nat_list_nat(upt(I2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),M)) ) ) ).

% take_upt
tff(fact_9361_upt__rec__numeral,axiom,
    ! [M: num,N2: num] :
      aa_nat_list_nat(upt(aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,N2)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,N2)),cons_nat(aa_num_nat(numeral_numeral_nat,M),aa_nat_list_nat(upt(aa_nat_nat(suc,aa_num_nat(numeral_numeral_nat,M))),aa_num_nat(numeral_numeral_nat,N2))),nil_nat) ).

% upt_rec_numeral
tff(fact_9362_last__upt,axiom,
    ! [I2: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),J)
     => ( aa_list_nat_nat(last_nat,aa_nat_list_nat(upt(I2),J)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),one_one_nat) ) ) ).

% last_upt
tff(fact_9363_nth__upt,axiom,
    ! [I2: nat,K: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),K)),J)
     => ( aa_nat_nat(nth_nat(aa_nat_list_nat(upt(I2),J)),K) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),K) ) ) ).

% nth_upt
tff(fact_9364_sum__list__upt,axiom,
    ! [M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
     => ( aa_list_nat_nat(groups4561878855575611511st_nat,aa_nat_list_nat(upt(M),N2)) = aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_fu),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,M),N2)) ) ) ).

% sum_list_upt
tff(fact_9365_map__add__upt_H,axiom,
    ! [Ofs: nat,A: nat,B: nat] : map_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_hu,Ofs),aa_nat_list_nat(upt(A),B)) = aa_nat_list_nat(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_9366_upt__eq__append__conv,axiom,
    ! [I2: nat,J: nat,Xs: list_nat,Ys2: list_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),J)
     => ( ( aa_nat_list_nat(upt(I2),J) = append_nat(Xs,Ys2) )
      <=> ? [K3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),K3)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K3),J)
            & ( aa_nat_list_nat(upt(I2),K3) = Xs )
            & ( aa_nat_list_nat(upt(K3),J) = Ys2 ) ) ) ) ).

% upt_eq_append_conv
tff(fact_9367_upt__append,axiom,
    ! [I2: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),J)
     => ( append_nat(aa_nat_list_nat(upt(zero_zero_nat),I2),aa_nat_list_nat(upt(I2),J)) = aa_nat_list_nat(upt(zero_zero_nat),J) ) ) ).

% upt_append
tff(fact_9368_upt__rec,axiom,
    ! [I2: nat,J: nat] :
      aa_nat_list_nat(upt(I2),J) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),J),cons_nat(I2,aa_nat_list_nat(upt(aa_nat_nat(suc,I2)),J)),nil_nat) ).

% upt_rec
tff(fact_9369_upt__conv__Cons,axiom,
    ! [I2: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),J)
     => ( aa_nat_list_nat(upt(I2),J) = cons_nat(I2,aa_nat_list_nat(upt(aa_nat_nat(suc,I2)),J)) ) ) ).

% upt_conv_Cons
tff(fact_9370_upt__Suc,axiom,
    ! [I2: nat,J: nat] :
      aa_nat_list_nat(upt(I2),aa_nat_nat(suc,J)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),J),append_nat(aa_nat_list_nat(upt(I2),J),cons_nat(J,nil_nat)),nil_nat) ).

% upt_Suc
tff(fact_9371_upt__Suc__append,axiom,
    ! [I2: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),J)
     => ( aa_nat_list_nat(upt(I2),aa_nat_nat(suc,J)) = append_nat(aa_nat_list_nat(upt(I2),J),cons_nat(J,nil_nat)) ) ) ).

% upt_Suc_append
tff(fact_9372_upt__add__eq__append,axiom,
    ! [I2: nat,J: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),J)
     => ( aa_nat_list_nat(upt(I2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K)) = append_nat(aa_nat_list_nat(upt(I2),J),aa_nat_list_nat(upt(J),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K))) ) ) ).

% upt_add_eq_append
tff(fact_9373_butlast__upt,axiom,
    ! [M: nat,N2: nat] : butlast_nat(aa_nat_list_nat(upt(M),N2)) = aa_nat_list_nat(upt(M),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),one_one_nat)) ).

% butlast_upt
tff(fact_9374_sorted__wrt__upt,axiom,
    ! [M: nat,N2: nat] : aa_list_nat_o(sorted_wrt_nat(ord_less_nat),aa_nat_list_nat(upt(M),N2)) ).

% sorted_wrt_upt
tff(fact_9375_map__add__upt,axiom,
    ! [N2: nat,M: nat] : map_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_hu,N2),aa_nat_list_nat(upt(zero_zero_nat),M)) = aa_nat_list_nat(upt(N2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N2)) ).

% map_add_upt
tff(fact_9376_sorted__upt,axiom,
    ! [M: nat,N2: nat] : aa_list_nat_o(sorted_wrt_nat(ord_less_eq_nat),aa_nat_list_nat(upt(M),N2)) ).

% sorted_upt
tff(fact_9377_upt__eq__lel__conv,axiom,
    ! [L: nat,Ha: nat,Is1: list_nat,I2: nat,Is2: list_nat] :
      ( ( aa_nat_list_nat(upt(L),Ha) = append_nat(Is1,cons_nat(I2,Is2)) )
    <=> ( ( Is1 = aa_nat_list_nat(upt(L),I2) )
        & ( Is2 = aa_nat_list_nat(upt(aa_nat_nat(suc,I2)),Ha) )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,L),I2)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),Ha) ) ) ).

% upt_eq_lel_conv
tff(fact_9378_upt__eq__Cons__conv,axiom,
    ! [I2: nat,J: nat,X: nat,Xs: list_nat] :
      ( ( aa_nat_list_nat(upt(I2),J) = cons_nat(X,Xs) )
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),J)
        & ( I2 = X )
        & ( aa_nat_list_nat(upt(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),one_one_nat)),J) = Xs ) ) ) ).

% upt_eq_Cons_conv
tff(fact_9379_upt__filter__extend,axiom,
    ! [U: nat,U2: nat,Pa: fun_nat_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,U),U2)
     => ( ! [I3: nat] :
            ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,U),I3)
              & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),U2) )
           => ~ aa_nat_o(Pa,I3) )
       => ( filter_nat2(Pa,aa_nat_list_nat(upt(zero_zero_nat),U)) = filter_nat2(Pa,aa_nat_list_nat(upt(zero_zero_nat),U2)) ) ) ) ).

% upt_filter_extend
tff(fact_9380_map__decr__upt,axiom,
    ! [M: nat,N2: nat] : map_nat_nat(aTP_Lamm_ank,aa_nat_list_nat(upt(aa_nat_nat(suc,M)),aa_nat_nat(suc,N2))) = aa_nat_list_nat(upt(M),N2) ).

% map_decr_upt
tff(fact_9381_bezw_Oelims,axiom,
    ! [X: nat,Xa2: nat,Y: product_prod_int_int] :
      ( ( bezw(X,Xa2) = Y )
     => ( Y = $ite(Xa2 = 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(Xa2,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Xa2)))),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_Pro6799258434465214121nt_int(product_fst_int_int,bezw(Xa2,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Xa2)))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_Pro6799258434465214121nt_int(product_snd_int_int,bezw(Xa2,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Xa2)))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X),Xa2)))))) ) ) ).

% bezw.elims
tff(fact_9382_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_9383_bezw_Opelims,axiom,
    ! [X: nat,Xa2: nat,Y: product_prod_int_int] :
      ( ( bezw(X,Xa2) = Y )
     => ( aa_Pro6150637604445964947_nat_o(accp_P4275260045618599050at_nat(bezw_rel),aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,X),Xa2))
       => ~ ( ( Y = $ite(Xa2 = 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(Xa2,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Xa2)))),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_Pro6799258434465214121nt_int(product_fst_int_int,bezw(Xa2,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Xa2)))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_Pro6799258434465214121nt_int(product_snd_int_int,bezw(Xa2,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Xa2)))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X),Xa2)))))) )
           => ~ aa_Pro6150637604445964947_nat_o(accp_P4275260045618599050at_nat(bezw_rel),aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,X),Xa2)) ) ) ) ).

% bezw.pelims
tff(fact_9384_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_9385_mod__pure,axiom,
    ! [B: $o,Ha: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(rep_assn(aa_o_assn(pure_assn,(B))),Ha)
    <=> ( ( produc8586169260539613262et_nat(Ha) = bot_bot_set_nat )
        & (B) ) ) ).

% mod_pure
tff(fact_9386_mod__emp,axiom,
    ! [Ha: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(rep_assn(one_one_assn),Ha)
    <=> ( produc8586169260539613262et_nat(Ha) = bot_bot_set_nat ) ) ).

% mod_emp
tff(fact_9387_mod__star__trueE_H,axiom,
    ! [Pa: assn,Ha: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(rep_assn(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Pa),top_top_assn)),Ha)
     => ~ ! [H4: produc3658429121746597890et_nat] :
            ( ( produc1824681642469235216et_nat(H4) = produc1824681642469235216et_nat(Ha) )
           => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,produc8586169260539613262et_nat(H4)),produc8586169260539613262et_nat(Ha))
             => ~ aa_Pro653082453696054616_nat_o(rep_assn(Pa),H4) ) ) ) ).

% mod_star_trueE'
tff(fact_9388_natLess__def,axiom,
    bNF_Ca8459412986667044542atLess = collec3392354462482085612at_nat(produc6081775807080527818_nat_o(ord_less_nat)) ).

% natLess_def
tff(fact_9389_less__eq,axiom,
    ! [M: nat,N2: nat] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,M),N2)),transi6264000038957366511cl_nat(pred_nat))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N2) ) ).

% less_eq
tff(fact_9390_pred__nat__trancl__eq__le,axiom,
    ! [M: nat,N2: nat] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,M),N2)),transi2905341329935302413cl_nat(pred_nat))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2) ) ).

% pred_nat_trancl_eq_le
tff(fact_9391_min__0R,axiom,
    ! [N2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,N2),zero_zero_nat) = zero_zero_nat ).

% min_0R
tff(fact_9392_min__0L,axiom,
    ! [N2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,zero_zero_nat),N2) = zero_zero_nat ).

% min_0L
tff(fact_9393_min__Suc__Suc,axiom,
    ! [M: nat,N2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(suc,M)),aa_nat_nat(suc,N2)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,M),N2)) ).

% min_Suc_Suc
tff(fact_9394_min__Suc__gt_I2_J,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,B),aa_nat_nat(suc,A)) = aa_nat_nat(suc,A) ) ) ).

% min_Suc_gt(2)
tff(fact_9395_min__Suc__gt_I1_J,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(suc,A)),B) = aa_nat_nat(suc,A) ) ) ).

% min_Suc_gt(1)
tff(fact_9396_nat__mult__min__right,axiom,
    ! [M: nat,N2: nat,Q4: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,N2),Q4)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Q4)) ).

% nat_mult_min_right
tff(fact_9397_nat__mult__min__left,axiom,
    ! [M: nat,N2: nat,Q4: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,M),N2)),Q4) = aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Q4)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N2),Q4)) ).

% nat_mult_min_left
tff(fact_9398_min__diff,axiom,
    ! [M: nat,I2: nat,N2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),I2)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),I2)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,M),N2)),I2) ).

% min_diff
tff(fact_9399_inf__nat__def,axiom,
    inf_inf_nat = ord_min_nat ).

% inf_nat_def
tff(fact_9400_min__Suc2,axiom,
    ! [M: nat,N2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,M),aa_nat_nat(suc,N2)) = aa_nat_nat(case_nat_nat(zero_zero_nat,aa_nat_fun_nat_nat(aTP_Lamm_anl,N2)),M) ).

% min_Suc2
tff(fact_9401_min__Suc1,axiom,
    ! [N2: nat,M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(suc,N2)),M) = aa_nat_nat(case_nat_nat(zero_zero_nat,aa_nat_fun_nat_nat(aTP_Lamm_anm,N2)),M) ).

% min_Suc1
tff(fact_9402_of__nat__eq__id,axiom,
    semiri1316708129612266289at_nat = id_nat ).

% of_nat_eq_id
tff(fact_9403_nat__def,axiom,
    nat2 = map_fu2345160673673942751at_nat(rep_Integ,id_nat,produc6842872674320459806at_nat(minus_minus_nat)) ).

% nat_def
tff(fact_9404_less__int__def,axiom,
    ord_less_int = map_fu3191548434056052140_int_o(rep_Integ,map_fu4826362097070443709at_o_o(rep_Integ,id_o),aa_fun5912894404062470332_nat_o(produc6395135316500328296_nat_o,aTP_Lamm_amu)) ).

% less_int_def
tff(fact_9405_less__eq__int__def,axiom,
    ord_less_eq_int = map_fu3191548434056052140_int_o(rep_Integ,map_fu4826362097070443709at_o_o(rep_Integ,id_o),aa_fun5912894404062470332_nat_o(produc6395135316500328296_nat_o,aTP_Lamm_amw)) ).

% less_eq_int_def
tff(fact_9406_less__eq__integer__def,axiom,
    ord_le3102999989581377725nteger = map_fu4249723513525335408eger_o(code_int_of_integer,map_fu6957801986076833569nt_o_o(code_int_of_integer,id_o),ord_less_eq_int) ).

% less_eq_integer_def
tff(fact_9407_less__integer__def,axiom,
    ord_le6747313008572928689nteger = map_fu4249723513525335408eger_o(code_int_of_integer,map_fu6957801986076833569nt_o_o(code_int_of_integer,id_o),ord_less_int) ).

% less_integer_def
tff(fact_9408_max__nat_Oeq__neutr__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B) = zero_zero_nat )
    <=> ( ( A = zero_zero_nat )
        & ( B = zero_zero_nat ) ) ) ).

% max_nat.eq_neutr_iff
tff(fact_9409_max__nat_Oleft__neutral,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,zero_zero_nat),A) = A ).

% max_nat.left_neutral
tff(fact_9410_max__nat_Oneutr__eq__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( zero_zero_nat = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B) )
    <=> ( ( A = zero_zero_nat )
        & ( B = zero_zero_nat ) ) ) ).

% max_nat.neutr_eq_iff
tff(fact_9411_max__nat_Oright__neutral,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),zero_zero_nat) = A ).

% max_nat.right_neutral
tff(fact_9412_max__0L,axiom,
    ! [N2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,zero_zero_nat),N2) = N2 ).

% max_0L
tff(fact_9413_max__0R,axiom,
    ! [N2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,N2),zero_zero_nat) = N2 ).

% max_0R
tff(fact_9414_max__Suc__Suc,axiom,
    ! [M: nat,N2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(suc,M)),aa_nat_nat(suc,N2)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,M),N2)) ).

% max_Suc_Suc
tff(fact_9415_nat__mult__max__left,axiom,
    ! [M: nat,N2: nat,Q4: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,M),N2)),Q4) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Q4)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N2),Q4)) ).

% nat_mult_max_left
tff(fact_9416_nat__mult__max__right,axiom,
    ! [M: nat,N2: nat,Q4: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,N2),Q4)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Q4)) ).

% nat_mult_max_right
tff(fact_9417_nat__add__max__right,axiom,
    ! [M: nat,N2: nat,Q4: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,N2),Q4)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Q4)) ).

% nat_add_max_right
tff(fact_9418_nat__add__max__left,axiom,
    ! [M: nat,N2: nat,Q4: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,M),N2)),Q4) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Q4)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),Q4)) ).

% nat_add_max_left
tff(fact_9419_nat__minus__add__max,axiom,
    ! [N2: nat,M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),M)),M) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,N2),M) ).

% nat_minus_add_max
tff(fact_9420_max__Suc1,axiom,
    ! [N2: nat,M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(suc,N2)),M) = aa_nat_nat(case_nat_nat(aa_nat_nat(suc,N2),aa_nat_fun_nat_nat(aTP_Lamm_ann,N2)),M) ).

% max_Suc1
tff(fact_9421_max__Suc2,axiom,
    ! [M: nat,N2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,M),aa_nat_nat(suc,N2)) = aa_nat_nat(case_nat_nat(aa_nat_nat(suc,N2),aa_nat_fun_nat_nat(aTP_Lamm_ano,N2)),M) ).

% max_Suc2
tff(fact_9422_max__nat_Osemilattice__neutr__order__axioms,axiom,
    semila1623282765462674594er_nat(ord_max_nat,zero_zero_nat,aTP_Lamm_df,aTP_Lamm_dg) ).

% max_nat.semilattice_neutr_order_axioms
tff(fact_9423_sup__nat__def,axiom,
    sup_sup_nat = ord_max_nat ).

% sup_nat_def
tff(fact_9424_inj__Suc,axiom,
    ! [N7: set_nat] : aa_set_nat_o(inj_on_nat_nat(suc),N7) ).

% inj_Suc
tff(fact_9425_inj__on__diff__nat,axiom,
    ! [N7: set_nat,K: nat] :
      ( ! [N5: nat] :
          ( aa_set_nat_o(member_nat(N5),N7)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),N5) )
     => aa_set_nat_o(inj_on_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_amp,K)),N7) ) ).

% inj_on_diff_nat
tff(fact_9426_floor__rat__def,axiom,
    ! [X: rat] : aa_rat_int(archim3151403230148437115or_rat,X) = the_int(aTP_Lamm_anp(X)) ).

% floor_rat_def
tff(fact_9427_less__eq__rat__def,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 ) ) ) ).

% less_eq_rat_def
tff(fact_9428_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_9429_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_9430_normalize__negative,axiom,
    ! [Q4: int,P2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Q4),zero_zero_int)
     => ( aa_Pro6970504127699319548nt_int(normalize,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,P2),Q4)) = aa_Pro6970504127699319548nt_int(normalize,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(uminus_uminus_int,P2)),aa_int_int(uminus_uminus_int,Q4))) ) ) ).

% normalize_negative
tff(fact_9431_normalize__denom__zero,axiom,
    ! [P2: int] : aa_Pro6970504127699319548nt_int(normalize,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,P2),zero_zero_int)) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,zero_zero_int),one_one_int) ).

% normalize_denom_zero
tff(fact_9432_normalize__denom__pos,axiom,
    ! [Ra2: product_prod_int_int,P2: int,Q4: int] :
      ( ( aa_Pro6970504127699319548nt_int(normalize,Ra2) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,P2),Q4) )
     => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Q4) ) ).

% normalize_denom_pos
tff(fact_9433_normalize__crossproduct,axiom,
    ! [Q4: int,S2: int,P2: int,Ra2: int] :
      ( ( Q4 != zero_zero_int )
     => ( ( S2 != zero_zero_int )
       => ( ( aa_Pro6970504127699319548nt_int(normalize,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,P2),Q4)) = aa_Pro6970504127699319548nt_int(normalize,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Ra2),S2)) )
         => ( aa_int_int(aa_int_fun_int_int(times_times_int,P2),S2) = aa_int_int(aa_int_fun_int_int(times_times_int,Ra2),Q4) ) ) ) ) ).

% normalize_crossproduct
tff(fact_9434_normalize__def,axiom,
    ! [P2: product_prod_int_int] :
      aa_Pro6970504127699319548nt_int(normalize,P2) = $ite(
        aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_Pro6799258434465214121nt_int(product_snd_int_int,P2)),
        $let(
          a2: int,
          a2:= aa_int_int(aa_int_fun_int_int(gcd_gcd_int,aa_Pro6799258434465214121nt_int(product_fst_int_int,P2)),aa_Pro6799258434465214121nt_int(product_snd_int_int,P2)),
          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,P2)),a2)),aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_Pro6799258434465214121nt_int(product_snd_int_int,P2)),a2)) ),
        $ite(
          aa_Pro6799258434465214121nt_int(product_snd_int_int,P2) = zero_zero_int,
          aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,zero_zero_int),one_one_int),
          $let(
            a2: int,
            a2:= aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(gcd_gcd_int,aa_Pro6799258434465214121nt_int(product_fst_int_int,P2)),aa_Pro6799258434465214121nt_int(product_snd_int_int,P2))),
            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,P2)),a2)),aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_Pro6799258434465214121nt_int(product_snd_int_int,P2)),a2)) ) ) ) ).

% normalize_def
tff(fact_9435_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_9436_gcd__pos__int,axiom,
    ! [M: int,N2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(gcd_gcd_int,M),N2))
    <=> ( ( M != zero_zero_int )
        | ( N2 != zero_zero_int ) ) ) ).

% gcd_pos_int
tff(fact_9437_quotient__of__number_I3_J,axiom,
    ! [K: num] : quotient_of(aa_num_rat(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_9438_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_9439_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_9440_quotient__of__number_I5_J,axiom,
    ! [K: num] : quotient_of(aa_rat_rat(uminus_uminus_rat,aa_num_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_9441_gcd__ge__0__int,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(gcd_gcd_int,X),Y)) ).

% gcd_ge_0_int
tff(fact_9442_quotient__of__div,axiom,
    ! [Ra2: rat,N2: int,D2: int] :
      ( ( quotient_of(Ra2) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,N2),D2) )
     => ( Ra2 = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_int_rat(ring_1_of_int_rat,N2)),aa_int_rat(ring_1_of_int_rat,D2)) ) ) ).

% quotient_of_div
tff(fact_9443_gcd__le1__int,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(gcd_gcd_int,A),B)),A) ) ).

% gcd_le1_int
tff(fact_9444_gcd__le2__int,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(gcd_gcd_int,A),B)),B) ) ).

% gcd_le2_int
tff(fact_9445_gcd__cases__int,axiom,
    ! [X: int,Y: int,Pa: 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(Pa,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(Pa,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(Pa,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(Pa,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(Pa,aa_int_int(aa_int_fun_int_int(gcd_gcd_int,X),Y)) ) ) ) ) ).

% gcd_cases_int
tff(fact_9446_gcd__unique__int,axiom,
    ! [D2: int,A: int,B: int] :
      ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),D2)
        & aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D2),A)
        & aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D2),B)
        & ! [E3: int] :
            ( ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,E3),A)
              & aa_int_o(aa_int_fun_int_o(dvd_dvd_int,E3),B) )
           => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,E3),D2) ) )
    <=> ( D2 = aa_int_int(aa_int_fun_int_int(gcd_gcd_int,A),B) ) ) ).

% gcd_unique_int
tff(fact_9447_gcd__non__0__int,axiom,
    ! [Y: int,X: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Y)
     => ( aa_int_int(aa_int_fun_int_int(gcd_gcd_int,X),Y) = aa_int_int(aa_int_fun_int_int(gcd_gcd_int,Y),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,X),Y)) ) ) ).

% gcd_non_0_int
tff(fact_9448_quotient__of__denom__pos,axiom,
    ! [Ra2: rat,P2: int,Q4: int] :
      ( ( quotient_of(Ra2) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,P2),Q4) )
     => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Q4) ) ).

% quotient_of_denom_pos
tff(fact_9449_quotient__of__denom__pos_H,axiom,
    ! [Ra2: rat] : aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_Pro6799258434465214121nt_int(product_snd_int_int,quotient_of(Ra2))) ).

% quotient_of_denom_pos'
tff(fact_9450_gcd__is__Max__divisors__int,axiom,
    ! [N2: int,M: int] :
      ( ( N2 != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(gcd_gcd_int,M),N2) = lattic8263393255366662781ax_int(collect_int(aa_int_fun_int_o(aTP_Lamm_anq(N2),M))) ) ) ).

% gcd_is_Max_divisors_int
tff(fact_9451_rat__uminus__code,axiom,
    ! [P2: rat] : quotient_of(aa_rat_rat(uminus_uminus_rat,P2)) = aa_Pro6970504127699319548nt_int(aa_fun8026974734089330650nt_int(produc4245557441103728435nt_int,aTP_Lamm_anr),quotient_of(P2)) ).

% rat_uminus_code
tff(fact_9452_rat__times__code,axiom,
    ! [P2: rat,Q4: rat] : quotient_of(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,P2),Q4)) = aa_Pro6970504127699319548nt_int(aa_fun8026974734089330650nt_int(produc4245557441103728435nt_int,aTP_Lamm_ant(Q4)),quotient_of(P2)) ).

% rat_times_code
tff(fact_9453_rat__divide__code,axiom,
    ! [P2: rat,Q4: rat] : quotient_of(aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,P2),Q4)) = aa_Pro6970504127699319548nt_int(aa_fun8026974734089330650nt_int(produc4245557441103728435nt_int,aTP_Lamm_anv(Q4)),quotient_of(P2)) ).

% rat_divide_code
tff(fact_9454_rat__less__code,axiom,
    ! [P2: rat,Q4: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,P2),Q4)
    <=> aa_Pro4592068669816069595_int_o(aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,aTP_Lamm_anx(Q4)),quotient_of(P2)) ) ).

% rat_less_code
tff(fact_9455_rat__less__eq__code,axiom,
    ! [P2: rat,Q4: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,P2),Q4)
    <=> aa_Pro4592068669816069595_int_o(aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,aTP_Lamm_anz(Q4)),quotient_of(P2)) ) ).

% rat_less_eq_code
tff(fact_9456_rat__floor__code,axiom,
    ! [P2: rat] : aa_rat_int(archim3151403230148437115or_rat,P2) = aa_Pro6799258434465214121nt_int(aa_fun914008879417524284nt_int(produc8211389475949308722nt_int,divide_divide_int),quotient_of(P2)) ).

% rat_floor_code
tff(fact_9457_rat__abs__code,axiom,
    ! [P2: rat] : quotient_of(aa_rat_rat(abs_abs_rat,P2)) = aa_Pro6970504127699319548nt_int(aa_fun8026974734089330650nt_int(produc4245557441103728435nt_int,aTP_Lamm_aoa),quotient_of(P2)) ).

% rat_abs_code
tff(fact_9458_rat__plus__code,axiom,
    ! [P2: rat,Q4: rat] : quotient_of(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,P2),Q4)) = aa_Pro6970504127699319548nt_int(aa_fun8026974734089330650nt_int(produc4245557441103728435nt_int,aTP_Lamm_aoc(Q4)),quotient_of(P2)) ).

% rat_plus_code
tff(fact_9459_rat__minus__code,axiom,
    ! [P2: rat,Q4: rat] : quotient_of(aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,P2),Q4)) = aa_Pro6970504127699319548nt_int(aa_fun8026974734089330650nt_int(produc4245557441103728435nt_int,aTP_Lamm_aoe(Q4)),quotient_of(P2)) ).

% rat_minus_code
tff(fact_9460_rat__sgn__code,axiom,
    ! [P2: rat] : quotient_of(aa_rat_rat(sgn_sgn_rat,P2)) = 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(P2)))),one_one_int) ).

% rat_sgn_code
tff(fact_9461_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_9462_rat__inverse__code,axiom,
    ! [P2: rat] : quotient_of(aa_rat_rat(inverse_inverse_rat,P2)) = aa_Pro6970504127699319548nt_int(aa_fun8026974734089330650nt_int(produc4245557441103728435nt_int,aTP_Lamm_aof),quotient_of(P2)) ).

% rat_inverse_code
tff(fact_9463_gcd__pos__nat,axiom,
    ! [M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,M),N2))
    <=> ( ( M != zero_zero_nat )
        | ( N2 != zero_zero_nat ) ) ) ).

% gcd_pos_nat
tff(fact_9464_gcd__le1__nat,axiom,
    ! [A: nat,B: nat] :
      ( ( A != zero_zero_nat )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B)),A) ) ).

% gcd_le1_nat
tff(fact_9465_gcd__le2__nat,axiom,
    ! [B: nat,A: nat] :
      ( ( B != zero_zero_nat )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B)),B) ) ).

% gcd_le2_nat
tff(fact_9466_gcd__diff1__nat,axiom,
    ! [N2: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),M)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N2)),N2) = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,M),N2) ) ) ).

% gcd_diff1_nat
tff(fact_9467_gcd__diff2__nat,axiom,
    ! [M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),M)),N2) = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,M),N2) ) ) ).

% gcd_diff2_nat
tff(fact_9468_Gcd__in,axiom,
    ! [A3: set_nat] :
      ( ! [A4: nat,B4: nat] :
          ( aa_set_nat_o(member_nat(A4),A3)
         => ( aa_set_nat_o(member_nat(B4),A3)
           => aa_set_nat_o(member_nat(aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A4),B4)),A3) ) )
     => ( ( A3 != bot_bot_set_nat )
       => aa_set_nat_o(member_nat(aa_set_nat_nat(gcd_Gcd_nat,A3)),A3) ) ) ).

% Gcd_in
tff(fact_9469_bezout__gcd__nat_H,axiom,
    ! [B: nat,A: nat] :
    ? [X3: nat,Y3: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),Y3)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),X3))
        & ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),X3)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),Y3)) = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B) ) )
      | ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),Y3)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),X3))
        & ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),X3)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),Y3)) = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B) ) ) ) ).

% bezout_gcd_nat'
tff(fact_9470_gcd__nat_Osemilattice__neutr__order__axioms,axiom,
    semila1623282765462674594er_nat(gcd_gcd_nat,zero_zero_nat,dvd_dvd_nat,aTP_Lamm_adw) ).

% gcd_nat.semilattice_neutr_order_axioms
tff(fact_9471_gcd__is__Max__divisors__nat,axiom,
    ! [N2: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,M),N2) = aa_set_nat_nat(lattic8265883725875713057ax_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_aog(N2),M))) ) ) ).

% gcd_is_Max_divisors_nat
tff(fact_9472_gcd__nat_Opelims,axiom,
    ! [X: nat,Xa2: nat,Y: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,X),Xa2) = Y )
     => ( aa_Pro6150637604445964947_nat_o(accp_P4275260045618599050at_nat(gcd_nat_rel),aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,X),Xa2))
       => ~ ( ( Y = $ite(Xa2 = zero_zero_nat,X,aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Xa2))) )
           => ~ aa_Pro6150637604445964947_nat_o(accp_P4275260045618599050at_nat(gcd_nat_rel),aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,X),Xa2)) ) ) ) ).

% gcd_nat.pelims
tff(fact_9473_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),aa_num_rat(numeral_numeral_rat,K)) ).

% Frct_code_post(5)
tff(fact_9474_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_9475_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_9476_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_9477_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_9478_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_9479_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)) = aa_num_rat(numeral_numeral_rat,K) ).

% Frct_code_post(4)
tff(fact_9480_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,aa_num_rat(numeral_numeral_rat,K)),aa_num_rat(numeral_numeral_rat,L)) ).

% Frct_code_post(6)
tff(fact_9481_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_9482_less__rat,axiom,
    ! [B: int,D2: int,A: int,Ca: int] :
      ( ( B != zero_zero_int )
     => ( ( D2 != zero_zero_int )
       => ( 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(aa_int_fun_int_rat(fract,Ca),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(aa_int_fun_int_int(times_times_int,A),D2)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D2))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,Ca),B)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D2))) ) ) ) ).

% less_rat
tff(fact_9483_le__rat,axiom,
    ! [B: int,D2: int,A: int,Ca: int] :
      ( ( B != zero_zero_int )
     => ( ( D2 != zero_zero_int )
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(aa_int_fun_int_rat(fract,A),B)),aa_int_rat(aa_int_fun_int_rat(fract,Ca),D2))
        <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),D2)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D2))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,Ca),B)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D2))) ) ) ) ).

% le_rat
tff(fact_9484_quotient__of__eq,axiom,
    ! [A: int,B: int,P2: int,Q4: int] :
      ( ( quotient_of(aa_int_rat(aa_int_fun_int_rat(fract,A),B)) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,P2),Q4) )
     => ( aa_int_rat(aa_int_fun_int_rat(fract,P2),Q4) = aa_int_rat(aa_int_fun_int_rat(fract,A),B) ) ) ).

% quotient_of_eq
tff(fact_9485_Rat__induct__pos,axiom,
    ! [Pa: fun_rat_o,Q4: rat] :
      ( ! [A4: int,B4: int] :
          ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B4)
         => aa_rat_o(Pa,aa_int_rat(aa_int_fun_int_rat(fract,A4),B4)) )
     => aa_rat_o(Pa,Q4) ) ).

% Rat_induct_pos
tff(fact_9486_normalize__eq,axiom,
    ! [A: int,B: int,P2: int,Q4: 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,P2),Q4) )
     => ( aa_int_rat(aa_int_fun_int_rat(fract,P2),Q4) = aa_int_rat(aa_int_fun_int_rat(fract,A),B) ) ) ).

% normalize_eq
tff(fact_9487_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_9488_Fract__less__zero__iff,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(aa_int_fun_int_rat(fract,A),B)),zero_zero_rat)
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int) ) ) ).

% Fract_less_zero_iff
tff(fact_9489_zero__less__Fract__iff,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_int_rat(aa_int_fun_int_rat(fract,A),B))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A) ) ) ).

% zero_less_Fract_iff
tff(fact_9490_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_9491_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_9492_zero__le__Fract__iff,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_int_rat(aa_int_fun_int_rat(fract,A),B))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A) ) ) ).

% zero_le_Fract_iff
tff(fact_9493_Fract__le__zero__iff,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(aa_int_fun_int_rat(fract,A),B)),zero_zero_rat)
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int) ) ) ).

% Fract_le_zero_iff
tff(fact_9494_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_9495_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_9496_positive__rat,axiom,
    ! [A: int,B: int] :
      ( aa_rat_o(positive,aa_int_rat(aa_int_fun_int_rat(fract,A),B))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A),B)) ) ).

% positive_rat
tff(fact_9497_less__rat__def,axiom,
    ! [X: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y)
    <=> aa_rat_o(positive,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,Y),X)) ) ).

% less_rat_def
tff(fact_9498_positive__def,axiom,
    positive = map_fu898904425404107465nt_o_o(rep_Rat,id_o,aTP_Lamm_aoh) ).

% positive_def
tff(fact_9499_positive_Orep__eq,axiom,
    ! [X: rat] :
      ( aa_rat_o(positive,X)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_Pro6799258434465214121nt_int(product_fst_int_int,aa_rat7754553326175492643nt_int(rep_Rat,X))),aa_Pro6799258434465214121nt_int(product_snd_int_int,aa_rat7754553326175492643nt_int(rep_Rat,X)))) ) ).

% positive.rep_eq
tff(fact_9500_plus__rat__def,axiom,
    plus_plus_rat = map_fu5968964422797239547at_rat(rep_Rat,map_fu5673905371560938248nt_rat(rep_Rat,abs_Rat),aTP_Lamm_aoi) ).

% plus_rat_def
tff(fact_9501_inverse__rat__def,axiom,
    inverse_inverse_rat = aa_fun7041730090766645614at_rat(map_fu5673905371560938248nt_rat(rep_Rat,abs_Rat),aTP_Lamm_aoj) ).

% inverse_rat_def
tff(fact_9502_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_9503_Fract_Oabs__eq,axiom,
    ! [Xa2: int,X: int] :
      aa_int_rat(aa_int_fun_int_rat(fract,Xa2),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,Xa2),X))) ).

% Fract.abs_eq
tff(fact_9504_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_9505_Fract__def,axiom,
    fract = map_fu7945985449245211511nt_rat(id_int,map_fu3424225382358772769nt_rat(id_int,abs_Rat),aTP_Lamm_aok) ).

% Fract_def
tff(fact_9506_uminus__rat__def,axiom,
    uminus_uminus_rat = aa_fun7041730090766645614at_rat(map_fu5673905371560938248nt_rat(rep_Rat,abs_Rat),aTP_Lamm_aol) ).

% uminus_rat_def
tff(fact_9507_times__rat__def,axiom,
    times_times_rat = map_fu5968964422797239547at_rat(rep_Rat,map_fu5673905371560938248nt_rat(rep_Rat,abs_Rat),aTP_Lamm_aom) ).

% times_rat_def
tff(fact_9508_plus__rat_Oabs__eq,axiom,
    ! [Xa2: product_prod_int_int,X: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aa_Pro3952998155787061599_int_o(ratrel,Xa2),Xa2)
     => ( 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,Xa2)),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,Xa2)),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,Xa2)))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_Pro6799258434465214121nt_int(product_snd_int_int,Xa2)),aa_Pro6799258434465214121nt_int(product_snd_int_int,X)))) ) ) ) ).

% plus_rat.abs_eq
tff(fact_9509_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_9510_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_9511_ratrel__def,axiom,
    ! [X5: product_prod_int_int,Xa3: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aa_Pro3952998155787061599_int_o(ratrel,X5),Xa3)
    <=> ( ( aa_Pro6799258434465214121nt_int(product_snd_int_int,X5) != 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,X5)),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,X5)) ) ) ) ).

% ratrel_def
tff(fact_9512_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_9513_times__rat_Oabs__eq,axiom,
    ! [Xa2: product_prod_int_int,X: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aa_Pro3952998155787061599_int_o(ratrel,Xa2),Xa2)
     => ( 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,Xa2)),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,Xa2)),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,Xa2)),aa_Pro6799258434465214121nt_int(product_snd_int_int,X)))) ) ) ) ).

% times_rat.abs_eq
tff(fact_9514_positive_Oabs__eq,axiom,
    ! [X: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aa_Pro3952998155787061599_int_o(ratrel,X),X)
     => ( aa_rat_o(positive,aa_Pro6166618844887768661nt_rat(abs_Rat,X))
      <=> 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,X)),aa_Pro6799258434465214121nt_int(product_snd_int_int,X))) ) ) ).

% positive.abs_eq
tff(fact_9515_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_9516_cr__rat__def,axiom,
    ! [X5: product_prod_int_int,Xa3: rat] :
      ( aa_rat_o(cr_rat(X5),Xa3)
    <=> ( aa_Pro4592068669816069595_int_o(aa_Pro3952998155787061599_int_o(ratrel,X5),X5)
        & ( aa_Pro6166618844887768661nt_rat(abs_Rat,X5) = Xa3 ) ) ) ).

% cr_rat_def
tff(fact_9517_Code__Numeral_Odup__def,axiom,
    code_dup = map_fu2599414010547811884nteger(code_int_of_integer,code_integer_of_int,aTP_Lamm_aon) ).

% Code_Numeral.dup_def
tff(fact_9518_quotient__of__def,axiom,
    ! [X: rat] : quotient_of(X) = the_Pr4378521158711661632nt_int(aTP_Lamm_aoo(X)) ).

% quotient_of_def
tff(fact_9519_normalize__stable,axiom,
    ! [Q4: int,P2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Q4)
     => ( aa_int_o(aa_int_fun_int_o(algebr932160517623751201me_int,P2),Q4)
       => ( aa_Pro6970504127699319548nt_int(normalize,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,P2),Q4)) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,P2),Q4) ) ) ) ).

% normalize_stable
tff(fact_9520_quotient__of__coprime,axiom,
    ! [Ra2: rat,P2: int,Q4: int] :
      ( ( quotient_of(Ra2) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,P2),Q4) )
     => aa_int_o(aa_int_fun_int_o(algebr932160517623751201me_int,P2),Q4) ) ).

% quotient_of_coprime
tff(fact_9521_normalize__coprime,axiom,
    ! [Ra2: product_prod_int_int,P2: int,Q4: int] :
      ( ( aa_Pro6970504127699319548nt_int(normalize,Ra2) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,P2),Q4) )
     => aa_int_o(aa_int_fun_int_o(algebr932160517623751201me_int,P2),Q4) ) ).

% normalize_coprime
tff(fact_9522_Rat__induct,axiom,
    ! [Pa: fun_rat_o,Q4: rat] :
      ( ! [A4: int,B4: int] :
          ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B4)
         => ( aa_int_o(aa_int_fun_int_o(algebr932160517623751201me_int,A4),B4)
           => aa_rat_o(Pa,aa_int_rat(aa_int_fun_int_rat(fract,A4),B4)) ) )
     => aa_rat_o(Pa,Q4) ) ).

% Rat_induct
tff(fact_9523_Rat__cases,axiom,
    ! [Q4: rat] :
      ~ ! [A4: int,B4: int] :
          ( ( Q4 = aa_int_rat(aa_int_fun_int_rat(fract,A4),B4) )
         => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B4)
           => ~ aa_int_o(aa_int_fun_int_o(algebr932160517623751201me_int,A4),B4) ) ) ).

% Rat_cases
tff(fact_9524_Rat__cases__nonzero,axiom,
    ! [Q4: rat] :
      ( ! [A4: int,B4: int] :
          ( ( Q4 = aa_int_rat(aa_int_fun_int_rat(fract,A4),B4) )
         => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B4)
           => ( ( A4 != zero_zero_int )
             => ~ aa_int_o(aa_int_fun_int_o(algebr932160517623751201me_int,A4),B4) ) ) )
     => ( Q4 = zero_zero_rat ) ) ).

% Rat_cases_nonzero
tff(fact_9525_quotient__of__unique,axiom,
    ! [Ra2: rat] :
    ? [X3: product_prod_int_int] :
      ( ( Ra2 = aa_int_rat(aa_int_fun_int_rat(fract,aa_Pro6799258434465214121nt_int(product_fst_int_int,X3)),aa_Pro6799258434465214121nt_int(product_snd_int_int,X3)) )
      & aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_Pro6799258434465214121nt_int(product_snd_int_int,X3))
      & aa_int_o(aa_int_fun_int_o(algebr932160517623751201me_int,aa_Pro6799258434465214121nt_int(product_fst_int_int,X3)),aa_Pro6799258434465214121nt_int(product_snd_int_int,X3))
      & ! [Y4: product_prod_int_int] :
          ( ( ( Ra2 = aa_int_rat(aa_int_fun_int_rat(fract,aa_Pro6799258434465214121nt_int(product_fst_int_int,Y4)),aa_Pro6799258434465214121nt_int(product_snd_int_int,Y4)) )
            & aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_Pro6799258434465214121nt_int(product_snd_int_int,Y4))
            & aa_int_o(aa_int_fun_int_o(algebr932160517623751201me_int,aa_Pro6799258434465214121nt_int(product_fst_int_int,Y4)),aa_Pro6799258434465214121nt_int(product_snd_int_int,Y4)) )
         => ( Y4 = X3 ) ) ) ).

% quotient_of_unique
tff(fact_9526_coprime__diff__one__left__nat,axiom,
    ! [N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)
     => aa_nat_o(aa_nat_fun_nat_o(algebr934650988132801477me_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),one_one_nat)),N2) ) ).

% coprime_diff_one_left_nat
tff(fact_9527_coprime__diff__one__right__nat,axiom,
    ! [N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)
     => aa_nat_o(aa_nat_fun_nat_o(algebr934650988132801477me_nat,N2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),one_one_nat)) ) ).

% coprime_diff_one_right_nat
tff(fact_9528_positive_Orsp,axiom,
    bNF_re8699439704749558557nt_o_o(ratrel,fequal_o,aTP_Lamm_aoh,aTP_Lamm_aoh) ).

% positive.rsp
tff(fact_9529_sub_Orsp,axiom,
    bNF_re6378542964492772123um_int(fequal_num,bNF_re1822329894187522285nt_int(fequal_num,fequal_int),aTP_Lamm_aop,aTP_Lamm_aop) ).

% sub.rsp
tff(fact_9530_Fract_Orsp,axiom,
    bNF_re4808202430149235895nt_int(fequal_int,bNF_re6250860962936578807nt_int(fequal_int,ratrel),aTP_Lamm_aok,aTP_Lamm_aok) ).

% Fract.rsp
tff(fact_9531_dup_Orsp,axiom,
    bNF_re4712519889275205905nt_int(fequal_int,fequal_int,aTP_Lamm_aon,aTP_Lamm_aon) ).

% dup.rsp
tff(fact_9532_less__eq__integer_Orsp,axiom,
    bNF_re8654322613993568759_int_o(fequal_int,bNF_re5089333283451836215nt_o_o(fequal_int,fequal_o),ord_less_eq_int,ord_less_eq_int) ).

% less_eq_integer.rsp
tff(fact_9533_less__eq__natural_Orsp,axiom,
    bNF_re1448341924010609215_nat_o(fequal_nat,bNF_re4705727531993890431at_o_o(fequal_nat,fequal_o),ord_less_eq_nat,ord_less_eq_nat) ).

% less_eq_natural.rsp
tff(fact_9534_less__natural_Orsp,axiom,
    bNF_re1448341924010609215_nat_o(fequal_nat,bNF_re4705727531993890431at_o_o(fequal_nat,fequal_o),ord_less_nat,ord_less_nat) ).

% less_natural.rsp
tff(fact_9535_less__integer_Orsp,axiom,
    bNF_re8654322613993568759_int_o(fequal_int,bNF_re5089333283451836215nt_o_o(fequal_int,fequal_o),ord_less_int,ord_less_int) ).

% less_integer.rsp
tff(fact_9536_times__rat_Orsp,axiom,
    bNF_re4734134812813710757nt_int(ratrel,bNF_re7145576690424134365nt_int(ratrel,ratrel),aTP_Lamm_aom,aTP_Lamm_aom) ).

% times_rat.rsp
tff(fact_9537_plus__rat_Orsp,axiom,
    bNF_re4734134812813710757nt_int(ratrel,bNF_re7145576690424134365nt_int(ratrel,ratrel),aTP_Lamm_aoi,aTP_Lamm_aoi) ).

% plus_rat.rsp
tff(fact_9538_uminus__rat_Orsp,axiom,
    aa_fun2650603632653508759_int_o(aa_fun3734617511155415725_int_o(bNF_re7145576690424134365nt_int(ratrel,ratrel),aTP_Lamm_aol),aTP_Lamm_aol) ).

% uminus_rat.rsp
tff(fact_9539_inverse__rat_Orsp,axiom,
    aa_fun2650603632653508759_int_o(aa_fun3734617511155415725_int_o(bNF_re7145576690424134365nt_int(ratrel,ratrel),aTP_Lamm_aoj),aTP_Lamm_aoj) ).

% inverse_rat.rsp
tff(fact_9540_plus__rat_Otransfer,axiom,
    bNF_re6323452265856231590at_rat(pcr_rat,bNF_re8279943556446156061nt_rat(pcr_rat,pcr_rat),aTP_Lamm_aoi,plus_plus_rat) ).

% plus_rat.transfer
tff(fact_9541_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_9542_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_9543_Fract_Otransfer,axiom,
    bNF_re2486238095283335024nt_rat(fequal_int,bNF_re2214769303045360666nt_rat(fequal_int,pcr_rat),aTP_Lamm_aok,fract) ).

% Fract.transfer
tff(fact_9544_uminus__rat_Otransfer,axiom,
    aa_fun_rat_rat_o(aa_fun382765352565706371_rat_o(bNF_re8279943556446156061nt_rat(pcr_rat,pcr_rat),aTP_Lamm_aol),uminus_uminus_rat) ).

% uminus_rat.transfer
tff(fact_9545_times__rat_Otransfer,axiom,
    bNF_re6323452265856231590at_rat(pcr_rat,bNF_re8279943556446156061nt_rat(pcr_rat,pcr_rat),aTP_Lamm_aom,times_times_rat) ).

% times_rat.transfer
tff(fact_9546_positive_Otransfer,axiom,
    bNF_re1494630372529172596at_o_o(pcr_rat,fequal_o,aTP_Lamm_aoh,positive) ).

% positive.transfer
tff(fact_9547_inverse__rat_Otransfer,axiom,
    aa_fun_rat_rat_o(aa_fun382765352565706371_rat_o(bNF_re8279943556446156061nt_rat(pcr_rat,pcr_rat),aTP_Lamm_aoj),inverse_inverse_rat) ).

% inverse_rat.transfer
tff(fact_9548_times__int_Otransfer,axiom,
    bNF_re8869769828555196650nt_int(pcr_int,bNF_re7400052026677387805at_int(pcr_int,pcr_int),aa_fun6875833325056119164at_nat(produc4528391901023682385at_nat,aTP_Lamm_amr),times_times_int) ).

% times_int.transfer
tff(fact_9549_zero__int_Otransfer,axiom,
    aa_int_o(aa_Pro2912763917461432128_int_o(pcr_int,aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,zero_zero_nat),zero_zero_nat)),zero_zero_int) ).

% zero_int.transfer
tff(fact_9550_int__transfer,axiom,
    bNF_re6830278522597306478at_int(fequal_nat,pcr_int,aTP_Lamm_aoq,semiri1314217659103216013at_int) ).

% int_transfer
tff(fact_9551_uminus__int_Otransfer,axiom,
    aa_fun_int_int_o(aa_fun856177901615811547_int_o(bNF_re7400052026677387805at_int(pcr_int,pcr_int),produc2626176000494625587at_nat(aTP_Lamm_ams)),uminus_uminus_int) ).

% uminus_int.transfer
tff(fact_9552_nat_Otransfer,axiom,
    bNF_re4555766996558763186at_nat(pcr_int,fequal_nat,produc6842872674320459806at_nat(minus_minus_nat),nat2) ).

% nat.transfer
tff(fact_9553_one__int_Otransfer,axiom,
    aa_int_o(aa_Pro2912763917461432128_int_o(pcr_int,aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,one_one_nat),zero_zero_nat)),one_one_int) ).

% one_int.transfer
tff(fact_9554_less__int_Otransfer,axiom,
    bNF_re4195927319312321177_int_o(pcr_int,bNF_re6644619430987730960nt_o_o(pcr_int,fequal_o),aa_fun5912894404062470332_nat_o(produc6395135316500328296_nat_o,aTP_Lamm_amu),ord_less_int) ).

% less_int.transfer
tff(fact_9555_less__eq__int_Otransfer,axiom,
    bNF_re4195927319312321177_int_o(pcr_int,bNF_re6644619430987730960nt_o_o(pcr_int,fequal_o),aa_fun5912894404062470332_nat_o(produc6395135316500328296_nat_o,aTP_Lamm_amw),ord_less_eq_int) ).

% less_eq_int.transfer
tff(fact_9556_plus__int_Otransfer,axiom,
    bNF_re8869769828555196650nt_int(pcr_int,bNF_re7400052026677387805at_int(pcr_int,pcr_int),aa_fun6875833325056119164at_nat(produc4528391901023682385at_nat,aTP_Lamm_amy),plus_plus_int) ).

% plus_int.transfer
tff(fact_9557_minus__int_Otransfer,axiom,
    bNF_re8869769828555196650nt_int(pcr_int,bNF_re7400052026677387805at_int(pcr_int,pcr_int),aa_fun6875833325056119164at_nat(produc4528391901023682385at_nat,aTP_Lamm_ana),minus_minus_int) ).

% minus_int.transfer
tff(fact_9558_Sup__int__def,axiom,
    ! [X6: set_int] : complete_Sup_Sup_int(X6) = the_int(aTP_Lamm_aor(X6)) ).

% Sup_int_def
tff(fact_9559_times__int_Orsp,axiom,
    bNF_re6214786843070381989at_nat(intrel,bNF_re2241393799969408733at_nat(intrel,intrel),aa_fun6875833325056119164at_nat(produc4528391901023682385at_nat,aTP_Lamm_amr),aa_fun6875833325056119164at_nat(produc4528391901023682385at_nat,aTP_Lamm_amr)) ).

% times_int.rsp
tff(fact_9560_intrel__iff,axiom,
    ! [X: nat,Y: nat,U: nat,V: nat] :
      ( aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(intrel,aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,X),Y)),aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_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_9561_zero__int_Orsp,axiom,
    aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(intrel,aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,zero_zero_nat),zero_zero_nat)),aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,zero_zero_nat),zero_zero_nat)) ).

% zero_int.rsp
tff(fact_9562_uminus__int_Orsp,axiom,
    aa_fun3538214411680374423_nat_o(aa_fun6920661532540581549_nat_o(bNF_re2241393799969408733at_nat(intrel,intrel),produc2626176000494625587at_nat(aTP_Lamm_ams)),produc2626176000494625587at_nat(aTP_Lamm_ams)) ).

% uminus_int.rsp
tff(fact_9563_nat_Orsp,axiom,
    bNF_re8246922863344978751at_nat(intrel,fequal_nat,produc6842872674320459806at_nat(minus_minus_nat),produc6842872674320459806at_nat(minus_minus_nat)) ).

% nat.rsp
tff(fact_9564_one__int_Orsp,axiom,
    aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(intrel,aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,one_one_nat),zero_zero_nat)),aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,one_one_nat),zero_zero_nat)) ).

% one_int.rsp
tff(fact_9565_intrel__def,axiom,
    intrel = aa_fun5912894404062470332_nat_o(produc6395135316500328296_nat_o,aTP_Lamm_aot) ).

% intrel_def
tff(fact_9566_less__int_Orsp,axiom,
    bNF_re4103222659376994899_nat_o(intrel,bNF_re3666534408544137501at_o_o(intrel,fequal_o),aa_fun5912894404062470332_nat_o(produc6395135316500328296_nat_o,aTP_Lamm_amu),aa_fun5912894404062470332_nat_o(produc6395135316500328296_nat_o,aTP_Lamm_amu)) ).

% less_int.rsp
tff(fact_9567_less__eq__int_Orsp,axiom,
    bNF_re4103222659376994899_nat_o(intrel,bNF_re3666534408544137501at_o_o(intrel,fequal_o),aa_fun5912894404062470332_nat_o(produc6395135316500328296_nat_o,aTP_Lamm_amw),aa_fun5912894404062470332_nat_o(produc6395135316500328296_nat_o,aTP_Lamm_amw)) ).

% less_eq_int.rsp
tff(fact_9568_minus__int_Orsp,axiom,
    bNF_re6214786843070381989at_nat(intrel,bNF_re2241393799969408733at_nat(intrel,intrel),aa_fun6875833325056119164at_nat(produc4528391901023682385at_nat,aTP_Lamm_ana),aa_fun6875833325056119164at_nat(produc4528391901023682385at_nat,aTP_Lamm_ana)) ).

% minus_int.rsp
tff(fact_9569_plus__int_Orsp,axiom,
    bNF_re6214786843070381989at_nat(intrel,bNF_re2241393799969408733at_nat(intrel,intrel),aa_fun6875833325056119164at_nat(produc4528391901023682385at_nat,aTP_Lamm_amy),aa_fun6875833325056119164at_nat(produc4528391901023682385at_nat,aTP_Lamm_amy)) ).

% plus_int.rsp
tff(fact_9570_GreatestI__nat,axiom,
    ! [Pa: fun_nat_o,K: nat,B: nat] :
      ( aa_nat_o(Pa,K)
     => ( ! [Y3: nat] :
            ( aa_nat_o(Pa,Y3)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y3),B) )
       => aa_nat_o(Pa,order_Greatest_nat(Pa)) ) ) ).

% GreatestI_nat
tff(fact_9571_Greatest__le__nat,axiom,
    ! [Pa: fun_nat_o,K: nat,B: nat] :
      ( aa_nat_o(Pa,K)
     => ( ! [Y3: nat] :
            ( aa_nat_o(Pa,Y3)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y3),B) )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),order_Greatest_nat(Pa)) ) ) ).

% Greatest_le_nat
tff(fact_9572_GreatestI__ex__nat,axiom,
    ! [Pa: fun_nat_o,B: nat] :
      ( ? [X_13: nat] : aa_nat_o(Pa,X_13)
     => ( ! [Y3: nat] :
            ( aa_nat_o(Pa,Y3)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y3),B) )
       => aa_nat_o(Pa,order_Greatest_nat(Pa)) ) ) ).

% GreatestI_ex_nat
tff(fact_9573_Field__natLeq__on,axiom,
    ! [N2: nat] : field_nat(collec3392354462482085612at_nat(produc6081775807080527818_nat_o(aTP_Lamm_aou(N2)))) = collect_nat(aa_nat_fun_nat_o(aTP_Lamm_dg,N2)) ).

% Field_natLeq_on
tff(fact_9574_wf__less,axiom,
    aa_set3702741264012733747_nat_o(wf_nat,collec3392354462482085612at_nat(produc6081775807080527818_nat_o(ord_less_nat))) ).

% wf_less
tff(fact_9575_less__than__iff,axiom,
    ! [X: nat,Y: nat] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_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_9576_nth__sorted__list__of__set__greaterThanLessThan,axiom,
    ! [N2: nat,J: nat,I2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),aa_nat_nat(suc,I2)))
     => ( aa_nat_nat(nth_nat(linord2614967742042102400et_nat(aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or5834768355832116004an_nat,I2),J))),N2) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),N2)) ) ) ).

% nth_sorted_list_of_set_greaterThanLessThan
tff(fact_9577_le__enumerate,axiom,
    ! [S: set_nat,N2: nat] :
      ( ~ aa_set_nat_o(finite_finite_nat,S)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),aa_nat_nat(infini8530281810654367211te_nat(S),N2)) ) ).

% le_enumerate
tff(fact_9578_sorted__list__of__set__greaterThanLessThan,axiom,
    ! [I2: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,I2)),J)
     => ( linord2614967742042102400et_nat(aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or5834768355832116004an_nat,I2),J)) = cons_nat(aa_nat_nat(suc,I2),linord2614967742042102400et_nat(aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or5834768355832116004an_nat,aa_nat_nat(suc,I2)),J))) ) ) ).

% sorted_list_of_set_greaterThanLessThan
tff(fact_9579_finite__le__enumerate,axiom,
    ! [S: set_nat,N2: nat] :
      ( aa_set_nat_o(finite_finite_nat,S)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),aa_set_nat_nat(finite_card_nat,S))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),aa_nat_nat(infini8530281810654367211te_nat(S),N2)) ) ) ).

% finite_le_enumerate
tff(fact_9580_nth__sorted__list__of__set__greaterThanAtMost,axiom,
    ! [N2: nat,J: nat,I2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),I2))
     => ( aa_nat_nat(nth_nat(linord2614967742042102400et_nat(aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or6659071591806873216st_nat,I2),J))),N2) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),N2)) ) ) ).

% nth_sorted_list_of_set_greaterThanAtMost
tff(fact_9581_Least__eq__0,axiom,
    ! [Pa: fun_nat_o] :
      ( aa_nat_o(Pa,zero_zero_nat)
     => ( ord_Least_nat(Pa) = zero_zero_nat ) ) ).

% Least_eq_0
tff(fact_9582_finite__greaterThanLessThan__integer,axiom,
    ! [L: code_integer,U: code_integer] : aa_set5643387332196575111eger_o(finite6017078050557962740nteger,set_or4266950643985792945nteger(L,U)) ).

% finite_greaterThanLessThan_integer
tff(fact_9583_Inf__nat__def,axiom,
    ! [X6: set_nat] : aa_set_nat_nat(complete_Inf_Inf_nat,X6) = ord_Least_nat(aTP_Lamm_ae(X6)) ).

% Inf_nat_def
tff(fact_9584_Least__Suc2,axiom,
    ! [Pa: fun_nat_o,N2: nat,Qa: fun_nat_o,M: nat] :
      ( aa_nat_o(Pa,N2)
     => ( aa_nat_o(Qa,M)
       => ( ~ aa_nat_o(Pa,zero_zero_nat)
         => ( ! [K2: nat] :
                ( aa_nat_o(Pa,aa_nat_nat(suc,K2))
              <=> aa_nat_o(Qa,K2) )
           => ( ord_Least_nat(Pa) = aa_nat_nat(suc,ord_Least_nat(Qa)) ) ) ) ) ) ).

% Least_Suc2
tff(fact_9585_Least__Suc,axiom,
    ! [Pa: fun_nat_o,N2: nat] :
      ( aa_nat_o(Pa,N2)
     => ( ~ aa_nat_o(Pa,zero_zero_nat)
       => ( ord_Least_nat(Pa) = aa_nat_nat(suc,ord_Least_nat(aTP_Lamm_aov(Pa))) ) ) ) ).

% Least_Suc
tff(fact_9586_atLeastPlusOneLessThan__greaterThanLessThan__integer,axiom,
    ! [L: code_integer,U: code_integer] : set_or8404916559141939852nteger(aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(L),one_one_Code_integer),U) = set_or4266950643985792945nteger(L,U) ).

% atLeastPlusOneLessThan_greaterThanLessThan_integer
tff(fact_9587_sorted__list__of__set__greaterThanAtMost,axiom,
    ! [I2: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,I2)),J)
     => ( linord2614967742042102400et_nat(aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or6659071591806873216st_nat,I2),J)) = cons_nat(aa_nat_nat(suc,I2),linord2614967742042102400et_nat(aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or6659071591806873216st_nat,aa_nat_nat(suc,I2)),J))) ) ) ).

% sorted_list_of_set_greaterThanAtMost
tff(fact_9588_finite__greaterThanAtMost__integer,axiom,
    ! [L: code_integer,U: code_integer] : aa_set5643387332196575111eger_o(finite6017078050557962740nteger,set_or2715278749043346189nteger(L,U)) ).

% finite_greaterThanAtMost_integer
tff(fact_9589_atLeastPlusOneAtMost__greaterThanAtMost__integer,axiom,
    ! [L: code_integer,U: code_integer] : set_or189985376899183464nteger(aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(L),one_one_Code_integer),U) = set_or2715278749043346189nteger(L,U) ).

% atLeastPlusOneAtMost_greaterThanAtMost_integer
tff(fact_9590_atLeast__Suc,axiom,
    ! [K: nat] : aa_nat_set_nat(set_ord_atLeast_nat,aa_nat_nat(suc,K)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,aa_nat_set_nat(set_ord_atLeast_nat,K)),insert_nat(K,bot_bot_set_nat)) ).

% atLeast_Suc
tff(fact_9591_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_9592_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_9593_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_9594_filterlim__Suc,axiom,
    filterlim_nat_nat(suc,at_top_nat,at_top_nat) ).

% filterlim_Suc
tff(fact_9595_trivial__limit__sequentially,axiom,
    at_top_nat != bot_bot_filter_nat ).

% trivial_limit_sequentially
tff(fact_9596_bij__betw__Suc,axiom,
    ! [M6: set_nat,N7: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(bij_betw_nat_nat(suc),M6),N7)
    <=> ( image_nat_nat(suc,M6) = N7 ) ) ).

% bij_betw_Suc
tff(fact_9597_Rep__unit,axiom,
    ! [X: product_unit] : member_o(aa_Product_unit_o(product_Rep_unit,X),insert_o($true,bot_bot_set_o)) ).

% Rep_unit
tff(fact_9598_Rep__unit__inject,axiom,
    ! [X: product_unit,Y: product_unit] :
      ( ( aa_Product_unit_o(product_Rep_unit,X)
      <=> aa_Product_unit_o(product_Rep_unit,Y) )
    <=> ( X = Y ) ) ).

% Rep_unit_inject
tff(fact_9599_Rep__unit__induct,axiom,
    ! [Y: $o,Pa: fun_o_o] :
      ( member_o((Y),insert_o($true,bot_bot_set_o))
     => ( ! [X3: product_unit] : aa_o_o(Pa,aa_Product_unit_o(product_Rep_unit,X3))
       => aa_o_o(Pa,(Y)) ) ) ).

% Rep_unit_induct
tff(fact_9600_Rep__unit__cases,axiom,
    ! [Y: $o] :
      ( member_o((Y),insert_o($true,bot_bot_set_o))
     => ~ ! [X3: product_unit] :
            ( (Y)
          <=> ~ aa_Product_unit_o(product_Rep_unit,X3) ) ) ).

% Rep_unit_cases
tff(fact_9601_type__definition__unit,axiom,
    type_d6188575255521822967unit_o(product_Rep_unit,product_Abs_unit,insert_o($true,bot_bot_set_o)) ).

% type_definition_unit
tff(fact_9602_Abs__unit__inverse,axiom,
    ! [Y: $o] :
      ( member_o((Y),insert_o($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_9603_Rep__unit__inverse,axiom,
    ! [X: product_unit] : aa_o_Product_unit(product_Abs_unit,aa_Product_unit_o(product_Rep_unit,X)) = X ).

% Rep_unit_inverse
tff(fact_9604_Abs__unit__cases,axiom,
    ! [X: product_unit] :
      ~ ! [Y3: $o] :
          ( ( X = aa_o_Product_unit(product_Abs_unit,(Y3)) )
         => ~ member_o((Y3),insert_o($true,bot_bot_set_o)) ) ).

% Abs_unit_cases
tff(fact_9605_Abs__unit__induct,axiom,
    ! [Pa: fun_Product_unit_o,X: product_unit] :
      ( ! [Y3: $o] :
          ( member_o((Y3),insert_o($true,bot_bot_set_o))
         => aa_Product_unit_o(Pa,aa_o_Product_unit(product_Abs_unit,(Y3))) )
     => aa_Product_unit_o(Pa,X) ) ).

% Abs_unit_induct
tff(fact_9606_Abs__unit__inject,axiom,
    ! [X: $o,Y: $o] :
      ( member_o((X),insert_o($true,bot_bot_set_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_9607_eventually__sequentially__Suc,axiom,
    ! [Pa: fun_nat_o] :
      ( eventually_nat(aTP_Lamm_aov(Pa),at_top_nat)
    <=> eventually_nat(Pa,at_top_nat) ) ).

% eventually_sequentially_Suc
tff(fact_9608_eventually__sequentially__seg,axiom,
    ! [Pa: fun_nat_o,K: nat] :
      ( eventually_nat(aa_nat_fun_nat_o(aTP_Lamm_aow(Pa),K),at_top_nat)
    <=> eventually_nat(Pa,at_top_nat) ) ).

% eventually_sequentially_seg
tff(fact_9609_le__sequentially,axiom,
    ! [F3: filter_nat] :
      ( ord_le2510731241096832064er_nat(F3,at_top_nat)
    <=> ! [N9: nat] : eventually_nat(aa_nat_fun_nat_o(ord_less_eq_nat,N9),F3) ) ).

% le_sequentially
tff(fact_9610_eventually__sequentiallyI,axiom,
    ! [Ca: nat,Pa: fun_nat_o] :
      ( ! [X3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ca),X3)
         => aa_nat_o(Pa,X3) )
     => eventually_nat(Pa,at_top_nat) ) ).

% eventually_sequentiallyI
tff(fact_9611_eventually__sequentially,axiom,
    ! [Pa: fun_nat_o] :
      ( eventually_nat(Pa,at_top_nat)
    <=> ? [N9: nat] :
        ! [N: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N9),N)
         => aa_nat_o(Pa,N) ) ) ).

% eventually_sequentially
tff(fact_9612_eventually__False__sequentially,axiom,
    ~ eventually_nat(aTP_Lamm_bg,at_top_nat) ).

% eventually_False_sequentially
tff(fact_9613_natLeq__on__wo__rel,axiom,
    ! [N2: nat] : aa_set3702741264012733747_nat_o(bNF_We3818239936649020644el_nat,collec3392354462482085612at_nat(produc6081775807080527818_nat_o(aTP_Lamm_aou(N2)))) ).

% natLeq_on_wo_rel
tff(fact_9614_UNIV__unit,axiom,
    top_to1996260823553986621t_unit = insert_Product_unit(product_Unity,bot_bo3957492148770167129t_unit) ).

% UNIV_unit
tff(fact_9615_bot__unit__def,axiom,
    bot_bot_Product_unit = product_Unity ).

% bot_unit_def
tff(fact_9616_Unity__def,axiom,
    product_Unity = aa_o_Product_unit(product_Abs_unit,$true) ).

% Unity_def
tff(fact_9617_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_9618_Inf__unit__def,axiom,
    ! [Uu: set_Product_unit] : comple2584293577114468500t_unit(Uu) = product_Unity ).

% Inf_unit_def
tff(fact_9619_old_Ounit_Oexhaust,axiom,
    ! [Y: product_unit] : Y = product_Unity ).

% old.unit.exhaust
tff(fact_9620_top__unit__def,axiom,
    top_top_Product_unit = product_Unity ).

% top_unit_def
tff(fact_9621_uminus__unit__def,axiom,
    ! [Uu: product_unit] : uminus2952777764628376836t_unit(Uu) = product_Unity ).

% uminus_unit_def
tff(fact_9622_Sup__unit__def,axiom,
    ! [Uu: set_Product_unit] : comple4687483117567863418t_unit(Uu) = product_Unity ).

% Sup_unit_def
tff(fact_9623_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_9624_wait__def,axiom,
    ! [N2: nat] : heap_Time_wait(N2) = heap_T6183433275982383450t_unit(aTP_Lamm_aox(N2)) ).

% wait_def
tff(fact_9625_default__unit__def,axiom,
    defaul566961228789861419t_unit = product_Unity ).

% default_unit_def
tff(fact_9626_log_Osimps,axiom,
    ! [B: code_natural,I2: code_natural] :
      log(B,I2) = $ite(
        ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,B),one_one_Code_natural)
        | aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,I2),B) ),
        one_one_Code_natural,
        plus_p4538020629002901425atural(one_one_Code_natural,log(B,divide5121882707175180666atural(I2,B))) ) ).

% log.simps
tff(fact_9627_log_Oelims,axiom,
    ! [X: code_natural,Xa2: code_natural,Y: code_natural] :
      ( ( log(X,Xa2) = Y )
     => ( Y = $ite(
            ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,X),one_one_Code_natural)
            | aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,Xa2),X) ),
            one_one_Code_natural,
            plus_p4538020629002901425atural(one_one_Code_natural,log(X,divide5121882707175180666atural(Xa2,X))) ) ) ) ).

% log.elims
tff(fact_9628_log_Opelims,axiom,
    ! [X: code_natural,Xa2: code_natural,Y: code_natural] :
      ( ( log(X,Xa2) = Y )
     => ( accp_P8126237942716283194atural(log_rel,produc3574140220909816553atural(X,Xa2))
       => ~ ( ( Y = $ite(
                  ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,X),one_one_Code_natural)
                  | aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,Xa2),X) ),
                  one_one_Code_natural,
                  plus_p4538020629002901425atural(one_one_Code_natural,log(X,divide5121882707175180666atural(Xa2,X))) ) )
           => ~ accp_P8126237942716283194atural(log_rel,produc3574140220909816553atural(X,Xa2)) ) ) ) ).

% log.pelims
tff(fact_9629_minus__shift__def,axiom,
    ! [Ra2: code_natural,K: code_natural,L: code_natural] :
      minus_shift(Ra2,K,L) = $ite(aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,K),L),minus_7197305767214868737atural(plus_p4538020629002901425atural(Ra2,K),L),minus_7197305767214868737atural(K,L)) ).

% minus_shift_def
tff(fact_9630_exhaustive__natural_H_Ocases,axiom,
    ! [X: produc7476732711998919211atural] :
      ~ ! [F2: fun_Co6386927884524458043e_term,D: code_natural,I3: code_natural] : X != produc3379161622255323613atural(F2,produc3574140220909816553atural(D,I3)) ).

% exhaustive_natural'.cases
tff(fact_9631_full__exhaustive__natural_H_Ocases,axiom,
    ! [X: produc9140211490168418807atural] :
      ~ ! [F2: fun_Pr285225112257423215e_term,D: code_natural,I3: code_natural] : X != produc3399137497101226409atural(F2,produc3574140220909816553atural(D,I3)) ).

% full_exhaustive_natural'.cases
tff(fact_9632_log_Ocases,axiom,
    ! [X: produc7822875418678951345atural] :
      ~ ! [B4: code_natural,I3: code_natural] : X != produc3574140220909816553atural(B4,I3) ).

% log.cases
tff(fact_9633_next_Osimps,axiom,
    ! [V: code_natural,W2: code_natural] :
      aa_Pro7165575656943818355atural(next,produc3574140220909816553atural(V,W2)) = $let(
        v: code_natural,
        v:= minus_shift(aa_num_Code_natural(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,aa_num_Code_natural(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))))))))))))))))),aa_num_Code_natural(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,aa_num_Code_natural(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))))))))))))))))),aa_num_Code_natural(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(aa_num_Code_natural(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(W2,aa_num_Code_natural(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))))))))))))))))),aa_num_Code_natural(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(W2,aa_num_Code_natural(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))))))))))))))))),aa_num_Code_natural(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(aa_num_Code_natural(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_9634_split__seed__def,axiom,
    ! [S2: produc7822875418678951345atural] : split_seed(S2) = produc8282080750456430313atural(aTP_Lamm_aoz(S2),S2) ).

% split_seed_def
tff(fact_9635_Random_Orange__def,axiom,
    ! [K: code_natural] : range(K) = produc5538323210962509403atural(iterat8892046348760725948atural(log(aa_num_Code_natural(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_apb,one_one_Code_natural),aTP_Lamm_apc(K)) ).

% Random.range_def
tff(fact_9636_range,axiom,
    ! [K: code_natural,S2: produc7822875418678951345atural] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,zero_z2226904508553997617atural),K)
     => aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,produc497848011034438852atural(aa_Pro7165575656943818355atural(range(K),S2))),K) ) ).

% range
tff(fact_9637_less__natural_Oabs__eq,axiom,
    ! [Xa2: nat,X: nat] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,aa_nat_Code_natural(code_natural_of_nat,Xa2)),aa_nat_Code_natural(code_natural_of_nat,X))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa2),X) ) ).

% less_natural.abs_eq
tff(fact_9638_less__eq__natural_Oabs__eq,axiom,
    ! [Xa2: nat,X: nat] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,aa_nat_Code_natural(code_natural_of_nat,Xa2)),aa_nat_Code_natural(code_natural_of_nat,X))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa2),X) ) ).

% less_eq_natural.abs_eq
tff(fact_9639_less__eq__natural_Orep__eq,axiom,
    ! [X: code_natural,Xa2: code_natural] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,X),Xa2)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_Code_natural_nat(code_nat_of_natural,X)),aa_Code_natural_nat(code_nat_of_natural,Xa2)) ) ).

% less_eq_natural.rep_eq
tff(fact_9640_less__natural_Orep__eq,axiom,
    ! [X: code_natural,Xa2: code_natural] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,X),Xa2)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_Code_natural_nat(code_nat_of_natural,X)),aa_Code_natural_nat(code_nat_of_natural,Xa2)) ) ).

% less_natural.rep_eq
tff(fact_9641_less__natural__def,axiom,
    ord_le5570908160329646204atural = map_fu2458051687388636536ural_o(code_nat_of_natural,map_fu4892316939951275536at_o_o(code_nat_of_natural,id_o),ord_less_nat) ).

% less_natural_def
tff(fact_9642_less__eq__natural__def,axiom,
    ord_le1926595141338095240atural = map_fu2458051687388636536ural_o(code_nat_of_natural,map_fu4892316939951275536at_o_o(code_nat_of_natural,id_o),ord_less_eq_nat) ).

% less_eq_natural_def
tff(fact_9643_natural__decr,axiom,
    ! [N2: code_natural] :
      ( ( N2 != zero_z2226904508553997617atural )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_Code_natural_nat(code_nat_of_natural,N2)),aa_nat_nat(suc,zero_zero_nat))),aa_Code_natural_nat(code_nat_of_natural,N2)) ) ).

% natural_decr
tff(fact_9644_bit__concat__bit__iff,axiom,
    ! [M: nat,K: int,L: int,N2: nat] :
      ( aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,aa_int_int(aa_int_fun_int_int(bit_concat_bit(M),K),L)),N2)
    <=> ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),M)
          & aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,K),N2) )
        | ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
          & aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,L),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),M)) ) ) ) ).

% bit_concat_bit_iff
tff(fact_9645_concat__bit__nonnegative__iff,axiom,
    ! [N2: nat,K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(bit_concat_bit(N2),K),L))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),L) ) ).

% concat_bit_nonnegative_iff
tff(fact_9646_concat__bit__negative__iff,axiom,
    ! [N2: nat,K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(bit_concat_bit(N2),K),L)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,L),zero_zero_int) ) ).

% concat_bit_negative_iff
tff(fact_9647_pair__lessI2,axiom,
    ! [A: nat,B: nat,S2: nat,Ta: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,S2),Ta)
       => member8206827879206165904at_nat(produc6161850002892822231at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,A),S2),aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,B),Ta)),fun_pair_less) ) ) ).

% pair_lessI2
tff(fact_9648_total__pair__less,axiom,
    ! [A3: set_Pr1261947904930325089at_nat] : total_3592101749530773125at_nat(A3,fun_pair_less) ).

% total_pair_less
tff(fact_9649_trans__pair__less,axiom,
    trans_4347625901269045472at_nat(fun_pair_less) ).

% trans_pair_less
tff(fact_9650_pair__less__iff1,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( member8206827879206165904at_nat(produc6161850002892822231at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,X),Y),aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_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_9651_pair__less__def,axiom,
    fun_pair_less = lex_prod_nat_nat(less_than,less_than) ).

% pair_less_def
tff(fact_9652_wf__pair__less,axiom,
    wf_Pro7803398752247294826at_nat(fun_pair_less) ).

% wf_pair_less
tff(fact_9653_pair__lessI1,axiom,
    ! [A: nat,B: nat,S2: nat,Ta: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => member8206827879206165904at_nat(produc6161850002892822231at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,A),S2),aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,B),Ta)),fun_pair_less) ) ).

% pair_lessI1
tff(fact_9654_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)
       => ( member8757157785044589968at_nat(produc2922128104949294807at_nat(XS,YS),fun_min_strict)
         => member8757157785044589968at_nat(produc2922128104949294807at_nat(XS,insert8211810215607154385at_nat(Y,YS)),fun_min_strict) ) ) ) ).

% smin_insertI
tff(fact_9655_pair__leqI2,axiom,
    ! [A: nat,B: nat,S2: nat,Ta: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,S2),Ta)
       => member8206827879206165904at_nat(produc6161850002892822231at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,A),S2),aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,B),Ta)),fun_pair_leq) ) ) ).

% pair_leqI2
tff(fact_9656_pair__leq__def,axiom,
    fun_pair_leq = aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,fun_pair_less),id_Pro2258643101195443293at_nat) ).

% pair_leq_def
tff(fact_9657_smin__emptyI,axiom,
    ! [X6: set_Pr1261947904930325089at_nat] :
      ( ( X6 != bot_bo2099793752762293965at_nat )
     => member8757157785044589968at_nat(produc2922128104949294807at_nat(X6,bot_bo2099793752762293965at_nat),fun_min_strict) ) ).

% smin_emptyI
tff(fact_9658_min__strict__def,axiom,
    fun_min_strict = min_ex6901939911449802026at_nat(fun_pair_less) ).

% min_strict_def
tff(fact_9659_pair__leqI1,axiom,
    ! [A: nat,B: nat,S2: nat,Ta: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => member8206827879206165904at_nat(produc6161850002892822231at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,A),S2),aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,B),Ta)),fun_pair_leq) ) ).

% pair_leqI1
tff(fact_9660_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)
       => ( member8757157785044589968at_nat(produc2922128104949294807at_nat(XS,YS),fun_max_weak)
         => member8757157785044589968at_nat(produc2922128104949294807at_nat(insert8211810215607154385at_nat(X,XS),YS),fun_max_weak) ) ) ) ).

% wmax_insertI
tff(fact_9661_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)
       => ( member8757157785044589968at_nat(produc2922128104949294807at_nat(XS,YS),fun_min_weak)
         => member8757157785044589968at_nat(produc2922128104949294807at_nat(XS,insert8211810215607154385at_nat(Y,YS)),fun_min_weak) ) ) ) ).

% wmin_insertI
tff(fact_9662_wmax__emptyI,axiom,
    ! [X6: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,X6)
     => member8757157785044589968at_nat(produc2922128104949294807at_nat(bot_bo2099793752762293965at_nat,X6),fun_max_weak) ) ).

% wmax_emptyI
tff(fact_9663_wmin__emptyI,axiom,
    ! [X6: set_Pr1261947904930325089at_nat] : member8757157785044589968at_nat(produc2922128104949294807at_nat(X6,bot_bo2099793752762293965at_nat),fun_min_weak) ).

% wmin_emptyI
tff(fact_9664_min__rpair__set,axiom,
    fun_re2478310338295953701at_nat(produc9060074326276436823at_nat(fun_min_strict,fun_min_weak)) ).

% min_rpair_set
tff(fact_9665_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_9666_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_9667_smax__insertI,axiom,
    ! [Y: product_prod_nat_nat,Y6: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat,X6: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Y),Y6)
     => ( member8206827879206165904at_nat(produc6161850002892822231at_nat(X,Y),fun_pair_less)
       => ( member8757157785044589968at_nat(produc2922128104949294807at_nat(X6,Y6),fun_max_strict)
         => member8757157785044589968at_nat(produc2922128104949294807at_nat(insert8211810215607154385at_nat(X,X6),Y6),fun_max_strict) ) ) ) ).

% smax_insertI
tff(fact_9668_max__strict__def,axiom,
    fun_max_strict = max_ex8135407076693332796at_nat(fun_pair_less) ).

% max_strict_def
tff(fact_9669_max__rpair__set,axiom,
    fun_re2478310338295953701at_nat(produc9060074326276436823at_nat(fun_max_strict,fun_max_weak)) ).

% max_rpair_set
tff(fact_9670_smax__emptyI,axiom,
    ! [Y6: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,Y6)
     => ( ( Y6 != bot_bo2099793752762293965at_nat )
       => member8757157785044589968at_nat(produc2922128104949294807at_nat(bot_bo2099793752762293965at_nat,Y6),fun_max_strict) ) ) ).

% smax_emptyI
tff(fact_9671_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_9672_wmsI,axiom,
    ! [A3: multis2468970476368604999at_nat,B3: multis2468970476368604999at_nat,Z3: multis2468970476368604999at_nat] :
      ( ( member8757157785044589968at_nat(produc2922128104949294807at_nat(set_ms8126754132646256062at_nat(A3),set_ms8126754132646256062at_nat(B3)),fun_max_strict)
        | ( ( A3 = zero_z1048942125864253310at_nat )
          & ( B3 = zero_z1048942125864253310at_nat ) ) )
     => member6689249552917799696at_nat(produc4348348721325984599at_nat(aa_mul5754855110707352700at_nat(plus_p7104986032573967614at_nat(Z3),A3),aa_mul5754855110707352700at_nat(plus_p7104986032573967614at_nat(Z3),B3)),ms_weak) ) ).

% wmsI
tff(fact_9673_smsI,axiom,
    ! [A3: multis2468970476368604999at_nat,B3: multis2468970476368604999at_nat,Z3: multis2468970476368604999at_nat] :
      ( member8757157785044589968at_nat(produc2922128104949294807at_nat(set_ms8126754132646256062at_nat(A3),set_ms8126754132646256062at_nat(B3)),fun_max_strict)
     => member6689249552917799696at_nat(produc4348348721325984599at_nat(aa_mul5754855110707352700at_nat(plus_p7104986032573967614at_nat(Z3),A3),aa_mul5754855110707352700at_nat(plus_p7104986032573967614at_nat(Z3),B3)),ms_strict) ) ).

% smsI
tff(fact_9674_ms__reduction__pair,axiom,
    fun_re7357418987779152907at_nat(produc5245064249948416855at_nat(ms_strict,ms_weak)) ).

% ms_reduction_pair
tff(fact_9675_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_9676_ms__strictI,axiom,
    ! [Z3: multis2468970476368604999at_nat,Z7: multis2468970476368604999at_nat,A3: multis2468970476368604999at_nat,B3: multis2468970476368604999at_nat] :
      ( pw_leq(Z3,Z7)
     => ( member8757157785044589968at_nat(produc2922128104949294807at_nat(set_ms8126754132646256062at_nat(A3),set_ms8126754132646256062at_nat(B3)),fun_max_strict)
       => member6689249552917799696at_nat(produc4348348721325984599at_nat(aa_mul5754855110707352700at_nat(plus_p7104986032573967614at_nat(Z3),A3),aa_mul5754855110707352700at_nat(plus_p7104986032573967614at_nat(Z7),B3)),ms_strict) ) ) ).

% ms_strictI
tff(fact_9677_ms__weakI1,axiom,
    ! [Z3: multis2468970476368604999at_nat,Z7: multis2468970476368604999at_nat,A3: multis2468970476368604999at_nat,B3: multis2468970476368604999at_nat] :
      ( pw_leq(Z3,Z7)
     => ( member8757157785044589968at_nat(produc2922128104949294807at_nat(set_ms8126754132646256062at_nat(A3),set_ms8126754132646256062at_nat(B3)),fun_max_strict)
       => member6689249552917799696at_nat(produc4348348721325984599at_nat(aa_mul5754855110707352700at_nat(plus_p7104986032573967614at_nat(Z3),A3),aa_mul5754855110707352700at_nat(plus_p7104986032573967614at_nat(Z7),B3)),ms_weak) ) ) ).

% ms_weakI1
tff(fact_9678_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_9679_pw__leq__step,axiom,
    ! [X: product_prod_nat_nat,Y: product_prod_nat_nat,X6: multis2468970476368604999at_nat,Y6: multis2468970476368604999at_nat] :
      ( member8206827879206165904at_nat(produc6161850002892822231at_nat(X,Y),fun_pair_leq)
     => ( pw_leq(X6,Y6)
       => pw_leq(aa_mul5754855110707352700at_nat(plus_p7104986032573967614at_nat(add_ms2612439473150266591at_nat(X,zero_z1048942125864253310at_nat)),X6),aa_mul5754855110707352700at_nat(plus_p7104986032573967614at_nat(add_ms2612439473150266591at_nat(Y,zero_z1048942125864253310at_nat)),Y6)) ) ) ).

% pw_leq_step
tff(fact_9680_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,Y5: product_prod_nat_nat,X9: multis2468970476368604999at_nat,Y7: multis2468970476368604999at_nat] :
            ( ( A1 = aa_mul5754855110707352700at_nat(plus_p7104986032573967614at_nat(add_ms2612439473150266591at_nat(X4,zero_z1048942125864253310at_nat)),X9) )
            & ( A22 = aa_mul5754855110707352700at_nat(plus_p7104986032573967614at_nat(add_ms2612439473150266591at_nat(Y5,zero_z1048942125864253310at_nat)),Y7) )
            & member8206827879206165904at_nat(produc6161850002892822231at_nat(X4,Y5),fun_pair_leq)
            & pw_leq(X9,Y7) ) ) ) ).

% pw_leq.simps
tff(fact_9681_pw__leq_Ocases,axiom,
    ! [A1: multis2468970476368604999at_nat,A22: multis2468970476368604999at_nat] :
      ( pw_leq(A1,A22)
     => ( ( ( A1 = zero_z1048942125864253310at_nat )
         => ( A22 != zero_z1048942125864253310at_nat ) )
       => ~ ! [X3: product_prod_nat_nat,Y3: product_prod_nat_nat,X10: multis2468970476368604999at_nat] :
              ( ( A1 = aa_mul5754855110707352700at_nat(plus_p7104986032573967614at_nat(add_ms2612439473150266591at_nat(X3,zero_z1048942125864253310at_nat)),X10) )
             => ! [Y8: multis2468970476368604999at_nat] :
                  ( ( A22 = aa_mul5754855110707352700at_nat(plus_p7104986032573967614at_nat(add_ms2612439473150266591at_nat(Y3,zero_z1048942125864253310at_nat)),Y8) )
                 => ( member8206827879206165904at_nat(produc6161850002892822231at_nat(X3,Y3),fun_pair_leq)
                   => ~ pw_leq(X10,Y8) ) ) ) ) ) ).

% pw_leq.cases
tff(fact_9682_ms__weakI2,axiom,
    ! [Z3: multis2468970476368604999at_nat,Z7: multis2468970476368604999at_nat] :
      ( pw_leq(Z3,Z7)
     => member6689249552917799696at_nat(produc4348348721325984599at_nat(aa_mul5754855110707352700at_nat(plus_p7104986032573967614at_nat(Z3),zero_z1048942125864253310at_nat),aa_mul5754855110707352700at_nat(plus_p7104986032573967614at_nat(Z7),zero_z1048942125864253310at_nat)),ms_weak) ) ).

% ms_weakI2
tff(fact_9683_pw__leq__split,axiom,
    ! [X6: multis2468970476368604999at_nat,Y6: multis2468970476368604999at_nat] :
      ( pw_leq(X6,Y6)
     => ? [A6: multis2468970476368604999at_nat,B9: multis2468970476368604999at_nat,Z8: multis2468970476368604999at_nat] :
          ( ( X6 = aa_mul5754855110707352700at_nat(plus_p7104986032573967614at_nat(A6),Z8) )
          & ( Y6 = aa_mul5754855110707352700at_nat(plus_p7104986032573967614at_nat(B9),Z8) )
          & ( member8757157785044589968at_nat(produc2922128104949294807at_nat(set_ms8126754132646256062at_nat(A6),set_ms8126754132646256062at_nat(B9)),fun_max_strict)
            | ( ( B9 = zero_z1048942125864253310at_nat )
              & ( A6 = zero_z1048942125864253310at_nat ) ) ) ) ) ).

% pw_leq_split
tff(fact_9684_natLeq__on__well__order__on,axiom,
    ! [N2: nat] : aa_set3702741264012733747_nat_o(order_2888998067076097458on_nat(collect_nat(aa_nat_fun_nat_o(aTP_Lamm_dg,N2))),collec3392354462482085612at_nat(produc6081775807080527818_nat_o(aTP_Lamm_aou(N2)))) ).

% natLeq_on_well_order_on
tff(fact_9685_natLeq__on__Well__order,axiom,
    ! [N2: nat] : aa_set3702741264012733747_nat_o(order_2888998067076097458on_nat(field_nat(collec3392354462482085612at_nat(produc6081775807080527818_nat_o(aTP_Lamm_aou(N2))))),collec3392354462482085612at_nat(produc6081775807080527818_nat_o(aTP_Lamm_aou(N2)))) ).

% natLeq_on_Well_order
tff(fact_9686_cone__def,axiom,
    bNF_Cardinal_cone = bNF_Ca7083678892733797993t_unit(insert_Product_unit(product_Unity,bot_bo3957492148770167129t_unit)) ).

% cone_def
tff(fact_9687_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_9688_ctwo__ordLess__natLeq,axiom,
    member4095101504841534314at_nat(produc8517790099723286449at_nat(bNF_Cardinal_ctwo,bNF_Ca8665028551170535155natLeq),bNF_We8182288985678559134_o_nat) ).

% ctwo_ordLess_natLeq
tff(fact_9689_natLeq__def,axiom,
    bNF_Ca8665028551170535155natLeq = collec3392354462482085612at_nat(produc6081775807080527818_nat_o(ord_less_eq_nat)) ).

% natLeq_def
tff(fact_9690_natLeq__underS__less,axiom,
    ! [N2: nat] : aa_nat_set_nat(order_underS_nat(bNF_Ca8665028551170535155natLeq),N2) = collect_nat(aa_nat_fun_nat_o(aTP_Lamm_dg,N2)) ).

% natLeq_underS_less
tff(fact_9691_Restr__natLeq,axiom,
    ! [N2: nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,bNF_Ca8665028551170535155natLeq),produc457027306803732586at_nat(collect_nat(aa_nat_fun_nat_o(aTP_Lamm_dg,N2)),aa_nat1818925681583257441et_nat(aTP_Lamm_apd,N2))) = collec3392354462482085612at_nat(produc6081775807080527818_nat_o(aTP_Lamm_aou(N2))) ).

% Restr_natLeq
tff(fact_9692_ATP_Olambda__1,axiom,
    ! [Uu: product_prod_int_int] :
      aa_Pro6970504127699319548nt_int(aTP_Lamm_aoj,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_9693_ATP_Olambda__2,axiom,
    ! [Uu: nat] :
      ( aa_nat_o(aTP_Lamm_ag,Uu)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uu),lim_Product_unit(h2))
        & ~ aa_set_nat_o(member_nat(Uu),as1) ) ) ).

% ATP.lambda_2
tff(fact_9694_ATP_Olambda__3,axiom,
    ! [Uu: nat] :
      ( aa_nat_o(aTP_Lamm_ah,Uu)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uu),lim_Product_unit(h2))
        & ~ aa_set_nat_o(member_nat(Uu),as) ) ) ).

% ATP.lambda_3
tff(fact_9695_ATP_Olambda__4,axiom,
    ! [Uu: product_prod_int_int] : aa_Pro6970504127699319548nt_int(aTP_Lamm_aol,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_9696_ATP_Olambda__5,axiom,
    ! [Uu: nat] : aa_nat_nat(aTP_Lamm_ank,Uu) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),aa_nat_nat(suc,zero_zero_nat)) ).

% ATP.lambda_5
tff(fact_9697_ATP_Olambda__6,axiom,
    ! [Uu: int] : aa_int_int(aTP_Lamm_aon,Uu) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Uu),Uu) ).

% ATP.lambda_6
tff(fact_9698_ATP_Olambda__7,axiom,
    ! [Uu: code_integer] : aa_Cod1783974652533396728nteger(aTP_Lamm_gf,Uu) = aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(Uu),one_one_Code_integer) ).

% ATP.lambda_7
tff(fact_9699_ATP_Olambda__8,axiom,
    ! [Uu: rat] : aa_rat_rat(aTP_Lamm_gc,Uu) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Uu),one_one_rat) ).

% ATP.lambda_8
tff(fact_9700_ATP_Olambda__9,axiom,
    ! [Uu: nat] : aa_nat_nat(aTP_Lamm_ge,Uu) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),one_one_nat) ).

% ATP.lambda_9
tff(fact_9701_ATP_Olambda__10,axiom,
    ! [Uu: int] : aa_int_int(aTP_Lamm_gd,Uu) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Uu),one_one_int) ).

% ATP.lambda_10
tff(fact_9702_ATP_Olambda__11,axiom,
    ! [Uu: nat] : aa_nat7603221163087113075at_nat(aTP_Lamm_aoq,Uu) = aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Uu),zero_zero_nat) ).

% ATP.lambda_11
tff(fact_9703_ATP_Olambda__12,axiom,
    ! [Uu: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aTP_Lamm_aoh,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_12
tff(fact_9704_ATP_Olambda__13,axiom,
    ! [Uu: code_natural] : aa_Cod6352003912080677724atural(aTP_Lamm_apb,Uu) = produc5538323210962509403atural(next,aTP_Lamm_apa(Uu)) ).

% ATP.lambda_13
tff(fact_9705_ATP_Olambda__14,axiom,
    ! [Uu: nat] : aa_nat_set_nat(aTP_Lamm_anc,Uu) = collect_nat(aa_nat_fun_nat_o(aTP_Lamm_adt,Uu)) ).

% ATP.lambda_14
tff(fact_9706_ATP_Olambda__15,axiom,
    ! [Uu: num] : aa_num_option_num(aTP_Lamm_amo,Uu) = aa_num_option_num(some_num,aa_num_num(bit1,Uu)) ).

% ATP.lambda_15
tff(fact_9707_ATP_Olambda__16,axiom,
    ! [Uu: num] : aa_num_option_num(aTP_Lamm_agv,Uu) = aa_num_option_num(some_num,aa_num_num(bit0,Uu)) ).

% ATP.lambda_16
tff(fact_9708_ATP_Olambda__17,axiom,
    ! [Uu: int] : aa_int2565445551602696444nt_int(aTP_Lamm_anr,Uu) = aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(uminus_uminus_int,Uu)) ).

% ATP.lambda_17
tff(fact_9709_ATP_Olambda__18,axiom,
    ! [Uu: int] : aa_int2565445551602696444nt_int(aTP_Lamm_aoa,Uu) = aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(abs_abs_int,Uu)) ).

% ATP.lambda_18
tff(fact_9710_ATP_Olambda__19,axiom,
    ! [Uu: nat] : aa_nat8803788312118491388at_nat(aTP_Lamm_agq,Uu) = aa_nat8803788312118491388at_nat(product_Pair_nat_nat,aa_nat_nat(suc,Uu)) ).

% ATP.lambda_19
tff(fact_9711_ATP_Olambda__20,axiom,
    ! [Uu: int] : aa_int_nat(aTP_Lamm_and,Uu) = aa_int_nat(nat2,aa_int_int(abs_abs_int,Uu)) ).

% ATP.lambda_20
tff(fact_9712_ATP_Olambda__21,axiom,
    ! [Uu: num,Uua: nat] : aa_nat_option_num(aTP_Lamm_ajg(Uu),Uua) = aa_num_option_num(case_num_option_num(aa_num_option_num(some_num,one),aTP_Lamm_aje(Uua),aTP_Lamm_ajf(Uua)),Uu) ).

% ATP.lambda_21
tff(fact_9713_ATP_Olambda__22,axiom,
    ! [Uu: nat,Uua: nat] :
      aa_nat_Code_integer(aTP_Lamm_ads(Uu),Uua) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua),aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu),Uua)),zero_z3403309356797280102nteger) ).

% ATP.lambda_22
tff(fact_9714_ATP_Olambda__23,axiom,
    ! [Uu: nat,Uua: nat] :
      aa_nat_rat(aTP_Lamm_adq(Uu),Uua) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua),aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu),Uua)),zero_zero_rat) ).

% ATP.lambda_23
tff(fact_9715_ATP_Olambda__24,axiom,
    ! [Uu: nat,Uua: nat] :
      aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_adr,Uu),Uua) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu),Uua)),zero_zero_int) ).

% ATP.lambda_24
tff(fact_9716_ATP_Olambda__25,axiom,
    ! [Uu: int,Uua: int] :
      aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_aof,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_25
tff(fact_9717_ATP_Olambda__26,axiom,
    ! [Uu: int,Uua: int] :
      aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_aok,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_26
tff(fact_9718_ATP_Olambda__27,axiom,
    ! [Uu: nat,Uua: nat] :
      aa_nat_Code_integer(aTP_Lamm_adp(Uu),Uua) = $ite(~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua),aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu),Uua)),zero_z3403309356797280102nteger) ).

% ATP.lambda_27
tff(fact_9719_ATP_Olambda__28,axiom,
    ! [Uu: nat,Uua: nat] :
      aa_nat_rat(aTP_Lamm_adn(Uu),Uua) = $ite(~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua),aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu),Uua)),zero_zero_rat) ).

% ATP.lambda_28
tff(fact_9720_ATP_Olambda__29,axiom,
    ! [Uu: nat,Uua: nat] :
      aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_ado,Uu),Uua) = $ite(~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu),Uua)),zero_zero_int) ).

% ATP.lambda_29
tff(fact_9721_ATP_Olambda__30,axiom,
    ! [Uu: option8963830502488799655at_nat,Uua: set_Pr1261947904930325089at_nat] : aa_set2934564513904597954at_nat(aTP_Lamm_en(Uu),Uua) = case_o311359030874850053at_nat(none_s625347054029921090at_nat,aTP_Lamm_em(Uua),Uu) ).

% ATP.lambda_30
tff(fact_9722_ATP_Olambda__31,axiom,
    ! [Uu: option_Product_unit,Uua: product_unit] : aa_Pro7616171890175246658t_unit(aTP_Lamm_el(Uu),Uua) = case_o2985186191577459077t_unit(none_Product_unit,aTP_Lamm_ek(Uua),Uu) ).

% ATP.lambda_31
tff(fact_9723_ATP_Olambda__32,axiom,
    ! [Uu: option_nat,Uua: nat] : aa_nat_option_nat(aTP_Lamm_ej(Uu),Uua) = case_o7429725398727453821at_nat(none_nat,aTP_Lamm_ei(Uua),Uu) ).

% ATP.lambda_32
tff(fact_9724_ATP_Olambda__33,axiom,
    ! [Uu: option_assn,Uua: assn] : aa_assn_option_assn(aTP_Lamm_eh(Uu),Uua) = case_o4484465799723439917n_assn(none_assn,aTP_Lamm_eg(Uua),Uu) ).

% ATP.lambda_33
tff(fact_9725_ATP_Olambda__34,axiom,
    ! [Uu: option_set_nat,Uua: set_nat] : aa_set6119337032432945050et_nat(aTP_Lamm_ef(Uu),Uua) = case_o4054078431260844265et_nat(none_set_nat,aTP_Lamm_ee(Uua),Uu) ).

% ATP.lambda_34
tff(fact_9726_ATP_Olambda__35,axiom,
    ! [Uu: option_num,Uua: num] :
      ( aa_num_o(aTP_Lamm_fc(Uu),Uua)
    <=> aa_option_num_o(case_option_o_num($true,aa_num_fun_num_o(aTP_Lamm_fb,Uua)),Uu) ) ).

% ATP.lambda_35
tff(fact_9727_ATP_Olambda__36,axiom,
    ! [Uu: option_int,Uua: int] :
      ( aa_int_o(aTP_Lamm_fe(Uu),Uua)
    <=> case_option_o_int($true,aa_int_fun_int_o(aTP_Lamm_dk,Uua),Uu) ) ).

% ATP.lambda_36
tff(fact_9728_ATP_Olambda__37,axiom,
    ! [Uu: option_rat,Uua: rat] :
      ( aa_rat_o(aTP_Lamm_fa(Uu),Uua)
    <=> case_option_o_rat($true,aa_rat_fun_rat_o(aTP_Lamm_di,Uua),Uu) ) ).

% ATP.lambda_37
tff(fact_9729_ATP_Olambda__38,axiom,
    ! [Uu: option_nat,Uua: nat] :
      ( aa_nat_o(aTP_Lamm_fd(Uu),Uua)
    <=> case_option_o_nat($true,aa_nat_fun_nat_o(aTP_Lamm_dg,Uua),Uu) ) ).

% ATP.lambda_38
tff(fact_9730_ATP_Olambda__39,axiom,
    ! [Uu: option_assn,Uua: assn] :
      ( aa_assn_o(aTP_Lamm_ez(Uu),Uua)
    <=> case_option_o_assn($true,aa_assn_fun_assn_o(aTP_Lamm_dc,Uua),Uu) ) ).

% ATP.lambda_39
tff(fact_9731_ATP_Olambda__40,axiom,
    ! [Uu: nat,Uua: num] : aa_num_option_num(aa_nat3713591252005393295on_num(aTP_Lamm_ajh,Uu),Uua) = aa_nat_option_num(case_nat_option_num(none_num,aTP_Lamm_ajg(Uua)),Uu) ).

% ATP.lambda_40
tff(fact_9732_ATP_Olambda__41,axiom,
    ! [Uu: option_set_int,Uua: set_int] :
      ( aa_set_int_o(aTP_Lamm_eu(Uu),Uua)
    <=> case_o223999843215110191et_int($false,aa_set5425576183772390127_int_o(ord_less_eq_set_int,Uua),Uu) ) ).

% ATP.lambda_41
tff(fact_9733_ATP_Olambda__42,axiom,
    ! [Uu: option_rat,Uua: rat] :
      ( aa_rat_o(aTP_Lamm_ev(Uu),Uua)
    <=> case_option_o_rat($false,aa_rat_fun_rat_o(ord_less_eq_rat,Uua),Uu) ) ).

% ATP.lambda_42
tff(fact_9734_ATP_Olambda__43,axiom,
    ! [Uu: option_num,Uua: num] :
      ( aa_num_o(aTP_Lamm_ew(Uu),Uua)
    <=> aa_option_num_o(case_option_o_num($false,aa_num_fun_num_o(ord_less_eq_num,Uua)),Uu) ) ).

% ATP.lambda_43
tff(fact_9735_ATP_Olambda__44,axiom,
    ! [Uu: option_nat,Uua: nat] :
      ( aa_nat_o(aTP_Lamm_ex(Uu),Uua)
    <=> case_option_o_nat($false,aa_nat_fun_nat_o(ord_less_eq_nat,Uua),Uu) ) ).

% ATP.lambda_44
tff(fact_9736_ATP_Olambda__45,axiom,
    ! [Uu: option_int,Uua: int] :
      ( aa_int_o(aTP_Lamm_ey(Uu),Uua)
    <=> case_option_o_int($false,aa_int_fun_int_o(ord_less_eq_int,Uua),Uu) ) ).

% ATP.lambda_45
tff(fact_9737_ATP_Olambda__46,axiom,
    ! [Uu: nat,Uua: num] : aa_num_option_num(aTP_Lamm_aje(Uu),Uua) = aa_opt5369159329878751634on_num(case_o6005452278849405969um_num(none_num,aTP_Lamm_agv),aa_num_option_num(bit_take_bit_num(Uu),Uua)) ).

% ATP.lambda_46
tff(fact_9738_ATP_Olambda__47,axiom,
    ! [Uu: fun_nat_fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_ahc(Uu),Uua) = aa_set_nat_int(groups705719431365010083at_int(aa_nat_fun_nat_int(aTP_Lamm_ahb(Uu),Uua)),aa_nat_set_nat(set_ord_atMost_nat,Uua)) ).

% ATP.lambda_47
tff(fact_9739_ATP_Olambda__48,axiom,
    ! [Uu: fun_nat_fun_nat_nat,Uua: nat] : aa_nat_nat(aTP_Lamm_aha(Uu),Uua) = aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_agy(Uu),Uua)),aa_nat_set_nat(set_ord_atMost_nat,Uua)) ).

% ATP.lambda_48
tff(fact_9740_ATP_Olambda__49,axiom,
    ! [Uu: fun_nat_fun_nat_nat,Uua: nat] : aa_nat_nat(aTP_Lamm_agz(Uu),Uua) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_agy(Uu),Uua)),aa_nat_set_nat(set_ord_atMost_nat,Uua)) ).

% ATP.lambda_49
tff(fact_9741_ATP_Olambda__50,axiom,
    ! [Uu: product_prod_int_int,Uua: product_prod_int_int] : aa_Pro6970504127699319548nt_int(aa_Pro4124234152028750080nt_int(aTP_Lamm_aoi,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_50
tff(fact_9742_ATP_Olambda__51,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_Code_integer(aTP_Lamm_pe(Uu),Uua) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),Uua)),aa_nat_Code_integer(semiri4939895301339042750nteger,Uua))),aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu),Uua))) ).

% ATP.lambda_51
tff(fact_9743_ATP_Olambda__52,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_rat(aTP_Lamm_pc(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_52
tff(fact_9744_ATP_Olambda__53,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_pd,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_53
tff(fact_9745_ATP_Olambda__54,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_rat(aTP_Lamm_abw(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(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),Uua)) ).

% ATP.lambda_54
tff(fact_9746_ATP_Olambda__55,axiom,
    ! [Uu: code_integer,Uua: code_integer] :
      aa_Code_integer_int(aa_Cod1242632294114111009er_int(aTP_Lamm_amg,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_55
tff(fact_9747_ATP_Olambda__56,axiom,
    ! [Uu: product_prod_int_int,Uua: product_prod_int_int] : aa_Pro6970504127699319548nt_int(aa_Pro4124234152028750080nt_int(aTP_Lamm_aom,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_56
tff(fact_9748_ATP_Olambda__57,axiom,
    ! [Uu: rat,Uua: int] :
      ( aa_int_o(aTP_Lamm_anp(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_57
tff(fact_9749_ATP_Olambda__58,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_Code_integer(aTP_Lamm_pb(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_58
tff(fact_9750_ATP_Olambda__59,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_rat(aTP_Lamm_oz(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_59
tff(fact_9751_ATP_Olambda__60,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_60
tff(fact_9752_ATP_Olambda__61,axiom,
    ! [Uu: rat,Uua: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aTP_Lamm_aoo(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_61
tff(fact_9753_ATP_Olambda__62,axiom,
    ! [Uu: rat,Uua: nat] : aa_nat_rat(aTP_Lamm_oj(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_62
tff(fact_9754_ATP_Olambda__63,axiom,
    ! [Uu: rat,Uua: nat] : aa_nat_rat(aTP_Lamm_ack(Uu),Uua) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(gbinomial_rat(Uu),Uua)),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,Uu),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)))),aa_nat_rat(semiri681578069525770553at_rat,Uua))) ).

% ATP.lambda_63
tff(fact_9755_ATP_Olambda__64,axiom,
    ! [Uu: rat,Uua: nat] : aa_nat_rat(aTP_Lamm_os(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_64
tff(fact_9756_ATP_Olambda__65,axiom,
    ! [Uu: nat,Uua: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_adw,Uu),Uua)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Uu),Uua)
        & ( Uu != Uua ) ) ) ).

% ATP.lambda_65
tff(fact_9757_ATP_Olambda__66,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(power_power_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu),Uua)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% ATP.lambda_66
tff(fact_9758_ATP_Olambda__67,axiom,
    ! [Uu: set_int,Uua: int] :
      ( aa_int_o(aTP_Lamm_aor(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_67
tff(fact_9759_ATP_Olambda__68,axiom,
    ! [Uu: set_Code_integer,Uua: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(aa_set4044230130517860621eger_o(aTP_Lamm_ahm,Uu),Uua)
    <=> ( aa_set5643387332196575111eger_o(ord_le1307284697595431911nteger(Uu),Uua)
        & aa_set5643387332196575111eger_o(finite6017078050557962740nteger,Uua) ) ) ).

% ATP.lambda_68
tff(fact_9760_ATP_Olambda__69,axiom,
    ! [Uu: set_Pr1261947904930325089at_nat,Uua: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(aTP_Lamm_ahn,Uu),Uua)
    <=> ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le7866589430770878221at_nat,Uu),Uua)
        & aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,Uua) ) ) ).

% ATP.lambda_69
tff(fact_9761_ATP_Olambda__70,axiom,
    ! [Uu: set_nat,Uua: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(aTP_Lamm_ahk,Uu),Uua)
    <=> ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,Uu),Uua)
        & aa_set_nat_o(finite_finite_nat,Uua) ) ) ).

% ATP.lambda_70
tff(fact_9762_ATP_Olambda__71,axiom,
    ! [Uu: set_int,Uua: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(aTP_Lamm_ahl,Uu),Uua)
    <=> ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_set_int,Uu),Uua)
        & aa_set_int_o(finite_finite_int,Uua) ) ) ).

% ATP.lambda_71
tff(fact_9763_ATP_Olambda__72,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_anf,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_72
tff(fact_9764_ATP_Olambda__73,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_yg,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_73
tff(fact_9765_ATP_Olambda__74,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_yf,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_74
tff(fact_9766_ATP_Olambda__75,axiom,
    ! [Uu: rat,Uua: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(aTP_Lamm_abf,Uu),Uua)
    <=> ( member_rat(Uua,ring_1_Ints_rat)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,Uua)),Uu) ) ) ).

% ATP.lambda_75
tff(fact_9767_ATP_Olambda__76,axiom,
    ! [Uu: fun_nat_Code_integer,Uua: nat] : aa_nat_Code_integer(aTP_Lamm_ya(Uu),Uua) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(Uu,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua))),aa_nat_Code_integer(Uu,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)))) ).

% ATP.lambda_76
tff(fact_9768_ATP_Olambda__77,axiom,
    ! [Uu: fun_nat_assn,Uua: nat] : aa_nat_assn(aTP_Lamm_yb(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,aa_num_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,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)))) ).

% ATP.lambda_77
tff(fact_9769_ATP_Olambda__78,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_yc(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,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua))),aa_nat_rat(Uu,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)))) ).

% ATP.lambda_78
tff(fact_9770_ATP_Olambda__79,axiom,
    ! [Uu: fun_nat_nat,Uua: nat] : aa_nat_nat(aTP_Lamm_yd(Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(Uu,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua))),aa_nat_nat(Uu,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)))) ).

% ATP.lambda_79
tff(fact_9771_ATP_Olambda__80,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_ye(Uu),Uua) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(Uu,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua))),aa_nat_int(Uu,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)))) ).

% ATP.lambda_80
tff(fact_9772_ATP_Olambda__81,axiom,
    ! [Uu: fun_na2338080690716861136at_nat,Uua: nat] : aa_nat2240542744625857721at_nat(aTP_Lamm_xw(Uu),Uua) = aa_mul5754855110707352700at_nat(plus_p7104986032573967614at_nat(aa_nat2240542744625857721at_nat(Uu,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua))),aa_nat2240542744625857721at_nat(Uu,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)))) ).

% ATP.lambda_81
tff(fact_9773_ATP_Olambda__82,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_xx(Uu),Uua) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(Uu,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua))),aa_nat_rat(Uu,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)))) ).

% ATP.lambda_82
tff(fact_9774_ATP_Olambda__83,axiom,
    ! [Uu: fun_nat_nat,Uua: nat] : aa_nat_nat(aTP_Lamm_xz(Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(Uu,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua))),aa_nat_nat(Uu,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)))) ).

% ATP.lambda_83
tff(fact_9775_ATP_Olambda__84,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_xy(Uu),Uua) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(Uu,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua))),aa_nat_int(Uu,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)))) ).

% ATP.lambda_84
tff(fact_9776_ATP_Olambda__85,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_kd(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_85
tff(fact_9777_ATP_Olambda__86,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_ke(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_86
tff(fact_9778_ATP_Olambda__87,axiom,
    ! [Uu: fun_nat_fun_nat_nat,Uua: nat] : aa_nat_nat(aTP_Lamm_abs(Uu),Uua) = aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(Uu,Uua)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),Uua)) ).

% ATP.lambda_87
tff(fact_9779_ATP_Olambda__88,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(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),Uua)) ).

% ATP.lambda_88
tff(fact_9780_ATP_Olambda__89,axiom,
    ! [Uu: fun_nat_fun_nat_nat,Uua: nat] : aa_nat_nat(aTP_Lamm_abp(Uu),Uua) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(Uu,Uua)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),Uua)) ).

% ATP.lambda_89
tff(fact_9781_ATP_Olambda__90,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_qv(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_90
tff(fact_9782_ATP_Olambda__91,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_kq(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_91
tff(fact_9783_ATP_Olambda__92,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_kr(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_92
tff(fact_9784_ATP_Olambda__93,axiom,
    ! [Uu: fun_nat_Code_integer,Uua: nat] : aa_nat_Code_integer(aTP_Lamm_kn(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_93
tff(fact_9785_ATP_Olambda__94,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_ko(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_94
tff(fact_9786_ATP_Olambda__95,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_kp(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_95
tff(fact_9787_ATP_Olambda__96,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_oc(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_96
tff(fact_9788_ATP_Olambda__97,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_od(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_97
tff(fact_9789_ATP_Olambda__98,axiom,
    ! [Uu: fun_nat_fun_nat_nat,Uua: nat] : aa_nat_nat(aTP_Lamm_aij(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_98
tff(fact_9790_ATP_Olambda__99,axiom,
    ! [Uu: fun_nat_fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_aik(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_99
tff(fact_9791_ATP_Olambda__100,axiom,
    ! [Uu: fun_nat_fun_nat_nat,Uua: nat] : aa_nat_nat(aTP_Lamm_aii(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_100
tff(fact_9792_ATP_Olambda__101,axiom,
    ! [Uu: fun_Co8499238673658658103nteger,Uua: code_integer] :
      ( aa_Code_integer_o(aTP_Lamm_zl(Uu),Uua)
    <=> ( aa_Cod1783974652533396728nteger(Uu,Uua) = zero_z3403309356797280102nteger ) ) ).

% ATP.lambda_101
tff(fact_9793_ATP_Olambda__102,axiom,
    ! [Uu: fun_Code_integer_rat,Uua: code_integer] :
      ( aa_Code_integer_o(aTP_Lamm_zn(Uu),Uua)
    <=> ( aa_Code_integer_rat(Uu,Uua) = zero_zero_rat ) ) ).

% ATP.lambda_102
tff(fact_9794_ATP_Olambda__103,axiom,
    ! [Uu: fun_Code_integer_nat,Uua: code_integer] :
      ( aa_Code_integer_o(aTP_Lamm_zp(Uu),Uua)
    <=> ( aa_Code_integer_nat(Uu,Uua) = zero_zero_nat ) ) ).

% ATP.lambda_103
tff(fact_9795_ATP_Olambda__104,axiom,
    ! [Uu: fun_Code_integer_int,Uua: code_integer] :
      ( aa_Code_integer_o(aTP_Lamm_zq(Uu),Uua)
    <=> ( aa_Code_integer_int(Uu,Uua) = zero_zero_int ) ) ).

% ATP.lambda_104
tff(fact_9796_ATP_Olambda__105,axiom,
    ! [Uu: fun_nat_Code_integer,Uua: nat] :
      ( aa_nat_o(aTP_Lamm_zr(Uu),Uua)
    <=> ( aa_nat_Code_integer(Uu,Uua) = zero_z3403309356797280102nteger ) ) ).

% ATP.lambda_105
tff(fact_9797_ATP_Olambda__106,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] :
      ( aa_nat_o(aTP_Lamm_zs(Uu),Uua)
    <=> ( aa_nat_rat(Uu,Uua) = zero_zero_rat ) ) ).

% ATP.lambda_106
tff(fact_9798_ATP_Olambda__107,axiom,
    ! [Uu: fun_nat_int,Uua: nat] :
      ( aa_nat_o(aTP_Lamm_zt(Uu),Uua)
    <=> ( aa_nat_int(Uu,Uua) = zero_zero_int ) ) ).

% ATP.lambda_107
tff(fact_9799_ATP_Olambda__108,axiom,
    ! [Uu: fun_int_Code_integer,Uua: int] :
      ( aa_int_o(aTP_Lamm_zk(Uu),Uua)
    <=> ( aa_int_Code_integer(Uu,Uua) = zero_z3403309356797280102nteger ) ) ).

% ATP.lambda_108
tff(fact_9800_ATP_Olambda__109,axiom,
    ! [Uu: fun_int_rat,Uua: int] :
      ( aa_int_o(aTP_Lamm_zm(Uu),Uua)
    <=> ( aa_int_rat(Uu,Uua) = zero_zero_rat ) ) ).

% ATP.lambda_109
tff(fact_9801_ATP_Olambda__110,axiom,
    ! [Uu: fun_int_nat,Uua: int] :
      ( aa_int_o(aTP_Lamm_zo(Uu),Uua)
    <=> ( aa_int_nat(Uu,Uua) = zero_zero_nat ) ) ).

% ATP.lambda_110
tff(fact_9802_ATP_Olambda__111,axiom,
    ! [Uu: fun_Co8499238673658658103nteger,Uua: code_integer] :
      ( aa_Code_integer_o(aTP_Lamm_aaw(Uu),Uua)
    <=> ( aa_Cod1783974652533396728nteger(Uu,Uua) = one_one_Code_integer ) ) ).

% ATP.lambda_111
tff(fact_9803_ATP_Olambda__112,axiom,
    ! [Uu: fun_Co4490034186337852034r_assn,Uua: code_integer] :
      ( aa_Code_integer_o(aTP_Lamm_aay(Uu),Uua)
    <=> ( aa_Code_integer_assn(Uu,Uua) = one_one_assn ) ) ).

% ATP.lambda_112
tff(fact_9804_ATP_Olambda__113,axiom,
    ! [Uu: fun_Code_integer_rat,Uua: code_integer] :
      ( aa_Code_integer_o(aTP_Lamm_aba(Uu),Uua)
    <=> ( aa_Code_integer_rat(Uu,Uua) = one_one_rat ) ) ).

% ATP.lambda_113
tff(fact_9805_ATP_Olambda__114,axiom,
    ! [Uu: fun_Code_integer_nat,Uua: code_integer] :
      ( aa_Code_integer_o(aTP_Lamm_abc(Uu),Uua)
    <=> ( aa_Code_integer_nat(Uu,Uua) = one_one_nat ) ) ).

% ATP.lambda_114
tff(fact_9806_ATP_Olambda__115,axiom,
    ! [Uu: fun_Code_integer_int,Uua: code_integer] :
      ( aa_Code_integer_o(aTP_Lamm_abd(Uu),Uua)
    <=> ( aa_Code_integer_int(Uu,Uua) = one_one_int ) ) ).

% ATP.lambda_115
tff(fact_9807_ATP_Olambda__116,axiom,
    ! [Uu: fun_nat_Code_integer,Uua: nat] :
      ( aa_nat_o(aTP_Lamm_abe(Uu),Uua)
    <=> ( aa_nat_Code_integer(Uu,Uua) = one_one_Code_integer ) ) ).

% ATP.lambda_116
tff(fact_9808_ATP_Olambda__117,axiom,
    ! [Uu: fun_int_Code_integer,Uua: int] :
      ( aa_int_o(aTP_Lamm_aav(Uu),Uua)
    <=> ( aa_int_Code_integer(Uu,Uua) = one_one_Code_integer ) ) ).

% ATP.lambda_117
tff(fact_9809_ATP_Olambda__118,axiom,
    ! [Uu: fun_int_assn,Uua: int] :
      ( aa_int_o(aTP_Lamm_aax(Uu),Uua)
    <=> ( aa_int_assn(Uu,Uua) = one_one_assn ) ) ).

% ATP.lambda_118
tff(fact_9810_ATP_Olambda__119,axiom,
    ! [Uu: fun_int_rat,Uua: int] :
      ( aa_int_o(aTP_Lamm_aaz(Uu),Uua)
    <=> ( aa_int_rat(Uu,Uua) = one_one_rat ) ) ).

% ATP.lambda_119
tff(fact_9811_ATP_Olambda__120,axiom,
    ! [Uu: fun_int_nat,Uua: int] :
      ( aa_int_o(aTP_Lamm_abb(Uu),Uua)
    <=> ( aa_int_nat(Uu,Uua) = one_one_nat ) ) ).

% ATP.lambda_120
tff(fact_9812_ATP_Olambda__121,axiom,
    ! [Uu: num,Uua: num] : aa_num_int(aTP_Lamm_agr(Uu),Uua) = aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,aa_num_nat(numeral_numeral_nat,Uu)),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,Uu))),aa_num_int(numeral_numeral_int,Uua))) ).

% ATP.lambda_121
tff(fact_9813_ATP_Olambda__122,axiom,
    ! [Uu: assn,Uua: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(aTP_Lamm_gz(Uu),Uua)
    <=> ( aa_Pro653082453696054616_nat_o(in_range,Uua)
        & ~ aa_Pro653082453696054616_nat_o(rep_assn(Uu),Uua) ) ) ).

% ATP.lambda_122
tff(fact_9814_ATP_Olambda__123,axiom,
    ! [Uu: num,Uua: num] : aa_num_int(aa_num_fun_num_int(aTP_Lamm_aop,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_123
tff(fact_9815_ATP_Olambda__124,axiom,
    ! [Uu: code_integer,Uua: code_integer] :
      aa_Code_integer_num(aa_Cod6128727250572145423er_num(aTP_Lamm_amf,Uu),Uua) = $let(
        l2: num,
        l2:= code_num_of_integer(Uu),
        $let(
          l3: num,
          l3:= aa_num_num(aa_num_fun_num_num(plus_plus_num,l2),l2),
          $ite(Uua = zero_z3403309356797280102nteger,l3,aa_num_num(aa_num_fun_num_num(plus_plus_num,l3),one)) ) ) ).

% ATP.lambda_124
tff(fact_9816_ATP_Olambda__125,axiom,
    ! [Uu: code_integer,Uua: code_integer] :
      aa_Code_integer_nat(aa_Cod5420483313623307717er_nat(aTP_Lamm_amj,Uu),Uua) = $let(
        l2: nat,
        l2:= aa_Code_integer_nat(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_125
tff(fact_9817_ATP_Olambda__126,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_rat(aTP_Lamm_ks(Uu),Uua) = aa_nat_rat(gbinomial_rat(aa_nat_rat(semiri681578069525770553at_rat,Uua)),Uu) ).

% ATP.lambda_126
tff(fact_9818_ATP_Olambda__127,axiom,
    ! [Uu: code_integer,Uua: code_integer] : aa_Cod6977894654539884864nteger(aa_Cod8380929501330563836nteger(aTP_Lamm_ahy,Uu),Uua) = produc1086072967326762835nteger(Uu,aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Uua)),one_one_Code_integer)) ).

% ATP.lambda_127
tff(fact_9819_ATP_Olambda__128,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aTP_Lamm_ahx,Uu),Uua) = aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Uu),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)),one_one_nat)) ).

% ATP.lambda_128
tff(fact_9820_ATP_Olambda__129,axiom,
    ! [Uu: int,Uua: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_ahw,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_129
tff(fact_9821_ATP_Olambda__130,axiom,
    ! [Uu: code_integer,Uua: code_integer] : aa_Cod6977894654539884864nteger(aa_Cod8380929501330563836nteger(aTP_Lamm_agu,Uu),Uua) = produc1086072967326762835nteger(Uu,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Uua)) ).

% ATP.lambda_130
tff(fact_9822_ATP_Olambda__131,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aTP_Lamm_agt,Uu),Uua) = aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Uu),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)) ).

% ATP.lambda_131
tff(fact_9823_ATP_Olambda__132,axiom,
    ! [Uu: int,Uua: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_ags,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_132
tff(fact_9824_ATP_Olambda__133,axiom,
    ! [Uu: rat,Uua: nat] : aa_nat_rat(aTP_Lamm_pf(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)),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)))) ).

% ATP.lambda_133
tff(fact_9825_ATP_Olambda__134,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_tt,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_134
tff(fact_9826_ATP_Olambda__135,axiom,
    ! [Uu: int,Uua: int] : aa_int_int(aa_int_fun_int_int(aTP_Lamm_agw,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_135
tff(fact_9827_ATP_Olambda__136,axiom,
    ! [Uu: code_natural,Uua: nat] : aa_nat_Code_natural(aTP_Lamm_me(Uu),Uua) = minus_7197305767214868737atural(Uu,aa_nat_Code_natural(semiri3763490453095760265atural,Uua)) ).

% ATP.lambda_136
tff(fact_9828_ATP_Olambda__137,axiom,
    ! [Uu: code_integer,Uua: nat] : aa_nat_Code_integer(aTP_Lamm_mf(Uu),Uua) = aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(Uu),aa_nat_Code_integer(semiri4939895301339042750nteger,Uua)) ).

% ATP.lambda_137
tff(fact_9829_ATP_Olambda__138,axiom,
    ! [Uu: rat,Uua: nat] : aa_nat_rat(aTP_Lamm_md(Uu),Uua) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,Uu),aa_nat_rat(semiri681578069525770553at_rat,Uua)) ).

% ATP.lambda_138
tff(fact_9830_ATP_Olambda__139,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_mg,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),aa_nat_nat(semiri1316708129612266289at_nat,Uua)) ).

% ATP.lambda_139
tff(fact_9831_ATP_Olambda__140,axiom,
    ! [Uu: int,Uua: nat] : aa_nat_int(aa_int_fun_nat_int(aTP_Lamm_mh,Uu),Uua) = aa_int_int(aa_int_fun_int_int(minus_minus_int,Uu),aa_nat_int(semiri1314217659103216013at_int,Uua)) ).

% ATP.lambda_140
tff(fact_9832_ATP_Olambda__141,axiom,
    ! [Uu: code_integer,Uua: nat] : aa_nat_Code_integer(aTP_Lamm_ol(Uu),Uua) = aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(Uu),aa_nat_Code_integer(semiri4939895301339042750nteger,Uua)) ).

% ATP.lambda_141
tff(fact_9833_ATP_Olambda__142,axiom,
    ! [Uu: rat,Uua: nat] : aa_nat_rat(aTP_Lamm_ok(Uu),Uua) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Uu),aa_nat_rat(semiri681578069525770553at_rat,Uua)) ).

% ATP.lambda_142
tff(fact_9834_ATP_Olambda__143,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_om,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),aa_nat_nat(semiri1316708129612266289at_nat,Uua)) ).

% ATP.lambda_143
tff(fact_9835_ATP_Olambda__144,axiom,
    ! [Uu: int,Uua: nat] : aa_nat_int(aa_int_fun_nat_int(aTP_Lamm_on,Uu),Uua) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Uu),aa_nat_int(semiri1314217659103216013at_int,Uua)) ).

% ATP.lambda_144
tff(fact_9836_ATP_Olambda__145,axiom,
    ! [Uu: nat,Uua: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_anb,Uu),Uua)
    <=> ( Uua = aa_nat_nat(suc,Uu) ) ) ).

% ATP.lambda_145
tff(fact_9837_ATP_Olambda__146,axiom,
    ! [Uu: assn,Uua: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(aTP_Lamm_db,Uu),Uua)
    <=> aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,Uua),Uu) ) ).

% ATP.lambda_146
tff(fact_9838_ATP_Olambda__147,axiom,
    ! [Uu: set_Code_integer,Uua: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(aTP_Lamm_acz(Uu),Uua)
    <=> aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(Uua),Uu) ) ).

% ATP.lambda_147
tff(fact_9839_ATP_Olambda__148,axiom,
    ! [Uu: set_Pr8693737435421807431at_nat,Uua: set_Pr8693737435421807431at_nat] :
      ( aa_set8084006625977036621_nat_o(aa_set2831299186196423263_nat_o(aTP_Lamm_cp,Uu),Uua)
    <=> ord_le3000389064537975527at_nat(Uua,Uu) ) ).

% ATP.lambda_148
tff(fact_9840_ATP_Olambda__149,axiom,
    ! [Uu: set_Pr8551490117392284871at_nat,Uua: set_Pr8551490117392284871at_nat] :
      ( aa_set192106157127879885_nat_o(aa_set8206400607124912991_nat_o(aTP_Lamm_cr,Uu),Uua)
    <=> ord_le8081472938463900775at_nat(Uua,Uu) ) ).

% ATP.lambda_149
tff(fact_9841_ATP_Olambda__150,axiom,
    ! [Uu: set_Pr4329608150637261639at_nat,Uua: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(aa_set7096950279330728543_nat_o(aTP_Lamm_ct,Uu),Uua)
    <=> ord_le1268244103169919719at_nat(Uua,Uu) ) ).

% ATP.lambda_150
tff(fact_9842_ATP_Olambda__151,axiom,
    ! [Uu: set_Pr1261947904930325089at_nat,Uua: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(aTP_Lamm_cv,Uu),Uua)
    <=> aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,Uua),Uu) ) ).

% ATP.lambda_151
tff(fact_9843_ATP_Olambda__152,axiom,
    ! [Uu: set_o,Uua: set_o] :
      ( aa_set_o_o(aa_set_o_fun_set_o_o(aTP_Lamm_cx,Uu),Uua)
    <=> ord_less_eq_set_o(Uua,Uu) ) ).

% ATP.lambda_152
tff(fact_9844_ATP_Olambda__153,axiom,
    ! [Uu: set_nat,Uua: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(aTP_Lamm_cz,Uu),Uua)
    <=> aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Uua),Uu) ) ).

% ATP.lambda_153
tff(fact_9845_ATP_Olambda__154,axiom,
    ! [Uu: set_int,Uua: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(aTP_Lamm_dd,Uu),Uua)
    <=> aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,Uua),Uu) ) ).

% ATP.lambda_154
tff(fact_9846_ATP_Olambda__155,axiom,
    ! [Uu: rat,Uua: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(aTP_Lamm_dh,Uu),Uua)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Uua),Uu) ) ).

% ATP.lambda_155
tff(fact_9847_ATP_Olambda__156,axiom,
    ! [Uu: num,Uua: num] :
      ( aa_num_o(aa_num_fun_num_o(aTP_Lamm_nm,Uu),Uua)
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Uua),Uu) ) ).

% ATP.lambda_156
tff(fact_9848_ATP_Olambda__157,axiom,
    ! [Uu: nat,Uua: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_df,Uu),Uua)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Uua),Uu) ) ).

% ATP.lambda_157
tff(fact_9849_ATP_Olambda__158,axiom,
    ! [Uu: int,Uua: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_dj,Uu),Uua)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Uua),Uu) ) ).

% ATP.lambda_158
tff(fact_9850_ATP_Olambda__159,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_amn,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Uua),Uu) ).

% ATP.lambda_159
tff(fact_9851_ATP_Olambda__160,axiom,
    ! [Uu: rat,Uua: rat] : aa_rat_rat(aa_rat_fun_rat_rat(aTP_Lamm_ik,Uu),Uua) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,Uua),Uu) ).

% ATP.lambda_160
tff(fact_9852_ATP_Olambda__161,axiom,
    ! [Uu: assn,Uua: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(aTP_Lamm_dc,Uu),Uua)
    <=> aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,Uua),Uu) ) ).

% ATP.lambda_161
tff(fact_9853_ATP_Olambda__162,axiom,
    ! [Uu: set_Pr8693737435421807431at_nat,Uua: set_Pr8693737435421807431at_nat] :
      ( aa_set8084006625977036621_nat_o(aa_set2831299186196423263_nat_o(aTP_Lamm_cq,Uu),Uua)
    <=> ord_le6428140832669894131at_nat(Uua,Uu) ) ).

% ATP.lambda_162
tff(fact_9854_ATP_Olambda__163,axiom,
    ! [Uu: set_Pr8551490117392284871at_nat,Uua: set_Pr8551490117392284871at_nat] :
      ( aa_set192106157127879885_nat_o(aa_set8206400607124912991_nat_o(aTP_Lamm_cs,Uu),Uua)
    <=> ord_le7642048601412989811at_nat(Uua,Uu) ) ).

% ATP.lambda_163
tff(fact_9855_ATP_Olambda__164,axiom,
    ! [Uu: set_Pr4329608150637261639at_nat,Uua: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(aa_set7096950279330728543_nat_o(aTP_Lamm_cu,Uu),Uua)
    <=> ord_le2604355607129572851at_nat(Uua,Uu) ) ).

% ATP.lambda_164
tff(fact_9856_ATP_Olambda__165,axiom,
    ! [Uu: set_Pr1261947904930325089at_nat,Uua: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(aTP_Lamm_cw,Uu),Uua)
    <=> aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le7866589430770878221at_nat,Uua),Uu) ) ).

% ATP.lambda_165
tff(fact_9857_ATP_Olambda__166,axiom,
    ! [Uu: set_o,Uua: set_o] :
      ( aa_set_o_o(aa_set_o_fun_set_o_o(aTP_Lamm_cy,Uu),Uua)
    <=> ord_less_set_o(Uua,Uu) ) ).

% ATP.lambda_166
tff(fact_9858_ATP_Olambda__167,axiom,
    ! [Uu: set_nat,Uua: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(aTP_Lamm_da,Uu),Uua)
    <=> aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,Uua),Uu) ) ).

% ATP.lambda_167
tff(fact_9859_ATP_Olambda__168,axiom,
    ! [Uu: set_int,Uua: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(aTP_Lamm_de,Uu),Uua)
    <=> aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_set_int,Uua),Uu) ) ).

% ATP.lambda_168
tff(fact_9860_ATP_Olambda__169,axiom,
    ! [Uu: rat,Uua: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(aTP_Lamm_di,Uu),Uua)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Uua),Uu) ) ).

% ATP.lambda_169
tff(fact_9861_ATP_Olambda__170,axiom,
    ! [Uu: num,Uua: num] :
      ( aa_num_o(aa_num_fun_num_o(aTP_Lamm_fb,Uu),Uua)
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,Uua),Uu) ) ).

% ATP.lambda_170
tff(fact_9862_ATP_Olambda__171,axiom,
    ! [Uu: nat,Uua: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_dg,Uu),Uua)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uua),Uu) ) ).

% ATP.lambda_171
tff(fact_9863_ATP_Olambda__172,axiom,
    ! [Uu: int,Uua: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_dk,Uu),Uua)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Uua),Uu) ) ).

% ATP.lambda_172
tff(fact_9864_ATP_Olambda__173,axiom,
    ! [Uu: rat,Uua: rat] : aa_rat_rat(aa_rat_fun_rat_rat(aTP_Lamm_hr,Uu),Uua) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Uua),Uu) ).

% ATP.lambda_173
tff(fact_9865_ATP_Olambda__174,axiom,
    ! [Uu: code_integer,Uua: code_integer] : aa_Cod1783974652533396728nteger(aTP_Lamm_hy(Uu),Uua) = aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(Uua),Uu) ).

% ATP.lambda_174
tff(fact_9866_ATP_Olambda__175,axiom,
    ! [Uu: rat,Uua: rat] : aa_rat_rat(aa_rat_fun_rat_rat(aTP_Lamm_hw,Uu),Uua) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,Uua),Uu) ).

% ATP.lambda_175
tff(fact_9867_ATP_Olambda__176,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_amp,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uu) ).

% ATP.lambda_176
tff(fact_9868_ATP_Olambda__177,axiom,
    ! [Uu: int,Uua: int] : aa_int_int(aa_int_fun_int_int(aTP_Lamm_hx,Uu),Uua) = aa_int_int(aa_int_fun_int_int(minus_minus_int,Uua),Uu) ).

% ATP.lambda_177
tff(fact_9869_ATP_Olambda__178,axiom,
    ! [Uu: code_integer,Uua: code_integer] : aa_Cod1783974652533396728nteger(aTP_Lamm_hv(Uu),Uua) = aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(Uua),Uu) ).

% ATP.lambda_178
tff(fact_9870_ATP_Olambda__179,axiom,
    ! [Uu: rat,Uua: rat] : aa_rat_rat(aa_rat_fun_rat_rat(aTP_Lamm_hs,Uu),Uua) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Uua),Uu) ).

% ATP.lambda_179
tff(fact_9871_ATP_Olambda__180,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_hu,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),Uu) ).

% ATP.lambda_180
tff(fact_9872_ATP_Olambda__181,axiom,
    ! [Uu: int,Uua: int] : aa_int_int(aa_int_fun_int_int(aTP_Lamm_ht,Uu),Uua) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Uua),Uu) ).

% ATP.lambda_181
tff(fact_9873_ATP_Olambda__182,axiom,
    ! [Uu: code_integer,Uua: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(aTP_Lamm_adu,Uu),Uua)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Uua),Uu) ) ).

% ATP.lambda_182
tff(fact_9874_ATP_Olambda__183,axiom,
    ! [Uu: nat,Uua: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_adt,Uu),Uua)
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Uua),Uu) ) ).

% ATP.lambda_183
tff(fact_9875_ATP_Olambda__184,axiom,
    ! [Uu: int,Uua: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_adv,Uu),Uua)
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Uua),Uu) ) ).

% ATP.lambda_184
tff(fact_9876_ATP_Olambda__185,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aTP_Lamm_ams,Uu),Uua) = aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Uua),Uu) ).

% ATP.lambda_185
tff(fact_9877_ATP_Olambda__186,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_tu,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uua),Uu) ).

% ATP.lambda_186
tff(fact_9878_ATP_Olambda__187,axiom,
    ! [Uu: set_Pr8693737435421807431at_nat,Uua: produc859450856879609959at_nat] :
      ( aa_Pro602569247023830573_nat_o(aTP_Lamm_ai(Uu),Uua)
    <=> member8206827879206165904at_nat(Uua,Uu) ) ).

% ATP.lambda_187
tff(fact_9879_ATP_Olambda__188,axiom,
    ! [Uu: set_Pr8551490117392284871at_nat,Uua: produc4166570645942440679at_nat] :
      ( aa_Pro8143006071014924973_nat_o(aTP_Lamm_aj(Uu),Uua)
    <=> member6689249552917799696at_nat(Uua,Uu) ) ).

% ATP.lambda_188
tff(fact_9880_ATP_Olambda__189,axiom,
    ! [Uu: set_Pr4329608150637261639at_nat,Uua: produc3843707927480180839at_nat] :
      ( aa_Pro5947980084102686253_nat_o(aTP_Lamm_ak(Uu),Uua)
    <=> member8757157785044589968at_nat(Uua,Uu) ) ).

% ATP.lambda_189
tff(fact_9881_ATP_Olambda__190,axiom,
    ! [Uu: set_Pr1261947904930325089at_nat,Uua: product_prod_nat_nat] :
      ( aa_Pro6150637604445964947_nat_o(aTP_Lamm_a(Uu),Uua)
    <=> aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Uua),Uu) ) ).

% ATP.lambda_190
tff(fact_9882_ATP_Olambda__191,axiom,
    ! [Uu: set_Pr958786334691620121nt_int,Uua: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aTP_Lamm_ab(Uu),Uua)
    <=> member5262025264175285858nt_int(Uua,Uu) ) ).

% ATP.lambda_191
tff(fact_9883_ATP_Olambda__192,axiom,
    ! [Uu: set_list_nat,Uua: list_nat] :
      ( aa_list_nat_o(aTP_Lamm_ac(Uu),Uua)
    <=> member_list_nat(Uua,Uu) ) ).

% ATP.lambda_192
tff(fact_9884_ATP_Olambda__193,axiom,
    ! [Uu: set_o,Uua: $o] :
      ( aa_o_o(aTP_Lamm_aa(Uu),(Uua))
    <=> member_o((Uua),Uu) ) ).

% ATP.lambda_193
tff(fact_9885_ATP_Olambda__194,axiom,
    ! [Uu: set_set_nat,Uua: set_nat] :
      ( aa_set_nat_o(aTP_Lamm_ad(Uu),Uua)
    <=> aa_set_set_nat_o(member_set_nat(Uua),Uu) ) ).

% ATP.lambda_194
tff(fact_9886_ATP_Olambda__195,axiom,
    ! [Uu: set_nat,Uua: nat] :
      ( aa_nat_o(aTP_Lamm_ae(Uu),Uua)
    <=> aa_set_nat_o(member_nat(Uua),Uu) ) ).

% ATP.lambda_195
tff(fact_9887_ATP_Olambda__196,axiom,
    ! [Uu: set_int,Uua: int] :
      ( aa_int_o(aTP_Lamm_af(Uu),Uua)
    <=> aa_set_int_o(member_int(Uua),Uu) ) ).

% ATP.lambda_196
tff(fact_9888_ATP_Olambda__197,axiom,
    ! [Uu: produc8664842809031399944it_nat,Uua: produc8664842809031399944it_nat] :
      ( aa_Pro4744234731172327378_nat_o(aTP_Lamm_fh(Uu),Uua)
    <=> ( Uua = Uu ) ) ).

% ATP.lambda_197
tff(fact_9889_ATP_Olambda__198,axiom,
    ! [Uu: produc3260487557148687353it_nat,Uua: produc3260487557148687353it_nat] :
      ( aa_Pro6426884464393220769_nat_o(aTP_Lamm_ff(Uu),Uua)
    <=> ( Uua = Uu ) ) ).

% ATP.lambda_198
tff(fact_9890_ATP_Olambda__199,axiom,
    ! [Uu: num,Uua: num] :
      ( aa_num_o(aa_num_fun_num_o(aTP_Lamm_fg,Uu),Uua)
    <=> ( Uua = Uu ) ) ).

% ATP.lambda_199
tff(fact_9891_ATP_Olambda__200,axiom,
    ! [Uu: fun_Code_integer_nat,Uua: code_integer] :
      ( aa_Code_integer_o(aTP_Lamm_we(Uu),Uua)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_Code_integer_nat(Uu,Uua)) ) ).

% ATP.lambda_200
tff(fact_9892_ATP_Olambda__201,axiom,
    ! [Uu: fun_Pr2602117396471556196at_nat,Uua: product_prod_nat_nat] :
      ( aa_Pro6150637604445964947_nat_o(aTP_Lamm_wg(Uu),Uua)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_Pro4243173770044397973at_nat(Uu,Uua)) ) ).

% ATP.lambda_201
tff(fact_9893_ATP_Olambda__202,axiom,
    ! [Uu: fun_Pr5160692531401422620nt_nat,Uua: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aTP_Lamm_vu(Uu),Uua)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_Pro6801748904974264397nt_nat(Uu,Uua)) ) ).

% ATP.lambda_202
tff(fact_9894_ATP_Olambda__203,axiom,
    ! [Uu: fun_list_nat_nat,Uua: list_nat] :
      ( aa_list_nat_o(aTP_Lamm_vw(Uu),Uua)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_list_nat_nat(Uu,Uua)) ) ).

% ATP.lambda_203
tff(fact_9895_ATP_Olambda__204,axiom,
    ! [Uu: fun_set_nat_nat,Uua: set_nat] :
      ( aa_set_nat_o(aTP_Lamm_vy(Uu),Uua)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_set_nat_nat(Uu,Uua)) ) ).

% ATP.lambda_204
tff(fact_9896_ATP_Olambda__205,axiom,
    ! [Uu: fun_nat_nat,Uua: nat] :
      ( aa_nat_o(aTP_Lamm_wa(Uu),Uua)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(Uu,Uua)) ) ).

% ATP.lambda_205
tff(fact_9897_ATP_Olambda__206,axiom,
    ! [Uu: fun_int_nat,Uua: int] :
      ( aa_int_o(aTP_Lamm_wc(Uu),Uua)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_int_nat(Uu,Uua)) ) ).

% ATP.lambda_206
tff(fact_9898_ATP_Olambda__207,axiom,
    ! [Uu: fun_nat_Code_integer,Uua: nat] : aa_nat_Code_integer(aTP_Lamm_nk(Uu),Uua) = aa_nat_Code_integer(Uu,aa_nat_nat(suc,Uua)) ).

% ATP.lambda_207
tff(fact_9899_ATP_Olambda__208,axiom,
    ! [Uu: fun_na2338080690716861136at_nat,Uua: nat] : aa_nat2240542744625857721at_nat(aTP_Lamm_ka(Uu),Uua) = aa_nat2240542744625857721at_nat(Uu,aa_nat_nat(suc,Uua)) ).

% ATP.lambda_208
tff(fact_9900_ATP_Olambda__209,axiom,
    ! [Uu: fun_nat_assn,Uua: nat] : aa_nat_assn(aTP_Lamm_nl(Uu),Uua) = aa_nat_assn(Uu,aa_nat_nat(suc,Uua)) ).

% ATP.lambda_209
tff(fact_9901_ATP_Olambda__210,axiom,
    ! [Uu: fun_nat_o,Uua: nat] :
      ( aa_nat_o(aTP_Lamm_aov(Uu),Uua)
    <=> aa_nat_o(Uu,aa_nat_nat(suc,Uua)) ) ).

% ATP.lambda_210
tff(fact_9902_ATP_Olambda__211,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_kb(Uu),Uua) = aa_nat_rat(Uu,aa_nat_nat(suc,Uua)) ).

% ATP.lambda_211
tff(fact_9903_ATP_Olambda__212,axiom,
    ! [Uu: fun_nat_nat,Uua: nat] : aa_nat_nat(aTP_Lamm_jx(Uu),Uua) = aa_nat_nat(Uu,aa_nat_nat(suc,Uua)) ).

% ATP.lambda_212
tff(fact_9904_ATP_Olambda__213,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_kc(Uu),Uua) = aa_nat_int(Uu,aa_nat_nat(suc,Uua)) ).

% ATP.lambda_213
tff(fact_9905_ATP_Olambda__214,axiom,
    ! [Uu: nat,Uua: num] : aa_num_option_num(aTP_Lamm_ajf(Uu),Uua) = aa_num_option_num(some_num,aa_option_num_num(case_option_num_num(one,bit1),aa_num_option_num(bit_take_bit_num(Uu),Uua))) ).

% ATP.lambda_214
tff(fact_9906_ATP_Olambda__215,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(aTP_Lamm_aot,Uu),Uua) = produc6081775807080527818_nat_o(aa_nat8314798964848609180_nat_o(aTP_Lamm_aos(Uu),Uua)) ).

% ATP.lambda_215
tff(fact_9907_ATP_Olambda__216,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(aTP_Lamm_ana,Uu),Uua) = produc2626176000494625587at_nat(aa_nat4158579785191972357at_nat(aTP_Lamm_amz(Uu),Uua)) ).

% ATP.lambda_216
tff(fact_9908_ATP_Olambda__217,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(aTP_Lamm_amy,Uu),Uua) = produc2626176000494625587at_nat(aa_nat4158579785191972357at_nat(aTP_Lamm_amx(Uu),Uua)) ).

% ATP.lambda_217
tff(fact_9909_ATP_Olambda__218,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(aTP_Lamm_amw,Uu),Uua) = produc6081775807080527818_nat_o(aa_nat8314798964848609180_nat_o(aTP_Lamm_amv(Uu),Uua)) ).

% ATP.lambda_218
tff(fact_9910_ATP_Olambda__219,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(aTP_Lamm_amu,Uu),Uua) = produc6081775807080527818_nat_o(aa_nat8314798964848609180_nat_o(aTP_Lamm_amt(Uu),Uua)) ).

% ATP.lambda_219
tff(fact_9911_ATP_Olambda__220,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(aTP_Lamm_amr,Uu),Uua) = produc2626176000494625587at_nat(aa_nat4158579785191972357at_nat(aTP_Lamm_amq(Uu),Uua)) ).

% ATP.lambda_220
tff(fact_9912_ATP_Olambda__221,axiom,
    ! [Uu: fun_a_2990225006182978344t_unit,Uua: a] : aa_a_f2724909083865345251it_nat(aTP_Lamm_ahi(Uu),Uua) = produc7488178964372371019it_nat(aa_a_f112692477493148739it_nat(aTP_Lamm_ahh(Uu),Uua)) ).

% ATP.lambda_221
tff(fact_9913_ATP_Olambda__222,axiom,
    ! [Uu: fun_a_6093291656547882009Heap_a,Uua: a] : aa_a_f2042851359257671380it_nat(aTP_Lamm_ahe(Uu),Uua) = produc5645566021302314940it_nat(aa_a_f1761598956578211508it_nat(aTP_Lamm_ahd(Uu),Uua)) ).

% ATP.lambda_222
tff(fact_9914_ATP_Olambda__223,axiom,
    ! [Uu: fun_he5123621388056232515t_unit,Uua: heap_e7401611519738050253t_unit] : aa_hea684056619475940329it_nat(aTP_Lamm_gb(Uu),Uua) = aa_Pro8996696869555642526it_nat(some_P1914260805536162275it_nat,produc7133225469290080770it_nat(aa_hea4146273921685161788t_unit(Uu,Uua),produc584006145561248582it_nat(Uua,one_one_nat))) ).

% ATP.lambda_223
tff(fact_9915_ATP_Olambda__224,axiom,
    ! [Uu: fun_he6342020308413958580unit_a,Uua: heap_e7401611519738050253t_unit] : aa_hea4724376117875161946it_nat(aTP_Lamm_ga(Uu),Uua) = aa_Pro5302094286594872192it_nat(some_P7913643980934408916it_nat,aa_Pro8078911840264235907it_nat(aa_a_f2234720956197727492it_nat(produc9178034014595674355it_nat,aa_hea5100639427662539565unit_a(Uu,Uua)),produc584006145561248582it_nat(Uua,one_one_nat))) ).

% ATP.lambda_224
tff(fact_9916_ATP_Olambda__225,axiom,
    ! [Uu: nat,Uua: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_afb,Uu),Uua)
    <=> ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uu),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua))) ) ).

% ATP.lambda_225
tff(fact_9917_ATP_Olambda__226,axiom,
    ! [Uu: code_natural,Uua: code_natural] : aa_Cod6352003912080677724atural(aTP_Lamm_apa(Uu),Uua) = produc6639722614265839536atural(plus_p4538020629002901425atural(Uua,times_2397367101498566445atural(Uu,aa_num_Code_natural(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_226
tff(fact_9918_ATP_Olambda__227,axiom,
    ! [Uu: rat,Uua: nat] : aa_nat_fun_rat_rat(aTP_Lamm_hm(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_227
tff(fact_9919_ATP_Olambda__228,axiom,
    ! [Uu: code_integer,Uua: nat] : aa_nat6383147911832638789nteger(aTP_Lamm_gi(Uu),Uua) = aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(Uu),aa_nat_Code_integer(semiri4939895301339042750nteger,Uua))) ).

% ATP.lambda_228
tff(fact_9920_ATP_Olambda__229,axiom,
    ! [Uu: rat,Uua: nat] : aa_nat_fun_rat_rat(aTP_Lamm_gg(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_229
tff(fact_9921_ATP_Olambda__230,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_fun_nat_nat(aTP_Lamm_gj(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_230
tff(fact_9922_ATP_Olambda__231,axiom,
    ! [Uu: int,Uua: nat] : aa_nat_fun_int_int(aTP_Lamm_gh(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_231
tff(fact_9923_ATP_Olambda__232,axiom,
    ! [Uu: set_Pr8693737435421807431at_nat,Uua: set_Pr8693737435421807431at_nat] : aa_set6661686244540357314at_nat(aTP_Lamm_dl(Uu),Uua) = some_s579751585967276588at_nat(aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,Uu),Uua)) ).

% ATP.lambda_232
tff(fact_9924_ATP_Olambda__233,axiom,
    ! [Uu: set_Pr8551490117392284871at_nat,Uua: set_Pr8551490117392284871at_nat] : aa_set1897275755269116354at_nat(aTP_Lamm_dn(Uu),Uua) = some_s287724117700012716at_nat(aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,Uu),Uua)) ).

% ATP.lambda_233
tff(fact_9925_ATP_Olambda__234,axiom,
    ! [Uu: set_Pr4329608150637261639at_nat,Uua: set_Pr4329608150637261639at_nat] : aa_set7071694670514008770at_nat(aTP_Lamm_dp(Uu),Uua) = some_s5890477192898017836at_nat(aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,Uu),Uua)) ).

% ATP.lambda_234
tff(fact_9926_ATP_Olambda__235,axiom,
    ! [Uu: set_nat,Uua: set_nat] : aa_set6119337032432945050et_nat(aTP_Lamm_dr(Uu),Uua) = some_set_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Uu),Uua)) ).

% ATP.lambda_235
tff(fact_9927_ATP_Olambda__236,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_option_nat(aTP_Lamm_dt(Uu),Uua) = aa_nat_option_nat(some_nat,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,Uu),Uua)) ).

% ATP.lambda_236
tff(fact_9928_ATP_Olambda__237,axiom,
    ! [Uu: product_unit,Uua: product_unit] : aa_Pro7616171890175246658t_unit(aTP_Lamm_ek(Uu),Uua) = some_Product_unit(aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,Uu),Uua)) ).

% ATP.lambda_237
tff(fact_9929_ATP_Olambda__238,axiom,
    ! [Uu: assn,Uua: assn] : aa_assn_option_assn(aTP_Lamm_eg(Uu),Uua) = some_assn(aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Uu),Uua)) ).

% ATP.lambda_238
tff(fact_9930_ATP_Olambda__239,axiom,
    ! [Uu: set_Pr1261947904930325089at_nat,Uua: set_Pr1261947904930325089at_nat] : aa_set2934564513904597954at_nat(aTP_Lamm_em(Uu),Uua) = some_s147305329494351046at_nat(aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Uu),Uua)) ).

% ATP.lambda_239
tff(fact_9931_ATP_Olambda__240,axiom,
    ! [Uu: set_nat,Uua: set_nat] : aa_set6119337032432945050et_nat(aTP_Lamm_ee(Uu),Uua) = some_set_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Uu),Uua)) ).

% ATP.lambda_240
tff(fact_9932_ATP_Olambda__241,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_option_nat(aTP_Lamm_ei(Uu),Uua) = aa_nat_option_nat(some_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,Uu),Uua)) ).

% ATP.lambda_241
tff(fact_9933_ATP_Olambda__242,axiom,
    ! [Uu: code_natural,Uua: code_natural] : aa_Cod6352003912080677724atural(aTP_Lamm_apc(Uu),Uua) = produc6639722614265839536atural(modulo8411746178871703098atural(Uua,Uu)) ).

% ATP.lambda_242
tff(fact_9934_ATP_Olambda__243,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_anm,Uu),Uua) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,Uu),Uua)) ).

% ATP.lambda_243
tff(fact_9935_ATP_Olambda__244,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_anl,Uu),Uua) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,Uua),Uu)) ).

% ATP.lambda_244
tff(fact_9936_ATP_Olambda__245,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_ann,Uu),Uua) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Uu),Uua)) ).

% ATP.lambda_245
tff(fact_9937_ATP_Olambda__246,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_ano,Uu),Uua) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Uua),Uu)) ).

% ATP.lambda_246
tff(fact_9938_ATP_Olambda__247,axiom,
    ! [Uu: set_Pr1261947904930325089at_nat,Uua: product_prod_nat_nat] :
      ( aa_Pro6150637604445964947_nat_o(aTP_Lamm_hf(Uu),Uua)
    <=> ~ aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Uua),Uu) ) ).

% ATP.lambda_247
tff(fact_9939_ATP_Olambda__248,axiom,
    ! [Uu: set_Pr958786334691620121nt_int,Uua: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aTP_Lamm_hh(Uu),Uua)
    <=> ~ member5262025264175285858nt_int(Uua,Uu) ) ).

% ATP.lambda_248
tff(fact_9940_ATP_Olambda__249,axiom,
    ! [Uu: set_list_nat,Uua: list_nat] :
      ( aa_list_nat_o(aTP_Lamm_hi(Uu),Uua)
    <=> ~ member_list_nat(Uua,Uu) ) ).

% ATP.lambda_249
tff(fact_9941_ATP_Olambda__250,axiom,
    ! [Uu: set_o,Uua: $o] :
      ( aa_o_o(aTP_Lamm_hg(Uu),(Uua))
    <=> ~ member_o((Uua),Uu) ) ).

% ATP.lambda_250
tff(fact_9942_ATP_Olambda__251,axiom,
    ! [Uu: set_set_nat,Uua: set_nat] :
      ( aa_set_nat_o(aTP_Lamm_hj(Uu),Uua)
    <=> ~ aa_set_set_nat_o(member_set_nat(Uua),Uu) ) ).

% ATP.lambda_251
tff(fact_9943_ATP_Olambda__252,axiom,
    ! [Uu: set_nat,Uua: nat] :
      ( aa_nat_o(aTP_Lamm_hk(Uu),Uua)
    <=> ~ aa_set_nat_o(member_nat(Uua),Uu) ) ).

% ATP.lambda_252
tff(fact_9944_ATP_Olambda__253,axiom,
    ! [Uu: set_int,Uua: int] :
      ( aa_int_o(aTP_Lamm_hl(Uu),Uua)
    <=> ~ aa_set_int_o(member_int(Uua),Uu) ) ).

% ATP.lambda_253
tff(fact_9945_ATP_Olambda__254,axiom,
    ! [Uu: nat,Uua: heap_e7401611519738050253t_unit] : aa_hea684056619475940329it_nat(aTP_Lamm_aox(Uu),Uua) = aa_Pro8996696869555642526it_nat(some_P1914260805536162275it_nat,produc7133225469290080770it_nat(product_Unity,produc584006145561248582it_nat(Uua,Uu))) ).

% ATP.lambda_254
tff(fact_9946_ATP_Olambda__255,axiom,
    ! [Uu: fun_Code_integer_o,Uua: code_integer] : aa_Cod1783974652533396728nteger(aTP_Lamm_ake(Uu),Uua) = aa_o_Code_integer(zero_n356916108424825756nteger,aa_Code_integer_o(Uu,Uua)) ).

% ATP.lambda_255
tff(fact_9947_ATP_Olambda__256,axiom,
    ! [Uu: fun_Code_integer_o,Uua: code_integer] : aa_Code_integer_nat(aTP_Lamm_akh(Uu),Uua) = aa_o_nat(zero_n2687167440665602831ol_nat,aa_Code_integer_o(Uu,Uua)) ).

% ATP.lambda_256
tff(fact_9948_ATP_Olambda__257,axiom,
    ! [Uu: fun_Code_integer_o,Uua: code_integer] : aa_Code_integer_int(aTP_Lamm_aki(Uu),Uua) = aa_o_int(zero_n2684676970156552555ol_int,aa_Code_integer_o(Uu,Uua)) ).

% ATP.lambda_257
tff(fact_9949_ATP_Olambda__258,axiom,
    ! [Uu: fun_set_nat_o,Uua: set_nat] : aa_set871585252387937313nteger(aTP_Lamm_akm(Uu),Uua) = aa_o_Code_integer(zero_n356916108424825756nteger,aa_set_nat_o(Uu,Uua)) ).

% ATP.lambda_258
tff(fact_9950_ATP_Olambda__259,axiom,
    ! [Uu: fun_nat_o,Uua: nat] : aa_nat_Code_integer(aTP_Lamm_akf(Uu),Uua) = aa_o_Code_integer(zero_n356916108424825756nteger,aa_nat_o(Uu,Uua)) ).

% ATP.lambda_259
tff(fact_9951_ATP_Olambda__260,axiom,
    ! [Uu: fun_nat_o,Uua: nat] : aa_nat_nat(aTP_Lamm_akk(Uu),Uua) = aa_o_nat(zero_n2687167440665602831ol_nat,aa_nat_o(Uu,Uua)) ).

% ATP.lambda_260
tff(fact_9952_ATP_Olambda__261,axiom,
    ! [Uu: fun_nat_o,Uua: nat] : aa_nat_int(aTP_Lamm_akj(Uu),Uua) = aa_o_int(zero_n2684676970156552555ol_int,aa_nat_o(Uu,Uua)) ).

% ATP.lambda_261
tff(fact_9953_ATP_Olambda__262,axiom,
    ! [Uu: fun_int_o,Uua: int] : aa_int_Code_integer(aTP_Lamm_akd(Uu),Uua) = aa_o_Code_integer(zero_n356916108424825756nteger,aa_int_o(Uu,Uua)) ).

% ATP.lambda_262
tff(fact_9954_ATP_Olambda__263,axiom,
    ! [Uu: fun_int_o,Uua: int] : aa_int_nat(aTP_Lamm_akg(Uu),Uua) = aa_o_nat(zero_n2687167440665602831ol_nat,aa_int_o(Uu,Uua)) ).

% ATP.lambda_263
tff(fact_9955_ATP_Olambda__264,axiom,
    ! [Uu: fun_int_o,Uua: int] : aa_int_int(aTP_Lamm_akl(Uu),Uua) = aa_o_int(zero_n2684676970156552555ol_int,aa_int_o(Uu,Uua)) ).

% ATP.lambda_264
tff(fact_9956_ATP_Olambda__265,axiom,
    ! [Uu: fun_nat_nat,Uua: nat] : aa_nat_Code_integer(aTP_Lamm_jr(Uu),Uua) = aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(Uu,Uua)) ).

% ATP.lambda_265
tff(fact_9957_ATP_Olambda__266,axiom,
    ! [Uu: fun_nat_nat,Uua: nat] : aa_nat_nat(aTP_Lamm_js(Uu),Uua) = aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(Uu,Uua)) ).

% ATP.lambda_266
tff(fact_9958_ATP_Olambda__267,axiom,
    ! [Uu: fun_nat_nat,Uua: nat] : aa_nat_int(aTP_Lamm_jq(Uu),Uua) = aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(Uu,Uua)) ).

% ATP.lambda_267
tff(fact_9959_ATP_Olambda__268,axiom,
    ! [Uu: fun_int_nat,Uua: int] : aa_int_int(aTP_Lamm_jp(Uu),Uua) = aa_nat_int(semiri1314217659103216013at_int,aa_int_nat(Uu,Uua)) ).

% ATP.lambda_268
tff(fact_9960_ATP_Olambda__269,axiom,
    ! [Uu: fun_int_int,Uua: int] : aa_int_int(aTP_Lamm_ma(Uu),Uua) = aa_int_int(uminus_uminus_int,aa_int_int(Uu,Uua)) ).

% ATP.lambda_269
tff(fact_9961_ATP_Olambda__270,axiom,
    ! [Uu: fun_nat_Code_integer,Uua: nat] : aa_nat6383147911832638789nteger(aTP_Lamm_oe(Uu),Uua) = aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(Uu,Uua)) ).

% ATP.lambda_270
tff(fact_9962_ATP_Olambda__271,axiom,
    ! [Uu: fun_nat_assn,Uua: nat] : aa_nat_fun_assn_assn(aTP_Lamm_of(Uu),Uua) = aa_ass954072293579886259n_assn(times_times_assn,aa_nat_assn(Uu,Uua)) ).

% ATP.lambda_271
tff(fact_9963_ATP_Olambda__272,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : aa_nat_fun_rat_rat(aTP_Lamm_og(Uu),Uua) = aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(Uu,Uua)) ).

% ATP.lambda_272
tff(fact_9964_ATP_Olambda__273,axiom,
    ! [Uu: fun_nat_nat,Uua: nat] : aa_nat_fun_nat_nat(aTP_Lamm_oh(Uu),Uua) = aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(Uu,Uua)) ).

% ATP.lambda_273
tff(fact_9965_ATP_Olambda__274,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_fun_int_int(aTP_Lamm_oi(Uu),Uua) = aa_int_fun_int_int(times_times_int,aa_nat_int(Uu,Uua)) ).

% ATP.lambda_274
tff(fact_9966_ATP_Olambda__275,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_rat(aTP_Lamm_mm(Uu),Uua) = aa_int_rat(ring_1_of_int_rat,aa_nat_int(Uu,Uua)) ).

% ATP.lambda_275
tff(fact_9967_ATP_Olambda__276,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_mn(Uu),Uua) = aa_int_int(ring_1_of_int_int,aa_nat_int(Uu,Uua)) ).

% ATP.lambda_276
tff(fact_9968_ATP_Olambda__277,axiom,
    ! [Uu: fun_int_int,Uua: int] : aa_int_rat(aTP_Lamm_jt(Uu),Uua) = aa_int_rat(ring_1_of_int_rat,aa_int_int(Uu,Uua)) ).

% ATP.lambda_277
tff(fact_9969_ATP_Olambda__278,axiom,
    ! [Uu: fun_int_int,Uua: int] : aa_int_int(aTP_Lamm_ju(Uu),Uua) = aa_int_int(ring_1_of_int_int,aa_int_int(Uu,Uua)) ).

% ATP.lambda_278
tff(fact_9970_ATP_Olambda__279,axiom,
    ! [Uu: fun_nat_Code_integer,Uua: nat] : aa_nat6383147911832638789nteger(aTP_Lamm_ki(Uu),Uua) = plus_p5714425477246183910nteger(aa_nat_Code_integer(Uu,Uua)) ).

% ATP.lambda_279
tff(fact_9971_ATP_Olambda__280,axiom,
    ! [Uu: fun_na2338080690716861136at_nat,Uua: nat] : aa_nat1862996181841379217at_nat(aTP_Lamm_kj(Uu),Uua) = plus_p7104986032573967614at_nat(aa_nat2240542744625857721at_nat(Uu,Uua)) ).

% ATP.lambda_280
tff(fact_9972_ATP_Olambda__281,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : aa_nat_fun_rat_rat(aTP_Lamm_kk(Uu),Uua) = aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(Uu,Uua)) ).

% ATP.lambda_281
tff(fact_9973_ATP_Olambda__282,axiom,
    ! [Uu: fun_nat_nat,Uua: nat] : aa_nat_fun_nat_nat(aTP_Lamm_km(Uu),Uua) = aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(Uu,Uua)) ).

% ATP.lambda_282
tff(fact_9974_ATP_Olambda__283,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_fun_int_int(aTP_Lamm_kl(Uu),Uua) = aa_int_fun_int_int(plus_plus_int,aa_nat_int(Uu,Uua)) ).

% ATP.lambda_283
tff(fact_9975_ATP_Olambda__284,axiom,
    ! [Uu: fun_int_int,Uua: int] : aa_int_int(aTP_Lamm_kt(Uu),Uua) = aa_int_int(abs_abs_int,aa_int_int(Uu,Uua)) ).

% ATP.lambda_284
tff(fact_9976_ATP_Olambda__285,axiom,
    ! [Uu: fun_list_nat_nat,Uua: list_nat] : aa_list_nat_nat(aTP_Lamm_amb(Uu),Uua) = aa_nat_nat(suc,aa_list_nat_nat(Uu,Uua)) ).

% ATP.lambda_285
tff(fact_9977_ATP_Olambda__286,axiom,
    ! [Uu: fun_set_nat_nat,Uua: set_nat] : aa_set_nat_nat(aTP_Lamm_ama(Uu),Uua) = aa_nat_nat(suc,aa_set_nat_nat(Uu,Uua)) ).

% ATP.lambda_286
tff(fact_9978_ATP_Olambda__287,axiom,
    ! [Uu: fun_nat_nat,Uua: nat] : aa_nat_nat(aTP_Lamm_amc(Uu),Uua) = aa_nat_nat(suc,aa_nat_nat(Uu,Uua)) ).

% ATP.lambda_287
tff(fact_9979_ATP_Olambda__288,axiom,
    ! [Uu: fun_Pr4796943543331071514_nat_o,Uua: product_prod_nat_nat] :
      ( aa_Pro6150637604445964947_nat_o(aa_fun359341089595923442_nat_o(aTP_Lamm_ii,Uu),Uua)
    <=> ~ aa_Pro6150637604445964947_nat_o(Uu,Uua) ) ).

% ATP.lambda_288
tff(fact_9980_ATP_Olambda__289,axiom,
    ! [Uu: fun_Pr3238374608701176162_int_o,Uua: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aTP_Lamm_ha(Uu),Uua)
    <=> ~ aa_Pro4592068669816069595_int_o(Uu,Uua) ) ).

% ATP.lambda_289
tff(fact_9981_ATP_Olambda__290,axiom,
    ! [Uu: fun_list_nat_o,Uua: list_nat] :
      ( aa_list_nat_o(aTP_Lamm_hb(Uu),Uua)
    <=> ~ aa_list_nat_o(Uu,Uua) ) ).

% ATP.lambda_290
tff(fact_9982_ATP_Olambda__291,axiom,
    ! [Uu: fun_set_nat_o,Uua: set_nat] :
      ( aa_set_nat_o(aTP_Lamm_hc(Uu),Uua)
    <=> ~ aa_set_nat_o(Uu,Uua) ) ).

% ATP.lambda_291
tff(fact_9983_ATP_Olambda__292,axiom,
    ! [Uu: fun_nat_o,Uua: nat] :
      ( aa_nat_o(aTP_Lamm_hd(Uu),Uua)
    <=> ~ aa_nat_o(Uu,Uua) ) ).

% ATP.lambda_292
tff(fact_9984_ATP_Olambda__293,axiom,
    ! [Uu: fun_int_o,Uua: int] :
      ( aa_int_o(aTP_Lamm_he(Uu),Uua)
    <=> ~ aa_int_o(Uu,Uua) ) ).

% ATP.lambda_293
tff(fact_9985_ATP_Olambda__294,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_set_nat(aa_nat1818925681583257441et_nat(aTP_Lamm_apd,Uu),Uua) = collect_nat(aa_nat_fun_nat_o(aTP_Lamm_dg,Uu)) ).

% ATP.lambda_294
tff(fact_9986_ATP_Olambda__295,axiom,
    ! [Uu: num,Uua: code_integer,Uub: code_integer] :
      aa_Cod6977894654539884864nteger(aa_Cod8380929501330563836nteger(aTP_Lamm_afe(Uu),Uua),Uub) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_num_Code_integer(numera6620942414471956472nteger,Uu)),Uub),produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Uua)),one_one_Code_integer),aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(Uub),aa_num_Code_integer(numera6620942414471956472nteger,Uu))),produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Uua),Uub)) ).

% ATP.lambda_295
tff(fact_9987_ATP_Olambda__296,axiom,
    ! [Uu: num,Uua: nat,Uub: nat] :
      aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aTP_Lamm_afc(Uu),Uua),Uub) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,Uu)),Uub),aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),aa_num_nat(numeral_numeral_nat,Uu))),aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)),Uub)) ).

% ATP.lambda_296
tff(fact_9988_ATP_Olambda__297,axiom,
    ! [Uu: num,Uua: int,Uub: int] :
      aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_afd(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_297
tff(fact_9989_ATP_Olambda__298,axiom,
    ! [Uu: fun_Co8499238673658658103nteger,Uua: set_Code_integer,Uub: code_integer] :
      aa_Cod1783974652533396728nteger(aa_set4294055438810586376nteger(aTP_Lamm_yq(Uu),Uua),Uub) = $ite(aa_set5643387332196575111eger_o(member_Code_integer(Uub),Uua),aa_Cod1783974652533396728nteger(Uu,Uub),zero_z3403309356797280102nteger) ).

% ATP.lambda_298
tff(fact_9990_ATP_Olambda__299,axiom,
    ! [Uu: fun_Code_integer_rat,Uua: set_Code_integer,Uub: code_integer] :
      aa_Code_integer_rat(aa_set7344733238996822275er_rat(aTP_Lamm_yt(Uu),Uua),Uub) = $ite(aa_set5643387332196575111eger_o(member_Code_integer(Uub),Uua),aa_Code_integer_rat(Uu,Uub),zero_zero_rat) ).

% ATP.lambda_299
tff(fact_9991_ATP_Olambda__300,axiom,
    ! [Uu: fun_Code_integer_nat,Uua: set_Code_integer,Uub: code_integer] :
      aa_Code_integer_nat(aa_set1629589760535443195er_nat(aTP_Lamm_yw(Uu),Uua),Uub) = $ite(aa_set5643387332196575111eger_o(member_Code_integer(Uub),Uua),aa_Code_integer_nat(Uu,Uub),zero_zero_nat) ).

% ATP.lambda_300
tff(fact_9992_ATP_Olambda__301,axiom,
    ! [Uu: fun_o_Code_integer,Uua: set_o,Uub: $o] :
      aa_o_Code_integer(aa_set9049949509649958604nteger(aTP_Lamm_yo(Uu),Uua),(Uub)) = $ite(member_o((Uub),Uua),aa_o_Code_integer(Uu,(Uub)),zero_z3403309356797280102nteger) ).

% ATP.lambda_301
tff(fact_9993_ATP_Olambda__302,axiom,
    ! [Uu: fun_o_rat,Uua: set_o,Uub: $o] :
      aa_o_rat(aa_set_o_fun_o_rat(aTP_Lamm_yr(Uu),Uua),(Uub)) = $ite(member_o((Uub),Uua),aa_o_rat(Uu,(Uub)),zero_zero_rat) ).

% ATP.lambda_302
tff(fact_9994_ATP_Olambda__303,axiom,
    ! [Uu: fun_o_nat,Uua: set_o,Uub: $o] :
      aa_o_nat(aa_set_o_fun_o_nat(aTP_Lamm_yu(Uu),Uua),(Uub)) = $ite(member_o((Uub),Uua),aa_o_nat(Uu,(Uub)),zero_zero_nat) ).

% ATP.lambda_303
tff(fact_9995_ATP_Olambda__304,axiom,
    ! [Uu: fun_o_int,Uua: set_o,Uub: $o] :
      aa_o_int(aa_set_o_fun_o_int(aTP_Lamm_yx(Uu),Uua),(Uub)) = $ite(member_o((Uub),Uua),aa_o_int(Uu,(Uub)),zero_zero_int) ).

% ATP.lambda_304
tff(fact_9996_ATP_Olambda__305,axiom,
    ! [Uu: fun_int_Code_integer,Uua: set_int,Uub: int] :
      aa_int_Code_integer(aa_set5517701783735052070nteger(aTP_Lamm_yp(Uu),Uua),Uub) = $ite(aa_set_int_o(member_int(Uub),Uua),aa_int_Code_integer(Uu,Uub),zero_z3403309356797280102nteger) ).

% ATP.lambda_305
tff(fact_9997_ATP_Olambda__306,axiom,
    ! [Uu: fun_int_rat,Uua: set_int,Uub: int] :
      aa_int_rat(aa_set7979663334043413281nt_rat(aTP_Lamm_ys(Uu),Uua),Uub) = $ite(aa_set_int_o(member_int(Uub),Uua),aa_int_rat(Uu,Uub),zero_zero_rat) ).

% ATP.lambda_306
tff(fact_9998_ATP_Olambda__307,axiom,
    ! [Uu: fun_int_nat,Uua: set_int,Uub: int] :
      aa_int_nat(aa_set2264519855582034201nt_nat(aTP_Lamm_yv(Uu),Uua),Uub) = $ite(aa_set_int_o(member_int(Uub),Uua),aa_int_nat(Uu,Uub),zero_zero_nat) ).

% ATP.lambda_307
tff(fact_9999_ATP_Olambda__308,axiom,
    ! [Uu: fun_Co8499238673658658103nteger,Uua: set_Code_integer,Uub: code_integer] :
      aa_Cod1783974652533396728nteger(aa_set4294055438810586376nteger(aTP_Lamm_aan(Uu),Uua),Uub) = $ite(aa_set5643387332196575111eger_o(member_Code_integer(Uub),Uua),aa_Cod1783974652533396728nteger(Uu,Uub),one_one_Code_integer) ).

% ATP.lambda_308
tff(fact_10000_ATP_Olambda__309,axiom,
    ! [Uu: fun_Co4490034186337852034r_assn,Uua: set_Code_integer,Uub: code_integer] :
      aa_Code_integer_assn(aa_set2437433236263590995r_assn(aTP_Lamm_aaq(Uu),Uua),Uub) = $ite(aa_set5643387332196575111eger_o(member_Code_integer(Uub),Uua),aa_Code_integer_assn(Uu,Uub),one_one_assn) ).

% ATP.lambda_309
tff(fact_10001_ATP_Olambda__310,axiom,
    ! [Uu: fun_Code_integer_rat,Uua: set_Code_integer,Uub: code_integer] :
      aa_Code_integer_rat(aa_set7344733238996822275er_rat(aTP_Lamm_aat(Uu),Uua),Uub) = $ite(aa_set5643387332196575111eger_o(member_Code_integer(Uub),Uua),aa_Code_integer_rat(Uu,Uub),one_one_rat) ).

% ATP.lambda_310
tff(fact_10002_ATP_Olambda__311,axiom,
    ! [Uu: fun_o_Code_integer,Uua: set_o,Uub: $o] :
      aa_o_Code_integer(aa_set9049949509649958604nteger(aTP_Lamm_aal(Uu),Uua),(Uub)) = $ite(member_o((Uub),Uua),aa_o_Code_integer(Uu,(Uub)),one_one_Code_integer) ).

% ATP.lambda_311
tff(fact_10003_ATP_Olambda__312,axiom,
    ! [Uu: fun_o_assn,Uua: set_o,Uub: $o] :
      aa_o_assn(aa_set_o_fun_o_assn(aTP_Lamm_aao(Uu),Uua),(Uub)) = $ite(member_o((Uub),Uua),aa_o_assn(Uu,(Uub)),one_one_assn) ).

% ATP.lambda_312
tff(fact_10004_ATP_Olambda__313,axiom,
    ! [Uu: fun_o_rat,Uua: set_o,Uub: $o] :
      aa_o_rat(aa_set_o_fun_o_rat(aTP_Lamm_aar(Uu),Uua),(Uub)) = $ite(member_o((Uub),Uua),aa_o_rat(Uu,(Uub)),one_one_rat) ).

% ATP.lambda_313
tff(fact_10005_ATP_Olambda__314,axiom,
    ! [Uu: fun_o_nat,Uua: set_o,Uub: $o] :
      aa_o_nat(aa_set_o_fun_o_nat(aTP_Lamm_aau(Uu),Uua),(Uub)) = $ite(member_o((Uub),Uua),aa_o_nat(Uu,(Uub)),one_one_nat) ).

% ATP.lambda_314
tff(fact_10006_ATP_Olambda__315,axiom,
    ! [Uu: fun_int_Code_integer,Uua: set_int,Uub: int] :
      aa_int_Code_integer(aa_set5517701783735052070nteger(aTP_Lamm_aam(Uu),Uua),Uub) = $ite(aa_set_int_o(member_int(Uub),Uua),aa_int_Code_integer(Uu,Uub),one_one_Code_integer) ).

% ATP.lambda_315
tff(fact_10007_ATP_Olambda__316,axiom,
    ! [Uu: fun_int_assn,Uua: set_int,Uub: int] :
      aa_int_assn(aa_set5310648174735401585t_assn(aTP_Lamm_aap(Uu),Uua),Uub) = $ite(aa_set_int_o(member_int(Uub),Uua),aa_int_assn(Uu,Uub),one_one_assn) ).

% ATP.lambda_316
tff(fact_10008_ATP_Olambda__317,axiom,
    ! [Uu: fun_int_rat,Uua: set_int,Uub: int] :
      aa_int_rat(aa_set7979663334043413281nt_rat(aTP_Lamm_aas(Uu),Uua),Uub) = $ite(aa_set_int_o(member_int(Uub),Uua),aa_int_rat(Uu,Uub),one_one_rat) ).

% ATP.lambda_317
tff(fact_10009_ATP_Olambda__318,axiom,
    ! [Uu: code_integer,Uua: fun_Co8499238673658658103nteger,Uub: code_integer] :
      aa_Cod1783974652533396728nteger(aa_fun4435975922079263404nteger(aTP_Lamm_pk(Uu),Uua),Uub) = $ite(Uu = Uub,aa_Cod1783974652533396728nteger(Uua,Uub),zero_z3403309356797280102nteger) ).

% ATP.lambda_318
tff(fact_10010_ATP_Olambda__319,axiom,
    ! [Uu: code_integer,Uua: fun_Code_integer_rat,Uub: code_integer] :
      aa_Code_integer_rat(aa_fun8169474305930283170er_rat(aTP_Lamm_po(Uu),Uua),Uub) = $ite(Uu = Uub,aa_Code_integer_rat(Uua,Uub),zero_zero_rat) ).

% ATP.lambda_319
tff(fact_10011_ATP_Olambda__320,axiom,
    ! [Uu: $o,Uua: fun_o_Code_integer,Uub: $o] :
      aa_o_Code_integer(aa_fun8520278569782659900nteger(aTP_Lamm_ph((Uu)),Uua),(Uub)) = $ite((Uu) = (Uub),aa_o_Code_integer(Uua,(Uub)),zero_z3403309356797280102nteger) ).

% ATP.lambda_320
tff(fact_10012_ATP_Olambda__321,axiom,
    ! [Uu: $o,Uua: fun_o_rat,Uub: $o] :
      aa_o_rat(aa_fun4336163974892599420_o_rat(aTP_Lamm_pl((Uu)),Uua),(Uub)) = $ite((Uu) = (Uub),aa_o_rat(Uua,(Uub)),zero_zero_rat) ).

% ATP.lambda_321
tff(fact_10013_ATP_Olambda__322,axiom,
    ! [Uu: $o,Uua: fun_o_nat,Uub: $o] :
      aa_o_nat(aa_fun6176012476734143100_o_nat(aTP_Lamm_pp((Uu)),Uua),(Uub)) = $ite((Uu) = (Uub),aa_o_nat(Uua,(Uub)),zero_zero_nat) ).

% ATP.lambda_322
tff(fact_10014_ATP_Olambda__323,axiom,
    ! [Uu: nat,Uua: fun_nat_Code_integer,Uub: nat] :
      aa_nat_Code_integer(aa_fun589076786913273746nteger(aTP_Lamm_pi(Uu),Uua),Uub) = $ite(Uu = Uub,aa_nat_Code_integer(Uua,Uub),zero_z3403309356797280102nteger) ).

% ATP.lambda_323
tff(fact_10015_ATP_Olambda__324,axiom,
    ! [Uu: nat,Uua: fun_nat_rat,Uub: nat] :
      aa_nat_rat(aa_fun765758790203053448at_rat(aTP_Lamm_pm(Uu),Uua),Uub) = $ite(Uu = Uub,aa_nat_rat(Uua,Uub),zero_zero_rat) ).

% ATP.lambda_324
tff(fact_10016_ATP_Olambda__325,axiom,
    ! [Uu: int,Uua: fun_int_Code_integer,Uub: int] :
      aa_int_Code_integer(aa_fun5304498221708291146nteger(aTP_Lamm_pj(Uu),Uua),Uub) = $ite(Uu = Uub,aa_int_Code_integer(Uua,Uub),zero_z3403309356797280102nteger) ).

% ATP.lambda_325
tff(fact_10017_ATP_Olambda__326,axiom,
    ! [Uu: int,Uua: fun_int_rat,Uub: int] :
      aa_int_rat(aa_fun1481061427733998144nt_rat(aTP_Lamm_pn(Uu),Uua),Uub) = $ite(Uu = Uub,aa_int_rat(Uua,Uub),zero_zero_rat) ).

% ATP.lambda_326
tff(fact_10018_ATP_Olambda__327,axiom,
    ! [Uu: int,Uua: fun_int_nat,Uub: int] :
      aa_int_nat(aa_fun6506752150744536624nt_nat(aTP_Lamm_pq(Uu),Uua),Uub) = $ite(Uu = Uub,aa_int_nat(Uua,Uub),zero_zero_nat) ).

% ATP.lambda_327
tff(fact_10019_ATP_Olambda__328,axiom,
    ! [Uu: code_integer,Uua: fun_Co8499238673658658103nteger,Uub: code_integer] :
      aa_Cod1783974652533396728nteger(aa_fun4435975922079263404nteger(aTP_Lamm_qe(Uu),Uua),Uub) = $ite(Uu = Uub,aa_Cod1783974652533396728nteger(Uua,Uub),one_one_Code_integer) ).

% ATP.lambda_328
tff(fact_10020_ATP_Olambda__329,axiom,
    ! [Uu: code_integer,Uua: fun_Co4490034186337852034r_assn,Uub: code_integer] :
      aa_Code_integer_assn(aa_fun4119713415300323650r_assn(aTP_Lamm_qi(Uu),Uua),Uub) = $ite(Uu = Uub,aa_Code_integer_assn(Uua,Uub),one_one_assn) ).

% ATP.lambda_329
tff(fact_10021_ATP_Olambda__330,axiom,
    ! [Uu: $o,Uua: fun_o_Code_integer,Uub: $o] :
      aa_o_Code_integer(aa_fun8520278569782659900nteger(aTP_Lamm_qb((Uu)),Uua),(Uub)) = $ite((Uu) = (Uub),aa_o_Code_integer(Uua,(Uub)),one_one_Code_integer) ).

% ATP.lambda_330
tff(fact_10022_ATP_Olambda__331,axiom,
    ! [Uu: $o,Uua: fun_o_assn,Uub: $o] :
      aa_o_assn(aa_fun4536517337817438332o_assn(aTP_Lamm_qf((Uu)),Uua),(Uub)) = $ite((Uu) = (Uub),aa_o_assn(Uua,(Uub)),one_one_assn) ).

% ATP.lambda_331
tff(fact_10023_ATP_Olambda__332,axiom,
    ! [Uu: $o,Uua: fun_o_rat,Uub: $o] :
      aa_o_rat(aa_fun4336163974892599420_o_rat(aTP_Lamm_qj((Uu)),Uua),(Uub)) = $ite((Uu) = (Uub),aa_o_rat(Uua,(Uub)),one_one_rat) ).

% ATP.lambda_332
tff(fact_10024_ATP_Olambda__333,axiom,
    ! [Uu: nat,Uua: fun_nat_Code_integer,Uub: nat] :
      aa_nat_Code_integer(aa_fun589076786913273746nteger(aTP_Lamm_qc(Uu),Uua),Uub) = $ite(Uu = Uub,aa_nat_Code_integer(Uua,Uub),one_one_Code_integer) ).

% ATP.lambda_333
tff(fact_10025_ATP_Olambda__334,axiom,
    ! [Uu: nat,Uua: fun_nat_assn,Uub: nat] :
      aa_nat_assn(aa_fun785744391571715112t_assn(aTP_Lamm_qg(Uu),Uua),Uub) = $ite(Uu = Uub,aa_nat_assn(Uua,Uub),one_one_assn) ).

% ATP.lambda_334
tff(fact_10026_ATP_Olambda__335,axiom,
    ! [Uu: nat,Uua: fun_nat_rat,Uub: nat] :
      aa_nat_rat(aa_fun765758790203053448at_rat(aTP_Lamm_qk(Uu),Uua),Uub) = $ite(Uu = Uub,aa_nat_rat(Uua,Uub),one_one_rat) ).

% ATP.lambda_335
tff(fact_10027_ATP_Olambda__336,axiom,
    ! [Uu: int,Uua: fun_int_Code_integer,Uub: int] :
      aa_int_Code_integer(aa_fun5304498221708291146nteger(aTP_Lamm_qd(Uu),Uua),Uub) = $ite(Uu = Uub,aa_int_Code_integer(Uua,Uub),one_one_Code_integer) ).

% ATP.lambda_336
tff(fact_10028_ATP_Olambda__337,axiom,
    ! [Uu: int,Uua: fun_int_assn,Uub: int] :
      aa_int_assn(aa_fun919007281442365664t_assn(aTP_Lamm_qh(Uu),Uua),Uub) = $ite(Uu = Uub,aa_int_assn(Uua,Uub),one_one_assn) ).

% ATP.lambda_337
tff(fact_10029_ATP_Olambda__338,axiom,
    ! [Uu: code_integer,Uua: fun_Co8499238673658658103nteger,Uub: code_integer] :
      aa_Cod1783974652533396728nteger(aa_fun4435975922079263404nteger(aTP_Lamm_pu(Uu),Uua),Uub) = $ite(Uub = Uu,aa_Cod1783974652533396728nteger(Uua,Uub),zero_z3403309356797280102nteger) ).

% ATP.lambda_338
tff(fact_10030_ATP_Olambda__339,axiom,
    ! [Uu: code_integer,Uua: fun_Code_integer_rat,Uub: code_integer] :
      aa_Code_integer_rat(aa_fun8169474305930283170er_rat(aTP_Lamm_py(Uu),Uua),Uub) = $ite(Uub = Uu,aa_Code_integer_rat(Uua,Uub),zero_zero_rat) ).

% ATP.lambda_339
tff(fact_10031_ATP_Olambda__340,axiom,
    ! [Uu: $o,Uua: fun_o_Code_integer,Uub: $o] :
      aa_o_Code_integer(aa_fun8520278569782659900nteger(aTP_Lamm_pr((Uu)),Uua),(Uub)) = $ite((Uub) = (Uu),aa_o_Code_integer(Uua,(Uub)),zero_z3403309356797280102nteger) ).

% ATP.lambda_340
tff(fact_10032_ATP_Olambda__341,axiom,
    ! [Uu: $o,Uua: fun_o_rat,Uub: $o] :
      aa_o_rat(aa_fun4336163974892599420_o_rat(aTP_Lamm_pv((Uu)),Uua),(Uub)) = $ite((Uub) = (Uu),aa_o_rat(Uua,(Uub)),zero_zero_rat) ).

% ATP.lambda_341
tff(fact_10033_ATP_Olambda__342,axiom,
    ! [Uu: $o,Uua: fun_o_nat,Uub: $o] :
      aa_o_nat(aa_fun6176012476734143100_o_nat(aTP_Lamm_pz((Uu)),Uua),(Uub)) = $ite((Uub) = (Uu),aa_o_nat(Uua,(Uub)),zero_zero_nat) ).

% ATP.lambda_342
tff(fact_10034_ATP_Olambda__343,axiom,
    ! [Uu: nat,Uua: fun_nat_Code_integer,Uub: nat] :
      aa_nat_Code_integer(aa_fun589076786913273746nteger(aTP_Lamm_ps(Uu),Uua),Uub) = $ite(Uub = Uu,aa_nat_Code_integer(Uua,Uub),zero_z3403309356797280102nteger) ).

% ATP.lambda_343
tff(fact_10035_ATP_Olambda__344,axiom,
    ! [Uu: nat,Uua: fun_nat_rat,Uub: nat] :
      aa_nat_rat(aa_fun765758790203053448at_rat(aTP_Lamm_pw(Uu),Uua),Uub) = $ite(Uub = Uu,aa_nat_rat(Uua,Uub),zero_zero_rat) ).

% ATP.lambda_344
tff(fact_10036_ATP_Olambda__345,axiom,
    ! [Uu: int,Uua: fun_int_Code_integer,Uub: int] :
      aa_int_Code_integer(aa_fun5304498221708291146nteger(aTP_Lamm_pt(Uu),Uua),Uub) = $ite(Uub = Uu,aa_int_Code_integer(Uua,Uub),zero_z3403309356797280102nteger) ).

% ATP.lambda_345
tff(fact_10037_ATP_Olambda__346,axiom,
    ! [Uu: int,Uua: fun_int_rat,Uub: int] :
      aa_int_rat(aa_fun1481061427733998144nt_rat(aTP_Lamm_px(Uu),Uua),Uub) = $ite(Uub = Uu,aa_int_rat(Uua,Uub),zero_zero_rat) ).

% ATP.lambda_346
tff(fact_10038_ATP_Olambda__347,axiom,
    ! [Uu: int,Uua: fun_int_nat,Uub: int] :
      aa_int_nat(aa_fun6506752150744536624nt_nat(aTP_Lamm_qa(Uu),Uua),Uub) = $ite(Uub = Uu,aa_int_nat(Uua,Uub),zero_zero_nat) ).

% ATP.lambda_347
tff(fact_10039_ATP_Olambda__348,axiom,
    ! [Uu: code_integer,Uua: fun_Co8499238673658658103nteger,Uub: code_integer] :
      aa_Cod1783974652533396728nteger(aa_fun4435975922079263404nteger(aTP_Lamm_qo(Uu),Uua),Uub) = $ite(Uub = Uu,aa_Cod1783974652533396728nteger(Uua,Uub),one_one_Code_integer) ).

% ATP.lambda_348
tff(fact_10040_ATP_Olambda__349,axiom,
    ! [Uu: code_integer,Uua: fun_Co4490034186337852034r_assn,Uub: code_integer] :
      aa_Code_integer_assn(aa_fun4119713415300323650r_assn(aTP_Lamm_qs(Uu),Uua),Uub) = $ite(Uub = Uu,aa_Code_integer_assn(Uua,Uub),one_one_assn) ).

% ATP.lambda_349
tff(fact_10041_ATP_Olambda__350,axiom,
    ! [Uu: $o,Uua: fun_o_Code_integer,Uub: $o] :
      aa_o_Code_integer(aa_fun8520278569782659900nteger(aTP_Lamm_ql((Uu)),Uua),(Uub)) = $ite((Uub) = (Uu),aa_o_Code_integer(Uua,(Uub)),one_one_Code_integer) ).

% ATP.lambda_350
tff(fact_10042_ATP_Olambda__351,axiom,
    ! [Uu: $o,Uua: fun_o_assn,Uub: $o] :
      aa_o_assn(aa_fun4536517337817438332o_assn(aTP_Lamm_qp((Uu)),Uua),(Uub)) = $ite((Uub) = (Uu),aa_o_assn(Uua,(Uub)),one_one_assn) ).

% ATP.lambda_351
tff(fact_10043_ATP_Olambda__352,axiom,
    ! [Uu: $o,Uua: fun_o_rat,Uub: $o] :
      aa_o_rat(aa_fun4336163974892599420_o_rat(aTP_Lamm_qt((Uu)),Uua),(Uub)) = $ite((Uub) = (Uu),aa_o_rat(Uua,(Uub)),one_one_rat) ).

% ATP.lambda_352
tff(fact_10044_ATP_Olambda__353,axiom,
    ! [Uu: nat,Uua: fun_nat_Code_integer,Uub: nat] :
      aa_nat_Code_integer(aa_fun589076786913273746nteger(aTP_Lamm_qm(Uu),Uua),Uub) = $ite(Uub = Uu,aa_nat_Code_integer(Uua,Uub),one_one_Code_integer) ).

% ATP.lambda_353
tff(fact_10045_ATP_Olambda__354,axiom,
    ! [Uu: nat,Uua: fun_nat_assn,Uub: nat] :
      aa_nat_assn(aa_fun785744391571715112t_assn(aTP_Lamm_qq(Uu),Uua),Uub) = $ite(Uub = Uu,aa_nat_assn(Uua,Uub),one_one_assn) ).

% ATP.lambda_354
tff(fact_10046_ATP_Olambda__355,axiom,
    ! [Uu: nat,Uua: fun_nat_rat,Uub: nat] :
      aa_nat_rat(aa_fun765758790203053448at_rat(aTP_Lamm_qu(Uu),Uua),Uub) = $ite(Uub = Uu,aa_nat_rat(Uua,Uub),one_one_rat) ).

% ATP.lambda_355
tff(fact_10047_ATP_Olambda__356,axiom,
    ! [Uu: int,Uua: fun_int_Code_integer,Uub: int] :
      aa_int_Code_integer(aa_fun5304498221708291146nteger(aTP_Lamm_qn(Uu),Uua),Uub) = $ite(Uub = Uu,aa_int_Code_integer(Uua,Uub),one_one_Code_integer) ).

% ATP.lambda_356
tff(fact_10048_ATP_Olambda__357,axiom,
    ! [Uu: int,Uua: fun_int_assn,Uub: int] :
      aa_int_assn(aa_fun919007281442365664t_assn(aTP_Lamm_qr(Uu),Uua),Uub) = $ite(Uub = Uu,aa_int_assn(Uua,Uub),one_one_assn) ).

% ATP.lambda_357
tff(fact_10049_ATP_Olambda__358,axiom,
    ! [Uu: code_integer,Uua: code_integer,Uub: code_integer] :
      aa_Cod6977894654539884864nteger(aa_Cod8380929501330563836nteger(aTP_Lamm_aml(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_358
tff(fact_10050_ATP_Olambda__359,axiom,
    ! [Uu: code_integer,Uua: code_integer,Uub: code_integer] :
      aa_Cod6977894654539884864nteger(aa_Cod8380929501330563836nteger(aTP_Lamm_amm(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_359
tff(fact_10051_ATP_Olambda__360,axiom,
    ! [Uu: code_integer,Uua: code_integer,Uub: code_integer] :
      aa_Cod6977894654539884864nteger(aa_Cod8380929501330563836nteger(aTP_Lamm_amk(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_360
tff(fact_10052_ATP_Olambda__361,axiom,
    ! [Uu: fun_Co8499238673658658103nteger,Uua: fun_Code_integer_o,Uub: code_integer] :
      aa_Cod1783974652533396728nteger(aa_fun791458909507501263nteger(aTP_Lamm_vn(Uu),Uua),Uub) = $ite(aa_Code_integer_o(Uua,Uub),aa_Cod1783974652533396728nteger(Uu,Uub),zero_z3403309356797280102nteger) ).

% ATP.lambda_361
tff(fact_10053_ATP_Olambda__362,axiom,
    ! [Uu: fun_Code_integer_rat,Uua: fun_Code_integer_o,Uub: code_integer] :
      aa_Code_integer_rat(aa_fun25512833367352522er_rat(aTP_Lamm_vr(Uu),Uua),Uub) = $ite(aa_Code_integer_o(Uua,Uub),aa_Code_integer_rat(Uu,Uub),zero_zero_rat) ).

% ATP.lambda_362
tff(fact_10054_ATP_Olambda__363,axiom,
    ! [Uu: fun_o_Code_integer,Uua: fun_o_o,Uub: $o] :
      aa_o_Code_integer(aa_fun6542514109734600263nteger(aTP_Lamm_vj(Uu),Uua),(Uub)) = $ite(aa_o_o(Uua,(Uub)),aa_o_Code_integer(Uu,(Uub)),zero_z3403309356797280102nteger) ).

% ATP.lambda_363
tff(fact_10055_ATP_Olambda__364,axiom,
    ! [Uu: fun_o_rat,Uua: fun_o_o,Uub: $o] :
      aa_o_rat(aa_fun_o_o_fun_o_rat(aTP_Lamm_vo(Uu),Uua),(Uub)) = $ite(aa_o_o(Uua,(Uub)),aa_o_rat(Uu,(Uub)),zero_zero_rat) ).

% ATP.lambda_364
tff(fact_10056_ATP_Olambda__365,axiom,
    ! [Uu: fun_o_nat,Uua: fun_o_o,Uub: $o] :
      aa_o_nat(aa_fun_o_o_fun_o_nat(aTP_Lamm_vs(Uu),Uua),(Uub)) = $ite(aa_o_o(Uua,(Uub)),aa_o_nat(Uu,(Uub)),zero_zero_nat) ).

% ATP.lambda_365
tff(fact_10057_ATP_Olambda__366,axiom,
    ! [Uu: fun_nat_Code_integer,Uua: fun_nat_o,Uub: nat] :
      aa_nat_Code_integer(aa_fun3969837373441324047nteger(aTP_Lamm_vk(Uu),Uua),Uub) = $ite(aa_nat_o(Uua,Uub),aa_nat_Code_integer(Uu,Uub),zero_z3403309356797280102nteger) ).

% ATP.lambda_366
tff(fact_10058_ATP_Olambda__367,axiom,
    ! [Uu: fun_nat_rat,Uua: fun_nat_o,Uub: nat] :
      aa_nat_rat(aa_fun696632391729880074at_rat(aTP_Lamm_vp(Uu),Uua),Uub) = $ite(aa_nat_o(Uua,Uub),aa_nat_rat(Uu,Uub),zero_zero_rat) ).

% ATP.lambda_367
tff(fact_10059_ATP_Olambda__368,axiom,
    ! [Uu: fun_int_Code_integer,Uua: fun_int_o,Uub: int] :
      aa_int_Code_integer(aa_fun6019004165459918095nteger(aTP_Lamm_vl(Uu),Uua),Uub) = $ite(aa_int_o(Uua,Uub),aa_int_Code_integer(Uu,Uub),zero_z3403309356797280102nteger) ).

% ATP.lambda_368
tff(fact_10060_ATP_Olambda__369,axiom,
    ! [Uu: fun_int_rat,Uua: fun_int_o,Uub: int] :
      aa_int_rat(aa_fun268837275264525578nt_rat(aTP_Lamm_vq(Uu),Uua),Uub) = $ite(aa_int_o(Uua,Uub),aa_int_rat(Uu,Uub),zero_zero_rat) ).

% ATP.lambda_369
tff(fact_10061_ATP_Olambda__370,axiom,
    ! [Uu: fun_int_nat,Uua: fun_int_o,Uub: int] :
      aa_int_nat(aa_fun3777065833657922306nt_nat(aTP_Lamm_vt(Uu),Uua),Uub) = $ite(aa_int_o(Uua,Uub),aa_int_nat(Uu,Uub),zero_zero_nat) ).

% ATP.lambda_370
tff(fact_10062_ATP_Olambda__371,axiom,
    ! [Uu: fun_Co8499238673658658103nteger,Uua: fun_Code_integer_o,Uub: code_integer] :
      aa_Cod1783974652533396728nteger(aa_fun791458909507501263nteger(aTP_Lamm_xd(Uu),Uua),Uub) = $ite(aa_Code_integer_o(Uua,Uub),aa_Cod1783974652533396728nteger(Uu,Uub),one_one_Code_integer) ).

% ATP.lambda_371
tff(fact_10063_ATP_Olambda__372,axiom,
    ! [Uu: fun_Co4490034186337852034r_assn,Uua: fun_Code_integer_o,Uub: code_integer] :
      aa_Code_integer_assn(aa_fun6392223008197429786r_assn(aTP_Lamm_xh(Uu),Uua),Uub) = $ite(aa_Code_integer_o(Uua,Uub),aa_Code_integer_assn(Uu,Uub),one_one_assn) ).

% ATP.lambda_372
tff(fact_10064_ATP_Olambda__373,axiom,
    ! [Uu: fun_o_Code_integer,Uua: fun_o_o,Uub: $o] :
      aa_o_Code_integer(aa_fun6542514109734600263nteger(aTP_Lamm_xa(Uu),Uua),(Uub)) = $ite(aa_o_o(Uua,(Uub)),aa_o_Code_integer(Uu,(Uub)),one_one_Code_integer) ).

% ATP.lambda_373
tff(fact_10065_ATP_Olambda__374,axiom,
    ! [Uu: fun_o_assn,Uua: fun_o_o,Uub: $o] :
      aa_o_assn(aa_fun8560762093604944786o_assn(aTP_Lamm_xe(Uu),Uua),(Uub)) = $ite(aa_o_o(Uua,(Uub)),aa_o_assn(Uu,(Uub)),one_one_assn) ).

% ATP.lambda_374
tff(fact_10066_ATP_Olambda__375,axiom,
    ! [Uu: fun_o_rat,Uua: fun_o_o,Uub: $o] :
      aa_o_rat(aa_fun_o_o_fun_o_rat(aTP_Lamm_xi(Uu),Uua),(Uub)) = $ite(aa_o_o(Uua,(Uub)),aa_o_rat(Uu,(Uub)),one_one_rat) ).

% ATP.lambda_375
tff(fact_10067_ATP_Olambda__376,axiom,
    ! [Uu: fun_nat_Code_integer,Uua: fun_nat_o,Uub: nat] :
      aa_nat_Code_integer(aa_fun3969837373441324047nteger(aTP_Lamm_xb(Uu),Uua),Uub) = $ite(aa_nat_o(Uua,Uub),aa_nat_Code_integer(Uu,Uub),one_one_Code_integer) ).

% ATP.lambda_376
tff(fact_10068_ATP_Olambda__377,axiom,
    ! [Uu: fun_nat_assn,Uua: fun_nat_o,Uub: nat] :
      aa_nat_assn(aa_fun3990007403578212698t_assn(aTP_Lamm_xf(Uu),Uua),Uub) = $ite(aa_nat_o(Uua,Uub),aa_nat_assn(Uu,Uub),one_one_assn) ).

% ATP.lambda_377
tff(fact_10069_ATP_Olambda__378,axiom,
    ! [Uu: fun_nat_rat,Uua: fun_nat_o,Uub: nat] :
      aa_nat_rat(aa_fun696632391729880074at_rat(aTP_Lamm_xj(Uu),Uua),Uub) = $ite(aa_nat_o(Uua,Uub),aa_nat_rat(Uu,Uub),one_one_rat) ).

% ATP.lambda_378
tff(fact_10070_ATP_Olambda__379,axiom,
    ! [Uu: fun_int_Code_integer,Uua: fun_int_o,Uub: int] :
      aa_int_Code_integer(aa_fun6019004165459918095nteger(aTP_Lamm_xc(Uu),Uua),Uub) = $ite(aa_int_o(Uua,Uub),aa_int_Code_integer(Uu,Uub),one_one_Code_integer) ).

% ATP.lambda_379
tff(fact_10071_ATP_Olambda__380,axiom,
    ! [Uu: fun_int_assn,Uua: fun_int_o,Uub: int] :
      aa_int_assn(aa_fun9028805181350005338t_assn(aTP_Lamm_xg(Uu),Uua),Uub) = $ite(aa_int_o(Uua,Uub),aa_int_assn(Uu,Uub),one_one_assn) ).

% ATP.lambda_380
tff(fact_10072_ATP_Olambda__381,axiom,
    ! [Uu: fun_he8413527710729349838unit_o,Uua: fun_he4944555133437423336it_nat,Uub: heap_e7401611519738050253t_unit] :
      aa_hea684056619475940329it_nat(aa_fun2986507860694160780it_nat(aTP_Lamm_fj(Uu),Uua),Uub) = $ite(aa_hea7994211679605476167unit_o(Uu,Uub),aa_Pro8996696869555642526it_nat(some_P1914260805536162275it_nat,aa_hea256619718444519449it_nat(Uua,Uub)),none_P9117596204409417319it_nat) ).

% ATP.lambda_381
tff(fact_10073_ATP_Olambda__382,axiom,
    ! [Uu: fun_he8413527710729349838unit_o,Uua: fun_he8889679731534893401it_nat,Uub: heap_e7401611519738050253t_unit] :
      aa_hea4724376117875161946it_nat(aa_fun3370411987892275404it_nat(aTP_Lamm_fi(Uu),Uua),Uub) = $ite(aa_hea7994211679605476167unit_o(Uu,Uub),aa_Pro5302094286594872192it_nat(some_P7913643980934408916it_nat,aa_hea5645563107516622602it_nat(Uua,Uub)),none_P2651198173097904984it_nat) ).

% ATP.lambda_382
tff(fact_10074_ATP_Olambda__383,axiom,
    ! [Uu: fun_num_fun_num_num,Uua: option_num,Uub: num] : aa_num_option_num(aa_opt8899475237019873193on_num(aTP_Lamm_fq(Uu),Uua),Uub) = aa_opt5369159329878751634on_num(case_o6005452278849405969um_num(aa_num_option_num(some_num,Uub),aa_num6390622462783740633on_num(aTP_Lamm_fp(Uu),Uub)),Uua) ).

% ATP.lambda_383
tff(fact_10075_ATP_Olambda__384,axiom,
    ! [Uu: fun_Pr1003294667014919570it_nat,Uua: option8956607266484857688it_nat,Uub: produc8664842809031399944it_nat] : aa_Pro8996696869555642526it_nat(aa_opt7728438819757478003it_nat(aTP_Lamm_fo(Uu),Uua),Uub) = case_o2963978774867076333it_nat(aa_Pro8996696869555642526it_nat(some_P1914260805536162275it_nat,Uub),aa_Pro5635350972512700067it_nat(aTP_Lamm_fn(Uu),Uub),Uua) ).

% ATP.lambda_384
tff(fact_10076_ATP_Olambda__385,axiom,
    ! [Uu: fun_Pr7750636411365408485it_nat,Uua: option3562590408128118217it_nat,Uub: produc3260487557148687353it_nat] : aa_Pro5302094286594872192it_nat(aa_opt996909609682481350it_nat(aTP_Lamm_fm(Uu),Uua),Uub) = case_o6061097939050036047it_nat(aa_Pro5302094286594872192it_nat(some_P7913643980934408916it_nat,Uub),aa_Pro7038079567744603766it_nat(aTP_Lamm_fl(Uu),Uub),Uua) ).

% ATP.lambda_385
tff(fact_10077_ATP_Olambda__386,axiom,
    ! [Uu: option_nat,Uua: option_nat,Uub: nat] : aa_nat_option_nat(aa_opt1558076126722380235on_nat(aTP_Lamm_du(Uu),Uua),Uub) = case_o7429725398727453821at_nat(Uu,aTP_Lamm_dt(Uub),Uua) ).

% ATP.lambda_386
tff(fact_10078_ATP_Olambda__387,axiom,
    ! [Uu: option_set_nat,Uua: option_set_nat,Uub: set_nat] : aa_set6119337032432945050et_nat(aa_opt5648106117674462701et_nat(aTP_Lamm_ds(Uu),Uua),Uub) = case_o4054078431260844265et_nat(Uu,aTP_Lamm_dr(Uub),Uua) ).

% ATP.lambda_387
tff(fact_10079_ATP_Olambda__388,axiom,
    ! [Uu: option1583680563626158861at_nat,Uua: option1583680563626158861at_nat,Uub: set_Pr4329608150637261639at_nat] : aa_set7071694670514008770at_nat(aa_opt1027706798495913626at_nat(aTP_Lamm_dq(Uu),Uua),Uub) = case_o9080059180780454917at_nat(Uu,aTP_Lamm_dp(Uub),Uua) ).

% ATP.lambda_388
tff(fact_10080_ATP_Olambda__389,axiom,
    ! [Uu: option2498585697089621389at_nat,Uua: option2498585697089621389at_nat,Uub: set_Pr8551490117392284871at_nat] : aa_set1897275755269116354at_nat(aa_opt1741456117285354266at_nat(aTP_Lamm_do(Uu),Uua),Uub) = case_o8757249484648823045at_nat(Uu,aTP_Lamm_dn(Uub),Uua) ).

% ATP.lambda_389
tff(fact_10081_ATP_Olambda__390,axiom,
    ! [Uu: option2158460348403737357at_nat,Uua: option2158460348403737357at_nat,Uub: set_Pr8693737435421807431at_nat] : aa_set6661686244540357314at_nat(aa_opt2198001859980869274at_nat(aTP_Lamm_dm(Uu),Uua),Uub) = case_o7467001542679133189at_nat(Uu,aTP_Lamm_dl(Uub),Uua) ).

% ATP.lambda_390
tff(fact_10082_ATP_Olambda__391,axiom,
    ! [Uu: heap_Time_Heap_a,Uua: fun_a_2990225006182978344t_unit,Uub: heap_e7401611519738050253t_unit] : aa_hea684056619475940329it_nat(aa_fun372765568861188684it_nat(aTP_Lamm_ahj(Uu),Uua),Uub) = case_o7727800614283616222it_nat(none_P9117596204409417319it_nat,produc5648861294512138360it_nat(aTP_Lamm_ahi(Uua)),aa_hea4724376117875161946it_nat(heap_Time_execute_a(Uu),Uub)) ).

% ATP.lambda_391
tff(fact_10083_ATP_Olambda__392,axiom,
    ! [Uu: heap_Time_Heap_a,Uua: fun_a_6093291656547882009Heap_a,Uub: heap_e7401611519738050253t_unit] : aa_hea4724376117875161946it_nat(aa_fun6119220767042692876it_nat(aTP_Lamm_ahg(Uu),Uua),Uub) = case_o6061097939050036047it_nat(none_P2651198173097904984it_nat,produc7850007024774191849it_nat(aTP_Lamm_ahe(Uua)),aa_hea4724376117875161946it_nat(heap_Time_execute_a(Uu),Uub)) ).

% ATP.lambda_392
tff(fact_10084_ATP_Olambda__393,axiom,
    ! [Uu: fun_nat_fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_agy(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_393
tff(fact_10085_ATP_Olambda__394,axiom,
    ! [Uu: fun_nat_fun_nat_int,Uua: nat,Uub: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_ahb(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_394
tff(fact_10086_ATP_Olambda__395,axiom,
    ! [Uu: fun_Co3312552843764780407at_nat,Uua: nat,Uub: code_integer] : aa_Code_integer_nat(aa_nat5852537043073606456er_nat(aTP_Lamm_rp(Uu),Uua),Uub) = aa_nat_nat(aa_Cod8670293052248272184at_nat(Uu,Uub),Uua) ).

% ATP.lambda_395
tff(fact_10087_ATP_Olambda__396,axiom,
    ! [Uu: fun_Co822469698833053651at_int,Uua: nat,Uub: code_integer] : aa_Code_integer_int(aa_nat1674686023564409748er_int(aTP_Lamm_th(Uu),Uua),Uub) = aa_nat_int(aa_Cod4492442032739075476at_int(Uu,Uub),Uua) ).

% ATP.lambda_396
tff(fact_10088_ATP_Olambda__397,axiom,
    ! [Uu: fun_Co3009391273526075439nt_int,Uua: int,Uub: code_integer] : aa_Code_integer_int(aa_int1390634949194299888er_int(aTP_Lamm_sd(Uu),Uua),Uub) = aa_int_int(aa_Cod5491795744640130032nt_int(Uu,Uub),Uua) ).

% ATP.lambda_397
tff(fact_10089_ATP_Olambda__398,axiom,
    ! [Uu: fun_o_fun_nat_nat,Uua: nat,Uub: $o] : aa_o_nat(aa_nat_fun_o_nat(aTP_Lamm_rf(Uu),Uua),(Uub)) = aa_nat_nat(aa_o_fun_nat_nat(Uu,(Uub)),Uua) ).

% ATP.lambda_398
tff(fact_10090_ATP_Olambda__399,axiom,
    ! [Uu: fun_o_fun_nat_int,Uua: nat,Uub: $o] : aa_o_int(aa_nat_fun_o_int(aTP_Lamm_te(Uu),Uua),(Uub)) = aa_nat_int(aa_o_fun_nat_int(Uu,(Uub)),Uua) ).

% ATP.lambda_399
tff(fact_10091_ATP_Olambda__400,axiom,
    ! [Uu: fun_o_fun_int_int,Uua: int,Uub: $o] : aa_o_int(aa_int_fun_o_int(aTP_Lamm_ru(Uu),Uua),(Uub)) = aa_int_int(aa_o_fun_int_int(Uu,(Uub)),Uua) ).

% ATP.lambda_400
tff(fact_10092_ATP_Olambda__401,axiom,
    ! [Uu: fun_na5492297256294505719er_nat,Uua: code_integer,Uub: nat] : aa_nat_nat(aa_Cod8670293052248272184at_nat(aTP_Lamm_sq(Uu),Uua),Uub) = aa_Code_integer_nat(aa_nat5852537043073606456er_nat(Uu,Uub),Uua) ).

% ATP.lambda_401
tff(fact_10093_ATP_Olambda__402,axiom,
    ! [Uu: fun_nat_fun_o_nat,Uua: $o,Uub: nat] : aa_nat_nat(aa_o_fun_nat_nat(aTP_Lamm_si(Uu),(Uua)),Uub) = aa_o_nat(aa_nat_fun_o_nat(Uu,Uub),(Uua)) ).

% ATP.lambda_402
tff(fact_10094_ATP_Olambda__403,axiom,
    ! [Uu: fun_nat_fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_lk(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(Uu,Uub),Uua) ).

% ATP.lambda_403
tff(fact_10095_ATP_Olambda__404,axiom,
    ! [Uu: fun_nat_fun_nat_int,Uua: nat,Uub: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_mr(Uu),Uua),Uub) = aa_nat_int(aa_nat_fun_nat_int(Uu,Uub),Uua) ).

% ATP.lambda_404
tff(fact_10096_ATP_Olambda__405,axiom,
    ! [Uu: fun_nat_fun_int_nat,Uua: int,Uub: nat] : aa_nat_nat(aa_int_fun_nat_nat(aTP_Lamm_sm(Uu),Uua),Uub) = aa_int_nat(aa_nat_fun_int_nat(Uu,Uub),Uua) ).

% ATP.lambda_405
tff(fact_10097_ATP_Olambda__406,axiom,
    ! [Uu: fun_nat_fun_int_int,Uua: int,Uub: nat] : aa_nat_int(aa_int_fun_nat_int(aTP_Lamm_mu(Uu),Uua),Uub) = aa_int_int(aa_nat_fun_int_int(Uu,Uub),Uua) ).

% ATP.lambda_406
tff(fact_10098_ATP_Olambda__407,axiom,
    ! [Uu: fun_int_fun_nat_nat,Uua: nat,Uub: int] : aa_int_nat(aa_nat_fun_int_nat(aTP_Lamm_rk(Uu),Uua),Uub) = aa_nat_nat(aa_int_fun_nat_nat(Uu,Uub),Uua) ).

% ATP.lambda_407
tff(fact_10099_ATP_Olambda__408,axiom,
    ! [Uu: fun_int_fun_nat_int,Uua: nat,Uub: int] : aa_int_int(aa_nat_fun_int_int(aTP_Lamm_mx(Uu),Uua),Uub) = aa_nat_int(aa_int_fun_nat_int(Uu,Uub),Uua) ).

% ATP.lambda_408
tff(fact_10100_ATP_Olambda__409,axiom,
    ! [Uu: fun_int_fun_int_int,Uua: int,Uub: int] : aa_int_int(aa_int_fun_int_int(aTP_Lamm_ln(Uu),Uua),Uub) = aa_int_int(aa_int_fun_int_int(Uu,Uub),Uua) ).

% ATP.lambda_409
tff(fact_10101_ATP_Olambda__410,axiom,
    ! [Uu: code_integer,Uua: code_integer,Uub: code_integer] :
      aa_Cod3168588076859124107eger_o(aa_Cod1096751537565085541eger_o(aTP_Lamm_ami(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_410
tff(fact_10102_ATP_Olambda__411,axiom,
    ! [Uu: fun_nat_fun_nat_int,Uua: nat,Uub: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_abv(Uu),Uua),Uub) = aa_set_nat_int(groups705719431365010083at_int(aa_nat_fun_nat_int(aTP_Lamm_mr(Uu),Uub)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,aa_nat_nat(suc,Uub)),Uua)) ).

% ATP.lambda_411
tff(fact_10103_ATP_Olambda__412,axiom,
    ! [Uu: fun_nat_fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_abt(Uu),Uua),Uub) = aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_lk(Uu),Uub)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,aa_nat_nat(suc,Uub)),Uua)) ).

% ATP.lambda_412
tff(fact_10104_ATP_Olambda__413,axiom,
    ! [Uu: fun_nat_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(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_lk(Uu),Uub)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,aa_nat_nat(suc,Uub)),Uua)) ).

% ATP.lambda_413
tff(fact_10105_ATP_Olambda__414,axiom,
    ! [Uu: produc7822875418678951345atural,Uua: code_natural,Uub: code_natural] : aa_Cod6808739660676082561atural(aa_Cod4292056611156568242atural(aTP_Lamm_aoz(Uu),Uua),Uub) = produc8282080750456430313atural(aa_Cod8811915439870959971atural(aTP_Lamm_aoy(Uua),Uub),produc6591912806276919810atural(aa_Pro7165575656943818355atural(next,Uu))) ).

% ATP.lambda_414
tff(fact_10106_ATP_Olambda__415,axiom,
    ! [Uu: rat,Uua: int,Uub: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_aoe(Uu),Uua),Uub) = aa_Pro6970504127699319548nt_int(aa_fun8026974734089330650nt_int(produc4245557441103728435nt_int,aa_int1966403025003302569nt_int(aTP_Lamm_aod(Uua),Uub)),quotient_of(Uu)) ).

% ATP.lambda_415
tff(fact_10107_ATP_Olambda__416,axiom,
    ! [Uu: rat,Uua: int,Uub: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_aoc(Uu),Uua),Uub) = aa_Pro6970504127699319548nt_int(aa_fun8026974734089330650nt_int(produc4245557441103728435nt_int,aa_int1966403025003302569nt_int(aTP_Lamm_aob(Uua),Uub)),quotient_of(Uu)) ).

% ATP.lambda_416
tff(fact_10108_ATP_Olambda__417,axiom,
    ! [Uu: rat,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_anz(Uu),Uua),Uub)
    <=> aa_Pro4592068669816069595_int_o(aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,aa_int2464330468366073224_int_o(aTP_Lamm_any(Uua),Uub)),quotient_of(Uu)) ) ).

% ATP.lambda_417
tff(fact_10109_ATP_Olambda__418,axiom,
    ! [Uu: rat,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_anx(Uu),Uua),Uub)
    <=> aa_Pro4592068669816069595_int_o(aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,aa_int2464330468366073224_int_o(aTP_Lamm_anw(Uua),Uub)),quotient_of(Uu)) ) ).

% ATP.lambda_418
tff(fact_10110_ATP_Olambda__419,axiom,
    ! [Uu: rat,Uua: int,Uub: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_anv(Uu),Uua),Uub) = aa_Pro6970504127699319548nt_int(aa_fun8026974734089330650nt_int(produc4245557441103728435nt_int,aa_int1966403025003302569nt_int(aTP_Lamm_anu(Uua),Uub)),quotient_of(Uu)) ).

% ATP.lambda_419
tff(fact_10111_ATP_Olambda__420,axiom,
    ! [Uu: rat,Uua: int,Uub: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_ant(Uu),Uua),Uub) = aa_Pro6970504127699319548nt_int(aa_fun8026974734089330650nt_int(produc4245557441103728435nt_int,aa_int1966403025003302569nt_int(aTP_Lamm_ans(Uua),Uub)),quotient_of(Uu)) ).

% ATP.lambda_420
tff(fact_10112_ATP_Olambda__421,axiom,
    ! [Uu: fun_int_fun_nat_int,Uua: set_int,Uub: nat] : aa_nat_int(aa_set6310687161026558745at_int(aTP_Lamm_my(Uu),Uua),Uub) = groups1705073143266064639nt_int(aa_nat_fun_int_int(aTP_Lamm_mx(Uu),Uub),Uua) ).

% ATP.lambda_421
tff(fact_10113_ATP_Olambda__422,axiom,
    ! [Uu: fun_nat_fun_int_int,Uua: set_nat,Uub: int] : aa_int_int(aa_set3358596244511904025nt_int(aTP_Lamm_mv(Uu),Uua),Uub) = aa_set_nat_int(groups705719431365010083at_int(aa_int_fun_nat_int(aTP_Lamm_mu(Uu),Uub)),Uua) ).

% ATP.lambda_422
tff(fact_10114_ATP_Olambda__423,axiom,
    ! [Uu: fun_nat_fun_nat_int,Uua: set_nat,Uub: nat] : aa_nat_int(aa_set2359242532610849469at_int(aTP_Lamm_ms(Uu),Uua),Uub) = aa_set_nat_int(groups705719431365010083at_int(aa_nat_fun_nat_int(aTP_Lamm_mr(Uu),Uub)),Uua) ).

% ATP.lambda_423
tff(fact_10115_ATP_Olambda__424,axiom,
    ! [Uu: fun_int_fun_int_int,Uua: set_int,Uub: int] : aa_int_int(aa_set7310040872927613301nt_int(aTP_Lamm_na(Uu),Uua),Uub) = groups1705073143266064639nt_int(aa_int_fun_int_int(aTP_Lamm_ln(Uu),Uub),Uua) ).

% ATP.lambda_424
tff(fact_10116_ATP_Olambda__425,axiom,
    ! [Uu: fun_nat_fun_nat_nat,Uua: set_nat,Uub: nat] : aa_nat_nat(aa_set6537093552120046177at_nat(aTP_Lamm_mp(Uu),Uua),Uub) = aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_lk(Uu),Uub)),Uua) ).

% ATP.lambda_425
tff(fact_10117_ATP_Olambda__426,axiom,
    ! [Uu: fun_int_fun_int_int,Uua: set_int,Uub: int] : aa_int_int(aa_set7310040872927613301nt_int(aTP_Lamm_lo(Uu),Uua),Uub) = groups4538972089207619220nt_int(aa_int_fun_int_int(aTP_Lamm_ln(Uu),Uub),Uua) ).

% ATP.lambda_426
tff(fact_10118_ATP_Olambda__427,axiom,
    ! [Uu: fun_nat_fun_nat_nat,Uua: set_nat,Uub: nat] : aa_nat_nat(aa_set6537093552120046177at_nat(aTP_Lamm_ll(Uu),Uua),Uub) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_lk(Uu),Uub)),Uua) ).

% ATP.lambda_427
tff(fact_10119_ATP_Olambda__428,axiom,
    ! [Uu: fun_nat_rat,Uua: fun_nat_rat,Uub: nat] : aa_nat_rat(aa_fun765758790203053448at_rat(aTP_Lamm_pg(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_428
tff(fact_10120_ATP_Olambda__429,axiom,
    ! [Uu: nat,Uua: nat,Uub: list_nat] :
      ( aa_list_nat_o(aa_nat5883842100386203267_nat_o(aTP_Lamm_ani(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_429
tff(fact_10121_ATP_Olambda__430,axiom,
    ! [Uu: nat,Uua: nat,Uub: list_nat] :
      ( aa_list_nat_o(aa_nat5883842100386203267_nat_o(aTP_Lamm_anj(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_430
tff(fact_10122_ATP_Olambda__431,axiom,
    ! [Uu: nat,Uua: nat,Uub: list_nat] :
      ( aa_list_nat_o(aa_nat5883842100386203267_nat_o(aTP_Lamm_anh(Uu),Uua),Uub)
    <=> ( ( aa_list_nat_nat(size_size_list_nat,Uub) = Uu )
        & ( aa_list_nat_nat(groups4561878855575611511st_nat,Uub) = Uua ) ) ) ).

% ATP.lambda_431
tff(fact_10123_ATP_Olambda__432,axiom,
    ! [Uu: heap_e7401611519738050253t_unit,Uua: heap_e7401611519738050253t_unit,Uub: nat] :
      ( aa_nat_o(aa_hea9023553427033036112_nat_o(aTP_Lamm_cn(Uu),Uua),Uub)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,lim_Product_unit(Uu)),Uub)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uub),lim_Product_unit(Uua)) ) ) ).

% ATP.lambda_432
tff(fact_10124_ATP_Olambda__433,axiom,
    ! [Uu: set_nat,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_aly(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(member_nat(aa_nat_nat(suc,Uub)),Uu)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uub),Uua) ) ) ).

% ATP.lambda_433
tff(fact_10125_ATP_Olambda__434,axiom,
    ! [Uu: nat,Uua: nat,Uub: set_nat] :
      ( aa_set_nat_o(aa_nat_fun_set_nat_o(aTP_Lamm_ang(Uu),Uua),Uub)
    <=> ( aa_set_set_nat_o(member_set_nat(Uub),pow_nat(aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),Uu)))
        & ( aa_set_nat_nat(finite_card_nat,Uub) = Uua ) ) ) ).

% ATP.lambda_434
tff(fact_10126_ATP_Olambda__435,axiom,
    ! [Uu: rat,Uua: nat,Uub: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aTP_Lamm_acr(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_435
tff(fact_10127_ATP_Olambda__436,axiom,
    ! [Uu: heap_e7401611519738050253t_unit,Uua: set_nat,Uub: nat] :
      ( aa_nat_o(aa_set_nat_fun_nat_o(aTP_Lamm_co(Uu),Uua),Uub)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uub),lim_Product_unit(Uu))
        & ~ aa_set_nat_o(member_nat(Uub),Uua) ) ) ).

% ATP.lambda_436
tff(fact_10128_ATP_Olambda__437,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_aou(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_437
tff(fact_10129_ATP_Olambda__438,axiom,
    ! [Uu: set_Code_integer,Uua: nat,Uub: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(aa_nat2915507816091602832eger_o(aTP_Lamm_aky(Uu),Uua),Uub)
    <=> ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(Uub),Uu)
        & ( finite4902975817058060853nteger(Uub) = Uua ) ) ) ).

% ATP.lambda_438
tff(fact_10130_ATP_Olambda__439,axiom,
    ! [Uu: set_Pr1261947904930325089at_nat,Uua: nat,Uub: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_nat1727191234135941576_nat_o(aTP_Lamm_akz(Uu),Uua),Uub)
    <=> ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,Uub),Uu)
        & ( finite711546835091564841at_nat(Uub) = Uua ) ) ) ).

% ATP.lambda_439
tff(fact_10131_ATP_Olambda__440,axiom,
    ! [Uu: set_list_nat,Uua: nat,Uub: set_list_nat] :
      ( aa_set_list_nat_o(aa_nat7721107121239158477_nat_o(aTP_Lamm_akw(Uu),Uua),Uub)
    <=> ( ord_le6045566169113846134st_nat(Uub,Uu)
        & ( finite_card_list_nat(Uub) = Uua ) ) ) ).

% ATP.lambda_440
tff(fact_10132_ATP_Olambda__441,axiom,
    ! [Uu: set_set_nat,Uua: nat,Uub: set_set_nat] :
      ( aa_set_set_nat_o(aa_nat5160323483289836711_nat_o(aTP_Lamm_akv(Uu),Uua),Uub)
    <=> ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(Uub),Uu)
        & ( finite_card_set_nat(Uub) = Uua ) ) ) ).

% ATP.lambda_441
tff(fact_10133_ATP_Olambda__442,axiom,
    ! [Uu: set_nat,Uua: nat,Uub: set_nat] :
      ( aa_set_nat_o(aa_nat_fun_set_nat_o(aTP_Lamm_akx(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_442
tff(fact_10134_ATP_Olambda__443,axiom,
    ! [Uu: set_int,Uua: nat,Uub: set_int] :
      ( aa_set_int_o(aa_nat_fun_set_int_o(aTP_Lamm_ala(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_443
tff(fact_10135_ATP_Olambda__444,axiom,
    ! [Uu: set_nat,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_alz(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(member_nat(Uub),Uu)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uub),aa_nat_nat(suc,Uua)) ) ) ).

% ATP.lambda_444
tff(fact_10136_ATP_Olambda__445,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_abm(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_445
tff(fact_10137_ATP_Olambda__446,axiom,
    ! [Uu: assn,Uua: assn,Uub: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(aTP_Lamm_ep(Uu),Uua),Uub)
    <=> ( aa_Pro653082453696054616_nat_o(rep_assn(Uu),Uub)
        | aa_Pro653082453696054616_nat_o(rep_assn(Uua),Uub) ) ) ).

% ATP.lambda_446
tff(fact_10138_ATP_Olambda__447,axiom,
    ! [Uu: set_Pr8693737435421807431at_nat,Uua: set_Pr8693737435421807431at_nat,Uub: produc859450856879609959at_nat] :
      ( aa_Pro602569247023830573_nat_o(aa_set6735692237302977855_nat_o(aTP_Lamm_cb(Uu),Uua),Uub)
    <=> ( member8206827879206165904at_nat(Uub,Uu)
        | member8206827879206165904at_nat(Uub,Uua) ) ) ).

% ATP.lambda_447
tff(fact_10139_ATP_Olambda__448,axiom,
    ! [Uu: set_Pr8551490117392284871at_nat,Uua: set_Pr8551490117392284871at_nat,Uub: produc4166570645942440679at_nat] :
      ( aa_Pro8143006071014924973_nat_o(aa_set1219845518824332607_nat_o(aTP_Lamm_cc(Uu),Uua),Uub)
    <=> ( member6689249552917799696at_nat(Uub,Uu)
        | member6689249552917799696at_nat(Uub,Uua) ) ) ).

% ATP.lambda_448
tff(fact_10140_ATP_Olambda__449,axiom,
    ! [Uu: set_Pr4329608150637261639at_nat,Uua: set_Pr4329608150637261639at_nat,Uub: produc3843707927480180839at_nat] :
      ( aa_Pro5947980084102686253_nat_o(aa_set6353399069419443519_nat_o(aTP_Lamm_cd(Uu),Uua),Uub)
    <=> ( member8757157785044589968at_nat(Uub,Uu)
        | member8757157785044589968at_nat(Uub,Uua) ) ) ).

% ATP.lambda_449
tff(fact_10141_ATP_Olambda__450,axiom,
    ! [Uu: set_Pr1261947904930325089at_nat,Uua: set_Pr1261947904930325089at_nat,Uub: product_prod_nat_nat] :
      ( aa_Pro6150637604445964947_nat_o(aa_set1857767853180004543_nat_o(aTP_Lamm_bv(Uu),Uua),Uub)
    <=> ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Uub),Uu)
        | aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Uub),Uua) ) ) ).

% ATP.lambda_450
tff(fact_10142_ATP_Olambda__451,axiom,
    ! [Uu: set_Pr958786334691620121nt_int,Uua: set_Pr958786334691620121nt_int,Uub: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aa_set2224860389584585407_int_o(aTP_Lamm_bx(Uu),Uua),Uub)
    <=> ( member5262025264175285858nt_int(Uub,Uu)
        | member5262025264175285858nt_int(Uub,Uua) ) ) ).

% ATP.lambda_451
tff(fact_10143_ATP_Olambda__452,axiom,
    ! [Uu: set_list_nat,Uua: set_list_nat,Uub: list_nat] :
      ( aa_list_nat_o(aa_set6675653781276780861_nat_o(aTP_Lamm_by(Uu),Uua),Uub)
    <=> ( member_list_nat(Uub,Uu)
        | member_list_nat(Uub,Uua) ) ) ).

% ATP.lambda_452
tff(fact_10144_ATP_Olambda__453,axiom,
    ! [Uu: set_o,Uua: set_o,Uub: $o] :
      ( aa_o_o(aa_set_o_fun_o_o(aTP_Lamm_bw(Uu),Uua),(Uub))
    <=> ( member_o((Uub),Uu)
        | member_o((Uub),Uua) ) ) ).

% ATP.lambda_453
tff(fact_10145_ATP_Olambda__454,axiom,
    ! [Uu: set_set_nat,Uua: set_set_nat,Uub: set_nat] :
      ( aa_set_nat_o(aa_set4404905041378927473_nat_o(aTP_Lamm_bz(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_454
tff(fact_10146_ATP_Olambda__455,axiom,
    ! [Uu: set_nat,Uua: set_nat,Uub: nat] :
      ( aa_nat_o(aa_set_nat_fun_nat_o(aTP_Lamm_ce(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(member_nat(Uub),Uu)
        | aa_set_nat_o(member_nat(Uub),Uua) ) ) ).

% ATP.lambda_455
tff(fact_10147_ATP_Olambda__456,axiom,
    ! [Uu: set_int,Uua: set_int,Uub: int] :
      ( aa_int_o(aa_set_int_fun_int_o(aTP_Lamm_ca(Uu),Uua),Uub)
    <=> ( aa_set_int_o(member_int(Uub),Uu)
        | aa_set_int_o(member_int(Uub),Uua) ) ) ).

% ATP.lambda_456
tff(fact_10148_ATP_Olambda__457,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_aia(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_457
tff(fact_10149_ATP_Olambda__458,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_acv(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_458
tff(fact_10150_ATP_Olambda__459,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_ahz(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_459
tff(fact_10151_ATP_Olambda__460,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_acy(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_460
tff(fact_10152_ATP_Olambda__461,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_acx(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_461
tff(fact_10153_ATP_Olambda__462,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_acw(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_462
tff(fact_10154_ATP_Olambda__463,axiom,
    ! [Uu: assn,Uua: assn,Uub: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(aTP_Lamm_eq(Uu),Uua),Uub)
    <=> ( aa_Pro653082453696054616_nat_o(rep_assn(Uu),Uub)
        & aa_Pro653082453696054616_nat_o(rep_assn(Uua),Uub) ) ) ).

% ATP.lambda_463
tff(fact_10155_ATP_Olambda__464,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_aog(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_464
tff(fact_10156_ATP_Olambda__465,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_anq(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_465
tff(fact_10157_ATP_Olambda__466,axiom,
    ! [Uu: set_Pr1261947904930325089at_nat,Uua: set_Pr1261947904930325089at_nat,Uub: product_prod_nat_nat] :
      ( aa_Pro6150637604445964947_nat_o(aa_set1857767853180004543_nat_o(aTP_Lamm_bo(Uu),Uua),Uub)
    <=> ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Uub),Uu)
        & aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Uub),Uua) ) ) ).

% ATP.lambda_466
tff(fact_10158_ATP_Olambda__467,axiom,
    ! [Uu: set_Pr958786334691620121nt_int,Uua: set_Pr958786334691620121nt_int,Uub: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aa_set2224860389584585407_int_o(aTP_Lamm_bj(Uu),Uua),Uub)
    <=> ( member5262025264175285858nt_int(Uub,Uu)
        & member5262025264175285858nt_int(Uub,Uua) ) ) ).

% ATP.lambda_467
tff(fact_10159_ATP_Olambda__468,axiom,
    ! [Uu: set_list_nat,Uua: set_list_nat,Uub: list_nat] :
      ( aa_list_nat_o(aa_set6675653781276780861_nat_o(aTP_Lamm_bk(Uu),Uua),Uub)
    <=> ( member_list_nat(Uub,Uu)
        & member_list_nat(Uub,Uua) ) ) ).

% ATP.lambda_468
tff(fact_10160_ATP_Olambda__469,axiom,
    ! [Uu: set_o,Uua: set_o,Uub: $o] :
      ( aa_o_o(aa_set_o_fun_o_o(aTP_Lamm_bi(Uu),Uua),(Uub))
    <=> ( member_o((Uub),Uu)
        & member_o((Uub),Uua) ) ) ).

% ATP.lambda_469
tff(fact_10161_ATP_Olambda__470,axiom,
    ! [Uu: set_set_nat,Uua: set_set_nat,Uub: set_nat] :
      ( aa_set_nat_o(aa_set4404905041378927473_nat_o(aTP_Lamm_bl(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_470
tff(fact_10162_ATP_Olambda__471,axiom,
    ! [Uu: set_nat,Uua: set_nat,Uub: nat] :
      ( aa_nat_o(aa_set_nat_fun_nat_o(aTP_Lamm_bn(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(member_nat(Uub),Uu)
        & aa_set_nat_o(member_nat(Uub),Uua) ) ) ).

% ATP.lambda_471
tff(fact_10163_ATP_Olambda__472,axiom,
    ! [Uu: set_int,Uua: set_int,Uub: int] :
      ( aa_int_o(aa_set_int_fun_int_o(aTP_Lamm_bm(Uu),Uua),Uub)
    <=> ( aa_set_int_o(member_int(Uub),Uu)
        & aa_set_int_o(member_int(Uub),Uua) ) ) ).

% ATP.lambda_472
tff(fact_10164_ATP_Olambda__473,axiom,
    ! [Uu: set_Code_integer,Uua: set_Code_integer,Uub: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(aa_set4044230130517860621eger_o(aTP_Lamm_qx(Uu),Uua),Uub)
    <=> ( minus_2355218937544613996nteger(Uub,Uu) = minus_2355218937544613996nteger(Uua,Uu) ) ) ).

% ATP.lambda_473
tff(fact_10165_ATP_Olambda__474,axiom,
    ! [Uu: set_Pr1261947904930325089at_nat,Uua: set_Pr1261947904930325089at_nat,Uub: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(aTP_Lamm_qy(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_474
tff(fact_10166_ATP_Olambda__475,axiom,
    ! [Uu: set_nat,Uua: set_nat,Uub: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(aTP_Lamm_qz(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_475
tff(fact_10167_ATP_Olambda__476,axiom,
    ! [Uu: set_int,Uua: set_int,Uub: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(aTP_Lamm_qw(Uu),Uua),Uub)
    <=> ( aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,Uub),Uu) = aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,Uua),Uu) ) ) ).

% ATP.lambda_476
tff(fact_10168_ATP_Olambda__477,axiom,
    ! [Uu: set_Code_integer,Uua: fun_Code_integer_o,Uub: code_integer] :
      ( aa_Code_integer_o(aa_fun8214818745163655228eger_o(aTP_Lamm_vm(Uu),Uua),Uub)
    <=> ( aa_set5643387332196575111eger_o(member_Code_integer(Uub),Uu)
        & aa_Code_integer_o(Uua,Uub) ) ) ).

% ATP.lambda_477
tff(fact_10169_ATP_Olambda__478,axiom,
    ! [Uu: set_Pr1261947904930325089at_nat,Uua: fun_Pr4796943543331071514_nat_o,Uub: product_prod_nat_nat] :
      ( aa_Pro6150637604445964947_nat_o(aa_fun359341089595923442_nat_o(aTP_Lamm_al(Uu),Uua),Uub)
    <=> ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Uub),Uu)
        & aa_Pro6150637604445964947_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_478
tff(fact_10170_ATP_Olambda__479,axiom,
    ! [Uu: set_Pr958786334691620121nt_int,Uua: fun_Pr3238374608701176162_int_o,Uub: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aa_fun5349570537165229698_int_o(aTP_Lamm_an(Uu),Uua),Uub)
    <=> ( member5262025264175285858nt_int(Uub,Uu)
        & aa_Pro4592068669816069595_int_o(Uua,Uub) ) ) ).

% ATP.lambda_479
tff(fact_10171_ATP_Olambda__480,axiom,
    ! [Uu: set_list_nat,Uua: fun_list_nat_o,Uub: list_nat] :
      ( aa_list_nat_o(aa_fun6155608546934989052_nat_o(aTP_Lamm_ao(Uu),Uua),Uub)
    <=> ( member_list_nat(Uub,Uu)
        & aa_list_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_480
tff(fact_10172_ATP_Olambda__481,axiom,
    ! [Uu: set_o,Uua: fun_o_o,Uub: $o] :
      ( aa_o_o(aa_fun_o_o_fun_o_o(aTP_Lamm_am(Uu),Uua),(Uub))
    <=> ( member_o((Uub),Uu)
        & aa_o_o(Uua,(Uub)) ) ) ).

% ATP.lambda_481
tff(fact_10173_ATP_Olambda__482,axiom,
    ! [Uu: set_set_nat,Uua: fun_set_nat_o,Uub: set_nat] :
      ( aa_set_nat_o(aa_fun751667557118977660_nat_o(aTP_Lamm_ap(Uu),Uua),Uub)
    <=> ( aa_set_set_nat_o(member_set_nat(Uub),Uu)
        & aa_set_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_482
tff(fact_10174_ATP_Olambda__483,axiom,
    ! [Uu: set_nat,Uua: fun_nat_o,Uub: nat] :
      ( aa_nat_o(aa_fun5937193402907943164_nat_o(aTP_Lamm_aq(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(member_nat(Uub),Uu)
        & aa_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_483
tff(fact_10175_ATP_Olambda__484,axiom,
    ! [Uu: set_int,Uua: fun_int_o,Uub: int] :
      ( aa_int_o(aa_fun6243517787477317628_int_o(aTP_Lamm_ar(Uu),Uua),Uub)
    <=> ( aa_set_int_o(member_int(Uub),Uu)
        & aa_int_o(Uua,Uub) ) ) ).

% ATP.lambda_484
tff(fact_10176_ATP_Olambda__485,axiom,
    ! [Uu: set_Code_integer,Uua: fun_Co8499238673658658103nteger,Uub: code_integer] :
      ( aa_Code_integer_o(aa_fun7159535342743012639eger_o(aTP_Lamm_ub(Uu),Uua),Uub)
    <=> ( aa_set5643387332196575111eger_o(member_Code_integer(Uub),Uu)
        & ( aa_Cod1783974652533396728nteger(Uua,Uub) != zero_z3403309356797280102nteger ) ) ) ).

% ATP.lambda_485
tff(fact_10177_ATP_Olambda__486,axiom,
    ! [Uu: set_Code_integer,Uua: fun_Code_integer_rat,Uub: code_integer] :
      ( aa_Code_integer_o(aa_fun6395433285121891940eger_o(aTP_Lamm_uj(Uu),Uua),Uub)
    <=> ( aa_set5643387332196575111eger_o(member_Code_integer(Uub),Uu)
        & ( aa_Code_integer_rat(Uua,Uub) != zero_zero_rat ) ) ) ).

% ATP.lambda_486
tff(fact_10178_ATP_Olambda__487,axiom,
    ! [Uu: set_o,Uua: fun_o_Code_integer,Uub: $o] :
      ( aa_o_o(aa_fun8192762245619933327un_o_o(aTP_Lamm_tv(Uu),Uua),(Uub))
    <=> ( member_o((Uub),Uu)
        & ( aa_o_Code_integer(Uua,(Uub)) != zero_z3403309356797280102nteger ) ) ) ).

% ATP.lambda_487
tff(fact_10179_ATP_Olambda__488,axiom,
    ! [Uu: set_o,Uua: fun_o_rat,Uub: $o] :
      ( aa_o_o(aa_fun_o_rat_fun_o_o(aTP_Lamm_ud(Uu),Uua),(Uub))
    <=> ( member_o((Uub),Uu)
        & ( aa_o_rat(Uua,(Uub)) != zero_zero_rat ) ) ) ).

% ATP.lambda_488
tff(fact_10180_ATP_Olambda__489,axiom,
    ! [Uu: set_o,Uua: fun_o_nat,Uub: $o] :
      ( aa_o_o(aa_fun_o_nat_fun_o_o(aTP_Lamm_ul(Uu),Uua),(Uub))
    <=> ( member_o((Uub),Uu)
        & ( aa_o_nat(Uua,(Uub)) != zero_zero_nat ) ) ) ).

% ATP.lambda_489
tff(fact_10181_ATP_Olambda__490,axiom,
    ! [Uu: set_nat,Uua: fun_nat_Code_integer,Uub: nat] :
      ( aa_nat_o(aa_fun3480067038916266681_nat_o(aTP_Lamm_tx(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(member_nat(Uub),Uu)
        & ( aa_nat_Code_integer(Uua,Uub) != zero_z3403309356797280102nteger ) ) ) ).

% ATP.lambda_490
tff(fact_10182_ATP_Olambda__491,axiom,
    ! [Uu: set_nat,Uua: fun_nat_rat,Uub: nat] :
      ( aa_nat_o(aa_fun1118977351124560894_nat_o(aTP_Lamm_uf(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(member_nat(Uub),Uu)
        & ( aa_nat_rat(Uua,Uub) != zero_zero_rat ) ) ) ).

% ATP.lambda_491
tff(fact_10183_ATP_Olambda__492,axiom,
    ! [Uu: set_nat,Uua: fun_nat_nat,Uub: nat] :
      ( aa_nat_o(aa_fun8673969331427483654_nat_o(aTP_Lamm_un(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(member_nat(Uub),Uu)
        & ( aa_nat_nat(Uua,Uub) != zero_zero_nat ) ) ) ).

% ATP.lambda_492
tff(fact_10184_ATP_Olambda__493,axiom,
    ! [Uu: set_int,Uua: fun_int_Code_integer,Uub: int] :
      ( aa_int_o(aa_fun3336537925265328129_int_o(aTP_Lamm_tz(Uu),Uua),Uub)
    <=> ( aa_set_int_o(member_int(Uub),Uu)
        & ( aa_int_Code_integer(Uua,Uub) != zero_z3403309356797280102nteger ) ) ) ).

% ATP.lambda_493
tff(fact_10185_ATP_Olambda__494,axiom,
    ! [Uu: set_int,Uua: fun_int_rat,Uub: int] :
      ( aa_int_o(aa_fun7094158214180239686_int_o(aTP_Lamm_uh(Uu),Uua),Uub)
    <=> ( aa_set_int_o(member_int(Uub),Uu)
        & ( aa_int_rat(Uua,Uub) != zero_zero_rat ) ) ) ).

% ATP.lambda_494
tff(fact_10186_ATP_Olambda__495,axiom,
    ! [Uu: set_Code_integer,Uua: fun_Co8499238673658658103nteger,Uub: code_integer] :
      ( aa_Code_integer_o(aa_fun7159535342743012639eger_o(aTP_Lamm_uv(Uu),Uua),Uub)
    <=> ( aa_set5643387332196575111eger_o(member_Code_integer(Uub),Uu)
        & ( aa_Cod1783974652533396728nteger(Uua,Uub) != one_one_Code_integer ) ) ) ).

% ATP.lambda_495
tff(fact_10187_ATP_Olambda__496,axiom,
    ! [Uu: set_Code_integer,Uua: fun_Co4490034186337852034r_assn,Uub: code_integer] :
      ( aa_Code_integer_o(aa_fun6538703224075513620eger_o(aTP_Lamm_vd(Uu),Uua),Uub)
    <=> ( aa_set5643387332196575111eger_o(member_Code_integer(Uub),Uu)
        & ( aa_Code_integer_assn(Uua,Uub) != one_one_assn ) ) ) ).

% ATP.lambda_496
tff(fact_10188_ATP_Olambda__497,axiom,
    ! [Uu: set_o,Uua: fun_o_Code_integer,Uub: $o] :
      ( aa_o_o(aa_fun8192762245619933327un_o_o(aTP_Lamm_up(Uu),Uua),(Uub))
    <=> ( member_o((Uub),Uu)
        & ( aa_o_Code_integer(Uua,(Uub)) != one_one_Code_integer ) ) ) ).

% ATP.lambda_497
tff(fact_10189_ATP_Olambda__498,axiom,
    ! [Uu: set_o,Uua: fun_o_assn,Uub: $o] :
      ( aa_o_o(aa_fun8382565201914911066un_o_o(aTP_Lamm_ux(Uu),Uua),(Uub))
    <=> ( member_o((Uub),Uu)
        & ( aa_o_assn(Uua,(Uub)) != one_one_assn ) ) ) ).

% ATP.lambda_498
tff(fact_10190_ATP_Olambda__499,axiom,
    ! [Uu: set_o,Uua: fun_o_rat,Uub: $o] :
      ( aa_o_o(aa_fun_o_rat_fun_o_o(aTP_Lamm_vf(Uu),Uua),(Uub))
    <=> ( member_o((Uub),Uu)
        & ( aa_o_rat(Uua,(Uub)) != one_one_rat ) ) ) ).

% ATP.lambda_499
tff(fact_10191_ATP_Olambda__500,axiom,
    ! [Uu: set_nat,Uua: fun_nat_Code_integer,Uub: nat] :
      ( aa_nat_o(aa_fun3480067038916266681_nat_o(aTP_Lamm_ur(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(member_nat(Uub),Uu)
        & ( aa_nat_Code_integer(Uua,Uub) != one_one_Code_integer ) ) ) ).

% ATP.lambda_500
tff(fact_10192_ATP_Olambda__501,axiom,
    ! [Uu: set_nat,Uua: fun_nat_assn,Uub: nat] :
      ( aa_nat_o(aa_fun7183233715851567790_nat_o(aTP_Lamm_uz(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(member_nat(Uub),Uu)
        & ( aa_nat_assn(Uua,Uub) != one_one_assn ) ) ) ).

% ATP.lambda_501
tff(fact_10193_ATP_Olambda__502,axiom,
    ! [Uu: set_nat,Uua: fun_nat_rat,Uub: nat] :
      ( aa_nat_o(aa_fun1118977351124560894_nat_o(aTP_Lamm_vh(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(member_nat(Uub),Uu)
        & ( aa_nat_rat(Uua,Uub) != one_one_rat ) ) ) ).

% ATP.lambda_502
tff(fact_10194_ATP_Olambda__503,axiom,
    ! [Uu: set_int,Uua: fun_int_Code_integer,Uub: int] :
      ( aa_int_o(aa_fun3336537925265328129_int_o(aTP_Lamm_ut(Uu),Uua),Uub)
    <=> ( aa_set_int_o(member_int(Uub),Uu)
        & ( aa_int_Code_integer(Uua,Uub) != one_one_Code_integer ) ) ) ).

% ATP.lambda_503
tff(fact_10195_ATP_Olambda__504,axiom,
    ! [Uu: set_int,Uua: fun_int_assn,Uub: int] :
      ( aa_int_o(aa_fun202081646129719286_int_o(aTP_Lamm_vb(Uu),Uua),Uub)
    <=> ( aa_set_int_o(member_int(Uub),Uu)
        & ( aa_int_assn(Uua,Uub) != one_one_assn ) ) ) ).

% ATP.lambda_504
tff(fact_10196_ATP_Olambda__505,axiom,
    ! [Uu: set_Code_integer,Uua: fun_Co8499238673658658103nteger,Uub: code_integer] :
      ( aa_Code_integer_o(aa_fun7159535342743012639eger_o(aTP_Lamm_ajr(Uu),Uua),Uub)
    <=> ( aa_set5643387332196575111eger_o(member_Code_integer(Uub),Uu)
        & ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(Uua,Uub)) ) ) ).

% ATP.lambda_505
tff(fact_10197_ATP_Olambda__506,axiom,
    ! [Uu: set_Code_integer,Uua: fun_Code_integer_nat,Uub: code_integer] :
      ( aa_Code_integer_o(aa_fun4054834855839773292eger_o(aTP_Lamm_ajk(Uu),Uua),Uub)
    <=> ( aa_set5643387332196575111eger_o(member_Code_integer(Uub),Uu)
        & ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_Code_integer_nat(Uua,Uub)) ) ) ).

% ATP.lambda_506
tff(fact_10198_ATP_Olambda__507,axiom,
    ! [Uu: set_Code_integer,Uua: fun_Code_integer_int,Uub: code_integer] :
      ( aa_Code_integer_o(aa_fun3864128940607509264eger_o(aTP_Lamm_ajn(Uu),Uua),Uub)
    <=> ( aa_set5643387332196575111eger_o(member_Code_integer(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_Code_integer_int(Uua,Uub)) ) ) ).

% ATP.lambda_507
tff(fact_10199_ATP_Olambda__508,axiom,
    ! [Uu: set_o,Uua: fun_o_Code_integer,Uub: $o] :
      ( aa_o_o(aa_fun8192762245619933327un_o_o(aTP_Lamm_ajo(Uu),Uua),(Uub))
    <=> ( 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_508
tff(fact_10200_ATP_Olambda__509,axiom,
    ! [Uu: set_o,Uua: fun_o_nat,Uub: $o] :
      ( aa_o_o(aa_fun_o_nat_fun_o_o(aTP_Lamm_aji(Uu),Uua),(Uub))
    <=> ( member_o((Uub),Uu)
        & ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_o_nat(Uua,(Uub))) ) ) ).

% ATP.lambda_509
tff(fact_10201_ATP_Olambda__510,axiom,
    ! [Uu: set_o,Uua: fun_o_int,Uub: $o] :
      ( aa_o_o(aa_fun_o_int_fun_o_o(aTP_Lamm_ajl(Uu),Uua),(Uub))
    <=> ( 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_510
tff(fact_10202_ATP_Olambda__511,axiom,
    ! [Uu: set_nat,Uua: fun_nat_Code_integer,Uub: nat] :
      ( aa_nat_o(aa_fun3480067038916266681_nat_o(aTP_Lamm_ajp(Uu),Uua),Uub)
    <=> ( aa_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_511
tff(fact_10203_ATP_Olambda__512,axiom,
    ! [Uu: set_nat,Uua: fun_nat_int,Uub: nat] :
      ( aa_nat_o(aa_fun2180201864171524266_nat_o(aTP_Lamm_ajm(Uu),Uua),Uub)
    <=> ( aa_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_512
tff(fact_10204_ATP_Olambda__513,axiom,
    ! [Uu: set_int,Uua: fun_int_Code_integer,Uub: int] :
      ( aa_int_o(aa_fun3336537925265328129_int_o(aTP_Lamm_ajq(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_513
tff(fact_10205_ATP_Olambda__514,axiom,
    ! [Uu: set_int,Uua: fun_int_nat,Uub: int] :
      ( aa_int_o(aa_fun5425778157628386638_int_o(aTP_Lamm_ajj(Uu),Uua),Uub)
    <=> ( aa_set_int_o(member_int(Uub),Uu)
        & ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_int_nat(Uua,Uub)) ) ) ).

% ATP.lambda_514
tff(fact_10206_ATP_Olambda__515,axiom,
    ! [Uu: set_Pr1261947904930325089at_nat,Uua: set_Pr1261947904930325089at_nat,Uub: product_prod_nat_nat] :
      ( aa_Pro6150637604445964947_nat_o(aa_set1857767853180004543_nat_o(aTP_Lamm_gk(Uu),Uua),Uub)
    <=> ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Uub),Uu)
        & ~ aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Uub),Uua) ) ) ).

% ATP.lambda_515
tff(fact_10207_ATP_Olambda__516,axiom,
    ! [Uu: set_Pr958786334691620121nt_int,Uua: set_Pr958786334691620121nt_int,Uub: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aa_set2224860389584585407_int_o(aTP_Lamm_gm(Uu),Uua),Uub)
    <=> ( member5262025264175285858nt_int(Uub,Uu)
        & ~ member5262025264175285858nt_int(Uub,Uua) ) ) ).

% ATP.lambda_516
tff(fact_10208_ATP_Olambda__517,axiom,
    ! [Uu: set_list_nat,Uua: set_list_nat,Uub: list_nat] :
      ( aa_list_nat_o(aa_set6675653781276780861_nat_o(aTP_Lamm_gn(Uu),Uua),Uub)
    <=> ( member_list_nat(Uub,Uu)
        & ~ member_list_nat(Uub,Uua) ) ) ).

% ATP.lambda_517
tff(fact_10209_ATP_Olambda__518,axiom,
    ! [Uu: set_o,Uua: set_o,Uub: $o] :
      ( aa_o_o(aa_set_o_fun_o_o(aTP_Lamm_gl(Uu),Uua),(Uub))
    <=> ( member_o((Uub),Uu)
        & ~ member_o((Uub),Uua) ) ) ).

% ATP.lambda_518
tff(fact_10210_ATP_Olambda__519,axiom,
    ! [Uu: set_set_nat,Uua: set_set_nat,Uub: set_nat] :
      ( aa_set_nat_o(aa_set4404905041378927473_nat_o(aTP_Lamm_go(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_519
tff(fact_10211_ATP_Olambda__520,axiom,
    ! [Uu: set_nat,Uua: set_nat,Uub: nat] :
      ( aa_nat_o(aa_set_nat_fun_nat_o(aTP_Lamm_gq(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(member_nat(Uub),Uu)
        & ~ aa_set_nat_o(member_nat(Uub),Uua) ) ) ).

% ATP.lambda_520
tff(fact_10212_ATP_Olambda__521,axiom,
    ! [Uu: set_int,Uua: set_int,Uub: int] :
      ( aa_int_o(aa_set_int_fun_int_o(aTP_Lamm_gp(Uu),Uua),Uub)
    <=> ( aa_set_int_o(member_int(Uub),Uu)
        & ~ aa_set_int_o(member_int(Uub),Uua) ) ) ).

% ATP.lambda_521
tff(fact_10213_ATP_Olambda__522,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_ane(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_522
tff(fact_10214_ATP_Olambda__523,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_agx(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_523
tff(fact_10215_ATP_Olambda__524,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_aic(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_524
tff(fact_10216_ATP_Olambda__525,axiom,
    ! [Uu: rat,Uua: rat,Uub: rat] : aa_rat_rat(aa_rat_fun_rat_rat(aTP_Lamm_hn(Uu),Uua),Uub) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,Uub),Uu)),Uua) ).

% ATP.lambda_525
tff(fact_10217_ATP_Olambda__526,axiom,
    ! [Uu: rat,Uua: rat,Uub: rat] : aa_rat_rat(aa_rat_fun_rat_rat(aTP_Lamm_hp(Uu),Uua),Uub) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Uu),Uub)),Uua) ).

% ATP.lambda_526
tff(fact_10218_ATP_Olambda__527,axiom,
    ! [Uu: rat,Uua: rat,Uub: rat] : aa_rat_rat(aa_rat_fun_rat_rat(aTP_Lamm_ho(Uu),Uua),Uub) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,Uub),Uu)),Uua) ).

% ATP.lambda_527
tff(fact_10219_ATP_Olambda__528,axiom,
    ! [Uu: rat,Uua: rat,Uub: rat] : aa_rat_rat(aa_rat_fun_rat_rat(aTP_Lamm_hq(Uu),Uua),Uub) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Uu),Uub)),Uua) ).

% ATP.lambda_528
tff(fact_10220_ATP_Olambda__529,axiom,
    ! [Uu: set_Pr8693737435421807431at_nat,Uua: product_prod_nat_nat,Uub: product_prod_nat_nat] :
      ( aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(aTP_Lamm_ay(Uu),Uua),Uub)
    <=> member8206827879206165904at_nat(produc6161850002892822231at_nat(Uua,Uub),Uu) ) ).

% ATP.lambda_529
tff(fact_10221_ATP_Olambda__530,axiom,
    ! [Uu: set_Pr8551490117392284871at_nat,Uua: multis2468970476368604999at_nat,Uub: multis2468970476368604999at_nat] :
      ( aa_mul869302825175964237_nat_o(aa_mul3916465897682844511_nat_o(aTP_Lamm_az(Uu),Uua),Uub)
    <=> member6689249552917799696at_nat(produc4348348721325984599at_nat(Uua,Uub),Uu) ) ).

% ATP.lambda_530
tff(fact_10222_ATP_Olambda__531,axiom,
    ! [Uu: set_Pr4329608150637261639at_nat,Uua: set_Pr1261947904930325089at_nat,Uub: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(aTP_Lamm_ba(Uu),Uua),Uub)
    <=> member8757157785044589968at_nat(produc2922128104949294807at_nat(Uua,Uub),Uu) ) ).

% ATP.lambda_531
tff(fact_10223_ATP_Olambda__532,axiom,
    ! [Uu: set_Pr1261947904930325089at_nat,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_as(Uu),Uua),Uub)
    <=> aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Uua),Uub)),Uu) ) ).

% ATP.lambda_532
tff(fact_10224_ATP_Olambda__533,axiom,
    ! [Uu: set_Pr1312406793078187350et_nat,Uua: fun_Pr6460944699146269935_nat_o,Uub: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(aa_fun2638059832795814268_nat_o(aTP_Lamm_at(Uu),Uua),Uub)
    <=> member4513763875858039863et_nat(aa_Pro434340453207536480et_nat(aa_fun3926903090414666066et_nat(produc6017135503889078682et_nat,Uua),Uub),Uu) ) ).

% ATP.lambda_533
tff(fact_10225_ATP_Olambda__534,axiom,
    ! [Uu: set_Pr4040394263813967220et_nat,Uua: fun_Pr6460944699146269935_nat_o,Uub: produc8799533983703829792et_nat] :
      ( aa_Pro5643009357358181178_nat_o(aa_fun912680268133416158_nat_o(aTP_Lamm_au(Uu),Uua),Uub)
    <=> member9174161442226468181et_nat(aa_Pro1979504083478021276et_nat(aa_fun615918787490101902et_nat(produc7291855603632349880et_nat,Uua),Uub),Uu) ) ).

% ATP.lambda_534
tff(fact_10226_ATP_Olambda__535,axiom,
    ! [Uu: set_Pr6700895561065273913nt_int,Uua: fun_Pr7763680713840462443e_term,Uub: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aa_fun3538162476766722827_int_o(aTP_Lamm_av(Uu),Uua),Uub)
    <=> member753276645541486746nt_int(aa_Pro5748361610964203648nt_int(aa_fun3265472288063244902nt_int(produc7320969046608631221nt_int,Uua),Uub),Uu) ) ).

% ATP.lambda_535
tff(fact_10227_ATP_Olambda__536,axiom,
    ! [Uu: set_Pr4166680992137330533nt_int,Uua: fun_in5730379987311621695e_term,Uub: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aa_fun586359091965693151_int_o(aTP_Lamm_aw(Uu),Uua),Uub)
    <=> member5890090881338784710nt_int(aa_Pro8800328608558162860nt_int(aa_fun260840925927807678nt_int(produc4729913460834754785nt_int,Uua),Uub),Uu) ) ).

% ATP.lambda_536
tff(fact_10228_ATP_Olambda__537,axiom,
    ! [Uu: set_Pr7098220151150636591it_nat,Uua: a,Uub: produc6653097349344004940it_nat] :
      ( aa_Pro1202363709425262478_nat_o(aa_a_f337466037417904483_nat_o(aTP_Lamm_ax(Uu),Uua),Uub)
    <=> member4554412811331277712it_nat(aa_Pro8078911840264235907it_nat(aa_a_f2234720956197727492it_nat(produc9178034014595674355it_nat,Uua),Uub),Uu) ) ).

% ATP.lambda_537
tff(fact_10229_ATP_Olambda__538,axiom,
    ! [Uu: rat,Uua: rat,Uub: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(aTP_Lamm_xu(Uu),Uua),Uub)
    <=> ( member_rat(Uub,ring_1_Ints_rat)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Uu),Uub)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Uub),Uua) ) ) ).

% ATP.lambda_538
tff(fact_10230_ATP_Olambda__539,axiom,
    ! [Uu: fun_nat_o,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_rb(Uu),Uua),Uub)
    <=> ( aa_nat_o(Uu,Uub)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uub),Uua) ) ) ).

% ATP.lambda_539
tff(fact_10231_ATP_Olambda__540,axiom,
    ! [Uu: fun_nat_nat,Uua: fun_nat_nat,Uub: nat] : aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_nd(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_540
tff(fact_10232_ATP_Olambda__541,axiom,
    ! [Uu: fun_nat_int,Uua: fun_nat_int,Uub: nat] : aa_nat_int(aa_fun5565043122120104496at_int(aTP_Lamm_ne(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_541
tff(fact_10233_ATP_Olambda__542,axiom,
    ! [Uu: fun_int_int,Uua: fun_int_int,Uub: int] : aa_int_int(aa_fun6280345759651049192nt_int(aTP_Lamm_nf(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_542
tff(fact_10234_ATP_Olambda__543,axiom,
    ! [Uu: fun_o_Code_integer,Uua: fun_o_Code_integer,Uub: $o] : aa_o_Code_integer(aa_fun8520278569782659900nteger(aTP_Lamm_ku(Uu),Uua),(Uub)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_o_Code_integer(Uua,(Uub))),aa_o_Code_integer(Uu,(Uub))) ).

% ATP.lambda_543
tff(fact_10235_ATP_Olambda__544,axiom,
    ! [Uu: fun_o_rat,Uua: fun_o_rat,Uub: $o] : aa_o_rat(aa_fun4336163974892599420_o_rat(aTP_Lamm_kx(Uu),Uua),(Uub)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_o_rat(Uua,(Uub))),aa_o_rat(Uu,(Uub))) ).

% ATP.lambda_544
tff(fact_10236_ATP_Olambda__545,axiom,
    ! [Uu: fun_o_int,Uua: fun_o_int,Uub: $o] : aa_o_int(aa_fun4727766026823762812_o_int(aTP_Lamm_la(Uu),Uua),(Uub)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_o_int(Uua,(Uub))),aa_o_int(Uu,(Uub))) ).

% ATP.lambda_545
tff(fact_10237_ATP_Olambda__546,axiom,
    ! [Uu: fun_se7586849273513198688nteger,Uua: fun_se7586849273513198688nteger,Uub: set_nat] : aa_set871585252387937313nteger(aa_fun4587287018503165694nteger(aTP_Lamm_ld(Uu),Uua),Uub) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_set871585252387937313nteger(Uua,Uub)),aa_set871585252387937313nteger(Uu,Uub)) ).

% ATP.lambda_546
tff(fact_10238_ATP_Olambda__547,axiom,
    ! [Uu: fun_nat_Code_integer,Uua: fun_nat_Code_integer,Uub: nat] : aa_nat_Code_integer(aa_fun589076786913273746nteger(aTP_Lamm_kv(Uu),Uua),Uub) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(Uua,Uub)),aa_nat_Code_integer(Uu,Uub)) ).

% ATP.lambda_547
tff(fact_10239_ATP_Olambda__548,axiom,
    ! [Uu: fun_nat_rat,Uua: fun_nat_rat,Uub: nat] : aa_nat_rat(aa_fun765758790203053448at_rat(aTP_Lamm_ky(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_548
tff(fact_10240_ATP_Olambda__549,axiom,
    ! [Uu: fun_nat_int,Uua: fun_nat_int,Uub: nat] : aa_nat_int(aa_fun5565043122120104496at_int(aTP_Lamm_lb(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_549
tff(fact_10241_ATP_Olambda__550,axiom,
    ! [Uu: fun_int_Code_integer,Uua: fun_int_Code_integer,Uub: int] : aa_int_Code_integer(aa_fun5304498221708291146nteger(aTP_Lamm_kw(Uu),Uua),Uub) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_int_Code_integer(Uua,Uub)),aa_int_Code_integer(Uu,Uub)) ).

% ATP.lambda_550
tff(fact_10242_ATP_Olambda__551,axiom,
    ! [Uu: fun_int_rat,Uua: fun_int_rat,Uub: int] : aa_int_rat(aa_fun1481061427733998144nt_rat(aTP_Lamm_kz(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_551
tff(fact_10243_ATP_Olambda__552,axiom,
    ! [Uu: fun_int_int,Uua: fun_int_int,Uub: int] : aa_int_int(aa_fun6280345759651049192nt_int(aTP_Lamm_lc(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_552
tff(fact_10244_ATP_Olambda__553,axiom,
    ! [Uu: fun_int_int,Uua: fun_int_int,Uub: int] : aa_int_int(aa_fun6280345759651049192nt_int(aTP_Lamm_lz(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_553
tff(fact_10245_ATP_Olambda__554,axiom,
    ! [Uu: fun_Pr2602117396471556196at_nat,Uua: fun_Pr2602117396471556196at_nat,Uub: product_prod_nat_nat] : aa_Pro4243173770044397973at_nat(aa_fun2351638177499875132at_nat(aTP_Lamm_le(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_Pro4243173770044397973at_nat(Uua,Uub)),aa_Pro4243173770044397973at_nat(Uu,Uub)) ).

% ATP.lambda_554
tff(fact_10246_ATP_Olambda__555,axiom,
    ! [Uu: fun_o_nat,Uua: fun_o_nat,Uub: $o] : aa_o_nat(aa_fun6176012476734143100_o_nat(aTP_Lamm_lf(Uu),Uua),(Uub)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_o_nat(Uua,(Uub))),aa_o_nat(Uu,(Uub))) ).

% ATP.lambda_555
tff(fact_10247_ATP_Olambda__556,axiom,
    ! [Uu: fun_set_nat_nat,Uua: fun_set_nat_nat,Uub: set_nat] : aa_set_nat_nat(aa_fun3386010016858254564at_nat(aTP_Lamm_lg(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_556
tff(fact_10248_ATP_Olambda__557,axiom,
    ! [Uu: fun_nat_nat,Uua: fun_nat_nat,Uub: nat] : aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_li(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_557
tff(fact_10249_ATP_Olambda__558,axiom,
    ! [Uu: fun_int_nat,Uua: fun_int_nat,Uub: int] : aa_int_nat(aa_fun6506752150744536624nt_nat(aTP_Lamm_lh(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_558
tff(fact_10250_ATP_Olambda__559,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(plus_plus_nat,aa_nat_nat(Uu,Uub)),aa_nat_nat(Uua,Uub)) ).

% ATP.lambda_559
tff(fact_10251_ATP_Olambda__560,axiom,
    ! [Uu: fun_int_int,Uua: fun_int_int,Uub: int] : aa_int_int(aa_fun6280345759651049192nt_int(aTP_Lamm_lq(Uu),Uua),Uub) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(Uu,Uub)),aa_int_int(Uua,Uub)) ).

% ATP.lambda_560
tff(fact_10252_ATP_Olambda__561,axiom,
    ! [Uu: fun_Pr363642936170290612_nat_o,Uua: fun_Pr363642936170290612_nat_o,Uub: produc859450856879609959at_nat] :
      ( aa_Pro602569247023830573_nat_o(aa_fun6414004499577345958_nat_o(aTP_Lamm_gv(Uu),Uua),Uub)
    <=> ( aa_Pro602569247023830573_nat_o(Uu,Uub)
       => aa_Pro602569247023830573_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_561
tff(fact_10253_ATP_Olambda__562,axiom,
    ! [Uu: fun_Pr1099658227000185396_nat_o,Uua: fun_Pr1099658227000185396_nat_o,Uub: produc4166570645942440679at_nat] :
      ( aa_Pro8143006071014924973_nat_o(aa_fun9125434449681309094_nat_o(aTP_Lamm_gw(Uu),Uua),Uub)
    <=> ( aa_Pro8143006071014924973_nat_o(Uu,Uub)
       => aa_Pro8143006071014924973_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_562
tff(fact_10254_ATP_Olambda__563,axiom,
    ! [Uu: fun_Pr3059359313727237556_nat_o,Uua: fun_Pr3059359313727237556_nat_o,Uub: produc3843707927480180839at_nat] :
      ( aa_Pro5947980084102686253_nat_o(aa_fun2566793329686429606_nat_o(aTP_Lamm_gx(Uu),Uua),Uub)
    <=> ( aa_Pro5947980084102686253_nat_o(Uu,Uub)
       => aa_Pro5947980084102686253_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_563
tff(fact_10255_ATP_Olambda__564,axiom,
    ! [Uu: fun_Pr3238374608701176162_int_o,Uua: fun_Pr3238374608701176162_int_o,Uub: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aa_fun5349570537165229698_int_o(aTP_Lamm_gr(Uu),Uua),Uub)
    <=> ( aa_Pro4592068669816069595_int_o(Uu,Uub)
       => aa_Pro4592068669816069595_int_o(Uua,Uub) ) ) ).

% ATP.lambda_564
tff(fact_10256_ATP_Olambda__565,axiom,
    ! [Uu: fun_list_nat_o,Uua: fun_list_nat_o,Uub: list_nat] :
      ( aa_list_nat_o(aa_fun6155608546934989052_nat_o(aTP_Lamm_gs(Uu),Uua),Uub)
    <=> ( aa_list_nat_o(Uu,Uub)
       => aa_list_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_565
tff(fact_10257_ATP_Olambda__566,axiom,
    ! [Uu: fun_set_nat_o,Uua: fun_set_nat_o,Uub: set_nat] :
      ( aa_set_nat_o(aa_fun751667557118977660_nat_o(aTP_Lamm_gt(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(Uu,Uub)
       => aa_set_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_566
tff(fact_10258_ATP_Olambda__567,axiom,
    ! [Uu: fun_nat_o,Uua: fun_nat_o,Uub: nat] :
      ( aa_nat_o(aa_fun5937193402907943164_nat_o(aTP_Lamm_gy(Uu),Uua),Uub)
    <=> ( aa_nat_o(Uu,Uub)
       => aa_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_567
tff(fact_10259_ATP_Olambda__568,axiom,
    ! [Uu: fun_int_o,Uua: fun_int_o,Uub: int] :
      ( aa_int_o(aa_fun6243517787477317628_int_o(aTP_Lamm_gu(Uu),Uua),Uub)
    <=> ( aa_int_o(Uu,Uub)
       => aa_int_o(Uua,Uub) ) ) ).

% ATP.lambda_568
tff(fact_10260_ATP_Olambda__569,axiom,
    ! [Uu: fun_Code_integer_o,Uua: fun_Code_integer_o,Uub: code_integer] :
      ( aa_Code_integer_o(aa_fun8214818745163655228eger_o(aTP_Lamm_act(Uu),Uua),Uub)
    <=> ( aa_Code_integer_o(Uu,Uub)
        | aa_Code_integer_o(Uua,Uub) ) ) ).

% ATP.lambda_569
tff(fact_10261_ATP_Olambda__570,axiom,
    ! [Uu: fun_Pr363642936170290612_nat_o,Uua: fun_Pr363642936170290612_nat_o,Uub: produc859450856879609959at_nat] :
      ( aa_Pro602569247023830573_nat_o(aa_fun6414004499577345958_nat_o(aTP_Lamm_cj(Uu),Uua),Uub)
    <=> ( aa_Pro602569247023830573_nat_o(Uu,Uub)
        | aa_Pro602569247023830573_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_570
tff(fact_10262_ATP_Olambda__571,axiom,
    ! [Uu: fun_Pr1099658227000185396_nat_o,Uua: fun_Pr1099658227000185396_nat_o,Uub: produc4166570645942440679at_nat] :
      ( aa_Pro8143006071014924973_nat_o(aa_fun9125434449681309094_nat_o(aTP_Lamm_ck(Uu),Uua),Uub)
    <=> ( aa_Pro8143006071014924973_nat_o(Uu,Uub)
        | aa_Pro8143006071014924973_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_571
tff(fact_10263_ATP_Olambda__572,axiom,
    ! [Uu: fun_Pr3059359313727237556_nat_o,Uua: fun_Pr3059359313727237556_nat_o,Uub: produc3843707927480180839at_nat] :
      ( aa_Pro5947980084102686253_nat_o(aa_fun2566793329686429606_nat_o(aTP_Lamm_cl(Uu),Uua),Uub)
    <=> ( aa_Pro5947980084102686253_nat_o(Uu,Uub)
        | aa_Pro5947980084102686253_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_572
tff(fact_10264_ATP_Olambda__573,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_acu(Uu),Uua),Uub)
    <=> ( aa_Pro6150637604445964947_nat_o(Uu,Uub)
        | aa_Pro6150637604445964947_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_573
tff(fact_10265_ATP_Olambda__574,axiom,
    ! [Uu: fun_Pr3238374608701176162_int_o,Uua: fun_Pr3238374608701176162_int_o,Uub: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aa_fun5349570537165229698_int_o(aTP_Lamm_cf(Uu),Uua),Uub)
    <=> ( aa_Pro4592068669816069595_int_o(Uu,Uub)
        | aa_Pro4592068669816069595_int_o(Uua,Uub) ) ) ).

% ATP.lambda_574
tff(fact_10266_ATP_Olambda__575,axiom,
    ! [Uu: fun_list_nat_o,Uua: fun_list_nat_o,Uub: list_nat] :
      ( aa_list_nat_o(aa_fun6155608546934989052_nat_o(aTP_Lamm_cg(Uu),Uua),Uub)
    <=> ( aa_list_nat_o(Uu,Uub)
        | aa_list_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_575
tff(fact_10267_ATP_Olambda__576,axiom,
    ! [Uu: fun_set_nat_o,Uua: fun_set_nat_o,Uub: set_nat] :
      ( aa_set_nat_o(aa_fun751667557118977660_nat_o(aTP_Lamm_ch(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(Uu,Uub)
        | aa_set_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_576
tff(fact_10268_ATP_Olambda__577,axiom,
    ! [Uu: fun_nat_o,Uua: fun_nat_o,Uub: nat] :
      ( aa_nat_o(aa_fun5937193402907943164_nat_o(aTP_Lamm_cm(Uu),Uua),Uub)
    <=> ( aa_nat_o(Uu,Uub)
        | aa_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_577
tff(fact_10269_ATP_Olambda__578,axiom,
    ! [Uu: fun_int_o,Uua: fun_int_o,Uub: int] :
      ( aa_int_o(aa_fun6243517787477317628_int_o(aTP_Lamm_ci(Uu),Uua),Uub)
    <=> ( aa_int_o(Uu,Uub)
        | aa_int_o(Uua,Uub) ) ) ).

% ATP.lambda_578
tff(fact_10270_ATP_Olambda__579,axiom,
    ! [Uu: fun_Code_integer_o,Uua: fun_Code_integer_o,Uub: code_integer] :
      ( aa_Code_integer_o(aa_fun8214818745163655228eger_o(aTP_Lamm_acs(Uu),Uua),Uub)
    <=> ( aa_Code_integer_o(Uu,Uub)
        & aa_Code_integer_o(Uua,Uub) ) ) ).

% ATP.lambda_579
tff(fact_10271_ATP_Olambda__580,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_bu(Uu),Uua),Uub)
    <=> ( aa_Pro6150637604445964947_nat_o(Uu,Uub)
        & aa_Pro6150637604445964947_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_580
tff(fact_10272_ATP_Olambda__581,axiom,
    ! [Uu: fun_Pr3238374608701176162_int_o,Uua: fun_Pr3238374608701176162_int_o,Uub: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aa_fun5349570537165229698_int_o(aTP_Lamm_bp(Uu),Uua),Uub)
    <=> ( aa_Pro4592068669816069595_int_o(Uu,Uub)
        & aa_Pro4592068669816069595_int_o(Uua,Uub) ) ) ).

% ATP.lambda_581
tff(fact_10273_ATP_Olambda__582,axiom,
    ! [Uu: fun_list_nat_o,Uua: fun_list_nat_o,Uub: list_nat] :
      ( aa_list_nat_o(aa_fun6155608546934989052_nat_o(aTP_Lamm_bq(Uu),Uua),Uub)
    <=> ( aa_list_nat_o(Uu,Uub)
        & aa_list_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_582
tff(fact_10274_ATP_Olambda__583,axiom,
    ! [Uu: fun_set_nat_o,Uua: fun_set_nat_o,Uub: set_nat] :
      ( aa_set_nat_o(aa_fun751667557118977660_nat_o(aTP_Lamm_br(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(Uu,Uub)
        & aa_set_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_583
tff(fact_10275_ATP_Olambda__584,axiom,
    ! [Uu: fun_nat_o,Uua: fun_nat_o,Uub: nat] :
      ( aa_nat_o(aa_fun5937193402907943164_nat_o(aTP_Lamm_bt(Uu),Uua),Uub)
    <=> ( aa_nat_o(Uu,Uub)
        & aa_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_584
tff(fact_10276_ATP_Olambda__585,axiom,
    ! [Uu: fun_int_o,Uua: fun_int_o,Uub: int] :
      ( aa_int_o(aa_fun6243517787477317628_int_o(aTP_Lamm_bs(Uu),Uua),Uub)
    <=> ( aa_int_o(Uu,Uub)
        & aa_int_o(Uua,Uub) ) ) ).

% ATP.lambda_585
tff(fact_10277_ATP_Olambda__586,axiom,
    ! [Uu: fun_Co8499238673658658103nteger,Uua: fun_Code_integer_o,Uub: code_integer] : aa_Cod1783974652533396728nteger(aa_fun791458909507501263nteger(aTP_Lamm_afg(Uu),Uua),Uub) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(Uu,Uub)),aa_o_Code_integer(zero_n356916108424825756nteger,aa_Code_integer_o(Uua,Uub))) ).

% ATP.lambda_586
tff(fact_10278_ATP_Olambda__587,axiom,
    ! [Uu: fun_Code_integer_rat,Uua: fun_Code_integer_o,Uub: code_integer] : aa_Code_integer_rat(aa_fun25512833367352522er_rat(aTP_Lamm_afj(Uu),Uua),Uub) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_Code_integer_rat(Uu,Uub)),aa_o_rat(zero_n2052037380579107095ol_rat,aa_Code_integer_o(Uua,Uub))) ).

% ATP.lambda_587
tff(fact_10279_ATP_Olambda__588,axiom,
    ! [Uu: fun_Code_integer_nat,Uua: fun_Code_integer_o,Uub: code_integer] : aa_Code_integer_nat(aa_fun3533741391760749250er_nat(aTP_Lamm_afm(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_Code_integer_nat(Uu,Uub)),aa_o_nat(zero_n2687167440665602831ol_nat,aa_Code_integer_o(Uua,Uub))) ).

% ATP.lambda_588
tff(fact_10280_ATP_Olambda__589,axiom,
    ! [Uu: fun_Code_integer_int,Uua: fun_Code_integer_o,Uub: code_integer] : aa_Code_integer_int(aa_fun8579262409106328350er_int(aTP_Lamm_afn(Uu),Uua),Uub) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_Code_integer_int(Uu,Uub)),aa_o_int(zero_n2684676970156552555ol_int,aa_Code_integer_o(Uua,Uub))) ).

% ATP.lambda_589
tff(fact_10281_ATP_Olambda__590,axiom,
    ! [Uu: fun_nat_Code_integer,Uua: fun_nat_o,Uub: nat] : aa_nat_Code_integer(aa_fun3969837373441324047nteger(aTP_Lamm_afh(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_590
tff(fact_10282_ATP_Olambda__591,axiom,
    ! [Uu: fun_nat_rat,Uua: fun_nat_o,Uub: nat] : aa_nat_rat(aa_fun696632391729880074at_rat(aTP_Lamm_afk(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_591
tff(fact_10283_ATP_Olambda__592,axiom,
    ! [Uu: fun_nat_int,Uua: fun_nat_o,Uub: nat] : aa_nat_int(aa_fun27009930614080094at_int(aTP_Lamm_afo(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_592
tff(fact_10284_ATP_Olambda__593,axiom,
    ! [Uu: fun_int_Code_integer,Uua: fun_int_o,Uub: int] : aa_int_Code_integer(aa_fun6019004165459918095nteger(aTP_Lamm_aff(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_593
tff(fact_10285_ATP_Olambda__594,axiom,
    ! [Uu: fun_int_rat,Uua: fun_int_o,Uub: int] : aa_int_rat(aa_fun268837275264525578nt_rat(aTP_Lamm_afi(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_594
tff(fact_10286_ATP_Olambda__595,axiom,
    ! [Uu: fun_int_nat,Uua: fun_int_o,Uub: int] : aa_int_nat(aa_fun3777065833657922306nt_nat(aTP_Lamm_afl(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_595
tff(fact_10287_ATP_Olambda__596,axiom,
    ! [Uu: fun_int_option_int,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_ahf(Uu),Uua),Uub)
    <=> ( aa_int_option_int(Uu,Uua) = aa_int_option_int(some_int,Uub) ) ) ).

% ATP.lambda_596
tff(fact_10288_ATP_Olambda__597,axiom,
    ! [Uu: fun_nat_fun_nat_nat,Uua: set_nat,Uub: nat] : aa_nat_nat(aa_set6537093552120046177at_nat(aTP_Lamm_mo(Uu),Uua),Uub) = aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(Uu,Uub)),Uua) ).

% ATP.lambda_597
tff(fact_10289_ATP_Olambda__598,axiom,
    ! [Uu: fun_nat_fun_nat_int,Uua: set_nat,Uub: nat] : aa_nat_int(aa_set2359242532610849469at_int(aTP_Lamm_mq(Uu),Uua),Uub) = aa_set_nat_int(groups705719431365010083at_int(aa_nat_fun_nat_int(Uu,Uub)),Uua) ).

% ATP.lambda_598
tff(fact_10290_ATP_Olambda__599,axiom,
    ! [Uu: fun_nat_fun_int_int,Uua: set_int,Uub: nat] : aa_nat_int(aa_set6310687161026558745at_int(aTP_Lamm_mt(Uu),Uua),Uub) = groups1705073143266064639nt_int(aa_nat_fun_int_int(Uu,Uub),Uua) ).

% ATP.lambda_599
tff(fact_10291_ATP_Olambda__600,axiom,
    ! [Uu: fun_int_fun_nat_int,Uua: set_nat,Uub: int] : aa_int_int(aa_set3358596244511904025nt_int(aTP_Lamm_mw(Uu),Uua),Uub) = aa_set_nat_int(groups705719431365010083at_int(aa_int_fun_nat_int(Uu,Uub)),Uua) ).

% ATP.lambda_600
tff(fact_10292_ATP_Olambda__601,axiom,
    ! [Uu: fun_int_fun_int_int,Uua: set_int,Uub: int] : aa_int_int(aa_set7310040872927613301nt_int(aTP_Lamm_mz(Uu),Uua),Uub) = groups1705073143266064639nt_int(aa_int_fun_int_int(Uu,Uub),Uua) ).

% ATP.lambda_601
tff(fact_10293_ATP_Olambda__602,axiom,
    ! [Uu: fun_nat_fun_nat_nat,Uua: set_nat,Uub: nat] : aa_nat_nat(aa_set6537093552120046177at_nat(aTP_Lamm_lj(Uu),Uua),Uub) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(Uu,Uub)),Uua) ).

% ATP.lambda_602
tff(fact_10294_ATP_Olambda__603,axiom,
    ! [Uu: fun_int_fun_int_int,Uua: set_int,Uub: int] : aa_int_int(aa_set7310040872927613301nt_int(aTP_Lamm_lm(Uu),Uua),Uub) = groups4538972089207619220nt_int(aa_int_fun_int_int(Uu,Uub),Uua) ).

% ATP.lambda_603
tff(fact_10295_ATP_Olambda__604,axiom,
    ! [Uu: fun_nat_nat,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_rc(Uu),Uua),Uub)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(Uu,Uub)),Uua) ) ).

% ATP.lambda_604
tff(fact_10296_ATP_Olambda__605,axiom,
    ! [Uu: fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_jv(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(Uu,Uub)),Uua) ).

% ATP.lambda_605
tff(fact_10297_ATP_Olambda__606,axiom,
    ! [Uu: fun_nat_int,Uua: int,Uub: nat] : aa_nat_int(aa_int_fun_nat_int(aTP_Lamm_nj(Uu),Uua),Uub) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_nat_int(Uu,Uub)),Uua) ).

% ATP.lambda_606
tff(fact_10298_ATP_Olambda__607,axiom,
    ! [Uu: fun_int_int,Uua: int,Uub: int] : aa_int_int(aa_int_fun_int_int(aTP_Lamm_jw(Uu),Uua),Uub) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(Uu,Uub)),Uua) ).

% ATP.lambda_607
tff(fact_10299_ATP_Olambda__608,axiom,
    ! [Uu: fun_Co3734316457315071362atural,Uua: code_natural,Uub: code_integer] : aa_Cod607569804290114243atural(aa_Cod1476545362447692776atural(aTP_Lamm_aev(Uu),Uua),Uub) = divide5121882707175180666atural(aa_Cod607569804290114243atural(Uu,Uub),Uua) ).

% ATP.lambda_608
tff(fact_10300_ATP_Olambda__609,axiom,
    ! [Uu: fun_Co8499238673658658103nteger,Uua: code_integer,Uub: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(aTP_Lamm_aem(Uu),Uua),Uub) = aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(Uu,Uub)),Uua) ).

% ATP.lambda_609
tff(fact_10301_ATP_Olambda__610,axiom,
    ! [Uu: fun_Code_integer_nat,Uua: nat,Uub: code_integer] : aa_Code_integer_nat(aa_nat5852537043073606456er_nat(aTP_Lamm_aea(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_Code_integer_nat(Uu,Uub)),Uua) ).

% ATP.lambda_610
tff(fact_10302_ATP_Olambda__611,axiom,
    ! [Uu: fun_Code_integer_int,Uua: int,Uub: code_integer] : aa_Code_integer_int(aa_int1390634949194299888er_int(aTP_Lamm_aed(Uu),Uua),Uub) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_Code_integer_int(Uu,Uub)),Uua) ).

% ATP.lambda_611
tff(fact_10303_ATP_Olambda__612,axiom,
    ! [Uu: fun_nat_Code_natural,Uua: code_natural,Uub: nat] : aa_nat_Code_natural(aa_Cod5320533616963873627atural(aTP_Lamm_aey(Uu),Uua),Uub) = divide5121882707175180666atural(aa_nat_Code_natural(Uu,Uub),Uua) ).

% ATP.lambda_612
tff(fact_10304_ATP_Olambda__613,axiom,
    ! [Uu: fun_nat_Code_integer,Uua: code_integer,Uub: nat] : aa_nat_Code_integer(aa_Cod1578151519121547077nteger(aTP_Lamm_aep(Uu),Uua),Uub) = aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_nat_Code_integer(Uu,Uub)),Uua) ).

% ATP.lambda_613
tff(fact_10305_ATP_Olambda__614,axiom,
    ! [Uu: fun_nat_int,Uua: int,Uub: nat] : aa_nat_int(aa_int_fun_nat_int(aTP_Lamm_aeg(Uu),Uua),Uub) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_nat_int(Uu,Uub)),Uua) ).

% ATP.lambda_614
tff(fact_10306_ATP_Olambda__615,axiom,
    ! [Uu: fun_int_Code_natural,Uua: code_natural,Uub: int] : aa_int_Code_natural(aa_Cod5692187177200718263atural(aTP_Lamm_aes(Uu),Uua),Uub) = divide5121882707175180666atural(aa_int_Code_natural(Uu,Uub),Uua) ).

% ATP.lambda_615
tff(fact_10307_ATP_Olambda__616,axiom,
    ! [Uu: fun_int_Code_integer,Uua: code_integer,Uub: int] : aa_int_Code_integer(aa_Cod1949805079358391713nteger(aTP_Lamm_aej(Uu),Uua),Uub) = aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_int_Code_integer(Uu,Uub)),Uua) ).

% ATP.lambda_616
tff(fact_10308_ATP_Olambda__617,axiom,
    ! [Uu: fun_int_nat,Uua: nat,Uub: int] : aa_int_nat(aa_nat_fun_int_nat(aTP_Lamm_adx(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_int_nat(Uu,Uub)),Uua) ).

% ATP.lambda_617
tff(fact_10309_ATP_Olambda__618,axiom,
    ! [Uu: fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_lv(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(Uu,Uub)),Uua) ).

% ATP.lambda_618
tff(fact_10310_ATP_Olambda__619,axiom,
    ! [Uu: fun_int_int,Uua: int,Uub: int] : aa_int_int(aa_int_fun_int_int(aTP_Lamm_lw(Uu),Uua),Uub) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(Uu,Uub)),Uua) ).

% ATP.lambda_619
tff(fact_10311_ATP_Olambda__620,axiom,
    ! [Uu: fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_ng(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(Uu,Uub)),Uua) ).

% ATP.lambda_620
tff(fact_10312_ATP_Olambda__621,axiom,
    ! [Uu: fun_nat_int,Uua: nat,Uub: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_nh(Uu),Uua),Uub) = aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_nat_int(Uu,Uub)),Uua) ).

% ATP.lambda_621
tff(fact_10313_ATP_Olambda__622,axiom,
    ! [Uu: fun_int_int,Uua: nat,Uub: int] : aa_int_int(aa_nat_fun_int_int(aTP_Lamm_ni(Uu),Uua),Uub) = aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(Uu,Uub)),Uua) ).

% ATP.lambda_622
tff(fact_10314_ATP_Olambda__623,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aTP_Lamm_xv(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_623
tff(fact_10315_ATP_Olambda__624,axiom,
    ! [Uu: fun_Code_integer_o,Uua: fun_Co8499238673658658103nteger,Uub: code_integer] : aa_Cod1783974652533396728nteger(aa_fun4435975922079263404nteger(aTP_Lamm_afq(Uu),Uua),Uub) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_o_Code_integer(zero_n356916108424825756nteger,aa_Code_integer_o(Uu,Uub))),aa_Cod1783974652533396728nteger(Uua,Uub)) ).

% ATP.lambda_624
tff(fact_10316_ATP_Olambda__625,axiom,
    ! [Uu: fun_Code_integer_o,Uua: fun_Code_integer_rat,Uub: code_integer] : aa_Code_integer_rat(aa_fun8169474305930283170er_rat(aTP_Lamm_aft(Uu),Uua),Uub) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_o_rat(zero_n2052037380579107095ol_rat,aa_Code_integer_o(Uu,Uub))),aa_Code_integer_rat(Uua,Uub)) ).

% ATP.lambda_625
tff(fact_10317_ATP_Olambda__626,axiom,
    ! [Uu: fun_Code_integer_o,Uua: fun_Code_integer_nat,Uub: code_integer] : aa_Code_integer_nat(aa_fun3076867170244285586er_nat(aTP_Lamm_afw(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_o_nat(zero_n2687167440665602831ol_nat,aa_Code_integer_o(Uu,Uub))),aa_Code_integer_nat(Uua,Uub)) ).

% ATP.lambda_626
tff(fact_10318_ATP_Olambda__627,axiom,
    ! [Uu: fun_Code_integer_o,Uua: fun_Code_integer_int,Uub: code_integer] : aa_Code_integer_int(aa_fun3291479149765983562er_int(aTP_Lamm_afx(Uu),Uua),Uub) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_o_int(zero_n2684676970156552555ol_int,aa_Code_integer_o(Uu,Uub))),aa_Code_integer_int(Uua,Uub)) ).

% ATP.lambda_627
tff(fact_10319_ATP_Olambda__628,axiom,
    ! [Uu: fun_nat_o,Uua: fun_nat_Code_integer,Uub: nat] : aa_nat_Code_integer(aa_fun589076786913273746nteger(aTP_Lamm_afr(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_10320_ATP_Olambda__629,axiom,
    ! [Uu: fun_nat_o,Uua: fun_nat_rat,Uub: nat] : aa_nat_rat(aa_fun765758790203053448at_rat(aTP_Lamm_afu(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_10321_ATP_Olambda__630,axiom,
    ! [Uu: fun_nat_o,Uua: fun_nat_int,Uub: nat] : aa_nat_int(aa_fun5565043122120104496at_int(aTP_Lamm_afy(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_630
tff(fact_10322_ATP_Olambda__631,axiom,
    ! [Uu: fun_int_o,Uua: fun_int_Code_integer,Uub: int] : aa_int_Code_integer(aa_fun5304498221708291146nteger(aTP_Lamm_afp(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_631
tff(fact_10323_ATP_Olambda__632,axiom,
    ! [Uu: fun_int_o,Uua: fun_int_rat,Uub: int] : aa_int_rat(aa_fun1481061427733998144nt_rat(aTP_Lamm_afs(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_632
tff(fact_10324_ATP_Olambda__633,axiom,
    ! [Uu: fun_int_o,Uua: fun_int_nat,Uub: int] : aa_int_nat(aa_fun6506752150744536624nt_nat(aTP_Lamm_afv(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_633
tff(fact_10325_ATP_Olambda__634,axiom,
    ! [Uu: fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_aig(Uu),Uua),Uub) = aa_set_nat_nat(groups708209901874060359at_nat(Uu),aa_nat_set_nat(aa_nat1818925681583257441et_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_634
tff(fact_10326_ATP_Olambda__635,axiom,
    ! [Uu: fun_nat_int,Uua: nat,Uub: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_aih(Uu),Uua),Uub) = aa_set_nat_int(groups705719431365010083at_int(Uu),aa_nat_set_nat(aa_nat1818925681583257441et_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_635
tff(fact_10327_ATP_Olambda__636,axiom,
    ! [Uu: fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_aif(Uu),Uua),Uub) = aa_set_nat_nat(groups3542108847815614940at_nat(Uu),aa_nat_set_nat(aa_nat1818925681583257441et_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_10328_ATP_Olambda__637,axiom,
    ! [Uu: code_natural,Uua: code_natural,Uub: nat] : aa_nat_Code_natural(aa_Cod5320533616963873627atural(aTP_Lamm_abi(Uu),Uua),Uub) = plus_p4538020629002901425atural(Uu,times_2397367101498566445atural(aa_nat_Code_natural(semiri3763490453095760265atural,Uub),Uua)) ).

% ATP.lambda_637
tff(fact_10329_ATP_Olambda__638,axiom,
    ! [Uu: code_integer,Uua: code_integer,Uub: nat] : aa_nat_Code_integer(aa_Cod1578151519121547077nteger(aTP_Lamm_abk(Uu),Uua),Uub) = aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(Uu),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,Uub)),Uua)) ).

% ATP.lambda_638
tff(fact_10330_ATP_Olambda__639,axiom,
    ! [Uu: rat,Uua: rat,Uub: nat] : aa_nat_rat(aa_rat_fun_nat_rat(aTP_Lamm_abh(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_639
tff(fact_10331_ATP_Olambda__640,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_abl(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_640
tff(fact_10332_ATP_Olambda__641,axiom,
    ! [Uu: int,Uua: int,Uub: nat] : aa_nat_int(aa_int_fun_nat_int(aTP_Lamm_abj(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_641
tff(fact_10333_ATP_Olambda__642,axiom,
    ! [Uu: code_integer,Uua: nat,Uub: nat] : aa_nat_Code_integer(aa_nat2010205248571845816nteger(aTP_Lamm_air(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_642
tff(fact_10334_ATP_Olambda__643,axiom,
    ! [Uu: rat,Uua: nat,Uub: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aTP_Lamm_ais(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_643
tff(fact_10335_ATP_Olambda__644,axiom,
    ! [Uu: int,Uua: nat,Uub: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_ait(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_644
tff(fact_10336_ATP_Olambda__645,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_645
tff(fact_10337_ATP_Olambda__646,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_646
tff(fact_10338_ATP_Olambda__647,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_647
tff(fact_10339_ATP_Olambda__648,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_abg(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_648
tff(fact_10340_ATP_Olambda__649,axiom,
    ! [Uu: nat,Uua: fun_nat_nat,Uub: nat] : aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_lx(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uu),aa_nat_nat(Uua,Uub)) ).

% ATP.lambda_649
tff(fact_10341_ATP_Olambda__650,axiom,
    ! [Uu: int,Uua: fun_int_int,Uub: int] : aa_int_int(aa_fun6280345759651049192nt_int(aTP_Lamm_ly(Uu),Uua),Uub) = aa_int_int(aa_int_fun_int_int(times_times_int,Uu),aa_int_int(Uua,Uub)) ).

% ATP.lambda_650
tff(fact_10342_ATP_Olambda__651,axiom,
    ! [Uu: nat,Uua: fun_nat_nat,Uub: nat] : aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_nn(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Uu),aa_nat_nat(Uua,Uub)) ).

% ATP.lambda_651
tff(fact_10343_ATP_Olambda__652,axiom,
    ! [Uu: int,Uua: fun_nat_nat,Uub: nat] : aa_nat_int(aa_fun1613598493704395220at_int(aTP_Lamm_no(Uu),Uua),Uub) = aa_nat_int(aa_int_fun_nat_int(power_power_int,Uu),aa_nat_nat(Uua,Uub)) ).

% ATP.lambda_652
tff(fact_10344_ATP_Olambda__653,axiom,
    ! [Uu: int,Uua: fun_int_nat,Uub: int] : aa_int_int(aa_fun2328901131235339916nt_int(aTP_Lamm_np(Uu),Uua),Uub) = aa_nat_int(aa_int_fun_nat_int(power_power_int,Uu),aa_int_nat(Uua,Uub)) ).

% ATP.lambda_653
tff(fact_10345_ATP_Olambda__654,axiom,
    ! [Uu: fun_Co3734316457315071362atural,Uua: code_natural,Uub: code_integer] :
      ( aa_Code_integer_o(aa_Cod403035634466734574eger_o(aTP_Lamm_aew(Uu),Uua),Uub)
    <=> aa_Code_natural_o(dvd_dvd_Code_natural(Uua),aa_Cod607569804290114243atural(Uu,Uub)) ) ).

% ATP.lambda_654
tff(fact_10346_ATP_Olambda__655,axiom,
    ! [Uu: fun_Co8499238673658658103nteger,Uua: code_integer,Uub: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(aTP_Lamm_aen(Uu),Uua),Uub)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Uua),aa_Cod1783974652533396728nteger(Uu,Uub)) ) ).

% ATP.lambda_655
tff(fact_10347_ATP_Olambda__656,axiom,
    ! [Uu: fun_Code_integer_nat,Uua: nat,Uub: code_integer] :
      ( aa_Code_integer_o(aa_nat7420681472272698950eger_o(aTP_Lamm_aeb(Uu),Uua),Uub)
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Uua),aa_Code_integer_nat(Uu,Uub)) ) ).

% ATP.lambda_656
tff(fact_10348_ATP_Olambda__657,axiom,
    ! [Uu: fun_Code_integer_int,Uua: int,Uub: code_integer] :
      ( aa_Code_integer_o(aa_int5993532643906289898eger_o(aTP_Lamm_aee(Uu),Uua),Uub)
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Uua),aa_Code_integer_int(Uu,Uub)) ) ).

% ATP.lambda_657
tff(fact_10349_ATP_Olambda__658,axiom,
    ! [Uu: fun_nat_Code_natural,Uua: code_natural,Uub: nat] :
      ( aa_nat_o(aa_Cod9120328075896114107_nat_o(aTP_Lamm_aez(Uu),Uua),Uub)
    <=> aa_Code_natural_o(dvd_dvd_Code_natural(Uua),aa_nat_Code_natural(Uu,Uub)) ) ).

% ATP.lambda_658
tff(fact_10350_ATP_Olambda__659,axiom,
    ! [Uu: fun_nat_Code_integer,Uua: code_integer,Uub: nat] :
      ( aa_nat_o(aa_Cod1075288278386115654_nat_o(aTP_Lamm_aeq(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_10351_ATP_Olambda__660,axiom,
    ! [Uu: fun_nat_int,Uua: int,Uub: nat] :
      ( aa_nat_o(aa_int_fun_nat_o(aTP_Lamm_aeh(Uu),Uua),Uub)
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Uua),aa_nat_int(Uu,Uub)) ) ).

% ATP.lambda_660
tff(fact_10352_ATP_Olambda__661,axiom,
    ! [Uu: fun_int_Code_natural,Uua: code_natural,Uub: int] :
      ( aa_int_o(aa_Cod7299285730463426143_int_o(aTP_Lamm_aet(Uu),Uua),Uub)
    <=> aa_Code_natural_o(dvd_dvd_Code_natural(Uua),aa_int_Code_natural(Uu,Uub)) ) ).

% ATP.lambda_661
tff(fact_10353_ATP_Olambda__662,axiom,
    ! [Uu: fun_int_Code_integer,Uua: code_integer,Uub: int] :
      ( aa_int_o(aa_Cod8477617969808203498_int_o(aTP_Lamm_aek(Uu),Uua),Uub)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Uua),aa_int_Code_integer(Uu,Uub)) ) ).

% ATP.lambda_662
tff(fact_10354_ATP_Olambda__663,axiom,
    ! [Uu: fun_int_nat,Uua: nat,Uub: int] :
      ( aa_int_o(aa_nat_fun_int_o(aTP_Lamm_ady(Uu),Uua),Uub)
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Uua),aa_int_nat(Uu,Uub)) ) ).

% ATP.lambda_663
tff(fact_10355_ATP_Olambda__664,axiom,
    ! [Uu: code_integer,Uua: nat,Uub: nat] : aa_nat_Code_integer(aa_nat2010205248571845816nteger(aTP_Lamm_op(Uu),Uua),Uub) = aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(Uu),aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uub))) ).

% ATP.lambda_664
tff(fact_10356_ATP_Olambda__665,axiom,
    ! [Uu: rat,Uua: nat,Uub: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aTP_Lamm_oo(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_665
tff(fact_10357_ATP_Olambda__666,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_oq(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_666
tff(fact_10358_ATP_Olambda__667,axiom,
    ! [Uu: int,Uua: nat,Uub: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_or(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_667
tff(fact_10359_ATP_Olambda__668,axiom,
    ! [Uu: fun_Code_integer_nat,Uua: code_integer,Uub: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(aTP_Lamm_ym(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_Code_integer_nat(Uu,Uub)),aa_Code_integer_nat(Uu,Uub))) ) ).

% ATP.lambda_668
tff(fact_10360_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_yn(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_10361_ATP_Olambda__670,axiom,
    ! [Uu: fun_Pr5160692531401422620nt_nat,Uua: product_prod_int_int,Uub: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aa_Pro3952998155787061599_int_o(aTP_Lamm_yh(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_Pro6801748904974264397nt_nat(Uu,Uub)),aa_Pro6801748904974264397nt_nat(Uu,Uub))) ) ).

% ATP.lambda_670
tff(fact_10362_ATP_Olambda__671,axiom,
    ! [Uu: fun_list_nat_nat,Uua: list_nat,Uub: list_nat] :
      ( aa_list_nat_o(aa_lis557685189327716851_nat_o(aTP_Lamm_yi(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_list_nat_nat(Uu,Uub)),aa_list_nat_nat(Uu,Uub))) ) ).

% ATP.lambda_671
tff(fact_10363_ATP_Olambda__672,axiom,
    ! [Uu: fun_set_nat_nat,Uua: set_nat,Uub: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(aTP_Lamm_yj(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_672
tff(fact_10364_ATP_Olambda__673,axiom,
    ! [Uu: fun_nat_nat,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_yk(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_673
tff(fact_10365_ATP_Olambda__674,axiom,
    ! [Uu: fun_int_nat,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_yl(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_674
tff(fact_10366_ATP_Olambda__675,axiom,
    ! [Uu: fun_Code_integer_nat,Uua: fun_Code_integer_o,Uub: code_integer] :
      ( aa_Code_integer_o(aa_fun8214818745163655228eger_o(aTP_Lamm_wf(Uu),Uua),Uub)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),
          $ite(aa_Code_integer_o(Uua,Uub),aa_Code_integer_nat(Uu,Uub),zero_zero_nat)) ) ).

% ATP.lambda_675
tff(fact_10367_ATP_Olambda__676,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_wh(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_676
tff(fact_10368_ATP_Olambda__677,axiom,
    ! [Uu: fun_Pr5160692531401422620nt_nat,Uua: fun_Pr3238374608701176162_int_o,Uub: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aa_fun5349570537165229698_int_o(aTP_Lamm_vv(Uu),Uua),Uub)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),
          $ite(aa_Pro4592068669816069595_int_o(Uua,Uub),aa_Pro6801748904974264397nt_nat(Uu,Uub),zero_zero_nat)) ) ).

% ATP.lambda_677
tff(fact_10369_ATP_Olambda__678,axiom,
    ! [Uu: fun_list_nat_nat,Uua: fun_list_nat_o,Uub: list_nat] :
      ( aa_list_nat_o(aa_fun6155608546934989052_nat_o(aTP_Lamm_vx(Uu),Uua),Uub)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),
          $ite(aa_list_nat_o(Uua,Uub),aa_list_nat_nat(Uu,Uub),zero_zero_nat)) ) ).

% ATP.lambda_678
tff(fact_10370_ATP_Olambda__679,axiom,
    ! [Uu: fun_set_nat_nat,Uua: fun_set_nat_o,Uub: set_nat] :
      ( aa_set_nat_o(aa_fun751667557118977660_nat_o(aTP_Lamm_vz(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_679
tff(fact_10371_ATP_Olambda__680,axiom,
    ! [Uu: fun_nat_nat,Uua: fun_nat_o,Uub: nat] :
      ( aa_nat_o(aa_fun5937193402907943164_nat_o(aTP_Lamm_wb(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_680
tff(fact_10372_ATP_Olambda__681,axiom,
    ! [Uu: fun_int_nat,Uua: fun_int_o,Uub: int] :
      ( aa_int_o(aa_fun6243517787477317628_int_o(aTP_Lamm_wd(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_681
tff(fact_10373_ATP_Olambda__682,axiom,
    ! [Uu: fun_Code_integer_nat,Uua: fun_Code_integer_nat,Uub: code_integer] :
      ( aa_Code_integer_o(aa_fun4054834855839773292eger_o(aTP_Lamm_zz(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_Code_integer_nat(Uu,Uub)),aa_Code_integer_nat(Uua,Uub))) ) ).

% ATP.lambda_682
tff(fact_10374_ATP_Olambda__683,axiom,
    ! [Uu: fun_Pr2602117396471556196at_nat,Uua: fun_Pr2602117396471556196at_nat,Uub: product_prod_nat_nat] :
      ( aa_Pro6150637604445964947_nat_o(aa_fun741873263805325634_nat_o(aTP_Lamm_aaa(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_683
tff(fact_10375_ATP_Olambda__684,axiom,
    ! [Uu: fun_Pr5160692531401422620nt_nat,Uua: fun_Pr5160692531401422620nt_nat,Uub: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aa_fun6302685600795359042_int_o(aTP_Lamm_zu(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_Pro6801748904974264397nt_nat(Uu,Uub)),aa_Pro6801748904974264397nt_nat(Uua,Uub))) ) ).

% ATP.lambda_684
tff(fact_10376_ATP_Olambda__685,axiom,
    ! [Uu: fun_list_nat_nat,Uua: fun_list_nat_nat,Uub: list_nat] :
      ( aa_list_nat_o(aa_fun7915899554530022_nat_o(aTP_Lamm_zv(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_list_nat_nat(Uu,Uub)),aa_list_nat_nat(Uua,Uub))) ) ).

% ATP.lambda_685
tff(fact_10377_ATP_Olambda__686,axiom,
    ! [Uu: fun_set_nat_nat,Uua: fun_set_nat_nat,Uub: set_nat] :
      ( aa_set_nat_o(aa_fun1332161875654541722_nat_o(aTP_Lamm_zw(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_686
tff(fact_10378_ATP_Olambda__687,axiom,
    ! [Uu: fun_nat_nat,Uua: fun_nat_nat,Uub: nat] :
      ( aa_nat_o(aa_fun8673969331427483654_nat_o(aTP_Lamm_zx(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_687
tff(fact_10379_ATP_Olambda__688,axiom,
    ! [Uu: fun_int_nat,Uua: fun_int_nat,Uub: int] :
      ( aa_int_o(aa_fun5425778157628386638_int_o(aTP_Lamm_zy(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_688
tff(fact_10380_ATP_Olambda__689,axiom,
    ! [Uu: fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_aid(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_689
tff(fact_10381_ATP_Olambda__690,axiom,
    ! [Uu: fun_nat_int,Uua: nat,Uub: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_aie(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_690
tff(fact_10382_ATP_Olambda__691,axiom,
    ! [Uu: fun_nat_o,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_aow(Uu),Uua),Uub)
    <=> aa_nat_o(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)) ) ).

% ATP.lambda_691
tff(fact_10383_ATP_Olambda__692,axiom,
    ! [Uu: fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_jy(Uu),Uua),Uub) = aa_nat_nat(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)) ).

% ATP.lambda_692
tff(fact_10384_ATP_Olambda__693,axiom,
    ! [Uu: fun_nat_int,Uua: nat,Uub: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_nq(Uu),Uua),Uub) = aa_nat_int(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)) ).

% ATP.lambda_693
tff(fact_10385_ATP_Olambda__694,axiom,
    ! [Uu: fun_Pr2477239776880934807at_nat,Uua: nat,Uub: nat] : aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(aTP_Lamm_agl(Uu),Uua),Uub) = aa_Pro8023792847356852296at_nat(Uu,aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Uua),Uub)) ).

% ATP.lambda_694
tff(fact_10386_ATP_Olambda__695,axiom,
    ! [Uu: fun_Pr2028404973800837934_nat_o,Uua: nat,Uub: nat] : aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(aTP_Lamm_agm(Uu),Uua),Uub) = aa_Pro989595756709598047_nat_o(Uu,aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Uua),Uub)) ).

% ATP.lambda_695
tff(fact_10387_ATP_Olambda__696,axiom,
    ! [Uu: fun_Pr2478283820680522243nt_int,Uua: int,Uub: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_agn(Uu),Uua),Uub) = aa_Pro6970504127699319548nt_int(Uu,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Uua),Uub)) ).

% ATP.lambda_696
tff(fact_10388_ATP_Olambda__697,axiom,
    ! [Uu: fun_Pr3238374608701176162_int_o,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_ago(Uu),Uua),Uub)
    <=> aa_Pro4592068669816069595_int_o(Uu,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Uua),Uub)) ) ).

% ATP.lambda_697
tff(fact_10389_ATP_Olambda__698,axiom,
    ! [Uu: fun_Pr982841511892225912nt_int,Uua: int,Uub: int] : aa_int_int(aa_int_fun_int_int(aTP_Lamm_agp(Uu),Uua),Uub) = aa_Pro6799258434465214121nt_int(Uu,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Uua),Uub)) ).

% ATP.lambda_698
tff(fact_10390_ATP_Olambda__699,axiom,
    ! [Uu: fun_op8989173174992673105on_num,Uua: fun_num_option_num,Uub: num] : aa_num_option_num(aa_fun3636977833914307392on_num(aTP_Lamm_dz(Uu),Uua),Uub) = aa_opt5369159329878751634on_num(Uu,aa_num_option_num(Uua,Uub)) ).

% ATP.lambda_699
tff(fact_10391_ATP_Olambda__700,axiom,
    ! [Uu: fun_op8989173174992673105on_num,Uua: fun_nat_option_num,Uub: nat] : aa_nat_option_num(aa_fun7716400287237121964on_num(aTP_Lamm_akn(Uu),Uua),Uub) = aa_opt5369159329878751634on_num(Uu,aa_nat_option_num(Uua,Uub)) ).

% ATP.lambda_700
tff(fact_10392_ATP_Olambda__701,axiom,
    ! [Uu: fun_option_num_o,Uua: fun_nat_option_num,Uub: nat] :
      ( aa_nat_o(aa_fun1675964276245824108_nat_o(aTP_Lamm_ako(Uu),Uua),Uub)
    <=> aa_option_num_o(Uu,aa_nat_option_num(Uua,Uub)) ) ).

% ATP.lambda_701
tff(fact_10393_ATP_Olambda__702,axiom,
    ! [Uu: fun_option_num_num,Uua: fun_num_option_num,Uub: num] : aa_num_num(aa_fun7142274290070961008um_num(aTP_Lamm_ea(Uu),Uua),Uub) = aa_option_num_num(Uu,aa_num_option_num(Uua,Uub)) ).

% ATP.lambda_702
tff(fact_10394_ATP_Olambda__703,axiom,
    ! [Uu: fun_option_num_nat,Uua: fun_nat_option_num,Uub: nat] : aa_nat_nat(aa_fun7387201323300553874at_nat(aTP_Lamm_akp(Uu),Uua),Uub) = aa_option_num_nat(Uu,aa_nat_option_num(Uua,Uub)) ).

% ATP.lambda_703
tff(fact_10395_ATP_Olambda__704,axiom,
    ! [Uu: fun_option_num_int,Uua: fun_num_option_num,Uub: num] : aa_num_int(aa_fun2256179333612926594um_int(aTP_Lamm_dy(Uu),Uua),Uub) = aa_option_num_int(Uu,aa_num_option_num(Uua,Uub)) ).

% ATP.lambda_704
tff(fact_10396_ATP_Olambda__705,axiom,
    ! [Uu: fun_o_option_num,Uua: fun_nat_o,Uub: nat] : aa_nat_option_num(aa_fun6990899178391151900on_num(aTP_Lamm_akq(Uu),Uua),Uub) = aa_o_option_num(Uu,aa_nat_o(Uua,Uub)) ).

% ATP.lambda_705
tff(fact_10397_ATP_Olambda__706,axiom,
    ! [Uu: fun_o_o,Uua: fun_nat_o,Uub: nat] :
      ( aa_nat_o(aa_fun5937193402907943164_nat_o(aTP_Lamm_akr(Uu),Uua),Uub)
    <=> aa_o_o(Uu,aa_nat_o(Uua,Uub)) ) ).

% ATP.lambda_706
tff(fact_10398_ATP_Olambda__707,axiom,
    ! [Uu: fun_o_nat,Uua: fun_nat_o,Uub: nat] : aa_nat_nat(aa_fun4204860950123276802at_nat(aTP_Lamm_aks(Uu),Uua),Uub) = aa_o_nat(Uu,aa_nat_o(Uua,Uub)) ).

% ATP.lambda_707
tff(fact_10399_ATP_Olambda__708,axiom,
    ! [Uu: fun_set_nat_nat,Uua: fun_nat_set_nat,Uub: nat] : aa_nat_nat(aa_fun4820382421437094446at_nat(aTP_Lamm_it(Uu),Uua),Uub) = aa_set_nat_nat(Uu,aa_nat_set_nat(Uua,Uub)) ).

% ATP.lambda_708
tff(fact_10400_ATP_Olambda__709,axiom,
    ! [Uu: fun_set_nat_int,Uua: fun_nat_set_nat,Uub: nat] : aa_nat_int(aa_fun642531401927897738at_int(aTP_Lamm_iu(Uu),Uua),Uub) = aa_set_nat_int(Uu,aa_nat_set_nat(Uua,Uub)) ).

% ATP.lambda_709
tff(fact_10401_ATP_Olambda__710,axiom,
    ! [Uu: fun_num_option_num,Uua: fun_num_num,Uub: num] : aa_num_option_num(aa_fun612943910911948912on_num(aTP_Lamm_ec(Uu),Uua),Uub) = aa_num_option_num(Uu,aa_num_num(Uua,Uub)) ).

% ATP.lambda_710
tff(fact_10402_ATP_Olambda__711,axiom,
    ! [Uu: fun_num_num,Uua: fun_num_num,Uub: num] : aa_num_num(aa_fun8048995283964520608um_num(aTP_Lamm_ed(Uu),Uua),Uub) = aa_num_num(Uu,aa_num_num(Uua,Uub)) ).

% ATP.lambda_711
tff(fact_10403_ATP_Olambda__712,axiom,
    ! [Uu: fun_num_int,Uua: fun_num_num,Uub: num] : aa_num_int(aa_fun3162900327506486194um_int(aTP_Lamm_eb(Uu),Uua),Uub) = aa_num_int(Uu,aa_num_num(Uua,Uub)) ).

% ATP.lambda_712
tff(fact_10404_ATP_Olambda__713,axiom,
    ! [Uu: fun_nat_option_num,Uua: fun_nat_nat,Uub: nat] : aa_nat_option_num(aa_fun6517009837249254290on_num(aTP_Lamm_akt(Uu),Uua),Uub) = aa_nat_option_num(Uu,aa_nat_nat(Uua,Uub)) ).

% ATP.lambda_713
tff(fact_10405_ATP_Olambda__714,axiom,
    ! [Uu: fun_nat_o,Uua: fun_nat_nat,Uub: nat] :
      ( aa_nat_o(aa_fun8673969331427483654_nat_o(aTP_Lamm_aku(Uu),Uua),Uub)
    <=> aa_nat_o(Uu,aa_nat_nat(Uua,Uub)) ) ).

% ATP.lambda_714
tff(fact_10406_ATP_Olambda__715,axiom,
    ! [Uu: fun_nat_nat,Uua: fun_nat_nat,Uub: nat] : aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_il(Uu),Uua),Uub) = aa_nat_nat(Uu,aa_nat_nat(Uua,Uub)) ).

% ATP.lambda_715
tff(fact_10407_ATP_Olambda__716,axiom,
    ! [Uu: fun_nat_nat,Uua: fun_int_nat,Uub: int] : aa_int_nat(aa_fun6506752150744536624nt_nat(aTP_Lamm_im(Uu),Uua),Uub) = aa_nat_nat(Uu,aa_int_nat(Uua,Uub)) ).

% ATP.lambda_716
tff(fact_10408_ATP_Olambda__717,axiom,
    ! [Uu: fun_nat_int,Uua: fun_nat_nat,Uub: nat] : aa_nat_int(aa_fun1613598493704395220at_int(aTP_Lamm_in(Uu),Uua),Uub) = aa_nat_int(Uu,aa_nat_nat(Uua,Uub)) ).

% ATP.lambda_717
tff(fact_10409_ATP_Olambda__718,axiom,
    ! [Uu: fun_nat_int,Uua: fun_int_nat,Uub: int] : aa_int_int(aa_fun2328901131235339916nt_int(aTP_Lamm_io(Uu),Uua),Uub) = aa_nat_int(Uu,aa_int_nat(Uua,Uub)) ).

% ATP.lambda_718
tff(fact_10410_ATP_Olambda__719,axiom,
    ! [Uu: fun_int_option_num,Uua: fun_num_int,Uub: num] : aa_num_option_num(aa_fun3858197967482829954on_num(aTP_Lamm_dw(Uu),Uua),Uub) = aa_int_option_num(Uu,aa_num_int(Uua,Uub)) ).

% ATP.lambda_719
tff(fact_10411_ATP_Olambda__720,axiom,
    ! [Uu: fun_int_num,Uua: fun_num_int,Uub: num] : aa_num_num(aa_fun5215869348392759218um_num(aTP_Lamm_dx(Uu),Uua),Uub) = aa_int_num(Uu,aa_num_int(Uua,Uub)) ).

% ATP.lambda_720
tff(fact_10412_ATP_Olambda__721,axiom,
    ! [Uu: fun_int_nat,Uua: fun_nat_int,Uub: nat] : aa_nat_nat(aa_fun519522104774525396at_nat(aTP_Lamm_ip(Uu),Uua),Uub) = aa_int_nat(Uu,aa_nat_int(Uua,Uub)) ).

% ATP.lambda_721
tff(fact_10413_ATP_Olambda__722,axiom,
    ! [Uu: fun_int_nat,Uua: fun_int_int,Uub: int] : aa_int_nat(aa_fun1234824742305470092nt_nat(aTP_Lamm_iq(Uu),Uua),Uub) = aa_int_nat(Uu,aa_int_int(Uua,Uub)) ).

% ATP.lambda_722
tff(fact_10414_ATP_Olambda__723,axiom,
    ! [Uu: fun_int_int,Uua: fun_num_int,Uub: num] : aa_num_int(aa_fun329774391934724804um_int(aTP_Lamm_dv(Uu),Uua),Uub) = aa_int_int(Uu,aa_num_int(Uua,Uub)) ).

% ATP.lambda_723
tff(fact_10415_ATP_Olambda__724,axiom,
    ! [Uu: fun_int_int,Uua: fun_nat_int,Uub: nat] : aa_nat_int(aa_fun5565043122120104496at_int(aTP_Lamm_ir(Uu),Uua),Uub) = aa_int_int(Uu,aa_nat_int(Uua,Uub)) ).

% ATP.lambda_724
tff(fact_10416_ATP_Olambda__725,axiom,
    ! [Uu: fun_int_int,Uua: fun_int_int,Uub: int] : aa_int_int(aa_fun6280345759651049192nt_int(aTP_Lamm_is(Uu),Uua),Uub) = aa_int_int(Uu,aa_int_int(Uua,Uub)) ).

% ATP.lambda_725
tff(fact_10417_ATP_Olambda__726,axiom,
    ! [Uu: fun_Pr1003294667014919570it_nat,Uua: produc8664842809031399944it_nat,Uub: produc8664842809031399944it_nat] : aa_Pro8996696869555642526it_nat(aa_Pro5635350972512700067it_nat(aTP_Lamm_fn(Uu),Uua),Uub) = aa_Pro8996696869555642526it_nat(some_P1914260805536162275it_nat,aa_Pro550487673543749710it_nat(aa_Pro4096753284154920275it_nat(Uu,Uua),Uub)) ).

% ATP.lambda_726
tff(fact_10418_ATP_Olambda__727,axiom,
    ! [Uu: fun_Pr7750636411365408485it_nat,Uua: produc3260487557148687353it_nat,Uub: produc3260487557148687353it_nat] : aa_Pro5302094286594872192it_nat(aa_Pro7038079567744603766it_nat(aTP_Lamm_fl(Uu),Uua),Uub) = aa_Pro5302094286594872192it_nat(some_P7913643980934408916it_nat,aa_Pro4202338132779708464it_nat(aa_Pro2263720180353132454it_nat(Uu,Uua),Uub)) ).

% ATP.lambda_727
tff(fact_10419_ATP_Olambda__728,axiom,
    ! [Uu: fun_num_fun_num_num,Uua: num,Uub: num] : aa_num_option_num(aa_num6390622462783740633on_num(aTP_Lamm_fp(Uu),Uua),Uub) = aa_num_option_num(some_num,aa_num_num(aa_num_fun_num_num(Uu,Uua),Uub)) ).

% ATP.lambda_728
tff(fact_10420_ATP_Olambda__729,axiom,
    ! [Uu: fun_Co3734316457315071362atural,Uua: code_natural,Uub: code_integer] :
      ( aa_Code_integer_o(aa_Cod403035634466734574eger_o(aTP_Lamm_aex(Uu),Uua),Uub)
    <=> ~ aa_Code_natural_o(dvd_dvd_Code_natural(Uua),aa_Cod607569804290114243atural(Uu,Uub)) ) ).

% ATP.lambda_729
tff(fact_10421_ATP_Olambda__730,axiom,
    ! [Uu: fun_Co8499238673658658103nteger,Uua: code_integer,Uub: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(aTP_Lamm_aeo(Uu),Uua),Uub)
    <=> ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Uua),aa_Cod1783974652533396728nteger(Uu,Uub)) ) ).

% ATP.lambda_730
tff(fact_10422_ATP_Olambda__731,axiom,
    ! [Uu: fun_Code_integer_nat,Uua: nat,Uub: code_integer] :
      ( aa_Code_integer_o(aa_nat7420681472272698950eger_o(aTP_Lamm_aec(Uu),Uua),Uub)
    <=> ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Uua),aa_Code_integer_nat(Uu,Uub)) ) ).

% ATP.lambda_731
tff(fact_10423_ATP_Olambda__732,axiom,
    ! [Uu: fun_Code_integer_int,Uua: int,Uub: code_integer] :
      ( aa_Code_integer_o(aa_int5993532643906289898eger_o(aTP_Lamm_aef(Uu),Uua),Uub)
    <=> ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Uua),aa_Code_integer_int(Uu,Uub)) ) ).

% ATP.lambda_732
tff(fact_10424_ATP_Olambda__733,axiom,
    ! [Uu: fun_nat_Code_natural,Uua: code_natural,Uub: nat] :
      ( aa_nat_o(aa_Cod9120328075896114107_nat_o(aTP_Lamm_afa(Uu),Uua),Uub)
    <=> ~ aa_Code_natural_o(dvd_dvd_Code_natural(Uua),aa_nat_Code_natural(Uu,Uub)) ) ).

% ATP.lambda_733
tff(fact_10425_ATP_Olambda__734,axiom,
    ! [Uu: fun_nat_Code_integer,Uua: code_integer,Uub: nat] :
      ( aa_nat_o(aa_Cod1075288278386115654_nat_o(aTP_Lamm_aer(Uu),Uua),Uub)
    <=> ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Uua),aa_nat_Code_integer(Uu,Uub)) ) ).

% ATP.lambda_734
tff(fact_10426_ATP_Olambda__735,axiom,
    ! [Uu: fun_nat_int,Uua: int,Uub: nat] :
      ( aa_nat_o(aa_int_fun_nat_o(aTP_Lamm_aei(Uu),Uua),Uub)
    <=> ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Uua),aa_nat_int(Uu,Uub)) ) ).

% ATP.lambda_735
tff(fact_10427_ATP_Olambda__736,axiom,
    ! [Uu: fun_int_Code_natural,Uua: code_natural,Uub: int] :
      ( aa_int_o(aa_Cod7299285730463426143_int_o(aTP_Lamm_aeu(Uu),Uua),Uub)
    <=> ~ aa_Code_natural_o(dvd_dvd_Code_natural(Uua),aa_int_Code_natural(Uu,Uub)) ) ).

% ATP.lambda_736
tff(fact_10428_ATP_Olambda__737,axiom,
    ! [Uu: fun_int_Code_integer,Uua: code_integer,Uub: int] :
      ( aa_int_o(aa_Cod8477617969808203498_int_o(aTP_Lamm_ael(Uu),Uua),Uub)
    <=> ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Uua),aa_int_Code_integer(Uu,Uub)) ) ).

% ATP.lambda_737
tff(fact_10429_ATP_Olambda__738,axiom,
    ! [Uu: fun_int_nat,Uua: nat,Uub: int] :
      ( aa_int_o(aa_nat_fun_int_o(aTP_Lamm_adz(Uu),Uua),Uub)
    <=> ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Uua),aa_int_nat(Uu,Uub)) ) ).

% ATP.lambda_738
tff(fact_10430_ATP_Olambda__739,axiom,
    ! [Uu: set_Code_integer,Uua: fun_Co2081214437782444048eger_o,Uub: code_integer] : aa_Code_integer_nat(aa_fun8504338775591962366er_nat(aTP_Lamm_alu(Uu),Uua),Uub) = finite4902975817058060853nteger(collect_Code_integer(aa_Cod326357343816142457eger_o(aa_fun7683620812777033532eger_o(aTP_Lamm_alt(Uu),Uua),Uub))) ).

% ATP.lambda_739
tff(fact_10431_ATP_Olambda__740,axiom,
    ! [Uu: set_o,Uua: fun_Co480733329897759773un_o_o,Uub: code_integer] : aa_Code_integer_nat(aa_fun7670077909833392133er_nat(aTP_Lamm_alq(Uu),Uua),Uub) = finite_card_o(collect_o(aa_Cod6379977994291289310un_o_o(aa_fun8978408420733827192un_o_o(aTP_Lamm_alp(Uu),Uua),Uub))) ).

% ATP.lambda_740
tff(fact_10432_ATP_Olambda__741,axiom,
    ! [Uu: set_Code_integer,Uua: fun_in4891666191172872833eger_o,Uub: int] : aa_int_nat(aa_fun511150112467848510nt_nat(aTP_Lamm_aln(Uu),Uua),Uub) = finite4902975817058060853nteger(collect_Code_integer(aa_int5993532643906289898eger_o(aa_fun5642168435487881468eger_o(aTP_Lamm_alm(Uu),Uua),Uub))) ).

% ATP.lambda_741
tff(fact_10433_ATP_Olambda__742,axiom,
    ! [Uu: set_int,Uua: fun_int_fun_int_o,Uub: int] : aa_int_nat(aa_fun4460091444687325999nt_nat(aTP_Lamm_all(Uu),Uua),Uub) = finite_card_int(collect_int(aa_int_fun_int_o(aa_fun6133342665515662012_int_o(aTP_Lamm_alk(Uu),Uua),Uub))) ).

% ATP.lambda_742
tff(fact_10434_ATP_Olambda__743,axiom,
    ! [Uu: set_o,Uua: fun_int_fun_o_o,Uub: int] : aa_int_nat(aa_fun7603103345900947235nt_nat(aTP_Lamm_alj(Uu),Uua),Uub) = finite_card_o(collect_o(aa_int_fun_o_o(aa_fun3869439686586307222un_o_o(aTP_Lamm_ali(Uu),Uua),Uub))) ).

% ATP.lambda_743
tff(fact_10435_ATP_Olambda__744,axiom,
    ! [Uu: set_Code_integer,Uua: fun_o_4021313873839786239eger_o,Uub: $o] : aa_o_nat(aa_fun5400594822745170686_o_nat(aTP_Lamm_alg(Uu),Uua),(Uub)) = finite4902975817058060853nteger(collect_Code_integer(aa_o_f8517747772616379256eger_o(aa_fun3267419118247850812eger_o(aTP_Lamm_alf(Uu),Uua),(Uub)))) ).

% ATP.lambda_744
tff(fact_10436_ATP_Olambda__745,axiom,
    ! [Uu: set_o,Uua: fun_o_fun_o_o,Uub: $o] : aa_o_nat(aa_fun8807093888949732041_o_nat(aTP_Lamm_ald(Uu),Uua),(Uub)) = finite_card_o(collect_o(aa_o_fun_o_o(aa_fun6846671739589424444un_o_o(aTP_Lamm_alc(Uu),Uua),(Uub)))) ).

% ATP.lambda_745
tff(fact_10437_ATP_Olambda__746,axiom,
    ! [Uu: set_int,Uua: fun_Co8353275306710069249_int_o,Uub: code_integer] : aa_Code_integer_nat(aa_fun726042532089134319er_nat(aTP_Lamm_alr(Uu),Uua),Uub) = finite_card_int(collect_int(aa_Cod8477617969808203498_int_o(aa_fun1593725752699181308_int_o(aTP_Lamm_sb(Uu),Uua),Uub))) ).

% ATP.lambda_746
tff(fact_10438_ATP_Olambda__747,axiom,
    ! [Uu: set_int,Uua: fun_o_fun_int_o,Uub: $o] : aa_o_nat(aa_fun2417465416094144205_o_nat(aTP_Lamm_ale(Uu),Uua),(Uub)) = finite_card_int(collect_int(aa_o_fun_int_o(aa_fun1733087058159411102_int_o(aTP_Lamm_rs(Uu),Uua),(Uub)))) ).

% ATP.lambda_747
tff(fact_10439_ATP_Olambda__748,axiom,
    ! [Uu: set_nat,Uua: fun_Co1375048237525739357_nat_o,Uub: code_integer] : aa_Code_integer_nat(aa_fun6683192525214198859er_nat(aTP_Lamm_ame(Uu),Uua),Uub) = aa_set_nat_nat(finite_card_nat,collect_nat(aa_Cod1075288278386115654_nat_o(aa_fun2430183811687954940_nat_o(aTP_Lamm_rn(Uu),Uua),Uub))) ).

% ATP.lambda_748
tff(fact_10440_ATP_Olambda__749,axiom,
    ! [Uu: set_nat,Uua: fun_int_fun_nat_o,Uub: int] : aa_int_nat(aa_fun4650797359919590027nt_nat(aTP_Lamm_amd(Uu),Uua),Uub) = aa_set_nat_nat(finite_card_nat,collect_nat(aa_int_fun_nat_o(aa_fun16395659681819580_nat_o(aTP_Lamm_ri(Uu),Uua),Uub))) ).

% ATP.lambda_749
tff(fact_10441_ATP_Olambda__750,axiom,
    ! [Uu: set_nat,Uua: fun_o_fun_nat_o,Uub: $o] : aa_o_nat(aa_fun6229755271245457521_o_nat(aTP_Lamm_alv(Uu),Uua),(Uub)) = aa_set_nat_nat(finite_card_nat,collect_nat(aa_o_fun_nat_o(aa_fun9138341750119017558_nat_o(aTP_Lamm_rd(Uu),Uua),(Uub)))) ).

% ATP.lambda_750
tff(fact_10442_ATP_Olambda__751,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_afz(Uu),Uua),Uub),Uuc) = aa_Pro262675715343573244at_nat(aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(Uu,Uub),Uuc),Uua) ).

% ATP.lambda_751
tff(fact_10443_ATP_Olambda__752,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_aga(Uu),Uua),Uub),Uuc)
    <=> aa_Pro6150637604445964947_nat_o(aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(Uu,Uub),Uuc),Uua) ) ).

% ATP.lambda_752
tff(fact_10444_ATP_Olambda__753,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_ot(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_753
tff(fact_10445_ATP_Olambda__754,axiom,
    ! [Uu: nat,Uua: fun_na2338080690716861136at_nat,Uub: fun_na2338080690716861136at_nat,Uuc: nat] :
      aa_nat2240542744625857721at_nat(aa_fun4141531684813184444at_nat(aa_fun8230936523094545239at_nat(aTP_Lamm_ou(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_754
tff(fact_10446_ATP_Olambda__755,axiom,
    ! [Uu: nat,Uua: fun_nat_rat,Uub: fun_nat_rat,Uuc: nat] :
      aa_nat_rat(aa_fun765758790203053448at_rat(aa_fun3638354034882855594at_rat(aTP_Lamm_ow(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_755
tff(fact_10447_ATP_Olambda__756,axiom,
    ! [Uu: nat,Uua: fun_nat_nat,Uub: fun_nat_nat,Uuc: nat] :
      aa_nat_nat(aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(aTP_Lamm_oy(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_756
tff(fact_10448_ATP_Olambda__757,axiom,
    ! [Uu: nat,Uua: fun_nat_int,Uub: fun_nat_int,Uuc: nat] :
      aa_nat_int(aa_fun5565043122120104496at_int(aa_fun5140708168904641446at_int(aTP_Lamm_ox(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_757
tff(fact_10449_ATP_Olambda__758,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_ns(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_758
tff(fact_10450_ATP_Olambda__759,axiom,
    ! [Uu: nat,Uua: fun_nat_assn,Uub: fun_nat_assn,Uuc: nat] :
      aa_nat_assn(aa_fun785744391571715112t_assn(aa_fun6591345337174305434t_assn(aTP_Lamm_nu(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_759
tff(fact_10451_ATP_Olambda__760,axiom,
    ! [Uu: nat,Uua: fun_nat_rat,Uub: fun_nat_rat,Uuc: nat] :
      aa_nat_rat(aa_fun765758790203053448at_rat(aa_fun3638354034882855594at_rat(aTP_Lamm_nw(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_760
tff(fact_10452_ATP_Olambda__761,axiom,
    ! [Uu: nat,Uua: fun_nat_nat,Uub: fun_nat_nat,Uuc: nat] :
      aa_nat_nat(aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(aTP_Lamm_ny(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_761
tff(fact_10453_ATP_Olambda__762,axiom,
    ! [Uu: nat,Uua: fun_nat_int,Uub: fun_nat_int,Uuc: nat] :
      aa_nat_int(aa_fun5565043122120104496at_int(aa_fun5140708168904641446at_int(aTP_Lamm_oa(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_762
tff(fact_10454_ATP_Olambda__763,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_nt(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_763
tff(fact_10455_ATP_Olambda__764,axiom,
    ! [Uu: nat,Uua: fun_na2338080690716861136at_nat,Uub: fun_na2338080690716861136at_nat,Uuc: nat] :
      aa_nat2240542744625857721at_nat(aa_fun4141531684813184444at_nat(aa_fun8230936523094545239at_nat(aTP_Lamm_ov(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_764
tff(fact_10456_ATP_Olambda__765,axiom,
    ! [Uu: nat,Uua: fun_nat_assn,Uub: fun_nat_assn,Uuc: nat] :
      aa_nat_assn(aa_fun785744391571715112t_assn(aa_fun6591345337174305434t_assn(aTP_Lamm_nv(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_765
tff(fact_10457_ATP_Olambda__766,axiom,
    ! [Uu: nat,Uua: fun_nat_rat,Uub: fun_nat_rat,Uuc: nat] :
      aa_nat_rat(aa_fun765758790203053448at_rat(aa_fun3638354034882855594at_rat(aTP_Lamm_nx(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_766
tff(fact_10458_ATP_Olambda__767,axiom,
    ! [Uu: nat,Uua: fun_nat_nat,Uub: fun_nat_nat,Uuc: nat] :
      aa_nat_nat(aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(aTP_Lamm_nz(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_767
tff(fact_10459_ATP_Olambda__768,axiom,
    ! [Uu: nat,Uua: fun_nat_int,Uub: fun_nat_int,Uuc: nat] :
      aa_nat_int(aa_fun5565043122120104496at_int(aa_fun5140708168904641446at_int(aTP_Lamm_ob(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_768
tff(fact_10460_ATP_Olambda__769,axiom,
    ! [Uu: fun_Code_integer_o,Uua: fun_Co8499238673658658103nteger,Uub: fun_Co8499238673658658103nteger,Uuc: code_integer] :
      aa_Cod1783974652533396728nteger(aa_fun4435975922079263404nteger(aa_fun2957433868855875424nteger(aTP_Lamm_ace(Uu),Uua),Uub),Uuc) = $ite(aa_Code_integer_o(Uu,Uuc),aa_Cod1783974652533396728nteger(Uua,Uuc),aa_Cod1783974652533396728nteger(Uub,Uuc)) ).

% ATP.lambda_769
tff(fact_10461_ATP_Olambda__770,axiom,
    ! [Uu: fun_Code_integer_o,Uua: fun_Co4490034186337852034r_assn,Uub: fun_Co4490034186337852034r_assn,Uuc: code_integer] :
      aa_Code_integer_assn(aa_fun4119713415300323650r_assn(aa_fun8067317676786998721r_assn(aTP_Lamm_ach(Uu),Uua),Uub),Uuc) = $ite(aa_Code_integer_o(Uu,Uuc),aa_Code_integer_assn(Uua,Uuc),aa_Code_integer_assn(Uub,Uuc)) ).

% ATP.lambda_770
tff(fact_10462_ATP_Olambda__771,axiom,
    ! [Uu: fun_Code_integer_o,Uua: fun_Code_integer_rat,Uub: fun_Code_integer_rat,Uuc: code_integer] :
      aa_Code_integer_rat(aa_fun8169474305930283170er_rat(aa_fun1894662059565830609er_rat(aTP_Lamm_aby(Uu),Uua),Uub),Uuc) = $ite(aa_Code_integer_o(Uu,Uuc),aa_Code_integer_rat(Uua,Uuc),aa_Code_integer_rat(Uub,Uuc)) ).

% ATP.lambda_771
tff(fact_10463_ATP_Olambda__772,axiom,
    ! [Uu: fun_Code_integer_o,Uua: fun_Code_integer_nat,Uub: fun_Code_integer_nat,Uuc: code_integer] :
      aa_Code_integer_nat(aa_fun3076867170244285586er_nat(aa_fun7635586485349294009er_nat(aTP_Lamm_abz(Uu),Uua),Uub),Uuc) = $ite(aa_Code_integer_o(Uu,Uuc),aa_Code_integer_nat(Uua,Uuc),aa_Code_integer_nat(Uub,Uuc)) ).

% ATP.lambda_772
tff(fact_10464_ATP_Olambda__773,axiom,
    ! [Uu: fun_Code_integer_o,Uua: fun_Code_integer_int,Uub: fun_Code_integer_int,Uuc: code_integer] :
      aa_Code_integer_int(aa_fun3291479149765983562er_int(aa_fun4411173778969602765er_int(aTP_Lamm_aca(Uu),Uua),Uub),Uuc) = $ite(aa_Code_integer_o(Uu,Uuc),aa_Code_integer_int(Uua,Uuc),aa_Code_integer_int(Uub,Uuc)) ).

% ATP.lambda_773
tff(fact_10465_ATP_Olambda__774,axiom,
    ! [Uu: fun_Pr4796943543331071514_nat_o,Uua: fun_Pr2602117396471556196at_nat,Uub: fun_Pr2602117396471556196at_nat,Uuc: product_prod_nat_nat] :
      aa_Pro4243173770044397973at_nat(aa_fun2351638177499875132at_nat(aa_fun1768432373507419115at_nat(aTP_Lamm_ih(Uu),Uua),Uub),Uuc) = $ite(aa_Pro6150637604445964947_nat_o(Uu,Uuc),aa_Pro4243173770044397973at_nat(Uua,Uuc),aa_Pro4243173770044397973at_nat(Uub,Uuc)) ).

% ATP.lambda_774
tff(fact_10466_ATP_Olambda__775,axiom,
    ! [Uu: fun_Pr3238374608701176162_int_o,Uua: fun_Pr5160692531401422620nt_nat,Uub: fun_Pr5160692531401422620nt_nat,Uuc: product_prod_int_int] :
      aa_Pro6801748904974264397nt_nat(aa_fun6016190284160528700nt_nat(aa_fun1083184158259774627nt_nat(aTP_Lamm_ig(Uu),Uua),Uub),Uuc) = $ite(aa_Pro4592068669816069595_int_o(Uu,Uuc),aa_Pro6801748904974264397nt_nat(Uua,Uuc),aa_Pro6801748904974264397nt_nat(Uub,Uuc)) ).

% ATP.lambda_775
tff(fact_10467_ATP_Olambda__776,axiom,
    ! [Uu: fun_list_nat_o,Uua: fun_list_nat_rat,Uub: fun_list_nat_rat,Uuc: list_nat] :
      aa_list_nat_rat(aa_fun1630595088783574696at_rat(aa_fun6629217458047550682at_rat(aTP_Lamm_acc(Uu),Uua),Uub),Uuc) = $ite(aa_list_nat_o(Uu,Uuc),aa_list_nat_rat(Uua,Uuc),aa_list_nat_rat(Uub,Uuc)) ).

% ATP.lambda_776
tff(fact_10468_ATP_Olambda__777,axiom,
    ! [Uu: fun_list_nat_o,Uua: fun_list_nat_nat,Uub: fun_list_nat_nat,Uuc: list_nat] :
      aa_list_nat_nat(aa_fun8775324257704122008at_nat(aa_fun7930739032552464066at_nat(aTP_Lamm_ie(Uu),Uua),Uub),Uuc) = $ite(aa_list_nat_o(Uu,Uuc),aa_list_nat_nat(Uua,Uuc),aa_list_nat_nat(Uub,Uuc)) ).

% ATP.lambda_777
tff(fact_10469_ATP_Olambda__778,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_if(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_778
tff(fact_10470_ATP_Olambda__779,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_acf(Uu),Uua),Uub),Uuc) = $ite(aa_nat_o(Uu,Uuc),aa_nat_Code_integer(Uua,Uuc),aa_nat_Code_integer(Uub,Uuc)) ).

% ATP.lambda_779
tff(fact_10471_ATP_Olambda__780,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_780
tff(fact_10472_ATP_Olambda__781,axiom,
    ! [Uu: fun_nat_o,Uua: fun_nat_set_nat,Uub: fun_nat_set_nat,Uuc: nat] :
      aa_nat_set_nat(aa_fun850356524293783268et_nat(aa_fun2558789851993377716et_nat(aTP_Lamm_id(Uu),Uua),Uub),Uuc) = $ite(aa_nat_o(Uu,Uuc),aa_nat_set_nat(Uua,Uuc),aa_nat_set_nat(Uub,Uuc)) ).

% ATP.lambda_781
tff(fact_10473_ATP_Olambda__782,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_acb(Uu),Uua),Uub),Uuc) = $ite(aa_nat_o(Uu,Uuc),aa_nat_rat(Uua,Uuc),aa_nat_rat(Uub,Uuc)) ).

% ATP.lambda_782
tff(fact_10474_ATP_Olambda__783,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_ic(Uu),Uua),Uub),Uuc) = $ite(aa_nat_o(Uu,Uuc),aa_nat_nat(Uua,Uuc),aa_nat_nat(Uub,Uuc)) ).

% ATP.lambda_783
tff(fact_10475_ATP_Olambda__784,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_ia(Uu),Uua),Uub),Uuc) = $ite(aa_nat_o(Uu,Uuc),aa_nat_int(Uua,Uuc),aa_nat_int(Uub,Uuc)) ).

% ATP.lambda_784
tff(fact_10476_ATP_Olambda__785,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_acd(Uu),Uua),Uub),Uuc) = $ite(aa_int_o(Uu,Uuc),aa_int_Code_integer(Uua,Uuc),aa_int_Code_integer(Uub,Uuc)) ).

% ATP.lambda_785
tff(fact_10477_ATP_Olambda__786,axiom,
    ! [Uu: fun_int_o,Uua: fun_in5957238474111007380at_nat,Uub: fun_in5957238474111007380at_nat,Uuc: int] :
      aa_int7547510126839136381at_nat(aa_fun6444483342604292028at_nat(aa_fun5672742635520905243at_nat(aTP_Lamm_ij(Uu),Uua),Uub),Uuc) = $ite(aa_int_o(Uu,Uuc),aa_int7547510126839136381at_nat(Uua,Uuc),aa_int7547510126839136381at_nat(Uub,Uuc)) ).

% ATP.lambda_786
tff(fact_10478_ATP_Olambda__787,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_acg(Uu),Uua),Uub),Uuc) = $ite(aa_int_o(Uu,Uuc),aa_int_assn(Uua,Uuc),aa_int_assn(Uub,Uuc)) ).

% ATP.lambda_787
tff(fact_10479_ATP_Olambda__788,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_abx(Uu),Uua),Uub),Uuc) = $ite(aa_int_o(Uu,Uuc),aa_int_rat(Uua,Uuc),aa_int_rat(Uub,Uuc)) ).

% ATP.lambda_788
tff(fact_10480_ATP_Olambda__789,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_ib(Uu),Uua),Uub),Uuc) = $ite(aa_int_o(Uu,Uuc),aa_int_nat(Uua,Uuc),aa_int_nat(Uub,Uuc)) ).

% ATP.lambda_789
tff(fact_10481_ATP_Olambda__790,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_hz(Uu),Uua),Uub),Uuc) = $ite(aa_int_o(Uu,Uuc),aa_int_int(Uua,Uuc),aa_int_int(Uub,Uuc)) ).

% ATP.lambda_790
tff(fact_10482_ATP_Olambda__791,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_lu(Uu),Uua),Uub),Uuc) = groups4538972089207619220nt_int(aa_int_fun_int_int(aa_fun5911324870756489105nt_int(aTP_Lamm_lt(Uu),Uua),Uuc),Uub) ).

% ATP.lambda_791
tff(fact_10483_ATP_Olambda__792,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_ls(Uu),Uua),Uub),Uuc) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aa_fun6650658898577385413at_nat(aTP_Lamm_lr(Uu),Uua),Uuc)),Uub) ).

% ATP.lambda_792
tff(fact_10484_ATP_Olambda__793,axiom,
    ! [Uu: set_Code_integer,Uua: fun_Co822469698833053651at_int,Uub: fun_Co1375048237525739357_nat_o,Uuc: nat] : aa_nat_int(aa_fun3278309868433077274at_int(aa_fun1276769135904165912at_int(aTP_Lamm_ti(Uu),Uua),Uub),Uuc) = groups3188404863801439024er_int(aa_nat1674686023564409748er_int(aTP_Lamm_th(Uua),Uuc),collect_Code_integer(aa_nat7420681472272698950eger_o(aa_fun7373950593701497468eger_o(aTP_Lamm_rq(Uu),Uub),Uuc))) ).

% ATP.lambda_793
tff(fact_10485_ATP_Olambda__794,axiom,
    ! [Uu: set_o,Uua: fun_o_fun_nat_int,Uub: fun_o_fun_nat_o,Uuc: nat] : aa_nat_int(aa_fun6872859990233652739at_int(aa_fun8947132502270365866at_int(aTP_Lamm_tf(Uu),Uua),Uub),Uuc) = groups3502327434004483295_o_int(aa_nat_fun_o_int(aTP_Lamm_te(Uua),Uuc),collect_o(aa_nat_fun_o_o(aa_fun532071374735444762un_o_o(aTP_Lamm_rg(Uu),Uub),Uuc))) ).

% ATP.lambda_794
tff(fact_10486_ATP_Olambda__795,axiom,
    ! [Uu: set_nat,Uua: fun_na5492297256294505719er_nat,Uub: fun_na6318815019539281885eger_o,Uuc: code_integer] : aa_Code_integer_nat(aa_fun7264493923017702603er_nat(aa_fun7529132916787797541er_nat(aTP_Lamm_ts(Uu),Uua),Uub),Uuc) = aa_set_nat_nat(groups708209901874060359at_nat(aa_Cod8670293052248272184at_nat(aTP_Lamm_sq(Uua),Uuc)),collect_nat(aa_Cod1075288278386115654_nat_o(aa_fun3722190751602477436_nat_o(aTP_Lamm_sr(Uu),Uub),Uuc))) ).

% ATP.lambda_795
tff(fact_10487_ATP_Olambda__796,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_tq(Uu),Uua),Uub),Uuc) = aa_set_nat_nat(groups708209901874060359at_nat(aa_int_fun_nat_nat(aTP_Lamm_sm(Uua),Uuc)),collect_nat(aa_int_fun_nat_o(aa_fun7393466555336238524_nat_o(aTP_Lamm_rz(Uu),Uub),Uuc))) ).

% ATP.lambda_796
tff(fact_10488_ATP_Olambda__797,axiom,
    ! [Uu: set_nat,Uua: fun_nat_fun_o_nat,Uub: fun_nat_fun_o_o,Uuc: $o] : aa_o_nat(aa_fun5720900557306247341_o_nat(aa_fun4151889263603125372_o_nat(aTP_Lamm_to(Uu),Uua),Uub),(Uuc)) = aa_set_nat_nat(groups708209901874060359at_nat(aa_o_fun_nat_nat(aTP_Lamm_si(Uua),(Uuc))),collect_nat(aa_o_fun_nat_o(aa_fun2937962278134185242_nat_o(aTP_Lamm_sj(Uu),Uub),(Uuc)))) ).

% ATP.lambda_797
tff(fact_10489_ATP_Olambda__798,axiom,
    ! [Uu: set_Code_integer,Uua: fun_Co3009391273526075439nt_int,Uub: fun_Co8353275306710069249_int_o,Uuc: int] : aa_int_int(aa_fun4086957665101867802nt_int(aa_fun5679615576903115196nt_int(aTP_Lamm_tm(Uu),Uua),Uub),Uuc) = groups3188404863801439024er_int(aa_int1390634949194299888er_int(aTP_Lamm_sd(Uua),Uuc),collect_Code_integer(aa_int5993532643906289898eger_o(aa_fun7355488674016760700eger_o(aTP_Lamm_se(Uu),Uub),Uuc))) ).

% ATP.lambda_798
tff(fact_10490_ATP_Olambda__799,axiom,
    ! [Uu: set_o,Uua: fun_o_fun_int_int,Uub: fun_o_fun_int_o,Uuc: int] : aa_int_int(aa_fun7681507786902443267nt_int(aa_fun5020185059765800966nt_int(aTP_Lamm_tk(Uu),Uua),Uub),Uuc) = groups3502327434004483295_o_int(aa_int_fun_o_int(aTP_Lamm_ru(Uua),Uuc),collect_o(aa_int_fun_o_o(aa_fun250872857246597658un_o_o(aTP_Lamm_rv(Uu),Uub),Uuc))) ).

% ATP.lambda_799
tff(fact_10491_ATP_Olambda__800,axiom,
    ! [Uu: set_Code_integer,Uua: fun_Co3312552843764780407at_nat,Uub: fun_Co1375048237525739357_nat_o,Uuc: nat] : aa_nat_nat(aa_fun7456160887942273982at_nat(aa_fun4417447297828429592at_nat(aTP_Lamm_tc(Uu),Uua),Uub),Uuc) = groups3190895334310489300er_nat(aa_nat5852537043073606456er_nat(aTP_Lamm_rp(Uua),Uuc),collect_Code_integer(aa_nat7420681472272698950eger_o(aa_fun7373950593701497468eger_o(aTP_Lamm_rq(Uu),Uub),Uuc))) ).

% ATP.lambda_800
tff(fact_10492_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_ta(Uu),Uua),Uub),Uuc) = groups1707563613775114915nt_nat(aa_nat_fun_int_nat(aTP_Lamm_rk(Uua),Uuc),collect_int(aa_nat_fun_int_o(aa_fun5439836533389078972_int_o(aTP_Lamm_rl(Uu),Uub),Uuc))) ).

% ATP.lambda_801
tff(fact_10493_ATP_Olambda__802,axiom,
    ! [Uu: set_o,Uua: fun_o_fun_nat_nat,Uub: fun_o_fun_nat_o,Uuc: nat] : aa_nat_nat(aa_fun1827338972888073639at_nat(aa_fun390978558358138354at_nat(aTP_Lamm_sy(Uu),Uua),Uub),Uuc) = groups3504817904513533571_o_nat(aa_nat_fun_o_nat(aTP_Lamm_rf(Uua),Uuc),collect_o(aa_nat_fun_o_o(aa_fun532071374735444762un_o_o(aTP_Lamm_rg(Uu),Uub),Uuc))) ).

% ATP.lambda_802
tff(fact_10494_ATP_Olambda__803,axiom,
    ! [Uu: set_nat,Uua: fun_na5492297256294505719er_nat,Uub: fun_na6318815019539281885eger_o,Uuc: code_integer] : aa_Code_integer_nat(aa_fun7264493923017702603er_nat(aa_fun7529132916787797541er_nat(aTP_Lamm_ss(Uu),Uua),Uub),Uuc) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_Cod8670293052248272184at_nat(aTP_Lamm_sq(Uua),Uuc)),collect_nat(aa_Cod1075288278386115654_nat_o(aa_fun3722190751602477436_nat_o(aTP_Lamm_sr(Uu),Uub),Uuc))) ).

% ATP.lambda_803
tff(fact_10495_ATP_Olambda__804,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_sn(Uu),Uua),Uub),Uuc) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_int_fun_nat_nat(aTP_Lamm_sm(Uua),Uuc)),collect_nat(aa_int_fun_nat_o(aa_fun7393466555336238524_nat_o(aTP_Lamm_rz(Uu),Uub),Uuc))) ).

% ATP.lambda_804
tff(fact_10496_ATP_Olambda__805,axiom,
    ! [Uu: set_nat,Uua: fun_nat_fun_o_nat,Uub: fun_nat_fun_o_o,Uuc: $o] : aa_o_nat(aa_fun5720900557306247341_o_nat(aa_fun4151889263603125372_o_nat(aTP_Lamm_sk(Uu),Uua),Uub),(Uuc)) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_o_fun_nat_nat(aTP_Lamm_si(Uua),(Uuc))),collect_nat(aa_o_fun_nat_o(aa_fun2937962278134185242_nat_o(aTP_Lamm_sj(Uu),Uub),(Uuc)))) ).

% ATP.lambda_805
tff(fact_10497_ATP_Olambda__806,axiom,
    ! [Uu: set_Code_integer,Uua: fun_Co3009391273526075439nt_int,Uub: fun_Co8353275306710069249_int_o,Uuc: int] : aa_int_int(aa_fun4086957665101867802nt_int(aa_fun5679615576903115196nt_int(aTP_Lamm_sf(Uu),Uua),Uub),Uuc) = groups7234854612051535045er_int(aa_int1390634949194299888er_int(aTP_Lamm_sd(Uua),Uuc),collect_Code_integer(aa_int5993532643906289898eger_o(aa_fun7355488674016760700eger_o(aTP_Lamm_se(Uu),Uub),Uuc))) ).

% ATP.lambda_806
tff(fact_10498_ATP_Olambda__807,axiom,
    ! [Uu: set_o,Uua: fun_o_fun_int_int,Uub: fun_o_fun_int_o,Uuc: int] : aa_int_int(aa_fun7681507786902443267nt_int(aa_fun5020185059765800966nt_int(aTP_Lamm_rw(Uu),Uua),Uub),Uuc) = groups8505340233167759370_o_int(aa_int_fun_o_int(aTP_Lamm_ru(Uua),Uuc),collect_o(aa_int_fun_o_o(aa_fun250872857246597658un_o_o(aTP_Lamm_rv(Uu),Uub),Uuc))) ).

% ATP.lambda_807
tff(fact_10499_ATP_Olambda__808,axiom,
    ! [Uu: set_Code_integer,Uua: fun_Co3312552843764780407at_nat,Uub: fun_Co1375048237525739357_nat_o,Uuc: nat] : aa_nat_nat(aa_fun7456160887942273982at_nat(aa_fun4417447297828429592at_nat(aTP_Lamm_rr(Uu),Uua),Uub),Uuc) = groups7237345082560585321er_nat(aa_nat5852537043073606456er_nat(aTP_Lamm_rp(Uua),Uuc),collect_Code_integer(aa_nat7420681472272698950eger_o(aa_fun7373950593701497468eger_o(aTP_Lamm_rq(Uu),Uub),Uuc))) ).

% ATP.lambda_808
tff(fact_10500_ATP_Olambda__809,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_rm(Uu),Uua),Uub),Uuc) = groups4541462559716669496nt_nat(aa_nat_fun_int_nat(aTP_Lamm_rk(Uua),Uuc),collect_int(aa_nat_fun_int_o(aa_fun5439836533389078972_int_o(aTP_Lamm_rl(Uu),Uub),Uuc))) ).

% ATP.lambda_809
tff(fact_10501_ATP_Olambda__810,axiom,
    ! [Uu: set_o,Uua: fun_o_fun_nat_nat,Uub: fun_o_fun_nat_o,Uuc: nat] : aa_nat_nat(aa_fun1827338972888073639at_nat(aa_fun390978558358138354at_nat(aTP_Lamm_rh(Uu),Uua),Uub),Uuc) = groups8507830703676809646_o_nat(aa_nat_fun_o_nat(aTP_Lamm_rf(Uua),Uuc),collect_o(aa_nat_fun_o_o(aa_fun532071374735444762un_o_o(aTP_Lamm_rg(Uu),Uub),Uuc))) ).

% ATP.lambda_810
tff(fact_10502_ATP_Olambda__811,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_sa(Uu),Uua),Uub),Uuc) = aa_set_nat_int(groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_mu(Uua),Uuc)),collect_nat(aa_int_fun_nat_o(aa_fun7393466555336238524_nat_o(aTP_Lamm_rz(Uu),Uub),Uuc))) ).

% ATP.lambda_811
tff(fact_10503_ATP_Olambda__812,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_sw(Uu),Uua),Uub),Uuc) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_lk(Uua),Uuc)),collect_nat(aa_nat_fun_nat_o(aa_fun6699960423209655484_nat_o(aTP_Lamm_sv(Uu),Uub),Uuc))) ).

% ATP.lambda_812
tff(fact_10504_ATP_Olambda__813,axiom,
    ! [Uu: code_natural,Uua: code_natural,Uub: code_natural,Uuc: code_natural] : aa_Cod6808739660676082561atural(aa_Cod4292056611156568242atural(aa_Cod8811915439870959971atural(aTP_Lamm_aoy(Uu),Uua),Uub),Uuc) = produc4480994950612372183atural(produc3574140220909816553atural(inc_shift(aa_num_Code_natural(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(aa_num_Code_natural(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_813
tff(fact_10505_ATP_Olambda__814,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] : aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aa_nat4158579785191972357at_nat(aTP_Lamm_amq(Uu),Uua),Uub),Uuc) = aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_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_814
tff(fact_10506_ATP_Olambda__815,axiom,
    ! [Uu: fun_nat_nat,Uua: set_Pr1261947904930325089at_nat,Uub: nat,Uuc: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aa_set2634951030311766213_nat_o(aTP_Lamm_ahu(Uu),Uua),Uub),Uuc)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(Uu,Uub)),aa_nat_nat(Uu,Uuc))
        | ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(Uu,Uub)),aa_nat_nat(Uu,Uuc))
          & aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Uub),Uuc)),Uua) ) ) ) ).

% ATP.lambda_815
tff(fact_10507_ATP_Olambda__816,axiom,
    ! [Uu: fun_int_nat,Uua: set_Pr958786334691620121nt_int,Uub: int,Uuc: int] :
      ( aa_int_o(aa_int_fun_int_o(aa_set742265667465749333_int_o(aTP_Lamm_ahv(Uu),Uua),Uub),Uuc)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_int_nat(Uu,Uub)),aa_int_nat(Uu,Uuc))
        | ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(Uu,Uub)),aa_int_nat(Uu,Uuc))
          & member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Uub),Uuc),Uua) ) ) ) ).

% ATP.lambda_816
tff(fact_10508_ATP_Olambda__817,axiom,
    ! [Uu: fun_int_option_int,Uua: fun_Pr3238374608701176162_int_o,Uub: int,Uuc: int] :
      ( aa_int_o(aa_int_fun_int_o(aa_fun5140975492127840594_int_o(aTP_Lamm_aht(Uu),Uua),Uub),Uuc)
    <=> ( ( aa_int_option_int(Uu,Uub) = aa_int_option_int(some_int,Uuc) )
        & aa_Pro4592068669816069595_int_o(Uua,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Uub),Uuc)) ) ) ).

% ATP.lambda_817
tff(fact_10509_ATP_Olambda__818,axiom,
    ! [Uu: fun_fu2352890940381120596et_nat,Uua: fun_Pr2620521676128474833_nat_o,Uub: fun_Pr6460944699146269935_nat_o,Uuc: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(aa_fun2638059832795814268_nat_o(aa_fun1045796833200626456_nat_o(aTP_Lamm_aho(Uu),Uua),Uub),Uuc)
    <=> ( ( aa_fun5824947343344787589et_nat(Uu,Uub) = some_P624177172695371229et_nat(Uuc) )
        & aa_Pro5643009357358181178_nat_o(Uua,aa_Pro434340453207536480et_nat(aa_fun3926903090414666066et_nat(produc6017135503889078682et_nat,Uub),Uuc)) ) ) ).

% ATP.lambda_818
tff(fact_10510_ATP_Olambda__819,axiom,
    ! [Uu: fun_fu1905361174868762994et_nat,Uua: fun_Pr1227177680593602867_nat_o,Uub: fun_Pr6460944699146269935_nat_o,Uuc: produc8799533983703829792et_nat] :
      ( aa_Pro5643009357358181178_nat_o(aa_fun912680268133416158_nat_o(aa_fun4422562838209026524_nat_o(aTP_Lamm_ahp(Uu),Uua),Uub),Uuc)
    <=> ( ( aa_fun1716447855799474851et_nat(Uu,Uub) = some_P994283844982041979et_nat(Uuc) )
        & aa_Pro2671238023309943196_nat_o(Uua,aa_Pro1979504083478021276et_nat(aa_fun615918787490101902et_nat(produc7291855603632349880et_nat,Uub),Uuc)) ) ) ).

% ATP.lambda_819
tff(fact_10511_ATP_Olambda__820,axiom,
    ! [Uu: fun_fu4253467608928286509nt_int,Uua: fun_Pr3540825087398332590_int_o,Uub: fun_Pr7763680713840462443e_term,Uuc: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aa_fun3538162476766722827_int_o(aa_fun1054583131507368252_int_o(aTP_Lamm_ahq(Uu),Uua),Uub),Uuc)
    <=> ( ( aa_fun251074914522852630nt_int(Uu,Uub) = some_P4184893108420464158nt_int(Uuc) )
        & aa_Pro2601676399929874711_int_o(Uua,aa_Pro5748361610964203648nt_int(aa_fun3265472288063244902nt_int(produc7320969046608631221nt_int,Uub),Uuc)) ) ) ).

% ATP.lambda_820
tff(fact_10512_ATP_Olambda__821,axiom,
    ! [Uu: fun_fu5631754973857388633nt_int,Uua: fun_Pr1196100989263358594_int_o,Uub: fun_in5730379987311621695e_term,Uuc: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aa_fun586359091965693151_int_o(aa_fun8601777901039773924_int_o(aTP_Lamm_ahr(Uu),Uua),Uub),Uuc)
    <=> ( ( aa_fun3970441718214872642nt_int(Uu,Uub) = some_P4184893108420464158nt_int(Uuc) )
        & aa_Pro5451197003417746155_int_o(Uua,aa_Pro8800328608558162860nt_int(aa_fun260840925927807678nt_int(produc4729913460834754785nt_int,Uub),Uuc)) ) ) ).

% ATP.lambda_821
tff(fact_10513_ATP_Olambda__822,axiom,
    ! [Uu: fun_a_1313742400928293293it_nat,Uua: fun_Pr5746006151868627512_nat_o,Uub: a,Uuc: produc6653097349344004940it_nat] :
      ( aa_Pro1202363709425262478_nat_o(aa_a_f337466037417904483_nat_o(aa_fun7773039478287727974_nat_o(aTP_Lamm_ahs(Uu),Uua),Uub),Uuc)
    <=> ( ( aa_a_o5231022909227610846it_nat(Uu,Uub) = some_P1054239925786823975it_nat(Uuc) )
        & aa_Pro6426884464393220769_nat_o(Uua,aa_Pro8078911840264235907it_nat(aa_a_f2234720956197727492it_nat(produc9178034014595674355it_nat,Uub),Uuc)) ) ) ).

% ATP.lambda_822
tff(fact_10514_ATP_Olambda__823,axiom,
    ! [Uu: code_integer,Uua: code_integer,Uub: nat,Uuc: nat] : aa_nat_Code_integer(aa_nat2010205248571845816nteger(aa_Cod3140087502949435154nteger(aTP_Lamm_acq(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_823
tff(fact_10515_ATP_Olambda__824,axiom,
    ! [Uu: rat,Uua: rat,Uub: nat,Uuc: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_aco(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_824
tff(fact_10516_ATP_Olambda__825,axiom,
    ! [Uu: int,Uua: int,Uub: nat,Uuc: nat] : aa_nat_int(aa_nat_fun_nat_int(aa_int431785320988601264at_int(aTP_Lamm_acp(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_825
tff(fact_10517_ATP_Olambda__826,axiom,
    ! [Uu: code_integer,Uua: code_integer,Uub: nat,Uuc: nat] : aa_nat_Code_integer(aa_nat2010205248571845816nteger(aa_Cod3140087502949435154nteger(aTP_Lamm_acn(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_826
tff(fact_10518_ATP_Olambda__827,axiom,
    ! [Uu: rat,Uua: rat,Uub: nat,Uuc: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_acl(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_827
tff(fact_10519_ATP_Olambda__828,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_acj(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_828
tff(fact_10520_ATP_Olambda__829,axiom,
    ! [Uu: int,Uua: int,Uub: nat,Uuc: nat] : aa_nat_int(aa_nat_fun_nat_int(aa_int431785320988601264at_int(aTP_Lamm_acm(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_829
tff(fact_10521_ATP_Olambda__830,axiom,
    ! [Uu: code_integer,Uua: nat,Uub: code_integer,Uuc: nat] : aa_nat_Code_integer(aa_Cod1578151519121547077nteger(aa_nat2351529209232473362nteger(aTP_Lamm_aio(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_830
tff(fact_10522_ATP_Olambda__831,axiom,
    ! [Uu: rat,Uua: nat,Uub: rat,Uuc: nat] : aa_nat_rat(aa_rat_fun_nat_rat(aa_nat298952458717073160at_rat(aTP_Lamm_aip(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_831
tff(fact_10523_ATP_Olambda__832,axiom,
    ! [Uu: int,Uua: nat,Uub: int,Uuc: nat] : aa_nat_int(aa_int_fun_nat_int(aa_nat6704147566042551216at_int(aTP_Lamm_aiq(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_832
tff(fact_10524_ATP_Olambda__833,axiom,
    ! [Uu: fun_list_nat_o,Uua: set_list_nat,Uub: fun_o_o,Uuc: $o] :
      ( aa_o_o(aa_fun_o_o_fun_o_o(aa_set4082382446602818263un_o_o(aTP_Lamm_jn(Uu),Uua),Uub),(Uuc))
    <=> ( member_o((Uuc),image_list_nat_o(Uu,Uua))
        & aa_o_o(Uub,(Uuc)) ) ) ).

% ATP.lambda_833
tff(fact_10525_ATP_Olambda__834,axiom,
    ! [Uu: fun_o_o,Uua: set_o,Uub: fun_o_o,Uuc: $o] :
      ( aa_o_o(aa_fun_o_o_fun_o_o(aa_set8294393945046410815un_o_o(aTP_Lamm_iv(Uu),Uua),Uub),(Uuc))
    <=> ( member_o((Uuc),image_o_o(Uu,Uua))
        & aa_o_o(Uub,(Uuc)) ) ) ).

% ATP.lambda_834
tff(fact_10526_ATP_Olambda__835,axiom,
    ! [Uu: fun_o_nat,Uua: set_o,Uub: fun_nat_o,Uuc: nat] :
      ( aa_nat_o(aa_fun5937193402907943164_nat_o(aa_set1339306056044713023_nat_o(aTP_Lamm_jb(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_nat_o(member_nat(Uuc),image_o_nat(Uu,Uua))
        & aa_nat_o(Uub,Uuc) ) ) ).

% ATP.lambda_835
tff(fact_10527_ATP_Olambda__836,axiom,
    ! [Uu: fun_o_int,Uua: set_o,Uub: fun_int_o,Uuc: int] :
      ( aa_int_o(aa_fun6243517787477317628_int_o(aa_set1221527818376329535_int_o(aTP_Lamm_jh(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_int_o(member_int(Uuc),image_o_int(Uu,Uua))
        & aa_int_o(Uub,Uuc) ) ) ).

% ATP.lambda_836
tff(fact_10528_ATP_Olambda__837,axiom,
    ! [Uu: fun_nat_o,Uua: set_nat,Uub: fun_o_o,Uuc: $o] :
      ( aa_o_o(aa_fun_o_o_fun_o_o(aa_set1656415104011699143un_o_o(aTP_Lamm_ix(Uu),Uua),Uub),(Uuc))
    <=> ( member_o((Uuc),image_nat_o(Uu,Uua))
        & aa_o_o(Uub,(Uuc)) ) ) ).

% ATP.lambda_837
tff(fact_10529_ATP_Olambda__838,axiom,
    ! [Uu: fun_nat_nat,Uua: set_nat,Uub: fun_nat_o,Uuc: nat] :
      ( aa_nat_o(aa_fun5937193402907943164_nat_o(aa_set1405850572773641671_nat_o(aTP_Lamm_jd(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_nat_o(member_nat(Uuc),image_nat_nat(Uu,Uua))
        & aa_nat_o(Uub,Uuc) ) ) ).

% ATP.lambda_838
tff(fact_10530_ATP_Olambda__839,axiom,
    ! [Uu: fun_nat_int,Uua: set_nat,Uub: fun_int_o,Uuc: int] :
      ( aa_int_o(aa_fun6243517787477317628_int_o(aa_set1288072335105258183_int_o(aTP_Lamm_jj(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_int_o(member_int(Uuc),image_nat_int(Uu,Uua))
        & aa_int_o(Uub,Uuc) ) ) ).

% ATP.lambda_839
tff(fact_10531_ATP_Olambda__840,axiom,
    ! [Uu: fun_int_o,Uua: set_int,Uub: fun_o_o,Uuc: $o] :
      ( aa_o_o(aa_fun_o_o_fun_o_o(aa_set2864532236940079139un_o_o(aTP_Lamm_iz(Uu),Uua),Uub),(Uuc))
    <=> ( member_o((Uuc),image_int_o(Uu,Uua))
        & aa_o_o(Uub,(Uuc)) ) ) ).

% ATP.lambda_840
tff(fact_10532_ATP_Olambda__841,axiom,
    ! [Uu: fun_int_nat,Uua: set_int,Uub: fun_nat_o,Uuc: nat] :
      ( aa_nat_o(aa_fun5937193402907943164_nat_o(aa_set563577997438114851_nat_o(aTP_Lamm_jf(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_nat_o(member_nat(Uuc),image_int_nat(Uu,Uua))
        & aa_nat_o(Uub,Uuc) ) ) ).

% ATP.lambda_841
tff(fact_10533_ATP_Olambda__842,axiom,
    ! [Uu: fun_int_int,Uua: set_int,Uub: fun_int_o,Uuc: int] :
      ( aa_int_o(aa_fun6243517787477317628_int_o(aa_set445799759769731363_int_o(aTP_Lamm_jl(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_int_o(member_int(Uuc),image_int_int(Uu,Uua))
        & aa_int_o(Uub,Uuc) ) ) ).

% ATP.lambda_842
tff(fact_10534_ATP_Olambda__843,axiom,
    ! [Uu: set_Code_integer,Uua: fun_Co2081214437782444048eger_o,Uub: code_integer,Uuc: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(aa_fun7683620812777033532eger_o(aTP_Lamm_alt(Uu),Uua),Uub),Uuc)
    <=> ( aa_set5643387332196575111eger_o(member_Code_integer(Uuc),Uu)
        & aa_Code_integer_o(aa_Cod326357343816142457eger_o(Uua,Uub),Uuc) ) ) ).

% ATP.lambda_843
tff(fact_10535_ATP_Olambda__844,axiom,
    ! [Uu: set_Code_integer,Uua: fun_o_4021313873839786239eger_o,Uub: $o,Uuc: code_integer] :
      ( aa_Code_integer_o(aa_o_f8517747772616379256eger_o(aa_fun3267419118247850812eger_o(aTP_Lamm_alf(Uu),Uua),(Uub)),Uuc)
    <=> ( aa_set5643387332196575111eger_o(member_Code_integer(Uuc),Uu)
        & aa_Code_integer_o(aa_o_f8517747772616379256eger_o(Uua,(Uub)),Uuc) ) ) ).

% ATP.lambda_844
tff(fact_10536_ATP_Olambda__845,axiom,
    ! [Uu: set_Code_integer,Uua: fun_na6318815019539281885eger_o,Uub: nat,Uuc: code_integer] :
      ( aa_Code_integer_o(aa_nat7420681472272698950eger_o(aa_fun8665957533616019964eger_o(aTP_Lamm_so(Uu),Uua),Uub),Uuc)
    <=> ( aa_set5643387332196575111eger_o(member_Code_integer(Uuc),Uu)
        & aa_Code_integer_o(aa_nat7420681472272698950eger_o(Uua,Uub),Uuc) ) ) ).

% ATP.lambda_845
tff(fact_10537_ATP_Olambda__846,axiom,
    ! [Uu: set_Code_integer,Uua: fun_in4891666191172872833eger_o,Uub: int,Uuc: code_integer] :
      ( aa_Code_integer_o(aa_int5993532643906289898eger_o(aa_fun5642168435487881468eger_o(aTP_Lamm_alm(Uu),Uua),Uub),Uuc)
    <=> ( aa_set5643387332196575111eger_o(member_Code_integer(Uuc),Uu)
        & aa_Code_integer_o(aa_int5993532643906289898eger_o(Uua,Uub),Uuc) ) ) ).

% ATP.lambda_846
tff(fact_10538_ATP_Olambda__847,axiom,
    ! [Uu: set_o,Uua: fun_Co480733329897759773un_o_o,Uub: code_integer,Uuc: $o] :
      ( aa_o_o(aa_Cod6379977994291289310un_o_o(aa_fun8978408420733827192un_o_o(aTP_Lamm_alp(Uu),Uua),Uub),(Uuc))
    <=> ( member_o((Uuc),Uu)
        & aa_o_o(aa_Cod6379977994291289310un_o_o(Uua,Uub),(Uuc)) ) ) ).

% ATP.lambda_847
tff(fact_10539_ATP_Olambda__848,axiom,
    ! [Uu: set_o,Uua: fun_o_fun_o_o,Uub: $o,Uuc: $o] :
      ( aa_o_o(aa_o_fun_o_o(aa_fun6846671739589424444un_o_o(aTP_Lamm_alc(Uu),Uua),(Uub)),(Uuc))
    <=> ( member_o((Uuc),Uu)
        & aa_o_o(aa_o_fun_o_o(Uua,(Uub)),(Uuc)) ) ) ).

% ATP.lambda_848
tff(fact_10540_ATP_Olambda__849,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_sg(Uu),Uua),Uub),(Uuc))
    <=> ( member_o((Uuc),Uu)
        & aa_o_o(aa_nat_fun_o_o(Uua,Uub),(Uuc)) ) ) ).

% ATP.lambda_849
tff(fact_10541_ATP_Olambda__850,axiom,
    ! [Uu: set_o,Uua: fun_int_fun_o_o,Uub: int,Uuc: $o] :
      ( aa_o_o(aa_int_fun_o_o(aa_fun3869439686586307222un_o_o(aTP_Lamm_ali(Uu),Uua),Uub),(Uuc))
    <=> ( member_o((Uuc),Uu)
        & aa_o_o(aa_int_fun_o_o(Uua,Uub),(Uuc)) ) ) ).

% ATP.lambda_850
tff(fact_10542_ATP_Olambda__851,axiom,
    ! [Uu: set_nat,Uua: fun_Co1375048237525739357_nat_o,Uub: code_integer,Uuc: nat] :
      ( aa_nat_o(aa_Cod1075288278386115654_nat_o(aa_fun2430183811687954940_nat_o(aTP_Lamm_rn(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_nat_o(member_nat(Uuc),Uu)
        & aa_nat_o(aa_Cod1075288278386115654_nat_o(Uua,Uub),Uuc) ) ) ).

% ATP.lambda_851
tff(fact_10543_ATP_Olambda__852,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_rd(Uu),Uua),(Uub)),Uuc)
    <=> ( aa_set_nat_o(member_nat(Uuc),Uu)
        & aa_nat_o(aa_o_fun_nat_o(Uua,(Uub)),Uuc) ) ) ).

% ATP.lambda_852
tff(fact_10544_ATP_Olambda__853,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_st(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_nat_o(member_nat(Uuc),Uu)
        & aa_nat_o(aa_nat_fun_nat_o(Uua,Uub),Uuc) ) ) ).

% ATP.lambda_853
tff(fact_10545_ATP_Olambda__854,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_ri(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_nat_o(member_nat(Uuc),Uu)
        & aa_nat_o(aa_int_fun_nat_o(Uua,Uub),Uuc) ) ) ).

% ATP.lambda_854
tff(fact_10546_ATP_Olambda__855,axiom,
    ! [Uu: set_int,Uua: fun_Co8353275306710069249_int_o,Uub: code_integer,Uuc: int] :
      ( aa_int_o(aa_Cod8477617969808203498_int_o(aa_fun1593725752699181308_int_o(aTP_Lamm_sb(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_int_o(member_int(Uuc),Uu)
        & aa_int_o(aa_Cod8477617969808203498_int_o(Uua,Uub),Uuc) ) ) ).

% ATP.lambda_855
tff(fact_10547_ATP_Olambda__856,axiom,
    ! [Uu: set_int,Uua: fun_o_fun_int_o,Uub: $o,Uuc: int] :
      ( aa_int_o(aa_o_fun_int_o(aa_fun1733087058159411102_int_o(aTP_Lamm_rs(Uu),Uua),(Uub)),Uuc)
    <=> ( aa_set_int_o(member_int(Uuc),Uu)
        & aa_int_o(aa_o_fun_int_o(Uua,(Uub)),Uuc) ) ) ).

% ATP.lambda_856
tff(fact_10548_ATP_Olambda__857,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_rx(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_857
tff(fact_10549_ATP_Olambda__858,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_alk(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_858
tff(fact_10550_ATP_Olambda__859,axiom,
    ! [Uu: set_Code_integer,Uua: fun_Co2081214437782444048eger_o,Uub: code_integer,Uuc: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(aa_fun7683620812777033532eger_o(aTP_Lamm_als(Uu),Uua),Uub),Uuc)
    <=> ( aa_set5643387332196575111eger_o(member_Code_integer(Uuc),Uu)
        & aa_Code_integer_o(aa_Cod326357343816142457eger_o(Uua,Uuc),Uub) ) ) ).

% ATP.lambda_859
tff(fact_10551_ATP_Olambda__860,axiom,
    ! [Uu: set_Code_integer,Uua: fun_Co480733329897759773un_o_o,Uub: $o,Uuc: code_integer] :
      ( aa_Code_integer_o(aa_o_f8517747772616379256eger_o(aa_fun3295616927821077850eger_o(aTP_Lamm_alo(Uu),Uua),(Uub)),Uuc)
    <=> ( aa_set5643387332196575111eger_o(member_Code_integer(Uuc),Uu)
        & aa_o_o(aa_Cod6379977994291289310un_o_o(Uua,Uuc),(Uub)) ) ) ).

% ATP.lambda_860
tff(fact_10552_ATP_Olambda__861,axiom,
    ! [Uu: set_Code_integer,Uua: fun_Co1375048237525739357_nat_o,Uub: nat,Uuc: code_integer] :
      ( aa_Code_integer_o(aa_nat7420681472272698950eger_o(aa_fun7373950593701497468eger_o(aTP_Lamm_rq(Uu),Uua),Uub),Uuc)
    <=> ( aa_set5643387332196575111eger_o(member_Code_integer(Uuc),Uu)
        & aa_nat_o(aa_Cod1075288278386115654_nat_o(Uua,Uuc),Uub) ) ) ).

% ATP.lambda_861
tff(fact_10553_ATP_Olambda__862,axiom,
    ! [Uu: set_Code_integer,Uua: fun_Co8353275306710069249_int_o,Uub: int,Uuc: code_integer] :
      ( aa_Code_integer_o(aa_int5993532643906289898eger_o(aa_fun7355488674016760700eger_o(aTP_Lamm_se(Uu),Uua),Uub),Uuc)
    <=> ( aa_set5643387332196575111eger_o(member_Code_integer(Uuc),Uu)
        & aa_int_o(aa_Cod8477617969808203498_int_o(Uua,Uuc),Uub) ) ) ).

% ATP.lambda_862
tff(fact_10554_ATP_Olambda__863,axiom,
    ! [Uu: set_o,Uua: fun_o_4021313873839786239eger_o,Uub: code_integer,Uuc: $o] :
      ( aa_o_o(aa_Cod6379977994291289310un_o_o(aa_fun8950210611160600154un_o_o(aTP_Lamm_ada(Uu),Uua),Uub),(Uuc))
    <=> ( member_o((Uuc),Uu)
        & aa_Code_integer_o(aa_o_f8517747772616379256eger_o(Uua,(Uuc)),Uub) ) ) ).

% ATP.lambda_863
tff(fact_10555_ATP_Olambda__864,axiom,
    ! [Uu: set_o,Uua: fun_o_fun_o_o,Uub: $o,Uuc: $o] :
      ( aa_o_o(aa_o_fun_o_o(aa_fun6846671739589424444un_o_o(aTP_Lamm_alb(Uu),Uua),(Uub)),(Uuc))
    <=> ( member_o((Uuc),Uu)
        & aa_o_o(aa_o_fun_o_o(Uua,(Uuc)),(Uub)) ) ) ).

% ATP.lambda_864
tff(fact_10556_ATP_Olambda__865,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_rg(Uu),Uua),Uub),(Uuc))
    <=> ( member_o((Uuc),Uu)
        & aa_nat_o(aa_o_fun_nat_o(Uua,(Uuc)),Uub) ) ) ).

% ATP.lambda_865
tff(fact_10557_ATP_Olambda__866,axiom,
    ! [Uu: set_o,Uua: fun_o_fun_int_o,Uub: int,Uuc: $o] :
      ( aa_o_o(aa_int_fun_o_o(aa_fun250872857246597658un_o_o(aTP_Lamm_rv(Uu),Uua),Uub),(Uuc))
    <=> ( member_o((Uuc),Uu)
        & aa_int_o(aa_o_fun_int_o(Uua,(Uuc)),Uub) ) ) ).

% ATP.lambda_866
tff(fact_10558_ATP_Olambda__867,axiom,
    ! [Uu: set_nat,Uua: fun_na6318815019539281885eger_o,Uub: code_integer,Uuc: nat] :
      ( aa_nat_o(aa_Cod1075288278386115654_nat_o(aa_fun3722190751602477436_nat_o(aTP_Lamm_sr(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_nat_o(member_nat(Uuc),Uu)
        & aa_Code_integer_o(aa_nat7420681472272698950eger_o(Uua,Uuc),Uub) ) ) ).

% ATP.lambda_867
tff(fact_10559_ATP_Olambda__868,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_sj(Uu),Uua),(Uub)),Uuc)
    <=> ( aa_set_nat_o(member_nat(Uuc),Uu)
        & aa_o_o(aa_nat_fun_o_o(Uua,Uuc),(Uub)) ) ) ).

% ATP.lambda_868
tff(fact_10560_ATP_Olambda__869,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_sv(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_nat_o(member_nat(Uuc),Uu)
        & aa_nat_o(aa_nat_fun_nat_o(Uua,Uuc),Uub) ) ) ).

% ATP.lambda_869
tff(fact_10561_ATP_Olambda__870,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_rz(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_nat_o(member_nat(Uuc),Uu)
        & aa_int_o(aa_nat_fun_int_o(Uua,Uuc),Uub) ) ) ).

% ATP.lambda_870
tff(fact_10562_ATP_Olambda__871,axiom,
    ! [Uu: set_int,Uua: fun_in4891666191172872833eger_o,Uub: code_integer,Uuc: int] :
      ( aa_int_o(aa_Cod8477617969808203498_int_o(aa_fun9103777551025077884_int_o(aTP_Lamm_adc(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_int_o(member_int(Uuc),Uu)
        & aa_Code_integer_o(aa_int5993532643906289898eger_o(Uua,Uuc),Uub) ) ) ).

% ATP.lambda_871
tff(fact_10563_ATP_Olambda__872,axiom,
    ! [Uu: set_int,Uua: fun_int_fun_o_o,Uub: $o,Uuc: int] :
      ( aa_int_o(aa_o_fun_int_o(aa_fun5351653887499120666_int_o(aTP_Lamm_alh(Uu),Uua),(Uub)),Uuc)
    <=> ( aa_set_int_o(member_int(Uuc),Uu)
        & aa_o_o(aa_int_fun_o_o(Uua,Uuc),(Uub)) ) ) ).

% ATP.lambda_872
tff(fact_10564_ATP_Olambda__873,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_rl(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_873
tff(fact_10565_ATP_Olambda__874,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_adb(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_874
tff(fact_10566_ATP_Olambda__875,axiom,
    ! [Uu: set_Code_integer,Uua: fun_Code_integer_nat,Uub: code_integer,Uuc: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(aa_fun7938718647838309544eger_o(aTP_Lamm_adm(Uu),Uua),Uub),Uuc)
    <=> ( aa_set5643387332196575111eger_o(member_Code_integer(Uuc),Uu)
        & ( aa_Code_integer_nat(Uua,Uuc) = aa_Code_integer_nat(Uua,Uub) ) ) ) ).

% ATP.lambda_875
tff(fact_10567_ATP_Olambda__876,axiom,
    ! [Uu: set_o,Uua: fun_o_Code_integer,Uub: $o,Uuc: $o] :
      ( aa_o_o(aa_o_fun_o_o(aa_fun5562237597604540066un_o_o(aTP_Lamm_adf(Uu),Uua),(Uub)),(Uuc))
    <=> ( member_o((Uuc),Uu)
        & ( aa_o_Code_integer(Uua,(Uuc)) = aa_o_Code_integer(Uua,(Uub)) ) ) ) ).

% ATP.lambda_876
tff(fact_10568_ATP_Olambda__877,axiom,
    ! [Uu: set_o,Uua: fun_o_nat,Uub: $o,Uuc: $o] :
      ( aa_o_o(aa_o_fun_o_o(aa_fun8960405049393213551un_o_o(aTP_Lamm_add(Uu),Uua),(Uub)),(Uuc))
    <=> ( member_o((Uuc),Uu)
        & ( aa_o_nat(Uua,(Uuc)) = aa_o_nat(Uua,(Uub)) ) ) ) ).

% ATP.lambda_877
tff(fact_10569_ATP_Olambda__878,axiom,
    ! [Uu: set_o,Uua: fun_o_int,Uub: $o,Uuc: $o] :
      ( aa_o_o(aa_o_fun_o_o(aa_fun3248236805073677075un_o_o(aTP_Lamm_ade(Uu),Uua),(Uub)),(Uuc))
    <=> ( member_o((Uuc),Uu)
        & ( aa_o_int(Uua,(Uuc)) = aa_o_int(Uua,(Uub)) ) ) ) ).

% ATP.lambda_878
tff(fact_10570_ATP_Olambda__879,axiom,
    ! [Uu: set_nat,Uua: fun_nat_Code_integer,Uub: nat,Uuc: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aa_fun8436538915998086722_nat_o(aTP_Lamm_adi(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_nat_o(member_nat(Uuc),Uu)
        & ( aa_nat_Code_integer(Uua,Uuc) = aa_nat_Code_integer(Uua,Uub) ) ) ) ).

% ATP.lambda_879
tff(fact_10571_ATP_Olambda__880,axiom,
    ! [Uu: set_nat,Uua: fun_nat_nat,Uub: nat,Uuc: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aa_fun6824225949033397647_nat_o(aTP_Lamm_adg(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_nat_o(member_nat(Uuc),Uu)
        & ( aa_nat_nat(Uua,Uuc) = aa_nat_nat(Uua,Uub) ) ) ) ).

% ATP.lambda_880
tff(fact_10572_ATP_Olambda__881,axiom,
    ! [Uu: set_nat,Uua: fun_nat_int,Uub: nat,Uuc: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aa_fun4569312790843823667_nat_o(aTP_Lamm_adh(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_nat_o(member_nat(Uuc),Uu)
        & ( aa_nat_int(Uua,Uuc) = aa_nat_int(Uua,Uub) ) ) ) ).

% ATP.lambda_881
tff(fact_10573_ATP_Olambda__882,axiom,
    ! [Uu: set_int,Uua: fun_int_Code_integer,Uub: int,Uuc: int] :
      ( aa_int_o(aa_int_fun_int_o(aa_fun5199375812828433454_int_o(aTP_Lamm_adl(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_int_o(member_int(Uuc),Uu)
        & ( aa_int_Code_integer(Uua,Uuc) = aa_int_Code_integer(Uua,Uub) ) ) ) ).

% ATP.lambda_882
tff(fact_10574_ATP_Olambda__883,axiom,
    ! [Uu: set_int,Uua: fun_int_nat,Uub: int,Uuc: int] :
      ( aa_int_o(aa_int_fun_int_o(aa_fun927631379751134075_int_o(aTP_Lamm_adj(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_int_o(member_int(Uuc),Uu)
        & ( aa_int_nat(Uua,Uuc) = aa_int_nat(Uua,Uub) ) ) ) ).

% ATP.lambda_883
tff(fact_10575_ATP_Olambda__884,axiom,
    ! [Uu: set_int,Uua: fun_int_int,Uub: int,Uuc: int] :
      ( aa_int_o(aa_int_fun_int_o(aa_fun7896090258416335903_int_o(aTP_Lamm_adk(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_int_o(member_int(Uuc),Uu)
        & ( aa_int_int(Uua,Uuc) = aa_int_int(Uua,Uub) ) ) ) ).

% ATP.lambda_884
tff(fact_10576_ATP_Olambda__885,axiom,
    ! [Uu: set_Code_integer,Uua: fun_Code_integer_nat,Uub: nat,Uuc: code_integer] :
      ( aa_Code_integer_o(aa_nat7420681472272698950eger_o(aa_fun4176659983681010805eger_o(aTP_Lamm_wm(Uu),Uua),Uub),Uuc)
    <=> ( aa_set5643387332196575111eger_o(member_Code_integer(Uuc),Uu)
        & ( aa_Code_integer_nat(Uua,Uuc) = Uub ) ) ) ).

% ATP.lambda_885
tff(fact_10577_ATP_Olambda__886,axiom,
    ! [Uu: set_Code_integer,Uua: fun_Code_integer_int,Uub: int,Uuc: code_integer] :
      ( aa_Code_integer_o(aa_int5993532643906289898eger_o(aa_fun7732714517052602301eger_o(aTP_Lamm_ws(Uu),Uua),Uub),Uuc)
    <=> ( aa_set5643387332196575111eger_o(member_Code_integer(Uuc),Uu)
        & ( aa_Code_integer_int(Uua,Uuc) = Uub ) ) ) ).

% ATP.lambda_886
tff(fact_10578_ATP_Olambda__887,axiom,
    ! [Uu: set_o,Uua: fun_o_nat,Uub: nat,Uuc: $o] :
      ( aa_o_o(aa_nat_fun_o_o(aa_fun953500542417269327un_o_o(aTP_Lamm_wi(Uu),Uua),Uub),(Uuc))
    <=> ( member_o((Uuc),Uu)
        & ( aa_o_nat(Uua,(Uuc)) = Uub ) ) ) ).

% ATP.lambda_887
tff(fact_10579_ATP_Olambda__888,axiom,
    ! [Uu: set_o,Uua: fun_o_int,Uub: int,Uuc: $o] :
      ( aa_o_o(aa_int_fun_o_o(aa_fun3522593817124631815un_o_o(aTP_Lamm_wo(Uu),Uua),Uub),(Uuc))
    <=> ( member_o((Uuc),Uu)
        & ( aa_o_int(Uua,(Uuc)) = Uub ) ) ) ).

% ATP.lambda_888
tff(fact_10580_ATP_Olambda__889,axiom,
    ! [Uu: set_nat,Uua: fun_nat_Code_integer,Uub: code_integer,Uuc: nat] :
      ( aa_nat_o(aa_Cod1075288278386115654_nat_o(aa_fun262808356775936629_nat_o(aTP_Lamm_yy(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_nat_o(member_nat(Uuc),Uu)
        & ( aa_nat_Code_integer(Uua,Uuc) = Uub ) ) ) ).

% ATP.lambda_889
tff(fact_10581_ATP_Olambda__890,axiom,
    ! [Uu: set_nat,Uua: fun_nat_nat,Uub: nat,Uuc: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aa_fun6824225949033397647_nat_o(aTP_Lamm_wv(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_nat_o(member_nat(Uuc),Uu)
        & ( aa_nat_nat(Uua,Uuc) = Uub ) ) ) ).

% ATP.lambda_890
tff(fact_10582_ATP_Olambda__891,axiom,
    ! [Uu: set_nat,Uua: fun_nat_int,Uub: int,Uuc: nat] :
      ( aa_nat_o(aa_int_fun_nat_o(aa_fun6124098986320894167_nat_o(aTP_Lamm_wq(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_nat_o(member_nat(Uuc),Uu)
        & ( aa_nat_int(Uua,Uuc) = Uub ) ) ) ).

% ATP.lambda_891
tff(fact_10583_ATP_Olambda__892,axiom,
    ! [Uu: set_int,Uua: fun_int_Code_integer,Uub: code_integer,Uuc: int] :
      ( aa_int_o(aa_Cod8477617969808203498_int_o(aa_fun7746027916420482493_int_o(aTP_Lamm_zc(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_int_o(member_int(Uuc),Uu)
        & ( aa_int_Code_integer(Uua,Uuc) = Uub ) ) ) ).

% ATP.lambda_892
tff(fact_10584_ATP_Olambda__893,axiom,
    ! [Uu: set_int,Uua: fun_int_nat,Uub: nat,Uuc: int] :
      ( aa_int_o(aa_nat_fun_int_o(aa_fun8596217221128839383_int_o(aTP_Lamm_wk(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_int_o(member_int(Uuc),Uu)
        & ( aa_int_nat(Uua,Uuc) = Uub ) ) ) ).

% ATP.lambda_893
tff(fact_10585_ATP_Olambda__894,axiom,
    ! [Uu: set_int,Uua: fun_int_int,Uub: int,Uuc: int] :
      ( aa_int_o(aa_int_fun_int_o(aa_fun7896090258416335903_int_o(aTP_Lamm_wy(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_int_o(member_int(Uuc),Uu)
        & ( aa_int_int(Uua,Uuc) = Uub ) ) ) ).

% ATP.lambda_894
tff(fact_10586_ATP_Olambda__895,axiom,
    ! [Uu: code_integer,Uua: nat,Uub: code_integer,Uuc: nat] : aa_nat_Code_integer(aa_Cod1578151519121547077nteger(aa_nat2351529209232473362nteger(aTP_Lamm_ail(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_895
tff(fact_10587_ATP_Olambda__896,axiom,
    ! [Uu: rat,Uua: nat,Uub: rat,Uuc: nat] : aa_nat_rat(aa_rat_fun_nat_rat(aa_nat298952458717073160at_rat(aTP_Lamm_aim(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_896
tff(fact_10588_ATP_Olambda__897,axiom,
    ! [Uu: int,Uua: nat,Uub: int,Uuc: nat] : aa_nat_int(aa_int_fun_nat_int(aa_nat6704147566042551216at_int(aTP_Lamm_ain(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_897
tff(fact_10589_ATP_Olambda__898,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_abn(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_898
tff(fact_10590_ATP_Olambda__899,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] :
      ( aa_int_o(aa_int_fun_int_o(aa_int2464330468366073224_int_o(aTP_Lamm_any(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_899
tff(fact_10591_ATP_Olambda__900,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aa_nat8314798964848609180_nat_o(aTP_Lamm_amv(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_900
tff(fact_10592_ATP_Olambda__901,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] :
      ( aa_int_o(aa_int_fun_int_o(aa_int2464330468366073224_int_o(aTP_Lamm_anw(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_901
tff(fact_10593_ATP_Olambda__902,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aa_nat8314798964848609180_nat_o(aTP_Lamm_amt(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_902
tff(fact_10594_ATP_Olambda__903,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] : aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aa_nat4158579785191972357at_nat(aTP_Lamm_amx(Uu),Uua),Uub),Uuc) = aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_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_903
tff(fact_10595_ATP_Olambda__904,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] : aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aa_nat4158579785191972357at_nat(aTP_Lamm_amz(Uu),Uua),Uub),Uuc) = aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_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_904
tff(fact_10596_ATP_Olambda__905,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aa_nat8314798964848609180_nat_o(aTP_Lamm_aos(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_905
tff(fact_10597_ATP_Olambda__906,axiom,
    ! [Uu: fun_list_nat_o,Uua: set_list_nat,Uub: fun_o_o,Uuc: list_nat] :
      ( aa_list_nat_o(aa_fun3550071730982498768_nat_o(aa_set7739190917479332627_nat_o(aTP_Lamm_jo(Uu),Uua),Uub),Uuc)
    <=> ( member_list_nat(Uuc,Uua)
        & aa_o_o(Uub,aa_list_nat_o(Uu,Uuc)) ) ) ).

% ATP.lambda_906
tff(fact_10598_ATP_Olambda__907,axiom,
    ! [Uu: fun_o_o,Uua: set_o,Uub: fun_o_o,Uuc: $o] :
      ( aa_o_o(aa_fun_o_o_fun_o_o(aa_set8294393945046410815un_o_o(aTP_Lamm_iw(Uu),Uua),Uub),(Uuc))
    <=> ( member_o((Uuc),Uua)
        & aa_o_o(Uub,aa_o_o(Uu,(Uuc))) ) ) ).

% ATP.lambda_907
tff(fact_10599_ATP_Olambda__908,axiom,
    ! [Uu: fun_o_nat,Uua: set_o,Uub: fun_nat_o,Uuc: $o] :
      ( aa_o_o(aa_fun_nat_o_fun_o_o(aa_set26024703028173499un_o_o(aTP_Lamm_jc(Uu),Uua),Uub),(Uuc))
    <=> ( member_o((Uuc),Uua)
        & aa_nat_o(Uub,aa_o_nat(Uu,(Uuc))) ) ) ).

% ATP.lambda_908
tff(fact_10600_ATP_Olambda__909,axiom,
    ! [Uu: fun_o_int,Uua: set_o,Uub: fun_int_o,Uuc: $o] :
      ( aa_o_o(aa_fun_int_o_fun_o_o(aa_set6365818687085129567un_o_o(aTP_Lamm_ji(Uu),Uua),Uub),(Uuc))
    <=> ( member_o((Uuc),Uua)
        & aa_int_o(Uub,aa_o_int(Uu,(Uuc))) ) ) ).

% ATP.lambda_909
tff(fact_10601_ATP_Olambda__910,axiom,
    ! [Uu: fun_nat_o,Uua: set_nat,Uub: fun_o_o,Uuc: nat] :
      ( aa_nat_o(aa_fun_o_o_fun_nat_o(aa_set2987053916987590195_nat_o(aTP_Lamm_iy(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_nat_o(member_nat(Uuc),Uua)
        & aa_o_o(Uub,aa_nat_o(Uu,Uuc)) ) ) ).

% ATP.lambda_910
tff(fact_10602_ATP_Olambda__911,axiom,
    ! [Uu: fun_nat_nat,Uua: set_nat,Uub: fun_nat_o,Uuc: nat] :
      ( aa_nat_o(aa_fun5937193402907943164_nat_o(aa_set1405850572773641671_nat_o(aTP_Lamm_je(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_nat_o(member_nat(Uuc),Uua)
        & aa_nat_o(Uub,aa_nat_nat(Uu,Uuc)) ) ) ).

% ATP.lambda_911
tff(fact_10603_ATP_Olambda__912,axiom,
    ! [Uu: fun_nat_int,Uua: set_nat,Uub: fun_int_o,Uuc: nat] :
      ( aa_nat_o(aa_fun8064560132910005592_nat_o(aa_set3533217302775704099_nat_o(aTP_Lamm_jk(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_nat_o(member_nat(Uuc),Uua)
        & aa_int_o(Uub,aa_nat_int(Uu,Uuc)) ) ) ).

% ATP.lambda_912
tff(fact_10604_ATP_Olambda__913,axiom,
    ! [Uu: fun_int_o,Uua: set_int,Uub: fun_o_o,Uuc: int] :
      ( aa_int_o(aa_fun_o_o_fun_int_o(aa_set2419422181098893691_int_o(aTP_Lamm_ja(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_int_o(member_int(Uuc),Uua)
        & aa_o_o(Uub,aa_int_o(Uu,Uuc)) ) ) ).

% ATP.lambda_913
tff(fact_10605_ATP_Olambda__914,axiom,
    ! [Uu: fun_int_nat,Uua: set_int,Uub: fun_nat_o,Uuc: int] :
      ( aa_int_o(aa_fun4116151057475255200_int_o(aa_set7541805066622444743_int_o(aTP_Lamm_jg(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_int_o(member_int(Uuc),Uua)
        & aa_nat_o(Uub,aa_int_nat(Uu,Uuc)) ) ) ).

% ATP.lambda_914
tff(fact_10606_ATP_Olambda__915,axiom,
    ! [Uu: fun_int_int,Uua: set_int,Uub: fun_int_o,Uuc: int] :
      ( aa_int_o(aa_fun6243517787477317628_int_o(aa_set445799759769731363_int_o(aTP_Lamm_jm(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_int_o(member_int(Uuc),Uua)
        & aa_int_o(Uub,aa_int_int(Uu,Uuc)) ) ) ).

% ATP.lambda_915
tff(fact_10607_ATP_Olambda__916,axiom,
    ! [Uu: set_Code_integer,Uua: fun_Co8499238673658658103nteger,Uub: fun_Co8499238673658658103nteger,Uuc: code_integer] :
      ( aa_Code_integer_o(aa_fun7159535342743012639eger_o(aa_fun5296778454019918337eger_o(aTP_Lamm_uw(Uu),Uua),Uub),Uuc)
    <=> ( aa_set5643387332196575111eger_o(member_Code_integer(Uuc),Uu)
        & ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(Uua,Uuc)),aa_Cod1783974652533396728nteger(Uub,Uuc)) != one_one_Code_integer ) ) ) ).

% ATP.lambda_916
tff(fact_10608_ATP_Olambda__917,axiom,
    ! [Uu: set_Code_integer,Uua: fun_Co4490034186337852034r_assn,Uub: fun_Co4490034186337852034r_assn,Uuc: code_integer] :
      ( aa_Code_integer_o(aa_fun6538703224075513620eger_o(aa_fun4422309413433313771eger_o(aTP_Lamm_ve(Uu),Uua),Uub),Uuc)
    <=> ( aa_set5643387332196575111eger_o(member_Code_integer(Uuc),Uu)
        & ( aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_Code_integer_assn(Uua,Uuc)),aa_Code_integer_assn(Uub,Uuc)) != one_one_assn ) ) ) ).

% ATP.lambda_917
tff(fact_10609_ATP_Olambda__918,axiom,
    ! [Uu: set_o,Uua: fun_o_Code_integer,Uub: fun_o_Code_integer,Uuc: $o] :
      ( aa_o_o(aa_fun8192762245619933327un_o_o(aa_fun1865401534164435218un_o_o(aTP_Lamm_uq(Uu),Uua),Uub),(Uuc))
    <=> ( member_o((Uuc),Uu)
        & ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_o_Code_integer(Uua,(Uuc))),aa_o_Code_integer(Uub,(Uuc))) != one_one_Code_integer ) ) ) ).

% ATP.lambda_918
tff(fact_10610_ATP_Olambda__919,axiom,
    ! [Uu: set_o,Uua: fun_o_assn,Uub: fun_o_assn,Uuc: $o] :
      ( aa_o_o(aa_fun8382565201914911066un_o_o(aa_fun5794833117214906962un_o_o(aTP_Lamm_uy(Uu),Uua),Uub),(Uuc))
    <=> ( member_o((Uuc),Uu)
        & ( aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_o_assn(Uua,(Uuc))),aa_o_assn(Uub,(Uuc))) != one_one_assn ) ) ) ).

% ATP.lambda_919
tff(fact_10611_ATP_Olambda__920,axiom,
    ! [Uu: set_o,Uua: fun_o_rat,Uub: fun_o_rat,Uuc: $o] :
      ( aa_o_o(aa_fun_o_rat_fun_o_o(aa_fun5510981055783399506un_o_o(aTP_Lamm_vg(Uu),Uua),Uub),(Uuc))
    <=> ( member_o((Uuc),Uu)
        & ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_o_rat(Uua,(Uuc))),aa_o_rat(Uub,(Uuc))) != one_one_rat ) ) ) ).

% ATP.lambda_920
tff(fact_10612_ATP_Olambda__921,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_us(Uu),Uua),Uub),Uuc)
    <=> ( aa_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_921
tff(fact_10613_ATP_Olambda__922,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_va(Uu),Uua),Uub),Uuc)
    <=> ( aa_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_922
tff(fact_10614_ATP_Olambda__923,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_vi(Uu),Uua),Uub),Uuc)
    <=> ( aa_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_923
tff(fact_10615_ATP_Olambda__924,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_uu(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_924
tff(fact_10616_ATP_Olambda__925,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_vc(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_925
tff(fact_10617_ATP_Olambda__926,axiom,
    ! [Uu: set_Code_integer,Uua: fun_Co8499238673658658103nteger,Uub: fun_Co8499238673658658103nteger,Uuc: code_integer] :
      ( aa_Code_integer_o(aa_fun7159535342743012639eger_o(aa_fun5296778454019918337eger_o(aTP_Lamm_uc(Uu),Uua),Uub),Uuc)
    <=> ( aa_set5643387332196575111eger_o(member_Code_integer(Uuc),Uu)
        & ( aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(Uua,Uuc)),aa_Cod1783974652533396728nteger(Uub,Uuc)) != zero_z3403309356797280102nteger ) ) ) ).

% ATP.lambda_926
tff(fact_10618_ATP_Olambda__927,axiom,
    ! [Uu: set_Code_integer,Uua: fun_Code_integer_rat,Uub: fun_Code_integer_rat,Uuc: code_integer] :
      ( aa_Code_integer_o(aa_fun6395433285121891940eger_o(aa_fun4095556841599367307eger_o(aTP_Lamm_uk(Uu),Uua),Uub),Uuc)
    <=> ( aa_set5643387332196575111eger_o(member_Code_integer(Uuc),Uu)
        & ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_Code_integer_rat(Uua,Uuc)),aa_Code_integer_rat(Uub,Uuc)) != zero_zero_rat ) ) ) ).

% ATP.lambda_927
tff(fact_10619_ATP_Olambda__928,axiom,
    ! [Uu: set_o,Uua: fun_o_Code_integer,Uub: fun_o_Code_integer,Uuc: $o] :
      ( aa_o_o(aa_fun8192762245619933327un_o_o(aa_fun1865401534164435218un_o_o(aTP_Lamm_tw(Uu),Uua),Uub),(Uuc))
    <=> ( member_o((Uuc),Uu)
        & ( aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_o_Code_integer(Uua,(Uuc))),aa_o_Code_integer(Uub,(Uuc))) != zero_z3403309356797280102nteger ) ) ) ).

% ATP.lambda_928
tff(fact_10620_ATP_Olambda__929,axiom,
    ! [Uu: set_o,Uua: fun_o_rat,Uub: fun_o_rat,Uuc: $o] :
      ( aa_o_o(aa_fun_o_rat_fun_o_o(aa_fun5510981055783399506un_o_o(aTP_Lamm_ue(Uu),Uua),Uub),(Uuc))
    <=> ( member_o((Uuc),Uu)
        & ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_o_rat(Uua,(Uuc))),aa_o_rat(Uub,(Uuc))) != zero_zero_rat ) ) ) ).

% ATP.lambda_929
tff(fact_10621_ATP_Olambda__930,axiom,
    ! [Uu: set_o,Uua: fun_o_nat,Uub: fun_o_nat,Uuc: $o] :
      ( aa_o_o(aa_fun_o_nat_fun_o_o(aa_fun5833554292436378194un_o_o(aTP_Lamm_um(Uu),Uua),Uub),(Uuc))
    <=> ( member_o((Uuc),Uu)
        & ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_o_nat(Uua,(Uuc))),aa_o_nat(Uub,(Uuc))) != zero_zero_nat ) ) ) ).

% ATP.lambda_930
tff(fact_10622_ATP_Olambda__931,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_ty(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_nat_o(member_nat(Uuc),Uu)
        & ( aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_nat_Code_integer(Uua,Uuc)),aa_nat_Code_integer(Uub,Uuc)) != zero_z3403309356797280102nteger ) ) ) ).

% ATP.lambda_931
tff(fact_10623_ATP_Olambda__932,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_ug(Uu),Uua),Uub),Uuc)
    <=> ( aa_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_932
tff(fact_10624_ATP_Olambda__933,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_uo(Uu),Uua),Uub),Uuc)
    <=> ( aa_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_933
tff(fact_10625_ATP_Olambda__934,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_ua(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_int_o(member_int(Uuc),Uu)
        & ( aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_int_Code_integer(Uua,Uuc)),aa_int_Code_integer(Uub,Uuc)) != zero_z3403309356797280102nteger ) ) ) ).

% ATP.lambda_934
tff(fact_10626_ATP_Olambda__935,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_ui(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_935
tff(fact_10627_ATP_Olambda__936,axiom,
    ! [Uu: fun_o_Code_integer,Uua: code_integer,Uub: list_o,Uuc: nat] : aa_nat_Code_integer(aa_lis2656634407596002188nteger(aa_Cod901939952664962432nteger(aTP_Lamm_aiw(Uu),Uua),Uub),Uuc) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_o_Code_integer(Uu,aa_nat_o(nth_o(Uub),Uuc))),aa_nat_Code_integer(power_8256067586552552935nteger(Uua),Uuc)) ).

% ATP.lambda_936
tff(fact_10628_ATP_Olambda__937,axiom,
    ! [Uu: fun_o_rat,Uua: rat,Uub: list_o,Uuc: nat] : aa_nat_rat(aa_lis2932070769283852167at_rat(aa_rat5396404679109611328at_rat(aTP_Lamm_aiz(Uu),Uua),Uub),Uuc) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_o_rat(Uu,aa_nat_o(nth_o(Uub),Uuc))),aa_nat_rat(power_power_rat(Uua),Uuc)) ).

% ATP.lambda_937
tff(fact_10629_ATP_Olambda__938,axiom,
    ! [Uu: fun_nat_Code_integer,Uua: code_integer,Uub: list_nat,Uuc: nat] : aa_nat_Code_integer(aa_lis7000467330633774408nteger(aa_Cod9048065728759425570nteger(aTP_Lamm_aiv(Uu),Uua),Uub),Uuc) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(Uu,aa_nat_nat(nth_nat(Uub),Uuc))),aa_nat_Code_integer(power_8256067586552552935nteger(Uua),Uuc)) ).

% ATP.lambda_938
tff(fact_10630_ATP_Olambda__939,axiom,
    ! [Uu: fun_nat_rat,Uua: rat,Uub: list_nat,Uuc: nat] : aa_nat_rat(aa_lis7014846069874721091at_rat(aa_rat4396832037704353944at_rat(aTP_Lamm_aiy(Uu),Uua),Uub),Uuc) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(Uu,aa_nat_nat(nth_nat(Uub),Uuc))),aa_nat_rat(power_power_rat(Uua),Uuc)) ).

% ATP.lambda_939
tff(fact_10631_ATP_Olambda__940,axiom,
    ! [Uu: fun_nat_nat,Uua: nat,Uub: list_nat,Uuc: nat] : aa_nat_nat(aa_lis1299702591413342011at_nat(aa_nat9125522253416007816at_nat(aTP_Lamm_ajd(Uu),Uua),Uub),Uuc) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(Uu,aa_nat_nat(nth_nat(Uub),Uuc))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Uua),Uuc)) ).

% ATP.lambda_940
tff(fact_10632_ATP_Olambda__941,axiom,
    ! [Uu: fun_nat_int,Uua: int,Uub: list_nat,Uuc: nat] : aa_nat_int(aa_lis6345223608758921111at_int(aa_int7844622935458978624at_int(aTP_Lamm_ajb(Uu),Uua),Uub),Uuc) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(Uu,aa_nat_nat(nth_nat(Uub),Uuc))),aa_nat_int(aa_int_fun_nat_int(power_power_int,Uua),Uuc)) ).

% ATP.lambda_941
tff(fact_10633_ATP_Olambda__942,axiom,
    ! [Uu: fun_int_Code_integer,Uua: code_integer,Uub: list_int,Uuc: nat] : aa_nat_Code_integer(aa_lis2169558292809893284nteger(aa_Cod4217156690935544446nteger(aTP_Lamm_aiu(Uu),Uua),Uub),Uuc) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_int_Code_integer(Uu,aa_nat_int(nth_int(Uub),Uuc))),aa_nat_Code_integer(power_8256067586552552935nteger(Uua),Uuc)) ).

% ATP.lambda_942
tff(fact_10634_ATP_Olambda__943,axiom,
    ! [Uu: fun_int_rat,Uua: rat,Uub: list_int,Uuc: nat] : aa_nat_rat(aa_lis1742918661435654559at_rat(aa_rat8348276666120063220at_rat(aTP_Lamm_aix(Uu),Uua),Uub),Uuc) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_int_rat(Uu,aa_nat_int(nth_int(Uub),Uuc))),aa_nat_rat(power_power_rat(Uua),Uuc)) ).

% ATP.lambda_943
tff(fact_10635_ATP_Olambda__944,axiom,
    ! [Uu: fun_int_nat,Uua: nat,Uub: list_int,Uuc: nat] : aa_nat_nat(aa_lis5251147219829051287at_nat(aa_nat3853594844976941284at_nat(aTP_Lamm_ajc(Uu),Uua),Uub),Uuc) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_int_nat(Uu,aa_nat_int(nth_int(Uub),Uuc))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Uua),Uuc)) ).

% ATP.lambda_944
tff(fact_10636_ATP_Olambda__945,axiom,
    ! [Uu: fun_int_int,Uua: int,Uub: list_int,Uuc: nat] : aa_nat_int(aa_lis1073296200319854579at_int(aa_int2572695527019912092at_int(aTP_Lamm_aja(Uu),Uua),Uub),Uuc) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(Uu,aa_nat_int(nth_int(Uub),Uuc))),aa_nat_int(aa_int_fun_nat_int(power_power_int,Uua),Uuc)) ).

% ATP.lambda_945
tff(fact_10637_ATP_Olambda__946,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_lr(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_946
tff(fact_10638_ATP_Olambda__947,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_lt(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_947
tff(fact_10639_ATP_Olambda__948,axiom,
    ! [Uu: set_Code_integer,Uua: fun_na5492297256294505719er_nat,Uub: fun_na6318815019539281885eger_o,Uuc: nat] : aa_nat_nat(aa_fun2058600731543071806at_nat(aa_fun4864976072519958552at_nat(aTP_Lamm_tr(Uu),Uua),Uub),Uuc) = groups3190895334310489300er_nat(aa_nat5852537043073606456er_nat(Uua,Uuc),collect_Code_integer(aa_nat7420681472272698950eger_o(aa_fun8665957533616019964eger_o(aTP_Lamm_so(Uu),Uub),Uuc))) ).

% ATP.lambda_948
tff(fact_10640_ATP_Olambda__949,axiom,
    ! [Uu: set_o,Uua: fun_nat_fun_o_nat,Uub: fun_nat_fun_o_o,Uuc: nat] : aa_nat_nat(aa_fun4178193321845537899at_nat(aa_fun3391070971708407026at_nat(aTP_Lamm_tn(Uu),Uua),Uub),Uuc) = groups3504817904513533571_o_nat(aa_nat_fun_o_nat(Uua,Uuc),collect_o(aa_nat_fun_o_o(aa_fun3555063939605388254un_o_o(aTP_Lamm_sg(Uu),Uub),Uuc))) ).

% ATP.lambda_949
tff(fact_10641_ATP_Olambda__950,axiom,
    ! [Uu: set_int,Uua: fun_Co3009391273526075439nt_int,Uub: fun_Co8353275306710069249_int_o,Uuc: code_integer] : aa_Code_integer_int(aa_fun5771563549434713419er_int(aa_fun4265095070934345581er_int(aTP_Lamm_tl(Uu),Uua),Uub),Uuc) = groups1705073143266064639nt_int(aa_Cod5491795744640130032nt_int(Uua,Uuc),collect_int(aa_Cod8477617969808203498_int_o(aa_fun1593725752699181308_int_o(aTP_Lamm_sb(Uu),Uub),Uuc))) ).

% ATP.lambda_950
tff(fact_10642_ATP_Olambda__951,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_tp(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_rx(Uu),Uub),Uuc))) ).

% ATP.lambda_951
tff(fact_10643_ATP_Olambda__952,axiom,
    ! [Uu: set_int,Uua: fun_o_fun_int_int,Uub: fun_o_fun_int_o,Uuc: $o] : aa_o_int(aa_fun7462986433439723305_o_int(aa_fun4201161283678742396_o_int(aTP_Lamm_tj(Uu),Uua),Uub),(Uuc)) = groups1705073143266064639nt_int(aa_o_fun_int_int(Uua,(Uuc)),collect_int(aa_o_fun_int_o(aa_fun1733087058159411102_int_o(aTP_Lamm_rs(Uu),Uub),(Uuc)))) ).

% ATP.lambda_952
tff(fact_10644_ATP_Olambda__953,axiom,
    ! [Uu: set_nat,Uua: fun_Co3312552843764780407at_nat,Uub: fun_Co1375048237525739357_nat_o,Uuc: code_integer] : aa_Code_integer_nat(aa_fun6683192525214198859er_nat(aa_fun2143651634668453157er_nat(aTP_Lamm_tb(Uu),Uua),Uub),Uuc) = aa_set_nat_nat(groups708209901874060359at_nat(aa_Cod8670293052248272184at_nat(Uua,Uuc)),collect_nat(aa_Cod1075288278386115654_nat_o(aa_fun2430183811687954940_nat_o(aTP_Lamm_rn(Uu),Uub),Uuc))) ).

% ATP.lambda_953
tff(fact_10645_ATP_Olambda__954,axiom,
    ! [Uu: set_nat,Uua: fun_Co822469698833053651at_int,Uub: fun_Co1375048237525739357_nat_o,Uuc: code_integer] : aa_Code_integer_int(aa_fun2505341505705002151er_int(aa_fun493137163814837797er_int(aTP_Lamm_tg(Uu),Uua),Uub),Uuc) = aa_set_nat_int(groups705719431365010083at_int(aa_Cod4492442032739075476at_int(Uua,Uuc)),collect_nat(aa_Cod1075288278386115654_nat_o(aa_fun2430183811687954940_nat_o(aTP_Lamm_rn(Uu),Uub),Uuc))) ).

% ATP.lambda_954
tff(fact_10646_ATP_Olambda__955,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_sz(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_ri(Uu),Uub),Uuc))) ).

% ATP.lambda_955
tff(fact_10647_ATP_Olambda__956,axiom,
    ! [Uu: set_nat,Uua: fun_o_fun_nat_nat,Uub: fun_o_fun_nat_o,Uuc: $o] : aa_o_nat(aa_fun6229755271245457521_o_nat(aa_fun802073206315457404_o_nat(aTP_Lamm_sx(Uu),Uua),Uub),(Uuc)) = aa_set_nat_nat(groups708209901874060359at_nat(aa_o_fun_nat_nat(Uua,(Uuc))),collect_nat(aa_o_fun_nat_o(aa_fun9138341750119017558_nat_o(aTP_Lamm_rd(Uu),Uub),(Uuc)))) ).

% ATP.lambda_956
tff(fact_10648_ATP_Olambda__957,axiom,
    ! [Uu: set_nat,Uua: fun_o_fun_nat_int,Uub: fun_o_fun_nat_o,Uuc: $o] : aa_o_int(aa_fun2051904251736260813_o_int(aa_fun7508458648369839740_o_int(aTP_Lamm_td(Uu),Uua),Uub),(Uuc)) = aa_set_nat_int(groups705719431365010083at_int(aa_o_fun_nat_int(Uua,(Uuc))),collect_nat(aa_o_fun_nat_o(aa_fun9138341750119017558_nat_o(aTP_Lamm_rd(Uu),Uub),(Uuc)))) ).

% ATP.lambda_957
tff(fact_10649_ATP_Olambda__958,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_su(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_st(Uu),Uub),Uuc))) ).

% ATP.lambda_958
tff(fact_10650_ATP_Olambda__959,axiom,
    ! [Uu: set_Code_integer,Uua: fun_na5492297256294505719er_nat,Uub: fun_na6318815019539281885eger_o,Uuc: nat] : aa_nat_nat(aa_fun2058600731543071806at_nat(aa_fun4864976072519958552at_nat(aTP_Lamm_sp(Uu),Uua),Uub),Uuc) = groups7237345082560585321er_nat(aa_nat5852537043073606456er_nat(Uua,Uuc),collect_Code_integer(aa_nat7420681472272698950eger_o(aa_fun8665957533616019964eger_o(aTP_Lamm_so(Uu),Uub),Uuc))) ).

% ATP.lambda_959
tff(fact_10651_ATP_Olambda__960,axiom,
    ! [Uu: set_o,Uua: fun_nat_fun_o_nat,Uub: fun_nat_fun_o_o,Uuc: nat] : aa_nat_nat(aa_fun4178193321845537899at_nat(aa_fun3391070971708407026at_nat(aTP_Lamm_sh(Uu),Uua),Uub),Uuc) = groups8507830703676809646_o_nat(aa_nat_fun_o_nat(Uua,Uuc),collect_o(aa_nat_fun_o_o(aa_fun3555063939605388254un_o_o(aTP_Lamm_sg(Uu),Uub),Uuc))) ).

% ATP.lambda_960
tff(fact_10652_ATP_Olambda__961,axiom,
    ! [Uu: set_int,Uua: fun_Co3009391273526075439nt_int,Uub: fun_Co8353275306710069249_int_o,Uuc: code_integer] : aa_Code_integer_int(aa_fun5771563549434713419er_int(aa_fun4265095070934345581er_int(aTP_Lamm_sc(Uu),Uua),Uub),Uuc) = groups4538972089207619220nt_int(aa_Cod5491795744640130032nt_int(Uua,Uuc),collect_int(aa_Cod8477617969808203498_int_o(aa_fun1593725752699181308_int_o(aTP_Lamm_sb(Uu),Uub),Uuc))) ).

% ATP.lambda_961
tff(fact_10653_ATP_Olambda__962,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_sl(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_rx(Uu),Uub),Uuc))) ).

% ATP.lambda_962
tff(fact_10654_ATP_Olambda__963,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_ry(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_rx(Uu),Uub),Uuc))) ).

% ATP.lambda_963
tff(fact_10655_ATP_Olambda__964,axiom,
    ! [Uu: set_int,Uua: fun_o_fun_int_int,Uub: fun_o_fun_int_o,Uuc: $o] : aa_o_int(aa_fun7462986433439723305_o_int(aa_fun4201161283678742396_o_int(aTP_Lamm_rt(Uu),Uua),Uub),(Uuc)) = groups4538972089207619220nt_int(aa_o_fun_int_int(Uua,(Uuc)),collect_int(aa_o_fun_int_o(aa_fun1733087058159411102_int_o(aTP_Lamm_rs(Uu),Uub),(Uuc)))) ).

% ATP.lambda_964
tff(fact_10656_ATP_Olambda__965,axiom,
    ! [Uu: set_nat,Uua: fun_Co3312552843764780407at_nat,Uub: fun_Co1375048237525739357_nat_o,Uuc: code_integer] : aa_Code_integer_nat(aa_fun6683192525214198859er_nat(aa_fun2143651634668453157er_nat(aTP_Lamm_ro(Uu),Uua),Uub),Uuc) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_Cod8670293052248272184at_nat(Uua,Uuc)),collect_nat(aa_Cod1075288278386115654_nat_o(aa_fun2430183811687954940_nat_o(aTP_Lamm_rn(Uu),Uub),Uuc))) ).

% ATP.lambda_965
tff(fact_10657_ATP_Olambda__966,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_rj(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_ri(Uu),Uub),Uuc))) ).

% ATP.lambda_966
tff(fact_10658_ATP_Olambda__967,axiom,
    ! [Uu: set_nat,Uua: fun_o_fun_nat_nat,Uub: fun_o_fun_nat_o,Uuc: $o] : aa_o_nat(aa_fun6229755271245457521_o_nat(aa_fun802073206315457404_o_nat(aTP_Lamm_re(Uu),Uua),Uub),(Uuc)) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_o_fun_nat_nat(Uua,(Uuc))),collect_nat(aa_o_fun_nat_o(aa_fun9138341750119017558_nat_o(aTP_Lamm_rd(Uu),Uub),(Uuc)))) ).

% ATP.lambda_967
tff(fact_10659_ATP_Olambda__968,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_aib((Uu)),Uua),Uub),Uuc)
    <=> ( (Uu)
        & aa_int_o(aa_int_fun_int_o(Uua,Uub),Uuc) ) ) ).

% ATP.lambda_968
tff(fact_10660_ATP_Olambda__969,axiom,
    ! [Uu: fun_a_2990225006182978344t_unit,Uua: a,Uub: heap_e7401611519738050253t_unit,Uuc: nat] : aa_nat3804989708481012326it_nat(aa_hea5083147195637387702it_nat(aa_a_f112692477493148739it_nat(aTP_Lamm_ahh(Uu),Uua),Uub),Uuc) = heap_T3616969660504097270t_unit(Uuc,aa_hea684056619475940329it_nat(heap_T875086893843062177t_unit(aa_a_H4862568901524997081t_unit(Uu,Uua)),Uub)) ).

% ATP.lambda_969
tff(fact_10661_ATP_Olambda__970,axiom,
    ! [Uu: fun_a_6093291656547882009Heap_a,Uua: a,Uub: heap_e7401611519738050253t_unit,Uuc: nat] : aa_nat3190776082673941207it_nat(aa_hea414212936809421479it_nat(aa_a_f1761598956578211508it_nat(aTP_Lamm_ahd(Uu),Uua),Uub),Uuc) = heap_T7616092557645711335rame_a(Uuc,aa_hea4724376117875161946it_nat(heap_Time_execute_a(aa_a_H8829438773920479562Heap_a(Uu,Uua)),Uub)) ).

% ATP.lambda_970
tff(fact_10662_ATP_Olambda__971,axiom,
    ! [Uu: set_nat,Uua: fun_nat_nat,Uub: fun_nat_nat,Uuc: nat] : aa_nat_nat(aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(aTP_Lamm_xs(Uu),Uua),Uub),Uuc) = aa_set_nat_nat(groups708209901874060359at_nat(Uua),collect_nat(aa_nat_fun_nat_o(aa_fun6824225949033397647_nat_o(aTP_Lamm_wv(Uu),Uub),Uuc))) ).

% ATP.lambda_971
tff(fact_10663_ATP_Olambda__972,axiom,
    ! [Uu: set_nat,Uua: fun_nat_int,Uub: fun_nat_nat,Uuc: nat] : aa_nat_int(aa_fun1613598493704395220at_int(aa_fun1189263540488932170at_int(aTP_Lamm_xt(Uu),Uua),Uub),Uuc) = aa_set_nat_int(groups705719431365010083at_int(Uua),collect_nat(aa_nat_fun_nat_o(aa_fun6824225949033397647_nat_o(aTP_Lamm_wv(Uu),Uub),Uuc))) ).

% ATP.lambda_972
tff(fact_10664_ATP_Olambda__973,axiom,
    ! [Uu: set_Code_integer,Uua: fun_Code_integer_int,Uub: fun_Code_integer_int,Uuc: int] : aa_int_int(aa_fun2480750449653600025nt_int(aa_fun6168136379806246172nt_int(aTP_Lamm_xq(Uu),Uua),Uub),Uuc) = groups3188404863801439024er_int(Uua,collect_Code_integer(aa_int5993532643906289898eger_o(aa_fun7732714517052602301eger_o(aTP_Lamm_ws(Uu),Uub),Uuc))) ).

% ATP.lambda_973
tff(fact_10665_ATP_Olambda__974,axiom,
    ! [Uu: set_nat,Uua: fun_nat_nat,Uub: fun_nat_int,Uuc: int] : aa_int_nat(aa_fun1518875816675579952nt_nat(aa_fun3769689747080337738nt_nat(aTP_Lamm_xr(Uu),Uua),Uub),Uuc) = aa_set_nat_nat(groups708209901874060359at_nat(Uua),collect_nat(aa_int_fun_nat_o(aa_fun6124098986320894167_nat_o(aTP_Lamm_wq(Uu),Uub),Uuc))) ).

% ATP.lambda_974
tff(fact_10666_ATP_Olambda__975,axiom,
    ! [Uu: set_o,Uua: fun_o_int,Uub: fun_o_int,Uuc: int] : aa_int_int(aa_fun1998554717657234288nt_int(aa_fun5578626826553213156nt_int(aTP_Lamm_xp(Uu),Uua),Uub),Uuc) = groups3502327434004483295_o_int(Uua,collect_o(aa_int_fun_o_o(aa_fun3522593817124631815un_o_o(aTP_Lamm_wo(Uu),Uub),Uuc))) ).

% ATP.lambda_975
tff(fact_10667_ATP_Olambda__976,axiom,
    ! [Uu: set_Code_integer,Uua: fun_Code_integer_nat,Uub: fun_Code_integer_nat,Uuc: nat] : aa_nat_nat(aa_fun1707803128846032901at_nat(aa_fun2075405564639334828at_nat(aTP_Lamm_xm(Uu),Uua),Uub),Uuc) = groups3190895334310489300er_nat(Uua,collect_Code_integer(aa_nat7420681472272698950eger_o(aa_fun4176659983681010805eger_o(aTP_Lamm_wm(Uu),Uub),Uuc))) ).

% ATP.lambda_976
tff(fact_10668_ATP_Olambda__977,axiom,
    ! [Uu: set_Code_integer,Uua: fun_Code_integer_int,Uub: fun_Code_integer_nat,Uuc: nat] : aa_nat_int(aa_fun6753324146191612001at_int(aa_fun29770176697515108at_int(aTP_Lamm_xo(Uu),Uua),Uub),Uuc) = groups3188404863801439024er_int(Uua,collect_Code_integer(aa_nat7420681472272698950eger_o(aa_fun4176659983681010805eger_o(aTP_Lamm_wm(Uu),Uub),Uuc))) ).

% ATP.lambda_977
tff(fact_10669_ATP_Olambda__978,axiom,
    ! [Uu: set_int,Uua: fun_int_nat,Uub: fun_int_nat,Uuc: nat] : aa_nat_nat(aa_fun5507398438843482068at_nat(aa_fun3220669823809117514at_nat(aTP_Lamm_xl(Uu),Uua),Uub),Uuc) = groups1707563613775114915nt_nat(Uua,collect_int(aa_nat_fun_int_o(aa_fun8596217221128839383_int_o(aTP_Lamm_wk(Uu),Uub),Uuc))) ).

% ATP.lambda_978
tff(fact_10670_ATP_Olambda__979,axiom,
    ! [Uu: set_o,Uua: fun_o_nat,Uub: fun_o_nat,Uuc: nat] : aa_nat_nat(aa_fun1225607396849667164at_nat(aa_fun2442258407354871084at_nat(aTP_Lamm_xk(Uu),Uua),Uub),Uuc) = groups3504817904513533571_o_nat(Uua,collect_o(aa_nat_fun_o_o(aa_fun953500542417269327un_o_o(aTP_Lamm_wi(Uu),Uub),Uuc))) ).

% ATP.lambda_979
tff(fact_10671_ATP_Olambda__980,axiom,
    ! [Uu: set_o,Uua: fun_o_int,Uub: fun_o_nat,Uuc: nat] : aa_nat_int(aa_fun6271128414195246264at_int(aa_fun8663632660299257900at_int(aTP_Lamm_xn(Uu),Uua),Uub),Uuc) = groups3502327434004483295_o_int(Uua,collect_o(aa_nat_fun_o_o(aa_fun953500542417269327un_o_o(aTP_Lamm_wi(Uu),Uub),Uuc))) ).

% ATP.lambda_980
tff(fact_10672_ATP_Olambda__981,axiom,
    ! [Uu: set_int,Uua: fun_int_Code_integer,Uub: fun_int_int,Uuc: code_integer] : aa_Code_integer_int(aa_fun2092574682676764953er_int(aa_fun9146121993134396956er_int(aTP_Lamm_aae(Uu),Uua),Uub),Uuc) = groups1705073143266064639nt_int(Uub,collect_int(aa_Cod8477617969808203498_int_o(aa_fun7746027916420482493_int_o(aTP_Lamm_zc(Uu),Uua),Uuc))) ).

% ATP.lambda_981
tff(fact_10673_ATP_Olambda__982,axiom,
    ! [Uu: set_nat,Uua: fun_nat_Code_integer,Uub: fun_nat_nat,Uuc: code_integer] : aa_Code_integer_nat(aa_fun2720235278490593797er_nat(aa_fun676797813428960940er_nat(aTP_Lamm_aab(Uu),Uua),Uub),Uuc) = aa_set_nat_nat(groups708209901874060359at_nat(Uub),collect_nat(aa_Cod1075288278386115654_nat_o(aa_fun262808356775936629_nat_o(aTP_Lamm_yy(Uu),Uua),Uuc))) ).

% ATP.lambda_982
tff(fact_10674_ATP_Olambda__983,axiom,
    ! [Uu: set_nat,Uua: fun_nat_Code_integer,Uub: fun_nat_int,Uuc: code_integer] : aa_Code_integer_int(aa_fun2934847258012291773er_int(aa_fun2579177667528128868er_int(aTP_Lamm_aad(Uu),Uua),Uub),Uuc) = aa_set_nat_int(groups705719431365010083at_int(Uub),collect_nat(aa_Cod1075288278386115654_nat_o(aa_fun262808356775936629_nat_o(aTP_Lamm_yy(Uu),Uua),Uuc))) ).

% ATP.lambda_983
tff(fact_10675_ATP_Olambda__984,axiom,
    ! [Uu: set_nat,Uua: fun_nat_nat,Uub: fun_nat_nat,Uuc: nat] : aa_nat_nat(aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(aTP_Lamm_aai(Uu),Uua),Uub),Uuc) = aa_set_nat_nat(groups708209901874060359at_nat(Uub),collect_nat(aa_nat_fun_nat_o(aa_fun6824225949033397647_nat_o(aTP_Lamm_wv(Uu),Uua),Uuc))) ).

% ATP.lambda_984
tff(fact_10676_ATP_Olambda__985,axiom,
    ! [Uu: set_nat,Uua: fun_nat_int,Uub: fun_nat_nat,Uuc: int] : aa_int_nat(aa_fun6790803225114646484nt_nat(aa_fun5866268260113680714nt_nat(aTP_Lamm_aac(Uu),Uua),Uub),Uuc) = aa_set_nat_nat(groups708209901874060359at_nat(Uub),collect_nat(aa_int_fun_nat_o(aa_fun6124098986320894167_nat_o(aTP_Lamm_wq(Uu),Uua),Uuc))) ).

% ATP.lambda_985
tff(fact_10677_ATP_Olambda__986,axiom,
    ! [Uu: set_Code_integer,Uua: fun_Code_integer_nat,Uub: fun_Code_integer_nat,Uuc: nat] : aa_nat_nat(aa_fun1707803128846032901at_nat(aa_fun2075405564639334828at_nat(aTP_Lamm_aah(Uu),Uua),Uub),Uuc) = groups3190895334310489300er_nat(Uub,collect_Code_integer(aa_nat7420681472272698950eger_o(aa_fun4176659983681010805eger_o(aTP_Lamm_wm(Uu),Uua),Uuc))) ).

% ATP.lambda_986
tff(fact_10678_ATP_Olambda__987,axiom,
    ! [Uu: set_Code_integer,Uua: fun_Code_integer_nat,Uub: fun_Code_integer_int,Uuc: nat] : aa_nat_int(aa_fun1481396737752545469at_int(aa_fun3536767048123317348at_int(aTP_Lamm_aak(Uu),Uua),Uub),Uuc) = groups3188404863801439024er_int(Uub,collect_Code_integer(aa_nat7420681472272698950eger_o(aa_fun4176659983681010805eger_o(aTP_Lamm_wm(Uu),Uua),Uuc))) ).

% ATP.lambda_987
tff(fact_10679_ATP_Olambda__988,axiom,
    ! [Uu: set_int,Uua: fun_int_nat,Uub: fun_int_nat,Uuc: nat] : aa_nat_nat(aa_fun5507398438843482068at_nat(aa_fun3220669823809117514at_nat(aTP_Lamm_aag(Uu),Uua),Uub),Uuc) = groups1707563613775114915nt_nat(Uub,collect_int(aa_nat_fun_int_o(aa_fun8596217221128839383_int_o(aTP_Lamm_wk(Uu),Uua),Uuc))) ).

% ATP.lambda_988
tff(fact_10680_ATP_Olambda__989,axiom,
    ! [Uu: set_o,Uua: fun_o_nat,Uub: fun_o_nat,Uuc: nat] : aa_nat_nat(aa_fun1225607396849667164at_nat(aa_fun2442258407354871084at_nat(aTP_Lamm_aaf(Uu),Uua),Uub),Uuc) = groups3504817904513533571_o_nat(Uub,collect_o(aa_nat_fun_o_o(aa_fun953500542417269327un_o_o(aTP_Lamm_wi(Uu),Uua),Uuc))) ).

% ATP.lambda_989
tff(fact_10681_ATP_Olambda__990,axiom,
    ! [Uu: set_o,Uua: fun_o_nat,Uub: fun_o_int,Uuc: nat] : aa_nat_int(aa_fun999201005756179732at_int(aa_fun3903619890838853604at_int(aTP_Lamm_aaj(Uu),Uua),Uub),Uuc) = groups3502327434004483295_o_int(Uub,collect_o(aa_nat_fun_o_o(aa_fun953500542417269327un_o_o(aTP_Lamm_wi(Uu),Uua),Uuc))) ).

% ATP.lambda_990
tff(fact_10682_ATP_Olambda__991,axiom,
    ! [Uu: set_int,Uua: fun_int_int,Uub: fun_int_int,Uuc: int] : aa_int_int(aa_fun6280345759651049192nt_int(aa_fun3693603986580398266nt_int(aTP_Lamm_wz(Uu),Uua),Uub),Uuc) = groups4538972089207619220nt_int(Uua,collect_int(aa_int_fun_int_o(aa_fun7896090258416335903_int_o(aTP_Lamm_wy(Uu),Uub),Uuc))) ).

% ATP.lambda_991
tff(fact_10683_ATP_Olambda__992,axiom,
    ! [Uu: set_nat,Uua: fun_nat_nat,Uub: fun_nat_nat,Uuc: nat] : aa_nat_nat(aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(aTP_Lamm_ww(Uu),Uua),Uub),Uuc) = aa_set_nat_nat(groups3542108847815614940at_nat(Uua),collect_nat(aa_nat_fun_nat_o(aa_fun6824225949033397647_nat_o(aTP_Lamm_wv(Uu),Uub),Uuc))) ).

% ATP.lambda_992
tff(fact_10684_ATP_Olambda__993,axiom,
    ! [Uu: set_Code_integer,Uua: fun_Code_integer_int,Uub: fun_Code_integer_int,Uuc: int] : aa_int_int(aa_fun2480750449653600025nt_int(aa_fun6168136379806246172nt_int(aTP_Lamm_wt(Uu),Uua),Uub),Uuc) = groups7234854612051535045er_int(Uua,collect_Code_integer(aa_int5993532643906289898eger_o(aa_fun7732714517052602301eger_o(aTP_Lamm_ws(Uu),Uub),Uuc))) ).

% ATP.lambda_993
tff(fact_10685_ATP_Olambda__994,axiom,
    ! [Uu: set_nat,Uua: fun_nat_nat,Uub: fun_nat_int,Uuc: int] : aa_int_nat(aa_fun1518875816675579952nt_nat(aa_fun3769689747080337738nt_nat(aTP_Lamm_wu(Uu),Uua),Uub),Uuc) = aa_set_nat_nat(groups3542108847815614940at_nat(Uua),collect_nat(aa_int_fun_nat_o(aa_fun6124098986320894167_nat_o(aTP_Lamm_wq(Uu),Uub),Uuc))) ).

% ATP.lambda_994
tff(fact_10686_ATP_Olambda__995,axiom,
    ! [Uu: set_nat,Uua: fun_nat_int,Uub: fun_nat_int,Uuc: int] : aa_int_int(aa_fun6564396834021159052nt_int(aa_fun7327629743597663234nt_int(aTP_Lamm_wr(Uu),Uua),Uub),Uuc) = aa_set_nat_int(groups3539618377306564664at_int(Uua),collect_nat(aa_int_fun_nat_o(aa_fun6124098986320894167_nat_o(aTP_Lamm_wq(Uu),Uub),Uuc))) ).

% ATP.lambda_995
tff(fact_10687_ATP_Olambda__996,axiom,
    ! [Uu: set_o,Uua: fun_o_int,Uub: fun_o_int,Uuc: int] : aa_int_int(aa_fun1998554717657234288nt_int(aa_fun5578626826553213156nt_int(aTP_Lamm_wp(Uu),Uua),Uub),Uuc) = groups8505340233167759370_o_int(Uua,collect_o(aa_int_fun_o_o(aa_fun3522593817124631815un_o_o(aTP_Lamm_wo(Uu),Uub),Uuc))) ).

% ATP.lambda_996
tff(fact_10688_ATP_Olambda__997,axiom,
    ! [Uu: set_Code_integer,Uua: fun_Code_integer_nat,Uub: fun_Code_integer_nat,Uuc: nat] : aa_nat_nat(aa_fun1707803128846032901at_nat(aa_fun2075405564639334828at_nat(aTP_Lamm_wn(Uu),Uua),Uub),Uuc) = groups7237345082560585321er_nat(Uua,collect_Code_integer(aa_nat7420681472272698950eger_o(aa_fun4176659983681010805eger_o(aTP_Lamm_wm(Uu),Uub),Uuc))) ).

% ATP.lambda_997
tff(fact_10689_ATP_Olambda__998,axiom,
    ! [Uu: set_int,Uua: fun_int_nat,Uub: fun_int_nat,Uuc: nat] : aa_nat_nat(aa_fun5507398438843482068at_nat(aa_fun3220669823809117514at_nat(aTP_Lamm_wl(Uu),Uua),Uub),Uuc) = groups4541462559716669496nt_nat(Uua,collect_int(aa_nat_fun_int_o(aa_fun8596217221128839383_int_o(aTP_Lamm_wk(Uu),Uub),Uuc))) ).

% ATP.lambda_998
tff(fact_10690_ATP_Olambda__999,axiom,
    ! [Uu: set_int,Uua: fun_int_int,Uub: fun_int_nat,Uuc: nat] : aa_nat_int(aa_fun1329547419334285360at_int(aa_fun6778609820326443010at_int(aTP_Lamm_wx(Uu),Uua),Uub),Uuc) = groups4538972089207619220nt_int(Uua,collect_int(aa_nat_fun_int_o(aa_fun8596217221128839383_int_o(aTP_Lamm_wk(Uu),Uub),Uuc))) ).

% ATP.lambda_999
tff(fact_10691_ATP_Olambda__1000,axiom,
    ! [Uu: set_o,Uua: fun_o_nat,Uub: fun_o_nat,Uuc: nat] : aa_nat_nat(aa_fun1225607396849667164at_nat(aa_fun2442258407354871084at_nat(aTP_Lamm_wj(Uu),Uua),Uub),Uuc) = groups8507830703676809646_o_nat(Uua,collect_o(aa_nat_fun_o_o(aa_fun953500542417269327un_o_o(aTP_Lamm_wi(Uu),Uub),Uuc))) ).

% ATP.lambda_1000
tff(fact_10692_ATP_Olambda__1001,axiom,
    ! [Uu: set_int,Uua: fun_int_Code_integer,Uub: fun_int_int,Uuc: code_integer] : aa_Code_integer_int(aa_fun2092574682676764953er_int(aa_fun9146121993134396956er_int(aTP_Lamm_zd(Uu),Uua),Uub),Uuc) = groups4538972089207619220nt_int(Uub,collect_int(aa_Cod8477617969808203498_int_o(aa_fun7746027916420482493_int_o(aTP_Lamm_zc(Uu),Uua),Uuc))) ).

% ATP.lambda_1001
tff(fact_10693_ATP_Olambda__1002,axiom,
    ! [Uu: set_nat,Uua: fun_nat_Code_integer,Uub: fun_nat_nat,Uuc: code_integer] : aa_Code_integer_nat(aa_fun2720235278490593797er_nat(aa_fun676797813428960940er_nat(aTP_Lamm_yz(Uu),Uua),Uub),Uuc) = aa_set_nat_nat(groups3542108847815614940at_nat(Uub),collect_nat(aa_Cod1075288278386115654_nat_o(aa_fun262808356775936629_nat_o(aTP_Lamm_yy(Uu),Uua),Uuc))) ).

% ATP.lambda_1002
tff(fact_10694_ATP_Olambda__1003,axiom,
    ! [Uu: set_nat,Uua: fun_nat_nat,Uub: fun_nat_nat,Uuc: nat] : aa_nat_nat(aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(aTP_Lamm_zh(Uu),Uua),Uub),Uuc) = aa_set_nat_nat(groups3542108847815614940at_nat(Uub),collect_nat(aa_nat_fun_nat_o(aa_fun6824225949033397647_nat_o(aTP_Lamm_wv(Uu),Uua),Uuc))) ).

% ATP.lambda_1003
tff(fact_10695_ATP_Olambda__1004,axiom,
    ! [Uu: set_nat,Uua: fun_nat_int,Uub: fun_nat_nat,Uuc: int] : aa_int_nat(aa_fun6790803225114646484nt_nat(aa_fun5866268260113680714nt_nat(aTP_Lamm_za(Uu),Uua),Uub),Uuc) = aa_set_nat_nat(groups3542108847815614940at_nat(Uub),collect_nat(aa_int_fun_nat_o(aa_fun6124098986320894167_nat_o(aTP_Lamm_wq(Uu),Uua),Uuc))) ).

% ATP.lambda_1004
tff(fact_10696_ATP_Olambda__1005,axiom,
    ! [Uu: set_nat,Uua: fun_nat_int,Uub: fun_nat_int,Uuc: int] : aa_int_int(aa_fun6564396834021159052nt_int(aa_fun7327629743597663234nt_int(aTP_Lamm_zj(Uu),Uua),Uub),Uuc) = aa_set_nat_int(groups3539618377306564664at_int(Uub),collect_nat(aa_int_fun_nat_o(aa_fun6124098986320894167_nat_o(aTP_Lamm_wq(Uu),Uua),Uuc))) ).

% ATP.lambda_1005
tff(fact_10697_ATP_Olambda__1006,axiom,
    ! [Uu: set_o,Uua: fun_o_int,Uub: fun_o_int,Uuc: int] : aa_int_int(aa_fun1998554717657234288nt_int(aa_fun5578626826553213156nt_int(aTP_Lamm_zi(Uu),Uua),Uub),Uuc) = groups8505340233167759370_o_int(Uub,collect_o(aa_int_fun_o_o(aa_fun3522593817124631815un_o_o(aTP_Lamm_wo(Uu),Uua),Uuc))) ).

% ATP.lambda_1006
tff(fact_10698_ATP_Olambda__1007,axiom,
    ! [Uu: set_Code_integer,Uua: fun_Code_integer_nat,Uub: fun_Code_integer_nat,Uuc: nat] : aa_nat_nat(aa_fun1707803128846032901at_nat(aa_fun2075405564639334828at_nat(aTP_Lamm_zg(Uu),Uua),Uub),Uuc) = groups7237345082560585321er_nat(Uub,collect_Code_integer(aa_nat7420681472272698950eger_o(aa_fun4176659983681010805eger_o(aTP_Lamm_wm(Uu),Uua),Uuc))) ).

% ATP.lambda_1007
tff(fact_10699_ATP_Olambda__1008,axiom,
    ! [Uu: set_int,Uua: fun_int_nat,Uub: fun_int_nat,Uuc: nat] : aa_nat_nat(aa_fun5507398438843482068at_nat(aa_fun3220669823809117514at_nat(aTP_Lamm_zf(Uu),Uua),Uub),Uuc) = groups4541462559716669496nt_nat(Uub,collect_int(aa_nat_fun_int_o(aa_fun8596217221128839383_int_o(aTP_Lamm_wk(Uu),Uua),Uuc))) ).

% ATP.lambda_1008
tff(fact_10700_ATP_Olambda__1009,axiom,
    ! [Uu: set_int,Uua: fun_int_nat,Uub: fun_int_int,Uuc: nat] : aa_nat_int(aa_fun5280992047749994636at_int(aa_fun4682031307293100034at_int(aTP_Lamm_zb(Uu),Uua),Uub),Uuc) = groups4538972089207619220nt_int(Uub,collect_int(aa_nat_fun_int_o(aa_fun8596217221128839383_int_o(aTP_Lamm_wk(Uu),Uua),Uuc))) ).

% ATP.lambda_1009
tff(fact_10701_ATP_Olambda__1010,axiom,
    ! [Uu: set_o,Uua: fun_o_nat,Uub: fun_o_nat,Uuc: nat] : aa_nat_nat(aa_fun1225607396849667164at_nat(aa_fun2442258407354871084at_nat(aTP_Lamm_ze(Uu),Uua),Uub),Uuc) = groups8507830703676809646_o_nat(Uub,collect_o(aa_nat_fun_o_o(aa_fun953500542417269327un_o_o(aTP_Lamm_wi(Uu),Uua),Uuc))) ).

% ATP.lambda_1010
tff(fact_10702_ATP_Olambda__1011,axiom,
    ! [Uu: fun_Pr2478283820680522243nt_int,Uua: fun_in1696471567231710611nt_int,Uub: int,Uuc: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aa_fun4152872853588681340nt_int(aTP_Lamm_agj(Uu),Uua),Uub),Uuc) = aa_Pro6970504127699319548nt_int(Uu,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(Uua,Uub),Uuc)) ).

% ATP.lambda_1011
tff(fact_10703_ATP_Olambda__1012,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_agf(Uu),Uua),Uub),Uuc)
    <=> aa_Pro4592068669816069595_int_o(Uu,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(Uua,Uub),Uuc)) ) ).

% ATP.lambda_1012
tff(fact_10704_ATP_Olambda__1013,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_agg(Uu),Uua),Uub),Uuc) = aa_Pro6799258434465214121nt_int(Uu,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(Uua,Uub),Uuc)) ).

% ATP.lambda_1013
tff(fact_10705_ATP_Olambda__1014,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_agh(Uu),Uua),Uub),Uuc) = aa_o_P3134438092330336605nt_int(Uu,aa_int_o(aa_int_fun_int_o(Uua,Uub),Uuc)) ).

% ATP.lambda_1014
tff(fact_10706_ATP_Olambda__1015,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_agb(Uu),Uua),Uub),Uuc)
    <=> aa_o_o(Uu,aa_int_o(aa_int_fun_int_o(Uua,Uub),Uuc)) ) ).

% ATP.lambda_1015
tff(fact_10707_ATP_Olambda__1016,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_agc(Uu),Uua),Uub),Uuc) = aa_o_int(Uu,aa_int_o(aa_int_fun_int_o(Uua,Uub),Uuc)) ).

% ATP.lambda_1016
tff(fact_10708_ATP_Olambda__1017,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_agi(Uu),Uua),Uub),Uuc) = aa_int7462565939943104719nt_int(Uu,aa_int_int(aa_int_fun_int_int(Uua,Uub),Uuc)) ).

% ATP.lambda_1017
tff(fact_10709_ATP_Olambda__1018,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_agd(Uu),Uua),Uub),Uuc)
    <=> aa_int_o(Uu,aa_int_int(aa_int_fun_int_int(Uua,Uub),Uuc)) ) ).

% ATP.lambda_1018
tff(fact_10710_ATP_Olambda__1019,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_age(Uu),Uua),Uub),Uuc) = aa_int_int(Uu,aa_int_int(aa_int_fun_int_int(Uua,Uub),Uuc)) ).

% ATP.lambda_1019
tff(fact_10711_ATP_Olambda__1020,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_agk(Uu),Uua),Uub),Uuc) = aa_fun359341089595923442_nat_o(Uu,aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(Uua,Uub),Uuc)) ).

% ATP.lambda_1020
tff(fact_10712_ATP_Olambda__1021,axiom,
    ! [Uu: fun_nat_nat,Uua: nat,Uub: nat,Uuc: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_abo(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_1021
tff(fact_10713_ATP_Olambda__1022,axiom,
    ! [Uu: fun_nat_int,Uua: nat,Uub: nat,Uuc: nat] : aa_nat_int(aa_nat_fun_nat_int(aa_nat1293065384339088724at_int(aTP_Lamm_abr(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_1022
tff(fact_10714_ATP_Olambda__1023,axiom,
    ! [Uu: fun_nat_nat,Uua: nat,Uub: nat,Uuc: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_jz(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_1023
tff(fact_10715_ATP_Olambda__1024,axiom,
    ! [Uu: fun_nat_int,Uua: nat,Uub: nat,Uuc: nat] : aa_nat_int(aa_nat_fun_nat_int(aa_nat1293065384339088724at_int(aTP_Lamm_nr(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_1024
tff(fact_10716_ATP_Olambda__1025,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_aod(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_1025
tff(fact_10717_ATP_Olambda__1026,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_aob(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_1026
tff(fact_10718_ATP_Olambda__1027,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_ans(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_1027
tff(fact_10719_ATP_Olambda__1028,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_anu(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_1028
tff(fact_10720_ATP_Olambda__1029,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_mc(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_1029
tff(fact_10721_ATP_Olambda__1030,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_mb(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_1030
tff(fact_10722_ATP_Olambda__1031,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_mi(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_1031
tff(fact_10723_ATP_Olambda__1032,axiom,
    ! [Uu: code_integer,Uua: nat] : aa_nat_Code_integer(aTP_Lamm_aka(Uu),Uua) = Uu ).

% ATP.lambda_1032
tff(fact_10724_ATP_Olambda__1033,axiom,
    ! [Uu: rat,Uua: list_nat] : aa_list_nat_rat(aTP_Lamm_akc(Uu),Uua) = Uu ).

% ATP.lambda_1033
tff(fact_10725_ATP_Olambda__1034,axiom,
    ! [Uu: rat,Uua: set_nat] : aa_set_nat_rat(aTP_Lamm_akb(Uu),Uua) = Uu ).

% ATP.lambda_1034
tff(fact_10726_ATP_Olambda__1035,axiom,
    ! [Uu: rat,Uua: nat] : aa_nat_rat(aTP_Lamm_ajz(Uu),Uua) = Uu ).

% ATP.lambda_1035
tff(fact_10727_ATP_Olambda__1036,axiom,
    ! [Uu: nat,Uua: list_nat] : aa_list_nat_nat(aTP_Lamm_ajt(Uu),Uua) = Uu ).

% ATP.lambda_1036
tff(fact_10728_ATP_Olambda__1037,axiom,
    ! [Uu: nat,Uua: set_nat] : aa_set_nat_nat(aTP_Lamm_ajs(Uu),Uua) = Uu ).

% ATP.lambda_1037
tff(fact_10729_ATP_Olambda__1038,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_ajw,Uu),Uua) = Uu ).

% ATP.lambda_1038
tff(fact_10730_ATP_Olambda__1039,axiom,
    ! [Uu: int,Uua: list_nat] : aa_list_nat_int(aTP_Lamm_ajv(Uu),Uua) = Uu ).

% ATP.lambda_1039
tff(fact_10731_ATP_Olambda__1040,axiom,
    ! [Uu: int,Uua: set_nat] : aa_set_nat_int(aTP_Lamm_aju(Uu),Uua) = Uu ).

% ATP.lambda_1040
tff(fact_10732_ATP_Olambda__1041,axiom,
    ! [Uu: int,Uua: nat] : aa_nat_int(aa_int_fun_nat_int(aTP_Lamm_ajx,Uu),Uua) = Uu ).

% ATP.lambda_1041
tff(fact_10733_ATP_Olambda__1042,axiom,
    ! [Uu: int,Uua: int] : aa_int_int(aa_int_fun_int_int(aTP_Lamm_ajy,Uu),Uua) = Uu ).

% ATP.lambda_1042
tff(fact_10734_ATP_Olambda__1043,axiom,
    ! [Uu: nat,Uua: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_nb,Uu),Uua)
    <=> $true ) ).

% ATP.lambda_1043
tff(fact_10735_ATP_Olambda__1044,axiom,
    ! [Uu: int,Uua: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_nc,Uu),Uua)
    <=> $true ) ).

% ATP.lambda_1044
tff(fact_10736_ATP_Olambda__1045,axiom,
    ! [Uu: code_integer] : aa_Cod1783974652533396728nteger(aTP_Lamm_fr,Uu) = Uu ).

% ATP.lambda_1045
tff(fact_10737_ATP_Olambda__1046,axiom,
    ! [Uu: assn] : aa_assn_assn(aTP_Lamm_fs,Uu) = Uu ).

% ATP.lambda_1046
tff(fact_10738_ATP_Olambda__1047,axiom,
    ! [Uu: rat] : aa_rat_rat(aTP_Lamm_ft,Uu) = Uu ).

% ATP.lambda_1047
tff(fact_10739_ATP_Olambda__1048,axiom,
    ! [Uu: nat] : aa_nat_nat(aTP_Lamm_fu,Uu) = Uu ).

% ATP.lambda_1048
tff(fact_10740_ATP_Olambda__1049,axiom,
    ! [Uu: int] : aa_int_int(aTP_Lamm_fv,Uu) = Uu ).

% ATP.lambda_1049
tff(fact_10741_ATP_Olambda__1050,axiom,
    ! [Uu: code_integer] : aa_Cod1783974652533396728nteger(aTP_Lamm_fw,Uu) = zero_z3403309356797280102nteger ).

% ATP.lambda_1050
tff(fact_10742_ATP_Olambda__1051,axiom,
    ! [Uu: rat] : aa_rat_rat(aTP_Lamm_fx,Uu) = zero_zero_rat ).

% ATP.lambda_1051
tff(fact_10743_ATP_Olambda__1052,axiom,
    ! [Uu: nat] : aa_nat_nat(aTP_Lamm_fy,Uu) = zero_zero_nat ).

% ATP.lambda_1052
tff(fact_10744_ATP_Olambda__1053,axiom,
    ! [Uu: int] : aa_int_int(aTP_Lamm_fz,Uu) = zero_zero_int ).

% ATP.lambda_1053
tff(fact_10745_ATP_Olambda__1054,axiom,
    ! [Uu: list_nat] : aa_list_nat_nat(aTP_Lamm_alx,Uu) = one_one_nat ).

% ATP.lambda_1054
tff(fact_10746_ATP_Olambda__1055,axiom,
    ! [Uu: set_nat] : aa_set_nat_nat(aTP_Lamm_alw,Uu) = one_one_nat ).

% ATP.lambda_1055
tff(fact_10747_ATP_Olambda__1056,axiom,
    ! [Uu: nat] : aa_nat_nat(aTP_Lamm_mj,Uu) = one_one_nat ).

% ATP.lambda_1056
tff(fact_10748_ATP_Olambda__1057,axiom,
    ! [Uu: nat] : aa_nat_int(aTP_Lamm_mk,Uu) = one_one_int ).

% ATP.lambda_1057
tff(fact_10749_ATP_Olambda__1058,axiom,
    ! [Uu: int] : aa_int_int(aTP_Lamm_ml,Uu) = one_one_int ).

% ATP.lambda_1058
tff(fact_10750_ATP_Olambda__1059,axiom,
    ! [Uu: heap_e7401611519738050253t_unit] : aa_hea684056619475940329it_nat(aTP_Lamm_fk,Uu) = none_P9117596204409417319it_nat ).

% ATP.lambda_1059
tff(fact_10751_ATP_Olambda__1060,axiom,
    ! [Uu: nat] : aa_nat_option_nat(aTP_Lamm_er,Uu) = none_nat ).

% ATP.lambda_1060
tff(fact_10752_ATP_Olambda__1061,axiom,
    ! [Uu: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(aTP_Lamm_eo,Uu)
    <=> $false ) ).

% ATP.lambda_1061
tff(fact_10753_ATP_Olambda__1062,axiom,
    ! [Uu: product_prod_nat_nat] :
      ( aa_Pro6150637604445964947_nat_o(aTP_Lamm_be,Uu)
    <=> $false ) ).

% ATP.lambda_1062
tff(fact_10754_ATP_Olambda__1063,axiom,
    ! [Uu: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aTP_Lamm_bb,Uu)
    <=> $false ) ).

% ATP.lambda_1063
tff(fact_10755_ATP_Olambda__1064,axiom,
    ! [Uu: list_nat] :
      ( aa_list_nat_o(aTP_Lamm_bc,Uu)
    <=> $false ) ).

% ATP.lambda_1064
tff(fact_10756_ATP_Olambda__1065,axiom,
    ! [Uu: $o] :
      ( aa_o_o(aTP_Lamm_bf,(Uu))
    <=> $false ) ).

% ATP.lambda_1065
tff(fact_10757_ATP_Olambda__1066,axiom,
    ! [Uu: set_nat] :
      ( aa_set_nat_o(aTP_Lamm_bd,Uu)
    <=> $false ) ).

% ATP.lambda_1066
tff(fact_10758_ATP_Olambda__1067,axiom,
    ! [Uu: num] :
      ( aa_num_o(aTP_Lamm_et,Uu)
    <=> $false ) ).

% ATP.lambda_1067
tff(fact_10759_ATP_Olambda__1068,axiom,
    ! [Uu: nat] :
      ( aa_nat_o(aTP_Lamm_bg,Uu)
    <=> $false ) ).

% ATP.lambda_1068
tff(fact_10760_ATP_Olambda__1069,axiom,
    ! [Uu: int] :
      ( aa_int_o(aTP_Lamm_bh,Uu)
    <=> $false ) ).

% ATP.lambda_1069
tff(fact_10761_ATP_Olambda__1070,axiom,
    ! [Uu: num] :
      ( aa_num_o(aTP_Lamm_es,Uu)
    <=> $true ) ).

% ATP.lambda_1070
tff(fact_10762_ATP_Olambda__1071,axiom,
    ! [Uu: nat] :
      ( aa_nat_o(aTP_Lamm_amh,Uu)
    <=> $true ) ).

% ATP.lambda_1071

% Helper facts (12)
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__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J_T,axiom,
    ! [X: produc3260487557148687353it_nat,Y: produc3260487557148687353it_nat] :
      ( ( X != Y )
      | aa_Pro6426884464393220769_nat_o(fequal7405809337025681252it_nat(X),Y) ) ).

tff(help_fequal_1_1_fequal_001t__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J_T,axiom,
    ! [X: produc3260487557148687353it_nat,Y: produc3260487557148687353it_nat] :
      ( ~ aa_Pro6426884464393220769_nat_o(fequal7405809337025681252it_nat(X),Y)
      | ( X = Y ) ) ).

tff(help_fequal_2_1_fequal_001t__Product____Type__Oprod_It__Product____Type__Ounit_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J_T,axiom,
    ! [X: produc8664842809031399944it_nat,Y: produc8664842809031399944it_nat] :
      ( ( X != Y )
      | aa_Pro4744234731172327378_nat_o(fequal1407654770548522611it_nat(X),Y) ) ).

tff(help_fequal_1_1_fequal_001t__Product____Type__Oprod_It__Product____Type__Ounit_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J_T,axiom,
    ! [X: produc8664842809031399944it_nat,Y: produc8664842809031399944it_nat] :
      ( ~ aa_Pro4744234731172327378_nat_o(fequal1407654770548522611it_nat(X),Y)
      | ( X = Y ) ) ).

% Conjectures (1)
tff(conj_0,conjecture,
    aa_Pro653082453696054616_nat_o(rep_assn(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_a_assn(q,r2)),r)),produc7507926704131184380et_nat(h,hoare_new_addrs(h2,as,h))) ).

%------------------------------------------------------------------------------