%------------------------------------------------------------------------------
% File     : ITP286_3 : TPTP v8.2.0. Released v8.0.0.
% Domain   : Interactive Theorem Proving
% Problem  : Sledgehammer problem VEBT_SuccPredImperative 00347_018417
% Version  : [Des22] axioms.
% English  :

% Refs     : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
%          : [Des22] Desharnais (2022), Email to Geoff Sutcliffe
% Source   : [Des22]
% Names    : 0094_VEBT_SuccPredImperative_00347_018417 [Des22]

% Status   : Theorem
% Rating   : 1.00 v8.1.0
% Syntax   : Number of formulae    : 18185 (4733 unt;6568 typ;   0 def)
%            Number of atoms       : 29167 (10940 equ)
%            Maximal formula atoms :   73 (   2 avg)
%            Number of connectives : 18755 (2676   ~; 468   |;1444   &)
%                                         (2769 <=>;11398  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   35 (   5 avg)
%            Maximal term depth    :   36 (   2 avg)
%            Number of FOOLs       : 2772 (1310 fml;1462 var)
%            Number of X terms     : 1456 (   0  [];1141 ite; 315 let)
%            Number of types       : 1877 (1876 usr)
%            Number of type conns  : 6444 (4016   >;2428   *;   0   +;   0  <<)
%            Number of predicates  :  242 ( 239 usr;   2 prp; 0-3 aty)
%            Number of functors    : 4480 (4480 usr; 703 con; 0-7 aty)
%            Number of variables   : 33725 (32730   !; 995   ?;33725   :)
% SPC      : TX0_THM_EQU_NAR

% Comments : This file was generated by Isabelle (most likely Sledgehammer)
%            from the van Emde Boas Trees session in the Archive of Formal
%            proofs - 
%            www.isa-afp.org/browser_info/current/AFP/Van_Emde_Boas_Trees
%            2022-02-18 20:22:39.364
%------------------------------------------------------------------------------
% Could-be-implicit typings (1876)
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tff(ty_n_t__fun_It__Int__Oint_Mt__Rat__Orat_J,type,
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tff(ty_n_t__fun_It__Int__Oint_Mt__Num__Onum_J,type,
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tff(ty_n_t__fun_It__Int__Oint_Mt__Nat__Onat_J,type,
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tff(ty_n_t__fun_It__Int__Oint_Mt__Int__Oint_J,type,
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tff(ty_n_t__Set__Oset_It__Complex__Ocomplex_J,type,
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tff(ty_n_t__Set__Oset_It__Assertions__Oassn_J,type,
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tff(ty_n_t__Option__Ooption_It__Real__Oreal_J,type,
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tff(ty_n_t__List__Olist_It__Uint32__Ouint32_J,type,
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tff(ty_n_t__Filter__Ofilter_It__Real__Oreal_J,type,
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tff(ty_n_t__fun_It__Complex__Ocomplex_M_Eo_J,type,
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tff(ty_n_t__fun_It__Assertions__Oassn_M_Eo_J,type,
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tff(ty_n_t__fun_I_Eo_Mt__Assertions__Oassn_J,type,
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tff(ty_n_t__itself_It__Enum__Ofinite____3_J,type,
    itself_finite_3: $tType ).

tff(ty_n_t__itself_It__Enum__Ofinite____2_J,type,
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tff(ty_n_t__itself_It__Enum__Ofinite____1_J,type,
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tff(ty_n_t__Set__Oset_It__Uint32__Ouint32_J,type,
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tff(ty_n_t__Option__Ooption_It__Rat__Orat_J,type,
    option_rat: $tType ).

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

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

tff(ty_n_t__Option__Ooption_It__Int__Oint_J,type,
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tff(ty_n_t__fun_It__Uint32__Ouint32_M_Eo_J,type,
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tff(ty_n_t__fun_I_Eo_Mt__fun_I_Eo_M_Eo_J_J,type,
    fun_o_fun_o_o: $tType ).

tff(ty_n_t__fun_I_Eo_Mt__Uint32__Ouint32_J,type,
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tff(ty_n_t__VEBT____BuildupMemImp__OVEBTi,type,
    vEBT_VEBTi: $tType ).

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

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

tff(ty_n_t__fun_It__String__Ochar_M_Eo_J,type,
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tff(ty_n_t__fun_I_Eo_Mt__String__Ochar_J,type,
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tff(ty_n_t__Heap__Oarray_It__Nat__Onat_J,type,
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tff(ty_n_t__Heap__Oarray_It__Int__Oint_J,type,
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tff(ty_n_t__Set__Oset_It__Real__Oreal_J,type,
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tff(ty_n_t__List__Olist_It__Nat__Onat_J,type,
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tff(ty_n_t__List__Olist_It__Int__Oint_J,type,
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tff(ty_n_t__fun_It__Real__Oreal_M_Eo_J,type,
    fun_real_o: $tType ).

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

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

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

tff(ty_n_t__Set__Oset_It__Num__Onum_J,type,
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tff(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
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tff(ty_n_t__Set__Oset_It__Int__Oint_J,type,
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tff(ty_n_t__Code____Numeral__Ointeger,type,
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tff(ty_n_t__fun_It__Rat__Orat_M_Eo_J,type,
    fun_rat_o: $tType ).

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

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

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

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

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

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

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

tff(ty_n_t__Heap__Oarray_I_Eo_J,type,
    array_o: $tType ).

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

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

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

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

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

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

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

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

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

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

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

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

% Explicit typings (4692)
tff(sy_c_ATP_058Lamm__a____,type,
    aTP_Lamm_a: nat > fun_op4683986414774905957on_nat ).

tff(sy_c_ATP_058Lamm__aa____,type,
    aTP_Lamm_aa: option_nat > fun_op4683986414774905957on_nat ).

tff(sy_c_ATP_058Lamm__aaa____,type,
    aTP_Lamm_aaa: fun_Pr940977554174145406Heap_o > fun_na2699581809715174900Heap_o ).

tff(sy_c_ATP_058Lamm__aab____,type,
    aTP_Lamm_aab: fun_Pr940977554174145406Heap_o > fun_na853727736315331060Heap_o ).

tff(sy_c_ATP_058Lamm__aac____,type,
    aTP_Lamm_aac: fun_Pr940977554174145406Heap_o > fun_na3474800335600542119Heap_o ).

tff(sy_c_ATP_058Lamm__aad____,type,
    aTP_Lamm_aad: fun_Pr940977554174145406Heap_o > fun_na2622557336161645666Heap_o ).

tff(sy_c_ATP_058Lamm__aae____,type,
    aTP_Lamm_aae: fun_Pr940977554174145406Heap_o > fun_na7247921347214250394Heap_o ).

tff(sy_c_ATP_058Lamm__aaf____,type,
    aTP_Lamm_aaf: fun_Pr940977554174145406Heap_o > fun_na5958411666642256818Heap_o ).

tff(sy_c_ATP_058Lamm__aag____,type,
    aTP_Lamm_aag: fun_Pr940977554174145406Heap_o > fun_na7955433379684683584Heap_o ).

tff(sy_c_ATP_058Lamm__aah____,type,
    aTP_Lamm_aah: fun_Pr940977554174145406Heap_o > fun_VE567930495301233690Heap_o ).

tff(sy_c_ATP_058Lamm__aai____,type,
    aTP_Lamm_aai: fun_Pr940977554174145406Heap_o > fun_VE1467545974346792018Heap_o ).

tff(sy_c_ATP_058Lamm__aaj____,type,
    aTP_Lamm_aaj: fun_Pr940977554174145406Heap_o > fun_VE2348196291809386290Heap_o ).

tff(sy_c_ATP_058Lamm__aak____,type,
    aTP_Lamm_aak: fun_Pr940977554174145406Heap_o > fun_VE8181444127144381288Heap_o ).

tff(sy_c_ATP_058Lamm__aal____,type,
    aTP_Lamm_aal: produc3960855945107176009Ti_nat > fun_fu7243727008143648435Heap_o ).

tff(sy_c_ATP_058Lamm__aam____,type,
    aTP_Lamm_aam: fun_Pr342922516890243977on_nat > fun_na4155721213724604198on_nat ).

tff(sy_c_ATP_058Lamm__aan____,type,
    aTP_Lamm_aan: fun_Pr342922516890243977on_nat > fun_na1029252574726078829on_nat ).

tff(sy_c_ATP_058Lamm__aao____,type,
    aTP_Lamm_aao: fun_Pr342922516890243977on_nat > fun_na7271643239918768137on_nat ).

tff(sy_c_ATP_058Lamm__aap____,type,
    aTP_Lamm_aap: fun_Pr342922516890243977on_nat > fun_na2011684170944547840on_nat ).

tff(sy_c_ATP_058Lamm__aaq____,type,
    aTP_Lamm_aaq: fun_Pr342922516890243977on_nat > fun_na2990367117748150135on_nat ).

tff(sy_c_ATP_058Lamm__aar____,type,
    aTP_Lamm_aar: fun_Pr342922516890243977on_nat > fun_na6220387211928018449on_nat ).

tff(sy_c_ATP_058Lamm__aas____,type,
    aTP_Lamm_aas: fun_Pr342922516890243977on_nat > fun_VE8516060933855021849on_nat ).

tff(sy_c_ATP_058Lamm__aat____,type,
    aTP_Lamm_aat: produc3881548065746020326Ti_nat > fun_fu4400511211004959420on_nat ).

tff(sy_c_ATP_058Lamm__aau____,type,
    aTP_Lamm_aau: fun_Pr342922516890243977on_nat > fun_na3472305698324104669on_nat ).

tff(sy_c_ATP_058Lamm__aav____,type,
    aTP_Lamm_aav: fun_Pr342922516890243977on_nat > fun_na505420239896681636on_nat ).

tff(sy_c_ATP_058Lamm__aaw____,type,
    aTP_Lamm_aaw: fun_Pr342922516890243977on_nat > fun_na3776232666933163584on_nat ).

tff(sy_c_ATP_058Lamm__aax____,type,
    aTP_Lamm_aax: fun_Pr342922516890243977on_nat > fun_na3371937797455410308on_nat ).

tff(sy_c_ATP_058Lamm__aay____,type,
    aTP_Lamm_aay: fun_Pr342922516890243977on_nat > fun_na2990367117748150135on_nat ).

tff(sy_c_ATP_058Lamm__aaz____,type,
    aTP_Lamm_aaz: fun_Pr342922516890243977on_nat > fun_na6220387211928018449on_nat ).

tff(sy_c_ATP_058Lamm__ab____,type,
    aTP_Lamm_ab: option_nat > fun_VE2693554162914142864on_nat ).

tff(sy_c_ATP_058Lamm__aba____,type,
    aTP_Lamm_aba: fun_Pr342922516890243977on_nat > fun_VE8516060933855021849on_nat ).

tff(sy_c_ATP_058Lamm__abb____,type,
    aTP_Lamm_abb: produc3881548065746020326Ti_nat > fun_fu4400511211004959420on_nat ).

tff(sy_c_ATP_058Lamm__abc____,type,
    aTP_Lamm_abc: fun_Pr4171499318505651870_VEBTi > fun_na2890314705291535822_VEBTi ).

tff(sy_c_ATP_058Lamm__abd____,type,
    aTP_Lamm_abd: fun_Pr4171499318505651870_VEBTi > fun_na7797079703196089028_VEBTi ).

tff(sy_c_ATP_058Lamm__abe____,type,
    aTP_Lamm_abe: fun_Pr4171499318505651870_VEBTi > fun_na5084500599656708831_VEBTi ).

tff(sy_c_ATP_058Lamm__abf____,type,
    aTP_Lamm_abf: fun_Pr4171499318505651870_VEBTi > fun_na9009963084909443284_VEBTi ).

tff(sy_c_ATP_058Lamm__abg____,type,
    aTP_Lamm_abg: fun_Pr4171499318505651870_VEBTi > fun_na2291395713665613775_VEBTi ).

tff(sy_c_ATP_058Lamm__abh____,type,
    aTP_Lamm_abh: fun_Pr4171499318505651870_VEBTi > fun_na7682395145315030854_VEBTi ).

tff(sy_c_ATP_058Lamm__abi____,type,
    aTP_Lamm_abi: fun_Pr4171499318505651870_VEBTi > fun_na7174282198332753469_VEBTi ).

tff(sy_c_ATP_058Lamm__abj____,type,
    aTP_Lamm_abj: fun_Pr4171499318505651870_VEBTi > fun_na3224852944424307892_VEBTi ).

tff(sy_c_ATP_058Lamm__abk____,type,
    aTP_Lamm_abk: fun_Pr4171499318505651870_VEBTi > fun_na460406568315038379_VEBTi ).

tff(sy_c_ATP_058Lamm__abl____,type,
    aTP_Lamm_abl: fun_Pr4171499318505651870_VEBTi > fun_na2086927332677804066_VEBTi ).

tff(sy_c_ATP_058Lamm__abm____,type,
    aTP_Lamm_abm: fun_Pr4171499318505651870_VEBTi > fun_na2740539754339541653_VEBTi ).

tff(sy_c_ATP_058Lamm__abn____,type,
    aTP_Lamm_abn: fun_Pr4171499318505651870_VEBTi > fun_na278150576929200012_VEBTi ).

tff(sy_c_ATP_058Lamm__abo____,type,
    aTP_Lamm_abo: fun_Pr4171499318505651870_VEBTi > fun_na8762902223149059366_VEBTi ).

tff(sy_c_ATP_058Lamm__abp____,type,
    aTP_Lamm_abp: fun_Pr4171499318505651870_VEBTi > fun_VE8118064890490510894_VEBTi ).

tff(sy_c_ATP_058Lamm__abq____,type,
    aTP_Lamm_abq: produc3881548065746020326Ti_nat > fun_fu6595418844270726780_VEBTi ).

tff(sy_c_ATP_058Lamm__abr____,type,
    aTP_Lamm_abr: fun_fu1178290271985267629EBTi_o > fun_fu8912552214369285843EBTi_o ).

tff(sy_c_ATP_058Lamm__abs____,type,
    aTP_Lamm_abs: fun_VE8118064890490510894_VEBTi ).

tff(sy_c_ATP_058Lamm__abt____,type,
    aTP_Lamm_abt: fun_VE8118064890490510894_VEBTi > fun_VE8118064890490510894_VEBTi ).

tff(sy_c_ATP_058Lamm__abu____,type,
    aTP_Lamm_abu: fun_fu6711510597495717193_nat_o > fun_fu7108391812694488075_nat_o ).

tff(sy_c_ATP_058Lamm__abv____,type,
    aTP_Lamm_abv: fun_VE3775134097089128360on_nat ).

tff(sy_c_ATP_058Lamm__abw____,type,
    aTP_Lamm_abw: fun_VE3775134097089128360on_nat > fun_VE3775134097089128360on_nat ).

tff(sy_c_ATP_058Lamm__abx____,type,
    aTP_Lamm_abx: fun_VE3775134097089128360on_nat > fun_VE3775134097089128360on_nat ).

tff(sy_c_ATP_058Lamm__aby____,type,
    aTP_Lamm_aby: fun_fu2707964056798801139ap_o_o > fun_fu8879972437492397021ap_o_o ).

tff(sy_c_ATP_058Lamm__abz____,type,
    aTP_Lamm_abz: fun_VE8181444127144381288Heap_o ).

tff(sy_c_ATP_058Lamm__ac____,type,
    aTP_Lamm_ac: product_prod_nat_nat > fun_op8835855855430932543on_nat ).

tff(sy_c_ATP_058Lamm__aca____,type,
    aTP_Lamm_aca: fun_VE8181444127144381288Heap_o > fun_VE8181444127144381288Heap_o ).

tff(sy_c_ATP_058Lamm__acb____,type,
    aTP_Lamm_acb: fun_fu4750890421531948930_nat_o > fun_fu1312537534749845416_nat_o ).

tff(sy_c_ATP_058Lamm__acc____,type,
    aTP_Lamm_acc: fun_VE8516060933855021849on_nat ).

tff(sy_c_ATP_058Lamm__acd____,type,
    aTP_Lamm_acd: fun_VE8516060933855021849on_nat > fun_VE8516060933855021849on_nat ).

tff(sy_c_ATP_058Lamm__ace____,type,
    aTP_Lamm_ace: fun_VE8516060933855021849on_nat > fun_VE8516060933855021849on_nat ).

tff(sy_c_ATP_058Lamm__acf____,type,
    aTP_Lamm_acf: fun_fu8886734894937297424ap_o_o > fun_fu9110254849647837056ap_o_o ).

tff(sy_c_ATP_058Lamm__acg____,type,
    aTP_Lamm_acg: fun_VE7134199479003659425Heap_o ).

tff(sy_c_ATP_058Lamm__ach____,type,
    aTP_Lamm_ach: fun_VE7134199479003659425Heap_o > fun_VE7134199479003659425Heap_o ).

tff(sy_c_ATP_058Lamm__aci____,type,
    aTP_Lamm_aci: fun_he7283999071544186133_VEBTi > fun_he3045699446853741534it_nat ).

tff(sy_c_ATP_058Lamm__acj____,type,
    aTP_Lamm_acj: fun_he4876066172508077696on_nat > fun_he1353482274668092617it_nat ).

tff(sy_c_ATP_058Lamm__ack____,type,
    aTP_Lamm_ack: fun_he8413527710729349838unit_o > fun_he1183298388918465219it_nat ).

tff(sy_c_ATP_058Lamm__acl____,type,
    aTP_Lamm_acl: fun_fu411293875173739919on_nat > fun_Pr6807451334997518105on_nat ).

tff(sy_c_ATP_058Lamm__acm____,type,
    aTP_Lamm_acm: fun_fu2426956696497423235Heap_o > fun_Pr5859542034317231789Heap_o ).

tff(sy_c_ATP_058Lamm__acn____,type,
    aTP_Lamm_acn: fun_fu1151686773700218227_VEBTi > fun_Pr7000814954542638233_VEBTi ).

tff(sy_c_ATP_058Lamm__aco____,type,
    aTP_Lamm_aco: fun_fu3620910142932681on_nat > fun_Pr7105669390387666649on_nat ).

tff(sy_c_ATP_058Lamm__acp____,type,
    aTP_Lamm_acp: fun_fu8308473583445096899Heap_o > fun_Pr2479799759690407533Heap_o ).

tff(sy_c_ATP_058Lamm__acq____,type,
    aTP_Lamm_acq: set_Pr3486312109254157132nteger > fun_fu7695273699059480439eger_o ).

tff(sy_c_ATP_058Lamm__acr____,type,
    aTP_Lamm_acr: set_Pr1312406793078187350et_nat > fun_fu8419763957888848259_nat_o ).

tff(sy_c_ATP_058Lamm__acs____,type,
    aTP_Lamm_acs: set_Pr4040394263813967220et_nat > fun_fu2611741329380808165_nat_o ).

tff(sy_c_ATP_058Lamm__act____,type,
    aTP_Lamm_act: set_Pr6700895561065273913nt_int > fun_fu6865642474282033866_int_o ).

tff(sy_c_ATP_058Lamm__acu____,type,
    aTP_Lamm_acu: set_Pr4166680992137330533nt_int > fun_fu2978539155105838238_int_o ).

tff(sy_c_ATP_058Lamm__acv____,type,
    aTP_Lamm_acv: set_int > fun_se2051630309975736070_int_o ).

tff(sy_c_ATP_058Lamm__acw____,type,
    aTP_Lamm_acw: set_complex > fun_se823489619503225858plex_o ).

tff(sy_c_ATP_058Lamm__acx____,type,
    aTP_Lamm_acx: set_Code_integer > fun_se5529282185252607396eger_o ).

tff(sy_c_ATP_058Lamm__acy____,type,
    aTP_Lamm_acy: set_nat > fun_se2170385732985038782_nat_o ).

tff(sy_c_ATP_058Lamm__acz____,type,
    aTP_Lamm_acz: fun_nat_o > fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__ad____,type,
    aTP_Lamm_ad: product_prod_nat_nat > fun_na7530211986279412206on_nat ).

tff(sy_c_ATP_058Lamm__ada____,type,
    aTP_Lamm_ada: fun_nat_nat > fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__adb____,type,
    aTP_Lamm_adb: set_VEBT_VEBT > fun_na5873943405026509934VEBT_o ).

tff(sy_c_ATP_058Lamm__adc____,type,
    aTP_Lamm_adc: set_Pr5949110396991348497n_assn > fun_na674329079771440817assn_o ).

tff(sy_c_ATP_058Lamm__add____,type,
    aTP_Lamm_add: set_complex > fun_na2418085665696872764plex_o ).

tff(sy_c_ATP_058Lamm__ade____,type,
    aTP_Lamm_ade: set_Code_integer > fun_na1863607095336112845eger_o ).

tff(sy_c_ATP_058Lamm__adf____,type,
    aTP_Lamm_adf: set_real > fun_na5739727354687878078real_o ).

tff(sy_c_ATP_058Lamm__adg____,type,
    aTP_Lamm_adg: set_o > fun_nat_fun_list_o_o ).

tff(sy_c_ATP_058Lamm__adh____,type,
    aTP_Lamm_adh: set_nat > fun_na7234455742697417626_nat_o ).

tff(sy_c_ATP_058Lamm__adi____,type,
    aTP_Lamm_adi: set_int > fun_na7306406234920378942_int_o ).

tff(sy_c_ATP_058Lamm__adj____,type,
    aTP_Lamm_adj: set_VEBT_VEBT > fun_na5873943405026509934VEBT_o ).

tff(sy_c_ATP_058Lamm__adk____,type,
    aTP_Lamm_adk: set_Pr5949110396991348497n_assn > fun_na674329079771440817assn_o ).

tff(sy_c_ATP_058Lamm__adl____,type,
    aTP_Lamm_adl: set_complex > fun_na2418085665696872764plex_o ).

tff(sy_c_ATP_058Lamm__adm____,type,
    aTP_Lamm_adm: set_Code_integer > fun_na1863607095336112845eger_o ).

tff(sy_c_ATP_058Lamm__adn____,type,
    aTP_Lamm_adn: set_real > fun_na5739727354687878078real_o ).

tff(sy_c_ATP_058Lamm__ado____,type,
    aTP_Lamm_ado: set_o > fun_nat_fun_list_o_o ).

tff(sy_c_ATP_058Lamm__adp____,type,
    aTP_Lamm_adp: set_nat > fun_na7234455742697417626_nat_o ).

tff(sy_c_ATP_058Lamm__adq____,type,
    aTP_Lamm_adq: set_int > fun_na7306406234920378942_int_o ).

tff(sy_c_ATP_058Lamm__adr____,type,
    aTP_Lamm_adr: fun_na1554462459066888140plex_o ).

tff(sy_c_ATP_058Lamm__ads____,type,
    aTP_Lamm_ads: fun_nat_fun_real_o ).

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: set_nat > fun_set_nat_o ).

tff(sy_c_ATP_058Lamm__adv____,type,
    aTP_Lamm_adv: set_complex > fun_set_complex_o ).

tff(sy_c_ATP_058Lamm__adw____,type,
    aTP_Lamm_adw: set_Code_integer > fun_se7289222521248229150eger_o ).

tff(sy_c_ATP_058Lamm__adx____,type,
    aTP_Lamm_adx: set_int > fun_set_int_o ).

tff(sy_c_ATP_058Lamm__ady____,type,
    aTP_Lamm_ady: fun_Pr3238374608701176162_int_o > fun_Pr3238374608701176162_int_o ).

tff(sy_c_ATP_058Lamm__adz____,type,
    aTP_Lamm_adz: fun_int_o > fun_int_o ).

tff(sy_c_ATP_058Lamm__ae____,type,
    aTP_Lamm_ae: product_prod_nat_nat > fun_na7879173167913326844on_nat ).

tff(sy_c_ATP_058Lamm__aea____,type,
    aTP_Lamm_aea: fun_complex_o > fun_complex_o ).

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

tff(sy_c_ATP_058Lamm__aec____,type,
    aTP_Lamm_aec: fun_real_o > fun_real_o ).

tff(sy_c_ATP_058Lamm__aed____,type,
    aTP_Lamm_aed: fun_nat_o > fun_nat_o ).

tff(sy_c_ATP_058Lamm__aee____,type,
    aTP_Lamm_aee: fun_char_o > fun_char_o ).

tff(sy_c_ATP_058Lamm__aef____,type,
    aTP_Lamm_aef: fun_Pr3238374608701176162_int_o > fun_fu3068557634114553409_int_o ).

tff(sy_c_ATP_058Lamm__aeg____,type,
    aTP_Lamm_aeg: fun_nat_o > fun_fu4001317426178549763_nat_o ).

tff(sy_c_ATP_058Lamm__aeh____,type,
    aTP_Lamm_aeh: fun_int_o > fun_fu3883539188510166275_int_o ).

tff(sy_c_ATP_058Lamm__aei____,type,
    aTP_Lamm_aei: fun_complex_o > fun_fu1358370060594899587plex_o ).

tff(sy_c_ATP_058Lamm__aej____,type,
    aTP_Lamm_aej: fun_Code_integer_o > fun_fu8094397175230428611eger_o ).

tff(sy_c_ATP_058Lamm__aek____,type,
    aTP_Lamm_aek: fun_Pr3238374608701176162_int_o > fun_fu3068557634114553409_int_o ).

tff(sy_c_ATP_058Lamm__ael____,type,
    aTP_Lamm_ael: fun_nat_o > fun_fu4001317426178549763_nat_o ).

tff(sy_c_ATP_058Lamm__aem____,type,
    aTP_Lamm_aem: fun_int_o > fun_fu3883539188510166275_int_o ).

tff(sy_c_ATP_058Lamm__aen____,type,
    aTP_Lamm_aen: fun_complex_o > fun_fu1358370060594899587plex_o ).

tff(sy_c_ATP_058Lamm__aeo____,type,
    aTP_Lamm_aeo: fun_Code_integer_o > fun_fu8094397175230428611eger_o ).

tff(sy_c_ATP_058Lamm__aep____,type,
    aTP_Lamm_aep: int > fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__aeq____,type,
    aTP_Lamm_aeq: int > fun_int_fun_int_o ).

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

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

tff(sy_c_ATP_058Lamm__aet____,type,
    aTP_Lamm_aet: set_VEBT_VEBT > fun_fu1493198444879918881VEBT_o ).

tff(sy_c_ATP_058Lamm__aeu____,type,
    aTP_Lamm_aeu: set_real > fun_fu4757868175734369219real_o ).

tff(sy_c_ATP_058Lamm__aev____,type,
    aTP_Lamm_aev: set_VEBT_VEBT > fun_fu6979804496544055329VEBT_o ).

tff(sy_c_ATP_058Lamm__aew____,type,
    aTP_Lamm_aew: set_real > fun_fu8602227297639178947real_o ).

tff(sy_c_ATP_058Lamm__aex____,type,
    aTP_Lamm_aex: set_VEBT_VEBT > fun_fu1633589921238504865VEBT_o ).

tff(sy_c_ATP_058Lamm__aey____,type,
    aTP_Lamm_aey: set_real > fun_fu4472351043248340291real_o ).

tff(sy_c_ATP_058Lamm__aez____,type,
    aTP_Lamm_aez: set_VEBT_VEBT > fun_fu8552322614197228513VEBT_o ).

tff(sy_c_ATP_058Lamm__af____,type,
    aTP_Lamm_af: product_prod_nat_nat > fun_na1616859549786149312on_nat ).

tff(sy_c_ATP_058Lamm__afa____,type,
    aTP_Lamm_afa: set_real > fun_fu3313596450186583811real_o ).

tff(sy_c_ATP_058Lamm__afb____,type,
    aTP_Lamm_afb: set_nat > fun_fu6436183194761443139_nat_o ).

tff(sy_c_ATP_058Lamm__afc____,type,
    aTP_Lamm_afc: set_nat > fun_fu8304507701009772099_nat_o ).

tff(sy_c_ATP_058Lamm__afd____,type,
    aTP_Lamm_afd: nat > fun_co1148497455491383662plex_o ).

tff(sy_c_ATP_058Lamm__afe____,type,
    aTP_Lamm_afe: fun_Pr5160692531401422620nt_nat > fun_Pr3238374608701176162_int_o ).

tff(sy_c_ATP_058Lamm__aff____,type,
    aTP_Lamm_aff: fun_Pr5160692531401422620nt_nat > fun_fu7601549123499533073_int_o ).

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

tff(sy_c_ATP_058Lamm__afh____,type,
    aTP_Lamm_afh: fun_nat_nat > fun_fu5477517762487726877_nat_o ).

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

tff(sy_c_ATP_058Lamm__afj____,type,
    aTP_Lamm_afj: fun_int_nat > fun_fu1805223966450871909_int_o ).

tff(sy_c_ATP_058Lamm__afk____,type,
    aTP_Lamm_afk: fun_complex_nat > fun_complex_o ).

tff(sy_c_ATP_058Lamm__afl____,type,
    aTP_Lamm_afl: fun_complex_nat > fun_fu6122667721776870753plex_o ).

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

tff(sy_c_ATP_058Lamm__afn____,type,
    aTP_Lamm_afn: fun_Code_integer_nat > fun_fu2182969643484777475eger_o ).

tff(sy_c_ATP_058Lamm__afo____,type,
    aTP_Lamm_afo: fun_Pr5160692531401422620nt_nat > fun_Pr9018117161326028078_int_o ).

tff(sy_c_ATP_058Lamm__afp____,type,
    aTP_Lamm_afp: fun_nat_nat > fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__afq____,type,
    aTP_Lamm_afq: fun_int_nat > fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__afr____,type,
    aTP_Lamm_afr: fun_complex_nat > fun_co1148497455491383662plex_o ).

tff(sy_c_ATP_058Lamm__afs____,type,
    aTP_Lamm_afs: fun_Code_integer_nat > fun_Co2081214437782444048eger_o ).

tff(sy_c_ATP_058Lamm__aft____,type,
    aTP_Lamm_aft: fun_Pr5160692531401422620nt_nat > fun_fu3068557634114553409_int_o ).

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

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

tff(sy_c_ATP_058Lamm__afw____,type,
    aTP_Lamm_afw: fun_complex_nat > fun_fu1358370060594899587plex_o ).

tff(sy_c_ATP_058Lamm__afx____,type,
    aTP_Lamm_afx: fun_Code_integer_nat > fun_fu8094397175230428611eger_o ).

tff(sy_c_ATP_058Lamm__afy____,type,
    aTP_Lamm_afy: assn > fun_fu8734043002210220995t_assn ).

tff(sy_c_ATP_058Lamm__afz____,type,
    aTP_Lamm_afz: assn > fun_fu8831737957336725536t_assn ).

tff(sy_c_ATP_058Lamm__ag____,type,
    aTP_Lamm_ag: product_prod_nat_nat > fun_na6892347620928151586on_nat ).

tff(sy_c_ATP_058Lamm__aga____,type,
    aTP_Lamm_aga: assn > fun_fu96848488177702211t_assn ).

tff(sy_c_ATP_058Lamm__agb____,type,
    aTP_Lamm_agb: assn > fun_fu1092876921774764064t_assn ).

tff(sy_c_ATP_058Lamm__agc____,type,
    aTP_Lamm_agc: assn > fun_fu9053108753108735937o_assn ).

tff(sy_c_ATP_058Lamm__agd____,type,
    aTP_Lamm_agd: assn > fun_fu7710348523913773264o_assn ).

tff(sy_c_ATP_058Lamm__age____,type,
    aTP_Lamm_age: assn > fun_fu7175514181539732267t_assn ).

tff(sy_c_ATP_058Lamm__agf____,type,
    aTP_Lamm_agf: assn > fun_fu1125806356063775034t_assn ).

tff(sy_c_ATP_058Lamm__agg____,type,
    aTP_Lamm_agg: assn > fun_fu8979060164380311043i_assn ).

tff(sy_c_ATP_058Lamm__agh____,type,
    aTP_Lamm_agh: assn > fun_fu8850137702660867424i_assn ).

tff(sy_c_ATP_058Lamm__agi____,type,
    aTP_Lamm_agi: vEBT_VEBT > fun_re2335940313395306534T_VEBT ).

tff(sy_c_ATP_058Lamm__agj____,type,
    aTP_Lamm_agj: vEBT_VEBT > fun_o_5254285122974450542T_VEBT ).

tff(sy_c_ATP_058Lamm__agk____,type,
    aTP_Lamm_agk: vEBT_VEBT > fun_na2623420418515794982T_VEBT ).

tff(sy_c_ATP_058Lamm__agl____,type,
    aTP_Lamm_agl: vEBT_VEBT > fun_in2974297243791461670T_VEBT ).

tff(sy_c_ATP_058Lamm__agm____,type,
    aTP_Lamm_agm: code_integer > fun_Co6597186368970283863nteger ).

tff(sy_c_ATP_058Lamm__agn____,type,
    aTP_Lamm_agn: nat > fun_VEBT_VEBT_nat ).

tff(sy_c_ATP_058Lamm__ago____,type,
    aTP_Lamm_ago: real > fun_VEBT_VEBT_real ).

tff(sy_c_ATP_058Lamm__agp____,type,
    aTP_Lamm_agp: assn > fun_Pr3816910165196019820n_assn ).

tff(sy_c_ATP_058Lamm__agq____,type,
    aTP_Lamm_agq: vEBT_VEBT > fun_real_VEBT_VEBT ).

tff(sy_c_ATP_058Lamm__agr____,type,
    aTP_Lamm_agr: vEBT_VEBT > fun_o_VEBT_VEBT ).

tff(sy_c_ATP_058Lamm__ags____,type,
    aTP_Lamm_ags: vEBT_VEBT > fun_nat_VEBT_VEBT ).

tff(sy_c_ATP_058Lamm__agt____,type,
    aTP_Lamm_agt: vEBT_VEBT > fun_int_VEBT_VEBT ).

tff(sy_c_ATP_058Lamm__agu____,type,
    aTP_Lamm_agu: array_VEBT_VEBTi > fun_li7822782016457402668i_assn ).

tff(sy_c_ATP_058Lamm__agv____,type,
    aTP_Lamm_agv: set_VEBT_VEBT > fun_fu6610162456291744429VEBT_o ).

tff(sy_c_ATP_058Lamm__agw____,type,
    aTP_Lamm_agw: set_VEBT_VEBT > fun_fu8270842325377534013VEBT_o ).

tff(sy_c_ATP_058Lamm__agx____,type,
    aTP_Lamm_agx: set_real > fun_fu3022329567491847689real_o ).

tff(sy_c_ATP_058Lamm__agy____,type,
    aTP_Lamm_agy: set_real > fun_fu5533088652784741581real_o ).

tff(sy_c_ATP_058Lamm__agz____,type,
    aTP_Lamm_agz: set_nat > fun_fu6621322437453427393_nat_o ).

tff(sy_c_ATP_058Lamm__ah____,type,
    aTP_Lamm_ah: product_prod_nat_nat > fun_na6214517451692987068on_nat ).

tff(sy_c_ATP_058Lamm__aha____,type,
    aTP_Lamm_aha: set_nat > fun_fu6813129056075673441_nat_o ).

tff(sy_c_ATP_058Lamm__ahb____,type,
    aTP_Lamm_ahb: set_int > fun_fu4185471554550717449_int_o ).

tff(sy_c_ATP_058Lamm__ahc____,type,
    aTP_Lamm_ahc: set_int > fun_fu4228744305898656589_int_o ).

tff(sy_c_ATP_058Lamm__ahd____,type,
    aTP_Lamm_ahd: set_complex > fun_fu806425790188207109plex_o ).

tff(sy_c_ATP_058Lamm__ahe____,type,
    aTP_Lamm_ahe: set_complex > fun_fu7523448966510524487plex_o ).

tff(sy_c_ATP_058Lamm__ahf____,type,
    aTP_Lamm_ahf: set_Code_integer > fun_fu8247212220001980967eger_o ).

tff(sy_c_ATP_058Lamm__ahg____,type,
    aTP_Lamm_ahg: set_Code_integer > fun_fu8248176645984958714eger_o ).

tff(sy_c_ATP_058Lamm__ahh____,type,
    aTP_Lamm_ahh: set_VEBT_VEBT > fun_fu3470462164674526041VEBT_o ).

tff(sy_c_ATP_058Lamm__ahi____,type,
    aTP_Lamm_ahi: set_VEBT_VEBT > fun_fu7917602684971597757VEBT_o ).

tff(sy_c_ATP_058Lamm__ahj____,type,
    aTP_Lamm_ahj: set_real > fun_fu6510078244439976541real_o ).

tff(sy_c_ATP_058Lamm__ahk____,type,
    aTP_Lamm_ahk: set_real > fun_fu5587483295140630005real_o ).

tff(sy_c_ATP_058Lamm__ahl____,type,
    aTP_Lamm_ahl: set_nat > fun_fu7145897819039579925_nat_o ).

tff(sy_c_ATP_058Lamm__ahm____,type,
    aTP_Lamm_ahm: set_nat > fun_fu1806770557662746761_nat_o ).

tff(sy_c_ATP_058Lamm__ahn____,type,
    aTP_Lamm_ahn: set_int > fun_fu3473604023002724957_int_o ).

tff(sy_c_ATP_058Lamm__aho____,type,
    aTP_Lamm_aho: set_int > fun_fu7714913403879666293_int_o ).

tff(sy_c_ATP_058Lamm__ahp____,type,
    aTP_Lamm_ahp: set_VEBT_VEBT > fun_fu6775762051927458541VEBT_o ).

tff(sy_c_ATP_058Lamm__ahq____,type,
    aTP_Lamm_ahq: set_VEBT_VEBT > fun_fu7387521734868505405VEBT_o ).

tff(sy_c_ATP_058Lamm__ahr____,type,
    aTP_Lamm_ahr: set_real > fun_fu1352006120924326345real_o ).

tff(sy_c_ATP_058Lamm__ahs____,type,
    aTP_Lamm_ahs: set_real > fun_fu9205927727435512653real_o ).

tff(sy_c_ATP_058Lamm__aht____,type,
    aTP_Lamm_aht: set_nat > fun_fu8157887890162707073_nat_o ).

tff(sy_c_ATP_058Lamm__ahu____,type,
    aTP_Lamm_ahu: set_nat > fun_fu1829967339744774625_nat_o ).

tff(sy_c_ATP_058Lamm__ahv____,type,
    aTP_Lamm_ahv: set_int > fun_fu4320861794972987337_int_o ).

tff(sy_c_ATP_058Lamm__ahw____,type,
    aTP_Lamm_ahw: set_int > fun_fu6214582427498676685_int_o ).

tff(sy_c_ATP_058Lamm__ahx____,type,
    aTP_Lamm_ahx: set_complex > fun_fu5921220358231560645plex_o ).

tff(sy_c_ATP_058Lamm__ahy____,type,
    aTP_Lamm_ahy: set_complex > fun_fu5587198656263613639plex_o ).

tff(sy_c_ATP_058Lamm__ahz____,type,
    aTP_Lamm_ahz: set_Code_integer > fun_fu2470101743130883815eger_o ).

tff(sy_c_ATP_058Lamm__ai____,type,
    aTP_Lamm_ai: product_prod_nat_nat > fun_na6214517451692987068on_nat ).

tff(sy_c_ATP_058Lamm__aia____,type,
    aTP_Lamm_aia: set_Code_integer > fun_fu2782498379308227194eger_o ).

tff(sy_c_ATP_058Lamm__aib____,type,
    aTP_Lamm_aib: set_VEBT_VEBT > fun_fu6610162456291744429VEBT_o ).

tff(sy_c_ATP_058Lamm__aic____,type,
    aTP_Lamm_aic: set_VEBT_VEBT > fun_fu8270842325377534013VEBT_o ).

tff(sy_c_ATP_058Lamm__aid____,type,
    aTP_Lamm_aid: set_real > fun_fu3022329567491847689real_o ).

tff(sy_c_ATP_058Lamm__aie____,type,
    aTP_Lamm_aie: set_real > fun_fu5533088652784741581real_o ).

tff(sy_c_ATP_058Lamm__aif____,type,
    aTP_Lamm_aif: set_nat > fun_fu6621322437453427393_nat_o ).

tff(sy_c_ATP_058Lamm__aig____,type,
    aTP_Lamm_aig: set_nat > fun_fu6813129056075673441_nat_o ).

tff(sy_c_ATP_058Lamm__aih____,type,
    aTP_Lamm_aih: set_int > fun_fu4185471554550717449_int_o ).

tff(sy_c_ATP_058Lamm__aii____,type,
    aTP_Lamm_aii: set_int > fun_fu4228744305898656589_int_o ).

tff(sy_c_ATP_058Lamm__aij____,type,
    aTP_Lamm_aij: fun_se2170385732985038782_nat_o ).

tff(sy_c_ATP_058Lamm__aik____,type,
    aTP_Lamm_aik: fun_se2051630309975736070_int_o ).

tff(sy_c_ATP_058Lamm__ail____,type,
    aTP_Lamm_ail: fun_se823489619503225858plex_o ).

tff(sy_c_ATP_058Lamm__aim____,type,
    aTP_Lamm_aim: fun_se5529282185252607396eger_o ).

tff(sy_c_ATP_058Lamm__ain____,type,
    aTP_Lamm_ain: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__aio____,type,
    aTP_Lamm_aio: fun_co3258106519541737638omplex ).

tff(sy_c_ATP_058Lamm__aip____,type,
    aTP_Lamm_aip: fun_complex_complex ).

tff(sy_c_ATP_058Lamm__aiq____,type,
    aTP_Lamm_aiq: fun_nat_real ).

tff(sy_c_ATP_058Lamm__air____,type,
    aTP_Lamm_air: fun_complex_nat > fun_complex_complex ).

tff(sy_c_ATP_058Lamm__ais____,type,
    aTP_Lamm_ais: fun_int_nat > fun_int_int ).

tff(sy_c_ATP_058Lamm__ait____,type,
    aTP_Lamm_ait: fun_nat_nat > fun_nat_int ).

tff(sy_c_ATP_058Lamm__aiu____,type,
    aTP_Lamm_aiu: fun_nat_nat > fun_nat_Code_integer ).

tff(sy_c_ATP_058Lamm__aiv____,type,
    aTP_Lamm_aiv: fun_nat_nat > fun_nat_nat ).

tff(sy_c_ATP_058Lamm__aiw____,type,
    aTP_Lamm_aiw: fun_complex_int > fun_complex_complex ).

tff(sy_c_ATP_058Lamm__aix____,type,
    aTP_Lamm_aix: fun_nat_int > fun_nat_real ).

tff(sy_c_ATP_058Lamm__aiy____,type,
    aTP_Lamm_aiy: fun_int_int > fun_int_real ).

tff(sy_c_ATP_058Lamm__aiz____,type,
    aTP_Lamm_aiz: fun_int_int > fun_int_rat ).

tff(sy_c_ATP_058Lamm__aj____,type,
    aTP_Lamm_aj: product_prod_nat_nat > fun_na6214517451692987068on_nat ).

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

tff(sy_c_ATP_058Lamm__ajb____,type,
    aTP_Lamm_ajb: vEBT_VEBT > fun_fu8380412309343763075uint32 ).

tff(sy_c_ATP_058Lamm__ajc____,type,
    aTP_Lamm_ajc: real > fun_fu3826616023661743271uint32 ).

tff(sy_c_ATP_058Lamm__ajd____,type,
    aTP_Lamm_ajd: nat > fun_fu2798604719624125935uint32 ).

tff(sy_c_ATP_058Lamm__aje____,type,
    aTP_Lamm_aje: int > fun_fu8432976761865684647uint32 ).

tff(sy_c_ATP_058Lamm__ajf____,type,
    aTP_Lamm_ajf: complex > fun_fu3643814932998673835uint32 ).

tff(sy_c_ATP_058Lamm__ajg____,type,
    aTP_Lamm_ajg: code_integer > fun_fu2584227209315667977uint32 ).

tff(sy_c_ATP_058Lamm__ajh____,type,
    aTP_Lamm_ajh: vEBT_VEBT > fun_fu8198533418994508931T_real ).

tff(sy_c_ATP_058Lamm__aji____,type,
    aTP_Lamm_aji: real > fun_fu3077872570260440615l_real ).

tff(sy_c_ATP_058Lamm__ajj____,type,
    aTP_Lamm_ajj: int > fun_fu121957152452960295t_real ).

tff(sy_c_ATP_058Lamm__ajk____,type,
    aTP_Lamm_ajk: complex > fun_fu7714627026455246123x_real ).

tff(sy_c_ATP_058Lamm__ajl____,type,
    aTP_Lamm_ajl: vEBT_VEBT > fun_fu8380412309343763075uint32 ).

tff(sy_c_ATP_058Lamm__ajm____,type,
    aTP_Lamm_ajm: real > fun_fu3826616023661743271uint32 ).

tff(sy_c_ATP_058Lamm__ajn____,type,
    aTP_Lamm_ajn: nat > fun_fu2798604719624125935uint32 ).

tff(sy_c_ATP_058Lamm__ajo____,type,
    aTP_Lamm_ajo: int > fun_fu8432976761865684647uint32 ).

tff(sy_c_ATP_058Lamm__ajp____,type,
    aTP_Lamm_ajp: complex > fun_fu3643814932998673835uint32 ).

tff(sy_c_ATP_058Lamm__ajq____,type,
    aTP_Lamm_ajq: code_integer > fun_fu2584227209315667977uint32 ).

tff(sy_c_ATP_058Lamm__ajr____,type,
    aTP_Lamm_ajr: vEBT_VEBT > fun_fu8198533418994508931T_real ).

tff(sy_c_ATP_058Lamm__ajs____,type,
    aTP_Lamm_ajs: real > fun_fu3077872570260440615l_real ).

tff(sy_c_ATP_058Lamm__ajt____,type,
    aTP_Lamm_ajt: int > fun_fu121957152452960295t_real ).

tff(sy_c_ATP_058Lamm__aju____,type,
    aTP_Lamm_aju: complex > fun_fu7714627026455246123x_real ).

tff(sy_c_ATP_058Lamm__ajv____,type,
    aTP_Lamm_ajv: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__ajw____,type,
    aTP_Lamm_ajw: fun_nat_rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__ajx____,type,
    aTP_Lamm_ajx: fun_nat_rat > fun_fu8328862064278946247at_rat ).

tff(sy_c_ATP_058Lamm__ajy____,type,
    aTP_Lamm_ajy: fun_nat_real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__ajz____,type,
    aTP_Lamm_ajz: fun_nat_fun_nat_nat > fun_se1179353343636554400at_nat ).

tff(sy_c_ATP_058Lamm__ak____,type,
    aTP_Lamm_ak: product_prod_nat_nat > fun_na5928314549591439550on_nat ).

tff(sy_c_ATP_058Lamm__aka____,type,
    aTP_Lamm_aka: fun_nat_fun_nat_nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__akb____,type,
    aTP_Lamm_akb: fun_nat_fun_nat_nat > fun_se1179353343636554400at_nat ).

tff(sy_c_ATP_058Lamm__akc____,type,
    aTP_Lamm_akc: fun_co7673397065115134084omplex > fun_se7512233253626865466omplex ).

tff(sy_c_ATP_058Lamm__akd____,type,
    aTP_Lamm_akd: fun_co7673397065115134084omplex > fun_co7673397065115134084omplex ).

tff(sy_c_ATP_058Lamm__ake____,type,
    aTP_Lamm_ake: fun_co7673397065115134084omplex > fun_se7512233253626865466omplex ).

tff(sy_c_ATP_058Lamm__akf____,type,
    aTP_Lamm_akf: fun_int_fun_int_int > fun_se4827636401813558708nt_int ).

tff(sy_c_ATP_058Lamm__akg____,type,
    aTP_Lamm_akg: fun_int_fun_int_int > fun_int_fun_int_int ).

tff(sy_c_ATP_058Lamm__akh____,type,
    aTP_Lamm_akh: fun_int_fun_int_int > fun_se4827636401813558708nt_int ).

tff(sy_c_ATP_058Lamm__aki____,type,
    aTP_Lamm_aki: fun_nat_fun_nat_real > fun_se1654316830156123260t_real ).

tff(sy_c_ATP_058Lamm__akj____,type,
    aTP_Lamm_akj: fun_nat_fun_nat_real > fun_nat_fun_nat_real ).

tff(sy_c_ATP_058Lamm__akk____,type,
    aTP_Lamm_akk: fun_nat_fun_nat_real > fun_se1654316830156123260t_real ).

tff(sy_c_ATP_058Lamm__akl____,type,
    aTP_Lamm_akl: fun_nat_complex > fun_nat_real ).

tff(sy_c_ATP_058Lamm__akm____,type,
    aTP_Lamm_akm: fun_complex_complex > fun_complex_real ).

tff(sy_c_ATP_058Lamm__akn____,type,
    aTP_Lamm_akn: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__ako____,type,
    aTP_Lamm_ako: fun_nat_nat > fun_fu7002938529518989751at_nat ).

tff(sy_c_ATP_058Lamm__akp____,type,
    aTP_Lamm_akp: fun_complex_complex > fun_fu5956788528830221615omplex ).

tff(sy_c_ATP_058Lamm__akq____,type,
    aTP_Lamm_akq: fun_int_int > fun_fu1143798476471108391nt_int ).

tff(sy_c_ATP_058Lamm__akr____,type,
    aTP_Lamm_akr: fun_nat_real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__aks____,type,
    aTP_Lamm_aks: nat > fun_fu7002938529518989751at_nat ).

tff(sy_c_ATP_058Lamm__akt____,type,
    aTP_Lamm_akt: complex > fun_fu5956788528830221615omplex ).

tff(sy_c_ATP_058Lamm__aku____,type,
    aTP_Lamm_aku: int > fun_fu1143798476471108391nt_int ).

tff(sy_c_ATP_058Lamm__akv____,type,
    aTP_Lamm_akv: real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__akw____,type,
    aTP_Lamm_akw: fun_nat_nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__akx____,type,
    aTP_Lamm_akx: fun_complex_complex > fun_co7673397065115134084omplex ).

tff(sy_c_ATP_058Lamm__aky____,type,
    aTP_Lamm_aky: fun_int_int > fun_int_fun_int_int ).

tff(sy_c_ATP_058Lamm__akz____,type,
    aTP_Lamm_akz: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__al____,type,
    aTP_Lamm_al: product_prod_nat_nat > fun_na5556145776640407093on_nat ).

tff(sy_c_ATP_058Lamm__ala____,type,
    aTP_Lamm_ala: fun_nat_nat > fun_fu4413992040155564420at_nat ).

tff(sy_c_ATP_058Lamm__alb____,type,
    aTP_Lamm_alb: fun_nat_nat > fun_fu2454420574033104442at_nat ).

tff(sy_c_ATP_058Lamm__alc____,type,
    aTP_Lamm_alc: fun_complex_complex > fun_fu8998703528969179354omplex ).

tff(sy_c_ATP_058Lamm__ald____,type,
    aTP_Lamm_ald: fun_complex_complex > fun_fu8084223252616112272omplex ).

tff(sy_c_ATP_058Lamm__ale____,type,
    aTP_Lamm_ale: fun_int_int > fun_fu923337204662071120nt_int ).

tff(sy_c_ATP_058Lamm__alf____,type,
    aTP_Lamm_alf: fun_int_int > fun_fu3739269683040691718nt_int ).

tff(sy_c_ATP_058Lamm__alg____,type,
    aTP_Lamm_alg: fun_nat_real > fun_fu163975356400620604t_real ).

tff(sy_c_ATP_058Lamm__alh____,type,
    aTP_Lamm_alh: fun_nat_real > fun_fu3306995184869731826t_real ).

tff(sy_c_ATP_058Lamm__ali____,type,
    aTP_Lamm_ali: fun_complex_complex > fun_fu5956788528830221615omplex ).

tff(sy_c_ATP_058Lamm__alj____,type,
    aTP_Lamm_alj: fun_int_int > fun_fu1143798476471108391nt_int ).

tff(sy_c_ATP_058Lamm__alk____,type,
    aTP_Lamm_alk: fun_nat_real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__all____,type,
    aTP_Lamm_all: fun_complex_complex > fun_co7673397065115134084omplex ).

tff(sy_c_ATP_058Lamm__alm____,type,
    aTP_Lamm_alm: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__aln____,type,
    aTP_Lamm_aln: set_nat > fun_fu2435797646934184285_nat_o ).

tff(sy_c_ATP_058Lamm__alo____,type,
    aTP_Lamm_alo: set_nat > fun_fu2828149587716756511BT_nat ).

tff(sy_c_ATP_058Lamm__alp____,type,
    aTP_Lamm_alp: fun_VE597728521808591204at_nat > fun_na6208021027755453088BT_nat ).

tff(sy_c_ATP_058Lamm__alq____,type,
    aTP_Lamm_alq: set_VEBT_VEBT > fun_fu4061198407372677825at_nat ).

tff(sy_c_ATP_058Lamm__alr____,type,
    aTP_Lamm_alr: set_nat > fun_fu5289434066392675523_nat_o ).

tff(sy_c_ATP_058Lamm__als____,type,
    aTP_Lamm_als: set_nat > fun_fu8695033588447641837al_nat ).

tff(sy_c_ATP_058Lamm__alt____,type,
    aTP_Lamm_alt: fun_real_fun_nat_nat > fun_nat_fun_real_nat ).

tff(sy_c_ATP_058Lamm__alu____,type,
    aTP_Lamm_alu: set_real > fun_fu5553606195750802321at_nat ).

tff(sy_c_ATP_058Lamm__alv____,type,
    aTP_Lamm_alv: set_nat > fun_fu927436805355353155_nat_o ).

tff(sy_c_ATP_058Lamm__alw____,type,
    aTP_Lamm_alw: set_nat > fun_fu6398909621167755501nt_nat ).

tff(sy_c_ATP_058Lamm__alx____,type,
    aTP_Lamm_alx: fun_int_fun_nat_nat > fun_nat_fun_int_nat ).

tff(sy_c_ATP_058Lamm__aly____,type,
    aTP_Lamm_aly: set_int > fun_fu7493154764834273859_int_o ).

tff(sy_c_ATP_058Lamm__alz____,type,
    aTP_Lamm_alz: set_int > fun_fu3180206027003385489at_nat ).

tff(sy_c_ATP_058Lamm__am____,type,
    aTP_Lamm_am: fun_Pr5461209604930428974on_nat ).

tff(sy_c_ATP_058Lamm__ama____,type,
    aTP_Lamm_ama: set_nat > fun_fu1580565752866428867_nat_o ).

tff(sy_c_ATP_058Lamm__amb____,type,
    aTP_Lamm_amb: set_nat > fun_fu8020931516743756395ex_nat ).

tff(sy_c_ATP_058Lamm__amc____,type,
    aTP_Lamm_amc: fun_co636942507509095880at_nat > fun_na8457291176974909128ex_nat ).

tff(sy_c_ATP_058Lamm__amd____,type,
    aTP_Lamm_amd: set_complex > fun_fu3074144567237359299plex_o ).

tff(sy_c_ATP_058Lamm__ame____,type,
    aTP_Lamm_ame: set_complex > fun_fu3351286688164852109at_nat ).

tff(sy_c_ATP_058Lamm__amf____,type,
    aTP_Lamm_amf: set_nat > fun_fu8096999557949053699_nat_o ).

tff(sy_c_ATP_058Lamm__amg____,type,
    aTP_Lamm_amg: set_nat > fun_fu4176442768802536380er_nat ).

tff(sy_c_ATP_058Lamm__amh____,type,
    aTP_Lamm_amh: fun_Co3312552843764780407at_nat > fun_na5492297256294505719er_nat ).

tff(sy_c_ATP_058Lamm__ami____,type,
    aTP_Lamm_ami: set_Code_integer > fun_fu6309186030181755779eger_o ).

tff(sy_c_ATP_058Lamm__amj____,type,
    aTP_Lamm_amj: set_Code_integer > fun_fu1095011443884336175at_nat ).

tff(sy_c_ATP_058Lamm__amk____,type,
    aTP_Lamm_amk: set_complex > fun_fu720419882121075617plex_o ).

tff(sy_c_ATP_058Lamm__aml____,type,
    aTP_Lamm_aml: set_complex > fun_fu8871020438547428127omplex ).

tff(sy_c_ATP_058Lamm__amm____,type,
    aTP_Lamm_amm: fun_VE2006399705683203104omplex > fun_co3541073130412562976omplex ).

tff(sy_c_ATP_058Lamm__amn____,type,
    aTP_Lamm_amn: set_VEBT_VEBT > fun_fu8266604298679398363omplex ).

tff(sy_c_ATP_058Lamm__amo____,type,
    aTP_Lamm_amo: set_complex > fun_fu3043630359712370243plex_o ).

tff(sy_c_ATP_058Lamm__amp____,type,
    aTP_Lamm_amp: set_complex > fun_fu7433039812794084273omplex ).

tff(sy_c_ATP_058Lamm__amq____,type,
    aTP_Lamm_amq: fun_re7208826350674946562omplex > fun_co286204877037288194omplex ).

tff(sy_c_ATP_058Lamm__amr____,type,
    aTP_Lamm_amr: set_real > fun_fu1853463631764337203omplex ).

tff(sy_c_ATP_058Lamm__ams____,type,
    aTP_Lamm_ams: set_complex > fun_fu5273811730930669507plex_o ).

tff(sy_c_ATP_058Lamm__amt____,type,
    aTP_Lamm_amt: set_complex > fun_fu6938252901338612493omplex ).

tff(sy_c_ATP_058Lamm__amu____,type,
    aTP_Lamm_amu: fun_na1591161823130732966omplex > fun_co3258106519541737638omplex ).

tff(sy_c_ATP_058Lamm__amv____,type,
    aTP_Lamm_amv: set_nat > fun_fu3780232916559739075_nat_o ).

tff(sy_c_ATP_058Lamm__amw____,type,
    aTP_Lamm_amw: set_nat > fun_fu3656911019502402539omplex ).

tff(sy_c_ATP_058Lamm__amx____,type,
    aTP_Lamm_amx: set_complex > fun_fu4149201653251398851plex_o ).

tff(sy_c_ATP_058Lamm__amy____,type,
    aTP_Lamm_amy: set_complex > fun_fu7724386853639975089omplex ).

tff(sy_c_ATP_058Lamm__amz____,type,
    aTP_Lamm_amz: fun_in1307110748760623106omplex > fun_co7673900812888889602omplex ).

tff(sy_c_ATP_058Lamm__an____,type,
    aTP_Lamm_an: fun_Pr293967910607515554on_nat ).

tff(sy_c_ATP_058Lamm__ana____,type,
    aTP_Lamm_ana: set_int > fun_fu2620374121714387907_int_o ).

tff(sy_c_ATP_058Lamm__anb____,type,
    aTP_Lamm_anb: set_int > fun_fu260611630652282675omplex ).

tff(sy_c_ATP_058Lamm__anc____,type,
    aTP_Lamm_anc: set_complex > fun_fu7012200369710426499plex_o ).

tff(sy_c_ATP_058Lamm__and____,type,
    aTP_Lamm_and: set_complex > fun_fu403589845504736640omplex ).

tff(sy_c_ATP_058Lamm__ane____,type,
    aTP_Lamm_ane: fun_Co8905303033605938227omplex > fun_co1305099417915878579omplex ).

tff(sy_c_ATP_058Lamm__anf____,type,
    aTP_Lamm_anf: set_Code_integer > fun_fu9120999634463783171eger_o ).

tff(sy_c_ATP_058Lamm__ang____,type,
    aTP_Lamm_ang: set_Code_integer > fun_fu243871637272208593omplex ).

tff(sy_c_ATP_058Lamm__anh____,type,
    aTP_Lamm_anh: fun_nat_nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__ani____,type,
    aTP_Lamm_ani: fun_int_int > fun_int_fun_int_int ).

tff(sy_c_ATP_058Lamm__anj____,type,
    aTP_Lamm_anj: fun_VEBT_VEBT_uint32 > fun_fu4239552856239149349uint32 ).

tff(sy_c_ATP_058Lamm__ank____,type,
    aTP_Lamm_ank: fun_real_uint32 > fun_fu5616893895707965165uint32 ).

tff(sy_c_ATP_058Lamm__anl____,type,
    aTP_Lamm_anl: fun_nat_uint32 > fun_fu5445312597681047533uint32 ).

tff(sy_c_ATP_058Lamm__anm____,type,
    aTP_Lamm_anm: fun_int_uint32 > fun_fu4828966935367951085uint32 ).

tff(sy_c_ATP_058Lamm__ann____,type,
    aTP_Lamm_ann: fun_complex_uint32 > fun_fu7155343958083200365uint32 ).

tff(sy_c_ATP_058Lamm__ano____,type,
    aTP_Lamm_ano: set_Code_integer > fun_fu8094397175230428611eger_o ).

tff(sy_c_ATP_058Lamm__anp____,type,
    aTP_Lamm_anp: fun_Co1554742298434871494uint32 > fun_fu7156106677247248941uint32 ).

tff(sy_c_ATP_058Lamm__anq____,type,
    aTP_Lamm_anq: fun_VEBT_VEBT_real > fun_fu4911191927098811621T_real ).

tff(sy_c_ATP_058Lamm__anr____,type,
    aTP_Lamm_anr: fun_real_real > fun_fu4478067837645853357l_real ).

tff(sy_c_ATP_058Lamm__ans____,type,
    aTP_Lamm_ans: fun_int_real > fun_fu7030473000426307245t_real ).

tff(sy_c_ATP_058Lamm__ant____,type,
    aTP_Lamm_ant: fun_complex_real > fun_fu3043978019632300333x_real ).

tff(sy_c_ATP_058Lamm__anu____,type,
    aTP_Lamm_anu: fun_nat_nat > fun_nat_nat ).

tff(sy_c_ATP_058Lamm__anv____,type,
    aTP_Lamm_anv: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__anw____,type,
    aTP_Lamm_anw: fun_nat_nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__anx____,type,
    aTP_Lamm_anx: fun_nat_real > fun_nat_fun_nat_real ).

tff(sy_c_ATP_058Lamm__any____,type,
    aTP_Lamm_any: fun_int_uint32 > fun_int_o ).

tff(sy_c_ATP_058Lamm__anz____,type,
    aTP_Lamm_anz: fun_complex_uint32 > fun_complex_o ).

tff(sy_c_ATP_058Lamm__ao____,type,
    aTP_Lamm_ao: fun_Pr293967910607515554on_nat ).

tff(sy_c_ATP_058Lamm__aoa____,type,
    aTP_Lamm_aoa: fun_Co1554742298434871494uint32 > fun_Code_integer_o ).

tff(sy_c_ATP_058Lamm__aob____,type,
    aTP_Lamm_aob: fun_int_real > fun_int_o ).

tff(sy_c_ATP_058Lamm__aoc____,type,
    aTP_Lamm_aoc: fun_complex_real > fun_complex_o ).

tff(sy_c_ATP_058Lamm__aod____,type,
    aTP_Lamm_aod: fun_Co3960896333048504454r_real > fun_Code_integer_o ).

tff(sy_c_ATP_058Lamm__aoe____,type,
    aTP_Lamm_aoe: fun_int_rat > fun_int_o ).

tff(sy_c_ATP_058Lamm__aof____,type,
    aTP_Lamm_aof: fun_complex_rat > fun_complex_o ).

tff(sy_c_ATP_058Lamm__aog____,type,
    aTP_Lamm_aog: fun_Code_integer_rat > fun_Code_integer_o ).

tff(sy_c_ATP_058Lamm__aoh____,type,
    aTP_Lamm_aoh: fun_int_nat > fun_int_o ).

tff(sy_c_ATP_058Lamm__aoi____,type,
    aTP_Lamm_aoi: uint32 > fun_na5363892732107265798uint32 ).

tff(sy_c_ATP_058Lamm__aoj____,type,
    aTP_Lamm_aoj: word_N3645301735248828278l_num1 > fun_na4844478536282698896l_num1 ).

tff(sy_c_ATP_058Lamm__aok____,type,
    aTP_Lamm_aok: rat > fun_nat_fun_nat_rat ).

tff(sy_c_ATP_058Lamm__aol____,type,
    aTP_Lamm_aol: int > fun_nat_fun_nat_int ).

tff(sy_c_ATP_058Lamm__aom____,type,
    aTP_Lamm_aom: real > fun_nat_fun_nat_real ).

tff(sy_c_ATP_058Lamm__aon____,type,
    aTP_Lamm_aon: fun_nat_nat > fun_na3244735409550470327at_nat ).

tff(sy_c_ATP_058Lamm__aoo____,type,
    aTP_Lamm_aoo: fun_nat_real > fun_na7718645245721260691t_real ).

tff(sy_c_ATP_058Lamm__aop____,type,
    aTP_Lamm_aop: fun_nat_rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__aoq____,type,
    aTP_Lamm_aoq: fun_nat_int > fun_nat_int ).

tff(sy_c_ATP_058Lamm__aor____,type,
    aTP_Lamm_aor: fun_nat_rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__aos____,type,
    aTP_Lamm_aos: fun_nat_int > fun_nat_int ).

tff(sy_c_ATP_058Lamm__aot____,type,
    aTP_Lamm_aot: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__aou____,type,
    aTP_Lamm_aou: fun_nat_uint32 > fun_na3857091055356329634uint32 ).

tff(sy_c_ATP_058Lamm__aov____,type,
    aTP_Lamm_aov: fun_nat_rat > fun_nat_fun_rat_rat ).

tff(sy_c_ATP_058Lamm__aow____,type,
    aTP_Lamm_aow: fun_nat_int > fun_nat_fun_int_int ).

tff(sy_c_ATP_058Lamm__aox____,type,
    aTP_Lamm_aox: fun_nat_nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__aoy____,type,
    aTP_Lamm_aoy: fun_nat_real > fun_na6724076748692969250l_real ).

tff(sy_c_ATP_058Lamm__aoz____,type,
    aTP_Lamm_aoz: vEBT_VEBT > fun_fu4141963935195952137T_real ).

tff(sy_c_ATP_058Lamm__ap____,type,
    aTP_Lamm_ap: fun_op8552747401449489031i_assn ).

tff(sy_c_ATP_058Lamm__apa____,type,
    aTP_Lamm_apa: complex > fun_fu709861064203714431x_real ).

tff(sy_c_ATP_058Lamm__apb____,type,
    aTP_Lamm_apb: code_integer > fun_fu8135725022974991116r_real ).

tff(sy_c_ATP_058Lamm__apc____,type,
    aTP_Lamm_apc: vEBT_VEBT > fun_fu4463953228700081077BT_rat ).

tff(sy_c_ATP_058Lamm__apd____,type,
    aTP_Lamm_apd: complex > fun_fu6638204120336560003ex_rat ).

tff(sy_c_ATP_058Lamm__ape____,type,
    aTP_Lamm_ape: code_integer > fun_fu8767785434844327952er_rat ).

tff(sy_c_ATP_058Lamm__apf____,type,
    aTP_Lamm_apf: vEBT_VEBT > fun_fu5586652545743564717BT_nat ).

tff(sy_c_ATP_058Lamm__apg____,type,
    aTP_Lamm_apg: complex > fun_fu2513798379308225899ex_nat ).

tff(sy_c_ATP_058Lamm__aph____,type,
    aTP_Lamm_aph: code_integer > fun_fu2498091903632033272er_nat ).

tff(sy_c_ATP_058Lamm__api____,type,
    aTP_Lamm_api: vEBT_VEBT > fun_fu7059288016784409353BT_int ).

tff(sy_c_ATP_058Lamm__apj____,type,
    aTP_Lamm_apj: fun_nat_uint32 > fun_nat_uint32 ).

tff(sy_c_ATP_058Lamm__apk____,type,
    aTP_Lamm_apk: fun_nat_rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__apl____,type,
    aTP_Lamm_apl: fun_nat_int > fun_nat_int ).

tff(sy_c_ATP_058Lamm__apm____,type,
    aTP_Lamm_apm: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__apn____,type,
    aTP_Lamm_apn: fun_nat_rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__apo____,type,
    aTP_Lamm_apo: fun_nat_int > fun_nat_int ).

tff(sy_c_ATP_058Lamm__app____,type,
    aTP_Lamm_app: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__apq____,type,
    aTP_Lamm_apq: fun_nat_rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__apr____,type,
    aTP_Lamm_apr: fun_nat_int > fun_nat_int ).

tff(sy_c_ATP_058Lamm__aps____,type,
    aTP_Lamm_aps: fun_nat_nat > fun_nat_nat ).

tff(sy_c_ATP_058Lamm__apt____,type,
    aTP_Lamm_apt: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__apu____,type,
    aTP_Lamm_apu: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__apv____,type,
    aTP_Lamm_apv: fun_nat_real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__apw____,type,
    aTP_Lamm_apw: word_N3645301735248828278l_num1 > fun_wo3147960520859867190l_num1 ).

tff(sy_c_ATP_058Lamm__apx____,type,
    aTP_Lamm_apx: rat > fun_rat_fun_nat_rat ).

tff(sy_c_ATP_058Lamm__apy____,type,
    aTP_Lamm_apy: int > fun_int_fun_nat_int ).

tff(sy_c_ATP_058Lamm__apz____,type,
    aTP_Lamm_apz: code_integer > fun_Co6226381647958292740nteger ).

tff(sy_c_ATP_058Lamm__aq____,type,
    aTP_Lamm_aq: fun_option_nat_assn ).

tff(sy_c_ATP_058Lamm__aqa____,type,
    aTP_Lamm_aqa: nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__aqb____,type,
    aTP_Lamm_aqb: real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__aqc____,type,
    aTP_Lamm_aqc: nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__aqd____,type,
    aTP_Lamm_aqd: fun_nat_real ).

tff(sy_c_ATP_058Lamm__aqe____,type,
    aTP_Lamm_aqe: fun_nat_int ).

tff(sy_c_ATP_058Lamm__aqf____,type,
    aTP_Lamm_aqf: fun_nat_real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__aqg____,type,
    aTP_Lamm_aqg: fun_nat_real > fun_se2100927751953696370t_real ).

tff(sy_c_ATP_058Lamm__aqh____,type,
    aTP_Lamm_aqh: nat > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__aqi____,type,
    aTP_Lamm_aqi: nat > fun_int_fun_nat_int ).

tff(sy_c_ATP_058Lamm__aqj____,type,
    aTP_Lamm_aqj: fun_VEBT_VEBT_nat > fun_fu5238640373165125763BT_nat ).

tff(sy_c_ATP_058Lamm__aqk____,type,
    aTP_Lamm_aqk: fun_real_nat > fun_fu3214020956574444655al_nat ).

tff(sy_c_ATP_058Lamm__aql____,type,
    aTP_Lamm_aql: fun_int_nat > fun_fu8905809029841901679nt_nat ).

tff(sy_c_ATP_058Lamm__aqm____,type,
    aTP_Lamm_aqm: fun_nat_nat > fun_fu7002938529518989751at_nat ).

tff(sy_c_ATP_058Lamm__aqn____,type,
    aTP_Lamm_aqn: fun_complex_complex ).

tff(sy_c_ATP_058Lamm__aqo____,type,
    aTP_Lamm_aqo: nat > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__aqp____,type,
    aTP_Lamm_aqp: nat > fun_fu7002938529518989751at_nat ).

tff(sy_c_ATP_058Lamm__aqq____,type,
    aTP_Lamm_aqq: nat > fun_fu8464300013002972271at_int ).

tff(sy_c_ATP_058Lamm__aqr____,type,
    aTP_Lamm_aqr: fun_nat_int > fun_fu8464300013002972271at_int ).

tff(sy_c_ATP_058Lamm__aqs____,type,
    aTP_Lamm_aqs: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__aqt____,type,
    aTP_Lamm_aqt: fun_VE2989161857680785728t_real > fun_na2714374887907562620T_real ).

tff(sy_c_ATP_058Lamm__aqu____,type,
    aTP_Lamm_aqu: set_VEBT_VEBT > fun_fu830247258886439116t_real ).

tff(sy_c_ATP_058Lamm__aqv____,type,
    aTP_Lamm_aqv: fun_re4238438383274954274t_real > fun_na6724076748692969250l_real ).

tff(sy_c_ATP_058Lamm__aqw____,type,
    aTP_Lamm_aqw: set_real > fun_fu5104398059618711576t_real ).

tff(sy_c_ATP_058Lamm__aqx____,type,
    aTP_Lamm_aqx: fun_int_fun_nat_real > fun_nat_fun_int_real ).

tff(sy_c_ATP_058Lamm__aqy____,type,
    aTP_Lamm_aqy: set_int > fun_fu7141299348810748056t_real ).

tff(sy_c_ATP_058Lamm__aqz____,type,
    aTP_Lamm_aqz: set_nat > fun_fu7979797992566127108at_nat ).

tff(sy_c_ATP_058Lamm__ar____,type,
    aTP_Lamm_ar: vEBT_VEBT > fun_VE6841497200278198903t_assn ).

tff(sy_c_ATP_058Lamm__ara____,type,
    aTP_Lamm_ara: fun_co3258106519541737638omplex > fun_na1591161823130732966omplex ).

tff(sy_c_ATP_058Lamm__arb____,type,
    aTP_Lamm_arb: set_complex > fun_fu689363623238371998omplex ).

tff(sy_c_ATP_058Lamm__arc____,type,
    aTP_Lamm_arc: fun_int_fun_nat_int > fun_nat_fun_int_int ).

tff(sy_c_ATP_058Lamm__ard____,type,
    aTP_Lamm_ard: set_int > fun_fu688336635129761048at_int ).

tff(sy_c_ATP_058Lamm__are____,type,
    aTP_Lamm_are: set_nat > fun_fu2115079063411676732t_real ).

tff(sy_c_ATP_058Lamm__arf____,type,
    aTP_Lamm_arf: real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__arg____,type,
    aTP_Lamm_arg: nat > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__arh____,type,
    aTP_Lamm_arh: fun_nat_o > fun_fu8464300013002972271at_int ).

tff(sy_c_ATP_058Lamm__ari____,type,
    aTP_Lamm_ari: fun_nat_o > fun_fu7002938529518989751at_nat ).

tff(sy_c_ATP_058Lamm__arj____,type,
    aTP_Lamm_arj: fun_nat_o > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__ark____,type,
    aTP_Lamm_ark: set_nat > fun_fu8464300013002972271at_int ).

tff(sy_c_ATP_058Lamm__arl____,type,
    aTP_Lamm_arl: set_nat > fun_fu7002938529518989751at_nat ).

tff(sy_c_ATP_058Lamm__arm____,type,
    aTP_Lamm_arm: set_nat > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__arn____,type,
    aTP_Lamm_arn: rat > fun_ra5513060666784058567at_rat ).

tff(sy_c_ATP_058Lamm__aro____,type,
    aTP_Lamm_aro: rat > fun_ra5513060666784058567at_rat ).

tff(sy_c_ATP_058Lamm__arp____,type,
    aTP_Lamm_arp: real > fun_re4756235295228751087t_real ).

tff(sy_c_ATP_058Lamm__arq____,type,
    aTP_Lamm_arq: real > fun_re4756235295228751087t_real ).

tff(sy_c_ATP_058Lamm__arr____,type,
    aTP_Lamm_arr: fun_VE7674639578305544598at_nat > fun_fu424623322819101878EBTi_o ).

tff(sy_c_ATP_058Lamm__ars____,type,
    aTP_Lamm_ars: fun_VE7674639578305544598at_nat > fun_fu7108391812694488075_nat_o ).

tff(sy_c_ATP_058Lamm__art____,type,
    aTP_Lamm_art: fun_VE7674639578305544598at_nat > fun_fu8879972437492397021ap_o_o ).

tff(sy_c_ATP_058Lamm__aru____,type,
    aTP_Lamm_aru: fun_wo7490464530504204009l_num1 ).

tff(sy_c_ATP_058Lamm__arv____,type,
    aTP_Lamm_arv: fun_uint32_uint32 ).

tff(sy_c_ATP_058Lamm__arw____,type,
    aTP_Lamm_arw: fun_rat_rat ).

tff(sy_c_ATP_058Lamm__arx____,type,
    aTP_Lamm_arx: fun_real_real ).

tff(sy_c_ATP_058Lamm__ary____,type,
    aTP_Lamm_ary: fun_int_int ).

tff(sy_c_ATP_058Lamm__arz____,type,
    aTP_Lamm_arz: fun_nat_nat ).

tff(sy_c_ATP_058Lamm__as____,type,
    aTP_Lamm_as: vEBT_VEBT > fun_VE6841497200278198903t_assn ).

tff(sy_c_ATP_058Lamm__asa____,type,
    aTP_Lamm_asa: fun_Co8499238673658658103nteger ).

tff(sy_c_ATP_058Lamm__asb____,type,
    aTP_Lamm_asb: set_VEBT_VEBT > fun_se2994566860605507364VEBT_o ).

tff(sy_c_ATP_058Lamm__asc____,type,
    aTP_Lamm_asc: set_VEBT_VEBT > fun_se8577372796259920304real_o ).

tff(sy_c_ATP_058Lamm__asd____,type,
    aTP_Lamm_asd: set_real > fun_se7507094010934790064VEBT_o ).

tff(sy_c_ATP_058Lamm__ase____,type,
    aTP_Lamm_ase: set_real > fun_se7053787797352540032real_o ).

tff(sy_c_ATP_058Lamm__asf____,type,
    aTP_Lamm_asf: set_VEBT_VEBT > fun_se1984261111898847700_nat_o ).

tff(sy_c_ATP_058Lamm__asg____,type,
    aTP_Lamm_asg: set_real > fun_se3507568288292146196_nat_o ).

tff(sy_c_ATP_058Lamm__ash____,type,
    aTP_Lamm_ash: set_VEBT_VEBT > fun_se1347286066314442544_int_o ).

tff(sy_c_ATP_058Lamm__asi____,type,
    aTP_Lamm_asi: set_real > fun_se5930322593162487296_int_o ).

tff(sy_c_ATP_058Lamm__asj____,type,
    aTP_Lamm_asj: set_VEBT_VEBT > fun_se4615211112688433714plex_o ).

tff(sy_c_ATP_058Lamm__ask____,type,
    aTP_Lamm_ask: set_real > fun_se6597659941194369018plex_o ).

tff(sy_c_ATP_058Lamm__asl____,type,
    aTP_Lamm_asl: fun_Pr6004456690014451917_nat_o > fun_fu1312537534749845416_nat_o ).

tff(sy_c_ATP_058Lamm__asm____,type,
    aTP_Lamm_asm: fun_Pr7132087832597382491_nat_o > fun_fu9110254849647837056ap_o_o ).

tff(sy_c_ATP_058Lamm__asn____,type,
    aTP_Lamm_asn: fun_Pr4018130278567771477_nat_o > fun_fu424623322819101878EBTi_o ).

tff(sy_c_ATP_058Lamm__aso____,type,
    aTP_Lamm_aso: fun_Pr4917118001609057450_nat_o > fun_fu7108391812694488075_nat_o ).

tff(sy_c_ATP_058Lamm__asp____,type,
    aTP_Lamm_asp: fun_Pr3008164681728764094_nat_o > fun_fu8879972437492397021ap_o_o ).

tff(sy_c_ATP_058Lamm__asq____,type,
    aTP_Lamm_asq: fun_nat_nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__asr____,type,
    aTP_Lamm_asr: fun_nat_real > fun_nat_fun_nat_real ).

tff(sy_c_ATP_058Lamm__ass____,type,
    aTP_Lamm_ass: fun_nat_rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__ast____,type,
    aTP_Lamm_ast: fun_nat_int > fun_nat_int ).

tff(sy_c_ATP_058Lamm__asu____,type,
    aTP_Lamm_asu: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__asv____,type,
    aTP_Lamm_asv: fun_nat_rat > fun_rat_fun_nat_rat ).

tff(sy_c_ATP_058Lamm__asw____,type,
    aTP_Lamm_asw: fun_nat_int > fun_int_fun_nat_int ).

tff(sy_c_ATP_058Lamm__asx____,type,
    aTP_Lamm_asx: fun_nat_Code_integer > fun_Co6226381647958292740nteger ).

tff(sy_c_ATP_058Lamm__asy____,type,
    aTP_Lamm_asy: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__asz____,type,
    aTP_Lamm_asz: uint32 > fun_ui359189924029289583uint32 ).

tff(sy_c_ATP_058Lamm__at____,type,
    aTP_Lamm_at: nat > fun_VE6841497200278198903t_assn ).

tff(sy_c_ATP_058Lamm__ata____,type,
    aTP_Lamm_ata: uint32 > fun_ui359189924029289583uint32 ).

tff(sy_c_ATP_058Lamm__atb____,type,
    aTP_Lamm_atb: word_N3645301735248828278l_num1 > fun_wo8382523896974462723l_num1 ).

tff(sy_c_ATP_058Lamm__atc____,type,
    aTP_Lamm_atc: word_N3645301735248828278l_num1 > fun_wo8382523896974462723l_num1 ).

tff(sy_c_ATP_058Lamm__atd____,type,
    aTP_Lamm_atd: rat > fun_ra5513060666784058567at_rat ).

tff(sy_c_ATP_058Lamm__ate____,type,
    aTP_Lamm_ate: rat > fun_ra5513060666784058567at_rat ).

tff(sy_c_ATP_058Lamm__atf____,type,
    aTP_Lamm_atf: int > fun_in6793278964266839919at_int ).

tff(sy_c_ATP_058Lamm__atg____,type,
    aTP_Lamm_atg: int > fun_in6793278964266839919at_int ).

tff(sy_c_ATP_058Lamm__ath____,type,
    aTP_Lamm_ath: real > fun_re4756235295228751087t_real ).

tff(sy_c_ATP_058Lamm__ati____,type,
    aTP_Lamm_ati: real > fun_re4756235295228751087t_real ).

tff(sy_c_ATP_058Lamm__atj____,type,
    aTP_Lamm_atj: uint32 > fun_na7952645553354447727uint32 ).

tff(sy_c_ATP_058Lamm__atk____,type,
    aTP_Lamm_atk: word_N3645301735248828278l_num1 > fun_na8877676674051794947l_num1 ).

tff(sy_c_ATP_058Lamm__atl____,type,
    aTP_Lamm_atl: rat > fun_na1148497095348402375at_rat ).

tff(sy_c_ATP_058Lamm__atm____,type,
    aTP_Lamm_atm: int > fun_na2382631619178260847at_int ).

tff(sy_c_ATP_058Lamm__atn____,type,
    aTP_Lamm_atn: real > fun_na5225841814499866095t_real ).

tff(sy_c_ATP_058Lamm__ato____,type,
    aTP_Lamm_ato: uint32 > fun_na7952645553354447727uint32 ).

tff(sy_c_ATP_058Lamm__atp____,type,
    aTP_Lamm_atp: word_N3645301735248828278l_num1 > fun_na8877676674051794947l_num1 ).

tff(sy_c_ATP_058Lamm__atq____,type,
    aTP_Lamm_atq: rat > fun_na1148497095348402375at_rat ).

tff(sy_c_ATP_058Lamm__atr____,type,
    aTP_Lamm_atr: int > fun_na2382631619178260847at_int ).

tff(sy_c_ATP_058Lamm__ats____,type,
    aTP_Lamm_ats: real > fun_na5225841814499866095t_real ).

tff(sy_c_ATP_058Lamm__att____,type,
    aTP_Lamm_att: uint32 > fun_na5363892732107265798uint32 ).

tff(sy_c_ATP_058Lamm__atu____,type,
    aTP_Lamm_atu: word_N3645301735248828278l_num1 > fun_na4844478536282698896l_num1 ).

tff(sy_c_ATP_058Lamm__atv____,type,
    aTP_Lamm_atv: rat > fun_nat_fun_nat_rat ).

tff(sy_c_ATP_058Lamm__atw____,type,
    aTP_Lamm_atw: int > fun_nat_fun_nat_int ).

tff(sy_c_ATP_058Lamm__atx____,type,
    aTP_Lamm_atx: real > fun_nat_fun_nat_real ).

tff(sy_c_ATP_058Lamm__aty____,type,
    aTP_Lamm_aty: fun_Pr8246910697231053897at_nat > fun_fu1312537534749845416_nat_o ).

tff(sy_c_ATP_058Lamm__atz____,type,
    aTP_Lamm_atz: fun_Pr8246910697231053897at_nat > fun_fu9110254849647837056ap_o_o ).

tff(sy_c_ATP_058Lamm__au____,type,
    aTP_Lamm_au: nat > fun_na4576486074062635204t_assn ).

tff(sy_c_ATP_058Lamm__aua____,type,
    aTP_Lamm_aua: fun_Pr3615232783394198956at_nat > fun_fu424623322819101878EBTi_o ).

tff(sy_c_ATP_058Lamm__aub____,type,
    aTP_Lamm_aub: fun_Pr3615232783394198956at_nat > fun_fu7108391812694488075_nat_o ).

tff(sy_c_ATP_058Lamm__auc____,type,
    aTP_Lamm_auc: fun_Pr3615232783394198956at_nat > fun_fu8879972437492397021ap_o_o ).

tff(sy_c_ATP_058Lamm__aud____,type,
    aTP_Lamm_aud: fun_fu7962274784768186808_nat_o ).

tff(sy_c_ATP_058Lamm__aue____,type,
    aTP_Lamm_aue: fun_fu2183070276626753582ap_o_o ).

tff(sy_c_ATP_058Lamm__auf____,type,
    aTP_Lamm_auf: fun_fu3145514261243071828EBTi_o ).

tff(sy_c_ATP_058Lamm__aug____,type,
    aTP_Lamm_aug: fun_fu5284765496190745342_nat_o ).

tff(sy_c_ATP_058Lamm__auh____,type,
    aTP_Lamm_auh: fun_fu6850233865993928942ap_o_o ).

tff(sy_c_ATP_058Lamm__aui____,type,
    aTP_Lamm_aui: fun_nat_real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__auj____,type,
    aTP_Lamm_auj: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__auk____,type,
    aTP_Lamm_auk: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__aul____,type,
    aTP_Lamm_aul: word_N3645301735248828278l_num1 > fun_na6674401012815655222l_num1 ).

tff(sy_c_ATP_058Lamm__aum____,type,
    aTP_Lamm_aum: rat > fun_nat_fun_rat_rat ).

tff(sy_c_ATP_058Lamm__aun____,type,
    aTP_Lamm_aun: real > fun_na6724076748692969250l_real ).

tff(sy_c_ATP_058Lamm__auo____,type,
    aTP_Lamm_auo: int > fun_nat_fun_int_int ).

tff(sy_c_ATP_058Lamm__aup____,type,
    aTP_Lamm_aup: code_integer > fun_na7078387940644931716nteger ).

tff(sy_c_ATP_058Lamm__auq____,type,
    aTP_Lamm_auq: nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__aur____,type,
    aTP_Lamm_aur: fun_VE4436863797057549263at_nat > fun_fu1312537534749845416_nat_o ).

tff(sy_c_ATP_058Lamm__aus____,type,
    aTP_Lamm_aus: fun_VE4436863797057549263at_nat > fun_fu9110254849647837056ap_o_o ).

tff(sy_c_ATP_058Lamm__aut____,type,
    aTP_Lamm_aut: fun_Pr2564325226415181874at_nat > fun_fu424623322819101878EBTi_o ).

tff(sy_c_ATP_058Lamm__auu____,type,
    aTP_Lamm_auu: fun_Pr2564325226415181874at_nat > fun_fu7108391812694488075_nat_o ).

tff(sy_c_ATP_058Lamm__auv____,type,
    aTP_Lamm_auv: fun_Pr2564325226415181874at_nat > fun_fu8879972437492397021ap_o_o ).

tff(sy_c_ATP_058Lamm__auw____,type,
    aTP_Lamm_auw: rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__aux____,type,
    aTP_Lamm_aux: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__auy____,type,
    aTP_Lamm_auy: fun_nat_real ).

tff(sy_c_ATP_058Lamm__auz____,type,
    aTP_Lamm_auz: fun_nat_int ).

tff(sy_c_ATP_058Lamm__av____,type,
    aTP_Lamm_av: set_VEBT_VEBT > fun_VEBT_VEBT_o ).

tff(sy_c_ATP_058Lamm__ava____,type,
    aTP_Lamm_ava: fun_int_int ).

tff(sy_c_ATP_058Lamm__avb____,type,
    aTP_Lamm_avb: fun_nat_nat ).

tff(sy_c_ATP_058Lamm__avc____,type,
    aTP_Lamm_avc: fun_nat_int > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__avd____,type,
    aTP_Lamm_avd: fun_nat_int > fun_nat_int ).

tff(sy_c_ATP_058Lamm__ave____,type,
    aTP_Lamm_ave: vEBT_VEBT > fun_fu8198533418994508931T_real ).

tff(sy_c_ATP_058Lamm__avf____,type,
    aTP_Lamm_avf: real > fun_fu3077872570260440615l_real ).

tff(sy_c_ATP_058Lamm__avg____,type,
    aTP_Lamm_avg: nat > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__avh____,type,
    aTP_Lamm_avh: int > fun_fu121957152452960295t_real ).

tff(sy_c_ATP_058Lamm__avi____,type,
    aTP_Lamm_avi: complex > fun_fu7714627026455246123x_real ).

tff(sy_c_ATP_058Lamm__avj____,type,
    aTP_Lamm_avj: code_integer > fun_fu555566762037042313r_real ).

tff(sy_c_ATP_058Lamm__avk____,type,
    aTP_Lamm_avk: vEBT_VEBT > fun_fu2188483506383969923BT_rat ).

tff(sy_c_ATP_058Lamm__avl____,type,
    aTP_Lamm_avl: real > fun_fu2598743018846156927al_rat ).

tff(sy_c_ATP_058Lamm__avm____,type,
    aTP_Lamm_avm: nat > fun_fu8328862064278946247at_rat ).

tff(sy_c_ATP_058Lamm__avn____,type,
    aTP_Lamm_avn: int > fun_fu1008360527747082367nt_rat ).

tff(sy_c_ATP_058Lamm__avo____,type,
    aTP_Lamm_avo: vEBT_VEBT > fun_fu8198533418994508931T_real ).

tff(sy_c_ATP_058Lamm__avp____,type,
    aTP_Lamm_avp: real > fun_fu3077872570260440615l_real ).

tff(sy_c_ATP_058Lamm__avq____,type,
    aTP_Lamm_avq: nat > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__avr____,type,
    aTP_Lamm_avr: int > fun_fu121957152452960295t_real ).

tff(sy_c_ATP_058Lamm__avs____,type,
    aTP_Lamm_avs: complex > fun_fu7714627026455246123x_real ).

tff(sy_c_ATP_058Lamm__avt____,type,
    aTP_Lamm_avt: code_integer > fun_fu555566762037042313r_real ).

tff(sy_c_ATP_058Lamm__avu____,type,
    aTP_Lamm_avu: vEBT_VEBT > fun_fu2188483506383969923BT_rat ).

tff(sy_c_ATP_058Lamm__avv____,type,
    aTP_Lamm_avv: real > fun_fu2598743018846156927al_rat ).

tff(sy_c_ATP_058Lamm__avw____,type,
    aTP_Lamm_avw: nat > fun_fu8328862064278946247at_rat ).

tff(sy_c_ATP_058Lamm__avx____,type,
    aTP_Lamm_avx: int > fun_fu1008360527747082367nt_rat ).

tff(sy_c_ATP_058Lamm__avy____,type,
    aTP_Lamm_avy: fun_nat_fun_nat_int > fun_se7912642235559603452at_int ).

tff(sy_c_ATP_058Lamm__avz____,type,
    aTP_Lamm_avz: fun_nat_fun_nat_int > fun_nat_fun_nat_int ).

tff(sy_c_ATP_058Lamm__aw____,type,
    aTP_Lamm_aw: set_real > fun_real_o ).

tff(sy_c_ATP_058Lamm__awa____,type,
    aTP_Lamm_awa: fun_nat_fun_nat_int > fun_se7912642235559603452at_int ).

tff(sy_c_ATP_058Lamm__awb____,type,
    aTP_Lamm_awb: fun_nat_fun_int_int > fun_se2640714827120536920at_int ).

tff(sy_c_ATP_058Lamm__awc____,type,
    aTP_Lamm_awc: fun_nat_fun_int_int > fun_int_fun_nat_int ).

tff(sy_c_ATP_058Lamm__awd____,type,
    aTP_Lamm_awd: fun_nat_fun_int_int > fun_se876191773397849432nt_int ).

tff(sy_c_ATP_058Lamm__awe____,type,
    aTP_Lamm_awe: fun_int_fun_nat_int > fun_se876191773397849432nt_int ).

tff(sy_c_ATP_058Lamm__awf____,type,
    aTP_Lamm_awf: fun_int_fun_nat_int > fun_se2640714827120536920at_int ).

tff(sy_c_ATP_058Lamm__awg____,type,
    aTP_Lamm_awg: fun_int_fun_int_int > fun_se4827636401813558708nt_int ).

tff(sy_c_ATP_058Lamm__awh____,type,
    aTP_Lamm_awh: fun_int_fun_int_int > fun_se4827636401813558708nt_int ).

tff(sy_c_ATP_058Lamm__awi____,type,
    aTP_Lamm_awi: fun_nat_fun_nat_nat > fun_se1179353343636554400at_nat ).

tff(sy_c_ATP_058Lamm__awj____,type,
    aTP_Lamm_awj: fun_nat_fun_nat_nat > fun_se1179353343636554400at_nat ).

tff(sy_c_ATP_058Lamm__awk____,type,
    aTP_Lamm_awk: fun_nat_int > fun_fu8464300013002972271at_int ).

tff(sy_c_ATP_058Lamm__awl____,type,
    aTP_Lamm_awl: fun_int_int > fun_fu1143798476471108391nt_int ).

tff(sy_c_ATP_058Lamm__awm____,type,
    aTP_Lamm_awm: fun_nat_nat > fun_fu7002938529518989751at_nat ).

tff(sy_c_ATP_058Lamm__awn____,type,
    aTP_Lamm_awn: fun_nat_int > fun_nat_fun_nat_int ).

tff(sy_c_ATP_058Lamm__awo____,type,
    aTP_Lamm_awo: fun_int_int > fun_nat_fun_int_int ).

tff(sy_c_ATP_058Lamm__awp____,type,
    aTP_Lamm_awp: fun_nat_nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__awq____,type,
    aTP_Lamm_awq: set_nat > fun_fu8251827778467837215BT_int ).

tff(sy_c_ATP_058Lamm__awr____,type,
    aTP_Lamm_awr: fun_VE7331017413731640256at_int > fun_na3717937882823726332BT_int ).

tff(sy_c_ATP_058Lamm__aws____,type,
    aTP_Lamm_aws: set_VEBT_VEBT > fun_fu6630916859057521017at_int ).

tff(sy_c_ATP_058Lamm__awt____,type,
    aTP_Lamm_awt: set_nat > fun_fu3562227594288721901al_int ).

tff(sy_c_ATP_058Lamm__awu____,type,
    aTP_Lamm_awu: fun_real_fun_nat_int > fun_nat_fun_real_int ).

tff(sy_c_ATP_058Lamm__awv____,type,
    aTP_Lamm_awv: set_real > fun_fu3518430294936654993at_int ).

tff(sy_c_ATP_058Lamm__aww____,type,
    aTP_Lamm_aww: set_nat > fun_fu2556788080963300715ex_int ).

tff(sy_c_ATP_058Lamm__awx____,type,
    aTP_Lamm_awx: fun_co7370231399432144932at_int > fun_na5967208032043182372ex_int ).

tff(sy_c_ATP_058Lamm__awy____,type,
    aTP_Lamm_awy: set_complex > fun_fu3636980702898289805at_int ).

tff(sy_c_ATP_058Lamm__awz____,type,
    aTP_Lamm_awz: set_nat > fun_fu202463024092729020er_int ).

tff(sy_c_ATP_058Lamm__ax____,type,
    aTP_Lamm_ax: set_nat > fun_nat_o ).

tff(sy_c_ATP_058Lamm__axa____,type,
    aTP_Lamm_axa: fun_Co822469698833053651at_int > fun_na3002214111362778963er_int ).

tff(sy_c_ATP_058Lamm__axb____,type,
    aTP_Lamm_axb: set_Code_integer > fun_fu4854240044958656303at_int ).

tff(sy_c_ATP_058Lamm__axc____,type,
    aTP_Lamm_axc: set_int > fun_fu1995833500178483365_int_o ).

tff(sy_c_ATP_058Lamm__axd____,type,
    aTP_Lamm_axd: set_int > fun_fu3330635997124576799BT_int ).

tff(sy_c_ATP_058Lamm__axe____,type,
    aTP_Lamm_axe: fun_VE294566951569886236nt_int > fun_in1463024724634152352BT_int ).

tff(sy_c_ATP_058Lamm__axf____,type,
    aTP_Lamm_axf: set_VEBT_VEBT > fun_fu6200680875500743381nt_int ).

tff(sy_c_ATP_058Lamm__axg____,type,
    aTP_Lamm_axg: set_int > fun_fu7328815047503086531_int_o ).

tff(sy_c_ATP_058Lamm__axh____,type,
    aTP_Lamm_axh: set_int > fun_fu8404755328348021045al_int ).

tff(sy_c_ATP_058Lamm__axi____,type,
    aTP_Lamm_axi: fun_real_fun_int_int > fun_int_fun_real_int ).

tff(sy_c_ATP_058Lamm__axj____,type,
    aTP_Lamm_axj: set_real > fun_fu6324116013759063093nt_int ).

tff(sy_c_ATP_058Lamm__axk____,type,
    aTP_Lamm_axk: set_int > fun_fu810243669753863363_int_o ).

tff(sy_c_ATP_058Lamm__axl____,type,
    aTP_Lamm_axl: set_int > fun_fu8001260946965626035ex_int ).

tff(sy_c_ATP_058Lamm__axm____,type,
    aTP_Lamm_axm: fun_co333780937270390912nt_int > fun_in3657390099950231424ex_int ).

tff(sy_c_ATP_058Lamm__axn____,type,
    aTP_Lamm_axn: set_complex > fun_fu2339071201290874307plex_o ).

tff(sy_c_ATP_058Lamm__axo____,type,
    aTP_Lamm_axo: set_complex > fun_fu7424888833511495217nt_int ).

tff(sy_c_ATP_058Lamm__axp____,type,
    aTP_Lamm_axp: set_int > fun_fu354264921998911491_int_o ).

tff(sy_c_ATP_058Lamm__axq____,type,
    aTP_Lamm_axq: set_int > fun_fu8759299743271765508er_int ).

tff(sy_c_ATP_058Lamm__axr____,type,
    aTP_Lamm_axr: fun_Co3009391273526075439nt_int > fun_in2718163036992669103er_int ).

tff(sy_c_ATP_058Lamm__axs____,type,
    aTP_Lamm_axs: set_Code_integer > fun_fu7527167023631164035eger_o ).

tff(sy_c_ATP_058Lamm__axt____,type,
    aTP_Lamm_axt: set_Code_integer > fun_fu1493779525457017043nt_int ).

tff(sy_c_ATP_058Lamm__axu____,type,
    aTP_Lamm_axu: set_nat > fun_fu2828149587716756511BT_nat ).

tff(sy_c_ATP_058Lamm__axv____,type,
    aTP_Lamm_axv: set_VEBT_VEBT > fun_fu4061198407372677825at_nat ).

tff(sy_c_ATP_058Lamm__axw____,type,
    aTP_Lamm_axw: set_nat > fun_fu8695033588447641837al_nat ).

tff(sy_c_ATP_058Lamm__axx____,type,
    aTP_Lamm_axx: set_real > fun_fu5553606195750802321at_nat ).

tff(sy_c_ATP_058Lamm__axy____,type,
    aTP_Lamm_axy: fun_nat_int > fun_int_fun_nat_int ).

tff(sy_c_ATP_058Lamm__axz____,type,
    aTP_Lamm_axz: fun_VEBT_VEBT_real > fun_fu4911191927098811621T_real ).

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

tff(sy_c_ATP_058Lamm__aya____,type,
    aTP_Lamm_aya: fun_real_real > fun_fu4478067837645853357l_real ).

tff(sy_c_ATP_058Lamm__ayb____,type,
    aTP_Lamm_ayb: fun_nat_real > fun_fu4002475321494201261t_real ).

tff(sy_c_ATP_058Lamm__ayc____,type,
    aTP_Lamm_ayc: fun_int_real > fun_fu7030473000426307245t_real ).

tff(sy_c_ATP_058Lamm__ayd____,type,
    aTP_Lamm_ayd: fun_complex_real > fun_fu3043978019632300333x_real ).

tff(sy_c_ATP_058Lamm__aye____,type,
    aTP_Lamm_aye: fun_Co3960896333048504454r_real > fun_fu8352104517646067181r_real ).

tff(sy_c_ATP_058Lamm__ayf____,type,
    aTP_Lamm_ayf: fun_VEBT_VEBT_rat > fun_fu14152356423011729BT_rat ).

tff(sy_c_ATP_058Lamm__ayg____,type,
    aTP_Lamm_ayg: fun_real_rat > fun_fu5551332503302594393al_rat ).

tff(sy_c_ATP_058Lamm__ayh____,type,
    aTP_Lamm_ayh: fun_nat_rat > fun_fu3162257917499486297at_rat ).

tff(sy_c_ATP_058Lamm__ayi____,type,
    aTP_Lamm_ayi: fun_int_rat > fun_fu3922030663826099033nt_rat ).

tff(sy_c_ATP_058Lamm__ayj____,type,
    aTP_Lamm_ayj: uint32 > fun_fu3025721337920667219uint32 ).

tff(sy_c_ATP_058Lamm__ayk____,type,
    aTP_Lamm_ayk: word_N3645301735248828278l_num1 > fun_fu7479362964929962845l_num1 ).

tff(sy_c_ATP_058Lamm__ayl____,type,
    aTP_Lamm_ayl: real > fun_fu3106432010075488275t_real ).

tff(sy_c_ATP_058Lamm__aym____,type,
    aTP_Lamm_aym: int > fun_fu4512855384587262995at_int ).

tff(sy_c_ATP_058Lamm__ayn____,type,
    aTP_Lamm_ayn: int > fun_fu6415725884910174923nt_int ).

tff(sy_c_ATP_058Lamm__ayo____,type,
    aTP_Lamm_ayo: nat > fun_fu7002938529518989751at_nat ).

tff(sy_c_ATP_058Lamm__ayp____,type,
    aTP_Lamm_ayp: fun_nat_int > fun_nat_fun_nat_int ).

tff(sy_c_ATP_058Lamm__ayq____,type,
    aTP_Lamm_ayq: fun_nat_int > fun_nat_o ).

tff(sy_c_ATP_058Lamm__ayr____,type,
    aTP_Lamm_ayr: fun_int_int > fun_int_o ).

tff(sy_c_ATP_058Lamm__ays____,type,
    aTP_Lamm_ays: fun_nat_nat > fun_nat_o ).

tff(sy_c_ATP_058Lamm__ayt____,type,
    aTP_Lamm_ayt: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__ayu____,type,
    aTP_Lamm_ayu: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__ayv____,type,
    aTP_Lamm_ayv: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__ayw____,type,
    aTP_Lamm_ayw: nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__ayx____,type,
    aTP_Lamm_ayx: nat > fun_na3244735409550470327at_nat ).

tff(sy_c_ATP_058Lamm__ayy____,type,
    aTP_Lamm_ayy: nat > fun_na3244735409550470327at_nat ).

tff(sy_c_ATP_058Lamm__ayz____,type,
    aTP_Lamm_ayz: fun_nat_nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__az____,type,
    aTP_Lamm_az: set_complex > fun_complex_o ).

tff(sy_c_ATP_058Lamm__aza____,type,
    aTP_Lamm_aza: fun_nat_nat > fun_fu4413992040155564420at_nat ).

tff(sy_c_ATP_058Lamm__azb____,type,
    aTP_Lamm_azb: fun_nat_nat > fun_fu4413992040155564420at_nat ).

tff(sy_c_ATP_058Lamm__azc____,type,
    aTP_Lamm_azc: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__azd____,type,
    aTP_Lamm_azd: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__aze____,type,
    aTP_Lamm_aze: real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__azf____,type,
    aTP_Lamm_azf: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__azg____,type,
    aTP_Lamm_azg: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__azh____,type,
    aTP_Lamm_azh: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__azi____,type,
    aTP_Lamm_azi: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__azj____,type,
    aTP_Lamm_azj: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__azk____,type,
    aTP_Lamm_azk: fun_nat_real ).

tff(sy_c_ATP_058Lamm__azl____,type,
    aTP_Lamm_azl: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__azm____,type,
    aTP_Lamm_azm: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__azn____,type,
    aTP_Lamm_azn: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__azo____,type,
    aTP_Lamm_azo: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__azp____,type,
    aTP_Lamm_azp: fun_nat_real > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__azq____,type,
    aTP_Lamm_azq: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__azr____,type,
    aTP_Lamm_azr: nat > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__azs____,type,
    aTP_Lamm_azs: complex > fun_na1554462459066888140plex_o ).

tff(sy_c_ATP_058Lamm__azt____,type,
    aTP_Lamm_azt: int > fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__azu____,type,
    aTP_Lamm_azu: int > fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__azv____,type,
    aTP_Lamm_azv: fun_int_fun_int_int ).

tff(sy_c_ATP_058Lamm__azw____,type,
    aTP_Lamm_azw: fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__azx____,type,
    aTP_Lamm_azx: fun_num_option_num ).

tff(sy_c_ATP_058Lamm__azy____,type,
    aTP_Lamm_azy: num > fun_num_int ).

tff(sy_c_ATP_058Lamm__azz____,type,
    aTP_Lamm_azz: fun_num_option_num ).

tff(sy_c_ATP_058Lamm__ba____,type,
    aTP_Lamm_ba: set_Pr958786334691620121nt_int > fun_Pr3238374608701176162_int_o ).

tff(sy_c_ATP_058Lamm__baa____,type,
    aTP_Lamm_baa: num > fun_nat_option_num ).

tff(sy_c_ATP_058Lamm__bab____,type,
    aTP_Lamm_bab: fun_real_fun_real_o ).

tff(sy_c_ATP_058Lamm__bac____,type,
    aTP_Lamm_bac: fun_real_fun_real_o ).

tff(sy_c_ATP_058Lamm__bad____,type,
    aTP_Lamm_bad: fun_real_fun_real_o ).

tff(sy_c_ATP_058Lamm__bae____,type,
    aTP_Lamm_bae: fun_nat_option_num ).

tff(sy_c_ATP_058Lamm__baf____,type,
    aTP_Lamm_baf: fun_real_o ).

tff(sy_c_ATP_058Lamm__bag____,type,
    aTP_Lamm_bag: num > fun_nat_option_num ).

tff(sy_c_ATP_058Lamm__bah____,type,
    aTP_Lamm_bah: fun_real_fun_real_o ).

tff(sy_c_ATP_058Lamm__bai____,type,
    aTP_Lamm_bai: nat > fun_num_option_num ).

tff(sy_c_ATP_058Lamm__baj____,type,
    aTP_Lamm_baj: nat > fun_num_option_num ).

tff(sy_c_ATP_058Lamm__bak____,type,
    aTP_Lamm_bak: num > fun_nat_option_num ).

tff(sy_c_ATP_058Lamm__bal____,type,
    aTP_Lamm_bal: fun_na7595719129879163854on_num ).

tff(sy_c_ATP_058Lamm__bam____,type,
    aTP_Lamm_bam: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__ban____,type,
    aTP_Lamm_ban: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__bao____,type,
    aTP_Lamm_bao: fun_real_fun_int_o ).

tff(sy_c_ATP_058Lamm__bap____,type,
    aTP_Lamm_bap: rat > fun_int_o ).

tff(sy_c_ATP_058Lamm__baq____,type,
    aTP_Lamm_baq: nat > fun_real_real ).

tff(sy_c_ATP_058Lamm__bar____,type,
    aTP_Lamm_bar: complex > fun_real_o ).

tff(sy_c_ATP_058Lamm__bas____,type,
    aTP_Lamm_bas: fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__bat____,type,
    aTP_Lamm_bat: fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__bau____,type,
    aTP_Lamm_bau: fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__bav____,type,
    aTP_Lamm_bav: int > fun_in5570076352459579423_int_o ).

tff(sy_c_ATP_058Lamm__baw____,type,
    aTP_Lamm_baw: rat > fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__bax____,type,
    aTP_Lamm_bax: int > fun_in5570076352459579423_int_o ).

tff(sy_c_ATP_058Lamm__bay____,type,
    aTP_Lamm_bay: rat > fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__baz____,type,
    aTP_Lamm_baz: int > fun_in2204242111582549184nt_int ).

tff(sy_c_ATP_058Lamm__bb____,type,
    aTP_Lamm_bb: fun_option_nat_assn ).

tff(sy_c_ATP_058Lamm__bba____,type,
    aTP_Lamm_bba: rat > fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__bbb____,type,
    aTP_Lamm_bbb: int > fun_in2204242111582549184nt_int ).

tff(sy_c_ATP_058Lamm__bbc____,type,
    aTP_Lamm_bbc: rat > fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__bbd____,type,
    aTP_Lamm_bbd: int > fun_in2204242111582549184nt_int ).

tff(sy_c_ATP_058Lamm__bbe____,type,
    aTP_Lamm_bbe: rat > fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__bbf____,type,
    aTP_Lamm_bbf: int > fun_in2204242111582549184nt_int ).

tff(sy_c_ATP_058Lamm__bbg____,type,
    aTP_Lamm_bbg: rat > fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__bbh____,type,
    aTP_Lamm_bbh: nat > fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__bbi____,type,
    aTP_Lamm_bbi: int > fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__bbj____,type,
    aTP_Lamm_bbj: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__bbk____,type,
    aTP_Lamm_bbk: nat > fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__bbl____,type,
    aTP_Lamm_bbl: fun_int_fun_int_int ).

tff(sy_c_ATP_058Lamm__bbm____,type,
    aTP_Lamm_bbm: code_integer > fun_Co8499238673658658103nteger ).

tff(sy_c_ATP_058Lamm__bbn____,type,
    aTP_Lamm_bbn: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__bbo____,type,
    aTP_Lamm_bbo: fun_Pr6626665806141890376t_assn ).

tff(sy_c_ATP_058Lamm__bbp____,type,
    aTP_Lamm_bbp: fun_int_o ).

tff(sy_c_ATP_058Lamm__bbq____,type,
    aTP_Lamm_bbq: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__bbr____,type,
    aTP_Lamm_bbr: fun_nat_nat ).

tff(sy_c_ATP_058Lamm__bbs____,type,
    aTP_Lamm_bbs: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__bbt____,type,
    aTP_Lamm_bbt: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__bbu____,type,
    aTP_Lamm_bbu: fun_nat_o > fun_nat_o ).

tff(sy_c_ATP_058Lamm__bbv____,type,
    aTP_Lamm_bbv: fun_Co3583066429914935172er_nat ).

tff(sy_c_ATP_058Lamm__bbw____,type,
    aTP_Lamm_bbw: fun_Co8521658056209057742er_num ).

tff(sy_c_ATP_058Lamm__bbx____,type,
    aTP_Lamm_bbx: fun_Co1092983284983208416er_int ).

tff(sy_c_ATP_058Lamm__bby____,type,
    aTP_Lamm_bby: code_integer > fun_Co2470855966376689427nteger ).

tff(sy_c_ATP_058Lamm__bbz____,type,
    aTP_Lamm_bbz: code_integer > fun_Co2470855966376689427nteger ).

tff(sy_c_ATP_058Lamm__bc____,type,
    aTP_Lamm_bc: nat > fun_nat_fun_nat_assn ).

tff(sy_c_ATP_058Lamm__bca____,type,
    aTP_Lamm_bca: code_integer > fun_Co2470855966376689427nteger ).

tff(sy_c_ATP_058Lamm__bcb____,type,
    aTP_Lamm_bcb: code_integer > fun_Co3544761417380347684eger_o ).

tff(sy_c_ATP_058Lamm__bcc____,type,
    aTP_Lamm_bcc: fun_nat_o > fun_nat_o ).

tff(sy_c_ATP_058Lamm__bcd____,type,
    aTP_Lamm_bcd: fun_nat_o > fun_nat_o ).

tff(sy_c_ATP_058Lamm__bce____,type,
    aTP_Lamm_bce: fun_nat_o > fun_nat_o ).

tff(sy_c_ATP_058Lamm__bcf____,type,
    aTP_Lamm_bcf: nat > fun_li6914387844523535256_nat_o ).

tff(sy_c_ATP_058Lamm__bcg____,type,
    aTP_Lamm_bcg: fun_o_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__bch____,type,
    aTP_Lamm_bch: $o > fun_o_5488095450495718970o_char ).

tff(sy_c_ATP_058Lamm__bci____,type,
    aTP_Lamm_bci: $o > fun_o_6360995121869435791o_char ).

tff(sy_c_ATP_058Lamm__bcj____,type,
    aTP_Lamm_bcj: $o > fun_o_2683029531876921444o_char ).

tff(sy_c_ATP_058Lamm__bck____,type,
    aTP_Lamm_bck: $o > fun_o_3037882179896959673o_char ).

tff(sy_c_ATP_058Lamm__bcl____,type,
    aTP_Lamm_bcl: $o > fun_o_335877185394996366o_char ).

tff(sy_c_ATP_058Lamm__bcm____,type,
    aTP_Lamm_bcm: $o > fun_o_2016307092959984611o_char ).

tff(sy_c_ATP_058Lamm__bcn____,type,
    aTP_Lamm_bcn: $o > fun_Co681828205884031672o_char ).

tff(sy_c_ATP_058Lamm__bco____,type,
    aTP_Lamm_bco: fun_Co681828205884031672o_char ).

tff(sy_c_ATP_058Lamm__bcp____,type,
    aTP_Lamm_bcp: nat > fun_real_real ).

tff(sy_c_ATP_058Lamm__bcq____,type,
    aTP_Lamm_bcq: fun_real_real > fun_real_real ).

tff(sy_c_ATP_058Lamm__bcr____,type,
    aTP_Lamm_bcr: fun_real_real > fun_na6724076748692969250l_real ).

tff(sy_c_ATP_058Lamm__bd____,type,
    aTP_Lamm_bd: nat > fun_nat_fun_nat_assn ).

tff(sy_c_ATP_058Lamm__be____,type,
    aTP_Lamm_be: vEBT_VEBT > fun_VE7792148054706007876t_assn ).

tff(sy_c_ATP_058Lamm__bf____,type,
    aTP_Lamm_bf: list_VEBT_VEBT > fun_ar1453094831611173103i_assn ).

tff(sy_c_ATP_058Lamm__bg____,type,
    aTP_Lamm_bg: nat > fun_li8672845860709154922t_assn ).

tff(sy_c_ATP_058Lamm__bh____,type,
    aTP_Lamm_bh: nat > fun_li7646410839599160030t_assn ).

tff(sy_c_ATP_058Lamm__bi____,type,
    aTP_Lamm_bi: nat > fun_li6911922195568332558t_assn ).

tff(sy_c_ATP_058Lamm__bj____,type,
    aTP_Lamm_bj: nat > fun_li4611866707255273568o_assn ).

tff(sy_c_ATP_058Lamm__bk____,type,
    aTP_Lamm_bk: nat > fun_li6689927339917241139i_assn ).

tff(sy_c_ATP_058Lamm__bl____,type,
    aTP_Lamm_bl: nat > fun_li124330075790197867t_assn ).

tff(sy_c_ATP_058Lamm__bm____,type,
    aTP_Lamm_bm: nat > fun_li3871283220525210964t_assn ).

tff(sy_c_ATP_058Lamm__bn____,type,
    aTP_Lamm_bn: nat > fun_li3871283220525210964t_assn ).

tff(sy_c_ATP_058Lamm__bo____,type,
    aTP_Lamm_bo: vEBT_VEBT > fun_VE1035329831157012416o_assn ).

tff(sy_c_ATP_058Lamm__bp____,type,
    aTP_Lamm_bp: fun_option_nat_assn > fun_as2043416024874063466t_assn ).

tff(sy_c_ATP_058Lamm__bq____,type,
    aTP_Lamm_bq: fun_nat_assn > fun_as8240436605320707994t_assn ).

tff(sy_c_ATP_058Lamm__br____,type,
    aTP_Lamm_br: fun_o_assn > fun_assn_fun_o_assn ).

tff(sy_c_ATP_058Lamm__bs____,type,
    aTP_Lamm_bs: fun_VEBT_VEBTi_assn > fun_as315122898794150911i_assn ).

tff(sy_c_ATP_058Lamm__bt____,type,
    aTP_Lamm_bt: fun_Pr6626665806141890376t_assn > fun_as4019953064511417593t_assn ).

tff(sy_c_ATP_058Lamm__bu____,type,
    aTP_Lamm_bu: fun_op8552747401449489031i_assn > fun_li3309467822378783623t_assn ).

tff(sy_c_ATP_058Lamm__bv____,type,
    aTP_Lamm_bv: fun_op8552747401449489031i_assn > fun_option_nat_assn ).

tff(sy_c_ATP_058Lamm__bw____,type,
    aTP_Lamm_bw: fun_na8005944528675429559i_assn > fun_li7023630697154767927t_assn ).

tff(sy_c_ATP_058Lamm__bx____,type,
    aTP_Lamm_bx: fun_na8005944528675429559i_assn > fun_nat_assn ).

tff(sy_c_ATP_058Lamm__by____,type,
    aTP_Lamm_by: fun_o_4874018796564477717i_assn > fun_li5930421888187426471o_assn ).

tff(sy_c_ATP_058Lamm__bz____,type,
    aTP_Lamm_bz: fun_o_4874018796564477717i_assn > fun_o_assn ).

tff(sy_c_ATP_058Lamm__ca____,type,
    aTP_Lamm_ca: fun_VE3708348504534315036i_assn > fun_li8825114373765265308i_assn ).

tff(sy_c_ATP_058Lamm__cb____,type,
    aTP_Lamm_cb: fun_VE3708348504534315036i_assn > fun_VEBT_VEBTi_assn ).

tff(sy_c_ATP_058Lamm__cc____,type,
    aTP_Lamm_cc: fun_Pr6039563626565002634i_assn > fun_li6155761431698120348t_assn ).

tff(sy_c_ATP_058Lamm__cd____,type,
    aTP_Lamm_cd: fun_Pr6039563626565002634i_assn > fun_Pr6626665806141890376t_assn ).

tff(sy_c_ATP_058Lamm__ce____,type,
    aTP_Lamm_ce: assn > fun_op6483474040057813218t_assn ).

tff(sy_c_ATP_058Lamm__cf____,type,
    aTP_Lamm_cf: assn > fun_nat_fun_nat_assn ).

tff(sy_c_ATP_058Lamm__cg____,type,
    aTP_Lamm_cg: assn > fun_o_fun_o_assn ).

tff(sy_c_ATP_058Lamm__ch____,type,
    aTP_Lamm_ch: assn > fun_VE2561185598974572172i_assn ).

tff(sy_c_ATP_058Lamm__ci____,type,
    aTP_Lamm_ci: assn > fun_Pr2477578491906815678t_assn ).

tff(sy_c_ATP_058Lamm__cj____,type,
    aTP_Lamm_cj: fun_num_num > fun_fu8155524643419131615um_num ).

tff(sy_c_ATP_058Lamm__ck____,type,
    aTP_Lamm_ck: fun_num_int > fun_fu726849872193282289um_int ).

tff(sy_c_ATP_058Lamm__cl____,type,
    aTP_Lamm_cl: fun_int_num > fun_fu5322398707847370225um_num ).

tff(sy_c_ATP_058Lamm__cm____,type,
    aTP_Lamm_cm: fun_int_int > fun_fu7117095973476296707um_int ).

tff(sy_c_ATP_058Lamm__cn____,type,
    aTP_Lamm_cn: fun_option_num_num > fun_fu152868422178311599um_num ).

tff(sy_c_ATP_058Lamm__co____,type,
    aTP_Lamm_co: fun_option_num_int > fun_fu1947565687807238081um_int ).

tff(sy_c_ATP_058Lamm__cp____,type,
    aTP_Lamm_cp: fun_num_option_num > fun_fu4719417816141531311on_num ).

tff(sy_c_ATP_058Lamm__cq____,type,
    aTP_Lamm_cq: fun_int_option_num > fun_fu7964671872712412353on_num ).

tff(sy_c_ATP_058Lamm__cr____,type,
    aTP_Lamm_cr: fun_op8989173174992673105on_num > fun_fu2987389051837124479on_num ).

tff(sy_c_ATP_058Lamm__cs____,type,
    aTP_Lamm_cs: fun_o_o > fun_fu3275390434952296369_nat_o ).

tff(sy_c_ATP_058Lamm__ct____,type,
    aTP_Lamm_ct: assn > fun_fu5741754899130534993i_assn ).

tff(sy_c_ATP_058Lamm__cu____,type,
    aTP_Lamm_cu: fun_li3352006551604695530i_assn > fun_as5323700489645809679i_assn ).

tff(sy_c_ATP_058Lamm__cv____,type,
    aTP_Lamm_cv: fun_Pr7755520105644788814_nat_o ).

tff(sy_c_ATP_058Lamm__cw____,type,
    aTP_Lamm_cw: fun_Pr3989439477023515934_nat_o ).

tff(sy_c_ATP_058Lamm__cx____,type,
    aTP_Lamm_cx: fun_num_o ).

tff(sy_c_ATP_058Lamm__cy____,type,
    aTP_Lamm_cy: fun_nat_o ).

tff(sy_c_ATP_058Lamm__cz____,type,
    aTP_Lamm_cz: fun_Pr4796943543331071514_nat_o ).

tff(sy_c_ATP_058Lamm__da____,type,
    aTP_Lamm_da: fun_Pr7755520105644788814_nat_o ).

tff(sy_c_ATP_058Lamm__db____,type,
    aTP_Lamm_db: fun_Pr3989439477023515934_nat_o ).

tff(sy_c_ATP_058Lamm__dc____,type,
    aTP_Lamm_dc: fun_num_o ).

tff(sy_c_ATP_058Lamm__dd____,type,
    aTP_Lamm_dd: fun_nat_o ).

tff(sy_c_ATP_058Lamm__de____,type,
    aTP_Lamm_de: fun_Pr4796943543331071514_nat_o ).

tff(sy_c_ATP_058Lamm__df____,type,
    aTP_Lamm_df: fun_wo7490464530504204009l_num1 ).

tff(sy_c_ATP_058Lamm__dg____,type,
    aTP_Lamm_dg: fun_real_real ).

tff(sy_c_ATP_058Lamm__dh____,type,
    aTP_Lamm_dh: fun_rat_rat ).

tff(sy_c_ATP_058Lamm__di____,type,
    aTP_Lamm_di: fun_nat_nat ).

tff(sy_c_ATP_058Lamm__dj____,type,
    aTP_Lamm_dj: fun_int_int ).

tff(sy_c_ATP_058Lamm__dk____,type,
    aTP_Lamm_dk: fun_option_nat_assn > fun_as2043416024874063466t_assn ).

tff(sy_c_ATP_058Lamm__dl____,type,
    aTP_Lamm_dl: fun_nat_assn > fun_as8240436605320707994t_assn ).

tff(sy_c_ATP_058Lamm__dm____,type,
    aTP_Lamm_dm: fun_o_assn > fun_assn_fun_o_assn ).

tff(sy_c_ATP_058Lamm__dn____,type,
    aTP_Lamm_dn: fun_VEBT_VEBTi_assn > fun_as315122898794150911i_assn ).

tff(sy_c_ATP_058Lamm__do____,type,
    aTP_Lamm_do: fun_Pr6626665806141890376t_assn > fun_as4019953064511417593t_assn ).

tff(sy_c_ATP_058Lamm__dp____,type,
    aTP_Lamm_dp: fun_real_fun_real_o ).

tff(sy_c_ATP_058Lamm__dq____,type,
    aTP_Lamm_dq: option_real > fun_real_o ).

tff(sy_c_ATP_058Lamm__dr____,type,
    aTP_Lamm_dr: fun_rat_fun_rat_o ).

tff(sy_c_ATP_058Lamm__ds____,type,
    aTP_Lamm_ds: option_rat > fun_rat_o ).

tff(sy_c_ATP_058Lamm__dt____,type,
    aTP_Lamm_dt: num > fun_num_o ).

tff(sy_c_ATP_058Lamm__du____,type,
    aTP_Lamm_du: option_num > fun_num_o ).

tff(sy_c_ATP_058Lamm__dv____,type,
    aTP_Lamm_dv: fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__dw____,type,
    aTP_Lamm_dw: option_nat > fun_nat_o ).

tff(sy_c_ATP_058Lamm__dx____,type,
    aTP_Lamm_dx: fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__dy____,type,
    aTP_Lamm_dy: option_int > fun_int_o ).

tff(sy_c_ATP_058Lamm__dz____,type,
    aTP_Lamm_dz: fun_re7681423839302404990l_real ).

tff(sy_c_ATP_058Lamm__ea____,type,
    aTP_Lamm_ea: rat > fun_rat_rat ).

tff(sy_c_ATP_058Lamm__eb____,type,
    aTP_Lamm_eb: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__ec____,type,
    aTP_Lamm_ec: fun_int_fun_int_int ).

tff(sy_c_ATP_058Lamm__ed____,type,
    aTP_Lamm_ed: option_set_int > fun_set_int_o ).

tff(sy_c_ATP_058Lamm__ee____,type,
    aTP_Lamm_ee: option_rat > fun_rat_o ).

tff(sy_c_ATP_058Lamm__ef____,type,
    aTP_Lamm_ef: option_num > fun_num_o ).

tff(sy_c_ATP_058Lamm__eg____,type,
    aTP_Lamm_eg: option_nat > fun_nat_o ).

tff(sy_c_ATP_058Lamm__eh____,type,
    aTP_Lamm_eh: option_int > fun_int_o ).

tff(sy_c_ATP_058Lamm__ei____,type,
    aTP_Lamm_ei: assn > fun_li3352006551604695530i_assn ).

tff(sy_c_ATP_058Lamm__ej____,type,
    aTP_Lamm_ej: vEBT_VEBT > fun_VE9201042017749490615o_assn ).

tff(sy_c_ATP_058Lamm__ek____,type,
    aTP_Lamm_ek: fun_uint32_uint32 ).

tff(sy_c_ATP_058Lamm__el____,type,
    aTP_Lamm_el: fun_real_real ).

tff(sy_c_ATP_058Lamm__em____,type,
    aTP_Lamm_em: fun_rat_rat ).

tff(sy_c_ATP_058Lamm__en____,type,
    aTP_Lamm_en: fun_nat_nat ).

tff(sy_c_ATP_058Lamm__eo____,type,
    aTP_Lamm_eo: fun_int_int ).

tff(sy_c_ATP_058Lamm__ep____,type,
    aTP_Lamm_ep: nat > fun_na7530211986279412206on_nat ).

tff(sy_c_ATP_058Lamm__eq____,type,
    aTP_Lamm_eq: nat > fun_op8929431053563478046on_nat ).

tff(sy_c_ATP_058Lamm__er____,type,
    aTP_Lamm_er: nat > fun_op3734645406857789385on_nat ).

tff(sy_c_ATP_058Lamm__es____,type,
    aTP_Lamm_es: nat > fun_na8204034076925005301on_nat ).

tff(sy_c_ATP_058Lamm__et____,type,
    aTP_Lamm_et: nat > fun_na6142010823788121245on_nat ).

tff(sy_c_ATP_058Lamm__eu____,type,
    aTP_Lamm_eu: nat > fun_na554821608089334116on_nat ).

tff(sy_c_ATP_058Lamm__ev____,type,
    aTP_Lamm_ev: nat > fun_na8711077528228054016on_nat ).

tff(sy_c_ATP_058Lamm__ew____,type,
    aTP_Lamm_ew: nat > fun_na4693688529567471479on_nat ).

tff(sy_c_ATP_058Lamm__ex____,type,
    aTP_Lamm_ex: nat > fun_na3458768066340930158on_nat ).

tff(sy_c_ATP_058Lamm__ey____,type,
    aTP_Lamm_ey: nat > fun_op5783775701855621896on_nat ).

tff(sy_c_ATP_058Lamm__ez____,type,
    aTP_Lamm_ez: nat > fun_o_4049874799390722306on_nat ).

tff(sy_c_ATP_058Lamm__fa____,type,
    aTP_Lamm_fa: nat > fun_op3734645406857789385on_nat ).

tff(sy_c_ATP_058Lamm__fb____,type,
    aTP_Lamm_fb: nat > fun_ar5553829433130746806on_nat ).

tff(sy_c_ATP_058Lamm__fc____,type,
    aTP_Lamm_fc: nat > fun_ar5028184689208713744on_nat ).

tff(sy_c_ATP_058Lamm__fd____,type,
    aTP_Lamm_fd: nat > fun_ar3067344226597455191on_nat ).

tff(sy_c_ATP_058Lamm__fe____,type,
    aTP_Lamm_fe: nat > fun_ar3350189613669807859on_nat ).

tff(sy_c_ATP_058Lamm__ff____,type,
    aTP_Lamm_ff: nat > fun_na3723828179601053495on_nat ).

tff(sy_c_ATP_058Lamm__fg____,type,
    aTP_Lamm_fg: nat > fun_na3458768066340930158on_nat ).

tff(sy_c_ATP_058Lamm__fh____,type,
    aTP_Lamm_fh: nat > fun_op5783775701855621896on_nat ).

tff(sy_c_ATP_058Lamm__fi____,type,
    aTP_Lamm_fi: nat > fun_o_4049874799390722306on_nat ).

tff(sy_c_ATP_058Lamm__fj____,type,
    aTP_Lamm_fj: fun_He8640631688564546883on_nat > fun_fu8409376595709561667on_nat ).

tff(sy_c_ATP_058Lamm__fk____,type,
    aTP_Lamm_fk: fun_He8640631688564546883on_nat > fun_fu7388674814006075715on_nat ).

tff(sy_c_ATP_058Lamm__fl____,type,
    aTP_Lamm_fl: fun_He2623223239018661208_VEBTi > fun_fu6559571188541174616_VEBTi ).

tff(sy_c_ATP_058Lamm__fm____,type,
    aTP_Lamm_fm: fun_He2623223239018661208_VEBTi > fun_fu4739529698945370456_VEBTi ).

tff(sy_c_ATP_058Lamm__fn____,type,
    aTP_Lamm_fn: fun_He5988046180898962017Heap_o > fun_fu1930685970884637881Heap_o ).

tff(sy_c_ATP_058Lamm__fo____,type,
    aTP_Lamm_fo: fun_He5988046180898962017Heap_o > fun_fu2310530280156192909Heap_o ).

tff(sy_c_ATP_058Lamm__fp____,type,
    aTP_Lamm_fp: fun_He1783186893022644846on_nat > fun_fu8147167295154171246on_nat ).

tff(sy_c_ATP_058Lamm__fq____,type,
    aTP_Lamm_fq: fun_He1783186893022644846on_nat > fun_fu3089959412332122350on_nat ).

tff(sy_c_ATP_058Lamm__fr____,type,
    aTP_Lamm_fr: fun_He8247729123535230979_VEBTi > fun_fu230338336617200387_VEBTi ).

tff(sy_c_ATP_058Lamm__fs____,type,
    aTP_Lamm_fs: fun_He8247729123535230979_VEBTi > fun_fu6035134072308932227_VEBTi ).

tff(sy_c_ATP_058Lamm__ft____,type,
    aTP_Lamm_ft: fun_He4624006784692404214Heap_o > fun_fu6351278744920802638Heap_o ).

tff(sy_c_ATP_058Lamm__fu____,type,
    aTP_Lamm_fu: fun_He4624006784692404214Heap_o > fun_fu5088205192217346978Heap_o ).

tff(sy_c_ATP_058Lamm__fv____,type,
    aTP_Lamm_fv: fun_He6140748070426431971on_nat > fun_fu2747026234331107907on_nat ).

tff(sy_c_ATP_058Lamm__fw____,type,
    aTP_Lamm_fw: fun_He6140748070426431971on_nat > fun_fu8544618812552386275on_nat ).

tff(sy_c_ATP_058Lamm__fx____,type,
    aTP_Lamm_fx: fun_He6517945810935171576_VEBTi > fun_fu4397613994831067736_VEBTi ).

tff(sy_c_ATP_058Lamm__fy____,type,
    aTP_Lamm_fy: fun_He6517945810935171576_VEBTi > fun_fu4945982929356410616_VEBTi ).

tff(sy_c_ATP_058Lamm__fz____,type,
    aTP_Lamm_fz: fun_He6597989805181038017Heap_o > fun_fu7474633291259637177Heap_o ).

tff(sy_c_ATP_058Lamm__ga____,type,
    aTP_Lamm_ga: fun_He6597989805181038017Heap_o > fun_fu144732360711459565Heap_o ).

tff(sy_c_ATP_058Lamm__gb____,type,
    aTP_Lamm_gb: vEBT_VEBT > fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__gc____,type,
    aTP_Lamm_gc: vEBT_VEBT > fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__gd____,type,
    aTP_Lamm_gd: fun_complex_o ).

tff(sy_c_ATP_058Lamm__ge____,type,
    aTP_Lamm_ge: fun_Pr3238374608701176162_int_o ).

tff(sy_c_ATP_058Lamm__gf____,type,
    aTP_Lamm_gf: fun_int_o ).

tff(sy_c_ATP_058Lamm__gg____,type,
    aTP_Lamm_gg: fun_real_o ).

tff(sy_c_ATP_058Lamm__gh____,type,
    aTP_Lamm_gh: set_VEBT_VEBT > fun_fu6329627000648617163VEBT_o ).

tff(sy_c_ATP_058Lamm__gi____,type,
    aTP_Lamm_gi: set_real > fun_fu9050450510664235267real_o ).

tff(sy_c_ATP_058Lamm__gj____,type,
    aTP_Lamm_gj: set_nat > fun_fu4001317426178549763_nat_o ).

tff(sy_c_ATP_058Lamm__gk____,type,
    aTP_Lamm_gk: set_complex > fun_fu1358370060594899587plex_o ).

tff(sy_c_ATP_058Lamm__gl____,type,
    aTP_Lamm_gl: set_Pr958786334691620121nt_int > fun_fu3068557634114553409_int_o ).

tff(sy_c_ATP_058Lamm__gm____,type,
    aTP_Lamm_gm: set_int > fun_fu3883539188510166275_int_o ).

tff(sy_c_ATP_058Lamm__gn____,type,
    aTP_Lamm_gn: fun_li3309467822378783623t_assn > fun_op8552747401449489031i_assn ).

tff(sy_c_ATP_058Lamm__go____,type,
    aTP_Lamm_go: fun_li3309467822378783623t_assn > fun_option_nat_assn ).

tff(sy_c_ATP_058Lamm__gp____,type,
    aTP_Lamm_gp: fun_li7023630697154767927t_assn > fun_na8005944528675429559i_assn ).

tff(sy_c_ATP_058Lamm__gq____,type,
    aTP_Lamm_gq: fun_li7023630697154767927t_assn > fun_nat_assn ).

tff(sy_c_ATP_058Lamm__gr____,type,
    aTP_Lamm_gr: fun_li8825114373765265308i_assn > fun_VE3708348504534315036i_assn ).

tff(sy_c_ATP_058Lamm__gs____,type,
    aTP_Lamm_gs: fun_li8825114373765265308i_assn > fun_VEBT_VEBTi_assn ).

tff(sy_c_ATP_058Lamm__gt____,type,
    aTP_Lamm_gt: fun_li5930421888187426471o_assn > fun_o_4874018796564477717i_assn ).

tff(sy_c_ATP_058Lamm__gu____,type,
    aTP_Lamm_gu: fun_li5930421888187426471o_assn > fun_o_assn ).

tff(sy_c_ATP_058Lamm__gv____,type,
    aTP_Lamm_gv: heap_Time_Heap_o > fun_Pr4445936646954502082Heap_o ).

tff(sy_c_ATP_058Lamm__gw____,type,
    aTP_Lamm_gw: heap_Time_Heap_nat > fun_Pr6201033369230528594ap_nat ).

tff(sy_c_ATP_058Lamm__gx____,type,
    aTP_Lamm_gx: heap_T2636463487746394924on_nat > fun_Pr293967910607515554on_nat ).

tff(sy_c_ATP_058Lamm__gy____,type,
    aTP_Lamm_gy: heap_T8145700208782473153_VEBTi > fun_Pr7231383449453139767_VEBTi ).

tff(sy_c_ATP_058Lamm__gz____,type,
    aTP_Lamm_gz: set_VEBT_VEBT > fun_se4683600015404540110VEBT_o ).

tff(sy_c_ATP_058Lamm__ha____,type,
    aTP_Lamm_ha: set_real > fun_se2161501677553633980real_o ).

tff(sy_c_ATP_058Lamm__hb____,type,
    aTP_Lamm_hb: set_int > fun_se867122708194164540_int_o ).

tff(sy_c_ATP_058Lamm__hc____,type,
    aTP_Lamm_hc: set_complex > fun_se9162176452187397432plex_o ).

tff(sy_c_ATP_058Lamm__hd____,type,
    aTP_Lamm_hd: set_Pr958786334691620121nt_int > fun_se4767462807232413326_int_o ).

tff(sy_c_ATP_058Lamm__he____,type,
    aTP_Lamm_he: set_nat > fun_se382663106265794036_nat_o ).

tff(sy_c_ATP_058Lamm__hf____,type,
    aTP_Lamm_hf: heap_T5738788834812785303t_unit > fun_Pr292408699144966967t_unit ).

tff(sy_c_ATP_058Lamm__hg____,type,
    aTP_Lamm_hg: assn > fun_Pr6626665806141890376t_assn ).

tff(sy_c_ATP_058Lamm__hh____,type,
    aTP_Lamm_hh: vEBT_VEBT > fun_VEBT_VEBT_o ).

tff(sy_c_ATP_058Lamm__hi____,type,
    aTP_Lamm_hi: complex > fun_complex_o ).

tff(sy_c_ATP_058Lamm__hj____,type,
    aTP_Lamm_hj: fun_Pr9018117161326028078_int_o ).

tff(sy_c_ATP_058Lamm__hk____,type,
    aTP_Lamm_hk: fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__hl____,type,
    aTP_Lamm_hl: fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__hm____,type,
    aTP_Lamm_hm: fun_real_fun_real_o ).

tff(sy_c_ATP_058Lamm__hn____,type,
    aTP_Lamm_hn: fun_real_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__ho____,type,
    aTP_Lamm_ho: fun_o_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__hp____,type,
    aTP_Lamm_hp: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__hq____,type,
    aTP_Lamm_hq: fun_int_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__hr____,type,
    aTP_Lamm_hr: vEBT_VEBT > fun_fu6329627000648617163VEBT_o ).

tff(sy_c_ATP_058Lamm__hs____,type,
    aTP_Lamm_hs: real > fun_fu9050450510664235267real_o ).

tff(sy_c_ATP_058Lamm__ht____,type,
    aTP_Lamm_ht: nat > fun_fu4001317426178549763_nat_o ).

tff(sy_c_ATP_058Lamm__hu____,type,
    aTP_Lamm_hu: int > fun_fu3883539188510166275_int_o ).

tff(sy_c_ATP_058Lamm__hv____,type,
    aTP_Lamm_hv: complex > fun_fu1358370060594899587plex_o ).

tff(sy_c_ATP_058Lamm__hw____,type,
    aTP_Lamm_hw: product_prod_int_int > fun_fu3068557634114553409_int_o ).

tff(sy_c_ATP_058Lamm__hx____,type,
    aTP_Lamm_hx: vEBT_VEBT > fun_se4683600015404540110VEBT_o ).

tff(sy_c_ATP_058Lamm__hy____,type,
    aTP_Lamm_hy: real > fun_se2161501677553633980real_o ).

tff(sy_c_ATP_058Lamm__hz____,type,
    aTP_Lamm_hz: nat > fun_se382663106265794036_nat_o ).

tff(sy_c_ATP_058Lamm__ia____,type,
    aTP_Lamm_ia: int > fun_se867122708194164540_int_o ).

tff(sy_c_ATP_058Lamm__ib____,type,
    aTP_Lamm_ib: complex > fun_se9162176452187397432plex_o ).

tff(sy_c_ATP_058Lamm__ic____,type,
    aTP_Lamm_ic: product_prod_int_int > fun_se4767462807232413326_int_o ).

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

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

tff(sy_c_ATP_058Lamm__if____,type,
    aTP_Lamm_if: product_prod_int_int > fun_fu3068557634114553409_int_o ).

tff(sy_c_ATP_058Lamm__ig____,type,
    aTP_Lamm_ig: nat > fun_fu4001317426178549763_nat_o ).

tff(sy_c_ATP_058Lamm__ih____,type,
    aTP_Lamm_ih: int > fun_fu3883539188510166275_int_o ).

tff(sy_c_ATP_058Lamm__ii____,type,
    aTP_Lamm_ii: real > fun_fu9050450510664235267real_o ).

tff(sy_c_ATP_058Lamm__ij____,type,
    aTP_Lamm_ij: vEBT_VEBT > fun_fu6329627000648617163VEBT_o ).

tff(sy_c_ATP_058Lamm__ik____,type,
    aTP_Lamm_ik: complex > fun_fu1358370060594899587plex_o ).

tff(sy_c_ATP_058Lamm__il____,type,
    aTP_Lamm_il: product_prod_int_int > fun_fu3068557634114553409_int_o ).

tff(sy_c_ATP_058Lamm__im____,type,
    aTP_Lamm_im: nat > fun_fu4001317426178549763_nat_o ).

tff(sy_c_ATP_058Lamm__in____,type,
    aTP_Lamm_in: int > fun_fu3883539188510166275_int_o ).

tff(sy_c_ATP_058Lamm__io____,type,
    aTP_Lamm_io: real > fun_fu9050450510664235267real_o ).

tff(sy_c_ATP_058Lamm__ip____,type,
    aTP_Lamm_ip: nat > fun_na2880281091838646604_VEBTi ).

tff(sy_c_ATP_058Lamm__iq____,type,
    aTP_Lamm_iq: nat > fun_na4683941076921484125_VEBTi ).

tff(sy_c_ATP_058Lamm__ir____,type,
    aTP_Lamm_ir: nat > fun_VE3458738137785649989_VEBTi ).

tff(sy_c_ATP_058Lamm__is____,type,
    aTP_Lamm_is: nat > fun_ar5136084541013886267_VEBTi ).

tff(sy_c_ATP_058Lamm__it____,type,
    aTP_Lamm_it: nat > fun_ar1785979704382658262_VEBTi ).

tff(sy_c_ATP_058Lamm__iu____,type,
    aTP_Lamm_iu: nat > fun_ar1077903036952521863_VEBTi ).

tff(sy_c_ATP_058Lamm__iv____,type,
    aTP_Lamm_iv: nat > fun_na2161844862588497666_VEBTi ).

tff(sy_c_ATP_058Lamm__iw____,type,
    aTP_Lamm_iw: nat > fun_na7209158589592815097_VEBTi ).

tff(sy_c_ATP_058Lamm__ix____,type,
    aTP_Lamm_ix: nat > fun_na5705991245886026864_VEBTi ).

tff(sy_c_ATP_058Lamm__iy____,type,
    aTP_Lamm_iy: nat > fun_na4396779847561378407_VEBTi ).

tff(sy_c_ATP_058Lamm__iz____,type,
    aTP_Lamm_iz: nat > fun_na4714263456849926110_VEBTi ).

tff(sy_c_ATP_058Lamm__ja____,type,
    aTP_Lamm_ja: nat > fun_na9015037352111213781_VEBTi ).

tff(sy_c_ATP_058Lamm__jb____,type,
    aTP_Lamm_jb: nat > fun_na561812334728403340_VEBTi ).

tff(sy_c_ATP_058Lamm__jc____,type,
    aTP_Lamm_jc: nat > fun_op4795736666101542719_VEBTi ).

tff(sy_c_ATP_058Lamm__jd____,type,
    aTP_Lamm_jd: nat > fun_op5127434927081926045_VEBTi ).

tff(sy_c_ATP_058Lamm__je____,type,
    aTP_Lamm_je: nat > fun_o_6972675622308511895_VEBTi ).

tff(sy_c_ATP_058Lamm__jf____,type,
    aTP_Lamm_jf: nat > fun_li189207710930753934o_assn ).

tff(sy_c_ATP_058Lamm__jg____,type,
    aTP_Lamm_jg: nat > fun_li714799757257233060t_assn ).

tff(sy_c_ATP_058Lamm__jh____,type,
    aTP_Lamm_jh: nat > fun_li92960653718430884t_assn ).

tff(sy_c_ATP_058Lamm__ji____,type,
    aTP_Lamm_ji: nat > fun_li3423855071176226084i_assn ).

tff(sy_c_ATP_058Lamm__jj____,type,
    aTP_Lamm_jj: array_o > fun_li3807767012653957926t_assn ).

tff(sy_c_ATP_058Lamm__jk____,type,
    aTP_Lamm_jk: array_nat > fun_li4928566814967809234t_assn ).

tff(sy_c_ATP_058Lamm__jl____,type,
    aTP_Lamm_jl: array_int > fun_li1703715468755832110t_assn ).

tff(sy_c_ATP_058Lamm__jm____,type,
    aTP_Lamm_jm: array_VEBT_VEBTi > fun_li7023630697154767927t_assn ).

tff(sy_c_ATP_058Lamm__jn____,type,
    aTP_Lamm_jn: fun_VEBT_VEBT_nat > fun_VEBT_VEBT_real ).

tff(sy_c_ATP_058Lamm__jo____,type,
    aTP_Lamm_jo: fun_nat_nat > fun_nat_real ).

tff(sy_c_ATP_058Lamm__jp____,type,
    aTP_Lamm_jp: nat > fun_VE7038733415900436564Heap_o ).

tff(sy_c_ATP_058Lamm__jq____,type,
    aTP_Lamm_jq: array_VEBT_VEBTi > fun_na24089148966465545Heap_o ).

tff(sy_c_ATP_058Lamm__jr____,type,
    aTP_Lamm_jr: array_VEBT_VEBTi > fun_na1260656955978387516Heap_o ).

tff(sy_c_ATP_058Lamm__js____,type,
    aTP_Lamm_js: nat > fun_na7155415448731535229Heap_o ).

tff(sy_c_ATP_058Lamm__jt____,type,
    aTP_Lamm_jt: nat > fun_na5646030511792458954Heap_o ).

tff(sy_c_ATP_058Lamm__ju____,type,
    aTP_Lamm_ju: nat > fun_op1763656958002597022Heap_o ).

tff(sy_c_ATP_058Lamm__jv____,type,
    aTP_Lamm_jv: nat > fun_o_1207901889202581752Heap_o ).

tff(sy_c_ATP_058Lamm__jw____,type,
    aTP_Lamm_jw: produc8923325533196201883nteger > fun_fu8902681562042749453nteger ).

tff(sy_c_ATP_058Lamm__jx____,type,
    aTP_Lamm_jx: produc3658429121746597890et_nat > fun_fu8710561278413400866et_nat ).

tff(sy_c_ATP_058Lamm__jy____,type,
    aTP_Lamm_jy: produc8799533983703829792et_nat > fun_fu6196531292930346304et_nat ).

tff(sy_c_ATP_058Lamm__jz____,type,
    aTP_Lamm_jz: product_prod_int_int > fun_fu5813366174458671467nt_int ).

tff(sy_c_ATP_058Lamm__ka____,type,
    aTP_Lamm_ka: product_prod_int_int > fun_fu4545303443005895275nt_int ).

tff(sy_c_ATP_058Lamm__kb____,type,
    aTP_Lamm_kb: set_nat > fun_he3132782743728456162et_nat ).

tff(sy_c_ATP_058Lamm__kc____,type,
    aTP_Lamm_kc: assn > fun_as229090903528795618n_assn ).

tff(sy_c_ATP_058Lamm__kd____,type,
    aTP_Lamm_kd: uint32 > fun_ui2440373670664278182uint32 ).

tff(sy_c_ATP_058Lamm__ke____,type,
    aTP_Lamm_ke: fun_na4138050610385705875at_nat ).

tff(sy_c_ATP_058Lamm__kf____,type,
    aTP_Lamm_kf: fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__kg____,type,
    aTP_Lamm_kg: fun_o_o > fun_fu6436183194761443139_nat_o ).

tff(sy_c_ATP_058Lamm__kh____,type,
    aTP_Lamm_kh: fun_o_o > fun_fu138107234227827011_int_o ).

tff(sy_c_ATP_058Lamm__ki____,type,
    aTP_Lamm_ki: fun_o_int > fun_fu6453395667394816515nt_int ).

tff(sy_c_ATP_058Lamm__kj____,type,
    aTP_Lamm_kj: fun_int_o > fun_fu1481857817119020493_int_o ).

tff(sy_c_ATP_058Lamm__kk____,type,
    aTP_Lamm_kk: fun_int_int > fun_fu6092417073193843705nt_int ).

tff(sy_c_ATP_058Lamm__kl____,type,
    aTP_Lamm_kl: fun_Pr4796943543331071514_nat_o > fun_fu3843450905092635034_nat_o ).

tff(sy_c_ATP_058Lamm__km____,type,
    aTP_Lamm_km: fun_o_67843798866261420at_nat > fun_fu4108308376836824364at_nat ).

tff(sy_c_ATP_058Lamm__kn____,type,
    aTP_Lamm_kn: fun_Pr3238374608701176162_int_o > fun_fu4592178885959468770_int_o ).

tff(sy_c_ATP_058Lamm__ko____,type,
    aTP_Lamm_ko: fun_Pr982841511892225912nt_int > fun_fu717549361983437348nt_int ).

tff(sy_c_ATP_058Lamm__kp____,type,
    aTP_Lamm_kp: fun_o_8988054265482332260nt_int > fun_fu8427683764667544036nt_int ).

tff(sy_c_ATP_058Lamm__kq____,type,
    aTP_Lamm_kq: fun_Pr2118491707810114563at_nat > fun_na4138050610385705875at_nat ).

tff(sy_c_ATP_058Lamm__kr____,type,
    aTP_Lamm_kr: fun_Pr4796943543331071514_nat_o > fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__ks____,type,
    aTP_Lamm_ks: fun_Pr2478283820680522243nt_int > fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__kt____,type,
    aTP_Lamm_kt: fun_Pr3238374608701176162_int_o > fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__ku____,type,
    aTP_Lamm_ku: fun_Pr982841511892225912nt_int > fun_int_fun_int_int ).

tff(sy_c_ATP_058Lamm__kv____,type,
    aTP_Lamm_kv: fun_na1476630412504255114at_nat > fun_Pr2118491707810114563at_nat ).

tff(sy_c_ATP_058Lamm__kw____,type,
    aTP_Lamm_kw: fun_na1476630412504255114at_nat > fun_na4138050610385705875at_nat ).

tff(sy_c_ATP_058Lamm__kx____,type,
    aTP_Lamm_kx: fun_nat_o > fun_Pr4796943543331071514_nat_o ).

tff(sy_c_ATP_058Lamm__ky____,type,
    aTP_Lamm_ky: fun_nat_o > fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__kz____,type,
    aTP_Lamm_kz: fun_in4211310926729683942nt_int > fun_Pr2478283820680522243nt_int ).

tff(sy_c_ATP_058Lamm__la____,type,
    aTP_Lamm_la: fun_in4211310926729683942nt_int > fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__lb____,type,
    aTP_Lamm_lb: fun_int_o > fun_Pr3238374608701176162_int_o ).

tff(sy_c_ATP_058Lamm__lc____,type,
    aTP_Lamm_lc: fun_int_o > fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__ld____,type,
    aTP_Lamm_ld: fun_int_int > fun_Pr982841511892225912nt_int ).

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

tff(sy_c_ATP_058Lamm__lf____,type,
    aTP_Lamm_lf: fun_he6286135909418734550t_unit ).

tff(sy_c_ATP_058Lamm__lg____,type,
    aTP_Lamm_lg: fun_as3983628580203722482n_assn ).

tff(sy_c_ATP_058Lamm__lh____,type,
    aTP_Lamm_lh: fun_ui2637848667930195262uint32 ).

tff(sy_c_ATP_058Lamm__li____,type,
    aTP_Lamm_li: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__lj____,type,
    aTP_Lamm_lj: fun_int_fun_int_int ).

tff(sy_c_ATP_058Lamm__lk____,type,
    aTP_Lamm_lk: fun_na1476630412504255114at_nat > fun_Pr2118491707810114563at_nat ).

tff(sy_c_ATP_058Lamm__ll____,type,
    aTP_Lamm_ll: fun_na1476630412504255114at_nat > fun_na4138050610385705875at_nat ).

tff(sy_c_ATP_058Lamm__lm____,type,
    aTP_Lamm_lm: fun_nat_o > fun_Pr4796943543331071514_nat_o ).

tff(sy_c_ATP_058Lamm__ln____,type,
    aTP_Lamm_ln: fun_nat_o > fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__lo____,type,
    aTP_Lamm_lo: fun_in4211310926729683942nt_int > fun_Pr2478283820680522243nt_int ).

tff(sy_c_ATP_058Lamm__lp____,type,
    aTP_Lamm_lp: fun_in4211310926729683942nt_int > fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__lq____,type,
    aTP_Lamm_lq: fun_int_o > fun_Pr3238374608701176162_int_o ).

tff(sy_c_ATP_058Lamm__lr____,type,
    aTP_Lamm_lr: fun_int_o > fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__ls____,type,
    aTP_Lamm_ls: fun_int_int > fun_Pr982841511892225912nt_int ).

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

tff(sy_c_ATP_058Lamm__lu____,type,
    aTP_Lamm_lu: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__lv____,type,
    aTP_Lamm_lv: fun_he6520342654808555881et_nat ).

tff(sy_c_ATP_058Lamm__lw____,type,
    aTP_Lamm_lw: fun_Co1981900195452921617nteger ).

tff(sy_c_ATP_058Lamm__lx____,type,
    aTP_Lamm_lx: fun_as3983628580203722482n_assn ).

tff(sy_c_ATP_058Lamm__ly____,type,
    aTP_Lamm_ly: fun_int_fun_int_int ).

tff(sy_c_ATP_058Lamm__lz____,type,
    aTP_Lamm_lz: fun_na5363892732107265798uint32 ).

tff(sy_c_ATP_058Lamm__ma____,type,
    aTP_Lamm_ma: fun_na4844478536282698896l_num1 ).

tff(sy_c_ATP_058Lamm__mb____,type,
    aTP_Lamm_mb: fun_nat_fun_nat_rat ).

tff(sy_c_ATP_058Lamm__mc____,type,
    aTP_Lamm_mc: fun_nat_fun_nat_real ).

tff(sy_c_ATP_058Lamm__md____,type,
    aTP_Lamm_md: fun_nat_fun_nat_int ).

tff(sy_c_ATP_058Lamm__me____,type,
    aTP_Lamm_me: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__mf____,type,
    aTP_Lamm_mf: fun_na8135612474337863287nteger ).

tff(sy_c_ATP_058Lamm__mg____,type,
    aTP_Lamm_mg: $o > fun_complex_o ).

tff(sy_c_ATP_058Lamm__mh____,type,
    aTP_Lamm_mh: $o > fun_Pr3238374608701176162_int_o ).

tff(sy_c_ATP_058Lamm__mi____,type,
    aTP_Lamm_mi: $o > fun_int_o ).

tff(sy_c_ATP_058Lamm__mj____,type,
    aTP_Lamm_mj: $o > fun_real_o ).

tff(sy_c_ATP_058Lamm__mk____,type,
    aTP_Lamm_mk: fun_o_fun_nat_o ).

tff(sy_c_ATP_058Lamm__ml____,type,
    aTP_Lamm_ml: $o > fun_char_o ).

tff(sy_c_ATP_058Lamm__mm____,type,
    aTP_Lamm_mm: $o > fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__mn____,type,
    aTP_Lamm_mn: $o > fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__mo____,type,
    aTP_Lamm_mo: fun_na4138050610385705875at_nat ).

tff(sy_c_ATP_058Lamm__mp____,type,
    aTP_Lamm_mp: fun_fu4810197545401214784nteger > fun_fu4530937843762191077eger_o ).

tff(sy_c_ATP_058Lamm__mq____,type,
    aTP_Lamm_mq: fun_fu2352890940381120596et_nat > fun_fu7582738562687708775_nat_o ).

tff(sy_c_ATP_058Lamm__mr____,type,
    aTP_Lamm_mr: fun_fu1905361174868762994et_nat > fun_fu4927615759029954091_nat_o ).

tff(sy_c_ATP_058Lamm__ms____,type,
    aTP_Lamm_ms: fun_fu4253467608928286509nt_int > fun_fu4695829907076200203_int_o ).

tff(sy_c_ATP_058Lamm__mt____,type,
    aTP_Lamm_mt: fun_fu5631754973857388633nt_int > fun_fu6286765928754279603_int_o ).

tff(sy_c_ATP_058Lamm__mu____,type,
    aTP_Lamm_mu: fun_nat_option_nat > fun_fu994687079535449577_nat_o ).

tff(sy_c_ATP_058Lamm__mv____,type,
    aTP_Lamm_mv: fun_int_option_int > fun_fu7409062071265970153_int_o ).

tff(sy_c_ATP_058Lamm__mw____,type,
    aTP_Lamm_mw: num > fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__mx____,type,
    aTP_Lamm_mx: num > fun_na4138050610385705875at_nat ).

tff(sy_c_ATP_058Lamm__my____,type,
    aTP_Lamm_my: num > fun_Co2470855966376689427nteger ).

tff(sy_c_ATP_058Lamm__mz____,type,
    aTP_Lamm_mz: fun_VE8118064890490510894_VEBTi > fun_na2890314705291535822_VEBTi ).

tff(sy_c_ATP_058Lamm__na____,type,
    aTP_Lamm_na: fun_VE8118064890490510894_VEBTi > fun_na7797079703196089028_VEBTi ).

tff(sy_c_ATP_058Lamm__nb____,type,
    aTP_Lamm_nb: fun_VE8118064890490510894_VEBTi > fun_na5084500599656708831_VEBTi ).

tff(sy_c_ATP_058Lamm__nc____,type,
    aTP_Lamm_nc: fun_VE8118064890490510894_VEBTi > fun_na9009963084909443284_VEBTi ).

tff(sy_c_ATP_058Lamm__nd____,type,
    aTP_Lamm_nd: fun_VE8118064890490510894_VEBTi > fun_na2291395713665613775_VEBTi ).

tff(sy_c_ATP_058Lamm__ne____,type,
    aTP_Lamm_ne: fun_VE8118064890490510894_VEBTi > fun_na7682395145315030854_VEBTi ).

tff(sy_c_ATP_058Lamm__nf____,type,
    aTP_Lamm_nf: fun_VE8118064890490510894_VEBTi > fun_na7174282198332753469_VEBTi ).

tff(sy_c_ATP_058Lamm__ng____,type,
    aTP_Lamm_ng: fun_VE8118064890490510894_VEBTi > fun_na3224852944424307892_VEBTi ).

tff(sy_c_ATP_058Lamm__nh____,type,
    aTP_Lamm_nh: fun_VE8118064890490510894_VEBTi > fun_na460406568315038379_VEBTi ).

tff(sy_c_ATP_058Lamm__ni____,type,
    aTP_Lamm_ni: fun_VE8118064890490510894_VEBTi > fun_na2086927332677804066_VEBTi ).

tff(sy_c_ATP_058Lamm__nj____,type,
    aTP_Lamm_nj: fun_VE8118064890490510894_VEBTi > fun_na2740539754339541653_VEBTi ).

tff(sy_c_ATP_058Lamm__nk____,type,
    aTP_Lamm_nk: fun_VE8118064890490510894_VEBTi > fun_na278150576929200012_VEBTi ).

tff(sy_c_ATP_058Lamm__nl____,type,
    aTP_Lamm_nl: fun_VE8118064890490510894_VEBTi > fun_na8762902223149059366_VEBTi ).

tff(sy_c_ATP_058Lamm__nm____,type,
    aTP_Lamm_nm: fun_VE8516060933855021849on_nat > fun_na4155721213724604198on_nat ).

tff(sy_c_ATP_058Lamm__nn____,type,
    aTP_Lamm_nn: fun_VE8516060933855021849on_nat > fun_na1029252574726078829on_nat ).

tff(sy_c_ATP_058Lamm__no____,type,
    aTP_Lamm_no: fun_VE8516060933855021849on_nat > fun_na7271643239918768137on_nat ).

tff(sy_c_ATP_058Lamm__np____,type,
    aTP_Lamm_np: fun_VE8516060933855021849on_nat > fun_na2011684170944547840on_nat ).

tff(sy_c_ATP_058Lamm__nq____,type,
    aTP_Lamm_nq: fun_VE8516060933855021849on_nat > fun_na2990367117748150135on_nat ).

tff(sy_c_ATP_058Lamm__nr____,type,
    aTP_Lamm_nr: fun_VE8516060933855021849on_nat > fun_na6220387211928018449on_nat ).

tff(sy_c_ATP_058Lamm__ns____,type,
    aTP_Lamm_ns: fun_VE8516060933855021849on_nat > fun_na3472305698324104669on_nat ).

tff(sy_c_ATP_058Lamm__nt____,type,
    aTP_Lamm_nt: fun_VE8516060933855021849on_nat > fun_na505420239896681636on_nat ).

tff(sy_c_ATP_058Lamm__nu____,type,
    aTP_Lamm_nu: fun_VE8516060933855021849on_nat > fun_na3776232666933163584on_nat ).

tff(sy_c_ATP_058Lamm__nv____,type,
    aTP_Lamm_nv: fun_VE8516060933855021849on_nat > fun_na3371937797455410308on_nat ).

tff(sy_c_ATP_058Lamm__nw____,type,
    aTP_Lamm_nw: fun_VE8516060933855021849on_nat > fun_na2990367117748150135on_nat ).

tff(sy_c_ATP_058Lamm__nx____,type,
    aTP_Lamm_nx: fun_VE8516060933855021849on_nat > fun_na6220387211928018449on_nat ).

tff(sy_c_ATP_058Lamm__ny____,type,
    aTP_Lamm_ny: fun_VE7134199479003659425Heap_o > fun_na8576134899853594145Heap_o ).

tff(sy_c_ATP_058Lamm__nz____,type,
    aTP_Lamm_nz: fun_VE7134199479003659425Heap_o > fun_ar6383509519145165198Heap_o ).

tff(sy_c_ATP_058Lamm__oa____,type,
    aTP_Lamm_oa: fun_VE7134199479003659425Heap_o > fun_ar8039747222344348737Heap_o ).

tff(sy_c_ATP_058Lamm__ob____,type,
    aTP_Lamm_ob: fun_VE7134199479003659425Heap_o > fun_na60875040783188486Heap_o ).

tff(sy_c_ATP_058Lamm__oc____,type,
    aTP_Lamm_oc: fun_VE7134199479003659425Heap_o > fun_na6148071205614320851Heap_o ).

tff(sy_c_ATP_058Lamm__od____,type,
    aTP_Lamm_od: fun_VE7134199479003659425Heap_o > fun_na1841732828589024619Heap_o ).

tff(sy_c_ATP_058Lamm__oe____,type,
    aTP_Lamm_oe: nat > fun_ar7864286482443943364on_nat ).

tff(sy_c_ATP_058Lamm__of____,type,
    aTP_Lamm_of: nat > fun_ar7575598148623397506on_nat ).

tff(sy_c_ATP_058Lamm__og____,type,
    aTP_Lamm_og: nat > fun_ar5515879994638340162on_nat ).

tff(sy_c_ATP_058Lamm__oh____,type,
    aTP_Lamm_oh: nat > fun_ar4372988750007754556on_nat ).

tff(sy_c_ATP_058Lamm__oi____,type,
    aTP_Lamm_oi: nat > fun_ar6283851350152303155on_nat ).

tff(sy_c_ATP_058Lamm__oj____,type,
    aTP_Lamm_oj: nat > fun_ar3874946511942987587on_nat ).

tff(sy_c_ATP_058Lamm__ok____,type,
    aTP_Lamm_ok: nat > fun_ar3874946511942987587on_nat ).

tff(sy_c_ATP_058Lamm__ol____,type,
    aTP_Lamm_ol: nat > fun_na4376472396080214220on_nat ).

tff(sy_c_ATP_058Lamm__om____,type,
    aTP_Lamm_om: nat > fun_na5759534986367084828on_nat ).

tff(sy_c_ATP_058Lamm__on____,type,
    aTP_Lamm_on: nat > fun_na7014419191514879763on_nat ).

tff(sy_c_ATP_058Lamm__oo____,type,
    aTP_Lamm_oo: nat > fun_na2272093518330687934on_nat ).

tff(sy_c_ATP_058Lamm__op____,type,
    aTP_Lamm_op: nat > fun_op1686896281538429038on_nat ).

tff(sy_c_ATP_058Lamm__oq____,type,
    aTP_Lamm_oq: vEBT_VEBT > fun_na8849172711643138838on_nat ).

tff(sy_c_ATP_058Lamm__or____,type,
    aTP_Lamm_or: vEBT_VEBT > fun_na1749477412626277658on_nat ).

tff(sy_c_ATP_058Lamm__os____,type,
    aTP_Lamm_os: vEBT_VEBT > fun_na3054862186897204036on_nat ).

tff(sy_c_ATP_058Lamm__ot____,type,
    aTP_Lamm_ot: fun_op8229906066235313459T_VEBT ).

tff(sy_c_ATP_058Lamm__ou____,type,
    aTP_Lamm_ou: fun_o_8142510167905321106T_VEBT ).

tff(sy_c_ATP_058Lamm__ov____,type,
    aTP_Lamm_ov: vEBT_VEBT > fun_na5992628573614119403on_nat ).

tff(sy_c_ATP_058Lamm__ow____,type,
    aTP_Lamm_ow: vEBT_VEBT > fun_na6220387211928018449on_nat ).

tff(sy_c_ATP_058Lamm__ox____,type,
    aTP_Lamm_ox: $o > fun_fu6436183194761443139_nat_o ).

tff(sy_c_ATP_058Lamm__oy____,type,
    aTP_Lamm_oy: $o > fun_fu138107234227827011_int_o ).

tff(sy_c_ATP_058Lamm__oz____,type,
    aTP_Lamm_oz: fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__pa____,type,
    aTP_Lamm_pa: fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__pb____,type,
    aTP_Lamm_pb: fun_int_o ).

tff(sy_c_ATP_058Lamm__pc____,type,
    aTP_Lamm_pc: fun_complex_o ).

tff(sy_c_ATP_058Lamm__pd____,type,
    aTP_Lamm_pd: fun_Pr3238374608701176162_int_o ).

tff(sy_c_ATP_058Lamm__pe____,type,
    aTP_Lamm_pe: fun_real_o ).

tff(sy_c_ATP_058Lamm__pf____,type,
    aTP_Lamm_pf: fun_char_o ).

tff(sy_c_ATP_058Lamm__pg____,type,
    aTP_Lamm_pg: fun_Pr3650993782763339253T_VEBT > fun_fu493888963134383235T_VEBT ).

tff(sy_c_ATP_058Lamm__ph____,type,
    aTP_Lamm_ph: fun_Pr3650993782763339253T_VEBT > fun_fu4137589202531160609T_VEBT ).

tff(sy_c_ATP_058Lamm__pi____,type,
    aTP_Lamm_pi: fun_Pr7471953115884902709VEBT_o > fun_fu4430894231949415039VEBT_o ).

tff(sy_c_ATP_058Lamm__pj____,type,
    aTP_Lamm_pj: fun_Pr7471953115884902709VEBT_o > fun_fu5205589992830698549un_o_o ).

tff(sy_c_ATP_058Lamm__pk____,type,
    aTP_Lamm_pk: fun_o_3001263932818058983T_VEBT > fun_fu327035696625524243T_VEBT ).

tff(sy_c_ATP_058Lamm__pl____,type,
    aTP_Lamm_pl: fun_o_3001263932818058983T_VEBT > fun_fu1082429846158404371T_VEBT ).

tff(sy_c_ATP_058Lamm__pm____,type,
    aTP_Lamm_pm: fun_o_o > fun_fu4591712890356382959VEBT_o ).

tff(sy_c_ATP_058Lamm__pn____,type,
    aTP_Lamm_pn: fun_o_o > fun_fu4605530428521801667un_o_o ).

tff(sy_c_ATP_058Lamm__po____,type,
    aTP_Lamm_po: fun_fu6311994054701583983uint32 ).

tff(sy_c_ATP_058Lamm__pp____,type,
    aTP_Lamm_pp: fun_fu8568212769178095465eger_o ).

tff(sy_c_ATP_058Lamm__pq____,type,
    aTP_Lamm_pq: fun_fu5407763515825486313uint32 ).

tff(sy_c_ATP_058Lamm__pr____,type,
    aTP_Lamm_pr: fun_fu713059014528672409uint32 ).

tff(sy_c_ATP_058Lamm__ps____,type,
    aTP_Lamm_ps: fun_fu5055813439476669081uint32 ).

tff(sy_c_ATP_058Lamm__pt____,type,
    aTP_Lamm_pt: fun_Pr2118491707810114563at_nat ).

tff(sy_c_ATP_058Lamm__pu____,type,
    aTP_Lamm_pu: fun_num_num ).

tff(sy_c_ATP_058Lamm__pv____,type,
    aTP_Lamm_pv: fun_VE8181444127144381288Heap_o > fun_li5500215499729237691Heap_o ).

tff(sy_c_ATP_058Lamm__pw____,type,
    aTP_Lamm_pw: fun_VE8181444127144381288Heap_o > fun_ar8795608905921525346Heap_o ).

tff(sy_c_ATP_058Lamm__px____,type,
    aTP_Lamm_px: fun_VE8181444127144381288Heap_o > fun_na8963100177754871293Heap_o ).

tff(sy_c_ATP_058Lamm__py____,type,
    aTP_Lamm_py: fun_VE8181444127144381288Heap_o > fun_na6370613109607759809Heap_o ).

tff(sy_c_ATP_058Lamm__pz____,type,
    aTP_Lamm_pz: fun_VE8181444127144381288Heap_o > fun_na2699581809715174900Heap_o ).

tff(sy_c_ATP_058Lamm__qa____,type,
    aTP_Lamm_qa: fun_VE8181444127144381288Heap_o > fun_na853727736315331060Heap_o ).

tff(sy_c_ATP_058Lamm__qb____,type,
    aTP_Lamm_qb: fun_VE8181444127144381288Heap_o > fun_na3474800335600542119Heap_o ).

tff(sy_c_ATP_058Lamm__qc____,type,
    aTP_Lamm_qc: fun_VE8181444127144381288Heap_o > fun_na2622557336161645666Heap_o ).

tff(sy_c_ATP_058Lamm__qd____,type,
    aTP_Lamm_qd: fun_VE8181444127144381288Heap_o > fun_na7247921347214250394Heap_o ).

tff(sy_c_ATP_058Lamm__qe____,type,
    aTP_Lamm_qe: fun_VE8181444127144381288Heap_o > fun_na5958411666642256818Heap_o ).

tff(sy_c_ATP_058Lamm__qf____,type,
    aTP_Lamm_qf: fun_VE8181444127144381288Heap_o > fun_na7955433379684683584Heap_o ).

tff(sy_c_ATP_058Lamm__qg____,type,
    aTP_Lamm_qg: fun_VE8181444127144381288Heap_o > fun_VE567930495301233690Heap_o ).

tff(sy_c_ATP_058Lamm__qh____,type,
    aTP_Lamm_qh: fun_VE8181444127144381288Heap_o > fun_VE1467545974346792018Heap_o ).

tff(sy_c_ATP_058Lamm__qi____,type,
    aTP_Lamm_qi: fun_VE8181444127144381288Heap_o > fun_VE2348196291809386290Heap_o ).

tff(sy_c_ATP_058Lamm__qj____,type,
    aTP_Lamm_qj: list_VEBT_VEBT > fun_na8277067783256924526Heap_o ).

tff(sy_c_ATP_058Lamm__qk____,type,
    aTP_Lamm_qk: array_VEBT_VEBTi > fun_li6502840675107701469Heap_o ).

tff(sy_c_ATP_058Lamm__ql____,type,
    aTP_Lamm_ql: nat > fun_na2926107884465348340Heap_o ).

tff(sy_c_ATP_058Lamm__qm____,type,
    aTP_Lamm_qm: nat > fun_na1856392443187364084Heap_o ).

tff(sy_c_ATP_058Lamm__qn____,type,
    aTP_Lamm_qn: nat > fun_na8953503613173268075Heap_o ).

tff(sy_c_ATP_058Lamm__qo____,type,
    aTP_Lamm_qo: nat > fun_na2536804662393741479Heap_o ).

tff(sy_c_ATP_058Lamm__qp____,type,
    aTP_Lamm_qp: nat > fun_na8215782167631191514Heap_o ).

tff(sy_c_ATP_058Lamm__qq____,type,
    aTP_Lamm_qq: nat > fun_na89412549627184089Heap_o ).

tff(sy_c_ATP_058Lamm__qr____,type,
    aTP_Lamm_qr: nat > fun_op218661150302373393Heap_o ).

tff(sy_c_ATP_058Lamm__qs____,type,
    aTP_Lamm_qs: nat > fun_op6476899888854096997Heap_o ).

tff(sy_c_ATP_058Lamm__qt____,type,
    aTP_Lamm_qt: nat > fun_op1410506891405710263Heap_o ).

tff(sy_c_ATP_058Lamm__qu____,type,
    aTP_Lamm_qu: vEBT_VEBT > fun_na6537123900254388819Heap_o ).

tff(sy_c_ATP_058Lamm__qv____,type,
    aTP_Lamm_qv: vEBT_VEBT > fun_na7998299765372931779Heap_o ).

tff(sy_c_ATP_058Lamm__qw____,type,
    aTP_Lamm_qw: vEBT_VEBT > fun_na1841732828589024619Heap_o ).

tff(sy_c_ATP_058Lamm__qx____,type,
    aTP_Lamm_qx: fun_VE2018681615261037117_VEBTi > fun_na2548587511393692655_VEBTi ).

tff(sy_c_ATP_058Lamm__qy____,type,
    aTP_Lamm_qy: fun_VE2018681615261037117_VEBTi > fun_na7679953648084048589_VEBTi ).

tff(sy_c_ATP_058Lamm__qz____,type,
    aTP_Lamm_qz: fun_VE2018681615261037117_VEBTi > fun_na2840261909164627793_VEBTi ).

tff(sy_c_ATP_058Lamm__ra____,type,
    aTP_Lamm_ra: fun_VE2018681615261037117_VEBTi > fun_na7137134861304782383_VEBTi ).

tff(sy_c_ATP_058Lamm__rb____,type,
    aTP_Lamm_rb: fun_VE2018681615261037117_VEBTi > fun_na7565023553309985968_VEBTi ).

tff(sy_c_ATP_058Lamm__rc____,type,
    aTP_Lamm_rc: fun_VE2018681615261037117_VEBTi > fun_na1347330265571094344_VEBTi ).

tff(sy_c_ATP_058Lamm__rd____,type,
    aTP_Lamm_rd: fun_VE2018681615261037117_VEBTi > fun_na1347330265571094344_VEBTi ).

tff(sy_c_ATP_058Lamm__re____,type,
    aTP_Lamm_re: fun_VE2018681615261037117_VEBTi > fun_na1659118370164772413_VEBTi ).

tff(sy_c_ATP_058Lamm__rf____,type,
    aTP_Lamm_rf: fun_VE2018681615261037117_VEBTi > fun_na2169200058431587252_VEBTi ).

tff(sy_c_ATP_058Lamm__rg____,type,
    aTP_Lamm_rg: fun_VE2018681615261037117_VEBTi > fun_na719414919122706795_VEBTi ).

tff(sy_c_ATP_058Lamm__rh____,type,
    aTP_Lamm_rh: fun_VE2018681615261037117_VEBTi > fun_na9095497340641005739_VEBTi ).

tff(sy_c_ATP_058Lamm__ri____,type,
    aTP_Lamm_ri: fun_VE2018681615261037117_VEBTi > fun_na4748561171398030626_VEBTi ).

tff(sy_c_ATP_058Lamm__rj____,type,
    aTP_Lamm_rj: fun_VE2018681615261037117_VEBTi > fun_na817248180520022297_VEBTi ).

tff(sy_c_ATP_058Lamm__rk____,type,
    aTP_Lamm_rk: fun_VE2018681615261037117_VEBTi > fun_na5699700385611646096_VEBTi ).

tff(sy_c_ATP_058Lamm__rl____,type,
    aTP_Lamm_rl: fun_VE2018681615261037117_VEBTi > fun_na113775696040589571_VEBTi ).

tff(sy_c_ATP_058Lamm__rm____,type,
    aTP_Lamm_rm: fun_VE2018681615261037117_VEBTi > fun_na113775696040589571_VEBTi ).

tff(sy_c_ATP_058Lamm__rn____,type,
    aTP_Lamm_rn: fun_VE2018681615261037117_VEBTi > fun_na653084535925934713_VEBTi ).

tff(sy_c_ATP_058Lamm__ro____,type,
    aTP_Lamm_ro: fun_VE2018681615261037117_VEBTi > fun_na993102950844727003_VEBTi ).

tff(sy_c_ATP_058Lamm__rp____,type,
    aTP_Lamm_rp: fun_VE2018681615261037117_VEBTi > fun_na4691356797839745173_VEBTi ).

tff(sy_c_ATP_058Lamm__rq____,type,
    aTP_Lamm_rq: fun_VE2018681615261037117_VEBTi > fun_na7353129812761597833_VEBTi ).

tff(sy_c_ATP_058Lamm__rr____,type,
    aTP_Lamm_rr: fun_VE2018681615261037117_VEBTi > fun_VE8810447512154760383_VEBTi ).

tff(sy_c_ATP_058Lamm__rs____,type,
    aTP_Lamm_rs: fun_VE2018681615261037117_VEBTi > fun_VE91144239014113491_VEBTi ).

tff(sy_c_ATP_058Lamm__rt____,type,
    aTP_Lamm_rt: fun_VE2018681615261037117_VEBTi > fun_VE728358326801380661_VEBTi ).

tff(sy_c_ATP_058Lamm__ru____,type,
    aTP_Lamm_ru: nat > fun_na8307603164994189425on_nat ).

tff(sy_c_ATP_058Lamm__rv____,type,
    aTP_Lamm_rv: nat > fun_na7745806701448065237on_nat ).

tff(sy_c_ATP_058Lamm__rw____,type,
    aTP_Lamm_rw: fun_VE3775134097089128360on_nat > fun_na607092491672936340on_nat ).

tff(sy_c_ATP_058Lamm__rx____,type,
    aTP_Lamm_rx: fun_VE3775134097089128360on_nat > fun_na4347519868517868204on_nat ).

tff(sy_c_ATP_058Lamm__ry____,type,
    aTP_Lamm_ry: fun_VE3775134097089128360on_nat > fun_na7760356811779829437on_nat ).

tff(sy_c_ATP_058Lamm__rz____,type,
    aTP_Lamm_rz: fun_VE3775134097089128360on_nat > fun_na1922817992914042619on_nat ).

tff(sy_c_ATP_058Lamm__sa____,type,
    aTP_Lamm_sa: fun_VE3775134097089128360on_nat > fun_na1922817992914042619on_nat ).

tff(sy_c_ATP_058Lamm__sb____,type,
    aTP_Lamm_sb: fun_VE3775134097089128360on_nat > fun_na1922817992914042619on_nat ).

tff(sy_c_ATP_058Lamm__sc____,type,
    aTP_Lamm_sc: fun_VE3775134097089128360on_nat > fun_na6896436734642855769on_nat ).

tff(sy_c_ATP_058Lamm__sd____,type,
    aTP_Lamm_sd: fun_VE3775134097089128360on_nat > fun_na56368789070356560on_nat ).

tff(sy_c_ATP_058Lamm__se____,type,
    aTP_Lamm_se: fun_VE3775134097089128360on_nat > fun_na3160842509525366983on_nat ).

tff(sy_c_ATP_058Lamm__sf____,type,
    aTP_Lamm_sf: fun_VE3775134097089128360on_nat > fun_na7284882202938760507on_nat ).

tff(sy_c_ATP_058Lamm__sg____,type,
    aTP_Lamm_sg: fun_VE3775134097089128360on_nat > fun_VE5053670003584447325on_nat ).

tff(sy_c_ATP_058Lamm__sh____,type,
    aTP_Lamm_sh: fun_VE3775134097089128360on_nat > fun_VE205885719767202089on_nat ).

tff(sy_c_ATP_058Lamm__si____,type,
    aTP_Lamm_si: fun_VE3775134097089128360on_nat > fun_VE8661257954422131339on_nat ).

tff(sy_c_ATP_058Lamm__sj____,type,
    aTP_Lamm_sj: fun_VE3775134097089128360on_nat > fun_VE7418359107739301810on_nat ).

tff(sy_c_ATP_058Lamm__sk____,type,
    aTP_Lamm_sk: fun_VE3775134097089128360on_nat > fun_VE8881588537313513120on_nat ).

tff(sy_c_ATP_058Lamm__sl____,type,
    aTP_Lamm_sl: fun_VE3775134097089128360on_nat > fun_na7821417838835476043on_nat ).

tff(sy_c_ATP_058Lamm__sm____,type,
    aTP_Lamm_sm: fun_VE3775134097089128360on_nat > fun_na4144732571386696713on_nat ).

tff(sy_c_ATP_058Lamm__sn____,type,
    aTP_Lamm_sn: fun_VE3775134097089128360on_nat > fun_na5199945083401365888on_nat ).

tff(sy_c_ATP_058Lamm__so____,type,
    aTP_Lamm_so: fun_VE3775134097089128360on_nat > fun_na4376472396080214220on_nat ).

tff(sy_c_ATP_058Lamm__sp____,type,
    aTP_Lamm_sp: fun_VE3775134097089128360on_nat > fun_na4376472396080214220on_nat ).

tff(sy_c_ATP_058Lamm__sq____,type,
    aTP_Lamm_sq: fun_VE3775134097089128360on_nat > fun_na6803271923218958805on_nat ).

tff(sy_c_ATP_058Lamm__sr____,type,
    aTP_Lamm_sr: fun_VE3775134097089128360on_nat > fun_na6506281770025666025on_nat ).

tff(sy_c_ATP_058Lamm__ss____,type,
    aTP_Lamm_ss: fun_VE3775134097089128360on_nat > fun_na223517484589429856on_nat ).

tff(sy_c_ATP_058Lamm__st____,type,
    aTP_Lamm_st: fun_VE3775134097089128360on_nat > fun_na3160842509525366983on_nat ).

tff(sy_c_ATP_058Lamm__su____,type,
    aTP_Lamm_su: fun_VE3775134097089128360on_nat > fun_na7284882202938760507on_nat ).

tff(sy_c_ATP_058Lamm__sv____,type,
    aTP_Lamm_sv: fun_VE3775134097089128360on_nat > fun_VE5053670003584447325on_nat ).

tff(sy_c_ATP_058Lamm__sw____,type,
    aTP_Lamm_sw: fun_VE3775134097089128360on_nat > fun_VE205885719767202089on_nat ).

tff(sy_c_ATP_058Lamm__sx____,type,
    aTP_Lamm_sx: fun_VE3775134097089128360on_nat > fun_VE8661257954422131339on_nat ).

tff(sy_c_ATP_058Lamm__sy____,type,
    aTP_Lamm_sy: fun_VE3775134097089128360on_nat > fun_VE7418359107739301810on_nat ).

tff(sy_c_ATP_058Lamm__sz____,type,
    aTP_Lamm_sz: fun_VE3775134097089128360on_nat > fun_VE8881588537313513120on_nat ).

tff(sy_c_ATP_058Lamm__ta____,type,
    aTP_Lamm_ta: nat > fun_VE8714614573866374946_VEBTi ).

tff(sy_c_ATP_058Lamm__tb____,type,
    aTP_Lamm_tb: nat > fun_ar2561552978615356160_VEBTi ).

tff(sy_c_ATP_058Lamm__tc____,type,
    aTP_Lamm_tc: nat > fun_ar5696067373849871748_VEBTi ).

tff(sy_c_ATP_058Lamm__td____,type,
    aTP_Lamm_td: nat > fun_ar4938665239794946086_VEBTi ).

tff(sy_c_ATP_058Lamm__te____,type,
    aTP_Lamm_te: nat > fun_ar1504418214855557219_VEBTi ).

tff(sy_c_ATP_058Lamm__tf____,type,
    aTP_Lamm_tf: nat > fun_ar9185468459080841407_VEBTi ).

tff(sy_c_ATP_058Lamm__tg____,type,
    aTP_Lamm_tg: nat > fun_ar9185468459080841407_VEBTi ).

tff(sy_c_ATP_058Lamm__th____,type,
    aTP_Lamm_th: nat > fun_na6954118198501441136_VEBTi ).

tff(sy_c_ATP_058Lamm__ti____,type,
    aTP_Lamm_ti: nat > fun_na3147167826068894567_VEBTi ).

tff(sy_c_ATP_058Lamm__tj____,type,
    aTP_Lamm_tj: nat > fun_na1791409581363865698_VEBTi ).

tff(sy_c_ATP_058Lamm__tk____,type,
    aTP_Lamm_tk: nat > fun_na2269434147276521438_VEBTi ).

tff(sy_c_ATP_058Lamm__tl____,type,
    aTP_Lamm_tl: nat > fun_na235080101556577237_VEBTi ).

tff(sy_c_ATP_058Lamm__tm____,type,
    aTP_Lamm_tm: nat > fun_na8695323739542216524_VEBTi ).

tff(sy_c_ATP_058Lamm__tn____,type,
    aTP_Lamm_tn: nat > fun_na2998378723532091971_VEBTi ).

tff(sy_c_ATP_058Lamm__to____,type,
    aTP_Lamm_to: nat > fun_na1238378468414741754_VEBTi ).

tff(sy_c_ATP_058Lamm__tp____,type,
    aTP_Lamm_tp: nat > fun_na1238378468414741754_VEBTi ).

tff(sy_c_ATP_058Lamm__tq____,type,
    aTP_Lamm_tq: nat > fun_na4431491620761096816_VEBTi ).

tff(sy_c_ATP_058Lamm__tr____,type,
    aTP_Lamm_tr: nat > fun_op246729168895827410_VEBTi ).

tff(sy_c_ATP_058Lamm__ts____,type,
    aTP_Lamm_ts: nat > fun_op3511852844878897352_VEBTi ).

tff(sy_c_ATP_058Lamm__tt____,type,
    aTP_Lamm_tt: nat > fun_op8407643897966791296_VEBTi ).

tff(sy_c_ATP_058Lamm__tu____,type,
    aTP_Lamm_tu: vEBT_VEBT > fun_na8722835333295262128_VEBTi ).

tff(sy_c_ATP_058Lamm__tv____,type,
    aTP_Lamm_tv: vEBT_VEBT > fun_na278150576929200012_VEBTi ).

tff(sy_c_ATP_058Lamm__tw____,type,
    aTP_Lamm_tw: vEBT_VEBT > fun_na8762902223149059366_VEBTi ).

tff(sy_c_ATP_058Lamm__tx____,type,
    aTP_Lamm_tx: nat > fun_na9055242642580439111on_nat ).

tff(sy_c_ATP_058Lamm__ty____,type,
    aTP_Lamm_ty: nat > fun_na7511075313506320931on_nat ).

tff(sy_c_ATP_058Lamm__tz____,type,
    aTP_Lamm_tz: nat > fun_na8752921509028558004on_nat ).

tff(sy_c_ATP_058Lamm__ua____,type,
    aTP_Lamm_ua: nat > fun_na5280544852190576430on_nat ).

tff(sy_c_ATP_058Lamm__ub____,type,
    aTP_Lamm_ub: nat > fun_na5280544852190576430on_nat ).

tff(sy_c_ATP_058Lamm__uc____,type,
    aTP_Lamm_uc: nat > fun_na5280544852190576430on_nat ).

tff(sy_c_ATP_058Lamm__ud____,type,
    aTP_Lamm_ud: nat > fun_na6850568596825991504on_nat ).

tff(sy_c_ATP_058Lamm__ue____,type,
    aTP_Lamm_ue: nat > fun_na8435634892762120647on_nat ).

tff(sy_c_ATP_058Lamm__uf____,type,
    aTP_Lamm_uf: nat > fun_na2272093518330687934on_nat ).

tff(sy_c_ATP_058Lamm__ug____,type,
    aTP_Lamm_ug: nat > fun_op1686896281538429038on_nat ).

tff(sy_c_ATP_058Lamm__uh____,type,
    aTP_Lamm_uh: vEBT_VEBT > fun_na8849172711643138838on_nat ).

tff(sy_c_ATP_058Lamm__ui____,type,
    aTP_Lamm_ui: vEBT_VEBT > fun_na1749477412626277658on_nat ).

tff(sy_c_ATP_058Lamm__uj____,type,
    aTP_Lamm_uj: vEBT_VEBT > fun_na3054862186897204036on_nat ).

tff(sy_c_ATP_058Lamm__uk____,type,
    aTP_Lamm_uk: vEBT_VEBT > fun_na5992628573614119403on_nat ).

tff(sy_c_ATP_058Lamm__ul____,type,
    aTP_Lamm_ul: vEBT_VEBT > fun_na6220387211928018449on_nat ).

tff(sy_c_ATP_058Lamm__um____,type,
    aTP_Lamm_um: fun_na4138050610385705875at_nat ).

tff(sy_c_ATP_058Lamm__un____,type,
    aTP_Lamm_un: fun_Co2470855966376689427nteger ).

tff(sy_c_ATP_058Lamm__uo____,type,
    aTP_Lamm_uo: fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__up____,type,
    aTP_Lamm_up: fun_na4138050610385705875at_nat ).

tff(sy_c_ATP_058Lamm__uq____,type,
    aTP_Lamm_uq: fun_Co2470855966376689427nteger ).

tff(sy_c_ATP_058Lamm__ur____,type,
    aTP_Lamm_ur: fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__us____,type,
    aTP_Lamm_us: fun_op541075054904862521VEBT_o ).

tff(sy_c_ATP_058Lamm__ut____,type,
    aTP_Lamm_ut: fun_o_fun_o_o ).

tff(sy_c_ATP_058Lamm__uu____,type,
    aTP_Lamm_uu: fun_Pr4404027214751104635_VEBTi > fun_na2548587511393692655_VEBTi ).

tff(sy_c_ATP_058Lamm__uv____,type,
    aTP_Lamm_uv: fun_Pr4404027214751104635_VEBTi > fun_na7679953648084048589_VEBTi ).

tff(sy_c_ATP_058Lamm__uw____,type,
    aTP_Lamm_uw: fun_Pr4404027214751104635_VEBTi > fun_na2840261909164627793_VEBTi ).

tff(sy_c_ATP_058Lamm__ux____,type,
    aTP_Lamm_ux: fun_Pr4404027214751104635_VEBTi > fun_na7137134861304782383_VEBTi ).

tff(sy_c_ATP_058Lamm__uy____,type,
    aTP_Lamm_uy: fun_Pr4404027214751104635_VEBTi > fun_na7565023553309985968_VEBTi ).

tff(sy_c_ATP_058Lamm__uz____,type,
    aTP_Lamm_uz: fun_Pr4404027214751104635_VEBTi > fun_na1347330265571094344_VEBTi ).

tff(sy_c_ATP_058Lamm__va____,type,
    aTP_Lamm_va: fun_Pr4404027214751104635_VEBTi > fun_na1347330265571094344_VEBTi ).

tff(sy_c_ATP_058Lamm__vb____,type,
    aTP_Lamm_vb: fun_Pr4404027214751104635_VEBTi > fun_na1659118370164772413_VEBTi ).

tff(sy_c_ATP_058Lamm__vc____,type,
    aTP_Lamm_vc: fun_Pr4404027214751104635_VEBTi > fun_na2169200058431587252_VEBTi ).

tff(sy_c_ATP_058Lamm__vd____,type,
    aTP_Lamm_vd: fun_Pr4404027214751104635_VEBTi > fun_na719414919122706795_VEBTi ).

tff(sy_c_ATP_058Lamm__ve____,type,
    aTP_Lamm_ve: fun_Pr4404027214751104635_VEBTi > fun_na9095497340641005739_VEBTi ).

tff(sy_c_ATP_058Lamm__vf____,type,
    aTP_Lamm_vf: fun_Pr4404027214751104635_VEBTi > fun_na4748561171398030626_VEBTi ).

tff(sy_c_ATP_058Lamm__vg____,type,
    aTP_Lamm_vg: fun_Pr4404027214751104635_VEBTi > fun_na817248180520022297_VEBTi ).

tff(sy_c_ATP_058Lamm__vh____,type,
    aTP_Lamm_vh: fun_Pr4404027214751104635_VEBTi > fun_na5699700385611646096_VEBTi ).

tff(sy_c_ATP_058Lamm__vi____,type,
    aTP_Lamm_vi: fun_Pr4404027214751104635_VEBTi > fun_na113775696040589571_VEBTi ).

tff(sy_c_ATP_058Lamm__vj____,type,
    aTP_Lamm_vj: fun_Pr4404027214751104635_VEBTi > fun_na113775696040589571_VEBTi ).

tff(sy_c_ATP_058Lamm__vk____,type,
    aTP_Lamm_vk: fun_Pr4404027214751104635_VEBTi > fun_na653084535925934713_VEBTi ).

tff(sy_c_ATP_058Lamm__vl____,type,
    aTP_Lamm_vl: fun_Pr4404027214751104635_VEBTi > fun_na993102950844727003_VEBTi ).

tff(sy_c_ATP_058Lamm__vm____,type,
    aTP_Lamm_vm: fun_Pr4404027214751104635_VEBTi > fun_na4691356797839745173_VEBTi ).

tff(sy_c_ATP_058Lamm__vn____,type,
    aTP_Lamm_vn: fun_Pr4404027214751104635_VEBTi > fun_na7353129812761597833_VEBTi ).

tff(sy_c_ATP_058Lamm__vo____,type,
    aTP_Lamm_vo: fun_Pr4404027214751104635_VEBTi > fun_VE8810447512154760383_VEBTi ).

tff(sy_c_ATP_058Lamm__vp____,type,
    aTP_Lamm_vp: fun_Pr4404027214751104635_VEBTi > fun_VE91144239014113491_VEBTi ).

tff(sy_c_ATP_058Lamm__vq____,type,
    aTP_Lamm_vq: fun_Pr4404027214751104635_VEBTi > fun_VE728358326801380661_VEBTi ).

tff(sy_c_ATP_058Lamm__vr____,type,
    aTP_Lamm_vr: fun_Pr4404027214751104635_VEBTi > fun_VE2018681615261037117_VEBTi ).

tff(sy_c_ATP_058Lamm__vs____,type,
    aTP_Lamm_vs: produc3960855945107176009Ti_nat > fun_fu6714510707232648287_VEBTi ).

tff(sy_c_ATP_058Lamm__vt____,type,
    aTP_Lamm_vt: fun_Pr633315466666964198on_nat > fun_na7821417838835476043on_nat ).

tff(sy_c_ATP_058Lamm__vu____,type,
    aTP_Lamm_vu: fun_Pr633315466666964198on_nat > fun_na4144732571386696713on_nat ).

tff(sy_c_ATP_058Lamm__vv____,type,
    aTP_Lamm_vv: fun_Pr633315466666964198on_nat > fun_na5199945083401365888on_nat ).

tff(sy_c_ATP_058Lamm__vw____,type,
    aTP_Lamm_vw: fun_Pr633315466666964198on_nat > fun_na4376472396080214220on_nat ).

tff(sy_c_ATP_058Lamm__vx____,type,
    aTP_Lamm_vx: fun_Pr633315466666964198on_nat > fun_na4376472396080214220on_nat ).

tff(sy_c_ATP_058Lamm__vy____,type,
    aTP_Lamm_vy: fun_Pr633315466666964198on_nat > fun_na6803271923218958805on_nat ).

tff(sy_c_ATP_058Lamm__vz____,type,
    aTP_Lamm_vz: fun_Pr633315466666964198on_nat > fun_na6506281770025666025on_nat ).

tff(sy_c_ATP_058Lamm__wa____,type,
    aTP_Lamm_wa: fun_Pr633315466666964198on_nat > fun_na223517484589429856on_nat ).

tff(sy_c_ATP_058Lamm__wb____,type,
    aTP_Lamm_wb: fun_Pr633315466666964198on_nat > fun_na3160842509525366983on_nat ).

tff(sy_c_ATP_058Lamm__wc____,type,
    aTP_Lamm_wc: fun_Pr633315466666964198on_nat > fun_na7284882202938760507on_nat ).

tff(sy_c_ATP_058Lamm__wd____,type,
    aTP_Lamm_wd: fun_Pr633315466666964198on_nat > fun_VE5053670003584447325on_nat ).

tff(sy_c_ATP_058Lamm__we____,type,
    aTP_Lamm_we: fun_Pr633315466666964198on_nat > fun_VE205885719767202089on_nat ).

tff(sy_c_ATP_058Lamm__wf____,type,
    aTP_Lamm_wf: fun_Pr633315466666964198on_nat > fun_VE8661257954422131339on_nat ).

tff(sy_c_ATP_058Lamm__wg____,type,
    aTP_Lamm_wg: fun_Pr633315466666964198on_nat > fun_VE7418359107739301810on_nat ).

tff(sy_c_ATP_058Lamm__wh____,type,
    aTP_Lamm_wh: fun_Pr633315466666964198on_nat > fun_VE8881588537313513120on_nat ).

tff(sy_c_ATP_058Lamm__wi____,type,
    aTP_Lamm_wi: fun_Pr633315466666964198on_nat > fun_VE3775134097089128360on_nat ).

tff(sy_c_ATP_058Lamm__wj____,type,
    aTP_Lamm_wj: produc3960855945107176009Ti_nat > fun_fu8904838144393377183on_nat ).

tff(sy_c_ATP_058Lamm__wk____,type,
    aTP_Lamm_wk: fun_na5556145776640407093on_nat > fun_Pr5956696566359781750on_nat ).

tff(sy_c_ATP_058Lamm__wl____,type,
    aTP_Lamm_wl: fun_na2343073368905879023Heap_o > fun_Pr4877903221429077380Heap_o ).

tff(sy_c_ATP_058Lamm__wm____,type,
    aTP_Lamm_wm: heap_T8145700208782473153_VEBTi > fun_Pr4404027214751104635_VEBTi ).

tff(sy_c_ATP_058Lamm__wn____,type,
    aTP_Lamm_wn: heap_T2636463487746394924on_nat > fun_Pr633315466666964198on_nat ).

tff(sy_c_ATP_058Lamm__wo____,type,
    aTP_Lamm_wo: heap_Time_Heap_o > fun_Pr940977554174145406Heap_o ).

tff(sy_c_ATP_058Lamm__wp____,type,
    aTP_Lamm_wp: complex > fun_complex_o ).

tff(sy_c_ATP_058Lamm__wq____,type,
    aTP_Lamm_wq: fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__wr____,type,
    aTP_Lamm_wr: fun_ui3478071520421546738nt32_o ).

tff(sy_c_ATP_058Lamm__ws____,type,
    aTP_Lamm_ws: word_N3645301735248828278l_num1 > fun_wo7859709073469921915num1_o ).

tff(sy_c_ATP_058Lamm__wt____,type,
    aTP_Lamm_wt: fun_Co2081214437782444048eger_o ).

tff(sy_c_ATP_058Lamm__wu____,type,
    aTP_Lamm_wu: fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__wv____,type,
    aTP_Lamm_wv: fun_Pr633315466666964198on_nat > fun_na607092491672936340on_nat ).

tff(sy_c_ATP_058Lamm__ww____,type,
    aTP_Lamm_ww: fun_Pr633315466666964198on_nat > fun_na4347519868517868204on_nat ).

tff(sy_c_ATP_058Lamm__wx____,type,
    aTP_Lamm_wx: fun_Pr633315466666964198on_nat > fun_na7760356811779829437on_nat ).

tff(sy_c_ATP_058Lamm__wy____,type,
    aTP_Lamm_wy: fun_Pr633315466666964198on_nat > fun_na1922817992914042619on_nat ).

tff(sy_c_ATP_058Lamm__wz____,type,
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tff(sy_c_ATP_058Lamm__xc____,type,
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tff(sy_c_ATP_058Lamm__xd____,type,
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tff(sy_c_ATP_058Lamm__xe____,type,
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    bit_se8570568707652914677it_nat: fun_nat_fun_nat_nat ).

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    bit_se5176125413884933531l_num1: ( nat * word_N3645301735248828278l_num1 ) > word_N3645301735248828278l_num1 ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oflip__bit_001t__Code____Numeral__Ointeger,type,
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tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oflip__bit_001t__Nat__Onat,type,
    bit_se2161824704523386999it_nat: fun_nat_fun_nat_nat ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oflip__bit_001t__Uint32__Ouint32,type,
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tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oflip__bit_001t__Word__Oword_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J_J_J_J_J_J,type,
    bit_se4491814353640558621l_num1: ( nat * word_N3645301735248828278l_num1 ) > word_N3645301735248828278l_num1 ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Omask_001t__Int__Oint,type,
    bit_se2000444600071755411sk_int: fun_nat_int ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Omask_001t__Nat__Onat,type,
    bit_se2002935070580805687sk_nat: fun_nat_nat ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oor_001t__Code____Numeral__Ointeger,type,
    bit_se1080825931792720795nteger: code_integer > fun_Co8499238673658658103nteger ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oor_001t__Int__Oint,type,
    bit_se1409905431419307370or_int: fun_int_fun_int_int ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oor_001t__Nat__Onat,type,
    bit_se1412395901928357646or_nat: fun_nat_fun_nat_nat ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oor_001t__Uint32__Ouint32,type,
    bit_se2966626333419230250uint32: ( uint32 * uint32 ) > uint32 ).

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    bit_se547839408752420682it_nat: fun_nat_fun_nat_nat ).

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tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Otake__bit_001t__Int__Oint,type,
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tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Ounset__bit_001t__Nat__Onat,type,
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tff(sy_c_Bit__Operations_Osemiring__bits__class_Obit_001t__Int__Oint,type,
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tff(sy_c_Bit__Operations_Osemiring__bits__class_Obit_001t__Nat__Onat,type,
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tff(sy_c_Bit__Operations_Osemiring__bits__class_Obit_001t__Uint32__Ouint32,type,
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tff(sy_c_Bit__Operations_Otake__bit__num,type,
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tff(sy_c_Bit__Shifts__Infix__Syntax_Osemiring__bit__operations__class_Oshiftl_001t__Nat__Onat,type,
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tff(sy_c_Bits__Integer_OBit__integer,type,
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tff(sy_c_Bits__Integer_Obin__last__integer,type,
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tff(sy_c_Bits__Integer_Obin__rest__integer,type,
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tff(sy_c_Bits__Integer_Ointeger__set__bit,type,
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tff(sy_c_Code__Numeral_Obit__cut__integer,type,
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tff(sy_c_Code__Numeral_Odivmod__abs,type,
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tff(sy_c_Code__Numeral_Odivmod__integer,type,
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tff(sy_c_Code__Numeral_Odup,type,
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tff(sy_c_Code__Numeral_Ointeger_Oint__of__integer,type,
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tff(sy_c_Code__Numeral_Ointeger_Ointeger__of__int,type,
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tff(sy_c_Code__Numeral_Ointeger__of__nat,type,
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tff(sy_c_Code__Numeral_Onat__of__integer,type,
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tff(sy_c_Code__Numeral_Onegative,type,
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tff(sy_c_Code__Numeral_Onum__of__integer,type,
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tff(sy_c_Code__Numeral_Opositive,type,
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tff(sy_c_Code__Target__Int_Onegative,type,
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tff(sy_c_Code__Target__Word__Base_Oquickcheck__narrowing__samples_Opartial__term__of__sample_001t__Code____Numeral__Ointeger,type,
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tff(sy_c_Code__Target__Word__Base_Oquickcheck__narrowing__samples_Opartial__term__of__sample_001t__Int__Oint,type,
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tff(sy_c_Code__Target__Word__Base_Oquickcheck__narrowing__samples_Opartial__term__of__sample_001t__Nat__Onat,type,
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tff(sy_c_Code__Target__Word__Base_Oquickcheck__narrowing__samples_Opartial__term__of__sample_001t__Uint32__Ouint32,type,
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tff(sy_c_Complete__Partial__Order_Occpo_Oadmissible_001t__fun_It__Product____Type__Oprod_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__VEBT____BuildupMemImp__OVEBTi_J_Mt__Nat__Onat_J_Mt__Heap____Time____Monad__OHeap_I_Eo_J_J,type,
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tff(sy_c_Complete__Partial__Order_Occpo_Oadmissible_001t__fun_It__Product____Type__Oprod_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__VEBT____BuildupMemImp__OVEBTi_J_Mt__Nat__Onat_J_Mt__Heap____Time____Monad__OHeap_It__Option__Ooption_It__Nat__Onat_J_J_J,type,
    comple5664458124825824345on_nat: ( fun_se1756668854628305023on_nat * fun_fu5284765496190745342_nat_o * fun_fu7108391812694488075_nat_o ) > $o ).

tff(sy_c_Complete__Partial__Order_Occpo_Oadmissible_001t__fun_It__Product____Type__Oprod_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__VEBT____BuildupMemImp__OVEBTi_J_Mt__Nat__Onat_J_Mt__Heap____Time____Monad__OHeap_It__VEBT____BuildupMemImp__OVEBTi_J_J,type,
    comple8556566605269262190_VEBTi: ( fun_se3949293115670521129_VEBTi * fun_fu3145514261243071828EBTi_o * fun_fu424623322819101878EBTi_o ) > $o ).

tff(sy_c_Complete__Partial__Order_Occpo_Oadmissible_001t__fun_It__Product____Type__Oprod_It__VEBT____BuildupMemImp__OVEBTi_Mt__Nat__Onat_J_Mt__Heap____Time____Monad__OHeap_I_Eo_J_J,type,
    comple8003709893544916456Heap_o: ( fun_se7475304312357925539Heap_o * fun_fu2183070276626753582ap_o_o * fun_fu9110254849647837056ap_o_o ) > $o ).

tff(sy_c_Complete__Partial__Order_Occpo_Oadmissible_001t__fun_It__Product____Type__Oprod_It__VEBT____BuildupMemImp__OVEBTi_Mt__Nat__Onat_J_Mt__Heap____Time____Monad__OHeap_It__Option__Ooption_It__Nat__Onat_J_J_J,type,
    comple7368800650122905596on_nat: ( fun_se772746884452211909on_nat * fun_fu7962274784768186808_nat_o * fun_fu1312537534749845416_nat_o ) > $o ).

tff(sy_c_Complete__Partial__Order_Occpo_Oadmissible_001t__fun_It__Product____Type__Oprod_It__VEBT____BuildupMemImp__OVEBTi_Mt__Nat__Onat_J_Mt__Heap____Time____Monad__OHeap_It__VEBT____BuildupMemImp__OVEBTi_J_J,type,
    comple7878442886591315089_VEBTi: ( fun_se5764602393054361839_VEBTi * fun_fu175107047954615438EBTi_o * fun_fu8912552214369285843EBTi_o ) > $o ).

tff(sy_c_Complete__Partial__Order_Occpo_Ofixp_001t__fun_It__Product____Type__Oprod_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__VEBT____BuildupMemImp__OVEBTi_J_Mt__Nat__Onat_J_Mt__Heap____Time____Monad__OHeap_I_Eo_J_J,type,
    comple5362527178602408641Heap_o: ( fun_se3420632608416671075Heap_o * fun_fu6850233865993928942ap_o_o * fun_fu8308473583445096899Heap_o ) > fun_Pr940977554174145406Heap_o ).

tff(sy_c_Complete__Partial__Order_Occpo_Ofixp_001t__fun_It__Product____Type__Oprod_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__VEBT____BuildupMemImp__OVEBTi_J_Mt__Nat__Onat_J_Mt__Heap____Time____Monad__OHeap_It__Option__Ooption_It__Nat__Onat_J_J_J,type,
    comple4104493882897929443on_nat: ( fun_se1756668854628305023on_nat * fun_fu5284765496190745342_nat_o * fun_fu3620910142932681on_nat ) > fun_Pr633315466666964198on_nat ).

tff(sy_c_Complete__Partial__Order_Occpo_Ofixp_001t__fun_It__Product____Type__Oprod_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__VEBT____BuildupMemImp__OVEBTi_J_Mt__Nat__Onat_J_Mt__Heap____Time____Monad__OHeap_It__VEBT____BuildupMemImp__OVEBTi_J_J,type,
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tff(sy_c_Complete__Partial__Order_Occpo_Ofixp_001t__fun_It__Product____Type__Oprod_It__VEBT____BuildupMemImp__OVEBTi_Mt__Nat__Onat_J_Mt__Heap____Time____Monad__OHeap_I_Eo_J_J,type,
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tff(sy_c_Complete__Partial__Order_Occpo_Ofixp_001t__fun_It__Product____Type__Oprod_It__VEBT____BuildupMemImp__OVEBTi_Mt__Nat__Onat_J_Mt__Heap____Time____Monad__OHeap_It__Option__Ooption_It__Nat__Onat_J_J_J,type,
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tff(sy_c_Complete__Partial__Order_Omonotone_001t__fun_It__Product____Type__Oprod_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__VEBT____BuildupMemImp__OVEBTi_J_Mt__Nat__Onat_J_Mt__Heap____Time____Monad__OHeap_I_Eo_J_J_001t__Heap____Time____Monad__OHeap_I_Eo_J,type,
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tff(sy_c_Complete__Partial__Order_Omonotone_001t__fun_It__Product____Type__Oprod_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__VEBT____BuildupMemImp__OVEBTi_J_Mt__Nat__Onat_J_Mt__Heap____Time____Monad__OHeap_It__Option__Ooption_It__Nat__Onat_J_J_J_001t__Heap____Time____Monad__OHeap_It__Option__Ooption_It__Nat__Onat_J_J,type,
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tff(sy_c_Complete__Partial__Order_Omonotone_001t__fun_It__Product____Type__Oprod_It__VEBT____BuildupMemImp__OVEBTi_Mt__Nat__Onat_J_Mt__Heap____Time____Monad__OHeap_I_Eo_J_J_001t__Heap____Time____Monad__OHeap_I_Eo_J,type,
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tff(sy_c_Complete__Partial__Order_Omonotone_001t__fun_It__Product____Type__Oprod_It__VEBT____BuildupMemImp__OVEBTi_Mt__Nat__Onat_J_Mt__Heap____Time____Monad__OHeap_I_Eo_J_J_001t__Heap____Time____Monad__OHeap_It__Nat__Onat_J,type,
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tff(sy_c_Complete__Partial__Order_Omonotone_001t__fun_It__Product____Type__Oprod_It__VEBT____BuildupMemImp__OVEBTi_Mt__Nat__Onat_J_Mt__Heap____Time____Monad__OHeap_It__Option__Ooption_It__Nat__Onat_J_J_J_001t__Heap____Time____Monad__OHeap_I_Eo_J,type,
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tff(sy_c_List_Olist__update_001t__Int__Oint,type,
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tff(sy_c_List_Olist__update_001t__Nat__Onat,type,
    list_update_nat: ( list_nat * nat * nat ) > list_nat ).

tff(sy_c_List_Olist__update_001t__Product____Type__Oprod_It__Assertions__Oassn_Mt__Assertions__Oassn_J,type,
    list_u4534839942911652127n_assn: ( list_P8527749157015355191n_assn * nat * produc6575502325842934193n_assn ) > list_P8527749157015355191n_assn ).

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

tff(sy_c_List_Olist__update_001t__VEBT____BuildupMemImp__OVEBTi,type,
    list_u6098035379799741383_VEBTi: ( list_VEBT_VEBTi * nat * vEBT_VEBTi ) > list_VEBT_VEBTi ).

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

tff(sy_c_List_Onth_001_Eo,type,
    nth_o: list_o > fun_nat_o ).

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

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

tff(sy_c_List_Onth_001t__Option__Ooption_It__Nat__Onat_J,type,
    nth_option_nat: list_option_nat > fun_nat_option_nat ).

tff(sy_c_List_Onth_001t__Product____Type__Oprod_It__Assertions__Oassn_Mt__Assertions__Oassn_J,type,
    nth_Pr1769885009046257848n_assn: ( list_P8527749157015355191n_assn * nat ) > produc6575502325842934193n_assn ).

tff(sy_c_List_Onth_001t__Product____Type__Oprod_It__Nat__Onat_M_Eo_J,type,
    nth_Pr112076138515278198_nat_o: ( list_P7333126701944960589_nat_o * nat ) > product_prod_nat_o ).

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

tff(sy_c_List_Onth_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    nth_Pr7617993195940197384at_nat: list_P6011104703257516679at_nat > fun_na1476630412504255114at_nat ).

tff(sy_c_List_Onth_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Real__Oreal_J,type,
    nth_Pr7767817059697521252t_real: ( list_P3644420460460130531t_real * nat ) > produc7716430852924023517t_real ).

tff(sy_c_List_Onth_001t__Product____Type__Oprod_It__Nat__Onat_Mt__VEBT____BuildupMemImp__OVEBTi_J,type,
    nth_Pr3244165891152107629_VEBTi: ( list_P2320588648998582380_VEBTi * nat ) > produc214224863196444774_VEBTi ).

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

tff(sy_c_List_Onth_001t__Product____Type__Oprod_It__VEBT____BuildupMemImp__OVEBTi_M_Eo_J,type,
    nth_Pr3306050735993963089EBTi_o: ( list_P8833571063612306856EBTi_o * nat ) > produc5014006835512566296EBTi_o ).

tff(sy_c_List_Onth_001t__Product____Type__Oprod_It__VEBT____BuildupMemImp__OVEBTi_Mt__Nat__Onat_J,type,
    nth_Pr6911489093701683181Ti_nat: ( list_P659468882601404396Ti_nat * nat ) > produc3881548065746020326Ti_nat ).

tff(sy_c_List_Onth_001t__Product____Type__Oprod_It__VEBT____BuildupMemImp__OVEBTi_Mt__Real__Oreal_J,type,
    nth_Pr3433448822664029129i_real: ( list_P8536626330812492744i_real * nat ) > produc6680258955013199682i_real ).

tff(sy_c_List_Onth_001t__Product____Type__Oprod_It__VEBT____BuildupMemImp__OVEBTi_Mt__VEBT____BuildupMemImp__OVEBTi_J,type,
    nth_Pr6329974346453275474_VEBTi: ( list_P785718909624839377_VEBTi * nat ) > produc3777764054643897931_VEBTi ).

tff(sy_c_List_Onth_001t__Product____Type__Oprod_It__VEBT____BuildupMemImp__OVEBTi_Mt__VEBT____Definitions__OVEBT_J,type,
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tff(sy_c_List_Onth_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_M_Eo_J,type,
    nth_Pr4606735188037164562VEBT_o: ( list_P3126845725202233233VEBT_o * nat ) > produc334124729049499915VEBT_o ).

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

tff(sy_c_List_Onth_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Real__Oreal_J,type,
    nth_Pr6842391030413306568T_real: ( list_P2623026923184700063T_real * nat ) > produc5170161368751668367T_real ).

tff(sy_c_List_Onth_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__VEBT____BuildupMemImp__OVEBTi_J,type,
    nth_Pr316670251186196177_VEBTi: ( list_P735349106241217576_VEBTi * nat ) > produc3625547720036274456_VEBTi ).

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

tff(sy_c_List_Onth_001t__Product____Type__Ounit,type,
    nth_Product_unit: ( list_Product_unit * nat ) > product_unit ).

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

tff(sy_c_List_Onth_001t__VEBT____BuildupMemImp__OVEBTi,type,
    nth_VEBT_VEBTi: ( list_VEBT_VEBTi * nat ) > vEBT_VEBTi ).

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

tff(sy_c_List_Oproduct_001_Eo_001_Eo,type,
    product_o_o: ( list_o * list_o ) > list_P4002435161011370285od_o_o ).

tff(sy_c_List_Oproduct_001_Eo_001t__Int__Oint,type,
    product_o_int: ( list_o * list_int ) > list_P3795440434834930179_o_int ).

tff(sy_c_List_Oproduct_001_Eo_001t__Nat__Onat,type,
    product_o_nat: ( list_o * list_nat ) > list_P6285523579766656935_o_nat ).

tff(sy_c_List_Oproduct_001_Eo_001t__Real__Oreal,type,
    product_o_real: ( list_o * list_real ) > list_P5232166724548748803o_real ).

tff(sy_c_List_Oproduct_001t__Nat__Onat_001_Eo,type,
    product_nat_o: ( list_nat * list_o ) > list_P7333126701944960589_nat_o ).

tff(sy_c_List_Oproduct_001t__Nat__Onat_001t__Real__Oreal,type,
    product_nat_real: ( list_nat * list_real ) > list_P3644420460460130531t_real ).

tff(sy_c_List_Oproduct_001t__Real__Oreal_001_Eo,type,
    product_real_o: ( list_real * list_o ) > list_P3595434254542482545real_o ).

tff(sy_c_List_Oproduct_001t__Real__Oreal_001t__Int__Oint,type,
    product_real_int: ( list_real * list_int ) > list_P4344331454722006975al_int ).

tff(sy_c_List_Oproduct_001t__Real__Oreal_001t__Nat__Onat,type,
    product_real_nat: ( list_real * list_nat ) > list_P6834414599653733731al_nat ).

tff(sy_c_List_Oproduct_001t__Real__Oreal_001t__Real__Oreal,type,
    product_real_real: ( list_real * list_real ) > list_P8689742595348180415l_real ).

tff(sy_c_List_Oproduct_001t__VEBT____BuildupMemImp__OVEBTi_001_Eo,type,
    product_VEBT_VEBTi_o: ( list_VEBT_VEBTi * list_o ) > list_P8833571063612306856EBTi_o ).

tff(sy_c_List_Oproduct_001t__VEBT____BuildupMemImp__OVEBTi_001t__Nat__Onat,type,
    produc2282297823089607884Ti_nat: ( list_VEBT_VEBTi * list_nat ) > list_P659468882601404396Ti_nat ).

tff(sy_c_List_Oproduct_001t__VEBT____BuildupMemImp__OVEBTi_001t__Real__Oreal,type,
    produc5476717833281694120i_real: ( list_VEBT_VEBTi * list_real ) > list_P8536626330812492744i_real ).

tff(sy_c_List_Oproduct_001t__VEBT____BuildupMemImp__OVEBTi_001t__VEBT____BuildupMemImp__OVEBTi,type,
    produc194614972289024177_VEBTi: ( list_VEBT_VEBTi * list_VEBT_VEBTi ) > list_P785718909624839377_VEBTi ).

tff(sy_c_List_Oproduct_001t__VEBT____BuildupMemImp__OVEBTi_001t__VEBT____Definitions__OVEBT,type,
    produc1285381384045549624T_VEBT: ( list_VEBT_VEBTi * list_VEBT_VEBT ) > list_P5988454224134618948T_VEBT ).

tff(sy_c_List_Oproduct_001t__VEBT____Definitions__OVEBT_001_Eo,type,
    product_VEBT_VEBT_o: ( list_VEBT_VEBT * list_o ) > list_P3126845725202233233VEBT_o ).

tff(sy_c_List_Oproduct_001t__VEBT____Definitions__OVEBT_001t__Nat__Onat,type,
    produc7295137177222721919BT_nat: ( list_VEBT_VEBT * list_nat ) > list_P7037539587688870467BT_nat ).

tff(sy_c_List_Oproduct_001t__VEBT____Definitions__OVEBT_001t__Real__Oreal,type,
    produc4908677263432625371T_real: ( list_VEBT_VEBT * list_real ) > list_P2623026923184700063T_real ).

tff(sy_c_List_Oproduct_001t__VEBT____Definitions__OVEBT_001t__VEBT____BuildupMemImp__OVEBTi,type,
    produc316462671093861988_VEBTi: ( list_VEBT_VEBT * list_VEBT_VEBTi ) > list_P735349106241217576_VEBTi ).

tff(sy_c_List_Oproduct_001t__VEBT____Definitions__OVEBT_001t__VEBT____Definitions__OVEBT,type,
    produc4743750530478302277T_VEBT: ( list_VEBT_VEBT * list_VEBT_VEBT ) > list_P7413028617227757229T_VEBT ).

tff(sy_c_List_Oreplicate_001_Eo,type,
    replicate_o: ( nat * $o ) > list_o ).

tff(sy_c_List_Oreplicate_001t__Assertions__Oassn,type,
    replicate_assn: ( nat * assn ) > list_assn ).

tff(sy_c_List_Oreplicate_001t__Code____Numeral__Ointeger,type,
    replic7707675349574490269nteger: ( nat * code_integer ) > list_Code_integer ).

tff(sy_c_List_Oreplicate_001t__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J,type,
    replic5388364368018022029t_unit: ( nat * heap_e7401611519738050253t_unit ) > list_h2745205591237434579t_unit ).

tff(sy_c_List_Oreplicate_001t__Int__Oint,type,
    replicate_int: ( nat * int ) > list_int ).

tff(sy_c_List_Oreplicate_001t__Nat__Onat,type,
    replicate_nat: ( nat * nat ) > list_nat ).

tff(sy_c_List_Oreplicate_001t__Product____Type__Oprod_It__Assertions__Oassn_Mt__Assertions__Oassn_J,type,
    replic3825545231534752113n_assn: ( nat * produc6575502325842934193n_assn ) > list_P8527749157015355191n_assn ).

tff(sy_c_List_Oreplicate_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    replic1057375728873637753nt_int: ( nat * product_prod_int_int ) > list_P5707943133018811711nt_int ).

tff(sy_c_List_Oreplicate_001t__Real__Oreal,type,
    replicate_real: ( nat * real ) > list_real ).

tff(sy_c_List_Oreplicate_001t__Set__Oset_It__Nat__Onat_J,type,
    replicate_set_nat: ( nat * set_nat ) > list_set_nat ).

tff(sy_c_List_Oreplicate_001t__Uint32__Ouint32,type,
    replicate_uint32: ( nat * uint32 ) > list_uint32 ).

tff(sy_c_List_Oreplicate_001t__VEBT____BuildupMemImp__OVEBTi,type,
    replicate_VEBT_VEBTi: ( nat * vEBT_VEBTi ) > list_VEBT_VEBTi ).

tff(sy_c_List_Oreplicate_001t__VEBT____Definitions__OVEBT,type,
    replicate_VEBT_VEBT: ( nat * vEBT_VEBT ) > list_VEBT_VEBT ).

tff(sy_c_List_Oreplicate_001t__fun_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J,type,
    replic8593248471952391807e_term: ( nat * fun_in5730379987311621695e_term ) > list_f628289730585369423e_term ).

tff(sy_c_List_Oupt,type,
    upt: ( nat * nat ) > list_nat ).

tff(sy_c_List_Oupto,type,
    upto: ( int * int ) > list_int ).

tff(sy_c_List_Oupto__aux,type,
    upto_aux: ( int * int * list_int ) > list_int ).

tff(sy_c_List_Oupto__rel,type,
    upto_rel: fun_Pr9018117161326028078_int_o ).

tff(sy_c_Misc_Orel__of_001t__Int__Oint_001t__Int__Oint,type,
    rel_of_int_int: ( fun_int_option_int * fun_Pr3238374608701176162_int_o ) > set_Pr958786334691620121nt_int ).

tff(sy_c_Misc_Orel__of_001t__Nat__Onat_001t__Nat__Onat,type,
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tff(sy_c_Misc_Orel__of_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_Orel__of_001t__fun_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
    rel_of3240586176920340263nteger: ( fun_fu4810197545401214784nteger * fun_Pr7015759342434529115eger_o ) > set_Pr3486312109254157132nteger ).

tff(sy_c_Misc_Orel__of_001t__fun_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_001t__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J,type,
    rel_of7233497478102412281et_nat: ( fun_fu2352890940381120596et_nat * fun_Pr2620521676128474833_nat_o ) > set_Pr1312406793078187350et_nat ).

tff(sy_c_Misc_Orel__of_001t__fun_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_001t__Product____Type__Oprod_It__fun_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_J,type,
    rel_of4573248530116435351et_nat: ( fun_fu1905361174868762994et_nat * fun_Pr1227177680593602867_nat_o ) > set_Pr4040394263813967220et_nat ).

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tff(sy_c_Most__significant__bit_Omsb__class_Omsb_001t__Int__Oint,type,
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tff(sy_c_Most__significant__bit_Omsb__class_Omsb_001t__Uint32__Ouint32,type,
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tff(sy_c_Nat_OSuc,type,
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tff(sy_c_Nat_Onat_Ocase__nat_001_Eo,type,
    case_nat_o: ( $o * fun_nat_o ) > fun_nat_o ).

tff(sy_c_Nat_Onat_Ocase__nat_001t__Nat__Onat,type,
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tff(sy_c_Nat_Onat_Ocase__nat_001t__Option__Ooption_It__Num__Onum_J,type,
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tff(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Complex__Ocomplex,type,
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tff(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Int__Oint,type,
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tff(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Nat__Onat,type,
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tff(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Rat__Orat,type,
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tff(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Real__Oreal,type,
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tff(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Uint32__Ouint32,type,
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tff(sy_c_Nat_Osemiring__1__class_Oof__nat__aux_001t__Nat__Onat,type,
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tff(sy_c_Nat_Osemiring__1__class_Oof__nat__aux_001t__Rat__Orat,type,
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tff(sy_c_Nat_Osize__class_Osize_001t__Heap____Time____Monad__OHeap_I_Eo_J,type,
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tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_I_Eo_J,type,
    size_size_list_o: list_o > nat ).

tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Assertions__Oassn_J,type,
    size_size_list_assn: list_assn > nat ).

tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Code____Numeral__Ointeger_J,type,
    size_s3445333598471063425nteger: list_Code_integer > nat ).

tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Complex__Ocomplex_J,type,
    size_s3451745648224563538omplex: list_complex > nat ).

tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_J,type,
    size_s7011499401410611007t_unit: list_h2745205591237434579t_unit > nat ).

tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Int__Oint_J,type,
    size_size_list_int: list_int > nat ).

tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Nat__Onat_J,type,
    size_size_list_nat: list_nat > nat ).

tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Option__Ooption_It__Nat__Onat_J_J,type,
    size_s6086282163384603972on_nat: list_option_nat > nat ).

tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_I_Eo_M_Eo_J_J,type,
    size_s1515746228057227161od_o_o: list_P4002435161011370285od_o_o > nat ).

tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_I_Eo_Mt__Int__Oint_J_J,type,
    size_s2953683556165314199_o_int: list_P3795440434834930179_o_int > nat ).

tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_I_Eo_Mt__Nat__Onat_J_J,type,
    size_s5443766701097040955_o_nat: list_P6285523579766656935_o_nat > nat ).

tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_I_Eo_Mt__Real__Oreal_J_J,type,
    size_s2624279037499656343o_real: list_P5232166724548748803o_real > nat ).

tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Assertions__Oassn_Mt__Assertions__Oassn_J_J,type,
    size_s6829681357464350627n_assn: list_P8527749157015355191n_assn > nat ).

tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
    size_s5157815400016825771nt_int: list_P5707943133018811711nt_int > nat ).

tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_M_Eo_J_J,type,
    size_s6491369823275344609_nat_o: list_P7333126701944960589_nat_o > nat ).

tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Int__Oint_J_J,type,
    size_s2970893825323803983at_int: list_P3521021558325789923at_int > nat ).

tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    size_s5460976970255530739at_nat: list_P6011104703257516679at_nat > nat ).

tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Real__Oreal_J_J,type,
    size_s7910714270633306959t_real: list_P3644420460460130531t_real > nat ).

tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Real__Oreal_M_Eo_J_J,type,
    size_s987546567493390085real_o: list_P3595434254542482545real_o > nat ).

tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Real__Oreal_Mt__Int__Oint_J_J,type,
    size_s8610625264895183403al_int: list_P4344331454722006975al_int > nat ).

tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Real__Oreal_Mt__Nat__Onat_J_J,type,
    size_s1877336372972134351al_nat: list_P6834414599653733731al_nat > nat ).

tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J_J,type,
    size_s3932428310213730859l_real: list_P8689742595348180415l_real > nat ).

tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Ounit_J,type,
    size_s245203480648594047t_unit: list_Product_unit > nat ).

tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Real__Oreal_J,type,
    size_size_list_real: list_real > nat ).

tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Set__Oset_It__Nat__Onat_J_J,type,
    size_s3254054031482475050et_nat: list_set_nat > nat ).

tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Uint32__Ouint32_J,type,
    size_s4844771616002835472uint32: list_uint32 > nat ).

tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__VEBT____BuildupMemImp__OVEBTi_J,type,
    size_s7982070591426661849_VEBTi: list_VEBT_VEBTi > nat ).

tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__VEBT____Definitions__OVEBT_J,type,
    size_s6755466524823107622T_VEBT: list_VEBT_VEBT > nat ).

tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_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_J,type,
    size_s4003082270698862563e_term: list_f628289730585369423e_term > nat ).

tff(sy_c_Nat_Osize__class_Osize_001t__Num__Onum,type,
    size_size_num: fun_num_nat ).

tff(sy_c_Nat_Osize__class_Osize_001t__Option__Ooption_It__Nat__Onat_J,type,
    size_size_option_nat: option_nat > nat ).

tff(sy_c_Nat_Osize__class_Osize_001t__Option__Ooption_It__Num__Onum_J,type,
    size_size_option_num: option_num > nat ).

tff(sy_c_Nat_Osize__class_Osize_001t__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J_J,type,
    size_s364044314319911927it_nat: option7339022715339332451it_nat > nat ).

tff(sy_c_Nat_Osize__class_Osize_001t__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    size_s170228958280169651at_nat: option4927543243414619207at_nat > nat ).

tff(sy_c_Nat_Osize__class_Osize_001t__Option__Ooption_It__Product____Type__Oprod_It__Option__Ooption_It__Nat__Onat_J_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J_J,type,
    size_s3991424295186984831it_nat: option2621746655072343315it_nat > nat ).

tff(sy_c_Nat_Osize__class_Osize_001t__String__Ochar,type,
    size_size_char: char > nat ).

tff(sy_c_Nat_Osize__class_Osize_001t__Uint32__Ouint32,type,
    size_size_uint32: uint32 > nat ).

tff(sy_c_Nat_Osize__class_Osize_001t__VEBT____BuildupMemImp__OVEBTi,type,
    size_size_VEBT_VEBTi: fun_VEBT_VEBTi_nat ).

tff(sy_c_Nat_Osize__class_Osize_001t__VEBT____Definitions__OVEBT,type,
    size_size_VEBT_VEBT: fun_VEBT_VEBT_nat ).

tff(sy_c_Nat_Osize__class_Osize_001t__Word__Oword_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J_J_J_J_J_J,type,
    size_s8261804613246490634l_num1: word_N3645301735248828278l_num1 > nat ).

tff(sy_c_Nat__Bijection_Oset__decode,type,
    nat_set_decode: nat > set_nat ).

tff(sy_c_Nat__Bijection_Oset__encode,type,
    nat_set_encode: fun_set_nat_nat ).

tff(sy_c_Nat__Bijection_Otriangle,type,
    nat_triangle: fun_nat_nat ).

tff(sy_c_NthRoot_Oroot,type,
    root: nat > fun_real_real ).

tff(sy_c_NthRoot_Osqrt,type,
    sqrt: fun_real_real ).

tff(sy_c_Num_OBitM,type,
    bitM: fun_num_num ).

tff(sy_c_Num_Oinc,type,
    inc: fun_num_num ).

tff(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Int__Oint,type,
    neg_nu5851722552734809277nc_int: fun_int_int ).

tff(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Rat__Orat,type,
    neg_nu5219082963157363817nc_rat: rat > rat ).

tff(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Real__Oreal,type,
    neg_nu8295874005876285629c_real: fun_real_real ).

tff(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Uint32__Ouint32,type,
    neg_nu4269007558841261821uint32: uint32 > uint32 ).

tff(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Word__Oword_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J_J_J_J_J_J,type,
    neg_nu8115118780965096967l_num1: word_N3645301735248828278l_num1 > word_N3645301735248828278l_num1 ).

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_Osize__num,type,
    size_num: fun_num_nat ).

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__Complex__Ocomplex,type,
    numera6690914467698888265omplex: num > complex ).

tff(sy_c_Num_Onumeral__class_Onumeral_001t__Extended____Nat__Oenat,type,
    numera1916890842035813515d_enat: num > extended_enat ).

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: num > rat ).

tff(sy_c_Num_Onumeral__class_Onumeral_001t__Real__Oreal,type,
    numeral_numeral_real: num > real ).

tff(sy_c_Num_Onumeral__class_Onumeral_001t__Uint32__Ouint32,type,
    numera9087168376688890119uint32: num > uint32 ).

tff(sy_c_Num_Onumeral__class_Onumeral_001t__Word__Oword_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J_J_J_J_J_J,type,
    numera7442385471795722001l_num1: num > word_N3645301735248828278l_num1 ).

tff(sy_c_Num_Opow,type,
    pow: fun_num_fun_num_num ).

tff(sy_c_Num_Opred__numeral,type,
    pred_numeral: fun_num_nat ).

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_I_Eo_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J,type,
    none_P7668321371905463026it_nat: option7339022715339332451it_nat ).

tff(sy_c_Option_Ooption_ONone_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    none_P5556105721700978146at_nat: option4927543243414619207at_nat ).

tff(sy_c_Option_Ooption_ONone_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J,type,
    none_P281974696781278558it_nat: option7211493094183709123it_nat ).

tff(sy_c_Option_Ooption_ONone_001t__Product____Type__Oprod_It__Option__Ooption_It__Nat__Onat_J_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J,type,
    none_P1551326421579882414it_nat: option2621746655072343315it_nat ).

tff(sy_c_Option_Ooption_ONone_001t__Product____Type__Oprod_It__VEBT____BuildupMemImp__OVEBTi_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J,type,
    none_P7832717587476222275it_nat: option5408194888911472936it_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,
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tff(sy_c_Option_Ooption_ONone_001t__Product____Type__Oprod_It__fun_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J,type,
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tff(sy_c_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,
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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,
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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,
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tff(sy_c_Option_Ooption_OSome_001t__Assertions__Oassn,type,
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tff(sy_c_Option_Ooption_OSome_001t__Int__Oint,type,
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tff(sy_c_Option_Ooption_OSome_001t__Nat__Onat,type,
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tff(sy_c_Option_Ooption_OSome_001t__Num__Onum,type,
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tff(sy_c_Option_Ooption_OSome_001t__Product____Type__Oprod_I_Eo_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J,type,
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tff(sy_c_Option_Ooption_OSome_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
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tff(sy_c_Option_Ooption_OSome_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
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tff(sy_c_Option_Ooption_OSome_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
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tff(sy_c_Option_Ooption_OSome_001t__Product____Type__Oprod_It__Option__Ooption_It__Nat__Onat_J_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J,type,
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tff(sy_c_Option_Ooption_OSome_001t__Product____Type__Oprod_It__VEBT____BuildupMemImp__OVEBTi_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J,type,
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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,
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tff(sy_c_Option_Ooption_OSome_001t__Product____Type__Oprod_It__fun_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J,type,
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tff(sy_c_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,
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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,
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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__Rat__Orat,type,
    some_rat: rat > option_rat ).

tff(sy_c_Option_Ooption_OSome_001t__Real__Oreal,type,
    some_real: real > option_real ).

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__Real__Oreal_J,type,
    some_set_real: set_real > option_set_real ).

tff(sy_c_Option_Ooption_OSome_001t__Set__Oset_It__String__Ochar_J,type,
    some_set_char: set_char > option_set_char ).

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 ) > fun_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_I_Eo_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J,type,
    case_o1358941076187788256it_nat: ( $o * fun_Pr3989439477023515934_nat_o ) > fun_op7459662026604166478_nat_o ).

tff(sy_c_Option_Ooption_Ocase__option_001_Eo_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    case_o184042715313410164at_nat: ( $o * fun_Pr4796943543331071514_nat_o ) > fun_op8028897214545316308_nat_o ).

tff(sy_c_Option_Ooption_Ocase__option_001_Eo_001t__Product____Type__Oprod_It__Option__Ooption_It__Nat__Onat_J_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J,type,
    case_o535201446637900608it_nat: ( $o * fun_Pr7755520105644788814_nat_o ) > fun_op2107615217426680968_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__Real__Oreal,type,
    case_option_o_real: ( $o * fun_real_o * option_real ) > $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__Heap____Time____Monad__OHeap_I_Eo_J_001t__Nat__Onat,type,
    case_o6892868863119666303_o_nat: ( heap_Time_Heap_o * fun_na2343073368905879023Heap_o ) > fun_op3694375823608013759Heap_o ).

tff(sy_c_Option_Ooption_Ocase__option_001t__Heap____Time____Monad__OHeap_I_Eo_J_001t__Num__Onum,type,
    case_o3450200649275444937_o_num: ( heap_Time_Heap_o * fun_nu9127643932893349689Heap_o * option_num ) > heap_Time_Heap_o ).

tff(sy_c_Option_Ooption_Ocase__option_001t__Heap____Time____Monad__OHeap_I_Eo_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    case_o1442776274061689234at_nat: ( heap_Time_Heap_o * fun_Pr1329037631615965750Heap_o * option4927543243414619207at_nat ) > heap_Time_Heap_o ).

tff(sy_c_Option_Ooption_Ocase__option_001t__Heap____Time____Monad__OHeap_It__Nat__Onat_J_001t__Nat__Onat,type,
    case_o6609685678014844897at_nat: ( heap_Time_Heap_nat * fun_na3533371890617757541ap_nat * option_nat ) > heap_Time_Heap_nat ).

tff(sy_c_Option_Ooption_Ocase__option_001t__Heap____Time____Monad__OHeap_It__Nat__Onat_J_001t__Num__Onum,type,
    case_o3167017464170623531at_num: ( heap_Time_Heap_nat * fun_nu8176864466684091547ap_nat * option_num ) > heap_Time_Heap_nat ).

tff(sy_c_Option_Ooption_Ocase__option_001t__Heap____Time____Monad__OHeap_It__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    case_o3959993630158478256at_nat: ( heap_Time_Heap_nat * fun_Pr7768898707182944990ap_nat * option4927543243414619207at_nat ) > heap_Time_Heap_nat ).

tff(sy_c_Option_Ooption_Ocase__option_001t__Heap____Time____Monad__OHeap_It__Option__Ooption_It__Nat__Onat_J_J_001t__Nat__Onat,type,
    case_o2256915875499652529at_nat: ( heap_T2636463487746394924on_nat * fun_na5556145776640407093on_nat * option_nat ) > heap_T2636463487746394924on_nat ).

tff(sy_c_Option_Ooption_Ocase__option_001t__Heap____Time____Monad__OHeap_It__Option__Ooption_It__Nat__Onat_J_J_001t__Num__Onum,type,
    case_o8037619698510206971at_num: ( heap_T2636463487746394924on_nat * fun_nu1337165112811199083on_nat * option_num ) > heap_T2636463487746394924on_nat ).

tff(sy_c_Option_Ooption_Ocase__option_001t__Heap____Time____Monad__OHeap_It__Option__Ooption_It__Nat__Onat_J_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    case_o8344607093967974880at_nat: ( heap_T2636463487746394924on_nat * fun_Pr5461209604930428974on_nat * option4927543243414619207at_nat ) > heap_T2636463487746394924on_nat ).

tff(sy_c_Option_Ooption_Ocase__option_001t__Heap____Time____Monad__OHeap_It__Product____Type__Ounit_J_001t__Nat__Onat,type,
    case_o8044209130310455284it_nat: ( heap_T5738788834812785303t_unit * fun_na5725305022723494372t_unit * option_nat ) > heap_T5738788834812785303t_unit ).

tff(sy_c_Option_Ooption_Ocase__option_001t__Heap____Time____Monad__OHeap_It__Product____Type__Ounit_J_001t__Num__Onum,type,
    case_o4601540916466233918it_num: ( heap_T5738788834812785303t_unit * fun_nu1618770902802125870t_unit * option_num ) > heap_T5738788834812785303t_unit ).

tff(sy_c_Option_Ooption_Ocase__option_001t__Heap____Time____Monad__OHeap_It__VEBT____BuildupMemImp__OVEBTi_J_001t__Nat__Onat,type,
    case_o3780387683879180358Ti_nat: ( heap_T8145700208782473153_VEBTi * fun_na7848093160509028170_VEBTi ) > fun_op1509349959255590266_VEBTi ).

tff(sy_c_Option_Ooption_Ocase__option_001t__Heap____Time____Monad__OHeap_It__VEBT____BuildupMemImp__OVEBTi_J_001t__Num__Onum,type,
    case_o337719470034958992Ti_num: ( heap_T8145700208782473153_VEBTi * fun_nu5987414290159654528_VEBTi * option_num ) > heap_T8145700208782473153_VEBTi ).

tff(sy_c_Option_Ooption_Ocase__option_001t__Heap____Time____Monad__OHeap_It__VEBT____BuildupMemImp__OVEBTi_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    case_o1356590567247012107at_nat: ( heap_T8145700208782473153_VEBTi * fun_Pr45531988981583299_VEBTi * option4927543243414619207at_nat ) > heap_T8145700208782473153_VEBTi ).

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__Nat__Onat,type,
    case_option_nat_nat: ( nat * fun_nat_nat * option_nat ) > nat ).

tff(sy_c_Option_Ooption_Ocase__option_001t__Nat__Onat_001t__Num__Onum,type,
    case_option_nat_num: ( nat * fun_num_nat * option_num ) > nat ).

tff(sy_c_Option_Ooption_Ocase__option_001t__Nat__Onat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    case_o2098746482150326116at_nat: ( nat * fun_Pr2602117396471556196at_nat * option4927543243414619207at_nat ) > 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__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__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    case_o7430979018509204427at_nat: ( product_prod_nat_nat * fun_Pr2118491707810114563at_nat * option4927543243414619207at_nat ) > product_prod_nat_nat ).

tff(sy_c_Option_Ooption_Ocase__option_001t__fun_It__fun_It__Nat__Onat_Mt__fun_It__Uint32__Ouint32_Mt__Uint32__Ouint32_J_J_Mt__fun_It__Code____Numeral__Ointeger_Mt__fun_It__Uint32__Ouint32_Mt__Uint32__Ouint32_J_J_J_001t__fun_It__fun_It__Nat__Onat_Mt__fun_It__Uint32__Ouint32_Mt__Uint32__Ouint32_J_J_Mt__fun_It__Code____Numeral__Ointeger_Mt__fun_It__Uint32__Ouint32_Mt__Uint32__Ouint32_J_J_J,type,
    case_o5757947405571428393uint32: ( fun_fu1303640425391265102uint32 * fun_fu713059014528672409uint32 * option3426696969397718942uint32 ) > fun_fu1303640425391265102uint32 ).

tff(sy_c_Option_Ooption_Ocase__option_001t__fun_It__fun_It__Nat__Onat_Mt__fun_It__Uint32__Ouint32_Mt__Uint32__Ouint32_J_J_Mt__fun_It__Uint32__Ouint32_Mt__fun_It__Code____Numeral__Ointeger_Mt__Uint32__Ouint32_J_J_J_001t__fun_It__fun_It__Nat__Onat_Mt__fun_It__Uint32__Ouint32_Mt__Uint32__Ouint32_J_J_Mt__fun_It__Uint32__Ouint32_Mt__fun_It__Code____Numeral__Ointeger_Mt__Uint32__Ouint32_J_J_J,type,
    case_o3262472310887385129uint32: ( fun_fu5030138910357066062uint32 * fun_fu5055813439476669081uint32 * option4768052937140784030uint32 ) > fun_fu5030138910357066062uint32 ).

tff(sy_c_Option_Ooption_Ocase__option_001t__fun_It__fun_It__Uint32__Ouint32_Mt__fun_It__Nat__Onat_M_Eo_J_J_Mt__fun_It__Uint32__Ouint32_Mt__fun_It__Code____Numeral__Ointeger_M_Eo_J_J_J_001t__fun_It__fun_It__Uint32__Ouint32_Mt__fun_It__Nat__Onat_M_Eo_J_J_Mt__fun_It__Uint32__Ouint32_Mt__fun_It__Code____Numeral__Ointeger_M_Eo_J_J_J,type,
    case_o2694340534094624761eger_o: ( fun_fu3466578569608634614eger_o * fun_fu8568212769178095465eger_o * option3423447615326530758eger_o ) > fun_fu3466578569608634614eger_o ).

tff(sy_c_Option_Ooption_Ocase__option_001t__fun_It__fun_It__Uint32__Ouint32_Mt__fun_It__Nat__Onat_Mt__fun_I_Eo_Mt__Uint32__Ouint32_J_J_J_Mt__fun_It__Uint32__Ouint32_Mt__fun_It__Code____Numeral__Ointeger_Mt__fun_I_Eo_Mt__Uint32__Ouint32_J_J_J_J_001t__fun_It__fun_It__Uint32__Ouint32_Mt__fun_It__Nat__Onat_Mt__fun_I_Eo_Mt__Uint32__Ouint32_J_J_J_Mt__fun_It__Uint32__Ouint32_Mt__fun_It__Code____Numeral__Ointeger_Mt__fun_I_Eo_Mt__Uint32__Ouint32_J_J_J_J,type,
    case_o61842737470731385uint32: ( fun_fu8139443716560175030uint32 * fun_fu5407763515825486313uint32 * option5345528073251348614uint32 ) > fun_fu8139443716560175030uint32 ).

tff(sy_c_Option_Ooption_Ocase__option_001t__fun_It__fun_It__Uint32__Ouint32_Mt__fun_It__Uint32__Ouint32_Mt__Uint32__Ouint32_J_J_Mt__fun_It__Uint32__Ouint32_Mt__fun_It__Uint32__Ouint32_Mt__Uint32__Ouint32_J_J_J_001t__fun_It__fun_It__Uint32__Ouint32_Mt__fun_It__Uint32__Ouint32_Mt__Uint32__Ouint32_J_J_Mt__fun_It__Uint32__Ouint32_Mt__fun_It__Uint32__Ouint32_Mt__Uint32__Ouint32_J_J_J,type,
    case_o1863524157202330495uint32: ( fun_fu6749172196426063865uint32 * fun_fu6311994054701583983uint32 * option6094739215472523721uint32 ) > fun_fu6749172196426063865uint32 ).

tff(sy_c_Option_Ooption_Othe_001t__Nat__Onat,type,
    the_nat: option_nat > nat ).

tff(sy_c_Option_Ooption_Othe_001t__Num__Onum,type,
    the_num: fun_option_num_num ).

tff(sy_c_Option_Ooption_Othe_001t__Product____Type__Oprod_I_Eo_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J,type,
    the_Pr5838048819577852031it_nat: option7339022715339332451it_nat > produc8047831477865546771it_nat ).

tff(sy_c_Option_Ooption_Othe_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    the_Pr8591224930841456533at_nat: option4927543243414619207at_nat > product_prod_nat_nat ).

tff(sy_c_Option_Ooption_Othe_001t__Product____Type__Oprod_It__Option__Ooption_It__Nat__Onat_J_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J,type,
    the_Pr3501439614016493281it_nat: option2621746655072343315it_nat > produc120671012495760973it_nat ).

tff(sy_c_Option_Ooption_Othe_001t__fun_It__fun_It__Nat__Onat_Mt__fun_It__Uint32__Ouint32_Mt__Uint32__Ouint32_J_J_Mt__fun_It__Code____Numeral__Ointeger_Mt__fun_It__Uint32__Ouint32_Mt__Uint32__Ouint32_J_J_J,type,
    the_fu1630297056608640954uint32: option3426696969397718942uint32 > fun_fu1303640425391265102uint32 ).

tff(sy_c_Option_Ooption_Othe_001t__fun_It__fun_It__Nat__Onat_Mt__fun_It__Uint32__Ouint32_Mt__Uint32__Ouint32_J_J_Mt__fun_It__Uint32__Ouint32_Mt__fun_It__Code____Numeral__Ointeger_Mt__Uint32__Ouint32_J_J_J,type,
    the_fu5356795541574441914uint32: option4768052937140784030uint32 > fun_fu5030138910357066062uint32 ).

tff(sy_c_Option_Ooption_Othe_001t__fun_It__fun_It__Uint32__Ouint32_Mt__fun_It__Nat__Onat_M_Eo_J_J_Mt__fun_It__Uint32__Ouint32_Mt__fun_It__Code____Numeral__Ointeger_M_Eo_J_J_J,type,
    the_fu7763965076112332130eger_o: option3423447615326530758eger_o > fun_fu3466578569608634614eger_o ).

tff(sy_c_Option_Ooption_Othe_001t__fun_It__fun_It__Uint32__Ouint32_Mt__fun_It__Nat__Onat_Mt__fun_I_Eo_Mt__Uint32__Ouint32_J_J_J_Mt__fun_It__Uint32__Ouint32_Mt__fun_It__Code____Numeral__Ointeger_Mt__fun_I_Eo_Mt__Uint32__Ouint32_J_J_J_J,type,
    the_fu4420311046302572578uint32: option5345528073251348614uint32 > fun_fu8139443716560175030uint32 ).

tff(sy_c_Option_Ooption_Othe_001t__fun_It__fun_It__Uint32__Ouint32_Mt__fun_It__Uint32__Ouint32_Mt__Uint32__Ouint32_J_J_Mt__fun_It__Uint32__Ouint32_Mt__fun_It__Uint32__Ouint32_Mt__Uint32__Ouint32_J_J_J,type,
    the_fu3257833805496833637uint32: option6094739215472523721uint32 > fun_fu6749172196426063865uint32 ).

tff(sy_c_Orderings_Obot__class_Obot_001t__Assertions__Oassn,type,
    bot_bot_assn: assn ).

tff(sy_c_Orderings_Obot__class_Obot_001t__Nat__Onat,type,
    bot_bot_nat: nat ).

tff(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Code____Numeral__Ointeger_J,type,
    bot_bo3990330152332043303nteger: set_Code_integer ).

tff(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Complex__Ocomplex_J,type,
    bot_bot_set_complex: set_complex ).

tff(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Heap____Time____Monad__OHeap_I_Eo_J_J,type,
    bot_bo3236126332025433324Heap_o: set_Heap_Time_Heap_o ).

tff(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Heap____Time____Monad__OHeap_It__Option__Ooption_It__Nat__Onat_J_J_J,type,
    bot_bo8932748503833948152on_nat: set_He5367250461312314764on_nat ).

tff(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Heap____Time____Monad__OHeap_It__VEBT____BuildupMemImp__OVEBTi_J_J,type,
    bot_bo3125955617464001165_VEBTi: set_He5684063546058238497_VEBTi ).

tff(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Int__Oint_J,type,
    bot_bot_set_int: set_int ).

tff(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Nat__Onat_J,type,
    bot_bot_set_nat: set_nat ).

tff(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Num__Onum_J,type,
    bot_bot_set_num: set_num ).

tff(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Product____Type__Oprod_It__Assertions__Oassn_Mt__Assertions__Oassn_J_J,type,
    bot_bo1176836662018730877n_assn: set_Pr5949110396991348497n_assn ).

tff(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
    bot_bo1796632182523588997nt_int: set_Pr958786334691620121nt_int ).

tff(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    bot_bo2099793752762293965at_nat: set_Pr1261947904930325089at_nat ).

tff(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Rat__Orat_J,type,
    bot_bot_set_rat: set_rat ).

tff(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Real__Oreal_J,type,
    bot_bot_set_real: set_real ).

tff(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__Int__Oint_J_J,type,
    bot_bot_set_set_int: set_set_int ).

tff(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__String__Ochar_J,type,
    bot_bot_set_char: set_char ).

tff(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__VEBT____Definitions__OVEBT_J,type,
    bot_bo8194388402131092736T_VEBT: set_VEBT_VEBT ).

tff(sy_c_Orderings_Oord__class_OLeast_001t__Nat__Onat,type,
    ord_Least_nat: fun_nat_o > nat ).

tff(sy_c_Orderings_Oord__class_Oless_001t__Code____Numeral__Ointeger,type,
    ord_le6747313008572928689nteger: fun_Co2081214437782444048eger_o ).

tff(sy_c_Orderings_Oord__class_Oless_001t__Extended____Nat__Oenat,type,
    ord_le72135733267957522d_enat: ( extended_enat * extended_enat ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless_001t__Int__Oint,type,
    ord_less_int: fun_int_fun_int_o ).

tff(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
    ord_less_nat: fun_nat_fun_nat_o ).

tff(sy_c_Orderings_Oord__class_Oless_001t__Num__Onum,type,
    ord_less_num: num > fun_num_o ).

tff(sy_c_Orderings_Oord__class_Oless_001t__Option__Ooption_It__Int__Oint_J,type,
    ord_less_option_int: ( option_int * option_int ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless_001t__Option__Ooption_It__Nat__Onat_J,type,
    ord_less_option_nat: fun_op5794009454290973322_nat_o ).

tff(sy_c_Orderings_Oord__class_Oless_001t__Option__Ooption_It__Num__Onum_J,type,
    ord_less_option_num: fun_op493924057549976310_num_o ).

tff(sy_c_Orderings_Oord__class_Oless_001t__Option__Ooption_It__Rat__Orat_J,type,
    ord_less_option_rat: ( option_rat * option_rat ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless_001t__Option__Ooption_It__Real__Oreal_J,type,
    ord_less_option_real: ( option_real * option_real ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless_001t__Rat__Orat,type,
    ord_less_rat: fun_rat_fun_rat_o ).

tff(sy_c_Orderings_Oord__class_Oless_001t__Real__Oreal,type,
    ord_less_real: fun_real_fun_real_o ).

tff(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Code____Numeral__Ointeger_J,type,
    ord_le1307284697595431911nteger: set_Code_integer > fun_se7289222521248229150eger_o ).

tff(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Complex__Ocomplex_J,type,
    ord_less_set_complex: set_complex > fun_set_complex_o ).

tff(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Int__Oint_J,type,
    ord_less_set_int: set_int > fun_set_int_o ).

tff(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Nat__Onat_J,type,
    ord_less_set_nat: set_nat > fun_set_nat_o ).

tff(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Num__Onum_J,type,
    ord_less_set_num: set_num > fun_set_num_o ).

tff(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Rat__Orat_J,type,
    ord_less_set_rat: set_rat > fun_set_rat_o ).

tff(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Real__Oreal_J,type,
    ord_less_set_real: set_real > fun_set_real_o ).

tff(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Set__Oset_It__Int__Oint_J_J,type,
    ord_less_set_set_int: ( set_set_int * set_set_int ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Uint32__Ouint32_J,type,
    ord_less_set_uint32: ( set_uint32 * set_uint32 ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__VEBT____Definitions__OVEBT_J,type,
    ord_le3480810397992357184T_VEBT: set_VEBT_VEBT > fun_set_VEBT_VEBT_o ).

tff(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Word__Oword_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J_J_J_J_J_J_J,type,
    ord_le6726900395242856064l_num1: ( set_wo3913738467083021356l_num1 * set_wo3913738467083021356l_num1 ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless_001t__String__Ochar,type,
    ord_less_char: char > fun_char_o ).

tff(sy_c_Orderings_Oord__class_Oless_001t__Uint32__Ouint32,type,
    ord_less_uint32: fun_ui3478071520421546738nt32_o ).

tff(sy_c_Orderings_Oord__class_Oless_001t__Word__Oword_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J_J_J_J_J_J,type,
    ord_le750835935415966154l_num1: word_N3645301735248828278l_num1 > fun_wo7859709073469921915num1_o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Code____Numeral__Ointeger,type,
    ord_le3102999989581377725nteger: fun_Co2081214437782444048eger_o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Extended____Nat__Oenat,type,
    ord_le2932123472753598470d_enat: ( extended_enat * extended_enat ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Int__Oint,type,
    ord_less_eq_int: fun_int_fun_int_o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
    ord_less_eq_nat: fun_nat_fun_nat_o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Num__Onum,type,
    ord_less_eq_num: num > fun_num_o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Option__Ooption_It__Int__Oint_J,type,
    ord_le1736525451366464988on_int: ( option_int * option_int ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Option__Ooption_It__Nat__Onat_J,type,
    ord_le5914376470875661696on_nat: fun_op5794009454290973322_nat_o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Option__Ooption_It__Num__Onum_J,type,
    ord_le6622620407824499402on_num: fun_op493924057549976310_num_o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Option__Ooption_It__Rat__Orat_J,type,
    ord_le2406147912482264968on_rat: ( option_rat * option_rat ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Option__Ooption_It__Set__Oset_It__Int__Oint_J_J,type,
    ord_le353528952715127954et_int: ( option_set_int * option_set_int ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Rat__Orat,type,
    ord_less_eq_rat: fun_rat_fun_rat_o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Real__Oreal,type,
    ord_less_eq_real: fun_real_fun_real_o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_Eo_J,type,
    ord_less_eq_set_o: ( set_o * set_o ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Code____Numeral__Ointeger_J,type,
    ord_le7084787975880047091nteger: set_Code_integer > fun_se7289222521248229150eger_o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Complex__Ocomplex_J,type,
    ord_le211207098394363844omplex: set_complex > fun_set_complex_o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Int__Oint_J,type,
    ord_less_eq_set_int: set_int > fun_set_int_o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
    ord_less_eq_set_nat: set_nat > fun_set_nat_o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Num__Onum_J,type,
    ord_less_eq_set_num: set_num > fun_set_num_o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_It__Assertions__Oassn_Mt__Assertions__Oassn_J_J,type,
    ord_le171416862856029873n_assn: ( set_Pr5949110396991348497n_assn * set_Pr5949110396991348497n_assn ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
    ord_le2843351958646193337nt_int: ( set_Pr958786334691620121nt_int * set_Pr958786334691620121nt_int ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    ord_le3146513528884898305at_nat: ( set_Pr1261947904930325089at_nat * set_Pr1261947904930325089at_nat ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_It__fun_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_J,type,
    ord_le3197784583289274309nt_int: ( set_Pr4166680992137330533nt_int * set_Pr4166680992137330533nt_int ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_It__fun_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J_J,type,
    ord_le2932375828722672556nteger: ( set_Pr3486312109254157132nteger * set_Pr3486312109254157132nteger ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_It__fun_It__Product____Type__Oprod_It__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,
    ord_le3919382008627836342et_nat: ( set_Pr1312406793078187350et_nat * set_Pr1312406793078187350et_nat ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__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_J,type,
    ord_le5189643068899110356et_nat: ( set_Pr4040394263813967220et_nat * set_Pr4040394263813967220et_nat ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_It__fun_It__Product____Type__Oprod_It__Int__Oint_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_J,type,
    ord_le3990897393281686169nt_int: ( set_Pr6700895561065273913nt_int * set_Pr6700895561065273913nt_int ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Rat__Orat_J,type,
    ord_less_eq_set_rat: set_rat > fun_set_rat_o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Real__Oreal_J,type,
    ord_less_eq_set_real: set_real > fun_set_real_o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Int__Oint_J_J,type,
    ord_le4403425263959731960et_int: ( set_set_int * set_set_int ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__String__Ochar_J,type,
    ord_less_eq_set_char: ( set_char * set_char ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Uint32__Ouint32_J,type,
    ord_le2219237028632753026uint32: ( set_uint32 * set_uint32 ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__VEBT____BuildupMemImp__OVEBTi_J,type,
    ord_le6592769550269828683_VEBTi: ( set_VEBT_VEBTi * set_VEBT_VEBTi ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__VEBT____Definitions__OVEBT_J,type,
    ord_le4337996190870823476T_VEBT: set_VEBT_VEBT > fun_set_VEBT_VEBT_o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Word__Oword_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J_J_J_J_J_J_J,type,
    ord_le5203802739334966412l_num1: ( set_wo3913738467083021356l_num1 * set_wo3913738467083021356l_num1 ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__String__Ochar,type,
    ord_less_eq_char: char > fun_char_o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Uint32__Ouint32,type,
    ord_less_eq_uint32: fun_ui3478071520421546738nt32_o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Word__Oword_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J_J_J_J_J_J,type,
    ord_le3335648743751981014l_num1: word_N3645301735248828278l_num1 > fun_wo7859709073469921915num1_o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__fun_It__Int__Oint_M_Eo_J,type,
    ord_le5418975059635765733_int_o: ( fun_int_o * fun_int_o ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__fun_It__Int__Oint_Mt__fun_It__Int__Oint_M_Eo_J_J,type,
    ord_le77980682512674322_int_o: ( fun_int_fun_int_o * fun_int_fun_int_o ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__fun_It__Nat__Onat_M_Eo_J,type,
    ord_le7240017405068453697_nat_o: ( fun_nat_o * fun_nat_o ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__fun_It__Nat__Onat_Mt__fun_It__Nat__Onat_M_Eo_J_J,type,
    ord_le768339454706049738_nat_o: ( fun_nat_fun_nat_o * fun_nat_fun_nat_o ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__fun_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_M_Eo_J,type,
    ord_le14110778710936002_int_o: ( fun_Pr3238374608701176162_int_o * fun_Pr3238374608701176162_int_o ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__fun_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_Mt__fun_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_M_Eo_J_J,type,
    ord_le1416746762259324110_int_o: ( fun_Pr9018117161326028078_int_o * fun_Pr9018117161326028078_int_o ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__fun_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J,type,
    ord_le1572679713340831354_nat_o: ( fun_Pr4796943543331071514_nat_o * fun_Pr4796943543331071514_nat_o ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__fun_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__fun_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J_J,type,
    ord_le3650406611588909774_nat_o: ( fun_Pr2028404973800837934_nat_o * fun_Pr2028404973800837934_nat_o ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__fun_It__Product____Type__Oprod_It__Num__Onum_Mt__Num__Onum_J_M_Eo_J,type,
    ord_le3107050023622517222_num_o: ( fun_Pr6331313853612757382_num_o * fun_Pr6331313853612757382_num_o ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__fun_It__Product____Type__Oprod_It__Num__Onum_Mt__Num__Onum_J_Mt__fun_It__Product____Type__Oprod_It__Num__Onum_Mt__Num__Onum_J_M_Eo_J_J,type,
    ord_le6421466325430216142_num_o: ( fun_Pr4799464687642144302_num_o * fun_Pr4799464687642144302_num_o ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__fun_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Nat__Onat_J_M_Eo_J,type,
    ord_le1769015658490769310_nat_o: ( fun_Pr7546709192626087934_nat_o * fun_Pr7546709192626087934_nat_o ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__fun_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Nat__Onat_J_Mt__fun_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Nat__Onat_J_M_Eo_J_J,type,
    ord_le4038806025055212036_nat_o: ( fun_Pr8877776288780692068_nat_o * fun_Pr8877776288780692068_nat_o ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__fun_It__Real__Oreal_M_Eo_J,type,
    ord_le3605727316730466277real_o: ( fun_real_o * fun_real_o ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__fun_It__VEBT____Definitions__OVEBT_M_Eo_J,type,
    ord_le88303844413636359VEBT_o: ( fun_VEBT_VEBT_o * fun_VEBT_VEBT_o ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__fun_It__fun_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__fun_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_M_Eo_J_J,type,
    ord_le2876595303158420734_int_o: ( fun_fu2978539155105838238_int_o * fun_fu2978539155105838238_int_o ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__fun_It__fun_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__fun_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_M_Eo_J_J,type,
    ord_le4825817444062779351eger_o: ( fun_fu7695273699059480439eger_o * fun_fu7695273699059480439eger_o ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__fun_It__fun_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_Mt__fun_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_J,type,
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tff(sy_c_Orderings_Oord__class_Oless__eq_001t__fun_It__fun_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_Mt__fun_It__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_M_Eo_J_J,type,
    ord_le2834457679758710341_nat_o: ( fun_fu2611741329380808165_nat_o * fun_fu2611741329380808165_nat_o ) > $o ).

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tff(sy_c_Orderings_Oord__class_Omax_001t__Int__Oint,type,
    ord_max_int: fun_int_fun_int_int ).

tff(sy_c_Orderings_Oord__class_Omax_001t__Nat__Onat,type,
    ord_max_nat: fun_nat_fun_nat_nat ).

tff(sy_c_Orderings_Oord__class_Omax_001t__Num__Onum,type,
    ord_max_num: fun_num_fun_num_num ).

tff(sy_c_Orderings_Oord__class_Omax_001t__Rat__Orat,type,
    ord_max_rat: ( rat * rat ) > rat ).

tff(sy_c_Orderings_Oord__class_Omax_001t__Real__Oreal,type,
    ord_max_real: fun_re7681423839302404990l_real ).

tff(sy_c_Orderings_Oord__class_Omax_001t__Set__Oset_It__Int__Oint_J,type,
    ord_max_set_int: ( set_int * set_int ) > set_int ).

tff(sy_c_Orderings_Oord__class_Omax_001t__Word__Oword_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J_J_J_J_J_J,type,
    ord_ma8239519435860878689l_num1: ( word_N3645301735248828278l_num1 * word_N3645301735248828278l_num1 ) > word_N3645301735248828278l_num1 ).

tff(sy_c_Orderings_Oord__class_Omin_001t__Nat__Onat,type,
    ord_min_nat: fun_nat_fun_nat_nat ).

tff(sy_c_Orderings_Oorder__class_OGreatest_001t__Nat__Onat,type,
    order_Greatest_nat: fun_nat_o > nat ).

tff(sy_c_Orderings_Otop__class_Otop_001t__Assertions__Oassn,type,
    top_top_assn: assn ).

tff(sy_c_Orderings_Otop__class_Otop_001t__Option__Ooption_It__Assertions__Oassn_J,type,
    top_top_option_assn: option_assn ).

tff(sy_c_Orderings_Otop__class_Otop_001t__Option__Ooption_It__Set__Oset_It__Nat__Onat_J_J,type,
    top_to4826455019444611206et_nat: option_set_nat ).

tff(sy_c_Orderings_Otop__class_Otop_001t__Option__Ooption_It__Set__Oset_It__Real__Oreal_J_J,type,
    top_to1083748111577038690t_real: option_set_real ).

tff(sy_c_Orderings_Otop__class_Otop_001t__Option__Ooption_It__Set__Oset_It__String__Ochar_J_J,type,
    top_to6745749650031393671t_char: option_set_char ).

tff(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Code____Numeral__Ointeger_J,type,
    top_to4645266643341252675nteger: set_Code_integer ).

tff(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Complex__Ocomplex_J,type,
    top_top_set_complex: set_complex ).

tff(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Int__Oint_J,type,
    top_top_set_int: set_int ).

tff(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Nat__Onat_J,type,
    top_top_set_nat: set_nat ).

tff(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Option__Ooption_It__Code____Numeral__Ointeger_J_J,type,
    top_to5929521628599800467nteger: set_op687863988967635939nteger ).

tff(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Option__Ooption_It__Complex__Ocomplex_J_J,type,
    top_to6180147692022559204omplex: set_option_complex ).

tff(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Option__Ooption_It__Int__Oint_J_J,type,
    top_to6430115241214627170on_int: set_option_int ).

tff(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Option__Ooption_It__Nat__Onat_J_J,type,
    top_to8920198386146353926on_nat: set_option_nat ).

tff(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Option__Ooption_It__Real__Oreal_J_J,type,
    top_to853713521313446370n_real: set_option_real ).

tff(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Option__Ooption_It__String__Ochar_J_J,type,
    top_to576892095687177735n_char: set_option_char ).

tff(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
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tff(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    top_to4669805908274784177at_nat: set_Pr1261947904930325089at_nat ).

tff(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Product____Type__Oprod_It__fun_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_J,type,
    top_to1212741241995312661nt_int: set_Pr4166680992137330533nt_int ).

tff(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Product____Type__Oprod_It__fun_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J_J,type,
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tff(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__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,
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tff(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__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_J,type,
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tff(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Product____Type__Oprod_It__fun_It__Product____Type__Oprod_It__Int__Oint_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_J,type,
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tff(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Real__Oreal_J,type,
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tff(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__String__Ochar_J,type,
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tff(sy_c_Orderings_Otop__class_Otop_001t__fun_It__fun_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__fun_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_M_Eo_J_J,type,
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tff(sy_c_Orderings_Otop__class_Otop_001t__fun_It__fun_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__fun_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_M_Eo_J_J,type,
    top_to2784585791485011495eger_o: fun_fu7695273699059480439eger_o ).

tff(sy_c_Orderings_Otop__class_Otop_001t__fun_It__fun_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_Mt__fun_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_J,type,
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tff(sy_c_Orderings_Otop__class_Otop_001t__fun_It__fun_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_Mt__fun_It__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_M_Eo_J_J,type,
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tff(sy_c_Orderings_Otop__class_Otop_001t__fun_It__fun_It__Product____Type__Oprod_It__Int__Oint_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__fun_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_M_Eo_J_J,type,
    top_to1218662295448195450_int_o: fun_fu6865642474282033866_int_o ).

tff(sy_c_Partial__Function_Ofun__lub_001t__Heap____Time____Monad__OHeap_I_Eo_J_001t__Heap____Time____Monad__OHeap_I_Eo_J_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__VEBT____BuildupMemImp__OVEBTi_J_Mt__Nat__Onat_J,type,
    partia6726927458685305659Ti_nat: fun_se788439010262289911Heap_o > fun_se3420632608416671075Heap_o ).

tff(sy_c_Partial__Function_Ofun__lub_001t__Heap____Time____Monad__OHeap_I_Eo_J_001t__Heap____Time____Monad__OHeap_I_Eo_J_001t__Product____Type__Oprod_It__VEBT____BuildupMemImp__OVEBTi_Mt__Nat__Onat_J,type,
    partia5551857090987368152Ti_nat: fun_se788439010262289911Heap_o > fun_se7475304312357925539Heap_o ).

tff(sy_c_Partial__Function_Ofun__lub_001t__Heap____Time____Monad__OHeap_It__Option__Ooption_It__Nat__Onat_J_J_001t__Heap____Time____Monad__OHeap_It__Option__Ooption_It__Nat__Onat_J_J_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__VEBT____BuildupMemImp__OVEBTi_J_Mt__Nat__Onat_J,type,
    partia6039416512482706817Ti_nat: fun_se3101692492007276259on_nat > fun_se1756668854628305023on_nat ).

tff(sy_c_Partial__Function_Ofun__lub_001t__Heap____Time____Monad__OHeap_It__Option__Ooption_It__Nat__Onat_J_J_001t__Heap____Time____Monad__OHeap_It__Option__Ooption_It__Nat__Onat_J_J_001t__Product____Type__Oprod_It__VEBT____BuildupMemImp__OVEBTi_Mt__Nat__Onat_J,type,
    partia2080987842261039902Ti_nat: fun_se3101692492007276259on_nat > fun_se772746884452211909on_nat ).

tff(sy_c_Partial__Function_Ofun__lub_001t__Heap____Time____Monad__OHeap_It__VEBT____BuildupMemImp__OVEBTi_J_001t__Heap____Time____Monad__OHeap_It__VEBT____BuildupMemImp__OVEBTi_J_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__VEBT____BuildupMemImp__OVEBTi_J_Mt__Nat__Onat_J,type,
    partia7782936097874681665Ti_nat: fun_se3996372049210177955_VEBTi > fun_se3949293115670521129_VEBTi ).

tff(sy_c_Partial__Function_Ofun__lub_001t__Heap____Time____Monad__OHeap_It__VEBT____BuildupMemImp__OVEBTi_J_001t__Heap____Time____Monad__OHeap_It__VEBT____BuildupMemImp__OVEBTi_J_001t__Product____Type__Oprod_It__VEBT____BuildupMemImp__OVEBTi_Mt__Nat__Onat_J,type,
    partia6972460264168101086Ti_nat: fun_se3996372049210177955_VEBTi > fun_se5764602393054361839_VEBTi ).

tff(sy_c_Partial__Function_Ofun__ord_001t__Heap____Time____Monad__OHeap_I_Eo_J_001t__Heap____Time____Monad__OHeap_I_Eo_J_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__VEBT____BuildupMemImp__OVEBTi_J_Mt__Nat__Onat_J,type,
    partia3290229181235258227Ti_nat: fun_He2061078463025973126ap_o_o > fun_fu6850233865993928942ap_o_o ).

tff(sy_c_Partial__Function_Ofun__ord_001t__Heap____Time____Monad__OHeap_I_Eo_J_001t__Heap____Time____Monad__OHeap_I_Eo_J_001t__Product____Type__Oprod_It__VEBT____BuildupMemImp__OVEBTi_Mt__Nat__Onat_J,type,
    partia2925774515620677392Ti_nat: fun_He2061078463025973126ap_o_o > fun_fu2183070276626753582ap_o_o ).

tff(sy_c_Partial__Function_Ofun__ord_001t__Heap____Time____Monad__OHeap_It__Option__Ooption_It__Nat__Onat_J_J_001t__Heap____Time____Monad__OHeap_It__Option__Ooption_It__Nat__Onat_J_J_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__VEBT____BuildupMemImp__OVEBTi_J_Mt__Nat__Onat_J,type,
    partia7778075537949639097Ti_nat: fun_He5905523610488218222_nat_o > fun_fu5284765496190745342_nat_o ).

tff(sy_c_Partial__Function_Ofun__ord_001t__Heap____Time____Monad__OHeap_It__Option__Ooption_It__Nat__Onat_J_J_001t__Heap____Time____Monad__OHeap_It__Option__Ooption_It__Nat__Onat_J_J_001t__Product____Type__Oprod_It__VEBT____BuildupMemImp__OVEBTi_Mt__Nat__Onat_J,type,
    partia8535946980722491222Ti_nat: fun_He5905523610488218222_nat_o > fun_fu7962274784768186808_nat_o ).

tff(sy_c_Partial__Function_Ofun__ord_001t__Heap____Time____Monad__OHeap_It__VEBT____BuildupMemImp__OVEBTi_J_001t__Heap____Time____Monad__OHeap_It__VEBT____BuildupMemImp__OVEBTi_J_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__VEBT____BuildupMemImp__OVEBTi_J_Mt__Nat__Onat_J,type,
    partia1868168049876374393Ti_nat: fun_He5596775886410745902EBTi_o > fun_fu3145514261243071828EBTi_o ).

tff(sy_c_Partial__Function_Ofun__ord_001t__Heap____Time____Monad__OHeap_It__VEBT____BuildupMemImp__OVEBTi_J_001t__Heap____Time____Monad__OHeap_It__VEBT____BuildupMemImp__OVEBTi_J_001t__Product____Type__Oprod_It__VEBT____BuildupMemImp__OVEBTi_Mt__Nat__Onat_J,type,
    partia6690842624828592406Ti_nat: fun_He5596775886410745902EBTi_o > fun_fu175107047954615438EBTi_o ).

tff(sy_c_Power_Opower__class_Opower_001t__Code____Numeral__Ointeger,type,
    power_8256067586552552935nteger: code_integer > fun_nat_Code_integer ).

tff(sy_c_Power_Opower__class_Opower_001t__Complex__Ocomplex,type,
    power_power_complex: fun_co3258106519541737638omplex ).

tff(sy_c_Power_Opower__class_Opower_001t__Int__Oint,type,
    power_power_int: fun_int_fun_nat_int ).

tff(sy_c_Power_Opower__class_Opower_001t__Nat__Onat,type,
    power_power_nat: fun_nat_fun_nat_nat ).

tff(sy_c_Power_Opower__class_Opower_001t__Rat__Orat,type,
    power_power_rat: rat > fun_nat_rat ).

tff(sy_c_Power_Opower__class_Opower_001t__Real__Oreal,type,
    power_power_real: fun_re4238438383274954274t_real ).

tff(sy_c_Power_Opower__class_Opower_001t__Uint32__Ouint32,type,
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tff(sy_c_Power_Opower__class_Opower_001t__Word__Oword_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J_J_J_J_J_J,type,
    power_2184487114949457152l_num1: word_N3645301735248828278l_num1 > fun_na2984781812408219459l_num1 ).

tff(sy_c_Product__Type_OPair_001_Eo_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_Product__Type_OPair_001_Eo_001t__VEBT____Definitions__OVEBT,type,
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tff(sy_c_Product__Type_OPair_001t__Assertions__Oassn_001t__Assertions__Oassn,type,
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tff(sy_c_Product__Type_OPair_001t__Code____Numeral__Ointeger_001_Eo,type,
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tff(sy_c_Product__Type_OPair_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger,type,
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tff(sy_c_Product__Type_OPair_001t__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_001t__Nat__Onat,type,
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tff(sy_c_Product__Type_OPair_001t__Int__Oint_001t__Int__Oint,type,
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tff(sy_c_Product__Type_OPair_001t__Int__Oint_001t__VEBT____Definitions__OVEBT,type,
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tff(sy_c_Product__Type_OPair_001t__List__Olist_It__VEBT____Definitions__OVEBT_J_001t__VEBT____Definitions__OVEBT,type,
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tff(sy_c_Product__Type_OPair_001t__Nat__Onat_001_Eo,type,
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tff(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Int__Oint,type,
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tff(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Nat__Onat,type,
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tff(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Num__Onum,type,
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tff(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Real__Oreal,type,
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tff(sy_c_Product__Type_Oprod_Ofst_001t__Nat__Onat_001t__Nat__Onat,type,
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tff(sy_c_Rat_OFrct,type,
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tff(sy_c_Rat_Oquotient__of,type,
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tff(sy_c_Real__Vector__Spaces_OscaleR__class_OscaleR_001t__Complex__Ocomplex,type,
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tff(sy_c_Refine__Imp__Hol_Oassert_H,type,
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tff(sy_c_Refine__Imp__Hol_Orefines_001_Eo,type,
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tff(sy_c_Refine__Imp__Hol_Orefines_001t__List__Olist_I_Eo_J,type,
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tff(sy_c_Refine__Imp__Hol_Orefines_001t__List__Olist_It__Option__Ooption_It__Nat__Onat_J_J,type,
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tff(sy_c_Refine__Imp__Hol_Orefines_001t__Nat__Onat,type,
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tff(sy_c_Refine__Imp__Hol_Orefines_001t__Option__Ooption_It__Nat__Onat_J,type,
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tff(sy_c_Rings_Odivide__class_Odivide_001t__Int__Oint,type,
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tff(sy_c_Rings_Odivide__class_Odivide_001t__Nat__Onat,type,
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tff(sy_c_Rings_Odivide__class_Odivide_001t__Rat__Orat,type,
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tff(sy_c_Rings_Odvd__class_Odvd_001t__Rat__Orat,type,
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tff(sy_c_Rings_Odvd__class_Odvd_001t__Real__Oreal,type,
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tff(sy_c_Rings_Omodulo__class_Omodulo_001t__Nat__Onat,type,
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tff(sy_c_Syntax__Match_Osyntax__fo__nomatch_001t__fun_It__fun_It__Uint32__Ouint32_Mt__fun_It__Nat__Onat_M_Eo_J_J_Mt__fun_It__Uint32__Ouint32_Mt__fun_It__Code____Numeral__Ointeger_M_Eo_J_J_J_001t__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J,type,
    syntax3231729521921793939t_unit: fun_fu3466578569608634614eger_o > fun_he8413527710729349838unit_o ).

tff(sy_c_Syntax__Match_Osyntax__fo__nomatch_001t__fun_It__fun_It__Uint32__Ouint32_Mt__fun_It__Nat__Onat_Mt__fun_I_Eo_Mt__Uint32__Ouint32_J_J_J_Mt__fun_It__Uint32__Ouint32_Mt__fun_It__Code____Numeral__Ointeger_Mt__fun_I_Eo_Mt__Uint32__Ouint32_J_J_J_J_001t__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J,type,
    syntax2407925819610226131t_unit: fun_fu8139443716560175030uint32 > fun_he8413527710729349838unit_o ).

tff(sy_c_Syntax__Match_Osyntax__fo__nomatch_001t__fun_It__fun_It__Uint32__Ouint32_Mt__fun_It__Uint32__Ouint32_Mt__Uint32__Ouint32_J_J_Mt__fun_It__Uint32__Ouint32_Mt__fun_It__Uint32__Ouint32_Mt__Uint32__Ouint32_J_J_J_001t__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J,type,
    syntax732013874993758672t_unit: fun_fu6749172196426063865uint32 > fun_he8413527710729349838unit_o ).

tff(sy_c_Time__Reasoning_OTBOUND_001_Eo,type,
    time_TBOUND_o: heap_Time_Heap_o > fun_nat_o ).

tff(sy_c_Time__Reasoning_OTBOUND_001t__Heap__Oarray_I_Eo_J,type,
    time_TBOUND_array_o: heap_T5660665574680485309rray_o > fun_nat_o ).

tff(sy_c_Time__Reasoning_OTBOUND_001t__Heap__Oarray_It__Int__Oint_J,type,
    time_T8667231999468956601ay_int: heap_T1346037964561226099ay_int > fun_nat_o ).

tff(sy_c_Time__Reasoning_OTBOUND_001t__Heap__Oarray_It__Nat__Onat_J,type,
    time_T3621710982123377501ay_nat: heap_T3836121109492952855ay_nat > fun_nat_o ).

tff(sy_c_Time__Reasoning_OTBOUND_001t__Heap__Oarray_It__VEBT____BuildupMemImp__OVEBTi_J,type,
    time_T6070283812100419266_VEBTi: heap_T8822477325091257596_VEBTi > fun_nat_o ).

tff(sy_c_Time__Reasoning_OTBOUND_001t__List__Olist_I_Eo_J,type,
    time_TBOUND_list_o: heap_T844314716496656296list_o > fun_nat_o ).

tff(sy_c_Time__Reasoning_OTBOUND_001t__List__Olist_It__Nat__Onat_J,type,
    time_TBOUND_list_nat: heap_T290393402774840812st_nat > fun_nat_o ).

tff(sy_c_Time__Reasoning_OTBOUND_001t__List__Olist_It__Option__Ooption_It__Nat__Onat_J_J,type,
    time_T3808005469503390304on_nat: heap_T5317711798761887292on_nat > fun_nat_o ).

tff(sy_c_Time__Reasoning_OTBOUND_001t__List__Olist_It__VEBT____BuildupMemImp__OVEBTi_J,type,
    time_T8149879359713347829_VEBTi: heap_T4980287057938770641_VEBTi > fun_nat_o ).

tff(sy_c_Time__Reasoning_OTBOUND_001t__Nat__Onat,type,
    time_TBOUND_nat: heap_Time_Heap_nat > fun_nat_o ).

tff(sy_c_Time__Reasoning_OTBOUND_001t__Option__Ooption_It__Nat__Onat_J,type,
    time_T8353473612707095248on_nat: heap_T2636463487746394924on_nat > fun_nat_o ).

tff(sy_c_Time__Reasoning_OTBOUND_001t__Product____Type__Ounit,type,
    time_T7469515765551943773t_unit: heap_T5738788834812785303t_unit > fun_nat_o ).

tff(sy_c_Time__Reasoning_OTBOUND_001t__VEBT____BuildupMemImp__OVEBTi,type,
    time_T5737551269749752165_VEBTi: heap_T8145700208782473153_VEBTi > fun_nat_o ).

tff(sy_c_Time__Reasoning_Ofails_001t__Product____Type__Ounit,type,
    time_f8834461667527620124t_unit: heap_T5738788834812785303t_unit > fun_he8413527710729349838unit_o ).

tff(sy_c_Time__Reasoning_Ohtt_001_Eo,type,
    time_htt_o: ( assn * heap_Time_Heap_o * fun_o_assn ) > fun_nat_o ).

tff(sy_c_Time__Reasoning_Ohtt_001t__Nat__Onat,type,
    time_htt_nat: ( assn * heap_Time_Heap_nat * fun_nat_assn ) > fun_nat_o ).

tff(sy_c_Time__Reasoning_Ohtt_001t__Option__Ooption_It__Nat__Onat_J,type,
    time_htt_option_nat: ( assn * heap_T2636463487746394924on_nat * fun_option_nat_assn ) > fun_nat_o ).

tff(sy_c_Time__Reasoning_Ohtt_001t__Product____Type__Ounit,type,
    time_h7375078312994015913t_unit: ( assn * heap_T5738788834812785303t_unit * fun_Pr6626665806141890376t_assn ) > fun_nat_o ).

tff(sy_c_Time__Reasoning_Ohtt_001t__VEBT____BuildupMemImp__OVEBTi,type,
    time_htt_VEBT_VEBTi: ( assn * heap_T8145700208782473153_VEBTi * fun_VEBT_VEBTi_assn ) > fun_nat_o ).

tff(sy_c_Time__Reasoning_Otime_001_Eo,type,
    time_time_o: ( heap_Time_Heap_o * heap_e7401611519738050253t_unit ) > nat ).

tff(sy_c_Time__Reasoning_Otime_001t__Heap__Oarray_I_Eo_J,type,
    time_time_array_o: ( heap_T5660665574680485309rray_o * heap_e7401611519738050253t_unit ) > nat ).

tff(sy_c_Time__Reasoning_Otime_001t__Heap__Oarray_It__Int__Oint_J,type,
    time_time_array_int: ( heap_T1346037964561226099ay_int * heap_e7401611519738050253t_unit ) > nat ).

tff(sy_c_Time__Reasoning_Otime_001t__Heap__Oarray_It__Nat__Onat_J,type,
    time_time_array_nat: ( heap_T3836121109492952855ay_nat * heap_e7401611519738050253t_unit ) > nat ).

tff(sy_c_Time__Reasoning_Otime_001t__Heap__Oarray_It__VEBT____BuildupMemImp__OVEBTi_J,type,
    time_t9122064381910598399_VEBTi: ( heap_T8822477325091257596_VEBTi * heap_e7401611519738050253t_unit ) > nat ).

tff(sy_c_Time__Reasoning_Otime_001t__List__Olist_It__Product____Type__Ounit_J,type,
    time_t4781937132199089312t_unit: ( heap_T7268547540234007069t_unit * heap_e7401611519738050253t_unit ) > nat ).

tff(sy_c_Time__Reasoning_Otime_001t__List__Olist_It__VEBT____BuildupMemImp__OVEBTi_J,type,
    time_t3534373299052942712_VEBTi: ( heap_T4980287057938770641_VEBTi * heap_e7401611519738050253t_unit ) > nat ).

tff(sy_c_Time__Reasoning_Otime_001t__Nat__Onat,type,
    time_time_nat: ( heap_Time_Heap_nat * heap_e7401611519738050253t_unit ) > nat ).

tff(sy_c_Time__Reasoning_Otime_001t__Option__Ooption_It__Nat__Onat_J,type,
    time_time_option_nat: ( heap_T2636463487746394924on_nat * heap_e7401611519738050253t_unit ) > nat ).

tff(sy_c_Time__Reasoning_Otime_001t__Product____Type__Ounit,type,
    time_t4224138285095624986t_unit: ( heap_T5738788834812785303t_unit * heap_e7401611519738050253t_unit ) > nat ).

tff(sy_c_Time__Reasoning_Otime_001t__VEBT____BuildupMemImp__OVEBTi,type,
    time_time_VEBT_VEBTi: ( heap_T8145700208782473153_VEBTi * heap_e7401611519738050253t_unit ) > nat ).

tff(sy_c_Topological__Spaces_Omonoseq_001t__Real__Oreal,type,
    topolo6980174941875973593q_real: fun_nat_real > $o ).

tff(sy_c_Topological__Spaces_Otopological__space__class_Oat__within_001t__Real__Oreal,type,
    topolo2177554685111907308n_real: ( real * set_real ) > filter_real ).

tff(sy_c_Topological__Spaces_Ouniform__space__class_OCauchy_001t__Real__Oreal,type,
    topolo4055970368930404560y_real: fun_nat_real > $o ).

tff(sy_c_Transcendental_Oarccos,type,
    arccos: fun_real_real ).

tff(sy_c_Transcendental_Oarcosh_001t__Real__Oreal,type,
    arcosh_real: fun_real_real ).

tff(sy_c_Transcendental_Oarcsin,type,
    arcsin: fun_real_real ).

tff(sy_c_Transcendental_Oarctan,type,
    arctan: fun_real_real ).

tff(sy_c_Transcendental_Oarsinh_001t__Real__Oreal,type,
    arsinh_real: fun_real_real ).

tff(sy_c_Transcendental_Oartanh_001t__Real__Oreal,type,
    artanh_real: fun_real_real ).

tff(sy_c_Transcendental_Ocos_001t__Real__Oreal,type,
    cos_real: fun_real_real ).

tff(sy_c_Transcendental_Ocos__coeff,type,
    cos_coeff: fun_nat_real ).

tff(sy_c_Transcendental_Ocosh_001t__Real__Oreal,type,
    cosh_real: fun_real_real ).

tff(sy_c_Transcendental_Ocot_001t__Real__Oreal,type,
    cot_real: fun_real_real ).

tff(sy_c_Transcendental_Oexp_001t__Complex__Ocomplex,type,
    exp_complex: fun_complex_complex ).

tff(sy_c_Transcendental_Oexp_001t__Real__Oreal,type,
    exp_real: fun_real_real ).

tff(sy_c_Transcendental_Oln__class_Oln_001t__Real__Oreal,type,
    ln_ln_real: fun_real_real ).

tff(sy_c_Transcendental_Olog,type,
    log: fun_re7681423839302404990l_real ).

tff(sy_c_Transcendental_Opi,type,
    pi: real ).

tff(sy_c_Transcendental_Opowr_001t__Real__Oreal,type,
    powr_real: fun_re7681423839302404990l_real ).

tff(sy_c_Transcendental_Osin_001t__Real__Oreal,type,
    sin_real: fun_real_real ).

tff(sy_c_Transcendental_Osin__coeff,type,
    sin_coeff: fun_nat_real ).

tff(sy_c_Transcendental_Osinh_001t__Real__Oreal,type,
    sinh_real: fun_real_real ).

tff(sy_c_Transcendental_Otan_001t__Real__Oreal,type,
    tan_real: fun_real_real ).

tff(sy_c_Transcendental_Otanh_001t__Real__Oreal,type,
    tanh_real: fun_real_real ).

tff(sy_c_Type__Length_Olen0__class_Olen__of_001t__Enum__Ofinite____1,type,
    type_l31302759751748491nite_1: itself_finite_1 > nat ).

tff(sy_c_Type__Length_Olen0__class_Olen__of_001t__Enum__Ofinite____2,type,
    type_l31302759751748492nite_2: itself_finite_2 > nat ).

tff(sy_c_Type__Length_Olen0__class_Olen__of_001t__Enum__Ofinite____3,type,
    type_l31302759751748493nite_3: itself_finite_3 > nat ).

tff(sy_c_Type__Length_Olen0__class_Olen__of_001t__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J_J_J_J_J,type,
    type_l796852477590012082l_num1: itself8794530163899892676l_num1 > nat ).

tff(sy_c_Type__Length_Olen0__class_Olen__of_001t__Numeral____Type__Onum1,type,
    type_l4264026598287037465l_num1: itself_Numeral_num1 > nat ).

tff(sy_c_Uint32_ORep__uint32_H,type,
    rep_uint32: uint32 > word_N3645301735248828278l_num1 ).

tff(sy_c_Uint32_Odiv0__uint32,type,
    div0_uint32: uint32 > uint32 ).

tff(sy_c_Uint32_Ointeger__of__uint32,type,
    integer_of_uint32: fun_ui1560626348547333958nteger ).

tff(sy_c_Uint32_Ointeger__of__uint32__signed,type,
    intege5370686899274169573signed: uint32 > code_integer ).

tff(sy_c_Uint32_Omod0__uint32,type,
    mod0_uint32: uint32 > uint32 ).

tff(sy_c_Uint32_Oset__bits__aux__uint32,type,
    set_bits_aux_uint32: ( fun_nat_o * nat * uint32 ) > uint32 ).

tff(sy_c_Uint32_Osigned__drop__bit__uint32,type,
    signed489701013188660438uint32: fun_na3857091055356329634uint32 ).

tff(sy_c_Uint32_Ouint32_OAbs__uint32,type,
    abs_uint32: word_N3645301735248828278l_num1 > uint32 ).

tff(sy_c_Uint32_Ouint32_ORep__uint32,type,
    rep_uint322: uint32 > word_N3645301735248828278l_num1 ).

tff(sy_c_Uint32_Ouint32__div,type,
    uint32_div: ( uint32 * uint32 ) > uint32 ).

tff(sy_c_Uint32_Ouint32__divmod,type,
    uint32_divmod: ( uint32 * uint32 ) > produc827990862158126777uint32 ).

tff(sy_c_Uint32_Ouint32__sdiv,type,
    uint32_sdiv: ( uint32 * uint32 ) > uint32 ).

tff(sy_c_Uint32_Ouint32__set__bit,type,
    uint32_set_bit: ( uint32 * code_integer * $o ) > uint32 ).

tff(sy_c_Uint32_Ouint32__shiftl,type,
    uint32_shiftl: uint32 > fun_Co1554742298434871494uint32 ).

tff(sy_c_Uint32_Ouint32__shiftr,type,
    uint32_shiftr: uint32 > fun_Co1554742298434871494uint32 ).

tff(sy_c_Uint32_Ouint32__sshiftr,type,
    uint32_sshiftr: uint32 > fun_Co1554742298434871494uint32 ).

tff(sy_c_Uint32_Ouint32__test__bit,type,
    uint32_test_bit: uint32 > fun_Code_integer_o ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t,type,
    vEBT_T_i_n_s_e_r_t: fun_VE597728521808591204at_nat ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H,type,
    vEBT_T_i_n_s_e_r_t2: fun_VE597728521808591204at_nat ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H__rel,type,
    vEBT_T5076183648494686801_t_rel: fun_Pr8877776288780692068_nat_o ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t__rel,type,
    vEBT_T9217963907923527482_t_rel: fun_Pr8877776288780692068_nat_o ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062x_092_060_094sub_062t,type,
    vEBT_T_m_a_x_t: fun_VEBT_VEBT_nat ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062x_092_060_094sub_062t__rel,type,
    vEBT_T_m_a_x_t_rel: fun_VE2962686792283380846VEBT_o ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r,type,
    vEBT_T_m_e_m_b_e_r: fun_VE597728521808591204at_nat ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H,type,
    vEBT_T_m_e_m_b_e_r2: fun_VE597728521808591204at_nat ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H__rel,type,
    vEBT_T8099345112685741742_r_rel: fun_Pr8877776288780692068_nat_o ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r__rel,type,
    vEBT_T5837161174952499735_r_rel: fun_Pr8877776288780692068_nat_o ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062N_092_060_094sub_062u_092_060_094sub_062l_092_060_094sub_062l,type,
    vEBT_T_m_i_n_N_u_l_l: fun_VEBT_VEBT_nat ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062N_092_060_094sub_062u_092_060_094sub_062l_092_060_094sub_062l__rel,type,
    vEBT_T5462971552011256508_l_rel: fun_VE2962686792283380846VEBT_o ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t,type,
    vEBT_T_m_i_n_t: fun_VEBT_VEBT_nat ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t__rel,type,
    vEBT_T_m_i_n_t_rel: fun_VE2962686792283380846VEBT_o ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d,type,
    vEBT_T_p_r_e_d: fun_VE597728521808591204at_nat ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H,type,
    vEBT_T_p_r_e_d2: fun_VE597728521808591204at_nat ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H__rel,type,
    vEBT_T_p_r_e_d_rel: fun_Pr8877776288780692068_nat_o ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d__rel,type,
    vEBT_T_p_r_e_d_rel2: fun_Pr8877776288780692068_nat_o ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c,type,
    vEBT_T_s_u_c_c: fun_VE597728521808591204at_nat ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H,type,
    vEBT_T_s_u_c_c2: fun_VE597728521808591204at_nat ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H__rel,type,
    vEBT_T_s_u_c_c_rel: fun_Pr8877776288780692068_nat_o ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c__rel,type,
    vEBT_T_s_u_c_c_rel2: fun_Pr8877776288780692068_nat_o ).

tff(sy_c_VEBT__BuildupMemImp_OVEBT__internal_OT__vebt__buildupi,type,
    vEBT_V441764108873111860ildupi: fun_nat_nat ).

tff(sy_c_VEBT__BuildupMemImp_OVEBT__internal_OT__vebt__buildupi_H,type,
    vEBT_V9176841429113362141ildupi: fun_nat_int ).

tff(sy_c_VEBT__BuildupMemImp_OVEBT__internal_OT__vebt__buildupi_H__rel,type,
    vEBT_V3352910403632780892pi_rel: fun_nat_fun_nat_o ).

tff(sy_c_VEBT__BuildupMemImp_OVEBT__internal_OT__vebt__buildupi__rel,type,
    vEBT_V2957053500504383685pi_rel: fun_nat_fun_nat_o ).

tff(sy_c_VEBT__BuildupMemImp_OVEBT__internal_OTb,type,
    vEBT_VEBT_Tb: fun_nat_int ).

tff(sy_c_VEBT__BuildupMemImp_OVEBT__internal_OTb_H,type,
    vEBT_VEBT_Tb2: fun_nat_nat ).

tff(sy_c_VEBT__BuildupMemImp_OVEBT__internal_OTb_H__rel,type,
    vEBT_VEBT_Tb_rel: fun_nat_fun_nat_o ).

tff(sy_c_VEBT__BuildupMemImp_OVEBT__internal_OTb__rel,type,
    vEBT_VEBT_Tb_rel2: fun_nat_fun_nat_o ).

tff(sy_c_VEBT__BuildupMemImp_OVEBT__internal_Ohighi,type,
    vEBT_VEBT_highi: nat > fun_na3533371890617757541ap_nat ).

tff(sy_c_VEBT__BuildupMemImp_OVEBT__internal_Olowi,type,
    vEBT_VEBT_lowi: nat > fun_na3533371890617757541ap_nat ).

tff(sy_c_VEBT__BuildupMemImp_OVEBT__internal_OminNulli,type,
    vEBT_VEBT_minNulli: fun_VE7038733415900436564Heap_o ).

tff(sy_c_VEBT__BuildupMemImp_OVEBT__internal_OminNulli__rel,type,
    vEBT_V5740978063120863272li_rel: fun_VE8636387394336989792EBTi_o ).

tff(sy_c_VEBT__BuildupMemImp_OVEBT__internal_Oreplicatei_001_Eo,type,
    vEBT_V2326993469660664182atei_o: ( nat * heap_Time_Heap_o ) > heap_T844314716496656296list_o ).

tff(sy_c_VEBT__BuildupMemImp_OVEBT__internal_Oreplicatei_001t__Nat__Onat,type,
    vEBT_V7726092123322077554ei_nat: ( nat * heap_Time_Heap_nat ) > heap_T290393402774840812st_nat ).

tff(sy_c_VEBT__BuildupMemImp_OVEBT__internal_Oreplicatei_001t__Option__Ooption_It__Nat__Onat_J,type,
    vEBT_V792416675989592002on_nat: ( nat * heap_T2636463487746394924on_nat ) > heap_T5317711798761887292on_nat ).

tff(sy_c_VEBT__BuildupMemImp_OVEBT__internal_Oreplicatei_001t__Product____Type__Ounit,type,
    vEBT_V7483891112628345579t_unit: ( nat * heap_T5738788834812785303t_unit ) > heap_T7268547540234007069t_unit ).

tff(sy_c_VEBT__BuildupMemImp_OVEBT__internal_Oreplicatei_001t__VEBT____BuildupMemImp__OVEBTi,type,
    vEBT_V1859673955506687831_VEBTi: ( nat * heap_T8145700208782473153_VEBTi ) > heap_T4980287057938770641_VEBTi ).

tff(sy_c_VEBT__BuildupMemImp_OVEBT__internal_Ovebt__buildupi_H,type,
    vEBT_V739175172307565963ildupi: fun_na7848093160509028170_VEBTi ).

tff(sy_c_VEBT__BuildupMemImp_OVEBT__internal_Ovebt__buildupi_H__rel,type,
    vEBT_V254170901696579886pi_rel: fun_nat_fun_nat_o ).

tff(sy_c_VEBT__BuildupMemImp_OVEBT__internal_Ovebt__inserti_H,type,
    vEBT_V3964819847710782039nserti: fun_VE2018681615261037117_VEBTi ).

tff(sy_c_VEBT__BuildupMemImp_OVEBT__internal_Ovebt__memberi_H,type,
    vEBT_V854960066525838166emberi: fun_VE8181444127144381288Heap_o ).

tff(sy_c_VEBT__BuildupMemImp_OVEBTi_OLeafi,type,
    vEBT_Leafi: ( $o * $o ) > vEBT_VEBTi ).

tff(sy_c_VEBT__BuildupMemImp_OVEBTi_ONodei,type,
    vEBT_Nodei: ( option4927543243414619207at_nat * nat * array_VEBT_VEBTi ) > fun_VE8449866691444886503_VEBTi ).

tff(sy_c_VEBT__BuildupMemImp_OVEBTi_Ocase__VEBTi_001t__Heap____Time____Monad__OHeap_I_Eo_J,type,
    vEBT_c6104975476656191286Heap_o: ( fun_op1763656958002597022Heap_o * fun_o_1207901889202581752Heap_o ) > fun_VE7038733415900436564Heap_o ).

tff(sy_c_VEBT__BuildupMemImp_OVEBTi_Ocase__VEBTi_001t__Heap____Time____Monad__OHeap_It__Nat__Onat_J,type,
    vEBT_c1335663792808957512ap_nat: ( fun_op2425996437410981688ap_nat * fun_o_7009304396571837746ap_nat ) > fun_VE2791623396910532672ap_nat ).

tff(sy_c_VEBT__BuildupMemImp_OVEBTi_Ocase__VEBTi_001t__Heap____Time____Monad__OHeap_It__Option__Ooption_It__Nat__Onat_J_J,type,
    vEBT_c6250501799366334488on_nat: ( fun_op5783775701855621896on_nat * fun_o_4049874799390722306on_nat ) > fun_VE2693554162914142864on_nat ).

tff(sy_c_VEBT__BuildupMemImp_OVEBTi_Ocase__VEBTi_001t__Heap____Time____Monad__OHeap_It__VEBT____BuildupMemImp__OVEBTi_J,type,
    vEBT_c6028912655521741485_VEBTi: ( fun_op5127434927081926045_VEBTi * fun_o_6972675622308511895_VEBTi ) > fun_VE4765431424076933413_VEBTi ).

tff(sy_c_VEBT__BuildupMemImp_OVEBTi_Ocase__VEBTi_001t__Nat__Onat,type,
    vEBT_case_VEBTi_nat: ( fun_op7256438278201533260Ti_nat * fun_o_fun_o_nat ) > fun_VEBT_VEBTi_nat ).

tff(sy_c_VEBT__BuildupMemImp_OVEBTi_Osize__VEBTi,type,
    vEBT_size_VEBTi: fun_VEBT_VEBTi_nat ).

tff(sy_c_VEBT__BuildupMemImp_Ovebt__assn__raw,type,
    vEBT_vebt_assn_raw: fun_VE905575822219544865i_assn ).

tff(sy_c_VEBT__BuildupMemImp_Ovebt__assn__raw__rel,type,
    vEBT_v8524038756793281170aw_rel: fun_Pr3720955412185915546EBTi_o ).

tff(sy_c_VEBT__BuildupMemImp_Ovebt__buildupi,type,
    vEBT_vebt_buildupi: fun_na7848093160509028170_VEBTi ).

tff(sy_c_VEBT__BuildupMemImp_Ovebt__buildupi__rel,type,
    vEBT_v1230518104690509829pi_rel: fun_nat_fun_nat_o ).

tff(sy_c_VEBT__BuildupMemImp_Ovebt__inserti,type,
    vEBT_vebt_inserti: fun_VE8118064890490510894_VEBTi ).

tff(sy_c_VEBT__BuildupMemImp_Ovebt__maxti,type,
    vEBT_vebt_maxti: fun_VE2693554162914142864on_nat ).

tff(sy_c_VEBT__BuildupMemImp_Ovebt__maxti__rel,type,
    vEBT_vebt_maxti_rel: fun_VE8636387394336989792EBTi_o ).

tff(sy_c_VEBT__BuildupMemImp_Ovebt__memberi,type,
    vEBT_vebt_memberi: fun_VE7134199479003659425Heap_o ).

tff(sy_c_VEBT__BuildupMemImp_Ovebt__minti,type,
    vEBT_vebt_minti: fun_VE2693554162914142864on_nat ).

tff(sy_c_VEBT__BuildupMemImp_Ovebt__minti__rel,type,
    vEBT_vebt_minti_rel: fun_VE8636387394336989792EBTi_o ).

tff(sy_c_VEBT__Definitions_OVEBT_OLeaf,type,
    vEBT_Leaf: ( $o * $o ) > vEBT_VEBT ).

tff(sy_c_VEBT__Definitions_OVEBT_ONode,type,
    vEBT_Node: ( option4927543243414619207at_nat * nat * list_VEBT_VEBT ) > fun_VE3165209603783583043T_VEBT ).

tff(sy_c_VEBT__Definitions_OVEBT_Ocase__VEBT_001_Eo,type,
    vEBT_case_VEBT_o: ( fun_op541075054904862521VEBT_o * fun_o_fun_o_o ) > fun_VEBT_VEBT_o ).

tff(sy_c_VEBT__Definitions_OVEBT_Ocase__VEBT_001t__Product____Type__Oprod_It__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__List__Olist_It__VEBT____Definitions__OVEBT_J_Mt__VEBT____Definitions__OVEBT_J_J_J,type,
    vEBT_c634343235235684882T_VEBT: ( fun_op8229906066235313459T_VEBT * fun_o_8142510167905321106T_VEBT * vEBT_VEBT ) > produc819165548630102716T_VEBT ).

tff(sy_c_VEBT__Definitions_OVEBT_Ois__Node,type,
    vEBT_is_Node: fun_VEBT_VEBT_o ).

tff(sy_c_VEBT__Definitions_OVEBT_Osize__VEBT,type,
    vEBT_size_VEBT: fun_VEBT_VEBT_nat ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Oboth__member__options,type,
    vEBT_V8194947554948674370ptions: fun_VE6480751511521678896_nat_o ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Ohigh,type,
    vEBT_VEBT_high: fun_nat_fun_nat_nat ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Oin__children,type,
    vEBT_V5917875025757280293ildren: ( nat * list_VEBT_VEBT ) > fun_nat_o ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Olow,type,
    vEBT_VEBT_low: fun_nat_fun_nat_nat ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Omembermima,type,
    vEBT_VEBT_membermima: fun_VE6480751511521678896_nat_o ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Omembermima__rel,type,
    vEBT_V4351362008482014158ma_rel: fun_Pr8877776288780692068_nat_o ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Onaive__member,type,
    vEBT_V5719532721284313246member: fun_VE6480751511521678896_nat_o ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Onaive__member__rel,type,
    vEBT_V5765760719290551771er_rel: fun_Pr8877776288780692068_nat_o ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Ovalid_H,type,
    vEBT_VEBT_valid: fun_VE6480751511521678896_nat_o ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Ovalid_H__rel,type,
    vEBT_VEBT_valid_rel: fun_Pr8877776288780692068_nat_o ).

tff(sy_c_VEBT__Definitions_Oinvar__vebt,type,
    vEBT_invar_vebt: fun_VE6480751511521678896_nat_o ).

tff(sy_c_VEBT__Definitions_Oset__vebt,type,
    vEBT_set_vebt: vEBT_VEBT > set_nat ).

tff(sy_c_VEBT__Definitions_Ovebt__buildup,type,
    vEBT_vebt_buildup: nat > vEBT_VEBT ).

tff(sy_c_VEBT__Definitions_Ovebt__buildup__rel,type,
    vEBT_v4011308405150292612up_rel: fun_nat_fun_nat_o ).

tff(sy_c_VEBT__DeleteBounds_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e,type,
    vEBT_T_d_e_l_e_t_e: fun_VE597728521808591204at_nat ).

tff(sy_c_VEBT__DeleteBounds_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e__rel,type,
    vEBT_T8441311223069195367_e_rel: fun_Pr8877776288780692068_nat_o ).

tff(sy_c_VEBT__DeleteBounds_OVEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H,type,
    vEBT_V1232361888498592333_e_t_e: fun_VE597728521808591204at_nat ).

tff(sy_c_VEBT__DeleteBounds_OVEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H__rel,type,
    vEBT_V6368547301243506412_e_rel: fun_Pr8877776288780692068_nat_o ).

tff(sy_c_VEBT__Delete_Ovebt__delete,type,
    vEBT_vebt_delete: ( vEBT_VEBT * nat ) > vEBT_VEBT ).

tff(sy_c_VEBT__Delete_Ovebt__delete__rel,type,
    vEBT_vebt_delete_rel: fun_Pr8877776288780692068_nat_o ).

tff(sy_c_VEBT__Height_OVEBT__internal_Oheight,type,
    vEBT_VEBT_height: fun_VEBT_VEBT_nat ).

tff(sy_c_VEBT__Height_OVEBT__internal_Oheight__rel,type,
    vEBT_VEBT_height_rel: fun_VE2962686792283380846VEBT_o ).

tff(sy_c_VEBT__Insert_Ovebt__insert,type,
    vEBT_vebt_insert: ( vEBT_VEBT * nat ) > vEBT_VEBT ).

tff(sy_c_VEBT__Insert_Ovebt__insert__rel,type,
    vEBT_vebt_insert_rel: fun_Pr8877776288780692068_nat_o ).

tff(sy_c_VEBT__List__Assn_OlistI__assn_001t__VEBT____Definitions__OVEBT_001t__VEBT____BuildupMemImp__OVEBTi,type,
    vEBT_L1528199826722428489_VEBTi: ( set_nat * fun_VE905575822219544865i_assn * list_VEBT_VEBT ) > fun_li3352006551604695530i_assn ).

tff(sy_c_VEBT__List__Assn_Olist__assn_001_Eo_001_Eo,type,
    vEBT_L7363604446928714179sn_o_o: ( fun_o_fun_o_assn * list_o * list_o ) > assn ).

tff(sy_c_VEBT__List__Assn_Olist__assn_001_Eo_001t__Int__Oint,type,
    vEBT_L4782520652837395649_o_int: ( fun_o_fun_int_assn * list_o * list_int ) > assn ).

tff(sy_c_VEBT__List__Assn_Olist__assn_001_Eo_001t__Nat__Onat,type,
    vEBT_L4785011123346445925_o_nat: ( fun_o_fun_nat_assn * list_o * list_nat ) > assn ).

tff(sy_c_VEBT__List__Assn_Olist__assn_001_Eo_001t__Real__Oreal,type,
    vEBT_L4725278957065240257o_real: ( fun_o_fun_real_assn * list_o * list_real ) > assn ).

tff(sy_c_VEBT__List__Assn_Olist__assn_001t__Int__Oint_001_Eo,type,
    vEBT_L6066640139021943271_int_o: ( fun_int_fun_o_assn * list_int * list_o ) > assn ).

tff(sy_c_VEBT__List__Assn_Olist__assn_001t__Int__Oint_001t__Real__Oreal,type,
    vEBT_L8288995350762215837t_real: ( fun_in6446958447386763770l_assn * list_int * list_real ) > assn ).

tff(sy_c_VEBT__List__Assn_Olist__assn_001t__Nat__Onat_001_Eo,type,
    vEBT_L7887682484454631235_nat_o: ( fun_nat_fun_o_assn * list_nat * list_o ) > assn ).

tff(sy_c_VEBT__List__Assn_Olist__assn_001t__Nat__Onat_001t__Int__Oint,type,
    vEBT_L8298612041380073281at_int: ( fun_nat_fun_int_assn * list_nat * list_int ) > assn ).

tff(sy_c_VEBT__List__Assn_Olist__assn_001t__Nat__Onat_001t__Nat__Onat,type,
    vEBT_L8301102511889123557at_nat: ( fun_nat_fun_nat_assn * list_nat * list_nat ) > assn ).

tff(sy_c_VEBT__List__Assn_Olist__assn_001t__Nat__Onat_001t__Real__Oreal,type,
    vEBT_L6102073776069194049t_real: ( fun_na448437756743965086l_assn * list_nat * list_real ) > assn ).

tff(sy_c_VEBT__List__Assn_Olist__assn_001t__Nat__Onat_001t__VEBT____Definitions__OVEBT,type,
    vEBT_L8158188754432654943T_VEBT: ( fun_na4489259242115493880T_assn * list_nat * list_VEBT_VEBT ) > assn ).

tff(sy_c_VEBT__List__Assn_Olist__assn_001t__Real__Oreal_001_Eo,type,
    vEBT_L6234343332106409831real_o: ( fun_real_fun_o_assn * list_real * list_o ) > assn ).

tff(sy_c_VEBT__List__Assn_Olist__assn_001t__Real__Oreal_001t__Int__Oint,type,
    vEBT_L1443519841834266653al_int: ( fun_re4990660043736809722t_assn * list_real * list_int ) > assn ).

tff(sy_c_VEBT__List__Assn_Olist__assn_001t__Real__Oreal_001t__Nat__Onat,type,
    vEBT_L1446010312343316929al_nat: ( fun_re574865750389657758t_assn * list_real * list_nat ) > assn ).

tff(sy_c_VEBT__List__Assn_Olist__assn_001t__Real__Oreal_001t__Real__Oreal,type,
    vEBT_L1930518968523514909l_real: ( fun_re6303261851654292474l_assn * list_real * list_real ) > assn ).

tff(sy_c_VEBT__List__Assn_Olist__assn_001t__Real__Oreal_001t__VEBT____Definitions__OVEBT,type,
    vEBT_L4595930785310033027T_VEBT: ( fun_re8050108611700907676T_assn * list_real * list_VEBT_VEBT ) > assn ).

tff(sy_c_VEBT__List__Assn_Olist__assn_001t__VEBT____Definitions__OVEBT_001_Eo,type,
    vEBT_L7489408758114837031VEBT_o: ( fun_VE2104784059010244834o_assn * list_VEBT_VEBT ) > fun_list_o_assn ).

tff(sy_c_VEBT__List__Assn_Olist__assn_001t__VEBT____Definitions__OVEBT_001t__Int__Oint,type,
    vEBT_L8294436054247626077BT_int: ( fun_VE123837653852366616t_assn * list_VEBT_VEBT * list_int ) > assn ).

tff(sy_c_VEBT__List__Assn_Olist__assn_001t__VEBT____Definitions__OVEBT_001t__Nat__Onat,type,
    vEBT_L8296926524756676353BT_nat: ( fun_VE4931415397359990460t_assn * list_VEBT_VEBT ) > fun_list_nat_assn ).

tff(sy_c_VEBT__List__Assn_Olist__assn_001t__VEBT____Definitions__OVEBT_001t__Option__Ooption_It__Nat__Onat_J,type,
    vEBT_L8010285020845282001on_nat: ( fun_VE6334927249922962060t_assn * list_VEBT_VEBT ) > fun_li338670087962420949t_assn ).

tff(sy_c_VEBT__List__Assn_Olist__assn_001t__VEBT____Definitions__OVEBT_001t__Product____Type__Ounit,type,
    vEBT_L8068554427805421084t_unit: ( fun_VE8866928305347306583t_assn * list_VEBT_VEBT ) > fun_li5772175206745016270t_assn ).

tff(sy_c_VEBT__List__Assn_Olist__assn_001t__VEBT____Definitions__OVEBT_001t__Real__Oreal,type,
    vEBT_L5781919052683127133T_real: ( fun_VE6910866152645720856l_assn * list_VEBT_VEBT * list_real ) > assn ).

tff(sy_c_VEBT__List__Assn_Olist__assn_001t__VEBT____Definitions__OVEBT_001t__VEBT____BuildupMemImp__OVEBTi,type,
    vEBT_L6296928887356842470_VEBTi: ( fun_VE905575822219544865i_assn * list_VEBT_VEBT ) > fun_li3352006551604695530i_assn ).

tff(sy_c_VEBT__List__Assn_Olist__assn_001t__VEBT____Definitions__OVEBT_001t__VEBT____Definitions__OVEBT,type,
    vEBT_L1279224858307276611T_VEBT: ( fun_VE1571388752190420990T_assn * list_VEBT_VEBT * list_VEBT_VEBT ) > assn ).

tff(sy_c_VEBT__Member_OVEBT__internal_Obit__concat,type,
    vEBT_VEBT_bit_concat: nat > fun_nat_fun_nat_nat ).

tff(sy_c_VEBT__Member_OVEBT__internal_OminNull,type,
    vEBT_VEBT_minNull: fun_VEBT_VEBT_o ).

tff(sy_c_VEBT__Member_OVEBT__internal_OminNull__rel,type,
    vEBT_V6963167321098673237ll_rel: fun_VE2962686792283380846VEBT_o ).

tff(sy_c_VEBT__Member_OVEBT__internal_Oset__vebt_H,type,
    vEBT_VEBT_set_vebt: vEBT_VEBT > set_nat ).

tff(sy_c_VEBT__Member_Ovebt__member,type,
    vEBT_vebt_member: fun_VE6480751511521678896_nat_o ).

tff(sy_c_VEBT__Member_Ovebt__member__rel,type,
    vEBT_vebt_member_rel: fun_Pr8877776288780692068_nat_o ).

tff(sy_c_VEBT__MinMax_OVEBT__internal_Oadd,type,
    vEBT_VEBT_add: fun_op7210469962420818650on_nat ).

tff(sy_c_VEBT__MinMax_OVEBT__internal_Ogreater,type,
    vEBT_VEBT_greater: fun_op5794009454290973322_nat_o ).

tff(sy_c_VEBT__MinMax_OVEBT__internal_Oless,type,
    vEBT_VEBT_less: fun_op5794009454290973322_nat_o ).

tff(sy_c_VEBT__MinMax_OVEBT__internal_Olesseq,type,
    vEBT_VEBT_lesseq: fun_op5794009454290973322_nat_o ).

tff(sy_c_VEBT__MinMax_OVEBT__internal_Omax__in__set,type,
    vEBT_VEBT_max_in_set: set_nat > fun_nat_o ).

tff(sy_c_VEBT__MinMax_OVEBT__internal_Omin__in__set,type,
    vEBT_VEBT_min_in_set: set_nat > fun_nat_o ).

tff(sy_c_VEBT__MinMax_OVEBT__internal_Omul,type,
    vEBT_VEBT_mul: fun_op7210469962420818650on_nat ).

tff(sy_c_VEBT__MinMax_OVEBT__internal_Ooption__shift_001t__Nat__Onat,type,
    vEBT_V4262088993061758097ft_nat: fun_nat_fun_nat_nat > fun_op7210469962420818650on_nat ).

tff(sy_c_VEBT__MinMax_OVEBT__internal_Ooption__shift_001t__Num__Onum,type,
    vEBT_V819420779217536731ft_num: ( fun_num_fun_num_num * option_num ) > fun_op8989173174992673105on_num ).

tff(sy_c_VEBT__MinMax_OVEBT__internal_Ooption__shift_001t__Product____Type__Oprod_I_Eo_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J,type,
    vEBT_V613753007643960916it_nat: ( fun_Pr1313076919630728883it_nat * option7339022715339332451it_nat * option7339022715339332451it_nat ) > option7339022715339332451it_nat ).

tff(sy_c_VEBT__MinMax_OVEBT__internal_Ooption__shift_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    vEBT_V1502963449132264192at_nat: ( fun_Pr2477239776880934807at_nat * option4927543243414619207at_nat * option4927543243414619207at_nat ) > option4927543243414619207at_nat ).

tff(sy_c_VEBT__MinMax_OVEBT__internal_Ooption__shift_001t__Product____Type__Oprod_It__Option__Ooption_It__Nat__Onat_J_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J,type,
    vEBT_V819568868292977612it_nat: ( fun_Pr3118856740240616547it_nat * option2621746655072343315it_nat * option2621746655072343315it_nat ) > option2621746655072343315it_nat ).

tff(sy_c_VEBT__MinMax_OVEBT__internal_Ooption__shift__rel_001t__Nat__Onat,type,
    vEBT_V3895251965096974666el_nat: fun_Pr1283576600272279410_nat_o ).

tff(sy_c_VEBT__MinMax_OVEBT__internal_Ooption__shift__rel_001t__Num__Onum,type,
    vEBT_V452583751252753300el_num: fun_Pr4822914654802553862_num_o ).

tff(sy_c_VEBT__MinMax_OVEBT__internal_Ooption__shift__rel_001t__Product____Type__Oprod_I_Eo_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J,type,
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tff(sy_c_VEBT__MinMax_OVEBT__internal_Ooption__shift__rel_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
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tff(sy_c_VEBT__MinMax_OVEBT__internal_Ooption__shift__rel_001t__Product____Type__Oprod_It__Option__Ooption_It__Nat__Onat_J_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J,type,
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tff(sy_c_VEBT__MinMax_OVEBT__internal_Opower,type,
    vEBT_VEBT_power: fun_op7210469962420818650on_nat ).

tff(sy_c_VEBT__MinMax_Ovebt__maxt,type,
    vEBT_vebt_maxt: vEBT_VEBT > option_nat ).

tff(sy_c_VEBT__MinMax_Ovebt__maxt__rel,type,
    vEBT_vebt_maxt_rel: fun_VE2962686792283380846VEBT_o ).

tff(sy_c_VEBT__MinMax_Ovebt__mint,type,
    vEBT_vebt_mint: vEBT_VEBT > option_nat ).

tff(sy_c_VEBT__MinMax_Ovebt__mint__rel,type,
    vEBT_vebt_mint_rel: fun_VE2962686792283380846VEBT_o ).

tff(sy_c_VEBT__Pred_Ois__pred__in__set,type,
    vEBT_is_pred_in_set: set_nat > fun_nat_fun_nat_o ).

tff(sy_c_VEBT__Pred_Ovebt__pred,type,
    vEBT_vebt_pred: vEBT_VEBT > fun_nat_option_nat ).

tff(sy_c_VEBT__Pred_Ovebt__pred__rel,type,
    vEBT_vebt_pred_rel: fun_Pr8877776288780692068_nat_o ).

tff(sy_c_VEBT__Space_OVEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d,type,
    vEBT_V8646137997579335489_i_l_d: fun_nat_nat ).

tff(sy_c_VEBT__Space_OVEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_092_060_094sub_062u_092_060_094sub_062p,type,
    vEBT_V8346862874174094_d_u_p: fun_nat_nat ).

tff(sy_c_VEBT__Space_OVEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_092_060_094sub_062u_092_060_094sub_062p__rel,type,
    vEBT_V1247956027447740395_p_rel: fun_nat_fun_nat_o ).

tff(sy_c_VEBT__Space_OVEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d__rel,type,
    vEBT_V5144397997797733112_d_rel: fun_nat_fun_nat_o ).

tff(sy_c_VEBT__Space_OVEBT__internal_Ocnt,type,
    vEBT_VEBT_cnt: fun_VEBT_VEBT_real ).

tff(sy_c_VEBT__Space_OVEBT__internal_Ocnt_H,type,
    vEBT_VEBT_cnt2: fun_VEBT_VEBT_nat ).

tff(sy_c_VEBT__Space_OVEBT__internal_Ocnt_H__rel,type,
    vEBT_VEBT_cnt_rel: fun_VE2962686792283380846VEBT_o ).

tff(sy_c_VEBT__Space_OVEBT__internal_Ocnt__rel,type,
    vEBT_VEBT_cnt_rel2: fun_VE2962686792283380846VEBT_o ).

tff(sy_c_VEBT__Space_OVEBT__internal_Ospace,type,
    vEBT_VEBT_space: fun_VEBT_VEBT_nat ).

tff(sy_c_VEBT__Space_OVEBT__internal_Ospace_H,type,
    vEBT_VEBT_space2: fun_VEBT_VEBT_nat ).

tff(sy_c_VEBT__Space_OVEBT__internal_Ospace_H__rel,type,
    vEBT_VEBT_space_rel: fun_VE2962686792283380846VEBT_o ).

tff(sy_c_VEBT__Space_OVEBT__internal_Ospace__rel,type,
    vEBT_VEBT_space_rel2: fun_VE2962686792283380846VEBT_o ).

tff(sy_c_VEBT__SuccPredImperative_OVEBT__internal_Ovebt__predi_H,type,
    vEBT_VEBT_vebt_predi: fun_VE3775134097089128360on_nat ).

tff(sy_c_VEBT__SuccPredImperative_OVEBT__internal_Ovebt__succi_H,type,
    vEBT_VEBT_vebt_succi: fun_VE3775134097089128360on_nat ).

tff(sy_c_VEBT__SuccPredImperative_Ovebt__predi,type,
    vEBT_vebt_predi: fun_VE8516060933855021849on_nat ).

tff(sy_c_VEBT__SuccPredImperative_Ovebt__succi,type,
    vEBT_vebt_succi: fun_VE8516060933855021849on_nat ).

tff(sy_c_VEBT__Succ_Ois__succ__in__set,type,
    vEBT_is_succ_in_set: set_nat > fun_nat_fun_nat_o ).

tff(sy_c_VEBT__Succ_Ovebt__succ,type,
    vEBT_vebt_succ: vEBT_VEBT > fun_nat_option_nat ).

tff(sy_c_VEBT__Succ_Ovebt__succ__rel,type,
    vEBT_vebt_succ_rel: fun_Pr8877776288780692068_nat_o ).

tff(sy_c_Wellfounded_Oaccp_001t__Nat__Onat,type,
    accp_nat: fun_nat_fun_nat_o > fun_nat_o ).

tff(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
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tff(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
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tff(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Num__Onum_Mt__Num__Onum_J,type,
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tff(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Nat__Onat_J,type,
    accp_P2887432264394892906BT_nat: fun_Pr8877776288780692068_nat_o > fun_Pr7546709192626087934_nat_o ).

tff(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__VEBT____BuildupMemImp__OVEBTi_J,type,
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tff(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__fun_It__Nat__Onat_Mt__fun_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
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tff(sy_c_Wellfounded_Oaccp_001t__VEBT____BuildupMemImp__OVEBTi,type,
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tff(sy_c_Wellfounded_Oaccp_001t__VEBT____Definitions__OVEBT,type,
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tff(sy_c_Wellfounded_Ofinite__psubset_001t__Code____Numeral__Ointeger,type,
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tff(sy_c_Wellfounded_Ofinite__psubset_001t__Complex__Ocomplex,type,
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tff(sy_c_Wellfounded_Ofinite__psubset_001t__Int__Oint,type,
    finite_psubset_int: set_Pr2522554150109002629et_int ).

tff(sy_c_Wellfounded_Ofinite__psubset_001t__Nat__Onat,type,
    finite_psubset_nat: set_Pr5488025237498180813et_nat ).

tff(sy_c_Word_Oeven__word_001t__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J_J_J_J_J,type,
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tff(sy_c_Word_Osemiring__1__class_Ounsigned_001t__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J_J_J_J_J_001t__Int__Oint,type,
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tff(sy_c_Word_Osigned__drop__bit_001t__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J_J_J_J_J,type,
    signed5000768011106662067l_num1: ( nat * word_N3645301735248828278l_num1 ) > word_N3645301735248828278l_num1 ).

tff(sy_c_aa_001_Eo_001_Eo,type,
    aa_o_o: ( fun_o_o * $o ) > $o ).

tff(sy_c_aa_001_Eo_001t__Assertions__Oassn,type,
    aa_o_assn: ( fun_o_assn * $o ) > assn ).

tff(sy_c_aa_001_Eo_001t__Code____Numeral__Ointeger,type,
    aa_o_Code_integer: ( fun_o_Code_integer * $o ) > code_integer ).

tff(sy_c_aa_001_Eo_001t__Heap____Time____Monad__OHeap_I_Eo_J,type,
    aa_o_H7399033255965945470Heap_o: ( fun_o_8325599348076409421Heap_o * $o ) > heap_Time_Heap_o ).

tff(sy_c_aa_001_Eo_001t__Heap____Time____Monad__OHeap_It__Nat__Onat_J,type,
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tff(sy_c_aa_001_Eo_001t__Heap____Time____Monad__OHeap_It__Option__Ooption_It__Nat__Onat_J_J,type,
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tff(sy_c_aa_001_Eo_001t__Heap____Time____Monad__OHeap_It__Product____Type__Ounit_J,type,
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tff(sy_c_aa_001_Eo_001t__Heap____Time____Monad__OHeap_It__VEBT____BuildupMemImp__OVEBTi_J,type,
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tff(sy_c_aa_001_Eo_001t__Int__Oint,type,
    aa_o_int: ( fun_o_int * $o ) > int ).

tff(sy_c_aa_001_Eo_001t__Nat__Onat,type,
    aa_o_nat: ( fun_o_nat * $o ) > nat ).

tff(sy_c_aa_001_Eo_001t__Product____Type__Oprod_I_Eo_Mt__VEBT____Definitions__OVEBT_J,type,
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tff(sy_c_aa_001_Eo_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
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tff(sy_c_aa_001_Eo_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
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tff(sy_c_aa_001_Eo_001t__Product____Type__Oprod_It__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__List__Olist_It__VEBT____Definitions__OVEBT_J_Mt__VEBT____Definitions__OVEBT_J_J_J,type,
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tff(sy_c_aa_001_Eo_001t__Real__Oreal,type,
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tff(sy_c_aa_001_Eo_001t__String__Ochar,type,
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tff(sy_c_aa_001_Eo_001t__Uint32__Ouint32,type,
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tff(sy_c_aa_001_Eo_001t__VEBT____Definitions__OVEBT,type,
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tff(sy_c_aa_001_Eo_001t__fun_I_Eo_M_Eo_J,type,
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tff(sy_c_aa_001_Eo_001t__fun_I_Eo_Mt__Assertions__Oassn_J,type,
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tff(sy_c_aa_001_Eo_001t__fun_I_Eo_Mt__Heap____Time____Monad__OHeap_I_Eo_J_J,type,
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tff(sy_c_aa_001_Eo_001t__fun_I_Eo_Mt__Heap____Time____Monad__OHeap_It__Nat__Onat_J_J,type,
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tff(sy_c_aa_001_Eo_001t__fun_I_Eo_Mt__Heap____Time____Monad__OHeap_It__Option__Ooption_It__Nat__Onat_J_J_J,type,
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tff(sy_c_aa_001_Eo_001t__fun_I_Eo_Mt__Heap____Time____Monad__OHeap_It__VEBT____BuildupMemImp__OVEBTi_J_J,type,
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tff(sy_c_aa_001_Eo_001t__fun_I_Eo_Mt__Nat__Onat_J,type,
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tff(sy_c_aa_001_Eo_001t__fun_I_Eo_Mt__Product____Type__Oprod_It__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__List__Olist_It__VEBT____Definitions__OVEBT_J_Mt__VEBT____Definitions__OVEBT_J_J_J_J,type,
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tff(sy_c_aa_001t__fun_It__fun_It__Uint32__Ouint32_Mt__fun_It__Nat__Onat_Mt__fun_I_Eo_Mt__Uint32__Ouint32_J_J_J_Mt__fun_It__Uint32__Ouint32_Mt__fun_It__Code____Numeral__Ointeger_Mt__fun_I_Eo_Mt__Uint32__Ouint32_J_J_J_J_001t__fun_It__fun_It__Uint32__Ouint32_Mt__fun_It__Nat__Onat_Mt__fun_I_Eo_Mt__Uint32__Ouint32_J_J_J_Mt__fun_It__Uint32__Ouint32_Mt__fun_It__Code____Numeral__Ointeger_Mt__fun_I_Eo_Mt__Uint32__Ouint32_J_J_J_J,type,
    aa_fun3432716020469233706uint32: ( fun_fu5407763515825486313uint32 * fun_fu8139443716560175030uint32 ) > fun_fu8139443716560175030uint32 ).

tff(sy_c_aa_001t__fun_It__fun_It__Uint32__Ouint32_Mt__fun_It__Uint32__Ouint32_Mt__Uint32__Ouint32_J_J_Mt__fun_It__Uint32__Ouint32_Mt__fun_It__Uint32__Ouint32_Mt__Uint32__Ouint32_J_J_J_001t__fun_It__fun_It__Uint32__Ouint32_Mt__fun_It__Uint32__Ouint32_Mt__Uint32__Ouint32_J_J_Mt__fun_It__Uint32__Ouint32_Mt__fun_It__Uint32__Ouint32_Mt__Uint32__Ouint32_J_J_J,type,
    aa_fun3431294132209178672uint32: ( fun_fu6311994054701583983uint32 * fun_fu6749172196426063865uint32 ) > fun_fu6749172196426063865uint32 ).

tff(sy_c_fChoice_001t__Real__Oreal,type,
    fChoice_real: fun_real_o > real ).

tff(sy_c_fequal_001t__Complex__Ocomplex,type,
    fequal_complex: complex > fun_complex_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__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    fequal6010348034681699726nt_int: fun_Pr9018117161326028078_int_o ).

tff(sy_c_fequal_001t__Real__Oreal,type,
    fequal_real: fun_real_fun_real_o ).

tff(sy_c_fequal_001t__VEBT____Definitions__OVEBT,type,
    fequal_VEBT_VEBT: vEBT_VEBT > fun_VEBT_VEBT_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__Complex__Ocomplex,type,
    member_complex: complex > fun_set_complex_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__Int__Oint_J,type,
    member_list_int: ( list_int * set_list_int ) > $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__List__Olist_It__Real__Oreal_J,type,
    member_list_real: ( list_real * set_list_real ) > $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 > fun_set_num_o ).

tff(sy_c_member_001t__Product____Type__Oprod_It__Assertions__Oassn_Mt__Assertions__Oassn_J,type,
    member7957490590177025114n_assn: ( produc6575502325842934193n_assn * set_Pr5949110396991348497n_assn ) > $o ).

tff(sy_c_member_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
    member157494554546826820nteger: ( produc8923325533196201883nteger * set_Pr4811707699266497531nteger ) > $o ).

tff(sy_c_member_001t__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J,type,
    member6260224972018164377et_nat: ( produc3658429121746597890et_nat * set_Pr3948176798113811640et_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__Nat__Onat_M_Eo_J,type,
    member6310962623043647828_nat_o: ( product_prod_nat_o * set_Pr3149072824959771635_nat_o ) > $o ).

tff(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Int__Oint_J,type,
    member4262671552274231302at_int: ( product_prod_nat_int * set_Pr7995236796853374141at_int ) > $o ).

tff(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    member8440522571783428010at_nat: ( product_prod_nat_nat * set_Pr1261947904930325089at_nat ) > $o ).

tff(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Real__Oreal_J,type,
    member557208447399453958t_real: ( produc7716430852924023517t_real * set_Pr320017278500174781t_real ) > $o ).

tff(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mt__VEBT____BuildupMemImp__OVEBTi_J,type,
    member763447850065367567_VEBTi: ( produc214224863196444774_VEBTi * set_Pr1938536134445252166_VEBTi ) > $o ).

tff(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mt__VEBT____Definitions__OVEBT_J,type,
    member8549952807677709168T_VEBT: ( produc8025551001238799321T_VEBT * set_Pr6167073792073659919T_VEBT ) > $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__Complex__Ocomplex_J_Mt__Set__Oset_It__Complex__Ocomplex_J_J,type,
    member351165363924911826omplex: ( produc8064648209034914857omplex * set_Pr6308028481084910985omplex ) > $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__Uint32__Ouint32_Mt__Uint32__Ouint32_J,type,
    member8027108493173000802uint32: ( produc827990862158126777uint32 * set_Pr1773385645901665561uint32 ) > $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__Code____Numeral__Ointeger_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J,type,
    member8978266922497061677nteger: ( produc2453742332425399446nteger * set_Pr3486312109254157132nteger ) > $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__Rat__Orat,type,
    member_rat: rat > fun_set_rat_o ).

tff(sy_c_member_001t__Real__Oreal,type,
    member_real: real > fun_set_real_o ).

tff(sy_c_member_001t__Set__Oset_It__Int__Oint_J,type,
    member_set_int: ( set_int * set_set_int ) > $o ).

tff(sy_c_member_001t__VEBT____BuildupMemImp__OVEBTi,type,
    member_VEBT_VEBTi: ( vEBT_VEBTi * set_VEBT_VEBTi ) > $o ).

tff(sy_c_member_001t__VEBT____Definitions__OVEBT,type,
    member_VEBT_VEBT: vEBT_VEBT > fun_set_VEBT_VEBT_o ).

tff(sy_v_ma____,type,
    ma: nat ).

tff(sy_v_mi____,type,
    mi: nat ).

tff(sy_v_na____,type,
    na: nat ).

tff(sy_v_summary____,type,
    summary: vEBT_VEBT ).

tff(sy_v_tia____,type,
    tia: vEBT_VEBTi ).

tff(sy_v_treeList____,type,
    treeList: list_VEBT_VEBT ).

tff(sy_v_tree__is,type,
    tree_is: list_VEBT_VEBTi ).

tff(sy_v_va____,type,
    va: nat ).

tff(sy_v_x11______,type,
    x11: option4927543243414619207at_nat ).

tff(sy_v_x12______,type,
    x12: nat ).

tff(sy_v_x13______,type,
    x13: array_VEBT_VEBTi ).

tff(sy_v_x14______,type,
    x14: vEBT_VEBTi ).

tff(sy_v_xa____,type,
    xa: nat ).

% Relevant facts (11601)
tff(fact_0_greater__shift,axiom,
    ! [Y: nat,Xb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),Xb)
    <=> aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_greater,aa_nat_option_nat(some_nat,Xb)),aa_nat_option_nat(some_nat,Y)) ) ).

% greater_shift
tff(fact_1_max__in__set__def,axiom,
    ! [Xs: set_nat,Xb: nat] :
      ( aa_nat_o(vEBT_VEBT_max_in_set(Xs),Xb)
    <=> ( aa_set_nat_o(member_nat(Xb),Xs)
        & ! [X: nat] :
            ( aa_set_nat_o(member_nat(X),Xs)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Xb) ) ) ) ).

% max_in_set_def
tff(fact_2_min__in__set__def,axiom,
    ! [Xs: set_nat,Xb: nat] :
      ( aa_nat_o(vEBT_VEBT_min_in_set(Xs),Xb)
    <=> ( aa_set_nat_o(member_nat(Xb),Xs)
        & ! [X: nat] :
            ( aa_set_nat_o(member_nat(X),Xs)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),X) ) ) ) ).

% min_in_set_def
tff(fact_3_power__shift,axiom,
    ! [Xb: nat,Y: nat,Z: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Xb),Y) = Z )
    <=> ( aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_power,aa_nat_option_nat(some_nat,Xb)),aa_nat_option_nat(some_nat,Y)) = aa_nat_option_nat(some_nat,Z) ) ) ).

% power_shift
tff(fact_4_bit__split__inv,axiom,
    ! [Xb: nat,D: nat] : aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_bit_concat(aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),D)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xb),D)),D) = Xb ).

% bit_split_inv
tff(fact_5_mimapr,axiom,
    aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,mi),ma) ).

% mimapr
tff(fact_6_high__def,axiom,
    ! [Xb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ).

% high_def
tff(fact_7_vebt__minti__h,axiom,
    ! [Ta: vEBT_VEBT,Tib: vEBT_VEBTi] : hoare_7629718768684598413on_nat(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Ta),Tib),aa_VEB2711320435582112505on_nat(vEBT_vebt_minti,Tib),aa_VEB7611521244314475576t_assn(aTP_Lamm_ar(Ta),Tib)) ).

% vebt_minti_h
tff(fact_8_vebt__maxti__h,axiom,
    ! [Ta: vEBT_VEBT,Tib: vEBT_VEBTi] : hoare_7629718768684598413on_nat(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Ta),Tib),aa_VEB2711320435582112505on_nat(vEBT_vebt_maxti,Tib),aa_VEB7611521244314475576t_assn(aTP_Lamm_as(Ta),Tib)) ).

% vebt_maxti_h
tff(fact_9_lesseq__shift,axiom,
    ! [Xb: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Y)
    <=> aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_lesseq,aa_nat_option_nat(some_nat,Xb)),aa_nat_option_nat(some_nat,Y)) ) ).

% lesseq_shift
tff(fact_10__C7_OIH_C_I2_J,axiom,
    ! [Xaa: nat,Xb: nat,Xc: option_nat,Nb: nat,Tib: vEBT_VEBTi] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,ma),xa)
     => ( ( Xaa = aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) )
       => ( ( Xb = aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) )
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),size_s6755466524823107622T_VEBT(treeList))
           => ( ( Xc = vEBT_vebt_mint(nth_VEBT_VEBT(treeList,Xb)) )
             => ( ~ ( ( Xc != none_nat )
                    & aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_greater,aa_nat_option_nat(some_nat,Xaa)),Xc) )
               => ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,summary),Nb)
                 => hoare_7629718768684598413on_nat(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,summary),Tib),aa_nat947733127081326238on_nat(aa_VEB3566417204837707778on_nat(aa_VEB1030146501360450209on_nat(vEBT_VEBT_vebt_predi,summary),Tib),Xb),aa_VEB7611521244314475576t_assn(aTP_Lamm_at(Xb),Tib)) ) ) ) ) ) ) ) ).

% "7.IH"(2)
tff(fact_11__C7_OIH_C_I1_J,axiom,
    ! [Xaa: nat,Xb: nat,Xc: option_nat,Nb: nat,Tib: vEBT_VEBTi] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,ma),xa)
     => ( ( Xaa = aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) )
       => ( ( Xb = aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) )
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),size_s6755466524823107622T_VEBT(treeList))
           => ( ( Xc = vEBT_vebt_mint(nth_VEBT_VEBT(treeList,Xb)) )
             => ( ( ( Xc != none_nat )
                  & aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_greater,aa_nat_option_nat(some_nat,Xaa)),Xc) )
               => ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,nth_VEBT_VEBT(treeList,Xb)),Nb)
                 => hoare_7629718768684598413on_nat(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,nth_VEBT_VEBT(treeList,Xb)),Tib),aa_nat947733127081326238on_nat(aa_VEB3566417204837707778on_nat(aa_VEB1030146501360450209on_nat(vEBT_VEBT_vebt_predi,nth_VEBT_VEBT(treeList,Xb)),Tib),Xaa),aa_VEB7611521244314475576t_assn(aa_nat2954700136012447621t_assn(aTP_Lamm_au(Xaa),Xb),Tib)) ) ) ) ) ) ) ) ).

% "7.IH"(1)
tff(fact_12_pred__less__length__list,axiom,
    ! [Deg: nat,Xb: nat,Maa: nat,TreeLista: list_VEBT_VEBT,Mia: nat,Summarya: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Maa)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeLista))
         => ( aa_nat_option_nat(vEBT_vebt_pred(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mia),Maa)),Deg,TreeLista),Summarya)),Xb) = $let(
                l: nat,
                l:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                $let(
                  h: nat,
                  h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                  $let(
                    minlow: option_nat,
                    minlow:= vEBT_vebt_mint(nth_VEBT_VEBT(TreeLista,h)),
                    $ite(
                      ( ( minlow != none_nat )
                      & aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_greater,aa_nat_option_nat(some_nat,l)),minlow) ),
                      aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,aa_nat_option_nat(some_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_option_nat(some_nat,h))),aa_nat_option_nat(vEBT_vebt_pred(nth_VEBT_VEBT(TreeLista,h)),l)),
                      $let(
                        pr: option_nat,
                        pr:= aa_nat_option_nat(vEBT_vebt_pred(Summarya),h),
                        $ite(
                          pr = none_nat,
                          $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mia),Xb),aa_nat_option_nat(some_nat,Mia),none_nat),
                          aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,aa_nat_option_nat(some_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),pr)),vEBT_vebt_maxt(nth_VEBT_VEBT(TreeLista,the_nat(pr)))) ) ) ) ) ) ) ) ) ) ) ).

% pred_less_length_list
tff(fact_13_pred__lesseq__max,axiom,
    ! [Deg: nat,Xb: nat,Maa: nat,Mia: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Maa)
       => ( aa_nat_option_nat(vEBT_vebt_pred(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mia),Maa)),Deg,TreeLista),Summarya)),Xb) = $let(
              l: nat,
              l:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
              $let(
                h: nat,
                h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                $ite(
                  aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeLista)),
                  $let(
                    minlow: option_nat,
                    minlow:= vEBT_vebt_mint(nth_VEBT_VEBT(TreeLista,h)),
                    $ite(
                      ( ( minlow != none_nat )
                      & aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_greater,aa_nat_option_nat(some_nat,l)),minlow) ),
                      aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,aa_nat_option_nat(some_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_option_nat(some_nat,h))),aa_nat_option_nat(vEBT_vebt_pred(nth_VEBT_VEBT(TreeLista,h)),l)),
                      $let(
                        pr: option_nat,
                        pr:= aa_nat_option_nat(vEBT_vebt_pred(Summarya),h),
                        $ite(
                          pr = none_nat,
                          $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mia),Xb),aa_nat_option_nat(some_nat,Mia),none_nat),
                          aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,aa_nat_option_nat(some_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),pr)),vEBT_vebt_maxt(nth_VEBT_VEBT(TreeLista,the_nat(pr)))) ) ) ) ),
                  none_nat ) ) ) ) ) ) ).

% pred_lesseq_max
tff(fact_14_vebt__maxti__hT,axiom,
    ! [Ta: vEBT_VEBT,Tib: vEBT_VEBTi] : aa_nat_o(time_htt_option_nat(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Ta),Tib),aa_VEB2711320435582112505on_nat(vEBT_vebt_maxti,Tib),aa_VEB7611521244314475576t_assn(aTP_Lamm_as(Ta),Tib)),one_one_nat) ).

% vebt_maxti_hT
tff(fact_15_vebt__minti__hT,axiom,
    ! [Ta: vEBT_VEBT,Tib: vEBT_VEBTi] : aa_nat_o(time_htt_option_nat(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Ta),Tib),aa_VEB2711320435582112505on_nat(vEBT_vebt_minti,Tib),aa_VEB7611521244314475576t_assn(aTP_Lamm_ar(Ta),Tib)),one_one_nat) ).

% vebt_minti_hT
tff(fact_16_div2__Suc__Suc,axiom,
    ! [M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,M))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% div2_Suc_Suc
tff(fact_17_Suc__1,axiom,
    aa_nat_nat(suc,one_one_nat) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) ).

% Suc_1
tff(fact_18_power__increasing__iff,axiom,
    ! [B: real,Xb: nat,Y: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Xb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Y))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Y) ) ) ).

% power_increasing_iff
tff(fact_19_power__increasing__iff,axiom,
    ! [B: rat,Xb: 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),Xb)),aa_nat_rat(power_power_rat(B),Y))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Y) ) ) ).

% power_increasing_iff
tff(fact_20_power__increasing__iff,axiom,
    ! [B: nat,Xb: 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),Xb)),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,Xb),Y) ) ) ).

% power_increasing_iff
tff(fact_21_power__increasing__iff,axiom,
    ! [B: int,Xb: 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),Xb)),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,Xb),Y) ) ) ).

% power_increasing_iff
tff(fact_22_power__strict__increasing__iff,axiom,
    ! [B: real,Xb: nat,Y: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Xb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Y))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y) ) ) ).

% power_strict_increasing_iff
tff(fact_23_power__strict__increasing__iff,axiom,
    ! [B: rat,Xb: 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),Xb)),aa_nat_rat(power_power_rat(B),Y))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y) ) ) ).

% power_strict_increasing_iff
tff(fact_24_power__strict__increasing__iff,axiom,
    ! [B: nat,Xb: 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),Xb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Y))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y) ) ) ).

% power_strict_increasing_iff
tff(fact_25_power__strict__increasing__iff,axiom,
    ! [B: int,Xb: 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),Xb)),aa_nat_int(aa_int_fun_nat_int(power_power_int,B),Y))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y) ) ) ).

% power_strict_increasing_iff
tff(fact_26_divide__less__eq__numeral1_I1_J,axiom,
    ! [B: real,W: num,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),numeral_numeral_real(W))),A)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),numeral_numeral_real(W))) ) ).

% divide_less_eq_numeral1(1)
tff(fact_27_divide__less__eq__numeral1_I1_J,axiom,
    ! [B: rat,W: num,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,divide_divide_rat(B,numeral_numeral_rat(W))),A)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),aa_rat_rat(times_times_rat(A),numeral_numeral_rat(W))) ) ).

% divide_less_eq_numeral1(1)
tff(fact_28_less__divide__eq__numeral1_I1_J,axiom,
    ! [A: real,B: real,W: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),numeral_numeral_real(W)))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),numeral_numeral_real(W))),B) ) ).

% less_divide_eq_numeral1(1)
tff(fact_29_less__divide__eq__numeral1_I1_J,axiom,
    ! [A: rat,B: rat,W: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),divide_divide_rat(B,numeral_numeral_rat(W)))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A),numeral_numeral_rat(W))),B) ) ).

% less_divide_eq_numeral1(1)
tff(fact_30_deg__deg__n,axiom,
    ! [Info: option4927543243414619207at_nat,Deg: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Nb: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info,Deg,TreeLista),Summarya)),Nb)
     => ( Deg = Nb ) ) ).

% deg_deg_n
tff(fact_31_mul__shift,axiom,
    ! [Xb: nat,Y: nat,Z: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Xb),Y) = Z )
    <=> ( aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,aa_nat_option_nat(some_nat,Xb)),aa_nat_option_nat(some_nat,Y)) = aa_nat_option_nat(some_nat,Z) ) ) ).

% mul_shift
tff(fact_32_deg__SUcn__Node,axiom,
    ! [Tree: vEBT_VEBT,Nb: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Tree),aa_nat_nat(suc,aa_nat_nat(suc,Nb)))
     => ? [Info2: option4927543243414619207at_nat,TreeList: list_VEBT_VEBT,S: vEBT_VEBT] : Tree = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,aa_nat_nat(suc,aa_nat_nat(suc,Nb)),TreeList),S) ) ).

% deg_SUcn_Node
tff(fact_33_numeral__eq__iff,axiom,
    ! [M: num,Nb: num] :
      ( ( numeral_numeral_real(M) = numeral_numeral_real(Nb) )
    <=> ( M = Nb ) ) ).

% numeral_eq_iff
tff(fact_34_numeral__eq__iff,axiom,
    ! [M: num,Nb: num] :
      ( ( numeral_numeral_rat(M) = numeral_numeral_rat(Nb) )
    <=> ( M = Nb ) ) ).

% numeral_eq_iff
tff(fact_35_numeral__eq__iff,axiom,
    ! [M: num,Nb: num] :
      ( ( aa_num_nat(numeral_numeral_nat,M) = aa_num_nat(numeral_numeral_nat,Nb) )
    <=> ( M = Nb ) ) ).

% numeral_eq_iff
tff(fact_36_numeral__eq__iff,axiom,
    ! [M: num,Nb: num] :
      ( ( aa_num_int(numeral_numeral_int,M) = aa_num_int(numeral_numeral_int,Nb) )
    <=> ( M = Nb ) ) ).

% numeral_eq_iff
tff(fact_37_numeral__le__iff,axiom,
    ! [M: num,Nb: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,numeral_numeral_real(M)),numeral_numeral_real(Nb))
    <=> aa_num_o(ord_less_eq_num(M),Nb) ) ).

% numeral_le_iff
tff(fact_38_numeral__le__iff,axiom,
    ! [M: num,Nb: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,numeral_numeral_rat(M)),numeral_numeral_rat(Nb))
    <=> aa_num_o(ord_less_eq_num(M),Nb) ) ).

% numeral_le_iff
tff(fact_39_numeral__le__iff,axiom,
    ! [M: num,Nb: 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,Nb))
    <=> aa_num_o(ord_less_eq_num(M),Nb) ) ).

% numeral_le_iff
tff(fact_40_numeral__le__iff,axiom,
    ! [M: num,Nb: 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,Nb))
    <=> aa_num_o(ord_less_eq_num(M),Nb) ) ).

% numeral_le_iff
tff(fact_41_numeral__less__iff,axiom,
    ! [M: num,Nb: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,numeral_numeral_real(M)),numeral_numeral_real(Nb))
    <=> aa_num_o(ord_less_num(M),Nb) ) ).

% numeral_less_iff
tff(fact_42_numeral__less__iff,axiom,
    ! [M: num,Nb: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,numeral_numeral_rat(M)),numeral_numeral_rat(Nb))
    <=> aa_num_o(ord_less_num(M),Nb) ) ).

% numeral_less_iff
tff(fact_43_numeral__less__iff,axiom,
    ! [M: num,Nb: 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,Nb))
    <=> aa_num_o(ord_less_num(M),Nb) ) ).

% numeral_less_iff
tff(fact_44_numeral__less__iff,axiom,
    ! [M: num,Nb: 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,Nb))
    <=> aa_num_o(ord_less_num(M),Nb) ) ).

% numeral_less_iff
tff(fact_45_mult__numeral__left__semiring__numeral,axiom,
    ! [V: num,W: num,Z: word_N3645301735248828278l_num1] : aa_wor7274019993978516266l_num1(times_7065122842183080059l_num1(numera7442385471795722001l_num1(V)),aa_wor7274019993978516266l_num1(times_7065122842183080059l_num1(numera7442385471795722001l_num1(W)),Z)) = aa_wor7274019993978516266l_num1(times_7065122842183080059l_num1(numera7442385471795722001l_num1(aa_num_num(aa_num_fun_num_num(times_times_num,V),W))),Z) ).

% mult_numeral_left_semiring_numeral
tff(fact_46_mult__numeral__left__semiring__numeral,axiom,
    ! [V: num,W: num,Z: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(V)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(W)),Z)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(aa_num_fun_num_num(times_times_num,V),W))),Z) ).

% mult_numeral_left_semiring_numeral
tff(fact_47_mult__numeral__left__semiring__numeral,axiom,
    ! [V: num,W: num,Z: rat] : aa_rat_rat(times_times_rat(numeral_numeral_rat(V)),aa_rat_rat(times_times_rat(numeral_numeral_rat(W)),Z)) = aa_rat_rat(times_times_rat(numeral_numeral_rat(aa_num_num(aa_num_fun_num_num(times_times_num,V),W))),Z) ).

% mult_numeral_left_semiring_numeral
tff(fact_48_mult__numeral__left__semiring__numeral,axiom,
    ! [V: num,W: 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,W)),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),W))),Z) ).

% mult_numeral_left_semiring_numeral
tff(fact_49_mult__numeral__left__semiring__numeral,axiom,
    ! [V: num,W: num,Z: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,V)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,W)),Z)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(times_times_num,V),W))),Z) ).

% mult_numeral_left_semiring_numeral
tff(fact_50_numeral__times__numeral,axiom,
    ! [M: num,Nb: num] : aa_wor7274019993978516266l_num1(times_7065122842183080059l_num1(numera7442385471795722001l_num1(M)),numera7442385471795722001l_num1(Nb)) = numera7442385471795722001l_num1(aa_num_num(aa_num_fun_num_num(times_times_num,M),Nb)) ).

% numeral_times_numeral
tff(fact_51_numeral__times__numeral,axiom,
    ! [M: num,Nb: num] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(M)),numeral_numeral_real(Nb)) = numeral_numeral_real(aa_num_num(aa_num_fun_num_num(times_times_num,M),Nb)) ).

% numeral_times_numeral
tff(fact_52_numeral__times__numeral,axiom,
    ! [M: num,Nb: num] : aa_rat_rat(times_times_rat(numeral_numeral_rat(M)),numeral_numeral_rat(Nb)) = numeral_numeral_rat(aa_num_num(aa_num_fun_num_num(times_times_num,M),Nb)) ).

% numeral_times_numeral
tff(fact_53_numeral__times__numeral,axiom,
    ! [M: num,Nb: 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,Nb)) = aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(times_times_num,M),Nb)) ).

% numeral_times_numeral
tff(fact_54_numeral__times__numeral,axiom,
    ! [M: num,Nb: 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,Nb)) = aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(times_times_num,M),Nb)) ).

% numeral_times_numeral
tff(fact_55_num__double,axiom,
    ! [Nb: num] : aa_num_num(aa_num_fun_num_num(times_times_num,aa_num_num(bit0,one)),Nb) = aa_num_num(bit0,Nb) ).

% num_double
tff(fact_56_misiz,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,M: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Ta),Nb)
     => ( ( aa_nat_option_nat(some_nat,M) = vEBT_vebt_mint(Ta) )
       => 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))),Nb)) ) ) ).

% misiz
tff(fact_57_power__one,axiom,
    ! [Nb: nat] : aa_nat_rat(power_power_rat(one_one_rat),Nb) = one_one_rat ).

% power_one
tff(fact_58_power__one,axiom,
    ! [Nb: nat] : aa_nat_uint32(power_power_uint32(one_one_uint32),Nb) = one_one_uint32 ).

% power_one
tff(fact_59_power__one,axiom,
    ! [Nb: nat] : aa_nat5599461682457136004l_num1(power_2184487114949457152l_num1(one_on7727431528512463931l_num1),Nb) = one_on7727431528512463931l_num1 ).

% power_one
tff(fact_60_power__one,axiom,
    ! [Nb: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,one_one_real),Nb) = one_one_real ).

% power_one
tff(fact_61_power__one,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,one_one_nat),Nb) = one_one_nat ).

% power_one
tff(fact_62_power__one,axiom,
    ! [Nb: nat] : aa_nat_int(aa_int_fun_nat_int(power_power_int,one_one_int),Nb) = one_one_int ).

% power_one
tff(fact_63_power__mult__numeral,axiom,
    ! [A: uint32,M: num,Nb: num] : aa_nat_uint32(power_power_uint32(aa_nat_uint32(power_power_uint32(A),aa_num_nat(numeral_numeral_nat,M))),aa_num_nat(numeral_numeral_nat,Nb)) = aa_nat_uint32(power_power_uint32(A),aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(times_times_num,M),Nb))) ).

% power_mult_numeral
tff(fact_64_power__mult__numeral,axiom,
    ! [A: word_N3645301735248828278l_num1,M: num,Nb: num] : aa_nat5599461682457136004l_num1(power_2184487114949457152l_num1(aa_nat5599461682457136004l_num1(power_2184487114949457152l_num1(A),aa_num_nat(numeral_numeral_nat,M))),aa_num_nat(numeral_numeral_nat,Nb)) = aa_nat5599461682457136004l_num1(power_2184487114949457152l_num1(A),aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(times_times_num,M),Nb))) ).

% power_mult_numeral
tff(fact_65_power__mult__numeral,axiom,
    ! [A: real,M: num,Nb: num] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,M))),aa_num_nat(numeral_numeral_nat,Nb)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(times_times_num,M),Nb))) ).

% power_mult_numeral
tff(fact_66_power__mult__numeral,axiom,
    ! [A: nat,M: num,Nb: num] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_num_nat(numeral_numeral_nat,M))),aa_num_nat(numeral_numeral_nat,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(times_times_num,M),Nb))) ).

% power_mult_numeral
tff(fact_67_power__mult__numeral,axiom,
    ! [A: int,M: num,Nb: num] : aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A),aa_num_nat(numeral_numeral_nat,M))),aa_num_nat(numeral_numeral_nat,Nb)) = 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(times_times_num,M),Nb))) ).

% power_mult_numeral
tff(fact_68_helpypredd,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xb: nat,Y: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Ta),Nb)
     => ( ( aa_nat_option_nat(vEBT_vebt_pred(Ta),Xb) = aa_nat_option_nat(some_nat,Y) )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ) ) ).

% helpypredd
tff(fact_69_power__one__right,axiom,
    ! [A: uint32] : aa_nat_uint32(power_power_uint32(A),one_one_nat) = A ).

% power_one_right
tff(fact_70_power__one__right,axiom,
    ! [A: word_N3645301735248828278l_num1] : aa_nat5599461682457136004l_num1(power_2184487114949457152l_num1(A),one_one_nat) = A ).

% power_one_right
tff(fact_71_power__one__right,axiom,
    ! [A: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,A),one_one_nat) = A ).

% power_one_right
tff(fact_72_power__one__right,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),one_one_nat) = A ).

% power_one_right
tff(fact_73_power__one__right,axiom,
    ! [A: int] : aa_nat_int(aa_int_fun_nat_int(power_power_int,A),one_one_nat) = A ).

% power_one_right
tff(fact_74_mi__ma__2__deg,axiom,
    ! [Mia: nat,Maa: nat,Deg: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Nb: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mia),Maa)),Deg,TreeLista),Summarya)),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mia),Maa)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Maa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg)) ) ) ).

% mi_ma_2_deg
tff(fact_75_pred__max,axiom,
    ! [Deg: nat,Maa: nat,Xb: nat,Mia: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Maa),Xb)
       => ( aa_nat_option_nat(vEBT_vebt_pred(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mia),Maa)),Deg,TreeLista),Summarya)),Xb) = aa_nat_option_nat(some_nat,Maa) ) ) ) ).

% pred_max
tff(fact_76_summaxma,axiom,
    ! [Mia: nat,Maa: nat,Deg: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mia),Maa)),Deg,TreeLista),Summarya)),Deg)
     => ( ( Mia != Maa )
       => ( the_nat(vEBT_vebt_maxt(Summarya)) = aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Maa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ) ) ).

% summaxma
tff(fact_77_one__eq__numeral__iff,axiom,
    ! [Nb: num] :
      ( ( one_one_real = numeral_numeral_real(Nb) )
    <=> ( one = Nb ) ) ).

% one_eq_numeral_iff
tff(fact_78_one__eq__numeral__iff,axiom,
    ! [Nb: num] :
      ( ( one_one_rat = numeral_numeral_rat(Nb) )
    <=> ( one = Nb ) ) ).

% one_eq_numeral_iff
tff(fact_79_one__eq__numeral__iff,axiom,
    ! [Nb: num] :
      ( ( one_one_nat = aa_num_nat(numeral_numeral_nat,Nb) )
    <=> ( one = Nb ) ) ).

% one_eq_numeral_iff
tff(fact_80_one__eq__numeral__iff,axiom,
    ! [Nb: num] :
      ( ( one_one_int = aa_num_int(numeral_numeral_int,Nb) )
    <=> ( one = Nb ) ) ).

% one_eq_numeral_iff
tff(fact_81_numeral__eq__one__iff,axiom,
    ! [Nb: num] :
      ( ( numeral_numeral_real(Nb) = one_one_real )
    <=> ( Nb = one ) ) ).

% numeral_eq_one_iff
tff(fact_82_numeral__eq__one__iff,axiom,
    ! [Nb: num] :
      ( ( numeral_numeral_rat(Nb) = one_one_rat )
    <=> ( Nb = one ) ) ).

% numeral_eq_one_iff
tff(fact_83_numeral__eq__one__iff,axiom,
    ! [Nb: num] :
      ( ( aa_num_nat(numeral_numeral_nat,Nb) = one_one_nat )
    <=> ( Nb = one ) ) ).

% numeral_eq_one_iff
tff(fact_84_numeral__eq__one__iff,axiom,
    ! [Nb: num] :
      ( ( aa_num_int(numeral_numeral_int,Nb) = one_one_int )
    <=> ( Nb = one ) ) ).

% numeral_eq_one_iff
tff(fact_85_power__inject__exp,axiom,
    ! [A: real,M: nat,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,A),M) = aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb) )
      <=> ( M = Nb ) ) ) ).

% power_inject_exp
tff(fact_86_power__inject__exp,axiom,
    ! [A: rat,M: nat,Nb: 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),Nb) )
      <=> ( M = Nb ) ) ) ).

% power_inject_exp
tff(fact_87_power__inject__exp,axiom,
    ! [A: nat,M: nat,Nb: 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),Nb) )
      <=> ( M = Nb ) ) ) ).

% power_inject_exp
tff(fact_88_power__inject__exp,axiom,
    ! [A: int,M: nat,Nb: 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),Nb) )
      <=> ( M = Nb ) ) ) ).

% power_inject_exp
tff(fact_89_pred__list__to__short,axiom,
    ! [Deg: nat,Xb: nat,Maa: nat,TreeLista: list_VEBT_VEBT,Mia: nat,Summarya: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Maa)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_s6755466524823107622T_VEBT(TreeLista)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
         => ( aa_nat_option_nat(vEBT_vebt_pred(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mia),Maa)),Deg,TreeLista),Summarya)),Xb) = none_nat ) ) ) ) ).

% pred_list_to_short
tff(fact_90_numeral__le__one__iff,axiom,
    ! [Nb: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,numeral_numeral_real(Nb)),one_one_real)
    <=> aa_num_o(ord_less_eq_num(Nb),one) ) ).

% numeral_le_one_iff
tff(fact_91_numeral__le__one__iff,axiom,
    ! [Nb: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,numeral_numeral_rat(Nb)),one_one_rat)
    <=> aa_num_o(ord_less_eq_num(Nb),one) ) ).

% numeral_le_one_iff
tff(fact_92_numeral__le__one__iff,axiom,
    ! [Nb: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,Nb)),one_one_nat)
    <=> aa_num_o(ord_less_eq_num(Nb),one) ) ).

% numeral_le_one_iff
tff(fact_93_numeral__le__one__iff,axiom,
    ! [Nb: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,Nb)),one_one_int)
    <=> aa_num_o(ord_less_eq_num(Nb),one) ) ).

% numeral_le_one_iff
tff(fact_94_mem__Collect__eq,axiom,
    ! [A: vEBT_VEBT,P: fun_VEBT_VEBT_o] :
      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A),collect_VEBT_VEBT(P))
    <=> aa_VEBT_VEBT_o(P,A) ) ).

% mem_Collect_eq
tff(fact_95_mem__Collect__eq,axiom,
    ! [A: real,P: fun_real_o] :
      ( aa_set_real_o(member_real(A),collect_real(P))
    <=> aa_real_o(P,A) ) ).

% mem_Collect_eq
tff(fact_96_mem__Collect__eq,axiom,
    ! [A: nat,P: fun_nat_o] :
      ( aa_set_nat_o(member_nat(A),collect_nat(P))
    <=> aa_nat_o(P,A) ) ).

% mem_Collect_eq
tff(fact_97_mem__Collect__eq,axiom,
    ! [A: int,P: fun_int_o] :
      ( aa_set_int_o(member_int(A),collect_int(P))
    <=> aa_int_o(P,A) ) ).

% mem_Collect_eq
tff(fact_98_mem__Collect__eq,axiom,
    ! [A: complex,P: fun_complex_o] :
      ( aa_set_complex_o(member_complex(A),collect_complex(P))
    <=> aa_complex_o(P,A) ) ).

% mem_Collect_eq
tff(fact_99_mem__Collect__eq,axiom,
    ! [A: product_prod_int_int,P: fun_Pr3238374608701176162_int_o] :
      ( member5262025264175285858nt_int(A,collec213857154873943460nt_int(P))
    <=> aa_Pro4592068669816069595_int_o(P,A) ) ).

% mem_Collect_eq
tff(fact_100_Collect__mem__eq,axiom,
    ! [A2: set_VEBT_VEBT] : collect_VEBT_VEBT(aTP_Lamm_av(A2)) = A2 ).

% Collect_mem_eq
tff(fact_101_Collect__mem__eq,axiom,
    ! [A2: set_real] : collect_real(aTP_Lamm_aw(A2)) = A2 ).

% Collect_mem_eq
tff(fact_102_Collect__mem__eq,axiom,
    ! [A2: set_nat] : collect_nat(aTP_Lamm_ax(A2)) = A2 ).

% Collect_mem_eq
tff(fact_103_Collect__mem__eq,axiom,
    ! [A2: set_int] : collect_int(aTP_Lamm_ay(A2)) = A2 ).

% Collect_mem_eq
tff(fact_104_Collect__mem__eq,axiom,
    ! [A2: set_complex] : collect_complex(aTP_Lamm_az(A2)) = A2 ).

% Collect_mem_eq
tff(fact_105_Collect__mem__eq,axiom,
    ! [A2: set_Pr958786334691620121nt_int] : collec213857154873943460nt_int(aTP_Lamm_ba(A2)) = A2 ).

% Collect_mem_eq
tff(fact_106_Collect__cong,axiom,
    ! [P: fun_nat_o,Q: fun_nat_o] :
      ( ! [X2: nat] :
          ( aa_nat_o(P,X2)
        <=> aa_nat_o(Q,X2) )
     => ( collect_nat(P) = collect_nat(Q) ) ) ).

% Collect_cong
tff(fact_107_Collect__cong,axiom,
    ! [P: fun_int_o,Q: fun_int_o] :
      ( ! [X2: int] :
          ( aa_int_o(P,X2)
        <=> aa_int_o(Q,X2) )
     => ( collect_int(P) = collect_int(Q) ) ) ).

% Collect_cong
tff(fact_108_Collect__cong,axiom,
    ! [P: fun_complex_o,Q: fun_complex_o] :
      ( ! [X2: complex] :
          ( aa_complex_o(P,X2)
        <=> aa_complex_o(Q,X2) )
     => ( collect_complex(P) = collect_complex(Q) ) ) ).

% Collect_cong
tff(fact_109_Collect__cong,axiom,
    ! [P: fun_Pr3238374608701176162_int_o,Q: fun_Pr3238374608701176162_int_o] :
      ( ! [X2: product_prod_int_int] :
          ( aa_Pro4592068669816069595_int_o(P,X2)
        <=> aa_Pro4592068669816069595_int_o(Q,X2) )
     => ( collec213857154873943460nt_int(P) = collec213857154873943460nt_int(Q) ) ) ).

% Collect_cong
tff(fact_110_one__less__numeral__iff,axiom,
    ! [Nb: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),numeral_numeral_real(Nb))
    <=> aa_num_o(ord_less_num(one),Nb) ) ).

% one_less_numeral_iff
tff(fact_111_one__less__numeral__iff,axiom,
    ! [Nb: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),numeral_numeral_rat(Nb))
    <=> aa_num_o(ord_less_num(one),Nb) ) ).

% one_less_numeral_iff
tff(fact_112_one__less__numeral__iff,axiom,
    ! [Nb: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,Nb))
    <=> aa_num_o(ord_less_num(one),Nb) ) ).

% one_less_numeral_iff
tff(fact_113_one__less__numeral__iff,axiom,
    ! [Nb: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_num_int(numeral_numeral_int,Nb))
    <=> aa_num_o(ord_less_num(one),Nb) ) ).

% one_less_numeral_iff
tff(fact_114_le__divide__eq__numeral1_I1_J,axiom,
    ! [A: real,B: real,W: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),numeral_numeral_real(W)))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),numeral_numeral_real(W))),B) ) ).

% le_divide_eq_numeral1(1)
tff(fact_115_le__divide__eq__numeral1_I1_J,axiom,
    ! [A: rat,B: rat,W: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),divide_divide_rat(B,numeral_numeral_rat(W)))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),numeral_numeral_rat(W))),B) ) ).

% le_divide_eq_numeral1(1)
tff(fact_116_divide__le__eq__numeral1_I1_J,axiom,
    ! [B: real,W: num,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),numeral_numeral_real(W))),A)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),numeral_numeral_real(W))) ) ).

% divide_le_eq_numeral1(1)
tff(fact_117_divide__le__eq__numeral1_I1_J,axiom,
    ! [B: rat,W: num,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,divide_divide_rat(B,numeral_numeral_rat(W))),A)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),aa_rat_rat(times_times_rat(A),numeral_numeral_rat(W))) ) ).

% divide_le_eq_numeral1(1)
tff(fact_118_mul__def,axiom,
    vEBT_VEBT_mul = vEBT_V4262088993061758097ft_nat(times_times_nat) ).

% mul_def
tff(fact_119__C7_Oprems_C,axiom,
    aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,mi),ma)),aa_nat_nat(suc,aa_nat_nat(suc,va)),treeList),summary)),na) ).

% "7.prems"
tff(fact_120_local_Opower__def,axiom,
    vEBT_VEBT_power = vEBT_V4262088993061758097ft_nat(power_power_nat) ).

% local.power_def
tff(fact_121_le__num__One__iff,axiom,
    ! [Xb: num] :
      ( aa_num_o(ord_less_eq_num(Xb),one)
    <=> ( Xb = one ) ) ).

% le_num_One_iff
tff(fact_122_power__mult,axiom,
    ! [A: uint32,M: nat,Nb: nat] : aa_nat_uint32(power_power_uint32(A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Nb)) = aa_nat_uint32(power_power_uint32(aa_nat_uint32(power_power_uint32(A),M)),Nb) ).

% power_mult
tff(fact_123_power__mult,axiom,
    ! [A: word_N3645301735248828278l_num1,M: nat,Nb: nat] : aa_nat5599461682457136004l_num1(power_2184487114949457152l_num1(A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Nb)) = aa_nat5599461682457136004l_num1(power_2184487114949457152l_num1(aa_nat5599461682457136004l_num1(power_2184487114949457152l_num1(A),M)),Nb) ).

% power_mult
tff(fact_124_power__mult,axiom,
    ! [A: real,M: nat,Nb: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Nb)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),M)),Nb) ).

% power_mult
tff(fact_125_power__mult,axiom,
    ! [A: nat,M: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),M)),Nb) ).

% power_mult
tff(fact_126_power__mult,axiom,
    ! [A: int,M: nat,Nb: nat] : aa_nat_int(aa_int_fun_nat_int(power_power_int,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Nb)) = aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A),M)),Nb) ).

% power_mult
tff(fact_127_div__mult2__eq,axiom,
    ! [M: nat,Nb: nat,Q2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),Q2)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Nb)),Q2) ).

% div_mult2_eq
tff(fact_128_div__mult2__numeral__eq,axiom,
    ! [A: nat,K: num,L: num] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,K))),aa_num_nat(numeral_numeral_nat,L)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(times_times_num,K),L))) ).

% div_mult2_numeral_eq
tff(fact_129_div__mult2__numeral__eq,axiom,
    ! [A: int,K: num,L: num] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,K))),aa_num_int(numeral_numeral_int,L)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(times_times_num,K),L))) ).

% div_mult2_numeral_eq
tff(fact_130_less__mult__imp__div__less,axiom,
    ! [M: nat,I: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),Nb))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Nb)),I) ) ).

% less_mult_imp_div_less
tff(fact_131_times__div__less__eq__dividend,axiom,
    ! [Nb: 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,Nb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Nb))),M) ).

% times_div_less_eq_dividend
tff(fact_132_div__times__less__eq__dividend,axiom,
    ! [M: nat,Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Nb)),Nb)),M) ).

% div_times_less_eq_dividend
tff(fact_133_power__even__eq,axiom,
    ! [A: uint32,Nb: nat] : aa_nat_uint32(power_power_uint32(A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = aa_nat_uint32(power_power_uint32(aa_nat_uint32(power_power_uint32(A),Nb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power_even_eq
tff(fact_134_power__even__eq,axiom,
    ! [A: word_N3645301735248828278l_num1,Nb: nat] : aa_nat5599461682457136004l_num1(power_2184487114949457152l_num1(A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = aa_nat5599461682457136004l_num1(power_2184487114949457152l_num1(aa_nat5599461682457136004l_num1(power_2184487114949457152l_num1(A),Nb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power_even_eq
tff(fact_135_power__even__eq,axiom,
    ! [A: real,Nb: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power_even_eq
tff(fact_136_power__even__eq,axiom,
    ! [A: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power_even_eq
tff(fact_137_power__even__eq,axiom,
    ! [A: int,Nb: nat] : 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))),Nb)) = aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A),Nb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power_even_eq
tff(fact_138_div__nat__eqI,axiom,
    ! [Nb: nat,Q2: 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,Nb),Q2)),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),aa_nat_nat(suc,Q2)))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Nb) = Q2 ) ) ) ).

% div_nat_eqI
tff(fact_139_le__numeral__extra_I4_J,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),one_one_real) ).

% le_numeral_extra(4)
tff(fact_140_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_141_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_142_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_143_less__numeral__extra_I4_J,axiom,
    ~ aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),one_one_real) ).

% less_numeral_extra(4)
tff(fact_144_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_145_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_146_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_147_power__commuting__commutes,axiom,
    ! [Xb: uint32,Y: uint32,Nb: nat] :
      ( ( aa_uint32_uint32(times_times_uint32(Xb),Y) = aa_uint32_uint32(times_times_uint32(Y),Xb) )
     => ( aa_uint32_uint32(times_times_uint32(aa_nat_uint32(power_power_uint32(Xb),Nb)),Y) = aa_uint32_uint32(times_times_uint32(Y),aa_nat_uint32(power_power_uint32(Xb),Nb)) ) ) ).

% power_commuting_commutes
tff(fact_148_power__commuting__commutes,axiom,
    ! [Xb: word_N3645301735248828278l_num1,Y: word_N3645301735248828278l_num1,Nb: nat] :
      ( ( aa_wor7274019993978516266l_num1(times_7065122842183080059l_num1(Xb),Y) = aa_wor7274019993978516266l_num1(times_7065122842183080059l_num1(Y),Xb) )
     => ( aa_wor7274019993978516266l_num1(times_7065122842183080059l_num1(aa_nat5599461682457136004l_num1(power_2184487114949457152l_num1(Xb),Nb)),Y) = aa_wor7274019993978516266l_num1(times_7065122842183080059l_num1(Y),aa_nat5599461682457136004l_num1(power_2184487114949457152l_num1(Xb),Nb)) ) ) ).

% power_commuting_commutes
tff(fact_149_power__commuting__commutes,axiom,
    ! [Xb: real,Y: real,Nb: nat] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Y) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Xb) )
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb)),Y) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb)) ) ) ).

% power_commuting_commutes
tff(fact_150_power__commuting__commutes,axiom,
    ! [Xb: rat,Y: rat,Nb: nat] :
      ( ( aa_rat_rat(times_times_rat(Xb),Y) = aa_rat_rat(times_times_rat(Y),Xb) )
     => ( aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(Xb),Nb)),Y) = aa_rat_rat(times_times_rat(Y),aa_nat_rat(power_power_rat(Xb),Nb)) ) ) ).

% power_commuting_commutes
tff(fact_151_power__commuting__commutes,axiom,
    ! [Xb: nat,Y: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Xb),Y) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Y),Xb) )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Xb),Nb)),Y) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Xb),Nb)) ) ) ).

% power_commuting_commutes
tff(fact_152_power__commuting__commutes,axiom,
    ! [Xb: int,Y: int,Nb: nat] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,Xb),Y) = aa_int_int(aa_int_fun_int_int(times_times_int,Y),Xb) )
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,Xb),Nb)),Y) = aa_int_int(aa_int_fun_int_int(times_times_int,Y),aa_nat_int(aa_int_fun_nat_int(power_power_int,Xb),Nb)) ) ) ).

% power_commuting_commutes
tff(fact_153_power__mult__distrib,axiom,
    ! [A: uint32,B: uint32,Nb: nat] : aa_nat_uint32(power_power_uint32(aa_uint32_uint32(times_times_uint32(A),B)),Nb) = aa_uint32_uint32(times_times_uint32(aa_nat_uint32(power_power_uint32(A),Nb)),aa_nat_uint32(power_power_uint32(B),Nb)) ).

% power_mult_distrib
tff(fact_154_power__mult__distrib,axiom,
    ! [A: word_N3645301735248828278l_num1,B: word_N3645301735248828278l_num1,Nb: nat] : aa_nat5599461682457136004l_num1(power_2184487114949457152l_num1(aa_wor7274019993978516266l_num1(times_7065122842183080059l_num1(A),B)),Nb) = aa_wor7274019993978516266l_num1(times_7065122842183080059l_num1(aa_nat5599461682457136004l_num1(power_2184487114949457152l_num1(A),Nb)),aa_nat5599461682457136004l_num1(power_2184487114949457152l_num1(B),Nb)) ).

% power_mult_distrib
tff(fact_155_power__mult__distrib,axiom,
    ! [A: real,B: real,Nb: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),Nb) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Nb)) ).

% power_mult_distrib
tff(fact_156_power__mult__distrib,axiom,
    ! [A: rat,B: rat,Nb: nat] : aa_nat_rat(power_power_rat(aa_rat_rat(times_times_rat(A),B)),Nb) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(A),Nb)),aa_nat_rat(power_power_rat(B),Nb)) ).

% power_mult_distrib
tff(fact_157_power__mult__distrib,axiom,
    ! [A: nat,B: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Nb)) ).

% power_mult_distrib
tff(fact_158_power__mult__distrib,axiom,
    ! [A: int,B: int,Nb: 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)),Nb) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A),Nb)),aa_nat_int(aa_int_fun_nat_int(power_power_int,B),Nb)) ).

% power_mult_distrib
tff(fact_159_power__commutes,axiom,
    ! [A: uint32,Nb: nat] : aa_uint32_uint32(times_times_uint32(aa_nat_uint32(power_power_uint32(A),Nb)),A) = aa_uint32_uint32(times_times_uint32(A),aa_nat_uint32(power_power_uint32(A),Nb)) ).

% power_commutes
tff(fact_160_power__commutes,axiom,
    ! [A: word_N3645301735248828278l_num1,Nb: nat] : aa_wor7274019993978516266l_num1(times_7065122842183080059l_num1(aa_nat5599461682457136004l_num1(power_2184487114949457152l_num1(A),Nb)),A) = aa_wor7274019993978516266l_num1(times_7065122842183080059l_num1(A),aa_nat5599461682457136004l_num1(power_2184487114949457152l_num1(A),Nb)) ).

% power_commutes
tff(fact_161_power__commutes,axiom,
    ! [A: real,Nb: nat] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)),A) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)) ).

% power_commutes
tff(fact_162_power__commutes,axiom,
    ! [A: rat,Nb: nat] : aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(A),Nb)),A) = aa_rat_rat(times_times_rat(A),aa_nat_rat(power_power_rat(A),Nb)) ).

% power_commutes
tff(fact_163_power__commutes,axiom,
    ! [A: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)),A) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)) ).

% power_commutes
tff(fact_164_power__commutes,axiom,
    ! [A: int,Nb: nat] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A),Nb)),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),Nb)) ).

% power_commutes
tff(fact_165_power__divide,axiom,
    ! [A: real,B: real,Nb: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B)),Nb) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Nb)) ).

% power_divide
tff(fact_166_power__divide,axiom,
    ! [A: rat,B: rat,Nb: nat] : aa_nat_rat(power_power_rat(divide_divide_rat(A,B)),Nb) = divide_divide_rat(aa_nat_rat(power_power_rat(A),Nb),aa_nat_rat(power_power_rat(B),Nb)) ).

% power_divide
tff(fact_167_div__le__dividend,axiom,
    ! [M: nat,Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Nb)),M) ).

% div_le_dividend
tff(fact_168_div__le__mono,axiom,
    ! [M: nat,Nb: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
     => 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,Nb),K)) ) ).

% div_le_mono
tff(fact_169_one__le__numeral,axiom,
    ! [Nb: num] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),numeral_numeral_real(Nb)) ).

% one_le_numeral
tff(fact_170_one__le__numeral,axiom,
    ! [Nb: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),numeral_numeral_rat(Nb)) ).

% one_le_numeral
tff(fact_171_one__le__numeral,axiom,
    ! [Nb: num] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,Nb)) ).

% one_le_numeral
tff(fact_172_one__le__numeral,axiom,
    ! [Nb: num] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_num_int(numeral_numeral_int,Nb)) ).

% one_le_numeral
tff(fact_173_not__numeral__less__one,axiom,
    ! [Nb: num] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,numeral_numeral_real(Nb)),one_one_real) ).

% not_numeral_less_one
tff(fact_174_not__numeral__less__one,axiom,
    ! [Nb: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,numeral_numeral_rat(Nb)),one_one_rat) ).

% not_numeral_less_one
tff(fact_175_not__numeral__less__one,axiom,
    ! [Nb: num] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_num_nat(numeral_numeral_nat,Nb)),one_one_nat) ).

% not_numeral_less_one
tff(fact_176_not__numeral__less__one,axiom,
    ! [Nb: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,Nb)),one_one_int) ).

% not_numeral_less_one
tff(fact_177_numeral__One,axiom,
    numera7442385471795722001l_num1(one) = one_on7727431528512463931l_num1 ).

% numeral_One
tff(fact_178_numeral__One,axiom,
    numeral_numeral_real(one) = one_one_real ).

% numeral_One
tff(fact_179_numeral__One,axiom,
    numeral_numeral_rat(one) = one_one_rat ).

% numeral_One
tff(fact_180_numeral__One,axiom,
    aa_num_nat(numeral_numeral_nat,one) = one_one_nat ).

% numeral_One
tff(fact_181_numeral__One,axiom,
    aa_num_int(numeral_numeral_int,one) = one_one_int ).

% numeral_One
tff(fact_182_mult__numeral__1__right,axiom,
    ! [A: word_N3645301735248828278l_num1] : aa_wor7274019993978516266l_num1(times_7065122842183080059l_num1(A),numera7442385471795722001l_num1(one)) = A ).

% mult_numeral_1_right
tff(fact_183_mult__numeral__1__right,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),numeral_numeral_real(one)) = A ).

% mult_numeral_1_right
tff(fact_184_mult__numeral__1__right,axiom,
    ! [A: rat] : aa_rat_rat(times_times_rat(A),numeral_numeral_rat(one)) = A ).

% mult_numeral_1_right
tff(fact_185_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_186_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_187_mult__numeral__1,axiom,
    ! [A: word_N3645301735248828278l_num1] : aa_wor7274019993978516266l_num1(times_7065122842183080059l_num1(numera7442385471795722001l_num1(one)),A) = A ).

% mult_numeral_1
tff(fact_188_mult__numeral__1,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(one)),A) = A ).

% mult_numeral_1
tff(fact_189_mult__numeral__1,axiom,
    ! [A: rat] : aa_rat_rat(times_times_rat(numeral_numeral_rat(one)),A) = A ).

% mult_numeral_1
tff(fact_190_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_191_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_192_one__le__power,axiom,
    ! [A: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),A)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)) ) ).

% one_le_power
tff(fact_193_one__le__power,axiom,
    ! [A: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),A)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_nat_rat(power_power_rat(A),Nb)) ) ).

% one_le_power
tff(fact_194_one__le__power,axiom,
    ! [A: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),A)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)) ) ).

% one_le_power
tff(fact_195_one__le__power,axiom,
    ! [A: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_nat_int(aa_int_fun_nat_int(power_power_int,A),Nb)) ) ).

% one_le_power
tff(fact_196_divide__numeral__1,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),numeral_numeral_real(one)) = A ).

% divide_numeral_1
tff(fact_197_divide__numeral__1,axiom,
    ! [A: rat] : divide_divide_rat(A,numeral_numeral_rat(one)) = A ).

% divide_numeral_1
tff(fact_198_left__right__inverse__power,axiom,
    ! [Xb: uint32,Y: uint32,Nb: nat] :
      ( ( aa_uint32_uint32(times_times_uint32(Xb),Y) = one_one_uint32 )
     => ( aa_uint32_uint32(times_times_uint32(aa_nat_uint32(power_power_uint32(Xb),Nb)),aa_nat_uint32(power_power_uint32(Y),Nb)) = one_one_uint32 ) ) ).

% left_right_inverse_power
tff(fact_199_left__right__inverse__power,axiom,
    ! [Xb: word_N3645301735248828278l_num1,Y: word_N3645301735248828278l_num1,Nb: nat] :
      ( ( aa_wor7274019993978516266l_num1(times_7065122842183080059l_num1(Xb),Y) = one_on7727431528512463931l_num1 )
     => ( aa_wor7274019993978516266l_num1(times_7065122842183080059l_num1(aa_nat5599461682457136004l_num1(power_2184487114949457152l_num1(Xb),Nb)),aa_nat5599461682457136004l_num1(power_2184487114949457152l_num1(Y),Nb)) = one_on7727431528512463931l_num1 ) ) ).

% left_right_inverse_power
tff(fact_200_left__right__inverse__power,axiom,
    ! [Xb: real,Y: real,Nb: nat] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Y) = one_one_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),Nb)) = one_one_real ) ) ).

% left_right_inverse_power
tff(fact_201_left__right__inverse__power,axiom,
    ! [Xb: rat,Y: rat,Nb: nat] :
      ( ( aa_rat_rat(times_times_rat(Xb),Y) = one_one_rat )
     => ( aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(Xb),Nb)),aa_nat_rat(power_power_rat(Y),Nb)) = one_one_rat ) ) ).

% left_right_inverse_power
tff(fact_202_left__right__inverse__power,axiom,
    ! [Xb: nat,Y: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Xb),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,Xb),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Y),Nb)) = one_one_nat ) ) ).

% left_right_inverse_power
tff(fact_203_left__right__inverse__power,axiom,
    ! [Xb: int,Y: int,Nb: nat] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,Xb),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,Xb),Nb)),aa_nat_int(aa_int_fun_nat_int(power_power_int,Y),Nb)) = one_one_int ) ) ).

% left_right_inverse_power
tff(fact_204_power__Suc2,axiom,
    ! [A: uint32,Nb: nat] : aa_nat_uint32(power_power_uint32(A),aa_nat_nat(suc,Nb)) = aa_uint32_uint32(times_times_uint32(aa_nat_uint32(power_power_uint32(A),Nb)),A) ).

% power_Suc2
tff(fact_205_power__Suc2,axiom,
    ! [A: word_N3645301735248828278l_num1,Nb: nat] : aa_nat5599461682457136004l_num1(power_2184487114949457152l_num1(A),aa_nat_nat(suc,Nb)) = aa_wor7274019993978516266l_num1(times_7065122842183080059l_num1(aa_nat5599461682457136004l_num1(power_2184487114949457152l_num1(A),Nb)),A) ).

% power_Suc2
tff(fact_206_power__Suc2,axiom,
    ! [A: real,Nb: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(suc,Nb)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)),A) ).

% power_Suc2
tff(fact_207_power__Suc2,axiom,
    ! [A: rat,Nb: nat] : aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,Nb)) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(A),Nb)),A) ).

% power_Suc2
tff(fact_208_power__Suc2,axiom,
    ! [A: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(suc,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)),A) ).

% power_Suc2
tff(fact_209_power__Suc2,axiom,
    ! [A: int,Nb: nat] : aa_nat_int(aa_int_fun_nat_int(power_power_int,A),aa_nat_nat(suc,Nb)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A),Nb)),A) ).

% power_Suc2
tff(fact_210_power__Suc,axiom,
    ! [A: uint32,Nb: nat] : aa_nat_uint32(power_power_uint32(A),aa_nat_nat(suc,Nb)) = aa_uint32_uint32(times_times_uint32(A),aa_nat_uint32(power_power_uint32(A),Nb)) ).

% power_Suc
tff(fact_211_power__Suc,axiom,
    ! [A: word_N3645301735248828278l_num1,Nb: nat] : aa_nat5599461682457136004l_num1(power_2184487114949457152l_num1(A),aa_nat_nat(suc,Nb)) = aa_wor7274019993978516266l_num1(times_7065122842183080059l_num1(A),aa_nat5599461682457136004l_num1(power_2184487114949457152l_num1(A),Nb)) ).

% power_Suc
tff(fact_212_power__Suc,axiom,
    ! [A: real,Nb: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(suc,Nb)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)) ).

% power_Suc
tff(fact_213_power__Suc,axiom,
    ! [A: rat,Nb: nat] : aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,Nb)) = aa_rat_rat(times_times_rat(A),aa_nat_rat(power_power_rat(A),Nb)) ).

% power_Suc
tff(fact_214_power__Suc,axiom,
    ! [A: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(suc,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)) ).

% power_Suc
tff(fact_215_power__Suc,axiom,
    ! [A: int,Nb: nat] : aa_nat_int(aa_int_fun_nat_int(power_power_int,A),aa_nat_nat(suc,Nb)) = aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_nat_int(aa_int_fun_nat_int(power_power_int,A),Nb)) ).

% power_Suc
tff(fact_216_power__one__over,axiom,
    ! [A: real,Nb: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),A)),Nb) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)) ).

% power_one_over
tff(fact_217_power__one__over,axiom,
    ! [A: rat,Nb: nat] : aa_nat_rat(power_power_rat(divide_divide_rat(one_one_rat,A)),Nb) = divide_divide_rat(one_one_rat,aa_nat_rat(power_power_rat(A),Nb)) ).

% power_one_over
tff(fact_218_numerals_I1_J,axiom,
    aa_num_nat(numeral_numeral_nat,one) = one_one_nat ).

% numerals(1)
tff(fact_219_Suc__div__le__mono,axiom,
    ! [M: nat,Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,M)),Nb)) ).

% Suc_div_le_mono
tff(fact_220_numeral__Bit0__div__2,axiom,
    ! [Nb: num] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Nb))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_num_nat(numeral_numeral_nat,Nb) ).

% numeral_Bit0_div_2
tff(fact_221_numeral__Bit0__div__2,axiom,
    ! [Nb: num] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = aa_num_int(numeral_numeral_int,Nb) ).

% numeral_Bit0_div_2
tff(fact_222_power__less__power__Suc,axiom,
    ! [A: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb))) ) ).

% power_less_power_Suc
tff(fact_223_power__less__power__Suc,axiom,
    ! [A: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),A)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(A),Nb)),aa_rat_rat(times_times_rat(A),aa_nat_rat(power_power_rat(A),Nb))) ) ).

% power_less_power_Suc
tff(fact_224_power__less__power__Suc,axiom,
    ! [A: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),A)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb))) ) ).

% power_less_power_Suc
tff(fact_225_power__less__power__Suc,axiom,
    ! [A: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A),Nb)),aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_nat_int(aa_int_fun_nat_int(power_power_int,A),Nb))) ) ).

% power_less_power_Suc
tff(fact_226_power__gt1__lemma,axiom,
    ! [A: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb))) ) ).

% power_gt1_lemma
tff(fact_227_power__gt1__lemma,axiom,
    ! [A: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),A)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_rat_rat(times_times_rat(A),aa_nat_rat(power_power_rat(A),Nb))) ) ).

% power_gt1_lemma
tff(fact_228_power__gt1__lemma,axiom,
    ! [A: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),A)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb))) ) ).

% power_gt1_lemma
tff(fact_229_power__gt1__lemma,axiom,
    ! [A: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_nat_int(aa_int_fun_nat_int(power_power_int,A),Nb))) ) ).

% power_gt1_lemma
tff(fact_230_power__gt1,axiom,
    ! [A: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(suc,Nb))) ) ).

% power_gt1
tff(fact_231_power__gt1,axiom,
    ! [A: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),A)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,Nb))) ) ).

% power_gt1
tff(fact_232_power__gt1,axiom,
    ! [A: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),A)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(suc,Nb))) ) ).

% power_gt1
tff(fact_233_power__gt1,axiom,
    ! [A: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_nat_int(aa_int_fun_nat_int(power_power_int,A),aa_nat_nat(suc,Nb))) ) ).

% power_gt1
tff(fact_234_power__strict__increasing,axiom,
    ! [Nb: nat,N: nat,A: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),N)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N)) ) ) ).

% power_strict_increasing
tff(fact_235_power__strict__increasing,axiom,
    ! [Nb: nat,N: nat,A: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),N)
     => ( 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),Nb)),aa_nat_rat(power_power_rat(A),N)) ) ) ).

% power_strict_increasing
tff(fact_236_power__strict__increasing,axiom,
    ! [Nb: nat,N: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),N)
     => ( 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),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)) ) ) ).

% power_strict_increasing
tff(fact_237_power__strict__increasing,axiom,
    ! [Nb: nat,N: nat,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),N)
     => ( 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),Nb)),aa_nat_int(aa_int_fun_nat_int(power_power_int,A),N)) ) ) ).

% power_strict_increasing
tff(fact_238_power__less__imp__less__exp,axiom,
    ! [A: real,M: nat,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),M)),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb) ) ) ).

% power_less_imp_less_exp
tff(fact_239_power__less__imp__less__exp,axiom,
    ! [A: rat,M: nat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(A),M)),aa_nat_rat(power_power_rat(A),Nb))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb) ) ) ).

% power_less_imp_less_exp
tff(fact_240_power__less__imp__less__exp,axiom,
    ! [A: nat,M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,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),Nb))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb) ) ) ).

% power_less_imp_less_exp
tff(fact_241_power__less__imp__less__exp,axiom,
    ! [A: int,M: nat,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_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),Nb))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb) ) ) ).

% power_less_imp_less_exp
tff(fact_242_power__increasing,axiom,
    ! [Nb: nat,N: nat,A: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),A)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N)) ) ) ).

% power_increasing
tff(fact_243_power__increasing,axiom,
    ! [Nb: nat,N: nat,A: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),A)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),Nb)),aa_nat_rat(power_power_rat(A),N)) ) ) ).

% power_increasing
tff(fact_244_power__increasing,axiom,
    ! [Nb: nat,N: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),A)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)) ) ) ).

% power_increasing
tff(fact_245_power__increasing,axiom,
    ! [Nb: nat,N: nat,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,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),Nb)),aa_nat_int(aa_int_fun_nat_int(power_power_int,A),N)) ) ) ).

% power_increasing
tff(fact_246_power__numeral__even,axiom,
    ! [Z: uint32,W: num] :
      aa_nat_uint32(power_power_uint32(Z),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,W))) = $let(
        w: uint32,
        w:= aa_nat_uint32(power_power_uint32(Z),aa_num_nat(numeral_numeral_nat,W)),
        aa_uint32_uint32(times_times_uint32(w),w) ) ).

% power_numeral_even
tff(fact_247_power__numeral__even,axiom,
    ! [Z: word_N3645301735248828278l_num1,W: num] :
      aa_nat5599461682457136004l_num1(power_2184487114949457152l_num1(Z),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,W))) = $let(
        w: word_N3645301735248828278l_num1,
        w:= aa_nat5599461682457136004l_num1(power_2184487114949457152l_num1(Z),aa_num_nat(numeral_numeral_nat,W)),
        aa_wor7274019993978516266l_num1(times_7065122842183080059l_num1(w),w) ) ).

% power_numeral_even
tff(fact_248_power__numeral__even,axiom,
    ! [Z: real,W: num] :
      aa_nat_real(aa_real_fun_nat_real(power_power_real,Z),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,W))) = $let(
        w: real,
        w:= aa_nat_real(aa_real_fun_nat_real(power_power_real,Z),aa_num_nat(numeral_numeral_nat,W)),
        aa_real_real(aa_rea7340605660300375871l_real(times_times_real,w),w) ) ).

% power_numeral_even
tff(fact_249_power__numeral__even,axiom,
    ! [Z: rat,W: num] :
      aa_nat_rat(power_power_rat(Z),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,W))) = $let(
        w: rat,
        w:= aa_nat_rat(power_power_rat(Z),aa_num_nat(numeral_numeral_nat,W)),
        aa_rat_rat(times_times_rat(w),w) ) ).

% power_numeral_even
tff(fact_250_power__numeral__even,axiom,
    ! [Z: nat,W: num] :
      aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Z),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,W))) = $let(
        w: nat,
        w:= aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Z),aa_num_nat(numeral_numeral_nat,W)),
        aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,w),w) ) ).

% power_numeral_even
tff(fact_251_power__numeral__even,axiom,
    ! [Z: int,W: num] :
      aa_nat_int(aa_int_fun_nat_int(power_power_int,Z),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,W))) = $let(
        w: int,
        w:= aa_nat_int(aa_int_fun_nat_int(power_power_int,Z),aa_num_nat(numeral_numeral_nat,W)),
        aa_int_int(aa_int_fun_int_int(times_times_int,w),w) ) ).

% power_numeral_even
tff(fact_252_one__power2,axiom,
    aa_nat_rat(power_power_rat(one_one_rat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_rat ).

% one_power2
tff(fact_253_one__power2,axiom,
    aa_nat_uint32(power_power_uint32(one_one_uint32),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_uint32 ).

% one_power2
tff(fact_254_one__power2,axiom,
    aa_nat5599461682457136004l_num1(power_2184487114949457152l_num1(one_on7727431528512463931l_num1),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_on7727431528512463931l_num1 ).

% one_power2
tff(fact_255_one__power2,axiom,
    aa_nat_real(aa_real_fun_nat_real(power_power_real,one_one_real),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_real ).

% one_power2
tff(fact_256_one__power2,axiom,
    aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_nat ).

% one_power2
tff(fact_257_one__power2,axiom,
    aa_nat_int(aa_int_fun_nat_int(power_power_int,one_one_int),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_int ).

% one_power2
tff(fact_258_power2__eq__square,axiom,
    ! [A: uint32] : aa_nat_uint32(power_power_uint32(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_uint32_uint32(times_times_uint32(A),A) ).

% power2_eq_square
tff(fact_259_power2__eq__square,axiom,
    ! [A: word_N3645301735248828278l_num1] : aa_nat5599461682457136004l_num1(power_2184487114949457152l_num1(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_wor7274019993978516266l_num1(times_7065122842183080059l_num1(A),A) ).

% power2_eq_square
tff(fact_260_power2__eq__square,axiom,
    ! [A: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),A) ).

% power2_eq_square
tff(fact_261_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(times_times_rat(A),A) ).

% power2_eq_square
tff(fact_262_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_263_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_264_power4__eq__xxxx,axiom,
    ! [Xb: uint32] : aa_nat_uint32(power_power_uint32(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))) = aa_uint32_uint32(times_times_uint32(aa_uint32_uint32(times_times_uint32(aa_uint32_uint32(times_times_uint32(Xb),Xb)),Xb)),Xb) ).

% power4_eq_xxxx
tff(fact_265_power4__eq__xxxx,axiom,
    ! [Xb: word_N3645301735248828278l_num1] : aa_nat5599461682457136004l_num1(power_2184487114949457152l_num1(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))) = aa_wor7274019993978516266l_num1(times_7065122842183080059l_num1(aa_wor7274019993978516266l_num1(times_7065122842183080059l_num1(aa_wor7274019993978516266l_num1(times_7065122842183080059l_num1(Xb),Xb)),Xb)),Xb) ).

% power4_eq_xxxx
tff(fact_266_power4__eq__xxxx,axiom,
    ! [Xb: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Xb)),Xb)),Xb) ).

% power4_eq_xxxx
tff(fact_267_power4__eq__xxxx,axiom,
    ! [Xb: rat] : aa_nat_rat(power_power_rat(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))) = aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(Xb),Xb)),Xb)),Xb) ).

% power4_eq_xxxx
tff(fact_268_power4__eq__xxxx,axiom,
    ! [Xb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Xb),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,Xb),Xb)),Xb)),Xb) ).

% power4_eq_xxxx
tff(fact_269_power4__eq__xxxx,axiom,
    ! [Xb: int] : aa_nat_int(aa_int_fun_nat_int(power_power_int,Xb),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,Xb),Xb)),Xb)),Xb) ).

% power4_eq_xxxx
tff(fact_270_power__le__imp__le__exp,axiom,
    ! [A: real,M: nat,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),M)),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb) ) ) ).

% power_le_imp_le_exp
tff(fact_271_power__le__imp__le__exp,axiom,
    ! [A: rat,M: nat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),M)),aa_nat_rat(power_power_rat(A),Nb))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb) ) ) ).

% power_le_imp_le_exp
tff(fact_272_power__le__imp__le__exp,axiom,
    ! [A: nat,M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb) ) ) ).

% power_le_imp_le_exp
tff(fact_273_power__le__imp__le__exp,axiom,
    ! [A: int,M: nat,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A),M)),aa_nat_int(aa_int_fun_nat_int(power_power_int,A),Nb))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb) ) ) ).

% power_le_imp_le_exp
tff(fact_274_power2__nat__le__imp__le,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb) ) ).

% power2_nat_le_imp_le
tff(fact_275_power2__nat__le__eq__le,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb) ) ).

% power2_nat_le_eq_le
tff(fact_276_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_277_power__odd__eq,axiom,
    ! [A: uint32,Nb: nat] : aa_nat_uint32(power_power_uint32(A),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))) = aa_uint32_uint32(times_times_uint32(A),aa_nat_uint32(power_power_uint32(aa_nat_uint32(power_power_uint32(A),Nb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% power_odd_eq
tff(fact_278_power__odd__eq,axiom,
    ! [A: word_N3645301735248828278l_num1,Nb: nat] : aa_nat5599461682457136004l_num1(power_2184487114949457152l_num1(A),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))) = aa_wor7274019993978516266l_num1(times_7065122842183080059l_num1(A),aa_nat5599461682457136004l_num1(power_2184487114949457152l_num1(aa_nat5599461682457136004l_num1(power_2184487114949457152l_num1(A),Nb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% power_odd_eq
tff(fact_279_power__odd__eq,axiom,
    ! [A: real,Nb: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% power_odd_eq
tff(fact_280_power__odd__eq,axiom,
    ! [A: rat,Nb: nat] : aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))) = aa_rat_rat(times_times_rat(A),aa_nat_rat(power_power_rat(aa_nat_rat(power_power_rat(A),Nb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% power_odd_eq
tff(fact_281_power__odd__eq,axiom,
    ! [A: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% power_odd_eq
tff(fact_282_power__odd__eq,axiom,
    ! [A: int,Nb: 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))),Nb))) = 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),Nb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% power_odd_eq
tff(fact_283_vebt__pred_Osimps_I7_J,axiom,
    ! [Mia: nat,Maa: nat,Vaa: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Xb: nat] :
      aa_nat_option_nat(vEBT_vebt_pred(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mia),Maa)),aa_nat_nat(suc,aa_nat_nat(suc,Vaa)),TreeLista),Summarya)),Xb) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Maa),Xb),
        aa_nat_option_nat(some_nat,Maa),
        $let(
          l: nat,
          l:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Vaa))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
          $let(
            h: nat,
            h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Vaa))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
            $ite(
              aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeLista)),
              $let(
                minlow: option_nat,
                minlow:= vEBT_vebt_mint(nth_VEBT_VEBT(TreeLista,h)),
                $ite(
                  ( ( minlow != none_nat )
                  & aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_greater,aa_nat_option_nat(some_nat,l)),minlow) ),
                  aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,aa_nat_option_nat(some_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Vaa))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_option_nat(some_nat,h))),aa_nat_option_nat(vEBT_vebt_pred(nth_VEBT_VEBT(TreeLista,h)),l)),
                  $let(
                    pr: option_nat,
                    pr:= aa_nat_option_nat(vEBT_vebt_pred(Summarya),h),
                    $ite(
                      pr = none_nat,
                      $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mia),Xb),aa_nat_option_nat(some_nat,Mia),none_nat),
                      aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,aa_nat_option_nat(some_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Vaa))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),pr)),vEBT_vebt_maxt(nth_VEBT_VEBT(TreeLista,the_nat(pr)))) ) ) ) ),
              none_nat ) ) ) ) ).

% vebt_pred.simps(7)
tff(fact_284_tdeletemimi_H,axiom,
    ! [Deg: nat,Mia: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Xb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_V1232361888498592333_e_t_e,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mia),Mia)),Deg,TreeLista),Summarya)),Xb)),one_one_nat) ) ).

% tdeletemimi'
tff(fact_285__092_060open_062_092_060And_062x22_Ax21_O_Ati_A_061_ALeafi_Ax21_Ax22_A_092_060Longrightarrow_062_A_060vebt__assn__raw_A_INode_A_ISome_A_Imi_M_Ama_J_J_A_ISuc_A_ISuc_Ava_J_J_AtreeList_Asummary_J_Ati_062_Avebt__predi_H_A_INode_A_ISome_A_Imi_M_Ama_J_J_A_ISuc_A_ISuc_Ava_J_J_AtreeList_Asummary_J_Ati_Ax_A_060_092_060lambda_062r_O_Avebt__assn__raw_A_INode_A_ISome_A_Imi_M_Ama_J_J_A_ISuc_A_ISuc_Ava_J_J_AtreeList_Asummary_J_Ati_A_K_A_092_060up_062_A_Ir_A_061_Avebt__pred_A_INode_A_ISome_A_Imi_M_Ama_J_J_A_ISuc_A_ISuc_Ava_J_J_AtreeList_Asummary_J_Ax_J_062_092_060close_062,axiom,
    ! [X21: $o,X22: $o] :
      ( ( tia = vEBT_Leafi((X21),(X22)) )
     => hoare_7629718768684598413on_nat(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,mi),ma)),aa_nat_nat(suc,aa_nat_nat(suc,va)),treeList),summary)),tia),aa_nat947733127081326238on_nat(aa_VEB3566417204837707778on_nat(aa_VEB1030146501360450209on_nat(vEBT_VEBT_vebt_predi,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,mi),ma)),aa_nat_nat(suc,aa_nat_nat(suc,va)),treeList),summary)),tia),xa),aTP_Lamm_bb) ) ).

% \<open>\<And>x22 x21. ti = Leafi x21 x22 \<Longrightarrow> <vebt_assn_raw (Node (Some (mi, ma)) (Suc (Suc va)) treeList summary) ti> vebt_predi' (Node (Some (mi, ma)) (Suc (Suc va)) treeList summary) ti x <\<lambda>r. vebt_assn_raw (Node (Some (mi, ma)) (Suc (Suc va)) treeList summary) ti * \<up> (r = vebt_pred (Node (Some (mi, ma)) (Suc (Suc va)) treeList summary) x)>\<close>
tff(fact_286_maxt__sound,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xb: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Ta),Nb)
     => ( aa_nat_o(vEBT_VEBT_max_in_set(vEBT_VEBT_set_vebt(Ta)),Xb)
       => ( vEBT_vebt_maxt(Ta) = aa_nat_option_nat(some_nat,Xb) ) ) ) ).

% maxt_sound
tff(fact_287_maxt__corr,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xb: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Ta),Nb)
     => ( ( vEBT_vebt_maxt(Ta) = aa_nat_option_nat(some_nat,Xb) )
       => aa_nat_o(vEBT_VEBT_max_in_set(vEBT_VEBT_set_vebt(Ta)),Xb) ) ) ).

% maxt_corr
tff(fact_288_mint__sound,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xb: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Ta),Nb)
     => ( aa_nat_o(vEBT_VEBT_min_in_set(vEBT_VEBT_set_vebt(Ta)),Xb)
       => ( vEBT_vebt_mint(Ta) = aa_nat_option_nat(some_nat,Xb) ) ) ) ).

% mint_sound
tff(fact_289_mint__corr,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xb: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Ta),Nb)
     => ( ( vEBT_vebt_mint(Ta) = aa_nat_option_nat(some_nat,Xb) )
       => aa_nat_o(vEBT_VEBT_min_in_set(vEBT_VEBT_set_vebt(Ta)),Xb) ) ) ).

% mint_corr
tff(fact_290_nested__mint,axiom,
    ! [Mia: nat,Maa: nat,Deg: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Nb: nat,Vaa: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mia),Maa)),Deg,TreeLista),Summarya)),Nb)
     => ( ( Nb = aa_nat_nat(suc,aa_nat_nat(suc,Vaa)) )
       => ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Maa),Mia)
         => ( ( Maa != Mia )
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summarya))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Vaa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeLista,the_nat(vEBT_vebt_mint(Summarya))))))),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Vaa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),size_s6755466524823107622T_VEBT(TreeLista)) ) ) ) ) ).

% nested_mint
tff(fact_291_member__inv,axiom,
    ! [Mia: nat,Maa: nat,Deg: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Xb: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mia),Maa)),Deg,TreeLista),Summarya)),Xb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg)
        & ( ( Xb = Mia )
          | ( Xb = Maa )
          | ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Maa)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mia),Xb)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeLista))
            & aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,nth_VEBT_VEBT(TreeLista,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ) ) ) ).

% member_inv
tff(fact_292_succ__list__to__short,axiom,
    ! [Deg: nat,Mia: nat,Xb: nat,TreeLista: list_VEBT_VEBT,Maa: nat,Summarya: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mia),Xb)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_s6755466524823107622T_VEBT(TreeLista)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
         => ( aa_nat_option_nat(vEBT_vebt_succ(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mia),Maa)),Deg,TreeLista),Summarya)),Xb) = none_nat ) ) ) ) ).

% succ_list_to_short
tff(fact_293_lowi__hT,axiom,
    ! [Xb: nat,Nb: nat] : aa_nat_o(time_htt_nat(one_one_assn,aa_nat5821310369566504398ap_nat(vEBT_VEBT_lowi(Xb),Nb),aa_nat_fun_nat_assn(aTP_Lamm_bc(Xb),Nb)),one_one_nat) ).

% lowi_hT
tff(fact_294_highi__hT,axiom,
    ! [Xb: nat,Nb: nat] : aa_nat_o(time_htt_nat(one_one_assn,aa_nat5821310369566504398ap_nat(vEBT_VEBT_highi(Xb),Nb),aa_nat_fun_nat_assn(aTP_Lamm_bd(Xb),Nb)),one_one_nat) ).

% highi_hT
tff(fact_295_even__odd__cases,axiom,
    ! [Xb: nat] :
      ( ! [N2: nat] : Xb != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),N2)
     => ~ ! [N2: nat] : Xb != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),aa_nat_nat(suc,N2)) ) ).

% even_odd_cases
tff(fact_296_set__vebt_H__def,axiom,
    ! [Ta: vEBT_VEBT] : vEBT_VEBT_set_vebt(Ta) = collect_nat(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,Ta)) ).

% set_vebt'_def
tff(fact_297_add__shift,axiom,
    ! [Xb: nat,Y: nat,Z: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xb),Y) = Z )
    <=> ( aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_nat_option_nat(some_nat,Xb)),aa_nat_option_nat(some_nat,Y)) = aa_nat_option_nat(some_nat,Z) ) ) ).

% add_shift
tff(fact_298_add__def,axiom,
    vEBT_VEBT_add = vEBT_V4262088993061758097ft_nat(plus_plus_nat) ).

% add_def
tff(fact_299_mint__member,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Maxi: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Ta),Nb)
     => ( ( vEBT_vebt_mint(Ta) = aa_nat_option_nat(some_nat,Maxi) )
       => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,Ta),Maxi) ) ) ).

% mint_member
tff(fact_300_maxt__member,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Maxi: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Ta),Nb)
     => ( ( vEBT_vebt_maxt(Ta) = aa_nat_option_nat(some_nat,Maxi) )
       => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,Ta),Maxi) ) ) ).

% maxt_member
tff(fact_301_mint__corr__help,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Mini: nat,Xb: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Ta),Nb)
     => ( ( vEBT_vebt_mint(Ta) = aa_nat_option_nat(some_nat,Mini) )
       => ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,Ta),Xb)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mini),Xb) ) ) ) ).

% mint_corr_help
tff(fact_302_maxt__corr__help,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Maxi: nat,Xb: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Ta),Nb)
     => ( ( vEBT_vebt_maxt(Ta) = aa_nat_option_nat(some_nat,Maxi) )
       => ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,Ta),Xb)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Maxi) ) ) ) ).

% maxt_corr_help
tff(fact_303_pow__sum,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B) ).

% pow_sum
tff(fact_304_high__bound__aux,axiom,
    ! [Maa: nat,Nb: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Maa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),M)))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Maa),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)) ) ).

% high_bound_aux
tff(fact_305_member__bound,axiom,
    ! [Tree: vEBT_VEBT,Xb: nat,Nb: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,Tree),Xb)
     => ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Tree),Nb)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ) ) ).

% member_bound
tff(fact_306_geqmaxNone,axiom,
    ! [Mia: nat,Maa: nat,Deg: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Nb: nat,Xb: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mia),Maa)),Deg,TreeLista),Summarya)),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Maa),Xb)
       => ( aa_nat_option_nat(vEBT_vebt_succ(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mia),Maa)),Deg,TreeLista),Summarya)),Xb) = none_nat ) ) ) ).

% geqmaxNone
tff(fact_307_high__inv,axiom,
    ! [Xb: nat,Nb: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))),Xb)),Nb) = Y ) ) ).

% high_inv
tff(fact_308_low__inv,axiom,
    ! [Xb: nat,Nb: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))),Xb)),Nb) = Xb ) ) ).

% low_inv
tff(fact_309_add__numeral__left,axiom,
    ! [V: num,W: num,Z: word_N3645301735248828278l_num1] : plus_p361126936061061375l_num1(numera7442385471795722001l_num1(V),plus_p361126936061061375l_num1(numera7442385471795722001l_num1(W),Z)) = plus_p361126936061061375l_num1(numera7442385471795722001l_num1(aa_num_num(aa_num_fun_num_num(plus_plus_num,V),W)),Z) ).

% add_numeral_left
tff(fact_310_add__numeral__left,axiom,
    ! [V: num,W: num,Z: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,numeral_numeral_real(V)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,numeral_numeral_real(W)),Z)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,numeral_numeral_real(aa_num_num(aa_num_fun_num_num(plus_plus_num,V),W))),Z) ).

% add_numeral_left
tff(fact_311_add__numeral__left,axiom,
    ! [V: num,W: num,Z: rat] : aa_rat_rat(plus_plus_rat(numeral_numeral_rat(V)),aa_rat_rat(plus_plus_rat(numeral_numeral_rat(W)),Z)) = aa_rat_rat(plus_plus_rat(numeral_numeral_rat(aa_num_num(aa_num_fun_num_num(plus_plus_num,V),W))),Z) ).

% add_numeral_left
tff(fact_312_add__numeral__left,axiom,
    ! [V: num,W: num,Z: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,V)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,W)),Z)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,V),W))),Z) ).

% add_numeral_left
tff(fact_313_add__numeral__left,axiom,
    ! [V: num,W: num,Z: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,V)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,W)),Z)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(plus_plus_num,V),W))),Z) ).

% add_numeral_left
tff(fact_314_numeral__plus__numeral,axiom,
    ! [M: num,Nb: num] : plus_p361126936061061375l_num1(numera7442385471795722001l_num1(M),numera7442385471795722001l_num1(Nb)) = numera7442385471795722001l_num1(aa_num_num(aa_num_fun_num_num(plus_plus_num,M),Nb)) ).

% numeral_plus_numeral
tff(fact_315_numeral__plus__numeral,axiom,
    ! [M: num,Nb: num] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,numeral_numeral_real(M)),numeral_numeral_real(Nb)) = numeral_numeral_real(aa_num_num(aa_num_fun_num_num(plus_plus_num,M),Nb)) ).

% numeral_plus_numeral
tff(fact_316_numeral__plus__numeral,axiom,
    ! [M: num,Nb: num] : aa_rat_rat(plus_plus_rat(numeral_numeral_rat(M)),numeral_numeral_rat(Nb)) = numeral_numeral_rat(aa_num_num(aa_num_fun_num_num(plus_plus_num,M),Nb)) ).

% numeral_plus_numeral
tff(fact_317_numeral__plus__numeral,axiom,
    ! [M: num,Nb: num] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,Nb)) = aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,M),Nb)) ).

% numeral_plus_numeral
tff(fact_318_numeral__plus__numeral,axiom,
    ! [M: num,Nb: num] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,Nb)) = aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(plus_plus_num,M),Nb)) ).

% numeral_plus_numeral
tff(fact_319_bit__concat__def,axiom,
    ! [H: nat,L: nat,D: nat] : aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_bit_concat(H),L),D) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,H),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),D))),L) ).

% bit_concat_def
tff(fact_320_helpyd,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xb: nat,Y: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Ta),Nb)
     => ( ( aa_nat_option_nat(vEBT_vebt_succ(Ta),Xb) = aa_nat_option_nat(some_nat,Y) )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ) ) ).

% helpyd
tff(fact_321_succ__min,axiom,
    ! [Deg: nat,Xb: nat,Mia: nat,Maa: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Mia)
       => ( aa_nat_option_nat(vEBT_vebt_succ(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mia),Maa)),Deg,TreeLista),Summarya)),Xb) = aa_nat_option_nat(some_nat,Mia) ) ) ) ).

% succ_min
tff(fact_322_member__correct,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xb: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Ta),Nb)
     => ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,Ta),Xb)
      <=> aa_set_nat_o(member_nat(Xb),vEBT_set_vebt(Ta)) ) ) ).

% member_correct
tff(fact_323_set__vebt__set__vebt_H__valid,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Ta),Nb)
     => ( vEBT_set_vebt(Ta) = vEBT_VEBT_set_vebt(Ta) ) ) ).

% set_vebt_set_vebt'_valid
tff(fact_324_distrib__left__numeral,axiom,
    ! [V: num,B: word_N3645301735248828278l_num1,C: word_N3645301735248828278l_num1] : aa_wor7274019993978516266l_num1(times_7065122842183080059l_num1(numera7442385471795722001l_num1(V)),plus_p361126936061061375l_num1(B,C)) = plus_p361126936061061375l_num1(aa_wor7274019993978516266l_num1(times_7065122842183080059l_num1(numera7442385471795722001l_num1(V)),B),aa_wor7274019993978516266l_num1(times_7065122842183080059l_num1(numera7442385471795722001l_num1(V)),C)) ).

% distrib_left_numeral
tff(fact_325_distrib__left__numeral,axiom,
    ! [V: num,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(V)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(V)),B)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(V)),C)) ).

% distrib_left_numeral
tff(fact_326_distrib__left__numeral,axiom,
    ! [V: num,B: rat,C: rat] : aa_rat_rat(times_times_rat(numeral_numeral_rat(V)),aa_rat_rat(plus_plus_rat(B),C)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(numeral_numeral_rat(V)),B)),aa_rat_rat(times_times_rat(numeral_numeral_rat(V)),C)) ).

% distrib_left_numeral
tff(fact_327_distrib__left__numeral,axiom,
    ! [V: num,B: nat,C: 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),C)) = 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)),C)) ).

% distrib_left_numeral
tff(fact_328_distrib__left__numeral,axiom,
    ! [V: num,B: int,C: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,V)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,V)),B)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,V)),C)) ).

% distrib_left_numeral
tff(fact_329_distrib__right__numeral,axiom,
    ! [A: word_N3645301735248828278l_num1,B: word_N3645301735248828278l_num1,V: num] : aa_wor7274019993978516266l_num1(times_7065122842183080059l_num1(plus_p361126936061061375l_num1(A,B)),numera7442385471795722001l_num1(V)) = plus_p361126936061061375l_num1(aa_wor7274019993978516266l_num1(times_7065122842183080059l_num1(A),numera7442385471795722001l_num1(V)),aa_wor7274019993978516266l_num1(times_7065122842183080059l_num1(B),numera7442385471795722001l_num1(V))) ).

% distrib_right_numeral
tff(fact_330_distrib__right__numeral,axiom,
    ! [A: real,B: real,V: num] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),numeral_numeral_real(V)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),numeral_numeral_real(V))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),numeral_numeral_real(V))) ).

% distrib_right_numeral
tff(fact_331_distrib__right__numeral,axiom,
    ! [A: rat,B: rat,V: num] : aa_rat_rat(times_times_rat(aa_rat_rat(plus_plus_rat(A),B)),numeral_numeral_rat(V)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(A),numeral_numeral_rat(V))),aa_rat_rat(times_times_rat(B),numeral_numeral_rat(V))) ).

% distrib_right_numeral
tff(fact_332_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_333_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_334_numeral__plus__one,axiom,
    ! [Nb: num] : plus_p361126936061061375l_num1(numera7442385471795722001l_num1(Nb),one_on7727431528512463931l_num1) = numera7442385471795722001l_num1(aa_num_num(aa_num_fun_num_num(plus_plus_num,Nb),one)) ).

% numeral_plus_one
tff(fact_335_numeral__plus__one,axiom,
    ! [Nb: num] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,numeral_numeral_real(Nb)),one_one_real) = numeral_numeral_real(aa_num_num(aa_num_fun_num_num(plus_plus_num,Nb),one)) ).

% numeral_plus_one
tff(fact_336_numeral__plus__one,axiom,
    ! [Nb: num] : aa_rat_rat(plus_plus_rat(numeral_numeral_rat(Nb)),one_one_rat) = numeral_numeral_rat(aa_num_num(aa_num_fun_num_num(plus_plus_num,Nb),one)) ).

% numeral_plus_one
tff(fact_337_numeral__plus__one,axiom,
    ! [Nb: num] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,Nb)),one_one_nat) = aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,Nb),one)) ).

% numeral_plus_one
tff(fact_338_numeral__plus__one,axiom,
    ! [Nb: num] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,Nb)),one_one_int) = aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(plus_plus_num,Nb),one)) ).

% numeral_plus_one
tff(fact_339_one__plus__numeral,axiom,
    ! [Nb: num] : plus_p361126936061061375l_num1(one_on7727431528512463931l_num1,numera7442385471795722001l_num1(Nb)) = numera7442385471795722001l_num1(aa_num_num(aa_num_fun_num_num(plus_plus_num,one),Nb)) ).

% one_plus_numeral
tff(fact_340_one__plus__numeral,axiom,
    ! [Nb: num] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),numeral_numeral_real(Nb)) = numeral_numeral_real(aa_num_num(aa_num_fun_num_num(plus_plus_num,one),Nb)) ).

% one_plus_numeral
tff(fact_341_one__plus__numeral,axiom,
    ! [Nb: num] : aa_rat_rat(plus_plus_rat(one_one_rat),numeral_numeral_rat(Nb)) = numeral_numeral_rat(aa_num_num(aa_num_fun_num_num(plus_plus_num,one),Nb)) ).

% one_plus_numeral
tff(fact_342_one__plus__numeral,axiom,
    ! [Nb: num] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,Nb)) = aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,one),Nb)) ).

% one_plus_numeral
tff(fact_343_one__plus__numeral,axiom,
    ! [Nb: num] : aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_num_int(numeral_numeral_int,Nb)) = aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(plus_plus_num,one),Nb)) ).

% one_plus_numeral
tff(fact_344_one__add__one,axiom,
    plus_p361126936061061375l_num1(one_on7727431528512463931l_num1,one_on7727431528512463931l_num1) = numera7442385471795722001l_num1(aa_num_num(bit0,one)) ).

% one_add_one
tff(fact_345_one__add__one,axiom,
    aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),one_one_real) = numeral_numeral_real(aa_num_num(bit0,one)) ).

% one_add_one
tff(fact_346_one__add__one,axiom,
    aa_rat_rat(plus_plus_rat(one_one_rat),one_one_rat) = numeral_numeral_rat(aa_num_num(bit0,one)) ).

% one_add_one
tff(fact_347_one__add__one,axiom,
    aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) ).

% one_add_one
tff(fact_348_one__add__one,axiom,
    aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),one_one_int) = aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)) ).

% one_add_one
tff(fact_349_add__2__eq__Suc,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb) = aa_nat_nat(suc,aa_nat_nat(suc,Nb)) ).

% add_2_eq_Suc
tff(fact_350_add__2__eq__Suc_H,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(suc,aa_nat_nat(suc,Nb)) ).

% add_2_eq_Suc'
tff(fact_351_add__self__div__2,axiom,
    ! [M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),M)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = M ).

% add_self_div_2
tff(fact_352_pred__member,axiom,
    ! [Ta: vEBT_VEBT,Xb: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(vEBT_is_pred_in_set(vEBT_VEBT_set_vebt(Ta)),Xb),Y)
    <=> ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,Ta),Y)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),Xb)
        & ! [Z2: nat] :
            ( ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,Ta),Z2)
              & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z2),Xb) )
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Z2),Y) ) ) ) ).

% pred_member
tff(fact_353_succ__member,axiom,
    ! [Ta: vEBT_VEBT,Xb: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(vEBT_is_succ_in_set(vEBT_VEBT_set_vebt(Ta)),Xb),Y)
    <=> ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,Ta),Y)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y)
        & ! [Z2: nat] :
            ( ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,Ta),Z2)
              & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Z2) )
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),Z2) ) ) ) ).

% succ_member
tff(fact_354_pred__corr,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xb: nat,Px: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Ta),Nb)
     => ( ( aa_nat_option_nat(vEBT_vebt_pred(Ta),Xb) = aa_nat_option_nat(some_nat,Px) )
      <=> aa_nat_o(aa_nat_fun_nat_o(vEBT_is_pred_in_set(vEBT_VEBT_set_vebt(Ta)),Xb),Px) ) ) ).

% pred_corr
tff(fact_355_succ__corr,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xb: nat,Sx: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Ta),Nb)
     => ( ( aa_nat_option_nat(vEBT_vebt_succ(Ta),Xb) = aa_nat_option_nat(some_nat,Sx) )
      <=> aa_nat_o(aa_nat_fun_nat_o(vEBT_is_succ_in_set(vEBT_VEBT_set_vebt(Ta)),Xb),Sx) ) ) ).

% succ_corr
tff(fact_356_vebt__succi_H__rf__abstr,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Tib: vEBT_VEBTi,Xb: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Ta),Nb)
     => hoare_7629718768684598413on_nat(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Ta),Tib),aa_nat947733127081326238on_nat(aa_VEB3566417204837707778on_nat(aa_VEB1030146501360450209on_nat(vEBT_VEBT_vebt_succi,Ta),Tib),Xb),aa_nat435388790147128531t_assn(aa_VEB1697702717351837573t_assn(aTP_Lamm_be(Ta),Tib),Xb)) ) ).

% vebt_succi'_rf_abstr
tff(fact_357_is__num__normalize_I1_J,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),C) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C)) ).

% is_num_normalize(1)
tff(fact_358_is__num__normalize_I1_J,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(plus_plus_rat(aa_rat_rat(plus_plus_rat(A),B)),C) = aa_rat_rat(plus_plus_rat(A),aa_rat_rat(plus_plus_rat(B),C)) ).

% is_num_normalize(1)
tff(fact_359_is__num__normalize_I1_J,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C)) ).

% is_num_normalize(1)
tff(fact_360_numeral__Bit0,axiom,
    ! [Nb: num] : numera7442385471795722001l_num1(aa_num_num(bit0,Nb)) = plus_p361126936061061375l_num1(numera7442385471795722001l_num1(Nb),numera7442385471795722001l_num1(Nb)) ).

% numeral_Bit0
tff(fact_361_numeral__Bit0,axiom,
    ! [Nb: num] : numeral_numeral_real(aa_num_num(bit0,Nb)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,numeral_numeral_real(Nb)),numeral_numeral_real(Nb)) ).

% numeral_Bit0
tff(fact_362_numeral__Bit0,axiom,
    ! [Nb: num] : numeral_numeral_rat(aa_num_num(bit0,Nb)) = aa_rat_rat(plus_plus_rat(numeral_numeral_rat(Nb)),numeral_numeral_rat(Nb)) ).

% numeral_Bit0
tff(fact_363_numeral__Bit0,axiom,
    ! [Nb: num] : aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,Nb)),aa_num_nat(numeral_numeral_nat,Nb)) ).

% numeral_Bit0
tff(fact_364_numeral__Bit0,axiom,
    ! [Nb: num] : aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,Nb)),aa_num_int(numeral_numeral_int,Nb)) ).

% numeral_Bit0
tff(fact_365_one__plus__numeral__commute,axiom,
    ! [Xb: num] : plus_p361126936061061375l_num1(one_on7727431528512463931l_num1,numera7442385471795722001l_num1(Xb)) = plus_p361126936061061375l_num1(numera7442385471795722001l_num1(Xb),one_on7727431528512463931l_num1) ).

% one_plus_numeral_commute
tff(fact_366_one__plus__numeral__commute,axiom,
    ! [Xb: num] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),numeral_numeral_real(Xb)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,numeral_numeral_real(Xb)),one_one_real) ).

% one_plus_numeral_commute
tff(fact_367_one__plus__numeral__commute,axiom,
    ! [Xb: num] : aa_rat_rat(plus_plus_rat(one_one_rat),numeral_numeral_rat(Xb)) = aa_rat_rat(plus_plus_rat(numeral_numeral_rat(Xb)),one_one_rat) ).

% one_plus_numeral_commute
tff(fact_368_one__plus__numeral__commute,axiom,
    ! [Xb: num] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,Xb)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,Xb)),one_one_nat) ).

% one_plus_numeral_commute
tff(fact_369_one__plus__numeral__commute,axiom,
    ! [Xb: num] : aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_num_int(numeral_numeral_int,Xb)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,Xb)),one_one_int) ).

% one_plus_numeral_commute
tff(fact_370_power__add,axiom,
    ! [A: uint32,M: nat,Nb: nat] : aa_nat_uint32(power_power_uint32(A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Nb)) = aa_uint32_uint32(times_times_uint32(aa_nat_uint32(power_power_uint32(A),M)),aa_nat_uint32(power_power_uint32(A),Nb)) ).

% power_add
tff(fact_371_power__add,axiom,
    ! [A: word_N3645301735248828278l_num1,M: nat,Nb: nat] : aa_nat5599461682457136004l_num1(power_2184487114949457152l_num1(A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Nb)) = aa_wor7274019993978516266l_num1(times_7065122842183080059l_num1(aa_nat5599461682457136004l_num1(power_2184487114949457152l_num1(A),M)),aa_nat5599461682457136004l_num1(power_2184487114949457152l_num1(A),Nb)) ).

% power_add
tff(fact_372_power__add,axiom,
    ! [A: real,M: nat,Nb: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Nb)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),M)),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)) ).

% power_add
tff(fact_373_power__add,axiom,
    ! [A: rat,M: nat,Nb: nat] : aa_nat_rat(power_power_rat(A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Nb)) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(A),M)),aa_nat_rat(power_power_rat(A),Nb)) ).

% power_add
tff(fact_374_power__add,axiom,
    ! [A: nat,M: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Nb)) = 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),Nb)) ).

% power_add
tff(fact_375_power__add,axiom,
    ! [A: int,M: nat,Nb: 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),Nb)) = 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),Nb)) ).

% power_add
tff(fact_376_numeral__code_I2_J,axiom,
    ! [Nb: num] :
      numera7442385471795722001l_num1(aa_num_num(bit0,Nb)) = $let(
        m: word_N3645301735248828278l_num1,
        m:= numera7442385471795722001l_num1(Nb),
        plus_p361126936061061375l_num1(m,m) ) ).

% numeral_code(2)
tff(fact_377_numeral__code_I2_J,axiom,
    ! [Nb: num] :
      numeral_numeral_real(aa_num_num(bit0,Nb)) = $let(
        m: real,
        m:= numeral_numeral_real(Nb),
        aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,m),m) ) ).

% numeral_code(2)
tff(fact_378_numeral__code_I2_J,axiom,
    ! [Nb: num] :
      numeral_numeral_rat(aa_num_num(bit0,Nb)) = $let(
        m: rat,
        m:= numeral_numeral_rat(Nb),
        aa_rat_rat(plus_plus_rat(m),m) ) ).

% numeral_code(2)
tff(fact_379_numeral__code_I2_J,axiom,
    ! [Nb: num] :
      aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Nb)) = $let(
        m: nat,
        m:= aa_num_nat(numeral_numeral_nat,Nb),
        aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,m),m) ) ).

% numeral_code(2)
tff(fact_380_numeral__code_I2_J,axiom,
    ! [Nb: num] :
      aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb)) = $let(
        m: int,
        m:= aa_num_int(numeral_numeral_int,Nb),
        aa_int_int(aa_int_fun_int_int(plus_plus_int,m),m) ) ).

% numeral_code(2)
tff(fact_381_mult__2,axiom,
    ! [Z: word_N3645301735248828278l_num1] : aa_wor7274019993978516266l_num1(times_7065122842183080059l_num1(numera7442385471795722001l_num1(aa_num_num(bit0,one))),Z) = plus_p361126936061061375l_num1(Z,Z) ).

% mult_2
tff(fact_382_mult__2,axiom,
    ! [Z: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),Z) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Z),Z) ).

% mult_2
tff(fact_383_mult__2,axiom,
    ! [Z: rat] : aa_rat_rat(times_times_rat(numeral_numeral_rat(aa_num_num(bit0,one))),Z) = aa_rat_rat(plus_plus_rat(Z),Z) ).

% mult_2
tff(fact_384_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_385_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_386_mult__2__right,axiom,
    ! [Z: word_N3645301735248828278l_num1] : aa_wor7274019993978516266l_num1(times_7065122842183080059l_num1(Z),numera7442385471795722001l_num1(aa_num_num(bit0,one))) = plus_p361126936061061375l_num1(Z,Z) ).

% mult_2_right
tff(fact_387_mult__2__right,axiom,
    ! [Z: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z),numeral_numeral_real(aa_num_num(bit0,one))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Z),Z) ).

% mult_2_right
tff(fact_388_mult__2__right,axiom,
    ! [Z: rat] : aa_rat_rat(times_times_rat(Z),numeral_numeral_rat(aa_num_num(bit0,one))) = aa_rat_rat(plus_plus_rat(Z),Z) ).

% mult_2_right
tff(fact_389_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_390_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_391_left__add__twice,axiom,
    ! [A: word_N3645301735248828278l_num1,B: word_N3645301735248828278l_num1] : plus_p361126936061061375l_num1(A,plus_p361126936061061375l_num1(A,B)) = plus_p361126936061061375l_num1(aa_wor7274019993978516266l_num1(times_7065122842183080059l_num1(numera7442385471795722001l_num1(aa_num_num(bit0,one))),A),B) ).

% left_add_twice
tff(fact_392_left__add__twice,axiom,
    ! [A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),A)),B) ).

% left_add_twice
tff(fact_393_left__add__twice,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(plus_plus_rat(A),aa_rat_rat(plus_plus_rat(A),B)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(numeral_numeral_rat(aa_num_num(bit0,one))),A)),B) ).

% left_add_twice
tff(fact_394_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_395_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_396_nat__1__add__1,axiom,
    aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) ).

% nat_1_add_1
tff(fact_397_power2__sum,axiom,
    ! [Xb: uint32,Y: uint32] : aa_nat_uint32(power_power_uint32(aa_uint32_uint32(plus_plus_uint32(Xb),Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_uint32_uint32(plus_plus_uint32(aa_uint32_uint32(plus_plus_uint32(aa_nat_uint32(power_power_uint32(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_uint32(power_power_uint32(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_uint32_uint32(times_times_uint32(aa_uint32_uint32(times_times_uint32(numera9087168376688890119uint32(aa_num_num(bit0,one))),Xb)),Y)) ).

% power2_sum
tff(fact_398_power2__sum,axiom,
    ! [Xb: word_N3645301735248828278l_num1,Y: word_N3645301735248828278l_num1] : aa_nat5599461682457136004l_num1(power_2184487114949457152l_num1(plus_p361126936061061375l_num1(Xb,Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = plus_p361126936061061375l_num1(plus_p361126936061061375l_num1(aa_nat5599461682457136004l_num1(power_2184487114949457152l_num1(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat5599461682457136004l_num1(power_2184487114949457152l_num1(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_wor7274019993978516266l_num1(times_7065122842183080059l_num1(aa_wor7274019993978516266l_num1(times_7065122842183080059l_num1(numera7442385471795722001l_num1(aa_num_num(bit0,one))),Xb)),Y)) ).

% power2_sum
tff(fact_399_power2__sum,axiom,
    ! [Xb: real,Y: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),Xb)),Y)) ).

% power2_sum
tff(fact_400_power2__sum,axiom,
    ! [Xb: rat,Y: rat] : aa_nat_rat(power_power_rat(aa_rat_rat(plus_plus_rat(Xb),Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_rat_rat(plus_plus_rat(aa_rat_rat(plus_plus_rat(aa_nat_rat(power_power_rat(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(numeral_numeral_rat(aa_num_num(bit0,one))),Xb)),Y)) ).

% power2_sum
tff(fact_401_power2__sum,axiom,
    ! [Xb: 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,Xb),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,Xb),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))),Xb)),Y)) ).

% power2_sum
tff(fact_402_power2__sum,axiom,
    ! [Xb: 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,Xb),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,Xb),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))),Xb)),Y)) ).

% power2_sum
tff(fact_403_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)
       => ? [N2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),N2)),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,N2),one_one_nat))) ) ) ) ).

% ex_power_ivl1
tff(fact_404_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)
       => ? [N2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),N2)),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,N2),one_one_nat))) ) ) ) ).

% ex_power_ivl2
tff(fact_405_succ__greatereq__min,axiom,
    ! [Deg: nat,Mia: nat,Xb: nat,Maa: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mia),Xb)
       => ( aa_nat_option_nat(vEBT_vebt_succ(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mia),Maa)),Deg,TreeLista),Summarya)),Xb) = $let(
              l: nat,
              l:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
              $let(
                h: nat,
                h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                $ite(
                  aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeLista)),
                  $let(
                    maxlow: option_nat,
                    maxlow:= vEBT_vebt_maxt(nth_VEBT_VEBT(TreeLista,h)),
                    $ite(
                      ( ( maxlow != none_nat )
                      & aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_less,aa_nat_option_nat(some_nat,l)),maxlow) ),
                      aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,aa_nat_option_nat(some_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_option_nat(some_nat,h))),aa_nat_option_nat(vEBT_vebt_succ(nth_VEBT_VEBT(TreeLista,h)),l)),
                      $let(
                        sc: option_nat,
                        sc:= aa_nat_option_nat(vEBT_vebt_succ(Summarya),h),
                        $ite(sc = none_nat,none_nat,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,aa_nat_option_nat(some_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),sc)),vEBT_vebt_mint(nth_VEBT_VEBT(TreeLista,the_nat(sc))))) ) ) ),
                  none_nat ) ) ) ) ) ) ).

% succ_greatereq_min
tff(fact_406_succ__less__length__list,axiom,
    ! [Deg: nat,Mia: nat,Xb: nat,TreeLista: list_VEBT_VEBT,Maa: nat,Summarya: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mia),Xb)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeLista))
         => ( aa_nat_option_nat(vEBT_vebt_succ(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mia),Maa)),Deg,TreeLista),Summarya)),Xb) = $let(
                l: nat,
                l:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                $let(
                  h: nat,
                  h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                  $let(
                    maxlow: option_nat,
                    maxlow:= vEBT_vebt_maxt(nth_VEBT_VEBT(TreeLista,h)),
                    $ite(
                      ( ( maxlow != none_nat )
                      & aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_less,aa_nat_option_nat(some_nat,l)),maxlow) ),
                      aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,aa_nat_option_nat(some_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_option_nat(some_nat,h))),aa_nat_option_nat(vEBT_vebt_succ(nth_VEBT_VEBT(TreeLista,h)),l)),
                      $let(
                        sc: option_nat,
                        sc:= aa_nat_option_nat(vEBT_vebt_succ(Summarya),h),
                        $ite(sc = none_nat,none_nat,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,aa_nat_option_nat(some_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),sc)),vEBT_vebt_mint(nth_VEBT_VEBT(TreeLista,the_nat(sc))))) ) ) ) ) ) ) ) ) ) ).

% succ_less_length_list
tff(fact_407_vebt__member_Osimps_I5_J,axiom,
    ! [Mia: nat,Maa: nat,Vaa: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Xb: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mia),Maa)),aa_nat_nat(suc,aa_nat_nat(suc,Vaa)),TreeLista),Summarya)),Xb)
    <=> $ite(
          Xb = Mia,
          $true,
          $ite(
            Xb = Maa,
            $true,
            $ite(
              aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Mia),
              $false,
              $ite(
                aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Maa),Xb),
                $false,
                $let(
                  h: nat,
                  h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Vaa))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                  $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeLista)),aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,nth_VEBT_VEBT(TreeLista,h)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Vaa))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ).

% vebt_member.simps(5)
tff(fact_408_post__member__pre__member,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xb: nat,Y: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Ta),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))
         => ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,vEBT_vebt_insert(Ta,Xb)),Y)
           => ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,Ta),Y)
              | ( Xb = Y ) ) ) ) ) ) ).

% post_member_pre_member
tff(fact_409_vebt__assn__raw_Osimps_I2_J,axiom,
    ! [Mmo: option4927543243414619207at_nat,Deg: nat,Tree_list: list_VEBT_VEBT,Summarya: vEBT_VEBT,Mmoi: option4927543243414619207at_nat,Degi: nat,Tree_array: array_VEBT_VEBTi,Summaryi: vEBT_VEBTi] :
      aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,aa_VEB5832953974749077948T_VEBT(vEBT_Node(Mmo,Deg,Tree_list),Summarya)),aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(Mmoi,Degi,Tree_array),Summaryi)) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,
          aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,
              aa_o_assn(pure_assn,
                ( ( Mmoi = Mmo )
                & ( Degi = Deg ) ))),aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Summarya),Summaryi))),ex_ass463751140784270563_VEBTi(aa_arr7909941341466502432i_assn(aTP_Lamm_bf(Tree_list),Tree_array))) ).

% vebt_assn_raw.simps(2)
tff(fact_410_insert__simp__mima,axiom,
    ! [Xb: nat,Mia: nat,Maa: nat,Deg: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT] :
      ( ( ( Xb = Mia )
        | ( Xb = Maa ) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg)
       => ( vEBT_vebt_insert(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mia),Maa)),Deg,TreeLista),Summarya),Xb) = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mia),Maa)),Deg,TreeLista),Summarya) ) ) ) ).

% insert_simp_mima
tff(fact_411_mintlistlength,axiom,
    ! [Mia: nat,Maa: nat,Deg: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Nb: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mia),Maa)),Deg,TreeLista),Summarya)),Nb)
     => ( ( Mia != Maa )
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mia),Maa)
          & ? [M2: nat] :
              ( ( aa_nat_option_nat(some_nat,M2) = vEBT_vebt_mint(Summarya) )
              & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ) ) ) ).

% mintlistlength
tff(fact_412_nth__rule,axiom,
    ! [I: nat,Xs: list_int,A: array_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_int(Xs))
     => hoare_3065115510600077593le_int(snga_assn_int(A,Xs),array_nth_int(A,I),aa_arr1589534196239669886t_assn(aa_lis459371148865260371t_assn(aTP_Lamm_bg(I),Xs),A)) ) ).

% nth_rule
tff(fact_413_nth__rule,axiom,
    ! [I: nat,Xs: list_option_nat,A: array_option_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s6086282163384603972on_nat(Xs))
     => hoare_7629718768684598413on_nat(snga_assn_option_nat(A,Xs),aa_nat947733127081326238on_nat(array_nth_option_nat(A),I),aa_arr7483714166967448038t_assn(aa_lis7582702936096254407t_assn(aTP_Lamm_bh(I),Xs),A)) ) ).

% nth_rule
tff(fact_414_nth__rule,axiom,
    ! [I: nat,Xs: list_nat,A: array_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_nat(Xs))
     => hoare_3067605981109127869le_nat(snga_assn_nat(A,Xs),aa_nat5821310369566504398ap_nat(array_nth_nat(A),I),aa_arr3838850077245469638t_assn(aa_lis668879630561721847t_assn(aTP_Lamm_bi(I),Xs),A)) ) ).

% nth_rule
tff(fact_415_nth__rule,axiom,
    ! [I: nat,Xs: list_o,A: array_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_o(Xs))
     => hoare_hoare_triple_o(snga_assn_o(A,Xs),aa_nat2910048432523430448Heap_o(array_nth_o(A),I),aa_arr2544151649598861476o_assn(aa_lis966790859614740569o_assn(aTP_Lamm_bj(I),Xs),A)) ) ).

% nth_rule
tff(fact_416_nth__rule,axiom,
    ! [I: nat,Xs: list_VEBT_VEBTi,A: array_VEBT_VEBTi] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s7982070591426661849_VEBTi(Xs))
     => hoare_1429296392585015714_VEBTi(aa_lis2508312872377092907i_assn(snga_assn_VEBT_VEBTi(A),Xs),aa_nat5408703056993926195_VEBTi(array_nth_VEBT_VEBTi(A),I),aa_arr8081119733880211216i_assn(aa_lis5880058344902901532i_assn(aTP_Lamm_bk(I),Xs),A)) ) ).

% nth_rule
tff(fact_417_nth__rule,axiom,
    ! [I: nat,Xs: list_Product_unit,A: array_Product_unit] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s245203480648594047t_unit(Xs))
     => hoare_8945653483474564448t_unit(snga_a4522542871529764173t_unit(A,Xs),aa_nat5861795706794010277t_unit(array_7872002506669749220t_unit(A),I),aa_arr8761957006411741156t_assn(aa_lis5628126305658872932t_assn(aTP_Lamm_bl(I),Xs),A)) ) ).

% nth_rule
tff(fact_418_power__2__mult__step__le,axiom,
    ! [N3: nat,Nb: nat,K2: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N3),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N3)),K2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)),K))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N3)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K2),one_one_nat))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)),K)) ) ) ).

% power_2_mult_step_le
tff(fact_419_norm__pre__pure__iff__sng,axiom,
    ! [B: $o,F: heap_T2636463487746394924on_nat,Q: fun_option_nat_assn] :
      ( hoare_7629718768684598413on_nat(aa_o_assn(pure_assn,(B)),F,Q)
    <=> ( (B)
       => hoare_7629718768684598413on_nat(one_one_assn,F,Q) ) ) ).

% norm_pre_pure_iff_sng
tff(fact_420_norm__pre__pure__iff__sng,axiom,
    ! [B: $o,F: heap_Time_Heap_nat,Q: fun_nat_assn] :
      ( hoare_3067605981109127869le_nat(aa_o_assn(pure_assn,(B)),F,Q)
    <=> ( (B)
       => hoare_3067605981109127869le_nat(one_one_assn,F,Q) ) ) ).

% norm_pre_pure_iff_sng
tff(fact_421_norm__pre__pure__iff__sng,axiom,
    ! [B: $o,F: heap_Time_Heap_o,Q: fun_o_assn] :
      ( hoare_hoare_triple_o(aa_o_assn(pure_assn,(B)),F,Q)
    <=> ( (B)
       => hoare_hoare_triple_o(one_one_assn,F,Q) ) ) ).

% norm_pre_pure_iff_sng
tff(fact_422_norm__pre__pure__iff__sng,axiom,
    ! [B: $o,F: heap_T8145700208782473153_VEBTi,Q: fun_VEBT_VEBTi_assn] :
      ( hoare_1429296392585015714_VEBTi(aa_o_assn(pure_assn,(B)),F,Q)
    <=> ( (B)
       => hoare_1429296392585015714_VEBTi(one_one_assn,F,Q) ) ) ).

% norm_pre_pure_iff_sng
tff(fact_423_norm__pre__pure__iff__sng,axiom,
    ! [B: $o,F: heap_T5738788834812785303t_unit,Q: fun_Pr6626665806141890376t_assn] :
      ( hoare_8945653483474564448t_unit(aa_o_assn(pure_assn,(B)),F,Q)
    <=> ( (B)
       => hoare_8945653483474564448t_unit(one_one_assn,F,Q) ) ) ).

% norm_pre_pure_iff_sng
tff(fact_424_both__member__options__from__chilf__to__complete__tree,axiom,
    ! [Xb: nat,Deg: nat,TreeLista: list_VEBT_VEBT,Mia: nat,Maa: nat,Summarya: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeLista))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),Deg)
       => ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeLista,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
         => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mia),Maa)),Deg,TreeLista),Summarya)),Xb) ) ) ) ).

% both_member_options_from_chilf_to_complete_tree
tff(fact_425_norm__pre__pure__iff,axiom,
    ! [P: assn,B: $o,F: heap_T2636463487746394924on_nat,Q: fun_option_nat_assn] :
      ( hoare_7629718768684598413on_nat(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,P),aa_o_assn(pure_assn,(B))),F,Q)
    <=> ( (B)
       => hoare_7629718768684598413on_nat(P,F,Q) ) ) ).

% norm_pre_pure_iff
tff(fact_426_norm__pre__pure__iff,axiom,
    ! [P: assn,B: $o,F: heap_Time_Heap_nat,Q: fun_nat_assn] :
      ( hoare_3067605981109127869le_nat(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,P),aa_o_assn(pure_assn,(B))),F,Q)
    <=> ( (B)
       => hoare_3067605981109127869le_nat(P,F,Q) ) ) ).

% norm_pre_pure_iff
tff(fact_427_norm__pre__pure__iff,axiom,
    ! [P: assn,B: $o,F: heap_Time_Heap_o,Q: fun_o_assn] :
      ( hoare_hoare_triple_o(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,P),aa_o_assn(pure_assn,(B))),F,Q)
    <=> ( (B)
       => hoare_hoare_triple_o(P,F,Q) ) ) ).

% norm_pre_pure_iff
tff(fact_428_norm__pre__pure__iff,axiom,
    ! [P: assn,B: $o,F: heap_T8145700208782473153_VEBTi,Q: fun_VEBT_VEBTi_assn] :
      ( hoare_1429296392585015714_VEBTi(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,P),aa_o_assn(pure_assn,(B))),F,Q)
    <=> ( (B)
       => hoare_1429296392585015714_VEBTi(P,F,Q) ) ) ).

% norm_pre_pure_iff
tff(fact_429_norm__pre__pure__iff,axiom,
    ! [P: assn,B: $o,F: heap_T5738788834812785303t_unit,Q: fun_Pr6626665806141890376t_assn] :
      ( hoare_8945653483474564448t_unit(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,P),aa_o_assn(pure_assn,(B))),F,Q)
    <=> ( (B)
       => hoare_8945653483474564448t_unit(P,F,Q) ) ) ).

% norm_pre_pure_iff
tff(fact_430_less__shift,axiom,
    ! [Xb: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y)
    <=> aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_less,aa_nat_option_nat(some_nat,Xb)),aa_nat_option_nat(some_nat,Y)) ) ).

% less_shift
tff(fact_431_maxbmo,axiom,
    ! [Ta: vEBT_VEBT,Xb: nat] :
      ( ( vEBT_vebt_maxt(Ta) = aa_nat_option_nat(some_nat,Xb) )
     => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,Ta),Xb) ) ).

% maxbmo
tff(fact_432_both__member__options__equiv__member,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xb: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Ta),Nb)
     => ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,Ta),Xb)
      <=> aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,Ta),Xb) ) ) ).

% both_member_options_equiv_member
tff(fact_433_valid__member__both__member__options,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xb: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Ta),Nb)
     => ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,Ta),Xb)
       => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,Ta),Xb) ) ) ).

% valid_member_both_member_options
tff(fact_434_highi__h,axiom,
    ! [Xb: nat,Nb: nat] : hoare_3067605981109127869le_nat(one_one_assn,aa_nat5821310369566504398ap_nat(vEBT_VEBT_highi(Xb),Nb),aa_nat_fun_nat_assn(aTP_Lamm_bd(Xb),Nb)) ).

% highi_h
tff(fact_435_lowi__h,axiom,
    ! [Xb: nat,Nb: nat] : hoare_3067605981109127869le_nat(one_one_assn,aa_nat5821310369566504398ap_nat(vEBT_VEBT_lowi(Xb),Nb),aa_nat_fun_nat_assn(aTP_Lamm_bc(Xb),Nb)) ).

% lowi_h
tff(fact_436_VEBTi_Oinject_I1_J,axiom,
    ! [X11a: option4927543243414619207at_nat,X12a: nat,X13a: array_VEBT_VEBTi,X14a: vEBT_VEBTi,Y11: option4927543243414619207at_nat,Y12: nat,Y13: array_VEBT_VEBTi,Y14: vEBT_VEBTi] :
      ( ( aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(X11a,X12a,X13a),X14a) = aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(Y11,Y12,Y13),Y14) )
    <=> ( ( X11a = Y11 )
        & ( X12a = Y12 )
        & ( X13a = Y13 )
        & ( X14a = Y14 ) ) ) ).

% VEBTi.inject(1)
tff(fact_437_VEBTi_Oinject_I2_J,axiom,
    ! [X21: $o,X22: $o,Y21: $o,Y22: $o] :
      ( ( vEBT_Leafi((X21),(X22)) = vEBT_Leafi((Y21),(Y22)) )
    <=> ( ( (X21)
        <=> (Y21) )
        & ( (X22)
        <=> (Y22) ) ) ) ).

% VEBTi.inject(2)
tff(fact_438_pred__correct,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xb: nat,Sx: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Ta),Nb)
     => ( ( aa_nat_option_nat(vEBT_vebt_pred(Ta),Xb) = aa_nat_option_nat(some_nat,Sx) )
      <=> aa_nat_o(aa_nat_fun_nat_o(vEBT_is_pred_in_set(vEBT_set_vebt(Ta)),Xb),Sx) ) ) ).

% pred_correct
tff(fact_439_succ__correct,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xb: nat,Sx: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Ta),Nb)
     => ( ( aa_nat_option_nat(vEBT_vebt_succ(Ta),Xb) = aa_nat_option_nat(some_nat,Sx) )
      <=> aa_nat_o(aa_nat_fun_nat_o(vEBT_is_succ_in_set(vEBT_set_vebt(Ta)),Xb),Sx) ) ) ).

% succ_correct
tff(fact_440_valid__insert__both__member__options__pres,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xb: nat,Y: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Ta),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))
         => ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,Ta),Xb)
           => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,vEBT_vebt_insert(Ta,Y)),Xb) ) ) ) ) ).

% valid_insert_both_member_options_pres
tff(fact_441_valid__insert__both__member__options__add,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xb: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Ta),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))
       => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,vEBT_vebt_insert(Ta,Xb)),Xb) ) ) ).

% valid_insert_both_member_options_add
tff(fact_442_sumprop,axiom,
    aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,summary),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,na),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% sumprop
tff(fact_443_listlength,axiom,
    size_s6755466524823107622T_VEBT(treeList) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,na),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% listlength
tff(fact_444_right__diff__distrib__numeral,axiom,
    ! [V: num,B: word_N3645301735248828278l_num1,C: word_N3645301735248828278l_num1] : aa_wor7274019993978516266l_num1(times_7065122842183080059l_num1(numera7442385471795722001l_num1(V)),minus_4019991460397169231l_num1(B,C)) = minus_4019991460397169231l_num1(aa_wor7274019993978516266l_num1(times_7065122842183080059l_num1(numera7442385471795722001l_num1(V)),B),aa_wor7274019993978516266l_num1(times_7065122842183080059l_num1(numera7442385471795722001l_num1(V)),C)) ).

% right_diff_distrib_numeral
tff(fact_445_right__diff__distrib__numeral,axiom,
    ! [V: num,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(V)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),C)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(V)),B)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(V)),C)) ).

% right_diff_distrib_numeral
tff(fact_446_right__diff__distrib__numeral,axiom,
    ! [V: num,B: rat,C: rat] : aa_rat_rat(times_times_rat(numeral_numeral_rat(V)),minus_minus_rat(B,C)) = minus_minus_rat(aa_rat_rat(times_times_rat(numeral_numeral_rat(V)),B),aa_rat_rat(times_times_rat(numeral_numeral_rat(V)),C)) ).

% right_diff_distrib_numeral
tff(fact_447_right__diff__distrib__numeral,axiom,
    ! [V: num,B: int,C: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,V)),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),C)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,V)),B)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,V)),C)) ).

% right_diff_distrib_numeral
tff(fact_448_left__diff__distrib__numeral,axiom,
    ! [A: word_N3645301735248828278l_num1,B: word_N3645301735248828278l_num1,V: num] : aa_wor7274019993978516266l_num1(times_7065122842183080059l_num1(minus_4019991460397169231l_num1(A,B)),numera7442385471795722001l_num1(V)) = minus_4019991460397169231l_num1(aa_wor7274019993978516266l_num1(times_7065122842183080059l_num1(A),numera7442385471795722001l_num1(V)),aa_wor7274019993978516266l_num1(times_7065122842183080059l_num1(B),numera7442385471795722001l_num1(V))) ).

% left_diff_distrib_numeral
tff(fact_449_left__diff__distrib__numeral,axiom,
    ! [A: real,B: real,V: num] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)),numeral_numeral_real(V)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),numeral_numeral_real(V))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),numeral_numeral_real(V))) ).

% left_diff_distrib_numeral
tff(fact_450_left__diff__distrib__numeral,axiom,
    ! [A: rat,B: rat,V: num] : aa_rat_rat(times_times_rat(minus_minus_rat(A,B)),numeral_numeral_rat(V)) = minus_minus_rat(aa_rat_rat(times_times_rat(A),numeral_numeral_rat(V)),aa_rat_rat(times_times_rat(B),numeral_numeral_rat(V))) ).

% left_diff_distrib_numeral
tff(fact_451_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_452_vebt__mintilist,axiom,
    ! [I: nat,Ts: list_VEBT_VEBT,Tsi: list_VEBT_VEBTi] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s6755466524823107622T_VEBT(Ts))
     => hoare_7629718768684598413on_nat(aa_lis2508312872377092907i_assn(vEBT_L6296928887356842470_VEBTi(vEBT_vebt_assn_raw,Ts),Tsi),aa_VEB2711320435582112505on_nat(vEBT_vebt_minti,nth_VEBT_VEBTi(Tsi,I)),aa_lis7387135221810377032t_assn(aa_lis7458637400188059141t_assn(aTP_Lamm_bm(I),Ts),Tsi)) ) ).

% vebt_mintilist
tff(fact_453_vebt__maxtilist,axiom,
    ! [I: nat,Ts: list_VEBT_VEBT,Tsi: list_VEBT_VEBTi] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s6755466524823107622T_VEBT(Ts))
     => hoare_7629718768684598413on_nat(aa_lis2508312872377092907i_assn(vEBT_L6296928887356842470_VEBTi(vEBT_vebt_assn_raw,Ts),Tsi),aa_VEB2711320435582112505on_nat(vEBT_vebt_maxti,nth_VEBT_VEBTi(Tsi,I)),aa_lis7387135221810377032t_assn(aa_lis7458637400188059141t_assn(aTP_Lamm_bn(I),Ts),Tsi)) ) ).

% vebt_maxtilist
tff(fact_454_both__member__options__ding,axiom,
    ! [Info: option4927543243414619207at_nat,Deg: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Nb: nat,Xb: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info,Deg,TreeLista),Summarya)),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg))
       => ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeLista,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
         => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info,Deg,TreeLista),Summarya)),Xb) ) ) ) ).

% both_member_options_ding
tff(fact_455_both__member__options__from__complete__tree__to__child,axiom,
    ! [Deg: nat,Mia: nat,Maa: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Xb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),Deg)
     => ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mia),Maa)),Deg,TreeLista),Summarya)),Xb)
       => ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeLista,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
          | ( Xb = Mia )
          | ( Xb = Maa ) ) ) ) ).

% both_member_options_from_complete_tree_to_child
tff(fact_456_Suc__numeral,axiom,
    ! [Nb: num] : aa_nat_nat(suc,aa_num_nat(numeral_numeral_nat,Nb)) = aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,Nb),one)) ).

% Suc_numeral
tff(fact_457_power__add__numeral2,axiom,
    ! [A: uint32,M: num,Nb: num,B: uint32] : aa_uint32_uint32(times_times_uint32(aa_nat_uint32(power_power_uint32(A),aa_num_nat(numeral_numeral_nat,M))),aa_uint32_uint32(times_times_uint32(aa_nat_uint32(power_power_uint32(A),aa_num_nat(numeral_numeral_nat,Nb))),B)) = aa_uint32_uint32(times_times_uint32(aa_nat_uint32(power_power_uint32(A),aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,M),Nb)))),B) ).

% power_add_numeral2
tff(fact_458_power__add__numeral2,axiom,
    ! [A: word_N3645301735248828278l_num1,M: num,Nb: num,B: word_N3645301735248828278l_num1] : aa_wor7274019993978516266l_num1(times_7065122842183080059l_num1(aa_nat5599461682457136004l_num1(power_2184487114949457152l_num1(A),aa_num_nat(numeral_numeral_nat,M))),aa_wor7274019993978516266l_num1(times_7065122842183080059l_num1(aa_nat5599461682457136004l_num1(power_2184487114949457152l_num1(A),aa_num_nat(numeral_numeral_nat,Nb))),B)) = aa_wor7274019993978516266l_num1(times_7065122842183080059l_num1(aa_nat5599461682457136004l_num1(power_2184487114949457152l_num1(A),aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,M),Nb)))),B) ).

% power_add_numeral2
tff(fact_459_power__add__numeral2,axiom,
    ! [A: real,M: num,Nb: num,B: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,M))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,Nb))),B)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,M),Nb)))),B) ).

% power_add_numeral2
tff(fact_460_power__add__numeral2,axiom,
    ! [A: rat,M: num,Nb: num,B: rat] : aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,M))),aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,Nb))),B)) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,M),Nb)))),B) ).

% power_add_numeral2
tff(fact_461_power__add__numeral2,axiom,
    ! [A: nat,M: num,Nb: num,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_num_nat(numeral_numeral_nat,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,Nb))),B)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,M),Nb)))),B) ).

% power_add_numeral2
tff(fact_462_power__add__numeral2,axiom,
    ! [A: int,M: num,Nb: 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,Nb))),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),Nb)))),B) ).

% power_add_numeral2
tff(fact_463_power__add__numeral,axiom,
    ! [A: uint32,M: num,Nb: num] : aa_uint32_uint32(times_times_uint32(aa_nat_uint32(power_power_uint32(A),aa_num_nat(numeral_numeral_nat,M))),aa_nat_uint32(power_power_uint32(A),aa_num_nat(numeral_numeral_nat,Nb))) = aa_nat_uint32(power_power_uint32(A),aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,M),Nb))) ).

% power_add_numeral
tff(fact_464_power__add__numeral,axiom,
    ! [A: word_N3645301735248828278l_num1,M: num,Nb: num] : aa_wor7274019993978516266l_num1(times_7065122842183080059l_num1(aa_nat5599461682457136004l_num1(power_2184487114949457152l_num1(A),aa_num_nat(numeral_numeral_nat,M))),aa_nat5599461682457136004l_num1(power_2184487114949457152l_num1(A),aa_num_nat(numeral_numeral_nat,Nb))) = aa_nat5599461682457136004l_num1(power_2184487114949457152l_num1(A),aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,M),Nb))) ).

% power_add_numeral
tff(fact_465_power__add__numeral,axiom,
    ! [A: real,M: num,Nb: num] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,M))),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,Nb))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,M),Nb))) ).

% power_add_numeral
tff(fact_466_power__add__numeral,axiom,
    ! [A: rat,M: num,Nb: num] : aa_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,Nb))) = 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),Nb))) ).

% power_add_numeral
tff(fact_467_power__add__numeral,axiom,
    ! [A: nat,M: num,Nb: num] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_num_nat(numeral_numeral_nat,M))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_num_nat(numeral_numeral_nat,Nb))) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,M),Nb))) ).

% power_add_numeral
tff(fact_468_power__add__numeral,axiom,
    ! [A: int,M: num,Nb: 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,Nb))) = 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),Nb))) ).

% power_add_numeral
tff(fact_469_vebt__memberi_H__rf__abstr,axiom,
    ! [Ta: vEBT_VEBT,Tib: vEBT_VEBTi,Xb: nat] : hoare_hoare_triple_o(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Ta),Tib),aa_nat2910048432523430448Heap_o(aa_VEB1455453220190609250Heap_o(aa_VEB8672304581938048793Heap_o(vEBT_V854960066525838166emberi,Ta),Tib),Xb),aa_nat_fun_o_assn(aa_VEB8176250900481067561o_assn(aTP_Lamm_bo(Ta),Tib),Xb)) ).

% vebt_memberi'_rf_abstr
tff(fact_470_vebt__inserti_H__rf__abstr,axiom,
    ! [Ta: vEBT_VEBT,Tib: vEBT_VEBTi,Xb: nat] : hoare_1429296392585015714_VEBTi(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Ta),Tib),aa_nat5408703056993926195_VEBTi(aa_VEB4833742232818206615_VEBTi(aa_VEB6292075424854053302_VEBTi(vEBT_V3964819847710782039nserti,Ta),Tib),Xb),aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,vEBT_vebt_insert(Ta,Xb))) ).

% vebt_inserti'_rf_abstr
tff(fact_471_diff__diff__less,axiom,
    ! [I: nat,M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Nb)))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),M)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),Nb) ) ) ).

% diff_diff_less
tff(fact_472_add__One__commute,axiom,
    ! [Nb: num] : aa_num_num(aa_num_fun_num_num(plus_plus_num,one),Nb) = aa_num_num(aa_num_fun_num_num(plus_plus_num,Nb),one) ).

% add_One_commute
tff(fact_473_if__rule,axiom,
    ! [B: $o,P: assn,F: heap_T2636463487746394924on_nat,Q: fun_option_nat_assn,G: heap_T2636463487746394924on_nat] :
      ( ( (B)
       => hoare_7629718768684598413on_nat(P,F,Q) )
     => ( ( ~ (B)
         => hoare_7629718768684598413on_nat(P,G,Q) )
       => hoare_7629718768684598413on_nat(P,
            $ite((B),F,G),
            Q) ) ) ).

% if_rule
tff(fact_474_if__rule,axiom,
    ! [B: $o,P: assn,F: heap_Time_Heap_nat,Q: fun_nat_assn,G: heap_Time_Heap_nat] :
      ( ( (B)
       => hoare_3067605981109127869le_nat(P,F,Q) )
     => ( ( ~ (B)
         => hoare_3067605981109127869le_nat(P,G,Q) )
       => hoare_3067605981109127869le_nat(P,
            $ite((B),F,G),
            Q) ) ) ).

% if_rule
tff(fact_475_if__rule,axiom,
    ! [B: $o,P: assn,F: heap_Time_Heap_o,Q: fun_o_assn,G: heap_Time_Heap_o] :
      ( ( (B)
       => hoare_hoare_triple_o(P,F,Q) )
     => ( ( ~ (B)
         => hoare_hoare_triple_o(P,G,Q) )
       => hoare_hoare_triple_o(P,
            $ite((B),F,G),
            Q) ) ) ).

% if_rule
tff(fact_476_if__rule,axiom,
    ! [B: $o,P: assn,F: heap_T8145700208782473153_VEBTi,Q: fun_VEBT_VEBTi_assn,G: heap_T8145700208782473153_VEBTi] :
      ( ( (B)
       => hoare_1429296392585015714_VEBTi(P,F,Q) )
     => ( ( ~ (B)
         => hoare_1429296392585015714_VEBTi(P,G,Q) )
       => hoare_1429296392585015714_VEBTi(P,
            $ite((B),F,G),
            Q) ) ) ).

% if_rule
tff(fact_477_if__rule,axiom,
    ! [B: $o,P: assn,F: heap_T5738788834812785303t_unit,Q: fun_Pr6626665806141890376t_assn,G: heap_T5738788834812785303t_unit] :
      ( ( (B)
       => hoare_8945653483474564448t_unit(P,F,Q) )
     => ( ( ~ (B)
         => hoare_8945653483474564448t_unit(P,G,Q) )
       => hoare_8945653483474564448t_unit(P,
            $ite((B),F,G),
            Q) ) ) ).

% if_rule
tff(fact_478_power2__commute,axiom,
    ! [Xb: uint32,Y: uint32] : aa_nat_uint32(power_power_uint32(minus_minus_uint32(Xb,Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_uint32(power_power_uint32(minus_minus_uint32(Y,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power2_commute
tff(fact_479_power2__commute,axiom,
    ! [Xb: word_N3645301735248828278l_num1,Y: word_N3645301735248828278l_num1] : aa_nat5599461682457136004l_num1(power_2184487114949457152l_num1(minus_4019991460397169231l_num1(Xb,Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat5599461682457136004l_num1(power_2184487114949457152l_num1(minus_4019991460397169231l_num1(Y,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power2_commute
tff(fact_480_power2__commute,axiom,
    ! [Xb: real,Y: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Y),Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power2_commute
tff(fact_481_power2__commute,axiom,
    ! [Xb: rat,Y: rat] : aa_nat_rat(power_power_rat(minus_minus_rat(Xb,Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_rat(power_power_rat(minus_minus_rat(Y,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power2_commute
tff(fact_482_power2__commute,axiom,
    ! [Xb: 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,Xb),Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Y),Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power2_commute
tff(fact_483_Suc__nat__number__of__add,axiom,
    ! [V: num,Nb: nat] : aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,V)),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,V),one))),Nb) ).

% Suc_nat_number_of_add
tff(fact_484_nat__power__less__diff,axiom,
    ! [Nb: nat,Q2: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)),Q2)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Q2),aa_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),Nb))) ) ).

% nat_power_less_diff
tff(fact_485_nat__le__power__trans,axiom,
    ! [Nb: nat,M: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),K)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),M)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K)),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)) ) ) ).

% nat_le_power_trans
tff(fact_486_is__pred__in__set__def,axiom,
    ! [Xs: set_nat,Xb: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(vEBT_is_pred_in_set(Xs),Xb),Y)
    <=> ( aa_set_nat_o(member_nat(Y),Xs)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),Xb)
        & ! [X: nat] :
            ( aa_set_nat_o(member_nat(X),Xs)
           => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Xb)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y) ) ) ) ) ).

% is_pred_in_set_def
tff(fact_487_diff__le__diff__pow,axiom,
    ! [K: nat,M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),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),Nb)),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),Nb))) ) ).

% diff_le_diff_pow
tff(fact_488_nat__less__power__trans,axiom,
    ! [Nb: nat,M: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),K)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),M)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K)),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)) ) ) ).

% nat_less_power_trans
tff(fact_489_power2__diff,axiom,
    ! [Xb: uint32,Y: uint32] : aa_nat_uint32(power_power_uint32(minus_minus_uint32(Xb,Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = minus_minus_uint32(aa_uint32_uint32(plus_plus_uint32(aa_nat_uint32(power_power_uint32(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_uint32(power_power_uint32(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_uint32_uint32(times_times_uint32(aa_uint32_uint32(times_times_uint32(numera9087168376688890119uint32(aa_num_num(bit0,one))),Xb)),Y)) ).

% power2_diff
tff(fact_490_power2__diff,axiom,
    ! [Xb: word_N3645301735248828278l_num1,Y: word_N3645301735248828278l_num1] : aa_nat5599461682457136004l_num1(power_2184487114949457152l_num1(minus_4019991460397169231l_num1(Xb,Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = minus_4019991460397169231l_num1(plus_p361126936061061375l_num1(aa_nat5599461682457136004l_num1(power_2184487114949457152l_num1(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat5599461682457136004l_num1(power_2184487114949457152l_num1(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_wor7274019993978516266l_num1(times_7065122842183080059l_num1(aa_wor7274019993978516266l_num1(times_7065122842183080059l_num1(numera7442385471795722001l_num1(aa_num_num(bit0,one))),Xb)),Y)) ).

% power2_diff
tff(fact_491_power2__diff,axiom,
    ! [Xb: real,Y: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),Xb)),Y)) ).

% power2_diff
tff(fact_492_power2__diff,axiom,
    ! [Xb: rat,Y: rat] : aa_nat_rat(power_power_rat(minus_minus_rat(Xb,Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = minus_minus_rat(aa_rat_rat(plus_plus_rat(aa_nat_rat(power_power_rat(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(numeral_numeral_rat(aa_num_num(bit0,one))),Xb)),Y)) ).

% power2_diff
tff(fact_493_power2__diff,axiom,
    ! [Xb: 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,Xb),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,Xb),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))),Xb)),Y)) ).

% power2_diff
tff(fact_494_return__wp__rule,axiom,
    ! [Q: fun_option_nat_assn,Xb: option_nat] : hoare_7629718768684598413on_nat(aa_option_nat_assn(Q,Xb),heap_T3487192422709364219on_nat(Xb),Q) ).

% return_wp_rule
tff(fact_495_return__wp__rule,axiom,
    ! [Q: fun_nat_assn,Xb: nat] : hoare_3067605981109127869le_nat(aa_nat_assn(Q,Xb),aa_nat5821310369566504398ap_nat(heap_Time_return_nat,Xb),Q) ).

% return_wp_rule
tff(fact_496_return__wp__rule,axiom,
    ! [Q: fun_o_assn,Xb: $o] : hoare_hoare_triple_o(aa_o_assn(Q,(Xb)),aa_o_H7399033255965945470Heap_o(heap_Time_return_o,(Xb)),Q) ).

% return_wp_rule
tff(fact_497_return__wp__rule,axiom,
    ! [Q: fun_VEBT_VEBTi_assn,Xb: vEBT_VEBTi] : hoare_1429296392585015714_VEBTi(aa_VEBT_VEBTi_assn(Q,Xb),aa_VEB836174949890266638_VEBTi(heap_T3630416162098727440_VEBTi,Xb),Q) ).

% return_wp_rule
tff(fact_498_return__wp__rule,axiom,
    ! [Q: fun_Pr6626665806141890376t_assn,Xb: product_unit] : hoare_8945653483474564448t_unit(aa_Product_unit_assn(Q,Xb),heap_T7507251653302230130t_unit(Xb),Q) ).

% return_wp_rule
tff(fact_499_bind__rule,axiom,
    ! [P: assn,F: heap_Time_Heap_nat,R: fun_nat_assn,G: fun_na3533371890617757541ap_nat,Q: fun_nat_assn] :
      ( hoare_3067605981109127869le_nat(P,F,R)
     => ( ! [X2: nat] : hoare_3067605981109127869le_nat(aa_nat_assn(R,X2),aa_nat5821310369566504398ap_nat(G,X2),Q)
       => hoare_3067605981109127869le_nat(P,heap_T7049098217575491753at_nat(F,G),Q) ) ) ).

% bind_rule
tff(fact_500_bind__rule,axiom,
    ! [P: assn,F: heap_Time_Heap_nat,R: fun_nat_assn,G: fun_na2343073368905879023Heap_o,Q: fun_o_assn] :
      ( hoare_3067605981109127869le_nat(P,F,R)
     => ( ! [X2: nat] : hoare_hoare_triple_o(aa_nat_assn(R,X2),aa_nat2910048432523430448Heap_o(G,X2),Q)
       => hoare_hoare_triple_o(P,heap_Time_bind_nat_o(F,G),Q) ) ) ).

% bind_rule
tff(fact_501_bind__rule,axiom,
    ! [P: assn,F: heap_Time_Heap_nat,R: fun_nat_assn,G: fun_na7848093160509028170_VEBTi,Q: fun_VEBT_VEBTi_assn] :
      ( hoare_3067605981109127869le_nat(P,F,R)
     => ( ! [X2: nat] : hoare_1429296392585015714_VEBTi(aa_nat_assn(R,X2),aa_nat5408703056993926195_VEBTi(G,X2),Q)
       => hoare_1429296392585015714_VEBTi(P,heap_T844888390831797134_VEBTi(F,G),Q) ) ) ).

% bind_rule
tff(fact_502_bind__rule,axiom,
    ! [P: assn,F: heap_Time_Heap_nat,R: fun_nat_assn,G: fun_na5725305022723494372t_unit,Q: fun_Pr6626665806141890376t_assn] :
      ( hoare_3067605981109127869le_nat(P,F,R)
     => ( ! [X2: nat] : hoare_8945653483474564448t_unit(aa_nat_assn(R,X2),aa_nat5861795706794010277t_unit(G,X2),Q)
       => hoare_8945653483474564448t_unit(P,heap_T2412598413086283380t_unit(F,G),Q) ) ) ).

% bind_rule
tff(fact_503_bind__rule,axiom,
    ! [P: assn,F: heap_Time_Heap_o,R: fun_o_assn,G: fun_o_4158591058828729991ap_nat,Q: fun_nat_assn] :
      ( hoare_hoare_triple_o(P,F,R)
     => ( ! [X2: $o] : hoare_3067605981109127869le_nat(aa_o_assn(R,(X2)),aa_o_H8739796202006397184ap_nat(G,(X2)),Q)
       => hoare_3067605981109127869le_nat(P,heap_Time_bind_o_nat(F,G),Q) ) ) ).

% bind_rule
tff(fact_504_bind__rule,axiom,
    ! [P: assn,F: heap_Time_Heap_o,R: fun_o_assn,G: fun_o_8325599348076409421Heap_o,Q: fun_o_assn] :
      ( hoare_hoare_triple_o(P,F,R)
     => ( ! [X2: $o] : hoare_hoare_triple_o(aa_o_assn(R,(X2)),aa_o_H7399033255965945470Heap_o(G,(X2)),Q)
       => hoare_hoare_triple_o(P,heap_Time_bind_o_o(F,G),Q) ) ) ).

% bind_rule
tff(fact_505_bind__rule,axiom,
    ! [P: assn,F: heap_Time_Heap_o,R: fun_o_assn,G: fun_o_2238389867833408620_VEBTi,Q: fun_VEBT_VEBTi_assn] :
      ( hoare_hoare_triple_o(P,F,R)
     => ( ! [X2: $o] : hoare_1429296392585015714_VEBTi(aa_o_assn(R,(X2)),aa_o_H7577119287286566245_VEBTi(G,(X2)),Q)
       => hoare_1429296392585015714_VEBTi(P,heap_T5998771940306268294_VEBTi(F,G),Q) ) ) ).

% bind_rule
tff(fact_506_bind__rule,axiom,
    ! [P: assn,F: heap_Time_Heap_o,R: fun_o_assn,G: fun_o_6448842521939546434t_unit,Q: fun_Pr6626665806141890376t_assn] :
      ( hoare_hoare_triple_o(P,F,R)
     => ( ! [X2: $o] : hoare_8945653483474564448t_unit(aa_o_assn(R,(X2)),aa_o_H2459438917461573107t_unit(G,(X2)),Q)
       => hoare_8945653483474564448t_unit(P,heap_T2991970491176150908t_unit(F,G),Q) ) ) ).

% bind_rule
tff(fact_507_bind__rule,axiom,
    ! [P: assn,F: heap_T8145700208782473153_VEBTi,R: fun_VEBT_VEBTi_assn,G: fun_VE2791623396910532672ap_nat,Q: fun_nat_assn] :
      ( hoare_1429296392585015714_VEBTi(P,F,R)
     => ( ! [X2: vEBT_VEBTi] : hoare_3067605981109127869le_nat(aa_VEBT_VEBTi_assn(R,X2),aa_VEB183921879236458153ap_nat(G,X2),Q)
       => hoare_3067605981109127869le_nat(P,heap_T5387808048883414158Ti_nat(F,G),Q) ) ) ).

% bind_rule
tff(fact_508_bind__rule,axiom,
    ! [P: assn,F: heap_T8145700208782473153_VEBTi,R: fun_VEBT_VEBTi_assn,G: fun_VE7038733415900436564Heap_o,Q: fun_o_assn] :
      ( hoare_1429296392585015714_VEBTi(P,F,R)
     => ( ! [X2: vEBT_VEBTi] : hoare_hoare_triple_o(aa_VEBT_VEBTi_assn(R,X2),aa_VEB8194921341634505877Heap_o(G,X2),Q)
       => hoare_hoare_triple_o(P,heap_T3040810144269856602EBTi_o(F,G),Q) ) ) ).

% bind_rule
tff(fact_509_VEBTi_Oexhaust,axiom,
    ! [Y: vEBT_VEBTi] :
      ( ! [X11: option4927543243414619207at_nat,X12: nat,X13: array_VEBT_VEBTi,X14: vEBT_VEBTi] : Y != aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(X11,X12,X13),X14)
     => ~ ! [X212: $o,X222: $o] : Y != vEBT_Leafi((X212),(X222)) ) ).

% VEBTi.exhaust
tff(fact_510_VEBTi_Odistinct_I1_J,axiom,
    ! [X11a: option4927543243414619207at_nat,X12a: nat,X13a: array_VEBT_VEBTi,X14a: vEBT_VEBTi,X21: $o,X22: $o] : aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(X11a,X12a,X13a),X14a) != vEBT_Leafi((X21),(X22)) ).

% VEBTi.distinct(1)
tff(fact_511_frame__rule,axiom,
    ! [P: assn,C: heap_T2636463487746394924on_nat,Q: fun_option_nat_assn,R: assn] :
      ( hoare_7629718768684598413on_nat(P,C,Q)
     => hoare_7629718768684598413on_nat(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,P),R),C,aa_ass4112278883933122603t_assn(aTP_Lamm_bp(Q),R)) ) ).

% frame_rule
tff(fact_512_frame__rule,axiom,
    ! [P: assn,C: heap_Time_Heap_nat,Q: fun_nat_assn,R: assn] :
      ( hoare_3067605981109127869le_nat(P,C,Q)
     => hoare_3067605981109127869le_nat(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,P),R),C,aa_assn_fun_nat_assn(aTP_Lamm_bq(Q),R)) ) ).

% frame_rule
tff(fact_513_frame__rule,axiom,
    ! [P: assn,C: heap_Time_Heap_o,Q: fun_o_assn,R: assn] :
      ( hoare_hoare_triple_o(P,C,Q)
     => hoare_hoare_triple_o(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,P),R),C,aa_assn_fun_o_assn(aTP_Lamm_br(Q),R)) ) ).

% frame_rule
tff(fact_514_frame__rule,axiom,
    ! [P: assn,C: heap_T8145700208782473153_VEBTi,Q: fun_VEBT_VEBTi_assn,R: assn] :
      ( hoare_1429296392585015714_VEBTi(P,C,Q)
     => hoare_1429296392585015714_VEBTi(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,P),R),C,aa_ass4389913640333755968i_assn(aTP_Lamm_bs(Q),R)) ) ).

% frame_rule
tff(fact_515_frame__rule,axiom,
    ! [P: assn,C: heap_T5738788834812785303t_unit,Q: fun_Pr6626665806141890376t_assn,R: assn] :
      ( hoare_8945653483474564448t_unit(P,C,Q)
     => hoare_8945653483474564448t_unit(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,P),R),C,aa_ass3177796706412234594t_assn(aTP_Lamm_bt(Q),R)) ) ).

% frame_rule
tff(fact_516_norm__pre__ex__rule,axiom,
    ! [P: fun_li3352006551604695530i_assn,F: heap_T2636463487746394924on_nat,Q: fun_option_nat_assn] :
      ( ! [X2: list_VEBT_VEBTi] : hoare_7629718768684598413on_nat(aa_lis2508312872377092907i_assn(P,X2),F,Q)
     => hoare_7629718768684598413on_nat(ex_ass463751140784270563_VEBTi(P),F,Q) ) ).

% norm_pre_ex_rule
tff(fact_517_norm__pre__ex__rule,axiom,
    ! [P: fun_li3352006551604695530i_assn,F: heap_Time_Heap_nat,Q: fun_nat_assn] :
      ( ! [X2: list_VEBT_VEBTi] : hoare_3067605981109127869le_nat(aa_lis2508312872377092907i_assn(P,X2),F,Q)
     => hoare_3067605981109127869le_nat(ex_ass463751140784270563_VEBTi(P),F,Q) ) ).

% norm_pre_ex_rule
tff(fact_518_norm__pre__ex__rule,axiom,
    ! [P: fun_li3352006551604695530i_assn,F: heap_Time_Heap_o,Q: fun_o_assn] :
      ( ! [X2: list_VEBT_VEBTi] : hoare_hoare_triple_o(aa_lis2508312872377092907i_assn(P,X2),F,Q)
     => hoare_hoare_triple_o(ex_ass463751140784270563_VEBTi(P),F,Q) ) ).

% norm_pre_ex_rule
tff(fact_519_norm__pre__ex__rule,axiom,
    ! [P: fun_li3352006551604695530i_assn,F: heap_T8145700208782473153_VEBTi,Q: fun_VEBT_VEBTi_assn] :
      ( ! [X2: list_VEBT_VEBTi] : hoare_1429296392585015714_VEBTi(aa_lis2508312872377092907i_assn(P,X2),F,Q)
     => hoare_1429296392585015714_VEBTi(ex_ass463751140784270563_VEBTi(P),F,Q) ) ).

% norm_pre_ex_rule
tff(fact_520_norm__pre__ex__rule,axiom,
    ! [P: fun_li3352006551604695530i_assn,F: heap_T5738788834812785303t_unit,Q: fun_Pr6626665806141890376t_assn] :
      ( ! [X2: list_VEBT_VEBTi] : hoare_8945653483474564448t_unit(aa_lis2508312872377092907i_assn(P,X2),F,Q)
     => hoare_8945653483474564448t_unit(ex_ass463751140784270563_VEBTi(P),F,Q) ) ).

% norm_pre_ex_rule
tff(fact_521_post__exI__rule,axiom,
    ! [P: assn,C: heap_T2636463487746394924on_nat,Q: fun_op8552747401449489031i_assn,Xb: list_VEBT_VEBTi] :
      ( hoare_7629718768684598413on_nat(P,C,aa_lis7387135221810377032t_assn(aTP_Lamm_bu(Q),Xb))
     => hoare_7629718768684598413on_nat(P,C,aTP_Lamm_bv(Q)) ) ).

% post_exI_rule
tff(fact_522_post__exI__rule,axiom,
    ! [P: assn,C: heap_Time_Heap_nat,Q: fun_na8005944528675429559i_assn,Xb: list_VEBT_VEBTi] :
      ( hoare_3067605981109127869le_nat(P,C,aa_lis7348694692495225848t_assn(aTP_Lamm_bw(Q),Xb))
     => hoare_3067605981109127869le_nat(P,C,aTP_Lamm_bx(Q)) ) ).

% post_exI_rule
tff(fact_523_post__exI__rule,axiom,
    ! [P: assn,C: heap_Time_Heap_o,Q: fun_o_4874018796564477717i_assn,Xb: list_VEBT_VEBTi] :
      ( hoare_hoare_triple_o(P,C,aa_lis3121611389386064016o_assn(aTP_Lamm_by(Q),Xb))
     => hoare_hoare_triple_o(P,C,aTP_Lamm_bz(Q)) ) ).

% post_exI_rule
tff(fact_524_post__exI__rule,axiom,
    ! [P: assn,C: heap_T8145700208782473153_VEBTi,Q: fun_VE3708348504534315036i_assn,Xb: list_VEBT_VEBTi] :
      ( hoare_1429296392585015714_VEBTi(P,C,aa_lis5319508155010942429i_assn(aTP_Lamm_ca(Q),Xb))
     => hoare_1429296392585015714_VEBTi(P,C,aTP_Lamm_cb(Q)) ) ).

% post_exI_rule
tff(fact_525_post__exI__rule,axiom,
    ! [P: assn,C: heap_T5738788834812785303t_unit,Q: fun_Pr6039563626565002634i_assn,Xb: list_VEBT_VEBTi] :
      ( hoare_8945653483474564448t_unit(P,C,aa_lis1661100524205913349t_assn(aTP_Lamm_cc(Q),Xb))
     => hoare_8945653483474564448t_unit(P,C,aTP_Lamm_cd(Q)) ) ).

% post_exI_rule
tff(fact_526_norm__pre__pure__rule1,axiom,
    ! [B: $o,P: assn,F: heap_T2636463487746394924on_nat,Q: fun_option_nat_assn] :
      ( ( (B)
       => hoare_7629718768684598413on_nat(P,F,Q) )
     => hoare_7629718768684598413on_nat(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,P),aa_o_assn(pure_assn,(B))),F,Q) ) ).

% norm_pre_pure_rule1
tff(fact_527_norm__pre__pure__rule1,axiom,
    ! [B: $o,P: assn,F: heap_Time_Heap_nat,Q: fun_nat_assn] :
      ( ( (B)
       => hoare_3067605981109127869le_nat(P,F,Q) )
     => hoare_3067605981109127869le_nat(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,P),aa_o_assn(pure_assn,(B))),F,Q) ) ).

% norm_pre_pure_rule1
tff(fact_528_norm__pre__pure__rule1,axiom,
    ! [B: $o,P: assn,F: heap_Time_Heap_o,Q: fun_o_assn] :
      ( ( (B)
       => hoare_hoare_triple_o(P,F,Q) )
     => hoare_hoare_triple_o(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,P),aa_o_assn(pure_assn,(B))),F,Q) ) ).

% norm_pre_pure_rule1
tff(fact_529_norm__pre__pure__rule1,axiom,
    ! [B: $o,P: assn,F: heap_T8145700208782473153_VEBTi,Q: fun_VEBT_VEBTi_assn] :
      ( ( (B)
       => hoare_1429296392585015714_VEBTi(P,F,Q) )
     => hoare_1429296392585015714_VEBTi(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,P),aa_o_assn(pure_assn,(B))),F,Q) ) ).

% norm_pre_pure_rule1
tff(fact_530_norm__pre__pure__rule1,axiom,
    ! [B: $o,P: assn,F: heap_T5738788834812785303t_unit,Q: fun_Pr6626665806141890376t_assn] :
      ( ( (B)
       => hoare_8945653483474564448t_unit(P,F,Q) )
     => hoare_8945653483474564448t_unit(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,P),aa_o_assn(pure_assn,(B))),F,Q) ) ).

% norm_pre_pure_rule1
tff(fact_531_norm__pre__pure__rule2,axiom,
    ! [B: $o,F: heap_T2636463487746394924on_nat,Q: fun_option_nat_assn] :
      ( ( (B)
       => hoare_7629718768684598413on_nat(one_one_assn,F,Q) )
     => hoare_7629718768684598413on_nat(aa_o_assn(pure_assn,(B)),F,Q) ) ).

% norm_pre_pure_rule2
tff(fact_532_norm__pre__pure__rule2,axiom,
    ! [B: $o,F: heap_Time_Heap_nat,Q: fun_nat_assn] :
      ( ( (B)
       => hoare_3067605981109127869le_nat(one_one_assn,F,Q) )
     => hoare_3067605981109127869le_nat(aa_o_assn(pure_assn,(B)),F,Q) ) ).

% norm_pre_pure_rule2
tff(fact_533_norm__pre__pure__rule2,axiom,
    ! [B: $o,F: heap_Time_Heap_o,Q: fun_o_assn] :
      ( ( (B)
       => hoare_hoare_triple_o(one_one_assn,F,Q) )
     => hoare_hoare_triple_o(aa_o_assn(pure_assn,(B)),F,Q) ) ).

% norm_pre_pure_rule2
tff(fact_534_norm__pre__pure__rule2,axiom,
    ! [B: $o,F: heap_T8145700208782473153_VEBTi,Q: fun_VEBT_VEBTi_assn] :
      ( ( (B)
       => hoare_1429296392585015714_VEBTi(one_one_assn,F,Q) )
     => hoare_1429296392585015714_VEBTi(aa_o_assn(pure_assn,(B)),F,Q) ) ).

% norm_pre_pure_rule2
tff(fact_535_norm__pre__pure__rule2,axiom,
    ! [B: $o,F: heap_T5738788834812785303t_unit,Q: fun_Pr6626665806141890376t_assn] :
      ( ( (B)
       => hoare_8945653483474564448t_unit(one_one_assn,F,Q) )
     => hoare_8945653483474564448t_unit(aa_o_assn(pure_assn,(B)),F,Q) ) ).

% norm_pre_pure_rule2
tff(fact_536_case__option__rule,axiom,
    ! [V: option_nat,P: assn,Fn: heap_Time_Heap_nat,Q: fun_nat_assn,Fs: fun_na3533371890617757541ap_nat] :
      ( ( ( V = none_nat )
       => hoare_3067605981109127869le_nat(P,Fn,Q) )
     => ( ! [X2: nat] :
            ( ( V = aa_nat_option_nat(some_nat,X2) )
           => hoare_3067605981109127869le_nat(P,aa_nat5821310369566504398ap_nat(Fs,X2),Q) )
       => hoare_3067605981109127869le_nat(P,case_o6609685678014844897at_nat(Fn,Fs,V),Q) ) ) ).

% case_option_rule
tff(fact_537_case__option__rule,axiom,
    ! [V: option_num,P: assn,Fn: heap_Time_Heap_nat,Q: fun_nat_assn,Fs: fun_nu8176864466684091547ap_nat] :
      ( ( ( V = none_num )
       => hoare_3067605981109127869le_nat(P,Fn,Q) )
     => ( ! [X2: num] :
            ( ( V = aa_num_option_num(some_num,X2) )
           => hoare_3067605981109127869le_nat(P,aa_num1241430908778062596ap_nat(Fs,X2),Q) )
       => hoare_3067605981109127869le_nat(P,case_o3167017464170623531at_num(Fn,Fs,V),Q) ) ) ).

% case_option_rule
tff(fact_538_case__option__rule,axiom,
    ! [V: option_nat,P: assn,Fn: heap_Time_Heap_o,Q: fun_o_assn,Fs: fun_na2343073368905879023Heap_o] :
      ( ( ( V = none_nat )
       => hoare_hoare_triple_o(P,Fn,Q) )
     => ( ! [X2: nat] :
            ( ( V = aa_nat_option_nat(some_nat,X2) )
           => hoare_hoare_triple_o(P,aa_nat2910048432523430448Heap_o(Fs,X2),Q) )
       => hoare_hoare_triple_o(P,aa_opt6753654531493786624Heap_o(case_o6892868863119666303_o_nat(Fn,Fs),V),Q) ) ) ).

% case_option_rule
tff(fact_539_case__option__rule,axiom,
    ! [V: option_num,P: assn,Fn: heap_Time_Heap_o,Q: fun_o_assn,Fs: fun_nu9127643932893349689Heap_o] :
      ( ( ( V = none_num )
       => hoare_hoare_triple_o(P,Fn,Q) )
     => ( ! [X2: num] :
            ( ( V = aa_num_option_num(some_num,X2) )
           => hoare_hoare_triple_o(P,aa_num471246959656125306Heap_o(Fs,X2),Q) )
       => hoare_hoare_triple_o(P,case_o3450200649275444937_o_num(Fn,Fs,V),Q) ) ) ).

% case_option_rule
tff(fact_540_case__option__rule,axiom,
    ! [V: option_nat,P: assn,Fn: heap_T8145700208782473153_VEBTi,Q: fun_VEBT_VEBTi_assn,Fs: fun_na7848093160509028170_VEBTi] :
      ( ( ( V = none_nat )
       => hoare_1429296392585015714_VEBTi(P,Fn,Q) )
     => ( ! [X2: nat] :
            ( ( V = aa_nat_option_nat(some_nat,X2) )
           => hoare_1429296392585015714_VEBTi(P,aa_nat5408703056993926195_VEBTi(Fs,X2),Q) )
       => hoare_1429296392585015714_VEBTi(P,aa_opt6483592524530923107_VEBTi(case_o3780387683879180358Ti_nat(Fn,Fs),V),Q) ) ) ).

% case_option_rule
tff(fact_541_case__option__rule,axiom,
    ! [V: option_num,P: assn,Fn: heap_T8145700208782473153_VEBTi,Q: fun_VEBT_VEBTi_assn,Fs: fun_nu5987414290159654528_VEBTi] :
      ( ( ( V = none_num )
       => hoare_1429296392585015714_VEBTi(P,Fn,Q) )
     => ( ! [X2: num] :
            ( ( V = aa_num_option_num(some_num,X2) )
           => hoare_1429296392585015714_VEBTi(P,aa_num3548024186644552553_VEBTi(Fs,X2),Q) )
       => hoare_1429296392585015714_VEBTi(P,case_o337719470034958992Ti_num(Fn,Fs,V),Q) ) ) ).

% case_option_rule
tff(fact_542_case__option__rule,axiom,
    ! [V: option_nat,P: assn,Fn: heap_T5738788834812785303t_unit,Q: fun_Pr6626665806141890376t_assn,Fs: fun_na5725305022723494372t_unit] :
      ( ( ( V = none_nat )
       => hoare_8945653483474564448t_unit(P,Fn,Q) )
     => ( ! [X2: nat] :
            ( ( V = aa_nat_option_nat(some_nat,X2) )
           => hoare_8945653483474564448t_unit(P,aa_nat5861795706794010277t_unit(Fs,X2),Q) )
       => hoare_8945653483474564448t_unit(P,case_o8044209130310455284it_nat(Fn,Fs,V),Q) ) ) ).

% case_option_rule
tff(fact_543_case__option__rule,axiom,
    ! [V: option_num,P: assn,Fn: heap_T5738788834812785303t_unit,Q: fun_Pr6626665806141890376t_assn,Fs: fun_nu1618770902802125870t_unit] :
      ( ( ( V = none_num )
       => hoare_8945653483474564448t_unit(P,Fn,Q) )
     => ( ! [X2: num] :
            ( ( V = aa_num_option_num(some_num,X2) )
           => hoare_8945653483474564448t_unit(P,aa_num1755261586872641775t_unit(Fs,X2),Q) )
       => hoare_8945653483474564448t_unit(P,case_o4601540916466233918it_num(Fn,Fs,V),Q) ) ) ).

% case_option_rule
tff(fact_544_case__option__rule,axiom,
    ! [V: option_nat,P: assn,Fn: heap_T2636463487746394924on_nat,Q: fun_option_nat_assn,Fs: fun_na5556145776640407093on_nat] :
      ( ( ( V = none_nat )
       => hoare_7629718768684598413on_nat(P,Fn,Q) )
     => ( ! [X2: nat] :
            ( ( V = aa_nat_option_nat(some_nat,X2) )
           => hoare_7629718768684598413on_nat(P,aa_nat947733127081326238on_nat(Fs,X2),Q) )
       => hoare_7629718768684598413on_nat(P,case_o2256915875499652529at_nat(Fn,Fs,V),Q) ) ) ).

% case_option_rule
tff(fact_545_case__option__rule,axiom,
    ! [V: option_num,P: assn,Fn: heap_T2636463487746394924on_nat,Q: fun_option_nat_assn,Fs: fun_nu1337165112811199083on_nat] :
      ( ( ( V = none_num )
       => hoare_7629718768684598413on_nat(P,Fn,Q) )
     => ( ! [X2: num] :
            ( ( V = aa_num_option_num(some_num,X2) )
           => hoare_7629718768684598413on_nat(P,aa_num5952124500106894036on_nat(Fs,X2),Q) )
       => hoare_7629718768684598413on_nat(P,case_o8037619698510206971at_num(Fn,Fs,V),Q) ) ) ).

% case_option_rule
tff(fact_546_return__sp__rule,axiom,
    ! [P: assn,Xb: option_nat] : hoare_7629718768684598413on_nat(P,heap_T3487192422709364219on_nat(Xb),aa_opt3861602675778688675t_assn(aTP_Lamm_ce(P),Xb)) ).

% return_sp_rule
tff(fact_547_return__sp__rule,axiom,
    ! [P: assn,Xb: nat] : hoare_3067605981109127869le_nat(P,aa_nat5821310369566504398ap_nat(heap_Time_return_nat,Xb),aa_nat_fun_nat_assn(aTP_Lamm_cf(P),Xb)) ).

% return_sp_rule
tff(fact_548_return__sp__rule,axiom,
    ! [P: assn,Xb: $o] : hoare_hoare_triple_o(P,aa_o_H7399033255965945470Heap_o(heap_Time_return_o,(Xb)),aa_o_fun_o_assn(aTP_Lamm_cg(P),(Xb))) ).

% return_sp_rule
tff(fact_549_return__sp__rule,axiom,
    ! [P: assn,Xb: vEBT_VEBTi] : hoare_1429296392585015714_VEBTi(P,aa_VEB836174949890266638_VEBTi(heap_T3630416162098727440_VEBTi,Xb),aa_VEB1545114916007340237i_assn(aTP_Lamm_ch(P),Xb)) ).

% return_sp_rule
tff(fact_550_return__sp__rule,axiom,
    ! [P: assn,Xb: product_unit] : hoare_8945653483474564448t_unit(P,heap_T7507251653302230130t_unit(Xb),aa_Pro6018021088111102263t_assn(aTP_Lamm_ci(P),Xb)) ).

% return_sp_rule
tff(fact_551_n__less__equal__power__2,axiom,
    ! [Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ).

% n_less_equal_power_2
tff(fact_552_vebt__minti_Osimps_I3_J,axiom,
    ! [Mia: nat,Maa: nat,Ux: nat,Uy: array_VEBT_VEBTi,Uz: vEBT_VEBTi] : aa_VEB2711320435582112505on_nat(vEBT_vebt_minti,aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mia),Maa)),Ux,Uy),Uz)) = heap_T3487192422709364219on_nat(aa_nat_option_nat(some_nat,Mia)) ).

% vebt_minti.simps(3)
tff(fact_553_vebt__maxti_Osimps_I3_J,axiom,
    ! [Mia: nat,Maa: nat,Ux: nat,Uy: array_VEBT_VEBTi,Uz: vEBT_VEBTi] : aa_VEB2711320435582112505on_nat(vEBT_vebt_maxti,aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mia),Maa)),Ux,Uy),Uz)) = heap_T3487192422709364219on_nat(aa_nat_option_nat(some_nat,Maa)) ).

% vebt_maxti.simps(3)
tff(fact_554_nat__add__offset__less,axiom,
    ! [Y: nat,Nb: nat,Xb: nat,M: nat,Sz: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M))
       => ( ( Sz = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Nb) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))),Y)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Sz)) ) ) ) ).

% nat_add_offset_less
tff(fact_555_vebt__succ_Osimps_I6_J,axiom,
    ! [Mia: nat,Maa: nat,Vaa: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Xb: nat] :
      aa_nat_option_nat(vEBT_vebt_succ(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mia),Maa)),aa_nat_nat(suc,aa_nat_nat(suc,Vaa)),TreeLista),Summarya)),Xb) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Mia),
        aa_nat_option_nat(some_nat,Mia),
        $let(
          l: nat,
          l:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Vaa))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
          $let(
            h: nat,
            h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Vaa))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
            $ite(
              aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeLista)),
              $let(
                maxlow: option_nat,
                maxlow:= vEBT_vebt_maxt(nth_VEBT_VEBT(TreeLista,h)),
                $ite(
                  ( ( maxlow != none_nat )
                  & aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_less,aa_nat_option_nat(some_nat,l)),maxlow) ),
                  aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,aa_nat_option_nat(some_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Vaa))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_option_nat(some_nat,h))),aa_nat_option_nat(vEBT_vebt_succ(nth_VEBT_VEBT(TreeLista,h)),l)),
                  $let(
                    sc: option_nat,
                    sc:= aa_nat_option_nat(vEBT_vebt_succ(Summarya),h),
                    $ite(sc = none_nat,none_nat,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,aa_nat_option_nat(some_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Vaa))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),sc)),vEBT_vebt_mint(nth_VEBT_VEBT(TreeLista,the_nat(sc))))) ) ) ),
              none_nat ) ) ) ) ).

% vebt_succ.simps(6)
tff(fact_556_in__children__def,axiom,
    ! [Nb: nat,TreeLista: list_VEBT_VEBT,Xb: nat] :
      ( aa_nat_o(vEBT_V5917875025757280293ildren(Nb,TreeLista),Xb)
    <=> aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeLista,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),Nb))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xb),Nb)) ) ).

% in_children_def
tff(fact_557_Suc__diff,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
     => ( 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,Nb),M)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat)) ) ) ) ).

% Suc_diff
tff(fact_558_diff__Suc__diff__eq2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),J)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K))),I) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,J)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),I)) ) ) ).

% diff_Suc_diff_eq2
tff(fact_559_diff__Suc__diff__eq1,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),J)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(suc,J)) ) ) ).

% diff_Suc_diff_eq1
tff(fact_560_semiring__norm_I76_J,axiom,
    ! [Nb: num] : aa_num_o(ord_less_num(one),aa_num_num(bit0,Nb)) ).

% semiring_norm(76)
tff(fact_561_semiring__norm_I69_J,axiom,
    ! [M: num] : ~ aa_num_o(ord_less_eq_num(aa_num_num(bit0,M)),one) ).

% semiring_norm(69)
tff(fact_562_less__two__pow__divI,axiom,
    ! [Xb: nat,Nb: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),M)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),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))),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M))) ) ) ).

% less_two_pow_divI
tff(fact_563_less__two__pow__divD,axiom,
    ! [Xb: nat,Nb: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),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))),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),M))) ) ) ).

% less_two_pow_divD
tff(fact_564_sum__squares__bound,axiom,
    ! [Xb: real,Y: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),Xb)),Y)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% sum_squares_bound
tff(fact_565_sum__squares__bound,axiom,
    ! [Xb: rat,Y: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(numeral_numeral_rat(aa_num_num(bit0,one))),Xb)),Y)),aa_rat_rat(plus_plus_rat(aa_nat_rat(power_power_rat(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% sum_squares_bound
tff(fact_566_option_Ocollapse,axiom,
    ! [Option: option2621746655072343315it_nat] :
      ( ( Option != none_P1551326421579882414it_nat )
     => ( some_P2407035485129114418it_nat(the_Pr3501439614016493281it_nat(Option)) = Option ) ) ).

% option.collapse
tff(fact_567_option_Ocollapse,axiom,
    ! [Option: option7339022715339332451it_nat] :
      ( ( Option != none_P7668321371905463026it_nat )
     => ( some_P468703482102919278it_nat(the_Pr5838048819577852031it_nat(Option)) = Option ) ) ).

% option.collapse
tff(fact_568_option_Ocollapse,axiom,
    ! [Option: option4927543243414619207at_nat] :
      ( ( Option != none_P5556105721700978146at_nat )
     => ( some_P7363390416028606310at_nat(the_Pr8591224930841456533at_nat(Option)) = Option ) ) ).

% option.collapse
tff(fact_569_option_Ocollapse,axiom,
    ! [Option: option_nat] :
      ( ( Option != none_nat )
     => ( aa_nat_option_nat(some_nat,the_nat(Option)) = Option ) ) ).

% option.collapse
tff(fact_570_option_Ocollapse,axiom,
    ! [Option: option_num] :
      ( ( Option != none_num )
     => ( aa_num_option_num(some_num,aa_option_num_num(the_num,Option)) = Option ) ) ).

% option.collapse
tff(fact_571_highsimp,axiom,
    ! [Xb: nat,Nb: nat] : aa_nat5821310369566504398ap_nat(heap_Time_return_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),Nb)) = aa_nat5821310369566504398ap_nat(vEBT_VEBT_highi(Xb),Nb) ).

% highsimp
tff(fact_572_lowsimp,axiom,
    ! [Xb: nat,Nb: nat] : aa_nat5821310369566504398ap_nat(heap_Time_return_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xb),Nb)) = aa_nat5821310369566504398ap_nat(vEBT_VEBT_lowi(Xb),Nb) ).

% lowsimp
tff(fact_573_semiring__norm_I87_J,axiom,
    ! [M: num,Nb: num] :
      ( ( aa_num_num(bit0,M) = aa_num_num(bit0,Nb) )
    <=> ( M = Nb ) ) ).

% semiring_norm(87)
tff(fact_574_nat_Oinject,axiom,
    ! [X23: nat,Y2: nat] :
      ( ( aa_nat_nat(suc,X23) = aa_nat_nat(suc,Y2) )
    <=> ( X23 = Y2 ) ) ).

% nat.inject
tff(fact_575_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_576_option_Oinject,axiom,
    ! [X23: product_prod_nat_nat,Y2: product_prod_nat_nat] :
      ( ( some_P7363390416028606310at_nat(X23) = some_P7363390416028606310at_nat(Y2) )
    <=> ( X23 = Y2 ) ) ).

% option.inject
tff(fact_577_option_Oinject,axiom,
    ! [X23: nat,Y2: nat] :
      ( ( aa_nat_option_nat(some_nat,X23) = aa_nat_option_nat(some_nat,Y2) )
    <=> ( X23 = Y2 ) ) ).

% option.inject
tff(fact_578_option_Oinject,axiom,
    ! [X23: num,Y2: num] :
      ( ( aa_num_option_num(some_num,X23) = aa_num_option_num(some_num,Y2) )
    <=> ( X23 = Y2 ) ) ).

% option.inject
tff(fact_579_highi__def,axiom,
    ! [Xb: nat,Nb: nat] : aa_nat5821310369566504398ap_nat(vEBT_VEBT_highi(Xb),Nb) = aa_nat5821310369566504398ap_nat(heap_Time_return_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))) ).

% highi_def
tff(fact_580_semiring__norm_I85_J,axiom,
    ! [M: num] : aa_num_num(bit0,M) != one ).

% semiring_norm(85)
tff(fact_581_semiring__norm_I83_J,axiom,
    ! [Nb: num] : one != aa_num_num(bit0,Nb) ).

% semiring_norm(83)
tff(fact_582_lessI,axiom,
    ! [Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(suc,Nb)) ).

% lessI
tff(fact_583_Suc__mono,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,M)),aa_nat_nat(suc,Nb)) ) ).

% Suc_mono
tff(fact_584_Suc__less__eq,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,M)),aa_nat_nat(suc,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb) ) ).

% Suc_less_eq
tff(fact_585_add__Suc__right,axiom,
    ! [M: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),aa_nat_nat(suc,Nb)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Nb)) ).

% add_Suc_right
tff(fact_586_Suc__le__mono,axiom,
    ! [Nb: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Nb)),aa_nat_nat(suc,M))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),M) ) ).

% Suc_le_mono
tff(fact_587_nat__add__left__cancel__less,axiom,
    ! [K: nat,M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb) ) ).

% nat_add_left_cancel_less
tff(fact_588_diff__Suc__Suc,axiom,
    ! [M: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,M)),aa_nat_nat(suc,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Nb) ).

% diff_Suc_Suc
tff(fact_589_Suc__diff__diff,axiom,
    ! [M: nat,Nb: 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)),Nb)),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),Nb)),K) ).

% Suc_diff_diff
tff(fact_590_nat__add__left__cancel__le,axiom,
    ! [K: nat,M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb) ) ).

% nat_add_left_cancel_le
tff(fact_591_not__Some__eq2,axiom,
    ! [V: option983905398058327270nteger] :
      ( ! [X: fun_Pr5098669352780829690e_term,Y3: produc8923325533196201883nteger] : V != some_P7453818414240362097nteger(aa_Pro4291442876242880245nteger(produc565448451787809096nteger(X),Y3))
    <=> ( V = none_P612323165605163765nteger ) ) ).

% not_Some_eq2
tff(fact_592_not__Some__eq2,axiom,
    ! [V: option4955102875562834416et_nat] :
      ( ! [X: fun_Pr6460944699146269935_nat_o,Y3: produc3658429121746597890et_nat] : V != some_P994283844982041979et_nat(aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(X),Y3))
    <=> ( V = none_P7891021755743879167et_nat ) ) ).

% not_Some_eq2
tff(fact_593_not__Some__eq2,axiom,
    ! [V: option5899113203645146382et_nat] :
      ( ! [X: fun_Pr6460944699146269935_nat_o,Y3: produc8799533983703829792et_nat] : V != some_P3006888012174695577et_nat(aa_Pro1979504083478021276et_nat(produc7291855603632349880et_nat(X),Y3))
    <=> ( V = none_P2758821087663475997et_nat ) ) ).

% not_Some_eq2
tff(fact_594_not__Some__eq2,axiom,
    ! [V: option1544258867535432659nt_int] :
      ( ! [X: fun_Pr7763680713840462443e_term,Y3: product_prod_int_int] : V != some_P9145269820779039710nt_int(aa_Pro5748361610964203648nt_int(produc7320969046608631221nt_int(X),Y3))
    <=> ( V = none_P3112835510759997538nt_int ) ) ).

% not_Some_eq2
tff(fact_595_not__Some__eq2,axiom,
    ! [V: option6186025235080601855nt_int] :
      ( ! [X: fun_in5730379987311621695e_term,Y3: product_prod_int_int] : V != some_P8798507657793832714nt_int(aa_Pro8800328608558162860nt_int(produc4729913460834754785nt_int(X),Y3))
    <=> ( V = none_P1716269551890668430nt_int ) ) ).

% not_Some_eq2
tff(fact_596_not__Some__eq2,axiom,
    ! [V: option2621746655072343315it_nat] :
      ( ! [X: option_nat,Y3: produc6653097349344004940it_nat] : V != some_P2407035485129114418it_nat(produc61566615109097733it_nat(X,Y3))
    <=> ( V = none_P1551326421579882414it_nat ) ) ).

% not_Some_eq2
tff(fact_597_not__Some__eq2,axiom,
    ! [V: option7339022715339332451it_nat] :
      ( ! [X: $o,Y3: produc6653097349344004940it_nat] : V != some_P468703482102919278it_nat(produc6655106138504972685it_nat((X),Y3))
    <=> ( V = none_P7668321371905463026it_nat ) ) ).

% not_Some_eq2
tff(fact_598_not__Some__eq2,axiom,
    ! [V: option4927543243414619207at_nat] :
      ( ! [X: nat,Y3: nat] : V != some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,X),Y3))
    <=> ( V = none_P5556105721700978146at_nat ) ) ).

% not_Some_eq2
tff(fact_599_diff__diff__left,axiom,
    ! [I: nat,J: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I),J)),K) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K)) ).

% diff_diff_left
tff(fact_600_diff__diff__cancel,axiom,
    ! [I: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),I)) = I ) ) ).

% diff_diff_cancel
tff(fact_601_not__None__eq,axiom,
    ! [Xb: option2621746655072343315it_nat] :
      ( ( Xb != none_P1551326421579882414it_nat )
    <=> ? [Y3: produc120671012495760973it_nat] : Xb = some_P2407035485129114418it_nat(Y3) ) ).

% not_None_eq
tff(fact_602_not__None__eq,axiom,
    ! [Xb: option7339022715339332451it_nat] :
      ( ( Xb != none_P7668321371905463026it_nat )
    <=> ? [Y3: produc8047831477865546771it_nat] : Xb = some_P468703482102919278it_nat(Y3) ) ).

% not_None_eq
tff(fact_603_not__None__eq,axiom,
    ! [Xb: option4927543243414619207at_nat] :
      ( ( Xb != none_P5556105721700978146at_nat )
    <=> ? [Y3: product_prod_nat_nat] : Xb = some_P7363390416028606310at_nat(Y3) ) ).

% not_None_eq
tff(fact_604_not__None__eq,axiom,
    ! [Xb: option_nat] :
      ( ( Xb != none_nat )
    <=> ? [Y3: nat] : Xb = aa_nat_option_nat(some_nat,Y3) ) ).

% not_None_eq
tff(fact_605_not__None__eq,axiom,
    ! [Xb: option_num] :
      ( ( Xb != none_num )
    <=> ? [Y3: num] : Xb = aa_num_option_num(some_num,Y3) ) ).

% not_None_eq
tff(fact_606_not__Some__eq,axiom,
    ! [Xb: option2621746655072343315it_nat] :
      ( ! [Y3: produc120671012495760973it_nat] : Xb != some_P2407035485129114418it_nat(Y3)
    <=> ( Xb = none_P1551326421579882414it_nat ) ) ).

% not_Some_eq
tff(fact_607_not__Some__eq,axiom,
    ! [Xb: option7339022715339332451it_nat] :
      ( ! [Y3: produc8047831477865546771it_nat] : Xb != some_P468703482102919278it_nat(Y3)
    <=> ( Xb = none_P7668321371905463026it_nat ) ) ).

% not_Some_eq
tff(fact_608_not__Some__eq,axiom,
    ! [Xb: option4927543243414619207at_nat] :
      ( ! [Y3: product_prod_nat_nat] : Xb != some_P7363390416028606310at_nat(Y3)
    <=> ( Xb = none_P5556105721700978146at_nat ) ) ).

% not_Some_eq
tff(fact_609_not__Some__eq,axiom,
    ! [Xb: option_nat] :
      ( ! [Y3: nat] : Xb != aa_nat_option_nat(some_nat,Y3)
    <=> ( Xb = none_nat ) ) ).

% not_Some_eq
tff(fact_610_not__Some__eq,axiom,
    ! [Xb: option_num] :
      ( ! [Y3: num] : Xb != aa_num_option_num(some_num,Y3)
    <=> ( Xb = none_num ) ) ).

% not_Some_eq
tff(fact_611_nat__1__eq__mult__iff,axiom,
    ! [M: nat,Nb: nat] :
      ( ( one_one_nat = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Nb) )
    <=> ( ( M = one_one_nat )
        & ( Nb = one_one_nat ) ) ) ).

% nat_1_eq_mult_iff
tff(fact_612_nat__mult__eq__1__iff,axiom,
    ! [M: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Nb) = one_one_nat )
    <=> ( ( M = one_one_nat )
        & ( Nb = one_one_nat ) ) ) ).

% nat_mult_eq_1_iff
tff(fact_613_semiring__norm_I6_J,axiom,
    ! [M: num,Nb: num] : aa_num_num(aa_num_fun_num_num(plus_plus_num,aa_num_num(bit0,M)),aa_num_num(bit0,Nb)) = aa_num_num(bit0,aa_num_num(aa_num_fun_num_num(plus_plus_num,M),Nb)) ).

% semiring_norm(6)
tff(fact_614_semiring__norm_I13_J,axiom,
    ! [M: num,Nb: num] : aa_num_num(aa_num_fun_num_num(times_times_num,aa_num_num(bit0,M)),aa_num_num(bit0,Nb)) = aa_num_num(bit0,aa_num_num(bit0,aa_num_num(aa_num_fun_num_num(times_times_num,M),Nb))) ).

% semiring_norm(13)
tff(fact_615_semiring__norm_I12_J,axiom,
    ! [Nb: num] : aa_num_num(aa_num_fun_num_num(times_times_num,one),Nb) = Nb ).

% semiring_norm(12)
tff(fact_616_semiring__norm_I11_J,axiom,
    ! [M: num] : aa_num_num(aa_num_fun_num_num(times_times_num,M),one) = M ).

% semiring_norm(11)
tff(fact_617_semiring__norm_I71_J,axiom,
    ! [M: num,Nb: num] :
      ( aa_num_o(ord_less_eq_num(aa_num_num(bit0,M)),aa_num_num(bit0,Nb))
    <=> aa_num_o(ord_less_eq_num(M),Nb) ) ).

% semiring_norm(71)
tff(fact_618_semiring__norm_I78_J,axiom,
    ! [M: num,Nb: num] :
      ( aa_num_o(ord_less_num(aa_num_num(bit0,M)),aa_num_num(bit0,Nb))
    <=> aa_num_o(ord_less_num(M),Nb) ) ).

% semiring_norm(78)
tff(fact_619_semiring__norm_I68_J,axiom,
    ! [Nb: num] : aa_num_o(ord_less_eq_num(one),Nb) ).

% semiring_norm(68)
tff(fact_620_semiring__norm_I75_J,axiom,
    ! [M: num] : ~ aa_num_o(ord_less_num(M),one) ).

% semiring_norm(75)
tff(fact_621_diff__Suc__1,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,Nb)),one_one_nat) = Nb ).

% diff_Suc_1
tff(fact_622_mult__Suc__right,axiom,
    ! [M: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),aa_nat_nat(suc,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Nb)) ).

% mult_Suc_right
tff(fact_623_Nat_Oadd__diff__assoc,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),J)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),J)),K) ) ) ).

% Nat.add_diff_assoc
tff(fact_624_Nat_Oadd__diff__assoc2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),J)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K)),I) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),I)),K) ) ) ).

% Nat.add_diff_assoc2
tff(fact_625_Nat_Odiff__diff__right,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),J)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),J) ) ) ).

% Nat.diff_diff_right
tff(fact_626_semiring__norm_I2_J,axiom,
    aa_num_num(aa_num_fun_num_num(plus_plus_num,one),one) = aa_num_num(bit0,one) ).

% semiring_norm(2)
tff(fact_627_vebt__succ_Osimps_I3_J,axiom,
    ! [Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT,Vaa: nat] : aa_nat_option_nat(vEBT_vebt_succ(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Ux,Uy),Uz)),Vaa) = none_nat ).

% vebt_succ.simps(3)
tff(fact_628_vebt__member_Osimps_I2_J,axiom,
    ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT,Xb: nat] : ~ aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv),Uw)),Xb) ).

% vebt_member.simps(2)
tff(fact_629_ord__eq__le__eq__trans,axiom,
    ! [A: set_int,B: set_int,C: set_int,D: set_int] :
      ( ( A = B )
     => ( aa_set_int_o(ord_less_eq_set_int(B),C)
       => ( ( C = D )
         => aa_set_int_o(ord_less_eq_set_int(A),D) ) ) ) ).

% ord_eq_le_eq_trans
tff(fact_630_ord__eq__le__eq__trans,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( A = B )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C)
       => ( ( C = D )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),D) ) ) ) ).

% ord_eq_le_eq_trans
tff(fact_631_ord__eq__le__eq__trans,axiom,
    ! [A: num,B: num,C: num,D: num] :
      ( ( A = B )
     => ( aa_num_o(ord_less_eq_num(B),C)
       => ( ( C = D )
         => aa_num_o(ord_less_eq_num(A),D) ) ) ) ).

% ord_eq_le_eq_trans
tff(fact_632_ord__eq__le__eq__trans,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( A = B )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),C)
       => ( ( C = D )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),D) ) ) ) ).

% ord_eq_le_eq_trans
tff(fact_633_ord__eq__le__eq__trans,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( A = B )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),C)
       => ( ( C = D )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),D) ) ) ) ).

% ord_eq_le_eq_trans
tff(fact_634_pairself_Ocases,axiom,
    ! [Xb: produc4788131091073775919nt_int] :
      ~ ! [F2: fun_in5730379987311621695e_term,A3: int,B2: int] : Xb != aa_Pro8800328608558162860nt_int(produc4729913460834754785nt_int(F2),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A3),B2)) ).

% pairself.cases
tff(fact_635_bex2I,axiom,
    ! [A: fun_Pr5098669352780829690e_term,B: produc8923325533196201883nteger,S2: set_Pr3486312109254157132nteger,P: fun_fu7695273699059480439eger_o] :
      ( member8978266922497061677nteger(aa_Pro4291442876242880245nteger(produc565448451787809096nteger(A),B),S2)
     => ( ( member8978266922497061677nteger(aa_Pro4291442876242880245nteger(produc565448451787809096nteger(A),B),S2)
         => aa_Pro4926521453222654329eger_o(aa_fun4505422719058378424eger_o(P,A),B) )
       => ? [A3: fun_Pr5098669352780829690e_term,B2: produc8923325533196201883nteger] :
            ( member8978266922497061677nteger(aa_Pro4291442876242880245nteger(produc565448451787809096nteger(A3),B2),S2)
            & aa_Pro4926521453222654329eger_o(aa_fun4505422719058378424eger_o(P,A3),B2) ) ) ) ).

% bex2I
tff(fact_636_bex2I,axiom,
    ! [A: fun_Pr6460944699146269935_nat_o,B: produc3658429121746597890et_nat,S2: set_Pr1312406793078187350et_nat,P: fun_fu8419763957888848259_nat_o] :
      ( member4513763875858039863et_nat(aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(A),B),S2)
     => ( ( member4513763875858039863et_nat(aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(A),B),S2)
         => aa_Pro653082453696054616_nat_o(aa_fun2638059832795814268_nat_o(P,A),B) )
       => ? [A3: fun_Pr6460944699146269935_nat_o,B2: produc3658429121746597890et_nat] :
            ( member4513763875858039863et_nat(aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(A3),B2),S2)
            & aa_Pro653082453696054616_nat_o(aa_fun2638059832795814268_nat_o(P,A3),B2) ) ) ) ).

% bex2I
tff(fact_637_bex2I,axiom,
    ! [A: fun_Pr6460944699146269935_nat_o,B: produc8799533983703829792et_nat,S2: set_Pr4040394263813967220et_nat,P: fun_fu2611741329380808165_nat_o] :
      ( member9174161442226468181et_nat(aa_Pro1979504083478021276et_nat(produc7291855603632349880et_nat(A),B),S2)
     => ( ( member9174161442226468181et_nat(aa_Pro1979504083478021276et_nat(produc7291855603632349880et_nat(A),B),S2)
         => aa_Pro5643009357358181178_nat_o(aa_fun912680268133416158_nat_o(P,A),B) )
       => ? [A3: fun_Pr6460944699146269935_nat_o,B2: produc8799533983703829792et_nat] :
            ( member9174161442226468181et_nat(aa_Pro1979504083478021276et_nat(produc7291855603632349880et_nat(A3),B2),S2)
            & aa_Pro5643009357358181178_nat_o(aa_fun912680268133416158_nat_o(P,A3),B2) ) ) ) ).

% bex2I
tff(fact_638_bex2I,axiom,
    ! [A: fun_Pr7763680713840462443e_term,B: product_prod_int_int,S2: set_Pr6700895561065273913nt_int,P: fun_fu6865642474282033866_int_o] :
      ( member753276645541486746nt_int(aa_Pro5748361610964203648nt_int(produc7320969046608631221nt_int(A),B),S2)
     => ( ( member753276645541486746nt_int(aa_Pro5748361610964203648nt_int(produc7320969046608631221nt_int(A),B),S2)
         => aa_Pro4592068669816069595_int_o(aa_fun3538162476766722827_int_o(P,A),B) )
       => ? [A3: fun_Pr7763680713840462443e_term,B2: product_prod_int_int] :
            ( member753276645541486746nt_int(aa_Pro5748361610964203648nt_int(produc7320969046608631221nt_int(A3),B2),S2)
            & aa_Pro4592068669816069595_int_o(aa_fun3538162476766722827_int_o(P,A3),B2) ) ) ) ).

% bex2I
tff(fact_639_bex2I,axiom,
    ! [A: fun_in5730379987311621695e_term,B: product_prod_int_int,S2: set_Pr4166680992137330533nt_int,P: fun_fu2978539155105838238_int_o] :
      ( member5890090881338784710nt_int(aa_Pro8800328608558162860nt_int(produc4729913460834754785nt_int(A),B),S2)
     => ( ( member5890090881338784710nt_int(aa_Pro8800328608558162860nt_int(produc4729913460834754785nt_int(A),B),S2)
         => aa_Pro4592068669816069595_int_o(aa_fun586359091965693151_int_o(P,A),B) )
       => ? [A3: fun_in5730379987311621695e_term,B2: product_prod_int_int] :
            ( member5890090881338784710nt_int(aa_Pro8800328608558162860nt_int(produc4729913460834754785nt_int(A3),B2),S2)
            & aa_Pro4592068669816069595_int_o(aa_fun586359091965693151_int_o(P,A3),B2) ) ) ) ).

% bex2I
tff(fact_640_Suc__inject,axiom,
    ! [Xb: nat,Y: nat] :
      ( ( aa_nat_nat(suc,Xb) = aa_nat_nat(suc,Y) )
     => ( Xb = Y ) ) ).

% Suc_inject
tff(fact_641_n__not__Suc__n,axiom,
    ! [Nb: nat] : Nb != aa_nat_nat(suc,Nb) ).

% n_not_Suc_n
tff(fact_642_nat__neq__iff,axiom,
    ! [M: nat,Nb: nat] :
      ( ( M != Nb )
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb)
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),M) ) ) ).

% nat_neq_iff
tff(fact_643_less__not__refl,axiom,
    ! [Nb: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Nb) ).

% less_not_refl
tff(fact_644_less__not__refl2,axiom,
    ! [Nb: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),M)
     => ( M != Nb ) ) ).

% less_not_refl2
tff(fact_645_less__not__refl3,axiom,
    ! [S3: nat,Ta: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,S3),Ta)
     => ( S3 != Ta ) ) ).

% less_not_refl3
tff(fact_646_less__irrefl__nat,axiom,
    ! [Nb: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Nb) ).

% less_irrefl_nat
tff(fact_647_nat__less__induct,axiom,
    ! [P: fun_nat_o,Nb: nat] :
      ( ! [N2: nat] :
          ( ! [M3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M3),N2)
             => aa_nat_o(P,M3) )
         => aa_nat_o(P,N2) )
     => aa_nat_o(P,Nb) ) ).

% nat_less_induct
tff(fact_648_infinite__descent,axiom,
    ! [P: fun_nat_o,Nb: nat] :
      ( ! [N2: nat] :
          ( ~ aa_nat_o(P,N2)
         => ? [M3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M3),N2)
              & ~ aa_nat_o(P,M3) ) )
     => aa_nat_o(P,Nb) ) ).

% infinite_descent
tff(fact_649_linorder__neqE__nat,axiom,
    ! [Xb: nat,Y: nat] :
      ( ( Xb != Y )
     => ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),Xb) ) ) ).

% linorder_neqE_nat
tff(fact_650_size__neq__size__imp__neq,axiom,
    ! [Xb: list_real,Y: list_real] :
      ( ( size_size_list_real(Xb) != size_size_list_real(Y) )
     => ( Xb != Y ) ) ).

% size_neq_size_imp_neq
tff(fact_651_size__neq__size__imp__neq,axiom,
    ! [Xb: list_o,Y: list_o] :
      ( ( size_size_list_o(Xb) != size_size_list_o(Y) )
     => ( Xb != Y ) ) ).

% size_neq_size_imp_neq
tff(fact_652_size__neq__size__imp__neq,axiom,
    ! [Xb: list_nat,Y: list_nat] :
      ( ( size_size_list_nat(Xb) != size_size_list_nat(Y) )
     => ( Xb != Y ) ) ).

% size_neq_size_imp_neq
tff(fact_653_size__neq__size__imp__neq,axiom,
    ! [Xb: list_int,Y: list_int] :
      ( ( size_size_list_int(Xb) != size_size_list_int(Y) )
     => ( Xb != Y ) ) ).

% size_neq_size_imp_neq
tff(fact_654_size__neq__size__imp__neq,axiom,
    ! [Xb: num,Y: num] :
      ( ( aa_num_nat(size_size_num,Xb) != aa_num_nat(size_size_num,Y) )
     => ( Xb != Y ) ) ).

% size_neq_size_imp_neq
tff(fact_655_le__refl,axiom,
    ! [Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Nb) ).

% le_refl
tff(fact_656_le__trans,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,J),K)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),K) ) ) ).

% le_trans
tff(fact_657_eq__imp__le,axiom,
    ! [M: nat,Nb: nat] :
      ( ( M = Nb )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb) ) ).

% eq_imp_le
tff(fact_658_le__antisym,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),M)
       => ( M = Nb ) ) ) ).

% le_antisym
tff(fact_659_nat__le__linear,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
      | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),M) ) ).

% nat_le_linear
tff(fact_660_Nat_Oex__has__greatest__nat,axiom,
    ! [P: fun_nat_o,K: nat,B: nat] :
      ( aa_nat_o(P,K)
     => ( ! [Y4: nat] :
            ( aa_nat_o(P,Y4)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y4),B) )
       => ? [X2: nat] :
            ( aa_nat_o(P,X2)
            & ! [Y5: nat] :
                ( aa_nat_o(P,Y5)
               => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y5),X2) ) ) ) ) ).

% Nat.ex_has_greatest_nat
tff(fact_661_diff__commute,axiom,
    ! [I: nat,J: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I),J)),K) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I),K)),J) ).

% diff_commute
tff(fact_662_vebt__pred_Osimps_I4_J,axiom,
    ! [Uy: nat,Uz: list_VEBT_VEBT,Vaa: vEBT_VEBT,Vb: nat] : aa_nat_option_nat(vEBT_vebt_pred(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uy,Uz),Vaa)),Vb) = none_nat ).

% vebt_pred.simps(4)
tff(fact_663_VEBT__internal_OminNulli_Ocases,axiom,
    ! [Xb: vEBT_VEBTi] :
      ( ( Xb != vEBT_Leafi($false,$false) )
     => ( ! [Uv2: $o] : Xb != vEBT_Leafi($true,(Uv2))
       => ( ! [Uu2: $o] : Xb != vEBT_Leafi((Uu2),$true)
         => ( ! [Uw2: nat,Ux2: array_VEBT_VEBTi,Uy2: vEBT_VEBTi] : Xb != aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(none_P5556105721700978146at_nat,Uw2,Ux2),Uy2)
           => ~ ! [Uz2: product_prod_nat_nat,Va: nat,Vb2: array_VEBT_VEBTi,Vc: vEBT_VEBTi] : Xb != aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(some_P7363390416028606310at_nat(Uz2),Va,Vb2),Vc) ) ) ) ) ).

% VEBT_internal.minNulli.cases
tff(fact_664_vebt__minti_Ocases,axiom,
    ! [Xb: vEBT_VEBTi] :
      ( ! [A3: $o,B2: $o] : Xb != vEBT_Leafi((A3),(B2))
     => ( ! [Uu2: nat,Uv2: array_VEBT_VEBTi,Uw2: vEBT_VEBTi] : Xb != aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(none_P5556105721700978146at_nat,Uu2,Uv2),Uw2)
       => ~ ! [Mi: nat,Ma: nat,Ux2: nat,Uy2: array_VEBT_VEBTi,Uz2: vEBT_VEBTi] : Xb != aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),Ux2,Uy2),Uz2) ) ) ).

% vebt_minti.cases
tff(fact_665_vebt__maxti_Osimps_I2_J,axiom,
    ! [Uu: nat,Uv: array_VEBT_VEBTi,Uw: vEBT_VEBTi] : aa_VEB2711320435582112505on_nat(vEBT_vebt_maxti,aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(none_P5556105721700978146at_nat,Uu,Uv),Uw)) = heap_T3487192422709364219on_nat(none_nat) ).

% vebt_maxti.simps(2)
tff(fact_666_vebt__minti_Osimps_I2_J,axiom,
    ! [Uu: nat,Uv: array_VEBT_VEBTi,Uw: vEBT_VEBTi] : aa_VEB2711320435582112505on_nat(vEBT_vebt_minti,aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(none_P5556105721700978146at_nat,Uu,Uv),Uw)) = heap_T3487192422709364219on_nat(none_nat) ).

% vebt_minti.simps(2)
tff(fact_667_option_Ocase__distrib,axiom,
    ! [H: fun_num_num,F1: num,F22: fun_num_num,Option: option_num] : aa_num_num(H,aa_option_num_num(case_option_num_num(F1,F22),Option)) = aa_option_num_num(case_option_num_num(aa_num_num(H,F1),aa_fun8048995283964520608um_num(aTP_Lamm_cj(H),F22)),Option) ).

% option.case_distrib
tff(fact_668_option_Ocase__distrib,axiom,
    ! [H: fun_num_int,F1: num,F22: fun_num_num,Option: option_num] : aa_num_int(H,aa_option_num_num(case_option_num_num(F1,F22),Option)) = aa_option_num_int(case_option_int_num(aa_num_int(H,F1),aa_fun3162900327506486194um_int(aTP_Lamm_ck(H),F22)),Option) ).

% option.case_distrib
tff(fact_669_option_Ocase__distrib,axiom,
    ! [H: fun_int_num,F1: int,F22: fun_num_int,Option: option_num] : aa_int_num(H,aa_option_num_int(case_option_int_num(F1,F22),Option)) = aa_option_num_num(case_option_num_num(aa_int_num(H,F1),aa_fun5215869348392759218um_num(aTP_Lamm_cl(H),F22)),Option) ).

% option.case_distrib
tff(fact_670_option_Ocase__distrib,axiom,
    ! [H: fun_int_int,F1: int,F22: fun_num_int,Option: option_num] : aa_int_int(H,aa_option_num_int(case_option_int_num(F1,F22),Option)) = aa_option_num_int(case_option_int_num(aa_int_int(H,F1),aa_fun329774391934724804um_int(aTP_Lamm_cm(H),F22)),Option) ).

% option.case_distrib
tff(fact_671_option_Ocase__distrib,axiom,
    ! [H: fun_option_num_num,F1: option_num,F22: fun_num_option_num,Option: option_num] : aa_option_num_num(H,aa_opt5369159329878751634on_num(case_o6005452278849405969um_num(F1,F22),Option)) = aa_option_num_num(case_option_num_num(aa_option_num_num(H,F1),aa_fun7142274290070961008um_num(aTP_Lamm_cn(H),F22)),Option) ).

% option.case_distrib
tff(fact_672_option_Ocase__distrib,axiom,
    ! [H: fun_option_num_int,F1: option_num,F22: fun_num_option_num,Option: option_num] : aa_option_num_int(H,aa_opt5369159329878751634on_num(case_o6005452278849405969um_num(F1,F22),Option)) = aa_option_num_int(case_option_int_num(aa_option_num_int(H,F1),aa_fun2256179333612926594um_int(aTP_Lamm_co(H),F22)),Option) ).

% option.case_distrib
tff(fact_673_option_Ocase__distrib,axiom,
    ! [H: fun_num_option_num,F1: num,F22: fun_num_num,Option: option_num] : aa_num_option_num(H,aa_option_num_num(case_option_num_num(F1,F22),Option)) = aa_opt5369159329878751634on_num(case_o6005452278849405969um_num(aa_num_option_num(H,F1),aa_fun612943910911948912on_num(aTP_Lamm_cp(H),F22)),Option) ).

% option.case_distrib
tff(fact_674_option_Ocase__distrib,axiom,
    ! [H: fun_int_option_num,F1: int,F22: fun_num_int,Option: option_num] : aa_int_option_num(H,aa_option_num_int(case_option_int_num(F1,F22),Option)) = aa_opt5369159329878751634on_num(case_o6005452278849405969um_num(aa_int_option_num(H,F1),aa_fun3858197967482829954on_num(aTP_Lamm_cq(H),F22)),Option) ).

% option.case_distrib
tff(fact_675_option_Ocase__distrib,axiom,
    ! [H: fun_op8989173174992673105on_num,F1: option_num,F22: fun_num_option_num,Option: option_num] : aa_opt5369159329878751634on_num(H,aa_opt5369159329878751634on_num(case_o6005452278849405969um_num(F1,F22),Option)) = aa_opt5369159329878751634on_num(case_o6005452278849405969um_num(aa_opt5369159329878751634on_num(H,F1),aa_fun3636977833914307392on_num(aTP_Lamm_cr(H),F22)),Option) ).

% option.case_distrib
tff(fact_676_option_Ocase__distrib,axiom,
    ! [H: fun_o_o,F1: $o,F22: fun_Pr4796943543331071514_nat_o,Option: option4927543243414619207at_nat] :
      ( aa_o_o(H,aa_opt7112393969400857165_nat_o(case_o184042715313410164at_nat((F1),F22),Option))
    <=> aa_opt7112393969400857165_nat_o(case_o184042715313410164at_nat(aa_o_o(H,(F1)),aa_fun359341089595923442_nat_o(aTP_Lamm_cs(H),F22)),Option) ) ).

% option.case_distrib
tff(fact_677_z1pdiv2,axiom,
    ! [B: 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,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B)),one_one_int)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = B ).

% z1pdiv2
tff(fact_678_le__some__optE,axiom,
    ! [M: set_int,Xb: option_set_int] :
      ( ord_le353528952715127954et_int(some_set_int(M),Xb)
     => ~ ! [M4: set_int] :
            ( ( Xb = some_set_int(M4) )
           => ~ aa_set_int_o(ord_less_eq_set_int(M),M4) ) ) ).

% le_some_optE
tff(fact_679_le__some__optE,axiom,
    ! [M: rat,Xb: option_rat] :
      ( ord_le2406147912482264968on_rat(some_rat(M),Xb)
     => ~ ! [M4: rat] :
            ( ( Xb = some_rat(M4) )
           => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,M),M4) ) ) ).

% le_some_optE
tff(fact_680_le__some__optE,axiom,
    ! [M: num,Xb: option_num] :
      ( aa_option_num_o(aa_opt8674010985164783839_num_o(ord_le6622620407824499402on_num,aa_num_option_num(some_num,M)),Xb)
     => ~ ! [M4: num] :
            ( ( Xb = aa_num_option_num(some_num,M4) )
           => ~ aa_num_o(ord_less_eq_num(M),M4) ) ) ).

% le_some_optE
tff(fact_681_le__some__optE,axiom,
    ! [M: nat,Xb: option_nat] :
      ( aa_option_nat_o(aa_opt4319133605541840243_nat_o(ord_le5914376470875661696on_nat,aa_nat_option_nat(some_nat,M)),Xb)
     => ~ ! [M4: nat] :
            ( ( Xb = aa_nat_option_nat(some_nat,M4) )
           => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),M4) ) ) ).

% le_some_optE
tff(fact_682_le__some__optE,axiom,
    ! [M: int,Xb: option_int] :
      ( ord_le1736525451366464988on_int(aa_int_option_int(some_int,M),Xb)
     => ~ ! [M4: int] :
            ( ( Xb = aa_int_option_int(some_int,M4) )
           => ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,M),M4) ) ) ).

% le_some_optE
tff(fact_683_Nat_OlessE,axiom,
    ! [I: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),K)
     => ( ( K != aa_nat_nat(suc,I) )
       => ~ ! [J2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J2)
             => ( K != aa_nat_nat(suc,J2) ) ) ) ) ).

% Nat.lessE
tff(fact_684_Suc__lessD,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,M)),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb) ) ).

% Suc_lessD
tff(fact_685_Suc__lessE,axiom,
    ! [I: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,I)),K)
     => ~ ! [J2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J2)
           => ( K != aa_nat_nat(suc,J2) ) ) ) ).

% Suc_lessE
tff(fact_686_Suc__lessI,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb)
     => ( ( aa_nat_nat(suc,M) != Nb )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,M)),Nb) ) ) ).

% Suc_lessI
tff(fact_687_less__SucE,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),aa_nat_nat(suc,Nb))
     => ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb)
       => ( M = Nb ) ) ) ).

% less_SucE
tff(fact_688_less__SucI,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),aa_nat_nat(suc,Nb)) ) ).

% less_SucI
tff(fact_689_Ex__less__Suc,axiom,
    ! [Nb: nat,P: fun_nat_o] :
      ( ? [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),aa_nat_nat(suc,Nb))
          & aa_nat_o(P,I2) )
    <=> ( aa_nat_o(P,Nb)
        | ? [I2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),Nb)
            & aa_nat_o(P,I2) ) ) ) ).

% Ex_less_Suc
tff(fact_690_less__Suc__eq,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),aa_nat_nat(suc,Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb)
        | ( M = Nb ) ) ) ).

% less_Suc_eq
tff(fact_691_not__less__eq,axiom,
    ! [M: nat,Nb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(suc,M)) ) ).

% not_less_eq
tff(fact_692_Nat_OAll__less__Suc,axiom,
    ! [Nb: nat,P: fun_nat_o] :
      ( ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),aa_nat_nat(suc,Nb))
         => aa_nat_o(P,I2) )
    <=> ( aa_nat_o(P,Nb)
        & ! [I2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),Nb)
           => aa_nat_o(P,I2) ) ) ) ).

% Nat.All_less_Suc
tff(fact_693_Suc__less__eq2,axiom,
    ! [Nb: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Nb)),M)
    <=> ? [M5: nat] :
          ( ( M = aa_nat_nat(suc,M5) )
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),M5) ) ) ).

% Suc_less_eq2
tff(fact_694_less__antisym,axiom,
    ! [Nb: nat,M: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(suc,M))
       => ( M = Nb ) ) ) ).

% less_antisym
tff(fact_695_Suc__less__SucD,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,M)),aa_nat_nat(suc,Nb))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb) ) ).

% Suc_less_SucD
tff(fact_696_less__trans__Suc,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J),K)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,I)),K) ) ) ).

% less_trans_Suc
tff(fact_697_less__Suc__induct,axiom,
    ! [I: nat,J: nat,P: fun_nat_fun_nat_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
     => ( ! [I3: nat] : aa_nat_o(aa_nat_fun_nat_o(P,I3),aa_nat_nat(suc,I3))
       => ( ! [I3: nat,J2: nat,K3: 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),K3)
               => ( aa_nat_o(aa_nat_fun_nat_o(P,I3),J2)
                 => ( aa_nat_o(aa_nat_fun_nat_o(P,J2),K3)
                   => aa_nat_o(aa_nat_fun_nat_o(P,I3),K3) ) ) ) )
         => aa_nat_o(aa_nat_fun_nat_o(P,I),J) ) ) ) ).

% less_Suc_induct
tff(fact_698_strict__inc__induct,axiom,
    ! [I: nat,J: nat,P: fun_nat_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
     => ( ! [I3: nat] :
            ( ( J = aa_nat_nat(suc,I3) )
           => aa_nat_o(P,I3) )
       => ( ! [I3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),J)
             => ( aa_nat_o(P,aa_nat_nat(suc,I3))
               => aa_nat_o(P,I3) ) )
         => aa_nat_o(P,I) ) ) ) ).

% strict_inc_induct
tff(fact_699_not__less__less__Suc__eq,axiom,
    ! [Nb: nat,M: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(suc,M))
      <=> ( Nb = M ) ) ) ).

% not_less_less_Suc_eq
tff(fact_700_nat__arith_Osuc1,axiom,
    ! [A2: nat,K: nat,A: nat] :
      ( ( A2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),A) )
     => ( aa_nat_nat(suc,A2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),aa_nat_nat(suc,A)) ) ) ).

% nat_arith.suc1
tff(fact_701_add__Suc,axiom,
    ! [M: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(suc,M)),Nb) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Nb)) ).

% add_Suc
tff(fact_702_add__Suc__shift,axiom,
    ! [M: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(suc,M)),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),aa_nat_nat(suc,Nb)) ).

% add_Suc_shift
tff(fact_703_Suc__leD,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,M)),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb) ) ).

% Suc_leD
tff(fact_704_le__SucE,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(suc,Nb))
     => ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
       => ( M = aa_nat_nat(suc,Nb) ) ) ) ).

% le_SucE
tff(fact_705_le__SucI,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(suc,Nb)) ) ).

% le_SucI
tff(fact_706_Suc__le__D,axiom,
    ! [Nb: nat,M6: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Nb)),M6)
     => ? [M2: nat] : M6 = aa_nat_nat(suc,M2) ) ).

% Suc_le_D
tff(fact_707_le__Suc__eq,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(suc,Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
        | ( M = aa_nat_nat(suc,Nb) ) ) ) ).

% le_Suc_eq
tff(fact_708_Suc__n__not__le__n,axiom,
    ! [Nb: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Nb)),Nb) ).

% Suc_n_not_le_n
tff(fact_709_not__less__eq__eq,axiom,
    ! [M: nat,Nb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Nb)),M) ) ).

% not_less_eq_eq
tff(fact_710_full__nat__induct,axiom,
    ! [P: fun_nat_o,Nb: nat] :
      ( ! [N2: nat] :
          ( ! [M3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,M3)),N2)
             => aa_nat_o(P,M3) )
         => aa_nat_o(P,N2) )
     => aa_nat_o(P,Nb) ) ).

% full_nat_induct
tff(fact_711_nat__induct__at__least,axiom,
    ! [M: nat,Nb: nat,P: fun_nat_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
     => ( aa_nat_o(P,M)
       => ( ! [N2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
             => ( aa_nat_o(P,N2)
               => aa_nat_o(P,aa_nat_nat(suc,N2)) ) )
         => aa_nat_o(P,Nb) ) ) ) ).

% nat_induct_at_least
tff(fact_712_transitive__stepwise__le,axiom,
    ! [M: nat,Nb: nat,R: fun_nat_fun_nat_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
     => ( ! [X2: nat] : aa_nat_o(aa_nat_fun_nat_o(R,X2),X2)
       => ( ! [X2: nat,Y4: nat,Z3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(R,X2),Y4)
             => ( aa_nat_o(aa_nat_fun_nat_o(R,Y4),Z3)
               => aa_nat_o(aa_nat_fun_nat_o(R,X2),Z3) ) )
         => ( ! [N2: nat] : aa_nat_o(aa_nat_fun_nat_o(R,N2),aa_nat_nat(suc,N2))
           => aa_nat_o(aa_nat_fun_nat_o(R,M),Nb) ) ) ) ) ).

% transitive_stepwise_le
tff(fact_713_add__lessD1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),J)),K)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),K) ) ).

% add_lessD1
tff(fact_714_add__less__mono,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),L)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L)) ) ) ).

% add_less_mono
tff(fact_715_not__add__less1,axiom,
    ! [I: nat,J: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),J)),I) ).

% not_add_less1
tff(fact_716_not__add__less2,axiom,
    ! [J: nat,I: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),I)),I) ).

% not_add_less2
tff(fact_717_add__less__mono1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K)) ) ).

% add_less_mono1
tff(fact_718_trans__less__add1,axiom,
    ! [I: nat,J: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),M)) ) ).

% trans_less_add1
tff(fact_719_trans__less__add2,axiom,
    ! [I: nat,J: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),J)) ) ).

% trans_less_add2
tff(fact_720_less__add__eq__less,axiom,
    ! [K: nat,L: nat,M: nat,Nb: 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),Nb) )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb) ) ) ).

% less_add_eq_less
tff(fact_721_exists__leI,axiom,
    ! [Nb: nat,P: fun_nat_o] :
      ( ( ! [N4: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N4),Nb)
           => ~ aa_nat_o(P,N4) )
       => aa_nat_o(P,Nb) )
     => ? [N5: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N5),Nb)
          & aa_nat_o(P,N5) ) ) ).

% exists_leI
tff(fact_722_nat__less__le,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
        & ( M != Nb ) ) ) ).

% nat_less_le
tff(fact_723_less__imp__le__nat,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb) ) ).

% less_imp_le_nat
tff(fact_724_le__eq__less__or__eq,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb)
        | ( M = Nb ) ) ) ).

% le_eq_less_or_eq
tff(fact_725_less__or__eq__imp__le,axiom,
    ! [M: nat,Nb: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb)
        | ( M = Nb ) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb) ) ).

% less_or_eq_imp_le
tff(fact_726_le__neq__implies__less,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
     => ( ( M != Nb )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb) ) ) ).

% le_neq_implies_less
tff(fact_727_less__mono__imp__le__mono,axiom,
    ! [F: fun_nat_nat,I: 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,I),J)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F,I)),aa_nat_nat(F,J)) ) ) ).

% less_mono_imp_le_mono
tff(fact_728_zero__induct__lemma,axiom,
    ! [P: fun_nat_o,K: nat,I: nat] :
      ( aa_nat_o(P,K)
     => ( ! [N2: nat] :
            ( aa_nat_o(P,aa_nat_nat(suc,N2))
           => aa_nat_o(P,N2) )
       => aa_nat_o(P,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),I)) ) ) ).

% zero_induct_lemma
tff(fact_729_diff__less__mono2,axiom,
    ! [M: nat,Nb: nat,L: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb)
     => ( 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),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,L),M)) ) ) ).

% diff_less_mono2
tff(fact_730_less__imp__diff__less,axiom,
    ! [J: nat,K: nat,Nb: 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),Nb)),K) ) ).

% less_imp_diff_less
tff(fact_731_Suc__mult__cancel1,axiom,
    ! [K: nat,M: nat,Nb: 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)),Nb) )
    <=> ( M = Nb ) ) ).

% Suc_mult_cancel1
tff(fact_732_add__leE,axiom,
    ! [M: nat,K: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K)),Nb)
     => ~ ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
         => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb) ) ) ).

% add_leE
tff(fact_733_le__add1,axiom,
    ! [Nb: nat,M: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),M)) ).

% le_add1
tff(fact_734_le__add2,axiom,
    ! [Nb: nat,M: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Nb)) ).

% le_add2
tff(fact_735_add__leD1,axiom,
    ! [M: nat,K: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K)),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb) ) ).

% add_leD1
tff(fact_736_add__leD2,axiom,
    ! [M: nat,K: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K)),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb) ) ).

% add_leD2
tff(fact_737_le__Suc__ex,axiom,
    ! [K: nat,L: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),L)
     => ? [N2: nat] : L = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),N2) ) ).

% le_Suc_ex
tff(fact_738_add__le__mono,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),L)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L)) ) ) ).

% add_le_mono
tff(fact_739_add__le__mono1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K)) ) ).

% add_le_mono1
tff(fact_740_trans__le__add1,axiom,
    ! [I: nat,J: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),M)) ) ).

% trans_le_add1
tff(fact_741_trans__le__add2,axiom,
    ! [I: nat,J: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),J)) ) ).

% trans_le_add2
tff(fact_742_nat__le__iff__add,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
    <=> ? [K4: nat] : Nb = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K4) ) ).

% nat_le_iff_add
tff(fact_743_fstE,axiom,
    ! [Xb: produc2453742332425399446nteger,A: fun_Pr5098669352780829690e_term,B: produc8923325533196201883nteger,P: fun_fu4736795817735658359term_o] :
      ( ( Xb = aa_Pro4291442876242880245nteger(produc565448451787809096nteger(A),B) )
     => ( aa_fun7993445754732355552term_o(P,aa_Pro4276937008533432526e_term(produc2481150572478215260nteger,Xb))
       => aa_fun7993445754732355552term_o(P,A) ) ) ).

% fstE
tff(fact_744_fstE,axiom,
    ! [Xb: produc8799533983703829792et_nat,A: fun_Pr6460944699146269935_nat_o,B: produc3658429121746597890et_nat,P: fun_fu5135969500543783852at_o_o] :
      ( ( Xb = aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(A),B) )
     => ( aa_fun1675921005893230629at_o_o(P,aa_Pro8262445894397805713_nat_o(produc5325900173150335662et_nat,Xb))
       => aa_fun1675921005893230629at_o_o(P,A) ) ) ).

% fstE
tff(fact_745_fstE,axiom,
    ! [Xb: produc3112576180687729086et_nat,A: fun_Pr6460944699146269935_nat_o,B: produc8799533983703829792et_nat,P: fun_fu5135969500543783852at_o_o] :
      ( ( Xb = aa_Pro1979504083478021276et_nat(produc7291855603632349880et_nat(A),B) )
     => ( aa_fun1675921005893230629at_o_o(P,aa_Pro2837773213478468339_nat_o(produc4078497350495916492et_nat,Xb))
       => aa_fun1675921005893230629at_o_o(P,A) ) ) ).

% fstE
tff(fact_746_fstE,axiom,
    ! [Xb: produc3043004577601930755nt_int,A: fun_Pr7763680713840462443e_term,B: product_prod_int_int,P: fun_fu6378075912484669126term_o] :
      ( ( Xb = aa_Pro5748361610964203648nt_int(produc7320969046608631221nt_int(A),B) )
     => ( aa_fun7650857933016935215term_o(P,aa_Pro4010435699744810540e_term(produc3130796691802200265nt_int,Xb))
       => aa_fun7650857933016935215term_o(P,A) ) ) ).

% fstE
tff(fact_747_fstE,axiom,
    ! [Xb: produc4788131091073775919nt_int,A: fun_in5730379987311621695e_term,B: product_prod_int_int,P: fun_fu3265955482920998386term_o] :
      ( ( Xb = aa_Pro8800328608558162860nt_int(produc4729913460834754785nt_int(A),B) )
     => ( aa_fun3020711759848831579term_o(P,aa_Pro2551989255418117932e_term(produc4463008262350210037nt_int,Xb))
       => aa_fun3020711759848831579term_o(P,A) ) ) ).

% fstE
tff(fact_748_fstE,axiom,
    ! [Xb: produc3658429121746597890et_nat,A: heap_e7401611519738050253t_unit,B: set_nat,P: fun_he8413527710729349838unit_o] :
      ( ( Xb = aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(A),B) )
     => ( aa_hea7994211679605476167unit_o(P,aa_Pro1219312880135979149t_unit(produc1824681642469235216et_nat,Xb))
       => aa_hea7994211679605476167unit_o(P,A) ) ) ).

% fstE
tff(fact_749_fstE,axiom,
    ! [Xb: produc6575502325842934193n_assn,A: assn,B: assn,P: fun_assn_o] :
      ( ( Xb = aa_ass5585057816575862475n_assn(produc118845697133431529n_assn(A),B) )
     => ( aa_assn_o(P,aa_Pro2979818503705714077n_assn(produc9167289414957590229n_assn,Xb))
       => aa_assn_o(P,A) ) ) ).

% fstE
tff(fact_750_fstE,axiom,
    ! [Xb: produc827990862158126777uint32,A: uint32,B: uint32,P: fun_uint32_o] :
      ( ( Xb = aa_uin6954824371948101007uint32(produc1400373151660368625uint32(A),B) )
     => ( aa_uint32_o(P,aa_Pro3737690986720249065uint32(produc9004433772639906525uint32,Xb))
       => aa_uint32_o(P,A) ) ) ).

% fstE
tff(fact_751_fstE,axiom,
    ! [Xb: product_prod_nat_nat,A: nat,B: nat,P: fun_nat_o] :
      ( ( Xb = aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,A),B) )
     => ( aa_nat_o(P,aa_Pro4243173770044397973at_nat(product_fst_nat_nat,Xb))
       => aa_nat_o(P,A) ) ) ).

% fstE
tff(fact_752_fstE,axiom,
    ! [Xb: product_prod_int_int,A: int,B: int,P: fun_int_o] :
      ( ( Xb = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),B) )
     => ( aa_int_o(P,aa_Pro6799258434465214121nt_int(product_fst_int_int,Xb))
       => aa_int_o(P,A) ) ) ).

% fstE
tff(fact_753_diff__add__inverse2,axiom,
    ! [M: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Nb)),Nb) = M ).

% diff_add_inverse2
tff(fact_754_diff__add__inverse,axiom,
    ! [Nb: 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,Nb),M)),Nb) = M ).

% diff_add_inverse
tff(fact_755_diff__cancel2,axiom,
    ! [M: nat,K: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),K)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Nb) ).

% diff_cancel2
tff(fact_756_Nat_Odiff__cancel,axiom,
    ! [K: nat,M: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Nb) ).

% Nat.diff_cancel
tff(fact_757_eq__diff__iff,axiom,
    ! [K: nat,M: nat,Nb: 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),Nb)
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),K) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K) )
        <=> ( M = Nb ) ) ) ) ).

% eq_diff_iff
tff(fact_758_le__diff__iff,axiom,
    ! [K: nat,M: nat,Nb: 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),Nb)
       => ( 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,Nb),K))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb) ) ) ) ).

% le_diff_iff
tff(fact_759_Nat_Odiff__diff__eq,axiom,
    ! [K: nat,M: nat,Nb: 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),Nb)
       => ( 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,Nb),K)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Nb) ) ) ) ).

% Nat.diff_diff_eq
tff(fact_760_diff__le__mono,axiom,
    ! [M: nat,Nb: nat,L: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
     => 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,Nb),L)) ) ).

% diff_le_mono
tff(fact_761_diff__le__self,axiom,
    ! [M: nat,Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Nb)),M) ).

% diff_le_self
tff(fact_762_le__diff__iff_H,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),C)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),C)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,C),B))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A) ) ) ) ).

% le_diff_iff'
tff(fact_763_diff__le__mono2,axiom,
    ! [M: nat,Nb: nat,L: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,L),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,L),M)) ) ).

% diff_le_mono2
tff(fact_764_option_Odistinct_I1_J,axiom,
    ! [X23: produc120671012495760973it_nat] : none_P1551326421579882414it_nat != some_P2407035485129114418it_nat(X23) ).

% option.distinct(1)
tff(fact_765_option_Odistinct_I1_J,axiom,
    ! [X23: produc8047831477865546771it_nat] : none_P7668321371905463026it_nat != some_P468703482102919278it_nat(X23) ).

% option.distinct(1)
tff(fact_766_option_Odistinct_I1_J,axiom,
    ! [X23: product_prod_nat_nat] : none_P5556105721700978146at_nat != some_P7363390416028606310at_nat(X23) ).

% option.distinct(1)
tff(fact_767_option_Odistinct_I1_J,axiom,
    ! [X23: nat] : none_nat != aa_nat_option_nat(some_nat,X23) ).

% option.distinct(1)
tff(fact_768_option_Odistinct_I1_J,axiom,
    ! [X23: num] : none_num != aa_num_option_num(some_num,X23) ).

% option.distinct(1)
tff(fact_769_option_OdiscI,axiom,
    ! [Option: option2621746655072343315it_nat,X23: produc120671012495760973it_nat] :
      ( ( Option = some_P2407035485129114418it_nat(X23) )
     => ( Option != none_P1551326421579882414it_nat ) ) ).

% option.discI
tff(fact_770_option_OdiscI,axiom,
    ! [Option: option7339022715339332451it_nat,X23: produc8047831477865546771it_nat] :
      ( ( Option = some_P468703482102919278it_nat(X23) )
     => ( Option != none_P7668321371905463026it_nat ) ) ).

% option.discI
tff(fact_771_option_OdiscI,axiom,
    ! [Option: option4927543243414619207at_nat,X23: product_prod_nat_nat] :
      ( ( Option = some_P7363390416028606310at_nat(X23) )
     => ( Option != none_P5556105721700978146at_nat ) ) ).

% option.discI
tff(fact_772_option_OdiscI,axiom,
    ! [Option: option_nat,X23: nat] :
      ( ( Option = aa_nat_option_nat(some_nat,X23) )
     => ( Option != none_nat ) ) ).

% option.discI
tff(fact_773_option_OdiscI,axiom,
    ! [Option: option_num,X23: num] :
      ( ( Option = aa_num_option_num(some_num,X23) )
     => ( Option != none_num ) ) ).

% option.discI
tff(fact_774_option_Oexhaust,axiom,
    ! [Y: option2621746655072343315it_nat] :
      ( ( Y != none_P1551326421579882414it_nat )
     => ~ ! [X24: produc120671012495760973it_nat] : Y != some_P2407035485129114418it_nat(X24) ) ).

% option.exhaust
tff(fact_775_option_Oexhaust,axiom,
    ! [Y: option7339022715339332451it_nat] :
      ( ( Y != none_P7668321371905463026it_nat )
     => ~ ! [X24: produc8047831477865546771it_nat] : Y != some_P468703482102919278it_nat(X24) ) ).

% option.exhaust
tff(fact_776_option_Oexhaust,axiom,
    ! [Y: option4927543243414619207at_nat] :
      ( ( Y != none_P5556105721700978146at_nat )
     => ~ ! [X24: product_prod_nat_nat] : Y != some_P7363390416028606310at_nat(X24) ) ).

% option.exhaust
tff(fact_777_option_Oexhaust,axiom,
    ! [Y: option_nat] :
      ( ( Y != none_nat )
     => ~ ! [X24: nat] : Y != aa_nat_option_nat(some_nat,X24) ) ).

% option.exhaust
tff(fact_778_option_Oexhaust,axiom,
    ! [Y: option_num] :
      ( ( Y != none_num )
     => ~ ! [X24: num] : Y != aa_num_option_num(some_num,X24) ) ).

% option.exhaust
tff(fact_779_split__option__ex,axiom,
    ! [P: fun_op2107615217426680968_nat_o] :
      ( ? [X_1: option2621746655072343315it_nat] : aa_opt424737865643518209_nat_o(P,X_1)
    <=> ( aa_opt424737865643518209_nat_o(P,none_P1551326421579882414it_nat)
        | ? [X: produc120671012495760973it_nat] : aa_opt424737865643518209_nat_o(P,some_P2407035485129114418it_nat(X)) ) ) ).

% split_option_ex
tff(fact_780_split__option__ex,axiom,
    ! [P: fun_op7459662026604166478_nat_o] :
      ( ? [X_1: option7339022715339332451it_nat] : aa_opt4171472420520861111_nat_o(P,X_1)
    <=> ( aa_opt4171472420520861111_nat_o(P,none_P7668321371905463026it_nat)
        | ? [X: produc8047831477865546771it_nat] : aa_opt4171472420520861111_nat_o(P,some_P468703482102919278it_nat(X)) ) ) ).

% split_option_ex
tff(fact_781_split__option__ex,axiom,
    ! [P: fun_op8028897214545316308_nat_o] :
      ( ? [X_1: option4927543243414619207at_nat] : aa_opt7112393969400857165_nat_o(P,X_1)
    <=> ( aa_opt7112393969400857165_nat_o(P,none_P5556105721700978146at_nat)
        | ? [X: product_prod_nat_nat] : aa_opt7112393969400857165_nat_o(P,some_P7363390416028606310at_nat(X)) ) ) ).

% split_option_ex
tff(fact_782_split__option__ex,axiom,
    ! [P: fun_option_nat_o] :
      ( ? [X_1: option_nat] : aa_option_nat_o(P,X_1)
    <=> ( aa_option_nat_o(P,none_nat)
        | ? [X: nat] : aa_option_nat_o(P,aa_nat_option_nat(some_nat,X)) ) ) ).

% split_option_ex
tff(fact_783_split__option__ex,axiom,
    ! [P: fun_option_num_o] :
      ( ? [X_1: option_num] : aa_option_num_o(P,X_1)
    <=> ( aa_option_num_o(P,none_num)
        | ? [X: num] : aa_option_num_o(P,aa_num_option_num(some_num,X)) ) ) ).

% split_option_ex
tff(fact_784_split__option__all,axiom,
    ! [P: fun_op2107615217426680968_nat_o] :
      ( ! [X_1: option2621746655072343315it_nat] : aa_opt424737865643518209_nat_o(P,X_1)
    <=> ( aa_opt424737865643518209_nat_o(P,none_P1551326421579882414it_nat)
        & ! [X: produc120671012495760973it_nat] : aa_opt424737865643518209_nat_o(P,some_P2407035485129114418it_nat(X)) ) ) ).

% split_option_all
tff(fact_785_split__option__all,axiom,
    ! [P: fun_op7459662026604166478_nat_o] :
      ( ! [X_1: option7339022715339332451it_nat] : aa_opt4171472420520861111_nat_o(P,X_1)
    <=> ( aa_opt4171472420520861111_nat_o(P,none_P7668321371905463026it_nat)
        & ! [X: produc8047831477865546771it_nat] : aa_opt4171472420520861111_nat_o(P,some_P468703482102919278it_nat(X)) ) ) ).

% split_option_all
tff(fact_786_split__option__all,axiom,
    ! [P: fun_op8028897214545316308_nat_o] :
      ( ! [X_1: option4927543243414619207at_nat] : aa_opt7112393969400857165_nat_o(P,X_1)
    <=> ( aa_opt7112393969400857165_nat_o(P,none_P5556105721700978146at_nat)
        & ! [X: product_prod_nat_nat] : aa_opt7112393969400857165_nat_o(P,some_P7363390416028606310at_nat(X)) ) ) ).

% split_option_all
tff(fact_787_split__option__all,axiom,
    ! [P: fun_option_nat_o] :
      ( ! [X_1: option_nat] : aa_option_nat_o(P,X_1)
    <=> ( aa_option_nat_o(P,none_nat)
        & ! [X: nat] : aa_option_nat_o(P,aa_nat_option_nat(some_nat,X)) ) ) ).

% split_option_all
tff(fact_788_split__option__all,axiom,
    ! [P: fun_option_num_o] :
      ( ! [X_1: option_num] : aa_option_num_o(P,X_1)
    <=> ( aa_option_num_o(P,none_num)
        & ! [X: num] : aa_option_num_o(P,aa_num_option_num(some_num,X)) ) ) ).

% split_option_all
tff(fact_789_combine__options__cases,axiom,
    ! [Xb: option_nat,P: fun_op5794009454290973322_nat_o,Y: option_nat] :
      ( ( ( Xb = none_nat )
       => aa_option_nat_o(aa_opt4319133605541840243_nat_o(P,Xb),Y) )
     => ( ( ( Y = none_nat )
         => aa_option_nat_o(aa_opt4319133605541840243_nat_o(P,Xb),Y) )
       => ( ! [A3: nat,B2: nat] :
              ( ( Xb = aa_nat_option_nat(some_nat,A3) )
             => ( ( Y = aa_nat_option_nat(some_nat,B2) )
               => aa_option_nat_o(aa_opt4319133605541840243_nat_o(P,Xb),Y) ) )
         => aa_option_nat_o(aa_opt4319133605541840243_nat_o(P,Xb),Y) ) ) ) ).

% combine_options_cases
tff(fact_790_combine__options__cases,axiom,
    ! [Xb: option_nat,P: fun_op6120599277751319488_num_o,Y: option_num] :
      ( ( ( Xb = none_nat )
       => aa_option_num_o(aa_opt5077314168511351209_num_o(P,Xb),Y) )
     => ( ( ( Y = none_num )
         => aa_option_num_o(aa_opt5077314168511351209_num_o(P,Xb),Y) )
       => ( ! [A3: nat,B2: num] :
              ( ( Xb = aa_nat_option_nat(some_nat,A3) )
             => ( ( Y = aa_num_option_num(some_num,B2) )
               => aa_option_num_o(aa_opt5077314168511351209_num_o(P,Xb),Y) ) )
         => aa_option_num_o(aa_opt5077314168511351209_num_o(P,Xb),Y) ) ) ) ).

% combine_options_cases
tff(fact_791_combine__options__cases,axiom,
    ! [Xb: option_num,P: fun_op167334234089630144_nat_o,Y: option_nat] :
      ( ( ( Xb = none_num )
       => aa_option_nat_o(aa_opt7915830422195272873_nat_o(P,Xb),Y) )
     => ( ( ( Y = none_nat )
         => aa_option_nat_o(aa_opt7915830422195272873_nat_o(P,Xb),Y) )
       => ( ! [A3: num,B2: nat] :
              ( ( Xb = aa_num_option_num(some_num,A3) )
             => ( ( Y = aa_nat_option_nat(some_nat,B2) )
               => aa_option_nat_o(aa_opt7915830422195272873_nat_o(P,Xb),Y) ) )
         => aa_option_nat_o(aa_opt7915830422195272873_nat_o(P,Xb),Y) ) ) ) ).

% combine_options_cases
tff(fact_792_combine__options__cases,axiom,
    ! [Xb: option_num,P: fun_op493924057549976310_num_o,Y: option_num] :
      ( ( ( Xb = none_num )
       => aa_option_num_o(aa_opt8674010985164783839_num_o(P,Xb),Y) )
     => ( ( ( Y = none_num )
         => aa_option_num_o(aa_opt8674010985164783839_num_o(P,Xb),Y) )
       => ( ! [A3: num,B2: num] :
              ( ( Xb = aa_num_option_num(some_num,A3) )
             => ( ( Y = aa_num_option_num(some_num,B2) )
               => aa_option_num_o(aa_opt8674010985164783839_num_o(P,Xb),Y) ) )
         => aa_option_num_o(aa_opt8674010985164783839_num_o(P,Xb),Y) ) ) ) ).

% combine_options_cases
tff(fact_793_combine__options__cases,axiom,
    ! [Xb: option4927543243414619207at_nat,P: fun_op7326680082016885773_nat_o,Y: option_nat] :
      ( ( ( Xb = none_P5556105721700978146at_nat )
       => aa_option_nat_o(aa_opt7110402928079307526_nat_o(P,Xb),Y) )
     => ( ( ( Y = none_nat )
         => aa_option_nat_o(aa_opt7110402928079307526_nat_o(P,Xb),Y) )
       => ( ! [A3: product_prod_nat_nat,B2: nat] :
              ( ( Xb = some_P7363390416028606310at_nat(A3) )
             => ( ( Y = aa_nat_option_nat(some_nat,B2) )
               => aa_option_nat_o(aa_opt7110402928079307526_nat_o(P,Xb),Y) ) )
         => aa_option_nat_o(aa_opt7110402928079307526_nat_o(P,Xb),Y) ) ) ) ).

% combine_options_cases
tff(fact_794_combine__options__cases,axiom,
    ! [Xb: option4927543243414619207at_nat,P: fun_op7653269905477231939_num_o,Y: option_num] :
      ( ( ( Xb = none_P5556105721700978146at_nat )
       => aa_option_num_o(aa_opt7868583491048818492_num_o(P,Xb),Y) )
     => ( ( ( Y = none_num )
         => aa_option_num_o(aa_opt7868583491048818492_num_o(P,Xb),Y) )
       => ( ! [A3: product_prod_nat_nat,B2: num] :
              ( ( Xb = some_P7363390416028606310at_nat(A3) )
             => ( ( Y = aa_num_option_num(some_num,B2) )
               => aa_option_num_o(aa_opt7868583491048818492_num_o(P,Xb),Y) ) )
         => aa_option_num_o(aa_opt7868583491048818492_num_o(P,Xb),Y) ) ) ) ).

% combine_options_cases
tff(fact_795_combine__options__cases,axiom,
    ! [Xb: option_nat,P: fun_op3832135881594366065_nat_o,Y: option4927543243414619207at_nat] :
      ( ( ( Xb = none_nat )
       => aa_opt7112393969400857165_nat_o(aa_opt120907569613471666_nat_o(P,Xb),Y) )
     => ( ( ( Y = none_P5556105721700978146at_nat )
         => aa_opt7112393969400857165_nat_o(aa_opt120907569613471666_nat_o(P,Xb),Y) )
       => ( ! [A3: nat,B2: product_prod_nat_nat] :
              ( ( Xb = aa_nat_option_nat(some_nat,A3) )
             => ( ( Y = some_P7363390416028606310at_nat(B2) )
               => aa_opt7112393969400857165_nat_o(aa_opt120907569613471666_nat_o(P,Xb),Y) ) )
         => aa_opt7112393969400857165_nat_o(aa_opt120907569613471666_nat_o(P,Xb),Y) ) ) ) ).

% combine_options_cases
tff(fact_796_combine__options__cases,axiom,
    ! [Xb: option_num,P: fun_op5373868528577402299_nat_o,Y: option4927543243414619207at_nat] :
      ( ( ( Xb = none_num )
       => aa_opt7112393969400857165_nat_o(aa_opt1662640216596507900_nat_o(P,Xb),Y) )
     => ( ( ( Y = none_P5556105721700978146at_nat )
         => aa_opt7112393969400857165_nat_o(aa_opt1662640216596507900_nat_o(P,Xb),Y) )
       => ( ! [A3: num,B2: product_prod_nat_nat] :
              ( ( Xb = aa_num_option_num(some_num,A3) )
             => ( ( Y = some_P7363390416028606310at_nat(B2) )
               => aa_opt7112393969400857165_nat_o(aa_opt1662640216596507900_nat_o(P,Xb),Y) ) )
         => aa_opt7112393969400857165_nat_o(aa_opt1662640216596507900_nat_o(P,Xb),Y) ) ) ) ).

% combine_options_cases
tff(fact_797_combine__options__cases,axiom,
    ! [Xb: option4927543243414619207at_nat,P: fun_op1221328669798608686_nat_o,Y: option4927543243414619207at_nat] :
      ( ( ( Xb = none_P5556105721700978146at_nat )
       => aa_opt7112393969400857165_nat_o(aa_opt9039718665754746463_nat_o(P,Xb),Y) )
     => ( ( ( Y = none_P5556105721700978146at_nat )
         => aa_opt7112393969400857165_nat_o(aa_opt9039718665754746463_nat_o(P,Xb),Y) )
       => ( ! [A3: product_prod_nat_nat,B2: product_prod_nat_nat] :
              ( ( Xb = some_P7363390416028606310at_nat(A3) )
             => ( ( Y = some_P7363390416028606310at_nat(B2) )
               => aa_opt7112393969400857165_nat_o(aa_opt9039718665754746463_nat_o(P,Xb),Y) ) )
         => aa_opt7112393969400857165_nat_o(aa_opt9039718665754746463_nat_o(P,Xb),Y) ) ) ) ).

% combine_options_cases
tff(fact_798_combine__options__cases,axiom,
    ! [Xb: option7339022715339332451it_nat,P: fun_op2280021900175422599_nat_o,Y: option_nat] :
      ( ( ( Xb = none_P7668321371905463026it_nat )
       => aa_option_nat_o(aa_opt9210023198028561264_nat_o(P,Xb),Y) )
     => ( ( ( Y = none_nat )
         => aa_option_nat_o(aa_opt9210023198028561264_nat_o(P,Xb),Y) )
       => ( ! [A3: produc8047831477865546771it_nat,B2: nat] :
              ( ( Xb = some_P468703482102919278it_nat(A3) )
             => ( ( Y = aa_nat_option_nat(some_nat,B2) )
               => aa_option_nat_o(aa_opt9210023198028561264_nat_o(P,Xb),Y) ) )
         => aa_option_nat_o(aa_opt9210023198028561264_nat_o(P,Xb),Y) ) ) ) ).

% combine_options_cases
tff(fact_799_sndE,axiom,
    ! [Xb: produc2453742332425399446nteger,A: fun_Pr5098669352780829690e_term,B: produc8923325533196201883nteger,P: fun_Pr7211993375558241536eger_o] :
      ( ( Xb = aa_Pro4291442876242880245nteger(produc565448451787809096nteger(A),B) )
     => ( aa_Pro4926521453222654329eger_o(P,aa_Pro4625778721341478855nteger(produc6589491691745714842nteger,Xb))
       => aa_Pro4926521453222654329eger_o(P,B) ) ) ).

% sndE
tff(fact_800_sndE,axiom,
    ! [Xb: produc8799533983703829792et_nat,A: fun_Pr6460944699146269935_nat_o,B: produc3658429121746597890et_nat,P: fun_Pr6460944699146269935_nat_o] :
      ( ( Xb = aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(A),B) )
     => ( aa_Pro653082453696054616_nat_o(P,aa_Pro5541488491759012064et_nat(produc4960056325182568940et_nat,Xb))
       => aa_Pro653082453696054616_nat_o(P,B) ) ) ).

% sndE
tff(fact_801_sndE,axiom,
    ! [Xb: produc3112576180687729086et_nat,A: fun_Pr6460944699146269935_nat_o,B: produc8799533983703829792et_nat,P: fun_Pr2620521676128474833_nat_o] :
      ( ( Xb = aa_Pro1979504083478021276et_nat(produc7291855603632349880et_nat(A),B) )
     => ( aa_Pro5643009357358181178_nat_o(P,aa_Pro7798056222271281692et_nat(produc9081944055658670602et_nat,Xb))
       => aa_Pro5643009357358181178_nat_o(P,B) ) ) ).

% sndE
tff(fact_802_sndE,axiom,
    ! [Xb: produc3043004577601930755nt_int,A: fun_Pr7763680713840462443e_term,B: product_prod_int_int,P: fun_Pr3238374608701176162_int_o] :
      ( ( Xb = aa_Pro5748361610964203648nt_int(produc7320969046608631221nt_int(A),B) )
     => ( aa_Pro4592068669816069595_int_o(P,aa_Pro2675344531184151096nt_int(produc1557020426977177351nt_int,Xb))
       => aa_Pro4592068669816069595_int_o(P,B) ) ) ).

% sndE
tff(fact_803_sndE,axiom,
    ! [Xb: produc4788131091073775919nt_int,A: fun_in5730379987311621695e_term,B: product_prod_int_int,P: fun_Pr3238374608701176162_int_o] :
      ( ( Xb = aa_Pro8800328608558162860nt_int(produc4729913460834754785nt_int(A),B) )
     => ( aa_Pro4592068669816069595_int_o(P,aa_Pro4437156595432605708nt_int(produc7629796867588368947nt_int,Xb))
       => aa_Pro4592068669816069595_int_o(P,B) ) ) ).

% sndE
tff(fact_804_sndE,axiom,
    ! [Xb: product_prod_nat_nat,A: nat,B: nat,P: fun_nat_o] :
      ( ( Xb = aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,A),B) )
     => ( aa_nat_o(P,aa_Pro4243173770044397973at_nat(product_snd_nat_nat,Xb))
       => aa_nat_o(P,B) ) ) ).

% sndE
tff(fact_805_sndE,axiom,
    ! [Xb: produc3658429121746597890et_nat,A: heap_e7401611519738050253t_unit,B: set_nat,P: fun_set_nat_o] :
      ( ( Xb = aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(A),B) )
     => ( aa_set_nat_o(P,aa_Pro3416150927818442182et_nat(produc8586169260539613262et_nat,Xb))
       => aa_set_nat_o(P,B) ) ) ).

% sndE
tff(fact_806_sndE,axiom,
    ! [Xb: product_prod_int_int,A: int,B: int,P: fun_int_o] :
      ( ( Xb = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),B) )
     => ( aa_int_o(P,aa_Pro6799258434465214121nt_int(product_snd_int_int,Xb))
       => aa_int_o(P,B) ) ) ).

% sndE
tff(fact_807_sndE,axiom,
    ! [Xb: produc8923325533196201883nteger,A: code_integer,B: code_integer,P: fun_Code_integer_o] :
      ( ( Xb = aa_Cod6977894654539884864nteger(produc1086072967326762835nteger(A),B) )
     => ( aa_Code_integer_o(P,aa_Pro5245656335456409788nteger(produc6174133586879617921nteger,Xb))
       => aa_Code_integer_o(P,B) ) ) ).

% sndE
tff(fact_808_sndE,axiom,
    ! [Xb: produc6575502325842934193n_assn,A: assn,B: assn,P: fun_assn_o] :
      ( ( Xb = aa_ass5585057816575862475n_assn(produc118845697133431529n_assn(A),B) )
     => ( aa_assn_o(P,aa_Pro2979818503705714077n_assn(produc2051961928117032727n_assn,Xb))
       => aa_assn_o(P,B) ) ) ).

% sndE
tff(fact_809_add__mult__distrib,axiom,
    ! [M: nat,Nb: 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),Nb)),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,Nb),K)) ).

% add_mult_distrib
tff(fact_810_add__mult__distrib2,axiom,
    ! [K: nat,M: nat,Nb: 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),Nb)) = 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),Nb)) ).

% add_mult_distrib2
tff(fact_811_left__add__mult__distrib,axiom,
    ! [I: nat,U: nat,J: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),U)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),K)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),J)),U)),K) ).

% left_add_mult_distrib
tff(fact_812_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_813_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_814_mult__le__mono,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),L)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),L)) ) ) ).

% mult_le_mono
tff(fact_815_mult__le__mono1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),K)) ) ).

% mult_le_mono1
tff(fact_816_mult__le__mono2,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),I)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),J)) ) ).

% mult_le_mono2
tff(fact_817_nat__mult__1,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,one_one_nat),Nb) = Nb ).

% nat_mult_1
tff(fact_818_nat__mult__1__right,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),one_one_nat) = Nb ).

% nat_mult_1_right
tff(fact_819_diff__mult__distrib,axiom,
    ! [M: nat,Nb: 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),Nb)),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,Nb),K)) ).

% diff_mult_distrib
tff(fact_820_diff__mult__distrib2,axiom,
    ! [K: nat,M: nat,Nb: 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),Nb)) = 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),Nb)) ).

% diff_mult_distrib2
tff(fact_821_option_Osel,axiom,
    ! [X23: product_prod_nat_nat] : the_Pr8591224930841456533at_nat(some_P7363390416028606310at_nat(X23)) = X23 ).

% option.sel
tff(fact_822_option_Osel,axiom,
    ! [X23: nat] : the_nat(aa_nat_option_nat(some_nat,X23)) = X23 ).

% option.sel
tff(fact_823_option_Osel,axiom,
    ! [X23: num] : aa_option_num_num(the_num,aa_num_option_num(some_num,X23)) = X23 ).

% option.sel
tff(fact_824_Ex__prod__contract,axiom,
    ! [P: fun_ui3478071520421546738nt32_o] :
      ( ? [A4: uint32,X_1: uint32] : aa_uint32_o(aa_uin5924812838015166427nt32_o(P,A4),X_1)
    <=> ? [Z2: produc827990862158126777uint32] : aa_uint32_o(aa_uin5924812838015166427nt32_o(P,aa_Pro3737690986720249065uint32(produc9004433772639906525uint32,Z2)),aa_Pro3737690986720249065uint32(produc1510406741064981791uint32,Z2)) ) ).

% Ex_prod_contract
tff(fact_825_Ex__prod__contract,axiom,
    ! [P: fun_nat_fun_nat_o] :
      ( ? [A4: nat,X_1: nat] : aa_nat_o(aa_nat_fun_nat_o(P,A4),X_1)
    <=> ? [Z2: product_prod_nat_nat] : aa_nat_o(aa_nat_fun_nat_o(P,aa_Pro4243173770044397973at_nat(product_fst_nat_nat,Z2)),aa_Pro4243173770044397973at_nat(product_snd_nat_nat,Z2)) ) ).

% Ex_prod_contract
tff(fact_826_Ex__prod__contract,axiom,
    ! [P: fun_he5746958392356810273_nat_o] :
      ( ? [A4: heap_e7401611519738050253t_unit,X_1: set_nat] : aa_set_nat_o(aa_hea8622544166608216090_nat_o(P,A4),X_1)
    <=> ? [Z2: produc3658429121746597890et_nat] : aa_set_nat_o(aa_hea8622544166608216090_nat_o(P,aa_Pro1219312880135979149t_unit(produc1824681642469235216et_nat,Z2)),aa_Pro3416150927818442182et_nat(produc8586169260539613262et_nat,Z2)) ) ).

% Ex_prod_contract
tff(fact_827_Ex__prod__contract,axiom,
    ! [P: fun_int_fun_int_o] :
      ( ? [A4: int,X_1: int] : aa_int_o(aa_int_fun_int_o(P,A4),X_1)
    <=> ? [Z2: product_prod_int_int] : aa_int_o(aa_int_fun_int_o(P,aa_Pro6799258434465214121nt_int(product_fst_int_int,Z2)),aa_Pro6799258434465214121nt_int(product_snd_int_int,Z2)) ) ).

% Ex_prod_contract
tff(fact_828_Ex__prod__contract,axiom,
    ! [P: fun_Co2081214437782444048eger_o] :
      ( ? [A4: code_integer,X_1: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(P,A4),X_1)
    <=> ? [Z2: produc8923325533196201883nteger] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(P,aa_Pro5245656335456409788nteger(produc8508995932063986495nteger,Z2)),aa_Pro5245656335456409788nteger(produc6174133586879617921nteger,Z2)) ) ).

% Ex_prod_contract
tff(fact_829_Ex__prod__contract,axiom,
    ! [P: fun_assn_fun_assn_o] :
      ( ? [A4: assn,X_1: assn] : aa_assn_o(aa_assn_fun_assn_o(P,A4),X_1)
    <=> ? [Z2: produc6575502325842934193n_assn] : aa_assn_o(aa_assn_fun_assn_o(P,aa_Pro2979818503705714077n_assn(produc9167289414957590229n_assn,Z2)),aa_Pro2979818503705714077n_assn(produc2051961928117032727n_assn,Z2)) ) ).

% Ex_prod_contract
tff(fact_830_All__prod__contract,axiom,
    ! [P: fun_ui3478071520421546738nt32_o] :
      ( ! [A4: uint32,X_1: uint32] : aa_uint32_o(aa_uin5924812838015166427nt32_o(P,A4),X_1)
    <=> ! [Z2: produc827990862158126777uint32] : aa_uint32_o(aa_uin5924812838015166427nt32_o(P,aa_Pro3737690986720249065uint32(produc9004433772639906525uint32,Z2)),aa_Pro3737690986720249065uint32(produc1510406741064981791uint32,Z2)) ) ).

% All_prod_contract
tff(fact_831_All__prod__contract,axiom,
    ! [P: fun_nat_fun_nat_o] :
      ( ! [A4: nat,X_1: nat] : aa_nat_o(aa_nat_fun_nat_o(P,A4),X_1)
    <=> ! [Z2: product_prod_nat_nat] : aa_nat_o(aa_nat_fun_nat_o(P,aa_Pro4243173770044397973at_nat(product_fst_nat_nat,Z2)),aa_Pro4243173770044397973at_nat(product_snd_nat_nat,Z2)) ) ).

% All_prod_contract
tff(fact_832_All__prod__contract,axiom,
    ! [P: fun_he5746958392356810273_nat_o] :
      ( ! [A4: heap_e7401611519738050253t_unit,X_1: set_nat] : aa_set_nat_o(aa_hea8622544166608216090_nat_o(P,A4),X_1)
    <=> ! [Z2: produc3658429121746597890et_nat] : aa_set_nat_o(aa_hea8622544166608216090_nat_o(P,aa_Pro1219312880135979149t_unit(produc1824681642469235216et_nat,Z2)),aa_Pro3416150927818442182et_nat(produc8586169260539613262et_nat,Z2)) ) ).

% All_prod_contract
tff(fact_833_All__prod__contract,axiom,
    ! [P: fun_int_fun_int_o] :
      ( ! [A4: int,X_1: int] : aa_int_o(aa_int_fun_int_o(P,A4),X_1)
    <=> ! [Z2: product_prod_int_int] : aa_int_o(aa_int_fun_int_o(P,aa_Pro6799258434465214121nt_int(product_fst_int_int,Z2)),aa_Pro6799258434465214121nt_int(product_snd_int_int,Z2)) ) ).

% All_prod_contract
tff(fact_834_All__prod__contract,axiom,
    ! [P: fun_Co2081214437782444048eger_o] :
      ( ! [A4: code_integer,X_1: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(P,A4),X_1)
    <=> ! [Z2: produc8923325533196201883nteger] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(P,aa_Pro5245656335456409788nteger(produc8508995932063986495nteger,Z2)),aa_Pro5245656335456409788nteger(produc6174133586879617921nteger,Z2)) ) ).

% All_prod_contract
tff(fact_835_All__prod__contract,axiom,
    ! [P: fun_assn_fun_assn_o] :
      ( ! [A4: assn,X_1: assn] : aa_assn_o(aa_assn_fun_assn_o(P,A4),X_1)
    <=> ! [Z2: produc6575502325842934193n_assn] : aa_assn_o(aa_assn_fun_assn_o(P,aa_Pro2979818503705714077n_assn(produc9167289414957590229n_assn,Z2)),aa_Pro2979818503705714077n_assn(produc2051961928117032727n_assn,Z2)) ) ).

% All_prod_contract
tff(fact_836_option_Oexpand,axiom,
    ! [Option: option2621746655072343315it_nat,Option2: option2621746655072343315it_nat] :
      ( ( ( Option = none_P1551326421579882414it_nat )
      <=> ( Option2 = none_P1551326421579882414it_nat ) )
     => ( ( ( Option != none_P1551326421579882414it_nat )
         => ( ( Option2 != none_P1551326421579882414it_nat )
           => ( the_Pr3501439614016493281it_nat(Option) = the_Pr3501439614016493281it_nat(Option2) ) ) )
       => ( Option = Option2 ) ) ) ).

% option.expand
tff(fact_837_option_Oexpand,axiom,
    ! [Option: option7339022715339332451it_nat,Option2: option7339022715339332451it_nat] :
      ( ( ( Option = none_P7668321371905463026it_nat )
      <=> ( Option2 = none_P7668321371905463026it_nat ) )
     => ( ( ( Option != none_P7668321371905463026it_nat )
         => ( ( Option2 != none_P7668321371905463026it_nat )
           => ( the_Pr5838048819577852031it_nat(Option) = the_Pr5838048819577852031it_nat(Option2) ) ) )
       => ( Option = Option2 ) ) ) ).

% option.expand
tff(fact_838_option_Oexpand,axiom,
    ! [Option: option4927543243414619207at_nat,Option2: option4927543243414619207at_nat] :
      ( ( ( Option = none_P5556105721700978146at_nat )
      <=> ( Option2 = none_P5556105721700978146at_nat ) )
     => ( ( ( Option != none_P5556105721700978146at_nat )
         => ( ( Option2 != none_P5556105721700978146at_nat )
           => ( the_Pr8591224930841456533at_nat(Option) = the_Pr8591224930841456533at_nat(Option2) ) ) )
       => ( Option = Option2 ) ) ) ).

% option.expand
tff(fact_839_option_Oexpand,axiom,
    ! [Option: option_num,Option2: option_num] :
      ( ( ( Option = none_num )
      <=> ( Option2 = none_num ) )
     => ( ( ( Option != none_num )
         => ( ( Option2 != none_num )
           => ( aa_option_num_num(the_num,Option) = aa_option_num_num(the_num,Option2) ) ) )
       => ( Option = Option2 ) ) ) ).

% option.expand
tff(fact_840_option_Oexpand,axiom,
    ! [Option: option_nat,Option2: option_nat] :
      ( ( ( Option = none_nat )
      <=> ( Option2 = none_nat ) )
     => ( ( ( Option != none_nat )
         => ( ( Option2 != none_nat )
           => ( the_nat(Option) = the_nat(Option2) ) ) )
       => ( Option = Option2 ) ) ) ).

% option.expand
tff(fact_841_option_Osimps_I5_J,axiom,
    ! [F1: heap_Time_Heap_o,F22: fun_Pr1329037631615965750Heap_o,X23: product_prod_nat_nat] : case_o1442776274061689234at_nat(F1,F22,some_P7363390416028606310at_nat(X23)) = aa_Pro691548381575413351Heap_o(F22,X23) ).

% option.simps(5)
tff(fact_842_option_Osimps_I5_J,axiom,
    ! [F1: option_num,F22: fun_num_option_num,X23: num] : aa_opt5369159329878751634on_num(case_o6005452278849405969um_num(F1,F22),aa_num_option_num(some_num,X23)) = aa_num_option_num(F22,X23) ).

% option.simps(5)
tff(fact_843_option_Osimps_I5_J,axiom,
    ! [F1: $o,F22: fun_Pr4796943543331071514_nat_o,X23: product_prod_nat_nat] :
      ( aa_opt7112393969400857165_nat_o(case_o184042715313410164at_nat((F1),F22),some_P7363390416028606310at_nat(X23))
    <=> aa_Pro6150637604445964947_nat_o(F22,X23) ) ).

% option.simps(5)
tff(fact_844_option_Osimps_I5_J,axiom,
    ! [F1: num,F22: fun_num_num,X23: num] : aa_option_num_num(case_option_num_num(F1,F22),aa_num_option_num(some_num,X23)) = aa_num_num(F22,X23) ).

% option.simps(5)
tff(fact_845_option_Osimps_I5_J,axiom,
    ! [F1: int,F22: fun_num_int,X23: num] : aa_option_num_int(case_option_int_num(F1,F22),aa_num_option_num(some_num,X23)) = aa_num_int(F22,X23) ).

% option.simps(5)
tff(fact_846_option_Osimps_I4_J,axiom,
    ! [F1: heap_Time_Heap_o,F22: fun_Pr1329037631615965750Heap_o] : case_o1442776274061689234at_nat(F1,F22,none_P5556105721700978146at_nat) = F1 ).

% option.simps(4)
tff(fact_847_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_848_option_Osimps_I4_J,axiom,
    ! [F1: $o,F22: fun_Pr4796943543331071514_nat_o] :
      ( aa_opt7112393969400857165_nat_o(case_o184042715313410164at_nat((F1),F22),none_P5556105721700978146at_nat)
    <=> (F1) ) ).

% option.simps(4)
tff(fact_849_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_850_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_851_four__x__squared,axiom,
    ! [Xb: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% four_x_squared
tff(fact_852_L2__set__mult__ineq__lemma,axiom,
    ! [A: real,C: real,B: real,D: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),D))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,D),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,B),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,C),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% L2_set_mult_ineq_lemma
tff(fact_853_lift__Suc__mono__less,axiom,
    ! [F: fun_nat_real,Nb: nat,N3: nat] :
      ( ! [N2: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(F,N2)),aa_nat_real(F,aa_nat_nat(suc,N2)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),N3)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(F,Nb)),aa_nat_real(F,N3)) ) ) ).

% lift_Suc_mono_less
tff(fact_854_lift__Suc__mono__less,axiom,
    ! [F: fun_nat_rat,Nb: nat,N3: nat] :
      ( ! [N2: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(F,N2)),aa_nat_rat(F,aa_nat_nat(suc,N2)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),N3)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(F,Nb)),aa_nat_rat(F,N3)) ) ) ).

% lift_Suc_mono_less
tff(fact_855_lift__Suc__mono__less,axiom,
    ! [F: fun_nat_num,Nb: nat,N3: nat] :
      ( ! [N2: nat] : aa_num_o(ord_less_num(aa_nat_num(F,N2)),aa_nat_num(F,aa_nat_nat(suc,N2)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),N3)
       => aa_num_o(ord_less_num(aa_nat_num(F,Nb)),aa_nat_num(F,N3)) ) ) ).

% lift_Suc_mono_less
tff(fact_856_lift__Suc__mono__less,axiom,
    ! [F: fun_nat_nat,Nb: nat,N3: nat] :
      ( ! [N2: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(F,N2)),aa_nat_nat(F,aa_nat_nat(suc,N2)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),N3)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(F,Nb)),aa_nat_nat(F,N3)) ) ) ).

% lift_Suc_mono_less
tff(fact_857_lift__Suc__mono__less,axiom,
    ! [F: fun_nat_int,Nb: nat,N3: nat] :
      ( ! [N2: nat] : aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(F,N2)),aa_nat_int(F,aa_nat_nat(suc,N2)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),N3)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(F,Nb)),aa_nat_int(F,N3)) ) ) ).

% lift_Suc_mono_less
tff(fact_858_lift__Suc__mono__less__iff,axiom,
    ! [F: fun_nat_real,Nb: nat,M: nat] :
      ( ! [N2: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(F,N2)),aa_nat_real(F,aa_nat_nat(suc,N2)))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(F,Nb)),aa_nat_real(F,M))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),M) ) ) ).

% lift_Suc_mono_less_iff
tff(fact_859_lift__Suc__mono__less__iff,axiom,
    ! [F: fun_nat_rat,Nb: nat,M: nat] :
      ( ! [N2: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(F,N2)),aa_nat_rat(F,aa_nat_nat(suc,N2)))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(F,Nb)),aa_nat_rat(F,M))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),M) ) ) ).

% lift_Suc_mono_less_iff
tff(fact_860_lift__Suc__mono__less__iff,axiom,
    ! [F: fun_nat_num,Nb: nat,M: nat] :
      ( ! [N2: nat] : aa_num_o(ord_less_num(aa_nat_num(F,N2)),aa_nat_num(F,aa_nat_nat(suc,N2)))
     => ( aa_num_o(ord_less_num(aa_nat_num(F,Nb)),aa_nat_num(F,M))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),M) ) ) ).

% lift_Suc_mono_less_iff
tff(fact_861_lift__Suc__mono__less__iff,axiom,
    ! [F: fun_nat_nat,Nb: nat,M: nat] :
      ( ! [N2: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(F,N2)),aa_nat_nat(F,aa_nat_nat(suc,N2)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(F,Nb)),aa_nat_nat(F,M))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),M) ) ) ).

% lift_Suc_mono_less_iff
tff(fact_862_lift__Suc__mono__less__iff,axiom,
    ! [F: fun_nat_int,Nb: nat,M: nat] :
      ( ! [N2: nat] : aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(F,N2)),aa_nat_int(F,aa_nat_nat(suc,N2)))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(F,Nb)),aa_nat_int(F,M))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),M) ) ) ).

% lift_Suc_mono_less_iff
tff(fact_863_lift__Suc__mono__le,axiom,
    ! [F: fun_nat_set_int,Nb: nat,N3: nat] :
      ( ! [N2: nat] : aa_set_int_o(ord_less_eq_set_int(aa_nat_set_int(F,N2)),aa_nat_set_int(F,aa_nat_nat(suc,N2)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N3)
       => aa_set_int_o(ord_less_eq_set_int(aa_nat_set_int(F,Nb)),aa_nat_set_int(F,N3)) ) ) ).

% lift_Suc_mono_le
tff(fact_864_lift__Suc__mono__le,axiom,
    ! [F: fun_nat_rat,Nb: nat,N3: nat] :
      ( ! [N2: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,N2)),aa_nat_rat(F,aa_nat_nat(suc,N2)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N3)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,Nb)),aa_nat_rat(F,N3)) ) ) ).

% lift_Suc_mono_le
tff(fact_865_lift__Suc__mono__le,axiom,
    ! [F: fun_nat_num,Nb: nat,N3: nat] :
      ( ! [N2: nat] : aa_num_o(ord_less_eq_num(aa_nat_num(F,N2)),aa_nat_num(F,aa_nat_nat(suc,N2)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N3)
       => aa_num_o(ord_less_eq_num(aa_nat_num(F,Nb)),aa_nat_num(F,N3)) ) ) ).

% lift_Suc_mono_le
tff(fact_866_lift__Suc__mono__le,axiom,
    ! [F: fun_nat_nat,Nb: nat,N3: nat] :
      ( ! [N2: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F,N2)),aa_nat_nat(F,aa_nat_nat(suc,N2)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N3)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F,Nb)),aa_nat_nat(F,N3)) ) ) ).

% lift_Suc_mono_le
tff(fact_867_lift__Suc__mono__le,axiom,
    ! [F: fun_nat_int,Nb: nat,N3: nat] :
      ( ! [N2: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,N2)),aa_nat_int(F,aa_nat_nat(suc,N2)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N3)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,Nb)),aa_nat_int(F,N3)) ) ) ).

% lift_Suc_mono_le
tff(fact_868_lift__Suc__antimono__le,axiom,
    ! [F: fun_nat_set_int,Nb: nat,N3: nat] :
      ( ! [N2: nat] : aa_set_int_o(ord_less_eq_set_int(aa_nat_set_int(F,aa_nat_nat(suc,N2))),aa_nat_set_int(F,N2))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N3)
       => aa_set_int_o(ord_less_eq_set_int(aa_nat_set_int(F,N3)),aa_nat_set_int(F,Nb)) ) ) ).

% lift_Suc_antimono_le
tff(fact_869_lift__Suc__antimono__le,axiom,
    ! [F: fun_nat_rat,Nb: nat,N3: nat] :
      ( ! [N2: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,aa_nat_nat(suc,N2))),aa_nat_rat(F,N2))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N3)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,N3)),aa_nat_rat(F,Nb)) ) ) ).

% lift_Suc_antimono_le
tff(fact_870_lift__Suc__antimono__le,axiom,
    ! [F: fun_nat_num,Nb: nat,N3: nat] :
      ( ! [N2: nat] : aa_num_o(ord_less_eq_num(aa_nat_num(F,aa_nat_nat(suc,N2))),aa_nat_num(F,N2))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N3)
       => aa_num_o(ord_less_eq_num(aa_nat_num(F,N3)),aa_nat_num(F,Nb)) ) ) ).

% lift_Suc_antimono_le
tff(fact_871_lift__Suc__antimono__le,axiom,
    ! [F: fun_nat_nat,Nb: nat,N3: nat] :
      ( ! [N2: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F,aa_nat_nat(suc,N2))),aa_nat_nat(F,N2))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N3)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F,N3)),aa_nat_nat(F,Nb)) ) ) ).

% lift_Suc_antimono_le
tff(fact_872_lift__Suc__antimono__le,axiom,
    ! [F: fun_nat_int,Nb: nat,N3: nat] :
      ( ! [N2: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,aa_nat_nat(suc,N2))),aa_nat_int(F,N2))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N3)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,N3)),aa_nat_int(F,Nb)) ) ) ).

% lift_Suc_antimono_le
tff(fact_873_less__natE,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb)
     => ~ ! [Q3: nat] : Nb != aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Q3)) ) ).

% less_natE
tff(fact_874_less__add__Suc1,axiom,
    ! [I: nat,M: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),M))) ).

% less_add_Suc1
tff(fact_875_less__add__Suc2,axiom,
    ! [I: nat,M: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),I))) ).

% less_add_Suc2
tff(fact_876_less__iff__Suc__add,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb)
    <=> ? [K4: nat] : Nb = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K4)) ) ).

% less_iff_Suc_add
tff(fact_877_less__imp__Suc__add,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb)
     => ? [K3: nat] : Nb = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K3)) ) ).

% less_imp_Suc_add
tff(fact_878_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_879_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_880_Suc__leI,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,M)),Nb) ) ).

% Suc_leI
tff(fact_881_Suc__le__eq,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,M)),Nb)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb) ) ).

% Suc_le_eq
tff(fact_882_dec__induct,axiom,
    ! [I: nat,J: nat,P: fun_nat_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => ( aa_nat_o(P,I)
       => ( ! [N2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),N2)
             => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),J)
               => ( aa_nat_o(P,N2)
                 => aa_nat_o(P,aa_nat_nat(suc,N2)) ) ) )
         => aa_nat_o(P,J) ) ) ) ).

% dec_induct
tff(fact_883_inc__induct,axiom,
    ! [I: nat,J: nat,P: fun_nat_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => ( aa_nat_o(P,J)
       => ( ! [N2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),N2)
             => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),J)
               => ( aa_nat_o(P,aa_nat_nat(suc,N2))
                 => aa_nat_o(P,N2) ) ) )
         => aa_nat_o(P,I) ) ) ) ).

% inc_induct
tff(fact_884_Suc__le__lessD,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,M)),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb) ) ).

% Suc_le_lessD
tff(fact_885_le__less__Suc__eq,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(suc,M))
      <=> ( Nb = M ) ) ) ).

% le_less_Suc_eq
tff(fact_886_less__Suc__eq__le,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),aa_nat_nat(suc,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb) ) ).

% less_Suc_eq_le
tff(fact_887_less__eq__Suc__le,axiom,
    ! [Nb: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),M)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Nb)),M) ) ).

% less_eq_Suc_le
tff(fact_888_le__imp__less__Suc,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),aa_nat_nat(suc,Nb)) ) ).

% le_imp_less_Suc
tff(fact_889_Suc__diff__Suc,axiom,
    ! [Nb: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),M)
     => ( aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),aa_nat_nat(suc,Nb))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Nb) ) ) ).

% Suc_diff_Suc
tff(fact_890_diff__less__Suc,axiom,
    ! [M: nat,Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Nb)),aa_nat_nat(suc,M)) ).

% diff_less_Suc
tff(fact_891_Suc__eq__plus1,axiom,
    ! [Nb: nat] : aa_nat_nat(suc,Nb) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat) ).

% Suc_eq_plus1
tff(fact_892_plus__1__eq__Suc,axiom,
    aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat) = suc ).

% plus_1_eq_Suc
tff(fact_893_Suc__eq__plus1__left,axiom,
    ! [Nb: nat] : aa_nat_nat(suc,Nb) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),Nb) ).

% Suc_eq_plus1_left
tff(fact_894_Suc__mult__less__cancel1,axiom,
    ! [K: nat,M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K)),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K)),Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb) ) ).

% Suc_mult_less_cancel1
tff(fact_895_mono__nat__linear__lb,axiom,
    ! [F: fun_nat_nat,M: nat,K: nat] :
      ( ! [M2: nat,N2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),N2)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(F,M2)),aa_nat_nat(F,N2)) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(F,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_896_Suc__diff__le,axiom,
    ! [Nb: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),M)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,M)),Nb) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Nb)) ) ) ).

% Suc_diff_le
tff(fact_897_diff__Suc__eq__diff__pred,axiom,
    ! [M: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),aa_nat_nat(suc,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat)),Nb) ).

% diff_Suc_eq_diff_pred
tff(fact_898_less__diff__conv,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),J) ) ).

% less_diff_conv
tff(fact_899_add__diff__inverse__nat,axiom,
    ! [M: nat,Nb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Nb)) = M ) ) ).

% add_diff_inverse_nat
tff(fact_900_less__diff__iff,axiom,
    ! [K: nat,M: nat,Nb: 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),Nb)
       => ( 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,Nb),K))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb) ) ) ) ).

% less_diff_iff
tff(fact_901_diff__less__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),A)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),C)) ) ) ).

% diff_less_mono
tff(fact_902_mult__Suc,axiom,
    ! [M: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,M)),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Nb)) ).

% mult_Suc
tff(fact_903_Suc__mult__le__cancel1,axiom,
    ! [K: nat,M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K)),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K)),Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb) ) ).

% Suc_mult_le_cancel1
tff(fact_904_mlex__snd__decrI,axiom,
    ! [A: nat,A5: nat,B: nat,B3: nat,N: nat] :
      ( ( A = A5 )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),B3)
       => 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),N)),B)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A5),N)),B3)) ) ) ).

% mlex_snd_decrI
tff(fact_905_mlex__fst__decrI,axiom,
    ! [A: nat,A5: nat,B: nat,N: nat,B3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),A5)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),N)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B3),N)
         => 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),N)),B)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A5),N)),B3)) ) ) ) ).

% mlex_fst_decrI
tff(fact_906_mlex__bound,axiom,
    ! [A: nat,A2: nat,B: nat,N: 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),N)
       => 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),N)),B)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),N)) ) ) ).

% mlex_bound
tff(fact_907_le__diff__conv,axiom,
    ! [J: nat,K: nat,I: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K)),I)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,J),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)) ) ).

% le_diff_conv
tff(fact_908_Nat_Ole__diff__conv2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),J)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),J) ) ) ).

% Nat.le_diff_conv2
tff(fact_909_Nat_Odiff__add__assoc,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),J)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),J)),K) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K)) ) ) ).

% Nat.diff_add_assoc
tff(fact_910_Nat_Odiff__add__assoc2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),J)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),I)),K) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K)),I) ) ) ).

% Nat.diff_add_assoc2
tff(fact_911_Nat_Ole__imp__diff__is__add,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),I) = K )
      <=> ( J = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),I) ) ) ) ).

% Nat.le_imp_diff_is_add
tff(fact_912_obtain__list__from__elements,axiom,
    ! [Nb: nat,P: fun_VE6480751511521678896_nat_o] :
      ( ! [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),Nb)
         => ? [Li: vEBT_VEBT] : aa_nat_o(aa_VEB2314620471664397097_nat_o(P,Li),I3) )
     => ~ ! [L2: list_VEBT_VEBT] :
            ( ( size_s6755466524823107622T_VEBT(L2) = Nb )
           => ~ ! [I4: nat] :
                  ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),Nb)
                 => aa_nat_o(aa_VEB2314620471664397097_nat_o(P,nth_VEBT_VEBT(L2,I4)),I4) ) ) ) ).

% obtain_list_from_elements
tff(fact_913_obtain__list__from__elements,axiom,
    ! [Nb: nat,P: fun_VE6346650636051888901_nat_o] :
      ( ! [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),Nb)
         => ? [Li: vEBT_VEBTi] : aa_nat_o(aa_VEB8883426185042993134_nat_o(P,Li),I3) )
     => ~ ! [L2: list_VEBT_VEBTi] :
            ( ( size_s7982070591426661849_VEBTi(L2) = Nb )
           => ~ ! [I4: nat] :
                  ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),Nb)
                 => aa_nat_o(aa_VEB8883426185042993134_nat_o(P,nth_VEBT_VEBTi(L2,I4)),I4) ) ) ) ).

% obtain_list_from_elements
tff(fact_914_obtain__list__from__elements,axiom,
    ! [Nb: nat,P: fun_real_fun_nat_o] :
      ( ! [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),Nb)
         => ? [Li: real] : aa_nat_o(aa_real_fun_nat_o(P,Li),I3) )
     => ~ ! [L2: list_real] :
            ( ( size_size_list_real(L2) = Nb )
           => ~ ! [I4: nat] :
                  ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),Nb)
                 => aa_nat_o(aa_real_fun_nat_o(P,aa_nat_real(nth_real(L2),I4)),I4) ) ) ) ).

% obtain_list_from_elements
tff(fact_915_obtain__list__from__elements,axiom,
    ! [Nb: nat,P: fun_o_fun_nat_o] :
      ( ! [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),Nb)
         => ? [Li: $o] : aa_nat_o(aa_o_fun_nat_o(P,(Li)),I3) )
     => ~ ! [L2: list_o] :
            ( ( size_size_list_o(L2) = Nb )
           => ~ ! [I4: nat] :
                  ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),Nb)
                 => aa_nat_o(aa_o_fun_nat_o(P,aa_nat_o(nth_o(L2),I4)),I4) ) ) ) ).

% obtain_list_from_elements
tff(fact_916_obtain__list__from__elements,axiom,
    ! [Nb: nat,P: fun_nat_fun_nat_o] :
      ( ! [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),Nb)
         => ? [Li: nat] : aa_nat_o(aa_nat_fun_nat_o(P,Li),I3) )
     => ~ ! [L2: list_nat] :
            ( ( size_size_list_nat(L2) = Nb )
           => ~ ! [I4: nat] :
                  ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),Nb)
                 => aa_nat_o(aa_nat_fun_nat_o(P,aa_nat_nat(nth_nat(L2),I4)),I4) ) ) ) ).

% obtain_list_from_elements
tff(fact_917_obtain__list__from__elements,axiom,
    ! [Nb: nat,P: fun_int_fun_nat_o] :
      ( ! [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),Nb)
         => ? [Li: int] : aa_nat_o(aa_int_fun_nat_o(P,Li),I3) )
     => ~ ! [L2: list_int] :
            ( ( size_size_list_int(L2) = Nb )
           => ~ ! [I4: nat] :
                  ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),Nb)
                 => aa_nat_o(aa_int_fun_nat_o(P,aa_nat_int(nth_int(L2),I4)),I4) ) ) ) ).

% obtain_list_from_elements
tff(fact_918_mlex__leI,axiom,
    ! [A: nat,A5: nat,B: nat,B3: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),A5)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),B3)
       => 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),N)),B)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A5),N)),B3)) ) ) ).

% mlex_leI
tff(fact_919_div__mult__le,axiom,
    ! [A: nat,B: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),B)),A) ).

% div_mult_le
tff(fact_920_option_Oexhaust__sel,axiom,
    ! [Option: option2621746655072343315it_nat] :
      ( ( Option != none_P1551326421579882414it_nat )
     => ( Option = some_P2407035485129114418it_nat(the_Pr3501439614016493281it_nat(Option)) ) ) ).

% option.exhaust_sel
tff(fact_921_option_Oexhaust__sel,axiom,
    ! [Option: option7339022715339332451it_nat] :
      ( ( Option != none_P7668321371905463026it_nat )
     => ( Option = some_P468703482102919278it_nat(the_Pr5838048819577852031it_nat(Option)) ) ) ).

% option.exhaust_sel
tff(fact_922_option_Oexhaust__sel,axiom,
    ! [Option: option4927543243414619207at_nat] :
      ( ( Option != none_P5556105721700978146at_nat )
     => ( Option = some_P7363390416028606310at_nat(the_Pr8591224930841456533at_nat(Option)) ) ) ).

% option.exhaust_sel
tff(fact_923_option_Oexhaust__sel,axiom,
    ! [Option: option_nat] :
      ( ( Option != none_nat )
     => ( Option = aa_nat_option_nat(some_nat,the_nat(Option)) ) ) ).

% option.exhaust_sel
tff(fact_924_option_Oexhaust__sel,axiom,
    ! [Option: option_num] :
      ( ( Option != none_num )
     => ( Option = aa_num_option_num(some_num,aa_option_num_num(the_num,Option)) ) ) ).

% option.exhaust_sel
tff(fact_925_option_Ocase__eq__if,axiom,
    ! [F1: heap_Time_Heap_o,F22: fun_Pr1329037631615965750Heap_o,Option: option4927543243414619207at_nat] :
      case_o1442776274061689234at_nat(F1,F22,Option) = $ite(Option = none_P5556105721700978146at_nat,F1,aa_Pro691548381575413351Heap_o(F22,the_Pr8591224930841456533at_nat(Option))) ).

% option.case_eq_if
tff(fact_926_option_Ocase__eq__if,axiom,
    ! [F1: option_num,F22: fun_num_option_num,Option: option_num] :
      aa_opt5369159329878751634on_num(case_o6005452278849405969um_num(F1,F22),Option) = $ite(Option = none_num,F1,aa_num_option_num(F22,aa_option_num_num(the_num,Option))) ).

% option.case_eq_if
tff(fact_927_option_Ocase__eq__if,axiom,
    ! [F1: $o,F22: fun_Pr4796943543331071514_nat_o,Option: option4927543243414619207at_nat] :
      ( aa_opt7112393969400857165_nat_o(case_o184042715313410164at_nat((F1),F22),Option)
    <=> $ite(Option = none_P5556105721700978146at_nat,(F1),aa_Pro6150637604445964947_nat_o(F22,the_Pr8591224930841456533at_nat(Option))) ) ).

% option.case_eq_if
tff(fact_928_option_Ocase__eq__if,axiom,
    ! [F1: num,F22: fun_num_num,Option: option_num] :
      aa_option_num_num(case_option_num_num(F1,F22),Option) = $ite(Option = none_num,F1,aa_num_num(F22,aa_option_num_num(the_num,Option))) ).

% option.case_eq_if
tff(fact_929_option_Ocase__eq__if,axiom,
    ! [F1: int,F22: fun_num_int,Option: option_num] :
      aa_option_num_int(case_option_int_num(F1,F22),Option) = $ite(Option = none_num,F1,aa_num_int(F22,aa_option_num_num(the_num,Option))) ).

% option.case_eq_if
tff(fact_930_is__succ__in__set__def,axiom,
    ! [Xs: set_nat,Xb: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(vEBT_is_succ_in_set(Xs),Xb),Y)
    <=> ( aa_set_nat_o(member_nat(Y),Xs)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y)
        & ! [X: nat] :
            ( aa_set_nat_o(member_nat(X),Xs)
           => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),X)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),X) ) ) ) ) ).

% is_succ_in_set_def
tff(fact_931_less__diff__conv2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),J)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K)),I)
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)) ) ) ).

% less_diff_conv2
tff(fact_932_nat__eq__add__iff1,axiom,
    ! [J: nat,I: nat,U: nat,M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,J),I)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),U)),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)),Nb) )
      <=> ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I),J)),U)),M) = Nb ) ) ) ).

% nat_eq_add_iff1
tff(fact_933_nat__eq__add__iff2,axiom,
    ! [I: nat,J: nat,U: nat,M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),U)),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)),Nb) )
      <=> ( M = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),I)),U)),Nb) ) ) ) ).

% nat_eq_add_iff2
tff(fact_934_nat__le__add__iff1,axiom,
    ! [J: nat,I: nat,U: nat,M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,J),I)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),U)),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)),Nb))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I),J)),U)),M)),Nb) ) ) ).

% nat_le_add_iff1
tff(fact_935_nat__le__add__iff2,axiom,
    ! [I: nat,J: nat,U: nat,M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),U)),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)),Nb))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),I)),U)),Nb)) ) ) ).

% nat_le_add_iff2
tff(fact_936_nat__diff__add__eq1,axiom,
    ! [J: nat,I: nat,U: nat,M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,J),I)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),U)),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)),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I),J)),U)),M)),Nb) ) ) ).

% nat_diff_add_eq1
tff(fact_937_nat__diff__add__eq2,axiom,
    ! [I: nat,J: nat,U: nat,M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),U)),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)),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),I)),U)),Nb)) ) ) ).

% nat_diff_add_eq2
tff(fact_938_option_Osplit__sel,axiom,
    ! [P: fun_He8247548780604731983ap_o_o,F1: heap_Time_Heap_o,F22: fun_Pr1329037631615965750Heap_o,Option: option4927543243414619207at_nat] :
      ( aa_Hea4522166450600650424ap_o_o(P,case_o1442776274061689234at_nat(F1,F22,Option))
    <=> ( ( ( Option = none_P5556105721700978146at_nat )
         => aa_Hea4522166450600650424ap_o_o(P,F1) )
        & ( ( Option = some_P7363390416028606310at_nat(the_Pr8591224930841456533at_nat(Option)) )
         => aa_Hea4522166450600650424ap_o_o(P,aa_Pro691548381575413351Heap_o(F22,the_Pr8591224930841456533at_nat(Option))) ) ) ) ).

% option.split_sel
tff(fact_939_option_Osplit__sel,axiom,
    ! [P: fun_option_num_o,F1: option_num,F22: fun_num_option_num,Option: option_num] :
      ( aa_option_num_o(P,aa_opt5369159329878751634on_num(case_o6005452278849405969um_num(F1,F22),Option))
    <=> ( ( ( Option = none_num )
         => aa_option_num_o(P,F1) )
        & ( ( Option = aa_num_option_num(some_num,aa_option_num_num(the_num,Option)) )
         => aa_option_num_o(P,aa_num_option_num(F22,aa_option_num_num(the_num,Option))) ) ) ) ).

% option.split_sel
tff(fact_940_option_Osplit__sel,axiom,
    ! [P: fun_o_o,F1: $o,F22: fun_Pr4796943543331071514_nat_o,Option: option4927543243414619207at_nat] :
      ( aa_o_o(P,aa_opt7112393969400857165_nat_o(case_o184042715313410164at_nat((F1),F22),Option))
    <=> ( ( ( Option = none_P5556105721700978146at_nat )
         => aa_o_o(P,(F1)) )
        & ( ( Option = some_P7363390416028606310at_nat(the_Pr8591224930841456533at_nat(Option)) )
         => aa_o_o(P,aa_Pro6150637604445964947_nat_o(F22,the_Pr8591224930841456533at_nat(Option))) ) ) ) ).

% option.split_sel
tff(fact_941_option_Osplit__sel,axiom,
    ! [P: fun_num_o,F1: num,F22: fun_num_num,Option: option_num] :
      ( aa_num_o(P,aa_option_num_num(case_option_num_num(F1,F22),Option))
    <=> ( ( ( Option = none_num )
         => aa_num_o(P,F1) )
        & ( ( Option = aa_num_option_num(some_num,aa_option_num_num(the_num,Option)) )
         => aa_num_o(P,aa_num_num(F22,aa_option_num_num(the_num,Option))) ) ) ) ).

% option.split_sel
tff(fact_942_option_Osplit__sel,axiom,
    ! [P: fun_int_o,F1: int,F22: fun_num_int,Option: option_num] :
      ( aa_int_o(P,aa_option_num_int(case_option_int_num(F1,F22),Option))
    <=> ( ( ( Option = none_num )
         => aa_int_o(P,F1) )
        & ( ( Option = aa_num_option_num(some_num,aa_option_num_num(the_num,Option)) )
         => aa_int_o(P,aa_num_int(F22,aa_option_num_num(the_num,Option))) ) ) ) ).

% option.split_sel
tff(fact_943_option_Osplit__sel__asm,axiom,
    ! [P: fun_He8247548780604731983ap_o_o,F1: heap_Time_Heap_o,F22: fun_Pr1329037631615965750Heap_o,Option: option4927543243414619207at_nat] :
      ( aa_Hea4522166450600650424ap_o_o(P,case_o1442776274061689234at_nat(F1,F22,Option))
    <=> ~ ( ( ( Option = none_P5556105721700978146at_nat )
            & ~ aa_Hea4522166450600650424ap_o_o(P,F1) )
          | ( ( Option = some_P7363390416028606310at_nat(the_Pr8591224930841456533at_nat(Option)) )
            & ~ aa_Hea4522166450600650424ap_o_o(P,aa_Pro691548381575413351Heap_o(F22,the_Pr8591224930841456533at_nat(Option))) ) ) ) ).

% option.split_sel_asm
tff(fact_944_option_Osplit__sel__asm,axiom,
    ! [P: fun_option_num_o,F1: option_num,F22: fun_num_option_num,Option: option_num] :
      ( aa_option_num_o(P,aa_opt5369159329878751634on_num(case_o6005452278849405969um_num(F1,F22),Option))
    <=> ~ ( ( ( Option = none_num )
            & ~ aa_option_num_o(P,F1) )
          | ( ( Option = aa_num_option_num(some_num,aa_option_num_num(the_num,Option)) )
            & ~ aa_option_num_o(P,aa_num_option_num(F22,aa_option_num_num(the_num,Option))) ) ) ) ).

% option.split_sel_asm
tff(fact_945_option_Osplit__sel__asm,axiom,
    ! [P: fun_o_o,F1: $o,F22: fun_Pr4796943543331071514_nat_o,Option: option4927543243414619207at_nat] :
      ( aa_o_o(P,aa_opt7112393969400857165_nat_o(case_o184042715313410164at_nat((F1),F22),Option))
    <=> ~ ( ( ( Option = none_P5556105721700978146at_nat )
            & ~ aa_o_o(P,(F1)) )
          | ( ( Option = some_P7363390416028606310at_nat(the_Pr8591224930841456533at_nat(Option)) )
            & ~ aa_o_o(P,aa_Pro6150637604445964947_nat_o(F22,the_Pr8591224930841456533at_nat(Option))) ) ) ) ).

% option.split_sel_asm
tff(fact_946_option_Osplit__sel__asm,axiom,
    ! [P: fun_num_o,F1: num,F22: fun_num_num,Option: option_num] :
      ( aa_num_o(P,aa_option_num_num(case_option_num_num(F1,F22),Option))
    <=> ~ ( ( ( Option = none_num )
            & ~ aa_num_o(P,F1) )
          | ( ( Option = aa_num_option_num(some_num,aa_option_num_num(the_num,Option)) )
            & ~ aa_num_o(P,aa_num_num(F22,aa_option_num_num(the_num,Option))) ) ) ) ).

% option.split_sel_asm
tff(fact_947_option_Osplit__sel__asm,axiom,
    ! [P: fun_int_o,F1: int,F22: fun_num_int,Option: option_num] :
      ( aa_int_o(P,aa_option_num_int(case_option_int_num(F1,F22),Option))
    <=> ~ ( ( ( Option = none_num )
            & ~ aa_int_o(P,F1) )
          | ( ( Option = aa_num_option_num(some_num,aa_option_num_num(the_num,Option)) )
            & ~ aa_int_o(P,aa_num_int(F22,aa_option_num_num(the_num,Option))) ) ) ) ).

% option.split_sel_asm
tff(fact_948_Suc__n__minus__m__eq,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
     => ( 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,Nb),M)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat)) ) ) ) ).

% Suc_n_minus_m_eq
tff(fact_949_nat__less__add__iff1,axiom,
    ! [J: nat,I: nat,U: nat,M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,J),I)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),U)),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)),Nb))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I),J)),U)),M)),Nb) ) ) ).

% nat_less_add_iff1
tff(fact_950_nat__less__add__iff2,axiom,
    ! [I: nat,J: nat,U: nat,M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),U)),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)),Nb))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),I)),U)),Nb)) ) ) ).

% nat_less_add_iff2
tff(fact_951_two__pow__div__gt__le,axiom,
    ! [V: nat,Nb: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,V),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb) ) ).

% two_pow_div_gt_le
tff(fact_952_power__minus__is__div,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B)) ) ) ).

% power_minus_is_div
tff(fact_953_cnt__bound_H,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Ta),Nb)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_VEBT_VEBT_real(vEBT_VEBT_cnt,Ta)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,numeral_numeral_real(aa_num_num(bit0,one))),Nb)),one_one_real))) ) ).

% cnt_bound'
tff(fact_954_prod_Ocollapse,axiom,
    ! [Prod: produc827990862158126777uint32] : aa_uin6954824371948101007uint32(produc1400373151660368625uint32(aa_Pro3737690986720249065uint32(produc9004433772639906525uint32,Prod)),aa_Pro3737690986720249065uint32(produc1510406741064981791uint32,Prod)) = Prod ).

% prod.collapse
tff(fact_955_prod_Ocollapse,axiom,
    ! [Prod: product_prod_nat_nat] : aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,aa_Pro4243173770044397973at_nat(product_fst_nat_nat,Prod)),aa_Pro4243173770044397973at_nat(product_snd_nat_nat,Prod)) = Prod ).

% prod.collapse
tff(fact_956_prod_Ocollapse,axiom,
    ! [Prod: product_prod_int_int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_Pro6799258434465214121nt_int(product_fst_int_int,Prod)),aa_Pro6799258434465214121nt_int(product_snd_int_int,Prod)) = Prod ).

% prod.collapse
tff(fact_957_prod_Ocollapse,axiom,
    ! [Prod: produc8923325533196201883nteger] : aa_Cod6977894654539884864nteger(produc1086072967326762835nteger(aa_Pro5245656335456409788nteger(produc8508995932063986495nteger,Prod)),aa_Pro5245656335456409788nteger(produc6174133586879617921nteger,Prod)) = Prod ).

% prod.collapse
tff(fact_958_prod_Ocollapse,axiom,
    ! [Prod: produc6575502325842934193n_assn] : aa_ass5585057816575862475n_assn(produc118845697133431529n_assn(aa_Pro2979818503705714077n_assn(produc9167289414957590229n_assn,Prod)),aa_Pro2979818503705714077n_assn(produc2051961928117032727n_assn,Prod)) = Prod ).

% prod.collapse
tff(fact_959_prod_Ocollapse,axiom,
    ! [Prod: produc3658429121746597890et_nat] : aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(aa_Pro1219312880135979149t_unit(produc1824681642469235216et_nat,Prod)),aa_Pro3416150927818442182et_nat(produc8586169260539613262et_nat,Prod)) = Prod ).

% prod.collapse
tff(fact_960_prod_Ocollapse,axiom,
    ! [Prod: produc4788131091073775919nt_int] : aa_Pro8800328608558162860nt_int(produc4729913460834754785nt_int(aa_Pro2551989255418117932e_term(produc4463008262350210037nt_int,Prod)),aa_Pro4437156595432605708nt_int(produc7629796867588368947nt_int,Prod)) = Prod ).

% prod.collapse
tff(fact_961_prod_Ocollapse,axiom,
    ! [Prod: produc8799533983703829792et_nat] : aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(aa_Pro8262445894397805713_nat_o(produc5325900173150335662et_nat,Prod)),aa_Pro5541488491759012064et_nat(produc4960056325182568940et_nat,Prod)) = Prod ).

% prod.collapse
tff(fact_962_prod_Ocollapse,axiom,
    ! [Prod: produc2453742332425399446nteger] : aa_Pro4291442876242880245nteger(produc565448451787809096nteger(aa_Pro4276937008533432526e_term(produc2481150572478215260nteger,Prod)),aa_Pro4625778721341478855nteger(produc6589491691745714842nteger,Prod)) = Prod ).

% prod.collapse
tff(fact_963_prod_Ocollapse,axiom,
    ! [Prod: produc3043004577601930755nt_int] : aa_Pro5748361610964203648nt_int(produc7320969046608631221nt_int(aa_Pro4010435699744810540e_term(produc3130796691802200265nt_int,Prod)),aa_Pro2675344531184151096nt_int(produc1557020426977177351nt_int,Prod)) = Prod ).

% prod.collapse
tff(fact_964_le__add__diff__inverse,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A)
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)) = A ) ) ).

% le_add_diff_inverse
tff(fact_965_le__add__diff__inverse,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
     => ( aa_rat_rat(plus_plus_rat(B),minus_minus_rat(A,B)) = A ) ) ).

% le_add_diff_inverse
tff(fact_966_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_967_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_968_le__add__diff__inverse2,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A)
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)),B) = A ) ) ).

% le_add_diff_inverse2
tff(fact_969_le__add__diff__inverse2,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
     => ( aa_rat_rat(plus_plus_rat(minus_minus_rat(A,B)),B) = A ) ) ).

% le_add_diff_inverse2
tff(fact_970_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_971_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_972_two__powr__height__bound__deg,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Ta),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_VEBT_VEBT_nat(vEBT_VEBT_height,Ta))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ) ).

% two_powr_height_bound_deg
tff(fact_973_setprop,axiom,
    ! [Ta: vEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Ta),set_VEBT_VEBT2(treeList))
     => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Ta),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ).

% setprop
tff(fact_974_norm__assertion__simps_I17_J,axiom,
    ! [R: assn,Q: fun_li3352006551604695530i_assn] : aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,R),ex_ass463751140784270563_VEBTi(Q)) = ex_ass463751140784270563_VEBTi(aa_fun6795544665460849810i_assn(aTP_Lamm_ct(R),Q)) ).

% norm_assertion_simps(17)
tff(fact_975_norm__assertion__simps_I16_J,axiom,
    ! [Q: fun_li3352006551604695530i_assn,R: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,ex_ass463751140784270563_VEBTi(Q)),R) = ex_ass463751140784270563_VEBTi(aa_ass1897699608592665680i_assn(aTP_Lamm_cu(Q),R)) ).

% norm_assertion_simps(16)
tff(fact_976_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Osimps_I6_J,axiom,
    ! [Mia: nat,Maa: nat,Vaa: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Xb: nat] :
      aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_s_u_c_c2,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mia),Maa)),aa_nat_nat(suc,aa_nat_nat(suc,Vaa)),TreeLista),Summarya)),Xb) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Mia),
        one_one_nat,
        $let(
          l: nat,
          l:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Vaa))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
          $let(
            h: nat,
            h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Vaa))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
            $ite(
              aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeLista)),
              $let(
                maxlow: option_nat,
                maxlow:= vEBT_vebt_maxt(nth_VEBT_VEBT(TreeLista,h)),
                $ite(
                  ( ( maxlow != none_nat )
                  & aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_less,aa_nat_option_nat(some_nat,l)),maxlow) ),
                  aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_s_u_c_c2,nth_VEBT_VEBT(TreeLista,h)),l)),
                  aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_s_u_c_c2,Summarya),h)),one_one_nat) ) ),
              one_one_nat ) ) ) ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.simps(6)
tff(fact_977_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Osimps_I7_J,axiom,
    ! [Mia: nat,Maa: nat,Vaa: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Xb: nat] :
      aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_p_r_e_d2,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mia),Maa)),aa_nat_nat(suc,aa_nat_nat(suc,Vaa)),TreeLista),Summarya)),Xb) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Maa),Xb),
        one_one_nat,
        $let(
          l: nat,
          l:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Vaa))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
          $let(
            h: nat,
            h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Vaa))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
            $ite(
              aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeLista)),
              $let(
                minlow: option_nat,
                minlow:= vEBT_vebt_mint(nth_VEBT_VEBT(TreeLista,h)),
                $ite(
                  ( ( minlow != none_nat )
                  & aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_greater,aa_nat_option_nat(some_nat,l)),minlow) ),
                  aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_p_r_e_d2,nth_VEBT_VEBT(TreeLista,h)),l)),
                  aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_p_r_e_d2,Summarya),h)),one_one_nat) ) ),
              one_one_nat ) ) ) ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.simps(7)
tff(fact_978_set__n__deg__not__0,axiom,
    ! [TreeLista: list_VEBT_VEBT,Nb: nat,M: nat] :
      ( ! [X2: vEBT_VEBT] :
          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X2),set_VEBT_VEBT2(TreeLista))
         => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,X2),Nb) )
     => ( ( size_s6755466524823107622T_VEBT(TreeLista) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M) )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),Nb) ) ) ).

% set_n_deg_not_0
tff(fact_979_inthall,axiom,
    ! [Xs: list_VEBT_VEBTi,P: fun_VEBT_VEBTi_o,Nb: nat] :
      ( ! [X2: vEBT_VEBTi] :
          ( member_VEBT_VEBTi(X2,set_VEBT_VEBTi2(Xs))
         => aa_VEBT_VEBTi_o(P,X2) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_s7982070591426661849_VEBTi(Xs))
       => aa_VEBT_VEBTi_o(P,nth_VEBT_VEBTi(Xs,Nb)) ) ) ).

% inthall
tff(fact_980_inthall,axiom,
    ! [Xs: list_VEBT_VEBT,P: fun_VEBT_VEBT_o,Nb: nat] :
      ( ! [X2: vEBT_VEBT] :
          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X2),set_VEBT_VEBT2(Xs))
         => aa_VEBT_VEBT_o(P,X2) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_s6755466524823107622T_VEBT(Xs))
       => aa_VEBT_VEBT_o(P,nth_VEBT_VEBT(Xs,Nb)) ) ) ).

% inthall
tff(fact_981_inthall,axiom,
    ! [Xs: list_P8527749157015355191n_assn,P: fun_Pr3571187171380813162assn_o,Nb: nat] :
      ( ! [X2: produc6575502325842934193n_assn] :
          ( member7957490590177025114n_assn(X2,set_Pr1139785259514867910n_assn(Xs))
         => aa_Pro2491721552408928739assn_o(P,X2) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_s6829681357464350627n_assn(Xs))
       => aa_Pro2491721552408928739assn_o(P,nth_Pr1769885009046257848n_assn(Xs,Nb)) ) ) ).

% inthall
tff(fact_982_inthall,axiom,
    ! [Xs: list_real,P: fun_real_o,Nb: nat] :
      ( ! [X2: real] :
          ( aa_set_real_o(member_real(X2),set_real2(Xs))
         => aa_real_o(P,X2) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_size_list_real(Xs))
       => aa_real_o(P,aa_nat_real(nth_real(Xs),Nb)) ) ) ).

% inthall
tff(fact_983_inthall,axiom,
    ! [Xs: list_o,P: fun_o_o,Nb: nat] :
      ( ! [X2: $o] :
          ( member_o((X2),set_o2(Xs))
         => aa_o_o(P,(X2)) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_size_list_o(Xs))
       => aa_o_o(P,aa_nat_o(nth_o(Xs),Nb)) ) ) ).

% inthall
tff(fact_984_inthall,axiom,
    ! [Xs: list_nat,P: fun_nat_o,Nb: nat] :
      ( ! [X2: nat] :
          ( aa_set_nat_o(member_nat(X2),set_nat2(Xs))
         => aa_nat_o(P,X2) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_size_list_nat(Xs))
       => aa_nat_o(P,aa_nat_nat(nth_nat(Xs),Nb)) ) ) ).

% inthall
tff(fact_985_inthall,axiom,
    ! [Xs: list_int,P: fun_int_o,Nb: nat] :
      ( ! [X2: int] :
          ( aa_set_int_o(member_int(X2),set_int2(Xs))
         => aa_int_o(P,X2) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_size_list_int(Xs))
       => aa_int_o(P,aa_nat_int(nth_int(Xs),Nb)) ) ) ).

% inthall
tff(fact_986_old_Oprod_Oinject,axiom,
    ! [A: fun_Pr5098669352780829690e_term,B: produc8923325533196201883nteger,A5: fun_Pr5098669352780829690e_term,B3: produc8923325533196201883nteger] :
      ( ( aa_Pro4291442876242880245nteger(produc565448451787809096nteger(A),B) = aa_Pro4291442876242880245nteger(produc565448451787809096nteger(A5),B3) )
    <=> ( ( A = A5 )
        & ( B = B3 ) ) ) ).

% old.prod.inject
tff(fact_987_old_Oprod_Oinject,axiom,
    ! [A: fun_Pr6460944699146269935_nat_o,B: produc3658429121746597890et_nat,A5: fun_Pr6460944699146269935_nat_o,B3: produc3658429121746597890et_nat] :
      ( ( aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(A),B) = aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(A5),B3) )
    <=> ( ( A = A5 )
        & ( B = B3 ) ) ) ).

% old.prod.inject
tff(fact_988_old_Oprod_Oinject,axiom,
    ! [A: fun_Pr6460944699146269935_nat_o,B: produc8799533983703829792et_nat,A5: fun_Pr6460944699146269935_nat_o,B3: produc8799533983703829792et_nat] :
      ( ( aa_Pro1979504083478021276et_nat(produc7291855603632349880et_nat(A),B) = aa_Pro1979504083478021276et_nat(produc7291855603632349880et_nat(A5),B3) )
    <=> ( ( A = A5 )
        & ( B = B3 ) ) ) ).

% old.prod.inject
tff(fact_989_old_Oprod_Oinject,axiom,
    ! [A: fun_Pr7763680713840462443e_term,B: product_prod_int_int,A5: fun_Pr7763680713840462443e_term,B3: product_prod_int_int] :
      ( ( aa_Pro5748361610964203648nt_int(produc7320969046608631221nt_int(A),B) = aa_Pro5748361610964203648nt_int(produc7320969046608631221nt_int(A5),B3) )
    <=> ( ( A = A5 )
        & ( B = B3 ) ) ) ).

% old.prod.inject
tff(fact_990_old_Oprod_Oinject,axiom,
    ! [A: fun_in5730379987311621695e_term,B: product_prod_int_int,A5: fun_in5730379987311621695e_term,B3: product_prod_int_int] :
      ( ( aa_Pro8800328608558162860nt_int(produc4729913460834754785nt_int(A),B) = aa_Pro8800328608558162860nt_int(produc4729913460834754785nt_int(A5),B3) )
    <=> ( ( A = A5 )
        & ( B = B3 ) ) ) ).

% old.prod.inject
tff(fact_991_prod_Oinject,axiom,
    ! [X1: fun_Pr5098669352780829690e_term,X23: produc8923325533196201883nteger,Y1: fun_Pr5098669352780829690e_term,Y2: produc8923325533196201883nteger] :
      ( ( aa_Pro4291442876242880245nteger(produc565448451787809096nteger(X1),X23) = aa_Pro4291442876242880245nteger(produc565448451787809096nteger(Y1),Y2) )
    <=> ( ( X1 = Y1 )
        & ( X23 = Y2 ) ) ) ).

% prod.inject
tff(fact_992_prod_Oinject,axiom,
    ! [X1: fun_Pr6460944699146269935_nat_o,X23: produc3658429121746597890et_nat,Y1: fun_Pr6460944699146269935_nat_o,Y2: produc3658429121746597890et_nat] :
      ( ( aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(X1),X23) = aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(Y1),Y2) )
    <=> ( ( X1 = Y1 )
        & ( X23 = Y2 ) ) ) ).

% prod.inject
tff(fact_993_prod_Oinject,axiom,
    ! [X1: fun_Pr6460944699146269935_nat_o,X23: produc8799533983703829792et_nat,Y1: fun_Pr6460944699146269935_nat_o,Y2: produc8799533983703829792et_nat] :
      ( ( aa_Pro1979504083478021276et_nat(produc7291855603632349880et_nat(X1),X23) = aa_Pro1979504083478021276et_nat(produc7291855603632349880et_nat(Y1),Y2) )
    <=> ( ( X1 = Y1 )
        & ( X23 = Y2 ) ) ) ).

% prod.inject
tff(fact_994_prod_Oinject,axiom,
    ! [X1: fun_Pr7763680713840462443e_term,X23: product_prod_int_int,Y1: fun_Pr7763680713840462443e_term,Y2: product_prod_int_int] :
      ( ( aa_Pro5748361610964203648nt_int(produc7320969046608631221nt_int(X1),X23) = aa_Pro5748361610964203648nt_int(produc7320969046608631221nt_int(Y1),Y2) )
    <=> ( ( X1 = Y1 )
        & ( X23 = Y2 ) ) ) ).

% prod.inject
tff(fact_995_prod_Oinject,axiom,
    ! [X1: fun_in5730379987311621695e_term,X23: product_prod_int_int,Y1: fun_in5730379987311621695e_term,Y2: product_prod_int_int] :
      ( ( aa_Pro8800328608558162860nt_int(produc4729913460834754785nt_int(X1),X23) = aa_Pro8800328608558162860nt_int(produc4729913460834754785nt_int(Y1),Y2) )
    <=> ( ( X1 = Y1 )
        & ( X23 = Y2 ) ) ) ).

% prod.inject
tff(fact_996_real__divide__square__eq,axiom,
    ! [R2: real,A: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R2),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R2),R2)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),R2) ).

% real_divide_square_eq
tff(fact_997_height__compose__summary,axiom,
    ! [Summarya: vEBT_VEBT,Info: option4927543243414619207at_nat,Deg: nat,TreeLista: list_VEBT_VEBT] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_VEBT_height,Summarya))),aa_VEBT_VEBT_nat(vEBT_VEBT_height,aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info,Deg,TreeLista),Summarya))) ).

% height_compose_summary
tff(fact_998_height__compose__child,axiom,
    ! [Ta: vEBT_VEBT,TreeLista: list_VEBT_VEBT,Info: option4927543243414619207at_nat,Deg: nat,Summarya: vEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Ta),set_VEBT_VEBT2(TreeLista))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_VEBT_height,Ta))),aa_VEBT_VEBT_nat(vEBT_VEBT_height,aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info,Deg,TreeLista),Summarya))) ) ).

% height_compose_child
tff(fact_999_mi__eq__ma__no__ch,axiom,
    ! [Mia: nat,Maa: nat,Deg: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mia),Maa)),Deg,TreeLista),Summarya)),Deg)
     => ( ( Mia = Maa )
       => ( ! [X3: vEBT_VEBT] :
              ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X3),set_VEBT_VEBT2(TreeLista))
             => ~ ? [X_12: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,X3),X_12) )
          & ~ ? [X_12: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,Summarya),X_12) ) ) ) ).

% mi_eq_ma_no_ch
tff(fact_1000_delete__bound__height_H,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xb: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Ta),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_V1232361888498592333_e_t_e,Ta),Xb)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_VEBT_height,Ta))) ) ).

% delete_bound_height'
tff(fact_1001_div__by__1,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),one_one_real) = A ).

% div_by_1
tff(fact_1002_div__by__1,axiom,
    ! [A: rat] : divide_divide_rat(A,one_one_rat) = A ).

% div_by_1
tff(fact_1003_div__by__1,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),one_one_nat) = A ).

% div_by_1
tff(fact_1004_div__by__1,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,A),one_one_int) = A ).

% div_by_1
tff(fact_1005_option_Odisc__eq__case_I1_J,axiom,
    ! [Option: option2621746655072343315it_nat] :
      ( ( Option = none_P1551326421579882414it_nat )
    <=> aa_opt424737865643518209_nat_o(case_o535201446637900608it_nat($true,aTP_Lamm_cv),Option) ) ).

% option.disc_eq_case(1)
tff(fact_1006_option_Odisc__eq__case_I1_J,axiom,
    ! [Option: option7339022715339332451it_nat] :
      ( ( Option = none_P7668321371905463026it_nat )
    <=> aa_opt4171472420520861111_nat_o(case_o1358941076187788256it_nat($true,aTP_Lamm_cw),Option) ) ).

% option.disc_eq_case(1)
tff(fact_1007_option_Odisc__eq__case_I1_J,axiom,
    ! [Option: option_num] :
      ( ( Option = none_num )
    <=> aa_option_num_o(case_option_o_num($true,aTP_Lamm_cx),Option) ) ).

% option.disc_eq_case(1)
tff(fact_1008_option_Odisc__eq__case_I1_J,axiom,
    ! [Option: option_nat] :
      ( ( Option = none_nat )
    <=> aa_option_nat_o(case_option_o_nat($true,aTP_Lamm_cy),Option) ) ).

% option.disc_eq_case(1)
tff(fact_1009_option_Odisc__eq__case_I1_J,axiom,
    ! [Option: option4927543243414619207at_nat] :
      ( ( Option = none_P5556105721700978146at_nat )
    <=> aa_opt7112393969400857165_nat_o(case_o184042715313410164at_nat($true,aTP_Lamm_cz),Option) ) ).

% option.disc_eq_case(1)
tff(fact_1010_option_Odisc__eq__case_I2_J,axiom,
    ! [Option: option2621746655072343315it_nat] :
      ( ( Option != none_P1551326421579882414it_nat )
    <=> aa_opt424737865643518209_nat_o(case_o535201446637900608it_nat($false,aTP_Lamm_da),Option) ) ).

% option.disc_eq_case(2)
tff(fact_1011_option_Odisc__eq__case_I2_J,axiom,
    ! [Option: option7339022715339332451it_nat] :
      ( ( Option != none_P7668321371905463026it_nat )
    <=> aa_opt4171472420520861111_nat_o(case_o1358941076187788256it_nat($false,aTP_Lamm_db),Option) ) ).

% option.disc_eq_case(2)
tff(fact_1012_option_Odisc__eq__case_I2_J,axiom,
    ! [Option: option_num] :
      ( ( Option != none_num )
    <=> aa_option_num_o(case_option_o_num($false,aTP_Lamm_dc),Option) ) ).

% option.disc_eq_case(2)
tff(fact_1013_option_Odisc__eq__case_I2_J,axiom,
    ! [Option: option_nat] :
      ( ( Option != none_nat )
    <=> aa_option_nat_o(case_option_o_nat($false,aTP_Lamm_dd),Option) ) ).

% option.disc_eq_case(2)
tff(fact_1014_option_Odisc__eq__case_I2_J,axiom,
    ! [Option: option4927543243414619207at_nat] :
      ( ( Option != none_P5556105721700978146at_nat )
    <=> aa_opt7112393969400857165_nat_o(case_o184042715313410164at_nat($false,aTP_Lamm_de),Option) ) ).

% option.disc_eq_case(2)
tff(fact_1015_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Osimps_I4_J,axiom,
    ! [Uy: nat,Uz: list_VEBT_VEBT,Vaa: vEBT_VEBT,Vb: nat] : aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_p_r_e_d2,aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uy,Uz),Vaa)),Vb) = one_one_nat ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.simps(4)
tff(fact_1016_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Osimps_I3_J,axiom,
    ! [Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT,Vaa: nat] : aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_s_u_c_c2,aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Ux,Uy),Uz)),Vaa) = one_one_nat ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.simps(3)
tff(fact_1017_pred__bound__height_H,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xb: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Ta),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_p_r_e_d2,Ta),Xb)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_VEBT_height,Ta))) ) ).

% pred_bound_height'
tff(fact_1018_succ_H__bound__height,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xb: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Ta),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_s_u_c_c2,Ta),Xb)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_VEBT_height,Ta))) ) ).

% succ'_bound_height
tff(fact_1019_case__optionE,axiom,
    ! [P: $o,Q: fun_Pr7755520105644788814_nat_o,Xb: option2621746655072343315it_nat] :
      ( aa_opt424737865643518209_nat_o(case_o535201446637900608it_nat((P),Q),Xb)
     => ( ( ( Xb = none_P1551326421579882414it_nat )
         => ~ (P) )
       => ~ ! [Y4: produc120671012495760973it_nat] :
              ( ( Xb = some_P2407035485129114418it_nat(Y4) )
             => ~ aa_Pro2986953523225089223_nat_o(Q,Y4) ) ) ) ).

% case_optionE
tff(fact_1020_case__optionE,axiom,
    ! [P: $o,Q: fun_Pr3989439477023515934_nat_o,Xb: option7339022715339332451it_nat] :
      ( aa_opt4171472420520861111_nat_o(case_o1358941076187788256it_nat((P),Q),Xb)
     => ( ( ( Xb = none_P7668321371905463026it_nat )
         => ~ (P) )
       => ~ ! [Y4: produc8047831477865546771it_nat] :
              ( ( Xb = some_P468703482102919278it_nat(Y4) )
             => ~ aa_Pro6395689520723908999_nat_o(Q,Y4) ) ) ) ).

% case_optionE
tff(fact_1021_case__optionE,axiom,
    ! [P: $o,Q: fun_nat_o,Xb: option_nat] :
      ( aa_option_nat_o(case_option_o_nat((P),Q),Xb)
     => ( ( ( Xb = none_nat )
         => ~ (P) )
       => ~ ! [Y4: nat] :
              ( ( Xb = aa_nat_option_nat(some_nat,Y4) )
             => ~ aa_nat_o(Q,Y4) ) ) ) ).

% case_optionE
tff(fact_1022_case__optionE,axiom,
    ! [P: $o,Q: fun_num_o,Xb: option_num] :
      ( aa_option_num_o(case_option_o_num((P),Q),Xb)
     => ( ( ( Xb = none_num )
         => ~ (P) )
       => ~ ! [Y4: num] :
              ( ( Xb = aa_num_option_num(some_num,Y4) )
             => ~ aa_num_o(Q,Y4) ) ) ) ).

% case_optionE
tff(fact_1023_case__optionE,axiom,
    ! [P: $o,Q: fun_Pr4796943543331071514_nat_o,Xb: option4927543243414619207at_nat] :
      ( aa_opt7112393969400857165_nat_o(case_o184042715313410164at_nat((P),Q),Xb)
     => ( ( ( Xb = none_P5556105721700978146at_nat )
         => ~ (P) )
       => ~ ! [Y4: product_prod_nat_nat] :
              ( ( Xb = some_P7363390416028606310at_nat(Y4) )
             => ~ aa_Pro6150637604445964947_nat_o(Q,Y4) ) ) ) ).

% case_optionE
tff(fact_1024_all__set__conv__nth,axiom,
    ! [L: list_VEBT_VEBTi,P: fun_VEBT_VEBTi_o] :
      ( ! [X: vEBT_VEBTi] :
          ( member_VEBT_VEBTi(X,set_VEBT_VEBTi2(L))
         => aa_VEBT_VEBTi_o(P,X) )
    <=> ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_s7982070591426661849_VEBTi(L))
         => aa_VEBT_VEBTi_o(P,nth_VEBT_VEBTi(L,I2)) ) ) ).

% all_set_conv_nth
tff(fact_1025_all__set__conv__nth,axiom,
    ! [L: list_VEBT_VEBT,P: fun_VEBT_VEBT_o] :
      ( ! [X: vEBT_VEBT] :
          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),set_VEBT_VEBT2(L))
         => aa_VEBT_VEBT_o(P,X) )
    <=> ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_s6755466524823107622T_VEBT(L))
         => aa_VEBT_VEBT_o(P,nth_VEBT_VEBT(L,I2)) ) ) ).

% all_set_conv_nth
tff(fact_1026_all__set__conv__nth,axiom,
    ! [L: list_P8527749157015355191n_assn,P: fun_Pr3571187171380813162assn_o] :
      ( ! [X: produc6575502325842934193n_assn] :
          ( member7957490590177025114n_assn(X,set_Pr1139785259514867910n_assn(L))
         => aa_Pro2491721552408928739assn_o(P,X) )
    <=> ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_s6829681357464350627n_assn(L))
         => aa_Pro2491721552408928739assn_o(P,nth_Pr1769885009046257848n_assn(L,I2)) ) ) ).

% all_set_conv_nth
tff(fact_1027_all__set__conv__nth,axiom,
    ! [L: list_real,P: fun_real_o] :
      ( ! [X: real] :
          ( aa_set_real_o(member_real(X),set_real2(L))
         => aa_real_o(P,X) )
    <=> ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_size_list_real(L))
         => aa_real_o(P,aa_nat_real(nth_real(L),I2)) ) ) ).

% all_set_conv_nth
tff(fact_1028_all__set__conv__nth,axiom,
    ! [L: list_o,P: fun_o_o] :
      ( ! [X: $o] :
          ( member_o((X),set_o2(L))
         => aa_o_o(P,(X)) )
    <=> ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_size_list_o(L))
         => aa_o_o(P,aa_nat_o(nth_o(L),I2)) ) ) ).

% all_set_conv_nth
tff(fact_1029_all__set__conv__nth,axiom,
    ! [L: list_nat,P: fun_nat_o] :
      ( ! [X: nat] :
          ( aa_set_nat_o(member_nat(X),set_nat2(L))
         => aa_nat_o(P,X) )
    <=> ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_size_list_nat(L))
         => aa_nat_o(P,aa_nat_nat(nth_nat(L),I2)) ) ) ).

% all_set_conv_nth
tff(fact_1030_all__set__conv__nth,axiom,
    ! [L: list_int,P: fun_int_o] :
      ( ! [X: int] :
          ( aa_set_int_o(member_int(X),set_int2(L))
         => aa_int_o(P,X) )
    <=> ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_size_list_int(L))
         => aa_int_o(P,aa_nat_int(nth_int(L),I2)) ) ) ).

% all_set_conv_nth
tff(fact_1031_linorder__neqE__linordered__idom,axiom,
    ! [Xb: real,Y: real] :
      ( ( Xb != Y )
     => ( ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Xb) ) ) ).

% linorder_neqE_linordered_idom
tff(fact_1032_linorder__neqE__linordered__idom,axiom,
    ! [Xb: rat,Y: rat] :
      ( ( Xb != Y )
     => ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Y)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),Xb) ) ) ).

% linorder_neqE_linordered_idom
tff(fact_1033_linorder__neqE__linordered__idom,axiom,
    ! [Xb: int,Y: int] :
      ( ( Xb != Y )
     => ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Y)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,Y),Xb) ) ) ).

% linorder_neqE_linordered_idom
tff(fact_1034_prod__induct4,axiom,
    ! [P: fun_Pr1227177680593602867_nat_o,Xb: produc3112576180687729086et_nat] :
      ( ! [A3: fun_Pr6460944699146269935_nat_o,B2: fun_Pr6460944699146269935_nat_o,C2: heap_e7401611519738050253t_unit,D2: set_nat] : aa_Pro2671238023309943196_nat_o(P,aa_Pro1979504083478021276et_nat(produc7291855603632349880et_nat(A3),aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(B2),aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(C2),D2))))
     => aa_Pro2671238023309943196_nat_o(P,Xb) ) ).

% prod_induct4
tff(fact_1035_prod__induct3,axiom,
    ! [P: fun_Pr7015759342434529115eger_o,Xb: produc2453742332425399446nteger] :
      ( ! [A3: fun_Pr5098669352780829690e_term,B2: code_integer,C2: code_integer] : aa_Pro4148064003102580676eger_o(P,aa_Pro4291442876242880245nteger(produc565448451787809096nteger(A3),aa_Cod6977894654539884864nteger(produc1086072967326762835nteger(B2),C2)))
     => aa_Pro4148064003102580676eger_o(P,Xb) ) ).

% prod_induct3
tff(fact_1036_prod__induct3,axiom,
    ! [P: fun_Pr2620521676128474833_nat_o,Xb: produc8799533983703829792et_nat] :
      ( ! [A3: fun_Pr6460944699146269935_nat_o,B2: heap_e7401611519738050253t_unit,C2: set_nat] : aa_Pro5643009357358181178_nat_o(P,aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(A3),aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(B2),C2)))
     => aa_Pro5643009357358181178_nat_o(P,Xb) ) ).

% prod_induct3
tff(fact_1037_prod__induct3,axiom,
    ! [P: fun_Pr1227177680593602867_nat_o,Xb: produc3112576180687729086et_nat] :
      ( ! [A3: fun_Pr6460944699146269935_nat_o,B2: fun_Pr6460944699146269935_nat_o,C2: produc3658429121746597890et_nat] : aa_Pro2671238023309943196_nat_o(P,aa_Pro1979504083478021276et_nat(produc7291855603632349880et_nat(A3),aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(B2),C2)))
     => aa_Pro2671238023309943196_nat_o(P,Xb) ) ).

% prod_induct3
tff(fact_1038_prod__induct3,axiom,
    ! [P: fun_Pr3540825087398332590_int_o,Xb: produc3043004577601930755nt_int] :
      ( ! [A3: fun_Pr7763680713840462443e_term,B2: int,C2: int] : aa_Pro2601676399929874711_int_o(P,aa_Pro5748361610964203648nt_int(produc7320969046608631221nt_int(A3),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,B2),C2)))
     => aa_Pro2601676399929874711_int_o(P,Xb) ) ).

% prod_induct3
tff(fact_1039_prod__induct3,axiom,
    ! [P: fun_Pr1196100989263358594_int_o,Xb: produc4788131091073775919nt_int] :
      ( ! [A3: fun_in5730379987311621695e_term,B2: int,C2: int] : aa_Pro5451197003417746155_int_o(P,aa_Pro8800328608558162860nt_int(produc4729913460834754785nt_int(A3),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,B2),C2)))
     => aa_Pro5451197003417746155_int_o(P,Xb) ) ).

% prod_induct3
tff(fact_1040_prod__cases4,axiom,
    ! [Y: produc3112576180687729086et_nat] :
      ~ ! [A3: fun_Pr6460944699146269935_nat_o,B2: fun_Pr6460944699146269935_nat_o,C2: heap_e7401611519738050253t_unit,D2: set_nat] : Y != aa_Pro1979504083478021276et_nat(produc7291855603632349880et_nat(A3),aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(B2),aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(C2),D2))) ).

% prod_cases4
tff(fact_1041_prod__cases3,axiom,
    ! [Y: produc2453742332425399446nteger] :
      ~ ! [A3: fun_Pr5098669352780829690e_term,B2: code_integer,C2: code_integer] : Y != aa_Pro4291442876242880245nteger(produc565448451787809096nteger(A3),aa_Cod6977894654539884864nteger(produc1086072967326762835nteger(B2),C2)) ).

% prod_cases3
tff(fact_1042_prod__cases3,axiom,
    ! [Y: produc8799533983703829792et_nat] :
      ~ ! [A3: fun_Pr6460944699146269935_nat_o,B2: heap_e7401611519738050253t_unit,C2: set_nat] : Y != aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(A3),aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(B2),C2)) ).

% prod_cases3
tff(fact_1043_prod__cases3,axiom,
    ! [Y: produc3112576180687729086et_nat] :
      ~ ! [A3: fun_Pr6460944699146269935_nat_o,B2: fun_Pr6460944699146269935_nat_o,C2: produc3658429121746597890et_nat] : Y != aa_Pro1979504083478021276et_nat(produc7291855603632349880et_nat(A3),aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(B2),C2)) ).

% prod_cases3
tff(fact_1044_prod__cases3,axiom,
    ! [Y: produc3043004577601930755nt_int] :
      ~ ! [A3: fun_Pr7763680713840462443e_term,B2: int,C2: int] : Y != aa_Pro5748361610964203648nt_int(produc7320969046608631221nt_int(A3),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,B2),C2)) ).

% prod_cases3
tff(fact_1045_prod__cases3,axiom,
    ! [Y: produc4788131091073775919nt_int] :
      ~ ! [A3: fun_in5730379987311621695e_term,B2: int,C2: int] : Y != aa_Pro8800328608558162860nt_int(produc4729913460834754785nt_int(A3),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,B2),C2)) ).

% prod_cases3
tff(fact_1046_Pair__inject,axiom,
    ! [A: fun_Pr5098669352780829690e_term,B: produc8923325533196201883nteger,A5: fun_Pr5098669352780829690e_term,B3: produc8923325533196201883nteger] :
      ( ( aa_Pro4291442876242880245nteger(produc565448451787809096nteger(A),B) = aa_Pro4291442876242880245nteger(produc565448451787809096nteger(A5),B3) )
     => ~ ( ( A = A5 )
         => ( B != B3 ) ) ) ).

% Pair_inject
tff(fact_1047_Pair__inject,axiom,
    ! [A: fun_Pr6460944699146269935_nat_o,B: produc3658429121746597890et_nat,A5: fun_Pr6460944699146269935_nat_o,B3: produc3658429121746597890et_nat] :
      ( ( aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(A),B) = aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(A5),B3) )
     => ~ ( ( A = A5 )
         => ( B != B3 ) ) ) ).

% Pair_inject
tff(fact_1048_Pair__inject,axiom,
    ! [A: fun_Pr6460944699146269935_nat_o,B: produc8799533983703829792et_nat,A5: fun_Pr6460944699146269935_nat_o,B3: produc8799533983703829792et_nat] :
      ( ( aa_Pro1979504083478021276et_nat(produc7291855603632349880et_nat(A),B) = aa_Pro1979504083478021276et_nat(produc7291855603632349880et_nat(A5),B3) )
     => ~ ( ( A = A5 )
         => ( B != B3 ) ) ) ).

% Pair_inject
tff(fact_1049_Pair__inject,axiom,
    ! [A: fun_Pr7763680713840462443e_term,B: product_prod_int_int,A5: fun_Pr7763680713840462443e_term,B3: product_prod_int_int] :
      ( ( aa_Pro5748361610964203648nt_int(produc7320969046608631221nt_int(A),B) = aa_Pro5748361610964203648nt_int(produc7320969046608631221nt_int(A5),B3) )
     => ~ ( ( A = A5 )
         => ( B != B3 ) ) ) ).

% Pair_inject
tff(fact_1050_Pair__inject,axiom,
    ! [A: fun_in5730379987311621695e_term,B: product_prod_int_int,A5: fun_in5730379987311621695e_term,B3: product_prod_int_int] :
      ( ( aa_Pro8800328608558162860nt_int(produc4729913460834754785nt_int(A),B) = aa_Pro8800328608558162860nt_int(produc4729913460834754785nt_int(A5),B3) )
     => ~ ( ( A = A5 )
         => ( B != B3 ) ) ) ).

% Pair_inject
tff(fact_1051_prod__cases,axiom,
    ! [P: fun_Pr7015759342434529115eger_o,P2: produc2453742332425399446nteger] :
      ( ! [A3: fun_Pr5098669352780829690e_term,B2: produc8923325533196201883nteger] : aa_Pro4148064003102580676eger_o(P,aa_Pro4291442876242880245nteger(produc565448451787809096nteger(A3),B2))
     => aa_Pro4148064003102580676eger_o(P,P2) ) ).

% prod_cases
tff(fact_1052_prod__cases,axiom,
    ! [P: fun_Pr2620521676128474833_nat_o,P2: produc8799533983703829792et_nat] :
      ( ! [A3: fun_Pr6460944699146269935_nat_o,B2: produc3658429121746597890et_nat] : aa_Pro5643009357358181178_nat_o(P,aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(A3),B2))
     => aa_Pro5643009357358181178_nat_o(P,P2) ) ).

% prod_cases
tff(fact_1053_prod__cases,axiom,
    ! [P: fun_Pr1227177680593602867_nat_o,P2: produc3112576180687729086et_nat] :
      ( ! [A3: fun_Pr6460944699146269935_nat_o,B2: produc8799533983703829792et_nat] : aa_Pro2671238023309943196_nat_o(P,aa_Pro1979504083478021276et_nat(produc7291855603632349880et_nat(A3),B2))
     => aa_Pro2671238023309943196_nat_o(P,P2) ) ).

% prod_cases
tff(fact_1054_prod__cases,axiom,
    ! [P: fun_Pr3540825087398332590_int_o,P2: produc3043004577601930755nt_int] :
      ( ! [A3: fun_Pr7763680713840462443e_term,B2: product_prod_int_int] : aa_Pro2601676399929874711_int_o(P,aa_Pro5748361610964203648nt_int(produc7320969046608631221nt_int(A3),B2))
     => aa_Pro2601676399929874711_int_o(P,P2) ) ).

% prod_cases
tff(fact_1055_prod__cases,axiom,
    ! [P: fun_Pr1196100989263358594_int_o,P2: produc4788131091073775919nt_int] :
      ( ! [A3: fun_in5730379987311621695e_term,B2: product_prod_int_int] : aa_Pro5451197003417746155_int_o(P,aa_Pro8800328608558162860nt_int(produc4729913460834754785nt_int(A3),B2))
     => aa_Pro5451197003417746155_int_o(P,P2) ) ).

% prod_cases
tff(fact_1056_surj__pair,axiom,
    ! [P2: produc2453742332425399446nteger] :
    ? [X2: fun_Pr5098669352780829690e_term,Y4: produc8923325533196201883nteger] : P2 = aa_Pro4291442876242880245nteger(produc565448451787809096nteger(X2),Y4) ).

% surj_pair
tff(fact_1057_surj__pair,axiom,
    ! [P2: produc8799533983703829792et_nat] :
    ? [X2: fun_Pr6460944699146269935_nat_o,Y4: produc3658429121746597890et_nat] : P2 = aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(X2),Y4) ).

% surj_pair
tff(fact_1058_surj__pair,axiom,
    ! [P2: produc3112576180687729086et_nat] :
    ? [X2: fun_Pr6460944699146269935_nat_o,Y4: produc8799533983703829792et_nat] : P2 = aa_Pro1979504083478021276et_nat(produc7291855603632349880et_nat(X2),Y4) ).

% surj_pair
tff(fact_1059_surj__pair,axiom,
    ! [P2: produc3043004577601930755nt_int] :
    ? [X2: fun_Pr7763680713840462443e_term,Y4: product_prod_int_int] : P2 = aa_Pro5748361610964203648nt_int(produc7320969046608631221nt_int(X2),Y4) ).

% surj_pair
tff(fact_1060_surj__pair,axiom,
    ! [P2: produc4788131091073775919nt_int] :
    ? [X2: fun_in5730379987311621695e_term,Y4: product_prod_int_int] : P2 = aa_Pro8800328608558162860nt_int(produc4729913460834754785nt_int(X2),Y4) ).

% surj_pair
tff(fact_1061_old_Oprod_Oexhaust,axiom,
    ! [Y: produc2453742332425399446nteger] :
      ~ ! [A3: fun_Pr5098669352780829690e_term,B2: produc8923325533196201883nteger] : Y != aa_Pro4291442876242880245nteger(produc565448451787809096nteger(A3),B2) ).

% old.prod.exhaust
tff(fact_1062_old_Oprod_Oexhaust,axiom,
    ! [Y: produc8799533983703829792et_nat] :
      ~ ! [A3: fun_Pr6460944699146269935_nat_o,B2: produc3658429121746597890et_nat] : Y != aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(A3),B2) ).

% old.prod.exhaust
tff(fact_1063_old_Oprod_Oexhaust,axiom,
    ! [Y: produc3112576180687729086et_nat] :
      ~ ! [A3: fun_Pr6460944699146269935_nat_o,B2: produc8799533983703829792et_nat] : Y != aa_Pro1979504083478021276et_nat(produc7291855603632349880et_nat(A3),B2) ).

% old.prod.exhaust
tff(fact_1064_old_Oprod_Oexhaust,axiom,
    ! [Y: produc3043004577601930755nt_int] :
      ~ ! [A3: fun_Pr7763680713840462443e_term,B2: product_prod_int_int] : Y != aa_Pro5748361610964203648nt_int(produc7320969046608631221nt_int(A3),B2) ).

% old.prod.exhaust
tff(fact_1065_old_Oprod_Oexhaust,axiom,
    ! [Y: produc4788131091073775919nt_int] :
      ~ ! [A3: fun_in5730379987311621695e_term,B2: product_prod_int_int] : Y != aa_Pro8800328608558162860nt_int(produc4729913460834754785nt_int(A3),B2) ).

% old.prod.exhaust
tff(fact_1066_star__assoc,axiom,
    ! [A: assn,B: assn,C: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A),B)),C) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,B),C)) ).

% star_assoc
tff(fact_1067_star__aci_I2_J,axiom,
    ! [A: assn,B: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A),B) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,B),A) ).

% star_aci(2)
tff(fact_1068_star__aci_I3_J,axiom,
    ! [A: assn,B: assn,C: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,B),C)) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,B),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A),C)) ).

% star_aci(3)
tff(fact_1069_assn__aci_I10_J,axiom,
    ! [A: assn,B: assn,C: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A),B)),C) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A),C)),B) ).

% assn_aci(10)
tff(fact_1070_is__hoare__triple,axiom,
    ! [P: assn,C: heap_T2636463487746394924on_nat,Q: fun_option_nat_assn] :
      ( hoare_7629718768684598413on_nat(P,C,Q)
     => hoare_7629718768684598413on_nat(P,C,Q) ) ).

% is_hoare_triple
tff(fact_1071_is__hoare__triple,axiom,
    ! [P: assn,C: heap_Time_Heap_nat,Q: fun_nat_assn] :
      ( hoare_3067605981109127869le_nat(P,C,Q)
     => hoare_3067605981109127869le_nat(P,C,Q) ) ).

% is_hoare_triple
tff(fact_1072_is__hoare__triple,axiom,
    ! [P: assn,C: heap_Time_Heap_o,Q: fun_o_assn] :
      ( hoare_hoare_triple_o(P,C,Q)
     => hoare_hoare_triple_o(P,C,Q) ) ).

% is_hoare_triple
tff(fact_1073_is__hoare__triple,axiom,
    ! [P: assn,C: heap_T8145700208782473153_VEBTi,Q: fun_VEBT_VEBTi_assn] :
      ( hoare_1429296392585015714_VEBTi(P,C,Q)
     => hoare_1429296392585015714_VEBTi(P,C,Q) ) ).

% is_hoare_triple
tff(fact_1074_is__hoare__triple,axiom,
    ! [P: assn,C: heap_T5738788834812785303t_unit,Q: fun_Pr6626665806141890376t_assn] :
      ( hoare_8945653483474564448t_unit(P,C,Q)
     => hoare_8945653483474564448t_unit(P,C,Q) ) ).

% is_hoare_triple
tff(fact_1075_height__node,axiom,
    ! [Mia: nat,Maa: nat,Deg: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Nb: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mia),Maa)),Deg,TreeLista),Summarya)),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_VEBT_height,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mia),Maa)),Deg,TreeLista),Summarya))) ) ).

% height_node
tff(fact_1076_combine__common__factor,axiom,
    ! [A: real,E: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),E)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),E)),C)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),E)),C) ).

% combine_common_factor
tff(fact_1077_combine__common__factor,axiom,
    ! [A: rat,E: rat,B: rat,C: rat] : aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(A),E)),aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(B),E)),C)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(aa_rat_rat(plus_plus_rat(A),B)),E)),C) ).

% combine_common_factor
tff(fact_1078_combine__common__factor,axiom,
    ! [A: nat,E: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),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)),C)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),E)),C) ).

% combine_common_factor
tff(fact_1079_combine__common__factor,axiom,
    ! [A: int,E: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),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)),C)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),E)),C) ).

% combine_common_factor
tff(fact_1080_distrib__right,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),C) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)) ).

% distrib_right
tff(fact_1081_distrib__right,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(plus_plus_rat(A),B)),C) = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),C)) ).

% distrib_right
tff(fact_1082_distrib__right,axiom,
    ! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) ).

% distrib_right
tff(fact_1083_distrib__right,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ).

% distrib_right
tff(fact_1084_distrib__left,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)) ).

% distrib_left
tff(fact_1085_distrib__left,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(times_times_rat(A),aa_rat_rat(plus_plus_rat(B),C)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(A),B)),aa_rat_rat(times_times_rat(A),C)) ).

% distrib_left
tff(fact_1086_distrib__left,axiom,
    ! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)) ).

% distrib_left
tff(fact_1087_distrib__left,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),aa_int_int(aa_int_fun_int_int(times_times_int,A),C)) ).

% distrib_left
tff(fact_1088_comm__semiring__class_Odistrib,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),C) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)) ).

% comm_semiring_class.distrib
tff(fact_1089_comm__semiring__class_Odistrib,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(plus_plus_rat(A),B)),C) = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),C)) ).

% comm_semiring_class.distrib
tff(fact_1090_comm__semiring__class_Odistrib,axiom,
    ! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) ).

% comm_semiring_class.distrib
tff(fact_1091_comm__semiring__class_Odistrib,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ).

% comm_semiring_class.distrib
tff(fact_1092_ring__class_Oring__distribs_I1_J,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)) ).

% ring_class.ring_distribs(1)
tff(fact_1093_ring__class_Oring__distribs_I1_J,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(times_times_rat(A),aa_rat_rat(plus_plus_rat(B),C)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(A),B)),aa_rat_rat(times_times_rat(A),C)) ).

% ring_class.ring_distribs(1)
tff(fact_1094_ring__class_Oring__distribs_I1_J,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),aa_int_int(aa_int_fun_int_int(times_times_int,A),C)) ).

% ring_class.ring_distribs(1)
tff(fact_1095_ring__class_Oring__distribs_I2_J,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),C) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)) ).

% ring_class.ring_distribs(2)
tff(fact_1096_ring__class_Oring__distribs_I2_J,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(plus_plus_rat(A),B)),C) = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),C)) ).

% ring_class.ring_distribs(2)
tff(fact_1097_ring__class_Oring__distribs_I2_J,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ).

% ring_class.ring_distribs(2)
tff(fact_1098_right__diff__distrib_H,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),C)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)) ).

% right_diff_distrib'
tff(fact_1099_right__diff__distrib_H,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(times_times_rat(A),minus_minus_rat(B,C)) = minus_minus_rat(aa_rat_rat(times_times_rat(A),B),aa_rat_rat(times_times_rat(A),C)) ).

% right_diff_distrib'
tff(fact_1100_right__diff__distrib_H,axiom,
    ! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)) ).

% right_diff_distrib'
tff(fact_1101_right__diff__distrib_H,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),C)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),aa_int_int(aa_int_fun_int_int(times_times_int,A),C)) ).

% right_diff_distrib'
tff(fact_1102_left__diff__distrib_H,axiom,
    ! [B: real,C: real,A: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),C)),A) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)) ).

% left_diff_distrib'
tff(fact_1103_left__diff__distrib_H,axiom,
    ! [B: rat,C: rat,A: rat] : aa_rat_rat(times_times_rat(minus_minus_rat(B,C)),A) = minus_minus_rat(aa_rat_rat(times_times_rat(B),A),aa_rat_rat(times_times_rat(C),A)) ).

% left_diff_distrib'
tff(fact_1104_left__diff__distrib_H,axiom,
    ! [B: nat,C: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),C)),A) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)) ).

% left_diff_distrib'
tff(fact_1105_left__diff__distrib_H,axiom,
    ! [B: int,C: int,A: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B),C)),A) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),A)) ).

% left_diff_distrib'
tff(fact_1106_right__diff__distrib,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),C)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)) ).

% right_diff_distrib
tff(fact_1107_right__diff__distrib,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(times_times_rat(A),minus_minus_rat(B,C)) = minus_minus_rat(aa_rat_rat(times_times_rat(A),B),aa_rat_rat(times_times_rat(A),C)) ).

% right_diff_distrib
tff(fact_1108_right__diff__distrib,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),C)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),aa_int_int(aa_int_fun_int_int(times_times_int,A),C)) ).

% right_diff_distrib
tff(fact_1109_left__diff__distrib,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)),C) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)) ).

% left_diff_distrib
tff(fact_1110_left__diff__distrib,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(times_times_rat(minus_minus_rat(A,B)),C) = minus_minus_rat(aa_rat_rat(times_times_rat(A),C),aa_rat_rat(times_times_rat(B),C)) ).

% left_diff_distrib
tff(fact_1111_left__diff__distrib,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ).

% left_diff_distrib
tff(fact_1112_fst__conv,axiom,
    ! [X1: fun_Pr5098669352780829690e_term,X23: produc8923325533196201883nteger] : aa_Pro4276937008533432526e_term(produc2481150572478215260nteger,aa_Pro4291442876242880245nteger(produc565448451787809096nteger(X1),X23)) = X1 ).

% fst_conv
tff(fact_1113_fst__conv,axiom,
    ! [X1: fun_Pr6460944699146269935_nat_o,X23: produc3658429121746597890et_nat] : aa_Pro8262445894397805713_nat_o(produc5325900173150335662et_nat,aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(X1),X23)) = X1 ).

% fst_conv
tff(fact_1114_fst__conv,axiom,
    ! [X1: fun_Pr6460944699146269935_nat_o,X23: produc8799533983703829792et_nat] : aa_Pro2837773213478468339_nat_o(produc4078497350495916492et_nat,aa_Pro1979504083478021276et_nat(produc7291855603632349880et_nat(X1),X23)) = X1 ).

% fst_conv
tff(fact_1115_fst__conv,axiom,
    ! [X1: fun_Pr7763680713840462443e_term,X23: product_prod_int_int] : aa_Pro4010435699744810540e_term(produc3130796691802200265nt_int,aa_Pro5748361610964203648nt_int(produc7320969046608631221nt_int(X1),X23)) = X1 ).

% fst_conv
tff(fact_1116_fst__conv,axiom,
    ! [X1: fun_in5730379987311621695e_term,X23: product_prod_int_int] : aa_Pro2551989255418117932e_term(produc4463008262350210037nt_int,aa_Pro8800328608558162860nt_int(produc4729913460834754785nt_int(X1),X23)) = X1 ).

% fst_conv
tff(fact_1117_fst__conv,axiom,
    ! [X1: heap_e7401611519738050253t_unit,X23: set_nat] : aa_Pro1219312880135979149t_unit(produc1824681642469235216et_nat,aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(X1),X23)) = X1 ).

% fst_conv
tff(fact_1118_fst__conv,axiom,
    ! [X1: assn,X23: assn] : aa_Pro2979818503705714077n_assn(produc9167289414957590229n_assn,aa_ass5585057816575862475n_assn(produc118845697133431529n_assn(X1),X23)) = X1 ).

% fst_conv
tff(fact_1119_fst__conv,axiom,
    ! [X1: uint32,X23: uint32] : aa_Pro3737690986720249065uint32(produc9004433772639906525uint32,aa_uin6954824371948101007uint32(produc1400373151660368625uint32(X1),X23)) = X1 ).

% fst_conv
tff(fact_1120_fst__conv,axiom,
    ! [X1: nat,X23: nat] : aa_Pro4243173770044397973at_nat(product_fst_nat_nat,aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,X1),X23)) = X1 ).

% fst_conv
tff(fact_1121_fst__conv,axiom,
    ! [X1: int,X23: int] : aa_Pro6799258434465214121nt_int(product_fst_int_int,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X1),X23)) = X1 ).

% fst_conv
tff(fact_1122_fst__eqD,axiom,
    ! [Xb: fun_Pr5098669352780829690e_term,Y: produc8923325533196201883nteger,A: fun_Pr5098669352780829690e_term] :
      ( ( aa_Pro4276937008533432526e_term(produc2481150572478215260nteger,aa_Pro4291442876242880245nteger(produc565448451787809096nteger(Xb),Y)) = A )
     => ( Xb = A ) ) ).

% fst_eqD
tff(fact_1123_fst__eqD,axiom,
    ! [Xb: fun_Pr6460944699146269935_nat_o,Y: produc3658429121746597890et_nat,A: fun_Pr6460944699146269935_nat_o] :
      ( ( aa_Pro8262445894397805713_nat_o(produc5325900173150335662et_nat,aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(Xb),Y)) = A )
     => ( Xb = A ) ) ).

% fst_eqD
tff(fact_1124_fst__eqD,axiom,
    ! [Xb: fun_Pr6460944699146269935_nat_o,Y: produc8799533983703829792et_nat,A: fun_Pr6460944699146269935_nat_o] :
      ( ( aa_Pro2837773213478468339_nat_o(produc4078497350495916492et_nat,aa_Pro1979504083478021276et_nat(produc7291855603632349880et_nat(Xb),Y)) = A )
     => ( Xb = A ) ) ).

% fst_eqD
tff(fact_1125_fst__eqD,axiom,
    ! [Xb: fun_Pr7763680713840462443e_term,Y: product_prod_int_int,A: fun_Pr7763680713840462443e_term] :
      ( ( aa_Pro4010435699744810540e_term(produc3130796691802200265nt_int,aa_Pro5748361610964203648nt_int(produc7320969046608631221nt_int(Xb),Y)) = A )
     => ( Xb = A ) ) ).

% fst_eqD
tff(fact_1126_fst__eqD,axiom,
    ! [Xb: fun_in5730379987311621695e_term,Y: product_prod_int_int,A: fun_in5730379987311621695e_term] :
      ( ( aa_Pro2551989255418117932e_term(produc4463008262350210037nt_int,aa_Pro8800328608558162860nt_int(produc4729913460834754785nt_int(Xb),Y)) = A )
     => ( Xb = A ) ) ).

% fst_eqD
tff(fact_1127_fst__eqD,axiom,
    ! [Xb: heap_e7401611519738050253t_unit,Y: set_nat,A: heap_e7401611519738050253t_unit] :
      ( ( aa_Pro1219312880135979149t_unit(produc1824681642469235216et_nat,aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(Xb),Y)) = A )
     => ( Xb = A ) ) ).

% fst_eqD
tff(fact_1128_fst__eqD,axiom,
    ! [Xb: assn,Y: assn,A: assn] :
      ( ( aa_Pro2979818503705714077n_assn(produc9167289414957590229n_assn,aa_ass5585057816575862475n_assn(produc118845697133431529n_assn(Xb),Y)) = A )
     => ( Xb = A ) ) ).

% fst_eqD
tff(fact_1129_fst__eqD,axiom,
    ! [Xb: uint32,Y: uint32,A: uint32] :
      ( ( aa_Pro3737690986720249065uint32(produc9004433772639906525uint32,aa_uin6954824371948101007uint32(produc1400373151660368625uint32(Xb),Y)) = A )
     => ( Xb = A ) ) ).

% fst_eqD
tff(fact_1130_fst__eqD,axiom,
    ! [Xb: nat,Y: nat,A: nat] :
      ( ( aa_Pro4243173770044397973at_nat(product_fst_nat_nat,aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Xb),Y)) = A )
     => ( Xb = A ) ) ).

% fst_eqD
tff(fact_1131_fst__eqD,axiom,
    ! [Xb: int,Y: int,A: int] :
      ( ( aa_Pro6799258434465214121nt_int(product_fst_int_int,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Xb),Y)) = A )
     => ( Xb = A ) ) ).

% fst_eqD
tff(fact_1132_snd__conv,axiom,
    ! [X1: fun_Pr5098669352780829690e_term,X23: produc8923325533196201883nteger] : aa_Pro4625778721341478855nteger(produc6589491691745714842nteger,aa_Pro4291442876242880245nteger(produc565448451787809096nteger(X1),X23)) = X23 ).

% snd_conv
tff(fact_1133_snd__conv,axiom,
    ! [X1: fun_Pr6460944699146269935_nat_o,X23: produc3658429121746597890et_nat] : aa_Pro5541488491759012064et_nat(produc4960056325182568940et_nat,aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(X1),X23)) = X23 ).

% snd_conv
tff(fact_1134_snd__conv,axiom,
    ! [X1: fun_Pr6460944699146269935_nat_o,X23: produc8799533983703829792et_nat] : aa_Pro7798056222271281692et_nat(produc9081944055658670602et_nat,aa_Pro1979504083478021276et_nat(produc7291855603632349880et_nat(X1),X23)) = X23 ).

% snd_conv
tff(fact_1135_snd__conv,axiom,
    ! [X1: fun_Pr7763680713840462443e_term,X23: product_prod_int_int] : aa_Pro2675344531184151096nt_int(produc1557020426977177351nt_int,aa_Pro5748361610964203648nt_int(produc7320969046608631221nt_int(X1),X23)) = X23 ).

% snd_conv
tff(fact_1136_snd__conv,axiom,
    ! [X1: fun_in5730379987311621695e_term,X23: product_prod_int_int] : aa_Pro4437156595432605708nt_int(produc7629796867588368947nt_int,aa_Pro8800328608558162860nt_int(produc4729913460834754785nt_int(X1),X23)) = X23 ).

% snd_conv
tff(fact_1137_snd__conv,axiom,
    ! [X1: nat,X23: nat] : aa_Pro4243173770044397973at_nat(product_snd_nat_nat,aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,X1),X23)) = X23 ).

% snd_conv
tff(fact_1138_snd__conv,axiom,
    ! [X1: heap_e7401611519738050253t_unit,X23: set_nat] : aa_Pro3416150927818442182et_nat(produc8586169260539613262et_nat,aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(X1),X23)) = X23 ).

% snd_conv
tff(fact_1139_snd__conv,axiom,
    ! [X1: int,X23: int] : aa_Pro6799258434465214121nt_int(product_snd_int_int,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X1),X23)) = X23 ).

% snd_conv
tff(fact_1140_snd__conv,axiom,
    ! [X1: code_integer,X23: code_integer] : aa_Pro5245656335456409788nteger(produc6174133586879617921nteger,aa_Cod6977894654539884864nteger(produc1086072967326762835nteger(X1),X23)) = X23 ).

% snd_conv
tff(fact_1141_snd__conv,axiom,
    ! [X1: assn,X23: assn] : aa_Pro2979818503705714077n_assn(produc2051961928117032727n_assn,aa_ass5585057816575862475n_assn(produc118845697133431529n_assn(X1),X23)) = X23 ).

% snd_conv
tff(fact_1142_snd__eqD,axiom,
    ! [Xb: fun_Pr5098669352780829690e_term,Y: produc8923325533196201883nteger,A: produc8923325533196201883nteger] :
      ( ( aa_Pro4625778721341478855nteger(produc6589491691745714842nteger,aa_Pro4291442876242880245nteger(produc565448451787809096nteger(Xb),Y)) = A )
     => ( Y = A ) ) ).

% snd_eqD
tff(fact_1143_snd__eqD,axiom,
    ! [Xb: fun_Pr6460944699146269935_nat_o,Y: produc3658429121746597890et_nat,A: produc3658429121746597890et_nat] :
      ( ( aa_Pro5541488491759012064et_nat(produc4960056325182568940et_nat,aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(Xb),Y)) = A )
     => ( Y = A ) ) ).

% snd_eqD
tff(fact_1144_snd__eqD,axiom,
    ! [Xb: fun_Pr6460944699146269935_nat_o,Y: produc8799533983703829792et_nat,A: produc8799533983703829792et_nat] :
      ( ( aa_Pro7798056222271281692et_nat(produc9081944055658670602et_nat,aa_Pro1979504083478021276et_nat(produc7291855603632349880et_nat(Xb),Y)) = A )
     => ( Y = A ) ) ).

% snd_eqD
tff(fact_1145_snd__eqD,axiom,
    ! [Xb: fun_Pr7763680713840462443e_term,Y: product_prod_int_int,A: product_prod_int_int] :
      ( ( aa_Pro2675344531184151096nt_int(produc1557020426977177351nt_int,aa_Pro5748361610964203648nt_int(produc7320969046608631221nt_int(Xb),Y)) = A )
     => ( Y = A ) ) ).

% snd_eqD
tff(fact_1146_snd__eqD,axiom,
    ! [Xb: fun_in5730379987311621695e_term,Y: product_prod_int_int,A: product_prod_int_int] :
      ( ( aa_Pro4437156595432605708nt_int(produc7629796867588368947nt_int,aa_Pro8800328608558162860nt_int(produc4729913460834754785nt_int(Xb),Y)) = A )
     => ( Y = A ) ) ).

% snd_eqD
tff(fact_1147_snd__eqD,axiom,
    ! [Xb: nat,Y: nat,A: nat] :
      ( ( aa_Pro4243173770044397973at_nat(product_snd_nat_nat,aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Xb),Y)) = A )
     => ( Y = A ) ) ).

% snd_eqD
tff(fact_1148_snd__eqD,axiom,
    ! [Xb: heap_e7401611519738050253t_unit,Y: set_nat,A: set_nat] :
      ( ( aa_Pro3416150927818442182et_nat(produc8586169260539613262et_nat,aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(Xb),Y)) = A )
     => ( Y = A ) ) ).

% snd_eqD
tff(fact_1149_snd__eqD,axiom,
    ! [Xb: int,Y: int,A: int] :
      ( ( aa_Pro6799258434465214121nt_int(product_snd_int_int,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Xb),Y)) = A )
     => ( Y = A ) ) ).

% snd_eqD
tff(fact_1150_snd__eqD,axiom,
    ! [Xb: code_integer,Y: code_integer,A: code_integer] :
      ( ( aa_Pro5245656335456409788nteger(produc6174133586879617921nteger,aa_Cod6977894654539884864nteger(produc1086072967326762835nteger(Xb),Y)) = A )
     => ( Y = A ) ) ).

% snd_eqD
tff(fact_1151_snd__eqD,axiom,
    ! [Xb: assn,Y: assn,A: assn] :
      ( ( aa_Pro2979818503705714077n_assn(produc2051961928117032727n_assn,aa_ass5585057816575862475n_assn(produc118845697133431529n_assn(Xb),Y)) = A )
     => ( Y = A ) ) ).

% snd_eqD
tff(fact_1152_prod__eq__iff,axiom,
    ! [S3: produc827990862158126777uint32,Ta: produc827990862158126777uint32] :
      ( ( S3 = Ta )
    <=> ( ( aa_Pro3737690986720249065uint32(produc9004433772639906525uint32,S3) = aa_Pro3737690986720249065uint32(produc9004433772639906525uint32,Ta) )
        & ( aa_Pro3737690986720249065uint32(produc1510406741064981791uint32,S3) = aa_Pro3737690986720249065uint32(produc1510406741064981791uint32,Ta) ) ) ) ).

% prod_eq_iff
tff(fact_1153_prod__eq__iff,axiom,
    ! [S3: product_prod_nat_nat,Ta: product_prod_nat_nat] :
      ( ( S3 = Ta )
    <=> ( ( aa_Pro4243173770044397973at_nat(product_fst_nat_nat,S3) = aa_Pro4243173770044397973at_nat(product_fst_nat_nat,Ta) )
        & ( aa_Pro4243173770044397973at_nat(product_snd_nat_nat,S3) = aa_Pro4243173770044397973at_nat(product_snd_nat_nat,Ta) ) ) ) ).

% prod_eq_iff
tff(fact_1154_prod__eq__iff,axiom,
    ! [S3: produc3658429121746597890et_nat,Ta: produc3658429121746597890et_nat] :
      ( ( S3 = Ta )
    <=> ( ( aa_Pro1219312880135979149t_unit(produc1824681642469235216et_nat,S3) = aa_Pro1219312880135979149t_unit(produc1824681642469235216et_nat,Ta) )
        & ( aa_Pro3416150927818442182et_nat(produc8586169260539613262et_nat,S3) = aa_Pro3416150927818442182et_nat(produc8586169260539613262et_nat,Ta) ) ) ) ).

% prod_eq_iff
tff(fact_1155_prod__eq__iff,axiom,
    ! [S3: product_prod_int_int,Ta: product_prod_int_int] :
      ( ( S3 = Ta )
    <=> ( ( aa_Pro6799258434465214121nt_int(product_fst_int_int,S3) = aa_Pro6799258434465214121nt_int(product_fst_int_int,Ta) )
        & ( aa_Pro6799258434465214121nt_int(product_snd_int_int,S3) = aa_Pro6799258434465214121nt_int(product_snd_int_int,Ta) ) ) ) ).

% prod_eq_iff
tff(fact_1156_prod__eq__iff,axiom,
    ! [S3: produc8923325533196201883nteger,Ta: produc8923325533196201883nteger] :
      ( ( S3 = Ta )
    <=> ( ( aa_Pro5245656335456409788nteger(produc8508995932063986495nteger,S3) = aa_Pro5245656335456409788nteger(produc8508995932063986495nteger,Ta) )
        & ( aa_Pro5245656335456409788nteger(produc6174133586879617921nteger,S3) = aa_Pro5245656335456409788nteger(produc6174133586879617921nteger,Ta) ) ) ) ).

% prod_eq_iff
tff(fact_1157_prod__eq__iff,axiom,
    ! [S3: produc6575502325842934193n_assn,Ta: produc6575502325842934193n_assn] :
      ( ( S3 = Ta )
    <=> ( ( aa_Pro2979818503705714077n_assn(produc9167289414957590229n_assn,S3) = aa_Pro2979818503705714077n_assn(produc9167289414957590229n_assn,Ta) )
        & ( aa_Pro2979818503705714077n_assn(produc2051961928117032727n_assn,S3) = aa_Pro2979818503705714077n_assn(produc2051961928117032727n_assn,Ta) ) ) ) ).

% prod_eq_iff
tff(fact_1158_prod__eqI,axiom,
    ! [P2: produc827990862158126777uint32,Q2: produc827990862158126777uint32] :
      ( ( aa_Pro3737690986720249065uint32(produc9004433772639906525uint32,P2) = aa_Pro3737690986720249065uint32(produc9004433772639906525uint32,Q2) )
     => ( ( aa_Pro3737690986720249065uint32(produc1510406741064981791uint32,P2) = aa_Pro3737690986720249065uint32(produc1510406741064981791uint32,Q2) )
       => ( P2 = Q2 ) ) ) ).

% prod_eqI
tff(fact_1159_prod__eqI,axiom,
    ! [P2: product_prod_nat_nat,Q2: product_prod_nat_nat] :
      ( ( aa_Pro4243173770044397973at_nat(product_fst_nat_nat,P2) = aa_Pro4243173770044397973at_nat(product_fst_nat_nat,Q2) )
     => ( ( aa_Pro4243173770044397973at_nat(product_snd_nat_nat,P2) = aa_Pro4243173770044397973at_nat(product_snd_nat_nat,Q2) )
       => ( P2 = Q2 ) ) ) ).

% prod_eqI
tff(fact_1160_prod__eqI,axiom,
    ! [P2: produc3658429121746597890et_nat,Q2: produc3658429121746597890et_nat] :
      ( ( aa_Pro1219312880135979149t_unit(produc1824681642469235216et_nat,P2) = aa_Pro1219312880135979149t_unit(produc1824681642469235216et_nat,Q2) )
     => ( ( aa_Pro3416150927818442182et_nat(produc8586169260539613262et_nat,P2) = aa_Pro3416150927818442182et_nat(produc8586169260539613262et_nat,Q2) )
       => ( P2 = Q2 ) ) ) ).

% prod_eqI
tff(fact_1161_prod__eqI,axiom,
    ! [P2: product_prod_int_int,Q2: product_prod_int_int] :
      ( ( aa_Pro6799258434465214121nt_int(product_fst_int_int,P2) = aa_Pro6799258434465214121nt_int(product_fst_int_int,Q2) )
     => ( ( aa_Pro6799258434465214121nt_int(product_snd_int_int,P2) = aa_Pro6799258434465214121nt_int(product_snd_int_int,Q2) )
       => ( P2 = Q2 ) ) ) ).

% prod_eqI
tff(fact_1162_prod__eqI,axiom,
    ! [P2: produc8923325533196201883nteger,Q2: produc8923325533196201883nteger] :
      ( ( aa_Pro5245656335456409788nteger(produc8508995932063986495nteger,P2) = aa_Pro5245656335456409788nteger(produc8508995932063986495nteger,Q2) )
     => ( ( aa_Pro5245656335456409788nteger(produc6174133586879617921nteger,P2) = aa_Pro5245656335456409788nteger(produc6174133586879617921nteger,Q2) )
       => ( P2 = Q2 ) ) ) ).

% prod_eqI
tff(fact_1163_prod__eqI,axiom,
    ! [P2: produc6575502325842934193n_assn,Q2: produc6575502325842934193n_assn] :
      ( ( aa_Pro2979818503705714077n_assn(produc9167289414957590229n_assn,P2) = aa_Pro2979818503705714077n_assn(produc9167289414957590229n_assn,Q2) )
     => ( ( aa_Pro2979818503705714077n_assn(produc2051961928117032727n_assn,P2) = aa_Pro2979818503705714077n_assn(produc2051961928117032727n_assn,Q2) )
       => ( P2 = Q2 ) ) ) ).

% prod_eqI
tff(fact_1164_prod_Oexpand,axiom,
    ! [Prod: produc827990862158126777uint32,Prod2: produc827990862158126777uint32] :
      ( ( ( aa_Pro3737690986720249065uint32(produc9004433772639906525uint32,Prod) = aa_Pro3737690986720249065uint32(produc9004433772639906525uint32,Prod2) )
        & ( aa_Pro3737690986720249065uint32(produc1510406741064981791uint32,Prod) = aa_Pro3737690986720249065uint32(produc1510406741064981791uint32,Prod2) ) )
     => ( Prod = Prod2 ) ) ).

% prod.expand
tff(fact_1165_prod_Oexpand,axiom,
    ! [Prod: product_prod_nat_nat,Prod2: product_prod_nat_nat] :
      ( ( ( aa_Pro4243173770044397973at_nat(product_fst_nat_nat,Prod) = aa_Pro4243173770044397973at_nat(product_fst_nat_nat,Prod2) )
        & ( aa_Pro4243173770044397973at_nat(product_snd_nat_nat,Prod) = aa_Pro4243173770044397973at_nat(product_snd_nat_nat,Prod2) ) )
     => ( Prod = Prod2 ) ) ).

% prod.expand
tff(fact_1166_prod_Oexpand,axiom,
    ! [Prod: produc3658429121746597890et_nat,Prod2: produc3658429121746597890et_nat] :
      ( ( ( aa_Pro1219312880135979149t_unit(produc1824681642469235216et_nat,Prod) = aa_Pro1219312880135979149t_unit(produc1824681642469235216et_nat,Prod2) )
        & ( aa_Pro3416150927818442182et_nat(produc8586169260539613262et_nat,Prod) = aa_Pro3416150927818442182et_nat(produc8586169260539613262et_nat,Prod2) ) )
     => ( Prod = Prod2 ) ) ).

% prod.expand
tff(fact_1167_prod_Oexpand,axiom,
    ! [Prod: product_prod_int_int,Prod2: product_prod_int_int] :
      ( ( ( aa_Pro6799258434465214121nt_int(product_fst_int_int,Prod) = aa_Pro6799258434465214121nt_int(product_fst_int_int,Prod2) )
        & ( aa_Pro6799258434465214121nt_int(product_snd_int_int,Prod) = aa_Pro6799258434465214121nt_int(product_snd_int_int,Prod2) ) )
     => ( Prod = Prod2 ) ) ).

% prod.expand
tff(fact_1168_prod_Oexpand,axiom,
    ! [Prod: produc8923325533196201883nteger,Prod2: produc8923325533196201883nteger] :
      ( ( ( aa_Pro5245656335456409788nteger(produc8508995932063986495nteger,Prod) = aa_Pro5245656335456409788nteger(produc8508995932063986495nteger,Prod2) )
        & ( aa_Pro5245656335456409788nteger(produc6174133586879617921nteger,Prod) = aa_Pro5245656335456409788nteger(produc6174133586879617921nteger,Prod2) ) )
     => ( Prod = Prod2 ) ) ).

% prod.expand
tff(fact_1169_prod_Oexpand,axiom,
    ! [Prod: produc6575502325842934193n_assn,Prod2: produc6575502325842934193n_assn] :
      ( ( ( aa_Pro2979818503705714077n_assn(produc9167289414957590229n_assn,Prod) = aa_Pro2979818503705714077n_assn(produc9167289414957590229n_assn,Prod2) )
        & ( aa_Pro2979818503705714077n_assn(produc2051961928117032727n_assn,Prod) = aa_Pro2979818503705714077n_assn(produc2051961928117032727n_assn,Prod2) ) )
     => ( Prod = Prod2 ) ) ).

% prod.expand
tff(fact_1170_norm__assertion__simps_I1_J,axiom,
    ! [A: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,one_one_assn),A) = A ).

% norm_assertion_simps(1)
tff(fact_1171_norm__assertion__simps_I2_J,axiom,
    ! [A: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A),one_one_assn) = A ).

% norm_assertion_simps(2)
tff(fact_1172_lambda__one,axiom,
    aTP_Lamm_df = times_7065122842183080059l_num1(one_on7727431528512463931l_num1) ).

% lambda_one
tff(fact_1173_lambda__one,axiom,
    aTP_Lamm_dg = aa_rea7340605660300375871l_real(times_times_real,one_one_real) ).

% lambda_one
tff(fact_1174_lambda__one,axiom,
    aTP_Lamm_dh = times_times_rat(one_one_rat) ).

% lambda_one
tff(fact_1175_lambda__one,axiom,
    aTP_Lamm_di = aa_nat_fun_nat_nat(times_times_nat,one_one_nat) ).

% lambda_one
tff(fact_1176_lambda__one,axiom,
    aTP_Lamm_dj = aa_int_fun_int_int(times_times_int,one_one_int) ).

% lambda_one
tff(fact_1177_frame__rule__left,axiom,
    ! [P: assn,C: heap_T2636463487746394924on_nat,Q: fun_option_nat_assn,R: assn] :
      ( hoare_7629718768684598413on_nat(P,C,Q)
     => hoare_7629718768684598413on_nat(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,R),P),C,aa_ass4112278883933122603t_assn(aTP_Lamm_dk(Q),R)) ) ).

% frame_rule_left
tff(fact_1178_frame__rule__left,axiom,
    ! [P: assn,C: heap_Time_Heap_nat,Q: fun_nat_assn,R: assn] :
      ( hoare_3067605981109127869le_nat(P,C,Q)
     => hoare_3067605981109127869le_nat(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,R),P),C,aa_assn_fun_nat_assn(aTP_Lamm_dl(Q),R)) ) ).

% frame_rule_left
tff(fact_1179_frame__rule__left,axiom,
    ! [P: assn,C: heap_Time_Heap_o,Q: fun_o_assn,R: assn] :
      ( hoare_hoare_triple_o(P,C,Q)
     => hoare_hoare_triple_o(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,R),P),C,aa_assn_fun_o_assn(aTP_Lamm_dm(Q),R)) ) ).

% frame_rule_left
tff(fact_1180_frame__rule__left,axiom,
    ! [P: assn,C: heap_T8145700208782473153_VEBTi,Q: fun_VEBT_VEBTi_assn,R: assn] :
      ( hoare_1429296392585015714_VEBTi(P,C,Q)
     => hoare_1429296392585015714_VEBTi(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,R),P),C,aa_ass4389913640333755968i_assn(aTP_Lamm_dn(Q),R)) ) ).

% frame_rule_left
tff(fact_1181_frame__rule__left,axiom,
    ! [P: assn,C: heap_T5738788834812785303t_unit,Q: fun_Pr6626665806141890376t_assn,R: assn] :
      ( hoare_8945653483474564448t_unit(P,C,Q)
     => hoare_8945653483474564448t_unit(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,R),P),C,aa_ass3177796706412234594t_assn(aTP_Lamm_do(Q),R)) ) ).

% frame_rule_left
tff(fact_1182_add__mono1,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),one_one_real)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),one_one_real)) ) ).

% add_mono1
tff(fact_1183_add__mono1,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(A),one_one_rat)),aa_rat_rat(plus_plus_rat(B),one_one_rat)) ) ).

% add_mono1
tff(fact_1184_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_1185_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_1186_less__add__one,axiom,
    ! [A: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),one_one_real)) ).

% less_add_one
tff(fact_1187_less__add__one,axiom,
    ! [A: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(plus_plus_rat(A),one_one_rat)) ).

% less_add_one
tff(fact_1188_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_1189_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_1190_less__1__mult,axiom,
    ! [M: real,Nb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),M)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Nb)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,M),Nb)) ) ) ).

% less_1_mult
tff(fact_1191_less__1__mult,axiom,
    ! [M: rat,Nb: 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),Nb)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_rat_rat(times_times_rat(M),Nb)) ) ) ).

% less_1_mult
tff(fact_1192_less__1__mult,axiom,
    ! [M: nat,Nb: 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),Nb)
       => 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),Nb)) ) ) ).

% less_1_mult
tff(fact_1193_less__1__mult,axiom,
    ! [M: int,Nb: 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),Nb)
       => 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),Nb)) ) ) ).

% less_1_mult
tff(fact_1194_add__le__add__imp__diff__le,axiom,
    ! [I: real,K: real,Nb: real,J: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K)),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Nb),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J),K))
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K)),Nb)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Nb),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J),K))
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Nb),K)),J) ) ) ) ) ).

% add_le_add_imp_diff_le
tff(fact_1195_add__le__add__imp__diff__le,axiom,
    ! [I: rat,K: rat,Nb: rat,J: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(I),K)),Nb)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Nb),aa_rat_rat(plus_plus_rat(J),K))
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(I),K)),Nb)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Nb),aa_rat_rat(plus_plus_rat(J),K))
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,minus_minus_rat(Nb,K)),J) ) ) ) ) ).

% add_le_add_imp_diff_le
tff(fact_1196_add__le__add__imp__diff__le,axiom,
    ! [I: nat,K: nat,Nb: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K))
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),Nb)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K))
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K)),J) ) ) ) ) ).

% add_le_add_imp_diff_le
tff(fact_1197_add__le__add__imp__diff__le,axiom,
    ! [I: int,K: int,Nb: int,J: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K)),Nb)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Nb),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),K))
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K)),Nb)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Nb),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),K))
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Nb),K)),J) ) ) ) ) ).

% add_le_add_imp_diff_le
tff(fact_1198_add__le__imp__le__diff,axiom,
    ! [I: real,K: real,Nb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K)),Nb)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,I),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Nb),K)) ) ).

% add_le_imp_le_diff
tff(fact_1199_add__le__imp__le__diff,axiom,
    ! [I: rat,K: rat,Nb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(I),K)),Nb)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,I),minus_minus_rat(Nb,K)) ) ).

% add_le_imp_le_diff
tff(fact_1200_add__le__imp__le__diff,axiom,
    ! [I: nat,K: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K)) ) ).

% add_le_imp_le_diff
tff(fact_1201_add__le__imp__le__diff,axiom,
    ! [I: int,K: int,Nb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K)),Nb)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),aa_int_int(aa_int_fun_int_int(minus_minus_int,Nb),K)) ) ).

% add_le_imp_le_diff
tff(fact_1202_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [A: real,B: real] :
      ( ~ aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)) = A ) ) ).

% linordered_semidom_class.add_diff_inverse
tff(fact_1203_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [A: rat,B: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_rat(plus_plus_rat(B),minus_minus_rat(A,B)) = A ) ) ).

% linordered_semidom_class.add_diff_inverse
tff(fact_1204_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_1205_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_1206_square__diff__square__factored,axiom,
    ! [Xb: real,Y: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Xb)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Y)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Y)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),Y)) ).

% square_diff_square_factored
tff(fact_1207_square__diff__square__factored,axiom,
    ! [Xb: rat,Y: rat] : minus_minus_rat(aa_rat_rat(times_times_rat(Xb),Xb),aa_rat_rat(times_times_rat(Y),Y)) = aa_rat_rat(times_times_rat(aa_rat_rat(plus_plus_rat(Xb),Y)),minus_minus_rat(Xb,Y)) ).

% square_diff_square_factored
tff(fact_1208_square__diff__square__factored,axiom,
    ! [Xb: 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,Xb),Xb)),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,Xb),Y)),aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb),Y)) ).

% square_diff_square_factored
tff(fact_1209_eq__add__iff2,axiom,
    ! [A: real,E: real,C: real,B: real,D: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),E)),C) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),E)),D) )
    <=> ( C = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),A)),E)),D) ) ) ).

% eq_add_iff2
tff(fact_1210_eq__add__iff2,axiom,
    ! [A: rat,E: rat,C: rat,B: rat,D: rat] :
      ( ( aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(A),E)),C) = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(B),E)),D) )
    <=> ( C = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(minus_minus_rat(B,A)),E)),D) ) ) ).

% eq_add_iff2
tff(fact_1211_eq__add__iff2,axiom,
    ! [A: int,E: int,C: int,B: int,D: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),E)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),E)),D) )
    <=> ( C = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B),A)),E)),D) ) ) ).

% eq_add_iff2
tff(fact_1212_eq__add__iff1,axiom,
    ! [A: real,E: real,C: real,B: real,D: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),E)),C) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),E)),D) )
    <=> ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)),E)),C) = D ) ) ).

% eq_add_iff1
tff(fact_1213_eq__add__iff1,axiom,
    ! [A: rat,E: rat,C: rat,B: rat,D: rat] :
      ( ( aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(A),E)),C) = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(B),E)),D) )
    <=> ( aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(minus_minus_rat(A,B)),E)),C) = D ) ) ).

% eq_add_iff1
tff(fact_1214_eq__add__iff1,axiom,
    ! [A: int,E: int,C: int,B: int,D: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),E)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),E)),D) )
    <=> ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),E)),C) = D ) ) ).

% eq_add_iff1
tff(fact_1215_surjective__pairing,axiom,
    ! [Ta: produc827990862158126777uint32] : Ta = aa_uin6954824371948101007uint32(produc1400373151660368625uint32(aa_Pro3737690986720249065uint32(produc9004433772639906525uint32,Ta)),aa_Pro3737690986720249065uint32(produc1510406741064981791uint32,Ta)) ).

% surjective_pairing
tff(fact_1216_surjective__pairing,axiom,
    ! [Ta: product_prod_nat_nat] : Ta = aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,aa_Pro4243173770044397973at_nat(product_fst_nat_nat,Ta)),aa_Pro4243173770044397973at_nat(product_snd_nat_nat,Ta)) ).

% surjective_pairing
tff(fact_1217_surjective__pairing,axiom,
    ! [Ta: product_prod_int_int] : Ta = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_Pro6799258434465214121nt_int(product_fst_int_int,Ta)),aa_Pro6799258434465214121nt_int(product_snd_int_int,Ta)) ).

% surjective_pairing
tff(fact_1218_surjective__pairing,axiom,
    ! [Ta: produc8923325533196201883nteger] : Ta = aa_Cod6977894654539884864nteger(produc1086072967326762835nteger(aa_Pro5245656335456409788nteger(produc8508995932063986495nteger,Ta)),aa_Pro5245656335456409788nteger(produc6174133586879617921nteger,Ta)) ).

% surjective_pairing
tff(fact_1219_surjective__pairing,axiom,
    ! [Ta: produc6575502325842934193n_assn] : Ta = aa_ass5585057816575862475n_assn(produc118845697133431529n_assn(aa_Pro2979818503705714077n_assn(produc9167289414957590229n_assn,Ta)),aa_Pro2979818503705714077n_assn(produc2051961928117032727n_assn,Ta)) ).

% surjective_pairing
tff(fact_1220_surjective__pairing,axiom,
    ! [Ta: produc3658429121746597890et_nat] : Ta = aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(aa_Pro1219312880135979149t_unit(produc1824681642469235216et_nat,Ta)),aa_Pro3416150927818442182et_nat(produc8586169260539613262et_nat,Ta)) ).

% surjective_pairing
tff(fact_1221_surjective__pairing,axiom,
    ! [Ta: produc4788131091073775919nt_int] : Ta = aa_Pro8800328608558162860nt_int(produc4729913460834754785nt_int(aa_Pro2551989255418117932e_term(produc4463008262350210037nt_int,Ta)),aa_Pro4437156595432605708nt_int(produc7629796867588368947nt_int,Ta)) ).

% surjective_pairing
tff(fact_1222_surjective__pairing,axiom,
    ! [Ta: produc8799533983703829792et_nat] : Ta = aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(aa_Pro8262445894397805713_nat_o(produc5325900173150335662et_nat,Ta)),aa_Pro5541488491759012064et_nat(produc4960056325182568940et_nat,Ta)) ).

% surjective_pairing
tff(fact_1223_surjective__pairing,axiom,
    ! [Ta: produc2453742332425399446nteger] : Ta = aa_Pro4291442876242880245nteger(produc565448451787809096nteger(aa_Pro4276937008533432526e_term(produc2481150572478215260nteger,Ta)),aa_Pro4625778721341478855nteger(produc6589491691745714842nteger,Ta)) ).

% surjective_pairing
tff(fact_1224_surjective__pairing,axiom,
    ! [Ta: produc3043004577601930755nt_int] : Ta = aa_Pro5748361610964203648nt_int(produc7320969046608631221nt_int(aa_Pro4010435699744810540e_term(produc3130796691802200265nt_int,Ta)),aa_Pro2675344531184151096nt_int(produc1557020426977177351nt_int,Ta)) ).

% surjective_pairing
tff(fact_1225_prod_Oexhaust__sel,axiom,
    ! [Prod: produc827990862158126777uint32] : Prod = aa_uin6954824371948101007uint32(produc1400373151660368625uint32(aa_Pro3737690986720249065uint32(produc9004433772639906525uint32,Prod)),aa_Pro3737690986720249065uint32(produc1510406741064981791uint32,Prod)) ).

% prod.exhaust_sel
tff(fact_1226_prod_Oexhaust__sel,axiom,
    ! [Prod: product_prod_nat_nat] : Prod = aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,aa_Pro4243173770044397973at_nat(product_fst_nat_nat,Prod)),aa_Pro4243173770044397973at_nat(product_snd_nat_nat,Prod)) ).

% prod.exhaust_sel
tff(fact_1227_prod_Oexhaust__sel,axiom,
    ! [Prod: product_prod_int_int] : Prod = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_Pro6799258434465214121nt_int(product_fst_int_int,Prod)),aa_Pro6799258434465214121nt_int(product_snd_int_int,Prod)) ).

% prod.exhaust_sel
tff(fact_1228_prod_Oexhaust__sel,axiom,
    ! [Prod: produc8923325533196201883nteger] : Prod = aa_Cod6977894654539884864nteger(produc1086072967326762835nteger(aa_Pro5245656335456409788nteger(produc8508995932063986495nteger,Prod)),aa_Pro5245656335456409788nteger(produc6174133586879617921nteger,Prod)) ).

% prod.exhaust_sel
tff(fact_1229_prod_Oexhaust__sel,axiom,
    ! [Prod: produc6575502325842934193n_assn] : Prod = aa_ass5585057816575862475n_assn(produc118845697133431529n_assn(aa_Pro2979818503705714077n_assn(produc9167289414957590229n_assn,Prod)),aa_Pro2979818503705714077n_assn(produc2051961928117032727n_assn,Prod)) ).

% prod.exhaust_sel
tff(fact_1230_prod_Oexhaust__sel,axiom,
    ! [Prod: produc3658429121746597890et_nat] : Prod = aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(aa_Pro1219312880135979149t_unit(produc1824681642469235216et_nat,Prod)),aa_Pro3416150927818442182et_nat(produc8586169260539613262et_nat,Prod)) ).

% prod.exhaust_sel
tff(fact_1231_prod_Oexhaust__sel,axiom,
    ! [Prod: produc4788131091073775919nt_int] : Prod = aa_Pro8800328608558162860nt_int(produc4729913460834754785nt_int(aa_Pro2551989255418117932e_term(produc4463008262350210037nt_int,Prod)),aa_Pro4437156595432605708nt_int(produc7629796867588368947nt_int,Prod)) ).

% prod.exhaust_sel
tff(fact_1232_prod_Oexhaust__sel,axiom,
    ! [Prod: produc8799533983703829792et_nat] : Prod = aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(aa_Pro8262445894397805713_nat_o(produc5325900173150335662et_nat,Prod)),aa_Pro5541488491759012064et_nat(produc4960056325182568940et_nat,Prod)) ).

% prod.exhaust_sel
tff(fact_1233_prod_Oexhaust__sel,axiom,
    ! [Prod: produc2453742332425399446nteger] : Prod = aa_Pro4291442876242880245nteger(produc565448451787809096nteger(aa_Pro4276937008533432526e_term(produc2481150572478215260nteger,Prod)),aa_Pro4625778721341478855nteger(produc6589491691745714842nteger,Prod)) ).

% prod.exhaust_sel
tff(fact_1234_prod_Oexhaust__sel,axiom,
    ! [Prod: produc3043004577601930755nt_int] : Prod = aa_Pro5748361610964203648nt_int(produc7320969046608631221nt_int(aa_Pro4010435699744810540e_term(produc3130796691802200265nt_int,Prod)),aa_Pro2675344531184151096nt_int(produc1557020426977177351nt_int,Prod)) ).

% prod.exhaust_sel
tff(fact_1235_ordered__ring__class_Ole__add__iff2,axiom,
    ! [A: real,E: real,C: real,B: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),E)),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),E)),D))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),A)),E)),D)) ) ).

% ordered_ring_class.le_add_iff2
tff(fact_1236_ordered__ring__class_Ole__add__iff2,axiom,
    ! [A: rat,E: rat,C: rat,B: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(A),E)),C)),aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(B),E)),D))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(minus_minus_rat(B,A)),E)),D)) ) ).

% ordered_ring_class.le_add_iff2
tff(fact_1237_ordered__ring__class_Ole__add__iff2,axiom,
    ! [A: int,E: int,C: int,B: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),E)),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),E)),D))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B),A)),E)),D)) ) ).

% ordered_ring_class.le_add_iff2
tff(fact_1238_ordered__ring__class_Ole__add__iff1,axiom,
    ! [A: real,E: real,C: real,B: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),E)),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),E)),D))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)),E)),C)),D) ) ).

% ordered_ring_class.le_add_iff1
tff(fact_1239_ordered__ring__class_Ole__add__iff1,axiom,
    ! [A: rat,E: rat,C: rat,B: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(A),E)),C)),aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(B),E)),D))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(minus_minus_rat(A,B)),E)),C)),D) ) ).

% ordered_ring_class.le_add_iff1
tff(fact_1240_ordered__ring__class_Ole__add__iff1,axiom,
    ! [A: int,E: int,C: int,B: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),E)),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),E)),D))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),E)),C)),D) ) ).

% ordered_ring_class.le_add_iff1
tff(fact_1241_less__add__iff2,axiom,
    ! [A: real,E: real,C: real,B: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),E)),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),E)),D))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,C),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),A)),E)),D)) ) ).

% less_add_iff2
tff(fact_1242_less__add__iff2,axiom,
    ! [A: rat,E: rat,C: rat,B: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(A),E)),C)),aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(B),E)),D))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(minus_minus_rat(B,A)),E)),D)) ) ).

% less_add_iff2
tff(fact_1243_less__add__iff2,axiom,
    ! [A: int,E: int,C: int,B: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),E)),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),E)),D))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,C),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B),A)),E)),D)) ) ).

% less_add_iff2
tff(fact_1244_less__add__iff1,axiom,
    ! [A: real,E: real,C: real,B: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),E)),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),E)),D))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)),E)),C)),D) ) ).

% less_add_iff1
tff(fact_1245_less__add__iff1,axiom,
    ! [A: rat,E: rat,C: rat,B: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(A),E)),C)),aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(B),E)),D))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(minus_minus_rat(A,B)),E)),C)),D) ) ).

% less_add_iff1
tff(fact_1246_less__add__iff1,axiom,
    ! [A: int,E: int,C: int,B: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),E)),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),E)),D))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),E)),C)),D) ) ).

% less_add_iff1
tff(fact_1247_square__diff__one__factored,axiom,
    ! [Xb: word_N3645301735248828278l_num1] : minus_4019991460397169231l_num1(aa_wor7274019993978516266l_num1(times_7065122842183080059l_num1(Xb),Xb),one_on7727431528512463931l_num1) = aa_wor7274019993978516266l_num1(times_7065122842183080059l_num1(plus_p361126936061061375l_num1(Xb,one_on7727431528512463931l_num1)),minus_4019991460397169231l_num1(Xb,one_on7727431528512463931l_num1)) ).

% square_diff_one_factored
tff(fact_1248_square__diff__one__factored,axiom,
    ! [Xb: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Xb)),one_one_real) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),one_one_real)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),one_one_real)) ).

% square_diff_one_factored
tff(fact_1249_square__diff__one__factored,axiom,
    ! [Xb: rat] : minus_minus_rat(aa_rat_rat(times_times_rat(Xb),Xb),one_one_rat) = aa_rat_rat(times_times_rat(aa_rat_rat(plus_plus_rat(Xb),one_one_rat)),minus_minus_rat(Xb,one_one_rat)) ).

% square_diff_one_factored
tff(fact_1250_square__diff__one__factored,axiom,
    ! [Xb: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Xb),Xb)),one_one_int) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb),one_one_int)),aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb),one_one_int)) ).

% square_diff_one_factored
tff(fact_1251_invar__vebt_Ointros_I5_J,axiom,
    ! [TreeLista: list_VEBT_VEBT,Nb: nat,Summarya: vEBT_VEBT,M: nat,Deg: nat,Mia: nat,Maa: nat] :
      ( ! [X2: vEBT_VEBT] :
          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X2),set_VEBT_VEBT2(TreeLista))
         => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,X2),Nb) )
     => ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Summarya),M)
       => ( ( size_s6755466524823107622T_VEBT(TreeLista) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M) )
         => ( ( M = aa_nat_nat(suc,Nb) )
           => ( ( Deg = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),M) )
             => ( ! [I3: nat] :
                    ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M))
                   => ( ? [X_1: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeLista,I3)),X_1)
                    <=> aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,Summarya),I3) ) )
               => ( ( ( Mia = Maa )
                   => ! [X2: vEBT_VEBT] :
                        ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X2),set_VEBT_VEBT2(TreeLista))
                       => ~ ? [X_13: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,X2),X_13) ) )
                 => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mia),Maa)
                   => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Maa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg))
                     => ( ( ( Mia != Maa )
                         => ! [I3: nat] :
                              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M))
                             => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Maa),Nb) = I3 )
                                 => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeLista,I3)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Maa),Nb)) )
                                & ! [X2: nat] :
                                    ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X2),Nb) = I3 )
                                      & aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeLista,I3)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X2),Nb)) )
                                   => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mia),X2)
                                      & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X2),Maa) ) ) ) ) )
                       => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mia),Maa)),Deg,TreeLista),Summarya)),Deg) ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.intros(5)
tff(fact_1252_invar__vebt_Ointros_I4_J,axiom,
    ! [TreeLista: list_VEBT_VEBT,Nb: nat,Summarya: vEBT_VEBT,M: nat,Deg: nat,Mia: nat,Maa: nat] :
      ( ! [X2: vEBT_VEBT] :
          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X2),set_VEBT_VEBT2(TreeLista))
         => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,X2),Nb) )
     => ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Summarya),M)
       => ( ( size_s6755466524823107622T_VEBT(TreeLista) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M) )
         => ( ( M = Nb )
           => ( ( Deg = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),M) )
             => ( ! [I3: nat] :
                    ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M))
                   => ( ? [X_1: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeLista,I3)),X_1)
                    <=> aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,Summarya),I3) ) )
               => ( ( ( Mia = Maa )
                   => ! [X2: vEBT_VEBT] :
                        ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X2),set_VEBT_VEBT2(TreeLista))
                       => ~ ? [X_13: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,X2),X_13) ) )
                 => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mia),Maa)
                   => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Maa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg))
                     => ( ( ( Mia != Maa )
                         => ! [I3: nat] :
                              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M))
                             => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Maa),Nb) = I3 )
                                 => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeLista,I3)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Maa),Nb)) )
                                & ! [X2: nat] :
                                    ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X2),Nb) = I3 )
                                      & aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeLista,I3)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X2),Nb)) )
                                   => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mia),X2)
                                      & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X2),Maa) ) ) ) ) )
                       => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mia),Maa)),Deg,TreeLista),Summarya)),Deg) ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.intros(4)
tff(fact_1253_real__average__minus__first,axiom,
    ! [A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),numeral_numeral_real(aa_num_num(bit0,one)))),A) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),A)),numeral_numeral_real(aa_num_num(bit0,one))) ).

% real_average_minus_first
tff(fact_1254_real__average__minus__second,axiom,
    ! [B: real,A: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),A)),numeral_numeral_real(aa_num_num(bit0,one)))),A) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),A)),numeral_numeral_real(aa_num_num(bit0,one))) ).

% real_average_minus_second
tff(fact_1255_invar__vebt_Ointros_I3_J,axiom,
    ! [TreeLista: list_VEBT_VEBT,Nb: nat,Summarya: vEBT_VEBT,M: nat,Deg: nat] :
      ( ! [X2: vEBT_VEBT] :
          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X2),set_VEBT_VEBT2(TreeLista))
         => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,X2),Nb) )
     => ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Summarya),M)
       => ( ( size_s6755466524823107622T_VEBT(TreeLista) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M) )
         => ( ( M = aa_nat_nat(suc,Nb) )
           => ( ( Deg = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),M) )
             => ( ~ ? [X_13: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,Summarya),X_13)
               => ( ! [X2: vEBT_VEBT] :
                      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X2),set_VEBT_VEBT2(TreeLista))
                     => ~ ? [X_13: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,X2),X_13) )
                 => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Deg,TreeLista),Summarya)),Deg) ) ) ) ) ) ) ) ).

% invar_vebt.intros(3)
tff(fact_1256_invar__vebt_Ointros_I2_J,axiom,
    ! [TreeLista: list_VEBT_VEBT,Nb: nat,Summarya: vEBT_VEBT,M: nat,Deg: nat] :
      ( ! [X2: vEBT_VEBT] :
          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X2),set_VEBT_VEBT2(TreeLista))
         => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,X2),Nb) )
     => ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Summarya),M)
       => ( ( size_s6755466524823107622T_VEBT(TreeLista) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M) )
         => ( ( M = Nb )
           => ( ( Deg = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),M) )
             => ( ~ ? [X_13: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,Summarya),X_13)
               => ( ! [X2: vEBT_VEBT] :
                      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X2),set_VEBT_VEBT2(TreeLista))
                     => ~ ? [X_13: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,X2),X_13) )
                 => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Deg,TreeLista),Summarya)),Deg) ) ) ) ) ) ) ) ).

% invar_vebt.intros(2)
tff(fact_1257_less__eq__option__Some__None,axiom,
    ! [Xb: nat] : ~ aa_option_nat_o(aa_opt4319133605541840243_nat_o(ord_le5914376470875661696on_nat,aa_nat_option_nat(some_nat,Xb)),none_nat) ).

% less_eq_option_Some_None
tff(fact_1258_less__eq__option__Some__None,axiom,
    ! [Xb: num] : ~ aa_option_num_o(aa_opt8674010985164783839_num_o(ord_le6622620407824499402on_num,aa_num_option_num(some_num,Xb)),none_num) ).

% less_eq_option_Some_None
tff(fact_1259_count__buildup,axiom,
    ! [Nb: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_VEBT_VEBT_real(vEBT_VEBT_cnt,vEBT_vebt_buildup(Nb))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),aa_nat_real(aa_real_fun_nat_real(power_power_real,numeral_numeral_real(aa_num_num(bit0,one))),Nb))) ).

% count_buildup
tff(fact_1260_less__eq__option__Some,axiom,
    ! [Xb: set_int,Y: set_int] :
      ( ord_le353528952715127954et_int(some_set_int(Xb),some_set_int(Y))
    <=> aa_set_int_o(ord_less_eq_set_int(Xb),Y) ) ).

% less_eq_option_Some
tff(fact_1261_less__eq__option__Some,axiom,
    ! [Xb: rat,Y: rat] :
      ( ord_le2406147912482264968on_rat(some_rat(Xb),some_rat(Y))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Y) ) ).

% less_eq_option_Some
tff(fact_1262_less__eq__option__Some,axiom,
    ! [Xb: num,Y: num] :
      ( aa_option_num_o(aa_opt8674010985164783839_num_o(ord_le6622620407824499402on_num,aa_num_option_num(some_num,Xb)),aa_num_option_num(some_num,Y))
    <=> aa_num_o(ord_less_eq_num(Xb),Y) ) ).

% less_eq_option_Some
tff(fact_1263_less__eq__option__Some,axiom,
    ! [Xb: nat,Y: nat] :
      ( aa_option_nat_o(aa_opt4319133605541840243_nat_o(ord_le5914376470875661696on_nat,aa_nat_option_nat(some_nat,Xb)),aa_nat_option_nat(some_nat,Y))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Y) ) ).

% less_eq_option_Some
tff(fact_1264_less__eq__option__Some,axiom,
    ! [Xb: int,Y: int] :
      ( ord_le1736525451366464988on_int(aa_int_option_int(some_int,Xb),aa_int_option_int(some_int,Y))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Y) ) ).

% less_eq_option_Some
tff(fact_1265_zdiv__numeral__Bit0,axiom,
    ! [V: num,W: num] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,V))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,W))) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_num_int(numeral_numeral_int,V)),aa_num_int(numeral_numeral_int,W)) ).

% zdiv_numeral_Bit0
tff(fact_1266_less__option__None,axiom,
    ! [Xb: option_num] : ~ aa_option_num_o(aa_opt8674010985164783839_num_o(ord_less_option_num,Xb),none_num) ).

% less_option_None
tff(fact_1267_less__option__None,axiom,
    ! [Xb: option_nat] : ~ aa_option_nat_o(aa_opt4319133605541840243_nat_o(ord_less_option_nat,Xb),none_nat) ).

% less_option_None
tff(fact_1268_VEBT_Oinject_I1_J,axiom,
    ! [X11a: option4927543243414619207at_nat,X12a: nat,X13a: list_VEBT_VEBT,X14a: vEBT_VEBT,Y11: option4927543243414619207at_nat,Y12: nat,Y13: list_VEBT_VEBT,Y14: vEBT_VEBT] :
      ( ( aa_VEB5832953974749077948T_VEBT(vEBT_Node(X11a,X12a,X13a),X14a) = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Y11,Y12,Y13),Y14) )
    <=> ( ( X11a = Y11 )
        & ( X12a = Y12 )
        & ( X13a = Y13 )
        & ( X14a = Y14 ) ) ) ).

% VEBT.inject(1)
tff(fact_1269_less__option__Some,axiom,
    ! [Xb: real,Y: real] :
      ( ord_less_option_real(some_real(Xb),some_real(Y))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y) ) ).

% less_option_Some
tff(fact_1270_less__option__Some,axiom,
    ! [Xb: rat,Y: rat] :
      ( ord_less_option_rat(some_rat(Xb),some_rat(Y))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Y) ) ).

% less_option_Some
tff(fact_1271_less__option__Some,axiom,
    ! [Xb: num,Y: num] :
      ( aa_option_num_o(aa_opt8674010985164783839_num_o(ord_less_option_num,aa_num_option_num(some_num,Xb)),aa_num_option_num(some_num,Y))
    <=> aa_num_o(ord_less_num(Xb),Y) ) ).

% less_option_Some
tff(fact_1272_less__option__Some,axiom,
    ! [Xb: nat,Y: nat] :
      ( aa_option_nat_o(aa_opt4319133605541840243_nat_o(ord_less_option_nat,aa_nat_option_nat(some_nat,Xb)),aa_nat_option_nat(some_nat,Y))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y) ) ).

% less_option_Some
tff(fact_1273_less__option__Some,axiom,
    ! [Xb: int,Y: int] :
      ( ord_less_option_int(aa_int_option_int(some_int,Xb),aa_int_option_int(some_int,Y))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Y) ) ).

% less_option_Some
tff(fact_1274_less__option__None__Some__code,axiom,
    ! [Xb: nat] : aa_option_nat_o(aa_opt4319133605541840243_nat_o(ord_less_option_nat,none_nat),aa_nat_option_nat(some_nat,Xb)) ).

% less_option_None_Some_code
tff(fact_1275_less__option__None__Some__code,axiom,
    ! [Xb: num] : aa_option_num_o(aa_opt8674010985164783839_num_o(ord_less_option_num,none_num),aa_num_option_num(some_num,Xb)) ).

% less_option_None_Some_code
tff(fact_1276_less__eq__option__None__code,axiom,
    ! [Xb: option_num] : aa_option_num_o(aa_opt8674010985164783839_num_o(ord_le6622620407824499402on_num,none_num),Xb) ).

% less_eq_option_None_code
tff(fact_1277_less__eq__option__None__code,axiom,
    ! [Xb: option_nat] : aa_option_nat_o(aa_opt4319133605541840243_nat_o(ord_le5914376470875661696on_nat,none_nat),Xb) ).

% less_eq_option_None_code
tff(fact_1278_less__option__def,axiom,
    ! [Xb: option_real,Y: option_real] :
      ( ord_less_option_real(Xb,Y)
    <=> case_option_o_real($false,aTP_Lamm_dq(Xb),Y) ) ).

% less_option_def
tff(fact_1279_less__option__def,axiom,
    ! [Xb: option_rat,Y: option_rat] :
      ( ord_less_option_rat(Xb,Y)
    <=> case_option_o_rat($false,aTP_Lamm_ds(Xb),Y) ) ).

% less_option_def
tff(fact_1280_less__option__def,axiom,
    ! [Xb: option_num,Y: option_num] :
      ( aa_option_num_o(aa_opt8674010985164783839_num_o(ord_less_option_num,Xb),Y)
    <=> aa_option_num_o(case_option_o_num($false,aTP_Lamm_du(Xb)),Y) ) ).

% less_option_def
tff(fact_1281_less__option__def,axiom,
    ! [Xb: option_nat,Y: option_nat] :
      ( aa_option_nat_o(aa_opt4319133605541840243_nat_o(ord_less_option_nat,Xb),Y)
    <=> aa_option_nat_o(case_option_o_nat($false,aTP_Lamm_dw(Xb)),Y) ) ).

% less_option_def
tff(fact_1282_less__option__def,axiom,
    ! [Xb: option_int,Y: option_int] :
      ( ord_less_option_int(Xb,Y)
    <=> case_option_o_int($false,aTP_Lamm_dy(Xb),Y) ) ).

% less_option_def
tff(fact_1283_mult__commute__abs,axiom,
    ! [C: real] : aa_rea7340605660300375871l_real(aTP_Lamm_dz,C) = aa_rea7340605660300375871l_real(times_times_real,C) ).

% mult_commute_abs
tff(fact_1284_mult__commute__abs,axiom,
    ! [C: rat] : aTP_Lamm_ea(C) = times_times_rat(C) ).

% mult_commute_abs
tff(fact_1285_mult__commute__abs,axiom,
    ! [C: nat] : aa_nat_fun_nat_nat(aTP_Lamm_eb,C) = aa_nat_fun_nat_nat(times_times_nat,C) ).

% mult_commute_abs
tff(fact_1286_mult__commute__abs,axiom,
    ! [C: int] : aa_int_fun_int_int(aTP_Lamm_ec,C) = aa_int_fun_int_int(times_times_int,C) ).

% mult_commute_abs
tff(fact_1287_less__option__None__Some,axiom,
    ! [Xb: nat] : aa_option_nat_o(aa_opt4319133605541840243_nat_o(ord_less_option_nat,none_nat),aa_nat_option_nat(some_nat,Xb)) ).

% less_option_None_Some
tff(fact_1288_less__option__None__Some,axiom,
    ! [Xb: num] : aa_option_num_o(aa_opt8674010985164783839_num_o(ord_less_option_num,none_num),aa_num_option_num(some_num,Xb)) ).

% less_option_None_Some
tff(fact_1289_less__option__None__is__Some,axiom,
    ! [Xb: option_nat] :
      ( aa_option_nat_o(aa_opt4319133605541840243_nat_o(ord_less_option_nat,none_nat),Xb)
     => ? [Z3: nat] : Xb = aa_nat_option_nat(some_nat,Z3) ) ).

% less_option_None_is_Some
tff(fact_1290_less__option__None__is__Some,axiom,
    ! [Xb: option_num] :
      ( aa_option_num_o(aa_opt8674010985164783839_num_o(ord_less_option_num,none_num),Xb)
     => ? [Z3: num] : Xb = aa_num_option_num(some_num,Z3) ) ).

% less_option_None_is_Some
tff(fact_1291_less__eq__option__None,axiom,
    ! [Xb: option_num] : aa_option_num_o(aa_opt8674010985164783839_num_o(ord_le6622620407824499402on_num,none_num),Xb) ).

% less_eq_option_None
tff(fact_1292_less__eq__option__None,axiom,
    ! [Xb: option_nat] : aa_option_nat_o(aa_opt4319133605541840243_nat_o(ord_le5914376470875661696on_nat,none_nat),Xb) ).

% less_eq_option_None
tff(fact_1293_less__eq__option__None__is__None,axiom,
    ! [Xb: option_num] :
      ( aa_option_num_o(aa_opt8674010985164783839_num_o(ord_le6622620407824499402on_num,Xb),none_num)
     => ( Xb = none_num ) ) ).

% less_eq_option_None_is_None
tff(fact_1294_less__eq__option__None__is__None,axiom,
    ! [Xb: option_nat] :
      ( aa_option_nat_o(aa_opt4319133605541840243_nat_o(ord_le5914376470875661696on_nat,Xb),none_nat)
     => ( Xb = none_nat ) ) ).

% less_eq_option_None_is_None
tff(fact_1295_less__eq__option__def,axiom,
    ! [Xb: option_set_int,Y: option_set_int] :
      ( ord_le353528952715127954et_int(Xb,Y)
    <=> case_o223999843215110191et_int($true,aTP_Lamm_ed(Y),Xb) ) ).

% less_eq_option_def
tff(fact_1296_less__eq__option__def,axiom,
    ! [Xb: option_rat,Y: option_rat] :
      ( ord_le2406147912482264968on_rat(Xb,Y)
    <=> case_option_o_rat($true,aTP_Lamm_ee(Y),Xb) ) ).

% less_eq_option_def
tff(fact_1297_less__eq__option__def,axiom,
    ! [Xb: option_num,Y: option_num] :
      ( aa_option_num_o(aa_opt8674010985164783839_num_o(ord_le6622620407824499402on_num,Xb),Y)
    <=> aa_option_num_o(case_option_o_num($true,aTP_Lamm_ef(Y)),Xb) ) ).

% less_eq_option_def
tff(fact_1298_less__eq__option__def,axiom,
    ! [Xb: option_nat,Y: option_nat] :
      ( aa_option_nat_o(aa_opt4319133605541840243_nat_o(ord_le5914376470875661696on_nat,Xb),Y)
    <=> aa_option_nat_o(case_option_o_nat($true,aTP_Lamm_eg(Y)),Xb) ) ).

% less_eq_option_def
tff(fact_1299_less__eq__option__def,axiom,
    ! [Xb: option_int,Y: option_int] :
      ( ord_le1736525451366464988on_int(Xb,Y)
    <=> case_option_o_int($true,aTP_Lamm_eh(Y),Xb) ) ).

% less_eq_option_def
tff(fact_1300_set__vebt__def,axiom,
    ! [Ta: vEBT_VEBT] : vEBT_set_vebt(Ta) = collect_nat(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,Ta)) ).

% set_vebt_def
tff(fact_1301_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_1302_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_1303_builupi_Hcorr,axiom,
    ! [Nb: nat] : hoare_1429296392585015714_VEBTi(one_one_assn,aa_nat5408703056993926195_VEBTi(vEBT_V739175172307565963ildupi,Nb),aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,vEBT_vebt_buildup(Nb))) ).

% builupi'corr
tff(fact_1304_builupicorr,axiom,
    ! [Nb: nat] : hoare_1429296392585015714_VEBTi(one_one_assn,aa_nat5408703056993926195_VEBTi(vEBT_vebt_buildupi,Nb),aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,vEBT_vebt_buildup(Nb))) ).

% builupicorr
tff(fact_1305_divmod__step__eq,axiom,
    ! [L: num,Q2: int,R2: int] :
      aa_Pro6970504127699319548nt_int(unique5024387138958732305ep_int(L),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q2),R2)) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,L)),R2),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Q2)),one_one_int)),aa_int_int(aa_int_fun_int_int(minus_minus_int,R2),aa_num_int(numeral_numeral_int,L))),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Q2)),R2)) ).

% divmod_step_eq
tff(fact_1306_divmod__step__eq,axiom,
    ! [L: num,Q2: nat,R2: nat] :
      aa_Pro262675715343573244at_nat(unique5026877609467782581ep_nat(L),aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Q2),R2)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,L)),R2),aa_nat7603221163087113075at_nat(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))),Q2)),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,R2),aa_num_nat(numeral_numeral_nat,L))),aa_nat7603221163087113075at_nat(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))),Q2)),R2)) ).

% divmod_step_eq
tff(fact_1307_divmod__step__eq,axiom,
    ! [L: num,Q2: code_integer,R2: code_integer] :
      unique4921790084139445826nteger(L,aa_Cod6977894654539884864nteger(produc1086072967326762835nteger(Q2),R2)) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_num_Code_integer(numera6620942414471956472nteger,L)),R2),aa_Cod6977894654539884864nteger(produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Q2)),one_one_Code_integer)),aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(R2),aa_num_Code_integer(numera6620942414471956472nteger,L))),aa_Cod6977894654539884864nteger(produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Q2)),R2)) ).

% divmod_step_eq
tff(fact_1308_cnt__bound,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Ta),Nb)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_VEBT_VEBT_real(vEBT_VEBT_cnt,Ta)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,numeral_numeral_real(aa_num_num(bit0,one))),Nb)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,numeral_numeral_real(aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,one))))),numeral_numeral_real(aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,one)))))))) ) ).

% cnt_bound
tff(fact_1309_TBOUND__vebt__memberi,axiom,
    ! [Ta: vEBT_VEBT,Tib: vEBT_VEBTi,Xb: nat] : aa_nat_o(time_TBOUND_o(aa_nat2910048432523430448Heap_o(aa_VEB1455453220190609250Heap_o(aa_VEB8672304581938048793Heap_o(vEBT_V854960066525838166emberi,Ta),Tib),Xb)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_VEBT_height,Ta)))) ).

% TBOUND_vebt_memberi
tff(fact_1310_vebt__insert_Osimps_I4_J,axiom,
    ! [V: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Xb: nat] : vEBT_vebt_insert(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V)),TreeLista),Summarya),Xb) = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Xb),Xb)),aa_nat_nat(suc,aa_nat_nat(suc,V)),TreeLista),Summarya) ).

% vebt_insert.simps(4)
tff(fact_1311_two__realpow__ge__one,axiom,
    ! [Nb: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,numeral_numeral_real(aa_num_num(bit0,one))),Nb)) ).

% two_realpow_ge_one
tff(fact_1312_count__buildup_H,axiom,
    ! [Nb: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_VEBT_VEBT_real(vEBT_VEBT_cnt,vEBT_vebt_buildup(Nb))),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)))) ).

% count_buildup'
tff(fact_1313_heaphelp,axiom,
    ! [Xaa: array_VEBT_VEBTi,Tree_is: list_VEBT_VEBTi,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Xb: vEBT_VEBTi,Nb: nat,Xc: vEBT_VEBTi,H: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(rep_assn(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,
                aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_lis2508312872377092907i_assn(snga_assn_VEBT_VEBTi(Xaa),Tree_is)),aa_lis2508312872377092907i_assn(vEBT_L6296928887356842470_VEBTi(vEBT_vebt_assn_raw,TreeLista),Tree_is))),aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Summarya),Xb))),
                  aa_o_assn(pure_assn,
                    ( ( none_P1551326421579882414it_nat = none_P1551326421579882414it_nat )
                    & ( Nb = Nb ) )))),aa_o_assn(pure_assn,Xc = aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(none_P5556105721700978146at_nat,Nb,Xaa),Xb)))),H)
     => aa_Pro653082453696054616_nat_o(rep_assn(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Nb,TreeLista),Summarya)),Xc)),H) ) ).

% heaphelp
tff(fact_1314_heaphelp,axiom,
    ! [Xaa: array_VEBT_VEBTi,Tree_is: list_VEBT_VEBTi,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Xb: vEBT_VEBTi,Nb: nat,Xc: vEBT_VEBTi,H: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(rep_assn(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,
                aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_lis2508312872377092907i_assn(snga_assn_VEBT_VEBTi(Xaa),Tree_is)),aa_lis2508312872377092907i_assn(vEBT_L6296928887356842470_VEBTi(vEBT_vebt_assn_raw,TreeLista),Tree_is))),aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Summarya),Xb))),
                  aa_o_assn(pure_assn,
                    ( ( none_P7668321371905463026it_nat = none_P7668321371905463026it_nat )
                    & ( Nb = Nb ) )))),aa_o_assn(pure_assn,Xc = aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(none_P5556105721700978146at_nat,Nb,Xaa),Xb)))),H)
     => aa_Pro653082453696054616_nat_o(rep_assn(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Nb,TreeLista),Summarya)),Xc)),H) ) ).

% heaphelp
tff(fact_1315_heaphelp,axiom,
    ! [Xaa: array_VEBT_VEBTi,Tree_is: list_VEBT_VEBTi,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Xb: vEBT_VEBTi,Nb: nat,Xc: vEBT_VEBTi,H: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(rep_assn(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,
                aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_lis2508312872377092907i_assn(snga_assn_VEBT_VEBTi(Xaa),Tree_is)),aa_lis2508312872377092907i_assn(vEBT_L6296928887356842470_VEBTi(vEBT_vebt_assn_raw,TreeLista),Tree_is))),aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Summarya),Xb))),
                  aa_o_assn(pure_assn,
                    ( ( none_P5556105721700978146at_nat = none_P5556105721700978146at_nat )
                    & ( Nb = Nb ) )))),aa_o_assn(pure_assn,Xc = aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(none_P5556105721700978146at_nat,Nb,Xaa),Xb)))),H)
     => aa_Pro653082453696054616_nat_o(rep_assn(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Nb,TreeLista),Summarya)),Xc)),H) ) ).

% heaphelp
tff(fact_1316_heaphelp,axiom,
    ! [Xaa: array_VEBT_VEBTi,Tree_is: list_VEBT_VEBTi,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Xb: vEBT_VEBTi,Nb: nat,Xc: vEBT_VEBTi,H: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(rep_assn(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,
                aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_lis2508312872377092907i_assn(snga_assn_VEBT_VEBTi(Xaa),Tree_is)),aa_lis2508312872377092907i_assn(vEBT_L6296928887356842470_VEBTi(vEBT_vebt_assn_raw,TreeLista),Tree_is))),aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Summarya),Xb))),
                  aa_o_assn(pure_assn,
                    ( ( none_num = none_num )
                    & ( Nb = Nb ) )))),aa_o_assn(pure_assn,Xc = aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(none_P5556105721700978146at_nat,Nb,Xaa),Xb)))),H)
     => aa_Pro653082453696054616_nat_o(rep_assn(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Nb,TreeLista),Summarya)),Xc)),H) ) ).

% heaphelp
tff(fact_1317_heaphelp,axiom,
    ! [Xaa: array_VEBT_VEBTi,Tree_is: list_VEBT_VEBTi,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Xb: vEBT_VEBTi,Nb: nat,Xc: vEBT_VEBTi,H: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(rep_assn(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,
                aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_lis2508312872377092907i_assn(snga_assn_VEBT_VEBTi(Xaa),Tree_is)),aa_lis2508312872377092907i_assn(vEBT_L6296928887356842470_VEBTi(vEBT_vebt_assn_raw,TreeLista),Tree_is))),aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Summarya),Xb))),
                  aa_o_assn(pure_assn,
                    ( ( none_nat = none_nat )
                    & ( Nb = Nb ) )))),aa_o_assn(pure_assn,Xc = aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(none_P5556105721700978146at_nat,Nb,Xaa),Xb)))),H)
     => aa_Pro653082453696054616_nat_o(rep_assn(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Nb,TreeLista),Summarya)),Xc)),H) ) ).

% heaphelp
tff(fact_1318_div__exp__eq,axiom,
    ! [A: uint32,M: nat,Nb: nat] : aa_uint32_uint32(aa_uin3778660748068847871uint32(divide_divide_uint32,aa_uint32_uint32(aa_uin3778660748068847871uint32(divide_divide_uint32,A),aa_nat_uint32(power_power_uint32(numera9087168376688890119uint32(aa_num_num(bit0,one))),M))),aa_nat_uint32(power_power_uint32(numera9087168376688890119uint32(aa_num_num(bit0,one))),Nb)) = aa_uint32_uint32(aa_uin3778660748068847871uint32(divide_divide_uint32,A),aa_nat_uint32(power_power_uint32(numera9087168376688890119uint32(aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Nb))) ).

% div_exp_eq
tff(fact_1319_div__exp__eq,axiom,
    ! [A: word_N3645301735248828278l_num1,M: nat,Nb: nat] : divide1791077408188789448l_num1(divide1791077408188789448l_num1(A,aa_nat5599461682457136004l_num1(power_2184487114949457152l_num1(numera7442385471795722001l_num1(aa_num_num(bit0,one))),M)),aa_nat5599461682457136004l_num1(power_2184487114949457152l_num1(numera7442385471795722001l_num1(aa_num_num(bit0,one))),Nb)) = divide1791077408188789448l_num1(A,aa_nat5599461682457136004l_num1(power_2184487114949457152l_num1(numera7442385471795722001l_num1(aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Nb))) ).

% div_exp_eq
tff(fact_1320_div__exp__eq,axiom,
    ! [A: nat,M: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Nb))) ).

% div_exp_eq
tff(fact_1321_div__exp__eq,axiom,
    ! [A: int,M: nat,Nb: nat] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_nat_int(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))),Nb)) = 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(plus_plus_nat,M),Nb))) ).

% div_exp_eq
tff(fact_1322_semiring__norm_I90_J,axiom,
    ! [M: num,Nb: num] :
      ( ( aa_num_num(bit1,M) = aa_num_num(bit1,Nb) )
    <=> ( M = Nb ) ) ).

% semiring_norm(90)
tff(fact_1323_of__nat__eq__iff,axiom,
    ! [M: nat,Nb: nat] :
      ( ( aa_nat_real(semiri5074537144036343181t_real,M) = aa_nat_real(semiri5074537144036343181t_real,Nb) )
    <=> ( M = Nb ) ) ).

% of_nat_eq_iff
tff(fact_1324_of__nat__eq__iff,axiom,
    ! [M: nat,Nb: nat] :
      ( ( aa_nat_int(semiri1314217659103216013at_int,M) = aa_nat_int(semiri1314217659103216013at_int,Nb) )
    <=> ( M = Nb ) ) ).

% of_nat_eq_iff
tff(fact_1325_of__nat__eq__iff,axiom,
    ! [M: nat,Nb: nat] :
      ( ( aa_nat_nat(semiri1316708129612266289at_nat,M) = aa_nat_nat(semiri1316708129612266289at_nat,Nb) )
    <=> ( M = Nb ) ) ).

% of_nat_eq_iff
tff(fact_1326_of__nat__eq__iff,axiom,
    ! [M: nat,Nb: nat] :
      ( ( aa_nat_Code_integer(semiri4939895301339042750nteger,M) = aa_nat_Code_integer(semiri4939895301339042750nteger,Nb) )
    <=> ( M = Nb ) ) ).

% of_nat_eq_iff
tff(fact_1327_two__realpow__ge__two,axiom,
    ! [Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_nat_real(semiri5074537144036343181t_real,Nb))
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,numeral_numeral_real(aa_num_num(bit0,one))),aa_nat_real(aa_real_fun_nat_real(power_power_real,numeral_numeral_real(aa_num_num(bit0,one))),Nb)) ) ).

% two_realpow_ge_two
tff(fact_1328_bits__div__by__1,axiom,
    ! [A: word_N3645301735248828278l_num1] : divide1791077408188789448l_num1(A,one_on7727431528512463931l_num1) = A ).

% bits_div_by_1
tff(fact_1329_bits__div__by__1,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),one_one_nat) = A ).

% bits_div_by_1
tff(fact_1330_bits__div__by__1,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,A),one_one_int) = A ).

% bits_div_by_1
tff(fact_1331_semiring__norm_I88_J,axiom,
    ! [M: num,Nb: num] : aa_num_num(bit0,M) != aa_num_num(bit1,Nb) ).

% semiring_norm(88)
tff(fact_1332_semiring__norm_I89_J,axiom,
    ! [M: num,Nb: num] : aa_num_num(bit1,M) != aa_num_num(bit0,Nb) ).

% semiring_norm(89)
tff(fact_1333_semiring__norm_I84_J,axiom,
    ! [Nb: num] : one != aa_num_num(bit1,Nb) ).

% semiring_norm(84)
tff(fact_1334_semiring__norm_I86_J,axiom,
    ! [M: num] : aa_num_num(bit1,M) != one ).

% semiring_norm(86)
tff(fact_1335_semiring__norm_I73_J,axiom,
    ! [M: num,Nb: num] :
      ( aa_num_o(ord_less_eq_num(aa_num_num(bit1,M)),aa_num_num(bit1,Nb))
    <=> aa_num_o(ord_less_eq_num(M),Nb) ) ).

% semiring_norm(73)
tff(fact_1336_semiring__norm_I80_J,axiom,
    ! [M: num,Nb: num] :
      ( aa_num_o(ord_less_num(aa_num_num(bit1,M)),aa_num_num(bit1,Nb))
    <=> aa_num_o(ord_less_num(M),Nb) ) ).

% semiring_norm(80)
tff(fact_1337_of__nat__numeral,axiom,
    ! [Nb: num] : aa_nat5599461682457136004l_num1(semiri8819519690708144855l_num1,aa_num_nat(numeral_numeral_nat,Nb)) = numera7442385471795722001l_num1(Nb) ).

% of_nat_numeral
tff(fact_1338_of__nat__numeral,axiom,
    ! [Nb: num] : aa_nat_rat(semiri681578069525770553at_rat,aa_num_nat(numeral_numeral_nat,Nb)) = numeral_numeral_rat(Nb) ).

% of_nat_numeral
tff(fact_1339_of__nat__numeral,axiom,
    ! [Nb: num] : aa_nat_real(semiri5074537144036343181t_real,aa_num_nat(numeral_numeral_nat,Nb)) = numeral_numeral_real(Nb) ).

% of_nat_numeral
tff(fact_1340_of__nat__numeral,axiom,
    ! [Nb: num] : aa_nat_int(semiri1314217659103216013at_int,aa_num_nat(numeral_numeral_nat,Nb)) = aa_num_int(numeral_numeral_int,Nb) ).

% of_nat_numeral
tff(fact_1341_of__nat__numeral,axiom,
    ! [Nb: num] : aa_nat_nat(semiri1316708129612266289at_nat,aa_num_nat(numeral_numeral_nat,Nb)) = aa_num_nat(numeral_numeral_nat,Nb) ).

% of_nat_numeral
tff(fact_1342_of__nat__numeral,axiom,
    ! [Nb: num] : aa_nat_Code_integer(semiri4939895301339042750nteger,aa_num_nat(numeral_numeral_nat,Nb)) = aa_num_Code_integer(numera6620942414471956472nteger,Nb) ).

% of_nat_numeral
tff(fact_1343_of__nat__less__iff,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,M)),aa_nat_rat(semiri681578069525770553at_rat,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb) ) ).

% of_nat_less_iff
tff(fact_1344_of__nat__less__iff,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,M)),aa_nat_real(semiri5074537144036343181t_real,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb) ) ).

% of_nat_less_iff
tff(fact_1345_of__nat__less__iff,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb) ) ).

% of_nat_less_iff
tff(fact_1346_of__nat__less__iff,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb) ) ).

% of_nat_less_iff
tff(fact_1347_of__nat__less__iff,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,M)),aa_nat_Code_integer(semiri4939895301339042750nteger,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb) ) ).

% of_nat_less_iff
tff(fact_1348_of__nat__le__iff,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,M)),aa_nat_real(semiri5074537144036343181t_real,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb) ) ).

% of_nat_le_iff
tff(fact_1349_of__nat__le__iff,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,M)),aa_nat_Code_integer(semiri4939895301339042750nteger,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb) ) ).

% of_nat_le_iff
tff(fact_1350_of__nat__le__iff,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(semiri681578069525770553at_rat,M)),aa_nat_rat(semiri681578069525770553at_rat,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb) ) ).

% of_nat_le_iff
tff(fact_1351_of__nat__le__iff,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb) ) ).

% of_nat_le_iff
tff(fact_1352_of__nat__le__iff,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb) ) ).

% of_nat_le_iff
tff(fact_1353_of__nat__add,axiom,
    ! [M: nat,Nb: nat] : aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Nb)) = aa_rat_rat(plus_plus_rat(aa_nat_rat(semiri681578069525770553at_rat,M)),aa_nat_rat(semiri681578069525770553at_rat,Nb)) ).

% of_nat_add
tff(fact_1354_of__nat__add,axiom,
    ! [M: nat,Nb: nat] : aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Nb)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(semiri5074537144036343181t_real,M)),aa_nat_real(semiri5074537144036343181t_real,Nb)) ).

% of_nat_add
tff(fact_1355_of__nat__add,axiom,
    ! [M: nat,Nb: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Nb)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,Nb)) ).

% of_nat_add
tff(fact_1356_of__nat__add,axiom,
    ! [M: nat,Nb: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,Nb)) ).

% of_nat_add
tff(fact_1357_of__nat__add,axiom,
    ! [M: nat,Nb: nat] : aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Nb)) = aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,M)),aa_nat_Code_integer(semiri4939895301339042750nteger,Nb)) ).

% of_nat_add
tff(fact_1358_of__nat__eq__1__iff,axiom,
    ! [Nb: nat] :
      ( ( aa_nat_rat(semiri681578069525770553at_rat,Nb) = one_one_rat )
    <=> ( Nb = one_one_nat ) ) ).

% of_nat_eq_1_iff
tff(fact_1359_of__nat__eq__1__iff,axiom,
    ! [Nb: nat] :
      ( ( aa_nat_real(semiri5074537144036343181t_real,Nb) = one_one_real )
    <=> ( Nb = one_one_nat ) ) ).

% of_nat_eq_1_iff
tff(fact_1360_of__nat__eq__1__iff,axiom,
    ! [Nb: nat] :
      ( ( aa_nat_int(semiri1314217659103216013at_int,Nb) = one_one_int )
    <=> ( Nb = one_one_nat ) ) ).

% of_nat_eq_1_iff
tff(fact_1361_of__nat__eq__1__iff,axiom,
    ! [Nb: nat] :
      ( ( aa_nat_nat(semiri1316708129612266289at_nat,Nb) = one_one_nat )
    <=> ( Nb = one_one_nat ) ) ).

% of_nat_eq_1_iff
tff(fact_1362_of__nat__eq__1__iff,axiom,
    ! [Nb: nat] :
      ( ( aa_nat_Code_integer(semiri4939895301339042750nteger,Nb) = one_one_Code_integer )
    <=> ( Nb = one_one_nat ) ) ).

% of_nat_eq_1_iff
tff(fact_1363_of__nat__1__eq__iff,axiom,
    ! [Nb: nat] :
      ( ( one_one_rat = aa_nat_rat(semiri681578069525770553at_rat,Nb) )
    <=> ( Nb = one_one_nat ) ) ).

% of_nat_1_eq_iff
tff(fact_1364_of__nat__1__eq__iff,axiom,
    ! [Nb: nat] :
      ( ( one_one_real = aa_nat_real(semiri5074537144036343181t_real,Nb) )
    <=> ( Nb = one_one_nat ) ) ).

% of_nat_1_eq_iff
tff(fact_1365_of__nat__1__eq__iff,axiom,
    ! [Nb: nat] :
      ( ( one_one_int = aa_nat_int(semiri1314217659103216013at_int,Nb) )
    <=> ( Nb = one_one_nat ) ) ).

% of_nat_1_eq_iff
tff(fact_1366_of__nat__1__eq__iff,axiom,
    ! [Nb: nat] :
      ( ( one_one_nat = aa_nat_nat(semiri1316708129612266289at_nat,Nb) )
    <=> ( Nb = one_one_nat ) ) ).

% of_nat_1_eq_iff
tff(fact_1367_of__nat__1__eq__iff,axiom,
    ! [Nb: nat] :
      ( ( one_one_Code_integer = aa_nat_Code_integer(semiri4939895301339042750nteger,Nb) )
    <=> ( Nb = one_one_nat ) ) ).

% of_nat_1_eq_iff
tff(fact_1368_of__nat__1,axiom,
    aa_nat5599461682457136004l_num1(semiri8819519690708144855l_num1,one_one_nat) = one_on7727431528512463931l_num1 ).

% of_nat_1
tff(fact_1369_of__nat__1,axiom,
    aa_nat_rat(semiri681578069525770553at_rat,one_one_nat) = one_one_rat ).

% of_nat_1
tff(fact_1370_of__nat__1,axiom,
    aa_nat_real(semiri5074537144036343181t_real,one_one_nat) = one_one_real ).

% of_nat_1
tff(fact_1371_of__nat__1,axiom,
    aa_nat_int(semiri1314217659103216013at_int,one_one_nat) = one_one_int ).

% of_nat_1
tff(fact_1372_of__nat__1,axiom,
    aa_nat_nat(semiri1316708129612266289at_nat,one_one_nat) = one_one_nat ).

% of_nat_1
tff(fact_1373_of__nat__1,axiom,
    aa_nat_Code_integer(semiri4939895301339042750nteger,one_one_nat) = one_one_Code_integer ).

% of_nat_1
tff(fact_1374_of__nat__mult,axiom,
    ! [M: nat,Nb: nat] : aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Nb)) = aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,M)),aa_nat_rat(semiri681578069525770553at_rat,Nb)) ).

% of_nat_mult
tff(fact_1375_of__nat__mult,axiom,
    ! [M: nat,Nb: nat] : aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Nb)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,M)),aa_nat_real(semiri5074537144036343181t_real,Nb)) ).

% of_nat_mult
tff(fact_1376_of__nat__mult,axiom,
    ! [M: nat,Nb: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Nb)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,Nb)) ).

% of_nat_mult
tff(fact_1377_of__nat__mult,axiom,
    ! [M: nat,Nb: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,Nb)) ).

% of_nat_mult
tff(fact_1378_of__nat__mult,axiom,
    ! [M: nat,Nb: nat] : aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Nb)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,M)),aa_nat_Code_integer(semiri4939895301339042750nteger,Nb)) ).

% of_nat_mult
tff(fact_1379_of__nat__power__eq__of__nat__cancel__iff,axiom,
    ! [Xb: nat,B: nat,W: nat] :
      ( ( aa_nat_real(semiri5074537144036343181t_real,Xb) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(semiri5074537144036343181t_real,B)),W) )
    <=> ( Xb = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W) ) ) ).

% of_nat_power_eq_of_nat_cancel_iff
tff(fact_1380_of__nat__power__eq__of__nat__cancel__iff,axiom,
    ! [Xb: nat,B: nat,W: nat] :
      ( ( aa_nat_int(semiri1314217659103216013at_int,Xb) = aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_nat_int(semiri1314217659103216013at_int,B)),W) )
    <=> ( Xb = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W) ) ) ).

% of_nat_power_eq_of_nat_cancel_iff
tff(fact_1381_of__nat__power__eq__of__nat__cancel__iff,axiom,
    ! [Xb: nat,B: nat,W: nat] :
      ( ( aa_nat_nat(semiri1316708129612266289at_nat,Xb) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(semiri1316708129612266289at_nat,B)),W) )
    <=> ( Xb = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W) ) ) ).

% of_nat_power_eq_of_nat_cancel_iff
tff(fact_1382_of__nat__power__eq__of__nat__cancel__iff,axiom,
    ! [Xb: nat,B: nat,W: nat] :
      ( ( aa_nat_Code_integer(semiri4939895301339042750nteger,Xb) = aa_nat_Code_integer(power_8256067586552552935nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,B)),W) )
    <=> ( Xb = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W) ) ) ).

% of_nat_power_eq_of_nat_cancel_iff
tff(fact_1383_of__nat__eq__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,Xb: nat] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(semiri5074537144036343181t_real,B)),W) = aa_nat_real(semiri5074537144036343181t_real,Xb) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W) = Xb ) ) ).

% of_nat_eq_of_nat_power_cancel_iff
tff(fact_1384_of__nat__eq__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,Xb: nat] :
      ( ( aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_nat_int(semiri1314217659103216013at_int,B)),W) = aa_nat_int(semiri1314217659103216013at_int,Xb) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W) = Xb ) ) ).

% of_nat_eq_of_nat_power_cancel_iff
tff(fact_1385_of__nat__eq__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,Xb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(semiri1316708129612266289at_nat,B)),W) = aa_nat_nat(semiri1316708129612266289at_nat,Xb) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W) = Xb ) ) ).

% of_nat_eq_of_nat_power_cancel_iff
tff(fact_1386_of__nat__eq__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,Xb: nat] :
      ( ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,B)),W) = aa_nat_Code_integer(semiri4939895301339042750nteger,Xb) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W) = Xb ) ) ).

% of_nat_eq_of_nat_power_cancel_iff
tff(fact_1387_semiring__1__class_Oof__nat__power,axiom,
    ! [M: nat,Nb: nat] : aa_nat_uint32(semiri2565882477558803405uint32,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,M),Nb)) = aa_nat_uint32(power_power_uint32(aa_nat_uint32(semiri2565882477558803405uint32,M)),Nb) ).

% semiring_1_class.of_nat_power
tff(fact_1388_semiring__1__class_Oof__nat__power,axiom,
    ! [M: nat,Nb: nat] : aa_nat5599461682457136004l_num1(semiri8819519690708144855l_num1,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,M),Nb)) = aa_nat5599461682457136004l_num1(power_2184487114949457152l_num1(aa_nat5599461682457136004l_num1(semiri8819519690708144855l_num1,M)),Nb) ).

% semiring_1_class.of_nat_power
tff(fact_1389_semiring__1__class_Oof__nat__power,axiom,
    ! [M: nat,Nb: nat] : aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,M),Nb)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(semiri5074537144036343181t_real,M)),Nb) ).

% semiring_1_class.of_nat_power
tff(fact_1390_semiring__1__class_Oof__nat__power,axiom,
    ! [M: nat,Nb: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,M),Nb)) = aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_nat_int(semiri1314217659103216013at_int,M)),Nb) ).

% semiring_1_class.of_nat_power
tff(fact_1391_semiring__1__class_Oof__nat__power,axiom,
    ! [M: nat,Nb: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,M),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),Nb) ).

% semiring_1_class.of_nat_power
tff(fact_1392_semiring__1__class_Oof__nat__power,axiom,
    ! [M: nat,Nb: nat] : aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,M),Nb)) = aa_nat_Code_integer(power_8256067586552552935nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,M)),Nb) ).

% semiring_1_class.of_nat_power
tff(fact_1393_semiring__norm_I7_J,axiom,
    ! [M: num,Nb: num] : aa_num_num(aa_num_fun_num_num(plus_plus_num,aa_num_num(bit0,M)),aa_num_num(bit1,Nb)) = aa_num_num(bit1,aa_num_num(aa_num_fun_num_num(plus_plus_num,M),Nb)) ).

% semiring_norm(7)
tff(fact_1394_semiring__norm_I9_J,axiom,
    ! [M: num,Nb: num] : aa_num_num(aa_num_fun_num_num(plus_plus_num,aa_num_num(bit1,M)),aa_num_num(bit0,Nb)) = aa_num_num(bit1,aa_num_num(aa_num_fun_num_num(plus_plus_num,M),Nb)) ).

% semiring_norm(9)
tff(fact_1395_semiring__norm_I15_J,axiom,
    ! [M: num,Nb: num] : aa_num_num(aa_num_fun_num_num(times_times_num,aa_num_num(bit1,M)),aa_num_num(bit0,Nb)) = aa_num_num(bit0,aa_num_num(aa_num_fun_num_num(times_times_num,aa_num_num(bit1,M)),Nb)) ).

% semiring_norm(15)
tff(fact_1396_semiring__norm_I14_J,axiom,
    ! [M: num,Nb: num] : aa_num_num(aa_num_fun_num_num(times_times_num,aa_num_num(bit0,M)),aa_num_num(bit1,Nb)) = aa_num_num(bit0,aa_num_num(aa_num_fun_num_num(times_times_num,M),aa_num_num(bit1,Nb))) ).

% semiring_norm(14)
tff(fact_1397_semiring__norm_I81_J,axiom,
    ! [M: num,Nb: num] :
      ( aa_num_o(ord_less_num(aa_num_num(bit1,M)),aa_num_num(bit0,Nb))
    <=> aa_num_o(ord_less_num(M),Nb) ) ).

% semiring_norm(81)
tff(fact_1398_semiring__norm_I72_J,axiom,
    ! [M: num,Nb: num] :
      ( aa_num_o(ord_less_eq_num(aa_num_num(bit0,M)),aa_num_num(bit1,Nb))
    <=> aa_num_o(ord_less_eq_num(M),Nb) ) ).

% semiring_norm(72)
tff(fact_1399_semiring__norm_I77_J,axiom,
    ! [Nb: num] : aa_num_o(ord_less_num(one),aa_num_num(bit1,Nb)) ).

% semiring_norm(77)
tff(fact_1400_semiring__norm_I70_J,axiom,
    ! [M: num] : ~ aa_num_o(ord_less_eq_num(aa_num_num(bit1,M)),one) ).

% semiring_norm(70)
tff(fact_1401_of__nat__Suc,axiom,
    ! [M: nat] : aa_nat5599461682457136004l_num1(semiri8819519690708144855l_num1,aa_nat_nat(suc,M)) = plus_p361126936061061375l_num1(one_on7727431528512463931l_num1,aa_nat5599461682457136004l_num1(semiri8819519690708144855l_num1,M)) ).

% of_nat_Suc
tff(fact_1402_of__nat__Suc,axiom,
    ! [M: nat] : aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,M)) = aa_rat_rat(plus_plus_rat(one_one_rat),aa_nat_rat(semiri681578069525770553at_rat,M)) ).

% of_nat_Suc
tff(fact_1403_of__nat__Suc,axiom,
    ! [M: nat] : aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,M)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_nat_real(semiri5074537144036343181t_real,M)) ).

% of_nat_Suc
tff(fact_1404_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_1405_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_1406_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_1407_zdiv__numeral__Bit1,axiom,
    ! [V: num,W: num] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,V))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,W))) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_num_int(numeral_numeral_int,V)),aa_num_int(numeral_numeral_int,W)) ).

% zdiv_numeral_Bit1
tff(fact_1408_real__of__nat__less__numeral__iff,axiom,
    ! [Nb: nat,W: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),numeral_numeral_real(W))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_num_nat(numeral_numeral_nat,W)) ) ).

% real_of_nat_less_numeral_iff
tff(fact_1409_numeral__less__real__of__nat__iff,axiom,
    ! [W: num,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,numeral_numeral_real(W)),aa_nat_real(semiri5074537144036343181t_real,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_num_nat(numeral_numeral_nat,W)),Nb) ) ).

% numeral_less_real_of_nat_iff
tff(fact_1410_semiring__norm_I3_J,axiom,
    ! [Nb: num] : aa_num_num(aa_num_fun_num_num(plus_plus_num,one),aa_num_num(bit0,Nb)) = aa_num_num(bit1,Nb) ).

% semiring_norm(3)
tff(fact_1411_semiring__norm_I4_J,axiom,
    ! [Nb: num] : aa_num_num(aa_num_fun_num_num(plus_plus_num,one),aa_num_num(bit1,Nb)) = aa_num_num(bit0,aa_num_num(aa_num_fun_num_num(plus_plus_num,Nb),one)) ).

% semiring_norm(4)
tff(fact_1412_semiring__norm_I5_J,axiom,
    ! [M: num] : aa_num_num(aa_num_fun_num_num(plus_plus_num,aa_num_num(bit0,M)),one) = aa_num_num(bit1,M) ).

% semiring_norm(5)
tff(fact_1413_semiring__norm_I8_J,axiom,
    ! [M: num] : aa_num_num(aa_num_fun_num_num(plus_plus_num,aa_num_num(bit1,M)),one) = aa_num_num(bit0,aa_num_num(aa_num_fun_num_num(plus_plus_num,M),one)) ).

% semiring_norm(8)
tff(fact_1414_semiring__norm_I10_J,axiom,
    ! [M: num,Nb: num] : aa_num_num(aa_num_fun_num_num(plus_plus_num,aa_num_num(bit1,M)),aa_num_num(bit1,Nb)) = aa_num_num(bit0,aa_num_num(aa_num_fun_num_num(plus_plus_num,aa_num_num(aa_num_fun_num_num(plus_plus_num,M),Nb)),one)) ).

% semiring_norm(10)
tff(fact_1415_semiring__norm_I16_J,axiom,
    ! [M: num,Nb: num] : aa_num_num(aa_num_fun_num_num(times_times_num,aa_num_num(bit1,M)),aa_num_num(bit1,Nb)) = aa_num_num(bit1,aa_num_num(aa_num_fun_num_num(plus_plus_num,aa_num_num(aa_num_fun_num_num(plus_plus_num,M),Nb)),aa_num_num(bit0,aa_num_num(aa_num_fun_num_num(times_times_num,M),Nb)))) ).

% semiring_norm(16)
tff(fact_1416_semiring__norm_I79_J,axiom,
    ! [M: num,Nb: num] :
      ( aa_num_o(ord_less_num(aa_num_num(bit0,M)),aa_num_num(bit1,Nb))
    <=> aa_num_o(ord_less_eq_num(M),Nb) ) ).

% semiring_norm(79)
tff(fact_1417_semiring__norm_I74_J,axiom,
    ! [M: num,Nb: num] :
      ( aa_num_o(ord_less_eq_num(aa_num_num(bit1,M)),aa_num_num(bit0,Nb))
    <=> aa_num_o(ord_less_num(M),Nb) ) ).

% semiring_norm(74)
tff(fact_1418_real__of__nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y: nat,Xb: num,Nb: nat] :
      ( ( aa_nat_rat(semiri681578069525770553at_rat,Y) = aa_nat_rat(power_power_rat(numeral_numeral_rat(Xb)),Nb) )
    <=> ( Y = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xb)),Nb) ) ) ).

% real_of_nat_eq_numeral_power_cancel_iff
tff(fact_1419_real__of__nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y: nat,Xb: num,Nb: nat] :
      ( ( aa_nat_real(semiri5074537144036343181t_real,Y) = aa_nat_real(aa_real_fun_nat_real(power_power_real,numeral_numeral_real(Xb)),Nb) )
    <=> ( Y = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xb)),Nb) ) ) ).

% real_of_nat_eq_numeral_power_cancel_iff
tff(fact_1420_real__of__nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y: nat,Xb: num,Nb: nat] :
      ( ( aa_nat_int(semiri1314217659103216013at_int,Y) = aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,Xb)),Nb) )
    <=> ( Y = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xb)),Nb) ) ) ).

% real_of_nat_eq_numeral_power_cancel_iff
tff(fact_1421_real__of__nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y: nat,Xb: num,Nb: nat] :
      ( ( aa_nat_nat(semiri1316708129612266289at_nat,Y) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xb)),Nb) )
    <=> ( Y = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xb)),Nb) ) ) ).

% real_of_nat_eq_numeral_power_cancel_iff
tff(fact_1422_real__of__nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y: nat,Xb: num,Nb: nat] :
      ( ( aa_nat_Code_integer(semiri4939895301339042750nteger,Y) = aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,Xb)),Nb) )
    <=> ( Y = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xb)),Nb) ) ) ).

% real_of_nat_eq_numeral_power_cancel_iff
tff(fact_1423_numeral__power__eq__of__nat__cancel__iff,axiom,
    ! [Xb: num,Nb: nat,Y: nat] :
      ( ( aa_nat_rat(power_power_rat(numeral_numeral_rat(Xb)),Nb) = aa_nat_rat(semiri681578069525770553at_rat,Y) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xb)),Nb) = Y ) ) ).

% numeral_power_eq_of_nat_cancel_iff
tff(fact_1424_numeral__power__eq__of__nat__cancel__iff,axiom,
    ! [Xb: num,Nb: nat,Y: nat] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,numeral_numeral_real(Xb)),Nb) = aa_nat_real(semiri5074537144036343181t_real,Y) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xb)),Nb) = Y ) ) ).

% numeral_power_eq_of_nat_cancel_iff
tff(fact_1425_numeral__power__eq__of__nat__cancel__iff,axiom,
    ! [Xb: num,Nb: nat,Y: nat] :
      ( ( aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,Xb)),Nb) = aa_nat_int(semiri1314217659103216013at_int,Y) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xb)),Nb) = Y ) ) ).

% numeral_power_eq_of_nat_cancel_iff
tff(fact_1426_numeral__power__eq__of__nat__cancel__iff,axiom,
    ! [Xb: num,Nb: nat,Y: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xb)),Nb) = aa_nat_nat(semiri1316708129612266289at_nat,Y) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xb)),Nb) = Y ) ) ).

% numeral_power_eq_of_nat_cancel_iff
tff(fact_1427_numeral__power__eq__of__nat__cancel__iff,axiom,
    ! [Xb: num,Nb: nat,Y: nat] :
      ( ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,Xb)),Nb) = aa_nat_Code_integer(semiri4939895301339042750nteger,Y) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xb)),Nb) = Y ) ) ).

% numeral_power_eq_of_nat_cancel_iff
tff(fact_1428_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [Xb: nat,B: nat,W: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,Xb)),aa_nat_rat(power_power_rat(aa_nat_rat(semiri681578069525770553at_rat,B)),W))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W)) ) ).

% of_nat_power_less_of_nat_cancel_iff
tff(fact_1429_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [Xb: nat,B: nat,W: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,Xb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(semiri5074537144036343181t_real,B)),W))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W)) ) ).

% of_nat_power_less_of_nat_cancel_iff
tff(fact_1430_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [Xb: nat,B: nat,W: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,Xb)),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_nat_int(semiri1314217659103216013at_int,B)),W))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W)) ) ).

% of_nat_power_less_of_nat_cancel_iff
tff(fact_1431_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [Xb: nat,B: nat,W: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(semiri1316708129612266289at_nat,Xb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(semiri1316708129612266289at_nat,B)),W))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W)) ) ).

% of_nat_power_less_of_nat_cancel_iff
tff(fact_1432_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [Xb: nat,B: nat,W: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,Xb)),aa_nat_Code_integer(power_8256067586552552935nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,B)),W))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W)) ) ).

% of_nat_power_less_of_nat_cancel_iff
tff(fact_1433_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,Xb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(aa_nat_rat(semiri681578069525770553at_rat,B)),W)),aa_nat_rat(semiri681578069525770553at_rat,Xb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W)),Xb) ) ).

% of_nat_less_of_nat_power_cancel_iff
tff(fact_1434_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,Xb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(semiri5074537144036343181t_real,B)),W)),aa_nat_real(semiri5074537144036343181t_real,Xb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W)),Xb) ) ).

% of_nat_less_of_nat_power_cancel_iff
tff(fact_1435_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,Xb: 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)),W)),aa_nat_int(semiri1314217659103216013at_int,Xb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W)),Xb) ) ).

% of_nat_less_of_nat_power_cancel_iff
tff(fact_1436_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,Xb: 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)),W)),aa_nat_nat(semiri1316708129612266289at_nat,Xb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W)),Xb) ) ).

% of_nat_less_of_nat_power_cancel_iff
tff(fact_1437_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,Xb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,B)),W)),aa_nat_Code_integer(semiri4939895301339042750nteger,Xb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W)),Xb) ) ).

% of_nat_less_of_nat_power_cancel_iff
tff(fact_1438_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [Xb: nat,B: nat,W: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,Xb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(semiri5074537144036343181t_real,B)),W))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W)) ) ).

% of_nat_power_le_of_nat_cancel_iff
tff(fact_1439_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [Xb: nat,B: nat,W: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,Xb)),aa_nat_Code_integer(power_8256067586552552935nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,B)),W))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W)) ) ).

% of_nat_power_le_of_nat_cancel_iff
tff(fact_1440_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [Xb: nat,B: nat,W: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(semiri681578069525770553at_rat,Xb)),aa_nat_rat(power_power_rat(aa_nat_rat(semiri681578069525770553at_rat,B)),W))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W)) ) ).

% of_nat_power_le_of_nat_cancel_iff
tff(fact_1441_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [Xb: nat,B: nat,W: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(semiri1316708129612266289at_nat,Xb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(semiri1316708129612266289at_nat,B)),W))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W)) ) ).

% of_nat_power_le_of_nat_cancel_iff
tff(fact_1442_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [Xb: nat,B: nat,W: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,Xb)),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_nat_int(semiri1314217659103216013at_int,B)),W))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W)) ) ).

% of_nat_power_le_of_nat_cancel_iff
tff(fact_1443_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,Xb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(semiri5074537144036343181t_real,B)),W)),aa_nat_real(semiri5074537144036343181t_real,Xb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W)),Xb) ) ).

% of_nat_le_of_nat_power_cancel_iff
tff(fact_1444_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,Xb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,B)),W)),aa_nat_Code_integer(semiri4939895301339042750nteger,Xb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W)),Xb) ) ).

% of_nat_le_of_nat_power_cancel_iff
tff(fact_1445_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,Xb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(aa_nat_rat(semiri681578069525770553at_rat,B)),W)),aa_nat_rat(semiri681578069525770553at_rat,Xb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W)),Xb) ) ).

% of_nat_le_of_nat_power_cancel_iff
tff(fact_1446_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,Xb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(semiri1316708129612266289at_nat,B)),W)),aa_nat_nat(semiri1316708129612266289at_nat,Xb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W)),Xb) ) ).

% of_nat_le_of_nat_power_cancel_iff
tff(fact_1447_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,Xb: 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)),W)),aa_nat_int(semiri1314217659103216013at_int,Xb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W)),Xb) ) ).

% of_nat_le_of_nat_power_cancel_iff
tff(fact_1448_numeral__le__real__of__nat__iff,axiom,
    ! [Nb: num,M: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,numeral_numeral_real(Nb)),aa_nat_real(semiri5074537144036343181t_real,M))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,Nb)),M) ) ).

% numeral_le_real_of_nat_iff
tff(fact_1449_Suc__div__eq__add3__div__numeral,axiom,
    ! [M: nat,V: num] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,M)))),aa_num_nat(numeral_numeral_nat,V)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),M)),aa_num_nat(numeral_numeral_nat,V)) ).

% Suc_div_eq_add3_div_numeral
tff(fact_1450_div__Suc__eq__div__add3,axiom,
    ! [M: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,Nb)))) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),Nb)) ).

% div_Suc_eq_div_add3
tff(fact_1451_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [Xb: nat,I: num,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,Xb)),aa_nat_rat(power_power_rat(numeral_numeral_rat(I)),Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Nb)) ) ).

% of_nat_less_numeral_power_cancel_iff
tff(fact_1452_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [Xb: nat,I: num,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,Xb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,numeral_numeral_real(I)),Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Nb)) ) ).

% of_nat_less_numeral_power_cancel_iff
tff(fact_1453_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [Xb: nat,I: num,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,Xb)),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,I)),Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Nb)) ) ).

% of_nat_less_numeral_power_cancel_iff
tff(fact_1454_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [Xb: nat,I: num,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(semiri1316708129612266289at_nat,Xb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Nb)) ) ).

% of_nat_less_numeral_power_cancel_iff
tff(fact_1455_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [Xb: nat,I: num,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,Xb)),aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,I)),Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Nb)) ) ).

% of_nat_less_numeral_power_cancel_iff
tff(fact_1456_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [I: num,Nb: nat,Xb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(numeral_numeral_rat(I)),Nb)),aa_nat_rat(semiri681578069525770553at_rat,Xb))
    <=> 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,I)),Nb)),Xb) ) ).

% numeral_power_less_of_nat_cancel_iff
tff(fact_1457_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [I: num,Nb: nat,Xb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,numeral_numeral_real(I)),Nb)),aa_nat_real(semiri5074537144036343181t_real,Xb))
    <=> 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,I)),Nb)),Xb) ) ).

% numeral_power_less_of_nat_cancel_iff
tff(fact_1458_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [I: num,Nb: nat,Xb: 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,I)),Nb)),aa_nat_int(semiri1314217659103216013at_int,Xb))
    <=> 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,I)),Nb)),Xb) ) ).

% numeral_power_less_of_nat_cancel_iff
tff(fact_1459_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [I: num,Nb: nat,Xb: 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,I)),Nb)),aa_nat_nat(semiri1316708129612266289at_nat,Xb))
    <=> 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,I)),Nb)),Xb) ) ).

% numeral_power_less_of_nat_cancel_iff
tff(fact_1460_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [I: num,Nb: nat,Xb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,I)),Nb)),aa_nat_Code_integer(semiri4939895301339042750nteger,Xb))
    <=> 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,I)),Nb)),Xb) ) ).

% numeral_power_less_of_nat_cancel_iff
tff(fact_1461_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [Xb: nat,I: num,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,Xb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,numeral_numeral_real(I)),Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Nb)) ) ).

% of_nat_le_numeral_power_cancel_iff
tff(fact_1462_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [Xb: nat,I: num,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,Xb)),aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,I)),Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Nb)) ) ).

% of_nat_le_numeral_power_cancel_iff
tff(fact_1463_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [Xb: nat,I: num,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(semiri681578069525770553at_rat,Xb)),aa_nat_rat(power_power_rat(numeral_numeral_rat(I)),Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Nb)) ) ).

% of_nat_le_numeral_power_cancel_iff
tff(fact_1464_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [Xb: nat,I: num,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(semiri1316708129612266289at_nat,Xb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Nb)) ) ).

% of_nat_le_numeral_power_cancel_iff
tff(fact_1465_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [Xb: nat,I: num,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,Xb)),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,I)),Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Nb)) ) ).

% of_nat_le_numeral_power_cancel_iff
tff(fact_1466_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I: num,Nb: nat,Xb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,numeral_numeral_real(I)),Nb)),aa_nat_real(semiri5074537144036343181t_real,Xb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Nb)),Xb) ) ).

% numeral_power_le_of_nat_cancel_iff
tff(fact_1467_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I: num,Nb: nat,Xb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,I)),Nb)),aa_nat_Code_integer(semiri4939895301339042750nteger,Xb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Nb)),Xb) ) ).

% numeral_power_le_of_nat_cancel_iff
tff(fact_1468_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I: num,Nb: nat,Xb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(numeral_numeral_rat(I)),Nb)),aa_nat_rat(semiri681578069525770553at_rat,Xb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Nb)),Xb) ) ).

% numeral_power_le_of_nat_cancel_iff
tff(fact_1469_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I: num,Nb: nat,Xb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Nb)),aa_nat_nat(semiri1316708129612266289at_nat,Xb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Nb)),Xb) ) ).

% numeral_power_le_of_nat_cancel_iff
tff(fact_1470_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I: num,Nb: nat,Xb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,I)),Nb)),aa_nat_int(semiri1314217659103216013at_int,Xb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Nb)),Xb) ) ).

% numeral_power_le_of_nat_cancel_iff
tff(fact_1471_mult__of__nat__commute,axiom,
    ! [Xb: nat,Y: rat] : aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,Xb)),Y) = aa_rat_rat(times_times_rat(Y),aa_nat_rat(semiri681578069525770553at_rat,Xb)) ).

% mult_of_nat_commute
tff(fact_1472_mult__of__nat__commute,axiom,
    ! [Xb: nat,Y: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Xb)),Y) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),aa_nat_real(semiri5074537144036343181t_real,Xb)) ).

% mult_of_nat_commute
tff(fact_1473_mult__of__nat__commute,axiom,
    ! [Xb: nat,Y: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,Xb)),Y) = aa_int_int(aa_int_fun_int_int(times_times_int,Y),aa_nat_int(semiri1314217659103216013at_int,Xb)) ).

% mult_of_nat_commute
tff(fact_1474_mult__of__nat__commute,axiom,
    ! [Xb: nat,Y: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,Xb)),Y) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Y),aa_nat_nat(semiri1316708129612266289at_nat,Xb)) ).

% mult_of_nat_commute
tff(fact_1475_mult__of__nat__commute,axiom,
    ! [Xb: nat,Y: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,Xb)),Y) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Y),aa_nat_Code_integer(semiri4939895301339042750nteger,Xb)) ).

% mult_of_nat_commute
tff(fact_1476_hoare__triple__preI,axiom,
    ! [P: assn,C: heap_T2636463487746394924on_nat,Q: fun_option_nat_assn] :
      ( ! [H2: produc3658429121746597890et_nat] :
          ( aa_Pro653082453696054616_nat_o(rep_assn(P),H2)
         => hoare_7629718768684598413on_nat(P,C,Q) )
     => hoare_7629718768684598413on_nat(P,C,Q) ) ).

% hoare_triple_preI
tff(fact_1477_hoare__triple__preI,axiom,
    ! [P: assn,C: heap_Time_Heap_nat,Q: fun_nat_assn] :
      ( ! [H2: produc3658429121746597890et_nat] :
          ( aa_Pro653082453696054616_nat_o(rep_assn(P),H2)
         => hoare_3067605981109127869le_nat(P,C,Q) )
     => hoare_3067605981109127869le_nat(P,C,Q) ) ).

% hoare_triple_preI
tff(fact_1478_hoare__triple__preI,axiom,
    ! [P: assn,C: heap_Time_Heap_o,Q: fun_o_assn] :
      ( ! [H2: produc3658429121746597890et_nat] :
          ( aa_Pro653082453696054616_nat_o(rep_assn(P),H2)
         => hoare_hoare_triple_o(P,C,Q) )
     => hoare_hoare_triple_o(P,C,Q) ) ).

% hoare_triple_preI
tff(fact_1479_hoare__triple__preI,axiom,
    ! [P: assn,C: heap_T8145700208782473153_VEBTi,Q: fun_VEBT_VEBTi_assn] :
      ( ! [H2: produc3658429121746597890et_nat] :
          ( aa_Pro653082453696054616_nat_o(rep_assn(P),H2)
         => hoare_1429296392585015714_VEBTi(P,C,Q) )
     => hoare_1429296392585015714_VEBTi(P,C,Q) ) ).

% hoare_triple_preI
tff(fact_1480_hoare__triple__preI,axiom,
    ! [P: assn,C: heap_T5738788834812785303t_unit,Q: fun_Pr6626665806141890376t_assn] :
      ( ! [H2: produc3658429121746597890et_nat] :
          ( aa_Pro653082453696054616_nat_o(rep_assn(P),H2)
         => hoare_8945653483474564448t_unit(P,C,Q) )
     => hoare_8945653483474564448t_unit(P,C,Q) ) ).

% hoare_triple_preI
tff(fact_1481_xor__num_Ocases,axiom,
    ! [Xb: product_prod_num_num] :
      ( ( Xb != product_Pair_num_num(one,one) )
     => ( ! [N2: num] : Xb != product_Pair_num_num(one,aa_num_num(bit0,N2))
       => ( ! [N2: num] : Xb != product_Pair_num_num(one,aa_num_num(bit1,N2))
         => ( ! [M2: num] : Xb != product_Pair_num_num(aa_num_num(bit0,M2),one)
           => ( ! [M2: num,N2: num] : Xb != product_Pair_num_num(aa_num_num(bit0,M2),aa_num_num(bit0,N2))
             => ( ! [M2: num,N2: num] : Xb != product_Pair_num_num(aa_num_num(bit0,M2),aa_num_num(bit1,N2))
               => ( ! [M2: num] : Xb != product_Pair_num_num(aa_num_num(bit1,M2),one)
                 => ( ! [M2: num,N2: num] : Xb != product_Pair_num_num(aa_num_num(bit1,M2),aa_num_num(bit0,N2))
                   => ~ ! [M2: num,N2: num] : Xb != product_Pair_num_num(aa_num_num(bit1,M2),aa_num_num(bit1,N2)) ) ) ) ) ) ) ) ) ).

% xor_num.cases
tff(fact_1482_of__nat__less__imp__less,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,M)),aa_nat_rat(semiri681578069525770553at_rat,Nb))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb) ) ).

% of_nat_less_imp_less
tff(fact_1483_of__nat__less__imp__less,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,M)),aa_nat_real(semiri5074537144036343181t_real,Nb))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb) ) ).

% of_nat_less_imp_less
tff(fact_1484_of__nat__less__imp__less,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,Nb))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb) ) ).

% of_nat_less_imp_less
tff(fact_1485_of__nat__less__imp__less,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,Nb))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb) ) ).

% of_nat_less_imp_less
tff(fact_1486_of__nat__less__imp__less,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,M)),aa_nat_Code_integer(semiri4939895301339042750nteger,Nb))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb) ) ).

% of_nat_less_imp_less
tff(fact_1487_less__imp__of__nat__less,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,M)),aa_nat_rat(semiri681578069525770553at_rat,Nb)) ) ).

% less_imp_of_nat_less
tff(fact_1488_less__imp__of__nat__less,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,M)),aa_nat_real(semiri5074537144036343181t_real,Nb)) ) ).

% less_imp_of_nat_less
tff(fact_1489_less__imp__of__nat__less,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,Nb)) ) ).

% less_imp_of_nat_less
tff(fact_1490_less__imp__of__nat__less,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,Nb)) ) ).

% less_imp_of_nat_less
tff(fact_1491_less__imp__of__nat__less,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,M)),aa_nat_Code_integer(semiri4939895301339042750nteger,Nb)) ) ).

% less_imp_of_nat_less
tff(fact_1492_div__mult2__eq_H,axiom,
    ! [A: int,M: nat,Nb: 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,Nb))) = 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,Nb)) ).

% div_mult2_eq'
tff(fact_1493_div__mult2__eq_H,axiom,
    ! [A: nat,M: nat,Nb: 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,Nb))) = 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,Nb)) ).

% div_mult2_eq'
tff(fact_1494_div__mult2__eq_H,axiom,
    ! [A: code_integer,M: nat,Nb: nat] : aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,M)),aa_nat_Code_integer(semiri4939895301339042750nteger,Nb))) = aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),aa_nat_Code_integer(semiri4939895301339042750nteger,M))),aa_nat_Code_integer(semiri4939895301339042750nteger,Nb)) ).

% div_mult2_eq'
tff(fact_1495_of__nat__mono,axiom,
    ! [I: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,I)),aa_nat_real(semiri5074537144036343181t_real,J)) ) ).

% of_nat_mono
tff(fact_1496_of__nat__mono,axiom,
    ! [I: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,I)),aa_nat_Code_integer(semiri4939895301339042750nteger,J)) ) ).

% of_nat_mono
tff(fact_1497_of__nat__mono,axiom,
    ! [I: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(semiri681578069525770553at_rat,I)),aa_nat_rat(semiri681578069525770553at_rat,J)) ) ).

% of_nat_mono
tff(fact_1498_of__nat__mono,axiom,
    ! [I: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(semiri1316708129612266289at_nat,I)),aa_nat_nat(semiri1316708129612266289at_nat,J)) ) ).

% of_nat_mono
tff(fact_1499_of__nat__mono,axiom,
    ! [I: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,I)),aa_nat_int(semiri1314217659103216013at_int,J)) ) ).

% of_nat_mono
tff(fact_1500_unique__euclidean__semiring__with__nat__class_Oof__nat__div,axiom,
    ! [M: nat,Nb: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Nb)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,Nb)) ).

% unique_euclidean_semiring_with_nat_class.of_nat_div
tff(fact_1501_unique__euclidean__semiring__with__nat__class_Oof__nat__div,axiom,
    ! [M: nat,Nb: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,Nb)) ).

% unique_euclidean_semiring_with_nat_class.of_nat_div
tff(fact_1502_unique__euclidean__semiring__with__nat__class_Oof__nat__div,axiom,
    ! [M: nat,Nb: nat] : aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Nb)) = aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,M)),aa_nat_Code_integer(semiri4939895301339042750nteger,Nb)) ).

% unique_euclidean_semiring_with_nat_class.of_nat_div
tff(fact_1503_num_Oexhaust,axiom,
    ! [Y: num] :
      ( ( Y != one )
     => ( ! [X24: num] : Y != aa_num_num(bit0,X24)
       => ~ ! [X32: num] : Y != aa_num_num(bit1,X32) ) ) ).

% num.exhaust
tff(fact_1504_real__of__nat__div4,axiom,
    ! [Nb: nat,Xb: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),Xb))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_nat_real(semiri5074537144036343181t_real,Xb))) ).

% real_of_nat_div4
tff(fact_1505_nat__le__real__less,axiom,
    ! [Nb: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),M)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(semiri5074537144036343181t_real,M)),one_one_real)) ) ).

% nat_le_real_less
tff(fact_1506_nat__less__real__le,axiom,
    ! [Nb: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),M)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),one_one_real)),aa_nat_real(semiri5074537144036343181t_real,M)) ) ).

% nat_less_real_le
tff(fact_1507_of__nat__diff,axiom,
    ! [Nb: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),M)
     => ( aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Nb)) = minus_minus_rat(aa_nat_rat(semiri681578069525770553at_rat,M),aa_nat_rat(semiri681578069525770553at_rat,Nb)) ) ) ).

% of_nat_diff
tff(fact_1508_of__nat__diff,axiom,
    ! [Nb: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),M)
     => ( aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Nb)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(semiri5074537144036343181t_real,M)),aa_nat_real(semiri5074537144036343181t_real,Nb)) ) ) ).

% of_nat_diff
tff(fact_1509_of__nat__diff,axiom,
    ! [Nb: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),M)
     => ( aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Nb)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,Nb)) ) ) ).

% of_nat_diff
tff(fact_1510_of__nat__diff,axiom,
    ! [Nb: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),M)
     => ( aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,Nb)) ) ) ).

% of_nat_diff
tff(fact_1511_of__nat__diff,axiom,
    ! [Nb: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),M)
     => ( aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Nb)) = aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,M)),aa_nat_Code_integer(semiri4939895301339042750nteger,Nb)) ) ) ).

% of_nat_diff
tff(fact_1512_numeral__Bit1,axiom,
    ! [Nb: num] : numera7442385471795722001l_num1(aa_num_num(bit1,Nb)) = plus_p361126936061061375l_num1(plus_p361126936061061375l_num1(numera7442385471795722001l_num1(Nb),numera7442385471795722001l_num1(Nb)),one_on7727431528512463931l_num1) ).

% numeral_Bit1
tff(fact_1513_numeral__Bit1,axiom,
    ! [Nb: num] : numeral_numeral_real(aa_num_num(bit1,Nb)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,numeral_numeral_real(Nb)),numeral_numeral_real(Nb))),one_one_real) ).

% numeral_Bit1
tff(fact_1514_numeral__Bit1,axiom,
    ! [Nb: num] : numeral_numeral_rat(aa_num_num(bit1,Nb)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(plus_plus_rat(numeral_numeral_rat(Nb)),numeral_numeral_rat(Nb))),one_one_rat) ).

% numeral_Bit1
tff(fact_1515_numeral__Bit1,axiom,
    ! [Nb: num] : aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,Nb)),aa_num_nat(numeral_numeral_nat,Nb))),one_one_nat) ).

% numeral_Bit1
tff(fact_1516_numeral__Bit1,axiom,
    ! [Nb: num] : aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,Nb)),aa_num_int(numeral_numeral_int,Nb))),one_one_int) ).

% numeral_Bit1
tff(fact_1517_eval__nat__numeral_I3_J,axiom,
    ! [Nb: num] : aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Nb)) = aa_nat_nat(suc,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Nb))) ).

% eval_nat_numeral(3)
tff(fact_1518_complete__real,axiom,
    ! [S2: set_real] :
      ( ? [X3: real] : aa_set_real_o(member_real(X3),S2)
     => ( ? [Z4: real] :
          ! [X2: real] :
            ( aa_set_real_o(member_real(X2),S2)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X2),Z4) )
       => ? [Y4: real] :
            ( ! [X3: real] :
                ( aa_set_real_o(member_real(X3),S2)
               => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X3),Y4) )
            & ! [Z4: real] :
                ( ! [X2: real] :
                    ( aa_set_real_o(member_real(X2),S2)
                   => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X2),Z4) )
               => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y4),Z4) ) ) ) ) ).

% complete_real
tff(fact_1519_less__eq__real__def,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y)
        | ( Xb = Y ) ) ) ).

% less_eq_real_def
tff(fact_1520_of__nat__less__two__power,axiom,
    ! [Nb: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,Nb)),aa_nat_rat(power_power_rat(numeral_numeral_rat(aa_num_num(bit0,one))),Nb)) ).

% of_nat_less_two_power
tff(fact_1521_of__nat__less__two__power,axiom,
    ! [Nb: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,numeral_numeral_real(aa_num_num(bit0,one))),Nb)) ).

% of_nat_less_two_power
tff(fact_1522_of__nat__less__two__power,axiom,
    ! [Nb: nat] : aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) ).

% of_nat_less_two_power
tff(fact_1523_of__nat__less__two__power,axiom,
    ! [Nb: nat] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,Nb)),aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Nb)) ).

% of_nat_less_two_power
tff(fact_1524_numeral__code_I3_J,axiom,
    ! [Nb: num] :
      numera7442385471795722001l_num1(aa_num_num(bit1,Nb)) = $let(
        m: word_N3645301735248828278l_num1,
        m:= numera7442385471795722001l_num1(Nb),
        plus_p361126936061061375l_num1(plus_p361126936061061375l_num1(m,m),one_on7727431528512463931l_num1) ) ).

% numeral_code(3)
tff(fact_1525_numeral__code_I3_J,axiom,
    ! [Nb: num] :
      numeral_numeral_real(aa_num_num(bit1,Nb)) = $let(
        m: real,
        m:= numeral_numeral_real(Nb),
        aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,m),m)),one_one_real) ) ).

% numeral_code(3)
tff(fact_1526_numeral__code_I3_J,axiom,
    ! [Nb: num] :
      numeral_numeral_rat(aa_num_num(bit1,Nb)) = $let(
        m: rat,
        m:= numeral_numeral_rat(Nb),
        aa_rat_rat(plus_plus_rat(aa_rat_rat(plus_plus_rat(m),m)),one_one_rat) ) ).

% numeral_code(3)
tff(fact_1527_numeral__code_I3_J,axiom,
    ! [Nb: num] :
      aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Nb)) = $let(
        m: nat,
        m:= aa_num_nat(numeral_numeral_nat,Nb),
        aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,m),m)),one_one_nat) ) ).

% numeral_code(3)
tff(fact_1528_numeral__code_I3_J,axiom,
    ! [Nb: num] :
      aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb)) = $let(
        m: int,
        m:= aa_num_int(numeral_numeral_int,Nb),
        aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,m),m)),one_one_int) ) ).

% numeral_code(3)
tff(fact_1529_power__numeral__odd,axiom,
    ! [Z: uint32,W: num] :
      aa_nat_uint32(power_power_uint32(Z),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,W))) = $let(
        w: uint32,
        w:= aa_nat_uint32(power_power_uint32(Z),aa_num_nat(numeral_numeral_nat,W)),
        aa_uint32_uint32(times_times_uint32(aa_uint32_uint32(times_times_uint32(Z),w)),w) ) ).

% power_numeral_odd
tff(fact_1530_power__numeral__odd,axiom,
    ! [Z: word_N3645301735248828278l_num1,W: num] :
      aa_nat5599461682457136004l_num1(power_2184487114949457152l_num1(Z),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,W))) = $let(
        w: word_N3645301735248828278l_num1,
        w:= aa_nat5599461682457136004l_num1(power_2184487114949457152l_num1(Z),aa_num_nat(numeral_numeral_nat,W)),
        aa_wor7274019993978516266l_num1(times_7065122842183080059l_num1(aa_wor7274019993978516266l_num1(times_7065122842183080059l_num1(Z),w)),w) ) ).

% power_numeral_odd
tff(fact_1531_power__numeral__odd,axiom,
    ! [Z: real,W: num] :
      aa_nat_real(aa_real_fun_nat_real(power_power_real,Z),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,W))) = $let(
        w: real,
        w:= aa_nat_real(aa_real_fun_nat_real(power_power_real,Z),aa_num_nat(numeral_numeral_nat,W)),
        aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z),w)),w) ) ).

% power_numeral_odd
tff(fact_1532_power__numeral__odd,axiom,
    ! [Z: rat,W: num] :
      aa_nat_rat(power_power_rat(Z),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,W))) = $let(
        w: rat,
        w:= aa_nat_rat(power_power_rat(Z),aa_num_nat(numeral_numeral_nat,W)),
        aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(Z),w)),w) ) ).

% power_numeral_odd
tff(fact_1533_power__numeral__odd,axiom,
    ! [Z: nat,W: num] :
      aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Z),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,W))) = $let(
        w: nat,
        w:= aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Z),aa_num_nat(numeral_numeral_nat,W)),
        aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Z),w)),w) ) ).

% power_numeral_odd
tff(fact_1534_power__numeral__odd,axiom,
    ! [Z: int,W: num] :
      aa_nat_int(aa_int_fun_nat_int(power_power_int,Z),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,W))) = $let(
        w: int,
        w:= aa_nat_int(aa_int_fun_nat_int(power_power_int,Z),aa_num_nat(numeral_numeral_nat,W)),
        aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,Z),w)),w) ) ).

% power_numeral_odd
tff(fact_1535_real__of__nat__div3,axiom,
    ! [Nb: nat,Xb: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_nat_real(semiri5074537144036343181t_real,Xb))),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),Xb)))),one_one_real) ).

% real_of_nat_div3
tff(fact_1536_numeral__Bit1__div__2,axiom,
    ! [Nb: num] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Nb))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_num_nat(numeral_numeral_nat,Nb) ).

% numeral_Bit1_div_2
tff(fact_1537_numeral__Bit1__div__2,axiom,
    ! [Nb: num] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = aa_num_int(numeral_numeral_int,Nb) ).

% numeral_Bit1_div_2
tff(fact_1538_power3__eq__cube,axiom,
    ! [A: uint32] : aa_nat_uint32(power_power_uint32(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))) = aa_uint32_uint32(times_times_uint32(aa_uint32_uint32(times_times_uint32(A),A)),A) ).

% power3_eq_cube
tff(fact_1539_power3__eq__cube,axiom,
    ! [A: word_N3645301735248828278l_num1] : aa_nat5599461682457136004l_num1(power_2184487114949457152l_num1(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))) = aa_wor7274019993978516266l_num1(times_7065122842183080059l_num1(aa_wor7274019993978516266l_num1(times_7065122842183080059l_num1(A),A)),A) ).

% power3_eq_cube
tff(fact_1540_power3__eq__cube,axiom,
    ! [A: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),A)),A) ).

% power3_eq_cube
tff(fact_1541_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(times_times_rat(aa_rat_rat(times_times_rat(A),A)),A) ).

% power3_eq_cube
tff(fact_1542_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_1543_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_1544_Suc3__eq__add__3,axiom,
    ! [Nb: nat] : aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,Nb))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),Nb) ).

% Suc3_eq_add_3
tff(fact_1545_Suc__div__eq__add3__div,axiom,
    ! [M: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,M)))),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),M)),Nb) ).

% Suc_div_eq_add3_div
tff(fact_1546_add__diff__add,axiom,
    ! [A: real,C: real,B: real,D: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),D)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C),D)) ).

% add_diff_add
tff(fact_1547_add__diff__add,axiom,
    ! [A: rat,C: rat,B: rat,D: rat] : minus_minus_rat(aa_rat_rat(plus_plus_rat(A),C),aa_rat_rat(plus_plus_rat(B),D)) = aa_rat_rat(plus_plus_rat(minus_minus_rat(A,B)),minus_minus_rat(C,D)) ).

% add_diff_add
tff(fact_1548_add__diff__add,axiom,
    ! [A: int,C: int,B: int,D: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),D)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),aa_int_int(aa_int_fun_int_int(minus_minus_int,C),D)) ).

% add_diff_add
tff(fact_1549_real__arch__pow,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xb)
     => ? [N2: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,Y),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),N2)) ) ).

% real_arch_pow
tff(fact_1550_small__powers__of__2,axiom,
    ! [Xb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),Xb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Xb),one_one_nat))) ) ).

% small_powers_of_2
tff(fact_1551_mult__diff__mult,axiom,
    ! [Xb: real,Y: real,A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Y)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Y),B))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),A)),B)) ).

% mult_diff_mult
tff(fact_1552_mult__diff__mult,axiom,
    ! [Xb: rat,Y: rat,A: rat,B: rat] : minus_minus_rat(aa_rat_rat(times_times_rat(Xb),Y),aa_rat_rat(times_times_rat(A),B)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(Xb),minus_minus_rat(Y,B))),aa_rat_rat(times_times_rat(minus_minus_rat(Xb,A)),B)) ).

% mult_diff_mult
tff(fact_1553_mult__diff__mult,axiom,
    ! [Xb: int,Y: int,A: int,B: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Xb),Y)),aa_int_int(aa_int_fun_int_int(times_times_int,A),B)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Xb),aa_int_int(aa_int_fun_int_int(minus_minus_int,Y),B))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb),A)),B)) ).

% mult_diff_mult
tff(fact_1554_field__sum__of__halves,axiom,
    ! [Xb: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),numeral_numeral_real(aa_num_num(bit0,one)))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),numeral_numeral_real(aa_num_num(bit0,one)))) = Xb ).

% field_sum_of_halves
tff(fact_1555_field__sum__of__halves,axiom,
    ! [Xb: rat] : aa_rat_rat(plus_plus_rat(divide_divide_rat(Xb,numeral_numeral_rat(aa_num_num(bit0,one)))),divide_divide_rat(Xb,numeral_numeral_rat(aa_num_num(bit0,one)))) = Xb ).

% field_sum_of_halves
tff(fact_1556_field__less__half__sum,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Y)),numeral_numeral_real(aa_num_num(bit0,one)))) ) ).

% field_less_half_sum
tff(fact_1557_field__less__half__sum,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Y)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),divide_divide_rat(aa_rat_rat(plus_plus_rat(Xb),Y),numeral_numeral_rat(aa_num_num(bit0,one)))) ) ).

% field_less_half_sum
tff(fact_1558_space__bound,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,U: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Ta),Nb)
     => ( ( U = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb) )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_VEBT_VEBT_nat(vEBT_VEBT_space,Ta)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,one))))),U)) ) ) ).

% space_bound
tff(fact_1559_cnt__cnt__eq,axiom,
    ! [Ta: vEBT_VEBT] : aa_VEBT_VEBT_real(vEBT_VEBT_cnt,Ta) = aa_nat_real(semiri5074537144036343181t_real,aa_VEBT_VEBT_nat(vEBT_VEBT_cnt2,Ta)) ).

% cnt_cnt_eq
tff(fact_1560_space__2__pow__bound,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Ta),Nb)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,aa_VEBT_VEBT_nat(vEBT_VEBT_space2,Ta))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,one))))),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,numeral_numeral_real(aa_num_num(bit0,one))),Nb)),one_one_real))) ) ).

% space_2_pow_bound
tff(fact_1561_space__cnt,axiom,
    ! [Ta: vEBT_VEBT] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,aa_VEBT_VEBT_nat(vEBT_VEBT_space2,Ta))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,aa_num_num(bit1,one)))),aa_VEBT_VEBT_real(vEBT_VEBT_cnt,Ta))) ).

% space_cnt
tff(fact_1562_space_H__bound,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,U: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Ta),Nb)
     => ( ( U = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb) )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_VEBT_VEBT_nat(vEBT_VEBT_space2,Ta)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,one))))),U)) ) ) ).

% space'_bound
tff(fact_1563_TBOUND__replicate,axiom,
    ! [Xb: heap_Time_Heap_o,C: nat,Nb: nat] :
      ( aa_nat_o(time_TBOUND_o(Xb),C)
     => aa_nat_o(time_TBOUND_list_o(vEBT_V2326993469660664182atei_o(Nb,Xb)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),C)),Nb))) ) ).

% TBOUND_replicate
tff(fact_1564_TBOUND__replicate,axiom,
    ! [Xb: heap_Time_Heap_nat,C: nat,Nb: nat] :
      ( aa_nat_o(time_TBOUND_nat(Xb),C)
     => aa_nat_o(time_TBOUND_list_nat(vEBT_V7726092123322077554ei_nat(Nb,Xb)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),C)),Nb))) ) ).

% TBOUND_replicate
tff(fact_1565_TBOUND__replicate,axiom,
    ! [Xb: heap_T2636463487746394924on_nat,C: nat,Nb: nat] :
      ( aa_nat_o(time_T8353473612707095248on_nat(Xb),C)
     => aa_nat_o(time_T3808005469503390304on_nat(vEBT_V792416675989592002on_nat(Nb,Xb)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),C)),Nb))) ) ).

% TBOUND_replicate
tff(fact_1566_TBOUND__replicate,axiom,
    ! [Xb: heap_T8145700208782473153_VEBTi,C: nat,Nb: nat] :
      ( aa_nat_o(time_T5737551269749752165_VEBTi(Xb),C)
     => aa_nat_o(time_T8149879359713347829_VEBTi(vEBT_V1859673955506687831_VEBTi(Nb,Xb)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),C)),Nb))) ) ).

% TBOUND_replicate
tff(fact_1567_delete__bound__height,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xb: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Ta),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_d_e_l_e_t_e,Ta),Xb)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_VEBT_height,Ta))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,one))))))))) ) ).

% delete_bound_height
tff(fact_1568_t__build__cnt,axiom,
    ! [Nb: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(vEBT_V8646137997579335489_i_l_d,Nb))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_VEBT_VEBT_real(vEBT_VEBT_cnt,vEBT_vebt_buildup(Nb))),numeral_numeral_real(aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,one)))))) ).

% t_build_cnt
tff(fact_1569_t__buildup__cnt,axiom,
    ! [Nb: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(vEBT_V8346862874174094_d_u_p,Nb))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_VEBT_VEBT_real(vEBT_VEBT_cnt,vEBT_vebt_buildup(Nb))),numeral_numeral_real(aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,one)))))) ).

% t_buildup_cnt
tff(fact_1570_tdeletemimi,axiom,
    ! [Deg: nat,Mia: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Xb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_d_e_l_e_t_e,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mia),Mia)),Deg,TreeLista),Summarya)),Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,one))))) ) ).

% tdeletemimi
tff(fact_1571_TBOUND__highi,axiom,
    ! [Xb: nat,Nb: nat] : aa_nat_o(time_TBOUND_nat(aa_nat5821310369566504398ap_nat(vEBT_VEBT_highi(Xb),Nb)),one_one_nat) ).

% TBOUND_highi
tff(fact_1572_TBOUND__lowi,axiom,
    ! [Xb: nat,Nb: nat] : aa_nat_o(time_TBOUND_nat(aa_nat5821310369566504398ap_nat(vEBT_VEBT_lowi(Xb),Nb)),one_one_nat) ).

% TBOUND_lowi
tff(fact_1573_TBOUND__vebt__maxti,axiom,
    ! [Ta: vEBT_VEBTi] : aa_nat_o(time_T8353473612707095248on_nat(aa_VEB2711320435582112505on_nat(vEBT_vebt_maxti,Ta)),one_one_nat) ).

% TBOUND_vebt_maxti
tff(fact_1574_TBOUND__vebt__minti,axiom,
    ! [Ta: vEBT_VEBTi] : aa_nat_o(time_T8353473612707095248on_nat(aa_VEB2711320435582112505on_nat(vEBT_vebt_minti,Ta)),one_one_nat) ).

% TBOUND_vebt_minti
tff(fact_1575_buildup__build__time,axiom,
    ! [Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(vEBT_V8346862874174094_d_u_p,Nb)),aa_nat_nat(vEBT_V8646137997579335489_i_l_d,Nb)) ).

% buildup_build_time
tff(fact_1576_space__space_H,axiom,
    ! [Ta: vEBT_VEBT] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_VEBT_VEBT_nat(vEBT_VEBT_space,Ta)),aa_VEBT_VEBT_nat(vEBT_VEBT_space2,Ta)) ).

% space_space'
tff(fact_1577_vebt__buildup__bound,axiom,
    ! [U: nat,Nb: nat] :
      ( ( U = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(vEBT_V8346862874174094_d_u_p,Nb)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,one)))))),U)) ) ).

% vebt_buildup_bound
tff(fact_1578_TBOUND__vebt__succi,axiom,
    ! [Ta: vEBT_VEBT,Tib: vEBT_VEBTi,Xb: nat] : aa_nat_o(time_T8353473612707095248on_nat(aa_nat947733127081326238on_nat(aa_VEB3566417204837707778on_nat(aa_VEB1030146501360450209on_nat(vEBT_VEBT_vebt_succi,Ta),Tib),Xb)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit1,one)))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_VEBT_height,Ta)))) ).

% TBOUND_vebt_succi
tff(fact_1579_zdiv__int,axiom,
    ! [A: nat,B: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_nat_int(semiri1314217659103216013at_int,A)),aa_nat_int(semiri1314217659103216013at_int,B)) ).

% zdiv_int
tff(fact_1580_Tb_H__cnt,axiom,
    ! [Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(vEBT_VEBT_Tb2,Nb)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit0,one)))),aa_VEBT_VEBT_nat(vEBT_VEBT_cnt2,vEBT_vebt_buildup(Nb)))) ).

% Tb'_cnt
tff(fact_1581_htt__vebt__buildupi_H__univ,axiom,
    ! [U: nat,Nb: nat] :
      ( ( U = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb) )
     => aa_nat_o(time_htt_VEBT_VEBTi(one_one_assn,aa_nat5408703056993926195_VEBTi(vEBT_V739175172307565963ildupi,Nb),aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,vEBT_vebt_buildup(Nb))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,one))))),U)) ) ).

% htt_vebt_buildupi'_univ
tff(fact_1582_htt__vebt__buildupi__univ,axiom,
    ! [U: nat,Nb: nat] :
      ( ( U = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb) )
     => aa_nat_o(time_htt_VEBT_VEBTi(one_one_assn,aa_nat5408703056993926195_VEBTi(vEBT_vebt_buildupi,Nb),aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,vEBT_vebt_buildup(Nb))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,one))))),U)) ) ).

% htt_vebt_buildupi_univ
tff(fact_1583_T__vebt__buildupi__cnt_H,axiom,
    ! [Nb: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(vEBT_V441764108873111860ildupi,Nb))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit1,aa_num_num(bit0,one)))),aa_VEBT_VEBT_real(vEBT_VEBT_cnt,vEBT_vebt_buildup(Nb)))) ).

% T_vebt_buildupi_cnt'
tff(fact_1584_T__vebt__buildupi__univ,axiom,
    ! [U: nat,Nb: nat] :
      ( ( U = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(vEBT_V441764108873111860ildupi,Nb)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,one))))),U)) ) ).

% T_vebt_buildupi_univ
tff(fact_1585_TBOUND__minNulli,axiom,
    ! [Ta: vEBT_VEBTi] : aa_nat_o(time_TBOUND_o(aa_VEB8194921341634505877Heap_o(vEBT_VEBT_minNulli,Ta)),one_one_nat) ).

% TBOUND_minNulli
tff(fact_1586_mod__pure__star__dist,axiom,
    ! [P: assn,B: $o,H: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(rep_assn(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,P),aa_o_assn(pure_assn,(B)))),H)
    <=> ( aa_Pro653082453696054616_nat_o(rep_assn(P),H)
        & (B) ) ) ).

% mod_pure_star_dist
tff(fact_1587_TBOUND__buildupi,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => aa_nat_o(time_T5737551269749752165_VEBTi(aa_nat5408703056993926195_VEBTi(vEBT_vebt_buildupi,Nb)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))) ) ).

% TBOUND_buildupi
tff(fact_1588_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_Osimps_I5_J,axiom,
    ! [Mia: nat,Maa: nat,Vaa: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Xb: nat] :
      aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_m_e_m_b_e_r,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mia),Maa)),aa_nat_nat(suc,aa_nat_nat(suc,Vaa)),TreeLista),Summarya)),Xb) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
        $ite(
          Xb = Mia,
          one_one_nat,
          aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
            $ite(
              Xb = Maa,
              one_one_nat,
              aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
                $ite(
                  aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Mia),
                  one_one_nat,
                  aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
                    $ite(
                      aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Maa),Xb),
                      one_one_nat,
                      aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,one))))),
                        $let(
                          h: nat,
                          h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Vaa))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                          $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeLista)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_m_e_m_b_e_r,nth_VEBT_VEBT(TreeLista,h)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Vaa))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),one_one_nat) )) )) )) )) )) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.simps(5)
tff(fact_1589_TBOUND__vebt__inserti,axiom,
    ! [Ta: vEBT_VEBT,Tib: vEBT_VEBTi,Xb: nat] :
      aa_nat_o(time_T5737551269749752165_VEBTi(aa_nat5408703056993926195_VEBTi(aa_VEB4833742232818206615_VEBTi(aa_VEB6292075424854053302_VEBTi(vEBT_V3964819847710782039nserti,Ta),Tib),Xb)),
        $ite(aa_VEBT_VEBT_o(vEBT_VEBT_minNull,Ta),one_one_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,one))))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_VEBT_height,Ta))))) ).

% TBOUND_vebt_inserti
tff(fact_1590_valid__0__not,axiom,
    ! [Ta: vEBT_VEBT] : ~ aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Ta),zero_zero_nat) ).

% valid_0_not
tff(fact_1591_valid__tree__deg__neq__0,axiom,
    ! [Ta: vEBT_VEBT] : ~ aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Ta),zero_zero_nat) ).

% valid_tree_deg_neq_0
tff(fact_1592_not__min__Null__member,axiom,
    ! [Ta: vEBT_VEBT] :
      ( ~ aa_VEBT_VEBT_o(vEBT_VEBT_minNull,Ta)
     => ? [X_13: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,Ta),X_13) ) ).

% not_min_Null_member
tff(fact_1593_min__Null__member,axiom,
    ! [Ta: vEBT_VEBT,Xb: nat] :
      ( aa_VEBT_VEBT_o(vEBT_VEBT_minNull,Ta)
     => ~ aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,Ta),Xb) ) ).

% min_Null_member
tff(fact_1594_deg__not__0,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Ta),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ).

% deg_not_0
tff(fact_1595_T__vebt__buildupi__gq__0,axiom,
    ! [Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(vEBT_V441764108873111860ildupi,Nb)) ).

% T_vebt_buildupi_gq_0
tff(fact_1596_minminNull,axiom,
    ! [Ta: vEBT_VEBT] :
      ( ( vEBT_vebt_mint(Ta) = none_nat )
     => aa_VEBT_VEBT_o(vEBT_VEBT_minNull,Ta) ) ).

% minminNull
tff(fact_1597_minNullmin,axiom,
    ! [Ta: vEBT_VEBT] :
      ( aa_VEBT_VEBT_o(vEBT_VEBT_minNull,Ta)
     => ( vEBT_vebt_mint(Ta) = none_nat ) ) ).

% minNullmin
tff(fact_1598_TBOUND__vebt__buildupi,axiom,
    ! [Nb: nat] : aa_nat_o(time_T5737551269749752165_VEBTi(aa_nat5408703056993926195_VEBTi(vEBT_V739175172307565963ildupi,Nb)),aa_nat_nat(vEBT_V441764108873111860ildupi,Nb)) ).

% TBOUND_vebt_buildupi
tff(fact_1599_buildup__gives__valid,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,vEBT_vebt_buildup(Nb)),Nb) ) ).

% buildup_gives_valid
tff(fact_1600_pure__assn__eq__conv,axiom,
    ! [P: $o,Q: $o] :
      ( ( aa_o_assn(pure_assn,(P)) = aa_o_assn(pure_assn,(Q)) )
    <=> ( (P)
      <=> (Q) ) ) ).

% pure_assn_eq_conv
tff(fact_1601_TBOUND__minNull,axiom,
    ! [Ta: vEBT_VEBT,Tib: vEBT_VEBTi,Xb: nat] :
      ( aa_VEBT_VEBT_o(vEBT_VEBT_minNull,Ta)
     => aa_nat_o(time_T5737551269749752165_VEBTi(aa_nat5408703056993926195_VEBTi(aa_VEB4833742232818206615_VEBTi(aa_VEB6292075424854053302_VEBTi(vEBT_V3964819847710782039nserti,Ta),Tib),Xb)),one_one_nat) ) ).

% TBOUND_minNull
tff(fact_1602_ex__assn__const,axiom,
    ! [C: assn] : ex_ass463751140784270563_VEBTi(aTP_Lamm_ei(C)) = C ).

% ex_assn_const
tff(fact_1603_Tb__T__vebt__buildupi_H_H,axiom,
    ! [Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(vEBT_V441764108873111860ildupi,Nb)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(vEBT_VEBT_Tb2,Nb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% Tb_T_vebt_buildupi''
tff(fact_1604_htt__vebt__buildupi,axiom,
    ! [Nb: nat] : aa_nat_o(time_htt_VEBT_VEBTi(one_one_assn,aa_nat5408703056993926195_VEBTi(vEBT_vebt_buildupi,Nb),aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,vEBT_vebt_buildup(Nb))),aa_nat_nat(vEBT_V441764108873111860ildupi,Nb)) ).

% htt_vebt_buildupi
tff(fact_1605_htt__vebt__buildupi_H,axiom,
    ! [Nb: nat] : aa_nat_o(time_htt_VEBT_VEBTi(one_one_assn,aa_nat5408703056993926195_VEBTi(vEBT_V739175172307565963ildupi,Nb),aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,vEBT_vebt_buildup(Nb))),aa_nat_nat(vEBT_V441764108873111860ildupi,Nb)) ).

% htt_vebt_buildupi'
tff(fact_1606_mult__cancel__right,axiom,
    ! [A: real,C: real,B: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C) )
    <=> ( ( C = zero_zero_real )
        | ( A = B ) ) ) ).

% mult_cancel_right
tff(fact_1607_mult__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( aa_rat_rat(times_times_rat(A),C) = aa_rat_rat(times_times_rat(B),C) )
    <=> ( ( C = zero_zero_rat )
        | ( A = B ) ) ) ).

% mult_cancel_right
tff(fact_1608_mult__cancel__right,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C) )
    <=> ( ( C = zero_zero_nat )
        | ( A = B ) ) ) ).

% mult_cancel_right
tff(fact_1609_mult__cancel__right,axiom,
    ! [A: int,C: int,B: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,A),C) = aa_int_int(aa_int_fun_int_int(times_times_int,B),C) )
    <=> ( ( C = zero_zero_int )
        | ( A = B ) ) ) ).

% mult_cancel_right
tff(fact_1610_mult__cancel__left,axiom,
    ! [C: real,A: real,B: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B) )
    <=> ( ( C = zero_zero_real )
        | ( A = B ) ) ) ).

% mult_cancel_left
tff(fact_1611_mult__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( aa_rat_rat(times_times_rat(C),A) = aa_rat_rat(times_times_rat(C),B) )
    <=> ( ( C = zero_zero_rat )
        | ( A = B ) ) ) ).

% mult_cancel_left
tff(fact_1612_mult__cancel__left,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B) )
    <=> ( ( C = zero_zero_nat )
        | ( A = B ) ) ) ).

% mult_cancel_left
tff(fact_1613_mult__cancel__left,axiom,
    ! [C: int,A: int,B: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,C),A) = aa_int_int(aa_int_fun_int_int(times_times_int,C),B) )
    <=> ( ( C = zero_zero_int )
        | ( A = B ) ) ) ).

% mult_cancel_left
tff(fact_1614_mult__eq__0__iff,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B) = zero_zero_real )
    <=> ( ( A = zero_zero_real )
        | ( B = zero_zero_real ) ) ) ).

% mult_eq_0_iff
tff(fact_1615_mult__eq__0__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(times_times_rat(A),B) = zero_zero_rat )
    <=> ( ( A = zero_zero_rat )
        | ( B = zero_zero_rat ) ) ) ).

% mult_eq_0_iff
tff(fact_1616_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_1617_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_1618_mult__zero__right,axiom,
    ! [A: uint32] : aa_uint32_uint32(times_times_uint32(A),zero_zero_uint32) = zero_zero_uint32 ).

% mult_zero_right
tff(fact_1619_mult__zero__right,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),zero_zero_real) = zero_zero_real ).

% mult_zero_right
tff(fact_1620_mult__zero__right,axiom,
    ! [A: rat] : aa_rat_rat(times_times_rat(A),zero_zero_rat) = zero_zero_rat ).

% mult_zero_right
tff(fact_1621_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_1622_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_1623_mult__zero__left,axiom,
    ! [A: uint32] : aa_uint32_uint32(times_times_uint32(zero_zero_uint32),A) = zero_zero_uint32 ).

% mult_zero_left
tff(fact_1624_mult__zero__left,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,zero_zero_real),A) = zero_zero_real ).

% mult_zero_left
tff(fact_1625_mult__zero__left,axiom,
    ! [A: rat] : aa_rat_rat(times_times_rat(zero_zero_rat),A) = zero_zero_rat ).

% mult_zero_left
tff(fact_1626_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_1627_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_1628_bits__div__by__0,axiom,
    ! [A: uint32] : aa_uint32_uint32(aa_uin3778660748068847871uint32(divide_divide_uint32,A),zero_zero_uint32) = zero_zero_uint32 ).

% bits_div_by_0
tff(fact_1629_bits__div__by__0,axiom,
    ! [A: word_N3645301735248828278l_num1] : divide1791077408188789448l_num1(A,zero_z3563351764282998399l_num1) = zero_z3563351764282998399l_num1 ).

% bits_div_by_0
tff(fact_1630_bits__div__by__0,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),zero_zero_nat) = zero_zero_nat ).

% bits_div_by_0
tff(fact_1631_bits__div__by__0,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,A),zero_zero_int) = zero_zero_int ).

% bits_div_by_0
tff(fact_1632_bits__div__0,axiom,
    ! [A: uint32] : aa_uint32_uint32(aa_uin3778660748068847871uint32(divide_divide_uint32,zero_zero_uint32),A) = zero_zero_uint32 ).

% bits_div_0
tff(fact_1633_bits__div__0,axiom,
    ! [A: word_N3645301735248828278l_num1] : divide1791077408188789448l_num1(zero_z3563351764282998399l_num1,A) = zero_z3563351764282998399l_num1 ).

% bits_div_0
tff(fact_1634_bits__div__0,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,zero_zero_nat),A) = zero_zero_nat ).

% bits_div_0
tff(fact_1635_bits__div__0,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,zero_zero_int),A) = zero_zero_int ).

% bits_div_0
tff(fact_1636_div__by__0,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),zero_zero_real) = zero_zero_real ).

% div_by_0
tff(fact_1637_div__by__0,axiom,
    ! [A: rat] : divide_divide_rat(A,zero_zero_rat) = zero_zero_rat ).

% div_by_0
tff(fact_1638_div__by__0,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),zero_zero_nat) = zero_zero_nat ).

% div_by_0
tff(fact_1639_div__by__0,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,A),zero_zero_int) = zero_zero_int ).

% div_by_0
tff(fact_1640_div__0,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,zero_zero_real),A) = zero_zero_real ).

% div_0
tff(fact_1641_div__0,axiom,
    ! [A: rat] : divide_divide_rat(zero_zero_rat,A) = zero_zero_rat ).

% div_0
tff(fact_1642_div__0,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,zero_zero_nat),A) = zero_zero_nat ).

% div_0
tff(fact_1643_div__0,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,zero_zero_int),A) = zero_zero_int ).

% div_0
tff(fact_1644_less__nat__zero__code,axiom,
    ! [Nb: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),zero_zero_nat) ).

% less_nat_zero_code
tff(fact_1645_neq0__conv,axiom,
    ! [Nb: nat] :
      ( ( Nb != zero_zero_nat )
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ).

% neq0_conv
tff(fact_1646_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_1647_le0,axiom,
    ! [Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Nb) ).

% le0
tff(fact_1648_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_1649_add__is__0,axiom,
    ! [M: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Nb) = zero_zero_nat )
    <=> ( ( M = zero_zero_nat )
        & ( Nb = zero_zero_nat ) ) ) ).

% add_is_0
tff(fact_1650_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_1651_diff__0__eq__0,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,zero_zero_nat),Nb) = zero_zero_nat ).

% diff_0_eq_0
tff(fact_1652_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_1653_minNulli__rule,axiom,
    ! [Ta: vEBT_VEBT,Tib: vEBT_VEBTi] : hoare_hoare_triple_o(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Ta),Tib),aa_VEB8194921341634505877Heap_o(vEBT_VEBT_minNulli,Tib),aa_VEB6687300313434771872o_assn(aTP_Lamm_ej(Ta),Tib)) ).

% minNulli_rule
tff(fact_1654_mult__cancel2,axiom,
    ! [M: nat,K: nat,Nb: 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,Nb),K) )
    <=> ( ( M = Nb )
        | ( K = zero_zero_nat ) ) ) ).

% mult_cancel2
tff(fact_1655_mult__cancel1,axiom,
    ! [K: nat,M: nat,Nb: 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),Nb) )
    <=> ( ( M = Nb )
        | ( K = zero_zero_nat ) ) ) ).

% mult_cancel1
tff(fact_1656_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_1657_mult__is__0,axiom,
    ! [M: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Nb) = zero_zero_nat )
    <=> ( ( M = zero_zero_nat )
        | ( Nb = zero_zero_nat ) ) ) ).

% mult_is_0
tff(fact_1658_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_1659_pure__assn__eq__emp__iff,axiom,
    ! [P: $o] :
      ( ( aa_o_assn(pure_assn,(P)) = one_one_assn )
    <=> (P) ) ).

% pure_assn_eq_emp_iff
tff(fact_1660_pure__true,axiom,
    aa_o_assn(pure_assn,$true) = one_one_assn ).

% pure_true
tff(fact_1661_mod__ex__dist,axiom,
    ! [P: fun_li3352006551604695530i_assn,H: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(rep_assn(ex_ass463751140784270563_VEBTi(P)),H)
    <=> ? [X: list_VEBT_VEBTi] : aa_Pro653082453696054616_nat_o(rep_assn(aa_lis2508312872377092907i_assn(P,X)),H) ) ).

% mod_ex_dist
tff(fact_1662_zero__comp__diff__simps_I1_J,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A) ) ).

% zero_comp_diff_simps(1)
tff(fact_1663_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),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_1664_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_1665_zero__comp__diff__simps_I2_J,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,B),A) ) ).

% zero_comp_diff_simps(2)
tff(fact_1666_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),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_1667_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_1668_sum__squares__eq__zero__iff,axiom,
    ! [Xb: real,Y: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Xb)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Y)) = zero_zero_real )
    <=> ( ( Xb = zero_zero_real )
        & ( Y = zero_zero_real ) ) ) ).

% sum_squares_eq_zero_iff
tff(fact_1669_sum__squares__eq__zero__iff,axiom,
    ! [Xb: rat,Y: rat] :
      ( ( aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(Xb),Xb)),aa_rat_rat(times_times_rat(Y),Y)) = zero_zero_rat )
    <=> ( ( Xb = zero_zero_rat )
        & ( Y = zero_zero_rat ) ) ) ).

% sum_squares_eq_zero_iff
tff(fact_1670_sum__squares__eq__zero__iff,axiom,
    ! [Xb: 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,Xb),Xb)),aa_int_int(aa_int_fun_int_int(times_times_int,Y),Y)) = zero_zero_int )
    <=> ( ( Xb = zero_zero_int )
        & ( Y = zero_zero_int ) ) ) ).

% sum_squares_eq_zero_iff
tff(fact_1671_mult__cancel__right2,axiom,
    ! [A: real,C: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C) = C )
    <=> ( ( C = zero_zero_real )
        | ( A = one_one_real ) ) ) ).

% mult_cancel_right2
tff(fact_1672_mult__cancel__right2,axiom,
    ! [A: rat,C: rat] :
      ( ( aa_rat_rat(times_times_rat(A),C) = C )
    <=> ( ( C = zero_zero_rat )
        | ( A = one_one_rat ) ) ) ).

% mult_cancel_right2
tff(fact_1673_mult__cancel__right2,axiom,
    ! [A: int,C: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,A),C) = C )
    <=> ( ( C = zero_zero_int )
        | ( A = one_one_int ) ) ) ).

% mult_cancel_right2
tff(fact_1674_mult__cancel__right1,axiom,
    ! [C: real,B: real] :
      ( ( C = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C) )
    <=> ( ( C = zero_zero_real )
        | ( B = one_one_real ) ) ) ).

% mult_cancel_right1
tff(fact_1675_mult__cancel__right1,axiom,
    ! [C: rat,B: rat] :
      ( ( C = aa_rat_rat(times_times_rat(B),C) )
    <=> ( ( C = zero_zero_rat )
        | ( B = one_one_rat ) ) ) ).

% mult_cancel_right1
tff(fact_1676_mult__cancel__right1,axiom,
    ! [C: int,B: int] :
      ( ( C = aa_int_int(aa_int_fun_int_int(times_times_int,B),C) )
    <=> ( ( C = zero_zero_int )
        | ( B = one_one_int ) ) ) ).

% mult_cancel_right1
tff(fact_1677_mult__cancel__left2,axiom,
    ! [C: real,A: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A) = C )
    <=> ( ( C = zero_zero_real )
        | ( A = one_one_real ) ) ) ).

% mult_cancel_left2
tff(fact_1678_mult__cancel__left2,axiom,
    ! [C: rat,A: rat] :
      ( ( aa_rat_rat(times_times_rat(C),A) = C )
    <=> ( ( C = zero_zero_rat )
        | ( A = one_one_rat ) ) ) ).

% mult_cancel_left2
tff(fact_1679_mult__cancel__left2,axiom,
    ! [C: int,A: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,C),A) = C )
    <=> ( ( C = zero_zero_int )
        | ( A = one_one_int ) ) ) ).

% mult_cancel_left2
tff(fact_1680_mult__cancel__left1,axiom,
    ! [C: real,B: real] :
      ( ( C = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B) )
    <=> ( ( C = zero_zero_real )
        | ( B = one_one_real ) ) ) ).

% mult_cancel_left1
tff(fact_1681_mult__cancel__left1,axiom,
    ! [C: rat,B: rat] :
      ( ( C = aa_rat_rat(times_times_rat(C),B) )
    <=> ( ( C = zero_zero_rat )
        | ( B = one_one_rat ) ) ) ).

% mult_cancel_left1
tff(fact_1682_mult__cancel__left1,axiom,
    ! [C: int,B: int] :
      ( ( C = aa_int_int(aa_int_fun_int_int(times_times_int,C),B) )
    <=> ( ( C = zero_zero_int )
        | ( B = one_one_int ) ) ) ).

% mult_cancel_left1
tff(fact_1683_diff__numeral__special_I9_J,axiom,
    minus_4019991460397169231l_num1(one_on7727431528512463931l_num1,one_on7727431528512463931l_num1) = zero_z3563351764282998399l_num1 ).

% diff_numeral_special(9)
tff(fact_1684_diff__numeral__special_I9_J,axiom,
    minus_minus_uint32(one_one_uint32,one_one_uint32) = zero_zero_uint32 ).

% diff_numeral_special(9)
tff(fact_1685_diff__numeral__special_I9_J,axiom,
    aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),one_one_real) = zero_zero_real ).

% diff_numeral_special(9)
tff(fact_1686_diff__numeral__special_I9_J,axiom,
    minus_minus_rat(one_one_rat,one_one_rat) = zero_zero_rat ).

% diff_numeral_special(9)
tff(fact_1687_diff__numeral__special_I9_J,axiom,
    aa_int_int(aa_int_fun_int_int(minus_minus_int,one_one_int),one_one_int) = zero_zero_int ).

% diff_numeral_special(9)
tff(fact_1688_div__self,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),A) = one_one_real ) ) ).

% div_self
tff(fact_1689_div__self,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( divide_divide_rat(A,A) = one_one_rat ) ) ).

% div_self
tff(fact_1690_div__self,axiom,
    ! [A: nat] :
      ( ( A != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),A) = one_one_nat ) ) ).

% div_self
tff(fact_1691_div__self,axiom,
    ! [A: int] :
      ( ( A != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),A) = one_one_int ) ) ).

% div_self
tff(fact_1692_nonzero__mult__div__cancel__right,axiom,
    ! [B: real,A: real] :
      ( ( B != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),B) = A ) ) ).

% nonzero_mult_div_cancel_right
tff(fact_1693_nonzero__mult__div__cancel__right,axiom,
    ! [B: rat,A: rat] :
      ( ( B != zero_zero_rat )
     => ( divide_divide_rat(aa_rat_rat(times_times_rat(A),B),B) = A ) ) ).

% nonzero_mult_div_cancel_right
tff(fact_1694_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_1695_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_1696_nonzero__mult__div__cancel__left,axiom,
    ! [A: real,B: real] :
      ( ( A != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),A) = B ) ) ).

% nonzero_mult_div_cancel_left
tff(fact_1697_nonzero__mult__div__cancel__left,axiom,
    ! [A: rat,B: rat] :
      ( ( A != zero_zero_rat )
     => ( divide_divide_rat(aa_rat_rat(times_times_rat(A),B),A) = B ) ) ).

% nonzero_mult_div_cancel_left
tff(fact_1698_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_1699_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_1700_div__mult__mult1__if,axiom,
    ! [C: nat,A: nat,B: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B)) = $ite(C = zero_zero_nat,zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)) ).

% div_mult_mult1_if
tff(fact_1701_div__mult__mult1__if,axiom,
    ! [C: int,A: int,B: int] :
      aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B)) = $ite(C = zero_zero_int,zero_zero_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)) ).

% div_mult_mult1_if
tff(fact_1702_div__mult__mult2,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( C != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B) ) ) ).

% div_mult_mult2
tff(fact_1703_div__mult__mult2,axiom,
    ! [C: int,A: int,B: int] :
      ( ( C != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B) ) ) ).

% div_mult_mult2
tff(fact_1704_div__mult__mult1,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( C != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B) ) ) ).

% div_mult_mult1
tff(fact_1705_div__mult__mult1,axiom,
    ! [C: int,A: int,B: int] :
      ( ( C != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B) ) ) ).

% div_mult_mult1
tff(fact_1706_power__0__Suc,axiom,
    ! [Nb: nat] : aa_nat_rat(power_power_rat(zero_zero_rat),aa_nat_nat(suc,Nb)) = zero_zero_rat ).

% power_0_Suc
tff(fact_1707_power__0__Suc,axiom,
    ! [Nb: nat] : aa_nat_uint32(power_power_uint32(zero_zero_uint32),aa_nat_nat(suc,Nb)) = zero_zero_uint32 ).

% power_0_Suc
tff(fact_1708_power__0__Suc,axiom,
    ! [Nb: nat] : aa_nat5599461682457136004l_num1(power_2184487114949457152l_num1(zero_z3563351764282998399l_num1),aa_nat_nat(suc,Nb)) = zero_z3563351764282998399l_num1 ).

% power_0_Suc
tff(fact_1709_power__0__Suc,axiom,
    ! [Nb: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,zero_zero_real),aa_nat_nat(suc,Nb)) = zero_zero_real ).

% power_0_Suc
tff(fact_1710_power__0__Suc,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,zero_zero_nat),aa_nat_nat(suc,Nb)) = zero_zero_nat ).

% power_0_Suc
tff(fact_1711_power__0__Suc,axiom,
    ! [Nb: nat] : aa_nat_int(aa_int_fun_nat_int(power_power_int,zero_zero_int),aa_nat_nat(suc,Nb)) = zero_zero_int ).

% power_0_Suc
tff(fact_1712_power__zero__numeral,axiom,
    ! [K: num] : aa_nat_rat(power_power_rat(zero_zero_rat),aa_num_nat(numeral_numeral_nat,K)) = zero_zero_rat ).

% power_zero_numeral
tff(fact_1713_power__zero__numeral,axiom,
    ! [K: num] : aa_nat_uint32(power_power_uint32(zero_zero_uint32),aa_num_nat(numeral_numeral_nat,K)) = zero_zero_uint32 ).

% power_zero_numeral
tff(fact_1714_power__zero__numeral,axiom,
    ! [K: num] : aa_nat5599461682457136004l_num1(power_2184487114949457152l_num1(zero_z3563351764282998399l_num1),aa_num_nat(numeral_numeral_nat,K)) = zero_z3563351764282998399l_num1 ).

% power_zero_numeral
tff(fact_1715_power__zero__numeral,axiom,
    ! [K: num] : aa_nat_real(aa_real_fun_nat_real(power_power_real,zero_zero_real),aa_num_nat(numeral_numeral_nat,K)) = zero_zero_real ).

% power_zero_numeral
tff(fact_1716_power__zero__numeral,axiom,
    ! [K: num] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,K)) = zero_zero_nat ).

% power_zero_numeral
tff(fact_1717_power__zero__numeral,axiom,
    ! [K: num] : aa_nat_int(aa_int_fun_nat_int(power_power_int,zero_zero_int),aa_num_nat(numeral_numeral_nat,K)) = zero_zero_int ).

% power_zero_numeral
tff(fact_1718_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_1719_of__nat__eq__0__iff,axiom,
    ! [M: nat] :
      ( ( aa_nat_real(semiri5074537144036343181t_real,M) = zero_zero_real )
    <=> ( M = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
tff(fact_1720_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_1721_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_1722_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_1723_of__nat__0__eq__iff,axiom,
    ! [Nb: nat] :
      ( ( zero_zero_rat = aa_nat_rat(semiri681578069525770553at_rat,Nb) )
    <=> ( zero_zero_nat = Nb ) ) ).

% of_nat_0_eq_iff
tff(fact_1724_of__nat__0__eq__iff,axiom,
    ! [Nb: nat] :
      ( ( zero_zero_real = aa_nat_real(semiri5074537144036343181t_real,Nb) )
    <=> ( zero_zero_nat = Nb ) ) ).

% of_nat_0_eq_iff
tff(fact_1725_of__nat__0__eq__iff,axiom,
    ! [Nb: nat] :
      ( ( zero_zero_int = aa_nat_int(semiri1314217659103216013at_int,Nb) )
    <=> ( zero_zero_nat = Nb ) ) ).

% of_nat_0_eq_iff
tff(fact_1726_of__nat__0__eq__iff,axiom,
    ! [Nb: nat] :
      ( ( zero_zero_nat = aa_nat_nat(semiri1316708129612266289at_nat,Nb) )
    <=> ( zero_zero_nat = Nb ) ) ).

% of_nat_0_eq_iff
tff(fact_1727_of__nat__0__eq__iff,axiom,
    ! [Nb: nat] :
      ( ( zero_z3403309356797280102nteger = aa_nat_Code_integer(semiri4939895301339042750nteger,Nb) )
    <=> ( zero_zero_nat = Nb ) ) ).

% of_nat_0_eq_iff
tff(fact_1728_semiring__1__class_Oof__nat__0,axiom,
    aa_nat_uint32(semiri2565882477558803405uint32,zero_zero_nat) = zero_zero_uint32 ).

% semiring_1_class.of_nat_0
tff(fact_1729_semiring__1__class_Oof__nat__0,axiom,
    aa_nat_rat(semiri681578069525770553at_rat,zero_zero_nat) = zero_zero_rat ).

% semiring_1_class.of_nat_0
tff(fact_1730_semiring__1__class_Oof__nat__0,axiom,
    aa_nat_real(semiri5074537144036343181t_real,zero_zero_nat) = zero_zero_real ).

% semiring_1_class.of_nat_0
tff(fact_1731_semiring__1__class_Oof__nat__0,axiom,
    aa_nat_int(semiri1314217659103216013at_int,zero_zero_nat) = zero_zero_int ).

% semiring_1_class.of_nat_0
tff(fact_1732_semiring__1__class_Oof__nat__0,axiom,
    aa_nat_nat(semiri1316708129612266289at_nat,zero_zero_nat) = zero_zero_nat ).

% semiring_1_class.of_nat_0
tff(fact_1733_semiring__1__class_Oof__nat__0,axiom,
    aa_nat_Code_integer(semiri4939895301339042750nteger,zero_zero_nat) = zero_z3403309356797280102nteger ).

% semiring_1_class.of_nat_0
tff(fact_1734_power__Suc0__right,axiom,
    ! [A: uint32] : aa_nat_uint32(power_power_uint32(A),aa_nat_nat(suc,zero_zero_nat)) = A ).

% power_Suc0_right
tff(fact_1735_power__Suc0__right,axiom,
    ! [A: word_N3645301735248828278l_num1] : aa_nat5599461682457136004l_num1(power_2184487114949457152l_num1(A),aa_nat_nat(suc,zero_zero_nat)) = A ).

% power_Suc0_right
tff(fact_1736_power__Suc0__right,axiom,
    ! [A: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(suc,zero_zero_nat)) = A ).

% power_Suc0_right
tff(fact_1737_power__Suc0__right,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(suc,zero_zero_nat)) = A ).

% power_Suc0_right
tff(fact_1738_power__Suc0__right,axiom,
    ! [A: int] : aa_nat_int(aa_int_fun_nat_int(power_power_int,A),aa_nat_nat(suc,zero_zero_nat)) = A ).

% power_Suc0_right
tff(fact_1739_zero__less__Suc,axiom,
    ! [Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(suc,Nb)) ).

% zero_less_Suc
tff(fact_1740_less__Suc0,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(suc,zero_zero_nat))
    <=> ( Nb = zero_zero_nat ) ) ).

% less_Suc0
tff(fact_1741_add__gr__0,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Nb))
    <=> ( 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),Nb) ) ) ).

% add_gr_0
tff(fact_1742_less__one,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),one_one_nat)
    <=> ( Nb = zero_zero_nat ) ) ).

% less_one
tff(fact_1743_div__by__Suc__0,axiom,
    ! [M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),aa_nat_nat(suc,zero_zero_nat)) = M ).

% div_by_Suc_0
tff(fact_1744_zero__less__diff,axiom,
    ! [Nb: 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,Nb),M))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb) ) ).

% zero_less_diff
tff(fact_1745_mult__eq__1__iff,axiom,
    ! [M: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Nb) = aa_nat_nat(suc,zero_zero_nat) )
    <=> ( ( M = aa_nat_nat(suc,zero_zero_nat) )
        & ( Nb = aa_nat_nat(suc,zero_zero_nat) ) ) ) ).

% mult_eq_1_iff
tff(fact_1746_one__eq__mult__iff,axiom,
    ! [M: nat,Nb: nat] :
      ( ( aa_nat_nat(suc,zero_zero_nat) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Nb) )
    <=> ( ( M = aa_nat_nat(suc,zero_zero_nat) )
        & ( Nb = aa_nat_nat(suc,zero_zero_nat) ) ) ) ).

% one_eq_mult_iff
tff(fact_1747_div__less,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Nb) = zero_zero_nat ) ) ).

% div_less
tff(fact_1748_nat__power__eq__Suc__0__iff,axiom,
    ! [Xb: nat,M: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Xb),M) = aa_nat_nat(suc,zero_zero_nat) )
    <=> ( ( M = zero_zero_nat )
        | ( Xb = aa_nat_nat(suc,zero_zero_nat) ) ) ) ).

% nat_power_eq_Suc_0_iff
tff(fact_1749_power__Suc__0,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(suc,zero_zero_nat)),Nb) = aa_nat_nat(suc,zero_zero_nat) ).

% power_Suc_0
tff(fact_1750_mult__less__cancel2,axiom,
    ! [M: nat,K: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),K)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),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),Nb) ) ) ).

% mult_less_cancel2
tff(fact_1751_nat__0__less__mult__iff,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Nb))
    <=> ( 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),Nb) ) ) ).

% nat_0_less_mult_iff
tff(fact_1752_nat__mult__less__cancel__disj,axiom,
    ! [K: nat,M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Nb))
    <=> ( 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),Nb) ) ) ).

% nat_mult_less_cancel_disj
tff(fact_1753_diff__is__0__eq,axiom,
    ! [M: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Nb) = zero_zero_nat )
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb) ) ).

% diff_is_0_eq
tff(fact_1754_diff__is__0__eq_H,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Nb) = zero_zero_nat ) ) ).

% diff_is_0_eq'
tff(fact_1755_nat__zero__less__power__iff,axiom,
    ! [Xb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Xb),Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Xb)
        | ( Nb = zero_zero_nat ) ) ) ).

% nat_zero_less_power_iff
tff(fact_1756_nat__mult__div__cancel__disj,axiom,
    ! [K: nat,M: nat,Nb: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Nb)) = $ite(K = zero_zero_nat,zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Nb)) ).

% nat_mult_div_cancel_disj
tff(fact_1757_vebt__buildupi__rule,axiom,
    ! [Nb: nat] : aa_nat_o(time_htt_VEBT_VEBTi(aa_o_assn(pure_assn,aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)),aa_nat5408703056993926195_VEBTi(vEBT_vebt_buildupi,Nb),aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,vEBT_vebt_buildup(Nb))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))) ).

% vebt_buildupi_rule
tff(fact_1758_eq__divide__eq__numeral1_I1_J,axiom,
    ! [A: real,B: real,W: num] :
      ( ( A = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),numeral_numeral_real(W)) )
    <=> $ite(numeral_numeral_real(W) != zero_zero_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),numeral_numeral_real(W)) = B,A = zero_zero_real) ) ).

% eq_divide_eq_numeral1(1)
tff(fact_1759_eq__divide__eq__numeral1_I1_J,axiom,
    ! [A: rat,B: rat,W: num] :
      ( ( A = divide_divide_rat(B,numeral_numeral_rat(W)) )
    <=> $ite(numeral_numeral_rat(W) != zero_zero_rat,aa_rat_rat(times_times_rat(A),numeral_numeral_rat(W)) = B,A = zero_zero_rat) ) ).

% eq_divide_eq_numeral1(1)
tff(fact_1760_divide__eq__eq__numeral1_I1_J,axiom,
    ! [B: real,W: num,A: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),numeral_numeral_real(W)) = A )
    <=> $ite(numeral_numeral_real(W) != zero_zero_real,B = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),numeral_numeral_real(W)),A = zero_zero_real) ) ).

% divide_eq_eq_numeral1(1)
tff(fact_1761_divide__eq__eq__numeral1_I1_J,axiom,
    ! [B: rat,W: num,A: rat] :
      ( ( divide_divide_rat(B,numeral_numeral_rat(W)) = A )
    <=> $ite(numeral_numeral_rat(W) != zero_zero_rat,B = aa_rat_rat(times_times_rat(A),numeral_numeral_rat(W)),A = zero_zero_rat) ) ).

% divide_eq_eq_numeral1(1)
tff(fact_1762_div__mult__self4,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( ( B != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)),A)),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)) ) ) ).

% div_mult_self4
tff(fact_1763_div__mult__self4,axiom,
    ! [B: int,C: int,A: int] :
      ( ( B != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),C)),A)),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,C),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)) ) ) ).

% div_mult_self4
tff(fact_1764_div__mult__self3,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( ( B != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B)),A)),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)) ) ) ).

% div_mult_self3
tff(fact_1765_div__mult__self3,axiom,
    ! [B: int,C: int,A: int] :
      ( ( B != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),B)),A)),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,C),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)) ) ) ).

% div_mult_self3
tff(fact_1766_div__mult__self2,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( B != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C))),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)) ) ) ).

% div_mult_self2
tff(fact_1767_div__mult__self2,axiom,
    ! [B: int,A: int,C: int] :
      ( ( B != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C))),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,C),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)) ) ) ).

% div_mult_self2
tff(fact_1768_div__mult__self1,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( B != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B))),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)) ) ) ).

% div_mult_self1
tff(fact_1769_div__mult__self1,axiom,
    ! [B: int,A: int,C: int] :
      ( ( B != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,C),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)) ) ) ).

% div_mult_self1
tff(fact_1770_of__nat__le__0__iff,axiom,
    ! [M: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,M)),zero_zero_real)
    <=> ( M = zero_zero_nat ) ) ).

% of_nat_le_0_iff
tff(fact_1771_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_1772_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_1773_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_1774_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_1775_power__eq__0__iff,axiom,
    ! [A: rat,Nb: nat] :
      ( ( aa_nat_rat(power_power_rat(A),Nb) = zero_zero_rat )
    <=> ( ( A = zero_zero_rat )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ) ).

% power_eq_0_iff
tff(fact_1776_power__eq__0__iff,axiom,
    ! [A: real,Nb: nat] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb) = zero_zero_real )
    <=> ( ( A = zero_zero_real )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ) ).

% power_eq_0_iff
tff(fact_1777_power__eq__0__iff,axiom,
    ! [A: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb) = zero_zero_nat )
    <=> ( ( A = zero_zero_nat )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ) ).

% power_eq_0_iff
tff(fact_1778_power__eq__0__iff,axiom,
    ! [A: int,Nb: nat] :
      ( ( aa_nat_int(aa_int_fun_nat_int(power_power_int,A),Nb) = zero_zero_int )
    <=> ( ( A = zero_zero_int )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ) ).

% power_eq_0_iff
tff(fact_1779_Suc__pred,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat))) = Nb ) ) ).

% Suc_pred
tff(fact_1780_one__le__mult__iff,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Nb))
    <=> ( 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)),Nb) ) ) ).

% one_le_mult_iff
tff(fact_1781_div__eq__dividend__iff,axiom,
    ! [M: nat,Nb: 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),Nb) = M )
      <=> ( Nb = one_one_nat ) ) ) ).

% div_eq_dividend_iff
tff(fact_1782_nat__mult__le__cancel__disj,axiom,
    ! [K: nat,M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Nb))
    <=> ( 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),Nb) ) ) ).

% nat_mult_le_cancel_disj
tff(fact_1783_mult__le__cancel2,axiom,
    ! [M: nat,K: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),K)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),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),Nb) ) ) ).

% mult_le_cancel2
tff(fact_1784_div__mult__self1__is__m,axiom,
    ! [Nb: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),M)),Nb) = M ) ) ).

% div_mult_self1_is_m
tff(fact_1785_div__mult__self__is__m,axiom,
    ! [Nb: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Nb)),Nb) = M ) ) ).

% div_mult_self_is_m
tff(fact_1786_zero__eq__power2,axiom,
    ! [A: rat] :
      ( ( aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_rat )
    <=> ( A = zero_zero_rat ) ) ).

% zero_eq_power2
tff(fact_1787_zero__eq__power2,axiom,
    ! [A: real] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_real )
    <=> ( A = zero_zero_real ) ) ).

% zero_eq_power2
tff(fact_1788_zero__eq__power2,axiom,
    ! [A: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_nat )
    <=> ( A = zero_zero_nat ) ) ).

% zero_eq_power2
tff(fact_1789_zero__eq__power2,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))) = zero_zero_int )
    <=> ( A = zero_zero_int ) ) ).

% zero_eq_power2
tff(fact_1790_power__strict__decreasing__iff,axiom,
    ! [B: real,M: nat,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,B),M)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Nb))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),M) ) ) ) ).

% power_strict_decreasing_iff
tff(fact_1791_power__strict__decreasing__iff,axiom,
    ! [B: rat,M: nat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),one_one_rat)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(B),M)),aa_nat_rat(power_power_rat(B),Nb))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),M) ) ) ) ).

% power_strict_decreasing_iff
tff(fact_1792_power__strict__decreasing__iff,axiom,
    ! [B: nat,M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),one_one_nat)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Nb))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),M) ) ) ) ).

% power_strict_decreasing_iff
tff(fact_1793_power__strict__decreasing__iff,axiom,
    ! [B: int,M: nat,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),one_one_int)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,B),M)),aa_nat_int(aa_int_fun_nat_int(power_power_int,B),Nb))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),M) ) ) ) ).

% power_strict_decreasing_iff
tff(fact_1794_power__mono__iff,axiom,
    ! [A: real,B: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Nb))
          <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ) ) ) ).

% power_mono_iff
tff(fact_1795_power__mono__iff,axiom,
    ! [A: rat,B: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),Nb)),aa_nat_rat(power_power_rat(B),Nb))
          <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ) ) ) ).

% power_mono_iff
tff(fact_1796_power__mono__iff,axiom,
    ! [A: nat,B: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Nb))
          <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B) ) ) ) ) ).

% power_mono_iff
tff(fact_1797_power__mono__iff,axiom,
    ! [A: int,B: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A),Nb)),aa_nat_int(aa_int_fun_nat_int(power_power_int,B),Nb))
          <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ) ) ) ).

% power_mono_iff
tff(fact_1798_of__nat__0__less__iff,axiom,
    ! [Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_nat_rat(semiri681578069525770553at_rat,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ).

% of_nat_0_less_iff
tff(fact_1799_of__nat__0__less__iff,axiom,
    ! [Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_nat_real(semiri5074537144036343181t_real,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ).

% of_nat_0_less_iff
tff(fact_1800_of__nat__0__less__iff,axiom,
    ! [Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(semiri1314217659103216013at_int,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ).

% of_nat_0_less_iff
tff(fact_1801_of__nat__0__less__iff,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(semiri1316708129612266289at_nat,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ).

% of_nat_0_less_iff
tff(fact_1802_of__nat__0__less__iff,axiom,
    ! [Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(semiri4939895301339042750nteger,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ).

% of_nat_0_less_iff
tff(fact_1803_Suc__diff__1,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)) = Nb ) ) ).

% Suc_diff_1
tff(fact_1804_bits__1__div__2,axiom,
    aa_uint32_uint32(aa_uin3778660748068847871uint32(divide_divide_uint32,one_one_uint32),numera9087168376688890119uint32(aa_num_num(bit0,one))) = zero_zero_uint32 ).

% bits_1_div_2
tff(fact_1805_bits__1__div__2,axiom,
    divide1791077408188789448l_num1(one_on7727431528512463931l_num1,numera7442385471795722001l_num1(aa_num_num(bit0,one))) = zero_z3563351764282998399l_num1 ).

% bits_1_div_2
tff(fact_1806_bits__1__div__2,axiom,
    aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_nat ).

% bits_1_div_2
tff(fact_1807_bits__1__div__2,axiom,
    aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = zero_zero_int ).

% bits_1_div_2
tff(fact_1808_one__div__two__eq__zero,axiom,
    aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_nat ).

% one_div_two_eq_zero
tff(fact_1809_one__div__two__eq__zero,axiom,
    aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = zero_zero_int ).

% one_div_two_eq_zero
tff(fact_1810_power2__eq__iff__nonneg,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
       => ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
        <=> ( Xb = Y ) ) ) ) ).

% power2_eq_iff_nonneg
tff(fact_1811_power2__eq__iff__nonneg,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Xb)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Y)
       => ( ( aa_nat_rat(power_power_rat(Xb),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))) )
        <=> ( Xb = Y ) ) ) ) ).

% power2_eq_iff_nonneg
tff(fact_1812_power2__eq__iff__nonneg,axiom,
    ! [Xb: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Xb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Y)
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
        <=> ( Xb = Y ) ) ) ) ).

% power2_eq_iff_nonneg
tff(fact_1813_power2__eq__iff__nonneg,axiom,
    ! [Xb: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xb)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
       => ( ( aa_nat_int(aa_int_fun_nat_int(power_power_int,Xb),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))) )
        <=> ( Xb = Y ) ) ) ) ).

% power2_eq_iff_nonneg
tff(fact_1814_power2__less__eq__zero__iff,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),zero_zero_real)
    <=> ( A = zero_zero_real ) ) ).

% power2_less_eq_zero_iff
tff(fact_1815_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_1816_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_1817_zero__less__power2,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
    <=> ( A != zero_zero_real ) ) ).

% zero_less_power2
tff(fact_1818_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_1819_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_1820_sum__power2__eq__zero__iff,axiom,
    ! [Xb: rat,Y: rat] :
      ( ( aa_rat_rat(plus_plus_rat(aa_nat_rat(power_power_rat(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = zero_zero_rat )
    <=> ( ( Xb = zero_zero_rat )
        & ( Y = zero_zero_rat ) ) ) ).

% sum_power2_eq_zero_iff
tff(fact_1821_sum__power2__eq__zero__iff,axiom,
    ! [Xb: real,Y: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = zero_zero_real )
    <=> ( ( Xb = zero_zero_real )
        & ( Y = zero_zero_real ) ) ) ).

% sum_power2_eq_zero_iff
tff(fact_1822_sum__power2__eq__zero__iff,axiom,
    ! [Xb: int,Y: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,Xb),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 )
    <=> ( ( Xb = zero_zero_int )
        & ( Y = zero_zero_int ) ) ) ).

% sum_power2_eq_zero_iff
tff(fact_1823_power__decreasing__iff,axiom,
    ! [B: real,M: nat,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,B),M)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Nb))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),M) ) ) ) ).

% power_decreasing_iff
tff(fact_1824_power__decreasing__iff,axiom,
    ! [B: rat,M: nat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),one_one_rat)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(B),M)),aa_nat_rat(power_power_rat(B),Nb))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),M) ) ) ) ).

% power_decreasing_iff
tff(fact_1825_power__decreasing__iff,axiom,
    ! [B: nat,M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),one_one_nat)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Nb))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),M) ) ) ) ).

% power_decreasing_iff
tff(fact_1826_power__decreasing__iff,axiom,
    ! [B: int,M: nat,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),one_one_int)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,B),M)),aa_nat_int(aa_int_fun_nat_int(power_power_int,B),Nb))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),M) ) ) ) ).

% power_decreasing_iff
tff(fact_1827_of__nat__zero__less__power__iff,axiom,
    ! [Xb: nat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_nat_rat(power_power_rat(aa_nat_rat(semiri681578069525770553at_rat,Xb)),Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Xb)
        | ( Nb = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
tff(fact_1828_of__nat__zero__less__power__iff,axiom,
    ! [Xb: nat,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(semiri5074537144036343181t_real,Xb)),Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Xb)
        | ( Nb = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
tff(fact_1829_of__nat__zero__less__power__iff,axiom,
    ! [Xb: nat,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_nat_int(semiri1314217659103216013at_int,Xb)),Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Xb)
        | ( Nb = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
tff(fact_1830_of__nat__zero__less__power__iff,axiom,
    ! [Xb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(semiri1316708129612266289at_nat,Xb)),Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Xb)
        | ( Nb = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
tff(fact_1831_of__nat__zero__less__power__iff,axiom,
    ! [Xb: nat,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(power_8256067586552552935nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,Xb)),Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Xb)
        | ( Nb = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
tff(fact_1832_htt__vebt__inserti,axiom,
    ! [Ta: vEBT_VEBT,Tib: vEBT_VEBTi,Xb: nat] : aa_nat_o(time_htt_VEBT_VEBTi(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Ta),Tib),aa_nat5408703056993926195_VEBTi(aa_VEB4833742232818206615_VEBTi(vEBT_vebt_inserti,Tib),Xb),aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,vEBT_vebt_insert(Ta,Xb))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,one))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,one))))),aa_VEBT_VEBT_nat(vEBT_VEBT_height,Ta)))) ).

% htt_vebt_inserti
tff(fact_1833_VEBT__internal_OT__vebt__buildupi_Osimps_I2_J,axiom,
    aa_nat_nat(vEBT_V441764108873111860ildupi,aa_nat_nat(suc,zero_zero_nat)) = aa_nat_nat(suc,zero_zero_nat) ).

% VEBT_internal.T_vebt_buildupi.simps(2)
tff(fact_1834_VEBT__internal_OT__vebt__buildupi_Osimps_I1_J,axiom,
    aa_nat_nat(vEBT_V441764108873111860ildupi,zero_zero_nat) = aa_nat_nat(suc,zero_zero_nat) ).

% VEBT_internal.T_vebt_buildupi.simps(1)
tff(fact_1835_power__0__left,axiom,
    ! [Nb: nat] :
      aa_nat_rat(power_power_rat(zero_zero_rat),Nb) = $ite(Nb = zero_zero_nat,one_one_rat,zero_zero_rat) ).

% power_0_left
tff(fact_1836_power__0__left,axiom,
    ! [Nb: nat] :
      aa_nat_uint32(power_power_uint32(zero_zero_uint32),Nb) = $ite(Nb = zero_zero_nat,one_one_uint32,zero_zero_uint32) ).

% power_0_left
tff(fact_1837_power__0__left,axiom,
    ! [Nb: nat] :
      aa_nat5599461682457136004l_num1(power_2184487114949457152l_num1(zero_z3563351764282998399l_num1),Nb) = $ite(Nb = zero_zero_nat,one_on7727431528512463931l_num1,zero_z3563351764282998399l_num1) ).

% power_0_left
tff(fact_1838_power__0__left,axiom,
    ! [Nb: nat] :
      aa_nat_real(aa_real_fun_nat_real(power_power_real,zero_zero_real),Nb) = $ite(Nb = zero_zero_nat,one_one_real,zero_zero_real) ).

% power_0_left
tff(fact_1839_power__0__left,axiom,
    ! [Nb: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,zero_zero_nat),Nb) = $ite(Nb = zero_zero_nat,one_one_nat,zero_zero_nat) ).

% power_0_left
tff(fact_1840_power__0__left,axiom,
    ! [Nb: nat] :
      aa_nat_int(aa_int_fun_nat_int(power_power_int,zero_zero_int),Nb) = $ite(Nb = zero_zero_nat,one_one_int,zero_zero_int) ).

% power_0_left
tff(fact_1841_zero__power,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_rat(power_power_rat(zero_zero_rat),Nb) = zero_zero_rat ) ) ).

% zero_power
tff(fact_1842_zero__power,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_uint32(power_power_uint32(zero_zero_uint32),Nb) = zero_zero_uint32 ) ) ).

% zero_power
tff(fact_1843_zero__power,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat5599461682457136004l_num1(power_2184487114949457152l_num1(zero_z3563351764282998399l_num1),Nb) = zero_z3563351764282998399l_num1 ) ) ).

% zero_power
tff(fact_1844_zero__power,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,zero_zero_real),Nb) = zero_zero_real ) ) ).

% zero_power
tff(fact_1845_zero__power,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,zero_zero_nat),Nb) = zero_zero_nat ) ) ).

% zero_power
tff(fact_1846_zero__power,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_int(aa_int_fun_nat_int(power_power_int,zero_zero_int),Nb) = zero_zero_int ) ) ).

% zero_power
tff(fact_1847_le__numeral__extra_I3_J,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),zero_zero_real) ).

% le_numeral_extra(3)
tff(fact_1848_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_1849_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_1850_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_1851_field__lbound__gt__zero,axiom,
    ! [D1: real,D22: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D1)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D22)
       => ? [E2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),E2)
            & aa_real_o(aa_real_fun_real_o(ord_less_real,E2),D1)
            & aa_real_o(aa_real_fun_real_o(ord_less_real,E2),D22) ) ) ) ).

% field_lbound_gt_zero
tff(fact_1852_field__lbound__gt__zero,axiom,
    ! [D1: rat,D22: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),D1)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),D22)
       => ? [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_rat,E2),D1)
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,E2),D22) ) ) ) ).

% field_lbound_gt_zero
tff(fact_1853_less__numeral__extra_I3_J,axiom,
    ~ aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),zero_zero_real) ).

% less_numeral_extra(3)
tff(fact_1854_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_1855_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_1856_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_1857_zero__neq__numeral,axiom,
    ! [Nb: num] : zero_zero_real != numeral_numeral_real(Nb) ).

% zero_neq_numeral
tff(fact_1858_zero__neq__numeral,axiom,
    ! [Nb: num] : zero_zero_rat != numeral_numeral_rat(Nb) ).

% zero_neq_numeral
tff(fact_1859_zero__neq__numeral,axiom,
    ! [Nb: num] : zero_zero_nat != aa_num_nat(numeral_numeral_nat,Nb) ).

% zero_neq_numeral
tff(fact_1860_zero__neq__numeral,axiom,
    ! [Nb: num] : zero_zero_int != aa_num_int(numeral_numeral_int,Nb) ).

% zero_neq_numeral
tff(fact_1861_zero__neq__one,axiom,
    zero_z3563351764282998399l_num1 != one_on7727431528512463931l_num1 ).

% zero_neq_one
tff(fact_1862_zero__neq__one,axiom,
    zero_zero_uint32 != one_one_uint32 ).

% zero_neq_one
tff(fact_1863_zero__neq__one,axiom,
    zero_zero_real != one_one_real ).

% zero_neq_one
tff(fact_1864_zero__neq__one,axiom,
    zero_zero_rat != one_one_rat ).

% zero_neq_one
tff(fact_1865_zero__neq__one,axiom,
    zero_zero_nat != one_one_nat ).

% zero_neq_one
tff(fact_1866_zero__neq__one,axiom,
    zero_zero_int != one_one_int ).

% zero_neq_one
tff(fact_1867_mult__right__cancel,axiom,
    ! [C: real,A: real,B: real] :
      ( ( C != zero_zero_real )
     => ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C) )
      <=> ( A = B ) ) ) ).

% mult_right_cancel
tff(fact_1868_mult__right__cancel,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( ( aa_rat_rat(times_times_rat(A),C) = aa_rat_rat(times_times_rat(B),C) )
      <=> ( A = B ) ) ) ).

% mult_right_cancel
tff(fact_1869_mult__right__cancel,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( C != zero_zero_nat )
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C) )
      <=> ( A = B ) ) ) ).

% mult_right_cancel
tff(fact_1870_mult__right__cancel,axiom,
    ! [C: int,A: int,B: int] :
      ( ( C != zero_zero_int )
     => ( ( aa_int_int(aa_int_fun_int_int(times_times_int,A),C) = aa_int_int(aa_int_fun_int_int(times_times_int,B),C) )
      <=> ( A = B ) ) ) ).

% mult_right_cancel
tff(fact_1871_mult__left__cancel,axiom,
    ! [C: real,A: real,B: real] :
      ( ( C != zero_zero_real )
     => ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B) )
      <=> ( A = B ) ) ) ).

% mult_left_cancel
tff(fact_1872_mult__left__cancel,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( ( aa_rat_rat(times_times_rat(C),A) = aa_rat_rat(times_times_rat(C),B) )
      <=> ( A = B ) ) ) ).

% mult_left_cancel
tff(fact_1873_mult__left__cancel,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( C != zero_zero_nat )
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B) )
      <=> ( A = B ) ) ) ).

% mult_left_cancel
tff(fact_1874_mult__left__cancel,axiom,
    ! [C: int,A: int,B: int] :
      ( ( C != zero_zero_int )
     => ( ( aa_int_int(aa_int_fun_int_int(times_times_int,C),A) = aa_int_int(aa_int_fun_int_int(times_times_int,C),B) )
      <=> ( A = B ) ) ) ).

% mult_left_cancel
tff(fact_1875_no__zero__divisors,axiom,
    ! [A: real,B: real] :
      ( ( A != zero_zero_real )
     => ( ( B != zero_zero_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B) != zero_zero_real ) ) ) ).

% no_zero_divisors
tff(fact_1876_no__zero__divisors,axiom,
    ! [A: rat,B: rat] :
      ( ( A != zero_zero_rat )
     => ( ( B != zero_zero_rat )
       => ( aa_rat_rat(times_times_rat(A),B) != zero_zero_rat ) ) ) ).

% no_zero_divisors
tff(fact_1877_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_1878_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_1879_divisors__zero,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B) = zero_zero_real )
     => ( ( A = zero_zero_real )
        | ( B = zero_zero_real ) ) ) ).

% divisors_zero
tff(fact_1880_divisors__zero,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(times_times_rat(A),B) = zero_zero_rat )
     => ( ( A = zero_zero_rat )
        | ( B = zero_zero_rat ) ) ) ).

% divisors_zero
tff(fact_1881_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_1882_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_1883_mult__not__zero,axiom,
    ! [A: uint32,B: uint32] :
      ( ( aa_uint32_uint32(times_times_uint32(A),B) != zero_zero_uint32 )
     => ( ( A != zero_zero_uint32 )
        & ( B != zero_zero_uint32 ) ) ) ).

% mult_not_zero
tff(fact_1884_mult__not__zero,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B) != zero_zero_real )
     => ( ( A != zero_zero_real )
        & ( B != zero_zero_real ) ) ) ).

% mult_not_zero
tff(fact_1885_mult__not__zero,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(times_times_rat(A),B) != zero_zero_rat )
     => ( ( A != zero_zero_rat )
        & ( B != zero_zero_rat ) ) ) ).

% mult_not_zero
tff(fact_1886_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_1887_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_1888_semiring__1__no__zero__divisors__class_Opower__not__zero,axiom,
    ! [A: rat,Nb: nat] :
      ( ( A != zero_zero_rat )
     => ( aa_nat_rat(power_power_rat(A),Nb) != zero_zero_rat ) ) ).

% semiring_1_no_zero_divisors_class.power_not_zero
tff(fact_1889_semiring__1__no__zero__divisors__class_Opower__not__zero,axiom,
    ! [A: real,Nb: nat] :
      ( ( A != zero_zero_real )
     => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb) != zero_zero_real ) ) ).

% semiring_1_no_zero_divisors_class.power_not_zero
tff(fact_1890_semiring__1__no__zero__divisors__class_Opower__not__zero,axiom,
    ! [A: nat,Nb: nat] :
      ( ( A != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb) != zero_zero_nat ) ) ).

% semiring_1_no_zero_divisors_class.power_not_zero
tff(fact_1891_semiring__1__no__zero__divisors__class_Opower__not__zero,axiom,
    ! [A: int,Nb: nat] :
      ( ( A != zero_zero_int )
     => ( aa_nat_int(aa_int_fun_nat_int(power_power_int,A),Nb) != zero_zero_int ) ) ).

% semiring_1_no_zero_divisors_class.power_not_zero
tff(fact_1892_num_Osize_I4_J,axiom,
    aa_num_nat(size_size_num,one) = zero_zero_nat ).

% num.size(4)
tff(fact_1893_not0__implies__Suc,axiom,
    ! [Nb: nat] :
      ( ( Nb != zero_zero_nat )
     => ? [M2: nat] : Nb = aa_nat_nat(suc,M2) ) ).

% not0_implies_Suc
tff(fact_1894_Zero__not__Suc,axiom,
    ! [M: nat] : zero_zero_nat != aa_nat_nat(suc,M) ).

% Zero_not_Suc
tff(fact_1895_Zero__neq__Suc,axiom,
    ! [M: nat] : zero_zero_nat != aa_nat_nat(suc,M) ).

% Zero_neq_Suc
tff(fact_1896_Suc__neq__Zero,axiom,
    ! [M: nat] : aa_nat_nat(suc,M) != zero_zero_nat ).

% Suc_neq_Zero
tff(fact_1897_zero__induct,axiom,
    ! [P: fun_nat_o,K: nat] :
      ( aa_nat_o(P,K)
     => ( ! [N2: nat] :
            ( aa_nat_o(P,aa_nat_nat(suc,N2))
           => aa_nat_o(P,N2) )
       => aa_nat_o(P,zero_zero_nat) ) ) ).

% zero_induct
tff(fact_1898_diff__induct,axiom,
    ! [P: fun_nat_fun_nat_o,M: nat,Nb: nat] :
      ( ! [X2: nat] : aa_nat_o(aa_nat_fun_nat_o(P,X2),zero_zero_nat)
     => ( ! [Y4: nat] : aa_nat_o(aa_nat_fun_nat_o(P,zero_zero_nat),aa_nat_nat(suc,Y4))
       => ( ! [X2: nat,Y4: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(P,X2),Y4)
             => aa_nat_o(aa_nat_fun_nat_o(P,aa_nat_nat(suc,X2)),aa_nat_nat(suc,Y4)) )
         => aa_nat_o(aa_nat_fun_nat_o(P,M),Nb) ) ) ) ).

% diff_induct
tff(fact_1899_nat__induct,axiom,
    ! [P: fun_nat_o,Nb: nat] :
      ( aa_nat_o(P,zero_zero_nat)
     => ( ! [N2: nat] :
            ( aa_nat_o(P,N2)
           => aa_nat_o(P,aa_nat_nat(suc,N2)) )
       => aa_nat_o(P,Nb) ) ) ).

% nat_induct
tff(fact_1900_old_Onat_Oexhaust,axiom,
    ! [Y: nat] :
      ( ( Y != zero_zero_nat )
     => ~ ! [Nat3: nat] : Y != aa_nat_nat(suc,Nat3) ) ).

% old.nat.exhaust
tff(fact_1901_nat_OdiscI,axiom,
    ! [Nat: nat,X23: nat] :
      ( ( Nat = aa_nat_nat(suc,X23) )
     => ( Nat != zero_zero_nat ) ) ).

% nat.discI
tff(fact_1902_old_Onat_Odistinct_I1_J,axiom,
    ! [Nat2: nat] : zero_zero_nat != aa_nat_nat(suc,Nat2) ).

% old.nat.distinct(1)
tff(fact_1903_old_Onat_Odistinct_I2_J,axiom,
    ! [Nat2: nat] : aa_nat_nat(suc,Nat2) != zero_zero_nat ).

% old.nat.distinct(2)
tff(fact_1904_nat_Odistinct_I1_J,axiom,
    ! [X23: nat] : zero_zero_nat != aa_nat_nat(suc,X23) ).

% nat.distinct(1)
tff(fact_1905_infinite__descent0,axiom,
    ! [P: fun_nat_o,Nb: nat] :
      ( aa_nat_o(P,zero_zero_nat)
     => ( ! [N2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)
           => ( ~ aa_nat_o(P,N2)
             => ? [M3: nat] :
                  ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M3),N2)
                  & ~ aa_nat_o(P,M3) ) ) )
       => aa_nat_o(P,Nb) ) ) ).

% infinite_descent0
tff(fact_1906_gr__implies__not0,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb)
     => ( Nb != zero_zero_nat ) ) ).

% gr_implies_not0
tff(fact_1907_less__zeroE,axiom,
    ! [Nb: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),zero_zero_nat) ).

% less_zeroE
tff(fact_1908_not__less0,axiom,
    ! [Nb: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),zero_zero_nat) ).

% not_less0
tff(fact_1909_not__gr0,axiom,
    ! [Nb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
    <=> ( Nb = zero_zero_nat ) ) ).

% not_gr0
tff(fact_1910_gr0I,axiom,
    ! [Nb: nat] :
      ( ( Nb != zero_zero_nat )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ).

% gr0I
tff(fact_1911_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_1912_le__0__eq,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),zero_zero_nat)
    <=> ( Nb = zero_zero_nat ) ) ).

% le_0_eq
tff(fact_1913_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_1914_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_1915_less__eq__nat_Osimps_I1_J,axiom,
    ! [Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Nb) ).

% less_eq_nat.simps(1)
tff(fact_1916_plus__nat_Oadd__0,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,zero_zero_nat),Nb) = Nb ).

% plus_nat.add_0
tff(fact_1917_add__eq__self__zero,axiom,
    ! [M: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Nb) = M )
     => ( Nb = zero_zero_nat ) ) ).

% add_eq_self_zero
tff(fact_1918_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_1919_diffs0__imp__equal,axiom,
    ! [M: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Nb) = zero_zero_nat )
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),M) = zero_zero_nat )
       => ( M = Nb ) ) ) ).

% diffs0_imp_equal
tff(fact_1920_nat__mult__eq__cancel__disj,axiom,
    ! [K: nat,M: nat,Nb: 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),Nb) )
    <=> ( ( K = zero_zero_nat )
        | ( M = Nb ) ) ) ).

% nat_mult_eq_cancel_disj
tff(fact_1921_mult__0,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,zero_zero_nat),Nb) = zero_zero_nat ).

% mult_0
tff(fact_1922_VEBTi_Osize_I4_J,axiom,
    ! [X21: $o,X22: $o] : aa_VEBT_VEBTi_nat(size_size_VEBT_VEBTi,vEBT_Leafi((X21),(X22))) = zero_zero_nat ).

% VEBTi.size(4)
tff(fact_1923_power__eq__imp__eq__base,axiom,
    ! [A: real,Nb: nat,B: real] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb) = aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Nb) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
           => ( A = B ) ) ) ) ) ).

% power_eq_imp_eq_base
tff(fact_1924_power__eq__imp__eq__base,axiom,
    ! [A: rat,Nb: nat,B: rat] :
      ( ( aa_nat_rat(power_power_rat(A),Nb) = aa_nat_rat(power_power_rat(B),Nb) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
           => ( A = B ) ) ) ) ) ).

% power_eq_imp_eq_base
tff(fact_1925_power__eq__imp__eq__base,axiom,
    ! [A: nat,Nb: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Nb) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
           => ( A = B ) ) ) ) ) ).

% power_eq_imp_eq_base
tff(fact_1926_power__eq__imp__eq__base,axiom,
    ! [A: int,Nb: nat,B: int] :
      ( ( aa_nat_int(aa_int_fun_nat_int(power_power_int,A),Nb) = aa_nat_int(aa_int_fun_nat_int(power_power_int,B),Nb) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
           => ( A = B ) ) ) ) ) ).

% power_eq_imp_eq_base
tff(fact_1927_power__eq__iff__eq__base,axiom,
    ! [Nb: nat,A: real,B: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B)
         => ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb) = aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Nb) )
          <=> ( A = B ) ) ) ) ) ).

% power_eq_iff_eq_base
tff(fact_1928_power__eq__iff__eq__base,axiom,
    ! [Nb: nat,A: rat,B: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
         => ( ( aa_nat_rat(power_power_rat(A),Nb) = aa_nat_rat(power_power_rat(B),Nb) )
          <=> ( A = B ) ) ) ) ) ).

% power_eq_iff_eq_base
tff(fact_1929_power__eq__iff__eq__base,axiom,
    ! [Nb: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B)
         => ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Nb) )
          <=> ( A = B ) ) ) ) ) ).

% power_eq_iff_eq_base
tff(fact_1930_power__eq__iff__eq__base,axiom,
    ! [Nb: nat,A: int,B: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
         => ( ( aa_nat_int(aa_int_fun_nat_int(power_power_int,A),Nb) = aa_nat_int(aa_int_fun_nat_int(power_power_int,B),Nb) )
          <=> ( A = B ) ) ) ) ) ).

% power_eq_iff_eq_base
tff(fact_1931_lambda__zero,axiom,
    aTP_Lamm_ek = times_times_uint32(zero_zero_uint32) ).

% lambda_zero
tff(fact_1932_lambda__zero,axiom,
    aTP_Lamm_el = aa_rea7340605660300375871l_real(times_times_real,zero_zero_real) ).

% lambda_zero
tff(fact_1933_lambda__zero,axiom,
    aTP_Lamm_em = times_times_rat(zero_zero_rat) ).

% lambda_zero
tff(fact_1934_lambda__zero,axiom,
    aTP_Lamm_en = aa_nat_fun_nat_nat(times_times_nat,zero_zero_nat) ).

% lambda_zero
tff(fact_1935_lambda__zero,axiom,
    aTP_Lamm_eo = aa_int_fun_int_int(times_times_int,zero_zero_int) ).

% lambda_zero
tff(fact_1936_VEBT__internal_OTb_H_Osimps_I1_J,axiom,
    aa_nat_nat(vEBT_VEBT_Tb2,zero_zero_nat) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)) ).

% VEBT_internal.Tb'.simps(1)
tff(fact_1937_VEBT__internal_OminNulli_Osimps_I1_J,axiom,
    aa_VEB8194921341634505877Heap_o(vEBT_VEBT_minNulli,vEBT_Leafi($false,$false)) = aa_o_H7399033255965945470Heap_o(heap_Time_return_o,$true) ).

% VEBT_internal.minNulli.simps(1)
tff(fact_1938_VEBT__internal_OminNulli_Osimps_I2_J,axiom,
    ! [Uv: $o] : aa_VEB8194921341634505877Heap_o(vEBT_VEBT_minNulli,vEBT_Leafi($true,(Uv))) = aa_o_H7399033255965945470Heap_o(heap_Time_return_o,$false) ).

% VEBT_internal.minNulli.simps(2)
tff(fact_1939_VEBT__internal_OminNulli_Osimps_I3_J,axiom,
    ! [Uu: $o] : aa_VEB8194921341634505877Heap_o(vEBT_VEBT_minNulli,vEBT_Leafi((Uu),$true)) = aa_o_H7399033255965945470Heap_o(heap_Time_return_o,$false) ).

% VEBT_internal.minNulli.simps(3)
tff(fact_1940_power__strict__mono,axiom,
    ! [A: real,B: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Nb)) ) ) ) ).

% power_strict_mono
tff(fact_1941_power__strict__mono,axiom,
    ! [A: rat,B: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(A),Nb)),aa_nat_rat(power_power_rat(B),Nb)) ) ) ) ).

% power_strict_mono
tff(fact_1942_power__strict__mono,axiom,
    ! [A: nat,B: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Nb)) ) ) ) ).

% power_strict_mono
tff(fact_1943_power__strict__mono,axiom,
    ! [A: int,B: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A),Nb)),aa_nat_int(aa_int_fun_nat_int(power_power_int,B),Nb)) ) ) ) ).

% power_strict_mono
tff(fact_1944_not__numeral__le__zero,axiom,
    ! [Nb: num] : ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,numeral_numeral_real(Nb)),zero_zero_real) ).

% not_numeral_le_zero
tff(fact_1945_not__numeral__le__zero,axiom,
    ! [Nb: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,numeral_numeral_rat(Nb)),zero_zero_rat) ).

% not_numeral_le_zero
tff(fact_1946_not__numeral__le__zero,axiom,
    ! [Nb: num] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,Nb)),zero_zero_nat) ).

% not_numeral_le_zero
tff(fact_1947_not__numeral__le__zero,axiom,
    ! [Nb: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,Nb)),zero_zero_int) ).

% not_numeral_le_zero
tff(fact_1948_zero__le__numeral,axiom,
    ! [Nb: num] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),numeral_numeral_real(Nb)) ).

% zero_le_numeral
tff(fact_1949_zero__le__numeral,axiom,
    ! [Nb: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),numeral_numeral_rat(Nb)) ).

% zero_le_numeral
tff(fact_1950_zero__le__numeral,axiom,
    ! [Nb: num] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,Nb)) ).

% zero_le_numeral
tff(fact_1951_zero__le__numeral,axiom,
    ! [Nb: num] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_num_int(numeral_numeral_int,Nb)) ).

% zero_le_numeral
tff(fact_1952_VEBT__internal_OminNull_Osimps_I5_J,axiom,
    ! [Uz: product_prod_nat_nat,Vaa: nat,Vb: list_VEBT_VEBT,Vc2: vEBT_VEBT] : ~ aa_VEBT_VEBT_o(vEBT_VEBT_minNull,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(Uz),Vaa,Vb),Vc2)) ).

% VEBT_internal.minNull.simps(5)
tff(fact_1953_not__numeral__less__zero,axiom,
    ! [Nb: num] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,numeral_numeral_real(Nb)),zero_zero_real) ).

% not_numeral_less_zero
tff(fact_1954_not__numeral__less__zero,axiom,
    ! [Nb: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,numeral_numeral_rat(Nb)),zero_zero_rat) ).

% not_numeral_less_zero
tff(fact_1955_not__numeral__less__zero,axiom,
    ! [Nb: num] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_num_nat(numeral_numeral_nat,Nb)),zero_zero_nat) ).

% not_numeral_less_zero
tff(fact_1956_not__numeral__less__zero,axiom,
    ! [Nb: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,Nb)),zero_zero_int) ).

% not_numeral_less_zero
tff(fact_1957_zero__less__numeral,axiom,
    ! [Nb: num] : aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),numeral_numeral_real(Nb)) ).

% zero_less_numeral
tff(fact_1958_zero__less__numeral,axiom,
    ! [Nb: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),numeral_numeral_rat(Nb)) ).

% zero_less_numeral
tff(fact_1959_zero__less__numeral,axiom,
    ! [Nb: num] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,Nb)) ).

% zero_less_numeral
tff(fact_1960_zero__less__numeral,axiom,
    ! [Nb: num] : aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_num_int(numeral_numeral_int,Nb)) ).

% zero_less_numeral
tff(fact_1961_not__one__le__zero,axiom,
    ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),zero_zero_real) ).

% not_one_le_zero
tff(fact_1962_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_1963_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_1964_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_1965_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),one_one_real) ).

% linordered_nonzero_semiring_class.zero_le_one
tff(fact_1966_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_1967_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_1968_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_1969_zero__less__one__class_Ozero__le__one,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),one_one_real) ).

% zero_less_one_class.zero_le_one
tff(fact_1970_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_1971_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_1972_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_1973_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B)) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
tff(fact_1974_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B)) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
tff(fact_1975_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B)) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
tff(fact_1976_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B)) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
tff(fact_1977_zero__le__mult__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),zero_zero_real) ) ) ) ).

% zero_le_mult_iff
tff(fact_1978_zero__le__mult__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A),B))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),zero_zero_rat) ) ) ) ).

% zero_le_mult_iff
tff(fact_1979_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_1980_mult__nonneg__nonpos2,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),A)),zero_zero_real) ) ) ).

% mult_nonneg_nonpos2
tff(fact_1981_mult__nonneg__nonpos2,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(B),A)),zero_zero_rat) ) ) ).

% mult_nonneg_nonpos2
tff(fact_1982_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_1983_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_1984_mult__nonpos__nonneg,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),zero_zero_real) ) ) ).

% mult_nonpos_nonneg
tff(fact_1985_mult__nonpos__nonneg,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),B)),zero_zero_rat) ) ) ).

% mult_nonpos_nonneg
tff(fact_1986_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_1987_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_1988_mult__nonneg__nonpos,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),zero_zero_real) ) ) ).

% mult_nonneg_nonpos
tff(fact_1989_mult__nonneg__nonpos,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),B)),zero_zero_rat) ) ) ).

% mult_nonneg_nonpos
tff(fact_1990_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_1991_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_1992_mult__nonneg__nonneg,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)) ) ) ).

% mult_nonneg_nonneg
tff(fact_1993_mult__nonneg__nonneg,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A),B)) ) ) ).

% mult_nonneg_nonneg
tff(fact_1994_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_1995_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_1996_split__mult__neg__le,axiom,
    ! [A: real,B: real] :
      ( ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),zero_zero_real) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B) ) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),zero_zero_real) ) ).

% split_mult_neg_le
tff(fact_1997_split__mult__neg__le,axiom,
    ! [A: rat,B: rat] :
      ( ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),zero_zero_rat) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B) ) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),B)),zero_zero_rat) ) ).

% split_mult_neg_le
tff(fact_1998_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_1999_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_2000_mult__le__0__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),zero_zero_real)
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),zero_zero_real) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B) ) ) ) ).

% mult_le_0_iff
tff(fact_2001_mult__le__0__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),B)),zero_zero_rat)
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),zero_zero_rat) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B) ) ) ) ).

% mult_le_0_iff
tff(fact_2002_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_2003_mult__right__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)) ) ) ).

% mult_right_mono
tff(fact_2004_mult__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),C)) ) ) ).

% mult_right_mono
tff(fact_2005_mult__right__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) ) ) ).

% mult_right_mono
tff(fact_2006_mult__right__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ) ) ).

% mult_right_mono
tff(fact_2007_mult__right__mono__neg,axiom,
    ! [B: real,A: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)) ) ) ).

% mult_right_mono_neg
tff(fact_2008_mult__right__mono__neg,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),C)) ) ) ).

% mult_right_mono_neg
tff(fact_2009_mult__right__mono__neg,axiom,
    ! [B: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ) ) ).

% mult_right_mono_neg
tff(fact_2010_mult__left__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B)) ) ) ).

% mult_left_mono
tff(fact_2011_mult__left__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B)) ) ) ).

% mult_left_mono
tff(fact_2012_mult__left__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B)) ) ) ).

% mult_left_mono
tff(fact_2013_mult__left__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B)) ) ) ).

% mult_left_mono
tff(fact_2014_mult__nonpos__nonpos,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)) ) ) ).

% mult_nonpos_nonpos
tff(fact_2015_mult__nonpos__nonpos,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A),B)) ) ) ).

% mult_nonpos_nonpos
tff(fact_2016_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_2017_mult__left__mono__neg,axiom,
    ! [B: real,A: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B)) ) ) ).

% mult_left_mono_neg
tff(fact_2018_mult__left__mono__neg,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B)) ) ) ).

% mult_left_mono_neg
tff(fact_2019_mult__left__mono__neg,axiom,
    ! [B: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B)) ) ) ).

% mult_left_mono_neg
tff(fact_2020_split__mult__pos__le,axiom,
    ! [A: real,B: real] :
      ( ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),zero_zero_real) ) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)) ) ).

% split_mult_pos_le
tff(fact_2021_split__mult__pos__le,axiom,
    ! [A: rat,B: rat] :
      ( ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),zero_zero_rat) ) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A),B)) ) ).

% split_mult_pos_le
tff(fact_2022_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_2023_zero__le__square,axiom,
    ! [A: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),A)) ).

% zero_le_square
tff(fact_2024_zero__le__square,axiom,
    ! [A: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A),A)) ).

% zero_le_square
tff(fact_2025_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_2026_mult__mono_H,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),D)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),D)) ) ) ) ) ).

% mult_mono'
tff(fact_2027_mult__mono_H,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),D)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),D)) ) ) ) ) ).

% mult_mono'
tff(fact_2028_mult__mono_H,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),D)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D)) ) ) ) ) ).

% mult_mono'
tff(fact_2029_mult__mono_H,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),D)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D)) ) ) ) ) ).

% mult_mono'
tff(fact_2030_mult__mono,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),D)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),D)) ) ) ) ) ).

% mult_mono
tff(fact_2031_mult__mono,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),D)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),D)) ) ) ) ) ).

% mult_mono
tff(fact_2032_mult__mono,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),D)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D)) ) ) ) ) ).

% mult_mono
tff(fact_2033_mult__mono,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),D)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D)) ) ) ) ) ).

% mult_mono
tff(fact_2034_add__less__zeroD,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Y)),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),zero_zero_real)
        | aa_real_o(aa_real_fun_real_o(ord_less_real,Y),zero_zero_real) ) ) ).

% add_less_zeroD
tff(fact_2035_add__less__zeroD,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(Xb),Y)),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),zero_zero_rat)
        | aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),zero_zero_rat) ) ) ).

% add_less_zeroD
tff(fact_2036_add__less__zeroD,axiom,
    ! [Xb: 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,Xb),Y)),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),zero_zero_int)
        | aa_int_o(aa_int_fun_int_o(ord_less_int,Y),zero_zero_int) ) ) ).

% add_less_zeroD
tff(fact_2037_not__one__less__zero,axiom,
    ~ aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),zero_zero_real) ).

% not_one_less_zero
tff(fact_2038_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_2039_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_2040_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_2041_zero__less__one,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),one_one_real) ).

% zero_less_one
tff(fact_2042_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_2043_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_2044_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_2045_less__numeral__extra_I1_J,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),one_one_real) ).

% less_numeral_extra(1)
tff(fact_2046_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_2047_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_2048_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_2049_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B)) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
tff(fact_2050_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B)) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
tff(fact_2051_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B)) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
tff(fact_2052_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B)) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
tff(fact_2053_mult__less__cancel__right__disj,axiom,
    ! [A: real,C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,B),A) ) ) ) ).

% mult_less_cancel_right_disj
tff(fact_2054_mult__less__cancel__right__disj,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),C))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A) ) ) ) ).

% mult_less_cancel_right_disj
tff(fact_2055_mult__less__cancel__right__disj,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),zero_zero_int)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,B),A) ) ) ) ).

% mult_less_cancel_right_disj
tff(fact_2056_mult__strict__right__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)) ) ) ).

% mult_strict_right_mono
tff(fact_2057_mult__strict__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),C)) ) ) ).

% mult_strict_right_mono
tff(fact_2058_mult__strict__right__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) ) ) ).

% mult_strict_right_mono
tff(fact_2059_mult__strict__right__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ) ) ).

% mult_strict_right_mono
tff(fact_2060_mult__strict__right__mono__neg,axiom,
    ! [B: real,A: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)) ) ) ).

% mult_strict_right_mono_neg
tff(fact_2061_mult__strict__right__mono__neg,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),C)) ) ) ).

% mult_strict_right_mono_neg
tff(fact_2062_mult__strict__right__mono__neg,axiom,
    ! [B: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ) ) ).

% mult_strict_right_mono_neg
tff(fact_2063_mult__less__cancel__left__disj,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,B),A) ) ) ) ).

% mult_less_cancel_left_disj
tff(fact_2064_mult__less__cancel__left__disj,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A) ) ) ) ).

% mult_less_cancel_left_disj
tff(fact_2065_mult__less__cancel__left__disj,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),zero_zero_int)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,B),A) ) ) ) ).

% mult_less_cancel_left_disj
tff(fact_2066_mult__strict__left__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B)) ) ) ).

% mult_strict_left_mono
tff(fact_2067_mult__strict__left__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B)) ) ) ).

% mult_strict_left_mono
tff(fact_2068_mult__strict__left__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B)) ) ) ).

% mult_strict_left_mono
tff(fact_2069_mult__strict__left__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B)) ) ) ).

% mult_strict_left_mono
tff(fact_2070_mult__strict__left__mono__neg,axiom,
    ! [B: real,A: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B)) ) ) ).

% mult_strict_left_mono_neg
tff(fact_2071_mult__strict__left__mono__neg,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B)) ) ) ).

% mult_strict_left_mono_neg
tff(fact_2072_mult__strict__left__mono__neg,axiom,
    ! [B: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B)) ) ) ).

% mult_strict_left_mono_neg
tff(fact_2073_mult__less__cancel__left__pos,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) ) ) ).

% mult_less_cancel_left_pos
tff(fact_2074_mult__less__cancel__left__pos,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ) ).

% mult_less_cancel_left_pos
tff(fact_2075_mult__less__cancel__left__pos,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ) ).

% mult_less_cancel_left_pos
tff(fact_2076_mult__less__cancel__left__neg,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,B),A) ) ) ).

% mult_less_cancel_left_neg
tff(fact_2077_mult__less__cancel__left__neg,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A) ) ) ).

% mult_less_cancel_left_neg
tff(fact_2078_mult__less__cancel__left__neg,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,B),A) ) ) ).

% mult_less_cancel_left_neg
tff(fact_2079_zero__less__mult__pos2,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),A))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B) ) ) ).

% zero_less_mult_pos2
tff(fact_2080_zero__less__mult__pos2,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(times_times_rat(B),A))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B) ) ) ).

% zero_less_mult_pos2
tff(fact_2081_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_2082_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_2083_zero__less__mult__pos,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B) ) ) ).

% zero_less_mult_pos
tff(fact_2084_zero__less__mult__pos,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A),B))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B) ) ) ).

% zero_less_mult_pos
tff(fact_2085_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_2086_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_2087_zero__less__mult__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,B),zero_zero_real) ) ) ) ).

% zero_less_mult_iff
tff(fact_2088_zero__less__mult__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A),B))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),zero_zero_rat) ) ) ) ).

% zero_less_mult_iff
tff(fact_2089_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_2090_mult__pos__neg2,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),A)),zero_zero_real) ) ) ).

% mult_pos_neg2
tff(fact_2091_mult__pos__neg2,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(B),A)),zero_zero_rat) ) ) ).

% mult_pos_neg2
tff(fact_2092_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_2093_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_2094_mult__pos__pos,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)) ) ) ).

% mult_pos_pos
tff(fact_2095_mult__pos__pos,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A),B)) ) ) ).

% mult_pos_pos
tff(fact_2096_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_2097_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_2098_mult__pos__neg,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),zero_zero_real) ) ) ).

% mult_pos_neg
tff(fact_2099_mult__pos__neg,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A),B)),zero_zero_rat) ) ) ).

% mult_pos_neg
tff(fact_2100_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_2101_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_2102_mult__neg__pos,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),zero_zero_real) ) ) ).

% mult_neg_pos
tff(fact_2103_mult__neg__pos,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A),B)),zero_zero_rat) ) ) ).

% mult_neg_pos
tff(fact_2104_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_2105_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_2106_mult__less__0__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),zero_zero_real)
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,B),zero_zero_real) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B) ) ) ) ).

% mult_less_0_iff
tff(fact_2107_mult__less__0__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A),B)),zero_zero_rat)
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),zero_zero_rat) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B) ) ) ) ).

% mult_less_0_iff
tff(fact_2108_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_2109_not__square__less__zero,axiom,
    ! [A: real] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),A)),zero_zero_real) ).

% not_square_less_zero
tff(fact_2110_not__square__less__zero,axiom,
    ! [A: rat] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A),A)),zero_zero_rat) ).

% not_square_less_zero
tff(fact_2111_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_2112_mult__neg__neg,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)) ) ) ).

% mult_neg_neg
tff(fact_2113_mult__neg__neg,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A),B)) ) ) ).

% mult_neg_neg
tff(fact_2114_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_2115_zero__le__power,axiom,
    ! [A: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)) ) ).

% zero_le_power
tff(fact_2116_zero__le__power,axiom,
    ! [A: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),Nb)) ) ).

% zero_le_power
tff(fact_2117_zero__le__power,axiom,
    ! [A: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)) ) ).

% zero_le_power
tff(fact_2118_zero__le__power,axiom,
    ! [A: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(aa_int_fun_nat_int(power_power_int,A),Nb)) ) ).

% zero_le_power
tff(fact_2119_power__mono,axiom,
    ! [A: real,B: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Nb)) ) ) ).

% power_mono
tff(fact_2120_power__mono,axiom,
    ! [A: rat,B: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),Nb)),aa_nat_rat(power_power_rat(B),Nb)) ) ) ).

% power_mono
tff(fact_2121_power__mono,axiom,
    ! [A: nat,B: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Nb)) ) ) ).

% power_mono
tff(fact_2122_power__mono,axiom,
    ! [A: int,B: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A),Nb)),aa_nat_int(aa_int_fun_nat_int(power_power_int,B),Nb)) ) ) ).

% power_mono
tff(fact_2123_zero__less__power,axiom,
    ! [A: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)) ) ).

% zero_less_power
tff(fact_2124_zero__less__power,axiom,
    ! [A: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),Nb)) ) ).

% zero_less_power
tff(fact_2125_zero__less__power,axiom,
    ! [A: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)) ) ).

% zero_less_power
tff(fact_2126_zero__less__power,axiom,
    ! [A: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(aa_int_fun_nat_int(power_power_int,A),Nb)) ) ).

% zero_less_power
tff(fact_2127_VEBT__internal_OminNull_Osimps_I4_J,axiom,
    ! [Uw: nat,Ux: list_VEBT_VEBT,Uy: vEBT_VEBT] : aa_VEBT_VEBT_o(vEBT_VEBT_minNull,aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uw,Ux),Uy)) ).

% VEBT_internal.minNull.simps(4)
tff(fact_2128_of__nat__0__le__iff,axiom,
    ! [Nb: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(semiri5074537144036343181t_real,Nb)) ).

% of_nat_0_le_iff
tff(fact_2129_of__nat__0__le__iff,axiom,
    ! [Nb: nat] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(semiri4939895301339042750nteger,Nb)) ).

% of_nat_0_le_iff
tff(fact_2130_of__nat__0__le__iff,axiom,
    ! [Nb: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(semiri681578069525770553at_rat,Nb)) ).

% of_nat_0_le_iff
tff(fact_2131_of__nat__0__le__iff,axiom,
    ! [Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(semiri1316708129612266289at_nat,Nb)) ).

% of_nat_0_le_iff
tff(fact_2132_of__nat__0__le__iff,axiom,
    ! [Nb: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(semiri1314217659103216013at_int,Nb)) ).

% of_nat_0_le_iff
tff(fact_2133_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_2134_of__nat__less__0__iff,axiom,
    ! [M: nat] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,M)),zero_zero_real) ).

% of_nat_less_0_iff
tff(fact_2135_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_2136_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_2137_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_2138_semiring__char__0__class_Oof__nat__neq__0,axiom,
    ! [Nb: nat] : aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,Nb)) != zero_zero_rat ).

% semiring_char_0_class.of_nat_neq_0
tff(fact_2139_semiring__char__0__class_Oof__nat__neq__0,axiom,
    ! [Nb: nat] : aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Nb)) != zero_zero_real ).

% semiring_char_0_class.of_nat_neq_0
tff(fact_2140_semiring__char__0__class_Oof__nat__neq__0,axiom,
    ! [Nb: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,Nb)) != zero_zero_int ).

% semiring_char_0_class.of_nat_neq_0
tff(fact_2141_semiring__char__0__class_Oof__nat__neq__0,axiom,
    ! [Nb: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(suc,Nb)) != zero_zero_nat ).

% semiring_char_0_class.of_nat_neq_0
tff(fact_2142_semiring__char__0__class_Oof__nat__neq__0,axiom,
    ! [Nb: nat] : aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(suc,Nb)) != zero_z3403309356797280102nteger ).

% semiring_char_0_class.of_nat_neq_0
tff(fact_2143_power__0,axiom,
    ! [A: rat] : aa_nat_rat(power_power_rat(A),zero_zero_nat) = one_one_rat ).

% power_0
tff(fact_2144_power__0,axiom,
    ! [A: uint32] : aa_nat_uint32(power_power_uint32(A),zero_zero_nat) = one_one_uint32 ).

% power_0
tff(fact_2145_power__0,axiom,
    ! [A: word_N3645301735248828278l_num1] : aa_nat5599461682457136004l_num1(power_2184487114949457152l_num1(A),zero_zero_nat) = one_on7727431528512463931l_num1 ).

% power_0
tff(fact_2146_power__0,axiom,
    ! [A: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,A),zero_zero_nat) = one_one_real ).

% power_0
tff(fact_2147_power__0,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),zero_zero_nat) = one_one_nat ).

% power_0
tff(fact_2148_power__0,axiom,
    ! [A: int] : aa_nat_int(aa_int_fun_nat_int(power_power_int,A),zero_zero_nat) = one_one_int ).

% power_0
tff(fact_2149_less__Suc__eq__0__disj,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),aa_nat_nat(suc,Nb))
    <=> ( ( M = zero_zero_nat )
        | ? [J3: nat] :
            ( ( M = aa_nat_nat(suc,J3) )
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J3),Nb) ) ) ) ).

% less_Suc_eq_0_disj
tff(fact_2150_gr0__implies__Suc,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ? [M2: nat] : Nb = aa_nat_nat(suc,M2) ) ).

% gr0_implies_Suc
tff(fact_2151_All__less__Suc2,axiom,
    ! [Nb: nat,P: fun_nat_o] :
      ( ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),aa_nat_nat(suc,Nb))
         => aa_nat_o(P,I2) )
    <=> ( aa_nat_o(P,zero_zero_nat)
        & ! [I2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),Nb)
           => aa_nat_o(P,aa_nat_nat(suc,I2)) ) ) ) ).

% All_less_Suc2
tff(fact_2152_gr0__conv__Suc,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
    <=> ? [M7: nat] : Nb = aa_nat_nat(suc,M7) ) ).

% gr0_conv_Suc
tff(fact_2153_Ex__less__Suc2,axiom,
    ! [Nb: nat,P: fun_nat_o] :
      ( ? [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),aa_nat_nat(suc,Nb))
          & aa_nat_o(P,I2) )
    <=> ( aa_nat_o(P,zero_zero_nat)
        | ? [I2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),Nb)
            & aa_nat_o(P,aa_nat_nat(suc,I2)) ) ) ) ).

% Ex_less_Suc2
tff(fact_2154_nat__compl__induct_H,axiom,
    ! [P: fun_nat_o,Nb: nat] :
      ( aa_nat_o(P,zero_zero_nat)
     => ( ! [N2: nat] :
            ( ! [Nn: nat] :
                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nn),N2)
               => aa_nat_o(P,Nn) )
           => aa_nat_o(P,aa_nat_nat(suc,N2)) )
       => aa_nat_o(P,Nb) ) ) ).

% nat_compl_induct'
tff(fact_2155_nat__compl__induct,axiom,
    ! [P: fun_nat_o,Nb: nat] :
      ( aa_nat_o(P,zero_zero_nat)
     => ( ! [N2: nat] :
            ( ! [Nn: nat] :
                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nn),N2)
               => aa_nat_o(P,Nn) )
           => aa_nat_o(P,aa_nat_nat(suc,N2)) )
       => aa_nat_o(P,Nb) ) ) ).

% nat_compl_induct
tff(fact_2156_add__is__1,axiom,
    ! [M: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Nb) = aa_nat_nat(suc,zero_zero_nat) )
    <=> ( ( ( M = aa_nat_nat(suc,zero_zero_nat) )
          & ( Nb = zero_zero_nat ) )
        | ( ( M = zero_zero_nat )
          & ( Nb = aa_nat_nat(suc,zero_zero_nat) ) ) ) ) ).

% add_is_1
tff(fact_2157_one__is__add,axiom,
    ! [M: nat,Nb: nat] :
      ( ( aa_nat_nat(suc,zero_zero_nat) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Nb) )
    <=> ( ( ( M = aa_nat_nat(suc,zero_zero_nat) )
          & ( Nb = zero_zero_nat ) )
        | ( ( M = zero_zero_nat )
          & ( Nb = aa_nat_nat(suc,zero_zero_nat) ) ) ) ) ).

% one_is_add
tff(fact_2158_One__nat__def,axiom,
    one_one_nat = aa_nat_nat(suc,zero_zero_nat) ).

% One_nat_def
tff(fact_2159_option_Osize_I4_J,axiom,
    ! [X23: product_prod_nat_nat] : size_s170228958280169651at_nat(some_P7363390416028606310at_nat(X23)) = aa_nat_nat(suc,zero_zero_nat) ).

% option.size(4)
tff(fact_2160_option_Osize_I4_J,axiom,
    ! [X23: nat] : size_size_option_nat(aa_nat_option_nat(some_nat,X23)) = aa_nat_nat(suc,zero_zero_nat) ).

% option.size(4)
tff(fact_2161_option_Osize_I4_J,axiom,
    ! [X23: num] : size_size_option_num(aa_num_option_num(some_num,X23)) = aa_nat_nat(suc,zero_zero_nat) ).

% option.size(4)
tff(fact_2162_option_Osize_I3_J,axiom,
    size_s3991424295186984831it_nat(none_P1551326421579882414it_nat) = aa_nat_nat(suc,zero_zero_nat) ).

% option.size(3)
tff(fact_2163_option_Osize_I3_J,axiom,
    size_s364044314319911927it_nat(none_P7668321371905463026it_nat) = aa_nat_nat(suc,zero_zero_nat) ).

% option.size(3)
tff(fact_2164_option_Osize_I3_J,axiom,
    size_s170228958280169651at_nat(none_P5556105721700978146at_nat) = aa_nat_nat(suc,zero_zero_nat) ).

% option.size(3)
tff(fact_2165_option_Osize_I3_J,axiom,
    size_size_option_num(none_num) = aa_nat_nat(suc,zero_zero_nat) ).

% option.size(3)
tff(fact_2166_option_Osize_I3_J,axiom,
    size_size_option_nat(none_nat) = aa_nat_nat(suc,zero_zero_nat) ).

% option.size(3)
tff(fact_2167_ex__least__nat__le,axiom,
    ! [P: fun_nat_o,Nb: nat] :
      ( aa_nat_o(P,Nb)
     => ( ~ aa_nat_o(P,zero_zero_nat)
       => ? [K3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K3),Nb)
            & ! [I4: nat] :
                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),K3)
               => ~ aa_nat_o(P,I4) )
            & aa_nat_o(P,K3) ) ) ) ).

% ex_least_nat_le
tff(fact_2168_less__imp__add__positive,axiom,
    ! [I: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
     => ? [K3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K3)
          & ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K3) = J ) ) ) ).

% less_imp_add_positive
tff(fact_2169_Suc__to__right,axiom,
    ! [Nb: nat,M: nat] :
      ( ( aa_nat_nat(suc,Nb) = M )
     => ( Nb = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),aa_nat_nat(suc,zero_zero_nat)) ) ) ).

% Suc_to_right
tff(fact_2170_diff__less,axiom,
    ! [Nb: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Nb)),M) ) ) ).

% diff_less
tff(fact_2171_Euclidean__Division_Odiv__eq__0__iff,axiom,
    ! [M: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Nb) = zero_zero_nat )
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb)
        | ( Nb = zero_zero_nat ) ) ) ).

% Euclidean_Division.div_eq_0_iff
tff(fact_2172_nat__geq__1__eq__neqz,axiom,
    ! [Xb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),Xb)
    <=> ( Xb != zero_zero_nat ) ) ).

% nat_geq_1_eq_neqz
tff(fact_2173_VEBT__internal_OTb_H_Osimps_I2_J,axiom,
    aa_nat_nat(vEBT_VEBT_Tb2,aa_nat_nat(suc,zero_zero_nat)) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)) ).

% VEBT_internal.Tb'.simps(2)
tff(fact_2174_mult__less__mono1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),K)) ) ) ).

% mult_less_mono1
tff(fact_2175_mult__less__mono2,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),I)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),J)) ) ) ).

% mult_less_mono2
tff(fact_2176_nat__mult__eq__cancel1,axiom,
    ! [K: nat,M: nat,Nb: 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),Nb) )
      <=> ( M = Nb ) ) ) ).

% nat_mult_eq_cancel1
tff(fact_2177_nat__mult__less__cancel1,axiom,
    ! [K: nat,M: nat,Nb: 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),Nb))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb) ) ) ).

% nat_mult_less_cancel1
tff(fact_2178_diff__add__0,axiom,
    ! [Nb: nat,M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),M)) = zero_zero_nat ).

% diff_add_0
tff(fact_2179_nat__power__less__imp__less,axiom,
    ! [I: nat,M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),I)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,I),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,I),Nb))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb) ) ) ).

% nat_power_less_imp_less
tff(fact_2180_mult__eq__self__implies__10,axiom,
    ! [M: nat,Nb: nat] :
      ( ( M = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Nb) )
     => ( ( Nb = one_one_nat )
        | ( M = zero_zero_nat ) ) ) ).

% mult_eq_self_implies_10
tff(fact_2181_vebt__insert_Osimps_I2_J,axiom,
    ! [Info: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S3: vEBT_VEBT,Xb: nat] : vEBT_vebt_insert(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info,zero_zero_nat,Ts),S3),Xb) = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info,zero_zero_nat,Ts),S3) ).

% vebt_insert.simps(2)
tff(fact_2182_vebt__buildupi_Osimps_I1_J,axiom,
    aa_nat5408703056993926195_VEBTi(vEBT_vebt_buildupi,zero_zero_nat) = aa_VEB836174949890266638_VEBTi(heap_T3630416162098727440_VEBTi,vEBT_Leafi($false,$false)) ).

% vebt_buildupi.simps(1)
tff(fact_2183_VEBT__internal_Ovebt__buildupi_H_Osimps_I1_J,axiom,
    aa_nat5408703056993926195_VEBTi(vEBT_V739175172307565963ildupi,zero_zero_nat) = aa_VEB836174949890266638_VEBTi(heap_T3630416162098727440_VEBTi,vEBT_Leafi($false,$false)) ).

% VEBT_internal.vebt_buildupi'.simps(1)
tff(fact_2184_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_Osimps_I3_J,axiom,
    ! [V: product_prod_nat_nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT,Xb: nat] : aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_m_e_m_b_e_r,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V),zero_zero_nat,Uy),Uz)),Xb) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.simps(3)
tff(fact_2185_mult__less__le__imp__less,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),D)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
           => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),D)) ) ) ) ) ).

% mult_less_le_imp_less
tff(fact_2186_mult__less__le__imp__less,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),D)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),D)) ) ) ) ) ).

% mult_less_le_imp_less
tff(fact_2187_mult__less__le__imp__less,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),D)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),C)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D)) ) ) ) ) ).

% mult_less_le_imp_less
tff(fact_2188_mult__less__le__imp__less,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),D)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
           => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D)) ) ) ) ) ).

% mult_less_le_imp_less
tff(fact_2189_mult__le__less__imp__less,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),D)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
           => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),D)) ) ) ) ) ).

% mult_le_less_imp_less
tff(fact_2190_mult__le__less__imp__less,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),D)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),D)) ) ) ) ) ).

% mult_le_less_imp_less
tff(fact_2191_mult__le__less__imp__less,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),D)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D)) ) ) ) ) ).

% mult_le_less_imp_less
tff(fact_2192_mult__le__less__imp__less,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),D)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
           => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D)) ) ) ) ) ).

% mult_le_less_imp_less
tff(fact_2193_mult__right__le__imp__le,axiom,
    ! [A: real,C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ) ).

% mult_right_le_imp_le
tff(fact_2194_mult__right__le__imp__le,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),C))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ) ).

% mult_right_le_imp_le
tff(fact_2195_mult__right__le__imp__le,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B) ) ) ).

% mult_right_le_imp_le
tff(fact_2196_mult__right__le__imp__le,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ) ).

% mult_right_le_imp_le
tff(fact_2197_mult__left__le__imp__le,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ) ).

% mult_left_le_imp_le
tff(fact_2198_mult__left__le__imp__le,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ) ).

% mult_left_le_imp_le
tff(fact_2199_mult__left__le__imp__le,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B) ) ) ).

% mult_left_le_imp_le
tff(fact_2200_mult__left__le__imp__le,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ) ).

% mult_left_le_imp_le
tff(fact_2201_mult__le__cancel__left__pos,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ) ).

% mult_le_cancel_left_pos
tff(fact_2202_mult__le__cancel__left__pos,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ) ).

% mult_le_cancel_left_pos
tff(fact_2203_mult__le__cancel__left__pos,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ) ).

% mult_le_cancel_left_pos
tff(fact_2204_mult__le__cancel__left__neg,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A) ) ) ).

% mult_le_cancel_left_neg
tff(fact_2205_mult__le__cancel__left__neg,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A) ) ) ).

% mult_le_cancel_left_neg
tff(fact_2206_mult__le__cancel__left__neg,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A) ) ) ).

% mult_le_cancel_left_neg
tff(fact_2207_mult__less__cancel__right,axiom,
    ! [A: real,C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,B),A) ) ) ) ).

% mult_less_cancel_right
tff(fact_2208_mult__less__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),C))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A) ) ) ) ).

% mult_less_cancel_right
tff(fact_2209_mult__less__cancel__right,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,B),A) ) ) ) ).

% mult_less_cancel_right
tff(fact_2210_mult__strict__mono_H,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),D)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
           => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),D)) ) ) ) ) ).

% mult_strict_mono'
tff(fact_2211_mult__strict__mono_H,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),D)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),D)) ) ) ) ) ).

% mult_strict_mono'
tff(fact_2212_mult__strict__mono_H,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),D)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D)) ) ) ) ) ).

% mult_strict_mono'
tff(fact_2213_mult__strict__mono_H,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),D)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
           => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D)) ) ) ) ) ).

% mult_strict_mono'
tff(fact_2214_mult__right__less__imp__less,axiom,
    ! [A: real,C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) ) ) ).

% mult_right_less_imp_less
tff(fact_2215_mult__right__less__imp__less,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),C))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ) ).

% mult_right_less_imp_less
tff(fact_2216_mult__right__less__imp__less,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B) ) ) ).

% mult_right_less_imp_less
tff(fact_2217_mult__right__less__imp__less,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ) ).

% mult_right_less_imp_less
tff(fact_2218_mult__less__cancel__left,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,B),A) ) ) ) ).

% mult_less_cancel_left
tff(fact_2219_mult__less__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A) ) ) ) ).

% mult_less_cancel_left
tff(fact_2220_mult__less__cancel__left,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,B),A) ) ) ) ).

% mult_less_cancel_left
tff(fact_2221_mult__strict__mono,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),D)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
           => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),D)) ) ) ) ) ).

% mult_strict_mono
tff(fact_2222_mult__strict__mono,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),D)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),D)) ) ) ) ) ).

% mult_strict_mono
tff(fact_2223_mult__strict__mono,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),D)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D)) ) ) ) ) ).

% mult_strict_mono
tff(fact_2224_mult__strict__mono,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),D)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
           => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D)) ) ) ) ) ).

% mult_strict_mono
tff(fact_2225_mult__left__less__imp__less,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) ) ) ).

% mult_left_less_imp_less
tff(fact_2226_mult__left__less__imp__less,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ) ).

% mult_left_less_imp_less
tff(fact_2227_mult__left__less__imp__less,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B) ) ) ).

% mult_left_less_imp_less
tff(fact_2228_mult__left__less__imp__less,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ) ).

% mult_left_less_imp_less
tff(fact_2229_mult__le__cancel__right,axiom,
    ! [A: real,C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A) ) ) ) ).

% mult_le_cancel_right
tff(fact_2230_mult__le__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),C))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A) ) ) ) ).

% mult_le_cancel_right
tff(fact_2231_mult__le__cancel__right,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A) ) ) ) ).

% mult_le_cancel_right
tff(fact_2232_mult__le__cancel__left,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A) ) ) ) ).

% mult_le_cancel_left
tff(fact_2233_mult__le__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A) ) ) ) ).

% mult_le_cancel_left
tff(fact_2234_mult__le__cancel__left,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A) ) ) ) ).

% mult_le_cancel_left
tff(fact_2235_sum__squares__ge__zero,axiom,
    ! [Xb: real,Y: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Xb)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Y))) ).

% sum_squares_ge_zero
tff(fact_2236_sum__squares__ge__zero,axiom,
    ! [Xb: rat,Y: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(Xb),Xb)),aa_rat_rat(times_times_rat(Y),Y))) ).

% sum_squares_ge_zero
tff(fact_2237_sum__squares__ge__zero,axiom,
    ! [Xb: 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,Xb),Xb)),aa_int_int(aa_int_fun_int_int(times_times_int,Y),Y))) ).

% sum_squares_ge_zero
tff(fact_2238_sum__squares__le__zero__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Xb)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Y))),zero_zero_real)
    <=> ( ( Xb = zero_zero_real )
        & ( Y = zero_zero_real ) ) ) ).

% sum_squares_le_zero_iff
tff(fact_2239_sum__squares__le__zero__iff,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(Xb),Xb)),aa_rat_rat(times_times_rat(Y),Y))),zero_zero_rat)
    <=> ( ( Xb = zero_zero_rat )
        & ( Y = zero_zero_rat ) ) ) ).

% sum_squares_le_zero_iff
tff(fact_2240_sum__squares__le__zero__iff,axiom,
    ! [Xb: 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,Xb),Xb)),aa_int_int(aa_int_fun_int_int(times_times_int,Y),Y))),zero_zero_int)
    <=> ( ( Xb = zero_zero_int )
        & ( Y = zero_zero_int ) ) ) ).

% sum_squares_le_zero_iff
tff(fact_2241_mult__left__le,axiom,
    ! [C: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),one_one_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),A) ) ) ).

% mult_left_le
tff(fact_2242_mult__left__le,axiom,
    ! [C: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),one_one_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C)),A) ) ) ).

% mult_left_le
tff(fact_2243_mult__left__le,axiom,
    ! [C: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),one_one_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),A) ) ) ).

% mult_left_le
tff(fact_2244_mult__left__le,axiom,
    ! [C: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),one_one_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),A) ) ) ).

% mult_left_le
tff(fact_2245_mult__le__one,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),one_one_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),one_one_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),one_one_real) ) ) ) ).

% mult_le_one
tff(fact_2246_mult__le__one,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),one_one_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),one_one_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),B)),one_one_rat) ) ) ) ).

% mult_le_one
tff(fact_2247_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_2248_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_2249_mult__right__le__one__le,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),one_one_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Y)),Xb) ) ) ) ).

% mult_right_le_one_le
tff(fact_2250_mult__right__le__one__le,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Xb)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),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(times_times_rat(Xb),Y)),Xb) ) ) ) ).

% mult_right_le_one_le
tff(fact_2251_mult__right__le__one__le,axiom,
    ! [Xb: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xb)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),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,Xb),Y)),Xb) ) ) ) ).

% mult_right_le_one_le
tff(fact_2252_mult__left__le__one__le,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),one_one_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Xb)),Xb) ) ) ) ).

% mult_left_le_one_le
tff(fact_2253_mult__left__le__one__le,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Xb)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),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(times_times_rat(Y),Xb)),Xb) ) ) ) ).

% mult_left_le_one_le
tff(fact_2254_mult__left__le__one__le,axiom,
    ! [Xb: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xb)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),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),Xb)),Xb) ) ) ) ).

% mult_left_le_one_le
tff(fact_2255_zero__less__two,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),one_one_real)) ).

% zero_less_two
tff(fact_2256_zero__less__two,axiom,
    aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(plus_plus_rat(one_one_rat),one_one_rat)) ).

% zero_less_two
tff(fact_2257_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_2258_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_2259_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_2260_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_2261_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_2262_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_2263_not__sum__squares__lt__zero,axiom,
    ! [Xb: real,Y: real] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Xb)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Y))),zero_zero_real) ).

% not_sum_squares_lt_zero
tff(fact_2264_not__sum__squares__lt__zero,axiom,
    ! [Xb: rat,Y: rat] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(Xb),Xb)),aa_rat_rat(times_times_rat(Y),Y))),zero_zero_rat) ).

% not_sum_squares_lt_zero
tff(fact_2265_not__sum__squares__lt__zero,axiom,
    ! [Xb: 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,Xb),Xb)),aa_int_int(aa_int_fun_int_int(times_times_int,Y),Y))),zero_zero_int) ).

% not_sum_squares_lt_zero
tff(fact_2266_sum__squares__gt__zero__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Xb)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Y)))
    <=> ( ( Xb != zero_zero_real )
        | ( Y != zero_zero_real ) ) ) ).

% sum_squares_gt_zero_iff
tff(fact_2267_sum__squares__gt__zero__iff,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(Xb),Xb)),aa_rat_rat(times_times_rat(Y),Y)))
    <=> ( ( Xb != zero_zero_rat )
        | ( Y != zero_zero_rat ) ) ) ).

% sum_squares_gt_zero_iff
tff(fact_2268_sum__squares__gt__zero__iff,axiom,
    ! [Xb: 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,Xb),Xb)),aa_int_int(aa_int_fun_int_int(times_times_int,Y),Y)))
    <=> ( ( Xb != zero_zero_int )
        | ( Y != zero_zero_int ) ) ) ).

% sum_squares_gt_zero_iff
tff(fact_2269_power__less__imp__less__base,axiom,
    ! [A: real,Nb: nat,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Nb))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) ) ) ).

% power_less_imp_less_base
tff(fact_2270_power__less__imp__less__base,axiom,
    ! [A: rat,Nb: nat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(A),Nb)),aa_nat_rat(power_power_rat(B),Nb))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ) ).

% power_less_imp_less_base
tff(fact_2271_power__less__imp__less__base,axiom,
    ! [A: nat,Nb: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Nb))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B) ) ) ).

% power_less_imp_less_base
tff(fact_2272_power__less__imp__less__base,axiom,
    ! [A: int,Nb: nat,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A),Nb)),aa_nat_int(aa_int_fun_nat_int(power_power_int,B),Nb))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ) ).

% power_less_imp_less_base
tff(fact_2273_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),C) ) ) ).

% unique_euclidean_semiring_numeral_class.div_mult2_eq
tff(fact_2274_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),C) ) ) ).

% unique_euclidean_semiring_numeral_class.div_mult2_eq
tff(fact_2275_power__le__one,axiom,
    ! [A: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)),one_one_real) ) ) ).

% power_le_one
tff(fact_2276_power__le__one,axiom,
    ! [A: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),one_one_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),Nb)),one_one_rat) ) ) ).

% power_le_one
tff(fact_2277_power__le__one,axiom,
    ! [A: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),one_one_nat)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)),one_one_nat) ) ) ).

% power_le_one
tff(fact_2278_power__le__one,axiom,
    ! [A: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),one_one_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A),Nb)),one_one_int) ) ) ).

% power_le_one
tff(fact_2279_eq__divide__eq__numeral_I1_J,axiom,
    ! [W: num,B: real,C: real] :
      ( ( numeral_numeral_real(W) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C) )
    <=> $ite(C != zero_zero_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(W)),C) = B,numeral_numeral_real(W) = zero_zero_real) ) ).

% eq_divide_eq_numeral(1)
tff(fact_2280_eq__divide__eq__numeral_I1_J,axiom,
    ! [W: num,B: rat,C: rat] :
      ( ( numeral_numeral_rat(W) = divide_divide_rat(B,C) )
    <=> $ite(C != zero_zero_rat,aa_rat_rat(times_times_rat(numeral_numeral_rat(W)),C) = B,numeral_numeral_rat(W) = zero_zero_rat) ) ).

% eq_divide_eq_numeral(1)
tff(fact_2281_divide__eq__eq__numeral_I1_J,axiom,
    ! [B: real,C: real,W: num] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C) = numeral_numeral_real(W) )
    <=> $ite(C != zero_zero_real,B = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(W)),C),numeral_numeral_real(W) = zero_zero_real) ) ).

% divide_eq_eq_numeral(1)
tff(fact_2282_divide__eq__eq__numeral_I1_J,axiom,
    ! [B: rat,C: rat,W: num] :
      ( ( divide_divide_rat(B,C) = numeral_numeral_rat(W) )
    <=> $ite(C != zero_zero_rat,B = aa_rat_rat(times_times_rat(numeral_numeral_rat(W)),C),numeral_numeral_rat(W) = zero_zero_rat) ) ).

% divide_eq_eq_numeral(1)
tff(fact_2283_power__le__imp__le__base,axiom,
    ! [A: real,Nb: nat,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(suc,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),aa_nat_nat(suc,Nb)))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ) ).

% power_le_imp_le_base
tff(fact_2284_power__le__imp__le__base,axiom,
    ! [A: rat,Nb: nat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,Nb))),aa_nat_rat(power_power_rat(B),aa_nat_nat(suc,Nb)))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ) ).

% power_le_imp_le_base
tff(fact_2285_power__le__imp__le__base,axiom,
    ! [A: nat,Nb: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(suc,Nb))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),aa_nat_nat(suc,Nb)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B) ) ) ).

% power_le_imp_le_base
tff(fact_2286_power__le__imp__le__base,axiom,
    ! [A: int,Nb: nat,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A),aa_nat_nat(suc,Nb))),aa_nat_int(aa_int_fun_nat_int(power_power_int,B),aa_nat_nat(suc,Nb)))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ) ).

% power_le_imp_le_base
tff(fact_2287_power__inject__base,axiom,
    ! [A: real,Nb: nat,B: real] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(suc,Nb)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,B),aa_nat_nat(suc,Nb)) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B)
         => ( A = B ) ) ) ) ).

% power_inject_base
tff(fact_2288_power__inject__base,axiom,
    ! [A: rat,Nb: nat,B: rat] :
      ( ( aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,Nb)) = aa_nat_rat(power_power_rat(B),aa_nat_nat(suc,Nb)) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
         => ( A = B ) ) ) ) ).

% power_inject_base
tff(fact_2289_power__inject__base,axiom,
    ! [A: nat,Nb: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(suc,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),aa_nat_nat(suc,Nb)) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B)
         => ( A = B ) ) ) ) ).

% power_inject_base
tff(fact_2290_power__inject__base,axiom,
    ! [A: int,Nb: nat,B: int] :
      ( ( aa_nat_int(aa_int_fun_nat_int(power_power_int,A),aa_nat_nat(suc,Nb)) = aa_nat_int(aa_int_fun_nat_int(power_power_int,B),aa_nat_nat(suc,Nb)) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
         => ( A = B ) ) ) ) ).

% power_inject_base
tff(fact_2291_div__add__self2,axiom,
    ! [B: nat,A: nat] :
      ( ( B != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),one_one_nat) ) ) ).

% div_add_self2
tff(fact_2292_div__add__self2,axiom,
    ! [B: int,A: int] :
      ( ( B != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),one_one_int) ) ) ).

% div_add_self2
tff(fact_2293_div__add__self1,axiom,
    ! [B: nat,A: nat] :
      ( ( B != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A)),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),one_one_nat) ) ) ).

% div_add_self1
tff(fact_2294_div__add__self1,axiom,
    ! [B: int,A: int] :
      ( ( B != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A)),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),one_one_int) ) ) ).

% div_add_self1
tff(fact_2295_VEBT__internal_OminNulli_Osimps_I5_J,axiom,
    ! [Uz: product_prod_nat_nat,Vaa: nat,Vb: array_VEBT_VEBTi,Vc2: vEBT_VEBTi] : aa_VEB8194921341634505877Heap_o(vEBT_VEBT_minNulli,aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(some_P7363390416028606310at_nat(Uz),Vaa,Vb),Vc2)) = aa_o_H7399033255965945470Heap_o(heap_Time_return_o,$false) ).

% VEBT_internal.minNulli.simps(5)
tff(fact_2296_numeral__1__eq__Suc__0,axiom,
    aa_num_nat(numeral_numeral_nat,one) = aa_nat_nat(suc,zero_zero_nat) ).

% numeral_1_eq_Suc_0
tff(fact_2297_VEBT__internal_OminNulli_Osimps_I4_J,axiom,
    ! [Uw: nat,Ux: array_VEBT_VEBTi,Uy: vEBT_VEBTi] : aa_VEB8194921341634505877Heap_o(vEBT_VEBT_minNulli,aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(none_P5556105721700978146at_nat,Uw,Ux),Uy)) = aa_o_H7399033255965945470Heap_o(heap_Time_return_o,$true) ).

% VEBT_internal.minNulli.simps(4)
tff(fact_2298_num_Osize_I5_J,axiom,
    ! [X23: num] : aa_num_nat(size_size_num,aa_num_num(bit0,X23)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(size_size_num,X23)),aa_nat_nat(suc,zero_zero_nat)) ).

% num.size(5)
tff(fact_2299_ex__least__nat__less,axiom,
    ! [P: fun_nat_o,Nb: nat] :
      ( aa_nat_o(P,Nb)
     => ( ~ aa_nat_o(P,zero_zero_nat)
       => ? [K3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K3),Nb)
            & ! [I4: nat] :
                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I4),K3)
               => ~ aa_nat_o(P,I4) )
            & aa_nat_o(P,aa_nat_nat(suc,K3)) ) ) ) ).

% ex_least_nat_less
tff(fact_2300_nat__induct__non__zero,axiom,
    ! [Nb: nat,P: fun_nat_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_o(P,one_one_nat)
       => ( ! [N2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)
             => ( aa_nat_o(P,N2)
               => aa_nat_o(P,aa_nat_nat(suc,N2)) ) )
         => aa_nat_o(P,Nb) ) ) ) ).

% nat_induct_non_zero
tff(fact_2301_num_Osize_I6_J,axiom,
    ! [X33: num] : aa_num_nat(size_size_num,aa_num_num(bit1,X33)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(size_size_num,X33)),aa_nat_nat(suc,zero_zero_nat)) ).

% num.size(6)
tff(fact_2302_diff__Suc__less,axiom,
    ! [Nb: nat,I: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,I))),Nb) ) ).

% diff_Suc_less
tff(fact_2303_n__less__n__mult__m,axiom,
    ! [Nb: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),M)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),M)) ) ) ).

% n_less_n_mult_m
tff(fact_2304_n__less__m__mult__n,axiom,
    ! [Nb: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),M)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Nb)) ) ) ).

% n_less_m_mult_n
tff(fact_2305_one__less__mult,axiom,
    ! [Nb: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),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),Nb)) ) ) ).

% one_less_mult
tff(fact_2306_power__gt__expt,axiom,
    ! [Nb: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Nb),K)) ) ).

% power_gt_expt
tff(fact_2307_nat__diff__split__asm,axiom,
    ! [P: fun_nat_o,A: nat,B: nat] :
      ( aa_nat_o(P,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(P,zero_zero_nat) )
          | ? [D3: nat] :
              ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),D3) )
              & ~ aa_nat_o(P,D3) ) ) ) ).

% nat_diff_split_asm
tff(fact_2308_nat__diff__split,axiom,
    ! [P: fun_nat_o,A: nat,B: nat] :
      ( aa_nat_o(P,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(P,zero_zero_nat) )
        & ! [D3: nat] :
            ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),D3) )
           => aa_nat_o(P,D3) ) ) ) ).

% nat_diff_split
tff(fact_2309_div__greater__zero__iff,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),M)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ) ).

% div_greater_zero_iff
tff(fact_2310_div__le__mono2,axiom,
    ! [M: nat,Nb: 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),Nb)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,K),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,K),M)) ) ) ).

% div_le_mono2
tff(fact_2311_div__less__dividend,axiom,
    ! [Nb: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Nb)),M) ) ) ).

% div_less_dividend
tff(fact_2312_nat__one__le__power,axiom,
    ! [I: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),I)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,I),Nb)) ) ).

% nat_one_le_power
tff(fact_2313_nat__mult__le__cancel1,axiom,
    ! [K: nat,M: nat,Nb: 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),Nb))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb) ) ) ).

% nat_mult_le_cancel1
tff(fact_2314_div__less__iff__less__mult,axiom,
    ! [Q2: nat,M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Q2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Q2)),Nb)
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),Q2)) ) ) ).

% div_less_iff_less_mult
tff(fact_2315_nat__mult__div__cancel1,axiom,
    ! [K: nat,M: nat,Nb: 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),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Nb) ) ) ).

% nat_mult_div_cancel1
tff(fact_2316_td__gal__lt,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),C)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),C)),B) ) ) ).

% td_gal_lt
tff(fact_2317_assn__times__assoc,axiom,
    ! [P: assn,Q: assn,R: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,P),Q)),R) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,P),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Q),R)) ).

% assn_times_assoc
tff(fact_2318_assn__times__comm,axiom,
    ! [P: assn,Q: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,P),Q) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Q),P) ).

% assn_times_comm
tff(fact_2319_vebt__insert_Osimps_I3_J,axiom,
    ! [Info: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S3: vEBT_VEBT,Xb: nat] : vEBT_vebt_insert(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info,aa_nat_nat(suc,zero_zero_nat),Ts),S3),Xb) = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info,aa_nat_nat(suc,zero_zero_nat),Ts),S3) ).

% vebt_insert.simps(3)
tff(fact_2320_vebt__member_Osimps_I3_J,axiom,
    ! [V: product_prod_nat_nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT,Xb: nat] : ~ aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V),zero_zero_nat,Uy),Uz)),Xb) ).

% vebt_member.simps(3)
tff(fact_2321_vebt__buildupi_Osimps_I2_J,axiom,
    aa_nat5408703056993926195_VEBTi(vEBT_vebt_buildupi,aa_nat_nat(suc,zero_zero_nat)) = aa_VEB836174949890266638_VEBTi(heap_T3630416162098727440_VEBTi,vEBT_Leafi($false,$false)) ).

% vebt_buildupi.simps(2)
tff(fact_2322_VEBT__internal_Ovebt__buildupi_H_Osimps_I2_J,axiom,
    aa_nat5408703056993926195_VEBTi(vEBT_V739175172307565963ildupi,aa_nat_nat(suc,zero_zero_nat)) = aa_VEB836174949890266638_VEBTi(heap_T3630416162098727440_VEBTi,vEBT_Leafi($false,$false)) ).

% VEBT_internal.vebt_buildupi'.simps(2)
tff(fact_2323_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_Osimps_I4_J,axiom,
    ! [V: product_prod_nat_nat,Vb: list_VEBT_VEBT,Vc2: vEBT_VEBT,Xb: nat] : aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_m_e_m_b_e_r,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V),aa_nat_nat(suc,zero_zero_nat),Vb),Vc2)),Xb) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.simps(4)
tff(fact_2324_mult__less__cancel__right2,axiom,
    ! [A: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),C)
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A),one_one_real) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A) ) ) ) ).

% mult_less_cancel_right2
tff(fact_2325_mult__less__cancel__right2,axiom,
    ! [A: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),C)
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),one_one_rat) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),A) ) ) ) ).

% mult_less_cancel_right2
tff(fact_2326_mult__less__cancel__right2,axiom,
    ! [A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),C)
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,A),one_one_int) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A) ) ) ) ).

% mult_less_cancel_right2
tff(fact_2327_mult__less__cancel__right1,axiom,
    ! [C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,B),one_one_real) ) ) ) ).

% mult_less_cancel_right1
tff(fact_2328_mult__less__cancel__right1,axiom,
    ! [C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),aa_rat_rat(times_times_rat(B),C))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),B) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),one_one_rat) ) ) ) ).

% mult_less_cancel_right1
tff(fact_2329_mult__less__cancel__right1,axiom,
    ! [C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),aa_int_int(aa_int_fun_int_int(times_times_int,B),C))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),B) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,B),one_one_int) ) ) ) ).

% mult_less_cancel_right1
tff(fact_2330_mult__less__cancel__left2,axiom,
    ! [C: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),C)
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A),one_one_real) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A) ) ) ) ).

% mult_less_cancel_left2
tff(fact_2331_mult__less__cancel__left2,axiom,
    ! [C: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(C),A)),C)
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),one_one_rat) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),A) ) ) ) ).

% mult_less_cancel_left2
tff(fact_2332_mult__less__cancel__left2,axiom,
    ! [C: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),C)
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,A),one_one_int) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A) ) ) ) ).

% mult_less_cancel_left2
tff(fact_2333_mult__less__cancel__left1,axiom,
    ! [C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,B),one_one_real) ) ) ) ).

% mult_less_cancel_left1
tff(fact_2334_mult__less__cancel__left1,axiom,
    ! [C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),aa_rat_rat(times_times_rat(C),B))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),B) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),one_one_rat) ) ) ) ).

% mult_less_cancel_left1
tff(fact_2335_mult__less__cancel__left1,axiom,
    ! [C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),B) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,B),one_one_int) ) ) ) ).

% mult_less_cancel_left1
tff(fact_2336_mult__le__cancel__right2,axiom,
    ! [A: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),C)
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),one_one_real) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),A) ) ) ) ).

% mult_le_cancel_right2
tff(fact_2337_mult__le__cancel__right2,axiom,
    ! [A: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C)),C)
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),one_one_rat) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),A) ) ) ) ).

% mult_le_cancel_right2
tff(fact_2338_mult__le__cancel__right2,axiom,
    ! [A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),C)
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),one_one_int) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),A) ) ) ) ).

% mult_le_cancel_right2
tff(fact_2339_mult__le__cancel__right1,axiom,
    ! [C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),B) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),one_one_real) ) ) ) ).

% mult_le_cancel_right1
tff(fact_2340_mult__le__cancel__right1,axiom,
    ! [C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),aa_rat_rat(times_times_rat(B),C))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),B) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),one_one_rat) ) ) ) ).

% mult_le_cancel_right1
tff(fact_2341_mult__le__cancel__right1,axiom,
    ! [C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),aa_int_int(aa_int_fun_int_int(times_times_int,B),C))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),B) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),one_one_int) ) ) ) ).

% mult_le_cancel_right1
tff(fact_2342_mult__le__cancel__left2,axiom,
    ! [C: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),C)
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),one_one_real) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),A) ) ) ) ).

% mult_le_cancel_left2
tff(fact_2343_mult__le__cancel__left2,axiom,
    ! [C: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(C),A)),C)
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),one_one_rat) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),A) ) ) ) ).

% mult_le_cancel_left2
tff(fact_2344_mult__le__cancel__left2,axiom,
    ! [C: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),C)
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),one_one_int) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),A) ) ) ) ).

% mult_le_cancel_left2
tff(fact_2345_mult__le__cancel__left1,axiom,
    ! [C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),B) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),one_one_real) ) ) ) ).

% mult_le_cancel_left1
tff(fact_2346_mult__le__cancel__left1,axiom,
    ! [C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),aa_rat_rat(times_times_rat(C),B))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),B) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),one_one_rat) ) ) ) ).

% mult_le_cancel_left1
tff(fact_2347_mult__le__cancel__left1,axiom,
    ! [C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),B) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),one_one_int) ) ) ) ).

% mult_le_cancel_left1
tff(fact_2348_convex__bound__le,axiom,
    ! [Xb: real,A: real,Y: real,U: real,V: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),A)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),U)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),V)
           => ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,U),V) = one_one_real )
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,U),Xb)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,V),Y))),A) ) ) ) ) ) ).

% convex_bound_le
tff(fact_2349_convex__bound__le,axiom,
    ! [Xb: rat,A: rat,Y: rat,U: rat,V: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,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(plus_plus_rat(U),V) = one_one_rat )
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(U),Xb)),aa_rat_rat(times_times_rat(V),Y))),A) ) ) ) ) ) ).

% convex_bound_le
tff(fact_2350_convex__bound__le,axiom,
    ! [Xb: int,A: int,Y: int,U: int,V: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,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),Xb)),aa_int_int(aa_int_fun_int_int(times_times_int,V),Y))),A) ) ) ) ) ) ).

% convex_bound_le
tff(fact_2351_less__divide__eq__numeral_I1_J,axiom,
    ! [W: num,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,numeral_numeral_real(W)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C))
    <=> $ite(
          aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C),
          aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(W)),C)),B),
          $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real),aa_real_o(aa_real_fun_real_o(ord_less_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(W)),C)),aa_real_o(aa_real_fun_real_o(ord_less_real,numeral_numeral_real(W)),zero_zero_real)) ) ) ).

% less_divide_eq_numeral(1)
tff(fact_2352_less__divide__eq__numeral_I1_J,axiom,
    ! [W: num,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,numeral_numeral_rat(W)),divide_divide_rat(B,C))
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(numeral_numeral_rat(W)),C)),B),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),aa_rat_rat(times_times_rat(numeral_numeral_rat(W)),C)),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,numeral_numeral_rat(W)),zero_zero_rat)) ) ) ).

% less_divide_eq_numeral(1)
tff(fact_2353_divide__less__eq__numeral_I1_J,axiom,
    ! [B: real,C: real,W: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)),numeral_numeral_real(W))
    <=> $ite(
          aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C),
          aa_real_o(aa_real_fun_real_o(ord_less_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(W)),C)),
          $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real),aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(W)),C)),B),aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),numeral_numeral_real(W))) ) ) ).

% divide_less_eq_numeral(1)
tff(fact_2354_divide__less__eq__numeral_I1_J,axiom,
    ! [B: rat,C: rat,W: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,divide_divide_rat(B,C)),numeral_numeral_rat(W))
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),aa_rat_rat(times_times_rat(numeral_numeral_rat(W)),C)),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(numeral_numeral_rat(W)),C)),B),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),numeral_numeral_rat(W))) ) ) ).

% divide_less_eq_numeral(1)
tff(fact_2355_power__Suc__less,axiom,
    ! [A: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)) ) ) ).

% power_Suc_less
tff(fact_2356_power__Suc__less,axiom,
    ! [A: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),one_one_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A),aa_nat_rat(power_power_rat(A),Nb))),aa_nat_rat(power_power_rat(A),Nb)) ) ) ).

% power_Suc_less
tff(fact_2357_power__Suc__less,axiom,
    ! [A: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),one_one_nat)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)) ) ) ).

% power_Suc_less
tff(fact_2358_power__Suc__less,axiom,
    ! [A: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),one_one_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_nat_int(aa_int_fun_nat_int(power_power_int,A),Nb))),aa_nat_int(aa_int_fun_nat_int(power_power_int,A),Nb)) ) ) ).

% power_Suc_less
tff(fact_2359_power__Suc__le__self,axiom,
    ! [A: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(suc,Nb))),A) ) ) ).

% power_Suc_le_self
tff(fact_2360_power__Suc__le__self,axiom,
    ! [A: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),one_one_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,Nb))),A) ) ) ).

% power_Suc_le_self
tff(fact_2361_power__Suc__le__self,axiom,
    ! [A: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),one_one_nat)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(suc,Nb))),A) ) ) ).

% power_Suc_le_self
tff(fact_2362_power__Suc__le__self,axiom,
    ! [A: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),one_one_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A),aa_nat_nat(suc,Nb))),A) ) ) ).

% power_Suc_le_self
tff(fact_2363_power__Suc__less__one,axiom,
    ! [A: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(suc,Nb))),one_one_real) ) ) ).

% power_Suc_less_one
tff(fact_2364_power__Suc__less__one,axiom,
    ! [A: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),one_one_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,Nb))),one_one_rat) ) ) ).

% power_Suc_less_one
tff(fact_2365_power__Suc__less__one,axiom,
    ! [A: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),one_one_nat)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(suc,Nb))),one_one_nat) ) ) ).

% power_Suc_less_one
tff(fact_2366_power__Suc__less__one,axiom,
    ! [A: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),one_one_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A),aa_nat_nat(suc,Nb))),one_one_int) ) ) ).

% power_Suc_less_one
tff(fact_2367_zero__power2,axiom,
    aa_nat_rat(power_power_rat(zero_zero_rat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_rat ).

% zero_power2
tff(fact_2368_zero__power2,axiom,
    aa_nat_uint32(power_power_uint32(zero_zero_uint32),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_uint32 ).

% zero_power2
tff(fact_2369_zero__power2,axiom,
    aa_nat5599461682457136004l_num1(power_2184487114949457152l_num1(zero_z3563351764282998399l_num1),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_z3563351764282998399l_num1 ).

% zero_power2
tff(fact_2370_zero__power2,axiom,
    aa_nat_real(aa_real_fun_nat_real(power_power_real,zero_zero_real),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_real ).

% zero_power2
tff(fact_2371_zero__power2,axiom,
    aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_nat ).

% zero_power2
tff(fact_2372_zero__power2,axiom,
    aa_nat_int(aa_int_fun_nat_int(power_power_int,zero_zero_int),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_int ).

% zero_power2
tff(fact_2373_power__strict__decreasing,axiom,
    ! [Nb: nat,N: nat,A: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),N)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),one_one_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N)),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)) ) ) ) ).

% power_strict_decreasing
tff(fact_2374_power__strict__decreasing,axiom,
    ! [Nb: nat,N: nat,A: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),N)
     => ( 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),N)),aa_nat_rat(power_power_rat(A),Nb)) ) ) ) ).

% power_strict_decreasing
tff(fact_2375_power__strict__decreasing,axiom,
    ! [Nb: nat,N: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),N)
     => ( 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),N)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)) ) ) ) ).

% power_strict_decreasing
tff(fact_2376_power__strict__decreasing,axiom,
    ! [Nb: nat,N: nat,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),N)
     => ( 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),N)),aa_nat_int(aa_int_fun_nat_int(power_power_int,A),Nb)) ) ) ) ).

% power_strict_decreasing
tff(fact_2377_power__decreasing,axiom,
    ! [Nb: nat,N: nat,A: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),one_one_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N)),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)) ) ) ) ).

% power_decreasing
tff(fact_2378_power__decreasing,axiom,
    ! [Nb: nat,N: nat,A: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),one_one_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),N)),aa_nat_rat(power_power_rat(A),Nb)) ) ) ) ).

% power_decreasing
tff(fact_2379_power__decreasing,axiom,
    ! [Nb: nat,N: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),one_one_nat)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)) ) ) ) ).

% power_decreasing
tff(fact_2380_power__decreasing,axiom,
    ! [Nb: nat,N: nat,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),one_one_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A),N)),aa_nat_int(aa_int_fun_nat_int(power_power_int,A),Nb)) ) ) ) ).

% power_decreasing
tff(fact_2381_numeral__2__eq__2,axiom,
    aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) = aa_nat_nat(suc,aa_nat_nat(suc,zero_zero_nat)) ).

% numeral_2_eq_2
tff(fact_2382_self__le__power,axiom,
    ! [A: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)) ) ) ).

% self_le_power
tff(fact_2383_self__le__power,axiom,
    ! [A: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_nat_rat(power_power_rat(A),Nb)) ) ) ).

% self_le_power
tff(fact_2384_self__le__power,axiom,
    ! [A: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)) ) ) ).

% self_le_power
tff(fact_2385_self__le__power,axiom,
    ! [A: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_nat_int(aa_int_fun_nat_int(power_power_int,A),Nb)) ) ) ).

% self_le_power
tff(fact_2386_pos2,axiom,
    aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% pos2
tff(fact_2387_one__less__power,axiom,
    ! [A: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)) ) ) ).

% one_less_power
tff(fact_2388_one__less__power,axiom,
    ! [A: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_nat_rat(power_power_rat(A),Nb)) ) ) ).

% one_less_power
tff(fact_2389_one__less__power,axiom,
    ! [A: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)) ) ) ).

% one_less_power
tff(fact_2390_one__less__power,axiom,
    ! [A: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_nat_int(aa_int_fun_nat_int(power_power_int,A),Nb)) ) ) ).

% one_less_power
tff(fact_2391_power__diff,axiom,
    ! [A: real,Nb: nat,M: nat] :
      ( ( A != zero_zero_real )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),M)
       => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Nb)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),M)),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)) ) ) ) ).

% power_diff
tff(fact_2392_power__diff,axiom,
    ! [A: rat,Nb: nat,M: nat] :
      ( ( A != zero_zero_rat )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),M)
       => ( aa_nat_rat(power_power_rat(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Nb)) = divide_divide_rat(aa_nat_rat(power_power_rat(A),M),aa_nat_rat(power_power_rat(A),Nb)) ) ) ) ).

% power_diff
tff(fact_2393_power__diff,axiom,
    ! [A: nat,Nb: nat,M: nat] :
      ( ( A != zero_zero_nat )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),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),Nb)) = 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),Nb)) ) ) ) ).

% power_diff
tff(fact_2394_power__diff,axiom,
    ! [A: int,Nb: nat,M: nat] :
      ( ( A != zero_zero_int )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),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),Nb)) = 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),Nb)) ) ) ) ).

% power_diff
tff(fact_2395_numeral__3__eq__3,axiom,
    aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)) = aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,zero_zero_nat))) ).

% numeral_3_eq_3
tff(fact_2396_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_2397_Suc__diff__eq__diff__pred,axiom,
    ! [Nb: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,M)),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)) ) ) ).

% Suc_diff_eq_diff_pred
tff(fact_2398_Suc__pred_H,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( Nb = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)) ) ) ).

% Suc_pred'
tff(fact_2399_div__if,axiom,
    ! [M: nat,Nb: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Nb) = $ite(
        ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb)
        | ( Nb = zero_zero_nat ) ),
        zero_zero_nat,
        aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Nb)),Nb)) ) ).

% div_if
tff(fact_2400_div__geq,axiom,
    ! [Nb: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Nb) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Nb)),Nb)) ) ) ) ).

% div_geq
tff(fact_2401_add__eq__if,axiom,
    ! [M: nat,Nb: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Nb) = $ite(M = zero_zero_nat,Nb,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat)),Nb))) ).

% add_eq_if
tff(fact_2402_msrevs_I1_J,axiom,
    ! [Nb: nat,K: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Nb)),M)),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Nb)),K) ) ) ).

% msrevs(1)
tff(fact_2403_less__eq__div__iff__mult__less__eq,axiom,
    ! [Q2: nat,M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Q2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),Q2))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Q2)),Nb) ) ) ).

% less_eq_div_iff_mult_less_eq
tff(fact_2404_dividend__less__times__div,axiom,
    ! [Nb: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Nb)))) ) ).

% dividend_less_times_div
tff(fact_2405_dividend__less__div__times,axiom,
    ! [Nb: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Nb)),Nb))) ) ).

% dividend_less_div_times
tff(fact_2406_split__div,axiom,
    ! [P: fun_nat_o,M: nat,Nb: nat] :
      ( aa_nat_o(P,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Nb))
    <=> ( ( ( Nb = zero_zero_nat )
         => aa_nat_o(P,zero_zero_nat) )
        & ( ( Nb != zero_zero_nat )
         => ! [I2: nat,J3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J3),Nb)
             => ( ( M = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),I2)),J3) )
               => aa_nat_o(P,I2) ) ) ) ) ) ).

% split_div
tff(fact_2407_td__gal,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),C)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)),A)
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),C)) ) ) ).

% td_gal
tff(fact_2408_mult__eq__if,axiom,
    ! [M: nat,Nb: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Nb) = $ite(M = zero_zero_nat,zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat)),Nb))) ).

% mult_eq_if
tff(fact_2409_nat__mult__power__less__eq,axiom,
    ! [B: nat,A: nat,Nb: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Nb))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),M))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Nb))) ) ) ).

% nat_mult_power_less_eq
tff(fact_2410_vebt__member_Osimps_I4_J,axiom,
    ! [V: product_prod_nat_nat,Vb: list_VEBT_VEBT,Vc2: vEBT_VEBT,Xb: nat] : ~ aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V),aa_nat_nat(suc,zero_zero_nat),Vb),Vc2)),Xb) ).

% vebt_member.simps(4)
tff(fact_2411_vebt__pred_Osimps_I5_J,axiom,
    ! [V: product_prod_nat_nat,Vd: list_VEBT_VEBT,Ve: vEBT_VEBT,Vf: nat] : aa_nat_option_nat(vEBT_vebt_pred(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V),zero_zero_nat,Vd),Ve)),Vf) = none_nat ).

% vebt_pred.simps(5)
tff(fact_2412_vebt__succ_Osimps_I4_J,axiom,
    ! [V: product_prod_nat_nat,Vc2: list_VEBT_VEBT,Vd: vEBT_VEBT,Ve: nat] : aa_nat_option_nat(vEBT_vebt_succ(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V),zero_zero_nat,Vc2),Vd)),Ve) = none_nat ).

% vebt_succ.simps(4)
tff(fact_2413_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Osimps_I5_J,axiom,
    ! [V: product_prod_nat_nat,Vd: list_VEBT_VEBT,Ve: vEBT_VEBT,Vf: nat] : aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_p_r_e_d2,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V),zero_zero_nat,Vd),Ve)),Vf) = one_one_nat ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.simps(5)
tff(fact_2414_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Osimps_I4_J,axiom,
    ! [V: product_prod_nat_nat,Vc2: list_VEBT_VEBT,Vd: vEBT_VEBT,Ve: nat] : aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_s_u_c_c2,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V),zero_zero_nat,Vc2),Vd)),Ve) = one_one_nat ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.simps(4)
tff(fact_2415_convex__bound__lt,axiom,
    ! [Xb: real,A: real,Y: real,U: real,V: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),A)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),U)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),V)
           => ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,U),V) = one_one_real )
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,U),Xb)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,V),Y))),A) ) ) ) ) ) ).

% convex_bound_lt
tff(fact_2416_convex__bound__lt,axiom,
    ! [Xb: rat,A: rat,Y: rat,U: rat,V: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,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(plus_plus_rat(U),V) = one_one_rat )
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(U),Xb)),aa_rat_rat(times_times_rat(V),Y))),A) ) ) ) ) ) ).

% convex_bound_lt
tff(fact_2417_convex__bound__lt,axiom,
    ! [Xb: int,A: int,Y: int,U: int,V: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,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),Xb)),aa_int_int(aa_int_fun_int_int(times_times_int,V),Y))),A) ) ) ) ) ) ).

% convex_bound_lt
tff(fact_2418_le__divide__eq__numeral_I1_J,axiom,
    ! [W: num,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,numeral_numeral_real(W)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C))
    <=> $ite(
          aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C),
          aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(W)),C)),B),
          $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(W)),C)),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,numeral_numeral_real(W)),zero_zero_real)) ) ) ).

% le_divide_eq_numeral(1)
tff(fact_2419_le__divide__eq__numeral_I1_J,axiom,
    ! [W: num,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,numeral_numeral_rat(W)),divide_divide_rat(B,C))
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(numeral_numeral_rat(W)),C)),B),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),aa_rat_rat(times_times_rat(numeral_numeral_rat(W)),C)),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,numeral_numeral_rat(W)),zero_zero_rat)) ) ) ).

% le_divide_eq_numeral(1)
tff(fact_2420_divide__le__eq__numeral_I1_J,axiom,
    ! [B: real,C: real,W: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)),numeral_numeral_real(W))
    <=> $ite(
          aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C),
          aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(W)),C)),
          $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(W)),C)),B),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),numeral_numeral_real(W))) ) ) ).

% divide_le_eq_numeral(1)
tff(fact_2421_divide__le__eq__numeral_I1_J,axiom,
    ! [B: rat,C: rat,W: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,divide_divide_rat(B,C)),numeral_numeral_rat(W))
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),aa_rat_rat(times_times_rat(numeral_numeral_rat(W)),C)),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(numeral_numeral_rat(W)),C)),B),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),numeral_numeral_rat(W))) ) ) ).

% divide_le_eq_numeral(1)
tff(fact_2422_half__gt__zero__iff,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),numeral_numeral_real(aa_num_num(bit0,one))))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A) ) ).

% half_gt_zero_iff
tff(fact_2423_half__gt__zero__iff,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),divide_divide_rat(A,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_2424_half__gt__zero,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),numeral_numeral_real(aa_num_num(bit0,one)))) ) ).

% half_gt_zero
tff(fact_2425_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),divide_divide_rat(A,numeral_numeral_rat(aa_num_num(bit0,one)))) ) ).

% half_gt_zero
tff(fact_2426_power2__le__imp__le,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y) ) ) ).

% power2_le_imp_le
tff(fact_2427_power2__le__imp__le,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(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,Xb),Y) ) ) ).

% power2_le_imp_le
tff(fact_2428_power2__le__imp__le,axiom,
    ! [Xb: 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,Xb),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,Xb),Y) ) ) ).

% power2_le_imp_le
tff(fact_2429_power2__le__imp__le,axiom,
    ! [Xb: 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,Xb),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,Xb),Y) ) ) ).

% power2_le_imp_le
tff(fact_2430_power2__eq__imp__eq,axiom,
    ! [Xb: real,Y: real] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
         => ( Xb = Y ) ) ) ) ).

% power2_eq_imp_eq
tff(fact_2431_power2__eq__imp__eq,axiom,
    ! [Xb: rat,Y: rat] :
      ( ( aa_nat_rat(power_power_rat(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_rat(power_power_rat(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),Xb)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Y)
         => ( Xb = Y ) ) ) ) ).

% power2_eq_imp_eq
tff(fact_2432_power2__eq__imp__eq,axiom,
    ! [Xb: nat,Y: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,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),Xb)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Y)
         => ( Xb = Y ) ) ) ) ).

% power2_eq_imp_eq
tff(fact_2433_power2__eq__imp__eq,axiom,
    ! [Xb: int,Y: int] :
      ( ( aa_nat_int(aa_int_fun_nat_int(power_power_int,Xb),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),Xb)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
         => ( Xb = Y ) ) ) ) ).

% power2_eq_imp_eq
tff(fact_2434_zero__le__power2,axiom,
    ! [A: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% zero_le_power2
tff(fact_2435_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_2436_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_2437_power2__less__0,axiom,
    ! [A: real] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),zero_zero_real) ).

% power2_less_0
tff(fact_2438_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_2439_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_2440_exp__add__not__zero__imp__right,axiom,
    ! [M: nat,Nb: nat] :
      ( ( aa_nat_uint32(power_power_uint32(numera9087168376688890119uint32(aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Nb)) != zero_zero_uint32 )
     => ( aa_nat_uint32(power_power_uint32(numera9087168376688890119uint32(aa_num_num(bit0,one))),Nb) != zero_zero_uint32 ) ) ).

% exp_add_not_zero_imp_right
tff(fact_2441_exp__add__not__zero__imp__right,axiom,
    ! [M: nat,Nb: nat] :
      ( ( aa_nat5599461682457136004l_num1(power_2184487114949457152l_num1(numera7442385471795722001l_num1(aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Nb)) != zero_z3563351764282998399l_num1 )
     => ( aa_nat5599461682457136004l_num1(power_2184487114949457152l_num1(numera7442385471795722001l_num1(aa_num_num(bit0,one))),Nb) != zero_z3563351764282998399l_num1 ) ) ).

% exp_add_not_zero_imp_right
tff(fact_2442_exp__add__not__zero__imp__right,axiom,
    ! [M: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Nb)) != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb) != zero_zero_nat ) ) ).

% exp_add_not_zero_imp_right
tff(fact_2443_exp__add__not__zero__imp__right,axiom,
    ! [M: nat,Nb: nat] :
      ( ( 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(plus_plus_nat,M),Nb)) != zero_zero_int )
     => ( aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb) != zero_zero_int ) ) ).

% exp_add_not_zero_imp_right
tff(fact_2444_exp__add__not__zero__imp__left,axiom,
    ! [M: nat,Nb: nat] :
      ( ( aa_nat_uint32(power_power_uint32(numera9087168376688890119uint32(aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Nb)) != zero_zero_uint32 )
     => ( aa_nat_uint32(power_power_uint32(numera9087168376688890119uint32(aa_num_num(bit0,one))),M) != zero_zero_uint32 ) ) ).

% exp_add_not_zero_imp_left
tff(fact_2445_exp__add__not__zero__imp__left,axiom,
    ! [M: nat,Nb: nat] :
      ( ( aa_nat5599461682457136004l_num1(power_2184487114949457152l_num1(numera7442385471795722001l_num1(aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Nb)) != zero_z3563351764282998399l_num1 )
     => ( aa_nat5599461682457136004l_num1(power_2184487114949457152l_num1(numera7442385471795722001l_num1(aa_num_num(bit0,one))),M) != zero_z3563351764282998399l_num1 ) ) ).

% exp_add_not_zero_imp_left
tff(fact_2446_exp__add__not__zero__imp__left,axiom,
    ! [M: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Nb)) != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M) != zero_zero_nat ) ) ).

% exp_add_not_zero_imp_left
tff(fact_2447_exp__add__not__zero__imp__left,axiom,
    ! [M: nat,Nb: nat] :
      ( ( 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(plus_plus_nat,M),Nb)) != zero_zero_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 ) ) ).

% exp_add_not_zero_imp_left
tff(fact_2448_exp__not__zero__imp__exp__diff__not__zero,axiom,
    ! [Nb: nat,M: nat] :
      ( ( aa_nat_uint32(power_power_uint32(numera9087168376688890119uint32(aa_num_num(bit0,one))),Nb) != zero_zero_uint32 )
     => ( aa_nat_uint32(power_power_uint32(numera9087168376688890119uint32(aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),M)) != zero_zero_uint32 ) ) ).

% exp_not_zero_imp_exp_diff_not_zero
tff(fact_2449_exp__not__zero__imp__exp__diff__not__zero,axiom,
    ! [Nb: nat,M: nat] :
      ( ( aa_nat5599461682457136004l_num1(power_2184487114949457152l_num1(numera7442385471795722001l_num1(aa_num_num(bit0,one))),Nb) != zero_z3563351764282998399l_num1 )
     => ( aa_nat5599461682457136004l_num1(power_2184487114949457152l_num1(numera7442385471795722001l_num1(aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),M)) != zero_z3563351764282998399l_num1 ) ) ).

% exp_not_zero_imp_exp_diff_not_zero
tff(fact_2450_exp__not__zero__imp__exp__diff__not__zero,axiom,
    ! [Nb: nat,M: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb) != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),M)) != zero_zero_nat ) ) ).

% exp_not_zero_imp_exp_diff_not_zero
tff(fact_2451_exp__not__zero__imp__exp__diff__not__zero,axiom,
    ! [Nb: nat,M: nat] :
      ( ( aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb) != zero_zero_int )
     => ( aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),M)) != zero_zero_int ) ) ).

% exp_not_zero_imp_exp_diff_not_zero
tff(fact_2452_power__diff__power__eq,axiom,
    ! [A: nat,M: nat,Nb: nat] :
      ( ( A != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),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),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),M)))) ) ) ).

% power_diff_power_eq
tff(fact_2453_power__diff__power__eq,axiom,
    ! [A: int,M: nat,Nb: 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),Nb)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),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),Nb)),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,Nb),M)))) ) ) ).

% power_diff_power_eq
tff(fact_2454_less__2__cases__iff,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))
    <=> ( ( Nb = zero_zero_nat )
        | ( Nb = aa_nat_nat(suc,zero_zero_nat) ) ) ) ).

% less_2_cases_iff
tff(fact_2455_less__2__cases,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))
     => ( ( Nb = zero_zero_nat )
        | ( Nb = aa_nat_nat(suc,zero_zero_nat) ) ) ) ).

% less_2_cases
tff(fact_2456_inverse__of__nat__le,axiom,
    ! [Nb: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),M)
     => ( ( Nb != zero_zero_nat )
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_nat_real(semiri5074537144036343181t_real,M))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_nat_real(semiri5074537144036343181t_real,Nb))) ) ) ).

% inverse_of_nat_le
tff(fact_2457_inverse__of__nat__le,axiom,
    ! [Nb: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),M)
     => ( ( Nb != zero_zero_nat )
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,divide_divide_rat(one_one_rat,aa_nat_rat(semiri681578069525770553at_rat,M))),divide_divide_rat(one_one_rat,aa_nat_rat(semiri681578069525770553at_rat,Nb))) ) ) ).

% inverse_of_nat_le
tff(fact_2458_power__eq__if,axiom,
    ! [P2: uint32,M: nat] :
      aa_nat_uint32(power_power_uint32(P2),M) = $ite(M = zero_zero_nat,one_one_uint32,aa_uint32_uint32(times_times_uint32(P2),aa_nat_uint32(power_power_uint32(P2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat)))) ).

% power_eq_if
tff(fact_2459_power__eq__if,axiom,
    ! [P2: word_N3645301735248828278l_num1,M: nat] :
      aa_nat5599461682457136004l_num1(power_2184487114949457152l_num1(P2),M) = $ite(M = zero_zero_nat,one_on7727431528512463931l_num1,aa_wor7274019993978516266l_num1(times_7065122842183080059l_num1(P2),aa_nat5599461682457136004l_num1(power_2184487114949457152l_num1(P2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat)))) ).

% power_eq_if
tff(fact_2460_power__eq__if,axiom,
    ! [P2: real,M: nat] :
      aa_nat_real(aa_real_fun_nat_real(power_power_real,P2),M) = $ite(M = zero_zero_nat,one_one_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,P2),aa_nat_real(aa_real_fun_nat_real(power_power_real,P2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat)))) ).

% power_eq_if
tff(fact_2461_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(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_2462_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_2463_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_2464_power__minus__mult,axiom,
    ! [Nb: nat,A: uint32] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_uint32_uint32(times_times_uint32(aa_nat_uint32(power_power_uint32(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat))),A) = aa_nat_uint32(power_power_uint32(A),Nb) ) ) ).

% power_minus_mult
tff(fact_2465_power__minus__mult,axiom,
    ! [Nb: nat,A: word_N3645301735248828278l_num1] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_wor7274019993978516266l_num1(times_7065122842183080059l_num1(aa_nat5599461682457136004l_num1(power_2184487114949457152l_num1(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat))),A) = aa_nat5599461682457136004l_num1(power_2184487114949457152l_num1(A),Nb) ) ) ).

% power_minus_mult
tff(fact_2466_power__minus__mult,axiom,
    ! [Nb: nat,A: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat))),A) = aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb) ) ) ).

% power_minus_mult
tff(fact_2467_power__minus__mult,axiom,
    ! [Nb: nat,A: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat))),A) = aa_nat_rat(power_power_rat(A),Nb) ) ) ).

% power_minus_mult
tff(fact_2468_power__minus__mult,axiom,
    ! [Nb: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat))),A) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb) ) ) ).

% power_minus_mult
tff(fact_2469_power__minus__mult,axiom,
    ! [Nb: nat,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat))),A) = aa_nat_int(aa_int_fun_nat_int(power_power_int,A),Nb) ) ) ).

% power_minus_mult
tff(fact_2470_le__div__geq,axiom,
    ! [Nb: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),M)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Nb) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Nb)),Nb)) ) ) ) ).

% le_div_geq
tff(fact_2471_split__div_H,axiom,
    ! [P: fun_nat_o,M: nat,Nb: nat] :
      ( aa_nat_o(P,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Nb))
    <=> ( ( ( Nb = zero_zero_nat )
          & aa_nat_o(P,zero_zero_nat) )
        | ? [Q4: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),Q4)),M)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),aa_nat_nat(suc,Q4)))
            & aa_nat_o(P,Q4) ) ) ) ).

% split_div'
tff(fact_2472_power__sub,axiom,
    ! [Nb: nat,M: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Nb)) = 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),Nb)) ) ) ) ).

% power_sub
tff(fact_2473_vebt__pred_Osimps_I6_J,axiom,
    ! [V: product_prod_nat_nat,Vh: list_VEBT_VEBT,Vi: vEBT_VEBT,Vj: nat] : aa_nat_option_nat(vEBT_vebt_pred(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V),aa_nat_nat(suc,zero_zero_nat),Vh),Vi)),Vj) = none_nat ).

% vebt_pred.simps(6)
tff(fact_2474_vebt__succ_Osimps_I5_J,axiom,
    ! [V: product_prod_nat_nat,Vg: list_VEBT_VEBT,Vh: vEBT_VEBT,Vi: nat] : aa_nat_option_nat(vEBT_vebt_succ(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V),aa_nat_nat(suc,zero_zero_nat),Vg),Vh)),Vi) = none_nat ).

% vebt_succ.simps(5)
tff(fact_2475_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Osimps_I6_J,axiom,
    ! [V: product_prod_nat_nat,Vh: list_VEBT_VEBT,Vi: vEBT_VEBT,Vj: nat] : aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_p_r_e_d2,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V),aa_nat_nat(suc,zero_zero_nat),Vh),Vi)),Vj) = one_one_nat ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.simps(6)
tff(fact_2476_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Osimps_I5_J,axiom,
    ! [V: product_prod_nat_nat,Vg: list_VEBT_VEBT,Vh: vEBT_VEBT,Vi: nat] : aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_s_u_c_c2,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V),aa_nat_nat(suc,zero_zero_nat),Vg),Vh)),Vi) = one_one_nat ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.simps(5)
tff(fact_2477_power2__less__imp__less,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y) ) ) ).

% power2_less_imp_less
tff(fact_2478_power2__less__imp__less,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(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,Xb),Y) ) ) ).

% power2_less_imp_less
tff(fact_2479_power2__less__imp__less,axiom,
    ! [Xb: 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,Xb),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,Xb),Y) ) ) ).

% power2_less_imp_less
tff(fact_2480_power2__less__imp__less,axiom,
    ! [Xb: 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,Xb),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,Xb),Y) ) ) ).

% power2_less_imp_less
tff(fact_2481_sum__power2__le__zero__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),zero_zero_real)
    <=> ( ( Xb = zero_zero_real )
        & ( Y = zero_zero_real ) ) ) ).

% sum_power2_le_zero_iff
tff(fact_2482_sum__power2__le__zero__iff,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(aa_nat_rat(power_power_rat(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),zero_zero_rat)
    <=> ( ( Xb = zero_zero_rat )
        & ( Y = zero_zero_rat ) ) ) ).

% sum_power2_le_zero_iff
tff(fact_2483_sum__power2__le__zero__iff,axiom,
    ! [Xb: 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,Xb),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)
    <=> ( ( Xb = zero_zero_int )
        & ( Y = zero_zero_int ) ) ) ).

% sum_power2_le_zero_iff
tff(fact_2484_sum__power2__ge__zero,axiom,
    ! [Xb: real,Y: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% sum_power2_ge_zero
tff(fact_2485_sum__power2__ge__zero,axiom,
    ! [Xb: rat,Y: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(plus_plus_rat(aa_nat_rat(power_power_rat(Xb),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_2486_sum__power2__ge__zero,axiom,
    ! [Xb: 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,Xb),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_2487_sum__power2__gt__zero__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
    <=> ( ( Xb != zero_zero_real )
        | ( Y != zero_zero_real ) ) ) ).

% sum_power2_gt_zero_iff
tff(fact_2488_sum__power2__gt__zero__iff,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(plus_plus_rat(aa_nat_rat(power_power_rat(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
    <=> ( ( Xb != zero_zero_rat )
        | ( Y != zero_zero_rat ) ) ) ).

% sum_power2_gt_zero_iff
tff(fact_2489_sum__power2__gt__zero__iff,axiom,
    ! [Xb: 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,Xb),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)))))
    <=> ( ( Xb != zero_zero_int )
        | ( Y != zero_zero_int ) ) ) ).

% sum_power2_gt_zero_iff
tff(fact_2490_not__sum__power2__lt__zero,axiom,
    ! [Xb: real,Y: real] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),zero_zero_real) ).

% not_sum_power2_lt_zero
tff(fact_2491_not__sum__power2__lt__zero,axiom,
    ! [Xb: rat,Y: rat] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(aa_nat_rat(power_power_rat(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),zero_zero_rat) ).

% not_sum_power2_lt_zero
tff(fact_2492_not__sum__power2__lt__zero,axiom,
    ! [Xb: 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,Xb),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_2493_VEBT__internal_OminNulli_Oelims,axiom,
    ! [Xb: vEBT_VEBTi,Y: heap_Time_Heap_o] :
      ( ( aa_VEB8194921341634505877Heap_o(vEBT_VEBT_minNulli,Xb) = Y )
     => ( ( ( Xb = vEBT_Leafi($false,$false) )
         => ( Y != aa_o_H7399033255965945470Heap_o(heap_Time_return_o,$true) ) )
       => ( ( ? [Uv2: $o] : Xb = vEBT_Leafi($true,(Uv2))
           => ( Y != aa_o_H7399033255965945470Heap_o(heap_Time_return_o,$false) ) )
         => ( ( ? [Uu2: $o] : Xb = vEBT_Leafi((Uu2),$true)
             => ( Y != aa_o_H7399033255965945470Heap_o(heap_Time_return_o,$false) ) )
           => ( ( ? [Uw2: nat,Ux2: array_VEBT_VEBTi,Uy2: vEBT_VEBTi] : Xb = aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(none_P5556105721700978146at_nat,Uw2,Ux2),Uy2)
               => ( Y != aa_o_H7399033255965945470Heap_o(heap_Time_return_o,$true) ) )
             => ~ ( ? [Uz2: product_prod_nat_nat,Va: nat,Vb2: array_VEBT_VEBTi,Vc: vEBT_VEBTi] : Xb = aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(some_P7363390416028606310at_nat(Uz2),Va,Vb2),Vc)
                 => ( Y != aa_o_H7399033255965945470Heap_o(heap_Time_return_o,$false) ) ) ) ) ) ) ) ).

% VEBT_internal.minNulli.elims
tff(fact_2494_zero__le__even__power_H,axiom,
    ! [A: real,Nb: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))) ).

% zero_le_even_power'
tff(fact_2495_zero__le__even__power_H,axiom,
    ! [A: rat,Nb: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))) ).

% zero_le_even_power'
tff(fact_2496_zero__le__even__power_H,axiom,
    ! [A: int,Nb: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(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))),Nb))) ).

% zero_le_even_power'
tff(fact_2497_Suc__n__div__2__gt__zero,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,Nb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ).

% Suc_n_div_2_gt_zero
tff(fact_2498_div__2__gt__zero,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ).

% div_2_gt_zero
tff(fact_2499_nat__bit__induct,axiom,
    ! [P: fun_nat_o,Nb: nat] :
      ( aa_nat_o(P,zero_zero_nat)
     => ( ! [N2: nat] :
            ( aa_nat_o(P,N2)
           => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)
             => aa_nat_o(P,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2)) ) )
       => ( ! [N2: nat] :
              ( aa_nat_o(P,N2)
             => aa_nat_o(P,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2))) )
         => aa_nat_o(P,Nb) ) ) ) ).

% nat_bit_induct
tff(fact_2500_vebt__minti_Osimps_I1_J,axiom,
    ! [A: $o,B: $o] :
      aa_VEB2711320435582112505on_nat(vEBT_vebt_minti,vEBT_Leafi((A),(B))) = $ite(
        (A),
        heap_T3487192422709364219on_nat(aa_nat_option_nat(some_nat,zero_zero_nat)),
        $ite((B),heap_T3487192422709364219on_nat(aa_nat_option_nat(some_nat,one_one_nat)),heap_T3487192422709364219on_nat(none_nat)) ) ).

% vebt_minti.simps(1)
tff(fact_2501_vebt__maxti_Osimps_I1_J,axiom,
    ! [A: $o,B: $o] :
      aa_VEB2711320435582112505on_nat(vEBT_vebt_maxti,vEBT_Leafi((A),(B))) = $ite(
        (B),
        heap_T3487192422709364219on_nat(aa_nat_option_nat(some_nat,one_one_nat)),
        $ite((A),heap_T3487192422709364219on_nat(aa_nat_option_nat(some_nat,zero_zero_nat)),heap_T3487192422709364219on_nat(none_nat)) ) ).

% vebt_maxti.simps(1)
tff(fact_2502_mod__starE,axiom,
    ! [A: assn,B: assn,H: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(rep_assn(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A),B)),H)
     => ~ ( ? [X_13: produc3658429121746597890et_nat] : aa_Pro653082453696054616_nat_o(rep_assn(A),X_13)
         => ! [H_2: produc3658429121746597890et_nat] : ~ aa_Pro653082453696054616_nat_o(rep_assn(B),H_2) ) ) ).

% mod_starE
tff(fact_2503_mod__starD,axiom,
    ! [A2: assn,B4: assn,H: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(rep_assn(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A2),B4)),H)
     => ? [H1: produc3658429121746597890et_nat,H22: produc3658429121746597890et_nat] :
          ( aa_Pro653082453696054616_nat_o(rep_assn(A2),H1)
          & aa_Pro653082453696054616_nat_o(rep_assn(B4),H22) ) ) ).

% mod_starD
tff(fact_2504_odd__0__le__power__imp__0__le,axiom,
    ! [A: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))))
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A) ) ).

% odd_0_le_power_imp_0_le
tff(fact_2505_odd__0__le__power__imp__0__le,axiom,
    ! [A: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))))
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A) ) ).

% odd_0_le_power_imp_0_le
tff(fact_2506_odd__0__le__power__imp__0__le,axiom,
    ! [A: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(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))),Nb))))
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A) ) ).

% odd_0_le_power_imp_0_le
tff(fact_2507_odd__power__less__zero,axiom,
    ! [A: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)))),zero_zero_real) ) ).

% odd_power_less_zero
tff(fact_2508_odd__power__less__zero,axiom,
    ! [A: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)))),zero_zero_rat) ) ).

% odd_power_less_zero
tff(fact_2509_odd__power__less__zero,axiom,
    ! [A: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(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))),Nb)))),zero_zero_int) ) ).

% odd_power_less_zero
tff(fact_2510_assn__one__left,axiom,
    ! [P: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,one_one_assn),P) = P ).

% assn_one_left
tff(fact_2511_VEBT__internal_Oexp__split__high__low_I1_J,axiom,
    ! [Xb: nat,Nb: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),M)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)) ) ) ) ).

% VEBT_internal.exp_split_high_low(1)
tff(fact_2512_VEBT__internal_Oexp__split__high__low_I2_J,axiom,
    ! [Xb: nat,Nb: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),M)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xb),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ) ) ) ).

% VEBT_internal.exp_split_high_low(2)
tff(fact_2513_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_Osimps_I2_J,axiom,
    ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT,Xb: nat] : aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_m_e_m_b_e_r,aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv),Uw)),Xb) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.simps(2)
tff(fact_2514_nat__div__eq__Suc__0__iff,axiom,
    ! [Nb: nat,M: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),M) = aa_nat_nat(suc,zero_zero_nat) )
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)) ) ) ).

% nat_div_eq_Suc_0_iff
tff(fact_2515_ex__distrib__star,axiom,
    ! [P: fun_li3352006551604695530i_assn,Q: assn] : ex_ass463751140784270563_VEBTi(aa_ass1897699608592665680i_assn(aTP_Lamm_cu(P),Q)) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,ex_ass463751140784270563_VEBTi(P)),Q) ).

% ex_distrib_star
tff(fact_2516_ex__one__point__gen,axiom,
    ! [P: fun_li3352006551604695530i_assn,V: list_VEBT_VEBTi] :
      ( ! [H2: produc3658429121746597890et_nat,X2: list_VEBT_VEBTi] :
          ( aa_Pro653082453696054616_nat_o(rep_assn(aa_lis2508312872377092907i_assn(P,X2)),H2)
         => ( X2 = V ) )
     => ( ex_ass463751140784270563_VEBTi(P) = aa_lis2508312872377092907i_assn(P,V) ) ) ).

% ex_one_point_gen
tff(fact_2517_mod__exI,axiom,
    ! [P: fun_li3352006551604695530i_assn,H: produc3658429121746597890et_nat] :
      ( ? [X3: list_VEBT_VEBTi] : aa_Pro653082453696054616_nat_o(rep_assn(aa_lis2508312872377092907i_assn(P,X3)),H)
     => aa_Pro653082453696054616_nat_o(rep_assn(ex_ass463751140784270563_VEBTi(P)),H) ) ).

% mod_exI
tff(fact_2518_mod__exE,axiom,
    ! [P: fun_li3352006551604695530i_assn,H: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(rep_assn(ex_ass463751140784270563_VEBTi(P)),H)
     => ~ ! [X2: list_VEBT_VEBTi] : ~ aa_Pro653082453696054616_nat_o(rep_assn(aa_lis2508312872377092907i_assn(P,X2)),H) ) ).

% mod_exE
tff(fact_2519_arith__geo__mean,axiom,
    ! [U: real,Xb: real,Y: real] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,U),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Y) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,U),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Y)),numeral_numeral_real(aa_num_num(bit0,one)))) ) ) ) ).

% arith_geo_mean
tff(fact_2520_arith__geo__mean,axiom,
    ! [U: rat,Xb: rat,Y: rat] :
      ( ( aa_nat_rat(power_power_rat(U),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_rat_rat(times_times_rat(Xb),Y) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Xb)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Y)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,U),divide_divide_rat(aa_rat_rat(plus_plus_rat(Xb),Y),numeral_numeral_rat(aa_num_num(bit0,one)))) ) ) ) ).

% arith_geo_mean
tff(fact_2521_vebt__minti_Oelims,axiom,
    ! [Xb: vEBT_VEBTi,Y: heap_T2636463487746394924on_nat] :
      ( ( aa_VEB2711320435582112505on_nat(vEBT_vebt_minti,Xb) = Y )
     => ( ! [A3: $o,B2: $o] :
            ( ( Xb = vEBT_Leafi((A3),(B2)) )
           => ( Y != $ite(
                  (A3),
                  heap_T3487192422709364219on_nat(aa_nat_option_nat(some_nat,zero_zero_nat)),
                  $ite((B2),heap_T3487192422709364219on_nat(aa_nat_option_nat(some_nat,one_one_nat)),heap_T3487192422709364219on_nat(none_nat)) ) ) )
       => ( ( ? [Uu2: nat,Uv2: array_VEBT_VEBTi,Uw2: vEBT_VEBTi] : Xb = aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(none_P5556105721700978146at_nat,Uu2,Uv2),Uw2)
           => ( Y != heap_T3487192422709364219on_nat(none_nat) ) )
         => ~ ! [Mi: nat] :
                ( ? [Ma: nat,Ux2: nat,Uy2: array_VEBT_VEBTi,Uz2: vEBT_VEBTi] : Xb = aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),Ux2,Uy2),Uz2)
               => ( Y != heap_T3487192422709364219on_nat(aa_nat_option_nat(some_nat,Mi)) ) ) ) ) ) ).

% vebt_minti.elims
tff(fact_2522_vebt__maxti_Oelims,axiom,
    ! [Xb: vEBT_VEBTi,Y: heap_T2636463487746394924on_nat] :
      ( ( aa_VEB2711320435582112505on_nat(vEBT_vebt_maxti,Xb) = Y )
     => ( ! [A3: $o,B2: $o] :
            ( ( Xb = vEBT_Leafi((A3),(B2)) )
           => ( Y != $ite(
                  (B2),
                  heap_T3487192422709364219on_nat(aa_nat_option_nat(some_nat,one_one_nat)),
                  $ite((A3),heap_T3487192422709364219on_nat(aa_nat_option_nat(some_nat,zero_zero_nat)),heap_T3487192422709364219on_nat(none_nat)) ) ) )
       => ( ( ? [Uu2: nat,Uv2: array_VEBT_VEBTi,Uw2: vEBT_VEBTi] : Xb = aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(none_P5556105721700978146at_nat,Uu2,Uv2),Uw2)
           => ( Y != heap_T3487192422709364219on_nat(none_nat) ) )
         => ~ ! [Mi: nat,Ma: nat] :
                ( ? [Ux2: nat,Uy2: array_VEBT_VEBTi,Uz2: vEBT_VEBTi] : Xb = aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),Ux2,Uy2),Uz2)
               => ( Y != heap_T3487192422709364219on_nat(aa_nat_option_nat(some_nat,Ma)) ) ) ) ) ) ).

% vebt_maxti.elims
tff(fact_2523_member__bound__height,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xb: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Ta),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_m_e_m_b_e_r,Ta),Xb)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_VEBT_height,Ta))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,one)))))) ) ).

% member_bound_height
tff(fact_2524_minNrulli__ruleT,axiom,
    ! [Ta: vEBT_VEBT,Tib: vEBT_VEBTi] : aa_nat_o(time_htt_o(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Ta),Tib),aa_VEB8194921341634505877Heap_o(vEBT_VEBT_minNulli,Tib),aa_VEB6687300313434771872o_assn(aTP_Lamm_ej(Ta),Tib)),one_one_nat) ).

% minNrulli_ruleT
tff(fact_2525_Tb__T__vebt__buildupi,axiom,
    ! [Nb: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(vEBT_V441764108873111860ildupi,Nb))),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(vEBT_VEBT_Tb,Nb)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))) ).

% Tb_T_vebt_buildupi
tff(fact_2526_le__divide__eq__1__pos,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),A))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ) ).

% le_divide_eq_1_pos
tff(fact_2527_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),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_2528_le__divide__eq__1__neg,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),A))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A) ) ) ).

% le_divide_eq_1_neg
tff(fact_2529_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),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_2530_divide__le__eq__1__pos,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),A)),one_one_real)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A) ) ) ).

% divide_le_eq_1_pos
tff(fact_2531_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,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_2532_divide__le__eq__1__neg,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),A)),one_one_real)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ) ).

% divide_le_eq_1_neg
tff(fact_2533_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,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_2534_T__vebt__buildupi,axiom,
    ! [Nb: nat,H: heap_e7401611519738050253t_unit] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,time_time_VEBT_VEBTi(aa_nat5408703056993926195_VEBTi(vEBT_V739175172307565963ildupi,Nb),H)),aa_nat_nat(vEBT_V441764108873111860ildupi,Nb)) ).

% T_vebt_buildupi
tff(fact_2535_nonzero__divide__mult__cancel__right,axiom,
    ! [B: real,A: real] :
      ( ( B != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),A) ) ) ).

% nonzero_divide_mult_cancel_right
tff(fact_2536_nonzero__divide__mult__cancel__right,axiom,
    ! [B: rat,A: rat] :
      ( ( B != zero_zero_rat )
     => ( divide_divide_rat(B,aa_rat_rat(times_times_rat(A),B)) = divide_divide_rat(one_one_rat,A) ) ) ).

% nonzero_divide_mult_cancel_right
tff(fact_2537_nonzero__divide__mult__cancel__left,axiom,
    ! [A: real,B: real] :
      ( ( A != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),B) ) ) ).

% nonzero_divide_mult_cancel_left
tff(fact_2538_nonzero__divide__mult__cancel__left,axiom,
    ! [A: rat,B: rat] :
      ( ( A != zero_zero_rat )
     => ( divide_divide_rat(A,aa_rat_rat(times_times_rat(A),B)) = divide_divide_rat(one_one_rat,B) ) ) ).

% nonzero_divide_mult_cancel_left
tff(fact_2539_zero__less__divide__1__iff,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),A))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A) ) ).

% zero_less_divide_1_iff
tff(fact_2540_zero__less__divide__1__iff,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_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_2541_cnt__non__neg,axiom,
    ! [Ta: vEBT_VEBT] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_VEBT_VEBT_real(vEBT_VEBT_cnt,Ta)) ).

% cnt_non_neg
tff(fact_2542_Tb__Tb_H,axiom,
    ! [Ta: nat] : aa_nat_int(vEBT_VEBT_Tb,Ta) = aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(vEBT_VEBT_Tb2,Ta)) ).

% Tb_Tb'
tff(fact_2543_time__replicate,axiom,
    ! [Xb: heap_T5738788834812785303t_unit,C: nat,Nb: nat,H: heap_e7401611519738050253t_unit] :
      ( ! [H2: heap_e7401611519738050253t_unit] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,time_t4224138285095624986t_unit(Xb,H2)),C)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,time_t4781937132199089312t_unit(vEBT_V7483891112628345579t_unit(Nb,Xb),H)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),C)),Nb))) ) ).

% time_replicate
tff(fact_2544_time__replicate,axiom,
    ! [Xb: heap_T8145700208782473153_VEBTi,C: nat,Nb: nat,H: heap_e7401611519738050253t_unit] :
      ( ! [H2: heap_e7401611519738050253t_unit] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,time_time_VEBT_VEBTi(Xb,H2)),C)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,time_t3534373299052942712_VEBTi(vEBT_V1859673955506687831_VEBTi(Nb,Xb),H)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),C)),Nb))) ) ).

% time_replicate
tff(fact_2545_divide__eq__0__iff,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B) = zero_zero_real )
    <=> ( ( A = zero_zero_real )
        | ( B = zero_zero_real ) ) ) ).

% divide_eq_0_iff
tff(fact_2546_divide__eq__0__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( divide_divide_rat(A,B) = zero_zero_rat )
    <=> ( ( A = zero_zero_rat )
        | ( B = zero_zero_rat ) ) ) ).

% divide_eq_0_iff
tff(fact_2547_divide__cancel__left,axiom,
    ! [C: real,A: real,B: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,C),A) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,C),B) )
    <=> ( ( C = zero_zero_real )
        | ( A = B ) ) ) ).

% divide_cancel_left
tff(fact_2548_divide__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( divide_divide_rat(C,A) = divide_divide_rat(C,B) )
    <=> ( ( C = zero_zero_rat )
        | ( A = B ) ) ) ).

% divide_cancel_left
tff(fact_2549_divide__cancel__right,axiom,
    ! [A: real,C: real,B: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),C) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C) )
    <=> ( ( C = zero_zero_real )
        | ( A = B ) ) ) ).

% divide_cancel_right
tff(fact_2550_divide__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( divide_divide_rat(A,C) = divide_divide_rat(B,C) )
    <=> ( ( C = zero_zero_rat )
        | ( A = B ) ) ) ).

% divide_cancel_right
tff(fact_2551_division__ring__divide__zero,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),zero_zero_real) = zero_zero_real ).

% division_ring_divide_zero
tff(fact_2552_division__ring__divide__zero,axiom,
    ! [A: rat] : divide_divide_rat(A,zero_zero_rat) = zero_zero_rat ).

% division_ring_divide_zero
tff(fact_2553_times__divide__eq__left,axiom,
    ! [B: real,C: real,A: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)),A) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),A)),C) ).

% times_divide_eq_left
tff(fact_2554_times__divide__eq__left,axiom,
    ! [B: rat,C: rat,A: rat] : aa_rat_rat(times_times_rat(divide_divide_rat(B,C)),A) = divide_divide_rat(aa_rat_rat(times_times_rat(B),A),C) ).

% times_divide_eq_left
tff(fact_2555_divide__divide__eq__left,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B)),C) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)) ).

% divide_divide_eq_left
tff(fact_2556_divide__divide__eq__left,axiom,
    ! [A: rat,B: rat,C: rat] : divide_divide_rat(divide_divide_rat(A,B),C) = divide_divide_rat(A,aa_rat_rat(times_times_rat(B),C)) ).

% divide_divide_eq_left
tff(fact_2557_divide__divide__eq__right,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),B) ).

% divide_divide_eq_right
tff(fact_2558_divide__divide__eq__right,axiom,
    ! [A: rat,B: rat,C: rat] : divide_divide_rat(A,divide_divide_rat(B,C)) = divide_divide_rat(aa_rat_rat(times_times_rat(A),C),B) ).

% divide_divide_eq_right
tff(fact_2559_times__divide__eq__right,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),C) ).

% times_divide_eq_right
tff(fact_2560_times__divide__eq__right,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(times_times_rat(A),divide_divide_rat(B,C)) = divide_divide_rat(aa_rat_rat(times_times_rat(A),B),C) ).

% times_divide_eq_right
tff(fact_2561_not__real__square__gt__zero,axiom,
    ! [Xb: real] :
      ( ~ aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Xb))
    <=> ( Xb = zero_zero_real ) ) ).

% not_real_square_gt_zero
tff(fact_2562_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_2563_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_2564_int__div__same__is__1,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
     => ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B) = A )
      <=> ( B = one_one_int ) ) ) ).

% int_div_same_is_1
tff(fact_2565_divide__eq__1__iff,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B) = one_one_real )
    <=> ( ( B != zero_zero_real )
        & ( A = B ) ) ) ).

% divide_eq_1_iff
tff(fact_2566_divide__eq__1__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( divide_divide_rat(A,B) = one_one_rat )
    <=> ( ( B != zero_zero_rat )
        & ( A = B ) ) ) ).

% divide_eq_1_iff
tff(fact_2567_one__eq__divide__iff,axiom,
    ! [A: real,B: real] :
      ( ( one_one_real = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B) )
    <=> ( ( B != zero_zero_real )
        & ( A = B ) ) ) ).

% one_eq_divide_iff
tff(fact_2568_one__eq__divide__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( one_one_rat = divide_divide_rat(A,B) )
    <=> ( ( B != zero_zero_rat )
        & ( A = B ) ) ) ).

% one_eq_divide_iff
tff(fact_2569_divide__self,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),A) = one_one_real ) ) ).

% divide_self
tff(fact_2570_divide__self,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( divide_divide_rat(A,A) = one_one_rat ) ) ).

% divide_self
tff(fact_2571_divide__self__if,axiom,
    ! [A: real] :
      aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),A) = $ite(A = zero_zero_real,zero_zero_real,one_one_real) ).

% divide_self_if
tff(fact_2572_divide__self__if,axiom,
    ! [A: rat] :
      divide_divide_rat(A,A) = $ite(A = zero_zero_rat,zero_zero_rat,one_one_rat) ).

% divide_self_if
tff(fact_2573_divide__eq__eq__1,axiom,
    ! [B: real,A: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),A) = one_one_real )
    <=> ( ( A != zero_zero_real )
        & ( A = B ) ) ) ).

% divide_eq_eq_1
tff(fact_2574_divide__eq__eq__1,axiom,
    ! [B: rat,A: rat] :
      ( ( divide_divide_rat(B,A) = one_one_rat )
    <=> ( ( A != zero_zero_rat )
        & ( A = B ) ) ) ).

% divide_eq_eq_1
tff(fact_2575_eq__divide__eq__1,axiom,
    ! [B: real,A: real] :
      ( ( one_one_real = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),A) )
    <=> ( ( A != zero_zero_real )
        & ( A = B ) ) ) ).

% eq_divide_eq_1
tff(fact_2576_eq__divide__eq__1,axiom,
    ! [B: rat,A: rat] :
      ( ( one_one_rat = divide_divide_rat(B,A) )
    <=> ( ( A != zero_zero_rat )
        & ( A = B ) ) ) ).

% eq_divide_eq_1
tff(fact_2577_one__divide__eq__0__iff,axiom,
    ! [A: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),A) = zero_zero_real )
    <=> ( A = zero_zero_real ) ) ).

% one_divide_eq_0_iff
tff(fact_2578_one__divide__eq__0__iff,axiom,
    ! [A: rat] :
      ( ( divide_divide_rat(one_one_rat,A) = zero_zero_rat )
    <=> ( A = zero_zero_rat ) ) ).

% one_divide_eq_0_iff
tff(fact_2579_zero__eq__1__divide__iff,axiom,
    ! [A: real] :
      ( ( zero_zero_real = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),A) )
    <=> ( A = zero_zero_real ) ) ).

% zero_eq_1_divide_iff
tff(fact_2580_zero__eq__1__divide__iff,axiom,
    ! [A: rat] :
      ( ( zero_zero_rat = divide_divide_rat(one_one_rat,A) )
    <=> ( A = zero_zero_rat ) ) ).

% zero_eq_1_divide_iff
tff(fact_2581_mult__divide__mult__cancel__left__if,axiom,
    ! [C: real,A: real,B: real] :
      aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B)) = $ite(C = zero_zero_real,zero_zero_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B)) ).

% mult_divide_mult_cancel_left_if
tff(fact_2582_mult__divide__mult__cancel__left__if,axiom,
    ! [C: rat,A: rat,B: rat] :
      divide_divide_rat(aa_rat_rat(times_times_rat(C),A),aa_rat_rat(times_times_rat(C),B)) = $ite(C = zero_zero_rat,zero_zero_rat,divide_divide_rat(A,B)) ).

% mult_divide_mult_cancel_left_if
tff(fact_2583_nonzero__mult__divide__mult__cancel__left,axiom,
    ! [C: real,A: real,B: real] :
      ( ( C != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B) ) ) ).

% nonzero_mult_divide_mult_cancel_left
tff(fact_2584_nonzero__mult__divide__mult__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( divide_divide_rat(aa_rat_rat(times_times_rat(C),A),aa_rat_rat(times_times_rat(C),B)) = divide_divide_rat(A,B) ) ) ).

% nonzero_mult_divide_mult_cancel_left
tff(fact_2585_nonzero__mult__divide__mult__cancel__left2,axiom,
    ! [C: real,A: real,B: real] :
      ( ( C != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B) ) ) ).

% nonzero_mult_divide_mult_cancel_left2
tff(fact_2586_nonzero__mult__divide__mult__cancel__left2,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( divide_divide_rat(aa_rat_rat(times_times_rat(C),A),aa_rat_rat(times_times_rat(B),C)) = divide_divide_rat(A,B) ) ) ).

% nonzero_mult_divide_mult_cancel_left2
tff(fact_2587_nonzero__mult__divide__mult__cancel__right,axiom,
    ! [C: real,A: real,B: real] :
      ( ( C != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B) ) ) ).

% nonzero_mult_divide_mult_cancel_right
tff(fact_2588_nonzero__mult__divide__mult__cancel__right,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( divide_divide_rat(aa_rat_rat(times_times_rat(A),C),aa_rat_rat(times_times_rat(B),C)) = divide_divide_rat(A,B) ) ) ).

% nonzero_mult_divide_mult_cancel_right
tff(fact_2589_nonzero__mult__divide__mult__cancel__right2,axiom,
    ! [C: real,A: real,B: real] :
      ( ( C != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B) ) ) ).

% nonzero_mult_divide_mult_cancel_right2
tff(fact_2590_nonzero__mult__divide__mult__cancel__right2,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( divide_divide_rat(aa_rat_rat(times_times_rat(A),C),aa_rat_rat(times_times_rat(C),B)) = divide_divide_rat(A,B) ) ) ).

% nonzero_mult_divide_mult_cancel_right2
tff(fact_2591_divide__le__0__1__iff,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),A)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real) ) ).

% divide_le_0_1_iff
tff(fact_2592_divide__le__0__1__iff,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_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_2593_zero__le__divide__1__iff,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),A))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A) ) ).

% zero_le_divide_1_iff
tff(fact_2594_zero__le__divide__1__iff,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_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_2595_divide__less__0__1__iff,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),A)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real) ) ).

% divide_less_0_1_iff
tff(fact_2596_divide__less__0__1__iff,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_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_2597_divide__less__eq__1__neg,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),A)),one_one_real)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) ) ) ).

% divide_less_eq_1_neg
tff(fact_2598_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,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_2599_divide__less__eq__1__pos,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),A)),one_one_real)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,B),A) ) ) ).

% divide_less_eq_1_pos
tff(fact_2600_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,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_2601_less__divide__eq__1__neg,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),A))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,B),A) ) ) ).

% less_divide_eq_1_neg
tff(fact_2602_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),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_2603_less__divide__eq__1__pos,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),A))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) ) ) ).

% less_divide_eq_1_pos
tff(fact_2604_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),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_2605_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_2606_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_2607_htt__vebt__memberi,axiom,
    ! [Ta: vEBT_VEBT,Tib: vEBT_VEBTi,Xb: nat] : aa_nat_o(time_htt_o(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Ta),Tib),aa_nat2910048432523430448Heap_o(aa_VEB1455453220190609250Heap_o(vEBT_vebt_memberi,Tib),Xb),aa_nat_fun_o_assn(aa_VEB8176250900481067561o_assn(aTP_Lamm_bo(Ta),Tib),Xb)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit0,one)))),aa_VEBT_VEBT_nat(vEBT_VEBT_height,Ta)))) ).

% htt_vebt_memberi
tff(fact_2608_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_2609_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_2610_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_2611_pos__imp__zdiv__pos__iff,axiom,
    ! [K: int,I: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),K)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(divide_divide_int,I),K))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),I) ) ) ).

% pos_imp_zdiv_pos_iff
tff(fact_2612_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_2613_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_2614_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_2615_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_2616_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_2617_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_2618_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_2619_zdiv__mono2__neg,axiom,
    ! [A: int,B3: 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),B3)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B3),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),B3)),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)) ) ) ) ).

% zdiv_mono2_neg
tff(fact_2620_zdiv__mono1__neg,axiom,
    ! [A: int,A5: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),A5)
     => ( 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,A5),B)),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)) ) ) ).

% zdiv_mono1_neg
tff(fact_2621_zdiv__eq__0__iff,axiom,
    ! [I: int,K: int] :
      ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,I),K) = zero_zero_int )
    <=> ( ( K = zero_zero_int )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),I)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,I),K) )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),zero_zero_int)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,K),I) ) ) ) ).

% zdiv_eq_0_iff
tff(fact_2622_zdiv__mono2,axiom,
    ! [A: int,B3: 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),B3)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B3),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),B3)) ) ) ) ).

% zdiv_mono2
tff(fact_2623_zdiv__mono1,axiom,
    ! [A: int,A5: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),A5)
     => ( 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,A5),B)) ) ) ).

% zdiv_mono1
tff(fact_2624_zdiv__le__dividend,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),A) ) ) ).

% zdiv_le_dividend
tff(fact_2625_q__pos__lemma,axiom,
    ! [B3: int,Q5: int,R3: 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,B3),Q5)),R3))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,R3),B3)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B3)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Q5) ) ) ) ).

% q_pos_lemma
tff(fact_2626_zdiv__mono2__lemma,axiom,
    ! [B: int,Q2: int,R2: int,B3: int,Q5: int,R3: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),Q2)),R2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B3),Q5)),R3) )
     => ( 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,B3),Q5)),R3))
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,R3),B3)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),R2)
           => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B3)
             => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B3),B)
               => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Q2),Q5) ) ) ) ) ) ) ).

% zdiv_mono2_lemma
tff(fact_2627_zdiv__mono2__neg__lemma,axiom,
    ! [B: int,Q2: int,R2: int,B3: int,Q5: int,R3: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),Q2)),R2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B3),Q5)),R3) )
     => ( 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,B3),Q5)),R3)),zero_zero_int)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,R2),B)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),R3)
           => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B3)
             => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B3),B)
               => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Q5),Q2) ) ) ) ) ) ) ).

% zdiv_mono2_neg_lemma
tff(fact_2628_unique__quotient__lemma,axiom,
    ! [B: int,Q5: int,R3: int,Q2: int,R2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),Q5)),R3)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),Q2)),R2))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),R3)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,R3),B)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_int,R2),B)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Q5),Q2) ) ) ) ) ).

% unique_quotient_lemma
tff(fact_2629_unique__quotient__lemma__neg,axiom,
    ! [B: int,Q5: int,R3: int,Q2: int,R2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),Q5)),R3)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),Q2)),R2))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,R2),zero_zero_int)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),R2)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),R3)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Q2),Q5) ) ) ) ) ).

% unique_quotient_lemma_neg
tff(fact_2630_int__div__less__self,axiom,
    ! [Xb: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Xb)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),K)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,Xb),K)),Xb) ) ) ).

% int_div_less_self
tff(fact_2631_zdiv__zmult2__eq,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),C) ) ) ).

% zdiv_zmult2_eq
tff(fact_2632_linordered__field__no__lb,axiom,
    ! [X3: real] :
    ? [Y4: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,Y4),X3) ).

% linordered_field_no_lb
tff(fact_2633_linordered__field__no__lb,axiom,
    ! [X3: rat] :
    ? [Y4: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y4),X3) ).

% linordered_field_no_lb
tff(fact_2634_linordered__field__no__ub,axiom,
    ! [X3: real] :
    ? [X_13: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,X3),X_13) ).

% linordered_field_no_ub
tff(fact_2635_linordered__field__no__ub,axiom,
    ! [X3: rat] :
    ? [X_13: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),X_13) ).

% linordered_field_no_ub
tff(fact_2636_not__exp__less__eq__0__int,axiom,
    ! [Nb: 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))),Nb)),zero_zero_int) ).

% not_exp_less_eq_0_int
tff(fact_2637_realpow__pos__nth2,axiom,
    ! [A: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ? [R4: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),R4)
          & ( aa_nat_real(aa_real_fun_nat_real(power_power_real,R4),aa_nat_nat(suc,Nb)) = A ) ) ) ).

% realpow_pos_nth2
tff(fact_2638_real__arch__pow__inv,axiom,
    ! [Y: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),one_one_real)
       => ? [N2: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),N2)),Y) ) ) ).

% real_arch_pow_inv
tff(fact_2639_reals__Archimedean3,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ! [Y5: real] :
        ? [N2: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,Y5),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N2)),Xb)) ) ).

% reals_Archimedean3
tff(fact_2640_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_2641_split__zdiv,axiom,
    ! [P: fun_int_o,Nb: int,K: int] :
      ( aa_int_o(P,aa_int_int(aa_int_fun_int_int(divide_divide_int,Nb),K))
    <=> ( ( ( K = zero_zero_int )
         => aa_int_o(P,zero_zero_int) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),K)
         => ! [I2: 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)
                & ( Nb = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,K),I2)),J3) ) )
             => aa_int_o(P,I2) ) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int)
         => ! [I2: 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)
                & ( Nb = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,K),I2)),J3) ) )
             => aa_int_o(P,I2) ) ) ) ) ).

% split_zdiv
tff(fact_2642_int__div__neg__eq,axiom,
    ! [A: int,B: int,Q2: int,R2: int] :
      ( ( A = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),Q2)),R2) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,R2),zero_zero_int)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),R2)
         => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B) = Q2 ) ) ) ) ).

% int_div_neg_eq
tff(fact_2643_int__div__pos__eq,axiom,
    ! [A: int,B: int,Q2: int,R2: int] :
      ( ( A = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),Q2)),R2) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),R2)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,R2),B)
         => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B) = Q2 ) ) ) ) ).

% int_div_pos_eq
tff(fact_2644_zdiv__mult__self,axiom,
    ! [M: int,A: int,Nb: int] :
      ( ( M != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,M),Nb))),M) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),M)),Nb) ) ) ).

% zdiv_mult_self
tff(fact_2645_realpow__pos__nth__unique,axiom,
    ! [Nb: nat,A: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
       => ? [X2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X2)
            & ( aa_nat_real(aa_real_fun_nat_real(power_power_real,X2),Nb) = A )
            & ! [Y5: real] :
                ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y5)
                  & ( aa_nat_real(aa_real_fun_nat_real(power_power_real,Y5),Nb) = A ) )
               => ( Y5 = X2 ) ) ) ) ) ).

% realpow_pos_nth_unique
tff(fact_2646_realpow__pos__nth,axiom,
    ! [Nb: nat,A: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
       => ? [R4: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),R4)
            & ( aa_nat_real(aa_real_fun_nat_real(power_power_real,R4),Nb) = A ) ) ) ) ).

% realpow_pos_nth
tff(fact_2647_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_2648_VEBT__internal_OTb_Osimps_I1_J,axiom,
    aa_nat_int(vEBT_VEBT_Tb,zero_zero_nat) = aa_num_int(numeral_numeral_int,aa_num_num(bit1,one)) ).

% VEBT_internal.Tb.simps(1)
tff(fact_2649_real__archimedian__rdiv__eq__0,axiom,
    ! [Xb: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
       => ( ! [M2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M2)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,M2)),Xb)),C) )
         => ( Xb = zero_zero_real ) ) ) ) ).

% real_archimedian_rdiv_eq_0
tff(fact_2650_real__of__nat__div2,axiom,
    ! [Nb: nat,Xb: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_nat_real(semiri5074537144036343181t_real,Xb))),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),Xb)))) ).

% real_of_nat_div2
tff(fact_2651_axxdiv2,axiom,
    ! [Xb: int] :
      ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),Xb)),Xb)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = Xb )
      & ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,zero_zero_int),Xb)),Xb)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = Xb ) ) ).

% axxdiv2
tff(fact_2652_add__divide__distrib,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),C) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)) ).

% add_divide_distrib
tff(fact_2653_add__divide__distrib,axiom,
    ! [A: rat,B: rat,C: rat] : divide_divide_rat(aa_rat_rat(plus_plus_rat(A),B),C) = aa_rat_rat(plus_plus_rat(divide_divide_rat(A,C)),divide_divide_rat(B,C)) ).

% add_divide_distrib
tff(fact_2654_times__divide__times__eq,axiom,
    ! [Xb: real,Y: real,Z: real,W: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Y)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Z),W)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Z)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),W)) ).

% times_divide_times_eq
tff(fact_2655_times__divide__times__eq,axiom,
    ! [Xb: rat,Y: rat,Z: rat,W: rat] : aa_rat_rat(times_times_rat(divide_divide_rat(Xb,Y)),divide_divide_rat(Z,W)) = divide_divide_rat(aa_rat_rat(times_times_rat(Xb),Z),aa_rat_rat(times_times_rat(Y),W)) ).

% times_divide_times_eq
tff(fact_2656_divide__divide__times__eq,axiom,
    ! [Xb: real,Y: real,Z: real,W: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Y)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Z),W)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),W)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Z)) ).

% divide_divide_times_eq
tff(fact_2657_divide__divide__times__eq,axiom,
    ! [Xb: rat,Y: rat,Z: rat,W: rat] : divide_divide_rat(divide_divide_rat(Xb,Y),divide_divide_rat(Z,W)) = divide_divide_rat(aa_rat_rat(times_times_rat(Xb),W),aa_rat_rat(times_times_rat(Y),Z)) ).

% divide_divide_times_eq
tff(fact_2658_divide__divide__eq__left_H,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B)),C) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B)) ).

% divide_divide_eq_left'
tff(fact_2659_divide__divide__eq__left_H,axiom,
    ! [A: rat,B: rat,C: rat] : divide_divide_rat(divide_divide_rat(A,B),C) = divide_divide_rat(A,aa_rat_rat(times_times_rat(C),B)) ).

% divide_divide_eq_left'
tff(fact_2660_diff__divide__distrib,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)),C) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)) ).

% diff_divide_distrib
tff(fact_2661_diff__divide__distrib,axiom,
    ! [A: rat,B: rat,C: rat] : divide_divide_rat(minus_minus_rat(A,B),C) = minus_minus_rat(divide_divide_rat(A,C),divide_divide_rat(B,C)) ).

% diff_divide_distrib
tff(fact_2662_VEBT__internal_OTb_Osimps_I2_J,axiom,
    aa_nat_int(vEBT_VEBT_Tb,aa_nat_nat(suc,zero_zero_nat)) = aa_num_int(numeral_numeral_int,aa_num_num(bit1,one)) ).

% VEBT_internal.Tb.simps(2)
tff(fact_2663_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_2664_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_2665_divide__le__0__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B)),zero_zero_real)
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),zero_zero_real) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B) ) ) ) ).

% divide_le_0_iff
tff(fact_2666_divide__le__0__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_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_2667_divide__right__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)) ) ) ).

% divide_right_mono
tff(fact_2668_divide__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,divide_divide_rat(A,C)),divide_divide_rat(B,C)) ) ) ).

% divide_right_mono
tff(fact_2669_zero__le__divide__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),zero_zero_real) ) ) ) ).

% zero_le_divide_iff
tff(fact_2670_zero__le__divide__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_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_2671_divide__nonneg__nonneg,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Y)) ) ) ).

% divide_nonneg_nonneg
tff(fact_2672_divide__nonneg__nonneg,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Xb)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Y)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),divide_divide_rat(Xb,Y)) ) ) ).

% divide_nonneg_nonneg
tff(fact_2673_divide__nonneg__nonpos,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Y)),zero_zero_real) ) ) ).

% divide_nonneg_nonpos
tff(fact_2674_divide__nonneg__nonpos,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Xb)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,divide_divide_rat(Xb,Y)),zero_zero_rat) ) ) ).

% divide_nonneg_nonpos
tff(fact_2675_divide__nonpos__nonneg,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Y)),zero_zero_real) ) ) ).

% divide_nonpos_nonneg
tff(fact_2676_divide__nonpos__nonneg,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Y)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,divide_divide_rat(Xb,Y)),zero_zero_rat) ) ) ).

% divide_nonpos_nonneg
tff(fact_2677_divide__nonpos__nonpos,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Y)) ) ) ).

% divide_nonpos_nonpos
tff(fact_2678_divide__nonpos__nonpos,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),divide_divide_rat(Xb,Y)) ) ) ).

% divide_nonpos_nonpos
tff(fact_2679_divide__right__mono__neg,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),C)) ) ) ).

% divide_right_mono_neg
tff(fact_2680_divide__right__mono__neg,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,divide_divide_rat(B,C)),divide_divide_rat(A,C)) ) ) ).

% divide_right_mono_neg
tff(fact_2681_divide__neg__neg,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Y)) ) ) ).

% divide_neg_neg
tff(fact_2682_divide__neg__neg,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),divide_divide_rat(Xb,Y)) ) ) ).

% divide_neg_neg
tff(fact_2683_divide__neg__pos,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Y)),zero_zero_real) ) ) ).

% divide_neg_pos
tff(fact_2684_divide__neg__pos,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Y)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,divide_divide_rat(Xb,Y)),zero_zero_rat) ) ) ).

% divide_neg_pos
tff(fact_2685_divide__pos__neg,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Y)),zero_zero_real) ) ) ).

% divide_pos_neg
tff(fact_2686_divide__pos__neg,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Xb)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,divide_divide_rat(Xb,Y)),zero_zero_rat) ) ) ).

% divide_pos_neg
tff(fact_2687_divide__pos__pos,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Y)) ) ) ).

% divide_pos_pos
tff(fact_2688_divide__pos__pos,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Xb)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Y)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),divide_divide_rat(Xb,Y)) ) ) ).

% divide_pos_pos
tff(fact_2689_divide__less__0__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B)),zero_zero_real)
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,B),zero_zero_real) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B) ) ) ) ).

% divide_less_0_iff
tff(fact_2690_divide__less__0__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_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_2691_divide__less__cancel,axiom,
    ! [A: real,C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,B),A) )
        & ( C != zero_zero_real ) ) ) ).

% divide_less_cancel
tff(fact_2692_divide__less__cancel,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,divide_divide_rat(A,C)),divide_divide_rat(B,C))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A) )
        & ( C != zero_zero_rat ) ) ) ).

% divide_less_cancel
tff(fact_2693_zero__less__divide__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,B),zero_zero_real) ) ) ) ).

% zero_less_divide_iff
tff(fact_2694_zero__less__divide__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_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_2695_divide__strict__right__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)) ) ) ).

% divide_strict_right_mono
tff(fact_2696_divide__strict__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,divide_divide_rat(A,C)),divide_divide_rat(B,C)) ) ) ).

% divide_strict_right_mono
tff(fact_2697_divide__strict__right__mono__neg,axiom,
    ! [B: real,A: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)) ) ) ).

% divide_strict_right_mono_neg
tff(fact_2698_divide__strict__right__mono__neg,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,divide_divide_rat(A,C)),divide_divide_rat(B,C)) ) ) ).

% divide_strict_right_mono_neg
tff(fact_2699_right__inverse__eq,axiom,
    ! [B: real,A: real] :
      ( ( B != zero_zero_real )
     => ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B) = one_one_real )
      <=> ( A = B ) ) ) ).

% right_inverse_eq
tff(fact_2700_right__inverse__eq,axiom,
    ! [B: rat,A: rat] :
      ( ( B != zero_zero_rat )
     => ( ( divide_divide_rat(A,B) = one_one_rat )
      <=> ( A = B ) ) ) ).

% right_inverse_eq
tff(fact_2701_frac__eq__eq,axiom,
    ! [Y: real,Z: real,Xb: real,W: real] :
      ( ( Y != zero_zero_real )
     => ( ( Z != zero_zero_real )
       => ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Y) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,W),Z) )
        <=> ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Z) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,W),Y) ) ) ) ) ).

% frac_eq_eq
tff(fact_2702_frac__eq__eq,axiom,
    ! [Y: rat,Z: rat,Xb: rat,W: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( Z != zero_zero_rat )
       => ( ( divide_divide_rat(Xb,Y) = divide_divide_rat(W,Z) )
        <=> ( aa_rat_rat(times_times_rat(Xb),Z) = aa_rat_rat(times_times_rat(W),Y) ) ) ) ) ).

% frac_eq_eq
tff(fact_2703_divide__eq__eq,axiom,
    ! [B: real,C: real,A: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C) = A )
    <=> $ite(C != zero_zero_real,B = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C),A = zero_zero_real) ) ).

% divide_eq_eq
tff(fact_2704_divide__eq__eq,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( ( divide_divide_rat(B,C) = A )
    <=> $ite(C != zero_zero_rat,B = aa_rat_rat(times_times_rat(A),C),A = zero_zero_rat) ) ).

% divide_eq_eq
tff(fact_2705_eq__divide__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( ( A = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C) )
    <=> $ite(C != zero_zero_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C) = B,A = zero_zero_real) ) ).

% eq_divide_eq
tff(fact_2706_eq__divide__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( A = divide_divide_rat(B,C) )
    <=> $ite(C != zero_zero_rat,aa_rat_rat(times_times_rat(A),C) = B,A = zero_zero_rat) ) ).

% eq_divide_eq
tff(fact_2707_divide__eq__imp,axiom,
    ! [C: real,B: real,A: real] :
      ( ( C != zero_zero_real )
     => ( ( B = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C) )
       => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C) = A ) ) ) ).

% divide_eq_imp
tff(fact_2708_divide__eq__imp,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( ( C != zero_zero_rat )
     => ( ( B = aa_rat_rat(times_times_rat(A),C) )
       => ( divide_divide_rat(B,C) = A ) ) ) ).

% divide_eq_imp
tff(fact_2709_eq__divide__imp,axiom,
    ! [C: real,A: real,B: real] :
      ( ( C != zero_zero_real )
     => ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C) = B )
       => ( A = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C) ) ) ) ).

% eq_divide_imp
tff(fact_2710_eq__divide__imp,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( ( aa_rat_rat(times_times_rat(A),C) = B )
       => ( A = divide_divide_rat(B,C) ) ) ) ).

% eq_divide_imp
tff(fact_2711_nonzero__divide__eq__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( ( C != zero_zero_real )
     => ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C) = A )
      <=> ( B = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C) ) ) ) ).

% nonzero_divide_eq_eq
tff(fact_2712_nonzero__divide__eq__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( ( C != zero_zero_rat )
     => ( ( divide_divide_rat(B,C) = A )
      <=> ( B = aa_rat_rat(times_times_rat(A),C) ) ) ) ).

% nonzero_divide_eq_eq
tff(fact_2713_nonzero__eq__divide__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( ( C != zero_zero_real )
     => ( ( A = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C) )
      <=> ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C) = B ) ) ) ).

% nonzero_eq_divide_eq
tff(fact_2714_nonzero__eq__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( ( A = divide_divide_rat(B,C) )
      <=> ( aa_rat_rat(times_times_rat(A),C) = B ) ) ) ).

% nonzero_eq_divide_eq
tff(fact_2715_field__le__epsilon,axiom,
    ! [Xb: real,Y: real] :
      ( ! [E2: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),E2)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Y),E2)) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y) ) ).

% field_le_epsilon
tff(fact_2716_field__le__epsilon,axiom,
    ! [Xb: 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,Xb),aa_rat_rat(plus_plus_rat(Y),E2)) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Y) ) ).

% field_le_epsilon
tff(fact_2717_frac__le,axiom,
    ! [Y: real,Xb: real,W: real,Z: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),W)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,W),Z)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Z)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Y),W)) ) ) ) ) ).

% frac_le
tff(fact_2718_frac__le,axiom,
    ! [Y: rat,Xb: rat,W: 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,Xb),Y)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),W)
         => ( 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,divide_divide_rat(Xb,Z)),divide_divide_rat(Y,W)) ) ) ) ) ).

% frac_le
tff(fact_2719_frac__less,axiom,
    ! [Xb: real,Y: real,W: real,Z: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),W)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,W),Z)
           => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Z)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Y),W)) ) ) ) ) ).

% frac_less
tff(fact_2720_frac__less,axiom,
    ! [Xb: rat,Y: rat,W: rat,Z: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Xb)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Y)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),W)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,W),Z)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,divide_divide_rat(Xb,Z)),divide_divide_rat(Y,W)) ) ) ) ) ).

% frac_less
tff(fact_2721_frac__less2,axiom,
    ! [Xb: real,Y: real,W: real,Z: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),W)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,W),Z)
           => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Z)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Y),W)) ) ) ) ) ).

% frac_less2
tff(fact_2722_frac__less2,axiom,
    ! [Xb: rat,Y: rat,W: rat,Z: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Xb)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Y)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),W)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,W),Z)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,divide_divide_rat(Xb,Z)),divide_divide_rat(Y,W)) ) ) ) ) ).

% frac_less2
tff(fact_2723_divide__le__cancel,axiom,
    ! [A: real,C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A) ) ) ) ).

% divide_le_cancel
tff(fact_2724_divide__le__cancel,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,divide_divide_rat(A,C)),divide_divide_rat(B,C))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A) ) ) ) ).

% divide_le_cancel
tff(fact_2725_divide__nonneg__neg,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Y)),zero_zero_real) ) ) ).

% divide_nonneg_neg
tff(fact_2726_divide__nonneg__neg,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Xb)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,divide_divide_rat(Xb,Y)),zero_zero_rat) ) ) ).

% divide_nonneg_neg
tff(fact_2727_divide__nonneg__pos,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Y)) ) ) ).

% divide_nonneg_pos
tff(fact_2728_divide__nonneg__pos,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Xb)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Y)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),divide_divide_rat(Xb,Y)) ) ) ).

% divide_nonneg_pos
tff(fact_2729_divide__nonpos__neg,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Y)) ) ) ).

% divide_nonpos_neg
tff(fact_2730_divide__nonpos__neg,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),divide_divide_rat(Xb,Y)) ) ) ).

% divide_nonpos_neg
tff(fact_2731_divide__nonpos__pos,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Y)),zero_zero_real) ) ) ).

% divide_nonpos_pos
tff(fact_2732_divide__nonpos__pos,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Y)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,divide_divide_rat(Xb,Y)),zero_zero_rat) ) ) ).

% divide_nonpos_pos
tff(fact_2733_divide__less__eq__1,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),A)),one_one_real)
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,B),A) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) )
        | ( A = zero_zero_real ) ) ) ).

% divide_less_eq_1
tff(fact_2734_divide__less__eq__1,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_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_2735_less__divide__eq__1,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),A))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,B),A) ) ) ) ).

% less_divide_eq_1
tff(fact_2736_less__divide__eq__1,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_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_2737_divide__less__eq,axiom,
    ! [B: real,C: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)),A)
    <=> $ite(
          aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C),
          aa_real_o(aa_real_fun_real_o(ord_less_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),
          $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real),aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),B),aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)) ) ) ).

% divide_less_eq
tff(fact_2738_divide__less__eq,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,divide_divide_rat(B,C)),A)
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),aa_rat_rat(times_times_rat(A),C)),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),B),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)) ) ) ).

% divide_less_eq
tff(fact_2739_less__divide__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C))
    <=> $ite(
          aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C),
          aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),B),
          $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real),aa_real_o(aa_real_fun_real_o(ord_less_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)) ) ) ).

% less_divide_eq
tff(fact_2740_less__divide__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),divide_divide_rat(B,C))
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),B),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),aa_rat_rat(times_times_rat(A),C)),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)) ) ) ).

% less_divide_eq
tff(fact_2741_neg__divide__less__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)),A)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),B) ) ) ).

% neg_divide_less_eq
tff(fact_2742_neg__divide__less__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,divide_divide_rat(B,C)),A)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),B) ) ) ).

% neg_divide_less_eq
tff(fact_2743_neg__less__divide__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)) ) ) ).

% neg_less_divide_eq
tff(fact_2744_neg__less__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),divide_divide_rat(B,C))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),aa_rat_rat(times_times_rat(A),C)) ) ) ).

% neg_less_divide_eq
tff(fact_2745_pos__divide__less__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)),A)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)) ) ) ).

% pos_divide_less_eq
tff(fact_2746_pos__divide__less__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,divide_divide_rat(B,C)),A)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),aa_rat_rat(times_times_rat(A),C)) ) ) ).

% pos_divide_less_eq
tff(fact_2747_pos__less__divide__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),B) ) ) ).

% pos_less_divide_eq
tff(fact_2748_pos__less__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),divide_divide_rat(B,C))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),B) ) ) ).

% pos_less_divide_eq
tff(fact_2749_mult__imp__div__pos__less,axiom,
    ! [Y: real,Xb: real,Z: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z),Y))
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Y)),Z) ) ) ).

% mult_imp_div_pos_less
tff(fact_2750_mult__imp__div__pos__less,axiom,
    ! [Y: rat,Xb: 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,Xb),aa_rat_rat(times_times_rat(Z),Y))
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,divide_divide_rat(Xb,Y)),Z) ) ) ).

% mult_imp_div_pos_less
tff(fact_2751_mult__imp__less__div__pos,axiom,
    ! [Y: real,Z: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z),Y)),Xb)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,Z),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Y)) ) ) ).

% mult_imp_less_div_pos
tff(fact_2752_mult__imp__less__div__pos,axiom,
    ! [Y: rat,Z: rat,Xb: 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(times_times_rat(Z),Y)),Xb)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z),divide_divide_rat(Xb,Y)) ) ) ).

% mult_imp_less_div_pos
tff(fact_2753_divide__strict__left__mono,axiom,
    ! [B: real,A: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B))
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,C),B)) ) ) ) ).

% divide_strict_left_mono
tff(fact_2754_divide__strict__left__mono,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A),B))
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,divide_divide_rat(C,A)),divide_divide_rat(C,B)) ) ) ) ).

% divide_strict_left_mono
tff(fact_2755_divide__strict__left__mono__neg,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B))
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,C),B)) ) ) ) ).

% divide_strict_left_mono_neg
tff(fact_2756_divide__strict__left__mono__neg,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A),B))
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,divide_divide_rat(C,A)),divide_divide_rat(C,B)) ) ) ) ).

% divide_strict_left_mono_neg
tff(fact_2757_add__divide__eq__if__simps_I2_J,axiom,
    ! [A: real,Z: real,B: real] :
      aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),Z)),B) = $ite(Z = zero_zero_real,B,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),Z))),Z)) ).

% add_divide_eq_if_simps(2)
tff(fact_2758_add__divide__eq__if__simps_I2_J,axiom,
    ! [A: rat,Z: rat,B: rat] :
      aa_rat_rat(plus_plus_rat(divide_divide_rat(A,Z)),B) = $ite(Z = zero_zero_rat,B,divide_divide_rat(aa_rat_rat(plus_plus_rat(A),aa_rat_rat(times_times_rat(B),Z)),Z)) ).

% add_divide_eq_if_simps(2)
tff(fact_2759_add__divide__eq__if__simps_I1_J,axiom,
    ! [A: real,B: real,Z: real] :
      aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),Z)) = $ite(Z = zero_zero_real,A,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),Z)),B)),Z)) ).

% add_divide_eq_if_simps(1)
tff(fact_2760_add__divide__eq__if__simps_I1_J,axiom,
    ! [A: rat,B: rat,Z: rat] :
      aa_rat_rat(plus_plus_rat(A),divide_divide_rat(B,Z)) = $ite(Z = zero_zero_rat,A,divide_divide_rat(aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(A),Z)),B),Z)) ).

% add_divide_eq_if_simps(1)
tff(fact_2761_add__frac__eq,axiom,
    ! [Y: real,Z: real,Xb: real,W: real] :
      ( ( Y != zero_zero_real )
     => ( ( Z != zero_zero_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Y)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,W),Z)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Z)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,W),Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Z)) ) ) ) ).

% add_frac_eq
tff(fact_2762_add__frac__eq,axiom,
    ! [Y: rat,Z: rat,Xb: rat,W: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( Z != zero_zero_rat )
       => ( aa_rat_rat(plus_plus_rat(divide_divide_rat(Xb,Y)),divide_divide_rat(W,Z)) = divide_divide_rat(aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(Xb),Z)),aa_rat_rat(times_times_rat(W),Y)),aa_rat_rat(times_times_rat(Y),Z)) ) ) ) ).

% add_frac_eq
tff(fact_2763_add__frac__num,axiom,
    ! [Y: real,Xb: real,Z: real] :
      ( ( Y != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Y)),Z) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z),Y))),Y) ) ) ).

% add_frac_num
tff(fact_2764_add__frac__num,axiom,
    ! [Y: rat,Xb: rat,Z: rat] :
      ( ( Y != zero_zero_rat )
     => ( aa_rat_rat(plus_plus_rat(divide_divide_rat(Xb,Y)),Z) = divide_divide_rat(aa_rat_rat(plus_plus_rat(Xb),aa_rat_rat(times_times_rat(Z),Y)),Y) ) ) ).

% add_frac_num
tff(fact_2765_add__num__frac,axiom,
    ! [Y: real,Z: real,Xb: real] :
      ( ( Y != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Z),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Y)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z),Y))),Y) ) ) ).

% add_num_frac
tff(fact_2766_add__num__frac,axiom,
    ! [Y: rat,Z: rat,Xb: rat] :
      ( ( Y != zero_zero_rat )
     => ( aa_rat_rat(plus_plus_rat(Z),divide_divide_rat(Xb,Y)) = divide_divide_rat(aa_rat_rat(plus_plus_rat(Xb),aa_rat_rat(times_times_rat(Z),Y)),Y) ) ) ).

% add_num_frac
tff(fact_2767_add__divide__eq__iff,axiom,
    ! [Z: real,Xb: real,Y: real] :
      ( ( Z != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Y),Z)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Z)),Y)),Z) ) ) ).

% add_divide_eq_iff
tff(fact_2768_add__divide__eq__iff,axiom,
    ! [Z: rat,Xb: rat,Y: rat] :
      ( ( Z != zero_zero_rat )
     => ( aa_rat_rat(plus_plus_rat(Xb),divide_divide_rat(Y,Z)) = divide_divide_rat(aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(Xb),Z)),Y),Z) ) ) ).

% add_divide_eq_iff
tff(fact_2769_divide__add__eq__iff,axiom,
    ! [Z: real,Xb: real,Y: real] :
      ( ( Z != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Z)),Y) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Z))),Z) ) ) ).

% divide_add_eq_iff
tff(fact_2770_divide__add__eq__iff,axiom,
    ! [Z: rat,Xb: rat,Y: rat] :
      ( ( Z != zero_zero_rat )
     => ( aa_rat_rat(plus_plus_rat(divide_divide_rat(Xb,Z)),Y) = divide_divide_rat(aa_rat_rat(plus_plus_rat(Xb),aa_rat_rat(times_times_rat(Y),Z)),Z) ) ) ).

% divide_add_eq_iff
tff(fact_2771_gt__half__sum,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),one_one_real))),B) ) ).

% gt_half_sum
tff(fact_2772_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,divide_divide_rat(aa_rat_rat(plus_plus_rat(A),B),aa_rat_rat(plus_plus_rat(one_one_rat),one_one_rat))),B) ) ).

% gt_half_sum
tff(fact_2773_less__half__sum,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),one_one_real))) ) ).

% less_half_sum
tff(fact_2774_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),divide_divide_rat(aa_rat_rat(plus_plus_rat(A),B),aa_rat_rat(plus_plus_rat(one_one_rat),one_one_rat))) ) ).

% less_half_sum
tff(fact_2775_add__divide__eq__if__simps_I4_J,axiom,
    ! [A: real,B: real,Z: real] :
      aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),Z)) = $ite(Z = zero_zero_real,A,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),Z)),B)),Z)) ).

% add_divide_eq_if_simps(4)
tff(fact_2776_add__divide__eq__if__simps_I4_J,axiom,
    ! [A: rat,B: rat,Z: rat] :
      minus_minus_rat(A,divide_divide_rat(B,Z)) = $ite(Z = zero_zero_rat,A,divide_divide_rat(minus_minus_rat(aa_rat_rat(times_times_rat(A),Z),B),Z)) ).

% add_divide_eq_if_simps(4)
tff(fact_2777_diff__frac__eq,axiom,
    ! [Y: real,Z: real,Xb: real,W: real] :
      ( ( Y != zero_zero_real )
     => ( ( Z != zero_zero_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Y)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,W),Z)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Z)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,W),Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Z)) ) ) ) ).

% diff_frac_eq
tff(fact_2778_diff__frac__eq,axiom,
    ! [Y: rat,Z: rat,Xb: rat,W: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( Z != zero_zero_rat )
       => ( minus_minus_rat(divide_divide_rat(Xb,Y),divide_divide_rat(W,Z)) = divide_divide_rat(minus_minus_rat(aa_rat_rat(times_times_rat(Xb),Z),aa_rat_rat(times_times_rat(W),Y)),aa_rat_rat(times_times_rat(Y),Z)) ) ) ) ).

% diff_frac_eq
tff(fact_2779_diff__divide__eq__iff,axiom,
    ! [Z: real,Xb: real,Y: real] :
      ( ( Z != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Y),Z)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Z)),Y)),Z) ) ) ).

% diff_divide_eq_iff
tff(fact_2780_diff__divide__eq__iff,axiom,
    ! [Z: rat,Xb: rat,Y: rat] :
      ( ( Z != zero_zero_rat )
     => ( minus_minus_rat(Xb,divide_divide_rat(Y,Z)) = divide_divide_rat(minus_minus_rat(aa_rat_rat(times_times_rat(Xb),Z),Y),Z) ) ) ).

% diff_divide_eq_iff
tff(fact_2781_divide__diff__eq__iff,axiom,
    ! [Z: real,Xb: real,Y: real] :
      ( ( Z != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Z)),Y) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Z))),Z) ) ) ).

% divide_diff_eq_iff
tff(fact_2782_divide__diff__eq__iff,axiom,
    ! [Z: rat,Xb: rat,Y: rat] :
      ( ( Z != zero_zero_rat )
     => ( minus_minus_rat(divide_divide_rat(Xb,Z),Y) = divide_divide_rat(minus_minus_rat(Xb,aa_rat_rat(times_times_rat(Y),Z)),Z) ) ) ).

% divide_diff_eq_iff
tff(fact_2783_field__le__mult__one__interval,axiom,
    ! [Xb: real,Y: real] :
      ( ! [Z3: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Z3)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z3),one_one_real)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z3),Xb)),Y) ) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y) ) ).

% field_le_mult_one_interval
tff(fact_2784_field__le__mult__one__interval,axiom,
    ! [Xb: rat,Y: rat] :
      ( ! [Z3: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Z3)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z3),one_one_rat)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(Z3),Xb)),Y) ) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Y) ) ).

% field_le_mult_one_interval
tff(fact_2785_divide__le__eq__1,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),A)),one_one_real)
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) )
        | ( A = zero_zero_real ) ) ) ).

% divide_le_eq_1
tff(fact_2786_divide__le__eq__1,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_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_2787_le__divide__eq__1,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),A))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A) ) ) ) ).

% le_divide_eq_1
tff(fact_2788_le__divide__eq__1,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_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_2789_divide__le__eq,axiom,
    ! [B: real,C: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)),A)
    <=> $ite(
          aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C),
          aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),
          $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),B),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)) ) ) ).

% divide_le_eq
tff(fact_2790_divide__le__eq,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,divide_divide_rat(B,C)),A)
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),aa_rat_rat(times_times_rat(A),C)),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C)),B),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)) ) ) ).

% divide_le_eq
tff(fact_2791_le__divide__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C))
    <=> $ite(
          aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C),
          aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),B),
          $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)) ) ) ).

% le_divide_eq
tff(fact_2792_le__divide__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),divide_divide_rat(B,C))
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C)),B),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),aa_rat_rat(times_times_rat(A),C)),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)) ) ) ).

% le_divide_eq
tff(fact_2793_divide__left__mono,axiom,
    ! [B: real,A: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B))
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,C),B)) ) ) ) ).

% divide_left_mono
tff(fact_2794_divide__left__mono,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A),B))
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,divide_divide_rat(C,A)),divide_divide_rat(C,B)) ) ) ) ).

% divide_left_mono
tff(fact_2795_neg__divide__le__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)),A)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),B) ) ) ).

% neg_divide_le_eq
tff(fact_2796_neg__divide__le__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,divide_divide_rat(B,C)),A)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C)),B) ) ) ).

% neg_divide_le_eq
tff(fact_2797_neg__le__divide__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)) ) ) ).

% neg_le_divide_eq
tff(fact_2798_neg__le__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),divide_divide_rat(B,C))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),aa_rat_rat(times_times_rat(A),C)) ) ) ).

% neg_le_divide_eq
tff(fact_2799_pos__divide__le__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)),A)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)) ) ) ).

% pos_divide_le_eq
tff(fact_2800_pos__divide__le__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,divide_divide_rat(B,C)),A)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),aa_rat_rat(times_times_rat(A),C)) ) ) ).

% pos_divide_le_eq
tff(fact_2801_pos__le__divide__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),B) ) ) ).

% pos_le_divide_eq
tff(fact_2802_pos__le__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),divide_divide_rat(B,C))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C)),B) ) ) ).

% pos_le_divide_eq
tff(fact_2803_mult__imp__div__pos__le,axiom,
    ! [Y: real,Xb: real,Z: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z),Y))
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Y)),Z) ) ) ).

% mult_imp_div_pos_le
tff(fact_2804_mult__imp__div__pos__le,axiom,
    ! [Y: rat,Xb: 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,Xb),aa_rat_rat(times_times_rat(Z),Y))
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,divide_divide_rat(Xb,Y)),Z) ) ) ).

% mult_imp_div_pos_le
tff(fact_2805_mult__imp__le__div__pos,axiom,
    ! [Y: real,Z: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z),Y)),Xb)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Z),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Y)) ) ) ).

% mult_imp_le_div_pos
tff(fact_2806_mult__imp__le__div__pos,axiom,
    ! [Y: rat,Z: rat,Xb: 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(times_times_rat(Z),Y)),Xb)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Z),divide_divide_rat(Xb,Y)) ) ) ).

% mult_imp_le_div_pos
tff(fact_2807_divide__left__mono__neg,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),zero_zero_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B))
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,C),B)) ) ) ) ).

% divide_left_mono_neg
tff(fact_2808_divide__left__mono__neg,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),zero_zero_rat)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A),B))
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,divide_divide_rat(C,A)),divide_divide_rat(C,B)) ) ) ) ).

% divide_left_mono_neg
tff(fact_2809_frac__le__eq,axiom,
    ! [Y: real,Z: real,Xb: real,W: real] :
      ( ( Y != zero_zero_real )
     => ( ( Z != zero_zero_real )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Y)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,W),Z))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Z)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,W),Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Z))),zero_zero_real) ) ) ) ).

% frac_le_eq
tff(fact_2810_frac__le__eq,axiom,
    ! [Y: rat,Z: rat,Xb: rat,W: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( Z != zero_zero_rat )
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,divide_divide_rat(Xb,Y)),divide_divide_rat(W,Z))
        <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,divide_divide_rat(minus_minus_rat(aa_rat_rat(times_times_rat(Xb),Z),aa_rat_rat(times_times_rat(W),Y)),aa_rat_rat(times_times_rat(Y),Z))),zero_zero_rat) ) ) ) ).

% frac_le_eq
tff(fact_2811_frac__less__eq,axiom,
    ! [Y: real,Z: real,Xb: real,W: real] :
      ( ( Y != zero_zero_real )
     => ( ( Z != zero_zero_real )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Y)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,W),Z))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Z)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,W),Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Z))),zero_zero_real) ) ) ) ).

% frac_less_eq
tff(fact_2812_frac__less__eq,axiom,
    ! [Y: rat,Z: rat,Xb: rat,W: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( Z != zero_zero_rat )
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,divide_divide_rat(Xb,Y)),divide_divide_rat(W,Z))
        <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,divide_divide_rat(minus_minus_rat(aa_rat_rat(times_times_rat(Xb),Z),aa_rat_rat(times_times_rat(W),Y)),aa_rat_rat(times_times_rat(Y),Z))),zero_zero_rat) ) ) ) ).

% frac_less_eq
tff(fact_2813_scaling__mono,axiom,
    ! [U: real,V: real,R2: real,S3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,U),V)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),R2)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,R2),S3)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,U),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R2),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,V),U))),S3))),V) ) ) ) ).

% scaling_mono
tff(fact_2814_scaling__mono,axiom,
    ! [U: rat,V: rat,R2: rat,S3: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,U),V)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),R2)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,R2),S3)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(U),divide_divide_rat(aa_rat_rat(times_times_rat(R2),minus_minus_rat(V,U)),S3))),V) ) ) ) ).

% scaling_mono
tff(fact_2815_Tb__T__vebt__buildupi_H,axiom,
    ! [Nb: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(vEBT_V9176841429113362141ildupi,Nb)),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(vEBT_VEBT_Tb,Nb)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))) ).

% Tb_T_vebt_buildupi'
tff(fact_2816_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H_Osimps_I5_J,axiom,
    ! [Mia: nat,Maa: nat,Vaa: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Xb: nat] :
      aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_m_e_m_b_e_r2,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mia),Maa)),aa_nat_nat(suc,aa_nat_nat(suc,Vaa)),TreeLista),Summarya)),Xb) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
        $ite(
          Xb = Mia,
          zero_zero_nat,
          $ite(
            Xb = Maa,
            zero_zero_nat,
            $ite(
              aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Mia),
              zero_zero_nat,
              $ite(
                aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Maa),Xb),
                zero_zero_nat,
                $ite(
                  ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mia),Xb)
                  & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Maa) ),
                  $let(
                    h: nat,
                    h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Vaa))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                    $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeLista)),aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_m_e_m_b_e_r2,nth_VEBT_VEBT(TreeLista,h)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Vaa))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),zero_zero_nat) ),
                  zero_zero_nat ) ) ) ) )) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.simps(5)
tff(fact_2817_zle__diff1__eq,axiom,
    ! [W: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,W),aa_int_int(aa_int_fun_int_int(minus_minus_int,Z),one_one_int))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,W),Z) ) ).

% zle_diff1_eq
tff(fact_2818_zle__add1__eq__le,axiom,
    ! [W: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,W),aa_int_int(aa_int_fun_int_int(plus_plus_int,Z),one_one_int))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,W),Z) ) ).

% zle_add1_eq_le
tff(fact_2819_int__eq__iff__numeral,axiom,
    ! [M: nat,V: num] :
      ( ( aa_nat_int(semiri1314217659103216013at_int,M) = aa_num_int(numeral_numeral_int,V) )
    <=> ( M = aa_num_nat(numeral_numeral_nat,V) ) ) ).

% int_eq_iff_numeral
tff(fact_2820_vebt__succ_Oelims,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: option_nat] :
      ( ( aa_nat_option_nat(vEBT_vebt_succ(Xb),Xaa) = Y )
     => ( ! [Uu2: $o,B2: $o] :
            ( ( Xb = vEBT_Leaf((Uu2),(B2)) )
           => ( ( Xaa = zero_zero_nat )
             => ( Y != $ite((B2),aa_nat_option_nat(some_nat,one_one_nat),none_nat) ) ) )
       => ( ( ? [Uv2: $o,Uw2: $o] : Xb = vEBT_Leaf((Uv2),(Uw2))
           => ( ? [N2: nat] : Xaa = aa_nat_nat(suc,N2)
             => ( Y != none_nat ) ) )
         => ( ( ? [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] : Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Ux2,Uy2),Uz2)
             => ( Y != none_nat ) )
           => ( ( ? [V2: product_prod_nat_nat,Vc: list_VEBT_VEBT,Vd2: vEBT_VEBT] : Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Vc),Vd2)
               => ( Y != none_nat ) )
             => ( ( ? [V2: product_prod_nat_nat,Vg2: list_VEBT_VEBT,Vh2: vEBT_VEBT] : Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vg2),Vh2)
                 => ( Y != none_nat ) )
               => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                      ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList),Summary) )
                     => ( Y != $ite(
                            aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi),
                            aa_nat_option_nat(some_nat,Mi),
                            $let(
                              l: nat,
                              l:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                              $let(
                                h: nat,
                                h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                $ite(
                                  aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList)),
                                  $let(
                                    maxlow: option_nat,
                                    maxlow:= vEBT_vebt_maxt(nth_VEBT_VEBT(TreeList,h)),
                                    $ite(
                                      ( ( maxlow != none_nat )
                                      & aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_less,aa_nat_option_nat(some_nat,l)),maxlow) ),
                                      aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,aa_nat_option_nat(some_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_option_nat(some_nat,h))),aa_nat_option_nat(vEBT_vebt_succ(nth_VEBT_VEBT(TreeList,h)),l)),
                                      $let(
                                        sc: option_nat,
                                        sc:= aa_nat_option_nat(vEBT_vebt_succ(Summary),h),
                                        $ite(sc = none_nat,none_nat,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,aa_nat_option_nat(some_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),sc)),vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat(sc))))) ) ) ),
                                  none_nat ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_succ.elims
tff(fact_2821_vebt__pred_Oelims,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: option_nat] :
      ( ( aa_nat_option_nat(vEBT_vebt_pred(Xb),Xaa) = Y )
     => ( ( ? [Uu2: $o,Uv2: $o] : Xb = vEBT_Leaf((Uu2),(Uv2))
         => ( ( Xaa = zero_zero_nat )
           => ( Y != none_nat ) ) )
       => ( ! [A3: $o] :
              ( ? [Uw2: $o] : Xb = vEBT_Leaf((A3),(Uw2))
             => ( ( Xaa = aa_nat_nat(suc,zero_zero_nat) )
               => ( Y != $ite((A3),aa_nat_option_nat(some_nat,zero_zero_nat),none_nat) ) ) )
         => ( ! [A3: $o,B2: $o] :
                ( ( Xb = vEBT_Leaf((A3),(B2)) )
               => ( ? [Va2: nat] : Xaa = aa_nat_nat(suc,aa_nat_nat(suc,Va2))
                 => ( Y != $ite(
                        (B2),
                        aa_nat_option_nat(some_nat,one_one_nat),
                        $ite((A3),aa_nat_option_nat(some_nat,zero_zero_nat),none_nat) ) ) ) )
           => ( ( ? [Uy2: nat,Uz2: list_VEBT_VEBT,Va: vEBT_VEBT] : Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uy2,Uz2),Va)
               => ( Y != none_nat ) )
             => ( ( ? [V2: product_prod_nat_nat,Vd2: list_VEBT_VEBT,Ve2: vEBT_VEBT] : Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Vd2),Ve2)
                 => ( Y != none_nat ) )
               => ( ( ? [V2: product_prod_nat_nat,Vh2: list_VEBT_VEBT,Vi2: vEBT_VEBT] : Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vh2),Vi2)
                   => ( Y != none_nat ) )
                 => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                        ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList),Summary) )
                       => ( Y != $ite(
                              aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Xaa),
                              aa_nat_option_nat(some_nat,Ma),
                              $let(
                                l: nat,
                                l:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                $let(
                                  h: nat,
                                  h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                  $ite(
                                    aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList)),
                                    $let(
                                      minlow: option_nat,
                                      minlow:= vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,h)),
                                      $ite(
                                        ( ( minlow != none_nat )
                                        & aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_greater,aa_nat_option_nat(some_nat,l)),minlow) ),
                                        aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,aa_nat_option_nat(some_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_option_nat(some_nat,h))),aa_nat_option_nat(vEBT_vebt_pred(nth_VEBT_VEBT(TreeList,h)),l)),
                                        $let(
                                          pr: option_nat,
                                          pr:= aa_nat_option_nat(vEBT_vebt_pred(Summary),h),
                                          $ite(
                                            pr = none_nat,
                                            $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mi),Xaa),aa_nat_option_nat(some_nat,Mi),none_nat),
                                            aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,aa_nat_option_nat(some_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),pr)),vEBT_vebt_maxt(nth_VEBT_VEBT(TreeList,the_nat(pr)))) ) ) ) ),
                                    none_nat ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_pred.elims
tff(fact_2822_T_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_Osimps_I6_J,axiom,
    ! [Mia: nat,Maa: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Xb: nat] : aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_d_e_l_e_t_e,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mia),Maa)),aa_nat_nat(suc,zero_zero_nat),TreeLista),Summarya)),Xb) = one_one_nat ).

% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.simps(6)
tff(fact_2823_VEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H_Osimps_I6_J,axiom,
    ! [Mia: nat,Maa: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Xb: nat] : aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_V1232361888498592333_e_t_e,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mia),Maa)),aa_nat_nat(suc,zero_zero_nat),TreeLista),Summarya)),Xb) = one_one_nat ).

% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.simps(6)
tff(fact_2824_Leaf__0__not,axiom,
    ! [A: $o,B: $o] : ~ aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,vEBT_Leaf((A),(B))),zero_zero_nat) ).

% Leaf_0_not
tff(fact_2825_deg__1__Leafy,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Ta),Nb)
     => ( ( Nb = one_one_nat )
       => ? [A3: $o,B2: $o] : Ta = vEBT_Leaf((A3),(B2)) ) ) ).

% deg_1_Leafy
tff(fact_2826_deg__1__Leaf,axiom,
    ! [Ta: vEBT_VEBT] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Ta),one_one_nat)
     => ? [A3: $o,B2: $o] : Ta = vEBT_Leaf((A3),(B2)) ) ).

% deg_1_Leaf
tff(fact_2827_deg1Leaf,axiom,
    ! [Ta: vEBT_VEBT] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Ta),one_one_nat)
    <=> ? [A4: $o,B5: $o] : Ta = vEBT_Leaf((A4),(B5)) ) ).

% deg1Leaf
tff(fact_2828_Tbuildupi__buildupi_H,axiom,
    ! [Nb: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(vEBT_V441764108873111860ildupi,Nb)) = aa_nat_int(vEBT_V9176841429113362141ildupi,Nb) ).

% Tbuildupi_buildupi'
tff(fact_2829_double__eq__0__iff,axiom,
    ! [A: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),A) = zero_zero_real )
    <=> ( A = zero_zero_real ) ) ).

% double_eq_0_iff
tff(fact_2830_double__eq__0__iff,axiom,
    ! [A: rat] :
      ( ( aa_rat_rat(plus_plus_rat(A),A) = zero_zero_rat )
    <=> ( A = zero_zero_rat ) ) ).

% double_eq_0_iff
tff(fact_2831_double__eq__0__iff,axiom,
    ! [A: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),A) = zero_zero_int )
    <=> ( A = zero_zero_int ) ) ).

% double_eq_0_iff
tff(fact_2832_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H_Osimps_I1_J,axiom,
    ! [A: $o,B: $o,Xb: nat] : aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_m_e_m_b_e_r2,vEBT_Leaf((A),(B))),Xb) = one_one_nat ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.simps(1)
tff(fact_2833_VEBT_Osize_I4_J,axiom,
    ! [X21: $o,X22: $o] : aa_VEBT_VEBT_nat(size_size_VEBT_VEBT,vEBT_Leaf((X21),(X22))) = zero_zero_nat ).

% VEBT.size(4)
tff(fact_2834_VEBT__internal_Ovalid_H_Ocases,axiom,
    ! [Xb: produc9072475918466114483BT_nat] :
      ( ! [Uu2: $o,Uv2: $o,D2: nat] : Xb != aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(vEBT_Leaf((Uu2),(Uv2))),D2)
     => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,Deg3: nat] : Xb != aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Mima,Deg2,TreeList),Summary)),Deg3) ) ).

% VEBT_internal.valid'.cases
tff(fact_2835_VEBT_Odistinct_I1_J,axiom,
    ! [X11a: option4927543243414619207at_nat,X12a: nat,X13a: list_VEBT_VEBT,X14a: vEBT_VEBT,X21: $o,X22: $o] : aa_VEB5832953974749077948T_VEBT(vEBT_Node(X11a,X12a,X13a),X14a) != vEBT_Leaf((X21),(X22)) ).

% VEBT.distinct(1)
tff(fact_2836_VEBT_Oexhaust,axiom,
    ! [Y: vEBT_VEBT] :
      ( ! [X11: option4927543243414619207at_nat,X12: nat,X13: list_VEBT_VEBT,X14: vEBT_VEBT] : Y != aa_VEB5832953974749077948T_VEBT(vEBT_Node(X11,X12,X13),X14)
     => ~ ! [X212: $o,X222: $o] : Y != vEBT_Leaf((X212),(X222)) ) ).

% VEBT.exhaust
tff(fact_2837_VEBT__internal_OminNull_Osimps_I1_J,axiom,
    aa_VEBT_VEBT_o(vEBT_VEBT_minNull,vEBT_Leaf($false,$false)) ).

% VEBT_internal.minNull.simps(1)
tff(fact_2838_VEBT__internal_OminNull_Osimps_I2_J,axiom,
    ! [Uv: $o] : ~ aa_VEBT_VEBT_o(vEBT_VEBT_minNull,vEBT_Leaf($true,(Uv))) ).

% VEBT_internal.minNull.simps(2)
tff(fact_2839_VEBT__internal_OminNull_Osimps_I3_J,axiom,
    ! [Uu: $o] : ~ aa_VEBT_VEBT_o(vEBT_VEBT_minNull,vEBT_Leaf((Uu),$true)) ).

% VEBT_internal.minNull.simps(3)
tff(fact_2840_VEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H_Osimps_I3_J,axiom,
    ! [A: $o,B: $o,Nb: nat] : aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_V1232361888498592333_e_t_e,vEBT_Leaf((A),(B))),aa_nat_nat(suc,aa_nat_nat(suc,Nb))) = one_one_nat ).

% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.simps(3)
tff(fact_2841_T_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_Osimps_I3_J,axiom,
    ! [A: $o,B: $o,Nb: nat] : aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_d_e_l_e_t_e,vEBT_Leaf((A),(B))),aa_nat_nat(suc,aa_nat_nat(suc,Nb))) = one_one_nat ).

% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.simps(3)
tff(fact_2842_VEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H_Osimps_I1_J,axiom,
    ! [A: $o,B: $o] : aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_V1232361888498592333_e_t_e,vEBT_Leaf((A),(B))),zero_zero_nat) = one_one_nat ).

% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.simps(1)
tff(fact_2843_T_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_Osimps_I1_J,axiom,
    ! [A: $o,B: $o] : aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_d_e_l_e_t_e,vEBT_Leaf((A),(B))),zero_zero_nat) = one_one_nat ).

% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.simps(1)
tff(fact_2844_VEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H_Osimps_I2_J,axiom,
    ! [A: $o,B: $o] : aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_V1232361888498592333_e_t_e,vEBT_Leaf((A),(B))),aa_nat_nat(suc,zero_zero_nat)) = one_one_nat ).

% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.simps(2)
tff(fact_2845_T_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_Osimps_I2_J,axiom,
    ! [A: $o,B: $o] : aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_d_e_l_e_t_e,vEBT_Leaf((A),(B))),aa_nat_nat(suc,zero_zero_nat)) = one_one_nat ).

% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.simps(2)
tff(fact_2846_VEBT__internal_Onaive__member_Ocases,axiom,
    ! [Xb: produc9072475918466114483BT_nat] :
      ( ! [A3: $o,B2: $o,X2: nat] : Xb != aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(vEBT_Leaf((A3),(B2))),X2)
     => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT,Ux2: nat] : Xb != aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uu2,zero_zero_nat,Uv2),Uw2)),Ux2)
       => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList: list_VEBT_VEBT,S: vEBT_VEBT,X2: nat] : Xb != aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList),S)),X2) ) ) ).

% VEBT_internal.naive_member.cases
tff(fact_2847_invar__vebt_Ointros_I1_J,axiom,
    ! [A: $o,B: $o] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,vEBT_Leaf((A),(B))),aa_nat_nat(suc,zero_zero_nat)) ).

% invar_vebt.intros(1)
tff(fact_2848_VEBT__internal_OT__vebt__buildupi_H_Osimps_I1_J,axiom,
    aa_nat_int(vEBT_V9176841429113362141ildupi,zero_zero_nat) = one_one_int ).

% VEBT_internal.T_vebt_buildupi'.simps(1)
tff(fact_2849_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062N_092_060_094sub_062u_092_060_094sub_062l_092_060_094sub_062l_Ocases,axiom,
    ! [Xb: vEBT_VEBT] :
      ( ( Xb != vEBT_Leaf($false,$false) )
     => ( ! [Uv2: $o] : Xb != vEBT_Leaf($true,(Uv2))
       => ( ! [Uu2: $o] : Xb != vEBT_Leaf((Uu2),$true)
         => ( ! [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] : Xb != aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uw2,Ux2),Uy2)
           => ~ ! [Uz2: product_prod_nat_nat,Va: nat,Vb2: list_VEBT_VEBT,Vc: vEBT_VEBT] : Xb != aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(Uz2),Va,Vb2),Vc) ) ) ) ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.cases
tff(fact_2850_vebt__member_Osimps_I1_J,axiom,
    ! [A: $o,B: $o,Xb: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,vEBT_Leaf((A),(B))),Xb)
    <=> $ite(
          Xb = zero_zero_nat,
          (A),
          $ite(Xb = one_one_nat,(B),$false) ) ) ).

% vebt_member.simps(1)
tff(fact_2851_vebt__buildup_Osimps_I2_J,axiom,
    vEBT_vebt_buildup(aa_nat_nat(suc,zero_zero_nat)) = vEBT_Leaf($false,$false) ).

% vebt_buildup.simps(2)
tff(fact_2852_VEBT__internal_OminNull_Oelims_I3_J,axiom,
    ! [Xb: vEBT_VEBT] :
      ( ~ aa_VEBT_VEBT_o(vEBT_VEBT_minNull,Xb)
     => ( ! [Uv2: $o] : Xb != vEBT_Leaf($true,(Uv2))
       => ( ! [Uu2: $o] : Xb != vEBT_Leaf((Uu2),$true)
         => ~ ! [Uz2: product_prod_nat_nat,Va: nat,Vb2: list_VEBT_VEBT,Vc: vEBT_VEBT] : Xb != aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(Uz2),Va,Vb2),Vc) ) ) ) ).

% VEBT_internal.minNull.elims(3)
tff(fact_2853_vebt__insert_Osimps_I1_J,axiom,
    ! [A: $o,B: $o,Xb: nat] :
      vEBT_vebt_insert(vEBT_Leaf((A),(B)),Xb) = $ite(
        Xb = zero_zero_nat,
        vEBT_Leaf($true,(B)),
        $ite(Xb = one_one_nat,vEBT_Leaf((A),$true),vEBT_Leaf((A),(B))) ) ).

% vebt_insert.simps(1)
tff(fact_2854_vebt__succ_Osimps_I2_J,axiom,
    ! [Uv: $o,Uw: $o,Nb: nat] : aa_nat_option_nat(vEBT_vebt_succ(vEBT_Leaf((Uv),(Uw))),aa_nat_nat(suc,Nb)) = none_nat ).

% vebt_succ.simps(2)
tff(fact_2855_vebt__pred_Osimps_I1_J,axiom,
    ! [Uu: $o,Uv: $o] : aa_nat_option_nat(vEBT_vebt_pred(vEBT_Leaf((Uu),(Uv))),zero_zero_nat) = none_nat ).

% vebt_pred.simps(1)
tff(fact_2856_VEBT__internal_OminNull_Oelims_I2_J,axiom,
    ! [Xb: vEBT_VEBT] :
      ( aa_VEBT_VEBT_o(vEBT_VEBT_minNull,Xb)
     => ( ( Xb != vEBT_Leaf($false,$false) )
       => ~ ! [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] : Xb != aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uw2,Ux2),Uy2) ) ) ).

% VEBT_internal.minNull.elims(2)
tff(fact_2857_vebt__assn__raw_Ocases,axiom,
    ! [Xb: produc3625547720036274456_VEBTi] :
      ( ! [A3: $o,B2: $o,Ai: $o,Bi: $o] : Xb != produc6084888613844515218_VEBTi(vEBT_Leaf((A3),(B2)),vEBT_Leafi((Ai),(Bi)))
     => ( ! [Mmo2: option4927543243414619207at_nat,Deg2: nat,Tree_list2: list_VEBT_VEBT,Summary: vEBT_VEBT,Mmoi2: option4927543243414619207at_nat,Degi2: nat,Tree_array2: array_VEBT_VEBTi,Summaryi2: vEBT_VEBTi] : Xb != produc6084888613844515218_VEBTi(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Mmo2,Deg2,Tree_list2),Summary),aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(Mmoi2,Degi2,Tree_array2),Summaryi2))
       => ( ! [V2: option4927543243414619207at_nat,Va2: nat,Vb3: list_VEBT_VEBT,Vc3: vEBT_VEBT,Vd3: $o,Ve3: $o] : Xb != produc6084888613844515218_VEBTi(aa_VEB5832953974749077948T_VEBT(vEBT_Node(V2,Va2,Vb3),Vc3),vEBT_Leafi((Vd3),(Ve3)))
         => ~ ! [Vd3: $o,Ve3: $o,V2: option4927543243414619207at_nat,Va2: nat,Vb3: array_VEBT_VEBTi,Vc3: vEBT_VEBTi] : Xb != produc6084888613844515218_VEBTi(vEBT_Leaf((Vd3),(Ve3)),aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(V2,Va2,Vb3),Vc3)) ) ) ) ).

% vebt_assn_raw.cases
tff(fact_2858_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Osimps_I3_J,axiom,
    ! [A: $o,B: $o,Vaa: nat] : aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_p_r_e_d2,vEBT_Leaf((A),(B))),aa_nat_nat(suc,aa_nat_nat(suc,Vaa))) = one_one_nat ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.simps(3)
tff(fact_2859_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Osimps_I1_J,axiom,
    ! [Uu: $o,Uv: $o] : aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_p_r_e_d2,vEBT_Leaf((Uu),(Uv))),zero_zero_nat) = one_one_nat ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.simps(1)
tff(fact_2860_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Osimps_I2_J,axiom,
    ! [Uv: $o,Uw: $o,Nb: nat] : aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_s_u_c_c2,vEBT_Leaf((Uv),(Uw))),aa_nat_nat(suc,Nb)) = one_one_nat ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.simps(2)
tff(fact_2861_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Osimps_I1_J,axiom,
    ! [Uu: $o,B: $o] : aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_s_u_c_c2,vEBT_Leaf((Uu),(B))),zero_zero_nat) = one_one_nat ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.simps(1)
tff(fact_2862_vebt__assn__raw_Osimps_I1_J,axiom,
    ! [A: $o,B: $o,Ai2: $o,Bi2: $o] :
      aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,vEBT_Leaf((A),(B))),vEBT_Leafi((Ai2),(Bi2))) = aa_o_assn(pure_assn,
        ( ( (Ai2) = (A) )
        & ( (Bi2) = (B) ) )) ).

% vebt_assn_raw.simps(1)
tff(fact_2863_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_2864_plus__int__code_I2_J,axiom,
    ! [L: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,zero_zero_int),L) = L ).

% plus_int_code(2)
tff(fact_2865_plus__int__code_I1_J,axiom,
    ! [K: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,K),zero_zero_int) = K ).

% plus_int_code(1)
tff(fact_2866_times__int__code_I2_J,axiom,
    ! [L: int] : aa_int_int(aa_int_fun_int_int(times_times_int,zero_zero_int),L) = zero_zero_int ).

% times_int_code(2)
tff(fact_2867_times__int__code_I1_J,axiom,
    ! [K: int] : aa_int_int(aa_int_fun_int_int(times_times_int,K),zero_zero_int) = zero_zero_int ).

% times_int_code(1)
tff(fact_2868_zmult__zless__mono2,axiom,
    ! [I: int,J: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,I),J)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),K)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,K),I)),aa_int_int(aa_int_fun_int_int(times_times_int,K),J)) ) ) ).

% zmult_zless_mono2
tff(fact_2869_minus__int__code_I1_J,axiom,
    ! [K: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,K),zero_zero_int) = K ).

% minus_int_code(1)
tff(fact_2870_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t_Ocases,axiom,
    ! [Xb: vEBT_VEBT] :
      ( ! [A3: $o,B2: $o] : Xb != vEBT_Leaf((A3),(B2))
     => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] : Xb != aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2),Uw2)
       => ~ ! [Mi: nat,Ma: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] : Xb != aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),Ux2,Uy2),Uz2) ) ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>t.cases
tff(fact_2871_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Osimps_I2_J,axiom,
    ! [A: $o,Uw: $o] : aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_p_r_e_d2,vEBT_Leaf((A),(Uw))),aa_nat_nat(suc,zero_zero_nat)) = one_one_nat ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.simps(2)
tff(fact_2872_VEBT__internal_OminNull_Oelims_I1_J,axiom,
    ! [Xb: vEBT_VEBT,Y: $o] :
      ( ( aa_VEBT_VEBT_o(vEBT_VEBT_minNull,Xb)
      <=> (Y) )
     => ( ( ( Xb = vEBT_Leaf($false,$false) )
         => ~ (Y) )
       => ( ( ? [Uv2: $o] : Xb = vEBT_Leaf($true,(Uv2))
           => (Y) )
         => ( ( ? [Uu2: $o] : Xb = vEBT_Leaf((Uu2),$true)
             => (Y) )
           => ( ( ? [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] : Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uw2,Ux2),Uy2)
               => ~ (Y) )
             => ~ ( ? [Uz2: product_prod_nat_nat,Va: nat,Vb2: list_VEBT_VEBT,Vc: vEBT_VEBT] : Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(Uz2),Va,Vb2),Vc)
                 => (Y) ) ) ) ) ) ) ).

% VEBT_internal.minNull.elims(1)
tff(fact_2873_VEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H_Ocases,axiom,
    ! [Xb: produc9072475918466114483BT_nat] :
      ( ! [A3: $o,B2: $o] : Xb != aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(vEBT_Leaf((A3),(B2))),zero_zero_nat)
     => ( ! [A3: $o,B2: $o] : Xb != aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(vEBT_Leaf((A3),(B2))),aa_nat_nat(suc,zero_zero_nat))
       => ( ! [A3: $o,B2: $o,N2: nat] : Xb != aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(vEBT_Leaf((A3),(B2))),aa_nat_nat(suc,aa_nat_nat(suc,N2)))
         => ( ! [Deg2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,Uu2: nat] : Xb != aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Deg2,TreeList),Summary)),Uu2)
           => ( ! [Mi: nat,Ma: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X2: nat] : Xb != aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),zero_zero_nat,TreeList),Summary)),X2)
             => ( ! [Mi: nat,Ma: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X2: nat] : Xb != aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),aa_nat_nat(suc,zero_zero_nat),TreeList),Summary)),X2)
               => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X2: nat] : Xb != aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList),Summary)),X2) ) ) ) ) ) ) ).

% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.cases
tff(fact_2874_VEBT__internal_OT__vebt__buildupi_H_Osimps_I2_J,axiom,
    aa_nat_int(vEBT_V9176841429113362141ildupi,aa_nat_nat(suc,zero_zero_nat)) = one_one_int ).

% VEBT_internal.T_vebt_buildupi'.simps(2)
tff(fact_2875_vebt__pred_Osimps_I2_J,axiom,
    ! [A: $o,Uw: $o] :
      aa_nat_option_nat(vEBT_vebt_pred(vEBT_Leaf((A),(Uw))),aa_nat_nat(suc,zero_zero_nat)) = $ite((A),aa_nat_option_nat(some_nat,zero_zero_nat),none_nat) ).

% vebt_pred.simps(2)
tff(fact_2876_vebt__succ_Osimps_I1_J,axiom,
    ! [Uu: $o,B: $o] :
      aa_nat_option_nat(vEBT_vebt_succ(vEBT_Leaf((Uu),(B))),zero_zero_nat) = $ite((B),aa_nat_option_nat(some_nat,one_one_nat),none_nat) ).

% vebt_succ.simps(1)
tff(fact_2877_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H_Osimps_I2_J,axiom,
    ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT,Xb: nat] : aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_m_e_m_b_e_r2,aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv),Uw)),Xb) = one_one_nat ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.simps(2)
tff(fact_2878_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H_Ocases,axiom,
    ! [Xb: produc9072475918466114483BT_nat] :
      ( ! [A3: $o,B2: $o,X2: nat] : Xb != aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(vEBT_Leaf((A3),(B2))),X2)
     => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT,X2: nat] : Xb != aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2),Uw2)),X2)
       => ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT,X2: nat] : Xb != aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Uy2),Uz2)),X2)
         => ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc: vEBT_VEBT,X2: nat] : Xb != aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vb2),Vc)),X2)
           => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X2: nat] : Xb != aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList),Summary)),X2) ) ) ) ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.cases
tff(fact_2879_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H_Ocases,axiom,
    ! [Xb: produc9072475918466114483BT_nat] :
      ( ! [A3: $o,B2: $o,X2: nat] : Xb != aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(vEBT_Leaf((A3),(B2))),X2)
     => ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S: vEBT_VEBT,X2: nat] : Xb != aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,zero_zero_nat,Ts2),S)),X2)
       => ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S: vEBT_VEBT,X2: nat] : Xb != aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts2),S)),X2)
         => ( ! [V2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X2: nat] : Xb != aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V2)),TreeList),Summary)),X2)
           => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X2: nat] : Xb != aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList),Summary)),X2) ) ) ) ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.cases
tff(fact_2880_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Ocases,axiom,
    ! [Xb: produc9072475918466114483BT_nat] :
      ( ! [Uu2: $o,B2: $o] : Xb != aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(vEBT_Leaf((Uu2),(B2))),zero_zero_nat)
     => ( ! [Uv2: $o,Uw2: $o,N2: nat] : Xb != aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(vEBT_Leaf((Uv2),(Uw2))),aa_nat_nat(suc,N2))
       => ( ! [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT,Va: nat] : Xb != aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Ux2,Uy2),Uz2)),Va)
         => ( ! [V2: product_prod_nat_nat,Vc: list_VEBT_VEBT,Vd2: vEBT_VEBT,Ve2: nat] : Xb != aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Vc),Vd2)),Ve2)
           => ( ! [V2: product_prod_nat_nat,Vg2: list_VEBT_VEBT,Vh2: vEBT_VEBT,Vi2: nat] : Xb != aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vg2),Vh2)),Vi2)
             => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X2: nat] : Xb != aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList),Summary)),X2) ) ) ) ) ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.cases
tff(fact_2881_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Ocases,axiom,
    ! [Xb: produc9072475918466114483BT_nat] :
      ( ! [Uu2: $o,Uv2: $o] : Xb != aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(vEBT_Leaf((Uu2),(Uv2))),zero_zero_nat)
     => ( ! [A3: $o,Uw2: $o] : Xb != aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(vEBT_Leaf((A3),(Uw2))),aa_nat_nat(suc,zero_zero_nat))
       => ( ! [A3: $o,B2: $o,Va2: nat] : Xb != aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(vEBT_Leaf((A3),(B2))),aa_nat_nat(suc,aa_nat_nat(suc,Va2)))
         => ( ! [Uy2: nat,Uz2: list_VEBT_VEBT,Va: vEBT_VEBT,Vb2: nat] : Xb != aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uy2,Uz2),Va)),Vb2)
           => ( ! [V2: product_prod_nat_nat,Vd2: list_VEBT_VEBT,Ve2: vEBT_VEBT,Vf2: nat] : Xb != aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Vd2),Ve2)),Vf2)
             => ( ! [V2: product_prod_nat_nat,Vh2: list_VEBT_VEBT,Vi2: vEBT_VEBT,Vj2: nat] : Xb != aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vh2),Vi2)),Vj2)
               => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X2: nat] : Xb != aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList),Summary)),X2) ) ) ) ) ) ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.cases
tff(fact_2882_VEBT__internal_Omembermima_Ocases,axiom,
    ! [Xb: produc9072475918466114483BT_nat] :
      ( ! [Uu2: $o,Uv2: $o,Uw2: nat] : Xb != aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(vEBT_Leaf((Uu2),(Uv2))),Uw2)
     => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT,Uz2: nat] : Xb != aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux2),Uy2)),Uz2)
       => ( ! [Mi: nat,Ma: nat,Va: list_VEBT_VEBT,Vb2: vEBT_VEBT,X2: nat] : Xb != aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),zero_zero_nat,Va),Vb2)),X2)
         => ( ! [Mi: nat,Ma: nat,V2: nat,TreeList: list_VEBT_VEBT,Vc: vEBT_VEBT,X2: nat] : Xb != aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),aa_nat_nat(suc,V2),TreeList),Vc)),X2)
           => ~ ! [V2: nat,TreeList: list_VEBT_VEBT,Vd2: vEBT_VEBT,X2: nat] : Xb != aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList),Vd2)),X2) ) ) ) ) ).

% VEBT_internal.membermima.cases
tff(fact_2883_vebt__pred_Osimps_I3_J,axiom,
    ! [A: $o,B: $o,Vaa: nat] :
      aa_nat_option_nat(vEBT_vebt_pred(vEBT_Leaf((A),(B))),aa_nat_nat(suc,aa_nat_nat(suc,Vaa))) = $ite(
        (B),
        aa_nat_option_nat(some_nat,one_one_nat),
        $ite((A),aa_nat_option_nat(some_nat,zero_zero_nat),none_nat) ) ).

% vebt_pred.simps(3)
tff(fact_2884_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_Osimps_I1_J,axiom,
    ! [A: $o,B: $o,Xb: nat] :
      aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_m_e_m_b_e_r,vEBT_Leaf((A),(B))),Xb) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
        $ite(Xb = zero_zero_nat,one_one_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat))) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.simps(1)
tff(fact_2885_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H_Osimps_I3_J,axiom,
    ! [V: product_prod_nat_nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT,Xb: nat] : aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_m_e_m_b_e_r2,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V),zero_zero_nat,Uy),Uz)),Xb) = one_one_nat ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.simps(3)
tff(fact_2886_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_2887_zless__add1__eq,axiom,
    ! [W: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,W),aa_int_int(aa_int_fun_int_int(plus_plus_int,Z),one_one_int))
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,W),Z)
        | ( W = Z ) ) ) ).

% zless_add1_eq
tff(fact_2888_int__gr__induct,axiom,
    ! [K: int,I: int,P: fun_int_o] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),I)
     => ( aa_int_o(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,K),one_one_int))
       => ( ! [I3: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),I3)
             => ( aa_int_o(P,I3)
               => aa_int_o(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,I3),one_one_int)) ) )
         => aa_int_o(P,I) ) ) ) ).

% int_gr_induct
tff(fact_2889_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_2890_odd__nonzero,axiom,
    ! [Z: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),Z)),Z) != zero_zero_int ).

% odd_nonzero
tff(fact_2891_pos__zmult__eq__1__iff,axiom,
    ! [M: int,Nb: 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),Nb) = one_one_int )
      <=> ( ( M = one_one_int )
          & ( Nb = one_one_int ) ) ) ) ).

% pos_zmult_eq_1_iff
tff(fact_2892_zero__le__imp__eq__int,axiom,
    ! [K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K)
     => ? [N2: nat] : K = aa_nat_int(semiri1314217659103216013at_int,N2) ) ).

% zero_le_imp_eq_int
tff(fact_2893_nonneg__int__cases,axiom,
    ! [K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K)
     => ~ ! [N2: nat] : K != aa_nat_int(semiri1314217659103216013at_int,N2) ) ).

% nonneg_int_cases
tff(fact_2894_int__distrib_I1_J,axiom,
    ! [Z1: int,Z22: int,W: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Z1),Z22)),W) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Z1),W)),aa_int_int(aa_int_fun_int_int(times_times_int,Z22),W)) ).

% int_distrib(1)
tff(fact_2895_int__distrib_I2_J,axiom,
    ! [W: int,Z1: int,Z22: int] : aa_int_int(aa_int_fun_int_int(times_times_int,W),aa_int_int(aa_int_fun_int_int(plus_plus_int,Z1),Z22)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,W),Z1)),aa_int_int(aa_int_fun_int_int(times_times_int,W),Z22)) ).

% int_distrib(2)
tff(fact_2896_int__less__induct,axiom,
    ! [I: int,K: int,P: fun_int_o] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,I),K)
     => ( aa_int_o(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,K),one_one_int))
       => ( ! [I3: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,I3),K)
             => ( aa_int_o(P,I3)
               => aa_int_o(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,I3),one_one_int)) ) )
         => aa_int_o(P,I) ) ) ) ).

% int_less_induct
tff(fact_2897_int__distrib_I4_J,axiom,
    ! [W: int,Z1: int,Z22: int] : aa_int_int(aa_int_fun_int_int(times_times_int,W),aa_int_int(aa_int_fun_int_int(minus_minus_int,Z1),Z22)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,W),Z1)),aa_int_int(aa_int_fun_int_int(times_times_int,W),Z22)) ).

% int_distrib(4)
tff(fact_2898_int__distrib_I3_J,axiom,
    ! [Z1: int,Z22: int,W: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Z1),Z22)),W) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Z1),W)),aa_int_int(aa_int_fun_int_int(times_times_int,Z22),W)) ).

% int_distrib(3)
tff(fact_2899_int__diff__cases,axiom,
    ! [Z: int] :
      ~ ! [M2: nat,N2: nat] : Z != aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(semiri1314217659103216013at_int,M2)),aa_nat_int(semiri1314217659103216013at_int,N2)) ).

% int_diff_cases
tff(fact_2900_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H_Osimps_I4_J,axiom,
    ! [V: product_prod_nat_nat,Vb: list_VEBT_VEBT,Vc2: vEBT_VEBT,Xb: nat] : aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_m_e_m_b_e_r2,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V),aa_nat_nat(suc,zero_zero_nat),Vb),Vc2)),Xb) = one_one_nat ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.simps(4)
tff(fact_2901_member__bound__height_H,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xb: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Ta),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_m_e_m_b_e_r2,Ta),Xb)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_VEBT_height,Ta))) ) ).

% member_bound_height'
tff(fact_2902_pos__int__cases,axiom,
    ! [K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),K)
     => ~ ! [N2: nat] :
            ( ( K = aa_nat_int(semiri1314217659103216013at_int,N2) )
           => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2) ) ) ).

% pos_int_cases
tff(fact_2903_zero__less__imp__eq__int,axiom,
    ! [K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),K)
     => ? [N2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)
          & ( K = aa_nat_int(semiri1314217659103216013at_int,N2) ) ) ) ).

% zero_less_imp_eq_int
tff(fact_2904_zless__iff__Suc__zadd,axiom,
    ! [W: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,W),Z)
    <=> ? [N6: nat] : Z = aa_int_int(aa_int_fun_int_int(plus_plus_int,W),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,N6))) ) ).

% zless_iff_Suc_zadd
tff(fact_2905_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H_Oelims,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_m_e_m_b_e_r2,Xb),Xaa) = Y )
     => ( ( ? [A3: $o,B2: $o] : Xb = vEBT_Leaf((A3),(B2))
         => ( Y != one_one_nat ) )
       => ( ( ? [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] : Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2),Uw2)
           => ( Y != one_one_nat ) )
         => ( ( ? [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] : Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Uy2),Uz2)
             => ( Y != one_one_nat ) )
           => ( ( ? [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc: vEBT_VEBT] : Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vb2),Vc)
               => ( Y != one_one_nat ) )
             => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT] :
                    ( ? [Summary: vEBT_VEBT] : Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList),Summary)
                   => ( Y != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
                          $ite(
                            Xaa = Mi,
                            zero_zero_nat,
                            $ite(
                              Xaa = Ma,
                              zero_zero_nat,
                              $ite(
                                aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi),
                                zero_zero_nat,
                                $ite(
                                  aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Xaa),
                                  zero_zero_nat,
                                  $ite(
                                    ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mi),Xaa)
                                    & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Ma) ),
                                    $let(
                                      h: nat,
                                      h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                      $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_m_e_m_b_e_r2,nth_VEBT_VEBT(TreeList,h)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),zero_zero_nat) ),
                                    zero_zero_nat ) ) ) ) )) ) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.elims
tff(fact_2906_zle__int,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb) ) ).

% zle_int
tff(fact_2907_zadd__int__left,axiom,
    ! [M: nat,Nb: nat,Z: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),Z)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Nb))),Z) ).

% zadd_int_left
tff(fact_2908_zless__imp__add1__zle,axiom,
    ! [W: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,W),Z)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,W),one_one_int)),Z) ) ).

% zless_imp_add1_zle
tff(fact_2909_int__ge__induct,axiom,
    ! [K: int,I: int,P: fun_int_o] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),I)
     => ( aa_int_o(P,K)
       => ( ! [I3: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),I3)
             => ( aa_int_o(P,I3)
               => aa_int_o(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,I3),one_one_int)) ) )
         => aa_int_o(P,I) ) ) ) ).

% int_ge_induct
tff(fact_2910_add1__zle__eq,axiom,
    ! [W: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,W),one_one_int)),Z)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,W),Z) ) ).

% add1_zle_eq
tff(fact_2911_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_2912_zle__iff__zadd,axiom,
    ! [W: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,W),Z)
    <=> ? [N6: nat] : Z = aa_int_int(aa_int_fun_int_int(plus_plus_int,W),aa_nat_int(semiri1314217659103216013at_int,N6)) ) ).

% zle_iff_zadd
tff(fact_2913_int__le__induct,axiom,
    ! [I: int,K: int,P: fun_int_o] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),K)
     => ( aa_int_o(P,K)
       => ( ! [I3: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I3),K)
             => ( aa_int_o(P,I3)
               => aa_int_o(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,I3),one_one_int)) ) )
         => aa_int_o(P,I) ) ) ) ).

% int_le_induct
tff(fact_2914_zmult__zless__mono2__lemma,axiom,
    ! [I: int,J: int,K: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,I),J)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,K)),I)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,K)),J)) ) ) ).

% zmult_zless_mono2_lemma
tff(fact_2915_int__induct,axiom,
    ! [P: fun_int_o,K: int,I: int] :
      ( aa_int_o(P,K)
     => ( ! [I3: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),I3)
           => ( aa_int_o(P,I3)
             => aa_int_o(P,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(P,I3)
               => aa_int_o(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,I3),one_one_int)) ) )
         => aa_int_o(P,I) ) ) ) ).

% int_induct
tff(fact_2916_VEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H_Osimps_I4_J,axiom,
    ! [Deg: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Uu: nat] : aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_V1232361888498592333_e_t_e,aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Deg,TreeLista),Summarya)),Uu) = one_one_nat ).

% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.simps(4)
tff(fact_2917_T_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_Osimps_I4_J,axiom,
    ! [Deg: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Uu: nat] : aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_d_e_l_e_t_e,aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Deg,TreeLista),Summarya)),Uu) = one_one_nat ).

% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.simps(4)
tff(fact_2918_vebt__member_Oelims_I2_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,Xb),Xaa)
     => ( ! [A3: $o,B2: $o] :
            ( ( Xb = vEBT_Leaf((A3),(B2)) )
           => ~ $ite(
                  Xaa = zero_zero_nat,
                  (A3),
                  $ite(Xaa = one_one_nat,(B2),$false) ) )
       => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT] :
              ( ? [Summary: vEBT_VEBT] : Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList),Summary)
             => ~ $ite(
                    Xaa = Mi,
                    $true,
                    $ite(
                      Xaa = Ma,
                      $true,
                      $ite(
                        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi),
                        $false,
                        $ite(
                          aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Xaa),
                          $false,
                          $let(
                            h: nat,
                            h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                            $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,nth_VEBT_VEBT(TreeList,h)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ) ) ).

% vebt_member.elims(2)
tff(fact_2919_vebt__member_Oelims_I1_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: $o] :
      ( ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,Xb),Xaa)
      <=> (Y) )
     => ( ! [A3: $o,B2: $o] :
            ( ( Xb = vEBT_Leaf((A3),(B2)) )
           => ( (Y)
            <=> ~ $ite(
                    Xaa = zero_zero_nat,
                    (A3),
                    $ite(Xaa = one_one_nat,(B2),$false) ) ) )
       => ( ( ? [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] : Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2),Uw2)
           => (Y) )
         => ( ( ? [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] : Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Uy2),Uz2)
             => (Y) )
           => ( ( ? [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc: vEBT_VEBT] : Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vb2),Vc)
               => (Y) )
             => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT] :
                    ( ? [Summary: vEBT_VEBT] : Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList),Summary)
                   => ( (Y)
                    <=> ~ $ite(
                            Xaa = Mi,
                            $true,
                            $ite(
                              Xaa = Ma,
                              $true,
                              $ite(
                                aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi),
                                $false,
                                $ite(
                                  aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Xaa),
                                  $false,
                                  $let(
                                    h: nat,
                                    h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                    $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,nth_VEBT_VEBT(TreeList,h)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.elims(1)
tff(fact_2920_vebt__member_Oelims_I3_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( ~ aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,Xb),Xaa)
     => ( ! [A3: $o,B2: $o] :
            ( ( Xb = vEBT_Leaf((A3),(B2)) )
           => $ite(
                Xaa = zero_zero_nat,
                (A3),
                $ite(Xaa = one_one_nat,(B2),$false) ) )
       => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] : Xb != aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2),Uw2)
         => ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] : Xb != aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Uy2),Uz2)
           => ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc: vEBT_VEBT] : Xb != aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vb2),Vc)
             => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT] :
                    ( ? [Summary: vEBT_VEBT] : Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList),Summary)
                   => $ite(
                        Xaa = Mi,
                        $true,
                        $ite(
                          Xaa = Ma,
                          $true,
                          $ite(
                            aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi),
                            $false,
                            $ite(
                              aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Xaa),
                              $false,
                              $let(
                                h: nat,
                                h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,nth_VEBT_VEBT(TreeList,h)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.elims(3)
tff(fact_2921_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_Oelims,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_m_e_m_b_e_r,Xb),Xaa) = Y )
     => ( ( ? [A3: $o,B2: $o] : Xb = vEBT_Leaf((A3),(B2))
         => ( Y != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                $ite(Xaa = zero_zero_nat,one_one_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat))) ) )
       => ( ( ? [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] : Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2),Uw2)
           => ( Y != aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) ) )
         => ( ( ? [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] : Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Uy2),Uz2)
             => ( Y != aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) ) )
           => ( ( ? [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc: vEBT_VEBT] : Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vb2),Vc)
               => ( Y != aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) ) )
             => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT] :
                    ( ? [Summary: vEBT_VEBT] : Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList),Summary)
                   => ( Y != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                          $ite(
                            Xaa = Mi,
                            one_one_nat,
                            aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
                              $ite(
                                Xaa = Ma,
                                one_one_nat,
                                aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
                                  $ite(
                                    aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi),
                                    one_one_nat,
                                    aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
                                      $ite(
                                        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Xaa),
                                        one_one_nat,
                                        aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,one))))),
                                          $let(
                                            h: nat,
                                            h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                            $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_m_e_m_b_e_r,nth_VEBT_VEBT(TreeList,h)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),one_one_nat) )) )) )) )) )) ) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.elims
tff(fact_2922_invar__vebt_Osimps,axiom,
    ! [A1: vEBT_VEBT,A22: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,A1),A22)
    <=> ( ( ? [A4: $o,B5: $o] : A1 = vEBT_Leaf((A4),(B5))
          & ( A22 = aa_nat_nat(suc,zero_zero_nat) ) )
        | ? [TreeList2: list_VEBT_VEBT,N6: nat,Summary2: vEBT_VEBT] :
            ( ( A1 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,A22,TreeList2),Summary2) )
            & ! [X: vEBT_VEBT] :
                ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),set_VEBT_VEBT2(TreeList2))
               => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,X),N6) )
            & aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Summary2),N6)
            & ( size_s6755466524823107622T_VEBT(TreeList2) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N6) )
            & ( A22 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N6),N6) )
            & ~ ? [X_1: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,Summary2),X_1)
            & ! [X: vEBT_VEBT] :
                ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),set_VEBT_VEBT2(TreeList2))
               => ~ ? [X_1: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,X),X_1) ) )
        | ? [TreeList2: list_VEBT_VEBT,N6: nat,Summary2: vEBT_VEBT] :
            ( ( A1 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,A22,TreeList2),Summary2) )
            & ! [X: vEBT_VEBT] :
                ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),set_VEBT_VEBT2(TreeList2))
               => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,X),N6) )
            & aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Summary2),aa_nat_nat(suc,N6))
            & ( size_s6755466524823107622T_VEBT(TreeList2) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,N6)) )
            & ( A22 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N6),aa_nat_nat(suc,N6)) )
            & ~ ? [X_1: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,Summary2),X_1)
            & ! [X: vEBT_VEBT] :
                ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),set_VEBT_VEBT2(TreeList2))
               => ~ ? [X_1: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,X),X_1) ) )
        | ? [TreeList2: list_VEBT_VEBT,N6: nat,Summary2: vEBT_VEBT,Mi2: nat,Ma2: nat] :
            ( ( A1 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi2),Ma2)),A22,TreeList2),Summary2) )
            & ! [X: vEBT_VEBT] :
                ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),set_VEBT_VEBT2(TreeList2))
               => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,X),N6) )
            & aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Summary2),N6)
            & ( size_s6755466524823107622T_VEBT(TreeList2) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N6) )
            & ( A22 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N6),N6) )
            & ! [I2: nat] :
                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N6))
               => ( ? [X_1: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList2,I2)),X_1)
                <=> aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,Summary2),I2) ) )
            & ( ( Mi2 = Ma2 )
             => ! [X: vEBT_VEBT] :
                  ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),set_VEBT_VEBT2(TreeList2))
                 => ~ ? [X_1: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,X),X_1) ) )
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mi2),Ma2)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A22))
            & ( ( Mi2 != Ma2 )
             => ! [I2: nat] :
                  ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N6))
                 => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Ma2),N6) = I2 )
                     => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList2,I2)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Ma2),N6)) )
                    & ! [X: nat] :
                        ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),N6) = I2 )
                          & aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList2,I2)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),N6)) )
                       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mi2),X)
                          & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Ma2) ) ) ) ) ) )
        | ? [TreeList2: list_VEBT_VEBT,N6: nat,Summary2: vEBT_VEBT,Mi2: nat,Ma2: nat] :
            ( ( A1 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi2),Ma2)),A22,TreeList2),Summary2) )
            & ! [X: vEBT_VEBT] :
                ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),set_VEBT_VEBT2(TreeList2))
               => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,X),N6) )
            & aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Summary2),aa_nat_nat(suc,N6))
            & ( size_s6755466524823107622T_VEBT(TreeList2) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,N6)) )
            & ( A22 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N6),aa_nat_nat(suc,N6)) )
            & ! [I2: nat] :
                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,N6)))
               => ( ? [X_1: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList2,I2)),X_1)
                <=> aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,Summary2),I2) ) )
            & ( ( Mi2 = Ma2 )
             => ! [X: vEBT_VEBT] :
                  ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),set_VEBT_VEBT2(TreeList2))
                 => ~ ? [X_1: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,X),X_1) ) )
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mi2),Ma2)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A22))
            & ( ( Mi2 != Ma2 )
             => ! [I2: nat] :
                  ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,N6)))
                 => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Ma2),N6) = I2 )
                     => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList2,I2)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Ma2),N6)) )
                    & ! [X: nat] :
                        ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),N6) = I2 )
                          & aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList2,I2)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),N6)) )
                       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mi2),X)
                          & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Ma2) ) ) ) ) ) ) ) ) ).

% invar_vebt.simps
tff(fact_2923_invar__vebt_Ocases,axiom,
    ! [A1: vEBT_VEBT,A22: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,A1),A22)
     => ( ( ? [A3: $o,B2: $o] : A1 = vEBT_Leaf((A3),(B2))
         => ( A22 != aa_nat_nat(suc,zero_zero_nat) ) )
       => ( ! [TreeList: list_VEBT_VEBT,N2: nat,Summary: vEBT_VEBT,M2: nat,Deg2: nat] :
              ( ( A1 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Deg2,TreeList),Summary) )
             => ( ( A22 = Deg2 )
               => ( ! [X3: vEBT_VEBT] :
                      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X3),set_VEBT_VEBT2(TreeList))
                     => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,X3),N2) )
                 => ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Summary),M2)
                   => ( ( size_s6755466524823107622T_VEBT(TreeList) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M2) )
                     => ( ( M2 = N2 )
                       => ( ( Deg2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),M2) )
                         => ( ~ ? [X_12: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,Summary),X_12)
                           => ~ ! [X3: vEBT_VEBT] :
                                  ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X3),set_VEBT_VEBT2(TreeList))
                                 => ~ ? [X_12: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,X3),X_12) ) ) ) ) ) ) ) ) )
         => ( ! [TreeList: list_VEBT_VEBT,N2: nat,Summary: vEBT_VEBT,M2: nat,Deg2: nat] :
                ( ( A1 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Deg2,TreeList),Summary) )
               => ( ( A22 = Deg2 )
                 => ( ! [X3: vEBT_VEBT] :
                        ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X3),set_VEBT_VEBT2(TreeList))
                       => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,X3),N2) )
                   => ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Summary),M2)
                     => ( ( size_s6755466524823107622T_VEBT(TreeList) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M2) )
                       => ( ( M2 = aa_nat_nat(suc,N2) )
                         => ( ( Deg2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),M2) )
                           => ( ~ ? [X_12: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,Summary),X_12)
                             => ~ ! [X3: vEBT_VEBT] :
                                    ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X3),set_VEBT_VEBT2(TreeList))
                                   => ~ ? [X_12: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,X3),X_12) ) ) ) ) ) ) ) ) )
           => ( ! [TreeList: list_VEBT_VEBT,N2: nat,Summary: vEBT_VEBT,M2: nat,Deg2: nat,Mi: nat,Ma: nat] :
                  ( ( A1 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),Deg2,TreeList),Summary) )
                 => ( ( A22 = Deg2 )
                   => ( ! [X3: vEBT_VEBT] :
                          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X3),set_VEBT_VEBT2(TreeList))
                         => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,X3),N2) )
                     => ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Summary),M2)
                       => ( ( size_s6755466524823107622T_VEBT(TreeList) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M2) )
                         => ( ( M2 = N2 )
                           => ( ( Deg2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),M2) )
                             => ( ! [I4: nat] :
                                    ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M2))
                                   => ( ? [X_1: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList,I4)),X_1)
                                    <=> aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,Summary),I4) ) )
                               => ( ( ( Mi = Ma )
                                   => ! [X3: vEBT_VEBT] :
                                        ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X3),set_VEBT_VEBT2(TreeList))
                                       => ~ ? [X_12: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,X3),X_12) ) )
                                 => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mi),Ma)
                                   => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg2))
                                     => ~ ( ( Mi != Ma )
                                         => ! [I4: nat] :
                                              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M2))
                                             => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Ma),N2) = I4 )
                                                 => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList,I4)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Ma),N2)) )
                                                & ! [X3: nat] :
                                                    ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X3),N2) = I4 )
                                                      & aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList,I4)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X3),N2)) )
                                                   => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mi),X3)
                                                      & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),Ma) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
             => ~ ! [TreeList: list_VEBT_VEBT,N2: nat,Summary: vEBT_VEBT,M2: nat,Deg2: nat,Mi: nat,Ma: nat] :
                    ( ( A1 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),Deg2,TreeList),Summary) )
                   => ( ( A22 = Deg2 )
                     => ( ! [X3: vEBT_VEBT] :
                            ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X3),set_VEBT_VEBT2(TreeList))
                           => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,X3),N2) )
                       => ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Summary),M2)
                         => ( ( size_s6755466524823107622T_VEBT(TreeList) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M2) )
                           => ( ( M2 = aa_nat_nat(suc,N2) )
                             => ( ( Deg2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),M2) )
                               => ( ! [I4: nat] :
                                      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M2))
                                     => ( ? [X_1: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList,I4)),X_1)
                                      <=> aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,Summary),I4) ) )
                                 => ( ( ( Mi = Ma )
                                     => ! [X3: vEBT_VEBT] :
                                          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X3),set_VEBT_VEBT2(TreeList))
                                         => ~ ? [X_12: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,X3),X_12) ) )
                                   => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mi),Ma)
                                     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg2))
                                       => ~ ( ( Mi != Ma )
                                           => ! [I4: nat] :
                                                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M2))
                                               => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Ma),N2) = I4 )
                                                   => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList,I4)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Ma),N2)) )
                                                  & ! [X3: nat] :
                                                      ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X3),N2) = I4 )
                                                        & aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList,I4)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X3),N2)) )
                                                     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mi),X3)
                                                        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),Ma) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.cases
tff(fact_2924_VEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H_Osimps_I5_J,axiom,
    ! [Mia: nat,Maa: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Xb: nat] : aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_V1232361888498592333_e_t_e,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mia),Maa)),zero_zero_nat,TreeLista),Summarya)),Xb) = one_one_nat ).

% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.simps(5)
tff(fact_2925_T_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_Osimps_I5_J,axiom,
    ! [Mia: nat,Maa: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Xb: nat] : aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_d_e_l_e_t_e,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mia),Maa)),zero_zero_nat,TreeLista),Summarya)),Xb) = one_one_nat ).

% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.simps(5)
tff(fact_2926_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Oelims,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_p_r_e_d2,Xb),Xaa) = Y )
     => ( ( ? [Uu2: $o,Uv2: $o] : Xb = vEBT_Leaf((Uu2),(Uv2))
         => ( ( Xaa = zero_zero_nat )
           => ( Y != one_one_nat ) ) )
       => ( ( ? [A3: $o,Uw2: $o] : Xb = vEBT_Leaf((A3),(Uw2))
           => ( ( Xaa = aa_nat_nat(suc,zero_zero_nat) )
             => ( Y != one_one_nat ) ) )
         => ( ( ? [A3: $o,B2: $o] : Xb = vEBT_Leaf((A3),(B2))
             => ( ? [Va2: nat] : Xaa = aa_nat_nat(suc,aa_nat_nat(suc,Va2))
               => ( Y != one_one_nat ) ) )
           => ( ( ? [Uy2: nat,Uz2: list_VEBT_VEBT,Va: vEBT_VEBT] : Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uy2,Uz2),Va)
               => ( Y != one_one_nat ) )
             => ( ( ? [V2: product_prod_nat_nat,Vd2: list_VEBT_VEBT,Ve2: vEBT_VEBT] : Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Vd2),Ve2)
                 => ( Y != one_one_nat ) )
               => ( ( ? [V2: product_prod_nat_nat,Vh2: list_VEBT_VEBT,Vi2: vEBT_VEBT] : Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vh2),Vi2)
                   => ( Y != one_one_nat ) )
                 => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                        ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList),Summary) )
                       => ( Y != $ite(
                              aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Xaa),
                              one_one_nat,
                              $let(
                                l: nat,
                                l:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                $let(
                                  h: nat,
                                  h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                  $ite(
                                    aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList)),
                                    $let(
                                      minlow: option_nat,
                                      minlow:= vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,h)),
                                      $ite(
                                        ( ( minlow != none_nat )
                                        & aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_greater,aa_nat_option_nat(some_nat,l)),minlow) ),
                                        aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_p_r_e_d2,nth_VEBT_VEBT(TreeList,h)),l)),
                                        aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_p_r_e_d2,Summary),h)),one_one_nat) ) ),
                                    one_one_nat ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.elims
tff(fact_2927_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Oelims,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_s_u_c_c2,Xb),Xaa) = Y )
     => ( ( ? [Uu2: $o,B2: $o] : Xb = vEBT_Leaf((Uu2),(B2))
         => ( ( Xaa = zero_zero_nat )
           => ( Y != one_one_nat ) ) )
       => ( ( ? [Uv2: $o,Uw2: $o] : Xb = vEBT_Leaf((Uv2),(Uw2))
           => ( ? [N2: nat] : Xaa = aa_nat_nat(suc,N2)
             => ( Y != one_one_nat ) ) )
         => ( ( ? [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] : Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Ux2,Uy2),Uz2)
             => ( Y != one_one_nat ) )
           => ( ( ? [V2: product_prod_nat_nat,Vc: list_VEBT_VEBT,Vd2: vEBT_VEBT] : Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Vc),Vd2)
               => ( Y != one_one_nat ) )
             => ( ( ? [V2: product_prod_nat_nat,Vg2: list_VEBT_VEBT,Vh2: vEBT_VEBT] : Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vg2),Vh2)
                 => ( Y != one_one_nat ) )
               => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                      ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList),Summary) )
                     => ( Y != $ite(
                            aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi),
                            one_one_nat,
                            $let(
                              l: nat,
                              l:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                              $let(
                                h: nat,
                                h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                $ite(
                                  aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList)),
                                  $let(
                                    maxlow: option_nat,
                                    maxlow:= vEBT_vebt_maxt(nth_VEBT_VEBT(TreeList,h)),
                                    $ite(
                                      ( ( maxlow != none_nat )
                                      & aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_less,aa_nat_option_nat(some_nat,l)),maxlow) ),
                                      aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_s_u_c_c2,nth_VEBT_VEBT(TreeList,h)),l)),
                                      aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_s_u_c_c2,Summary),h)),one_one_nat) ) ),
                                  one_one_nat ) ) ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.elims
tff(fact_2928_vebt__maxt_Oelims,axiom,
    ! [Xb: vEBT_VEBT,Y: option_nat] :
      ( ( vEBT_vebt_maxt(Xb) = Y )
     => ( ! [A3: $o,B2: $o] :
            ( ( Xb = vEBT_Leaf((A3),(B2)) )
           => ( Y != $ite(
                  (B2),
                  aa_nat_option_nat(some_nat,one_one_nat),
                  $ite((A3),aa_nat_option_nat(some_nat,zero_zero_nat),none_nat) ) ) )
       => ( ( ? [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] : Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2),Uw2)
           => ( Y != none_nat ) )
         => ~ ! [Mi: nat,Ma: nat] :
                ( ? [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] : Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),Ux2,Uy2),Uz2)
               => ( Y != aa_nat_option_nat(some_nat,Ma) ) ) ) ) ) ).

% vebt_maxt.elims
tff(fact_2929_vebt__mint_Oelims,axiom,
    ! [Xb: vEBT_VEBT,Y: option_nat] :
      ( ( vEBT_vebt_mint(Xb) = Y )
     => ( ! [A3: $o,B2: $o] :
            ( ( Xb = vEBT_Leaf((A3),(B2)) )
           => ( Y != $ite(
                  (A3),
                  aa_nat_option_nat(some_nat,zero_zero_nat),
                  $ite((B2),aa_nat_option_nat(some_nat,one_one_nat),none_nat) ) ) )
       => ( ( ? [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] : Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2),Uw2)
           => ( Y != none_nat ) )
         => ~ ! [Mi: nat] :
                ( ? [Ma: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] : Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),Ux2,Uy2),Uz2)
               => ( Y != aa_nat_option_nat(some_nat,Mi) ) ) ) ) ) ).

% vebt_mint.elims
tff(fact_2930_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H_Oelims,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_i_n_s_e_r_t2,Xb),Xaa) = Y )
     => ( ( ? [A3: $o,B2: $o] : Xb = vEBT_Leaf((A3),(B2))
         => ( Y != one_one_nat ) )
       => ( ( ? [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S: vEBT_VEBT] : Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,zero_zero_nat,Ts2),S)
           => ( Y != one_one_nat ) )
         => ( ( ? [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S: vEBT_VEBT] : Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts2),S)
             => ( Y != one_one_nat ) )
           => ( ( ? [V2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] : Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V2)),TreeList),Summary)
               => ( Y != one_one_nat ) )
             => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                    ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList),Summary) )
                   => ( Y != $let(
                          xn: nat,
                          xn:= 
                            $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi),Mi,Xaa),
                          $let(
                            h: nat,
                            h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                            $ite(
                              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList))
                              & ~ ( ( Xaa = Mi )
                                  | ( Xaa = Ma ) ) ),
                              aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_i_n_s_e_r_t2,nth_VEBT_VEBT(TreeList,h)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),
                                $ite(aa_VEBT_VEBT_o(vEBT_VEBT_minNull,nth_VEBT_VEBT(TreeList,h)),aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_i_n_s_e_r_t2,Summary),h),one_one_nat)),
                              one_one_nat ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.elims
tff(fact_2931_linear__plus__1__le__power,axiom,
    ! [Xb: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),Xb)),one_one_real)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),one_one_real)),Nb)) ) ).

% linear_plus_1_le_power
tff(fact_2932_diff__add__zero,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)) = zero_zero_nat ).

% diff_add_zero
tff(fact_2933_diff__gt__0__iff__gt,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,B),A) ) ).

% diff_gt_0_iff_gt
tff(fact_2934_diff__gt__0__iff__gt,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_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_2935_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_2936_diff__ge__0__iff__ge,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A) ) ).

% diff_ge_0_iff_ge
tff(fact_2937_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),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_2938_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_2939_zero__less__double__add__iff__zero__less__single__add,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),A))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A) ) ).

% zero_less_double_add_iff_zero_less_single_add
tff(fact_2940_zero__less__double__add__iff__zero__less__single__add,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(plus_plus_rat(A),A))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A) ) ).

% zero_less_double_add_iff_zero_less_single_add
tff(fact_2941_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_2942_add__right__cancel,axiom,
    ! [B: real,A: real,C: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),A) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),A) )
    <=> ( B = C ) ) ).

% add_right_cancel
tff(fact_2943_add__right__cancel,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( aa_rat_rat(plus_plus_rat(B),A) = aa_rat_rat(plus_plus_rat(C),A) )
    <=> ( B = C ) ) ).

% add_right_cancel
tff(fact_2944_add__right__cancel,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A) )
    <=> ( B = C ) ) ).

% add_right_cancel
tff(fact_2945_add__right__cancel,axiom,
    ! [B: int,A: int,C: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,C),A) )
    <=> ( B = C ) ) ).

% add_right_cancel
tff(fact_2946_add__left__cancel,axiom,
    ! [A: real,B: real,C: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C) )
    <=> ( B = C ) ) ).

% add_left_cancel
tff(fact_2947_add__left__cancel,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( aa_rat_rat(plus_plus_rat(A),B) = aa_rat_rat(plus_plus_rat(A),C) )
    <=> ( B = C ) ) ).

% add_left_cancel
tff(fact_2948_add__left__cancel,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C) )
    <=> ( B = C ) ) ).

% add_left_cancel
tff(fact_2949_add__left__cancel,axiom,
    ! [A: int,B: int,C: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C) )
    <=> ( B = C ) ) ).

% add_left_cancel
tff(fact_2950_le__zero__eq,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),zero_zero_nat)
    <=> ( Nb = zero_zero_nat ) ) ).

% le_zero_eq
tff(fact_2951_not__gr__zero,axiom,
    ! [Nb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
    <=> ( Nb = zero_zero_nat ) ) ).

% not_gr_zero
tff(fact_2952_add__0,axiom,
    ! [A: uint32] : aa_uint32_uint32(plus_plus_uint32(zero_zero_uint32),A) = A ).

% add_0
tff(fact_2953_add__0,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,zero_zero_real),A) = A ).

% add_0
tff(fact_2954_add__0,axiom,
    ! [A: rat] : aa_rat_rat(plus_plus_rat(zero_zero_rat),A) = A ).

% add_0
tff(fact_2955_add__0,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,zero_zero_nat),A) = A ).

% add_0
tff(fact_2956_add__0,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,zero_zero_int),A) = A ).

% add_0
tff(fact_2957_zero__eq__add__iff__both__eq__0,axiom,
    ! [Xb: nat,Y: nat] :
      ( ( zero_zero_nat = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xb),Y) )
    <=> ( ( Xb = zero_zero_nat )
        & ( Y = zero_zero_nat ) ) ) ).

% zero_eq_add_iff_both_eq_0
tff(fact_2958_add__eq__0__iff__both__eq__0,axiom,
    ! [Xb: nat,Y: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xb),Y) = zero_zero_nat )
    <=> ( ( Xb = zero_zero_nat )
        & ( Y = zero_zero_nat ) ) ) ).

% add_eq_0_iff_both_eq_0
tff(fact_2959_add__cancel__right__right,axiom,
    ! [A: uint32,B: uint32] :
      ( ( A = aa_uint32_uint32(plus_plus_uint32(A),B) )
    <=> ( B = zero_zero_uint32 ) ) ).

% add_cancel_right_right
tff(fact_2960_add__cancel__right__right,axiom,
    ! [A: real,B: real] :
      ( ( A = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B) )
    <=> ( B = zero_zero_real ) ) ).

% add_cancel_right_right
tff(fact_2961_add__cancel__right__right,axiom,
    ! [A: rat,B: rat] :
      ( ( A = aa_rat_rat(plus_plus_rat(A),B) )
    <=> ( B = zero_zero_rat ) ) ).

% add_cancel_right_right
tff(fact_2962_add__cancel__right__right,axiom,
    ! [A: nat,B: nat] :
      ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B) )
    <=> ( B = zero_zero_nat ) ) ).

% add_cancel_right_right
tff(fact_2963_add__cancel__right__right,axiom,
    ! [A: int,B: int] :
      ( ( A = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B) )
    <=> ( B = zero_zero_int ) ) ).

% add_cancel_right_right
tff(fact_2964_add__cancel__right__left,axiom,
    ! [A: uint32,B: uint32] :
      ( ( A = aa_uint32_uint32(plus_plus_uint32(B),A) )
    <=> ( B = zero_zero_uint32 ) ) ).

% add_cancel_right_left
tff(fact_2965_add__cancel__right__left,axiom,
    ! [A: real,B: real] :
      ( ( A = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),A) )
    <=> ( B = zero_zero_real ) ) ).

% add_cancel_right_left
tff(fact_2966_add__cancel__right__left,axiom,
    ! [A: rat,B: rat] :
      ( ( A = aa_rat_rat(plus_plus_rat(B),A) )
    <=> ( B = zero_zero_rat ) ) ).

% add_cancel_right_left
tff(fact_2967_add__cancel__right__left,axiom,
    ! [A: nat,B: nat] :
      ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A) )
    <=> ( B = zero_zero_nat ) ) ).

% add_cancel_right_left
tff(fact_2968_add__cancel__right__left,axiom,
    ! [A: int,B: int] :
      ( ( A = aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A) )
    <=> ( B = zero_zero_int ) ) ).

% add_cancel_right_left
tff(fact_2969_add__cancel__left__right,axiom,
    ! [A: uint32,B: uint32] :
      ( ( aa_uint32_uint32(plus_plus_uint32(A),B) = A )
    <=> ( B = zero_zero_uint32 ) ) ).

% add_cancel_left_right
tff(fact_2970_add__cancel__left__right,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B) = A )
    <=> ( B = zero_zero_real ) ) ).

% add_cancel_left_right
tff(fact_2971_add__cancel__left__right,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(plus_plus_rat(A),B) = A )
    <=> ( B = zero_zero_rat ) ) ).

% add_cancel_left_right
tff(fact_2972_add__cancel__left__right,axiom,
    ! [A: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B) = A )
    <=> ( B = zero_zero_nat ) ) ).

% add_cancel_left_right
tff(fact_2973_add__cancel__left__right,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B) = A )
    <=> ( B = zero_zero_int ) ) ).

% add_cancel_left_right
tff(fact_2974_add__cancel__left__left,axiom,
    ! [B: uint32,A: uint32] :
      ( ( aa_uint32_uint32(plus_plus_uint32(B),A) = A )
    <=> ( B = zero_zero_uint32 ) ) ).

% add_cancel_left_left
tff(fact_2975_add__cancel__left__left,axiom,
    ! [B: real,A: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),A) = A )
    <=> ( B = zero_zero_real ) ) ).

% add_cancel_left_left
tff(fact_2976_add__cancel__left__left,axiom,
    ! [B: rat,A: rat] :
      ( ( aa_rat_rat(plus_plus_rat(B),A) = A )
    <=> ( B = zero_zero_rat ) ) ).

% add_cancel_left_left
tff(fact_2977_add__cancel__left__left,axiom,
    ! [B: nat,A: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A) = A )
    <=> ( B = zero_zero_nat ) ) ).

% add_cancel_left_left
tff(fact_2978_add__cancel__left__left,axiom,
    ! [B: int,A: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A) = A )
    <=> ( B = zero_zero_int ) ) ).

% add_cancel_left_left
tff(fact_2979_double__zero__sym,axiom,
    ! [A: real] :
      ( ( zero_zero_real = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),A) )
    <=> ( A = zero_zero_real ) ) ).

% double_zero_sym
tff(fact_2980_double__zero__sym,axiom,
    ! [A: rat] :
      ( ( zero_zero_rat = aa_rat_rat(plus_plus_rat(A),A) )
    <=> ( A = zero_zero_rat ) ) ).

% double_zero_sym
tff(fact_2981_double__zero__sym,axiom,
    ! [A: int] :
      ( ( zero_zero_int = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),A) )
    <=> ( A = zero_zero_int ) ) ).

% double_zero_sym
tff(fact_2982_add_Oright__neutral,axiom,
    ! [A: uint32] : aa_uint32_uint32(plus_plus_uint32(A),zero_zero_uint32) = A ).

% add.right_neutral
tff(fact_2983_add_Oright__neutral,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),zero_zero_real) = A ).

% add.right_neutral
tff(fact_2984_add_Oright__neutral,axiom,
    ! [A: rat] : aa_rat_rat(plus_plus_rat(A),zero_zero_rat) = A ).

% add.right_neutral
tff(fact_2985_add_Oright__neutral,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),zero_zero_nat) = A ).

% add.right_neutral
tff(fact_2986_add_Oright__neutral,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,A),zero_zero_int) = A ).

% add.right_neutral
tff(fact_2987_add__le__cancel__left,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),B))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ).

% add_le_cancel_left
tff(fact_2988_add__le__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(C),A)),aa_rat_rat(plus_plus_rat(C),B))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ).

% add_le_cancel_left
tff(fact_2989_add__le__cancel__left,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B) ) ).

% add_le_cancel_left
tff(fact_2990_add__le__cancel__left,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,C),A)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ).

% add_le_cancel_left
tff(fact_2991_add__le__cancel__right,axiom,
    ! [A: real,C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ).

% add_le_cancel_right
tff(fact_2992_add__le__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(A),C)),aa_rat_rat(plus_plus_rat(B),C))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ).

% add_le_cancel_right
tff(fact_2993_add__le__cancel__right,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B) ) ).

% add_le_cancel_right
tff(fact_2994_add__le__cancel__right,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ).

% add_le_cancel_right
tff(fact_2995_add__less__cancel__left,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),B))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) ) ).

% add_less_cancel_left
tff(fact_2996_add__less__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(C),A)),aa_rat_rat(plus_plus_rat(C),B))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ).

% add_less_cancel_left
tff(fact_2997_add__less__cancel__left,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B) ) ).

% add_less_cancel_left
tff(fact_2998_add__less__cancel__left,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,C),A)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ).

% add_less_cancel_left
tff(fact_2999_add__less__cancel__right,axiom,
    ! [A: real,C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) ) ).

% add_less_cancel_right
tff(fact_3000_add__less__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(A),C)),aa_rat_rat(plus_plus_rat(B),C))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ).

% add_less_cancel_right
tff(fact_3001_add__less__cancel__right,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B) ) ).

% add_less_cancel_right
tff(fact_3002_add__less__cancel__right,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ).

% add_less_cancel_right
tff(fact_3003_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A: uint32] : minus_minus_uint32(A,A) = zero_zero_uint32 ).

% cancel_comm_monoid_add_class.diff_cancel
tff(fact_3004_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),A) = zero_zero_real ).

% cancel_comm_monoid_add_class.diff_cancel
tff(fact_3005_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A: rat] : minus_minus_rat(A,A) = zero_zero_rat ).

% cancel_comm_monoid_add_class.diff_cancel
tff(fact_3006_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),A) = zero_zero_nat ).

% cancel_comm_monoid_add_class.diff_cancel
tff(fact_3007_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,A),A) = zero_zero_int ).

% cancel_comm_monoid_add_class.diff_cancel
tff(fact_3008_diff__zero,axiom,
    ! [A: uint32] : minus_minus_uint32(A,zero_zero_uint32) = A ).

% diff_zero
tff(fact_3009_diff__zero,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),zero_zero_real) = A ).

% diff_zero
tff(fact_3010_diff__zero,axiom,
    ! [A: rat] : minus_minus_rat(A,zero_zero_rat) = A ).

% diff_zero
tff(fact_3011_diff__zero,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),zero_zero_nat) = A ).

% diff_zero
tff(fact_3012_diff__zero,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,A),zero_zero_int) = A ).

% diff_zero
tff(fact_3013_zero__diff,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,zero_zero_nat),A) = zero_zero_nat ).

% zero_diff
tff(fact_3014_diff__0__right,axiom,
    ! [A: uint32] : minus_minus_uint32(A,zero_zero_uint32) = A ).

% diff_0_right
tff(fact_3015_diff__0__right,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),zero_zero_real) = A ).

% diff_0_right
tff(fact_3016_diff__0__right,axiom,
    ! [A: rat] : minus_minus_rat(A,zero_zero_rat) = A ).

% diff_0_right
tff(fact_3017_diff__0__right,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,A),zero_zero_int) = A ).

% diff_0_right
tff(fact_3018_diff__self,axiom,
    ! [A: uint32] : minus_minus_uint32(A,A) = zero_zero_uint32 ).

% diff_self
tff(fact_3019_diff__self,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),A) = zero_zero_real ).

% diff_self
tff(fact_3020_diff__self,axiom,
    ! [A: rat] : minus_minus_rat(A,A) = zero_zero_rat ).

% diff_self
tff(fact_3021_diff__self,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,A),A) = zero_zero_int ).

% diff_self
tff(fact_3022_mult_Oright__neutral,axiom,
    ! [A: word_N3645301735248828278l_num1] : aa_wor7274019993978516266l_num1(times_7065122842183080059l_num1(A),one_on7727431528512463931l_num1) = A ).

% mult.right_neutral
tff(fact_3023_mult_Oright__neutral,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),one_one_real) = A ).

% mult.right_neutral
tff(fact_3024_mult_Oright__neutral,axiom,
    ! [A: rat] : aa_rat_rat(times_times_rat(A),one_one_rat) = A ).

% mult.right_neutral
tff(fact_3025_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_3026_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_3027_mult__1,axiom,
    ! [A: word_N3645301735248828278l_num1] : aa_wor7274019993978516266l_num1(times_7065122842183080059l_num1(one_on7727431528512463931l_num1),A) = A ).

% mult_1
tff(fact_3028_mult__1,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,one_one_real),A) = A ).

% mult_1
tff(fact_3029_mult__1,axiom,
    ! [A: rat] : aa_rat_rat(times_times_rat(one_one_rat),A) = A ).

% mult_1
tff(fact_3030_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_3031_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_3032_add__diff__cancel__right_H,axiom,
    ! [A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),B) = A ).

% add_diff_cancel_right'
tff(fact_3033_add__diff__cancel__right_H,axiom,
    ! [A: rat,B: rat] : minus_minus_rat(aa_rat_rat(plus_plus_rat(A),B),B) = A ).

% add_diff_cancel_right'
tff(fact_3034_add__diff__cancel__right_H,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),B) = A ).

% add_diff_cancel_right'
tff(fact_3035_add__diff__cancel__right_H,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),B) = A ).

% add_diff_cancel_right'
tff(fact_3036_add__diff__cancel__right,axiom,
    ! [A: real,C: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B) ).

% add_diff_cancel_right
tff(fact_3037_add__diff__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] : minus_minus_rat(aa_rat_rat(plus_plus_rat(A),C),aa_rat_rat(plus_plus_rat(B),C)) = minus_minus_rat(A,B) ).

% add_diff_cancel_right
tff(fact_3038_add__diff__cancel__right,axiom,
    ! [A: nat,C: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B) ).

% add_diff_cancel_right
tff(fact_3039_add__diff__cancel__right,axiom,
    ! [A: int,C: int,B: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B) ).

% add_diff_cancel_right
tff(fact_3040_add__diff__cancel__left_H,axiom,
    ! [A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),A) = B ).

% add_diff_cancel_left'
tff(fact_3041_add__diff__cancel__left_H,axiom,
    ! [A: rat,B: rat] : minus_minus_rat(aa_rat_rat(plus_plus_rat(A),B),A) = B ).

% add_diff_cancel_left'
tff(fact_3042_add__diff__cancel__left_H,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),A) = B ).

% add_diff_cancel_left'
tff(fact_3043_add__diff__cancel__left_H,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),A) = B ).

% add_diff_cancel_left'
tff(fact_3044_add__diff__cancel__left,axiom,
    ! [C: real,A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),B)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B) ).

% add_diff_cancel_left
tff(fact_3045_add__diff__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] : minus_minus_rat(aa_rat_rat(plus_plus_rat(C),A),aa_rat_rat(plus_plus_rat(C),B)) = minus_minus_rat(A,B) ).

% add_diff_cancel_left
tff(fact_3046_add__diff__cancel__left,axiom,
    ! [C: nat,A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B) ).

% add_diff_cancel_left
tff(fact_3047_add__diff__cancel__left,axiom,
    ! [C: int,A: int,B: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,C),A)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B) ).

% add_diff_cancel_left
tff(fact_3048_diff__add__cancel,axiom,
    ! [A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)),B) = A ).

% diff_add_cancel
tff(fact_3049_diff__add__cancel,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(plus_plus_rat(minus_minus_rat(A,B)),B) = A ).

% diff_add_cancel
tff(fact_3050_diff__add__cancel,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),B) = A ).

% diff_add_cancel
tff(fact_3051_add__diff__cancel,axiom,
    ! [A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),B) = A ).

% add_diff_cancel
tff(fact_3052_add__diff__cancel,axiom,
    ! [A: rat,B: rat] : minus_minus_rat(aa_rat_rat(plus_plus_rat(A),B),B) = A ).

% add_diff_cancel
tff(fact_3053_add__diff__cancel,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),B) = A ).

% add_diff_cancel
tff(fact_3054_add__le__same__cancel1,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),A)),B)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real) ) ).

% add_le_same_cancel1
tff(fact_3055_add__le__same__cancel1,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(B),A)),B)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat) ) ).

% add_le_same_cancel1
tff(fact_3056_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_3057_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_3058_add__le__same__cancel2,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),B)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real) ) ).

% add_le_same_cancel2
tff(fact_3059_add__le__same__cancel2,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(A),B)),B)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat) ) ).

% add_le_same_cancel2
tff(fact_3060_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_3061_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_3062_le__add__same__cancel1,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B) ) ).

% le_add_same_cancel1
tff(fact_3063_le__add__same__cancel1,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(plus_plus_rat(A),B))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B) ) ).

% le_add_same_cancel1
tff(fact_3064_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_3065_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_3066_le__add__same__cancel2,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),A))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B) ) ).

% le_add_same_cancel2
tff(fact_3067_le__add__same__cancel2,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(plus_plus_rat(B),A))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B) ) ).

% le_add_same_cancel2
tff(fact_3068_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_3069_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_3070_double__add__le__zero__iff__single__add__le__zero,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),A)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real) ) ).

% double_add_le_zero_iff_single_add_le_zero
tff(fact_3071_double__add__le__zero__iff__single__add__le__zero,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(A),A)),zero_zero_rat)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat) ) ).

% double_add_le_zero_iff_single_add_le_zero
tff(fact_3072_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_3073_zero__le__double__add__iff__zero__le__single__add,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),A))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A) ) ).

% zero_le_double_add_iff_zero_le_single_add
tff(fact_3074_zero__le__double__add__iff__zero__le__single__add,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(plus_plus_rat(A),A))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A) ) ).

% zero_le_double_add_iff_zero_le_single_add
tff(fact_3075_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_3076_add__less__same__cancel1,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),A)),B)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real) ) ).

% add_less_same_cancel1
tff(fact_3077_add__less__same__cancel1,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(B),A)),B)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat) ) ).

% add_less_same_cancel1
tff(fact_3078_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_3079_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_3080_add__less__same__cancel2,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),B)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real) ) ).

% add_less_same_cancel2
tff(fact_3081_add__less__same__cancel2,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(A),B)),B)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat) ) ).

% add_less_same_cancel2
tff(fact_3082_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_3083_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_3084_less__add__same__cancel1,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B) ) ).

% less_add_same_cancel1
tff(fact_3085_less__add__same__cancel1,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(plus_plus_rat(A),B))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B) ) ).

% less_add_same_cancel1
tff(fact_3086_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_3087_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_3088_less__add__same__cancel2,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),A))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B) ) ).

% less_add_same_cancel2
tff(fact_3089_less__add__same__cancel2,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(plus_plus_rat(B),A))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B) ) ).

% less_add_same_cancel2
tff(fact_3090_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_3091_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_3092_double__add__less__zero__iff__single__add__less__zero,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),A)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real) ) ).

% double_add_less_zero_iff_single_add_less_zero
tff(fact_3093_double__add__less__zero__iff__single__add__less__zero,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(A),A)),zero_zero_rat)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat) ) ).

% double_add_less_zero_iff_single_add_less_zero
tff(fact_3094_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_3095_add__right__imp__eq,axiom,
    ! [B: real,A: real,C: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),A) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),A) )
     => ( B = C ) ) ).

% add_right_imp_eq
tff(fact_3096_add__right__imp__eq,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( aa_rat_rat(plus_plus_rat(B),A) = aa_rat_rat(plus_plus_rat(C),A) )
     => ( B = C ) ) ).

% add_right_imp_eq
tff(fact_3097_add__right__imp__eq,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A) )
     => ( B = C ) ) ).

% add_right_imp_eq
tff(fact_3098_add__right__imp__eq,axiom,
    ! [B: int,A: int,C: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,C),A) )
     => ( B = C ) ) ).

% add_right_imp_eq
tff(fact_3099_add__left__imp__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C) )
     => ( B = C ) ) ).

% add_left_imp_eq
tff(fact_3100_add__left__imp__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( aa_rat_rat(plus_plus_rat(A),B) = aa_rat_rat(plus_plus_rat(A),C) )
     => ( B = C ) ) ).

% add_left_imp_eq
tff(fact_3101_add__left__imp__eq,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C) )
     => ( B = C ) ) ).

% add_left_imp_eq
tff(fact_3102_add__left__imp__eq,axiom,
    ! [A: int,B: int,C: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C) )
     => ( B = C ) ) ).

% add_left_imp_eq
tff(fact_3103_ab__semigroup__add__class_Oadd_Oleft__commute,axiom,
    ! [B: real,A: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C)) ).

% ab_semigroup_add_class.add.left_commute
tff(fact_3104_ab__semigroup__add__class_Oadd_Oleft__commute,axiom,
    ! [B: rat,A: rat,C: rat] : aa_rat_rat(plus_plus_rat(B),aa_rat_rat(plus_plus_rat(A),C)) = aa_rat_rat(plus_plus_rat(A),aa_rat_rat(plus_plus_rat(B),C)) ).

% ab_semigroup_add_class.add.left_commute
tff(fact_3105_ab__semigroup__add__class_Oadd_Oleft__commute,axiom,
    ! [B: nat,A: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)) ).

% ab_semigroup_add_class.add.left_commute
tff(fact_3106_ab__semigroup__add__class_Oadd_Oleft__commute,axiom,
    ! [B: int,A: int,C: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,B),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C)) ).

% ab_semigroup_add_class.add.left_commute
tff(fact_3107_ab__semigroup__add__class_Oadd_Ocommute,axiom,
    ! [A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),A) ).

% ab_semigroup_add_class.add.commute
tff(fact_3108_ab__semigroup__add__class_Oadd_Ocommute,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(plus_plus_rat(A),B) = aa_rat_rat(plus_plus_rat(B),A) ).

% ab_semigroup_add_class.add.commute
tff(fact_3109_ab__semigroup__add__class_Oadd_Ocommute,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A) ).

% ab_semigroup_add_class.add.commute
tff(fact_3110_ab__semigroup__add__class_Oadd_Ocommute,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A) ).

% ab_semigroup_add_class.add.commute
tff(fact_3111_add_Oright__cancel,axiom,
    ! [B: real,A: real,C: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),A) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),A) )
    <=> ( B = C ) ) ).

% add.right_cancel
tff(fact_3112_add_Oright__cancel,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( aa_rat_rat(plus_plus_rat(B),A) = aa_rat_rat(plus_plus_rat(C),A) )
    <=> ( B = C ) ) ).

% add.right_cancel
tff(fact_3113_add_Oright__cancel,axiom,
    ! [B: int,A: int,C: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,C),A) )
    <=> ( B = C ) ) ).

% add.right_cancel
tff(fact_3114_add_Oleft__cancel,axiom,
    ! [A: real,B: real,C: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C) )
    <=> ( B = C ) ) ).

% add.left_cancel
tff(fact_3115_add_Oleft__cancel,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( aa_rat_rat(plus_plus_rat(A),B) = aa_rat_rat(plus_plus_rat(A),C) )
    <=> ( B = C ) ) ).

% add.left_cancel
tff(fact_3116_add_Oleft__cancel,axiom,
    ! [A: int,B: int,C: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C) )
    <=> ( B = C ) ) ).

% add.left_cancel
tff(fact_3117_add_Oassoc,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),C) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C)) ).

% add.assoc
tff(fact_3118_add_Oassoc,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(plus_plus_rat(aa_rat_rat(plus_plus_rat(A),B)),C) = aa_rat_rat(plus_plus_rat(A),aa_rat_rat(plus_plus_rat(B),C)) ).

% add.assoc
tff(fact_3119_add_Oassoc,axiom,
    ! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)) ).

% add.assoc
tff(fact_3120_add_Oassoc,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C)) ).

% add.assoc
tff(fact_3121_group__cancel_Oadd2,axiom,
    ! [B4: real,K: real,B: real,A: real] :
      ( ( B4 = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,K),B) )
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B4) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,K),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)) ) ) ).

% group_cancel.add2
tff(fact_3122_group__cancel_Oadd2,axiom,
    ! [B4: rat,K: rat,B: rat,A: rat] :
      ( ( B4 = aa_rat_rat(plus_plus_rat(K),B) )
     => ( aa_rat_rat(plus_plus_rat(A),B4) = aa_rat_rat(plus_plus_rat(K),aa_rat_rat(plus_plus_rat(A),B)) ) ) ).

% group_cancel.add2
tff(fact_3123_group__cancel_Oadd2,axiom,
    ! [B4: nat,K: nat,B: nat,A: nat] :
      ( ( B4 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),B) )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B4) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)) ) ) ).

% group_cancel.add2
tff(fact_3124_group__cancel_Oadd2,axiom,
    ! [B4: int,K: int,B: int,A: int] :
      ( ( B4 = aa_int_int(aa_int_fun_int_int(plus_plus_int,K),B) )
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B4) = aa_int_int(aa_int_fun_int_int(plus_plus_int,K),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)) ) ) ).

% group_cancel.add2
tff(fact_3125_group__cancel_Oadd1,axiom,
    ! [A2: real,K: real,A: real,B: real] :
      ( ( A2 = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,K),A) )
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),B) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,K),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)) ) ) ).

% group_cancel.add1
tff(fact_3126_group__cancel_Oadd1,axiom,
    ! [A2: rat,K: rat,A: rat,B: rat] :
      ( ( A2 = aa_rat_rat(plus_plus_rat(K),A) )
     => ( aa_rat_rat(plus_plus_rat(A2),B) = aa_rat_rat(plus_plus_rat(K),aa_rat_rat(plus_plus_rat(A),B)) ) ) ).

% group_cancel.add1
tff(fact_3127_group__cancel_Oadd1,axiom,
    ! [A2: nat,K: nat,A: nat,B: nat] :
      ( ( A2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),A) )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)) ) ) ).

% group_cancel.add1
tff(fact_3128_group__cancel_Oadd1,axiom,
    ! [A2: int,K: int,A: int,B: int] :
      ( ( A2 = aa_int_int(aa_int_fun_int_int(plus_plus_int,K),A) )
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,K),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)) ) ) ).

% group_cancel.add1
tff(fact_3129_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I: real,J: real,K: real,L: real] :
      ( ( ( I = J )
        & ( K = L ) )
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J),L) ) ) ).

% add_mono_thms_linordered_semiring(4)
tff(fact_3130_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( ( I = J )
        & ( K = L ) )
     => ( aa_rat_rat(plus_plus_rat(I),K) = aa_rat_rat(plus_plus_rat(J),L) ) ) ).

% add_mono_thms_linordered_semiring(4)
tff(fact_3131_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ( I = J )
        & ( K = L ) )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L) ) ) ).

% add_mono_thms_linordered_semiring(4)
tff(fact_3132_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( ( I = J )
        & ( K = L ) )
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K) = aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L) ) ) ).

% add_mono_thms_linordered_semiring(4)
tff(fact_3133_one__reorient,axiom,
    ! [Xb: word_N3645301735248828278l_num1] :
      ( ( one_on7727431528512463931l_num1 = Xb )
    <=> ( Xb = one_on7727431528512463931l_num1 ) ) ).

% one_reorient
tff(fact_3134_one__reorient,axiom,
    ! [Xb: real] :
      ( ( one_one_real = Xb )
    <=> ( Xb = one_one_real ) ) ).

% one_reorient
tff(fact_3135_one__reorient,axiom,
    ! [Xb: rat] :
      ( ( one_one_rat = Xb )
    <=> ( Xb = one_one_rat ) ) ).

% one_reorient
tff(fact_3136_one__reorient,axiom,
    ! [Xb: nat] :
      ( ( one_one_nat = Xb )
    <=> ( Xb = one_one_nat ) ) ).

% one_reorient
tff(fact_3137_one__reorient,axiom,
    ! [Xb: int] :
      ( ( one_one_int = Xb )
    <=> ( Xb = one_one_int ) ) ).

% one_reorient
tff(fact_3138_mult_Oassoc,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),C) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)) ).

% mult.assoc
tff(fact_3139_mult_Oassoc,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(A),B)),C) = aa_rat_rat(times_times_rat(A),aa_rat_rat(times_times_rat(B),C)) ).

% mult.assoc
tff(fact_3140_mult_Oassoc,axiom,
    ! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) ).

% mult.assoc
tff(fact_3141_mult_Oassoc,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ).

% mult.assoc
tff(fact_3142_ab__semigroup__mult__class_Omult_Ocommute,axiom,
    ! [A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),A) ).

% ab_semigroup_mult_class.mult.commute
tff(fact_3143_ab__semigroup__mult__class_Omult_Ocommute,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(times_times_rat(A),B) = aa_rat_rat(times_times_rat(B),A) ).

% ab_semigroup_mult_class.mult.commute
tff(fact_3144_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_3145_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_3146_ab__semigroup__mult__class_Omult_Oleft__commute,axiom,
    ! [B: real,A: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)) ).

% ab_semigroup_mult_class.mult.left_commute
tff(fact_3147_ab__semigroup__mult__class_Omult_Oleft__commute,axiom,
    ! [B: rat,A: rat,C: rat] : aa_rat_rat(times_times_rat(B),aa_rat_rat(times_times_rat(A),C)) = aa_rat_rat(times_times_rat(A),aa_rat_rat(times_times_rat(B),C)) ).

% ab_semigroup_mult_class.mult.left_commute
tff(fact_3148_ab__semigroup__mult__class_Omult_Oleft__commute,axiom,
    ! [B: nat,A: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) ).

% ab_semigroup_mult_class.mult.left_commute
tff(fact_3149_ab__semigroup__mult__class_Omult_Oleft__commute,axiom,
    ! [B: int,A: int,C: int] : aa_int_int(aa_int_fun_int_int(times_times_int,B),aa_int_int(aa_int_fun_int_int(times_times_int,A),C)) = aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ).

% ab_semigroup_mult_class.mult.left_commute
tff(fact_3150_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
    ! [A: real,C: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),C)),B) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)),C) ).

% cancel_ab_semigroup_add_class.diff_right_commute
tff(fact_3151_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
    ! [A: rat,C: rat,B: rat] : minus_minus_rat(minus_minus_rat(A,C),B) = minus_minus_rat(minus_minus_rat(A,B),C) ).

% cancel_ab_semigroup_add_class.diff_right_commute
tff(fact_3152_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
    ! [A: nat,C: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),C)),B) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B)),C) ).

% cancel_ab_semigroup_add_class.diff_right_commute
tff(fact_3153_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
    ! [A: int,C: int,B: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),C)),B) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),C) ).

% cancel_ab_semigroup_add_class.diff_right_commute
tff(fact_3154_diff__eq__diff__eq,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C),D) )
     => ( ( A = B )
      <=> ( C = D ) ) ) ).

% diff_eq_diff_eq
tff(fact_3155_diff__eq__diff__eq,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( minus_minus_rat(A,B) = minus_minus_rat(C,D) )
     => ( ( A = B )
      <=> ( C = D ) ) ) ).

% diff_eq_diff_eq
tff(fact_3156_diff__eq__diff__eq,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B) = aa_int_int(aa_int_fun_int_int(minus_minus_int,C),D) )
     => ( ( A = B )
      <=> ( C = D ) ) ) ).

% diff_eq_diff_eq
tff(fact_3157_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H_Osimps_I1_J,axiom,
    ! [A: $o,B: $o,Xb: nat] : aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_i_n_s_e_r_t2,vEBT_Leaf((A),(B))),Xb) = one_one_nat ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.simps(1)
tff(fact_3158_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H_Osimps_I2_J,axiom,
    ! [Info: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S3: vEBT_VEBT,Xb: nat] : aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_i_n_s_e_r_t2,aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info,zero_zero_nat,Ts),S3)),Xb) = one_one_nat ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.simps(2)
tff(fact_3159_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H_Osimps_I3_J,axiom,
    ! [Info: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S3: vEBT_VEBT,Xb: nat] : aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_i_n_s_e_r_t2,aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info,aa_nat_nat(suc,zero_zero_nat),Ts),S3)),Xb) = one_one_nat ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.simps(3)
tff(fact_3160_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H_Osimps_I4_J,axiom,
    ! [V: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Xb: nat] : aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_i_n_s_e_r_t2,aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V)),TreeLista),Summarya)),Xb) = one_one_nat ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.simps(4)
tff(fact_3161_insersimp_H,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Y: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Ta),Nb)
     => ( ~ ? [X_13: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,Ta),X_13)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_i_n_s_e_r_t2,Ta),Y)),one_one_nat) ) ) ).

% insersimp'
tff(fact_3162_insertsimp_H,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,L: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Ta),Nb)
     => ( aa_VEBT_VEBT_o(vEBT_VEBT_minNull,Ta)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_i_n_s_e_r_t2,Ta),L)),one_one_nat) ) ) ).

% insertsimp'
tff(fact_3163_zero__le,axiom,
    ! [Xb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Xb) ).

% zero_le
tff(fact_3164_gr__zeroI,axiom,
    ! [Nb: nat] :
      ( ( Nb != zero_zero_nat )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ).

% gr_zeroI
tff(fact_3165_not__less__zero,axiom,
    ! [Nb: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),zero_zero_nat) ).

% not_less_zero
tff(fact_3166_gr__implies__not__zero,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb)
     => ( Nb != zero_zero_nat ) ) ).

% gr_implies_not_zero
tff(fact_3167_zero__less__iff__neq__zero,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
    <=> ( Nb != zero_zero_nat ) ) ).

% zero_less_iff_neq_zero
tff(fact_3168_add_Ogroup__left__neutral,axiom,
    ! [A: uint32] : aa_uint32_uint32(plus_plus_uint32(zero_zero_uint32),A) = A ).

% add.group_left_neutral
tff(fact_3169_add_Ogroup__left__neutral,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,zero_zero_real),A) = A ).

% add.group_left_neutral
tff(fact_3170_add_Ogroup__left__neutral,axiom,
    ! [A: rat] : aa_rat_rat(plus_plus_rat(zero_zero_rat),A) = A ).

% add.group_left_neutral
tff(fact_3171_add_Ogroup__left__neutral,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,zero_zero_int),A) = A ).

% add.group_left_neutral
tff(fact_3172_add_Ocomm__neutral,axiom,
    ! [A: uint32] : aa_uint32_uint32(plus_plus_uint32(A),zero_zero_uint32) = A ).

% add.comm_neutral
tff(fact_3173_add_Ocomm__neutral,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),zero_zero_real) = A ).

% add.comm_neutral
tff(fact_3174_add_Ocomm__neutral,axiom,
    ! [A: rat] : aa_rat_rat(plus_plus_rat(A),zero_zero_rat) = A ).

% add.comm_neutral
tff(fact_3175_add_Ocomm__neutral,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),zero_zero_nat) = A ).

% add.comm_neutral
tff(fact_3176_add_Ocomm__neutral,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,A),zero_zero_int) = A ).

% add.comm_neutral
tff(fact_3177_comm__monoid__add__class_Oadd__0,axiom,
    ! [A: uint32] : aa_uint32_uint32(plus_plus_uint32(zero_zero_uint32),A) = A ).

% comm_monoid_add_class.add_0
tff(fact_3178_comm__monoid__add__class_Oadd__0,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,zero_zero_real),A) = A ).

% comm_monoid_add_class.add_0
tff(fact_3179_comm__monoid__add__class_Oadd__0,axiom,
    ! [A: rat] : aa_rat_rat(plus_plus_rat(zero_zero_rat),A) = A ).

% comm_monoid_add_class.add_0
tff(fact_3180_comm__monoid__add__class_Oadd__0,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,zero_zero_nat),A) = A ).

% comm_monoid_add_class.add_0
tff(fact_3181_comm__monoid__add__class_Oadd__0,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,zero_zero_int),A) = A ).

% comm_monoid_add_class.add_0
tff(fact_3182_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I: real,J: real,K: real,L: real] :
      ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,I),J)
        & ( K = L ) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J),L)) ) ).

% add_mono_thms_linordered_semiring(3)
tff(fact_3183_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,I),J)
        & ( K = L ) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(I),K)),aa_rat_rat(plus_plus_rat(J),L)) ) ).

% add_mono_thms_linordered_semiring(3)
tff(fact_3184_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
        & ( K = L ) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L)) ) ).

% add_mono_thms_linordered_semiring(3)
tff(fact_3185_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),J)
        & ( K = L ) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L)) ) ).

% add_mono_thms_linordered_semiring(3)
tff(fact_3186_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I: real,J: real,K: real,L: real] :
      ( ( ( I = J )
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,K),L) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J),L)) ) ).

% add_mono_thms_linordered_semiring(2)
tff(fact_3187_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( ( I = J )
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,K),L) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(I),K)),aa_rat_rat(plus_plus_rat(J),L)) ) ).

% add_mono_thms_linordered_semiring(2)
tff(fact_3188_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ( I = J )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),L) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L)) ) ).

% add_mono_thms_linordered_semiring(2)
tff(fact_3189_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( ( I = J )
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),L) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L)) ) ).

% add_mono_thms_linordered_semiring(2)
tff(fact_3190_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I: real,J: real,K: real,L: real] :
      ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,I),J)
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,K),L) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J),L)) ) ).

% add_mono_thms_linordered_semiring(1)
tff(fact_3191_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,I),J)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,K),L) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(I),K)),aa_rat_rat(plus_plus_rat(J),L)) ) ).

% add_mono_thms_linordered_semiring(1)
tff(fact_3192_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),L) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L)) ) ).

% add_mono_thms_linordered_semiring(1)
tff(fact_3193_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),J)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),L) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L)) ) ).

% add_mono_thms_linordered_semiring(1)
tff(fact_3194_add__mono,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),D)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),D)) ) ) ).

% add_mono
tff(fact_3195_add__mono,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),D)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(A),C)),aa_rat_rat(plus_plus_rat(B),D)) ) ) ).

% add_mono
tff(fact_3196_add__mono,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),D)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),D)) ) ) ).

% add_mono
tff(fact_3197_add__mono,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),D)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),D)) ) ) ).

% add_mono
tff(fact_3198_add__left__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),B)) ) ).

% add_left_mono
tff(fact_3199_add__left__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(C),A)),aa_rat_rat(plus_plus_rat(C),B)) ) ).

% add_left_mono
tff(fact_3200_add__left__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B)) ) ).

% add_left_mono
tff(fact_3201_add__left__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,C),A)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B)) ) ).

% add_left_mono
tff(fact_3202_less__eqE,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ~ ! [C2: nat] : B != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C2) ) ).

% less_eqE
tff(fact_3203_add__right__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C)) ) ).

% add_right_mono
tff(fact_3204_add__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(A),C)),aa_rat_rat(plus_plus_rat(B),C)) ) ).

% add_right_mono
tff(fact_3205_add__right__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)) ) ).

% add_right_mono
tff(fact_3206_add__right__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C)) ) ).

% add_right_mono
tff(fact_3207_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_3208_add__le__imp__le__left,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),B))
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ).

% add_le_imp_le_left
tff(fact_3209_add__le__imp__le__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(C),A)),aa_rat_rat(plus_plus_rat(C),B))
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ).

% add_le_imp_le_left
tff(fact_3210_add__le__imp__le__left,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B) ) ).

% add_le_imp_le_left
tff(fact_3211_add__le__imp__le__left,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,C),A)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B))
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ).

% add_le_imp_le_left
tff(fact_3212_add__le__imp__le__right,axiom,
    ! [A: real,C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C))
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ).

% add_le_imp_le_right
tff(fact_3213_add__le__imp__le__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(A),C)),aa_rat_rat(plus_plus_rat(B),C))
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ).

% add_le_imp_le_right
tff(fact_3214_add__le__imp__le__right,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B) ) ).

% add_le_imp_le_right
tff(fact_3215_add__le__imp__le__right,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C))
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ).

% add_le_imp_le_right
tff(fact_3216_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I: real,J: real,K: real,L: real] :
      ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,I),J)
        & aa_real_o(aa_real_fun_real_o(ord_less_real,K),L) )
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J),L)) ) ).

% add_mono_thms_linordered_field(5)
tff(fact_3217_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,I),J)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,K),L) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(I),K)),aa_rat_rat(plus_plus_rat(J),L)) ) ).

% add_mono_thms_linordered_field(5)
tff(fact_3218_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),L) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L)) ) ).

% add_mono_thms_linordered_field(5)
tff(fact_3219_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,I),J)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,K),L) )
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L)) ) ).

% add_mono_thms_linordered_field(5)
tff(fact_3220_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I: real,J: real,K: real,L: real] :
      ( ( ( I = J )
        & aa_real_o(aa_real_fun_real_o(ord_less_real,K),L) )
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J),L)) ) ).

% add_mono_thms_linordered_field(2)
tff(fact_3221_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( ( I = J )
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,K),L) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(I),K)),aa_rat_rat(plus_plus_rat(J),L)) ) ).

% add_mono_thms_linordered_field(2)
tff(fact_3222_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ( I = J )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),L) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L)) ) ).

% add_mono_thms_linordered_field(2)
tff(fact_3223_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( ( I = J )
        & aa_int_o(aa_int_fun_int_o(ord_less_int,K),L) )
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L)) ) ).

% add_mono_thms_linordered_field(2)
tff(fact_3224_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I: real,J: real,K: real,L: real] :
      ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,I),J)
        & ( K = L ) )
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J),L)) ) ).

% add_mono_thms_linordered_field(1)
tff(fact_3225_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,I),J)
        & ( K = L ) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(I),K)),aa_rat_rat(plus_plus_rat(J),L)) ) ).

% add_mono_thms_linordered_field(1)
tff(fact_3226_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
        & ( K = L ) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L)) ) ).

% add_mono_thms_linordered_field(1)
tff(fact_3227_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,I),J)
        & ( K = L ) )
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L)) ) ).

% add_mono_thms_linordered_field(1)
tff(fact_3228_add__strict__mono,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),D)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),D)) ) ) ).

% add_strict_mono
tff(fact_3229_add__strict__mono,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),D)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(A),C)),aa_rat_rat(plus_plus_rat(B),D)) ) ) ).

% add_strict_mono
tff(fact_3230_add__strict__mono,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),D)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),D)) ) ) ).

% add_strict_mono
tff(fact_3231_add__strict__mono,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),D)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),D)) ) ) ).

% add_strict_mono
tff(fact_3232_add__strict__left__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),B)) ) ).

% add_strict_left_mono
tff(fact_3233_add__strict__left__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(C),A)),aa_rat_rat(plus_plus_rat(C),B)) ) ).

% add_strict_left_mono
tff(fact_3234_add__strict__left__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B)) ) ).

% add_strict_left_mono
tff(fact_3235_add__strict__left__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,C),A)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B)) ) ).

% add_strict_left_mono
tff(fact_3236_add__strict__right__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C)) ) ).

% add_strict_right_mono
tff(fact_3237_add__strict__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(A),C)),aa_rat_rat(plus_plus_rat(B),C)) ) ).

% add_strict_right_mono
tff(fact_3238_add__strict__right__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)) ) ).

% add_strict_right_mono
tff(fact_3239_add__strict__right__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C)) ) ).

% add_strict_right_mono
tff(fact_3240_add__less__imp__less__left,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),B))
     => aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) ) ).

% add_less_imp_less_left
tff(fact_3241_add__less__imp__less__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(C),A)),aa_rat_rat(plus_plus_rat(C),B))
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ).

% add_less_imp_less_left
tff(fact_3242_add__less__imp__less__left,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B) ) ).

% add_less_imp_less_left
tff(fact_3243_add__less__imp__less__left,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,C),A)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B))
     => aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ).

% add_less_imp_less_left
tff(fact_3244_add__less__imp__less__right,axiom,
    ! [A: real,C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C))
     => aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) ) ).

% add_less_imp_less_right
tff(fact_3245_add__less__imp__less__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(A),C)),aa_rat_rat(plus_plus_rat(B),C))
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ).

% add_less_imp_less_right
tff(fact_3246_add__less__imp__less__right,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B) ) ).

% add_less_imp_less_right
tff(fact_3247_add__less__imp__less__right,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C))
     => aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ).

% add_less_imp_less_right
tff(fact_3248_eq__iff__diff__eq__0,axiom,
    ! [A: uint32,B: uint32] :
      ( ( A = B )
    <=> ( minus_minus_uint32(A,B) = zero_zero_uint32 ) ) ).

% eq_iff_diff_eq_0
tff(fact_3249_eq__iff__diff__eq__0,axiom,
    ! [A: real,B: real] :
      ( ( A = B )
    <=> ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B) = zero_zero_real ) ) ).

% eq_iff_diff_eq_0
tff(fact_3250_eq__iff__diff__eq__0,axiom,
    ! [A: rat,B: rat] :
      ( ( A = B )
    <=> ( minus_minus_rat(A,B) = zero_zero_rat ) ) ).

% eq_iff_diff_eq_0
tff(fact_3251_eq__iff__diff__eq__0,axiom,
    ! [A: int,B: int] :
      ( ( A = B )
    <=> ( aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B) = zero_zero_int ) ) ).

% eq_iff_diff_eq_0
tff(fact_3252_diff__mono,axiom,
    ! [A: real,B: real,D: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,D),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),D)) ) ) ).

% diff_mono
tff(fact_3253_diff__mono,axiom,
    ! [A: rat,B: rat,D: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,D),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,minus_minus_rat(A,C)),minus_minus_rat(B,D)) ) ) ).

% diff_mono
tff(fact_3254_diff__mono,axiom,
    ! [A: int,B: int,D: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,D),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),C)),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),D)) ) ) ).

% diff_mono
tff(fact_3255_diff__left__mono,axiom,
    ! [B: real,A: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C),B)) ) ).

% diff_left_mono
tff(fact_3256_diff__left__mono,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,minus_minus_rat(C,A)),minus_minus_rat(C,B)) ) ).

% diff_left_mono
tff(fact_3257_diff__left__mono,axiom,
    ! [B: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,C),A)),aa_int_int(aa_int_fun_int_int(minus_minus_int,C),B)) ) ).

% diff_left_mono
tff(fact_3258_diff__right__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),C)) ) ).

% diff_right_mono
tff(fact_3259_diff__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,minus_minus_rat(A,C)),minus_minus_rat(B,C)) ) ).

% diff_right_mono
tff(fact_3260_diff__right__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),C)),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),C)) ) ).

% diff_right_mono
tff(fact_3261_diff__eq__diff__less__eq,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C),D) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),D) ) ) ).

% diff_eq_diff_less_eq
tff(fact_3262_diff__eq__diff__less__eq,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( minus_minus_rat(A,B) = minus_minus_rat(C,D) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),D) ) ) ).

% diff_eq_diff_less_eq
tff(fact_3263_diff__eq__diff__less__eq,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B) = aa_int_int(aa_int_fun_int_int(minus_minus_int,C),D) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),D) ) ) ).

% diff_eq_diff_less_eq
tff(fact_3264_diff__strict__mono,axiom,
    ! [A: real,B: real,D: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,D),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),D)) ) ) ).

% diff_strict_mono
tff(fact_3265_diff__strict__mono,axiom,
    ! [A: rat,B: rat,D: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,D),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,minus_minus_rat(A,C)),minus_minus_rat(B,D)) ) ) ).

% diff_strict_mono
tff(fact_3266_diff__strict__mono,axiom,
    ! [A: int,B: int,D: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,D),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),C)),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),D)) ) ) ).

% diff_strict_mono
tff(fact_3267_diff__eq__diff__less,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C),D) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,C),D) ) ) ).

% diff_eq_diff_less
tff(fact_3268_diff__eq__diff__less,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( minus_minus_rat(A,B) = minus_minus_rat(C,D) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),D) ) ) ).

% diff_eq_diff_less
tff(fact_3269_diff__eq__diff__less,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B) = aa_int_int(aa_int_fun_int_int(minus_minus_int,C),D) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,C),D) ) ) ).

% diff_eq_diff_less
tff(fact_3270_diff__strict__left__mono,axiom,
    ! [B: real,A: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),A)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C),B)) ) ).

% diff_strict_left_mono
tff(fact_3271_diff__strict__left__mono,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,minus_minus_rat(C,A)),minus_minus_rat(C,B)) ) ).

% diff_strict_left_mono
tff(fact_3272_diff__strict__left__mono,axiom,
    ! [B: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,C),A)),aa_int_int(aa_int_fun_int_int(minus_minus_int,C),B)) ) ).

% diff_strict_left_mono
tff(fact_3273_diff__strict__right__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),C)) ) ).

% diff_strict_right_mono
tff(fact_3274_diff__strict__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,minus_minus_rat(A,C)),minus_minus_rat(B,C)) ) ).

% diff_strict_right_mono
tff(fact_3275_diff__strict__right__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),C)),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),C)) ) ).

% diff_strict_right_mono
tff(fact_3276_comm__monoid__mult__class_Omult__1,axiom,
    ! [A: word_N3645301735248828278l_num1] : aa_wor7274019993978516266l_num1(times_7065122842183080059l_num1(one_on7727431528512463931l_num1),A) = A ).

% comm_monoid_mult_class.mult_1
tff(fact_3277_comm__monoid__mult__class_Omult__1,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,one_one_real),A) = A ).

% comm_monoid_mult_class.mult_1
tff(fact_3278_comm__monoid__mult__class_Omult__1,axiom,
    ! [A: rat] : aa_rat_rat(times_times_rat(one_one_rat),A) = A ).

% comm_monoid_mult_class.mult_1
tff(fact_3279_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_3280_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_3281_mult_Ocomm__neutral,axiom,
    ! [A: word_N3645301735248828278l_num1] : aa_wor7274019993978516266l_num1(times_7065122842183080059l_num1(A),one_on7727431528512463931l_num1) = A ).

% mult.comm_neutral
tff(fact_3282_mult_Ocomm__neutral,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),one_one_real) = A ).

% mult.comm_neutral
tff(fact_3283_mult_Ocomm__neutral,axiom,
    ! [A: rat] : aa_rat_rat(times_times_rat(A),one_one_rat) = A ).

% mult.comm_neutral
tff(fact_3284_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_3285_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_3286_diff__diff__eq,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)),C) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C)) ).

% diff_diff_eq
tff(fact_3287_diff__diff__eq,axiom,
    ! [A: rat,B: rat,C: rat] : minus_minus_rat(minus_minus_rat(A,B),C) = minus_minus_rat(A,aa_rat_rat(plus_plus_rat(B),C)) ).

% diff_diff_eq
tff(fact_3288_diff__diff__eq,axiom,
    ! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)) ).

% diff_diff_eq
tff(fact_3289_diff__diff__eq,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(minus_minus_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C)) ).

% diff_diff_eq
tff(fact_3290_add__implies__diff,axiom,
    ! [C: real,B: real,A: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),B) = A )
     => ( C = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B) ) ) ).

% add_implies_diff
tff(fact_3291_add__implies__diff,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( ( aa_rat_rat(plus_plus_rat(C),B) = A )
     => ( C = minus_minus_rat(A,B) ) ) ).

% add_implies_diff
tff(fact_3292_add__implies__diff,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B) = A )
     => ( C = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B) ) ) ).

% add_implies_diff
tff(fact_3293_add__implies__diff,axiom,
    ! [C: int,B: int,A: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B) = A )
     => ( C = aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B) ) ) ).

% add_implies_diff
tff(fact_3294_diff__add__eq__diff__diff__swap,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),C)),B) ).

% diff_add_eq_diff_diff_swap
tff(fact_3295_diff__add__eq__diff__diff__swap,axiom,
    ! [A: rat,B: rat,C: rat] : minus_minus_rat(A,aa_rat_rat(plus_plus_rat(B),C)) = minus_minus_rat(minus_minus_rat(A,C),B) ).

% diff_add_eq_diff_diff_swap
tff(fact_3296_diff__add__eq__diff__diff__swap,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),C)),B) ).

% diff_add_eq_diff_diff_swap
tff(fact_3297_diff__add__eq,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)),C) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),B) ).

% diff_add_eq
tff(fact_3298_diff__add__eq,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(plus_plus_rat(minus_minus_rat(A,B)),C) = minus_minus_rat(aa_rat_rat(plus_plus_rat(A),C),B) ).

% diff_add_eq
tff(fact_3299_diff__add__eq,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),B) ).

% diff_add_eq
tff(fact_3300_diff__diff__eq2,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),C)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),B) ).

% diff_diff_eq2
tff(fact_3301_diff__diff__eq2,axiom,
    ! [A: rat,B: rat,C: rat] : minus_minus_rat(A,minus_minus_rat(B,C)) = minus_minus_rat(aa_rat_rat(plus_plus_rat(A),C),B) ).

% diff_diff_eq2
tff(fact_3302_diff__diff__eq2,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,A),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),C)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),B) ).

% diff_diff_eq2
tff(fact_3303_add__diff__eq,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),C)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),C) ).

% add_diff_eq
tff(fact_3304_add__diff__eq,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(plus_plus_rat(A),minus_minus_rat(B,C)) = minus_minus_rat(aa_rat_rat(plus_plus_rat(A),B),C) ).

% add_diff_eq
tff(fact_3305_add__diff__eq,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),C)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C) ).

% add_diff_eq
tff(fact_3306_eq__diff__eq,axiom,
    ! [A: real,C: real,B: real] :
      ( ( A = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C),B) )
    <=> ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B) = C ) ) ).

% eq_diff_eq
tff(fact_3307_eq__diff__eq,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( A = minus_minus_rat(C,B) )
    <=> ( aa_rat_rat(plus_plus_rat(A),B) = C ) ) ).

% eq_diff_eq
tff(fact_3308_eq__diff__eq,axiom,
    ! [A: int,C: int,B: int] :
      ( ( A = aa_int_int(aa_int_fun_int_int(minus_minus_int,C),B) )
    <=> ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B) = C ) ) ).

% eq_diff_eq
tff(fact_3309_diff__eq__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B) = C )
    <=> ( A = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),B) ) ) ).

% diff_eq_eq
tff(fact_3310_diff__eq__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( minus_minus_rat(A,B) = C )
    <=> ( A = aa_rat_rat(plus_plus_rat(C),B) ) ) ).

% diff_eq_eq
tff(fact_3311_diff__eq__eq,axiom,
    ! [A: int,B: int,C: int] :
      ( ( aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B) = C )
    <=> ( A = aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B) ) ) ).

% diff_eq_eq
tff(fact_3312_group__cancel_Osub1,axiom,
    ! [A2: real,K: real,A: real,B: real] :
      ( ( A2 = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,K),A) )
     => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A2),B) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,K),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)) ) ) ).

% group_cancel.sub1
tff(fact_3313_group__cancel_Osub1,axiom,
    ! [A2: rat,K: rat,A: rat,B: rat] :
      ( ( A2 = aa_rat_rat(plus_plus_rat(K),A) )
     => ( minus_minus_rat(A2,B) = aa_rat_rat(plus_plus_rat(K),minus_minus_rat(A,B)) ) ) ).

% group_cancel.sub1
tff(fact_3314_group__cancel_Osub1,axiom,
    ! [A2: int,K: int,A: int,B: int] :
      ( ( A2 = aa_int_int(aa_int_fun_int_int(plus_plus_int,K),A) )
     => ( aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,K),aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)) ) ) ).

% group_cancel.sub1
tff(fact_3315_VEBT__internal_Ooption__comp__shift_Ocases,axiom,
    ! [Xb: produc650719256901721312it_nat] :
      ( ! [Uu2: fun_Pr226060869445836462_nat_o,Uv2: option2621746655072343315it_nat] : Xb != produc4438071236943126352it_nat(Uu2,produc6851560022941992023it_nat(none_P1551326421579882414it_nat,Uv2))
     => ( ! [Uw2: fun_Pr226060869445836462_nat_o,V2: produc120671012495760973it_nat] : Xb != produc4438071236943126352it_nat(Uw2,produc6851560022941992023it_nat(some_P2407035485129114418it_nat(V2),none_P1551326421579882414it_nat))
       => ~ ! [F2: fun_Pr226060869445836462_nat_o,X2: produc120671012495760973it_nat,Y4: produc120671012495760973it_nat] : Xb != produc4438071236943126352it_nat(F2,produc6851560022941992023it_nat(some_P2407035485129114418it_nat(X2),some_P2407035485129114418it_nat(Y4))) ) ) ).

% VEBT_internal.option_comp_shift.cases
tff(fact_3316_VEBT__internal_Ooption__comp__shift_Ocases,axiom,
    ! [Xb: produc3124918679657246250it_nat] :
      ( ! [Uu2: fun_Pr5882989906983993892_nat_o,Uv2: option7339022715339332451it_nat] : Xb != produc5062306165551350682it_nat(Uu2,produc9206348758962449759it_nat(none_P7668321371905463026it_nat,Uv2))
     => ( ! [Uw2: fun_Pr5882989906983993892_nat_o,V2: produc8047831477865546771it_nat] : Xb != produc5062306165551350682it_nat(Uw2,produc9206348758962449759it_nat(some_P468703482102919278it_nat(V2),none_P7668321371905463026it_nat))
       => ~ ! [F2: fun_Pr5882989906983993892_nat_o,X2: produc8047831477865546771it_nat,Y4: produc8047831477865546771it_nat] : Xb != produc5062306165551350682it_nat(F2,produc9206348758962449759it_nat(some_P468703482102919278it_nat(X2),some_P468703482102919278it_nat(Y4))) ) ) ).

% VEBT_internal.option_comp_shift.cases
tff(fact_3317_VEBT__internal_Ooption__comp__shift_Ocases,axiom,
    ! [Xb: produc3739950874144486240at_nat] :
      ( ! [Uu2: fun_Pr2028404973800837934_nat_o,Uv2: option4927543243414619207at_nat] : Xb != produc3653971535690468304at_nat(Uu2,produc488173922507101015at_nat(none_P5556105721700978146at_nat,Uv2))
     => ( ! [Uw2: fun_Pr2028404973800837934_nat_o,V2: product_prod_nat_nat] : Xb != produc3653971535690468304at_nat(Uw2,produc488173922507101015at_nat(some_P7363390416028606310at_nat(V2),none_P5556105721700978146at_nat))
       => ~ ! [F2: fun_Pr2028404973800837934_nat_o,X2: product_prod_nat_nat,Y4: product_prod_nat_nat] : Xb != produc3653971535690468304at_nat(F2,produc488173922507101015at_nat(some_P7363390416028606310at_nat(X2),some_P7363390416028606310at_nat(Y4))) ) ) ).

% VEBT_internal.option_comp_shift.cases
tff(fact_3318_VEBT__internal_Ooption__comp__shift_Ocases,axiom,
    ! [Xb: produc1615844007061411358on_nat] :
      ( ! [Uu2: fun_nat_fun_nat_o,Uv2: option_nat] : Xb != produc7967433646104525454on_nat(Uu2,produc5098337634421038937on_nat(none_nat,Uv2))
     => ( ! [Uw2: fun_nat_fun_nat_o,V2: nat] : Xb != produc7967433646104525454on_nat(Uw2,produc5098337634421038937on_nat(aa_nat_option_nat(some_nat,V2),none_nat))
       => ~ ! [F2: fun_nat_fun_nat_o,X2: nat,Y4: nat] : Xb != produc7967433646104525454on_nat(F2,produc5098337634421038937on_nat(aa_nat_option_nat(some_nat,X2),aa_nat_option_nat(some_nat,Y4))) ) ) ).

% VEBT_internal.option_comp_shift.cases
tff(fact_3319_VEBT__internal_Ooption__comp__shift_Ocases,axiom,
    ! [Xb: produc50614890452615238on_num] :
      ( ! [Uu2: fun_num_fun_num_o,Uv2: option_num] : Xb != produc1140783415258738358on_num(Uu2,produc8585076106096196333on_num(none_num,Uv2))
     => ( ! [Uw2: fun_num_fun_num_o,V2: num] : Xb != produc1140783415258738358on_num(Uw2,produc8585076106096196333on_num(aa_num_option_num(some_num,V2),none_num))
       => ~ ! [F2: fun_num_fun_num_o,X2: num,Y4: num] : Xb != produc1140783415258738358on_num(F2,produc8585076106096196333on_num(aa_num_option_num(some_num,X2),aa_num_option_num(some_num,Y4))) ) ) ).

% VEBT_internal.option_comp_shift.cases
tff(fact_3320_VEBT__internal_Ooption__shift_Ocases,axiom,
    ! [Xb: produc1531648545593279979it_nat] :
      ( ! [Uu2: fun_Pr3118856740240616547it_nat,Uv2: option2621746655072343315it_nat] : Xb != produc2260137262690574555it_nat(Uu2,produc6851560022941992023it_nat(none_P1551326421579882414it_nat,Uv2))
     => ( ! [Uw2: fun_Pr3118856740240616547it_nat,V2: produc120671012495760973it_nat] : Xb != produc2260137262690574555it_nat(Uw2,produc6851560022941992023it_nat(some_P2407035485129114418it_nat(V2),none_P1551326421579882414it_nat))
       => ~ ! [F2: fun_Pr3118856740240616547it_nat,A3: produc120671012495760973it_nat,B2: produc120671012495760973it_nat] : Xb != produc2260137262690574555it_nat(F2,produc6851560022941992023it_nat(some_P2407035485129114418it_nat(A3),some_P2407035485129114418it_nat(B2))) ) ) ).

% VEBT_internal.option_shift.cases
tff(fact_3321_VEBT__internal_Ooption__shift_Ocases,axiom,
    ! [Xb: produc7847833613904905897it_nat] :
      ( ! [Uu2: fun_Pr1313076919630728883it_nat,Uv2: option7339022715339332451it_nat] : Xb != produc5529849720642941531it_nat(Uu2,produc9206348758962449759it_nat(none_P7668321371905463026it_nat,Uv2))
     => ( ! [Uw2: fun_Pr1313076919630728883it_nat,V2: produc8047831477865546771it_nat] : Xb != produc5529849720642941531it_nat(Uw2,produc9206348758962449759it_nat(some_P468703482102919278it_nat(V2),none_P7668321371905463026it_nat))
       => ~ ! [F2: fun_Pr1313076919630728883it_nat,A3: produc8047831477865546771it_nat,B2: produc8047831477865546771it_nat] : Xb != produc5529849720642941531it_nat(F2,produc9206348758962449759it_nat(some_P468703482102919278it_nat(A3),some_P468703482102919278it_nat(B2))) ) ) ).

% VEBT_internal.option_shift.cases
tff(fact_3322_VEBT__internal_Ooption__shift_Ocases,axiom,
    ! [Xb: produc1260151622200656695at_nat] :
      ( ! [Uu2: fun_Pr2477239776880934807at_nat,Uv2: option4927543243414619207at_nat] : Xb != produc7814092408141220007at_nat(Uu2,produc488173922507101015at_nat(none_P5556105721700978146at_nat,Uv2))
     => ( ! [Uw2: fun_Pr2477239776880934807at_nat,V2: product_prod_nat_nat] : Xb != produc7814092408141220007at_nat(Uw2,produc488173922507101015at_nat(some_P7363390416028606310at_nat(V2),none_P5556105721700978146at_nat))
       => ~ ! [F2: fun_Pr2477239776880934807at_nat,A3: product_prod_nat_nat,B2: product_prod_nat_nat] : Xb != produc7814092408141220007at_nat(F2,produc488173922507101015at_nat(some_P7363390416028606310at_nat(A3),some_P7363390416028606310at_nat(B2))) ) ) ).

% VEBT_internal.option_shift.cases
tff(fact_3323_VEBT__internal_Ooption__shift_Ocases,axiom,
    ! [Xb: produc7004362771917097900on_nat] :
      ( ! [Uu2: fun_nat_fun_nat_nat,Uv2: option_nat] : Xb != produc6699827698519632094on_nat(Uu2,produc5098337634421038937on_nat(none_nat,Uv2))
     => ( ! [Uw2: fun_nat_fun_nat_nat,V2: nat] : Xb != produc6699827698519632094on_nat(Uw2,produc5098337634421038937on_nat(aa_nat_option_nat(some_nat,V2),none_nat))
       => ~ ! [F2: fun_nat_fun_nat_nat,A3: nat,B2: nat] : Xb != produc6699827698519632094on_nat(F2,produc5098337634421038937on_nat(aa_nat_option_nat(some_nat,A3),aa_nat_option_nat(some_nat,B2))) ) ) ).

% VEBT_internal.option_shift.cases
tff(fact_3324_VEBT__internal_Ooption__shift_Ocases,axiom,
    ! [Xb: produc2703417854250991842on_num] :
      ( ! [Uu2: fun_num_fun_num_num,Uv2: option_num] : Xb != produc6360833703471310868on_num(Uu2,produc8585076106096196333on_num(none_num,Uv2))
     => ( ! [Uw2: fun_num_fun_num_num,V2: num] : Xb != produc6360833703471310868on_num(Uw2,produc8585076106096196333on_num(aa_num_option_num(some_num,V2),none_num))
       => ~ ! [F2: fun_num_fun_num_num,A3: num,B2: num] : Xb != produc6360833703471310868on_num(F2,produc8585076106096196333on_num(aa_num_option_num(some_num,A3),aa_num_option_num(some_num,B2))) ) ) ).

% VEBT_internal.option_shift.cases
tff(fact_3325_insert_H__bound__height,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xb: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Ta),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_i_n_s_e_r_t2,Ta),Xb)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_VEBT_height,Ta))) ) ).

% insert'_bound_height
tff(fact_3326_VEBT__internal_Ooption__shift_Osimps_I3_J,axiom,
    ! [F: fun_Pr2477239776880934807at_nat,A: product_prod_nat_nat,B: product_prod_nat_nat] : vEBT_V1502963449132264192at_nat(F,some_P7363390416028606310at_nat(A),some_P7363390416028606310at_nat(B)) = some_P7363390416028606310at_nat(aa_Pro262675715343573244at_nat(aa_Pro8023792847356852296at_nat(F,A),B)) ).

% VEBT_internal.option_shift.simps(3)
tff(fact_3327_VEBT__internal_Ooption__shift_Osimps_I3_J,axiom,
    ! [F: fun_num_fun_num_num,A: num,B: num] : aa_opt5369159329878751634on_num(vEBT_V819420779217536731ft_num(F,aa_num_option_num(some_num,A)),aa_num_option_num(some_num,B)) = aa_num_option_num(some_num,aa_num_num(aa_num_fun_num_num(F,A),B)) ).

% VEBT_internal.option_shift.simps(3)
tff(fact_3328_VEBT__internal_Ooption__shift_Osimps_I3_J,axiom,
    ! [F: fun_nat_fun_nat_nat,A: nat,B: nat] : aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_V4262088993061758097ft_nat(F),aa_nat_option_nat(some_nat,A)),aa_nat_option_nat(some_nat,B)) = aa_nat_option_nat(some_nat,aa_nat_nat(aa_nat_fun_nat_nat(F,A),B)) ).

% VEBT_internal.option_shift.simps(3)
tff(fact_3329_VEBT__internal_Ooption__shift_Osimps_I1_J,axiom,
    ! [Uu: fun_Pr3118856740240616547it_nat,Uv: option2621746655072343315it_nat] : vEBT_V819568868292977612it_nat(Uu,none_P1551326421579882414it_nat,Uv) = none_P1551326421579882414it_nat ).

% VEBT_internal.option_shift.simps(1)
tff(fact_3330_VEBT__internal_Ooption__shift_Osimps_I1_J,axiom,
    ! [Uu: fun_Pr1313076919630728883it_nat,Uv: option7339022715339332451it_nat] : vEBT_V613753007643960916it_nat(Uu,none_P7668321371905463026it_nat,Uv) = none_P7668321371905463026it_nat ).

% VEBT_internal.option_shift.simps(1)
tff(fact_3331_VEBT__internal_Ooption__shift_Osimps_I1_J,axiom,
    ! [Uu: fun_Pr2477239776880934807at_nat,Uv: option4927543243414619207at_nat] : vEBT_V1502963449132264192at_nat(Uu,none_P5556105721700978146at_nat,Uv) = none_P5556105721700978146at_nat ).

% VEBT_internal.option_shift.simps(1)
tff(fact_3332_VEBT__internal_Ooption__shift_Osimps_I1_J,axiom,
    ! [Uu: fun_num_fun_num_num,Uv: option_num] : aa_opt5369159329878751634on_num(vEBT_V819420779217536731ft_num(Uu,none_num),Uv) = none_num ).

% VEBT_internal.option_shift.simps(1)
tff(fact_3333_VEBT__internal_Ooption__shift_Osimps_I1_J,axiom,
    ! [Uu: fun_nat_fun_nat_nat,Uv: option_nat] : aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_V4262088993061758097ft_nat(Uu),none_nat),Uv) = none_nat ).

% VEBT_internal.option_shift.simps(1)
tff(fact_3334_add__decreasing,axiom,
    ! [A: real,C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),B) ) ) ).

% add_decreasing
tff(fact_3335_add__decreasing,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(A),C)),B) ) ) ).

% add_decreasing
tff(fact_3336_add__decreasing,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),zero_zero_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),B) ) ) ).

% add_decreasing
tff(fact_3337_add__decreasing,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),B) ) ) ).

% add_decreasing
tff(fact_3338_add__increasing,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)) ) ) ).

% add_increasing
tff(fact_3339_add__increasing,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),aa_rat_rat(plus_plus_rat(A),C)) ) ) ).

% add_increasing
tff(fact_3340_add__increasing,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)) ) ) ).

% add_increasing
tff(fact_3341_add__increasing,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)) ) ) ).

% add_increasing
tff(fact_3342_add__decreasing2,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),B) ) ) ).

% add_decreasing2
tff(fact_3343_add__decreasing2,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(A),C)),B) ) ) ).

% add_decreasing2
tff(fact_3344_add__decreasing2,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),zero_zero_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),B) ) ) ).

% add_decreasing2
tff(fact_3345_add__decreasing2,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),B) ) ) ).

% add_decreasing2
tff(fact_3346_add__increasing2,axiom,
    ! [C: real,B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)) ) ) ).

% add_increasing2
tff(fact_3347_add__increasing2,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),aa_rat_rat(plus_plus_rat(A),C)) ) ) ).

% add_increasing2
tff(fact_3348_add__increasing2,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)) ) ) ).

% add_increasing2
tff(fact_3349_add__increasing2,axiom,
    ! [C: int,B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)) ) ) ).

% add_increasing2
tff(fact_3350_add__nonneg__nonneg,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)) ) ) ).

% add_nonneg_nonneg
tff(fact_3351_add__nonneg__nonneg,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(plus_plus_rat(A),B)) ) ) ).

% add_nonneg_nonneg
tff(fact_3352_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_3353_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_3354_add__nonpos__nonpos,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),zero_zero_real) ) ) ).

% add_nonpos_nonpos
tff(fact_3355_add__nonpos__nonpos,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(A),B)),zero_zero_rat) ) ) ).

% add_nonpos_nonpos
tff(fact_3356_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_3357_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_3358_add__nonneg__eq__0__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
       => ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Y) = zero_zero_real )
        <=> ( ( Xb = zero_zero_real )
            & ( Y = zero_zero_real ) ) ) ) ) ).

% add_nonneg_eq_0_iff
tff(fact_3359_add__nonneg__eq__0__iff,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Xb)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Y)
       => ( ( aa_rat_rat(plus_plus_rat(Xb),Y) = zero_zero_rat )
        <=> ( ( Xb = zero_zero_rat )
            & ( Y = zero_zero_rat ) ) ) ) ) ).

% add_nonneg_eq_0_iff
tff(fact_3360_add__nonneg__eq__0__iff,axiom,
    ! [Xb: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Xb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Y)
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xb),Y) = zero_zero_nat )
        <=> ( ( Xb = zero_zero_nat )
            & ( Y = zero_zero_nat ) ) ) ) ) ).

% add_nonneg_eq_0_iff
tff(fact_3361_add__nonneg__eq__0__iff,axiom,
    ! [Xb: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xb)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
       => ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb),Y) = zero_zero_int )
        <=> ( ( Xb = zero_zero_int )
            & ( Y = zero_zero_int ) ) ) ) ) ).

% add_nonneg_eq_0_iff
tff(fact_3362_add__nonpos__eq__0__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),zero_zero_real)
       => ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Y) = zero_zero_real )
        <=> ( ( Xb = zero_zero_real )
            & ( Y = zero_zero_real ) ) ) ) ) ).

% add_nonpos_eq_0_iff
tff(fact_3363_add__nonpos__eq__0__iff,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),zero_zero_rat)
       => ( ( aa_rat_rat(plus_plus_rat(Xb),Y) = zero_zero_rat )
        <=> ( ( Xb = zero_zero_rat )
            & ( Y = zero_zero_rat ) ) ) ) ) ).

% add_nonpos_eq_0_iff
tff(fact_3364_add__nonpos__eq__0__iff,axiom,
    ! [Xb: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),zero_zero_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),zero_zero_nat)
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xb),Y) = zero_zero_nat )
        <=> ( ( Xb = zero_zero_nat )
            & ( Y = zero_zero_nat ) ) ) ) ) ).

% add_nonpos_eq_0_iff
tff(fact_3365_add__nonpos__eq__0__iff,axiom,
    ! [Xb: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),zero_zero_int)
       => ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb),Y) = zero_zero_int )
        <=> ( ( Xb = zero_zero_int )
            & ( Y = zero_zero_int ) ) ) ) ) ).

% add_nonpos_eq_0_iff
tff(fact_3366_add__neg__neg,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),zero_zero_real) ) ) ).

% add_neg_neg
tff(fact_3367_add__neg__neg,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(A),B)),zero_zero_rat) ) ) ).

% add_neg_neg
tff(fact_3368_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_3369_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_3370_add__pos__pos,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)) ) ) ).

% add_pos_pos
tff(fact_3371_add__pos__pos,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(plus_plus_rat(A),B)) ) ) ).

% add_pos_pos
tff(fact_3372_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_3373_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_3374_canonically__ordered__monoid__add__class_OlessE,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ~ ! [C2: nat] :
            ( ( B = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C2) )
           => ( C2 = zero_zero_nat ) ) ) ).

% canonically_ordered_monoid_add_class.lessE
tff(fact_3375_pos__add__strict,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,B),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)) ) ) ).

% pos_add_strict
tff(fact_3376_pos__add__strict,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),aa_rat_rat(plus_plus_rat(A),C)) ) ) ).

% pos_add_strict
tff(fact_3377_pos__add__strict,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)) ) ) ).

% pos_add_strict
tff(fact_3378_pos__add__strict,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,B),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)) ) ) ).

% pos_add_strict
tff(fact_3379_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I: real,J: real,K: real,L: real] :
      ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,I),J)
        & aa_real_o(aa_real_fun_real_o(ord_less_real,K),L) )
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J),L)) ) ).

% add_mono_thms_linordered_field(4)
tff(fact_3380_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,I),J)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,K),L) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(I),K)),aa_rat_rat(plus_plus_rat(J),L)) ) ).

% add_mono_thms_linordered_field(4)
tff(fact_3381_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),L) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L)) ) ).

% add_mono_thms_linordered_field(4)
tff(fact_3382_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),J)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,K),L) )
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L)) ) ).

% add_mono_thms_linordered_field(4)
tff(fact_3383_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I: real,J: real,K: real,L: real] :
      ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,I),J)
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,K),L) )
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J),L)) ) ).

% add_mono_thms_linordered_field(3)
tff(fact_3384_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,I),J)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,K),L) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(I),K)),aa_rat_rat(plus_plus_rat(J),L)) ) ).

% add_mono_thms_linordered_field(3)
tff(fact_3385_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),L) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L)) ) ).

% add_mono_thms_linordered_field(3)
tff(fact_3386_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,I),J)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),L) )
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L)) ) ).

% add_mono_thms_linordered_field(3)
tff(fact_3387_add__le__less__mono,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),D)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),D)) ) ) ).

% add_le_less_mono
tff(fact_3388_add__le__less__mono,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),D)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(A),C)),aa_rat_rat(plus_plus_rat(B),D)) ) ) ).

% add_le_less_mono
tff(fact_3389_add__le__less__mono,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),D)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),D)) ) ) ).

% add_le_less_mono
tff(fact_3390_add__le__less__mono,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),D)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),D)) ) ) ).

% add_le_less_mono
tff(fact_3391_add__less__le__mono,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),D)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),D)) ) ) ).

% add_less_le_mono
tff(fact_3392_add__less__le__mono,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),D)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(A),C)),aa_rat_rat(plus_plus_rat(B),D)) ) ) ).

% add_less_le_mono
tff(fact_3393_add__less__le__mono,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),D)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),D)) ) ) ).

% add_less_le_mono
tff(fact_3394_add__less__le__mono,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),D)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),D)) ) ) ).

% add_less_le_mono
tff(fact_3395_le__iff__diff__le__0,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)),zero_zero_real) ) ).

% le_iff_diff_le_0
tff(fact_3396_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,minus_minus_rat(A,B)),zero_zero_rat) ) ).

% le_iff_diff_le_0
tff(fact_3397_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_3398_less__iff__diff__less__0,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)),zero_zero_real) ) ).

% less_iff_diff_less_0
tff(fact_3399_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,minus_minus_rat(A,B)),zero_zero_rat) ) ).

% less_iff_diff_less_0
tff(fact_3400_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_3401_diff__le__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)),C)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),B)) ) ).

% diff_le_eq
tff(fact_3402_diff__le__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,minus_minus_rat(A,B)),C)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(plus_plus_rat(C),B)) ) ).

% diff_le_eq
tff(fact_3403_diff__le__eq,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),C)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B)) ) ).

% diff_le_eq
tff(fact_3404_le__diff__eq,axiom,
    ! [A: real,C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C),B))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),C) ) ).

% le_diff_eq
tff(fact_3405_le__diff__eq,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),minus_minus_rat(C,B))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(A),B)),C) ) ).

% le_diff_eq
tff(fact_3406_le__diff__eq,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_int_int(aa_int_fun_int_int(minus_minus_int,C),B))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C) ) ).

% le_diff_eq
tff(fact_3407_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_3408_le__add__diff,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)),A)) ) ).

% le_add_diff
tff(fact_3409_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A)),B) ) ) ).

% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
tff(fact_3410_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B)),A) ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
tff(fact_3411_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B)),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A)) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
tff(fact_3412_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A)),C) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)),A) ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
tff(fact_3413_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A)),C) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
tff(fact_3414_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A)),B) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_diff_right
tff(fact_3415_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_3416_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A) = C )
        <=> ( B = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A) ) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
tff(fact_3417_diff__less__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)),C)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),B)) ) ).

% diff_less_eq
tff(fact_3418_diff__less__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,minus_minus_rat(A,B)),C)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(plus_plus_rat(C),B)) ) ).

% diff_less_eq
tff(fact_3419_diff__less__eq,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),C)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B)) ) ).

% diff_less_eq
tff(fact_3420_less__diff__eq,axiom,
    ! [A: real,C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C),B))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),C) ) ).

% less_diff_eq
tff(fact_3421_less__diff__eq,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),minus_minus_rat(C,B))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(A),B)),C) ) ).

% less_diff_eq
tff(fact_3422_less__diff__eq,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_int_int(aa_int_fun_int_int(minus_minus_int,C),B))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C) ) ).

% less_diff_eq
tff(fact_3423_VEBT__internal_Ooption__shift_Osimps_I2_J,axiom,
    ! [Uw: fun_Pr3118856740240616547it_nat,V: produc120671012495760973it_nat] : vEBT_V819568868292977612it_nat(Uw,some_P2407035485129114418it_nat(V),none_P1551326421579882414it_nat) = none_P1551326421579882414it_nat ).

% VEBT_internal.option_shift.simps(2)
tff(fact_3424_VEBT__internal_Ooption__shift_Osimps_I2_J,axiom,
    ! [Uw: fun_Pr1313076919630728883it_nat,V: produc8047831477865546771it_nat] : vEBT_V613753007643960916it_nat(Uw,some_P468703482102919278it_nat(V),none_P7668321371905463026it_nat) = none_P7668321371905463026it_nat ).

% VEBT_internal.option_shift.simps(2)
tff(fact_3425_VEBT__internal_Ooption__shift_Osimps_I2_J,axiom,
    ! [Uw: fun_Pr2477239776880934807at_nat,V: product_prod_nat_nat] : vEBT_V1502963449132264192at_nat(Uw,some_P7363390416028606310at_nat(V),none_P5556105721700978146at_nat) = none_P5556105721700978146at_nat ).

% VEBT_internal.option_shift.simps(2)
tff(fact_3426_VEBT__internal_Ooption__shift_Osimps_I2_J,axiom,
    ! [Uw: fun_num_fun_num_num,V: num] : aa_opt5369159329878751634on_num(vEBT_V819420779217536731ft_num(Uw,aa_num_option_num(some_num,V)),none_num) = none_num ).

% VEBT_internal.option_shift.simps(2)
tff(fact_3427_VEBT__internal_Ooption__shift_Osimps_I2_J,axiom,
    ! [Uw: fun_nat_fun_nat_nat,V: nat] : aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_V4262088993061758097ft_nat(Uw),aa_nat_option_nat(some_nat,V)),none_nat) = none_nat ).

% VEBT_internal.option_shift.simps(2)
tff(fact_3428_VEBT__internal_Ooption__shift_Oelims,axiom,
    ! [Xb: fun_Pr3118856740240616547it_nat,Xaa: option2621746655072343315it_nat,Xba: option2621746655072343315it_nat,Y: option2621746655072343315it_nat] :
      ( ( vEBT_V819568868292977612it_nat(Xb,Xaa,Xba) = Y )
     => ( ( ( Xaa = none_P1551326421579882414it_nat )
         => ( Y != none_P1551326421579882414it_nat ) )
       => ( ( ? [V2: produc120671012495760973it_nat] : Xaa = some_P2407035485129114418it_nat(V2)
           => ( ( Xba = none_P1551326421579882414it_nat )
             => ( Y != none_P1551326421579882414it_nat ) ) )
         => ~ ! [A3: produc120671012495760973it_nat] :
                ( ( Xaa = some_P2407035485129114418it_nat(A3) )
               => ! [B2: produc120671012495760973it_nat] :
                    ( ( Xba = some_P2407035485129114418it_nat(B2) )
                   => ( Y != some_P2407035485129114418it_nat(aa_Pro6543753689527670780it_nat(aa_Pro329082998274276628it_nat(Xb,A3),B2)) ) ) ) ) ) ) ).

% VEBT_internal.option_shift.elims
tff(fact_3429_VEBT__internal_Ooption__shift_Oelims,axiom,
    ! [Xb: fun_Pr1313076919630728883it_nat,Xaa: option7339022715339332451it_nat,Xba: option7339022715339332451it_nat,Y: option7339022715339332451it_nat] :
      ( ( vEBT_V613753007643960916it_nat(Xb,Xaa,Xba) = Y )
     => ( ( ( Xaa = none_P7668321371905463026it_nat )
         => ( Y != none_P7668321371905463026it_nat ) )
       => ( ( ? [V2: produc8047831477865546771it_nat] : Xaa = some_P468703482102919278it_nat(V2)
           => ( ( Xba = none_P7668321371905463026it_nat )
             => ( Y != none_P7668321371905463026it_nat ) ) )
         => ~ ! [A3: produc8047831477865546771it_nat] :
                ( ( Xaa = some_P468703482102919278it_nat(A3) )
               => ! [B2: produc8047831477865546771it_nat] :
                    ( ( Xba = some_P468703482102919278it_nat(B2) )
                   => ( Y != some_P468703482102919278it_nat(aa_Pro7661135075737848164it_nat(aa_Pro52009495076012404it_nat(Xb,A3),B2)) ) ) ) ) ) ) ).

% VEBT_internal.option_shift.elims
tff(fact_3430_VEBT__internal_Ooption__shift_Oelims,axiom,
    ! [Xb: fun_Pr2477239776880934807at_nat,Xaa: option4927543243414619207at_nat,Xba: option4927543243414619207at_nat,Y: option4927543243414619207at_nat] :
      ( ( vEBT_V1502963449132264192at_nat(Xb,Xaa,Xba) = Y )
     => ( ( ( Xaa = none_P5556105721700978146at_nat )
         => ( Y != none_P5556105721700978146at_nat ) )
       => ( ( ? [V2: product_prod_nat_nat] : Xaa = some_P7363390416028606310at_nat(V2)
           => ( ( Xba = none_P5556105721700978146at_nat )
             => ( Y != none_P5556105721700978146at_nat ) ) )
         => ~ ! [A3: product_prod_nat_nat] :
                ( ( Xaa = some_P7363390416028606310at_nat(A3) )
               => ! [B2: product_prod_nat_nat] :
                    ( ( Xba = some_P7363390416028606310at_nat(B2) )
                   => ( Y != some_P7363390416028606310at_nat(aa_Pro262675715343573244at_nat(aa_Pro8023792847356852296at_nat(Xb,A3),B2)) ) ) ) ) ) ) ).

% VEBT_internal.option_shift.elims
tff(fact_3431_VEBT__internal_Ooption__shift_Oelims,axiom,
    ! [Xb: fun_num_fun_num_num,Xaa: option_num,Xba: option_num,Y: option_num] :
      ( ( aa_opt5369159329878751634on_num(vEBT_V819420779217536731ft_num(Xb,Xaa),Xba) = Y )
     => ( ( ( Xaa = none_num )
         => ( Y != none_num ) )
       => ( ( ? [V2: num] : Xaa = aa_num_option_num(some_num,V2)
           => ( ( Xba = none_num )
             => ( Y != none_num ) ) )
         => ~ ! [A3: num] :
                ( ( Xaa = aa_num_option_num(some_num,A3) )
               => ! [B2: num] :
                    ( ( Xba = aa_num_option_num(some_num,B2) )
                   => ( Y != aa_num_option_num(some_num,aa_num_num(aa_num_fun_num_num(Xb,A3),B2)) ) ) ) ) ) ) ).

% VEBT_internal.option_shift.elims
tff(fact_3432_VEBT__internal_Ooption__shift_Oelims,axiom,
    ! [Xb: fun_nat_fun_nat_nat,Xaa: option_nat,Xba: option_nat,Y: option_nat] :
      ( ( aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_V4262088993061758097ft_nat(Xb),Xaa),Xba) = Y )
     => ( ( ( Xaa = none_nat )
         => ( Y != none_nat ) )
       => ( ( ? [V2: nat] : Xaa = aa_nat_option_nat(some_nat,V2)
           => ( ( Xba = none_nat )
             => ( Y != none_nat ) ) )
         => ~ ! [A3: nat] :
                ( ( Xaa = aa_nat_option_nat(some_nat,A3) )
               => ! [B2: nat] :
                    ( ( Xba = aa_nat_option_nat(some_nat,B2) )
                   => ( Y != aa_nat_option_nat(some_nat,aa_nat_nat(aa_nat_fun_nat_nat(Xb,A3),B2)) ) ) ) ) ) ) ).

% VEBT_internal.option_shift.elims
tff(fact_3433_add__neg__nonpos,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),zero_zero_real) ) ) ).

% add_neg_nonpos
tff(fact_3434_add__neg__nonpos,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(A),B)),zero_zero_rat) ) ) ).

% add_neg_nonpos
tff(fact_3435_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_3436_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_3437_add__nonneg__pos,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)) ) ) ).

% add_nonneg_pos
tff(fact_3438_add__nonneg__pos,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(plus_plus_rat(A),B)) ) ) ).

% add_nonneg_pos
tff(fact_3439_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_3440_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_3441_add__nonpos__neg,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),zero_zero_real) ) ) ).

% add_nonpos_neg
tff(fact_3442_add__nonpos__neg,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(A),B)),zero_zero_rat) ) ) ).

% add_nonpos_neg
tff(fact_3443_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_3444_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_3445_add__pos__nonneg,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)) ) ) ).

% add_pos_nonneg
tff(fact_3446_add__pos__nonneg,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(plus_plus_rat(A),B)) ) ) ).

% add_pos_nonneg
tff(fact_3447_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_3448_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_3449_add__strict__increasing,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,B),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)) ) ) ).

% add_strict_increasing
tff(fact_3450_add__strict__increasing,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),aa_rat_rat(plus_plus_rat(A),C)) ) ) ).

% add_strict_increasing
tff(fact_3451_add__strict__increasing,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)) ) ) ).

% add_strict_increasing
tff(fact_3452_add__strict__increasing,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,B),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)) ) ) ).

% add_strict_increasing
tff(fact_3453_add__strict__increasing2,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,B),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)) ) ) ).

% add_strict_increasing2
tff(fact_3454_add__strict__increasing2,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),aa_rat_rat(plus_plus_rat(A),C)) ) ) ).

% add_strict_increasing2
tff(fact_3455_add__strict__increasing2,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)) ) ) ).

% add_strict_increasing2
tff(fact_3456_add__strict__increasing2,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,B),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)) ) ) ).

% add_strict_increasing2
tff(fact_3457_Bolzano,axiom,
    ! [A: real,B: real,P: fun_real_fun_real_o] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( ! [A3: real,B2: real,C2: real] :
            ( aa_real_o(aa_real_fun_real_o(P,A3),B2)
           => ( aa_real_o(aa_real_fun_real_o(P,B2),C2)
             => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A3),B2)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B2),C2)
                 => aa_real_o(aa_real_fun_real_o(P,A3),C2) ) ) ) )
       => ( ! [X2: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),X2)
             => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X2),B)
               => ? [D4: real] :
                    ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D4)
                    & ! [A3: real,B2: real] :
                        ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A3),X2)
                          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X2),B2)
                          & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B2),A3)),D4) )
                       => aa_real_o(aa_real_fun_real_o(P,A3),B2) ) ) ) )
         => aa_real_o(aa_real_fun_real_o(P,A),B) ) ) ) ).

% Bolzano
tff(fact_3458_vebt__mint_Osimps_I2_J,axiom,
    ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] : vEBT_vebt_mint(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv),Uw)) = none_nat ).

% vebt_mint.simps(2)
tff(fact_3459_vebt__maxt_Osimps_I2_J,axiom,
    ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] : vEBT_vebt_maxt(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv),Uw)) = none_nat ).

% vebt_maxt.simps(2)
tff(fact_3460_vebt__mint_Osimps_I3_J,axiom,
    ! [Mia: nat,Maa: nat,Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT] : vEBT_vebt_mint(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mia),Maa)),Ux,Uy),Uz)) = aa_nat_option_nat(some_nat,Mia) ).

% vebt_mint.simps(3)
tff(fact_3461_vebt__maxt_Osimps_I3_J,axiom,
    ! [Mia: nat,Maa: nat,Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT] : vEBT_vebt_maxt(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mia),Maa)),Ux,Uy),Uz)) = aa_nat_option_nat(some_nat,Maa) ).

% vebt_maxt.simps(3)
tff(fact_3462_vebt__mint_Osimps_I1_J,axiom,
    ! [A: $o,B: $o] :
      vEBT_vebt_mint(vEBT_Leaf((A),(B))) = $ite(
        (A),
        aa_nat_option_nat(some_nat,zero_zero_nat),
        $ite((B),aa_nat_option_nat(some_nat,one_one_nat),none_nat) ) ).

% vebt_mint.simps(1)
tff(fact_3463_vebt__maxt_Osimps_I1_J,axiom,
    ! [A: $o,B: $o] :
      vEBT_vebt_maxt(vEBT_Leaf((A),(B))) = $ite(
        (B),
        aa_nat_option_nat(some_nat,one_one_nat),
        $ite((A),aa_nat_option_nat(some_nat,zero_zero_nat),none_nat) ) ).

% vebt_maxt.simps(1)
tff(fact_3464_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H_Osimps_I5_J,axiom,
    ! [Mia: nat,Maa: nat,Vaa: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Xb: nat] :
      aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_i_n_s_e_r_t2,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mia),Maa)),aa_nat_nat(suc,aa_nat_nat(suc,Vaa)),TreeLista),Summarya)),Xb) = $let(
        xn: nat,
        xn:= 
          $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Mia),Mia,Xb),
        $let(
          h: nat,
          h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Vaa))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
          $ite(
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeLista))
            & ~ ( ( Xb = Mia )
                | ( Xb = Maa ) ) ),
            aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_i_n_s_e_r_t2,nth_VEBT_VEBT(TreeLista,h)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Vaa))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),
              $ite(aa_VEBT_VEBT_o(vEBT_VEBT_minNull,nth_VEBT_VEBT(TreeLista,h)),aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_i_n_s_e_r_t2,Summarya),h),one_one_nat)),
            one_one_nat ) ) ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.simps(5)
tff(fact_3465_vebt__predi_Osimps,axiom,
    ! [Ta: vEBT_VEBTi,Xb: nat] : aa_nat947733127081326238on_nat(aa_VEB3566417204837707778on_nat(vEBT_vebt_predi,Ta),Xb) = aa_VEB2711320435582112505on_nat(vEBT_c6250501799366334488on_nat(aTP_Lamm_ey(Xb),aTP_Lamm_ez(Xb)),Ta) ).

% vebt_predi.simps
tff(fact_3466_delete__bound__size__univ,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,U: real,Xb: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Ta),Nb)
     => ( ( U = aa_nat_real(aa_real_fun_nat_real(power_power_real,numeral_numeral_real(aa_num_num(bit0,one))),Nb) )
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_d_e_l_e_t_e,Ta),Xb))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,numeral_numeral_real(aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,one))))))))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,one)))))))),aa_real_real(aa_rea7340605660300375871l_real(log,numeral_numeral_real(aa_num_num(bit0,one))),aa_real_real(aa_rea7340605660300375871l_real(log,numeral_numeral_real(aa_num_num(bit0,one))),U))))) ) ) ).

% delete_bound_size_univ
tff(fact_3467_vebt__succi_Osimps,axiom,
    ! [Ta: vEBT_VEBTi,Xb: nat] : aa_nat947733127081326238on_nat(aa_VEB3566417204837707778on_nat(vEBT_vebt_succi,Ta),Xb) = aa_VEB2711320435582112505on_nat(vEBT_c6250501799366334488on_nat(aTP_Lamm_fh(Xb),aTP_Lamm_fi(Xb)),Ta) ).

% vebt_succi.simps
tff(fact_3468_inrange,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Ta),Nb)
     => aa_set_nat_o(ord_less_eq_set_nat(vEBT_VEBT_set_vebt(Ta)),set_or1269000886237332187st_nat(zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)),one_one_nat))) ) ).

% inrange
tff(fact_3469_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_092_060_094sub_062u_092_060_094sub_062p_Osimps_I2_J,axiom,
    aa_nat_nat(vEBT_V8346862874174094_d_u_p,aa_nat_nat(suc,zero_zero_nat)) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d\<^sub>u\<^sub>p.simps(2)
tff(fact_3470_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_Osimps_I2_J,axiom,
    aa_nat_nat(vEBT_V8646137997579335489_i_l_d,aa_nat_nat(suc,zero_zero_nat)) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one))) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d.simps(2)
tff(fact_3471_height__double__log__univ__size,axiom,
    ! [U: real,Deg: nat,Ta: vEBT_VEBT] :
      ( ( U = aa_nat_real(aa_real_fun_nat_real(power_power_real,numeral_numeral_real(aa_num_num(bit0,one))),Deg) )
     => ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Ta),Deg)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,aa_VEBT_VEBT_nat(vEBT_VEBT_height,Ta))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(log,numeral_numeral_real(aa_num_num(bit0,one))),aa_real_real(aa_rea7340605660300375871l_real(log,numeral_numeral_real(aa_num_num(bit0,one))),U)))) ) ) ).

% height_double_log_univ_size
tff(fact_3472_zdiff__int__split,axiom,
    ! [P: fun_int_o,Xb: nat,Y: nat] :
      ( aa_int_o(P,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Xb),Y)))
    <=> ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),Xb)
         => aa_int_o(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(semiri1314217659103216013at_int,Xb)),aa_nat_int(semiri1314217659103216013at_int,Y))) )
        & ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y)
         => aa_int_o(P,zero_zero_int) ) ) ) ).

% zdiff_int_split
tff(fact_3473_delete__bound__size__univ_H,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,U: real,Xb: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Ta),Nb)
     => ( ( U = aa_nat_real(aa_real_fun_nat_real(power_power_real,numeral_numeral_real(aa_num_num(bit0,one))),Nb) )
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_V1232361888498592333_e_t_e,Ta),Xb))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,numeral_numeral_real(aa_num_num(bit0,one))),aa_real_real(aa_rea7340605660300375871l_real(log,numeral_numeral_real(aa_num_num(bit0,one))),aa_real_real(aa_rea7340605660300375871l_real(log,numeral_numeral_real(aa_num_num(bit0,one))),U)))) ) ) ).

% delete_bound_size_univ'
tff(fact_3474_TBOUND__VEBT__case,axiom,
    ! [Tib: vEBT_VEBTi,F: fun_o_7009304396571837746ap_nat,Bnd: fun_o_fun_o_nat,F3: fun_op2425996437410981688ap_nat,Bnd2: fun_op7256438278201533260Ti_nat] :
      ( ! [A3: $o,B2: $o] :
          ( ( Tib = vEBT_Leafi((A3),(B2)) )
         => aa_nat_o(time_TBOUND_nat(aa_o_H8739796202006397184ap_nat(aa_o_f2078835126877633507ap_nat(F,(A3)),(B2))),aa_o_nat(aa_o_fun_o_nat(Bnd,(A3)),(B2))) )
     => ( ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeArray: array_VEBT_VEBTi,Summary: vEBT_VEBTi] :
            ( ( Tib = aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(Info2,Deg2,TreeArray),Summary) )
           => aa_nat_o(time_TBOUND_nat(aa_VEB183921879236458153ap_nat(aa_arr6934346397072338570ap_nat(aa_nat1288005622339309855ap_nat(aa_opt6697879614838309225ap_nat(F3,Info2),Deg2),TreeArray),Summary)),aa_VEBT_VEBTi_nat(aa_arr8157624991474932664Ti_nat(aa_nat7496786144979625977Ti_nat(aa_opt8327319410639121349Ti_nat(Bnd2,Info2),Deg2),TreeArray),Summary)) )
       => aa_nat_o(time_TBOUND_nat(aa_VEB183921879236458153ap_nat(vEBT_c1335663792808957512ap_nat(F3,F),Tib)),aa_VEBT_VEBTi_nat(vEBT_case_VEBTi_nat(Bnd2,Bnd),Tib)) ) ) ).

% TBOUND_VEBT_case
tff(fact_3475_TBOUND__VEBT__case,axiom,
    ! [Tib: vEBT_VEBTi,F: fun_o_4049874799390722306on_nat,Bnd: fun_o_fun_o_nat,F3: fun_op5783775701855621896on_nat,Bnd2: fun_op7256438278201533260Ti_nat] :
      ( ! [A3: $o,B2: $o] :
          ( ( Tib = vEBT_Leafi((A3),(B2)) )
         => aa_nat_o(time_T8353473612707095248on_nat(aa_o_H4215342640551218896on_nat(aa_o_f5606600903069576627on_nat(F,(A3)),(B2))),aa_o_nat(aa_o_fun_o_nat(Bnd,(A3)),(B2))) )
     => ( ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeArray: array_VEBT_VEBTi,Summary: vEBT_VEBTi] :
            ( ( Tib = aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(Info2,Deg2,TreeArray),Summary) )
           => aa_nat_o(time_T8353473612707095248on_nat(aa_VEB2711320435582112505on_nat(aa_arr2955191963898321370on_nat(aa_nat5317369504323972463on_nat(aa_opt8624683241416710969on_nat(F3,Info2),Deg2),TreeArray),Summary)),aa_VEBT_VEBTi_nat(aa_arr8157624991474932664Ti_nat(aa_nat7496786144979625977Ti_nat(aa_opt8327319410639121349Ti_nat(Bnd2,Info2),Deg2),TreeArray),Summary)) )
       => aa_nat_o(time_T8353473612707095248on_nat(aa_VEB2711320435582112505on_nat(vEBT_c6250501799366334488on_nat(F3,F),Tib)),aa_VEBT_VEBTi_nat(vEBT_case_VEBTi_nat(Bnd2,Bnd),Tib)) ) ) ).

% TBOUND_VEBT_case
tff(fact_3476_TBOUND__VEBT__case,axiom,
    ! [Tib: vEBT_VEBTi,F: fun_o_6972675622308511895_VEBTi,Bnd: fun_o_fun_o_nat,F3: fun_op5127434927081926045_VEBTi,Bnd2: fun_op7256438278201533260Ti_nat] :
      ( ! [A3: $o,B2: $o] :
          ( ( Tib = vEBT_Leafi((A3),(B2)) )
         => aa_nat_o(time_T5737551269749752165_VEBTi(aa_o_H7577119287286566245_VEBTi(aa_o_f5262348187908705736_VEBTi(F,(A3)),(B2))),aa_o_nat(aa_o_fun_o_nat(Bnd,(A3)),(B2))) )
     => ( ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeArray: array_VEBT_VEBTi,Summary: vEBT_VEBTi] :
            ( ( Tib = aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(Info2,Deg2,TreeArray),Summary) )
           => aa_nat_o(time_T5737551269749752165_VEBTi(aa_VEB836174949890266638_VEBTi(aa_arr2912372873376718191_VEBTi(aa_nat6445230840460233604_VEBTi(aa_opt5182041009420754510_VEBTi(F3,Info2),Deg2),TreeArray),Summary)),aa_VEBT_VEBTi_nat(aa_arr8157624991474932664Ti_nat(aa_nat7496786144979625977Ti_nat(aa_opt8327319410639121349Ti_nat(Bnd2,Info2),Deg2),TreeArray),Summary)) )
       => aa_nat_o(time_T5737551269749752165_VEBTi(aa_VEB836174949890266638_VEBTi(vEBT_c6028912655521741485_VEBTi(F3,F),Tib)),aa_VEBT_VEBTi_nat(vEBT_case_VEBTi_nat(Bnd2,Bnd),Tib)) ) ) ).

% TBOUND_VEBT_case
tff(fact_3477_TBOUND__VEBT__case,axiom,
    ! [Tib: vEBT_VEBTi,F: fun_o_1207901889202581752Heap_o,Bnd: fun_o_fun_o_nat,F3: fun_op1763656958002597022Heap_o,Bnd2: fun_op7256438278201533260Ti_nat] :
      ( ! [A3: $o,B2: $o] :
          ( ( Tib = vEBT_Leafi((A3),(B2)) )
         => aa_nat_o(time_TBOUND_o(aa_o_H7399033255965945470Heap_o(aa_o_f869145629572826609Heap_o(F,(A3)),(B2))),aa_o_nat(aa_o_fun_o_nat(Bnd,(A3)),(B2))) )
     => ( ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeArray: array_VEBT_VEBTi,Summary: vEBT_VEBTi] :
            ( ( Tib = aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(Info2,Deg2,TreeArray),Summary) )
           => aa_nat_o(time_TBOUND_o(aa_VEB8194921341634505877Heap_o(aa_arr2373296168907467658Heap_o(aa_nat8650107858160254923Heap_o(aa_opt6539295502271640599Heap_o(F3,Info2),Deg2),TreeArray),Summary)),aa_VEBT_VEBTi_nat(aa_arr8157624991474932664Ti_nat(aa_nat7496786144979625977Ti_nat(aa_opt8327319410639121349Ti_nat(Bnd2,Info2),Deg2),TreeArray),Summary)) )
       => aa_nat_o(time_TBOUND_o(aa_VEB8194921341634505877Heap_o(vEBT_c6104975476656191286Heap_o(F3,F),Tib)),aa_VEBT_VEBTi_nat(vEBT_case_VEBTi_nat(Bnd2,Bnd),Tib)) ) ) ).

% TBOUND_VEBT_case
tff(fact_3478_subset__Collect__conv,axiom,
    ! [S2: set_nat,P: fun_nat_o] :
      ( aa_set_nat_o(ord_less_eq_set_nat(S2),collect_nat(P))
    <=> ! [X: nat] :
          ( aa_set_nat_o(member_nat(X),S2)
         => aa_nat_o(P,X) ) ) ).

% subset_Collect_conv
tff(fact_3479_subset__Collect__conv,axiom,
    ! [S2: set_complex,P: fun_complex_o] :
      ( aa_set_complex_o(ord_le211207098394363844omplex(S2),collect_complex(P))
    <=> ! [X: complex] :
          ( aa_set_complex_o(member_complex(X),S2)
         => aa_complex_o(P,X) ) ) ).

% subset_Collect_conv
tff(fact_3480_subset__Collect__conv,axiom,
    ! [S2: set_Pr958786334691620121nt_int,P: fun_Pr3238374608701176162_int_o] :
      ( ord_le2843351958646193337nt_int(S2,collec213857154873943460nt_int(P))
    <=> ! [X: product_prod_int_int] :
          ( member5262025264175285858nt_int(X,S2)
         => aa_Pro4592068669816069595_int_o(P,X) ) ) ).

% subset_Collect_conv
tff(fact_3481_subset__Collect__conv,axiom,
    ! [S2: set_int,P: fun_int_o] :
      ( aa_set_int_o(ord_less_eq_set_int(S2),collect_int(P))
    <=> ! [X: int] :
          ( aa_set_int_o(member_int(X),S2)
         => aa_int_o(P,X) ) ) ).

% subset_Collect_conv
tff(fact_3482_VEBTi_Ocase__distrib,axiom,
    ! [H: fun_He8640631688564546883on_nat,F1: fun_op5783775701855621896on_nat,F22: fun_o_4049874799390722306on_nat,VEBTi: vEBT_VEBTi] : aa_Hea6597239020403563964on_nat(H,aa_VEB2711320435582112505on_nat(vEBT_c6250501799366334488on_nat(F1,F22),VEBTi)) = aa_VEB2711320435582112505on_nat(vEBT_c6250501799366334488on_nat(aa_fun5534811901068199356on_nat(aTP_Lamm_fj(H),F1),aa_fun4373765338990682812on_nat(aTP_Lamm_fk(H),F22)),VEBTi) ).

% VEBTi.case_distrib
tff(fact_3483_VEBTi_Ocase__distrib,axiom,
    ! [H: fun_He2623223239018661208_VEBTi,F1: fun_op5783775701855621896on_nat,F22: fun_o_4049874799390722306on_nat,VEBTi: vEBT_VEBTi] : aa_Hea2832269215923769937_VEBTi(H,aa_VEB2711320435582112505on_nat(vEBT_c6250501799366334488on_nat(F1,F22),VEBTi)) = aa_VEB836174949890266638_VEBTi(vEBT_c6028912655521741485_VEBTi(aa_fun8651712798330174033_VEBTi(aTP_Lamm_fl(H),F1),aa_fun8772555301277947729_VEBTi(aTP_Lamm_fm(H),F22)),VEBTi) ).

% VEBTi.case_distrib
tff(fact_3484_VEBTi_Ocase__distrib,axiom,
    ! [H: fun_He5988046180898962017Heap_o,F1: fun_op5783775701855621896on_nat,F22: fun_o_4049874799390722306on_nat,VEBTi: vEBT_VEBTi] : aa_Hea662945835469240978Heap_o(H,aa_VEB2711320435582112505on_nat(vEBT_c6250501799366334488on_nat(F1,F22),VEBTi)) = aa_VEB8194921341634505877Heap_o(vEBT_c6104975476656191286Heap_o(aa_fun836855539625404138Heap_o(aTP_Lamm_fn(H),F1),aa_fun4676301296891262398Heap_o(aTP_Lamm_fo(H),F22)),VEBTi) ).

% VEBTi.case_distrib
tff(fact_3485_VEBTi_Ocase__distrib,axiom,
    ! [H: fun_He1783186893022644846on_nat,F1: fun_op5127434927081926045_VEBTi,F22: fun_o_6972675622308511895_VEBTi,VEBTi: vEBT_VEBTi] : aa_Hea6259128614175166183on_nat(H,aa_VEB836174949890266638_VEBTi(vEBT_c6028912655521741485_VEBTi(F1,F22),VEBTi)) = aa_VEB2711320435582112505on_nat(vEBT_c6250501799366334488on_nat(aa_fun7829374822385380327on_nat(aTP_Lamm_fp(H),F1),aa_fun4045834922583578215on_nat(aTP_Lamm_fq(H),F22)),VEBTi) ).

% VEBTi.case_distrib
tff(fact_3486_VEBTi_Ocase__distrib,axiom,
    ! [H: fun_He8247729123535230979_VEBTi,F1: fun_op5127434927081926045_VEBTi,F22: fun_o_6972675622308511895_VEBTi,VEBTi: vEBT_VEBTi] : aa_Hea2545666323642607868_VEBTi(H,aa_VEB836174949890266638_VEBTi(vEBT_c6028912655521741485_VEBTi(F1,F22),VEBTi)) = aa_VEB836174949890266638_VEBTi(vEBT_c6028912655521741485_VEBTi(aa_fun4118786628394255868_VEBTi(aTP_Lamm_fr(H),F1),aa_fun5654756598358634620_VEBTi(aTP_Lamm_fs(H),F22)),VEBTi) ).

% VEBTi.case_distrib
tff(fact_3487_VEBTi_Ocase__distrib,axiom,
    ! [H: fun_He4624006784692404214Heap_o,F1: fun_op5127434927081926045_VEBTi,F22: fun_o_6972675622308511895_VEBTi,VEBTi: vEBT_VEBTi] : aa_Hea7612170549154561063Heap_o(H,aa_VEB836174949890266638_VEBTi(vEBT_c6028912655521741485_VEBTi(F1,F22),VEBTi)) = aa_VEB8194921341634505877Heap_o(vEBT_c6104975476656191286Heap_o(aa_fun7400618842136392575Heap_o(aTP_Lamm_ft(H),F1),aa_fun3457972572965721299Heap_o(aTP_Lamm_fu(H),F22)),VEBTi) ).

% VEBTi.case_distrib
tff(fact_3488_VEBTi_Ocase__distrib,axiom,
    ! [H: fun_He6140748070426431971on_nat,F1: fun_op1763656958002597022Heap_o,F22: fun_o_1207901889202581752Heap_o,VEBTi: vEBT_VEBTi] : aa_Hea41712926108436044on_nat(H,aa_VEB8194921341634505877Heap_o(vEBT_c6104975476656191286Heap_o(F1,F22),VEBTi)) = aa_VEB2711320435582112505on_nat(vEBT_c6250501799366334488on_nat(aa_fun878802566620225708on_nat(aTP_Lamm_fv(H),F1),aa_fun3614415378023021644on_nat(aTP_Lamm_fw(H),F22)),VEBTi) ).

% VEBTi.case_distrib
tff(fact_3489_VEBTi_Ocase__distrib,axiom,
    ! [H: fun_He6517945810935171576_VEBTi,F1: fun_op1763656958002597022Heap_o,F22: fun_o_1207901889202581752Heap_o,VEBTi: vEBT_VEBTi] : aa_Hea4465279032261640929_VEBTi(H,aa_VEB8194921341634505877Heap_o(vEBT_c6104975476656191286Heap_o(F1,F22),VEBTi)) = aa_VEB836174949890266638_VEBTi(vEBT_c6028912655521741485_VEBTi(aa_fun7082494938152799553_VEBTi(aTP_Lamm_fx(H),F1),aa_fun8320297987776248033_VEBTi(aTP_Lamm_fy(H),F22)),VEBTi) ).

% VEBTi.case_distrib
tff(fact_3490_VEBTi_Ocase__distrib,axiom,
    ! [H: fun_He6597989805181038017Heap_o,F1: fun_op1763656958002597022Heap_o,F22: fun_o_1207901889202581752Heap_o,VEBTi: vEBT_VEBTi] : aa_Hea4824394052061459970Heap_o(H,aa_VEB8194921341634505877Heap_o(vEBT_c6104975476656191286Heap_o(F1,F22),VEBTi)) = aa_VEB8194921341634505877Heap_o(vEBT_c6104975476656191286Heap_o(aa_fun2115832704338266106Heap_o(aTP_Lamm_fz(H),F1),aa_fun3804294760359788590Heap_o(aTP_Lamm_ga(H),F22)),VEBTi) ).

% VEBTi.case_distrib
tff(fact_3491_VEBTi_Osimps_I5_J,axiom,
    ! [F1: fun_op5783775701855621896on_nat,F22: fun_o_4049874799390722306on_nat,X11a: option4927543243414619207at_nat,X12a: nat,X13a: array_VEBT_VEBTi,X14a: vEBT_VEBTi] : aa_VEB2711320435582112505on_nat(vEBT_c6250501799366334488on_nat(F1,F22),aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(X11a,X12a,X13a),X14a)) = aa_VEB2711320435582112505on_nat(aa_arr2955191963898321370on_nat(aa_nat5317369504323972463on_nat(aa_opt8624683241416710969on_nat(F1,X11a),X12a),X13a),X14a) ).

% VEBTi.simps(5)
tff(fact_3492_VEBTi_Osimps_I5_J,axiom,
    ! [F1: fun_op5127434927081926045_VEBTi,F22: fun_o_6972675622308511895_VEBTi,X11a: option4927543243414619207at_nat,X12a: nat,X13a: array_VEBT_VEBTi,X14a: vEBT_VEBTi] : aa_VEB836174949890266638_VEBTi(vEBT_c6028912655521741485_VEBTi(F1,F22),aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(X11a,X12a,X13a),X14a)) = aa_VEB836174949890266638_VEBTi(aa_arr2912372873376718191_VEBTi(aa_nat6445230840460233604_VEBTi(aa_opt5182041009420754510_VEBTi(F1,X11a),X12a),X13a),X14a) ).

% VEBTi.simps(5)
tff(fact_3493_VEBTi_Osimps_I5_J,axiom,
    ! [F1: fun_op1763656958002597022Heap_o,F22: fun_o_1207901889202581752Heap_o,X11a: option4927543243414619207at_nat,X12a: nat,X13a: array_VEBT_VEBTi,X14a: vEBT_VEBTi] : aa_VEB8194921341634505877Heap_o(vEBT_c6104975476656191286Heap_o(F1,F22),aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(X11a,X12a,X13a),X14a)) = aa_VEB8194921341634505877Heap_o(aa_arr2373296168907467658Heap_o(aa_nat8650107858160254923Heap_o(aa_opt6539295502271640599Heap_o(F1,X11a),X12a),X13a),X14a) ).

% VEBTi.simps(5)
tff(fact_3494_VEBTi_Osimps_I6_J,axiom,
    ! [F1: fun_op5783775701855621896on_nat,F22: fun_o_4049874799390722306on_nat,X21: $o,X22: $o] : aa_VEB2711320435582112505on_nat(vEBT_c6250501799366334488on_nat(F1,F22),vEBT_Leafi((X21),(X22))) = aa_o_H4215342640551218896on_nat(aa_o_f5606600903069576627on_nat(F22,(X21)),(X22)) ).

% VEBTi.simps(6)
tff(fact_3495_VEBTi_Osimps_I6_J,axiom,
    ! [F1: fun_op5127434927081926045_VEBTi,F22: fun_o_6972675622308511895_VEBTi,X21: $o,X22: $o] : aa_VEB836174949890266638_VEBTi(vEBT_c6028912655521741485_VEBTi(F1,F22),vEBT_Leafi((X21),(X22))) = aa_o_H7577119287286566245_VEBTi(aa_o_f5262348187908705736_VEBTi(F22,(X21)),(X22)) ).

% VEBTi.simps(6)
tff(fact_3496_VEBTi_Osimps_I6_J,axiom,
    ! [F1: fun_op1763656958002597022Heap_o,F22: fun_o_1207901889202581752Heap_o,X21: $o,X22: $o] : aa_VEB8194921341634505877Heap_o(vEBT_c6104975476656191286Heap_o(F1,F22),vEBT_Leafi((X21),(X22))) = aa_o_H7399033255965945470Heap_o(aa_o_f869145629572826609Heap_o(F22,(X21)),(X22)) ).

% VEBTi.simps(6)
tff(fact_3497_pinf_I1_J,axiom,
    ! [P: fun_real_o,P3: fun_real_o,Q: fun_real_o,Q6: fun_real_o] :
      ( ? [Z4: real] :
        ! [X2: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z4),X2)
         => ( aa_real_o(P,X2)
          <=> aa_real_o(P3,X2) ) )
     => ( ? [Z4: real] :
          ! [X2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z4),X2)
           => ( aa_real_o(Q,X2)
            <=> aa_real_o(Q6,X2) ) )
       => ? [Z3: real] :
          ! [X3: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z3),X3)
           => ( ( aa_real_o(P,X3)
                & aa_real_o(Q,X3) )
            <=> ( aa_real_o(P3,X3)
                & aa_real_o(Q6,X3) ) ) ) ) ) ).

% pinf(1)
tff(fact_3498_pinf_I1_J,axiom,
    ! [P: fun_rat_o,P3: fun_rat_o,Q: fun_rat_o,Q6: fun_rat_o] :
      ( ? [Z4: rat] :
        ! [X2: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z4),X2)
         => ( aa_rat_o(P,X2)
          <=> aa_rat_o(P3,X2) ) )
     => ( ? [Z4: rat] :
          ! [X2: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z4),X2)
           => ( aa_rat_o(Q,X2)
            <=> aa_rat_o(Q6,X2) ) )
       => ? [Z3: rat] :
          ! [X3: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z3),X3)
           => ( ( aa_rat_o(P,X3)
                & aa_rat_o(Q,X3) )
            <=> ( aa_rat_o(P3,X3)
                & aa_rat_o(Q6,X3) ) ) ) ) ) ).

% pinf(1)
tff(fact_3499_pinf_I1_J,axiom,
    ! [P: fun_num_o,P3: fun_num_o,Q: fun_num_o,Q6: fun_num_o] :
      ( ? [Z4: num] :
        ! [X2: num] :
          ( aa_num_o(ord_less_num(Z4),X2)
         => ( aa_num_o(P,X2)
          <=> aa_num_o(P3,X2) ) )
     => ( ? [Z4: num] :
          ! [X2: num] :
            ( aa_num_o(ord_less_num(Z4),X2)
           => ( aa_num_o(Q,X2)
            <=> aa_num_o(Q6,X2) ) )
       => ? [Z3: num] :
          ! [X3: num] :
            ( aa_num_o(ord_less_num(Z3),X3)
           => ( ( aa_num_o(P,X3)
                & aa_num_o(Q,X3) )
            <=> ( aa_num_o(P3,X3)
                & aa_num_o(Q6,X3) ) ) ) ) ) ).

% pinf(1)
tff(fact_3500_pinf_I1_J,axiom,
    ! [P: fun_nat_o,P3: fun_nat_o,Q: fun_nat_o,Q6: fun_nat_o] :
      ( ? [Z4: nat] :
        ! [X2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z4),X2)
         => ( aa_nat_o(P,X2)
          <=> aa_nat_o(P3,X2) ) )
     => ( ? [Z4: nat] :
          ! [X2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z4),X2)
           => ( aa_nat_o(Q,X2)
            <=> aa_nat_o(Q6,X2) ) )
       => ? [Z3: nat] :
          ! [X3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z3),X3)
           => ( ( aa_nat_o(P,X3)
                & aa_nat_o(Q,X3) )
            <=> ( aa_nat_o(P3,X3)
                & aa_nat_o(Q6,X3) ) ) ) ) ) ).

% pinf(1)
tff(fact_3501_pinf_I1_J,axiom,
    ! [P: fun_int_o,P3: fun_int_o,Q: fun_int_o,Q6: fun_int_o] :
      ( ? [Z4: int] :
        ! [X2: int] :
          ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z4),X2)
         => ( aa_int_o(P,X2)
          <=> aa_int_o(P3,X2) ) )
     => ( ? [Z4: int] :
          ! [X2: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z4),X2)
           => ( aa_int_o(Q,X2)
            <=> aa_int_o(Q6,X2) ) )
       => ? [Z3: int] :
          ! [X3: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z3),X3)
           => ( ( aa_int_o(P,X3)
                & aa_int_o(Q,X3) )
            <=> ( aa_int_o(P3,X3)
                & aa_int_o(Q6,X3) ) ) ) ) ) ).

% pinf(1)
tff(fact_3502_pinf_I2_J,axiom,
    ! [P: fun_real_o,P3: fun_real_o,Q: fun_real_o,Q6: fun_real_o] :
      ( ? [Z4: real] :
        ! [X2: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z4),X2)
         => ( aa_real_o(P,X2)
          <=> aa_real_o(P3,X2) ) )
     => ( ? [Z4: real] :
          ! [X2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z4),X2)
           => ( aa_real_o(Q,X2)
            <=> aa_real_o(Q6,X2) ) )
       => ? [Z3: real] :
          ! [X3: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z3),X3)
           => ( ( aa_real_o(P,X3)
                | aa_real_o(Q,X3) )
            <=> ( aa_real_o(P3,X3)
                | aa_real_o(Q6,X3) ) ) ) ) ) ).

% pinf(2)
tff(fact_3503_pinf_I2_J,axiom,
    ! [P: fun_rat_o,P3: fun_rat_o,Q: fun_rat_o,Q6: fun_rat_o] :
      ( ? [Z4: rat] :
        ! [X2: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z4),X2)
         => ( aa_rat_o(P,X2)
          <=> aa_rat_o(P3,X2) ) )
     => ( ? [Z4: rat] :
          ! [X2: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z4),X2)
           => ( aa_rat_o(Q,X2)
            <=> aa_rat_o(Q6,X2) ) )
       => ? [Z3: rat] :
          ! [X3: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z3),X3)
           => ( ( aa_rat_o(P,X3)
                | aa_rat_o(Q,X3) )
            <=> ( aa_rat_o(P3,X3)
                | aa_rat_o(Q6,X3) ) ) ) ) ) ).

% pinf(2)
tff(fact_3504_pinf_I2_J,axiom,
    ! [P: fun_num_o,P3: fun_num_o,Q: fun_num_o,Q6: fun_num_o] :
      ( ? [Z4: num] :
        ! [X2: num] :
          ( aa_num_o(ord_less_num(Z4),X2)
         => ( aa_num_o(P,X2)
          <=> aa_num_o(P3,X2) ) )
     => ( ? [Z4: num] :
          ! [X2: num] :
            ( aa_num_o(ord_less_num(Z4),X2)
           => ( aa_num_o(Q,X2)
            <=> aa_num_o(Q6,X2) ) )
       => ? [Z3: num] :
          ! [X3: num] :
            ( aa_num_o(ord_less_num(Z3),X3)
           => ( ( aa_num_o(P,X3)
                | aa_num_o(Q,X3) )
            <=> ( aa_num_o(P3,X3)
                | aa_num_o(Q6,X3) ) ) ) ) ) ).

% pinf(2)
tff(fact_3505_pinf_I2_J,axiom,
    ! [P: fun_nat_o,P3: fun_nat_o,Q: fun_nat_o,Q6: fun_nat_o] :
      ( ? [Z4: nat] :
        ! [X2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z4),X2)
         => ( aa_nat_o(P,X2)
          <=> aa_nat_o(P3,X2) ) )
     => ( ? [Z4: nat] :
          ! [X2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z4),X2)
           => ( aa_nat_o(Q,X2)
            <=> aa_nat_o(Q6,X2) ) )
       => ? [Z3: nat] :
          ! [X3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z3),X3)
           => ( ( aa_nat_o(P,X3)
                | aa_nat_o(Q,X3) )
            <=> ( aa_nat_o(P3,X3)
                | aa_nat_o(Q6,X3) ) ) ) ) ) ).

% pinf(2)
tff(fact_3506_pinf_I2_J,axiom,
    ! [P: fun_int_o,P3: fun_int_o,Q: fun_int_o,Q6: fun_int_o] :
      ( ? [Z4: int] :
        ! [X2: int] :
          ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z4),X2)
         => ( aa_int_o(P,X2)
          <=> aa_int_o(P3,X2) ) )
     => ( ? [Z4: int] :
          ! [X2: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z4),X2)
           => ( aa_int_o(Q,X2)
            <=> aa_int_o(Q6,X2) ) )
       => ? [Z3: int] :
          ! [X3: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z3),X3)
           => ( ( aa_int_o(P,X3)
                | aa_int_o(Q,X3) )
            <=> ( aa_int_o(P3,X3)
                | aa_int_o(Q6,X3) ) ) ) ) ) ).

% pinf(2)
tff(fact_3507_pinf_I3_J,axiom,
    ! [Ta: real] :
    ? [Z3: real] :
    ! [X3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z3),X3)
     => ( X3 != Ta ) ) ).

% pinf(3)
tff(fact_3508_pinf_I3_J,axiom,
    ! [Ta: rat] :
    ? [Z3: rat] :
    ! [X3: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z3),X3)
     => ( X3 != Ta ) ) ).

% pinf(3)
tff(fact_3509_pinf_I3_J,axiom,
    ! [Ta: num] :
    ? [Z3: num] :
    ! [X3: num] :
      ( aa_num_o(ord_less_num(Z3),X3)
     => ( X3 != Ta ) ) ).

% pinf(3)
tff(fact_3510_pinf_I3_J,axiom,
    ! [Ta: nat] :
    ? [Z3: nat] :
    ! [X3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z3),X3)
     => ( X3 != Ta ) ) ).

% pinf(3)
tff(fact_3511_pinf_I3_J,axiom,
    ! [Ta: int] :
    ? [Z3: int] :
    ! [X3: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z3),X3)
     => ( X3 != Ta ) ) ).

% pinf(3)
tff(fact_3512_pinf_I4_J,axiom,
    ! [Ta: real] :
    ? [Z3: real] :
    ! [X3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z3),X3)
     => ( X3 != Ta ) ) ).

% pinf(4)
tff(fact_3513_pinf_I4_J,axiom,
    ! [Ta: rat] :
    ? [Z3: rat] :
    ! [X3: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z3),X3)
     => ( X3 != Ta ) ) ).

% pinf(4)
tff(fact_3514_pinf_I4_J,axiom,
    ! [Ta: num] :
    ? [Z3: num] :
    ! [X3: num] :
      ( aa_num_o(ord_less_num(Z3),X3)
     => ( X3 != Ta ) ) ).

% pinf(4)
tff(fact_3515_pinf_I4_J,axiom,
    ! [Ta: nat] :
    ? [Z3: nat] :
    ! [X3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z3),X3)
     => ( X3 != Ta ) ) ).

% pinf(4)
tff(fact_3516_pinf_I4_J,axiom,
    ! [Ta: int] :
    ? [Z3: int] :
    ! [X3: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z3),X3)
     => ( X3 != Ta ) ) ).

% pinf(4)
tff(fact_3517_pinf_I5_J,axiom,
    ! [Ta: real] :
    ? [Z3: real] :
    ! [X3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z3),X3)
     => ~ aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Ta) ) ).

% pinf(5)
tff(fact_3518_pinf_I5_J,axiom,
    ! [Ta: rat] :
    ? [Z3: rat] :
    ! [X3: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z3),X3)
     => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Ta) ) ).

% pinf(5)
tff(fact_3519_pinf_I5_J,axiom,
    ! [Ta: num] :
    ? [Z3: num] :
    ! [X3: num] :
      ( aa_num_o(ord_less_num(Z3),X3)
     => ~ aa_num_o(ord_less_num(X3),Ta) ) ).

% pinf(5)
tff(fact_3520_pinf_I5_J,axiom,
    ! [Ta: nat] :
    ? [Z3: nat] :
    ! [X3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z3),X3)
     => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Ta) ) ).

% pinf(5)
tff(fact_3521_pinf_I5_J,axiom,
    ! [Ta: int] :
    ? [Z3: int] :
    ! [X3: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z3),X3)
     => ~ aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Ta) ) ).

% pinf(5)
tff(fact_3522_pinf_I7_J,axiom,
    ! [Ta: real] :
    ? [Z3: real] :
    ! [X3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z3),X3)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,Ta),X3) ) ).

% pinf(7)
tff(fact_3523_pinf_I7_J,axiom,
    ! [Ta: rat] :
    ? [Z3: rat] :
    ! [X3: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z3),X3)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Ta),X3) ) ).

% pinf(7)
tff(fact_3524_pinf_I7_J,axiom,
    ! [Ta: num] :
    ? [Z3: num] :
    ! [X3: num] :
      ( aa_num_o(ord_less_num(Z3),X3)
     => aa_num_o(ord_less_num(Ta),X3) ) ).

% pinf(7)
tff(fact_3525_pinf_I7_J,axiom,
    ! [Ta: nat] :
    ? [Z3: nat] :
    ! [X3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z3),X3)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ta),X3) ) ).

% pinf(7)
tff(fact_3526_pinf_I7_J,axiom,
    ! [Ta: int] :
    ? [Z3: int] :
    ! [X3: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z3),X3)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,Ta),X3) ) ).

% pinf(7)
tff(fact_3527_minf_I1_J,axiom,
    ! [P: fun_real_o,P3: fun_real_o,Q: fun_real_o,Q6: fun_real_o] :
      ( ? [Z4: real] :
        ! [X2: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,X2),Z4)
         => ( aa_real_o(P,X2)
          <=> aa_real_o(P3,X2) ) )
     => ( ? [Z4: real] :
          ! [X2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,X2),Z4)
           => ( aa_real_o(Q,X2)
            <=> aa_real_o(Q6,X2) ) )
       => ? [Z3: real] :
          ! [X3: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Z3)
           => ( ( aa_real_o(P,X3)
                & aa_real_o(Q,X3) )
            <=> ( aa_real_o(P3,X3)
                & aa_real_o(Q6,X3) ) ) ) ) ) ).

% minf(1)
tff(fact_3528_minf_I1_J,axiom,
    ! [P: fun_rat_o,P3: fun_rat_o,Q: fun_rat_o,Q6: fun_rat_o] :
      ( ? [Z4: rat] :
        ! [X2: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X2),Z4)
         => ( aa_rat_o(P,X2)
          <=> aa_rat_o(P3,X2) ) )
     => ( ? [Z4: rat] :
          ! [X2: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X2),Z4)
           => ( aa_rat_o(Q,X2)
            <=> aa_rat_o(Q6,X2) ) )
       => ? [Z3: rat] :
          ! [X3: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Z3)
           => ( ( aa_rat_o(P,X3)
                & aa_rat_o(Q,X3) )
            <=> ( aa_rat_o(P3,X3)
                & aa_rat_o(Q6,X3) ) ) ) ) ) ).

% minf(1)
tff(fact_3529_minf_I1_J,axiom,
    ! [P: fun_num_o,P3: fun_num_o,Q: fun_num_o,Q6: fun_num_o] :
      ( ? [Z4: num] :
        ! [X2: num] :
          ( aa_num_o(ord_less_num(X2),Z4)
         => ( aa_num_o(P,X2)
          <=> aa_num_o(P3,X2) ) )
     => ( ? [Z4: num] :
          ! [X2: num] :
            ( aa_num_o(ord_less_num(X2),Z4)
           => ( aa_num_o(Q,X2)
            <=> aa_num_o(Q6,X2) ) )
       => ? [Z3: num] :
          ! [X3: num] :
            ( aa_num_o(ord_less_num(X3),Z3)
           => ( ( aa_num_o(P,X3)
                & aa_num_o(Q,X3) )
            <=> ( aa_num_o(P3,X3)
                & aa_num_o(Q6,X3) ) ) ) ) ) ).

% minf(1)
tff(fact_3530_minf_I1_J,axiom,
    ! [P: fun_nat_o,P3: fun_nat_o,Q: fun_nat_o,Q6: fun_nat_o] :
      ( ? [Z4: nat] :
        ! [X2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X2),Z4)
         => ( aa_nat_o(P,X2)
          <=> aa_nat_o(P3,X2) ) )
     => ( ? [Z4: nat] :
          ! [X2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X2),Z4)
           => ( aa_nat_o(Q,X2)
            <=> aa_nat_o(Q6,X2) ) )
       => ? [Z3: nat] :
          ! [X3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Z3)
           => ( ( aa_nat_o(P,X3)
                & aa_nat_o(Q,X3) )
            <=> ( aa_nat_o(P3,X3)
                & aa_nat_o(Q6,X3) ) ) ) ) ) ).

% minf(1)
tff(fact_3531_minf_I1_J,axiom,
    ! [P: fun_int_o,P3: fun_int_o,Q: fun_int_o,Q6: fun_int_o] :
      ( ? [Z4: int] :
        ! [X2: int] :
          ( aa_int_o(aa_int_fun_int_o(ord_less_int,X2),Z4)
         => ( aa_int_o(P,X2)
          <=> aa_int_o(P3,X2) ) )
     => ( ? [Z4: int] :
          ! [X2: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_int,X2),Z4)
           => ( aa_int_o(Q,X2)
            <=> aa_int_o(Q6,X2) ) )
       => ? [Z3: int] :
          ! [X3: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Z3)
           => ( ( aa_int_o(P,X3)
                & aa_int_o(Q,X3) )
            <=> ( aa_int_o(P3,X3)
                & aa_int_o(Q6,X3) ) ) ) ) ) ).

% minf(1)
tff(fact_3532_minf_I2_J,axiom,
    ! [P: fun_real_o,P3: fun_real_o,Q: fun_real_o,Q6: fun_real_o] :
      ( ? [Z4: real] :
        ! [X2: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,X2),Z4)
         => ( aa_real_o(P,X2)
          <=> aa_real_o(P3,X2) ) )
     => ( ? [Z4: real] :
          ! [X2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,X2),Z4)
           => ( aa_real_o(Q,X2)
            <=> aa_real_o(Q6,X2) ) )
       => ? [Z3: real] :
          ! [X3: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Z3)
           => ( ( aa_real_o(P,X3)
                | aa_real_o(Q,X3) )
            <=> ( aa_real_o(P3,X3)
                | aa_real_o(Q6,X3) ) ) ) ) ) ).

% minf(2)
tff(fact_3533_minf_I2_J,axiom,
    ! [P: fun_rat_o,P3: fun_rat_o,Q: fun_rat_o,Q6: fun_rat_o] :
      ( ? [Z4: rat] :
        ! [X2: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X2),Z4)
         => ( aa_rat_o(P,X2)
          <=> aa_rat_o(P3,X2) ) )
     => ( ? [Z4: rat] :
          ! [X2: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X2),Z4)
           => ( aa_rat_o(Q,X2)
            <=> aa_rat_o(Q6,X2) ) )
       => ? [Z3: rat] :
          ! [X3: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Z3)
           => ( ( aa_rat_o(P,X3)
                | aa_rat_o(Q,X3) )
            <=> ( aa_rat_o(P3,X3)
                | aa_rat_o(Q6,X3) ) ) ) ) ) ).

% minf(2)
tff(fact_3534_minf_I2_J,axiom,
    ! [P: fun_num_o,P3: fun_num_o,Q: fun_num_o,Q6: fun_num_o] :
      ( ? [Z4: num] :
        ! [X2: num] :
          ( aa_num_o(ord_less_num(X2),Z4)
         => ( aa_num_o(P,X2)
          <=> aa_num_o(P3,X2) ) )
     => ( ? [Z4: num] :
          ! [X2: num] :
            ( aa_num_o(ord_less_num(X2),Z4)
           => ( aa_num_o(Q,X2)
            <=> aa_num_o(Q6,X2) ) )
       => ? [Z3: num] :
          ! [X3: num] :
            ( aa_num_o(ord_less_num(X3),Z3)
           => ( ( aa_num_o(P,X3)
                | aa_num_o(Q,X3) )
            <=> ( aa_num_o(P3,X3)
                | aa_num_o(Q6,X3) ) ) ) ) ) ).

% minf(2)
tff(fact_3535_minf_I2_J,axiom,
    ! [P: fun_nat_o,P3: fun_nat_o,Q: fun_nat_o,Q6: fun_nat_o] :
      ( ? [Z4: nat] :
        ! [X2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X2),Z4)
         => ( aa_nat_o(P,X2)
          <=> aa_nat_o(P3,X2) ) )
     => ( ? [Z4: nat] :
          ! [X2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X2),Z4)
           => ( aa_nat_o(Q,X2)
            <=> aa_nat_o(Q6,X2) ) )
       => ? [Z3: nat] :
          ! [X3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Z3)
           => ( ( aa_nat_o(P,X3)
                | aa_nat_o(Q,X3) )
            <=> ( aa_nat_o(P3,X3)
                | aa_nat_o(Q6,X3) ) ) ) ) ) ).

% minf(2)
tff(fact_3536_minf_I2_J,axiom,
    ! [P: fun_int_o,P3: fun_int_o,Q: fun_int_o,Q6: fun_int_o] :
      ( ? [Z4: int] :
        ! [X2: int] :
          ( aa_int_o(aa_int_fun_int_o(ord_less_int,X2),Z4)
         => ( aa_int_o(P,X2)
          <=> aa_int_o(P3,X2) ) )
     => ( ? [Z4: int] :
          ! [X2: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_int,X2),Z4)
           => ( aa_int_o(Q,X2)
            <=> aa_int_o(Q6,X2) ) )
       => ? [Z3: int] :
          ! [X3: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Z3)
           => ( ( aa_int_o(P,X3)
                | aa_int_o(Q,X3) )
            <=> ( aa_int_o(P3,X3)
                | aa_int_o(Q6,X3) ) ) ) ) ) ).

% minf(2)
tff(fact_3537_minf_I3_J,axiom,
    ! [Ta: real] :
    ? [Z3: real] :
    ! [X3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Z3)
     => ( X3 != Ta ) ) ).

% minf(3)
tff(fact_3538_minf_I3_J,axiom,
    ! [Ta: rat] :
    ? [Z3: rat] :
    ! [X3: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Z3)
     => ( X3 != Ta ) ) ).

% minf(3)
tff(fact_3539_minf_I3_J,axiom,
    ! [Ta: num] :
    ? [Z3: num] :
    ! [X3: num] :
      ( aa_num_o(ord_less_num(X3),Z3)
     => ( X3 != Ta ) ) ).

% minf(3)
tff(fact_3540_minf_I3_J,axiom,
    ! [Ta: nat] :
    ? [Z3: nat] :
    ! [X3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Z3)
     => ( X3 != Ta ) ) ).

% minf(3)
tff(fact_3541_minf_I3_J,axiom,
    ! [Ta: int] :
    ? [Z3: int] :
    ! [X3: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Z3)
     => ( X3 != Ta ) ) ).

% minf(3)
tff(fact_3542_minf_I4_J,axiom,
    ! [Ta: real] :
    ? [Z3: real] :
    ! [X3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Z3)
     => ( X3 != Ta ) ) ).

% minf(4)
tff(fact_3543_minf_I4_J,axiom,
    ! [Ta: rat] :
    ? [Z3: rat] :
    ! [X3: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Z3)
     => ( X3 != Ta ) ) ).

% minf(4)
tff(fact_3544_minf_I4_J,axiom,
    ! [Ta: num] :
    ? [Z3: num] :
    ! [X3: num] :
      ( aa_num_o(ord_less_num(X3),Z3)
     => ( X3 != Ta ) ) ).

% minf(4)
tff(fact_3545_minf_I4_J,axiom,
    ! [Ta: nat] :
    ? [Z3: nat] :
    ! [X3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Z3)
     => ( X3 != Ta ) ) ).

% minf(4)
tff(fact_3546_minf_I4_J,axiom,
    ! [Ta: int] :
    ? [Z3: int] :
    ! [X3: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Z3)
     => ( X3 != Ta ) ) ).

% minf(4)
tff(fact_3547_minf_I5_J,axiom,
    ! [Ta: real] :
    ? [Z3: real] :
    ! [X3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Z3)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Ta) ) ).

% minf(5)
tff(fact_3548_minf_I5_J,axiom,
    ! [Ta: rat] :
    ? [Z3: rat] :
    ! [X3: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Z3)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Ta) ) ).

% minf(5)
tff(fact_3549_minf_I5_J,axiom,
    ! [Ta: num] :
    ? [Z3: num] :
    ! [X3: num] :
      ( aa_num_o(ord_less_num(X3),Z3)
     => aa_num_o(ord_less_num(X3),Ta) ) ).

% minf(5)
tff(fact_3550_minf_I5_J,axiom,
    ! [Ta: nat] :
    ? [Z3: nat] :
    ! [X3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Z3)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Ta) ) ).

% minf(5)
tff(fact_3551_minf_I5_J,axiom,
    ! [Ta: int] :
    ? [Z3: int] :
    ! [X3: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Z3)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Ta) ) ).

% minf(5)
tff(fact_3552_minf_I7_J,axiom,
    ! [Ta: real] :
    ? [Z3: real] :
    ! [X3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Z3)
     => ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Ta),X3) ) ).

% minf(7)
tff(fact_3553_minf_I7_J,axiom,
    ! [Ta: rat] :
    ? [Z3: rat] :
    ! [X3: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Z3)
     => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Ta),X3) ) ).

% minf(7)
tff(fact_3554_minf_I7_J,axiom,
    ! [Ta: num] :
    ? [Z3: num] :
    ! [X3: num] :
      ( aa_num_o(ord_less_num(X3),Z3)
     => ~ aa_num_o(ord_less_num(Ta),X3) ) ).

% minf(7)
tff(fact_3555_minf_I7_J,axiom,
    ! [Ta: nat] :
    ? [Z3: nat] :
    ! [X3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Z3)
     => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ta),X3) ) ).

% minf(7)
tff(fact_3556_minf_I7_J,axiom,
    ! [Ta: int] :
    ? [Z3: int] :
    ! [X3: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Z3)
     => ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Ta),X3) ) ).

% minf(7)
tff(fact_3557_pinf_I6_J,axiom,
    ! [Ta: real] :
    ? [Z3: real] :
    ! [X3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z3),X3)
     => ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X3),Ta) ) ).

% pinf(6)
tff(fact_3558_pinf_I6_J,axiom,
    ! [Ta: rat] :
    ? [Z3: rat] :
    ! [X3: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z3),X3)
     => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Ta) ) ).

% pinf(6)
tff(fact_3559_pinf_I6_J,axiom,
    ! [Ta: num] :
    ? [Z3: num] :
    ! [X3: num] :
      ( aa_num_o(ord_less_num(Z3),X3)
     => ~ aa_num_o(ord_less_eq_num(X3),Ta) ) ).

% pinf(6)
tff(fact_3560_pinf_I6_J,axiom,
    ! [Ta: nat] :
    ? [Z3: nat] :
    ! [X3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z3),X3)
     => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),Ta) ) ).

% pinf(6)
tff(fact_3561_pinf_I6_J,axiom,
    ! [Ta: int] :
    ? [Z3: int] :
    ! [X3: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z3),X3)
     => ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X3),Ta) ) ).

% pinf(6)
tff(fact_3562_pinf_I8_J,axiom,
    ! [Ta: real] :
    ? [Z3: real] :
    ! [X3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z3),X3)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Ta),X3) ) ).

% pinf(8)
tff(fact_3563_pinf_I8_J,axiom,
    ! [Ta: rat] :
    ? [Z3: rat] :
    ! [X3: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z3),X3)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Ta),X3) ) ).

% pinf(8)
tff(fact_3564_pinf_I8_J,axiom,
    ! [Ta: num] :
    ? [Z3: num] :
    ! [X3: num] :
      ( aa_num_o(ord_less_num(Z3),X3)
     => aa_num_o(ord_less_eq_num(Ta),X3) ) ).

% pinf(8)
tff(fact_3565_pinf_I8_J,axiom,
    ! [Ta: nat] :
    ? [Z3: nat] :
    ! [X3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z3),X3)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ta),X3) ) ).

% pinf(8)
tff(fact_3566_pinf_I8_J,axiom,
    ! [Ta: int] :
    ? [Z3: int] :
    ! [X3: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z3),X3)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ta),X3) ) ).

% pinf(8)
tff(fact_3567_minf_I6_J,axiom,
    ! [Ta: real] :
    ? [Z3: real] :
    ! [X3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Z3)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X3),Ta) ) ).

% minf(6)
tff(fact_3568_minf_I6_J,axiom,
    ! [Ta: rat] :
    ? [Z3: rat] :
    ! [X3: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Z3)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Ta) ) ).

% minf(6)
tff(fact_3569_minf_I6_J,axiom,
    ! [Ta: num] :
    ? [Z3: num] :
    ! [X3: num] :
      ( aa_num_o(ord_less_num(X3),Z3)
     => aa_num_o(ord_less_eq_num(X3),Ta) ) ).

% minf(6)
tff(fact_3570_minf_I6_J,axiom,
    ! [Ta: nat] :
    ? [Z3: nat] :
    ! [X3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Z3)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),Ta) ) ).

% minf(6)
tff(fact_3571_minf_I6_J,axiom,
    ! [Ta: int] :
    ? [Z3: int] :
    ! [X3: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Z3)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X3),Ta) ) ).

% minf(6)
tff(fact_3572_minf_I8_J,axiom,
    ! [Ta: real] :
    ? [Z3: real] :
    ! [X3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Z3)
     => ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Ta),X3) ) ).

% minf(8)
tff(fact_3573_minf_I8_J,axiom,
    ! [Ta: rat] :
    ? [Z3: rat] :
    ! [X3: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Z3)
     => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Ta),X3) ) ).

% minf(8)
tff(fact_3574_minf_I8_J,axiom,
    ! [Ta: num] :
    ? [Z3: num] :
    ! [X3: num] :
      ( aa_num_o(ord_less_num(X3),Z3)
     => ~ aa_num_o(ord_less_eq_num(Ta),X3) ) ).

% minf(8)
tff(fact_3575_minf_I8_J,axiom,
    ! [Ta: nat] :
    ? [Z3: nat] :
    ! [X3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Z3)
     => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ta),X3) ) ).

% minf(8)
tff(fact_3576_minf_I8_J,axiom,
    ! [Ta: int] :
    ? [Z3: int] :
    ! [X3: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Z3)
     => ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ta),X3) ) ).

% minf(8)
tff(fact_3577_inf__period_I1_J,axiom,
    ! [P: fun_real_o,D5: real,Q: fun_real_o] :
      ( ! [X2: real,K3: real] :
          ( aa_real_o(P,X2)
        <=> aa_real_o(P,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,K3),D5))) )
     => ( ! [X2: real,K3: real] :
            ( aa_real_o(Q,X2)
          <=> aa_real_o(Q,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,K3),D5))) )
       => ! [X3: real,K5: real] :
            ( ( aa_real_o(P,X3)
              & aa_real_o(Q,X3) )
          <=> ( aa_real_o(P,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X3),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,K5),D5)))
              & aa_real_o(Q,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X3),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,K5),D5))) ) ) ) ) ).

% inf_period(1)
tff(fact_3578_inf__period_I1_J,axiom,
    ! [P: fun_rat_o,D5: rat,Q: fun_rat_o] :
      ( ! [X2: rat,K3: rat] :
          ( aa_rat_o(P,X2)
        <=> aa_rat_o(P,minus_minus_rat(X2,aa_rat_rat(times_times_rat(K3),D5))) )
     => ( ! [X2: rat,K3: rat] :
            ( aa_rat_o(Q,X2)
          <=> aa_rat_o(Q,minus_minus_rat(X2,aa_rat_rat(times_times_rat(K3),D5))) )
       => ! [X3: rat,K5: rat] :
            ( ( aa_rat_o(P,X3)
              & aa_rat_o(Q,X3) )
          <=> ( aa_rat_o(P,minus_minus_rat(X3,aa_rat_rat(times_times_rat(K5),D5)))
              & aa_rat_o(Q,minus_minus_rat(X3,aa_rat_rat(times_times_rat(K5),D5))) ) ) ) ) ).

% inf_period(1)
tff(fact_3579_inf__period_I1_J,axiom,
    ! [P: fun_int_o,D5: int,Q: fun_int_o] :
      ( ! [X2: int,K3: int] :
          ( aa_int_o(P,X2)
        <=> aa_int_o(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X2),aa_int_int(aa_int_fun_int_int(times_times_int,K3),D5))) )
     => ( ! [X2: int,K3: int] :
            ( aa_int_o(Q,X2)
          <=> aa_int_o(Q,aa_int_int(aa_int_fun_int_int(minus_minus_int,X2),aa_int_int(aa_int_fun_int_int(times_times_int,K3),D5))) )
       => ! [X3: int,K5: int] :
            ( ( aa_int_o(P,X3)
              & aa_int_o(Q,X3) )
          <=> ( aa_int_o(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),aa_int_int(aa_int_fun_int_int(times_times_int,K5),D5)))
              & aa_int_o(Q,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),aa_int_int(aa_int_fun_int_int(times_times_int,K5),D5))) ) ) ) ) ).

% inf_period(1)
tff(fact_3580_inf__period_I2_J,axiom,
    ! [P: fun_real_o,D5: real,Q: fun_real_o] :
      ( ! [X2: real,K3: real] :
          ( aa_real_o(P,X2)
        <=> aa_real_o(P,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,K3),D5))) )
     => ( ! [X2: real,K3: real] :
            ( aa_real_o(Q,X2)
          <=> aa_real_o(Q,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,K3),D5))) )
       => ! [X3: real,K5: real] :
            ( ( aa_real_o(P,X3)
              | aa_real_o(Q,X3) )
          <=> ( aa_real_o(P,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X3),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,K5),D5)))
              | aa_real_o(Q,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X3),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,K5),D5))) ) ) ) ) ).

% inf_period(2)
tff(fact_3581_inf__period_I2_J,axiom,
    ! [P: fun_rat_o,D5: rat,Q: fun_rat_o] :
      ( ! [X2: rat,K3: rat] :
          ( aa_rat_o(P,X2)
        <=> aa_rat_o(P,minus_minus_rat(X2,aa_rat_rat(times_times_rat(K3),D5))) )
     => ( ! [X2: rat,K3: rat] :
            ( aa_rat_o(Q,X2)
          <=> aa_rat_o(Q,minus_minus_rat(X2,aa_rat_rat(times_times_rat(K3),D5))) )
       => ! [X3: rat,K5: rat] :
            ( ( aa_rat_o(P,X3)
              | aa_rat_o(Q,X3) )
          <=> ( aa_rat_o(P,minus_minus_rat(X3,aa_rat_rat(times_times_rat(K5),D5)))
              | aa_rat_o(Q,minus_minus_rat(X3,aa_rat_rat(times_times_rat(K5),D5))) ) ) ) ) ).

% inf_period(2)
tff(fact_3582_inf__period_I2_J,axiom,
    ! [P: fun_int_o,D5: int,Q: fun_int_o] :
      ( ! [X2: int,K3: int] :
          ( aa_int_o(P,X2)
        <=> aa_int_o(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X2),aa_int_int(aa_int_fun_int_int(times_times_int,K3),D5))) )
     => ( ! [X2: int,K3: int] :
            ( aa_int_o(Q,X2)
          <=> aa_int_o(Q,aa_int_int(aa_int_fun_int_int(minus_minus_int,X2),aa_int_int(aa_int_fun_int_int(times_times_int,K3),D5))) )
       => ! [X3: int,K5: int] :
            ( ( aa_int_o(P,X3)
              | aa_int_o(Q,X3) )
          <=> ( aa_int_o(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),aa_int_int(aa_int_fun_int_int(times_times_int,K5),D5)))
              | aa_int_o(Q,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),aa_int_int(aa_int_fun_int_int(times_times_int,K5),D5))) ) ) ) ) ).

% inf_period(2)
tff(fact_3583_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_092_060_094sub_062u_092_060_094sub_062p_Ocases,axiom,
    ! [Xb: nat] :
      ( ( Xb != zero_zero_nat )
     => ( ( Xb != aa_nat_nat(suc,zero_zero_nat) )
       => ~ ! [Va2: nat] : Xb != aa_nat_nat(suc,aa_nat_nat(suc,Va2)) ) ) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d\<^sub>u\<^sub>p.cases
tff(fact_3584_imp__le__cong,axiom,
    ! [Xb: int,X4: int,P: $o,P3: $o] :
      ( ( Xb = X4 )
     => ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X4)
         => ( (P)
          <=> (P3) ) )
       => ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xb)
           => (P) )
        <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X4)
           => (P3) ) ) ) ) ).

% imp_le_cong
tff(fact_3585_conj__le__cong,axiom,
    ! [Xb: int,X4: int,P: $o,P3: $o] :
      ( ( Xb = X4 )
     => ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X4)
         => ( (P)
          <=> (P3) ) )
       => ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xb)
            & (P) )
        <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X4)
            & (P3) ) ) ) ) ).

% conj_le_cong
tff(fact_3586_pred__bound__size__univ_H,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,U: real,Xb: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Ta),Nb)
     => ( ( U = aa_nat_real(aa_real_fun_nat_real(power_power_real,numeral_numeral_real(aa_num_num(bit0,one))),Nb) )
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_p_r_e_d2,Ta),Xb))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,numeral_numeral_real(aa_num_num(bit0,one))),aa_real_real(aa_rea7340605660300375871l_real(log,numeral_numeral_real(aa_num_num(bit0,one))),aa_real_real(aa_rea7340605660300375871l_real(log,numeral_numeral_real(aa_num_num(bit0,one))),U)))) ) ) ).

% pred_bound_size_univ'
tff(fact_3587_succ__bound__size__univ_H,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,U: real,Xb: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Ta),Nb)
     => ( ( U = aa_nat_real(aa_real_fun_nat_real(power_power_real,numeral_numeral_real(aa_num_num(bit0,one))),Nb) )
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_s_u_c_c2,Ta),Xb))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,numeral_numeral_real(aa_num_num(bit0,one))),aa_real_real(aa_rea7340605660300375871l_real(log,numeral_numeral_real(aa_num_num(bit0,one))),aa_real_real(aa_rea7340605660300375871l_real(log,numeral_numeral_real(aa_num_num(bit0,one))),U)))) ) ) ).

% succ_bound_size_univ'
tff(fact_3588_member__bound__size__univ,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,U: real,Xb: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Ta),Nb)
     => ( ( U = aa_nat_real(aa_real_fun_nat_real(power_power_real,numeral_numeral_real(aa_num_num(bit0,one))),Nb) )
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_m_e_m_b_e_r,Ta),Xb))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,numeral_numeral_real(aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,one)))))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,one))))),aa_real_real(aa_rea7340605660300375871l_real(log,numeral_numeral_real(aa_num_num(bit0,one))),aa_real_real(aa_rea7340605660300375871l_real(log,numeral_numeral_real(aa_num_num(bit0,one))),U))))) ) ) ).

% member_bound_size_univ
tff(fact_3589_VEBT__internal_Ocnt_Osimps_I1_J,axiom,
    ! [A: $o,B: $o] : aa_VEBT_VEBT_real(vEBT_VEBT_cnt,vEBT_Leaf((A),(B))) = one_one_real ).

% VEBT_internal.cnt.simps(1)
tff(fact_3590_VEBT__internal_Ocnt_H_Osimps_I1_J,axiom,
    ! [A: $o,B: $o] : aa_VEBT_VEBT_nat(vEBT_VEBT_cnt2,vEBT_Leaf((A),(B))) = one_one_nat ).

% VEBT_internal.cnt'.simps(1)
tff(fact_3591_plusinfinity,axiom,
    ! [D: int,P3: fun_int_o,P: fun_int_o] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D)
     => ( ! [X2: int,K3: int] :
            ( aa_int_o(P3,X2)
          <=> aa_int_o(P3,aa_int_int(aa_int_fun_int_int(minus_minus_int,X2),aa_int_int(aa_int_fun_int_int(times_times_int,K3),D))) )
       => ( ? [Z4: int] :
            ! [X2: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z4),X2)
             => ( aa_int_o(P,X2)
              <=> aa_int_o(P3,X2) ) )
         => ( ? [X_12: int] : aa_int_o(P3,X_12)
           => ? [X_13: int] : aa_int_o(P,X_13) ) ) ) ) ).

% plusinfinity
tff(fact_3592_minusinfinity,axiom,
    ! [D: int,P1: fun_int_o,P: fun_int_o] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D)
     => ( ! [X2: int,K3: int] :
            ( aa_int_o(P1,X2)
          <=> aa_int_o(P1,aa_int_int(aa_int_fun_int_int(minus_minus_int,X2),aa_int_int(aa_int_fun_int_int(times_times_int,K3),D))) )
       => ( ? [Z4: int] :
            ! [X2: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,X2),Z4)
             => ( aa_int_o(P,X2)
              <=> aa_int_o(P1,X2) ) )
         => ( ? [X_12: int] : aa_int_o(P1,X_12)
           => ? [X_13: int] : aa_int_o(P,X_13) ) ) ) ) ).

% minusinfinity
tff(fact_3593_incr__mult__lemma,axiom,
    ! [D: int,P: fun_int_o,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D)
     => ( ! [X2: int] :
            ( aa_int_o(P,X2)
           => aa_int_o(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,X2),D)) )
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K)
         => ! [X3: int] :
              ( aa_int_o(P,X3)
             => aa_int_o(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),aa_int_int(aa_int_fun_int_int(times_times_int,K),D))) ) ) ) ) ).

% incr_mult_lemma
tff(fact_3594_decr__mult__lemma,axiom,
    ! [D: int,P: fun_int_o,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D)
     => ( ! [X2: int] :
            ( aa_int_o(P,X2)
           => aa_int_o(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X2),D)) )
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K)
         => ! [X3: int] :
              ( aa_int_o(P,X3)
             => aa_int_o(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),aa_int_int(aa_int_fun_int_int(times_times_int,K),D))) ) ) ) ) ).

% decr_mult_lemma
tff(fact_3595_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_Osimps_I1_J,axiom,
    aa_nat_nat(vEBT_V8646137997579335489_i_l_d,zero_zero_nat) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one))) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d.simps(1)
tff(fact_3596_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_092_060_094sub_062u_092_060_094sub_062p_Osimps_I1_J,axiom,
    aa_nat_nat(vEBT_V8346862874174094_d_u_p,zero_zero_nat) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d\<^sub>u\<^sub>p.simps(1)
tff(fact_3597_VEBT__internal_Ospace_H_Osimps_I1_J,axiom,
    ! [A: $o,B: $o] : aa_VEBT_VEBT_nat(vEBT_VEBT_space2,vEBT_Leaf((A),(B))) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one))) ).

% VEBT_internal.space'.simps(1)
tff(fact_3598_VEBT__internal_Ospace_Osimps_I1_J,axiom,
    ! [A: $o,B: $o] : aa_VEBT_VEBT_nat(vEBT_VEBT_space,vEBT_Leaf((A),(B))) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)) ).

% VEBT_internal.space.simps(1)
tff(fact_3599_log__pow__cancel,axiom,
    ! [A: real,B: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( ( A != one_one_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(log,A),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),B)) = aa_nat_real(semiri5074537144036343181t_real,B) ) ) ) ).

% log_pow_cancel
tff(fact_3600_log__le__cancel__iff,axiom,
    ! [A: real,Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(log,A),Xb)),aa_real_real(aa_rea7340605660300375871l_real(log,A),Y))
          <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y) ) ) ) ) ).

% log_le_cancel_iff
tff(fact_3601_log__le__one__cancel__iff,axiom,
    ! [A: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(log,A),Xb)),one_one_real)
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),A) ) ) ) ).

% log_le_one_cancel_iff
tff(fact_3602_one__le__log__cancel__iff,axiom,
    ! [A: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(log,A),Xb))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),Xb) ) ) ) ).

% one_le_log_cancel_iff
tff(fact_3603_log__le__zero__cancel__iff,axiom,
    ! [A: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(log,A),Xb)),zero_zero_real)
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),one_one_real) ) ) ) ).

% log_le_zero_cancel_iff
tff(fact_3604_zero__le__log__cancel__iff,axiom,
    ! [A: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(log,A),Xb))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Xb) ) ) ) ).

% zero_le_log_cancel_iff
tff(fact_3605_log2__of__power__le,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(log,numeral_numeral_real(aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,M))),aa_nat_real(semiri5074537144036343181t_real,Nb)) ) ) ).

% log2_of_power_le
tff(fact_3606_log__eq__one,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( ( A != one_one_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(log,A),A) = one_one_real ) ) ) ).

% log_eq_one
tff(fact_3607_log__one,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(log,A),one_one_real) = zero_zero_real ).

% log_one
tff(fact_3608_zero__less__log__cancel__iff,axiom,
    ! [A: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(log,A),Xb))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xb) ) ) ) ).

% zero_less_log_cancel_iff
tff(fact_3609_log__less__zero__cancel__iff,axiom,
    ! [A: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(log,A),Xb)),zero_zero_real)
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),one_one_real) ) ) ) ).

% log_less_zero_cancel_iff
tff(fact_3610_one__less__log__cancel__iff,axiom,
    ! [A: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(log,A),Xb))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),Xb) ) ) ) ).

% one_less_log_cancel_iff
tff(fact_3611_log__less__one__cancel__iff,axiom,
    ! [A: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(log,A),Xb)),one_one_real)
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),A) ) ) ) ).

% log_less_one_cancel_iff
tff(fact_3612_log__less__cancel__iff,axiom,
    ! [A: real,Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(log,A),Xb)),aa_real_real(aa_rea7340605660300375871l_real(log,A),Y))
          <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y) ) ) ) ) ).

% log_less_cancel_iff
tff(fact_3613_log__base__change,axiom,
    ! [A: real,B: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( ( A != one_one_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(log,B),Xb) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(log,A),Xb)),aa_real_real(aa_rea7340605660300375871l_real(log,A),B)) ) ) ) ).

% log_base_change
tff(fact_3614_less__log__of__power,axiom,
    ! [B: real,Nb: nat,M: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Nb)),M)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_real_real(aa_rea7340605660300375871l_real(log,B),M)) ) ) ).

% less_log_of_power
tff(fact_3615_log__of__power__eq,axiom,
    ! [M: nat,B: real,Nb: nat] :
      ( ( aa_nat_real(semiri5074537144036343181t_real,M) = aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Nb) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B)
       => ( aa_nat_real(semiri5074537144036343181t_real,Nb) = aa_real_real(aa_rea7340605660300375871l_real(log,B),aa_nat_real(semiri5074537144036343181t_real,M)) ) ) ) ).

% log_of_power_eq
tff(fact_3616_log__mult,axiom,
    ! [A: real,Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( ( A != one_one_real )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
           => ( aa_real_real(aa_rea7340605660300375871l_real(log,A),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Y)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(log,A),Xb)),aa_real_real(aa_rea7340605660300375871l_real(log,A),Y)) ) ) ) ) ) ).

% log_mult
tff(fact_3617_le__log__of__power,axiom,
    ! [B: real,Nb: nat,M: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Nb)),M)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_real_real(aa_rea7340605660300375871l_real(log,B),M)) ) ) ).

% le_log_of_power
tff(fact_3618_log__divide,axiom,
    ! [A: real,Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( ( A != one_one_real )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
           => ( aa_real_real(aa_rea7340605660300375871l_real(log,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Y)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(log,A),Xb)),aa_real_real(aa_rea7340605660300375871l_real(log,A),Y)) ) ) ) ) ) ).

% log_divide
tff(fact_3619_log__base__pow,axiom,
    ! [A: real,Nb: nat,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( aa_real_real(aa_rea7340605660300375871l_real(log,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)),Xb) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(log,A),Xb)),aa_nat_real(semiri5074537144036343181t_real,Nb)) ) ) ).

% log_base_pow
tff(fact_3620_log__nat__power,axiom,
    ! [Xb: real,B: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_real(aa_rea7340605660300375871l_real(log,B),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_real_real(aa_rea7340605660300375871l_real(log,B),Xb)) ) ) ).

% log_nat_power
tff(fact_3621_log2__of__power__eq,axiom,
    ! [M: nat,Nb: nat] :
      ( ( M = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb) )
     => ( aa_nat_real(semiri5074537144036343181t_real,Nb) = aa_real_real(aa_rea7340605660300375871l_real(log,numeral_numeral_real(aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,M)) ) ) ).

% log2_of_power_eq
tff(fact_3622_log__of__power__less,axiom,
    ! [M: nat,B: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,M)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Nb))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(log,B),aa_nat_real(semiri5074537144036343181t_real,M))),aa_nat_real(semiri5074537144036343181t_real,Nb)) ) ) ) ).

% log_of_power_less
tff(fact_3623_log__of__power__le,axiom,
    ! [M: nat,B: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,M)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Nb))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(log,B),aa_nat_real(semiri5074537144036343181t_real,M))),aa_nat_real(semiri5074537144036343181t_real,Nb)) ) ) ) ).

% log_of_power_le
tff(fact_3624_less__log2__of__power,axiom,
    ! [Nb: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)),M)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_real_real(aa_rea7340605660300375871l_real(log,numeral_numeral_real(aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,M))) ) ).

% less_log2_of_power
tff(fact_3625_le__log2__of__power,axiom,
    ! [Nb: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)),M)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_real_real(aa_rea7340605660300375871l_real(log,numeral_numeral_real(aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,M))) ) ).

% le_log2_of_power
tff(fact_3626_log2__of__power__less,axiom,
    ! [M: nat,Nb: 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))),Nb))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(log,numeral_numeral_real(aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,M))),aa_nat_real(semiri5074537144036343181t_real,Nb)) ) ) ).

% log2_of_power_less
tff(fact_3627_atLeastatMost__subset__iff,axiom,
    ! [A: set_int,B: set_int,C: set_int,D: set_int] :
      ( ord_le4403425263959731960et_int(set_or370866239135849197et_int(A,B),set_or370866239135849197et_int(C,D))
    <=> ( ~ aa_set_int_o(ord_less_eq_set_int(A),B)
        | ( aa_set_int_o(ord_less_eq_set_int(C),A)
          & aa_set_int_o(ord_less_eq_set_int(B),D) ) ) ) ).

% atLeastatMost_subset_iff
tff(fact_3628_atLeastatMost__subset__iff,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( aa_set_rat_o(ord_less_eq_set_rat(set_or633870826150836451st_rat(A,B)),set_or633870826150836451st_rat(C,D))
    <=> ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),A)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),D) ) ) ) ).

% atLeastatMost_subset_iff
tff(fact_3629_atLeastatMost__subset__iff,axiom,
    ! [A: num,B: num,C: num,D: num] :
      ( aa_set_num_o(ord_less_eq_set_num(set_or7049704709247886629st_num(A,B)),set_or7049704709247886629st_num(C,D))
    <=> ( ~ aa_num_o(ord_less_eq_num(A),B)
        | ( aa_num_o(ord_less_eq_num(C),A)
          & aa_num_o(ord_less_eq_num(B),D) ) ) ) ).

% atLeastatMost_subset_iff
tff(fact_3630_atLeastatMost__subset__iff,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(set_or1269000886237332187st_nat(A,B)),set_or1269000886237332187st_nat(C,D))
    <=> ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
        | ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),A)
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),D) ) ) ) ).

% atLeastatMost_subset_iff
tff(fact_3631_atLeastatMost__subset__iff,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( aa_set_int_o(ord_less_eq_set_int(set_or1266510415728281911st_int(A,B)),set_or1266510415728281911st_int(C,D))
    <=> ( ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
        | ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),A)
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),D) ) ) ) ).

% atLeastatMost_subset_iff
tff(fact_3632_atLeastatMost__subset__iff,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer,D: code_integer] :
      ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(set_or189985376899183464nteger(A,B)),set_or189985376899183464nteger(C,D))
    <=> ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B)
        | ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,C),A)
          & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),D) ) ) ) ).

% atLeastatMost_subset_iff
tff(fact_3633_atLeastatMost__subset__iff,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( aa_set_real_o(ord_less_eq_set_real(set_or1222579329274155063t_real(A,B)),set_or1222579329274155063t_real(C,D))
    <=> ( ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
        | ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),A)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),D) ) ) ) ).

% atLeastatMost_subset_iff
tff(fact_3634_arcosh__1,axiom,
    aa_real_real(arcosh_real,one_one_real) = zero_zero_real ).

% arcosh_1
tff(fact_3635_minNull__delete__time__bound,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xb: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Ta),Nb)
     => ( aa_VEBT_VEBT_o(vEBT_VEBT_minNull,vEBT_vebt_delete(Ta,Xb))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_d_e_l_e_t_e,Ta),Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,one))))) ) ) ).

% minNull_delete_time_bound
tff(fact_3636_nat__induct2,axiom,
    ! [P: fun_nat_o,Nb: nat] :
      ( aa_nat_o(P,zero_zero_nat)
     => ( aa_nat_o(P,one_one_nat)
       => ( ! [N2: nat] :
              ( aa_nat_o(P,N2)
             => aa_nat_o(P,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) )
         => aa_nat_o(P,Nb) ) ) ) ).

% nat_induct2
tff(fact_3637_VEBT__internal_Omembermima_Oelims_I3_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( ~ aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_membermima,Xb),Xaa)
     => ( ! [Uu2: $o,Uv2: $o] : Xb != vEBT_Leaf((Uu2),(Uv2))
       => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] : Xb != aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux2),Uy2)
         => ( ! [Mi: nat,Ma: nat] :
                ( ? [Va: list_VEBT_VEBT,Vb2: vEBT_VEBT] : Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),zero_zero_nat,Va),Vb2)
               => ( ( Xaa = Mi )
                  | ( Xaa = Ma ) ) )
           => ( ! [Mi: nat,Ma: nat,V2: nat,TreeList: list_VEBT_VEBT] :
                  ( ? [Vc: vEBT_VEBT] : Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),aa_nat_nat(suc,V2),TreeList),Vc)
                 => ( ( Xaa = Mi )
                    | ( Xaa = Ma )
                    | $let(
                        pos: nat,
                        pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                        $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_membermima,nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) )
             => ~ ! [V2: nat,TreeList: list_VEBT_VEBT] :
                    ( ? [Vd2: vEBT_VEBT] : Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList),Vd2)
                   => $let(
                        pos: nat,
                        pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                        $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_membermima,nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.elims(3)
tff(fact_3638_VEBT__internal_Omembermima_Oelims_I1_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: $o] :
      ( ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_membermima,Xb),Xaa)
      <=> (Y) )
     => ( ( ? [Uu2: $o,Uv2: $o] : Xb = vEBT_Leaf((Uu2),(Uv2))
         => (Y) )
       => ( ( ? [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] : Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux2),Uy2)
           => (Y) )
         => ( ! [Mi: nat,Ma: nat] :
                ( ? [Va: list_VEBT_VEBT,Vb2: vEBT_VEBT] : Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),zero_zero_nat,Va),Vb2)
               => ( (Y)
                <=> ~ ( ( Xaa = Mi )
                      | ( Xaa = Ma ) ) ) )
           => ( ! [Mi: nat,Ma: nat,V2: nat,TreeList: list_VEBT_VEBT] :
                  ( ? [Vc: vEBT_VEBT] : Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),aa_nat_nat(suc,V2),TreeList),Vc)
                 => ( (Y)
                  <=> ~ ( ( Xaa = Mi )
                        | ( Xaa = Ma )
                        | $let(
                            pos: nat,
                            pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                            $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_membermima,nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) )
             => ~ ! [V2: nat,TreeList: list_VEBT_VEBT] :
                    ( ? [Vd2: vEBT_VEBT] : Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList),Vd2)
                   => ( (Y)
                    <=> ~ $let(
                            pos: nat,
                            pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                            $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_membermima,nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.elims(1)
tff(fact_3639_nat__approx__posE,axiom,
    ! [E: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),E)
     => ~ ! [N2: nat] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,divide_divide_rat(one_one_rat,aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,N2)))),E) ) ).

% nat_approx_posE
tff(fact_3640_nat__approx__posE,axiom,
    ! [E: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),E)
     => ~ ! [N2: nat] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,N2)))),E) ) ).

% nat_approx_posE
tff(fact_3641_del__single__cont,axiom,
    ! [Xb: nat,Mia: nat,Maa: nat,Deg: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT] :
      ( ( ( Xb = Mia )
        & ( Xb = Maa ) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg)
       => ( vEBT_vebt_delete(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mia),Maa)),Deg,TreeLista),Summarya),Xb) = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Deg,TreeLista),Summarya) ) ) ) ).

% del_single_cont
tff(fact_3642_delete__pres__valid,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xb: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Ta),Nb)
     => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,vEBT_vebt_delete(Ta,Xb)),Nb) ) ).

% delete_pres_valid
tff(fact_3643_buildup__nothing__in__min__max,axiom,
    ! [Nb: nat,Xb: nat] : ~ aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_membermima,vEBT_vebt_buildup(Nb)),Xb) ).

% buildup_nothing_in_min_max
tff(fact_3644_dele__bmo__cont__corr,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xb: nat,Y: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Ta),Nb)
     => ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,vEBT_vebt_delete(Ta,Xb)),Y)
      <=> ( ( Xb != Y )
          & aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,Ta),Y) ) ) ) ).

% dele_bmo_cont_corr
tff(fact_3645_dele__member__cont__corr,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xb: nat,Y: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Ta),Nb)
     => ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,vEBT_vebt_delete(Ta,Xb)),Y)
      <=> ( ( Xb != Y )
          & aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,Ta),Y) ) ) ) ).

% dele_member_cont_corr
tff(fact_3646_minNull__delete__time__bound_H,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xb: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Ta),Nb)
     => ( aa_VEBT_VEBT_o(vEBT_VEBT_minNull,vEBT_vebt_delete(Ta,Xb))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_V1232361888498592333_e_t_e,Ta),Xb)),one_one_nat) ) ) ).

% minNull_delete_time_bound'
tff(fact_3647_Icc__eq__Icc,axiom,
    ! [L: set_int,H: set_int,L3: set_int,H3: set_int] :
      ( ( set_or370866239135849197et_int(L,H) = set_or370866239135849197et_int(L3,H3) )
    <=> ( ( ( L = L3 )
          & ( H = H3 ) )
        | ( ~ aa_set_int_o(ord_less_eq_set_int(L),H)
          & ~ aa_set_int_o(ord_less_eq_set_int(L3),H3) ) ) ) ).

% Icc_eq_Icc
tff(fact_3648_Icc__eq__Icc,axiom,
    ! [L: rat,H: rat,L3: rat,H3: rat] :
      ( ( set_or633870826150836451st_rat(L,H) = set_or633870826150836451st_rat(L3,H3) )
    <=> ( ( ( L = L3 )
          & ( H = H3 ) )
        | ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,L),H)
          & ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,L3),H3) ) ) ) ).

% Icc_eq_Icc
tff(fact_3649_Icc__eq__Icc,axiom,
    ! [L: num,H: num,L3: num,H3: num] :
      ( ( set_or7049704709247886629st_num(L,H) = set_or7049704709247886629st_num(L3,H3) )
    <=> ( ( ( L = L3 )
          & ( H = H3 ) )
        | ( ~ aa_num_o(ord_less_eq_num(L),H)
          & ~ aa_num_o(ord_less_eq_num(L3),H3) ) ) ) ).

% Icc_eq_Icc
tff(fact_3650_Icc__eq__Icc,axiom,
    ! [L: nat,H: nat,L3: nat,H3: nat] :
      ( ( set_or1269000886237332187st_nat(L,H) = set_or1269000886237332187st_nat(L3,H3) )
    <=> ( ( ( L = L3 )
          & ( H = H3 ) )
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,L),H)
          & ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,L3),H3) ) ) ) ).

% Icc_eq_Icc
tff(fact_3651_Icc__eq__Icc,axiom,
    ! [L: int,H: int,L3: int,H3: int] :
      ( ( set_or1266510415728281911st_int(L,H) = set_or1266510415728281911st_int(L3,H3) )
    <=> ( ( ( L = L3 )
          & ( H = H3 ) )
        | ( ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,L),H)
          & ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,L3),H3) ) ) ) ).

% Icc_eq_Icc
tff(fact_3652_Icc__eq__Icc,axiom,
    ! [L: code_integer,H: code_integer,L3: code_integer,H3: code_integer] :
      ( ( set_or189985376899183464nteger(L,H) = set_or189985376899183464nteger(L3,H3) )
    <=> ( ( ( L = L3 )
          & ( H = H3 ) )
        | ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,L),H)
          & ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,L3),H3) ) ) ) ).

% Icc_eq_Icc
tff(fact_3653_Icc__eq__Icc,axiom,
    ! [L: real,H: real,L3: real,H3: real] :
      ( ( set_or1222579329274155063t_real(L,H) = set_or1222579329274155063t_real(L3,H3) )
    <=> ( ( ( L = L3 )
          & ( H = H3 ) )
        | ( ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,L),H)
          & ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,L3),H3) ) ) ) ).

% Icc_eq_Icc
tff(fact_3654_atLeastAtMost__iff,axiom,
    ! [I: set_int,L: set_int,U: set_int] :
      ( member_set_int(I,set_or370866239135849197et_int(L,U))
    <=> ( aa_set_int_o(ord_less_eq_set_int(L),I)
        & aa_set_int_o(ord_less_eq_set_int(I),U) ) ) ).

% atLeastAtMost_iff
tff(fact_3655_atLeastAtMost__iff,axiom,
    ! [I: rat,L: rat,U: rat] :
      ( aa_set_rat_o(member_rat(I),set_or633870826150836451st_rat(L,U))
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,L),I)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,I),U) ) ) ).

% atLeastAtMost_iff
tff(fact_3656_atLeastAtMost__iff,axiom,
    ! [I: num,L: num,U: num] :
      ( aa_set_num_o(member_num(I),set_or7049704709247886629st_num(L,U))
    <=> ( aa_num_o(ord_less_eq_num(L),I)
        & aa_num_o(ord_less_eq_num(I),U) ) ) ).

% atLeastAtMost_iff
tff(fact_3657_atLeastAtMost__iff,axiom,
    ! [I: nat,L: nat,U: nat] :
      ( aa_set_nat_o(member_nat(I),set_or1269000886237332187st_nat(L,U))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,L),I)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),U) ) ) ).

% atLeastAtMost_iff
tff(fact_3658_atLeastAtMost__iff,axiom,
    ! [I: int,L: int,U: int] :
      ( aa_set_int_o(member_int(I),set_or1266510415728281911st_int(L,U))
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,L),I)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),U) ) ) ).

% atLeastAtMost_iff
tff(fact_3659_atLeastAtMost__iff,axiom,
    ! [I: code_integer,L: code_integer,U: code_integer] :
      ( aa_set5643387332196575111eger_o(member_Code_integer(I),set_or189985376899183464nteger(L,U))
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,L),I)
        & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,I),U) ) ) ).

% atLeastAtMost_iff
tff(fact_3660_atLeastAtMost__iff,axiom,
    ! [I: real,L: real,U: real] :
      ( aa_set_real_o(member_real(I),set_or1222579329274155063t_real(L,U))
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,L),I)
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,I),U) ) ) ).

% atLeastAtMost_iff
tff(fact_3661_delt__out__of__range,axiom,
    ! [Xb: nat,Mia: nat,Maa: nat,Deg: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Mia)
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Maa),Xb) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg)
       => ( vEBT_vebt_delete(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mia),Maa)),Deg,TreeLista),Summarya),Xb) = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mia),Maa)),Deg,TreeLista),Summarya) ) ) ) ).

% delt_out_of_range
tff(fact_3662_aset_I2_J,axiom,
    ! [D5: int,A2: set_int,P: fun_int_o,Q: fun_int_o] :
      ( ! [X2: int] :
          ( ! [Xa: int] :
              ( aa_set_int_o(member_int(Xa),set_or1266510415728281911st_int(one_one_int,D5))
             => ! [Xb2: int] :
                  ( aa_set_int_o(member_int(Xb2),A2)
                 => ( X2 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb2),Xa) ) ) )
         => ( aa_int_o(P,X2)
           => aa_int_o(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,X2),D5)) ) )
     => ( ! [X2: int] :
            ( ! [Xa: int] :
                ( aa_set_int_o(member_int(Xa),set_or1266510415728281911st_int(one_one_int,D5))
               => ! [Xb2: int] :
                    ( aa_set_int_o(member_int(Xb2),A2)
                   => ( X2 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb2),Xa) ) ) )
           => ( aa_int_o(Q,X2)
             => aa_int_o(Q,aa_int_int(aa_int_fun_int_int(plus_plus_int,X2),D5)) ) )
       => ! [X3: int] :
            ( ! [Xa2: int] :
                ( aa_set_int_o(member_int(Xa2),set_or1266510415728281911st_int(one_one_int,D5))
               => ! [Xb3: int] :
                    ( aa_set_int_o(member_int(Xb3),A2)
                   => ( X3 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb3),Xa2) ) ) )
           => ( ( aa_int_o(P,X3)
                | aa_int_o(Q,X3) )
             => ( aa_int_o(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),D5))
                | aa_int_o(Q,aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),D5)) ) ) ) ) ) ).

% aset(2)
tff(fact_3663_aset_I1_J,axiom,
    ! [D5: int,A2: set_int,P: fun_int_o,Q: fun_int_o] :
      ( ! [X2: int] :
          ( ! [Xa: int] :
              ( aa_set_int_o(member_int(Xa),set_or1266510415728281911st_int(one_one_int,D5))
             => ! [Xb2: int] :
                  ( aa_set_int_o(member_int(Xb2),A2)
                 => ( X2 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb2),Xa) ) ) )
         => ( aa_int_o(P,X2)
           => aa_int_o(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,X2),D5)) ) )
     => ( ! [X2: int] :
            ( ! [Xa: int] :
                ( aa_set_int_o(member_int(Xa),set_or1266510415728281911st_int(one_one_int,D5))
               => ! [Xb2: int] :
                    ( aa_set_int_o(member_int(Xb2),A2)
                   => ( X2 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb2),Xa) ) ) )
           => ( aa_int_o(Q,X2)
             => aa_int_o(Q,aa_int_int(aa_int_fun_int_int(plus_plus_int,X2),D5)) ) )
       => ! [X3: int] :
            ( ! [Xa2: int] :
                ( aa_set_int_o(member_int(Xa2),set_or1266510415728281911st_int(one_one_int,D5))
               => ! [Xb3: int] :
                    ( aa_set_int_o(member_int(Xb3),A2)
                   => ( X3 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb3),Xa2) ) ) )
           => ( ( aa_int_o(P,X3)
                & aa_int_o(Q,X3) )
             => ( aa_int_o(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),D5))
                & aa_int_o(Q,aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),D5)) ) ) ) ) ) ).

% aset(1)
tff(fact_3664_bset_I2_J,axiom,
    ! [D5: int,B4: set_int,P: fun_int_o,Q: fun_int_o] :
      ( ! [X2: int] :
          ( ! [Xa: int] :
              ( aa_set_int_o(member_int(Xa),set_or1266510415728281911st_int(one_one_int,D5))
             => ! [Xb2: int] :
                  ( aa_set_int_o(member_int(Xb2),B4)
                 => ( X2 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb2),Xa) ) ) )
         => ( aa_int_o(P,X2)
           => aa_int_o(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X2),D5)) ) )
     => ( ! [X2: int] :
            ( ! [Xa: int] :
                ( aa_set_int_o(member_int(Xa),set_or1266510415728281911st_int(one_one_int,D5))
               => ! [Xb2: int] :
                    ( aa_set_int_o(member_int(Xb2),B4)
                   => ( X2 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb2),Xa) ) ) )
           => ( aa_int_o(Q,X2)
             => aa_int_o(Q,aa_int_int(aa_int_fun_int_int(minus_minus_int,X2),D5)) ) )
       => ! [X3: int] :
            ( ! [Xa2: int] :
                ( aa_set_int_o(member_int(Xa2),set_or1266510415728281911st_int(one_one_int,D5))
               => ! [Xb3: int] :
                    ( aa_set_int_o(member_int(Xb3),B4)
                   => ( X3 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb3),Xa2) ) ) )
           => ( ( aa_int_o(P,X3)
                | aa_int_o(Q,X3) )
             => ( aa_int_o(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),D5))
                | aa_int_o(Q,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),D5)) ) ) ) ) ) ).

% bset(2)
tff(fact_3665_bset_I1_J,axiom,
    ! [D5: int,B4: set_int,P: fun_int_o,Q: fun_int_o] :
      ( ! [X2: int] :
          ( ! [Xa: int] :
              ( aa_set_int_o(member_int(Xa),set_or1266510415728281911st_int(one_one_int,D5))
             => ! [Xb2: int] :
                  ( aa_set_int_o(member_int(Xb2),B4)
                 => ( X2 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb2),Xa) ) ) )
         => ( aa_int_o(P,X2)
           => aa_int_o(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X2),D5)) ) )
     => ( ! [X2: int] :
            ( ! [Xa: int] :
                ( aa_set_int_o(member_int(Xa),set_or1266510415728281911st_int(one_one_int,D5))
               => ! [Xb2: int] :
                    ( aa_set_int_o(member_int(Xb2),B4)
                   => ( X2 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb2),Xa) ) ) )
           => ( aa_int_o(Q,X2)
             => aa_int_o(Q,aa_int_int(aa_int_fun_int_int(minus_minus_int,X2),D5)) ) )
       => ! [X3: int] :
            ( ! [Xa2: int] :
                ( aa_set_int_o(member_int(Xa2),set_or1266510415728281911st_int(one_one_int,D5))
               => ! [Xb3: int] :
                    ( aa_set_int_o(member_int(Xb3),B4)
                   => ( X3 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb3),Xa2) ) ) )
           => ( ( aa_int_o(P,X3)
                & aa_int_o(Q,X3) )
             => ( aa_int_o(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),D5))
                & aa_int_o(Q,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),D5)) ) ) ) ) ) ).

% bset(1)
tff(fact_3666_bounded__Max__nat,axiom,
    ! [P: fun_nat_o,Xb: nat,M8: nat] :
      ( aa_nat_o(P,Xb)
     => ( ! [X2: nat] :
            ( aa_nat_o(P,X2)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X2),M8) )
       => ~ ! [M2: nat] :
              ( aa_nat_o(P,M2)
             => ~ ! [X3: nat] :
                    ( aa_nat_o(P,X3)
                   => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),M2) ) ) ) ) ).

% bounded_Max_nat
tff(fact_3667_VEBT__internal_Omembermima_Osimps_I2_J,axiom,
    ! [Ux: list_VEBT_VEBT,Uy: vEBT_VEBT,Uz: nat] : ~ aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_membermima,aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux),Uy)),Uz) ).

% VEBT_internal.membermima.simps(2)
tff(fact_3668_aset_I7_J,axiom,
    ! [D5: int,A2: set_int,Ta: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D5)
     => ! [X3: int] :
          ( ! [Xa2: int] :
              ( aa_set_int_o(member_int(Xa2),set_or1266510415728281911st_int(one_one_int,D5))
             => ! [Xb3: int] :
                  ( aa_set_int_o(member_int(Xb3),A2)
                 => ( X3 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb3),Xa2) ) ) )
         => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Ta),X3)
           => aa_int_o(aa_int_fun_int_o(ord_less_int,Ta),aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),D5)) ) ) ) ).

% aset(7)
tff(fact_3669_aset_I5_J,axiom,
    ! [D5: int,Ta: int,A2: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D5)
     => ( aa_set_int_o(member_int(Ta),A2)
       => ! [X3: int] :
            ( ! [Xa2: int] :
                ( aa_set_int_o(member_int(Xa2),set_or1266510415728281911st_int(one_one_int,D5))
               => ! [Xb3: int] :
                    ( aa_set_int_o(member_int(Xb3),A2)
                   => ( X3 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb3),Xa2) ) ) )
           => ( aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Ta)
             => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),D5)),Ta) ) ) ) ) ).

% aset(5)
tff(fact_3670_aset_I4_J,axiom,
    ! [D5: int,Ta: int,A2: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D5)
     => ( aa_set_int_o(member_int(Ta),A2)
       => ! [X3: int] :
            ( ! [Xa2: int] :
                ( aa_set_int_o(member_int(Xa2),set_or1266510415728281911st_int(one_one_int,D5))
               => ! [Xb3: int] :
                    ( aa_set_int_o(member_int(Xb3),A2)
                   => ( X3 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb3),Xa2) ) ) )
           => ( ( X3 != Ta )
             => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),D5) != Ta ) ) ) ) ) ).

% aset(4)
tff(fact_3671_aset_I3_J,axiom,
    ! [D5: int,Ta: int,A2: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D5)
     => ( aa_set_int_o(member_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,Ta),one_one_int)),A2)
       => ! [X3: int] :
            ( ! [Xa2: int] :
                ( aa_set_int_o(member_int(Xa2),set_or1266510415728281911st_int(one_one_int,D5))
               => ! [Xb3: int] :
                    ( aa_set_int_o(member_int(Xb3),A2)
                   => ( X3 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb3),Xa2) ) ) )
           => ( ( X3 = Ta )
             => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),D5) = Ta ) ) ) ) ) ).

% aset(3)
tff(fact_3672_bset_I7_J,axiom,
    ! [D5: int,Ta: int,B4: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D5)
     => ( aa_set_int_o(member_int(Ta),B4)
       => ! [X3: int] :
            ( ! [Xa2: int] :
                ( aa_set_int_o(member_int(Xa2),set_or1266510415728281911st_int(one_one_int,D5))
               => ! [Xb3: int] :
                    ( aa_set_int_o(member_int(Xb3),B4)
                   => ( X3 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb3),Xa2) ) ) )
           => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Ta),X3)
             => aa_int_o(aa_int_fun_int_o(ord_less_int,Ta),aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),D5)) ) ) ) ) ).

% bset(7)
tff(fact_3673_bset_I5_J,axiom,
    ! [D5: int,B4: set_int,Ta: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D5)
     => ! [X3: int] :
          ( ! [Xa2: int] :
              ( aa_set_int_o(member_int(Xa2),set_or1266510415728281911st_int(one_one_int,D5))
             => ! [Xb3: int] :
                  ( aa_set_int_o(member_int(Xb3),B4)
                 => ( X3 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb3),Xa2) ) ) )
         => ( aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Ta)
           => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),D5)),Ta) ) ) ) ).

% bset(5)
tff(fact_3674_bset_I4_J,axiom,
    ! [D5: int,Ta: int,B4: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D5)
     => ( aa_set_int_o(member_int(Ta),B4)
       => ! [X3: int] :
            ( ! [Xa2: int] :
                ( aa_set_int_o(member_int(Xa2),set_or1266510415728281911st_int(one_one_int,D5))
               => ! [Xb3: int] :
                    ( aa_set_int_o(member_int(Xb3),B4)
                   => ( X3 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb3),Xa2) ) ) )
           => ( ( X3 != Ta )
             => ( aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),D5) != Ta ) ) ) ) ) ).

% bset(4)
tff(fact_3675_bset_I3_J,axiom,
    ! [D5: int,Ta: int,B4: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D5)
     => ( aa_set_int_o(member_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,Ta),one_one_int)),B4)
       => ! [X3: int] :
            ( ! [Xa2: int] :
                ( aa_set_int_o(member_int(Xa2),set_or1266510415728281911st_int(one_one_int,D5))
               => ! [Xb3: int] :
                    ( aa_set_int_o(member_int(Xb3),B4)
                   => ( X3 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb3),Xa2) ) ) )
           => ( ( X3 = Ta )
             => ( aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),D5) = Ta ) ) ) ) ) ).

% bset(3)
tff(fact_3676_periodic__finite__ex,axiom,
    ! [D: int,P: fun_int_o] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D)
     => ( ! [X2: int,K3: int] :
            ( aa_int_o(P,X2)
          <=> aa_int_o(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X2),aa_int_int(aa_int_fun_int_int(times_times_int,K3),D))) )
       => ( ? [X_1: int] : aa_int_o(P,X_1)
        <=> ? [X: int] :
              ( aa_set_int_o(member_int(X),set_or1266510415728281911st_int(one_one_int,D))
              & aa_int_o(P,X) ) ) ) ) ).

% periodic_finite_ex
tff(fact_3677_VEBT__internal_Omembermima_Osimps_I3_J,axiom,
    ! [Mia: nat,Maa: nat,Vaa: list_VEBT_VEBT,Vb: vEBT_VEBT,Xb: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_membermima,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mia),Maa)),zero_zero_nat,Vaa),Vb)),Xb)
    <=> ( ( Xb = Mia )
        | ( Xb = Maa ) ) ) ).

% VEBT_internal.membermima.simps(3)
tff(fact_3678_aset_I8_J,axiom,
    ! [D5: int,A2: set_int,Ta: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D5)
     => ! [X3: int] :
          ( ! [Xa2: int] :
              ( aa_set_int_o(member_int(Xa2),set_or1266510415728281911st_int(one_one_int,D5))
             => ! [Xb3: int] :
                  ( aa_set_int_o(member_int(Xb3),A2)
                 => ( X3 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb3),Xa2) ) ) )
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ta),X3)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ta),aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),D5)) ) ) ) ).

% aset(8)
tff(fact_3679_aset_I6_J,axiom,
    ! [D5: int,Ta: int,A2: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D5)
     => ( aa_set_int_o(member_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,Ta),one_one_int)),A2)
       => ! [X3: int] :
            ( ! [Xa2: int] :
                ( aa_set_int_o(member_int(Xa2),set_or1266510415728281911st_int(one_one_int,D5))
               => ! [Xb3: int] :
                    ( aa_set_int_o(member_int(Xb3),A2)
                   => ( X3 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb3),Xa2) ) ) )
           => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X3),Ta)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),D5)),Ta) ) ) ) ) ).

% aset(6)
tff(fact_3680_bset_I8_J,axiom,
    ! [D5: int,Ta: int,B4: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D5)
     => ( aa_set_int_o(member_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,Ta),one_one_int)),B4)
       => ! [X3: int] :
            ( ! [Xa2: int] :
                ( aa_set_int_o(member_int(Xa2),set_or1266510415728281911st_int(one_one_int,D5))
               => ! [Xb3: int] :
                    ( aa_set_int_o(member_int(Xb3),B4)
                   => ( X3 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb3),Xa2) ) ) )
           => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ta),X3)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ta),aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),D5)) ) ) ) ) ).

% bset(8)
tff(fact_3681_bset_I6_J,axiom,
    ! [D5: int,B4: set_int,Ta: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D5)
     => ! [X3: int] :
          ( ! [Xa2: int] :
              ( aa_set_int_o(member_int(Xa2),set_or1266510415728281911st_int(one_one_int,D5))
             => ! [Xb3: int] :
                  ( aa_set_int_o(member_int(Xb3),B4)
                 => ( X3 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb3),Xa2) ) ) )
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X3),Ta)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),D5)),Ta) ) ) ) ).

% bset(6)
tff(fact_3682_cpmi,axiom,
    ! [D5: int,P: fun_int_o,P3: fun_int_o,B4: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D5)
     => ( ? [Z4: int] :
          ! [X2: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_int,X2),Z4)
           => ( aa_int_o(P,X2)
            <=> aa_int_o(P3,X2) ) )
       => ( ! [X2: int] :
              ( ! [Xa: int] :
                  ( aa_set_int_o(member_int(Xa),set_or1266510415728281911st_int(one_one_int,D5))
                 => ! [Xb2: int] :
                      ( aa_set_int_o(member_int(Xb2),B4)
                     => ( X2 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb2),Xa) ) ) )
             => ( aa_int_o(P,X2)
               => aa_int_o(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X2),D5)) ) )
         => ( ! [X2: int,K3: int] :
                ( aa_int_o(P3,X2)
              <=> aa_int_o(P3,aa_int_int(aa_int_fun_int_int(minus_minus_int,X2),aa_int_int(aa_int_fun_int_int(times_times_int,K3),D5))) )
           => ( ? [X_1: int] : aa_int_o(P,X_1)
            <=> ( ? [X: int] :
                    ( aa_set_int_o(member_int(X),set_or1266510415728281911st_int(one_one_int,D5))
                    & aa_int_o(P3,X) )
                | ? [X: int] :
                    ( aa_set_int_o(member_int(X),set_or1266510415728281911st_int(one_one_int,D5))
                    & ? [Xa3: int] :
                        ( aa_set_int_o(member_int(Xa3),B4)
                        & aa_int_o(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,Xa3),X)) ) ) ) ) ) ) ) ) ).

% cpmi
tff(fact_3683_cppi,axiom,
    ! [D5: int,P: fun_int_o,P3: fun_int_o,A2: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D5)
     => ( ? [Z4: int] :
          ! [X2: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z4),X2)
           => ( aa_int_o(P,X2)
            <=> aa_int_o(P3,X2) ) )
       => ( ! [X2: int] :
              ( ! [Xa: int] :
                  ( aa_set_int_o(member_int(Xa),set_or1266510415728281911st_int(one_one_int,D5))
                 => ! [Xb2: int] :
                      ( aa_set_int_o(member_int(Xb2),A2)
                     => ( X2 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb2),Xa) ) ) )
             => ( aa_int_o(P,X2)
               => aa_int_o(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,X2),D5)) ) )
         => ( ! [X2: int,K3: int] :
                ( aa_int_o(P3,X2)
              <=> aa_int_o(P3,aa_int_int(aa_int_fun_int_int(minus_minus_int,X2),aa_int_int(aa_int_fun_int_int(times_times_int,K3),D5))) )
           => ( ? [X_1: int] : aa_int_o(P,X_1)
            <=> ( ? [X: int] :
                    ( aa_set_int_o(member_int(X),set_or1266510415728281911st_int(one_one_int,D5))
                    & aa_int_o(P3,X) )
                | ? [X: int] :
                    ( aa_set_int_o(member_int(X),set_or1266510415728281911st_int(one_one_int,D5))
                    & ? [Xa3: int] :
                        ( aa_set_int_o(member_int(Xa3),A2)
                        & aa_int_o(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,Xa3),X)) ) ) ) ) ) ) ) ) ).

% cppi
tff(fact_3684_exists__least__lemma,axiom,
    ! [P: fun_nat_o] :
      ( ~ aa_nat_o(P,zero_zero_nat)
     => ( ? [X_12: nat] : aa_nat_o(P,X_12)
       => ? [N2: nat] :
            ( ~ aa_nat_o(P,N2)
            & aa_nat_o(P,aa_nat_nat(suc,N2)) ) ) ) ).

% exists_least_lemma
tff(fact_3685_real__arch__simple,axiom,
    ! [Xb: real] :
    ? [N2: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_nat_real(semiri5074537144036343181t_real,N2)) ).

% real_arch_simple
tff(fact_3686_real__arch__simple,axiom,
    ! [Xb: rat] :
    ? [N2: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),aa_nat_rat(semiri681578069525770553at_rat,N2)) ).

% real_arch_simple
tff(fact_3687_reals__Archimedean2,axiom,
    ! [Xb: rat] :
    ? [N2: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),aa_nat_rat(semiri681578069525770553at_rat,N2)) ).

% reals_Archimedean2
tff(fact_3688_reals__Archimedean2,axiom,
    ! [Xb: real] :
    ? [N2: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_nat_real(semiri5074537144036343181t_real,N2)) ).

% reals_Archimedean2
tff(fact_3689_atLeastatMost__psubset__iff,axiom,
    ! [A: set_int,B: set_int,C: set_int,D: set_int] :
      ( ord_less_set_set_int(set_or370866239135849197et_int(A,B),set_or370866239135849197et_int(C,D))
    <=> ( ( ~ aa_set_int_o(ord_less_eq_set_int(A),B)
          | ( aa_set_int_o(ord_less_eq_set_int(C),A)
            & aa_set_int_o(ord_less_eq_set_int(B),D)
            & ( aa_set_int_o(ord_less_set_int(C),A)
              | aa_set_int_o(ord_less_set_int(B),D) ) ) )
        & aa_set_int_o(ord_less_eq_set_int(C),D) ) ) ).

% atLeastatMost_psubset_iff
tff(fact_3690_atLeastatMost__psubset__iff,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( aa_set_rat_o(ord_less_set_rat(set_or633870826150836451st_rat(A,B)),set_or633870826150836451st_rat(C,D))
    <=> ( ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
          | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),A)
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),D)
            & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),A)
              | aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),D) ) ) )
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),D) ) ) ).

% atLeastatMost_psubset_iff
tff(fact_3691_atLeastatMost__psubset__iff,axiom,
    ! [A: num,B: num,C: num,D: num] :
      ( aa_set_num_o(ord_less_set_num(set_or7049704709247886629st_num(A,B)),set_or7049704709247886629st_num(C,D))
    <=> ( ( ~ aa_num_o(ord_less_eq_num(A),B)
          | ( aa_num_o(ord_less_eq_num(C),A)
            & aa_num_o(ord_less_eq_num(B),D)
            & ( aa_num_o(ord_less_num(C),A)
              | aa_num_o(ord_less_num(B),D) ) ) )
        & aa_num_o(ord_less_eq_num(C),D) ) ) ).

% atLeastatMost_psubset_iff
tff(fact_3692_atLeastatMost__psubset__iff,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( aa_set_nat_o(ord_less_set_nat(set_or1269000886237332187st_nat(A,B)),set_or1269000886237332187st_nat(C,D))
    <=> ( ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
          | ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),A)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),D)
            & ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),A)
              | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),D) ) ) )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),D) ) ) ).

% atLeastatMost_psubset_iff
tff(fact_3693_atLeastatMost__psubset__iff,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( aa_set_int_o(ord_less_set_int(set_or1266510415728281911st_int(A,B)),set_or1266510415728281911st_int(C,D))
    <=> ( ( ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
          | ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),A)
            & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),D)
            & ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),A)
              | aa_int_o(aa_int_fun_int_o(ord_less_int,B),D) ) ) )
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),D) ) ) ).

% atLeastatMost_psubset_iff
tff(fact_3694_atLeastatMost__psubset__iff,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer,D: code_integer] :
      ( aa_set5643387332196575111eger_o(ord_le1307284697595431911nteger(set_or189985376899183464nteger(A,B)),set_or189985376899183464nteger(C,D))
    <=> ( ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B)
          | ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,C),A)
            & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),D)
            & ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,C),A)
              | aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),D) ) ) )
        & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,C),D) ) ) ).

% atLeastatMost_psubset_iff
tff(fact_3695_atLeastatMost__psubset__iff,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( aa_set_real_o(ord_less_set_real(set_or1222579329274155063t_real(A,B)),set_or1222579329274155063t_real(C,D))
    <=> ( ( ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
          | ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),A)
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),D)
            & ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),A)
              | aa_real_o(aa_real_fun_real_o(ord_less_real,B),D) ) ) )
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),D) ) ) ).

% atLeastatMost_psubset_iff
tff(fact_3696_ex__less__of__nat__mult,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Xb)
     => ? [N2: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,N2)),Xb)) ) ).

% ex_less_of_nat_mult
tff(fact_3697_ex__less__of__nat__mult,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ? [N2: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,Y),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N2)),Xb)) ) ).

% ex_less_of_nat_mult
tff(fact_3698_VEBT__internal_Omembermima_Osimps_I5_J,axiom,
    ! [V: nat,TreeLista: list_VEBT_VEBT,Vd: vEBT_VEBT,Xb: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_membermima,aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V),TreeLista),Vd)),Xb)
    <=> $let(
          pos: nat,
          pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
          $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeLista)),aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_membermima,nth_VEBT_VEBT(TreeLista,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ).

% VEBT_internal.membermima.simps(5)
tff(fact_3699_VEBT__internal_Omembermima_Osimps_I4_J,axiom,
    ! [Mia: nat,Maa: nat,V: nat,TreeLista: list_VEBT_VEBT,Vc2: vEBT_VEBT,Xb: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_membermima,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mia),Maa)),aa_nat_nat(suc,V),TreeLista),Vc2)),Xb)
    <=> ( ( Xb = Mia )
        | ( Xb = Maa )
        | $let(
            pos: nat,
            pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
            $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeLista)),aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_membermima,nth_VEBT_VEBT(TreeLista,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ).

% VEBT_internal.membermima.simps(4)
tff(fact_3700_double__not__eq__Suc__double,axiom,
    ! [M: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),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))),Nb)) ).

% double_not_eq_Suc_double
tff(fact_3701_Suc__double__not__eq__double,axiom,
    ! [M: nat,Nb: nat] : aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)) != aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb) ).

% Suc_double_not_eq_double
tff(fact_3702_VEBT__internal_Omembermima_Oelims_I2_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_membermima,Xb),Xaa)
     => ( ! [Mi: nat,Ma: nat] :
            ( ? [Va: list_VEBT_VEBT,Vb2: vEBT_VEBT] : Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),zero_zero_nat,Va),Vb2)
           => ~ ( ( Xaa = Mi )
                | ( Xaa = Ma ) ) )
       => ( ! [Mi: nat,Ma: nat,V2: nat,TreeList: list_VEBT_VEBT] :
              ( ? [Vc: vEBT_VEBT] : Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),aa_nat_nat(suc,V2),TreeList),Vc)
             => ~ ( ( Xaa = Mi )
                  | ( Xaa = Ma )
                  | $let(
                      pos: nat,
                      pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                      $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_membermima,nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) )
         => ~ ! [V2: nat,TreeList: list_VEBT_VEBT] :
                ( ? [Vd2: vEBT_VEBT] : Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList),Vd2)
               => ~ $let(
                      pos: nat,
                      pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                      $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_membermima,nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ).

% VEBT_internal.membermima.elims(2)
tff(fact_3703_del__x__mia,axiom,
    ! [Xb: nat,Mia: nat,Maa: nat,Deg: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT] :
      ( ( ( Xb = Mia )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Maa) )
     => ( ( Mia != Maa )
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg)
         => ( vEBT_vebt_delete(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mia),Maa)),Deg,TreeLista),Summarya),Xb) = $let(
                xn: nat,
                xn:= aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summarya))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeLista,the_nat(vEBT_vebt_mint(Summarya)))))),
                $let(
                  h: nat,
                  h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                  $ite(
                    aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeLista)),
                    $let(
                      newnode: vEBT_VEBT,
                      newnode:= vEBT_vebt_delete(nth_VEBT_VEBT(TreeLista,h),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),
                      $let(
                        newlist: list_VEBT_VEBT,
                        newlist:= list_u1324408373059187874T_VEBT(TreeLista,h,newnode),
                        $ite(
                          aa_VEBT_VEBT_o(vEBT_VEBT_minNull,newnode),
                          $let(
                            sn: vEBT_VEBT,
                            sn:= vEBT_vebt_delete(Summarya,h),
                            aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,xn),
                                    $ite(
                                      xn = Maa,
                                      $let(
                                        maxs: option_nat,
                                        maxs:= vEBT_vebt_maxt(sn),
                                        $ite(maxs = none_nat,xn,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),the_nat(maxs))),the_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(newlist,the_nat(maxs)))))) ),
                                      Maa ))),Deg,newlist),sn) ),
                          aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,xn),
                                  $ite(xn = Maa,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,h),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(newlist,h)))),Maa))),Deg,newlist),Summarya) ) ) ),
                    aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mia),Maa)),Deg,TreeLista),Summarya) ) ) ) ) ) ) ) ).

% del_x_mia
tff(fact_3704_del__x__mi__lets__in__minNull,axiom,
    ! [Xb: nat,Mia: nat,Maa: nat,Deg: nat,Xn: nat,H: nat,Summarya: vEBT_VEBT,TreeLista: list_VEBT_VEBT,L: nat,Newnode: vEBT_VEBT,Newlist: list_VEBT_VEBT,Sn: vEBT_VEBT] :
      ( ( ( Xb = Mia )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Maa) )
     => ( ( Mia != Maa )
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg)
         => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = H )
           => ( ( Xn = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summarya))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeLista,the_nat(vEBT_vebt_mint(Summarya)))))) )
             => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = L )
               => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeLista))
                 => ( ( Newnode = vEBT_vebt_delete(nth_VEBT_VEBT(TreeLista,H),L) )
                   => ( ( Newlist = list_u1324408373059187874T_VEBT(TreeLista,H,Newnode) )
                     => ( aa_VEBT_VEBT_o(vEBT_VEBT_minNull,Newnode)
                       => ( ( Sn = vEBT_vebt_delete(Summarya,H) )
                         => ( vEBT_vebt_delete(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mia),Maa)),Deg,TreeLista),Summarya),Xb) = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Xn),
                                      $ite(
                                        Xn = Maa,
                                        $let(
                                          maxs: option_nat,
                                          maxs:= vEBT_vebt_maxt(Sn),
                                          $ite(maxs = none_nat,Xn,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),the_nat(maxs))),the_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(Newlist,the_nat(maxs)))))) ),
                                        Maa ))),Deg,Newlist),Sn) ) ) ) ) ) ) ) ) ) ) ) ) ).

% del_x_mi_lets_in_minNull
tff(fact_3705_del__x__mi__lets__in,axiom,
    ! [Xb: nat,Mia: nat,Maa: nat,Deg: nat,Xn: nat,H: nat,Summarya: vEBT_VEBT,TreeLista: list_VEBT_VEBT,L: nat,Newnode: vEBT_VEBT,Newlist: list_VEBT_VEBT] :
      ( ( ( Xb = Mia )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Maa) )
     => ( ( Mia != Maa )
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg)
         => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = H )
           => ( ( Xn = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summarya))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeLista,the_nat(vEBT_vebt_mint(Summarya)))))) )
             => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = L )
               => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeLista))
                 => ( ( Newnode = vEBT_vebt_delete(nth_VEBT_VEBT(TreeLista,H),L) )
                   => ( ( Newlist = list_u1324408373059187874T_VEBT(TreeLista,H,Newnode) )
                     => ( vEBT_vebt_delete(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mia),Maa)),Deg,TreeLista),Summarya),Xb) = $ite(
                            aa_VEBT_VEBT_o(vEBT_VEBT_minNull,Newnode),
                            $let(
                              sn: vEBT_VEBT,
                              sn:= vEBT_vebt_delete(Summarya,H),
                              aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Xn),
                                      $ite(
                                        Xn = Maa,
                                        $let(
                                          maxs: option_nat,
                                          maxs:= vEBT_vebt_maxt(sn),
                                          $ite(maxs = none_nat,Xn,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),the_nat(maxs))),the_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(Newlist,the_nat(maxs)))))) ),
                                        Maa ))),Deg,Newlist),sn) ),
                            aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Xn),
                                    $ite(Xn = Maa,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,H),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(Newlist,H)))),Maa))),Deg,Newlist),Summarya) ) ) ) ) ) ) ) ) ) ) ) ).

% del_x_mi_lets_in
tff(fact_3706_del__x__mi,axiom,
    ! [Xb: nat,Mia: nat,Maa: nat,Deg: nat,Xn: nat,H: nat,Summarya: vEBT_VEBT,TreeLista: list_VEBT_VEBT,L: nat] :
      ( ( ( Xb = Mia )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Maa) )
     => ( ( Mia != Maa )
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg)
         => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = H )
           => ( ( Xn = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summarya))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeLista,the_nat(vEBT_vebt_mint(Summarya)))))) )
             => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = L )
               => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeLista))
                 => ( vEBT_vebt_delete(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mia),Maa)),Deg,TreeLista),Summarya),Xb) = $let(
                        newnode: vEBT_VEBT,
                        newnode:= vEBT_vebt_delete(nth_VEBT_VEBT(TreeLista,H),L),
                        $let(
                          newlist: list_VEBT_VEBT,
                          newlist:= list_u1324408373059187874T_VEBT(TreeLista,H,newnode),
                          $ite(
                            aa_VEBT_VEBT_o(vEBT_VEBT_minNull,newnode),
                            $let(
                              sn: vEBT_VEBT,
                              sn:= vEBT_vebt_delete(Summarya,H),
                              aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Xn),
                                      $ite(
                                        Xn = Maa,
                                        $let(
                                          maxs: option_nat,
                                          maxs:= vEBT_vebt_maxt(sn),
                                          $ite(maxs = none_nat,Xn,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),the_nat(maxs))),the_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(newlist,the_nat(maxs)))))) ),
                                        Maa ))),Deg,newlist),sn) ),
                            aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Xn),
                                    $ite(Xn = Maa,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,H),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(newlist,H)))),Maa))),Deg,newlist),Summarya) ) ) ) ) ) ) ) ) ) ) ) ).

% del_x_mi
tff(fact_3707_del__in__range,axiom,
    ! [Mia: nat,Xb: nat,Maa: nat,Deg: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mia),Xb)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Maa) )
     => ( ( Mia != Maa )
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg)
         => ( vEBT_vebt_delete(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mia),Maa)),Deg,TreeLista),Summarya),Xb) = $let(
                xn: nat,
                xn:= 
                  $ite(Xb = Mia,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summarya))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeLista,the_nat(vEBT_vebt_mint(Summarya)))))),Xb),
                $let(
                  minn: nat,
                  minn:= 
                    $ite(Xb = Mia,xn,Mia),
                  $let(
                    h: nat,
                    h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                    $ite(
                      aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeLista)),
                      $let(
                        newnode: vEBT_VEBT,
                        newnode:= vEBT_vebt_delete(nth_VEBT_VEBT(TreeLista,h),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),
                        $let(
                          newlist: list_VEBT_VEBT,
                          newlist:= list_u1324408373059187874T_VEBT(TreeLista,h,newnode),
                          $ite(
                            aa_VEBT_VEBT_o(vEBT_VEBT_minNull,newnode),
                            $let(
                              sn: vEBT_VEBT,
                              sn:= vEBT_vebt_delete(Summarya,h),
                              aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,minn),
                                      $ite(
                                        xn = Maa,
                                        $let(
                                          maxs: option_nat,
                                          maxs:= vEBT_vebt_maxt(sn),
                                          $ite(maxs = none_nat,minn,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),the_nat(maxs))),the_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(newlist,the_nat(maxs)))))) ),
                                        Maa ))),Deg,newlist),sn) ),
                            aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,minn),
                                    $ite(xn = Maa,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,h),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(newlist,h)))),Maa))),Deg,newlist),Summarya) ) ) ),
                      aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mia),Maa)),Deg,TreeLista),Summarya) ) ) ) ) ) ) ) ) ).

% del_in_range
tff(fact_3708_del__x__not__mia,axiom,
    ! [Mia: nat,Xb: nat,Maa: nat,Deg: nat,H: nat,L: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mia),Xb)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Maa) )
     => ( ( Mia != Maa )
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg)
         => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = H )
           => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = L )
             => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeLista))
               => ( vEBT_vebt_delete(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mia),Maa)),Deg,TreeLista),Summarya),Xb) = $let(
                      newnode: vEBT_VEBT,
                      newnode:= vEBT_vebt_delete(nth_VEBT_VEBT(TreeLista,H),L),
                      $let(
                        newlist: list_VEBT_VEBT,
                        newlist:= list_u1324408373059187874T_VEBT(TreeLista,H,newnode),
                        $ite(
                          aa_VEBT_VEBT_o(vEBT_VEBT_minNull,newnode),
                          $let(
                            sn: vEBT_VEBT,
                            sn:= vEBT_vebt_delete(Summarya,H),
                            aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mia),
                                    $ite(
                                      Xb = Maa,
                                      $let(
                                        maxs: option_nat,
                                        maxs:= vEBT_vebt_maxt(sn),
                                        $ite(maxs = none_nat,Mia,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),the_nat(maxs))),the_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(newlist,the_nat(maxs)))))) ),
                                      Maa ))),Deg,newlist),sn) ),
                          aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mia),
                                  $ite(Xb = Maa,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,H),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(newlist,H)))),Maa))),Deg,newlist),Summarya) ) ) ) ) ) ) ) ) ) ) ).

% del_x_not_mia
tff(fact_3709_del__x__not__mi__new__node__nil,axiom,
    ! [Mia: nat,Xb: nat,Maa: nat,Deg: nat,H: nat,L: nat,Newnode: vEBT_VEBT,TreeLista: list_VEBT_VEBT,Sn: vEBT_VEBT,Summarya: vEBT_VEBT,Newlist: list_VEBT_VEBT] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mia),Xb)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Maa) )
     => ( ( Mia != Maa )
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg)
         => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = H )
           => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = L )
             => ( ( Newnode = vEBT_vebt_delete(nth_VEBT_VEBT(TreeLista,H),L) )
               => ( aa_VEBT_VEBT_o(vEBT_VEBT_minNull,Newnode)
                 => ( ( Sn = vEBT_vebt_delete(Summarya,H) )
                   => ( ( Newlist = list_u1324408373059187874T_VEBT(TreeLista,H,Newnode) )
                     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeLista))
                       => ( vEBT_vebt_delete(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mia),Maa)),Deg,TreeLista),Summarya),Xb) = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mia),
                                    $ite(
                                      Xb = Maa,
                                      $let(
                                        maxs: option_nat,
                                        maxs:= vEBT_vebt_maxt(Sn),
                                        $ite(maxs = none_nat,Mia,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),the_nat(maxs))),the_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(Newlist,the_nat(maxs)))))) ),
                                      Maa ))),Deg,Newlist),Sn) ) ) ) ) ) ) ) ) ) ) ) ).

% del_x_not_mi_new_node_nil
tff(fact_3710_del__x__not__mi,axiom,
    ! [Mia: nat,Xb: nat,Maa: nat,Deg: nat,H: nat,L: nat,Newnode: vEBT_VEBT,TreeLista: list_VEBT_VEBT,Newlist: list_VEBT_VEBT,Summarya: vEBT_VEBT] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mia),Xb)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Maa) )
     => ( ( Mia != Maa )
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg)
         => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = H )
           => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = L )
             => ( ( Newnode = vEBT_vebt_delete(nth_VEBT_VEBT(TreeLista,H),L) )
               => ( ( Newlist = list_u1324408373059187874T_VEBT(TreeLista,H,Newnode) )
                 => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeLista))
                   => ( vEBT_vebt_delete(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mia),Maa)),Deg,TreeLista),Summarya),Xb) = $ite(
                          aa_VEBT_VEBT_o(vEBT_VEBT_minNull,Newnode),
                          $let(
                            sn: vEBT_VEBT,
                            sn:= vEBT_vebt_delete(Summarya,H),
                            aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mia),
                                    $ite(
                                      Xb = Maa,
                                      $let(
                                        maxs: option_nat,
                                        maxs:= vEBT_vebt_maxt(sn),
                                        $ite(maxs = none_nat,Mia,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),the_nat(maxs))),the_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(Newlist,the_nat(maxs)))))) ),
                                      Maa ))),Deg,Newlist),sn) ),
                          aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mia),
                                  $ite(Xb = Maa,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,H),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(Newlist,H)))),Maa))),Deg,Newlist),Summarya) ) ) ) ) ) ) ) ) ) ) ).

% del_x_not_mi
tff(fact_3711_nth__update__invalid,axiom,
    ! [I: nat,L: list_VEBT_VEBT,J: nat,Xb: vEBT_VEBT] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s6755466524823107622T_VEBT(L))
     => ( nth_VEBT_VEBT(list_u1324408373059187874T_VEBT(L,J,Xb),I) = nth_VEBT_VEBT(L,I) ) ) ).

% nth_update_invalid
tff(fact_3712_nth__update__invalid,axiom,
    ! [I: nat,L: list_VEBT_VEBTi,J: nat,Xb: vEBT_VEBTi] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s7982070591426661849_VEBTi(L))
     => ( nth_VEBT_VEBTi(list_u6098035379799741383_VEBTi(L,J,Xb),I) = nth_VEBT_VEBTi(L,I) ) ) ).

% nth_update_invalid
tff(fact_3713_nth__update__invalid,axiom,
    ! [I: nat,L: list_real,J: nat,Xb: real] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_real(L))
     => ( aa_nat_real(nth_real(list_update_real(L,J,Xb)),I) = aa_nat_real(nth_real(L),I) ) ) ).

% nth_update_invalid
tff(fact_3714_nth__update__invalid,axiom,
    ! [I: nat,L: list_o,J: nat,Xb: $o] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_o(L))
     => ( aa_nat_o(nth_o(list_update_o(L,J,(Xb))),I)
      <=> aa_nat_o(nth_o(L),I) ) ) ).

% nth_update_invalid
tff(fact_3715_nth__update__invalid,axiom,
    ! [I: nat,L: list_nat,J: nat,Xb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_nat(L))
     => ( aa_nat_nat(nth_nat(list_update_nat(L,J,Xb)),I) = aa_nat_nat(nth_nat(L),I) ) ) ).

% nth_update_invalid
tff(fact_3716_nth__update__invalid,axiom,
    ! [I: nat,L: list_int,J: nat,Xb: int] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_int(L))
     => ( aa_nat_int(nth_int(list_update_int(L,J,Xb)),I) = aa_nat_int(nth_int(L),I) ) ) ).

% nth_update_invalid
tff(fact_3717_del__x__not__mi__newnode__not__nil,axiom,
    ! [Mia: nat,Xb: nat,Maa: nat,Deg: nat,H: nat,L: nat,Newnode: vEBT_VEBT,TreeLista: list_VEBT_VEBT,Newlist: list_VEBT_VEBT,Summarya: vEBT_VEBT] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mia),Xb)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Maa) )
     => ( ( Mia != Maa )
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg)
         => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = H )
           => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = L )
             => ( ( Newnode = vEBT_vebt_delete(nth_VEBT_VEBT(TreeLista,H),L) )
               => ( ~ aa_VEBT_VEBT_o(vEBT_VEBT_minNull,Newnode)
                 => ( ( Newlist = list_u1324408373059187874T_VEBT(TreeLista,H,Newnode) )
                   => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeLista))
                     => ( vEBT_vebt_delete(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mia),Maa)),Deg,TreeLista),Summarya),Xb) = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mia),
                                  $ite(Xb = Maa,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,H),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(Newlist,H)))),Maa))),Deg,Newlist),Summarya) ) ) ) ) ) ) ) ) ) ) ).

% del_x_not_mi_newnode_not_nil
tff(fact_3718_del__x__mi__lets__in__not__minNull,axiom,
    ! [Xb: nat,Mia: nat,Maa: nat,Deg: nat,Xn: nat,H: nat,Summarya: vEBT_VEBT,TreeLista: list_VEBT_VEBT,L: nat,Newnode: vEBT_VEBT,Newlist: list_VEBT_VEBT] :
      ( ( ( Xb = Mia )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Maa) )
     => ( ( Mia != Maa )
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg)
         => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = H )
           => ( ( Xn = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summarya))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeLista,the_nat(vEBT_vebt_mint(Summarya)))))) )
             => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = L )
               => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeLista))
                 => ( ( Newnode = vEBT_vebt_delete(nth_VEBT_VEBT(TreeLista,H),L) )
                   => ( ( Newlist = list_u1324408373059187874T_VEBT(TreeLista,H,Newnode) )
                     => ( ~ aa_VEBT_VEBT_o(vEBT_VEBT_minNull,Newnode)
                       => ( vEBT_vebt_delete(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mia),Maa)),Deg,TreeLista),Summarya),Xb) = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Xn),
                                    $ite(Xn = Maa,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,H),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(Newlist,H)))),Maa))),Deg,Newlist),Summarya) ) ) ) ) ) ) ) ) ) ) ) ).

% del_x_mi_lets_in_not_minNull
tff(fact_3719_in__set__upd__eq,axiom,
    ! [I: nat,L: list_P8527749157015355191n_assn,Xb: produc6575502325842934193n_assn,Y: produc6575502325842934193n_assn] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s6829681357464350627n_assn(L))
     => ( member7957490590177025114n_assn(Xb,set_Pr1139785259514867910n_assn(list_u4534839942911652127n_assn(L,I,Y)))
      <=> ( ( Xb = Y )
          | ( member7957490590177025114n_assn(Xb,set_Pr1139785259514867910n_assn(L))
            & ! [Y3: produc6575502325842934193n_assn] : member7957490590177025114n_assn(Xb,set_Pr1139785259514867910n_assn(list_u4534839942911652127n_assn(L,I,Y3))) ) ) ) ) ).

% in_set_upd_eq
tff(fact_3720_in__set__upd__eq,axiom,
    ! [I: nat,L: list_VEBT_VEBT,Xb: vEBT_VEBT,Y: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s6755466524823107622T_VEBT(L))
     => ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xb),set_VEBT_VEBT2(list_u1324408373059187874T_VEBT(L,I,Y)))
      <=> ( ( Xb = Y )
          | ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xb),set_VEBT_VEBT2(L))
            & ! [Y3: vEBT_VEBT] : aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xb),set_VEBT_VEBT2(list_u1324408373059187874T_VEBT(L,I,Y3))) ) ) ) ) ).

% in_set_upd_eq
tff(fact_3721_in__set__upd__eq,axiom,
    ! [I: nat,L: list_VEBT_VEBTi,Xb: vEBT_VEBTi,Y: vEBT_VEBTi] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s7982070591426661849_VEBTi(L))
     => ( member_VEBT_VEBTi(Xb,set_VEBT_VEBTi2(list_u6098035379799741383_VEBTi(L,I,Y)))
      <=> ( ( Xb = Y )
          | ( member_VEBT_VEBTi(Xb,set_VEBT_VEBTi2(L))
            & ! [Y3: vEBT_VEBTi] : member_VEBT_VEBTi(Xb,set_VEBT_VEBTi2(list_u6098035379799741383_VEBTi(L,I,Y3))) ) ) ) ) ).

% in_set_upd_eq
tff(fact_3722_in__set__upd__eq,axiom,
    ! [I: nat,L: list_real,Xb: real,Y: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_real(L))
     => ( aa_set_real_o(member_real(Xb),set_real2(list_update_real(L,I,Y)))
      <=> ( ( Xb = Y )
          | ( aa_set_real_o(member_real(Xb),set_real2(L))
            & ! [Y3: real] : aa_set_real_o(member_real(Xb),set_real2(list_update_real(L,I,Y3))) ) ) ) ) ).

% in_set_upd_eq
tff(fact_3723_in__set__upd__eq,axiom,
    ! [I: nat,L: list_o,Xb: $o,Y: $o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_o(L))
     => ( member_o((Xb),set_o2(list_update_o(L,I,(Y))))
      <=> ( ( (Xb)
          <=> (Y) )
          | ( member_o((Xb),set_o2(L))
            & ! [Y3: $o] : member_o((Xb),set_o2(list_update_o(L,I,(Y3)))) ) ) ) ) ).

% in_set_upd_eq
tff(fact_3724_in__set__upd__eq,axiom,
    ! [I: nat,L: list_nat,Xb: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_nat(L))
     => ( aa_set_nat_o(member_nat(Xb),set_nat2(list_update_nat(L,I,Y)))
      <=> ( ( Xb = Y )
          | ( aa_set_nat_o(member_nat(Xb),set_nat2(L))
            & ! [Y3: nat] : aa_set_nat_o(member_nat(Xb),set_nat2(list_update_nat(L,I,Y3))) ) ) ) ) ).

% in_set_upd_eq
tff(fact_3725_in__set__upd__eq,axiom,
    ! [I: nat,L: list_int,Xb: int,Y: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_int(L))
     => ( aa_set_int_o(member_int(Xb),set_int2(list_update_int(L,I,Y)))
      <=> ( ( Xb = Y )
          | ( aa_set_int_o(member_int(Xb),set_int2(L))
            & ! [Y3: int] : aa_set_int_o(member_int(Xb),set_int2(list_update_int(L,I,Y3))) ) ) ) ) ).

% in_set_upd_eq
tff(fact_3726_in__set__upd__cases,axiom,
    ! [Xb: produc6575502325842934193n_assn,L: list_P8527749157015355191n_assn,I: nat,Y: produc6575502325842934193n_assn] :
      ( member7957490590177025114n_assn(Xb,set_Pr1139785259514867910n_assn(list_u4534839942911652127n_assn(L,I,Y)))
     => ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s6829681357464350627n_assn(L))
         => ( Xb != Y ) )
       => member7957490590177025114n_assn(Xb,set_Pr1139785259514867910n_assn(L)) ) ) ).

% in_set_upd_cases
tff(fact_3727_in__set__upd__cases,axiom,
    ! [Xb: vEBT_VEBT,L: list_VEBT_VEBT,I: nat,Y: vEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xb),set_VEBT_VEBT2(list_u1324408373059187874T_VEBT(L,I,Y)))
     => ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s6755466524823107622T_VEBT(L))
         => ( Xb != Y ) )
       => aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xb),set_VEBT_VEBT2(L)) ) ) ).

% in_set_upd_cases
tff(fact_3728_in__set__upd__cases,axiom,
    ! [Xb: vEBT_VEBTi,L: list_VEBT_VEBTi,I: nat,Y: vEBT_VEBTi] :
      ( member_VEBT_VEBTi(Xb,set_VEBT_VEBTi2(list_u6098035379799741383_VEBTi(L,I,Y)))
     => ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s7982070591426661849_VEBTi(L))
         => ( Xb != Y ) )
       => member_VEBT_VEBTi(Xb,set_VEBT_VEBTi2(L)) ) ) ).

% in_set_upd_cases
tff(fact_3729_in__set__upd__cases,axiom,
    ! [Xb: real,L: list_real,I: nat,Y: real] :
      ( aa_set_real_o(member_real(Xb),set_real2(list_update_real(L,I,Y)))
     => ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_real(L))
         => ( Xb != Y ) )
       => aa_set_real_o(member_real(Xb),set_real2(L)) ) ) ).

% in_set_upd_cases
tff(fact_3730_in__set__upd__cases,axiom,
    ! [Xb: $o,L: list_o,I: nat,Y: $o] :
      ( member_o((Xb),set_o2(list_update_o(L,I,(Y))))
     => ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_o(L))
         => ( (Xb)
          <=> ~ (Y) ) )
       => member_o((Xb),set_o2(L)) ) ) ).

% in_set_upd_cases
tff(fact_3731_in__set__upd__cases,axiom,
    ! [Xb: nat,L: list_nat,I: nat,Y: nat] :
      ( aa_set_nat_o(member_nat(Xb),set_nat2(list_update_nat(L,I,Y)))
     => ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_nat(L))
         => ( Xb != Y ) )
       => aa_set_nat_o(member_nat(Xb),set_nat2(L)) ) ) ).

% in_set_upd_cases
tff(fact_3732_in__set__upd__cases,axiom,
    ! [Xb: int,L: list_int,I: nat,Y: int] :
      ( aa_set_int_o(member_int(Xb),set_int2(list_update_int(L,I,Y)))
     => ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_int(L))
         => ( Xb != Y ) )
       => aa_set_int_o(member_int(Xb),set_int2(L)) ) ) ).

% in_set_upd_cases
tff(fact_3733_in__set__upd__eq__aux,axiom,
    ! [I: nat,L: list_P8527749157015355191n_assn,Xb: produc6575502325842934193n_assn,Y: produc6575502325842934193n_assn] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s6829681357464350627n_assn(L))
     => ( member7957490590177025114n_assn(Xb,set_Pr1139785259514867910n_assn(list_u4534839942911652127n_assn(L,I,Y)))
      <=> ( ( Xb = Y )
          | ! [Y3: produc6575502325842934193n_assn] : member7957490590177025114n_assn(Xb,set_Pr1139785259514867910n_assn(list_u4534839942911652127n_assn(L,I,Y3))) ) ) ) ).

% in_set_upd_eq_aux
tff(fact_3734_in__set__upd__eq__aux,axiom,
    ! [I: nat,L: list_VEBT_VEBT,Xb: vEBT_VEBT,Y: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s6755466524823107622T_VEBT(L))
     => ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xb),set_VEBT_VEBT2(list_u1324408373059187874T_VEBT(L,I,Y)))
      <=> ( ( Xb = Y )
          | ! [Y3: vEBT_VEBT] : aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xb),set_VEBT_VEBT2(list_u1324408373059187874T_VEBT(L,I,Y3))) ) ) ) ).

% in_set_upd_eq_aux
tff(fact_3735_in__set__upd__eq__aux,axiom,
    ! [I: nat,L: list_VEBT_VEBTi,Xb: vEBT_VEBTi,Y: vEBT_VEBTi] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s7982070591426661849_VEBTi(L))
     => ( member_VEBT_VEBTi(Xb,set_VEBT_VEBTi2(list_u6098035379799741383_VEBTi(L,I,Y)))
      <=> ( ( Xb = Y )
          | ! [Y3: vEBT_VEBTi] : member_VEBT_VEBTi(Xb,set_VEBT_VEBTi2(list_u6098035379799741383_VEBTi(L,I,Y3))) ) ) ) ).

% in_set_upd_eq_aux
tff(fact_3736_in__set__upd__eq__aux,axiom,
    ! [I: nat,L: list_real,Xb: real,Y: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_real(L))
     => ( aa_set_real_o(member_real(Xb),set_real2(list_update_real(L,I,Y)))
      <=> ( ( Xb = Y )
          | ! [Y3: real] : aa_set_real_o(member_real(Xb),set_real2(list_update_real(L,I,Y3))) ) ) ) ).

% in_set_upd_eq_aux
tff(fact_3737_in__set__upd__eq__aux,axiom,
    ! [I: nat,L: list_o,Xb: $o,Y: $o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_o(L))
     => ( member_o((Xb),set_o2(list_update_o(L,I,(Y))))
      <=> ( ( (Xb)
          <=> (Y) )
          | ! [Y3: $o] : member_o((Xb),set_o2(list_update_o(L,I,(Y3)))) ) ) ) ).

% in_set_upd_eq_aux
tff(fact_3738_in__set__upd__eq__aux,axiom,
    ! [I: nat,L: list_nat,Xb: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_nat(L))
     => ( aa_set_nat_o(member_nat(Xb),set_nat2(list_update_nat(L,I,Y)))
      <=> ( ( Xb = Y )
          | ! [Y3: nat] : aa_set_nat_o(member_nat(Xb),set_nat2(list_update_nat(L,I,Y3))) ) ) ) ).

% in_set_upd_eq_aux
tff(fact_3739_in__set__upd__eq__aux,axiom,
    ! [I: nat,L: list_int,Xb: int,Y: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_int(L))
     => ( aa_set_int_o(member_int(Xb),set_int2(list_update_int(L,I,Y)))
      <=> ( ( Xb = Y )
          | ! [Y3: int] : aa_set_int_o(member_int(Xb),set_int2(list_update_int(L,I,Y3))) ) ) ) ).

% in_set_upd_eq_aux
tff(fact_3740_nth__list__update_H,axiom,
    ! [L: list_VEBT_VEBT,I: nat,Xb: vEBT_VEBT,J: nat] :
      nth_VEBT_VEBT(list_u1324408373059187874T_VEBT(L,I,Xb),J) = $ite(
        ( ( I = J )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s6755466524823107622T_VEBT(L)) ),
        Xb,
        nth_VEBT_VEBT(L,J) ) ).

% nth_list_update'
tff(fact_3741_nth__list__update_H,axiom,
    ! [L: list_VEBT_VEBTi,I: nat,Xb: vEBT_VEBTi,J: nat] :
      nth_VEBT_VEBTi(list_u6098035379799741383_VEBTi(L,I,Xb),J) = $ite(
        ( ( I = J )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s7982070591426661849_VEBTi(L)) ),
        Xb,
        nth_VEBT_VEBTi(L,J) ) ).

% nth_list_update'
tff(fact_3742_nth__list__update_H,axiom,
    ! [L: list_real,I: nat,Xb: real,J: nat] :
      aa_nat_real(nth_real(list_update_real(L,I,Xb)),J) = $ite(
        ( ( I = J )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_real(L)) ),
        Xb,
        aa_nat_real(nth_real(L),J) ) ).

% nth_list_update'
tff(fact_3743_nth__list__update_H,axiom,
    ! [L: list_o,I: nat,Xb: $o,J: nat] :
      ( aa_nat_o(nth_o(list_update_o(L,I,(Xb))),J)
    <=> $ite(
          ( ( I = J )
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_o(L)) ),
          (Xb),
          aa_nat_o(nth_o(L),J) ) ) ).

% nth_list_update'
tff(fact_3744_nth__list__update_H,axiom,
    ! [L: list_nat,I: nat,Xb: nat,J: nat] :
      aa_nat_nat(nth_nat(list_update_nat(L,I,Xb)),J) = $ite(
        ( ( I = J )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_nat(L)) ),
        Xb,
        aa_nat_nat(nth_nat(L),J) ) ).

% nth_list_update'
tff(fact_3745_nth__list__update_H,axiom,
    ! [L: list_int,I: nat,Xb: int,J: nat] :
      aa_nat_int(nth_int(list_update_int(L,I,Xb)),J) = $ite(
        ( ( I = J )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_int(L)) ),
        Xb,
        aa_nat_int(nth_int(L),J) ) ).

% nth_list_update'
tff(fact_3746_VEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H_Osimps_I7_J,axiom,
    ! [Mia: nat,Maa: nat,Vaa: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Xb: nat] :
      aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_V1232361888498592333_e_t_e,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mia),Maa)),aa_nat_nat(suc,aa_nat_nat(suc,Vaa)),TreeLista),Summarya)),Xb) = $ite(
        ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Mia)
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Maa),Xb) ),
        one_one_nat,
        $ite(
          ( ( Xb = Mia )
          & ( Xb = Maa ) ),
          one_one_nat,
          $let(
            xn: nat,
            xn:= 
              $ite(Xb = Mia,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summarya))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Vaa))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeLista,the_nat(vEBT_vebt_mint(Summarya)))))),Xb),
            $let(
              l: nat,
              l:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Vaa))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
              $let(
                h: nat,
                h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Vaa))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                $ite(
                  aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeLista)),
                  aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_V1232361888498592333_e_t_e,nth_VEBT_VEBT(TreeLista,h)),l)),
                    $ite(aa_VEBT_VEBT_o(vEBT_VEBT_minNull,vEBT_vebt_delete(nth_VEBT_VEBT(TreeLista,h),l)),aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_V1232361888498592333_e_t_e,Summarya),h),one_one_nat)),
                  one_one_nat ) ) ) ) ) ) ).

% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.simps(7)
tff(fact_3747_VEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H_Oelims,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_V1232361888498592333_e_t_e,Xb),Xaa) = Y )
     => ( ( ? [A3: $o,B2: $o] : Xb = vEBT_Leaf((A3),(B2))
         => ( ( Xaa = zero_zero_nat )
           => ( Y != one_one_nat ) ) )
       => ( ( ? [A3: $o,B2: $o] : Xb = vEBT_Leaf((A3),(B2))
           => ( ( Xaa = aa_nat_nat(suc,zero_zero_nat) )
             => ( Y != one_one_nat ) ) )
         => ( ( ? [A3: $o,B2: $o] : Xb = vEBT_Leaf((A3),(B2))
             => ( ? [N2: nat] : Xaa = aa_nat_nat(suc,aa_nat_nat(suc,N2))
               => ( Y != one_one_nat ) ) )
           => ( ( ? [Deg2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] : Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Deg2,TreeList),Summary)
               => ( Y != one_one_nat ) )
             => ( ( ? [Mi: nat,Ma: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] : Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),zero_zero_nat,TreeList),Summary)
                 => ( Y != one_one_nat ) )
               => ( ( ? [Mi: nat,Ma: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] : Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),aa_nat_nat(suc,zero_zero_nat),TreeList),Summary)
                   => ( Y != one_one_nat ) )
                 => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                        ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList),Summary) )
                       => ( Y != $ite(
                              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi)
                              | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Xaa) ),
                              one_one_nat,
                              $ite(
                                ( ( Xaa = Mi )
                                & ( Xaa = Ma ) ),
                                one_one_nat,
                                $let(
                                  xn: nat,
                                  xn:= 
                                    $ite(Xaa = Mi,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat(vEBT_vebt_mint(Summary)))))),Xaa),
                                  $let(
                                    l: nat,
                                    l:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                    $let(
                                      h: nat,
                                      h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                      $ite(
                                        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList)),
                                        aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_V1232361888498592333_e_t_e,nth_VEBT_VEBT(TreeList,h)),l)),
                                          $ite(aa_VEBT_VEBT_o(vEBT_VEBT_minNull,vEBT_vebt_delete(nth_VEBT_VEBT(TreeList,h),l)),aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_V1232361888498592333_e_t_e,Summary),h),one_one_nat)),
                                        one_one_nat ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.elims
tff(fact_3748_vebt__delete_Oelims,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: vEBT_VEBT] :
      ( ( vEBT_vebt_delete(Xb,Xaa) = Y )
     => ( ! [A3: $o,B2: $o] :
            ( ( Xb = vEBT_Leaf((A3),(B2)) )
           => ( ( Xaa = zero_zero_nat )
             => ( Y != vEBT_Leaf($false,(B2)) ) ) )
       => ( ! [A3: $o] :
              ( ? [B2: $o] : Xb = vEBT_Leaf((A3),(B2))
             => ( ( Xaa = aa_nat_nat(suc,zero_zero_nat) )
               => ( Y != vEBT_Leaf((A3),$false) ) ) )
         => ( ! [A3: $o,B2: $o] :
                ( ( Xb = vEBT_Leaf((A3),(B2)) )
               => ( ? [N2: nat] : Xaa = aa_nat_nat(suc,aa_nat_nat(suc,N2))
                 => ( Y != vEBT_Leaf((A3),(B2)) ) ) )
           => ( ! [Deg2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                  ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Deg2,TreeList),Summary) )
                 => ( Y != aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Deg2,TreeList),Summary) ) )
             => ( ! [Mi: nat,Ma: nat,TrLst: list_VEBT_VEBT,Smry: vEBT_VEBT] :
                    ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),zero_zero_nat,TrLst),Smry) )
                   => ( Y != aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),zero_zero_nat,TrLst),Smry) ) )
               => ( ! [Mi: nat,Ma: nat,Tr: list_VEBT_VEBT,Sm: vEBT_VEBT] :
                      ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),aa_nat_nat(suc,zero_zero_nat),Tr),Sm) )
                     => ( Y != aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),aa_nat_nat(suc,zero_zero_nat),Tr),Sm) ) )
                 => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                        ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList),Summary) )
                       => ( Y != $ite(
                              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi)
                              | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Xaa) ),
                              aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList),Summary),
                              $ite(
                                ( ( Xaa = Mi )
                                & ( Xaa = Ma ) ),
                                aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList),Summary),
                                $let(
                                  xn: nat,
                                  xn:= 
                                    $ite(Xaa = Mi,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat(vEBT_vebt_mint(Summary)))))),Xaa),
                                  $let(
                                    minn: nat,
                                    minn:= 
                                      $ite(Xaa = Mi,xn,Mi),
                                    $let(
                                      h: nat,
                                      h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                      $ite(
                                        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList)),
                                        $let(
                                          newnode: vEBT_VEBT,
                                          newnode:= vEBT_vebt_delete(nth_VEBT_VEBT(TreeList,h),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),
                                          $let(
                                            newlist: list_VEBT_VEBT,
                                            newlist:= list_u1324408373059187874T_VEBT(TreeList,h,newnode),
                                            $ite(
                                              aa_VEBT_VEBT_o(vEBT_VEBT_minNull,newnode),
                                              $let(
                                                sn: vEBT_VEBT,
                                                sn:= vEBT_vebt_delete(Summary,h),
                                                aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,minn),
                                                        $ite(
                                                          xn = Ma,
                                                          $let(
                                                            maxs: option_nat,
                                                            maxs:= vEBT_vebt_maxt(sn),
                                                            $ite(maxs = none_nat,minn,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),the_nat(maxs))),the_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(newlist,the_nat(maxs)))))) ),
                                                          Ma ))),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),newlist),sn) ),
                                              aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,minn),
                                                      $ite(xn = Ma,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,h),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(newlist,h)))),Ma))),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),newlist),Summary) ) ) ),
                                        aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList),Summary) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_delete.elims
tff(fact_3749_vebt__delete_Osimps_I7_J,axiom,
    ! [Mia: nat,Maa: nat,Vaa: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Xb: nat] :
      vEBT_vebt_delete(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mia),Maa)),aa_nat_nat(suc,aa_nat_nat(suc,Vaa)),TreeLista),Summarya),Xb) = $ite(
        ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Mia)
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Maa),Xb) ),
        aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mia),Maa)),aa_nat_nat(suc,aa_nat_nat(suc,Vaa)),TreeLista),Summarya),
        $ite(
          ( ( Xb = Mia )
          & ( Xb = Maa ) ),
          aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,Vaa)),TreeLista),Summarya),
          $let(
            xn: nat,
            xn:= 
              $ite(Xb = Mia,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summarya))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Vaa))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeLista,the_nat(vEBT_vebt_mint(Summarya)))))),Xb),
            $let(
              minn: nat,
              minn:= 
                $ite(Xb = Mia,xn,Mia),
              $let(
                h: nat,
                h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Vaa))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                $ite(
                  aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeLista)),
                  $let(
                    newnode: vEBT_VEBT,
                    newnode:= vEBT_vebt_delete(nth_VEBT_VEBT(TreeLista,h),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Vaa))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),
                    $let(
                      newlist: list_VEBT_VEBT,
                      newlist:= list_u1324408373059187874T_VEBT(TreeLista,h,newnode),
                      $ite(
                        aa_VEBT_VEBT_o(vEBT_VEBT_minNull,newnode),
                        $let(
                          sn: vEBT_VEBT,
                          sn:= vEBT_vebt_delete(Summarya,h),
                          aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,minn),
                                  $ite(
                                    xn = Maa,
                                    $let(
                                      maxs: option_nat,
                                      maxs:= vEBT_vebt_maxt(sn),
                                      $ite(maxs = none_nat,minn,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Vaa))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),the_nat(maxs))),the_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(newlist,the_nat(maxs)))))) ),
                                    Maa ))),aa_nat_nat(suc,aa_nat_nat(suc,Vaa)),newlist),sn) ),
                        aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,minn),
                                $ite(xn = Maa,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,h),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Vaa))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(newlist,h)))),Maa))),aa_nat_nat(suc,aa_nat_nat(suc,Vaa)),newlist),Summarya) ) ) ),
                  aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mia),Maa)),aa_nat_nat(suc,aa_nat_nat(suc,Vaa)),TreeLista),Summarya) ) ) ) ) ) ) ).

% vebt_delete.simps(7)
tff(fact_3750_set__swap,axiom,
    ! [I: nat,Xs: list_P8527749157015355191n_assn,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s6829681357464350627n_assn(Xs))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J),size_s6829681357464350627n_assn(Xs))
       => ( set_Pr1139785259514867910n_assn(list_u4534839942911652127n_assn(list_u4534839942911652127n_assn(Xs,I,nth_Pr1769885009046257848n_assn(Xs,J)),J,nth_Pr1769885009046257848n_assn(Xs,I))) = set_Pr1139785259514867910n_assn(Xs) ) ) ) ).

% set_swap
tff(fact_3751_set__swap,axiom,
    ! [I: nat,Xs: list_VEBT_VEBT,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J),size_s6755466524823107622T_VEBT(Xs))
       => ( set_VEBT_VEBT2(list_u1324408373059187874T_VEBT(list_u1324408373059187874T_VEBT(Xs,I,nth_VEBT_VEBT(Xs,J)),J,nth_VEBT_VEBT(Xs,I))) = set_VEBT_VEBT2(Xs) ) ) ) ).

% set_swap
tff(fact_3752_set__swap,axiom,
    ! [I: nat,Xs: list_VEBT_VEBTi,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s7982070591426661849_VEBTi(Xs))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J),size_s7982070591426661849_VEBTi(Xs))
       => ( set_VEBT_VEBTi2(list_u6098035379799741383_VEBTi(list_u6098035379799741383_VEBTi(Xs,I,nth_VEBT_VEBTi(Xs,J)),J,nth_VEBT_VEBTi(Xs,I))) = set_VEBT_VEBTi2(Xs) ) ) ) ).

% set_swap
tff(fact_3753_set__swap,axiom,
    ! [I: nat,Xs: list_real,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_real(Xs))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J),size_size_list_real(Xs))
       => ( set_real2(list_update_real(list_update_real(Xs,I,aa_nat_real(nth_real(Xs),J)),J,aa_nat_real(nth_real(Xs),I))) = set_real2(Xs) ) ) ) ).

% set_swap
tff(fact_3754_set__swap,axiom,
    ! [I: nat,Xs: list_o,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_o(Xs))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J),size_size_list_o(Xs))
       => ( set_o2(list_update_o(list_update_o(Xs,I,aa_nat_o(nth_o(Xs),J)),J,aa_nat_o(nth_o(Xs),I))) = set_o2(Xs) ) ) ) ).

% set_swap
tff(fact_3755_set__swap,axiom,
    ! [I: nat,Xs: list_nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_nat(Xs))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J),size_size_list_nat(Xs))
       => ( set_nat2(list_update_nat(list_update_nat(Xs,I,aa_nat_nat(nth_nat(Xs),J)),J,aa_nat_nat(nth_nat(Xs),I))) = set_nat2(Xs) ) ) ) ).

% set_swap
tff(fact_3756_set__swap,axiom,
    ! [I: nat,Xs: list_int,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_int(Xs))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J),size_size_list_int(Xs))
       => ( set_int2(list_update_int(list_update_int(Xs,I,aa_nat_int(nth_int(Xs),J)),J,aa_nat_int(nth_int(Xs),I))) = set_int2(Xs) ) ) ) ).

% set_swap
tff(fact_3757_insert__simp__excp,axiom,
    ! [Mia: nat,Deg: nat,TreeLista: list_VEBT_VEBT,Xb: nat,Maa: nat,Summarya: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Mia),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeLista))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Mia)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg)
         => ( ( Xb != Maa )
           => ( vEBT_vebt_insert(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mia),Maa)),Deg,TreeLista),Summarya),Xb) = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Mia),Maa))),Deg,list_u1324408373059187874T_VEBT(TreeLista,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Mia),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),vEBT_vebt_insert(nth_VEBT_VEBT(TreeLista,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Mia),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Mia),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),
                  $ite(aa_VEBT_VEBT_o(vEBT_VEBT_minNull,nth_VEBT_VEBT(TreeLista,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Mia),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),vEBT_vebt_insert(Summarya,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Mia),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),Summarya)) ) ) ) ) ) ).

% insert_simp_excp
tff(fact_3758_insert__simp__norm,axiom,
    ! [Xb: nat,Deg: nat,TreeLista: list_VEBT_VEBT,Mia: nat,Maa: nat,Summarya: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeLista))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mia),Xb)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg)
         => ( ( Xb != Maa )
           => ( vEBT_vebt_insert(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mia),Maa)),Deg,TreeLista),Summarya),Xb) = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mia),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Xb),Maa))),Deg,list_u1324408373059187874T_VEBT(TreeLista,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),vEBT_vebt_insert(nth_VEBT_VEBT(TreeLista,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),
                  $ite(aa_VEBT_VEBT_o(vEBT_VEBT_minNull,nth_VEBT_VEBT(TreeLista,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),vEBT_vebt_insert(Summarya,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),Summarya)) ) ) ) ) ) ).

% insert_simp_norm
tff(fact_3759_nth__list__update__eq,axiom,
    ! [I: nat,Xs: list_VEBT_VEBT,Xb: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs))
     => ( nth_VEBT_VEBT(list_u1324408373059187874T_VEBT(Xs,I,Xb),I) = Xb ) ) ).

% nth_list_update_eq
tff(fact_3760_nth__list__update__eq,axiom,
    ! [I: nat,Xs: list_VEBT_VEBTi,Xb: vEBT_VEBTi] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s7982070591426661849_VEBTi(Xs))
     => ( nth_VEBT_VEBTi(list_u6098035379799741383_VEBTi(Xs,I,Xb),I) = Xb ) ) ).

% nth_list_update_eq
tff(fact_3761_nth__list__update__eq,axiom,
    ! [I: nat,Xs: list_real,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_real(Xs))
     => ( aa_nat_real(nth_real(list_update_real(Xs,I,Xb)),I) = Xb ) ) ).

% nth_list_update_eq
tff(fact_3762_nth__list__update__eq,axiom,
    ! [I: nat,Xs: list_o,Xb: $o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_o(Xs))
     => ( aa_nat_o(nth_o(list_update_o(Xs,I,(Xb))),I)
      <=> (Xb) ) ) ).

% nth_list_update_eq
tff(fact_3763_nth__list__update__eq,axiom,
    ! [I: nat,Xs: list_nat,Xb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_nat(Xs))
     => ( aa_nat_nat(nth_nat(list_update_nat(Xs,I,Xb)),I) = Xb ) ) ).

% nth_list_update_eq
tff(fact_3764_nth__list__update__eq,axiom,
    ! [I: nat,Xs: list_int,Xb: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_int(Xs))
     => ( aa_nat_int(nth_int(list_update_int(Xs,I,Xb)),I) = Xb ) ) ).

% nth_list_update_eq
tff(fact_3765_member__valid__both__member__options,axiom,
    ! [Tree: vEBT_VEBT,Nb: nat,Xb: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Tree),Nb)
     => ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,Tree),Xb)
       => ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V5719532721284313246member,Tree),Xb)
          | aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_membermima,Tree),Xb) ) ) ) ).

% member_valid_both_member_options
tff(fact_3766_buildup__nothing__in__leaf,axiom,
    ! [Nb: nat,Xb: nat] : ~ aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V5719532721284313246member,vEBT_vebt_buildup(Nb)),Xb) ).

% buildup_nothing_in_leaf
tff(fact_3767_both__member__options__def,axiom,
    ! [Ta: vEBT_VEBT,Xb: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,Ta),Xb)
    <=> ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V5719532721284313246member,Ta),Xb)
        | aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_membermima,Ta),Xb) ) ) ).

% both_member_options_def
tff(fact_3768_length__list__update,axiom,
    ! [Xs: list_VEBT_VEBT,I: nat,Xb: vEBT_VEBT] : size_s6755466524823107622T_VEBT(list_u1324408373059187874T_VEBT(Xs,I,Xb)) = size_s6755466524823107622T_VEBT(Xs) ).

% length_list_update
tff(fact_3769_length__list__update,axiom,
    ! [Xs: list_VEBT_VEBTi,I: nat,Xb: vEBT_VEBTi] : size_s7982070591426661849_VEBTi(list_u6098035379799741383_VEBTi(Xs,I,Xb)) = size_s7982070591426661849_VEBTi(Xs) ).

% length_list_update
tff(fact_3770_length__list__update,axiom,
    ! [Xs: list_real,I: nat,Xb: real] : size_size_list_real(list_update_real(Xs,I,Xb)) = size_size_list_real(Xs) ).

% length_list_update
tff(fact_3771_length__list__update,axiom,
    ! [Xs: list_o,I: nat,Xb: $o] : size_size_list_o(list_update_o(Xs,I,(Xb))) = size_size_list_o(Xs) ).

% length_list_update
tff(fact_3772_length__list__update,axiom,
    ! [Xs: list_nat,I: nat,Xb: nat] : size_size_list_nat(list_update_nat(Xs,I,Xb)) = size_size_list_nat(Xs) ).

% length_list_update
tff(fact_3773_length__list__update,axiom,
    ! [Xs: list_int,I: nat,Xb: int] : size_size_list_int(list_update_int(Xs,I,Xb)) = size_size_list_int(Xs) ).

% length_list_update
tff(fact_3774_list__update__id,axiom,
    ! [Xs: list_nat,I: nat] : list_update_nat(Xs,I,aa_nat_nat(nth_nat(Xs),I)) = Xs ).

% list_update_id
tff(fact_3775_list__update__id,axiom,
    ! [Xs: list_int,I: nat] : list_update_int(Xs,I,aa_nat_int(nth_int(Xs),I)) = Xs ).

% list_update_id
tff(fact_3776_list__update__id,axiom,
    ! [Xs: list_VEBT_VEBT,I: nat] : list_u1324408373059187874T_VEBT(Xs,I,nth_VEBT_VEBT(Xs,I)) = Xs ).

% list_update_id
tff(fact_3777_list__update__id,axiom,
    ! [Xs: list_VEBT_VEBTi,I: nat] : list_u6098035379799741383_VEBTi(Xs,I,nth_VEBT_VEBTi(Xs,I)) = Xs ).

% list_update_id
tff(fact_3778_nth__list__update__neq,axiom,
    ! [I: nat,J: nat,Xs: list_nat,Xb: nat] :
      ( ( I != J )
     => ( aa_nat_nat(nth_nat(list_update_nat(Xs,I,Xb)),J) = aa_nat_nat(nth_nat(Xs),J) ) ) ).

% nth_list_update_neq
tff(fact_3779_nth__list__update__neq,axiom,
    ! [I: nat,J: nat,Xs: list_int,Xb: int] :
      ( ( I != J )
     => ( aa_nat_int(nth_int(list_update_int(Xs,I,Xb)),J) = aa_nat_int(nth_int(Xs),J) ) ) ).

% nth_list_update_neq
tff(fact_3780_nth__list__update__neq,axiom,
    ! [I: nat,J: nat,Xs: list_VEBT_VEBT,Xb: vEBT_VEBT] :
      ( ( I != J )
     => ( nth_VEBT_VEBT(list_u1324408373059187874T_VEBT(Xs,I,Xb),J) = nth_VEBT_VEBT(Xs,J) ) ) ).

% nth_list_update_neq
tff(fact_3781_nth__list__update__neq,axiom,
    ! [I: nat,J: nat,Xs: list_VEBT_VEBTi,Xb: vEBT_VEBTi] :
      ( ( I != J )
     => ( nth_VEBT_VEBTi(list_u6098035379799741383_VEBTi(Xs,I,Xb),J) = nth_VEBT_VEBTi(Xs,J) ) ) ).

% nth_list_update_neq
tff(fact_3782_max__Suc__Suc,axiom,
    ! [M: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(suc,M)),aa_nat_nat(suc,Nb)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,M),Nb)) ).

% max_Suc_Suc
tff(fact_3783_max__0R,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Nb),zero_zero_nat) = Nb ).

% max_0R
tff(fact_3784_max__0L,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,zero_zero_nat),Nb) = Nb ).

% max_0L
tff(fact_3785_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_3786_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_3787_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_3788_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_3789_max__0__1_I4_J,axiom,
    ! [Xb: num] : aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,numeral_numeral_real(Xb)),zero_zero_real) = numeral_numeral_real(Xb) ).

% max_0_1(4)
tff(fact_3790_max__0__1_I4_J,axiom,
    ! [Xb: num] : ord_max_rat(numeral_numeral_rat(Xb),zero_zero_rat) = numeral_numeral_rat(Xb) ).

% max_0_1(4)
tff(fact_3791_max__0__1_I4_J,axiom,
    ! [Xb: num] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_num_nat(numeral_numeral_nat,Xb)),zero_zero_nat) = aa_num_nat(numeral_numeral_nat,Xb) ).

% max_0_1(4)
tff(fact_3792_max__0__1_I4_J,axiom,
    ! [Xb: num] : aa_int_int(aa_int_fun_int_int(ord_max_int,aa_num_int(numeral_numeral_int,Xb)),zero_zero_int) = aa_num_int(numeral_numeral_int,Xb) ).

% max_0_1(4)
tff(fact_3793_max__0__1_I3_J,axiom,
    ! [Xb: num] : aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,zero_zero_real),numeral_numeral_real(Xb)) = numeral_numeral_real(Xb) ).

% max_0_1(3)
tff(fact_3794_max__0__1_I3_J,axiom,
    ! [Xb: num] : ord_max_rat(zero_zero_rat,numeral_numeral_rat(Xb)) = numeral_numeral_rat(Xb) ).

% max_0_1(3)
tff(fact_3795_max__0__1_I3_J,axiom,
    ! [Xb: num] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,Xb)) = aa_num_nat(numeral_numeral_nat,Xb) ).

% max_0_1(3)
tff(fact_3796_max__0__1_I3_J,axiom,
    ! [Xb: num] : aa_int_int(aa_int_fun_int_int(ord_max_int,zero_zero_int),aa_num_int(numeral_numeral_int,Xb)) = aa_num_int(numeral_numeral_int,Xb) ).

% max_0_1(3)
tff(fact_3797_max__number__of_I1_J,axiom,
    ! [U: num,V: num] :
      ord_ma8239519435860878689l_num1(numera7442385471795722001l_num1(U),numera7442385471795722001l_num1(V)) = $ite(aa_wor4452366487969921764num1_o(ord_le3335648743751981014l_num1(numera7442385471795722001l_num1(U)),numera7442385471795722001l_num1(V)),numera7442385471795722001l_num1(V),numera7442385471795722001l_num1(U)) ).

% max_number_of(1)
tff(fact_3798_max__number__of_I1_J,axiom,
    ! [U: num,V: num] :
      aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,numeral_numeral_real(U)),numeral_numeral_real(V)) = $ite(aa_real_o(aa_real_fun_real_o(ord_less_eq_real,numeral_numeral_real(U)),numeral_numeral_real(V)),numeral_numeral_real(V),numeral_numeral_real(U)) ).

% max_number_of(1)
tff(fact_3799_max__number__of_I1_J,axiom,
    ! [U: num,V: num] :
      ord_max_rat(numeral_numeral_rat(U),numeral_numeral_rat(V)) = $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,numeral_numeral_rat(U)),numeral_numeral_rat(V)),numeral_numeral_rat(V),numeral_numeral_rat(U)) ).

% max_number_of(1)
tff(fact_3800_max__number__of_I1_J,axiom,
    ! [U: num,V: num] :
      aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_num_nat(numeral_numeral_nat,U)),aa_num_nat(numeral_numeral_nat,V)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,U)),aa_num_nat(numeral_numeral_nat,V)),aa_num_nat(numeral_numeral_nat,V),aa_num_nat(numeral_numeral_nat,U)) ).

% max_number_of(1)
tff(fact_3801_max__number__of_I1_J,axiom,
    ! [U: num,V: num] :
      aa_int_int(aa_int_fun_int_int(ord_max_int,aa_num_int(numeral_numeral_int,U)),aa_num_int(numeral_numeral_int,V)) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,U)),aa_num_int(numeral_numeral_int,V)),aa_num_int(numeral_numeral_int,V),aa_num_int(numeral_numeral_int,U)) ).

% max_number_of(1)
tff(fact_3802_max__0__1_I2_J,axiom,
    aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,one_one_real),zero_zero_real) = one_one_real ).

% max_0_1(2)
tff(fact_3803_max__0__1_I2_J,axiom,
    ord_max_rat(one_one_rat,zero_zero_rat) = one_one_rat ).

% max_0_1(2)
tff(fact_3804_max__0__1_I2_J,axiom,
    aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,one_one_nat),zero_zero_nat) = one_one_nat ).

% max_0_1(2)
tff(fact_3805_max__0__1_I2_J,axiom,
    aa_int_int(aa_int_fun_int_int(ord_max_int,one_one_int),zero_zero_int) = one_one_int ).

% max_0_1(2)
tff(fact_3806_max__0__1_I1_J,axiom,
    aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,zero_zero_real),one_one_real) = one_one_real ).

% max_0_1(1)
tff(fact_3807_max__0__1_I1_J,axiom,
    ord_max_rat(zero_zero_rat,one_one_rat) = one_one_rat ).

% max_0_1(1)
tff(fact_3808_max__0__1_I1_J,axiom,
    aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,zero_zero_nat),one_one_nat) = one_one_nat ).

% max_0_1(1)
tff(fact_3809_max__0__1_I1_J,axiom,
    aa_int_int(aa_int_fun_int_int(ord_max_int,zero_zero_int),one_one_int) = one_one_int ).

% max_0_1(1)
tff(fact_3810_max__0__1_I5_J,axiom,
    ! [Xb: num] : aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,one_one_real),numeral_numeral_real(Xb)) = numeral_numeral_real(Xb) ).

% max_0_1(5)
tff(fact_3811_max__0__1_I5_J,axiom,
    ! [Xb: num] : ord_max_rat(one_one_rat,numeral_numeral_rat(Xb)) = numeral_numeral_rat(Xb) ).

% max_0_1(5)
tff(fact_3812_max__0__1_I5_J,axiom,
    ! [Xb: num] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,Xb)) = aa_num_nat(numeral_numeral_nat,Xb) ).

% max_0_1(5)
tff(fact_3813_max__0__1_I5_J,axiom,
    ! [Xb: num] : aa_int_int(aa_int_fun_int_int(ord_max_int,one_one_int),aa_num_int(numeral_numeral_int,Xb)) = aa_num_int(numeral_numeral_int,Xb) ).

% max_0_1(5)
tff(fact_3814_max__0__1_I6_J,axiom,
    ! [Xb: num] : aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,numeral_numeral_real(Xb)),one_one_real) = numeral_numeral_real(Xb) ).

% max_0_1(6)
tff(fact_3815_max__0__1_I6_J,axiom,
    ! [Xb: num] : ord_max_rat(numeral_numeral_rat(Xb),one_one_rat) = numeral_numeral_rat(Xb) ).

% max_0_1(6)
tff(fact_3816_max__0__1_I6_J,axiom,
    ! [Xb: num] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_num_nat(numeral_numeral_nat,Xb)),one_one_nat) = aa_num_nat(numeral_numeral_nat,Xb) ).

% max_0_1(6)
tff(fact_3817_max__0__1_I6_J,axiom,
    ! [Xb: num] : aa_int_int(aa_int_fun_int_int(ord_max_int,aa_num_int(numeral_numeral_int,Xb)),one_one_int) = aa_num_int(numeral_numeral_int,Xb) ).

% max_0_1(6)
tff(fact_3818_list__update__beyond,axiom,
    ! [Xs: list_VEBT_VEBT,I: nat,Xb: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_s6755466524823107622T_VEBT(Xs)),I)
     => ( list_u1324408373059187874T_VEBT(Xs,I,Xb) = Xs ) ) ).

% list_update_beyond
tff(fact_3819_list__update__beyond,axiom,
    ! [Xs: list_VEBT_VEBTi,I: nat,Xb: vEBT_VEBTi] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_s7982070591426661849_VEBTi(Xs)),I)
     => ( list_u6098035379799741383_VEBTi(Xs,I,Xb) = Xs ) ) ).

% list_update_beyond
tff(fact_3820_list__update__beyond,axiom,
    ! [Xs: list_real,I: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_size_list_real(Xs)),I)
     => ( list_update_real(Xs,I,Xb) = Xs ) ) ).

% list_update_beyond
tff(fact_3821_list__update__beyond,axiom,
    ! [Xs: list_o,I: nat,Xb: $o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_size_list_o(Xs)),I)
     => ( list_update_o(Xs,I,(Xb)) = Xs ) ) ).

% list_update_beyond
tff(fact_3822_list__update__beyond,axiom,
    ! [Xs: list_nat,I: nat,Xb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_size_list_nat(Xs)),I)
     => ( list_update_nat(Xs,I,Xb) = Xs ) ) ).

% list_update_beyond
tff(fact_3823_list__update__beyond,axiom,
    ! [Xs: list_int,I: nat,Xb: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_size_list_int(Xs)),I)
     => ( list_update_int(Xs,I,Xb) = Xs ) ) ).

% list_update_beyond
tff(fact_3824_of__nat__max,axiom,
    ! [Xb: nat,Y: nat] : aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Xb),Y)) = aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,aa_nat_real(semiri5074537144036343181t_real,Xb)),aa_nat_real(semiri5074537144036343181t_real,Y)) ).

% of_nat_max
tff(fact_3825_of__nat__max,axiom,
    ! [Xb: nat,Y: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Xb),Y)) = aa_int_int(aa_int_fun_int_int(ord_max_int,aa_nat_int(semiri1314217659103216013at_int,Xb)),aa_nat_int(semiri1314217659103216013at_int,Y)) ).

% of_nat_max
tff(fact_3826_of__nat__max,axiom,
    ! [Xb: nat,Y: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Xb),Y)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(semiri1316708129612266289at_nat,Xb)),aa_nat_nat(semiri1316708129612266289at_nat,Y)) ).

% of_nat_max
tff(fact_3827_of__nat__max,axiom,
    ! [Xb: nat,Y: nat] : aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Xb),Y)) = aa_Cod1783974652533396728nteger(ord_max_Code_integer(aa_nat_Code_integer(semiri4939895301339042750nteger,Xb)),aa_nat_Code_integer(semiri4939895301339042750nteger,Y)) ).

% of_nat_max
tff(fact_3828_max__add__distrib__right,axiom,
    ! [Xb: real,Y: real,Z: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,Y),Z)) = aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Y)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Z)) ).

% max_add_distrib_right
tff(fact_3829_max__add__distrib__right,axiom,
    ! [Xb: rat,Y: rat,Z: rat] : aa_rat_rat(plus_plus_rat(Xb),ord_max_rat(Y,Z)) = ord_max_rat(aa_rat_rat(plus_plus_rat(Xb),Y),aa_rat_rat(plus_plus_rat(Xb),Z)) ).

% max_add_distrib_right
tff(fact_3830_max__add__distrib__right,axiom,
    ! [Xb: nat,Y: nat,Z: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Y),Z)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xb),Y)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xb),Z)) ).

% max_add_distrib_right
tff(fact_3831_max__add__distrib__right,axiom,
    ! [Xb: int,Y: int,Z: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb),aa_int_int(aa_int_fun_int_int(ord_max_int,Y),Z)) = aa_int_int(aa_int_fun_int_int(ord_max_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb),Y)),aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb),Z)) ).

% max_add_distrib_right
tff(fact_3832_max__add__distrib__left,axiom,
    ! [Xb: real,Y: real,Z: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,Xb),Y)),Z) = aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Z)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Y),Z)) ).

% max_add_distrib_left
tff(fact_3833_max__add__distrib__left,axiom,
    ! [Xb: rat,Y: rat,Z: rat] : aa_rat_rat(plus_plus_rat(ord_max_rat(Xb,Y)),Z) = ord_max_rat(aa_rat_rat(plus_plus_rat(Xb),Z),aa_rat_rat(plus_plus_rat(Y),Z)) ).

% max_add_distrib_left
tff(fact_3834_max__add__distrib__left,axiom,
    ! [Xb: nat,Y: nat,Z: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Xb),Y)),Z) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xb),Z)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Y),Z)) ).

% max_add_distrib_left
tff(fact_3835_max__add__distrib__left,axiom,
    ! [Xb: int,Y: int,Z: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(ord_max_int,Xb),Y)),Z) = aa_int_int(aa_int_fun_int_int(ord_max_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb),Z)),aa_int_int(aa_int_fun_int_int(plus_plus_int,Y),Z)) ).

% max_add_distrib_left
tff(fact_3836_max__diff__distrib__left,axiom,
    ! [Xb: real,Y: real,Z: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,Xb),Y)),Z) = aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),Z)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Y),Z)) ).

% max_diff_distrib_left
tff(fact_3837_max__diff__distrib__left,axiom,
    ! [Xb: rat,Y: rat,Z: rat] : minus_minus_rat(ord_max_rat(Xb,Y),Z) = ord_max_rat(minus_minus_rat(Xb,Z),minus_minus_rat(Y,Z)) ).

% max_diff_distrib_left
tff(fact_3838_max__diff__distrib__left,axiom,
    ! [Xb: int,Y: int,Z: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(ord_max_int,Xb),Y)),Z) = aa_int_int(aa_int_fun_int_int(ord_max_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb),Z)),aa_int_int(aa_int_fun_int_int(minus_minus_int,Y),Z)) ).

% max_diff_distrib_left
tff(fact_3839_nat__add__max__left,axiom,
    ! [M: nat,Nb: nat,Q2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,M),Nb)),Q2) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Q2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Q2)) ).

% nat_add_max_left
tff(fact_3840_nat__add__max__right,axiom,
    ! [M: nat,Nb: nat,Q2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Nb),Q2)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Q2)) ).

% nat_add_max_right
tff(fact_3841_nat__mult__max__right,axiom,
    ! [M: nat,Nb: nat,Q2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Nb),Q2)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Q2)) ).

% nat_mult_max_right
tff(fact_3842_nat__mult__max__left,axiom,
    ! [M: nat,Nb: nat,Q2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,M),Nb)),Q2) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Q2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),Q2)) ).

% nat_mult_max_left
tff(fact_3843_nat__minus__add__max,axiom,
    ! [Nb: 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,Nb),M)),M) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Nb),M) ).

% nat_minus_add_max
tff(fact_3844_VEBT__internal_Onaive__member_Osimps_I2_J,axiom,
    ! [Uu: option4927543243414619207at_nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT,Ux: nat] : ~ aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V5719532721284313246member,aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uu,zero_zero_nat,Uv),Uw)),Ux) ).

% VEBT_internal.naive_member.simps(2)
tff(fact_3845_neq__if__length__neq,axiom,
    ! [Xs: list_real,Ys: list_real] :
      ( ( size_size_list_real(Xs) != size_size_list_real(Ys) )
     => ( Xs != Ys ) ) ).

% neq_if_length_neq
tff(fact_3846_neq__if__length__neq,axiom,
    ! [Xs: list_o,Ys: list_o] :
      ( ( size_size_list_o(Xs) != size_size_list_o(Ys) )
     => ( Xs != Ys ) ) ).

% neq_if_length_neq
tff(fact_3847_neq__if__length__neq,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( size_size_list_nat(Xs) != size_size_list_nat(Ys) )
     => ( Xs != Ys ) ) ).

% neq_if_length_neq
tff(fact_3848_neq__if__length__neq,axiom,
    ! [Xs: list_int,Ys: list_int] :
      ( ( size_size_list_int(Xs) != size_size_list_int(Ys) )
     => ( Xs != Ys ) ) ).

% neq_if_length_neq
tff(fact_3849_Ex__list__of__length,axiom,
    ! [Nb: nat] :
    ? [Xs2: list_real] : size_size_list_real(Xs2) = Nb ).

% Ex_list_of_length
tff(fact_3850_Ex__list__of__length,axiom,
    ! [Nb: nat] :
    ? [Xs2: list_o] : size_size_list_o(Xs2) = Nb ).

% Ex_list_of_length
tff(fact_3851_Ex__list__of__length,axiom,
    ! [Nb: nat] :
    ? [Xs2: list_nat] : size_size_list_nat(Xs2) = Nb ).

% Ex_list_of_length
tff(fact_3852_Ex__list__of__length,axiom,
    ! [Nb: nat] :
    ? [Xs2: list_int] : size_size_list_int(Xs2) = Nb ).

% Ex_list_of_length
tff(fact_3853_VEBT__internal_Onaive__member_Osimps_I1_J,axiom,
    ! [A: $o,B: $o,Xb: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V5719532721284313246member,vEBT_Leaf((A),(B))),Xb)
    <=> $ite(
          Xb = zero_zero_nat,
          (A),
          $ite(Xb = one_one_nat,(B),$false) ) ) ).

% VEBT_internal.naive_member.simps(1)
tff(fact_3854_length__induct,axiom,
    ! [P: fun_list_real_o,Xs: list_real] :
      ( ! [Xs2: list_real] :
          ( ! [Ys2: list_real] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,size_size_list_real(Ys2)),size_size_list_real(Xs2))
             => aa_list_real_o(P,Ys2) )
         => aa_list_real_o(P,Xs2) )
     => aa_list_real_o(P,Xs) ) ).

% length_induct
tff(fact_3855_length__induct,axiom,
    ! [P: fun_list_o_o,Xs: list_o] :
      ( ! [Xs2: list_o] :
          ( ! [Ys2: list_o] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,size_size_list_o(Ys2)),size_size_list_o(Xs2))
             => aa_list_o_o(P,Ys2) )
         => aa_list_o_o(P,Xs2) )
     => aa_list_o_o(P,Xs) ) ).

% length_induct
tff(fact_3856_length__induct,axiom,
    ! [P: fun_list_nat_o,Xs: list_nat] :
      ( ! [Xs2: list_nat] :
          ( ! [Ys2: list_nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,size_size_list_nat(Ys2)),size_size_list_nat(Xs2))
             => aa_list_nat_o(P,Ys2) )
         => aa_list_nat_o(P,Xs2) )
     => aa_list_nat_o(P,Xs) ) ).

% length_induct
tff(fact_3857_length__induct,axiom,
    ! [P: fun_list_int_o,Xs: list_int] :
      ( ! [Xs2: list_int] :
          ( ! [Ys2: list_int] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,size_size_list_int(Ys2)),size_size_list_int(Xs2))
             => aa_list_int_o(P,Ys2) )
         => aa_list_int_o(P,Xs2) )
     => aa_list_int_o(P,Xs) ) ).

% length_induct
tff(fact_3858_subset__code_I1_J,axiom,
    ! [Xs: list_VEBT_VEBT,B4: set_VEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(set_VEBT_VEBT2(Xs)),B4)
    <=> ! [X: vEBT_VEBT] :
          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),set_VEBT_VEBT2(Xs))
         => aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),B4) ) ) ).

% subset_code(1)
tff(fact_3859_subset__code_I1_J,axiom,
    ! [Xs: list_nat,B4: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(set_nat2(Xs)),B4)
    <=> ! [X: nat] :
          ( aa_set_nat_o(member_nat(X),set_nat2(Xs))
         => aa_set_nat_o(member_nat(X),B4) ) ) ).

% subset_code(1)
tff(fact_3860_subset__code_I1_J,axiom,
    ! [Xs: list_real,B4: set_real] :
      ( aa_set_real_o(ord_less_eq_set_real(set_real2(Xs)),B4)
    <=> ! [X: real] :
          ( aa_set_real_o(member_real(X),set_real2(Xs))
         => aa_set_real_o(member_real(X),B4) ) ) ).

% subset_code(1)
tff(fact_3861_subset__code_I1_J,axiom,
    ! [Xs: list_P8527749157015355191n_assn,B4: set_Pr5949110396991348497n_assn] :
      ( ord_le171416862856029873n_assn(set_Pr1139785259514867910n_assn(Xs),B4)
    <=> ! [X: produc6575502325842934193n_assn] :
          ( member7957490590177025114n_assn(X,set_Pr1139785259514867910n_assn(Xs))
         => member7957490590177025114n_assn(X,B4) ) ) ).

% subset_code(1)
tff(fact_3862_subset__code_I1_J,axiom,
    ! [Xs: list_int,B4: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(set_int2(Xs)),B4)
    <=> ! [X: int] :
          ( aa_set_int_o(member_int(X),set_int2(Xs))
         => aa_set_int_o(member_int(X),B4) ) ) ).

% subset_code(1)
tff(fact_3863_list__eq__iff__nth__eq,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
      ( ( Xs = Ys )
    <=> ( ( size_s6755466524823107622T_VEBT(Xs) = size_s6755466524823107622T_VEBT(Ys) )
        & ! [I2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_s6755466524823107622T_VEBT(Xs))
           => ( nth_VEBT_VEBT(Xs,I2) = nth_VEBT_VEBT(Ys,I2) ) ) ) ) ).

% list_eq_iff_nth_eq
tff(fact_3864_list__eq__iff__nth__eq,axiom,
    ! [Xs: list_VEBT_VEBTi,Ys: list_VEBT_VEBTi] :
      ( ( Xs = Ys )
    <=> ( ( size_s7982070591426661849_VEBTi(Xs) = size_s7982070591426661849_VEBTi(Ys) )
        & ! [I2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_s7982070591426661849_VEBTi(Xs))
           => ( nth_VEBT_VEBTi(Xs,I2) = nth_VEBT_VEBTi(Ys,I2) ) ) ) ) ).

% list_eq_iff_nth_eq
tff(fact_3865_list__eq__iff__nth__eq,axiom,
    ! [Xs: list_real,Ys: list_real] :
      ( ( Xs = Ys )
    <=> ( ( size_size_list_real(Xs) = size_size_list_real(Ys) )
        & ! [I2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_size_list_real(Xs))
           => ( aa_nat_real(nth_real(Xs),I2) = aa_nat_real(nth_real(Ys),I2) ) ) ) ) ).

% list_eq_iff_nth_eq
tff(fact_3866_list__eq__iff__nth__eq,axiom,
    ! [Xs: list_o,Ys: list_o] :
      ( ( Xs = Ys )
    <=> ( ( size_size_list_o(Xs) = size_size_list_o(Ys) )
        & ! [I2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_size_list_o(Xs))
           => ( aa_nat_o(nth_o(Xs),I2)
            <=> aa_nat_o(nth_o(Ys),I2) ) ) ) ) ).

% list_eq_iff_nth_eq
tff(fact_3867_list__eq__iff__nth__eq,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( Xs = Ys )
    <=> ( ( size_size_list_nat(Xs) = size_size_list_nat(Ys) )
        & ! [I2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_size_list_nat(Xs))
           => ( aa_nat_nat(nth_nat(Xs),I2) = aa_nat_nat(nth_nat(Ys),I2) ) ) ) ) ).

% list_eq_iff_nth_eq
tff(fact_3868_list__eq__iff__nth__eq,axiom,
    ! [Xs: list_int,Ys: list_int] :
      ( ( Xs = Ys )
    <=> ( ( size_size_list_int(Xs) = size_size_list_int(Ys) )
        & ! [I2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_size_list_int(Xs))
           => ( aa_nat_int(nth_int(Xs),I2) = aa_nat_int(nth_int(Ys),I2) ) ) ) ) ).

% list_eq_iff_nth_eq
tff(fact_3869_Skolem__list__nth,axiom,
    ! [K: nat,P: fun_na6887940067898170868VEBT_o] :
      ( ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),K)
         => ? [X_1: vEBT_VEBT] : aa_VEBT_VEBT_o(aa_nat1925708948643195573VEBT_o(P,I2),X_1) )
    <=> ? [Xs3: list_VEBT_VEBT] :
          ( ( size_s6755466524823107622T_VEBT(Xs3) = K )
          & ! [I2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),K)
             => aa_VEBT_VEBT_o(aa_nat1925708948643195573VEBT_o(P,I2),nth_VEBT_VEBT(Xs3,I2)) ) ) ) ).

% Skolem_list_nth
tff(fact_3870_Skolem__list__nth,axiom,
    ! [K: nat,P: fun_na154763851189438853EBTi_o] :
      ( ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),K)
         => ? [X_1: vEBT_VEBTi] : aa_VEBT_VEBTi_o(aa_nat4571447880046010350EBTi_o(P,I2),X_1) )
    <=> ? [Xs3: list_VEBT_VEBTi] :
          ( ( size_s7982070591426661849_VEBTi(Xs3) = K )
          & ! [I2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),K)
             => aa_VEBT_VEBTi_o(aa_nat4571447880046010350EBTi_o(P,I2),nth_VEBT_VEBTi(Xs3,I2)) ) ) ) ).

% Skolem_list_nth
tff(fact_3871_Skolem__list__nth,axiom,
    ! [K: nat,P: fun_nat_fun_real_o] :
      ( ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),K)
         => ? [X_1: real] : aa_real_o(aa_nat_fun_real_o(P,I2),X_1) )
    <=> ? [Xs3: list_real] :
          ( ( size_size_list_real(Xs3) = K )
          & ! [I2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),K)
             => aa_real_o(aa_nat_fun_real_o(P,I2),aa_nat_real(nth_real(Xs3),I2)) ) ) ) ).

% Skolem_list_nth
tff(fact_3872_Skolem__list__nth,axiom,
    ! [K: nat,P: fun_nat_fun_o_o] :
      ( ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),K)
         => ? [X_1: $o] : aa_o_o(aa_nat_fun_o_o(P,I2),(X_1)) )
    <=> ? [Xs3: list_o] :
          ( ( size_size_list_o(Xs3) = K )
          & ! [I2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),K)
             => aa_o_o(aa_nat_fun_o_o(P,I2),aa_nat_o(nth_o(Xs3),I2)) ) ) ) ).

% Skolem_list_nth
tff(fact_3873_Skolem__list__nth,axiom,
    ! [K: nat,P: fun_nat_fun_nat_o] :
      ( ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),K)
         => ? [X_1: nat] : aa_nat_o(aa_nat_fun_nat_o(P,I2),X_1) )
    <=> ? [Xs3: list_nat] :
          ( ( size_size_list_nat(Xs3) = K )
          & ! [I2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),K)
             => aa_nat_o(aa_nat_fun_nat_o(P,I2),aa_nat_nat(nth_nat(Xs3),I2)) ) ) ) ).

% Skolem_list_nth
tff(fact_3874_Skolem__list__nth,axiom,
    ! [K: nat,P: fun_nat_fun_int_o] :
      ( ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),K)
         => ? [X_1: int] : aa_int_o(aa_nat_fun_int_o(P,I2),X_1) )
    <=> ? [Xs3: list_int] :
          ( ( size_size_list_int(Xs3) = K )
          & ! [I2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),K)
             => aa_int_o(aa_nat_fun_int_o(P,I2),aa_nat_int(nth_int(Xs3),I2)) ) ) ) ).

% Skolem_list_nth
tff(fact_3875_nth__equalityI,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
      ( ( size_s6755466524823107622T_VEBT(Xs) = size_s6755466524823107622T_VEBT(Ys) )
     => ( ! [I3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),size_s6755466524823107622T_VEBT(Xs))
           => ( nth_VEBT_VEBT(Xs,I3) = nth_VEBT_VEBT(Ys,I3) ) )
       => ( Xs = Ys ) ) ) ).

% nth_equalityI
tff(fact_3876_nth__equalityI,axiom,
    ! [Xs: list_VEBT_VEBTi,Ys: list_VEBT_VEBTi] :
      ( ( size_s7982070591426661849_VEBTi(Xs) = size_s7982070591426661849_VEBTi(Ys) )
     => ( ! [I3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),size_s7982070591426661849_VEBTi(Xs))
           => ( nth_VEBT_VEBTi(Xs,I3) = nth_VEBT_VEBTi(Ys,I3) ) )
       => ( Xs = Ys ) ) ) ).

% nth_equalityI
tff(fact_3877_nth__equalityI,axiom,
    ! [Xs: list_real,Ys: list_real] :
      ( ( size_size_list_real(Xs) = size_size_list_real(Ys) )
     => ( ! [I3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),size_size_list_real(Xs))
           => ( aa_nat_real(nth_real(Xs),I3) = aa_nat_real(nth_real(Ys),I3) ) )
       => ( Xs = Ys ) ) ) ).

% nth_equalityI
tff(fact_3878_nth__equalityI,axiom,
    ! [Xs: list_o,Ys: list_o] :
      ( ( size_size_list_o(Xs) = size_size_list_o(Ys) )
     => ( ! [I3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),size_size_list_o(Xs))
           => ( aa_nat_o(nth_o(Xs),I3)
            <=> aa_nat_o(nth_o(Ys),I3) ) )
       => ( Xs = Ys ) ) ) ).

% nth_equalityI
tff(fact_3879_nth__equalityI,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( size_size_list_nat(Xs) = size_size_list_nat(Ys) )
     => ( ! [I3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),size_size_list_nat(Xs))
           => ( aa_nat_nat(nth_nat(Xs),I3) = aa_nat_nat(nth_nat(Ys),I3) ) )
       => ( Xs = Ys ) ) ) ).

% nth_equalityI
tff(fact_3880_nth__equalityI,axiom,
    ! [Xs: list_int,Ys: list_int] :
      ( ( size_size_list_int(Xs) = size_size_list_int(Ys) )
     => ( ! [I3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),size_size_list_int(Xs))
           => ( aa_nat_int(nth_int(Xs),I3) = aa_nat_int(nth_int(Ys),I3) ) )
       => ( Xs = Ys ) ) ) ).

% nth_equalityI
tff(fact_3881_all__nat__less,axiom,
    ! [Nb: nat,P: fun_nat_o] :
      ( ! [M7: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M7),Nb)
         => aa_nat_o(P,M7) )
    <=> ! [X: nat] :
          ( aa_set_nat_o(member_nat(X),set_or1269000886237332187st_nat(zero_zero_nat,Nb))
         => aa_nat_o(P,X) ) ) ).

% all_nat_less
tff(fact_3882_ex__nat__less,axiom,
    ! [Nb: nat,P: fun_nat_o] :
      ( ? [M7: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M7),Nb)
          & aa_nat_o(P,M7) )
    <=> ? [X: nat] :
          ( aa_set_nat_o(member_nat(X),set_or1269000886237332187st_nat(zero_zero_nat,Nb))
          & aa_nat_o(P,X) ) ) ).

% ex_nat_less
tff(fact_3883_set__update__subsetI,axiom,
    ! [Xs: list_nat,A2: set_nat,Xb: nat,I: nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(set_nat2(Xs)),A2)
     => ( aa_set_nat_o(member_nat(Xb),A2)
       => aa_set_nat_o(ord_less_eq_set_nat(set_nat2(list_update_nat(Xs,I,Xb))),A2) ) ) ).

% set_update_subsetI
tff(fact_3884_set__update__subsetI,axiom,
    ! [Xs: list_real,A2: set_real,Xb: real,I: nat] :
      ( aa_set_real_o(ord_less_eq_set_real(set_real2(Xs)),A2)
     => ( aa_set_real_o(member_real(Xb),A2)
       => aa_set_real_o(ord_less_eq_set_real(set_real2(list_update_real(Xs,I,Xb))),A2) ) ) ).

% set_update_subsetI
tff(fact_3885_set__update__subsetI,axiom,
    ! [Xs: list_P8527749157015355191n_assn,A2: set_Pr5949110396991348497n_assn,Xb: produc6575502325842934193n_assn,I: nat] :
      ( ord_le171416862856029873n_assn(set_Pr1139785259514867910n_assn(Xs),A2)
     => ( member7957490590177025114n_assn(Xb,A2)
       => ord_le171416862856029873n_assn(set_Pr1139785259514867910n_assn(list_u4534839942911652127n_assn(Xs,I,Xb)),A2) ) ) ).

% set_update_subsetI
tff(fact_3886_set__update__subsetI,axiom,
    ! [Xs: list_VEBT_VEBT,A2: set_VEBT_VEBT,Xb: vEBT_VEBT,I: nat] :
      ( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(set_VEBT_VEBT2(Xs)),A2)
     => ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xb),A2)
       => aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(set_VEBT_VEBT2(list_u1324408373059187874T_VEBT(Xs,I,Xb))),A2) ) ) ).

% set_update_subsetI
tff(fact_3887_set__update__subsetI,axiom,
    ! [Xs: list_VEBT_VEBTi,A2: set_VEBT_VEBTi,Xb: vEBT_VEBTi,I: nat] :
      ( ord_le6592769550269828683_VEBTi(set_VEBT_VEBTi2(Xs),A2)
     => ( member_VEBT_VEBTi(Xb,A2)
       => ord_le6592769550269828683_VEBTi(set_VEBT_VEBTi2(list_u6098035379799741383_VEBTi(Xs,I,Xb)),A2) ) ) ).

% set_update_subsetI
tff(fact_3888_set__update__subsetI,axiom,
    ! [Xs: list_int,A2: set_int,Xb: int,I: nat] :
      ( aa_set_int_o(ord_less_eq_set_int(set_int2(Xs)),A2)
     => ( aa_set_int_o(member_int(Xb),A2)
       => aa_set_int_o(ord_less_eq_set_int(set_int2(list_update_int(Xs,I,Xb))),A2) ) ) ).

% set_update_subsetI
tff(fact_3889_vebt__delete_Osimps_I3_J,axiom,
    ! [A: $o,B: $o,Nb: nat] : vEBT_vebt_delete(vEBT_Leaf((A),(B)),aa_nat_nat(suc,aa_nat_nat(suc,Nb))) = vEBT_Leaf((A),(B)) ).

% vebt_delete.simps(3)
tff(fact_3890_vebt__delete_Osimps_I4_J,axiom,
    ! [Deg: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Uu: nat] : vEBT_vebt_delete(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Deg,TreeLista),Summarya),Uu) = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Deg,TreeLista),Summarya) ).

% vebt_delete.simps(4)
tff(fact_3891_length__pos__if__in__set,axiom,
    ! [Xb: vEBT_VEBT,Xs: list_VEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xb),set_VEBT_VEBT2(Xs))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),size_s6755466524823107622T_VEBT(Xs)) ) ).

% length_pos_if_in_set
tff(fact_3892_length__pos__if__in__set,axiom,
    ! [Xb: produc6575502325842934193n_assn,Xs: list_P8527749157015355191n_assn] :
      ( member7957490590177025114n_assn(Xb,set_Pr1139785259514867910n_assn(Xs))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),size_s6829681357464350627n_assn(Xs)) ) ).

% length_pos_if_in_set
tff(fact_3893_length__pos__if__in__set,axiom,
    ! [Xb: real,Xs: list_real] :
      ( aa_set_real_o(member_real(Xb),set_real2(Xs))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),size_size_list_real(Xs)) ) ).

% length_pos_if_in_set
tff(fact_3894_length__pos__if__in__set,axiom,
    ! [Xb: $o,Xs: list_o] :
      ( member_o((Xb),set_o2(Xs))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),size_size_list_o(Xs)) ) ).

% length_pos_if_in_set
tff(fact_3895_length__pos__if__in__set,axiom,
    ! [Xb: nat,Xs: list_nat] :
      ( aa_set_nat_o(member_nat(Xb),set_nat2(Xs))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),size_size_list_nat(Xs)) ) ).

% length_pos_if_in_set
tff(fact_3896_length__pos__if__in__set,axiom,
    ! [Xb: int,Xs: list_int] :
      ( aa_set_int_o(member_int(Xb),set_int2(Xs))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),size_size_list_int(Xs)) ) ).

% length_pos_if_in_set
tff(fact_3897_all__set__conv__all__nth,axiom,
    ! [Xs: list_VEBT_VEBTi,P: fun_VEBT_VEBTi_o] :
      ( ! [X: vEBT_VEBTi] :
          ( member_VEBT_VEBTi(X,set_VEBT_VEBTi2(Xs))
         => aa_VEBT_VEBTi_o(P,X) )
    <=> ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_s7982070591426661849_VEBTi(Xs))
         => aa_VEBT_VEBTi_o(P,nth_VEBT_VEBTi(Xs,I2)) ) ) ).

% all_set_conv_all_nth
tff(fact_3898_all__set__conv__all__nth,axiom,
    ! [Xs: list_VEBT_VEBT,P: fun_VEBT_VEBT_o] :
      ( ! [X: vEBT_VEBT] :
          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),set_VEBT_VEBT2(Xs))
         => aa_VEBT_VEBT_o(P,X) )
    <=> ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_s6755466524823107622T_VEBT(Xs))
         => aa_VEBT_VEBT_o(P,nth_VEBT_VEBT(Xs,I2)) ) ) ).

% all_set_conv_all_nth
tff(fact_3899_all__set__conv__all__nth,axiom,
    ! [Xs: list_P8527749157015355191n_assn,P: fun_Pr3571187171380813162assn_o] :
      ( ! [X: produc6575502325842934193n_assn] :
          ( member7957490590177025114n_assn(X,set_Pr1139785259514867910n_assn(Xs))
         => aa_Pro2491721552408928739assn_o(P,X) )
    <=> ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_s6829681357464350627n_assn(Xs))
         => aa_Pro2491721552408928739assn_o(P,nth_Pr1769885009046257848n_assn(Xs,I2)) ) ) ).

% all_set_conv_all_nth
tff(fact_3900_all__set__conv__all__nth,axiom,
    ! [Xs: list_real,P: fun_real_o] :
      ( ! [X: real] :
          ( aa_set_real_o(member_real(X),set_real2(Xs))
         => aa_real_o(P,X) )
    <=> ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_size_list_real(Xs))
         => aa_real_o(P,aa_nat_real(nth_real(Xs),I2)) ) ) ).

% all_set_conv_all_nth
tff(fact_3901_all__set__conv__all__nth,axiom,
    ! [Xs: list_o,P: fun_o_o] :
      ( ! [X: $o] :
          ( member_o((X),set_o2(Xs))
         => aa_o_o(P,(X)) )
    <=> ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_size_list_o(Xs))
         => aa_o_o(P,aa_nat_o(nth_o(Xs),I2)) ) ) ).

% all_set_conv_all_nth
tff(fact_3902_all__set__conv__all__nth,axiom,
    ! [Xs: list_nat,P: fun_nat_o] :
      ( ! [X: nat] :
          ( aa_set_nat_o(member_nat(X),set_nat2(Xs))
         => aa_nat_o(P,X) )
    <=> ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_size_list_nat(Xs))
         => aa_nat_o(P,aa_nat_nat(nth_nat(Xs),I2)) ) ) ).

% all_set_conv_all_nth
tff(fact_3903_all__set__conv__all__nth,axiom,
    ! [Xs: list_int,P: fun_int_o] :
      ( ! [X: int] :
          ( aa_set_int_o(member_int(X),set_int2(Xs))
         => aa_int_o(P,X) )
    <=> ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_size_list_int(Xs))
         => aa_int_o(P,aa_nat_int(nth_int(Xs),I2)) ) ) ).

% all_set_conv_all_nth
tff(fact_3904_all__nth__imp__all__set,axiom,
    ! [Xs: list_VEBT_VEBTi,P: fun_VEBT_VEBTi_o,Xb: vEBT_VEBTi] :
      ( ! [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),size_s7982070591426661849_VEBTi(Xs))
         => aa_VEBT_VEBTi_o(P,nth_VEBT_VEBTi(Xs,I3)) )
     => ( member_VEBT_VEBTi(Xb,set_VEBT_VEBTi2(Xs))
       => aa_VEBT_VEBTi_o(P,Xb) ) ) ).

% all_nth_imp_all_set
tff(fact_3905_all__nth__imp__all__set,axiom,
    ! [Xs: list_VEBT_VEBT,P: fun_VEBT_VEBT_o,Xb: vEBT_VEBT] :
      ( ! [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),size_s6755466524823107622T_VEBT(Xs))
         => aa_VEBT_VEBT_o(P,nth_VEBT_VEBT(Xs,I3)) )
     => ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xb),set_VEBT_VEBT2(Xs))
       => aa_VEBT_VEBT_o(P,Xb) ) ) ).

% all_nth_imp_all_set
tff(fact_3906_all__nth__imp__all__set,axiom,
    ! [Xs: list_P8527749157015355191n_assn,P: fun_Pr3571187171380813162assn_o,Xb: produc6575502325842934193n_assn] :
      ( ! [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),size_s6829681357464350627n_assn(Xs))
         => aa_Pro2491721552408928739assn_o(P,nth_Pr1769885009046257848n_assn(Xs,I3)) )
     => ( member7957490590177025114n_assn(Xb,set_Pr1139785259514867910n_assn(Xs))
       => aa_Pro2491721552408928739assn_o(P,Xb) ) ) ).

% all_nth_imp_all_set
tff(fact_3907_all__nth__imp__all__set,axiom,
    ! [Xs: list_real,P: fun_real_o,Xb: real] :
      ( ! [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),size_size_list_real(Xs))
         => aa_real_o(P,aa_nat_real(nth_real(Xs),I3)) )
     => ( aa_set_real_o(member_real(Xb),set_real2(Xs))
       => aa_real_o(P,Xb) ) ) ).

% all_nth_imp_all_set
tff(fact_3908_all__nth__imp__all__set,axiom,
    ! [Xs: list_o,P: fun_o_o,Xb: $o] :
      ( ! [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),size_size_list_o(Xs))
         => aa_o_o(P,aa_nat_o(nth_o(Xs),I3)) )
     => ( member_o((Xb),set_o2(Xs))
       => aa_o_o(P,(Xb)) ) ) ).

% all_nth_imp_all_set
tff(fact_3909_all__nth__imp__all__set,axiom,
    ! [Xs: list_nat,P: fun_nat_o,Xb: nat] :
      ( ! [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),size_size_list_nat(Xs))
         => aa_nat_o(P,aa_nat_nat(nth_nat(Xs),I3)) )
     => ( aa_set_nat_o(member_nat(Xb),set_nat2(Xs))
       => aa_nat_o(P,Xb) ) ) ).

% all_nth_imp_all_set
tff(fact_3910_all__nth__imp__all__set,axiom,
    ! [Xs: list_int,P: fun_int_o,Xb: int] :
      ( ! [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),size_size_list_int(Xs))
         => aa_int_o(P,aa_nat_int(nth_int(Xs),I3)) )
     => ( aa_set_int_o(member_int(Xb),set_int2(Xs))
       => aa_int_o(P,Xb) ) ) ).

% all_nth_imp_all_set
tff(fact_3911_in__set__conv__nth,axiom,
    ! [Xb: vEBT_VEBTi,Xs: list_VEBT_VEBTi] :
      ( member_VEBT_VEBTi(Xb,set_VEBT_VEBTi2(Xs))
    <=> ? [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_s7982070591426661849_VEBTi(Xs))
          & ( nth_VEBT_VEBTi(Xs,I2) = Xb ) ) ) ).

% in_set_conv_nth
tff(fact_3912_in__set__conv__nth,axiom,
    ! [Xb: vEBT_VEBT,Xs: list_VEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xb),set_VEBT_VEBT2(Xs))
    <=> ? [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_s6755466524823107622T_VEBT(Xs))
          & ( nth_VEBT_VEBT(Xs,I2) = Xb ) ) ) ).

% in_set_conv_nth
tff(fact_3913_in__set__conv__nth,axiom,
    ! [Xb: produc6575502325842934193n_assn,Xs: list_P8527749157015355191n_assn] :
      ( member7957490590177025114n_assn(Xb,set_Pr1139785259514867910n_assn(Xs))
    <=> ? [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_s6829681357464350627n_assn(Xs))
          & ( nth_Pr1769885009046257848n_assn(Xs,I2) = Xb ) ) ) ).

% in_set_conv_nth
tff(fact_3914_in__set__conv__nth,axiom,
    ! [Xb: real,Xs: list_real] :
      ( aa_set_real_o(member_real(Xb),set_real2(Xs))
    <=> ? [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_size_list_real(Xs))
          & ( aa_nat_real(nth_real(Xs),I2) = Xb ) ) ) ).

% in_set_conv_nth
tff(fact_3915_in__set__conv__nth,axiom,
    ! [Xb: $o,Xs: list_o] :
      ( member_o((Xb),set_o2(Xs))
    <=> ? [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_size_list_o(Xs))
          & ( aa_nat_o(nth_o(Xs),I2)
          <=> (Xb) ) ) ) ).

% in_set_conv_nth
tff(fact_3916_in__set__conv__nth,axiom,
    ! [Xb: nat,Xs: list_nat] :
      ( aa_set_nat_o(member_nat(Xb),set_nat2(Xs))
    <=> ? [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_size_list_nat(Xs))
          & ( aa_nat_nat(nth_nat(Xs),I2) = Xb ) ) ) ).

% in_set_conv_nth
tff(fact_3917_in__set__conv__nth,axiom,
    ! [Xb: int,Xs: list_int] :
      ( aa_set_int_o(member_int(Xb),set_int2(Xs))
    <=> ? [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_size_list_int(Xs))
          & ( aa_nat_int(nth_int(Xs),I2) = Xb ) ) ) ).

% in_set_conv_nth
tff(fact_3918_list__ball__nth,axiom,
    ! [Nb: nat,Xs: list_VEBT_VEBTi,P: fun_VEBT_VEBTi_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_s7982070591426661849_VEBTi(Xs))
     => ( ! [X2: vEBT_VEBTi] :
            ( member_VEBT_VEBTi(X2,set_VEBT_VEBTi2(Xs))
           => aa_VEBT_VEBTi_o(P,X2) )
       => aa_VEBT_VEBTi_o(P,nth_VEBT_VEBTi(Xs,Nb)) ) ) ).

% list_ball_nth
tff(fact_3919_list__ball__nth,axiom,
    ! [Nb: nat,Xs: list_VEBT_VEBT,P: fun_VEBT_VEBT_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_s6755466524823107622T_VEBT(Xs))
     => ( ! [X2: vEBT_VEBT] :
            ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X2),set_VEBT_VEBT2(Xs))
           => aa_VEBT_VEBT_o(P,X2) )
       => aa_VEBT_VEBT_o(P,nth_VEBT_VEBT(Xs,Nb)) ) ) ).

% list_ball_nth
tff(fact_3920_list__ball__nth,axiom,
    ! [Nb: nat,Xs: list_P8527749157015355191n_assn,P: fun_Pr3571187171380813162assn_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_s6829681357464350627n_assn(Xs))
     => ( ! [X2: produc6575502325842934193n_assn] :
            ( member7957490590177025114n_assn(X2,set_Pr1139785259514867910n_assn(Xs))
           => aa_Pro2491721552408928739assn_o(P,X2) )
       => aa_Pro2491721552408928739assn_o(P,nth_Pr1769885009046257848n_assn(Xs,Nb)) ) ) ).

% list_ball_nth
tff(fact_3921_list__ball__nth,axiom,
    ! [Nb: nat,Xs: list_real,P: fun_real_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_size_list_real(Xs))
     => ( ! [X2: real] :
            ( aa_set_real_o(member_real(X2),set_real2(Xs))
           => aa_real_o(P,X2) )
       => aa_real_o(P,aa_nat_real(nth_real(Xs),Nb)) ) ) ).

% list_ball_nth
tff(fact_3922_list__ball__nth,axiom,
    ! [Nb: nat,Xs: list_o,P: fun_o_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_size_list_o(Xs))
     => ( ! [X2: $o] :
            ( member_o((X2),set_o2(Xs))
           => aa_o_o(P,(X2)) )
       => aa_o_o(P,aa_nat_o(nth_o(Xs),Nb)) ) ) ).

% list_ball_nth
tff(fact_3923_list__ball__nth,axiom,
    ! [Nb: nat,Xs: list_nat,P: fun_nat_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_size_list_nat(Xs))
     => ( ! [X2: nat] :
            ( aa_set_nat_o(member_nat(X2),set_nat2(Xs))
           => aa_nat_o(P,X2) )
       => aa_nat_o(P,aa_nat_nat(nth_nat(Xs),Nb)) ) ) ).

% list_ball_nth
tff(fact_3924_list__ball__nth,axiom,
    ! [Nb: nat,Xs: list_int,P: fun_int_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_size_list_int(Xs))
     => ( ! [X2: int] :
            ( aa_set_int_o(member_int(X2),set_int2(Xs))
           => aa_int_o(P,X2) )
       => aa_int_o(P,aa_nat_int(nth_int(Xs),Nb)) ) ) ).

% list_ball_nth
tff(fact_3925_nth__mem,axiom,
    ! [Nb: nat,Xs: list_VEBT_VEBTi] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_s7982070591426661849_VEBTi(Xs))
     => member_VEBT_VEBTi(nth_VEBT_VEBTi(Xs,Nb),set_VEBT_VEBTi2(Xs)) ) ).

% nth_mem
tff(fact_3926_nth__mem,axiom,
    ! [Nb: nat,Xs: list_VEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_s6755466524823107622T_VEBT(Xs))
     => aa_set_VEBT_VEBT_o(member_VEBT_VEBT(nth_VEBT_VEBT(Xs,Nb)),set_VEBT_VEBT2(Xs)) ) ).

% nth_mem
tff(fact_3927_nth__mem,axiom,
    ! [Nb: nat,Xs: list_P8527749157015355191n_assn] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_s6829681357464350627n_assn(Xs))
     => member7957490590177025114n_assn(nth_Pr1769885009046257848n_assn(Xs,Nb),set_Pr1139785259514867910n_assn(Xs)) ) ).

% nth_mem
tff(fact_3928_nth__mem,axiom,
    ! [Nb: nat,Xs: list_real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_size_list_real(Xs))
     => aa_set_real_o(member_real(aa_nat_real(nth_real(Xs),Nb)),set_real2(Xs)) ) ).

% nth_mem
tff(fact_3929_nth__mem,axiom,
    ! [Nb: nat,Xs: list_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_size_list_o(Xs))
     => member_o(aa_nat_o(nth_o(Xs),Nb),set_o2(Xs)) ) ).

% nth_mem
tff(fact_3930_nth__mem,axiom,
    ! [Nb: nat,Xs: list_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_size_list_nat(Xs))
     => aa_set_nat_o(member_nat(aa_nat_nat(nth_nat(Xs),Nb)),set_nat2(Xs)) ) ).

% nth_mem
tff(fact_3931_nth__mem,axiom,
    ! [Nb: nat,Xs: list_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_size_list_int(Xs))
     => aa_set_int_o(member_int(aa_nat_int(nth_int(Xs),Nb)),set_int2(Xs)) ) ).

% nth_mem
tff(fact_3932_set__update__memI,axiom,
    ! [Nb: nat,Xs: list_P8527749157015355191n_assn,Xb: produc6575502325842934193n_assn] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_s6829681357464350627n_assn(Xs))
     => member7957490590177025114n_assn(Xb,set_Pr1139785259514867910n_assn(list_u4534839942911652127n_assn(Xs,Nb,Xb))) ) ).

% set_update_memI
tff(fact_3933_set__update__memI,axiom,
    ! [Nb: nat,Xs: list_VEBT_VEBT,Xb: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_s6755466524823107622T_VEBT(Xs))
     => aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xb),set_VEBT_VEBT2(list_u1324408373059187874T_VEBT(Xs,Nb,Xb))) ) ).

% set_update_memI
tff(fact_3934_set__update__memI,axiom,
    ! [Nb: nat,Xs: list_VEBT_VEBTi,Xb: vEBT_VEBTi] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_s7982070591426661849_VEBTi(Xs))
     => member_VEBT_VEBTi(Xb,set_VEBT_VEBTi2(list_u6098035379799741383_VEBTi(Xs,Nb,Xb))) ) ).

% set_update_memI
tff(fact_3935_set__update__memI,axiom,
    ! [Nb: nat,Xs: list_real,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_size_list_real(Xs))
     => aa_set_real_o(member_real(Xb),set_real2(list_update_real(Xs,Nb,Xb))) ) ).

% set_update_memI
tff(fact_3936_set__update__memI,axiom,
    ! [Nb: nat,Xs: list_o,Xb: $o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_size_list_o(Xs))
     => member_o((Xb),set_o2(list_update_o(Xs,Nb,(Xb)))) ) ).

% set_update_memI
tff(fact_3937_set__update__memI,axiom,
    ! [Nb: nat,Xs: list_nat,Xb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_size_list_nat(Xs))
     => aa_set_nat_o(member_nat(Xb),set_nat2(list_update_nat(Xs,Nb,Xb))) ) ).

% set_update_memI
tff(fact_3938_set__update__memI,axiom,
    ! [Nb: nat,Xs: list_int,Xb: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_size_list_int(Xs))
     => aa_set_int_o(member_int(Xb),set_int2(list_update_int(Xs,Nb,Xb))) ) ).

% set_update_memI
tff(fact_3939_list__update__same__conv,axiom,
    ! [I: nat,Xs: list_VEBT_VEBT,Xb: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs))
     => ( ( list_u1324408373059187874T_VEBT(Xs,I,Xb) = Xs )
      <=> ( nth_VEBT_VEBT(Xs,I) = Xb ) ) ) ).

% list_update_same_conv
tff(fact_3940_list__update__same__conv,axiom,
    ! [I: nat,Xs: list_VEBT_VEBTi,Xb: vEBT_VEBTi] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s7982070591426661849_VEBTi(Xs))
     => ( ( list_u6098035379799741383_VEBTi(Xs,I,Xb) = Xs )
      <=> ( nth_VEBT_VEBTi(Xs,I) = Xb ) ) ) ).

% list_update_same_conv
tff(fact_3941_list__update__same__conv,axiom,
    ! [I: nat,Xs: list_real,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_real(Xs))
     => ( ( list_update_real(Xs,I,Xb) = Xs )
      <=> ( aa_nat_real(nth_real(Xs),I) = Xb ) ) ) ).

% list_update_same_conv
tff(fact_3942_list__update__same__conv,axiom,
    ! [I: nat,Xs: list_o,Xb: $o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_o(Xs))
     => ( ( list_update_o(Xs,I,(Xb)) = Xs )
      <=> ( aa_nat_o(nth_o(Xs),I)
        <=> (Xb) ) ) ) ).

% list_update_same_conv
tff(fact_3943_list__update__same__conv,axiom,
    ! [I: nat,Xs: list_nat,Xb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_nat(Xs))
     => ( ( list_update_nat(Xs,I,Xb) = Xs )
      <=> ( aa_nat_nat(nth_nat(Xs),I) = Xb ) ) ) ).

% list_update_same_conv
tff(fact_3944_list__update__same__conv,axiom,
    ! [I: nat,Xs: list_int,Xb: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_int(Xs))
     => ( ( list_update_int(Xs,I,Xb) = Xs )
      <=> ( aa_nat_int(nth_int(Xs),I) = Xb ) ) ) ).

% list_update_same_conv
tff(fact_3945_nth__list__update,axiom,
    ! [I: nat,Xs: list_VEBT_VEBT,Xb: vEBT_VEBT,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs))
     => ( nth_VEBT_VEBT(list_u1324408373059187874T_VEBT(Xs,I,Xb),J) = $ite(I = J,Xb,nth_VEBT_VEBT(Xs,J)) ) ) ).

% nth_list_update
tff(fact_3946_nth__list__update,axiom,
    ! [I: nat,Xs: list_VEBT_VEBTi,Xb: vEBT_VEBTi,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s7982070591426661849_VEBTi(Xs))
     => ( nth_VEBT_VEBTi(list_u6098035379799741383_VEBTi(Xs,I,Xb),J) = $ite(I = J,Xb,nth_VEBT_VEBTi(Xs,J)) ) ) ).

% nth_list_update
tff(fact_3947_nth__list__update,axiom,
    ! [I: nat,Xs: list_real,Xb: real,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_real(Xs))
     => ( aa_nat_real(nth_real(list_update_real(Xs,I,Xb)),J) = $ite(I = J,Xb,aa_nat_real(nth_real(Xs),J)) ) ) ).

% nth_list_update
tff(fact_3948_nth__list__update,axiom,
    ! [I: nat,Xs: list_o,Xb: $o,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_o(Xs))
     => ( aa_nat_o(nth_o(list_update_o(Xs,I,(Xb))),J)
      <=> $ite(I = J,(Xb),aa_nat_o(nth_o(Xs),J)) ) ) ).

% nth_list_update
tff(fact_3949_nth__list__update,axiom,
    ! [I: nat,Xs: list_nat,Xb: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_nat(Xs))
     => ( aa_nat_nat(nth_nat(list_update_nat(Xs,I,Xb)),J) = $ite(I = J,Xb,aa_nat_nat(nth_nat(Xs),J)) ) ) ).

% nth_list_update
tff(fact_3950_nth__list__update,axiom,
    ! [I: nat,Xs: list_int,Xb: int,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_int(Xs))
     => ( aa_nat_int(nth_int(list_update_int(Xs,I,Xb)),J) = $ite(I = J,Xb,aa_nat_int(nth_int(Xs),J)) ) ) ).

% nth_list_update
tff(fact_3951_vebt__delete_Osimps_I2_J,axiom,
    ! [A: $o,B: $o] : vEBT_vebt_delete(vEBT_Leaf((A),(B)),aa_nat_nat(suc,zero_zero_nat)) = vEBT_Leaf((A),$false) ).

% vebt_delete.simps(2)
tff(fact_3952_vebt__insert_Osimps_I5_J,axiom,
    ! [Mia: nat,Maa: nat,Vaa: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Xb: nat] :
      vEBT_vebt_insert(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mia),Maa)),aa_nat_nat(suc,aa_nat_nat(suc,Vaa)),TreeLista),Summarya),Xb) = $let(
        xn: nat,
        xn:= 
          $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Mia),Mia,Xb),
        $let(
          h: nat,
          h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Vaa))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
          $ite(
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeLista))
            & ~ ( ( Xb = Mia )
                | ( Xb = Maa ) ) ),
            aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,
                      $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Mia),Xb,Mia)),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,xn),Maa))),aa_nat_nat(suc,aa_nat_nat(suc,Vaa)),list_u1324408373059187874T_VEBT(TreeLista,h,vEBT_vebt_insert(nth_VEBT_VEBT(TreeLista,h),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Vaa))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),
              $ite(aa_VEBT_VEBT_o(vEBT_VEBT_minNull,nth_VEBT_VEBT(TreeLista,h)),vEBT_vebt_insert(Summarya,h),Summarya)),
            aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mia),Maa)),aa_nat_nat(suc,aa_nat_nat(suc,Vaa)),TreeLista),Summarya) ) ) ) ).

% vebt_insert.simps(5)
tff(fact_3953_VEBT__internal_Onaive__member_Osimps_I3_J,axiom,
    ! [Uy: option4927543243414619207at_nat,V: nat,TreeLista: list_VEBT_VEBT,S3: vEBT_VEBT,Xb: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V5719532721284313246member,aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uy,aa_nat_nat(suc,V),TreeLista),S3)),Xb)
    <=> $let(
          pos: nat,
          pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
          $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeLista)),aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V5719532721284313246member,nth_VEBT_VEBT(TreeLista,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ).

% VEBT_internal.naive_member.simps(3)
tff(fact_3954_vebt__delete_Osimps_I5_J,axiom,
    ! [Mia: nat,Maa: nat,TrLst2: list_VEBT_VEBT,Smry2: vEBT_VEBT,Xb: nat] : vEBT_vebt_delete(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mia),Maa)),zero_zero_nat,TrLst2),Smry2),Xb) = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mia),Maa)),zero_zero_nat,TrLst2),Smry2) ).

% vebt_delete.simps(5)
tff(fact_3955_vebt__insert_Oelims,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: vEBT_VEBT] :
      ( ( vEBT_vebt_insert(Xb,Xaa) = Y )
     => ( ! [A3: $o,B2: $o] :
            ( ( Xb = vEBT_Leaf((A3),(B2)) )
           => ( Y != $ite(
                  Xaa = zero_zero_nat,
                  vEBT_Leaf($true,(B2)),
                  $ite(Xaa = one_one_nat,vEBT_Leaf((A3),$true),vEBT_Leaf((A3),(B2))) ) ) )
       => ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S: vEBT_VEBT] :
              ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,zero_zero_nat,Ts2),S) )
             => ( Y != aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,zero_zero_nat,Ts2),S) ) )
         => ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S: vEBT_VEBT] :
                ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts2),S) )
               => ( Y != aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts2),S) ) )
           => ( ! [V2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                  ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V2)),TreeList),Summary) )
                 => ( Y != aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Xaa),Xaa)),aa_nat_nat(suc,aa_nat_nat(suc,V2)),TreeList),Summary) ) )
             => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                    ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList),Summary) )
                   => ( Y != $let(
                          xn: nat,
                          xn:= 
                            $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi),Mi,Xaa),
                          $let(
                            h: nat,
                            h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                            $ite(
                              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList))
                              & ~ ( ( Xaa = Mi )
                                  | ( Xaa = Ma ) ) ),
                              aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,
                                        $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi),Xaa,Mi)),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,xn),Ma))),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),list_u1324408373059187874T_VEBT(TreeList,h,vEBT_vebt_insert(nth_VEBT_VEBT(TreeList,h),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),
                                $ite(aa_VEBT_VEBT_o(vEBT_VEBT_minNull,nth_VEBT_VEBT(TreeList,h)),vEBT_vebt_insert(Summary,h),Summary)),
                              aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList),Summary) ) ) ) ) ) ) ) ) ) ) ).

% vebt_insert.elims
tff(fact_3956_VEBT__internal_Onaive__member_Oelims_I1_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: $o] :
      ( ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V5719532721284313246member,Xb),Xaa)
      <=> (Y) )
     => ( ! [A3: $o,B2: $o] :
            ( ( Xb = vEBT_Leaf((A3),(B2)) )
           => ( (Y)
            <=> ~ $ite(
                    Xaa = zero_zero_nat,
                    (A3),
                    $ite(Xaa = one_one_nat,(B2),$false) ) ) )
       => ( ( ? [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] : Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uu2,zero_zero_nat,Uv2),Uw2)
           => (Y) )
         => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList: list_VEBT_VEBT] :
                ( ? [S: vEBT_VEBT] : Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList),S)
               => ( (Y)
                <=> ~ $let(
                        pos: nat,
                        pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                        $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V5719532721284313246member,nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.elims(1)
tff(fact_3957_VEBT__internal_Onaive__member_Oelims_I2_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V5719532721284313246member,Xb),Xaa)
     => ( ! [A3: $o,B2: $o] :
            ( ( Xb = vEBT_Leaf((A3),(B2)) )
           => ~ $ite(
                  Xaa = zero_zero_nat,
                  (A3),
                  $ite(Xaa = one_one_nat,(B2),$false) ) )
       => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList: list_VEBT_VEBT] :
              ( ? [S: vEBT_VEBT] : Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList),S)
             => ~ $let(
                    pos: nat,
                    pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                    $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V5719532721284313246member,nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ).

% VEBT_internal.naive_member.elims(2)
tff(fact_3958_VEBT__internal_Onaive__member_Oelims_I3_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( ~ aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V5719532721284313246member,Xb),Xaa)
     => ( ! [A3: $o,B2: $o] :
            ( ( Xb = vEBT_Leaf((A3),(B2)) )
           => $ite(
                Xaa = zero_zero_nat,
                (A3),
                $ite(Xaa = one_one_nat,(B2),$false) ) )
       => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] : Xb != aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uu2,zero_zero_nat,Uv2),Uw2)
         => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList: list_VEBT_VEBT] :
                ( ? [S: vEBT_VEBT] : Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList),S)
               => $let(
                    pos: nat,
                    pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                    $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V5719532721284313246member,nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ).

% VEBT_internal.naive_member.elims(3)
tff(fact_3959_vebt__delete_Osimps_I6_J,axiom,
    ! [Mia: nat,Maa: nat,Tr2: list_VEBT_VEBT,Sm2: vEBT_VEBT,Xb: nat] : vEBT_vebt_delete(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mia),Maa)),aa_nat_nat(suc,zero_zero_nat),Tr2),Sm2),Xb) = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mia),Maa)),aa_nat_nat(suc,zero_zero_nat),Tr2),Sm2) ).

% vebt_delete.simps(6)
tff(fact_3960_httI,axiom,
    ! [P: assn,C: heap_T2636463487746394924on_nat,Q: fun_option_nat_assn,Ta: nat] :
      ( hoare_7629718768684598413on_nat(P,C,Q)
     => ( ! [H2: heap_e7401611519738050253t_unit,As: set_nat] :
            ( aa_Pro653082453696054616_nat_o(rep_assn(P),aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(H2),As))
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,time_time_option_nat(C,H2)),Ta) )
       => aa_nat_o(time_htt_option_nat(P,C,Q),Ta) ) ) ).

% httI
tff(fact_3961_httI,axiom,
    ! [P: assn,C: heap_Time_Heap_nat,Q: fun_nat_assn,Ta: nat] :
      ( hoare_3067605981109127869le_nat(P,C,Q)
     => ( ! [H2: heap_e7401611519738050253t_unit,As: set_nat] :
            ( aa_Pro653082453696054616_nat_o(rep_assn(P),aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(H2),As))
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,time_time_nat(C,H2)),Ta) )
       => aa_nat_o(time_htt_nat(P,C,Q),Ta) ) ) ).

% httI
tff(fact_3962_httI,axiom,
    ! [P: assn,C: heap_Time_Heap_o,Q: fun_o_assn,Ta: nat] :
      ( hoare_hoare_triple_o(P,C,Q)
     => ( ! [H2: heap_e7401611519738050253t_unit,As: set_nat] :
            ( aa_Pro653082453696054616_nat_o(rep_assn(P),aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(H2),As))
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,time_time_o(C,H2)),Ta) )
       => aa_nat_o(time_htt_o(P,C,Q),Ta) ) ) ).

% httI
tff(fact_3963_httI,axiom,
    ! [P: assn,C: heap_T8145700208782473153_VEBTi,Q: fun_VEBT_VEBTi_assn,Ta: nat] :
      ( hoare_1429296392585015714_VEBTi(P,C,Q)
     => ( ! [H2: heap_e7401611519738050253t_unit,As: set_nat] :
            ( aa_Pro653082453696054616_nat_o(rep_assn(P),aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(H2),As))
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,time_time_VEBT_VEBTi(C,H2)),Ta) )
       => aa_nat_o(time_htt_VEBT_VEBTi(P,C,Q),Ta) ) ) ).

% httI
tff(fact_3964_httI,axiom,
    ! [P: assn,C: heap_T5738788834812785303t_unit,Q: fun_Pr6626665806141890376t_assn,Ta: nat] :
      ( hoare_8945653483474564448t_unit(P,C,Q)
     => ( ! [H2: heap_e7401611519738050253t_unit,As: set_nat] :
            ( aa_Pro653082453696054616_nat_o(rep_assn(P),aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(H2),As))
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,time_t4224138285095624986t_unit(C,H2)),Ta) )
       => aa_nat_o(time_h7375078312994015913t_unit(P,C,Q),Ta) ) ) ).

% httI
tff(fact_3965_htt__def,axiom,
    ! [P: assn,C: heap_T2636463487746394924on_nat,Q: fun_option_nat_assn,Ta: nat] :
      ( aa_nat_o(time_htt_option_nat(P,C,Q),Ta)
    <=> ( hoare_7629718768684598413on_nat(P,C,Q)
        & ! [H4: heap_e7401611519738050253t_unit,As2: set_nat] :
            ( aa_Pro653082453696054616_nat_o(rep_assn(P),aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(H4),As2))
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,time_time_option_nat(C,H4)),Ta) ) ) ) ).

% htt_def
tff(fact_3966_htt__def,axiom,
    ! [P: assn,C: heap_Time_Heap_nat,Q: fun_nat_assn,Ta: nat] :
      ( aa_nat_o(time_htt_nat(P,C,Q),Ta)
    <=> ( hoare_3067605981109127869le_nat(P,C,Q)
        & ! [H4: heap_e7401611519738050253t_unit,As2: set_nat] :
            ( aa_Pro653082453696054616_nat_o(rep_assn(P),aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(H4),As2))
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,time_time_nat(C,H4)),Ta) ) ) ) ).

% htt_def
tff(fact_3967_htt__def,axiom,
    ! [P: assn,C: heap_Time_Heap_o,Q: fun_o_assn,Ta: nat] :
      ( aa_nat_o(time_htt_o(P,C,Q),Ta)
    <=> ( hoare_hoare_triple_o(P,C,Q)
        & ! [H4: heap_e7401611519738050253t_unit,As2: set_nat] :
            ( aa_Pro653082453696054616_nat_o(rep_assn(P),aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(H4),As2))
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,time_time_o(C,H4)),Ta) ) ) ) ).

% htt_def
tff(fact_3968_htt__def,axiom,
    ! [P: assn,C: heap_T8145700208782473153_VEBTi,Q: fun_VEBT_VEBTi_assn,Ta: nat] :
      ( aa_nat_o(time_htt_VEBT_VEBTi(P,C,Q),Ta)
    <=> ( hoare_1429296392585015714_VEBTi(P,C,Q)
        & ! [H4: heap_e7401611519738050253t_unit,As2: set_nat] :
            ( aa_Pro653082453696054616_nat_o(rep_assn(P),aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(H4),As2))
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,time_time_VEBT_VEBTi(C,H4)),Ta) ) ) ) ).

% htt_def
tff(fact_3969_htt__def,axiom,
    ! [P: assn,C: heap_T5738788834812785303t_unit,Q: fun_Pr6626665806141890376t_assn,Ta: nat] :
      ( aa_nat_o(time_h7375078312994015913t_unit(P,C,Q),Ta)
    <=> ( hoare_8945653483474564448t_unit(P,C,Q)
        & ! [H4: heap_e7401611519738050253t_unit,As2: set_nat] :
            ( aa_Pro653082453696054616_nat_o(rep_assn(P),aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(H4),As2))
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,time_t4224138285095624986t_unit(C,H4)),Ta) ) ) ) ).

% htt_def
tff(fact_3970_succ__empty,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xb: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Ta),Nb)
     => ( ( aa_nat_option_nat(vEBT_vebt_succ(Ta),Xb) = none_nat )
      <=> ( collect_nat(aa_nat_fun_nat_o(aTP_Lamm_gb(Ta),Xb)) = bot_bot_set_nat ) ) ) ).

% succ_empty
tff(fact_3971_pred__empty,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xb: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Ta),Nb)
     => ( ( aa_nat_option_nat(vEBT_vebt_pred(Ta),Xb) = none_nat )
      <=> ( collect_nat(aa_nat_fun_nat_o(aTP_Lamm_gc(Ta),Xb)) = bot_bot_set_nat ) ) ) ).

% pred_empty
tff(fact_3972_vebt__succ_Opelims,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: option_nat] :
      ( ( aa_nat_option_nat(vEBT_vebt_succ(Xb),Xaa) = Y )
     => ( aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_vebt_succ_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(Xb),Xaa))
       => ( ! [Uu2: $o,B2: $o] :
              ( ( Xb = vEBT_Leaf((Uu2),(B2)) )
             => ( ( Xaa = zero_zero_nat )
               => ( ( Y = $ite((B2),aa_nat_option_nat(some_nat,one_one_nat),none_nat) )
                 => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_vebt_succ_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(vEBT_Leaf((Uu2),(B2))),zero_zero_nat)) ) ) )
         => ( ! [Uv2: $o,Uw2: $o] :
                ( ( Xb = vEBT_Leaf((Uv2),(Uw2)) )
               => ! [N2: nat] :
                    ( ( Xaa = aa_nat_nat(suc,N2) )
                   => ( ( Y = none_nat )
                     => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_vebt_succ_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(vEBT_Leaf((Uv2),(Uw2))),aa_nat_nat(suc,N2))) ) ) )
           => ( ! [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Ux2,Uy2),Uz2) )
                 => ( ( Y = none_nat )
                   => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_vebt_succ_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Ux2,Uy2),Uz2)),Xaa)) ) )
             => ( ! [V2: product_prod_nat_nat,Vc: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
                    ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Vc),Vd2) )
                   => ( ( Y = none_nat )
                     => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_vebt_succ_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Vc),Vd2)),Xaa)) ) )
               => ( ! [V2: product_prod_nat_nat,Vg2: list_VEBT_VEBT,Vh2: vEBT_VEBT] :
                      ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vg2),Vh2) )
                     => ( ( Y = none_nat )
                       => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_vebt_succ_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vg2),Vh2)),Xaa)) ) )
                 => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                        ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList),Summary) )
                       => ( ( Y = $ite(
                                aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi),
                                aa_nat_option_nat(some_nat,Mi),
                                $let(
                                  l: nat,
                                  l:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                  $let(
                                    h: nat,
                                    h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                    $ite(
                                      aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList)),
                                      $let(
                                        maxlow: option_nat,
                                        maxlow:= vEBT_vebt_maxt(nth_VEBT_VEBT(TreeList,h)),
                                        $ite(
                                          ( ( maxlow != none_nat )
                                          & aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_less,aa_nat_option_nat(some_nat,l)),maxlow) ),
                                          aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,aa_nat_option_nat(some_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_option_nat(some_nat,h))),aa_nat_option_nat(vEBT_vebt_succ(nth_VEBT_VEBT(TreeList,h)),l)),
                                          $let(
                                            sc: option_nat,
                                            sc:= aa_nat_option_nat(vEBT_vebt_succ(Summary),h),
                                            $ite(sc = none_nat,none_nat,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,aa_nat_option_nat(some_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),sc)),vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat(sc))))) ) ) ),
                                      none_nat ) ) ) ) )
                         => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_vebt_succ_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList),Summary)),Xaa)) ) ) ) ) ) ) ) ) ) ).

% vebt_succ.pelims
tff(fact_3973_psubsetI,axiom,
    ! [A2: set_int,B4: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A2),B4)
     => ( ( A2 != B4 )
       => aa_set_int_o(ord_less_set_int(A2),B4) ) ) ).

% psubsetI
tff(fact_3974_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_3975_buildup__gives__empty,axiom,
    ! [Nb: nat] : vEBT_VEBT_set_vebt(vEBT_vebt_buildup(Nb)) = bot_bot_set_nat ).

% buildup_gives_empty
tff(fact_3976_verit__eq__simplify_I8_J,axiom,
    ! [X23: num,Y2: num] :
      ( ( aa_num_num(bit0,X23) = aa_num_num(bit0,Y2) )
    <=> ( X23 = Y2 ) ) ).

% verit_eq_simplify(8)
tff(fact_3977_Diff__cancel,axiom,
    ! [A2: set_int] : minus_minus_set_int(A2,A2) = bot_bot_set_int ).

% Diff_cancel
tff(fact_3978_Diff__cancel,axiom,
    ! [A2: set_real] : minus_minus_set_real(A2,A2) = bot_bot_set_real ).

% Diff_cancel
tff(fact_3979_Diff__cancel,axiom,
    ! [A2: set_nat] : minus_minus_set_nat(A2,A2) = bot_bot_set_nat ).

% Diff_cancel
tff(fact_3980_empty__Diff,axiom,
    ! [A2: set_int] : minus_minus_set_int(bot_bot_set_int,A2) = bot_bot_set_int ).

% empty_Diff
tff(fact_3981_empty__Diff,axiom,
    ! [A2: set_real] : minus_minus_set_real(bot_bot_set_real,A2) = bot_bot_set_real ).

% empty_Diff
tff(fact_3982_empty__Diff,axiom,
    ! [A2: set_nat] : minus_minus_set_nat(bot_bot_set_nat,A2) = bot_bot_set_nat ).

% empty_Diff
tff(fact_3983_Diff__empty,axiom,
    ! [A2: set_int] : minus_minus_set_int(A2,bot_bot_set_int) = A2 ).

% Diff_empty
tff(fact_3984_Diff__empty,axiom,
    ! [A2: set_real] : minus_minus_set_real(A2,bot_bot_set_real) = A2 ).

% Diff_empty
tff(fact_3985_Diff__empty,axiom,
    ! [A2: set_nat] : minus_minus_set_nat(A2,bot_bot_set_nat) = A2 ).

% Diff_empty
tff(fact_3986_subset__antisym,axiom,
    ! [A2: set_int,B4: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A2),B4)
     => ( aa_set_int_o(ord_less_eq_set_int(B4),A2)
       => ( A2 = B4 ) ) ) ).

% subset_antisym
tff(fact_3987_subsetI,axiom,
    ! [A2: set_nat,B4: set_nat] :
      ( ! [X2: nat] :
          ( aa_set_nat_o(member_nat(X2),A2)
         => aa_set_nat_o(member_nat(X2),B4) )
     => aa_set_nat_o(ord_less_eq_set_nat(A2),B4) ) ).

% subsetI
tff(fact_3988_subsetI,axiom,
    ! [A2: set_VEBT_VEBT,B4: set_VEBT_VEBT] :
      ( ! [X2: vEBT_VEBT] :
          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X2),A2)
         => aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X2),B4) )
     => aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(A2),B4) ) ).

% subsetI
tff(fact_3989_subsetI,axiom,
    ! [A2: set_real,B4: set_real] :
      ( ! [X2: real] :
          ( aa_set_real_o(member_real(X2),A2)
         => aa_set_real_o(member_real(X2),B4) )
     => aa_set_real_o(ord_less_eq_set_real(A2),B4) ) ).

% subsetI
tff(fact_3990_subsetI,axiom,
    ! [A2: set_int,B4: set_int] :
      ( ! [X2: int] :
          ( aa_set_int_o(member_int(X2),A2)
         => aa_set_int_o(member_int(X2),B4) )
     => aa_set_int_o(ord_less_eq_set_int(A2),B4) ) ).

% subsetI
tff(fact_3991_mint__corr__help__empty,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Ta),Nb)
     => ( ( vEBT_vebt_mint(Ta) = none_nat )
       => ( vEBT_VEBT_set_vebt(Ta) = bot_bot_set_nat ) ) ) ).

% mint_corr_help_empty
tff(fact_3992_maxt__corr__help__empty,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Ta),Nb)
     => ( ( vEBT_vebt_maxt(Ta) = none_nat )
       => ( vEBT_VEBT_set_vebt(Ta) = bot_bot_set_nat ) ) ) ).

% maxt_corr_help_empty
tff(fact_3993_Diff__eq__empty__iff,axiom,
    ! [A2: set_real,B4: set_real] :
      ( ( minus_minus_set_real(A2,B4) = bot_bot_set_real )
    <=> aa_set_real_o(ord_less_eq_set_real(A2),B4) ) ).

% Diff_eq_empty_iff
tff(fact_3994_Diff__eq__empty__iff,axiom,
    ! [A2: set_nat,B4: set_nat] :
      ( ( minus_minus_set_nat(A2,B4) = bot_bot_set_nat )
    <=> aa_set_nat_o(ord_less_eq_set_nat(A2),B4) ) ).

% Diff_eq_empty_iff
tff(fact_3995_Diff__eq__empty__iff,axiom,
    ! [A2: set_int,B4: set_int] :
      ( ( minus_minus_set_int(A2,B4) = bot_bot_set_int )
    <=> aa_set_int_o(ord_less_eq_set_int(A2),B4) ) ).

% Diff_eq_empty_iff
tff(fact_3996_subset__empty,axiom,
    ! [A2: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A2),bot_bot_set_nat)
    <=> ( A2 = bot_bot_set_nat ) ) ).

% subset_empty
tff(fact_3997_subset__empty,axiom,
    ! [A2: set_real] :
      ( aa_set_real_o(ord_less_eq_set_real(A2),bot_bot_set_real)
    <=> ( A2 = bot_bot_set_real ) ) ).

% subset_empty
tff(fact_3998_subset__empty,axiom,
    ! [A2: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A2),bot_bot_set_int)
    <=> ( A2 = bot_bot_set_int ) ) ).

% subset_empty
tff(fact_3999_empty__subsetI,axiom,
    ! [A2: set_nat] : aa_set_nat_o(ord_less_eq_set_nat(bot_bot_set_nat),A2) ).

% empty_subsetI
tff(fact_4000_empty__subsetI,axiom,
    ! [A2: set_real] : aa_set_real_o(ord_less_eq_set_real(bot_bot_set_real),A2) ).

% empty_subsetI
tff(fact_4001_empty__subsetI,axiom,
    ! [A2: set_int] : aa_set_int_o(ord_less_eq_set_int(bot_bot_set_int),A2) ).

% empty_subsetI
tff(fact_4002_atLeastatMost__empty__iff2,axiom,
    ! [A: set_int,B: set_int] :
      ( ( bot_bot_set_set_int = set_or370866239135849197et_int(A,B) )
    <=> ~ aa_set_int_o(ord_less_eq_set_int(A),B) ) ).

% atLeastatMost_empty_iff2
tff(fact_4003_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_4004_atLeastatMost__empty__iff2,axiom,
    ! [A: num,B: num] :
      ( ( bot_bot_set_num = set_or7049704709247886629st_num(A,B) )
    <=> ~ aa_num_o(ord_less_eq_num(A),B) ) ).

% atLeastatMost_empty_iff2
tff(fact_4005_atLeastatMost__empty__iff2,axiom,
    ! [A: nat,B: nat] :
      ( ( bot_bot_set_nat = set_or1269000886237332187st_nat(A,B) )
    <=> ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B) ) ).

% atLeastatMost_empty_iff2
tff(fact_4006_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_4007_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_4008_atLeastatMost__empty__iff2,axiom,
    ! [A: real,B: real] :
      ( ( bot_bot_set_real = set_or1222579329274155063t_real(A,B) )
    <=> ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ).

% atLeastatMost_empty_iff2
tff(fact_4009_atLeastatMost__empty__iff,axiom,
    ! [A: set_int,B: set_int] :
      ( ( set_or370866239135849197et_int(A,B) = bot_bot_set_set_int )
    <=> ~ aa_set_int_o(ord_less_eq_set_int(A),B) ) ).

% atLeastatMost_empty_iff
tff(fact_4010_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_4011_atLeastatMost__empty__iff,axiom,
    ! [A: num,B: num] :
      ( ( set_or7049704709247886629st_num(A,B) = bot_bot_set_num )
    <=> ~ aa_num_o(ord_less_eq_num(A),B) ) ).

% atLeastatMost_empty_iff
tff(fact_4012_atLeastatMost__empty__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( set_or1269000886237332187st_nat(A,B) = bot_bot_set_nat )
    <=> ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B) ) ).

% atLeastatMost_empty_iff
tff(fact_4013_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_4014_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_4015_atLeastatMost__empty__iff,axiom,
    ! [A: real,B: real] :
      ( ( set_or1222579329274155063t_real(A,B) = bot_bot_set_real )
    <=> ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ).

% atLeastatMost_empty_iff
tff(fact_4016_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_4017_atLeastatMost__empty,axiom,
    ! [B: num,A: num] :
      ( aa_num_o(ord_less_num(B),A)
     => ( set_or7049704709247886629st_num(A,B) = bot_bot_set_num ) ) ).

% atLeastatMost_empty
tff(fact_4018_atLeastatMost__empty,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A)
     => ( set_or1269000886237332187st_nat(A,B) = bot_bot_set_nat ) ) ).

% atLeastatMost_empty
tff(fact_4019_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_4020_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_4021_atLeastatMost__empty,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),A)
     => ( set_or1222579329274155063t_real(A,B) = bot_bot_set_real ) ) ).

% atLeastatMost_empty
tff(fact_4022_mod__pure,axiom,
    ! [B: $o,H: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(rep_assn(aa_o_assn(pure_assn,(B))),H)
    <=> ( ( aa_Pro3416150927818442182et_nat(produc8586169260539613262et_nat,H) = bot_bot_set_nat )
        & (B) ) ) ).

% mod_pure
tff(fact_4023_mod__h__bot__iff_I5_J,axiom,
    ! [P: assn,Q: assn,H: heap_e7401611519738050253t_unit] :
      ( aa_Pro653082453696054616_nat_o(rep_assn(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,P),Q)),aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(H),bot_bot_set_nat))
    <=> ( aa_Pro653082453696054616_nat_o(rep_assn(P),aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(H),bot_bot_set_nat))
        & aa_Pro653082453696054616_nat_o(rep_assn(Q),aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(H),bot_bot_set_nat)) ) ) ).

% mod_h_bot_iff(5)
tff(fact_4024_mod__h__bot__iff_I1_J,axiom,
    ! [B: $o,H: heap_e7401611519738050253t_unit] :
      ( aa_Pro653082453696054616_nat_o(rep_assn(aa_o_assn(pure_assn,(B))),aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(H),bot_bot_set_nat))
    <=> (B) ) ).

% mod_h_bot_iff(1)
tff(fact_4025_mod__h__bot__iff_I4_J,axiom,
    ! [Q2: array_VEBT_VEBTi,Y: list_VEBT_VEBTi,H: heap_e7401611519738050253t_unit] : ~ aa_Pro653082453696054616_nat_o(rep_assn(aa_lis2508312872377092907i_assn(snga_assn_VEBT_VEBTi(Q2),Y)),aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(H),bot_bot_set_nat)) ).

% mod_h_bot_iff(4)
tff(fact_4026_mod__h__bot__iff_I8_J,axiom,
    ! [R: fun_li3352006551604695530i_assn,H: heap_e7401611519738050253t_unit] :
      ( aa_Pro653082453696054616_nat_o(rep_assn(ex_ass463751140784270563_VEBTi(R)),aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(H),bot_bot_set_nat))
    <=> ? [X: list_VEBT_VEBTi] : aa_Pro653082453696054616_nat_o(rep_assn(aa_lis2508312872377092907i_assn(R,X)),aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(H),bot_bot_set_nat)) ) ).

% mod_h_bot_iff(8)
tff(fact_4027_Set_Oempty__def,axiom,
    bot_bot_set_complex = collect_complex(aTP_Lamm_gd) ).

% Set.empty_def
tff(fact_4028_Set_Oempty__def,axiom,
    bot_bo1796632182523588997nt_int = collec213857154873943460nt_int(aTP_Lamm_ge) ).

% Set.empty_def
tff(fact_4029_Set_Oempty__def,axiom,
    bot_bot_set_nat = collect_nat(aTP_Lamm_cy) ).

% Set.empty_def
tff(fact_4030_Set_Oempty__def,axiom,
    bot_bot_set_int = collect_int(aTP_Lamm_gf) ).

% Set.empty_def
tff(fact_4031_Set_Oempty__def,axiom,
    bot_bot_set_real = collect_real(aTP_Lamm_gg) ).

% Set.empty_def
tff(fact_4032_set__notEmptyE,axiom,
    ! [S2: set_VEBT_VEBT] :
      ( ( S2 != bot_bo8194388402131092736T_VEBT )
     => ~ ! [X2: vEBT_VEBT] : ~ aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X2),S2) ) ).

% set_notEmptyE
tff(fact_4033_set__notEmptyE,axiom,
    ! [S2: set_nat] :
      ( ( S2 != bot_bot_set_nat )
     => ~ ! [X2: nat] : ~ aa_set_nat_o(member_nat(X2),S2) ) ).

% set_notEmptyE
tff(fact_4034_set__notEmptyE,axiom,
    ! [S2: set_int] :
      ( ( S2 != bot_bot_set_int )
     => ~ ! [X2: int] : ~ aa_set_int_o(member_int(X2),S2) ) ).

% set_notEmptyE
tff(fact_4035_set__notEmptyE,axiom,
    ! [S2: set_real] :
      ( ( S2 != bot_bot_set_real )
     => ~ ! [X2: real] : ~ aa_set_real_o(member_real(X2),S2) ) ).

% set_notEmptyE
tff(fact_4036_memb__imp__not__empty,axiom,
    ! [Xb: vEBT_VEBT,S2: set_VEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xb),S2)
     => ( S2 != bot_bo8194388402131092736T_VEBT ) ) ).

% memb_imp_not_empty
tff(fact_4037_memb__imp__not__empty,axiom,
    ! [Xb: nat,S2: set_nat] :
      ( aa_set_nat_o(member_nat(Xb),S2)
     => ( S2 != bot_bot_set_nat ) ) ).

% memb_imp_not_empty
tff(fact_4038_memb__imp__not__empty,axiom,
    ! [Xb: int,S2: set_int] :
      ( aa_set_int_o(member_int(Xb),S2)
     => ( S2 != bot_bot_set_int ) ) ).

% memb_imp_not_empty
tff(fact_4039_memb__imp__not__empty,axiom,
    ! [Xb: real,S2: set_real] :
      ( aa_set_real_o(member_real(Xb),S2)
     => ( S2 != bot_bot_set_real ) ) ).

% memb_imp_not_empty
tff(fact_4040_times__assn__raw_Ocases,axiom,
    ! [Xb: produc3112576180687729086et_nat] :
      ~ ! [P4: fun_Pr6460944699146269935_nat_o,Q7: fun_Pr6460944699146269935_nat_o,H2: heap_e7401611519738050253t_unit,As: set_nat] : Xb != aa_Pro1979504083478021276et_nat(produc7291855603632349880et_nat(P4),aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(Q7),aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(H2),As))) ).

% times_assn_raw.cases
tff(fact_4041_one__assn__raw_Ocases,axiom,
    ! [Xb: produc3658429121746597890et_nat] :
      ~ ! [H2: heap_e7401611519738050253t_unit,As: set_nat] : Xb != aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(H2),As) ).

% one_assn_raw.cases
tff(fact_4042_subset__minus__empty,axiom,
    ! [A2: set_real,B4: set_real] :
      ( aa_set_real_o(ord_less_eq_set_real(A2),B4)
     => ( minus_minus_set_real(A2,B4) = bot_bot_set_real ) ) ).

% subset_minus_empty
tff(fact_4043_subset__minus__empty,axiom,
    ! [A2: set_nat,B4: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A2),B4)
     => ( minus_minus_set_nat(A2,B4) = bot_bot_set_nat ) ) ).

% subset_minus_empty
tff(fact_4044_subset__minus__empty,axiom,
    ! [A2: set_int,B4: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A2),B4)
     => ( minus_minus_set_int(A2,B4) = bot_bot_set_int ) ) ).

% subset_minus_empty
tff(fact_4045_mod__h__bot__indep,axiom,
    ! [P: assn,H: heap_e7401611519738050253t_unit,H3: heap_e7401611519738050253t_unit] :
      ( aa_Pro653082453696054616_nat_o(rep_assn(P),aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(H),bot_bot_set_nat))
    <=> aa_Pro653082453696054616_nat_o(rep_assn(P),aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(H3),bot_bot_set_nat)) ) ).

% mod_h_bot_indep
tff(fact_4046_mod__emp,axiom,
    ! [H: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(rep_assn(one_one_assn),H)
    <=> ( aa_Pro3416150927818442182et_nat(produc8586169260539613262et_nat,H) = bot_bot_set_nat ) ) ).

% mod_emp
tff(fact_4047_mod__emp__simp,axiom,
    ! [H: heap_e7401611519738050253t_unit] : aa_Pro653082453696054616_nat_o(rep_assn(one_one_assn),aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(H),bot_bot_set_nat)) ).

% mod_emp_simp
tff(fact_4048_verit__comp__simplify1_I2_J,axiom,
    ! [A: set_int] : aa_set_int_o(ord_less_eq_set_int(A),A) ).

% verit_comp_simplify1(2)
tff(fact_4049_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_4050_verit__comp__simplify1_I2_J,axiom,
    ! [A: num] : aa_num_o(ord_less_eq_num(A),A) ).

% verit_comp_simplify1(2)
tff(fact_4051_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_4052_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_4053_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_4054_verit__la__disequality,axiom,
    ! [A: num,B: num] :
      ( ( A = B )
      | ~ aa_num_o(ord_less_eq_num(A),B)
      | ~ aa_num_o(ord_less_eq_num(B),A) ) ).

% verit_la_disequality
tff(fact_4055_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_4056_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_4057_verit__comp__simplify1_I1_J,axiom,
    ! [A: real] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,A),A) ).

% verit_comp_simplify1(1)
tff(fact_4058_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_4059_verit__comp__simplify1_I1_J,axiom,
    ! [A: num] : ~ aa_num_o(ord_less_num(A),A) ).

% verit_comp_simplify1(1)
tff(fact_4060_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_4061_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_4062_Collect__mono__iff,axiom,
    ! [P: fun_nat_o,Q: fun_nat_o] :
      ( aa_set_nat_o(ord_less_eq_set_nat(collect_nat(P)),collect_nat(Q))
    <=> ! [X: nat] :
          ( aa_nat_o(P,X)
         => aa_nat_o(Q,X) ) ) ).

% Collect_mono_iff
tff(fact_4063_Collect__mono__iff,axiom,
    ! [P: fun_complex_o,Q: fun_complex_o] :
      ( aa_set_complex_o(ord_le211207098394363844omplex(collect_complex(P)),collect_complex(Q))
    <=> ! [X: complex] :
          ( aa_complex_o(P,X)
         => aa_complex_o(Q,X) ) ) ).

% Collect_mono_iff
tff(fact_4064_Collect__mono__iff,axiom,
    ! [P: fun_Pr3238374608701176162_int_o,Q: fun_Pr3238374608701176162_int_o] :
      ( ord_le2843351958646193337nt_int(collec213857154873943460nt_int(P),collec213857154873943460nt_int(Q))
    <=> ! [X: product_prod_int_int] :
          ( aa_Pro4592068669816069595_int_o(P,X)
         => aa_Pro4592068669816069595_int_o(Q,X) ) ) ).

% Collect_mono_iff
tff(fact_4065_Collect__mono__iff,axiom,
    ! [P: fun_int_o,Q: fun_int_o] :
      ( aa_set_int_o(ord_less_eq_set_int(collect_int(P)),collect_int(Q))
    <=> ! [X: int] :
          ( aa_int_o(P,X)
         => aa_int_o(Q,X) ) ) ).

% Collect_mono_iff
tff(fact_4066_set__eq__subset,axiom,
    ! [A2: set_int,B4: set_int] :
      ( ( A2 = B4 )
    <=> ( aa_set_int_o(ord_less_eq_set_int(A2),B4)
        & aa_set_int_o(ord_less_eq_set_int(B4),A2) ) ) ).

% set_eq_subset
tff(fact_4067_subset__trans,axiom,
    ! [A2: set_int,B4: set_int,C4: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A2),B4)
     => ( aa_set_int_o(ord_less_eq_set_int(B4),C4)
       => aa_set_int_o(ord_less_eq_set_int(A2),C4) ) ) ).

% subset_trans
tff(fact_4068_Collect__mono,axiom,
    ! [P: fun_nat_o,Q: fun_nat_o] :
      ( ! [X2: nat] :
          ( aa_nat_o(P,X2)
         => aa_nat_o(Q,X2) )
     => aa_set_nat_o(ord_less_eq_set_nat(collect_nat(P)),collect_nat(Q)) ) ).

% Collect_mono
tff(fact_4069_Collect__mono,axiom,
    ! [P: fun_complex_o,Q: fun_complex_o] :
      ( ! [X2: complex] :
          ( aa_complex_o(P,X2)
         => aa_complex_o(Q,X2) )
     => aa_set_complex_o(ord_le211207098394363844omplex(collect_complex(P)),collect_complex(Q)) ) ).

% Collect_mono
tff(fact_4070_Collect__mono,axiom,
    ! [P: fun_Pr3238374608701176162_int_o,Q: fun_Pr3238374608701176162_int_o] :
      ( ! [X2: product_prod_int_int] :
          ( aa_Pro4592068669816069595_int_o(P,X2)
         => aa_Pro4592068669816069595_int_o(Q,X2) )
     => ord_le2843351958646193337nt_int(collec213857154873943460nt_int(P),collec213857154873943460nt_int(Q)) ) ).

% Collect_mono
tff(fact_4071_Collect__mono,axiom,
    ! [P: fun_int_o,Q: fun_int_o] :
      ( ! [X2: int] :
          ( aa_int_o(P,X2)
         => aa_int_o(Q,X2) )
     => aa_set_int_o(ord_less_eq_set_int(collect_int(P)),collect_int(Q)) ) ).

% Collect_mono
tff(fact_4072_subset__refl,axiom,
    ! [A2: set_int] : aa_set_int_o(ord_less_eq_set_int(A2),A2) ).

% subset_refl
tff(fact_4073_subset__iff,axiom,
    ! [A2: set_nat,B4: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A2),B4)
    <=> ! [T: nat] :
          ( aa_set_nat_o(member_nat(T),A2)
         => aa_set_nat_o(member_nat(T),B4) ) ) ).

% subset_iff
tff(fact_4074_subset__iff,axiom,
    ! [A2: set_VEBT_VEBT,B4: set_VEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(A2),B4)
    <=> ! [T: vEBT_VEBT] :
          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(T),A2)
         => aa_set_VEBT_VEBT_o(member_VEBT_VEBT(T),B4) ) ) ).

% subset_iff
tff(fact_4075_subset__iff,axiom,
    ! [A2: set_real,B4: set_real] :
      ( aa_set_real_o(ord_less_eq_set_real(A2),B4)
    <=> ! [T: real] :
          ( aa_set_real_o(member_real(T),A2)
         => aa_set_real_o(member_real(T),B4) ) ) ).

% subset_iff
tff(fact_4076_subset__iff,axiom,
    ! [A2: set_int,B4: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A2),B4)
    <=> ! [T: int] :
          ( aa_set_int_o(member_int(T),A2)
         => aa_set_int_o(member_int(T),B4) ) ) ).

% subset_iff
tff(fact_4077_Set_OequalityD2,axiom,
    ! [A2: set_int,B4: set_int] :
      ( ( A2 = B4 )
     => aa_set_int_o(ord_less_eq_set_int(B4),A2) ) ).

% Set.equalityD2
tff(fact_4078_equalityD1,axiom,
    ! [A2: set_int,B4: set_int] :
      ( ( A2 = B4 )
     => aa_set_int_o(ord_less_eq_set_int(A2),B4) ) ).

% equalityD1
tff(fact_4079_subset__eq,axiom,
    ! [A2: set_nat,B4: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A2),B4)
    <=> ! [X: nat] :
          ( aa_set_nat_o(member_nat(X),A2)
         => aa_set_nat_o(member_nat(X),B4) ) ) ).

% subset_eq
tff(fact_4080_subset__eq,axiom,
    ! [A2: set_VEBT_VEBT,B4: set_VEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(A2),B4)
    <=> ! [X: vEBT_VEBT] :
          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),A2)
         => aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),B4) ) ) ).

% subset_eq
tff(fact_4081_subset__eq,axiom,
    ! [A2: set_real,B4: set_real] :
      ( aa_set_real_o(ord_less_eq_set_real(A2),B4)
    <=> ! [X: real] :
          ( aa_set_real_o(member_real(X),A2)
         => aa_set_real_o(member_real(X),B4) ) ) ).

% subset_eq
tff(fact_4082_subset__eq,axiom,
    ! [A2: set_int,B4: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A2),B4)
    <=> ! [X: int] :
          ( aa_set_int_o(member_int(X),A2)
         => aa_set_int_o(member_int(X),B4) ) ) ).

% subset_eq
tff(fact_4083_equalityE,axiom,
    ! [A2: set_int,B4: set_int] :
      ( ( A2 = B4 )
     => ~ ( aa_set_int_o(ord_less_eq_set_int(A2),B4)
         => ~ aa_set_int_o(ord_less_eq_set_int(B4),A2) ) ) ).

% equalityE
tff(fact_4084_subsetD,axiom,
    ! [A2: set_nat,B4: set_nat,C: nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A2),B4)
     => ( aa_set_nat_o(member_nat(C),A2)
       => aa_set_nat_o(member_nat(C),B4) ) ) ).

% subsetD
tff(fact_4085_subsetD,axiom,
    ! [A2: set_VEBT_VEBT,B4: set_VEBT_VEBT,C: vEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(A2),B4)
     => ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(C),A2)
       => aa_set_VEBT_VEBT_o(member_VEBT_VEBT(C),B4) ) ) ).

% subsetD
tff(fact_4086_subsetD,axiom,
    ! [A2: set_real,B4: set_real,C: real] :
      ( aa_set_real_o(ord_less_eq_set_real(A2),B4)
     => ( aa_set_real_o(member_real(C),A2)
       => aa_set_real_o(member_real(C),B4) ) ) ).

% subsetD
tff(fact_4087_subsetD,axiom,
    ! [A2: set_int,B4: set_int,C: int] :
      ( aa_set_int_o(ord_less_eq_set_int(A2),B4)
     => ( aa_set_int_o(member_int(C),A2)
       => aa_set_int_o(member_int(C),B4) ) ) ).

% subsetD
tff(fact_4088_in__mono,axiom,
    ! [A2: set_nat,B4: set_nat,Xb: nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A2),B4)
     => ( aa_set_nat_o(member_nat(Xb),A2)
       => aa_set_nat_o(member_nat(Xb),B4) ) ) ).

% in_mono
tff(fact_4089_in__mono,axiom,
    ! [A2: set_VEBT_VEBT,B4: set_VEBT_VEBT,Xb: vEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(A2),B4)
     => ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xb),A2)
       => aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xb),B4) ) ) ).

% in_mono
tff(fact_4090_in__mono,axiom,
    ! [A2: set_real,B4: set_real,Xb: real] :
      ( aa_set_real_o(ord_less_eq_set_real(A2),B4)
     => ( aa_set_real_o(member_real(Xb),A2)
       => aa_set_real_o(member_real(Xb),B4) ) ) ).

% in_mono
tff(fact_4091_in__mono,axiom,
    ! [A2: set_int,B4: set_int,Xb: int] :
      ( aa_set_int_o(ord_less_eq_set_int(A2),B4)
     => ( aa_set_int_o(member_int(Xb),A2)
       => aa_set_int_o(member_int(Xb),B4) ) ) ).

% in_mono
tff(fact_4092_double__diff,axiom,
    ! [A2: set_nat,B4: set_nat,C4: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A2),B4)
     => ( aa_set_nat_o(ord_less_eq_set_nat(B4),C4)
       => ( minus_minus_set_nat(B4,minus_minus_set_nat(C4,A2)) = A2 ) ) ) ).

% double_diff
tff(fact_4093_double__diff,axiom,
    ! [A2: set_int,B4: set_int,C4: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A2),B4)
     => ( aa_set_int_o(ord_less_eq_set_int(B4),C4)
       => ( minus_minus_set_int(B4,minus_minus_set_int(C4,A2)) = A2 ) ) ) ).

% double_diff
tff(fact_4094_Diff__subset,axiom,
    ! [A2: set_nat,B4: set_nat] : aa_set_nat_o(ord_less_eq_set_nat(minus_minus_set_nat(A2,B4)),A2) ).

% Diff_subset
tff(fact_4095_Diff__subset,axiom,
    ! [A2: set_int,B4: set_int] : aa_set_int_o(ord_less_eq_set_int(minus_minus_set_int(A2,B4)),A2) ).

% Diff_subset
tff(fact_4096_Diff__mono,axiom,
    ! [A2: set_nat,C4: set_nat,D5: set_nat,B4: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A2),C4)
     => ( aa_set_nat_o(ord_less_eq_set_nat(D5),B4)
       => aa_set_nat_o(ord_less_eq_set_nat(minus_minus_set_nat(A2,B4)),minus_minus_set_nat(C4,D5)) ) ) ).

% Diff_mono
tff(fact_4097_Diff__mono,axiom,
    ! [A2: set_int,C4: set_int,D5: set_int,B4: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A2),C4)
     => ( aa_set_int_o(ord_less_eq_set_int(D5),B4)
       => aa_set_int_o(ord_less_eq_set_int(minus_minus_set_int(A2,B4)),minus_minus_set_int(C4,D5)) ) ) ).

% Diff_mono
tff(fact_4098_verit__la__generic,axiom,
    ! [A: int,Xb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),Xb)
      | ( A = Xb )
      | aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),A) ) ).

% verit_la_generic
tff(fact_4099_psubset__imp__ex__mem,axiom,
    ! [A2: set_VEBT_VEBT,B4: set_VEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(ord_le3480810397992357184T_VEBT(A2),B4)
     => ? [B2: vEBT_VEBT] : aa_set_VEBT_VEBT_o(member_VEBT_VEBT(B2),minus_5127226145743854075T_VEBT(B4,A2)) ) ).

% psubset_imp_ex_mem
tff(fact_4100_psubset__imp__ex__mem,axiom,
    ! [A2: set_real,B4: set_real] :
      ( aa_set_real_o(ord_less_set_real(A2),B4)
     => ? [B2: real] : aa_set_real_o(member_real(B2),minus_minus_set_real(B4,A2)) ) ).

% psubset_imp_ex_mem
tff(fact_4101_psubset__imp__ex__mem,axiom,
    ! [A2: set_int,B4: set_int] :
      ( aa_set_int_o(ord_less_set_int(A2),B4)
     => ? [B2: int] : aa_set_int_o(member_int(B2),minus_minus_set_int(B4,A2)) ) ).

% psubset_imp_ex_mem
tff(fact_4102_psubset__imp__ex__mem,axiom,
    ! [A2: set_nat,B4: set_nat] :
      ( aa_set_nat_o(ord_less_set_nat(A2),B4)
     => ? [B2: nat] : aa_set_nat_o(member_nat(B2),minus_minus_set_nat(B4,A2)) ) ).

% psubset_imp_ex_mem
tff(fact_4103_less__eq__set__def,axiom,
    ! [A2: set_nat,B4: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A2),B4)
    <=> ord_le7240017405068453697_nat_o(aTP_Lamm_ax(A2),aTP_Lamm_ax(B4)) ) ).

% less_eq_set_def
tff(fact_4104_less__eq__set__def,axiom,
    ! [A2: set_VEBT_VEBT,B4: set_VEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(A2),B4)
    <=> ord_le88303844413636359VEBT_o(aTP_Lamm_av(A2),aTP_Lamm_av(B4)) ) ).

% less_eq_set_def
tff(fact_4105_less__eq__set__def,axiom,
    ! [A2: set_real,B4: set_real] :
      ( aa_set_real_o(ord_less_eq_set_real(A2),B4)
    <=> ord_le3605727316730466277real_o(aTP_Lamm_aw(A2),aTP_Lamm_aw(B4)) ) ).

% less_eq_set_def
tff(fact_4106_less__eq__set__def,axiom,
    ! [A2: set_int,B4: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A2),B4)
    <=> ord_le5418975059635765733_int_o(aTP_Lamm_ay(A2),aTP_Lamm_ay(B4)) ) ).

% less_eq_set_def
tff(fact_4107_Collect__subset,axiom,
    ! [A2: set_VEBT_VEBT,P: fun_VEBT_VEBT_o] : aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(collect_VEBT_VEBT(aa_fun8193769468198689164VEBT_o(aTP_Lamm_gh(A2),P))),A2) ).

% Collect_subset
tff(fact_4108_Collect__subset,axiom,
    ! [A2: set_real,P: fun_real_o] : aa_set_real_o(ord_less_eq_set_real(collect_real(aa_fun5868358341814070780real_o(aTP_Lamm_gi(A2),P))),A2) ).

% Collect_subset
tff(fact_4109_Collect__subset,axiom,
    ! [A2: set_nat,P: fun_nat_o] : aa_set_nat_o(ord_less_eq_set_nat(collect_nat(aa_fun5937193402907943164_nat_o(aTP_Lamm_gj(A2),P))),A2) ).

% Collect_subset
tff(fact_4110_Collect__subset,axiom,
    ! [A2: set_complex,P: fun_complex_o] : aa_set_complex_o(ord_le211207098394363844omplex(collect_complex(aa_fun2514317545224092796plex_o(aTP_Lamm_gk(A2),P))),A2) ).

% Collect_subset
tff(fact_4111_Collect__subset,axiom,
    ! [A2: set_Pr958786334691620121nt_int,P: fun_Pr3238374608701176162_int_o] : ord_le2843351958646193337nt_int(collec213857154873943460nt_int(aa_fun5349570537165229698_int_o(aTP_Lamm_gl(A2),P)),A2) ).

% Collect_subset
tff(fact_4112_Collect__subset,axiom,
    ! [A2: set_int,P: fun_int_o] : aa_set_int_o(ord_less_eq_set_int(collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_gm(A2),P))),A2) ).

% Collect_subset
tff(fact_4113_verit__comp__simplify1_I3_J,axiom,
    ! [B3: real,A5: real] :
      ( ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B3),A5)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A5),B3) ) ).

% verit_comp_simplify1(3)
tff(fact_4114_verit__comp__simplify1_I3_J,axiom,
    ! [B3: rat,A5: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B3),A5)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A5),B3) ) ).

% verit_comp_simplify1(3)
tff(fact_4115_verit__comp__simplify1_I3_J,axiom,
    ! [B3: num,A5: num] :
      ( ~ aa_num_o(ord_less_eq_num(B3),A5)
    <=> aa_num_o(ord_less_num(A5),B3) ) ).

% verit_comp_simplify1(3)
tff(fact_4116_verit__comp__simplify1_I3_J,axiom,
    ! [B3: nat,A5: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B3),A5)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A5),B3) ) ).

% verit_comp_simplify1(3)
tff(fact_4117_verit__comp__simplify1_I3_J,axiom,
    ! [B3: int,A5: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B3),A5)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A5),B3) ) ).

% verit_comp_simplify1(3)
tff(fact_4118_verit__sum__simplify,axiom,
    ! [A: uint32] : aa_uint32_uint32(plus_plus_uint32(A),zero_zero_uint32) = A ).

% verit_sum_simplify
tff(fact_4119_verit__sum__simplify,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),zero_zero_real) = A ).

% verit_sum_simplify
tff(fact_4120_verit__sum__simplify,axiom,
    ! [A: rat] : aa_rat_rat(plus_plus_rat(A),zero_zero_rat) = A ).

% verit_sum_simplify
tff(fact_4121_verit__sum__simplify,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),zero_zero_nat) = A ).

% verit_sum_simplify
tff(fact_4122_verit__sum__simplify,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,A),zero_zero_int) = A ).

% verit_sum_simplify
tff(fact_4123_verit__eq__simplify_I10_J,axiom,
    ! [X23: num] : one != aa_num_num(bit0,X23) ).

% verit_eq_simplify(10)
tff(fact_4124_verit__eq__simplify_I14_J,axiom,
    ! [X23: num,X33: num] : aa_num_num(bit0,X23) != aa_num_num(bit1,X33) ).

% verit_eq_simplify(14)
tff(fact_4125_verit__eq__simplify_I12_J,axiom,
    ! [X33: num] : one != aa_num_num(bit1,X33) ).

% verit_eq_simplify(12)
tff(fact_4126_psubsetE,axiom,
    ! [A2: set_int,B4: set_int] :
      ( aa_set_int_o(ord_less_set_int(A2),B4)
     => ~ ( aa_set_int_o(ord_less_eq_set_int(A2),B4)
         => aa_set_int_o(ord_less_eq_set_int(B4),A2) ) ) ).

% psubsetE
tff(fact_4127_psubset__eq,axiom,
    ! [A2: set_int,B4: set_int] :
      ( aa_set_int_o(ord_less_set_int(A2),B4)
    <=> ( aa_set_int_o(ord_less_eq_set_int(A2),B4)
        & ( A2 != B4 ) ) ) ).

% psubset_eq
tff(fact_4128_psubset__imp__subset,axiom,
    ! [A2: set_int,B4: set_int] :
      ( aa_set_int_o(ord_less_set_int(A2),B4)
     => aa_set_int_o(ord_less_eq_set_int(A2),B4) ) ).

% psubset_imp_subset
tff(fact_4129_psubset__subset__trans,axiom,
    ! [A2: set_int,B4: set_int,C4: set_int] :
      ( aa_set_int_o(ord_less_set_int(A2),B4)
     => ( aa_set_int_o(ord_less_eq_set_int(B4),C4)
       => aa_set_int_o(ord_less_set_int(A2),C4) ) ) ).

% psubset_subset_trans
tff(fact_4130_subset__not__subset__eq,axiom,
    ! [A2: set_int,B4: set_int] :
      ( aa_set_int_o(ord_less_set_int(A2),B4)
    <=> ( aa_set_int_o(ord_less_eq_set_int(A2),B4)
        & ~ aa_set_int_o(ord_less_eq_set_int(B4),A2) ) ) ).

% subset_not_subset_eq
tff(fact_4131_subset__psubset__trans,axiom,
    ! [A2: set_int,B4: set_int,C4: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A2),B4)
     => ( aa_set_int_o(ord_less_set_int(B4),C4)
       => aa_set_int_o(ord_less_set_int(A2),C4) ) ) ).

% subset_psubset_trans
tff(fact_4132_subset__iff__psubset__eq,axiom,
    ! [A2: set_int,B4: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A2),B4)
    <=> ( aa_set_int_o(ord_less_set_int(A2),B4)
        | ( A2 = B4 ) ) ) ).

% subset_iff_psubset_eq
tff(fact_4133_time__assert_H,axiom,
    ! [P: $o,H: heap_e7401611519738050253t_unit] : time_t4224138285095624986t_unit(aa_o_H2459438917461573107t_unit(refine_Imp_assert,(P)),H) = zero_zero_nat ).

% time_assert'
tff(fact_4134_TBOUND__mono,axiom,
    ! [C: heap_Time_Heap_o,Ta: nat,T2: nat] :
      ( aa_nat_o(time_TBOUND_o(C),Ta)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ta),T2)
       => aa_nat_o(time_TBOUND_o(C),T2) ) ) ).

% TBOUND_mono
tff(fact_4135_TBOUND__mono,axiom,
    ! [C: heap_Time_Heap_nat,Ta: nat,T2: nat] :
      ( aa_nat_o(time_TBOUND_nat(C),Ta)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ta),T2)
       => aa_nat_o(time_TBOUND_nat(C),T2) ) ) ).

% TBOUND_mono
tff(fact_4136_TBOUND__mono,axiom,
    ! [C: heap_T2636463487746394924on_nat,Ta: nat,T2: nat] :
      ( aa_nat_o(time_T8353473612707095248on_nat(C),Ta)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ta),T2)
       => aa_nat_o(time_T8353473612707095248on_nat(C),T2) ) ) ).

% TBOUND_mono
tff(fact_4137_TBOUND__mono,axiom,
    ! [C: heap_T8145700208782473153_VEBTi,Ta: nat,T2: nat] :
      ( aa_nat_o(time_T5737551269749752165_VEBTi(C),Ta)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ta),T2)
       => aa_nat_o(time_T5737551269749752165_VEBTi(C),T2) ) ) ).

% TBOUND_mono
tff(fact_4138_htt__htD,axiom,
    ! [P: assn,C: heap_T2636463487746394924on_nat,Q: fun_option_nat_assn,Ta: nat] :
      ( aa_nat_o(time_htt_option_nat(P,C,Q),Ta)
     => hoare_7629718768684598413on_nat(P,C,Q) ) ).

% htt_htD
tff(fact_4139_htt__htD,axiom,
    ! [P: assn,C: heap_Time_Heap_nat,Q: fun_nat_assn,Ta: nat] :
      ( aa_nat_o(time_htt_nat(P,C,Q),Ta)
     => hoare_3067605981109127869le_nat(P,C,Q) ) ).

% htt_htD
tff(fact_4140_htt__htD,axiom,
    ! [P: assn,C: heap_Time_Heap_o,Q: fun_o_assn,Ta: nat] :
      ( aa_nat_o(time_htt_o(P,C,Q),Ta)
     => hoare_hoare_triple_o(P,C,Q) ) ).

% htt_htD
tff(fact_4141_htt__htD,axiom,
    ! [P: assn,C: heap_T8145700208782473153_VEBTi,Q: fun_VEBT_VEBTi_assn,Ta: nat] :
      ( aa_nat_o(time_htt_VEBT_VEBTi(P,C,Q),Ta)
     => hoare_1429296392585015714_VEBTi(P,C,Q) ) ).

% htt_htD
tff(fact_4142_htt__htD,axiom,
    ! [P: assn,C: heap_T5738788834812785303t_unit,Q: fun_Pr6626665806141890376t_assn,Ta: nat] :
      ( aa_nat_o(time_h7375078312994015913t_unit(P,C,Q),Ta)
     => hoare_8945653483474564448t_unit(P,C,Q) ) ).

% htt_htD
tff(fact_4143_max__def__raw,axiom,
    ! [X3: set_int,Xa: set_int] :
      ord_max_set_int(X3,Xa) = $ite(aa_set_int_o(ord_less_eq_set_int(X3),Xa),Xa,X3) ).

% max_def_raw
tff(fact_4144_max__def__raw,axiom,
    ! [X3: rat,Xa: rat] :
      ord_max_rat(X3,Xa) = $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Xa),Xa,X3) ).

% max_def_raw
tff(fact_4145_max__def__raw,axiom,
    ! [X3: num,Xa: num] :
      aa_num_num(aa_num_fun_num_num(ord_max_num,X3),Xa) = $ite(aa_num_o(ord_less_eq_num(X3),Xa),Xa,X3) ).

% max_def_raw
tff(fact_4146_max__def__raw,axiom,
    ! [X3: nat,Xa: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,X3),Xa) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),Xa),Xa,X3) ).

% max_def_raw
tff(fact_4147_max__def__raw,axiom,
    ! [X3: int,Xa: int] :
      aa_int_int(aa_int_fun_int_int(ord_max_int,X3),Xa) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X3),Xa),Xa,X3) ).

% max_def_raw
tff(fact_4148_norm__pre__ex__rule__htt,axiom,
    ! [P: fun_li3352006551604695530i_assn,F: heap_T2636463487746394924on_nat,Q: fun_option_nat_assn,Ta: nat] :
      ( ! [X2: list_VEBT_VEBTi] : aa_nat_o(time_htt_option_nat(aa_lis2508312872377092907i_assn(P,X2),F,Q),Ta)
     => aa_nat_o(time_htt_option_nat(ex_ass463751140784270563_VEBTi(P),F,Q),Ta) ) ).

% norm_pre_ex_rule_htt
tff(fact_4149_norm__pre__ex__rule__htt,axiom,
    ! [P: fun_li3352006551604695530i_assn,F: heap_Time_Heap_nat,Q: fun_nat_assn,Ta: nat] :
      ( ! [X2: list_VEBT_VEBTi] : aa_nat_o(time_htt_nat(aa_lis2508312872377092907i_assn(P,X2),F,Q),Ta)
     => aa_nat_o(time_htt_nat(ex_ass463751140784270563_VEBTi(P),F,Q),Ta) ) ).

% norm_pre_ex_rule_htt
tff(fact_4150_norm__pre__ex__rule__htt,axiom,
    ! [P: fun_li3352006551604695530i_assn,F: heap_T8145700208782473153_VEBTi,Q: fun_VEBT_VEBTi_assn,Ta: nat] :
      ( ! [X2: list_VEBT_VEBTi] : aa_nat_o(time_htt_VEBT_VEBTi(aa_lis2508312872377092907i_assn(P,X2),F,Q),Ta)
     => aa_nat_o(time_htt_VEBT_VEBTi(ex_ass463751140784270563_VEBTi(P),F,Q),Ta) ) ).

% norm_pre_ex_rule_htt
tff(fact_4151_norm__pre__ex__rule__htt,axiom,
    ! [P: fun_li3352006551604695530i_assn,F: heap_Time_Heap_o,Q: fun_o_assn,Ta: nat] :
      ( ! [X2: list_VEBT_VEBTi] : aa_nat_o(time_htt_o(aa_lis2508312872377092907i_assn(P,X2),F,Q),Ta)
     => aa_nat_o(time_htt_o(ex_ass463751140784270563_VEBTi(P),F,Q),Ta) ) ).

% norm_pre_ex_rule_htt
tff(fact_4152_norm__post__ex__rule__htt,axiom,
    ! [P: assn,F: heap_T2636463487746394924on_nat,Q: fun_li3309467822378783623t_assn,Xb: list_VEBT_VEBTi,Ta: nat] :
      ( aa_nat_o(time_htt_option_nat(P,F,aa_lis7387135221810377032t_assn(Q,Xb)),Ta)
     => aa_nat_o(time_htt_option_nat(P,F,aTP_Lamm_go(Q)),Ta) ) ).

% norm_post_ex_rule_htt
tff(fact_4153_norm__post__ex__rule__htt,axiom,
    ! [P: assn,F: heap_Time_Heap_nat,Q: fun_li7023630697154767927t_assn,Xb: list_VEBT_VEBTi,Ta: nat] :
      ( aa_nat_o(time_htt_nat(P,F,aa_lis7348694692495225848t_assn(Q,Xb)),Ta)
     => aa_nat_o(time_htt_nat(P,F,aTP_Lamm_gq(Q)),Ta) ) ).

% norm_post_ex_rule_htt
tff(fact_4154_norm__post__ex__rule__htt,axiom,
    ! [P: assn,F: heap_T8145700208782473153_VEBTi,Q: fun_li8825114373765265308i_assn,Xb: list_VEBT_VEBTi,Ta: nat] :
      ( aa_nat_o(time_htt_VEBT_VEBTi(P,F,aa_lis5319508155010942429i_assn(Q,Xb)),Ta)
     => aa_nat_o(time_htt_VEBT_VEBTi(P,F,aTP_Lamm_gs(Q)),Ta) ) ).

% norm_post_ex_rule_htt
tff(fact_4155_norm__post__ex__rule__htt,axiom,
    ! [P: assn,F: heap_Time_Heap_o,Q: fun_li5930421888187426471o_assn,Xb: list_VEBT_VEBTi,Ta: nat] :
      ( aa_nat_o(time_htt_o(P,F,aa_lis3121611389386064016o_assn(Q,Xb)),Ta)
     => aa_nat_o(time_htt_o(P,F,aTP_Lamm_gu(Q)),Ta) ) ).

% norm_post_ex_rule_htt
tff(fact_4156_int__ops_I3_J,axiom,
    ! [Nb: num] : aa_nat_int(semiri1314217659103216013at_int,aa_num_nat(numeral_numeral_nat,Nb)) = aa_num_int(numeral_numeral_int,Nb) ).

% int_ops(3)
tff(fact_4157_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_4158_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_4159_int__ops_I2_J,axiom,
    aa_nat_int(semiri1314217659103216013at_int,one_one_nat) = one_one_int ).

% int_ops(2)
tff(fact_4160_int__ops_I5_J,axiom,
    ! [A: nat,B: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,A)),aa_nat_int(semiri1314217659103216013at_int,B)) ).

% int_ops(5)
tff(fact_4161_int__plus,axiom,
    ! [Nb: nat,M: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),M)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),aa_nat_int(semiri1314217659103216013at_int,M)) ).

% int_plus
tff(fact_4162_int__ops_I7_J,axiom,
    ! [A: nat,B: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,A)),aa_nat_int(semiri1314217659103216013at_int,B)) ).

% int_ops(7)
tff(fact_4163_TBOUND__return,axiom,
    ! [Xb: option_nat] : aa_nat_o(time_T8353473612707095248on_nat(heap_T3487192422709364219on_nat(Xb)),one_one_nat) ).

% TBOUND_return
tff(fact_4164_TBOUND__return,axiom,
    ! [Xb: nat] : aa_nat_o(time_TBOUND_nat(aa_nat5821310369566504398ap_nat(heap_Time_return_nat,Xb)),one_one_nat) ).

% TBOUND_return
tff(fact_4165_TBOUND__return,axiom,
    ! [Xb: $o] : aa_nat_o(time_TBOUND_o(aa_o_H7399033255965945470Heap_o(heap_Time_return_o,(Xb))),one_one_nat) ).

% TBOUND_return
tff(fact_4166_TBOUND__return,axiom,
    ! [Xb: vEBT_VEBTi] : aa_nat_o(time_T5737551269749752165_VEBTi(aa_VEB836174949890266638_VEBTi(heap_T3630416162098727440_VEBTi,Xb)),one_one_nat) ).

% TBOUND_return
tff(fact_4167_time__return,axiom,
    ! [Xb: product_unit,H: heap_e7401611519738050253t_unit] : time_t4224138285095624986t_unit(heap_T7507251653302230130t_unit(Xb),H) = one_one_nat ).

% time_return
tff(fact_4168_time__return,axiom,
    ! [Xb: option_nat,H: heap_e7401611519738050253t_unit] : time_time_option_nat(heap_T3487192422709364219on_nat(Xb),H) = one_one_nat ).

% time_return
tff(fact_4169_time__return,axiom,
    ! [Xb: nat,H: heap_e7401611519738050253t_unit] : time_time_nat(aa_nat5821310369566504398ap_nat(heap_Time_return_nat,Xb),H) = one_one_nat ).

% time_return
tff(fact_4170_time__return,axiom,
    ! [Xb: $o,H: heap_e7401611519738050253t_unit] : time_time_o(aa_o_H7399033255965945470Heap_o(heap_Time_return_o,(Xb)),H) = one_one_nat ).

% time_return
tff(fact_4171_time__return,axiom,
    ! [Xb: vEBT_VEBTi,H: heap_e7401611519738050253t_unit] : time_time_VEBT_VEBTi(aa_VEB836174949890266638_VEBTi(heap_T3630416162098727440_VEBTi,Xb),H) = one_one_nat ).

% time_return
tff(fact_4172_TBOUND__nth,axiom,
    ! [Xs: array_o,I: nat] : aa_nat_o(time_TBOUND_o(aa_nat2910048432523430448Heap_o(array_nth_o(Xs),I)),one_one_nat) ).

% TBOUND_nth
tff(fact_4173_TBOUND__nth,axiom,
    ! [Xs: array_nat,I: nat] : aa_nat_o(time_TBOUND_nat(aa_nat5821310369566504398ap_nat(array_nth_nat(Xs),I)),one_one_nat) ).

% TBOUND_nth
tff(fact_4174_TBOUND__nth,axiom,
    ! [Xs: array_option_nat,I: nat] : aa_nat_o(time_T8353473612707095248on_nat(aa_nat947733127081326238on_nat(array_nth_option_nat(Xs),I)),one_one_nat) ).

% TBOUND_nth
tff(fact_4175_TBOUND__nth,axiom,
    ! [Xs: array_VEBT_VEBTi,I: nat] : aa_nat_o(time_T5737551269749752165_VEBTi(aa_nat5408703056993926195_VEBTi(array_nth_VEBT_VEBTi(Xs),I)),one_one_nat) ).

% TBOUND_nth
tff(fact_4176_TBOUNDD,axiom,
    ! [M: heap_T5738788834812785303t_unit,Ta: nat,H: heap_e7401611519738050253t_unit] :
      ( aa_nat_o(time_T7469515765551943773t_unit(M),Ta)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,time_t4224138285095624986t_unit(M,H)),Ta) ) ).

% TBOUNDD
tff(fact_4177_TBOUNDD,axiom,
    ! [M: heap_Time_Heap_o,Ta: nat,H: heap_e7401611519738050253t_unit] :
      ( aa_nat_o(time_TBOUND_o(M),Ta)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,time_time_o(M,H)),Ta) ) ).

% TBOUNDD
tff(fact_4178_TBOUNDD,axiom,
    ! [M: heap_Time_Heap_nat,Ta: nat,H: heap_e7401611519738050253t_unit] :
      ( aa_nat_o(time_TBOUND_nat(M),Ta)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,time_time_nat(M,H)),Ta) ) ).

% TBOUNDD
tff(fact_4179_TBOUNDD,axiom,
    ! [M: heap_T2636463487746394924on_nat,Ta: nat,H: heap_e7401611519738050253t_unit] :
      ( aa_nat_o(time_T8353473612707095248on_nat(M),Ta)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,time_time_option_nat(M,H)),Ta) ) ).

% TBOUNDD
tff(fact_4180_TBOUNDD,axiom,
    ! [M: heap_T8145700208782473153_VEBTi,Ta: nat,H: heap_e7401611519738050253t_unit] :
      ( aa_nat_o(time_T5737551269749752165_VEBTi(M),Ta)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,time_time_VEBT_VEBTi(M,H)),Ta) ) ).

% TBOUNDD
tff(fact_4181_TBOUNDI,axiom,
    ! [M: heap_T5738788834812785303t_unit,Ta: nat] :
      ( ! [H2: heap_e7401611519738050253t_unit] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,time_t4224138285095624986t_unit(M,H2)),Ta)
     => aa_nat_o(time_T7469515765551943773t_unit(M),Ta) ) ).

% TBOUNDI
tff(fact_4182_TBOUNDI,axiom,
    ! [M: heap_Time_Heap_o,Ta: nat] :
      ( ! [H2: heap_e7401611519738050253t_unit] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,time_time_o(M,H2)),Ta)
     => aa_nat_o(time_TBOUND_o(M),Ta) ) ).

% TBOUNDI
tff(fact_4183_TBOUNDI,axiom,
    ! [M: heap_Time_Heap_nat,Ta: nat] :
      ( ! [H2: heap_e7401611519738050253t_unit] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,time_time_nat(M,H2)),Ta)
     => aa_nat_o(time_TBOUND_nat(M),Ta) ) ).

% TBOUNDI
tff(fact_4184_TBOUNDI,axiom,
    ! [M: heap_T2636463487746394924on_nat,Ta: nat] :
      ( ! [H2: heap_e7401611519738050253t_unit] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,time_time_option_nat(M,H2)),Ta)
     => aa_nat_o(time_T8353473612707095248on_nat(M),Ta) ) ).

% TBOUNDI
tff(fact_4185_TBOUNDI,axiom,
    ! [M: heap_T8145700208782473153_VEBTi,Ta: nat] :
      ( ! [H2: heap_e7401611519738050253t_unit] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,time_time_VEBT_VEBTi(M,H2)),Ta)
     => aa_nat_o(time_T5737551269749752165_VEBTi(M),Ta) ) ).

% TBOUNDI
tff(fact_4186_TBOUND__def,axiom,
    ! [M: heap_T5738788834812785303t_unit,Ta: nat] :
      ( aa_nat_o(time_T7469515765551943773t_unit(M),Ta)
    <=> ! [H4: heap_e7401611519738050253t_unit] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,time_t4224138285095624986t_unit(M,H4)),Ta) ) ).

% TBOUND_def
tff(fact_4187_TBOUND__def,axiom,
    ! [M: heap_Time_Heap_o,Ta: nat] :
      ( aa_nat_o(time_TBOUND_o(M),Ta)
    <=> ! [H4: heap_e7401611519738050253t_unit] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,time_time_o(M,H4)),Ta) ) ).

% TBOUND_def
tff(fact_4188_TBOUND__def,axiom,
    ! [M: heap_Time_Heap_nat,Ta: nat] :
      ( aa_nat_o(time_TBOUND_nat(M),Ta)
    <=> ! [H4: heap_e7401611519738050253t_unit] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,time_time_nat(M,H4)),Ta) ) ).

% TBOUND_def
tff(fact_4189_TBOUND__def,axiom,
    ! [M: heap_T2636463487746394924on_nat,Ta: nat] :
      ( aa_nat_o(time_T8353473612707095248on_nat(M),Ta)
    <=> ! [H4: heap_e7401611519738050253t_unit] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,time_time_option_nat(M,H4)),Ta) ) ).

% TBOUND_def
tff(fact_4190_TBOUND__def,axiom,
    ! [M: heap_T8145700208782473153_VEBTi,Ta: nat] :
      ( aa_nat_o(time_T5737551269749752165_VEBTi(M),Ta)
    <=> ! [H4: heap_e7401611519738050253t_unit] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,time_time_VEBT_VEBTi(M,H4)),Ta) ) ).

% TBOUND_def
tff(fact_4191_norm__pre__pure__iff__htt,axiom,
    ! [P: assn,B: $o,F: heap_T2636463487746394924on_nat,Q: fun_option_nat_assn,Ta: nat] :
      ( aa_nat_o(time_htt_option_nat(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,P),aa_o_assn(pure_assn,(B))),F,Q),Ta)
    <=> ( (B)
       => aa_nat_o(time_htt_option_nat(P,F,Q),Ta) ) ) ).

% norm_pre_pure_iff_htt
tff(fact_4192_norm__pre__pure__iff__htt,axiom,
    ! [P: assn,B: $o,F: heap_Time_Heap_nat,Q: fun_nat_assn,Ta: nat] :
      ( aa_nat_o(time_htt_nat(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,P),aa_o_assn(pure_assn,(B))),F,Q),Ta)
    <=> ( (B)
       => aa_nat_o(time_htt_nat(P,F,Q),Ta) ) ) ).

% norm_pre_pure_iff_htt
tff(fact_4193_norm__pre__pure__iff__htt,axiom,
    ! [P: assn,B: $o,F: heap_T8145700208782473153_VEBTi,Q: fun_VEBT_VEBTi_assn,Ta: nat] :
      ( aa_nat_o(time_htt_VEBT_VEBTi(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,P),aa_o_assn(pure_assn,(B))),F,Q),Ta)
    <=> ( (B)
       => aa_nat_o(time_htt_VEBT_VEBTi(P,F,Q),Ta) ) ) ).

% norm_pre_pure_iff_htt
tff(fact_4194_norm__pre__pure__iff__htt,axiom,
    ! [P: assn,B: $o,F: heap_Time_Heap_o,Q: fun_o_assn,Ta: nat] :
      ( aa_nat_o(time_htt_o(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,P),aa_o_assn(pure_assn,(B))),F,Q),Ta)
    <=> ( (B)
       => aa_nat_o(time_htt_o(P,F,Q),Ta) ) ) ).

% norm_pre_pure_iff_htt
tff(fact_4195_norm__pre__pure__iff__htt_H,axiom,
    ! [B: $o,P: assn,F: heap_T2636463487746394924on_nat,Q: fun_option_nat_assn,Ta: nat] :
      ( aa_nat_o(time_htt_option_nat(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_o_assn(pure_assn,(B))),P),F,Q),Ta)
    <=> ( (B)
       => aa_nat_o(time_htt_option_nat(P,F,Q),Ta) ) ) ).

% norm_pre_pure_iff_htt'
tff(fact_4196_norm__pre__pure__iff__htt_H,axiom,
    ! [B: $o,P: assn,F: heap_Time_Heap_nat,Q: fun_nat_assn,Ta: nat] :
      ( aa_nat_o(time_htt_nat(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_o_assn(pure_assn,(B))),P),F,Q),Ta)
    <=> ( (B)
       => aa_nat_o(time_htt_nat(P,F,Q),Ta) ) ) ).

% norm_pre_pure_iff_htt'
tff(fact_4197_norm__pre__pure__iff__htt_H,axiom,
    ! [B: $o,P: assn,F: heap_T8145700208782473153_VEBTi,Q: fun_VEBT_VEBTi_assn,Ta: nat] :
      ( aa_nat_o(time_htt_VEBT_VEBTi(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_o_assn(pure_assn,(B))),P),F,Q),Ta)
    <=> ( (B)
       => aa_nat_o(time_htt_VEBT_VEBTi(P,F,Q),Ta) ) ) ).

% norm_pre_pure_iff_htt'
tff(fact_4198_norm__pre__pure__iff__htt_H,axiom,
    ! [B: $o,P: assn,F: heap_Time_Heap_o,Q: fun_o_assn,Ta: nat] :
      ( aa_nat_o(time_htt_o(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_o_assn(pure_assn,(B))),P),F,Q),Ta)
    <=> ( (B)
       => aa_nat_o(time_htt_o(P,F,Q),Ta) ) ) ).

% norm_pre_pure_iff_htt'
tff(fact_4199_nat__less__as__int,axiom,
    ! [X3: nat,Xa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Xa)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,X3)),aa_nat_int(semiri1314217659103216013at_int,Xa)) ) ).

% nat_less_as_int
tff(fact_4200_nat__leq__as__int,axiom,
    ! [X3: nat,Xa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),Xa)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,X3)),aa_nat_int(semiri1314217659103216013at_int,Xa)) ) ).

% nat_leq_as_int
tff(fact_4201_httI__TBOUND,axiom,
    ! [P: assn,C: heap_T2636463487746394924on_nat,Q: fun_option_nat_assn,Ta: nat] :
      ( hoare_7629718768684598413on_nat(P,C,Q)
     => ( aa_nat_o(time_T8353473612707095248on_nat(C),Ta)
       => aa_nat_o(time_htt_option_nat(P,C,Q),Ta) ) ) ).

% httI_TBOUND
tff(fact_4202_httI__TBOUND,axiom,
    ! [P: assn,C: heap_Time_Heap_nat,Q: fun_nat_assn,Ta: nat] :
      ( hoare_3067605981109127869le_nat(P,C,Q)
     => ( aa_nat_o(time_TBOUND_nat(C),Ta)
       => aa_nat_o(time_htt_nat(P,C,Q),Ta) ) ) ).

% httI_TBOUND
tff(fact_4203_httI__TBOUND,axiom,
    ! [P: assn,C: heap_Time_Heap_o,Q: fun_o_assn,Ta: nat] :
      ( hoare_hoare_triple_o(P,C,Q)
     => ( aa_nat_o(time_TBOUND_o(C),Ta)
       => aa_nat_o(time_htt_o(P,C,Q),Ta) ) ) ).

% httI_TBOUND
tff(fact_4204_httI__TBOUND,axiom,
    ! [P: assn,C: heap_T8145700208782473153_VEBTi,Q: fun_VEBT_VEBTi_assn,Ta: nat] :
      ( hoare_1429296392585015714_VEBTi(P,C,Q)
     => ( aa_nat_o(time_T5737551269749752165_VEBTi(C),Ta)
       => aa_nat_o(time_htt_VEBT_VEBTi(P,C,Q),Ta) ) ) ).

% httI_TBOUND
tff(fact_4205_httI__TBOUND,axiom,
    ! [P: assn,C: heap_T5738788834812785303t_unit,Q: fun_Pr6626665806141890376t_assn,Ta: nat] :
      ( hoare_8945653483474564448t_unit(P,C,Q)
     => ( aa_nat_o(time_T7469515765551943773t_unit(C),Ta)
       => aa_nat_o(time_h7375078312994015913t_unit(P,C,Q),Ta) ) ) ).

% httI_TBOUND
tff(fact_4206_TBOUND__assert_H__weak,axiom,
    ! [P: $o,M: heap_Time_Heap_o,Ta: nat] :
      ( ( (P)
       => aa_nat_o(time_TBOUND_o(M),Ta) )
     => aa_nat_o(time_TBOUND_o(heap_T7577940988442287570unit_o(aa_o_H2459438917461573107t_unit(refine_Imp_assert,(P)),aTP_Lamm_gv(M))),Ta) ) ).

% TBOUND_assert'_weak
tff(fact_4207_TBOUND__assert_H__weak,axiom,
    ! [P: $o,M: heap_Time_Heap_nat,Ta: nat] :
      ( ( (P)
       => aa_nat_o(time_TBOUND_nat(M),Ta) )
     => aa_nat_o(time_TBOUND_nat(heap_T3781436268274291734it_nat(aa_o_H2459438917461573107t_unit(refine_Imp_assert,(P)),aTP_Lamm_gw(M))),Ta) ) ).

% TBOUND_assert'_weak
tff(fact_4208_TBOUND__assert_H__weak,axiom,
    ! [P: $o,M: heap_T2636463487746394924on_nat,Ta: nat] :
      ( ( (P)
       => aa_nat_o(time_T8353473612707095248on_nat(M),Ta) )
     => aa_nat_o(time_T8353473612707095248on_nat(heap_T5999496708990702694on_nat(aa_o_H2459438917461573107t_unit(refine_Imp_assert,(P)),aTP_Lamm_gx(M))),Ta) ) ).

% TBOUND_assert'_weak
tff(fact_4209_TBOUND__assert_H__weak,axiom,
    ! [P: $o,M: heap_T8145700208782473153_VEBTi,Ta: nat] :
      ( ( (P)
       => aa_nat_o(time_T5737551269749752165_VEBTi(M),Ta) )
     => aa_nat_o(time_T5737551269749752165_VEBTi(heap_T7982501707604696571_VEBTi(aa_o_H2459438917461573107t_unit(refine_Imp_assert,(P)),aTP_Lamm_gy(M))),Ta) ) ).

% TBOUND_assert'_weak
tff(fact_4210_int__Suc,axiom,
    ! [Nb: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,Nb)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),one_one_int) ).

% int_Suc
tff(fact_4211_int__ops_I4_J,axiom,
    ! [A: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,A)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,A)),one_one_int) ).

% int_ops(4)
tff(fact_4212_TBOUND__assert_H__bind__strong,axiom,
    ! [P: $o,M: heap_Time_Heap_o,Ta: nat] :
      ( ( (P)
       => aa_nat_o(time_TBOUND_o(M),Ta) )
     => aa_nat_o(time_TBOUND_o(heap_T7577940988442287570unit_o(aa_o_H2459438917461573107t_unit(refine_Imp_assert,(P)),aTP_Lamm_gv(M))),
          $ite((P),Ta,zero_zero_nat)) ) ).

% TBOUND_assert'_bind_strong
tff(fact_4213_TBOUND__assert_H__bind__strong,axiom,
    ! [P: $o,M: heap_Time_Heap_nat,Ta: nat] :
      ( ( (P)
       => aa_nat_o(time_TBOUND_nat(M),Ta) )
     => aa_nat_o(time_TBOUND_nat(heap_T3781436268274291734it_nat(aa_o_H2459438917461573107t_unit(refine_Imp_assert,(P)),aTP_Lamm_gw(M))),
          $ite((P),Ta,zero_zero_nat)) ) ).

% TBOUND_assert'_bind_strong
tff(fact_4214_TBOUND__assert_H__bind__strong,axiom,
    ! [P: $o,M: heap_T2636463487746394924on_nat,Ta: nat] :
      ( ( (P)
       => aa_nat_o(time_T8353473612707095248on_nat(M),Ta) )
     => aa_nat_o(time_T8353473612707095248on_nat(heap_T5999496708990702694on_nat(aa_o_H2459438917461573107t_unit(refine_Imp_assert,(P)),aTP_Lamm_gx(M))),
          $ite((P),Ta,zero_zero_nat)) ) ).

% TBOUND_assert'_bind_strong
tff(fact_4215_TBOUND__assert_H__bind__strong,axiom,
    ! [P: $o,M: heap_T8145700208782473153_VEBTi,Ta: nat] :
      ( ( (P)
       => aa_nat_o(time_T5737551269749752165_VEBTi(M),Ta) )
     => aa_nat_o(time_T5737551269749752165_VEBTi(heap_T7982501707604696571_VEBTi(aa_o_H2459438917461573107t_unit(refine_Imp_assert,(P)),aTP_Lamm_gy(M))),
          $ite((P),Ta,zero_zero_nat)) ) ).

% TBOUND_assert'_bind_strong
tff(fact_4216_TBOUND__option__case,axiom,
    ! [Ta: option_nat,F: heap_Time_Heap_o,Bnd: nat,F3: fun_na2343073368905879023Heap_o,Bnd2: fun_nat_nat] :
      ( ( ( Ta = none_nat )
       => aa_nat_o(time_TBOUND_o(F),Bnd) )
     => ( ! [X2: nat] :
            ( ( Ta = aa_nat_option_nat(some_nat,X2) )
           => aa_nat_o(time_TBOUND_o(aa_nat2910048432523430448Heap_o(F3,X2)),aa_nat_nat(Bnd2,X2)) )
       => aa_nat_o(time_TBOUND_o(aa_opt6753654531493786624Heap_o(case_o6892868863119666303_o_nat(F,F3),Ta)),case_option_nat_nat(Bnd,Bnd2,Ta)) ) ) ).

% TBOUND_option_case
tff(fact_4217_TBOUND__option__case,axiom,
    ! [Ta: option_num,F: heap_Time_Heap_o,Bnd: nat,F3: fun_nu9127643932893349689Heap_o,Bnd2: fun_num_nat] :
      ( ( ( Ta = none_num )
       => aa_nat_o(time_TBOUND_o(F),Bnd) )
     => ( ! [X2: num] :
            ( ( Ta = aa_num_option_num(some_num,X2) )
           => aa_nat_o(time_TBOUND_o(aa_num471246959656125306Heap_o(F3,X2)),aa_num_nat(Bnd2,X2)) )
       => aa_nat_o(time_TBOUND_o(case_o3450200649275444937_o_num(F,F3,Ta)),case_option_nat_num(Bnd,Bnd2,Ta)) ) ) ).

% TBOUND_option_case
tff(fact_4218_TBOUND__option__case,axiom,
    ! [Ta: option_nat,F: heap_Time_Heap_nat,Bnd: nat,F3: fun_na3533371890617757541ap_nat,Bnd2: fun_nat_nat] :
      ( ( ( Ta = none_nat )
       => aa_nat_o(time_TBOUND_nat(F),Bnd) )
     => ( ! [X2: nat] :
            ( ( Ta = aa_nat_option_nat(some_nat,X2) )
           => aa_nat_o(time_TBOUND_nat(aa_nat5821310369566504398ap_nat(F3,X2)),aa_nat_nat(Bnd2,X2)) )
       => aa_nat_o(time_TBOUND_nat(case_o6609685678014844897at_nat(F,F3,Ta)),case_option_nat_nat(Bnd,Bnd2,Ta)) ) ) ).

% TBOUND_option_case
tff(fact_4219_TBOUND__option__case,axiom,
    ! [Ta: option_num,F: heap_Time_Heap_nat,Bnd: nat,F3: fun_nu8176864466684091547ap_nat,Bnd2: fun_num_nat] :
      ( ( ( Ta = none_num )
       => aa_nat_o(time_TBOUND_nat(F),Bnd) )
     => ( ! [X2: num] :
            ( ( Ta = aa_num_option_num(some_num,X2) )
           => aa_nat_o(time_TBOUND_nat(aa_num1241430908778062596ap_nat(F3,X2)),aa_num_nat(Bnd2,X2)) )
       => aa_nat_o(time_TBOUND_nat(case_o3167017464170623531at_num(F,F3,Ta)),case_option_nat_num(Bnd,Bnd2,Ta)) ) ) ).

% TBOUND_option_case
tff(fact_4220_TBOUND__option__case,axiom,
    ! [Ta: option_nat,F: heap_T8145700208782473153_VEBTi,Bnd: nat,F3: fun_na7848093160509028170_VEBTi,Bnd2: fun_nat_nat] :
      ( ( ( Ta = none_nat )
       => aa_nat_o(time_T5737551269749752165_VEBTi(F),Bnd) )
     => ( ! [X2: nat] :
            ( ( Ta = aa_nat_option_nat(some_nat,X2) )
           => aa_nat_o(time_T5737551269749752165_VEBTi(aa_nat5408703056993926195_VEBTi(F3,X2)),aa_nat_nat(Bnd2,X2)) )
       => aa_nat_o(time_T5737551269749752165_VEBTi(aa_opt6483592524530923107_VEBTi(case_o3780387683879180358Ti_nat(F,F3),Ta)),case_option_nat_nat(Bnd,Bnd2,Ta)) ) ) ).

% TBOUND_option_case
tff(fact_4221_TBOUND__option__case,axiom,
    ! [Ta: option_num,F: heap_T8145700208782473153_VEBTi,Bnd: nat,F3: fun_nu5987414290159654528_VEBTi,Bnd2: fun_num_nat] :
      ( ( ( Ta = none_num )
       => aa_nat_o(time_T5737551269749752165_VEBTi(F),Bnd) )
     => ( ! [X2: num] :
            ( ( Ta = aa_num_option_num(some_num,X2) )
           => aa_nat_o(time_T5737551269749752165_VEBTi(aa_num3548024186644552553_VEBTi(F3,X2)),aa_num_nat(Bnd2,X2)) )
       => aa_nat_o(time_T5737551269749752165_VEBTi(case_o337719470034958992Ti_num(F,F3,Ta)),case_option_nat_num(Bnd,Bnd2,Ta)) ) ) ).

% TBOUND_option_case
tff(fact_4222_TBOUND__option__case,axiom,
    ! [Ta: option_nat,F: heap_T2636463487746394924on_nat,Bnd: nat,F3: fun_na5556145776640407093on_nat,Bnd2: fun_nat_nat] :
      ( ( ( Ta = none_nat )
       => aa_nat_o(time_T8353473612707095248on_nat(F),Bnd) )
     => ( ! [X2: nat] :
            ( ( Ta = aa_nat_option_nat(some_nat,X2) )
           => aa_nat_o(time_T8353473612707095248on_nat(aa_nat947733127081326238on_nat(F3,X2)),aa_nat_nat(Bnd2,X2)) )
       => aa_nat_o(time_T8353473612707095248on_nat(case_o2256915875499652529at_nat(F,F3,Ta)),case_option_nat_nat(Bnd,Bnd2,Ta)) ) ) ).

% TBOUND_option_case
tff(fact_4223_TBOUND__option__case,axiom,
    ! [Ta: option_num,F: heap_T2636463487746394924on_nat,Bnd: nat,F3: fun_nu1337165112811199083on_nat,Bnd2: fun_num_nat] :
      ( ( ( Ta = none_num )
       => aa_nat_o(time_T8353473612707095248on_nat(F),Bnd) )
     => ( ! [X2: num] :
            ( ( Ta = aa_num_option_num(some_num,X2) )
           => aa_nat_o(time_T8353473612707095248on_nat(aa_num5952124500106894036on_nat(F3,X2)),aa_num_nat(Bnd2,X2)) )
       => aa_nat_o(time_T8353473612707095248on_nat(case_o8037619698510206971at_num(F,F3,Ta)),case_option_nat_num(Bnd,Bnd2,Ta)) ) ) ).

% TBOUND_option_case
tff(fact_4224_TBOUND__option__case,axiom,
    ! [Ta: option4927543243414619207at_nat,F: heap_Time_Heap_nat,Bnd: nat,F3: fun_Pr7768898707182944990ap_nat,Bnd2: fun_Pr2602117396471556196at_nat] :
      ( ( ( Ta = none_P5556105721700978146at_nat )
       => aa_nat_o(time_TBOUND_nat(F),Bnd) )
     => ( ! [X2: product_prod_nat_nat] :
            ( ( Ta = some_P7363390416028606310at_nat(X2) )
           => aa_nat_o(time_TBOUND_nat(aa_Pro1370750447752133975ap_nat(F3,X2)),aa_Pro4243173770044397973at_nat(Bnd2,X2)) )
       => aa_nat_o(time_TBOUND_nat(case_o3959993630158478256at_nat(F,F3,Ta)),case_o2098746482150326116at_nat(Bnd,Bnd2,Ta)) ) ) ).

% TBOUND_option_case
tff(fact_4225_TBOUND__option__case,axiom,
    ! [Ta: option4927543243414619207at_nat,F: heap_T8145700208782473153_VEBTi,Bnd: nat,F3: fun_Pr45531988981583299_VEBTi,Bnd2: fun_Pr2602117396471556196at_nat] :
      ( ( ( Ta = none_P5556105721700978146at_nat )
       => aa_nat_o(time_T5737551269749752165_VEBTi(F),Bnd) )
     => ( ! [X2: product_prod_nat_nat] :
            ( ( Ta = some_P7363390416028606310at_nat(X2) )
           => aa_nat_o(time_T5737551269749752165_VEBTi(aa_Pro1395359565518501564_VEBTi(F3,X2)),aa_Pro4243173770044397973at_nat(Bnd2,X2)) )
       => aa_nat_o(time_T5737551269749752165_VEBTi(case_o1356590567247012107at_nat(F,F3,Ta)),case_o2098746482150326116at_nat(Bnd,Bnd2,Ta)) ) ) ).

% TBOUND_option_case
tff(fact_4226_vebt__pred_Opelims,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: option_nat] :
      ( ( aa_nat_option_nat(vEBT_vebt_pred(Xb),Xaa) = Y )
     => ( aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_vebt_pred_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(Xb),Xaa))
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( Xb = vEBT_Leaf((Uu2),(Uv2)) )
             => ( ( Xaa = zero_zero_nat )
               => ( ( Y = none_nat )
                 => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_vebt_pred_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(vEBT_Leaf((Uu2),(Uv2))),zero_zero_nat)) ) ) )
         => ( ! [A3: $o,Uw2: $o] :
                ( ( Xb = vEBT_Leaf((A3),(Uw2)) )
               => ( ( Xaa = aa_nat_nat(suc,zero_zero_nat) )
                 => ( ( Y = $ite((A3),aa_nat_option_nat(some_nat,zero_zero_nat),none_nat) )
                   => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_vebt_pred_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(vEBT_Leaf((A3),(Uw2))),aa_nat_nat(suc,zero_zero_nat))) ) ) )
           => ( ! [A3: $o,B2: $o] :
                  ( ( Xb = vEBT_Leaf((A3),(B2)) )
                 => ! [Va2: nat] :
                      ( ( Xaa = aa_nat_nat(suc,aa_nat_nat(suc,Va2)) )
                     => ( ( Y = $ite(
                              (B2),
                              aa_nat_option_nat(some_nat,one_one_nat),
                              $ite((A3),aa_nat_option_nat(some_nat,zero_zero_nat),none_nat) ) )
                       => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_vebt_pred_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(vEBT_Leaf((A3),(B2))),aa_nat_nat(suc,aa_nat_nat(suc,Va2)))) ) ) )
             => ( ! [Uy2: nat,Uz2: list_VEBT_VEBT,Va: vEBT_VEBT] :
                    ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uy2,Uz2),Va) )
                   => ( ( Y = none_nat )
                     => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_vebt_pred_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uy2,Uz2),Va)),Xaa)) ) )
               => ( ! [V2: product_prod_nat_nat,Vd2: list_VEBT_VEBT,Ve2: vEBT_VEBT] :
                      ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Vd2),Ve2) )
                     => ( ( Y = none_nat )
                       => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_vebt_pred_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Vd2),Ve2)),Xaa)) ) )
                 => ( ! [V2: product_prod_nat_nat,Vh2: list_VEBT_VEBT,Vi2: vEBT_VEBT] :
                        ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vh2),Vi2) )
                       => ( ( Y = none_nat )
                         => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_vebt_pred_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vh2),Vi2)),Xaa)) ) )
                   => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                          ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList),Summary) )
                         => ( ( Y = $ite(
                                  aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Xaa),
                                  aa_nat_option_nat(some_nat,Ma),
                                  $let(
                                    l: nat,
                                    l:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                    $let(
                                      h: nat,
                                      h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                      $ite(
                                        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList)),
                                        $let(
                                          minlow: option_nat,
                                          minlow:= vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,h)),
                                          $ite(
                                            ( ( minlow != none_nat )
                                            & aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_greater,aa_nat_option_nat(some_nat,l)),minlow) ),
                                            aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,aa_nat_option_nat(some_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_option_nat(some_nat,h))),aa_nat_option_nat(vEBT_vebt_pred(nth_VEBT_VEBT(TreeList,h)),l)),
                                            $let(
                                              pr: option_nat,
                                              pr:= aa_nat_option_nat(vEBT_vebt_pred(Summary),h),
                                              $ite(
                                                pr = none_nat,
                                                $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mi),Xaa),aa_nat_option_nat(some_nat,Mi),none_nat),
                                                aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,aa_nat_option_nat(some_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),pr)),vEBT_vebt_maxt(nth_VEBT_VEBT(TreeList,the_nat(pr)))) ) ) ) ),
                                        none_nat ) ) ) ) )
                           => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_vebt_pred_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList),Summary)),Xaa)) ) ) ) ) ) ) ) ) ) ) ).

% vebt_pred.pelims
tff(fact_4227_VEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H_Opelims,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_V1232361888498592333_e_t_e,Xb),Xaa) = Y )
     => ( aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_V6368547301243506412_e_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(Xb),Xaa))
       => ( ! [A3: $o,B2: $o] :
              ( ( Xb = vEBT_Leaf((A3),(B2)) )
             => ( ( Xaa = zero_zero_nat )
               => ( ( Y = one_one_nat )
                 => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_V6368547301243506412_e_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(vEBT_Leaf((A3),(B2))),zero_zero_nat)) ) ) )
         => ( ! [A3: $o,B2: $o] :
                ( ( Xb = vEBT_Leaf((A3),(B2)) )
               => ( ( Xaa = aa_nat_nat(suc,zero_zero_nat) )
                 => ( ( Y = one_one_nat )
                   => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_V6368547301243506412_e_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(vEBT_Leaf((A3),(B2))),aa_nat_nat(suc,zero_zero_nat))) ) ) )
           => ( ! [A3: $o,B2: $o] :
                  ( ( Xb = vEBT_Leaf((A3),(B2)) )
                 => ! [N2: nat] :
                      ( ( Xaa = aa_nat_nat(suc,aa_nat_nat(suc,N2)) )
                     => ( ( Y = one_one_nat )
                       => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_V6368547301243506412_e_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(vEBT_Leaf((A3),(B2))),aa_nat_nat(suc,aa_nat_nat(suc,N2)))) ) ) )
             => ( ! [Deg2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                    ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Deg2,TreeList),Summary) )
                   => ( ( Y = one_one_nat )
                     => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_V6368547301243506412_e_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Deg2,TreeList),Summary)),Xaa)) ) )
               => ( ! [Mi: nat,Ma: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                      ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),zero_zero_nat,TreeList),Summary) )
                     => ( ( Y = one_one_nat )
                       => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_V6368547301243506412_e_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),zero_zero_nat,TreeList),Summary)),Xaa)) ) )
                 => ( ! [Mi: nat,Ma: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                        ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),aa_nat_nat(suc,zero_zero_nat),TreeList),Summary) )
                       => ( ( Y = one_one_nat )
                         => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_V6368547301243506412_e_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),aa_nat_nat(suc,zero_zero_nat),TreeList),Summary)),Xaa)) ) )
                   => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                          ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList),Summary) )
                         => ( ( Y = $ite(
                                  ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi)
                                  | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Xaa) ),
                                  one_one_nat,
                                  $ite(
                                    ( ( Xaa = Mi )
                                    & ( Xaa = Ma ) ),
                                    one_one_nat,
                                    $let(
                                      xn: nat,
                                      xn:= 
                                        $ite(Xaa = Mi,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat(vEBT_vebt_mint(Summary)))))),Xaa),
                                      $let(
                                        l: nat,
                                        l:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                        $let(
                                          h: nat,
                                          h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                          $ite(
                                            aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList)),
                                            aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_V1232361888498592333_e_t_e,nth_VEBT_VEBT(TreeList,h)),l)),
                                              $ite(aa_VEBT_VEBT_o(vEBT_VEBT_minNull,vEBT_vebt_delete(nth_VEBT_VEBT(TreeList,h),l)),aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_V1232361888498592333_e_t_e,Summary),h),one_one_nat)),
                                            one_one_nat ) ) ) ) ) ) )
                           => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_V6368547301243506412_e_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList),Summary)),Xaa)) ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.pelims
tff(fact_4228_vebt__delete_Opelims,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: vEBT_VEBT] :
      ( ( vEBT_vebt_delete(Xb,Xaa) = Y )
     => ( aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_vebt_delete_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(Xb),Xaa))
       => ( ! [A3: $o,B2: $o] :
              ( ( Xb = vEBT_Leaf((A3),(B2)) )
             => ( ( Xaa = zero_zero_nat )
               => ( ( Y = vEBT_Leaf($false,(B2)) )
                 => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_vebt_delete_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(vEBT_Leaf((A3),(B2))),zero_zero_nat)) ) ) )
         => ( ! [A3: $o,B2: $o] :
                ( ( Xb = vEBT_Leaf((A3),(B2)) )
               => ( ( Xaa = aa_nat_nat(suc,zero_zero_nat) )
                 => ( ( Y = vEBT_Leaf((A3),$false) )
                   => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_vebt_delete_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(vEBT_Leaf((A3),(B2))),aa_nat_nat(suc,zero_zero_nat))) ) ) )
           => ( ! [A3: $o,B2: $o] :
                  ( ( Xb = vEBT_Leaf((A3),(B2)) )
                 => ! [N2: nat] :
                      ( ( Xaa = aa_nat_nat(suc,aa_nat_nat(suc,N2)) )
                     => ( ( Y = vEBT_Leaf((A3),(B2)) )
                       => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_vebt_delete_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(vEBT_Leaf((A3),(B2))),aa_nat_nat(suc,aa_nat_nat(suc,N2)))) ) ) )
             => ( ! [Deg2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                    ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Deg2,TreeList),Summary) )
                   => ( ( Y = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Deg2,TreeList),Summary) )
                     => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_vebt_delete_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Deg2,TreeList),Summary)),Xaa)) ) )
               => ( ! [Mi: nat,Ma: nat,TrLst: list_VEBT_VEBT,Smry: vEBT_VEBT] :
                      ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),zero_zero_nat,TrLst),Smry) )
                     => ( ( Y = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),zero_zero_nat,TrLst),Smry) )
                       => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_vebt_delete_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),zero_zero_nat,TrLst),Smry)),Xaa)) ) )
                 => ( ! [Mi: nat,Ma: nat,Tr: list_VEBT_VEBT,Sm: vEBT_VEBT] :
                        ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),aa_nat_nat(suc,zero_zero_nat),Tr),Sm) )
                       => ( ( Y = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),aa_nat_nat(suc,zero_zero_nat),Tr),Sm) )
                         => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_vebt_delete_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),aa_nat_nat(suc,zero_zero_nat),Tr),Sm)),Xaa)) ) )
                   => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                          ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList),Summary) )
                         => ( ( Y = $ite(
                                  ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi)
                                  | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Xaa) ),
                                  aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList),Summary),
                                  $ite(
                                    ( ( Xaa = Mi )
                                    & ( Xaa = Ma ) ),
                                    aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList),Summary),
                                    $let(
                                      xn: nat,
                                      xn:= 
                                        $ite(Xaa = Mi,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat(vEBT_vebt_mint(Summary)))))),Xaa),
                                      $let(
                                        minn: nat,
                                        minn:= 
                                          $ite(Xaa = Mi,xn,Mi),
                                        $let(
                                          h: nat,
                                          h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                          $ite(
                                            aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList)),
                                            $let(
                                              newnode: vEBT_VEBT,
                                              newnode:= vEBT_vebt_delete(nth_VEBT_VEBT(TreeList,h),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),
                                              $let(
                                                newlist: list_VEBT_VEBT,
                                                newlist:= list_u1324408373059187874T_VEBT(TreeList,h,newnode),
                                                $ite(
                                                  aa_VEBT_VEBT_o(vEBT_VEBT_minNull,newnode),
                                                  $let(
                                                    sn: vEBT_VEBT,
                                                    sn:= vEBT_vebt_delete(Summary,h),
                                                    aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,minn),
                                                            $ite(
                                                              xn = Ma,
                                                              $let(
                                                                maxs: option_nat,
                                                                maxs:= vEBT_vebt_maxt(sn),
                                                                $ite(maxs = none_nat,minn,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),the_nat(maxs))),the_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(newlist,the_nat(maxs)))))) ),
                                                              Ma ))),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),newlist),sn) ),
                                                  aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,minn),
                                                          $ite(xn = Ma,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,h),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(newlist,h)))),Ma))),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),newlist),Summary) ) ) ),
                                            aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList),Summary) ) ) ) ) ) ) )
                           => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_vebt_delete_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList),Summary)),Xaa)) ) ) ) ) ) ) ) ) ) ) ).

% vebt_delete.pelims
tff(fact_4229_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Opelims,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_s_u_c_c2,Xb),Xaa) = Y )
     => ( aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_T_s_u_c_c_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(Xb),Xaa))
       => ( ! [Uu2: $o,B2: $o] :
              ( ( Xb = vEBT_Leaf((Uu2),(B2)) )
             => ( ( Xaa = zero_zero_nat )
               => ( ( Y = one_one_nat )
                 => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_T_s_u_c_c_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(vEBT_Leaf((Uu2),(B2))),zero_zero_nat)) ) ) )
         => ( ! [Uv2: $o,Uw2: $o] :
                ( ( Xb = vEBT_Leaf((Uv2),(Uw2)) )
               => ! [N2: nat] :
                    ( ( Xaa = aa_nat_nat(suc,N2) )
                   => ( ( Y = one_one_nat )
                     => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_T_s_u_c_c_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(vEBT_Leaf((Uv2),(Uw2))),aa_nat_nat(suc,N2))) ) ) )
           => ( ! [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Ux2,Uy2),Uz2) )
                 => ( ( Y = one_one_nat )
                   => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_T_s_u_c_c_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Ux2,Uy2),Uz2)),Xaa)) ) )
             => ( ! [V2: product_prod_nat_nat,Vc: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
                    ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Vc),Vd2) )
                   => ( ( Y = one_one_nat )
                     => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_T_s_u_c_c_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Vc),Vd2)),Xaa)) ) )
               => ( ! [V2: product_prod_nat_nat,Vg2: list_VEBT_VEBT,Vh2: vEBT_VEBT] :
                      ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vg2),Vh2) )
                     => ( ( Y = one_one_nat )
                       => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_T_s_u_c_c_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vg2),Vh2)),Xaa)) ) )
                 => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                        ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList),Summary) )
                       => ( ( Y = $ite(
                                aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi),
                                one_one_nat,
                                $let(
                                  l: nat,
                                  l:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                  $let(
                                    h: nat,
                                    h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                    $ite(
                                      aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList)),
                                      $let(
                                        maxlow: option_nat,
                                        maxlow:= vEBT_vebt_maxt(nth_VEBT_VEBT(TreeList,h)),
                                        $ite(
                                          ( ( maxlow != none_nat )
                                          & aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_less,aa_nat_option_nat(some_nat,l)),maxlow) ),
                                          aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_s_u_c_c2,nth_VEBT_VEBT(TreeList,h)),l)),
                                          aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_s_u_c_c2,Summary),h)),one_one_nat) ) ),
                                      one_one_nat ) ) ) ) )
                         => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_T_s_u_c_c_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList),Summary)),Xaa)) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.pelims
tff(fact_4230_vebt__insert_Opelims,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: vEBT_VEBT] :
      ( ( vEBT_vebt_insert(Xb,Xaa) = Y )
     => ( aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_vebt_insert_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(Xb),Xaa))
       => ( ! [A3: $o,B2: $o] :
              ( ( Xb = vEBT_Leaf((A3),(B2)) )
             => ( ( Y = $ite(
                      Xaa = zero_zero_nat,
                      vEBT_Leaf($true,(B2)),
                      $ite(Xaa = one_one_nat,vEBT_Leaf((A3),$true),vEBT_Leaf((A3),(B2))) ) )
               => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_vebt_insert_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(vEBT_Leaf((A3),(B2))),Xaa)) ) )
         => ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S: vEBT_VEBT] :
                ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,zero_zero_nat,Ts2),S) )
               => ( ( Y = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,zero_zero_nat,Ts2),S) )
                 => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_vebt_insert_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,zero_zero_nat,Ts2),S)),Xaa)) ) )
           => ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S: vEBT_VEBT] :
                  ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts2),S) )
                 => ( ( Y = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts2),S) )
                   => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_vebt_insert_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts2),S)),Xaa)) ) )
             => ( ! [V2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                    ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V2)),TreeList),Summary) )
                   => ( ( Y = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Xaa),Xaa)),aa_nat_nat(suc,aa_nat_nat(suc,V2)),TreeList),Summary) )
                     => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_vebt_insert_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V2)),TreeList),Summary)),Xaa)) ) )
               => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                      ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList),Summary) )
                     => ( ( Y = $let(
                              xn: nat,
                              xn:= 
                                $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi),Mi,Xaa),
                              $let(
                                h: nat,
                                h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                $ite(
                                  ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList))
                                  & ~ ( ( Xaa = Mi )
                                      | ( Xaa = Ma ) ) ),
                                  aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,
                                            $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi),Xaa,Mi)),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,xn),Ma))),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),list_u1324408373059187874T_VEBT(TreeList,h,vEBT_vebt_insert(nth_VEBT_VEBT(TreeList,h),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),
                                    $ite(aa_VEBT_VEBT_o(vEBT_VEBT_minNull,nth_VEBT_VEBT(TreeList,h)),vEBT_vebt_insert(Summary,h),Summary)),
                                  aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList),Summary) ) ) ) )
                       => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_vebt_insert_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList),Summary)),Xaa)) ) ) ) ) ) ) ) ) ).

% vebt_insert.pelims
tff(fact_4231_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Opelims,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_p_r_e_d2,Xb),Xaa) = Y )
     => ( aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_T_p_r_e_d_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(Xb),Xaa))
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( Xb = vEBT_Leaf((Uu2),(Uv2)) )
             => ( ( Xaa = zero_zero_nat )
               => ( ( Y = one_one_nat )
                 => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_T_p_r_e_d_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(vEBT_Leaf((Uu2),(Uv2))),zero_zero_nat)) ) ) )
         => ( ! [A3: $o,Uw2: $o] :
                ( ( Xb = vEBT_Leaf((A3),(Uw2)) )
               => ( ( Xaa = aa_nat_nat(suc,zero_zero_nat) )
                 => ( ( Y = one_one_nat )
                   => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_T_p_r_e_d_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(vEBT_Leaf((A3),(Uw2))),aa_nat_nat(suc,zero_zero_nat))) ) ) )
           => ( ! [A3: $o,B2: $o] :
                  ( ( Xb = vEBT_Leaf((A3),(B2)) )
                 => ! [Va2: nat] :
                      ( ( Xaa = aa_nat_nat(suc,aa_nat_nat(suc,Va2)) )
                     => ( ( Y = one_one_nat )
                       => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_T_p_r_e_d_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(vEBT_Leaf((A3),(B2))),aa_nat_nat(suc,aa_nat_nat(suc,Va2)))) ) ) )
             => ( ! [Uy2: nat,Uz2: list_VEBT_VEBT,Va: vEBT_VEBT] :
                    ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uy2,Uz2),Va) )
                   => ( ( Y = one_one_nat )
                     => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_T_p_r_e_d_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uy2,Uz2),Va)),Xaa)) ) )
               => ( ! [V2: product_prod_nat_nat,Vd2: list_VEBT_VEBT,Ve2: vEBT_VEBT] :
                      ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Vd2),Ve2) )
                     => ( ( Y = one_one_nat )
                       => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_T_p_r_e_d_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Vd2),Ve2)),Xaa)) ) )
                 => ( ! [V2: product_prod_nat_nat,Vh2: list_VEBT_VEBT,Vi2: vEBT_VEBT] :
                        ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vh2),Vi2) )
                       => ( ( Y = one_one_nat )
                         => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_T_p_r_e_d_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vh2),Vi2)),Xaa)) ) )
                   => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                          ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList),Summary) )
                         => ( ( Y = $ite(
                                  aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Xaa),
                                  one_one_nat,
                                  $let(
                                    l: nat,
                                    l:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                    $let(
                                      h: nat,
                                      h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                      $ite(
                                        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList)),
                                        $let(
                                          minlow: option_nat,
                                          minlow:= vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,h)),
                                          $ite(
                                            ( ( minlow != none_nat )
                                            & aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_greater,aa_nat_option_nat(some_nat,l)),minlow) ),
                                            aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_p_r_e_d2,nth_VEBT_VEBT(TreeList,h)),l)),
                                            aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_p_r_e_d2,Summary),h)),one_one_nat) ) ),
                                        one_one_nat ) ) ) ) )
                           => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_T_p_r_e_d_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList),Summary)),Xaa)) ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.pelims
tff(fact_4232_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H_Opelims,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_i_n_s_e_r_t2,Xb),Xaa) = Y )
     => ( aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_T5076183648494686801_t_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(Xb),Xaa))
       => ( ! [A3: $o,B2: $o] :
              ( ( Xb = vEBT_Leaf((A3),(B2)) )
             => ( ( Y = one_one_nat )
               => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_T5076183648494686801_t_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(vEBT_Leaf((A3),(B2))),Xaa)) ) )
         => ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S: vEBT_VEBT] :
                ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,zero_zero_nat,Ts2),S) )
               => ( ( Y = one_one_nat )
                 => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_T5076183648494686801_t_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,zero_zero_nat,Ts2),S)),Xaa)) ) )
           => ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S: vEBT_VEBT] :
                  ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts2),S) )
                 => ( ( Y = one_one_nat )
                   => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_T5076183648494686801_t_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts2),S)),Xaa)) ) )
             => ( ! [V2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                    ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V2)),TreeList),Summary) )
                   => ( ( Y = one_one_nat )
                     => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_T5076183648494686801_t_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V2)),TreeList),Summary)),Xaa)) ) )
               => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                      ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList),Summary) )
                     => ( ( Y = $let(
                              xn: nat,
                              xn:= 
                                $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi),Mi,Xaa),
                              $let(
                                h: nat,
                                h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                $ite(
                                  ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList))
                                  & ~ ( ( Xaa = Mi )
                                      | ( Xaa = Ma ) ) ),
                                  aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_i_n_s_e_r_t2,nth_VEBT_VEBT(TreeList,h)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),
                                    $ite(aa_VEBT_VEBT_o(vEBT_VEBT_minNull,nth_VEBT_VEBT(TreeList,h)),aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_i_n_s_e_r_t2,Summary),h),one_one_nat)),
                                  one_one_nat ) ) ) )
                       => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_T5076183648494686801_t_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList),Summary)),Xaa)) ) ) ) ) ) ) ) ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.pelims
tff(fact_4233_Diff__idemp,axiom,
    ! [A2: set_nat,B4: set_nat] : minus_minus_set_nat(minus_minus_set_nat(A2,B4),B4) = minus_minus_set_nat(A2,B4) ).

% Diff_idemp
tff(fact_4234_Diff__iff,axiom,
    ! [C: vEBT_VEBT,A2: set_VEBT_VEBT,B4: set_VEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(C),minus_5127226145743854075T_VEBT(A2,B4))
    <=> ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(C),A2)
        & ~ aa_set_VEBT_VEBT_o(member_VEBT_VEBT(C),B4) ) ) ).

% Diff_iff
tff(fact_4235_Diff__iff,axiom,
    ! [C: real,A2: set_real,B4: set_real] :
      ( aa_set_real_o(member_real(C),minus_minus_set_real(A2,B4))
    <=> ( aa_set_real_o(member_real(C),A2)
        & ~ aa_set_real_o(member_real(C),B4) ) ) ).

% Diff_iff
tff(fact_4236_Diff__iff,axiom,
    ! [C: int,A2: set_int,B4: set_int] :
      ( aa_set_int_o(member_int(C),minus_minus_set_int(A2,B4))
    <=> ( aa_set_int_o(member_int(C),A2)
        & ~ aa_set_int_o(member_int(C),B4) ) ) ).

% Diff_iff
tff(fact_4237_Diff__iff,axiom,
    ! [C: nat,A2: set_nat,B4: set_nat] :
      ( aa_set_nat_o(member_nat(C),minus_minus_set_nat(A2,B4))
    <=> ( aa_set_nat_o(member_nat(C),A2)
        & ~ aa_set_nat_o(member_nat(C),B4) ) ) ).

% Diff_iff
tff(fact_4238_DiffI,axiom,
    ! [C: vEBT_VEBT,A2: set_VEBT_VEBT,B4: set_VEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(C),A2)
     => ( ~ aa_set_VEBT_VEBT_o(member_VEBT_VEBT(C),B4)
       => aa_set_VEBT_VEBT_o(member_VEBT_VEBT(C),minus_5127226145743854075T_VEBT(A2,B4)) ) ) ).

% DiffI
tff(fact_4239_DiffI,axiom,
    ! [C: real,A2: set_real,B4: set_real] :
      ( aa_set_real_o(member_real(C),A2)
     => ( ~ aa_set_real_o(member_real(C),B4)
       => aa_set_real_o(member_real(C),minus_minus_set_real(A2,B4)) ) ) ).

% DiffI
tff(fact_4240_DiffI,axiom,
    ! [C: int,A2: set_int,B4: set_int] :
      ( aa_set_int_o(member_int(C),A2)
     => ( ~ aa_set_int_o(member_int(C),B4)
       => aa_set_int_o(member_int(C),minus_minus_set_int(A2,B4)) ) ) ).

% DiffI
tff(fact_4241_DiffI,axiom,
    ! [C: nat,A2: set_nat,B4: set_nat] :
      ( aa_set_nat_o(member_nat(C),A2)
     => ( ~ aa_set_nat_o(member_nat(C),B4)
       => aa_set_nat_o(member_nat(C),minus_minus_set_nat(A2,B4)) ) ) ).

% DiffI
tff(fact_4242_minus__set__def,axiom,
    ! [A2: set_VEBT_VEBT,B4: set_VEBT_VEBT] : minus_5127226145743854075T_VEBT(A2,B4) = collect_VEBT_VEBT(minus_4582106842932979072VEBT_o(aTP_Lamm_av(A2),aTP_Lamm_av(B4))) ).

% minus_set_def
tff(fact_4243_minus__set__def,axiom,
    ! [A2: set_real,B4: set_real] : minus_minus_set_real(A2,B4) = collect_real(minus_404528632214381484real_o(aTP_Lamm_aw(A2),aTP_Lamm_aw(B4))) ).

% minus_set_def
tff(fact_4244_minus__set__def,axiom,
    ! [A2: set_int,B4: set_int] : minus_minus_set_int(A2,B4) = collect_int(minus_7151557278095657260_int_o(aTP_Lamm_ay(A2),aTP_Lamm_ay(B4))) ).

% minus_set_def
tff(fact_4245_minus__set__def,axiom,
    ! [A2: set_complex,B4: set_complex] : minus_811609699411566653omplex(A2,B4) = collect_complex(minus_8169481405280535082plex_o(aTP_Lamm_az(A2),aTP_Lamm_az(B4))) ).

% minus_set_def
tff(fact_4246_minus__set__def,axiom,
    ! [A2: set_Pr958786334691620121nt_int,B4: set_Pr958786334691620121nt_int] : minus_1052850069191792384nt_int(A2,B4) = collec213857154873943460nt_int(minus_827889751674334651_int_o(aTP_Lamm_ba(A2),aTP_Lamm_ba(B4))) ).

% minus_set_def
tff(fact_4247_minus__set__def,axiom,
    ! [A2: set_nat,B4: set_nat] : minus_minus_set_nat(A2,B4) = collect_nat(minus_8972599623528345224_nat_o(aTP_Lamm_ax(A2),aTP_Lamm_ax(B4))) ).

% minus_set_def
tff(fact_4248_set__diff__eq,axiom,
    ! [A2: set_VEBT_VEBT,B4: set_VEBT_VEBT] : minus_5127226145743854075T_VEBT(A2,B4) = collect_VEBT_VEBT(aa_set2497645856053385599VEBT_o(aTP_Lamm_gz(A2),B4)) ).

% set_diff_eq
tff(fact_4249_set__diff__eq,axiom,
    ! [A2: set_real,B4: set_real] : minus_minus_set_real(A2,B4) = collect_real(aa_set2323979645938365349real_o(aTP_Lamm_ha(A2),B4)) ).

% set_diff_eq
tff(fact_4250_set__diff__eq,axiom,
    ! [A2: set_int,B4: set_int] : minus_minus_set_int(A2,B4) = collect_int(aa_set_int_fun_int_o(aTP_Lamm_hb(A2),B4)) ).

% set_diff_eq
tff(fact_4251_set__diff__eq,axiom,
    ! [A2: set_complex,B4: set_complex] : minus_811609699411566653omplex(A2,B4) = collect_complex(aa_set1742385355640705825plex_o(aTP_Lamm_hc(A2),B4)) ).

% set_diff_eq
tff(fact_4252_set__diff__eq,axiom,
    ! [A2: set_Pr958786334691620121nt_int,B4: set_Pr958786334691620121nt_int] : minus_1052850069191792384nt_int(A2,B4) = collec213857154873943460nt_int(aa_set2224860389584585407_int_o(aTP_Lamm_hd(A2),B4)) ).

% set_diff_eq
tff(fact_4253_set__diff__eq,axiom,
    ! [A2: set_nat,B4: set_nat] : minus_minus_set_nat(A2,B4) = collect_nat(aa_set_nat_fun_nat_o(aTP_Lamm_he(A2),B4)) ).

% set_diff_eq
tff(fact_4254_DiffD2,axiom,
    ! [C: vEBT_VEBT,A2: set_VEBT_VEBT,B4: set_VEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(C),minus_5127226145743854075T_VEBT(A2,B4))
     => ~ aa_set_VEBT_VEBT_o(member_VEBT_VEBT(C),B4) ) ).

% DiffD2
tff(fact_4255_DiffD2,axiom,
    ! [C: real,A2: set_real,B4: set_real] :
      ( aa_set_real_o(member_real(C),minus_minus_set_real(A2,B4))
     => ~ aa_set_real_o(member_real(C),B4) ) ).

% DiffD2
tff(fact_4256_DiffD2,axiom,
    ! [C: int,A2: set_int,B4: set_int] :
      ( aa_set_int_o(member_int(C),minus_minus_set_int(A2,B4))
     => ~ aa_set_int_o(member_int(C),B4) ) ).

% DiffD2
tff(fact_4257_DiffD2,axiom,
    ! [C: nat,A2: set_nat,B4: set_nat] :
      ( aa_set_nat_o(member_nat(C),minus_minus_set_nat(A2,B4))
     => ~ aa_set_nat_o(member_nat(C),B4) ) ).

% DiffD2
tff(fact_4258_DiffD1,axiom,
    ! [C: vEBT_VEBT,A2: set_VEBT_VEBT,B4: set_VEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(C),minus_5127226145743854075T_VEBT(A2,B4))
     => aa_set_VEBT_VEBT_o(member_VEBT_VEBT(C),A2) ) ).

% DiffD1
tff(fact_4259_DiffD1,axiom,
    ! [C: real,A2: set_real,B4: set_real] :
      ( aa_set_real_o(member_real(C),minus_minus_set_real(A2,B4))
     => aa_set_real_o(member_real(C),A2) ) ).

% DiffD1
tff(fact_4260_DiffD1,axiom,
    ! [C: int,A2: set_int,B4: set_int] :
      ( aa_set_int_o(member_int(C),minus_minus_set_int(A2,B4))
     => aa_set_int_o(member_int(C),A2) ) ).

% DiffD1
tff(fact_4261_DiffD1,axiom,
    ! [C: nat,A2: set_nat,B4: set_nat] :
      ( aa_set_nat_o(member_nat(C),minus_minus_set_nat(A2,B4))
     => aa_set_nat_o(member_nat(C),A2) ) ).

% DiffD1
tff(fact_4262_DiffE,axiom,
    ! [C: vEBT_VEBT,A2: set_VEBT_VEBT,B4: set_VEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(C),minus_5127226145743854075T_VEBT(A2,B4))
     => ~ ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(C),A2)
         => aa_set_VEBT_VEBT_o(member_VEBT_VEBT(C),B4) ) ) ).

% DiffE
tff(fact_4263_DiffE,axiom,
    ! [C: real,A2: set_real,B4: set_real] :
      ( aa_set_real_o(member_real(C),minus_minus_set_real(A2,B4))
     => ~ ( aa_set_real_o(member_real(C),A2)
         => aa_set_real_o(member_real(C),B4) ) ) ).

% DiffE
tff(fact_4264_DiffE,axiom,
    ! [C: int,A2: set_int,B4: set_int] :
      ( aa_set_int_o(member_int(C),minus_minus_set_int(A2,B4))
     => ~ ( aa_set_int_o(member_int(C),A2)
         => aa_set_int_o(member_int(C),B4) ) ) ).

% DiffE
tff(fact_4265_DiffE,axiom,
    ! [C: nat,A2: set_nat,B4: set_nat] :
      ( aa_set_nat_o(member_nat(C),minus_minus_set_nat(A2,B4))
     => ~ ( aa_set_nat_o(member_nat(C),A2)
         => aa_set_nat_o(member_nat(C),B4) ) ) ).

% DiffE
tff(fact_4266_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_Opelims,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_m_e_m_b_e_r,Xb),Xaa) = Y )
     => ( aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_T5837161174952499735_r_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(Xb),Xaa))
       => ( ! [A3: $o,B2: $o] :
              ( ( Xb = vEBT_Leaf((A3),(B2)) )
             => ( ( Y = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                      $ite(Xaa = zero_zero_nat,one_one_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat))) )
               => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_T5837161174952499735_r_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(vEBT_Leaf((A3),(B2))),Xaa)) ) )
         => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2),Uw2) )
               => ( ( Y = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) )
                 => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_T5837161174952499735_r_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2),Uw2)),Xaa)) ) )
           => ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Uy2),Uz2) )
                 => ( ( Y = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) )
                   => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_T5837161174952499735_r_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Uy2),Uz2)),Xaa)) ) )
             => ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc: vEBT_VEBT] :
                    ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vb2),Vc) )
                   => ( ( Y = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) )
                     => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_T5837161174952499735_r_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vb2),Vc)),Xaa)) ) )
               => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                      ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList),Summary) )
                     => ( ( Y = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                              $ite(
                                Xaa = Mi,
                                one_one_nat,
                                aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
                                  $ite(
                                    Xaa = Ma,
                                    one_one_nat,
                                    aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
                                      $ite(
                                        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi),
                                        one_one_nat,
                                        aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
                                          $ite(
                                            aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Xaa),
                                            one_one_nat,
                                            aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,one))))),
                                              $let(
                                                h: nat,
                                                h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                                $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_m_e_m_b_e_r,nth_VEBT_VEBT(TreeList,h)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),one_one_nat) )) )) )) )) )) )
                       => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_T5837161174952499735_r_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList),Summary)),Xaa)) ) ) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.pelims
tff(fact_4267_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H_Opelims,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_m_e_m_b_e_r2,Xb),Xaa) = Y )
     => ( aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_T8099345112685741742_r_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(Xb),Xaa))
       => ( ! [A3: $o,B2: $o] :
              ( ( Xb = vEBT_Leaf((A3),(B2)) )
             => ( ( Y = one_one_nat )
               => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_T8099345112685741742_r_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(vEBT_Leaf((A3),(B2))),Xaa)) ) )
         => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2),Uw2) )
               => ( ( Y = one_one_nat )
                 => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_T8099345112685741742_r_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2),Uw2)),Xaa)) ) )
           => ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Uy2),Uz2) )
                 => ( ( Y = one_one_nat )
                   => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_T8099345112685741742_r_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Uy2),Uz2)),Xaa)) ) )
             => ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc: vEBT_VEBT] :
                    ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vb2),Vc) )
                   => ( ( Y = one_one_nat )
                     => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_T8099345112685741742_r_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vb2),Vc)),Xaa)) ) )
               => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                      ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList),Summary) )
                     => ( ( Y = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
                              $ite(
                                Xaa = Mi,
                                zero_zero_nat,
                                $ite(
                                  Xaa = Ma,
                                  zero_zero_nat,
                                  $ite(
                                    aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi),
                                    zero_zero_nat,
                                    $ite(
                                      aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Xaa),
                                      zero_zero_nat,
                                      $ite(
                                        ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mi),Xaa)
                                        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Ma) ),
                                        $let(
                                          h: nat,
                                          h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                          $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_m_e_m_b_e_r2,nth_VEBT_VEBT(TreeList,h)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),zero_zero_nat) ),
                                        zero_zero_nat ) ) ) ) )) )
                       => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_T8099345112685741742_r_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList),Summary)),Xaa)) ) ) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.pelims
tff(fact_4268_vebt__member_Opelims_I1_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: $o] :
      ( ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,Xb),Xaa)
      <=> (Y) )
     => ( aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(Xb),Xaa))
       => ( ! [A3: $o,B2: $o] :
              ( ( Xb = vEBT_Leaf((A3),(B2)) )
             => ( ( (Y)
                <=> $ite(
                      Xaa = zero_zero_nat,
                      (A3),
                      $ite(Xaa = one_one_nat,(B2),$false) ) )
               => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(vEBT_Leaf((A3),(B2))),Xaa)) ) )
         => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2),Uw2) )
               => ( ~ (Y)
                 => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2),Uw2)),Xaa)) ) )
           => ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Uy2),Uz2) )
                 => ( ~ (Y)
                   => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Uy2),Uz2)),Xaa)) ) )
             => ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc: vEBT_VEBT] :
                    ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vb2),Vc) )
                   => ( ~ (Y)
                     => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vb2),Vc)),Xaa)) ) )
               => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                      ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList),Summary) )
                     => ( ( (Y)
                        <=> $ite(
                              Xaa = Mi,
                              $true,
                              $ite(
                                Xaa = Ma,
                                $true,
                                $ite(
                                  aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi),
                                  $false,
                                  $ite(
                                    aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Xaa),
                                    $false,
                                    $let(
                                      h: nat,
                                      h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                      $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,nth_VEBT_VEBT(TreeList,h)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) )
                       => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList),Summary)),Xaa)) ) ) ) ) ) ) ) ) ).

% vebt_member.pelims(1)
tff(fact_4269_vebt__member_Opelims_I3_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( ~ aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,Xb),Xaa)
     => ( aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(Xb),Xaa))
       => ( ! [A3: $o,B2: $o] :
              ( ( Xb = vEBT_Leaf((A3),(B2)) )
             => ( aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(vEBT_Leaf((A3),(B2))),Xaa))
               => $ite(
                    Xaa = zero_zero_nat,
                    (A3),
                    $ite(Xaa = one_one_nat,(B2),$false) ) ) )
         => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2),Uw2) )
               => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2),Uw2)),Xaa)) )
           => ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Uy2),Uz2) )
                 => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Uy2),Uz2)),Xaa)) )
             => ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc: vEBT_VEBT] :
                    ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vb2),Vc) )
                   => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vb2),Vc)),Xaa)) )
               => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                      ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList),Summary) )
                     => ( aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList),Summary)),Xaa))
                       => $ite(
                            Xaa = Mi,
                            $true,
                            $ite(
                              Xaa = Ma,
                              $true,
                              $ite(
                                aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi),
                                $false,
                                $ite(
                                  aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Xaa),
                                  $false,
                                  $let(
                                    h: nat,
                                    h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                    $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,nth_VEBT_VEBT(TreeList,h)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.pelims(3)
tff(fact_4270_VEBT__internal_Onaive__member_Opelims_I1_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: $o] :
      ( ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V5719532721284313246member,Xb),Xaa)
      <=> (Y) )
     => ( aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(Xb),Xaa))
       => ( ! [A3: $o,B2: $o] :
              ( ( Xb = vEBT_Leaf((A3),(B2)) )
             => ( ( (Y)
                <=> $ite(
                      Xaa = zero_zero_nat,
                      (A3),
                      $ite(Xaa = one_one_nat,(B2),$false) ) )
               => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(vEBT_Leaf((A3),(B2))),Xaa)) ) )
         => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uu2,zero_zero_nat,Uv2),Uw2) )
               => ( ~ (Y)
                 => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uu2,zero_zero_nat,Uv2),Uw2)),Xaa)) ) )
           => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList: list_VEBT_VEBT,S: vEBT_VEBT] :
                  ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList),S) )
                 => ( ( (Y)
                    <=> $let(
                          pos: nat,
                          pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                          $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V5719532721284313246member,nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) )
                   => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList),S)),Xaa)) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(1)
tff(fact_4271_VEBT__internal_Onaive__member_Opelims_I2_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V5719532721284313246member,Xb),Xaa)
     => ( aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(Xb),Xaa))
       => ( ! [A3: $o,B2: $o] :
              ( ( Xb = vEBT_Leaf((A3),(B2)) )
             => ( aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(vEBT_Leaf((A3),(B2))),Xaa))
               => ~ $ite(
                      Xaa = zero_zero_nat,
                      (A3),
                      $ite(Xaa = one_one_nat,(B2),$false) ) ) )
         => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList: list_VEBT_VEBT,S: vEBT_VEBT] :
                ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList),S) )
               => ( aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList),S)),Xaa))
                 => ~ $let(
                        pos: nat,
                        pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                        $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V5719532721284313246member,nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(2)
tff(fact_4272_VEBT__internal_Onaive__member_Opelims_I3_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( ~ aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V5719532721284313246member,Xb),Xaa)
     => ( aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(Xb),Xaa))
       => ( ! [A3: $o,B2: $o] :
              ( ( Xb = vEBT_Leaf((A3),(B2)) )
             => ( aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(vEBT_Leaf((A3),(B2))),Xaa))
               => $ite(
                    Xaa = zero_zero_nat,
                    (A3),
                    $ite(Xaa = one_one_nat,(B2),$false) ) ) )
         => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uu2,zero_zero_nat,Uv2),Uw2) )
               => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uu2,zero_zero_nat,Uv2),Uw2)),Xaa)) )
           => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList: list_VEBT_VEBT,S: vEBT_VEBT] :
                  ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList),S) )
                 => ( aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList),S)),Xaa))
                   => $let(
                        pos: nat,
                        pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                        $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V5719532721284313246member,nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(3)
tff(fact_4273_vebt__member_Opelims_I2_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,Xb),Xaa)
     => ( aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(Xb),Xaa))
       => ( ! [A3: $o,B2: $o] :
              ( ( Xb = vEBT_Leaf((A3),(B2)) )
             => ( aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(vEBT_Leaf((A3),(B2))),Xaa))
               => ~ $ite(
                      Xaa = zero_zero_nat,
                      (A3),
                      $ite(Xaa = one_one_nat,(B2),$false) ) ) )
         => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList),Summary) )
               => ( aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList),Summary)),Xaa))
                 => ~ $ite(
                        Xaa = Mi,
                        $true,
                        $ite(
                          Xaa = Ma,
                          $true,
                          $ite(
                            aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi),
                            $false,
                            $ite(
                              aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Xaa),
                              $false,
                              $let(
                                h: nat,
                                h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,nth_VEBT_VEBT(TreeList,h)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.pelims(2)
tff(fact_4274_VEBT__internal_Omembermima_Opelims_I3_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( ~ aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_membermima,Xb),Xaa)
     => ( aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(Xb),Xaa))
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( Xb = vEBT_Leaf((Uu2),(Uv2)) )
             => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(vEBT_Leaf((Uu2),(Uv2))),Xaa)) )
         => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
                ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux2),Uy2) )
               => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux2),Uy2)),Xaa)) )
           => ( ! [Mi: nat,Ma: nat,Va: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
                  ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),zero_zero_nat,Va),Vb2) )
                 => ( aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),zero_zero_nat,Va),Vb2)),Xaa))
                   => ( ( Xaa = Mi )
                      | ( Xaa = Ma ) ) ) )
             => ( ! [Mi: nat,Ma: nat,V2: nat,TreeList: list_VEBT_VEBT,Vc: vEBT_VEBT] :
                    ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),aa_nat_nat(suc,V2),TreeList),Vc) )
                   => ( aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),aa_nat_nat(suc,V2),TreeList),Vc)),Xaa))
                     => ( ( Xaa = Mi )
                        | ( Xaa = Ma )
                        | $let(
                            pos: nat,
                            pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                            $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_membermima,nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) )
               => ~ ! [V2: nat,TreeList: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
                      ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList),Vd2) )
                     => ( aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList),Vd2)),Xaa))
                       => $let(
                            pos: nat,
                            pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                            $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_membermima,nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(3)
tff(fact_4275_VEBT__internal_Omembermima_Opelims_I1_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: $o] :
      ( ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_membermima,Xb),Xaa)
      <=> (Y) )
     => ( aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(Xb),Xaa))
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( Xb = vEBT_Leaf((Uu2),(Uv2)) )
             => ( ~ (Y)
               => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(vEBT_Leaf((Uu2),(Uv2))),Xaa)) ) )
         => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
                ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux2),Uy2) )
               => ( ~ (Y)
                 => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux2),Uy2)),Xaa)) ) )
           => ( ! [Mi: nat,Ma: nat,Va: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
                  ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),zero_zero_nat,Va),Vb2) )
                 => ( ( (Y)
                    <=> ( ( Xaa = Mi )
                        | ( Xaa = Ma ) ) )
                   => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),zero_zero_nat,Va),Vb2)),Xaa)) ) )
             => ( ! [Mi: nat,Ma: nat,V2: nat,TreeList: list_VEBT_VEBT,Vc: vEBT_VEBT] :
                    ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),aa_nat_nat(suc,V2),TreeList),Vc) )
                   => ( ( (Y)
                      <=> ( ( Xaa = Mi )
                          | ( Xaa = Ma )
                          | $let(
                              pos: nat,
                              pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                              $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_membermima,nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) )
                     => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),aa_nat_nat(suc,V2),TreeList),Vc)),Xaa)) ) )
               => ~ ! [V2: nat,TreeList: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
                      ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList),Vd2) )
                     => ( ( (Y)
                        <=> $let(
                              pos: nat,
                              pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                              $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_membermima,nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) )
                       => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList),Vd2)),Xaa)) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(1)
tff(fact_4276_VEBT__internal_Omembermima_Opelims_I2_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_membermima,Xb),Xaa)
     => ( aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(Xb),Xaa))
       => ( ! [Mi: nat,Ma: nat,Va: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
              ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),zero_zero_nat,Va),Vb2) )
             => ( aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),zero_zero_nat,Va),Vb2)),Xaa))
               => ~ ( ( Xaa = Mi )
                    | ( Xaa = Ma ) ) ) )
         => ( ! [Mi: nat,Ma: nat,V2: nat,TreeList: list_VEBT_VEBT,Vc: vEBT_VEBT] :
                ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),aa_nat_nat(suc,V2),TreeList),Vc) )
               => ( aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),aa_nat_nat(suc,V2),TreeList),Vc)),Xaa))
                 => ~ ( ( Xaa = Mi )
                      | ( Xaa = Ma )
                      | $let(
                          pos: nat,
                          pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                          $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_membermima,nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) )
           => ~ ! [V2: nat,TreeList: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
                  ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList),Vd2) )
                 => ( aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList),Vd2)),Xaa))
                   => ~ $let(
                          pos: nat,
                          pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                          $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_membermima,nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(2)
tff(fact_4277_pos__mult__pos__ge,axiom,
    ! [Xb: int,Nb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Xb)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Nb)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,Nb),one_one_int)),aa_int_int(aa_int_fun_int_int(times_times_int,Nb),Xb)) ) ) ).

% pos_mult_pos_ge
tff(fact_4278_max__less__iff__conj,axiom,
    ! [Xb: real,Y: real,Z: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,Xb),Y)),Z)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Z)
        & aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Z) ) ) ).

% max_less_iff_conj
tff(fact_4279_max__less__iff__conj,axiom,
    ! [Xb: rat,Y: rat,Z: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,ord_max_rat(Xb,Y)),Z)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Z)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),Z) ) ) ).

% max_less_iff_conj
tff(fact_4280_max__less__iff__conj,axiom,
    ! [Xb: num,Y: num,Z: num] :
      ( aa_num_o(ord_less_num(aa_num_num(aa_num_fun_num_num(ord_max_num,Xb),Y)),Z)
    <=> ( aa_num_o(ord_less_num(Xb),Z)
        & aa_num_o(ord_less_num(Y),Z) ) ) ).

% max_less_iff_conj
tff(fact_4281_max__less__iff__conj,axiom,
    ! [Xb: nat,Y: nat,Z: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Xb),Y)),Z)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Z)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),Z) ) ) ).

% max_less_iff_conj
tff(fact_4282_max__less__iff__conj,axiom,
    ! [Xb: int,Y: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(ord_max_int,Xb),Y)),Z)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Z)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,Y),Z) ) ) ).

% max_less_iff_conj
tff(fact_4283_max_Oabsorb4,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,A),B) = B ) ) ).

% max.absorb4
tff(fact_4284_max_Oabsorb4,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( ord_max_rat(A,B) = B ) ) ).

% max.absorb4
tff(fact_4285_max_Oabsorb4,axiom,
    ! [A: num,B: num] :
      ( aa_num_o(ord_less_num(A),B)
     => ( aa_num_num(aa_num_fun_num_num(ord_max_num,A),B) = B ) ) ).

% max.absorb4
tff(fact_4286_max_Oabsorb4,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B) = B ) ) ).

% max.absorb4
tff(fact_4287_max_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(ord_max_int,A),B) = B ) ) ).

% max.absorb4
tff(fact_4288_max_Oabsorb1,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
     => ( ord_max_rat(A,B) = A ) ) ).

% max.absorb1
tff(fact_4289_max_Oabsorb1,axiom,
    ! [B: num,A: num] :
      ( aa_num_o(ord_less_eq_num(B),A)
     => ( aa_num_num(aa_num_fun_num_num(ord_max_num,A),B) = A ) ) ).

% max.absorb1
tff(fact_4290_max_Oabsorb1,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B) = A ) ) ).

% max.absorb1
tff(fact_4291_max_Oabsorb1,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
     => ( aa_int_int(aa_int_fun_int_int(ord_max_int,A),B) = A ) ) ).

% max.absorb1
tff(fact_4292_max_Oabsorb2,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( ord_max_rat(A,B) = B ) ) ).

% max.absorb2
tff(fact_4293_max_Oabsorb2,axiom,
    ! [A: num,B: num] :
      ( aa_num_o(ord_less_eq_num(A),B)
     => ( aa_num_num(aa_num_fun_num_num(ord_max_num,A),B) = B ) ) ).

% max.absorb2
tff(fact_4294_max_Oabsorb2,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B) = B ) ) ).

% max.absorb2
tff(fact_4295_max_Oabsorb2,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_int(aa_int_fun_int_int(ord_max_int,A),B) = B ) ) ).

% max.absorb2
tff(fact_4296_max_Obounded__iff,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,ord_max_rat(B,C)),A)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),A) ) ) ).

% max.bounded_iff
tff(fact_4297_max_Obounded__iff,axiom,
    ! [B: num,C: num,A: num] :
      ( aa_num_o(ord_less_eq_num(aa_num_num(aa_num_fun_num_num(ord_max_num,B),C)),A)
    <=> ( aa_num_o(ord_less_eq_num(B),A)
        & aa_num_o(ord_less_eq_num(C),A) ) ) ).

% max.bounded_iff
tff(fact_4298_max_Obounded__iff,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,B),C)),A)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),A) ) ) ).

% max.bounded_iff
tff(fact_4299_max_Obounded__iff,axiom,
    ! [B: int,C: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(ord_max_int,B),C)),A)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),A) ) ) ).

% max.bounded_iff
tff(fact_4300_max_Oabsorb3,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),A)
     => ( aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,A),B) = A ) ) ).

% max.absorb3
tff(fact_4301_max_Oabsorb3,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A)
     => ( ord_max_rat(A,B) = A ) ) ).

% max.absorb3
tff(fact_4302_max_Oabsorb3,axiom,
    ! [B: num,A: num] :
      ( aa_num_o(ord_less_num(B),A)
     => ( aa_num_num(aa_num_fun_num_num(ord_max_num,A),B) = A ) ) ).

% max.absorb3
tff(fact_4303_max_Oabsorb3,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B) = A ) ) ).

% max.absorb3
tff(fact_4304_max_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(ord_max_int,A),B) = A ) ) ).

% max.absorb3
tff(fact_4305_max_OcoboundedI2,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),ord_max_rat(A,B)) ) ).

% max.coboundedI2
tff(fact_4306_max_OcoboundedI2,axiom,
    ! [C: num,B: num,A: num] :
      ( aa_num_o(ord_less_eq_num(C),B)
     => aa_num_o(ord_less_eq_num(C),aa_num_num(aa_num_fun_num_num(ord_max_num,A),B)) ) ).

% max.coboundedI2
tff(fact_4307_max_OcoboundedI2,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),B)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B)) ) ).

% max.coboundedI2
tff(fact_4308_max_OcoboundedI2,axiom,
    ! [C: int,B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),aa_int_int(aa_int_fun_int_int(ord_max_int,A),B)) ) ).

% max.coboundedI2
tff(fact_4309_max_OcoboundedI1,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),A)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),ord_max_rat(A,B)) ) ).

% max.coboundedI1
tff(fact_4310_max_OcoboundedI1,axiom,
    ! [C: num,A: num,B: num] :
      ( aa_num_o(ord_less_eq_num(C),A)
     => aa_num_o(ord_less_eq_num(C),aa_num_num(aa_num_fun_num_num(ord_max_num,A),B)) ) ).

% max.coboundedI1
tff(fact_4311_max_OcoboundedI1,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),A)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B)) ) ).

% max.coboundedI1
tff(fact_4312_max_OcoboundedI1,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),aa_int_int(aa_int_fun_int_int(ord_max_int,A),B)) ) ).

% max.coboundedI1
tff(fact_4313_max_Oabsorb__iff2,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
    <=> ( ord_max_rat(A,B) = B ) ) ).

% max.absorb_iff2
tff(fact_4314_max_Oabsorb__iff2,axiom,
    ! [A: num,B: num] :
      ( aa_num_o(ord_less_eq_num(A),B)
    <=> ( aa_num_num(aa_num_fun_num_num(ord_max_num,A),B) = B ) ) ).

% max.absorb_iff2
tff(fact_4315_max_Oabsorb__iff2,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B) = B ) ) ).

% max.absorb_iff2
tff(fact_4316_max_Oabsorb__iff2,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
    <=> ( aa_int_int(aa_int_fun_int_int(ord_max_int,A),B) = B ) ) ).

% max.absorb_iff2
tff(fact_4317_max_Oabsorb__iff1,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
    <=> ( ord_max_rat(A,B) = A ) ) ).

% max.absorb_iff1
tff(fact_4318_max_Oabsorb__iff1,axiom,
    ! [B: num,A: num] :
      ( aa_num_o(ord_less_eq_num(B),A)
    <=> ( aa_num_num(aa_num_fun_num_num(ord_max_num,A),B) = A ) ) ).

% max.absorb_iff1
tff(fact_4319_max_Oabsorb__iff1,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B) = A ) ) ).

% max.absorb_iff1
tff(fact_4320_max_Oabsorb__iff1,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
    <=> ( aa_int_int(aa_int_fun_int_int(ord_max_int,A),B) = A ) ) ).

% max.absorb_iff1
tff(fact_4321_le__max__iff__disj,axiom,
    ! [Z: rat,Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Z),ord_max_rat(Xb,Y))
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Z),Xb)
        | aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Z),Y) ) ) ).

% le_max_iff_disj
tff(fact_4322_le__max__iff__disj,axiom,
    ! [Z: num,Xb: num,Y: num] :
      ( aa_num_o(ord_less_eq_num(Z),aa_num_num(aa_num_fun_num_num(ord_max_num,Xb),Y))
    <=> ( aa_num_o(ord_less_eq_num(Z),Xb)
        | aa_num_o(ord_less_eq_num(Z),Y) ) ) ).

% le_max_iff_disj
tff(fact_4323_le__max__iff__disj,axiom,
    ! [Z: nat,Xb: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Z),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Xb),Y))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Z),Xb)
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Z),Y) ) ) ).

% le_max_iff_disj
tff(fact_4324_le__max__iff__disj,axiom,
    ! [Z: int,Xb: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),aa_int_int(aa_int_fun_int_int(ord_max_int,Xb),Y))
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),Xb)
        | aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),Y) ) ) ).

% le_max_iff_disj
tff(fact_4325_max_Ocobounded2,axiom,
    ! [B: rat,A: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),ord_max_rat(A,B)) ).

% max.cobounded2
tff(fact_4326_max_Ocobounded2,axiom,
    ! [B: num,A: num] : aa_num_o(ord_less_eq_num(B),aa_num_num(aa_num_fun_num_num(ord_max_num,A),B)) ).

% max.cobounded2
tff(fact_4327_max_Ocobounded2,axiom,
    ! [B: nat,A: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B)) ).

% max.cobounded2
tff(fact_4328_max_Ocobounded2,axiom,
    ! [B: int,A: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),aa_int_int(aa_int_fun_int_int(ord_max_int,A),B)) ).

% max.cobounded2
tff(fact_4329_max_Ocobounded1,axiom,
    ! [A: rat,B: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),ord_max_rat(A,B)) ).

% max.cobounded1
tff(fact_4330_max_Ocobounded1,axiom,
    ! [A: num,B: num] : aa_num_o(ord_less_eq_num(A),aa_num_num(aa_num_fun_num_num(ord_max_num,A),B)) ).

% max.cobounded1
tff(fact_4331_max_Ocobounded1,axiom,
    ! [A: nat,B: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B)) ).

% max.cobounded1
tff(fact_4332_max_Ocobounded1,axiom,
    ! [A: int,B: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_int_int(aa_int_fun_int_int(ord_max_int,A),B)) ).

% max.cobounded1
tff(fact_4333_max_Oorder__iff,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
    <=> ( A = ord_max_rat(A,B) ) ) ).

% max.order_iff
tff(fact_4334_max_Oorder__iff,axiom,
    ! [B: num,A: num] :
      ( aa_num_o(ord_less_eq_num(B),A)
    <=> ( A = aa_num_num(aa_num_fun_num_num(ord_max_num,A),B) ) ) ).

% max.order_iff
tff(fact_4335_max_Oorder__iff,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
    <=> ( A = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B) ) ) ).

% max.order_iff
tff(fact_4336_max_Oorder__iff,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
    <=> ( A = aa_int_int(aa_int_fun_int_int(ord_max_int,A),B) ) ) ).

% max.order_iff
tff(fact_4337_max_OboundedI,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),A)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,ord_max_rat(B,C)),A) ) ) ).

% max.boundedI
tff(fact_4338_max_OboundedI,axiom,
    ! [B: num,A: num,C: num] :
      ( aa_num_o(ord_less_eq_num(B),A)
     => ( aa_num_o(ord_less_eq_num(C),A)
       => aa_num_o(ord_less_eq_num(aa_num_num(aa_num_fun_num_num(ord_max_num,B),C)),A) ) ) ).

% max.boundedI
tff(fact_4339_max_OboundedI,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),A)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,B),C)),A) ) ) ).

% max.boundedI
tff(fact_4340_max_OboundedI,axiom,
    ! [B: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),A)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(ord_max_int,B),C)),A) ) ) ).

% max.boundedI
tff(fact_4341_max_OboundedE,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,ord_max_rat(B,C)),A)
     => ~ ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
         => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),A) ) ) ).

% max.boundedE
tff(fact_4342_max_OboundedE,axiom,
    ! [B: num,C: num,A: num] :
      ( aa_num_o(ord_less_eq_num(aa_num_num(aa_num_fun_num_num(ord_max_num,B),C)),A)
     => ~ ( aa_num_o(ord_less_eq_num(B),A)
         => ~ aa_num_o(ord_less_eq_num(C),A) ) ) ).

% max.boundedE
tff(fact_4343_max_OboundedE,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,B),C)),A)
     => ~ ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
         => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),A) ) ) ).

% max.boundedE
tff(fact_4344_max_OboundedE,axiom,
    ! [B: int,C: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(ord_max_int,B),C)),A)
     => ~ ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
         => ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),A) ) ) ).

% max.boundedE
tff(fact_4345_max_OorderI,axiom,
    ! [A: rat,B: rat] :
      ( ( A = ord_max_rat(A,B) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A) ) ).

% max.orderI
tff(fact_4346_max_OorderI,axiom,
    ! [A: num,B: num] :
      ( ( A = aa_num_num(aa_num_fun_num_num(ord_max_num,A),B) )
     => aa_num_o(ord_less_eq_num(B),A) ) ).

% max.orderI
tff(fact_4347_max_OorderI,axiom,
    ! [A: nat,B: nat] :
      ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A) ) ).

% max.orderI
tff(fact_4348_max_OorderI,axiom,
    ! [A: int,B: int] :
      ( ( A = aa_int_int(aa_int_fun_int_int(ord_max_int,A),B) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A) ) ).

% max.orderI
tff(fact_4349_max_OorderE,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
     => ( A = ord_max_rat(A,B) ) ) ).

% max.orderE
tff(fact_4350_max_OorderE,axiom,
    ! [B: num,A: num] :
      ( aa_num_o(ord_less_eq_num(B),A)
     => ( A = aa_num_num(aa_num_fun_num_num(ord_max_num,A),B) ) ) ).

% max.orderE
tff(fact_4351_max_OorderE,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
     => ( A = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B) ) ) ).

% max.orderE
tff(fact_4352_max_OorderE,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
     => ( A = aa_int_int(aa_int_fun_int_int(ord_max_int,A),B) ) ) ).

% max.orderE
tff(fact_4353_max_Omono,axiom,
    ! [C: rat,A: rat,D: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,D),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,ord_max_rat(C,D)),ord_max_rat(A,B)) ) ) ).

% max.mono
tff(fact_4354_max_Omono,axiom,
    ! [C: num,A: num,D: num,B: num] :
      ( aa_num_o(ord_less_eq_num(C),A)
     => ( aa_num_o(ord_less_eq_num(D),B)
       => aa_num_o(ord_less_eq_num(aa_num_num(aa_num_fun_num_num(ord_max_num,C),D)),aa_num_num(aa_num_fun_num_num(ord_max_num,A),B)) ) ) ).

% max.mono
tff(fact_4355_max_Omono,axiom,
    ! [C: nat,A: nat,D: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,D),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,C),D)),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B)) ) ) ).

% max.mono
tff(fact_4356_max_Omono,axiom,
    ! [C: int,A: int,D: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,D),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(ord_max_int,C),D)),aa_int_int(aa_int_fun_int_int(ord_max_int,A),B)) ) ) ).

% max.mono
tff(fact_4357_less__max__iff__disj,axiom,
    ! [Z: real,Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z),aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,Xb),Y))
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z),Xb)
        | aa_real_o(aa_real_fun_real_o(ord_less_real,Z),Y) ) ) ).

% less_max_iff_disj
tff(fact_4358_less__max__iff__disj,axiom,
    ! [Z: rat,Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z),ord_max_rat(Xb,Y))
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z),Xb)
        | aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z),Y) ) ) ).

% less_max_iff_disj
tff(fact_4359_less__max__iff__disj,axiom,
    ! [Z: num,Xb: num,Y: num] :
      ( aa_num_o(ord_less_num(Z),aa_num_num(aa_num_fun_num_num(ord_max_num,Xb),Y))
    <=> ( aa_num_o(ord_less_num(Z),Xb)
        | aa_num_o(ord_less_num(Z),Y) ) ) ).

% less_max_iff_disj
tff(fact_4360_less__max__iff__disj,axiom,
    ! [Z: nat,Xb: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Xb),Y))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z),Xb)
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z),Y) ) ) ).

% less_max_iff_disj
tff(fact_4361_less__max__iff__disj,axiom,
    ! [Z: int,Xb: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z),aa_int_int(aa_int_fun_int_int(ord_max_int,Xb),Y))
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z),Xb)
        | aa_int_o(aa_int_fun_int_o(ord_less_int,Z),Y) ) ) ).

% less_max_iff_disj
tff(fact_4362_max_Ostrict__boundedE,axiom,
    ! [B: real,C: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,B),C)),A)
     => ~ ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),A)
         => ~ aa_real_o(aa_real_fun_real_o(ord_less_real,C),A) ) ) ).

% max.strict_boundedE
tff(fact_4363_max_Ostrict__boundedE,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,ord_max_rat(B,C)),A)
     => ~ ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A)
         => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),A) ) ) ).

% max.strict_boundedE
tff(fact_4364_max_Ostrict__boundedE,axiom,
    ! [B: num,C: num,A: num] :
      ( aa_num_o(ord_less_num(aa_num_num(aa_num_fun_num_num(ord_max_num,B),C)),A)
     => ~ ( aa_num_o(ord_less_num(B),A)
         => ~ aa_num_o(ord_less_num(C),A) ) ) ).

% max.strict_boundedE
tff(fact_4365_max_Ostrict__boundedE,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,B),C)),A)
     => ~ ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A)
         => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),A) ) ) ).

% max.strict_boundedE
tff(fact_4366_max_Ostrict__boundedE,axiom,
    ! [B: int,C: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(ord_max_int,B),C)),A)
     => ~ ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
         => ~ aa_int_o(aa_int_fun_int_o(ord_less_int,C),A) ) ) ).

% max.strict_boundedE
tff(fact_4367_max_Ostrict__order__iff,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),A)
    <=> ( ( A = aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,A),B) )
        & ( A != B ) ) ) ).

% max.strict_order_iff
tff(fact_4368_max_Ostrict__order__iff,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A)
    <=> ( ( A = ord_max_rat(A,B) )
        & ( A != B ) ) ) ).

% max.strict_order_iff
tff(fact_4369_max_Ostrict__order__iff,axiom,
    ! [B: num,A: num] :
      ( aa_num_o(ord_less_num(B),A)
    <=> ( ( A = aa_num_num(aa_num_fun_num_num(ord_max_num,A),B) )
        & ( A != B ) ) ) ).

% max.strict_order_iff
tff(fact_4370_max_Ostrict__order__iff,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A)
    <=> ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B) )
        & ( A != B ) ) ) ).

% max.strict_order_iff
tff(fact_4371_max_Ostrict__order__iff,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
    <=> ( ( A = aa_int_int(aa_int_fun_int_int(ord_max_int,A),B) )
        & ( A != B ) ) ) ).

% max.strict_order_iff
tff(fact_4372_max_Ostrict__coboundedI1,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),A)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,C),aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,A),B)) ) ).

% max.strict_coboundedI1
tff(fact_4373_max_Ostrict__coboundedI1,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),A)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),ord_max_rat(A,B)) ) ).

% max.strict_coboundedI1
tff(fact_4374_max_Ostrict__coboundedI1,axiom,
    ! [C: num,A: num,B: num] :
      ( aa_num_o(ord_less_num(C),A)
     => aa_num_o(ord_less_num(C),aa_num_num(aa_num_fun_num_num(ord_max_num,A),B)) ) ).

% max.strict_coboundedI1
tff(fact_4375_max_Ostrict__coboundedI1,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),A)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B)) ) ).

% max.strict_coboundedI1
tff(fact_4376_max_Ostrict__coboundedI1,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,C),aa_int_int(aa_int_fun_int_int(ord_max_int,A),B)) ) ).

% max.strict_coboundedI1
tff(fact_4377_max_Ostrict__coboundedI2,axiom,
    ! [C: real,B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),B)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,C),aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,A),B)) ) ).

% max.strict_coboundedI2
tff(fact_4378_max_Ostrict__coboundedI2,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),ord_max_rat(A,B)) ) ).

% max.strict_coboundedI2
tff(fact_4379_max_Ostrict__coboundedI2,axiom,
    ! [C: num,B: num,A: num] :
      ( aa_num_o(ord_less_num(C),B)
     => aa_num_o(ord_less_num(C),aa_num_num(aa_num_fun_num_num(ord_max_num,A),B)) ) ).

% max.strict_coboundedI2
tff(fact_4380_max_Ostrict__coboundedI2,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),B)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B)) ) ).

% max.strict_coboundedI2
tff(fact_4381_max_Ostrict__coboundedI2,axiom,
    ! [C: int,B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,C),aa_int_int(aa_int_fun_int_int(ord_max_int,A),B)) ) ).

% max.strict_coboundedI2
tff(fact_4382_p2__eq__1,axiom,
    ! [Nb: nat] :
      ( ( aa_nat5599461682457136004l_num1(power_2184487114949457152l_num1(numera7442385471795722001l_num1(aa_num_num(bit0,one))),Nb) = one_on7727431528512463931l_num1 )
    <=> ( Nb = zero_zero_nat ) ) ).

% p2_eq_1
tff(fact_4383_word__unat__power,axiom,
    ! [Nb: nat] : aa_nat5599461682457136004l_num1(power_2184487114949457152l_num1(numera7442385471795722001l_num1(aa_num_num(bit0,one))),Nb) = aa_nat5599461682457136004l_num1(semiri8819519690708144855l_num1,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ).

% word_unat_power
tff(fact_4384_word__less__two__pow__divD,axiom,
    ! [Xb: word_N3645301735248828278l_num1,Nb: nat,M: nat] :
      ( aa_wor4452366487969921764num1_o(ord_le750835935415966154l_num1(Xb),divide1791077408188789448l_num1(aa_nat5599461682457136004l_num1(power_2184487114949457152l_num1(numera7442385471795722001l_num1(aa_num_num(bit0,one))),Nb),aa_nat5599461682457136004l_num1(power_2184487114949457152l_num1(numera7442385471795722001l_num1(aa_num_num(bit0,one))),M)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
        & aa_wor4452366487969921764num1_o(ord_le750835935415966154l_num1(Xb),aa_nat5599461682457136004l_num1(power_2184487114949457152l_num1(numera7442385471795722001l_num1(aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),M))) ) ) ).

% word_less_two_pow_divD
tff(fact_4385_less__1__helper,axiom,
    ! [Nb: int,M: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Nb),M)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Nb),one_one_int)),M) ) ).

% less_1_helper
tff(fact_4386_enat__ord__number_I1_J,axiom,
    ! [M: num,Nb: num] :
      ( ord_le2932123472753598470d_enat(numera1916890842035813515d_enat(M),numera1916890842035813515d_enat(Nb))
    <=> 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,Nb)) ) ).

% enat_ord_number(1)
tff(fact_4387_heigt__uplog__rel,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Ta),Nb)
     => ( aa_nat_int(semiri1314217659103216013at_int,aa_VEBT_VEBT_nat(vEBT_VEBT_height,Ta)) = aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(log,numeral_numeral_real(aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,Nb))) ) ) ).

% heigt_uplog_rel
tff(fact_4388_enat__ord__number_I2_J,axiom,
    ! [M: num,Nb: num] :
      ( ord_le72135733267957522d_enat(numera1916890842035813515d_enat(M),numera1916890842035813515d_enat(Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,Nb)) ) ).

% enat_ord_number(2)
tff(fact_4389_set__bit__0,axiom,
    ! [A: word_N3645301735248828278l_num1] : bit_se4894374433684937756l_num1(zero_zero_nat,A) = plus_p361126936061061375l_num1(one_on7727431528512463931l_num1,aa_wor7274019993978516266l_num1(times_7065122842183080059l_num1(numera7442385471795722001l_num1(aa_num_num(bit0,one))),divide1791077408188789448l_num1(A,numera7442385471795722001l_num1(aa_num_num(bit0,one))))) ).

% set_bit_0
tff(fact_4390_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_4391_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_4392_assert_H__bind__rule,axiom,
    ! [P: assn,Phi: $o,C: heap_T2636463487746394924on_nat,Q: fun_option_nat_assn] :
      ( ! [H2: produc3658429121746597890et_nat] :
          ( aa_Pro653082453696054616_nat_o(rep_assn(P),H2)
         => (Phi) )
     => ( ( (Phi)
         => hoare_7629718768684598413on_nat(P,C,Q) )
       => hoare_7629718768684598413on_nat(P,heap_T5999496708990702694on_nat(aa_o_H2459438917461573107t_unit(refine_Imp_assert,(Phi)),aTP_Lamm_gx(C)),Q) ) ) ).

% assert'_bind_rule
tff(fact_4393_assert_H__bind__rule,axiom,
    ! [P: assn,Phi: $o,C: heap_Time_Heap_nat,Q: fun_nat_assn] :
      ( ! [H2: produc3658429121746597890et_nat] :
          ( aa_Pro653082453696054616_nat_o(rep_assn(P),H2)
         => (Phi) )
     => ( ( (Phi)
         => hoare_3067605981109127869le_nat(P,C,Q) )
       => hoare_3067605981109127869le_nat(P,heap_T3781436268274291734it_nat(aa_o_H2459438917461573107t_unit(refine_Imp_assert,(Phi)),aTP_Lamm_gw(C)),Q) ) ) ).

% assert'_bind_rule
tff(fact_4394_assert_H__bind__rule,axiom,
    ! [P: assn,Phi: $o,C: heap_Time_Heap_o,Q: fun_o_assn] :
      ( ! [H2: produc3658429121746597890et_nat] :
          ( aa_Pro653082453696054616_nat_o(rep_assn(P),H2)
         => (Phi) )
     => ( ( (Phi)
         => hoare_hoare_triple_o(P,C,Q) )
       => hoare_hoare_triple_o(P,heap_T7577940988442287570unit_o(aa_o_H2459438917461573107t_unit(refine_Imp_assert,(Phi)),aTP_Lamm_gv(C)),Q) ) ) ).

% assert'_bind_rule
tff(fact_4395_assert_H__bind__rule,axiom,
    ! [P: assn,Phi: $o,C: heap_T8145700208782473153_VEBTi,Q: fun_VEBT_VEBTi_assn] :
      ( ! [H2: produc3658429121746597890et_nat] :
          ( aa_Pro653082453696054616_nat_o(rep_assn(P),H2)
         => (Phi) )
     => ( ( (Phi)
         => hoare_1429296392585015714_VEBTi(P,C,Q) )
       => hoare_1429296392585015714_VEBTi(P,heap_T7982501707604696571_VEBTi(aa_o_H2459438917461573107t_unit(refine_Imp_assert,(Phi)),aTP_Lamm_gy(C)),Q) ) ) ).

% assert'_bind_rule
tff(fact_4396_assert_H__bind__rule,axiom,
    ! [P: assn,Phi: $o,C: heap_T5738788834812785303t_unit,Q: fun_Pr6626665806141890376t_assn] :
      ( ! [H2: produc3658429121746597890et_nat] :
          ( aa_Pro653082453696054616_nat_o(rep_assn(P),H2)
         => (Phi) )
     => ( ( (Phi)
         => hoare_8945653483474564448t_unit(P,C,Q) )
       => hoare_8945653483474564448t_unit(P,heap_T2633723481742716231t_unit(aa_o_H2459438917461573107t_unit(refine_Imp_assert,(Phi)),aTP_Lamm_hf(C)),Q) ) ) ).

% assert'_bind_rule
tff(fact_4397_succ__bound__size__univ,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,U: real,Xb: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Ta),Nb)
     => ( ( U = aa_nat_real(aa_real_fun_nat_real(power_power_real,numeral_numeral_real(aa_num_num(bit0,one))),Nb) )
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_s_u_c_c,Ta),Xb))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,numeral_numeral_real(aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,one))))))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,one)))))),aa_real_real(aa_rea7340605660300375871l_real(log,numeral_numeral_real(aa_num_num(bit0,one))),aa_real_real(aa_rea7340605660300375871l_real(log,numeral_numeral_real(aa_num_num(bit0,one))),U))))) ) ) ).

% succ_bound_size_univ
tff(fact_4398_div__of__0__id,axiom,
    ! [Nb: word_N3645301735248828278l_num1] : divide1791077408188789448l_num1(zero_z3563351764282998399l_num1,Nb) = zero_z3563351764282998399l_num1 ).

% div_of_0_id
tff(fact_4399_set__bit__nonnegative__int__iff,axiom,
    ! [Nb: 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,Nb),K))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K) ) ).

% set_bit_nonnegative_int_iff
tff(fact_4400_word__le__sub1__numberof,axiom,
    ! [W: num] :
      ( ( numera7442385471795722001l_num1(W) != zero_z3563351764282998399l_num1 )
     => ( aa_wor4452366487969921764num1_o(ord_le3335648743751981014l_num1(one_on7727431528512463931l_num1),numera7442385471795722001l_num1(W))
      <=> aa_wor4452366487969921764num1_o(ord_le3335648743751981014l_num1(zero_z3563351764282998399l_num1),minus_4019991460397169231l_num1(numera7442385471795722001l_num1(W),one_on7727431528512463931l_num1)) ) ) ).

% word_le_sub1_numberof
tff(fact_4401_word__gt__0__no,axiom,
    ! [Y: num] :
      ( aa_wor4452366487969921764num1_o(ord_le750835935415966154l_num1(zero_z3563351764282998399l_num1),numera7442385471795722001l_num1(Y))
    <=> ( zero_z3563351764282998399l_num1 != numera7442385471795722001l_num1(Y) ) ) ).

% word_gt_0_no
tff(fact_4402_word__less__1,axiom,
    ! [Xb: word_N3645301735248828278l_num1] :
      ( aa_wor4452366487969921764num1_o(ord_le750835935415966154l_num1(Xb),one_on7727431528512463931l_num1)
    <=> ( Xb = zero_z3563351764282998399l_num1 ) ) ).

% word_less_1
tff(fact_4403_ceiling__numeral,axiom,
    ! [V: num] : aa_real_int(archim7802044766580827645g_real,numeral_numeral_real(V)) = aa_num_int(numeral_numeral_int,V) ).

% ceiling_numeral
tff(fact_4404_ceiling__numeral,axiom,
    ! [V: num] : archim2889992004027027881ng_rat(numeral_numeral_rat(V)) = aa_num_int(numeral_numeral_int,V) ).

% ceiling_numeral
tff(fact_4405_ceiling__one,axiom,
    archim2889992004027027881ng_rat(one_one_rat) = one_one_int ).

% ceiling_one
tff(fact_4406_ceiling__one,axiom,
    aa_real_int(archim7802044766580827645g_real,one_one_real) = one_one_int ).

% ceiling_one
tff(fact_4407_word__less__sub1__numberof,axiom,
    ! [W: num] :
      ( ( numera7442385471795722001l_num1(W) != zero_z3563351764282998399l_num1 )
     => ( aa_wor4452366487969921764num1_o(ord_le750835935415966154l_num1(one_on7727431528512463931l_num1),numera7442385471795722001l_num1(W))
      <=> aa_wor4452366487969921764num1_o(ord_le750835935415966154l_num1(zero_z3563351764282998399l_num1),minus_4019991460397169231l_num1(numera7442385471795722001l_num1(W),one_on7727431528512463931l_num1)) ) ) ).

% word_less_sub1_numberof
tff(fact_4408_ceiling__le__zero,axiom,
    ! [Xb: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(archim7802044766580827645g_real,Xb)),zero_zero_int)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),zero_zero_real) ) ).

% ceiling_le_zero
tff(fact_4409_ceiling__le__zero,axiom,
    ! [Xb: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,archim2889992004027027881ng_rat(Xb)),zero_zero_int)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),zero_zero_rat) ) ).

% ceiling_le_zero
tff(fact_4410_ceiling__le__numeral,axiom,
    ! [Xb: real,V: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(archim7802044766580827645g_real,Xb)),aa_num_int(numeral_numeral_int,V))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),numeral_numeral_real(V)) ) ).

% ceiling_le_numeral
tff(fact_4411_ceiling__le__numeral,axiom,
    ! [Xb: rat,V: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,archim2889992004027027881ng_rat(Xb)),aa_num_int(numeral_numeral_int,V))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),numeral_numeral_rat(V)) ) ).

% ceiling_le_numeral
tff(fact_4412_zero__less__ceiling,axiom,
    ! [Xb: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),archim2889992004027027881ng_rat(Xb))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Xb) ) ).

% zero_less_ceiling
tff(fact_4413_zero__less__ceiling,axiom,
    ! [Xb: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_real_int(archim7802044766580827645g_real,Xb))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb) ) ).

% zero_less_ceiling
tff(fact_4414_numeral__less__ceiling,axiom,
    ! [V: num,Xb: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,V)),aa_real_int(archim7802044766580827645g_real,Xb))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,numeral_numeral_real(V)),Xb) ) ).

% numeral_less_ceiling
tff(fact_4415_numeral__less__ceiling,axiom,
    ! [V: num,Xb: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,V)),archim2889992004027027881ng_rat(Xb))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,numeral_numeral_rat(V)),Xb) ) ).

% numeral_less_ceiling
tff(fact_4416_ceiling__less__one,axiom,
    ! [Xb: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(archim7802044766580827645g_real,Xb)),one_one_int)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),zero_zero_real) ) ).

% ceiling_less_one
tff(fact_4417_ceiling__less__one,axiom,
    ! [Xb: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,archim2889992004027027881ng_rat(Xb)),one_one_int)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),zero_zero_rat) ) ).

% ceiling_less_one
tff(fact_4418_one__le__ceiling,axiom,
    ! [Xb: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),archim2889992004027027881ng_rat(Xb))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Xb) ) ).

% one_le_ceiling
tff(fact_4419_one__le__ceiling,axiom,
    ! [Xb: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_real_int(archim7802044766580827645g_real,Xb))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb) ) ).

% one_le_ceiling
tff(fact_4420_ceiling__add__numeral,axiom,
    ! [Xb: real,V: num] : aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),numeral_numeral_real(V))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(archim7802044766580827645g_real,Xb)),aa_num_int(numeral_numeral_int,V)) ).

% ceiling_add_numeral
tff(fact_4421_ceiling__add__numeral,axiom,
    ! [Xb: rat,V: num] : archim2889992004027027881ng_rat(aa_rat_rat(plus_plus_rat(Xb),numeral_numeral_rat(V))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,archim2889992004027027881ng_rat(Xb)),aa_num_int(numeral_numeral_int,V)) ).

% ceiling_add_numeral
tff(fact_4422_ceiling__le__one,axiom,
    ! [Xb: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(archim7802044766580827645g_real,Xb)),one_one_int)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),one_one_real) ) ).

% ceiling_le_one
tff(fact_4423_ceiling__le__one,axiom,
    ! [Xb: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,archim2889992004027027881ng_rat(Xb)),one_one_int)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),one_one_rat) ) ).

% ceiling_le_one
tff(fact_4424_one__less__ceiling,axiom,
    ! [Xb: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),archim2889992004027027881ng_rat(Xb))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),Xb) ) ).

% one_less_ceiling
tff(fact_4425_one__less__ceiling,axiom,
    ! [Xb: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_real_int(archim7802044766580827645g_real,Xb))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xb) ) ).

% one_less_ceiling
tff(fact_4426_ceiling__add__one,axiom,
    ! [Xb: rat] : archim2889992004027027881ng_rat(aa_rat_rat(plus_plus_rat(Xb),one_one_rat)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,archim2889992004027027881ng_rat(Xb)),one_one_int) ).

% ceiling_add_one
tff(fact_4427_ceiling__add__one,axiom,
    ! [Xb: real] : aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),one_one_real)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(archim7802044766580827645g_real,Xb)),one_one_int) ).

% ceiling_add_one
tff(fact_4428_ceiling__diff__numeral,axiom,
    ! [Xb: real,V: num] : aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),numeral_numeral_real(V))) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_real_int(archim7802044766580827645g_real,Xb)),aa_num_int(numeral_numeral_int,V)) ).

% ceiling_diff_numeral
tff(fact_4429_ceiling__diff__numeral,axiom,
    ! [Xb: rat,V: num] : archim2889992004027027881ng_rat(minus_minus_rat(Xb,numeral_numeral_rat(V))) = aa_int_int(aa_int_fun_int_int(minus_minus_int,archim2889992004027027881ng_rat(Xb)),aa_num_int(numeral_numeral_int,V)) ).

% ceiling_diff_numeral
tff(fact_4430_ceiling__numeral__power,axiom,
    ! [Xb: num,Nb: nat] : aa_real_int(archim7802044766580827645g_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,numeral_numeral_real(Xb)),Nb)) = aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,Xb)),Nb) ).

% ceiling_numeral_power
tff(fact_4431_ceiling__numeral__power,axiom,
    ! [Xb: num,Nb: nat] : archim2889992004027027881ng_rat(aa_nat_rat(power_power_rat(numeral_numeral_rat(Xb)),Nb)) = aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,Xb)),Nb) ).

% ceiling_numeral_power
tff(fact_4432_ceiling__diff__one,axiom,
    ! [Xb: rat] : archim2889992004027027881ng_rat(minus_minus_rat(Xb,one_one_rat)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,archim2889992004027027881ng_rat(Xb)),one_one_int) ).

% ceiling_diff_one
tff(fact_4433_ceiling__diff__one,axiom,
    ! [Xb: real] : aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),one_one_real)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_real_int(archim7802044766580827645g_real,Xb)),one_one_int) ).

% ceiling_diff_one
tff(fact_4434_ceiling__less__numeral,axiom,
    ! [Xb: real,V: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(archim7802044766580827645g_real,Xb)),aa_num_int(numeral_numeral_int,V))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,numeral_numeral_real(V)),one_one_real)) ) ).

% ceiling_less_numeral
tff(fact_4435_ceiling__less__numeral,axiom,
    ! [Xb: rat,V: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,archim2889992004027027881ng_rat(Xb)),aa_num_int(numeral_numeral_int,V))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),minus_minus_rat(numeral_numeral_rat(V),one_one_rat)) ) ).

% ceiling_less_numeral
tff(fact_4436_numeral__le__ceiling,axiom,
    ! [V: num,Xb: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,V)),aa_real_int(archim7802044766580827645g_real,Xb))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,numeral_numeral_real(V)),one_one_real)),Xb) ) ).

% numeral_le_ceiling
tff(fact_4437_numeral__le__ceiling,axiom,
    ! [V: num,Xb: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,V)),archim2889992004027027881ng_rat(Xb))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,minus_minus_rat(numeral_numeral_rat(V),one_one_rat)),Xb) ) ).

% numeral_le_ceiling
tff(fact_4438_word__subset__less,axiom,
    ! [Xb: word_N3645301735248828278l_num1,R2: word_N3645301735248828278l_num1,Y: word_N3645301735248828278l_num1,S3: word_N3645301735248828278l_num1] :
      ( ord_le5203802739334966412l_num1(set_or6221694504095523457l_num1(Xb,minus_4019991460397169231l_num1(plus_p361126936061061375l_num1(Xb,R2),one_on7727431528512463931l_num1)),set_or6221694504095523457l_num1(Y,minus_4019991460397169231l_num1(plus_p361126936061061375l_num1(Y,S3),one_on7727431528512463931l_num1)))
     => ( aa_wor4452366487969921764num1_o(ord_le3335648743751981014l_num1(Xb),minus_4019991460397169231l_num1(plus_p361126936061061375l_num1(Xb,R2),one_on7727431528512463931l_num1))
       => ( aa_wor4452366487969921764num1_o(ord_le3335648743751981014l_num1(Y),minus_4019991460397169231l_num1(plus_p361126936061061375l_num1(Y,S3),one_on7727431528512463931l_num1))
         => ( ( S3 != zero_z3563351764282998399l_num1 )
           => aa_wor4452366487969921764num1_o(ord_le3335648743751981014l_num1(R2),S3) ) ) ) ) ).

% word_subset_less
tff(fact_4439_gt0__iff__gem1,axiom,
    ! [Xb: word_N3645301735248828278l_num1] :
      ( aa_wor4452366487969921764num1_o(ord_le750835935415966154l_num1(zero_z3563351764282998399l_num1),Xb)
    <=> aa_wor4452366487969921764num1_o(ord_le750835935415966154l_num1(minus_4019991460397169231l_num1(Xb,one_on7727431528512463931l_num1)),Xb) ) ).

% gt0_iff_gem1
tff(fact_4440_word__less__sub1,axiom,
    ! [Xb: word_N3645301735248828278l_num1] :
      ( ( Xb != zero_z3563351764282998399l_num1 )
     => ( aa_wor4452366487969921764num1_o(ord_le750835935415966154l_num1(one_on7727431528512463931l_num1),Xb)
      <=> aa_wor4452366487969921764num1_o(ord_le750835935415966154l_num1(zero_z3563351764282998399l_num1),minus_4019991460397169231l_num1(Xb,one_on7727431528512463931l_num1)) ) ) ).

% word_less_sub1
tff(fact_4441_word__less__cases,axiom,
    ! [Xb: word_N3645301735248828278l_num1,Y: word_N3645301735248828278l_num1] :
      ( aa_wor4452366487969921764num1_o(ord_le750835935415966154l_num1(Xb),Y)
     => ( ( Xb = minus_4019991460397169231l_num1(Y,one_on7727431528512463931l_num1) )
        | aa_wor4452366487969921764num1_o(ord_le750835935415966154l_num1(Xb),minus_4019991460397169231l_num1(Y,one_on7727431528512463931l_num1)) ) ) ).

% word_less_cases
tff(fact_4442_word__div__sub,axiom,
    ! [Y: word_N3645301735248828278l_num1,Xb: word_N3645301735248828278l_num1] :
      ( aa_wor4452366487969921764num1_o(ord_le3335648743751981014l_num1(Y),Xb)
     => ( aa_wor4452366487969921764num1_o(ord_le750835935415966154l_num1(zero_z3563351764282998399l_num1),Y)
       => ( divide1791077408188789448l_num1(minus_4019991460397169231l_num1(Xb,Y),Y) = minus_4019991460397169231l_num1(divide1791077408188789448l_num1(Xb,Y),one_on7727431528512463931l_num1) ) ) ) ).

% word_div_sub
tff(fact_4443_word__leq__minus__one__le,axiom,
    ! [Y: word_N3645301735248828278l_num1,Xb: word_N3645301735248828278l_num1] :
      ( ( Y != zero_z3563351764282998399l_num1 )
     => ( aa_wor4452366487969921764num1_o(ord_le3335648743751981014l_num1(Xb),minus_4019991460397169231l_num1(Y,one_on7727431528512463931l_num1))
       => aa_wor4452366487969921764num1_o(ord_le750835935415966154l_num1(Xb),Y) ) ) ).

% word_leq_minus_one_le
tff(fact_4444_word__leq__le__minus__one,axiom,
    ! [Xb: word_N3645301735248828278l_num1,Y: word_N3645301735248828278l_num1] :
      ( aa_wor4452366487969921764num1_o(ord_le3335648743751981014l_num1(Xb),Y)
     => ( ( Xb != zero_z3563351764282998399l_num1 )
       => aa_wor4452366487969921764num1_o(ord_le750835935415966154l_num1(minus_4019991460397169231l_num1(Xb,one_on7727431528512463931l_num1)),Y) ) ) ).

% word_leq_le_minus_one
tff(fact_4445_le__m1__iff__lt,axiom,
    ! [Xb: word_N3645301735248828278l_num1,Y: word_N3645301735248828278l_num1] :
      ( aa_wor4452366487969921764num1_o(ord_le750835935415966154l_num1(zero_z3563351764282998399l_num1),Xb)
    <=> ( aa_wor4452366487969921764num1_o(ord_le3335648743751981014l_num1(Y),minus_4019991460397169231l_num1(Xb,one_on7727431528512463931l_num1))
      <=> aa_wor4452366487969921764num1_o(ord_le750835935415966154l_num1(Y),Xb) ) ) ).

% le_m1_iff_lt
tff(fact_4446_less__1__simp,axiom,
    ! [Nb: word_N3645301735248828278l_num1,M: word_N3645301735248828278l_num1] :
      ( aa_wor4452366487969921764num1_o(ord_le750835935415966154l_num1(minus_4019991460397169231l_num1(Nb,one_on7727431528512463931l_num1)),M)
    <=> ( aa_wor4452366487969921764num1_o(ord_le3335648743751981014l_num1(Nb),M)
        & ( Nb != zero_z3563351764282998399l_num1 ) ) ) ).

% less_1_simp
tff(fact_4447_word__le__minus__one__leq,axiom,
    ! [Xb: word_N3645301735248828278l_num1,Y: word_N3645301735248828278l_num1] :
      ( aa_wor4452366487969921764num1_o(ord_le750835935415966154l_num1(Xb),Y)
     => aa_wor4452366487969921764num1_o(ord_le3335648743751981014l_num1(Xb),minus_4019991460397169231l_num1(Y,one_on7727431528512463931l_num1)) ) ).

% word_le_minus_one_leq
tff(fact_4448_word__minus__one__le__leq,axiom,
    ! [Xb: word_N3645301735248828278l_num1,Y: word_N3645301735248828278l_num1] :
      ( aa_wor4452366487969921764num1_o(ord_le750835935415966154l_num1(minus_4019991460397169231l_num1(Xb,one_on7727431528512463931l_num1)),Y)
     => aa_wor4452366487969921764num1_o(ord_le3335648743751981014l_num1(Xb),Y) ) ).

% word_minus_one_le_leq
tff(fact_4449_le__step__down__word,axiom,
    ! [I: word_N3645301735248828278l_num1,Nb: word_N3645301735248828278l_num1] :
      ( aa_wor4452366487969921764num1_o(ord_le3335648743751981014l_num1(I),Nb)
     => ( ( I != Nb )
       => aa_wor4452366487969921764num1_o(ord_le3335648743751981014l_num1(I),minus_4019991460397169231l_num1(Nb,one_on7727431528512463931l_num1)) ) ) ).

% le_step_down_word
tff(fact_4450_le__step__down__word__2,axiom,
    ! [Xb: word_N3645301735248828278l_num1,Y: word_N3645301735248828278l_num1] :
      ( aa_wor4452366487969921764num1_o(ord_le3335648743751981014l_num1(Xb),Y)
     => ( ( Xb != Y )
       => aa_wor4452366487969921764num1_o(ord_le3335648743751981014l_num1(Xb),minus_4019991460397169231l_num1(Y,one_on7727431528512463931l_num1)) ) ) ).

% le_step_down_word_2
tff(fact_4451_add__diff__assoc__enat,axiom,
    ! [Z: extended_enat,Y: extended_enat,Xb: extended_enat] :
      ( ord_le2932123472753598470d_enat(Z,Y)
     => ( plus_p3455044024723400733d_enat(Xb,minus_3235023915231533773d_enat(Y,Z)) = minus_3235023915231533773d_enat(plus_p3455044024723400733d_enat(Xb,Y),Z) ) ) ).

% add_diff_assoc_enat
tff(fact_4452_word__le__sub1,axiom,
    ! [Xb: word_N3645301735248828278l_num1] :
      ( ( Xb != zero_z3563351764282998399l_num1 )
     => ( aa_wor4452366487969921764num1_o(ord_le3335648743751981014l_num1(one_on7727431528512463931l_num1),Xb)
      <=> aa_wor4452366487969921764num1_o(ord_le3335648743751981014l_num1(zero_z3563351764282998399l_num1),minus_4019991460397169231l_num1(Xb,one_on7727431528512463931l_num1)) ) ) ).

% word_le_sub1
tff(fact_4453_word__sub__1__le,axiom,
    ! [Xb: word_N3645301735248828278l_num1] :
      ( ( Xb != zero_z3563351764282998399l_num1 )
     => aa_wor4452366487969921764num1_o(ord_le3335648743751981014l_num1(minus_4019991460397169231l_num1(Xb,one_on7727431528512463931l_num1)),Xb) ) ).

% word_sub_1_le
tff(fact_4454_word__must__wrap,axiom,
    ! [Xb: word_N3645301735248828278l_num1,Nb: word_N3645301735248828278l_num1] :
      ( aa_wor4452366487969921764num1_o(ord_le3335648743751981014l_num1(Xb),minus_4019991460397169231l_num1(Nb,one_on7727431528512463931l_num1))
     => ( aa_wor4452366487969921764num1_o(ord_le3335648743751981014l_num1(Nb),Xb)
       => ( Nb = zero_z3563351764282998399l_num1 ) ) ) ).

% word_must_wrap
tff(fact_4455_word__sub__plus__one__nonzero,axiom,
    ! [N3: word_N3645301735248828278l_num1,Nb: word_N3645301735248828278l_num1] :
      ( aa_wor4452366487969921764num1_o(ord_le3335648743751981014l_num1(N3),Nb)
     => ( ( N3 != zero_z3563351764282998399l_num1 )
       => ( plus_p361126936061061375l_num1(minus_4019991460397169231l_num1(Nb,N3),one_on7727431528512463931l_num1) != zero_z3563351764282998399l_num1 ) ) ) ).

% word_sub_plus_one_nonzero
tff(fact_4456_word__div__less,axiom,
    ! [W: word_N3645301735248828278l_num1,V: word_N3645301735248828278l_num1] :
      ( aa_wor4452366487969921764num1_o(ord_le750835935415966154l_num1(W),V)
     => ( divide1791077408188789448l_num1(W,V) = zero_z3563351764282998399l_num1 ) ) ).

% word_div_less
tff(fact_4457_More__Word_Oword__div__mult,axiom,
    ! [C: word_N3645301735248828278l_num1,A: word_N3645301735248828278l_num1,B: word_N3645301735248828278l_num1] :
      ( aa_wor4452366487969921764num1_o(ord_le750835935415966154l_num1(zero_z3563351764282998399l_num1),C)
     => ( aa_wor4452366487969921764num1_o(ord_le750835935415966154l_num1(A),aa_wor7274019993978516266l_num1(times_7065122842183080059l_num1(B),C))
       => aa_wor4452366487969921764num1_o(ord_le750835935415966154l_num1(divide1791077408188789448l_num1(A,C)),B) ) ) ).

% More_Word.word_div_mult
tff(fact_4458_word__less__div,axiom,
    ! [Xb: word_N3645301735248828278l_num1,Y: word_N3645301735248828278l_num1] :
      ( ( divide1791077408188789448l_num1(Xb,Y) = zero_z3563351764282998399l_num1 )
     => ( ( Y = zero_z3563351764282998399l_num1 )
        | aa_wor4452366487969921764num1_o(ord_le750835935415966154l_num1(Xb),Y) ) ) ).

% word_less_div
tff(fact_4459_word__div__lt__eq__0,axiom,
    ! [Xb: word_N3645301735248828278l_num1,Y: word_N3645301735248828278l_num1] :
      ( aa_wor4452366487969921764num1_o(ord_le750835935415966154l_num1(Xb),Y)
     => ( divide1791077408188789448l_num1(Xb,Y) = zero_z3563351764282998399l_num1 ) ) ).

% word_div_lt_eq_0
tff(fact_4460_div__less__dividend__word,axiom,
    ! [Xb: word_N3645301735248828278l_num1,Nb: word_N3645301735248828278l_num1] :
      ( ( Xb != zero_z3563351764282998399l_num1 )
     => ( ( Nb != one_on7727431528512463931l_num1 )
       => aa_wor4452366487969921764num1_o(ord_le750835935415966154l_num1(divide1791077408188789448l_num1(Xb,Nb)),Xb) ) ) ).

% div_less_dividend_word
tff(fact_4461_word__overflow,axiom,
    ! [Xb: word_N3645301735248828278l_num1] :
      ( aa_wor4452366487969921764num1_o(ord_le750835935415966154l_num1(Xb),plus_p361126936061061375l_num1(Xb,one_on7727431528512463931l_num1))
      | ( plus_p361126936061061375l_num1(Xb,one_on7727431528512463931l_num1) = zero_z3563351764282998399l_num1 ) ) ).

% word_overflow
tff(fact_4462_word__gr0__conv__Suc,axiom,
    ! [M: word_N3645301735248828278l_num1] :
      ( aa_wor4452366487969921764num1_o(ord_le750835935415966154l_num1(zero_z3563351764282998399l_num1),M)
     => ? [N2: word_N3645301735248828278l_num1] : M = plus_p361126936061061375l_num1(N2,one_on7727431528512463931l_num1) ) ).

% word_gr0_conv_Suc
tff(fact_4463_less__is__non__zero__p1,axiom,
    ! [A: word_N3645301735248828278l_num1,K: word_N3645301735248828278l_num1] :
      ( aa_wor4452366487969921764num1_o(ord_le750835935415966154l_num1(A),K)
     => ( plus_p361126936061061375l_num1(A,one_on7727431528512463931l_num1) != zero_z3563351764282998399l_num1 ) ) ).

% less_is_non_zero_p1
tff(fact_4464_word__div__1,axiom,
    ! [Nb: word_N3645301735248828278l_num1] : divide1791077408188789448l_num1(Nb,one_on7727431528512463931l_num1) = Nb ).

% word_div_1
tff(fact_4465_div__by__0__word,axiom,
    ! [Xb: word_N3645301735248828278l_num1] : divide1791077408188789448l_num1(Xb,zero_z3563351764282998399l_num1) = zero_z3563351764282998399l_num1 ).

% div_by_0_word
tff(fact_4466_word__plus__one__nonzero,axiom,
    ! [Xb: word_N3645301735248828278l_num1,Y: word_N3645301735248828278l_num1] :
      ( aa_wor4452366487969921764num1_o(ord_le3335648743751981014l_num1(Xb),plus_p361126936061061375l_num1(Xb,Y))
     => ( ( Y != zero_z3563351764282998399l_num1 )
       => ( plus_p361126936061061375l_num1(Xb,one_on7727431528512463931l_num1) != zero_z3563351764282998399l_num1 ) ) ) ).

% word_plus_one_nonzero
tff(fact_4467_lt1__neq0,axiom,
    ! [Xb: word_N3645301735248828278l_num1] :
      ( aa_wor4452366487969921764num1_o(ord_le3335648743751981014l_num1(one_on7727431528512463931l_num1),Xb)
    <=> ( Xb != zero_z3563351764282998399l_num1 ) ) ).

% lt1_neq0
tff(fact_4468_div__to__mult__word__lt,axiom,
    ! [Xb: word_N3645301735248828278l_num1,Y: word_N3645301735248828278l_num1,Z: word_N3645301735248828278l_num1] :
      ( aa_wor4452366487969921764num1_o(ord_le3335648743751981014l_num1(Xb),divide1791077408188789448l_num1(Y,Z))
     => aa_wor4452366487969921764num1_o(ord_le3335648743751981014l_num1(aa_wor7274019993978516266l_num1(times_7065122842183080059l_num1(Xb),Z)),Y) ) ).

% div_to_mult_word_lt
tff(fact_4469_word__le__plus__1,axiom,
    ! [Y: word_N3645301735248828278l_num1,Nb: word_N3645301735248828278l_num1,A: word_N3645301735248828278l_num1] :
      ( aa_wor4452366487969921764num1_o(ord_le750835935415966154l_num1(Y),plus_p361126936061061375l_num1(Y,Nb))
     => ( aa_wor4452366487969921764num1_o(ord_le750835935415966154l_num1(A),Nb)
       => aa_wor4452366487969921764num1_o(ord_le3335648743751981014l_num1(plus_p361126936061061375l_num1(Y,A)),plus_p361126936061061375l_num1(plus_p361126936061061375l_num1(Y,A),one_on7727431528512463931l_num1)) ) ) ).

% word_le_plus_1
tff(fact_4470_plus__one__helper,axiom,
    ! [Xb: word_N3645301735248828278l_num1,Nb: word_N3645301735248828278l_num1] :
      ( aa_wor4452366487969921764num1_o(ord_le750835935415966154l_num1(Xb),plus_p361126936061061375l_num1(Nb,one_on7727431528512463931l_num1))
     => aa_wor4452366487969921764num1_o(ord_le3335648743751981014l_num1(Xb),Nb) ) ).

% plus_one_helper
tff(fact_4471_plus__one__helper2,axiom,
    ! [Xb: word_N3645301735248828278l_num1,Nb: word_N3645301735248828278l_num1] :
      ( aa_wor4452366487969921764num1_o(ord_le3335648743751981014l_num1(Xb),Nb)
     => ( ( plus_p361126936061061375l_num1(Nb,one_on7727431528512463931l_num1) != zero_z3563351764282998399l_num1 )
       => aa_wor4452366487969921764num1_o(ord_le750835935415966154l_num1(Xb),plus_p361126936061061375l_num1(Nb,one_on7727431528512463931l_num1)) ) ) ).

% plus_one_helper2
tff(fact_4472_word__1__0,axiom,
    ! [A: word_N3645301735248828278l_num1,B: word_N3645301735248828278l_num1,Xb: nat] :
      ( aa_wor4452366487969921764num1_o(ord_le3335648743751981014l_num1(plus_p361126936061061375l_num1(A,one_on7727431528512463931l_num1)),B)
     => ( aa_wor4452366487969921764num1_o(ord_le750835935415966154l_num1(A),aa_nat5599461682457136004l_num1(semiri8819519690708144855l_num1,Xb))
       => aa_wor4452366487969921764num1_o(ord_le750835935415966154l_num1(A),B) ) ) ).

% word_1_0
tff(fact_4473_div__le__mult,axiom,
    ! [I: word_N3645301735248828278l_num1,K: word_N3645301735248828278l_num1,Xb: word_N3645301735248828278l_num1] :
      ( aa_wor4452366487969921764num1_o(ord_le3335648743751981014l_num1(I),divide1791077408188789448l_num1(K,Xb))
     => ( aa_wor4452366487969921764num1_o(ord_le750835935415966154l_num1(zero_z3563351764282998399l_num1),Xb)
       => aa_wor4452366487969921764num1_o(ord_le3335648743751981014l_num1(aa_wor7274019993978516266l_num1(times_7065122842183080059l_num1(I),Xb)),K) ) ) ).

% div_le_mult
tff(fact_4474_inc__le,axiom,
    ! [I: word_N3645301735248828278l_num1,M: word_N3645301735248828278l_num1] :
      ( aa_wor4452366487969921764num1_o(ord_le750835935415966154l_num1(I),M)
     => aa_wor4452366487969921764num1_o(ord_le3335648743751981014l_num1(plus_p361126936061061375l_num1(I,one_on7727431528512463931l_num1)),M) ) ).

% inc_le
tff(fact_4475_inc__i,axiom,
    ! [I: word_N3645301735248828278l_num1,M: word_N3645301735248828278l_num1] :
      ( aa_wor4452366487969921764num1_o(ord_le3335648743751981014l_num1(one_on7727431528512463931l_num1),I)
     => ( aa_wor4452366487969921764num1_o(ord_le750835935415966154l_num1(I),M)
       => ( aa_wor4452366487969921764num1_o(ord_le3335648743751981014l_num1(one_on7727431528512463931l_num1),plus_p361126936061061375l_num1(I,one_on7727431528512463931l_num1))
          & aa_wor4452366487969921764num1_o(ord_le3335648743751981014l_num1(plus_p361126936061061375l_num1(I,one_on7727431528512463931l_num1)),M) ) ) ) ).

% inc_i
tff(fact_4476_word__div__mult__le,axiom,
    ! [A: word_N3645301735248828278l_num1,B: word_N3645301735248828278l_num1] : aa_wor4452366487969921764num1_o(ord_le3335648743751981014l_num1(aa_wor7274019993978516266l_num1(times_7065122842183080059l_num1(divide1791077408188789448l_num1(A,B)),B)),A) ).

% word_div_mult_le
tff(fact_4477_i0__lb,axiom,
    ! [Nb: extended_enat] : ord_le2932123472753598470d_enat(zero_z5237406670263579293d_enat,Nb) ).

% i0_lb
tff(fact_4478_ile0__eq,axiom,
    ! [Nb: extended_enat] :
      ( ord_le2932123472753598470d_enat(Nb,zero_z5237406670263579293d_enat)
    <=> ( Nb = zero_z5237406670263579293d_enat ) ) ).

% ile0_eq
tff(fact_4479_div__word__self,axiom,
    ! [W: word_N3645301735248828278l_num1] :
      ( ( W != zero_z3563351764282998399l_num1 )
     => ( divide1791077408188789448l_num1(W,W) = one_on7727431528512463931l_num1 ) ) ).

% div_word_self
tff(fact_4480_word__induct,axiom,
    ! [P: fun_wo7859709073469921915num1_o,M: word_N3645301735248828278l_num1] :
      ( aa_wor4452366487969921764num1_o(P,zero_z3563351764282998399l_num1)
     => ( ! [N2: word_N3645301735248828278l_num1] :
            ( aa_wor4452366487969921764num1_o(P,N2)
           => aa_wor4452366487969921764num1_o(P,plus_p361126936061061375l_num1(one_on7727431528512463931l_num1,N2)) )
       => aa_wor4452366487969921764num1_o(P,M) ) ) ).

% word_induct
tff(fact_4481_word__induct2,axiom,
    ! [P: fun_wo7859709073469921915num1_o,Nb: word_N3645301735248828278l_num1] :
      ( aa_wor4452366487969921764num1_o(P,zero_z3563351764282998399l_num1)
     => ( ! [N2: word_N3645301735248828278l_num1] :
            ( ( plus_p361126936061061375l_num1(one_on7727431528512463931l_num1,N2) != zero_z3563351764282998399l_num1 )
           => ( aa_wor4452366487969921764num1_o(P,N2)
             => aa_wor4452366487969921764num1_o(P,plus_p361126936061061375l_num1(one_on7727431528512463931l_num1,N2)) ) )
       => aa_wor4452366487969921764num1_o(P,Nb) ) ) ).

% word_induct2
tff(fact_4482_enat__0__less__mult__iff,axiom,
    ! [M: extended_enat,Nb: extended_enat] :
      ( ord_le72135733267957522d_enat(zero_z5237406670263579293d_enat,times_7803423173614009249d_enat(M,Nb))
    <=> ( ord_le72135733267957522d_enat(zero_z5237406670263579293d_enat,M)
        & ord_le72135733267957522d_enat(zero_z5237406670263579293d_enat,Nb) ) ) ).

% enat_0_less_mult_iff
tff(fact_4483_word__induct__less,axiom,
    ! [P: fun_wo7859709073469921915num1_o,M: word_N3645301735248828278l_num1] :
      ( aa_wor4452366487969921764num1_o(P,zero_z3563351764282998399l_num1)
     => ( ! [N2: word_N3645301735248828278l_num1] :
            ( aa_wor4452366487969921764num1_o(ord_le750835935415966154l_num1(N2),M)
           => ( aa_wor4452366487969921764num1_o(P,N2)
             => aa_wor4452366487969921764num1_o(P,plus_p361126936061061375l_num1(one_on7727431528512463931l_num1,N2)) ) )
       => aa_wor4452366487969921764num1_o(P,M) ) ) ).

% word_induct_less
tff(fact_4484_div__lt__mult,axiom,
    ! [I: word_N3645301735248828278l_num1,K: word_N3645301735248828278l_num1,Xb: word_N3645301735248828278l_num1] :
      ( aa_wor4452366487969921764num1_o(ord_le750835935415966154l_num1(I),divide1791077408188789448l_num1(K,Xb))
     => ( aa_wor4452366487969921764num1_o(ord_le750835935415966154l_num1(zero_z3563351764282998399l_num1),Xb)
       => aa_wor4452366487969921764num1_o(ord_le750835935415966154l_num1(aa_wor7274019993978516266l_num1(times_7065122842183080059l_num1(I),Xb)),K) ) ) ).

% div_lt_mult
tff(fact_4485_set__bit__greater__eq,axiom,
    ! [K: int,Nb: 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,Nb),K)) ).

% set_bit_greater_eq
tff(fact_4486_ceiling__mono,axiom,
    ! [Y: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),Xb)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(archim7802044766580827645g_real,Y)),aa_real_int(archim7802044766580827645g_real,Xb)) ) ).

% ceiling_mono
tff(fact_4487_ceiling__mono,axiom,
    ! [Y: rat,Xb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),Xb)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,archim2889992004027027881ng_rat(Y)),archim2889992004027027881ng_rat(Xb)) ) ).

% ceiling_mono
tff(fact_4488_ceiling__less__cancel,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,archim2889992004027027881ng_rat(Xb)),archim2889992004027027881ng_rat(Y))
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Y) ) ).

% ceiling_less_cancel
tff(fact_4489_ceiling__less__cancel,axiom,
    ! [Xb: real,Y: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(archim7802044766580827645g_real,Xb)),aa_real_int(archim7802044766580827645g_real,Y))
     => aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y) ) ).

% ceiling_less_cancel
tff(fact_4490_ceiling__add__le,axiom,
    ! [Xb: rat,Y: rat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,archim2889992004027027881ng_rat(aa_rat_rat(plus_plus_rat(Xb),Y))),aa_int_int(aa_int_fun_int_int(plus_plus_int,archim2889992004027027881ng_rat(Xb)),archim2889992004027027881ng_rat(Y))) ).

% ceiling_add_le
tff(fact_4491_ceiling__add__le,axiom,
    ! [Xb: real,Y: real] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Y))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(archim7802044766580827645g_real,Xb)),aa_real_int(archim7802044766580827645g_real,Y))) ).

% ceiling_add_le
tff(fact_4492_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Osimps_I2_J,axiom,
    ! [Uv: $o,Uw: $o,Nb: nat] : aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_s_u_c_c,vEBT_Leaf((Uv),(Uw))),aa_nat_nat(suc,Nb)) = one_one_nat ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.simps(2)
tff(fact_4493_mult__ceiling__le,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_real_int(archim7802044766580827645g_real,A)),aa_real_int(archim7802044766580827645g_real,B))) ) ) ).

% mult_ceiling_le
tff(fact_4494_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,archim2889992004027027881ng_rat(aa_rat_rat(times_times_rat(A),B))),aa_int_int(aa_int_fun_int_int(times_times_int,archim2889992004027027881ng_rat(A)),archim2889992004027027881ng_rat(B))) ) ) ).

% mult_ceiling_le
tff(fact_4495_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Osimps_I3_J,axiom,
    ! [Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT,Vaa: nat] : aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_s_u_c_c,aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Ux,Uy),Uz)),Vaa) = one_one_nat ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.simps(3)
tff(fact_4496_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Osimps_I1_J,axiom,
    ! [Uu: $o,B: $o] : aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_s_u_c_c,vEBT_Leaf((Uu),(B))),zero_zero_nat) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.simps(1)
tff(fact_4497_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Osimps_I4_J,axiom,
    ! [V: product_prod_nat_nat,Vc2: list_VEBT_VEBT,Vd: vEBT_VEBT,Ve: nat] : aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_s_u_c_c,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V),zero_zero_nat,Vc2),Vd)),Ve) = one_one_nat ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.simps(4)
tff(fact_4498_Abs__fnat__hom__1,axiom,
    one_on7727431528512463931l_num1 = aa_nat5599461682457136004l_num1(semiri8819519690708144855l_num1,aa_nat_nat(suc,zero_zero_nat)) ).

% Abs_fnat_hom_1
tff(fact_4499_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Osimps_I5_J,axiom,
    ! [V: product_prod_nat_nat,Vg: list_VEBT_VEBT,Vh: vEBT_VEBT,Vi: nat] : aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_s_u_c_c,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V),aa_nat_nat(suc,zero_zero_nat),Vg),Vh)),Vi) = one_one_nat ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.simps(5)
tff(fact_4500_assert_H__rule,axiom,
    ! [P: assn,Phi: $o] :
      ( ! [H2: produc3658429121746597890et_nat] :
          ( aa_Pro653082453696054616_nat_o(rep_assn(P),H2)
         => (Phi) )
     => hoare_8945653483474564448t_unit(P,aa_o_H2459438917461573107t_unit(refine_Imp_assert,(Phi)),aTP_Lamm_hg(P)) ) ).

% assert'_rule
tff(fact_4501_Abs__fnat__hom__add,axiom,
    ! [A: nat,B: nat] : aa_rat_rat(plus_plus_rat(aa_nat_rat(semiri681578069525770553at_rat,A)),aa_nat_rat(semiri681578069525770553at_rat,B)) = aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)) ).

% Abs_fnat_hom_add
tff(fact_4502_Abs__fnat__hom__add,axiom,
    ! [A: nat,B: nat] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(semiri5074537144036343181t_real,A)),aa_nat_real(semiri5074537144036343181t_real,B)) = aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)) ).

% Abs_fnat_hom_add
tff(fact_4503_Abs__fnat__hom__add,axiom,
    ! [A: nat,B: nat] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,A)),aa_nat_int(semiri1314217659103216013at_int,B)) = aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)) ).

% Abs_fnat_hom_add
tff(fact_4504_Abs__fnat__hom__add,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(semiri1316708129612266289at_nat,A)),aa_nat_nat(semiri1316708129612266289at_nat,B)) = aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)) ).

% Abs_fnat_hom_add
tff(fact_4505_Abs__fnat__hom__add,axiom,
    ! [A: nat,B: nat] : aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,A)),aa_nat_Code_integer(semiri4939895301339042750nteger,B)) = aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)) ).

% Abs_fnat_hom_add
tff(fact_4506_ceiling__log__nat__eq__if,axiom,
    ! [B: nat,Nb: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Nb)),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,Nb),one_one_nat)))
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B)
         => ( aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(log,aa_nat_real(semiri5074537144036343181t_real,B)),aa_nat_real(semiri5074537144036343181t_real,K))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),one_one_int) ) ) ) ) ).

% ceiling_log_nat_eq_if
tff(fact_4507_ceiling__log2__div2,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(log,numeral_numeral_real(aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,Nb))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(log,numeral_numeral_real(aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_nat))))),one_one_int) ) ) ).

% ceiling_log2_div2
tff(fact_4508_ceiling__log__nat__eq__powr__iff,axiom,
    ! [B: nat,K: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
       => ( ( aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(log,aa_nat_real(semiri5074537144036343181t_real,B)),aa_nat_real(semiri5074537144036343181t_real,K))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),one_one_int) )
        <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Nb)),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,Nb),one_one_nat))) ) ) ) ) ).

% ceiling_log_nat_eq_powr_iff
tff(fact_4509_of__nat__gt__0,axiom,
    ! [K: nat] :
      ( ( aa_nat_uint32(semiri2565882477558803405uint32,K) != zero_zero_uint32 )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K) ) ).

% of_nat_gt_0
tff(fact_4510_of__nat__gt__0,axiom,
    ! [K: nat] :
      ( ( aa_nat_rat(semiri681578069525770553at_rat,K) != zero_zero_rat )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K) ) ).

% of_nat_gt_0
tff(fact_4511_of__nat__gt__0,axiom,
    ! [K: nat] :
      ( ( aa_nat_real(semiri5074537144036343181t_real,K) != zero_zero_real )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K) ) ).

% of_nat_gt_0
tff(fact_4512_of__nat__gt__0,axiom,
    ! [K: nat] :
      ( ( aa_nat_int(semiri1314217659103216013at_int,K) != zero_zero_int )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K) ) ).

% of_nat_gt_0
tff(fact_4513_of__nat__gt__0,axiom,
    ! [K: nat] :
      ( ( aa_nat_nat(semiri1316708129612266289at_nat,K) != zero_zero_nat )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K) ) ).

% of_nat_gt_0
tff(fact_4514_of__nat__gt__0,axiom,
    ! [K: nat] :
      ( ( aa_nat_Code_integer(semiri4939895301339042750nteger,K) != zero_z3403309356797280102nteger )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K) ) ).

% of_nat_gt_0
tff(fact_4515_succ__bound__height,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xb: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Ta),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_s_u_c_c,Ta),Xb)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_VEBT_height,Ta))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,one))))))) ) ).

% succ_bound_height
tff(fact_4516_htt__vebt__succi,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Tib: vEBT_VEBTi,Xb: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Ta),Nb)
     => aa_nat_o(time_htt_option_nat(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Ta),Tib),aa_nat947733127081326238on_nat(aa_VEB3566417204837707778on_nat(vEBT_vebt_succi,Tib),Xb),aa_nat435388790147128531t_assn(aa_VEB1697702717351837573t_assn(aTP_Lamm_be(Ta),Tib),Xb)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit1,one)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit1,one)))),aa_int_nat(nat2,aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(log,numeral_numeral_real(aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,Nb))))))) ) ).

% htt_vebt_succi
tff(fact_4517_htt__vebt__memberi__invar__vebt,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Tib: vEBT_VEBTi,Xb: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Ta),Nb)
     => aa_nat_o(time_htt_o(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Ta),Tib),aa_nat2910048432523430448Heap_o(aa_VEB1455453220190609250Heap_o(vEBT_vebt_memberi,Tib),Xb),aa_nat_fun_o_assn(aa_VEB8176250900481067561o_assn(aTP_Lamm_bo(Ta),Tib),Xb)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit0,one)))),aa_int_nat(nat2,aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(log,numeral_numeral_real(aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,Nb))))))) ) ).

% htt_vebt_memberi_invar_vebt
tff(fact_4518_htt__vebt__inserti__invar__vebt,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Tib: vEBT_VEBTi,Xb: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Ta),Nb)
     => aa_nat_o(time_htt_VEBT_VEBTi(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Ta),Tib),aa_nat5408703056993926195_VEBTi(aa_VEB4833742232818206615_VEBTi(vEBT_vebt_inserti,Tib),Xb),aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,vEBT_vebt_insert(Ta,Xb))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,one))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,one))))),aa_int_nat(nat2,aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(log,numeral_numeral_real(aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,Nb))))))) ) ).

% htt_vebt_inserti_invar_vebt
tff(fact_4519_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Oelims,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_s_u_c_c,Xb),Xaa) = Y )
     => ( ( ? [Uu2: $o,B2: $o] : Xb = vEBT_Leaf((Uu2),(B2))
         => ( ( Xaa = zero_zero_nat )
           => ( Y != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat) ) ) )
       => ( ( ? [Uv2: $o,Uw2: $o] : Xb = vEBT_Leaf((Uv2),(Uw2))
           => ( ? [N2: nat] : Xaa = aa_nat_nat(suc,N2)
             => ( Y != one_one_nat ) ) )
         => ( ( ? [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] : Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Ux2,Uy2),Uz2)
             => ( Y != one_one_nat ) )
           => ( ( ? [V2: product_prod_nat_nat,Vc: list_VEBT_VEBT,Vd2: vEBT_VEBT] : Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Vc),Vd2)
               => ( Y != one_one_nat ) )
             => ( ( ? [V2: product_prod_nat_nat,Vg2: list_VEBT_VEBT,Vh2: vEBT_VEBT] : Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vg2),Vh2)
                 => ( Y != one_one_nat ) )
               => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                      ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList),Summary) )
                     => ( Y != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
                            $ite(
                              aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi),
                              one_one_nat,
                              $let(
                                l: nat,
                                l:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                $let(
                                  h: nat,
                                  h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                  aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,one))))),
                                    $ite(
                                      aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList)),
                                      aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_T_m_a_x_t,nth_VEBT_VEBT(TreeList,h)))),
                                        $let(
                                          maxlow: option_nat,
                                          maxlow:= vEBT_vebt_maxt(nth_VEBT_VEBT(TreeList,h)),
                                          aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),
                                            $ite(
                                              ( ( maxlow != none_nat )
                                              & aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_less,aa_nat_option_nat(some_nat,l)),maxlow) ),
                                              aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))),aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_s_u_c_c,nth_VEBT_VEBT(TreeList,h)),l)),
                                              $let(
                                                sc: option_nat,
                                                sc:= aa_nat_option_nat(vEBT_vebt_succ(Summary),h),
                                                aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_s_u_c_c,Summary),h))),one_one_nat)),
                                                  $ite(sc = none_nat,one_one_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))),aa_VEBT_VEBT_nat(vEBT_T_m_i_n_t,nth_VEBT_VEBT(TreeList,the_nat(sc)))))) ) )) )),
                                      one_one_nat )) ) ) )) ) ) ) ) ) ) ) ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.elims
tff(fact_4520_log__ceil__idem,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Xb)
     => ( aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(log,numeral_numeral_real(aa_num_num(bit0,one))),Xb)) = aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(log,numeral_numeral_real(aa_num_num(bit0,one))),aa_int_real(ring_1_of_int_real,aa_real_int(archim7802044766580827645g_real,Xb)))) ) ) ).

% log_ceil_idem
tff(fact_4521_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Osimps_I6_J,axiom,
    ! [Mia: nat,Maa: nat,Vaa: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Xb: nat] :
      aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_s_u_c_c,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mia),Maa)),aa_nat_nat(suc,aa_nat_nat(suc,Vaa)),TreeLista),Summarya)),Xb) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
        $ite(
          aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Mia),
          one_one_nat,
          $let(
            l: nat,
            l:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Vaa))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
            $let(
              h: nat,
              h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Vaa))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
              aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,one))))),
                $ite(
                  aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeLista)),
                  aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_T_m_a_x_t,nth_VEBT_VEBT(TreeLista,h)))),
                    $let(
                      maxlow: option_nat,
                      maxlow:= vEBT_vebt_maxt(nth_VEBT_VEBT(TreeLista,h)),
                      aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),
                        $ite(
                          ( ( maxlow != none_nat )
                          & aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_less,aa_nat_option_nat(some_nat,l)),maxlow) ),
                          aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))),aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_s_u_c_c,nth_VEBT_VEBT(TreeLista,h)),l)),
                          $let(
                            sc: option_nat,
                            sc:= aa_nat_option_nat(vEBT_vebt_succ(Summarya),h),
                            aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_s_u_c_c,Summarya),h))),one_one_nat)),
                              $ite(sc = none_nat,one_one_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))),aa_VEBT_VEBT_nat(vEBT_T_m_i_n_t,nth_VEBT_VEBT(TreeLista,the_nat(sc)))))) ) )) )),
                  one_one_nat )) ) ) )) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.simps(6)
tff(fact_4522_insert__bound__size__univ,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,U: real,Xb: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Ta),Nb)
     => ( ( U = aa_nat_real(aa_real_fun_nat_real(power_power_real,numeral_numeral_real(aa_num_num(bit0,one))),Nb) )
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_i_n_s_e_r_t,Ta),Xb))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,numeral_numeral_real(aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,one))))))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,one)))))),aa_real_real(aa_rea7340605660300375871l_real(log,numeral_numeral_real(aa_num_num(bit0,one))),aa_real_real(aa_rea7340605660300375871l_real(log,numeral_numeral_real(aa_num_num(bit0,one))),U))))) ) ) ).

% insert_bound_size_univ
tff(fact_4523_idiff__0,axiom,
    ! [Nb: extended_enat] : minus_3235023915231533773d_enat(zero_z5237406670263579293d_enat,Nb) = zero_z5237406670263579293d_enat ).

% idiff_0
tff(fact_4524_idiff__0__right,axiom,
    ! [Nb: extended_enat] : minus_3235023915231533773d_enat(Nb,zero_z5237406670263579293d_enat) = Nb ).

% idiff_0_right
tff(fact_4525_of__int__eq__numeral__iff,axiom,
    ! [Z: int,Nb: num] :
      ( ( aa_int_real(ring_1_of_int_real,Z) = numeral_numeral_real(Nb) )
    <=> ( Z = aa_num_int(numeral_numeral_int,Nb) ) ) ).

% of_int_eq_numeral_iff
tff(fact_4526_of__int__eq__numeral__iff,axiom,
    ! [Z: int,Nb: num] :
      ( ( aa_int_rat(ring_1_of_int_rat,Z) = numeral_numeral_rat(Nb) )
    <=> ( Z = aa_num_int(numeral_numeral_int,Nb) ) ) ).

% of_int_eq_numeral_iff
tff(fact_4527_of__int__eq__numeral__iff,axiom,
    ! [Z: int,Nb: num] :
      ( ( aa_int_int(ring_1_of_int_int,Z) = aa_num_int(numeral_numeral_int,Nb) )
    <=> ( Z = aa_num_int(numeral_numeral_int,Nb) ) ) ).

% of_int_eq_numeral_iff
tff(fact_4528_of__int__numeral,axiom,
    ! [K: num] : ring_17408606157368542149l_num1(aa_num_int(numeral_numeral_int,K)) = numera7442385471795722001l_num1(K) ).

% of_int_numeral
tff(fact_4529_of__int__numeral,axiom,
    ! [K: num] : aa_int_real(ring_1_of_int_real,aa_num_int(numeral_numeral_int,K)) = numeral_numeral_real(K) ).

% of_int_numeral
tff(fact_4530_of__int__numeral,axiom,
    ! [K: num] : aa_int_rat(ring_1_of_int_rat,aa_num_int(numeral_numeral_int,K)) = numeral_numeral_rat(K) ).

% of_int_numeral
tff(fact_4531_of__int__numeral,axiom,
    ! [K: num] : aa_int_int(ring_1_of_int_int,aa_num_int(numeral_numeral_int,K)) = aa_num_int(numeral_numeral_int,K) ).

% of_int_numeral
tff(fact_4532_of__int__le__iff,axiom,
    ! [W: int,Z: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,W)),aa_int_real(ring_1_of_int_real,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,W),Z) ) ).

% of_int_le_iff
tff(fact_4533_of__int__le__iff,axiom,
    ! [W: int,Z: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,W)),aa_int_rat(ring_1_of_int_rat,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,W),Z) ) ).

% of_int_le_iff
tff(fact_4534_of__int__le__iff,axiom,
    ! [W: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(ring_1_of_int_int,W)),aa_int_int(ring_1_of_int_int,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,W),Z) ) ).

% of_int_le_iff
tff(fact_4535_of__int__less__iff,axiom,
    ! [W: int,Z: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(ring_1_of_int_real,W)),aa_int_real(ring_1_of_int_real,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,W),Z) ) ).

% of_int_less_iff
tff(fact_4536_of__int__less__iff,axiom,
    ! [W: int,Z: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(ring_1_of_int_rat,W)),aa_int_rat(ring_1_of_int_rat,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,W),Z) ) ).

% of_int_less_iff
tff(fact_4537_of__int__less__iff,axiom,
    ! [W: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(ring_1_of_int_int,W)),aa_int_int(ring_1_of_int_int,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,W),Z) ) ).

% of_int_less_iff
tff(fact_4538_of__int__eq__1__iff,axiom,
    ! [Z: int] :
      ( ( aa_int_int(ring_1_of_int_int,Z) = one_one_int )
    <=> ( Z = one_one_int ) ) ).

% of_int_eq_1_iff
tff(fact_4539_of__int__eq__1__iff,axiom,
    ! [Z: int] :
      ( ( aa_int_real(ring_1_of_int_real,Z) = one_one_real )
    <=> ( Z = one_one_int ) ) ).

% of_int_eq_1_iff
tff(fact_4540_of__int__eq__1__iff,axiom,
    ! [Z: int] :
      ( ( aa_int_rat(ring_1_of_int_rat,Z) = one_one_rat )
    <=> ( Z = one_one_int ) ) ).

% of_int_eq_1_iff
tff(fact_4541_of__int__1,axiom,
    ring_17408606157368542149l_num1(one_one_int) = one_on7727431528512463931l_num1 ).

% of_int_1
tff(fact_4542_of__int__1,axiom,
    aa_int_int(ring_1_of_int_int,one_one_int) = one_one_int ).

% of_int_1
tff(fact_4543_of__int__1,axiom,
    aa_int_real(ring_1_of_int_real,one_one_int) = one_one_real ).

% of_int_1
tff(fact_4544_of__int__1,axiom,
    aa_int_rat(ring_1_of_int_rat,one_one_int) = one_one_rat ).

% of_int_1
tff(fact_4545_of__int__add,axiom,
    ! [W: int,Z: int] : aa_int_int(ring_1_of_int_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,W),Z)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(ring_1_of_int_int,W)),aa_int_int(ring_1_of_int_int,Z)) ).

% of_int_add
tff(fact_4546_of__int__add,axiom,
    ! [W: int,Z: int] : aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(plus_plus_int,W),Z)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,W)),aa_int_real(ring_1_of_int_real,Z)) ).

% of_int_add
tff(fact_4547_of__int__add,axiom,
    ! [W: int,Z: int] : aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(plus_plus_int,W),Z)) = aa_rat_rat(plus_plus_rat(aa_int_rat(ring_1_of_int_rat,W)),aa_int_rat(ring_1_of_int_rat,Z)) ).

% of_int_add
tff(fact_4548_of__int__mult,axiom,
    ! [W: int,Z: int] : aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(times_times_int,W),Z)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(ring_1_of_int_real,W)),aa_int_real(ring_1_of_int_real,Z)) ).

% of_int_mult
tff(fact_4549_of__int__mult,axiom,
    ! [W: int,Z: int] : aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(times_times_int,W),Z)) = aa_rat_rat(times_times_rat(aa_int_rat(ring_1_of_int_rat,W)),aa_int_rat(ring_1_of_int_rat,Z)) ).

% of_int_mult
tff(fact_4550_of__int__mult,axiom,
    ! [W: int,Z: int] : aa_int_int(ring_1_of_int_int,aa_int_int(aa_int_fun_int_int(times_times_int,W),Z)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(ring_1_of_int_int,W)),aa_int_int(ring_1_of_int_int,Z)) ).

% of_int_mult
tff(fact_4551_nat__numeral,axiom,
    ! [K: num] : aa_int_nat(nat2,aa_num_int(numeral_numeral_int,K)) = aa_num_nat(numeral_numeral_nat,K) ).

% nat_numeral
tff(fact_4552_of__int__diff,axiom,
    ! [W: int,Z: int] : aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(minus_minus_int,W),Z)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_int_real(ring_1_of_int_real,W)),aa_int_real(ring_1_of_int_real,Z)) ).

% of_int_diff
tff(fact_4553_of__int__diff,axiom,
    ! [W: int,Z: int] : aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(minus_minus_int,W),Z)) = minus_minus_rat(aa_int_rat(ring_1_of_int_rat,W),aa_int_rat(ring_1_of_int_rat,Z)) ).

% of_int_diff
tff(fact_4554_of__int__diff,axiom,
    ! [W: int,Z: int] : aa_int_int(ring_1_of_int_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,W),Z)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(ring_1_of_int_int,W)),aa_int_int(ring_1_of_int_int,Z)) ).

% of_int_diff
tff(fact_4555_of__int__power__eq__of__int__cancel__iff,axiom,
    ! [Xb: int,B: int,W: nat] :
      ( ( aa_int_rat(ring_1_of_int_rat,Xb) = aa_nat_rat(power_power_rat(aa_int_rat(ring_1_of_int_rat,B)),W) )
    <=> ( Xb = aa_nat_int(aa_int_fun_nat_int(power_power_int,B),W) ) ) ).

% of_int_power_eq_of_int_cancel_iff
tff(fact_4556_of__int__power__eq__of__int__cancel__iff,axiom,
    ! [Xb: int,B: int,W: nat] :
      ( ( aa_int_real(ring_1_of_int_real,Xb) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_int_real(ring_1_of_int_real,B)),W) )
    <=> ( Xb = aa_nat_int(aa_int_fun_nat_int(power_power_int,B),W) ) ) ).

% of_int_power_eq_of_int_cancel_iff
tff(fact_4557_of__int__power__eq__of__int__cancel__iff,axiom,
    ! [Xb: int,B: int,W: nat] :
      ( ( aa_int_int(ring_1_of_int_int,Xb) = aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(ring_1_of_int_int,B)),W) )
    <=> ( Xb = aa_nat_int(aa_int_fun_nat_int(power_power_int,B),W) ) ) ).

% of_int_power_eq_of_int_cancel_iff
tff(fact_4558_of__int__eq__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,Xb: int] :
      ( ( aa_nat_rat(power_power_rat(aa_int_rat(ring_1_of_int_rat,B)),W) = aa_int_rat(ring_1_of_int_rat,Xb) )
    <=> ( aa_nat_int(aa_int_fun_nat_int(power_power_int,B),W) = Xb ) ) ).

% of_int_eq_of_int_power_cancel_iff
tff(fact_4559_of__int__eq__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,Xb: int] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_int_real(ring_1_of_int_real,B)),W) = aa_int_real(ring_1_of_int_real,Xb) )
    <=> ( aa_nat_int(aa_int_fun_nat_int(power_power_int,B),W) = Xb ) ) ).

% of_int_eq_of_int_power_cancel_iff
tff(fact_4560_of__int__eq__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,Xb: int] :
      ( ( aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(ring_1_of_int_int,B)),W) = aa_int_int(ring_1_of_int_int,Xb) )
    <=> ( aa_nat_int(aa_int_fun_nat_int(power_power_int,B),W) = Xb ) ) ).

% of_int_eq_of_int_power_cancel_iff
tff(fact_4561_of__int__power,axiom,
    ! [Z: int,Nb: nat] : aa_int_rat(ring_1_of_int_rat,aa_nat_int(aa_int_fun_nat_int(power_power_int,Z),Nb)) = aa_nat_rat(power_power_rat(aa_int_rat(ring_1_of_int_rat,Z)),Nb) ).

% of_int_power
tff(fact_4562_of__int__power,axiom,
    ! [Z: int,Nb: nat] : aa_int_uint32(ring_1_of_int_uint32,aa_nat_int(aa_int_fun_nat_int(power_power_int,Z),Nb)) = aa_nat_uint32(power_power_uint32(aa_int_uint32(ring_1_of_int_uint32,Z)),Nb) ).

% of_int_power
tff(fact_4563_of__int__power,axiom,
    ! [Z: int,Nb: nat] : ring_17408606157368542149l_num1(aa_nat_int(aa_int_fun_nat_int(power_power_int,Z),Nb)) = aa_nat5599461682457136004l_num1(power_2184487114949457152l_num1(ring_17408606157368542149l_num1(Z)),Nb) ).

% of_int_power
tff(fact_4564_of__int__power,axiom,
    ! [Z: int,Nb: nat] : aa_int_real(ring_1_of_int_real,aa_nat_int(aa_int_fun_nat_int(power_power_int,Z),Nb)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_int_real(ring_1_of_int_real,Z)),Nb) ).

% of_int_power
tff(fact_4565_of__int__power,axiom,
    ! [Z: int,Nb: nat] : aa_int_int(ring_1_of_int_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,Z),Nb)) = aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(ring_1_of_int_int,Z)),Nb) ).

% of_int_power
tff(fact_4566_nat__1,axiom,
    aa_int_nat(nat2,one_one_int) = aa_nat_nat(suc,zero_zero_nat) ).

% nat_1
tff(fact_4567_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_4568_nat__0__iff,axiom,
    ! [I: int] :
      ( ( aa_int_nat(nat2,I) = zero_zero_nat )
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),zero_zero_int) ) ).

% nat_0_iff
tff(fact_4569_zless__nat__conj,axiom,
    ! [W: int,Z: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_int_nat(nat2,W)),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,W),Z) ) ) ).

% zless_nat_conj
tff(fact_4570_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_4571_ceiling__add__of__int,axiom,
    ! [Xb: rat,Z: int] : archim2889992004027027881ng_rat(aa_rat_rat(plus_plus_rat(Xb),aa_int_rat(ring_1_of_int_rat,Z))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,archim2889992004027027881ng_rat(Xb)),Z) ).

% ceiling_add_of_int
tff(fact_4572_ceiling__add__of__int,axiom,
    ! [Xb: real,Z: int] : aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),aa_int_real(ring_1_of_int_real,Z))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(archim7802044766580827645g_real,Xb)),Z) ).

% ceiling_add_of_int
tff(fact_4573_ceiling__diff__of__int,axiom,
    ! [Xb: rat,Z: int] : archim2889992004027027881ng_rat(minus_minus_rat(Xb,aa_int_rat(ring_1_of_int_rat,Z))) = aa_int_int(aa_int_fun_int_int(minus_minus_int,archim2889992004027027881ng_rat(Xb)),Z) ).

% ceiling_diff_of_int
tff(fact_4574_ceiling__diff__of__int,axiom,
    ! [Xb: real,Z: int] : aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),aa_int_real(ring_1_of_int_real,Z))) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_real_int(archim7802044766580827645g_real,Xb)),Z) ).

% ceiling_diff_of_int
tff(fact_4575_of__int__le__0__iff,axiom,
    ! [Z: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Z)),zero_zero_real)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),zero_zero_int) ) ).

% of_int_le_0_iff
tff(fact_4576_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_4577_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_4578_of__int__0__le__iff,axiom,
    ! [Z: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_int_real(ring_1_of_int_real,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z) ) ).

% of_int_0_le_iff
tff(fact_4579_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_4580_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_4581_of__int__numeral__le__iff,axiom,
    ! [Nb: num,Z: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,numeral_numeral_real(Nb)),aa_int_real(ring_1_of_int_real,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,Nb)),Z) ) ).

% of_int_numeral_le_iff
tff(fact_4582_of__int__numeral__le__iff,axiom,
    ! [Nb: num,Z: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,numeral_numeral_rat(Nb)),aa_int_rat(ring_1_of_int_rat,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,Nb)),Z) ) ).

% of_int_numeral_le_iff
tff(fact_4583_of__int__numeral__le__iff,axiom,
    ! [Nb: num,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,Nb)),aa_int_int(ring_1_of_int_int,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,Nb)),Z) ) ).

% of_int_numeral_le_iff
tff(fact_4584_of__int__le__numeral__iff,axiom,
    ! [Z: int,Nb: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Z)),numeral_numeral_real(Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),aa_num_int(numeral_numeral_int,Nb)) ) ).

% of_int_le_numeral_iff
tff(fact_4585_of__int__le__numeral__iff,axiom,
    ! [Z: int,Nb: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Z)),numeral_numeral_rat(Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),aa_num_int(numeral_numeral_int,Nb)) ) ).

% of_int_le_numeral_iff
tff(fact_4586_of__int__le__numeral__iff,axiom,
    ! [Z: int,Nb: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(ring_1_of_int_int,Z)),aa_num_int(numeral_numeral_int,Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),aa_num_int(numeral_numeral_int,Nb)) ) ).

% of_int_le_numeral_iff
tff(fact_4587_of__int__0__less__iff,axiom,
    ! [Z: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_int_real(ring_1_of_int_real,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Z) ) ).

% of_int_0_less_iff
tff(fact_4588_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_4589_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_4590_of__int__less__0__iff,axiom,
    ! [Z: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(ring_1_of_int_real,Z)),zero_zero_real)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Z),zero_zero_int) ) ).

% of_int_less_0_iff
tff(fact_4591_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_4592_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_4593_of__int__numeral__less__iff,axiom,
    ! [Nb: num,Z: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,numeral_numeral_real(Nb)),aa_int_real(ring_1_of_int_real,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,Nb)),Z) ) ).

% of_int_numeral_less_iff
tff(fact_4594_of__int__numeral__less__iff,axiom,
    ! [Nb: num,Z: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,numeral_numeral_rat(Nb)),aa_int_rat(ring_1_of_int_rat,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,Nb)),Z) ) ).

% of_int_numeral_less_iff
tff(fact_4595_of__int__numeral__less__iff,axiom,
    ! [Nb: num,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,Nb)),aa_int_int(ring_1_of_int_int,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,Nb)),Z) ) ).

% of_int_numeral_less_iff
tff(fact_4596_of__int__less__numeral__iff,axiom,
    ! [Z: int,Nb: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(ring_1_of_int_real,Z)),numeral_numeral_real(Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Z),aa_num_int(numeral_numeral_int,Nb)) ) ).

% of_int_less_numeral_iff
tff(fact_4597_of__int__less__numeral__iff,axiom,
    ! [Z: int,Nb: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(ring_1_of_int_rat,Z)),numeral_numeral_rat(Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Z),aa_num_int(numeral_numeral_int,Nb)) ) ).

% of_int_less_numeral_iff
tff(fact_4598_of__int__less__numeral__iff,axiom,
    ! [Z: int,Nb: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(ring_1_of_int_int,Z)),aa_num_int(numeral_numeral_int,Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Z),aa_num_int(numeral_numeral_int,Nb)) ) ).

% of_int_less_numeral_iff
tff(fact_4599_of__int__le__1__iff,axiom,
    ! [Z: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Z)),one_one_real)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),one_one_int) ) ).

% of_int_le_1_iff
tff(fact_4600_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_4601_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_4602_of__int__1__le__iff,axiom,
    ! [Z: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_int_real(ring_1_of_int_real,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),Z) ) ).

% of_int_1_le_iff
tff(fact_4603_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_4604_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_4605_of__int__less__1__iff,axiom,
    ! [Z: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(ring_1_of_int_real,Z)),one_one_real)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Z),one_one_int) ) ).

% of_int_less_1_iff
tff(fact_4606_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_4607_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_4608_of__int__1__less__iff,axiom,
    ! [Z: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_int_real(ring_1_of_int_real,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),Z) ) ).

% of_int_1_less_iff
tff(fact_4609_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_4610_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_4611_of__int__eq__numeral__power__cancel__iff,axiom,
    ! [Y: int,Xb: num,Nb: nat] :
      ( ( aa_int_real(ring_1_of_int_real,Y) = aa_nat_real(aa_real_fun_nat_real(power_power_real,numeral_numeral_real(Xb)),Nb) )
    <=> ( Y = aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,Xb)),Nb) ) ) ).

% of_int_eq_numeral_power_cancel_iff
tff(fact_4612_of__int__eq__numeral__power__cancel__iff,axiom,
    ! [Y: int,Xb: num,Nb: nat] :
      ( ( aa_int_rat(ring_1_of_int_rat,Y) = aa_nat_rat(power_power_rat(numeral_numeral_rat(Xb)),Nb) )
    <=> ( Y = aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,Xb)),Nb) ) ) ).

% of_int_eq_numeral_power_cancel_iff
tff(fact_4613_of__int__eq__numeral__power__cancel__iff,axiom,
    ! [Y: int,Xb: num,Nb: nat] :
      ( ( aa_int_int(ring_1_of_int_int,Y) = aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,Xb)),Nb) )
    <=> ( Y = aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,Xb)),Nb) ) ) ).

% of_int_eq_numeral_power_cancel_iff
tff(fact_4614_numeral__power__eq__of__int__cancel__iff,axiom,
    ! [Xb: num,Nb: nat,Y: int] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,numeral_numeral_real(Xb)),Nb) = aa_int_real(ring_1_of_int_real,Y) )
    <=> ( aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,Xb)),Nb) = Y ) ) ).

% numeral_power_eq_of_int_cancel_iff
tff(fact_4615_numeral__power__eq__of__int__cancel__iff,axiom,
    ! [Xb: num,Nb: nat,Y: int] :
      ( ( aa_nat_rat(power_power_rat(numeral_numeral_rat(Xb)),Nb) = aa_int_rat(ring_1_of_int_rat,Y) )
    <=> ( aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,Xb)),Nb) = Y ) ) ).

% numeral_power_eq_of_int_cancel_iff
tff(fact_4616_numeral__power__eq__of__int__cancel__iff,axiom,
    ! [Xb: num,Nb: nat,Y: int] :
      ( ( aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,Xb)),Nb) = aa_int_int(ring_1_of_int_int,Y) )
    <=> ( aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,Xb)),Nb) = Y ) ) ).

% numeral_power_eq_of_int_cancel_iff
tff(fact_4617_of__int__power__le__of__int__cancel__iff,axiom,
    ! [Xb: int,B: int,W: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Xb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_int_real(ring_1_of_int_real,B)),W))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),aa_nat_int(aa_int_fun_nat_int(power_power_int,B),W)) ) ).

% of_int_power_le_of_int_cancel_iff
tff(fact_4618_of__int__power__le__of__int__cancel__iff,axiom,
    ! [Xb: int,B: int,W: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Xb)),aa_nat_rat(power_power_rat(aa_int_rat(ring_1_of_int_rat,B)),W))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),aa_nat_int(aa_int_fun_nat_int(power_power_int,B),W)) ) ).

% of_int_power_le_of_int_cancel_iff
tff(fact_4619_of__int__power__le__of__int__cancel__iff,axiom,
    ! [Xb: int,B: int,W: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(ring_1_of_int_int,Xb)),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(ring_1_of_int_int,B)),W))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),aa_nat_int(aa_int_fun_nat_int(power_power_int,B),W)) ) ).

% of_int_power_le_of_int_cancel_iff
tff(fact_4620_of__int__le__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,Xb: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_int_real(ring_1_of_int_real,B)),W)),aa_int_real(ring_1_of_int_real,Xb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,B),W)),Xb) ) ).

% of_int_le_of_int_power_cancel_iff
tff(fact_4621_of__int__le__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,Xb: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(aa_int_rat(ring_1_of_int_rat,B)),W)),aa_int_rat(ring_1_of_int_rat,Xb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,B),W)),Xb) ) ).

% of_int_le_of_int_power_cancel_iff
tff(fact_4622_of__int__le__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,Xb: 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)),W)),aa_int_int(ring_1_of_int_int,Xb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,B),W)),Xb) ) ).

% of_int_le_of_int_power_cancel_iff
tff(fact_4623_of__int__power__less__of__int__cancel__iff,axiom,
    ! [Xb: int,B: int,W: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(ring_1_of_int_real,Xb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_int_real(ring_1_of_int_real,B)),W))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),aa_nat_int(aa_int_fun_nat_int(power_power_int,B),W)) ) ).

% of_int_power_less_of_int_cancel_iff
tff(fact_4624_of__int__power__less__of__int__cancel__iff,axiom,
    ! [Xb: int,B: int,W: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(ring_1_of_int_rat,Xb)),aa_nat_rat(power_power_rat(aa_int_rat(ring_1_of_int_rat,B)),W))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),aa_nat_int(aa_int_fun_nat_int(power_power_int,B),W)) ) ).

% of_int_power_less_of_int_cancel_iff
tff(fact_4625_of__int__power__less__of__int__cancel__iff,axiom,
    ! [Xb: int,B: int,W: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(ring_1_of_int_int,Xb)),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(ring_1_of_int_int,B)),W))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),aa_nat_int(aa_int_fun_nat_int(power_power_int,B),W)) ) ).

% of_int_power_less_of_int_cancel_iff
tff(fact_4626_of__int__less__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,Xb: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_int_real(ring_1_of_int_real,B)),W)),aa_int_real(ring_1_of_int_real,Xb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,B),W)),Xb) ) ).

% of_int_less_of_int_power_cancel_iff
tff(fact_4627_of__int__less__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,Xb: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(aa_int_rat(ring_1_of_int_rat,B)),W)),aa_int_rat(ring_1_of_int_rat,Xb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,B),W)),Xb) ) ).

% of_int_less_of_int_power_cancel_iff
tff(fact_4628_of__int__less__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,Xb: 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)),W)),aa_int_int(ring_1_of_int_int,Xb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,B),W)),Xb) ) ).

% of_int_less_of_int_power_cancel_iff
tff(fact_4629_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_4630_diff__nat__numeral,axiom,
    ! [V: num,V3: num] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_num_nat(numeral_numeral_nat,V)),aa_num_nat(numeral_numeral_nat,V3)) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_num_int(numeral_numeral_int,V)),aa_num_int(numeral_numeral_int,V3))) ).

% diff_nat_numeral
tff(fact_4631_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_4632_of__nat__nat,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z)
     => ( aa_nat_real(semiri5074537144036343181t_real,aa_int_nat(nat2,Z)) = aa_int_real(ring_1_of_int_real,Z) ) ) ).

% of_nat_nat
tff(fact_4633_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_4634_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_4635_numeral__power__eq__nat__cancel__iff,axiom,
    ! [Xb: num,Nb: nat,Y: int] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xb)),Nb) = aa_int_nat(nat2,Y) )
    <=> ( aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,Xb)),Nb) = Y ) ) ).

% numeral_power_eq_nat_cancel_iff
tff(fact_4636_nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y: int,Xb: num,Nb: nat] :
      ( ( aa_int_nat(nat2,Y) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xb)),Nb) )
    <=> ( Y = aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,Xb)),Nb) ) ) ).

% nat_eq_numeral_power_cancel_iff
tff(fact_4637_nat__ceiling__le__eq,axiom,
    ! [Xb: real,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(nat2,aa_real_int(archim7802044766580827645g_real,Xb))),A)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_nat_real(semiri5074537144036343181t_real,A)) ) ).

% nat_ceiling_le_eq
tff(fact_4638_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_4639_of__int__le__numeral__power__cancel__iff,axiom,
    ! [A: int,Xb: num,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,A)),aa_nat_real(aa_real_fun_nat_real(power_power_real,numeral_numeral_real(Xb)),Nb))
    <=> 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,Xb)),Nb)) ) ).

% of_int_le_numeral_power_cancel_iff
tff(fact_4640_of__int__le__numeral__power__cancel__iff,axiom,
    ! [A: int,Xb: num,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,A)),aa_nat_rat(power_power_rat(numeral_numeral_rat(Xb)),Nb))
    <=> 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,Xb)),Nb)) ) ).

% of_int_le_numeral_power_cancel_iff
tff(fact_4641_of__int__le__numeral__power__cancel__iff,axiom,
    ! [A: int,Xb: num,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(ring_1_of_int_int,A)),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,Xb)),Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,Xb)),Nb)) ) ).

% of_int_le_numeral_power_cancel_iff
tff(fact_4642_numeral__power__le__of__int__cancel__iff,axiom,
    ! [Xb: num,Nb: nat,A: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,numeral_numeral_real(Xb)),Nb)),aa_int_real(ring_1_of_int_real,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,Xb)),Nb)),A) ) ).

% numeral_power_le_of_int_cancel_iff
tff(fact_4643_numeral__power__le__of__int__cancel__iff,axiom,
    ! [Xb: num,Nb: nat,A: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(numeral_numeral_rat(Xb)),Nb)),aa_int_rat(ring_1_of_int_rat,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,Xb)),Nb)),A) ) ).

% numeral_power_le_of_int_cancel_iff
tff(fact_4644_numeral__power__le__of__int__cancel__iff,axiom,
    ! [Xb: num,Nb: nat,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,Xb)),Nb)),aa_int_int(ring_1_of_int_int,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,Xb)),Nb)),A) ) ).

% numeral_power_le_of_int_cancel_iff
tff(fact_4645_of__int__less__numeral__power__cancel__iff,axiom,
    ! [A: int,Xb: num,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(ring_1_of_int_real,A)),aa_nat_real(aa_real_fun_nat_real(power_power_real,numeral_numeral_real(Xb)),Nb))
    <=> 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,Xb)),Nb)) ) ).

% of_int_less_numeral_power_cancel_iff
tff(fact_4646_of__int__less__numeral__power__cancel__iff,axiom,
    ! [A: int,Xb: num,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(ring_1_of_int_rat,A)),aa_nat_rat(power_power_rat(numeral_numeral_rat(Xb)),Nb))
    <=> 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,Xb)),Nb)) ) ).

% of_int_less_numeral_power_cancel_iff
tff(fact_4647_of__int__less__numeral__power__cancel__iff,axiom,
    ! [A: int,Xb: num,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(ring_1_of_int_int,A)),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,Xb)),Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,Xb)),Nb)) ) ).

% of_int_less_numeral_power_cancel_iff
tff(fact_4648_numeral__power__less__of__int__cancel__iff,axiom,
    ! [Xb: num,Nb: nat,A: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,numeral_numeral_real(Xb)),Nb)),aa_int_real(ring_1_of_int_real,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,Xb)),Nb)),A) ) ).

% numeral_power_less_of_int_cancel_iff
tff(fact_4649_numeral__power__less__of__int__cancel__iff,axiom,
    ! [Xb: num,Nb: nat,A: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(numeral_numeral_rat(Xb)),Nb)),aa_int_rat(ring_1_of_int_rat,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,Xb)),Nb)),A) ) ).

% numeral_power_less_of_int_cancel_iff
tff(fact_4650_numeral__power__less__of__int__cancel__iff,axiom,
    ! [Xb: num,Nb: nat,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,Xb)),Nb)),aa_int_int(ring_1_of_int_int,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,Xb)),Nb)),A) ) ).

% numeral_power_less_of_int_cancel_iff
tff(fact_4651_nat__numeral__diff__1,axiom,
    ! [V: num] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_num_nat(numeral_numeral_nat,V)),one_one_nat) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_num_int(numeral_numeral_int,V)),one_one_int)) ).

% nat_numeral_diff_1
tff(fact_4652_numeral__power__less__nat__cancel__iff,axiom,
    ! [Xb: num,Nb: nat,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xb)),Nb)),aa_int_nat(nat2,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,Xb)),Nb)),A) ) ).

% numeral_power_less_nat_cancel_iff
tff(fact_4653_nat__less__numeral__power__cancel__iff,axiom,
    ! [A: int,Xb: num,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_int_nat(nat2,A)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xb)),Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,Xb)),Nb)) ) ).

% nat_less_numeral_power_cancel_iff
tff(fact_4654_numeral__power__le__nat__cancel__iff,axiom,
    ! [Xb: num,Nb: nat,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xb)),Nb)),aa_int_nat(nat2,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,Xb)),Nb)),A) ) ).

% numeral_power_le_nat_cancel_iff
tff(fact_4655_nat__le__numeral__power__cancel__iff,axiom,
    ! [A: int,Xb: num,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(nat2,A)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xb)),Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,Xb)),Nb)) ) ).

% nat_le_numeral_power_cancel_iff
tff(fact_4656_iadd__is__0,axiom,
    ! [M: extended_enat,Nb: extended_enat] :
      ( ( plus_p3455044024723400733d_enat(M,Nb) = zero_z5237406670263579293d_enat )
    <=> ( ( M = zero_z5237406670263579293d_enat )
        & ( Nb = zero_z5237406670263579293d_enat ) ) ) ).

% iadd_is_0
tff(fact_4657_imult__is__0,axiom,
    ! [M: extended_enat,Nb: extended_enat] :
      ( ( times_7803423173614009249d_enat(M,Nb) = zero_z5237406670263579293d_enat )
    <=> ( ( M = zero_z5237406670263579293d_enat )
        | ( Nb = zero_z5237406670263579293d_enat ) ) ) ).

% imult_is_0
tff(fact_4658_zero__one__enat__neq_I1_J,axiom,
    zero_z5237406670263579293d_enat != one_on7984719198319812577d_enat ).

% zero_one_enat_neq(1)
tff(fact_4659_ex__le__of__int,axiom,
    ! [Xb: real] :
    ? [Z3: int] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_int_real(ring_1_of_int_real,Z3)) ).

% ex_le_of_int
tff(fact_4660_ex__le__of__int,axiom,
    ! [Xb: rat] :
    ? [Z3: int] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),aa_int_rat(ring_1_of_int_rat,Z3)) ).

% ex_le_of_int
tff(fact_4661_ex__of__int__less,axiom,
    ! [Xb: real] :
    ? [Z3: int] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(ring_1_of_int_real,Z3)),Xb) ).

% ex_of_int_less
tff(fact_4662_ex__of__int__less,axiom,
    ! [Xb: rat] :
    ? [Z3: int] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(ring_1_of_int_rat,Z3)),Xb) ).

% ex_of_int_less
tff(fact_4663_ex__less__of__int,axiom,
    ! [Xb: real] :
    ? [Z3: int] : aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_int_real(ring_1_of_int_real,Z3)) ).

% ex_less_of_int
tff(fact_4664_ex__less__of__int,axiom,
    ! [Xb: rat] :
    ? [Z3: int] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),aa_int_rat(ring_1_of_int_rat,Z3)) ).

% ex_less_of_int
tff(fact_4665_mult__of__int__commute,axiom,
    ! [Xb: int,Y: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(ring_1_of_int_real,Xb)),Y) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),aa_int_real(ring_1_of_int_real,Xb)) ).

% mult_of_int_commute
tff(fact_4666_mult__of__int__commute,axiom,
    ! [Xb: int,Y: rat] : aa_rat_rat(times_times_rat(aa_int_rat(ring_1_of_int_rat,Xb)),Y) = aa_rat_rat(times_times_rat(Y),aa_int_rat(ring_1_of_int_rat,Xb)) ).

% mult_of_int_commute
tff(fact_4667_mult__of__int__commute,axiom,
    ! [Xb: int,Y: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(ring_1_of_int_int,Xb)),Y) = aa_int_int(aa_int_fun_int_int(times_times_int,Y),aa_int_int(ring_1_of_int_int,Xb)) ).

% mult_of_int_commute
tff(fact_4668_nat__numeral__as__int,axiom,
    ! [X3: num] : aa_num_nat(numeral_numeral_nat,X3) = aa_int_nat(nat2,aa_num_int(numeral_numeral_int,X3)) ).

% nat_numeral_as_int
tff(fact_4669_nat__mono,axiom,
    ! [Xb: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Y)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(nat2,Xb)),aa_int_nat(nat2,Y)) ) ).

% nat_mono
tff(fact_4670_nat__one__as__int,axiom,
    one_one_nat = aa_int_nat(nat2,one_one_int) ).

% nat_one_as_int
tff(fact_4671_eq__nat__nat__iff,axiom,
    ! [Z: int,Z5: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z5)
       => ( ( aa_int_nat(nat2,Z) = aa_int_nat(nat2,Z5) )
        <=> ( Z = Z5 ) ) ) ) ).

% eq_nat_nat_iff
tff(fact_4672_all__nat,axiom,
    ! [P: fun_nat_o] :
      ( ! [X_1: nat] : aa_nat_o(P,X_1)
    <=> ! [X: int] :
          ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X)
         => aa_nat_o(P,aa_int_nat(nat2,X)) ) ) ).

% all_nat
tff(fact_4673_ex__nat,axiom,
    ! [P: fun_nat_o] :
      ( ? [X_1: nat] : aa_nat_o(P,X_1)
    <=> ? [X: int] :
          ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X)
          & aa_nat_o(P,aa_int_nat(nat2,X)) ) ) ).

% ex_nat
tff(fact_4674_le__of__int__ceiling,axiom,
    ! [Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_int_real(ring_1_of_int_real,aa_real_int(archim7802044766580827645g_real,Xb))) ).

% le_of_int_ceiling
tff(fact_4675_le__of__int__ceiling,axiom,
    ! [Xb: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),aa_int_rat(ring_1_of_int_rat,archim2889992004027027881ng_rat(Xb))) ).

% le_of_int_ceiling
tff(fact_4676_nat__mono__iff,axiom,
    ! [Z: int,W: 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,W)),aa_int_nat(nat2,Z))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,W),Z) ) ) ).

% nat_mono_iff
tff(fact_4677_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_4678_nat__le__iff,axiom,
    ! [Xb: int,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(nat2,Xb)),Nb)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),aa_nat_int(semiri1314217659103216013at_int,Nb)) ) ).

% nat_le_iff
tff(fact_4679_of__nat__ceiling,axiom,
    ! [R2: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,R2),aa_nat_real(semiri5074537144036343181t_real,aa_int_nat(nat2,aa_real_int(archim7802044766580827645g_real,R2)))) ).

% of_nat_ceiling
tff(fact_4680_of__nat__ceiling,axiom,
    ! [R2: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,R2),aa_nat_rat(semiri681578069525770553at_rat,aa_int_nat(nat2,archim2889992004027027881ng_rat(R2)))) ).

% of_nat_ceiling
tff(fact_4681_nat__int__add,axiom,
    ! [A: nat,B: nat] : aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,A)),aa_nat_int(semiri1314217659103216013at_int,B))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B) ).

% nat_int_add
tff(fact_4682_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_4683_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_4684_ceiling__le__iff,axiom,
    ! [Xb: real,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(archim7802044766580827645g_real,Xb)),Z)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_int_real(ring_1_of_int_real,Z)) ) ).

% ceiling_le_iff
tff(fact_4685_ceiling__le__iff,axiom,
    ! [Xb: rat,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,archim2889992004027027881ng_rat(Xb)),Z)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),aa_int_rat(ring_1_of_int_rat,Z)) ) ).

% ceiling_le_iff
tff(fact_4686_ceiling__le,axiom,
    ! [Xb: real,A: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_int_real(ring_1_of_int_real,A))
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(archim7802044766580827645g_real,Xb)),A) ) ).

% ceiling_le
tff(fact_4687_ceiling__le,axiom,
    ! [Xb: rat,A: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),aa_int_rat(ring_1_of_int_rat,A))
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,archim2889992004027027881ng_rat(Xb)),A) ) ).

% ceiling_le
tff(fact_4688_less__ceiling__iff,axiom,
    ! [Z: int,Xb: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z),archim2889992004027027881ng_rat(Xb))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(ring_1_of_int_rat,Z)),Xb) ) ).

% less_ceiling_iff
tff(fact_4689_less__ceiling__iff,axiom,
    ! [Z: int,Xb: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z),aa_real_int(archim7802044766580827645g_real,Xb))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(ring_1_of_int_real,Z)),Xb) ) ).

% less_ceiling_iff
tff(fact_4690_int__minus,axiom,
    ! [Nb: nat,M: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),M)) = aa_nat_int(semiri1314217659103216013at_int,aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),aa_nat_int(semiri1314217659103216013at_int,M)))) ).

% int_minus
tff(fact_4691_real__of__int__div4,axiom,
    ! [Nb: int,Xb: int] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(divide_divide_int,Nb),Xb))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_int_real(ring_1_of_int_real,Nb)),aa_int_real(ring_1_of_int_real,Xb))) ).

% real_of_int_div4
tff(fact_4692_real__nat__ceiling__ge,axiom,
    ! [Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_nat_real(semiri5074537144036343181t_real,aa_int_nat(nat2,aa_real_int(archim7802044766580827645g_real,Xb)))) ).

% real_nat_ceiling_ge
tff(fact_4693_nat__plus__as__int,axiom,
    ! [X3: nat,Xa: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X3),Xa) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,X3)),aa_nat_int(semiri1314217659103216013at_int,Xa))) ).

% nat_plus_as_int
tff(fact_4694_nat__times__as__int,axiom,
    ! [X3: nat,Xa: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,X3),Xa) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,X3)),aa_nat_int(semiri1314217659103216013at_int,Xa))) ).

% nat_times_as_int
tff(fact_4695_nat__minus__as__int,axiom,
    ! [X3: nat,Xa: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,X3),Xa) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(semiri1314217659103216013at_int,X3)),aa_nat_int(semiri1314217659103216013at_int,Xa))) ).

% nat_minus_as_int
tff(fact_4696_nat__div__as__int,axiom,
    ! [X3: nat,Xa: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X3),Xa) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_nat_int(semiri1314217659103216013at_int,X3)),aa_nat_int(semiri1314217659103216013at_int,Xa))) ).

% nat_div_as_int
tff(fact_4697_of__int__nonneg,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_int_real(ring_1_of_int_real,Z)) ) ).

% of_int_nonneg
tff(fact_4698_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_4699_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_4700_of__int__pos,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Z)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_int_real(ring_1_of_int_real,Z)) ) ).

% of_int_pos
tff(fact_4701_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_4702_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_4703_floor__exists1,axiom,
    ! [Xb: real] :
    ? [X2: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,X2)),Xb)
      & aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(plus_plus_int,X2),one_one_int)))
      & ! [Y5: int] :
          ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Y5)),Xb)
            & aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(plus_plus_int,Y5),one_one_int))) )
         => ( Y5 = X2 ) ) ) ).

% floor_exists1
tff(fact_4704_floor__exists1,axiom,
    ! [Xb: rat] :
    ? [X2: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,X2)),Xb)
      & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(plus_plus_int,X2),one_one_int)))
      & ! [Y5: int] :
          ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Y5)),Xb)
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(plus_plus_int,Y5),one_one_int))) )
         => ( Y5 = X2 ) ) ) ).

% floor_exists1
tff(fact_4705_floor__exists,axiom,
    ! [Xb: real] :
    ? [Z3: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Z3)),Xb)
      & aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(plus_plus_int,Z3),one_one_int))) ) ).

% floor_exists
tff(fact_4706_floor__exists,axiom,
    ! [Xb: rat] :
    ? [Z3: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Z3)),Xb)
      & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(plus_plus_int,Z3),one_one_int))) ) ).

% floor_exists
tff(fact_4707_of__int__ceiling__le__add__one,axiom,
    ! [R2: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,aa_real_int(archim7802044766580827645g_real,R2))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,R2),one_one_real)) ).

% of_int_ceiling_le_add_one
tff(fact_4708_of__int__ceiling__le__add__one,axiom,
    ! [R2: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,archim2889992004027027881ng_rat(R2))),aa_rat_rat(plus_plus_rat(R2),one_one_rat)) ).

% of_int_ceiling_le_add_one
tff(fact_4709_of__int__ceiling__diff__one__le,axiom,
    ! [R2: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_int_real(ring_1_of_int_real,aa_real_int(archim7802044766580827645g_real,R2))),one_one_real)),R2) ).

% of_int_ceiling_diff_one_le
tff(fact_4710_of__int__ceiling__diff__one__le,axiom,
    ! [R2: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,minus_minus_rat(aa_int_rat(ring_1_of_int_rat,archim2889992004027027881ng_rat(R2)),one_one_rat)),R2) ).

% of_int_ceiling_diff_one_le
tff(fact_4711_of__nat__less__of__int__iff,axiom,
    ! [Nb: nat,Xb: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,Nb)),aa_int_rat(ring_1_of_int_rat,Xb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),Xb) ) ).

% of_nat_less_of_int_iff
tff(fact_4712_of__nat__less__of__int__iff,axiom,
    ! [Nb: nat,Xb: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_int_real(ring_1_of_int_real,Xb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),Xb) ) ).

% of_nat_less_of_int_iff
tff(fact_4713_of__nat__less__of__int__iff,axiom,
    ! [Nb: nat,Xb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),aa_int_int(ring_1_of_int_int,Xb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),Xb) ) ).

% of_nat_less_of_int_iff
tff(fact_4714_of__nat__less__of__int__iff,axiom,
    ! [Nb: nat,Xb: int] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,Nb)),aa_int_Code_integer(ring_18347121197199848620nteger,Xb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),Xb) ) ).

% of_nat_less_of_int_iff
tff(fact_4715_nat__less__eq__zless,axiom,
    ! [W: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),W)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_int_nat(nat2,W)),aa_int_nat(nat2,Z))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,W),Z) ) ) ).

% nat_less_eq_zless
tff(fact_4716_nat__eq__iff2,axiom,
    ! [M: nat,W: int] :
      ( ( M = aa_int_nat(nat2,W) )
    <=> $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),W),W = aa_nat_int(semiri1314217659103216013at_int,M),M = zero_zero_nat) ) ).

% nat_eq_iff2
tff(fact_4717_nat__eq__iff,axiom,
    ! [W: int,M: nat] :
      ( ( aa_int_nat(nat2,W) = M )
    <=> $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),W),W = aa_nat_int(semiri1314217659103216013at_int,M),M = zero_zero_nat) ) ).

% nat_eq_iff
tff(fact_4718_nat__le__eq__zle,axiom,
    ! [W: int,Z: int] :
      ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),W)
        | 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,W)),aa_int_nat(nat2,Z))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,W),Z) ) ) ).

% nat_le_eq_zle
tff(fact_4719_nat__add__distrib,axiom,
    ! [Z: int,Z5: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z5)
       => ( aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(plus_plus_int,Z),Z5)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_int_nat(nat2,Z)),aa_int_nat(nat2,Z5)) ) ) ) ).

% nat_add_distrib
tff(fact_4720_le__nat__iff,axiom,
    ! [K: int,Nb: 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,Nb),aa_int_nat(nat2,K))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),K) ) ) ).

% le_nat_iff
tff(fact_4721_Suc__as__int,axiom,
    ! [X3: nat] : aa_nat_nat(suc,X3) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,X3)),one_one_int)) ).

% Suc_as_int
tff(fact_4722_nat__mult__distrib,axiom,
    ! [Z: int,Z5: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z)
     => ( aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(times_times_int,Z),Z5)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_int_nat(nat2,Z)),aa_int_nat(nat2,Z5)) ) ) ).

% nat_mult_distrib
tff(fact_4723_nat__diff__distrib,axiom,
    ! [Z5: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z5)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z5),Z)
       => ( aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(minus_minus_int,Z),Z5)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_int_nat(nat2,Z)),aa_int_nat(nat2,Z5)) ) ) ) ).

% nat_diff_distrib
tff(fact_4724_nat__diff__distrib_H,axiom,
    ! [Xb: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xb)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
       => ( aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb),Y)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_int_nat(nat2,Xb)),aa_int_nat(nat2,Y)) ) ) ) ).

% nat_diff_distrib'
tff(fact_4725_nat__div__distrib,axiom,
    ! [Xb: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xb)
     => ( aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(divide_divide_int,Xb),Y)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_int_nat(nat2,Xb)),aa_int_nat(nat2,Y)) ) ) ).

% nat_div_distrib
tff(fact_4726_nat__div__distrib_H,axiom,
    ! [Y: int,Xb: 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,Xb),Y)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_int_nat(nat2,Xb)),aa_int_nat(nat2,Y)) ) ) ).

% nat_div_distrib'
tff(fact_4727_nat__power__eq,axiom,
    ! [Z: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z)
     => ( aa_int_nat(nat2,aa_nat_int(aa_int_fun_nat_int(power_power_int,Z),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_int_nat(nat2,Z)),Nb) ) ) ).

% nat_power_eq
tff(fact_4728_int__le__real__less,axiom,
    ! [Nb: int,M: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Nb),M)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(ring_1_of_int_real,Nb)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,M)),one_one_real)) ) ).

% int_le_real_less
tff(fact_4729_int__less__real__le,axiom,
    ! [Nb: int,M: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Nb),M)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,Nb)),one_one_real)),aa_int_real(ring_1_of_int_real,M)) ) ).

% int_less_real_le
tff(fact_4730_T_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062x_092_060_094sub_062t_Osimps_I1_J,axiom,
    ! [A: $o,B: $o] :
      aa_VEBT_VEBT_nat(vEBT_T_m_a_x_t,vEBT_Leaf((A),(B))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
        $ite((B),one_one_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat))) ).

% T\<^sub>m\<^sub>a\<^sub>x\<^sub>t.simps(1)
tff(fact_4731_T_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062x_092_060_094sub_062t_Osimps_I2_J,axiom,
    ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] : aa_VEBT_VEBT_nat(vEBT_T_m_a_x_t,aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv),Uw)) = one_one_nat ).

% T\<^sub>m\<^sub>a\<^sub>x\<^sub>t.simps(2)
tff(fact_4732_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_Osimps_I2_J,axiom,
    ! [Info: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S3: vEBT_VEBT,Xb: nat] : aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_i_n_s_e_r_t,aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info,zero_zero_nat,Ts),S3)),Xb) = one_one_nat ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.simps(2)
tff(fact_4733_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t_Osimps_I2_J,axiom,
    ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] : aa_VEBT_VEBT_nat(vEBT_T_m_i_n_t,aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv),Uw)) = one_one_nat ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>t.simps(2)
tff(fact_4734_nat__2,axiom,
    aa_int_nat(nat2,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = aa_nat_nat(suc,aa_nat_nat(suc,zero_zero_nat)) ).

% nat_2
tff(fact_4735_ceiling__correct,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_int_real(ring_1_of_int_real,aa_real_int(archim7802044766580827645g_real,Xb))),one_one_real)),Xb)
      & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_int_real(ring_1_of_int_real,aa_real_int(archim7802044766580827645g_real,Xb))) ) ).

% ceiling_correct
tff(fact_4736_ceiling__correct,axiom,
    ! [Xb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,minus_minus_rat(aa_int_rat(ring_1_of_int_rat,archim2889992004027027881ng_rat(Xb)),one_one_rat)),Xb)
      & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),aa_int_rat(ring_1_of_int_rat,archim2889992004027027881ng_rat(Xb))) ) ).

% ceiling_correct
tff(fact_4737_ceiling__unique,axiom,
    ! [Z: int,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_int_real(ring_1_of_int_real,Z)),one_one_real)),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_int_real(ring_1_of_int_real,Z))
       => ( aa_real_int(archim7802044766580827645g_real,Xb) = Z ) ) ) ).

% ceiling_unique
tff(fact_4738_ceiling__unique,axiom,
    ! [Z: int,Xb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,minus_minus_rat(aa_int_rat(ring_1_of_int_rat,Z),one_one_rat)),Xb)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),aa_int_rat(ring_1_of_int_rat,Z))
       => ( archim2889992004027027881ng_rat(Xb) = Z ) ) ) ).

% ceiling_unique
tff(fact_4739_ceiling__eq__iff,axiom,
    ! [Xb: real,A: int] :
      ( ( aa_real_int(archim7802044766580827645g_real,Xb) = A )
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_int_real(ring_1_of_int_real,A)),one_one_real)),Xb)
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_int_real(ring_1_of_int_real,A)) ) ) ).

% ceiling_eq_iff
tff(fact_4740_ceiling__eq__iff,axiom,
    ! [Xb: rat,A: int] :
      ( ( archim2889992004027027881ng_rat(Xb) = A )
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,minus_minus_rat(aa_int_rat(ring_1_of_int_rat,A),one_one_rat)),Xb)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),aa_int_rat(ring_1_of_int_rat,A)) ) ) ).

% ceiling_eq_iff
tff(fact_4741_ceiling__split,axiom,
    ! [P: fun_int_o,Ta: real] :
      ( aa_int_o(P,aa_real_int(archim7802044766580827645g_real,Ta))
    <=> ! [I2: int] :
          ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_int_real(ring_1_of_int_real,I2)),one_one_real)),Ta)
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Ta),aa_int_real(ring_1_of_int_real,I2)) )
         => aa_int_o(P,I2) ) ) ).

% ceiling_split
tff(fact_4742_ceiling__split,axiom,
    ! [P: fun_int_o,Ta: rat] :
      ( aa_int_o(P,archim2889992004027027881ng_rat(Ta))
    <=> ! [I2: int] :
          ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,minus_minus_rat(aa_int_rat(ring_1_of_int_rat,I2),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,I2)) )
         => aa_int_o(P,I2) ) ) ).

% ceiling_split
tff(fact_4743_ceiling__less__iff,axiom,
    ! [Xb: real,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(archim7802044766580827645g_real,Xb)),Z)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_int_real(ring_1_of_int_real,Z)),one_one_real)) ) ).

% ceiling_less_iff
tff(fact_4744_ceiling__less__iff,axiom,
    ! [Xb: rat,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,archim2889992004027027881ng_rat(Xb)),Z)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),minus_minus_rat(aa_int_rat(ring_1_of_int_rat,Z),one_one_rat)) ) ).

% ceiling_less_iff
tff(fact_4745_le__ceiling__iff,axiom,
    ! [Z: int,Xb: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),archim2889992004027027881ng_rat(Xb))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,minus_minus_rat(aa_int_rat(ring_1_of_int_rat,Z),one_one_rat)),Xb) ) ).

% le_ceiling_iff
tff(fact_4746_le__ceiling__iff,axiom,
    ! [Z: int,Xb: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),aa_real_int(archim7802044766580827645g_real,Xb))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_int_real(ring_1_of_int_real,Z)),one_one_real)),Xb) ) ).

% le_ceiling_iff
tff(fact_4747_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_4748_nat__less__iff,axiom,
    ! [W: int,M: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),W)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_int_nat(nat2,W)),M)
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,W),aa_nat_int(semiri1314217659103216013at_int,M)) ) ) ).

% nat_less_iff
tff(fact_4749_real__of__int__div2,axiom,
    ! [Nb: int,Xb: int] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_int_real(ring_1_of_int_real,Nb)),aa_int_real(ring_1_of_int_real,Xb))),aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(divide_divide_int,Nb),Xb)))) ).

% real_of_int_div2
tff(fact_4750_real__of__int__div3,axiom,
    ! [Nb: int,Xb: int] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_int_real(ring_1_of_int_real,Nb)),aa_int_real(ring_1_of_int_real,Xb))),aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(divide_divide_int,Nb),Xb)))),one_one_real) ).

% real_of_int_div3
tff(fact_4751_diff__nat__eq__if,axiom,
    ! [Z: int,Z5: int] :
      aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_int_nat(nat2,Z)),aa_int_nat(nat2,Z5)) = $ite(
        aa_int_o(aa_int_fun_int_o(ord_less_int,Z5),zero_zero_int),
        aa_int_nat(nat2,Z),
        $let(
          d: int,
          d:= aa_int_int(aa_int_fun_int_int(minus_minus_int,Z),Z5),
          $ite(aa_int_o(aa_int_fun_int_o(ord_less_int,d),zero_zero_int),zero_zero_nat,aa_int_nat(nat2,d)) ) ) ).

% diff_nat_eq_if
tff(fact_4752_maxt__bound,axiom,
    ! [Ta: vEBT_VEBT] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_VEBT_VEBT_nat(vEBT_T_m_a_x_t,Ta)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))) ).

% maxt_bound
tff(fact_4753_ceiling__divide__upper,axiom,
    ! [Q2: real,P2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Q2)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,P2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(ring_1_of_int_real,aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,P2),Q2)))),Q2)) ) ).

% ceiling_divide_upper
tff(fact_4754_ceiling__divide__upper,axiom,
    ! [Q2: rat,P2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Q2)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,P2),aa_rat_rat(times_times_rat(aa_int_rat(ring_1_of_int_rat,archim2889992004027027881ng_rat(divide_divide_rat(P2,Q2)))),Q2)) ) ).

% ceiling_divide_upper
tff(fact_4755_T_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062x_092_060_094sub_062t_Osimps_I3_J,axiom,
    ! [Mia: nat,Maa: nat,Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT] : aa_VEBT_VEBT_nat(vEBT_T_m_a_x_t,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mia),Maa)),Ux,Uy),Uz)) = one_one_nat ).

% T\<^sub>m\<^sub>a\<^sub>x\<^sub>t.simps(3)
tff(fact_4756_mint__bound,axiom,
    ! [Ta: vEBT_VEBT] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_VEBT_VEBT_nat(vEBT_T_m_i_n_t,Ta)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))) ).

% mint_bound
tff(fact_4757_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t_Osimps_I1_J,axiom,
    ! [A: $o,B: $o] :
      aa_VEBT_VEBT_nat(vEBT_T_m_i_n_t,vEBT_Leaf((A),(B))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
        $ite((A),zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat))) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>t.simps(1)
tff(fact_4758_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_Osimps_I3_J,axiom,
    ! [Info: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S3: vEBT_VEBT,Xb: nat] : aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_i_n_s_e_r_t,aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info,aa_nat_nat(suc,zero_zero_nat),Ts),S3)),Xb) = one_one_nat ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.simps(3)
tff(fact_4759_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t_Osimps_I3_J,axiom,
    ! [Mia: nat,Maa: nat,Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT] : aa_VEBT_VEBT_nat(vEBT_T_m_i_n_t,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mia),Maa)),Ux,Uy),Uz)) = one_one_nat ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>t.simps(3)
tff(fact_4760_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_Osimps_I1_J,axiom,
    ! [A: $o,B: $o,Xb: nat] :
      aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_i_n_s_e_r_t,vEBT_Leaf((A),(B))),Xb) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
        $ite(Xb = zero_zero_nat,one_one_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat))) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.simps(1)
tff(fact_4761_ceiling__divide__lower,axiom,
    ! [Q2: real,P2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Q2)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_int_real(ring_1_of_int_real,aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,P2),Q2)))),one_one_real)),Q2)),P2) ) ).

% ceiling_divide_lower
tff(fact_4762_ceiling__divide__lower,axiom,
    ! [Q2: rat,P2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Q2)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(minus_minus_rat(aa_int_rat(ring_1_of_int_rat,archim2889992004027027881ng_rat(divide_divide_rat(P2,Q2))),one_one_rat)),Q2)),P2) ) ).

% ceiling_divide_lower
tff(fact_4763_ceiling__eq,axiom,
    ! [Nb: int,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(ring_1_of_int_real,Nb)),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,Nb)),one_one_real))
       => ( aa_real_int(archim7802044766580827645g_real,Xb) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Nb),one_one_int) ) ) ) ).

% ceiling_eq
tff(fact_4764_ceiling__eq,axiom,
    ! [Nb: int,Xb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(ring_1_of_int_rat,Nb)),Xb)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),aa_rat_rat(plus_plus_rat(aa_int_rat(ring_1_of_int_rat,Nb)),one_one_rat))
       => ( archim2889992004027027881ng_rat(Xb) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Nb),one_one_int) ) ) ) ).

% ceiling_eq
tff(fact_4765_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_Osimps_I4_J,axiom,
    ! [V: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Xb: nat] : aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_i_n_s_e_r_t,aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V)),TreeLista),Summarya)),Xb) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.simps(4)
tff(fact_4766_insersimp,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Y: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Ta),Nb)
     => ( ~ ? [X_13: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,Ta),X_13)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_i_n_s_e_r_t,Ta),Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))) ) ) ).

% insersimp
tff(fact_4767_insertsimp,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,L: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Ta),Nb)
     => ( aa_VEBT_VEBT_o(vEBT_VEBT_minNull,Ta)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_i_n_s_e_r_t,Ta),L)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))) ) ) ).

% insertsimp
tff(fact_4768_T_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062x_092_060_094sub_062t_Oelims,axiom,
    ! [Xb: vEBT_VEBT,Y: nat] :
      ( ( aa_VEBT_VEBT_nat(vEBT_T_m_a_x_t,Xb) = Y )
     => ( ! [A3: $o,B2: $o] :
            ( ( Xb = vEBT_Leaf((A3),(B2)) )
           => ( Y != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
                  $ite((B2),one_one_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat))) ) )
       => ( ( ? [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] : Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2),Uw2)
           => ( Y != one_one_nat ) )
         => ~ ( ? [Mi: nat,Ma: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] : Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),Ux2,Uy2),Uz2)
             => ( Y != one_one_nat ) ) ) ) ) ).

% T\<^sub>m\<^sub>a\<^sub>x\<^sub>t.elims
tff(fact_4769_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t_Oelims,axiom,
    ! [Xb: vEBT_VEBT,Y: nat] :
      ( ( aa_VEBT_VEBT_nat(vEBT_T_m_i_n_t,Xb) = Y )
     => ( ! [A3: $o] :
            ( ? [B2: $o] : Xb = vEBT_Leaf((A3),(B2))
           => ( Y != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
                  $ite((A3),zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat))) ) )
       => ( ( ? [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] : Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2),Uw2)
           => ( Y != one_one_nat ) )
         => ~ ( ? [Mi: nat,Ma: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] : Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),Ux2,Uy2),Uz2)
             => ( Y != one_one_nat ) ) ) ) ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>t.elims
tff(fact_4770_insert__bound__height,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xb: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Ta),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_i_n_s_e_r_t,Ta),Xb)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_VEBT_height,Ta))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,one))))))) ) ).

% insert_bound_height
tff(fact_4771_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Opelims,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_s_u_c_c,Xb),Xaa) = Y )
     => ( aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_T_s_u_c_c_rel2),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(Xb),Xaa))
       => ( ! [Uu2: $o,B2: $o] :
              ( ( Xb = vEBT_Leaf((Uu2),(B2)) )
             => ( ( Xaa = zero_zero_nat )
               => ( ( Y = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat) )
                 => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_T_s_u_c_c_rel2),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(vEBT_Leaf((Uu2),(B2))),zero_zero_nat)) ) ) )
         => ( ! [Uv2: $o,Uw2: $o] :
                ( ( Xb = vEBT_Leaf((Uv2),(Uw2)) )
               => ! [N2: nat] :
                    ( ( Xaa = aa_nat_nat(suc,N2) )
                   => ( ( Y = one_one_nat )
                     => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_T_s_u_c_c_rel2),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(vEBT_Leaf((Uv2),(Uw2))),aa_nat_nat(suc,N2))) ) ) )
           => ( ! [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Ux2,Uy2),Uz2) )
                 => ( ( Y = one_one_nat )
                   => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_T_s_u_c_c_rel2),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Ux2,Uy2),Uz2)),Xaa)) ) )
             => ( ! [V2: product_prod_nat_nat,Vc: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
                    ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Vc),Vd2) )
                   => ( ( Y = one_one_nat )
                     => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_T_s_u_c_c_rel2),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Vc),Vd2)),Xaa)) ) )
               => ( ! [V2: product_prod_nat_nat,Vg2: list_VEBT_VEBT,Vh2: vEBT_VEBT] :
                      ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vg2),Vh2) )
                     => ( ( Y = one_one_nat )
                       => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_T_s_u_c_c_rel2),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vg2),Vh2)),Xaa)) ) )
                 => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                        ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList),Summary) )
                       => ( ( Y = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
                                $ite(
                                  aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi),
                                  one_one_nat,
                                  $let(
                                    l: nat,
                                    l:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                    $let(
                                      h: nat,
                                      h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                      aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,one))))),
                                        $ite(
                                          aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList)),
                                          aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_T_m_a_x_t,nth_VEBT_VEBT(TreeList,h)))),
                                            $let(
                                              maxlow: option_nat,
                                              maxlow:= vEBT_vebt_maxt(nth_VEBT_VEBT(TreeList,h)),
                                              aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),
                                                $ite(
                                                  ( ( maxlow != none_nat )
                                                  & aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_less,aa_nat_option_nat(some_nat,l)),maxlow) ),
                                                  aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))),aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_s_u_c_c,nth_VEBT_VEBT(TreeList,h)),l)),
                                                  $let(
                                                    sc: option_nat,
                                                    sc:= aa_nat_option_nat(vEBT_vebt_succ(Summary),h),
                                                    aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_s_u_c_c,Summary),h))),one_one_nat)),
                                                      $ite(sc = none_nat,one_one_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))),aa_VEBT_VEBT_nat(vEBT_T_m_i_n_t,nth_VEBT_VEBT(TreeList,the_nat(sc)))))) ) )) )),
                                          one_one_nat )) ) ) )) )
                         => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_T_s_u_c_c_rel2),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList),Summary)),Xaa)) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.pelims
tff(fact_4772_T_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_Oelims,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_d_e_l_e_t_e,Xb),Xaa) = Y )
     => ( ( ? [A3: $o,B2: $o] : Xb = vEBT_Leaf((A3),(B2))
         => ( ( Xaa = zero_zero_nat )
           => ( Y != one_one_nat ) ) )
       => ( ( ? [A3: $o,B2: $o] : Xb = vEBT_Leaf((A3),(B2))
           => ( ( Xaa = aa_nat_nat(suc,zero_zero_nat) )
             => ( Y != one_one_nat ) ) )
         => ( ( ? [A3: $o,B2: $o] : Xb = vEBT_Leaf((A3),(B2))
             => ( ? [N2: nat] : Xaa = aa_nat_nat(suc,aa_nat_nat(suc,N2))
               => ( Y != one_one_nat ) ) )
           => ( ( ? [Deg2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] : Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Deg2,TreeList),Summary)
               => ( Y != one_one_nat ) )
             => ( ( ? [Mi: nat,Ma: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] : Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),zero_zero_nat,TreeList),Summary)
                 => ( Y != one_one_nat ) )
               => ( ( ? [Mi: nat,Ma: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] : Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),aa_nat_nat(suc,zero_zero_nat),TreeList),Summary)
                   => ( Y != one_one_nat ) )
                 => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                        ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList),Summary) )
                       => ( Y != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),
                              $ite(
                                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi)
                                | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Xaa) ),
                                one_one_nat,
                                aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),
                                  $ite(
                                    ( ( Xaa = Mi )
                                    & ( Xaa = Ma ) ),
                                    aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)),
                                    aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,
                                        aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,
                                            aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,one))))),
                                              $ite(Xaa = Mi,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_VEBT_VEBT_nat(vEBT_T_m_i_n_t,Summary)),aa_VEBT_VEBT_nat(vEBT_T_m_i_n_t,nth_VEBT_VEBT(TreeList,the_nat(vEBT_vebt_mint(Summary)))))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit1,one)))),one_one_nat))),one_one_nat)),
                                      $let(
                                        xn: nat,
                                        xn:= 
                                          $ite(Xaa = Mi,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat(vEBT_vebt_mint(Summary)))))),Xaa),
                                        $let(
                                          l: nat,
                                          l:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                          $let(
                                            h: nat,
                                            h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                            $ite(
                                              aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList)),
                                              aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))),aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_d_e_l_e_t_e,nth_VEBT_VEBT(TreeList,h)),l))),
                                                $let(
                                                  newnode: vEBT_VEBT,
                                                  newnode:= vEBT_vebt_delete(nth_VEBT_VEBT(TreeList,h),l),
                                                  $let(
                                                    newlist: list_VEBT_VEBT,
                                                    newlist:= list_u1324408373059187874T_VEBT(TreeList,h,newnode),
                                                    aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_T_m_i_n_N_u_l_l,newnode))),
                                                      $ite(
                                                        aa_VEBT_VEBT_o(vEBT_VEBT_minNull,newnode),
                                                        aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_d_e_l_e_t_e,Summary),h))),
                                                          $let(
                                                            sn: vEBT_VEBT,
                                                            sn:= vEBT_vebt_delete(Summary,h),
                                                            aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                                                              $ite(
                                                                xn = Ma,
                                                                aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_T_m_a_x_t,sn))),
                                                                  $let(
                                                                    maxs: option_nat,
                                                                    maxs:= vEBT_vebt_maxt(sn),
                                                                    aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
                                                                      $ite(maxs = none_nat,one_one_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,one))))),aa_VEBT_VEBT_nat(vEBT_T_m_a_x_t,nth_VEBT_VEBT(newlist,the_nat(maxs)))))) )),
                                                                one_one_nat )) )),
                                                        aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                                                          $ite(xn = Ma,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit1,one)))),aa_VEBT_VEBT_nat(vEBT_T_m_a_x_t,nth_VEBT_VEBT(newlist,h))),one_one_nat)) )) ) )),
                                              one_one_nat ) ) ) )) )) )) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.elims
tff(fact_4773_T_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_Osimps_I7_J,axiom,
    ! [Mia: nat,Maa: nat,Vaa: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Xb: nat] :
      aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_d_e_l_e_t_e,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mia),Maa)),aa_nat_nat(suc,aa_nat_nat(suc,Vaa)),TreeLista),Summarya)),Xb) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),
        $ite(
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Mia)
          | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Maa),Xb) ),
          one_one_nat,
          aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),
            $ite(
              ( ( Xb = Mia )
              & ( Xb = Maa ) ),
              aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)),
              aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,
                  aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,
                      aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,one))))),
                        $ite(Xb = Mia,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_VEBT_VEBT_nat(vEBT_T_m_i_n_t,Summarya)),aa_VEBT_VEBT_nat(vEBT_T_m_i_n_t,nth_VEBT_VEBT(TreeLista,the_nat(vEBT_vebt_mint(Summarya)))))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit1,one)))),one_one_nat))),one_one_nat)),
                $let(
                  xn: nat,
                  xn:= 
                    $ite(Xb = Mia,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summarya))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Vaa))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeLista,the_nat(vEBT_vebt_mint(Summarya)))))),Xb),
                  $let(
                    l: nat,
                    l:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Vaa))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                    $let(
                      h: nat,
                      h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Vaa))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                      $ite(
                        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeLista)),
                        aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))),aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_d_e_l_e_t_e,nth_VEBT_VEBT(TreeLista,h)),l))),
                          $let(
                            newnode: vEBT_VEBT,
                            newnode:= vEBT_vebt_delete(nth_VEBT_VEBT(TreeLista,h),l),
                            $let(
                              newlist: list_VEBT_VEBT,
                              newlist:= list_u1324408373059187874T_VEBT(TreeLista,h,newnode),
                              aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_T_m_i_n_N_u_l_l,newnode))),
                                $ite(
                                  aa_VEBT_VEBT_o(vEBT_VEBT_minNull,newnode),
                                  aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_d_e_l_e_t_e,Summarya),h))),
                                    $let(
                                      sn: vEBT_VEBT,
                                      sn:= vEBT_vebt_delete(Summarya,h),
                                      aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                                        $ite(
                                          xn = Maa,
                                          aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_T_m_a_x_t,sn))),
                                            $let(
                                              maxs: option_nat,
                                              maxs:= vEBT_vebt_maxt(sn),
                                              aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
                                                $ite(maxs = none_nat,one_one_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,one))))),aa_VEBT_VEBT_nat(vEBT_T_m_a_x_t,nth_VEBT_VEBT(newlist,the_nat(maxs)))))) )),
                                          one_one_nat )) )),
                                  aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                                    $ite(xn = Maa,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit1,one)))),aa_VEBT_VEBT_nat(vEBT_T_m_a_x_t,nth_VEBT_VEBT(newlist,h))),one_one_nat)) )) ) )),
                        one_one_nat ) ) ) )) )) )) ).

% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.simps(7)
tff(fact_4774_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Oelims,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_p_r_e_d,Xb),Xaa) = Y )
     => ( ( ? [Uu2: $o,Uv2: $o] : Xb = vEBT_Leaf((Uu2),(Uv2))
         => ( ( Xaa = zero_zero_nat )
           => ( Y != one_one_nat ) ) )
       => ( ( ? [A3: $o,Uw2: $o] : Xb = vEBT_Leaf((A3),(Uw2))
           => ( ( Xaa = aa_nat_nat(suc,zero_zero_nat) )
             => ( Y != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat) ) ) )
         => ( ! [A3: $o,B2: $o] :
                ( ( Xb = vEBT_Leaf((A3),(B2)) )
               => ( ? [Va2: nat] : Xaa = aa_nat_nat(suc,aa_nat_nat(suc,Va2))
                 => ( Y != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
                        $ite((B2),one_one_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat))) ) ) )
           => ( ( ? [Uy2: nat,Uz2: list_VEBT_VEBT,Va: vEBT_VEBT] : Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uy2,Uz2),Va)
               => ( Y != one_one_nat ) )
             => ( ( ? [V2: product_prod_nat_nat,Vd2: list_VEBT_VEBT,Ve2: vEBT_VEBT] : Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Vd2),Ve2)
                 => ( Y != one_one_nat ) )
               => ( ( ? [V2: product_prod_nat_nat,Vh2: list_VEBT_VEBT,Vi2: vEBT_VEBT] : Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vh2),Vi2)
                   => ( Y != one_one_nat ) )
                 => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                        ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList),Summary) )
                       => ( Y != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
                              $ite(
                                aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Xaa),
                                one_one_nat,
                                $let(
                                  l: nat,
                                  l:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                  $let(
                                    h: nat,
                                    h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                    aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,one))))),one_one_nat)),
                                      $ite(
                                        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList)),
                                        $let(
                                          minlow: option_nat,
                                          minlow:= vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,h)),
                                          aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_VEBT_VEBT_nat(vEBT_T_m_i_n_t,nth_VEBT_VEBT(TreeList,h)))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)))),
                                            $ite(
                                              ( ( minlow != none_nat )
                                              & aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_greater,aa_nat_option_nat(some_nat,l)),minlow) ),
                                              aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))),aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_p_r_e_d,nth_VEBT_VEBT(TreeList,h)),l)),
                                              $let(
                                                pr: option_nat,
                                                pr:= aa_nat_option_nat(vEBT_vebt_pred(Summary),h),
                                                aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_p_r_e_d,Summary),h))),one_one_nat)),
                                                  $ite(pr = none_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))),aa_VEBT_VEBT_nat(vEBT_T_m_a_x_t,nth_VEBT_VEBT(TreeList,the_nat(pr)))))) ) )) ),
                                        one_one_nat )) ) ) )) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.elims
tff(fact_4775_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Osimps_I7_J,axiom,
    ! [Mia: nat,Maa: nat,Vaa: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Xb: nat] :
      aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_p_r_e_d,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mia),Maa)),aa_nat_nat(suc,aa_nat_nat(suc,Vaa)),TreeLista),Summarya)),Xb) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
        $ite(
          aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Maa),Xb),
          one_one_nat,
          $let(
            l: nat,
            l:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Vaa))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
            $let(
              h: nat,
              h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Vaa))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
              aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,one))))),one_one_nat)),
                $ite(
                  aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeLista)),
                  $let(
                    minlow: option_nat,
                    minlow:= vEBT_vebt_mint(nth_VEBT_VEBT(TreeLista,h)),
                    aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_VEBT_VEBT_nat(vEBT_T_m_i_n_t,nth_VEBT_VEBT(TreeLista,h)))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)))),
                      $ite(
                        ( ( minlow != none_nat )
                        & aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_greater,aa_nat_option_nat(some_nat,l)),minlow) ),
                        aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))),aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_p_r_e_d,nth_VEBT_VEBT(TreeLista,h)),l)),
                        $let(
                          pr: option_nat,
                          pr:= aa_nat_option_nat(vEBT_vebt_pred(Summarya),h),
                          aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_p_r_e_d,Summarya),h))),one_one_nat)),
                            $ite(pr = none_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))),aa_VEBT_VEBT_nat(vEBT_T_m_a_x_t,nth_VEBT_VEBT(TreeLista,the_nat(pr)))))) ) )) ),
                  one_one_nat )) ) ) )) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.simps(7)
tff(fact_4776_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_Oelims,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_i_n_s_e_r_t,Xb),Xaa) = Y )
     => ( ( ? [A3: $o,B2: $o] : Xb = vEBT_Leaf((A3),(B2))
         => ( Y != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
                $ite(Xaa = zero_zero_nat,one_one_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat))) ) )
       => ( ( ? [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S: vEBT_VEBT] : Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,zero_zero_nat,Ts2),S)
           => ( Y != one_one_nat ) )
         => ( ( ? [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S: vEBT_VEBT] : Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts2),S)
             => ( Y != one_one_nat ) )
           => ( ( ? [V2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] : Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V2)),TreeList),Summary)
               => ( Y != aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) ) )
             => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                    ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList),Summary) )
                   => ( Y != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,one)))))),
                          $let(
                            xn: nat,
                            xn:= 
                              $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi),Mi,Xaa),
                            $let(
                              h: nat,
                              h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                              $ite(
                                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList))
                                & ~ ( ( Xaa = Mi )
                                    | ( Xaa = Ma ) ) ),
                                aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_i_n_s_e_r_t,nth_VEBT_VEBT(TreeList,h)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_VEBT_VEBT_nat(vEBT_T_m_i_n_N_u_l_l,nth_VEBT_VEBT(TreeList,h)))),
                                  $ite(aa_VEBT_VEBT_o(vEBT_VEBT_minNull,nth_VEBT_VEBT(TreeList,h)),aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_i_n_s_e_r_t,Summary),h),one_one_nat)),
                                one_one_nat ) ) )) ) ) ) ) ) ) ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.elims
tff(fact_4777_T_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_Opelims,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_d_e_l_e_t_e,Xb),Xaa) = Y )
     => ( aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_T8441311223069195367_e_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(Xb),Xaa))
       => ( ! [A3: $o,B2: $o] :
              ( ( Xb = vEBT_Leaf((A3),(B2)) )
             => ( ( Xaa = zero_zero_nat )
               => ( ( Y = one_one_nat )
                 => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_T8441311223069195367_e_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(vEBT_Leaf((A3),(B2))),zero_zero_nat)) ) ) )
         => ( ! [A3: $o,B2: $o] :
                ( ( Xb = vEBT_Leaf((A3),(B2)) )
               => ( ( Xaa = aa_nat_nat(suc,zero_zero_nat) )
                 => ( ( Y = one_one_nat )
                   => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_T8441311223069195367_e_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(vEBT_Leaf((A3),(B2))),aa_nat_nat(suc,zero_zero_nat))) ) ) )
           => ( ! [A3: $o,B2: $o] :
                  ( ( Xb = vEBT_Leaf((A3),(B2)) )
                 => ! [N2: nat] :
                      ( ( Xaa = aa_nat_nat(suc,aa_nat_nat(suc,N2)) )
                     => ( ( Y = one_one_nat )
                       => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_T8441311223069195367_e_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(vEBT_Leaf((A3),(B2))),aa_nat_nat(suc,aa_nat_nat(suc,N2)))) ) ) )
             => ( ! [Deg2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                    ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Deg2,TreeList),Summary) )
                   => ( ( Y = one_one_nat )
                     => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_T8441311223069195367_e_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Deg2,TreeList),Summary)),Xaa)) ) )
               => ( ! [Mi: nat,Ma: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                      ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),zero_zero_nat,TreeList),Summary) )
                     => ( ( Y = one_one_nat )
                       => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_T8441311223069195367_e_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),zero_zero_nat,TreeList),Summary)),Xaa)) ) )
                 => ( ! [Mi: nat,Ma: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                        ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),aa_nat_nat(suc,zero_zero_nat),TreeList),Summary) )
                       => ( ( Y = one_one_nat )
                         => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_T8441311223069195367_e_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),aa_nat_nat(suc,zero_zero_nat),TreeList),Summary)),Xaa)) ) )
                   => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                          ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList),Summary) )
                         => ( ( Y = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),
                                  $ite(
                                    ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi)
                                    | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Xaa) ),
                                    one_one_nat,
                                    aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),
                                      $ite(
                                        ( ( Xaa = Mi )
                                        & ( Xaa = Ma ) ),
                                        aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)),
                                        aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,
                                            aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,
                                                aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,one))))),
                                                  $ite(Xaa = Mi,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_VEBT_VEBT_nat(vEBT_T_m_i_n_t,Summary)),aa_VEBT_VEBT_nat(vEBT_T_m_i_n_t,nth_VEBT_VEBT(TreeList,the_nat(vEBT_vebt_mint(Summary)))))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit1,one)))),one_one_nat))),one_one_nat)),
                                          $let(
                                            xn: nat,
                                            xn:= 
                                              $ite(Xaa = Mi,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat(vEBT_vebt_mint(Summary)))))),Xaa),
                                            $let(
                                              l: nat,
                                              l:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                              $let(
                                                h: nat,
                                                h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                                $ite(
                                                  aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList)),
                                                  aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))),aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_d_e_l_e_t_e,nth_VEBT_VEBT(TreeList,h)),l))),
                                                    $let(
                                                      newnode: vEBT_VEBT,
                                                      newnode:= vEBT_vebt_delete(nth_VEBT_VEBT(TreeList,h),l),
                                                      $let(
                                                        newlist: list_VEBT_VEBT,
                                                        newlist:= list_u1324408373059187874T_VEBT(TreeList,h,newnode),
                                                        aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_T_m_i_n_N_u_l_l,newnode))),
                                                          $ite(
                                                            aa_VEBT_VEBT_o(vEBT_VEBT_minNull,newnode),
                                                            aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_d_e_l_e_t_e,Summary),h))),
                                                              $let(
                                                                sn: vEBT_VEBT,
                                                                sn:= vEBT_vebt_delete(Summary,h),
                                                                aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                                                                  $ite(
                                                                    xn = Ma,
                                                                    aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_T_m_a_x_t,sn))),
                                                                      $let(
                                                                        maxs: option_nat,
                                                                        maxs:= vEBT_vebt_maxt(sn),
                                                                        aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
                                                                          $ite(maxs = none_nat,one_one_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,one))))),aa_VEBT_VEBT_nat(vEBT_T_m_a_x_t,nth_VEBT_VEBT(newlist,the_nat(maxs)))))) )),
                                                                    one_one_nat )) )),
                                                            aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                                                              $ite(xn = Ma,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit1,one)))),aa_VEBT_VEBT_nat(vEBT_T_m_a_x_t,nth_VEBT_VEBT(newlist,h))),one_one_nat)) )) ) )),
                                                  one_one_nat ) ) ) )) )) )) )
                           => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_T8441311223069195367_e_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList),Summary)),Xaa)) ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.pelims
tff(fact_4778_word__of__int__numeral,axiom,
    ! [Bin: num] : ring_17408606157368542149l_num1(aa_num_int(numeral_numeral_int,Bin)) = numera7442385471795722001l_num1(Bin) ).

% word_of_int_numeral
tff(fact_4779_word__of__int__1,axiom,
    ring_17408606157368542149l_num1(one_one_int) = one_on7727431528512463931l_num1 ).

% word_of_int_1
tff(fact_4780_word__numeral__alt,axiom,
    ! [B: num] : numera7442385471795722001l_num1(B) = ring_17408606157368542149l_num1(aa_num_int(numeral_numeral_int,B)) ).

% word_numeral_alt
tff(fact_4781_word__of__int__power__hom,axiom,
    ! [A: int,Nb: nat] : aa_nat5599461682457136004l_num1(power_2184487114949457152l_num1(ring_17408606157368542149l_num1(A)),Nb) = ring_17408606157368542149l_num1(aa_nat_int(aa_int_fun_nat_int(power_power_int,A),Nb)) ).

% word_of_int_power_hom
tff(fact_4782_minNull__bound,axiom,
    ! [Ta: vEBT_VEBT] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_VEBT_VEBT_nat(vEBT_T_m_i_n_N_u_l_l,Ta)),one_one_nat) ).

% minNull_bound
tff(fact_4783_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062N_092_060_094sub_062u_092_060_094sub_062l_092_060_094sub_062l_Osimps_I1_J,axiom,
    aa_VEBT_VEBT_nat(vEBT_T_m_i_n_N_u_l_l,vEBT_Leaf($false,$false)) = one_one_nat ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.simps(1)
tff(fact_4784_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062N_092_060_094sub_062u_092_060_094sub_062l_092_060_094sub_062l_Osimps_I2_J,axiom,
    ! [Uv: $o] : aa_VEBT_VEBT_nat(vEBT_T_m_i_n_N_u_l_l,vEBT_Leaf($true,(Uv))) = one_one_nat ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.simps(2)
tff(fact_4785_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062N_092_060_094sub_062u_092_060_094sub_062l_092_060_094sub_062l_Osimps_I3_J,axiom,
    ! [Uu: $o] : aa_VEBT_VEBT_nat(vEBT_T_m_i_n_N_u_l_l,vEBT_Leaf((Uu),$true)) = one_one_nat ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.simps(3)
tff(fact_4786_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Osimps_I1_J,axiom,
    ! [Uu: $o,Uv: $o] : aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_p_r_e_d,vEBT_Leaf((Uu),(Uv))),zero_zero_nat) = one_one_nat ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.simps(1)
tff(fact_4787_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Osimps_I4_J,axiom,
    ! [Uy: nat,Uz: list_VEBT_VEBT,Vaa: vEBT_VEBT,Vb: nat] : aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_p_r_e_d,aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uy,Uz),Vaa)),Vb) = one_one_nat ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.simps(4)
tff(fact_4788_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062N_092_060_094sub_062u_092_060_094sub_062l_092_060_094sub_062l_Osimps_I5_J,axiom,
    ! [Uz: product_prod_nat_nat,Vaa: nat,Vb: list_VEBT_VEBT,Vc2: vEBT_VEBT] : aa_VEBT_VEBT_nat(vEBT_T_m_i_n_N_u_l_l,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(Uz),Vaa,Vb),Vc2)) = one_one_nat ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.simps(5)
tff(fact_4789_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062N_092_060_094sub_062u_092_060_094sub_062l_092_060_094sub_062l_Osimps_I4_J,axiom,
    ! [Uw: nat,Ux: list_VEBT_VEBT,Uy: vEBT_VEBT] : aa_VEBT_VEBT_nat(vEBT_T_m_i_n_N_u_l_l,aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uw,Ux),Uy)) = one_one_nat ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.simps(4)
tff(fact_4790_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Osimps_I3_J,axiom,
    ! [A: $o,B: $o,Vaa: nat] :
      aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_p_r_e_d,vEBT_Leaf((A),(B))),aa_nat_nat(suc,aa_nat_nat(suc,Vaa))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
        $ite((B),one_one_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat))) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.simps(3)
tff(fact_4791_word__of__int__2p,axiom,
    ! [Nb: nat] : ring_17408606157368542149l_num1(aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) = aa_nat5599461682457136004l_num1(power_2184487114949457152l_num1(numera7442385471795722001l_num1(aa_num_num(bit0,one))),Nb) ).

% word_of_int_2p
tff(fact_4792_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Osimps_I5_J,axiom,
    ! [V: product_prod_nat_nat,Vd: list_VEBT_VEBT,Ve: vEBT_VEBT,Vf: nat] : aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_p_r_e_d,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V),zero_zero_nat,Vd),Ve)),Vf) = one_one_nat ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.simps(5)
tff(fact_4793_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Osimps_I2_J,axiom,
    ! [A: $o,Uw: $o] : aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_p_r_e_d,vEBT_Leaf((A),(Uw))),aa_nat_nat(suc,zero_zero_nat)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.simps(2)
tff(fact_4794_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Osimps_I6_J,axiom,
    ! [V: product_prod_nat_nat,Vh: list_VEBT_VEBT,Vi: vEBT_VEBT,Vj: nat] : aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_p_r_e_d,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V),aa_nat_nat(suc,zero_zero_nat),Vh),Vi)),Vj) = one_one_nat ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.simps(6)
tff(fact_4795_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062N_092_060_094sub_062u_092_060_094sub_062l_092_060_094sub_062l_Oelims,axiom,
    ! [Xb: vEBT_VEBT,Y: nat] :
      ( ( aa_VEBT_VEBT_nat(vEBT_T_m_i_n_N_u_l_l,Xb) = Y )
     => ( ( ( Xb = vEBT_Leaf($false,$false) )
         => ( Y != one_one_nat ) )
       => ( ( ? [Uv2: $o] : Xb = vEBT_Leaf($true,(Uv2))
           => ( Y != one_one_nat ) )
         => ( ( ? [Uu2: $o] : Xb = vEBT_Leaf((Uu2),$true)
             => ( Y != one_one_nat ) )
           => ( ( ? [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] : Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uw2,Ux2),Uy2)
               => ( Y != one_one_nat ) )
             => ~ ( ? [Uz2: product_prod_nat_nat,Va: nat,Vb2: list_VEBT_VEBT,Vc: vEBT_VEBT] : Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(Uz2),Va,Vb2),Vc)
                 => ( Y != one_one_nat ) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.elims
tff(fact_4796_pred__bound__height,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xb: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Ta),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_p_r_e_d,Ta),Xb)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_VEBT_height,Ta))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,one))))))) ) ).

% pred_bound_height
tff(fact_4797_pred__bound__size__univ,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,U: real,Xb: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Ta),Nb)
     => ( ( U = aa_nat_real(aa_real_fun_nat_real(power_power_real,numeral_numeral_real(aa_num_num(bit0,one))),Nb) )
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_p_r_e_d,Ta),Xb))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,numeral_numeral_real(aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,one))))))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,one)))))),aa_real_real(aa_rea7340605660300375871l_real(log,numeral_numeral_real(aa_num_num(bit0,one))),aa_real_real(aa_rea7340605660300375871l_real(log,numeral_numeral_real(aa_num_num(bit0,one))),U))))) ) ) ).

% pred_bound_size_univ
tff(fact_4798_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_Osimps_I5_J,axiom,
    ! [Mia: nat,Maa: nat,Vaa: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Xb: nat] :
      aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_i_n_s_e_r_t,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mia),Maa)),aa_nat_nat(suc,aa_nat_nat(suc,Vaa)),TreeLista),Summarya)),Xb) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,one)))))),
        $let(
          xn: nat,
          xn:= 
            $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Mia),Mia,Xb),
          $let(
            h: nat,
            h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Vaa))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
            $ite(
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeLista))
              & ~ ( ( Xb = Mia )
                  | ( Xb = Maa ) ) ),
              aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_i_n_s_e_r_t,nth_VEBT_VEBT(TreeLista,h)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Vaa))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_VEBT_VEBT_nat(vEBT_T_m_i_n_N_u_l_l,nth_VEBT_VEBT(TreeLista,h)))),
                $ite(aa_VEBT_VEBT_o(vEBT_VEBT_minNull,nth_VEBT_VEBT(TreeLista,h)),aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_i_n_s_e_r_t,Summarya),h),one_one_nat)),
              one_one_nat ) ) )) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.simps(5)
tff(fact_4799_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Opelims,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_p_r_e_d,Xb),Xaa) = Y )
     => ( aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_T_p_r_e_d_rel2),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(Xb),Xaa))
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( Xb = vEBT_Leaf((Uu2),(Uv2)) )
             => ( ( Xaa = zero_zero_nat )
               => ( ( Y = one_one_nat )
                 => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_T_p_r_e_d_rel2),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(vEBT_Leaf((Uu2),(Uv2))),zero_zero_nat)) ) ) )
         => ( ! [A3: $o,Uw2: $o] :
                ( ( Xb = vEBT_Leaf((A3),(Uw2)) )
               => ( ( Xaa = aa_nat_nat(suc,zero_zero_nat) )
                 => ( ( Y = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat) )
                   => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_T_p_r_e_d_rel2),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(vEBT_Leaf((A3),(Uw2))),aa_nat_nat(suc,zero_zero_nat))) ) ) )
           => ( ! [A3: $o,B2: $o] :
                  ( ( Xb = vEBT_Leaf((A3),(B2)) )
                 => ! [Va2: nat] :
                      ( ( Xaa = aa_nat_nat(suc,aa_nat_nat(suc,Va2)) )
                     => ( ( Y = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
                              $ite((B2),one_one_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat))) )
                       => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_T_p_r_e_d_rel2),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(vEBT_Leaf((A3),(B2))),aa_nat_nat(suc,aa_nat_nat(suc,Va2)))) ) ) )
             => ( ! [Uy2: nat,Uz2: list_VEBT_VEBT,Va: vEBT_VEBT] :
                    ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uy2,Uz2),Va) )
                   => ( ( Y = one_one_nat )
                     => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_T_p_r_e_d_rel2),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uy2,Uz2),Va)),Xaa)) ) )
               => ( ! [V2: product_prod_nat_nat,Vd2: list_VEBT_VEBT,Ve2: vEBT_VEBT] :
                      ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Vd2),Ve2) )
                     => ( ( Y = one_one_nat )
                       => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_T_p_r_e_d_rel2),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Vd2),Ve2)),Xaa)) ) )
                 => ( ! [V2: product_prod_nat_nat,Vh2: list_VEBT_VEBT,Vi2: vEBT_VEBT] :
                        ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vh2),Vi2) )
                       => ( ( Y = one_one_nat )
                         => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_T_p_r_e_d_rel2),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vh2),Vi2)),Xaa)) ) )
                   => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                          ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList),Summary) )
                         => ( ( Y = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
                                  $ite(
                                    aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Xaa),
                                    one_one_nat,
                                    $let(
                                      l: nat,
                                      l:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                      $let(
                                        h: nat,
                                        h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                        aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,one))))),one_one_nat)),
                                          $ite(
                                            aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList)),
                                            $let(
                                              minlow: option_nat,
                                              minlow:= vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,h)),
                                              aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_VEBT_VEBT_nat(vEBT_T_m_i_n_t,nth_VEBT_VEBT(TreeList,h)))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)))),
                                                $ite(
                                                  ( ( minlow != none_nat )
                                                  & aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_greater,aa_nat_option_nat(some_nat,l)),minlow) ),
                                                  aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))),aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_p_r_e_d,nth_VEBT_VEBT(TreeList,h)),l)),
                                                  $let(
                                                    pr: option_nat,
                                                    pr:= aa_nat_option_nat(vEBT_vebt_pred(Summary),h),
                                                    aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_p_r_e_d,Summary),h))),one_one_nat)),
                                                      $ite(pr = none_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))),aa_VEBT_VEBT_nat(vEBT_T_m_a_x_t,nth_VEBT_VEBT(TreeList,the_nat(pr)))))) ) )) ),
                                            one_one_nat )) ) ) )) )
                           => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_T_p_r_e_d_rel2),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList),Summary)),Xaa)) ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.pelims
tff(fact_4800_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_Opelims,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_i_n_s_e_r_t,Xb),Xaa) = Y )
     => ( aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_T9217963907923527482_t_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(Xb),Xaa))
       => ( ! [A3: $o,B2: $o] :
              ( ( Xb = vEBT_Leaf((A3),(B2)) )
             => ( ( Y = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
                      $ite(Xaa = zero_zero_nat,one_one_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat))) )
               => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_T9217963907923527482_t_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(vEBT_Leaf((A3),(B2))),Xaa)) ) )
         => ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S: vEBT_VEBT] :
                ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,zero_zero_nat,Ts2),S) )
               => ( ( Y = one_one_nat )
                 => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_T9217963907923527482_t_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,zero_zero_nat,Ts2),S)),Xaa)) ) )
           => ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S: vEBT_VEBT] :
                  ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts2),S) )
                 => ( ( Y = one_one_nat )
                   => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_T9217963907923527482_t_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts2),S)),Xaa)) ) )
             => ( ! [V2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                    ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V2)),TreeList),Summary) )
                   => ( ( Y = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) )
                     => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_T9217963907923527482_t_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V2)),TreeList),Summary)),Xaa)) ) )
               => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                      ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList),Summary) )
                     => ( ( Y = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,one)))))),
                              $let(
                                xn: nat,
                                xn:= 
                                  $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi),Mi,Xaa),
                                $let(
                                  h: nat,
                                  h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                  $ite(
                                    ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList))
                                    & ~ ( ( Xaa = Mi )
                                        | ( Xaa = Ma ) ) ),
                                    aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_i_n_s_e_r_t,nth_VEBT_VEBT(TreeList,h)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_VEBT_VEBT_nat(vEBT_T_m_i_n_N_u_l_l,nth_VEBT_VEBT(TreeList,h)))),
                                      $ite(aa_VEBT_VEBT_o(vEBT_VEBT_minNull,nth_VEBT_VEBT(TreeList,h)),aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_i_n_s_e_r_t,Summary),h),one_one_nat)),
                                    one_one_nat ) ) )) )
                       => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_T9217963907923527482_t_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList),Summary)),Xaa)) ) ) ) ) ) ) ) ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.pelims
tff(fact_4801_delete__correct,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xb: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Ta),Nb)
     => ( vEBT_VEBT_set_vebt(vEBT_vebt_delete(Ta,Xb)) = minus_minus_set_nat(vEBT_set_vebt(Ta),insert_nat(Xb,bot_bot_set_nat)) ) ) ).

% delete_correct
tff(fact_4802_lemma__termdiff3,axiom,
    ! [H: real,Z: real,K6: real,Nb: nat] :
      ( ( H != zero_zero_real )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,Z)),K6)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Z),H))),K6)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Z),H)),Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Z),Nb))),H)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Z),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat))))))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat))))),aa_nat_real(aa_real_fun_nat_real(power_power_real,K6),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_real_real(real_V7735802525324610683m_real,H))) ) ) ) ).

% lemma_termdiff3
tff(fact_4803_lemma__termdiff3,axiom,
    ! [H: complex,Z: complex,K6: real,Nb: nat] :
      ( ( H != zero_zero_complex )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,Z)),K6)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,Z),H))),K6)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,Z),H)),Nb)),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Z),Nb))),H)),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_nat_complex(semiri8010041392384452111omplex,Nb)),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Z),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat))))))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat))))),aa_nat_real(aa_real_fun_nat_real(power_power_real,K6),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_complex_real(real_V1022390504157884413omplex,H))) ) ) ) ).

% lemma_termdiff3
tff(fact_4804_lowi__def,axiom,
    ! [Xb: nat,Nb: nat] : aa_nat5821310369566504398ap_nat(vEBT_VEBT_lowi(Xb),Nb) = aa_nat5821310369566504398ap_nat(heap_Time_return_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))) ).

% lowi_def
tff(fact_4805_foldr__zero,axiom,
    ! [Xs: list_nat,D: nat] :
      ( ! [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),size_size_list_nat(Xs))
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(nth_nat(Xs),I3)) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_size_list_nat(Xs)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(foldr_nat_nat(plus_plus_nat,Xs),D)),D)) ) ).

% foldr_zero
tff(fact_4806_foldr__one,axiom,
    ! [D: nat,Ys: list_nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,D),aa_nat_nat(foldr_nat_nat(plus_plus_nat,Ys),D)) ).

% foldr_one
tff(fact_4807_mod__mod__trivial,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)),B) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) ).

% mod_mod_trivial
tff(fact_4808_mod__mod__trivial,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)),B) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) ).

% mod_mod_trivial
tff(fact_4809_mod__mod__trivial,axiom,
    ! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A),B)),B) = aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A),B) ).

% mod_mod_trivial
tff(fact_4810_foldr__same__int,axiom,
    ! [Xs: list_nat,Y: nat] :
      ( ! [X2: nat,Y4: nat] :
          ( aa_set_nat_o(member_nat(X2),set_nat2(Xs))
         => ( aa_set_nat_o(member_nat(Y4),set_nat2(Xs))
           => ( X2 = Y4 ) ) )
     => ( ! [X2: nat] :
            ( aa_set_nat_o(member_nat(X2),set_nat2(Xs))
           => ( X2 = Y ) )
       => ( aa_nat_nat(foldr_nat_nat(plus_plus_nat,Xs),zero_zero_nat) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_nat(Xs)),Y) ) ) ) ).

% foldr_same_int
tff(fact_4811_low__def,axiom,
    ! [Xb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xb),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ).

% low_def
tff(fact_4812_delete__correct_H,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xb: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Ta),Nb)
     => ( vEBT_VEBT_set_vebt(vEBT_vebt_delete(Ta,Xb)) = minus_minus_set_nat(vEBT_VEBT_set_vebt(Ta),insert_nat(Xb,bot_bot_set_nat)) ) ) ).

% delete_correct'
tff(fact_4813_foldr__mono,axiom,
    ! [Xs: list_nat,Ys: list_nat,C: nat,D: nat] :
      ( ( size_size_list_nat(Xs) = size_size_list_nat(Ys) )
     => ( ! [I3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),size_size_list_nat(Xs))
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(nth_nat(Xs),I3)),aa_nat_nat(nth_nat(Ys),I3)) )
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),D)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(foldr_nat_nat(plus_plus_nat,Xs),C)),size_size_list_nat(Ys))),aa_nat_nat(foldr_nat_nat(plus_plus_nat,Ys),D)) ) ) ) ).

% foldr_mono
tff(fact_4814_mod__self,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),A) = zero_zero_nat ).

% mod_self
tff(fact_4815_mod__self,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),A) = zero_zero_int ).

% mod_self
tff(fact_4816_mod__self,axiom,
    ! [A: code_integer] : aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A),A) = zero_z3403309356797280102nteger ).

% mod_self
tff(fact_4817_mod__by__0,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),zero_zero_nat) = A ).

% mod_by_0
tff(fact_4818_mod__by__0,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),zero_zero_int) = A ).

% mod_by_0
tff(fact_4819_mod__by__0,axiom,
    ! [A: code_integer] : aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A),zero_z3403309356797280102nteger) = A ).

% mod_by_0
tff(fact_4820_mod__0,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,zero_zero_nat),A) = zero_zero_nat ).

% mod_0
tff(fact_4821_mod__0,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,zero_zero_int),A) = zero_zero_int ).

% mod_0
tff(fact_4822_mod__0,axiom,
    ! [A: code_integer] : aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(zero_z3403309356797280102nteger),A) = zero_z3403309356797280102nteger ).

% mod_0
tff(fact_4823_mod__add__self2,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),B) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) ).

% mod_add_self2
tff(fact_4824_mod__add__self2,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),B) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) ).

% mod_add_self2
tff(fact_4825_mod__add__self2,axiom,
    ! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(A),B)),B) = aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A),B) ).

% mod_add_self2
tff(fact_4826_mod__add__self1,axiom,
    ! [B: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A)),B) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) ).

% mod_add_self1
tff(fact_4827_mod__add__self1,axiom,
    ! [B: int,A: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A)),B) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) ).

% mod_add_self1
tff(fact_4828_mod__add__self1,axiom,
    ! [B: code_integer,A: code_integer] : aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(B),A)),B) = aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A),B) ).

% mod_add_self1
tff(fact_4829_minus__mod__self2,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),B) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) ).

% minus_mod_self2
tff(fact_4830_minus__mod__self2,axiom,
    ! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(A),B)),B) = aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A),B) ).

% minus_mod_self2
tff(fact_4831_insert__subset,axiom,
    ! [Xb: nat,A2: set_nat,B4: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(insert_nat(Xb,A2)),B4)
    <=> ( aa_set_nat_o(member_nat(Xb),B4)
        & aa_set_nat_o(ord_less_eq_set_nat(A2),B4) ) ) ).

% insert_subset
tff(fact_4832_insert__subset,axiom,
    ! [Xb: vEBT_VEBT,A2: set_VEBT_VEBT,B4: set_VEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(insert_VEBT_VEBT(Xb,A2)),B4)
    <=> ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xb),B4)
        & aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(A2),B4) ) ) ).

% insert_subset
tff(fact_4833_insert__subset,axiom,
    ! [Xb: real,A2: set_real,B4: set_real] :
      ( aa_set_real_o(ord_less_eq_set_real(insert_real(Xb,A2)),B4)
    <=> ( aa_set_real_o(member_real(Xb),B4)
        & aa_set_real_o(ord_less_eq_set_real(A2),B4) ) ) ).

% insert_subset
tff(fact_4834_insert__subset,axiom,
    ! [Xb: int,A2: set_int,B4: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(insert_int(Xb,A2)),B4)
    <=> ( aa_set_int_o(member_int(Xb),B4)
        & aa_set_int_o(ord_less_eq_set_int(A2),B4) ) ) ).

% insert_subset
tff(fact_4835_insert__Diff1,axiom,
    ! [Xb: vEBT_VEBT,B4: set_VEBT_VEBT,A2: set_VEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xb),B4)
     => ( minus_5127226145743854075T_VEBT(insert_VEBT_VEBT(Xb,A2),B4) = minus_5127226145743854075T_VEBT(A2,B4) ) ) ).

% insert_Diff1
tff(fact_4836_insert__Diff1,axiom,
    ! [Xb: real,B4: set_real,A2: set_real] :
      ( aa_set_real_o(member_real(Xb),B4)
     => ( minus_minus_set_real(insert_real(Xb,A2),B4) = minus_minus_set_real(A2,B4) ) ) ).

% insert_Diff1
tff(fact_4837_insert__Diff1,axiom,
    ! [Xb: int,B4: set_int,A2: set_int] :
      ( aa_set_int_o(member_int(Xb),B4)
     => ( minus_minus_set_int(insert_int(Xb,A2),B4) = minus_minus_set_int(A2,B4) ) ) ).

% insert_Diff1
tff(fact_4838_insert__Diff1,axiom,
    ! [Xb: nat,B4: set_nat,A2: set_nat] :
      ( aa_set_nat_o(member_nat(Xb),B4)
     => ( minus_minus_set_nat(insert_nat(Xb,A2),B4) = minus_minus_set_nat(A2,B4) ) ) ).

% insert_Diff1
tff(fact_4839_Diff__insert0,axiom,
    ! [Xb: vEBT_VEBT,A2: set_VEBT_VEBT,B4: set_VEBT_VEBT] :
      ( ~ aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xb),A2)
     => ( minus_5127226145743854075T_VEBT(A2,insert_VEBT_VEBT(Xb,B4)) = minus_5127226145743854075T_VEBT(A2,B4) ) ) ).

% Diff_insert0
tff(fact_4840_Diff__insert0,axiom,
    ! [Xb: real,A2: set_real,B4: set_real] :
      ( ~ aa_set_real_o(member_real(Xb),A2)
     => ( minus_minus_set_real(A2,insert_real(Xb,B4)) = minus_minus_set_real(A2,B4) ) ) ).

% Diff_insert0
tff(fact_4841_Diff__insert0,axiom,
    ! [Xb: int,A2: set_int,B4: set_int] :
      ( ~ aa_set_int_o(member_int(Xb),A2)
     => ( minus_minus_set_int(A2,insert_int(Xb,B4)) = minus_minus_set_int(A2,B4) ) ) ).

% Diff_insert0
tff(fact_4842_Diff__insert0,axiom,
    ! [Xb: nat,A2: set_nat,B4: set_nat] :
      ( ~ aa_set_nat_o(member_nat(Xb),A2)
     => ( minus_minus_set_nat(A2,insert_nat(Xb,B4)) = minus_minus_set_nat(A2,B4) ) ) ).

% Diff_insert0
tff(fact_4843_mod__less,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Nb) = M ) ) ).

% mod_less
tff(fact_4844_nat__mod__eq_H,axiom,
    ! [A: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),Nb) = A ) ) ).

% nat_mod_eq'
tff(fact_4845_singleton__conv,axiom,
    ! [A: vEBT_VEBT] : collect_VEBT_VEBT(aTP_Lamm_hh(A)) = insert_VEBT_VEBT(A,bot_bo8194388402131092736T_VEBT) ).

% singleton_conv
tff(fact_4846_singleton__conv,axiom,
    ! [A: complex] : collect_complex(aTP_Lamm_hi(A)) = insert_complex(A,bot_bot_set_complex) ).

% singleton_conv
tff(fact_4847_singleton__conv,axiom,
    ! [A: product_prod_int_int] : collec213857154873943460nt_int(aa_Pro3952998155787061599_int_o(aTP_Lamm_hj,A)) = insert5033312907999012233nt_int(A,bot_bo1796632182523588997nt_int) ).

% singleton_conv
tff(fact_4848_singleton__conv,axiom,
    ! [A: nat] : collect_nat(aa_nat_fun_nat_o(aTP_Lamm_hk,A)) = insert_nat(A,bot_bot_set_nat) ).

% singleton_conv
tff(fact_4849_singleton__conv,axiom,
    ! [A: int] : collect_int(aa_int_fun_int_o(aTP_Lamm_hl,A)) = insert_int(A,bot_bot_set_int) ).

% singleton_conv
tff(fact_4850_singleton__conv,axiom,
    ! [A: real] : collect_real(aa_real_fun_real_o(aTP_Lamm_hm,A)) = insert_real(A,bot_bot_set_real) ).

% singleton_conv
tff(fact_4851_singleton__conv2,axiom,
    ! [A: vEBT_VEBT] : collect_VEBT_VEBT(fequal_VEBT_VEBT(A)) = insert_VEBT_VEBT(A,bot_bo8194388402131092736T_VEBT) ).

% singleton_conv2
tff(fact_4852_singleton__conv2,axiom,
    ! [A: complex] : collect_complex(fequal_complex(A)) = insert_complex(A,bot_bot_set_complex) ).

% singleton_conv2
tff(fact_4853_singleton__conv2,axiom,
    ! [A: product_prod_int_int] : collec213857154873943460nt_int(aa_Pro3952998155787061599_int_o(fequal6010348034681699726nt_int,A)) = insert5033312907999012233nt_int(A,bot_bo1796632182523588997nt_int) ).

% singleton_conv2
tff(fact_4854_singleton__conv2,axiom,
    ! [A: nat] : collect_nat(aa_nat_fun_nat_o(fequal_nat,A)) = insert_nat(A,bot_bot_set_nat) ).

% singleton_conv2
tff(fact_4855_singleton__conv2,axiom,
    ! [A: int] : collect_int(aa_int_fun_int_o(fequal_int,A)) = insert_int(A,bot_bot_set_int) ).

% singleton_conv2
tff(fact_4856_singleton__conv2,axiom,
    ! [A: real] : collect_real(aa_real_fun_real_o(fequal_real,A)) = insert_real(A,bot_bot_set_real) ).

% singleton_conv2
tff(fact_4857_bits__mod__by__1,axiom,
    ! [A: word_N3645301735248828278l_num1] : modulo1504961113040953224l_num1(A,one_on7727431528512463931l_num1) = zero_z3563351764282998399l_num1 ).

% bits_mod_by_1
tff(fact_4858_bits__mod__by__1,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),one_one_nat) = zero_zero_nat ).

% bits_mod_by_1
tff(fact_4859_bits__mod__by__1,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),one_one_int) = zero_zero_int ).

% bits_mod_by_1
tff(fact_4860_bits__mod__by__1,axiom,
    ! [A: code_integer] : aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A),one_one_Code_integer) = zero_z3403309356797280102nteger ).

% bits_mod_by_1
tff(fact_4861_bits__mod__by__1,axiom,
    ! [A: uint32] : aa_uint32_uint32(aa_uin3778660748068847871uint32(modulo_modulo_uint32,A),one_one_uint32) = zero_zero_uint32 ).

% bits_mod_by_1
tff(fact_4862_mod__by__1,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),one_one_nat) = zero_zero_nat ).

% mod_by_1
tff(fact_4863_mod__by__1,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),one_one_int) = zero_zero_int ).

% mod_by_1
tff(fact_4864_mod__by__1,axiom,
    ! [A: code_integer] : aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A),one_one_Code_integer) = zero_z3403309356797280102nteger ).

% mod_by_1
tff(fact_4865_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_4866_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_4867_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_4868_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_4869_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_4870_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_4871_bits__mod__div__trivial,axiom,
    ! [A: word_N3645301735248828278l_num1,B: word_N3645301735248828278l_num1] : divide1791077408188789448l_num1(modulo1504961113040953224l_num1(A,B),B) = zero_z3563351764282998399l_num1 ).

% bits_mod_div_trivial
tff(fact_4872_bits__mod__div__trivial,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)),B) = zero_zero_nat ).

% bits_mod_div_trivial
tff(fact_4873_bits__mod__div__trivial,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)),B) = zero_zero_int ).

% bits_mod_div_trivial
tff(fact_4874_bits__mod__div__trivial,axiom,
    ! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A),B)),B) = zero_z3403309356797280102nteger ).

% bits_mod_div_trivial
tff(fact_4875_bits__mod__div__trivial,axiom,
    ! [A: uint32,B: uint32] : aa_uint32_uint32(aa_uin3778660748068847871uint32(divide_divide_uint32,aa_uint32_uint32(aa_uin3778660748068847871uint32(modulo_modulo_uint32,A),B)),B) = zero_zero_uint32 ).

% bits_mod_div_trivial
tff(fact_4876_mod__div__trivial,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)),B) = zero_zero_nat ).

% mod_div_trivial
tff(fact_4877_mod__div__trivial,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)),B) = zero_zero_int ).

% mod_div_trivial
tff(fact_4878_mod__div__trivial,axiom,
    ! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A),B)),B) = zero_z3403309356797280102nteger ).

% mod_div_trivial
tff(fact_4879_mod__mult__self1,axiom,
    ! [A: nat,C: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B))),B) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) ).

% mod_mult_self1
tff(fact_4880_mod__mult__self1,axiom,
    ! [A: int,C: int,B: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))),B) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) ).

% mod_mult_self1
tff(fact_4881_mod__mult__self1,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B))),B) = aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A),B) ).

% mod_mult_self1
tff(fact_4882_mod__mult__self2,axiom,
    ! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C))),B) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) ).

% mod_mult_self2
tff(fact_4883_mod__mult__self2,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C))),B) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) ).

% mod_mult_self2
tff(fact_4884_mod__mult__self2,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] : aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C))),B) = aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A),B) ).

% mod_mult_self2
tff(fact_4885_mod__mult__self3,axiom,
    ! [C: nat,B: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B)),A)),B) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) ).

% mod_mult_self3
tff(fact_4886_mod__mult__self3,axiom,
    ! [C: int,B: int,A: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),B)),A)),B) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) ).

% mod_mult_self3
tff(fact_4887_mod__mult__self3,axiom,
    ! [C: code_integer,B: code_integer,A: code_integer] : aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B)),A)),B) = aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A),B) ).

% mod_mult_self3
tff(fact_4888_mod__mult__self4,axiom,
    ! [B: nat,C: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)),A)),B) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) ).

% mod_mult_self4
tff(fact_4889_mod__mult__self4,axiom,
    ! [B: int,C: int,A: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),C)),A)),B) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) ).

% mod_mult_self4
tff(fact_4890_mod__mult__self4,axiom,
    ! [B: code_integer,C: code_integer,A: code_integer] : aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)),A)),B) = aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A),B) ).

% mod_mult_self4
tff(fact_4891_singleton__insert__inj__eq,axiom,
    ! [B: vEBT_VEBT,A: vEBT_VEBT,A2: set_VEBT_VEBT] :
      ( ( insert_VEBT_VEBT(B,bot_bo8194388402131092736T_VEBT) = insert_VEBT_VEBT(A,A2) )
    <=> ( ( A = B )
        & aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(A2),insert_VEBT_VEBT(B,bot_bo8194388402131092736T_VEBT)) ) ) ).

% singleton_insert_inj_eq
tff(fact_4892_singleton__insert__inj__eq,axiom,
    ! [B: nat,A: nat,A2: set_nat] :
      ( ( insert_nat(B,bot_bot_set_nat) = insert_nat(A,A2) )
    <=> ( ( A = B )
        & aa_set_nat_o(ord_less_eq_set_nat(A2),insert_nat(B,bot_bot_set_nat)) ) ) ).

% singleton_insert_inj_eq
tff(fact_4893_singleton__insert__inj__eq,axiom,
    ! [B: real,A: real,A2: set_real] :
      ( ( insert_real(B,bot_bot_set_real) = insert_real(A,A2) )
    <=> ( ( A = B )
        & aa_set_real_o(ord_less_eq_set_real(A2),insert_real(B,bot_bot_set_real)) ) ) ).

% singleton_insert_inj_eq
tff(fact_4894_singleton__insert__inj__eq,axiom,
    ! [B: int,A: int,A2: set_int] :
      ( ( insert_int(B,bot_bot_set_int) = insert_int(A,A2) )
    <=> ( ( A = B )
        & aa_set_int_o(ord_less_eq_set_int(A2),insert_int(B,bot_bot_set_int)) ) ) ).

% singleton_insert_inj_eq
tff(fact_4895_singleton__insert__inj__eq_H,axiom,
    ! [A: vEBT_VEBT,A2: set_VEBT_VEBT,B: vEBT_VEBT] :
      ( ( insert_VEBT_VEBT(A,A2) = insert_VEBT_VEBT(B,bot_bo8194388402131092736T_VEBT) )
    <=> ( ( A = B )
        & aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(A2),insert_VEBT_VEBT(B,bot_bo8194388402131092736T_VEBT)) ) ) ).

% singleton_insert_inj_eq'
tff(fact_4896_singleton__insert__inj__eq_H,axiom,
    ! [A: nat,A2: set_nat,B: nat] :
      ( ( insert_nat(A,A2) = insert_nat(B,bot_bot_set_nat) )
    <=> ( ( A = B )
        & aa_set_nat_o(ord_less_eq_set_nat(A2),insert_nat(B,bot_bot_set_nat)) ) ) ).

% singleton_insert_inj_eq'
tff(fact_4897_singleton__insert__inj__eq_H,axiom,
    ! [A: real,A2: set_real,B: real] :
      ( ( insert_real(A,A2) = insert_real(B,bot_bot_set_real) )
    <=> ( ( A = B )
        & aa_set_real_o(ord_less_eq_set_real(A2),insert_real(B,bot_bot_set_real)) ) ) ).

% singleton_insert_inj_eq'
tff(fact_4898_singleton__insert__inj__eq_H,axiom,
    ! [A: int,A2: set_int,B: int] :
      ( ( insert_int(A,A2) = insert_int(B,bot_bot_set_int) )
    <=> ( ( A = B )
        & aa_set_int_o(ord_less_eq_set_int(A2),insert_int(B,bot_bot_set_int)) ) ) ).

% singleton_insert_inj_eq'
tff(fact_4899_mod__by__Suc__0,axiom,
    ! [M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_nat_nat(suc,zero_zero_nat)) = zero_zero_nat ).

% mod_by_Suc_0
tff(fact_4900_insert__Diff__single,axiom,
    ! [A: vEBT_VEBT,A2: set_VEBT_VEBT] : insert_VEBT_VEBT(A,minus_5127226145743854075T_VEBT(A2,insert_VEBT_VEBT(A,bot_bo8194388402131092736T_VEBT))) = insert_VEBT_VEBT(A,A2) ).

% insert_Diff_single
tff(fact_4901_insert__Diff__single,axiom,
    ! [A: int,A2: set_int] : insert_int(A,minus_minus_set_int(A2,insert_int(A,bot_bot_set_int))) = insert_int(A,A2) ).

% insert_Diff_single
tff(fact_4902_insert__Diff__single,axiom,
    ! [A: real,A2: set_real] : insert_real(A,minus_minus_set_real(A2,insert_real(A,bot_bot_set_real))) = insert_real(A,A2) ).

% insert_Diff_single
tff(fact_4903_insert__Diff__single,axiom,
    ! [A: nat,A2: set_nat] : insert_nat(A,minus_minus_set_nat(A2,insert_nat(A,bot_bot_set_nat))) = insert_nat(A,A2) ).

% insert_Diff_single
tff(fact_4904_Suc__mod__mult__self1,axiom,
    ! [M: nat,K: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Nb)))),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,M)),Nb) ).

% Suc_mod_mult_self1
tff(fact_4905_Suc__mod__mult__self2,axiom,
    ! [M: nat,Nb: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),K)))),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,M)),Nb) ).

% Suc_mod_mult_self2
tff(fact_4906_Suc__mod__mult__self3,axiom,
    ! [K: nat,Nb: nat,M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Nb)),M))),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,M)),Nb) ).

% Suc_mod_mult_self3
tff(fact_4907_Suc__mod__mult__self4,axiom,
    ! [Nb: nat,K: nat,M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),K)),M))),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,M)),Nb) ).

% Suc_mod_mult_self4
tff(fact_4908_foldr__length,axiom,
    ! [L: list_real] : aa_nat_nat(foldr_real_nat(aTP_Lamm_hn,L),zero_zero_nat) = size_size_list_real(L) ).

% foldr_length
tff(fact_4909_foldr__length,axiom,
    ! [L: list_o] : aa_nat_nat(foldr_o_nat(aTP_Lamm_ho,L),zero_zero_nat) = size_size_list_o(L) ).

% foldr_length
tff(fact_4910_foldr__length,axiom,
    ! [L: list_nat] : aa_nat_nat(foldr_nat_nat(aTP_Lamm_hp,L),zero_zero_nat) = size_size_list_nat(L) ).

% foldr_length
tff(fact_4911_foldr__length,axiom,
    ! [L: list_int] : aa_nat_nat(foldr_int_nat(aTP_Lamm_hq,L),zero_zero_nat) = size_size_list_int(L) ).

% foldr_length
tff(fact_4912_bits__one__mod__two__eq__one,axiom,
    modulo1504961113040953224l_num1(one_on7727431528512463931l_num1,numera7442385471795722001l_num1(aa_num_num(bit0,one))) = one_on7727431528512463931l_num1 ).

% bits_one_mod_two_eq_one
tff(fact_4913_bits__one__mod__two__eq__one,axiom,
    aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_nat ).

% bits_one_mod_two_eq_one
tff(fact_4914_bits__one__mod__two__eq__one,axiom,
    aa_int_int(aa_int_fun_int_int(modulo_modulo_int,one_one_int),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = one_one_int ).

% bits_one_mod_two_eq_one
tff(fact_4915_bits__one__mod__two__eq__one,axiom,
    aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(one_one_Code_integer),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) = one_one_Code_integer ).

% bits_one_mod_two_eq_one
tff(fact_4916_bits__one__mod__two__eq__one,axiom,
    aa_uint32_uint32(aa_uin3778660748068847871uint32(modulo_modulo_uint32,one_one_uint32),numera9087168376688890119uint32(aa_num_num(bit0,one))) = one_one_uint32 ).

% bits_one_mod_two_eq_one
tff(fact_4917_one__mod__two__eq__one,axiom,
    aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_nat ).

% one_mod_two_eq_one
tff(fact_4918_one__mod__two__eq__one,axiom,
    aa_int_int(aa_int_fun_int_int(modulo_modulo_int,one_one_int),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = one_one_int ).

% one_mod_two_eq_one
tff(fact_4919_one__mod__two__eq__one,axiom,
    aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(one_one_Code_integer),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) = one_one_Code_integer ).

% one_mod_two_eq_one
tff(fact_4920_mod2__Suc__Suc,axiom,
    ! [M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(suc,M))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% mod2_Suc_Suc
tff(fact_4921_Suc__times__numeral__mod__eq,axiom,
    ! [K: num,Nb: nat] :
      ( ( aa_num_nat(numeral_numeral_nat,K) != one_one_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,K)),Nb))),aa_num_nat(numeral_numeral_nat,K)) = one_one_nat ) ) ).

% Suc_times_numeral_mod_eq
tff(fact_4922_not__mod__2__eq__1__eq__0,axiom,
    ! [A: word_N3645301735248828278l_num1] :
      ( ( modulo1504961113040953224l_num1(A,numera7442385471795722001l_num1(aa_num_num(bit0,one))) != one_on7727431528512463931l_num1 )
    <=> ( modulo1504961113040953224l_num1(A,numera7442385471795722001l_num1(aa_num_num(bit0,one))) = zero_z3563351764282998399l_num1 ) ) ).

% not_mod_2_eq_1_eq_0
tff(fact_4923_not__mod__2__eq__1__eq__0,axiom,
    ! [A: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) != one_one_nat )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_nat ) ) ).

% not_mod_2_eq_1_eq_0
tff(fact_4924_not__mod__2__eq__1__eq__0,axiom,
    ! [A: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) != one_one_int )
    <=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = zero_zero_int ) ) ).

% not_mod_2_eq_1_eq_0
tff(fact_4925_not__mod__2__eq__1__eq__0,axiom,
    ! [A: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) != one_one_Code_integer )
    <=> ( aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) = zero_z3403309356797280102nteger ) ) ).

% not_mod_2_eq_1_eq_0
tff(fact_4926_not__mod__2__eq__1__eq__0,axiom,
    ! [A: uint32] :
      ( ( aa_uint32_uint32(aa_uin3778660748068847871uint32(modulo_modulo_uint32,A),numera9087168376688890119uint32(aa_num_num(bit0,one))) != one_one_uint32 )
    <=> ( aa_uint32_uint32(aa_uin3778660748068847871uint32(modulo_modulo_uint32,A),numera9087168376688890119uint32(aa_num_num(bit0,one))) = zero_zero_uint32 ) ) ).

% not_mod_2_eq_1_eq_0
tff(fact_4927_not__mod__2__eq__0__eq__1,axiom,
    ! [A: word_N3645301735248828278l_num1] :
      ( ( modulo1504961113040953224l_num1(A,numera7442385471795722001l_num1(aa_num_num(bit0,one))) != zero_z3563351764282998399l_num1 )
    <=> ( modulo1504961113040953224l_num1(A,numera7442385471795722001l_num1(aa_num_num(bit0,one))) = one_on7727431528512463931l_num1 ) ) ).

% not_mod_2_eq_0_eq_1
tff(fact_4928_not__mod__2__eq__0__eq__1,axiom,
    ! [A: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) != zero_zero_nat )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_nat ) ) ).

% not_mod_2_eq_0_eq_1
tff(fact_4929_not__mod__2__eq__0__eq__1,axiom,
    ! [A: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) != zero_zero_int )
    <=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = one_one_int ) ) ).

% not_mod_2_eq_0_eq_1
tff(fact_4930_not__mod__2__eq__0__eq__1,axiom,
    ! [A: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) != zero_z3403309356797280102nteger )
    <=> ( aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) = one_one_Code_integer ) ) ).

% not_mod_2_eq_0_eq_1
tff(fact_4931_not__mod__2__eq__0__eq__1,axiom,
    ! [A: uint32] :
      ( ( aa_uint32_uint32(aa_uin3778660748068847871uint32(modulo_modulo_uint32,A),numera9087168376688890119uint32(aa_num_num(bit0,one))) != zero_zero_uint32 )
    <=> ( aa_uint32_uint32(aa_uin3778660748068847871uint32(modulo_modulo_uint32,A),numera9087168376688890119uint32(aa_num_num(bit0,one))) = one_one_uint32 ) ) ).

% not_mod_2_eq_0_eq_1
tff(fact_4932_not__mod2__eq__Suc__0__eq__0,axiom,
    ! [Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) != aa_nat_nat(suc,zero_zero_nat) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_nat ) ) ).

% not_mod2_eq_Suc_0_eq_0
tff(fact_4933_add__self__mod__2,axiom,
    ! [M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),M)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_nat ).

% add_self_mod_2
tff(fact_4934_Suc__mod__eq__add3__mod__numeral,axiom,
    ! [M: nat,V: num] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,M)))),aa_num_nat(numeral_numeral_nat,V)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),M)),aa_num_nat(numeral_numeral_nat,V)) ).

% Suc_mod_eq_add3_mod_numeral
tff(fact_4935_mod__Suc__eq__mod__add3,axiom,
    ! [M: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,Nb)))) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),Nb)) ).

% mod_Suc_eq_mod_add3
tff(fact_4936_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_4937_of__nat__mod,axiom,
    ! [M: nat,Nb: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Nb)) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,Nb)) ).

% of_nat_mod
tff(fact_4938_of__nat__mod,axiom,
    ! [M: nat,Nb: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,Nb)) ).

% of_nat_mod
tff(fact_4939_of__nat__mod,axiom,
    ! [M: nat,Nb: nat] : aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Nb)) = aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,M)),aa_nat_Code_integer(semiri4939895301339042750nteger,Nb)) ).

% of_nat_mod
tff(fact_4940_insert__Collect,axiom,
    ! [A: vEBT_VEBT,P: fun_VEBT_VEBT_o] : insert_VEBT_VEBT(A,collect_VEBT_VEBT(P)) = collect_VEBT_VEBT(aa_fun8193769468198689164VEBT_o(aTP_Lamm_hr(A),P)) ).

% insert_Collect
tff(fact_4941_insert__Collect,axiom,
    ! [A: real,P: fun_real_o] : insert_real(A,collect_real(P)) = collect_real(aa_fun5868358341814070780real_o(aTP_Lamm_hs(A),P)) ).

% insert_Collect
tff(fact_4942_insert__Collect,axiom,
    ! [A: nat,P: fun_nat_o] : insert_nat(A,collect_nat(P)) = collect_nat(aa_fun5937193402907943164_nat_o(aTP_Lamm_ht(A),P)) ).

% insert_Collect
tff(fact_4943_insert__Collect,axiom,
    ! [A: int,P: fun_int_o] : insert_int(A,collect_int(P)) = collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_hu(A),P)) ).

% insert_Collect
tff(fact_4944_insert__Collect,axiom,
    ! [A: complex,P: fun_complex_o] : insert_complex(A,collect_complex(P)) = collect_complex(aa_fun2514317545224092796plex_o(aTP_Lamm_hv(A),P)) ).

% insert_Collect
tff(fact_4945_insert__Collect,axiom,
    ! [A: product_prod_int_int,P: fun_Pr3238374608701176162_int_o] : insert5033312907999012233nt_int(A,collec213857154873943460nt_int(P)) = collec213857154873943460nt_int(aa_fun5349570537165229698_int_o(aTP_Lamm_hw(A),P)) ).

% insert_Collect
tff(fact_4946_insert__compr,axiom,
    ! [A: vEBT_VEBT,B4: set_VEBT_VEBT] : insert_VEBT_VEBT(A,B4) = collect_VEBT_VEBT(aa_set2497645856053385599VEBT_o(aTP_Lamm_hx(A),B4)) ).

% insert_compr
tff(fact_4947_insert__compr,axiom,
    ! [A: real,B4: set_real] : insert_real(A,B4) = collect_real(aa_set2323979645938365349real_o(aTP_Lamm_hy(A),B4)) ).

% insert_compr
tff(fact_4948_insert__compr,axiom,
    ! [A: nat,B4: set_nat] : insert_nat(A,B4) = collect_nat(aa_set_nat_fun_nat_o(aTP_Lamm_hz(A),B4)) ).

% insert_compr
tff(fact_4949_insert__compr,axiom,
    ! [A: int,B4: set_int] : insert_int(A,B4) = collect_int(aa_set_int_fun_int_o(aTP_Lamm_ia(A),B4)) ).

% insert_compr
tff(fact_4950_insert__compr,axiom,
    ! [A: complex,B4: set_complex] : insert_complex(A,B4) = collect_complex(aa_set1742385355640705825plex_o(aTP_Lamm_ib(A),B4)) ).

% insert_compr
tff(fact_4951_insert__compr,axiom,
    ! [A: product_prod_int_int,B4: set_Pr958786334691620121nt_int] : insert5033312907999012233nt_int(A,B4) = collec213857154873943460nt_int(aa_set2224860389584585407_int_o(aTP_Lamm_ic(A),B4)) ).

% insert_compr
tff(fact_4952_mod__add__right__eq,axiom,
    ! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B),C))),C) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),C) ).

% mod_add_right_eq
tff(fact_4953_mod__add__right__eq,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B),C))),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C) ).

% mod_add_right_eq
tff(fact_4954_mod__add__right__eq,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] : aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(A),aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(B),C))),C) = aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(A),B)),C) ).

% mod_add_right_eq
tff(fact_4955_mod__add__left__eq,axiom,
    ! [A: nat,C: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),C)),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),C) ).

% mod_add_left_eq
tff(fact_4956_mod__add__left__eq,axiom,
    ! [A: int,C: int,B: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),C)),B)),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C) ).

% mod_add_left_eq
tff(fact_4957_mod__add__left__eq,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A),C)),B)),C) = aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(A),B)),C) ).

% mod_add_left_eq
tff(fact_4958_mod__add__cong,axiom,
    ! [A: nat,C: nat,A5: nat,B: nat,B3: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),C) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A5),C) )
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B),C) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B3),C) )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A5),B3)),C) ) ) ) ).

% mod_add_cong
tff(fact_4959_mod__add__cong,axiom,
    ! [A: int,C: int,A5: int,B: int,B3: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A5),C) )
     => ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B3),C) )
       => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A5),B3)),C) ) ) ) ).

% mod_add_cong
tff(fact_4960_mod__add__cong,axiom,
    ! [A: code_integer,C: code_integer,A5: code_integer,B: code_integer,B3: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A),C) = aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A5),C) )
     => ( ( aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(B),C) = aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(B3),C) )
       => ( aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(A),B)),C) = aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(A5),B3)),C) ) ) ) ).

% mod_add_cong
tff(fact_4961_mod__add__eq,axiom,
    ! [A: nat,C: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B),C))),C) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),C) ).

% mod_add_eq
tff(fact_4962_mod__add__eq,axiom,
    ! [A: int,C: int,B: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),C)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B),C))),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C) ).

% mod_add_eq
tff(fact_4963_mod__add__eq,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A),C)),aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(B),C))),C) = aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(A),B)),C) ).

% mod_add_eq
tff(fact_4964_mod__mult__right__eq,axiom,
    ! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B),C))),C) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C) ).

% mod_mult_right_eq
tff(fact_4965_mod__mult__right__eq,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B),C))),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),C) ).

% mod_mult_right_eq
tff(fact_4966_mod__mult__right__eq,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] : aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(B),C))),C) = aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),C) ).

% mod_mult_right_eq
tff(fact_4967_mod__mult__left__eq,axiom,
    ! [A: nat,C: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),C)),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C) ).

% mod_mult_left_eq
tff(fact_4968_mod__mult__left__eq,axiom,
    ! [A: int,C: int,B: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),C)),B)),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),C) ).

% mod_mult_left_eq
tff(fact_4969_mod__mult__left__eq,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A),C)),B)),C) = aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),C) ).

% mod_mult_left_eq
tff(fact_4970_mult__mod__right,axiom,
    ! [C: nat,A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B)) ).

% mult_mod_right
tff(fact_4971_mult__mod__right,axiom,
    ! [C: int,A: int,B: int] : aa_int_int(aa_int_fun_int_int(times_times_int,C),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B)) ).

% mult_mod_right
tff(fact_4972_mult__mod__right,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A),B)) = aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B)) ).

% mult_mod_right
tff(fact_4973_mod__mult__mult2,axiom,
    ! [A: nat,C: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)),C) ).

% mod_mult_mult2
tff(fact_4974_mod__mult__mult2,axiom,
    ! [A: int,C: int,B: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)),C) ).

% mod_mult_mult2
tff(fact_4975_mod__mult__mult2,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A),B)),C) ).

% mod_mult_mult2
tff(fact_4976_mod__mult__cong,axiom,
    ! [A: nat,C: nat,A5: nat,B: nat,B3: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),C) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A5),C) )
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B),C) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B3),C) )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A5),B3)),C) ) ) ) ).

% mod_mult_cong
tff(fact_4977_mod__mult__cong,axiom,
    ! [A: int,C: int,A5: int,B: int,B3: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A5),C) )
     => ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B3),C) )
       => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(times_times_int,A5),B3)),C) ) ) ) ).

% mod_mult_cong
tff(fact_4978_mod__mult__cong,axiom,
    ! [A: code_integer,C: code_integer,A5: code_integer,B: code_integer,B3: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A),C) = aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A5),C) )
     => ( ( aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(B),C) = aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(B3),C) )
       => ( aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),C) = aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A5),B3)),C) ) ) ) ).

% mod_mult_cong
tff(fact_4979_mod__mult__eq,axiom,
    ! [A: nat,C: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B),C))),C) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C) ).

% mod_mult_eq
tff(fact_4980_mod__mult__eq,axiom,
    ! [A: int,C: int,B: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),C)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B),C))),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),C) ).

% mod_mult_eq
tff(fact_4981_mod__mult__eq,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A),C)),aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(B),C))),C) = aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),C) ).

% mod_mult_eq
tff(fact_4982_mod__diff__right__eq,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B),C))),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),C) ).

% mod_diff_right_eq
tff(fact_4983_mod__diff__right__eq,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] : aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(A),aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(B),C))),C) = aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(A),B)),C) ).

% mod_diff_right_eq
tff(fact_4984_mod__diff__left__eq,axiom,
    ! [A: int,C: int,B: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),C)),B)),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),C) ).

% mod_diff_left_eq
tff(fact_4985_mod__diff__left__eq,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A),C)),B)),C) = aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(A),B)),C) ).

% mod_diff_left_eq
tff(fact_4986_mod__diff__cong,axiom,
    ! [A: int,C: int,A5: int,B: int,B3: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A5),C) )
     => ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B3),C) )
       => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A5),B3)),C) ) ) ) ).

% mod_diff_cong
tff(fact_4987_mod__diff__cong,axiom,
    ! [A: code_integer,C: code_integer,A5: code_integer,B: code_integer,B3: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A),C) = aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A5),C) )
     => ( ( aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(B),C) = aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(B3),C) )
       => ( aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(A),B)),C) = aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(A5),B3)),C) ) ) ) ).

% mod_diff_cong
tff(fact_4988_mod__diff__eq,axiom,
    ! [A: int,C: int,B: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),C)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B),C))),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),C) ).

% mod_diff_eq
tff(fact_4989_mod__diff__eq,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A),C)),aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(B),C))),C) = aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(A),B)),C) ).

% mod_diff_eq
tff(fact_4990_power__mod,axiom,
    ! [A: nat,B: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)),Nb)),B) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)),B) ).

% power_mod
tff(fact_4991_power__mod,axiom,
    ! [A: int,B: int,Nb: nat] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)),Nb)),B) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A),Nb)),B) ).

% power_mod
tff(fact_4992_power__mod,axiom,
    ! [A: code_integer,B: code_integer,Nb: nat] : aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A),B)),Nb)),B) = aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_nat_Code_integer(power_8256067586552552935nteger(A),Nb)),B) ).

% power_mod
tff(fact_4993_mod__Suc__Suc__eq,axiom,
    ! [M: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Nb)))),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(suc,M))),Nb) ).

% mod_Suc_Suc_eq
tff(fact_4994_mod__Suc__eq,axiom,
    ! [M: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Nb))),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,M)),Nb) ).

% mod_Suc_eq
tff(fact_4995_nat__mod__eq,axiom,
    ! [B: nat,Nb: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),Nb)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B),Nb) )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),Nb) = B ) ) ) ).

% nat_mod_eq
tff(fact_4996_mod__less__eq__dividend,axiom,
    ! [M: nat,Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Nb)),M) ).

% mod_less_eq_dividend
tff(fact_4997_mod__plus__right,axiom,
    ! [A: nat,Xb: nat,M: 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),Xb)),M) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),Xb)),M) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),M) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B),M) ) ) ).

% mod_plus_right
tff(fact_4998_insert__mono,axiom,
    ! [C4: set_nat,D5: set_nat,A: nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(C4),D5)
     => aa_set_nat_o(ord_less_eq_set_nat(insert_nat(A,C4)),insert_nat(A,D5)) ) ).

% insert_mono
tff(fact_4999_insert__mono,axiom,
    ! [C4: set_VEBT_VEBT,D5: set_VEBT_VEBT,A: vEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(C4),D5)
     => aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(insert_VEBT_VEBT(A,C4)),insert_VEBT_VEBT(A,D5)) ) ).

% insert_mono
tff(fact_5000_insert__mono,axiom,
    ! [C4: set_real,D5: set_real,A: real] :
      ( aa_set_real_o(ord_less_eq_set_real(C4),D5)
     => aa_set_real_o(ord_less_eq_set_real(insert_real(A,C4)),insert_real(A,D5)) ) ).

% insert_mono
tff(fact_5001_insert__mono,axiom,
    ! [C4: set_int,D5: set_int,A: int] :
      ( aa_set_int_o(ord_less_eq_set_int(C4),D5)
     => aa_set_int_o(ord_less_eq_set_int(insert_int(A,C4)),insert_int(A,D5)) ) ).

% insert_mono
tff(fact_5002_subset__insert,axiom,
    ! [Xb: nat,A2: set_nat,B4: set_nat] :
      ( ~ aa_set_nat_o(member_nat(Xb),A2)
     => ( aa_set_nat_o(ord_less_eq_set_nat(A2),insert_nat(Xb,B4))
      <=> aa_set_nat_o(ord_less_eq_set_nat(A2),B4) ) ) ).

% subset_insert
tff(fact_5003_subset__insert,axiom,
    ! [Xb: vEBT_VEBT,A2: set_VEBT_VEBT,B4: set_VEBT_VEBT] :
      ( ~ aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xb),A2)
     => ( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(A2),insert_VEBT_VEBT(Xb,B4))
      <=> aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(A2),B4) ) ) ).

% subset_insert
tff(fact_5004_subset__insert,axiom,
    ! [Xb: real,A2: set_real,B4: set_real] :
      ( ~ aa_set_real_o(member_real(Xb),A2)
     => ( aa_set_real_o(ord_less_eq_set_real(A2),insert_real(Xb,B4))
      <=> aa_set_real_o(ord_less_eq_set_real(A2),B4) ) ) ).

% subset_insert
tff(fact_5005_subset__insert,axiom,
    ! [Xb: int,A2: set_int,B4: set_int] :
      ( ~ aa_set_int_o(member_int(Xb),A2)
     => ( aa_set_int_o(ord_less_eq_set_int(A2),insert_int(Xb,B4))
      <=> aa_set_int_o(ord_less_eq_set_int(A2),B4) ) ) ).

% subset_insert
tff(fact_5006_subset__insertI,axiom,
    ! [B4: set_nat,A: nat] : aa_set_nat_o(ord_less_eq_set_nat(B4),insert_nat(A,B4)) ).

% subset_insertI
tff(fact_5007_subset__insertI,axiom,
    ! [B4: set_VEBT_VEBT,A: vEBT_VEBT] : aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(B4),insert_VEBT_VEBT(A,B4)) ).

% subset_insertI
tff(fact_5008_subset__insertI,axiom,
    ! [B4: set_real,A: real] : aa_set_real_o(ord_less_eq_set_real(B4),insert_real(A,B4)) ).

% subset_insertI
tff(fact_5009_subset__insertI,axiom,
    ! [B4: set_int,A: int] : aa_set_int_o(ord_less_eq_set_int(B4),insert_int(A,B4)) ).

% subset_insertI
tff(fact_5010_subset__insertI2,axiom,
    ! [A2: set_nat,B4: set_nat,B: nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A2),B4)
     => aa_set_nat_o(ord_less_eq_set_nat(A2),insert_nat(B,B4)) ) ).

% subset_insertI2
tff(fact_5011_subset__insertI2,axiom,
    ! [A2: set_VEBT_VEBT,B4: set_VEBT_VEBT,B: vEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(A2),B4)
     => aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(A2),insert_VEBT_VEBT(B,B4)) ) ).

% subset_insertI2
tff(fact_5012_subset__insertI2,axiom,
    ! [A2: set_real,B4: set_real,B: real] :
      ( aa_set_real_o(ord_less_eq_set_real(A2),B4)
     => aa_set_real_o(ord_less_eq_set_real(A2),insert_real(B,B4)) ) ).

% subset_insertI2
tff(fact_5013_subset__insertI2,axiom,
    ! [A2: set_int,B4: set_int,B: int] :
      ( aa_set_int_o(ord_less_eq_set_int(A2),B4)
     => aa_set_int_o(ord_less_eq_set_int(A2),insert_int(B,B4)) ) ).

% subset_insertI2
tff(fact_5014_insert__Diff__if,axiom,
    ! [Xb: vEBT_VEBT,A2: set_VEBT_VEBT,B4: set_VEBT_VEBT] :
      minus_5127226145743854075T_VEBT(insert_VEBT_VEBT(Xb,A2),B4) = $ite(aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xb),B4),minus_5127226145743854075T_VEBT(A2,B4),insert_VEBT_VEBT(Xb,minus_5127226145743854075T_VEBT(A2,B4))) ).

% insert_Diff_if
tff(fact_5015_insert__Diff__if,axiom,
    ! [Xb: real,A2: set_real,B4: set_real] :
      minus_minus_set_real(insert_real(Xb,A2),B4) = $ite(aa_set_real_o(member_real(Xb),B4),minus_minus_set_real(A2,B4),insert_real(Xb,minus_minus_set_real(A2,B4))) ).

% insert_Diff_if
tff(fact_5016_insert__Diff__if,axiom,
    ! [Xb: int,A2: set_int,B4: set_int] :
      minus_minus_set_int(insert_int(Xb,A2),B4) = $ite(aa_set_int_o(member_int(Xb),B4),minus_minus_set_int(A2,B4),insert_int(Xb,minus_minus_set_int(A2,B4))) ).

% insert_Diff_if
tff(fact_5017_insert__Diff__if,axiom,
    ! [Xb: nat,A2: set_nat,B4: set_nat] :
      minus_minus_set_nat(insert_nat(Xb,A2),B4) = $ite(aa_set_nat_o(member_nat(Xb),B4),minus_minus_set_nat(A2,B4),insert_nat(Xb,minus_minus_set_nat(A2,B4))) ).

% insert_Diff_if
tff(fact_5018_foldr__cong,axiom,
    ! [A: assn,B: assn,L: list_assn,K: list_assn,F: fun_as3983628580203722482n_assn,G: fun_as3983628580203722482n_assn] :
      ( ( A = B )
     => ( ( L = K )
       => ( ! [A3: assn,X2: assn] :
              ( member_assn(X2,set_assn2(L))
             => ( aa_assn_assn(aa_ass954072293579886259n_assn(F,X2),A3) = aa_assn_assn(aa_ass954072293579886259n_assn(G,X2),A3) ) )
         => ( foldr_assn_assn(F,L,A) = foldr_assn_assn(G,K,B) ) ) ) ) ).

% foldr_cong
tff(fact_5019_foldr__cong,axiom,
    ! [A: nat,B: nat,L: list_o,K: list_o,F: fun_o_fun_nat_nat,G: fun_o_fun_nat_nat] :
      ( ( A = B )
     => ( ( L = K )
       => ( ! [A3: nat,X2: $o] :
              ( member_o((X2),set_o2(L))
             => ( aa_nat_nat(aa_o_fun_nat_nat(F,(X2)),A3) = aa_nat_nat(aa_o_fun_nat_nat(G,(X2)),A3) ) )
         => ( aa_nat_nat(foldr_o_nat(F,L),A) = aa_nat_nat(foldr_o_nat(G,K),B) ) ) ) ) ).

% foldr_cong
tff(fact_5020_foldr__cong,axiom,
    ! [A: nat,B: nat,L: list_nat,K: list_nat,F: fun_nat_fun_nat_nat,G: fun_nat_fun_nat_nat] :
      ( ( A = B )
     => ( ( L = K )
       => ( ! [A3: nat,X2: nat] :
              ( aa_set_nat_o(member_nat(X2),set_nat2(L))
             => ( aa_nat_nat(aa_nat_fun_nat_nat(F,X2),A3) = aa_nat_nat(aa_nat_fun_nat_nat(G,X2),A3) ) )
         => ( aa_nat_nat(foldr_nat_nat(F,L),A) = aa_nat_nat(foldr_nat_nat(G,K),B) ) ) ) ) ).

% foldr_cong
tff(fact_5021_foldr__cong,axiom,
    ! [A: real,B: real,L: list_real,K: list_real,F: fun_re7681423839302404990l_real,G: fun_re7681423839302404990l_real] :
      ( ( A = B )
     => ( ( L = K )
       => ( ! [A3: real,X2: real] :
              ( aa_set_real_o(member_real(X2),set_real2(L))
             => ( aa_real_real(aa_rea7340605660300375871l_real(F,X2),A3) = aa_real_real(aa_rea7340605660300375871l_real(G,X2),A3) ) )
         => ( aa_real_real(foldr_real_real(F,L),A) = aa_real_real(foldr_real_real(G,K),B) ) ) ) ) ).

% foldr_cong
tff(fact_5022_Collect__conv__if,axiom,
    ! [A: vEBT_VEBT,P: fun_VEBT_VEBT_o] :
      collect_VEBT_VEBT(aa_fun8193769468198689164VEBT_o(aTP_Lamm_id(A),P)) = $ite(aa_VEBT_VEBT_o(P,A),insert_VEBT_VEBT(A,bot_bo8194388402131092736T_VEBT),bot_bo8194388402131092736T_VEBT) ).

% Collect_conv_if
tff(fact_5023_Collect__conv__if,axiom,
    ! [A: complex,P: fun_complex_o] :
      collect_complex(aa_fun2514317545224092796plex_o(aTP_Lamm_ie(A),P)) = $ite(aa_complex_o(P,A),insert_complex(A,bot_bot_set_complex),bot_bot_set_complex) ).

% Collect_conv_if
tff(fact_5024_Collect__conv__if,axiom,
    ! [A: product_prod_int_int,P: fun_Pr3238374608701176162_int_o] :
      collec213857154873943460nt_int(aa_fun5349570537165229698_int_o(aTP_Lamm_if(A),P)) = $ite(aa_Pro4592068669816069595_int_o(P,A),insert5033312907999012233nt_int(A,bot_bo1796632182523588997nt_int),bot_bo1796632182523588997nt_int) ).

% Collect_conv_if
tff(fact_5025_Collect__conv__if,axiom,
    ! [A: nat,P: fun_nat_o] :
      collect_nat(aa_fun5937193402907943164_nat_o(aTP_Lamm_ig(A),P)) = $ite(aa_nat_o(P,A),insert_nat(A,bot_bot_set_nat),bot_bot_set_nat) ).

% Collect_conv_if
tff(fact_5026_Collect__conv__if,axiom,
    ! [A: int,P: fun_int_o] :
      collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_ih(A),P)) = $ite(aa_int_o(P,A),insert_int(A,bot_bot_set_int),bot_bot_set_int) ).

% Collect_conv_if
tff(fact_5027_Collect__conv__if,axiom,
    ! [A: real,P: fun_real_o] :
      collect_real(aa_fun5868358341814070780real_o(aTP_Lamm_ii(A),P)) = $ite(aa_real_o(P,A),insert_real(A,bot_bot_set_real),bot_bot_set_real) ).

% Collect_conv_if
tff(fact_5028_Collect__conv__if2,axiom,
    ! [A: vEBT_VEBT,P: fun_VEBT_VEBT_o] :
      collect_VEBT_VEBT(aa_fun8193769468198689164VEBT_o(aTP_Lamm_ij(A),P)) = $ite(aa_VEBT_VEBT_o(P,A),insert_VEBT_VEBT(A,bot_bo8194388402131092736T_VEBT),bot_bo8194388402131092736T_VEBT) ).

% Collect_conv_if2
tff(fact_5029_Collect__conv__if2,axiom,
    ! [A: complex,P: fun_complex_o] :
      collect_complex(aa_fun2514317545224092796plex_o(aTP_Lamm_ik(A),P)) = $ite(aa_complex_o(P,A),insert_complex(A,bot_bot_set_complex),bot_bot_set_complex) ).

% Collect_conv_if2
tff(fact_5030_Collect__conv__if2,axiom,
    ! [A: product_prod_int_int,P: fun_Pr3238374608701176162_int_o] :
      collec213857154873943460nt_int(aa_fun5349570537165229698_int_o(aTP_Lamm_il(A),P)) = $ite(aa_Pro4592068669816069595_int_o(P,A),insert5033312907999012233nt_int(A,bot_bo1796632182523588997nt_int),bot_bo1796632182523588997nt_int) ).

% Collect_conv_if2
tff(fact_5031_Collect__conv__if2,axiom,
    ! [A: nat,P: fun_nat_o] :
      collect_nat(aa_fun5937193402907943164_nat_o(aTP_Lamm_im(A),P)) = $ite(aa_nat_o(P,A),insert_nat(A,bot_bot_set_nat),bot_bot_set_nat) ).

% Collect_conv_if2
tff(fact_5032_Collect__conv__if2,axiom,
    ! [A: int,P: fun_int_o] :
      collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_in(A),P)) = $ite(aa_int_o(P,A),insert_int(A,bot_bot_set_int),bot_bot_set_int) ).

% Collect_conv_if2
tff(fact_5033_Collect__conv__if2,axiom,
    ! [A: real,P: fun_real_o] :
      collect_real(aa_fun5868358341814070780real_o(aTP_Lamm_io(A),P)) = $ite(aa_real_o(P,A),insert_real(A,bot_bot_set_real),bot_bot_set_real) ).

% Collect_conv_if2
tff(fact_5034_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_5035_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_5036_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_5037_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_5038_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_5039_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_5040_cong__exp__iff__simps_I9_J,axiom,
    ! [M: num,Q2: num,Nb: num] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,M))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q2))) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Nb))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q2))) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,Q2)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,Nb)),aa_num_nat(numeral_numeral_nat,Q2)) ) ) ).

% cong_exp_iff_simps(9)
tff(fact_5041_cong__exp__iff__simps_I9_J,axiom,
    ! [M: num,Q2: num,Nb: num] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,M))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q2))) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q2))) )
    <=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,Q2)) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,Nb)),aa_num_int(numeral_numeral_int,Q2)) ) ) ).

% cong_exp_iff_simps(9)
tff(fact_5042_cong__exp__iff__simps_I9_J,axiom,
    ! [M: num,Q2: num,Nb: num] :
      ( ( aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,M))),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,Q2))) = aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,Nb))),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,Q2))) )
    <=> ( aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_num_Code_integer(numera6620942414471956472nteger,M)),aa_num_Code_integer(numera6620942414471956472nteger,Q2)) = aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_num_Code_integer(numera6620942414471956472nteger,Nb)),aa_num_Code_integer(numera6620942414471956472nteger,Q2)) ) ) ).

% cong_exp_iff_simps(9)
tff(fact_5043_cong__exp__iff__simps_I4_J,axiom,
    ! [M: num,Nb: num] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,one)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,Nb)),aa_num_nat(numeral_numeral_nat,one)) ).

% cong_exp_iff_simps(4)
tff(fact_5044_cong__exp__iff__simps_I4_J,axiom,
    ! [M: num,Nb: num] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,one)) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,Nb)),aa_num_int(numeral_numeral_int,one)) ).

% cong_exp_iff_simps(4)
tff(fact_5045_cong__exp__iff__simps_I4_J,axiom,
    ! [M: num,Nb: num] : aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_num_Code_integer(numera6620942414471956472nteger,M)),aa_num_Code_integer(numera6620942414471956472nteger,one)) = aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_num_Code_integer(numera6620942414471956472nteger,Nb)),aa_num_Code_integer(numera6620942414471956472nteger,one)) ).

% cong_exp_iff_simps(4)
tff(fact_5046_mod__eq__self__iff__div__eq__0,axiom,
    ! [A: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) = A )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B) = zero_zero_nat ) ) ).

% mod_eq_self_iff_div_eq_0
tff(fact_5047_mod__eq__self__iff__div__eq__0,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) = A )
    <=> ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B) = zero_zero_int ) ) ).

% mod_eq_self_iff_div_eq_0
tff(fact_5048_mod__eq__self__iff__div__eq__0,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A),B) = A )
    <=> ( aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),B) = zero_z3403309356797280102nteger ) ) ).

% mod_eq_self_iff_div_eq_0
tff(fact_5049_mod__eqE,axiom,
    ! [A: int,C: int,B: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B),C) )
     => ~ ! [D2: int] : B != aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,C),D2)) ) ).

% mod_eqE
tff(fact_5050_mod__eqE,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A),C) = aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(B),C) )
     => ~ ! [D2: code_integer] : B != aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),D2)) ) ).

% mod_eqE
tff(fact_5051_div__add1__eq,axiom,
    ! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B),C))),C)) ).

% div_add1_eq
tff(fact_5052_div__add1__eq,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),C)),aa_int_int(aa_int_fun_int_int(divide_divide_int,B),C))),aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),C)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B),C))),C)) ).

% div_add1_eq
tff(fact_5053_div__add1__eq,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] : aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(A),B)),C) = aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),C)),aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(B),C))),aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A),C)),aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(B),C))),C)) ).

% div_add1_eq
tff(fact_5054_mod__Suc,axiom,
    ! [M: nat,Nb: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,M)),Nb) = $ite(aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Nb)) = Nb,zero_zero_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Nb))) ).

% mod_Suc
tff(fact_5055_mod__induct,axiom,
    ! [P: fun_nat_o,Nb: nat,P2: nat,M: nat] :
      ( aa_nat_o(P,Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),P2)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),P2)
         => ( ! [N2: nat] :
                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),P2)
               => ( aa_nat_o(P,N2)
                 => aa_nat_o(P,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,N2)),P2)) ) )
           => aa_nat_o(P,M) ) ) ) ) ).

% mod_induct
tff(fact_5056_nat__mod__lem,axiom,
    ! [Nb: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),Nb)
      <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B),Nb) = B ) ) ) ).

% nat_mod_lem
tff(fact_5057_mod__less__divisor,axiom,
    ! [Nb: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Nb)),Nb) ) ).

% mod_less_divisor
tff(fact_5058_mod__Suc__le__divisor,axiom,
    ! [M: nat,Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_nat_nat(suc,Nb))),Nb) ).

% mod_Suc_le_divisor
tff(fact_5059_word__rot__lem,axiom,
    ! [L: nat,K: nat,D: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,L),K) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,K),L)) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),L)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,D),Nb)),L) = Nb ) ) ) ).

% word_rot_lem
tff(fact_5060_nat__minus__mod,axiom,
    ! [Nb: nat,M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),M))),M) = zero_zero_nat ).

% nat_minus_mod
tff(fact_5061_mod__if,axiom,
    ! [M: nat,Nb: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Nb) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb),M,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Nb)),Nb)) ).

% mod_if
tff(fact_5062_mod__nat__sub,axiom,
    ! [Xb: nat,Z: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Z)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Xb),Y)),Z) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Xb),Y) ) ) ).

% mod_nat_sub
tff(fact_5063_mod__geq,axiom,
    ! [M: nat,Nb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Nb)),Nb) ) ) ).

% mod_geq
tff(fact_5064_mod__eq__0D,axiom,
    ! [M: nat,D: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),D) = zero_zero_nat )
     => ? [Q3: nat] : M = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,D),Q3) ) ).

% mod_eq_0D
tff(fact_5065_le__mod__geq,axiom,
    ! [Nb: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),M)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Nb)),Nb) ) ) ).

% le_mod_geq
tff(fact_5066_nat__minus__mod__plus__right,axiom,
    ! [Nb: nat,Xb: nat,M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Xb)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),M))),M) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xb),M) ).

% nat_minus_mod_plus_right
tff(fact_5067_msrevs_I2_J,axiom,
    ! [K: nat,Nb: nat,M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Nb)),M)),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Nb) ).

% msrevs(2)
tff(fact_5068_nat__mod__eq__iff,axiom,
    ! [Xb: nat,Nb: nat,Y: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xb),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Y),Nb) )
    <=> ? [Q1: nat,Q22: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),Q1)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),Q22)) ) ).

% nat_mod_eq_iff
tff(fact_5069_subset__singletonD,axiom,
    ! [A2: set_VEBT_VEBT,Xb: vEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(A2),insert_VEBT_VEBT(Xb,bot_bo8194388402131092736T_VEBT))
     => ( ( A2 = bot_bo8194388402131092736T_VEBT )
        | ( A2 = insert_VEBT_VEBT(Xb,bot_bo8194388402131092736T_VEBT) ) ) ) ).

% subset_singletonD
tff(fact_5070_subset__singletonD,axiom,
    ! [A2: set_nat,Xb: nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A2),insert_nat(Xb,bot_bot_set_nat))
     => ( ( A2 = bot_bot_set_nat )
        | ( A2 = insert_nat(Xb,bot_bot_set_nat) ) ) ) ).

% subset_singletonD
tff(fact_5071_subset__singletonD,axiom,
    ! [A2: set_real,Xb: real] :
      ( aa_set_real_o(ord_less_eq_set_real(A2),insert_real(Xb,bot_bot_set_real))
     => ( ( A2 = bot_bot_set_real )
        | ( A2 = insert_real(Xb,bot_bot_set_real) ) ) ) ).

% subset_singletonD
tff(fact_5072_subset__singletonD,axiom,
    ! [A2: set_int,Xb: int] :
      ( aa_set_int_o(ord_less_eq_set_int(A2),insert_int(Xb,bot_bot_set_int))
     => ( ( A2 = bot_bot_set_int )
        | ( A2 = insert_int(Xb,bot_bot_set_int) ) ) ) ).

% subset_singletonD
tff(fact_5073_subset__singleton__iff,axiom,
    ! [X5: set_VEBT_VEBT,A: vEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(X5),insert_VEBT_VEBT(A,bot_bo8194388402131092736T_VEBT))
    <=> ( ( X5 = bot_bo8194388402131092736T_VEBT )
        | ( X5 = insert_VEBT_VEBT(A,bot_bo8194388402131092736T_VEBT) ) ) ) ).

% subset_singleton_iff
tff(fact_5074_subset__singleton__iff,axiom,
    ! [X5: set_nat,A: nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(X5),insert_nat(A,bot_bot_set_nat))
    <=> ( ( X5 = bot_bot_set_nat )
        | ( X5 = insert_nat(A,bot_bot_set_nat) ) ) ) ).

% subset_singleton_iff
tff(fact_5075_subset__singleton__iff,axiom,
    ! [X5: set_real,A: real] :
      ( aa_set_real_o(ord_less_eq_set_real(X5),insert_real(A,bot_bot_set_real))
    <=> ( ( X5 = bot_bot_set_real )
        | ( X5 = insert_real(A,bot_bot_set_real) ) ) ) ).

% subset_singleton_iff
tff(fact_5076_subset__singleton__iff,axiom,
    ! [X5: set_int,A: int] :
      ( aa_set_int_o(ord_less_eq_set_int(X5),insert_int(A,bot_bot_set_int))
    <=> ( ( X5 = bot_bot_set_int )
        | ( X5 = insert_int(A,bot_bot_set_int) ) ) ) ).

% subset_singleton_iff
tff(fact_5077_Diff__insert,axiom,
    ! [A2: set_VEBT_VEBT,A: vEBT_VEBT,B4: set_VEBT_VEBT] : minus_5127226145743854075T_VEBT(A2,insert_VEBT_VEBT(A,B4)) = minus_5127226145743854075T_VEBT(minus_5127226145743854075T_VEBT(A2,B4),insert_VEBT_VEBT(A,bot_bo8194388402131092736T_VEBT)) ).

% Diff_insert
tff(fact_5078_Diff__insert,axiom,
    ! [A2: set_int,A: int,B4: set_int] : minus_minus_set_int(A2,insert_int(A,B4)) = minus_minus_set_int(minus_minus_set_int(A2,B4),insert_int(A,bot_bot_set_int)) ).

% Diff_insert
tff(fact_5079_Diff__insert,axiom,
    ! [A2: set_real,A: real,B4: set_real] : minus_minus_set_real(A2,insert_real(A,B4)) = minus_minus_set_real(minus_minus_set_real(A2,B4),insert_real(A,bot_bot_set_real)) ).

% Diff_insert
tff(fact_5080_Diff__insert,axiom,
    ! [A2: set_nat,A: nat,B4: set_nat] : minus_minus_set_nat(A2,insert_nat(A,B4)) = minus_minus_set_nat(minus_minus_set_nat(A2,B4),insert_nat(A,bot_bot_set_nat)) ).

% Diff_insert
tff(fact_5081_insert__Diff,axiom,
    ! [A: vEBT_VEBT,A2: set_VEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A),A2)
     => ( insert_VEBT_VEBT(A,minus_5127226145743854075T_VEBT(A2,insert_VEBT_VEBT(A,bot_bo8194388402131092736T_VEBT))) = A2 ) ) ).

% insert_Diff
tff(fact_5082_insert__Diff,axiom,
    ! [A: int,A2: set_int] :
      ( aa_set_int_o(member_int(A),A2)
     => ( insert_int(A,minus_minus_set_int(A2,insert_int(A,bot_bot_set_int))) = A2 ) ) ).

% insert_Diff
tff(fact_5083_insert__Diff,axiom,
    ! [A: real,A2: set_real] :
      ( aa_set_real_o(member_real(A),A2)
     => ( insert_real(A,minus_minus_set_real(A2,insert_real(A,bot_bot_set_real))) = A2 ) ) ).

% insert_Diff
tff(fact_5084_insert__Diff,axiom,
    ! [A: nat,A2: set_nat] :
      ( aa_set_nat_o(member_nat(A),A2)
     => ( insert_nat(A,minus_minus_set_nat(A2,insert_nat(A,bot_bot_set_nat))) = A2 ) ) ).

% insert_Diff
tff(fact_5085_Diff__insert2,axiom,
    ! [A2: set_VEBT_VEBT,A: vEBT_VEBT,B4: set_VEBT_VEBT] : minus_5127226145743854075T_VEBT(A2,insert_VEBT_VEBT(A,B4)) = minus_5127226145743854075T_VEBT(minus_5127226145743854075T_VEBT(A2,insert_VEBT_VEBT(A,bot_bo8194388402131092736T_VEBT)),B4) ).

% Diff_insert2
tff(fact_5086_Diff__insert2,axiom,
    ! [A2: set_int,A: int,B4: set_int] : minus_minus_set_int(A2,insert_int(A,B4)) = minus_minus_set_int(minus_minus_set_int(A2,insert_int(A,bot_bot_set_int)),B4) ).

% Diff_insert2
tff(fact_5087_Diff__insert2,axiom,
    ! [A2: set_real,A: real,B4: set_real] : minus_minus_set_real(A2,insert_real(A,B4)) = minus_minus_set_real(minus_minus_set_real(A2,insert_real(A,bot_bot_set_real)),B4) ).

% Diff_insert2
tff(fact_5088_Diff__insert2,axiom,
    ! [A2: set_nat,A: nat,B4: set_nat] : minus_minus_set_nat(A2,insert_nat(A,B4)) = minus_minus_set_nat(minus_minus_set_nat(A2,insert_nat(A,bot_bot_set_nat)),B4) ).

% Diff_insert2
tff(fact_5089_insert__minus__eq,axiom,
    ! [Xb: vEBT_VEBT,Y: vEBT_VEBT,A2: set_VEBT_VEBT] :
      ( ( Xb != Y )
     => ( minus_5127226145743854075T_VEBT(insert_VEBT_VEBT(Xb,A2),insert_VEBT_VEBT(Y,bot_bo8194388402131092736T_VEBT)) = insert_VEBT_VEBT(Xb,minus_5127226145743854075T_VEBT(A2,insert_VEBT_VEBT(Y,bot_bo8194388402131092736T_VEBT))) ) ) ).

% insert_minus_eq
tff(fact_5090_insert__minus__eq,axiom,
    ! [Xb: int,Y: int,A2: set_int] :
      ( ( Xb != Y )
     => ( minus_minus_set_int(insert_int(Xb,A2),insert_int(Y,bot_bot_set_int)) = insert_int(Xb,minus_minus_set_int(A2,insert_int(Y,bot_bot_set_int))) ) ) ).

% insert_minus_eq
tff(fact_5091_insert__minus__eq,axiom,
    ! [Xb: real,Y: real,A2: set_real] :
      ( ( Xb != Y )
     => ( minus_minus_set_real(insert_real(Xb,A2),insert_real(Y,bot_bot_set_real)) = insert_real(Xb,minus_minus_set_real(A2,insert_real(Y,bot_bot_set_real))) ) ) ).

% insert_minus_eq
tff(fact_5092_insert__minus__eq,axiom,
    ! [Xb: nat,Y: nat,A2: set_nat] :
      ( ( Xb != Y )
     => ( minus_minus_set_nat(insert_nat(Xb,A2),insert_nat(Y,bot_bot_set_nat)) = insert_nat(Xb,minus_minus_set_nat(A2,insert_nat(Y,bot_bot_set_nat))) ) ) ).

% insert_minus_eq
tff(fact_5093_Diff__insert__absorb,axiom,
    ! [Xb: vEBT_VEBT,A2: set_VEBT_VEBT] :
      ( ~ aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xb),A2)
     => ( minus_5127226145743854075T_VEBT(insert_VEBT_VEBT(Xb,A2),insert_VEBT_VEBT(Xb,bot_bo8194388402131092736T_VEBT)) = A2 ) ) ).

% Diff_insert_absorb
tff(fact_5094_Diff__insert__absorb,axiom,
    ! [Xb: int,A2: set_int] :
      ( ~ aa_set_int_o(member_int(Xb),A2)
     => ( minus_minus_set_int(insert_int(Xb,A2),insert_int(Xb,bot_bot_set_int)) = A2 ) ) ).

% Diff_insert_absorb
tff(fact_5095_Diff__insert__absorb,axiom,
    ! [Xb: real,A2: set_real] :
      ( ~ aa_set_real_o(member_real(Xb),A2)
     => ( minus_minus_set_real(insert_real(Xb,A2),insert_real(Xb,bot_bot_set_real)) = A2 ) ) ).

% Diff_insert_absorb
tff(fact_5096_Diff__insert__absorb,axiom,
    ! [Xb: nat,A2: set_nat] :
      ( ~ aa_set_nat_o(member_nat(Xb),A2)
     => ( minus_minus_set_nat(insert_nat(Xb,A2),insert_nat(Xb,bot_bot_set_nat)) = A2 ) ) ).

% Diff_insert_absorb
tff(fact_5097_set__minus__singleton__eq,axiom,
    ! [Xb: vEBT_VEBT,X5: set_VEBT_VEBT] :
      ( ~ aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xb),X5)
     => ( minus_5127226145743854075T_VEBT(X5,insert_VEBT_VEBT(Xb,bot_bo8194388402131092736T_VEBT)) = X5 ) ) ).

% set_minus_singleton_eq
tff(fact_5098_set__minus__singleton__eq,axiom,
    ! [Xb: int,X5: set_int] :
      ( ~ aa_set_int_o(member_int(Xb),X5)
     => ( minus_minus_set_int(X5,insert_int(Xb,bot_bot_set_int)) = X5 ) ) ).

% set_minus_singleton_eq
tff(fact_5099_set__minus__singleton__eq,axiom,
    ! [Xb: real,X5: set_real] :
      ( ~ aa_set_real_o(member_real(Xb),X5)
     => ( minus_minus_set_real(X5,insert_real(Xb,bot_bot_set_real)) = X5 ) ) ).

% set_minus_singleton_eq
tff(fact_5100_set__minus__singleton__eq,axiom,
    ! [Xb: nat,X5: set_nat] :
      ( ~ aa_set_nat_o(member_nat(Xb),X5)
     => ( minus_minus_set_nat(X5,insert_nat(Xb,bot_bot_set_nat)) = X5 ) ) ).

% set_minus_singleton_eq
tff(fact_5101_subset__Diff__insert,axiom,
    ! [A2: set_VEBT_VEBT,B4: set_VEBT_VEBT,Xb: vEBT_VEBT,C4: set_VEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(A2),minus_5127226145743854075T_VEBT(B4,insert_VEBT_VEBT(Xb,C4)))
    <=> ( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(A2),minus_5127226145743854075T_VEBT(B4,C4))
        & ~ aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xb),A2) ) ) ).

% subset_Diff_insert
tff(fact_5102_subset__Diff__insert,axiom,
    ! [A2: set_real,B4: set_real,Xb: real,C4: set_real] :
      ( aa_set_real_o(ord_less_eq_set_real(A2),minus_minus_set_real(B4,insert_real(Xb,C4)))
    <=> ( aa_set_real_o(ord_less_eq_set_real(A2),minus_minus_set_real(B4,C4))
        & ~ aa_set_real_o(member_real(Xb),A2) ) ) ).

% subset_Diff_insert
tff(fact_5103_subset__Diff__insert,axiom,
    ! [A2: set_nat,B4: set_nat,Xb: nat,C4: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A2),minus_minus_set_nat(B4,insert_nat(Xb,C4)))
    <=> ( aa_set_nat_o(ord_less_eq_set_nat(A2),minus_minus_set_nat(B4,C4))
        & ~ aa_set_nat_o(member_nat(Xb),A2) ) ) ).

% subset_Diff_insert
tff(fact_5104_subset__Diff__insert,axiom,
    ! [A2: set_int,B4: set_int,Xb: int,C4: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A2),minus_minus_set_int(B4,insert_int(Xb,C4)))
    <=> ( aa_set_int_o(ord_less_eq_set_int(A2),minus_minus_set_int(B4,C4))
        & ~ aa_set_int_o(member_int(Xb),A2) ) ) ).

% subset_Diff_insert
tff(fact_5105_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_5106_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_5107_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_5108_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_5109_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_5110_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_5111_cong__exp__iff__simps_I2_J,axiom,
    ! [Nb: num,Q2: num] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Nb))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q2))) = zero_zero_nat )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,Nb)),aa_num_nat(numeral_numeral_nat,Q2)) = zero_zero_nat ) ) ).

% cong_exp_iff_simps(2)
tff(fact_5112_cong__exp__iff__simps_I2_J,axiom,
    ! [Nb: num,Q2: num] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q2))) = zero_zero_int )
    <=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,Nb)),aa_num_int(numeral_numeral_int,Q2)) = zero_zero_int ) ) ).

% cong_exp_iff_simps(2)
tff(fact_5113_cong__exp__iff__simps_I2_J,axiom,
    ! [Nb: num,Q2: num] :
      ( ( aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,Nb))),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,Q2))) = zero_z3403309356797280102nteger )
    <=> ( aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_num_Code_integer(numera6620942414471956472nteger,Nb)),aa_num_Code_integer(numera6620942414471956472nteger,Q2)) = zero_z3403309356797280102nteger ) ) ).

% cong_exp_iff_simps(2)
tff(fact_5114_cong__exp__iff__simps_I1_J,axiom,
    ! [Nb: num] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,Nb)),aa_num_nat(numeral_numeral_nat,one)) = zero_zero_nat ).

% cong_exp_iff_simps(1)
tff(fact_5115_cong__exp__iff__simps_I1_J,axiom,
    ! [Nb: num] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,Nb)),aa_num_int(numeral_numeral_int,one)) = zero_zero_int ).

% cong_exp_iff_simps(1)
tff(fact_5116_cong__exp__iff__simps_I1_J,axiom,
    ! [Nb: num] : aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_num_Code_integer(numera6620942414471956472nteger,Nb)),aa_num_Code_integer(numera6620942414471956472nteger,one)) = zero_z3403309356797280102nteger ).

% cong_exp_iff_simps(1)
tff(fact_5117_cong__exp__iff__simps_I6_J,axiom,
    ! [Q2: num,Nb: num] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,one)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q2))) != aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Nb))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q2))) ).

% cong_exp_iff_simps(6)
tff(fact_5118_cong__exp__iff__simps_I6_J,axiom,
    ! [Q2: num,Nb: num] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,one)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q2))) != aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q2))) ).

% cong_exp_iff_simps(6)
tff(fact_5119_cong__exp__iff__simps_I6_J,axiom,
    ! [Q2: num,Nb: num] : aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_num_Code_integer(numera6620942414471956472nteger,one)),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,Q2))) != aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,Nb))),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,Q2))) ).

% cong_exp_iff_simps(6)
tff(fact_5120_cong__exp__iff__simps_I8_J,axiom,
    ! [M: num,Q2: num] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,M))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q2))) != aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,one)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q2))) ).

% cong_exp_iff_simps(8)
tff(fact_5121_cong__exp__iff__simps_I8_J,axiom,
    ! [M: num,Q2: num] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,M))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q2))) != aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,one)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q2))) ).

% cong_exp_iff_simps(8)
tff(fact_5122_cong__exp__iff__simps_I8_J,axiom,
    ! [M: num,Q2: num] : aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,M))),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,Q2))) != aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_num_Code_integer(numera6620942414471956472nteger,one)),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,Q2))) ).

% cong_exp_iff_simps(8)
tff(fact_5123_cong__exp__iff__simps_I10_J,axiom,
    ! [M: num,Q2: num,Nb: num] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,M))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q2))) != aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Nb))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q2))) ).

% cong_exp_iff_simps(10)
tff(fact_5124_cong__exp__iff__simps_I10_J,axiom,
    ! [M: num,Q2: num,Nb: num] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,M))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q2))) != aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q2))) ).

% cong_exp_iff_simps(10)
tff(fact_5125_cong__exp__iff__simps_I10_J,axiom,
    ! [M: num,Q2: num,Nb: num] : aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,M))),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,Q2))) != aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit1,Nb))),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,Q2))) ).

% cong_exp_iff_simps(10)
tff(fact_5126_cong__exp__iff__simps_I12_J,axiom,
    ! [M: num,Q2: num,Nb: num] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,M))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q2))) != aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Nb))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q2))) ).

% cong_exp_iff_simps(12)
tff(fact_5127_cong__exp__iff__simps_I12_J,axiom,
    ! [M: num,Q2: num,Nb: num] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,M))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q2))) != aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q2))) ).

% cong_exp_iff_simps(12)
tff(fact_5128_cong__exp__iff__simps_I12_J,axiom,
    ! [M: num,Q2: num,Nb: num] : aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit1,M))),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,Q2))) != aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,Nb))),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,Q2))) ).

% cong_exp_iff_simps(12)
tff(fact_5129_cong__exp__iff__simps_I13_J,axiom,
    ! [M: num,Q2: num,Nb: num] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,M))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q2))) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Nb))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q2))) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,Q2)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,Nb)),aa_num_nat(numeral_numeral_nat,Q2)) ) ) ).

% cong_exp_iff_simps(13)
tff(fact_5130_cong__exp__iff__simps_I13_J,axiom,
    ! [M: num,Q2: num,Nb: num] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,M))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q2))) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q2))) )
    <=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,Q2)) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,Nb)),aa_num_int(numeral_numeral_int,Q2)) ) ) ).

% cong_exp_iff_simps(13)
tff(fact_5131_cong__exp__iff__simps_I13_J,axiom,
    ! [M: num,Q2: num,Nb: num] :
      ( ( aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit1,M))),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,Q2))) = aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit1,Nb))),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,Q2))) )
    <=> ( aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_num_Code_integer(numera6620942414471956472nteger,M)),aa_num_Code_integer(numera6620942414471956472nteger,Q2)) = aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_num_Code_integer(numera6620942414471956472nteger,Nb)),aa_num_Code_integer(numera6620942414471956472nteger,Q2)) ) ) ).

% cong_exp_iff_simps(13)
tff(fact_5132_mult__div__mod__eq,axiom,
    ! [B: word_N3645301735248828278l_num1,A: word_N3645301735248828278l_num1] : plus_p361126936061061375l_num1(aa_wor7274019993978516266l_num1(times_7065122842183080059l_num1(B),divide1791077408188789448l_num1(A,B)),modulo1504961113040953224l_num1(A,B)) = A ).

% mult_div_mod_eq
tff(fact_5133_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_5134_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_5135_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_5136_mult__div__mod__eq,axiom,
    ! [B: uint32,A: uint32] : aa_uint32_uint32(plus_plus_uint32(aa_uint32_uint32(times_times_uint32(B),aa_uint32_uint32(aa_uin3778660748068847871uint32(divide_divide_uint32,A),B))),aa_uint32_uint32(aa_uin3778660748068847871uint32(modulo_modulo_uint32,A),B)) = A ).

% mult_div_mod_eq
tff(fact_5137_mod__mult__div__eq,axiom,
    ! [A: word_N3645301735248828278l_num1,B: word_N3645301735248828278l_num1] : plus_p361126936061061375l_num1(modulo1504961113040953224l_num1(A,B),aa_wor7274019993978516266l_num1(times_7065122842183080059l_num1(B),divide1791077408188789448l_num1(A,B))) = A ).

% mod_mult_div_eq
tff(fact_5138_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_5139_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_5140_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_5141_mod__mult__div__eq,axiom,
    ! [A: uint32,B: uint32] : aa_uint32_uint32(plus_plus_uint32(aa_uint32_uint32(aa_uin3778660748068847871uint32(modulo_modulo_uint32,A),B)),aa_uint32_uint32(times_times_uint32(B),aa_uint32_uint32(aa_uin3778660748068847871uint32(divide_divide_uint32,A),B))) = A ).

% mod_mult_div_eq
tff(fact_5142_mod__div__mult__eq,axiom,
    ! [A: word_N3645301735248828278l_num1,B: word_N3645301735248828278l_num1] : plus_p361126936061061375l_num1(modulo1504961113040953224l_num1(A,B),aa_wor7274019993978516266l_num1(times_7065122842183080059l_num1(divide1791077408188789448l_num1(A,B)),B)) = A ).

% mod_div_mult_eq
tff(fact_5143_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_5144_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_5145_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_5146_mod__div__mult__eq,axiom,
    ! [A: uint32,B: uint32] : aa_uint32_uint32(plus_plus_uint32(aa_uint32_uint32(aa_uin3778660748068847871uint32(modulo_modulo_uint32,A),B)),aa_uint32_uint32(times_times_uint32(aa_uint32_uint32(aa_uin3778660748068847871uint32(divide_divide_uint32,A),B)),B)) = A ).

% mod_div_mult_eq
tff(fact_5147_div__mult__mod__eq,axiom,
    ! [A: word_N3645301735248828278l_num1,B: word_N3645301735248828278l_num1] : plus_p361126936061061375l_num1(aa_wor7274019993978516266l_num1(times_7065122842183080059l_num1(divide1791077408188789448l_num1(A,B)),B),modulo1504961113040953224l_num1(A,B)) = A ).

% div_mult_mod_eq
tff(fact_5148_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_5149_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_5150_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_5151_div__mult__mod__eq,axiom,
    ! [A: uint32,B: uint32] : aa_uint32_uint32(plus_plus_uint32(aa_uint32_uint32(times_times_uint32(aa_uint32_uint32(aa_uin3778660748068847871uint32(divide_divide_uint32,A),B)),B)),aa_uint32_uint32(aa_uin3778660748068847871uint32(modulo_modulo_uint32,A),B)) = A ).

% div_mult_mod_eq
tff(fact_5152_mod__div__decomp,axiom,
    ! [A: word_N3645301735248828278l_num1,B: word_N3645301735248828278l_num1] : A = plus_p361126936061061375l_num1(aa_wor7274019993978516266l_num1(times_7065122842183080059l_num1(divide1791077408188789448l_num1(A,B)),B),modulo1504961113040953224l_num1(A,B)) ).

% mod_div_decomp
tff(fact_5153_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_5154_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_5155_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_5156_mod__div__decomp,axiom,
    ! [A: uint32,B: uint32] : A = aa_uint32_uint32(plus_plus_uint32(aa_uint32_uint32(times_times_uint32(aa_uint32_uint32(aa_uin3778660748068847871uint32(divide_divide_uint32,A),B)),B)),aa_uint32_uint32(aa_uin3778660748068847871uint32(modulo_modulo_uint32,A),B)) ).

% mod_div_decomp
tff(fact_5157_cancel__div__mod__rules_I1_J,axiom,
    ! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),B)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B))),C) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C) ).

% cancel_div_mod_rules(1)
tff(fact_5158_cancel__div__mod__rules_I1_J,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),B)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B))),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C) ).

% cancel_div_mod_rules(1)
tff(fact_5159_cancel__div__mod__rules_I1_J,axiom,
    ! [A: code_integer,B: code_integer,C: 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))),C) = aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(A),C) ).

% cancel_div_mod_rules(1)
tff(fact_5160_cancel__div__mod__rules_I2_J,axiom,
    ! [B: nat,A: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B))),C) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C) ).

% cancel_div_mod_rules(2)
tff(fact_5161_cancel__div__mod__rules_I2_J,axiom,
    ! [B: int,A: int,C: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B))),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C) ).

% cancel_div_mod_rules(2)
tff(fact_5162_cancel__div__mod__rules_I2_J,axiom,
    ! [B: code_integer,A: code_integer,C: 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))),C) = aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(A),C) ).

% cancel_div_mod_rules(2)
tff(fact_5163_div__mult1__eq,axiom,
    ! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B),C))),C)) ).

% div_mult1_eq
tff(fact_5164_div__mult1__eq,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(divide_divide_int,B),C))),aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B),C))),C)) ).

% div_mult1_eq
tff(fact_5165_div__mult1__eq,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] : aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),C) = aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(B),C))),aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(B),C))),C)) ).

% div_mult1_eq
tff(fact_5166_minus__mult__div__eq__mod,axiom,
    ! [A: word_N3645301735248828278l_num1,B: word_N3645301735248828278l_num1] : minus_4019991460397169231l_num1(A,aa_wor7274019993978516266l_num1(times_7065122842183080059l_num1(B),divide1791077408188789448l_num1(A,B))) = modulo1504961113040953224l_num1(A,B) ).

% minus_mult_div_eq_mod
tff(fact_5167_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_5168_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_5169_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_5170_minus__mult__div__eq__mod,axiom,
    ! [A: uint32,B: uint32] : minus_minus_uint32(A,aa_uint32_uint32(times_times_uint32(B),aa_uint32_uint32(aa_uin3778660748068847871uint32(divide_divide_uint32,A),B))) = aa_uint32_uint32(aa_uin3778660748068847871uint32(modulo_modulo_uint32,A),B) ).

% minus_mult_div_eq_mod
tff(fact_5171_minus__mod__eq__mult__div,axiom,
    ! [A: word_N3645301735248828278l_num1,B: word_N3645301735248828278l_num1] : minus_4019991460397169231l_num1(A,modulo1504961113040953224l_num1(A,B)) = aa_wor7274019993978516266l_num1(times_7065122842183080059l_num1(B),divide1791077408188789448l_num1(A,B)) ).

% minus_mod_eq_mult_div
tff(fact_5172_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_5173_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_5174_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_5175_minus__mod__eq__mult__div,axiom,
    ! [A: uint32,B: uint32] : minus_minus_uint32(A,aa_uint32_uint32(aa_uin3778660748068847871uint32(modulo_modulo_uint32,A),B)) = aa_uint32_uint32(times_times_uint32(B),aa_uint32_uint32(aa_uin3778660748068847871uint32(divide_divide_uint32,A),B)) ).

% minus_mod_eq_mult_div
tff(fact_5176_minus__mod__eq__div__mult,axiom,
    ! [A: word_N3645301735248828278l_num1,B: word_N3645301735248828278l_num1] : minus_4019991460397169231l_num1(A,modulo1504961113040953224l_num1(A,B)) = aa_wor7274019993978516266l_num1(times_7065122842183080059l_num1(divide1791077408188789448l_num1(A,B)),B) ).

% minus_mod_eq_div_mult
tff(fact_5177_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_5178_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_5179_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_5180_minus__mod__eq__div__mult,axiom,
    ! [A: uint32,B: uint32] : minus_minus_uint32(A,aa_uint32_uint32(aa_uin3778660748068847871uint32(modulo_modulo_uint32,A),B)) = aa_uint32_uint32(times_times_uint32(aa_uint32_uint32(aa_uin3778660748068847871uint32(divide_divide_uint32,A),B)),B) ).

% minus_mod_eq_div_mult
tff(fact_5181_minus__div__mult__eq__mod,axiom,
    ! [A: word_N3645301735248828278l_num1,B: word_N3645301735248828278l_num1] : minus_4019991460397169231l_num1(A,aa_wor7274019993978516266l_num1(times_7065122842183080059l_num1(divide1791077408188789448l_num1(A,B)),B)) = modulo1504961113040953224l_num1(A,B) ).

% minus_div_mult_eq_mod
tff(fact_5182_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_5183_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_5184_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_5185_minus__div__mult__eq__mod,axiom,
    ! [A: uint32,B: uint32] : minus_minus_uint32(A,aa_uint32_uint32(times_times_uint32(aa_uint32_uint32(aa_uin3778660748068847871uint32(divide_divide_uint32,A),B)),B)) = aa_uint32_uint32(aa_uin3778660748068847871uint32(modulo_modulo_uint32,A),B) ).

% minus_div_mult_eq_mod
tff(fact_5186_zmde,axiom,
    ! [B: word_N3645301735248828278l_num1,A: word_N3645301735248828278l_num1] : aa_wor7274019993978516266l_num1(times_7065122842183080059l_num1(B),divide1791077408188789448l_num1(A,B)) = minus_4019991460397169231l_num1(A,modulo1504961113040953224l_num1(A,B)) ).

% zmde
tff(fact_5187_zmde,axiom,
    ! [B: int,A: 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(minus_minus_int,A),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)) ).

% zmde
tff(fact_5188_zmde,axiom,
    ! [B: code_integer,A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),B)) = aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(A),aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A),B)) ).

% zmde
tff(fact_5189_zmde,axiom,
    ! [B: uint32,A: uint32] : aa_uint32_uint32(times_times_uint32(B),aa_uint32_uint32(aa_uin3778660748068847871uint32(divide_divide_uint32,A),B)) = minus_minus_uint32(A,aa_uint32_uint32(aa_uin3778660748068847871uint32(modulo_modulo_uint32,A),B)) ).

% zmde
tff(fact_5190_mod__le__divisor,axiom,
    ! [Nb: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Nb)),Nb) ) ).

% mod_le_divisor
tff(fact_5191_div__less__mono,axiom,
    ! [A2: nat,B4: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),B4)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),Nb) = zero_zero_nat )
         => ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B4),Nb) = zero_zero_nat )
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B4),Nb)) ) ) ) ) ).

% div_less_mono
tff(fact_5192_mod__nat__add,axiom,
    ! [Xb: nat,Z: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Z)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),Z)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xb),Y)),Z) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xb),Y)),Z),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xb),Y),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xb),Y)),Z)) ) ) ) ).

% mod_nat_add
tff(fact_5193_nat__mod__eq__lemma,axiom,
    ! [Xb: nat,Nb: nat,Y: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xb),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Y),Nb) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),Xb)
       => ? [Q3: nat] : Xb = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),Q3)) ) ) ).

% nat_mod_eq_lemma
tff(fact_5194_mod__eq__nat2E,axiom,
    ! [M: nat,Q2: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Q2) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),Q2) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
       => ~ ! [S: nat] : Nb != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Q2),S)) ) ) ).

% mod_eq_nat2E
tff(fact_5195_mod__eq__nat1E,axiom,
    ! [M: nat,Q2: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Q2) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),Q2) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),M)
       => ~ ! [S: nat] : M != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Q2),S)) ) ) ).

% mod_eq_nat1E
tff(fact_5196_mod__mult2__eq,axiom,
    ! [M: nat,Nb: nat,Q2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),Q2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Nb)),Q2))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Nb)) ).

% mod_mult2_eq
tff(fact_5197_div__mod__decomp,axiom,
    ! [A2: nat,Nb: nat] : A2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),Nb)),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),Nb)) ).

% div_mod_decomp
tff(fact_5198_modulo__nat__def,axiom,
    ! [M: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Nb)),Nb)) ).

% modulo_nat_def
tff(fact_5199_subset__insert__iff,axiom,
    ! [A2: set_VEBT_VEBT,Xb: vEBT_VEBT,B4: set_VEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(A2),insert_VEBT_VEBT(Xb,B4))
    <=> $ite(aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xb),A2),aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(minus_5127226145743854075T_VEBT(A2,insert_VEBT_VEBT(Xb,bot_bo8194388402131092736T_VEBT))),B4),aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(A2),B4)) ) ).

% subset_insert_iff
tff(fact_5200_subset__insert__iff,axiom,
    ! [A2: set_real,Xb: real,B4: set_real] :
      ( aa_set_real_o(ord_less_eq_set_real(A2),insert_real(Xb,B4))
    <=> $ite(aa_set_real_o(member_real(Xb),A2),aa_set_real_o(ord_less_eq_set_real(minus_minus_set_real(A2,insert_real(Xb,bot_bot_set_real))),B4),aa_set_real_o(ord_less_eq_set_real(A2),B4)) ) ).

% subset_insert_iff
tff(fact_5201_subset__insert__iff,axiom,
    ! [A2: set_nat,Xb: nat,B4: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A2),insert_nat(Xb,B4))
    <=> $ite(aa_set_nat_o(member_nat(Xb),A2),aa_set_nat_o(ord_less_eq_set_nat(minus_minus_set_nat(A2,insert_nat(Xb,bot_bot_set_nat))),B4),aa_set_nat_o(ord_less_eq_set_nat(A2),B4)) ) ).

% subset_insert_iff
tff(fact_5202_subset__insert__iff,axiom,
    ! [A2: set_int,Xb: int,B4: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A2),insert_int(Xb,B4))
    <=> $ite(aa_set_int_o(member_int(Xb),A2),aa_set_int_o(ord_less_eq_set_int(minus_minus_set_int(A2,insert_int(Xb,bot_bot_set_int))),B4),aa_set_int_o(ord_less_eq_set_int(A2),B4)) ) ).

% subset_insert_iff
tff(fact_5203_Diff__single__insert,axiom,
    ! [A2: set_VEBT_VEBT,Xb: vEBT_VEBT,B4: set_VEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(minus_5127226145743854075T_VEBT(A2,insert_VEBT_VEBT(Xb,bot_bo8194388402131092736T_VEBT))),B4)
     => aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(A2),insert_VEBT_VEBT(Xb,B4)) ) ).

% Diff_single_insert
tff(fact_5204_Diff__single__insert,axiom,
    ! [A2: set_real,Xb: real,B4: set_real] :
      ( aa_set_real_o(ord_less_eq_set_real(minus_minus_set_real(A2,insert_real(Xb,bot_bot_set_real))),B4)
     => aa_set_real_o(ord_less_eq_set_real(A2),insert_real(Xb,B4)) ) ).

% Diff_single_insert
tff(fact_5205_Diff__single__insert,axiom,
    ! [A2: set_nat,Xb: nat,B4: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(minus_minus_set_nat(A2,insert_nat(Xb,bot_bot_set_nat))),B4)
     => aa_set_nat_o(ord_less_eq_set_nat(A2),insert_nat(Xb,B4)) ) ).

% Diff_single_insert
tff(fact_5206_Diff__single__insert,axiom,
    ! [A2: set_int,Xb: int,B4: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(minus_minus_set_int(A2,insert_int(Xb,bot_bot_set_int))),B4)
     => aa_set_int_o(ord_less_eq_set_int(A2),insert_int(Xb,B4)) ) ).

% Diff_single_insert
tff(fact_5207_atLeast0__atMost__Suc,axiom,
    ! [Nb: nat] : set_or1269000886237332187st_nat(zero_zero_nat,aa_nat_nat(suc,Nb)) = insert_nat(aa_nat_nat(suc,Nb),set_or1269000886237332187st_nat(zero_zero_nat,Nb)) ).

% atLeast0_atMost_Suc
tff(fact_5208_Icc__eq__insert__lb__nat,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
     => ( set_or1269000886237332187st_nat(M,Nb) = insert_nat(M,set_or1269000886237332187st_nat(aa_nat_nat(suc,M),Nb)) ) ) ).

% Icc_eq_insert_lb_nat
tff(fact_5209_atLeastAtMostSuc__conv,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(suc,Nb))
     => ( set_or1269000886237332187st_nat(M,aa_nat_nat(suc,Nb)) = insert_nat(aa_nat_nat(suc,Nb),set_or1269000886237332187st_nat(M,Nb)) ) ) ).

% atLeastAtMostSuc_conv
tff(fact_5210_atLeastAtMost__insertL,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
     => ( insert_nat(M,set_or1269000886237332187st_nat(aa_nat_nat(suc,M),Nb)) = set_or1269000886237332187st_nat(M,Nb) ) ) ).

% atLeastAtMost_insertL
tff(fact_5211_remove__subset,axiom,
    ! [Xb: vEBT_VEBT,S2: set_VEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xb),S2)
     => aa_set_VEBT_VEBT_o(ord_le3480810397992357184T_VEBT(minus_5127226145743854075T_VEBT(S2,insert_VEBT_VEBT(Xb,bot_bo8194388402131092736T_VEBT))),S2) ) ).

% remove_subset
tff(fact_5212_remove__subset,axiom,
    ! [Xb: int,S2: set_int] :
      ( aa_set_int_o(member_int(Xb),S2)
     => aa_set_int_o(ord_less_set_int(minus_minus_set_int(S2,insert_int(Xb,bot_bot_set_int))),S2) ) ).

% remove_subset
tff(fact_5213_remove__subset,axiom,
    ! [Xb: real,S2: set_real] :
      ( aa_set_real_o(member_real(Xb),S2)
     => aa_set_real_o(ord_less_set_real(minus_minus_set_real(S2,insert_real(Xb,bot_bot_set_real))),S2) ) ).

% remove_subset
tff(fact_5214_remove__subset,axiom,
    ! [Xb: nat,S2: set_nat] :
      ( aa_set_nat_o(member_nat(Xb),S2)
     => aa_set_nat_o(ord_less_set_nat(minus_minus_set_nat(S2,insert_nat(Xb,bot_bot_set_nat))),S2) ) ).

% remove_subset
tff(fact_5215_set__update__subset__insert,axiom,
    ! [Xs: list_nat,I: nat,Xb: nat] : aa_set_nat_o(ord_less_eq_set_nat(set_nat2(list_update_nat(Xs,I,Xb))),insert_nat(Xb,set_nat2(Xs))) ).

% set_update_subset_insert
tff(fact_5216_set__update__subset__insert,axiom,
    ! [Xs: list_real,I: nat,Xb: real] : aa_set_real_o(ord_less_eq_set_real(set_real2(list_update_real(Xs,I,Xb))),insert_real(Xb,set_real2(Xs))) ).

% set_update_subset_insert
tff(fact_5217_set__update__subset__insert,axiom,
    ! [Xs: list_P8527749157015355191n_assn,I: nat,Xb: produc6575502325842934193n_assn] : ord_le171416862856029873n_assn(set_Pr1139785259514867910n_assn(list_u4534839942911652127n_assn(Xs,I,Xb)),insert5290817439147925377n_assn(Xb,set_Pr1139785259514867910n_assn(Xs))) ).

% set_update_subset_insert
tff(fact_5218_set__update__subset__insert,axiom,
    ! [Xs: list_VEBT_VEBT,I: nat,Xb: vEBT_VEBT] : aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(set_VEBT_VEBT2(list_u1324408373059187874T_VEBT(Xs,I,Xb))),insert_VEBT_VEBT(Xb,set_VEBT_VEBT2(Xs))) ).

% set_update_subset_insert
tff(fact_5219_set__update__subset__insert,axiom,
    ! [Xs: list_VEBT_VEBTi,I: nat,Xb: vEBT_VEBTi] : ord_le6592769550269828683_VEBTi(set_VEBT_VEBTi2(list_u6098035379799741383_VEBTi(Xs,I,Xb)),insert_VEBT_VEBTi(Xb,set_VEBT_VEBTi2(Xs))) ).

% set_update_subset_insert
tff(fact_5220_set__update__subset__insert,axiom,
    ! [Xs: list_int,I: nat,Xb: int] : aa_set_int_o(ord_less_eq_set_int(set_int2(list_update_int(Xs,I,Xb))),insert_int(Xb,set_int2(Xs))) ).

% set_update_subset_insert
tff(fact_5221_cong__exp__iff__simps_I3_J,axiom,
    ! [Nb: num,Q2: num] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Nb))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q2))) != zero_zero_nat ).

% cong_exp_iff_simps(3)
tff(fact_5222_cong__exp__iff__simps_I3_J,axiom,
    ! [Nb: num,Q2: num] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q2))) != zero_zero_int ).

% cong_exp_iff_simps(3)
tff(fact_5223_cong__exp__iff__simps_I3_J,axiom,
    ! [Nb: num,Q2: num] : aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit1,Nb))),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,Q2))) != zero_z3403309356797280102nteger ).

% cong_exp_iff_simps(3)
tff(fact_5224_mod__mult2__eq_H,axiom,
    ! [A: int,M: nat,Nb: nat] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,Nb))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,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,Nb)))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_nat_int(semiri1314217659103216013at_int,M))) ).

% mod_mult2_eq'
tff(fact_5225_mod__mult2__eq_H,axiom,
    ! [A: nat,M: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,Nb))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,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,Nb)))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(semiri1316708129612266289at_nat,M))) ).

% mod_mult2_eq'
tff(fact_5226_mod__mult2__eq_H,axiom,
    ! [A: code_integer,M: nat,Nb: nat] : aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,M)),aa_nat_Code_integer(semiri4939895301339042750nteger,Nb))) = 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,Nb)))),aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A),aa_nat_Code_integer(semiri4939895301339042750nteger,M))) ).

% mod_mult2_eq'
tff(fact_5227_field__char__0__class_Oof__nat__div,axiom,
    ! [M: nat,Nb: nat] : aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Nb)) = divide_divide_rat(minus_minus_rat(aa_nat_rat(semiri681578069525770553at_rat,M),aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Nb))),aa_nat_rat(semiri681578069525770553at_rat,Nb)) ).

% field_char_0_class.of_nat_div
tff(fact_5228_field__char__0__class_Oof__nat__div,axiom,
    ! [M: nat,Nb: nat] : aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Nb)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(semiri5074537144036343181t_real,M)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Nb)))),aa_nat_real(semiri5074537144036343181t_real,Nb)) ).

% field_char_0_class.of_nat_div
tff(fact_5229_split__mod,axiom,
    ! [P: fun_nat_o,M: nat,Nb: nat] :
      ( aa_nat_o(P,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Nb))
    <=> ( ( ( Nb = zero_zero_nat )
         => aa_nat_o(P,M) )
        & ( ( Nb != zero_zero_nat )
         => ! [I2: nat,J3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J3),Nb)
             => ( ( M = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),I2)),J3) )
               => aa_nat_o(P,J3) ) ) ) ) ) ).

% split_mod
tff(fact_5230_mod__lemma,axiom,
    ! [C: nat,R2: nat,B: nat,Q2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),C)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,R2),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Q2),C))),R2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) ) ) ).

% mod_lemma
tff(fact_5231_foldr__length__aux,axiom,
    ! [L: list_real,A: nat] : aa_nat_nat(foldr_real_nat(aTP_Lamm_hn,L),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),size_size_list_real(L)) ).

% foldr_length_aux
tff(fact_5232_foldr__length__aux,axiom,
    ! [L: list_o,A: nat] : aa_nat_nat(foldr_o_nat(aTP_Lamm_ho,L),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),size_size_list_o(L)) ).

% foldr_length_aux
tff(fact_5233_foldr__length__aux,axiom,
    ! [L: list_nat,A: nat] : aa_nat_nat(foldr_nat_nat(aTP_Lamm_hp,L),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),size_size_list_nat(L)) ).

% foldr_length_aux
tff(fact_5234_foldr__length__aux,axiom,
    ! [L: list_int,A: nat] : aa_nat_nat(foldr_int_nat(aTP_Lamm_hq,L),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),size_size_list_int(L)) ).

% foldr_length_aux
tff(fact_5235_psubset__insert__iff,axiom,
    ! [A2: set_VEBT_VEBT,Xb: vEBT_VEBT,B4: set_VEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(ord_le3480810397992357184T_VEBT(A2),insert_VEBT_VEBT(Xb,B4))
    <=> $ite(
          aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xb),B4),
          aa_set_VEBT_VEBT_o(ord_le3480810397992357184T_VEBT(A2),B4),
          $ite(aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xb),A2),aa_set_VEBT_VEBT_o(ord_le3480810397992357184T_VEBT(minus_5127226145743854075T_VEBT(A2,insert_VEBT_VEBT(Xb,bot_bo8194388402131092736T_VEBT))),B4),aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(A2),B4)) ) ) ).

% psubset_insert_iff
tff(fact_5236_psubset__insert__iff,axiom,
    ! [A2: set_real,Xb: real,B4: set_real] :
      ( aa_set_real_o(ord_less_set_real(A2),insert_real(Xb,B4))
    <=> $ite(
          aa_set_real_o(member_real(Xb),B4),
          aa_set_real_o(ord_less_set_real(A2),B4),
          $ite(aa_set_real_o(member_real(Xb),A2),aa_set_real_o(ord_less_set_real(minus_minus_set_real(A2,insert_real(Xb,bot_bot_set_real))),B4),aa_set_real_o(ord_less_eq_set_real(A2),B4)) ) ) ).

% psubset_insert_iff
tff(fact_5237_psubset__insert__iff,axiom,
    ! [A2: set_nat,Xb: nat,B4: set_nat] :
      ( aa_set_nat_o(ord_less_set_nat(A2),insert_nat(Xb,B4))
    <=> $ite(
          aa_set_nat_o(member_nat(Xb),B4),
          aa_set_nat_o(ord_less_set_nat(A2),B4),
          $ite(aa_set_nat_o(member_nat(Xb),A2),aa_set_nat_o(ord_less_set_nat(minus_minus_set_nat(A2,insert_nat(Xb,bot_bot_set_nat))),B4),aa_set_nat_o(ord_less_eq_set_nat(A2),B4)) ) ) ).

% psubset_insert_iff
tff(fact_5238_psubset__insert__iff,axiom,
    ! [A2: set_int,Xb: int,B4: set_int] :
      ( aa_set_int_o(ord_less_set_int(A2),insert_int(Xb,B4))
    <=> $ite(
          aa_set_int_o(member_int(Xb),B4),
          aa_set_int_o(ord_less_set_int(A2),B4),
          $ite(aa_set_int_o(member_int(Xb),A2),aa_set_int_o(ord_less_set_int(minus_minus_set_int(A2,insert_int(Xb,bot_bot_set_int))),B4),aa_set_int_o(ord_less_eq_set_int(A2),B4)) ) ) ).

% psubset_insert_iff
tff(fact_5239_real__of__nat__div__aux,axiom,
    ! [Xb: nat,D: nat] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(semiri5074537144036343181t_real,Xb)),aa_nat_real(semiri5074537144036343181t_real,D)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Xb),D))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xb),D))),aa_nat_real(semiri5074537144036343181t_real,D))) ).

% real_of_nat_div_aux
tff(fact_5240_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),C)
     => ( aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)) = aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),B)),C))),aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A),B)) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_mult2_eq
tff(fact_5241_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),C))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_mult2_eq
tff(fact_5242_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
     => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),C))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_mult2_eq
tff(fact_5243_cong__exp__iff__simps_I7_J,axiom,
    ! [Q2: num,Nb: num] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,one)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q2))) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Nb))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q2))) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,Nb)),aa_num_nat(numeral_numeral_nat,Q2)) = zero_zero_nat ) ) ).

% cong_exp_iff_simps(7)
tff(fact_5244_cong__exp__iff__simps_I7_J,axiom,
    ! [Q2: num,Nb: num] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,one)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q2))) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q2))) )
    <=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,Nb)),aa_num_int(numeral_numeral_int,Q2)) = zero_zero_int ) ) ).

% cong_exp_iff_simps(7)
tff(fact_5245_cong__exp__iff__simps_I7_J,axiom,
    ! [Q2: num,Nb: num] :
      ( ( aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_num_Code_integer(numera6620942414471956472nteger,one)),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,Q2))) = aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit1,Nb))),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,Q2))) )
    <=> ( aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_num_Code_integer(numera6620942414471956472nteger,Nb)),aa_num_Code_integer(numera6620942414471956472nteger,Q2)) = zero_z3403309356797280102nteger ) ) ).

% cong_exp_iff_simps(7)
tff(fact_5246_cong__exp__iff__simps_I11_J,axiom,
    ! [M: num,Q2: num] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,M))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q2))) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,one)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q2))) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,Q2)) = zero_zero_nat ) ) ).

% cong_exp_iff_simps(11)
tff(fact_5247_cong__exp__iff__simps_I11_J,axiom,
    ! [M: num,Q2: num] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,M))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q2))) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,one)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q2))) )
    <=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,Q2)) = zero_zero_int ) ) ).

% cong_exp_iff_simps(11)
tff(fact_5248_cong__exp__iff__simps_I11_J,axiom,
    ! [M: num,Q2: num] :
      ( ( aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit1,M))),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,Q2))) = aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_num_Code_integer(numera6620942414471956472nteger,one)),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,Q2))) )
    <=> ( aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_num_Code_integer(numera6620942414471956472nteger,M)),aa_num_Code_integer(numera6620942414471956472nteger,Q2)) = zero_z3403309356797280102nteger ) ) ).

% cong_exp_iff_simps(11)
tff(fact_5249_Suc__mod__eq__add3__mod,axiom,
    ! [M: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,M)))),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),M)),Nb) ).

% Suc_mod_eq_add3_mod
tff(fact_5250_Suc__times__mod__eq,axiom,
    ! [M: nat,Nb: 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),Nb))),M) = one_one_nat ) ) ).

% Suc_times_mod_eq
tff(fact_5251_insert__swap__set__eq,axiom,
    ! [I: nat,L: list_P8527749157015355191n_assn,Xb: produc6575502325842934193n_assn] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s6829681357464350627n_assn(L))
     => ( insert5290817439147925377n_assn(nth_Pr1769885009046257848n_assn(L,I),set_Pr1139785259514867910n_assn(list_u4534839942911652127n_assn(L,I,Xb))) = insert5290817439147925377n_assn(Xb,set_Pr1139785259514867910n_assn(L)) ) ) ).

% insert_swap_set_eq
tff(fact_5252_insert__swap__set__eq,axiom,
    ! [I: nat,L: list_VEBT_VEBT,Xb: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s6755466524823107622T_VEBT(L))
     => ( insert_VEBT_VEBT(nth_VEBT_VEBT(L,I),set_VEBT_VEBT2(list_u1324408373059187874T_VEBT(L,I,Xb))) = insert_VEBT_VEBT(Xb,set_VEBT_VEBT2(L)) ) ) ).

% insert_swap_set_eq
tff(fact_5253_insert__swap__set__eq,axiom,
    ! [I: nat,L: list_VEBT_VEBTi,Xb: vEBT_VEBTi] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s7982070591426661849_VEBTi(L))
     => ( insert_VEBT_VEBTi(nth_VEBT_VEBTi(L,I),set_VEBT_VEBTi2(list_u6098035379799741383_VEBTi(L,I,Xb))) = insert_VEBT_VEBTi(Xb,set_VEBT_VEBTi2(L)) ) ) ).

% insert_swap_set_eq
tff(fact_5254_insert__swap__set__eq,axiom,
    ! [I: nat,L: list_real,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_real(L))
     => ( insert_real(aa_nat_real(nth_real(L),I),set_real2(list_update_real(L,I,Xb))) = insert_real(Xb,set_real2(L)) ) ) ).

% insert_swap_set_eq
tff(fact_5255_insert__swap__set__eq,axiom,
    ! [I: nat,L: list_o,Xb: $o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_o(L))
     => ( insert_o(aa_nat_o(nth_o(L),I),set_o2(list_update_o(L,I,(Xb)))) = insert_o((Xb),set_o2(L)) ) ) ).

% insert_swap_set_eq
tff(fact_5256_insert__swap__set__eq,axiom,
    ! [I: nat,L: list_nat,Xb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_nat(L))
     => ( insert_nat(aa_nat_nat(nth_nat(L),I),set_nat2(list_update_nat(L,I,Xb))) = insert_nat(Xb,set_nat2(L)) ) ) ).

% insert_swap_set_eq
tff(fact_5257_insert__swap__set__eq,axiom,
    ! [I: nat,L: list_int,Xb: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_int(L))
     => ( insert_int(aa_nat_int(nth_int(L),I),set_int2(list_update_int(L,I,Xb))) = insert_int(Xb,set_int2(L)) ) ) ).

% insert_swap_set_eq
tff(fact_5258_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_5259_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_5260_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_5261_bits__stable__imp__add__self,axiom,
    ! [A: word_N3645301735248828278l_num1] :
      ( ( divide1791077408188789448l_num1(A,numera7442385471795722001l_num1(aa_num_num(bit0,one))) = A )
     => ( plus_p361126936061061375l_num1(A,modulo1504961113040953224l_num1(A,numera7442385471795722001l_num1(aa_num_num(bit0,one)))) = zero_z3563351764282998399l_num1 ) ) ).

% bits_stable_imp_add_self
tff(fact_5262_bits__stable__imp__add__self,axiom,
    ! [A: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = A )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = zero_zero_nat ) ) ).

% bits_stable_imp_add_self
tff(fact_5263_bits__stable__imp__add__self,axiom,
    ! [A: int] :
      ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = A )
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))) = zero_zero_int ) ) ).

% bits_stable_imp_add_self
tff(fact_5264_bits__stable__imp__add__self,axiom,
    ! [A: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) = A )
     => ( aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(A),aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)))) = zero_z3403309356797280102nteger ) ) ).

% bits_stable_imp_add_self
tff(fact_5265_bits__stable__imp__add__self,axiom,
    ! [A: uint32] :
      ( ( aa_uint32_uint32(aa_uin3778660748068847871uint32(divide_divide_uint32,A),numera9087168376688890119uint32(aa_num_num(bit0,one))) = A )
     => ( aa_uint32_uint32(plus_plus_uint32(A),aa_uint32_uint32(aa_uin3778660748068847871uint32(modulo_modulo_uint32,A),numera9087168376688890119uint32(aa_num_num(bit0,one)))) = zero_zero_uint32 ) ) ).

% bits_stable_imp_add_self
tff(fact_5266_div__exp__mod__exp__eq,axiom,
    ! [A: word_N3645301735248828278l_num1,Nb: nat,M: nat] : modulo1504961113040953224l_num1(divide1791077408188789448l_num1(A,aa_nat5599461682457136004l_num1(power_2184487114949457152l_num1(numera7442385471795722001l_num1(aa_num_num(bit0,one))),Nb)),aa_nat5599461682457136004l_num1(power_2184487114949457152l_num1(numera7442385471795722001l_num1(aa_num_num(bit0,one))),M)) = divide1791077408188789448l_num1(modulo1504961113040953224l_num1(A,aa_nat5599461682457136004l_num1(power_2184487114949457152l_num1(numera7442385471795722001l_num1(aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),M))),aa_nat5599461682457136004l_num1(power_2184487114949457152l_num1(numera7442385471795722001l_num1(aa_num_num(bit0,one))),Nb)) ).

% div_exp_mod_exp_eq
tff(fact_5267_div__exp__mod__exp__eq,axiom,
    ! [A: nat,Nb: nat,M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),M)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ).

% div_exp_mod_exp_eq
tff(fact_5268_div__exp__mod__exp__eq,axiom,
    ! [A: int,Nb: nat,M: nat] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb))),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),M)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_nat_int(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(plus_plus_nat,Nb),M)))),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) ).

% div_exp_mod_exp_eq
tff(fact_5269_div__exp__mod__exp__eq,axiom,
    ! [A: code_integer,Nb: nat,M: nat] : aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Nb))),aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),M)) = aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(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(plus_plus_nat,Nb),M)))),aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Nb)) ).

% div_exp_mod_exp_eq
tff(fact_5270_div__exp__mod__exp__eq,axiom,
    ! [A: uint32,Nb: nat,M: nat] : aa_uint32_uint32(aa_uin3778660748068847871uint32(modulo_modulo_uint32,aa_uint32_uint32(aa_uin3778660748068847871uint32(divide_divide_uint32,A),aa_nat_uint32(power_power_uint32(numera9087168376688890119uint32(aa_num_num(bit0,one))),Nb))),aa_nat_uint32(power_power_uint32(numera9087168376688890119uint32(aa_num_num(bit0,one))),M)) = aa_uint32_uint32(aa_uin3778660748068847871uint32(divide_divide_uint32,aa_uint32_uint32(aa_uin3778660748068847871uint32(modulo_modulo_uint32,A),aa_nat_uint32(power_power_uint32(numera9087168376688890119uint32(aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),M)))),aa_nat_uint32(power_power_uint32(numera9087168376688890119uint32(aa_num_num(bit0,one))),Nb)) ).

% div_exp_mod_exp_eq
tff(fact_5271_power__mod__div,axiom,
    ! [Xb: nat,Nb: nat,M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))),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(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),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,Nb),M))) ).

% power_mod_div
tff(fact_5272_verit__le__mono__div,axiom,
    ! [A2: nat,B4: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),B4)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,
              aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),Nb)),
                $ite(aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B4),Nb) = zero_zero_nat,one_one_nat,zero_zero_nat))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B4),Nb)) ) ) ).

% verit_le_mono_div
tff(fact_5273_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_5274_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_5275_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_5276_mult__exp__mod__exp__eq,axiom,
    ! [M: nat,Nb: nat,A: word_N3645301735248828278l_num1] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
     => ( modulo1504961113040953224l_num1(aa_wor7274019993978516266l_num1(times_7065122842183080059l_num1(A),aa_nat5599461682457136004l_num1(power_2184487114949457152l_num1(numera7442385471795722001l_num1(aa_num_num(bit0,one))),M)),aa_nat5599461682457136004l_num1(power_2184487114949457152l_num1(numera7442385471795722001l_num1(aa_num_num(bit0,one))),Nb)) = aa_wor7274019993978516266l_num1(times_7065122842183080059l_num1(modulo1504961113040953224l_num1(A,aa_nat5599461682457136004l_num1(power_2184487114949457152l_num1(numera7442385471795722001l_num1(aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),M)))),aa_nat5599461682457136004l_num1(power_2184487114949457152l_num1(numera7442385471795722001l_num1(aa_num_num(bit0,one))),M)) ) ) ).

% mult_exp_mod_exp_eq
tff(fact_5277_mult__exp__mod__exp__eq,axiom,
    ! [M: nat,Nb: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),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_5278_mult__exp__mod__exp__eq,axiom,
    ! [M: nat,Nb: nat,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
     => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_nat_int(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))),Nb)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_nat_int(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,Nb),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_5279_mult__exp__mod__exp__eq,axiom,
    ! [M: nat,Nb: nat,A: code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
     => ( 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))),Nb)) = 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,Nb),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_5280_mult__exp__mod__exp__eq,axiom,
    ! [M: nat,Nb: nat,A: uint32] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
     => ( aa_uint32_uint32(aa_uin3778660748068847871uint32(modulo_modulo_uint32,aa_uint32_uint32(times_times_uint32(A),aa_nat_uint32(power_power_uint32(numera9087168376688890119uint32(aa_num_num(bit0,one))),M))),aa_nat_uint32(power_power_uint32(numera9087168376688890119uint32(aa_num_num(bit0,one))),Nb)) = aa_uint32_uint32(times_times_uint32(aa_uint32_uint32(aa_uin3778660748068847871uint32(modulo_modulo_uint32,A),aa_nat_uint32(power_power_uint32(numera9087168376688890119uint32(aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),M)))),aa_nat_uint32(power_power_uint32(numera9087168376688890119uint32(aa_num_num(bit0,one))),M)) ) ) ).

% mult_exp_mod_exp_eq
tff(fact_5281_mod__double__modulus,axiom,
    ! [M: code_integer,Xb: 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),Xb)
       => ( ( aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(Xb),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),M)) = aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(Xb),M) )
          | ( aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(Xb),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),M)) = aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(Xb),M)),M) ) ) ) ) ).

% mod_double_modulus
tff(fact_5282_mod__double__modulus,axiom,
    ! [M: nat,Xb: 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),Xb)
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xb),M) )
          | ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xb),M)),M) ) ) ) ) ).

% mod_double_modulus
tff(fact_5283_mod__double__modulus,axiom,
    ! [M: int,Xb: 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),Xb)
       => ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Xb),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),M)) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Xb),M) )
          | ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Xb),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),M)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Xb),M)),M) ) ) ) ) ).

% mod_double_modulus
tff(fact_5284_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_5285_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_5286_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_5287_set__bit__Suc,axiom,
    ! [Nb: nat,A: word_N3645301735248828278l_num1] : bit_se4894374433684937756l_num1(aa_nat_nat(suc,Nb),A) = plus_p361126936061061375l_num1(modulo1504961113040953224l_num1(A,numera7442385471795722001l_num1(aa_num_num(bit0,one))),aa_wor7274019993978516266l_num1(times_7065122842183080059l_num1(numera7442385471795722001l_num1(aa_num_num(bit0,one))),bit_se4894374433684937756l_num1(Nb,divide1791077408188789448l_num1(A,numera7442385471795722001l_num1(aa_num_num(bit0,one)))))) ).

% set_bit_Suc
tff(fact_5288_set__bit__Suc,axiom,
    ! [Nb: nat,A: code_integer] : aa_Cod1783974652533396728nteger(bit_se2793503036327961859nteger(aa_nat_nat(suc,Nb)),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(Nb),aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)))))) ).

% set_bit_Suc
tff(fact_5289_set__bit__Suc,axiom,
    ! [Nb: nat,A: uint32] : bit_se6647067497041451410uint32(aa_nat_nat(suc,Nb),A) = aa_uint32_uint32(plus_plus_uint32(aa_uint32_uint32(aa_uin3778660748068847871uint32(modulo_modulo_uint32,A),numera9087168376688890119uint32(aa_num_num(bit0,one)))),aa_uint32_uint32(times_times_uint32(numera9087168376688890119uint32(aa_num_num(bit0,one))),bit_se6647067497041451410uint32(Nb,aa_uint32_uint32(aa_uin3778660748068847871uint32(divide_divide_uint32,A),numera9087168376688890119uint32(aa_num_num(bit0,one)))))) ).

% set_bit_Suc
tff(fact_5290_set__bit__Suc,axiom,
    ! [Nb: nat,A: int] : aa_int_int(aa_nat_fun_int_int(bit_se7879613467334960850it_int,aa_nat_nat(suc,Nb)),A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_nat_fun_int_int(bit_se7879613467334960850it_int,Nb),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ).

% set_bit_Suc
tff(fact_5291_set__bit__Suc,axiom,
    ! [Nb: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se7882103937844011126it_nat,aa_nat_nat(suc,Nb)),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se7882103937844011126it_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% set_bit_Suc
tff(fact_5292_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_5293_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_5294_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_5295_norm__divide__numeral,axiom,
    ! [A: real,W: num] : aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),numeral_numeral_real(W))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(real_V7735802525324610683m_real,A)),numeral_numeral_real(W)) ).

% norm_divide_numeral
tff(fact_5296_norm__divide__numeral,axiom,
    ! [A: complex,W: num] : aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,A),numera6690914467698888265omplex(W))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_complex_real(real_V1022390504157884413omplex,A)),numeral_numeral_real(W)) ).

% norm_divide_numeral
tff(fact_5297_norm__mult__numeral2,axiom,
    ! [A: real,W: num] : aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),numeral_numeral_real(W))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(real_V7735802525324610683m_real,A)),numeral_numeral_real(W)) ).

% norm_mult_numeral2
tff(fact_5298_norm__mult__numeral2,axiom,
    ! [A: complex,W: num] : aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,A),numera6690914467698888265omplex(W))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(real_V1022390504157884413omplex,A)),numeral_numeral_real(W)) ).

% norm_mult_numeral2
tff(fact_5299_norm__mult__numeral1,axiom,
    ! [W: num,A: real] : aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(W)),A)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(W)),aa_real_real(real_V7735802525324610683m_real,A)) ).

% norm_mult_numeral1
tff(fact_5300_norm__mult__numeral1,axiom,
    ! [W: num,A: complex] : aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,numera6690914467698888265omplex(W)),A)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(W)),aa_complex_real(real_V1022390504157884413omplex,A)) ).

% norm_mult_numeral1
tff(fact_5301_div__half__nat,axiom,
    ! [Y: nat,Xb: nat] :
      ( ( Y != zero_zero_nat )
     => ( aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Xb),Y)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xb),Y)) = $let(
            q: nat,
            q:= aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),Y)),
            $let(
              r: nat,
              r:= aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,q),Y)),
              $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),r),aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,q),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,r),Y)),aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,q),r)) ) ) ) ) ).

% div_half_nat
tff(fact_5302_norm__le__zero__iff,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,Xb)),zero_zero_real)
    <=> ( Xb = zero_zero_real ) ) ).

% norm_le_zero_iff
tff(fact_5303_norm__le__zero__iff,axiom,
    ! [Xb: complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,Xb)),zero_zero_real)
    <=> ( Xb = zero_zero_complex ) ) ).

% norm_le_zero_iff
tff(fact_5304_norm__numeral,axiom,
    ! [W: num] : aa_real_real(real_V7735802525324610683m_real,numeral_numeral_real(W)) = numeral_numeral_real(W) ).

% norm_numeral
tff(fact_5305_norm__numeral,axiom,
    ! [W: num] : aa_complex_real(real_V1022390504157884413omplex,numera6690914467698888265omplex(W)) = numeral_numeral_real(W) ).

% norm_numeral
tff(fact_5306_foldr0,axiom,
    ! [Xs: list_real,C: real,D: real] : aa_real_real(foldr_real_real(plus_plus_real,Xs),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),D)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(foldr_real_real(plus_plus_real,Xs),D)),C) ).

% foldr0
tff(fact_5307_foldr__same,axiom,
    ! [Xs: list_real,Y: real] :
      ( ! [X2: real,Y4: real] :
          ( aa_set_real_o(member_real(X2),set_real2(Xs))
         => ( aa_set_real_o(member_real(Y4),set_real2(Xs))
           => ( X2 = Y4 ) ) )
     => ( ! [X2: real] :
            ( aa_set_real_o(member_real(X2),set_real2(Xs))
           => ( X2 = Y ) )
       => ( aa_real_real(foldr_real_real(plus_plus_real,Xs),zero_zero_real) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,size_size_list_real(Xs))),Y) ) ) ) ).

% foldr_same
tff(fact_5308_norm__one,axiom,
    aa_real_real(real_V7735802525324610683m_real,one_one_real) = one_one_real ).

% norm_one
tff(fact_5309_norm__one,axiom,
    aa_complex_real(real_V1022390504157884413omplex,one_one_complex) = one_one_real ).

% norm_one
tff(fact_5310_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_5311_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_5312_zmod__numeral__Bit0,axiom,
    ! [V: num,W: num] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,V))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,W))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,V)),aa_num_int(numeral_numeral_int,W))) ).

% zmod_numeral_Bit0
tff(fact_5313_one__mod__exp__eq__one,axiom,
    ! [Nb: nat] : aa_int_int(aa_int_fun_int_int(modulo_modulo_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(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb))) = one_one_int ).

% one_mod_exp_eq_one
tff(fact_5314_zmod__numeral__Bit1,axiom,
    ! [V: num,W: num] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,V))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,W))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,V)),aa_num_int(numeral_numeral_int,W)))),one_one_int) ).

% zmod_numeral_Bit1
tff(fact_5315_mod__plus__cong,axiom,
    ! [B: int,B3: int,Xb: int,X4: int,Y: int,Y6: int,Z5: int] :
      ( ( B = B3 )
     => ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Xb),B3) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,X4),B3) )
       => ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Y),B3) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Y6),B3) )
         => ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,X4),Y6) = Z5 )
           => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb),Y)),B) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Z5),B3) ) ) ) ) ) ).

% mod_plus_cong
tff(fact_5316_zmod__helper,axiom,
    ! [Nb: int,M: int,K: int,A: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Nb),M) = K )
     => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Nb),A)),M) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,K),A)),M) ) ) ).

% zmod_helper
tff(fact_5317_Word_Omod__minus__cong,axiom,
    ! [B: int,B3: int,Xb: int,X4: int,Y: int,Y6: int,Z5: int] :
      ( ( B = B3 )
     => ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Xb),B3) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,X4),B3) )
       => ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Y),B3) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Y6),B3) )
         => ( ( aa_int_int(aa_int_fun_int_int(minus_minus_int,X4),Y6) = Z5 )
           => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb),Y)),B) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Z5),B3) ) ) ) ) ) ).

% Word.mod_minus_cong
tff(fact_5318_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_5319_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_5320_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_5321_zmod__eq__0D,axiom,
    ! [M: int,D: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,M),D) = zero_zero_int )
     => ? [Q3: int] : M = aa_int_int(aa_int_fun_int_int(times_times_int,D),Q3) ) ).

% zmod_eq_0D
tff(fact_5322_zmod__eq__0__iff,axiom,
    ! [M: int,D: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,M),D) = zero_zero_int )
    <=> ? [Q4: int] : M = aa_int_int(aa_int_fun_int_int(times_times_int,D),Q4) ) ).

% zmod_eq_0_iff
tff(fact_5323_int__mod__ge,axiom,
    ! [A: int,Nb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),Nb)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Nb)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),Nb)) ) ) ).

% int_mod_ge
tff(fact_5324_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_5325_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_5326_zmod__trivial__iff,axiom,
    ! [I: int,K: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,I),K) = I )
    <=> ( ( K = zero_zero_int )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),I)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,I),K) )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),zero_zero_int)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,K),I) ) ) ) ).

% zmod_trivial_iff
tff(fact_5327_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_5328_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_5329_int__mod__eq,axiom,
    ! [B: int,Nb: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),Nb)
       => ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),Nb) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B),Nb) )
         => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),Nb) = B ) ) ) ) ).

% int_mod_eq
tff(fact_5330_int__mod__lem,axiom,
    ! [Nb: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Nb)
     => ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,B),Nb) )
      <=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B),Nb) = B ) ) ) ).

% int_mod_lem
tff(fact_5331_int__mod__le_H,axiom,
    ! [B: int,Nb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),Nb))
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B),Nb)),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),Nb)) ) ).

% int_mod_le'
tff(fact_5332_nonneg__mod__div,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B))
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)) ) ) ) ).

% nonneg_mod_div
tff(fact_5333_zdiv__mono__strict,axiom,
    ! [A2: int,B4: int,Nb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A2),B4)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Nb)
       => ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),Nb) = zero_zero_int )
         => ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B4),Nb) = zero_zero_int )
           => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),Nb)),aa_int_int(aa_int_fun_int_int(divide_divide_int,B4),Nb)) ) ) ) ) ).

% zdiv_mono_strict
tff(fact_5334_div__mod__decomp__int,axiom,
    ! [A2: int,Nb: int] : A2 = 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,A2),Nb)),Nb)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),Nb)) ).

% div_mod_decomp_int
tff(fact_5335_mod__div__equality__div__eq,axiom,
    ! [A: int,B: 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(minus_minus_int,A),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)) ).

% mod_div_equality_div_eq
tff(fact_5336_word__mod__div__equality,axiom,
    ! [Nb: word_N3645301735248828278l_num1,B: word_N3645301735248828278l_num1] : plus_p361126936061061375l_num1(aa_wor7274019993978516266l_num1(times_7065122842183080059l_num1(divide1791077408188789448l_num1(Nb,B)),B),modulo1504961113040953224l_num1(Nb,B)) = Nb ).

% word_mod_div_equality
tff(fact_5337_nat__mod__as__int,axiom,
    ! [X3: nat,Xa: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X3),Xa) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_nat_int(semiri1314217659103216013at_int,X3)),aa_nat_int(semiri1314217659103216013at_int,Xa))) ).

% nat_mod_as_int
tff(fact_5338_pos__mod__bound2,axiom,
    ! [A: int] : aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% pos_mod_bound2
tff(fact_5339_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_5340_int__mod__ge_H,axiom,
    ! [B: int,Nb: 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),Nb)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,B),Nb)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B),Nb)) ) ) ).

% int_mod_ge'
tff(fact_5341_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_5342_nat__mod__distrib,axiom,
    ! [Xb: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xb)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
       => ( aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Xb),Y)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_int_nat(nat2,Xb)),aa_int_nat(nat2,Y)) ) ) ) ).

% nat_mod_distrib
tff(fact_5343_real__of__int__div__aux,axiom,
    ! [Xb: int,D: int] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_int_real(ring_1_of_int_real,Xb)),aa_int_real(ring_1_of_int_real,D)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(divide_divide_int,Xb),D))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Xb),D))),aa_int_real(ring_1_of_int_real,D))) ).

% real_of_int_div_aux
tff(fact_5344_atLeastAtMostPlus1__int__conv,axiom,
    ! [M: int,Nb: 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),Nb))
     => ( set_or1266510415728281911st_int(M,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),Nb)) = insert_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),Nb),set_or1266510415728281911st_int(M,Nb)) ) ) ).

% atLeastAtMostPlus1_int_conv
tff(fact_5345_simp__from__to,axiom,
    ! [I: int,J: int] :
      set_or1266510415728281911st_int(I,J) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_int,J),I),bot_bot_set_int,insert_int(I,set_or1266510415728281911st_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,I),one_one_int),J))) ).

% simp_from_to
tff(fact_5346_pos__mod__sign2,axiom,
    ! [A: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))) ).

% pos_mod_sign2
tff(fact_5347_mod__2__neq__1__eq__eq__0,axiom,
    ! [K: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) != one_one_int )
    <=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = zero_zero_int ) ) ).

% mod_2_neq_1_eq_eq_0
tff(fact_5348_nmod2,axiom,
    ! [Nb: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Nb),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = zero_zero_int )
      | ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Nb),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = one_one_int ) ) ).

% nmod2
tff(fact_5349_mod__exp__less__eq__exp,axiom,
    ! [A: int,Nb: nat] : aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb))),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) ).

% mod_exp_less_eq_exp
tff(fact_5350_mod__power__lem,axiom,
    ! [A: int,Nb: nat,M: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A)
     => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A),Nb)),aa_nat_int(aa_int_fun_nat_int(power_power_int,A),M)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb),zero_zero_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A),Nb)) ) ) ).

% mod_power_lem
tff(fact_5351_split__zmod,axiom,
    ! [P: fun_int_o,Nb: int,K: int] :
      ( aa_int_o(P,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Nb),K))
    <=> ( ( ( K = zero_zero_int )
         => aa_int_o(P,Nb) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),K)
         => ! [I2: 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)
                & ( Nb = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,K),I2)),J3) ) )
             => aa_int_o(P,J3) ) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int)
         => ! [I2: 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)
                & ( Nb = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,K),I2)),J3) ) )
             => aa_int_o(P,J3) ) ) ) ) ).

% split_zmod
tff(fact_5352_int__mod__neg__eq,axiom,
    ! [A: int,B: int,Q2: int,R2: int] :
      ( ( A = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),Q2)),R2) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,R2),zero_zero_int)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),R2)
         => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) = R2 ) ) ) ) ).

% int_mod_neg_eq
tff(fact_5353_int__mod__pos__eq,axiom,
    ! [A: int,B: int,Q2: int,R2: int] :
      ( ( A = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),Q2)),R2) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),R2)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,R2),B)
         => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) = R2 ) ) ) ) ).

% int_mod_pos_eq
tff(fact_5354_mod__sub__if__z,axiom,
    ! [Xb: int,Z: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Z)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Y),Z)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xb)
           => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z)
             => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb),Y)),Z) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),Xb),aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb),Y),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb),Y)),Z)) ) ) ) ) ) ) ).

% mod_sub_if_z
tff(fact_5355_mod__add__if__z,axiom,
    ! [Xb: int,Z: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Z)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Y),Z)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xb)
           => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z)
             => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb),Y)),Z) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb),Y)),Z),aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb),Y),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb),Y)),Z)) ) ) ) ) ) ) ).

% mod_add_if_z
tff(fact_5356_zmod__zmult2__eq,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
     => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),C))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)) ) ) ).

% zmod_zmult2_eq
tff(fact_5357_axxmod2,axiom,
    ! [Xb: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),Xb)),Xb)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = one_one_int )
      & ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,zero_zero_int),Xb)),Xb)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = zero_zero_int ) ) ).

% axxmod2
tff(fact_5358_z1pmod2,axiom,
    ! [B: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B)),one_one_int)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = one_one_int ).

% z1pmod2
tff(fact_5359_verit__le__mono__div__int,axiom,
    ! [A2: int,B4: int,Nb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A2),B4)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Nb)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,
              aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),Nb)),
                $ite(aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B4),Nb) = zero_zero_int,one_one_int,zero_zero_int))),aa_int_int(aa_int_fun_int_int(divide_divide_int,B4),Nb)) ) ) ).

% verit_le_mono_div_int
tff(fact_5360_split__neg__lemma,axiom,
    ! [K: int,P: fun_int_fun_int_o,Nb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(P,aa_int_int(aa_int_fun_int_int(divide_divide_int,Nb),K)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Nb),K))
      <=> ! [I2: 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)
              & ( Nb = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,K),I2)),J3) ) )
           => aa_int_o(aa_int_fun_int_o(P,I2),J3) ) ) ) ).

% split_neg_lemma
tff(fact_5361_split__pos__lemma,axiom,
    ! [K: int,P: fun_int_fun_int_o,Nb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),K)
     => ( aa_int_o(aa_int_fun_int_o(P,aa_int_int(aa_int_fun_int_int(divide_divide_int,Nb),K)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Nb),K))
      <=> ! [I2: 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)
              & ( Nb = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,K),I2)),J3) ) )
           => aa_int_o(aa_int_fun_int_o(P,I2),J3) ) ) ) ).

% split_pos_lemma
tff(fact_5362_norm__minus__commute,axiom,
    ! [A: real,B: real] : aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)) = aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),A)) ).

% norm_minus_commute
tff(fact_5363_norm__minus__commute,axiom,
    ! [A: complex,B: complex] : aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,A),B)) = aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,B),A)) ).

% norm_minus_commute
tff(fact_5364_p1mod22k_H,axiom,
    ! [B: int,Nb: nat] : 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))),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(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),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)))) ).

% p1mod22k'
tff(fact_5365_p1mod22k,axiom,
    ! [B: int,Nb: nat] : 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,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B)),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(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,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),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)))),one_one_int) ).

% p1mod22k
tff(fact_5366_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_5367_sb__inc__lem,axiom,
    ! [A: int,K: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K))),zero_zero_int)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),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,K)))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),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,K)))) ) ).

% sb_inc_lem
tff(fact_5368_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_5369_norm__mult,axiom,
    ! [Xb: real,Y: real] : aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Y)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(real_V7735802525324610683m_real,Xb)),aa_real_real(real_V7735802525324610683m_real,Y)) ).

% norm_mult
tff(fact_5370_norm__mult,axiom,
    ! [Xb: complex,Y: complex] : aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,Xb),Y)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(real_V1022390504157884413omplex,Xb)),aa_complex_real(real_V1022390504157884413omplex,Y)) ).

% norm_mult
tff(fact_5371_norm__ge__zero,axiom,
    ! [Xb: complex] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(real_V1022390504157884413omplex,Xb)) ).

% norm_ge_zero
tff(fact_5372_norm__divide,axiom,
    ! [A: real,B: real] : aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(real_V7735802525324610683m_real,A)),aa_real_real(real_V7735802525324610683m_real,B)) ).

% norm_divide
tff(fact_5373_norm__divide,axiom,
    ! [A: complex,B: complex] : aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,A),B)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_complex_real(real_V1022390504157884413omplex,A)),aa_complex_real(real_V1022390504157884413omplex,B)) ).

% norm_divide
tff(fact_5374_norm__power,axiom,
    ! [Xb: real,Nb: nat] : aa_real_real(real_V7735802525324610683m_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(real_V7735802525324610683m_real,Xb)),Nb) ).

% norm_power
tff(fact_5375_norm__power,axiom,
    ! [Xb: complex,Nb: nat] : aa_complex_real(real_V1022390504157884413omplex,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Xb),Nb)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(real_V1022390504157884413omplex,Xb)),Nb) ).

% norm_power
tff(fact_5376_power__eq__imp__eq__norm,axiom,
    ! [W: real,Nb: nat,Z: real] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,W),Nb) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Z),Nb) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => ( aa_real_real(real_V7735802525324610683m_real,W) = aa_real_real(real_V7735802525324610683m_real,Z) ) ) ) ).

% power_eq_imp_eq_norm
tff(fact_5377_power__eq__imp__eq__norm,axiom,
    ! [W: complex,Nb: nat,Z: complex] :
      ( ( aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,W),Nb) = aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Z),Nb) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => ( aa_complex_real(real_V1022390504157884413omplex,W) = aa_complex_real(real_V1022390504157884413omplex,Z) ) ) ) ).

% power_eq_imp_eq_norm
tff(fact_5378_nonzero__norm__divide,axiom,
    ! [B: real,A: real] :
      ( ( B != zero_zero_real )
     => ( aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(real_V7735802525324610683m_real,A)),aa_real_real(real_V7735802525324610683m_real,B)) ) ) ).

% nonzero_norm_divide
tff(fact_5379_nonzero__norm__divide,axiom,
    ! [B: complex,A: complex] :
      ( ( B != zero_zero_complex )
     => ( aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,A),B)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_complex_real(real_V1022390504157884413omplex,A)),aa_complex_real(real_V1022390504157884413omplex,B)) ) ) ).

% nonzero_norm_divide
tff(fact_5380_norm__mult__less,axiom,
    ! [Xb: real,R2: real,Y: real,S3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,Xb)),R2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,Y)),S3)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R2),S3)) ) ) ).

% norm_mult_less
tff(fact_5381_norm__mult__less,axiom,
    ! [Xb: complex,R2: real,Y: complex,S3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,Xb)),R2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,Y)),S3)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,Xb),Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R2),S3)) ) ) ).

% norm_mult_less
tff(fact_5382_norm__mult__ineq,axiom,
    ! [Xb: real,Y: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(real_V7735802525324610683m_real,Xb)),aa_real_real(real_V7735802525324610683m_real,Y))) ).

% norm_mult_ineq
tff(fact_5383_norm__mult__ineq,axiom,
    ! [Xb: complex,Y: complex] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,Xb),Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(real_V1022390504157884413omplex,Xb)),aa_complex_real(real_V1022390504157884413omplex,Y))) ).

% norm_mult_ineq
tff(fact_5384_norm__add__less,axiom,
    ! [Xb: real,R2: real,Y: real,S3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,Xb)),R2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,Y)),S3)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Y))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,R2),S3)) ) ) ).

% norm_add_less
tff(fact_5385_norm__add__less,axiom,
    ! [Xb: complex,R2: real,Y: complex,S3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,Xb)),R2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,Y)),S3)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,Xb),Y))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,R2),S3)) ) ) ).

% norm_add_less
tff(fact_5386_norm__triangle__lt,axiom,
    ! [Xb: real,Y: real,E: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(real_V7735802525324610683m_real,Xb)),aa_real_real(real_V7735802525324610683m_real,Y))),E)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Y))),E) ) ).

% norm_triangle_lt
tff(fact_5387_norm__triangle__lt,axiom,
    ! [Xb: complex,Y: complex,E: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(real_V1022390504157884413omplex,Xb)),aa_complex_real(real_V1022390504157884413omplex,Y))),E)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,Xb),Y))),E) ) ).

% norm_triangle_lt
tff(fact_5388_norm__add__leD,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B))),C)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,B)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(real_V7735802525324610683m_real,A)),C)) ) ).

% norm_add_leD
tff(fact_5389_norm__add__leD,axiom,
    ! [A: complex,B: complex,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,A),B))),C)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,B)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(real_V1022390504157884413omplex,A)),C)) ) ).

% norm_add_leD
tff(fact_5390_norm__triangle__le,axiom,
    ! [Xb: real,Y: real,E: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(real_V7735802525324610683m_real,Xb)),aa_real_real(real_V7735802525324610683m_real,Y))),E)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Y))),E) ) ).

% norm_triangle_le
tff(fact_5391_norm__triangle__le,axiom,
    ! [Xb: complex,Y: complex,E: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(real_V1022390504157884413omplex,Xb)),aa_complex_real(real_V1022390504157884413omplex,Y))),E)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,Xb),Y))),E) ) ).

% norm_triangle_le
tff(fact_5392_norm__triangle__ineq,axiom,
    ! [Xb: real,Y: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Y))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(real_V7735802525324610683m_real,Xb)),aa_real_real(real_V7735802525324610683m_real,Y))) ).

% norm_triangle_ineq
tff(fact_5393_norm__triangle__ineq,axiom,
    ! [Xb: complex,Y: complex] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,Xb),Y))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(real_V1022390504157884413omplex,Xb)),aa_complex_real(real_V1022390504157884413omplex,Y))) ).

% norm_triangle_ineq
tff(fact_5394_norm__triangle__mono,axiom,
    ! [A: real,R2: real,B: real,S3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,A)),R2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,B)),S3)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,R2),S3)) ) ) ).

% norm_triangle_mono
tff(fact_5395_norm__triangle__mono,axiom,
    ! [A: complex,R2: real,B: complex,S3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,A)),R2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,B)),S3)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,A),B))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,R2),S3)) ) ) ).

% norm_triangle_mono
tff(fact_5396_norm__diff__triangle__less,axiom,
    ! [Xb: real,Y: real,E1: real,Z: real,E22: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),Y))),E1)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Y),Z))),E22)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),Z))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,E1),E22)) ) ) ).

% norm_diff_triangle_less
tff(fact_5397_norm__diff__triangle__less,axiom,
    ! [Xb: complex,Y: complex,E1: real,Z: complex,E22: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,Xb),Y))),E1)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,Y),Z))),E22)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,Xb),Z))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,E1),E22)) ) ) ).

% norm_diff_triangle_less
tff(fact_5398_norm__power__ineq,axiom,
    ! [Xb: real,Nb: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(real_V7735802525324610683m_real,Xb)),Nb)) ).

% norm_power_ineq
tff(fact_5399_norm__power__ineq,axiom,
    ! [Xb: complex,Nb: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Xb),Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(real_V1022390504157884413omplex,Xb)),Nb)) ).

% norm_power_ineq
tff(fact_5400_norm__triangle__le__diff,axiom,
    ! [Xb: real,Y: real,E: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(real_V7735802525324610683m_real,Xb)),aa_real_real(real_V7735802525324610683m_real,Y))),E)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),Y))),E) ) ).

% norm_triangle_le_diff
tff(fact_5401_norm__triangle__le__diff,axiom,
    ! [Xb: complex,Y: complex,E: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(real_V1022390504157884413omplex,Xb)),aa_complex_real(real_V1022390504157884413omplex,Y))),E)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,Xb),Y))),E) ) ).

% norm_triangle_le_diff
tff(fact_5402_norm__diff__triangle__le,axiom,
    ! [Xb: real,Y: real,E1: real,Z: real,E22: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),Y))),E1)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Y),Z))),E22)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),Z))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,E1),E22)) ) ) ).

% norm_diff_triangle_le
tff(fact_5403_norm__diff__triangle__le,axiom,
    ! [Xb: complex,Y: complex,E1: real,Z: complex,E22: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,Xb),Y))),E1)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,Y),Z))),E22)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,Xb),Z))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,E1),E22)) ) ) ).

% norm_diff_triangle_le
tff(fact_5404_norm__triangle__ineq4,axiom,
    ! [A: real,B: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(real_V7735802525324610683m_real,A)),aa_real_real(real_V7735802525324610683m_real,B))) ).

% norm_triangle_ineq4
tff(fact_5405_norm__triangle__ineq4,axiom,
    ! [A: complex,B: complex] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,A),B))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(real_V1022390504157884413omplex,A)),aa_complex_real(real_V1022390504157884413omplex,B))) ).

% norm_triangle_ineq4
tff(fact_5406_norm__triangle__sub,axiom,
    ! [Xb: real,Y: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,Xb)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(real_V7735802525324610683m_real,Y)),aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),Y)))) ).

% norm_triangle_sub
tff(fact_5407_norm__triangle__sub,axiom,
    ! [Xb: complex,Y: complex] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,Xb)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(real_V1022390504157884413omplex,Y)),aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,Xb),Y)))) ).

% norm_triangle_sub
tff(fact_5408_norm__diff__ineq,axiom,
    ! [A: real,B: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(real_V7735802525324610683m_real,A)),aa_real_real(real_V7735802525324610683m_real,B))),aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B))) ).

% norm_diff_ineq
tff(fact_5409_norm__diff__ineq,axiom,
    ! [A: complex,B: complex] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_complex_real(real_V1022390504157884413omplex,A)),aa_complex_real(real_V1022390504157884413omplex,B))),aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,A),B))) ).

% norm_diff_ineq
tff(fact_5410_norm__triangle__ineq2,axiom,
    ! [A: real,B: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(real_V7735802525324610683m_real,A)),aa_real_real(real_V7735802525324610683m_real,B))),aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B))) ).

% norm_triangle_ineq2
tff(fact_5411_norm__triangle__ineq2,axiom,
    ! [A: complex,B: complex] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_complex_real(real_V1022390504157884413omplex,A)),aa_complex_real(real_V1022390504157884413omplex,B))),aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,A),B))) ).

% norm_triangle_ineq2
tff(fact_5412_power__eq__1__iff,axiom,
    ! [W: real,Nb: nat] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,W),Nb) = one_one_real )
     => ( ( aa_real_real(real_V7735802525324610683m_real,W) = one_one_real )
        | ( Nb = zero_zero_nat ) ) ) ).

% power_eq_1_iff
tff(fact_5413_power__eq__1__iff,axiom,
    ! [W: complex,Nb: nat] :
      ( ( aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,W),Nb) = one_one_complex )
     => ( ( aa_complex_real(real_V1022390504157884413omplex,W) = one_one_real )
        | ( Nb = zero_zero_nat ) ) ) ).

% power_eq_1_iff
tff(fact_5414_norm__diff__triangle__ineq,axiom,
    ! [A: real,B: real,C: real,D: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),D)))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),C))),aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),D)))) ).

% norm_diff_triangle_ineq
tff(fact_5415_norm__diff__triangle__ineq,axiom,
    ! [A: complex,B: complex,C: complex,D: complex] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,A),B)),aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,C),D)))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,A),C))),aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,B),D)))) ).

% norm_diff_triangle_ineq
tff(fact_5416_square__norm__one,axiom,
    ! [Xb: real] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_real )
     => ( aa_real_real(real_V7735802525324610683m_real,Xb) = one_one_real ) ) ).

% square_norm_one
tff(fact_5417_square__norm__one,axiom,
    ! [Xb: complex] :
      ( ( aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_complex )
     => ( aa_complex_real(real_V1022390504157884413omplex,Xb) = one_one_real ) ) ).

% square_norm_one
tff(fact_5418_norm__power__diff,axiom,
    ! [Z: real,W: real,M: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,Z)),one_one_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,W)),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Z),M)),aa_nat_real(aa_real_fun_nat_real(power_power_real,W),M)))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,M)),aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Z),W)))) ) ) ).

% norm_power_diff
tff(fact_5419_norm__power__diff,axiom,
    ! [Z: complex,W: complex,M: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,Z)),one_one_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,W)),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Z),M)),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,W),M)))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,M)),aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,Z),W)))) ) ) ).

% norm_power_diff
tff(fact_5420_mod__exhaust__less__4,axiom,
    ! [M: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))) = zero_zero_nat )
      | ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))) = one_one_nat )
      | ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) )
      | ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)) ) ) ).

% mod_exhaust_less_4
tff(fact_5421_vebt__inserti_Osimps,axiom,
    ! [Ta: vEBT_VEBTi,Xb: nat] : aa_nat5408703056993926195_VEBTi(aa_VEB4833742232818206615_VEBTi(vEBT_vebt_inserti,Ta),Xb) = aa_VEB836174949890266638_VEBTi(vEBT_c6028912655521741485_VEBTi(aTP_Lamm_jd(Xb),aTP_Lamm_je(Xb)),Ta) ).

% vebt_inserti.simps
tff(fact_5422_VEBTi_Osize_I3_J,axiom,
    ! [X11a: option4927543243414619207at_nat,X12a: nat,X13a: array_VEBT_VEBTi,X14a: vEBT_VEBTi] : aa_VEBT_VEBTi_nat(size_size_VEBT_VEBTi,aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(X11a,X12a,X13a),X14a)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,size_a6397454172108246045_VEBTi(size_size_VEBT_VEBTi,X13a)),aa_VEBT_VEBTi_nat(size_size_VEBT_VEBTi,X14a))),aa_nat_nat(suc,zero_zero_nat)) ).

% VEBTi.size(3)
tff(fact_5423_list__every__elemnt__bound__sum__bound,axiom,
    ! [Xs: list_VEBT_VEBT,F: fun_VEBT_VEBT_nat,Bound: nat,I: nat] :
      ( ! [X2: vEBT_VEBT] :
          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X2),set_VEBT_VEBT2(Xs))
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_VEBT_VEBT_nat(F,X2)),Bound) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(foldr_nat_nat(plus_plus_nat,map_VEBT_VEBT_nat(F,Xs)),I)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_s6755466524823107622T_VEBT(Xs)),Bound)),I)) ) ).

% list_every_elemnt_bound_sum_bound
tff(fact_5424_list__every__elemnt__bound__sum__bound,axiom,
    ! [Xs: list_P8527749157015355191n_assn,F: fun_Pr7403558292872094996sn_nat,Bound: nat,I: nat] :
      ( ! [X2: produc6575502325842934193n_assn] :
          ( member7957490590177025114n_assn(X2,set_Pr1139785259514867910n_assn(Xs))
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_Pro8055486230010607173sn_nat(F,X2)),Bound) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(foldr_nat_nat(plus_plus_nat,map_Pr7570552894071451325sn_nat(F,Xs)),I)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_s6829681357464350627n_assn(Xs)),Bound)),I)) ) ).

% list_every_elemnt_bound_sum_bound
tff(fact_5425_list__every__elemnt__bound__sum__bound,axiom,
    ! [Xs: list_real,F: fun_real_nat,Bound: nat,I: nat] :
      ( ! [X2: real] :
          ( aa_set_real_o(member_real(X2),set_real2(Xs))
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_real_nat(F,X2)),Bound) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(foldr_nat_nat(plus_plus_nat,map_real_nat(F,Xs)),I)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_real(Xs)),Bound)),I)) ) ).

% list_every_elemnt_bound_sum_bound
tff(fact_5426_list__every__elemnt__bound__sum__bound,axiom,
    ! [Xs: list_o,F: fun_o_nat,Bound: nat,I: nat] :
      ( ! [X2: $o] :
          ( member_o((X2),set_o2(Xs))
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_o_nat(F,(X2))),Bound) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(foldr_nat_nat(plus_plus_nat,map_o_nat(F,Xs)),I)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_o(Xs)),Bound)),I)) ) ).

% list_every_elemnt_bound_sum_bound
tff(fact_5427_list__every__elemnt__bound__sum__bound,axiom,
    ! [Xs: list_nat,F: fun_nat_nat,Bound: nat,I: nat] :
      ( ! [X2: nat] :
          ( aa_set_nat_o(member_nat(X2),set_nat2(Xs))
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F,X2)),Bound) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(foldr_nat_nat(plus_plus_nat,map_nat_nat(F,Xs)),I)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_nat(Xs)),Bound)),I)) ) ).

% list_every_elemnt_bound_sum_bound
tff(fact_5428_list__every__elemnt__bound__sum__bound,axiom,
    ! [Xs: list_int,F: fun_int_nat,Bound: nat,I: nat] :
      ( ! [X2: int] :
          ( aa_set_int_o(member_int(X2),set_int2(Xs))
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(F,X2)),Bound) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(foldr_nat_nat(plus_plus_nat,map_int_nat(F,Xs)),I)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_int(Xs)),Bound)),I)) ) ).

% list_every_elemnt_bound_sum_bound
tff(fact_5429_map__ident,axiom,
    ! [X3: list_nat] : map_nat_nat(aTP_Lamm_di,X3) = X3 ).

% map_ident
tff(fact_5430_length__map,axiom,
    ! [F: fun_VEBT_VEBT_real,Xs: list_VEBT_VEBT] : size_size_list_real(map_VEBT_VEBT_real(F,Xs)) = size_s6755466524823107622T_VEBT(Xs) ).

% length_map
tff(fact_5431_length__map,axiom,
    ! [F: fun_real_real,Xs: list_real] : size_size_list_real(map_real_real(F,Xs)) = size_size_list_real(Xs) ).

% length_map
tff(fact_5432_length__map,axiom,
    ! [F: fun_o_real,Xs: list_o] : size_size_list_real(map_o_real(F,Xs)) = size_size_list_o(Xs) ).

% length_map
tff(fact_5433_length__map,axiom,
    ! [F: fun_nat_real,Xs: list_nat] : size_size_list_real(map_nat_real(F,Xs)) = size_size_list_nat(Xs) ).

% length_map
tff(fact_5434_length__map,axiom,
    ! [F: fun_int_real,Xs: list_int] : size_size_list_real(map_int_real(F,Xs)) = size_size_list_int(Xs) ).

% length_map
tff(fact_5435_length__map,axiom,
    ! [F: fun_real_o,Xs: list_real] : size_size_list_o(map_real_o(F,Xs)) = size_size_list_real(Xs) ).

% length_map
tff(fact_5436_length__map,axiom,
    ! [F: fun_o_o,Xs: list_o] : size_size_list_o(map_o_o(F,Xs)) = size_size_list_o(Xs) ).

% length_map
tff(fact_5437_length__map,axiom,
    ! [F: fun_nat_o,Xs: list_nat] : size_size_list_o(map_nat_o(F,Xs)) = size_size_list_nat(Xs) ).

% length_map
tff(fact_5438_length__map,axiom,
    ! [F: fun_int_o,Xs: list_int] : size_size_list_o(map_int_o(F,Xs)) = size_size_list_int(Xs) ).

% length_map
tff(fact_5439_length__map,axiom,
    ! [F: fun_VEBT_VEBT_nat,Xs: list_VEBT_VEBT] : size_size_list_nat(map_VEBT_VEBT_nat(F,Xs)) = size_s6755466524823107622T_VEBT(Xs) ).

% length_map
tff(fact_5440_map__eq__conv,axiom,
    ! [F: fun_VEBT_VEBT_nat,Xs: list_VEBT_VEBT,G: fun_VEBT_VEBT_nat] :
      ( ( map_VEBT_VEBT_nat(F,Xs) = map_VEBT_VEBT_nat(G,Xs) )
    <=> ! [X: vEBT_VEBT] :
          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),set_VEBT_VEBT2(Xs))
         => ( aa_VEBT_VEBT_nat(F,X) = aa_VEBT_VEBT_nat(G,X) ) ) ) ).

% map_eq_conv
tff(fact_5441_map__eq__conv,axiom,
    ! [F: fun_VEBT_VEBT_real,Xs: list_VEBT_VEBT,G: fun_VEBT_VEBT_real] :
      ( ( map_VEBT_VEBT_real(F,Xs) = map_VEBT_VEBT_real(G,Xs) )
    <=> ! [X: vEBT_VEBT] :
          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),set_VEBT_VEBT2(Xs))
         => ( aa_VEBT_VEBT_real(F,X) = aa_VEBT_VEBT_real(G,X) ) ) ) ).

% map_eq_conv
tff(fact_5442_map__eq__conv,axiom,
    ! [F: fun_nat_nat,Xs: list_nat,G: fun_nat_nat] :
      ( ( map_nat_nat(F,Xs) = map_nat_nat(G,Xs) )
    <=> ! [X: nat] :
          ( aa_set_nat_o(member_nat(X),set_nat2(Xs))
         => ( aa_nat_nat(F,X) = aa_nat_nat(G,X) ) ) ) ).

% map_eq_conv
tff(fact_5443_map__eq__conv,axiom,
    ! [F: fun_nat_o,Xs: list_nat,G: fun_nat_o] :
      ( ( map_nat_o(F,Xs) = map_nat_o(G,Xs) )
    <=> ! [X: nat] :
          ( aa_set_nat_o(member_nat(X),set_nat2(Xs))
         => ( aa_nat_o(F,X)
          <=> aa_nat_o(G,X) ) ) ) ).

% map_eq_conv
tff(fact_5444_map__eq__conv,axiom,
    ! [F: fun_Pr3816910165196019820n_assn,Xs: list_P8527749157015355191n_assn,G: fun_Pr3816910165196019820n_assn] :
      ( ( map_Pr8991440229025900053n_assn(F,Xs) = map_Pr8991440229025900053n_assn(G,Xs) )
    <=> ! [X: produc6575502325842934193n_assn] :
          ( member7957490590177025114n_assn(X,set_Pr1139785259514867910n_assn(Xs))
         => ( aa_Pro2979818503705714077n_assn(F,X) = aa_Pro2979818503705714077n_assn(G,X) ) ) ) ).

% map_eq_conv
tff(fact_5445_nth__map,axiom,
    ! [Nb: nat,Xs: list_VEBT_VEBT,F: fun_VE3165209603783583043T_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_s6755466524823107622T_VEBT(Xs))
     => ( nth_VEBT_VEBT(map_VE8901447254227204932T_VEBT(F,Xs),Nb) = aa_VEB5832953974749077948T_VEBT(F,nth_VEBT_VEBT(Xs,Nb)) ) ) ).

% nth_map
tff(fact_5446_nth__map,axiom,
    ! [Nb: nat,Xs: list_VEBT_VEBTi,F: fun_VE9120655849212850072T_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_s7982070591426661849_VEBTi(Xs))
     => ( nth_VEBT_VEBT(map_VE7998069337340375161T_VEBT(F,Xs),Nb) = aa_VEB4941942288398148609T_VEBT(F,nth_VEBT_VEBTi(Xs,Nb)) ) ) ).

% nth_map
tff(fact_5447_nth__map,axiom,
    ! [Nb: nat,Xs: list_VEBT_VEBT,F: fun_VE712148701811721852_VEBTi] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_s6755466524823107622T_VEBT(Xs))
     => ( nth_VEBT_VEBTi(map_VE7029150624388687525_VEBTi(F,Xs),Nb) = aa_VEB3973023575446460973_VEBTi(F,nth_VEBT_VEBT(Xs,Nb)) ) ) ).

% nth_map
tff(fact_5448_nth__map,axiom,
    ! [Nb: nat,Xs: list_VEBT_VEBTi,F: fun_VE8449866691444886503_VEBTi] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_s7982070591426661849_VEBTi(Xs))
     => ( nth_VEBT_VEBTi(map_VE483055756984248624_VEBTi(F,Xs),Nb) = aa_VEB5586453056816421800_VEBTi(F,nth_VEBT_VEBTi(Xs,Nb)) ) ) ).

% nth_map
tff(fact_5449_nth__map,axiom,
    ! [Nb: nat,Xs: list_VEBT_VEBTi,F: fun_VEBT_VEBTi_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_s7982070591426661849_VEBTi(Xs))
     => ( aa_nat_nat(nth_nat(map_VEBT_VEBTi_nat(F,Xs)),Nb) = aa_VEBT_VEBTi_nat(F,nth_VEBT_VEBTi(Xs,Nb)) ) ) ).

% nth_map
tff(fact_5450_nth__map,axiom,
    ! [Nb: nat,Xs: list_VEBT_VEBT,F: fun_VEBT_VEBT_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_s6755466524823107622T_VEBT(Xs))
     => ( aa_nat_int(nth_int(map_VEBT_VEBT_int(F,Xs)),Nb) = aa_VEBT_VEBT_int(F,nth_VEBT_VEBT(Xs,Nb)) ) ) ).

% nth_map
tff(fact_5451_nth__map,axiom,
    ! [Nb: nat,Xs: list_VEBT_VEBTi,F: fun_VEBT_VEBTi_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_s7982070591426661849_VEBTi(Xs))
     => ( aa_nat_int(nth_int(map_VEBT_VEBTi_int(F,Xs)),Nb) = aa_VEBT_VEBTi_int(F,nth_VEBT_VEBTi(Xs,Nb)) ) ) ).

% nth_map
tff(fact_5452_nth__map,axiom,
    ! [Nb: nat,Xs: list_VEBT_VEBT,F: fun_VEBT_VEBT_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_s6755466524823107622T_VEBT(Xs))
     => ( aa_nat_nat(nth_nat(map_VEBT_VEBT_nat(F,Xs)),Nb) = aa_VEBT_VEBT_nat(F,nth_VEBT_VEBT(Xs,Nb)) ) ) ).

% nth_map
tff(fact_5453_nth__map,axiom,
    ! [Nb: nat,Xs: list_VEBT_VEBT,F: fun_VEBT_VEBT_real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_s6755466524823107622T_VEBT(Xs))
     => ( aa_nat_real(nth_real(map_VEBT_VEBT_real(F,Xs)),Nb) = aa_VEBT_VEBT_real(F,nth_VEBT_VEBT(Xs,Nb)) ) ) ).

% nth_map
tff(fact_5454_nth__map,axiom,
    ! [Nb: nat,Xs: list_real,F: fun_real_VEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_size_list_real(Xs))
     => ( nth_VEBT_VEBT(map_real_VEBT_VEBT(F,Xs),Nb) = aa_real_VEBT_VEBT(F,aa_nat_real(nth_real(Xs),Nb)) ) ) ).

% nth_map
tff(fact_5455_list_Omap__ident,axiom,
    ! [Ta: list_nat] : map_nat_nat(aTP_Lamm_di,Ta) = Ta ).

% list.map_ident
tff(fact_5456_map__eq__imp__length__eq,axiom,
    ! [F: fun_VEBT_VEBT_nat,Xs: list_VEBT_VEBT,G: fun_VEBT_VEBT_nat,Ys: list_VEBT_VEBT] :
      ( ( map_VEBT_VEBT_nat(F,Xs) = map_VEBT_VEBT_nat(G,Ys) )
     => ( size_s6755466524823107622T_VEBT(Xs) = size_s6755466524823107622T_VEBT(Ys) ) ) ).

% map_eq_imp_length_eq
tff(fact_5457_map__eq__imp__length__eq,axiom,
    ! [F: fun_VEBT_VEBT_real,Xs: list_VEBT_VEBT,G: fun_VEBT_VEBT_real,Ys: list_VEBT_VEBT] :
      ( ( map_VEBT_VEBT_real(F,Xs) = map_VEBT_VEBT_real(G,Ys) )
     => ( size_s6755466524823107622T_VEBT(Xs) = size_s6755466524823107622T_VEBT(Ys) ) ) ).

% map_eq_imp_length_eq
tff(fact_5458_map__eq__imp__length__eq,axiom,
    ! [F: fun_VEBT_VEBT_nat,Xs: list_VEBT_VEBT,G: fun_real_nat,Ys: list_real] :
      ( ( map_VEBT_VEBT_nat(F,Xs) = map_real_nat(G,Ys) )
     => ( size_s6755466524823107622T_VEBT(Xs) = size_size_list_real(Ys) ) ) ).

% map_eq_imp_length_eq
tff(fact_5459_map__eq__imp__length__eq,axiom,
    ! [F: fun_VEBT_VEBT_real,Xs: list_VEBT_VEBT,G: fun_real_real,Ys: list_real] :
      ( ( map_VEBT_VEBT_real(F,Xs) = map_real_real(G,Ys) )
     => ( size_s6755466524823107622T_VEBT(Xs) = size_size_list_real(Ys) ) ) ).

% map_eq_imp_length_eq
tff(fact_5460_map__eq__imp__length__eq,axiom,
    ! [F: fun_VEBT_VEBT_nat,Xs: list_VEBT_VEBT,G: fun_o_nat,Ys: list_o] :
      ( ( map_VEBT_VEBT_nat(F,Xs) = map_o_nat(G,Ys) )
     => ( size_s6755466524823107622T_VEBT(Xs) = size_size_list_o(Ys) ) ) ).

% map_eq_imp_length_eq
tff(fact_5461_map__eq__imp__length__eq,axiom,
    ! [F: fun_VEBT_VEBT_real,Xs: list_VEBT_VEBT,G: fun_o_real,Ys: list_o] :
      ( ( map_VEBT_VEBT_real(F,Xs) = map_o_real(G,Ys) )
     => ( size_s6755466524823107622T_VEBT(Xs) = size_size_list_o(Ys) ) ) ).

% map_eq_imp_length_eq
tff(fact_5462_map__eq__imp__length__eq,axiom,
    ! [F: fun_VEBT_VEBT_nat,Xs: list_VEBT_VEBT,G: fun_nat_nat,Ys: list_nat] :
      ( ( map_VEBT_VEBT_nat(F,Xs) = map_nat_nat(G,Ys) )
     => ( size_s6755466524823107622T_VEBT(Xs) = size_size_list_nat(Ys) ) ) ).

% map_eq_imp_length_eq
tff(fact_5463_map__eq__imp__length__eq,axiom,
    ! [F: fun_VEBT_VEBT_real,Xs: list_VEBT_VEBT,G: fun_nat_real,Ys: list_nat] :
      ( ( map_VEBT_VEBT_real(F,Xs) = map_nat_real(G,Ys) )
     => ( size_s6755466524823107622T_VEBT(Xs) = size_size_list_nat(Ys) ) ) ).

% map_eq_imp_length_eq
tff(fact_5464_map__eq__imp__length__eq,axiom,
    ! [F: fun_VEBT_VEBT_nat,Xs: list_VEBT_VEBT,G: fun_int_nat,Ys: list_int] :
      ( ( map_VEBT_VEBT_nat(F,Xs) = map_int_nat(G,Ys) )
     => ( size_s6755466524823107622T_VEBT(Xs) = size_size_list_int(Ys) ) ) ).

% map_eq_imp_length_eq
tff(fact_5465_map__eq__imp__length__eq,axiom,
    ! [F: fun_VEBT_VEBT_real,Xs: list_VEBT_VEBT,G: fun_int_real,Ys: list_int] :
      ( ( map_VEBT_VEBT_real(F,Xs) = map_int_real(G,Ys) )
     => ( size_s6755466524823107622T_VEBT(Xs) = size_size_list_int(Ys) ) ) ).

% map_eq_imp_length_eq
tff(fact_5466_list_Omap__cong,axiom,
    ! [Xb: list_VEBT_VEBT,Ya: list_VEBT_VEBT,F: fun_VEBT_VEBT_nat,G: fun_VEBT_VEBT_nat] :
      ( ( Xb = Ya )
     => ( ! [Z3: vEBT_VEBT] :
            ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Z3),set_VEBT_VEBT2(Ya))
           => ( aa_VEBT_VEBT_nat(F,Z3) = aa_VEBT_VEBT_nat(G,Z3) ) )
       => ( map_VEBT_VEBT_nat(F,Xb) = map_VEBT_VEBT_nat(G,Ya) ) ) ) ).

% list.map_cong
tff(fact_5467_list_Omap__cong,axiom,
    ! [Xb: list_VEBT_VEBT,Ya: list_VEBT_VEBT,F: fun_VEBT_VEBT_real,G: fun_VEBT_VEBT_real] :
      ( ( Xb = Ya )
     => ( ! [Z3: vEBT_VEBT] :
            ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Z3),set_VEBT_VEBT2(Ya))
           => ( aa_VEBT_VEBT_real(F,Z3) = aa_VEBT_VEBT_real(G,Z3) ) )
       => ( map_VEBT_VEBT_real(F,Xb) = map_VEBT_VEBT_real(G,Ya) ) ) ) ).

% list.map_cong
tff(fact_5468_list_Omap__cong,axiom,
    ! [Xb: list_nat,Ya: list_nat,F: fun_nat_nat,G: fun_nat_nat] :
      ( ( Xb = Ya )
     => ( ! [Z3: nat] :
            ( aa_set_nat_o(member_nat(Z3),set_nat2(Ya))
           => ( aa_nat_nat(F,Z3) = aa_nat_nat(G,Z3) ) )
       => ( map_nat_nat(F,Xb) = map_nat_nat(G,Ya) ) ) ) ).

% list.map_cong
tff(fact_5469_list_Omap__cong,axiom,
    ! [Xb: list_nat,Ya: list_nat,F: fun_nat_o,G: fun_nat_o] :
      ( ( Xb = Ya )
     => ( ! [Z3: nat] :
            ( aa_set_nat_o(member_nat(Z3),set_nat2(Ya))
           => ( aa_nat_o(F,Z3)
            <=> aa_nat_o(G,Z3) ) )
       => ( map_nat_o(F,Xb) = map_nat_o(G,Ya) ) ) ) ).

% list.map_cong
tff(fact_5470_list_Omap__cong,axiom,
    ! [Xb: list_P8527749157015355191n_assn,Ya: list_P8527749157015355191n_assn,F: fun_Pr3816910165196019820n_assn,G: fun_Pr3816910165196019820n_assn] :
      ( ( Xb = Ya )
     => ( ! [Z3: produc6575502325842934193n_assn] :
            ( member7957490590177025114n_assn(Z3,set_Pr1139785259514867910n_assn(Ya))
           => ( aa_Pro2979818503705714077n_assn(F,Z3) = aa_Pro2979818503705714077n_assn(G,Z3) ) )
       => ( map_Pr8991440229025900053n_assn(F,Xb) = map_Pr8991440229025900053n_assn(G,Ya) ) ) ) ).

% list.map_cong
tff(fact_5471_list_Omap__cong0,axiom,
    ! [Xb: list_VEBT_VEBT,F: fun_VEBT_VEBT_nat,G: fun_VEBT_VEBT_nat] :
      ( ! [Z3: vEBT_VEBT] :
          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Z3),set_VEBT_VEBT2(Xb))
         => ( aa_VEBT_VEBT_nat(F,Z3) = aa_VEBT_VEBT_nat(G,Z3) ) )
     => ( map_VEBT_VEBT_nat(F,Xb) = map_VEBT_VEBT_nat(G,Xb) ) ) ).

% list.map_cong0
tff(fact_5472_list_Omap__cong0,axiom,
    ! [Xb: list_VEBT_VEBT,F: fun_VEBT_VEBT_real,G: fun_VEBT_VEBT_real] :
      ( ! [Z3: vEBT_VEBT] :
          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Z3),set_VEBT_VEBT2(Xb))
         => ( aa_VEBT_VEBT_real(F,Z3) = aa_VEBT_VEBT_real(G,Z3) ) )
     => ( map_VEBT_VEBT_real(F,Xb) = map_VEBT_VEBT_real(G,Xb) ) ) ).

% list.map_cong0
tff(fact_5473_list_Omap__cong0,axiom,
    ! [Xb: list_nat,F: fun_nat_nat,G: fun_nat_nat] :
      ( ! [Z3: nat] :
          ( aa_set_nat_o(member_nat(Z3),set_nat2(Xb))
         => ( aa_nat_nat(F,Z3) = aa_nat_nat(G,Z3) ) )
     => ( map_nat_nat(F,Xb) = map_nat_nat(G,Xb) ) ) ).

% list.map_cong0
tff(fact_5474_list_Omap__cong0,axiom,
    ! [Xb: list_nat,F: fun_nat_o,G: fun_nat_o] :
      ( ! [Z3: nat] :
          ( aa_set_nat_o(member_nat(Z3),set_nat2(Xb))
         => ( aa_nat_o(F,Z3)
          <=> aa_nat_o(G,Z3) ) )
     => ( map_nat_o(F,Xb) = map_nat_o(G,Xb) ) ) ).

% list.map_cong0
tff(fact_5475_list_Omap__cong0,axiom,
    ! [Xb: list_P8527749157015355191n_assn,F: fun_Pr3816910165196019820n_assn,G: fun_Pr3816910165196019820n_assn] :
      ( ! [Z3: produc6575502325842934193n_assn] :
          ( member7957490590177025114n_assn(Z3,set_Pr1139785259514867910n_assn(Xb))
         => ( aa_Pro2979818503705714077n_assn(F,Z3) = aa_Pro2979818503705714077n_assn(G,Z3) ) )
     => ( map_Pr8991440229025900053n_assn(F,Xb) = map_Pr8991440229025900053n_assn(G,Xb) ) ) ).

% list.map_cong0
tff(fact_5476_list_Oinj__map__strong,axiom,
    ! [Xb: list_VEBT_VEBT,Xaa: list_VEBT_VEBT,F: fun_VEBT_VEBT_nat,Fa: fun_VEBT_VEBT_nat] :
      ( ! [Z3: vEBT_VEBT,Za: vEBT_VEBT] :
          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Z3),set_VEBT_VEBT2(Xb))
         => ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Za),set_VEBT_VEBT2(Xaa))
           => ( ( aa_VEBT_VEBT_nat(F,Z3) = aa_VEBT_VEBT_nat(Fa,Za) )
             => ( Z3 = Za ) ) ) )
     => ( ( map_VEBT_VEBT_nat(F,Xb) = map_VEBT_VEBT_nat(Fa,Xaa) )
       => ( Xb = Xaa ) ) ) ).

% list.inj_map_strong
tff(fact_5477_list_Oinj__map__strong,axiom,
    ! [Xb: list_VEBT_VEBT,Xaa: list_VEBT_VEBT,F: fun_VEBT_VEBT_real,Fa: fun_VEBT_VEBT_real] :
      ( ! [Z3: vEBT_VEBT,Za: vEBT_VEBT] :
          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Z3),set_VEBT_VEBT2(Xb))
         => ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Za),set_VEBT_VEBT2(Xaa))
           => ( ( aa_VEBT_VEBT_real(F,Z3) = aa_VEBT_VEBT_real(Fa,Za) )
             => ( Z3 = Za ) ) ) )
     => ( ( map_VEBT_VEBT_real(F,Xb) = map_VEBT_VEBT_real(Fa,Xaa) )
       => ( Xb = Xaa ) ) ) ).

% list.inj_map_strong
tff(fact_5478_list_Oinj__map__strong,axiom,
    ! [Xb: list_nat,Xaa: list_nat,F: fun_nat_nat,Fa: fun_nat_nat] :
      ( ! [Z3: nat,Za: nat] :
          ( aa_set_nat_o(member_nat(Z3),set_nat2(Xb))
         => ( aa_set_nat_o(member_nat(Za),set_nat2(Xaa))
           => ( ( aa_nat_nat(F,Z3) = aa_nat_nat(Fa,Za) )
             => ( Z3 = Za ) ) ) )
     => ( ( map_nat_nat(F,Xb) = map_nat_nat(Fa,Xaa) )
       => ( Xb = Xaa ) ) ) ).

% list.inj_map_strong
tff(fact_5479_list_Oinj__map__strong,axiom,
    ! [Xb: list_nat,Xaa: list_nat,F: fun_nat_o,Fa: fun_nat_o] :
      ( ! [Z3: nat,Za: nat] :
          ( aa_set_nat_o(member_nat(Z3),set_nat2(Xb))
         => ( aa_set_nat_o(member_nat(Za),set_nat2(Xaa))
           => ( ( aa_nat_o(F,Z3)
              <=> aa_nat_o(Fa,Za) )
             => ( Z3 = Za ) ) ) )
     => ( ( map_nat_o(F,Xb) = map_nat_o(Fa,Xaa) )
       => ( Xb = Xaa ) ) ) ).

% list.inj_map_strong
tff(fact_5480_list_Oinj__map__strong,axiom,
    ! [Xb: list_P8527749157015355191n_assn,Xaa: list_P8527749157015355191n_assn,F: fun_Pr3816910165196019820n_assn,Fa: fun_Pr3816910165196019820n_assn] :
      ( ! [Z3: produc6575502325842934193n_assn,Za: produc6575502325842934193n_assn] :
          ( member7957490590177025114n_assn(Z3,set_Pr1139785259514867910n_assn(Xb))
         => ( member7957490590177025114n_assn(Za,set_Pr1139785259514867910n_assn(Xaa))
           => ( ( aa_Pro2979818503705714077n_assn(F,Z3) = aa_Pro2979818503705714077n_assn(Fa,Za) )
             => ( Z3 = Za ) ) ) )
     => ( ( map_Pr8991440229025900053n_assn(F,Xb) = map_Pr8991440229025900053n_assn(Fa,Xaa) )
       => ( Xb = Xaa ) ) ) ).

% list.inj_map_strong
tff(fact_5481_map__ext,axiom,
    ! [Xs: list_VEBT_VEBT,F: fun_VEBT_VEBT_nat,G: fun_VEBT_VEBT_nat] :
      ( ! [X2: vEBT_VEBT] :
          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X2),set_VEBT_VEBT2(Xs))
         => ( aa_VEBT_VEBT_nat(F,X2) = aa_VEBT_VEBT_nat(G,X2) ) )
     => ( map_VEBT_VEBT_nat(F,Xs) = map_VEBT_VEBT_nat(G,Xs) ) ) ).

% map_ext
tff(fact_5482_map__ext,axiom,
    ! [Xs: list_VEBT_VEBT,F: fun_VEBT_VEBT_real,G: fun_VEBT_VEBT_real] :
      ( ! [X2: vEBT_VEBT] :
          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X2),set_VEBT_VEBT2(Xs))
         => ( aa_VEBT_VEBT_real(F,X2) = aa_VEBT_VEBT_real(G,X2) ) )
     => ( map_VEBT_VEBT_real(F,Xs) = map_VEBT_VEBT_real(G,Xs) ) ) ).

% map_ext
tff(fact_5483_map__ext,axiom,
    ! [Xs: list_nat,F: fun_nat_nat,G: fun_nat_nat] :
      ( ! [X2: nat] :
          ( aa_set_nat_o(member_nat(X2),set_nat2(Xs))
         => ( aa_nat_nat(F,X2) = aa_nat_nat(G,X2) ) )
     => ( map_nat_nat(F,Xs) = map_nat_nat(G,Xs) ) ) ).

% map_ext
tff(fact_5484_map__ext,axiom,
    ! [Xs: list_nat,F: fun_nat_o,G: fun_nat_o] :
      ( ! [X2: nat] :
          ( aa_set_nat_o(member_nat(X2),set_nat2(Xs))
         => ( aa_nat_o(F,X2)
          <=> aa_nat_o(G,X2) ) )
     => ( map_nat_o(F,Xs) = map_nat_o(G,Xs) ) ) ).

% map_ext
tff(fact_5485_map__ext,axiom,
    ! [Xs: list_P8527749157015355191n_assn,F: fun_Pr3816910165196019820n_assn,G: fun_Pr3816910165196019820n_assn] :
      ( ! [X2: produc6575502325842934193n_assn] :
          ( member7957490590177025114n_assn(X2,set_Pr1139785259514867910n_assn(Xs))
         => ( aa_Pro2979818503705714077n_assn(F,X2) = aa_Pro2979818503705714077n_assn(G,X2) ) )
     => ( map_Pr8991440229025900053n_assn(F,Xs) = map_Pr8991440229025900053n_assn(G,Xs) ) ) ).

% map_ext
tff(fact_5486_map__idI,axiom,
    ! [Xs: list_VEBT_VEBT,F: fun_VE3165209603783583043T_VEBT] :
      ( ! [X2: vEBT_VEBT] :
          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X2),set_VEBT_VEBT2(Xs))
         => ( aa_VEB5832953974749077948T_VEBT(F,X2) = X2 ) )
     => ( map_VE8901447254227204932T_VEBT(F,Xs) = Xs ) ) ).

% map_idI
tff(fact_5487_map__idI,axiom,
    ! [Xs: list_nat,F: fun_nat_nat] :
      ( ! [X2: nat] :
          ( aa_set_nat_o(member_nat(X2),set_nat2(Xs))
         => ( aa_nat_nat(F,X2) = X2 ) )
     => ( map_nat_nat(F,Xs) = Xs ) ) ).

% map_idI
tff(fact_5488_map__idI,axiom,
    ! [Xs: list_real,F: fun_real_real] :
      ( ! [X2: real] :
          ( aa_set_real_o(member_real(X2),set_real2(Xs))
         => ( aa_real_real(F,X2) = X2 ) )
     => ( map_real_real(F,Xs) = Xs ) ) ).

% map_idI
tff(fact_5489_map__idI,axiom,
    ! [Xs: list_int,F: fun_int_int] :
      ( ! [X2: int] :
          ( aa_set_int_o(member_int(X2),set_int2(Xs))
         => ( aa_int_int(F,X2) = X2 ) )
     => ( map_int_int(F,Xs) = Xs ) ) ).

% map_idI
tff(fact_5490_map__idI,axiom,
    ! [Xs: list_P8527749157015355191n_assn,F: fun_Pr5530265736160528387n_assn] :
      ( ! [X2: produc6575502325842934193n_assn] :
          ( member7957490590177025114n_assn(X2,set_Pr1139785259514867910n_assn(Xs))
         => ( aa_Pro2851486235043456252n_assn(F,X2) = X2 ) )
     => ( map_Pr7925354932063753860n_assn(F,Xs) = Xs ) ) ).

% map_idI
tff(fact_5491_map__cong,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT,F: fun_VEBT_VEBT_nat,G: fun_VEBT_VEBT_nat] :
      ( ( Xs = Ys )
     => ( ! [X2: vEBT_VEBT] :
            ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X2),set_VEBT_VEBT2(Ys))
           => ( aa_VEBT_VEBT_nat(F,X2) = aa_VEBT_VEBT_nat(G,X2) ) )
       => ( map_VEBT_VEBT_nat(F,Xs) = map_VEBT_VEBT_nat(G,Ys) ) ) ) ).

% map_cong
tff(fact_5492_map__cong,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT,F: fun_VEBT_VEBT_real,G: fun_VEBT_VEBT_real] :
      ( ( Xs = Ys )
     => ( ! [X2: vEBT_VEBT] :
            ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X2),set_VEBT_VEBT2(Ys))
           => ( aa_VEBT_VEBT_real(F,X2) = aa_VEBT_VEBT_real(G,X2) ) )
       => ( map_VEBT_VEBT_real(F,Xs) = map_VEBT_VEBT_real(G,Ys) ) ) ) ).

% map_cong
tff(fact_5493_map__cong,axiom,
    ! [Xs: list_nat,Ys: list_nat,F: fun_nat_nat,G: fun_nat_nat] :
      ( ( Xs = Ys )
     => ( ! [X2: nat] :
            ( aa_set_nat_o(member_nat(X2),set_nat2(Ys))
           => ( aa_nat_nat(F,X2) = aa_nat_nat(G,X2) ) )
       => ( map_nat_nat(F,Xs) = map_nat_nat(G,Ys) ) ) ) ).

% map_cong
tff(fact_5494_map__cong,axiom,
    ! [Xs: list_nat,Ys: list_nat,F: fun_nat_o,G: fun_nat_o] :
      ( ( Xs = Ys )
     => ( ! [X2: nat] :
            ( aa_set_nat_o(member_nat(X2),set_nat2(Ys))
           => ( aa_nat_o(F,X2)
            <=> aa_nat_o(G,X2) ) )
       => ( map_nat_o(F,Xs) = map_nat_o(G,Ys) ) ) ) ).

% map_cong
tff(fact_5495_map__cong,axiom,
    ! [Xs: list_P8527749157015355191n_assn,Ys: list_P8527749157015355191n_assn,F: fun_Pr3816910165196019820n_assn,G: fun_Pr3816910165196019820n_assn] :
      ( ( Xs = Ys )
     => ( ! [X2: produc6575502325842934193n_assn] :
            ( member7957490590177025114n_assn(X2,set_Pr1139785259514867910n_assn(Ys))
           => ( aa_Pro2979818503705714077n_assn(F,X2) = aa_Pro2979818503705714077n_assn(G,X2) ) )
       => ( map_Pr8991440229025900053n_assn(F,Xs) = map_Pr8991440229025900053n_assn(G,Ys) ) ) ) ).

% map_cong
tff(fact_5496_ex__map__conv,axiom,
    ! [Ys: list_o,F: fun_nat_o] :
      ( ? [Xs3: list_nat] : Ys = map_nat_o(F,Xs3)
    <=> ! [X: $o] :
          ( member_o((X),set_o2(Ys))
         => ? [Xa3: nat] :
              ( (X)
            <=> aa_nat_o(F,Xa3) ) ) ) ).

% ex_map_conv
tff(fact_5497_ex__map__conv,axiom,
    ! [Ys: list_assn,F: fun_Pr3816910165196019820n_assn] :
      ( ? [Xs3: list_P8527749157015355191n_assn] : Ys = map_Pr8991440229025900053n_assn(F,Xs3)
    <=> ! [X: assn] :
          ( member_assn(X,set_assn2(Ys))
         => ? [Xa3: produc6575502325842934193n_assn] : X = aa_Pro2979818503705714077n_assn(F,Xa3) ) ) ).

% ex_map_conv
tff(fact_5498_ex__map__conv,axiom,
    ! [Ys: list_nat,F: fun_VEBT_VEBT_nat] :
      ( ? [Xs3: list_VEBT_VEBT] : Ys = map_VEBT_VEBT_nat(F,Xs3)
    <=> ! [X: nat] :
          ( aa_set_nat_o(member_nat(X),set_nat2(Ys))
         => ? [Xa3: vEBT_VEBT] : X = aa_VEBT_VEBT_nat(F,Xa3) ) ) ).

% ex_map_conv
tff(fact_5499_ex__map__conv,axiom,
    ! [Ys: list_nat,F: fun_nat_nat] :
      ( ? [Xs3: list_nat] : Ys = map_nat_nat(F,Xs3)
    <=> ! [X: nat] :
          ( aa_set_nat_o(member_nat(X),set_nat2(Ys))
         => ? [Xa3: nat] : X = aa_nat_nat(F,Xa3) ) ) ).

% ex_map_conv
tff(fact_5500_ex__map__conv,axiom,
    ! [Ys: list_real,F: fun_VEBT_VEBT_real] :
      ( ? [Xs3: list_VEBT_VEBT] : Ys = map_VEBT_VEBT_real(F,Xs3)
    <=> ! [X: real] :
          ( aa_set_real_o(member_real(X),set_real2(Ys))
         => ? [Xa3: vEBT_VEBT] : X = aa_VEBT_VEBT_real(F,Xa3) ) ) ).

% ex_map_conv
tff(fact_5501_map__eq__nth__eq,axiom,
    ! [F: fun_VEBT_VEBT_nat,L: list_VEBT_VEBT,L3: list_VEBT_VEBT,I: nat] :
      ( ( map_VEBT_VEBT_nat(F,L) = map_VEBT_VEBT_nat(F,L3) )
     => ( aa_VEBT_VEBT_nat(F,nth_VEBT_VEBT(L,I)) = aa_VEBT_VEBT_nat(F,nth_VEBT_VEBT(L3,I)) ) ) ).

% map_eq_nth_eq
tff(fact_5502_map__eq__nth__eq,axiom,
    ! [F: fun_VEBT_VEBT_real,L: list_VEBT_VEBT,L3: list_VEBT_VEBT,I: nat] :
      ( ( map_VEBT_VEBT_real(F,L) = map_VEBT_VEBT_real(F,L3) )
     => ( aa_VEBT_VEBT_real(F,nth_VEBT_VEBT(L,I)) = aa_VEBT_VEBT_real(F,nth_VEBT_VEBT(L3,I)) ) ) ).

% map_eq_nth_eq
tff(fact_5503_map__eq__nth__eq,axiom,
    ! [F: fun_nat_nat,L: list_nat,L3: list_nat,I: nat] :
      ( ( map_nat_nat(F,L) = map_nat_nat(F,L3) )
     => ( aa_nat_nat(F,aa_nat_nat(nth_nat(L),I)) = aa_nat_nat(F,aa_nat_nat(nth_nat(L3),I)) ) ) ).

% map_eq_nth_eq
tff(fact_5504_map__eq__nth__eq,axiom,
    ! [F: fun_nat_o,L: list_nat,L3: list_nat,I: nat] :
      ( ( map_nat_o(F,L) = map_nat_o(F,L3) )
     => ( aa_nat_o(F,aa_nat_nat(nth_nat(L),I))
      <=> aa_nat_o(F,aa_nat_nat(nth_nat(L3),I)) ) ) ).

% map_eq_nth_eq
tff(fact_5505_map__eq__nth__eq,axiom,
    ! [F: fun_Pr3816910165196019820n_assn,L: list_P8527749157015355191n_assn,L3: list_P8527749157015355191n_assn,I: nat] :
      ( ( map_Pr8991440229025900053n_assn(F,L) = map_Pr8991440229025900053n_assn(F,L3) )
     => ( aa_Pro2979818503705714077n_assn(F,nth_Pr1769885009046257848n_assn(L,I)) = aa_Pro2979818503705714077n_assn(F,nth_Pr1769885009046257848n_assn(L3,I)) ) ) ).

% map_eq_nth_eq
tff(fact_5506_TBOUND__upd,axiom,
    ! [Xs: nat,I: vEBT_VEBTi,Xb: array_VEBT_VEBTi] : aa_nat_o(time_T6070283812100419266_VEBTi(array_upd_VEBT_VEBTi(Xs,I,Xb)),one_one_nat) ).

% TBOUND_upd
tff(fact_5507_time__array__len,axiom,
    ! [P2: array_VEBT_VEBTi,H: heap_e7401611519738050253t_unit] : time_time_nat(array_len_VEBT_VEBTi(P2),H) = one_one_nat ).

% time_array_len
tff(fact_5508_map__upd__eq,axiom,
    ! [I: nat,L: list_P8527749157015355191n_assn,F: fun_Pr3816910165196019820n_assn,Xb: produc6575502325842934193n_assn] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s6829681357464350627n_assn(L))
       => ( aa_Pro2979818503705714077n_assn(F,nth_Pr1769885009046257848n_assn(L,I)) = aa_Pro2979818503705714077n_assn(F,Xb) ) )
     => ( map_Pr8991440229025900053n_assn(F,list_u4534839942911652127n_assn(L,I,Xb)) = map_Pr8991440229025900053n_assn(F,L) ) ) ).

% map_upd_eq
tff(fact_5509_map__upd__eq,axiom,
    ! [I: nat,L: list_VEBT_VEBT,F: fun_VEBT_VEBT_nat,Xb: vEBT_VEBT] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s6755466524823107622T_VEBT(L))
       => ( aa_VEBT_VEBT_nat(F,nth_VEBT_VEBT(L,I)) = aa_VEBT_VEBT_nat(F,Xb) ) )
     => ( map_VEBT_VEBT_nat(F,list_u1324408373059187874T_VEBT(L,I,Xb)) = map_VEBT_VEBT_nat(F,L) ) ) ).

% map_upd_eq
tff(fact_5510_map__upd__eq,axiom,
    ! [I: nat,L: list_VEBT_VEBT,F: fun_VEBT_VEBT_real,Xb: vEBT_VEBT] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s6755466524823107622T_VEBT(L))
       => ( aa_VEBT_VEBT_real(F,nth_VEBT_VEBT(L,I)) = aa_VEBT_VEBT_real(F,Xb) ) )
     => ( map_VEBT_VEBT_real(F,list_u1324408373059187874T_VEBT(L,I,Xb)) = map_VEBT_VEBT_real(F,L) ) ) ).

% map_upd_eq
tff(fact_5511_map__upd__eq,axiom,
    ! [I: nat,L: list_nat,F: fun_nat_nat,Xb: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_nat(L))
       => ( aa_nat_nat(F,aa_nat_nat(nth_nat(L),I)) = aa_nat_nat(F,Xb) ) )
     => ( map_nat_nat(F,list_update_nat(L,I,Xb)) = map_nat_nat(F,L) ) ) ).

% map_upd_eq
tff(fact_5512_map__upd__eq,axiom,
    ! [I: nat,L: list_nat,F: fun_nat_o,Xb: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_nat(L))
       => ( aa_nat_o(F,aa_nat_nat(nth_nat(L),I))
        <=> aa_nat_o(F,Xb) ) )
     => ( map_nat_o(F,list_update_nat(L,I,Xb)) = map_nat_o(F,L) ) ) ).

% map_upd_eq
tff(fact_5513_TBOUND__len,axiom,
    ! [Xs: array_VEBT_VEBTi] : aa_nat_o(time_TBOUND_nat(array_len_VEBT_VEBTi(Xs)),one_one_nat) ).

% TBOUND_len
tff(fact_5514_VEBT__internal_Ocnt_H_Osimps_I2_J,axiom,
    ! [Info: option4927543243414619207at_nat,Deg: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT] : aa_VEBT_VEBT_nat(vEBT_VEBT_cnt2,aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info,Deg,TreeLista),Summarya)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_VEBT_cnt2,Summarya))),aa_nat_nat(foldr_nat_nat(plus_plus_nat,map_VEBT_VEBT_nat(vEBT_VEBT_cnt2,TreeLista)),zero_zero_nat)) ).

% VEBT_internal.cnt'.simps(2)
tff(fact_5515_VEBT__internal_Ocnt_H_Oelims,axiom,
    ! [Xb: vEBT_VEBT,Y: nat] :
      ( ( aa_VEBT_VEBT_nat(vEBT_VEBT_cnt2,Xb) = Y )
     => ( ( ? [A3: $o,B2: $o] : Xb = vEBT_Leaf((A3),(B2))
         => ( Y != one_one_nat ) )
       => ~ ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
              ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,Deg2,TreeList),Summary) )
             => ( Y != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_VEBT_cnt2,Summary))),aa_nat_nat(foldr_nat_nat(plus_plus_nat,map_VEBT_VEBT_nat(vEBT_VEBT_cnt2,TreeList)),zero_zero_nat)) ) ) ) ) ).

% VEBT_internal.cnt'.elims
tff(fact_5516_eq__diff__eq_H,axiom,
    ! [Xb: real,Y: real,Z: real] :
      ( ( Xb = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Y),Z) )
    <=> ( Y = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Z) ) ) ).

% eq_diff_eq'
tff(fact_5517_VEBT__internal_Ospace_H_Osimps_I2_J,axiom,
    ! [Info: option4927543243414619207at_nat,Deg: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT] : aa_VEBT_VEBT_nat(vEBT_VEBT_space2,aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info,Deg,TreeLista),Summarya)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit1,one)))),aa_VEBT_VEBT_nat(vEBT_VEBT_space2,Summarya))),aa_nat_nat(foldr_nat_nat(plus_plus_nat,map_VEBT_VEBT_nat(vEBT_VEBT_space2,TreeLista)),zero_zero_nat)) ).

% VEBT_internal.space'.simps(2)
tff(fact_5518_VEBT__internal_Ospace_H_Oelims,axiom,
    ! [Xb: vEBT_VEBT,Y: nat] :
      ( ( aa_VEBT_VEBT_nat(vEBT_VEBT_space2,Xb) = Y )
     => ( ( ? [A3: $o,B2: $o] : Xb = vEBT_Leaf((A3),(B2))
         => ( Y != aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one))) ) )
       => ~ ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
              ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,Deg2,TreeList),Summary) )
             => ( Y != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit1,one)))),aa_VEBT_VEBT_nat(vEBT_VEBT_space2,Summary))),aa_nat_nat(foldr_nat_nat(plus_plus_nat,map_VEBT_VEBT_nat(vEBT_VEBT_space2,TreeList)),zero_zero_nat)) ) ) ) ) ).

% VEBT_internal.space'.elims
tff(fact_5519_VEBT__internal_Ospace_Osimps_I2_J,axiom,
    ! [Info: option4927543243414619207at_nat,Deg: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT] : aa_VEBT_VEBT_nat(vEBT_VEBT_space,aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info,Deg,TreeLista),Summarya)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit0,one)))),aa_VEBT_VEBT_nat(vEBT_VEBT_space,Summarya))),size_s6755466524823107622T_VEBT(TreeLista))),aa_nat_nat(foldr_nat_nat(plus_plus_nat,map_VEBT_VEBT_nat(vEBT_VEBT_space,TreeLista)),zero_zero_nat)) ).

% VEBT_internal.space.simps(2)
tff(fact_5520_VEBT__internal_Ospace_Oelims,axiom,
    ! [Xb: vEBT_VEBT,Y: nat] :
      ( ( aa_VEBT_VEBT_nat(vEBT_VEBT_space,Xb) = Y )
     => ( ( ? [A3: $o,B2: $o] : Xb = vEBT_Leaf((A3),(B2))
         => ( Y != aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)) ) )
       => ~ ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
              ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,Deg2,TreeList),Summary) )
             => ( Y != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit0,one)))),aa_VEBT_VEBT_nat(vEBT_VEBT_space,Summary))),size_s6755466524823107622T_VEBT(TreeList))),aa_nat_nat(foldr_nat_nat(plus_plus_nat,map_VEBT_VEBT_nat(vEBT_VEBT_space,TreeList)),zero_zero_nat)) ) ) ) ) ).

% VEBT_internal.space.elims
tff(fact_5521_upd__rule,axiom,
    ! [I: nat,Xs: list_o,A: array_o,Xb: $o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_o(Xs))
     => hoare_6478655245392655262rray_o(snga_assn_o(A,Xs),array_upd_o(I,(Xb),A),aa_o_f8424462421971924018o_assn(aa_arr7807244294037787346o_assn(aa_lis9214373194942273159o_assn(aTP_Lamm_jf(I),Xs),A),(Xb))) ) ).

% upd_rule
tff(fact_5522_upd__rule,axiom,
    ! [I: nat,Xs: list_nat,A: array_nat,Xb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_nat(Xs))
     => hoare_6807272225193264096ay_nat(snga_assn_nat(A,Xs),array_upd_nat(I,Xb,A),aa_nat8392338049721622586t_assn(aa_arr4802002299793635008t_assn(aa_lis5007279772662053221t_assn(aTP_Lamm_jg(I),Xs),A),Xb)) ) ).

% upd_rule
tff(fact_5523_upd__rule,axiom,
    ! [I: nat,Xs: list_int,A: array_int,Xb: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_int(Xs))
     => hoare_2629421205684067388ay_int(snga_assn_int(A,Xs),array_upd_int(I,Xb,A),aa_int2412489125148462906t_assn(aa_arr8066481932369824868t_assn(aa_lis2681267339321482597t_assn(aTP_Lamm_jh(I),Xs),A),Xb)) ) ).

% upd_rule
tff(fact_5524_upd__rule,axiom,
    ! [I: nat,Xs: list_VEBT_VEBTi,A: array_VEBT_VEBTi,Xb: vEBT_VEBTi] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s7982070591426661849_VEBTi(Xs))
     => hoare_3353465787467722821_VEBTi(aa_lis2508312872377092907i_assn(snga_assn_VEBT_VEBTi(A),Xs),array_upd_VEBT_VEBTi(I,Xb,A),aa_VEB7841561841333234298i_assn(aa_arr6331738488200487899i_assn(aa_lis1967528460011584869i_assn(aTP_Lamm_ji(I),Xs),A),Xb)) ) ).

% upd_rule
tff(fact_5525_length__rule,axiom,
    ! [A: array_o,Xs: list_o] : hoare_3067605981109127869le_nat(snga_assn_o(A,Xs),array_len_o(A),aa_lis7996121028773434071t_assn(aTP_Lamm_jj(A),Xs)) ).

% length_rule
tff(fact_5526_length__rule,axiom,
    ! [A: array_nat,Xs: list_nat] : hoare_3067605981109127869le_nat(snga_assn_nat(A,Xs),array_len_nat(A),aa_lis6041139530851742867t_assn(aTP_Lamm_jk(A),Xs)) ).

% length_rule
tff(fact_5527_length__rule,axiom,
    ! [A: array_int,Xs: list_int] : hoare_3067605981109127869le_nat(snga_assn_int(A,Xs),array_len_int(A),aa_lis2816288184639765743t_assn(aTP_Lamm_jl(A),Xs)) ).

% length_rule
tff(fact_5528_length__rule,axiom,
    ! [A: array_VEBT_VEBTi,Xs: list_VEBT_VEBTi] : hoare_3067605981109127869le_nat(aa_lis2508312872377092907i_assn(snga_assn_VEBT_VEBTi(A),Xs),array_len_VEBT_VEBTi(A),aa_lis7348694692495225848t_assn(aTP_Lamm_jm(A),Xs)) ).

% length_rule
tff(fact_5529_VEBTi_Osize__gen_I1_J,axiom,
    ! [X11a: option4927543243414619207at_nat,X12a: nat,X13a: array_VEBT_VEBTi,X14a: vEBT_VEBTi] : aa_VEBT_VEBTi_nat(vEBT_size_VEBTi,aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(X11a,X12a,X13a),X14a)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,size_a6397454172108246045_VEBTi(vEBT_size_VEBTi,X13a)),aa_VEBT_VEBTi_nat(vEBT_size_VEBTi,X14a))),aa_nat_nat(suc,zero_zero_nat)) ).

% VEBTi.size_gen(1)
tff(fact_5530_length__corresp,axiom,
    ! [Tree_array: array_VEBT_VEBTi,Tree_is: list_real] :
      ( ( ex_ass463751140784270563_VEBTi(snga_assn_VEBT_VEBTi(Tree_array)) = top_top_assn )
     => ( aa_nat5821310369566504398ap_nat(heap_Time_return_nat,size_size_list_real(Tree_is)) = array_len_VEBT_VEBTi(Tree_array) ) ) ).

% length_corresp
tff(fact_5531_length__corresp,axiom,
    ! [Tree_array: array_VEBT_VEBTi,Tree_is: list_o] :
      ( ( ex_ass463751140784270563_VEBTi(snga_assn_VEBT_VEBTi(Tree_array)) = top_top_assn )
     => ( aa_nat5821310369566504398ap_nat(heap_Time_return_nat,size_size_list_o(Tree_is)) = array_len_VEBT_VEBTi(Tree_array) ) ) ).

% length_corresp
tff(fact_5532_length__corresp,axiom,
    ! [Tree_array: array_VEBT_VEBTi,Tree_is: list_nat] :
      ( ( ex_ass463751140784270563_VEBTi(snga_assn_VEBT_VEBTi(Tree_array)) = top_top_assn )
     => ( aa_nat5821310369566504398ap_nat(heap_Time_return_nat,size_size_list_nat(Tree_is)) = array_len_VEBT_VEBTi(Tree_array) ) ) ).

% length_corresp
tff(fact_5533_length__corresp,axiom,
    ! [Tree_array: array_VEBT_VEBTi,Tree_is: list_int] :
      ( ( ex_ass463751140784270563_VEBTi(snga_assn_VEBT_VEBTi(Tree_array)) = top_top_assn )
     => ( aa_nat5821310369566504398ap_nat(heap_Time_return_nat,size_size_list_int(Tree_is)) = array_len_VEBT_VEBTi(Tree_array) ) ) ).

% length_corresp
tff(fact_5534_list__every__elemnt__bound__sum__bound__real,axiom,
    ! [Xs: list_VEBT_VEBT,F: fun_VEBT_VEBT_real,Bound: real,I: real] :
      ( ! [X2: vEBT_VEBT] :
          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X2),set_VEBT_VEBT2(Xs))
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_VEBT_VEBT_real(F,X2)),Bound) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(foldr_real_real(plus_plus_real,map_VEBT_VEBT_real(F,Xs)),I)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,size_s6755466524823107622T_VEBT(Xs))),Bound)),I)) ) ).

% list_every_elemnt_bound_sum_bound_real
tff(fact_5535_list__every__elemnt__bound__sum__bound__real,axiom,
    ! [Xs: list_P8527749157015355191n_assn,F: fun_Pr4198752369477672304n_real,Bound: real,I: real] :
      ( ! [X2: produc6575502325842934193n_assn] :
          ( member7957490590177025114n_assn(X2,set_Pr1139785259514867910n_assn(Xs))
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_Pro6225755720812867233n_real(F,X2)),Bound) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(foldr_real_real(plus_plus_real,map_Pr5018725648611240729n_real(F,Xs)),I)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,size_s6829681357464350627n_assn(Xs))),Bound)),I)) ) ).

% list_every_elemnt_bound_sum_bound_real
tff(fact_5536_list__every__elemnt__bound__sum__bound__real,axiom,
    ! [Xs: list_real,F: fun_real_real,Bound: real,I: real] :
      ( ! [X2: real] :
          ( aa_set_real_o(member_real(X2),set_real2(Xs))
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(F,X2)),Bound) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(foldr_real_real(plus_plus_real,map_real_real(F,Xs)),I)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,size_size_list_real(Xs))),Bound)),I)) ) ).

% list_every_elemnt_bound_sum_bound_real
tff(fact_5537_list__every__elemnt__bound__sum__bound__real,axiom,
    ! [Xs: list_o,F: fun_o_real,Bound: real,I: real] :
      ( ! [X2: $o] :
          ( member_o((X2),set_o2(Xs))
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_o_real(F,(X2))),Bound) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(foldr_real_real(plus_plus_real,map_o_real(F,Xs)),I)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,size_size_list_o(Xs))),Bound)),I)) ) ).

% list_every_elemnt_bound_sum_bound_real
tff(fact_5538_list__every__elemnt__bound__sum__bound__real,axiom,
    ! [Xs: list_nat,F: fun_nat_real,Bound: real,I: real] :
      ( ! [X2: nat] :
          ( aa_set_nat_o(member_nat(X2),set_nat2(Xs))
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(F,X2)),Bound) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(foldr_real_real(plus_plus_real,map_nat_real(F,Xs)),I)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,size_size_list_nat(Xs))),Bound)),I)) ) ).

% list_every_elemnt_bound_sum_bound_real
tff(fact_5539_list__every__elemnt__bound__sum__bound__real,axiom,
    ! [Xs: list_int,F: fun_int_real,Bound: real,I: real] :
      ( ! [X2: int] :
          ( aa_set_int_o(member_int(X2),set_int2(Xs))
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(F,X2)),Bound) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(foldr_real_real(plus_plus_real,map_int_real(F,Xs)),I)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,size_size_list_int(Xs))),Bound)),I)) ) ).

% list_every_elemnt_bound_sum_bound_real
tff(fact_5540_real__nat__list,axiom,
    ! [F: fun_VEBT_VEBT_nat,Xs: list_VEBT_VEBT,C: nat] : aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(foldr_nat_nat(plus_plus_nat,map_VEBT_VEBT_nat(F,Xs)),C)) = aa_real_real(foldr_real_real(plus_plus_real,map_VEBT_VEBT_real(aTP_Lamm_jn(F),Xs)),aa_nat_real(semiri5074537144036343181t_real,C)) ).

% real_nat_list
tff(fact_5541_real__nat__list,axiom,
    ! [F: fun_nat_nat,Xs: list_nat,C: nat] : aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(foldr_nat_nat(plus_plus_nat,map_nat_nat(F,Xs)),C)) = aa_real_real(foldr_real_real(plus_plus_real,map_nat_real(aTP_Lamm_jo(F),Xs)),aa_nat_real(semiri5074537144036343181t_real,C)) ).

% real_nat_list
tff(fact_5542_f__g__map__foldr__bound,axiom,
    ! [Xs: list_VEBT_VEBT,F: fun_VEBT_VEBT_real,C: real,G: fun_VEBT_VEBT_real,D: real] :
      ( ! [X2: vEBT_VEBT] :
          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X2),set_VEBT_VEBT2(Xs))
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_VEBT_VEBT_real(F,X2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),aa_VEBT_VEBT_real(G,X2))) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(foldr_real_real(plus_plus_real,map_VEBT_VEBT_real(F,Xs)),D)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),aa_real_real(foldr_real_real(plus_plus_real,map_VEBT_VEBT_real(G,Xs)),zero_zero_real))),D)) ) ).

% f_g_map_foldr_bound
tff(fact_5543_f__g__map__foldr__bound,axiom,
    ! [Xs: list_nat,F: fun_nat_real,C: real,G: fun_nat_real,D: real] :
      ( ! [X2: nat] :
          ( aa_set_nat_o(member_nat(X2),set_nat2(Xs))
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(F,X2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),aa_nat_real(G,X2))) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(foldr_real_real(plus_plus_real,map_nat_real(F,Xs)),D)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),aa_real_real(foldr_real_real(plus_plus_real,map_nat_real(G,Xs)),zero_zero_real))),D)) ) ).

% f_g_map_foldr_bound
tff(fact_5544_f__g__map__foldr__bound,axiom,
    ! [Xs: list_real,F: fun_real_real,C: real,G: fun_real_real,D: real] :
      ( ! [X2: real] :
          ( aa_set_real_o(member_real(X2),set_real2(Xs))
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(F,X2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),aa_real_real(G,X2))) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(foldr_real_real(plus_plus_real,map_real_real(F,Xs)),D)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),aa_real_real(foldr_real_real(plus_plus_real,map_real_real(G,Xs)),zero_zero_real))),D)) ) ).

% f_g_map_foldr_bound
tff(fact_5545_f__g__map__foldr__bound,axiom,
    ! [Xs: list_int,F: fun_int_real,C: real,G: fun_int_real,D: real] :
      ( ! [X2: int] :
          ( aa_set_int_o(member_int(X2),set_int2(Xs))
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(F,X2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),aa_int_real(G,X2))) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(foldr_real_real(plus_plus_real,map_int_real(F,Xs)),D)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),aa_real_real(foldr_real_real(plus_plus_real,map_int_real(G,Xs)),zero_zero_real))),D)) ) ).

% f_g_map_foldr_bound
tff(fact_5546_f__g__map__foldr__bound,axiom,
    ! [Xs: list_P8527749157015355191n_assn,F: fun_Pr4198752369477672304n_real,C: real,G: fun_Pr4198752369477672304n_real,D: real] :
      ( ! [X2: produc6575502325842934193n_assn] :
          ( member7957490590177025114n_assn(X2,set_Pr1139785259514867910n_assn(Xs))
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_Pro6225755720812867233n_real(F,X2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),aa_Pro6225755720812867233n_real(G,X2))) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(foldr_real_real(plus_plus_real,map_Pr5018725648611240729n_real(F,Xs)),D)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),aa_real_real(foldr_real_real(plus_plus_real,map_Pr5018725648611240729n_real(G,Xs)),zero_zero_real))),D)) ) ).

% f_g_map_foldr_bound
tff(fact_5547_vebt__memberi_Osimps,axiom,
    ! [Ta: vEBT_VEBTi,Xb: nat] : aa_nat2910048432523430448Heap_o(aa_VEB1455453220190609250Heap_o(vEBT_vebt_memberi,Ta),Xb) = aa_VEB8194921341634505877Heap_o(vEBT_c6104975476656191286Heap_o(aTP_Lamm_ju(Xb),aTP_Lamm_jv(Xb)),Ta) ).

% vebt_memberi.simps
tff(fact_5548_listsum__bound,axiom,
    ! [Xs: list_VEBT_VEBT,F: fun_VEBT_VEBT_real,Y: real] :
      ( ! [X2: vEBT_VEBT] :
          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X2),set_VEBT_VEBT2(Xs))
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_VEBT_VEBT_real(F,X2)) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),aa_real_real(foldr_real_real(plus_plus_real,map_VEBT_VEBT_real(F,Xs)),Y)) ) ).

% listsum_bound
tff(fact_5549_listsum__bound,axiom,
    ! [Xs: list_nat,F: fun_nat_real,Y: real] :
      ( ! [X2: nat] :
          ( aa_set_nat_o(member_nat(X2),set_nat2(Xs))
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(F,X2)) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),aa_real_real(foldr_real_real(plus_plus_real,map_nat_real(F,Xs)),Y)) ) ).

% listsum_bound
tff(fact_5550_listsum__bound,axiom,
    ! [Xs: list_real,F: fun_real_real,Y: real] :
      ( ! [X2: real] :
          ( aa_set_real_o(member_real(X2),set_real2(Xs))
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(F,X2)) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),aa_real_real(foldr_real_real(plus_plus_real,map_real_real(F,Xs)),Y)) ) ).

% listsum_bound
tff(fact_5551_listsum__bound,axiom,
    ! [Xs: list_int,F: fun_int_real,Y: real] :
      ( ! [X2: int] :
          ( aa_set_int_o(member_int(X2),set_int2(Xs))
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_int_real(F,X2)) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),aa_real_real(foldr_real_real(plus_plus_real,map_int_real(F,Xs)),Y)) ) ).

% listsum_bound
tff(fact_5552_listsum__bound,axiom,
    ! [Xs: list_P8527749157015355191n_assn,F: fun_Pr4198752369477672304n_real,Y: real] :
      ( ! [X2: produc6575502325842934193n_assn] :
          ( member7957490590177025114n_assn(X2,set_Pr1139785259514867910n_assn(Xs))
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_Pro6225755720812867233n_real(F,X2)) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),aa_real_real(foldr_real_real(plus_plus_real,map_Pr5018725648611240729n_real(F,Xs)),Y)) ) ).

% listsum_bound
tff(fact_5553_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_5554_assn__basic__inequalities_I1_J,axiom,
    top_top_assn != one_one_assn ).

% assn_basic_inequalities(1)
tff(fact_5555_case__prod__conv,axiom,
    ! [F: fun_na4138050610385705875at_nat,A: nat,B: nat] : aa_Pro262675715343573244at_nat(produc2626176000494625587at_nat(F),aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,A),B)) = aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(F,A),B) ).

% case_prod_conv
tff(fact_5556_case__prod__conv,axiom,
    ! [F: fun_nat_fun_nat_o,A: nat,B: nat] :
      ( aa_Pro6150637604445964947_nat_o(produc6081775807080527818_nat_o(F),aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,A),B))
    <=> aa_nat_o(aa_nat_fun_nat_o(F,A),B) ) ).

% case_prod_conv
tff(fact_5557_case__prod__conv,axiom,
    ! [F: fun_in1696471567231710611nt_int,A: int,B: int] : aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(F),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),B)) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(F,A),B) ).

% case_prod_conv
tff(fact_5558_case__prod__conv,axiom,
    ! [F: fun_int_fun_int_o,A: int,B: int] :
      ( aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(F),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),B))
    <=> aa_int_o(aa_int_fun_int_o(F,A),B) ) ).

% case_prod_conv
tff(fact_5559_case__prod__conv,axiom,
    ! [F: fun_int_fun_int_int,A: int,B: int] : aa_Pro6799258434465214121nt_int(produc8211389475949308722nt_int(F),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),B)) = aa_int_int(aa_int_fun_int_int(F,A),B) ).

% case_prod_conv
tff(fact_5560_map__fst__mk__snd,axiom,
    ! [K: produc8923325533196201883nteger,L: list_f4306079564415238026e_term] : map_Pr5355680216903570006e_term(produc2481150572478215260nteger,map_fu281021956687668182nteger(aTP_Lamm_jw(K),L)) = L ).

% map_fst_mk_snd
tff(fact_5561_map__fst__mk__snd,axiom,
    ! [K: produc3658429121746597890et_nat,L: list_f5222602229612168821_nat_o] : map_Pr1849485910839101705_nat_o(produc5325900173150335662et_nat,map_fu2989592291956603595et_nat(aTP_Lamm_jx(K),L)) = L ).

% map_fst_mk_snd
tff(fact_5562_map__fst__mk__snd,axiom,
    ! [K: produc8799533983703829792et_nat,L: list_f5222602229612168821_nat_o] : map_Pr249061351429473131_nat_o(produc4078497350495916492et_nat,map_fu6093296949760934377et_nat(aTP_Lamm_jy(K),L)) = L ).

% map_fst_mk_snd
tff(fact_5563_map__fst__mk__snd,axiom,
    ! [K: product_prod_int_int,L: list_f280343417242313083e_term] : map_Pr6544494440361398196e_term(produc3130796691802200265nt_int,map_fu2350696724278860212nt_int(aTP_Lamm_jz(K),L)) = L ).

% map_fst_mk_snd
tff(fact_5564_map__fst__mk__snd,axiom,
    ! [K: product_prod_int_int,L: list_f628289730585369423e_term] : map_Pr6740350451330705588e_term(produc4463008262350210037nt_int,map_fu7709190425054843060nt_int(aTP_Lamm_ka(K),L)) = L ).

% map_fst_mk_snd
tff(fact_5565_map__fst__mk__snd,axiom,
    ! [K: set_nat,L: list_h2745205591237434579t_unit] : map_Pr7042224834074979589t_unit(produc1824681642469235216et_nat,map_he4567745686773187723et_nat(aTP_Lamm_kb(K),L)) = L ).

% map_fst_mk_snd
tff(fact_5566_map__fst__mk__snd,axiom,
    ! [K: assn,L: list_assn] : map_Pr8991440229025900053n_assn(produc9167289414957590229n_assn,map_as2373307505041272643n_assn(aTP_Lamm_kc(K),L)) = L ).

% map_fst_mk_snd
tff(fact_5567_map__fst__mk__snd,axiom,
    ! [K: uint32,L: list_uint32] : map_Pr2530660914518622561uint32(produc9004433772639906525uint32,map_ui5747794299746474503uint32(aTP_Lamm_kd(K),L)) = L ).

% map_fst_mk_snd
tff(fact_5568_map__fst__mk__snd,axiom,
    ! [K: nat,L: list_nat] : map_Pr3938374229010428429at_nat(product_fst_nat_nat,map_na7298421622053143531at_nat(aa_nat8803788312118491388at_nat(aTP_Lamm_ke,K),L)) = L ).

% map_fst_mk_snd
tff(fact_5569_map__fst__mk__snd,axiom,
    ! [K: int,L: list_int] : map_Pr6494458893431244577nt_int(product_fst_int_int,map_in7157766398909135175nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_kf,K),L)) = L ).

% map_fst_mk_snd
tff(fact_5570_map__snd__mk__fst,axiom,
    ! [K: fun_Pr5098669352780829690e_term,L: list_P5578671422887162913nteger] : map_Pr6499330355454566463nteger(produc6589491691745714842nteger,map_Pr6164994510355967853nteger(produc565448451787809096nteger(K),L)) = L ).

% map_snd_mk_fst
tff(fact_5571_map__snd__mk__fst,axiom,
    ! [K: fun_Pr6460944699146269935_nat_o,L: list_P9062070895058802706et_nat] : map_Pr505308552747352680et_nat(produc4960056325182568940et_nat,map_Pr4621532551050652904et_nat(produc6017135503889078682et_nat(K),L)) = L ).

% map_snd_mk_fst
tff(fact_5572_map__snd__mk__fst,axiom,
    ! [K: fun_Pr6460944699146269935_nat_o,L: list_P8836981409019564976et_nat] : map_Pr1145936293888461732et_nat(produc9081944055658670602et_nat,map_Pr4550756191949977124et_nat(produc7291855603632349880et_nat(K),L)) = L ).

% map_snd_mk_fst
tff(fact_5573_map__snd__mk__fst,axiom,
    ! [K: fun_Pr7763680713840462443e_term,L: list_P5707943133018811711nt_int] : map_Pr349481598175089328nt_int(produc1557020426977177351nt_int,map_Pr3422498677955141880nt_int(produc7320969046608631221nt_int(K),L)) = L ).

% map_snd_mk_fst
tff(fact_5574_map__snd__mk__fst,axiom,
    ! [K: fun_in5730379987311621695e_term,L: list_P5707943133018811711nt_int] : map_Pr4441656247296274564nt_int(produc7629796867588368947nt_int,map_Pr8804828260421831716nt_int(produc4729913460834754785nt_int(K),L)) = L ).

% map_snd_mk_fst
tff(fact_5575_map__snd__mk__fst,axiom,
    ! [K: nat,L: list_nat] : map_Pr3938374229010428429at_nat(product_snd_nat_nat,map_na7298421622053143531at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,K),L)) = L ).

% map_snd_mk_fst
tff(fact_5576_map__snd__mk__fst,axiom,
    ! [K: heap_e7401611519738050253t_unit,L: list_set_nat] : map_Pr4871828589360411470et_nat(produc8586169260539613262et_nat,map_se5435525824004667726et_nat(produc7507926704131184380et_nat(K),L)) = L ).

% map_snd_mk_fst
tff(fact_5577_map__snd__mk__fst,axiom,
    ! [K: int,L: list_int] : map_Pr6494458893431244577nt_int(product_snd_int_int,map_in7157766398909135175nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,K),L)) = L ).

% map_snd_mk_fst
tff(fact_5578_map__snd__mk__fst,axiom,
    ! [K: code_integer,L: list_Code_integer] : map_Pr1857711230949937460nteger(produc6174133586879617921nteger,map_Co3589949550033412536nteger(produc1086072967326762835nteger(K),L)) = L ).

% map_snd_mk_fst
tff(fact_5579_map__snd__mk__fst,axiom,
    ! [K: assn,L: list_assn] : map_Pr8991440229025900053n_assn(produc2051961928117032727n_assn,map_as2373307505041272643n_assn(produc118845697133431529n_assn(K),L)) = L ).

% map_snd_mk_fst
tff(fact_5580_top__option__def,axiom,
    top_top_option_assn = some_assn(top_top_assn) ).

% top_option_def
tff(fact_5581_top__option__def,axiom,
    top_to1083748111577038690t_real = some_set_real(top_top_set_real) ).

% top_option_def
tff(fact_5582_top__option__def,axiom,
    top_to4826455019444611206et_nat = some_set_nat(top_top_set_nat) ).

% top_option_def
tff(fact_5583_top__option__def,axiom,
    top_to6745749650031393671t_char = some_set_char(top_top_set_char) ).

% top_option_def
tff(fact_5584_prod_Ocase__distrib,axiom,
    ! [H: fun_o_o,F: fun_nat_fun_nat_o,Prod: product_prod_nat_nat] :
      ( aa_o_o(H,aa_Pro6150637604445964947_nat_o(produc6081775807080527818_nat_o(F),Prod))
    <=> aa_Pro6150637604445964947_nat_o(produc6081775807080527818_nat_o(aa_fun6699960423209655484_nat_o(aTP_Lamm_kg(H),F)),Prod) ) ).

% prod.case_distrib
tff(fact_5585_prod_Ocase__distrib,axiom,
    ! [H: fun_o_o,F: fun_int_fun_int_o,Prod: product_prod_int_int] :
      ( aa_o_o(H,aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(F),Prod))
    <=> aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(aa_fun6133342665515662012_int_o(aTP_Lamm_kh(H),F)),Prod) ) ).

% prod.case_distrib
tff(fact_5586_prod_Ocase__distrib,axiom,
    ! [H: fun_o_int,F: fun_int_fun_int_o,Prod: product_prod_int_int] : aa_o_int(H,aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(F),Prod)) = aa_Pro6799258434465214121nt_int(produc8211389475949308722nt_int(aa_fun7323606162024495668nt_int(aTP_Lamm_ki(H),F)),Prod) ).

% prod.case_distrib
tff(fact_5587_prod_Ocase__distrib,axiom,
    ! [H: fun_int_o,F: fun_int_fun_int_int,Prod: product_prod_int_int] :
      ( aa_int_o(H,aa_Pro6799258434465214121nt_int(produc8211389475949308722nt_int(F),Prod))
    <=> aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(aa_fun82482788810740534_int_o(aTP_Lamm_kj(H),F)),Prod) ) ).

% prod.case_distrib
tff(fact_5588_prod_Ocase__distrib,axiom,
    ! [H: fun_int_int,F: fun_int_fun_int_int,Prod: product_prod_int_int] : aa_int_int(H,aa_Pro6799258434465214121nt_int(produc8211389475949308722nt_int(F),Prod)) = aa_Pro6799258434465214121nt_int(produc8211389475949308722nt_int(aa_fun6119123525526402106nt_int(aTP_Lamm_kk(H),F)),Prod) ).

% prod.case_distrib
tff(fact_5589_prod_Ocase__distrib,axiom,
    ! [H: fun_Pr4796943543331071514_nat_o,F: fun_na4138050610385705875at_nat,Prod: product_prod_nat_nat] :
      ( aa_Pro6150637604445964947_nat_o(H,aa_Pro262675715343573244at_nat(produc2626176000494625587at_nat(F),Prod))
    <=> aa_Pro6150637604445964947_nat_o(produc6081775807080527818_nat_o(aa_fun7288258519458182931_nat_o(aTP_Lamm_kl(H),F)),Prod) ) ).

% prod.case_distrib
tff(fact_5590_prod_Ocase__distrib,axiom,
    ! [H: fun_o_67843798866261420at_nat,F: fun_nat_fun_nat_o,Prod: product_prod_nat_nat] : aa_o_P6312935399938478757at_nat(H,aa_Pro6150637604445964947_nat_o(produc6081775807080527818_nat_o(F),Prod)) = aa_Pro262675715343573244at_nat(produc2626176000494625587at_nat(aa_fun9180491248050787109at_nat(aTP_Lamm_km(H),F)),Prod) ).

% prod.case_distrib
tff(fact_5591_prod_Ocase__distrib,axiom,
    ! [H: fun_Pr3238374608701176162_int_o,F: fun_in1696471567231710611nt_int,Prod: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(H,aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(F),Prod))
    <=> aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(aa_fun4545072666309863515_int_o(aTP_Lamm_kn(H),F)),Prod) ) ).

% prod.case_distrib
tff(fact_5592_prod_Ocase__distrib,axiom,
    ! [H: fun_Pr982841511892225912nt_int,F: fun_in1696471567231710611nt_int,Prod: product_prod_int_int] : aa_Pro6799258434465214121nt_int(H,aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(F),Prod)) = aa_Pro6799258434465214121nt_int(produc8211389475949308722nt_int(aa_fun7461304428735181397nt_int(aTP_Lamm_ko(H),F)),Prod) ).

% prod.case_distrib
tff(fact_5593_prod_Ocase__distrib,axiom,
    ! [H: fun_o_8988054265482332260nt_int,F: fun_int_fun_int_o,Prod: product_prod_int_int] : aa_o_P3134438092330336605nt_int(H,aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(F),Prod)) = aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(aa_fun4023131440832735197nt_int(aTP_Lamm_kp(H),F)),Prod) ).

% prod.case_distrib
tff(fact_5594_old_Oprod_Ocase,axiom,
    ! [F: fun_na4138050610385705875at_nat,X1: nat,X23: nat] : aa_Pro262675715343573244at_nat(produc2626176000494625587at_nat(F),aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,X1),X23)) = aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(F,X1),X23) ).

% old.prod.case
tff(fact_5595_old_Oprod_Ocase,axiom,
    ! [F: fun_nat_fun_nat_o,X1: nat,X23: nat] :
      ( aa_Pro6150637604445964947_nat_o(produc6081775807080527818_nat_o(F),aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,X1),X23))
    <=> aa_nat_o(aa_nat_fun_nat_o(F,X1),X23) ) ).

% old.prod.case
tff(fact_5596_old_Oprod_Ocase,axiom,
    ! [F: fun_in1696471567231710611nt_int,X1: int,X23: int] : aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(F),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X1),X23)) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(F,X1),X23) ).

% old.prod.case
tff(fact_5597_old_Oprod_Ocase,axiom,
    ! [F: fun_int_fun_int_o,X1: int,X23: int] :
      ( aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(F),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X1),X23))
    <=> aa_int_o(aa_int_fun_int_o(F,X1),X23) ) ).

% old.prod.case
tff(fact_5598_old_Oprod_Ocase,axiom,
    ! [F: fun_int_fun_int_int,X1: int,X23: int] : aa_Pro6799258434465214121nt_int(produc8211389475949308722nt_int(F),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X1),X23)) = aa_int_int(aa_int_fun_int_int(F,X1),X23) ).

% old.prod.case
tff(fact_5599_merge__true__star__ctx,axiom,
    ! [P: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,top_top_assn),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,top_top_assn),P)) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,top_top_assn),P) ).

% merge_true_star_ctx
tff(fact_5600_case__prodE2,axiom,
    ! [Q: fun_Pr4796943543331071514_nat_o,P: fun_na4138050610385705875at_nat,Z: product_prod_nat_nat] :
      ( aa_Pro6150637604445964947_nat_o(Q,aa_Pro262675715343573244at_nat(produc2626176000494625587at_nat(P),Z))
     => ~ ! [X2: nat,Y4: nat] :
            ( ( Z = aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,X2),Y4) )
           => ~ aa_Pro6150637604445964947_nat_o(Q,aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(P,X2),Y4)) ) ) ).

% case_prodE2
tff(fact_5601_case__prodE2,axiom,
    ! [Q: fun_o_o,P: fun_nat_fun_nat_o,Z: product_prod_nat_nat] :
      ( aa_o_o(Q,aa_Pro6150637604445964947_nat_o(produc6081775807080527818_nat_o(P),Z))
     => ~ ! [X2: nat,Y4: nat] :
            ( ( Z = aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,X2),Y4) )
           => ~ aa_o_o(Q,aa_nat_o(aa_nat_fun_nat_o(P,X2),Y4)) ) ) ).

% case_prodE2
tff(fact_5602_case__prodE2,axiom,
    ! [Q: fun_Pr3238374608701176162_int_o,P: fun_in1696471567231710611nt_int,Z: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(Q,aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(P),Z))
     => ~ ! [X2: int,Y4: int] :
            ( ( Z = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X2),Y4) )
           => ~ aa_Pro4592068669816069595_int_o(Q,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(P,X2),Y4)) ) ) ).

% case_prodE2
tff(fact_5603_case__prodE2,axiom,
    ! [Q: fun_o_o,P: fun_int_fun_int_o,Z: product_prod_int_int] :
      ( aa_o_o(Q,aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(P),Z))
     => ~ ! [X2: int,Y4: int] :
            ( ( Z = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X2),Y4) )
           => ~ aa_o_o(Q,aa_int_o(aa_int_fun_int_o(P,X2),Y4)) ) ) ).

% case_prodE2
tff(fact_5604_case__prodE2,axiom,
    ! [Q: fun_int_o,P: fun_int_fun_int_int,Z: product_prod_int_int] :
      ( aa_int_o(Q,aa_Pro6799258434465214121nt_int(produc8211389475949308722nt_int(P),Z))
     => ~ ! [X2: int,Y4: int] :
            ( ( Z = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X2),Y4) )
           => ~ aa_int_o(Q,aa_int_int(aa_int_fun_int_int(P,X2),Y4)) ) ) ).

% case_prodE2
tff(fact_5605_case__prod__eta,axiom,
    ! [F: fun_Pr2118491707810114563at_nat] : produc2626176000494625587at_nat(aTP_Lamm_kq(F)) = F ).

% case_prod_eta
tff(fact_5606_case__prod__eta,axiom,
    ! [F: fun_Pr4796943543331071514_nat_o] : produc6081775807080527818_nat_o(aTP_Lamm_kr(F)) = F ).

% case_prod_eta
tff(fact_5607_case__prod__eta,axiom,
    ! [F: fun_Pr2478283820680522243nt_int] : produc4245557441103728435nt_int(aTP_Lamm_ks(F)) = F ).

% case_prod_eta
tff(fact_5608_case__prod__eta,axiom,
    ! [F: fun_Pr3238374608701176162_int_o] : produc4947309494688390418_int_o(aTP_Lamm_kt(F)) = F ).

% case_prod_eta
tff(fact_5609_case__prod__eta,axiom,
    ! [F: fun_Pr982841511892225912nt_int] : produc8211389475949308722nt_int(aTP_Lamm_ku(F)) = F ).

% case_prod_eta
tff(fact_5610_cond__case__prod__eta,axiom,
    ! [F: fun_na4138050610385705875at_nat,G: fun_Pr2118491707810114563at_nat] :
      ( ! [X2: nat,Y4: nat] : aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(F,X2),Y4) = aa_Pro262675715343573244at_nat(G,aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,X2),Y4))
     => ( produc2626176000494625587at_nat(F) = G ) ) ).

% cond_case_prod_eta
tff(fact_5611_cond__case__prod__eta,axiom,
    ! [F: fun_nat_fun_nat_o,G: fun_Pr4796943543331071514_nat_o] :
      ( ! [X2: nat,Y4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(F,X2),Y4)
        <=> aa_Pro6150637604445964947_nat_o(G,aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,X2),Y4)) )
     => ( produc6081775807080527818_nat_o(F) = G ) ) ).

% cond_case_prod_eta
tff(fact_5612_cond__case__prod__eta,axiom,
    ! [F: fun_in1696471567231710611nt_int,G: fun_Pr2478283820680522243nt_int] :
      ( ! [X2: int,Y4: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(F,X2),Y4) = aa_Pro6970504127699319548nt_int(G,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X2),Y4))
     => ( produc4245557441103728435nt_int(F) = G ) ) ).

% cond_case_prod_eta
tff(fact_5613_cond__case__prod__eta,axiom,
    ! [F: fun_int_fun_int_o,G: fun_Pr3238374608701176162_int_o] :
      ( ! [X2: int,Y4: int] :
          ( aa_int_o(aa_int_fun_int_o(F,X2),Y4)
        <=> aa_Pro4592068669816069595_int_o(G,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X2),Y4)) )
     => ( produc4947309494688390418_int_o(F) = G ) ) ).

% cond_case_prod_eta
tff(fact_5614_cond__case__prod__eta,axiom,
    ! [F: fun_int_fun_int_int,G: fun_Pr982841511892225912nt_int] :
      ( ! [X2: int,Y4: int] : aa_int_int(aa_int_fun_int_int(F,X2),Y4) = aa_Pro6799258434465214121nt_int(G,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X2),Y4))
     => ( produc8211389475949308722nt_int(F) = G ) ) ).

% cond_case_prod_eta
tff(fact_5615_fn__fst__conv,axiom,
    ! [F: fun_na1476630412504255114at_nat] : aTP_Lamm_kv(F) = produc2626176000494625587at_nat(aTP_Lamm_kw(F)) ).

% fn_fst_conv
tff(fact_5616_fn__fst__conv,axiom,
    ! [F: fun_nat_o] : aTP_Lamm_kx(F) = produc6081775807080527818_nat_o(aTP_Lamm_ky(F)) ).

% fn_fst_conv
tff(fact_5617_fn__fst__conv,axiom,
    ! [F: fun_in4211310926729683942nt_int] : aTP_Lamm_kz(F) = produc4245557441103728435nt_int(aTP_Lamm_la(F)) ).

% fn_fst_conv
tff(fact_5618_fn__fst__conv,axiom,
    ! [F: fun_int_o] : aTP_Lamm_lb(F) = produc4947309494688390418_int_o(aTP_Lamm_lc(F)) ).

% fn_fst_conv
tff(fact_5619_fn__fst__conv,axiom,
    ! [F: fun_int_int] : aTP_Lamm_ld(F) = produc8211389475949308722nt_int(aTP_Lamm_le(F)) ).

% fn_fst_conv
tff(fact_5620_fst__def,axiom,
    ! [Prod: produc3658429121746597890et_nat] : aa_Pro1219312880135979149t_unit(produc1824681642469235216et_nat,Prod) = produc8289649279824296164t_unit(aTP_Lamm_lf,Prod) ).

% fst_def
tff(fact_5621_fst__def,axiom,
    ! [Prod: produc6575502325842934193n_assn] : aa_Pro2979818503705714077n_assn(produc9167289414957590229n_assn,Prod) = aa_Pro2979818503705714077n_assn(produc2152611005075324454n_assn(aTP_Lamm_lg),Prod) ).

% fst_def
tff(fact_5622_fst__def,axiom,
    ! [Prod: produc827990862158126777uint32] : aa_Pro3737690986720249065uint32(produc9004433772639906525uint32,Prod) = produc3183254182024347122uint32(aTP_Lamm_lh,Prod) ).

% fst_def
tff(fact_5623_fst__def,axiom,
    ! [Prod: product_prod_nat_nat] : aa_Pro4243173770044397973at_nat(product_fst_nat_nat,Prod) = aa_Pro4243173770044397973at_nat(produc6842872674320459806at_nat(aTP_Lamm_li),Prod) ).

% fst_def
tff(fact_5624_fst__def,axiom,
    ! [Prod: product_prod_int_int] : aa_Pro6799258434465214121nt_int(product_fst_int_int,Prod) = aa_Pro6799258434465214121nt_int(produc8211389475949308722nt_int(aTP_Lamm_lj),Prod) ).

% fst_def
tff(fact_5625_fn__snd__conv,axiom,
    ! [F: fun_na1476630412504255114at_nat] : aTP_Lamm_lk(F) = produc2626176000494625587at_nat(aTP_Lamm_ll(F)) ).

% fn_snd_conv
tff(fact_5626_fn__snd__conv,axiom,
    ! [F: fun_nat_o] : aTP_Lamm_lm(F) = produc6081775807080527818_nat_o(aTP_Lamm_ln(F)) ).

% fn_snd_conv
tff(fact_5627_fn__snd__conv,axiom,
    ! [F: fun_in4211310926729683942nt_int] : aTP_Lamm_lo(F) = produc4245557441103728435nt_int(aTP_Lamm_lp(F)) ).

% fn_snd_conv
tff(fact_5628_fn__snd__conv,axiom,
    ! [F: fun_int_o] : aTP_Lamm_lq(F) = produc4947309494688390418_int_o(aTP_Lamm_lr(F)) ).

% fn_snd_conv
tff(fact_5629_fn__snd__conv,axiom,
    ! [F: fun_int_int] : aTP_Lamm_ls(F) = produc8211389475949308722nt_int(aTP_Lamm_lt(F)) ).

% fn_snd_conv
tff(fact_5630_snd__def,axiom,
    ! [Prod: product_prod_nat_nat] : aa_Pro4243173770044397973at_nat(product_snd_nat_nat,Prod) = aa_Pro4243173770044397973at_nat(produc6842872674320459806at_nat(aTP_Lamm_lu),Prod) ).

% snd_def
tff(fact_5631_snd__def,axiom,
    ! [Prod: produc3658429121746597890et_nat] : aa_Pro3416150927818442182et_nat(produc8586169260539613262et_nat,Prod) = produc7138653640177796783et_nat(aTP_Lamm_lv,Prod) ).

% snd_def
tff(fact_5632_snd__def,axiom,
    ! [Prod: produc8923325533196201883nteger] : aa_Pro5245656335456409788nteger(produc6174133586879617921nteger,Prod) = produc7469247390737344965nteger(aTP_Lamm_lw,Prod) ).

% snd_def
tff(fact_5633_snd__def,axiom,
    ! [Prod: produc6575502325842934193n_assn] : aa_Pro2979818503705714077n_assn(produc2051961928117032727n_assn,Prod) = aa_Pro2979818503705714077n_assn(produc2152611005075324454n_assn(aTP_Lamm_lx),Prod) ).

% snd_def
tff(fact_5634_snd__def,axiom,
    ! [Prod: product_prod_int_int] : aa_Pro6799258434465214121nt_int(product_snd_int_int,Prod) = aa_Pro6799258434465214121nt_int(produc8211389475949308722nt_int(aTP_Lamm_ly),Prod) ).

% snd_def
tff(fact_5635_pair__list__eqI,axiom,
    ! [Xs: list_P3069071885182933823uint32,Ys: list_P3069071885182933823uint32] :
      ( ( map_Pr2530660914518622561uint32(produc9004433772639906525uint32,Xs) = map_Pr2530660914518622561uint32(produc9004433772639906525uint32,Ys) )
     => ( ( map_Pr2530660914518622561uint32(produc1510406741064981791uint32,Xs) = map_Pr2530660914518622561uint32(produc1510406741064981791uint32,Ys) )
       => ( Xs = Ys ) ) ) ).

% pair_list_eqI
tff(fact_5636_pair__list__eqI,axiom,
    ! [Xs: list_P6011104703257516679at_nat,Ys: list_P6011104703257516679at_nat] :
      ( ( map_Pr3938374229010428429at_nat(product_fst_nat_nat,Xs) = map_Pr3938374229010428429at_nat(product_fst_nat_nat,Ys) )
     => ( ( map_Pr3938374229010428429at_nat(product_snd_nat_nat,Xs) = map_Pr3938374229010428429at_nat(product_snd_nat_nat,Ys) )
       => ( Xs = Ys ) ) ) ).

% pair_list_eqI
tff(fact_5637_pair__list__eqI,axiom,
    ! [Xs: list_P9062070895058802706et_nat,Ys: list_P9062070895058802706et_nat] :
      ( ( map_Pr7042224834074979589t_unit(produc1824681642469235216et_nat,Xs) = map_Pr7042224834074979589t_unit(produc1824681642469235216et_nat,Ys) )
     => ( ( map_Pr4871828589360411470et_nat(produc8586169260539613262et_nat,Xs) = map_Pr4871828589360411470et_nat(produc8586169260539613262et_nat,Ys) )
       => ( Xs = Ys ) ) ) ).

% pair_list_eqI
tff(fact_5638_pair__list__eqI,axiom,
    ! [Xs: list_P5707943133018811711nt_int,Ys: list_P5707943133018811711nt_int] :
      ( ( map_Pr6494458893431244577nt_int(product_fst_int_int,Xs) = map_Pr6494458893431244577nt_int(product_fst_int_int,Ys) )
     => ( ( map_Pr6494458893431244577nt_int(product_snd_int_int,Xs) = map_Pr6494458893431244577nt_int(product_snd_int_int,Ys) )
       => ( Xs = Ys ) ) ) ).

% pair_list_eqI
tff(fact_5639_pair__list__eqI,axiom,
    ! [Xs: list_P5578671422887162913nteger,Ys: list_P5578671422887162913nteger] :
      ( ( map_Pr1857711230949937460nteger(produc8508995932063986495nteger,Xs) = map_Pr1857711230949937460nteger(produc8508995932063986495nteger,Ys) )
     => ( ( map_Pr1857711230949937460nteger(produc6174133586879617921nteger,Xs) = map_Pr1857711230949937460nteger(produc6174133586879617921nteger,Ys) )
       => ( Xs = Ys ) ) ) ).

% pair_list_eqI
tff(fact_5640_pair__list__eqI,axiom,
    ! [Xs: list_P8527749157015355191n_assn,Ys: list_P8527749157015355191n_assn] :
      ( ( map_Pr8991440229025900053n_assn(produc9167289414957590229n_assn,Xs) = map_Pr8991440229025900053n_assn(produc9167289414957590229n_assn,Ys) )
     => ( ( map_Pr8991440229025900053n_assn(produc2051961928117032727n_assn,Xs) = map_Pr8991440229025900053n_assn(produc2051961928117032727n_assn,Ys) )
       => ( Xs = Ys ) ) ) ).

% pair_list_eqI
tff(fact_5641_split__beta,axiom,
    ! [F: fun_na4138050610385705875at_nat,Prod: product_prod_nat_nat] : aa_Pro262675715343573244at_nat(produc2626176000494625587at_nat(F),Prod) = aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(F,aa_Pro4243173770044397973at_nat(product_fst_nat_nat,Prod)),aa_Pro4243173770044397973at_nat(product_snd_nat_nat,Prod)) ).

% split_beta
tff(fact_5642_split__beta,axiom,
    ! [F: fun_nat_fun_nat_o,Prod: product_prod_nat_nat] :
      ( aa_Pro6150637604445964947_nat_o(produc6081775807080527818_nat_o(F),Prod)
    <=> aa_nat_o(aa_nat_fun_nat_o(F,aa_Pro4243173770044397973at_nat(product_fst_nat_nat,Prod)),aa_Pro4243173770044397973at_nat(product_snd_nat_nat,Prod)) ) ).

% split_beta
tff(fact_5643_split__beta,axiom,
    ! [F: fun_in1696471567231710611nt_int,Prod: product_prod_int_int] : aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(F),Prod) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(F,aa_Pro6799258434465214121nt_int(product_fst_int_int,Prod)),aa_Pro6799258434465214121nt_int(product_snd_int_int,Prod)) ).

% split_beta
tff(fact_5644_split__beta,axiom,
    ! [F: fun_int_fun_int_o,Prod: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(F),Prod)
    <=> aa_int_o(aa_int_fun_int_o(F,aa_Pro6799258434465214121nt_int(product_fst_int_int,Prod)),aa_Pro6799258434465214121nt_int(product_snd_int_int,Prod)) ) ).

% split_beta
tff(fact_5645_split__beta,axiom,
    ! [F: fun_int_fun_int_int,Prod: product_prod_int_int] : aa_Pro6799258434465214121nt_int(produc8211389475949308722nt_int(F),Prod) = aa_int_int(aa_int_fun_int_int(F,aa_Pro6799258434465214121nt_int(product_fst_int_int,Prod)),aa_Pro6799258434465214121nt_int(product_snd_int_int,Prod)) ).

% split_beta
tff(fact_5646_case__prod__beta,axiom,
    ! [F: fun_na4138050610385705875at_nat,P2: product_prod_nat_nat] : aa_Pro262675715343573244at_nat(produc2626176000494625587at_nat(F),P2) = aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(F,aa_Pro4243173770044397973at_nat(product_fst_nat_nat,P2)),aa_Pro4243173770044397973at_nat(product_snd_nat_nat,P2)) ).

% case_prod_beta
tff(fact_5647_case__prod__beta,axiom,
    ! [F: fun_nat_fun_nat_o,P2: product_prod_nat_nat] :
      ( aa_Pro6150637604445964947_nat_o(produc6081775807080527818_nat_o(F),P2)
    <=> aa_nat_o(aa_nat_fun_nat_o(F,aa_Pro4243173770044397973at_nat(product_fst_nat_nat,P2)),aa_Pro4243173770044397973at_nat(product_snd_nat_nat,P2)) ) ).

% case_prod_beta
tff(fact_5648_case__prod__beta,axiom,
    ! [F: fun_in1696471567231710611nt_int,P2: product_prod_int_int] : aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(F),P2) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(F,aa_Pro6799258434465214121nt_int(product_fst_int_int,P2)),aa_Pro6799258434465214121nt_int(product_snd_int_int,P2)) ).

% case_prod_beta
tff(fact_5649_case__prod__beta,axiom,
    ! [F: fun_int_fun_int_o,P2: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(F),P2)
    <=> aa_int_o(aa_int_fun_int_o(F,aa_Pro6799258434465214121nt_int(product_fst_int_int,P2)),aa_Pro6799258434465214121nt_int(product_snd_int_int,P2)) ) ).

% case_prod_beta
tff(fact_5650_case__prod__beta,axiom,
    ! [F: fun_int_fun_int_int,P2: product_prod_int_int] : aa_Pro6799258434465214121nt_int(produc8211389475949308722nt_int(F),P2) = aa_int_int(aa_int_fun_int_int(F,aa_Pro6799258434465214121nt_int(product_fst_int_int,P2)),aa_Pro6799258434465214121nt_int(product_snd_int_int,P2)) ).

% case_prod_beta
tff(fact_5651_mod__star__trueE,axiom,
    ! [P: assn,H: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(rep_assn(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,P),top_top_assn)),H)
     => ~ ! [H5: produc3658429121746597890et_nat] : ~ aa_Pro653082453696054616_nat_o(rep_assn(P),H5) ) ).

% mod_star_trueE
tff(fact_5652_mod__star__trueI,axiom,
    ! [P: assn,H: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(rep_assn(P),H)
     => aa_Pro653082453696054616_nat_o(rep_assn(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,P),top_top_assn)),H) ) ).

% mod_star_trueI
tff(fact_5653_case__prod__rule,axiom,
    ! [Xb: produc4788131091073775919nt_int,P: assn,F: fun_fu136016935472200802ap_nat,Q: fun_nat_assn] :
      ( ! [A3: fun_in5730379987311621695e_term,B2: product_prod_int_int] :
          ( ( Xb = aa_Pro8800328608558162860nt_int(produc4729913460834754785nt_int(A3),B2) )
         => hoare_3067605981109127869le_nat(P,aa_Pro7829970176022100127ap_nat(aa_fun6257319592574587555ap_nat(F,A3),B2),Q) )
     => hoare_3067605981109127869le_nat(P,produc6431702567826247886ap_nat(F,Xb),Q) ) ).

% case_prod_rule
tff(fact_5654_case__prod__rule,axiom,
    ! [Xb: produc4788131091073775919nt_int,P: assn,F: fun_fu5406852889748164040Heap_o,Q: fun_o_assn] :
      ( ! [A3: fun_in5730379987311621695e_term,B2: product_prod_int_int] :
          ( ( Xb = aa_Pro8800328608558162860nt_int(produc4729913460834754785nt_int(A3),B2) )
         => hoare_hoare_triple_o(P,aa_Pro8097318296274772255Heap_o(aa_fun717413422199439153Heap_o(F,A3),B2),Q) )
     => hoare_hoare_triple_o(P,produc6980626716110614320Heap_o(F,Xb),Q) ) ).

% case_prod_rule
tff(fact_5655_case__prod__rule,axiom,
    ! [Xb: produc4788131091073775919nt_int,P: assn,F: fun_fu7458074433144620999_VEBTi,Q: fun_VEBT_VEBTi_assn] :
      ( ! [A3: fun_in5730379987311621695e_term,B2: product_prod_int_int] :
          ( ( Xb = aa_Pro8800328608558162860nt_int(produc4729913460834754785nt_int(A3),B2) )
         => hoare_1429296392585015714_VEBTi(P,aa_Pro3831758596280224260_VEBTi(aa_fun8292477295915508872_VEBTi(F,A3),B2),Q) )
     => hoare_1429296392585015714_VEBTi(P,produc6514969129755825971_VEBTi(F,Xb),Q) ) ).

% case_prod_rule
tff(fact_5656_case__prod__rule,axiom,
    ! [Xb: produc4788131091073775919nt_int,P: assn,F: fun_fu2816820776361429565t_unit,Q: fun_Pr6626665806141890376t_assn] :
      ( ! [A3: fun_in5730379987311621695e_term,B2: product_prod_int_int] :
          ( ( Xb = aa_Pro8800328608558162860nt_int(produc4729913460834754785nt_int(A3),B2) )
         => hoare_8945653483474564448t_unit(P,aa_Pro5189747760640633620t_unit(aa_fun2226484538892810278t_unit(F,A3),B2),Q) )
     => hoare_8945653483474564448t_unit(P,produc1338229256628767653t_unit(F,Xb),Q) ) ).

% case_prod_rule
tff(fact_5657_case__prod__rule,axiom,
    ! [Xb: produc4788131091073775919nt_int,P: assn,F: fun_fu1311000841445275698on_nat,Q: fun_option_nat_assn] :
      ( ! [A3: fun_in5730379987311621695e_term,B2: product_prod_int_int] :
          ( ( Xb = aa_Pro8800328608558162860nt_int(produc4729913460834754785nt_int(A3),B2) )
         => hoare_7629718768684598413on_nat(P,aa_Pro185635222820580335on_nat(aa_fun2956517174233687155on_nat(F,A3),B2),Q) )
     => hoare_7629718768684598413on_nat(P,produc7781486621487665566on_nat(F,Xb),Q) ) ).

% case_prod_rule
tff(fact_5658_case__prod__rule,axiom,
    ! [Xb: produc8799533983703829792et_nat,P: assn,F: fun_fu3935500115053699655ap_nat,Q: fun_nat_assn] :
      ( ! [A3: fun_Pr6460944699146269935_nat_o,B2: produc3658429121746597890et_nat] :
          ( ( Xb = aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(A3),B2) )
         => hoare_3067605981109127869le_nat(P,aa_Pro8061417612792635676ap_nat(aa_fun4763701871219163200ap_nat(F,A3),B2),Q) )
     => hoare_3067605981109127869le_nat(P,produc5378093055270544597ap_nat(F,Xb),Q) ) ).

% case_prod_rule
tff(fact_5659_case__prod__rule,axiom,
    ! [Xb: produc8799533983703829792et_nat,P: assn,F: fun_fu4848728041736279203Heap_o,Q: fun_o_assn] :
      ( ! [A3: fun_Pr6460944699146269935_nat_o,B2: produc3658429121746597890et_nat] :
          ( ( Xb = aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(A3),B2) )
         => hoare_hoare_triple_o(P,aa_Pro4765807838566354786Heap_o(aa_fun4444287950952374356Heap_o(F,A3),B2),Q) )
     => hoare_hoare_triple_o(P,produc2235927729741437801Heap_o(F,Xb),Q) ) ).

% case_prod_rule
tff(fact_5660_case__prod__rule,axiom,
    ! [Xb: produc8799533983703829792et_nat,P: assn,F: fun_fu7290308691160913068_VEBTi,Q: fun_VEBT_VEBTi_assn] :
      ( ! [A3: fun_Pr6460944699146269935_nat_o,B2: produc3658429121746597890et_nat] :
          ( ( Xb = aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(A3),B2) )
         => hoare_1429296392585015714_VEBTi(P,aa_Pro5789314108598051201_VEBTi(aa_fun7882737301604475173_VEBTi(F,A3),B2),Q) )
     => hoare_1429296392585015714_VEBTi(P,produc2318796205394716730_VEBTi(F,Xb),Q) ) ).

% case_prod_rule
tff(fact_5661_case__prod__rule,axiom,
    ! [Xb: produc8799533983703829792et_nat,P: assn,F: fun_fu1555771386458762392t_unit,Q: fun_Pr6626665806141890376t_assn] :
      ( ! [A3: fun_Pr6460944699146269935_nat_o,B2: produc3658429121746597890et_nat] :
          ( ( Xb = aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(A3),B2) )
         => hoare_8945653483474564448t_unit(P,aa_Pro5593527670638803159t_unit(aa_fun3290195026996439241t_unit(F,A3),B2),Q) )
     => hoare_8945653483474564448t_unit(P,produc8701330506106629726t_unit(F,Xb),Q) ) ).

% case_prod_rule
tff(fact_5662_case__prod__rule,axiom,
    ! [Xb: produc8799533983703829792et_nat,P: assn,F: fun_fu5645185405084508823on_nat,Q: fun_option_nat_assn] :
      ( ! [A3: fun_Pr6460944699146269935_nat_o,B2: produc3658429121746597890et_nat] :
          ( ( Xb = aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(A3),B2) )
         => hoare_7629718768684598413on_nat(P,aa_Pro620577756486784236on_nat(aa_fun1757556057354134160on_nat(F,A3),B2),Q) )
     => hoare_7629718768684598413on_nat(P,produc5392750298291506981on_nat(F,Xb),Q) ) ).

% case_prod_rule
tff(fact_5663_case__prod__beta_H,axiom,
    ! [F: fun_na4138050610385705875at_nat,X3: product_prod_nat_nat] : aa_Pro262675715343573244at_nat(produc2626176000494625587at_nat(F),X3) = aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(F,aa_Pro4243173770044397973at_nat(product_fst_nat_nat,X3)),aa_Pro4243173770044397973at_nat(product_snd_nat_nat,X3)) ).

% case_prod_beta'
tff(fact_5664_case__prod__beta_H,axiom,
    ! [F: fun_nat_fun_nat_o,X3: product_prod_nat_nat] :
      ( aa_Pro6150637604445964947_nat_o(produc6081775807080527818_nat_o(F),X3)
    <=> aa_nat_o(aa_nat_fun_nat_o(F,aa_Pro4243173770044397973at_nat(product_fst_nat_nat,X3)),aa_Pro4243173770044397973at_nat(product_snd_nat_nat,X3)) ) ).

% case_prod_beta'
tff(fact_5665_case__prod__beta_H,axiom,
    ! [F: fun_in1696471567231710611nt_int,X3: product_prod_int_int] : aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(F),X3) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(F,aa_Pro6799258434465214121nt_int(product_fst_int_int,X3)),aa_Pro6799258434465214121nt_int(product_snd_int_int,X3)) ).

% case_prod_beta'
tff(fact_5666_case__prod__beta_H,axiom,
    ! [F: fun_int_fun_int_o,X3: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(F),X3)
    <=> aa_int_o(aa_int_fun_int_o(F,aa_Pro6799258434465214121nt_int(product_fst_int_int,X3)),aa_Pro6799258434465214121nt_int(product_snd_int_int,X3)) ) ).

% case_prod_beta'
tff(fact_5667_case__prod__beta_H,axiom,
    ! [F: fun_int_fun_int_int,X3: product_prod_int_int] : aa_Pro6799258434465214121nt_int(produc8211389475949308722nt_int(F),X3) = aa_int_int(aa_int_fun_int_int(F,aa_Pro6799258434465214121nt_int(product_fst_int_int,X3)),aa_Pro6799258434465214121nt_int(product_snd_int_int,X3)) ).

% case_prod_beta'
tff(fact_5668_case__prod__unfold,axiom,
    ! [X3: fun_na4138050610385705875at_nat,Xa: product_prod_nat_nat] : aa_Pro262675715343573244at_nat(produc2626176000494625587at_nat(X3),Xa) = aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(X3,aa_Pro4243173770044397973at_nat(product_fst_nat_nat,Xa)),aa_Pro4243173770044397973at_nat(product_snd_nat_nat,Xa)) ).

% case_prod_unfold
tff(fact_5669_case__prod__unfold,axiom,
    ! [X3: fun_nat_fun_nat_o,Xa: product_prod_nat_nat] :
      ( aa_Pro6150637604445964947_nat_o(produc6081775807080527818_nat_o(X3),Xa)
    <=> aa_nat_o(aa_nat_fun_nat_o(X3,aa_Pro4243173770044397973at_nat(product_fst_nat_nat,Xa)),aa_Pro4243173770044397973at_nat(product_snd_nat_nat,Xa)) ) ).

% case_prod_unfold
tff(fact_5670_case__prod__unfold,axiom,
    ! [X3: fun_in1696471567231710611nt_int,Xa: product_prod_int_int] : aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(X3),Xa) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(X3,aa_Pro6799258434465214121nt_int(product_fst_int_int,Xa)),aa_Pro6799258434465214121nt_int(product_snd_int_int,Xa)) ).

% case_prod_unfold
tff(fact_5671_case__prod__unfold,axiom,
    ! [X3: fun_int_fun_int_o,Xa: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(X3),Xa)
    <=> aa_int_o(aa_int_fun_int_o(X3,aa_Pro6799258434465214121nt_int(product_fst_int_int,Xa)),aa_Pro6799258434465214121nt_int(product_snd_int_int,Xa)) ) ).

% case_prod_unfold
tff(fact_5672_case__prod__unfold,axiom,
    ! [X3: fun_int_fun_int_int,Xa: product_prod_int_int] : aa_Pro6799258434465214121nt_int(produc8211389475949308722nt_int(X3),Xa) = aa_int_int(aa_int_fun_int_int(X3,aa_Pro6799258434465214121nt_int(product_fst_int_int,Xa)),aa_Pro6799258434465214121nt_int(product_snd_int_int,Xa)) ).

% case_prod_unfold
tff(fact_5673_prod_Osplit__sel,axiom,
    ! [P: fun_Pr4796943543331071514_nat_o,F: fun_na4138050610385705875at_nat,Prod: product_prod_nat_nat] :
      ( aa_Pro6150637604445964947_nat_o(P,aa_Pro262675715343573244at_nat(produc2626176000494625587at_nat(F),Prod))
    <=> ( ( Prod = aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,aa_Pro4243173770044397973at_nat(product_fst_nat_nat,Prod)),aa_Pro4243173770044397973at_nat(product_snd_nat_nat,Prod)) )
       => aa_Pro6150637604445964947_nat_o(P,aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(F,aa_Pro4243173770044397973at_nat(product_fst_nat_nat,Prod)),aa_Pro4243173770044397973at_nat(product_snd_nat_nat,Prod))) ) ) ).

% prod.split_sel
tff(fact_5674_prod_Osplit__sel,axiom,
    ! [P: fun_o_o,F: fun_nat_fun_nat_o,Prod: product_prod_nat_nat] :
      ( aa_o_o(P,aa_Pro6150637604445964947_nat_o(produc6081775807080527818_nat_o(F),Prod))
    <=> ( ( Prod = aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,aa_Pro4243173770044397973at_nat(product_fst_nat_nat,Prod)),aa_Pro4243173770044397973at_nat(product_snd_nat_nat,Prod)) )
       => aa_o_o(P,aa_nat_o(aa_nat_fun_nat_o(F,aa_Pro4243173770044397973at_nat(product_fst_nat_nat,Prod)),aa_Pro4243173770044397973at_nat(product_snd_nat_nat,Prod))) ) ) ).

% prod.split_sel
tff(fact_5675_prod_Osplit__sel,axiom,
    ! [P: fun_Pr3238374608701176162_int_o,F: fun_in1696471567231710611nt_int,Prod: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(P,aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(F),Prod))
    <=> ( ( Prod = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_Pro6799258434465214121nt_int(product_fst_int_int,Prod)),aa_Pro6799258434465214121nt_int(product_snd_int_int,Prod)) )
       => aa_Pro4592068669816069595_int_o(P,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(F,aa_Pro6799258434465214121nt_int(product_fst_int_int,Prod)),aa_Pro6799258434465214121nt_int(product_snd_int_int,Prod))) ) ) ).

% prod.split_sel
tff(fact_5676_prod_Osplit__sel,axiom,
    ! [P: fun_o_o,F: fun_int_fun_int_o,Prod: product_prod_int_int] :
      ( aa_o_o(P,aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(F),Prod))
    <=> ( ( Prod = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_Pro6799258434465214121nt_int(product_fst_int_int,Prod)),aa_Pro6799258434465214121nt_int(product_snd_int_int,Prod)) )
       => aa_o_o(P,aa_int_o(aa_int_fun_int_o(F,aa_Pro6799258434465214121nt_int(product_fst_int_int,Prod)),aa_Pro6799258434465214121nt_int(product_snd_int_int,Prod))) ) ) ).

% prod.split_sel
tff(fact_5677_prod_Osplit__sel,axiom,
    ! [P: fun_int_o,F: fun_int_fun_int_int,Prod: product_prod_int_int] :
      ( aa_int_o(P,aa_Pro6799258434465214121nt_int(produc8211389475949308722nt_int(F),Prod))
    <=> ( ( Prod = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_Pro6799258434465214121nt_int(product_fst_int_int,Prod)),aa_Pro6799258434465214121nt_int(product_snd_int_int,Prod)) )
       => aa_int_o(P,aa_int_int(aa_int_fun_int_int(F,aa_Pro6799258434465214121nt_int(product_fst_int_int,Prod)),aa_Pro6799258434465214121nt_int(product_snd_int_int,Prod))) ) ) ).

% prod.split_sel
tff(fact_5678_prod_Osplit__sel__asm,axiom,
    ! [P: fun_Pr4796943543331071514_nat_o,F: fun_na4138050610385705875at_nat,Prod: product_prod_nat_nat] :
      ( aa_Pro6150637604445964947_nat_o(P,aa_Pro262675715343573244at_nat(produc2626176000494625587at_nat(F),Prod))
    <=> ~ ( ( Prod = aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,aa_Pro4243173770044397973at_nat(product_fst_nat_nat,Prod)),aa_Pro4243173770044397973at_nat(product_snd_nat_nat,Prod)) )
          & ~ aa_Pro6150637604445964947_nat_o(P,aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(F,aa_Pro4243173770044397973at_nat(product_fst_nat_nat,Prod)),aa_Pro4243173770044397973at_nat(product_snd_nat_nat,Prod))) ) ) ).

% prod.split_sel_asm
tff(fact_5679_prod_Osplit__sel__asm,axiom,
    ! [P: fun_o_o,F: fun_nat_fun_nat_o,Prod: product_prod_nat_nat] :
      ( aa_o_o(P,aa_Pro6150637604445964947_nat_o(produc6081775807080527818_nat_o(F),Prod))
    <=> ~ ( ( Prod = aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,aa_Pro4243173770044397973at_nat(product_fst_nat_nat,Prod)),aa_Pro4243173770044397973at_nat(product_snd_nat_nat,Prod)) )
          & ~ aa_o_o(P,aa_nat_o(aa_nat_fun_nat_o(F,aa_Pro4243173770044397973at_nat(product_fst_nat_nat,Prod)),aa_Pro4243173770044397973at_nat(product_snd_nat_nat,Prod))) ) ) ).

% prod.split_sel_asm
tff(fact_5680_prod_Osplit__sel__asm,axiom,
    ! [P: fun_Pr3238374608701176162_int_o,F: fun_in1696471567231710611nt_int,Prod: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(P,aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(F),Prod))
    <=> ~ ( ( Prod = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_Pro6799258434465214121nt_int(product_fst_int_int,Prod)),aa_Pro6799258434465214121nt_int(product_snd_int_int,Prod)) )
          & ~ aa_Pro4592068669816069595_int_o(P,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(F,aa_Pro6799258434465214121nt_int(product_fst_int_int,Prod)),aa_Pro6799258434465214121nt_int(product_snd_int_int,Prod))) ) ) ).

% prod.split_sel_asm
tff(fact_5681_prod_Osplit__sel__asm,axiom,
    ! [P: fun_o_o,F: fun_int_fun_int_o,Prod: product_prod_int_int] :
      ( aa_o_o(P,aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(F),Prod))
    <=> ~ ( ( Prod = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_Pro6799258434465214121nt_int(product_fst_int_int,Prod)),aa_Pro6799258434465214121nt_int(product_snd_int_int,Prod)) )
          & ~ aa_o_o(P,aa_int_o(aa_int_fun_int_o(F,aa_Pro6799258434465214121nt_int(product_fst_int_int,Prod)),aa_Pro6799258434465214121nt_int(product_snd_int_int,Prod))) ) ) ).

% prod.split_sel_asm
tff(fact_5682_prod_Osplit__sel__asm,axiom,
    ! [P: fun_int_o,F: fun_int_fun_int_int,Prod: product_prod_int_int] :
      ( aa_int_o(P,aa_Pro6799258434465214121nt_int(produc8211389475949308722nt_int(F),Prod))
    <=> ~ ( ( Prod = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_Pro6799258434465214121nt_int(product_fst_int_int,Prod)),aa_Pro6799258434465214121nt_int(product_snd_int_int,Prod)) )
          & ~ aa_int_o(P,aa_int_int(aa_int_fun_int_int(F,aa_Pro6799258434465214121nt_int(product_fst_int_int,Prod)),aa_Pro6799258434465214121nt_int(product_snd_int_int,Prod))) ) ) ).

% prod.split_sel_asm
tff(fact_5683_mod__h__bot__iff_I2_J,axiom,
    ! [H: heap_e7401611519738050253t_unit] : aa_Pro653082453696054616_nat_o(rep_assn(top_top_assn),aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(H),bot_bot_set_nat)) ).

% mod_h_bot_iff(2)
tff(fact_5684_mod__star__trueE_H,axiom,
    ! [P: assn,H: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(rep_assn(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,P),top_top_assn)),H)
     => ~ ! [H5: produc3658429121746597890et_nat] :
            ( ( aa_Pro1219312880135979149t_unit(produc1824681642469235216et_nat,H5) = aa_Pro1219312880135979149t_unit(produc1824681642469235216et_nat,H) )
           => ( aa_set_nat_o(ord_less_eq_set_nat(aa_Pro3416150927818442182et_nat(produc8586169260539613262et_nat,H5)),aa_Pro3416150927818442182et_nat(produc8586169260539613262et_nat,H))
             => ~ aa_Pro653082453696054616_nat_o(rep_assn(P),H5) ) ) ) ).

% mod_star_trueE'
tff(fact_5685_VEBTi_Osize__gen_I2_J,axiom,
    ! [X21: $o,X22: $o] : aa_VEBT_VEBTi_nat(vEBT_size_VEBTi,vEBT_Leafi((X21),(X22))) = zero_zero_nat ).

% VEBTi.size_gen(2)
tff(fact_5686_VEBT__internal_Ocnt_Osimps_I2_J,axiom,
    ! [Info: option4927543243414619207at_nat,Deg: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT] : aa_VEBT_VEBT_real(vEBT_VEBT_cnt,aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info,Deg,TreeLista),Summarya)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_VEBT_VEBT_real(vEBT_VEBT_cnt,Summarya))),aa_real_real(foldr_real_real(plus_plus_real,map_VEBT_VEBT_real(vEBT_VEBT_cnt,TreeLista)),zero_zero_real)) ).

% VEBT_internal.cnt.simps(2)
tff(fact_5687_VEBT__internal_Ocnt_Oelims,axiom,
    ! [Xb: vEBT_VEBT,Y: real] :
      ( ( aa_VEBT_VEBT_real(vEBT_VEBT_cnt,Xb) = Y )
     => ( ( ? [A3: $o,B2: $o] : Xb = vEBT_Leaf((A3),(B2))
         => ( Y != one_one_real ) )
       => ~ ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
              ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,Deg2,TreeList),Summary) )
             => ( Y != aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_VEBT_VEBT_real(vEBT_VEBT_cnt,Summary))),aa_real_real(foldr_real_real(plus_plus_real,map_VEBT_VEBT_real(vEBT_VEBT_cnt,TreeList)),zero_zero_real)) ) ) ) ) ).

% VEBT_internal.cnt.elims
tff(fact_5688_vebt__memberi__refines,axiom,
    ! [Tib: vEBT_VEBTi,Xb: nat,Ta: vEBT_VEBT] : aa_Hea4522166450600650424ap_o_o(refine_Imp_refines_o(aa_nat2910048432523430448Heap_o(aa_VEB1455453220190609250Heap_o(vEBT_vebt_memberi,Tib),Xb)),aa_nat2910048432523430448Heap_o(aa_VEB1455453220190609250Heap_o(aa_VEB8672304581938048793Heap_o(vEBT_V854960066525838166emberi,Ta),Tib),Xb)) ).

% vebt_memberi_refines
tff(fact_5689_of__nat__code__if,axiom,
    ! [Nb: nat] :
      aa_nat_uint32(semiri2565882477558803405uint32,Nb) = $ite(Nb = zero_zero_nat,zero_zero_uint32,produc2417093276151063866uint32(aTP_Lamm_lz,aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(divmod_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% of_nat_code_if
tff(fact_5690_of__nat__code__if,axiom,
    ! [Nb: nat] :
      aa_nat5599461682457136004l_num1(semiri8819519690708144855l_num1,Nb) = $ite(Nb = zero_zero_nat,zero_z3563351764282998399l_num1,produc6192303373133366212l_num1(aTP_Lamm_ma,aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(divmod_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% of_nat_code_if
tff(fact_5691_of__nat__code__if,axiom,
    ! [Nb: nat] :
      aa_nat_rat(semiri681578069525770553at_rat,Nb) = $ite(Nb = zero_zero_nat,zero_zero_rat,produc6207742614233964070at_rat(aTP_Lamm_mb,aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(divmod_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% of_nat_code_if
tff(fact_5692_of__nat__code__if,axiom,
    ! [Nb: nat] :
      aa_nat_real(semiri5074537144036343181t_real,Nb) = $ite(Nb = zero_zero_nat,zero_zero_real,produc1703576794950452218t_real(aTP_Lamm_mc,aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(divmod_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% of_nat_code_if
tff(fact_5693_of__nat__code__if,axiom,
    ! [Nb: nat] :
      aa_nat_int(semiri1314217659103216013at_int,Nb) = $ite(Nb = zero_zero_nat,zero_zero_int,produc6840382203811409530at_int(aTP_Lamm_md,aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(divmod_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% of_nat_code_if
tff(fact_5694_of__nat__code__if,axiom,
    ! [Nb: nat] :
      aa_nat_nat(semiri1316708129612266289at_nat,Nb) = $ite(Nb = zero_zero_nat,zero_zero_nat,aa_Pro4243173770044397973at_nat(produc6842872674320459806at_nat(aTP_Lamm_me),aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(divmod_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% of_nat_code_if
tff(fact_5695_of__nat__code__if,axiom,
    ! [Nb: nat] :
      aa_nat_Code_integer(semiri4939895301339042750nteger,Nb) = $ite(Nb = zero_zero_nat,zero_z3403309356797280102nteger,produc1830744345554046123nteger(aTP_Lamm_mf,aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(divmod_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% of_nat_code_if
tff(fact_5696_vebt__succi__refines,axiom,
    ! [Tib: vEBT_VEBTi,Xb: nat,Ta: vEBT_VEBT] : refine7594492741263601813on_nat(aa_nat947733127081326238on_nat(aa_VEB3566417204837707778on_nat(vEBT_vebt_succi,Tib),Xb),aa_nat947733127081326238on_nat(aa_VEB3566417204837707778on_nat(aa_VEB1030146501360450209on_nat(vEBT_VEBT_vebt_succi,Ta),Tib),Xb)) ).

% vebt_succi_refines
tff(fact_5697_round__unique,axiom,
    ! [Xb: real,Y: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(aa_num_num(bit0,one))))),aa_int_real(ring_1_of_int_real,Y))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Y)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(aa_num_num(bit0,one)))))
       => ( aa_real_int(archim8280529875227126926d_real,Xb) = Y ) ) ) ).

% round_unique
tff(fact_5698_round__unique,axiom,
    ! [Xb: rat,Y: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,minus_minus_rat(Xb,divide_divide_rat(one_one_rat,numeral_numeral_rat(aa_num_num(bit0,one))))),aa_int_rat(ring_1_of_int_rat,Y))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Y)),aa_rat_rat(plus_plus_rat(Xb),divide_divide_rat(one_one_rat,numeral_numeral_rat(aa_num_num(bit0,one)))))
       => ( archim7778729529865785530nd_rat(Xb) = Y ) ) ) ).

% round_unique
tff(fact_5699_mult__le__cancel__iff1,axiom,
    ! [Z: real,Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Z)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Z)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Z))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y) ) ) ).

% mult_le_cancel_iff1
tff(fact_5700_mult__le__cancel__iff1,axiom,
    ! [Z: rat,Xb: 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(times_times_rat(Xb),Z)),aa_rat_rat(times_times_rat(Y),Z))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Y) ) ) ).

% mult_le_cancel_iff1
tff(fact_5701_mult__le__cancel__iff1,axiom,
    ! [Z: int,Xb: 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,Xb),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,Xb),Y) ) ) ).

% mult_le_cancel_iff1
tff(fact_5702_refines__replicate,axiom,
    ! [F: heap_Time_Heap_o,F3: heap_Time_Heap_o,Nb: nat] :
      ( aa_Hea4522166450600650424ap_o_o(refine_Imp_refines_o(F),F3)
     => refine5896690332125372649list_o(vEBT_V2326993469660664182atei_o(Nb,F),vEBT_V2326993469660664182atei_o(Nb,F3)) ) ).

% refines_replicate
tff(fact_5703_refines__replicate,axiom,
    ! [F: heap_T2636463487746394924on_nat,F3: heap_T2636463487746394924on_nat,Nb: nat] :
      ( refine7594492741263601813on_nat(F,F3)
     => refine1935026298455697829on_nat(vEBT_V792416675989592002on_nat(Nb,F),vEBT_V792416675989592002on_nat(Nb,F3)) ) ).

% refines_replicate
tff(fact_5704_refines__replicate,axiom,
    ! [F: heap_T8145700208782473153_VEBTi,F3: heap_T8145700208782473153_VEBTi,Nb: nat] :
      ( refine5565527176597971370_VEBTi(F,F3)
     => refine3700189196150522554_VEBTi(vEBT_V1859673955506687831_VEBTi(Nb,F),vEBT_V1859673955506687831_VEBTi(Nb,F3)) ) ).

% refines_replicate
tff(fact_5705_refines__case__VEBTi,axiom,
    ! [Tib: vEBT_VEBTi,Ti: vEBT_VEBTi,F1: fun_o_4049874799390722306on_nat,F12: fun_o_4049874799390722306on_nat,F22: fun_op5783775701855621896on_nat,F23: fun_op5783775701855621896on_nat] :
      ( ( Tib = Ti )
     => ( ! [A3: $o,B2: $o] : refine7594492741263601813on_nat(aa_o_H4215342640551218896on_nat(aa_o_f5606600903069576627on_nat(F1,(A3)),(B2)),aa_o_H4215342640551218896on_nat(aa_o_f5606600903069576627on_nat(F12,(A3)),(B2)))
       => ( ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeArray: array_VEBT_VEBTi,Summary: vEBT_VEBTi] : refine7594492741263601813on_nat(aa_VEB2711320435582112505on_nat(aa_arr2955191963898321370on_nat(aa_nat5317369504323972463on_nat(aa_opt8624683241416710969on_nat(F22,Info2),Deg2),TreeArray),Summary),aa_VEB2711320435582112505on_nat(aa_arr2955191963898321370on_nat(aa_nat5317369504323972463on_nat(aa_opt8624683241416710969on_nat(F23,Info2),Deg2),TreeArray),Summary))
         => refine7594492741263601813on_nat(aa_VEB2711320435582112505on_nat(vEBT_c6250501799366334488on_nat(F22,F1),Tib),aa_VEB2711320435582112505on_nat(vEBT_c6250501799366334488on_nat(F23,F12),Ti)) ) ) ) ).

% refines_case_VEBTi
tff(fact_5706_refines__case__VEBTi,axiom,
    ! [Tib: vEBT_VEBTi,Ti: vEBT_VEBTi,F1: fun_o_6972675622308511895_VEBTi,F12: fun_o_6972675622308511895_VEBTi,F22: fun_op5127434927081926045_VEBTi,F23: fun_op5127434927081926045_VEBTi] :
      ( ( Tib = Ti )
     => ( ! [A3: $o,B2: $o] : refine5565527176597971370_VEBTi(aa_o_H7577119287286566245_VEBTi(aa_o_f5262348187908705736_VEBTi(F1,(A3)),(B2)),aa_o_H7577119287286566245_VEBTi(aa_o_f5262348187908705736_VEBTi(F12,(A3)),(B2)))
       => ( ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeArray: array_VEBT_VEBTi,Summary: vEBT_VEBTi] : refine5565527176597971370_VEBTi(aa_VEB836174949890266638_VEBTi(aa_arr2912372873376718191_VEBTi(aa_nat6445230840460233604_VEBTi(aa_opt5182041009420754510_VEBTi(F22,Info2),Deg2),TreeArray),Summary),aa_VEB836174949890266638_VEBTi(aa_arr2912372873376718191_VEBTi(aa_nat6445230840460233604_VEBTi(aa_opt5182041009420754510_VEBTi(F23,Info2),Deg2),TreeArray),Summary))
         => refine5565527176597971370_VEBTi(aa_VEB836174949890266638_VEBTi(vEBT_c6028912655521741485_VEBTi(F22,F1),Tib),aa_VEB836174949890266638_VEBTi(vEBT_c6028912655521741485_VEBTi(F23,F12),Ti)) ) ) ) ).

% refines_case_VEBTi
tff(fact_5707_refines__case__VEBTi,axiom,
    ! [Tib: vEBT_VEBTi,Ti: vEBT_VEBTi,F1: fun_o_1207901889202581752Heap_o,F12: fun_o_1207901889202581752Heap_o,F22: fun_op1763656958002597022Heap_o,F23: fun_op1763656958002597022Heap_o] :
      ( ( Tib = Ti )
     => ( ! [A3: $o,B2: $o] : aa_Hea4522166450600650424ap_o_o(refine_Imp_refines_o(aa_o_H7399033255965945470Heap_o(aa_o_f869145629572826609Heap_o(F1,(A3)),(B2))),aa_o_H7399033255965945470Heap_o(aa_o_f869145629572826609Heap_o(F12,(A3)),(B2)))
       => ( ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeArray: array_VEBT_VEBTi,Summary: vEBT_VEBTi] : aa_Hea4522166450600650424ap_o_o(refine_Imp_refines_o(aa_VEB8194921341634505877Heap_o(aa_arr2373296168907467658Heap_o(aa_nat8650107858160254923Heap_o(aa_opt6539295502271640599Heap_o(F22,Info2),Deg2),TreeArray),Summary)),aa_VEB8194921341634505877Heap_o(aa_arr2373296168907467658Heap_o(aa_nat8650107858160254923Heap_o(aa_opt6539295502271640599Heap_o(F23,Info2),Deg2),TreeArray),Summary))
         => aa_Hea4522166450600650424ap_o_o(refine_Imp_refines_o(aa_VEB8194921341634505877Heap_o(vEBT_c6104975476656191286Heap_o(F22,F1),Tib)),aa_VEB8194921341634505877Heap_o(vEBT_c6104975476656191286Heap_o(F23,F12),Ti)) ) ) ) ).

% refines_case_VEBTi
tff(fact_5708_case__prodI2,axiom,
    ! [P2: produc2453742332425399446nteger,C: fun_fu7695273699059480439eger_o] :
      ( ! [A3: fun_Pr5098669352780829690e_term,B2: produc8923325533196201883nteger] :
          ( ( P2 = aa_Pro4291442876242880245nteger(produc565448451787809096nteger(A3),B2) )
         => aa_Pro4926521453222654329eger_o(aa_fun4505422719058378424eger_o(C,A3),B2) )
     => aa_Pro4148064003102580676eger_o(produc7373287791920631523eger_o(C),P2) ) ).

% case_prodI2
tff(fact_5709_case__prodI2,axiom,
    ! [P2: produc8799533983703829792et_nat,C: fun_fu8419763957888848259_nat_o] :
      ( ! [A3: fun_Pr6460944699146269935_nat_o,B2: produc3658429121746597890et_nat] :
          ( ( P2 = aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(A3),B2) )
         => aa_Pro653082453696054616_nat_o(aa_fun2638059832795814268_nat_o(C,A3),B2) )
     => aa_Pro5643009357358181178_nat_o(produc2915205735004773393_nat_o(C),P2) ) ).

% case_prodI2
tff(fact_5710_case__prodI2,axiom,
    ! [P2: produc3112576180687729086et_nat,C: fun_fu2611741329380808165_nat_o] :
      ( ! [A3: fun_Pr6460944699146269935_nat_o,B2: produc8799533983703829792et_nat] :
          ( ( P2 = aa_Pro1979504083478021276et_nat(produc7291855603632349880et_nat(A3),B2) )
         => aa_Pro5643009357358181178_nat_o(aa_fun912680268133416158_nat_o(C,A3),B2) )
     => aa_Pro2671238023309943196_nat_o(produc4465912260561112435_nat_o(C),P2) ) ).

% case_prodI2
tff(fact_5711_case__prodI2,axiom,
    ! [P2: produc3043004577601930755nt_int,C: fun_fu6865642474282033866_int_o] :
      ( ! [A3: fun_Pr7763680713840462443e_term,B2: product_prod_int_int] :
          ( ( P2 = aa_Pro5748361610964203648nt_int(produc7320969046608631221nt_int(A3),B2) )
         => aa_Pro4592068669816069595_int_o(aa_fun3538162476766722827_int_o(C,A3),B2) )
     => aa_Pro2601676399929874711_int_o(produc2207054017359051062_int_o(C),P2) ) ).

% case_prodI2
tff(fact_5712_case__prodI2,axiom,
    ! [P2: produc4788131091073775919nt_int,C: fun_fu2978539155105838238_int_o] :
      ( ! [A3: fun_in5730379987311621695e_term,B2: product_prod_int_int] :
          ( ( P2 = aa_Pro8800328608558162860nt_int(produc4729913460834754785nt_int(A3),B2) )
         => aa_Pro4592068669816069595_int_o(aa_fun586359091965693151_int_o(C,A3),B2) )
     => aa_Pro5451197003417746155_int_o(produc7271678899836001034_int_o(C),P2) ) ).

% case_prodI2
tff(fact_5713_case__prodI2,axiom,
    ! [P2: product_prod_nat_nat,C: fun_nat_fun_nat_o] :
      ( ! [A3: nat,B2: nat] :
          ( ( P2 = aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,A3),B2) )
         => aa_nat_o(aa_nat_fun_nat_o(C,A3),B2) )
     => aa_Pro6150637604445964947_nat_o(produc6081775807080527818_nat_o(C),P2) ) ).

% case_prodI2
tff(fact_5714_case__prodI2,axiom,
    ! [P2: product_prod_int_int,C: fun_int_fun_int_o] :
      ( ! [A3: int,B2: int] :
          ( ( P2 = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A3),B2) )
         => aa_int_o(aa_int_fun_int_o(C,A3),B2) )
     => aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(C),P2) ) ).

% case_prodI2
tff(fact_5715_case__prodI,axiom,
    ! [F: fun_fu7695273699059480439eger_o,A: fun_Pr5098669352780829690e_term,B: produc8923325533196201883nteger] :
      ( aa_Pro4926521453222654329eger_o(aa_fun4505422719058378424eger_o(F,A),B)
     => aa_Pro4148064003102580676eger_o(produc7373287791920631523eger_o(F),aa_Pro4291442876242880245nteger(produc565448451787809096nteger(A),B)) ) ).

% case_prodI
tff(fact_5716_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(produc6017135503889078682et_nat(A),B)) ) ).

% case_prodI
tff(fact_5717_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(produc7291855603632349880et_nat(A),B)) ) ).

% case_prodI
tff(fact_5718_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(produc7320969046608631221nt_int(A),B)) ) ).

% case_prodI
tff(fact_5719_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(produc4729913460834754785nt_int(A),B)) ) ).

% case_prodI
tff(fact_5720_case__prodI,axiom,
    ! [F: fun_nat_fun_nat_o,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(F,A),B)
     => aa_Pro6150637604445964947_nat_o(produc6081775807080527818_nat_o(F),aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,A),B)) ) ).

% case_prodI
tff(fact_5721_case__prodI,axiom,
    ! [F: fun_int_fun_int_o,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(F,A),B)
     => aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(F),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),B)) ) ).

% case_prodI
tff(fact_5722_mem__case__prodI2,axiom,
    ! [P2: produc4788131091073775919nt_int,Z: nat,C: fun_fu9079770784915425324et_nat] :
      ( ! [A3: fun_in5730379987311621695e_term,B2: product_prod_int_int] :
          ( ( P2 = aa_Pro8800328608558162860nt_int(produc4729913460834754785nt_int(A3),B2) )
         => aa_set_nat_o(member_nat(Z),aa_Pro3599192995643711491et_nat(aa_fun8409204002232819093et_nat(C,A3),B2)) )
     => aa_set_nat_o(member_nat(Z),produc5011043943518292116et_nat(C,P2)) ) ).

% mem_case_prodI2
tff(fact_5723_mem__case__prodI2,axiom,
    ! [P2: produc4788131091073775919nt_int,Z: vEBT_VEBT,C: fun_fu7491663322100281242T_VEBT] :
      ( ! [A3: fun_in5730379987311621695e_term,B2: product_prod_int_int] :
          ( ( P2 = aa_Pro8800328608558162860nt_int(produc4729913460834754785nt_int(A3),B2) )
         => aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Z),aa_Pro2768979630923332951T_VEBT(aa_fun2798302631922751451T_VEBT(C,A3),B2)) )
     => aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Z),produc9026699715758669062T_VEBT(C,P2)) ) ).

% mem_case_prodI2
tff(fact_5724_mem__case__prodI2,axiom,
    ! [P2: produc4788131091073775919nt_int,Z: real,C: fun_fu604837885242254216t_real] :
      ( ! [A3: fun_in5730379987311621695e_term,B2: product_prod_int_int] :
          ( ( P2 = aa_Pro8800328608558162860nt_int(produc4729913460834754785nt_int(A3),B2) )
         => aa_set_real_o(member_real(Z),aa_Pro7989103684195733727t_real(aa_fun5197189019902991601t_real(C,A3),B2)) )
     => aa_set_real_o(member_real(Z),produc5863038274793111792t_real(C,P2)) ) ).

% mem_case_prodI2
tff(fact_5725_mem__case__prodI2,axiom,
    ! [P2: produc4788131091073775919nt_int,Z: int,C: fun_fu4266222366127506568et_int] :
      ( ! [A3: fun_in5730379987311621695e_term,B2: product_prod_int_int] :
          ( ( P2 = aa_Pro8800328608558162860nt_int(produc4729913460834754785nt_int(A3),B2) )
         => aa_set_int_o(member_int(Z),aa_Pro8644714012989290591et_int(aa_fun5919120857301092337et_int(C,A3),B2)) )
     => aa_set_int_o(member_int(Z),produc833192924009095408et_int(C,P2)) ) ).

% mem_case_prodI2
tff(fact_5726_mem__case__prodI2,axiom,
    ! [P2: produc8799533983703829792et_nat,Z: nat,C: fun_fu86038545275896455et_nat] :
      ( ! [A3: fun_Pr6460944699146269935_nat_o,B2: produc3658429121746597890et_nat] :
          ( ( P2 = aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(A3),B2) )
         => aa_set_nat_o(member_nat(Z),aa_Pro3416150927818442182et_nat(aa_fun6477889762090788408et_nat(C,A3),B2)) )
     => aa_set_nat_o(member_nat(Z),produc6247649352436904781et_nat(C,P2)) ) ).

% mem_case_prodI2
tff(fact_5727_mem__case__prodI2,axiom,
    ! [P2: produc8799533983703829792et_nat,Z: vEBT_VEBT,C: fun_fu6298409707154110463T_VEBT] :
      ( ! [A3: fun_Pr6460944699146269935_nat_o,B2: produc3658429121746597890et_nat] :
          ( ( P2 = aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(A3),B2) )
         => aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Z),aa_Pro5551276291186086996T_VEBT(aa_fun6431552983927604216T_VEBT(C,A3),B2)) )
     => aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Z),produc1803589779646358669T_VEBT(C,P2)) ) ).

% mem_case_prodI2
tff(fact_5728_mem__case__prodI2,axiom,
    ! [P2: produc8799533983703829792et_nat,Z: real,C: fun_fu2523205402392354915t_real] :
      ( ! [A3: fun_Pr6460944699146269935_nat_o,B2: produc3658429121746597890et_nat] :
          ( ( P2 = aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(A3),B2) )
         => aa_set_real_o(member_real(Z),aa_Pro5145766922317858082t_real(aa_fun7171570044062598164t_real(C,A3),B2)) )
     => aa_set_real_o(member_real(Z),produc2103481464906571561t_real(C,P2)) ) ).

% mem_case_prodI2
tff(fact_5729_mem__case__prodI2,axiom,
    ! [P2: produc8799533983703829792et_nat,Z: int,C: fun_fu4495862163342753507et_int] :
      ( ! [A3: fun_Pr6460944699146269935_nat_o,B2: produc3658429121746597890et_nat] :
          ( ( P2 = aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(A3),B2) )
         => aa_set_int_o(member_int(Z),aa_Pro8461671945164021282et_int(aa_fun3987806617159061652et_int(C,A3),B2)) )
     => aa_set_int_o(member_int(Z),produc2069798332927708073et_int(C,P2)) ) ).

% mem_case_prodI2
tff(fact_5730_mem__case__prodI2,axiom,
    ! [P2: produc2453742332425399446nteger,Z: nat,C: fun_fu2531416508075921811et_nat] :
      ( ! [A3: fun_Pr5098669352780829690e_term,B2: produc8923325533196201883nteger] :
          ( ( P2 = aa_Pro4291442876242880245nteger(produc565448451787809096nteger(A3),B2) )
         => aa_set_nat_o(member_nat(Z),aa_Pro381016180347496421et_nat(aa_fun4088370000251959292et_nat(C,A3),B2)) )
     => aa_set_nat_o(member_nat(Z),produc6164657169593426171et_nat(C,P2)) ) ).

% mem_case_prodI2
tff(fact_5731_mem__case__prodI2,axiom,
    ! [P2: produc2453742332425399446nteger,Z: vEBT_VEBT,C: fun_fu3339394088549833715T_VEBT] :
      ( ! [A3: fun_Pr5098669352780829690e_term,B2: produc8923325533196201883nteger] :
          ( ( P2 = aa_Pro4291442876242880245nteger(produc565448451787809096nteger(A3),B2) )
         => aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Z),aa_Pro2857938020663670261T_VEBT(aa_fun4954175298189341492T_VEBT(C,A3),B2)) )
     => aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Z),produc8725738732899457631T_VEBT(C,P2)) ) ).

% mem_case_prodI2
tff(fact_5732_mem__case__prodI,axiom,
    ! [Z: nat,C: 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(C,A),B))
     => aa_set_nat_o(member_nat(Z),produc5011043943518292116et_nat(C,aa_Pro8800328608558162860nt_int(produc4729913460834754785nt_int(A),B))) ) ).

% mem_case_prodI
tff(fact_5733_mem__case__prodI,axiom,
    ! [Z: vEBT_VEBT,C: fun_fu7491663322100281242T_VEBT,A: fun_in5730379987311621695e_term,B: product_prod_int_int] :
      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Z),aa_Pro2768979630923332951T_VEBT(aa_fun2798302631922751451T_VEBT(C,A),B))
     => aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Z),produc9026699715758669062T_VEBT(C,aa_Pro8800328608558162860nt_int(produc4729913460834754785nt_int(A),B))) ) ).

% mem_case_prodI
tff(fact_5734_mem__case__prodI,axiom,
    ! [Z: real,C: fun_fu604837885242254216t_real,A: fun_in5730379987311621695e_term,B: product_prod_int_int] :
      ( aa_set_real_o(member_real(Z),aa_Pro7989103684195733727t_real(aa_fun5197189019902991601t_real(C,A),B))
     => aa_set_real_o(member_real(Z),produc5863038274793111792t_real(C,aa_Pro8800328608558162860nt_int(produc4729913460834754785nt_int(A),B))) ) ).

% mem_case_prodI
tff(fact_5735_mem__case__prodI,axiom,
    ! [Z: int,C: fun_fu4266222366127506568et_int,A: fun_in5730379987311621695e_term,B: product_prod_int_int] :
      ( aa_set_int_o(member_int(Z),aa_Pro8644714012989290591et_int(aa_fun5919120857301092337et_int(C,A),B))
     => aa_set_int_o(member_int(Z),produc833192924009095408et_int(C,aa_Pro8800328608558162860nt_int(produc4729913460834754785nt_int(A),B))) ) ).

% mem_case_prodI
tff(fact_5736_mem__case__prodI,axiom,
    ! [Z: nat,C: fun_fu86038545275896455et_nat,A: fun_Pr6460944699146269935_nat_o,B: produc3658429121746597890et_nat] :
      ( aa_set_nat_o(member_nat(Z),aa_Pro3416150927818442182et_nat(aa_fun6477889762090788408et_nat(C,A),B))
     => aa_set_nat_o(member_nat(Z),produc6247649352436904781et_nat(C,aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(A),B))) ) ).

% mem_case_prodI
tff(fact_5737_mem__case__prodI,axiom,
    ! [Z: vEBT_VEBT,C: fun_fu6298409707154110463T_VEBT,A: fun_Pr6460944699146269935_nat_o,B: produc3658429121746597890et_nat] :
      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Z),aa_Pro5551276291186086996T_VEBT(aa_fun6431552983927604216T_VEBT(C,A),B))
     => aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Z),produc1803589779646358669T_VEBT(C,aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(A),B))) ) ).

% mem_case_prodI
tff(fact_5738_mem__case__prodI,axiom,
    ! [Z: real,C: fun_fu2523205402392354915t_real,A: fun_Pr6460944699146269935_nat_o,B: produc3658429121746597890et_nat] :
      ( aa_set_real_o(member_real(Z),aa_Pro5145766922317858082t_real(aa_fun7171570044062598164t_real(C,A),B))
     => aa_set_real_o(member_real(Z),produc2103481464906571561t_real(C,aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(A),B))) ) ).

% mem_case_prodI
tff(fact_5739_mem__case__prodI,axiom,
    ! [Z: int,C: fun_fu4495862163342753507et_int,A: fun_Pr6460944699146269935_nat_o,B: produc3658429121746597890et_nat] :
      ( aa_set_int_o(member_int(Z),aa_Pro8461671945164021282et_int(aa_fun3987806617159061652et_int(C,A),B))
     => aa_set_int_o(member_int(Z),produc2069798332927708073et_int(C,aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(A),B))) ) ).

% mem_case_prodI
tff(fact_5740_mem__case__prodI,axiom,
    ! [Z: nat,C: fun_fu2531416508075921811et_nat,A: fun_Pr5098669352780829690e_term,B: produc8923325533196201883nteger] :
      ( aa_set_nat_o(member_nat(Z),aa_Pro381016180347496421et_nat(aa_fun4088370000251959292et_nat(C,A),B))
     => aa_set_nat_o(member_nat(Z),produc6164657169593426171et_nat(C,aa_Pro4291442876242880245nteger(produc565448451787809096nteger(A),B))) ) ).

% mem_case_prodI
tff(fact_5741_mem__case__prodI,axiom,
    ! [Z: vEBT_VEBT,C: fun_fu3339394088549833715T_VEBT,A: fun_Pr5098669352780829690e_term,B: produc8923325533196201883nteger] :
      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Z),aa_Pro2857938020663670261T_VEBT(aa_fun4954175298189341492T_VEBT(C,A),B))
     => aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Z),produc8725738732899457631T_VEBT(C,aa_Pro4291442876242880245nteger(produc565448451787809096nteger(A),B))) ) ).

% mem_case_prodI
tff(fact_5742_Collect__const,axiom,
    ! [P: $o] :
      collect_complex(aTP_Lamm_mg((P))) = $ite((P),top_top_set_complex,bot_bot_set_complex) ).

% Collect_const
tff(fact_5743_Collect__const,axiom,
    ! [P: $o] :
      collec213857154873943460nt_int(aTP_Lamm_mh((P))) = $ite((P),top_to4366644338036079209nt_int,bot_bo1796632182523588997nt_int) ).

% Collect_const
tff(fact_5744_Collect__const,axiom,
    ! [P: $o] :
      collect_int(aTP_Lamm_mi((P))) = $ite((P),top_top_set_int,bot_bot_set_int) ).

% Collect_const
tff(fact_5745_Collect__const,axiom,
    ! [P: $o] :
      collect_real(aTP_Lamm_mj((P))) = $ite((P),top_top_set_real,bot_bot_set_real) ).

% Collect_const
tff(fact_5746_Collect__const,axiom,
    ! [P: $o] :
      collect_nat(aa_o_fun_nat_o(aTP_Lamm_mk,(P))) = $ite((P),top_top_set_nat,bot_bot_set_nat) ).

% Collect_const
tff(fact_5747_Collect__const,axiom,
    ! [P: $o] :
      collect_char(aTP_Lamm_ml((P))) = $ite((P),top_top_set_char,bot_bot_set_char) ).

% Collect_const
tff(fact_5748_Collect__const__case__prod,axiom,
    ! [P: $o] :
      collec3392354462482085612at_nat(produc6081775807080527818_nat_o(aTP_Lamm_mm((P)))) = $ite((P),top_to4669805908274784177at_nat,bot_bo2099793752762293965at_nat) ).

% Collect_const_case_prod
tff(fact_5749_Collect__const__case__prod,axiom,
    ! [P: $o] :
      collec213857154873943460nt_int(produc4947309494688390418_int_o(aTP_Lamm_mn((P)))) = $ite((P),top_to4366644338036079209nt_int,bot_bo1796632182523588997nt_int) ).

% Collect_const_case_prod
tff(fact_5750_Diff__UNIV,axiom,
    ! [A2: set_int] : minus_minus_set_int(A2,top_top_set_int) = bot_bot_set_int ).

% Diff_UNIV
tff(fact_5751_Diff__UNIV,axiom,
    ! [A2: set_real] : minus_minus_set_real(A2,top_top_set_real) = bot_bot_set_real ).

% Diff_UNIV
tff(fact_5752_Diff__UNIV,axiom,
    ! [A2: set_char] : minus_minus_set_char(A2,top_top_set_char) = bot_bot_set_char ).

% Diff_UNIV
tff(fact_5753_Diff__UNIV,axiom,
    ! [A2: set_nat] : minus_minus_set_nat(A2,top_top_set_nat) = bot_bot_set_nat ).

% Diff_UNIV
tff(fact_5754_round__numeral,axiom,
    ! [Nb: num] : aa_real_int(archim8280529875227126926d_real,numeral_numeral_real(Nb)) = aa_num_int(numeral_numeral_int,Nb) ).

% round_numeral
tff(fact_5755_round__numeral,axiom,
    ! [Nb: num] : archim7778729529865785530nd_rat(numeral_numeral_rat(Nb)) = aa_num_int(numeral_numeral_int,Nb) ).

% round_numeral
tff(fact_5756_round__1,axiom,
    aa_real_int(archim8280529875227126926d_real,one_one_real) = one_one_int ).

% round_1
tff(fact_5757_round__1,axiom,
    archim7778729529865785530nd_rat(one_one_rat) = one_one_int ).

% round_1
tff(fact_5758_fst__divmod__nat,axiom,
    ! [M: nat,Nb: nat] : aa_Pro4243173770044397973at_nat(product_fst_nat_nat,aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(divmod_nat,M),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Nb) ).

% fst_divmod_nat
tff(fact_5759_snd__divmod__nat,axiom,
    ! [M: nat,Nb: nat] : aa_Pro4243173770044397973at_nat(product_snd_nat_nat,aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(divmod_nat,M),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Nb) ).

% snd_divmod_nat
tff(fact_5760_mem__case__prodE,axiom,
    ! [Z: nat,C: fun_fu9079770784915425324et_nat,P2: produc4788131091073775919nt_int] :
      ( aa_set_nat_o(member_nat(Z),produc5011043943518292116et_nat(C,P2))
     => ~ ! [X2: fun_in5730379987311621695e_term,Y4: product_prod_int_int] :
            ( ( P2 = aa_Pro8800328608558162860nt_int(produc4729913460834754785nt_int(X2),Y4) )
           => ~ aa_set_nat_o(member_nat(Z),aa_Pro3599192995643711491et_nat(aa_fun8409204002232819093et_nat(C,X2),Y4)) ) ) ).

% mem_case_prodE
tff(fact_5761_mem__case__prodE,axiom,
    ! [Z: vEBT_VEBT,C: fun_fu7491663322100281242T_VEBT,P2: produc4788131091073775919nt_int] :
      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Z),produc9026699715758669062T_VEBT(C,P2))
     => ~ ! [X2: fun_in5730379987311621695e_term,Y4: product_prod_int_int] :
            ( ( P2 = aa_Pro8800328608558162860nt_int(produc4729913460834754785nt_int(X2),Y4) )
           => ~ aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Z),aa_Pro2768979630923332951T_VEBT(aa_fun2798302631922751451T_VEBT(C,X2),Y4)) ) ) ).

% mem_case_prodE
tff(fact_5762_mem__case__prodE,axiom,
    ! [Z: real,C: fun_fu604837885242254216t_real,P2: produc4788131091073775919nt_int] :
      ( aa_set_real_o(member_real(Z),produc5863038274793111792t_real(C,P2))
     => ~ ! [X2: fun_in5730379987311621695e_term,Y4: product_prod_int_int] :
            ( ( P2 = aa_Pro8800328608558162860nt_int(produc4729913460834754785nt_int(X2),Y4) )
           => ~ aa_set_real_o(member_real(Z),aa_Pro7989103684195733727t_real(aa_fun5197189019902991601t_real(C,X2),Y4)) ) ) ).

% mem_case_prodE
tff(fact_5763_mem__case__prodE,axiom,
    ! [Z: int,C: fun_fu4266222366127506568et_int,P2: produc4788131091073775919nt_int] :
      ( aa_set_int_o(member_int(Z),produc833192924009095408et_int(C,P2))
     => ~ ! [X2: fun_in5730379987311621695e_term,Y4: product_prod_int_int] :
            ( ( P2 = aa_Pro8800328608558162860nt_int(produc4729913460834754785nt_int(X2),Y4) )
           => ~ aa_set_int_o(member_int(Z),aa_Pro8644714012989290591et_int(aa_fun5919120857301092337et_int(C,X2),Y4)) ) ) ).

% mem_case_prodE
tff(fact_5764_mem__case__prodE,axiom,
    ! [Z: nat,C: fun_fu86038545275896455et_nat,P2: produc8799533983703829792et_nat] :
      ( aa_set_nat_o(member_nat(Z),produc6247649352436904781et_nat(C,P2))
     => ~ ! [X2: fun_Pr6460944699146269935_nat_o,Y4: produc3658429121746597890et_nat] :
            ( ( P2 = aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(X2),Y4) )
           => ~ aa_set_nat_o(member_nat(Z),aa_Pro3416150927818442182et_nat(aa_fun6477889762090788408et_nat(C,X2),Y4)) ) ) ).

% mem_case_prodE
tff(fact_5765_mem__case__prodE,axiom,
    ! [Z: vEBT_VEBT,C: fun_fu6298409707154110463T_VEBT,P2: produc8799533983703829792et_nat] :
      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Z),produc1803589779646358669T_VEBT(C,P2))
     => ~ ! [X2: fun_Pr6460944699146269935_nat_o,Y4: produc3658429121746597890et_nat] :
            ( ( P2 = aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(X2),Y4) )
           => ~ aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Z),aa_Pro5551276291186086996T_VEBT(aa_fun6431552983927604216T_VEBT(C,X2),Y4)) ) ) ).

% mem_case_prodE
tff(fact_5766_mem__case__prodE,axiom,
    ! [Z: real,C: fun_fu2523205402392354915t_real,P2: produc8799533983703829792et_nat] :
      ( aa_set_real_o(member_real(Z),produc2103481464906571561t_real(C,P2))
     => ~ ! [X2: fun_Pr6460944699146269935_nat_o,Y4: produc3658429121746597890et_nat] :
            ( ( P2 = aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(X2),Y4) )
           => ~ aa_set_real_o(member_real(Z),aa_Pro5145766922317858082t_real(aa_fun7171570044062598164t_real(C,X2),Y4)) ) ) ).

% mem_case_prodE
tff(fact_5767_mem__case__prodE,axiom,
    ! [Z: int,C: fun_fu4495862163342753507et_int,P2: produc8799533983703829792et_nat] :
      ( aa_set_int_o(member_int(Z),produc2069798332927708073et_int(C,P2))
     => ~ ! [X2: fun_Pr6460944699146269935_nat_o,Y4: produc3658429121746597890et_nat] :
            ( ( P2 = aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(X2),Y4) )
           => ~ aa_set_int_o(member_int(Z),aa_Pro8461671945164021282et_int(aa_fun3987806617159061652et_int(C,X2),Y4)) ) ) ).

% mem_case_prodE
tff(fact_5768_mem__case__prodE,axiom,
    ! [Z: nat,C: fun_fu2531416508075921811et_nat,P2: produc2453742332425399446nteger] :
      ( aa_set_nat_o(member_nat(Z),produc6164657169593426171et_nat(C,P2))
     => ~ ! [X2: fun_Pr5098669352780829690e_term,Y4: produc8923325533196201883nteger] :
            ( ( P2 = aa_Pro4291442876242880245nteger(produc565448451787809096nteger(X2),Y4) )
           => ~ aa_set_nat_o(member_nat(Z),aa_Pro381016180347496421et_nat(aa_fun4088370000251959292et_nat(C,X2),Y4)) ) ) ).

% mem_case_prodE
tff(fact_5769_mem__case__prodE,axiom,
    ! [Z: vEBT_VEBT,C: fun_fu3339394088549833715T_VEBT,P2: produc2453742332425399446nteger] :
      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Z),produc8725738732899457631T_VEBT(C,P2))
     => ~ ! [X2: fun_Pr5098669352780829690e_term,Y4: produc8923325533196201883nteger] :
            ( ( P2 = aa_Pro4291442876242880245nteger(produc565448451787809096nteger(X2),Y4) )
           => ~ aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Z),aa_Pro2857938020663670261T_VEBT(aa_fun4954175298189341492T_VEBT(C,X2),Y4)) ) ) ).

% mem_case_prodE
tff(fact_5770_subset__UNIV,axiom,
    ! [A2: set_real] : aa_set_real_o(ord_less_eq_set_real(A2),top_top_set_real) ).

% subset_UNIV
tff(fact_5771_subset__UNIV,axiom,
    ! [A2: set_nat] : aa_set_nat_o(ord_less_eq_set_nat(A2),top_top_set_nat) ).

% subset_UNIV
tff(fact_5772_subset__UNIV,axiom,
    ! [A2: set_char] : ord_less_eq_set_char(A2,top_top_set_char) ).

% subset_UNIV
tff(fact_5773_subset__UNIV,axiom,
    ! [A2: set_int] : aa_set_int_o(ord_less_eq_set_int(A2),top_top_set_int) ).

% subset_UNIV
tff(fact_5774_case__prodE,axiom,
    ! [C: fun_fu7695273699059480439eger_o,P2: produc2453742332425399446nteger] :
      ( aa_Pro4148064003102580676eger_o(produc7373287791920631523eger_o(C),P2)
     => ~ ! [X2: fun_Pr5098669352780829690e_term,Y4: produc8923325533196201883nteger] :
            ( ( P2 = aa_Pro4291442876242880245nteger(produc565448451787809096nteger(X2),Y4) )
           => ~ aa_Pro4926521453222654329eger_o(aa_fun4505422719058378424eger_o(C,X2),Y4) ) ) ).

% case_prodE
tff(fact_5775_case__prodE,axiom,
    ! [C: fun_fu8419763957888848259_nat_o,P2: produc8799533983703829792et_nat] :
      ( aa_Pro5643009357358181178_nat_o(produc2915205735004773393_nat_o(C),P2)
     => ~ ! [X2: fun_Pr6460944699146269935_nat_o,Y4: produc3658429121746597890et_nat] :
            ( ( P2 = aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(X2),Y4) )
           => ~ aa_Pro653082453696054616_nat_o(aa_fun2638059832795814268_nat_o(C,X2),Y4) ) ) ).

% case_prodE
tff(fact_5776_case__prodE,axiom,
    ! [C: fun_fu2611741329380808165_nat_o,P2: produc3112576180687729086et_nat] :
      ( aa_Pro2671238023309943196_nat_o(produc4465912260561112435_nat_o(C),P2)
     => ~ ! [X2: fun_Pr6460944699146269935_nat_o,Y4: produc8799533983703829792et_nat] :
            ( ( P2 = aa_Pro1979504083478021276et_nat(produc7291855603632349880et_nat(X2),Y4) )
           => ~ aa_Pro5643009357358181178_nat_o(aa_fun912680268133416158_nat_o(C,X2),Y4) ) ) ).

% case_prodE
tff(fact_5777_case__prodE,axiom,
    ! [C: fun_fu6865642474282033866_int_o,P2: produc3043004577601930755nt_int] :
      ( aa_Pro2601676399929874711_int_o(produc2207054017359051062_int_o(C),P2)
     => ~ ! [X2: fun_Pr7763680713840462443e_term,Y4: product_prod_int_int] :
            ( ( P2 = aa_Pro5748361610964203648nt_int(produc7320969046608631221nt_int(X2),Y4) )
           => ~ aa_Pro4592068669816069595_int_o(aa_fun3538162476766722827_int_o(C,X2),Y4) ) ) ).

% case_prodE
tff(fact_5778_case__prodE,axiom,
    ! [C: fun_fu2978539155105838238_int_o,P2: produc4788131091073775919nt_int] :
      ( aa_Pro5451197003417746155_int_o(produc7271678899836001034_int_o(C),P2)
     => ~ ! [X2: fun_in5730379987311621695e_term,Y4: product_prod_int_int] :
            ( ( P2 = aa_Pro8800328608558162860nt_int(produc4729913460834754785nt_int(X2),Y4) )
           => ~ aa_Pro4592068669816069595_int_o(aa_fun586359091965693151_int_o(C,X2),Y4) ) ) ).

% case_prodE
tff(fact_5779_case__prodE,axiom,
    ! [C: fun_nat_fun_nat_o,P2: product_prod_nat_nat] :
      ( aa_Pro6150637604445964947_nat_o(produc6081775807080527818_nat_o(C),P2)
     => ~ ! [X2: nat,Y4: nat] :
            ( ( P2 = aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,X2),Y4) )
           => ~ aa_nat_o(aa_nat_fun_nat_o(C,X2),Y4) ) ) ).

% case_prodE
tff(fact_5780_case__prodE,axiom,
    ! [C: fun_int_fun_int_o,P2: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(C),P2)
     => ~ ! [X2: int,Y4: int] :
            ( ( P2 = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X2),Y4) )
           => ~ aa_int_o(aa_int_fun_int_o(C,X2),Y4) ) ) ).

% case_prodE
tff(fact_5781_case__prodD,axiom,
    ! [F: fun_fu7695273699059480439eger_o,A: fun_Pr5098669352780829690e_term,B: produc8923325533196201883nteger] :
      ( aa_Pro4148064003102580676eger_o(produc7373287791920631523eger_o(F),aa_Pro4291442876242880245nteger(produc565448451787809096nteger(A),B))
     => aa_Pro4926521453222654329eger_o(aa_fun4505422719058378424eger_o(F,A),B) ) ).

% case_prodD
tff(fact_5782_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(produc6017135503889078682et_nat(A),B))
     => aa_Pro653082453696054616_nat_o(aa_fun2638059832795814268_nat_o(F,A),B) ) ).

% case_prodD
tff(fact_5783_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(produc7291855603632349880et_nat(A),B))
     => aa_Pro5643009357358181178_nat_o(aa_fun912680268133416158_nat_o(F,A),B) ) ).

% case_prodD
tff(fact_5784_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(produc7320969046608631221nt_int(A),B))
     => aa_Pro4592068669816069595_int_o(aa_fun3538162476766722827_int_o(F,A),B) ) ).

% case_prodD
tff(fact_5785_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(produc4729913460834754785nt_int(A),B))
     => aa_Pro4592068669816069595_int_o(aa_fun586359091965693151_int_o(F,A),B) ) ).

% case_prodD
tff(fact_5786_case__prodD,axiom,
    ! [F: fun_nat_fun_nat_o,A: nat,B: nat] :
      ( aa_Pro6150637604445964947_nat_o(produc6081775807080527818_nat_o(F),aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,A),B))
     => aa_nat_o(aa_nat_fun_nat_o(F,A),B) ) ).

% case_prodD
tff(fact_5787_case__prodD,axiom,
    ! [F: fun_int_fun_int_o,A: int,B: int] :
      ( aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(F),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),B))
     => aa_int_o(aa_int_fun_int_o(F,A),B) ) ).

% case_prodD
tff(fact_5788_hoare__triple__refines,axiom,
    ! [P: assn,C: heap_T2636463487746394924on_nat,Q: fun_option_nat_assn,C5: heap_T2636463487746394924on_nat] :
      ( hoare_7629718768684598413on_nat(P,C,Q)
     => ( refine7594492741263601813on_nat(C5,C)
       => hoare_7629718768684598413on_nat(P,C5,Q) ) ) ).

% hoare_triple_refines
tff(fact_5789_hoare__triple__refines,axiom,
    ! [P: assn,C: heap_Time_Heap_nat,Q: fun_nat_assn,C5: heap_Time_Heap_nat] :
      ( hoare_3067605981109127869le_nat(P,C,Q)
     => ( refine1365783493865988805es_nat(C5,C)
       => hoare_3067605981109127869le_nat(P,C5,Q) ) ) ).

% hoare_triple_refines
tff(fact_5790_hoare__triple__refines,axiom,
    ! [P: assn,C: heap_Time_Heap_o,Q: fun_o_assn,C5: heap_Time_Heap_o] :
      ( hoare_hoare_triple_o(P,C,Q)
     => ( aa_Hea4522166450600650424ap_o_o(refine_Imp_refines_o(C5),C)
       => hoare_hoare_triple_o(P,C5,Q) ) ) ).

% hoare_triple_refines
tff(fact_5791_hoare__triple__refines,axiom,
    ! [P: assn,C: heap_T8145700208782473153_VEBTi,Q: fun_VEBT_VEBTi_assn,C5: heap_T8145700208782473153_VEBTi] :
      ( hoare_1429296392585015714_VEBTi(P,C,Q)
     => ( refine5565527176597971370_VEBTi(C5,C)
       => hoare_1429296392585015714_VEBTi(P,C5,Q) ) ) ).

% hoare_triple_refines
tff(fact_5792_hoare__triple__refines,axiom,
    ! [P: assn,C: heap_T5738788834812785303t_unit,Q: fun_Pr6626665806141890376t_assn,C5: heap_T5738788834812785303t_unit] :
      ( hoare_8945653483474564448t_unit(P,C,Q)
     => ( refine451278484176762712t_unit(C5,C)
       => hoare_8945653483474564448t_unit(P,C5,Q) ) ) ).

% hoare_triple_refines
tff(fact_5793_htt__refine,axiom,
    ! [P: assn,C: heap_Time_Heap_nat,Q: fun_nat_assn,Ta: nat,C5: heap_Time_Heap_nat] :
      ( aa_nat_o(time_htt_nat(P,C,Q),Ta)
     => ( refine1365783493865988805es_nat(C5,C)
       => aa_nat_o(time_htt_nat(P,C5,Q),Ta) ) ) ).

% htt_refine
tff(fact_5794_htt__refine,axiom,
    ! [P: assn,C: heap_Time_Heap_o,Q: fun_o_assn,Ta: nat,C5: heap_Time_Heap_o] :
      ( aa_nat_o(time_htt_o(P,C,Q),Ta)
     => ( aa_Hea4522166450600650424ap_o_o(refine_Imp_refines_o(C5),C)
       => aa_nat_o(time_htt_o(P,C5,Q),Ta) ) ) ).

% htt_refine
tff(fact_5795_htt__refine,axiom,
    ! [P: assn,C: heap_T2636463487746394924on_nat,Q: fun_option_nat_assn,Ta: nat,C5: heap_T2636463487746394924on_nat] :
      ( aa_nat_o(time_htt_option_nat(P,C,Q),Ta)
     => ( refine7594492741263601813on_nat(C5,C)
       => aa_nat_o(time_htt_option_nat(P,C5,Q),Ta) ) ) ).

% htt_refine
tff(fact_5796_htt__refine,axiom,
    ! [P: assn,C: heap_T8145700208782473153_VEBTi,Q: fun_VEBT_VEBTi_assn,Ta: nat,C5: heap_T8145700208782473153_VEBTi] :
      ( aa_nat_o(time_htt_VEBT_VEBTi(P,C,Q),Ta)
     => ( refine5565527176597971370_VEBTi(C5,C)
       => aa_nat_o(time_htt_VEBT_VEBTi(P,C5,Q),Ta) ) ) ).

% htt_refine
tff(fact_5797_refines__bind,axiom,
    ! [M: heap_Time_Heap_nat,M6: heap_Time_Heap_nat,F: fun_na2343073368905879023Heap_o,F3: fun_na2343073368905879023Heap_o] :
      ( refine1365783493865988805es_nat(M,M6)
     => ( ! [X2: nat] : aa_Hea4522166450600650424ap_o_o(refine_Imp_refines_o(aa_nat2910048432523430448Heap_o(F,X2)),aa_nat2910048432523430448Heap_o(F3,X2))
       => aa_Hea4522166450600650424ap_o_o(refine_Imp_refines_o(heap_Time_bind_nat_o(M,F)),heap_Time_bind_nat_o(M6,F3)) ) ) ).

% refines_bind
tff(fact_5798_refines__bind,axiom,
    ! [M: heap_Time_Heap_nat,M6: heap_Time_Heap_nat,F: fun_na7848093160509028170_VEBTi,F3: fun_na7848093160509028170_VEBTi] :
      ( refine1365783493865988805es_nat(M,M6)
     => ( ! [X2: nat] : refine5565527176597971370_VEBTi(aa_nat5408703056993926195_VEBTi(F,X2),aa_nat5408703056993926195_VEBTi(F3,X2))
       => refine5565527176597971370_VEBTi(heap_T844888390831797134_VEBTi(M,F),heap_T844888390831797134_VEBTi(M6,F3)) ) ) ).

% refines_bind
tff(fact_5799_refines__bind,axiom,
    ! [M: heap_Time_Heap_o,M6: heap_Time_Heap_o,F: fun_o_8325599348076409421Heap_o,F3: fun_o_8325599348076409421Heap_o] :
      ( aa_Hea4522166450600650424ap_o_o(refine_Imp_refines_o(M),M6)
     => ( ! [X2: $o] : aa_Hea4522166450600650424ap_o_o(refine_Imp_refines_o(aa_o_H7399033255965945470Heap_o(F,(X2))),aa_o_H7399033255965945470Heap_o(F3,(X2)))
       => aa_Hea4522166450600650424ap_o_o(refine_Imp_refines_o(heap_Time_bind_o_o(M,F)),heap_Time_bind_o_o(M6,F3)) ) ) ).

% refines_bind
tff(fact_5800_refines__bind,axiom,
    ! [M: heap_Time_Heap_o,M6: heap_Time_Heap_o,F: fun_o_2238389867833408620_VEBTi,F3: fun_o_2238389867833408620_VEBTi] :
      ( aa_Hea4522166450600650424ap_o_o(refine_Imp_refines_o(M),M6)
     => ( ! [X2: $o] : refine5565527176597971370_VEBTi(aa_o_H7577119287286566245_VEBTi(F,(X2)),aa_o_H7577119287286566245_VEBTi(F3,(X2)))
       => refine5565527176597971370_VEBTi(heap_T5998771940306268294_VEBTi(M,F),heap_T5998771940306268294_VEBTi(M6,F3)) ) ) ).

% refines_bind
tff(fact_5801_refines__bind,axiom,
    ! [M: heap_T8145700208782473153_VEBTi,M6: heap_T8145700208782473153_VEBTi,F: fun_VE7038733415900436564Heap_o,F3: fun_VE7038733415900436564Heap_o] :
      ( refine5565527176597971370_VEBTi(M,M6)
     => ( ! [X2: vEBT_VEBTi] : aa_Hea4522166450600650424ap_o_o(refine_Imp_refines_o(aa_VEB8194921341634505877Heap_o(F,X2)),aa_VEB8194921341634505877Heap_o(F3,X2))
       => aa_Hea4522166450600650424ap_o_o(refine_Imp_refines_o(heap_T3040810144269856602EBTi_o(M,F)),heap_T3040810144269856602EBTi_o(M6,F3)) ) ) ).

% refines_bind
tff(fact_5802_refines__bind,axiom,
    ! [M: heap_T8145700208782473153_VEBTi,M6: heap_T8145700208782473153_VEBTi,F: fun_VE4765431424076933413_VEBTi,F3: fun_VE4765431424076933413_VEBTi] :
      ( refine5565527176597971370_VEBTi(M,M6)
     => ( ! [X2: vEBT_VEBTi] : refine5565527176597971370_VEBTi(aa_VEB836174949890266638_VEBTi(F,X2),aa_VEB836174949890266638_VEBTi(F3,X2))
       => refine5565527176597971370_VEBTi(heap_T1006145433769338483_VEBTi(M,F),heap_T1006145433769338483_VEBTi(M6,F3)) ) ) ).

% refines_bind
tff(fact_5803_refines__bind,axiom,
    ! [M: heap_Time_Heap_nat,M6: heap_Time_Heap_nat,F: fun_na5556145776640407093on_nat,F3: fun_na5556145776640407093on_nat] :
      ( refine1365783493865988805es_nat(M,M6)
     => ( ! [X2: nat] : refine7594492741263601813on_nat(aa_nat947733127081326238on_nat(F,X2),aa_nat947733127081326238on_nat(F3,X2))
       => refine7594492741263601813on_nat(heap_T8222160169144143993on_nat(M,F),heap_T8222160169144143993on_nat(M6,F3)) ) ) ).

% refines_bind
tff(fact_5804_refines__bind,axiom,
    ! [M: heap_T4980287057938770641_VEBTi,M6: heap_T4980287057938770641_VEBTi,F: fun_li7499208429157053461_VEBTi,F3: fun_li7499208429157053461_VEBTi] :
      ( refine3700189196150522554_VEBTi(M,M6)
     => ( ! [X2: list_VEBT_VEBTi] : refine5565527176597971370_VEBTi(aa_lis1839255523752068350_VEBTi(F,X2),aa_lis1839255523752068350_VEBTi(F3,X2))
       => refine5565527176597971370_VEBTi(heap_T5877712393672139267_VEBTi(M,F),heap_T5877712393672139267_VEBTi(M6,F3)) ) ) ).

% refines_bind
tff(fact_5805_refines__bind,axiom,
    ! [M: heap_Time_Heap_o,M6: heap_Time_Heap_o,F: fun_o_1929998731553720919on_nat,F3: fun_o_1929998731553720919on_nat] :
      ( aa_Hea4522166450600650424ap_o_o(refine_Imp_refines_o(M),M6)
     => ( ! [X2: $o] : refine7594492741263601813on_nat(aa_o_H4215342640551218896on_nat(F,(X2)),aa_o_H4215342640551218896on_nat(F3,(X2)))
       => refine7594492741263601813on_nat(heap_T6306279297776390513on_nat(M,F),heap_T6306279297776390513on_nat(M6,F3)) ) ) ).

% refines_bind
tff(fact_5806_refines__bind,axiom,
    ! [M: heap_T2636463487746394924on_nat,M6: heap_T2636463487746394924on_nat,F: fun_op3694375823608013759Heap_o,F3: fun_op3694375823608013759Heap_o] :
      ( refine7594492741263601813on_nat(M,M6)
     => ( ! [X2: option_nat] : aa_Hea4522166450600650424ap_o_o(refine_Imp_refines_o(aa_opt6753654531493786624Heap_o(F,X2)),aa_opt6753654531493786624Heap_o(F3,X2))
       => aa_Hea4522166450600650424ap_o_o(refine_Imp_refines_o(heap_T6471384023045698863_nat_o(M,F)),heap_T6471384023045698863_nat_o(M6,F3)) ) ) ).

% refines_bind
tff(fact_5807_Product__Type_OCollect__case__prodD,axiom,
    ! [Xb: produc827990862158126777uint32,A2: fun_ui3478071520421546738nt32_o] :
      ( member8027108493173000802uint32(Xb,collec104543260748735396uint32(produc6247642326928354578nt32_o(A2)))
     => aa_uint32_o(aa_uin5924812838015166427nt32_o(A2,aa_Pro3737690986720249065uint32(produc9004433772639906525uint32,Xb)),aa_Pro3737690986720249065uint32(produc1510406741064981791uint32,Xb)) ) ).

% Product_Type.Collect_case_prodD
tff(fact_5808_Product__Type_OCollect__case__prodD,axiom,
    ! [Xb: produc3658429121746597890et_nat,A2: fun_he5746958392356810273_nat_o] :
      ( member6260224972018164377et_nat(Xb,collec6777745107090321879et_nat(produc6968281743908733999_nat_o(A2)))
     => aa_set_nat_o(aa_hea8622544166608216090_nat_o(A2,aa_Pro1219312880135979149t_unit(produc1824681642469235216et_nat,Xb)),aa_Pro3416150927818442182et_nat(produc8586169260539613262et_nat,Xb)) ) ).

% Product_Type.Collect_case_prodD
tff(fact_5809_Product__Type_OCollect__case__prodD,axiom,
    ! [Xb: produc8923325533196201883nteger,A2: fun_Co2081214437782444048eger_o] :
      ( member157494554546826820nteger(Xb,collec7766031446232907142nteger(produc2066375834425727024eger_o(A2)))
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(A2,aa_Pro5245656335456409788nteger(produc8508995932063986495nteger,Xb)),aa_Pro5245656335456409788nteger(produc6174133586879617921nteger,Xb)) ) ).

% Product_Type.Collect_case_prodD
tff(fact_5810_Product__Type_OCollect__case__prodD,axiom,
    ! [Xb: produc6575502325842934193n_assn,A2: fun_assn_fun_assn_o] :
      ( member7957490590177025114n_assn(Xb,collec1604292450182575004n_assn(produc7274209992780475162assn_o(A2)))
     => aa_assn_o(aa_assn_fun_assn_o(A2,aa_Pro2979818503705714077n_assn(produc9167289414957590229n_assn,Xb)),aa_Pro2979818503705714077n_assn(produc2051961928117032727n_assn,Xb)) ) ).

% Product_Type.Collect_case_prodD
tff(fact_5811_Product__Type_OCollect__case__prodD,axiom,
    ! [Xb: product_prod_nat_nat,A2: fun_nat_fun_nat_o] :
      ( member8440522571783428010at_nat(Xb,collec3392354462482085612at_nat(produc6081775807080527818_nat_o(A2)))
     => aa_nat_o(aa_nat_fun_nat_o(A2,aa_Pro4243173770044397973at_nat(product_fst_nat_nat,Xb)),aa_Pro4243173770044397973at_nat(product_snd_nat_nat,Xb)) ) ).

% Product_Type.Collect_case_prodD
tff(fact_5812_Product__Type_OCollect__case__prodD,axiom,
    ! [Xb: product_prod_int_int,A2: fun_int_fun_int_o] :
      ( member5262025264175285858nt_int(Xb,collec213857154873943460nt_int(produc4947309494688390418_int_o(A2)))
     => aa_int_o(aa_int_fun_int_o(A2,aa_Pro6799258434465214121nt_int(product_fst_int_int,Xb)),aa_Pro6799258434465214121nt_int(product_snd_int_int,Xb)) ) ).

% Product_Type.Collect_case_prodD
tff(fact_5813_refines__option,axiom,
    ! [A: option4927543243414619207at_nat,A5: option4927543243414619207at_nat,M1: heap_Time_Heap_o,M12: heap_Time_Heap_o,M22: fun_Pr1329037631615965750Heap_o,M23: fun_Pr1329037631615965750Heap_o] :
      ( ( A = A5 )
     => ( aa_Hea4522166450600650424ap_o_o(refine_Imp_refines_o(M1),M12)
       => ( ! [X2: product_prod_nat_nat] : aa_Hea4522166450600650424ap_o_o(refine_Imp_refines_o(aa_Pro691548381575413351Heap_o(M22,X2)),aa_Pro691548381575413351Heap_o(M23,X2))
         => aa_Hea4522166450600650424ap_o_o(refine_Imp_refines_o(case_o1442776274061689234at_nat(M1,M22,A)),case_o1442776274061689234at_nat(M12,M23,A5)) ) ) ) ).

% refines_option
tff(fact_5814_not__UNIV__le__Icc,axiom,
    ! [L: nat,H: nat] : ~ aa_set_nat_o(ord_less_eq_set_nat(top_top_set_nat),set_or1269000886237332187st_nat(L,H)) ).

% not_UNIV_le_Icc
tff(fact_5815_not__UNIV__le__Icc,axiom,
    ! [L: int,H: int] : ~ aa_set_int_o(ord_less_eq_set_int(top_top_set_int),set_or1266510415728281911st_int(L,H)) ).

% not_UNIV_le_Icc
tff(fact_5816_not__UNIV__le__Icc,axiom,
    ! [L: real,H: real] : ~ aa_set_real_o(ord_less_eq_set_real(top_top_set_real),set_or1222579329274155063t_real(L,H)) ).

% not_UNIV_le_Icc
tff(fact_5817_round__mono,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Y)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,archim7778729529865785530nd_rat(Xb)),archim7778729529865785530nd_rat(Y)) ) ).

% round_mono
tff(fact_5818_divmod__nat__if,axiom,
    ! [M: nat,Nb: nat] :
      aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(divmod_nat,M),Nb) = $ite(
        ( ( Nb = zero_zero_nat )
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb) ),
        aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,zero_zero_nat),M),
        aa_Pro262675715343573244at_nat(produc2626176000494625587at_nat(aTP_Lamm_mo),aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(divmod_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Nb)),Nb)) ) ).

% divmod_nat_if
tff(fact_5819_ceiling__ge__round,axiom,
    ! [Xb: real] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(archim8280529875227126926d_real,Xb)),aa_real_int(archim7802044766580827645g_real,Xb)) ).

% ceiling_ge_round
tff(fact_5820_refines__assert_H__bind,axiom,
    ! [P2: heap_Time_Heap_o,Q2: heap_Time_Heap_o,Phi: $o] :
      ( aa_Hea4522166450600650424ap_o_o(refine_Imp_refines_o(P2),Q2)
     => aa_Hea4522166450600650424ap_o_o(refine_Imp_refines_o(P2),heap_T7577940988442287570unit_o(aa_o_H2459438917461573107t_unit(refine_Imp_assert,(Phi)),aTP_Lamm_gv(Q2))) ) ).

% refines_assert'_bind
tff(fact_5821_refines__assert_H__bind,axiom,
    ! [P2: heap_T2636463487746394924on_nat,Q2: heap_T2636463487746394924on_nat,Phi: $o] :
      ( refine7594492741263601813on_nat(P2,Q2)
     => refine7594492741263601813on_nat(P2,heap_T5999496708990702694on_nat(aa_o_H2459438917461573107t_unit(refine_Imp_assert,(Phi)),aTP_Lamm_gx(Q2))) ) ).

% refines_assert'_bind
tff(fact_5822_refines__assert_H__bind,axiom,
    ! [P2: heap_T8145700208782473153_VEBTi,Q2: heap_T8145700208782473153_VEBTi,Phi: $o] :
      ( refine5565527176597971370_VEBTi(P2,Q2)
     => refine5565527176597971370_VEBTi(P2,heap_T7982501707604696571_VEBTi(aa_o_H2459438917461573107t_unit(refine_Imp_assert,(Phi)),aTP_Lamm_gy(Q2))) ) ).

% refines_assert'_bind
tff(fact_5823_TBOUND__prod__case,axiom,
    ! [Ta: produc4788131091073775919nt_int,F: fun_fu5406852889748164040Heap_o,Bnd: fun_fu4121049827101807222nt_nat] :
      ( ! [A3: fun_in5730379987311621695e_term,B2: product_prod_int_int] :
          ( ( Ta = aa_Pro8800328608558162860nt_int(produc4729913460834754785nt_int(A3),B2) )
         => aa_nat_o(time_TBOUND_o(aa_Pro8097318296274772255Heap_o(aa_fun717413422199439153Heap_o(F,A3),B2)),aa_Pro6801748904974264397nt_nat(aa_fun262305326913600735nt_nat(Bnd,A3),B2)) )
     => aa_nat_o(time_TBOUND_o(produc6980626716110614320Heap_o(F,Ta)),produc6483382825394656478nt_nat(Bnd,Ta)) ) ).

% TBOUND_prod_case
tff(fact_5824_TBOUND__prod__case,axiom,
    ! [Ta: produc4788131091073775919nt_int,F: fun_fu136016935472200802ap_nat,Bnd: fun_fu4121049827101807222nt_nat] :
      ( ! [A3: fun_in5730379987311621695e_term,B2: product_prod_int_int] :
          ( ( Ta = aa_Pro8800328608558162860nt_int(produc4729913460834754785nt_int(A3),B2) )
         => aa_nat_o(time_TBOUND_nat(aa_Pro7829970176022100127ap_nat(aa_fun6257319592574587555ap_nat(F,A3),B2)),aa_Pro6801748904974264397nt_nat(aa_fun262305326913600735nt_nat(Bnd,A3),B2)) )
     => aa_nat_o(time_TBOUND_nat(produc6431702567826247886ap_nat(F,Ta)),produc6483382825394656478nt_nat(Bnd,Ta)) ) ).

% TBOUND_prod_case
tff(fact_5825_TBOUND__prod__case,axiom,
    ! [Ta: produc4788131091073775919nt_int,F: fun_fu7458074433144620999_VEBTi,Bnd: fun_fu4121049827101807222nt_nat] :
      ( ! [A3: fun_in5730379987311621695e_term,B2: product_prod_int_int] :
          ( ( Ta = aa_Pro8800328608558162860nt_int(produc4729913460834754785nt_int(A3),B2) )
         => aa_nat_o(time_T5737551269749752165_VEBTi(aa_Pro3831758596280224260_VEBTi(aa_fun8292477295915508872_VEBTi(F,A3),B2)),aa_Pro6801748904974264397nt_nat(aa_fun262305326913600735nt_nat(Bnd,A3),B2)) )
     => aa_nat_o(time_T5737551269749752165_VEBTi(produc6514969129755825971_VEBTi(F,Ta)),produc6483382825394656478nt_nat(Bnd,Ta)) ) ).

% TBOUND_prod_case
tff(fact_5826_TBOUND__prod__case,axiom,
    ! [Ta: produc4788131091073775919nt_int,F: fun_fu1311000841445275698on_nat,Bnd: fun_fu4121049827101807222nt_nat] :
      ( ! [A3: fun_in5730379987311621695e_term,B2: product_prod_int_int] :
          ( ( Ta = aa_Pro8800328608558162860nt_int(produc4729913460834754785nt_int(A3),B2) )
         => aa_nat_o(time_T8353473612707095248on_nat(aa_Pro185635222820580335on_nat(aa_fun2956517174233687155on_nat(F,A3),B2)),aa_Pro6801748904974264397nt_nat(aa_fun262305326913600735nt_nat(Bnd,A3),B2)) )
     => aa_nat_o(time_T8353473612707095248on_nat(produc7781486621487665566on_nat(F,Ta)),produc6483382825394656478nt_nat(Bnd,Ta)) ) ).

% TBOUND_prod_case
tff(fact_5827_TBOUND__prod__case,axiom,
    ! [Ta: produc8799533983703829792et_nat,F: fun_fu4848728041736279203Heap_o,Bnd: fun_fu725033328880924881at_nat] :
      ( ! [A3: fun_Pr6460944699146269935_nat_o,B2: produc3658429121746597890et_nat] :
          ( ( Ta = aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(A3),B2) )
         => aa_nat_o(time_TBOUND_o(aa_Pro4765807838566354786Heap_o(aa_fun4444287950952374356Heap_o(F,A3),B2)),aa_Pro8901340430247369232at_nat(aa_fun2746862352154169730at_nat(Bnd,A3),B2)) )
     => aa_nat_o(time_TBOUND_o(produc2235927729741437801Heap_o(F,Ta)),produc8538887819002602391at_nat(Bnd,Ta)) ) ).

% TBOUND_prod_case
tff(fact_5828_TBOUND__prod__case,axiom,
    ! [Ta: produc8799533983703829792et_nat,F: fun_fu3935500115053699655ap_nat,Bnd: fun_fu725033328880924881at_nat] :
      ( ! [A3: fun_Pr6460944699146269935_nat_o,B2: produc3658429121746597890et_nat] :
          ( ( Ta = aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(A3),B2) )
         => aa_nat_o(time_TBOUND_nat(aa_Pro8061417612792635676ap_nat(aa_fun4763701871219163200ap_nat(F,A3),B2)),aa_Pro8901340430247369232at_nat(aa_fun2746862352154169730at_nat(Bnd,A3),B2)) )
     => aa_nat_o(time_TBOUND_nat(produc5378093055270544597ap_nat(F,Ta)),produc8538887819002602391at_nat(Bnd,Ta)) ) ).

% TBOUND_prod_case
tff(fact_5829_TBOUND__prod__case,axiom,
    ! [Ta: produc8799533983703829792et_nat,F: fun_fu7290308691160913068_VEBTi,Bnd: fun_fu725033328880924881at_nat] :
      ( ! [A3: fun_Pr6460944699146269935_nat_o,B2: produc3658429121746597890et_nat] :
          ( ( Ta = aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(A3),B2) )
         => aa_nat_o(time_T5737551269749752165_VEBTi(aa_Pro5789314108598051201_VEBTi(aa_fun7882737301604475173_VEBTi(F,A3),B2)),aa_Pro8901340430247369232at_nat(aa_fun2746862352154169730at_nat(Bnd,A3),B2)) )
     => aa_nat_o(time_T5737551269749752165_VEBTi(produc2318796205394716730_VEBTi(F,Ta)),produc8538887819002602391at_nat(Bnd,Ta)) ) ).

% TBOUND_prod_case
tff(fact_5830_TBOUND__prod__case,axiom,
    ! [Ta: produc8799533983703829792et_nat,F: fun_fu5645185405084508823on_nat,Bnd: fun_fu725033328880924881at_nat] :
      ( ! [A3: fun_Pr6460944699146269935_nat_o,B2: produc3658429121746597890et_nat] :
          ( ( Ta = aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(A3),B2) )
         => aa_nat_o(time_T8353473612707095248on_nat(aa_Pro620577756486784236on_nat(aa_fun1757556057354134160on_nat(F,A3),B2)),aa_Pro8901340430247369232at_nat(aa_fun2746862352154169730at_nat(Bnd,A3),B2)) )
     => aa_nat_o(time_T8353473612707095248on_nat(produc5392750298291506981on_nat(F,Ta)),produc8538887819002602391at_nat(Bnd,Ta)) ) ).

% TBOUND_prod_case
tff(fact_5831_TBOUND__prod__case,axiom,
    ! [Ta: produc2453742332425399446nteger,F: fun_fu8697565728819266991Heap_o,Bnd: fun_fu335419639924914141er_nat] :
      ( ! [A3: fun_Pr5098669352780829690e_term,B2: produc8923325533196201883nteger] :
          ( ( Ta = aa_Pro4291442876242880245nteger(produc565448451787809096nteger(A3),B2) )
         => aa_nat_o(time_TBOUND_o(aa_Pro7031731416721101825Heap_o(aa_fun5631953093563615256Heap_o(F,A3),B2)),aa_Pro2825113747721566255er_nat(aa_fun6177767732178989894er_nat(Bnd,A3),B2)) )
     => aa_nat_o(time_TBOUND_o(produc5501110786615227927Heap_o(F,Ta)),produc7257083654846286149er_nat(Bnd,Ta)) ) ).

% TBOUND_prod_case
tff(fact_5832_TBOUND__prod__case,axiom,
    ! [Ta: produc3043004577601930755nt_int,F: fun_fu8834287435715338396Heap_o,Bnd: fun_fu2721829822554326858nt_nat] :
      ( ! [A3: fun_Pr7763680713840462443e_term,B2: product_prod_int_int] :
          ( ( Ta = aa_Pro5748361610964203648nt_int(produc7320969046608631221nt_int(A3),B2) )
         => aa_nat_o(time_TBOUND_o(aa_Pro8097318296274772255Heap_o(aa_fun3782508219895828997Heap_o(F,A3),B2)),aa_Pro6801748904974264397nt_nat(aa_fun6581587715306107315nt_nat(Bnd,A3),B2)) )
     => aa_nat_o(time_TBOUND_o(produc5412044208144667140Heap_o(F,Ta)),produc8877986901729335730nt_nat(Bnd,Ta)) ) ).

% TBOUND_prod_case
tff(fact_5833_rel__of__def,axiom,
    ! [M: fun_fu4810197545401214784nteger,P: fun_Pr7015759342434529115eger_o] : rel_of3240586176920340263nteger(M,P) = collec2950617477150063211nteger(produc7373287791920631523eger_o(aa_fun5185621348090534806eger_o(aTP_Lamm_mp(M),P))) ).

% rel_of_def
tff(fact_5834_rel__of__def,axiom,
    ! [M: fun_fu2352890940381120596et_nat,P: fun_Pr2620521676128474833_nat_o] : rel_of7233497478102412281et_nat(M,P) = collec6193113176214168693et_nat(produc2915205735004773393_nat_o(aa_fun1045796833200626456_nat_o(aTP_Lamm_mq(M),P))) ).

% rel_of_def
tff(fact_5835_rel__of__def,axiom,
    ! [M: fun_fu1905361174868762994et_nat,P: fun_Pr1227177680593602867_nat_o] : rel_of4573248530116435351et_nat(M,P) = collec5916191906451531411et_nat(produc4465912260561112435_nat_o(aa_fun4422562838209026524_nat_o(aTP_Lamm_mr(M),P))) ).

% rel_of_def
tff(fact_5836_rel__of__def,axiom,
    ! [M: fun_fu4253467608928286509nt_int,P: fun_Pr3540825087398332590_int_o] : rel_of9206080981073604500nt_int(M,P) = collec2586211367037964248nt_int(produc2207054017359051062_int_o(aa_fun1054583131507368252_int_o(aTP_Lamm_ms(M),P))) ).

% rel_of_def
tff(fact_5837_rel__of__def,axiom,
    ! [M: fun_fu5631754973857388633nt_int,P: fun_Pr1196100989263358594_int_o] : rel_of6825086619541906112nt_int(M,P) = collec8113046454796679940nt_int(produc7271678899836001034_int_o(aa_fun8601777901039773924_int_o(aTP_Lamm_mt(M),P))) ).

% rel_of_def
tff(fact_5838_rel__of__def,axiom,
    ! [M: fun_nat_option_nat,P: fun_Pr4796943543331071514_nat_o] : rel_of_nat_nat(M,P) = collec3392354462482085612at_nat(produc6081775807080527818_nat_o(aa_fun2218514334412473170_nat_o(aTP_Lamm_mu(M),P))) ).

% rel_of_def
tff(fact_5839_rel__of__def,axiom,
    ! [M: fun_int_option_int,P: fun_Pr3238374608701176162_int_o] : rel_of_int_int(M,P) = collec213857154873943460nt_int(produc4947309494688390418_int_o(aa_fun5140975492127840594_int_o(aTP_Lamm_mv(M),P))) ).

% rel_of_def
tff(fact_5840_divmod__nat__def,axiom,
    ! [M: nat,Nb: nat] : aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(divmod_nat,M),Nb) = aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Nb)) ).

% divmod_nat_def
tff(fact_5841_divmod__step__int__def,axiom,
    ! [L: num,Qr: product_prod_int_int] : aa_Pro6970504127699319548nt_int(unique5024387138958732305ep_int(L),Qr) = aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(aTP_Lamm_mw(L)),Qr) ).

% divmod_step_int_def
tff(fact_5842_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_mx(L)),Qr) ).

% divmod_step_nat_def
tff(fact_5843_divmod__step__def,axiom,
    ! [L: num,Qr: product_prod_int_int] : aa_Pro6970504127699319548nt_int(unique5024387138958732305ep_int(L),Qr) = aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(aTP_Lamm_mw(L)),Qr) ).

% divmod_step_def
tff(fact_5844_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_mx(L)),Qr) ).

% divmod_step_def
tff(fact_5845_divmod__step__def,axiom,
    ! [L: num,Qr: produc8923325533196201883nteger] : unique4921790084139445826nteger(L,Qr) = produc6916734918728496179nteger(aTP_Lamm_my(L),Qr) ).

% divmod_step_def
tff(fact_5846_mult__less__iff1,axiom,
    ! [Z: real,Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Z)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Z)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Z))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y) ) ) ).

% mult_less_iff1
tff(fact_5847_mult__less__iff1,axiom,
    ! [Z: rat,Xb: 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(times_times_rat(Xb),Z)),aa_rat_rat(times_times_rat(Y),Z))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Y) ) ) ).

% mult_less_iff1
tff(fact_5848_mult__less__iff1,axiom,
    ! [Z: int,Xb: 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,Xb),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,Xb),Y) ) ) ).

% mult_less_iff1
tff(fact_5849_of__int__round__le,axiom,
    ! [Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,aa_real_int(archim8280529875227126926d_real,Xb))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(aa_num_num(bit0,one))))) ).

% of_int_round_le
tff(fact_5850_of__int__round__le,axiom,
    ! [Xb: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,archim7778729529865785530nd_rat(Xb))),aa_rat_rat(plus_plus_rat(Xb),divide_divide_rat(one_one_rat,numeral_numeral_rat(aa_num_num(bit0,one))))) ).

% of_int_round_le
tff(fact_5851_of__int__round__ge,axiom,
    ! [Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(aa_num_num(bit0,one))))),aa_int_real(ring_1_of_int_real,aa_real_int(archim8280529875227126926d_real,Xb))) ).

% of_int_round_ge
tff(fact_5852_of__int__round__ge,axiom,
    ! [Xb: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,minus_minus_rat(Xb,divide_divide_rat(one_one_rat,numeral_numeral_rat(aa_num_num(bit0,one))))),aa_int_rat(ring_1_of_int_rat,archim7778729529865785530nd_rat(Xb))) ).

% of_int_round_ge
tff(fact_5853_of__int__round__gt,axiom,
    ! [Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(aa_num_num(bit0,one))))),aa_int_real(ring_1_of_int_real,aa_real_int(archim8280529875227126926d_real,Xb))) ).

% of_int_round_gt
tff(fact_5854_of__int__round__gt,axiom,
    ! [Xb: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,minus_minus_rat(Xb,divide_divide_rat(one_one_rat,numeral_numeral_rat(aa_num_num(bit0,one))))),aa_int_rat(ring_1_of_int_rat,archim7778729529865785530nd_rat(Xb))) ).

% of_int_round_gt
tff(fact_5855_mult__le__cancel__iff2,axiom,
    ! [Z: real,Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Z)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z),Xb)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z),Y))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y) ) ) ).

% mult_le_cancel_iff2
tff(fact_5856_mult__le__cancel__iff2,axiom,
    ! [Z: rat,Xb: 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(times_times_rat(Z),Xb)),aa_rat_rat(times_times_rat(Z),Y))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Y) ) ) ).

% mult_le_cancel_iff2
tff(fact_5857_mult__le__cancel__iff2,axiom,
    ! [Z: int,Xb: 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),Xb)),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,Xb),Y) ) ) ).

% mult_le_cancel_iff2
tff(fact_5858_vebt__inserti_Oraw__induct,axiom,
    ! [P: fun_VE8508601879845319688_nat_o,Xaa: produc3881548065746020326Ti_nat,H: heap_e7401611519738050253t_unit,H3: heap_e7401611519738050253t_unit,R2: vEBT_VEBTi,Nb: nat] :
      ( ! [Vebt_inserti: fun_VE8118064890490510894_VEBTi] :
          ( ! [A6: vEBT_VEBTi,B6: nat,H6: heap_e7401611519738050253t_unit,H7: heap_e7401611519738050253t_unit,R5: vEBT_VEBTi,N7: nat] :
              ( aa_nat_o(aa_VEB8883426185042993134_nat_o(heap_T2071195472996403633_VEBTi(aa_nat5408703056993926195_VEBTi(aa_VEB4833742232818206615_VEBTi(Vebt_inserti,A6),B6),H6,H7),R5),N7)
             => aa_nat_o(aa_VEB8883426185042993134_nat_o(aa_hea5256493101677210548_nat_o(aa_hea3862133550498874060_nat_o(aa_nat3537556108769025421_nat_o(aa_VEB2168324858650567793_nat_o(P,A6),B6),H6),H7),R5),N7) )
         => ! [T3: vEBT_VEBTi,X2: nat,Ta2: heap_e7401611519738050253t_unit,Xa2: heap_e7401611519738050253t_unit,R4: vEBT_VEBTi,N2: nat] :
              ( aa_nat_o(aa_VEB8883426185042993134_nat_o(heap_T2071195472996403633_VEBTi(aa_VEB836174949890266638_VEBTi(vEBT_c6028912655521741485_VEBTi(aa_nat3009484633486537743_VEBTi(aTP_Lamm_nl(Vebt_inserti),X2),aTP_Lamm_je(X2)),T3),Ta2,Xa2),R4),N2)
             => aa_nat_o(aa_VEB8883426185042993134_nat_o(aa_hea5256493101677210548_nat_o(aa_hea3862133550498874060_nat_o(aa_nat3537556108769025421_nat_o(aa_VEB2168324858650567793_nat_o(P,T3),X2),Ta2),Xa2),R4),N2) ) )
     => ( aa_nat_o(aa_VEB8883426185042993134_nat_o(heap_T2071195472996403633_VEBTi(aa_Pro1482084341531967383_VEBTi(produc3255295512018472142_VEBTi(vEBT_vebt_inserti),Xaa),H,H3),R2),Nb)
       => aa_nat_o(aa_VEB8883426185042993134_nat_o(aa_hea5256493101677210548_nat_o(aa_hea3862133550498874060_nat_o(produc2033849612716375976_nat_o(P,Xaa),H),H3),R2),Nb) ) ) ).

% vebt_inserti.raw_induct
tff(fact_5859_vebt__predi_Oraw__induct,axiom,
    ! [P: fun_VE2372197504167591133_nat_o,Xaa: produc3881548065746020326Ti_nat,H: heap_e7401611519738050253t_unit,H3: heap_e7401611519738050253t_unit,R2: option_nat,Nb: nat] :
      ( ! [Vebt_predi: fun_VE8516060933855021849on_nat] :
          ( ! [A6: vEBT_VEBTi,B6: nat,H6: heap_e7401611519738050253t_unit,H7: heap_e7401611519738050253t_unit,R5: option_nat,N7: nat] :
              ( aa_nat_o(heap_T306965388786959644on_nat(aa_nat947733127081326238on_nat(aa_VEB3566417204837707778on_nat(Vebt_predi,A6),B6),H6,H7,R5),N7)
             => aa_nat_o(aa_opt7444464167971770947_nat_o(aa_hea2436540527373857417_nat_o(aa_hea5443102814884448417_nat_o(aa_nat3515811770666409442_nat_o(aa_VEB4657881624497257926_nat_o(P,A6),B6),H6),H7),R5),N7) )
         => ! [T3: vEBT_VEBTi,X2: nat,Ta2: heap_e7401611519738050253t_unit,Xa2: heap_e7401611519738050253t_unit,R4: option_nat,N2: nat] :
              ( aa_nat_o(heap_T306965388786959644on_nat(aa_VEB2711320435582112505on_nat(vEBT_c6250501799366334488on_nat(aa_nat3457676333800472698on_nat(aTP_Lamm_nr(Vebt_predi),X2),aTP_Lamm_ez(X2)),T3),Ta2,Xa2,R4),N2)
             => aa_nat_o(aa_opt7444464167971770947_nat_o(aa_hea2436540527373857417_nat_o(aa_hea5443102814884448417_nat_o(aa_nat3515811770666409442_nat_o(aa_VEB4657881624497257926_nat_o(P,T3),X2),Ta2),Xa2),R4),N2) ) )
     => ( aa_nat_o(heap_T306965388786959644on_nat(aa_Pro2606730266749126658on_nat(produc8911080112929139129on_nat(vEBT_vebt_predi),Xaa),H,H3,R2),Nb)
       => aa_nat_o(aa_opt7444464167971770947_nat_o(aa_hea2436540527373857417_nat_o(aa_hea5443102814884448417_nat_o(aa_Pro190224997532733766_nat_o(produc4472639064241447549_nat_o(P),Xaa),H),H3),R2),Nb) ) ) ).

% vebt_predi.raw_induct
tff(fact_5860_vebt__succi_Oraw__induct,axiom,
    ! [P: fun_VE2372197504167591133_nat_o,Xaa: produc3881548065746020326Ti_nat,H: heap_e7401611519738050253t_unit,H3: heap_e7401611519738050253t_unit,R2: option_nat,Nb: nat] :
      ( ! [Vebt_succi: fun_VE8516060933855021849on_nat] :
          ( ! [A6: vEBT_VEBTi,B6: nat,H6: heap_e7401611519738050253t_unit,H7: heap_e7401611519738050253t_unit,R5: option_nat,N7: nat] :
              ( aa_nat_o(heap_T306965388786959644on_nat(aa_nat947733127081326238on_nat(aa_VEB3566417204837707778on_nat(Vebt_succi,A6),B6),H6,H7,R5),N7)
             => aa_nat_o(aa_opt7444464167971770947_nat_o(aa_hea2436540527373857417_nat_o(aa_hea5443102814884448417_nat_o(aa_nat3515811770666409442_nat_o(aa_VEB4657881624497257926_nat_o(P,A6),B6),H6),H7),R5),N7) )
         => ! [T3: vEBT_VEBTi,X2: nat,Ta2: heap_e7401611519738050253t_unit,Xa2: heap_e7401611519738050253t_unit,R4: option_nat,N2: nat] :
              ( aa_nat_o(heap_T306965388786959644on_nat(aa_VEB2711320435582112505on_nat(vEBT_c6250501799366334488on_nat(aa_nat3457676333800472698on_nat(aTP_Lamm_nx(Vebt_succi),X2),aTP_Lamm_fi(X2)),T3),Ta2,Xa2,R4),N2)
             => aa_nat_o(aa_opt7444464167971770947_nat_o(aa_hea2436540527373857417_nat_o(aa_hea5443102814884448417_nat_o(aa_nat3515811770666409442_nat_o(aa_VEB4657881624497257926_nat_o(P,T3),X2),Ta2),Xa2),R4),N2) ) )
     => ( aa_nat_o(heap_T306965388786959644on_nat(aa_Pro2606730266749126658on_nat(produc8911080112929139129on_nat(vEBT_vebt_succi),Xaa),H,H3,R2),Nb)
       => aa_nat_o(aa_opt7444464167971770947_nat_o(aa_hea2436540527373857417_nat_o(aa_hea5443102814884448417_nat_o(aa_Pro190224997532733766_nat_o(produc4472639064241447549_nat_o(P),Xaa),H),H3),R2),Nb) ) ) ).

% vebt_succi.raw_induct
tff(fact_5861_vebt__memberi_Oraw__induct,axiom,
    ! [P: fun_VE8999033711980490209_nat_o,Xaa: produc3881548065746020326Ti_nat,H: heap_e7401611519738050253t_unit,H3: heap_e7401611519738050253t_unit,R2: $o,Nb: nat] :
      ( ! [Vebt_memberi: fun_VE7134199479003659425Heap_o] :
          ( ! [A6: vEBT_VEBTi,B6: nat,H6: heap_e7401611519738050253t_unit,H7: heap_e7401611519738050253t_unit,R5: $o,N7: nat] :
              ( aa_nat_o(aa_o_fun_nat_o(heap_Time_effect_o(aa_nat2910048432523430448Heap_o(aa_VEB1455453220190609250Heap_o(Vebt_memberi,A6),B6),H6,H7),(R5)),N7)
             => aa_nat_o(aa_o_fun_nat_o(aa_hea6792647905190977245_nat_o(aa_hea8977266003532396635_nat_o(aa_nat1261834975400290800_nat_o(aa_VEB7158420292131534498_nat_o(P,A6),B6),H6),H7),(R5)),N7) )
         => ! [T3: vEBT_VEBTi,X2: nat,Ta2: heap_e7401611519738050253t_unit,Xa2: heap_e7401611519738050253t_unit,R4: $o,N2: nat] :
              ( aa_nat_o(aa_o_fun_nat_o(heap_Time_effect_o(aa_VEB8194921341634505877Heap_o(vEBT_c6104975476656191286Heap_o(aa_nat1227502447850154924Heap_o(aTP_Lamm_od(Vebt_memberi),X2),aTP_Lamm_jv(X2)),T3),Ta2,Xa2),(R4)),N2)
             => aa_nat_o(aa_o_fun_nat_o(aa_hea6792647905190977245_nat_o(aa_hea8977266003532396635_nat_o(aa_nat1261834975400290800_nat_o(aa_VEB7158420292131534498_nat_o(P,T3),X2),Ta2),Xa2),(R4)),N2) ) )
     => ( aa_nat_o(aa_o_fun_nat_o(heap_Time_effect_o(aa_Pro3916655985231737292Heap_o(produc770043135277712853Heap_o(vEBT_vebt_memberi),Xaa),H,H3),(R2)),Nb)
       => aa_nat_o(aa_o_fun_nat_o(aa_hea6792647905190977245_nat_o(aa_hea8977266003532396635_nat_o(aa_Pro3522642011054043020_nat_o(produc8468262472054731797_nat_o(P),Xaa),H),H3),(R2)),Nb) ) ) ).

% vebt_memberi.raw_induct
tff(fact_5862_divmod__step__integer__def,axiom,
    ! [L: num,Qr: produc8923325533196201883nteger] : unique4921790084139445826nteger(L,Qr) = produc6916734918728496179nteger(aTP_Lamm_my(L),Qr) ).

% divmod_step_integer_def
tff(fact_5863_VEBT__internal_Ovebt__succi_H_Osimps,axiom,
    ! [Ta: vEBT_VEBT,Tib: vEBT_VEBTi,Xb: nat] : aa_nat947733127081326238on_nat(aa_VEB3566417204837707778on_nat(aa_VEB1030146501360450209on_nat(vEBT_VEBT_vebt_succi,Ta),Tib),Xb) = aa_VEB2711320435582112505on_nat(vEBT_c6250501799366334488on_nat(aa_nat3457676333800472698on_nat(aTP_Lamm_ow(Ta),Xb),aTP_Lamm_fi(Xb)),Tib) ).

% VEBT_internal.vebt_succi'.simps
tff(fact_5864_vebt__buildupi__refines,axiom,
    ! [Nb: nat] : refine5565527176597971370_VEBTi(aa_nat5408703056993926195_VEBTi(vEBT_vebt_buildupi,Nb),aa_nat5408703056993926195_VEBTi(vEBT_V739175172307565963ildupi,Nb)) ).

% vebt_buildupi_refines
tff(fact_5865_vebt__inserti__refines,axiom,
    ! [Tib: vEBT_VEBTi,Xb: nat,Ta: vEBT_VEBT] : refine5565527176597971370_VEBTi(aa_nat5408703056993926195_VEBTi(aa_VEB4833742232818206615_VEBTi(vEBT_vebt_inserti,Tib),Xb),aa_nat5408703056993926195_VEBTi(aa_VEB4833742232818206615_VEBTi(aa_VEB6292075424854053302_VEBTi(vEBT_V3964819847710782039nserti,Ta),Tib),Xb)) ).

% vebt_inserti_refines
tff(fact_5866_split__part,axiom,
    ! [P: $o,Q: fun_nat_fun_nat_o,X3: product_prod_nat_nat] :
      ( aa_Pro6150637604445964947_nat_o(produc6081775807080527818_nat_o(aa_fun6699960423209655484_nat_o(aTP_Lamm_ox((P)),Q)),X3)
    <=> ( (P)
        & aa_Pro6150637604445964947_nat_o(produc6081775807080527818_nat_o(Q),X3) ) ) ).

% split_part
tff(fact_5867_split__part,axiom,
    ! [P: $o,Q: fun_int_fun_int_o,X3: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(aa_fun6133342665515662012_int_o(aTP_Lamm_oy((P)),Q)),X3)
    <=> ( (P)
        & aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(Q),X3) ) ) ).

% split_part
tff(fact_5868_Collect__case__prod__mono,axiom,
    ! [A2: fun_nat_fun_nat_o,B4: fun_nat_fun_nat_o] :
      ( ord_le768339454706049738_nat_o(A2,B4)
     => ord_le3146513528884898305at_nat(collec3392354462482085612at_nat(produc6081775807080527818_nat_o(A2)),collec3392354462482085612at_nat(produc6081775807080527818_nat_o(B4))) ) ).

% Collect_case_prod_mono
tff(fact_5869_Collect__case__prod__mono,axiom,
    ! [A2: fun_int_fun_int_o,B4: fun_int_fun_int_o] :
      ( ord_le77980682512674322_int_o(A2,B4)
     => ord_le2843351958646193337nt_int(collec213857154873943460nt_int(produc4947309494688390418_int_o(A2)),collec213857154873943460nt_int(produc4947309494688390418_int_o(B4))) ) ).

% Collect_case_prod_mono
tff(fact_5870_prod_Odisc__eq__case,axiom,
    ! [Prod: product_prod_nat_nat] : aa_Pro6150637604445964947_nat_o(produc6081775807080527818_nat_o(aTP_Lamm_oz),Prod) ).

% prod.disc_eq_case
tff(fact_5871_prod_Odisc__eq__case,axiom,
    ! [Prod: product_prod_int_int] : aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(aTP_Lamm_pa),Prod) ).

% prod.disc_eq_case
tff(fact_5872_UNIV__def,axiom,
    top_top_set_int = collect_int(aTP_Lamm_pb) ).

% UNIV_def
tff(fact_5873_UNIV__def,axiom,
    top_top_set_complex = collect_complex(aTP_Lamm_pc) ).

% UNIV_def
tff(fact_5874_UNIV__def,axiom,
    top_to4366644338036079209nt_int = collec213857154873943460nt_int(aTP_Lamm_pd) ).

% UNIV_def
tff(fact_5875_UNIV__def,axiom,
    top_top_set_real = collect_real(aTP_Lamm_pe) ).

% UNIV_def
tff(fact_5876_UNIV__def,axiom,
    top_top_set_nat = collect_nat(aTP_Lamm_dd) ).

% UNIV_def
tff(fact_5877_UNIV__def,axiom,
    top_top_set_char = collect_char(aTP_Lamm_pf) ).

% UNIV_def
tff(fact_5878_VEBT_Ocase__distrib,axiom,
    ! [H: fun_Pr3650993782763339253T_VEBT,F1: fun_op8229906066235313459T_VEBT,F22: fun_o_8142510167905321106T_VEBT,VEBT: vEBT_VEBT] : aa_Pro5421721056952420150T_VEBT(H,vEBT_c634343235235684882T_VEBT(F1,F22,VEBT)) = vEBT_c634343235235684882T_VEBT(aa_fun7544404396881244796T_VEBT(aTP_Lamm_pg(H),F1),aa_fun5380946117177639010T_VEBT(aTP_Lamm_ph(H),F22),VEBT) ).

% VEBT.case_distrib
tff(fact_5879_VEBT_Ocase__distrib,axiom,
    ! [H: fun_Pr7471953115884902709VEBT_o,F1: fun_op8229906066235313459T_VEBT,F22: fun_o_8142510167905321106T_VEBT,VEBT: vEBT_VEBT] :
      ( aa_Pro5475924895142249886VEBT_o(H,vEBT_c634343235235684882T_VEBT(F1,F22,VEBT))
    <=> aa_VEBT_VEBT_o(vEBT_case_VEBT_o(aa_fun5124765266653172784VEBT_o(aTP_Lamm_pi(H),F1),aa_fun449888776569364894un_o_o(aTP_Lamm_pj(H),F22)),VEBT) ) ).

% VEBT.case_distrib
tff(fact_5880_VEBT_Ocase__distrib,axiom,
    ! [H: fun_o_3001263932818058983T_VEBT,F1: fun_op541075054904862521VEBT_o,F22: fun_o_fun_o_o,VEBT: vEBT_VEBT] : aa_o_P8223745691486049304T_VEBT(H,aa_VEBT_VEBT_o(vEBT_case_VEBT_o(F1,F22),VEBT)) = vEBT_c634343235235684882T_VEBT(aa_fun8362160726713724924T_VEBT(aTP_Lamm_pk(H),F1),aa_fun4185041498930275652T_VEBT(aTP_Lamm_pl(H),F22),VEBT) ).

% VEBT.case_distrib
tff(fact_5881_VEBT_Ocase__distrib,axiom,
    ! [H: fun_o_o,F1: fun_op541075054904862521VEBT_o,F22: fun_o_fun_o_o,VEBT: vEBT_VEBT] :
      ( aa_o_o(H,aa_VEBT_VEBT_o(vEBT_case_VEBT_o(F1,F22),VEBT))
    <=> aa_VEBT_VEBT_o(vEBT_case_VEBT_o(aa_fun2455880738235528880VEBT_o(aTP_Lamm_pm(H),F1),aa_fun6846671739589424444un_o_o(aTP_Lamm_pn(H),F22)),VEBT) ) ).

% VEBT.case_distrib
tff(fact_5882_is__Node__def,axiom,
    ! [VEBT: vEBT_VEBT] :
      ( aa_VEBT_VEBT_o(vEBT_is_Node,VEBT)
    <=> ? [X112: option4927543243414619207at_nat,X122: nat,X132: list_VEBT_VEBT,X142: vEBT_VEBT] : VEBT = aa_VEB5832953974749077948T_VEBT(vEBT_Node(X112,X122,X132),X142) ) ).

% is_Node_def
tff(fact_5883_VEBT_OdiscI_I1_J,axiom,
    ! [VEBT: vEBT_VEBT,X11a: option4927543243414619207at_nat,X12a: nat,X13a: list_VEBT_VEBT,X14a: vEBT_VEBT] :
      ( ( VEBT = aa_VEB5832953974749077948T_VEBT(vEBT_Node(X11a,X12a,X13a),X14a) )
     => aa_VEBT_VEBT_o(vEBT_is_Node,VEBT) ) ).

% VEBT.discI(1)
tff(fact_5884_VEBT_Odisc_I1_J,axiom,
    ! [X11a: option4927543243414619207at_nat,X12a: nat,X13a: list_VEBT_VEBT,X14a: vEBT_VEBT] : aa_VEBT_VEBT_o(vEBT_is_Node,aa_VEB5832953974749077948T_VEBT(vEBT_Node(X11a,X12a,X13a),X14a)) ).

% VEBT.disc(1)
tff(fact_5885_VEBT_Osimps_I5_J,axiom,
    ! [F1: fun_op8229906066235313459T_VEBT,F22: fun_o_8142510167905321106T_VEBT,X11a: option4927543243414619207at_nat,X12a: nat,X13a: list_VEBT_VEBT,X14a: vEBT_VEBT] : vEBT_c634343235235684882T_VEBT(F1,F22,aa_VEB5832953974749077948T_VEBT(vEBT_Node(X11a,X12a,X13a),X14a)) = aa_VEB8692087746923902260T_VEBT(aa_lis2966823273985285125T_VEBT(aa_nat6705503460730637082T_VEBT(aa_opt2902932101092860132T_VEBT(F1,X11a),X12a),X13a),X14a) ).

% VEBT.simps(5)
tff(fact_5886_VEBT_Osimps_I5_J,axiom,
    ! [F1: fun_op541075054904862521VEBT_o,F22: fun_o_fun_o_o,X11a: option4927543243414619207at_nat,X12a: nat,X13a: list_VEBT_VEBT,X14a: vEBT_VEBT] :
      ( aa_VEBT_VEBT_o(vEBT_case_VEBT_o(F1,F22),aa_VEB5832953974749077948T_VEBT(vEBT_Node(X11a,X12a,X13a),X14a))
    <=> aa_VEBT_VEBT_o(aa_lis4184305612057247525VEBT_o(aa_nat8798449810042143974VEBT_o(aa_opt7072554309604550194VEBT_o(F1,X11a),X12a),X13a),X14a) ) ).

% VEBT.simps(5)
tff(fact_5887_option_Othe__def,axiom,
    ! [Option: option6094739215472523721uint32] : the_fu3257833805496833637uint32(Option) = case_o1863524157202330495uint32(undefi5883245521155754897uint32,aTP_Lamm_po,Option) ).

% option.the_def
tff(fact_5888_option_Othe__def,axiom,
    ! [Option: option3423447615326530758eger_o] : the_fu7763965076112332130eger_o(Option) = case_o2694340534094624761eger_o(undefi9030187962712549518eger_o,aTP_Lamm_pp,Option) ).

% option.the_def
tff(fact_5889_option_Othe__def,axiom,
    ! [Option: option5345528073251348614uint32] : the_fu4420311046302572578uint32(Option) = case_o61842737470731385uint32(undefi8935238026407452494uint32,aTP_Lamm_pq,Option) ).

% option.the_def
tff(fact_5890_option_Othe__def,axiom,
    ! [Option: option3426696969397718942uint32] : the_fu1630297056608640954uint32(Option) = case_o5757947405571428393uint32(undefi424500106705823974uint32,aTP_Lamm_pr,Option) ).

% option.the_def
tff(fact_5891_option_Othe__def,axiom,
    ! [Option: option4768052937140784030uint32] : the_fu5356795541574441914uint32(Option) = case_o3262472310887385129uint32(undefi4150998591671624934uint32,aTP_Lamm_ps,Option) ).

% option.the_def
tff(fact_5892_option_Othe__def,axiom,
    ! [Option: option_nat] : the_nat(Option) = case_option_nat_nat(undefined_nat,aTP_Lamm_di,Option) ).

% option.the_def
tff(fact_5893_option_Othe__def,axiom,
    ! [Option: option4927543243414619207at_nat] : the_Pr8591224930841456533at_nat(Option) = case_o7430979018509204427at_nat(undefi3946296454836805481at_nat,aTP_Lamm_pt,Option) ).

% option.the_def
tff(fact_5894_option_Othe__def,axiom,
    ! [Option: option_num] : aa_option_num_num(the_num,Option) = aa_option_num_num(case_option_num_num(undefined_num,aTP_Lamm_pu),Option) ).

% option.the_def
tff(fact_5895_one__natural_Orsp,axiom,
    one_one_nat = one_one_nat ).

% one_natural.rsp
tff(fact_5896_one__integer_Orsp,axiom,
    one_one_int = one_one_int ).

% one_integer.rsp
tff(fact_5897_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_5898_times__integer__code_I2_J,axiom,
    ! [L: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,zero_z3403309356797280102nteger),L) = zero_z3403309356797280102nteger ).

% times_integer_code(2)
tff(fact_5899_times__integer__code_I1_J,axiom,
    ! [K: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,K),zero_z3403309356797280102nteger) = zero_z3403309356797280102nteger ).

% times_integer_code(1)
tff(fact_5900_plus__integer__code_I2_J,axiom,
    ! [L: code_integer] : aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(zero_z3403309356797280102nteger),L) = L ).

% plus_integer_code(2)
tff(fact_5901_plus__integer__code_I1_J,axiom,
    ! [K: code_integer] : aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(K),zero_z3403309356797280102nteger) = K ).

% plus_integer_code(1)
tff(fact_5902_VEBT__internal_Ovebt__memberi_H_Oraw__induct,axiom,
    ! [P: fun_VE6326492262185073320_nat_o,Xaa: produc3960855945107176009Ti_nat,H: heap_e7401611519738050253t_unit,H3: heap_e7401611519738050253t_unit,R2: $o,Nb: nat] :
      ( ! [Vebt_memberi2: fun_VE8181444127144381288Heap_o] :
          ( ! [A6: vEBT_VEBT,B6: vEBT_VEBTi,Ba: nat,H6: heap_e7401611519738050253t_unit,H7: heap_e7401611519738050253t_unit,R5: $o,N7: nat] :
              ( aa_nat_o(aa_o_fun_nat_o(heap_Time_effect_o(aa_nat2910048432523430448Heap_o(aa_VEB1455453220190609250Heap_o(aa_VEB8672304581938048793Heap_o(Vebt_memberi2,A6),B6),Ba),H6,H7),(R5)),N7)
             => aa_nat_o(aa_o_fun_nat_o(aa_hea6792647905190977245_nat_o(aa_hea8977266003532396635_nat_o(aa_nat1261834975400290800_nat_o(aa_VEB7158420292131534498_nat_o(aa_VEB1179214148959353945_nat_o(P,A6),B6),Ba),H6),H7),(R5)),N7) )
         => ! [T3: vEBT_VEBT,Ti2: vEBT_VEBTi,X2: nat,Ta2: heap_e7401611519738050253t_unit,Tia: heap_e7401611519738050253t_unit,Xa2: $o,N2: nat] :
              ( aa_nat_o(aa_o_fun_nat_o(heap_Time_effect_o(aa_VEB8194921341634505877Heap_o(vEBT_c6104975476656191286Heap_o(aa_nat1227502447850154924Heap_o(aa_VEB6636772557416003555Heap_o(aTP_Lamm_qi(Vebt_memberi2),T3),X2),aTP_Lamm_jv(X2)),Ti2),Ta2,Tia),(Xa2)),N2)
             => aa_nat_o(aa_o_fun_nat_o(aa_hea6792647905190977245_nat_o(aa_hea8977266003532396635_nat_o(aa_nat1261834975400290800_nat_o(aa_VEB7158420292131534498_nat_o(aa_VEB1179214148959353945_nat_o(P,T3),Ti2),X2),Ta2),Tia),(Xa2)),N2) ) )
     => ( aa_nat_o(aa_o_fun_nat_o(heap_Time_effect_o(aa_Pro6173831900156065711Heap_o(produc5872130906356439992Heap_o(produc844971584604743726Heap_o(vEBT_V854960066525838166emberi)),Xaa),H,H3),(R2)),Nb)
       => aa_nat_o(aa_o_fun_nat_o(aa_hea6792647905190977245_nat_o(aa_hea8977266003532396635_nat_o(aa_Pro5855601548611308271_nat_o(produc4720546487998029688_nat_o(produc8632027548060743022_nat_o(P)),Xaa),H),H3),(R2)),Nb) ) ) ).

% VEBT_internal.vebt_memberi'.raw_induct
tff(fact_5903_minus__integer__code_I1_J,axiom,
    ! [K: code_integer] : aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(K),zero_z3403309356797280102nteger) = K ).

% minus_integer_code(1)
tff(fact_5904_VEBT__internal_Ovebt__memberi_H_Osimps,axiom,
    ! [Ta: vEBT_VEBT,Tib: vEBT_VEBTi,Xb: nat] : aa_nat2910048432523430448Heap_o(aa_VEB1455453220190609250Heap_o(aa_VEB8672304581938048793Heap_o(vEBT_V854960066525838166emberi,Ta),Tib),Xb) = aa_VEB8194921341634505877Heap_o(vEBT_c6104975476656191286Heap_o(aa_nat1227502447850154924Heap_o(aTP_Lamm_qw(Ta),Xb),aTP_Lamm_jv(Xb)),Tib) ).

% VEBT_internal.vebt_memberi'.simps
tff(fact_5905_VEBT__internal_Ovebt__inserti_H_Oraw__induct,axiom,
    ! [P: fun_VE7055638694157540631_nat_o,Xaa: produc3960855945107176009Ti_nat,H: heap_e7401611519738050253t_unit,H3: heap_e7401611519738050253t_unit,R2: vEBT_VEBTi,Nb: nat] :
      ( ! [Vebt_inserti2: fun_VE2018681615261037117_VEBTi] :
          ( ! [A6: vEBT_VEBT,B6: vEBT_VEBTi,Ba: nat,H6: heap_e7401611519738050253t_unit,H7: heap_e7401611519738050253t_unit,R5: vEBT_VEBTi,N7: nat] :
              ( aa_nat_o(aa_VEB8883426185042993134_nat_o(heap_T2071195472996403633_VEBTi(aa_nat5408703056993926195_VEBTi(aa_VEB4833742232818206615_VEBTi(aa_VEB6292075424854053302_VEBTi(Vebt_inserti2,A6),B6),Ba),H6,H7),R5),N7)
             => aa_nat_o(aa_VEB8883426185042993134_nat_o(aa_hea5256493101677210548_nat_o(aa_hea3862133550498874060_nat_o(aa_nat3537556108769025421_nat_o(aa_VEB2168324858650567793_nat_o(aa_VEB4449710241565182864_nat_o(P,A6),B6),Ba),H6),H7),R5),N7) )
         => ! [T3: vEBT_VEBT,Ti2: vEBT_VEBTi,X2: nat,Ta2: heap_e7401611519738050253t_unit,Tia: heap_e7401611519738050253t_unit,Xa2: vEBT_VEBTi,N2: nat] :
              ( aa_nat_o(aa_VEB8883426185042993134_nat_o(heap_T2071195472996403633_VEBTi(aa_VEB836174949890266638_VEBTi(vEBT_c6028912655521741485_VEBTi(aa_nat3009484633486537743_VEBTi(aa_VEB600817258366935214_VEBTi(aTP_Lamm_rt(Vebt_inserti2),T3),X2),aTP_Lamm_je(X2)),Ti2),Ta2,Tia),Xa2),N2)
             => aa_nat_o(aa_VEB8883426185042993134_nat_o(aa_hea5256493101677210548_nat_o(aa_hea3862133550498874060_nat_o(aa_nat3537556108769025421_nat_o(aa_VEB2168324858650567793_nat_o(aa_VEB4449710241565182864_nat_o(P,T3),Ti2),X2),Ta2),Tia),Xa2),N2) ) )
     => ( aa_nat_o(aa_VEB8883426185042993134_nat_o(heap_T2071195472996403633_VEBTi(aa_Pro721367001817565044_VEBTi(produc2943724498215716011_VEBTi(produc3258509807451162059_VEBTi(vEBT_V3964819847710782039nserti)),Xaa),H,H3),R2),Nb)
       => aa_nat_o(aa_VEB8883426185042993134_nat_o(aa_hea5256493101677210548_nat_o(aa_hea3862133550498874060_nat_o(aa_Pro9055585097396655438_nat_o(produc7249335055465628165_nat_o(produc1140538389304814757_nat_o(P)),Xaa),H),H3),R2),Nb) ) ) ).

% VEBT_internal.vebt_inserti'.raw_induct
tff(fact_5906_VEBT__internal_Ovebt__predi_H_Oraw__induct,axiom,
    ! [P: fun_VE3300235029744507244_nat_o,Xaa: produc3960855945107176009Ti_nat,H: heap_e7401611519738050253t_unit,H3: heap_e7401611519738050253t_unit,R2: option_nat,Nb: nat] :
      ( ! [Vebt_predi2: fun_VE3775134097089128360on_nat] :
          ( ! [A6: vEBT_VEBT,B6: vEBT_VEBTi,Ba: nat,H6: heap_e7401611519738050253t_unit,H7: heap_e7401611519738050253t_unit,R5: option_nat,N7: nat] :
              ( aa_nat_o(heap_T306965388786959644on_nat(aa_nat947733127081326238on_nat(aa_VEB3566417204837707778on_nat(aa_VEB1030146501360450209on_nat(Vebt_predi2,A6),B6),Ba),H6,H7,R5),N7)
             => aa_nat_o(aa_opt7444464167971770947_nat_o(aa_hea2436540527373857417_nat_o(aa_hea5443102814884448417_nat_o(aa_nat3515811770666409442_nat_o(aa_VEB4657881624497257926_nat_o(aa_VEB1759150482313007205_nat_o(P,A6),B6),Ba),H6),H7),R5),N7) )
         => ! [T3: vEBT_VEBT,Ti2: vEBT_VEBTi,X2: nat,Ta2: heap_e7401611519738050253t_unit,Tia: heap_e7401611519738050253t_unit,Xa2: option_nat,N2: nat] :
              ( aa_nat_o(heap_T306965388786959644on_nat(aa_VEB2711320435582112505on_nat(vEBT_c6250501799366334488on_nat(aa_nat3457676333800472698on_nat(aa_VEB6681700655871274393on_nat(aTP_Lamm_sk(Vebt_predi2),T3),X2),aTP_Lamm_ez(X2)),Ti2),Ta2,Tia,Xa2),N2)
             => aa_nat_o(aa_opt7444464167971770947_nat_o(aa_hea2436540527373857417_nat_o(aa_hea5443102814884448417_nat_o(aa_nat3515811770666409442_nat_o(aa_VEB4657881624497257926_nat_o(aa_VEB1759150482313007205_nat_o(P,T3),Ti2),X2),Ta2),Tia),Xa2),N2) ) )
     => ( aa_nat_o(heap_T306965388786959644on_nat(aa_Pro2727408364141080927on_nat(produc183673358652719894on_nat(produc3521483158534984758on_nat(vEBT_VEBT_vebt_predi)),Xaa),H,H3,R2),Nb)
       => aa_nat_o(aa_opt7444464167971770947_nat_o(aa_hea2436540527373857417_nat_o(aa_hea5443102814884448417_nat_o(aa_Pro3370481601460356387_nat_o(produc8196320604420998746_nat_o(produc1944727488097889274_nat_o(P)),Xaa),H),H3),R2),Nb) ) ) ).

% VEBT_internal.vebt_predi'.raw_induct
tff(fact_5907_VEBT__internal_Ovebt__succi_H_Oraw__induct,axiom,
    ! [P: fun_VE3300235029744507244_nat_o,Xaa: produc3960855945107176009Ti_nat,H: heap_e7401611519738050253t_unit,H3: heap_e7401611519738050253t_unit,R2: option_nat,Nb: nat] :
      ( ! [Vebt_succi2: fun_VE3775134097089128360on_nat] :
          ( ! [A6: vEBT_VEBT,B6: vEBT_VEBTi,Ba: nat,H6: heap_e7401611519738050253t_unit,H7: heap_e7401611519738050253t_unit,R5: option_nat,N7: nat] :
              ( aa_nat_o(heap_T306965388786959644on_nat(aa_nat947733127081326238on_nat(aa_VEB3566417204837707778on_nat(aa_VEB1030146501360450209on_nat(Vebt_succi2,A6),B6),Ba),H6,H7,R5),N7)
             => aa_nat_o(aa_opt7444464167971770947_nat_o(aa_hea2436540527373857417_nat_o(aa_hea5443102814884448417_nat_o(aa_nat3515811770666409442_nat_o(aa_VEB4657881624497257926_nat_o(aa_VEB1759150482313007205_nat_o(P,A6),B6),Ba),H6),H7),R5),N7) )
         => ! [T3: vEBT_VEBT,Ti2: vEBT_VEBTi,X2: nat,Ta2: heap_e7401611519738050253t_unit,Tia: heap_e7401611519738050253t_unit,Xa2: option_nat,N2: nat] :
              ( aa_nat_o(heap_T306965388786959644on_nat(aa_VEB2711320435582112505on_nat(vEBT_c6250501799366334488on_nat(aa_nat3457676333800472698on_nat(aa_VEB6681700655871274393on_nat(aTP_Lamm_sz(Vebt_succi2),T3),X2),aTP_Lamm_fi(X2)),Ti2),Ta2,Tia,Xa2),N2)
             => aa_nat_o(aa_opt7444464167971770947_nat_o(aa_hea2436540527373857417_nat_o(aa_hea5443102814884448417_nat_o(aa_nat3515811770666409442_nat_o(aa_VEB4657881624497257926_nat_o(aa_VEB1759150482313007205_nat_o(P,T3),Ti2),X2),Ta2),Tia),Xa2),N2) ) )
     => ( aa_nat_o(heap_T306965388786959644on_nat(aa_Pro2727408364141080927on_nat(produc183673358652719894on_nat(produc3521483158534984758on_nat(vEBT_VEBT_vebt_succi)),Xaa),H,H3,R2),Nb)
       => aa_nat_o(aa_opt7444464167971770947_nat_o(aa_hea2436540527373857417_nat_o(aa_hea5443102814884448417_nat_o(aa_Pro3370481601460356387_nat_o(produc8196320604420998746_nat_o(produc1944727488097889274_nat_o(P)),Xaa),H),H3),R2),Nb) ) ) ).

% VEBT_internal.vebt_succi'.raw_induct
tff(fact_5908_VEBT__internal_Ovebt__inserti_H_Osimps,axiom,
    ! [Ta: vEBT_VEBT,Tib: vEBT_VEBTi,Xb: nat] : aa_nat5408703056993926195_VEBTi(aa_VEB4833742232818206615_VEBTi(aa_VEB6292075424854053302_VEBTi(vEBT_V3964819847710782039nserti,Ta),Tib),Xb) = aa_VEB836174949890266638_VEBTi(vEBT_c6028912655521741485_VEBTi(aa_nat3009484633486537743_VEBTi(aTP_Lamm_tw(Ta),Xb),aTP_Lamm_je(Xb)),Tib) ).

% VEBT_internal.vebt_inserti'.simps
tff(fact_5909_VEBT__internal_Ovebt__predi_H_Osimps,axiom,
    ! [Ta: vEBT_VEBT,Tib: vEBT_VEBTi,Xb: nat] : aa_nat947733127081326238on_nat(aa_VEB3566417204837707778on_nat(aa_VEB1030146501360450209on_nat(vEBT_VEBT_vebt_predi,Ta),Tib),Xb) = aa_VEB2711320435582112505on_nat(vEBT_c6250501799366334488on_nat(aa_nat3457676333800472698on_nat(aTP_Lamm_ul(Ta),Xb),aTP_Lamm_ez(Xb)),Tib) ).

% VEBT_internal.vebt_predi'.simps
tff(fact_5910_quickcheck__narrowing__samples_Opartial__term__of__sample__def,axiom,
    ! [A_of_integer: fun_Co4342131667711709301uint32,Zero: uint32,I: code_integer] :
      aa_Cod8401701657498852871uint32(code_T8223577488154910793uint32(A_of_integer,Zero),I) = $ite(
        aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,I),zero_z3403309356797280102nteger),
        undefined_uint32,
        $ite(
          I = zero_z3403309356797280102nteger,
          Zero,
          $ite(aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(I),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) = zero_z3403309356797280102nteger,aa_Pro3737690986720249065uint32(produc1510406741064981791uint32,aa_Cod1324813365842506078uint32(A_of_integer,aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(I),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))))),aa_Pro3737690986720249065uint32(produc9004433772639906525uint32,aa_Cod1324813365842506078uint32(A_of_integer,aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(I),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)))),one_one_Code_integer)))) ) ) ).

% quickcheck_narrowing_samples.partial_term_of_sample_def
tff(fact_5911_quickcheck__narrowing__samples_Opartial__term__of__sample__def,axiom,
    ! [A_of_integer: fun_Co4416855287806144957at_nat,Zero: nat,I: code_integer] :
      aa_Code_integer_nat(code_T3174267988120873389le_nat(A_of_integer,Zero),I) = $ite(
        aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,I),zero_z3403309356797280102nteger),
        undefined_nat,
        $ite(
          I = zero_z3403309356797280102nteger,
          Zero,
          $ite(aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(I),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) = zero_z3403309356797280102nteger,aa_Pro4243173770044397973at_nat(product_snd_nat_nat,aa_Cod330945565099897510at_nat(A_of_integer,aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(I),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))))),aa_Pro4243173770044397973at_nat(product_fst_nat_nat,aa_Cod330945565099897510at_nat(A_of_integer,aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(I),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)))),one_one_Code_integer)))) ) ) ).

% quickcheck_narrowing_samples.partial_term_of_sample_def
tff(fact_5912_quickcheck__narrowing__samples_Opartial__term__of__sample__def,axiom,
    ! [A_of_integer: fun_Co4113693717567439989nt_int,Zero: int,I: code_integer] :
      aa_Code_integer_int(code_T3171777517611823113le_int(A_of_integer,Zero),I) = $ite(
        aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,I),zero_z3403309356797280102nteger),
        undefined_int,
        $ite(
          I = zero_z3403309356797280102nteger,
          Zero,
          $ite(aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(I),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) = zero_z3403309356797280102nteger,aa_Pro6799258434465214121nt_int(product_snd_int_int,aa_Cod6375820294346531166nt_int(A_of_integer,aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(I),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))))),aa_Pro6799258434465214121nt_int(product_fst_int_int,aa_Cod6375820294346531166nt_int(A_of_integer,aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(I),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)))),one_one_Code_integer)))) ) ) ).

% quickcheck_narrowing_samples.partial_term_of_sample_def
tff(fact_5913_quickcheck__narrowing__samples_Opartial__term__of__sample__def,axiom,
    ! [A_of_integer: fun_Co6597186368970283863nteger,Zero: code_integer,I: code_integer] :
      aa_Cod1783974652533396728nteger(code_T5410181022262596154nteger(A_of_integer,Zero),I) = $ite(
        aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,I),zero_z3403309356797280102nteger),
        undefi8781568570243851253nteger,
        $ite(
          I = zero_z3403309356797280102nteger,
          Zero,
          $ite(aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(I),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) = zero_z3403309356797280102nteger,aa_Pro5245656335456409788nteger(produc6174133586879617921nteger,aa_Cod6977894654539884864nteger(A_of_integer,aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(I),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))))),aa_Pro5245656335456409788nteger(produc8508995932063986495nteger,aa_Cod6977894654539884864nteger(A_of_integer,aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(I),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)))),one_one_Code_integer)))) ) ) ).

% quickcheck_narrowing_samples.partial_term_of_sample_def
tff(fact_5914_mod__h__bot__normalize,axiom,
    ! [H: heap_e7401611519738050253t_unit,P: assn] :
      ( aa_hea7994211679605476167unit_o(syntax732013874993758672t_unit(undefi5883245521155754897uint32),H)
     => ( aa_Pro653082453696054616_nat_o(rep_assn(P),aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(H),bot_bot_set_nat))
      <=> aa_Pro653082453696054616_nat_o(rep_assn(P),aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(undefi8805113437130903861t_unit),bot_bot_set_nat)) ) ) ).

% mod_h_bot_normalize
tff(fact_5915_mod__h__bot__normalize,axiom,
    ! [H: heap_e7401611519738050253t_unit,P: assn] :
      ( aa_hea7994211679605476167unit_o(syntax3231729521921793939t_unit(undefi9030187962712549518eger_o),H)
     => ( aa_Pro653082453696054616_nat_o(rep_assn(P),aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(H),bot_bot_set_nat))
      <=> aa_Pro653082453696054616_nat_o(rep_assn(P),aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(undefi8805113437130903861t_unit),bot_bot_set_nat)) ) ) ).

% mod_h_bot_normalize
tff(fact_5916_mod__h__bot__normalize,axiom,
    ! [H: heap_e7401611519738050253t_unit,P: assn] :
      ( aa_hea7994211679605476167unit_o(syntax2407925819610226131t_unit(undefi8935238026407452494uint32),H)
     => ( aa_Pro653082453696054616_nat_o(rep_assn(P),aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(H),bot_bot_set_nat))
      <=> aa_Pro653082453696054616_nat_o(rep_assn(P),aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(undefi8805113437130903861t_unit),bot_bot_set_nat)) ) ) ).

% mod_h_bot_normalize
tff(fact_5917_mod__h__bot__normalize,axiom,
    ! [H: heap_e7401611519738050253t_unit,P: assn] :
      ( aa_hea7994211679605476167unit_o(syntax953652980939177915t_unit(undefi424500106705823974uint32),H)
     => ( aa_Pro653082453696054616_nat_o(rep_assn(P),aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(H),bot_bot_set_nat))
      <=> aa_Pro653082453696054616_nat_o(rep_assn(P),aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(undefi8805113437130903861t_unit),bot_bot_set_nat)) ) ) ).

% mod_h_bot_normalize
tff(fact_5918_mod__h__bot__normalize,axiom,
    ! [H: heap_e7401611519738050253t_unit,P: assn] :
      ( aa_hea7994211679605476167unit_o(syntax3922806061600818107t_unit(undefi4150998591671624934uint32),H)
     => ( aa_Pro653082453696054616_nat_o(rep_assn(P),aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(H),bot_bot_set_nat))
      <=> aa_Pro653082453696054616_nat_o(rep_assn(P),aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(undefi8805113437130903861t_unit),bot_bot_set_nat)) ) ) ).

% mod_h_bot_normalize
tff(fact_5919_divmod__algorithm__code_I6_J,axiom,
    ! [M: num,Nb: num] : unique5055182867167087721od_nat(aa_num_num(bit1,M),aa_num_num(bit0,Nb)) = aa_Pro262675715343573244at_nat(produc2626176000494625587at_nat(aTP_Lamm_um),unique5055182867167087721od_nat(M,Nb)) ).

% divmod_algorithm_code(6)
tff(fact_5920_divmod__algorithm__code_I6_J,axiom,
    ! [M: num,Nb: num] : unique3479559517661332726nteger(aa_num_num(bit1,M),aa_num_num(bit0,Nb)) = produc6916734918728496179nteger(aTP_Lamm_un,unique3479559517661332726nteger(M,Nb)) ).

% divmod_algorithm_code(6)
tff(fact_5921_divmod__algorithm__code_I6_J,axiom,
    ! [M: num,Nb: num] : unique5052692396658037445od_int(aa_num_num(bit1,M),aa_num_num(bit0,Nb)) = aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(aTP_Lamm_uo),unique5052692396658037445od_int(M,Nb)) ).

% divmod_algorithm_code(6)
tff(fact_5922_neg__eucl__rel__int__mult__2,axiom,
    ! [B: int,A: int,Q2: int,R2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),zero_zero_int)
     => ( aa_Pro4592068669816069595_int_o(eucl_rel_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,A),one_one_int),B),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q2),R2))
       => aa_Pro4592068669816069595_int_o(eucl_rel_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B)),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q2),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),R2)),one_one_int))) ) ) ).

% neg_eucl_rel_int_mult_2
tff(fact_5923_divmod__algorithm__code_I7_J,axiom,
    ! [M: num,Nb: num] :
      unique5052692396658037445od_int(aa_num_num(bit0,M),aa_num_num(bit1,Nb)) = $ite(aa_num_o(ord_less_eq_num(M),Nb),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,Nb)),unique5052692396658037445od_int(aa_num_num(bit0,M),aa_num_num(bit0,aa_num_num(bit1,Nb))))) ).

% divmod_algorithm_code(7)
tff(fact_5924_divmod__algorithm__code_I7_J,axiom,
    ! [M: num,Nb: num] :
      unique5055182867167087721od_nat(aa_num_num(bit0,M),aa_num_num(bit1,Nb)) = $ite(aa_num_o(ord_less_eq_num(M),Nb),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,Nb)),unique5055182867167087721od_nat(aa_num_num(bit0,M),aa_num_num(bit0,aa_num_num(bit1,Nb))))) ).

% divmod_algorithm_code(7)
tff(fact_5925_divmod__algorithm__code_I7_J,axiom,
    ! [M: num,Nb: num] :
      unique3479559517661332726nteger(aa_num_num(bit0,M),aa_num_num(bit1,Nb)) = $ite(aa_num_o(ord_less_eq_num(M),Nb),aa_Cod6977894654539884864nteger(produc1086072967326762835nteger(zero_z3403309356797280102nteger),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,M))),unique4921790084139445826nteger(aa_num_num(bit1,Nb),unique3479559517661332726nteger(aa_num_num(bit0,M),aa_num_num(bit0,aa_num_num(bit1,Nb))))) ).

% divmod_algorithm_code(7)
tff(fact_5926_divmod__algorithm__code_I8_J,axiom,
    ! [M: num,Nb: num] :
      unique5052692396658037445od_int(aa_num_num(bit1,M),aa_num_num(bit1,Nb)) = $ite(aa_num_o(ord_less_num(M),Nb),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,Nb)),unique5052692396658037445od_int(aa_num_num(bit1,M),aa_num_num(bit0,aa_num_num(bit1,Nb))))) ).

% divmod_algorithm_code(8)
tff(fact_5927_divmod__algorithm__code_I8_J,axiom,
    ! [M: num,Nb: num] :
      unique5055182867167087721od_nat(aa_num_num(bit1,M),aa_num_num(bit1,Nb)) = $ite(aa_num_o(ord_less_num(M),Nb),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,Nb)),unique5055182867167087721od_nat(aa_num_num(bit1,M),aa_num_num(bit0,aa_num_num(bit1,Nb))))) ).

% divmod_algorithm_code(8)
tff(fact_5928_divmod__algorithm__code_I8_J,axiom,
    ! [M: num,Nb: num] :
      unique3479559517661332726nteger(aa_num_num(bit1,M),aa_num_num(bit1,Nb)) = $ite(aa_num_o(ord_less_num(M),Nb),aa_Cod6977894654539884864nteger(produc1086072967326762835nteger(zero_z3403309356797280102nteger),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit1,M))),unique4921790084139445826nteger(aa_num_num(bit1,Nb),unique3479559517661332726nteger(aa_num_num(bit1,M),aa_num_num(bit0,aa_num_num(bit1,Nb))))) ).

% divmod_algorithm_code(8)
tff(fact_5929_numeral__div__numeral,axiom,
    ! [K: num,L: num] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_num_nat(numeral_numeral_nat,K)),aa_num_nat(numeral_numeral_nat,L)) = aa_Pro4243173770044397973at_nat(product_fst_nat_nat,unique5055182867167087721od_nat(K,L)) ).

% numeral_div_numeral
tff(fact_5930_numeral__div__numeral,axiom,
    ! [K: num,L: num] : aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_num_Code_integer(numera6620942414471956472nteger,K)),aa_num_Code_integer(numera6620942414471956472nteger,L)) = aa_Pro5245656335456409788nteger(produc8508995932063986495nteger,unique3479559517661332726nteger(K,L)) ).

% numeral_div_numeral
tff(fact_5931_numeral__div__numeral,axiom,
    ! [K: num,L: num] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_num_int(numeral_numeral_int,K)),aa_num_int(numeral_numeral_int,L)) = aa_Pro6799258434465214121nt_int(product_fst_int_int,unique5052692396658037445od_int(K,L)) ).

% numeral_div_numeral
tff(fact_5932_numeral__mod__numeral,axiom,
    ! [K: num,L: num] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,K)),aa_num_nat(numeral_numeral_nat,L)) = aa_Pro4243173770044397973at_nat(product_snd_nat_nat,unique5055182867167087721od_nat(K,L)) ).

% numeral_mod_numeral
tff(fact_5933_numeral__mod__numeral,axiom,
    ! [K: num,L: num] : aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_num_Code_integer(numera6620942414471956472nteger,K)),aa_num_Code_integer(numera6620942414471956472nteger,L)) = aa_Pro5245656335456409788nteger(produc6174133586879617921nteger,unique3479559517661332726nteger(K,L)) ).

% numeral_mod_numeral
tff(fact_5934_numeral__mod__numeral,axiom,
    ! [K: num,L: num] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,K)),aa_num_int(numeral_numeral_int,L)) = aa_Pro6799258434465214121nt_int(product_snd_int_int,unique5052692396658037445od_int(K,L)) ).

% numeral_mod_numeral
tff(fact_5935_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_5936_divmod__algorithm__code_I2_J,axiom,
    ! [M: num] : unique3479559517661332726nteger(M,one) = aa_Cod6977894654539884864nteger(produc1086072967326762835nteger(aa_num_Code_integer(numera6620942414471956472nteger,M)),zero_z3403309356797280102nteger) ).

% divmod_algorithm_code(2)
tff(fact_5937_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_5938_divmod__algorithm__code_I3_J,axiom,
    ! [Nb: num] : unique5055182867167087721od_nat(one,aa_num_num(bit0,Nb)) = 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_5939_divmod__algorithm__code_I3_J,axiom,
    ! [Nb: num] : unique3479559517661332726nteger(one,aa_num_num(bit0,Nb)) = aa_Cod6977894654539884864nteger(produc1086072967326762835nteger(zero_z3403309356797280102nteger),aa_num_Code_integer(numera6620942414471956472nteger,one)) ).

% divmod_algorithm_code(3)
tff(fact_5940_divmod__algorithm__code_I3_J,axiom,
    ! [Nb: num] : unique5052692396658037445od_int(one,aa_num_num(bit0,Nb)) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,zero_zero_int),aa_num_int(numeral_numeral_int,one)) ).

% divmod_algorithm_code(3)
tff(fact_5941_divmod__algorithm__code_I4_J,axiom,
    ! [Nb: num] : unique5055182867167087721od_nat(one,aa_num_num(bit1,Nb)) = 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_5942_divmod__algorithm__code_I4_J,axiom,
    ! [Nb: num] : unique3479559517661332726nteger(one,aa_num_num(bit1,Nb)) = aa_Cod6977894654539884864nteger(produc1086072967326762835nteger(zero_z3403309356797280102nteger),aa_num_Code_integer(numera6620942414471956472nteger,one)) ).

% divmod_algorithm_code(4)
tff(fact_5943_divmod__algorithm__code_I4_J,axiom,
    ! [Nb: num] : unique5052692396658037445od_int(one,aa_num_num(bit1,Nb)) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,zero_zero_int),aa_num_int(numeral_numeral_int,one)) ).

% divmod_algorithm_code(4)
tff(fact_5944_Suc__0__div__numeral,axiom,
    ! [K: num] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,zero_zero_nat)),aa_num_nat(numeral_numeral_nat,K)) = aa_Pro4243173770044397973at_nat(product_fst_nat_nat,unique5055182867167087721od_nat(one,K)) ).

% Suc_0_div_numeral
tff(fact_5945_one__div__numeral,axiom,
    ! [Nb: num] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,Nb)) = aa_Pro4243173770044397973at_nat(product_fst_nat_nat,unique5055182867167087721od_nat(one,Nb)) ).

% one_div_numeral
tff(fact_5946_one__div__numeral,axiom,
    ! [Nb: num] : aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(one_one_Code_integer),aa_num_Code_integer(numera6620942414471956472nteger,Nb)) = aa_Pro5245656335456409788nteger(produc8508995932063986495nteger,unique3479559517661332726nteger(one,Nb)) ).

% one_div_numeral
tff(fact_5947_one__div__numeral,axiom,
    ! [Nb: num] : aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),aa_num_int(numeral_numeral_int,Nb)) = aa_Pro6799258434465214121nt_int(product_fst_int_int,unique5052692396658037445od_int(one,Nb)) ).

% one_div_numeral
tff(fact_5948_one__mod__numeral,axiom,
    ! [Nb: num] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,Nb)) = aa_Pro4243173770044397973at_nat(product_snd_nat_nat,unique5055182867167087721od_nat(one,Nb)) ).

% one_mod_numeral
tff(fact_5949_one__mod__numeral,axiom,
    ! [Nb: num] : aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(one_one_Code_integer),aa_num_Code_integer(numera6620942414471956472nteger,Nb)) = aa_Pro5245656335456409788nteger(produc6174133586879617921nteger,unique3479559517661332726nteger(one,Nb)) ).

% one_mod_numeral
tff(fact_5950_one__mod__numeral,axiom,
    ! [Nb: num] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,one_one_int),aa_num_int(numeral_numeral_int,Nb)) = aa_Pro6799258434465214121nt_int(product_snd_int_int,unique5052692396658037445od_int(one,Nb)) ).

% one_mod_numeral
tff(fact_5951_Suc__0__mod__numeral,axiom,
    ! [K: num] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,zero_zero_nat)),aa_num_nat(numeral_numeral_nat,K)) = aa_Pro4243173770044397973at_nat(product_snd_nat_nat,unique5055182867167087721od_nat(one,K)) ).

% Suc_0_mod_numeral
tff(fact_5952_divmod__algorithm__code_I5_J,axiom,
    ! [M: num,Nb: num] : unique5055182867167087721od_nat(aa_num_num(bit0,M),aa_num_num(bit0,Nb)) = aa_Pro262675715343573244at_nat(produc2626176000494625587at_nat(aTP_Lamm_up),unique5055182867167087721od_nat(M,Nb)) ).

% divmod_algorithm_code(5)
tff(fact_5953_divmod__algorithm__code_I5_J,axiom,
    ! [M: num,Nb: num] : unique3479559517661332726nteger(aa_num_num(bit0,M),aa_num_num(bit0,Nb)) = produc6916734918728496179nteger(aTP_Lamm_uq,unique3479559517661332726nteger(M,Nb)) ).

% divmod_algorithm_code(5)
tff(fact_5954_divmod__algorithm__code_I5_J,axiom,
    ! [M: num,Nb: num] : unique5052692396658037445od_int(aa_num_num(bit0,M),aa_num_num(bit0,Nb)) = aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(aTP_Lamm_ur),unique5052692396658037445od_int(M,Nb)) ).

% divmod_algorithm_code(5)
tff(fact_5955_unique__remainder,axiom,
    ! [A: int,B: int,Q2: int,R2: int,Q5: int,R3: int] :
      ( aa_Pro4592068669816069595_int_o(eucl_rel_int(A,B),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q2),R2))
     => ( aa_Pro4592068669816069595_int_o(eucl_rel_int(A,B),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q5),R3))
       => ( R2 = R3 ) ) ) ).

% unique_remainder
tff(fact_5956_unique__quotient,axiom,
    ! [A: int,B: int,Q2: int,R2: int,Q5: int,R3: int] :
      ( aa_Pro4592068669816069595_int_o(eucl_rel_int(A,B),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q2),R2))
     => ( aa_Pro4592068669816069595_int_o(eucl_rel_int(A,B),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q5),R3))
       => ( Q2 = Q5 ) ) ) ).

% unique_quotient
tff(fact_5957_VEBT_Odisc__eq__case_I1_J,axiom,
    ! [VEBT: vEBT_VEBT] :
      ( aa_VEBT_VEBT_o(vEBT_is_Node,VEBT)
    <=> aa_VEBT_VEBT_o(vEBT_case_VEBT_o(aTP_Lamm_us,aTP_Lamm_ut),VEBT) ) ).

% VEBT.disc_eq_case(1)
tff(fact_5958_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_5959_mod__int__unique,axiom,
    ! [K: int,L: int,Q2: int,R2: int] :
      ( aa_Pro4592068669816069595_int_o(eucl_rel_int(K,L),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q2),R2))
     => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),L) = R2 ) ) ).

% mod_int_unique
tff(fact_5960_div__int__unique,axiom,
    ! [K: int,L: int,Q2: int,R2: int] :
      ( aa_Pro4592068669816069595_int_o(eucl_rel_int(K,L),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q2),R2))
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,K),L) = Q2 ) ) ).

% div_int_unique
tff(fact_5961_divmod__integer_H__def,axiom,
    ! [M: num,Nb: num] : unique3479559517661332726nteger(M,Nb) = aa_Cod6977894654539884864nteger(produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_num_Code_integer(numera6620942414471956472nteger,M)),aa_num_Code_integer(numera6620942414471956472nteger,Nb))),aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_num_Code_integer(numera6620942414471956472nteger,M)),aa_num_Code_integer(numera6620942414471956472nteger,Nb))) ).

% divmod_integer'_def
tff(fact_5962_fst__divmod,axiom,
    ! [M: num,Nb: num] : aa_Pro4243173770044397973at_nat(product_fst_nat_nat,unique5055182867167087721od_nat(M,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,Nb)) ).

% fst_divmod
tff(fact_5963_fst__divmod,axiom,
    ! [M: num,Nb: num] : aa_Pro5245656335456409788nteger(produc8508995932063986495nteger,unique3479559517661332726nteger(M,Nb)) = aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_num_Code_integer(numera6620942414471956472nteger,M)),aa_num_Code_integer(numera6620942414471956472nteger,Nb)) ).

% fst_divmod
tff(fact_5964_fst__divmod,axiom,
    ! [M: num,Nb: num] : aa_Pro6799258434465214121nt_int(product_fst_int_int,unique5052692396658037445od_int(M,Nb)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,Nb)) ).

% fst_divmod
tff(fact_5965_snd__divmod,axiom,
    ! [M: num,Nb: num] : aa_Pro4243173770044397973at_nat(product_snd_nat_nat,unique5055182867167087721od_nat(M,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,Nb)) ).

% snd_divmod
tff(fact_5966_snd__divmod,axiom,
    ! [M: num,Nb: num] : aa_Pro5245656335456409788nteger(produc6174133586879617921nteger,unique3479559517661332726nteger(M,Nb)) = aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_num_Code_integer(numera6620942414471956472nteger,M)),aa_num_Code_integer(numera6620942414471956472nteger,Nb)) ).

% snd_divmod
tff(fact_5967_snd__divmod,axiom,
    ! [M: num,Nb: num] : aa_Pro6799258434465214121nt_int(product_snd_int_int,unique5052692396658037445od_int(M,Nb)) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,Nb)) ).

% snd_divmod
tff(fact_5968_eucl__rel__int__dividesI,axiom,
    ! [L: int,K: int,Q2: int] :
      ( ( L != zero_zero_int )
     => ( ( K = aa_int_int(aa_int_fun_int_int(times_times_int,Q2),L) )
       => aa_Pro4592068669816069595_int_o(eucl_rel_int(K,L),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q2),zero_zero_int)) ) ) ).

% eucl_rel_int_dividesI
tff(fact_5969_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_5970_divmod__int__def,axiom,
    ! [M: num,Nb: num] : unique5052692396658037445od_int(M,Nb) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,Nb))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,Nb))) ).

% divmod_int_def
tff(fact_5971_divmod__def,axiom,
    ! [M: num,Nb: num] : unique5055182867167087721od_nat(M,Nb) = 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,Nb))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,Nb))) ).

% divmod_def
tff(fact_5972_divmod__def,axiom,
    ! [M: num,Nb: num] : unique3479559517661332726nteger(M,Nb) = aa_Cod6977894654539884864nteger(produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_num_Code_integer(numera6620942414471956472nteger,M)),aa_num_Code_integer(numera6620942414471956472nteger,Nb))),aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_num_Code_integer(numera6620942414471956472nteger,M)),aa_num_Code_integer(numera6620942414471956472nteger,Nb))) ).

% divmod_def
tff(fact_5973_divmod__def,axiom,
    ! [M: num,Nb: num] : unique5052692396658037445od_int(M,Nb) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,Nb))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,Nb))) ).

% divmod_def
tff(fact_5974_divmod_H__nat__def,axiom,
    ! [M: num,Nb: num] : unique5055182867167087721od_nat(M,Nb) = 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,Nb))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,Nb))) ).

% divmod'_nat_def
tff(fact_5975_eucl__rel__int__iff,axiom,
    ! [K: int,L: int,Q2: int,R2: int] :
      ( aa_Pro4592068669816069595_int_o(eucl_rel_int(K,L),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q2),R2))
    <=> ( ( K = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,L),Q2)),R2) )
        & $ite(
            aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),L),
            ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),R2)
            & aa_int_o(aa_int_fun_int_o(ord_less_int,R2),L) ),
            $ite(
              aa_int_o(aa_int_fun_int_o(ord_less_int,L),zero_zero_int),
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,L),R2)
              & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,R2),zero_zero_int) ),
              Q2 = zero_zero_int ) ) ) ) ).

% eucl_rel_int_iff
tff(fact_5976_divmod__divmod__step,axiom,
    ! [M: num,Nb: num] :
      unique5052692396658037445od_int(M,Nb) = $ite(aa_num_o(ord_less_num(M),Nb),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(Nb),unique5052692396658037445od_int(M,aa_num_num(bit0,Nb)))) ).

% divmod_divmod_step
tff(fact_5977_divmod__divmod__step,axiom,
    ! [M: num,Nb: num] :
      unique5055182867167087721od_nat(M,Nb) = $ite(aa_num_o(ord_less_num(M),Nb),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(Nb),unique5055182867167087721od_nat(M,aa_num_num(bit0,Nb)))) ).

% divmod_divmod_step
tff(fact_5978_divmod__divmod__step,axiom,
    ! [M: num,Nb: num] :
      unique3479559517661332726nteger(M,Nb) = $ite(aa_num_o(ord_less_num(M),Nb),aa_Cod6977894654539884864nteger(produc1086072967326762835nteger(zero_z3403309356797280102nteger),aa_num_Code_integer(numera6620942414471956472nteger,M)),unique4921790084139445826nteger(Nb,unique3479559517661332726nteger(M,aa_num_num(bit0,Nb)))) ).

% divmod_divmod_step
tff(fact_5979_pos__eucl__rel__int__mult__2,axiom,
    ! [B: int,A: int,Q2: int,R2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
     => ( aa_Pro4592068669816069595_int_o(eucl_rel_int(A,B),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q2),R2))
       => aa_Pro4592068669816069595_int_o(eucl_rel_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B)),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q2),aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),R2)))) ) ) ).

% pos_eucl_rel_int_mult_2
tff(fact_5980_VEBT__internal_Ovebt__inserti_H_Omono,axiom,
    ! [Xb: produc3960855945107176009Ti_nat] : comple422786390015765981_VEBTi(partia1868168049876374393Ti_nat(heap_T7173139186834293313_VEBTi),heap_T7173139186834293313_VEBTi,aTP_Lamm_vs(Xb)) ).

% VEBT_internal.vebt_inserti'.mono
tff(fact_5981_divides__aux__eq,axiom,
    ! [Q2: nat,R2: nat] :
      ( aa_Pro6150637604445964947_nat_o(unique6322359934112328802ux_nat,aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Q2),R2))
    <=> ( R2 = zero_zero_nat ) ) ).

% divides_aux_eq
tff(fact_5982_divides__aux__eq,axiom,
    ! [Q2: int,R2: int] :
      ( aa_Pro4592068669816069595_int_o(unique6319869463603278526ux_int,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q2),R2))
    <=> ( R2 = zero_zero_int ) ) ).

% divides_aux_eq
tff(fact_5983_VEBT__internal_OT__vebt__buildupi_H_Oelims,axiom,
    ! [Xb: nat,Y: int] :
      ( ( aa_nat_int(vEBT_V9176841429113362141ildupi,Xb) = Y )
     => ( ( ( Xb = zero_zero_nat )
         => ( Y != one_one_int ) )
       => ( ( ( Xb = aa_nat_nat(suc,zero_zero_nat) )
           => ( Y != one_one_int ) )
         => ~ ! [N2: nat] :
                ( ( Xb = aa_nat_nat(suc,aa_nat_nat(suc,N2)) )
               => ( Y != $ite(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_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,one))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(vEBT_V9176841429113362141ildupi,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,aa_num_num(bit0,one)))),aa_nat_int(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(divide_divide_nat,N2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(vEBT_V9176841429113362141ildupi,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_int(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(divide_divide_nat,N2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,one))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(vEBT_V9176841429113362141ildupi,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,one))))),aa_nat_int(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(divide_divide_nat,N2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(vEBT_V9176841429113362141ildupi,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_int(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(divide_divide_nat,N2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))))) ) ) ) ) ) ).

% VEBT_internal.T_vebt_buildupi'.elims
tff(fact_5984_flip__bit__Suc,axiom,
    ! [Nb: nat,A: word_N3645301735248828278l_num1] : bit_se4491814353640558621l_num1(aa_nat_nat(suc,Nb),A) = plus_p361126936061061375l_num1(modulo1504961113040953224l_num1(A,numera7442385471795722001l_num1(aa_num_num(bit0,one))),aa_wor7274019993978516266l_num1(times_7065122842183080059l_num1(numera7442385471795722001l_num1(aa_num_num(bit0,one))),bit_se4491814353640558621l_num1(Nb,divide1791077408188789448l_num1(A,numera7442385471795722001l_num1(aa_num_num(bit0,one)))))) ).

% flip_bit_Suc
tff(fact_5985_flip__bit__Suc,axiom,
    ! [Nb: nat,A: code_integer] : aa_Cod1783974652533396728nteger(bit_se1345352211410354436nteger(aa_nat_nat(suc,Nb)),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(Nb),aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)))))) ).

% flip_bit_Suc
tff(fact_5986_flip__bit__Suc,axiom,
    ! [Nb: nat,A: uint32] : bit_se7025624438249859091uint32(aa_nat_nat(suc,Nb),A) = aa_uint32_uint32(plus_plus_uint32(aa_uint32_uint32(aa_uin3778660748068847871uint32(modulo_modulo_uint32,A),numera9087168376688890119uint32(aa_num_num(bit0,one)))),aa_uint32_uint32(times_times_uint32(numera9087168376688890119uint32(aa_num_num(bit0,one))),bit_se7025624438249859091uint32(Nb,aa_uint32_uint32(aa_uin3778660748068847871uint32(divide_divide_uint32,A),numera9087168376688890119uint32(aa_num_num(bit0,one)))))) ).

% flip_bit_Suc
tff(fact_5987_flip__bit__Suc,axiom,
    ! [Nb: nat,A: int] : aa_int_int(aa_nat_fun_int_int(bit_se2159334234014336723it_int,aa_nat_nat(suc,Nb)),A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_nat_fun_int_int(bit_se2159334234014336723it_int,Nb),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ).

% flip_bit_Suc
tff(fact_5988_flip__bit__Suc,axiom,
    ! [Nb: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2161824704523386999it_nat,aa_nat_nat(suc,Nb)),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2161824704523386999it_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% flip_bit_Suc
tff(fact_5989_unset__bit__0,axiom,
    ! [A: word_N3645301735248828278l_num1] : bit_se5331074070815623765l_num1(zero_zero_nat,A) = aa_wor7274019993978516266l_num1(times_7065122842183080059l_num1(numera7442385471795722001l_num1(aa_num_num(bit0,one))),divide1791077408188789448l_num1(A,numera7442385471795722001l_num1(aa_num_num(bit0,one)))) ).

% unset_bit_0
tff(fact_5990_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_5991_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_5992_VEBT__internal_Ovebt__succi_H_Omono,axiom,
    ! [Xb: produc3960855945107176009Ti_nat] : comple4513117127444775325on_nat(partia7778075537949639097Ti_nat(heap_T7875578835903804844on_nat),heap_T7875578835903804844on_nat,aTP_Lamm_wj(Xb)) ).

% VEBT_internal.vebt_succi'.mono
tff(fact_5993_dvd__0__right,axiom,
    ! [A: word_N3645301735248828278l_num1] : aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(A),zero_z3563351764282998399l_num1) ).

% dvd_0_right
tff(fact_5994_dvd__0__right,axiom,
    ! [A: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),zero_z3403309356797280102nteger) ).

% dvd_0_right
tff(fact_5995_dvd__0__right,axiom,
    ! [A: uint32] : aa_uint32_o(aa_uin5924812838015166427nt32_o(dvd_dvd_uint32,A),zero_zero_uint32) ).

% dvd_0_right
tff(fact_5996_dvd__0__right,axiom,
    ! [A: real] : aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),zero_zero_real) ).

% dvd_0_right
tff(fact_5997_dvd__0__right,axiom,
    ! [A: rat] : aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),zero_zero_rat) ).

% dvd_0_right
tff(fact_5998_dvd__0__right,axiom,
    ! [A: nat] : aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),zero_zero_nat) ).

% dvd_0_right
tff(fact_5999_dvd__0__right,axiom,
    ! [A: int] : aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),zero_zero_int) ).

% dvd_0_right
tff(fact_6000_dvd__0__left__iff,axiom,
    ! [A: word_N3645301735248828278l_num1] :
      ( aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(zero_z3563351764282998399l_num1),A)
    <=> ( A = zero_z3563351764282998399l_num1 ) ) ).

% dvd_0_left_iff
tff(fact_6001_dvd__0__left__iff,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,zero_z3403309356797280102nteger),A)
    <=> ( A = zero_z3403309356797280102nteger ) ) ).

% dvd_0_left_iff
tff(fact_6002_dvd__0__left__iff,axiom,
    ! [A: uint32] :
      ( aa_uint32_o(aa_uin5924812838015166427nt32_o(dvd_dvd_uint32,zero_zero_uint32),A)
    <=> ( A = zero_zero_uint32 ) ) ).

% dvd_0_left_iff
tff(fact_6003_dvd__0__left__iff,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,zero_zero_real),A)
    <=> ( A = zero_zero_real ) ) ).

% dvd_0_left_iff
tff(fact_6004_dvd__0__left__iff,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,zero_zero_rat),A)
    <=> ( A = zero_zero_rat ) ) ).

% dvd_0_left_iff
tff(fact_6005_dvd__0__left__iff,axiom,
    ! [A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,zero_zero_nat),A)
    <=> ( A = zero_zero_nat ) ) ).

% dvd_0_left_iff
tff(fact_6006_dvd__0__left__iff,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,zero_zero_int),A)
    <=> ( A = zero_zero_int ) ) ).

% dvd_0_left_iff
tff(fact_6007_dvd__add__triv__right__iff,axiom,
    ! [A: uint32,B: uint32] :
      ( aa_uint32_o(aa_uin5924812838015166427nt32_o(dvd_dvd_uint32,A),aa_uint32_uint32(plus_plus_uint32(B),A))
    <=> aa_uint32_o(aa_uin5924812838015166427nt32_o(dvd_dvd_uint32,A),B) ) ).

% dvd_add_triv_right_iff
tff(fact_6008_dvd__add__triv__right__iff,axiom,
    ! [A: word_N3645301735248828278l_num1,B: word_N3645301735248828278l_num1] :
      ( aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(A),plus_p361126936061061375l_num1(B,A))
    <=> aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(A),B) ) ).

% dvd_add_triv_right_iff
tff(fact_6009_dvd__add__triv__right__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(B),A))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B) ) ).

% dvd_add_triv_right_iff
tff(fact_6010_dvd__add__triv__right__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),A))
    <=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),B) ) ).

% dvd_add_triv_right_iff
tff(fact_6011_dvd__add__triv__right__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),aa_rat_rat(plus_plus_rat(B),A))
    <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),B) ) ).

% dvd_add_triv_right_iff
tff(fact_6012_dvd__add__triv__right__iff,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A))
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B) ) ).

% dvd_add_triv_right_iff
tff(fact_6013_dvd__add__triv__right__iff,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B) ) ).

% dvd_add_triv_right_iff
tff(fact_6014_dvd__add__triv__left__iff,axiom,
    ! [A: uint32,B: uint32] :
      ( aa_uint32_o(aa_uin5924812838015166427nt32_o(dvd_dvd_uint32,A),aa_uint32_uint32(plus_plus_uint32(A),B))
    <=> aa_uint32_o(aa_uin5924812838015166427nt32_o(dvd_dvd_uint32,A),B) ) ).

% dvd_add_triv_left_iff
tff(fact_6015_dvd__add__triv__left__iff,axiom,
    ! [A: word_N3645301735248828278l_num1,B: word_N3645301735248828278l_num1] :
      ( aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(A),plus_p361126936061061375l_num1(A,B))
    <=> aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(A),B) ) ).

% dvd_add_triv_left_iff
tff(fact_6016_dvd__add__triv__left__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(A),B))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B) ) ).

% dvd_add_triv_left_iff
tff(fact_6017_dvd__add__triv__left__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B))
    <=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),B) ) ).

% dvd_add_triv_left_iff
tff(fact_6018_dvd__add__triv__left__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),aa_rat_rat(plus_plus_rat(A),B))
    <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),B) ) ).

% dvd_add_triv_left_iff
tff(fact_6019_dvd__add__triv__left__iff,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B))
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B) ) ).

% dvd_add_triv_left_iff
tff(fact_6020_dvd__add__triv__left__iff,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B) ) ).

% dvd_add_triv_left_iff
tff(fact_6021_div__dvd__div,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),C)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(B),A)),aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(C),A))
        <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),C) ) ) ) ).

% div_dvd_div
tff(fact_6022_div__dvd__div,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),C)
       => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),A)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,C),A))
        <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),C) ) ) ) ).

% div_dvd_div
tff(fact_6023_div__dvd__div,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),C)
       => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,B),A)),aa_int_int(aa_int_fun_int_int(divide_divide_int,C),A))
        <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),C) ) ) ) ).

% div_dvd_div
tff(fact_6024_unset__bit__nonnegative__int__iff,axiom,
    ! [Nb: 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,Nb),K))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K) ) ).

% unset_bit_nonnegative_int_iff
tff(fact_6025_flip__bit__nonnegative__int__iff,axiom,
    ! [Nb: 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,Nb),K))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K) ) ).

% flip_bit_nonnegative_int_iff
tff(fact_6026_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_6027_curry__conv,axiom,
    ! [F: fun_Pr5956696566359781750on_nat,A: vEBT_VEBT,B: vEBT_VEBTi] : aa_VEB3566417204837707778on_nat(aa_VEB1030146501360450209on_nat(produc9121605208400662357on_nat(F),A),B) = aa_Pro1290670013494283871on_nat(F,produc6084888613844515218_VEBTi(A,B)) ).

% curry_conv
tff(fact_6028_curry__conv,axiom,
    ! [F: fun_Pr4877903221429077380Heap_o,A: vEBT_VEBT,B: vEBT_VEBTi] : aa_VEB1455453220190609250Heap_o(aa_VEB8672304581938048793Heap_o(produc8906325494159015119Heap_o(F),A),B) = aa_Pro259339148018954309Heap_o(F,produc6084888613844515218_VEBTi(A,B)) ).

% curry_conv
tff(fact_6029_curry__conv,axiom,
    ! [F: fun_Pr4404027214751104635_VEBTi,A: produc3625547720036274456_VEBTi,B: nat] : aa_nat5408703056993926195_VEBTi(aa_Pro284253941917238388_VEBTi(produc2164094337957399884_VEBTi(F),A),B) = aa_Pro721367001817565044_VEBTi(F,produc1853644041309157249Ti_nat(A,B)) ).

% curry_conv
tff(fact_6030_curry__conv,axiom,
    ! [F: fun_Pr633315466666964198on_nat,A: produc3625547720036274456_VEBTi,B: nat] : aa_nat947733127081326238on_nat(aa_Pro1290670013494283871on_nat(produc1757988346207259447on_nat(F),A),B) = aa_Pro2727408364141080927on_nat(F,produc1853644041309157249Ti_nat(A,B)) ).

% curry_conv
tff(fact_6031_curry__conv,axiom,
    ! [F: fun_Pr940977554174145406Heap_o,A: produc3625547720036274456_VEBTi,B: nat] : aa_nat2910048432523430448Heap_o(aa_Pro259339148018954309Heap_o(produc8381543706267210711Heap_o(F),A),B) = aa_Pro6173831900156065711Heap_o(F,produc1853644041309157249Ti_nat(A,B)) ).

% curry_conv
tff(fact_6032_case__prod__curry,axiom,
    ! [F: fun_Pr5956696566359781750on_nat] : produc3521483158534984758on_nat(produc9121605208400662357on_nat(F)) = F ).

% case_prod_curry
tff(fact_6033_case__prod__curry,axiom,
    ! [F: fun_Pr4877903221429077380Heap_o] : produc844971584604743726Heap_o(produc8906325494159015119Heap_o(F)) = F ).

% case_prod_curry
tff(fact_6034_case__prod__curry,axiom,
    ! [F: fun_Pr4404027214751104635_VEBTi] : produc2943724498215716011_VEBTi(produc2164094337957399884_VEBTi(F)) = F ).

% case_prod_curry
tff(fact_6035_case__prod__curry,axiom,
    ! [F: fun_Pr633315466666964198on_nat] : produc183673358652719894on_nat(produc1757988346207259447on_nat(F)) = F ).

% case_prod_curry
tff(fact_6036_case__prod__curry,axiom,
    ! [F: fun_Pr940977554174145406Heap_o] : produc5872130906356439992Heap_o(produc8381543706267210711Heap_o(F)) = F ).

% case_prod_curry
tff(fact_6037_case__prod__curry,axiom,
    ! [F: fun_Pr2118491707810114563at_nat] : produc2626176000494625587at_nat(produc6629854527392350932at_nat(F)) = F ).

% case_prod_curry
tff(fact_6038_case__prod__curry,axiom,
    ! [F: fun_Pr4796943543331071514_nat_o] : produc6081775807080527818_nat_o(produc1310100445399344235_nat_o(F)) = F ).

% case_prod_curry
tff(fact_6039_case__prod__curry,axiom,
    ! [F: fun_Pr2478283820680522243nt_int] : produc4245557441103728435nt_int(produc8249235968001453780nt_int(F)) = F ).

% case_prod_curry
tff(fact_6040_case__prod__curry,axiom,
    ! [F: fun_Pr3238374608701176162_int_o] : produc4947309494688390418_int_o(produc175634133007206835_int_o(F)) = F ).

% case_prod_curry
tff(fact_6041_case__prod__curry,axiom,
    ! [F: fun_Pr982841511892225912nt_int] : produc8211389475949308722nt_int(produc1016772743285680337nt_int(F)) = F ).

% case_prod_curry
tff(fact_6042_curry__case__prod,axiom,
    ! [F: fun_VE3775134097089128360on_nat] : produc9121605208400662357on_nat(produc3521483158534984758on_nat(F)) = F ).

% curry_case_prod
tff(fact_6043_curry__case__prod,axiom,
    ! [F: fun_VE8181444127144381288Heap_o] : produc8906325494159015119Heap_o(produc844971584604743726Heap_o(F)) = F ).

% curry_case_prod
tff(fact_6044_curry__case__prod,axiom,
    ! [F: fun_Pr7184143894984516363_VEBTi] : produc2164094337957399884_VEBTi(produc2943724498215716011_VEBTi(F)) = F ).

% curry_case_prod
tff(fact_6045_curry__case__prod,axiom,
    ! [F: fun_Pr5956696566359781750on_nat] : produc1757988346207259447on_nat(produc183673358652719894on_nat(F)) = F ).

% curry_case_prod
tff(fact_6046_curry__case__prod,axiom,
    ! [F: fun_Pr4877903221429077380Heap_o] : produc8381543706267210711Heap_o(produc5872130906356439992Heap_o(F)) = F ).

% curry_case_prod
tff(fact_6047_curry__case__prod,axiom,
    ! [F: fun_na4138050610385705875at_nat] : produc6629854527392350932at_nat(produc2626176000494625587at_nat(F)) = F ).

% curry_case_prod
tff(fact_6048_curry__case__prod,axiom,
    ! [F: fun_nat_fun_nat_o] : produc1310100445399344235_nat_o(produc6081775807080527818_nat_o(F)) = F ).

% curry_case_prod
tff(fact_6049_curry__case__prod,axiom,
    ! [F: fun_in1696471567231710611nt_int] : produc8249235968001453780nt_int(produc4245557441103728435nt_int(F)) = F ).

% curry_case_prod
tff(fact_6050_curry__case__prod,axiom,
    ! [F: fun_int_fun_int_o] : produc175634133007206835_int_o(produc4947309494688390418_int_o(F)) = F ).

% curry_case_prod
tff(fact_6051_curry__case__prod,axiom,
    ! [F: fun_int_fun_int_int] : produc1016772743285680337nt_int(produc8211389475949308722nt_int(F)) = F ).

% curry_case_prod
tff(fact_6052_dvd__times__right__cancel__iff,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),C) ) ) ).

% dvd_times_right_cancel_iff
tff(fact_6053_dvd__times__right__cancel__iff,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( A != zero_zero_nat )
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),C) ) ) ).

% dvd_times_right_cancel_iff
tff(fact_6054_dvd__times__right__cancel__iff,axiom,
    ! [A: int,B: int,C: int] :
      ( ( A != zero_zero_int )
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),A))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),C) ) ) ).

% dvd_times_right_cancel_iff
tff(fact_6055_dvd__times__left__cancel__iff,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),C) ) ) ).

% dvd_times_left_cancel_iff
tff(fact_6056_dvd__times__left__cancel__iff,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( A != zero_zero_nat )
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),C) ) ) ).

% dvd_times_left_cancel_iff
tff(fact_6057_dvd__times__left__cancel__iff,axiom,
    ! [A: int,B: int,C: int] :
      ( ( A != zero_zero_int )
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),aa_int_int(aa_int_fun_int_int(times_times_int,A),C))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),C) ) ) ).

% dvd_times_left_cancel_iff
tff(fact_6058_dvd__mult__cancel__right,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C))
    <=> ( ( C = zero_z3403309356797280102nteger )
        | aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B) ) ) ).

% dvd_mult_cancel_right
tff(fact_6059_dvd__mult__cancel__right,axiom,
    ! [A: real,C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C))
    <=> ( ( C = zero_zero_real )
        | aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),B) ) ) ).

% dvd_mult_cancel_right
tff(fact_6060_dvd__mult__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),C))
    <=> ( ( C = zero_zero_rat )
        | aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),B) ) ) ).

% dvd_mult_cancel_right
tff(fact_6061_dvd__mult__cancel__right,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C))
    <=> ( ( C = zero_zero_int )
        | aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B) ) ) ).

% dvd_mult_cancel_right
tff(fact_6062_dvd__mult__cancel__left,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B))
    <=> ( ( C = zero_z3403309356797280102nteger )
        | aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B) ) ) ).

% dvd_mult_cancel_left
tff(fact_6063_dvd__mult__cancel__left,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B))
    <=> ( ( C = zero_zero_real )
        | aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),B) ) ) ).

% dvd_mult_cancel_left
tff(fact_6064_dvd__mult__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B))
    <=> ( ( C = zero_zero_rat )
        | aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),B) ) ) ).

% dvd_mult_cancel_left
tff(fact_6065_dvd__mult__cancel__left,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))
    <=> ( ( C = zero_zero_int )
        | aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B) ) ) ).

% dvd_mult_cancel_left
tff(fact_6066_dvd__add__times__triv__left__iff,axiom,
    ! [A: uint32,C: uint32,B: uint32] :
      ( aa_uint32_o(aa_uin5924812838015166427nt32_o(dvd_dvd_uint32,A),aa_uint32_uint32(plus_plus_uint32(aa_uint32_uint32(times_times_uint32(C),A)),B))
    <=> aa_uint32_o(aa_uin5924812838015166427nt32_o(dvd_dvd_uint32,A),B) ) ).

% dvd_add_times_triv_left_iff
tff(fact_6067_dvd__add__times__triv__left__iff,axiom,
    ! [A: word_N3645301735248828278l_num1,C: word_N3645301735248828278l_num1,B: word_N3645301735248828278l_num1] :
      ( aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(A),plus_p361126936061061375l_num1(aa_wor7274019993978516266l_num1(times_7065122842183080059l_num1(C),A),B))
    <=> aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(A),B) ) ).

% dvd_add_times_triv_left_iff
tff(fact_6068_dvd__add__times__triv__left__iff,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A)),B))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B) ) ).

% dvd_add_times_triv_left_iff
tff(fact_6069_dvd__add__times__triv__left__iff,axiom,
    ! [A: real,C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),B))
    <=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),B) ) ).

% dvd_add_times_triv_left_iff
tff(fact_6070_dvd__add__times__triv__left__iff,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(C),A)),B))
    <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),B) ) ).

% dvd_add_times_triv_left_iff
tff(fact_6071_dvd__add__times__triv__left__iff,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)),B))
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B) ) ).

% dvd_add_times_triv_left_iff
tff(fact_6072_dvd__add__times__triv__left__iff,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),B))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B) ) ).

% dvd_add_times_triv_left_iff
tff(fact_6073_dvd__add__times__triv__right__iff,axiom,
    ! [A: uint32,B: uint32,C: uint32] :
      ( aa_uint32_o(aa_uin5924812838015166427nt32_o(dvd_dvd_uint32,A),aa_uint32_uint32(plus_plus_uint32(B),aa_uint32_uint32(times_times_uint32(C),A)))
    <=> aa_uint32_o(aa_uin5924812838015166427nt32_o(dvd_dvd_uint32,A),B) ) ).

% dvd_add_times_triv_right_iff
tff(fact_6074_dvd__add__times__triv__right__iff,axiom,
    ! [A: word_N3645301735248828278l_num1,B: word_N3645301735248828278l_num1,C: word_N3645301735248828278l_num1] :
      ( aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(A),plus_p361126936061061375l_num1(B,aa_wor7274019993978516266l_num1(times_7065122842183080059l_num1(C),A)))
    <=> aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(A),B) ) ).

% dvd_add_times_triv_right_iff
tff(fact_6075_dvd__add__times__triv__right__iff,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(B),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A)))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B) ) ).

% dvd_add_times_triv_right_iff
tff(fact_6076_dvd__add__times__triv__right__iff,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)))
    <=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),B) ) ).

% dvd_add_times_triv_right_iff
tff(fact_6077_dvd__add__times__triv__right__iff,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),aa_rat_rat(plus_plus_rat(B),aa_rat_rat(times_times_rat(C),A)))
    <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),B) ) ).

% dvd_add_times_triv_right_iff
tff(fact_6078_dvd__add__times__triv__right__iff,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)))
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B) ) ).

% dvd_add_times_triv_right_iff
tff(fact_6079_dvd__add__times__triv__right__iff,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),aa_int_int(aa_int_fun_int_int(times_times_int,C),A)))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B) ) ).

% dvd_add_times_triv_right_iff
tff(fact_6080_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_6081_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_6082_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_6083_div__add,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),A)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),B)
       => ( aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(A),B)),C) = aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),C)),aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(B),C)) ) ) ) ).

% div_add
tff(fact_6084_div__add,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),B)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C)) ) ) ) ).

% div_add
tff(fact_6085_div__add,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),A)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),B)
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),C)),aa_int_int(aa_int_fun_int_int(divide_divide_int,B),C)) ) ) ) ).

% div_add
tff(fact_6086_unit__div,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),one_one_Code_integer)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),B)),one_one_Code_integer) ) ) ).

% unit_div
tff(fact_6087_unit__div,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),one_one_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),one_one_nat) ) ) ).

% unit_div
tff(fact_6088_unit__div,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),one_one_int) ) ) ).

% unit_div
tff(fact_6089_unit__div__1__unit,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),one_one_Code_integer)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(one_one_Code_integer),A)),one_one_Code_integer) ) ).

% unit_div_1_unit
tff(fact_6090_unit__div__1__unit,axiom,
    ! [A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),one_one_nat)
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),A)),one_one_nat) ) ).

% unit_div_1_unit
tff(fact_6091_unit__div__1__unit,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int)
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),A)),one_one_int) ) ).

% unit_div_1_unit
tff(fact_6092_unit__div__1__div__1,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),one_one_Code_integer)
     => ( aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(one_one_Code_integer),aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(one_one_Code_integer),A)) = A ) ) ).

% unit_div_1_div_1
tff(fact_6093_unit__div__1__div__1,axiom,
    ! [A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),one_one_nat)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),A)) = A ) ) ).

% unit_div_1_div_1
tff(fact_6094_unit__div__1__div__1,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int)
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),A)) = A ) ) ).

% unit_div_1_div_1
tff(fact_6095_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_6096_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_6097_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_6098_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_6099_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_6100_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_6101_div__diff,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),A)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),B)
       => ( aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(A),B)),C) = aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),C)),aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(B),C)) ) ) ) ).

% div_diff
tff(fact_6102_div__diff,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),A)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),B)
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),C)),aa_int_int(aa_int_fun_int_int(divide_divide_int,B),C)) ) ) ) ).

% div_diff
tff(fact_6103_dvd__imp__mod__0,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B),A) = zero_zero_nat ) ) ).

% dvd_imp_mod_0
tff(fact_6104_dvd__imp__mod__0,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B)
     => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B),A) = zero_zero_int ) ) ).

% dvd_imp_mod_0
tff(fact_6105_dvd__imp__mod__0,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B)
     => ( aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(B),A) = zero_z3403309356797280102nteger ) ) ).

% dvd_imp_mod_0
tff(fact_6106_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_6107_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_6108_nat__mult__dvd__cancel__disj,axiom,
    ! [K: nat,M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Nb))
    <=> ( ( K = zero_zero_nat )
        | aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,M),Nb) ) ) ).

% nat_mult_dvd_cancel_disj
tff(fact_6109_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_6110_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_6111_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_6112_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_6113_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_6114_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_6115_odd__add,axiom,
    ! [A: uint32,B: uint32] :
      ( ~ aa_uint32_o(aa_uin5924812838015166427nt32_o(dvd_dvd_uint32,numera9087168376688890119uint32(aa_num_num(bit0,one))),aa_uint32_uint32(plus_plus_uint32(A),B))
    <=> ~ ( ~ aa_uint32_o(aa_uin5924812838015166427nt32_o(dvd_dvd_uint32,numera9087168376688890119uint32(aa_num_num(bit0,one))),A)
        <=> ~ aa_uint32_o(aa_uin5924812838015166427nt32_o(dvd_dvd_uint32,numera9087168376688890119uint32(aa_num_num(bit0,one))),B) ) ) ).

% odd_add
tff(fact_6116_odd__add,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(plus_p5714425477246183910nteger(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) ) ) ).

% odd_add
tff(fact_6117_odd__add,axiom,
    ! [A: word_N3645301735248828278l_num1,B: word_N3645301735248828278l_num1] :
      ( ~ aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(numera7442385471795722001l_num1(aa_num_num(bit0,one))),plus_p361126936061061375l_num1(A,B))
    <=> ~ ( ~ aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(numera7442385471795722001l_num1(aa_num_num(bit0,one))),A)
        <=> ~ aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(numera7442385471795722001l_num1(aa_num_num(bit0,one))),B) ) ) ).

% odd_add
tff(fact_6118_odd__add,axiom,
    ! [A: nat,B: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B))
    <=> ~ ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
        <=> ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B) ) ) ).

% odd_add
tff(fact_6119_odd__add,axiom,
    ! [A: int,B: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B))
    <=> ~ ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
        <=> ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B) ) ) ).

% odd_add
tff(fact_6120_even__add,axiom,
    ! [A: uint32,B: uint32] :
      ( aa_uint32_o(aa_uin5924812838015166427nt32_o(dvd_dvd_uint32,numera9087168376688890119uint32(aa_num_num(bit0,one))),aa_uint32_uint32(plus_plus_uint32(A),B))
    <=> ( aa_uint32_o(aa_uin5924812838015166427nt32_o(dvd_dvd_uint32,numera9087168376688890119uint32(aa_num_num(bit0,one))),A)
      <=> aa_uint32_o(aa_uin5924812838015166427nt32_o(dvd_dvd_uint32,numera9087168376688890119uint32(aa_num_num(bit0,one))),B) ) ) ).

% even_add
tff(fact_6121_even__add,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(plus_p5714425477246183910nteger(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_add
tff(fact_6122_even__add,axiom,
    ! [A: word_N3645301735248828278l_num1,B: word_N3645301735248828278l_num1] :
      ( aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(numera7442385471795722001l_num1(aa_num_num(bit0,one))),plus_p361126936061061375l_num1(A,B))
    <=> ( aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(numera7442385471795722001l_num1(aa_num_num(bit0,one))),A)
      <=> aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(numera7442385471795722001l_num1(aa_num_num(bit0,one))),B) ) ) ).

% even_add
tff(fact_6123_even__add,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B) ) ) ).

% even_add
tff(fact_6124_even__add,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B))
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B) ) ) ).

% even_add
tff(fact_6125_even__mult__iff,axiom,
    ! [A: uint32,B: uint32] :
      ( aa_uint32_o(aa_uin5924812838015166427nt32_o(dvd_dvd_uint32,numera9087168376688890119uint32(aa_num_num(bit0,one))),aa_uint32_uint32(times_times_uint32(A),B))
    <=> ( aa_uint32_o(aa_uin5924812838015166427nt32_o(dvd_dvd_uint32,numera9087168376688890119uint32(aa_num_num(bit0,one))),A)
        | aa_uint32_o(aa_uin5924812838015166427nt32_o(dvd_dvd_uint32,numera9087168376688890119uint32(aa_num_num(bit0,one))),B) ) ) ).

% even_mult_iff
tff(fact_6126_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_6127_even__mult__iff,axiom,
    ! [A: word_N3645301735248828278l_num1,B: word_N3645301735248828278l_num1] :
      ( aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(numera7442385471795722001l_num1(aa_num_num(bit0,one))),aa_wor7274019993978516266l_num1(times_7065122842183080059l_num1(A),B))
    <=> ( aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(numera7442385471795722001l_num1(aa_num_num(bit0,one))),A)
        | aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(numera7442385471795722001l_num1(aa_num_num(bit0,one))),B) ) ) ).

% even_mult_iff
tff(fact_6128_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_6129_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_6130_even__mod__2__iff,axiom,
    ! [A: word_N3645301735248828278l_num1] :
      ( aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(numera7442385471795722001l_num1(aa_num_num(bit0,one))),modulo1504961113040953224l_num1(A,numera7442385471795722001l_num1(aa_num_num(bit0,one))))
    <=> aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(numera7442385471795722001l_num1(aa_num_num(bit0,one))),A) ) ).

% even_mod_2_iff
tff(fact_6131_even__mod__2__iff,axiom,
    ! [A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A) ) ).

% even_mod_2_iff
tff(fact_6132_even__mod__2__iff,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A) ) ).

% even_mod_2_iff
tff(fact_6133_even__mod__2__iff,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A),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))),A) ) ).

% even_mod_2_iff
tff(fact_6134_even__mod__2__iff,axiom,
    ! [A: uint32] :
      ( aa_uint32_o(aa_uin5924812838015166427nt32_o(dvd_dvd_uint32,numera9087168376688890119uint32(aa_num_num(bit0,one))),aa_uint32_uint32(aa_uin3778660748068847871uint32(modulo_modulo_uint32,A),numera9087168376688890119uint32(aa_num_num(bit0,one))))
    <=> aa_uint32_o(aa_uin5924812838015166427nt32_o(dvd_dvd_uint32,numera9087168376688890119uint32(aa_num_num(bit0,one))),A) ) ).

% even_mod_2_iff
tff(fact_6135_even__Suc,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,Nb))
    <=> ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb) ) ).

% even_Suc
tff(fact_6136_even__Suc__Suc__iff,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,Nb)))
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb) ) ).

% even_Suc_Suc_iff
tff(fact_6137_dvd__numeral__simp,axiom,
    ! [M: num,Nb: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,Nb))
    <=> aa_Pro6150637604445964947_nat_o(unique6322359934112328802ux_nat,unique5055182867167087721od_nat(Nb,M)) ) ).

% dvd_numeral_simp
tff(fact_6138_dvd__numeral__simp,axiom,
    ! [M: num,Nb: num] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,M)),aa_num_Code_integer(numera6620942414471956472nteger,Nb))
    <=> aa_Pro4926521453222654329eger_o(unique5706413561485394159nteger,unique3479559517661332726nteger(Nb,M)) ) ).

% dvd_numeral_simp
tff(fact_6139_dvd__numeral__simp,axiom,
    ! [M: num,Nb: num] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,Nb))
    <=> aa_Pro4592068669816069595_int_o(unique6319869463603278526ux_int,unique5052692396658037445od_int(Nb,M)) ) ).

% dvd_numeral_simp
tff(fact_6140_even__plus__one__iff,axiom,
    ! [A: uint32] :
      ( aa_uint32_o(aa_uin5924812838015166427nt32_o(dvd_dvd_uint32,numera9087168376688890119uint32(aa_num_num(bit0,one))),aa_uint32_uint32(plus_plus_uint32(A),one_one_uint32))
    <=> ~ aa_uint32_o(aa_uin5924812838015166427nt32_o(dvd_dvd_uint32,numera9087168376688890119uint32(aa_num_num(bit0,one))),A) ) ).

% even_plus_one_iff
tff(fact_6141_even__plus__one__iff,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(A),one_one_Code_integer))
    <=> ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A) ) ).

% even_plus_one_iff
tff(fact_6142_even__plus__one__iff,axiom,
    ! [A: word_N3645301735248828278l_num1] :
      ( aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(numera7442385471795722001l_num1(aa_num_num(bit0,one))),plus_p361126936061061375l_num1(A,one_on7727431528512463931l_num1))
    <=> ~ aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(numera7442385471795722001l_num1(aa_num_num(bit0,one))),A) ) ).

% even_plus_one_iff
tff(fact_6143_even__plus__one__iff,axiom,
    ! [A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),one_one_nat))
    <=> ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A) ) ).

% even_plus_one_iff
tff(fact_6144_even__plus__one__iff,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),one_one_int))
    <=> ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A) ) ).

% even_plus_one_iff
tff(fact_6145_even__diff,axiom,
    ! [A: uint32,B: uint32] :
      ( aa_uint32_o(aa_uin5924812838015166427nt32_o(dvd_dvd_uint32,numera9087168376688890119uint32(aa_num_num(bit0,one))),minus_minus_uint32(A,B))
    <=> aa_uint32_o(aa_uin5924812838015166427nt32_o(dvd_dvd_uint32,numera9087168376688890119uint32(aa_num_num(bit0,one))),aa_uint32_uint32(plus_plus_uint32(A),B)) ) ).

% even_diff
tff(fact_6146_even__diff,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(minus_8373710615458151222nteger(A),B))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(A),B)) ) ).

% even_diff
tff(fact_6147_even__diff,axiom,
    ! [A: word_N3645301735248828278l_num1,B: word_N3645301735248828278l_num1] :
      ( aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(numera7442385471795722001l_num1(aa_num_num(bit0,one))),minus_4019991460397169231l_num1(A,B))
    <=> aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(numera7442385471795722001l_num1(aa_num_num(bit0,one))),plus_p361126936061061375l_num1(A,B)) ) ).

% even_diff
tff(fact_6148_even__diff,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)) ) ).

% even_diff
tff(fact_6149_odd__Suc__div__two,axiom,
    ! [Nb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,Nb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ) ).

% odd_Suc_div_two
tff(fact_6150_even__Suc__div__two,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,Nb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ) ) ).

% even_Suc_div_two
tff(fact_6151_even__succ__div__2,axiom,
    ! [A: uint32] :
      ( aa_uint32_o(aa_uin5924812838015166427nt32_o(dvd_dvd_uint32,numera9087168376688890119uint32(aa_num_num(bit0,one))),A)
     => ( aa_uint32_uint32(aa_uin3778660748068847871uint32(divide_divide_uint32,aa_uint32_uint32(plus_plus_uint32(one_one_uint32),A)),numera9087168376688890119uint32(aa_num_num(bit0,one))) = aa_uint32_uint32(aa_uin3778660748068847871uint32(divide_divide_uint32,A),numera9087168376688890119uint32(aa_num_num(bit0,one))) ) ) ).

% even_succ_div_2
tff(fact_6152_even__succ__div__2,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A)
     => ( aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(one_one_Code_integer),A)),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) = aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) ) ) ).

% even_succ_div_2
tff(fact_6153_even__succ__div__2,axiom,
    ! [A: word_N3645301735248828278l_num1] :
      ( aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(numera7442385471795722001l_num1(aa_num_num(bit0,one))),A)
     => ( divide1791077408188789448l_num1(plus_p361126936061061375l_num1(one_on7727431528512463931l_num1,A),numera7442385471795722001l_num1(aa_num_num(bit0,one))) = divide1791077408188789448l_num1(A,numera7442385471795722001l_num1(aa_num_num(bit0,one))) ) ) ).

% even_succ_div_2
tff(fact_6154_even__succ__div__2,axiom,
    ! [A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),A)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ) ) ).

% even_succ_div_2
tff(fact_6155_even__succ__div__2,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),A)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ) ) ).

% even_succ_div_2
tff(fact_6156_odd__succ__div__two,axiom,
    ! [A: code_integer] :
      ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A)
     => ( aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(A),one_one_Code_integer)),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) = aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)))),one_one_Code_integer) ) ) ).

% odd_succ_div_two
tff(fact_6157_odd__succ__div__two,axiom,
    ! [A: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),one_one_nat)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_nat) ) ) ).

% odd_succ_div_two
tff(fact_6158_odd__succ__div__two,axiom,
    ! [A: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),one_one_int)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),one_one_int) ) ) ).

% odd_succ_div_two
tff(fact_6159_even__succ__div__two,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A)
     => ( aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(A),one_one_Code_integer)),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) = aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) ) ) ).

% even_succ_div_two
tff(fact_6160_even__succ__div__two,axiom,
    ! [A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),one_one_nat)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ) ) ).

% even_succ_div_two
tff(fact_6161_even__succ__div__two,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),one_one_int)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ) ) ).

% even_succ_div_two
tff(fact_6162_even__power,axiom,
    ! [A: code_integer,Nb: 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),Nb))
    <=> ( 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),Nb) ) ) ).

% even_power
tff(fact_6163_even__power,axiom,
    ! [A: uint32,Nb: nat] :
      ( aa_uint32_o(aa_uin5924812838015166427nt32_o(dvd_dvd_uint32,numera9087168376688890119uint32(aa_num_num(bit0,one))),aa_nat_uint32(power_power_uint32(A),Nb))
    <=> ( aa_uint32_o(aa_uin5924812838015166427nt32_o(dvd_dvd_uint32,numera9087168376688890119uint32(aa_num_num(bit0,one))),A)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ) ).

% even_power
tff(fact_6164_even__power,axiom,
    ! [A: word_N3645301735248828278l_num1,Nb: nat] :
      ( aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(numera7442385471795722001l_num1(aa_num_num(bit0,one))),aa_nat5599461682457136004l_num1(power_2184487114949457152l_num1(A),Nb))
    <=> ( aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(numera7442385471795722001l_num1(aa_num_num(bit0,one))),A)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ) ).

% even_power
tff(fact_6165_even__power,axiom,
    ! [A: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ) ).

% even_power
tff(fact_6166_even__power,axiom,
    ! [A: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_int(aa_int_fun_nat_int(power_power_int,A),Nb))
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ) ).

% even_power
tff(fact_6167_zero__le__power__eq__numeral,axiom,
    ! [A: real,W: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,W)))
    <=> ( 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,W))
        | ( ~ 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,W))
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A) ) ) ) ).

% zero_le_power_eq_numeral
tff(fact_6168_zero__le__power__eq__numeral,axiom,
    ! [A: rat,W: 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,W)))
    <=> ( 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,W))
        | ( ~ 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,W))
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A) ) ) ) ).

% zero_le_power_eq_numeral
tff(fact_6169_zero__le__power__eq__numeral,axiom,
    ! [A: int,W: 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,W)))
    <=> ( 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,W))
        | ( ~ 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,W))
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A) ) ) ) ).

% zero_le_power_eq_numeral
tff(fact_6170_power__less__zero__eq,axiom,
    ! [A: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)),zero_zero_real)
    <=> ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
        & aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real) ) ) ).

% power_less_zero_eq
tff(fact_6171_power__less__zero__eq,axiom,
    ! [A: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(A),Nb)),zero_zero_rat)
    <=> ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat) ) ) ).

% power_less_zero_eq
tff(fact_6172_power__less__zero__eq,axiom,
    ! [A: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A),Nb)),zero_zero_int)
    <=> ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int) ) ) ).

% power_less_zero_eq
tff(fact_6173_power__less__zero__eq__numeral,axiom,
    ! [A: real,W: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,W))),zero_zero_real)
    <=> ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
        & aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real) ) ) ).

% power_less_zero_eq_numeral
tff(fact_6174_power__less__zero__eq__numeral,axiom,
    ! [A: rat,W: 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,W))),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,W))
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat) ) ) ).

% power_less_zero_eq_numeral
tff(fact_6175_power__less__zero__eq__numeral,axiom,
    ! [A: int,W: 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,W))),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,W))
        & aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int) ) ) ).

% power_less_zero_eq_numeral
tff(fact_6176_even__of__nat,axiom,
    ! [Nb: nat] :
      ( aa_uint32_o(aa_uin5924812838015166427nt32_o(dvd_dvd_uint32,numera9087168376688890119uint32(aa_num_num(bit0,one))),aa_nat_uint32(semiri2565882477558803405uint32,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb) ) ).

% even_of_nat
tff(fact_6177_even__of__nat,axiom,
    ! [Nb: nat] :
      ( aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(numera7442385471795722001l_num1(aa_num_num(bit0,one))),aa_nat5599461682457136004l_num1(semiri8819519690708144855l_num1,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb) ) ).

% even_of_nat
tff(fact_6178_even__of__nat,axiom,
    ! [Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_int(semiri1314217659103216013at_int,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb) ) ).

% even_of_nat
tff(fact_6179_even__of__nat,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(semiri1316708129612266289at_nat,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb) ) ).

% even_of_nat
tff(fact_6180_even__of__nat,axiom,
    ! [Nb: 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(semiri4939895301339042750nteger,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb) ) ).

% even_of_nat
tff(fact_6181_odd__Suc__minus__one,axiom,
    ! [Nb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat))) = Nb ) ) ).

% odd_Suc_minus_one
tff(fact_6182_even__diff__nat,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb)
        | aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Nb)) ) ) ).

% even_diff_nat
tff(fact_6183_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_6184_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_6185_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_6186_semiring__parity__class_Oeven__mask__iff,axiom,
    ! [Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Nb)),one_one_Code_integer))
    <=> ( Nb = zero_zero_nat ) ) ).

% semiring_parity_class.even_mask_iff
tff(fact_6187_semiring__parity__class_Oeven__mask__iff,axiom,
    ! [Nb: nat] :
      ( aa_uint32_o(aa_uin5924812838015166427nt32_o(dvd_dvd_uint32,numera9087168376688890119uint32(aa_num_num(bit0,one))),minus_minus_uint32(aa_nat_uint32(power_power_uint32(numera9087168376688890119uint32(aa_num_num(bit0,one))),Nb),one_one_uint32))
    <=> ( Nb = zero_zero_nat ) ) ).

% semiring_parity_class.even_mask_iff
tff(fact_6188_semiring__parity__class_Oeven__mask__iff,axiom,
    ! [Nb: nat] :
      ( aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(numera7442385471795722001l_num1(aa_num_num(bit0,one))),minus_4019991460397169231l_num1(aa_nat5599461682457136004l_num1(power_2184487114949457152l_num1(numera7442385471795722001l_num1(aa_num_num(bit0,one))),Nb),one_on7727431528512463931l_num1))
    <=> ( Nb = zero_zero_nat ) ) ).

% semiring_parity_class.even_mask_iff
tff(fact_6189_semiring__parity__class_Oeven__mask__iff,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)),one_one_nat))
    <=> ( Nb = zero_zero_nat ) ) ).

% semiring_parity_class.even_mask_iff
tff(fact_6190_semiring__parity__class_Oeven__mask__iff,axiom,
    ! [Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)),one_one_int))
    <=> ( Nb = zero_zero_nat ) ) ).

% semiring_parity_class.even_mask_iff
tff(fact_6191_zero__less__power__eq__numeral,axiom,
    ! [A: real,W: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,W)))
    <=> ( ( aa_num_nat(numeral_numeral_nat,W) = 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,W))
          & ( A != zero_zero_real ) )
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
          & aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A) ) ) ) ).

% zero_less_power_eq_numeral
tff(fact_6192_zero__less__power__eq__numeral,axiom,
    ! [A: rat,W: 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,W)))
    <=> ( ( aa_num_nat(numeral_numeral_nat,W) = 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,W))
          & ( 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,W))
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A) ) ) ) ).

% zero_less_power_eq_numeral
tff(fact_6193_zero__less__power__eq__numeral,axiom,
    ! [A: int,W: 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,W)))
    <=> ( ( aa_num_nat(numeral_numeral_nat,W) = 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,W))
          & ( 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,W))
          & aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A) ) ) ) ).

% zero_less_power_eq_numeral
tff(fact_6194_odd__two__times__div__two__nat,axiom,
    ! [Nb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(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,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat) ) ) ).

% odd_two_times_div_two_nat
tff(fact_6195_power__le__zero__eq__numeral,axiom,
    ! [A: real,W: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,W))),zero_zero_real)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,W))
        & ( ( ~ 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,W))
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real) )
          | ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
            & ( A = zero_zero_real ) ) ) ) ) ).

% power_le_zero_eq_numeral
tff(fact_6196_power__le__zero__eq__numeral,axiom,
    ! [A: rat,W: 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,W))),zero_zero_rat)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,W))
        & ( ( ~ 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,W))
            & 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,W))
            & ( A = zero_zero_rat ) ) ) ) ) ).

% power_le_zero_eq_numeral
tff(fact_6197_power__le__zero__eq__numeral,axiom,
    ! [A: int,W: 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,W))),zero_zero_int)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,W))
        & ( ( ~ 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,W))
            & 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,W))
            & ( A = zero_zero_int ) ) ) ) ) ).

% power_le_zero_eq_numeral
tff(fact_6198_even__succ__div__exp,axiom,
    ! [A: code_integer,Nb: 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),Nb)
       => ( 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))),Nb)) = aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Nb)) ) ) ) ).

% even_succ_div_exp
tff(fact_6199_even__succ__div__exp,axiom,
    ! [A: uint32,Nb: nat] :
      ( aa_uint32_o(aa_uin5924812838015166427nt32_o(dvd_dvd_uint32,numera9087168376688890119uint32(aa_num_num(bit0,one))),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => ( aa_uint32_uint32(aa_uin3778660748068847871uint32(divide_divide_uint32,aa_uint32_uint32(plus_plus_uint32(one_one_uint32),A)),aa_nat_uint32(power_power_uint32(numera9087168376688890119uint32(aa_num_num(bit0,one))),Nb)) = aa_uint32_uint32(aa_uin3778660748068847871uint32(divide_divide_uint32,A),aa_nat_uint32(power_power_uint32(numera9087168376688890119uint32(aa_num_num(bit0,one))),Nb)) ) ) ) ).

% even_succ_div_exp
tff(fact_6200_even__succ__div__exp,axiom,
    ! [A: word_N3645301735248828278l_num1,Nb: nat] :
      ( aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(numera7442385471795722001l_num1(aa_num_num(bit0,one))),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => ( divide1791077408188789448l_num1(plus_p361126936061061375l_num1(one_on7727431528512463931l_num1,A),aa_nat5599461682457136004l_num1(power_2184487114949457152l_num1(numera7442385471795722001l_num1(aa_num_num(bit0,one))),Nb)) = divide1791077408188789448l_num1(A,aa_nat5599461682457136004l_num1(power_2184487114949457152l_num1(numera7442385471795722001l_num1(aa_num_num(bit0,one))),Nb)) ) ) ) ).

% even_succ_div_exp
tff(fact_6201_even__succ__div__exp,axiom,
    ! [A: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),A)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ) ) ) ).

% even_succ_div_exp
tff(fact_6202_even__succ__div__exp,axiom,
    ! [A: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),A)),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) ) ) ) ).

% even_succ_div_exp
tff(fact_6203_even__succ__mod__exp,axiom,
    ! [A: word_N3645301735248828278l_num1,Nb: nat] :
      ( aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(numera7442385471795722001l_num1(aa_num_num(bit0,one))),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => ( modulo1504961113040953224l_num1(plus_p361126936061061375l_num1(one_on7727431528512463931l_num1,A),aa_nat5599461682457136004l_num1(power_2184487114949457152l_num1(numera7442385471795722001l_num1(aa_num_num(bit0,one))),Nb)) = plus_p361126936061061375l_num1(one_on7727431528512463931l_num1,modulo1504961113040953224l_num1(A,aa_nat5599461682457136004l_num1(power_2184487114949457152l_num1(numera7442385471795722001l_num1(aa_num_num(bit0,one))),Nb))) ) ) ) ).

% even_succ_mod_exp
tff(fact_6204_even__succ__mod__exp,axiom,
    ! [A: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),A)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))) ) ) ) ).

% even_succ_mod_exp
tff(fact_6205_even__succ__mod__exp,axiom,
    ! [A: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),A)),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb))) ) ) ) ).

% even_succ_mod_exp
tff(fact_6206_even__succ__mod__exp,axiom,
    ! [A: code_integer,Nb: 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),Nb)
       => ( 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))),Nb)) = 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))),Nb))) ) ) ) ).

% even_succ_mod_exp
tff(fact_6207_even__succ__mod__exp,axiom,
    ! [A: uint32,Nb: nat] :
      ( aa_uint32_o(aa_uin5924812838015166427nt32_o(dvd_dvd_uint32,numera9087168376688890119uint32(aa_num_num(bit0,one))),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => ( aa_uint32_uint32(aa_uin3778660748068847871uint32(modulo_modulo_uint32,aa_uint32_uint32(plus_plus_uint32(one_one_uint32),A)),aa_nat_uint32(power_power_uint32(numera9087168376688890119uint32(aa_num_num(bit0,one))),Nb)) = aa_uint32_uint32(plus_plus_uint32(one_one_uint32),aa_uint32_uint32(aa_uin3778660748068847871uint32(modulo_modulo_uint32,A),aa_nat_uint32(power_power_uint32(numera9087168376688890119uint32(aa_num_num(bit0,one))),Nb))) ) ) ) ).

% even_succ_mod_exp
tff(fact_6208_dvd__refl,axiom,
    ! [A: nat] : aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),A) ).

% dvd_refl
tff(fact_6209_dvd__refl,axiom,
    ! [A: int] : aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),A) ).

% dvd_refl
tff(fact_6210_dvd__refl,axiom,
    ! [A: uint32] : aa_uint32_o(aa_uin5924812838015166427nt32_o(dvd_dvd_uint32,A),A) ).

% dvd_refl
tff(fact_6211_dvd__refl,axiom,
    ! [A: word_N3645301735248828278l_num1] : aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(A),A) ).

% dvd_refl
tff(fact_6212_dvd__refl,axiom,
    ! [A: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),A) ).

% dvd_refl
tff(fact_6213_dvd__trans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),C)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),C) ) ) ).

% dvd_trans
tff(fact_6214_dvd__trans,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),C)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),C) ) ) ).

% dvd_trans
tff(fact_6215_dvd__trans,axiom,
    ! [A: uint32,B: uint32,C: uint32] :
      ( aa_uint32_o(aa_uin5924812838015166427nt32_o(dvd_dvd_uint32,A),B)
     => ( aa_uint32_o(aa_uin5924812838015166427nt32_o(dvd_dvd_uint32,B),C)
       => aa_uint32_o(aa_uin5924812838015166427nt32_o(dvd_dvd_uint32,A),C) ) ) ).

% dvd_trans
tff(fact_6216_dvd__trans,axiom,
    ! [A: word_N3645301735248828278l_num1,B: word_N3645301735248828278l_num1,C: word_N3645301735248828278l_num1] :
      ( aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(A),B)
     => ( aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(B),C)
       => aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(A),C) ) ) ).

% dvd_trans
tff(fact_6217_dvd__trans,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),C)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),C) ) ) ).

% dvd_trans
tff(fact_6218_dvd__antisym,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,M),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Nb),M)
       => ( M = Nb ) ) ) ).

% dvd_antisym
tff(fact_6219_of__nat__dvd__iff,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,M),Nb) ) ).

% of_nat_dvd_iff
tff(fact_6220_of__nat__dvd__iff,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,M),Nb) ) ).

% of_nat_dvd_iff
tff(fact_6221_of__nat__dvd__iff,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_nat_Code_integer(semiri4939895301339042750nteger,M)),aa_nat_Code_integer(semiri4939895301339042750nteger,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,M),Nb) ) ).

% of_nat_dvd_iff
tff(fact_6222_curry__K,axiom,
    ! [C: fun_na5556145776640407093on_nat,X3: vEBT_VEBT,Xa: vEBT_VEBTi] : aa_VEB3566417204837707778on_nat(aa_VEB1030146501360450209on_nat(produc9121605208400662357on_nat(aTP_Lamm_wk(C)),X3),Xa) = C ).

% curry_K
tff(fact_6223_curry__K,axiom,
    ! [C: fun_na2343073368905879023Heap_o,X3: vEBT_VEBT,Xa: vEBT_VEBTi] : aa_VEB1455453220190609250Heap_o(aa_VEB8672304581938048793Heap_o(produc8906325494159015119Heap_o(aTP_Lamm_wl(C)),X3),Xa) = C ).

% curry_K
tff(fact_6224_curry__K,axiom,
    ! [C: heap_T8145700208782473153_VEBTi,X3: produc3625547720036274456_VEBTi,Xa: nat] : aa_nat5408703056993926195_VEBTi(aa_Pro284253941917238388_VEBTi(produc2164094337957399884_VEBTi(aTP_Lamm_wm(C)),X3),Xa) = C ).

% curry_K
tff(fact_6225_curry__K,axiom,
    ! [C: heap_T2636463487746394924on_nat,X3: produc3625547720036274456_VEBTi,Xa: nat] : aa_nat947733127081326238on_nat(aa_Pro1290670013494283871on_nat(produc1757988346207259447on_nat(aTP_Lamm_wn(C)),X3),Xa) = C ).

% curry_K
tff(fact_6226_curry__K,axiom,
    ! [C: heap_Time_Heap_o,X3: produc3625547720036274456_VEBTi,Xa: nat] : aa_nat2910048432523430448Heap_o(aa_Pro259339148018954309Heap_o(produc8381543706267210711Heap_o(aTP_Lamm_wo(C)),X3),Xa) = C ).

% curry_K
tff(fact_6227_dvd__0__left,axiom,
    ! [A: word_N3645301735248828278l_num1] :
      ( aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(zero_z3563351764282998399l_num1),A)
     => ( A = zero_z3563351764282998399l_num1 ) ) ).

% dvd_0_left
tff(fact_6228_dvd__0__left,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,zero_z3403309356797280102nteger),A)
     => ( A = zero_z3403309356797280102nteger ) ) ).

% dvd_0_left
tff(fact_6229_dvd__0__left,axiom,
    ! [A: uint32] :
      ( aa_uint32_o(aa_uin5924812838015166427nt32_o(dvd_dvd_uint32,zero_zero_uint32),A)
     => ( A = zero_zero_uint32 ) ) ).

% dvd_0_left
tff(fact_6230_dvd__0__left,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,zero_zero_real),A)
     => ( A = zero_zero_real ) ) ).

% dvd_0_left
tff(fact_6231_dvd__0__left,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,zero_zero_rat),A)
     => ( A = zero_zero_rat ) ) ).

% dvd_0_left
tff(fact_6232_dvd__0__left,axiom,
    ! [A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,zero_zero_nat),A)
     => ( A = zero_zero_nat ) ) ).

% dvd_0_left
tff(fact_6233_dvd__0__left,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,zero_zero_int),A)
     => ( A = zero_zero_int ) ) ).

% dvd_0_left
tff(fact_6234_dvd__add__right__iff,axiom,
    ! [A: uint32,B: uint32,C: uint32] :
      ( aa_uint32_o(aa_uin5924812838015166427nt32_o(dvd_dvd_uint32,A),B)
     => ( aa_uint32_o(aa_uin5924812838015166427nt32_o(dvd_dvd_uint32,A),aa_uint32_uint32(plus_plus_uint32(B),C))
      <=> aa_uint32_o(aa_uin5924812838015166427nt32_o(dvd_dvd_uint32,A),C) ) ) ).

% dvd_add_right_iff
tff(fact_6235_dvd__add__right__iff,axiom,
    ! [A: word_N3645301735248828278l_num1,B: word_N3645301735248828278l_num1,C: word_N3645301735248828278l_num1] :
      ( aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(A),B)
     => ( aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(A),plus_p361126936061061375l_num1(B,C))
      <=> aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(A),C) ) ) ).

% dvd_add_right_iff
tff(fact_6236_dvd__add__right__iff,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(B),C))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),C) ) ) ).

% dvd_add_right_iff
tff(fact_6237_dvd__add__right__iff,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C))
      <=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),C) ) ) ).

% dvd_add_right_iff
tff(fact_6238_dvd__add__right__iff,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),aa_rat_rat(plus_plus_rat(B),C))
      <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),C) ) ) ).

% dvd_add_right_iff
tff(fact_6239_dvd__add__right__iff,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),C) ) ) ).

% dvd_add_right_iff
tff(fact_6240_dvd__add__right__iff,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),C) ) ) ).

% dvd_add_right_iff
tff(fact_6241_dvd__add__left__iff,axiom,
    ! [A: uint32,C: uint32,B: uint32] :
      ( aa_uint32_o(aa_uin5924812838015166427nt32_o(dvd_dvd_uint32,A),C)
     => ( aa_uint32_o(aa_uin5924812838015166427nt32_o(dvd_dvd_uint32,A),aa_uint32_uint32(plus_plus_uint32(B),C))
      <=> aa_uint32_o(aa_uin5924812838015166427nt32_o(dvd_dvd_uint32,A),B) ) ) ).

% dvd_add_left_iff
tff(fact_6242_dvd__add__left__iff,axiom,
    ! [A: word_N3645301735248828278l_num1,C: word_N3645301735248828278l_num1,B: word_N3645301735248828278l_num1] :
      ( aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(A),C)
     => ( aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(A),plus_p361126936061061375l_num1(B,C))
      <=> aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(A),B) ) ) ).

% dvd_add_left_iff
tff(fact_6243_dvd__add__left__iff,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),C)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(B),C))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B) ) ) ).

% dvd_add_left_iff
tff(fact_6244_dvd__add__left__iff,axiom,
    ! [A: real,C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),C)
     => ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C))
      <=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),B) ) ) ).

% dvd_add_left_iff
tff(fact_6245_dvd__add__left__iff,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),C)
     => ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),aa_rat_rat(plus_plus_rat(B),C))
      <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),B) ) ) ).

% dvd_add_left_iff
tff(fact_6246_dvd__add__left__iff,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),C)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B) ) ) ).

% dvd_add_left_iff
tff(fact_6247_dvd__add__left__iff,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),C)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B) ) ) ).

% dvd_add_left_iff
tff(fact_6248_dvd__add,axiom,
    ! [A: uint32,B: uint32,C: uint32] :
      ( aa_uint32_o(aa_uin5924812838015166427nt32_o(dvd_dvd_uint32,A),B)
     => ( aa_uint32_o(aa_uin5924812838015166427nt32_o(dvd_dvd_uint32,A),C)
       => aa_uint32_o(aa_uin5924812838015166427nt32_o(dvd_dvd_uint32,A),aa_uint32_uint32(plus_plus_uint32(B),C)) ) ) ).

% dvd_add
tff(fact_6249_dvd__add,axiom,
    ! [A: word_N3645301735248828278l_num1,B: word_N3645301735248828278l_num1,C: word_N3645301735248828278l_num1] :
      ( aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(A),B)
     => ( aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(A),C)
       => aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(A),plus_p361126936061061375l_num1(B,C)) ) ) ).

% dvd_add
tff(fact_6250_dvd__add,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),C)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(B),C)) ) ) ).

% dvd_add
tff(fact_6251_dvd__add,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),C)
       => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C)) ) ) ).

% dvd_add
tff(fact_6252_dvd__add,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),C)
       => aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),aa_rat_rat(plus_plus_rat(B),C)) ) ) ).

% dvd_add
tff(fact_6253_dvd__add,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),C)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)) ) ) ).

% dvd_add
tff(fact_6254_dvd__add,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),C)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C)) ) ) ).

% dvd_add
tff(fact_6255_one__dvd,axiom,
    ! [A: uint32] : aa_uint32_o(aa_uin5924812838015166427nt32_o(dvd_dvd_uint32,one_one_uint32),A) ).

% one_dvd
tff(fact_6256_one__dvd,axiom,
    ! [A: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,one_one_Code_integer),A) ).

% one_dvd
tff(fact_6257_one__dvd,axiom,
    ! [A: word_N3645301735248828278l_num1] : aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(one_on7727431528512463931l_num1),A) ).

% one_dvd
tff(fact_6258_one__dvd,axiom,
    ! [A: real] : aa_real_o(aa_real_fun_real_o(dvd_dvd_real,one_one_real),A) ).

% one_dvd
tff(fact_6259_one__dvd,axiom,
    ! [A: rat] : aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,one_one_rat),A) ).

% one_dvd
tff(fact_6260_one__dvd,axiom,
    ! [A: nat] : aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,one_one_nat),A) ).

% one_dvd
tff(fact_6261_one__dvd,axiom,
    ! [A: int] : aa_int_o(aa_int_fun_int_o(dvd_dvd_int,one_one_int),A) ).

% one_dvd
tff(fact_6262_unit__imp__dvd,axiom,
    ! [B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),A) ) ).

% unit_imp_dvd
tff(fact_6263_unit__imp__dvd,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),A) ) ).

% unit_imp_dvd
tff(fact_6264_unit__imp__dvd,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),A) ) ).

% unit_imp_dvd
tff(fact_6265_dvd__unit__imp__unit,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),one_one_Code_integer) ) ) ).

% dvd_unit_imp_unit
tff(fact_6266_dvd__unit__imp__unit,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),one_one_nat) ) ) ).

% dvd_unit_imp_unit
tff(fact_6267_dvd__unit__imp__unit,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int) ) ) ).

% dvd_unit_imp_unit
tff(fact_6268_dvdE,axiom,
    ! [B: uint32,A: uint32] :
      ( aa_uint32_o(aa_uin5924812838015166427nt32_o(dvd_dvd_uint32,B),A)
     => ~ ! [K3: uint32] : A != aa_uint32_uint32(times_times_uint32(B),K3) ) ).

% dvdE
tff(fact_6269_dvdE,axiom,
    ! [B: word_N3645301735248828278l_num1,A: word_N3645301735248828278l_num1] :
      ( aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(B),A)
     => ~ ! [K3: word_N3645301735248828278l_num1] : A != aa_wor7274019993978516266l_num1(times_7065122842183080059l_num1(B),K3) ) ).

% dvdE
tff(fact_6270_dvdE,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) ) ).

% dvdE
tff(fact_6271_dvdE,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,B),A)
     => ~ ! [K3: real] : A != aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),K3) ) ).

% dvdE
tff(fact_6272_dvdE,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,B),A)
     => ~ ! [K3: rat] : A != aa_rat_rat(times_times_rat(B),K3) ) ).

% dvdE
tff(fact_6273_dvdE,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) ) ).

% dvdE
tff(fact_6274_dvdE,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) ) ).

% dvdE
tff(fact_6275_dvdI,axiom,
    ! [A: uint32,B: uint32,K: uint32] :
      ( ( A = aa_uint32_uint32(times_times_uint32(B),K) )
     => aa_uint32_o(aa_uin5924812838015166427nt32_o(dvd_dvd_uint32,B),A) ) ).

% dvdI
tff(fact_6276_dvdI,axiom,
    ! [A: word_N3645301735248828278l_num1,B: word_N3645301735248828278l_num1,K: word_N3645301735248828278l_num1] :
      ( ( A = aa_wor7274019993978516266l_num1(times_7065122842183080059l_num1(B),K) )
     => aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(B),A) ) ).

% dvdI
tff(fact_6277_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_6278_dvdI,axiom,
    ! [A: real,B: real,K: real] :
      ( ( A = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),K) )
     => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,B),A) ) ).

% dvdI
tff(fact_6279_dvdI,axiom,
    ! [A: rat,B: rat,K: rat] :
      ( ( A = aa_rat_rat(times_times_rat(B),K) )
     => aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,B),A) ) ).

% dvdI
tff(fact_6280_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_6281_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_6282_dvd__def,axiom,
    ! [B: uint32,A: uint32] :
      ( aa_uint32_o(aa_uin5924812838015166427nt32_o(dvd_dvd_uint32,B),A)
    <=> ? [K4: uint32] : A = aa_uint32_uint32(times_times_uint32(B),K4) ) ).

% dvd_def
tff(fact_6283_dvd__def,axiom,
    ! [B: word_N3645301735248828278l_num1,A: word_N3645301735248828278l_num1] :
      ( aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(B),A)
    <=> ? [K4: word_N3645301735248828278l_num1] : A = aa_wor7274019993978516266l_num1(times_7065122842183080059l_num1(B),K4) ) ).

% dvd_def
tff(fact_6284_dvd__def,axiom,
    ! [B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),A)
    <=> ? [K4: code_integer] : A = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),K4) ) ).

% dvd_def
tff(fact_6285_dvd__def,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,B),A)
    <=> ? [K4: real] : A = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),K4) ) ).

% dvd_def
tff(fact_6286_dvd__def,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,B),A)
    <=> ? [K4: rat] : A = aa_rat_rat(times_times_rat(B),K4) ) ).

% dvd_def
tff(fact_6287_dvd__def,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),A)
    <=> ? [K4: nat] : A = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),K4) ) ).

% dvd_def
tff(fact_6288_dvd__def,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),A)
    <=> ? [K4: int] : A = aa_int_int(aa_int_fun_int_int(times_times_int,B),K4) ) ).

% dvd_def
tff(fact_6289_dvd__mult,axiom,
    ! [A: uint32,C: uint32,B: uint32] :
      ( aa_uint32_o(aa_uin5924812838015166427nt32_o(dvd_dvd_uint32,A),C)
     => aa_uint32_o(aa_uin5924812838015166427nt32_o(dvd_dvd_uint32,A),aa_uint32_uint32(times_times_uint32(B),C)) ) ).

% dvd_mult
tff(fact_6290_dvd__mult,axiom,
    ! [A: word_N3645301735248828278l_num1,C: word_N3645301735248828278l_num1,B: word_N3645301735248828278l_num1] :
      ( aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(A),C)
     => aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(A),aa_wor7274019993978516266l_num1(times_7065122842183080059l_num1(B),C)) ) ).

% dvd_mult
tff(fact_6291_dvd__mult,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),C)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)) ) ).

% dvd_mult
tff(fact_6292_dvd__mult,axiom,
    ! [A: real,C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),C)
     => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)) ) ).

% dvd_mult
tff(fact_6293_dvd__mult,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),C)
     => aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),aa_rat_rat(times_times_rat(B),C)) ) ).

% dvd_mult
tff(fact_6294_dvd__mult,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),C)
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) ) ).

% dvd_mult
tff(fact_6295_dvd__mult,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),C)
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ) ).

% dvd_mult
tff(fact_6296_dvd__mult2,axiom,
    ! [A: uint32,B: uint32,C: uint32] :
      ( aa_uint32_o(aa_uin5924812838015166427nt32_o(dvd_dvd_uint32,A),B)
     => aa_uint32_o(aa_uin5924812838015166427nt32_o(dvd_dvd_uint32,A),aa_uint32_uint32(times_times_uint32(B),C)) ) ).

% dvd_mult2
tff(fact_6297_dvd__mult2,axiom,
    ! [A: word_N3645301735248828278l_num1,B: word_N3645301735248828278l_num1,C: word_N3645301735248828278l_num1] :
      ( aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(A),B)
     => aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(A),aa_wor7274019993978516266l_num1(times_7065122842183080059l_num1(B),C)) ) ).

% dvd_mult2
tff(fact_6298_dvd__mult2,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)) ) ).

% dvd_mult2
tff(fact_6299_dvd__mult2,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),B)
     => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)) ) ).

% dvd_mult2
tff(fact_6300_dvd__mult2,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),B)
     => aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),aa_rat_rat(times_times_rat(B),C)) ) ).

% dvd_mult2
tff(fact_6301_dvd__mult2,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) ) ).

% dvd_mult2
tff(fact_6302_dvd__mult2,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B)
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ) ).

% dvd_mult2
tff(fact_6303_dvd__mult__left,axiom,
    ! [A: uint32,B: uint32,C: uint32] :
      ( aa_uint32_o(aa_uin5924812838015166427nt32_o(dvd_dvd_uint32,aa_uint32_uint32(times_times_uint32(A),B)),C)
     => aa_uint32_o(aa_uin5924812838015166427nt32_o(dvd_dvd_uint32,A),C) ) ).

% dvd_mult_left
tff(fact_6304_dvd__mult__left,axiom,
    ! [A: word_N3645301735248828278l_num1,B: word_N3645301735248828278l_num1,C: word_N3645301735248828278l_num1] :
      ( aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(aa_wor7274019993978516266l_num1(times_7065122842183080059l_num1(A),B)),C)
     => aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(A),C) ) ).

% dvd_mult_left
tff(fact_6305_dvd__mult__left,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),C)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),C) ) ).

% dvd_mult_left
tff(fact_6306_dvd__mult__left,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),C)
     => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),C) ) ).

% dvd_mult_left
tff(fact_6307_dvd__mult__left,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,aa_rat_rat(times_times_rat(A),B)),C)
     => aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),C) ) ).

% dvd_mult_left
tff(fact_6308_dvd__mult__left,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C)
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),C) ) ).

% dvd_mult_left
tff(fact_6309_dvd__mult__left,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),C)
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),C) ) ).

% dvd_mult_left
tff(fact_6310_dvd__triv__left,axiom,
    ! [A: uint32,B: uint32] : aa_uint32_o(aa_uin5924812838015166427nt32_o(dvd_dvd_uint32,A),aa_uint32_uint32(times_times_uint32(A),B)) ).

% dvd_triv_left
tff(fact_6311_dvd__triv__left,axiom,
    ! [A: word_N3645301735248828278l_num1,B: word_N3645301735248828278l_num1] : aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(A),aa_wor7274019993978516266l_num1(times_7065122842183080059l_num1(A),B)) ).

% dvd_triv_left
tff(fact_6312_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_6313_dvd__triv__left,axiom,
    ! [A: real,B: real] : aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)) ).

% dvd_triv_left
tff(fact_6314_dvd__triv__left,axiom,
    ! [A: rat,B: rat] : aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),aa_rat_rat(times_times_rat(A),B)) ).

% dvd_triv_left
tff(fact_6315_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_6316_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_6317_mult__dvd__mono,axiom,
    ! [A: uint32,B: uint32,C: uint32,D: uint32] :
      ( aa_uint32_o(aa_uin5924812838015166427nt32_o(dvd_dvd_uint32,A),B)
     => ( aa_uint32_o(aa_uin5924812838015166427nt32_o(dvd_dvd_uint32,C),D)
       => aa_uint32_o(aa_uin5924812838015166427nt32_o(dvd_dvd_uint32,aa_uint32_uint32(times_times_uint32(A),C)),aa_uint32_uint32(times_times_uint32(B),D)) ) ) ).

% mult_dvd_mono
tff(fact_6318_mult__dvd__mono,axiom,
    ! [A: word_N3645301735248828278l_num1,B: word_N3645301735248828278l_num1,C: word_N3645301735248828278l_num1,D: word_N3645301735248828278l_num1] :
      ( aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(A),B)
     => ( aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(C),D)
       => aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(aa_wor7274019993978516266l_num1(times_7065122842183080059l_num1(A),C)),aa_wor7274019993978516266l_num1(times_7065122842183080059l_num1(B),D)) ) ) ).

% mult_dvd_mono
tff(fact_6319_mult__dvd__mono,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer,D: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),D)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),D)) ) ) ).

% mult_dvd_mono
tff(fact_6320_mult__dvd__mono,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,C),D)
       => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),D)) ) ) ).

% mult_dvd_mono
tff(fact_6321_mult__dvd__mono,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,C),D)
       => aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),D)) ) ) ).

% mult_dvd_mono
tff(fact_6322_mult__dvd__mono,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),D)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D)) ) ) ).

% mult_dvd_mono
tff(fact_6323_mult__dvd__mono,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),D)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D)) ) ) ).

% mult_dvd_mono
tff(fact_6324_dvd__mult__right,axiom,
    ! [A: uint32,B: uint32,C: uint32] :
      ( aa_uint32_o(aa_uin5924812838015166427nt32_o(dvd_dvd_uint32,aa_uint32_uint32(times_times_uint32(A),B)),C)
     => aa_uint32_o(aa_uin5924812838015166427nt32_o(dvd_dvd_uint32,B),C) ) ).

% dvd_mult_right
tff(fact_6325_dvd__mult__right,axiom,
    ! [A: word_N3645301735248828278l_num1,B: word_N3645301735248828278l_num1,C: word_N3645301735248828278l_num1] :
      ( aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(aa_wor7274019993978516266l_num1(times_7065122842183080059l_num1(A),B)),C)
     => aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(B),C) ) ).

% dvd_mult_right
tff(fact_6326_dvd__mult__right,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),C)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),C) ) ).

% dvd_mult_right
tff(fact_6327_dvd__mult__right,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),C)
     => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,B),C) ) ).

% dvd_mult_right
tff(fact_6328_dvd__mult__right,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,aa_rat_rat(times_times_rat(A),B)),C)
     => aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,B),C) ) ).

% dvd_mult_right
tff(fact_6329_dvd__mult__right,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C)
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),C) ) ).

% dvd_mult_right
tff(fact_6330_dvd__mult__right,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),C)
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),C) ) ).

% dvd_mult_right
tff(fact_6331_dvd__triv__right,axiom,
    ! [A: uint32,B: uint32] : aa_uint32_o(aa_uin5924812838015166427nt32_o(dvd_dvd_uint32,A),aa_uint32_uint32(times_times_uint32(B),A)) ).

% dvd_triv_right
tff(fact_6332_dvd__triv__right,axiom,
    ! [A: word_N3645301735248828278l_num1,B: word_N3645301735248828278l_num1] : aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(A),aa_wor7274019993978516266l_num1(times_7065122842183080059l_num1(B),A)) ).

% dvd_triv_right
tff(fact_6333_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_6334_dvd__triv__right,axiom,
    ! [A: real,B: real] : aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),A)) ).

% dvd_triv_right
tff(fact_6335_dvd__triv__right,axiom,
    ! [A: rat,B: rat] : aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),aa_rat_rat(times_times_rat(B),A)) ).

% dvd_triv_right
tff(fact_6336_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_6337_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_6338_dvd__diff__commute,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(C),B))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(B),C)) ) ).

% dvd_diff_commute
tff(fact_6339_dvd__diff__commute,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(minus_minus_int,C),B))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),C)) ) ).

% dvd_diff_commute
tff(fact_6340_dvd__diff,axiom,
    ! [Xb: uint32,Y: uint32,Z: uint32] :
      ( aa_uint32_o(aa_uin5924812838015166427nt32_o(dvd_dvd_uint32,Xb),Y)
     => ( aa_uint32_o(aa_uin5924812838015166427nt32_o(dvd_dvd_uint32,Xb),Z)
       => aa_uint32_o(aa_uin5924812838015166427nt32_o(dvd_dvd_uint32,Xb),minus_minus_uint32(Y,Z)) ) ) ).

% dvd_diff
tff(fact_6341_dvd__diff,axiom,
    ! [Xb: word_N3645301735248828278l_num1,Y: word_N3645301735248828278l_num1,Z: word_N3645301735248828278l_num1] :
      ( aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(Xb),Y)
     => ( aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(Xb),Z)
       => aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(Xb),minus_4019991460397169231l_num1(Y,Z)) ) ) ).

% dvd_diff
tff(fact_6342_dvd__diff,axiom,
    ! [Xb: code_integer,Y: code_integer,Z: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Xb),Y)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Xb),Z)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Xb),aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(Y),Z)) ) ) ).

% dvd_diff
tff(fact_6343_dvd__diff,axiom,
    ! [Xb: real,Y: real,Z: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,Xb),Y)
     => ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,Xb),Z)
       => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Y),Z)) ) ) ).

% dvd_diff
tff(fact_6344_dvd__diff,axiom,
    ! [Xb: rat,Y: rat,Z: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,Xb),Y)
     => ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,Xb),Z)
       => aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,Xb),minus_minus_rat(Y,Z)) ) ) ).

% dvd_diff
tff(fact_6345_dvd__diff,axiom,
    ! [Xb: int,Y: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Xb),Y)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Xb),Z)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Xb),aa_int_int(aa_int_fun_int_int(minus_minus_int,Y),Z)) ) ) ).

% dvd_diff
tff(fact_6346_dvd__div__eq__iff,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),A)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),B)
       => ( ( aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),C) = aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(B),C) )
        <=> ( A = B ) ) ) ) ).

% dvd_div_eq_iff
tff(fact_6347_dvd__div__eq__iff,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,C),A)
     => ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,C),B)
       => ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),C) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C) )
        <=> ( A = B ) ) ) ) ).

% dvd_div_eq_iff
tff(fact_6348_dvd__div__eq__iff,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,C),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,C),B)
       => ( ( divide_divide_rat(A,C) = divide_divide_rat(B,C) )
        <=> ( A = B ) ) ) ) ).

% dvd_div_eq_iff
tff(fact_6349_dvd__div__eq__iff,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),B)
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),C) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C) )
        <=> ( A = B ) ) ) ) ).

% dvd_div_eq_iff
tff(fact_6350_dvd__div__eq__iff,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),A)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),B)
       => ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),C) = aa_int_int(aa_int_fun_int_int(divide_divide_int,B),C) )
        <=> ( A = B ) ) ) ) ).

% dvd_div_eq_iff
tff(fact_6351_dvd__div__eq__cancel,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),C) = aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(B),C) )
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),A)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),B)
         => ( A = B ) ) ) ) ).

% dvd_div_eq_cancel
tff(fact_6352_dvd__div__eq__cancel,axiom,
    ! [A: real,C: real,B: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),C) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C) )
     => ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,C),A)
       => ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,C),B)
         => ( A = B ) ) ) ) ).

% dvd_div_eq_cancel
tff(fact_6353_dvd__div__eq__cancel,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( divide_divide_rat(A,C) = divide_divide_rat(B,C) )
     => ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,C),A)
       => ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,C),B)
         => ( A = B ) ) ) ) ).

% dvd_div_eq_cancel
tff(fact_6354_dvd__div__eq__cancel,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),C) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C) )
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),A)
       => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),B)
         => ( A = B ) ) ) ) ).

% dvd_div_eq_cancel
tff(fact_6355_dvd__div__eq__cancel,axiom,
    ! [A: int,C: int,B: int] :
      ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),C) = aa_int_int(aa_int_fun_int_int(divide_divide_int,B),C) )
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),A)
       => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),B)
         => ( A = B ) ) ) ) ).

% dvd_div_eq_cancel
tff(fact_6356_div__div__div__same,axiom,
    ! [D: code_integer,B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),A)
       => ( aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),D)),aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(B),D)) = aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),B) ) ) ) ).

% div_div_div_same
tff(fact_6357_div__div__div__same,axiom,
    ! [D: nat,B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),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,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),D)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),D)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B) ) ) ) ).

% div_div_div_same
tff(fact_6358_div__div__div__same,axiom,
    ! [D: int,B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),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,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),D)),aa_int_int(aa_int_fun_int_int(divide_divide_int,B),D)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B) ) ) ) ).

% div_div_div_same
tff(fact_6359_dvd__power__same,axiom,
    ! [Xb: code_integer,Y: code_integer,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Xb),Y)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(Xb),Nb)),aa_nat_Code_integer(power_8256067586552552935nteger(Y),Nb)) ) ).

% dvd_power_same
tff(fact_6360_dvd__power__same,axiom,
    ! [Xb: uint32,Y: uint32,Nb: nat] :
      ( aa_uint32_o(aa_uin5924812838015166427nt32_o(dvd_dvd_uint32,Xb),Y)
     => aa_uint32_o(aa_uin5924812838015166427nt32_o(dvd_dvd_uint32,aa_nat_uint32(power_power_uint32(Xb),Nb)),aa_nat_uint32(power_power_uint32(Y),Nb)) ) ).

% dvd_power_same
tff(fact_6361_dvd__power__same,axiom,
    ! [Xb: word_N3645301735248828278l_num1,Y: word_N3645301735248828278l_num1,Nb: nat] :
      ( aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(Xb),Y)
     => aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(aa_nat5599461682457136004l_num1(power_2184487114949457152l_num1(Xb),Nb)),aa_nat5599461682457136004l_num1(power_2184487114949457152l_num1(Y),Nb)) ) ).

% dvd_power_same
tff(fact_6362_dvd__power__same,axiom,
    ! [Xb: real,Y: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,Xb),Y)
     => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),Nb)) ) ).

% dvd_power_same
tff(fact_6363_dvd__power__same,axiom,
    ! [Xb: nat,Y: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Xb),Y)
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Xb),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Y),Nb)) ) ).

% dvd_power_same
tff(fact_6364_dvd__power__same,axiom,
    ! [Xb: int,Y: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Xb),Y)
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,Xb),Nb)),aa_nat_int(aa_int_fun_nat_int(power_power_int,Y),Nb)) ) ).

% dvd_power_same
tff(fact_6365_dvd__mod__iff,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),A) ) ) ).

% dvd_mod_iff
tff(fact_6366_dvd__mod__iff,axiom,
    ! [C: int,B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),B)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),A) ) ) ).

% dvd_mod_iff
tff(fact_6367_dvd__mod__iff,axiom,
    ! [C: code_integer,B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A),B))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),A) ) ) ).

% dvd_mod_iff
tff(fact_6368_dvd__mod__imp__dvd,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B))
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),B)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),A) ) ) ).

% dvd_mod_imp_dvd
tff(fact_6369_dvd__mod__imp__dvd,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B))
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),B)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),A) ) ) ).

% dvd_mod_imp_dvd
tff(fact_6370_dvd__mod__imp__dvd,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A),B))
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),B)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),A) ) ) ).

% dvd_mod_imp_dvd
tff(fact_6371_dvd__mod,axiom,
    ! [K: nat,M: nat,Nb: 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),Nb)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Nb)) ) ) ).

% dvd_mod
tff(fact_6372_dvd__mod,axiom,
    ! [K: int,M: int,Nb: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,K),M)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,K),Nb)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,K),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,M),Nb)) ) ) ).

% dvd_mod
tff(fact_6373_dvd__mod,axiom,
    ! [K: code_integer,M: code_integer,Nb: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,K),M)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,K),Nb)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,K),aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(M),Nb)) ) ) ).

% dvd_mod
tff(fact_6374_mod__mod__cancel,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),C) ) ) ).

% mod_mod_cancel
tff(fact_6375_mod__mod__cancel,axiom,
    ! [C: int,B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),B)
     => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),C) ) ) ).

% mod_mod_cancel
tff(fact_6376_mod__mod__cancel,axiom,
    ! [C: code_integer,B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),B)
     => ( aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A),B)),C) = aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A),C) ) ) ).

% mod_mod_cancel
tff(fact_6377_dvd__diff__nat,axiom,
    ! [K: nat,M: nat,Nb: 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),Nb)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Nb)) ) ) ).

% dvd_diff_nat
tff(fact_6378_unset__bit__less__eq,axiom,
    ! [Nb: 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,Nb),K)),K) ).

% unset_bit_less_eq
tff(fact_6379_subset__divisors__dvd,axiom,
    ! [A: complex,B: complex] :
      ( aa_set_complex_o(ord_le211207098394363844omplex(collect_complex(aTP_Lamm_wp(A))),collect_complex(aTP_Lamm_wp(B)))
    <=> aa_complex_o(dvd_dvd_complex(A),B) ) ).

% subset_divisors_dvd
tff(fact_6380_subset__divisors__dvd,axiom,
    ! [A: nat,B: nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(collect_nat(aa_nat_fun_nat_o(aTP_Lamm_wq,A))),collect_nat(aa_nat_fun_nat_o(aTP_Lamm_wq,B)))
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B) ) ).

% subset_divisors_dvd
tff(fact_6381_subset__divisors__dvd,axiom,
    ! [A: uint32,B: uint32] :
      ( ord_le2219237028632753026uint32(collect_uint32(aa_uin5924812838015166427nt32_o(aTP_Lamm_wr,A)),collect_uint32(aa_uin5924812838015166427nt32_o(aTP_Lamm_wr,B)))
    <=> aa_uint32_o(aa_uin5924812838015166427nt32_o(dvd_dvd_uint32,A),B) ) ).

% subset_divisors_dvd
tff(fact_6382_subset__divisors__dvd,axiom,
    ! [A: word_N3645301735248828278l_num1,B: word_N3645301735248828278l_num1] :
      ( ord_le5203802739334966412l_num1(collec7814023847061821259l_num1(aTP_Lamm_ws(A)),collec7814023847061821259l_num1(aTP_Lamm_ws(B)))
    <=> aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(A),B) ) ).

% subset_divisors_dvd
tff(fact_6383_subset__divisors__dvd,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(collect_Code_integer(aa_Cod326357343816142457eger_o(aTP_Lamm_wt,A))),collect_Code_integer(aa_Cod326357343816142457eger_o(aTP_Lamm_wt,B)))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B) ) ).

% subset_divisors_dvd
tff(fact_6384_subset__divisors__dvd,axiom,
    ! [A: int,B: int] :
      ( aa_set_int_o(ord_less_eq_set_int(collect_int(aa_int_fun_int_o(aTP_Lamm_wu,A))),collect_int(aa_int_fun_int_o(aTP_Lamm_wu,B)))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B) ) ).

% subset_divisors_dvd
tff(fact_6385_strict__subset__divisors__dvd,axiom,
    ! [A: complex,B: complex] :
      ( aa_set_complex_o(ord_less_set_complex(collect_complex(aTP_Lamm_wp(A))),collect_complex(aTP_Lamm_wp(B)))
    <=> ( aa_complex_o(dvd_dvd_complex(A),B)
        & ~ aa_complex_o(dvd_dvd_complex(B),A) ) ) ).

% strict_subset_divisors_dvd
tff(fact_6386_strict__subset__divisors__dvd,axiom,
    ! [A: nat,B: nat] :
      ( aa_set_nat_o(ord_less_set_nat(collect_nat(aa_nat_fun_nat_o(aTP_Lamm_wq,A))),collect_nat(aa_nat_fun_nat_o(aTP_Lamm_wq,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_6387_strict__subset__divisors__dvd,axiom,
    ! [A: int,B: int] :
      ( aa_set_int_o(ord_less_set_int(collect_int(aa_int_fun_int_o(aTP_Lamm_wu,A))),collect_int(aa_int_fun_int_o(aTP_Lamm_wu,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_6388_strict__subset__divisors__dvd,axiom,
    ! [A: uint32,B: uint32] :
      ( ord_less_set_uint32(collect_uint32(aa_uin5924812838015166427nt32_o(aTP_Lamm_wr,A)),collect_uint32(aa_uin5924812838015166427nt32_o(aTP_Lamm_wr,B)))
    <=> ( aa_uint32_o(aa_uin5924812838015166427nt32_o(dvd_dvd_uint32,A),B)
        & ~ aa_uint32_o(aa_uin5924812838015166427nt32_o(dvd_dvd_uint32,B),A) ) ) ).

% strict_subset_divisors_dvd
tff(fact_6389_strict__subset__divisors__dvd,axiom,
    ! [A: word_N3645301735248828278l_num1,B: word_N3645301735248828278l_num1] :
      ( ord_le6726900395242856064l_num1(collec7814023847061821259l_num1(aTP_Lamm_ws(A)),collec7814023847061821259l_num1(aTP_Lamm_ws(B)))
    <=> ( aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(A),B)
        & ~ aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(B),A) ) ) ).

% strict_subset_divisors_dvd
tff(fact_6390_strict__subset__divisors__dvd,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_set5643387332196575111eger_o(ord_le1307284697595431911nteger(collect_Code_integer(aa_Cod326357343816142457eger_o(aTP_Lamm_wt,A))),collect_Code_integer(aa_Cod326357343816142457eger_o(aTP_Lamm_wt,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_6391_not__is__unit__0,axiom,
    ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,zero_z3403309356797280102nteger),one_one_Code_integer) ).

% not_is_unit_0
tff(fact_6392_not__is__unit__0,axiom,
    ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,zero_zero_nat),one_one_nat) ).

% not_is_unit_0
tff(fact_6393_not__is__unit__0,axiom,
    ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,zero_zero_int),one_one_int) ).

% not_is_unit_0
tff(fact_6394_minf_I10_J,axiom,
    ! [D: uint32,S3: uint32] :
    ? [Z3: uint32] :
    ! [X3: uint32] :
      ( aa_uint32_o(aa_uin5924812838015166427nt32_o(ord_less_uint32,X3),Z3)
     => ( ~ aa_uint32_o(aa_uin5924812838015166427nt32_o(dvd_dvd_uint32,D),aa_uint32_uint32(plus_plus_uint32(X3),S3))
      <=> ~ aa_uint32_o(aa_uin5924812838015166427nt32_o(dvd_dvd_uint32,D),aa_uint32_uint32(plus_plus_uint32(X3),S3)) ) ) ).

% minf(10)
tff(fact_6395_minf_I10_J,axiom,
    ! [D: word_N3645301735248828278l_num1,S3: word_N3645301735248828278l_num1] :
    ? [Z3: word_N3645301735248828278l_num1] :
    ! [X3: word_N3645301735248828278l_num1] :
      ( aa_wor4452366487969921764num1_o(ord_le750835935415966154l_num1(X3),Z3)
     => ( ~ aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(D),plus_p361126936061061375l_num1(X3,S3))
      <=> ~ aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(D),plus_p361126936061061375l_num1(X3,S3)) ) ) ).

% minf(10)
tff(fact_6396_minf_I10_J,axiom,
    ! [D: code_integer,S3: code_integer] :
    ? [Z3: code_integer] :
    ! [X3: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,X3),Z3)
     => ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(X3),S3))
      <=> ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(X3),S3)) ) ) ).

% minf(10)
tff(fact_6397_minf_I10_J,axiom,
    ! [D: real,S3: real] :
    ? [Z3: real] :
    ! [X3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Z3)
     => ( ~ aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X3),S3))
      <=> ~ aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X3),S3)) ) ) ).

% minf(10)
tff(fact_6398_minf_I10_J,axiom,
    ! [D: rat,S3: rat] :
    ? [Z3: rat] :
    ! [X3: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Z3)
     => ( ~ aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),aa_rat_rat(plus_plus_rat(X3),S3))
      <=> ~ aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),aa_rat_rat(plus_plus_rat(X3),S3)) ) ) ).

% minf(10)
tff(fact_6399_minf_I10_J,axiom,
    ! [D: nat,S3: nat] :
    ? [Z3: nat] :
    ! [X3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Z3)
     => ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X3),S3))
      <=> ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X3),S3)) ) ) ).

% minf(10)
tff(fact_6400_minf_I10_J,axiom,
    ! [D: int,S3: int] :
    ? [Z3: int] :
    ! [X3: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Z3)
     => ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),S3))
      <=> ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),S3)) ) ) ).

% minf(10)
tff(fact_6401_minf_I9_J,axiom,
    ! [D: uint32,S3: uint32] :
    ? [Z3: uint32] :
    ! [X3: uint32] :
      ( aa_uint32_o(aa_uin5924812838015166427nt32_o(ord_less_uint32,X3),Z3)
     => ( aa_uint32_o(aa_uin5924812838015166427nt32_o(dvd_dvd_uint32,D),aa_uint32_uint32(plus_plus_uint32(X3),S3))
      <=> aa_uint32_o(aa_uin5924812838015166427nt32_o(dvd_dvd_uint32,D),aa_uint32_uint32(plus_plus_uint32(X3),S3)) ) ) ).

% minf(9)
tff(fact_6402_minf_I9_J,axiom,
    ! [D: word_N3645301735248828278l_num1,S3: word_N3645301735248828278l_num1] :
    ? [Z3: word_N3645301735248828278l_num1] :
    ! [X3: word_N3645301735248828278l_num1] :
      ( aa_wor4452366487969921764num1_o(ord_le750835935415966154l_num1(X3),Z3)
     => ( aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(D),plus_p361126936061061375l_num1(X3,S3))
      <=> aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(D),plus_p361126936061061375l_num1(X3,S3)) ) ) ).

% minf(9)
tff(fact_6403_minf_I9_J,axiom,
    ! [D: code_integer,S3: code_integer] :
    ? [Z3: code_integer] :
    ! [X3: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,X3),Z3)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(X3),S3))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(X3),S3)) ) ) ).

% minf(9)
tff(fact_6404_minf_I9_J,axiom,
    ! [D: real,S3: real] :
    ? [Z3: real] :
    ! [X3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Z3)
     => ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X3),S3))
      <=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X3),S3)) ) ) ).

% minf(9)
tff(fact_6405_minf_I9_J,axiom,
    ! [D: rat,S3: rat] :
    ? [Z3: rat] :
    ! [X3: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Z3)
     => ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),aa_rat_rat(plus_plus_rat(X3),S3))
      <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),aa_rat_rat(plus_plus_rat(X3),S3)) ) ) ).

% minf(9)
tff(fact_6406_minf_I9_J,axiom,
    ! [D: nat,S3: nat] :
    ? [Z3: nat] :
    ! [X3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Z3)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X3),S3))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X3),S3)) ) ) ).

% minf(9)
tff(fact_6407_minf_I9_J,axiom,
    ! [D: int,S3: int] :
    ? [Z3: int] :
    ! [X3: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Z3)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),S3))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),S3)) ) ) ).

% minf(9)
tff(fact_6408_pinf_I10_J,axiom,
    ! [D: uint32,S3: uint32] :
    ? [Z3: uint32] :
    ! [X3: uint32] :
      ( aa_uint32_o(aa_uin5924812838015166427nt32_o(ord_less_uint32,Z3),X3)
     => ( ~ aa_uint32_o(aa_uin5924812838015166427nt32_o(dvd_dvd_uint32,D),aa_uint32_uint32(plus_plus_uint32(X3),S3))
      <=> ~ aa_uint32_o(aa_uin5924812838015166427nt32_o(dvd_dvd_uint32,D),aa_uint32_uint32(plus_plus_uint32(X3),S3)) ) ) ).

% pinf(10)
tff(fact_6409_pinf_I10_J,axiom,
    ! [D: word_N3645301735248828278l_num1,S3: word_N3645301735248828278l_num1] :
    ? [Z3: word_N3645301735248828278l_num1] :
    ! [X3: word_N3645301735248828278l_num1] :
      ( aa_wor4452366487969921764num1_o(ord_le750835935415966154l_num1(Z3),X3)
     => ( ~ aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(D),plus_p361126936061061375l_num1(X3,S3))
      <=> ~ aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(D),plus_p361126936061061375l_num1(X3,S3)) ) ) ).

% pinf(10)
tff(fact_6410_pinf_I10_J,axiom,
    ! [D: code_integer,S3: code_integer] :
    ? [Z3: code_integer] :
    ! [X3: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Z3),X3)
     => ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(X3),S3))
      <=> ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(X3),S3)) ) ) ).

% pinf(10)
tff(fact_6411_pinf_I10_J,axiom,
    ! [D: real,S3: real] :
    ? [Z3: real] :
    ! [X3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z3),X3)
     => ( ~ aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X3),S3))
      <=> ~ aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X3),S3)) ) ) ).

% pinf(10)
tff(fact_6412_pinf_I10_J,axiom,
    ! [D: rat,S3: rat] :
    ? [Z3: rat] :
    ! [X3: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z3),X3)
     => ( ~ aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),aa_rat_rat(plus_plus_rat(X3),S3))
      <=> ~ aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),aa_rat_rat(plus_plus_rat(X3),S3)) ) ) ).

% pinf(10)
tff(fact_6413_pinf_I10_J,axiom,
    ! [D: nat,S3: nat] :
    ? [Z3: nat] :
    ! [X3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z3),X3)
     => ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X3),S3))
      <=> ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X3),S3)) ) ) ).

% pinf(10)
tff(fact_6414_pinf_I10_J,axiom,
    ! [D: int,S3: int] :
    ? [Z3: int] :
    ! [X3: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z3),X3)
     => ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),S3))
      <=> ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),S3)) ) ) ).

% pinf(10)
tff(fact_6415_pinf_I9_J,axiom,
    ! [D: uint32,S3: uint32] :
    ? [Z3: uint32] :
    ! [X3: uint32] :
      ( aa_uint32_o(aa_uin5924812838015166427nt32_o(ord_less_uint32,Z3),X3)
     => ( aa_uint32_o(aa_uin5924812838015166427nt32_o(dvd_dvd_uint32,D),aa_uint32_uint32(plus_plus_uint32(X3),S3))
      <=> aa_uint32_o(aa_uin5924812838015166427nt32_o(dvd_dvd_uint32,D),aa_uint32_uint32(plus_plus_uint32(X3),S3)) ) ) ).

% pinf(9)
tff(fact_6416_pinf_I9_J,axiom,
    ! [D: word_N3645301735248828278l_num1,S3: word_N3645301735248828278l_num1] :
    ? [Z3: word_N3645301735248828278l_num1] :
    ! [X3: word_N3645301735248828278l_num1] :
      ( aa_wor4452366487969921764num1_o(ord_le750835935415966154l_num1(Z3),X3)
     => ( aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(D),plus_p361126936061061375l_num1(X3,S3))
      <=> aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(D),plus_p361126936061061375l_num1(X3,S3)) ) ) ).

% pinf(9)
tff(fact_6417_pinf_I9_J,axiom,
    ! [D: code_integer,S3: code_integer] :
    ? [Z3: code_integer] :
    ! [X3: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Z3),X3)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(X3),S3))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(X3),S3)) ) ) ).

% pinf(9)
tff(fact_6418_pinf_I9_J,axiom,
    ! [D: real,S3: real] :
    ? [Z3: real] :
    ! [X3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z3),X3)
     => ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X3),S3))
      <=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X3),S3)) ) ) ).

% pinf(9)
tff(fact_6419_pinf_I9_J,axiom,
    ! [D: rat,S3: rat] :
    ? [Z3: rat] :
    ! [X3: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z3),X3)
     => ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),aa_rat_rat(plus_plus_rat(X3),S3))
      <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),aa_rat_rat(plus_plus_rat(X3),S3)) ) ) ).

% pinf(9)
tff(fact_6420_pinf_I9_J,axiom,
    ! [D: nat,S3: nat] :
    ? [Z3: nat] :
    ! [X3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z3),X3)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X3),S3))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X3),S3)) ) ) ).

% pinf(9)
tff(fact_6421_pinf_I9_J,axiom,
    ! [D: int,S3: int] :
    ? [Z3: int] :
    ! [X3: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z3),X3)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),S3))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),S3)) ) ) ).

% pinf(9)
tff(fact_6422_dvd__div__eq__0__iff,axiom,
    ! [B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),A)
     => ( ( aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),B) = zero_z3403309356797280102nteger )
      <=> ( A = zero_z3403309356797280102nteger ) ) ) ).

% dvd_div_eq_0_iff
tff(fact_6423_dvd__div__eq__0__iff,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,B),A)
     => ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B) = zero_zero_real )
      <=> ( A = zero_zero_real ) ) ) ).

% dvd_div_eq_0_iff
tff(fact_6424_dvd__div__eq__0__iff,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,B),A)
     => ( ( divide_divide_rat(A,B) = zero_zero_rat )
      <=> ( A = zero_zero_rat ) ) ) ).

% dvd_div_eq_0_iff
tff(fact_6425_dvd__div__eq__0__iff,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),A)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B) = zero_zero_nat )
      <=> ( A = zero_zero_nat ) ) ) ).

% dvd_div_eq_0_iff
tff(fact_6426_dvd__div__eq__0__iff,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),A)
     => ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B) = zero_zero_int )
      <=> ( A = zero_zero_int ) ) ) ).

% dvd_div_eq_0_iff
tff(fact_6427_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_6428_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_6429_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_6430_dvd__mult__unit__iff,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),C) ) ) ).

% dvd_mult_unit_iff
tff(fact_6431_dvd__mult__unit__iff,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),C) ) ) ).

% dvd_mult_unit_iff
tff(fact_6432_dvd__mult__unit__iff,axiom,
    ! [B: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),C) ) ) ).

% dvd_mult_unit_iff
tff(fact_6433_mult__unit__dvd__iff,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),C)
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),C) ) ) ).

% mult_unit_dvd_iff
tff(fact_6434_mult__unit__dvd__iff,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C)
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),C) ) ) ).

% mult_unit_dvd_iff
tff(fact_6435_mult__unit__dvd__iff,axiom,
    ! [B: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),C)
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),C) ) ) ).

% mult_unit_dvd_iff
tff(fact_6436_dvd__mult__unit__iff_H,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),C) ) ) ).

% dvd_mult_unit_iff'
tff(fact_6437_dvd__mult__unit__iff_H,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),C) ) ) ).

% dvd_mult_unit_iff'
tff(fact_6438_dvd__mult__unit__iff_H,axiom,
    ! [B: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),C) ) ) ).

% dvd_mult_unit_iff'
tff(fact_6439_mult__unit__dvd__iff_H,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),one_one_Code_integer)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),C)
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),C) ) ) ).

% mult_unit_dvd_iff'
tff(fact_6440_mult__unit__dvd__iff_H,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),one_one_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C)
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),C) ) ) ).

% mult_unit_dvd_iff'
tff(fact_6441_mult__unit__dvd__iff_H,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),C)
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),C) ) ) ).

% mult_unit_dvd_iff'
tff(fact_6442_unit__mult__left__cancel,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),one_one_Code_integer)
     => ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C) )
      <=> ( B = C ) ) ) ).

% unit_mult_left_cancel
tff(fact_6443_unit__mult__left__cancel,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),one_one_nat)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C) )
      <=> ( B = C ) ) ) ).

% unit_mult_left_cancel
tff(fact_6444_unit__mult__left__cancel,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int)
     => ( ( aa_int_int(aa_int_fun_int_int(times_times_int,A),B) = aa_int_int(aa_int_fun_int_int(times_times_int,A),C) )
      <=> ( B = C ) ) ) ).

% unit_mult_left_cancel
tff(fact_6445_unit__mult__right__cancel,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),one_one_Code_integer)
     => ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),A) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A) )
      <=> ( B = C ) ) ) ).

% unit_mult_right_cancel
tff(fact_6446_unit__mult__right__cancel,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),one_one_nat)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),A) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A) )
      <=> ( B = C ) ) ) ).

% unit_mult_right_cancel
tff(fact_6447_unit__mult__right__cancel,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int)
     => ( ( aa_int_int(aa_int_fun_int_int(times_times_int,B),A) = aa_int_int(aa_int_fun_int_int(times_times_int,C),A) )
      <=> ( B = C ) ) ) ).

% unit_mult_right_cancel
tff(fact_6448_div__plus__div__distrib__dvd__right,axiom,
    ! [C: code_integer,B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),B)
     => ( aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(A),B)),C) = aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),C)),aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(B),C)) ) ) ).

% div_plus_div_distrib_dvd_right
tff(fact_6449_div__plus__div__distrib__dvd__right,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C)) ) ) ).

% div_plus_div_distrib_dvd_right
tff(fact_6450_div__plus__div__distrib__dvd__right,axiom,
    ! [C: int,B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),B)
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),C)),aa_int_int(aa_int_fun_int_int(divide_divide_int,B),C)) ) ) ).

% div_plus_div_distrib_dvd_right
tff(fact_6451_div__plus__div__distrib__dvd__left,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),A)
     => ( aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(A),B)),C) = aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),C)),aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(B),C)) ) ) ).

% div_plus_div_distrib_dvd_left
tff(fact_6452_div__plus__div__distrib__dvd__left,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),A)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C)) ) ) ).

% div_plus_div_distrib_dvd_left
tff(fact_6453_div__plus__div__distrib__dvd__left,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),A)
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),C)),aa_int_int(aa_int_fun_int_int(divide_divide_int,B),C)) ) ) ).

% div_plus_div_distrib_dvd_left
tff(fact_6454_unit__div__cancel,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),one_one_Code_integer)
     => ( ( aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(B),A) = aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(C),A) )
      <=> ( B = C ) ) ) ).

% unit_div_cancel
tff(fact_6455_unit__div__cancel,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),one_one_nat)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),A) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,C),A) )
      <=> ( B = C ) ) ) ).

% unit_div_cancel
tff(fact_6456_unit__div__cancel,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int)
     => ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,B),A) = aa_int_int(aa_int_fun_int_int(divide_divide_int,C),A) )
      <=> ( B = C ) ) ) ).

% unit_div_cancel
tff(fact_6457_div__unit__dvd__iff,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),B)),C)
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),C) ) ) ).

% div_unit_dvd_iff
tff(fact_6458_div__unit__dvd__iff,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),C)
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),C) ) ) ).

% div_unit_dvd_iff
tff(fact_6459_div__unit__dvd__iff,axiom,
    ! [B: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),C)
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),C) ) ) ).

% div_unit_dvd_iff
tff(fact_6460_dvd__div__unit__iff,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(C),B))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),C) ) ) ).

% dvd_div_unit_iff
tff(fact_6461_dvd__div__unit__iff,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,C),B))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),C) ) ) ).

% dvd_div_unit_iff
tff(fact_6462_dvd__div__unit__iff,axiom,
    ! [B: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(divide_divide_int,C),B))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),C) ) ) ).

% dvd_div_unit_iff
tff(fact_6463_dvd__div__mult,axiom,
    ! [C: code_integer,B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),B)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(B),C)),A) = aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),A)),C) ) ) ).

% dvd_div_mult
tff(fact_6464_dvd__div__mult,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C)),A) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),A)),C) ) ) ).

% dvd_div_mult
tff(fact_6465_dvd__div__mult,axiom,
    ! [C: int,B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),B)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,B),C)),A) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),A)),C) ) ) ).

% dvd_div_mult
tff(fact_6466_div__mult__swap,axiom,
    ! [C: code_integer,B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),B)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(B),C)) = aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),C) ) ) ).

% div_mult_swap
tff(fact_6467_div__mult__swap,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C) ) ) ).

% div_mult_swap
tff(fact_6468_div__mult__swap,axiom,
    ! [C: int,B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),B)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(divide_divide_int,B),C)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),C) ) ) ).

% div_mult_swap
tff(fact_6469_div__div__eq__right,axiom,
    ! [C: code_integer,B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),A)
       => ( aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(B),C)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),B)),C) ) ) ) ).

% div_div_eq_right
tff(fact_6470_div__div__eq__right,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),A)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),C) ) ) ) ).

% div_div_eq_right
tff(fact_6471_div__div__eq__right,axiom,
    ! [C: int,B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),B)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),A)
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_int_int(aa_int_fun_int_int(divide_divide_int,B),C)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),C) ) ) ) ).

% div_div_eq_right
tff(fact_6472_dvd__div__mult2__eq,axiom,
    ! [B: code_integer,C: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)),A)
     => ( aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)) = aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),B)),C) ) ) ).

% dvd_div_mult2_eq
tff(fact_6473_dvd__div__mult2__eq,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)),A)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),C) ) ) ).

% dvd_div_mult2_eq
tff(fact_6474_dvd__div__mult2__eq,axiom,
    ! [B: int,C: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),C)),A)
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),C) ) ) ).

% dvd_div_mult2_eq
tff(fact_6475_dvd__mult__imp__div,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),B)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(B),C)) ) ).

% dvd_mult_imp_div
tff(fact_6476_dvd__mult__imp__div,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),B)
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C)) ) ).

% dvd_mult_imp_div
tff(fact_6477_dvd__mult__imp__div,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),B)
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(divide_divide_int,B),C)) ) ).

% dvd_mult_imp_div
tff(fact_6478_div__mult__div__if__dvd,axiom,
    ! [B: code_integer,A: code_integer,D: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),A)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),C)
       => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),B)),aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(C),D)) = aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),D)) ) ) ) ).

% div_mult_div_if_dvd
tff(fact_6479_div__mult__div__if__dvd,axiom,
    ! [B: nat,A: nat,D: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),C)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,C),D)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D)) ) ) ) ).

% div_mult_div_if_dvd
tff(fact_6480_div__mult__div__if__dvd,axiom,
    ! [B: int,A: int,D: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),A)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),C)
       => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),aa_int_int(aa_int_fun_int_int(divide_divide_int,C),D)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D)) ) ) ) ).

% div_mult_div_if_dvd
tff(fact_6481_div__power,axiom,
    ! [B: code_integer,A: code_integer,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),A)
     => ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),B)),Nb) = aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_nat_Code_integer(power_8256067586552552935nteger(A),Nb)),aa_nat_Code_integer(power_8256067586552552935nteger(B),Nb)) ) ) ).

% div_power
tff(fact_6482_div__power,axiom,
    ! [B: nat,A: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),A)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Nb)) ) ) ).

% div_power
tff(fact_6483_div__power,axiom,
    ! [B: int,A: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),A)
     => ( aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),Nb) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A),Nb)),aa_nat_int(aa_int_fun_nat_int(power_power_int,B),Nb)) ) ) ).

% div_power
tff(fact_6484_mod__0__imp__dvd,axiom,
    ! [A: word_N3645301735248828278l_num1,B: word_N3645301735248828278l_num1] :
      ( ( modulo1504961113040953224l_num1(A,B) = zero_z3563351764282998399l_num1 )
     => aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(B),A) ) ).

% mod_0_imp_dvd
tff(fact_6485_mod__0__imp__dvd,axiom,
    ! [A: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) = zero_zero_nat )
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),A) ) ).

% mod_0_imp_dvd
tff(fact_6486_mod__0__imp__dvd,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) = zero_zero_int )
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),A) ) ).

% mod_0_imp_dvd
tff(fact_6487_mod__0__imp__dvd,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A),B) = zero_z3403309356797280102nteger )
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),A) ) ).

% mod_0_imp_dvd
tff(fact_6488_mod__0__imp__dvd,axiom,
    ! [A: uint32,B: uint32] :
      ( ( aa_uint32_uint32(aa_uin3778660748068847871uint32(modulo_modulo_uint32,A),B) = zero_zero_uint32 )
     => aa_uint32_o(aa_uin5924812838015166427nt32_o(dvd_dvd_uint32,B),A) ) ).

% mod_0_imp_dvd
tff(fact_6489_dvd__eq__mod__eq__0,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B),A) = zero_zero_nat ) ) ).

% dvd_eq_mod_eq_0
tff(fact_6490_dvd__eq__mod__eq__0,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B)
    <=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B),A) = zero_zero_int ) ) ).

% dvd_eq_mod_eq_0
tff(fact_6491_dvd__eq__mod__eq__0,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B)
    <=> ( aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(B),A) = zero_z3403309356797280102nteger ) ) ).

% dvd_eq_mod_eq_0
tff(fact_6492_mod__eq__0__iff__dvd,axiom,
    ! [A: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) = zero_zero_nat )
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),A) ) ).

% mod_eq_0_iff_dvd
tff(fact_6493_mod__eq__0__iff__dvd,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) = zero_zero_int )
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),A) ) ).

% mod_eq_0_iff_dvd
tff(fact_6494_mod__eq__0__iff__dvd,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A),B) = zero_z3403309356797280102nteger )
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),A) ) ).

% mod_eq_0_iff_dvd
tff(fact_6495_dvd__power__le,axiom,
    ! [Xb: code_integer,Y: code_integer,Nb: nat,M: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Xb),Y)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),M)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(Xb),Nb)),aa_nat_Code_integer(power_8256067586552552935nteger(Y),M)) ) ) ).

% dvd_power_le
tff(fact_6496_dvd__power__le,axiom,
    ! [Xb: uint32,Y: uint32,Nb: nat,M: nat] :
      ( aa_uint32_o(aa_uin5924812838015166427nt32_o(dvd_dvd_uint32,Xb),Y)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),M)
       => aa_uint32_o(aa_uin5924812838015166427nt32_o(dvd_dvd_uint32,aa_nat_uint32(power_power_uint32(Xb),Nb)),aa_nat_uint32(power_power_uint32(Y),M)) ) ) ).

% dvd_power_le
tff(fact_6497_dvd__power__le,axiom,
    ! [Xb: word_N3645301735248828278l_num1,Y: word_N3645301735248828278l_num1,Nb: nat,M: nat] :
      ( aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(Xb),Y)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),M)
       => aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(aa_nat5599461682457136004l_num1(power_2184487114949457152l_num1(Xb),Nb)),aa_nat5599461682457136004l_num1(power_2184487114949457152l_num1(Y),M)) ) ) ).

% dvd_power_le
tff(fact_6498_dvd__power__le,axiom,
    ! [Xb: real,Y: real,Nb: nat,M: nat] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,Xb),Y)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),M)
       => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),M)) ) ) ).

% dvd_power_le
tff(fact_6499_dvd__power__le,axiom,
    ! [Xb: nat,Y: nat,Nb: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Xb),Y)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),M)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Xb),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Y),M)) ) ) ).

% dvd_power_le
tff(fact_6500_dvd__power__le,axiom,
    ! [Xb: int,Y: int,Nb: nat,M: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Xb),Y)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),M)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,Xb),Nb)),aa_nat_int(aa_int_fun_nat_int(power_power_int,Y),M)) ) ) ).

% dvd_power_le
tff(fact_6501_power__le__dvd,axiom,
    ! [A: code_integer,Nb: nat,B: code_integer,M: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(A),Nb)),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
       => 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_6502_power__le__dvd,axiom,
    ! [A: uint32,Nb: nat,B: uint32,M: nat] :
      ( aa_uint32_o(aa_uin5924812838015166427nt32_o(dvd_dvd_uint32,aa_nat_uint32(power_power_uint32(A),Nb)),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
       => aa_uint32_o(aa_uin5924812838015166427nt32_o(dvd_dvd_uint32,aa_nat_uint32(power_power_uint32(A),M)),B) ) ) ).

% power_le_dvd
tff(fact_6503_power__le__dvd,axiom,
    ! [A: word_N3645301735248828278l_num1,Nb: nat,B: word_N3645301735248828278l_num1,M: nat] :
      ( aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(aa_nat5599461682457136004l_num1(power_2184487114949457152l_num1(A),Nb)),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
       => aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(aa_nat5599461682457136004l_num1(power_2184487114949457152l_num1(A),M)),B) ) ) ).

% power_le_dvd
tff(fact_6504_power__le__dvd,axiom,
    ! [A: real,Nb: nat,B: real,M: nat] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
       => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),M)),B) ) ) ).

% power_le_dvd
tff(fact_6505_power__le__dvd,axiom,
    ! [A: nat,Nb: 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),Nb)),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
       => 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_6506_power__le__dvd,axiom,
    ! [A: int,Nb: 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),Nb)),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
       => 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_6507_le__imp__power__dvd,axiom,
    ! [M: nat,Nb: nat,A: code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
     => 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),Nb)) ) ).

% le_imp_power_dvd
tff(fact_6508_le__imp__power__dvd,axiom,
    ! [M: nat,Nb: nat,A: uint32] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
     => aa_uint32_o(aa_uin5924812838015166427nt32_o(dvd_dvd_uint32,aa_nat_uint32(power_power_uint32(A),M)),aa_nat_uint32(power_power_uint32(A),Nb)) ) ).

% le_imp_power_dvd
tff(fact_6509_le__imp__power__dvd,axiom,
    ! [M: nat,Nb: nat,A: word_N3645301735248828278l_num1] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
     => aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(aa_nat5599461682457136004l_num1(power_2184487114949457152l_num1(A),M)),aa_nat5599461682457136004l_num1(power_2184487114949457152l_num1(A),Nb)) ) ).

% le_imp_power_dvd
tff(fact_6510_le__imp__power__dvd,axiom,
    ! [M: nat,Nb: nat,A: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
     => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),M)),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)) ) ).

% le_imp_power_dvd
tff(fact_6511_le__imp__power__dvd,axiom,
    ! [M: nat,Nb: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
     => 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),Nb)) ) ).

% le_imp_power_dvd
tff(fact_6512_le__imp__power__dvd,axiom,
    ! [M: nat,Nb: nat,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
     => 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),Nb)) ) ).

% le_imp_power_dvd
tff(fact_6513_mod__eq__dvd__iff,axiom,
    ! [A: int,C: int,B: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B),C) )
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)) ) ).

% mod_eq_dvd_iff
tff(fact_6514_mod__eq__dvd__iff,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A),C) = aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(B),C) )
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(A),B)) ) ).

% mod_eq_dvd_iff
tff(fact_6515_dvd__minus__mod,axiom,
    ! [B: nat,A: nat] : aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),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))) ).

% dvd_minus_mod
tff(fact_6516_dvd__minus__mod,axiom,
    ! [B: int,A: int] : aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),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))) ).

% dvd_minus_mod
tff(fact_6517_dvd__minus__mod,axiom,
    ! [B: code_integer,A: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(A),aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A),B))) ).

% dvd_minus_mod
tff(fact_6518_nat__dvd__not__less,axiom,
    ! [M: nat,Nb: 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),Nb)
       => ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Nb),M) ) ) ).

% nat_dvd_not_less
tff(fact_6519_dvd__minus__self,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),M))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),M)
        | aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,M),Nb) ) ) ).

% dvd_minus_self
tff(fact_6520_less__eq__dvd__minus,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,M),Nb)
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),M)) ) ) ).

% less_eq_dvd_minus
tff(fact_6521_dvd__diffD1,axiom,
    ! [K: nat,M: nat,Nb: 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),Nb))
     => ( 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,Nb),M)
         => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K),Nb) ) ) ) ).

% dvd_diffD1
tff(fact_6522_dvd__diffD,axiom,
    ! [K: nat,M: nat,Nb: 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),Nb))
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K),Nb)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),M)
         => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K),M) ) ) ) ).

% dvd_diffD
tff(fact_6523_even__unset__bit__iff,axiom,
    ! [M: nat,A: uint32] :
      ( aa_uint32_o(aa_uin5924812838015166427nt32_o(dvd_dvd_uint32,numera9087168376688890119uint32(aa_num_num(bit0,one))),bit_se4315839071623982667uint32(M,A))
    <=> ( aa_uint32_o(aa_uin5924812838015166427nt32_o(dvd_dvd_uint32,numera9087168376688890119uint32(aa_num_num(bit0,one))),A)
        | ( M = zero_zero_nat ) ) ) ).

% even_unset_bit_iff
tff(fact_6524_even__unset__bit__iff,axiom,
    ! [M: nat,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(bit_se8260200283734997820nteger(M),A))
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A)
        | ( M = zero_zero_nat ) ) ) ).

% even_unset_bit_iff
tff(fact_6525_even__unset__bit__iff,axiom,
    ! [M: nat,A: word_N3645301735248828278l_num1] :
      ( aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(numera7442385471795722001l_num1(aa_num_num(bit0,one))),bit_se5331074070815623765l_num1(M,A))
    <=> ( aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(numera7442385471795722001l_num1(aa_num_num(bit0,one))),A)
        | ( M = zero_zero_nat ) ) ) ).

% even_unset_bit_iff
tff(fact_6526_even__unset__bit__iff,axiom,
    ! [M: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se4205575877204974255it_nat,M),A))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
        | ( M = zero_zero_nat ) ) ) ).

% even_unset_bit_iff
tff(fact_6527_even__unset__bit__iff,axiom,
    ! [M: nat,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_nat_fun_int_int(bit_se4203085406695923979it_int,M),A))
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
        | ( M = zero_zero_nat ) ) ) ).

% even_unset_bit_iff
tff(fact_6528_even__flip__bit__iff,axiom,
    ! [M: nat,A: uint32] :
      ( aa_uint32_o(aa_uin5924812838015166427nt32_o(dvd_dvd_uint32,numera9087168376688890119uint32(aa_num_num(bit0,one))),bit_se7025624438249859091uint32(M,A))
    <=> ~ ( aa_uint32_o(aa_uin5924812838015166427nt32_o(dvd_dvd_uint32,numera9087168376688890119uint32(aa_num_num(bit0,one))),A)
        <=> ( M = zero_zero_nat ) ) ) ).

% even_flip_bit_iff
tff(fact_6529_even__flip__bit__iff,axiom,
    ! [M: nat,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(bit_se1345352211410354436nteger(M),A))
    <=> ~ ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A)
        <=> ( M = zero_zero_nat ) ) ) ).

% even_flip_bit_iff
tff(fact_6530_even__flip__bit__iff,axiom,
    ! [M: nat,A: word_N3645301735248828278l_num1] :
      ( aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(numera7442385471795722001l_num1(aa_num_num(bit0,one))),bit_se4491814353640558621l_num1(M,A))
    <=> ~ ( aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(numera7442385471795722001l_num1(aa_num_num(bit0,one))),A)
        <=> ( M = zero_zero_nat ) ) ) ).

% even_flip_bit_iff
tff(fact_6531_even__flip__bit__iff,axiom,
    ! [M: nat,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_nat_fun_int_int(bit_se2159334234014336723it_int,M),A))
    <=> ~ ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
        <=> ( M = zero_zero_nat ) ) ) ).

% even_flip_bit_iff
tff(fact_6532_even__flip__bit__iff,axiom,
    ! [M: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2161824704523386999it_nat,M),A))
    <=> ~ ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
        <=> ( M = zero_zero_nat ) ) ) ).

% even_flip_bit_iff
tff(fact_6533_even__of__int__iff,axiom,
    ! [K: int] :
      ( aa_uint32_o(aa_uin5924812838015166427nt32_o(dvd_dvd_uint32,numera9087168376688890119uint32(aa_num_num(bit0,one))),aa_int_uint32(ring_1_of_int_uint32,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_of_int_iff
tff(fact_6534_even__of__int__iff,axiom,
    ! [K: int] :
      ( 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(ring_18347121197199848620nteger,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_of_int_iff
tff(fact_6535_even__of__int__iff,axiom,
    ! [K: int] :
      ( aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(numera7442385471795722001l_num1(aa_num_num(bit0,one))),ring_17408606157368542149l_num1(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_of_int_iff
tff(fact_6536_even__of__int__iff,axiom,
    ! [K: 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(ring_1_of_int_int,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_of_int_iff
tff(fact_6537_curry__def,axiom,
    ! [X3: fun_Pr5956696566359781750on_nat,Xa: vEBT_VEBT,Xb2: vEBT_VEBTi] : aa_VEB3566417204837707778on_nat(aa_VEB1030146501360450209on_nat(produc9121605208400662357on_nat(X3),Xa),Xb2) = aa_Pro1290670013494283871on_nat(X3,produc6084888613844515218_VEBTi(Xa,Xb2)) ).

% curry_def
tff(fact_6538_curry__def,axiom,
    ! [X3: fun_Pr4877903221429077380Heap_o,Xa: vEBT_VEBT,Xb2: vEBT_VEBTi] : aa_VEB1455453220190609250Heap_o(aa_VEB8672304581938048793Heap_o(produc8906325494159015119Heap_o(X3),Xa),Xb2) = aa_Pro259339148018954309Heap_o(X3,produc6084888613844515218_VEBTi(Xa,Xb2)) ).

% curry_def
tff(fact_6539_curry__def,axiom,
    ! [X3: fun_Pr4404027214751104635_VEBTi,Xa: produc3625547720036274456_VEBTi,Xb2: nat] : aa_nat5408703056993926195_VEBTi(aa_Pro284253941917238388_VEBTi(produc2164094337957399884_VEBTi(X3),Xa),Xb2) = aa_Pro721367001817565044_VEBTi(X3,produc1853644041309157249Ti_nat(Xa,Xb2)) ).

% curry_def
tff(fact_6540_curry__def,axiom,
    ! [X3: fun_Pr633315466666964198on_nat,Xa: produc3625547720036274456_VEBTi,Xb2: nat] : aa_nat947733127081326238on_nat(aa_Pro1290670013494283871on_nat(produc1757988346207259447on_nat(X3),Xa),Xb2) = aa_Pro2727408364141080927on_nat(X3,produc1853644041309157249Ti_nat(Xa,Xb2)) ).

% curry_def
tff(fact_6541_curry__def,axiom,
    ! [X3: fun_Pr940977554174145406Heap_o,Xa: produc3625547720036274456_VEBTi,Xb2: nat] : aa_nat2910048432523430448Heap_o(aa_Pro259339148018954309Heap_o(produc8381543706267210711Heap_o(X3),Xa),Xb2) = aa_Pro6173831900156065711Heap_o(X3,produc1853644041309157249Ti_nat(Xa,Xb2)) ).

% curry_def
tff(fact_6542_even__numeral,axiom,
    ! [Nb: num] : aa_uint32_o(aa_uin5924812838015166427nt32_o(dvd_dvd_uint32,numera9087168376688890119uint32(aa_num_num(bit0,one))),numera9087168376688890119uint32(aa_num_num(bit0,Nb))) ).

% even_numeral
tff(fact_6543_even__numeral,axiom,
    ! [Nb: num] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,Nb))) ).

% even_numeral
tff(fact_6544_even__numeral,axiom,
    ! [Nb: num] : aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(numera7442385471795722001l_num1(aa_num_num(bit0,one))),numera7442385471795722001l_num1(aa_num_num(bit0,Nb))) ).

% even_numeral
tff(fact_6545_even__numeral,axiom,
    ! [Nb: num] : 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,aa_num_num(bit0,Nb))) ).

% even_numeral
tff(fact_6546_even__numeral,axiom,
    ! [Nb: num] : aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb))) ).

% even_numeral
tff(fact_6547_unity__coeff__ex,axiom,
    ! [P: fun_wo7859709073469921915num1_o,L: word_N3645301735248828278l_num1] :
      ( ? [X: word_N3645301735248828278l_num1] : aa_wor4452366487969921764num1_o(P,aa_wor7274019993978516266l_num1(times_7065122842183080059l_num1(L),X))
    <=> ? [X: word_N3645301735248828278l_num1] :
          ( aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(L),plus_p361126936061061375l_num1(X,zero_z3563351764282998399l_num1))
          & aa_wor4452366487969921764num1_o(P,X) ) ) ).

% unity_coeff_ex
tff(fact_6548_unity__coeff__ex,axiom,
    ! [P: fun_Code_integer_o,L: code_integer] :
      ( ? [X: code_integer] : aa_Code_integer_o(P,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,L),X))
    <=> ? [X: code_integer] :
          ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,L),aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(X),zero_z3403309356797280102nteger))
          & aa_Code_integer_o(P,X) ) ) ).

% unity_coeff_ex
tff(fact_6549_unity__coeff__ex,axiom,
    ! [P: fun_uint32_o,L: uint32] :
      ( ? [X: uint32] : aa_uint32_o(P,aa_uint32_uint32(times_times_uint32(L),X))
    <=> ? [X: uint32] :
          ( aa_uint32_o(aa_uin5924812838015166427nt32_o(dvd_dvd_uint32,L),aa_uint32_uint32(plus_plus_uint32(X),zero_zero_uint32))
          & aa_uint32_o(P,X) ) ) ).

% unity_coeff_ex
tff(fact_6550_unity__coeff__ex,axiom,
    ! [P: fun_real_o,L: real] :
      ( ? [X: real] : aa_real_o(P,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,L),X))
    <=> ? [X: real] :
          ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,L),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),zero_zero_real))
          & aa_real_o(P,X) ) ) ).

% unity_coeff_ex
tff(fact_6551_unity__coeff__ex,axiom,
    ! [P: fun_rat_o,L: rat] :
      ( ? [X: rat] : aa_rat_o(P,aa_rat_rat(times_times_rat(L),X))
    <=> ? [X: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,L),aa_rat_rat(plus_plus_rat(X),zero_zero_rat))
          & aa_rat_o(P,X) ) ) ).

% unity_coeff_ex
tff(fact_6552_unity__coeff__ex,axiom,
    ! [P: fun_nat_o,L: nat] :
      ( ? [X: nat] : aa_nat_o(P,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,L),X))
    <=> ? [X: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,L),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),zero_zero_nat))
          & aa_nat_o(P,X) ) ) ).

% unity_coeff_ex
tff(fact_6553_unity__coeff__ex,axiom,
    ! [P: fun_int_o,L: int] :
      ( ? [X: int] : aa_int_o(P,aa_int_int(aa_int_fun_int_int(times_times_int,L),X))
    <=> ? [X: int] :
          ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,L),aa_int_int(aa_int_fun_int_int(plus_plus_int,X),zero_zero_int))
          & aa_int_o(P,X) ) ) ).

% unity_coeff_ex
tff(fact_6554_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 )
         => ! [C2: code_integer] : B != aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C2) ) ) ).

% unit_dvdE
tff(fact_6555_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 )
         => ! [C2: nat] : B != aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C2) ) ) ).

% unit_dvdE
tff(fact_6556_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 )
         => ! [C2: int] : B != aa_int_int(aa_int_fun_int_int(times_times_int,A),C2) ) ) ).

% unit_dvdE
tff(fact_6557_unit__div__eq__0__iff,axiom,
    ! [B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer)
     => ( ( aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),B) = zero_z3403309356797280102nteger )
      <=> ( A = zero_z3403309356797280102nteger ) ) ) ).

% unit_div_eq_0_iff
tff(fact_6558_unit__div__eq__0__iff,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B) = zero_zero_nat )
      <=> ( A = zero_zero_nat ) ) ) ).

% unit_div_eq_0_iff
tff(fact_6559_unit__div__eq__0__iff,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
     => ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B) = zero_zero_int )
      <=> ( A = zero_zero_int ) ) ) ).

% unit_div_eq_0_iff
tff(fact_6560_dvd__div__eq__mult,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B)
       => ( ( aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(B),A) = C )
        <=> ( B = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A) ) ) ) ) ).

% dvd_div_eq_mult
tff(fact_6561_dvd__div__eq__mult,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( A != zero_zero_nat )
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),A) = C )
        <=> ( B = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A) ) ) ) ) ).

% dvd_div_eq_mult
tff(fact_6562_dvd__div__eq__mult,axiom,
    ! [A: int,B: int,C: int] :
      ( ( A != zero_zero_int )
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B)
       => ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,B),A) = C )
        <=> ( B = aa_int_int(aa_int_fun_int_int(times_times_int,C),A) ) ) ) ) ).

% dvd_div_eq_mult
tff(fact_6563_div__dvd__iff__mult,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( ( B != zero_z3403309356797280102nteger )
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),A)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),B)),C)
        <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B)) ) ) ) ).

% div_dvd_iff_mult
tff(fact_6564_div__dvd__iff__mult,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( B != zero_zero_nat )
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),A)
       => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),C)
        <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B)) ) ) ) ).

% div_dvd_iff_mult
tff(fact_6565_div__dvd__iff__mult,axiom,
    ! [B: int,A: int,C: int] :
      ( ( B != zero_zero_int )
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),A)
       => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),C)
        <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,C),B)) ) ) ) ).

% div_dvd_iff_mult
tff(fact_6566_dvd__div__iff__mult,axiom,
    ! [C: code_integer,B: code_integer,A: code_integer] :
      ( ( C != zero_z3403309356797280102nteger )
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),B)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(B),C))
        <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),B) ) ) ) ).

% dvd_div_iff_mult
tff(fact_6567_dvd__div__iff__mult,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( ( C != zero_zero_nat )
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),B)
       => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C))
        <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),B) ) ) ) ).

% dvd_div_iff_mult
tff(fact_6568_dvd__div__iff__mult,axiom,
    ! [C: int,B: int,A: int] :
      ( ( C != zero_zero_int )
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),B)
       => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(divide_divide_int,B),C))
        <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),B) ) ) ) ).

% dvd_div_iff_mult
tff(fact_6569_dvd__div__div__eq__mult,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer,D: code_integer] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( ( C != zero_z3403309356797280102nteger )
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B)
         => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),D)
           => ( ( aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(B),A) = aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(D),C) )
            <=> ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),D) ) ) ) ) ) ) ).

% dvd_div_div_eq_mult
tff(fact_6570_dvd__div__div__eq__mult,axiom,
    ! [A: nat,C: nat,B: nat,D: nat] :
      ( ( A != zero_zero_nat )
     => ( ( C != zero_zero_nat )
       => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
         => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),D)
           => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),A) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,D),C) )
            <=> ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),D) ) ) ) ) ) ) ).

% dvd_div_div_eq_mult
tff(fact_6571_dvd__div__div__eq__mult,axiom,
    ! [A: int,C: int,B: int,D: int] :
      ( ( A != zero_zero_int )
     => ( ( C != zero_zero_int )
       => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B)
         => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),D)
           => ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,B),A) = aa_int_int(aa_int_fun_int_int(divide_divide_int,D),C) )
            <=> ( aa_int_int(aa_int_fun_int_int(times_times_int,B),C) = aa_int_int(aa_int_fun_int_int(times_times_int,A),D) ) ) ) ) ) ) ).

% dvd_div_div_eq_mult
tff(fact_6572_inf__period_I4_J,axiom,
    ! [D: uint32,D5: uint32,Ta: uint32] :
      ( aa_uint32_o(aa_uin5924812838015166427nt32_o(dvd_dvd_uint32,D),D5)
     => ! [X3: uint32,K5: uint32] :
          ( ~ aa_uint32_o(aa_uin5924812838015166427nt32_o(dvd_dvd_uint32,D),aa_uint32_uint32(plus_plus_uint32(X3),Ta))
        <=> ~ aa_uint32_o(aa_uin5924812838015166427nt32_o(dvd_dvd_uint32,D),aa_uint32_uint32(plus_plus_uint32(minus_minus_uint32(X3,aa_uint32_uint32(times_times_uint32(K5),D5))),Ta)) ) ) ).

% inf_period(4)
tff(fact_6573_inf__period_I4_J,axiom,
    ! [D: word_N3645301735248828278l_num1,D5: word_N3645301735248828278l_num1,Ta: word_N3645301735248828278l_num1] :
      ( aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(D),D5)
     => ! [X3: word_N3645301735248828278l_num1,K5: word_N3645301735248828278l_num1] :
          ( ~ aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(D),plus_p361126936061061375l_num1(X3,Ta))
        <=> ~ aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(D),plus_p361126936061061375l_num1(minus_4019991460397169231l_num1(X3,aa_wor7274019993978516266l_num1(times_7065122842183080059l_num1(K5),D5)),Ta)) ) ) ).

% inf_period(4)
tff(fact_6574_inf__period_I4_J,axiom,
    ! [D: code_integer,D5: code_integer,Ta: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),D5)
     => ! [X3: code_integer,K5: code_integer] :
          ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(X3),Ta))
        <=> ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(X3),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,K5),D5))),Ta)) ) ) ).

% inf_period(4)
tff(fact_6575_inf__period_I4_J,axiom,
    ! [D: real,D5: real,Ta: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),D5)
     => ! [X3: real,K5: real] :
          ( ~ aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X3),Ta))
        <=> ~ aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X3),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,K5),D5))),Ta)) ) ) ).

% inf_period(4)
tff(fact_6576_inf__period_I4_J,axiom,
    ! [D: rat,D5: rat,Ta: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),D5)
     => ! [X3: rat,K5: rat] :
          ( ~ aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),aa_rat_rat(plus_plus_rat(X3),Ta))
        <=> ~ aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),aa_rat_rat(plus_plus_rat(minus_minus_rat(X3,aa_rat_rat(times_times_rat(K5),D5))),Ta)) ) ) ).

% inf_period(4)
tff(fact_6577_inf__period_I4_J,axiom,
    ! [D: int,D5: int,Ta: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),D5)
     => ! [X3: int,K5: int] :
          ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),Ta))
        <=> ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),aa_int_int(aa_int_fun_int_int(times_times_int,K5),D5))),Ta)) ) ) ).

% inf_period(4)
tff(fact_6578_inf__period_I3_J,axiom,
    ! [D: uint32,D5: uint32,Ta: uint32] :
      ( aa_uint32_o(aa_uin5924812838015166427nt32_o(dvd_dvd_uint32,D),D5)
     => ! [X3: uint32,K5: uint32] :
          ( aa_uint32_o(aa_uin5924812838015166427nt32_o(dvd_dvd_uint32,D),aa_uint32_uint32(plus_plus_uint32(X3),Ta))
        <=> aa_uint32_o(aa_uin5924812838015166427nt32_o(dvd_dvd_uint32,D),aa_uint32_uint32(plus_plus_uint32(minus_minus_uint32(X3,aa_uint32_uint32(times_times_uint32(K5),D5))),Ta)) ) ) ).

% inf_period(3)
tff(fact_6579_inf__period_I3_J,axiom,
    ! [D: word_N3645301735248828278l_num1,D5: word_N3645301735248828278l_num1,Ta: word_N3645301735248828278l_num1] :
      ( aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(D),D5)
     => ! [X3: word_N3645301735248828278l_num1,K5: word_N3645301735248828278l_num1] :
          ( aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(D),plus_p361126936061061375l_num1(X3,Ta))
        <=> aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(D),plus_p361126936061061375l_num1(minus_4019991460397169231l_num1(X3,aa_wor7274019993978516266l_num1(times_7065122842183080059l_num1(K5),D5)),Ta)) ) ) ).

% inf_period(3)
tff(fact_6580_inf__period_I3_J,axiom,
    ! [D: code_integer,D5: code_integer,Ta: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),D5)
     => ! [X3: code_integer,K5: code_integer] :
          ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(X3),Ta))
        <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(X3),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,K5),D5))),Ta)) ) ) ).

% inf_period(3)
tff(fact_6581_inf__period_I3_J,axiom,
    ! [D: real,D5: real,Ta: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),D5)
     => ! [X3: real,K5: real] :
          ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X3),Ta))
        <=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X3),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,K5),D5))),Ta)) ) ) ).

% inf_period(3)
tff(fact_6582_inf__period_I3_J,axiom,
    ! [D: rat,D5: rat,Ta: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),D5)
     => ! [X3: rat,K5: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),aa_rat_rat(plus_plus_rat(X3),Ta))
        <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),aa_rat_rat(plus_plus_rat(minus_minus_rat(X3,aa_rat_rat(times_times_rat(K5),D5))),Ta)) ) ) ).

% inf_period(3)
tff(fact_6583_inf__period_I3_J,axiom,
    ! [D: int,D5: int,Ta: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),D5)
     => ! [X3: int,K5: int] :
          ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),Ta))
        <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),aa_int_int(aa_int_fun_int_int(times_times_int,K5),D5))),Ta)) ) ) ).

% inf_period(3)
tff(fact_6584_unit__eq__div1,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer)
     => ( ( aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),B) = C )
      <=> ( A = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B) ) ) ) ).

% unit_eq_div1
tff(fact_6585_unit__eq__div1,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B) = C )
      <=> ( A = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B) ) ) ) ).

% unit_eq_div1
tff(fact_6586_unit__eq__div1,axiom,
    ! [B: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
     => ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B) = C )
      <=> ( A = aa_int_int(aa_int_fun_int_int(times_times_int,C),B) ) ) ) ).

% unit_eq_div1
tff(fact_6587_unit__eq__div2,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer)
     => ( ( A = aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(C),B) )
      <=> ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B) = C ) ) ) ).

% unit_eq_div2
tff(fact_6588_unit__eq__div2,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
     => ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,C),B) )
      <=> ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B) = C ) ) ) ).

% unit_eq_div2
tff(fact_6589_unit__eq__div2,axiom,
    ! [B: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
     => ( ( A = aa_int_int(aa_int_fun_int_int(divide_divide_int,C),B) )
      <=> ( aa_int_int(aa_int_fun_int_int(times_times_int,A),B) = C ) ) ) ).

% unit_eq_div2
tff(fact_6590_div__mult__unit2,axiom,
    ! [C: code_integer,B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),one_one_Code_integer)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),A)
       => ( aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)) = aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),B)),C) ) ) ) ).

% div_mult_unit2
tff(fact_6591_div__mult__unit2,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),one_one_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),A)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),C) ) ) ) ).

% div_mult_unit2
tff(fact_6592_div__mult__unit2,axiom,
    ! [C: int,B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),one_one_int)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),A)
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),C) ) ) ) ).

% div_mult_unit2
tff(fact_6593_unit__div__commute,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),B)),C) = aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),B) ) ) ).

% unit_div_commute
tff(fact_6594_unit__div__commute,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),B) ) ) ).

% unit_div_commute
tff(fact_6595_unit__div__commute,axiom,
    ! [B: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),B) ) ) ).

% unit_div_commute
tff(fact_6596_unit__div__mult__swap,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),one_one_Code_integer)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(B),C)) = aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),C) ) ) ).

% unit_div_mult_swap
tff(fact_6597_unit__div__mult__swap,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),one_one_nat)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C) ) ) ).

% unit_div_mult_swap
tff(fact_6598_unit__div__mult__swap,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),one_one_int)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(divide_divide_int,B),C)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),C) ) ) ).

% unit_div_mult_swap
tff(fact_6599_is__unit__div__mult2__eq,axiom,
    ! [B: code_integer,C: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),one_one_Code_integer)
       => ( aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)) = aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),B)),C) ) ) ) ).

% is_unit_div_mult2_eq
tff(fact_6600_is__unit__div__mult2__eq,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),one_one_nat)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),C) ) ) ) ).

% is_unit_div_mult2_eq
tff(fact_6601_is__unit__div__mult2__eq,axiom,
    ! [B: int,C: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),one_one_int)
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),C) ) ) ) ).

% is_unit_div_mult2_eq
tff(fact_6602_unit__imp__mod__eq__0,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) = zero_zero_nat ) ) ).

% unit_imp_mod_eq_0
tff(fact_6603_unit__imp__mod__eq__0,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
     => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) = zero_zero_int ) ) ).

% unit_imp_mod_eq_0
tff(fact_6604_unit__imp__mod__eq__0,axiom,
    ! [B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer)
     => ( aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A),B) = zero_z3403309356797280102nteger ) ) ).

% unit_imp_mod_eq_0
tff(fact_6605_is__unit__power__iff,axiom,
    ! [A: code_integer,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(A),Nb)),one_one_Code_integer)
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),one_one_Code_integer)
        | ( Nb = zero_zero_nat ) ) ) ).

% is_unit_power_iff
tff(fact_6606_is__unit__power__iff,axiom,
    ! [A: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)),one_one_nat)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),one_one_nat)
        | ( Nb = zero_zero_nat ) ) ) ).

% is_unit_power_iff
tff(fact_6607_is__unit__power__iff,axiom,
    ! [A: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A),Nb)),one_one_int)
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int)
        | ( Nb = zero_zero_nat ) ) ) ).

% is_unit_power_iff
tff(fact_6608_dvd__imp__le,axiom,
    ! [K: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb) ) ) ).

% dvd_imp_le
tff(fact_6609_dvd__mult__cancel,axiom,
    ! [K: nat,M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Nb))
     => ( 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),Nb) ) ) ).

% dvd_mult_cancel
tff(fact_6610_nat__mult__dvd__cancel1,axiom,
    ! [K: nat,M: nat,Nb: 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),Nb))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,M),Nb) ) ) ).

% nat_mult_dvd_cancel1
tff(fact_6611_mod__greater__zero__iff__not__dvd,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Nb))
    <=> ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Nb),M) ) ).

% mod_greater_zero_iff_not_dvd
tff(fact_6612_mod__eq__dvd__iff__nat,axiom,
    ! [Nb: nat,M: nat,Q2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),M)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Q2) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),Q2) )
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Q2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Nb)) ) ) ).

% mod_eq_dvd_iff_nat
tff(fact_6613_real__of__nat__div,axiom,
    ! [D: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),Nb)
     => ( aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),D)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_nat_real(semiri5074537144036343181t_real,D)) ) ) ).

% real_of_nat_div
tff(fact_6614_even__zero,axiom,
    aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),zero_z3403309356797280102nteger) ).

% even_zero
tff(fact_6615_even__zero,axiom,
    aa_uint32_o(aa_uin5924812838015166427nt32_o(dvd_dvd_uint32,numera9087168376688890119uint32(aa_num_num(bit0,one))),zero_zero_uint32) ).

% even_zero
tff(fact_6616_even__zero,axiom,
    aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(numera7442385471795722001l_num1(aa_num_num(bit0,one))),zero_z3563351764282998399l_num1) ).

% even_zero
tff(fact_6617_even__zero,axiom,
    aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),zero_zero_nat) ).

% even_zero
tff(fact_6618_even__zero,axiom,
    aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),zero_zero_int) ).

% even_zero
tff(fact_6619_odd__even__add,axiom,
    ! [A: uint32,B: uint32] :
      ( ~ aa_uint32_o(aa_uin5924812838015166427nt32_o(dvd_dvd_uint32,numera9087168376688890119uint32(aa_num_num(bit0,one))),A)
     => ( ~ aa_uint32_o(aa_uin5924812838015166427nt32_o(dvd_dvd_uint32,numera9087168376688890119uint32(aa_num_num(bit0,one))),B)
       => aa_uint32_o(aa_uin5924812838015166427nt32_o(dvd_dvd_uint32,numera9087168376688890119uint32(aa_num_num(bit0,one))),aa_uint32_uint32(plus_plus_uint32(A),B)) ) ) ).

% odd_even_add
tff(fact_6620_odd__even__add,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))),A)
     => ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),B)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(A),B)) ) ) ).

% odd_even_add
tff(fact_6621_odd__even__add,axiom,
    ! [A: word_N3645301735248828278l_num1,B: word_N3645301735248828278l_num1] :
      ( ~ aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(numera7442385471795722001l_num1(aa_num_num(bit0,one))),A)
     => ( ~ aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(numera7442385471795722001l_num1(aa_num_num(bit0,one))),B)
       => aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(numera7442385471795722001l_num1(aa_num_num(bit0,one))),plus_p361126936061061375l_num1(A,B)) ) ) ).

% odd_even_add
tff(fact_6622_odd__even__add,axiom,
    ! [A: nat,B: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
     => ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)) ) ) ).

% odd_even_add
tff(fact_6623_odd__even__add,axiom,
    ! [A: int,B: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
     => ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)) ) ) ).

% odd_even_add
tff(fact_6624_odd__one,axiom,
    ~ aa_uint32_o(aa_uin5924812838015166427nt32_o(dvd_dvd_uint32,numera9087168376688890119uint32(aa_num_num(bit0,one))),one_one_uint32) ).

% odd_one
tff(fact_6625_odd__one,axiom,
    ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),one_one_Code_integer) ).

% odd_one
tff(fact_6626_odd__one,axiom,
    ~ aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(numera7442385471795722001l_num1(aa_num_num(bit0,one))),one_on7727431528512463931l_num1) ).

% odd_one
tff(fact_6627_odd__one,axiom,
    ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),one_one_nat) ).

% odd_one
tff(fact_6628_odd__one,axiom,
    ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),one_one_int) ).

% odd_one
tff(fact_6629_evenE,axiom,
    ! [A: uint32] :
      ( aa_uint32_o(aa_uin5924812838015166427nt32_o(dvd_dvd_uint32,numera9087168376688890119uint32(aa_num_num(bit0,one))),A)
     => ~ ! [B2: uint32] : A != aa_uint32_uint32(times_times_uint32(numera9087168376688890119uint32(aa_num_num(bit0,one))),B2) ) ).

% evenE
tff(fact_6630_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)
     => ~ ! [B2: code_integer] : A != aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),B2) ) ).

% evenE
tff(fact_6631_evenE,axiom,
    ! [A: word_N3645301735248828278l_num1] :
      ( aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(numera7442385471795722001l_num1(aa_num_num(bit0,one))),A)
     => ~ ! [B2: word_N3645301735248828278l_num1] : A != aa_wor7274019993978516266l_num1(times_7065122842183080059l_num1(numera7442385471795722001l_num1(aa_num_num(bit0,one))),B2) ) ).

% evenE
tff(fact_6632_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)
     => ~ ! [B2: nat] : A != aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B2) ) ).

% evenE
tff(fact_6633_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)
     => ~ ! [B2: int] : A != aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B2) ) ).

% evenE
tff(fact_6634_bit__eq__rec,axiom,
    ! [A: uint32,B: uint32] :
      ( ( A = B )
    <=> ( ( aa_uint32_o(aa_uin5924812838015166427nt32_o(dvd_dvd_uint32,numera9087168376688890119uint32(aa_num_num(bit0,one))),A)
        <=> aa_uint32_o(aa_uin5924812838015166427nt32_o(dvd_dvd_uint32,numera9087168376688890119uint32(aa_num_num(bit0,one))),B) )
        & ( aa_uint32_uint32(aa_uin3778660748068847871uint32(divide_divide_uint32,A),numera9087168376688890119uint32(aa_num_num(bit0,one))) = aa_uint32_uint32(aa_uin3778660748068847871uint32(divide_divide_uint32,B),numera9087168376688890119uint32(aa_num_num(bit0,one))) ) ) ) ).

% bit_eq_rec
tff(fact_6635_bit__eq__rec,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( 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) )
        & ( aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) = aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(B),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) ) ) ) ).

% bit_eq_rec
tff(fact_6636_bit__eq__rec,axiom,
    ! [A: word_N3645301735248828278l_num1,B: word_N3645301735248828278l_num1] :
      ( ( A = B )
    <=> ( ( aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(numera7442385471795722001l_num1(aa_num_num(bit0,one))),A)
        <=> aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(numera7442385471795722001l_num1(aa_num_num(bit0,one))),B) )
        & ( divide1791077408188789448l_num1(A,numera7442385471795722001l_num1(aa_num_num(bit0,one))) = divide1791077408188789448l_num1(B,numera7442385471795722001l_num1(aa_num_num(bit0,one))) ) ) ) ).

% bit_eq_rec
tff(fact_6637_bit__eq__rec,axiom,
    ! [A: nat,B: 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) )
        & ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ) ) ) ).

% bit_eq_rec
tff(fact_6638_bit__eq__rec,axiom,
    ! [A: int,B: 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) )
        & ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = aa_int_int(aa_int_fun_int_int(divide_divide_int,B),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ) ) ) ).

% bit_eq_rec
tff(fact_6639_is__unitE,axiom,
    ! [A: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),one_one_Code_integer)
     => ~ ( ( A != zero_z3403309356797280102nteger )
         => ! [B2: code_integer] :
              ( ( B2 != zero_z3403309356797280102nteger )
             => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B2),one_one_Code_integer)
               => ( ( aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(one_one_Code_integer),A) = B2 )
                 => ( ( aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(one_one_Code_integer),B2) = A )
                   => ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B2) = one_one_Code_integer )
                     => ( aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(C),A) != aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B2) ) ) ) ) ) ) ) ) ).

% is_unitE
tff(fact_6640_is__unitE,axiom,
    ! [A: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),one_one_nat)
     => ~ ( ( A != zero_zero_nat )
         => ! [B2: nat] :
              ( ( B2 != zero_zero_nat )
             => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B2),one_one_nat)
               => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),A) = B2 )
                 => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),B2) = A )
                   => ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B2) = one_one_nat )
                     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,C),A) != aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B2) ) ) ) ) ) ) ) ) ).

% is_unitE
tff(fact_6641_is__unitE,axiom,
    ! [A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int)
     => ~ ( ( A != zero_zero_int )
         => ! [B2: int] :
              ( ( B2 != zero_zero_int )
             => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B2),one_one_int)
               => ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),A) = B2 )
                 => ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),B2) = A )
                   => ( ( aa_int_int(aa_int_fun_int_int(times_times_int,A),B2) = one_one_int )
                     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,C),A) != aa_int_int(aa_int_fun_int_int(times_times_int,C),B2) ) ) ) ) ) ) ) ) ).

% is_unitE
tff(fact_6642_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_6643_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_6644_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_6645_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_6646_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_6647_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_6648_odd__numeral,axiom,
    ! [Nb: num] : ~ aa_uint32_o(aa_uin5924812838015166427nt32_o(dvd_dvd_uint32,numera9087168376688890119uint32(aa_num_num(bit0,one))),numera9087168376688890119uint32(aa_num_num(bit1,Nb))) ).

% odd_numeral
tff(fact_6649_odd__numeral,axiom,
    ! [Nb: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit1,Nb))) ).

% odd_numeral
tff(fact_6650_odd__numeral,axiom,
    ! [Nb: num] : ~ aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(numera7442385471795722001l_num1(aa_num_num(bit0,one))),numera7442385471795722001l_num1(aa_num_num(bit1,Nb))) ).

% odd_numeral
tff(fact_6651_odd__numeral,axiom,
    ! [Nb: num] : ~ 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,aa_num_num(bit1,Nb))) ).

% odd_numeral
tff(fact_6652_odd__numeral,axiom,
    ! [Nb: num] : ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb))) ).

% odd_numeral
tff(fact_6653_dvd__power__iff,axiom,
    ! [Xb: code_integer,M: nat,Nb: nat] :
      ( ( Xb != zero_z3403309356797280102nteger )
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(Xb),M)),aa_nat_Code_integer(power_8256067586552552935nteger(Xb),Nb))
      <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Xb),one_one_Code_integer)
          | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb) ) ) ) ).

% dvd_power_iff
tff(fact_6654_dvd__power__iff,axiom,
    ! [Xb: nat,M: nat,Nb: nat] :
      ( ( Xb != zero_zero_nat )
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Xb),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Xb),Nb))
      <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Xb),one_one_nat)
          | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb) ) ) ) ).

% dvd_power_iff
tff(fact_6655_dvd__power__iff,axiom,
    ! [Xb: int,M: nat,Nb: nat] :
      ( ( Xb != 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,Xb),M)),aa_nat_int(aa_int_fun_nat_int(power_power_int,Xb),Nb))
      <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Xb),one_one_int)
          | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb) ) ) ) ).

% dvd_power_iff
tff(fact_6656_dvd__power,axiom,
    ! [Nb: nat,Xb: code_integer] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
        | ( Xb = one_one_Code_integer ) )
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Xb),aa_nat_Code_integer(power_8256067586552552935nteger(Xb),Nb)) ) ).

% dvd_power
tff(fact_6657_dvd__power,axiom,
    ! [Nb: nat,Xb: rat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
        | ( Xb = one_one_rat ) )
     => aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,Xb),aa_nat_rat(power_power_rat(Xb),Nb)) ) ).

% dvd_power
tff(fact_6658_dvd__power,axiom,
    ! [Nb: nat,Xb: uint32] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
        | ( Xb = one_one_uint32 ) )
     => aa_uint32_o(aa_uin5924812838015166427nt32_o(dvd_dvd_uint32,Xb),aa_nat_uint32(power_power_uint32(Xb),Nb)) ) ).

% dvd_power
tff(fact_6659_dvd__power,axiom,
    ! [Nb: nat,Xb: word_N3645301735248828278l_num1] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
        | ( Xb = one_on7727431528512463931l_num1 ) )
     => aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(Xb),aa_nat5599461682457136004l_num1(power_2184487114949457152l_num1(Xb),Nb)) ) ).

% dvd_power
tff(fact_6660_dvd__power,axiom,
    ! [Nb: nat,Xb: real] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
        | ( Xb = one_one_real ) )
     => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,Xb),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb)) ) ).

% dvd_power
tff(fact_6661_dvd__power,axiom,
    ! [Nb: nat,Xb: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
        | ( Xb = one_one_nat ) )
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Xb),Nb)) ) ).

% dvd_power
tff(fact_6662_dvd__power,axiom,
    ! [Nb: nat,Xb: int] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
        | ( Xb = one_one_int ) )
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Xb),aa_nat_int(aa_int_fun_nat_int(power_power_int,Xb),Nb)) ) ).

% dvd_power
tff(fact_6663_even__even__mod__4__iff,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one))))) ) ).

% even_even_mod_4_iff
tff(fact_6664_dvd__mult__cancel1,axiom,
    ! [M: nat,Nb: 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),Nb)),M)
      <=> ( Nb = one_one_nat ) ) ) ).

% dvd_mult_cancel1
tff(fact_6665_dvd__mult__cancel2,axiom,
    ! [M: nat,Nb: 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,Nb),M)),M)
      <=> ( Nb = one_one_nat ) ) ) ).

% dvd_mult_cancel2
tff(fact_6666_power__dvd__imp__le,axiom,
    ! [I: nat,M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,I),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,I),Nb))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),I)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb) ) ) ).

% power_dvd_imp_le
tff(fact_6667_dvd__minus__add,axiom,
    ! [Q2: nat,Nb: nat,R2: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Q2),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Q2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,R2),M))
       => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Q2))
        <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,R2),M)),Q2))) ) ) ) ).

% dvd_minus_add
tff(fact_6668_diff__mod__le,axiom,
    ! [A: nat,D: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),D)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),D)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,D),B)) ) ) ).

% diff_mod_le
tff(fact_6669_mod__nat__eqI,axiom,
    ! [R2: nat,Nb: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,R2),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,R2),M)
       => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),R2))
         => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Nb) = R2 ) ) ) ) ).

% mod_nat_eqI
tff(fact_6670_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_6671_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_6672_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_6673_even__iff__mod__2__eq__zero,axiom,
    ! [A: word_N3645301735248828278l_num1] :
      ( aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(numera7442385471795722001l_num1(aa_num_num(bit0,one))),A)
    <=> ( modulo1504961113040953224l_num1(A,numera7442385471795722001l_num1(aa_num_num(bit0,one))) = zero_z3563351764282998399l_num1 ) ) ).

% even_iff_mod_2_eq_zero
tff(fact_6674_even__iff__mod__2__eq__zero,axiom,
    ! [A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_nat ) ) ).

% even_iff_mod_2_eq_zero
tff(fact_6675_even__iff__mod__2__eq__zero,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
    <=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = zero_zero_int ) ) ).

% even_iff_mod_2_eq_zero
tff(fact_6676_even__iff__mod__2__eq__zero,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A)
    <=> ( aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) = zero_z3403309356797280102nteger ) ) ).

% even_iff_mod_2_eq_zero
tff(fact_6677_even__iff__mod__2__eq__zero,axiom,
    ! [A: uint32] :
      ( aa_uint32_o(aa_uin5924812838015166427nt32_o(dvd_dvd_uint32,numera9087168376688890119uint32(aa_num_num(bit0,one))),A)
    <=> ( aa_uint32_uint32(aa_uin3778660748068847871uint32(modulo_modulo_uint32,A),numera9087168376688890119uint32(aa_num_num(bit0,one))) = zero_zero_uint32 ) ) ).

% even_iff_mod_2_eq_zero
tff(fact_6678_odd__iff__mod__2__eq__one,axiom,
    ! [A: word_N3645301735248828278l_num1] :
      ( ~ aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(numera7442385471795722001l_num1(aa_num_num(bit0,one))),A)
    <=> ( modulo1504961113040953224l_num1(A,numera7442385471795722001l_num1(aa_num_num(bit0,one))) = one_on7727431528512463931l_num1 ) ) ).

% odd_iff_mod_2_eq_one
tff(fact_6679_odd__iff__mod__2__eq__one,axiom,
    ! [A: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_nat ) ) ).

% odd_iff_mod_2_eq_one
tff(fact_6680_odd__iff__mod__2__eq__one,axiom,
    ! [A: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
    <=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = one_one_int ) ) ).

% odd_iff_mod_2_eq_one
tff(fact_6681_odd__iff__mod__2__eq__one,axiom,
    ! [A: code_integer] :
      ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A)
    <=> ( aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) = one_one_Code_integer ) ) ).

% odd_iff_mod_2_eq_one
tff(fact_6682_odd__iff__mod__2__eq__one,axiom,
    ! [A: uint32] :
      ( ~ aa_uint32_o(aa_uin5924812838015166427nt32_o(dvd_dvd_uint32,numera9087168376688890119uint32(aa_num_num(bit0,one))),A)
    <=> ( aa_uint32_uint32(aa_uin3778660748068847871uint32(modulo_modulo_uint32,A),numera9087168376688890119uint32(aa_num_num(bit0,one))) = one_one_uint32 ) ) ).

% odd_iff_mod_2_eq_one
tff(fact_6683_power__mono__odd,axiom,
    ! [Nb: nat,A: real,B: real] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Nb)) ) ) ).

% power_mono_odd
tff(fact_6684_power__mono__odd,axiom,
    ! [Nb: nat,A: rat,B: rat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),Nb)),aa_nat_rat(power_power_rat(B),Nb)) ) ) ).

% power_mono_odd
tff(fact_6685_power__mono__odd,axiom,
    ! [Nb: nat,A: int,B: int] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A),Nb)),aa_nat_int(aa_int_fun_nat_int(power_power_int,B),Nb)) ) ) ).

% power_mono_odd
tff(fact_6686_odd__pos,axiom,
    ! [Nb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ).

% odd_pos
tff(fact_6687_even__set__bit__iff,axiom,
    ! [M: nat,A: uint32] :
      ( aa_uint32_o(aa_uin5924812838015166427nt32_o(dvd_dvd_uint32,numera9087168376688890119uint32(aa_num_num(bit0,one))),bit_se6647067497041451410uint32(M,A))
    <=> ( aa_uint32_o(aa_uin5924812838015166427nt32_o(dvd_dvd_uint32,numera9087168376688890119uint32(aa_num_num(bit0,one))),A)
        & ( M != zero_zero_nat ) ) ) ).

% even_set_bit_iff
tff(fact_6688_even__set__bit__iff,axiom,
    ! [M: nat,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(bit_se2793503036327961859nteger(M),A))
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A)
        & ( M != zero_zero_nat ) ) ) ).

% even_set_bit_iff
tff(fact_6689_even__set__bit__iff,axiom,
    ! [M: nat,A: word_N3645301735248828278l_num1] :
      ( aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(numera7442385471795722001l_num1(aa_num_num(bit0,one))),bit_se4894374433684937756l_num1(M,A))
    <=> ( aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(numera7442385471795722001l_num1(aa_num_num(bit0,one))),A)
        & ( M != zero_zero_nat ) ) ) ).

% even_set_bit_iff
tff(fact_6690_even__set__bit__iff,axiom,
    ! [M: nat,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_nat_fun_int_int(bit_se7879613467334960850it_int,M),A))
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
        & ( M != zero_zero_nat ) ) ) ).

% even_set_bit_iff
tff(fact_6691_even__set__bit__iff,axiom,
    ! [M: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se7882103937844011126it_nat,M),A))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
        & ( M != zero_zero_nat ) ) ) ).

% even_set_bit_iff
tff(fact_6692_dvd__power__iff__le,axiom,
    ! [K: nat,M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),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),Nb))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb) ) ) ).

% dvd_power_iff_le
tff(fact_6693_oddE,axiom,
    ! [A: uint32] :
      ( ~ aa_uint32_o(aa_uin5924812838015166427nt32_o(dvd_dvd_uint32,numera9087168376688890119uint32(aa_num_num(bit0,one))),A)
     => ~ ! [B2: uint32] : A != aa_uint32_uint32(plus_plus_uint32(aa_uint32_uint32(times_times_uint32(numera9087168376688890119uint32(aa_num_num(bit0,one))),B2)),one_one_uint32) ) ).

% oddE
tff(fact_6694_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)
     => ~ ! [B2: code_integer] : A != aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),B2)),one_one_Code_integer) ) ).

% oddE
tff(fact_6695_oddE,axiom,
    ! [A: word_N3645301735248828278l_num1] :
      ( ~ aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(numera7442385471795722001l_num1(aa_num_num(bit0,one))),A)
     => ~ ! [B2: word_N3645301735248828278l_num1] : A != plus_p361126936061061375l_num1(aa_wor7274019993978516266l_num1(times_7065122842183080059l_num1(numera7442385471795722001l_num1(aa_num_num(bit0,one))),B2),one_on7727431528512463931l_num1) ) ).

% oddE
tff(fact_6696_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)
     => ~ ! [B2: 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))),B2)),one_one_nat) ) ).

% oddE
tff(fact_6697_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)
     => ~ ! [B2: 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))),B2)),one_one_int) ) ).

% oddE
tff(fact_6698_mod2__eq__if,axiom,
    ! [A: word_N3645301735248828278l_num1] :
      modulo1504961113040953224l_num1(A,numera7442385471795722001l_num1(aa_num_num(bit0,one))) = $ite(aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(numera7442385471795722001l_num1(aa_num_num(bit0,one))),A),zero_z3563351764282998399l_num1,one_on7727431528512463931l_num1) ).

% mod2_eq_if
tff(fact_6699_mod2__eq__if,axiom,
    ! [A: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A),zero_zero_nat,one_one_nat) ).

% mod2_eq_if
tff(fact_6700_mod2__eq__if,axiom,
    ! [A: int] :
      aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = $ite(aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A),zero_zero_int,one_one_int) ).

% mod2_eq_if
tff(fact_6701_mod2__eq__if,axiom,
    ! [A: code_integer] :
      aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A),zero_z3403309356797280102nteger,one_one_Code_integer) ).

% mod2_eq_if
tff(fact_6702_mod2__eq__if,axiom,
    ! [A: uint32] :
      aa_uint32_uint32(aa_uin3778660748068847871uint32(modulo_modulo_uint32,A),numera9087168376688890119uint32(aa_num_num(bit0,one))) = $ite(aa_uint32_o(aa_uin5924812838015166427nt32_o(dvd_dvd_uint32,numera9087168376688890119uint32(aa_num_num(bit0,one))),A),zero_zero_uint32,one_one_uint32) ).

% mod2_eq_if
tff(fact_6703_parity__cases,axiom,
    ! [A: word_N3645301735248828278l_num1] :
      ( ( aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(numera7442385471795722001l_num1(aa_num_num(bit0,one))),A)
       => ( modulo1504961113040953224l_num1(A,numera7442385471795722001l_num1(aa_num_num(bit0,one))) != zero_z3563351764282998399l_num1 ) )
     => ~ ( ~ aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(numera7442385471795722001l_num1(aa_num_num(bit0,one))),A)
         => ( modulo1504961113040953224l_num1(A,numera7442385471795722001l_num1(aa_num_num(bit0,one))) != one_on7727431528512463931l_num1 ) ) ) ).

% parity_cases
tff(fact_6704_parity__cases,axiom,
    ! [A: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) != zero_zero_nat ) )
     => ~ ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
         => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) != one_one_nat ) ) ) ).

% parity_cases
tff(fact_6705_parity__cases,axiom,
    ! [A: int] :
      ( ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
       => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) != zero_zero_int ) )
     => ~ ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
         => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) != one_one_int ) ) ) ).

% parity_cases
tff(fact_6706_parity__cases,axiom,
    ! [A: code_integer] :
      ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A)
       => ( aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) != zero_z3403309356797280102nteger ) )
     => ~ ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A)
         => ( aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) != one_one_Code_integer ) ) ) ).

% parity_cases
tff(fact_6707_parity__cases,axiom,
    ! [A: uint32] :
      ( ( aa_uint32_o(aa_uin5924812838015166427nt32_o(dvd_dvd_uint32,numera9087168376688890119uint32(aa_num_num(bit0,one))),A)
       => ( aa_uint32_uint32(aa_uin3778660748068847871uint32(modulo_modulo_uint32,A),numera9087168376688890119uint32(aa_num_num(bit0,one))) != zero_zero_uint32 ) )
     => ~ ( ~ aa_uint32_o(aa_uin5924812838015166427nt32_o(dvd_dvd_uint32,numera9087168376688890119uint32(aa_num_num(bit0,one))),A)
         => ( aa_uint32_uint32(aa_uin3778660748068847871uint32(modulo_modulo_uint32,A),numera9087168376688890119uint32(aa_num_num(bit0,one))) != one_one_uint32 ) ) ) ).

% parity_cases
tff(fact_6708_zero__le__power__eq,axiom,
    ! [A: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A) ) ) ) ).

% zero_le_power_eq
tff(fact_6709_zero__le__power__eq,axiom,
    ! [A: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A) ) ) ) ).

% zero_le_power_eq
tff(fact_6710_zero__le__power__eq,axiom,
    ! [A: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(aa_int_fun_nat_int(power_power_int,A),Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A) ) ) ) ).

% zero_le_power_eq
tff(fact_6711_zero__le__odd__power,axiom,
    ! [Nb: nat,A: real] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A) ) ) ).

% zero_le_odd_power
tff(fact_6712_zero__le__odd__power,axiom,
    ! [Nb: nat,A: rat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),Nb))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A) ) ) ).

% zero_le_odd_power
tff(fact_6713_zero__le__odd__power,axiom,
    ! [Nb: nat,A: int] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(aa_int_fun_nat_int(power_power_int,A),Nb))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A) ) ) ).

% zero_le_odd_power
tff(fact_6714_zero__le__even__power,axiom,
    ! [Nb: nat,A: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)) ) ).

% zero_le_even_power
tff(fact_6715_zero__le__even__power,axiom,
    ! [Nb: nat,A: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),Nb)) ) ).

% zero_le_even_power
tff(fact_6716_zero__le__even__power,axiom,
    ! [Nb: nat,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(aa_int_fun_nat_int(power_power_int,A),Nb)) ) ).

% zero_le_even_power
tff(fact_6717_divides__aux__def,axiom,
    ! [Qr: produc8923325533196201883nteger] :
      ( aa_Pro4926521453222654329eger_o(unique5706413561485394159nteger,Qr)
    <=> ( aa_Pro5245656335456409788nteger(produc6174133586879617921nteger,Qr) = zero_z3403309356797280102nteger ) ) ).

% divides_aux_def
tff(fact_6718_divides__aux__def,axiom,
    ! [Qr: product_prod_nat_nat] :
      ( aa_Pro6150637604445964947_nat_o(unique6322359934112328802ux_nat,Qr)
    <=> ( aa_Pro4243173770044397973at_nat(product_snd_nat_nat,Qr) = zero_zero_nat ) ) ).

% divides_aux_def
tff(fact_6719_divides__aux__def,axiom,
    ! [Qr: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(unique6319869463603278526ux_int,Qr)
    <=> ( aa_Pro6799258434465214121nt_int(product_snd_int_int,Qr) = zero_zero_int ) ) ).

% divides_aux_def
tff(fact_6720_zero__less__power__eq,axiom,
    ! [A: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb))
    <=> ( ( Nb = zero_zero_nat )
        | ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
          & ( A != zero_zero_real ) )
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A) ) ) ) ).

% zero_less_power_eq
tff(fact_6721_zero__less__power__eq,axiom,
    ! [A: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),Nb))
    <=> ( ( Nb = zero_zero_nat )
        | ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
          & ( A != zero_zero_rat ) )
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A) ) ) ) ).

% zero_less_power_eq
tff(fact_6722_zero__less__power__eq,axiom,
    ! [A: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(aa_int_fun_nat_int(power_power_int,A),Nb))
    <=> ( ( Nb = zero_zero_nat )
        | ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
          & ( A != zero_zero_int ) )
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A) ) ) ) ).

% zero_less_power_eq
tff(fact_6723_even__mask__div__iff_H,axiom,
    ! [M: nat,Nb: 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))),Nb)))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb) ) ).

% even_mask_div_iff'
tff(fact_6724_even__mask__div__iff_H,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),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))),Nb)))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb) ) ).

% even_mask_div_iff'
tff(fact_6725_even__mask__div__iff_H,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(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))),Nb)))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb) ) ).

% even_mask_div_iff'
tff(fact_6726_power__le__zero__eq,axiom,
    ! [A: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)),zero_zero_real)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
        & ( ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real) )
          | ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
            & ( A = zero_zero_real ) ) ) ) ) ).

% power_le_zero_eq
tff(fact_6727_power__le__zero__eq,axiom,
    ! [A: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),Nb)),zero_zero_rat)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
        & ( ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat) )
          | ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
            & ( A = zero_zero_rat ) ) ) ) ) ).

% power_le_zero_eq
tff(fact_6728_power__le__zero__eq,axiom,
    ! [A: int,Nb: 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),Nb)),zero_zero_int)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
        & ( ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
            & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int) )
          | ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
            & ( A = zero_zero_int ) ) ) ) ) ).

% power_le_zero_eq
tff(fact_6729_even__mod__4__div__2,axiom,
    ! [Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))) = aa_nat_nat(suc,zero_zero_nat) )
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ).

% even_mod_4_div_2
tff(fact_6730_even__mask__div__iff,axiom,
    ! [M: nat,Nb: 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))),Nb)))
    <=> ( ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Nb) = zero_z3403309356797280102nteger )
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb) ) ) ).

% even_mask_div_iff
tff(fact_6731_even__mask__div__iff,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_uint32_o(aa_uin5924812838015166427nt32_o(dvd_dvd_uint32,numera9087168376688890119uint32(aa_num_num(bit0,one))),aa_uint32_uint32(aa_uin3778660748068847871uint32(divide_divide_uint32,minus_minus_uint32(aa_nat_uint32(power_power_uint32(numera9087168376688890119uint32(aa_num_num(bit0,one))),M),one_one_uint32)),aa_nat_uint32(power_power_uint32(numera9087168376688890119uint32(aa_num_num(bit0,one))),Nb)))
    <=> ( ( aa_nat_uint32(power_power_uint32(numera9087168376688890119uint32(aa_num_num(bit0,one))),Nb) = zero_zero_uint32 )
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb) ) ) ).

% even_mask_div_iff
tff(fact_6732_even__mask__div__iff,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(numera7442385471795722001l_num1(aa_num_num(bit0,one))),divide1791077408188789448l_num1(minus_4019991460397169231l_num1(aa_nat5599461682457136004l_num1(power_2184487114949457152l_num1(numera7442385471795722001l_num1(aa_num_num(bit0,one))),M),one_on7727431528512463931l_num1),aa_nat5599461682457136004l_num1(power_2184487114949457152l_num1(numera7442385471795722001l_num1(aa_num_num(bit0,one))),Nb)))
    <=> ( ( aa_nat5599461682457136004l_num1(power_2184487114949457152l_num1(numera7442385471795722001l_num1(aa_num_num(bit0,one))),Nb) = zero_z3563351764282998399l_num1 )
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb) ) ) ).

% even_mask_div_iff
tff(fact_6733_even__mask__div__iff,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),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))),Nb)))
    <=> ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb) = zero_zero_nat )
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb) ) ) ).

% even_mask_div_iff
tff(fact_6734_even__mask__div__iff,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(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))),Nb)))
    <=> ( ( aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb) = zero_zero_int )
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb) ) ) ).

% even_mask_div_iff
tff(fact_6735_odd__mod__4__div__2,axiom,
    ! [Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)) )
     => ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ).

% odd_mod_4_div_2
tff(fact_6736_Bernoulli__inequality__even,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),Xb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),Xb)),Nb)) ) ).

% Bernoulli_inequality_even
tff(fact_6737_VEBT__internal_OT__vebt__buildupi_Osimps_I3_J,axiom,
    ! [Nb: nat] :
      aa_nat_nat(vEBT_V441764108873111860ildupi,aa_nat_nat(suc,aa_nat_nat(suc,Nb))) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb),aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(vEBT_V441764108873111860ildupi,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(vEBT_V441764108873111860ildupi,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))))))),aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(vEBT_V441764108873111860ildupi,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(vEBT_V441764108873111860ildupi,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))))))) ).

% VEBT_internal.T_vebt_buildupi.simps(3)
tff(fact_6738_even__mult__exp__div__exp__iff,axiom,
    ! [A: code_integer,M: nat,Nb: 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))),Nb)))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),M)
        | ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Nb) = zero_z3403309356797280102nteger )
        | ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
          & 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,Nb),M)))) ) ) ) ).

% even_mult_exp_div_exp_iff
tff(fact_6739_even__mult__exp__div__exp__iff,axiom,
    ! [A: uint32,M: nat,Nb: nat] :
      ( aa_uint32_o(aa_uin5924812838015166427nt32_o(dvd_dvd_uint32,numera9087168376688890119uint32(aa_num_num(bit0,one))),aa_uint32_uint32(aa_uin3778660748068847871uint32(divide_divide_uint32,aa_uint32_uint32(times_times_uint32(A),aa_nat_uint32(power_power_uint32(numera9087168376688890119uint32(aa_num_num(bit0,one))),M))),aa_nat_uint32(power_power_uint32(numera9087168376688890119uint32(aa_num_num(bit0,one))),Nb)))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),M)
        | ( aa_nat_uint32(power_power_uint32(numera9087168376688890119uint32(aa_num_num(bit0,one))),Nb) = zero_zero_uint32 )
        | ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
          & aa_uint32_o(aa_uin5924812838015166427nt32_o(dvd_dvd_uint32,numera9087168376688890119uint32(aa_num_num(bit0,one))),aa_uint32_uint32(aa_uin3778660748068847871uint32(divide_divide_uint32,A),aa_nat_uint32(power_power_uint32(numera9087168376688890119uint32(aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),M)))) ) ) ) ).

% even_mult_exp_div_exp_iff
tff(fact_6740_even__mult__exp__div__exp__iff,axiom,
    ! [A: word_N3645301735248828278l_num1,M: nat,Nb: nat] :
      ( aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(numera7442385471795722001l_num1(aa_num_num(bit0,one))),divide1791077408188789448l_num1(aa_wor7274019993978516266l_num1(times_7065122842183080059l_num1(A),aa_nat5599461682457136004l_num1(power_2184487114949457152l_num1(numera7442385471795722001l_num1(aa_num_num(bit0,one))),M)),aa_nat5599461682457136004l_num1(power_2184487114949457152l_num1(numera7442385471795722001l_num1(aa_num_num(bit0,one))),Nb)))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),M)
        | ( aa_nat5599461682457136004l_num1(power_2184487114949457152l_num1(numera7442385471795722001l_num1(aa_num_num(bit0,one))),Nb) = zero_z3563351764282998399l_num1 )
        | ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
          & aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(numera7442385471795722001l_num1(aa_num_num(bit0,one))),divide1791077408188789448l_num1(A,aa_nat5599461682457136004l_num1(power_2184487114949457152l_num1(numera7442385471795722001l_num1(aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),M)))) ) ) ) ).

% even_mult_exp_div_exp_iff
tff(fact_6741_even__mult__exp__div__exp__iff,axiom,
    ! [A: nat,M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),M)
        | ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb) = zero_zero_nat )
        | ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
          & aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),M)))) ) ) ) ).

% even_mult_exp_div_exp_iff
tff(fact_6742_even__mult__exp__div__exp__iff,axiom,
    ! [A: int,M: nat,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_nat_int(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))),Nb)))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),M)
        | ( aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb) = zero_zero_int )
        | ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
          & aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_nat_int(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,Nb),M)))) ) ) ) ).

% even_mult_exp_div_exp_iff
tff(fact_6743_VEBT__internal_OT__vebt__buildupi_Oelims,axiom,
    ! [Xb: nat,Y: nat] :
      ( ( aa_nat_nat(vEBT_V441764108873111860ildupi,Xb) = Y )
     => ( ( ( Xb = zero_zero_nat )
         => ( Y != aa_nat_nat(suc,zero_zero_nat) ) )
       => ( ( ( Xb = aa_nat_nat(suc,zero_zero_nat) )
           => ( Y != aa_nat_nat(suc,zero_zero_nat) ) )
         => ~ ! [N2: nat] :
                ( ( Xb = aa_nat_nat(suc,aa_nat_nat(suc,N2)) )
               => ( Y != $ite(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_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(vEBT_V441764108873111860ildupi,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(vEBT_V441764108873111860ildupi,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))))))),aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(vEBT_V441764108873111860ildupi,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(vEBT_V441764108873111860ildupi,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))))))) ) ) ) ) ) ).

% VEBT_internal.T_vebt_buildupi.elims
tff(fact_6744_VEBT__internal_OTb_H_Osimps_I3_J,axiom,
    ! [Nb: nat] :
      aa_nat_nat(vEBT_VEBT_Tb2,aa_nat_nat(suc,aa_nat_nat(suc,Nb))) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit0,one)))),aa_nat_nat(vEBT_VEBT_Tb2,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(vEBT_VEBT_Tb2,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit0,one)))),aa_nat_nat(vEBT_VEBT_Tb2,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(vEBT_VEBT_Tb2,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))))) ).

% VEBT_internal.Tb'.simps(3)
tff(fact_6745_VEBT__internal_OTb_H_Oelims,axiom,
    ! [Xb: nat,Y: nat] :
      ( ( aa_nat_nat(vEBT_VEBT_Tb2,Xb) = Y )
     => ( ( ( Xb = zero_zero_nat )
         => ( Y != aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)) ) )
       => ( ( ( Xb = aa_nat_nat(suc,zero_zero_nat) )
           => ( Y != aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)) ) )
         => ~ ! [N2: nat] :
                ( ( Xb = aa_nat_nat(suc,aa_nat_nat(suc,N2)) )
               => ( Y != $ite(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_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit0,one)))),aa_nat_nat(vEBT_VEBT_Tb2,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(vEBT_VEBT_Tb2,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit0,one)))),aa_nat_nat(vEBT_VEBT_Tb2,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(vEBT_VEBT_Tb2,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))))) ) ) ) ) ) ).

% VEBT_internal.Tb'.elims
tff(fact_6746_VEBT__internal_OTb_Osimps_I3_J,axiom,
    ! [Nb: nat] :
      aa_nat_int(vEBT_VEBT_Tb,aa_nat_nat(suc,aa_nat_nat(suc,Nb))) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,aa_num_num(bit0,one)))),aa_nat_int(vEBT_VEBT_Tb,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(vEBT_VEBT_Tb,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),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,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,aa_num_num(bit0,one)))),aa_nat_int(vEBT_VEBT_Tb,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(vEBT_VEBT_Tb,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),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,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))))) ).

% VEBT_internal.Tb.simps(3)
tff(fact_6747_VEBT__internal_OT__vebt__buildupi_H_Osimps_I3_J,axiom,
    ! [Nb: nat] :
      aa_nat_int(vEBT_V9176841429113362141ildupi,aa_nat_nat(suc,aa_nat_nat(suc,Nb))) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,one))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(vEBT_V9176841429113362141ildupi,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,aa_num_num(bit0,one)))),aa_nat_int(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(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(vEBT_V9176841429113362141ildupi,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),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(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,one))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(vEBT_V9176841429113362141ildupi,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,one))))),aa_nat_int(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(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(vEBT_V9176841429113362141ildupi,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),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(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))))) ).

% VEBT_internal.T_vebt_buildupi'.simps(3)
tff(fact_6748_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_092_060_094sub_062u_092_060_094sub_062p_Osimps_I3_J,axiom,
    ! [Vaa: nat] :
      aa_nat_nat(vEBT_V8346862874174094_d_u_p,aa_nat_nat(suc,aa_nat_nat(suc,Vaa))) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,Vaa))),
        aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
          $let(
            half: nat,
            half:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Vaa))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
            aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,one))))),aa_nat_nat(vEBT_V8346862874174094_d_u_p,half))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),half)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(vEBT_V8346862874174094_d_u_p,half)),one_one_nat))) )),
        $let(
          half: nat,
          half:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Vaa))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
          aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,one))))),aa_nat_nat(vEBT_V8346862874174094_d_u_p,aa_nat_nat(suc,half)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,half))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(vEBT_V8346862874174094_d_u_p,half)),one_one_nat))) ) ) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d\<^sub>u\<^sub>p.simps(3)
tff(fact_6749_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_Osimps_I3_J,axiom,
    ! [Vaa: nat] :
      aa_nat_nat(vEBT_V8646137997579335489_i_l_d,aa_nat_nat(suc,aa_nat_nat(suc,Vaa))) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,Vaa))),
        aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
          $let(
            half: nat,
            half:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Vaa))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
            aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,one))))),aa_nat_nat(vEBT_V8646137997579335489_i_l_d,half))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),half)),aa_nat_nat(vEBT_V8646137997579335489_i_l_d,half))) )),
        $let(
          half: nat,
          half:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Vaa))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
          aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,one))))),aa_nat_nat(vEBT_V8646137997579335489_i_l_d,aa_nat_nat(suc,half)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,half))),aa_nat_nat(vEBT_V8646137997579335489_i_l_d,half))) ) ) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d.simps(3)
tff(fact_6750_VEBT__internal_OTb_Oelims,axiom,
    ! [Xb: nat,Y: int] :
      ( ( aa_nat_int(vEBT_VEBT_Tb,Xb) = Y )
     => ( ( ( Xb = zero_zero_nat )
         => ( Y != aa_num_int(numeral_numeral_int,aa_num_num(bit1,one)) ) )
       => ( ( ( Xb = aa_nat_nat(suc,zero_zero_nat) )
           => ( Y != aa_num_int(numeral_numeral_int,aa_num_num(bit1,one)) ) )
         => ~ ! [N2: nat] :
                ( ( Xb = aa_nat_nat(suc,aa_nat_nat(suc,N2)) )
               => ( Y != $ite(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_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,aa_num_num(bit0,one)))),aa_nat_int(vEBT_VEBT_Tb,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(vEBT_VEBT_Tb,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,aa_num_num(bit0,one)))),aa_nat_int(vEBT_VEBT_Tb,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(vEBT_VEBT_Tb,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))))) ) ) ) ) ) ).

% VEBT_internal.Tb.elims
tff(fact_6751_unset__bit__Suc,axiom,
    ! [Nb: nat,A: word_N3645301735248828278l_num1] : bit_se5331074070815623765l_num1(aa_nat_nat(suc,Nb),A) = plus_p361126936061061375l_num1(modulo1504961113040953224l_num1(A,numera7442385471795722001l_num1(aa_num_num(bit0,one))),aa_wor7274019993978516266l_num1(times_7065122842183080059l_num1(numera7442385471795722001l_num1(aa_num_num(bit0,one))),bit_se5331074070815623765l_num1(Nb,divide1791077408188789448l_num1(A,numera7442385471795722001l_num1(aa_num_num(bit0,one)))))) ).

% unset_bit_Suc
tff(fact_6752_unset__bit__Suc,axiom,
    ! [Nb: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se4205575877204974255it_nat,aa_nat_nat(suc,Nb)),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se4205575877204974255it_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% unset_bit_Suc
tff(fact_6753_unset__bit__Suc,axiom,
    ! [Nb: nat,A: code_integer] : aa_Cod1783974652533396728nteger(bit_se8260200283734997820nteger(aa_nat_nat(suc,Nb)),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(Nb),aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)))))) ).

% unset_bit_Suc
tff(fact_6754_unset__bit__Suc,axiom,
    ! [Nb: nat,A: uint32] : bit_se4315839071623982667uint32(aa_nat_nat(suc,Nb),A) = aa_uint32_uint32(plus_plus_uint32(aa_uint32_uint32(aa_uin3778660748068847871uint32(modulo_modulo_uint32,A),numera9087168376688890119uint32(aa_num_num(bit0,one)))),aa_uint32_uint32(times_times_uint32(numera9087168376688890119uint32(aa_num_num(bit0,one))),bit_se4315839071623982667uint32(Nb,aa_uint32_uint32(aa_uin3778660748068847871uint32(divide_divide_uint32,A),numera9087168376688890119uint32(aa_num_num(bit0,one)))))) ).

% unset_bit_Suc
tff(fact_6755_unset__bit__Suc,axiom,
    ! [Nb: nat,A: int] : aa_int_int(aa_nat_fun_int_int(bit_se4203085406695923979it_int,aa_nat_nat(suc,Nb)),A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_nat_fun_int_int(bit_se4203085406695923979it_int,Nb),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ).

% unset_bit_Suc
tff(fact_6756_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_092_060_094sub_062u_092_060_094sub_062p_Oelims,axiom,
    ! [Xb: nat,Y: nat] :
      ( ( aa_nat_nat(vEBT_V8346862874174094_d_u_p,Xb) = Y )
     => ( ( ( Xb = zero_zero_nat )
         => ( Y != aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)) ) )
       => ( ( ( Xb = aa_nat_nat(suc,zero_zero_nat) )
           => ( Y != aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)) ) )
         => ~ ! [Va2: nat] :
                ( ( Xb = aa_nat_nat(suc,aa_nat_nat(suc,Va2)) )
               => ( Y != $ite(
                      aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,Va2))),
                      aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
                        $let(
                          half: nat,
                          half:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                          aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,one))))),aa_nat_nat(vEBT_V8346862874174094_d_u_p,half))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),half)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(vEBT_V8346862874174094_d_u_p,half)),one_one_nat))) )),
                      $let(
                        half: nat,
                        half:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                        aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,one))))),aa_nat_nat(vEBT_V8346862874174094_d_u_p,aa_nat_nat(suc,half)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,half))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(vEBT_V8346862874174094_d_u_p,half)),one_one_nat))) ) ) ) ) ) ) ) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d\<^sub>u\<^sub>p.elims
tff(fact_6757_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_Oelims,axiom,
    ! [Xb: nat,Y: nat] :
      ( ( aa_nat_nat(vEBT_V8646137997579335489_i_l_d,Xb) = Y )
     => ( ( ( Xb = zero_zero_nat )
         => ( Y != aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one))) ) )
       => ( ( ( Xb = aa_nat_nat(suc,zero_zero_nat) )
           => ( Y != aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one))) ) )
         => ~ ! [Va2: nat] :
                ( ( Xb = aa_nat_nat(suc,aa_nat_nat(suc,Va2)) )
               => ( Y != $ite(
                      aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,Va2))),
                      aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
                        $let(
                          half: nat,
                          half:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                          aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,one))))),aa_nat_nat(vEBT_V8646137997579335489_i_l_d,half))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),half)),aa_nat_nat(vEBT_V8646137997579335489_i_l_d,half))) )),
                      $let(
                        half: nat,
                        half:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                        aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,one))))),aa_nat_nat(vEBT_V8646137997579335489_i_l_d,aa_nat_nat(suc,half)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,half))),aa_nat_nat(vEBT_V8646137997579335489_i_l_d,half))) ) ) ) ) ) ) ) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d.elims
tff(fact_6758_VEBT__internal_Ovebt__predi_H_Omono,axiom,
    ! [Xb: produc3960855945107176009Ti_nat] : comple4513117127444775325on_nat(partia7778075537949639097Ti_nat(heap_T7875578835903804844on_nat),heap_T7875578835903804844on_nat,aTP_Lamm_xl(Xb)) ).

% VEBT_internal.vebt_predi'.mono
tff(fact_6759_pow__divides__pow__iff,axiom,
    ! [Nb: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Nb))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B) ) ) ).

% pow_divides_pow_iff
tff(fact_6760_pow__divides__pow__iff,axiom,
    ! [Nb: nat,A: int,B: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A),Nb)),aa_nat_int(aa_int_fun_nat_int(power_power_int,B),Nb))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B) ) ) ).

% pow_divides_pow_iff
tff(fact_6761_VEBT__internal_Ovebt__buildupi_H_Oelims,axiom,
    ! [Xb: nat,Y: heap_T8145700208782473153_VEBTi] :
      ( ( aa_nat5408703056993926195_VEBTi(vEBT_V739175172307565963ildupi,Xb) = Y )
     => ( ( ( Xb = zero_zero_nat )
         => ( Y != aa_VEB836174949890266638_VEBTi(heap_T3630416162098727440_VEBTi,vEBT_Leafi($false,$false)) ) )
       => ( ( ( Xb = aa_nat_nat(suc,zero_zero_nat) )
           => ( Y != aa_VEB836174949890266638_VEBTi(heap_T3630416162098727440_VEBTi,vEBT_Leafi($false,$false)) ) )
         => ~ ! [Va2: nat] :
                ( ( Xb = aa_nat_nat(suc,aa_nat_nat(suc,Va2)) )
               => ( Y != $ite(
                      aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,Va2))),
                      $let(
                        half: nat,
                        half:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                        heap_T5877712393672139267_VEBTi(vEBT_V1859673955506687831_VEBTi(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),half),aa_nat5408703056993926195_VEBTi(vEBT_V739175172307565963ildupi,half)),aa_nat4125268785154830727_VEBTi(aTP_Lamm_xp(Va2),half)) ),
                      $let(
                        half: nat,
                        half:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                        heap_T5877712393672139267_VEBTi(vEBT_V1859673955506687831_VEBTi(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,half)),aa_nat5408703056993926195_VEBTi(vEBT_V739175172307565963ildupi,half)),aa_nat4125268785154830727_VEBTi(aTP_Lamm_xs(Va2),half)) ) ) ) ) ) ) ) ).

% VEBT_internal.vebt_buildupi'.elims
tff(fact_6762_VEBT__internal_Ovebt__buildupi_H_Osimps_I3_J,axiom,
    ! [Vaa: nat] :
      aa_nat5408703056993926195_VEBTi(vEBT_V739175172307565963ildupi,aa_nat_nat(suc,aa_nat_nat(suc,Vaa))) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,Vaa))),
        $let(
          half: nat,
          half:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Vaa))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
          heap_T5877712393672139267_VEBTi(vEBT_V1859673955506687831_VEBTi(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),half),aa_nat5408703056993926195_VEBTi(vEBT_V739175172307565963ildupi,half)),aa_nat4125268785154830727_VEBTi(aTP_Lamm_xp(Vaa),half)) ),
        $let(
          half: nat,
          half:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Vaa))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
          heap_T5877712393672139267_VEBTi(vEBT_V1859673955506687831_VEBTi(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,half)),aa_nat5408703056993926195_VEBTi(vEBT_V739175172307565963ildupi,half)),aa_nat4125268785154830727_VEBTi(aTP_Lamm_xs(Vaa),half)) ) ) ).

% VEBT_internal.vebt_buildupi'.simps(3)
tff(fact_6763_vebt__buildupi_Oelims,axiom,
    ! [Xb: nat,Y: heap_T8145700208782473153_VEBTi] :
      ( ( aa_nat5408703056993926195_VEBTi(vEBT_vebt_buildupi,Xb) = Y )
     => ( ( ( Xb = zero_zero_nat )
         => ( Y != aa_VEB836174949890266638_VEBTi(heap_T3630416162098727440_VEBTi,vEBT_Leafi($false,$false)) ) )
       => ( ( ( Xb = aa_nat_nat(suc,zero_zero_nat) )
           => ( Y != aa_VEB836174949890266638_VEBTi(heap_T3630416162098727440_VEBTi,vEBT_Leafi($false,$false)) ) )
         => ~ ! [Va2: nat] :
                ( ( Xb = aa_nat_nat(suc,aa_nat_nat(suc,Va2)) )
               => ( Y != $ite(
                      aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,Va2))),
                      $let(
                        half: nat,
                        half:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                        heap_T5877712393672139267_VEBTi(vEBT_V1859673955506687831_VEBTi(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),half),aa_nat5408703056993926195_VEBTi(vEBT_vebt_buildupi,half)),aa_nat4125268785154830727_VEBTi(aTP_Lamm_xu(Va2),half)) ),
                      $let(
                        half: nat,
                        half:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                        heap_T5877712393672139267_VEBTi(vEBT_V1859673955506687831_VEBTi(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,half)),aa_nat5408703056993926195_VEBTi(vEBT_vebt_buildupi,half)),aa_nat4125268785154830727_VEBTi(aTP_Lamm_xw(Va2),half)) ) ) ) ) ) ) ) ).

% vebt_buildupi.elims
tff(fact_6764_Heap__Time__Monad_Obind__mono,axiom,
    ! [B4: fun_fu5786974620624140868ap_nat,C4: fun_na5845961453008041334_VEBTi] :
      ( comple3617025000840655848ap_nat(partia2925774515620677392Ti_nat(heap_Time_Heap_ord_o),heap_T645984214031783516rd_nat,B4)
     => ( ! [Y4: nat] : comple7548324300139837517_VEBTi(partia2925774515620677392Ti_nat(heap_Time_Heap_ord_o),heap_T7173139186834293313_VEBTi,aa_nat4209467000370889527_VEBTi(C4,Y4))
       => comple7548324300139837517_VEBTi(partia2925774515620677392Ti_nat(heap_Time_Heap_ord_o),heap_T7173139186834293313_VEBTi,aa_fun90388479857626845_VEBTi(aTP_Lamm_xy(B4),C4)) ) ) ).

% Heap_Time_Monad.bind_mono
tff(fact_6765_Heap__Time__Monad_Obind__mono,axiom,
    ! [B4: fun_fu4424605311360796457_VEBTi,C4: fun_VE6491116402277797979_VEBTi] :
      ( comple7548324300139837517_VEBTi(partia2925774515620677392Ti_nat(heap_Time_Heap_ord_o),heap_T7173139186834293313_VEBTi,B4)
     => ( ! [Y4: vEBT_VEBTi] : comple7548324300139837517_VEBTi(partia2925774515620677392Ti_nat(heap_Time_Heap_ord_o),heap_T7173139186834293313_VEBTi,aa_VEB5816640871196325404_VEBTi(C4,Y4))
       => comple7548324300139837517_VEBTi(partia2925774515620677392Ti_nat(heap_Time_Heap_ord_o),heap_T7173139186834293313_VEBTi,aa_fun4276395798348166338_VEBTi(aTP_Lamm_ya(B4),C4)) ) ) ).

% Heap_Time_Monad.bind_mono
tff(fact_6766_Heap__Time__Monad_Obind__mono,axiom,
    ! [B4: fun_fu5786974620624140868ap_nat,C4: fun_na8517577583218006873Heap_o] :
      ( comple3617025000840655848ap_nat(partia2925774515620677392Ti_nat(heap_Time_Heap_ord_o),heap_T645984214031783516rd_nat,B4)
     => ( ! [Y4: nat] : comple6933591415923782038Heap_o(partia2925774515620677392Ti_nat(heap_Time_Heap_ord_o),heap_Time_Heap_ord_o,aa_nat205861980368755650Heap_o(C4,Y4))
       => comple6933591415923782038Heap_o(partia2925774515620677392Ti_nat(heap_Time_Heap_ord_o),heap_Time_Heap_ord_o,aa_fun3258944081180477171Heap_o(aTP_Lamm_yc(B4),C4)) ) ) ).

% Heap_Time_Monad.bind_mono
tff(fact_6767_Heap__Time__Monad_Obind__mono,axiom,
    ! [B4: fun_fu4424605311360796457_VEBTi,C4: fun_VE6943415934277719092Heap_o] :
      ( comple7548324300139837517_VEBTi(partia2925774515620677392Ti_nat(heap_Time_Heap_ord_o),heap_T7173139186834293313_VEBTi,B4)
     => ( ! [Y4: vEBT_VEBTi] : comple6933591415923782038Heap_o(partia2925774515620677392Ti_nat(heap_Time_Heap_ord_o),heap_Time_Heap_ord_o,aa_VEB6642844428796722333Heap_o(C4,Y4))
       => comple6933591415923782038Heap_o(partia2925774515620677392Ti_nat(heap_Time_Heap_ord_o),heap_Time_Heap_ord_o,aa_fun5382224708498380120Heap_o(aTP_Lamm_ye(B4),C4)) ) ) ).

% Heap_Time_Monad.bind_mono
tff(fact_6768_Heap__Time__Monad_Obind__mono,axiom,
    ! [B4: fun_fu335202653675969872Heap_o,C4: fun_o_7171189317104958676_VEBTi] :
      ( comple6933591415923782038Heap_o(partia2925774515620677392Ti_nat(heap_Time_Heap_ord_o),heap_Time_Heap_ord_o,B4)
     => ( ! [Y4: $o] : comple7548324300139837517_VEBTi(partia2925774515620677392Ti_nat(heap_Time_Heap_ord_o),heap_T7173139186834293313_VEBTi,aa_o_f8811364424157995653_VEBTi(C4,(Y4)))
       => comple7548324300139837517_VEBTi(partia2925774515620677392Ti_nat(heap_Time_Heap_ord_o),heap_T7173139186834293313_VEBTi,aa_fun2042045670669332929_VEBTi(aTP_Lamm_yg(B4),C4)) ) ) ).

% Heap_Time_Monad.bind_mono
tff(fact_6769_Heap__Time__Monad_Obind__mono,axiom,
    ! [B4: fun_fu335202653675969872Heap_o,C4: fun_o_2223814739247024763Heap_o] :
      ( comple6933591415923782038Heap_o(partia2925774515620677392Ti_nat(heap_Time_Heap_ord_o),heap_Time_Heap_ord_o,B4)
     => ( ! [Y4: $o] : comple6933591415923782038Heap_o(partia2925774515620677392Ti_nat(heap_Time_Heap_ord_o),heap_Time_Heap_ord_o,aa_o_f6784619210364167924Heap_o(C4,(Y4)))
       => comple6933591415923782038Heap_o(partia2925774515620677392Ti_nat(heap_Time_Heap_ord_o),heap_Time_Heap_ord_o,aa_fun4354595424708498391Heap_o(aTP_Lamm_yi(B4),C4)) ) ) ).

% Heap_Time_Monad.bind_mono
tff(fact_6770_Heap__Time__Monad_Obind__mono,axiom,
    ! [B4: fun_fu2606674006500944492ap_nat,C4: fun_na7267038673063590618_VEBTi] :
      ( comple6879160010309051114ap_nat(partia8535946980722491222Ti_nat(heap_T7875578835903804844on_nat),heap_T645984214031783516rd_nat,B4)
     => ( ! [Y4: nat] : comple2136407771590434127_VEBTi(partia8535946980722491222Ti_nat(heap_T7875578835903804844on_nat),heap_T7173139186834293313_VEBTi,aa_nat5169694059731795907_VEBTi(C4,Y4))
       => comple2136407771590434127_VEBTi(partia8535946980722491222Ti_nat(heap_T7875578835903804844on_nat),heap_T7173139186834293313_VEBTi,aa_fun3907257510474286387_VEBTi(aTP_Lamm_yk(B4),C4)) ) ) ).

% Heap_Time_Monad.bind_mono
tff(fact_6771_Heap__Time__Monad_Obind__mono,axiom,
    ! [B4: fun_fu2606674006500944492ap_nat,C4: fun_na979242554834862325Heap_o] :
      ( comple6879160010309051114ap_nat(partia8535946980722491222Ti_nat(heap_T7875578835903804844on_nat),heap_T645984214031783516rd_nat,B4)
     => ( ! [Y4: nat] : comple3937103672976186900Heap_o(partia8535946980722491222Ti_nat(heap_T7875578835903804844on_nat),heap_Time_Heap_ord_o,aa_nat5421218410817480758Heap_o(C4,Y4))
       => comple3937103672976186900Heap_o(partia8535946980722491222Ti_nat(heap_T7875578835903804844on_nat),heap_Time_Heap_ord_o,aa_fun7768191950043460187Heap_o(aTP_Lamm_ym(B4),C4)) ) ) ).

% Heap_Time_Monad.bind_mono
tff(fact_6772_Heap__Time__Monad_Obind__mono,axiom,
    ! [B4: fun_fu9179568866395467601_VEBTi,C4: fun_VE4144918862296792757_VEBTi] :
      ( comple2136407771590434127_VEBTi(partia8535946980722491222Ti_nat(heap_T7875578835903804844on_nat),heap_T7173139186834293313_VEBTi,B4)
     => ( ! [Y4: vEBT_VEBTi] : comple2136407771590434127_VEBTi(partia8535946980722491222Ti_nat(heap_T7875578835903804844on_nat),heap_T7173139186834293313_VEBTi,aa_VEB3118098574329375646_VEBTi(C4,Y4))
       => comple2136407771590434127_VEBTi(partia8535946980722491222Ti_nat(heap_T7875578835903804844on_nat),heap_T7173139186834293313_VEBTi,aa_fun1557956796225944984_VEBTi(aTP_Lamm_yo(B4),C4)) ) ) ).

% Heap_Time_Monad.bind_mono
tff(fact_6773_Heap__Time__Monad_Obind__mono,axiom,
    ! [B4: fun_fu9179568866395467601_VEBTi,C4: fun_VE6677632329911286106Heap_o] :
      ( comple2136407771590434127_VEBTi(partia8535946980722491222Ti_nat(heap_T7875578835903804844on_nat),heap_T7173139186834293313_VEBTi,B4)
     => ( ! [Y4: vEBT_VEBTi] : comple3937103672976186900Heap_o(partia8535946980722491222Ti_nat(heap_T7875578835903804844on_nat),heap_Time_Heap_ord_o,aa_VEB5612184156023240859Heap_o(C4,Y4))
       => comple3937103672976186900Heap_o(partia8535946980722491222Ti_nat(heap_T7875578835903804844on_nat),heap_Time_Heap_ord_o,aa_fun1993422656822580032Heap_o(aTP_Lamm_yq(B4),C4)) ) ) ).

% Heap_Time_Monad.bind_mono
tff(fact_6774_VEBT__internal_Ovebt__inserti_H_Ofixp__induct,axiom,
    ! [P: fun_fu1421233379579251956EBTi_o] :
      ( comple8556566605269262190_VEBTi(partia7782936097874681665Ti_nat(heap_T3112222404744780921_VEBTi),partia1868168049876374393Ti_nat(heap_T7173139186834293313_VEBTi),aTP_Lamm_yr(P))
     => ( aa_fun2485637723310249309EBTi_o(P,aTP_Lamm_yt)
       => ( ! [F2: fun_VE2018681615261037117_VEBTi] :
              ( aa_fun2485637723310249309EBTi_o(P,F2)
             => aa_fun2485637723310249309EBTi_o(P,aTP_Lamm_yu(F2)) )
         => aa_fun2485637723310249309EBTi_o(P,vEBT_V3964819847710782039nserti) ) ) ) ).

% VEBT_internal.vebt_inserti'.fixp_induct
tff(fact_6775_curryI,axiom,
    ! [F: fun_Pr7015759342434529115eger_o,A: fun_Pr5098669352780829690e_term,B: produc8923325533196201883nteger] :
      ( aa_Pro4148064003102580676eger_o(F,aa_Pro4291442876242880245nteger(produc565448451787809096nteger(A),B))
     => aa_Pro4926521453222654329eger_o(aa_fun4505422719058378424eger_o(produc177509963858859394eger_o(F),A),B) ) ).

% curryI
tff(fact_6776_curryI,axiom,
    ! [F: fun_Pr2620521676128474833_nat_o,A: fun_Pr6460944699146269935_nat_o,B: produc3658429121746597890et_nat] :
      ( aa_Pro5643009357358181178_nat_o(F,aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(A),B))
     => aa_Pro653082453696054616_nat_o(aa_fun2638059832795814268_nat_o(produc4701769847872139056_nat_o(F),A),B) ) ).

% curryI
tff(fact_6777_curryI,axiom,
    ! [F: fun_Pr1227177680593602867_nat_o,A: fun_Pr6460944699146269935_nat_o,B: produc8799533983703829792et_nat] :
      ( aa_Pro2671238023309943196_nat_o(F,aa_Pro1979504083478021276et_nat(produc7291855603632349880et_nat(A),B))
     => aa_Pro5643009357358181178_nat_o(aa_fun912680268133416158_nat_o(produc3333590387107866898_nat_o(F),A),B) ) ).

% curryI
tff(fact_6778_curryI,axiom,
    ! [F: fun_Pr3540825087398332590_int_o,A: fun_Pr7763680713840462443e_term,B: product_prod_int_int] :
      ( aa_Pro2601676399929874711_int_o(F,aa_Pro5748361610964203648nt_int(produc7320969046608631221nt_int(A),B))
     => aa_Pro4592068669816069595_int_o(aa_fun3538162476766722827_int_o(produc443495763101479893_int_o(F),A),B) ) ).

% curryI
tff(fact_6779_curryI,axiom,
    ! [F: fun_Pr1196100989263358594_int_o,A: fun_in5730379987311621695e_term,B: product_prod_int_int] :
      ( aa_Pro5451197003417746155_int_o(F,aa_Pro8800328608558162860nt_int(produc4729913460834754785nt_int(A),B))
     => aa_Pro4592068669816069595_int_o(aa_fun586359091965693151_int_o(produc2168430993391648169_int_o(F),A),B) ) ).

% curryI
tff(fact_6780_Heap__Time__Monad_Obind__bind,axiom,
    ! [F: heap_Time_Heap_o,G: fun_o_2238389867833408620_VEBTi,K: fun_VE4765431424076933413_VEBTi] : heap_T1006145433769338483_VEBTi(heap_T5998771940306268294_VEBTi(F,G),K) = heap_T5998771940306268294_VEBTi(F,aa_fun432179723158315733_VEBTi(aTP_Lamm_yv(G),K)) ).

% Heap_Time_Monad.bind_bind
tff(fact_6781_Heap__Time__Monad_Obind__bind,axiom,
    ! [F: heap_Time_Heap_nat,G: fun_na7848093160509028170_VEBTi,K: fun_VE4765431424076933413_VEBTi] : heap_T1006145433769338483_VEBTi(heap_T844888390831797134_VEBTi(F,G),K) = heap_T844888390831797134_VEBTi(F,aa_fun6311797650338102561_VEBTi(aTP_Lamm_yw(G),K)) ).

% Heap_Time_Monad.bind_bind
tff(fact_6782_Heap__Time__Monad_Obind__bind,axiom,
    ! [F: heap_Time_Heap_nat,G: fun_na7848093160509028170_VEBTi,K: fun_VE7038733415900436564Heap_o] : heap_T3040810144269856602EBTi_o(heap_T844888390831797134_VEBTi(F,G),K) = heap_Time_bind_nat_o(F,aa_fun3725866701989745153Heap_o(aTP_Lamm_yx(G),K)) ).

% Heap_Time_Monad.bind_bind
tff(fact_6783_Heap__Time__Monad_Obind__bind,axiom,
    ! [F: heap_Time_Heap_nat,G: fun_na2343073368905879023Heap_o,K: fun_o_8325599348076409421Heap_o] : heap_Time_bind_o_o(heap_Time_bind_nat_o(F,G),K) = heap_Time_bind_nat_o(F,aa_fun1108420279618462208Heap_o(aTP_Lamm_yy(G),K)) ).

% Heap_Time_Monad.bind_bind
tff(fact_6784_Heap__Time__Monad_Obind__bind,axiom,
    ! [F: heap_Time_Heap_o,G: fun_o_8325599348076409421Heap_o,K: fun_o_2238389867833408620_VEBTi] : heap_T5998771940306268294_VEBTi(heap_Time_bind_o_o(F,G),K) = heap_T5998771940306268294_VEBTi(F,aa_fun5660287117450118806_VEBTi(aTP_Lamm_yz(G),K)) ).

% Heap_Time_Monad.bind_bind
tff(fact_6785_Heap__Time__Monad_Obind__bind,axiom,
    ! [F: heap_Time_Heap_nat,G: fun_na2343073368905879023Heap_o,K: fun_o_2238389867833408620_VEBTi] : heap_T5998771940306268294_VEBTi(heap_Time_bind_nat_o(F,G),K) = heap_T844888390831797134_VEBTi(F,aa_fun5487288040762657696_VEBTi(aTP_Lamm_za(G),K)) ).

% Heap_Time_Monad.bind_bind
tff(fact_6786_Heap__Time__Monad_Obind__bind,axiom,
    ! [F: heap_Time_Heap_o,G: fun_o_4158591058828729991ap_nat,K: fun_na7848093160509028170_VEBTi] : heap_T844888390831797134_VEBTi(heap_Time_bind_o_nat(F,G),K) = heap_T5998771940306268294_VEBTi(F,aa_fun1800422248244663418_VEBTi(aTP_Lamm_zb(G),K)) ).

% Heap_Time_Monad.bind_bind
tff(fact_6787_Heap__Time__Monad_Obind__bind,axiom,
    ! [F: heap_Time_Heap_nat,G: fun_na3533371890617757541ap_nat,K: fun_na7848093160509028170_VEBTi] : heap_T844888390831797134_VEBTi(heap_T7049098217575491753at_nat(F,G),K) = heap_T844888390831797134_VEBTi(F,aa_fun49764924992962236_VEBTi(aTP_Lamm_zc(G),K)) ).

% Heap_Time_Monad.bind_bind
tff(fact_6788_Heap__Time__Monad_Obind__bind,axiom,
    ! [F: heap_Time_Heap_nat,G: fun_na3533371890617757541ap_nat,K: fun_na2343073368905879023Heap_o] : heap_Time_bind_nat_o(heap_T7049098217575491753at_nat(F,G),K) = heap_Time_bind_nat_o(F,aa_fun3424352875297450780Heap_o(aTP_Lamm_zd(G),K)) ).

% Heap_Time_Monad.bind_bind
tff(fact_6789_Heap__Time__Monad_Obind__bind,axiom,
    ! [F: heap_T4980287057938770641_VEBTi,G: fun_li7499208429157053461_VEBTi,K: fun_VE4765431424076933413_VEBTi] : heap_T1006145433769338483_VEBTi(heap_T5877712393672139267_VEBTi(F,G),K) = heap_T5877712393672139267_VEBTi(F,aa_fun2525796888956276588_VEBTi(aTP_Lamm_ze(G),K)) ).

% Heap_Time_Monad.bind_bind
tff(fact_6790_Heap__lub__empty,axiom,
    aa_set8609060500495870620_VEBTi(heap_T3112222404744780921_VEBTi,bot_bo3125955617464001165_VEBTi) = heap_T1489671670754571048_VEBTi(aTP_Lamm_ys) ).

% Heap_lub_empty
tff(fact_6791_Heap__lub__empty,axiom,
    aa_set4319781776900185948on_nat(heap_T7048022066654196708on_nat,bot_bo8932748503833948152on_nat) = heap_T5286843759275942675on_nat(aTP_Lamm_zf) ).

% Heap_lub_empty
tff(fact_6792_Heap__lub__empty,axiom,
    aa_set2179536208438818360Heap_o(heap_Time_Heap_lub_o,bot_bo3236126332025433324Heap_o) = heap_Time_Heap_o2(aTP_Lamm_zg) ).

% Heap_lub_empty
tff(fact_6793_timeFrame_Ocases,axiom,
    ! [Xb: produc2970534958541360184it_nat] :
      ( ! [N2: nat,R4: option_nat,H2: heap_e7401611519738050253t_unit,N5: nat] : Xb != produc1430306010071928938it_nat(N2,some_P2407035485129114418it_nat(produc61566615109097733it_nat(R4,produc584006145561248582it_nat(H2,N5))))
     => ~ ! [N2: nat] : Xb != produc1430306010071928938it_nat(N2,none_P1551326421579882414it_nat) ) ).

% timeFrame.cases
tff(fact_6794_timeFrame_Ocases,axiom,
    ! [Xb: produc78474840892876564it_nat] :
      ( ! [N2: nat,R4: $o,H2: heap_e7401611519738050253t_unit,N5: nat] : Xb != produc2174794033045126604it_nat(N2,some_P468703482102919278it_nat(produc6655106138504972685it_nat((R4),produc584006145561248582it_nat(H2,N5))))
     => ~ ! [N2: nat] : Xb != produc2174794033045126604it_nat(N2,none_P7668321371905463026it_nat) ) ).

% timeFrame.cases
tff(fact_6795_zdvd__zdiffD,axiom,
    ! [K: int,M: int,Nb: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,K),aa_int_int(aa_int_fun_int_int(minus_minus_int,M),Nb))
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,K),Nb)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,K),M) ) ) ).

% zdvd_zdiffD
tff(fact_6796_Heap_Osize_I2_J,axiom,
    ! [Xb: fun_he3045699446853741534it_nat] : size_s8425857057747876397_VEBTi(heap_T1489671670754571048_VEBTi(Xb)) = aa_nat_nat(suc,zero_zero_nat) ).

% Heap.size(2)
tff(fact_6797_Heap_Osize_I2_J,axiom,
    ! [Xb: fun_he1353482274668092617it_nat] : size_s6287829766004316056on_nat(heap_T5286843759275942675on_nat(Xb)) = aa_nat_nat(suc,zero_zero_nat) ).

% Heap.size(2)
tff(fact_6798_Heap_Osize_I2_J,axiom,
    ! [Xb: fun_he1183298388918465219it_nat] : size_s2700093152935483318Heap_o(heap_Time_Heap_o2(Xb)) = aa_nat_nat(suc,zero_zero_nat) ).

% Heap.size(2)
tff(fact_6799_curryE,axiom,
    ! [F: fun_Pr7015759342434529115eger_o,A: fun_Pr5098669352780829690e_term,B: produc8923325533196201883nteger] :
      ( aa_Pro4926521453222654329eger_o(aa_fun4505422719058378424eger_o(produc177509963858859394eger_o(F),A),B)
     => aa_Pro4148064003102580676eger_o(F,aa_Pro4291442876242880245nteger(produc565448451787809096nteger(A),B)) ) ).

% curryE
tff(fact_6800_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(produc6017135503889078682et_nat(A),B)) ) ).

% curryE
tff(fact_6801_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(produc7291855603632349880et_nat(A),B)) ) ).

% curryE
tff(fact_6802_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(produc7320969046608631221nt_int(A),B)) ) ).

% curryE
tff(fact_6803_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(produc4729913460834754785nt_int(A),B)) ) ).

% curryE
tff(fact_6804_curryD,axiom,
    ! [F: fun_Pr7015759342434529115eger_o,A: fun_Pr5098669352780829690e_term,B: produc8923325533196201883nteger] :
      ( aa_Pro4926521453222654329eger_o(aa_fun4505422719058378424eger_o(produc177509963858859394eger_o(F),A),B)
     => aa_Pro4148064003102580676eger_o(F,aa_Pro4291442876242880245nteger(produc565448451787809096nteger(A),B)) ) ).

% curryD
tff(fact_6805_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(produc6017135503889078682et_nat(A),B)) ) ).

% curryD
tff(fact_6806_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(produc7291855603632349880et_nat(A),B)) ) ).

% curryD
tff(fact_6807_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(produc7320969046608631221nt_int(A),B)) ) ).

% curryD
tff(fact_6808_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(produc4729913460834754785nt_int(A),B)) ) ).

% curryD
tff(fact_6809_TBOUND__empty,axiom,
    ! [Ta: nat] : aa_nat_o(time_TBOUND_nat(heap_Time_Heap_nat2(aTP_Lamm_zh)),Ta) ).

% TBOUND_empty
tff(fact_6810_TBOUND__empty,axiom,
    ! [Ta: nat] : aa_nat_o(time_T5737551269749752165_VEBTi(heap_T1489671670754571048_VEBTi(aTP_Lamm_ys)),Ta) ).

% TBOUND_empty
tff(fact_6811_TBOUND__empty,axiom,
    ! [Ta: nat] : aa_nat_o(time_T8353473612707095248on_nat(heap_T5286843759275942675on_nat(aTP_Lamm_zf)),Ta) ).

% TBOUND_empty
tff(fact_6812_TBOUND__empty,axiom,
    ! [Ta: nat] : aa_nat_o(time_TBOUND_o(heap_Time_Heap_o2(aTP_Lamm_zg)),Ta) ).

% TBOUND_empty
tff(fact_6813_refines__empty,axiom,
    ! [M: heap_T8145700208782473153_VEBTi] : refine5565527176597971370_VEBTi(M,heap_T1489671670754571048_VEBTi(aTP_Lamm_ys)) ).

% refines_empty
tff(fact_6814_refines__empty,axiom,
    ! [M: heap_T2636463487746394924on_nat] : refine7594492741263601813on_nat(M,heap_T5286843759275942675on_nat(aTP_Lamm_zf)) ).

% refines_empty
tff(fact_6815_refines__empty,axiom,
    ! [M: heap_Time_Heap_o] : aa_Hea4522166450600650424ap_o_o(refine_Imp_refines_o(M),heap_Time_Heap_o2(aTP_Lamm_zg)) ).

% refines_empty
tff(fact_6816_zdvd__antisym__nonneg,axiom,
    ! [M: int,Nb: 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),Nb)
       => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,M),Nb)
         => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Nb),M)
           => ( M = Nb ) ) ) ) ) ).

% zdvd_antisym_nonneg
tff(fact_6817_zdvd__mono,axiom,
    ! [K: int,M: int,Ta: int] :
      ( ( K != zero_zero_int )
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,M),Ta)
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,K),M)),aa_int_int(aa_int_fun_int_int(times_times_int,K),Ta)) ) ) ).

% zdvd_mono
tff(fact_6818_zdvd__mult__cancel,axiom,
    ! [K: int,M: int,Nb: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,K),M)),aa_int_int(aa_int_fun_int_int(times_times_int,K),Nb))
     => ( ( K != zero_zero_int )
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,M),Nb) ) ) ).

% zdvd_mult_cancel
tff(fact_6819_zdvd__reduce,axiom,
    ! [K: int,Nb: int,M: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,K),aa_int_int(aa_int_fun_int_int(plus_plus_int,Nb),aa_int_int(aa_int_fun_int_int(times_times_int,K),M)))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,K),Nb) ) ).

% zdvd_reduce
tff(fact_6820_zdvd__period,axiom,
    ! [A: int,D: int,Xb: int,Ta: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),D)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb),Ta))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb),aa_int_int(aa_int_fun_int_int(times_times_int,C),D))),Ta)) ) ) ).

% zdvd_period
tff(fact_6821_zdvd__imp__le,axiom,
    ! [Z: int,Nb: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Z),Nb)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Nb)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),Nb) ) ) ).

% zdvd_imp_le
tff(fact_6822_real__of__int__div,axiom,
    ! [D: int,Nb: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),Nb)
     => ( aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(divide_divide_int,Nb),D)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_int_real(ring_1_of_int_real,Nb)),aa_int_real(ring_1_of_int_real,D)) ) ) ).

% real_of_int_div
tff(fact_6823_fold__if__return,axiom,
    ! [B: $o,C: option_nat,D: option_nat] :
      $ite((B),heap_T3487192422709364219on_nat(C),heap_T3487192422709364219on_nat(D)) = heap_T3487192422709364219on_nat($ite((B),C,D)) ).

% fold_if_return
tff(fact_6824_fold__if__return,axiom,
    ! [B: $o,C: nat,D: nat] :
      $ite((B),aa_nat5821310369566504398ap_nat(heap_Time_return_nat,C),aa_nat5821310369566504398ap_nat(heap_Time_return_nat,D)) = aa_nat5821310369566504398ap_nat(heap_Time_return_nat,
        $ite((B),C,D)) ).

% fold_if_return
tff(fact_6825_fold__if__return,axiom,
    ! [B: $o,C: $o,D: $o] :
      $ite((B),aa_o_H7399033255965945470Heap_o(heap_Time_return_o,(C)),aa_o_H7399033255965945470Heap_o(heap_Time_return_o,(D))) = aa_o_H7399033255965945470Heap_o(heap_Time_return_o,
        $ite((B),(C),(D))) ).

% fold_if_return
tff(fact_6826_fold__if__return,axiom,
    ! [B: $o,C: vEBT_VEBTi,D: vEBT_VEBTi] :
      $ite((B),aa_VEB836174949890266638_VEBTi(heap_T3630416162098727440_VEBTi,C),aa_VEB836174949890266638_VEBTi(heap_T3630416162098727440_VEBTi,D)) = aa_VEB836174949890266638_VEBTi(heap_T3630416162098727440_VEBTi,
        $ite((B),C,D)) ).

% fold_if_return
tff(fact_6827_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_6828_int__div__sub__1,axiom,
    ! [M: int,Nb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),M)
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Nb),one_one_int)),M) = $ite(aa_int_o(aa_int_fun_int_o(dvd_dvd_int,M),Nb),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,Nb),M)),one_one_int),aa_int_int(aa_int_fun_int_int(divide_divide_int,Nb),M)) ) ) ).

% int_div_sub_1
tff(fact_6829_aset_I10_J,axiom,
    ! [D: int,D5: int,A2: set_int,Ta: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),D5)
     => ! [X3: int] :
          ( ! [Xa2: int] :
              ( aa_set_int_o(member_int(Xa2),set_or1266510415728281911st_int(one_one_int,D5))
             => ! [Xb3: int] :
                  ( aa_set_int_o(member_int(Xb3),A2)
                 => ( X3 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb3),Xa2) ) ) )
         => ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),Ta))
           => ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),D5)),Ta)) ) ) ) ).

% aset(10)
tff(fact_6830_aset_I9_J,axiom,
    ! [D: int,D5: int,A2: set_int,Ta: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),D5)
     => ! [X3: int] :
          ( ! [Xa2: int] :
              ( aa_set_int_o(member_int(Xa2),set_or1266510415728281911st_int(one_one_int,D5))
             => ! [Xb3: int] :
                  ( aa_set_int_o(member_int(Xb3),A2)
                 => ( X3 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb3),Xa2) ) ) )
         => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),Ta))
           => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),D5)),Ta)) ) ) ) ).

% aset(9)
tff(fact_6831_bset_I10_J,axiom,
    ! [D: int,D5: int,B4: set_int,Ta: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),D5)
     => ! [X3: int] :
          ( ! [Xa2: int] :
              ( aa_set_int_o(member_int(Xa2),set_or1266510415728281911st_int(one_one_int,D5))
             => ! [Xb3: int] :
                  ( aa_set_int_o(member_int(Xb3),B4)
                 => ( X3 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb3),Xa2) ) ) )
         => ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),Ta))
           => ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),D5)),Ta)) ) ) ) ).

% bset(10)
tff(fact_6832_bset_I9_J,axiom,
    ! [D: int,D5: int,B4: set_int,Ta: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),D5)
     => ! [X3: int] :
          ( ! [Xa2: int] :
              ( aa_set_int_o(member_int(Xa2),set_or1266510415728281911st_int(one_one_int,D5))
             => ! [Xb3: int] :
                  ( aa_set_int_o(member_int(Xb3),B4)
                 => ( X3 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb3),Xa2) ) ) )
         => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),Ta))
           => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),D5)),Ta)) ) ) ) ).

% bset(9)
tff(fact_6833_distrib__if__bind,axiom,
    ! [B: $o,C: heap_T4980287057938770641_VEBTi,D: heap_T4980287057938770641_VEBTi,F: fun_li7499208429157053461_VEBTi] :
      heap_T5877712393672139267_VEBTi($ite((B),C,D),F) = $ite((B),heap_T5877712393672139267_VEBTi(C,F),heap_T5877712393672139267_VEBTi(D,F)) ).

% distrib_if_bind
tff(fact_6834_distrib__if__bind,axiom,
    ! [B: $o,C: heap_Time_Heap_o,D: heap_Time_Heap_o,F: fun_o_2238389867833408620_VEBTi] :
      heap_T5998771940306268294_VEBTi($ite((B),C,D),F) = $ite((B),heap_T5998771940306268294_VEBTi(C,F),heap_T5998771940306268294_VEBTi(D,F)) ).

% distrib_if_bind
tff(fact_6835_distrib__if__bind,axiom,
    ! [B: $o,C: heap_Time_Heap_nat,D: heap_Time_Heap_nat,F: fun_na7848093160509028170_VEBTi] :
      heap_T844888390831797134_VEBTi($ite((B),C,D),F) = $ite((B),heap_T844888390831797134_VEBTi(C,F),heap_T844888390831797134_VEBTi(D,F)) ).

% distrib_if_bind
tff(fact_6836_distrib__if__bind,axiom,
    ! [B: $o,C: heap_Time_Heap_nat,D: heap_Time_Heap_nat,F: fun_na5556145776640407093on_nat] :
      heap_T8222160169144143993on_nat($ite((B),C,D),F) = $ite((B),heap_T8222160169144143993on_nat(C,F),heap_T8222160169144143993on_nat(D,F)) ).

% distrib_if_bind
tff(fact_6837_distrib__if__bind,axiom,
    ! [B: $o,C: heap_Time_Heap_nat,D: heap_Time_Heap_nat,F: fun_na2343073368905879023Heap_o] :
      heap_Time_bind_nat_o($ite((B),C,D),F) = $ite((B),heap_Time_bind_nat_o(C,F),heap_Time_bind_nat_o(D,F)) ).

% distrib_if_bind
tff(fact_6838_even__diff__iff,axiom,
    ! [K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(minus_minus_int,K),L))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(plus_plus_int,K),L)) ) ).

% even_diff_iff
tff(fact_6839_time__array__of__list,axiom,
    ! [Xs: list_VEBT_VEBTi,H: heap_e7401611519738050253t_unit] : time_t9122064381910598399_VEBTi(array_615059503499738533_VEBTi(Xs),H) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,size_s7982070591426661849_VEBTi(Xs)),one_one_nat) ).

% time_array_of_list
tff(fact_6840_time__array__of__list,axiom,
    ! [Xs: list_o,H: heap_e7401611519738050253t_unit] : time_time_array_o(array_of_list_o(Xs),H) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,size_size_list_o(Xs)),one_one_nat) ).

% time_array_of_list
tff(fact_6841_time__array__of__list,axiom,
    ! [Xs: list_nat,H: heap_e7401611519738050253t_unit] : time_time_array_nat(array_of_list_nat(Xs),H) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,size_size_list_nat(Xs)),one_one_nat) ).

% time_array_of_list
tff(fact_6842_time__array__of__list,axiom,
    ! [Xs: list_int,H: heap_e7401611519738050253t_unit] : time_time_array_int(array_of_list_int(Xs),H) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,size_size_list_int(Xs)),one_one_nat) ).

% time_array_of_list
tff(fact_6843_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_6844_vebt__memberi_Omono,axiom,
    ! [Xb: produc3881548065746020326Ti_nat] : comple6933591415923782038Heap_o(partia2925774515620677392Ti_nat(heap_Time_Heap_ord_o),heap_Time_Heap_ord_o,aTP_Lamm_zp(Xb)) ).

% vebt_memberi.mono
tff(fact_6845_TBOUND__of__list,axiom,
    ! [Xs: list_VEBT_VEBTi] : aa_nat_o(time_T6070283812100419266_VEBTi(array_615059503499738533_VEBTi(Xs)),aa_nat_nat(suc,size_s7982070591426661849_VEBTi(Xs))) ).

% TBOUND_of_list
tff(fact_6846_TBOUND__of__list,axiom,
    ! [Xs: list_o] : aa_nat_o(time_TBOUND_array_o(array_of_list_o(Xs)),aa_nat_nat(suc,size_size_list_o(Xs))) ).

% TBOUND_of_list
tff(fact_6847_TBOUND__of__list,axiom,
    ! [Xs: list_nat] : aa_nat_o(time_T3621710982123377501ay_nat(array_of_list_nat(Xs)),aa_nat_nat(suc,size_size_list_nat(Xs))) ).

% TBOUND_of_list
tff(fact_6848_TBOUND__of__list,axiom,
    ! [Xs: list_int] : aa_nat_o(time_T8667231999468956601ay_int(array_of_list_int(Xs)),aa_nat_nat(suc,size_size_list_int(Xs))) ).

% TBOUND_of_list
tff(fact_6849_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))
     => ~ ! [X2: nat,Y4: nat] :
            ( ( P2 = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,X2),Y4) )
           => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,X2),A)
             => ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Y4),B) ) ) ) ).

% dvd_productE
tff(fact_6850_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))
     => ~ ! [X2: int,Y4: int] :
            ( ( P2 = aa_int_int(aa_int_fun_int_int(times_times_int,X2),Y4) )
           => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,X2),A)
             => ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Y4),B) ) ) ) ).

% dvd_productE
tff(fact_6851_division__decomp,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C))
     => ? [B7: nat,C6: nat] :
          ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B7),C6) )
          & aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B7),B)
          & aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C6),C) ) ) ).

% division_decomp
tff(fact_6852_division__decomp,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C))
     => ? [B7: int,C6: int] :
          ( ( A = aa_int_int(aa_int_fun_int_int(times_times_int,B7),C6) )
          & aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B7),B)
          & aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C6),C) ) ) ).

% division_decomp
tff(fact_6853_Euclid__induct,axiom,
    ! [P: fun_nat_fun_nat_o,A: nat,B: nat] :
      ( ! [A3: nat,B2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(P,A3),B2)
        <=> aa_nat_o(aa_nat_fun_nat_o(P,B2),A3) )
     => ( ! [A3: nat] : aa_nat_o(aa_nat_fun_nat_o(P,A3),zero_zero_nat)
       => ( ! [A3: nat,B2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(P,A3),B2)
             => aa_nat_o(aa_nat_fun_nat_o(P,A3),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A3),B2)) )
         => aa_nat_o(aa_nat_fun_nat_o(P,A),B) ) ) ) ).

% Euclid_induct
tff(fact_6854_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_6855_emep1,axiom,
    ! [Nb: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),D)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),D)
         => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Nb),one_one_int)),D) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Nb),D)),one_one_int) ) ) ) ) ).

% emep1
tff(fact_6856_eme1p,axiom,
    ! [Nb: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),D)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),D)
         => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),Nb)),D) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Nb),D)) ) ) ) ) ).

% eme1p
tff(fact_6857_of__list__rule,axiom,
    ! [Xs: list_VEBT_VEBTi] : hoare_3353465787467722821_VEBTi(one_one_assn,array_615059503499738533_VEBTi(Xs),aTP_Lamm_zq(Xs)) ).

% of_list_rule
tff(fact_6858_heap_Oconst__mono,axiom,
    ! [Ord: fun_fu7962274784768186808_nat_o,C: heap_T2636463487746394924on_nat] : comple7427593846244870586on_nat(Ord,heap_T7875578835903804844on_nat,aTP_Lamm_zr(C)) ).

% heap.const_mono
tff(fact_6859_heap_Oconst__mono,axiom,
    ! [Ord: fun_fu2183070276626753582ap_o_o,C: heap_Time_Heap_o] : comple6933591415923782038Heap_o(Ord,heap_Time_Heap_ord_o,aTP_Lamm_zs(C)) ).

% heap.const_mono
tff(fact_6860_heap_Oconst__mono,axiom,
    ! [Ord: fun_fu3145514261243071828EBTi_o,C: heap_T8145700208782473153_VEBTi] : comple422786390015765981_VEBTi(Ord,heap_T7173139186834293313_VEBTi,aTP_Lamm_zt(C)) ).

% heap.const_mono
tff(fact_6861_heap_Oconst__mono,axiom,
    ! [Ord: fun_fu5284765496190745342_nat_o,C: heap_T2636463487746394924on_nat] : comple4513117127444775325on_nat(Ord,heap_T7875578835903804844on_nat,aTP_Lamm_zu(C)) ).

% heap.const_mono
tff(fact_6862_heap_Oconst__mono,axiom,
    ! [Ord: fun_fu6850233865993928942ap_o_o,C: heap_Time_Heap_o] : comple5241396456474302457Heap_o(Ord,heap_Time_Heap_ord_o,aTP_Lamm_zv(C)) ).

% heap.const_mono
tff(fact_6863_dvd__pos__nat,axiom,
    ! [Nb: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,M),Nb)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M) ) ) ).

% dvd_pos_nat
tff(fact_6864_gcd__nat__induct,axiom,
    ! [P: fun_nat_fun_nat_o,M: nat,Nb: nat] :
      ( ! [M2: nat] : aa_nat_o(aa_nat_fun_nat_o(P,M2),zero_zero_nat)
     => ( ! [M2: nat,N2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)
           => ( aa_nat_o(aa_nat_fun_nat_o(P,N2),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M2),N2))
             => aa_nat_o(aa_nat_fun_nat_o(P,M2),N2) ) )
       => aa_nat_o(aa_nat_fun_nat_o(P,M),Nb) ) ) ).

% gcd_nat_induct
tff(fact_6865_VEBT__internal_Ovebt__memberi_H_Omono,axiom,
    ! [Xb: produc3960855945107176009Ti_nat] : comple5241396456474302457Heap_o(partia3290229181235258227Ti_nat(heap_Time_Heap_ord_o),heap_Time_Heap_ord_o,aTP_Lamm_aal(Xb)) ).

% VEBT_internal.vebt_memberi'.mono
tff(fact_6866_bezout__lemma__nat,axiom,
    ! [D: nat,A: nat,B: nat,Xb: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),B)
       => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),Xb) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),Y)),D) )
            | ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),Xb) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),Y)),D) ) )
         => ? [X2: nat,Y4: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),A)
              & aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B))
              & ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),X2) = 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)),Y4)),D) )
                | ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),X2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),Y4)),D) ) ) ) ) ) ) ).

% bezout_lemma_nat
tff(fact_6867_bezout__add__nat,axiom,
    ! [A: nat,B: nat] :
    ? [D2: nat,X2: nat,Y4: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D2),A)
      & aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D2),B)
      & ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),X2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),Y4)),D2) )
        | ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),X2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),Y4)),D2) ) ) ) ).

% bezout_add_nat
tff(fact_6868_bezout1__nat,axiom,
    ! [A: nat,B: nat] :
    ? [D2: nat,X2: nat,Y4: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D2),A)
      & aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D2),B)
      & ( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),X2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),Y4)) = D2 )
        | ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),X2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),Y4)) = D2 ) ) ) ).

% bezout1_nat
tff(fact_6869_effect__returnI,axiom,
    ! [H: heap_e7401611519738050253t_unit,H3: heap_e7401611519738050253t_unit,Xb: option_nat] :
      ( ( H = H3 )
     => aa_nat_o(heap_T306965388786959644on_nat(heap_T3487192422709364219on_nat(Xb),H,H3,Xb),one_one_nat) ) ).

% effect_returnI
tff(fact_6870_effect__returnI,axiom,
    ! [H: heap_e7401611519738050253t_unit,H3: heap_e7401611519738050253t_unit,Xb: nat] :
      ( ( H = H3 )
     => aa_nat_o(aa_nat_fun_nat_o(heap_Time_effect_nat(aa_nat5821310369566504398ap_nat(heap_Time_return_nat,Xb),H,H3),Xb),one_one_nat) ) ).

% effect_returnI
tff(fact_6871_effect__returnI,axiom,
    ! [H: heap_e7401611519738050253t_unit,H3: heap_e7401611519738050253t_unit,Xb: $o] :
      ( ( H = H3 )
     => aa_nat_o(aa_o_fun_nat_o(heap_Time_effect_o(aa_o_H7399033255965945470Heap_o(heap_Time_return_o,(Xb)),H,H3),(Xb)),one_one_nat) ) ).

% effect_returnI
tff(fact_6872_effect__returnI,axiom,
    ! [H: heap_e7401611519738050253t_unit,H3: heap_e7401611519738050253t_unit,Xb: vEBT_VEBTi] :
      ( ( H = H3 )
     => aa_nat_o(aa_VEB8883426185042993134_nat_o(heap_T2071195472996403633_VEBTi(aa_VEB836174949890266638_VEBTi(heap_T3630416162098727440_VEBTi,Xb),H,H3),Xb),one_one_nat) ) ).

% effect_returnI
tff(fact_6873_effect__returnE,axiom,
    ! [Xb: option_nat,H: heap_e7401611519738050253t_unit,H3: heap_e7401611519738050253t_unit,R2: option_nat,Nb: nat] :
      ( aa_nat_o(heap_T306965388786959644on_nat(heap_T3487192422709364219on_nat(Xb),H,H3,R2),Nb)
     => ~ ( ( R2 = Xb )
         => ( ( H3 = H )
           => ( Nb != one_one_nat ) ) ) ) ).

% effect_returnE
tff(fact_6874_effect__returnE,axiom,
    ! [Xb: nat,H: heap_e7401611519738050253t_unit,H3: heap_e7401611519738050253t_unit,R2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(heap_Time_effect_nat(aa_nat5821310369566504398ap_nat(heap_Time_return_nat,Xb),H,H3),R2),Nb)
     => ~ ( ( R2 = Xb )
         => ( ( H3 = H )
           => ( Nb != one_one_nat ) ) ) ) ).

% effect_returnE
tff(fact_6875_effect__returnE,axiom,
    ! [Xb: $o,H: heap_e7401611519738050253t_unit,H3: heap_e7401611519738050253t_unit,R2: $o,Nb: nat] :
      ( aa_nat_o(aa_o_fun_nat_o(heap_Time_effect_o(aa_o_H7399033255965945470Heap_o(heap_Time_return_o,(Xb)),H,H3),(R2)),Nb)
     => ~ ( ( (R2)
          <=> (Xb) )
         => ( ( H3 = H )
           => ( Nb != one_one_nat ) ) ) ) ).

% effect_returnE
tff(fact_6876_effect__returnE,axiom,
    ! [Xb: vEBT_VEBTi,H: heap_e7401611519738050253t_unit,H3: heap_e7401611519738050253t_unit,R2: vEBT_VEBTi,Nb: nat] :
      ( aa_nat_o(aa_VEB8883426185042993134_nat_o(heap_T2071195472996403633_VEBTi(aa_VEB836174949890266638_VEBTi(heap_T3630416162098727440_VEBTi,Xb),H,H3),R2),Nb)
     => ~ ( ( R2 = Xb )
         => ( ( H3 = H )
           => ( Nb != one_one_nat ) ) ) ) ).

% effect_returnE
tff(fact_6877_effect__bindE,axiom,
    ! [F: heap_Time_Heap_nat,G: fun_na7848093160509028170_VEBTi,H: heap_e7401611519738050253t_unit,H8: heap_e7401611519738050253t_unit,R3: vEBT_VEBTi,Nb: nat] :
      ( aa_nat_o(aa_VEB8883426185042993134_nat_o(heap_T2071195472996403633_VEBTi(heap_T844888390831797134_VEBTi(F,G),H,H8),R3),Nb)
     => ~ ! [H5: heap_e7401611519738050253t_unit,R4: nat,N1: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(heap_Time_effect_nat(F,H,H5),R4),N1)
           => ! [N22: nat] :
                ( aa_nat_o(aa_VEB8883426185042993134_nat_o(heap_T2071195472996403633_VEBTi(aa_nat5408703056993926195_VEBTi(G,R4),H5,H8),R3),N22)
               => ( Nb != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N1),N22) ) ) ) ) ).

% effect_bindE
tff(fact_6878_effect__bindE,axiom,
    ! [F: heap_T8145700208782473153_VEBTi,G: fun_VE4765431424076933413_VEBTi,H: heap_e7401611519738050253t_unit,H8: heap_e7401611519738050253t_unit,R3: vEBT_VEBTi,Nb: nat] :
      ( aa_nat_o(aa_VEB8883426185042993134_nat_o(heap_T2071195472996403633_VEBTi(heap_T1006145433769338483_VEBTi(F,G),H,H8),R3),Nb)
     => ~ ! [H5: heap_e7401611519738050253t_unit,R4: vEBT_VEBTi,N1: nat] :
            ( aa_nat_o(aa_VEB8883426185042993134_nat_o(heap_T2071195472996403633_VEBTi(F,H,H5),R4),N1)
           => ! [N22: nat] :
                ( aa_nat_o(aa_VEB8883426185042993134_nat_o(heap_T2071195472996403633_VEBTi(aa_VEB836174949890266638_VEBTi(G,R4),H5,H8),R3),N22)
               => ( Nb != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N1),N22) ) ) ) ) ).

% effect_bindE
tff(fact_6879_effect__bindE,axiom,
    ! [F: heap_Time_Heap_o,G: fun_o_2238389867833408620_VEBTi,H: heap_e7401611519738050253t_unit,H8: heap_e7401611519738050253t_unit,R3: vEBT_VEBTi,Nb: nat] :
      ( aa_nat_o(aa_VEB8883426185042993134_nat_o(heap_T2071195472996403633_VEBTi(heap_T5998771940306268294_VEBTi(F,G),H,H8),R3),Nb)
     => ~ ! [H5: heap_e7401611519738050253t_unit,R4: $o,N1: nat] :
            ( aa_nat_o(aa_o_fun_nat_o(heap_Time_effect_o(F,H,H5),(R4)),N1)
           => ! [N22: nat] :
                ( aa_nat_o(aa_VEB8883426185042993134_nat_o(heap_T2071195472996403633_VEBTi(aa_o_H7577119287286566245_VEBTi(G,(R4)),H5,H8),R3),N22)
               => ( Nb != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N1),N22) ) ) ) ) ).

% effect_bindE
tff(fact_6880_effect__bindE,axiom,
    ! [F: heap_Time_Heap_nat,G: fun_na2343073368905879023Heap_o,H: heap_e7401611519738050253t_unit,H8: heap_e7401611519738050253t_unit,R3: $o,Nb: nat] :
      ( aa_nat_o(aa_o_fun_nat_o(heap_Time_effect_o(heap_Time_bind_nat_o(F,G),H,H8),(R3)),Nb)
     => ~ ! [H5: heap_e7401611519738050253t_unit,R4: nat,N1: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(heap_Time_effect_nat(F,H,H5),R4),N1)
           => ! [N22: nat] :
                ( aa_nat_o(aa_o_fun_nat_o(heap_Time_effect_o(aa_nat2910048432523430448Heap_o(G,R4),H5,H8),(R3)),N22)
               => ( Nb != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N1),N22) ) ) ) ) ).

% effect_bindE
tff(fact_6881_effect__bindE,axiom,
    ! [F: heap_T8145700208782473153_VEBTi,G: fun_VE7038733415900436564Heap_o,H: heap_e7401611519738050253t_unit,H8: heap_e7401611519738050253t_unit,R3: $o,Nb: nat] :
      ( aa_nat_o(aa_o_fun_nat_o(heap_Time_effect_o(heap_T3040810144269856602EBTi_o(F,G),H,H8),(R3)),Nb)
     => ~ ! [H5: heap_e7401611519738050253t_unit,R4: vEBT_VEBTi,N1: nat] :
            ( aa_nat_o(aa_VEB8883426185042993134_nat_o(heap_T2071195472996403633_VEBTi(F,H,H5),R4),N1)
           => ! [N22: nat] :
                ( aa_nat_o(aa_o_fun_nat_o(heap_Time_effect_o(aa_VEB8194921341634505877Heap_o(G,R4),H5,H8),(R3)),N22)
               => ( Nb != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N1),N22) ) ) ) ) ).

% effect_bindE
tff(fact_6882_effect__bindE,axiom,
    ! [F: heap_Time_Heap_o,G: fun_o_8325599348076409421Heap_o,H: heap_e7401611519738050253t_unit,H8: heap_e7401611519738050253t_unit,R3: $o,Nb: nat] :
      ( aa_nat_o(aa_o_fun_nat_o(heap_Time_effect_o(heap_Time_bind_o_o(F,G),H,H8),(R3)),Nb)
     => ~ ! [H5: heap_e7401611519738050253t_unit,R4: $o,N1: nat] :
            ( aa_nat_o(aa_o_fun_nat_o(heap_Time_effect_o(F,H,H5),(R4)),N1)
           => ! [N22: nat] :
                ( aa_nat_o(aa_o_fun_nat_o(heap_Time_effect_o(aa_o_H7399033255965945470Heap_o(G,(R4)),H5,H8),(R3)),N22)
               => ( Nb != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N1),N22) ) ) ) ) ).

% effect_bindE
tff(fact_6883_effect__bindE,axiom,
    ! [F: heap_T4980287057938770641_VEBTi,G: fun_li7499208429157053461_VEBTi,H: heap_e7401611519738050253t_unit,H8: heap_e7401611519738050253t_unit,R3: vEBT_VEBTi,Nb: nat] :
      ( aa_nat_o(aa_VEB8883426185042993134_nat_o(heap_T2071195472996403633_VEBTi(heap_T5877712393672139267_VEBTi(F,G),H,H8),R3),Nb)
     => ~ ! [H5: heap_e7401611519738050253t_unit,R4: list_VEBT_VEBTi,N1: nat] :
            ( aa_nat_o(heap_T33481931004607297_VEBTi(F,H,H5,R4),N1)
           => ! [N22: nat] :
                ( aa_nat_o(aa_VEB8883426185042993134_nat_o(heap_T2071195472996403633_VEBTi(aa_lis1839255523752068350_VEBTi(G,R4),H5,H8),R3),N22)
               => ( Nb != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N1),N22) ) ) ) ) ).

% effect_bindE
tff(fact_6884_effect__bindE,axiom,
    ! [F: heap_T2636463487746394924on_nat,G: fun_op1509349959255590266_VEBTi,H: heap_e7401611519738050253t_unit,H8: heap_e7401611519738050253t_unit,R3: vEBT_VEBTi,Nb: nat] :
      ( aa_nat_o(aa_VEB8883426185042993134_nat_o(heap_T2071195472996403633_VEBTi(heap_T5661892481228163294_VEBTi(F,G),H,H8),R3),Nb)
     => ~ ! [H5: heap_e7401611519738050253t_unit,R4: option_nat,N1: nat] :
            ( aa_nat_o(heap_T306965388786959644on_nat(F,H,H5,R4),N1)
           => ! [N22: nat] :
                ( aa_nat_o(aa_VEB8883426185042993134_nat_o(heap_T2071195472996403633_VEBTi(aa_opt6483592524530923107_VEBTi(G,R4),H5,H8),R3),N22)
               => ( Nb != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N1),N22) ) ) ) ) ).

% effect_bindE
tff(fact_6885_effect__bindE,axiom,
    ! [F: heap_Time_Heap_nat,G: fun_na5556145776640407093on_nat,H: heap_e7401611519738050253t_unit,H8: heap_e7401611519738050253t_unit,R3: option_nat,Nb: nat] :
      ( aa_nat_o(heap_T306965388786959644on_nat(heap_T8222160169144143993on_nat(F,G),H,H8,R3),Nb)
     => ~ ! [H5: heap_e7401611519738050253t_unit,R4: nat,N1: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(heap_Time_effect_nat(F,H,H5),R4),N1)
           => ! [N22: nat] :
                ( aa_nat_o(heap_T306965388786959644on_nat(aa_nat947733127081326238on_nat(G,R4),H5,H8,R3),N22)
               => ( Nb != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N1),N22) ) ) ) ) ).

% effect_bindE
tff(fact_6886_effect__bindE,axiom,
    ! [F: heap_T8145700208782473153_VEBTi,G: fun_VE2693554162914142864on_nat,H: heap_e7401611519738050253t_unit,H8: heap_e7401611519738050253t_unit,R3: option_nat,Nb: nat] :
      ( aa_nat_o(heap_T306965388786959644on_nat(heap_T2868974464944644318on_nat(F,G),H,H8,R3),Nb)
     => ~ ! [H5: heap_e7401611519738050253t_unit,R4: vEBT_VEBTi,N1: nat] :
            ( aa_nat_o(aa_VEB8883426185042993134_nat_o(heap_T2071195472996403633_VEBTi(F,H,H5),R4),N1)
           => ! [N22: nat] :
                ( aa_nat_o(heap_T306965388786959644on_nat(aa_VEB2711320435582112505on_nat(G,R4),H5,H8,R3),N22)
               => ( Nb != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N1),N22) ) ) ) ) ).

% effect_bindE
tff(fact_6887_effect__bindI,axiom,
    ! [F: heap_Time_Heap_nat,H: heap_e7401611519738050253t_unit,H3: heap_e7401611519738050253t_unit,R2: nat,Nb: nat,G: fun_na7848093160509028170_VEBTi,H8: heap_e7401611519738050253t_unit,R3: vEBT_VEBTi,N3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(heap_Time_effect_nat(F,H,H3),R2),Nb)
     => ( aa_nat_o(aa_VEB8883426185042993134_nat_o(heap_T2071195472996403633_VEBTi(aa_nat5408703056993926195_VEBTi(G,R2),H3,H8),R3),N3)
       => aa_nat_o(aa_VEB8883426185042993134_nat_o(heap_T2071195472996403633_VEBTi(heap_T844888390831797134_VEBTi(F,G),H,H8),R3),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),N3)) ) ) ).

% effect_bindI
tff(fact_6888_effect__bindI,axiom,
    ! [F: heap_Time_Heap_nat,H: heap_e7401611519738050253t_unit,H3: heap_e7401611519738050253t_unit,R2: nat,Nb: nat,G: fun_na2343073368905879023Heap_o,H8: heap_e7401611519738050253t_unit,R3: $o,N3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(heap_Time_effect_nat(F,H,H3),R2),Nb)
     => ( aa_nat_o(aa_o_fun_nat_o(heap_Time_effect_o(aa_nat2910048432523430448Heap_o(G,R2),H3,H8),(R3)),N3)
       => aa_nat_o(aa_o_fun_nat_o(heap_Time_effect_o(heap_Time_bind_nat_o(F,G),H,H8),(R3)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),N3)) ) ) ).

% effect_bindI
tff(fact_6889_effect__bindI,axiom,
    ! [F: heap_T8145700208782473153_VEBTi,H: heap_e7401611519738050253t_unit,H3: heap_e7401611519738050253t_unit,R2: vEBT_VEBTi,Nb: nat,G: fun_VE4765431424076933413_VEBTi,H8: heap_e7401611519738050253t_unit,R3: vEBT_VEBTi,N3: nat] :
      ( aa_nat_o(aa_VEB8883426185042993134_nat_o(heap_T2071195472996403633_VEBTi(F,H,H3),R2),Nb)
     => ( aa_nat_o(aa_VEB8883426185042993134_nat_o(heap_T2071195472996403633_VEBTi(aa_VEB836174949890266638_VEBTi(G,R2),H3,H8),R3),N3)
       => aa_nat_o(aa_VEB8883426185042993134_nat_o(heap_T2071195472996403633_VEBTi(heap_T1006145433769338483_VEBTi(F,G),H,H8),R3),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),N3)) ) ) ).

% effect_bindI
tff(fact_6890_effect__bindI,axiom,
    ! [F: heap_T8145700208782473153_VEBTi,H: heap_e7401611519738050253t_unit,H3: heap_e7401611519738050253t_unit,R2: vEBT_VEBTi,Nb: nat,G: fun_VE7038733415900436564Heap_o,H8: heap_e7401611519738050253t_unit,R3: $o,N3: nat] :
      ( aa_nat_o(aa_VEB8883426185042993134_nat_o(heap_T2071195472996403633_VEBTi(F,H,H3),R2),Nb)
     => ( aa_nat_o(aa_o_fun_nat_o(heap_Time_effect_o(aa_VEB8194921341634505877Heap_o(G,R2),H3,H8),(R3)),N3)
       => aa_nat_o(aa_o_fun_nat_o(heap_Time_effect_o(heap_T3040810144269856602EBTi_o(F,G),H,H8),(R3)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),N3)) ) ) ).

% effect_bindI
tff(fact_6891_effect__bindI,axiom,
    ! [F: heap_Time_Heap_o,H: heap_e7401611519738050253t_unit,H3: heap_e7401611519738050253t_unit,R2: $o,Nb: nat,G: fun_o_2238389867833408620_VEBTi,H8: heap_e7401611519738050253t_unit,R3: vEBT_VEBTi,N3: nat] :
      ( aa_nat_o(aa_o_fun_nat_o(heap_Time_effect_o(F,H,H3),(R2)),Nb)
     => ( aa_nat_o(aa_VEB8883426185042993134_nat_o(heap_T2071195472996403633_VEBTi(aa_o_H7577119287286566245_VEBTi(G,(R2)),H3,H8),R3),N3)
       => aa_nat_o(aa_VEB8883426185042993134_nat_o(heap_T2071195472996403633_VEBTi(heap_T5998771940306268294_VEBTi(F,G),H,H8),R3),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),N3)) ) ) ).

% effect_bindI
tff(fact_6892_effect__bindI,axiom,
    ! [F: heap_Time_Heap_o,H: heap_e7401611519738050253t_unit,H3: heap_e7401611519738050253t_unit,R2: $o,Nb: nat,G: fun_o_8325599348076409421Heap_o,H8: heap_e7401611519738050253t_unit,R3: $o,N3: nat] :
      ( aa_nat_o(aa_o_fun_nat_o(heap_Time_effect_o(F,H,H3),(R2)),Nb)
     => ( aa_nat_o(aa_o_fun_nat_o(heap_Time_effect_o(aa_o_H7399033255965945470Heap_o(G,(R2)),H3,H8),(R3)),N3)
       => aa_nat_o(aa_o_fun_nat_o(heap_Time_effect_o(heap_Time_bind_o_o(F,G),H,H8),(R3)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),N3)) ) ) ).

% effect_bindI
tff(fact_6893_effect__bindI,axiom,
    ! [F: heap_T4980287057938770641_VEBTi,H: heap_e7401611519738050253t_unit,H3: heap_e7401611519738050253t_unit,R2: list_VEBT_VEBTi,Nb: nat,G: fun_li7499208429157053461_VEBTi,H8: heap_e7401611519738050253t_unit,R3: vEBT_VEBTi,N3: nat] :
      ( aa_nat_o(heap_T33481931004607297_VEBTi(F,H,H3,R2),Nb)
     => ( aa_nat_o(aa_VEB8883426185042993134_nat_o(heap_T2071195472996403633_VEBTi(aa_lis1839255523752068350_VEBTi(G,R2),H3,H8),R3),N3)
       => aa_nat_o(aa_VEB8883426185042993134_nat_o(heap_T2071195472996403633_VEBTi(heap_T5877712393672139267_VEBTi(F,G),H,H8),R3),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),N3)) ) ) ).

% effect_bindI
tff(fact_6894_effect__bindI,axiom,
    ! [F: heap_Time_Heap_nat,H: heap_e7401611519738050253t_unit,H3: heap_e7401611519738050253t_unit,R2: nat,Nb: nat,G: fun_na5556145776640407093on_nat,H8: heap_e7401611519738050253t_unit,R3: option_nat,N3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(heap_Time_effect_nat(F,H,H3),R2),Nb)
     => ( aa_nat_o(heap_T306965388786959644on_nat(aa_nat947733127081326238on_nat(G,R2),H3,H8,R3),N3)
       => aa_nat_o(heap_T306965388786959644on_nat(heap_T8222160169144143993on_nat(F,G),H,H8,R3),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),N3)) ) ) ).

% effect_bindI
tff(fact_6895_effect__bindI,axiom,
    ! [F: heap_T8145700208782473153_VEBTi,H: heap_e7401611519738050253t_unit,H3: heap_e7401611519738050253t_unit,R2: vEBT_VEBTi,Nb: nat,G: fun_VE2693554162914142864on_nat,H8: heap_e7401611519738050253t_unit,R3: option_nat,N3: nat] :
      ( aa_nat_o(aa_VEB8883426185042993134_nat_o(heap_T2071195472996403633_VEBTi(F,H,H3),R2),Nb)
     => ( aa_nat_o(heap_T306965388786959644on_nat(aa_VEB2711320435582112505on_nat(G,R2),H3,H8,R3),N3)
       => aa_nat_o(heap_T306965388786959644on_nat(heap_T2868974464944644318on_nat(F,G),H,H8,R3),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),N3)) ) ) ).

% effect_bindI
tff(fact_6896_effect__bindI,axiom,
    ! [F: heap_T2636463487746394924on_nat,H: heap_e7401611519738050253t_unit,H3: heap_e7401611519738050253t_unit,R2: option_nat,Nb: nat,G: fun_op1509349959255590266_VEBTi,H8: heap_e7401611519738050253t_unit,R3: vEBT_VEBTi,N3: nat] :
      ( aa_nat_o(heap_T306965388786959644on_nat(F,H,H3,R2),Nb)
     => ( aa_nat_o(aa_VEB8883426185042993134_nat_o(heap_T2071195472996403633_VEBTi(aa_opt6483592524530923107_VEBTi(G,R2),H3,H8),R3),N3)
       => aa_nat_o(aa_VEB8883426185042993134_nat_o(heap_T2071195472996403633_VEBTi(heap_T5661892481228163294_VEBTi(F,G),H,H8),R3),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),N3)) ) ) ).

% effect_bindI
tff(fact_6897_vebt__predi_Omono,axiom,
    ! [Xb: produc3881548065746020326Ti_nat] : comple7427593846244870586on_nat(partia8535946980722491222Ti_nat(heap_T7875578835903804844on_nat),heap_T7875578835903804844on_nat,aTP_Lamm_aat(Xb)) ).

% vebt_predi.mono
tff(fact_6898_vebt__succi_Omono,axiom,
    ! [Xb: produc3881548065746020326Ti_nat] : comple7427593846244870586on_nat(partia8535946980722491222Ti_nat(heap_T7875578835903804844on_nat),heap_T7875578835903804844on_nat,aTP_Lamm_abb(Xb)) ).

% vebt_succi.mono
tff(fact_6899_vebt__inserti_Omono,axiom,
    ! [Xb: produc3881548065746020326Ti_nat] : comple6664421360520557306_VEBTi(partia6690842624828592406Ti_nat(heap_T7173139186834293313_VEBTi),heap_T7173139186834293313_VEBTi,aTP_Lamm_abq(Xb)) ).

% vebt_inserti.mono
tff(fact_6900_bezout__add__strong__nat,axiom,
    ! [A: nat,B: nat] :
      ( ( A != zero_zero_nat )
     => ? [D2: nat,X2: nat,Y4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D2),A)
          & aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D2),B)
          & ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),X2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),Y4)),D2) ) ) ) ).

% bezout_add_strong_nat
tff(fact_6901_vebt__buildupi_Osimps_I3_J,axiom,
    ! [Vaa: nat] :
      aa_nat5408703056993926195_VEBTi(vEBT_vebt_buildupi,aa_nat_nat(suc,aa_nat_nat(suc,Vaa))) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,Vaa))),
        $let(
          half: nat,
          half:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Vaa))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
          heap_T5877712393672139267_VEBTi(vEBT_V1859673955506687831_VEBTi(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),half),aa_nat5408703056993926195_VEBTi(vEBT_vebt_buildupi,half)),aa_nat4125268785154830727_VEBTi(aTP_Lamm_xu(Vaa),half)) ),
        $let(
          half: nat,
          half:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Vaa))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
          heap_T5877712393672139267_VEBTi(vEBT_V1859673955506687831_VEBTi(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,half)),aa_nat5408703056993926195_VEBTi(vEBT_vebt_buildupi,half)),aa_nat4125268785154830727_VEBTi(aTP_Lamm_xw(Vaa),half)) ) ) ).

% vebt_buildupi.simps(3)
tff(fact_6902_vebt__inserti_Ofixp__induct,axiom,
    ! [P: fun_fu1178290271985267629EBTi_o] :
      ( comple7878442886591315089_VEBTi(partia6972460264168101086Ti_nat(heap_T3112222404744780921_VEBTi),partia6690842624828592406Ti_nat(heap_T7173139186834293313_VEBTi),aTP_Lamm_abr(P))
     => ( aa_fun6333388201673717798EBTi_o(P,aTP_Lamm_abs)
       => ( ! [F2: fun_VE8118064890490510894_VEBTi] :
              ( aa_fun6333388201673717798EBTi_o(P,F2)
             => aa_fun6333388201673717798EBTi_o(P,aTP_Lamm_abt(F2)) )
         => aa_fun6333388201673717798EBTi_o(P,vEBT_vebt_inserti) ) ) ) ).

% vebt_inserti.fixp_induct
tff(fact_6903_div2__even__ext__nat,axiom,
    ! [Xb: nat,Y: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
     => ( ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Xb)
        <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Y) )
       => ( Xb = Y ) ) ) ).

% div2_even_ext_nat
tff(fact_6904_VEBT__internal_Ovebt__succi_H_Ofixp__induct,axiom,
    ! [P: fun_fu6711510597495717193_nat_o] :
      ( comple5664458124825824345on_nat(partia6039416512482706817Ti_nat(heap_T7048022066654196708on_nat),partia7778075537949639097Ti_nat(heap_T7875578835903804844on_nat),aTP_Lamm_abu(P))
     => ( aa_fun5369152942363543986_nat_o(P,aTP_Lamm_abv)
       => ( ! [F2: fun_VE3775134097089128360on_nat] :
              ( aa_fun5369152942363543986_nat_o(P,F2)
             => aa_fun5369152942363543986_nat_o(P,aTP_Lamm_abw(F2)) )
         => aa_fun5369152942363543986_nat_o(P,vEBT_VEBT_vebt_succi) ) ) ) ).

% VEBT_internal.vebt_succi'.fixp_induct
tff(fact_6905_VEBT__internal_Ovebt__predi_H_Ofixp__induct,axiom,
    ! [P: fun_fu6711510597495717193_nat_o] :
      ( comple5664458124825824345on_nat(partia6039416512482706817Ti_nat(heap_T7048022066654196708on_nat),partia7778075537949639097Ti_nat(heap_T7875578835903804844on_nat),aTP_Lamm_abu(P))
     => ( aa_fun5369152942363543986_nat_o(P,aTP_Lamm_abv)
       => ( ! [F2: fun_VE3775134097089128360on_nat] :
              ( aa_fun5369152942363543986_nat_o(P,F2)
             => aa_fun5369152942363543986_nat_o(P,aTP_Lamm_abx(F2)) )
         => aa_fun5369152942363543986_nat_o(P,vEBT_VEBT_vebt_predi) ) ) ) ).

% VEBT_internal.vebt_predi'.fixp_induct
tff(fact_6906_VEBT__internal_Ovebt__buildupi_H_Opelims,axiom,
    ! [Xb: nat,Y: heap_T8145700208782473153_VEBTi] :
      ( ( aa_nat5408703056993926195_VEBTi(vEBT_V739175172307565963ildupi,Xb) = Y )
     => ( aa_nat_o(accp_nat(vEBT_V254170901696579886pi_rel),Xb)
       => ( ( ( Xb = zero_zero_nat )
           => ( ( Y = aa_VEB836174949890266638_VEBTi(heap_T3630416162098727440_VEBTi,vEBT_Leafi($false,$false)) )
             => ~ aa_nat_o(accp_nat(vEBT_V254170901696579886pi_rel),zero_zero_nat) ) )
         => ( ( ( Xb = aa_nat_nat(suc,zero_zero_nat) )
             => ( ( Y = aa_VEB836174949890266638_VEBTi(heap_T3630416162098727440_VEBTi,vEBT_Leafi($false,$false)) )
               => ~ aa_nat_o(accp_nat(vEBT_V254170901696579886pi_rel),aa_nat_nat(suc,zero_zero_nat)) ) )
           => ~ ! [Va2: nat] :
                  ( ( Xb = aa_nat_nat(suc,aa_nat_nat(suc,Va2)) )
                 => ( ( Y = $ite(
                          aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,Va2))),
                          $let(
                            half: nat,
                            half:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                            heap_T5877712393672139267_VEBTi(vEBT_V1859673955506687831_VEBTi(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),half),aa_nat5408703056993926195_VEBTi(vEBT_V739175172307565963ildupi,half)),aa_nat4125268785154830727_VEBTi(aTP_Lamm_xp(Va2),half)) ),
                          $let(
                            half: nat,
                            half:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                            heap_T5877712393672139267_VEBTi(vEBT_V1859673955506687831_VEBTi(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,half)),aa_nat5408703056993926195_VEBTi(vEBT_V739175172307565963ildupi,half)),aa_nat4125268785154830727_VEBTi(aTP_Lamm_xs(Va2),half)) ) ) )
                   => ~ aa_nat_o(accp_nat(vEBT_V254170901696579886pi_rel),aa_nat_nat(suc,aa_nat_nat(suc,Va2))) ) ) ) ) ) ) ).

% VEBT_internal.vebt_buildupi'.pelims
tff(fact_6907_prod__decode__aux_Ocases,axiom,
    ! [Xb: product_prod_nat_nat] :
      ~ ! [K3: nat,M2: nat] : Xb != aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,K3),M2) ).

% prod_decode_aux.cases
tff(fact_6908_list__decode_Ocases,axiom,
    ! [Xb: nat] :
      ( ( Xb != zero_zero_nat )
     => ~ ! [N2: nat] : Xb != aa_nat_nat(suc,N2) ) ).

% list_decode.cases
tff(fact_6909_vebt__buildupi_Opelims,axiom,
    ! [Xb: nat,Y: heap_T8145700208782473153_VEBTi] :
      ( ( aa_nat5408703056993926195_VEBTi(vEBT_vebt_buildupi,Xb) = Y )
     => ( aa_nat_o(accp_nat(vEBT_v1230518104690509829pi_rel),Xb)
       => ( ( ( Xb = zero_zero_nat )
           => ( ( Y = aa_VEB836174949890266638_VEBTi(heap_T3630416162098727440_VEBTi,vEBT_Leafi($false,$false)) )
             => ~ aa_nat_o(accp_nat(vEBT_v1230518104690509829pi_rel),zero_zero_nat) ) )
         => ( ( ( Xb = aa_nat_nat(suc,zero_zero_nat) )
             => ( ( Y = aa_VEB836174949890266638_VEBTi(heap_T3630416162098727440_VEBTi,vEBT_Leafi($false,$false)) )
               => ~ aa_nat_o(accp_nat(vEBT_v1230518104690509829pi_rel),aa_nat_nat(suc,zero_zero_nat)) ) )
           => ~ ! [Va2: nat] :
                  ( ( Xb = aa_nat_nat(suc,aa_nat_nat(suc,Va2)) )
                 => ( ( Y = $ite(
                          aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,Va2))),
                          $let(
                            half: nat,
                            half:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                            heap_T5877712393672139267_VEBTi(vEBT_V1859673955506687831_VEBTi(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),half),aa_nat5408703056993926195_VEBTi(vEBT_vebt_buildupi,half)),aa_nat4125268785154830727_VEBTi(aTP_Lamm_xu(Va2),half)) ),
                          $let(
                            half: nat,
                            half:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                            heap_T5877712393672139267_VEBTi(vEBT_V1859673955506687831_VEBTi(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,half)),aa_nat5408703056993926195_VEBTi(vEBT_vebt_buildupi,half)),aa_nat4125268785154830727_VEBTi(aTP_Lamm_xw(Va2),half)) ) ) )
                   => ~ aa_nat_o(accp_nat(vEBT_v1230518104690509829pi_rel),aa_nat_nat(suc,aa_nat_nat(suc,Va2))) ) ) ) ) ) ) ).

% vebt_buildupi.pelims
tff(fact_6910_VEBT__internal_OT__vebt__buildupi_H_Opelims,axiom,
    ! [Xb: nat,Y: int] :
      ( ( aa_nat_int(vEBT_V9176841429113362141ildupi,Xb) = Y )
     => ( aa_nat_o(accp_nat(vEBT_V3352910403632780892pi_rel),Xb)
       => ( ( ( Xb = zero_zero_nat )
           => ( ( Y = one_one_int )
             => ~ aa_nat_o(accp_nat(vEBT_V3352910403632780892pi_rel),zero_zero_nat) ) )
         => ( ( ( Xb = aa_nat_nat(suc,zero_zero_nat) )
             => ( ( Y = one_one_int )
               => ~ aa_nat_o(accp_nat(vEBT_V3352910403632780892pi_rel),aa_nat_nat(suc,zero_zero_nat)) ) )
           => ~ ! [N2: nat] :
                  ( ( Xb = aa_nat_nat(suc,aa_nat_nat(suc,N2)) )
                 => ( ( Y = $ite(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_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,one))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(vEBT_V9176841429113362141ildupi,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,aa_num_num(bit0,one)))),aa_nat_int(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(divide_divide_nat,N2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(vEBT_V9176841429113362141ildupi,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_int(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(divide_divide_nat,N2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,one))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(vEBT_V9176841429113362141ildupi,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,one))))),aa_nat_int(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(divide_divide_nat,N2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(vEBT_V9176841429113362141ildupi,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_int(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(divide_divide_nat,N2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))))) )
                   => ~ aa_nat_o(accp_nat(vEBT_V3352910403632780892pi_rel),aa_nat_nat(suc,aa_nat_nat(suc,N2))) ) ) ) ) ) ) ).

% VEBT_internal.T_vebt_buildupi'.pelims
tff(fact_6911_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_Opelims,axiom,
    ! [Xb: nat,Y: nat] :
      ( ( aa_nat_nat(vEBT_V8646137997579335489_i_l_d,Xb) = Y )
     => ( aa_nat_o(accp_nat(vEBT_V5144397997797733112_d_rel),Xb)
       => ( ( ( Xb = zero_zero_nat )
           => ( ( Y = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one))) )
             => ~ aa_nat_o(accp_nat(vEBT_V5144397997797733112_d_rel),zero_zero_nat) ) )
         => ( ( ( Xb = aa_nat_nat(suc,zero_zero_nat) )
             => ( ( Y = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one))) )
               => ~ aa_nat_o(accp_nat(vEBT_V5144397997797733112_d_rel),aa_nat_nat(suc,zero_zero_nat)) ) )
           => ~ ! [Va2: nat] :
                  ( ( Xb = aa_nat_nat(suc,aa_nat_nat(suc,Va2)) )
                 => ( ( Y = $ite(
                          aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,Va2))),
                          aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
                            $let(
                              half: nat,
                              half:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                              aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,one))))),aa_nat_nat(vEBT_V8646137997579335489_i_l_d,half))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),half)),aa_nat_nat(vEBT_V8646137997579335489_i_l_d,half))) )),
                          $let(
                            half: nat,
                            half:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                            aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,one))))),aa_nat_nat(vEBT_V8646137997579335489_i_l_d,aa_nat_nat(suc,half)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,half))),aa_nat_nat(vEBT_V8646137997579335489_i_l_d,half))) ) ) )
                   => ~ aa_nat_o(accp_nat(vEBT_V5144397997797733112_d_rel),aa_nat_nat(suc,aa_nat_nat(suc,Va2))) ) ) ) ) ) ) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d.pelims
tff(fact_6912_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_092_060_094sub_062u_092_060_094sub_062p_Opelims,axiom,
    ! [Xb: nat,Y: nat] :
      ( ( aa_nat_nat(vEBT_V8346862874174094_d_u_p,Xb) = Y )
     => ( aa_nat_o(accp_nat(vEBT_V1247956027447740395_p_rel),Xb)
       => ( ( ( Xb = zero_zero_nat )
           => ( ( Y = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)) )
             => ~ aa_nat_o(accp_nat(vEBT_V1247956027447740395_p_rel),zero_zero_nat) ) )
         => ( ( ( Xb = aa_nat_nat(suc,zero_zero_nat) )
             => ( ( Y = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)) )
               => ~ aa_nat_o(accp_nat(vEBT_V1247956027447740395_p_rel),aa_nat_nat(suc,zero_zero_nat)) ) )
           => ~ ! [Va2: nat] :
                  ( ( Xb = aa_nat_nat(suc,aa_nat_nat(suc,Va2)) )
                 => ( ( Y = $ite(
                          aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,Va2))),
                          aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
                            $let(
                              half: nat,
                              half:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                              aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,one))))),aa_nat_nat(vEBT_V8346862874174094_d_u_p,half))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),half)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(vEBT_V8346862874174094_d_u_p,half)),one_one_nat))) )),
                          $let(
                            half: nat,
                            half:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                            aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,one))))),aa_nat_nat(vEBT_V8346862874174094_d_u_p,aa_nat_nat(suc,half)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,half))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(vEBT_V8346862874174094_d_u_p,half)),one_one_nat))) ) ) )
                   => ~ aa_nat_o(accp_nat(vEBT_V1247956027447740395_p_rel),aa_nat_nat(suc,aa_nat_nat(suc,Va2))) ) ) ) ) ) ) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d\<^sub>u\<^sub>p.pelims
tff(fact_6913_VEBT__internal_OTb_Opelims,axiom,
    ! [Xb: nat,Y: int] :
      ( ( aa_nat_int(vEBT_VEBT_Tb,Xb) = Y )
     => ( aa_nat_o(accp_nat(vEBT_VEBT_Tb_rel2),Xb)
       => ( ( ( Xb = zero_zero_nat )
           => ( ( Y = aa_num_int(numeral_numeral_int,aa_num_num(bit1,one)) )
             => ~ aa_nat_o(accp_nat(vEBT_VEBT_Tb_rel2),zero_zero_nat) ) )
         => ( ( ( Xb = aa_nat_nat(suc,zero_zero_nat) )
             => ( ( Y = aa_num_int(numeral_numeral_int,aa_num_num(bit1,one)) )
               => ~ aa_nat_o(accp_nat(vEBT_VEBT_Tb_rel2),aa_nat_nat(suc,zero_zero_nat)) ) )
           => ~ ! [N2: nat] :
                  ( ( Xb = aa_nat_nat(suc,aa_nat_nat(suc,N2)) )
                 => ( ( Y = $ite(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_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,aa_num_num(bit0,one)))),aa_nat_int(vEBT_VEBT_Tb,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(vEBT_VEBT_Tb,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,aa_num_num(bit0,one)))),aa_nat_int(vEBT_VEBT_Tb,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(vEBT_VEBT_Tb,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))))) )
                   => ~ aa_nat_o(accp_nat(vEBT_VEBT_Tb_rel2),aa_nat_nat(suc,aa_nat_nat(suc,N2))) ) ) ) ) ) ) ).

% VEBT_internal.Tb.pelims
tff(fact_6914_VEBT__internal_OTb_H_Opelims,axiom,
    ! [Xb: nat,Y: nat] :
      ( ( aa_nat_nat(vEBT_VEBT_Tb2,Xb) = Y )
     => ( aa_nat_o(accp_nat(vEBT_VEBT_Tb_rel),Xb)
       => ( ( ( Xb = zero_zero_nat )
           => ( ( Y = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)) )
             => ~ aa_nat_o(accp_nat(vEBT_VEBT_Tb_rel),zero_zero_nat) ) )
         => ( ( ( Xb = aa_nat_nat(suc,zero_zero_nat) )
             => ( ( Y = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)) )
               => ~ aa_nat_o(accp_nat(vEBT_VEBT_Tb_rel),aa_nat_nat(suc,zero_zero_nat)) ) )
           => ~ ! [N2: nat] :
                  ( ( Xb = aa_nat_nat(suc,aa_nat_nat(suc,N2)) )
                 => ( ( Y = $ite(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_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit0,one)))),aa_nat_nat(vEBT_VEBT_Tb2,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(vEBT_VEBT_Tb2,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit0,one)))),aa_nat_nat(vEBT_VEBT_Tb2,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(vEBT_VEBT_Tb2,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))))) )
                   => ~ aa_nat_o(accp_nat(vEBT_VEBT_Tb_rel),aa_nat_nat(suc,aa_nat_nat(suc,N2))) ) ) ) ) ) ) ).

% VEBT_internal.Tb'.pelims
tff(fact_6915_VEBT__internal_OT__vebt__buildupi_Opelims,axiom,
    ! [Xb: nat,Y: nat] :
      ( ( aa_nat_nat(vEBT_V441764108873111860ildupi,Xb) = Y )
     => ( aa_nat_o(accp_nat(vEBT_V2957053500504383685pi_rel),Xb)
       => ( ( ( Xb = zero_zero_nat )
           => ( ( Y = aa_nat_nat(suc,zero_zero_nat) )
             => ~ aa_nat_o(accp_nat(vEBT_V2957053500504383685pi_rel),zero_zero_nat) ) )
         => ( ( ( Xb = aa_nat_nat(suc,zero_zero_nat) )
             => ( ( Y = aa_nat_nat(suc,zero_zero_nat) )
               => ~ aa_nat_o(accp_nat(vEBT_V2957053500504383685pi_rel),aa_nat_nat(suc,zero_zero_nat)) ) )
           => ~ ! [N2: nat] :
                  ( ( Xb = aa_nat_nat(suc,aa_nat_nat(suc,N2)) )
                 => ( ( Y = $ite(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_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(vEBT_V441764108873111860ildupi,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(vEBT_V441764108873111860ildupi,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))))))),aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(vEBT_V441764108873111860ildupi,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(vEBT_V441764108873111860ildupi,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))))))) )
                   => ~ aa_nat_o(accp_nat(vEBT_V2957053500504383685pi_rel),aa_nat_nat(suc,aa_nat_nat(suc,N2))) ) ) ) ) ) ) ).

% VEBT_internal.T_vebt_buildupi.pelims
tff(fact_6916_VEBT__internal_Ovebt__memberi_H_Ofixp__induct,axiom,
    ! [P: fun_fu2707964056798801139ap_o_o] :
      ( comple5871978649864577227Heap_o(partia6726927458685305659Ti_nat(heap_Time_Heap_lub_o),partia3290229181235258227Ti_nat(heap_Time_Heap_ord_o),aTP_Lamm_aby(P))
     => ( aa_fun339294776471787884ap_o_o(P,aTP_Lamm_abz)
       => ( ! [F2: fun_VE8181444127144381288Heap_o] :
              ( aa_fun339294776471787884ap_o_o(P,F2)
             => aa_fun339294776471787884ap_o_o(P,aTP_Lamm_aca(F2)) )
         => aa_fun339294776471787884ap_o_o(P,vEBT_V854960066525838166emberi) ) ) ) ).

% VEBT_internal.vebt_memberi'.fixp_induct
tff(fact_6917_vebt__predi_Ofixp__induct,axiom,
    ! [P: fun_fu4750890421531948930_nat_o] :
      ( comple7368800650122905596on_nat(partia2080987842261039902Ti_nat(heap_T7048022066654196708on_nat),partia8535946980722491222Ti_nat(heap_T7875578835903804844on_nat),aTP_Lamm_acb(P))
     => ( aa_fun7244059194882734075_nat_o(P,aTP_Lamm_acc)
       => ( ! [F2: fun_VE8516060933855021849on_nat] :
              ( aa_fun7244059194882734075_nat_o(P,F2)
             => aa_fun7244059194882734075_nat_o(P,aTP_Lamm_acd(F2)) )
         => aa_fun7244059194882734075_nat_o(P,vEBT_vebt_predi) ) ) ) ).

% vebt_predi.fixp_induct
tff(fact_6918_vebt__succi_Ofixp__induct,axiom,
    ! [P: fun_fu4750890421531948930_nat_o] :
      ( comple7368800650122905596on_nat(partia2080987842261039902Ti_nat(heap_T7048022066654196708on_nat),partia8535946980722491222Ti_nat(heap_T7875578835903804844on_nat),aTP_Lamm_acb(P))
     => ( aa_fun7244059194882734075_nat_o(P,aTP_Lamm_acc)
       => ( ! [F2: fun_VE8516060933855021849on_nat] :
              ( aa_fun7244059194882734075_nat_o(P,F2)
             => aa_fun7244059194882734075_nat_o(P,aTP_Lamm_ace(F2)) )
         => aa_fun7244059194882734075_nat_o(P,vEBT_vebt_succi) ) ) ) ).

% vebt_succi.fixp_induct
tff(fact_6919_vebt__memberi_Ofixp__induct,axiom,
    ! [P: fun_fu8886734894937297424ap_o_o] :
      ( comple8003709893544916456Heap_o(partia5551857090987368152Ti_nat(heap_Time_Heap_lub_o),partia2925774515620677392Ti_nat(heap_Time_Heap_ord_o),aTP_Lamm_acf(P))
     => ( aa_fun4656092467790859897ap_o_o(P,aTP_Lamm_acg)
       => ( ! [F2: fun_VE7134199479003659425Heap_o] :
              ( aa_fun4656092467790859897ap_o_o(P,F2)
             => aa_fun4656092467790859897ap_o_o(P,aTP_Lamm_ach(F2)) )
         => aa_fun4656092467790859897ap_o_o(P,vEBT_vebt_memberi) ) ) ) ).

% vebt_memberi.fixp_induct
tff(fact_6920_bezw_Oelims,axiom,
    ! [Xb: nat,Xaa: nat,Y: product_prod_int_int] :
      ( ( bezw(Xb,Xaa) = Y )
     => ( Y = $ite(Xaa = zero_zero_nat,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,one_one_int),zero_zero_int),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_Pro6799258434465214121nt_int(product_snd_int_int,bezw(Xaa,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xb),Xaa)))),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_Pro6799258434465214121nt_int(product_fst_int_int,bezw(Xaa,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xb),Xaa)))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_Pro6799258434465214121nt_int(product_snd_int_int,bezw(Xaa,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xb),Xaa)))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Xb),Xaa)))))) ) ) ).

% bezw.elims
tff(fact_6921_bezw_Osimps,axiom,
    ! [Xb: nat,Y: nat] :
      bezw(Xb,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,Xb),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,Xb),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,Xb),Y)))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Xb),Y)))))) ).

% bezw.simps
tff(fact_6922_tap__def,axiom,
    ! [F: fun_he7283999071544186133_VEBTi] : heap_T861569056895978319_VEBTi(F) = heap_T1489671670754571048_VEBTi(aTP_Lamm_aci(F)) ).

% tap_def
tff(fact_6923_tap__def,axiom,
    ! [F: fun_he4876066172508077696on_nat] : heap_T3993282329578657722on_nat(F) = heap_T5286843759275942675on_nat(aTP_Lamm_acj(F)) ).

% tap_def
tff(fact_6924_tap__def,axiom,
    ! [F: fun_he8413527710729349838unit_o] : heap_Time_tap_o(F) = heap_Time_Heap_o2(aTP_Lamm_ack(F)) ).

% tap_def
tff(fact_6925_bezw__0,axiom,
    ! [Xb: nat] : bezw(Xb,zero_zero_nat) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,one_one_int),zero_zero_int) ).

% bezw_0
tff(fact_6926_effect__tapI,axiom,
    ! [H3: heap_e7401611519738050253t_unit,H: heap_e7401611519738050253t_unit,R2: vEBT_VEBTi,F: fun_he7283999071544186133_VEBTi] :
      ( ( H3 = H )
     => ( ( R2 = aa_hea7585900348394676422_VEBTi(F,H) )
       => aa_nat_o(aa_VEB8883426185042993134_nat_o(heap_T2071195472996403633_VEBTi(heap_T861569056895978319_VEBTi(F),H,H3),R2),one_one_nat) ) ) ).

% effect_tapI
tff(fact_6927_effect__tapI,axiom,
    ! [H3: heap_e7401611519738050253t_unit,H: heap_e7401611519738050253t_unit,R2: option_nat,F: fun_he4876066172508077696on_nat] :
      ( ( H3 = H )
     => ( ( R2 = aa_hea6610145743731876273on_nat(F,H) )
       => aa_nat_o(heap_T306965388786959644on_nat(heap_T3993282329578657722on_nat(F),H,H3,R2),one_one_nat) ) ) ).

% effect_tapI
tff(fact_6928_effect__tapI,axiom,
    ! [H3: heap_e7401611519738050253t_unit,H: heap_e7401611519738050253t_unit,R2: $o,F: fun_he8413527710729349838unit_o] :
      ( ( H3 = H )
     => ( ( (R2)
        <=> aa_hea7994211679605476167unit_o(F,H) )
       => aa_nat_o(aa_o_fun_nat_o(heap_Time_effect_o(heap_Time_tap_o(F),H,H3),(R2)),one_one_nat) ) ) ).

% effect_tapI
tff(fact_6929_effect__tapE,axiom,
    ! [F: fun_he7283999071544186133_VEBTi,H: heap_e7401611519738050253t_unit,H3: heap_e7401611519738050253t_unit,R2: vEBT_VEBTi,Nb: nat] :
      ( aa_nat_o(aa_VEB8883426185042993134_nat_o(heap_T2071195472996403633_VEBTi(heap_T861569056895978319_VEBTi(F),H,H3),R2),Nb)
     => ~ ( ( H3 = H )
         => ( ( R2 = aa_hea7585900348394676422_VEBTi(F,H) )
           => ( Nb != one_one_nat ) ) ) ) ).

% effect_tapE
tff(fact_6930_effect__tapE,axiom,
    ! [F: fun_he4876066172508077696on_nat,H: heap_e7401611519738050253t_unit,H3: heap_e7401611519738050253t_unit,R2: option_nat,Nb: nat] :
      ( aa_nat_o(heap_T306965388786959644on_nat(heap_T3993282329578657722on_nat(F),H,H3,R2),Nb)
     => ~ ( ( H3 = H )
         => ( ( R2 = aa_hea6610145743731876273on_nat(F,H) )
           => ( Nb != one_one_nat ) ) ) ) ).

% effect_tapE
tff(fact_6931_effect__tapE,axiom,
    ! [F: fun_he8413527710729349838unit_o,H: heap_e7401611519738050253t_unit,H3: heap_e7401611519738050253t_unit,R2: $o,Nb: nat] :
      ( aa_nat_o(aa_o_fun_nat_o(heap_Time_effect_o(heap_Time_tap_o(F),H,H3),(R2)),Nb)
     => ~ ( ( H3 = H )
         => ( ( (R2)
            <=> aa_hea7994211679605476167unit_o(F,H) )
           => ( Nb != one_one_nat ) ) ) ) ).

% effect_tapE
tff(fact_6932_heap_Omono__body__fixp,axiom,
    ! [F4: fun_fu411293875173739919on_nat] :
      ( ! [X2: produc3881548065746020326Ti_nat] : comple7427593846244870586on_nat(partia8535946980722491222Ti_nat(heap_T7875578835903804844on_nat),heap_T7875578835903804844on_nat,aa_Pro8654539709784947602on_nat(aTP_Lamm_acl(F4),X2))
     => ( comple6035428378238101126on_nat(partia2080987842261039902Ti_nat(heap_T7048022066654196708on_nat),partia8535946980722491222Ti_nat(heap_T7875578835903804844on_nat),F4) = aa_fun943833532922116944on_nat(F4,comple6035428378238101126on_nat(partia2080987842261039902Ti_nat(heap_T7048022066654196708on_nat),partia8535946980722491222Ti_nat(heap_T7875578835903804844on_nat),F4)) ) ) ).

% heap.mono_body_fixp
tff(fact_6933_heap_Omono__body__fixp,axiom,
    ! [F4: fun_fu2426956696497423235Heap_o] :
      ( ! [X2: produc3881548065746020326Ti_nat] : comple6933591415923782038Heap_o(partia2925774515620677392Ti_nat(heap_Time_Heap_ord_o),heap_Time_Heap_ord_o,aa_Pro5165804456746783526Heap_o(aTP_Lamm_acm(F4),X2))
     => ( comple8693199471913871838Heap_o(partia5551857090987368152Ti_nat(heap_Time_Heap_lub_o),partia2925774515620677392Ti_nat(heap_Time_Heap_ord_o),F4) = aa_fun5049358401508434812Heap_o(F4,comple8693199471913871838Heap_o(partia5551857090987368152Ti_nat(heap_Time_Heap_lub_o),partia2925774515620677392Ti_nat(heap_Time_Heap_ord_o),F4)) ) ) ).

% heap.mono_body_fixp
tff(fact_6934_heap_Omono__body__fixp,axiom,
    ! [F4: fun_fu1151686773700218227_VEBTi] :
      ( ! [X2: produc3960855945107176009Ti_nat] : comple422786390015765981_VEBTi(partia1868168049876374393Ti_nat(heap_T7173139186834293313_VEBTi),heap_T7173139186834293313_VEBTi,aa_Pro7504462699960347282_VEBTi(aTP_Lamm_acn(F4),X2))
     => ( comple5861331542469577976_VEBTi(partia7782936097874681665Ti_nat(heap_T3112222404744780921_VEBTi),partia1868168049876374393Ti_nat(heap_T7173139186834293313_VEBTi),F4) = aa_fun5250146550359294516_VEBTi(F4,comple5861331542469577976_VEBTi(partia7782936097874681665Ti_nat(heap_T3112222404744780921_VEBTi),partia1868168049876374393Ti_nat(heap_T7173139186834293313_VEBTi),F4)) ) ) ).

% heap.mono_body_fixp
tff(fact_6935_heap_Omono__body__fixp,axiom,
    ! [F4: fun_fu3620910142932681on_nat] :
      ( ! [X2: produc3960855945107176009Ti_nat] : comple4513117127444775325on_nat(partia7778075537949639097Ti_nat(heap_T7875578835903804844on_nat),heap_T7875578835903804844on_nat,aa_Pro1535628157467025618on_nat(aTP_Lamm_aco(F4),X2))
     => ( comple4104493882897929443on_nat(partia6039416512482706817Ti_nat(heap_T7048022066654196708on_nat),partia7778075537949639097Ti_nat(heap_T7875578835903804844on_nat),F4) = aa_fun7861206497237518090on_nat(F4,comple4104493882897929443on_nat(partia6039416512482706817Ti_nat(heap_T7048022066654196708on_nat),partia7778075537949639097Ti_nat(heap_T7875578835903804844on_nat),F4)) ) ) ).

% heap.mono_body_fixp
tff(fact_6936_heap_Omono__body__fixp,axiom,
    ! [F4: fun_fu8308473583445096899Heap_o] :
      ( ! [X2: produc3960855945107176009Ti_nat] : comple5241396456474302457Heap_o(partia3290229181235258227Ti_nat(heap_Time_Heap_ord_o),heap_Time_Heap_ord_o,aa_Pro9147008720614807654Heap_o(aTP_Lamm_acp(F4),X2))
     => ( comple5362527178602408641Heap_o(partia6726927458685305659Ti_nat(heap_Time_Heap_lub_o),partia3290229181235258227Ti_nat(heap_Time_Heap_ord_o),F4) = aa_fun55487300827498300Heap_o(F4,comple5362527178602408641Heap_o(partia6726927458685305659Ti_nat(heap_Time_Heap_lub_o),partia3290229181235258227Ti_nat(heap_Time_Heap_ord_o),F4)) ) ) ).

% heap.mono_body_fixp
tff(fact_6937_bezw__non__0,axiom,
    ! [Y: nat,Xb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Y)
     => ( bezw(Xb,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,Xb),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,Xb),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,Xb),Y)))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Xb),Y))))) ) ) ).

% bezw_non_0
tff(fact_6938_product__nth,axiom,
    ! [Nb: nat,Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_s6755466524823107622T_VEBT(Xs)),size_s6755466524823107622T_VEBT(Ys)))
     => ( nth_Pr4953567300277697838T_VEBT(produc4743750530478302277T_VEBT(Xs,Ys),Nb) = produc537772716801021591T_VEBT(nth_VEBT_VEBT(Xs,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),size_s6755466524823107622T_VEBT(Ys))),nth_VEBT_VEBT(Ys,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),size_s6755466524823107622T_VEBT(Ys)))) ) ) ).

% product_nth
tff(fact_6939_product__nth,axiom,
    ! [Nb: nat,Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBTi] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_s6755466524823107622T_VEBT(Xs)),size_s7982070591426661849_VEBTi(Ys)))
     => ( nth_Pr316670251186196177_VEBTi(produc316462671093861988_VEBTi(Xs,Ys),Nb) = produc6084888613844515218_VEBTi(nth_VEBT_VEBT(Xs,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),size_s7982070591426661849_VEBTi(Ys))),nth_VEBT_VEBTi(Ys,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),size_s7982070591426661849_VEBTi(Ys)))) ) ) ).

% product_nth
tff(fact_6940_product__nth,axiom,
    ! [Nb: nat,Xs: list_VEBT_VEBTi,Ys: list_VEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_s7982070591426661849_VEBTi(Xs)),size_s6755466524823107622T_VEBT(Ys)))
     => ( nth_Pr8725177398587324397T_VEBT(produc1285381384045549624T_VEBT(Xs,Ys),Nb) = produc7053807326796202854T_VEBT(nth_VEBT_VEBTi(Xs,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),size_s6755466524823107622T_VEBT(Ys))),nth_VEBT_VEBT(Ys,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),size_s6755466524823107622T_VEBT(Ys)))) ) ) ).

% product_nth
tff(fact_6941_product__nth,axiom,
    ! [Nb: nat,Xs: list_VEBT_VEBTi,Ys: list_VEBT_VEBTi] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_s7982070591426661849_VEBTi(Xs)),size_s7982070591426661849_VEBTi(Ys)))
     => ( nth_Pr6329974346453275474_VEBTi(produc194614972289024177_VEBTi(Xs,Ys),Nb) = produc436343169921013763_VEBTi(nth_VEBT_VEBTi(Xs,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),size_s7982070591426661849_VEBTi(Ys))),nth_VEBT_VEBTi(Ys,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),size_s7982070591426661849_VEBTi(Ys)))) ) ) ).

% product_nth
tff(fact_6942_product__nth,axiom,
    ! [Nb: nat,Xs: list_VEBT_VEBT,Ys: list_real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_s6755466524823107622T_VEBT(Xs)),size_size_list_real(Ys)))
     => ( nth_Pr6842391030413306568T_real(produc4908677263432625371T_real(Xs,Ys),Nb) = aa_rea4189226818901770745T_real(produc8117437818029410057T_real(nth_VEBT_VEBT(Xs,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),size_size_list_real(Ys)))),aa_nat_real(nth_real(Ys),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),size_size_list_real(Ys)))) ) ) ).

% product_nth
tff(fact_6943_product__nth,axiom,
    ! [Nb: nat,Xs: list_VEBT_VEBTi,Ys: list_real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_s7982070591426661849_VEBTi(Xs)),size_size_list_real(Ys)))
     => ( nth_Pr3433448822664029129i_real(produc5476717833281694120i_real(Xs,Ys),Nb) = produc8457151488442208762i_real(nth_VEBT_VEBTi(Xs,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),size_size_list_real(Ys))),aa_nat_real(nth_real(Ys),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),size_size_list_real(Ys)))) ) ) ).

% product_nth
tff(fact_6944_product__nth,axiom,
    ! [Nb: nat,Xs: list_VEBT_VEBT,Ys: list_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_s6755466524823107622T_VEBT(Xs)),size_size_list_o(Ys)))
     => ( nth_Pr4606735188037164562VEBT_o(product_VEBT_VEBT_o(Xs,Ys),Nb) = aa_o_P2543913824914997935VEBT_o(produc8721562602347293563VEBT_o(nth_VEBT_VEBT(Xs,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),size_size_list_o(Ys)))),aa_nat_o(nth_o(Ys),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),size_size_list_o(Ys)))) ) ) ).

% product_nth
tff(fact_6945_product__nth,axiom,
    ! [Nb: nat,Xs: list_VEBT_VEBTi,Ys: list_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_s7982070591426661849_VEBTi(Xs)),size_size_list_o(Ys)))
     => ( nth_Pr3306050735993963089EBTi_o(product_VEBT_VEBTi_o(Xs,Ys),Nb) = produc8194178580519725514EBTi_o(nth_VEBT_VEBTi(Xs,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),size_size_list_o(Ys))),aa_nat_o(nth_o(Ys),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),size_size_list_o(Ys)))) ) ) ).

% product_nth
tff(fact_6946_product__nth,axiom,
    ! [Nb: nat,Xs: list_VEBT_VEBT,Ys: list_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_s6755466524823107622T_VEBT(Xs)),size_size_list_nat(Ys)))
     => ( nth_Pr1791586995822124652BT_nat(produc7295137177222721919BT_nat(Xs,Ys),Nb) = aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(nth_VEBT_VEBT(Xs,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),size_size_list_nat(Ys)))),aa_nat_nat(nth_nat(Ys),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),size_size_list_nat(Ys)))) ) ) ).

% product_nth
tff(fact_6947_product__nth,axiom,
    ! [Nb: nat,Xs: list_VEBT_VEBTi,Ys: list_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_s7982070591426661849_VEBTi(Xs)),size_size_list_nat(Ys)))
     => ( nth_Pr6911489093701683181Ti_nat(produc2282297823089607884Ti_nat(Xs,Ys),Nb) = produc7192665754729510430Ti_nat(nth_VEBT_VEBTi(Xs,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),size_size_list_nat(Ys))),aa_nat_nat(nth_nat(Ys),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),size_size_list_nat(Ys)))) ) ) ).

% product_nth
tff(fact_6948_pred__subset__eq2,axiom,
    ! [R: set_Pr3486312109254157132nteger,S2: set_Pr3486312109254157132nteger] :
      ( ord_le4825817444062779351eger_o(aTP_Lamm_acq(R),aTP_Lamm_acq(S2))
    <=> ord_le2932375828722672556nteger(R,S2) ) ).

% pred_subset_eq2
tff(fact_6949_pred__subset__eq2,axiom,
    ! [R: set_Pr1312406793078187350et_nat,S2: set_Pr1312406793078187350et_nat] :
      ( ord_le3580378406483493347_nat_o(aTP_Lamm_acr(R),aTP_Lamm_acr(S2))
    <=> ord_le3919382008627836342et_nat(R,S2) ) ).

% pred_subset_eq2
tff(fact_6950_pred__subset__eq2,axiom,
    ! [R: set_Pr4040394263813967220et_nat,S2: set_Pr4040394263813967220et_nat] :
      ( ord_le2834457679758710341_nat_o(aTP_Lamm_acs(R),aTP_Lamm_acs(S2))
    <=> ord_le5189643068899110356et_nat(R,S2) ) ).

% pred_subset_eq2
tff(fact_6951_pred__subset__eq2,axiom,
    ! [R: set_Pr6700895561065273913nt_int,S2: set_Pr6700895561065273913nt_int] :
      ( ord_le4806730579060245290_int_o(aTP_Lamm_act(R),aTP_Lamm_act(S2))
    <=> ord_le3990897393281686169nt_int(R,S2) ) ).

% pred_subset_eq2
tff(fact_6952_pred__subset__eq2,axiom,
    ! [R: set_Pr4166680992137330533nt_int,S2: set_Pr4166680992137330533nt_int] :
      ( ord_le2876595303158420734_int_o(aTP_Lamm_acu(R),aTP_Lamm_acu(S2))
    <=> ord_le3197784583289274309nt_int(R,S2) ) ).

% pred_subset_eq2
tff(fact_6953_even__word__def,axiom,
    even_w9054469088133485505l_num1 = dvd_dv6812691276156420380l_num1(numera7442385471795722001l_num1(aa_num_num(bit0,one))) ).

% even_word_def
tff(fact_6954_triangle__def,axiom,
    ! [Nb: nat] : aa_nat_nat(nat_triangle,Nb) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),aa_nat_nat(suc,Nb))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% triangle_def
tff(fact_6955_length__product,axiom,
    ! [Xs: list_real,Ys: list_real] : size_s3932428310213730859l_real(product_real_real(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_real(Xs)),size_size_list_real(Ys)) ).

% length_product
tff(fact_6956_length__product,axiom,
    ! [Xs: list_real,Ys: list_o] : size_s987546567493390085real_o(product_real_o(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_real(Xs)),size_size_list_o(Ys)) ).

% length_product
tff(fact_6957_length__product,axiom,
    ! [Xs: list_real,Ys: list_nat] : size_s1877336372972134351al_nat(product_real_nat(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_real(Xs)),size_size_list_nat(Ys)) ).

% length_product
tff(fact_6958_length__product,axiom,
    ! [Xs: list_real,Ys: list_int] : size_s8610625264895183403al_int(product_real_int(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_real(Xs)),size_size_list_int(Ys)) ).

% length_product
tff(fact_6959_length__product,axiom,
    ! [Xs: list_o,Ys: list_real] : size_s2624279037499656343o_real(product_o_real(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_o(Xs)),size_size_list_real(Ys)) ).

% length_product
tff(fact_6960_length__product,axiom,
    ! [Xs: list_o,Ys: list_o] : size_s1515746228057227161od_o_o(product_o_o(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_o(Xs)),size_size_list_o(Ys)) ).

% length_product
tff(fact_6961_length__product,axiom,
    ! [Xs: list_o,Ys: list_nat] : size_s5443766701097040955_o_nat(product_o_nat(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_o(Xs)),size_size_list_nat(Ys)) ).

% length_product
tff(fact_6962_length__product,axiom,
    ! [Xs: list_o,Ys: list_int] : size_s2953683556165314199_o_int(product_o_int(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_o(Xs)),size_size_list_int(Ys)) ).

% length_product
tff(fact_6963_length__product,axiom,
    ! [Xs: list_nat,Ys: list_real] : size_s7910714270633306959t_real(product_nat_real(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_nat(Xs)),size_size_list_real(Ys)) ).

% length_product
tff(fact_6964_length__product,axiom,
    ! [Xs: list_nat,Ys: list_o] : size_s6491369823275344609_nat_o(product_nat_o(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_nat(Xs)),size_size_list_o(Ys)) ).

% length_product
tff(fact_6965_triangle__Suc,axiom,
    ! [Nb: nat] : aa_nat_nat(nat_triangle,aa_nat_nat(suc,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(nat_triangle,Nb)),aa_nat_nat(suc,Nb)) ).

% triangle_Suc
tff(fact_6966_exhaustive__int_H_Ocases,axiom,
    ! [Xb: produc4788131091073775919nt_int] :
      ~ ! [F2: fun_in5730379987311621695e_term,D2: int,I3: int] : Xb != aa_Pro8800328608558162860nt_int(produc4729913460834754785nt_int(F2),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,D2),I3)) ).

% exhaustive_int'.cases
tff(fact_6967_full__exhaustive__int_H_Ocases,axiom,
    ! [Xb: produc3043004577601930755nt_int] :
      ~ ! [F2: fun_Pr7763680713840462443e_term,D2: int,I3: int] : Xb != aa_Pro5748361610964203648nt_int(produc7320969046608631221nt_int(F2),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,D2),I3)) ).

% full_exhaustive_int'.cases
tff(fact_6968_exhaustive__integer_H_Ocases,axiom,
    ! [Xb: produc7509114972854644512nteger] :
      ~ ! [F2: fun_Co3460408610320954864e_term,D2: code_integer,I3: code_integer] : Xb != produc7264323995540650962nteger(F2,aa_Cod6977894654539884864nteger(produc1086072967326762835nteger(D2),I3)) ).

% exhaustive_integer'.cases
tff(fact_6969_full__exhaustive__integer_H_Ocases,axiom,
    ! [Xb: produc2453742332425399446nteger] :
      ~ ! [F2: fun_Pr5098669352780829690e_term,D2: code_integer,I3: code_integer] : Xb != aa_Pro4291442876242880245nteger(produc565448451787809096nteger(F2),aa_Cod6977894654539884864nteger(produc1086072967326762835nteger(D2),I3)) ).

% full_exhaustive_integer'.cases
tff(fact_6970_pred__equals__eq2,axiom,
    ! [R: set_Pr3486312109254157132nteger,S2: set_Pr3486312109254157132nteger] :
      ( ! [X: fun_Pr5098669352780829690e_term,Xa3: produc8923325533196201883nteger] :
          ( member8978266922497061677nteger(aa_Pro4291442876242880245nteger(produc565448451787809096nteger(X),Xa3),R)
        <=> member8978266922497061677nteger(aa_Pro4291442876242880245nteger(produc565448451787809096nteger(X),Xa3),S2) )
    <=> ( R = S2 ) ) ).

% pred_equals_eq2
tff(fact_6971_pred__equals__eq2,axiom,
    ! [R: set_Pr1312406793078187350et_nat,S2: set_Pr1312406793078187350et_nat] :
      ( ! [X: fun_Pr6460944699146269935_nat_o,Xa3: produc3658429121746597890et_nat] :
          ( member4513763875858039863et_nat(aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(X),Xa3),R)
        <=> member4513763875858039863et_nat(aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(X),Xa3),S2) )
    <=> ( R = S2 ) ) ).

% pred_equals_eq2
tff(fact_6972_pred__equals__eq2,axiom,
    ! [R: set_Pr4040394263813967220et_nat,S2: set_Pr4040394263813967220et_nat] :
      ( ! [X: fun_Pr6460944699146269935_nat_o,Xa3: produc8799533983703829792et_nat] :
          ( member9174161442226468181et_nat(aa_Pro1979504083478021276et_nat(produc7291855603632349880et_nat(X),Xa3),R)
        <=> member9174161442226468181et_nat(aa_Pro1979504083478021276et_nat(produc7291855603632349880et_nat(X),Xa3),S2) )
    <=> ( R = S2 ) ) ).

% pred_equals_eq2
tff(fact_6973_pred__equals__eq2,axiom,
    ! [R: set_Pr6700895561065273913nt_int,S2: set_Pr6700895561065273913nt_int] :
      ( ! [X: fun_Pr7763680713840462443e_term,Xa3: product_prod_int_int] :
          ( member753276645541486746nt_int(aa_Pro5748361610964203648nt_int(produc7320969046608631221nt_int(X),Xa3),R)
        <=> member753276645541486746nt_int(aa_Pro5748361610964203648nt_int(produc7320969046608631221nt_int(X),Xa3),S2) )
    <=> ( R = S2 ) ) ).

% pred_equals_eq2
tff(fact_6974_pred__equals__eq2,axiom,
    ! [R: set_Pr4166680992137330533nt_int,S2: set_Pr4166680992137330533nt_int] :
      ( ! [X: fun_in5730379987311621695e_term,Xa3: product_prod_int_int] :
          ( member5890090881338784710nt_int(aa_Pro8800328608558162860nt_int(produc4729913460834754785nt_int(X),Xa3),R)
        <=> member5890090881338784710nt_int(aa_Pro8800328608558162860nt_int(produc4729913460834754785nt_int(X),Xa3),S2) )
    <=> ( R = S2 ) ) ).

% pred_equals_eq2
tff(fact_6975_subrelI,axiom,
    ! [R2: set_Pr3486312109254157132nteger,S3: set_Pr3486312109254157132nteger] :
      ( ! [X2: fun_Pr5098669352780829690e_term,Y4: produc8923325533196201883nteger] :
          ( member8978266922497061677nteger(aa_Pro4291442876242880245nteger(produc565448451787809096nteger(X2),Y4),R2)
         => member8978266922497061677nteger(aa_Pro4291442876242880245nteger(produc565448451787809096nteger(X2),Y4),S3) )
     => ord_le2932375828722672556nteger(R2,S3) ) ).

% subrelI
tff(fact_6976_subrelI,axiom,
    ! [R2: set_Pr1312406793078187350et_nat,S3: set_Pr1312406793078187350et_nat] :
      ( ! [X2: fun_Pr6460944699146269935_nat_o,Y4: produc3658429121746597890et_nat] :
          ( member4513763875858039863et_nat(aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(X2),Y4),R2)
         => member4513763875858039863et_nat(aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(X2),Y4),S3) )
     => ord_le3919382008627836342et_nat(R2,S3) ) ).

% subrelI
tff(fact_6977_subrelI,axiom,
    ! [R2: set_Pr4040394263813967220et_nat,S3: set_Pr4040394263813967220et_nat] :
      ( ! [X2: fun_Pr6460944699146269935_nat_o,Y4: produc8799533983703829792et_nat] :
          ( member9174161442226468181et_nat(aa_Pro1979504083478021276et_nat(produc7291855603632349880et_nat(X2),Y4),R2)
         => member9174161442226468181et_nat(aa_Pro1979504083478021276et_nat(produc7291855603632349880et_nat(X2),Y4),S3) )
     => ord_le5189643068899110356et_nat(R2,S3) ) ).

% subrelI
tff(fact_6978_subrelI,axiom,
    ! [R2: set_Pr6700895561065273913nt_int,S3: set_Pr6700895561065273913nt_int] :
      ( ! [X2: fun_Pr7763680713840462443e_term,Y4: product_prod_int_int] :
          ( member753276645541486746nt_int(aa_Pro5748361610964203648nt_int(produc7320969046608631221nt_int(X2),Y4),R2)
         => member753276645541486746nt_int(aa_Pro5748361610964203648nt_int(produc7320969046608631221nt_int(X2),Y4),S3) )
     => ord_le3990897393281686169nt_int(R2,S3) ) ).

% subrelI
tff(fact_6979_subrelI,axiom,
    ! [R2: set_Pr4166680992137330533nt_int,S3: set_Pr4166680992137330533nt_int] :
      ( ! [X2: fun_in5730379987311621695e_term,Y4: product_prod_int_int] :
          ( member5890090881338784710nt_int(aa_Pro8800328608558162860nt_int(produc4729913460834754785nt_int(X2),Y4),R2)
         => member5890090881338784710nt_int(aa_Pro8800328608558162860nt_int(produc4729913460834754785nt_int(X2),Y4),S3) )
     => ord_le3197784583289274309nt_int(R2,S3) ) ).

% subrelI
tff(fact_6980_top__empty__eq2,axiom,
    ! [X3: fun_Pr5098669352780829690e_term,Xa: produc8923325533196201883nteger] :
      ( aa_Pro4926521453222654329eger_o(aa_fun4505422719058378424eger_o(top_to2784585791485011495eger_o,X3),Xa)
    <=> member8978266922497061677nteger(aa_Pro4291442876242880245nteger(produc565448451787809096nteger(X3),Xa),top_to7525899545246753276nteger) ) ).

% top_empty_eq2
tff(fact_6981_top__empty__eq2,axiom,
    ! [X3: fun_Pr6460944699146269935_nat_o,Xa: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(aa_fun2638059832795814268_nat_o(top_to2712223222597888051_nat_o,X3),Xa)
    <=> member4513763875858039863et_nat(aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(X3),Xa),top_to2987642378616949766et_nat) ) ).

% top_empty_eq2
tff(fact_6982_top__empty__eq2,axiom,
    ! [X3: fun_Pr6460944699146269935_nat_o,Xa: produc8799533983703829792et_nat] :
      ( aa_Pro5643009357358181178_nat_o(aa_fun912680268133416158_nat_o(top_to3067105749023872149_nat_o,X3),Xa)
    <=> member9174161442226468181et_nat(aa_Pro1979504083478021276et_nat(produc7291855603632349880et_nat(X3),Xa),top_to2648389616742571044et_nat) ) ).

% top_empty_eq2
tff(fact_6983_top__empty__eq2,axiom,
    ! [X3: fun_Pr7763680713840462443e_term,Xa: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aa_fun3538162476766722827_int_o(top_to1218662295448195450_int_o,X3),Xa)
    <=> member753276645541486746nt_int(aa_Pro5748361610964203648nt_int(produc7320969046608631221nt_int(X3),Xa),top_to6107225058432391401nt_int) ) ).

% top_empty_eq2
tff(fact_6984_top__empty__eq2,axiom,
    ! [X3: fun_in5730379987311621695e_term,Xa: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aa_fun586359091965693151_int_o(top_to1089409542825796430_int_o,X3),Xa)
    <=> member5890090881338784710nt_int(aa_Pro8800328608558162860nt_int(produc4729913460834754785nt_int(X3),Xa),top_to1212741241995312661nt_int) ) ).

% top_empty_eq2
tff(fact_6985_pred__subset__eq,axiom,
    ! [R: set_nat,S2: set_nat] :
      ( ord_le7240017405068453697_nat_o(aTP_Lamm_ax(R),aTP_Lamm_ax(S2))
    <=> aa_set_nat_o(ord_less_eq_set_nat(R),S2) ) ).

% pred_subset_eq
tff(fact_6986_pred__subset__eq,axiom,
    ! [R: set_VEBT_VEBT,S2: set_VEBT_VEBT] :
      ( ord_le88303844413636359VEBT_o(aTP_Lamm_av(R),aTP_Lamm_av(S2))
    <=> aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(R),S2) ) ).

% pred_subset_eq
tff(fact_6987_pred__subset__eq,axiom,
    ! [R: set_real,S2: set_real] :
      ( ord_le3605727316730466277real_o(aTP_Lamm_aw(R),aTP_Lamm_aw(S2))
    <=> aa_set_real_o(ord_less_eq_set_real(R),S2) ) ).

% pred_subset_eq
tff(fact_6988_pred__subset__eq,axiom,
    ! [R: set_int,S2: set_int] :
      ( ord_le5418975059635765733_int_o(aTP_Lamm_ay(R),aTP_Lamm_ay(S2))
    <=> aa_set_int_o(ord_less_eq_set_int(R),S2) ) ).

% pred_subset_eq
tff(fact_6989_exE__realizer,axiom,
    ! [P: fun_nat_fun_nat_o,P2: product_prod_nat_nat,Q: fun_Pr4796943543331071514_nat_o,F: fun_na4138050610385705875at_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(P,aa_Pro4243173770044397973at_nat(product_snd_nat_nat,P2)),aa_Pro4243173770044397973at_nat(product_fst_nat_nat,P2))
     => ( ! [X2: nat,Y4: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(P,Y4),X2)
           => aa_Pro6150637604445964947_nat_o(Q,aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(F,X2),Y4)) )
       => aa_Pro6150637604445964947_nat_o(Q,aa_Pro262675715343573244at_nat(produc2626176000494625587at_nat(F),P2)) ) ) ).

% exE_realizer
tff(fact_6990_exE__realizer,axiom,
    ! [P: fun_nat_fun_nat_o,P2: product_prod_nat_nat,Q: fun_o_o,F: fun_nat_fun_nat_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(P,aa_Pro4243173770044397973at_nat(product_snd_nat_nat,P2)),aa_Pro4243173770044397973at_nat(product_fst_nat_nat,P2))
     => ( ! [X2: nat,Y4: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(P,Y4),X2)
           => aa_o_o(Q,aa_nat_o(aa_nat_fun_nat_o(F,X2),Y4)) )
       => aa_o_o(Q,aa_Pro6150637604445964947_nat_o(produc6081775807080527818_nat_o(F),P2)) ) ) ).

% exE_realizer
tff(fact_6991_exE__realizer,axiom,
    ! [P: fun_int_fun_int_o,P2: product_prod_int_int,Q: fun_Pr3238374608701176162_int_o,F: fun_in1696471567231710611nt_int] :
      ( aa_int_o(aa_int_fun_int_o(P,aa_Pro6799258434465214121nt_int(product_snd_int_int,P2)),aa_Pro6799258434465214121nt_int(product_fst_int_int,P2))
     => ( ! [X2: int,Y4: int] :
            ( aa_int_o(aa_int_fun_int_o(P,Y4),X2)
           => aa_Pro4592068669816069595_int_o(Q,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(F,X2),Y4)) )
       => aa_Pro4592068669816069595_int_o(Q,aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(F),P2)) ) ) ).

% exE_realizer
tff(fact_6992_exE__realizer,axiom,
    ! [P: fun_int_fun_int_o,P2: product_prod_int_int,Q: fun_o_o,F: fun_int_fun_int_o] :
      ( aa_int_o(aa_int_fun_int_o(P,aa_Pro6799258434465214121nt_int(product_snd_int_int,P2)),aa_Pro6799258434465214121nt_int(product_fst_int_int,P2))
     => ( ! [X2: int,Y4: int] :
            ( aa_int_o(aa_int_fun_int_o(P,Y4),X2)
           => aa_o_o(Q,aa_int_o(aa_int_fun_int_o(F,X2),Y4)) )
       => aa_o_o(Q,aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(F),P2)) ) ) ).

% exE_realizer
tff(fact_6993_exE__realizer,axiom,
    ! [P: fun_int_fun_int_o,P2: product_prod_int_int,Q: fun_int_o,F: fun_int_fun_int_int] :
      ( aa_int_o(aa_int_fun_int_o(P,aa_Pro6799258434465214121nt_int(product_snd_int_int,P2)),aa_Pro6799258434465214121nt_int(product_fst_int_int,P2))
     => ( ! [X2: int,Y4: int] :
            ( aa_int_o(aa_int_fun_int_o(P,Y4),X2)
           => aa_int_o(Q,aa_int_int(aa_int_fun_int_int(F,X2),Y4)) )
       => aa_int_o(Q,aa_Pro6799258434465214121nt_int(produc8211389475949308722nt_int(F),P2)) ) ) ).

% exE_realizer
tff(fact_6994_vebt__assn__raw_Oelims,axiom,
    ! [Xb: vEBT_VEBT,Xaa: vEBT_VEBTi,Y: assn] :
      ( ( aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Xb),Xaa) = Y )
     => ( ! [A3: $o,B2: $o] :
            ( ( Xb = vEBT_Leaf((A3),(B2)) )
           => ! [Ai: $o,Bi: $o] :
                ( ( Xaa = vEBT_Leafi((Ai),(Bi)) )
               => ( Y != aa_o_assn(pure_assn,
                      ( ( (Ai) = (A3) )
                      & ( (Bi) = (B2) ) )) ) ) )
       => ( ! [Mmo2: option4927543243414619207at_nat,Deg2: nat,Tree_list2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
              ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Mmo2,Deg2,Tree_list2),Summary) )
             => ! [Mmoi2: option4927543243414619207at_nat,Degi2: nat,Tree_array2: array_VEBT_VEBTi,Summaryi2: vEBT_VEBTi] :
                  ( ( Xaa = aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(Mmoi2,Degi2,Tree_array2),Summaryi2) )
                 => ( Y != aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,
                          aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,
                              aa_o_assn(pure_assn,
                                ( ( Mmoi2 = Mmo2 )
                                & ( Degi2 = Deg2 ) ))),aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Summary),Summaryi2))),ex_ass463751140784270563_VEBTi(aa_arr7909941341466502432i_assn(aTP_Lamm_bf(Tree_list2),Tree_array2))) ) ) )
         => ( ( ? [V2: option4927543243414619207at_nat,Va2: nat,Vb3: list_VEBT_VEBT,Vc3: vEBT_VEBT] : Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(V2,Va2,Vb3),Vc3)
             => ( ? [Vd3: $o,Ve3: $o] : Xaa = vEBT_Leafi((Vd3),(Ve3))
               => ( Y != bot_bot_assn ) ) )
           => ~ ( ? [Vd3: $o,Ve3: $o] : Xb = vEBT_Leaf((Vd3),(Ve3))
               => ( ? [V2: option4927543243414619207at_nat,Va2: nat,Vb3: array_VEBT_VEBTi,Vc3: vEBT_VEBTi] : Xaa = aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(V2,Va2,Vb3),Vc3)
                 => ( Y != bot_bot_assn ) ) ) ) ) ) ) ).

% vebt_assn_raw.elims
tff(fact_6995_in__set__enumerate__eq,axiom,
    ! [P2: produc8025551001238799321T_VEBT,Nb: nat,Xs: list_VEBT_VEBT] :
      ( member8549952807677709168T_VEBT(P2,set_Pr5984661752051438084T_VEBT(enumerate_VEBT_VEBT(Nb,Xs)))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),produc8575180428842422559T_VEBT(P2))
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,produc8575180428842422559T_VEBT(P2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,size_s6755466524823107622T_VEBT(Xs)),Nb))
        & ( nth_VEBT_VEBT(Xs,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,produc8575180428842422559T_VEBT(P2)),Nb)) = aa_Pro5446849830759474909T_VEBT(produc8172668247895388509T_VEBT,P2) ) ) ) ).

% in_set_enumerate_eq
tff(fact_6996_in__set__enumerate__eq,axiom,
    ! [P2: produc214224863196444774_VEBTi,Nb: nat,Xs: list_VEBT_VEBTi] :
      ( member763447850065367567_VEBTi(P2,set_Pr4207466110102731387_VEBTi(enumerate_VEBT_VEBTi(Nb,Xs)))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),produc8252055991070844170_VEBTi(P2))
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,produc8252055991070844170_VEBTi(P2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,size_s7982070591426661849_VEBTi(Xs)),Nb))
        & ( nth_VEBT_VEBTi(Xs,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,produc8252055991070844170_VEBTi(P2)),Nb)) = produc271786961351840588_VEBTi(P2) ) ) ) ).

% in_set_enumerate_eq
tff(fact_6997_in__set__enumerate__eq,axiom,
    ! [P2: produc7716430852924023517t_real,Nb: nat,Xs: list_real] :
      ( member557208447399453958t_real(P2,set_Pr7149346036329476978t_real(enumerate_real(Nb,Xs)))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),product_fst_nat_real(P2))
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,product_fst_nat_real(P2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,size_size_list_real(Xs)),Nb))
        & ( aa_nat_real(nth_real(Xs),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,product_fst_nat_real(P2)),Nb)) = product_snd_nat_real(P2) ) ) ) ).

% in_set_enumerate_eq
tff(fact_6998_in__set__enumerate__eq,axiom,
    ! [P2: product_prod_nat_o,Nb: nat,Xs: list_o] :
      ( member6310962623043647828_nat_o(P2,set_Pr1291962091234853352_nat_o(enumerate_o(Nb,Xs)))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),product_fst_nat_o(P2))
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,product_fst_nat_o(P2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,size_size_list_o(Xs)),Nb))
        & ( aa_nat_o(nth_o(Xs),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,product_fst_nat_o(P2)),Nb))
        <=> product_snd_nat_o(P2) ) ) ) ).

% in_set_enumerate_eq
tff(fact_6999_in__set__enumerate__eq,axiom,
    ! [P2: product_prod_nat_int,Nb: nat,Xs: list_int] :
      ( member4262671552274231302at_int(P2,set_Pr1470767568048878706at_int(enumerate_int(Nb,Xs)))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),product_fst_nat_int(P2))
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,product_fst_nat_int(P2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,size_size_list_int(Xs)),Nb))
        & ( aa_nat_int(nth_int(Xs),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,product_fst_nat_int(P2)),Nb)) = product_snd_nat_int(P2) ) ) ) ).

% in_set_enumerate_eq
tff(fact_7000_in__set__enumerate__eq,axiom,
    ! [P2: product_prod_nat_nat,Nb: nat,Xs: list_nat] :
      ( member8440522571783428010at_nat(P2,set_Pr5648618587558075414at_nat(enumerate_nat(Nb,Xs)))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),aa_Pro4243173770044397973at_nat(product_fst_nat_nat,P2))
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_Pro4243173770044397973at_nat(product_fst_nat_nat,P2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,size_size_list_nat(Xs)),Nb))
        & ( aa_nat_nat(nth_nat(Xs),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_Pro4243173770044397973at_nat(product_fst_nat_nat,P2)),Nb)) = aa_Pro4243173770044397973at_nat(product_snd_nat_nat,P2) ) ) ) ).

% in_set_enumerate_eq
tff(fact_7001_obtain__set__pred,axiom,
    ! [Z: nat,Xb: nat,A2: set_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z),Xb)
     => ( aa_nat_o(vEBT_VEBT_min_in_set(A2),Z)
       => ( aa_set_nat_o(finite_finite_nat,A2)
         => ? [X_13: nat] : aa_nat_o(aa_nat_fun_nat_o(vEBT_is_pred_in_set(A2),Xb),X_13) ) ) ) ).

% obtain_set_pred
tff(fact_7002_obtain__set__succ,axiom,
    ! [Xb: nat,Z: nat,A2: set_nat,B4: set_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Z)
     => ( aa_nat_o(vEBT_VEBT_max_in_set(A2),Z)
       => ( aa_set_nat_o(finite_finite_nat,B4)
         => ( ( A2 = B4 )
           => ? [X_13: nat] : aa_nat_o(aa_nat_fun_nat_o(vEBT_is_succ_in_set(A2),Xb),X_13) ) ) ) ) ).

% obtain_set_succ
tff(fact_7003_set__vebt__finite,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Ta),Nb)
     => aa_set_nat_o(finite_finite_nat,vEBT_VEBT_set_vebt(Ta)) ) ).

% set_vebt_finite
tff(fact_7004_succ__none__empty,axiom,
    ! [Xs: set_nat,A: nat] :
      ( ~ ? [X_13: nat] : aa_nat_o(aa_nat_fun_nat_o(vEBT_is_succ_in_set(Xs),A),X_13)
     => ( aa_set_nat_o(finite_finite_nat,Xs)
       => ~ ? [X3: nat] :
              ( aa_set_nat_o(member_nat(X3),Xs)
              & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),X3) ) ) ) ).

% succ_none_empty
tff(fact_7005_pred__none__empty,axiom,
    ! [Xs: set_nat,A: nat] :
      ( ~ ? [X_13: nat] : aa_nat_o(aa_nat_fun_nat_o(vEBT_is_pred_in_set(Xs),A),X_13)
     => ( aa_set_nat_o(finite_finite_nat,Xs)
       => ~ ? [X3: nat] :
              ( aa_set_nat_o(member_nat(X3),Xs)
              & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),A) ) ) ) ).

% pred_none_empty
tff(fact_7006_finite__option__UNIV,axiom,
    ( finite1345302120164226195on_int(top_to6430115241214627170on_int)
  <=> aa_set_int_o(finite_finite_int,top_top_set_int) ) ).

% finite_option_UNIV
tff(fact_7007_finite__option__UNIV,axiom,
    ( finite2569390945932476949omplex(top_to6180147692022559204omplex)
  <=> aa_set_complex_o(finite3207457112153483333omplex,top_top_set_complex) ) ).

% finite_option_UNIV
tff(fact_7008_finite__option__UNIV,axiom,
    ( finite6785661671136154180nteger(top_to5929521628599800467nteger)
  <=> aa_set5643387332196575111eger_o(finite6017078050557962740nteger,top_to4645266643341252675nteger) ) ).

% finite_option_UNIV
tff(fact_7009_finite__option__UNIV,axiom,
    ( finite5731380839103853331n_real(top_to853713521313446370n_real)
  <=> aa_set_real_o(finite_finite_real,top_top_set_real) ) ).

% finite_option_UNIV
tff(fact_7010_finite__option__UNIV,axiom,
    ( finite5523153139673422903on_nat(top_to8920198386146353926on_nat)
  <=> aa_set_nat_o(finite_finite_nat,top_top_set_nat) ) ).

% finite_option_UNIV
tff(fact_7011_finite__option__UNIV,axiom,
    ( finite2053776701332276152n_char(top_to576892095687177735n_char)
  <=> finite_finite_char(top_top_set_char) ) ).

% finite_option_UNIV
tff(fact_7012_List_Ofinite__set,axiom,
    ! [Xs: list_VEBT_VEBT] : aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,set_VEBT_VEBT2(Xs)) ).

% List.finite_set
tff(fact_7013_List_Ofinite__set,axiom,
    ! [Xs: list_real] : aa_set_real_o(finite_finite_real,set_real2(Xs)) ).

% List.finite_set
tff(fact_7014_List_Ofinite__set,axiom,
    ! [Xs: list_P8527749157015355191n_assn] : finite5137929494490007386n_assn(set_Pr1139785259514867910n_assn(Xs)) ).

% List.finite_set
tff(fact_7015_List_Ofinite__set,axiom,
    ! [Xs: list_nat] : aa_set_nat_o(finite_finite_nat,set_nat2(Xs)) ).

% List.finite_set
tff(fact_7016_List_Ofinite__set,axiom,
    ! [Xs: list_int] : aa_set_int_o(finite_finite_int,set_int2(Xs)) ).

% List.finite_set
tff(fact_7017_List_Ofinite__set,axiom,
    ! [Xs: list_complex] : aa_set_complex_o(finite3207457112153483333omplex,set_complex2(Xs)) ).

% List.finite_set
tff(fact_7018_List_Ofinite__set,axiom,
    ! [Xs: list_Code_integer] : aa_set5643387332196575111eger_o(finite6017078050557962740nteger,set_Code_integer2(Xs)) ).

% List.finite_set
tff(fact_7019_star__false__right,axiom,
    ! [P: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,P),bot_bot_assn) = bot_bot_assn ).

% star_false_right
tff(fact_7020_star__false__left,axiom,
    ! [P: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,bot_bot_assn),P) = bot_bot_assn ).

% star_false_left
tff(fact_7021_false__rule,axiom,
    ! [C: heap_T2636463487746394924on_nat,Q: fun_option_nat_assn] : hoare_7629718768684598413on_nat(bot_bot_assn,C,Q) ).

% false_rule
tff(fact_7022_false__rule,axiom,
    ! [C: heap_Time_Heap_nat,Q: fun_nat_assn] : hoare_3067605981109127869le_nat(bot_bot_assn,C,Q) ).

% false_rule
tff(fact_7023_false__rule,axiom,
    ! [C: heap_Time_Heap_o,Q: fun_o_assn] : hoare_hoare_triple_o(bot_bot_assn,C,Q) ).

% false_rule
tff(fact_7024_false__rule,axiom,
    ! [C: heap_T8145700208782473153_VEBTi,Q: fun_VEBT_VEBTi_assn] : hoare_1429296392585015714_VEBTi(bot_bot_assn,C,Q) ).

% false_rule
tff(fact_7025_false__rule,axiom,
    ! [C: heap_T5738788834812785303t_unit,Q: fun_Pr6626665806141890376t_assn] : hoare_8945653483474564448t_unit(bot_bot_assn,C,Q) ).

% false_rule
tff(fact_7026_length__enumerate,axiom,
    ! [Nb: nat,Xs: list_real] : size_s7910714270633306959t_real(enumerate_real(Nb,Xs)) = size_size_list_real(Xs) ).

% length_enumerate
tff(fact_7027_length__enumerate,axiom,
    ! [Nb: nat,Xs: list_o] : size_s6491369823275344609_nat_o(enumerate_o(Nb,Xs)) = size_size_list_o(Xs) ).

% length_enumerate
tff(fact_7028_length__enumerate,axiom,
    ! [Nb: nat,Xs: list_nat] : size_s5460976970255530739at_nat(enumerate_nat(Nb,Xs)) = size_size_list_nat(Xs) ).

% length_enumerate
tff(fact_7029_length__enumerate,axiom,
    ! [Nb: nat,Xs: list_int] : size_s2970893825323803983at_int(enumerate_int(Nb,Xs)) = size_size_list_int(Xs) ).

% length_enumerate
tff(fact_7030_pure__assn__eq__false__iff,axiom,
    ! [P: $o] :
      ( ( aa_o_assn(pure_assn,(P)) = bot_bot_assn )
    <=> ~ (P) ) ).

% pure_assn_eq_false_iff
tff(fact_7031_pure__false,axiom,
    aa_o_assn(pure_assn,$false) = bot_bot_assn ).

% pure_false
tff(fact_7032_assn__basic__inequalities_I3_J,axiom,
    bot_bot_assn != one_one_assn ).

% assn_basic_inequalities(3)
tff(fact_7033_map__snd__enumerate,axiom,
    ! [Nb: nat,Xs: list_nat] : map_Pr3938374229010428429at_nat(product_snd_nat_nat,enumerate_nat(Nb,Xs)) = Xs ).

% map_snd_enumerate
tff(fact_7034_infinite__Icc__iff,axiom,
    ! [A: rat,B: rat] :
      ( ~ aa_set_rat_o(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_7035_infinite__Icc__iff,axiom,
    ! [A: real,B: real] :
      ( ~ aa_set_real_o(finite_finite_real,set_or1222579329274155063t_real(A,B))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) ) ).

% infinite_Icc_iff
tff(fact_7036_snga__same__false,axiom,
    ! [P2: array_VEBT_VEBTi,Xb: list_VEBT_VEBTi,Y: list_VEBT_VEBTi] : aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_lis2508312872377092907i_assn(snga_assn_VEBT_VEBTi(P2),Xb)),aa_lis2508312872377092907i_assn(snga_assn_VEBT_VEBTi(P2),Y)) = bot_bot_assn ).

% snga_same_false
tff(fact_7037_finite__if__eq__beyond__finite,axiom,
    ! [S2: set_int,S4: set_int] :
      ( aa_set_int_o(finite_finite_int,S2)
     => finite6197958912794628473et_int(collect_set_int(aa_set5425576183772390127_int_o(aTP_Lamm_acv(S2),S4))) ) ).

% finite_if_eq_beyond_finite
tff(fact_7038_finite__if__eq__beyond__finite,axiom,
    ! [S2: set_complex,S4: set_complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S2)
     => finite6551019134538273531omplex(collect_set_complex(aa_set8643480048548153067plex_o(aTP_Lamm_acw(S2),S4))) ) ).

% finite_if_eq_beyond_finite
tff(fact_7039_finite__if__eq__beyond__finite,axiom,
    ! [S2: set_Code_integer,S4: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,S2)
     => finite6931041176100689706nteger(collec574505750873337192nteger(aa_set4044230130517860621eger_o(aTP_Lamm_acx(S2),S4))) ) ).

% finite_if_eq_beyond_finite
tff(fact_7040_finite__if__eq__beyond__finite,axiom,
    ! [S2: set_nat,S4: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,S2)
     => finite1152437895449049373et_nat(collect_set_nat(aa_set7861427066675100071_nat_o(aTP_Lamm_acy(S2),S4))) ) ).

% finite_if_eq_beyond_finite
tff(fact_7041_finite__nat__set__iff__bounded,axiom,
    ! [N: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,N)
    <=> ? [M7: nat] :
        ! [X: nat] :
          ( aa_set_nat_o(member_nat(X),N)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),M7) ) ) ).

% finite_nat_set_iff_bounded
tff(fact_7042_bounded__nat__set__is__finite,axiom,
    ! [N: set_nat,Nb: nat] :
      ( ! [X2: nat] :
          ( aa_set_nat_o(member_nat(X2),N)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X2),Nb) )
     => aa_set_nat_o(finite_finite_nat,N) ) ).

% bounded_nat_set_is_finite
tff(fact_7043_finite__nat__set__iff__bounded__le,axiom,
    ! [N: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,N)
    <=> ? [M7: nat] :
        ! [X: nat] :
          ( aa_set_nat_o(member_nat(X),N)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),M7) ) ) ).

% finite_nat_set_iff_bounded_le
tff(fact_7044_finite__list,axiom,
    ! [A2: set_VEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,A2)
     => ? [Xs2: list_VEBT_VEBT] : set_VEBT_VEBT2(Xs2) = A2 ) ).

% finite_list
tff(fact_7045_finite__list,axiom,
    ! [A2: set_real] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ? [Xs2: list_real] : set_real2(Xs2) = A2 ) ).

% finite_list
tff(fact_7046_finite__list,axiom,
    ! [A2: set_Pr5949110396991348497n_assn] :
      ( finite5137929494490007386n_assn(A2)
     => ? [Xs2: list_P8527749157015355191n_assn] : set_Pr1139785259514867910n_assn(Xs2) = A2 ) ).

% finite_list
tff(fact_7047_finite__list,axiom,
    ! [A2: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ? [Xs2: list_nat] : set_nat2(Xs2) = A2 ) ).

% finite_list
tff(fact_7048_finite__list,axiom,
    ! [A2: set_int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ? [Xs2: list_int] : set_int2(Xs2) = A2 ) ).

% finite_list
tff(fact_7049_finite__list,axiom,
    ! [A2: set_complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ? [Xs2: list_complex] : set_complex2(Xs2) = A2 ) ).

% finite_list
tff(fact_7050_finite__list,axiom,
    ! [A2: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A2)
     => ? [Xs2: list_Code_integer] : set_Code_integer2(Xs2) = A2 ) ).

% finite_list
tff(fact_7051_finite__M__bounded__by__nat,axiom,
    ! [P: fun_nat_o,I: nat] : aa_set_nat_o(finite_finite_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_acz(P),I))) ).

% finite_M_bounded_by_nat
tff(fact_7052_finite__less__ub,axiom,
    ! [F: fun_nat_nat,U: nat] :
      ( ! [N2: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),aa_nat_nat(F,N2))
     => aa_set_nat_o(finite_finite_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_ada(F),U))) ) ).

% finite_less_ub
tff(fact_7053_finite__lists__length__eq,axiom,
    ! [A2: set_VEBT_VEBT,Nb: nat] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,A2)
     => finite3004134309566078307T_VEBT(collec5608196760682091941T_VEBT(aa_nat1274389614848952367VEBT_o(aTP_Lamm_adb(A2),Nb))) ) ).

% finite_lists_length_eq
tff(fact_7054_finite__lists__length__eq,axiom,
    ! [A2: set_Pr5949110396991348497n_assn,Nb: nat] :
      ( finite5137929494490007386n_assn(A2)
     => finite1351478129840809056n_assn(collec8177951099088521122n_assn(aa_nat2893684320935325170assn_o(aTP_Lamm_adc(A2),Nb))) ) ).

% finite_lists_length_eq
tff(fact_7055_finite__lists__length__eq,axiom,
    ! [A2: set_complex,Nb: nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => finite8712137658972009173omplex(collect_list_complex(aa_nat5940423528545770789plex_o(aTP_Lamm_add(A2),Nb))) ) ).

% finite_lists_length_eq
tff(fact_7056_finite__lists__length__eq,axiom,
    ! [A2: set_Code_integer,Nb: nat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A2)
     => finite1283093830868386564nteger(collec3483841146883906114nteger(aa_nat2697030861539126eger_o(aTP_Lamm_ade(A2),Nb))) ) ).

% finite_lists_length_eq
tff(fact_7057_finite__lists__length__eq,axiom,
    ! [A2: set_real,Nb: nat] :
      ( aa_set_real_o(finite_finite_real,A2)
     => finite306553202115118035t_real(collect_list_real(aa_nat3629845878562903719real_o(aTP_Lamm_adf(A2),Nb))) ) ).

% finite_lists_length_eq
tff(fact_7058_finite__lists__length__eq,axiom,
    ! [A2: set_o,Nb: nat] :
      ( finite_finite_o(A2)
     => finite_finite_list_o(collect_list_o(aa_nat_fun_list_o_o(aTP_Lamm_adg(A2),Nb))) ) ).

% finite_lists_length_eq
tff(fact_7059_finite__lists__length__eq,axiom,
    ! [A2: set_nat,Nb: nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => finite8100373058378681591st_nat(collect_list_nat(aa_nat5883842100386203267_nat_o(aTP_Lamm_adh(A2),Nb))) ) ).

% finite_lists_length_eq
tff(fact_7060_finite__lists__length__eq,axiom,
    ! [A2: set_int,Nb: nat] :
      ( aa_set_int_o(finite_finite_int,A2)
     => finite3922522038869484883st_int(collect_list_int(aa_nat3638697132715757351_int_o(aTP_Lamm_adi(A2),Nb))) ) ).

% finite_lists_length_eq
tff(fact_7061_infinite__Icc,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ~ aa_set_rat_o(finite_finite_rat,set_or633870826150836451st_rat(A,B)) ) ).

% infinite_Icc
tff(fact_7062_infinite__Icc,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ~ aa_set_real_o(finite_finite_real,set_or1222579329274155063t_real(A,B)) ) ).

% infinite_Icc
tff(fact_7063_finite__lists__length__le,axiom,
    ! [A2: set_VEBT_VEBT,Nb: nat] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,A2)
     => finite3004134309566078307T_VEBT(collec5608196760682091941T_VEBT(aa_nat1274389614848952367VEBT_o(aTP_Lamm_adj(A2),Nb))) ) ).

% finite_lists_length_le
tff(fact_7064_finite__lists__length__le,axiom,
    ! [A2: set_Pr5949110396991348497n_assn,Nb: nat] :
      ( finite5137929494490007386n_assn(A2)
     => finite1351478129840809056n_assn(collec8177951099088521122n_assn(aa_nat2893684320935325170assn_o(aTP_Lamm_adk(A2),Nb))) ) ).

% finite_lists_length_le
tff(fact_7065_finite__lists__length__le,axiom,
    ! [A2: set_complex,Nb: nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => finite8712137658972009173omplex(collect_list_complex(aa_nat5940423528545770789plex_o(aTP_Lamm_adl(A2),Nb))) ) ).

% finite_lists_length_le
tff(fact_7066_finite__lists__length__le,axiom,
    ! [A2: set_Code_integer,Nb: nat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A2)
     => finite1283093830868386564nteger(collec3483841146883906114nteger(aa_nat2697030861539126eger_o(aTP_Lamm_adm(A2),Nb))) ) ).

% finite_lists_length_le
tff(fact_7067_finite__lists__length__le,axiom,
    ! [A2: set_real,Nb: nat] :
      ( aa_set_real_o(finite_finite_real,A2)
     => finite306553202115118035t_real(collect_list_real(aa_nat3629845878562903719real_o(aTP_Lamm_adn(A2),Nb))) ) ).

% finite_lists_length_le
tff(fact_7068_finite__lists__length__le,axiom,
    ! [A2: set_o,Nb: nat] :
      ( finite_finite_o(A2)
     => finite_finite_list_o(collect_list_o(aa_nat_fun_list_o_o(aTP_Lamm_ado(A2),Nb))) ) ).

% finite_lists_length_le
tff(fact_7069_finite__lists__length__le,axiom,
    ! [A2: set_nat,Nb: nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => finite8100373058378681591st_nat(collect_list_nat(aa_nat5883842100386203267_nat_o(aTP_Lamm_adp(A2),Nb))) ) ).

% finite_lists_length_le
tff(fact_7070_finite__lists__length__le,axiom,
    ! [A2: set_int,Nb: nat] :
      ( aa_set_int_o(finite_finite_int,A2)
     => finite3922522038869484883st_int(collect_list_int(aa_nat3638697132715757351_int_o(aTP_Lamm_adq(A2),Nb))) ) ).

% finite_lists_length_le
tff(fact_7071_vebt__assn__raw_Osimps_I3_J,axiom,
    ! [V: option4927543243414619207at_nat,Vaa: nat,Vb: list_VEBT_VEBT,Vc2: vEBT_VEBT,Vd: $o,Ve: $o] : aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,aa_VEB5832953974749077948T_VEBT(vEBT_Node(V,Vaa,Vb),Vc2)),vEBT_Leafi((Vd),(Ve))) = bot_bot_assn ).

% vebt_assn_raw.simps(3)
tff(fact_7072_vebt__assn__raw_Osimps_I4_J,axiom,
    ! [Vd: $o,Ve: $o,V: option4927543243414619207at_nat,Vaa: nat,Vb: array_VEBT_VEBTi,Vc2: vEBT_VEBTi] : aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,vEBT_Leaf((Vd),(Ve))),aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(V,Vaa,Vb),Vc2)) = bot_bot_assn ).

% vebt_assn_raw.simps(4)
tff(fact_7073_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_wq,M))) ) ).

% finite_divisors_nat
tff(fact_7074_subset__eq__atLeast0__atMost__finite,axiom,
    ! [N: set_nat,Nb: nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(N),set_or1269000886237332187st_nat(zero_zero_nat,Nb))
     => aa_set_nat_o(finite_finite_nat,N) ) ).

% subset_eq_atLeast0_atMost_finite
tff(fact_7075_finite__roots__unity,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),Nb)
     => aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aa_nat_fun_complex_o(aTP_Lamm_adr,Nb))) ) ).

% finite_roots_unity
tff(fact_7076_finite__roots__unity,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),Nb)
     => aa_set_real_o(finite_finite_real,collect_real(aa_nat_fun_real_o(aTP_Lamm_ads,Nb))) ) ).

% finite_roots_unity
tff(fact_7077_exE__realizer_H,axiom,
    ! [P: fun_ui3478071520421546738nt32_o,P2: produc827990862158126777uint32] :
      ( aa_uint32_o(aa_uin5924812838015166427nt32_o(P,aa_Pro3737690986720249065uint32(produc1510406741064981791uint32,P2)),aa_Pro3737690986720249065uint32(produc9004433772639906525uint32,P2))
     => ~ ! [X2: uint32,Y4: uint32] : ~ aa_uint32_o(aa_uin5924812838015166427nt32_o(P,Y4),X2) ) ).

% exE_realizer'
tff(fact_7078_exE__realizer_H,axiom,
    ! [P: fun_nat_fun_nat_o,P2: product_prod_nat_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(P,aa_Pro4243173770044397973at_nat(product_snd_nat_nat,P2)),aa_Pro4243173770044397973at_nat(product_fst_nat_nat,P2))
     => ~ ! [X2: nat,Y4: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(P,Y4),X2) ) ).

% exE_realizer'
tff(fact_7079_exE__realizer_H,axiom,
    ! [P: fun_se9038067516210351697unit_o,P2: produc3658429121746597890et_nat] :
      ( aa_hea7994211679605476167unit_o(aa_set6435858302011484818unit_o(P,aa_Pro3416150927818442182et_nat(produc8586169260539613262et_nat,P2)),aa_Pro1219312880135979149t_unit(produc1824681642469235216et_nat,P2))
     => ~ ! [X2: heap_e7401611519738050253t_unit,Y4: set_nat] : ~ aa_hea7994211679605476167unit_o(aa_set6435858302011484818unit_o(P,Y4),X2) ) ).

% exE_realizer'
tff(fact_7080_exE__realizer_H,axiom,
    ! [P: fun_int_fun_int_o,P2: product_prod_int_int] :
      ( aa_int_o(aa_int_fun_int_o(P,aa_Pro6799258434465214121nt_int(product_snd_int_int,P2)),aa_Pro6799258434465214121nt_int(product_fst_int_int,P2))
     => ~ ! [X2: int,Y4: int] : ~ aa_int_o(aa_int_fun_int_o(P,Y4),X2) ) ).

% exE_realizer'
tff(fact_7081_exE__realizer_H,axiom,
    ! [P: fun_Co2081214437782444048eger_o,P2: produc8923325533196201883nteger] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(P,aa_Pro5245656335456409788nteger(produc6174133586879617921nteger,P2)),aa_Pro5245656335456409788nteger(produc8508995932063986495nteger,P2))
     => ~ ! [X2: code_integer,Y4: code_integer] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(P,Y4),X2) ) ).

% exE_realizer'
tff(fact_7082_exE__realizer_H,axiom,
    ! [P: fun_assn_fun_assn_o,P2: produc6575502325842934193n_assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(P,aa_Pro2979818503705714077n_assn(produc2051961928117032727n_assn,P2)),aa_Pro2979818503705714077n_assn(produc9167289414957590229n_assn,P2))
     => ~ ! [X2: assn,Y4: assn] : ~ aa_assn_o(aa_assn_fun_assn_o(P,Y4),X2) ) ).

% exE_realizer'
tff(fact_7083_nth__enumerate__eq,axiom,
    ! [M: nat,Xs: list_VEBT_VEBT,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),size_s6755466524823107622T_VEBT(Xs))
     => ( nth_Pr744662078594809490T_VEBT(enumerate_VEBT_VEBT(Nb,Xs),M) = produc599794634098209291T_VEBT(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),M),nth_VEBT_VEBT(Xs,M)) ) ) ).

% nth_enumerate_eq
tff(fact_7084_nth__enumerate__eq,axiom,
    ! [M: nat,Xs: list_VEBT_VEBTi,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),size_s7982070591426661849_VEBTi(Xs))
     => ( nth_Pr3244165891152107629_VEBTi(enumerate_VEBT_VEBTi(Nb,Xs),M) = produc2649746096677893406_VEBTi(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),M),nth_VEBT_VEBTi(Xs,M)) ) ) ).

% nth_enumerate_eq
tff(fact_7085_nth__enumerate__eq,axiom,
    ! [M: nat,Xs: list_real,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),size_size_list_real(Xs))
     => ( nth_Pr7767817059697521252t_real(enumerate_real(Nb,Xs),M) = produc7837566107596912789t_real(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),M),aa_nat_real(nth_real(Xs),M)) ) ) ).

% nth_enumerate_eq
tff(fact_7086_nth__enumerate__eq,axiom,
    ! [M: nat,Xs: list_o,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),size_size_list_o(Xs))
     => ( nth_Pr112076138515278198_nat_o(enumerate_o(Nb,Xs),M) = product_Pair_nat_o(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),M),aa_nat_o(nth_o(Xs),M)) ) ) ).

% nth_enumerate_eq
tff(fact_7087_nth__enumerate__eq,axiom,
    ! [M: nat,Xs: list_nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),size_size_list_nat(Xs))
     => ( aa_nat7603221163087113075at_nat(nth_Pr7617993195940197384at_nat(enumerate_nat(Nb,Xs)),M) = aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),M)),aa_nat_nat(nth_nat(Xs),M)) ) ) ).

% nth_enumerate_eq
tff(fact_7088_nth__enumerate__eq,axiom,
    ! [M: nat,Xs: list_int,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),size_size_list_int(Xs))
     => ( nth_Pr3440142176431000676at_int(enumerate_int(Nb,Xs),M) = product_Pair_nat_int(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),M),aa_nat_int(nth_int(Xs),M)) ) ) ).

% nth_enumerate_eq
tff(fact_7089_disjE__realizer2,axiom,
    ! [P: $o,Q: fun_num_o,Xb: option_num,R: fun_option_num_o,F: option_num,G: fun_num_option_num] :
      ( aa_option_num_o(case_option_o_num((P),Q),Xb)
     => ( ( (P)
         => aa_option_num_o(R,F) )
       => ( ! [Q3: num] :
              ( aa_num_o(Q,Q3)
             => aa_option_num_o(R,aa_num_option_num(G,Q3)) )
         => aa_option_num_o(R,aa_opt5369159329878751634on_num(case_o6005452278849405969um_num(F,G),Xb)) ) ) ) ).

% disjE_realizer2
tff(fact_7090_disjE__realizer2,axiom,
    ! [P: $o,Q: fun_num_o,Xb: option_num,R: fun_num_o,F: num,G: fun_num_num] :
      ( aa_option_num_o(case_option_o_num((P),Q),Xb)
     => ( ( (P)
         => aa_num_o(R,F) )
       => ( ! [Q3: num] :
              ( aa_num_o(Q,Q3)
             => aa_num_o(R,aa_num_num(G,Q3)) )
         => aa_num_o(R,aa_option_num_num(case_option_num_num(F,G),Xb)) ) ) ) ).

% disjE_realizer2
tff(fact_7091_disjE__realizer2,axiom,
    ! [P: $o,Q: fun_num_o,Xb: option_num,R: fun_int_o,F: int,G: fun_num_int] :
      ( aa_option_num_o(case_option_o_num((P),Q),Xb)
     => ( ( (P)
         => aa_int_o(R,F) )
       => ( ! [Q3: num] :
              ( aa_num_o(Q,Q3)
             => aa_int_o(R,aa_num_int(G,Q3)) )
         => aa_int_o(R,aa_option_num_int(case_option_int_num(F,G),Xb)) ) ) ) ).

% disjE_realizer2
tff(fact_7092_disjE__realizer2,axiom,
    ! [P: $o,Q: fun_Pr4796943543331071514_nat_o,Xb: option4927543243414619207at_nat,R: fun_He8247548780604731983ap_o_o,F: heap_Time_Heap_o,G: fun_Pr1329037631615965750Heap_o] :
      ( aa_opt7112393969400857165_nat_o(case_o184042715313410164at_nat((P),Q),Xb)
     => ( ( (P)
         => aa_Hea4522166450600650424ap_o_o(R,F) )
       => ( ! [Q3: product_prod_nat_nat] :
              ( aa_Pro6150637604445964947_nat_o(Q,Q3)
             => aa_Hea4522166450600650424ap_o_o(R,aa_Pro691548381575413351Heap_o(G,Q3)) )
         => aa_Hea4522166450600650424ap_o_o(R,case_o1442776274061689234at_nat(F,G,Xb)) ) ) ) ).

% disjE_realizer2
tff(fact_7093_disjE__realizer2,axiom,
    ! [P: $o,Q: fun_Pr4796943543331071514_nat_o,Xb: option4927543243414619207at_nat,R: fun_o_o,F: $o,G: fun_Pr4796943543331071514_nat_o] :
      ( aa_opt7112393969400857165_nat_o(case_o184042715313410164at_nat((P),Q),Xb)
     => ( ( (P)
         => aa_o_o(R,(F)) )
       => ( ! [Q3: product_prod_nat_nat] :
              ( aa_Pro6150637604445964947_nat_o(Q,Q3)
             => aa_o_o(R,aa_Pro6150637604445964947_nat_o(G,Q3)) )
         => aa_o_o(R,aa_opt7112393969400857165_nat_o(case_o184042715313410164at_nat((F),G),Xb)) ) ) ) ).

% disjE_realizer2
tff(fact_7094_conjI__realizer,axiom,
    ! [P: fun_uint32_o,P2: uint32,Q: fun_uint32_o,Q2: uint32] :
      ( aa_uint32_o(P,P2)
     => ( aa_uint32_o(Q,Q2)
       => ( aa_uint32_o(P,aa_Pro3737690986720249065uint32(produc9004433772639906525uint32,aa_uin6954824371948101007uint32(produc1400373151660368625uint32(P2),Q2)))
          & aa_uint32_o(Q,aa_Pro3737690986720249065uint32(produc1510406741064981791uint32,aa_uin6954824371948101007uint32(produc1400373151660368625uint32(P2),Q2))) ) ) ) ).

% conjI_realizer
tff(fact_7095_conjI__realizer,axiom,
    ! [P: fun_nat_o,P2: nat,Q: fun_nat_o,Q2: nat] :
      ( aa_nat_o(P,P2)
     => ( aa_nat_o(Q,Q2)
       => ( aa_nat_o(P,aa_Pro4243173770044397973at_nat(product_fst_nat_nat,aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,P2),Q2)))
          & aa_nat_o(Q,aa_Pro4243173770044397973at_nat(product_snd_nat_nat,aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,P2),Q2))) ) ) ) ).

% conjI_realizer
tff(fact_7096_conjI__realizer,axiom,
    ! [P: fun_int_o,P2: int,Q: fun_int_o,Q2: int] :
      ( aa_int_o(P,P2)
     => ( aa_int_o(Q,Q2)
       => ( aa_int_o(P,aa_Pro6799258434465214121nt_int(product_fst_int_int,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,P2),Q2)))
          & aa_int_o(Q,aa_Pro6799258434465214121nt_int(product_snd_int_int,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,P2),Q2))) ) ) ) ).

% conjI_realizer
tff(fact_7097_conjI__realizer,axiom,
    ! [P: fun_Code_integer_o,P2: code_integer,Q: fun_Code_integer_o,Q2: code_integer] :
      ( aa_Code_integer_o(P,P2)
     => ( aa_Code_integer_o(Q,Q2)
       => ( aa_Code_integer_o(P,aa_Pro5245656335456409788nteger(produc8508995932063986495nteger,aa_Cod6977894654539884864nteger(produc1086072967326762835nteger(P2),Q2)))
          & aa_Code_integer_o(Q,aa_Pro5245656335456409788nteger(produc6174133586879617921nteger,aa_Cod6977894654539884864nteger(produc1086072967326762835nteger(P2),Q2))) ) ) ) ).

% conjI_realizer
tff(fact_7098_conjI__realizer,axiom,
    ! [P: fun_assn_o,P2: assn,Q: fun_assn_o,Q2: assn] :
      ( aa_assn_o(P,P2)
     => ( aa_assn_o(Q,Q2)
       => ( aa_assn_o(P,aa_Pro2979818503705714077n_assn(produc9167289414957590229n_assn,aa_ass5585057816575862475n_assn(produc118845697133431529n_assn(P2),Q2)))
          & aa_assn_o(Q,aa_Pro2979818503705714077n_assn(produc2051961928117032727n_assn,aa_ass5585057816575862475n_assn(produc118845697133431529n_assn(P2),Q2))) ) ) ) ).

% conjI_realizer
tff(fact_7099_conjI__realizer,axiom,
    ! [P: fun_he8413527710729349838unit_o,P2: heap_e7401611519738050253t_unit,Q: fun_set_nat_o,Q2: set_nat] :
      ( aa_hea7994211679605476167unit_o(P,P2)
     => ( aa_set_nat_o(Q,Q2)
       => ( aa_hea7994211679605476167unit_o(P,aa_Pro1219312880135979149t_unit(produc1824681642469235216et_nat,aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(P2),Q2)))
          & aa_set_nat_o(Q,aa_Pro3416150927818442182et_nat(produc8586169260539613262et_nat,aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(P2),Q2))) ) ) ) ).

% conjI_realizer
tff(fact_7100_conjI__realizer,axiom,
    ! [P: fun_fu3265955482920998386term_o,P2: fun_in5730379987311621695e_term,Q: fun_Pr3238374608701176162_int_o,Q2: product_prod_int_int] :
      ( aa_fun3020711759848831579term_o(P,P2)
     => ( aa_Pro4592068669816069595_int_o(Q,Q2)
       => ( aa_fun3020711759848831579term_o(P,aa_Pro2551989255418117932e_term(produc4463008262350210037nt_int,aa_Pro8800328608558162860nt_int(produc4729913460834754785nt_int(P2),Q2)))
          & aa_Pro4592068669816069595_int_o(Q,aa_Pro4437156595432605708nt_int(produc7629796867588368947nt_int,aa_Pro8800328608558162860nt_int(produc4729913460834754785nt_int(P2),Q2))) ) ) ) ).

% conjI_realizer
tff(fact_7101_conjI__realizer,axiom,
    ! [P: fun_fu5135969500543783852at_o_o,P2: fun_Pr6460944699146269935_nat_o,Q: fun_Pr6460944699146269935_nat_o,Q2: produc3658429121746597890et_nat] :
      ( aa_fun1675921005893230629at_o_o(P,P2)
     => ( aa_Pro653082453696054616_nat_o(Q,Q2)
       => ( aa_fun1675921005893230629at_o_o(P,aa_Pro8262445894397805713_nat_o(produc5325900173150335662et_nat,aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(P2),Q2)))
          & aa_Pro653082453696054616_nat_o(Q,aa_Pro5541488491759012064et_nat(produc4960056325182568940et_nat,aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(P2),Q2))) ) ) ) ).

% conjI_realizer
tff(fact_7102_conjI__realizer,axiom,
    ! [P: fun_fu4736795817735658359term_o,P2: fun_Pr5098669352780829690e_term,Q: fun_Pr7211993375558241536eger_o,Q2: produc8923325533196201883nteger] :
      ( aa_fun7993445754732355552term_o(P,P2)
     => ( aa_Pro4926521453222654329eger_o(Q,Q2)
       => ( aa_fun7993445754732355552term_o(P,aa_Pro4276937008533432526e_term(produc2481150572478215260nteger,aa_Pro4291442876242880245nteger(produc565448451787809096nteger(P2),Q2)))
          & aa_Pro4926521453222654329eger_o(Q,aa_Pro4625778721341478855nteger(produc6589491691745714842nteger,aa_Pro4291442876242880245nteger(produc565448451787809096nteger(P2),Q2))) ) ) ) ).

% conjI_realizer
tff(fact_7103_conjI__realizer,axiom,
    ! [P: fun_fu6378075912484669126term_o,P2: fun_Pr7763680713840462443e_term,Q: fun_Pr3238374608701176162_int_o,Q2: product_prod_int_int] :
      ( aa_fun7650857933016935215term_o(P,P2)
     => ( aa_Pro4592068669816069595_int_o(Q,Q2)
       => ( aa_fun7650857933016935215term_o(P,aa_Pro4010435699744810540e_term(produc3130796691802200265nt_int,aa_Pro5748361610964203648nt_int(produc7320969046608631221nt_int(P2),Q2)))
          & aa_Pro4592068669816069595_int_o(Q,aa_Pro2675344531184151096nt_int(produc1557020426977177351nt_int,aa_Pro5748361610964203648nt_int(produc7320969046608631221nt_int(P2),Q2))) ) ) ) ).

% conjI_realizer
tff(fact_7104_exI__realizer,axiom,
    ! [P: fun_ui3478071520421546738nt32_o,Y: uint32,Xb: uint32] :
      ( aa_uint32_o(aa_uin5924812838015166427nt32_o(P,Y),Xb)
     => aa_uint32_o(aa_uin5924812838015166427nt32_o(P,aa_Pro3737690986720249065uint32(produc1510406741064981791uint32,aa_uin6954824371948101007uint32(produc1400373151660368625uint32(Xb),Y))),aa_Pro3737690986720249065uint32(produc9004433772639906525uint32,aa_uin6954824371948101007uint32(produc1400373151660368625uint32(Xb),Y))) ) ).

% exI_realizer
tff(fact_7105_exI__realizer,axiom,
    ! [P: fun_nat_fun_nat_o,Y: nat,Xb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(P,Y),Xb)
     => aa_nat_o(aa_nat_fun_nat_o(P,aa_Pro4243173770044397973at_nat(product_snd_nat_nat,aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Xb),Y))),aa_Pro4243173770044397973at_nat(product_fst_nat_nat,aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Xb),Y))) ) ).

% exI_realizer
tff(fact_7106_exI__realizer,axiom,
    ! [P: fun_int_fun_int_o,Y: int,Xb: int] :
      ( aa_int_o(aa_int_fun_int_o(P,Y),Xb)
     => aa_int_o(aa_int_fun_int_o(P,aa_Pro6799258434465214121nt_int(product_snd_int_int,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Xb),Y))),aa_Pro6799258434465214121nt_int(product_fst_int_int,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Xb),Y))) ) ).

% exI_realizer
tff(fact_7107_exI__realizer,axiom,
    ! [P: fun_Co2081214437782444048eger_o,Y: code_integer,Xb: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(P,Y),Xb)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(P,aa_Pro5245656335456409788nteger(produc6174133586879617921nteger,aa_Cod6977894654539884864nteger(produc1086072967326762835nteger(Xb),Y))),aa_Pro5245656335456409788nteger(produc8508995932063986495nteger,aa_Cod6977894654539884864nteger(produc1086072967326762835nteger(Xb),Y))) ) ).

% exI_realizer
tff(fact_7108_exI__realizer,axiom,
    ! [P: fun_assn_fun_assn_o,Y: assn,Xb: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(P,Y),Xb)
     => aa_assn_o(aa_assn_fun_assn_o(P,aa_Pro2979818503705714077n_assn(produc2051961928117032727n_assn,aa_ass5585057816575862475n_assn(produc118845697133431529n_assn(Xb),Y))),aa_Pro2979818503705714077n_assn(produc9167289414957590229n_assn,aa_ass5585057816575862475n_assn(produc118845697133431529n_assn(Xb),Y))) ) ).

% exI_realizer
tff(fact_7109_exI__realizer,axiom,
    ! [P: fun_se9038067516210351697unit_o,Y: set_nat,Xb: heap_e7401611519738050253t_unit] :
      ( aa_hea7994211679605476167unit_o(aa_set6435858302011484818unit_o(P,Y),Xb)
     => aa_hea7994211679605476167unit_o(aa_set6435858302011484818unit_o(P,aa_Pro3416150927818442182et_nat(produc8586169260539613262et_nat,aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(Xb),Y))),aa_Pro1219312880135979149t_unit(produc1824681642469235216et_nat,aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(Xb),Y))) ) ).

% exI_realizer
tff(fact_7110_exI__realizer,axiom,
    ! [P: fun_Pr7145817153658775868term_o,Y: product_prod_int_int,Xb: fun_in5730379987311621695e_term] :
      ( aa_fun3020711759848831579term_o(aa_Pro1235490124264145077term_o(P,Y),Xb)
     => aa_fun3020711759848831579term_o(aa_Pro1235490124264145077term_o(P,aa_Pro4437156595432605708nt_int(produc7629796867588368947nt_int,aa_Pro8800328608558162860nt_int(produc4729913460834754785nt_int(Xb),Y))),aa_Pro2551989255418117932e_term(produc4463008262350210037nt_int,aa_Pro8800328608558162860nt_int(produc4729913460834754785nt_int(Xb),Y))) ) ).

% exI_realizer
tff(fact_7111_exI__realizer,axiom,
    ! [P: fun_Pr5843349257357397163at_o_o,Y: produc3658429121746597890et_nat,Xb: fun_Pr6460944699146269935_nat_o] :
      ( aa_fun1675921005893230629at_o_o(aa_Pro3588693547670092268at_o_o(P,Y),Xb)
     => aa_fun1675921005893230629at_o_o(aa_Pro3588693547670092268at_o_o(P,aa_Pro5541488491759012064et_nat(produc4960056325182568940et_nat,aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(Xb),Y))),aa_Pro8262445894397805713_nat_o(produc5325900173150335662et_nat,aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(Xb),Y))) ) ).

% exI_realizer
tff(fact_7112_exI__realizer,axiom,
    ! [P: fun_Pr4680327687137491167term_o,Y: produc8923325533196201883nteger,Xb: fun_Pr5098669352780829690e_term] :
      ( aa_fun7993445754732355552term_o(aa_Pro9035772811636298456term_o(P,Y),Xb)
     => aa_fun7993445754732355552term_o(aa_Pro9035772811636298456term_o(P,aa_Pro4625778721341478855nteger(produc6589491691745714842nteger,aa_Pro4291442876242880245nteger(produc565448451787809096nteger(Xb),Y))),aa_Pro4276937008533432526e_term(produc2481150572478215260nteger,aa_Pro4291442876242880245nteger(produc565448451787809096nteger(Xb),Y))) ) ).

% exI_realizer
tff(fact_7113_exI__realizer,axiom,
    ! [P: fun_Pr5526332658708858896term_o,Y: product_prod_int_int,Xb: fun_Pr7763680713840462443e_term] :
      ( aa_fun7650857933016935215term_o(aa_Pro1023942307601852809term_o(P,Y),Xb)
     => aa_fun7650857933016935215term_o(aa_Pro1023942307601852809term_o(P,aa_Pro2675344531184151096nt_int(produc1557020426977177351nt_int,aa_Pro5748361610964203648nt_int(produc7320969046608631221nt_int(Xb),Y))),aa_Pro4010435699744810540e_term(produc3130796691802200265nt_int,aa_Pro5748361610964203648nt_int(produc7320969046608631221nt_int(Xb),Y))) ) ).

% exI_realizer
tff(fact_7114_finite__Diff__insert,axiom,
    ! [A2: set_VEBT_VEBT,A: vEBT_VEBT,B4: set_VEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,minus_5127226145743854075T_VEBT(A2,insert_VEBT_VEBT(A,B4)))
    <=> aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,minus_5127226145743854075T_VEBT(A2,B4)) ) ).

% finite_Diff_insert
tff(fact_7115_finite__Diff__insert,axiom,
    ! [A2: set_real,A: real,B4: set_real] :
      ( aa_set_real_o(finite_finite_real,minus_minus_set_real(A2,insert_real(A,B4)))
    <=> aa_set_real_o(finite_finite_real,minus_minus_set_real(A2,B4)) ) ).

% finite_Diff_insert
tff(fact_7116_finite__Diff__insert,axiom,
    ! [A2: set_int,A: int,B4: set_int] :
      ( aa_set_int_o(finite_finite_int,minus_minus_set_int(A2,insert_int(A,B4)))
    <=> aa_set_int_o(finite_finite_int,minus_minus_set_int(A2,B4)) ) ).

% finite_Diff_insert
tff(fact_7117_finite__Diff__insert,axiom,
    ! [A2: set_complex,A: complex,B4: set_complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,minus_811609699411566653omplex(A2,insert_complex(A,B4)))
    <=> aa_set_complex_o(finite3207457112153483333omplex,minus_811609699411566653omplex(A2,B4)) ) ).

% finite_Diff_insert
tff(fact_7118_finite__Diff__insert,axiom,
    ! [A2: set_Code_integer,A: code_integer,B4: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,minus_2355218937544613996nteger(A2,insert_Code_integer(A,B4)))
    <=> aa_set5643387332196575111eger_o(finite6017078050557962740nteger,minus_2355218937544613996nteger(A2,B4)) ) ).

% finite_Diff_insert
tff(fact_7119_finite__Diff__insert,axiom,
    ! [A2: set_nat,A: nat,B4: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,minus_minus_set_nat(A2,insert_nat(A,B4)))
    <=> aa_set_nat_o(finite_finite_nat,minus_minus_set_nat(A2,B4)) ) ).

% finite_Diff_insert
tff(fact_7120_finite__Collect__le__nat,axiom,
    ! [K: nat] : aa_set_nat_o(finite_finite_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_adt,K))) ).

% finite_Collect_le_nat
tff(fact_7121_finite__Collect__less__nat,axiom,
    ! [K: nat] : aa_set_nat_o(finite_finite_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_dv,K))) ).

% finite_Collect_less_nat
tff(fact_7122_finite__Collect__subsets,axiom,
    ! [A2: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => finite1152437895449049373et_nat(collect_set_nat(aTP_Lamm_adu(A2))) ) ).

% finite_Collect_subsets
tff(fact_7123_finite__Collect__subsets,axiom,
    ! [A2: set_complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => finite6551019134538273531omplex(collect_set_complex(aTP_Lamm_adv(A2))) ) ).

% finite_Collect_subsets
tff(fact_7124_finite__Collect__subsets,axiom,
    ! [A2: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A2)
     => finite6931041176100689706nteger(collec574505750873337192nteger(aTP_Lamm_adw(A2))) ) ).

% finite_Collect_subsets
tff(fact_7125_finite__Collect__subsets,axiom,
    ! [A2: set_int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => finite6197958912794628473et_int(collect_set_int(aTP_Lamm_adx(A2))) ) ).

% finite_Collect_subsets
tff(fact_7126_finite__Collect__not,axiom,
    ! [P: fun_Pr3238374608701176162_int_o] :
      ( finite2998713641127702882nt_int(collec213857154873943460nt_int(P))
     => ( finite2998713641127702882nt_int(collec213857154873943460nt_int(aTP_Lamm_ady(P)))
      <=> finite2998713641127702882nt_int(top_to4366644338036079209nt_int) ) ) ).

% finite_Collect_not
tff(fact_7127_finite__Collect__not,axiom,
    ! [P: fun_int_o] :
      ( aa_set_int_o(finite_finite_int,collect_int(P))
     => ( aa_set_int_o(finite_finite_int,collect_int(aTP_Lamm_adz(P)))
      <=> aa_set_int_o(finite_finite_int,top_top_set_int) ) ) ).

% finite_Collect_not
tff(fact_7128_finite__Collect__not,axiom,
    ! [P: fun_complex_o] :
      ( aa_set_complex_o(finite3207457112153483333omplex,collect_complex(P))
     => ( aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aTP_Lamm_aea(P)))
      <=> aa_set_complex_o(finite3207457112153483333omplex,top_top_set_complex) ) ) ).

% finite_Collect_not
tff(fact_7129_finite__Collect__not,axiom,
    ! [P: fun_Code_integer_o] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,collect_Code_integer(P))
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,collect_Code_integer(aTP_Lamm_aeb(P)))
      <=> aa_set5643387332196575111eger_o(finite6017078050557962740nteger,top_to4645266643341252675nteger) ) ) ).

% finite_Collect_not
tff(fact_7130_finite__Collect__not,axiom,
    ! [P: fun_real_o] :
      ( aa_set_real_o(finite_finite_real,collect_real(P))
     => ( aa_set_real_o(finite_finite_real,collect_real(aTP_Lamm_aec(P)))
      <=> aa_set_real_o(finite_finite_real,top_top_set_real) ) ) ).

% finite_Collect_not
tff(fact_7131_finite__Collect__not,axiom,
    ! [P: fun_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,collect_nat(P))
     => ( aa_set_nat_o(finite_finite_nat,collect_nat(aTP_Lamm_aed(P)))
      <=> aa_set_nat_o(finite_finite_nat,top_top_set_nat) ) ) ).

% finite_Collect_not
tff(fact_7132_finite__Collect__not,axiom,
    ! [P: fun_char_o] :
      ( finite_finite_char(collect_char(P))
     => ( finite_finite_char(collect_char(aTP_Lamm_aee(P)))
      <=> finite_finite_char(top_top_set_char) ) ) ).

% finite_Collect_not
tff(fact_7133_finite__Collect__conjI,axiom,
    ! [P: fun_Pr3238374608701176162_int_o,Q: fun_Pr3238374608701176162_int_o] :
      ( ( finite2998713641127702882nt_int(collec213857154873943460nt_int(P))
        | finite2998713641127702882nt_int(collec213857154873943460nt_int(Q)) )
     => finite2998713641127702882nt_int(collec213857154873943460nt_int(aa_fun5349570537165229698_int_o(aTP_Lamm_aef(P),Q))) ) ).

% finite_Collect_conjI
tff(fact_7134_finite__Collect__conjI,axiom,
    ! [P: fun_nat_o,Q: fun_nat_o] :
      ( ( aa_set_nat_o(finite_finite_nat,collect_nat(P))
        | aa_set_nat_o(finite_finite_nat,collect_nat(Q)) )
     => aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun5937193402907943164_nat_o(aTP_Lamm_aeg(P),Q))) ) ).

% finite_Collect_conjI
tff(fact_7135_finite__Collect__conjI,axiom,
    ! [P: fun_int_o,Q: fun_int_o] :
      ( ( aa_set_int_o(finite_finite_int,collect_int(P))
        | aa_set_int_o(finite_finite_int,collect_int(Q)) )
     => aa_set_int_o(finite_finite_int,collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_aeh(P),Q))) ) ).

% finite_Collect_conjI
tff(fact_7136_finite__Collect__conjI,axiom,
    ! [P: fun_complex_o,Q: fun_complex_o] :
      ( ( aa_set_complex_o(finite3207457112153483333omplex,collect_complex(P))
        | aa_set_complex_o(finite3207457112153483333omplex,collect_complex(Q)) )
     => aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aa_fun2514317545224092796plex_o(aTP_Lamm_aei(P),Q))) ) ).

% finite_Collect_conjI
tff(fact_7137_finite__Collect__conjI,axiom,
    ! [P: fun_Code_integer_o,Q: fun_Code_integer_o] :
      ( ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,collect_Code_integer(P))
        | aa_set5643387332196575111eger_o(finite6017078050557962740nteger,collect_Code_integer(Q)) )
     => aa_set5643387332196575111eger_o(finite6017078050557962740nteger,collect_Code_integer(aa_fun8214818745163655228eger_o(aTP_Lamm_aej(P),Q))) ) ).

% finite_Collect_conjI
tff(fact_7138_finite__Collect__disjI,axiom,
    ! [P: fun_Pr3238374608701176162_int_o,Q: fun_Pr3238374608701176162_int_o] :
      ( finite2998713641127702882nt_int(collec213857154873943460nt_int(aa_fun5349570537165229698_int_o(aTP_Lamm_aek(P),Q)))
    <=> ( finite2998713641127702882nt_int(collec213857154873943460nt_int(P))
        & finite2998713641127702882nt_int(collec213857154873943460nt_int(Q)) ) ) ).

% finite_Collect_disjI
tff(fact_7139_finite__Collect__disjI,axiom,
    ! [P: fun_nat_o,Q: fun_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun5937193402907943164_nat_o(aTP_Lamm_ael(P),Q)))
    <=> ( aa_set_nat_o(finite_finite_nat,collect_nat(P))
        & aa_set_nat_o(finite_finite_nat,collect_nat(Q)) ) ) ).

% finite_Collect_disjI
tff(fact_7140_finite__Collect__disjI,axiom,
    ! [P: fun_int_o,Q: fun_int_o] :
      ( aa_set_int_o(finite_finite_int,collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_aem(P),Q)))
    <=> ( aa_set_int_o(finite_finite_int,collect_int(P))
        & aa_set_int_o(finite_finite_int,collect_int(Q)) ) ) ).

% finite_Collect_disjI
tff(fact_7141_finite__Collect__disjI,axiom,
    ! [P: fun_complex_o,Q: fun_complex_o] :
      ( aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aa_fun2514317545224092796plex_o(aTP_Lamm_aen(P),Q)))
    <=> ( aa_set_complex_o(finite3207457112153483333omplex,collect_complex(P))
        & aa_set_complex_o(finite3207457112153483333omplex,collect_complex(Q)) ) ) ).

% finite_Collect_disjI
tff(fact_7142_finite__Collect__disjI,axiom,
    ! [P: fun_Code_integer_o,Q: fun_Code_integer_o] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,collect_Code_integer(aa_fun8214818745163655228eger_o(aTP_Lamm_aeo(P),Q)))
    <=> ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,collect_Code_integer(P))
        & aa_set5643387332196575111eger_o(finite6017078050557962740nteger,collect_Code_integer(Q)) ) ) ).

% finite_Collect_disjI
tff(fact_7143_finite__interval__int1,axiom,
    ! [A: int,B: int] : aa_set_int_o(finite_finite_int,collect_int(aa_int_fun_int_o(aTP_Lamm_aep(A),B))) ).

% finite_interval_int1
tff(fact_7144_finite__interval__int4,axiom,
    ! [A: int,B: int] : aa_set_int_o(finite_finite_int,collect_int(aa_int_fun_int_o(aTP_Lamm_aeq(A),B))) ).

% finite_interval_int4
tff(fact_7145_finite__Diff2,axiom,
    ! [B4: set_int,A2: set_int] :
      ( aa_set_int_o(finite_finite_int,B4)
     => ( aa_set_int_o(finite_finite_int,minus_minus_set_int(A2,B4))
      <=> aa_set_int_o(finite_finite_int,A2) ) ) ).

% finite_Diff2
tff(fact_7146_finite__Diff2,axiom,
    ! [B4: set_complex,A2: set_complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,B4)
     => ( aa_set_complex_o(finite3207457112153483333omplex,minus_811609699411566653omplex(A2,B4))
      <=> aa_set_complex_o(finite3207457112153483333omplex,A2) ) ) ).

% finite_Diff2
tff(fact_7147_finite__Diff2,axiom,
    ! [B4: set_Code_integer,A2: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B4)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,minus_2355218937544613996nteger(A2,B4))
      <=> aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A2) ) ) ).

% finite_Diff2
tff(fact_7148_finite__Diff2,axiom,
    ! [B4: set_nat,A2: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,B4)
     => ( aa_set_nat_o(finite_finite_nat,minus_minus_set_nat(A2,B4))
      <=> aa_set_nat_o(finite_finite_nat,A2) ) ) ).

% finite_Diff2
tff(fact_7149_finite__Diff,axiom,
    ! [A2: set_int,B4: set_int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => aa_set_int_o(finite_finite_int,minus_minus_set_int(A2,B4)) ) ).

% finite_Diff
tff(fact_7150_finite__Diff,axiom,
    ! [A2: set_complex,B4: set_complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => aa_set_complex_o(finite3207457112153483333omplex,minus_811609699411566653omplex(A2,B4)) ) ).

% finite_Diff
tff(fact_7151_finite__Diff,axiom,
    ! [A2: set_Code_integer,B4: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A2)
     => aa_set5643387332196575111eger_o(finite6017078050557962740nteger,minus_2355218937544613996nteger(A2,B4)) ) ).

% finite_Diff
tff(fact_7152_finite__Diff,axiom,
    ! [A2: set_nat,B4: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => aa_set_nat_o(finite_finite_nat,minus_minus_set_nat(A2,B4)) ) ).

% finite_Diff
tff(fact_7153_finite__interval__int2,axiom,
    ! [A: int,B: int] : aa_set_int_o(finite_finite_int,collect_int(aa_int_fun_int_o(aTP_Lamm_aer(A),B))) ).

% finite_interval_int2
tff(fact_7154_finite__interval__int3,axiom,
    ! [A: int,B: int] : aa_set_int_o(finite_finite_int,collect_int(aa_int_fun_int_o(aTP_Lamm_aes(A),B))) ).

% finite_interval_int3
tff(fact_7155_finite__maxlen,axiom,
    ! [M8: set_list_real] :
      ( finite306553202115118035t_real(M8)
     => ? [N2: nat] :
        ! [X3: list_real] :
          ( member_list_real(X3,M8)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,size_size_list_real(X3)),N2) ) ) ).

% finite_maxlen
tff(fact_7156_finite__maxlen,axiom,
    ! [M8: set_list_o] :
      ( finite_finite_list_o(M8)
     => ? [N2: nat] :
        ! [X3: list_o] :
          ( member_list_o(X3,M8)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,size_size_list_o(X3)),N2) ) ) ).

% finite_maxlen
tff(fact_7157_finite__maxlen,axiom,
    ! [M8: set_list_nat] :
      ( finite8100373058378681591st_nat(M8)
     => ? [N2: nat] :
        ! [X3: list_nat] :
          ( member_list_nat(X3,M8)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,size_size_list_nat(X3)),N2) ) ) ).

% finite_maxlen
tff(fact_7158_finite__maxlen,axiom,
    ! [M8: set_list_int] :
      ( finite3922522038869484883st_int(M8)
     => ? [N2: nat] :
        ! [X3: list_int] :
          ( member_list_int(X3,M8)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,size_size_list_int(X3)),N2) ) ) ).

% finite_maxlen
tff(fact_7159_finite__divisors__int,axiom,
    ! [I: int] :
      ( ( I != zero_zero_int )
     => aa_set_int_o(finite_finite_int,collect_int(aa_int_fun_int_o(aTP_Lamm_wu,I))) ) ).

% finite_divisors_int
tff(fact_7160_not__finite__existsD,axiom,
    ! [P: fun_Pr3238374608701176162_int_o] :
      ( ~ finite2998713641127702882nt_int(collec213857154873943460nt_int(P))
     => ? [X_13: product_prod_int_int] : aa_Pro4592068669816069595_int_o(P,X_13) ) ).

% not_finite_existsD
tff(fact_7161_not__finite__existsD,axiom,
    ! [P: fun_nat_o] :
      ( ~ aa_set_nat_o(finite_finite_nat,collect_nat(P))
     => ? [X_13: nat] : aa_nat_o(P,X_13) ) ).

% not_finite_existsD
tff(fact_7162_not__finite__existsD,axiom,
    ! [P: fun_int_o] :
      ( ~ aa_set_int_o(finite_finite_int,collect_int(P))
     => ? [X_13: int] : aa_int_o(P,X_13) ) ).

% not_finite_existsD
tff(fact_7163_not__finite__existsD,axiom,
    ! [P: fun_complex_o] :
      ( ~ aa_set_complex_o(finite3207457112153483333omplex,collect_complex(P))
     => ? [X_13: complex] : aa_complex_o(P,X_13) ) ).

% not_finite_existsD
tff(fact_7164_not__finite__existsD,axiom,
    ! [P: fun_Code_integer_o] :
      ( ~ aa_set5643387332196575111eger_o(finite6017078050557962740nteger,collect_Code_integer(P))
     => ? [X_13: code_integer] : aa_Code_integer_o(P,X_13) ) ).

% not_finite_existsD
tff(fact_7165_pigeonhole__infinite__rel,axiom,
    ! [A2: set_VEBT_VEBT,B4: set_nat,R: fun_VE6480751511521678896_nat_o] :
      ( ~ aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,A2)
     => ( aa_set_nat_o(finite_finite_nat,B4)
       => ( ! [X2: vEBT_VEBT] :
              ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X2),A2)
             => ? [Xa: nat] :
                  ( aa_set_nat_o(member_nat(Xa),B4)
                  & aa_nat_o(aa_VEB2314620471664397097_nat_o(R,X2),Xa) ) )
         => ? [X2: nat] :
              ( aa_set_nat_o(member_nat(X2),B4)
              & ~ aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,collect_VEBT_VEBT(aa_nat1925708948643195573VEBT_o(aa_fun2405540735884763746VEBT_o(aTP_Lamm_aet(A2),R),X2))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_7166_pigeonhole__infinite__rel,axiom,
    ! [A2: set_real,B4: set_nat,R: fun_real_fun_nat_o] :
      ( ~ aa_set_real_o(finite_finite_real,A2)
     => ( aa_set_nat_o(finite_finite_nat,B4)
       => ( ! [X2: real] :
              ( aa_set_real_o(member_real(X2),A2)
             => ? [Xa: nat] :
                  ( aa_set_nat_o(member_nat(Xa),B4)
                  & aa_nat_o(aa_real_fun_nat_o(R,X2),Xa) ) )
         => ? [X2: nat] :
              ( aa_set_nat_o(member_nat(X2),B4)
              & ~ aa_set_real_o(finite_finite_real,collect_real(aa_nat_fun_real_o(aa_fun6917832400337718588real_o(aTP_Lamm_aeu(A2),R),X2))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_7167_pigeonhole__infinite__rel,axiom,
    ! [A2: set_VEBT_VEBT,B4: set_int,R: fun_VE4235606543851232980_int_o] :
      ( ~ aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,A2)
     => ( aa_set_int_o(finite_finite_int,B4)
       => ( ! [X2: vEBT_VEBT] :
              ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X2),A2)
             => ? [Xa: int] :
                  ( aa_set_int_o(member_int(Xa),B4)
                  & aa_int_o(aa_VEB493578126231709133_int_o(R,X2),Xa) ) )
         => ? [X2: int] :
              ( aa_set_int_o(member_int(X2),B4)
              & ~ aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,collect_VEBT_VEBT(aa_int8839263053405020433VEBT_o(aa_fun8807180201667926370VEBT_o(aTP_Lamm_aev(A2),R),X2))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_7168_pigeonhole__infinite__rel,axiom,
    ! [A2: set_real,B4: set_int,R: fun_real_fun_int_o] :
      ( ~ aa_set_real_o(finite_finite_real,A2)
     => ( aa_set_int_o(finite_finite_int,B4)
       => ( ! [X2: real] :
              ( aa_set_real_o(member_real(X2),A2)
             => ? [Xa: int] :
                  ( aa_set_int_o(member_int(Xa),B4)
                  & aa_int_o(aa_real_fun_int_o(R,X2),Xa) ) )
         => ? [X2: int] :
              ( aa_set_int_o(member_int(X2),B4)
              & ~ aa_set_real_o(finite_finite_real,collect_real(aa_int_fun_real_o(aa_fun2391495421969208892real_o(aTP_Lamm_aew(A2),R),X2))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_7169_pigeonhole__infinite__rel,axiom,
    ! [A2: set_VEBT_VEBT,B4: set_complex,R: fun_VE7631951351884037842plex_o] :
      ( ~ aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,A2)
     => ( aa_set_complex_o(finite3207457112153483333omplex,B4)
       => ( ! [X2: vEBT_VEBT] :
              ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X2),A2)
             => ? [Xa: complex] :
                  ( aa_set_complex_o(member_complex(Xa),B4)
                  & aa_complex_o(aa_VEB8036846806599048395plex_o(R,X2),Xa) ) )
         => ? [X2: complex] :
              ( aa_set_complex_o(member_complex(X2),B4)
              & ~ aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,collect_VEBT_VEBT(aa_com3651963455866140051VEBT_o(aa_fun7268111043052431330VEBT_o(aTP_Lamm_aex(A2),R),X2))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_7170_pigeonhole__infinite__rel,axiom,
    ! [A2: set_real,B4: set_complex,R: fun_re7319081385389557872plex_o] :
      ( ~ aa_set_real_o(finite_finite_real,A2)
     => ( aa_set_complex_o(finite3207457112153483333omplex,B4)
       => ( ! [X2: real] :
              ( aa_set_real_o(member_real(X2),A2)
             => ? [Xa: complex] :
                  ( aa_set_complex_o(member_complex(Xa),B4)
                  & aa_complex_o(aa_rea2382570592068293209plex_o(R,X2),Xa) ) )
         => ? [X2: complex] :
              ( aa_set_complex_o(member_complex(X2),B4)
              & ~ aa_set_real_o(finite_finite_real,collect_real(aa_com1951324360330529881real_o(aa_fun7692540709613205436real_o(aTP_Lamm_aey(A2),R),X2))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_7171_pigeonhole__infinite__rel,axiom,
    ! [A2: set_VEBT_VEBT,B4: set_Code_integer,R: fun_VE1909044406494083811eger_o] :
      ( ~ aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,A2)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B4)
       => ( ! [X2: vEBT_VEBT] :
              ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X2),A2)
             => ? [Xa: code_integer] :
                  ( aa_set5643387332196575111eger_o(member_Code_integer(Xa),B4)
                  & aa_Code_integer_o(aa_VEB1409644639000594780eger_o(R,X2),Xa) ) )
         => ? [X2: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X2),B4)
              & ~ aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,collect_VEBT_VEBT(aa_Cod5040069908416755522VEBT_o(aa_fun1637734041313256866VEBT_o(aTP_Lamm_aez(A2),R),X2))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_7172_pigeonhole__infinite__rel,axiom,
    ! [A2: set_real,B4: set_Code_integer,R: fun_re3424108668441675905eger_o] :
      ( ~ aa_set_real_o(finite_finite_real,A2)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B4)
       => ( ! [X2: real] :
              ( aa_set_real_o(member_real(X2),A2)
             => ? [Xa: code_integer] :
                  ( aa_set5643387332196575111eger_o(member_Code_integer(Xa),B4)
                  & aa_Code_integer_o(aa_rea4090721512283939562eger_o(R,X2),Xa) ) )
         => ? [X2: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X2),B4)
              & ~ aa_set_real_o(finite_finite_real,collect_real(aa_Cod1315720040876898922real_o(aa_fun4247632250071955964real_o(aTP_Lamm_afa(A2),R),X2))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_7173_pigeonhole__infinite__rel,axiom,
    ! [A2: set_nat,B4: set_nat,R: fun_nat_fun_nat_o] :
      ( ~ aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_nat_o(finite_finite_nat,B4)
       => ( ! [X2: nat] :
              ( aa_set_nat_o(member_nat(X2),A2)
             => ? [Xa: nat] :
                  ( aa_set_nat_o(member_nat(Xa),B4)
                  & aa_nat_o(aa_nat_fun_nat_o(R,X2),Xa) ) )
         => ? [X2: nat] :
              ( aa_set_nat_o(member_nat(X2),B4)
              & ~ aa_set_nat_o(finite_finite_nat,collect_nat(aa_nat_fun_nat_o(aa_fun6699960423209655484_nat_o(aTP_Lamm_afb(A2),R),X2))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_7174_pigeonhole__infinite__rel,axiom,
    ! [A2: set_nat,B4: set_int,R: fun_nat_fun_int_o] :
      ( ~ aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_int_o(finite_finite_int,B4)
       => ( ! [X2: nat] :
              ( aa_set_nat_o(member_nat(X2),A2)
             => ? [Xa: int] :
                  ( aa_set_int_o(member_int(Xa),B4)
                  & aa_int_o(aa_nat_fun_int_o(R,X2),Xa) ) )
         => ? [X2: int] :
              ( aa_set_int_o(member_int(X2),B4)
              & ~ aa_set_nat_o(finite_finite_nat,collect_nat(aa_int_fun_nat_o(aa_fun7393466555336238524_nat_o(aTP_Lamm_afc(A2),R),X2))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_7175_finite__has__maximal2,axiom,
    ! [A2: set_real,A: real] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( aa_set_real_o(member_real(A),A2)
       => ? [X2: real] :
            ( aa_set_real_o(member_real(X2),A2)
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),X2)
            & ! [Xa: real] :
                ( aa_set_real_o(member_real(Xa),A2)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X2),Xa)
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
tff(fact_7176_finite__has__maximal2,axiom,
    ! [A2: set_Code_integer,A: code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A2)
     => ( aa_set5643387332196575111eger_o(member_Code_integer(A),A2)
       => ? [X2: code_integer] :
            ( aa_set5643387332196575111eger_o(member_Code_integer(X2),A2)
            & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),X2)
            & ! [Xa: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(Xa),A2)
               => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,X2),Xa)
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
tff(fact_7177_finite__has__maximal2,axiom,
    ! [A2: set_set_int,A: set_int] :
      ( finite6197958912794628473et_int(A2)
     => ( member_set_int(A,A2)
       => ? [X2: set_int] :
            ( member_set_int(X2,A2)
            & aa_set_int_o(ord_less_eq_set_int(A),X2)
            & ! [Xa: set_int] :
                ( member_set_int(Xa,A2)
               => ( aa_set_int_o(ord_less_eq_set_int(X2),Xa)
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
tff(fact_7178_finite__has__maximal2,axiom,
    ! [A2: set_rat,A: rat] :
      ( aa_set_rat_o(finite_finite_rat,A2)
     => ( aa_set_rat_o(member_rat(A),A2)
       => ? [X2: rat] :
            ( aa_set_rat_o(member_rat(X2),A2)
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),X2)
            & ! [Xa: rat] :
                ( aa_set_rat_o(member_rat(Xa),A2)
               => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X2),Xa)
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
tff(fact_7179_finite__has__maximal2,axiom,
    ! [A2: set_num,A: num] :
      ( aa_set_num_o(finite_finite_num,A2)
     => ( aa_set_num_o(member_num(A),A2)
       => ? [X2: num] :
            ( aa_set_num_o(member_num(X2),A2)
            & aa_num_o(ord_less_eq_num(A),X2)
            & ! [Xa: num] :
                ( aa_set_num_o(member_num(Xa),A2)
               => ( aa_num_o(ord_less_eq_num(X2),Xa)
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
tff(fact_7180_finite__has__maximal2,axiom,
    ! [A2: set_nat,A: nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_nat_o(member_nat(A),A2)
       => ? [X2: nat] :
            ( aa_set_nat_o(member_nat(X2),A2)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),X2)
            & ! [Xa: nat] :
                ( aa_set_nat_o(member_nat(Xa),A2)
               => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X2),Xa)
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
tff(fact_7181_finite__has__maximal2,axiom,
    ! [A2: set_int,A: int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( aa_set_int_o(member_int(A),A2)
       => ? [X2: int] :
            ( aa_set_int_o(member_int(X2),A2)
            & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),X2)
            & ! [Xa: int] :
                ( aa_set_int_o(member_int(Xa),A2)
               => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X2),Xa)
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
tff(fact_7182_finite__has__minimal2,axiom,
    ! [A2: set_real,A: real] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( aa_set_real_o(member_real(A),A2)
       => ? [X2: real] :
            ( aa_set_real_o(member_real(X2),A2)
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X2),A)
            & ! [Xa: real] :
                ( aa_set_real_o(member_real(Xa),A2)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),X2)
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
tff(fact_7183_finite__has__minimal2,axiom,
    ! [A2: set_Code_integer,A: code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A2)
     => ( aa_set5643387332196575111eger_o(member_Code_integer(A),A2)
       => ? [X2: code_integer] :
            ( aa_set5643387332196575111eger_o(member_Code_integer(X2),A2)
            & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,X2),A)
            & ! [Xa: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(Xa),A2)
               => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Xa),X2)
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
tff(fact_7184_finite__has__minimal2,axiom,
    ! [A2: set_set_int,A: set_int] :
      ( finite6197958912794628473et_int(A2)
     => ( member_set_int(A,A2)
       => ? [X2: set_int] :
            ( member_set_int(X2,A2)
            & aa_set_int_o(ord_less_eq_set_int(X2),A)
            & ! [Xa: set_int] :
                ( member_set_int(Xa,A2)
               => ( aa_set_int_o(ord_less_eq_set_int(Xa),X2)
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
tff(fact_7185_finite__has__minimal2,axiom,
    ! [A2: set_rat,A: rat] :
      ( aa_set_rat_o(finite_finite_rat,A2)
     => ( aa_set_rat_o(member_rat(A),A2)
       => ? [X2: rat] :
            ( aa_set_rat_o(member_rat(X2),A2)
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X2),A)
            & ! [Xa: rat] :
                ( aa_set_rat_o(member_rat(Xa),A2)
               => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),X2)
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
tff(fact_7186_finite__has__minimal2,axiom,
    ! [A2: set_num,A: num] :
      ( aa_set_num_o(finite_finite_num,A2)
     => ( aa_set_num_o(member_num(A),A2)
       => ? [X2: num] :
            ( aa_set_num_o(member_num(X2),A2)
            & aa_num_o(ord_less_eq_num(X2),A)
            & ! [Xa: num] :
                ( aa_set_num_o(member_num(Xa),A2)
               => ( aa_num_o(ord_less_eq_num(Xa),X2)
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
tff(fact_7187_finite__has__minimal2,axiom,
    ! [A2: set_nat,A: nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_nat_o(member_nat(A),A2)
       => ? [X2: nat] :
            ( aa_set_nat_o(member_nat(X2),A2)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X2),A)
            & ! [Xa: nat] :
                ( aa_set_nat_o(member_nat(Xa),A2)
               => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),X2)
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
tff(fact_7188_finite__has__minimal2,axiom,
    ! [A2: set_int,A: int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( aa_set_int_o(member_int(A),A2)
       => ? [X2: int] :
            ( aa_set_int_o(member_int(X2),A2)
            & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X2),A)
            & ! [Xa: int] :
                ( aa_set_int_o(member_int(Xa),A2)
               => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),X2)
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
tff(fact_7189_finite__subset,axiom,
    ! [A2: set_nat,B4: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A2),B4)
     => ( aa_set_nat_o(finite_finite_nat,B4)
       => aa_set_nat_o(finite_finite_nat,A2) ) ) ).

% finite_subset
tff(fact_7190_finite__subset,axiom,
    ! [A2: set_complex,B4: set_complex] :
      ( aa_set_complex_o(ord_le211207098394363844omplex(A2),B4)
     => ( aa_set_complex_o(finite3207457112153483333omplex,B4)
       => aa_set_complex_o(finite3207457112153483333omplex,A2) ) ) ).

% finite_subset
tff(fact_7191_finite__subset,axiom,
    ! [A2: set_Code_integer,B4: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(A2),B4)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B4)
       => aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A2) ) ) ).

% finite_subset
tff(fact_7192_finite__subset,axiom,
    ! [A2: set_int,B4: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A2),B4)
     => ( aa_set_int_o(finite_finite_int,B4)
       => aa_set_int_o(finite_finite_int,A2) ) ) ).

% finite_subset
tff(fact_7193_infinite__super,axiom,
    ! [S2: set_nat,T4: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(S2),T4)
     => ( ~ aa_set_nat_o(finite_finite_nat,S2)
       => ~ aa_set_nat_o(finite_finite_nat,T4) ) ) ).

% infinite_super
tff(fact_7194_infinite__super,axiom,
    ! [S2: set_complex,T4: set_complex] :
      ( aa_set_complex_o(ord_le211207098394363844omplex(S2),T4)
     => ( ~ aa_set_complex_o(finite3207457112153483333omplex,S2)
       => ~ aa_set_complex_o(finite3207457112153483333omplex,T4) ) ) ).

% infinite_super
tff(fact_7195_infinite__super,axiom,
    ! [S2: set_Code_integer,T4: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(S2),T4)
     => ( ~ aa_set5643387332196575111eger_o(finite6017078050557962740nteger,S2)
       => ~ aa_set5643387332196575111eger_o(finite6017078050557962740nteger,T4) ) ) ).

% infinite_super
tff(fact_7196_infinite__super,axiom,
    ! [S2: set_int,T4: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(S2),T4)
     => ( ~ aa_set_int_o(finite_finite_int,S2)
       => ~ aa_set_int_o(finite_finite_int,T4) ) ) ).

% infinite_super
tff(fact_7197_rev__finite__subset,axiom,
    ! [B4: set_nat,A2: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,B4)
     => ( aa_set_nat_o(ord_less_eq_set_nat(A2),B4)
       => aa_set_nat_o(finite_finite_nat,A2) ) ) ).

% rev_finite_subset
tff(fact_7198_rev__finite__subset,axiom,
    ! [B4: set_complex,A2: set_complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,B4)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(A2),B4)
       => aa_set_complex_o(finite3207457112153483333omplex,A2) ) ) ).

% rev_finite_subset
tff(fact_7199_rev__finite__subset,axiom,
    ! [B4: set_Code_integer,A2: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B4)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(A2),B4)
       => aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A2) ) ) ).

% rev_finite_subset
tff(fact_7200_rev__finite__subset,axiom,
    ! [B4: set_int,A2: set_int] :
      ( aa_set_int_o(finite_finite_int,B4)
     => ( aa_set_int_o(ord_less_eq_set_int(A2),B4)
       => aa_set_int_o(finite_finite_int,A2) ) ) ).

% rev_finite_subset
tff(fact_7201_Diff__infinite__finite,axiom,
    ! [T4: set_int,S2: set_int] :
      ( aa_set_int_o(finite_finite_int,T4)
     => ( ~ aa_set_int_o(finite_finite_int,S2)
       => ~ aa_set_int_o(finite_finite_int,minus_minus_set_int(S2,T4)) ) ) ).

% Diff_infinite_finite
tff(fact_7202_Diff__infinite__finite,axiom,
    ! [T4: set_complex,S2: set_complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T4)
     => ( ~ aa_set_complex_o(finite3207457112153483333omplex,S2)
       => ~ aa_set_complex_o(finite3207457112153483333omplex,minus_811609699411566653omplex(S2,T4)) ) ) ).

% Diff_infinite_finite
tff(fact_7203_Diff__infinite__finite,axiom,
    ! [T4: set_Code_integer,S2: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,T4)
     => ( ~ aa_set5643387332196575111eger_o(finite6017078050557962740nteger,S2)
       => ~ aa_set5643387332196575111eger_o(finite6017078050557962740nteger,minus_2355218937544613996nteger(S2,T4)) ) ) ).

% Diff_infinite_finite
tff(fact_7204_Diff__infinite__finite,axiom,
    ! [T4: set_nat,S2: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,T4)
     => ( ~ aa_set_nat_o(finite_finite_nat,S2)
       => ~ aa_set_nat_o(finite_finite_nat,minus_minus_set_nat(S2,T4)) ) ) ).

% Diff_infinite_finite
tff(fact_7205_finite__has__minimal,axiom,
    ! [A2: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A2)
     => ( ( A2 != bot_bo3990330152332043303nteger )
       => ? [X2: code_integer] :
            ( aa_set5643387332196575111eger_o(member_Code_integer(X2),A2)
            & ! [Xa: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(Xa),A2)
               => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Xa),X2)
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_7206_finite__has__minimal,axiom,
    ! [A2: set_real] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( ( A2 != bot_bot_set_real )
       => ? [X2: real] :
            ( aa_set_real_o(member_real(X2),A2)
            & ! [Xa: real] :
                ( aa_set_real_o(member_real(Xa),A2)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),X2)
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_7207_finite__has__minimal,axiom,
    ! [A2: set_set_int] :
      ( finite6197958912794628473et_int(A2)
     => ( ( A2 != bot_bot_set_set_int )
       => ? [X2: set_int] :
            ( member_set_int(X2,A2)
            & ! [Xa: set_int] :
                ( member_set_int(Xa,A2)
               => ( aa_set_int_o(ord_less_eq_set_int(Xa),X2)
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_7208_finite__has__minimal,axiom,
    ! [A2: set_rat] :
      ( aa_set_rat_o(finite_finite_rat,A2)
     => ( ( A2 != bot_bot_set_rat )
       => ? [X2: rat] :
            ( aa_set_rat_o(member_rat(X2),A2)
            & ! [Xa: rat] :
                ( aa_set_rat_o(member_rat(Xa),A2)
               => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),X2)
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_7209_finite__has__minimal,axiom,
    ! [A2: set_num] :
      ( aa_set_num_o(finite_finite_num,A2)
     => ( ( A2 != bot_bot_set_num )
       => ? [X2: num] :
            ( aa_set_num_o(member_num(X2),A2)
            & ! [Xa: num] :
                ( aa_set_num_o(member_num(Xa),A2)
               => ( aa_num_o(ord_less_eq_num(Xa),X2)
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_7210_finite__has__minimal,axiom,
    ! [A2: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( ( A2 != bot_bot_set_nat )
       => ? [X2: nat] :
            ( aa_set_nat_o(member_nat(X2),A2)
            & ! [Xa: nat] :
                ( aa_set_nat_o(member_nat(Xa),A2)
               => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),X2)
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_7211_finite__has__minimal,axiom,
    ! [A2: set_int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( ( A2 != bot_bot_set_int )
       => ? [X2: int] :
            ( aa_set_int_o(member_int(X2),A2)
            & ! [Xa: int] :
                ( aa_set_int_o(member_int(Xa),A2)
               => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),X2)
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_7212_finite__has__maximal,axiom,
    ! [A2: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A2)
     => ( ( A2 != bot_bo3990330152332043303nteger )
       => ? [X2: code_integer] :
            ( aa_set5643387332196575111eger_o(member_Code_integer(X2),A2)
            & ! [Xa: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(Xa),A2)
               => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,X2),Xa)
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_7213_finite__has__maximal,axiom,
    ! [A2: set_real] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( ( A2 != bot_bot_set_real )
       => ? [X2: real] :
            ( aa_set_real_o(member_real(X2),A2)
            & ! [Xa: real] :
                ( aa_set_real_o(member_real(Xa),A2)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X2),Xa)
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_7214_finite__has__maximal,axiom,
    ! [A2: set_set_int] :
      ( finite6197958912794628473et_int(A2)
     => ( ( A2 != bot_bot_set_set_int )
       => ? [X2: set_int] :
            ( member_set_int(X2,A2)
            & ! [Xa: set_int] :
                ( member_set_int(Xa,A2)
               => ( aa_set_int_o(ord_less_eq_set_int(X2),Xa)
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_7215_finite__has__maximal,axiom,
    ! [A2: set_rat] :
      ( aa_set_rat_o(finite_finite_rat,A2)
     => ( ( A2 != bot_bot_set_rat )
       => ? [X2: rat] :
            ( aa_set_rat_o(member_rat(X2),A2)
            & ! [Xa: rat] :
                ( aa_set_rat_o(member_rat(Xa),A2)
               => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X2),Xa)
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_7216_finite__has__maximal,axiom,
    ! [A2: set_num] :
      ( aa_set_num_o(finite_finite_num,A2)
     => ( ( A2 != bot_bot_set_num )
       => ? [X2: num] :
            ( aa_set_num_o(member_num(X2),A2)
            & ! [Xa: num] :
                ( aa_set_num_o(member_num(Xa),A2)
               => ( aa_num_o(ord_less_eq_num(X2),Xa)
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_7217_finite__has__maximal,axiom,
    ! [A2: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( ( A2 != bot_bot_set_nat )
       => ? [X2: nat] :
            ( aa_set_nat_o(member_nat(X2),A2)
            & ! [Xa: nat] :
                ( aa_set_nat_o(member_nat(Xa),A2)
               => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X2),Xa)
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_7218_finite__has__maximal,axiom,
    ! [A2: set_int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( ( A2 != bot_bot_set_int )
       => ? [X2: int] :
            ( aa_set_int_o(member_int(X2),A2)
            & ! [Xa: int] :
                ( aa_set_int_o(member_int(Xa),A2)
               => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X2),Xa)
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_7219_finite__subset__induct_H,axiom,
    ! [F4: set_VEBT_VEBT,A2: set_VEBT_VEBT,P: fun_set_VEBT_VEBT_o] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,F4)
     => ( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(F4),A2)
       => ( aa_set_VEBT_VEBT_o(P,bot_bo8194388402131092736T_VEBT)
         => ( ! [A3: vEBT_VEBT,F5: set_VEBT_VEBT] :
                ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,F5)
               => ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A3),A2)
                 => ( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(F5),A2)
                   => ( ~ aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A3),F5)
                     => ( aa_set_VEBT_VEBT_o(P,F5)
                       => aa_set_VEBT_VEBT_o(P,insert_VEBT_VEBT(A3,F5)) ) ) ) ) )
           => aa_set_VEBT_VEBT_o(P,F4) ) ) ) ) ).

% finite_subset_induct'
tff(fact_7220_finite__subset__induct_H,axiom,
    ! [F4: set_complex,A2: set_complex,P: fun_set_complex_o] :
      ( aa_set_complex_o(finite3207457112153483333omplex,F4)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(F4),A2)
       => ( aa_set_complex_o(P,bot_bot_set_complex)
         => ( ! [A3: complex,F5: set_complex] :
                ( aa_set_complex_o(finite3207457112153483333omplex,F5)
               => ( aa_set_complex_o(member_complex(A3),A2)
                 => ( aa_set_complex_o(ord_le211207098394363844omplex(F5),A2)
                   => ( ~ aa_set_complex_o(member_complex(A3),F5)
                     => ( aa_set_complex_o(P,F5)
                       => aa_set_complex_o(P,insert_complex(A3,F5)) ) ) ) ) )
           => aa_set_complex_o(P,F4) ) ) ) ) ).

% finite_subset_induct'
tff(fact_7221_finite__subset__induct_H,axiom,
    ! [F4: set_Code_integer,A2: set_Code_integer,P: fun_se7289222521248229150eger_o] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,F4)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(F4),A2)
       => ( aa_set5643387332196575111eger_o(P,bot_bo3990330152332043303nteger)
         => ( ! [A3: code_integer,F5: set_Code_integer] :
                ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,F5)
               => ( aa_set5643387332196575111eger_o(member_Code_integer(A3),A2)
                 => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(F5),A2)
                   => ( ~ aa_set5643387332196575111eger_o(member_Code_integer(A3),F5)
                     => ( aa_set5643387332196575111eger_o(P,F5)
                       => aa_set5643387332196575111eger_o(P,insert_Code_integer(A3,F5)) ) ) ) ) )
           => aa_set5643387332196575111eger_o(P,F4) ) ) ) ) ).

% finite_subset_induct'
tff(fact_7222_finite__subset__induct_H,axiom,
    ! [F4: set_nat,A2: set_nat,P: fun_set_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,F4)
     => ( aa_set_nat_o(ord_less_eq_set_nat(F4),A2)
       => ( aa_set_nat_o(P,bot_bot_set_nat)
         => ( ! [A3: nat,F5: set_nat] :
                ( aa_set_nat_o(finite_finite_nat,F5)
               => ( aa_set_nat_o(member_nat(A3),A2)
                 => ( aa_set_nat_o(ord_less_eq_set_nat(F5),A2)
                   => ( ~ aa_set_nat_o(member_nat(A3),F5)
                     => ( aa_set_nat_o(P,F5)
                       => aa_set_nat_o(P,insert_nat(A3,F5)) ) ) ) ) )
           => aa_set_nat_o(P,F4) ) ) ) ) ).

% finite_subset_induct'
tff(fact_7223_finite__subset__induct_H,axiom,
    ! [F4: set_real,A2: set_real,P: fun_set_real_o] :
      ( aa_set_real_o(finite_finite_real,F4)
     => ( aa_set_real_o(ord_less_eq_set_real(F4),A2)
       => ( aa_set_real_o(P,bot_bot_set_real)
         => ( ! [A3: real,F5: set_real] :
                ( aa_set_real_o(finite_finite_real,F5)
               => ( aa_set_real_o(member_real(A3),A2)
                 => ( aa_set_real_o(ord_less_eq_set_real(F5),A2)
                   => ( ~ aa_set_real_o(member_real(A3),F5)
                     => ( aa_set_real_o(P,F5)
                       => aa_set_real_o(P,insert_real(A3,F5)) ) ) ) ) )
           => aa_set_real_o(P,F4) ) ) ) ) ).

% finite_subset_induct'
tff(fact_7224_finite__subset__induct_H,axiom,
    ! [F4: set_int,A2: set_int,P: fun_set_int_o] :
      ( aa_set_int_o(finite_finite_int,F4)
     => ( aa_set_int_o(ord_less_eq_set_int(F4),A2)
       => ( aa_set_int_o(P,bot_bot_set_int)
         => ( ! [A3: int,F5: set_int] :
                ( aa_set_int_o(finite_finite_int,F5)
               => ( aa_set_int_o(member_int(A3),A2)
                 => ( aa_set_int_o(ord_less_eq_set_int(F5),A2)
                   => ( ~ aa_set_int_o(member_int(A3),F5)
                     => ( aa_set_int_o(P,F5)
                       => aa_set_int_o(P,insert_int(A3,F5)) ) ) ) ) )
           => aa_set_int_o(P,F4) ) ) ) ) ).

% finite_subset_induct'
tff(fact_7225_finite__subset__induct,axiom,
    ! [F4: set_VEBT_VEBT,A2: set_VEBT_VEBT,P: fun_set_VEBT_VEBT_o] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,F4)
     => ( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(F4),A2)
       => ( aa_set_VEBT_VEBT_o(P,bot_bo8194388402131092736T_VEBT)
         => ( ! [A3: vEBT_VEBT,F5: set_VEBT_VEBT] :
                ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,F5)
               => ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A3),A2)
                 => ( ~ aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A3),F5)
                   => ( aa_set_VEBT_VEBT_o(P,F5)
                     => aa_set_VEBT_VEBT_o(P,insert_VEBT_VEBT(A3,F5)) ) ) ) )
           => aa_set_VEBT_VEBT_o(P,F4) ) ) ) ) ).

% finite_subset_induct
tff(fact_7226_finite__subset__induct,axiom,
    ! [F4: set_complex,A2: set_complex,P: fun_set_complex_o] :
      ( aa_set_complex_o(finite3207457112153483333omplex,F4)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(F4),A2)
       => ( aa_set_complex_o(P,bot_bot_set_complex)
         => ( ! [A3: complex,F5: set_complex] :
                ( aa_set_complex_o(finite3207457112153483333omplex,F5)
               => ( aa_set_complex_o(member_complex(A3),A2)
                 => ( ~ aa_set_complex_o(member_complex(A3),F5)
                   => ( aa_set_complex_o(P,F5)
                     => aa_set_complex_o(P,insert_complex(A3,F5)) ) ) ) )
           => aa_set_complex_o(P,F4) ) ) ) ) ).

% finite_subset_induct
tff(fact_7227_finite__subset__induct,axiom,
    ! [F4: set_Code_integer,A2: set_Code_integer,P: fun_se7289222521248229150eger_o] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,F4)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(F4),A2)
       => ( aa_set5643387332196575111eger_o(P,bot_bo3990330152332043303nteger)
         => ( ! [A3: code_integer,F5: set_Code_integer] :
                ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,F5)
               => ( aa_set5643387332196575111eger_o(member_Code_integer(A3),A2)
                 => ( ~ aa_set5643387332196575111eger_o(member_Code_integer(A3),F5)
                   => ( aa_set5643387332196575111eger_o(P,F5)
                     => aa_set5643387332196575111eger_o(P,insert_Code_integer(A3,F5)) ) ) ) )
           => aa_set5643387332196575111eger_o(P,F4) ) ) ) ) ).

% finite_subset_induct
tff(fact_7228_finite__subset__induct,axiom,
    ! [F4: set_nat,A2: set_nat,P: fun_set_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,F4)
     => ( aa_set_nat_o(ord_less_eq_set_nat(F4),A2)
       => ( aa_set_nat_o(P,bot_bot_set_nat)
         => ( ! [A3: nat,F5: set_nat] :
                ( aa_set_nat_o(finite_finite_nat,F5)
               => ( aa_set_nat_o(member_nat(A3),A2)
                 => ( ~ aa_set_nat_o(member_nat(A3),F5)
                   => ( aa_set_nat_o(P,F5)
                     => aa_set_nat_o(P,insert_nat(A3,F5)) ) ) ) )
           => aa_set_nat_o(P,F4) ) ) ) ) ).

% finite_subset_induct
tff(fact_7229_finite__subset__induct,axiom,
    ! [F4: set_real,A2: set_real,P: fun_set_real_o] :
      ( aa_set_real_o(finite_finite_real,F4)
     => ( aa_set_real_o(ord_less_eq_set_real(F4),A2)
       => ( aa_set_real_o(P,bot_bot_set_real)
         => ( ! [A3: real,F5: set_real] :
                ( aa_set_real_o(finite_finite_real,F5)
               => ( aa_set_real_o(member_real(A3),A2)
                 => ( ~ aa_set_real_o(member_real(A3),F5)
                   => ( aa_set_real_o(P,F5)
                     => aa_set_real_o(P,insert_real(A3,F5)) ) ) ) )
           => aa_set_real_o(P,F4) ) ) ) ) ).

% finite_subset_induct
tff(fact_7230_finite__subset__induct,axiom,
    ! [F4: set_int,A2: set_int,P: fun_set_int_o] :
      ( aa_set_int_o(finite_finite_int,F4)
     => ( aa_set_int_o(ord_less_eq_set_int(F4),A2)
       => ( aa_set_int_o(P,bot_bot_set_int)
         => ( ! [A3: int,F5: set_int] :
                ( aa_set_int_o(finite_finite_int,F5)
               => ( aa_set_int_o(member_int(A3),A2)
                 => ( ~ aa_set_int_o(member_int(A3),F5)
                   => ( aa_set_int_o(P,F5)
                     => aa_set_int_o(P,insert_int(A3,F5)) ) ) ) )
           => aa_set_int_o(P,F4) ) ) ) ) ).

% finite_subset_induct
tff(fact_7231_infinite__remove,axiom,
    ! [S2: set_VEBT_VEBT,A: vEBT_VEBT] :
      ( ~ aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,S2)
     => ~ aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,minus_5127226145743854075T_VEBT(S2,insert_VEBT_VEBT(A,bot_bo8194388402131092736T_VEBT))) ) ).

% infinite_remove
tff(fact_7232_infinite__remove,axiom,
    ! [S2: set_complex,A: complex] :
      ( ~ aa_set_complex_o(finite3207457112153483333omplex,S2)
     => ~ aa_set_complex_o(finite3207457112153483333omplex,minus_811609699411566653omplex(S2,insert_complex(A,bot_bot_set_complex))) ) ).

% infinite_remove
tff(fact_7233_infinite__remove,axiom,
    ! [S2: set_Code_integer,A: code_integer] :
      ( ~ aa_set5643387332196575111eger_o(finite6017078050557962740nteger,S2)
     => ~ aa_set5643387332196575111eger_o(finite6017078050557962740nteger,minus_2355218937544613996nteger(S2,insert_Code_integer(A,bot_bo3990330152332043303nteger))) ) ).

% infinite_remove
tff(fact_7234_infinite__remove,axiom,
    ! [S2: set_int,A: int] :
      ( ~ aa_set_int_o(finite_finite_int,S2)
     => ~ aa_set_int_o(finite_finite_int,minus_minus_set_int(S2,insert_int(A,bot_bot_set_int))) ) ).

% infinite_remove
tff(fact_7235_infinite__remove,axiom,
    ! [S2: set_real,A: real] :
      ( ~ aa_set_real_o(finite_finite_real,S2)
     => ~ aa_set_real_o(finite_finite_real,minus_minus_set_real(S2,insert_real(A,bot_bot_set_real))) ) ).

% infinite_remove
tff(fact_7236_infinite__remove,axiom,
    ! [S2: set_nat,A: nat] :
      ( ~ aa_set_nat_o(finite_finite_nat,S2)
     => ~ aa_set_nat_o(finite_finite_nat,minus_minus_set_nat(S2,insert_nat(A,bot_bot_set_nat))) ) ).

% infinite_remove
tff(fact_7237_infinite__coinduct,axiom,
    ! [X5: fun_set_VEBT_VEBT_o,A2: set_VEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(X5,A2)
     => ( ! [A7: set_VEBT_VEBT] :
            ( aa_set_VEBT_VEBT_o(X5,A7)
           => ? [X3: vEBT_VEBT] :
                ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X3),A7)
                & ( aa_set_VEBT_VEBT_o(X5,minus_5127226145743854075T_VEBT(A7,insert_VEBT_VEBT(X3,bot_bo8194388402131092736T_VEBT)))
                  | ~ aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,minus_5127226145743854075T_VEBT(A7,insert_VEBT_VEBT(X3,bot_bo8194388402131092736T_VEBT))) ) ) )
       => ~ aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,A2) ) ) ).

% infinite_coinduct
tff(fact_7238_infinite__coinduct,axiom,
    ! [X5: fun_set_complex_o,A2: set_complex] :
      ( aa_set_complex_o(X5,A2)
     => ( ! [A7: set_complex] :
            ( aa_set_complex_o(X5,A7)
           => ? [X3: complex] :
                ( aa_set_complex_o(member_complex(X3),A7)
                & ( aa_set_complex_o(X5,minus_811609699411566653omplex(A7,insert_complex(X3,bot_bot_set_complex)))
                  | ~ aa_set_complex_o(finite3207457112153483333omplex,minus_811609699411566653omplex(A7,insert_complex(X3,bot_bot_set_complex))) ) ) )
       => ~ aa_set_complex_o(finite3207457112153483333omplex,A2) ) ) ).

% infinite_coinduct
tff(fact_7239_infinite__coinduct,axiom,
    ! [X5: fun_se7289222521248229150eger_o,A2: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(X5,A2)
     => ( ! [A7: set_Code_integer] :
            ( aa_set5643387332196575111eger_o(X5,A7)
           => ? [X3: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(X3),A7)
                & ( aa_set5643387332196575111eger_o(X5,minus_2355218937544613996nteger(A7,insert_Code_integer(X3,bot_bo3990330152332043303nteger)))
                  | ~ aa_set5643387332196575111eger_o(finite6017078050557962740nteger,minus_2355218937544613996nteger(A7,insert_Code_integer(X3,bot_bo3990330152332043303nteger))) ) ) )
       => ~ aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A2) ) ) ).

% infinite_coinduct
tff(fact_7240_infinite__coinduct,axiom,
    ! [X5: fun_set_int_o,A2: set_int] :
      ( aa_set_int_o(X5,A2)
     => ( ! [A7: set_int] :
            ( aa_set_int_o(X5,A7)
           => ? [X3: int] :
                ( aa_set_int_o(member_int(X3),A7)
                & ( aa_set_int_o(X5,minus_minus_set_int(A7,insert_int(X3,bot_bot_set_int)))
                  | ~ aa_set_int_o(finite_finite_int,minus_minus_set_int(A7,insert_int(X3,bot_bot_set_int))) ) ) )
       => ~ aa_set_int_o(finite_finite_int,A2) ) ) ).

% infinite_coinduct
tff(fact_7241_infinite__coinduct,axiom,
    ! [X5: fun_set_real_o,A2: set_real] :
      ( aa_set_real_o(X5,A2)
     => ( ! [A7: set_real] :
            ( aa_set_real_o(X5,A7)
           => ? [X3: real] :
                ( aa_set_real_o(member_real(X3),A7)
                & ( aa_set_real_o(X5,minus_minus_set_real(A7,insert_real(X3,bot_bot_set_real)))
                  | ~ aa_set_real_o(finite_finite_real,minus_minus_set_real(A7,insert_real(X3,bot_bot_set_real))) ) ) )
       => ~ aa_set_real_o(finite_finite_real,A2) ) ) ).

% infinite_coinduct
tff(fact_7242_infinite__coinduct,axiom,
    ! [X5: fun_set_nat_o,A2: set_nat] :
      ( aa_set_nat_o(X5,A2)
     => ( ! [A7: set_nat] :
            ( aa_set_nat_o(X5,A7)
           => ? [X3: nat] :
                ( aa_set_nat_o(member_nat(X3),A7)
                & ( aa_set_nat_o(X5,minus_minus_set_nat(A7,insert_nat(X3,bot_bot_set_nat)))
                  | ~ aa_set_nat_o(finite_finite_nat,minus_minus_set_nat(A7,insert_nat(X3,bot_bot_set_nat))) ) ) )
       => ~ aa_set_nat_o(finite_finite_nat,A2) ) ) ).

% infinite_coinduct
tff(fact_7243_finite__empty__induct,axiom,
    ! [A2: set_VEBT_VEBT,P: fun_set_VEBT_VEBT_o] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,A2)
     => ( aa_set_VEBT_VEBT_o(P,A2)
       => ( ! [A3: vEBT_VEBT,A7: set_VEBT_VEBT] :
              ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,A7)
             => ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A3),A7)
               => ( aa_set_VEBT_VEBT_o(P,A7)
                 => aa_set_VEBT_VEBT_o(P,minus_5127226145743854075T_VEBT(A7,insert_VEBT_VEBT(A3,bot_bo8194388402131092736T_VEBT))) ) ) )
         => aa_set_VEBT_VEBT_o(P,bot_bo8194388402131092736T_VEBT) ) ) ) ).

% finite_empty_induct
tff(fact_7244_finite__empty__induct,axiom,
    ! [A2: set_complex,P: fun_set_complex_o] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( aa_set_complex_o(P,A2)
       => ( ! [A3: complex,A7: set_complex] :
              ( aa_set_complex_o(finite3207457112153483333omplex,A7)
             => ( aa_set_complex_o(member_complex(A3),A7)
               => ( aa_set_complex_o(P,A7)
                 => aa_set_complex_o(P,minus_811609699411566653omplex(A7,insert_complex(A3,bot_bot_set_complex))) ) ) )
         => aa_set_complex_o(P,bot_bot_set_complex) ) ) ) ).

% finite_empty_induct
tff(fact_7245_finite__empty__induct,axiom,
    ! [A2: set_Code_integer,P: fun_se7289222521248229150eger_o] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A2)
     => ( aa_set5643387332196575111eger_o(P,A2)
       => ( ! [A3: code_integer,A7: set_Code_integer] :
              ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A7)
             => ( aa_set5643387332196575111eger_o(member_Code_integer(A3),A7)
               => ( aa_set5643387332196575111eger_o(P,A7)
                 => aa_set5643387332196575111eger_o(P,minus_2355218937544613996nteger(A7,insert_Code_integer(A3,bot_bo3990330152332043303nteger))) ) ) )
         => aa_set5643387332196575111eger_o(P,bot_bo3990330152332043303nteger) ) ) ) ).

% finite_empty_induct
tff(fact_7246_finite__empty__induct,axiom,
    ! [A2: set_int,P: fun_set_int_o] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( aa_set_int_o(P,A2)
       => ( ! [A3: int,A7: set_int] :
              ( aa_set_int_o(finite_finite_int,A7)
             => ( aa_set_int_o(member_int(A3),A7)
               => ( aa_set_int_o(P,A7)
                 => aa_set_int_o(P,minus_minus_set_int(A7,insert_int(A3,bot_bot_set_int))) ) ) )
         => aa_set_int_o(P,bot_bot_set_int) ) ) ) ).

% finite_empty_induct
tff(fact_7247_finite__empty__induct,axiom,
    ! [A2: set_real,P: fun_set_real_o] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( aa_set_real_o(P,A2)
       => ( ! [A3: real,A7: set_real] :
              ( aa_set_real_o(finite_finite_real,A7)
             => ( aa_set_real_o(member_real(A3),A7)
               => ( aa_set_real_o(P,A7)
                 => aa_set_real_o(P,minus_minus_set_real(A7,insert_real(A3,bot_bot_set_real))) ) ) )
         => aa_set_real_o(P,bot_bot_set_real) ) ) ) ).

% finite_empty_induct
tff(fact_7248_finite__empty__induct,axiom,
    ! [A2: set_nat,P: fun_set_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_nat_o(P,A2)
       => ( ! [A3: nat,A7: set_nat] :
              ( aa_set_nat_o(finite_finite_nat,A7)
             => ( aa_set_nat_o(member_nat(A3),A7)
               => ( aa_set_nat_o(P,A7)
                 => aa_set_nat_o(P,minus_minus_set_nat(A7,insert_nat(A3,bot_bot_set_nat))) ) ) )
         => aa_set_nat_o(P,bot_bot_set_nat) ) ) ) ).

% finite_empty_induct
tff(fact_7249_remove__induct,axiom,
    ! [P: fun_set_VEBT_VEBT_o,B4: set_VEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(P,bot_bo8194388402131092736T_VEBT)
     => ( ( ~ aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,B4)
         => aa_set_VEBT_VEBT_o(P,B4) )
       => ( ! [A7: set_VEBT_VEBT] :
              ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,A7)
             => ( ( A7 != bot_bo8194388402131092736T_VEBT )
               => ( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(A7),B4)
                 => ( ! [X3: vEBT_VEBT] :
                        ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X3),A7)
                       => aa_set_VEBT_VEBT_o(P,minus_5127226145743854075T_VEBT(A7,insert_VEBT_VEBT(X3,bot_bo8194388402131092736T_VEBT))) )
                   => aa_set_VEBT_VEBT_o(P,A7) ) ) ) )
         => aa_set_VEBT_VEBT_o(P,B4) ) ) ) ).

% remove_induct
tff(fact_7250_remove__induct,axiom,
    ! [P: fun_set_complex_o,B4: set_complex] :
      ( aa_set_complex_o(P,bot_bot_set_complex)
     => ( ( ~ aa_set_complex_o(finite3207457112153483333omplex,B4)
         => aa_set_complex_o(P,B4) )
       => ( ! [A7: set_complex] :
              ( aa_set_complex_o(finite3207457112153483333omplex,A7)
             => ( ( A7 != bot_bot_set_complex )
               => ( aa_set_complex_o(ord_le211207098394363844omplex(A7),B4)
                 => ( ! [X3: complex] :
                        ( aa_set_complex_o(member_complex(X3),A7)
                       => aa_set_complex_o(P,minus_811609699411566653omplex(A7,insert_complex(X3,bot_bot_set_complex))) )
                   => aa_set_complex_o(P,A7) ) ) ) )
         => aa_set_complex_o(P,B4) ) ) ) ).

% remove_induct
tff(fact_7251_remove__induct,axiom,
    ! [P: fun_se7289222521248229150eger_o,B4: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(P,bot_bo3990330152332043303nteger)
     => ( ( ~ aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B4)
         => aa_set5643387332196575111eger_o(P,B4) )
       => ( ! [A7: set_Code_integer] :
              ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A7)
             => ( ( A7 != bot_bo3990330152332043303nteger )
               => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(A7),B4)
                 => ( ! [X3: code_integer] :
                        ( aa_set5643387332196575111eger_o(member_Code_integer(X3),A7)
                       => aa_set5643387332196575111eger_o(P,minus_2355218937544613996nteger(A7,insert_Code_integer(X3,bot_bo3990330152332043303nteger))) )
                   => aa_set5643387332196575111eger_o(P,A7) ) ) ) )
         => aa_set5643387332196575111eger_o(P,B4) ) ) ) ).

% remove_induct
tff(fact_7252_remove__induct,axiom,
    ! [P: fun_set_real_o,B4: set_real] :
      ( aa_set_real_o(P,bot_bot_set_real)
     => ( ( ~ aa_set_real_o(finite_finite_real,B4)
         => aa_set_real_o(P,B4) )
       => ( ! [A7: set_real] :
              ( aa_set_real_o(finite_finite_real,A7)
             => ( ( A7 != bot_bot_set_real )
               => ( aa_set_real_o(ord_less_eq_set_real(A7),B4)
                 => ( ! [X3: real] :
                        ( aa_set_real_o(member_real(X3),A7)
                       => aa_set_real_o(P,minus_minus_set_real(A7,insert_real(X3,bot_bot_set_real))) )
                   => aa_set_real_o(P,A7) ) ) ) )
         => aa_set_real_o(P,B4) ) ) ) ).

% remove_induct
tff(fact_7253_remove__induct,axiom,
    ! [P: fun_set_nat_o,B4: set_nat] :
      ( aa_set_nat_o(P,bot_bot_set_nat)
     => ( ( ~ aa_set_nat_o(finite_finite_nat,B4)
         => aa_set_nat_o(P,B4) )
       => ( ! [A7: set_nat] :
              ( aa_set_nat_o(finite_finite_nat,A7)
             => ( ( A7 != bot_bot_set_nat )
               => ( aa_set_nat_o(ord_less_eq_set_nat(A7),B4)
                 => ( ! [X3: nat] :
                        ( aa_set_nat_o(member_nat(X3),A7)
                       => aa_set_nat_o(P,minus_minus_set_nat(A7,insert_nat(X3,bot_bot_set_nat))) )
                   => aa_set_nat_o(P,A7) ) ) ) )
         => aa_set_nat_o(P,B4) ) ) ) ).

% remove_induct
tff(fact_7254_remove__induct,axiom,
    ! [P: fun_set_int_o,B4: set_int] :
      ( aa_set_int_o(P,bot_bot_set_int)
     => ( ( ~ aa_set_int_o(finite_finite_int,B4)
         => aa_set_int_o(P,B4) )
       => ( ! [A7: set_int] :
              ( aa_set_int_o(finite_finite_int,A7)
             => ( ( A7 != bot_bot_set_int )
               => ( aa_set_int_o(ord_less_eq_set_int(A7),B4)
                 => ( ! [X3: int] :
                        ( aa_set_int_o(member_int(X3),A7)
                       => aa_set_int_o(P,minus_minus_set_int(A7,insert_int(X3,bot_bot_set_int))) )
                   => aa_set_int_o(P,A7) ) ) ) )
         => aa_set_int_o(P,B4) ) ) ) ).

% remove_induct
tff(fact_7255_finite__remove__induct,axiom,
    ! [B4: set_VEBT_VEBT,P: fun_set_VEBT_VEBT_o] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,B4)
     => ( aa_set_VEBT_VEBT_o(P,bot_bo8194388402131092736T_VEBT)
       => ( ! [A7: set_VEBT_VEBT] :
              ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,A7)
             => ( ( A7 != bot_bo8194388402131092736T_VEBT )
               => ( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(A7),B4)
                 => ( ! [X3: vEBT_VEBT] :
                        ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X3),A7)
                       => aa_set_VEBT_VEBT_o(P,minus_5127226145743854075T_VEBT(A7,insert_VEBT_VEBT(X3,bot_bo8194388402131092736T_VEBT))) )
                   => aa_set_VEBT_VEBT_o(P,A7) ) ) ) )
         => aa_set_VEBT_VEBT_o(P,B4) ) ) ) ).

% finite_remove_induct
tff(fact_7256_finite__remove__induct,axiom,
    ! [B4: set_complex,P: fun_set_complex_o] :
      ( aa_set_complex_o(finite3207457112153483333omplex,B4)
     => ( aa_set_complex_o(P,bot_bot_set_complex)
       => ( ! [A7: set_complex] :
              ( aa_set_complex_o(finite3207457112153483333omplex,A7)
             => ( ( A7 != bot_bot_set_complex )
               => ( aa_set_complex_o(ord_le211207098394363844omplex(A7),B4)
                 => ( ! [X3: complex] :
                        ( aa_set_complex_o(member_complex(X3),A7)
                       => aa_set_complex_o(P,minus_811609699411566653omplex(A7,insert_complex(X3,bot_bot_set_complex))) )
                   => aa_set_complex_o(P,A7) ) ) ) )
         => aa_set_complex_o(P,B4) ) ) ) ).

% finite_remove_induct
tff(fact_7257_finite__remove__induct,axiom,
    ! [B4: set_Code_integer,P: fun_se7289222521248229150eger_o] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B4)
     => ( aa_set5643387332196575111eger_o(P,bot_bo3990330152332043303nteger)
       => ( ! [A7: set_Code_integer] :
              ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A7)
             => ( ( A7 != bot_bo3990330152332043303nteger )
               => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(A7),B4)
                 => ( ! [X3: code_integer] :
                        ( aa_set5643387332196575111eger_o(member_Code_integer(X3),A7)
                       => aa_set5643387332196575111eger_o(P,minus_2355218937544613996nteger(A7,insert_Code_integer(X3,bot_bo3990330152332043303nteger))) )
                   => aa_set5643387332196575111eger_o(P,A7) ) ) ) )
         => aa_set5643387332196575111eger_o(P,B4) ) ) ) ).

% finite_remove_induct
tff(fact_7258_finite__remove__induct,axiom,
    ! [B4: set_real,P: fun_set_real_o] :
      ( aa_set_real_o(finite_finite_real,B4)
     => ( aa_set_real_o(P,bot_bot_set_real)
       => ( ! [A7: set_real] :
              ( aa_set_real_o(finite_finite_real,A7)
             => ( ( A7 != bot_bot_set_real )
               => ( aa_set_real_o(ord_less_eq_set_real(A7),B4)
                 => ( ! [X3: real] :
                        ( aa_set_real_o(member_real(X3),A7)
                       => aa_set_real_o(P,minus_minus_set_real(A7,insert_real(X3,bot_bot_set_real))) )
                   => aa_set_real_o(P,A7) ) ) ) )
         => aa_set_real_o(P,B4) ) ) ) ).

% finite_remove_induct
tff(fact_7259_finite__remove__induct,axiom,
    ! [B4: set_nat,P: fun_set_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,B4)
     => ( aa_set_nat_o(P,bot_bot_set_nat)
       => ( ! [A7: set_nat] :
              ( aa_set_nat_o(finite_finite_nat,A7)
             => ( ( A7 != bot_bot_set_nat )
               => ( aa_set_nat_o(ord_less_eq_set_nat(A7),B4)
                 => ( ! [X3: nat] :
                        ( aa_set_nat_o(member_nat(X3),A7)
                       => aa_set_nat_o(P,minus_minus_set_nat(A7,insert_nat(X3,bot_bot_set_nat))) )
                   => aa_set_nat_o(P,A7) ) ) ) )
         => aa_set_nat_o(P,B4) ) ) ) ).

% finite_remove_induct
tff(fact_7260_finite__remove__induct,axiom,
    ! [B4: set_int,P: fun_set_int_o] :
      ( aa_set_int_o(finite_finite_int,B4)
     => ( aa_set_int_o(P,bot_bot_set_int)
       => ( ! [A7: set_int] :
              ( aa_set_int_o(finite_finite_int,A7)
             => ( ( A7 != bot_bot_set_int )
               => ( aa_set_int_o(ord_less_eq_set_int(A7),B4)
                 => ( ! [X3: int] :
                        ( aa_set_int_o(member_int(X3),A7)
                       => aa_set_int_o(P,minus_minus_set_int(A7,insert_int(X3,bot_bot_set_int))) )
                   => aa_set_int_o(P,A7) ) ) ) )
         => aa_set_int_o(P,B4) ) ) ) ).

% finite_remove_induct
tff(fact_7261_finite__induct__select,axiom,
    ! [S2: set_VEBT_VEBT,P: fun_set_VEBT_VEBT_o] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,S2)
     => ( aa_set_VEBT_VEBT_o(P,bot_bo8194388402131092736T_VEBT)
       => ( ! [T5: set_VEBT_VEBT] :
              ( aa_set_VEBT_VEBT_o(ord_le3480810397992357184T_VEBT(T5),S2)
             => ( aa_set_VEBT_VEBT_o(P,T5)
               => ? [X3: vEBT_VEBT] :
                    ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X3),minus_5127226145743854075T_VEBT(S2,T5))
                    & aa_set_VEBT_VEBT_o(P,insert_VEBT_VEBT(X3,T5)) ) ) )
         => aa_set_VEBT_VEBT_o(P,S2) ) ) ) ).

% finite_induct_select
tff(fact_7262_finite__induct__select,axiom,
    ! [S2: set_complex,P: fun_set_complex_o] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S2)
     => ( aa_set_complex_o(P,bot_bot_set_complex)
       => ( ! [T5: set_complex] :
              ( aa_set_complex_o(ord_less_set_complex(T5),S2)
             => ( aa_set_complex_o(P,T5)
               => ? [X3: complex] :
                    ( aa_set_complex_o(member_complex(X3),minus_811609699411566653omplex(S2,T5))
                    & aa_set_complex_o(P,insert_complex(X3,T5)) ) ) )
         => aa_set_complex_o(P,S2) ) ) ) ).

% finite_induct_select
tff(fact_7263_finite__induct__select,axiom,
    ! [S2: set_Code_integer,P: fun_se7289222521248229150eger_o] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,S2)
     => ( aa_set5643387332196575111eger_o(P,bot_bo3990330152332043303nteger)
       => ( ! [T5: set_Code_integer] :
              ( aa_set5643387332196575111eger_o(ord_le1307284697595431911nteger(T5),S2)
             => ( aa_set5643387332196575111eger_o(P,T5)
               => ? [X3: code_integer] :
                    ( aa_set5643387332196575111eger_o(member_Code_integer(X3),minus_2355218937544613996nteger(S2,T5))
                    & aa_set5643387332196575111eger_o(P,insert_Code_integer(X3,T5)) ) ) )
         => aa_set5643387332196575111eger_o(P,S2) ) ) ) ).

% finite_induct_select
tff(fact_7264_finite__induct__select,axiom,
    ! [S2: set_int,P: fun_set_int_o] :
      ( aa_set_int_o(finite_finite_int,S2)
     => ( aa_set_int_o(P,bot_bot_set_int)
       => ( ! [T5: set_int] :
              ( aa_set_int_o(ord_less_set_int(T5),S2)
             => ( aa_set_int_o(P,T5)
               => ? [X3: int] :
                    ( aa_set_int_o(member_int(X3),minus_minus_set_int(S2,T5))
                    & aa_set_int_o(P,insert_int(X3,T5)) ) ) )
         => aa_set_int_o(P,S2) ) ) ) ).

% finite_induct_select
tff(fact_7265_finite__induct__select,axiom,
    ! [S2: set_real,P: fun_set_real_o] :
      ( aa_set_real_o(finite_finite_real,S2)
     => ( aa_set_real_o(P,bot_bot_set_real)
       => ( ! [T5: set_real] :
              ( aa_set_real_o(ord_less_set_real(T5),S2)
             => ( aa_set_real_o(P,T5)
               => ? [X3: real] :
                    ( aa_set_real_o(member_real(X3),minus_minus_set_real(S2,T5))
                    & aa_set_real_o(P,insert_real(X3,T5)) ) ) )
         => aa_set_real_o(P,S2) ) ) ) ).

% finite_induct_select
tff(fact_7266_finite__induct__select,axiom,
    ! [S2: set_nat,P: fun_set_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,S2)
     => ( aa_set_nat_o(P,bot_bot_set_nat)
       => ( ! [T5: set_nat] :
              ( aa_set_nat_o(ord_less_set_nat(T5),S2)
             => ( aa_set_nat_o(P,T5)
               => ? [X3: nat] :
                    ( aa_set_nat_o(member_nat(X3),minus_minus_set_nat(S2,T5))
                    & aa_set_nat_o(P,insert_nat(X3,T5)) ) ) )
         => aa_set_nat_o(P,S2) ) ) ) ).

% finite_induct_select
tff(fact_7267_finite__nth__roots,axiom,
    ! [Nb: nat,C: complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aa_com5808801942499186519plex_o(aTP_Lamm_afd(Nb),C))) ) ).

% finite_nth_roots
tff(fact_7268_set__encode__insert,axiom,
    ! [A2: set_nat,Nb: nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( ~ aa_set_nat_o(member_nat(Nb),A2)
       => ( aa_set_nat_nat(nat_set_encode,insert_nat(Nb,A2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)),aa_set_nat_nat(nat_set_encode,A2)) ) ) ) ).

% set_encode_insert
tff(fact_7269_diff__preserves__multiset,axiom,
    ! [M8: fun_Pr5160692531401422620nt_nat,N: fun_Pr5160692531401422620nt_nat] :
      ( finite2998713641127702882nt_int(collec213857154873943460nt_int(aTP_Lamm_afe(M8)))
     => finite2998713641127702882nt_int(collec213857154873943460nt_int(aa_fun6302685600795359042_int_o(aTP_Lamm_aff(M8),N))) ) ).

% diff_preserves_multiset
tff(fact_7270_diff__preserves__multiset,axiom,
    ! [M8: fun_nat_nat,N: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,collect_nat(aTP_Lamm_afg(M8)))
     => aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun8673969331427483654_nat_o(aTP_Lamm_afh(M8),N))) ) ).

% diff_preserves_multiset
tff(fact_7271_diff__preserves__multiset,axiom,
    ! [M8: fun_int_nat,N: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,collect_int(aTP_Lamm_afi(M8)))
     => aa_set_int_o(finite_finite_int,collect_int(aa_fun5425778157628386638_int_o(aTP_Lamm_afj(M8),N))) ) ).

% diff_preserves_multiset
tff(fact_7272_diff__preserves__multiset,axiom,
    ! [M8: fun_complex_nat,N: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aTP_Lamm_afk(M8)))
     => aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aa_fun8723867720292580170plex_o(aTP_Lamm_afl(M8),N))) ) ).

% diff_preserves_multiset
tff(fact_7273_diff__preserves__multiset,axiom,
    ! [M8: fun_Code_integer_nat,N: fun_Code_integer_nat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,collect_Code_integer(aTP_Lamm_afm(M8)))
     => aa_set5643387332196575111eger_o(finite6017078050557962740nteger,collect_Code_integer(aa_fun4054834855839773292eger_o(aTP_Lamm_afn(M8),N))) ) ).

% diff_preserves_multiset
tff(fact_7274_add__mset__in__multiset,axiom,
    ! [M8: fun_Pr5160692531401422620nt_nat,A: product_prod_int_int] :
      ( finite2998713641127702882nt_int(collec213857154873943460nt_int(aTP_Lamm_afe(M8)))
     => finite2998713641127702882nt_int(collec213857154873943460nt_int(aa_Pro3952998155787061599_int_o(aTP_Lamm_afo(M8),A))) ) ).

% add_mset_in_multiset
tff(fact_7275_add__mset__in__multiset,axiom,
    ! [M8: fun_nat_nat,A: nat] :
      ( aa_set_nat_o(finite_finite_nat,collect_nat(aTP_Lamm_afg(M8)))
     => aa_set_nat_o(finite_finite_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_afp(M8),A))) ) ).

% add_mset_in_multiset
tff(fact_7276_add__mset__in__multiset,axiom,
    ! [M8: fun_int_nat,A: int] :
      ( aa_set_int_o(finite_finite_int,collect_int(aTP_Lamm_afi(M8)))
     => aa_set_int_o(finite_finite_int,collect_int(aa_int_fun_int_o(aTP_Lamm_afq(M8),A))) ) ).

% add_mset_in_multiset
tff(fact_7277_add__mset__in__multiset,axiom,
    ! [M8: fun_complex_nat,A: complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aTP_Lamm_afk(M8)))
     => aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aa_com5808801942499186519plex_o(aTP_Lamm_afr(M8),A))) ) ).

% add_mset_in_multiset
tff(fact_7278_add__mset__in__multiset,axiom,
    ! [M8: fun_Code_integer_nat,A: code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,collect_Code_integer(aTP_Lamm_afm(M8)))
     => aa_set5643387332196575111eger_o(finite6017078050557962740nteger,collect_Code_integer(aa_Cod326357343816142457eger_o(aTP_Lamm_afs(M8),A))) ) ).

% add_mset_in_multiset
tff(fact_7279_finite__linorder__max__induct,axiom,
    ! [A2: set_Code_integer,P: fun_se7289222521248229150eger_o] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A2)
     => ( aa_set5643387332196575111eger_o(P,bot_bo3990330152332043303nteger)
       => ( ! [B2: code_integer,A7: set_Code_integer] :
              ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A7)
             => ( ! [X3: code_integer] :
                    ( aa_set5643387332196575111eger_o(member_Code_integer(X3),A7)
                   => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,X3),B2) )
               => ( aa_set5643387332196575111eger_o(P,A7)
                 => aa_set5643387332196575111eger_o(P,insert_Code_integer(B2,A7)) ) ) )
         => aa_set5643387332196575111eger_o(P,A2) ) ) ) ).

% finite_linorder_max_induct
tff(fact_7280_finite__linorder__max__induct,axiom,
    ! [A2: set_real,P: fun_set_real_o] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( aa_set_real_o(P,bot_bot_set_real)
       => ( ! [B2: real,A7: set_real] :
              ( aa_set_real_o(finite_finite_real,A7)
             => ( ! [X3: real] :
                    ( aa_set_real_o(member_real(X3),A7)
                   => aa_real_o(aa_real_fun_real_o(ord_less_real,X3),B2) )
               => ( aa_set_real_o(P,A7)
                 => aa_set_real_o(P,insert_real(B2,A7)) ) ) )
         => aa_set_real_o(P,A2) ) ) ) ).

% finite_linorder_max_induct
tff(fact_7281_finite__linorder__max__induct,axiom,
    ! [A2: set_rat,P: fun_set_rat_o] :
      ( aa_set_rat_o(finite_finite_rat,A2)
     => ( aa_set_rat_o(P,bot_bot_set_rat)
       => ( ! [B2: rat,A7: set_rat] :
              ( aa_set_rat_o(finite_finite_rat,A7)
             => ( ! [X3: rat] :
                    ( aa_set_rat_o(member_rat(X3),A7)
                   => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),B2) )
               => ( aa_set_rat_o(P,A7)
                 => aa_set_rat_o(P,insert_rat(B2,A7)) ) ) )
         => aa_set_rat_o(P,A2) ) ) ) ).

% finite_linorder_max_induct
tff(fact_7282_finite__linorder__max__induct,axiom,
    ! [A2: set_num,P: fun_set_num_o] :
      ( aa_set_num_o(finite_finite_num,A2)
     => ( aa_set_num_o(P,bot_bot_set_num)
       => ( ! [B2: num,A7: set_num] :
              ( aa_set_num_o(finite_finite_num,A7)
             => ( ! [X3: num] :
                    ( aa_set_num_o(member_num(X3),A7)
                   => aa_num_o(ord_less_num(X3),B2) )
               => ( aa_set_num_o(P,A7)
                 => aa_set_num_o(P,insert_num(B2,A7)) ) ) )
         => aa_set_num_o(P,A2) ) ) ) ).

% finite_linorder_max_induct
tff(fact_7283_finite__linorder__max__induct,axiom,
    ! [A2: set_nat,P: fun_set_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_nat_o(P,bot_bot_set_nat)
       => ( ! [B2: nat,A7: set_nat] :
              ( aa_set_nat_o(finite_finite_nat,A7)
             => ( ! [X3: nat] :
                    ( aa_set_nat_o(member_nat(X3),A7)
                   => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),B2) )
               => ( aa_set_nat_o(P,A7)
                 => aa_set_nat_o(P,insert_nat(B2,A7)) ) ) )
         => aa_set_nat_o(P,A2) ) ) ) ).

% finite_linorder_max_induct
tff(fact_7284_finite__linorder__max__induct,axiom,
    ! [A2: set_int,P: fun_set_int_o] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( aa_set_int_o(P,bot_bot_set_int)
       => ( ! [B2: int,A7: set_int] :
              ( aa_set_int_o(finite_finite_int,A7)
             => ( ! [X3: int] :
                    ( aa_set_int_o(member_int(X3),A7)
                   => aa_int_o(aa_int_fun_int_o(ord_less_int,X3),B2) )
               => ( aa_set_int_o(P,A7)
                 => aa_set_int_o(P,insert_int(B2,A7)) ) ) )
         => aa_set_int_o(P,A2) ) ) ) ).

% finite_linorder_max_induct
tff(fact_7285_complex__mod__triangle__ineq2,axiom,
    ! [B: complex,A: complex] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,B),A))),aa_complex_real(real_V1022390504157884413omplex,B))),aa_complex_real(real_V1022390504157884413omplex,A)) ).

% complex_mod_triangle_ineq2
tff(fact_7286_even__set__encode__iff,axiom,
    ! [A2: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_set_nat_nat(nat_set_encode,A2))
      <=> ~ aa_set_nat_o(member_nat(zero_zero_nat),A2) ) ) ).

% even_set_encode_iff
tff(fact_7287_infinite__growing,axiom,
    ! [X5: set_Code_integer] :
      ( ( X5 != bot_bo3990330152332043303nteger )
     => ( ! [X2: code_integer] :
            ( aa_set5643387332196575111eger_o(member_Code_integer(X2),X5)
           => ? [Xa: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(Xa),X5)
                & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,X2),Xa) ) )
       => ~ aa_set5643387332196575111eger_o(finite6017078050557962740nteger,X5) ) ) ).

% infinite_growing
tff(fact_7288_infinite__growing,axiom,
    ! [X5: set_real] :
      ( ( X5 != bot_bot_set_real )
     => ( ! [X2: real] :
            ( aa_set_real_o(member_real(X2),X5)
           => ? [Xa: real] :
                ( aa_set_real_o(member_real(Xa),X5)
                & aa_real_o(aa_real_fun_real_o(ord_less_real,X2),Xa) ) )
       => ~ aa_set_real_o(finite_finite_real,X5) ) ) ).

% infinite_growing
tff(fact_7289_infinite__growing,axiom,
    ! [X5: set_rat] :
      ( ( X5 != bot_bot_set_rat )
     => ( ! [X2: rat] :
            ( aa_set_rat_o(member_rat(X2),X5)
           => ? [Xa: rat] :
                ( aa_set_rat_o(member_rat(Xa),X5)
                & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X2),Xa) ) )
       => ~ aa_set_rat_o(finite_finite_rat,X5) ) ) ).

% infinite_growing
tff(fact_7290_infinite__growing,axiom,
    ! [X5: set_num] :
      ( ( X5 != bot_bot_set_num )
     => ( ! [X2: num] :
            ( aa_set_num_o(member_num(X2),X5)
           => ? [Xa: num] :
                ( aa_set_num_o(member_num(Xa),X5)
                & aa_num_o(ord_less_num(X2),Xa) ) )
       => ~ aa_set_num_o(finite_finite_num,X5) ) ) ).

% infinite_growing
tff(fact_7291_infinite__growing,axiom,
    ! [X5: set_nat] :
      ( ( X5 != bot_bot_set_nat )
     => ( ! [X2: nat] :
            ( aa_set_nat_o(member_nat(X2),X5)
           => ? [Xa: nat] :
                ( aa_set_nat_o(member_nat(Xa),X5)
                & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X2),Xa) ) )
       => ~ aa_set_nat_o(finite_finite_nat,X5) ) ) ).

% infinite_growing
tff(fact_7292_infinite__growing,axiom,
    ! [X5: set_int] :
      ( ( X5 != bot_bot_set_int )
     => ( ! [X2: int] :
            ( aa_set_int_o(member_int(X2),X5)
           => ? [Xa: int] :
                ( aa_set_int_o(member_int(Xa),X5)
                & aa_int_o(aa_int_fun_int_o(ord_less_int,X2),Xa) ) )
       => ~ aa_set_int_o(finite_finite_int,X5) ) ) ).

% infinite_growing
tff(fact_7293_ex__min__if__finite,axiom,
    ! [S2: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,S2)
     => ( ( S2 != bot_bo3990330152332043303nteger )
       => ? [X2: code_integer] :
            ( aa_set5643387332196575111eger_o(member_Code_integer(X2),S2)
            & ~ ? [Xa: code_integer] :
                  ( aa_set5643387332196575111eger_o(member_Code_integer(Xa),S2)
                  & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Xa),X2) ) ) ) ) ).

% ex_min_if_finite
tff(fact_7294_ex__min__if__finite,axiom,
    ! [S2: set_real] :
      ( aa_set_real_o(finite_finite_real,S2)
     => ( ( S2 != bot_bot_set_real )
       => ? [X2: real] :
            ( aa_set_real_o(member_real(X2),S2)
            & ~ ? [Xa: real] :
                  ( aa_set_real_o(member_real(Xa),S2)
                  & aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),X2) ) ) ) ) ).

% ex_min_if_finite
tff(fact_7295_ex__min__if__finite,axiom,
    ! [S2: set_rat] :
      ( aa_set_rat_o(finite_finite_rat,S2)
     => ( ( S2 != bot_bot_set_rat )
       => ? [X2: rat] :
            ( aa_set_rat_o(member_rat(X2),S2)
            & ~ ? [Xa: rat] :
                  ( aa_set_rat_o(member_rat(Xa),S2)
                  & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),X2) ) ) ) ) ).

% ex_min_if_finite
tff(fact_7296_ex__min__if__finite,axiom,
    ! [S2: set_num] :
      ( aa_set_num_o(finite_finite_num,S2)
     => ( ( S2 != bot_bot_set_num )
       => ? [X2: num] :
            ( aa_set_num_o(member_num(X2),S2)
            & ~ ? [Xa: num] :
                  ( aa_set_num_o(member_num(Xa),S2)
                  & aa_num_o(ord_less_num(Xa),X2) ) ) ) ) ).

% ex_min_if_finite
tff(fact_7297_ex__min__if__finite,axiom,
    ! [S2: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,S2)
     => ( ( S2 != bot_bot_set_nat )
       => ? [X2: nat] :
            ( aa_set_nat_o(member_nat(X2),S2)
            & ~ ? [Xa: nat] :
                  ( aa_set_nat_o(member_nat(Xa),S2)
                  & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),X2) ) ) ) ) ).

% ex_min_if_finite
tff(fact_7298_ex__min__if__finite,axiom,
    ! [S2: set_int] :
      ( aa_set_int_o(finite_finite_int,S2)
     => ( ( S2 != bot_bot_set_int )
       => ? [X2: int] :
            ( aa_set_int_o(member_int(X2),S2)
            & ~ ? [Xa: int] :
                  ( aa_set_int_o(member_int(Xa),S2)
                  & aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),X2) ) ) ) ) ).

% ex_min_if_finite
tff(fact_7299_filter__preserves__multiset,axiom,
    ! [M8: fun_Pr5160692531401422620nt_nat,P: fun_Pr3238374608701176162_int_o] :
      ( finite2998713641127702882nt_int(collec213857154873943460nt_int(aTP_Lamm_afe(M8)))
     => finite2998713641127702882nt_int(collec213857154873943460nt_int(aa_fun5349570537165229698_int_o(aTP_Lamm_aft(M8),P))) ) ).

% filter_preserves_multiset
tff(fact_7300_filter__preserves__multiset,axiom,
    ! [M8: fun_nat_nat,P: fun_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,collect_nat(aTP_Lamm_afg(M8)))
     => aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun5937193402907943164_nat_o(aTP_Lamm_afu(M8),P))) ) ).

% filter_preserves_multiset
tff(fact_7301_filter__preserves__multiset,axiom,
    ! [M8: fun_int_nat,P: fun_int_o] :
      ( aa_set_int_o(finite_finite_int,collect_int(aTP_Lamm_afi(M8)))
     => aa_set_int_o(finite_finite_int,collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_afv(M8),P))) ) ).

% filter_preserves_multiset
tff(fact_7302_filter__preserves__multiset,axiom,
    ! [M8: fun_complex_nat,P: fun_complex_o] :
      ( aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aTP_Lamm_afk(M8)))
     => aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aa_fun2514317545224092796plex_o(aTP_Lamm_afw(M8),P))) ) ).

% filter_preserves_multiset
tff(fact_7303_filter__preserves__multiset,axiom,
    ! [M8: fun_Code_integer_nat,P: fun_Code_integer_o] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,collect_Code_integer(aTP_Lamm_afm(M8)))
     => aa_set5643387332196575111eger_o(finite6017078050557962740nteger,collect_Code_integer(aa_fun8214818745163655228eger_o(aTP_Lamm_afx(M8),P))) ) ).

% filter_preserves_multiset
tff(fact_7304_finite__ranking__induct,axiom,
    ! [S2: set_VEBT_VEBT,P: fun_set_VEBT_VEBT_o,F: fun_VEBT_VEBT_rat] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,S2)
     => ( aa_set_VEBT_VEBT_o(P,bot_bo8194388402131092736T_VEBT)
       => ( ! [X2: vEBT_VEBT,S5: set_VEBT_VEBT] :
              ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,S5)
             => ( ! [Y5: vEBT_VEBT] :
                    ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Y5),S5)
                   => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_VEBT_VEBT_rat(F,Y5)),aa_VEBT_VEBT_rat(F,X2)) )
               => ( aa_set_VEBT_VEBT_o(P,S5)
                 => aa_set_VEBT_VEBT_o(P,insert_VEBT_VEBT(X2,S5)) ) ) )
         => aa_set_VEBT_VEBT_o(P,S2) ) ) ) ).

% finite_ranking_induct
tff(fact_7305_finite__ranking__induct,axiom,
    ! [S2: set_complex,P: fun_set_complex_o,F: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S2)
     => ( aa_set_complex_o(P,bot_bot_set_complex)
       => ( ! [X2: complex,S5: set_complex] :
              ( aa_set_complex_o(finite3207457112153483333omplex,S5)
             => ( ! [Y5: complex] :
                    ( aa_set_complex_o(member_complex(Y5),S5)
                   => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_complex_rat(F,Y5)),aa_complex_rat(F,X2)) )
               => ( aa_set_complex_o(P,S5)
                 => aa_set_complex_o(P,insert_complex(X2,S5)) ) ) )
         => aa_set_complex_o(P,S2) ) ) ) ).

% finite_ranking_induct
tff(fact_7306_finite__ranking__induct,axiom,
    ! [S2: set_Code_integer,P: fun_se7289222521248229150eger_o,F: fun_Code_integer_rat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,S2)
     => ( aa_set5643387332196575111eger_o(P,bot_bo3990330152332043303nteger)
       => ( ! [X2: code_integer,S5: set_Code_integer] :
              ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,S5)
             => ( ! [Y5: code_integer] :
                    ( aa_set5643387332196575111eger_o(member_Code_integer(Y5),S5)
                   => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_Code_integer_rat(F,Y5)),aa_Code_integer_rat(F,X2)) )
               => ( aa_set5643387332196575111eger_o(P,S5)
                 => aa_set5643387332196575111eger_o(P,insert_Code_integer(X2,S5)) ) ) )
         => aa_set5643387332196575111eger_o(P,S2) ) ) ) ).

% finite_ranking_induct
tff(fact_7307_finite__ranking__induct,axiom,
    ! [S2: set_nat,P: fun_set_nat_o,F: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,S2)
     => ( aa_set_nat_o(P,bot_bot_set_nat)
       => ( ! [X2: nat,S5: set_nat] :
              ( aa_set_nat_o(finite_finite_nat,S5)
             => ( ! [Y5: nat] :
                    ( aa_set_nat_o(member_nat(Y5),S5)
                   => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,Y5)),aa_nat_rat(F,X2)) )
               => ( aa_set_nat_o(P,S5)
                 => aa_set_nat_o(P,insert_nat(X2,S5)) ) ) )
         => aa_set_nat_o(P,S2) ) ) ) ).

% finite_ranking_induct
tff(fact_7308_finite__ranking__induct,axiom,
    ! [S2: set_int,P: fun_set_int_o,F: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,S2)
     => ( aa_set_int_o(P,bot_bot_set_int)
       => ( ! [X2: int,S5: set_int] :
              ( aa_set_int_o(finite_finite_int,S5)
             => ( ! [Y5: int] :
                    ( aa_set_int_o(member_int(Y5),S5)
                   => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(F,Y5)),aa_int_rat(F,X2)) )
               => ( aa_set_int_o(P,S5)
                 => aa_set_int_o(P,insert_int(X2,S5)) ) ) )
         => aa_set_int_o(P,S2) ) ) ) ).

% finite_ranking_induct
tff(fact_7309_finite__ranking__induct,axiom,
    ! [S2: set_real,P: fun_set_real_o,F: fun_real_rat] :
      ( aa_set_real_o(finite_finite_real,S2)
     => ( aa_set_real_o(P,bot_bot_set_real)
       => ( ! [X2: real,S5: set_real] :
              ( aa_set_real_o(finite_finite_real,S5)
             => ( ! [Y5: real] :
                    ( aa_set_real_o(member_real(Y5),S5)
                   => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_real_rat(F,Y5)),aa_real_rat(F,X2)) )
               => ( aa_set_real_o(P,S5)
                 => aa_set_real_o(P,insert_real(X2,S5)) ) ) )
         => aa_set_real_o(P,S2) ) ) ) ).

% finite_ranking_induct
tff(fact_7310_finite__ranking__induct,axiom,
    ! [S2: set_VEBT_VEBT,P: fun_set_VEBT_VEBT_o,F: fun_VEBT_VEBT_num] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,S2)
     => ( aa_set_VEBT_VEBT_o(P,bot_bo8194388402131092736T_VEBT)
       => ( ! [X2: vEBT_VEBT,S5: set_VEBT_VEBT] :
              ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,S5)
             => ( ! [Y5: vEBT_VEBT] :
                    ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Y5),S5)
                   => aa_num_o(ord_less_eq_num(aa_VEBT_VEBT_num(F,Y5)),aa_VEBT_VEBT_num(F,X2)) )
               => ( aa_set_VEBT_VEBT_o(P,S5)
                 => aa_set_VEBT_VEBT_o(P,insert_VEBT_VEBT(X2,S5)) ) ) )
         => aa_set_VEBT_VEBT_o(P,S2) ) ) ) ).

% finite_ranking_induct
tff(fact_7311_finite__ranking__induct,axiom,
    ! [S2: set_complex,P: fun_set_complex_o,F: fun_complex_num] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S2)
     => ( aa_set_complex_o(P,bot_bot_set_complex)
       => ( ! [X2: complex,S5: set_complex] :
              ( aa_set_complex_o(finite3207457112153483333omplex,S5)
             => ( ! [Y5: complex] :
                    ( aa_set_complex_o(member_complex(Y5),S5)
                   => aa_num_o(ord_less_eq_num(aa_complex_num(F,Y5)),aa_complex_num(F,X2)) )
               => ( aa_set_complex_o(P,S5)
                 => aa_set_complex_o(P,insert_complex(X2,S5)) ) ) )
         => aa_set_complex_o(P,S2) ) ) ) ).

% finite_ranking_induct
tff(fact_7312_finite__ranking__induct,axiom,
    ! [S2: set_Code_integer,P: fun_se7289222521248229150eger_o,F: fun_Code_integer_num] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,S2)
     => ( aa_set5643387332196575111eger_o(P,bot_bo3990330152332043303nteger)
       => ( ! [X2: code_integer,S5: set_Code_integer] :
              ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,S5)
             => ( ! [Y5: code_integer] :
                    ( aa_set5643387332196575111eger_o(member_Code_integer(Y5),S5)
                   => aa_num_o(ord_less_eq_num(aa_Code_integer_num(F,Y5)),aa_Code_integer_num(F,X2)) )
               => ( aa_set5643387332196575111eger_o(P,S5)
                 => aa_set5643387332196575111eger_o(P,insert_Code_integer(X2,S5)) ) ) )
         => aa_set5643387332196575111eger_o(P,S2) ) ) ) ).

% finite_ranking_induct
tff(fact_7313_finite__ranking__induct,axiom,
    ! [S2: set_nat,P: fun_set_nat_o,F: fun_nat_num] :
      ( aa_set_nat_o(finite_finite_nat,S2)
     => ( aa_set_nat_o(P,bot_bot_set_nat)
       => ( ! [X2: nat,S5: set_nat] :
              ( aa_set_nat_o(finite_finite_nat,S5)
             => ( ! [Y5: nat] :
                    ( aa_set_nat_o(member_nat(Y5),S5)
                   => aa_num_o(ord_less_eq_num(aa_nat_num(F,Y5)),aa_nat_num(F,X2)) )
               => ( aa_set_nat_o(P,S5)
                 => aa_set_nat_o(P,insert_nat(X2,S5)) ) ) )
         => aa_set_nat_o(P,S2) ) ) ) ).

% finite_ranking_induct
tff(fact_7314_finite__linorder__min__induct,axiom,
    ! [A2: set_Code_integer,P: fun_se7289222521248229150eger_o] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A2)
     => ( aa_set5643387332196575111eger_o(P,bot_bo3990330152332043303nteger)
       => ( ! [B2: code_integer,A7: set_Code_integer] :
              ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A7)
             => ( ! [X3: code_integer] :
                    ( aa_set5643387332196575111eger_o(member_Code_integer(X3),A7)
                   => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B2),X3) )
               => ( aa_set5643387332196575111eger_o(P,A7)
                 => aa_set5643387332196575111eger_o(P,insert_Code_integer(B2,A7)) ) ) )
         => aa_set5643387332196575111eger_o(P,A2) ) ) ) ).

% finite_linorder_min_induct
tff(fact_7315_finite__linorder__min__induct,axiom,
    ! [A2: set_real,P: fun_set_real_o] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( aa_set_real_o(P,bot_bot_set_real)
       => ( ! [B2: real,A7: set_real] :
              ( aa_set_real_o(finite_finite_real,A7)
             => ( ! [X3: real] :
                    ( aa_set_real_o(member_real(X3),A7)
                   => aa_real_o(aa_real_fun_real_o(ord_less_real,B2),X3) )
               => ( aa_set_real_o(P,A7)
                 => aa_set_real_o(P,insert_real(B2,A7)) ) ) )
         => aa_set_real_o(P,A2) ) ) ) ).

% finite_linorder_min_induct
tff(fact_7316_finite__linorder__min__induct,axiom,
    ! [A2: set_rat,P: fun_set_rat_o] :
      ( aa_set_rat_o(finite_finite_rat,A2)
     => ( aa_set_rat_o(P,bot_bot_set_rat)
       => ( ! [B2: rat,A7: set_rat] :
              ( aa_set_rat_o(finite_finite_rat,A7)
             => ( ! [X3: rat] :
                    ( aa_set_rat_o(member_rat(X3),A7)
                   => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),X3) )
               => ( aa_set_rat_o(P,A7)
                 => aa_set_rat_o(P,insert_rat(B2,A7)) ) ) )
         => aa_set_rat_o(P,A2) ) ) ) ).

% finite_linorder_min_induct
tff(fact_7317_finite__linorder__min__induct,axiom,
    ! [A2: set_num,P: fun_set_num_o] :
      ( aa_set_num_o(finite_finite_num,A2)
     => ( aa_set_num_o(P,bot_bot_set_num)
       => ( ! [B2: num,A7: set_num] :
              ( aa_set_num_o(finite_finite_num,A7)
             => ( ! [X3: num] :
                    ( aa_set_num_o(member_num(X3),A7)
                   => aa_num_o(ord_less_num(B2),X3) )
               => ( aa_set_num_o(P,A7)
                 => aa_set_num_o(P,insert_num(B2,A7)) ) ) )
         => aa_set_num_o(P,A2) ) ) ) ).

% finite_linorder_min_induct
tff(fact_7318_finite__linorder__min__induct,axiom,
    ! [A2: set_nat,P: fun_set_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_nat_o(P,bot_bot_set_nat)
       => ( ! [B2: nat,A7: set_nat] :
              ( aa_set_nat_o(finite_finite_nat,A7)
             => ( ! [X3: nat] :
                    ( aa_set_nat_o(member_nat(X3),A7)
                   => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B2),X3) )
               => ( aa_set_nat_o(P,A7)
                 => aa_set_nat_o(P,insert_nat(B2,A7)) ) ) )
         => aa_set_nat_o(P,A2) ) ) ) ).

% finite_linorder_min_induct
tff(fact_7319_finite__linorder__min__induct,axiom,
    ! [A2: set_int,P: fun_set_int_o] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( aa_set_int_o(P,bot_bot_set_int)
       => ( ! [B2: int,A7: set_int] :
              ( aa_set_int_o(finite_finite_int,A7)
             => ( ! [X3: int] :
                    ( aa_set_int_o(member_int(X3),A7)
                   => aa_int_o(aa_int_fun_int_o(ord_less_int,B2),X3) )
               => ( aa_set_int_o(P,A7)
                 => aa_set_int_o(P,insert_int(B2,A7)) ) ) )
         => aa_set_int_o(P,A2) ) ) ) ).

% finite_linorder_min_induct
tff(fact_7320_bezw_Opelims,axiom,
    ! [Xb: nat,Xaa: nat,Y: product_prod_int_int] :
      ( ( bezw(Xb,Xaa) = Y )
     => ( aa_Pro6150637604445964947_nat_o(accp_P4275260045618599050at_nat(bezw_rel),aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Xb),Xaa))
       => ~ ( ( Y = $ite(Xaa = zero_zero_nat,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,one_one_int),zero_zero_int),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_Pro6799258434465214121nt_int(product_snd_int_int,bezw(Xaa,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xb),Xaa)))),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_Pro6799258434465214121nt_int(product_fst_int_int,bezw(Xaa,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xb),Xaa)))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_Pro6799258434465214121nt_int(product_snd_int_int,bezw(Xaa,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xb),Xaa)))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Xb),Xaa)))))) )
           => ~ aa_Pro6150637604445964947_nat_o(accp_P4275260045618599050at_nat(bezw_rel),aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Xb),Xaa)) ) ) ) ).

% bezw.pelims
tff(fact_7321_list__assn__aux__ineq__len,axiom,
    ! [L: list_real,Li2: list_real,A2: fun_re6303261851654292474l_assn] :
      ( ( size_size_list_real(L) != size_size_list_real(Li2) )
     => ( vEBT_L1930518968523514909l_real(A2,L,Li2) = bot_bot_assn ) ) ).

% list_assn_aux_ineq_len
tff(fact_7322_list__assn__aux__ineq__len,axiom,
    ! [L: list_real,Li2: list_o,A2: fun_real_fun_o_assn] :
      ( ( size_size_list_real(L) != size_size_list_o(Li2) )
     => ( vEBT_L6234343332106409831real_o(A2,L,Li2) = bot_bot_assn ) ) ).

% list_assn_aux_ineq_len
tff(fact_7323_list__assn__aux__ineq__len,axiom,
    ! [L: list_real,Li2: list_nat,A2: fun_re574865750389657758t_assn] :
      ( ( size_size_list_real(L) != size_size_list_nat(Li2) )
     => ( vEBT_L1446010312343316929al_nat(A2,L,Li2) = bot_bot_assn ) ) ).

% list_assn_aux_ineq_len
tff(fact_7324_list__assn__aux__ineq__len,axiom,
    ! [L: list_real,Li2: list_int,A2: fun_re4990660043736809722t_assn] :
      ( ( size_size_list_real(L) != size_size_list_int(Li2) )
     => ( vEBT_L1443519841834266653al_int(A2,L,Li2) = bot_bot_assn ) ) ).

% list_assn_aux_ineq_len
tff(fact_7325_list__assn__aux__ineq__len,axiom,
    ! [L: list_o,Li2: list_real,A2: fun_o_fun_real_assn] :
      ( ( size_size_list_o(L) != size_size_list_real(Li2) )
     => ( vEBT_L4725278957065240257o_real(A2,L,Li2) = bot_bot_assn ) ) ).

% list_assn_aux_ineq_len
tff(fact_7326_list__assn__aux__ineq__len,axiom,
    ! [L: list_o,Li2: list_o,A2: fun_o_fun_o_assn] :
      ( ( size_size_list_o(L) != size_size_list_o(Li2) )
     => ( vEBT_L7363604446928714179sn_o_o(A2,L,Li2) = bot_bot_assn ) ) ).

% list_assn_aux_ineq_len
tff(fact_7327_list__assn__aux__ineq__len,axiom,
    ! [L: list_o,Li2: list_nat,A2: fun_o_fun_nat_assn] :
      ( ( size_size_list_o(L) != size_size_list_nat(Li2) )
     => ( vEBT_L4785011123346445925_o_nat(A2,L,Li2) = bot_bot_assn ) ) ).

% list_assn_aux_ineq_len
tff(fact_7328_list__assn__aux__ineq__len,axiom,
    ! [L: list_o,Li2: list_int,A2: fun_o_fun_int_assn] :
      ( ( size_size_list_o(L) != size_size_list_int(Li2) )
     => ( vEBT_L4782520652837395649_o_int(A2,L,Li2) = bot_bot_assn ) ) ).

% list_assn_aux_ineq_len
tff(fact_7329_list__assn__aux__ineq__len,axiom,
    ! [L: list_nat,Li2: list_real,A2: fun_na448437756743965086l_assn] :
      ( ( size_size_list_nat(L) != size_size_list_real(Li2) )
     => ( vEBT_L6102073776069194049t_real(A2,L,Li2) = bot_bot_assn ) ) ).

% list_assn_aux_ineq_len
tff(fact_7330_list__assn__aux__ineq__len,axiom,
    ! [L: list_nat,Li2: list_o,A2: fun_nat_fun_o_assn] :
      ( ( size_size_list_nat(L) != size_size_list_o(Li2) )
     => ( vEBT_L7887682484454631235_nat_o(A2,L,Li2) = bot_bot_assn ) ) ).

% list_assn_aux_ineq_len
tff(fact_7331_accp__subset,axiom,
    ! [R1: fun_Pr8877776288780692068_nat_o,R22: fun_Pr8877776288780692068_nat_o] :
      ( ord_le4038806025055212036_nat_o(R1,R22)
     => ord_le1769015658490769310_nat_o(accp_P2887432264394892906BT_nat(R22),accp_P2887432264394892906BT_nat(R1)) ) ).

% accp_subset
tff(fact_7332_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_7333_accp__subset,axiom,
    ! [R1: fun_Pr2028404973800837934_nat_o,R22: fun_Pr2028404973800837934_nat_o] :
      ( ord_le3650406611588909774_nat_o(R1,R22)
     => ord_le1572679713340831354_nat_o(accp_P4275260045618599050at_nat(R22),accp_P4275260045618599050at_nat(R1)) ) ).

% accp_subset
tff(fact_7334_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_7335_accp__subset,axiom,
    ! [R1: fun_nat_fun_nat_o,R22: fun_nat_fun_nat_o] :
      ( ord_le768339454706049738_nat_o(R1,R22)
     => ord_le7240017405068453697_nat_o(accp_nat(R22),accp_nat(R1)) ) ).

% accp_subset
tff(fact_7336_vebt__buildup_Oelims,axiom,
    ! [Xb: nat,Y: vEBT_VEBT] :
      ( ( vEBT_vebt_buildup(Xb) = Y )
     => ( ( ( Xb = zero_zero_nat )
         => ( Y != vEBT_Leaf($false,$false) ) )
       => ( ( ( Xb = aa_nat_nat(suc,zero_zero_nat) )
           => ( Y != vEBT_Leaf($false,$false) ) )
         => ~ ! [Va2: nat] :
                ( ( Xb = aa_nat_nat(suc,aa_nat_nat(suc,Va2)) )
               => ( Y != $ite(
                      aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,Va2))),
                      $let(
                        half: nat,
                        half:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                        aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2)),replicate_VEBT_VEBT(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),half),vEBT_vebt_buildup(half))),vEBT_vebt_buildup(half)) ),
                      $let(
                        half: nat,
                        half:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                        aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2)),replicate_VEBT_VEBT(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,half)),vEBT_vebt_buildup(half))),vEBT_vebt_buildup(aa_nat_nat(suc,half))) ) ) ) ) ) ) ) ).

% vebt_buildup.elims
tff(fact_7337_intind,axiom,
    ! [I: nat,Nb: nat,P: fun_VEBT_VEBTi_o,Xb: vEBT_VEBTi] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),Nb)
     => ( aa_VEBT_VEBTi_o(P,Xb)
       => aa_VEBT_VEBTi_o(P,nth_VEBT_VEBTi(replicate_VEBT_VEBTi(Nb,Xb),I)) ) ) ).

% intind
tff(fact_7338_intind,axiom,
    ! [I: nat,Nb: nat,P: fun_nat_o,Xb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),Nb)
     => ( aa_nat_o(P,Xb)
       => aa_nat_o(P,aa_nat_nat(nth_nat(replicate_nat(Nb,Xb)),I)) ) ) ).

% intind
tff(fact_7339_intind,axiom,
    ! [I: nat,Nb: nat,P: fun_int_o,Xb: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),Nb)
     => ( aa_int_o(P,Xb)
       => aa_int_o(P,aa_nat_int(nth_int(replicate_int(Nb,Xb)),I)) ) ) ).

% intind
tff(fact_7340_intind,axiom,
    ! [I: nat,Nb: nat,P: fun_VEBT_VEBT_o,Xb: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),Nb)
     => ( aa_VEBT_VEBT_o(P,Xb)
       => aa_VEBT_VEBT_o(P,nth_VEBT_VEBT(replicate_VEBT_VEBT(Nb,Xb),I)) ) ) ).

% intind
tff(fact_7341_repli__cons__repl,axiom,
    ! [Q: assn,Xb: heap_T2636463487746394924on_nat,A2: fun_VE6334927249922962060t_assn,Y: vEBT_VEBT,Nb: nat] :
      ( hoare_7629718768684598413on_nat(Q,Xb,aa_VEB931861714549014589t_assn(aa_fun1240582285364316732t_assn(aTP_Lamm_afy(Q),A2),Y))
     => hoare_6480275734082232733on_nat(Q,vEBT_V792416675989592002on_nat(Nb,Xb),aa_nat5248546159757461219t_assn(aa_VEB2926451386496479642t_assn(aa_fun7129419364150962969t_assn(aTP_Lamm_afz(Q),A2),Y),Nb)) ) ).

% repli_cons_repl
tff(fact_7342_repli__cons__repl,axiom,
    ! [Q: assn,Xb: heap_Time_Heap_nat,A2: fun_VE4931415397359990460t_assn,Y: vEBT_VEBT,Nb: nat] :
      ( hoare_3067605981109127869le_nat(Q,Xb,aa_VEB8640158415945897581t_assn(aa_fun7634423771582967228t_assn(aTP_Lamm_aga(Q),A2),Y))
     => hoare_7964568885773372237st_nat(Q,vEBT_V7726092123322077554ei_nat(Nb,Xb),aa_nat7967224596217534867t_assn(aa_VEB8402767592425156170t_assn(aa_fun6018615163312975129t_assn(aTP_Lamm_agb(Q),A2),Y),Nb)) ) ).

% repli_cons_repl
tff(fact_7343_repli__cons__repl,axiom,
    ! [Q: assn,Xb: heap_Time_Heap_o,A2: fun_VE2104784059010244834o_assn,Y: vEBT_VEBT,Nb: nat] :
      ( hoare_hoare_triple_o(Q,Xb,aa_VEB5185540838851036891o_assn(aa_fun6967398642911097346o_assn(aTP_Lamm_agc(Q),A2),Y))
     => hoare_9089481587091695345list_o(Q,vEBT_V2326993469660664182atei_o(Nb,Xb),aa_nat5137362611963950667o_assn(aa_VEB7710612541095692010o_assn(aa_fun380857135891842193o_assn(aTP_Lamm_agd(Q),A2),Y),Nb)) ) ).

% repli_cons_repl
tff(fact_7344_repli__cons__repl,axiom,
    ! [Q: assn,Xb: heap_T5738788834812785303t_unit,A2: fun_VE8866928305347306583t_assn,Y: vEBT_VEBT,Nb: nat] :
      ( hoare_8945653483474564448t_unit(Q,Xb,aa_VEB4153095751742533328t_assn(aa_fun1364258190791680492t_assn(aTP_Lamm_age(Q),A2),Y))
     => hoare_8193881092815343078t_unit(Q,vEBT_V7483891112628345579t_unit(Nb,Xb),aa_nat5137481719278757952t_assn(aa_VEB7218668862568288351t_assn(aa_fun8456556664957245819t_assn(aTP_Lamm_agf(Q),A2),Y),Nb)) ) ).

% repli_cons_repl
tff(fact_7345_repli__cons__repl,axiom,
    ! [Q: assn,Xb: heap_T8145700208782473153_VEBTi,A2: fun_VE905575822219544865i_assn,Y: vEBT_VEBT,Nb: nat] :
      ( hoare_1429296392585015714_VEBTi(Q,Xb,aa_VEB3717855479479287634i_assn(aa_fun4659777589554175228i_assn(aTP_Lamm_agg(Q),A2),Y))
     => hoare_3904069481286416050_VEBTi(Q,vEBT_V1859673955506687831_VEBTi(Nb,Xb),aa_nat6700791848136694008i_assn(aa_VEB320677594755043119i_assn(aa_fun8635824850342594777i_assn(aTP_Lamm_agh(Q),A2),Y),Nb)) ) ).

% repli_cons_repl
tff(fact_7346_repli__emp,axiom,
    ! [Xb: heap_T2636463487746394924on_nat,A2: fun_VE6334927249922962060t_assn,Y: vEBT_VEBT,Nb: nat] :
      ( hoare_7629718768684598413on_nat(one_one_assn,Xb,aa_VEB931861714549014589t_assn(A2,Y))
     => hoare_6480275734082232733on_nat(one_one_assn,vEBT_V792416675989592002on_nat(Nb,Xb),vEBT_L8010285020845282001on_nat(A2,replicate_VEBT_VEBT(Nb,Y))) ) ).

% repli_emp
tff(fact_7347_repli__emp,axiom,
    ! [Xb: heap_Time_Heap_nat,A2: fun_VE4931415397359990460t_assn,Y: vEBT_VEBT,Nb: nat] :
      ( hoare_3067605981109127869le_nat(one_one_assn,Xb,aa_VEB8640158415945897581t_assn(A2,Y))
     => hoare_7964568885773372237st_nat(one_one_assn,vEBT_V7726092123322077554ei_nat(Nb,Xb),vEBT_L8296926524756676353BT_nat(A2,replicate_VEBT_VEBT(Nb,Y))) ) ).

% repli_emp
tff(fact_7348_repli__emp,axiom,
    ! [Xb: heap_Time_Heap_o,A2: fun_VE2104784059010244834o_assn,Y: vEBT_VEBT,Nb: nat] :
      ( hoare_hoare_triple_o(one_one_assn,Xb,aa_VEB5185540838851036891o_assn(A2,Y))
     => hoare_9089481587091695345list_o(one_one_assn,vEBT_V2326993469660664182atei_o(Nb,Xb),vEBT_L7489408758114837031VEBT_o(A2,replicate_VEBT_VEBT(Nb,Y))) ) ).

% repli_emp
tff(fact_7349_repli__emp,axiom,
    ! [Xb: heap_T5738788834812785303t_unit,A2: fun_VE8866928305347306583t_assn,Y: vEBT_VEBT,Nb: nat] :
      ( hoare_8945653483474564448t_unit(one_one_assn,Xb,aa_VEB4153095751742533328t_assn(A2,Y))
     => hoare_8193881092815343078t_unit(one_one_assn,vEBT_V7483891112628345579t_unit(Nb,Xb),vEBT_L8068554427805421084t_unit(A2,replicate_VEBT_VEBT(Nb,Y))) ) ).

% repli_emp
tff(fact_7350_repli__emp,axiom,
    ! [Xb: heap_T8145700208782473153_VEBTi,A2: fun_VE905575822219544865i_assn,Y: vEBT_VEBT,Nb: nat] :
      ( hoare_1429296392585015714_VEBTi(one_one_assn,Xb,aa_VEB3717855479479287634i_assn(A2,Y))
     => hoare_3904069481286416050_VEBTi(one_one_assn,vEBT_V1859673955506687831_VEBTi(Nb,Xb),vEBT_L6296928887356842470_VEBTi(A2,replicate_VEBT_VEBT(Nb,Y))) ) ).

% repli_emp
tff(fact_7351_length__replicate,axiom,
    ! [Nb: nat,Xb: vEBT_VEBT] : size_s6755466524823107622T_VEBT(replicate_VEBT_VEBT(Nb,Xb)) = Nb ).

% length_replicate
tff(fact_7352_length__replicate,axiom,
    ! [Nb: nat,Xb: real] : size_size_list_real(replicate_real(Nb,Xb)) = Nb ).

% length_replicate
tff(fact_7353_length__replicate,axiom,
    ! [Nb: nat,Xb: $o] : size_size_list_o(replicate_o(Nb,(Xb))) = Nb ).

% length_replicate
tff(fact_7354_length__replicate,axiom,
    ! [Nb: nat,Xb: nat] : size_size_list_nat(replicate_nat(Nb,Xb)) = Nb ).

% length_replicate
tff(fact_7355_length__replicate,axiom,
    ! [Nb: nat,Xb: int] : size_size_list_int(replicate_int(Nb,Xb)) = Nb ).

% length_replicate
tff(fact_7356_Ball__set__replicate,axiom,
    ! [Nb: nat,A: nat,P: fun_nat_o] :
      ( ! [X: nat] :
          ( aa_set_nat_o(member_nat(X),set_nat2(replicate_nat(Nb,A)))
         => aa_nat_o(P,X) )
    <=> ( aa_nat_o(P,A)
        | ( Nb = zero_zero_nat ) ) ) ).

% Ball_set_replicate
tff(fact_7357_Ball__set__replicate,axiom,
    ! [Nb: nat,A: real,P: fun_real_o] :
      ( ! [X: real] :
          ( aa_set_real_o(member_real(X),set_real2(replicate_real(Nb,A)))
         => aa_real_o(P,X) )
    <=> ( aa_real_o(P,A)
        | ( Nb = zero_zero_nat ) ) ) ).

% Ball_set_replicate
tff(fact_7358_Ball__set__replicate,axiom,
    ! [Nb: nat,A: int,P: fun_int_o] :
      ( ! [X: int] :
          ( aa_set_int_o(member_int(X),set_int2(replicate_int(Nb,A)))
         => aa_int_o(P,X) )
    <=> ( aa_int_o(P,A)
        | ( Nb = zero_zero_nat ) ) ) ).

% Ball_set_replicate
tff(fact_7359_Ball__set__replicate,axiom,
    ! [Nb: nat,A: produc6575502325842934193n_assn,P: fun_Pr3571187171380813162assn_o] :
      ( ! [X: produc6575502325842934193n_assn] :
          ( member7957490590177025114n_assn(X,set_Pr1139785259514867910n_assn(replic3825545231534752113n_assn(Nb,A)))
         => aa_Pro2491721552408928739assn_o(P,X) )
    <=> ( aa_Pro2491721552408928739assn_o(P,A)
        | ( Nb = zero_zero_nat ) ) ) ).

% Ball_set_replicate
tff(fact_7360_Ball__set__replicate,axiom,
    ! [Nb: nat,A: vEBT_VEBT,P: fun_VEBT_VEBT_o] :
      ( ! [X: vEBT_VEBT] :
          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),set_VEBT_VEBT2(replicate_VEBT_VEBT(Nb,A)))
         => aa_VEBT_VEBT_o(P,X) )
    <=> ( aa_VEBT_VEBT_o(P,A)
        | ( Nb = zero_zero_nat ) ) ) ).

% Ball_set_replicate
tff(fact_7361_Bex__set__replicate,axiom,
    ! [Nb: nat,A: nat,P: fun_nat_o] :
      ( ? [X: nat] :
          ( aa_set_nat_o(member_nat(X),set_nat2(replicate_nat(Nb,A)))
          & aa_nat_o(P,X) )
    <=> ( aa_nat_o(P,A)
        & ( Nb != zero_zero_nat ) ) ) ).

% Bex_set_replicate
tff(fact_7362_Bex__set__replicate,axiom,
    ! [Nb: nat,A: real,P: fun_real_o] :
      ( ? [X: real] :
          ( aa_set_real_o(member_real(X),set_real2(replicate_real(Nb,A)))
          & aa_real_o(P,X) )
    <=> ( aa_real_o(P,A)
        & ( Nb != zero_zero_nat ) ) ) ).

% Bex_set_replicate
tff(fact_7363_Bex__set__replicate,axiom,
    ! [Nb: nat,A: int,P: fun_int_o] :
      ( ? [X: int] :
          ( aa_set_int_o(member_int(X),set_int2(replicate_int(Nb,A)))
          & aa_int_o(P,X) )
    <=> ( aa_int_o(P,A)
        & ( Nb != zero_zero_nat ) ) ) ).

% Bex_set_replicate
tff(fact_7364_Bex__set__replicate,axiom,
    ! [Nb: nat,A: produc6575502325842934193n_assn,P: fun_Pr3571187171380813162assn_o] :
      ( ? [X: produc6575502325842934193n_assn] :
          ( member7957490590177025114n_assn(X,set_Pr1139785259514867910n_assn(replic3825545231534752113n_assn(Nb,A)))
          & aa_Pro2491721552408928739assn_o(P,X) )
    <=> ( aa_Pro2491721552408928739assn_o(P,A)
        & ( Nb != zero_zero_nat ) ) ) ).

% Bex_set_replicate
tff(fact_7365_Bex__set__replicate,axiom,
    ! [Nb: nat,A: vEBT_VEBT,P: fun_VEBT_VEBT_o] :
      ( ? [X: vEBT_VEBT] :
          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),set_VEBT_VEBT2(replicate_VEBT_VEBT(Nb,A)))
          & aa_VEBT_VEBT_o(P,X) )
    <=> ( aa_VEBT_VEBT_o(P,A)
        & ( Nb != zero_zero_nat ) ) ) ).

% Bex_set_replicate
tff(fact_7366_in__set__replicate,axiom,
    ! [Xb: nat,Nb: nat,Y: nat] :
      ( aa_set_nat_o(member_nat(Xb),set_nat2(replicate_nat(Nb,Y)))
    <=> ( ( Xb = Y )
        & ( Nb != zero_zero_nat ) ) ) ).

% in_set_replicate
tff(fact_7367_in__set__replicate,axiom,
    ! [Xb: real,Nb: nat,Y: real] :
      ( aa_set_real_o(member_real(Xb),set_real2(replicate_real(Nb,Y)))
    <=> ( ( Xb = Y )
        & ( Nb != zero_zero_nat ) ) ) ).

% in_set_replicate
tff(fact_7368_in__set__replicate,axiom,
    ! [Xb: int,Nb: nat,Y: int] :
      ( aa_set_int_o(member_int(Xb),set_int2(replicate_int(Nb,Y)))
    <=> ( ( Xb = Y )
        & ( Nb != zero_zero_nat ) ) ) ).

% in_set_replicate
tff(fact_7369_in__set__replicate,axiom,
    ! [Xb: produc6575502325842934193n_assn,Nb: nat,Y: produc6575502325842934193n_assn] :
      ( member7957490590177025114n_assn(Xb,set_Pr1139785259514867910n_assn(replic3825545231534752113n_assn(Nb,Y)))
    <=> ( ( Xb = Y )
        & ( Nb != zero_zero_nat ) ) ) ).

% in_set_replicate
tff(fact_7370_in__set__replicate,axiom,
    ! [Xb: vEBT_VEBT,Nb: nat,Y: vEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xb),set_VEBT_VEBT2(replicate_VEBT_VEBT(Nb,Y)))
    <=> ( ( Xb = Y )
        & ( Nb != zero_zero_nat ) ) ) ).

% in_set_replicate
tff(fact_7371_nth__replicate,axiom,
    ! [I: nat,Nb: nat,Xb: vEBT_VEBTi] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),Nb)
     => ( nth_VEBT_VEBTi(replicate_VEBT_VEBTi(Nb,Xb),I) = Xb ) ) ).

% nth_replicate
tff(fact_7372_nth__replicate,axiom,
    ! [I: nat,Nb: nat,Xb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),Nb)
     => ( aa_nat_nat(nth_nat(replicate_nat(Nb,Xb)),I) = Xb ) ) ).

% nth_replicate
tff(fact_7373_nth__replicate,axiom,
    ! [I: nat,Nb: nat,Xb: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),Nb)
     => ( aa_nat_int(nth_int(replicate_int(Nb,Xb)),I) = Xb ) ) ).

% nth_replicate
tff(fact_7374_nth__replicate,axiom,
    ! [I: nat,Nb: nat,Xb: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),Nb)
     => ( nth_VEBT_VEBT(replicate_VEBT_VEBT(Nb,Xb),I) = Xb ) ) ).

% nth_replicate
tff(fact_7375_set__replicate,axiom,
    ! [Nb: nat,Xb: produc6575502325842934193n_assn] :
      ( ( Nb != zero_zero_nat )
     => ( set_Pr1139785259514867910n_assn(replic3825545231534752113n_assn(Nb,Xb)) = insert5290817439147925377n_assn(Xb,bot_bo1176836662018730877n_assn) ) ) ).

% set_replicate
tff(fact_7376_set__replicate,axiom,
    ! [Nb: nat,Xb: vEBT_VEBT] :
      ( ( Nb != zero_zero_nat )
     => ( set_VEBT_VEBT2(replicate_VEBT_VEBT(Nb,Xb)) = insert_VEBT_VEBT(Xb,bot_bo8194388402131092736T_VEBT) ) ) ).

% set_replicate
tff(fact_7377_set__replicate,axiom,
    ! [Nb: nat,Xb: nat] :
      ( ( Nb != zero_zero_nat )
     => ( set_nat2(replicate_nat(Nb,Xb)) = insert_nat(Xb,bot_bot_set_nat) ) ) ).

% set_replicate
tff(fact_7378_set__replicate,axiom,
    ! [Nb: nat,Xb: int] :
      ( ( Nb != zero_zero_nat )
     => ( set_int2(replicate_int(Nb,Xb)) = insert_int(Xb,bot_bot_set_int) ) ) ).

% set_replicate
tff(fact_7379_set__replicate,axiom,
    ! [Nb: nat,Xb: real] :
      ( ( Nb != zero_zero_nat )
     => ( set_real2(replicate_real(Nb,Xb)) = insert_real(Xb,bot_bot_set_real) ) ) ).

% set_replicate
tff(fact_7380_map__fst__mk__fst,axiom,
    ! [K: vEBT_VEBT,L: list_real] : map_Pr6195879527588727455T_VEBT(produc8110914911036349469T_real,map_re8618229306769252225T_real(produc8117437818029410057T_real(K),L)) = replicate_VEBT_VEBT(size_size_list_real(L),K) ).

% map_fst_mk_fst
tff(fact_7381_map__fst__mk__fst,axiom,
    ! [K: vEBT_VEBT,L: list_o] : map_Pr4868735216952053677T_VEBT(produc4993121158135996263VEBT_o,map_o_6754667662019005495VEBT_o(produc8721562602347293563VEBT_o(K),L)) = replicate_VEBT_VEBT(size_size_list_o(L),K) ).

% map_fst_mk_fst
tff(fact_7382_map__fst__mk__fst,axiom,
    ! [K: vEBT_VEBT,L: list_nat] : map_Pr1380729192516676091T_VEBT(produc8713918199166443969BT_nat,map_na4631810538828370761BT_nat(produc738532404422230701BT_nat(K),L)) = replicate_VEBT_VEBT(size_size_list_nat(L),K) ).

% map_fst_mk_fst
tff(fact_7383_map__fst__mk__fst,axiom,
    ! [K: vEBT_VEBT,L: list_int] : map_Pr3257657825534036127T_VEBT(produc8711427728657393693BT_int,map_in8151279748432256513BT_int(produc736041933913180425BT_int(K),L)) = replicate_VEBT_VEBT(size_size_list_int(L),K) ).

% map_fst_mk_fst
tff(fact_7384_map__fst__mk__fst,axiom,
    ! [K: assn,L: list_assn] : map_Pr8991440229025900053n_assn(produc9167289414957590229n_assn,map_as2373307505041272643n_assn(produc118845697133431529n_assn(K),L)) = replicate_assn(size_size_list_assn(L),K) ).

% map_fst_mk_fst
tff(fact_7385_map__fst__mk__fst,axiom,
    ! [K: uint32,L: list_uint32] : map_Pr2530660914518622561uint32(produc9004433772639906525uint32,map_ui5747794299746474503uint32(produc1400373151660368625uint32(K),L)) = replicate_uint32(size_s4844771616002835472uint32(L),K) ).

% map_fst_mk_fst
tff(fact_7386_map__fst__mk__fst,axiom,
    ! [K: nat,L: list_nat] : map_Pr3938374229010428429at_nat(product_fst_nat_nat,map_na7298421622053143531at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,K),L)) = replicate_nat(size_size_list_nat(L),K) ).

% map_fst_mk_fst
tff(fact_7387_map__fst__mk__fst,axiom,
    ! [K: int,L: list_int] : map_Pr6494458893431244577nt_int(product_fst_int_int,map_in7157766398909135175nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,K),L)) = replicate_int(size_size_list_int(L),K) ).

% map_fst_mk_fst
tff(fact_7388_map__fst__mk__fst,axiom,
    ! [K: heap_e7401611519738050253t_unit,L: list_set_nat] : map_Pr7042224834074979589t_unit(produc1824681642469235216et_nat,map_se5435525824004667726et_nat(produc7507926704131184380et_nat(K),L)) = replic5388364368018022029t_unit(size_s3254054031482475050et_nat(L),K) ).

% map_fst_mk_fst
tff(fact_7389_map__fst__mk__fst,axiom,
    ! [K: fun_in5730379987311621695e_term,L: list_P5707943133018811711nt_int] : map_Pr6740350451330705588e_term(produc4463008262350210037nt_int,map_Pr8804828260421831716nt_int(produc4729913460834754785nt_int(K),L)) = replic8593248471952391807e_term(size_s5157815400016825771nt_int(L),K) ).

% map_fst_mk_fst
tff(fact_7390_map__snd__mk__snd,axiom,
    ! [K: vEBT_VEBT,L: list_real] : map_Pr6147841162850987569T_VEBT(produc5083336317046741121T_VEBT,map_re7205069664741861231T_VEBT(aTP_Lamm_agi(K),L)) = replicate_VEBT_VEBT(size_size_list_real(L),K) ).

% map_snd_mk_snd
tff(fact_7391_map__snd__mk__snd,axiom,
    ! [K: vEBT_VEBT,L: list_o] : map_Pr7652832201708611317T_VEBT(produc7938581201502569057T_VEBT,map_o_8925299737569714927T_VEBT(aTP_Lamm_agj(K),L)) = replicate_VEBT_VEBT(size_size_list_o(L),K) ).

% map_snd_mk_snd
tff(fact_7392_map__snd__mk__snd,axiom,
    ! [K: vEBT_VEBT,L: list_nat] : map_Pr8570210702748812117T_VEBT(produc8172668247895388509T_VEBT,map_na3584885621601055599T_VEBT(aTP_Lamm_agk(K),L)) = replicate_VEBT_VEBT(size_size_list_nat(L),K) ).

% map_snd_mk_snd
tff(fact_7393_map__snd__mk__snd,axiom,
    ! [K: vEBT_VEBT,L: list_int] : map_Pr1314269154781486001T_VEBT(produc1678900780639429121T_VEBT,map_in4788438383458178671T_VEBT(aTP_Lamm_agl(K),L)) = replicate_VEBT_VEBT(size_size_list_int(L),K) ).

% map_snd_mk_snd
tff(fact_7394_map__snd__mk__snd,axiom,
    ! [K: nat,L: list_nat] : map_Pr3938374229010428429at_nat(product_snd_nat_nat,map_na7298421622053143531at_nat(aa_nat8803788312118491388at_nat(aTP_Lamm_ke,K),L)) = replicate_nat(size_size_list_nat(L),K) ).

% map_snd_mk_snd
tff(fact_7395_map__snd__mk__snd,axiom,
    ! [K: int,L: list_int] : map_Pr6494458893431244577nt_int(product_snd_int_int,map_in7157766398909135175nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_kf,K),L)) = replicate_int(size_size_list_int(L),K) ).

% map_snd_mk_snd
tff(fact_7396_map__snd__mk__snd,axiom,
    ! [K: code_integer,L: list_Code_integer] : map_Pr1857711230949937460nteger(produc6174133586879617921nteger,map_Co3589949550033412536nteger(aTP_Lamm_agm(K),L)) = replic7707675349574490269nteger(size_s3445333598471063425nteger(L),K) ).

% map_snd_mk_snd
tff(fact_7397_map__snd__mk__snd,axiom,
    ! [K: assn,L: list_assn] : map_Pr8991440229025900053n_assn(produc2051961928117032727n_assn,map_as2373307505041272643n_assn(aTP_Lamm_kc(K),L)) = replicate_assn(size_size_list_assn(L),K) ).

% map_snd_mk_snd
tff(fact_7398_map__snd__mk__snd,axiom,
    ! [K: set_nat,L: list_h2745205591237434579t_unit] : map_Pr4871828589360411470et_nat(produc8586169260539613262et_nat,map_he4567745686773187723et_nat(aTP_Lamm_kb(K),L)) = replicate_set_nat(size_s7011499401410611007t_unit(L),K) ).

% map_snd_mk_snd
tff(fact_7399_map__snd__mk__snd,axiom,
    ! [K: product_prod_int_int,L: list_f628289730585369423e_term] : map_Pr4441656247296274564nt_int(produc7629796867588368947nt_int,map_fu7709190425054843060nt_int(aTP_Lamm_ka(K),L)) = replic1057375728873637753nt_int(size_s4003082270698862563e_term(L),K) ).

% map_snd_mk_snd
tff(fact_7400_replicate__eqI,axiom,
    ! [Xs: list_P8527749157015355191n_assn,Nb: nat,Xb: produc6575502325842934193n_assn] :
      ( ( size_s6829681357464350627n_assn(Xs) = Nb )
     => ( ! [Y4: produc6575502325842934193n_assn] :
            ( member7957490590177025114n_assn(Y4,set_Pr1139785259514867910n_assn(Xs))
           => ( Y4 = Xb ) )
       => ( Xs = replic3825545231534752113n_assn(Nb,Xb) ) ) ) ).

% replicate_eqI
tff(fact_7401_replicate__eqI,axiom,
    ! [Xs: list_VEBT_VEBT,Nb: nat,Xb: vEBT_VEBT] :
      ( ( size_s6755466524823107622T_VEBT(Xs) = Nb )
     => ( ! [Y4: vEBT_VEBT] :
            ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Y4),set_VEBT_VEBT2(Xs))
           => ( Y4 = Xb ) )
       => ( Xs = replicate_VEBT_VEBT(Nb,Xb) ) ) ) ).

% replicate_eqI
tff(fact_7402_replicate__eqI,axiom,
    ! [Xs: list_real,Nb: nat,Xb: real] :
      ( ( size_size_list_real(Xs) = Nb )
     => ( ! [Y4: real] :
            ( aa_set_real_o(member_real(Y4),set_real2(Xs))
           => ( Y4 = Xb ) )
       => ( Xs = replicate_real(Nb,Xb) ) ) ) ).

% replicate_eqI
tff(fact_7403_replicate__eqI,axiom,
    ! [Xs: list_o,Nb: nat,Xb: $o] :
      ( ( size_size_list_o(Xs) = Nb )
     => ( ! [Y4: $o] :
            ( member_o((Y4),set_o2(Xs))
           => ( (Y4)
            <=> (Xb) ) )
       => ( Xs = replicate_o(Nb,(Xb)) ) ) ) ).

% replicate_eqI
tff(fact_7404_replicate__eqI,axiom,
    ! [Xs: list_nat,Nb: nat,Xb: nat] :
      ( ( size_size_list_nat(Xs) = Nb )
     => ( ! [Y4: nat] :
            ( aa_set_nat_o(member_nat(Y4),set_nat2(Xs))
           => ( Y4 = Xb ) )
       => ( Xs = replicate_nat(Nb,Xb) ) ) ) ).

% replicate_eqI
tff(fact_7405_replicate__eqI,axiom,
    ! [Xs: list_int,Nb: nat,Xb: int] :
      ( ( size_size_list_int(Xs) = Nb )
     => ( ! [Y4: int] :
            ( aa_set_int_o(member_int(Y4),set_int2(Xs))
           => ( Y4 = Xb ) )
       => ( Xs = replicate_int(Nb,Xb) ) ) ) ).

% replicate_eqI
tff(fact_7406_replicate__length__same,axiom,
    ! [Xs: list_P8527749157015355191n_assn,Xb: produc6575502325842934193n_assn] :
      ( ! [X2: produc6575502325842934193n_assn] :
          ( member7957490590177025114n_assn(X2,set_Pr1139785259514867910n_assn(Xs))
         => ( X2 = Xb ) )
     => ( replic3825545231534752113n_assn(size_s6829681357464350627n_assn(Xs),Xb) = Xs ) ) ).

% replicate_length_same
tff(fact_7407_replicate__length__same,axiom,
    ! [Xs: list_VEBT_VEBT,Xb: vEBT_VEBT] :
      ( ! [X2: vEBT_VEBT] :
          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X2),set_VEBT_VEBT2(Xs))
         => ( X2 = Xb ) )
     => ( replicate_VEBT_VEBT(size_s6755466524823107622T_VEBT(Xs),Xb) = Xs ) ) ).

% replicate_length_same
tff(fact_7408_replicate__length__same,axiom,
    ! [Xs: list_real,Xb: real] :
      ( ! [X2: real] :
          ( aa_set_real_o(member_real(X2),set_real2(Xs))
         => ( X2 = Xb ) )
     => ( replicate_real(size_size_list_real(Xs),Xb) = Xs ) ) ).

% replicate_length_same
tff(fact_7409_replicate__length__same,axiom,
    ! [Xs: list_o,Xb: $o] :
      ( ! [X2: $o] :
          ( member_o((X2),set_o2(Xs))
         => ( (X2)
          <=> (Xb) ) )
     => ( replicate_o(size_size_list_o(Xs),(Xb)) = Xs ) ) ).

% replicate_length_same
tff(fact_7410_replicate__length__same,axiom,
    ! [Xs: list_nat,Xb: nat] :
      ( ! [X2: nat] :
          ( aa_set_nat_o(member_nat(X2),set_nat2(Xs))
         => ( X2 = Xb ) )
     => ( replicate_nat(size_size_list_nat(Xs),Xb) = Xs ) ) ).

% replicate_length_same
tff(fact_7411_replicate__length__same,axiom,
    ! [Xs: list_int,Xb: int] :
      ( ! [X2: int] :
          ( aa_set_int_o(member_int(X2),set_int2(Xs))
         => ( X2 = Xb ) )
     => ( replicate_int(size_size_list_int(Xs),Xb) = Xs ) ) ).

% replicate_length_same
tff(fact_7412_map__replicate__const,axiom,
    ! [K: nat,Lst: list_VEBT_VEBT] : map_VEBT_VEBT_nat(aTP_Lamm_agn(K),Lst) = replicate_nat(size_s6755466524823107622T_VEBT(Lst),K) ).

% map_replicate_const
tff(fact_7413_map__replicate__const,axiom,
    ! [K: real,Lst: list_VEBT_VEBT] : map_VEBT_VEBT_real(aTP_Lamm_ago(K),Lst) = replicate_real(size_s6755466524823107622T_VEBT(Lst),K) ).

% map_replicate_const
tff(fact_7414_map__replicate__const,axiom,
    ! [K: assn,Lst: list_P8527749157015355191n_assn] : map_Pr8991440229025900053n_assn(aTP_Lamm_agp(K),Lst) = replicate_assn(size_s6829681357464350627n_assn(Lst),K) ).

% map_replicate_const
tff(fact_7415_map__replicate__const,axiom,
    ! [K: vEBT_VEBT,Lst: list_real] : map_real_VEBT_VEBT(aTP_Lamm_agq(K),Lst) = replicate_VEBT_VEBT(size_size_list_real(Lst),K) ).

% map_replicate_const
tff(fact_7416_map__replicate__const,axiom,
    ! [K: vEBT_VEBT,Lst: list_o] : map_o_VEBT_VEBT(aTP_Lamm_agr(K),Lst) = replicate_VEBT_VEBT(size_size_list_o(Lst),K) ).

% map_replicate_const
tff(fact_7417_map__replicate__const,axiom,
    ! [K: nat,Lst: list_nat] : map_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_li,K),Lst) = replicate_nat(size_size_list_nat(Lst),K) ).

% map_replicate_const
tff(fact_7418_map__replicate__const,axiom,
    ! [K: $o,Lst: list_nat] : map_nat_o(aa_o_fun_nat_o(aTP_Lamm_mk,(K)),Lst) = replicate_o(size_size_list_nat(Lst),(K)) ).

% map_replicate_const
tff(fact_7419_map__replicate__const,axiom,
    ! [K: vEBT_VEBT,Lst: list_nat] : map_nat_VEBT_VEBT(aTP_Lamm_ags(K),Lst) = replicate_VEBT_VEBT(size_size_list_nat(Lst),K) ).

% map_replicate_const
tff(fact_7420_map__replicate__const,axiom,
    ! [K: vEBT_VEBT,Lst: list_int] : map_int_VEBT_VEBT(aTP_Lamm_agt(K),Lst) = replicate_VEBT_VEBT(size_size_list_int(Lst),K) ).

% map_replicate_const
tff(fact_7421_set__replicate__Suc,axiom,
    ! [Nb: nat,Xb: produc6575502325842934193n_assn] : set_Pr1139785259514867910n_assn(replic3825545231534752113n_assn(aa_nat_nat(suc,Nb),Xb)) = insert5290817439147925377n_assn(Xb,bot_bo1176836662018730877n_assn) ).

% set_replicate_Suc
tff(fact_7422_set__replicate__Suc,axiom,
    ! [Nb: nat,Xb: vEBT_VEBT] : set_VEBT_VEBT2(replicate_VEBT_VEBT(aa_nat_nat(suc,Nb),Xb)) = insert_VEBT_VEBT(Xb,bot_bo8194388402131092736T_VEBT) ).

% set_replicate_Suc
tff(fact_7423_set__replicate__Suc,axiom,
    ! [Nb: nat,Xb: nat] : set_nat2(replicate_nat(aa_nat_nat(suc,Nb),Xb)) = insert_nat(Xb,bot_bot_set_nat) ).

% set_replicate_Suc
tff(fact_7424_set__replicate__Suc,axiom,
    ! [Nb: nat,Xb: int] : set_int2(replicate_int(aa_nat_nat(suc,Nb),Xb)) = insert_int(Xb,bot_bot_set_int) ).

% set_replicate_Suc
tff(fact_7425_set__replicate__Suc,axiom,
    ! [Nb: nat,Xb: real] : set_real2(replicate_real(aa_nat_nat(suc,Nb),Xb)) = insert_real(Xb,bot_bot_set_real) ).

% set_replicate_Suc
tff(fact_7426_set__replicate__conv__if,axiom,
    ! [Nb: nat,Xb: produc6575502325842934193n_assn] :
      set_Pr1139785259514867910n_assn(replic3825545231534752113n_assn(Nb,Xb)) = $ite(Nb = zero_zero_nat,bot_bo1176836662018730877n_assn,insert5290817439147925377n_assn(Xb,bot_bo1176836662018730877n_assn)) ).

% set_replicate_conv_if
tff(fact_7427_set__replicate__conv__if,axiom,
    ! [Nb: nat,Xb: vEBT_VEBT] :
      set_VEBT_VEBT2(replicate_VEBT_VEBT(Nb,Xb)) = $ite(Nb = zero_zero_nat,bot_bo8194388402131092736T_VEBT,insert_VEBT_VEBT(Xb,bot_bo8194388402131092736T_VEBT)) ).

% set_replicate_conv_if
tff(fact_7428_set__replicate__conv__if,axiom,
    ! [Nb: nat,Xb: nat] :
      set_nat2(replicate_nat(Nb,Xb)) = $ite(Nb = zero_zero_nat,bot_bot_set_nat,insert_nat(Xb,bot_bot_set_nat)) ).

% set_replicate_conv_if
tff(fact_7429_set__replicate__conv__if,axiom,
    ! [Nb: nat,Xb: int] :
      set_int2(replicate_int(Nb,Xb)) = $ite(Nb = zero_zero_nat,bot_bot_set_int,insert_int(Xb,bot_bot_set_int)) ).

% set_replicate_conv_if
tff(fact_7430_set__replicate__conv__if,axiom,
    ! [Nb: nat,Xb: real] :
      set_real2(replicate_real(Nb,Xb)) = $ite(Nb = zero_zero_nat,bot_bot_set_real,insert_real(Xb,bot_bot_set_real)) ).

% set_replicate_conv_if
tff(fact_7431_list__assn__cong,axiom,
    ! [Xs: list_VEBT_VEBT,Xs4: list_VEBT_VEBT,Xsi: list_VEBT_VEBT,Xsi2: list_VEBT_VEBT,A2: fun_VE1571388752190420990T_assn,A8: fun_VE1571388752190420990T_assn] :
      ( ( Xs = Xs4 )
     => ( ( Xsi = Xsi2 )
       => ( ! [X2: vEBT_VEBT,Xi: vEBT_VEBT] :
              ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X2),set_VEBT_VEBT2(Xs4))
             => ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xi),set_VEBT_VEBT2(Xsi2))
               => ( aa_VEBT_VEBT_assn(aa_VEB115603943081001463T_assn(A2,X2),Xi) = aa_VEBT_VEBT_assn(aa_VEB115603943081001463T_assn(A8,X2),Xi) ) ) )
         => ( vEBT_L1279224858307276611T_VEBT(A2,Xs,Xsi) = vEBT_L1279224858307276611T_VEBT(A8,Xs4,Xsi2) ) ) ) ) ).

% list_assn_cong
tff(fact_7432_list__assn__cong,axiom,
    ! [Xs: list_VEBT_VEBT,Xs4: list_VEBT_VEBT,Xsi: list_nat,Xsi2: list_nat,A2: fun_VE4931415397359990460t_assn,A8: fun_VE4931415397359990460t_assn] :
      ( ( Xs = Xs4 )
     => ( ( Xsi = Xsi2 )
       => ( ! [X2: vEBT_VEBT,Xi: nat] :
              ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X2),set_VEBT_VEBT2(Xs4))
             => ( aa_set_nat_o(member_nat(Xi),set_nat2(Xsi2))
               => ( aa_nat_assn(aa_VEB8640158415945897581t_assn(A2,X2),Xi) = aa_nat_assn(aa_VEB8640158415945897581t_assn(A8,X2),Xi) ) ) )
         => ( aa_list_nat_assn(vEBT_L8296926524756676353BT_nat(A2,Xs),Xsi) = aa_list_nat_assn(vEBT_L8296926524756676353BT_nat(A8,Xs4),Xsi2) ) ) ) ) ).

% list_assn_cong
tff(fact_7433_list__assn__cong,axiom,
    ! [Xs: list_VEBT_VEBT,Xs4: list_VEBT_VEBT,Xsi: list_real,Xsi2: list_real,A2: fun_VE6910866152645720856l_assn,A8: fun_VE6910866152645720856l_assn] :
      ( ( Xs = Xs4 )
     => ( ( Xsi = Xsi2 )
       => ( ! [X2: vEBT_VEBT,Xi: real] :
              ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X2),set_VEBT_VEBT2(Xs4))
             => ( aa_set_real_o(member_real(Xi),set_real2(Xsi2))
               => ( aa_real_assn(aa_VEB8575627266564790985l_assn(A2,X2),Xi) = aa_real_assn(aa_VEB8575627266564790985l_assn(A8,X2),Xi) ) ) )
         => ( vEBT_L5781919052683127133T_real(A2,Xs,Xsi) = vEBT_L5781919052683127133T_real(A8,Xs4,Xsi2) ) ) ) ) ).

% list_assn_cong
tff(fact_7434_list__assn__cong,axiom,
    ! [Xs: list_VEBT_VEBT,Xs4: list_VEBT_VEBT,Xsi: list_int,Xsi2: list_int,A2: fun_VE123837653852366616t_assn,A8: fun_VE123837653852366616t_assn] :
      ( ( Xs = Xs4 )
     => ( ( Xsi = Xsi2 )
       => ( ! [X2: vEBT_VEBT,Xi: int] :
              ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X2),set_VEBT_VEBT2(Xs4))
             => ( aa_set_int_o(member_int(Xi),set_int2(Xsi2))
               => ( aa_int_assn(aa_VEB392321844297299145t_assn(A2,X2),Xi) = aa_int_assn(aa_VEB392321844297299145t_assn(A8,X2),Xi) ) ) )
         => ( vEBT_L8294436054247626077BT_int(A2,Xs,Xsi) = vEBT_L8294436054247626077BT_int(A8,Xs4,Xsi2) ) ) ) ) ).

% list_assn_cong
tff(fact_7435_list__assn__cong,axiom,
    ! [Xs: list_nat,Xs4: list_nat,Xsi: list_VEBT_VEBT,Xsi2: list_VEBT_VEBT,A2: fun_na4489259242115493880T_assn,A8: fun_na4489259242115493880T_assn] :
      ( ( Xs = Xs4 )
     => ( ( Xsi = Xsi2 )
       => ( ! [X2: nat,Xi: vEBT_VEBT] :
              ( aa_set_nat_o(member_nat(X2),set_nat2(Xs4))
             => ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xi),set_VEBT_VEBT2(Xsi2))
               => ( aa_VEBT_VEBT_assn(aa_nat6835651039071750625T_assn(A2,X2),Xi) = aa_VEBT_VEBT_assn(aa_nat6835651039071750625T_assn(A8,X2),Xi) ) ) )
         => ( vEBT_L8158188754432654943T_VEBT(A2,Xs,Xsi) = vEBT_L8158188754432654943T_VEBT(A8,Xs4,Xsi2) ) ) ) ) ).

% list_assn_cong
tff(fact_7436_list__assn__cong,axiom,
    ! [Xs: list_nat,Xs4: list_nat,Xsi: list_nat,Xsi2: list_nat,A2: fun_nat_fun_nat_assn,A8: fun_nat_fun_nat_assn] :
      ( ( Xs = Xs4 )
     => ( ( Xsi = Xsi2 )
       => ( ! [X2: nat,Xi: nat] :
              ( aa_set_nat_o(member_nat(X2),set_nat2(Xs4))
             => ( aa_set_nat_o(member_nat(Xi),set_nat2(Xsi2))
               => ( aa_nat_assn(aa_nat_fun_nat_assn(A2,X2),Xi) = aa_nat_assn(aa_nat_fun_nat_assn(A8,X2),Xi) ) ) )
         => ( vEBT_L8301102511889123557at_nat(A2,Xs,Xsi) = vEBT_L8301102511889123557at_nat(A8,Xs4,Xsi2) ) ) ) ) ).

% list_assn_cong
tff(fact_7437_list__assn__cong,axiom,
    ! [Xs: list_nat,Xs4: list_nat,Xsi: list_real,Xsi2: list_real,A2: fun_na448437756743965086l_assn,A8: fun_na448437756743965086l_assn] :
      ( ( Xs = Xs4 )
     => ( ( Xsi = Xsi2 )
       => ( ! [X2: nat,Xi: real] :
              ( aa_set_nat_o(member_nat(X2),set_nat2(Xs4))
             => ( aa_set_real_o(member_real(Xi),set_real2(Xsi2))
               => ( aa_real_assn(aa_nat_fun_real_assn(A2,X2),Xi) = aa_real_assn(aa_nat_fun_real_assn(A8,X2),Xi) ) ) )
         => ( vEBT_L6102073776069194049t_real(A2,Xs,Xsi) = vEBT_L6102073776069194049t_real(A8,Xs4,Xsi2) ) ) ) ) ).

% list_assn_cong
tff(fact_7438_list__assn__cong,axiom,
    ! [Xs: list_nat,Xs4: list_nat,Xsi: list_int,Xsi2: list_int,A2: fun_nat_fun_int_assn,A8: fun_nat_fun_int_assn] :
      ( ( Xs = Xs4 )
     => ( ( Xsi = Xsi2 )
       => ( ! [X2: nat,Xi: int] :
              ( aa_set_nat_o(member_nat(X2),set_nat2(Xs4))
             => ( aa_set_int_o(member_int(Xi),set_int2(Xsi2))
               => ( aa_int_assn(aa_nat_fun_int_assn(A2,X2),Xi) = aa_int_assn(aa_nat_fun_int_assn(A8,X2),Xi) ) ) )
         => ( vEBT_L8298612041380073281at_int(A2,Xs,Xsi) = vEBT_L8298612041380073281at_int(A8,Xs4,Xsi2) ) ) ) ) ).

% list_assn_cong
tff(fact_7439_list__assn__cong,axiom,
    ! [Xs: list_real,Xs4: list_real,Xsi: list_VEBT_VEBT,Xsi2: list_VEBT_VEBT,A2: fun_re8050108611700907676T_assn,A8: fun_re8050108611700907676T_assn] :
      ( ( Xs = Xs4 )
     => ( ( Xsi = Xsi2 )
       => ( ! [X2: real,Xi: vEBT_VEBT] :
              ( aa_set_real_o(member_real(X2),set_real2(Xs4))
             => ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xi),set_VEBT_VEBT2(Xsi2))
               => ( aa_VEBT_VEBT_assn(aa_rea1965931645776870533T_assn(A2,X2),Xi) = aa_VEBT_VEBT_assn(aa_rea1965931645776870533T_assn(A8,X2),Xi) ) ) )
         => ( vEBT_L4595930785310033027T_VEBT(A2,Xs,Xsi) = vEBT_L4595930785310033027T_VEBT(A8,Xs4,Xsi2) ) ) ) ) ).

% list_assn_cong
tff(fact_7440_list__assn__cong,axiom,
    ! [Xs: list_real,Xs4: list_real,Xsi: list_nat,Xsi2: list_nat,A2: fun_re574865750389657758t_assn,A8: fun_re574865750389657758t_assn] :
      ( ( Xs = Xs4 )
     => ( ( Xsi = Xsi2 )
       => ( ! [X2: real,Xi: nat] :
              ( aa_set_real_o(member_real(X2),set_real2(Xs4))
             => ( aa_set_nat_o(member_nat(Xi),set_nat2(Xsi2))
               => ( aa_nat_assn(aa_real_fun_nat_assn(A2,X2),Xi) = aa_nat_assn(aa_real_fun_nat_assn(A8,X2),Xi) ) ) )
         => ( vEBT_L1446010312343316929al_nat(A2,Xs,Xsi) = vEBT_L1446010312343316929al_nat(A8,Xs4,Xsi2) ) ) ) ) ).

% list_assn_cong
tff(fact_7441_accp__subset__induct,axiom,
    ! [D5: fun_Pr7546709192626087934_nat_o,R: fun_Pr8877776288780692068_nat_o,Xb: produc9072475918466114483BT_nat,P: fun_Pr7546709192626087934_nat_o] :
      ( ord_le1769015658490769310_nat_o(D5,accp_P2887432264394892906BT_nat(R))
     => ( ! [X2: produc9072475918466114483BT_nat,Z3: produc9072475918466114483BT_nat] :
            ( aa_Pro632845113475883111_nat_o(D5,X2)
           => ( aa_Pro632845113475883111_nat_o(aa_Pro1578556086376436173_nat_o(R,Z3),X2)
             => aa_Pro632845113475883111_nat_o(D5,Z3) ) )
       => ( aa_Pro632845113475883111_nat_o(D5,Xb)
         => ( ! [X2: produc9072475918466114483BT_nat] :
                ( aa_Pro632845113475883111_nat_o(D5,X2)
               => ( ! [Z4: produc9072475918466114483BT_nat] :
                      ( aa_Pro632845113475883111_nat_o(aa_Pro1578556086376436173_nat_o(R,Z4),X2)
                     => aa_Pro632845113475883111_nat_o(P,Z4) )
                 => aa_Pro632845113475883111_nat_o(P,X2) ) )
           => aa_Pro632845113475883111_nat_o(P,Xb) ) ) ) ) ).

% accp_subset_induct
tff(fact_7442_accp__subset__induct,axiom,
    ! [D5: fun_Pr6331313853612757382_num_o,R: fun_Pr4799464687642144302_num_o,Xb: product_prod_num_num,P: fun_Pr6331313853612757382_num_o] :
      ( ord_le3107050023622517222_num_o(D5,accp_P3113834385874906142um_num(R))
     => ( ! [X2: product_prod_num_num,Z3: product_prod_num_num] :
            ( aa_Pro7685007914727650815_num_o(D5,X2)
           => ( aa_Pro7685007914727650815_num_o(aa_Pro4218333719626769503_num_o(R,Z3),X2)
             => aa_Pro7685007914727650815_num_o(D5,Z3) ) )
       => ( aa_Pro7685007914727650815_num_o(D5,Xb)
         => ( ! [X2: product_prod_num_num] :
                ( aa_Pro7685007914727650815_num_o(D5,X2)
               => ( ! [Z4: product_prod_num_num] :
                      ( aa_Pro7685007914727650815_num_o(aa_Pro4218333719626769503_num_o(R,Z4),X2)
                     => aa_Pro7685007914727650815_num_o(P,Z4) )
                 => aa_Pro7685007914727650815_num_o(P,X2) ) )
           => aa_Pro7685007914727650815_num_o(P,Xb) ) ) ) ) ).

% accp_subset_induct
tff(fact_7443_accp__subset__induct,axiom,
    ! [D5: fun_Pr4796943543331071514_nat_o,R: fun_Pr2028404973800837934_nat_o,Xb: product_prod_nat_nat,P: fun_Pr4796943543331071514_nat_o] :
      ( ord_le1572679713340831354_nat_o(D5,accp_P4275260045618599050at_nat(R))
     => ( ! [X2: product_prod_nat_nat,Z3: product_prod_nat_nat] :
            ( aa_Pro6150637604445964947_nat_o(D5,X2)
           => ( aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(R,Z3),X2)
             => aa_Pro6150637604445964947_nat_o(D5,Z3) ) )
       => ( aa_Pro6150637604445964947_nat_o(D5,Xb)
         => ( ! [X2: product_prod_nat_nat] :
                ( aa_Pro6150637604445964947_nat_o(D5,X2)
               => ( ! [Z4: product_prod_nat_nat] :
                      ( aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(R,Z4),X2)
                     => aa_Pro6150637604445964947_nat_o(P,Z4) )
                 => aa_Pro6150637604445964947_nat_o(P,X2) ) )
           => aa_Pro6150637604445964947_nat_o(P,Xb) ) ) ) ) ).

% accp_subset_induct
tff(fact_7444_accp__subset__induct,axiom,
    ! [D5: fun_Pr3238374608701176162_int_o,R: fun_Pr9018117161326028078_int_o,Xb: product_prod_int_int,P: fun_Pr3238374608701176162_int_o] :
      ( ord_le14110778710936002_int_o(D5,accp_P1096762738010456898nt_int(R))
     => ( ! [X2: product_prod_int_int,Z3: product_prod_int_int] :
            ( aa_Pro4592068669816069595_int_o(D5,X2)
           => ( aa_Pro4592068669816069595_int_o(aa_Pro3952998155787061599_int_o(R,Z3),X2)
             => aa_Pro4592068669816069595_int_o(D5,Z3) ) )
       => ( aa_Pro4592068669816069595_int_o(D5,Xb)
         => ( ! [X2: product_prod_int_int] :
                ( aa_Pro4592068669816069595_int_o(D5,X2)
               => ( ! [Z4: product_prod_int_int] :
                      ( aa_Pro4592068669816069595_int_o(aa_Pro3952998155787061599_int_o(R,Z4),X2)
                     => aa_Pro4592068669816069595_int_o(P,Z4) )
                 => aa_Pro4592068669816069595_int_o(P,X2) ) )
           => aa_Pro4592068669816069595_int_o(P,Xb) ) ) ) ) ).

% accp_subset_induct
tff(fact_7445_accp__subset__induct,axiom,
    ! [D5: fun_nat_o,R: fun_nat_fun_nat_o,Xb: nat,P: fun_nat_o] :
      ( ord_le7240017405068453697_nat_o(D5,accp_nat(R))
     => ( ! [X2: nat,Z3: nat] :
            ( aa_nat_o(D5,X2)
           => ( aa_nat_o(aa_nat_fun_nat_o(R,Z3),X2)
             => aa_nat_o(D5,Z3) ) )
       => ( aa_nat_o(D5,Xb)
         => ( ! [X2: nat] :
                ( aa_nat_o(D5,X2)
               => ( ! [Z4: nat] :
                      ( aa_nat_o(aa_nat_fun_nat_o(R,Z4),X2)
                     => aa_nat_o(P,Z4) )
                 => aa_nat_o(P,X2) ) )
           => aa_nat_o(P,Xb) ) ) ) ) ).

% accp_subset_induct
tff(fact_7446_vebt__buildup_Osimps_I3_J,axiom,
    ! [Vaa: nat] :
      vEBT_vebt_buildup(aa_nat_nat(suc,aa_nat_nat(suc,Vaa))) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,Vaa))),
        $let(
          half: nat,
          half:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Vaa))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
          aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,Vaa)),replicate_VEBT_VEBT(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),half),vEBT_vebt_buildup(half))),vEBT_vebt_buildup(half)) ),
        $let(
          half: nat,
          half:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Vaa))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
          aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,Vaa)),replicate_VEBT_VEBT(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,half)),vEBT_vebt_buildup(half))),vEBT_vebt_buildup(aa_nat_nat(suc,half))) ) ) ).

% vebt_buildup.simps(3)
tff(fact_7447_extract__pre__list__assn__lengthD,axiom,
    ! [A2: fun_re6303261851654292474l_assn,Xs: list_real,Xsi: list_real,H: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(rep_assn(vEBT_L1930518968523514909l_real(A2,Xs,Xsi)),H)
     => ( size_size_list_real(Xsi) = size_size_list_real(Xs) ) ) ).

% extract_pre_list_assn_lengthD
tff(fact_7448_extract__pre__list__assn__lengthD,axiom,
    ! [A2: fun_o_fun_real_assn,Xs: list_o,Xsi: list_real,H: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(rep_assn(vEBT_L4725278957065240257o_real(A2,Xs,Xsi)),H)
     => ( size_size_list_real(Xsi) = size_size_list_o(Xs) ) ) ).

% extract_pre_list_assn_lengthD
tff(fact_7449_extract__pre__list__assn__lengthD,axiom,
    ! [A2: fun_na448437756743965086l_assn,Xs: list_nat,Xsi: list_real,H: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(rep_assn(vEBT_L6102073776069194049t_real(A2,Xs,Xsi)),H)
     => ( size_size_list_real(Xsi) = size_size_list_nat(Xs) ) ) ).

% extract_pre_list_assn_lengthD
tff(fact_7450_extract__pre__list__assn__lengthD,axiom,
    ! [A2: fun_in6446958447386763770l_assn,Xs: list_int,Xsi: list_real,H: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(rep_assn(vEBT_L8288995350762215837t_real(A2,Xs,Xsi)),H)
     => ( size_size_list_real(Xsi) = size_size_list_int(Xs) ) ) ).

% extract_pre_list_assn_lengthD
tff(fact_7451_extract__pre__list__assn__lengthD,axiom,
    ! [A2: fun_real_fun_o_assn,Xs: list_real,Xsi: list_o,H: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(rep_assn(vEBT_L6234343332106409831real_o(A2,Xs,Xsi)),H)
     => ( size_size_list_o(Xsi) = size_size_list_real(Xs) ) ) ).

% extract_pre_list_assn_lengthD
tff(fact_7452_extract__pre__list__assn__lengthD,axiom,
    ! [A2: fun_o_fun_o_assn,Xs: list_o,Xsi: list_o,H: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(rep_assn(vEBT_L7363604446928714179sn_o_o(A2,Xs,Xsi)),H)
     => ( size_size_list_o(Xsi) = size_size_list_o(Xs) ) ) ).

% extract_pre_list_assn_lengthD
tff(fact_7453_extract__pre__list__assn__lengthD,axiom,
    ! [A2: fun_nat_fun_o_assn,Xs: list_nat,Xsi: list_o,H: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(rep_assn(vEBT_L7887682484454631235_nat_o(A2,Xs,Xsi)),H)
     => ( size_size_list_o(Xsi) = size_size_list_nat(Xs) ) ) ).

% extract_pre_list_assn_lengthD
tff(fact_7454_extract__pre__list__assn__lengthD,axiom,
    ! [A2: fun_int_fun_o_assn,Xs: list_int,Xsi: list_o,H: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(rep_assn(vEBT_L6066640139021943271_int_o(A2,Xs,Xsi)),H)
     => ( size_size_list_o(Xsi) = size_size_list_int(Xs) ) ) ).

% extract_pre_list_assn_lengthD
tff(fact_7455_extract__pre__list__assn__lengthD,axiom,
    ! [A2: fun_re574865750389657758t_assn,Xs: list_real,Xsi: list_nat,H: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(rep_assn(vEBT_L1446010312343316929al_nat(A2,Xs,Xsi)),H)
     => ( size_size_list_nat(Xsi) = size_size_list_real(Xs) ) ) ).

% extract_pre_list_assn_lengthD
tff(fact_7456_extract__pre__list__assn__lengthD,axiom,
    ! [A2: fun_o_fun_nat_assn,Xs: list_o,Xsi: list_nat,H: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(rep_assn(vEBT_L4785011123346445925_o_nat(A2,Xs,Xsi)),H)
     => ( size_size_list_nat(Xsi) = size_size_list_o(Xs) ) ) ).

% extract_pre_list_assn_lengthD
tff(fact_7457_vebt__buildup_Opelims,axiom,
    ! [Xb: nat,Y: vEBT_VEBT] :
      ( ( vEBT_vebt_buildup(Xb) = Y )
     => ( aa_nat_o(accp_nat(vEBT_v4011308405150292612up_rel),Xb)
       => ( ( ( Xb = zero_zero_nat )
           => ( ( Y = vEBT_Leaf($false,$false) )
             => ~ aa_nat_o(accp_nat(vEBT_v4011308405150292612up_rel),zero_zero_nat) ) )
         => ( ( ( Xb = aa_nat_nat(suc,zero_zero_nat) )
             => ( ( Y = vEBT_Leaf($false,$false) )
               => ~ aa_nat_o(accp_nat(vEBT_v4011308405150292612up_rel),aa_nat_nat(suc,zero_zero_nat)) ) )
           => ~ ! [Va2: nat] :
                  ( ( Xb = aa_nat_nat(suc,aa_nat_nat(suc,Va2)) )
                 => ( ( Y = $ite(
                          aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,Va2))),
                          $let(
                            half: nat,
                            half:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                            aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2)),replicate_VEBT_VEBT(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),half),vEBT_vebt_buildup(half))),vEBT_vebt_buildup(half)) ),
                          $let(
                            half: nat,
                            half:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                            aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2)),replicate_VEBT_VEBT(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,half)),vEBT_vebt_buildup(half))),vEBT_vebt_buildup(aa_nat_nat(suc,half))) ) ) )
                   => ~ aa_nat_o(accp_nat(vEBT_v4011308405150292612up_rel),aa_nat_nat(suc,aa_nat_nat(suc,Va2))) ) ) ) ) ) ) ).

% vebt_buildup.pelims
tff(fact_7458_VEBT__internal_Ospace_Opelims,axiom,
    ! [Xb: vEBT_VEBT,Y: nat] :
      ( ( aa_VEBT_VEBT_nat(vEBT_VEBT_space,Xb) = Y )
     => ( aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_VEBT_space_rel2),Xb)
       => ( ! [A3: $o,B2: $o] :
              ( ( Xb = vEBT_Leaf((A3),(B2)) )
             => ( ( Y = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)) )
               => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_VEBT_space_rel2),vEBT_Leaf((A3),(B2))) ) )
         => ~ ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,Deg2,TreeList),Summary) )
               => ( ( Y = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit0,one)))),aa_VEBT_VEBT_nat(vEBT_VEBT_space,Summary))),size_s6755466524823107622T_VEBT(TreeList))),aa_nat_nat(foldr_nat_nat(plus_plus_nat,map_VEBT_VEBT_nat(vEBT_VEBT_space,TreeList)),zero_zero_nat)) )
                 => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_VEBT_space_rel2),aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,Deg2,TreeList),Summary)) ) ) ) ) ) ).

% VEBT_internal.space.pelims
tff(fact_7459_VEBT__internal_Ospace_H_Opelims,axiom,
    ! [Xb: vEBT_VEBT,Y: nat] :
      ( ( aa_VEBT_VEBT_nat(vEBT_VEBT_space2,Xb) = Y )
     => ( aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_VEBT_space_rel),Xb)
       => ( ! [A3: $o,B2: $o] :
              ( ( Xb = vEBT_Leaf((A3),(B2)) )
             => ( ( Y = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one))) )
               => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_VEBT_space_rel),vEBT_Leaf((A3),(B2))) ) )
         => ~ ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,Deg2,TreeList),Summary) )
               => ( ( Y = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit1,one)))),aa_VEBT_VEBT_nat(vEBT_VEBT_space2,Summary))),aa_nat_nat(foldr_nat_nat(plus_plus_nat,map_VEBT_VEBT_nat(vEBT_VEBT_space2,TreeList)),zero_zero_nat)) )
                 => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_VEBT_space_rel),aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,Deg2,TreeList),Summary)) ) ) ) ) ) ).

% VEBT_internal.space'.pelims
tff(fact_7460_vebt__assn__raw_Opelims,axiom,
    ! [Xb: vEBT_VEBT,Xaa: vEBT_VEBTi,Y: assn] :
      ( ( aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Xb),Xaa) = Y )
     => ( accp_P7675410724331315407_VEBTi(vEBT_v8524038756793281170aw_rel,produc6084888613844515218_VEBTi(Xb,Xaa))
       => ( ! [A3: $o,B2: $o] :
              ( ( Xb = vEBT_Leaf((A3),(B2)) )
             => ! [Ai: $o,Bi: $o] :
                  ( ( Xaa = vEBT_Leafi((Ai),(Bi)) )
                 => ( ( Y = aa_o_assn(pure_assn,
                          ( ( (Ai) = (A3) )
                          & ( (Bi) = (B2) ) )) )
                   => ~ accp_P7675410724331315407_VEBTi(vEBT_v8524038756793281170aw_rel,produc6084888613844515218_VEBTi(vEBT_Leaf((A3),(B2)),vEBT_Leafi((Ai),(Bi)))) ) ) )
         => ( ! [Mmo2: option4927543243414619207at_nat,Deg2: nat,Tree_list2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Mmo2,Deg2,Tree_list2),Summary) )
               => ! [Mmoi2: option4927543243414619207at_nat,Degi2: nat,Tree_array2: array_VEBT_VEBTi,Summaryi2: vEBT_VEBTi] :
                    ( ( Xaa = aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(Mmoi2,Degi2,Tree_array2),Summaryi2) )
                   => ( ( Y = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,
                              aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,
                                  aa_o_assn(pure_assn,
                                    ( ( Mmoi2 = Mmo2 )
                                    & ( Degi2 = Deg2 ) ))),aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Summary),Summaryi2))),ex_ass463751140784270563_VEBTi(aa_arr7909941341466502432i_assn(aTP_Lamm_bf(Tree_list2),Tree_array2))) )
                     => ~ accp_P7675410724331315407_VEBTi(vEBT_v8524038756793281170aw_rel,produc6084888613844515218_VEBTi(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Mmo2,Deg2,Tree_list2),Summary),aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(Mmoi2,Degi2,Tree_array2),Summaryi2))) ) ) )
           => ( ! [V2: option4927543243414619207at_nat,Va2: nat,Vb3: list_VEBT_VEBT,Vc3: vEBT_VEBT] :
                  ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(V2,Va2,Vb3),Vc3) )
                 => ! [Vd3: $o,Ve3: $o] :
                      ( ( Xaa = vEBT_Leafi((Vd3),(Ve3)) )
                     => ( ( Y = bot_bot_assn )
                       => ~ accp_P7675410724331315407_VEBTi(vEBT_v8524038756793281170aw_rel,produc6084888613844515218_VEBTi(aa_VEB5832953974749077948T_VEBT(vEBT_Node(V2,Va2,Vb3),Vc3),vEBT_Leafi((Vd3),(Ve3)))) ) ) )
             => ~ ! [Vd3: $o,Ve3: $o] :
                    ( ( Xb = vEBT_Leaf((Vd3),(Ve3)) )
                   => ! [V2: option4927543243414619207at_nat,Va2: nat,Vb3: array_VEBT_VEBTi,Vc3: vEBT_VEBTi] :
                        ( ( Xaa = aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(V2,Va2,Vb3),Vc3) )
                       => ( ( Y = bot_bot_assn )
                         => ~ accp_P7675410724331315407_VEBTi(vEBT_v8524038756793281170aw_rel,produc6084888613844515218_VEBTi(vEBT_Leaf((Vd3),(Ve3)),aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(V2,Va2,Vb3),Vc3))) ) ) ) ) ) ) ) ) ).

% vebt_assn_raw.pelims
tff(fact_7461_vebt__maxti_Opelims,axiom,
    ! [Xb: vEBT_VEBTi,Y: heap_T2636463487746394924on_nat] :
      ( ( aa_VEB2711320435582112505on_nat(vEBT_vebt_maxti,Xb) = Y )
     => ( aa_VEBT_VEBTi_o(accp_VEBT_VEBTi(vEBT_vebt_maxti_rel),Xb)
       => ( ! [A3: $o,B2: $o] :
              ( ( Xb = vEBT_Leafi((A3),(B2)) )
             => ( ( Y = $ite(
                      (B2),
                      heap_T3487192422709364219on_nat(aa_nat_option_nat(some_nat,one_one_nat)),
                      $ite((A3),heap_T3487192422709364219on_nat(aa_nat_option_nat(some_nat,zero_zero_nat)),heap_T3487192422709364219on_nat(none_nat)) ) )
               => ~ aa_VEBT_VEBTi_o(accp_VEBT_VEBTi(vEBT_vebt_maxti_rel),vEBT_Leafi((A3),(B2))) ) )
         => ( ! [Uu2: nat,Uv2: array_VEBT_VEBTi,Uw2: vEBT_VEBTi] :
                ( ( Xb = aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(none_P5556105721700978146at_nat,Uu2,Uv2),Uw2) )
               => ( ( Y = heap_T3487192422709364219on_nat(none_nat) )
                 => ~ aa_VEBT_VEBTi_o(accp_VEBT_VEBTi(vEBT_vebt_maxti_rel),aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(none_P5556105721700978146at_nat,Uu2,Uv2),Uw2)) ) )
           => ~ ! [Mi: nat,Ma: nat,Ux2: nat,Uy2: array_VEBT_VEBTi,Uz2: vEBT_VEBTi] :
                  ( ( Xb = aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),Ux2,Uy2),Uz2) )
                 => ( ( Y = heap_T3487192422709364219on_nat(aa_nat_option_nat(some_nat,Ma)) )
                   => ~ aa_VEBT_VEBTi_o(accp_VEBT_VEBTi(vEBT_vebt_maxti_rel),aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),Ux2,Uy2),Uz2)) ) ) ) ) ) ) ).

% vebt_maxti.pelims
tff(fact_7462_vebt__minti_Opelims,axiom,
    ! [Xb: vEBT_VEBTi,Y: heap_T2636463487746394924on_nat] :
      ( ( aa_VEB2711320435582112505on_nat(vEBT_vebt_minti,Xb) = Y )
     => ( aa_VEBT_VEBTi_o(accp_VEBT_VEBTi(vEBT_vebt_minti_rel),Xb)
       => ( ! [A3: $o,B2: $o] :
              ( ( Xb = vEBT_Leafi((A3),(B2)) )
             => ( ( Y = $ite(
                      (A3),
                      heap_T3487192422709364219on_nat(aa_nat_option_nat(some_nat,zero_zero_nat)),
                      $ite((B2),heap_T3487192422709364219on_nat(aa_nat_option_nat(some_nat,one_one_nat)),heap_T3487192422709364219on_nat(none_nat)) ) )
               => ~ aa_VEBT_VEBTi_o(accp_VEBT_VEBTi(vEBT_vebt_minti_rel),vEBT_Leafi((A3),(B2))) ) )
         => ( ! [Uu2: nat,Uv2: array_VEBT_VEBTi,Uw2: vEBT_VEBTi] :
                ( ( Xb = aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(none_P5556105721700978146at_nat,Uu2,Uv2),Uw2) )
               => ( ( Y = heap_T3487192422709364219on_nat(none_nat) )
                 => ~ aa_VEBT_VEBTi_o(accp_VEBT_VEBTi(vEBT_vebt_minti_rel),aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(none_P5556105721700978146at_nat,Uu2,Uv2),Uw2)) ) )
           => ~ ! [Mi: nat,Ma: nat,Ux2: nat,Uy2: array_VEBT_VEBTi,Uz2: vEBT_VEBTi] :
                  ( ( Xb = aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),Ux2,Uy2),Uz2) )
                 => ( ( Y = heap_T3487192422709364219on_nat(aa_nat_option_nat(some_nat,Mi)) )
                   => ~ aa_VEBT_VEBTi_o(accp_VEBT_VEBTi(vEBT_vebt_minti_rel),aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),Ux2,Uy2),Uz2)) ) ) ) ) ) ) ).

% vebt_minti.pelims
tff(fact_7463_VEBT__internal_Ocnt_Opelims,axiom,
    ! [Xb: vEBT_VEBT,Y: real] :
      ( ( aa_VEBT_VEBT_real(vEBT_VEBT_cnt,Xb) = Y )
     => ( aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_VEBT_cnt_rel2),Xb)
       => ( ! [A3: $o,B2: $o] :
              ( ( Xb = vEBT_Leaf((A3),(B2)) )
             => ( ( Y = one_one_real )
               => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_VEBT_cnt_rel2),vEBT_Leaf((A3),(B2))) ) )
         => ~ ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,Deg2,TreeList),Summary) )
               => ( ( Y = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_VEBT_VEBT_real(vEBT_VEBT_cnt,Summary))),aa_real_real(foldr_real_real(plus_plus_real,map_VEBT_VEBT_real(vEBT_VEBT_cnt,TreeList)),zero_zero_real)) )
                 => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_VEBT_cnt_rel2),aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,Deg2,TreeList),Summary)) ) ) ) ) ) ).

% VEBT_internal.cnt.pelims
tff(fact_7464_VEBT__internal_Ocnt_H_Opelims,axiom,
    ! [Xb: vEBT_VEBT,Y: nat] :
      ( ( aa_VEBT_VEBT_nat(vEBT_VEBT_cnt2,Xb) = Y )
     => ( aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_VEBT_cnt_rel),Xb)
       => ( ! [A3: $o,B2: $o] :
              ( ( Xb = vEBT_Leaf((A3),(B2)) )
             => ( ( Y = one_one_nat )
               => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_VEBT_cnt_rel),vEBT_Leaf((A3),(B2))) ) )
         => ~ ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,Deg2,TreeList),Summary) )
               => ( ( Y = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_VEBT_cnt2,Summary))),aa_nat_nat(foldr_nat_nat(plus_plus_nat,map_VEBT_VEBT_nat(vEBT_VEBT_cnt2,TreeList)),zero_zero_nat)) )
                 => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_VEBT_cnt_rel),aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,Deg2,TreeList),Summary)) ) ) ) ) ) ).

% VEBT_internal.cnt'.pelims
tff(fact_7465_vebt__maxt_Opelims,axiom,
    ! [Xb: vEBT_VEBT,Y: option_nat] :
      ( ( vEBT_vebt_maxt(Xb) = Y )
     => ( aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_vebt_maxt_rel),Xb)
       => ( ! [A3: $o,B2: $o] :
              ( ( Xb = vEBT_Leaf((A3),(B2)) )
             => ( ( Y = $ite(
                      (B2),
                      aa_nat_option_nat(some_nat,one_one_nat),
                      $ite((A3),aa_nat_option_nat(some_nat,zero_zero_nat),none_nat) ) )
               => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_vebt_maxt_rel),vEBT_Leaf((A3),(B2))) ) )
         => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2),Uw2) )
               => ( ( Y = none_nat )
                 => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_vebt_maxt_rel),aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2),Uw2)) ) )
           => ~ ! [Mi: nat,Ma: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),Ux2,Uy2),Uz2) )
                 => ( ( Y = aa_nat_option_nat(some_nat,Ma) )
                   => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_vebt_maxt_rel),aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),Ux2,Uy2),Uz2)) ) ) ) ) ) ) ).

% vebt_maxt.pelims
tff(fact_7466_vebt__mint_Opelims,axiom,
    ! [Xb: vEBT_VEBT,Y: option_nat] :
      ( ( vEBT_vebt_mint(Xb) = Y )
     => ( aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_vebt_mint_rel),Xb)
       => ( ! [A3: $o,B2: $o] :
              ( ( Xb = vEBT_Leaf((A3),(B2)) )
             => ( ( Y = $ite(
                      (A3),
                      aa_nat_option_nat(some_nat,zero_zero_nat),
                      $ite((B2),aa_nat_option_nat(some_nat,one_one_nat),none_nat) ) )
               => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_vebt_mint_rel),vEBT_Leaf((A3),(B2))) ) )
         => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2),Uw2) )
               => ( ( Y = none_nat )
                 => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_vebt_mint_rel),aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2),Uw2)) ) )
           => ~ ! [Mi: nat,Ma: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),Ux2,Uy2),Uz2) )
                 => ( ( Y = aa_nat_option_nat(some_nat,Mi) )
                   => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_vebt_mint_rel),aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),Ux2,Uy2),Uz2)) ) ) ) ) ) ) ).

% vebt_mint.pelims
tff(fact_7467_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t_Opelims,axiom,
    ! [Xb: vEBT_VEBT,Y: nat] :
      ( ( aa_VEBT_VEBT_nat(vEBT_T_m_i_n_t,Xb) = Y )
     => ( aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_T_m_i_n_t_rel),Xb)
       => ( ! [A3: $o,B2: $o] :
              ( ( Xb = vEBT_Leaf((A3),(B2)) )
             => ( ( Y = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
                      $ite((A3),zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat))) )
               => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_T_m_i_n_t_rel),vEBT_Leaf((A3),(B2))) ) )
         => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2),Uw2) )
               => ( ( Y = one_one_nat )
                 => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_T_m_i_n_t_rel),aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2),Uw2)) ) )
           => ~ ! [Mi: nat,Ma: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),Ux2,Uy2),Uz2) )
                 => ( ( Y = one_one_nat )
                   => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_T_m_i_n_t_rel),aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),Ux2,Uy2),Uz2)) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>t.pelims
tff(fact_7468_T_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062x_092_060_094sub_062t_Opelims,axiom,
    ! [Xb: vEBT_VEBT,Y: nat] :
      ( ( aa_VEBT_VEBT_nat(vEBT_T_m_a_x_t,Xb) = Y )
     => ( aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_T_m_a_x_t_rel),Xb)
       => ( ! [A3: $o,B2: $o] :
              ( ( Xb = vEBT_Leaf((A3),(B2)) )
             => ( ( Y = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
                      $ite((B2),one_one_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat))) )
               => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_T_m_a_x_t_rel),vEBT_Leaf((A3),(B2))) ) )
         => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2),Uw2) )
               => ( ( Y = one_one_nat )
                 => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_T_m_a_x_t_rel),aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2),Uw2)) ) )
           => ~ ! [Mi: nat,Ma: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),Ux2,Uy2),Uz2) )
                 => ( ( Y = one_one_nat )
                   => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_T_m_a_x_t_rel),aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),Ux2,Uy2),Uz2)) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>a\<^sub>x\<^sub>t.pelims
tff(fact_7469_VEBT__internal_OminNulli_Opelims,axiom,
    ! [Xb: vEBT_VEBTi,Y: heap_Time_Heap_o] :
      ( ( aa_VEB8194921341634505877Heap_o(vEBT_VEBT_minNulli,Xb) = Y )
     => ( aa_VEBT_VEBTi_o(accp_VEBT_VEBTi(vEBT_V5740978063120863272li_rel),Xb)
       => ( ( ( Xb = vEBT_Leafi($false,$false) )
           => ( ( Y = aa_o_H7399033255965945470Heap_o(heap_Time_return_o,$true) )
             => ~ aa_VEBT_VEBTi_o(accp_VEBT_VEBTi(vEBT_V5740978063120863272li_rel),vEBT_Leafi($false,$false)) ) )
         => ( ! [Uv2: $o] :
                ( ( Xb = vEBT_Leafi($true,(Uv2)) )
               => ( ( Y = aa_o_H7399033255965945470Heap_o(heap_Time_return_o,$false) )
                 => ~ aa_VEBT_VEBTi_o(accp_VEBT_VEBTi(vEBT_V5740978063120863272li_rel),vEBT_Leafi($true,(Uv2))) ) )
           => ( ! [Uu2: $o] :
                  ( ( Xb = vEBT_Leafi((Uu2),$true) )
                 => ( ( Y = aa_o_H7399033255965945470Heap_o(heap_Time_return_o,$false) )
                   => ~ aa_VEBT_VEBTi_o(accp_VEBT_VEBTi(vEBT_V5740978063120863272li_rel),vEBT_Leafi((Uu2),$true)) ) )
             => ( ! [Uw2: nat,Ux2: array_VEBT_VEBTi,Uy2: vEBT_VEBTi] :
                    ( ( Xb = aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(none_P5556105721700978146at_nat,Uw2,Ux2),Uy2) )
                   => ( ( Y = aa_o_H7399033255965945470Heap_o(heap_Time_return_o,$true) )
                     => ~ aa_VEBT_VEBTi_o(accp_VEBT_VEBTi(vEBT_V5740978063120863272li_rel),aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(none_P5556105721700978146at_nat,Uw2,Ux2),Uy2)) ) )
               => ~ ! [Uz2: product_prod_nat_nat,Va: nat,Vb2: array_VEBT_VEBTi,Vc: vEBT_VEBTi] :
                      ( ( Xb = aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(some_P7363390416028606310at_nat(Uz2),Va,Vb2),Vc) )
                     => ( ( Y = aa_o_H7399033255965945470Heap_o(heap_Time_return_o,$false) )
                       => ~ aa_VEBT_VEBTi_o(accp_VEBT_VEBTi(vEBT_V5740978063120863272li_rel),aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(some_P7363390416028606310at_nat(Uz2),Va,Vb2),Vc)) ) ) ) ) ) ) ) ) ).

% VEBT_internal.minNulli.pelims
tff(fact_7470_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062N_092_060_094sub_062u_092_060_094sub_062l_092_060_094sub_062l_Opelims,axiom,
    ! [Xb: vEBT_VEBT,Y: nat] :
      ( ( aa_VEBT_VEBT_nat(vEBT_T_m_i_n_N_u_l_l,Xb) = Y )
     => ( aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_T5462971552011256508_l_rel),Xb)
       => ( ( ( Xb = vEBT_Leaf($false,$false) )
           => ( ( Y = one_one_nat )
             => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_T5462971552011256508_l_rel),vEBT_Leaf($false,$false)) ) )
         => ( ! [Uv2: $o] :
                ( ( Xb = vEBT_Leaf($true,(Uv2)) )
               => ( ( Y = one_one_nat )
                 => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_T5462971552011256508_l_rel),vEBT_Leaf($true,(Uv2))) ) )
           => ( ! [Uu2: $o] :
                  ( ( Xb = vEBT_Leaf((Uu2),$true) )
                 => ( ( Y = one_one_nat )
                   => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_T5462971552011256508_l_rel),vEBT_Leaf((Uu2),$true)) ) )
             => ( ! [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
                    ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uw2,Ux2),Uy2) )
                   => ( ( Y = one_one_nat )
                     => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_T5462971552011256508_l_rel),aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uw2,Ux2),Uy2)) ) )
               => ~ ! [Uz2: product_prod_nat_nat,Va: nat,Vb2: list_VEBT_VEBT,Vc: vEBT_VEBT] :
                      ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(Uz2),Va,Vb2),Vc) )
                     => ( ( Y = one_one_nat )
                       => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_T5462971552011256508_l_rel),aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(Uz2),Va,Vb2),Vc)) ) ) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.pelims
tff(fact_7471_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_7472_fold__atLeastAtMost__nat_Opelims,axiom,
    ! [Xb: fun_nat_fun_nat_nat,Xaa: nat,Xba: nat,Xc: nat,Y: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(Xb,Xaa),Xba),Xc) = Y )
     => ( accp_P4694664897847628410at_nat(set_fo3699595496184130361el_nat,produc2818902965508115497at_nat(Xb,produc487386426758144856at_nat(Xaa,aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Xba),Xc))))
       => ~ ( ( Y = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xba),Xaa),Xc,aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(Xb,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xaa),one_one_nat)),Xba),aa_nat_nat(aa_nat_fun_nat_nat(Xb,Xaa),Xc))) )
           => ~ accp_P4694664897847628410at_nat(set_fo3699595496184130361el_nat,produc2818902965508115497at_nat(Xb,produc487386426758144856at_nat(Xaa,aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Xba),Xc)))) ) ) ) ).

% fold_atLeastAtMost_nat.pelims
tff(fact_7473_VEBT__internal_Ooption__shift_Opelims,axiom,
    ! [Xb: fun_Pr3118856740240616547it_nat,Xaa: option2621746655072343315it_nat,Xba: option2621746655072343315it_nat,Y: option2621746655072343315it_nat] :
      ( ( vEBT_V819568868292977612it_nat(Xb,Xaa,Xba) = Y )
     => ( accp_P6111968099744563316it_nat(vEBT_V3802522976469930835it_nat,produc2260137262690574555it_nat(Xb,produc6851560022941992023it_nat(Xaa,Xba)))
       => ( ( ( Xaa = none_P1551326421579882414it_nat )
           => ( ( Y = none_P1551326421579882414it_nat )
             => ~ accp_P6111968099744563316it_nat(vEBT_V3802522976469930835it_nat,produc2260137262690574555it_nat(Xb,produc6851560022941992023it_nat(none_P1551326421579882414it_nat,Xba))) ) )
         => ( ! [V2: produc120671012495760973it_nat] :
                ( ( Xaa = some_P2407035485129114418it_nat(V2) )
               => ( ( Xba = none_P1551326421579882414it_nat )
                 => ( ( Y = none_P1551326421579882414it_nat )
                   => ~ accp_P6111968099744563316it_nat(vEBT_V3802522976469930835it_nat,produc2260137262690574555it_nat(Xb,produc6851560022941992023it_nat(some_P2407035485129114418it_nat(V2),none_P1551326421579882414it_nat))) ) ) )
           => ~ ! [A3: produc120671012495760973it_nat] :
                  ( ( Xaa = some_P2407035485129114418it_nat(A3) )
                 => ! [B2: produc120671012495760973it_nat] :
                      ( ( Xba = some_P2407035485129114418it_nat(B2) )
                     => ( ( Y = some_P2407035485129114418it_nat(aa_Pro6543753689527670780it_nat(aa_Pro329082998274276628it_nat(Xb,A3),B2)) )
                       => ~ accp_P6111968099744563316it_nat(vEBT_V3802522976469930835it_nat,produc2260137262690574555it_nat(Xb,produc6851560022941992023it_nat(some_P2407035485129114418it_nat(A3),some_P2407035485129114418it_nat(B2)))) ) ) ) ) ) ) ) ).

% VEBT_internal.option_shift.pelims
tff(fact_7474_VEBT__internal_Ooption__shift_Opelims,axiom,
    ! [Xb: fun_Pr1313076919630728883it_nat,Xaa: option7339022715339332451it_nat,Xba: option7339022715339332451it_nat,Y: option7339022715339332451it_nat] :
      ( ( vEBT_V613753007643960916it_nat(Xb,Xaa,Xba) = Y )
     => ( accp_P2096406429022662496it_nat(vEBT_V482137685244371085it_nat,produc5529849720642941531it_nat(Xb,produc9206348758962449759it_nat(Xaa,Xba)))
       => ( ( ( Xaa = none_P7668321371905463026it_nat )
           => ( ( Y = none_P7668321371905463026it_nat )
             => ~ accp_P2096406429022662496it_nat(vEBT_V482137685244371085it_nat,produc5529849720642941531it_nat(Xb,produc9206348758962449759it_nat(none_P7668321371905463026it_nat,Xba))) ) )
         => ( ! [V2: produc8047831477865546771it_nat] :
                ( ( Xaa = some_P468703482102919278it_nat(V2) )
               => ( ( Xba = none_P7668321371905463026it_nat )
                 => ( ( Y = none_P7668321371905463026it_nat )
                   => ~ accp_P2096406429022662496it_nat(vEBT_V482137685244371085it_nat,produc5529849720642941531it_nat(Xb,produc9206348758962449759it_nat(some_P468703482102919278it_nat(V2),none_P7668321371905463026it_nat))) ) ) )
           => ~ ! [A3: produc8047831477865546771it_nat] :
                  ( ( Xaa = some_P468703482102919278it_nat(A3) )
                 => ! [B2: produc8047831477865546771it_nat] :
                      ( ( Xba = some_P468703482102919278it_nat(B2) )
                     => ( ( Y = some_P468703482102919278it_nat(aa_Pro7661135075737848164it_nat(aa_Pro52009495076012404it_nat(Xb,A3),B2)) )
                       => ~ accp_P2096406429022662496it_nat(vEBT_V482137685244371085it_nat,produc5529849720642941531it_nat(Xb,produc9206348758962449759it_nat(some_P468703482102919278it_nat(A3),some_P468703482102919278it_nat(B2)))) ) ) ) ) ) ) ) ).

% VEBT_internal.option_shift.pelims
tff(fact_7475_VEBT__internal_Ooption__shift_Opelims,axiom,
    ! [Xb: fun_Pr2477239776880934807at_nat,Xaa: option4927543243414619207at_nat,Xba: option4927543243414619207at_nat,Y: option4927543243414619207at_nat] :
      ( ( vEBT_V1502963449132264192at_nat(Xb,Xaa,Xba) = Y )
     => ( accp_P4760198183577772864at_nat(vEBT_V7235779383477046023at_nat,produc7814092408141220007at_nat(Xb,produc488173922507101015at_nat(Xaa,Xba)))
       => ( ( ( Xaa = none_P5556105721700978146at_nat )
           => ( ( Y = none_P5556105721700978146at_nat )
             => ~ accp_P4760198183577772864at_nat(vEBT_V7235779383477046023at_nat,produc7814092408141220007at_nat(Xb,produc488173922507101015at_nat(none_P5556105721700978146at_nat,Xba))) ) )
         => ( ! [V2: product_prod_nat_nat] :
                ( ( Xaa = some_P7363390416028606310at_nat(V2) )
               => ( ( Xba = none_P5556105721700978146at_nat )
                 => ( ( Y = none_P5556105721700978146at_nat )
                   => ~ accp_P4760198183577772864at_nat(vEBT_V7235779383477046023at_nat,produc7814092408141220007at_nat(Xb,produc488173922507101015at_nat(some_P7363390416028606310at_nat(V2),none_P5556105721700978146at_nat))) ) ) )
           => ~ ! [A3: product_prod_nat_nat] :
                  ( ( Xaa = some_P7363390416028606310at_nat(A3) )
                 => ! [B2: product_prod_nat_nat] :
                      ( ( Xba = some_P7363390416028606310at_nat(B2) )
                     => ( ( Y = some_P7363390416028606310at_nat(aa_Pro262675715343573244at_nat(aa_Pro8023792847356852296at_nat(Xb,A3),B2)) )
                       => ~ accp_P4760198183577772864at_nat(vEBT_V7235779383477046023at_nat,produc7814092408141220007at_nat(Xb,produc488173922507101015at_nat(some_P7363390416028606310at_nat(A3),some_P7363390416028606310at_nat(B2)))) ) ) ) ) ) ) ) ).

% VEBT_internal.option_shift.pelims
tff(fact_7476_VEBT__internal_Ooption__shift_Opelims,axiom,
    ! [Xb: fun_num_fun_num_num,Xaa: option_num,Xba: option_num,Y: option_num] :
      ( ( aa_opt5369159329878751634on_num(vEBT_V819420779217536731ft_num(Xb,Xaa),Xba) = Y )
     => ( accp_P1225658569124232473on_num(vEBT_V452583751252753300el_num,produc6360833703471310868on_num(Xb,produc8585076106096196333on_num(Xaa,Xba)))
       => ( ( ( Xaa = none_num )
           => ( ( Y = none_num )
             => ~ accp_P1225658569124232473on_num(vEBT_V452583751252753300el_num,produc6360833703471310868on_num(Xb,produc8585076106096196333on_num(none_num,Xba))) ) )
         => ( ! [V2: num] :
                ( ( Xaa = aa_num_option_num(some_num,V2) )
               => ( ( Xba = none_num )
                 => ( ( Y = none_num )
                   => ~ accp_P1225658569124232473on_num(vEBT_V452583751252753300el_num,produc6360833703471310868on_num(Xb,produc8585076106096196333on_num(aa_num_option_num(some_num,V2),none_num))) ) ) )
           => ~ ! [A3: num] :
                  ( ( Xaa = aa_num_option_num(some_num,A3) )
                 => ! [B2: num] :
                      ( ( Xba = aa_num_option_num(some_num,B2) )
                     => ( ( Y = aa_num_option_num(some_num,aa_num_num(aa_num_fun_num_num(Xb,A3),B2)) )
                       => ~ accp_P1225658569124232473on_num(vEBT_V452583751252753300el_num,produc6360833703471310868on_num(Xb,produc8585076106096196333on_num(aa_num_option_num(some_num,A3),aa_num_option_num(some_num,B2)))) ) ) ) ) ) ) ) ).

% VEBT_internal.option_shift.pelims
tff(fact_7477_VEBT__internal_Ooption__shift_Opelims,axiom,
    ! [Xb: fun_nat_fun_nat_nat,Xaa: option_nat,Xba: option_nat,Y: option_nat] :
      ( ( aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_V4262088993061758097ft_nat(Xb),Xaa),Xba) = Y )
     => ( accp_P5526603486790338531on_nat(vEBT_V3895251965096974666el_nat,produc6699827698519632094on_nat(Xb,produc5098337634421038937on_nat(Xaa,Xba)))
       => ( ( ( Xaa = none_nat )
           => ( ( Y = none_nat )
             => ~ accp_P5526603486790338531on_nat(vEBT_V3895251965096974666el_nat,produc6699827698519632094on_nat(Xb,produc5098337634421038937on_nat(none_nat,Xba))) ) )
         => ( ! [V2: nat] :
                ( ( Xaa = aa_nat_option_nat(some_nat,V2) )
               => ( ( Xba = none_nat )
                 => ( ( Y = none_nat )
                   => ~ accp_P5526603486790338531on_nat(vEBT_V3895251965096974666el_nat,produc6699827698519632094on_nat(Xb,produc5098337634421038937on_nat(aa_nat_option_nat(some_nat,V2),none_nat))) ) ) )
           => ~ ! [A3: nat] :
                  ( ( Xaa = aa_nat_option_nat(some_nat,A3) )
                 => ! [B2: nat] :
                      ( ( Xba = aa_nat_option_nat(some_nat,B2) )
                     => ( ( Y = aa_nat_option_nat(some_nat,aa_nat_nat(aa_nat_fun_nat_nat(Xb,A3),B2)) )
                       => ~ accp_P5526603486790338531on_nat(vEBT_V3895251965096974666el_nat,produc6699827698519632094on_nat(Xb,produc5098337634421038937on_nat(aa_nat_option_nat(some_nat,A3),aa_nat_option_nat(some_nat,B2)))) ) ) ) ) ) ) ) ).

% VEBT_internal.option_shift.pelims
tff(fact_7478_fold__atLeastAtMost__nat_Oelims,axiom,
    ! [Xb: fun_nat_fun_nat_nat,Xaa: nat,Xba: nat,Xc: nat,Y: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(Xb,Xaa),Xba),Xc) = Y )
     => ( Y = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xba),Xaa),Xc,aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(Xb,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xaa),one_one_nat)),Xba),aa_nat_nat(aa_nat_fun_nat_nat(Xb,Xaa),Xc))) ) ) ).

% fold_atLeastAtMost_nat.elims
tff(fact_7479_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_7480_freeze__rule,axiom,
    ! [A: array_VEBT_VEBTi,Xs: list_VEBT_VEBTi] : hoare_3904069481286416050_VEBTi(aa_lis2508312872377092907i_assn(snga_assn_VEBT_VEBTi(A),Xs),array_8141364883501958055_VEBTi(A),aa_lis3169797078229555565i_assn(aTP_Lamm_agu(A),Xs)) ).

% freeze_rule
tff(fact_7481_in__finite__psubset,axiom,
    ! [A2: set_nat,B4: set_nat] :
      ( member8277197624267554838et_nat(produc4532415448927165861et_nat(A2,B4),finite_psubset_nat)
    <=> ( aa_set_nat_o(ord_less_set_nat(A2),B4)
        & aa_set_nat_o(finite_finite_nat,B4) ) ) ).

% in_finite_psubset
tff(fact_7482_in__finite__psubset,axiom,
    ! [A2: set_int,B4: set_int] :
      ( member2572552093476627150et_int(produc6363374080413544029et_int(A2,B4),finite_psubset_int)
    <=> ( aa_set_int_o(ord_less_set_int(A2),B4)
        & aa_set_int_o(finite_finite_int,B4) ) ) ).

% in_finite_psubset
tff(fact_7483_in__finite__psubset,axiom,
    ! [A2: set_complex,B4: set_complex] :
      ( member351165363924911826omplex(produc3790773574474814305omplex(A2,B4),finite8643634255014194347omplex)
    <=> ( aa_set_complex_o(ord_less_set_complex(A2),B4)
        & aa_set_complex_o(finite3207457112153483333omplex,B4) ) ) ).

% in_finite_psubset
tff(fact_7484_in__finite__psubset,axiom,
    ! [A2: set_Code_integer,B4: set_Code_integer] :
      ( member4307123515891402160nteger(produc7443773368509356479nteger(A2,B4),finite2416775604798480986nteger)
    <=> ( aa_set5643387332196575111eger_o(ord_le1307284697595431911nteger(A2),B4)
        & aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B4) ) ) ).

% in_finite_psubset
tff(fact_7485_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_VEBT_VEBT,Xb: fun_VEBT_VEBT_real,Y: fun_VEBT_VEBT_real] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,collect_VEBT_VEBT(aa_fun1965331558590768990VEBT_o(aTP_Lamm_agv(I5),Xb)))
     => ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,collect_VEBT_VEBT(aa_fun1965331558590768990VEBT_o(aTP_Lamm_agv(I5),Y)))
       => aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,collect_VEBT_VEBT(aa_fun1965331558590768990VEBT_o(aa_fun5637434397837810486VEBT_o(aTP_Lamm_agw(I5),Xb),Y))) ) ) ).

% prod.finite_Collect_op
tff(fact_7486_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_real,Xb: fun_real_real,Y: fun_real_real] :
      ( aa_set_real_o(finite_finite_real,collect_real(aa_fun2895569901277098738real_o(aTP_Lamm_agx(I5),Xb)))
     => ( aa_set_real_o(finite_finite_real,collect_real(aa_fun2895569901277098738real_o(aTP_Lamm_agx(I5),Y)))
       => aa_set_real_o(finite_finite_real,collect_real(aa_fun2895569901277098738real_o(aa_fun3693225987673163702real_o(aTP_Lamm_agy(I5),Xb),Y))) ) ) ).

% prod.finite_Collect_op
tff(fact_7487_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_nat,Xb: fun_nat_real,Y: fun_nat_real] :
      ( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun6456149469358400426_nat_o(aTP_Lamm_agz(I5),Xb)))
     => ( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun6456149469358400426_nat_o(aTP_Lamm_agz(I5),Y)))
       => aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun6456149469358400426_nat_o(aa_fun5464864432877286986_nat_o(aTP_Lamm_aha(I5),Xb),Y))) ) ) ).

% prod.finite_Collect_op
tff(fact_7488_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_int,Xb: fun_int_real,Y: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,collect_int(aa_fun4444401208693448434_int_o(aTP_Lamm_ahb(I5),Xb)))
     => ( aa_set_int_o(finite_finite_int,collect_int(aa_fun4444401208693448434_int_o(aTP_Lamm_ahb(I5),Y)))
       => aa_set_int_o(finite_finite_int,collect_int(aa_fun4444401208693448434_int_o(aa_fun135101165924023862_int_o(aTP_Lamm_ahc(I5),Xb),Y))) ) ) ).

% prod.finite_Collect_op
tff(fact_7489_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_complex,Xb: fun_complex_real,Y: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aa_fun6195079535267300846plex_o(aTP_Lamm_ahd(I5),Xb)))
     => ( aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aa_fun6195079535267300846plex_o(aTP_Lamm_ahd(I5),Y)))
       => aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aa_fun6195079535267300846plex_o(aa_fun805843229300662576plex_o(aTP_Lamm_ahe(I5),Xb),Y))) ) ) ).

% prod.finite_Collect_op
tff(fact_7490_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_Code_integer,Xb: fun_Co3960896333048504454r_real,Y: fun_Co3960896333048504454r_real] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,collect_Code_integer(aa_fun3640226575523069072eger_o(aTP_Lamm_ahf(I5),Xb)))
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,collect_Code_integer(aa_fun3640226575523069072eger_o(aTP_Lamm_ahf(I5),Y)))
       => aa_set5643387332196575111eger_o(finite6017078050557962740nteger,collect_Code_integer(aa_fun3640226575523069072eger_o(aa_fun5110011637356352227eger_o(aTP_Lamm_ahg(I5),Xb),Y))) ) ) ).

% prod.finite_Collect_op
tff(fact_7491_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_VEBT_VEBT,Xb: fun_VEBT_VEBT_rat,Y: fun_VEBT_VEBT_rat] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,collect_VEBT_VEBT(aa_fun5889122065779655690VEBT_o(aTP_Lamm_ahh(I5),Xb)))
     => ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,collect_VEBT_VEBT(aa_fun5889122065779655690VEBT_o(aTP_Lamm_ahh(I5),Y)))
       => aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,collect_VEBT_VEBT(aa_fun5889122065779655690VEBT_o(aa_fun1978406156034293430VEBT_o(aTP_Lamm_ahi(I5),Xb),Y))) ) ) ).

% prod.finite_Collect_op
tff(fact_7492_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_real,Xb: fun_real_rat,Y: fun_real_rat] :
      ( aa_set_real_o(finite_finite_real,collect_real(aa_fun3209485225100452678real_o(aTP_Lamm_ahj(I5),Xb)))
     => ( aa_set_real_o(finite_finite_real,collect_real(aa_fun3209485225100452678real_o(aTP_Lamm_ahj(I5),Y)))
       => aa_set_real_o(finite_finite_real,collect_real(aa_fun3209485225100452678real_o(aa_fun8459995508950063326real_o(aTP_Lamm_ahk(I5),Xb),Y))) ) ) ).

% prod.finite_Collect_op
tff(fact_7493_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_nat,Xb: fun_nat_rat,Y: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun1118977351124560894_nat_o(aTP_Lamm_ahl(I5),Xb)))
     => ( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun1118977351124560894_nat_o(aTP_Lamm_ahl(I5),Y)))
       => aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun1118977351124560894_nat_o(aa_fun8314661095172591474_nat_o(aTP_Lamm_ahm(I5),Xb),Y))) ) ) ).

% prod.finite_Collect_op
tff(fact_7494_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_int,Xb: fun_int_rat,Y: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,collect_int(aa_fun7094158214180239686_int_o(aTP_Lamm_ahn(I5),Xb)))
     => ( aa_set_int_o(finite_finite_int,collect_int(aa_fun7094158214180239686_int_o(aTP_Lamm_ahn(I5),Y)))
       => aa_set_int_o(finite_finite_int,collect_int(aa_fun7094158214180239686_int_o(aa_fun1445893531507182942_int_o(aTP_Lamm_aho(I5),Xb),Y))) ) ) ).

% prod.finite_Collect_op
tff(fact_7495_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_VEBT_VEBT,Xb: fun_VEBT_VEBT_uint32,Y: fun_VEBT_VEBT_uint32] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,collect_VEBT_VEBT(aa_fun6272599145770227614VEBT_o(aTP_Lamm_ahp(I5),Xb)))
     => ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,collect_VEBT_VEBT(aa_fun6272599145770227614VEBT_o(aTP_Lamm_ahp(I5),Y)))
       => aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,collect_VEBT_VEBT(aa_fun6272599145770227614VEBT_o(aa_fun1156574460560677430VEBT_o(aTP_Lamm_ahq(I5),Xb),Y))) ) ) ).

% sum.finite_Collect_op
tff(fact_7496_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_real,Xb: fun_real_uint32,Y: fun_real_uint32] :
      ( aa_set_real_o(finite_finite_real,collect_real(aa_fun7556296611544133298real_o(aTP_Lamm_ahr(I5),Xb)))
     => ( aa_set_real_o(finite_finite_real,collect_real(aa_fun7556296611544133298real_o(aTP_Lamm_ahr(I5),Y)))
       => aa_set_real_o(finite_finite_real,collect_real(aa_fun7556296611544133298real_o(aa_fun5572330763077102134real_o(aTP_Lamm_ahs(I5),Xb),Y))) ) ) ).

% sum.finite_Collect_op
tff(fact_7497_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_nat,Xb: fun_nat_uint32,Y: fun_nat_uint32] :
      ( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun1182053767699555178_nat_o(aTP_Lamm_aht(I5),Xb)))
     => ( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun1182053767699555178_nat_o(aTP_Lamm_aht(I5),Y)))
       => aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun1182053767699555178_nat_o(aa_fun6957237400125433034_nat_o(aTP_Lamm_ahu(I5),Xb),Y))) ) ) ).

% sum.finite_Collect_op
tff(fact_7498_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_int,Xb: fun_int_uint32,Y: fun_int_uint32] :
      ( aa_set_int_o(finite_finite_int,collect_int(aa_fun6992502331602369202_int_o(aTP_Lamm_ahv(I5),Xb)))
     => ( aa_set_int_o(finite_finite_int,collect_int(aa_fun6992502331602369202_int_o(aTP_Lamm_ahv(I5),Y)))
       => aa_set_int_o(finite_finite_int,collect_int(aa_fun6992502331602369202_int_o(aa_fun3369613587901038774_int_o(aTP_Lamm_ahw(I5),Xb),Y))) ) ) ).

% sum.finite_Collect_op
tff(fact_7499_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_complex,Xb: fun_complex_uint32,Y: fun_complex_uint32] :
      ( aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aa_fun436548534304416686plex_o(aTP_Lamm_ahx(I5),Xb)))
     => ( aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aa_fun436548534304416686plex_o(aTP_Lamm_ahx(I5),Y)))
       => aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aa_fun436548534304416686plex_o(aa_fun2751405167760845232plex_o(aTP_Lamm_ahy(I5),Xb),Y))) ) ) ).

% sum.finite_Collect_op
tff(fact_7500_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_Code_integer,Xb: fun_Co1554742298434871494uint32,Y: fun_Co1554742298434871494uint32] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,collect_Code_integer(aa_fun8926496376162642256eger_o(aTP_Lamm_ahz(I5),Xb)))
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,collect_Code_integer(aa_fun8926496376162642256eger_o(aTP_Lamm_ahz(I5),Y)))
       => aa_set5643387332196575111eger_o(finite6017078050557962740nteger,collect_Code_integer(aa_fun8926496376162642256eger_o(aa_fun1119956648780089443eger_o(aTP_Lamm_aia(I5),Xb),Y))) ) ) ).

% sum.finite_Collect_op
tff(fact_7501_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_VEBT_VEBT,Xb: fun_VEBT_VEBT_real,Y: fun_VEBT_VEBT_real] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,collect_VEBT_VEBT(aa_fun1965331558590768990VEBT_o(aTP_Lamm_aib(I5),Xb)))
     => ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,collect_VEBT_VEBT(aa_fun1965331558590768990VEBT_o(aTP_Lamm_aib(I5),Y)))
       => aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,collect_VEBT_VEBT(aa_fun1965331558590768990VEBT_o(aa_fun5637434397837810486VEBT_o(aTP_Lamm_aic(I5),Xb),Y))) ) ) ).

% sum.finite_Collect_op
tff(fact_7502_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_real,Xb: fun_real_real,Y: fun_real_real] :
      ( aa_set_real_o(finite_finite_real,collect_real(aa_fun2895569901277098738real_o(aTP_Lamm_aid(I5),Xb)))
     => ( aa_set_real_o(finite_finite_real,collect_real(aa_fun2895569901277098738real_o(aTP_Lamm_aid(I5),Y)))
       => aa_set_real_o(finite_finite_real,collect_real(aa_fun2895569901277098738real_o(aa_fun3693225987673163702real_o(aTP_Lamm_aie(I5),Xb),Y))) ) ) ).

% sum.finite_Collect_op
tff(fact_7503_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_nat,Xb: fun_nat_real,Y: fun_nat_real] :
      ( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun6456149469358400426_nat_o(aTP_Lamm_aif(I5),Xb)))
     => ( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun6456149469358400426_nat_o(aTP_Lamm_aif(I5),Y)))
       => aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun6456149469358400426_nat_o(aa_fun5464864432877286986_nat_o(aTP_Lamm_aig(I5),Xb),Y))) ) ) ).

% sum.finite_Collect_op
tff(fact_7504_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_int,Xb: fun_int_real,Y: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,collect_int(aa_fun4444401208693448434_int_o(aTP_Lamm_aih(I5),Xb)))
     => ( aa_set_int_o(finite_finite_int,collect_int(aa_fun4444401208693448434_int_o(aTP_Lamm_aih(I5),Y)))
       => aa_set_int_o(finite_finite_int,collect_int(aa_fun4444401208693448434_int_o(aa_fun135101165924023862_int_o(aTP_Lamm_aii(I5),Xb),Y))) ) ) ).

% sum.finite_Collect_op
tff(fact_7505_ceiling__log__eq__powr__iff,axiom,
    ! [Xb: real,B: real,K: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B)
       => ( ( aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(log,B),Xb)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,K)),one_one_int) )
        <=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),aa_nat_real(semiri5074537144036343181t_real,K))),Xb)
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),one_one_nat)))) ) ) ) ) ).

% ceiling_log_eq_powr_iff
tff(fact_7506_powr__one__eq__one,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(powr_real,one_one_real),A) = one_one_real ).

% powr_one_eq_one
tff(fact_7507_powr__zero__eq__one,axiom,
    ! [Xb: real] :
      aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),zero_zero_real) = $ite(Xb = zero_zero_real,zero_zero_real,one_one_real) ).

% powr_zero_eq_one
tff(fact_7508_powr__nonneg__iff,axiom,
    ! [A: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,A),Xb)),zero_zero_real)
    <=> ( A = zero_zero_real ) ) ).

% powr_nonneg_iff
tff(fact_7509_powr__less__cancel__iff,axiom,
    ! [Xb: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),A)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),B))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) ) ) ).

% powr_less_cancel_iff
tff(fact_7510_powr__eq__one__iff,axiom,
    ! [A: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => ( ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,A),Xb) = one_one_real )
      <=> ( Xb = zero_zero_real ) ) ) ).

% powr_eq_one_iff
tff(fact_7511_powr__one,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),one_one_real) = Xb ) ) ).

% powr_one
tff(fact_7512_powr__one__gt__zero__iff,axiom,
    ! [Xb: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),one_one_real) = Xb )
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb) ) ).

% powr_one_gt_zero_iff
tff(fact_7513_powr__le__cancel__iff,axiom,
    ! [Xb: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),A)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),B))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ) ).

% powr_le_cancel_iff
tff(fact_7514_numeral__powr__numeral__real,axiom,
    ! [M: num,Nb: num] : aa_real_real(aa_rea7340605660300375871l_real(powr_real,numeral_numeral_real(M)),numeral_numeral_real(Nb)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,numeral_numeral_real(M)),aa_num_nat(numeral_numeral_nat,Nb)) ).

% numeral_powr_numeral_real
tff(fact_7515_log__powr__cancel,axiom,
    ! [A: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( ( A != one_one_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(log,A),aa_real_real(aa_rea7340605660300375871l_real(powr_real,A),Y)) = Y ) ) ) ).

% log_powr_cancel
tff(fact_7516_powr__log__cancel,axiom,
    ! [A: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( ( A != one_one_real )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
         => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,A),aa_real_real(aa_rea7340605660300375871l_real(log,A),Xb)) = Xb ) ) ) ) ).

% powr_log_cancel
tff(fact_7517_powr__numeral,axiom,
    ! [Xb: real,Nb: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),numeral_numeral_real(Nb)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,Nb)) ) ) ).

% powr_numeral
tff(fact_7518_powr__powr,axiom,
    ! [Xb: real,A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(powr_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),A)),B) = aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)) ).

% powr_powr
tff(fact_7519_powr__ge__pzero,axiom,
    ! [Xb: real,Y: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),Y)) ).

% powr_ge_pzero
tff(fact_7520_powr__mono2,axiom,
    ! [A: real,Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),A)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Y),A)) ) ) ) ).

% powr_mono2
tff(fact_7521_powr__less__mono,axiom,
    ! [A: real,B: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xb)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),A)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),B)) ) ) ).

% powr_less_mono
tff(fact_7522_powr__less__cancel,axiom,
    ! [Xb: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),A)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),B))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xb)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) ) ) ).

% powr_less_cancel
tff(fact_7523_powr__mono,axiom,
    ! [A: real,B: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Xb)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),A)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),B)) ) ) ).

% powr_mono
tff(fact_7524_powr__mono2_H,axiom,
    ! [A: real,Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,Y),A)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),A)) ) ) ) ).

% powr_mono2'
tff(fact_7525_powr__less__mono2,axiom,
    ! [A: real,Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),A)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Y),A)) ) ) ) ).

% powr_less_mono2
tff(fact_7526_powr__inj,axiom,
    ! [A: real,Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( ( A != one_one_real )
       => ( ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,A),Xb) = aa_real_real(aa_rea7340605660300375871l_real(powr_real,A),Y) )
        <=> ( Xb = Y ) ) ) ) ).

% powr_inj
tff(fact_7527_gr__one__powr,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),Y)) ) ) ).

% gr_one_powr
tff(fact_7528_powr__le1,axiom,
    ! [A: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),one_one_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),A)),one_one_real) ) ) ) ).

% powr_le1
tff(fact_7529_powr__mono__both,axiom,
    ! [A: real,B: real,Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Xb)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),A)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Y),B)) ) ) ) ) ).

% powr_mono_both
tff(fact_7530_ge__one__powr__ge__zero,axiom,
    ! [Xb: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),A)) ) ) ).

% ge_one_powr_ge_zero
tff(fact_7531_powr__divide,axiom,
    ! [Xb: real,Y: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
       => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Y)),A) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),A)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Y),A)) ) ) ) ).

% powr_divide
tff(fact_7532_powr__mult,axiom,
    ! [Xb: real,Y: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
       => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Y)),A) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),A)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Y),A)) ) ) ) ).

% powr_mult
tff(fact_7533_log__base__powr,axiom,
    ! [A: real,B: real,Xb: real] :
      ( ( A != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(log,aa_real_real(aa_rea7340605660300375871l_real(powr_real,A),B)),Xb) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(log,A),Xb)),B) ) ) ).

% log_base_powr
tff(fact_7534_log__powr,axiom,
    ! [Xb: real,B: real,Y: real] :
      ( ( Xb != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(log,B),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),Y)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),aa_real_real(aa_rea7340605660300375871l_real(log,B),Xb)) ) ) ).

% log_powr
tff(fact_7535_powr__add,axiom,
    ! [Xb: real,A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),A)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),B)) ).

% powr_add
tff(fact_7536_powr__diff,axiom,
    ! [W: real,Z1: real,Z22: real] : aa_real_real(aa_rea7340605660300375871l_real(powr_real,W),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Z1),Z22)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,W),Z1)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,W),Z22)) ).

% powr_diff
tff(fact_7537_powr__realpow,axiom,
    ! [Xb: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),aa_nat_real(semiri5074537144036343181t_real,Nb)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb) ) ) ).

% powr_realpow
tff(fact_7538_less__log__iff,axiom,
    ! [B: real,Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),aa_real_real(aa_rea7340605660300375871l_real(log,B),Xb))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),Y)),Xb) ) ) ) ).

% less_log_iff
tff(fact_7539_log__less__iff,axiom,
    ! [B: real,Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(log,B),Xb)),Y)
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),Y)) ) ) ) ).

% log_less_iff
tff(fact_7540_less__powr__iff,axiom,
    ! [B: real,Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),Y))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(log,B),Xb)),Y) ) ) ) ).

% less_powr_iff
tff(fact_7541_powr__less__iff,axiom,
    ! [B: real,Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),Y)),Xb)
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Y),aa_real_real(aa_rea7340605660300375871l_real(log,B),Xb)) ) ) ) ).

% powr_less_iff
tff(fact_7542_powr__mult__base,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),Y)) = aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),Y)) ) ) ).

% powr_mult_base
tff(fact_7543_le__log__iff,axiom,
    ! [B: real,Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),aa_real_real(aa_rea7340605660300375871l_real(log,B),Xb))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),Y)),Xb) ) ) ) ).

% le_log_iff
tff(fact_7544_log__le__iff,axiom,
    ! [B: real,Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(log,B),Xb)),Y)
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),Y)) ) ) ) ).

% log_le_iff
tff(fact_7545_le__powr__iff,axiom,
    ! [B: real,Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),Y))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(log,B),Xb)),Y) ) ) ) ).

% le_powr_iff
tff(fact_7546_powr__le__iff,axiom,
    ! [B: real,Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),Y)),Xb)
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),aa_real_real(aa_rea7340605660300375871l_real(log,B),Xb)) ) ) ) ).

% powr_le_iff
tff(fact_7547_log__add__eq__powr,axiom,
    ! [B: real,Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B)
     => ( ( B != one_one_real )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
         => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(log,B),Xb)),Y) = aa_real_real(aa_rea7340605660300375871l_real(log,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),Y))) ) ) ) ) ).

% log_add_eq_powr
tff(fact_7548_add__log__eq__powr,axiom,
    ! [B: real,Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B)
     => ( ( B != one_one_real )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
         => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Y),aa_real_real(aa_rea7340605660300375871l_real(log,B),Xb)) = aa_real_real(aa_rea7340605660300375871l_real(log,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),Y)),Xb)) ) ) ) ) ).

% add_log_eq_powr
tff(fact_7549_minus__log__eq__powr,axiom,
    ! [B: real,Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B)
     => ( ( B != one_one_real )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
         => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Y),aa_real_real(aa_rea7340605660300375871l_real(log,B),Xb)) = aa_real_real(aa_rea7340605660300375871l_real(log,B),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),Y)),Xb)) ) ) ) ) ).

% minus_log_eq_powr
tff(fact_7550_finite__psubset__def,axiom,
    finite_psubset_nat = collec6662362479098859352et_nat(produc6247414631856714078_nat_o(aTP_Lamm_aij)) ).

% finite_psubset_def
tff(fact_7551_finite__psubset__def,axiom,
    finite_psubset_int = collec957716948307931664et_int(produc4109468873575309990_int_o(aTP_Lamm_aik)) ).

% finite_psubset_def
tff(fact_7552_finite__psubset__def,axiom,
    finite8643634255014194347omplex = collec5108298041176329748omplex(produc3914248068834153634plex_o(aTP_Lamm_ail)) ).

% finite_psubset_def
tff(fact_7553_finite__psubset__def,axiom,
    finite2416775604798480986nteger = collec2770208431294612722nteger(produc1658495936642081476eger_o(aTP_Lamm_aim)) ).

% finite_psubset_def
tff(fact_7554_sum__gp,axiom,
    ! [Xb: rat,M: nat,Nb: nat] :
      groups2906978787729119204at_rat(power_power_rat(Xb),set_or1269000886237332187st_nat(M,Nb)) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),M),
        zero_zero_rat,
        $ite(Xb = one_one_rat,aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat)),M)),divide_divide_rat(minus_minus_rat(aa_nat_rat(power_power_rat(Xb),M),aa_nat_rat(power_power_rat(Xb),aa_nat_nat(suc,Nb))),minus_minus_rat(one_one_rat,Xb))) ) ).

% sum_gp
tff(fact_7555_sum__gp,axiom,
    ! [Xb: real,M: nat,Nb: nat] :
      groups6591440286371151544t_real(aa_real_fun_nat_real(power_power_real,Xb),set_or1269000886237332187st_nat(M,Nb)) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),M),
        zero_zero_real,
        $ite(Xb = one_one_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat)),M)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),M)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_nat_nat(suc,Nb)))),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),Xb))) ) ).

% sum_gp
tff(fact_7556_dbl__inc__simps_I3_J,axiom,
    neg_nu8115118780965096967l_num1(one_on7727431528512463931l_num1) = numera7442385471795722001l_num1(aa_num_num(bit1,one)) ).

% dbl_inc_simps(3)
tff(fact_7557_dbl__inc__simps_I3_J,axiom,
    aa_real_real(neg_nu8295874005876285629c_real,one_one_real) = numeral_numeral_real(aa_num_num(bit1,one)) ).

% dbl_inc_simps(3)
tff(fact_7558_dbl__inc__simps_I3_J,axiom,
    neg_nu5219082963157363817nc_rat(one_one_rat) = numeral_numeral_rat(aa_num_num(bit1,one)) ).

% dbl_inc_simps(3)
tff(fact_7559_dbl__inc__simps_I3_J,axiom,
    aa_int_int(neg_nu5851722552734809277nc_int,one_one_int) = aa_num_int(numeral_numeral_int,aa_num_num(bit1,one)) ).

% dbl_inc_simps(3)
tff(fact_7560_geometric__deriv__sums,axiom,
    ! [Z: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,Z)),one_one_real)
     => aa_real_o(sums_real(aa_real_fun_nat_real(aTP_Lamm_ain,Z)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),Z)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ).

% geometric_deriv_sums
tff(fact_7561_geometric__deriv__sums,axiom,
    ! [Z: complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,Z)),one_one_real)
     => aa_complex_o(sums_complex(aa_com8181001469867462503omplex(aTP_Lamm_aio,Z)),aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,one_one_complex),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,one_one_complex),Z)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ).

% geometric_deriv_sums
tff(fact_7562_VEBT__internal_Oheight_Osimps_I1_J,axiom,
    ! [A: $o,B: $o] : aa_VEBT_VEBT_nat(vEBT_VEBT_height,vEBT_Leaf((A),(B))) = zero_zero_nat ).

% VEBT_internal.height.simps(1)
tff(fact_7563_sum_Oneutral__const,axiom,
    ! [A2: set_nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_en),A2) = zero_zero_nat ).

% sum.neutral_const
tff(fact_7564_sum_Oneutral__const,axiom,
    ! [A2: set_complex] : groups7754918857620584856omplex(aTP_Lamm_aip,A2) = zero_zero_complex ).

% sum.neutral_const
tff(fact_7565_sum_Oneutral__const,axiom,
    ! [A2: set_int] : groups4538972089207619220nt_int(aTP_Lamm_eo,A2) = zero_zero_int ).

% sum.neutral_const
tff(fact_7566_sum_Oneutral__const,axiom,
    ! [A2: set_nat] : groups6591440286371151544t_real(aTP_Lamm_aiq,A2) = zero_zero_real ).

% sum.neutral_const
tff(fact_7567_of__nat__sum,axiom,
    ! [F: fun_complex_nat,A2: set_complex] : aa_nat_complex(semiri8010041392384452111omplex,groups5693394587270226106ex_nat(F,A2)) = groups7754918857620584856omplex(aTP_Lamm_air(F),A2) ).

% of_nat_sum
tff(fact_7568_of__nat__sum,axiom,
    ! [F: fun_int_nat,A2: set_int] : aa_nat_int(semiri1314217659103216013at_int,groups4541462559716669496nt_nat(F,A2)) = groups4538972089207619220nt_int(aTP_Lamm_ais(F),A2) ).

% of_nat_sum
tff(fact_7569_of__nat__sum,axiom,
    ! [F: fun_nat_nat,A2: set_nat] : aa_nat_int(semiri1314217659103216013at_int,aa_set_nat_nat(groups3542108847815614940at_nat(F),A2)) = groups3539618377306564664at_int(aTP_Lamm_ait(F),A2) ).

% of_nat_sum
tff(fact_7570_of__nat__sum,axiom,
    ! [F: fun_nat_nat,A2: set_nat] : aa_nat_Code_integer(semiri4939895301339042750nteger,aa_set_nat_nat(groups3542108847815614940at_nat(F),A2)) = groups7501900531339628137nteger(aTP_Lamm_aiu(F),A2) ).

% of_nat_sum
tff(fact_7571_of__nat__sum,axiom,
    ! [F: fun_nat_nat,A2: set_nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),A2)) = aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_aiv(F)),A2) ).

% of_nat_sum
tff(fact_7572_of__nat__sum,axiom,
    ! [F: fun_nat_nat,A2: set_nat] : aa_nat_real(semiri5074537144036343181t_real,aa_set_nat_nat(groups3542108847815614940at_nat(F),A2)) = groups6591440286371151544t_real(aTP_Lamm_jo(F),A2) ).

% of_nat_sum
tff(fact_7573_of__int__sum,axiom,
    ! [F: fun_complex_int,A2: set_complex] : aa_int_complex(ring_17405671764205052669omplex,groups5690904116761175830ex_int(F,A2)) = groups7754918857620584856omplex(aTP_Lamm_aiw(F),A2) ).

% of_int_sum
tff(fact_7574_of__int__sum,axiom,
    ! [F: fun_nat_int,A2: set_nat] : aa_int_real(ring_1_of_int_real,groups3539618377306564664at_int(F,A2)) = groups6591440286371151544t_real(aTP_Lamm_aix(F),A2) ).

% of_int_sum
tff(fact_7575_of__int__sum,axiom,
    ! [F: fun_int_int,A2: set_int] : aa_int_real(ring_1_of_int_real,groups4538972089207619220nt_int(F,A2)) = groups8778361861064173332t_real(aTP_Lamm_aiy(F),A2) ).

% of_int_sum
tff(fact_7576_of__int__sum,axiom,
    ! [F: fun_int_int,A2: set_int] : aa_int_rat(ring_1_of_int_rat,groups4538972089207619220nt_int(F,A2)) = groups3906332499630173760nt_rat(aTP_Lamm_aiz(F),A2) ).

% of_int_sum
tff(fact_7577_of__int__sum,axiom,
    ! [F: fun_int_int,A2: set_int] : aa_int_int(ring_1_of_int_int,groups4538972089207619220nt_int(F,A2)) = groups4538972089207619220nt_int(aTP_Lamm_aja(F),A2) ).

% of_int_sum
tff(fact_7578_sum_Odelta_H,axiom,
    ! [S2: set_VEBT_VEBT,A: vEBT_VEBT,B: fun_VEBT_VEBT_uint32] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,S2)
     => ( groups8325533452322294502uint32(aa_fun7720134271958203516uint32(aTP_Lamm_ajb(A),B),S2) = $ite(aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A),S2),aa_VEBT_VEBT_uint32(B,A),zero_zero_uint32) ) ) ).

% sum.delta'
tff(fact_7579_sum_Odelta_H,axiom,
    ! [S2: set_real,A: real,B: fun_real_uint32] :
      ( aa_set_real_o(finite_finite_real,S2)
     => ( groups5944083974425963860uint32(aa_fun39000138016665704uint32(aTP_Lamm_ajc(A),B),S2) = $ite(aa_set_real_o(member_real(A),S2),aa_real_uint32(B,A),zero_zero_uint32) ) ) ).

% sum.delta'
tff(fact_7580_sum_Odelta_H,axiom,
    ! [S2: set_nat,A: nat,B: fun_nat_uint32] :
      ( aa_set_nat_o(finite_finite_nat,S2)
     => ( groups833757482993574392uint32(aa_fun7322500112117231024uint32(aTP_Lamm_ajd(A),B),S2) = $ite(aa_set_nat_o(member_nat(A),S2),aa_nat_uint32(B,A),zero_zero_uint32) ) ) ).

% sum.delta'
tff(fact_7581_sum_Odelta_H,axiom,
    ! [S2: set_int,A: int,B: fun_int_uint32] :
      ( aa_set_int_o(finite_finite_int,S2)
     => ( groups5712668689793887828uint32(aa_fun7636875859098149992uint32(aTP_Lamm_aje(A),B),S2) = $ite(aa_set_int_o(member_int(A),S2),aa_int_uint32(B,A),zero_zero_uint32) ) ) ).

% sum.delta'
tff(fact_7582_sum_Odelta_H,axiom,
    ! [S2: set_complex,A: complex,B: fun_complex_uint32] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S2)
     => ( groups8736914816313324502uint32(aa_fun947202747272686700uint32(aTP_Lamm_ajf(A),B),S2) = $ite(aa_set_complex_o(member_complex(A),S2),aa_complex_uint32(B,A),zero_zero_uint32) ) ) ).

% sum.delta'
tff(fact_7583_sum_Odelta_H,axiom,
    ! [S2: set_Code_integer,A: code_integer,B: fun_Co1554742298434871494uint32] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,S2)
     => ( groups8847630953604152069uint32(aa_fun8111848626357839434uint32(aTP_Lamm_ajg(A),B),S2) = $ite(aa_set5643387332196575111eger_o(member_Code_integer(A),S2),aa_Cod8401701657498852871uint32(B,A),zero_zero_uint32) ) ) ).

% sum.delta'
tff(fact_7584_sum_Odelta_H,axiom,
    ! [S2: set_VEBT_VEBT,A: vEBT_VEBT,B: fun_VEBT_VEBT_real] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,S2)
     => ( groups2240296850493347238T_real(aa_fun1825138017709814396T_real(aTP_Lamm_ajh(A),B),S2) = $ite(aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A),S2),aa_VEBT_VEBT_real(B,A),zero_zero_real) ) ) ).

% sum.delta'
tff(fact_7585_sum_Odelta_H,axiom,
    ! [S2: set_real,A: real,B: fun_real_real] :
      ( aa_set_real_o(finite_finite_real,S2)
     => ( groups8097168146408367636l_real(aa_fun6946558556880649704l_real(aTP_Lamm_aji(A),B),S2) = $ite(aa_set_real_o(member_real(A),S2),aa_real_real(B,A),zero_zero_real) ) ) ).

% sum.delta'
tff(fact_7586_sum_Odelta_H,axiom,
    ! [S2: set_int,A: int,B: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,S2)
     => ( groups8778361861064173332t_real(aa_fun3252798400008884712t_real(aTP_Lamm_ajj(A),B),S2) = $ite(aa_set_int_o(member_int(A),S2),aa_int_real(B,A),zero_zero_real) ) ) ).

% sum.delta'
tff(fact_7587_sum_Odelta_H,axiom,
    ! [S2: set_complex,A: complex,B: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S2)
     => ( groups5808333547571424918x_real(aa_fun8933909628229591020x_real(aTP_Lamm_ajk(A),B),S2) = $ite(aa_set_complex_o(member_complex(A),S2),aa_complex_real(B,A),zero_zero_real) ) ) ).

% sum.delta'
tff(fact_7588_sum_Odelta,axiom,
    ! [S2: set_VEBT_VEBT,A: vEBT_VEBT,B: fun_VEBT_VEBT_uint32] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,S2)
     => ( groups8325533452322294502uint32(aa_fun7720134271958203516uint32(aTP_Lamm_ajl(A),B),S2) = $ite(aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A),S2),aa_VEBT_VEBT_uint32(B,A),zero_zero_uint32) ) ) ).

% sum.delta
tff(fact_7589_sum_Odelta,axiom,
    ! [S2: set_real,A: real,B: fun_real_uint32] :
      ( aa_set_real_o(finite_finite_real,S2)
     => ( groups5944083974425963860uint32(aa_fun39000138016665704uint32(aTP_Lamm_ajm(A),B),S2) = $ite(aa_set_real_o(member_real(A),S2),aa_real_uint32(B,A),zero_zero_uint32) ) ) ).

% sum.delta
tff(fact_7590_sum_Odelta,axiom,
    ! [S2: set_nat,A: nat,B: fun_nat_uint32] :
      ( aa_set_nat_o(finite_finite_nat,S2)
     => ( groups833757482993574392uint32(aa_fun7322500112117231024uint32(aTP_Lamm_ajn(A),B),S2) = $ite(aa_set_nat_o(member_nat(A),S2),aa_nat_uint32(B,A),zero_zero_uint32) ) ) ).

% sum.delta
tff(fact_7591_sum_Odelta,axiom,
    ! [S2: set_int,A: int,B: fun_int_uint32] :
      ( aa_set_int_o(finite_finite_int,S2)
     => ( groups5712668689793887828uint32(aa_fun7636875859098149992uint32(aTP_Lamm_ajo(A),B),S2) = $ite(aa_set_int_o(member_int(A),S2),aa_int_uint32(B,A),zero_zero_uint32) ) ) ).

% sum.delta
tff(fact_7592_sum_Odelta,axiom,
    ! [S2: set_complex,A: complex,B: fun_complex_uint32] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S2)
     => ( groups8736914816313324502uint32(aa_fun947202747272686700uint32(aTP_Lamm_ajp(A),B),S2) = $ite(aa_set_complex_o(member_complex(A),S2),aa_complex_uint32(B,A),zero_zero_uint32) ) ) ).

% sum.delta
tff(fact_7593_sum_Odelta,axiom,
    ! [S2: set_Code_integer,A: code_integer,B: fun_Co1554742298434871494uint32] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,S2)
     => ( groups8847630953604152069uint32(aa_fun8111848626357839434uint32(aTP_Lamm_ajq(A),B),S2) = $ite(aa_set5643387332196575111eger_o(member_Code_integer(A),S2),aa_Cod8401701657498852871uint32(B,A),zero_zero_uint32) ) ) ).

% sum.delta
tff(fact_7594_sum_Odelta,axiom,
    ! [S2: set_VEBT_VEBT,A: vEBT_VEBT,B: fun_VEBT_VEBT_real] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,S2)
     => ( groups2240296850493347238T_real(aa_fun1825138017709814396T_real(aTP_Lamm_ajr(A),B),S2) = $ite(aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A),S2),aa_VEBT_VEBT_real(B,A),zero_zero_real) ) ) ).

% sum.delta
tff(fact_7595_sum_Odelta,axiom,
    ! [S2: set_real,A: real,B: fun_real_real] :
      ( aa_set_real_o(finite_finite_real,S2)
     => ( groups8097168146408367636l_real(aa_fun6946558556880649704l_real(aTP_Lamm_ajs(A),B),S2) = $ite(aa_set_real_o(member_real(A),S2),aa_real_real(B,A),zero_zero_real) ) ) ).

% sum.delta
tff(fact_7596_sum_Odelta,axiom,
    ! [S2: set_int,A: int,B: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,S2)
     => ( groups8778361861064173332t_real(aa_fun3252798400008884712t_real(aTP_Lamm_ajt(A),B),S2) = $ite(aa_set_int_o(member_int(A),S2),aa_int_real(B,A),zero_zero_real) ) ) ).

% sum.delta
tff(fact_7597_sum_Odelta,axiom,
    ! [S2: set_complex,A: complex,B: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S2)
     => ( groups5808333547571424918x_real(aa_fun8933909628229591020x_real(aTP_Lamm_aju(A),B),S2) = $ite(aa_set_complex_o(member_complex(A),S2),aa_complex_real(B,A),zero_zero_real) ) ) ).

% sum.delta
tff(fact_7598_dbl__inc__simps_I2_J,axiom,
    neg_nu8115118780965096967l_num1(zero_z3563351764282998399l_num1) = one_on7727431528512463931l_num1 ).

% dbl_inc_simps(2)
tff(fact_7599_dbl__inc__simps_I2_J,axiom,
    neg_nu4269007558841261821uint32(zero_zero_uint32) = one_one_uint32 ).

% dbl_inc_simps(2)
tff(fact_7600_dbl__inc__simps_I2_J,axiom,
    aa_real_real(neg_nu8295874005876285629c_real,zero_zero_real) = one_one_real ).

% dbl_inc_simps(2)
tff(fact_7601_dbl__inc__simps_I2_J,axiom,
    neg_nu5219082963157363817nc_rat(zero_zero_rat) = one_one_rat ).

% dbl_inc_simps(2)
tff(fact_7602_dbl__inc__simps_I2_J,axiom,
    aa_int_int(neg_nu5851722552734809277nc_int,zero_zero_int) = one_one_int ).

% dbl_inc_simps(2)
tff(fact_7603_dbl__inc__simps_I5_J,axiom,
    ! [K: num] : neg_nu8115118780965096967l_num1(numera7442385471795722001l_num1(K)) = numera7442385471795722001l_num1(aa_num_num(bit1,K)) ).

% dbl_inc_simps(5)
tff(fact_7604_dbl__inc__simps_I5_J,axiom,
    ! [K: num] : aa_real_real(neg_nu8295874005876285629c_real,numeral_numeral_real(K)) = numeral_numeral_real(aa_num_num(bit1,K)) ).

% dbl_inc_simps(5)
tff(fact_7605_dbl__inc__simps_I5_J,axiom,
    ! [K: num] : neg_nu5219082963157363817nc_rat(numeral_numeral_rat(K)) = numeral_numeral_rat(aa_num_num(bit1,K)) ).

% dbl_inc_simps(5)
tff(fact_7606_dbl__inc__simps_I5_J,axiom,
    ! [K: num] : aa_int_int(neg_nu5851722552734809277nc_int,aa_num_int(numeral_numeral_int,K)) = aa_num_int(numeral_numeral_int,aa_num_num(bit1,K)) ).

% dbl_inc_simps(5)
tff(fact_7607_sum_Oinsert,axiom,
    ! [A2: set_VEBT_VEBT,Xb: vEBT_VEBT,G: fun_VEBT_VEBT_real] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,A2)
     => ( ~ aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xb),A2)
       => ( groups2240296850493347238T_real(G,insert_VEBT_VEBT(Xb,A2)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_VEBT_VEBT_real(G,Xb)),groups2240296850493347238T_real(G,A2)) ) ) ) ).

% sum.insert
tff(fact_7608_sum_Oinsert,axiom,
    ! [A2: set_real,Xb: real,G: fun_real_real] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( ~ aa_set_real_o(member_real(Xb),A2)
       => ( groups8097168146408367636l_real(G,insert_real(Xb,A2)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(G,Xb)),groups8097168146408367636l_real(G,A2)) ) ) ) ).

% sum.insert
tff(fact_7609_sum_Oinsert,axiom,
    ! [A2: set_int,Xb: int,G: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( ~ aa_set_int_o(member_int(Xb),A2)
       => ( groups8778361861064173332t_real(G,insert_int(Xb,A2)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(G,Xb)),groups8778361861064173332t_real(G,A2)) ) ) ) ).

% sum.insert
tff(fact_7610_sum_Oinsert,axiom,
    ! [A2: set_complex,Xb: complex,G: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( ~ aa_set_complex_o(member_complex(Xb),A2)
       => ( groups5808333547571424918x_real(G,insert_complex(Xb,A2)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(G,Xb)),groups5808333547571424918x_real(G,A2)) ) ) ) ).

% sum.insert
tff(fact_7611_sum_Oinsert,axiom,
    ! [A2: set_Code_integer,Xb: code_integer,G: fun_Co3960896333048504454r_real] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A2)
     => ( ~ aa_set5643387332196575111eger_o(member_Code_integer(Xb),A2)
       => ( groups1270011288395367621r_real(G,insert_Code_integer(Xb,A2)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_Code_integer_real(G,Xb)),groups1270011288395367621r_real(G,A2)) ) ) ) ).

% sum.insert
tff(fact_7612_sum_Oinsert,axiom,
    ! [A2: set_VEBT_VEBT,Xb: vEBT_VEBT,G: fun_VEBT_VEBT_rat] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,A2)
     => ( ~ aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xb),A2)
       => ( groups136491112297645522BT_rat(G,insert_VEBT_VEBT(Xb,A2)) = aa_rat_rat(plus_plus_rat(aa_VEBT_VEBT_rat(G,Xb)),groups136491112297645522BT_rat(G,A2)) ) ) ) ).

% sum.insert
tff(fact_7613_sum_Oinsert,axiom,
    ! [A2: set_real,Xb: real,G: fun_real_rat] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( ~ aa_set_real_o(member_real(Xb),A2)
       => ( groups1300246762558778688al_rat(G,insert_real(Xb,A2)) = aa_rat_rat(plus_plus_rat(aa_real_rat(G,Xb)),groups1300246762558778688al_rat(G,A2)) ) ) ) ).

% sum.insert
tff(fact_7614_sum_Oinsert,axiom,
    ! [A2: set_nat,Xb: nat,G: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( ~ aa_set_nat_o(member_nat(Xb),A2)
       => ( groups2906978787729119204at_rat(G,insert_nat(Xb,A2)) = aa_rat_rat(plus_plus_rat(aa_nat_rat(G,Xb)),groups2906978787729119204at_rat(G,A2)) ) ) ) ).

% sum.insert
tff(fact_7615_sum_Oinsert,axiom,
    ! [A2: set_int,Xb: int,G: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( ~ aa_set_int_o(member_int(Xb),A2)
       => ( groups3906332499630173760nt_rat(G,insert_int(Xb,A2)) = aa_rat_rat(plus_plus_rat(aa_int_rat(G,Xb)),groups3906332499630173760nt_rat(G,A2)) ) ) ) ).

% sum.insert
tff(fact_7616_sum_Oinsert,axiom,
    ! [A2: set_complex,Xb: complex,G: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( ~ aa_set_complex_o(member_complex(Xb),A2)
       => ( groups5058264527183730370ex_rat(G,insert_complex(Xb,A2)) = aa_rat_rat(plus_plus_rat(aa_complex_rat(G,Xb)),groups5058264527183730370ex_rat(G,A2)) ) ) ) ).

% sum.insert
tff(fact_7617_powser__sums__zero__iff,axiom,
    ! [A: fun_nat_real,Xb: real] :
      ( aa_real_o(sums_real(aTP_Lamm_ajv(A)),Xb)
    <=> ( aa_nat_real(A,zero_zero_nat) = Xb ) ) ).

% powser_sums_zero_iff
tff(fact_7618_sum_Ocl__ivl__Suc,axiom,
    ! [G: fun_nat_uint32,M: nat,Nb: nat] :
      groups833757482993574392uint32(G,set_or1269000886237332187st_nat(M,aa_nat_nat(suc,Nb))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Nb)),M),zero_zero_uint32,aa_uint32_uint32(plus_plus_uint32(groups833757482993574392uint32(G,set_or1269000886237332187st_nat(M,Nb))),aa_nat_uint32(G,aa_nat_nat(suc,Nb)))) ).

% sum.cl_ivl_Suc
tff(fact_7619_sum_Ocl__ivl__Suc,axiom,
    ! [G: fun_nat_rat,M: nat,Nb: nat] :
      groups2906978787729119204at_rat(G,set_or1269000886237332187st_nat(M,aa_nat_nat(suc,Nb))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Nb)),M),zero_zero_rat,aa_rat_rat(plus_plus_rat(groups2906978787729119204at_rat(G,set_or1269000886237332187st_nat(M,Nb))),aa_nat_rat(G,aa_nat_nat(suc,Nb)))) ).

% sum.cl_ivl_Suc
tff(fact_7620_sum_Ocl__ivl__Suc,axiom,
    ! [G: fun_nat_int,M: nat,Nb: nat] :
      groups3539618377306564664at_int(G,set_or1269000886237332187st_nat(M,aa_nat_nat(suc,Nb))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Nb)),M),zero_zero_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,groups3539618377306564664at_int(G,set_or1269000886237332187st_nat(M,Nb))),aa_nat_int(G,aa_nat_nat(suc,Nb)))) ).

% sum.cl_ivl_Suc
tff(fact_7621_sum_Ocl__ivl__Suc,axiom,
    ! [G: fun_nat_nat,M: nat,Nb: nat] :
      aa_set_nat_nat(groups3542108847815614940at_nat(G),set_or1269000886237332187st_nat(M,aa_nat_nat(suc,Nb))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Nb)),M),zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(G),set_or1269000886237332187st_nat(M,Nb))),aa_nat_nat(G,aa_nat_nat(suc,Nb)))) ).

% sum.cl_ivl_Suc
tff(fact_7622_sum_Ocl__ivl__Suc,axiom,
    ! [G: fun_nat_real,M: nat,Nb: nat] :
      groups6591440286371151544t_real(G,set_or1269000886237332187st_nat(M,aa_nat_nat(suc,Nb))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Nb)),M),zero_zero_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(G,set_or1269000886237332187st_nat(M,Nb))),aa_nat_real(G,aa_nat_nat(suc,Nb)))) ).

% sum.cl_ivl_Suc
tff(fact_7623_sum__zero__power,axiom,
    ! [C: fun_nat_rat,A2: set_nat] :
      groups2906978787729119204at_rat(aTP_Lamm_ajw(C),A2) = $ite(
        ( aa_set_nat_o(finite_finite_nat,A2)
        & aa_set_nat_o(member_nat(zero_zero_nat),A2) ),
        aa_nat_rat(C,zero_zero_nat),
        zero_zero_rat ) ).

% sum_zero_power
tff(fact_7624_sum__zero__power,axiom,
    ! [C: fun_nat_real,A2: set_nat] :
      groups6591440286371151544t_real(aTP_Lamm_ajv(C),A2) = $ite(
        ( aa_set_nat_o(finite_finite_nat,A2)
        & aa_set_nat_o(member_nat(zero_zero_nat),A2) ),
        aa_nat_real(C,zero_zero_nat),
        zero_zero_real ) ).

% sum_zero_power
tff(fact_7625_sum__zero__power_H,axiom,
    ! [C: fun_nat_rat,D: fun_nat_rat,A2: set_nat] :
      groups2906978787729119204at_rat(aa_fun765758790203053448at_rat(aTP_Lamm_ajx(C),D),A2) = $ite(
        ( aa_set_nat_o(finite_finite_nat,A2)
        & aa_set_nat_o(member_nat(zero_zero_nat),A2) ),
        divide_divide_rat(aa_nat_rat(C,zero_zero_nat),aa_nat_rat(D,zero_zero_nat)),
        zero_zero_rat ) ).

% sum_zero_power'
tff(fact_7626_sum__zero__power_H,axiom,
    ! [C: fun_nat_real,D: fun_nat_real,A2: set_nat] :
      groups6591440286371151544t_real(aa_fun1839206346889086768t_real(aTP_Lamm_ajy(C),D),A2) = $ite(
        ( aa_set_nat_o(finite_finite_nat,A2)
        & aa_set_nat_o(member_nat(zero_zero_nat),A2) ),
        aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(C,zero_zero_nat)),aa_nat_real(D,zero_zero_nat)),
        zero_zero_real ) ).

% sum_zero_power'
tff(fact_7627_sum_Oswap,axiom,
    ! [G: fun_nat_fun_nat_nat,B4: set_nat,A2: set_nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_set6537093552120046177at_nat(aTP_Lamm_ajz(G),B4)),A2) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_set6537093552120046177at_nat(aTP_Lamm_akb(G),A2)),B4) ).

% sum.swap
tff(fact_7628_sum_Oswap,axiom,
    ! [G: fun_co7673397065115134084omplex,B4: set_complex,A2: set_complex] : groups7754918857620584856omplex(aa_set7743095458595710203omplex(aTP_Lamm_akc(G),B4),A2) = groups7754918857620584856omplex(aa_set7743095458595710203omplex(aTP_Lamm_ake(G),A2),B4) ).

% sum.swap
tff(fact_7629_sum_Oswap,axiom,
    ! [G: fun_int_fun_int_int,B4: set_int,A2: set_int] : groups4538972089207619220nt_int(aa_set7310040872927613301nt_int(aTP_Lamm_akf(G),B4),A2) = groups4538972089207619220nt_int(aa_set7310040872927613301nt_int(aTP_Lamm_akh(G),A2),B4) ).

% sum.swap
tff(fact_7630_sum_Oswap,axiom,
    ! [G: fun_nat_fun_nat_real,B4: set_nat,A2: set_nat] : groups6591440286371151544t_real(aa_set1166323925324055613t_real(aTP_Lamm_aki(G),B4),A2) = groups6591440286371151544t_real(aa_set1166323925324055613t_real(aTP_Lamm_akk(G),A2),B4) ).

% sum.swap
tff(fact_7631_norm__sum,axiom,
    ! [F: fun_nat_complex,A2: set_nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,groups2073611262835488442omplex(F,A2))),groups6591440286371151544t_real(aTP_Lamm_akl(F),A2)) ).

% norm_sum
tff(fact_7632_norm__sum,axiom,
    ! [F: fun_complex_complex,A2: set_complex] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,groups7754918857620584856omplex(F,A2))),groups5808333547571424918x_real(aTP_Lamm_akm(F),A2)) ).

% norm_sum
tff(fact_7633_norm__sum,axiom,
    ! [F: fun_nat_real,A2: set_nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,groups6591440286371151544t_real(F,A2))),groups6591440286371151544t_real(aTP_Lamm_akn(F),A2)) ).

% norm_sum
tff(fact_7634_sum__norm__le,axiom,
    ! [S2: set_VEBT_VEBT,F: fun_VE5569070535936029429omplex,G: fun_VEBT_VEBT_real] :
      ( ! [X2: vEBT_VEBT] :
          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X2),S2)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_VEBT_VEBT_complex(F,X2))),aa_VEBT_VEBT_real(G,X2)) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,groups1794756597179926696omplex(F,S2))),groups2240296850493347238T_real(G,S2)) ) ).

% sum_norm_le
tff(fact_7635_sum__norm__le,axiom,
    ! [S2: set_real,F: fun_real_complex,G: fun_real_real] :
      ( ! [X2: real] :
          ( aa_set_real_o(member_real(X2),S2)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_real_complex(F,X2))),aa_real_real(G,X2)) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,groups5754745047067104278omplex(F,S2))),groups8097168146408367636l_real(G,S2)) ) ).

% sum_norm_le
tff(fact_7636_sum__norm__le,axiom,
    ! [S2: set_int,F: fun_int_complex,G: fun_int_real] :
      ( ! [X2: int] :
          ( aa_set_int_o(member_int(X2),S2)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_int_complex(F,X2))),aa_int_real(G,X2)) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,groups3049146728041665814omplex(F,S2))),groups8778361861064173332t_real(G,S2)) ) ).

% sum_norm_le
tff(fact_7637_sum__norm__le,axiom,
    ! [S2: set_nat,F: fun_nat_complex,G: fun_nat_real] :
      ( ! [X2: nat] :
          ( aa_set_nat_o(member_nat(X2),S2)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_nat_complex(F,X2))),aa_nat_real(G,X2)) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,groups2073611262835488442omplex(F,S2))),groups6591440286371151544t_real(G,S2)) ) ).

% sum_norm_le
tff(fact_7638_sum__norm__le,axiom,
    ! [S2: set_complex,F: fun_complex_complex,G: fun_complex_real] :
      ( ! [X2: complex] :
          ( aa_set_complex_o(member_complex(X2),S2)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(F,X2))),aa_complex_real(G,X2)) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,groups7754918857620584856omplex(F,S2))),groups5808333547571424918x_real(G,S2)) ) ).

% sum_norm_le
tff(fact_7639_sum__norm__le,axiom,
    ! [S2: set_nat,F: fun_nat_real,G: fun_nat_real] :
      ( ! [X2: nat] :
          ( aa_set_nat_o(member_nat(X2),S2)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_nat_real(F,X2))),aa_nat_real(G,X2)) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,groups6591440286371151544t_real(F,S2))),groups6591440286371151544t_real(G,S2)) ) ).

% sum_norm_le
tff(fact_7640_sum__mono,axiom,
    ! [K6: set_nat,F: fun_nat_rat,G: fun_nat_rat] :
      ( ! [I3: nat] :
          ( aa_set_nat_o(member_nat(I3),K6)
         => 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,groups2906978787729119204at_rat(F,K6)),groups2906978787729119204at_rat(G,K6)) ) ).

% sum_mono
tff(fact_7641_sum__mono,axiom,
    ! [K6: set_VEBT_VEBT,F: fun_VEBT_VEBT_rat,G: fun_VEBT_VEBT_rat] :
      ( ! [I3: vEBT_VEBT] :
          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(I3),K6)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_VEBT_VEBT_rat(F,I3)),aa_VEBT_VEBT_rat(G,I3)) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups136491112297645522BT_rat(F,K6)),groups136491112297645522BT_rat(G,K6)) ) ).

% sum_mono
tff(fact_7642_sum__mono,axiom,
    ! [K6: set_real,F: fun_real_rat,G: fun_real_rat] :
      ( ! [I3: real] :
          ( aa_set_real_o(member_real(I3),K6)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_real_rat(F,I3)),aa_real_rat(G,I3)) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups1300246762558778688al_rat(F,K6)),groups1300246762558778688al_rat(G,K6)) ) ).

% sum_mono
tff(fact_7643_sum__mono,axiom,
    ! [K6: set_int,F: fun_int_rat,G: fun_int_rat] :
      ( ! [I3: int] :
          ( aa_set_int_o(member_int(I3),K6)
         => 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,K6)),groups3906332499630173760nt_rat(G,K6)) ) ).

% sum_mono
tff(fact_7644_sum__mono,axiom,
    ! [K6: set_VEBT_VEBT,F: fun_VEBT_VEBT_nat,G: fun_VEBT_VEBT_nat] :
      ( ! [I3: vEBT_VEBT] :
          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(I3),K6)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_VEBT_VEBT_nat(F,I3)),aa_VEBT_VEBT_nat(G,I3)) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,groups771621172384141258BT_nat(F,K6)),groups771621172384141258BT_nat(G,K6)) ) ).

% sum_mono
tff(fact_7645_sum__mono,axiom,
    ! [K6: set_real,F: fun_real_nat,G: fun_real_nat] :
      ( ! [I3: real] :
          ( aa_set_real_o(member_real(I3),K6)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_real_nat(F,I3)),aa_real_nat(G,I3)) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,groups1935376822645274424al_nat(F,K6)),groups1935376822645274424al_nat(G,K6)) ) ).

% sum_mono
tff(fact_7646_sum__mono,axiom,
    ! [K6: set_int,F: fun_int_nat,G: fun_int_nat] :
      ( ! [I3: int] :
          ( aa_set_int_o(member_int(I3),K6)
         => 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,K6)),groups4541462559716669496nt_nat(G,K6)) ) ).

% sum_mono
tff(fact_7647_sum__mono,axiom,
    ! [K6: set_nat,F: fun_nat_int,G: fun_nat_int] :
      ( ! [I3: nat] :
          ( aa_set_nat_o(member_nat(I3),K6)
         => 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,groups3539618377306564664at_int(F,K6)),groups3539618377306564664at_int(G,K6)) ) ).

% sum_mono
tff(fact_7648_sum__mono,axiom,
    ! [K6: set_VEBT_VEBT,F: fun_VEBT_VEBT_int,G: fun_VEBT_VEBT_int] :
      ( ! [I3: vEBT_VEBT] :
          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(I3),K6)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_VEBT_VEBT_int(F,I3)),aa_VEBT_VEBT_int(G,I3)) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,groups769130701875090982BT_int(F,K6)),groups769130701875090982BT_int(G,K6)) ) ).

% sum_mono
tff(fact_7649_sum__mono,axiom,
    ! [K6: set_real,F: fun_real_int,G: fun_real_int] :
      ( ! [I3: real] :
          ( aa_set_real_o(member_real(I3),K6)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(F,I3)),aa_real_int(G,I3)) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,groups1932886352136224148al_int(F,K6)),groups1932886352136224148al_int(G,K6)) ) ).

% sum_mono
tff(fact_7650_sum_Odistrib,axiom,
    ! [G: fun_nat_nat,H: fun_nat_nat,A2: set_nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_ako(G),H)),A2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(G),A2)),aa_set_nat_nat(groups3542108847815614940at_nat(H),A2)) ).

% sum.distrib
tff(fact_7651_sum_Odistrib,axiom,
    ! [G: fun_complex_complex,H: fun_complex_complex,A2: set_complex] : groups7754918857620584856omplex(aa_fun3187328805393345264omplex(aTP_Lamm_akp(G),H),A2) = aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,groups7754918857620584856omplex(G,A2)),groups7754918857620584856omplex(H,A2)) ).

% sum.distrib
tff(fact_7652_sum_Odistrib,axiom,
    ! [G: fun_int_int,H: fun_int_int,A2: set_int] : groups4538972089207619220nt_int(aa_fun6280345759651049192nt_int(aTP_Lamm_akq(G),H),A2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups4538972089207619220nt_int(G,A2)),groups4538972089207619220nt_int(H,A2)) ).

% sum.distrib
tff(fact_7653_sum_Odistrib,axiom,
    ! [G: fun_nat_real,H: fun_nat_real,A2: set_nat] : groups6591440286371151544t_real(aa_fun1839206346889086768t_real(aTP_Lamm_akr(G),H),A2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(G,A2)),groups6591440286371151544t_real(H,A2)) ).

% sum.distrib
tff(fact_7654_sum__distrib__left,axiom,
    ! [R2: nat,F: fun_nat_nat,A2: set_nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,R2),aa_set_nat_nat(groups3542108847815614940at_nat(F),A2)) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_aks(R2),F)),A2) ).

% sum_distrib_left
tff(fact_7655_sum__distrib__left,axiom,
    ! [R2: complex,F: fun_complex_complex,A2: set_complex] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,R2),groups7754918857620584856omplex(F,A2)) = groups7754918857620584856omplex(aa_fun3187328805393345264omplex(aTP_Lamm_akt(R2),F),A2) ).

% sum_distrib_left
tff(fact_7656_sum__distrib__left,axiom,
    ! [R2: int,F: fun_int_int,A2: set_int] : aa_int_int(aa_int_fun_int_int(times_times_int,R2),groups4538972089207619220nt_int(F,A2)) = groups4538972089207619220nt_int(aa_fun6280345759651049192nt_int(aTP_Lamm_aku(R2),F),A2) ).

% sum_distrib_left
tff(fact_7657_sum__distrib__left,axiom,
    ! [R2: real,F: fun_nat_real,A2: set_nat] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R2),groups6591440286371151544t_real(F,A2)) = groups6591440286371151544t_real(aa_fun1839206346889086768t_real(aTP_Lamm_akv(R2),F),A2) ).

% sum_distrib_left
tff(fact_7658_sum__distrib__right,axiom,
    ! [F: fun_nat_nat,A2: set_nat,R2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),A2)),R2) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_akw(F),R2)),A2) ).

% sum_distrib_right
tff(fact_7659_sum__distrib__right,axiom,
    ! [F: fun_complex_complex,A2: set_complex,R2: complex] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,groups7754918857620584856omplex(F,A2)),R2) = groups7754918857620584856omplex(aa_com3342990543769683525omplex(aTP_Lamm_akx(F),R2),A2) ).

% sum_distrib_right
tff(fact_7660_sum__distrib__right,axiom,
    ! [F: fun_int_int,A2: set_int,R2: int] : aa_int_int(aa_int_fun_int_int(times_times_int,groups4538972089207619220nt_int(F,A2)),R2) = groups4538972089207619220nt_int(aa_int_fun_int_int(aTP_Lamm_aky(F),R2),A2) ).

% sum_distrib_right
tff(fact_7661_sum__distrib__right,axiom,
    ! [F: fun_nat_real,A2: set_nat,R2: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,groups6591440286371151544t_real(F,A2)),R2) = groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_akz(F),R2),A2) ).

% sum_distrib_right
tff(fact_7662_sum__product,axiom,
    ! [F: fun_nat_nat,A2: set_nat,G: fun_nat_nat,B4: set_nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),A2)),aa_set_nat_nat(groups3542108847815614940at_nat(G),B4)) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_set6537093552120046177at_nat(aa_fun4561837934398617467at_nat(aTP_Lamm_alb(F),G),B4)),A2) ).

% sum_product
tff(fact_7663_sum__product,axiom,
    ! [F: fun_complex_complex,A2: set_complex,G: fun_complex_complex,B4: set_complex] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,groups7754918857620584856omplex(F,A2)),groups7754918857620584856omplex(G,B4)) = groups7754918857620584856omplex(aa_set7743095458595710203omplex(aa_fun1583801116089702097omplex(aTP_Lamm_ald(F),G),B4),A2) ).

% sum_product
tff(fact_7664_sum__product,axiom,
    ! [F: fun_int_int,A2: set_int,G: fun_int_int,B4: set_int] : aa_int_int(aa_int_fun_int_int(times_times_int,groups4538972089207619220nt_int(F,A2)),groups4538972089207619220nt_int(G,B4)) = groups4538972089207619220nt_int(aa_set7310040872927613301nt_int(aa_fun8076508010538293575nt_int(aTP_Lamm_alf(F),G),B4),A2) ).

% sum_product
tff(fact_7665_sum__product,axiom,
    ! [F: fun_nat_real,A2: set_nat,G: fun_nat_real,B4: set_nat] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,groups6591440286371151544t_real(F,A2)),groups6591440286371151544t_real(G,B4)) = groups6591440286371151544t_real(aa_set1166323925324055613t_real(aa_fun1400677770452585267t_real(aTP_Lamm_alh(F),G),B4),A2) ).

% sum_product
tff(fact_7666_sum__subtractf,axiom,
    ! [F: fun_complex_complex,G: fun_complex_complex,A2: set_complex] : groups7754918857620584856omplex(aa_fun3187328805393345264omplex(aTP_Lamm_ali(F),G),A2) = aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,groups7754918857620584856omplex(F,A2)),groups7754918857620584856omplex(G,A2)) ).

% sum_subtractf
tff(fact_7667_sum__subtractf,axiom,
    ! [F: fun_int_int,G: fun_int_int,A2: set_int] : groups4538972089207619220nt_int(aa_fun6280345759651049192nt_int(aTP_Lamm_alj(F),G),A2) = aa_int_int(aa_int_fun_int_int(minus_minus_int,groups4538972089207619220nt_int(F,A2)),groups4538972089207619220nt_int(G,A2)) ).

% sum_subtractf
tff(fact_7668_sum__subtractf,axiom,
    ! [F: fun_nat_real,G: fun_nat_real,A2: set_nat] : groups6591440286371151544t_real(aa_fun1839206346889086768t_real(aTP_Lamm_alk(F),G),A2) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,groups6591440286371151544t_real(F,A2)),groups6591440286371151544t_real(G,A2)) ).

% sum_subtractf
tff(fact_7669_sum__divide__distrib,axiom,
    ! [F: fun_complex_complex,A2: set_complex,R2: complex] : aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,groups7754918857620584856omplex(F,A2)),R2) = groups7754918857620584856omplex(aa_com3342990543769683525omplex(aTP_Lamm_all(F),R2),A2) ).

% sum_divide_distrib
tff(fact_7670_sum__divide__distrib,axiom,
    ! [F: fun_nat_real,A2: set_nat,R2: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,groups6591440286371151544t_real(F,A2)),R2) = groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_alm(F),R2),A2) ).

% sum_divide_distrib
tff(fact_7671_sum_Oswap__restrict,axiom,
    ! [A2: set_VEBT_VEBT,B4: set_nat,G: fun_VE597728521808591204at_nat,R: fun_VE6480751511521678896_nat_o] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,A2)
     => ( aa_set_nat_o(finite_finite_nat,B4)
       => ( groups771621172384141258BT_nat(aa_fun3621025987637128233BT_nat(aa_fun1603736613815373464BT_nat(aTP_Lamm_alo(B4),G),R),A2) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun6249118799011721995at_nat(aa_fun2705500642357200114at_nat(aTP_Lamm_alq(A2),G),R)),B4) ) ) ) ).

% sum.swap_restrict
tff(fact_7672_sum_Oswap__restrict,axiom,
    ! [A2: set_real,B4: set_nat,G: fun_real_fun_nat_nat,R: fun_real_fun_nat_o] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( aa_set_nat_o(finite_finite_nat,B4)
       => ( groups1935376822645274424al_nat(aa_fun2272639302901571595al_nat(aa_fun7039802393428226646al_nat(aTP_Lamm_als(B4),G),R),A2) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun7366046364592083119at_nat(aa_fun7718511627822915834at_nat(aTP_Lamm_alu(A2),G),R)),B4) ) ) ) ).

% sum.swap_restrict
tff(fact_7673_sum_Oswap__restrict,axiom,
    ! [A2: set_int,B4: set_nat,G: fun_int_fun_nat_nat,R: fun_int_fun_nat_o] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( aa_set_nat_o(finite_finite_nat,B4)
       => ( groups4541462559716669496nt_nat(aa_fun4650797359919590027nt_nat(aa_fun8303507722728657494nt_nat(aTP_Lamm_alw(B4),G),R),A2) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun3651443648018535471at_nat(aa_fun6116586148035635706at_nat(aTP_Lamm_alz(A2),G),R)),B4) ) ) ) ).

% sum.swap_restrict
tff(fact_7674_sum_Oswap__restrict,axiom,
    ! [A2: set_complex,B4: set_nat,G: fun_co636942507509095880at_nat,R: fun_co5676208753217392844_nat_o] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( aa_set_nat_o(finite_finite_nat,B4)
       => ( groups5693394587270226106ex_nat(aa_fun2265263913897893899ex_nat(aa_fun1127489917733235668ex_nat(aTP_Lamm_amb(B4),G),R),A2) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun5527826185635525933at_nat(aa_fun4894236392271416310at_nat(aTP_Lamm_ame(A2),G),R)),B4) ) ) ) ).

% sum.swap_restrict
tff(fact_7675_sum_Oswap__restrict,axiom,
    ! [A2: set_Code_integer,B4: set_nat,G: fun_Co3312552843764780407at_nat,R: fun_Co1375048237525739357_nat_o] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A2)
     => ( aa_set_nat_o(finite_finite_nat,B4)
       => ( groups7237345082560585321er_nat(aa_fun6683192525214198859er_nat(aa_fun2143651634668453157er_nat(aTP_Lamm_amg(B4),G),R),A2) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun7456160887942273982at_nat(aa_fun4417447297828429592at_nat(aTP_Lamm_amj(A2),G),R)),B4) ) ) ) ).

% sum.swap_restrict
tff(fact_7676_sum_Oswap__restrict,axiom,
    ! [A2: set_VEBT_VEBT,B4: set_complex,G: fun_VE2006399705683203104omplex,R: fun_VE7631951351884037842plex_o] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,A2)
     => ( aa_set_complex_o(finite3207457112153483333omplex,B4)
       => ( groups1794756597179926696omplex(aa_fun1953711888481404901omplex(aa_fun5829985485761010584omplex(aTP_Lamm_aml(B4),G),R),A2) = groups7754918857620584856omplex(aa_fun2592672125884331625omplex(aa_fun8556514703313836300omplex(aTP_Lamm_amn(A2),G),R),B4) ) ) ) ).

% sum.swap_restrict
tff(fact_7677_sum_Oswap__restrict,axiom,
    ! [A2: set_real,B4: set_complex,G: fun_re7208826350674946562omplex,R: fun_re7319081385389557872plex_o] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( aa_set_complex_o(finite3207457112153483333omplex,B4)
       => ( groups5754745047067104278omplex(aa_fun401690246322682763omplex(aa_fun7501666206714588442omplex(aTP_Lamm_amp(B4),G),R),A2) = groups7754918857620584856omplex(aa_fun9084337531551611917omplex(aa_fun6739043078600750748omplex(aTP_Lamm_amr(A2),G),R),B4) ) ) ) ).

% sum.swap_restrict
tff(fact_7678_sum_Oswap__restrict,axiom,
    ! [A2: set_nat,B4: set_complex,G: fun_na1591161823130732966omplex,R: fun_na1554462459066888140plex_o] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_complex_o(finite3207457112153483333omplex,B4)
       => ( groups2073611262835488442omplex(aa_fun4713049822795992587omplex(aa_fun3479084698764006518omplex(aTP_Lamm_amt(B4),G),R),A2) = groups7754918857620584856omplex(aa_fun1423577888295648489omplex(aa_fun601428455815558228omplex(aTP_Lamm_amw(A2),G),R),B4) ) ) ) ).

% sum.swap_restrict
tff(fact_7679_sum_Oswap__restrict,axiom,
    ! [A2: set_int,B4: set_complex,G: fun_in1307110748760623106omplex,R: fun_in5065666251602127472plex_o] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( aa_set_complex_o(finite3207457112153483333omplex,B4)
       => ( groups3049146728041665814omplex(aa_fun3141780957025892107omplex(aa_fun323909820831547930omplex(aTP_Lamm_amy(B4),G),R),A2) = groups7754918857620584856omplex(aa_fun7450476157521870733omplex(aa_fun6156949752960786844omplex(aTP_Lamm_anb(A2),G),R),B4) ) ) ) ).

% sum.swap_restrict
tff(fact_7680_sum_Oswap__restrict,axiom,
    ! [A2: set_Code_integer,B4: set_complex,G: fun_Co8905303033605938227omplex,R: fun_Co8975588572093281791plex_o] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A2)
     => ( aa_set_complex_o(finite3207457112153483333omplex,B4)
       => ( groups8024822376189712711omplex(aa_fun9008897831946212043omplex(aa_fun8319717443755388649omplex(aTP_Lamm_and(B4),G),R),A2) = groups7754918857620584856omplex(aa_fun7787660480616508700omplex(aa_fun963654016877434554omplex(aTP_Lamm_ang(A2),G),R),B4) ) ) ) ).

% sum.swap_restrict
tff(fact_7681_mod__sum__eq,axiom,
    ! [F: fun_nat_nat,A: nat,A2: set_nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_anh(F),A)),A2)),A) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),A2)),A) ).

% mod_sum_eq
tff(fact_7682_mod__sum__eq,axiom,
    ! [F: fun_int_int,A: int,A2: set_int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,groups4538972089207619220nt_int(aa_int_fun_int_int(aTP_Lamm_ani(F),A),A2)),A) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,groups4538972089207619220nt_int(F,A2)),A) ).

% mod_sum_eq
tff(fact_7683_sum__nonpos,axiom,
    ! [A2: set_VEBT_VEBT,F: fun_VEBT_VEBT_real] :
      ( ! [X2: vEBT_VEBT] :
          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X2),A2)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_VEBT_VEBT_real(F,X2)),zero_zero_real) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups2240296850493347238T_real(F,A2)),zero_zero_real) ) ).

% sum_nonpos
tff(fact_7684_sum__nonpos,axiom,
    ! [A2: set_real,F: fun_real_real] :
      ( ! [X2: real] :
          ( aa_set_real_o(member_real(X2),A2)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(F,X2)),zero_zero_real) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups8097168146408367636l_real(F,A2)),zero_zero_real) ) ).

% sum_nonpos
tff(fact_7685_sum__nonpos,axiom,
    ! [A2: set_int,F: fun_int_real] :
      ( ! [X2: int] :
          ( aa_set_int_o(member_int(X2),A2)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(F,X2)),zero_zero_real) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups8778361861064173332t_real(F,A2)),zero_zero_real) ) ).

% sum_nonpos
tff(fact_7686_sum__nonpos,axiom,
    ! [A2: set_nat,F: fun_nat_rat] :
      ( ! [X2: nat] :
          ( aa_set_nat_o(member_nat(X2),A2)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,X2)),zero_zero_rat) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups2906978787729119204at_rat(F,A2)),zero_zero_rat) ) ).

% sum_nonpos
tff(fact_7687_sum__nonpos,axiom,
    ! [A2: set_VEBT_VEBT,F: fun_VEBT_VEBT_rat] :
      ( ! [X2: vEBT_VEBT] :
          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X2),A2)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_VEBT_VEBT_rat(F,X2)),zero_zero_rat) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups136491112297645522BT_rat(F,A2)),zero_zero_rat) ) ).

% sum_nonpos
tff(fact_7688_sum__nonpos,axiom,
    ! [A2: set_real,F: fun_real_rat] :
      ( ! [X2: real] :
          ( aa_set_real_o(member_real(X2),A2)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_real_rat(F,X2)),zero_zero_rat) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups1300246762558778688al_rat(F,A2)),zero_zero_rat) ) ).

% sum_nonpos
tff(fact_7689_sum__nonpos,axiom,
    ! [A2: set_int,F: fun_int_rat] :
      ( ! [X2: int] :
          ( aa_set_int_o(member_int(X2),A2)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(F,X2)),zero_zero_rat) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups3906332499630173760nt_rat(F,A2)),zero_zero_rat) ) ).

% sum_nonpos
tff(fact_7690_sum__nonpos,axiom,
    ! [A2: set_VEBT_VEBT,F: fun_VEBT_VEBT_nat] :
      ( ! [X2: vEBT_VEBT] :
          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X2),A2)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_VEBT_VEBT_nat(F,X2)),zero_zero_nat) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,groups771621172384141258BT_nat(F,A2)),zero_zero_nat) ) ).

% sum_nonpos
tff(fact_7691_sum__nonpos,axiom,
    ! [A2: set_real,F: fun_real_nat] :
      ( ! [X2: real] :
          ( aa_set_real_o(member_real(X2),A2)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_real_nat(F,X2)),zero_zero_nat) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,groups1935376822645274424al_nat(F,A2)),zero_zero_nat) ) ).

% sum_nonpos
tff(fact_7692_sum__nonpos,axiom,
    ! [A2: set_int,F: fun_int_nat] :
      ( ! [X2: int] :
          ( aa_set_int_o(member_int(X2),A2)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(F,X2)),zero_zero_nat) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,groups4541462559716669496nt_nat(F,A2)),zero_zero_nat) ) ).

% sum_nonpos
tff(fact_7693_sum__nonneg,axiom,
    ! [A2: set_VEBT_VEBT,F: fun_VEBT_VEBT_real] :
      ( ! [X2: vEBT_VEBT] :
          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X2),A2)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_VEBT_VEBT_real(F,X2)) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),groups2240296850493347238T_real(F,A2)) ) ).

% sum_nonneg
tff(fact_7694_sum__nonneg,axiom,
    ! [A2: set_real,F: fun_real_real] :
      ( ! [X2: real] :
          ( aa_set_real_o(member_real(X2),A2)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(F,X2)) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),groups8097168146408367636l_real(F,A2)) ) ).

% sum_nonneg
tff(fact_7695_sum__nonneg,axiom,
    ! [A2: set_int,F: fun_int_real] :
      ( ! [X2: int] :
          ( aa_set_int_o(member_int(X2),A2)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_int_real(F,X2)) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),groups8778361861064173332t_real(F,A2)) ) ).

% sum_nonneg
tff(fact_7696_sum__nonneg,axiom,
    ! [A2: set_nat,F: fun_nat_rat] :
      ( ! [X2: nat] :
          ( aa_set_nat_o(member_nat(X2),A2)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(F,X2)) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),groups2906978787729119204at_rat(F,A2)) ) ).

% sum_nonneg
tff(fact_7697_sum__nonneg,axiom,
    ! [A2: set_VEBT_VEBT,F: fun_VEBT_VEBT_rat] :
      ( ! [X2: vEBT_VEBT] :
          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X2),A2)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_VEBT_VEBT_rat(F,X2)) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),groups136491112297645522BT_rat(F,A2)) ) ).

% sum_nonneg
tff(fact_7698_sum__nonneg,axiom,
    ! [A2: set_real,F: fun_real_rat] :
      ( ! [X2: real] :
          ( aa_set_real_o(member_real(X2),A2)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_real_rat(F,X2)) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),groups1300246762558778688al_rat(F,A2)) ) ).

% sum_nonneg
tff(fact_7699_sum__nonneg,axiom,
    ! [A2: set_int,F: fun_int_rat] :
      ( ! [X2: int] :
          ( aa_set_int_o(member_int(X2),A2)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_int_rat(F,X2)) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),groups3906332499630173760nt_rat(F,A2)) ) ).

% sum_nonneg
tff(fact_7700_sum__nonneg,axiom,
    ! [A2: set_VEBT_VEBT,F: fun_VEBT_VEBT_nat] :
      ( ! [X2: vEBT_VEBT] :
          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X2),A2)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_VEBT_VEBT_nat(F,X2)) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),groups771621172384141258BT_nat(F,A2)) ) ).

% sum_nonneg
tff(fact_7701_sum__nonneg,axiom,
    ! [A2: set_real,F: fun_real_nat] :
      ( ! [X2: real] :
          ( aa_set_real_o(member_real(X2),A2)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_real_nat(F,X2)) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),groups1935376822645274424al_nat(F,A2)) ) ).

% sum_nonneg
tff(fact_7702_sum__nonneg,axiom,
    ! [A2: set_int,F: fun_int_nat] :
      ( ! [X2: int] :
          ( aa_set_int_o(member_int(X2),A2)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_int_nat(F,X2)) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),groups4541462559716669496nt_nat(F,A2)) ) ).

% sum_nonneg
tff(fact_7703_sum__mono__inv,axiom,
    ! [F: fun_VEBT_VEBT_rat,I5: set_VEBT_VEBT,G: fun_VEBT_VEBT_rat,I: vEBT_VEBT] :
      ( ( groups136491112297645522BT_rat(F,I5) = groups136491112297645522BT_rat(G,I5) )
     => ( ! [I3: vEBT_VEBT] :
            ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(I3),I5)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_VEBT_VEBT_rat(F,I3)),aa_VEBT_VEBT_rat(G,I3)) )
       => ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(I),I5)
         => ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,I5)
           => ( aa_VEBT_VEBT_rat(F,I) = aa_VEBT_VEBT_rat(G,I) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_7704_sum__mono__inv,axiom,
    ! [F: fun_real_rat,I5: set_real,G: fun_real_rat,I: real] :
      ( ( groups1300246762558778688al_rat(F,I5) = groups1300246762558778688al_rat(G,I5) )
     => ( ! [I3: real] :
            ( aa_set_real_o(member_real(I3),I5)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_real_rat(F,I3)),aa_real_rat(G,I3)) )
       => ( aa_set_real_o(member_real(I),I5)
         => ( aa_set_real_o(finite_finite_real,I5)
           => ( aa_real_rat(F,I) = aa_real_rat(G,I) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_7705_sum__mono__inv,axiom,
    ! [F: fun_nat_rat,I5: set_nat,G: fun_nat_rat,I: nat] :
      ( ( groups2906978787729119204at_rat(F,I5) = groups2906978787729119204at_rat(G,I5) )
     => ( ! [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(I),I5)
         => ( aa_set_nat_o(finite_finite_nat,I5)
           => ( aa_nat_rat(F,I) = aa_nat_rat(G,I) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_7706_sum__mono__inv,axiom,
    ! [F: fun_int_rat,I5: set_int,G: fun_int_rat,I: int] :
      ( ( groups3906332499630173760nt_rat(F,I5) = groups3906332499630173760nt_rat(G,I5) )
     => ( ! [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(I),I5)
         => ( aa_set_int_o(finite_finite_int,I5)
           => ( aa_int_rat(F,I) = aa_int_rat(G,I) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_7707_sum__mono__inv,axiom,
    ! [F: fun_complex_rat,I5: set_complex,G: fun_complex_rat,I: complex] :
      ( ( groups5058264527183730370ex_rat(F,I5) = groups5058264527183730370ex_rat(G,I5) )
     => ( ! [I3: complex] :
            ( aa_set_complex_o(member_complex(I3),I5)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_complex_rat(F,I3)),aa_complex_rat(G,I3)) )
       => ( aa_set_complex_o(member_complex(I),I5)
         => ( aa_set_complex_o(finite3207457112153483333omplex,I5)
           => ( aa_complex_rat(F,I) = aa_complex_rat(G,I) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_7708_sum__mono__inv,axiom,
    ! [F: fun_Code_integer_rat,I5: set_Code_integer,G: fun_Code_integer_rat,I: code_integer] :
      ( ( groups6602215022474089585er_rat(F,I5) = groups6602215022474089585er_rat(G,I5) )
     => ( ! [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(I),I5)
         => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,I5)
           => ( aa_Code_integer_rat(F,I) = aa_Code_integer_rat(G,I) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_7709_sum__mono__inv,axiom,
    ! [F: fun_VEBT_VEBT_nat,I5: set_VEBT_VEBT,G: fun_VEBT_VEBT_nat,I: vEBT_VEBT] :
      ( ( groups771621172384141258BT_nat(F,I5) = groups771621172384141258BT_nat(G,I5) )
     => ( ! [I3: vEBT_VEBT] :
            ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(I3),I5)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_VEBT_VEBT_nat(F,I3)),aa_VEBT_VEBT_nat(G,I3)) )
       => ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(I),I5)
         => ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,I5)
           => ( aa_VEBT_VEBT_nat(F,I) = aa_VEBT_VEBT_nat(G,I) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_7710_sum__mono__inv,axiom,
    ! [F: fun_real_nat,I5: set_real,G: fun_real_nat,I: real] :
      ( ( groups1935376822645274424al_nat(F,I5) = groups1935376822645274424al_nat(G,I5) )
     => ( ! [I3: real] :
            ( aa_set_real_o(member_real(I3),I5)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_real_nat(F,I3)),aa_real_nat(G,I3)) )
       => ( aa_set_real_o(member_real(I),I5)
         => ( aa_set_real_o(finite_finite_real,I5)
           => ( aa_real_nat(F,I) = aa_real_nat(G,I) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_7711_sum__mono__inv,axiom,
    ! [F: fun_int_nat,I5: set_int,G: fun_int_nat,I: int] :
      ( ( groups4541462559716669496nt_nat(F,I5) = groups4541462559716669496nt_nat(G,I5) )
     => ( ! [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(I),I5)
         => ( aa_set_int_o(finite_finite_int,I5)
           => ( aa_int_nat(F,I) = aa_int_nat(G,I) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_7712_sum__mono__inv,axiom,
    ! [F: fun_complex_nat,I5: set_complex,G: fun_complex_nat,I: complex] :
      ( ( groups5693394587270226106ex_nat(F,I5) = groups5693394587270226106ex_nat(G,I5) )
     => ( ! [I3: complex] :
            ( aa_set_complex_o(member_complex(I3),I5)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_complex_nat(F,I3)),aa_complex_nat(G,I3)) )
       => ( aa_set_complex_o(member_complex(I),I5)
         => ( aa_set_complex_o(finite3207457112153483333omplex,I5)
           => ( aa_complex_nat(F,I) = aa_complex_nat(G,I) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_7713_sum__cong__Suc,axiom,
    ! [A2: set_nat,F: fun_nat_nat,G: fun_nat_nat] :
      ( ~ aa_set_nat_o(member_nat(zero_zero_nat),A2)
     => ( ! [X2: nat] :
            ( aa_set_nat_o(member_nat(aa_nat_nat(suc,X2)),A2)
           => ( aa_nat_nat(F,aa_nat_nat(suc,X2)) = aa_nat_nat(G,aa_nat_nat(suc,X2)) ) )
       => ( aa_set_nat_nat(groups3542108847815614940at_nat(F),A2) = aa_set_nat_nat(groups3542108847815614940at_nat(G),A2) ) ) ) ).

% sum_cong_Suc
tff(fact_7714_sum__cong__Suc,axiom,
    ! [A2: set_nat,F: fun_nat_real,G: fun_nat_real] :
      ( ~ aa_set_nat_o(member_nat(zero_zero_nat),A2)
     => ( ! [X2: nat] :
            ( aa_set_nat_o(member_nat(aa_nat_nat(suc,X2)),A2)
           => ( aa_nat_real(F,aa_nat_nat(suc,X2)) = aa_nat_real(G,aa_nat_nat(suc,X2)) ) )
       => ( groups6591440286371151544t_real(F,A2) = groups6591440286371151544t_real(G,A2) ) ) ) ).

% sum_cong_Suc
tff(fact_7715_sum_Ointer__filter,axiom,
    ! [A2: set_VEBT_VEBT,G: fun_VEBT_VEBT_uint32,P: fun_VEBT_VEBT_o] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,A2)
     => ( groups8325533452322294502uint32(G,collect_VEBT_VEBT(aa_fun8193769468198689164VEBT_o(aTP_Lamm_gh(A2),P))) = groups8325533452322294502uint32(aa_fun2089568865546376974uint32(aTP_Lamm_anj(G),P),A2) ) ) ).

% sum.inter_filter
tff(fact_7716_sum_Ointer__filter,axiom,
    ! [A2: set_real,G: fun_real_uint32,P: fun_real_o] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( groups5944083974425963860uint32(G,collect_real(aa_fun5868358341814070780real_o(aTP_Lamm_gi(A2),P))) = groups5944083974425963860uint32(aa_fun7747623534341301918uint32(aTP_Lamm_ank(G),P),A2) ) ) ).

% sum.inter_filter
tff(fact_7717_sum_Ointer__filter,axiom,
    ! [A2: set_nat,G: fun_nat_uint32,P: fun_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( groups833757482993574392uint32(G,collect_nat(aa_fun5937193402907943164_nat_o(aTP_Lamm_gj(A2),P))) = groups833757482993574392uint32(aa_fun7328656920723993502uint32(aTP_Lamm_anl(G),P),A2) ) ) ).

% sum.inter_filter
tff(fact_7718_sum_Ointer__filter,axiom,
    ! [A2: set_int,G: fun_int_uint32,P: fun_int_o] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( groups5712668689793887828uint32(G,collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_gm(A2),P))) = groups5712668689793887828uint32(aa_fun1392314963150257310uint32(aTP_Lamm_anm(G),P),A2) ) ) ).

% sum.inter_filter
tff(fact_7719_sum_Ointer__filter,axiom,
    ! [A2: set_complex,G: fun_complex_uint32,P: fun_complex_o] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( groups8736914816313324502uint32(G,collect_complex(aa_fun2514317545224092796plex_o(aTP_Lamm_gk(A2),P))) = groups8736914816313324502uint32(aa_fun9209306604415416350uint32(aTP_Lamm_ann(G),P),A2) ) ) ).

% sum.inter_filter
tff(fact_7720_sum_Ointer__filter,axiom,
    ! [A2: set_Code_integer,G: fun_Co1554742298434871494uint32,P: fun_Code_integer_o] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A2)
     => ( groups8847630953604152069uint32(G,collect_Code_integer(aa_fun8214818745163655228eger_o(aTP_Lamm_ano(A2),P))) = groups8847630953604152069uint32(aa_fun7787524580160642142uint32(aTP_Lamm_anp(G),P),A2) ) ) ).

% sum.inter_filter
tff(fact_7721_sum_Ointer__filter,axiom,
    ! [A2: set_VEBT_VEBT,G: fun_VEBT_VEBT_real,P: fun_VEBT_VEBT_o] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,A2)
     => ( groups2240296850493347238T_real(G,collect_VEBT_VEBT(aa_fun8193769468198689164VEBT_o(aTP_Lamm_gh(A2),P))) = groups2240296850493347238T_real(aa_fun3597767889293385422T_real(aTP_Lamm_anq(G),P),A2) ) ) ).

% sum.inter_filter
tff(fact_7722_sum_Ointer__filter,axiom,
    ! [A2: set_real,G: fun_real_real,P: fun_real_o] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( groups8097168146408367636l_real(G,collect_real(aa_fun5868358341814070780real_o(aTP_Lamm_gi(A2),P))) = groups8097168146408367636l_real(aa_fun4448911596306054750l_real(aTP_Lamm_anr(G),P),A2) ) ) ).

% sum.inter_filter
tff(fact_7723_sum_Ointer__filter,axiom,
    ! [A2: set_int,G: fun_int_real,P: fun_int_o] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( groups8778361861064173332t_real(G,collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_gm(A2),P))) = groups8778361861064173332t_real(aa_fun7506669883404309598t_real(aTP_Lamm_ans(G),P),A2) ) ) ).

% sum.inter_filter
tff(fact_7724_sum_Ointer__filter,axiom,
    ! [A2: set_complex,G: fun_complex_real,P: fun_complex_o] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( groups5808333547571424918x_real(G,collect_complex(aa_fun2514317545224092796plex_o(aTP_Lamm_gk(A2),P))) = groups5808333547571424918x_real(aa_fun8261057609414086622x_real(aTP_Lamm_ant(G),P),A2) ) ) ).

% sum.inter_filter
tff(fact_7725_sum_Oshift__bounds__cl__Suc__ivl,axiom,
    ! [G: fun_nat_nat,M: nat,Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),set_or1269000886237332187st_nat(aa_nat_nat(suc,M),aa_nat_nat(suc,Nb))) = aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_anu(G)),set_or1269000886237332187st_nat(M,Nb)) ).

% sum.shift_bounds_cl_Suc_ivl
tff(fact_7726_sum_Oshift__bounds__cl__Suc__ivl,axiom,
    ! [G: fun_nat_real,M: nat,Nb: nat] : groups6591440286371151544t_real(G,set_or1269000886237332187st_nat(aa_nat_nat(suc,M),aa_nat_nat(suc,Nb))) = groups6591440286371151544t_real(aTP_Lamm_anv(G),set_or1269000886237332187st_nat(M,Nb)) ).

% sum.shift_bounds_cl_Suc_ivl
tff(fact_7727_sum_Oshift__bounds__cl__nat__ivl,axiom,
    ! [G: fun_nat_nat,M: nat,K: nat,Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),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,Nb),K))) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_anw(G),K)),set_or1269000886237332187st_nat(M,Nb)) ).

% sum.shift_bounds_cl_nat_ivl
tff(fact_7728_sum_Oshift__bounds__cl__nat__ivl,axiom,
    ! [G: fun_nat_real,M: nat,K: nat,Nb: nat] : groups6591440286371151544t_real(G,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,Nb),K))) = groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_anx(G),K),set_or1269000886237332187st_nat(M,Nb)) ).

% sum.shift_bounds_cl_nat_ivl
tff(fact_7729_sum__le__included,axiom,
    ! [S3: set_int,Ta: set_int,G: fun_int_real,I: fun_int_int,F: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,S3)
     => ( aa_set_int_o(finite_finite_int,Ta)
       => ( ! [X2: int] :
              ( aa_set_int_o(member_int(X2),Ta)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_int_real(G,X2)) )
         => ( ! [X2: int] :
                ( aa_set_int_o(member_int(X2),S3)
               => ? [Xa: int] :
                    ( aa_set_int_o(member_int(Xa),Ta)
                    & ( aa_int_int(I,Xa) = X2 )
                    & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(F,X2)),aa_int_real(G,Xa)) ) )
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups8778361861064173332t_real(F,S3)),groups8778361861064173332t_real(G,Ta)) ) ) ) ) ).

% sum_le_included
tff(fact_7730_sum__le__included,axiom,
    ! [S3: set_int,Ta: set_complex,G: fun_complex_real,I: fun_complex_int,F: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,S3)
     => ( aa_set_complex_o(finite3207457112153483333omplex,Ta)
       => ( ! [X2: complex] :
              ( aa_set_complex_o(member_complex(X2),Ta)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(G,X2)) )
         => ( ! [X2: int] :
                ( aa_set_int_o(member_int(X2),S3)
               => ? [Xa: complex] :
                    ( aa_set_complex_o(member_complex(Xa),Ta)
                    & ( aa_complex_int(I,Xa) = X2 )
                    & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(F,X2)),aa_complex_real(G,Xa)) ) )
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups8778361861064173332t_real(F,S3)),groups5808333547571424918x_real(G,Ta)) ) ) ) ) ).

% sum_le_included
tff(fact_7731_sum__le__included,axiom,
    ! [S3: set_int,Ta: set_Code_integer,G: fun_Co3960896333048504454r_real,I: fun_Code_integer_int,F: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,S3)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,Ta)
       => ( ! [X2: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X2),Ta)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_Code_integer_real(G,X2)) )
         => ( ! [X2: int] :
                ( aa_set_int_o(member_int(X2),S3)
               => ? [Xa: code_integer] :
                    ( aa_set5643387332196575111eger_o(member_Code_integer(Xa),Ta)
                    & ( aa_Code_integer_int(I,Xa) = X2 )
                    & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(F,X2)),aa_Code_integer_real(G,Xa)) ) )
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups8778361861064173332t_real(F,S3)),groups1270011288395367621r_real(G,Ta)) ) ) ) ) ).

% sum_le_included
tff(fact_7732_sum__le__included,axiom,
    ! [S3: set_complex,Ta: set_int,G: fun_int_real,I: fun_int_complex,F: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S3)
     => ( aa_set_int_o(finite_finite_int,Ta)
       => ( ! [X2: int] :
              ( aa_set_int_o(member_int(X2),Ta)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_int_real(G,X2)) )
         => ( ! [X2: complex] :
                ( aa_set_complex_o(member_complex(X2),S3)
               => ? [Xa: int] :
                    ( aa_set_int_o(member_int(Xa),Ta)
                    & ( aa_int_complex(I,Xa) = X2 )
                    & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(F,X2)),aa_int_real(G,Xa)) ) )
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups5808333547571424918x_real(F,S3)),groups8778361861064173332t_real(G,Ta)) ) ) ) ) ).

% sum_le_included
tff(fact_7733_sum__le__included,axiom,
    ! [S3: set_complex,Ta: set_complex,G: fun_complex_real,I: fun_complex_complex,F: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S3)
     => ( aa_set_complex_o(finite3207457112153483333omplex,Ta)
       => ( ! [X2: complex] :
              ( aa_set_complex_o(member_complex(X2),Ta)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(G,X2)) )
         => ( ! [X2: complex] :
                ( aa_set_complex_o(member_complex(X2),S3)
               => ? [Xa: complex] :
                    ( aa_set_complex_o(member_complex(Xa),Ta)
                    & ( aa_complex_complex(I,Xa) = X2 )
                    & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(F,X2)),aa_complex_real(G,Xa)) ) )
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups5808333547571424918x_real(F,S3)),groups5808333547571424918x_real(G,Ta)) ) ) ) ) ).

% sum_le_included
tff(fact_7734_sum__le__included,axiom,
    ! [S3: set_complex,Ta: set_Code_integer,G: fun_Co3960896333048504454r_real,I: fun_Co1900716531650368264omplex,F: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S3)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,Ta)
       => ( ! [X2: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X2),Ta)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_Code_integer_real(G,X2)) )
         => ( ! [X2: complex] :
                ( aa_set_complex_o(member_complex(X2),S3)
               => ? [Xa: code_integer] :
                    ( aa_set5643387332196575111eger_o(member_Code_integer(Xa),Ta)
                    & ( aa_Cod5621350732091595593omplex(I,Xa) = X2 )
                    & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(F,X2)),aa_Code_integer_real(G,Xa)) ) )
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups5808333547571424918x_real(F,S3)),groups1270011288395367621r_real(G,Ta)) ) ) ) ) ).

% sum_le_included
tff(fact_7735_sum__le__included,axiom,
    ! [S3: set_Code_integer,Ta: set_int,G: fun_int_real,I: fun_int_Code_integer,F: fun_Co3960896333048504454r_real] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,S3)
     => ( aa_set_int_o(finite_finite_int,Ta)
       => ( ! [X2: int] :
              ( aa_set_int_o(member_int(X2),Ta)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_int_real(G,X2)) )
         => ( ! [X2: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(X2),S3)
               => ? [Xa: int] :
                    ( aa_set_int_o(member_int(Xa),Ta)
                    & ( aa_int_Code_integer(I,Xa) = X2 )
                    & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_Code_integer_real(F,X2)),aa_int_real(G,Xa)) ) )
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups1270011288395367621r_real(F,S3)),groups8778361861064173332t_real(G,Ta)) ) ) ) ) ).

% sum_le_included
tff(fact_7736_sum__le__included,axiom,
    ! [S3: set_Code_integer,Ta: set_complex,G: fun_complex_real,I: fun_co8153928105989009288nteger,F: fun_Co3960896333048504454r_real] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,S3)
     => ( aa_set_complex_o(finite3207457112153483333omplex,Ta)
       => ( ! [X2: complex] :
              ( aa_set_complex_o(member_complex(X2),Ta)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(G,X2)) )
         => ( ! [X2: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(X2),S3)
               => ? [Xa: complex] :
                    ( aa_set_complex_o(member_complex(Xa),Ta)
                    & ( aa_com4217951221296830281nteger(I,Xa) = X2 )
                    & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_Code_integer_real(F,X2)),aa_complex_real(G,Xa)) ) )
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups1270011288395367621r_real(F,S3)),groups5808333547571424918x_real(G,Ta)) ) ) ) ) ).

% sum_le_included
tff(fact_7737_sum__le__included,axiom,
    ! [S3: set_Code_integer,Ta: set_Code_integer,G: fun_Co3960896333048504454r_real,I: fun_Co8499238673658658103nteger,F: fun_Co3960896333048504454r_real] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,S3)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,Ta)
       => ( ! [X2: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X2),Ta)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_Code_integer_real(G,X2)) )
         => ( ! [X2: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(X2),S3)
               => ? [Xa: code_integer] :
                    ( aa_set5643387332196575111eger_o(member_Code_integer(Xa),Ta)
                    & ( aa_Cod1783974652533396728nteger(I,Xa) = X2 )
                    & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_Code_integer_real(F,X2)),aa_Code_integer_real(G,Xa)) ) )
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups1270011288395367621r_real(F,S3)),groups1270011288395367621r_real(G,Ta)) ) ) ) ) ).

% sum_le_included
tff(fact_7738_sum__le__included,axiom,
    ! [S3: set_nat,Ta: set_nat,G: fun_nat_rat,I: fun_nat_nat,F: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,S3)
     => ( aa_set_nat_o(finite_finite_nat,Ta)
       => ( ! [X2: nat] :
              ( aa_set_nat_o(member_nat(X2),Ta)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(G,X2)) )
         => ( ! [X2: nat] :
                ( aa_set_nat_o(member_nat(X2),S3)
               => ? [Xa: nat] :
                    ( aa_set_nat_o(member_nat(Xa),Ta)
                    & ( aa_nat_nat(I,Xa) = X2 )
                    & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,X2)),aa_nat_rat(G,Xa)) ) )
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups2906978787729119204at_rat(F,S3)),groups2906978787729119204at_rat(G,Ta)) ) ) ) ) ).

% sum_le_included
tff(fact_7739_sum__nonneg__eq__0__iff,axiom,
    ! [A2: set_VEBT_VEBT,F: fun_VEBT_VEBT_real] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,A2)
     => ( ! [X2: vEBT_VEBT] :
            ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X2),A2)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_VEBT_VEBT_real(F,X2)) )
       => ( ( groups2240296850493347238T_real(F,A2) = zero_zero_real )
        <=> ! [X: vEBT_VEBT] :
              ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),A2)
             => ( aa_VEBT_VEBT_real(F,X) = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_7740_sum__nonneg__eq__0__iff,axiom,
    ! [A2: set_real,F: fun_real_real] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( ! [X2: real] :
            ( aa_set_real_o(member_real(X2),A2)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(F,X2)) )
       => ( ( groups8097168146408367636l_real(F,A2) = zero_zero_real )
        <=> ! [X: real] :
              ( aa_set_real_o(member_real(X),A2)
             => ( aa_real_real(F,X) = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_7741_sum__nonneg__eq__0__iff,axiom,
    ! [A2: set_int,F: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( ! [X2: int] :
            ( aa_set_int_o(member_int(X2),A2)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_int_real(F,X2)) )
       => ( ( groups8778361861064173332t_real(F,A2) = zero_zero_real )
        <=> ! [X: int] :
              ( aa_set_int_o(member_int(X),A2)
             => ( aa_int_real(F,X) = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_7742_sum__nonneg__eq__0__iff,axiom,
    ! [A2: set_complex,F: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( ! [X2: complex] :
            ( aa_set_complex_o(member_complex(X2),A2)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(F,X2)) )
       => ( ( groups5808333547571424918x_real(F,A2) = zero_zero_real )
        <=> ! [X: complex] :
              ( aa_set_complex_o(member_complex(X),A2)
             => ( aa_complex_real(F,X) = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_7743_sum__nonneg__eq__0__iff,axiom,
    ! [A2: set_Code_integer,F: fun_Co3960896333048504454r_real] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A2)
     => ( ! [X2: code_integer] :
            ( aa_set5643387332196575111eger_o(member_Code_integer(X2),A2)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_Code_integer_real(F,X2)) )
       => ( ( groups1270011288395367621r_real(F,A2) = zero_zero_real )
        <=> ! [X: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X),A2)
             => ( aa_Code_integer_real(F,X) = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_7744_sum__nonneg__eq__0__iff,axiom,
    ! [A2: set_VEBT_VEBT,F: fun_VEBT_VEBT_rat] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,A2)
     => ( ! [X2: vEBT_VEBT] :
            ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X2),A2)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_VEBT_VEBT_rat(F,X2)) )
       => ( ( groups136491112297645522BT_rat(F,A2) = zero_zero_rat )
        <=> ! [X: vEBT_VEBT] :
              ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),A2)
             => ( aa_VEBT_VEBT_rat(F,X) = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_7745_sum__nonneg__eq__0__iff,axiom,
    ! [A2: set_real,F: fun_real_rat] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( ! [X2: real] :
            ( aa_set_real_o(member_real(X2),A2)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_real_rat(F,X2)) )
       => ( ( groups1300246762558778688al_rat(F,A2) = zero_zero_rat )
        <=> ! [X: real] :
              ( aa_set_real_o(member_real(X),A2)
             => ( aa_real_rat(F,X) = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_7746_sum__nonneg__eq__0__iff,axiom,
    ! [A2: set_nat,F: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( ! [X2: nat] :
            ( aa_set_nat_o(member_nat(X2),A2)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(F,X2)) )
       => ( ( groups2906978787729119204at_rat(F,A2) = zero_zero_rat )
        <=> ! [X: nat] :
              ( aa_set_nat_o(member_nat(X),A2)
             => ( aa_nat_rat(F,X) = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_7747_sum__nonneg__eq__0__iff,axiom,
    ! [A2: set_int,F: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( ! [X2: int] :
            ( aa_set_int_o(member_int(X2),A2)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_int_rat(F,X2)) )
       => ( ( groups3906332499630173760nt_rat(F,A2) = zero_zero_rat )
        <=> ! [X: int] :
              ( aa_set_int_o(member_int(X),A2)
             => ( aa_int_rat(F,X) = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_7748_sum__nonneg__eq__0__iff,axiom,
    ! [A2: set_complex,F: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( ! [X2: complex] :
            ( aa_set_complex_o(member_complex(X2),A2)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(F,X2)) )
       => ( ( groups5058264527183730370ex_rat(F,A2) = zero_zero_rat )
        <=> ! [X: complex] :
              ( aa_set_complex_o(member_complex(X),A2)
             => ( aa_complex_rat(F,X) = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_7749_sum__strict__mono__ex1,axiom,
    ! [A2: set_int,F: fun_int_real,G: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( ! [X2: int] :
            ( aa_set_int_o(member_int(X2),A2)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(F,X2)),aa_int_real(G,X2)) )
       => ( ? [X3: int] :
              ( aa_set_int_o(member_int(X3),A2)
              & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(F,X3)),aa_int_real(G,X3)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,groups8778361861064173332t_real(F,A2)),groups8778361861064173332t_real(G,A2)) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_7750_sum__strict__mono__ex1,axiom,
    ! [A2: set_complex,F: fun_complex_real,G: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( ! [X2: complex] :
            ( aa_set_complex_o(member_complex(X2),A2)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(F,X2)),aa_complex_real(G,X2)) )
       => ( ? [X3: complex] :
              ( aa_set_complex_o(member_complex(X3),A2)
              & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(F,X3)),aa_complex_real(G,X3)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,groups5808333547571424918x_real(F,A2)),groups5808333547571424918x_real(G,A2)) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_7751_sum__strict__mono__ex1,axiom,
    ! [A2: set_Code_integer,F: fun_Co3960896333048504454r_real,G: fun_Co3960896333048504454r_real] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A2)
     => ( ! [X2: code_integer] :
            ( aa_set5643387332196575111eger_o(member_Code_integer(X2),A2)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_Code_integer_real(F,X2)),aa_Code_integer_real(G,X2)) )
       => ( ? [X3: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X3),A2)
              & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_Code_integer_real(F,X3)),aa_Code_integer_real(G,X3)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,groups1270011288395367621r_real(F,A2)),groups1270011288395367621r_real(G,A2)) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_7752_sum__strict__mono__ex1,axiom,
    ! [A2: set_nat,F: fun_nat_rat,G: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( ! [X2: nat] :
            ( aa_set_nat_o(member_nat(X2),A2)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,X2)),aa_nat_rat(G,X2)) )
       => ( ? [X3: nat] :
              ( aa_set_nat_o(member_nat(X3),A2)
              & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(F,X3)),aa_nat_rat(G,X3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,groups2906978787729119204at_rat(F,A2)),groups2906978787729119204at_rat(G,A2)) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_7753_sum__strict__mono__ex1,axiom,
    ! [A2: set_int,F: fun_int_rat,G: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( ! [X2: int] :
            ( aa_set_int_o(member_int(X2),A2)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(F,X2)),aa_int_rat(G,X2)) )
       => ( ? [X3: int] :
              ( aa_set_int_o(member_int(X3),A2)
              & 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,A2)),groups3906332499630173760nt_rat(G,A2)) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_7754_sum__strict__mono__ex1,axiom,
    ! [A2: set_complex,F: fun_complex_rat,G: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( ! [X2: complex] :
            ( aa_set_complex_o(member_complex(X2),A2)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_complex_rat(F,X2)),aa_complex_rat(G,X2)) )
       => ( ? [X3: complex] :
              ( aa_set_complex_o(member_complex(X3),A2)
              & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_complex_rat(F,X3)),aa_complex_rat(G,X3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,groups5058264527183730370ex_rat(F,A2)),groups5058264527183730370ex_rat(G,A2)) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_7755_sum__strict__mono__ex1,axiom,
    ! [A2: set_Code_integer,F: fun_Code_integer_rat,G: fun_Code_integer_rat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A2)
     => ( ! [X2: code_integer] :
            ( aa_set5643387332196575111eger_o(member_Code_integer(X2),A2)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_Code_integer_rat(F,X2)),aa_Code_integer_rat(G,X2)) )
       => ( ? [X3: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X3),A2)
              & 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,A2)),groups6602215022474089585er_rat(G,A2)) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_7756_sum__strict__mono__ex1,axiom,
    ! [A2: set_int,F: fun_int_nat,G: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( ! [X2: int] :
            ( aa_set_int_o(member_int(X2),A2)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(F,X2)),aa_int_nat(G,X2)) )
       => ( ? [X3: int] :
              ( aa_set_int_o(member_int(X3),A2)
              & 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,A2)),groups4541462559716669496nt_nat(G,A2)) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_7757_sum__strict__mono__ex1,axiom,
    ! [A2: set_complex,F: fun_complex_nat,G: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( ! [X2: complex] :
            ( aa_set_complex_o(member_complex(X2),A2)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_complex_nat(F,X2)),aa_complex_nat(G,X2)) )
       => ( ? [X3: complex] :
              ( aa_set_complex_o(member_complex(X3),A2)
              & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_complex_nat(F,X3)),aa_complex_nat(G,X3)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,groups5693394587270226106ex_nat(F,A2)),groups5693394587270226106ex_nat(G,A2)) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_7758_sum__strict__mono__ex1,axiom,
    ! [A2: set_Code_integer,F: fun_Code_integer_nat,G: fun_Code_integer_nat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A2)
     => ( ! [X2: code_integer] :
            ( aa_set5643387332196575111eger_o(member_Code_integer(X2),A2)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_Code_integer_nat(F,X2)),aa_Code_integer_nat(G,X2)) )
       => ( ? [X3: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X3),A2)
              & 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,A2)),groups7237345082560585321er_nat(G,A2)) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_7759_sum_Orelated,axiom,
    ! [R: fun_ui3478071520421546738nt32_o,S2: set_nat,H: fun_nat_uint32,G: fun_nat_uint32] :
      ( aa_uint32_o(aa_uin5924812838015166427nt32_o(R,zero_zero_uint32),zero_zero_uint32)
     => ( ! [X15: uint32,Y15: uint32,X24: uint32,Y23: uint32] :
            ( ( aa_uint32_o(aa_uin5924812838015166427nt32_o(R,X15),X24)
              & aa_uint32_o(aa_uin5924812838015166427nt32_o(R,Y15),Y23) )
           => aa_uint32_o(aa_uin5924812838015166427nt32_o(R,aa_uint32_uint32(plus_plus_uint32(X15),Y15)),aa_uint32_uint32(plus_plus_uint32(X24),Y23)) )
       => ( aa_set_nat_o(finite_finite_nat,S2)
         => ( ! [X2: nat] :
                ( aa_set_nat_o(member_nat(X2),S2)
               => aa_uint32_o(aa_uin5924812838015166427nt32_o(R,aa_nat_uint32(H,X2)),aa_nat_uint32(G,X2)) )
           => aa_uint32_o(aa_uin5924812838015166427nt32_o(R,groups833757482993574392uint32(H,S2)),groups833757482993574392uint32(G,S2)) ) ) ) ) ).

% sum.related
tff(fact_7760_sum_Orelated,axiom,
    ! [R: fun_ui3478071520421546738nt32_o,S2: set_int,H: fun_int_uint32,G: fun_int_uint32] :
      ( aa_uint32_o(aa_uin5924812838015166427nt32_o(R,zero_zero_uint32),zero_zero_uint32)
     => ( ! [X15: uint32,Y15: uint32,X24: uint32,Y23: uint32] :
            ( ( aa_uint32_o(aa_uin5924812838015166427nt32_o(R,X15),X24)
              & aa_uint32_o(aa_uin5924812838015166427nt32_o(R,Y15),Y23) )
           => aa_uint32_o(aa_uin5924812838015166427nt32_o(R,aa_uint32_uint32(plus_plus_uint32(X15),Y15)),aa_uint32_uint32(plus_plus_uint32(X24),Y23)) )
       => ( aa_set_int_o(finite_finite_int,S2)
         => ( ! [X2: int] :
                ( aa_set_int_o(member_int(X2),S2)
               => aa_uint32_o(aa_uin5924812838015166427nt32_o(R,aa_int_uint32(H,X2)),aa_int_uint32(G,X2)) )
           => aa_uint32_o(aa_uin5924812838015166427nt32_o(R,groups5712668689793887828uint32(H,S2)),groups5712668689793887828uint32(G,S2)) ) ) ) ) ).

% sum.related
tff(fact_7761_sum_Orelated,axiom,
    ! [R: fun_ui3478071520421546738nt32_o,S2: set_complex,H: fun_complex_uint32,G: fun_complex_uint32] :
      ( aa_uint32_o(aa_uin5924812838015166427nt32_o(R,zero_zero_uint32),zero_zero_uint32)
     => ( ! [X15: uint32,Y15: uint32,X24: uint32,Y23: uint32] :
            ( ( aa_uint32_o(aa_uin5924812838015166427nt32_o(R,X15),X24)
              & aa_uint32_o(aa_uin5924812838015166427nt32_o(R,Y15),Y23) )
           => aa_uint32_o(aa_uin5924812838015166427nt32_o(R,aa_uint32_uint32(plus_plus_uint32(X15),Y15)),aa_uint32_uint32(plus_plus_uint32(X24),Y23)) )
       => ( aa_set_complex_o(finite3207457112153483333omplex,S2)
         => ( ! [X2: complex] :
                ( aa_set_complex_o(member_complex(X2),S2)
               => aa_uint32_o(aa_uin5924812838015166427nt32_o(R,aa_complex_uint32(H,X2)),aa_complex_uint32(G,X2)) )
           => aa_uint32_o(aa_uin5924812838015166427nt32_o(R,groups8736914816313324502uint32(H,S2)),groups8736914816313324502uint32(G,S2)) ) ) ) ) ).

% sum.related
tff(fact_7762_sum_Orelated,axiom,
    ! [R: fun_ui3478071520421546738nt32_o,S2: set_Code_integer,H: fun_Co1554742298434871494uint32,G: fun_Co1554742298434871494uint32] :
      ( aa_uint32_o(aa_uin5924812838015166427nt32_o(R,zero_zero_uint32),zero_zero_uint32)
     => ( ! [X15: uint32,Y15: uint32,X24: uint32,Y23: uint32] :
            ( ( aa_uint32_o(aa_uin5924812838015166427nt32_o(R,X15),X24)
              & aa_uint32_o(aa_uin5924812838015166427nt32_o(R,Y15),Y23) )
           => aa_uint32_o(aa_uin5924812838015166427nt32_o(R,aa_uint32_uint32(plus_plus_uint32(X15),Y15)),aa_uint32_uint32(plus_plus_uint32(X24),Y23)) )
       => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,S2)
         => ( ! [X2: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(X2),S2)
               => aa_uint32_o(aa_uin5924812838015166427nt32_o(R,aa_Cod8401701657498852871uint32(H,X2)),aa_Cod8401701657498852871uint32(G,X2)) )
           => aa_uint32_o(aa_uin5924812838015166427nt32_o(R,groups8847630953604152069uint32(H,S2)),groups8847630953604152069uint32(G,S2)) ) ) ) ) ).

% sum.related
tff(fact_7763_sum_Orelated,axiom,
    ! [R: fun_real_fun_real_o,S2: set_int,H: fun_int_real,G: fun_int_real] :
      ( aa_real_o(aa_real_fun_real_o(R,zero_zero_real),zero_zero_real)
     => ( ! [X15: real,Y15: real,X24: real,Y23: real] :
            ( ( aa_real_o(aa_real_fun_real_o(R,X15),X24)
              & aa_real_o(aa_real_fun_real_o(R,Y15),Y23) )
           => aa_real_o(aa_real_fun_real_o(R,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X15),Y15)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X24),Y23)) )
       => ( aa_set_int_o(finite_finite_int,S2)
         => ( ! [X2: int] :
                ( aa_set_int_o(member_int(X2),S2)
               => aa_real_o(aa_real_fun_real_o(R,aa_int_real(H,X2)),aa_int_real(G,X2)) )
           => aa_real_o(aa_real_fun_real_o(R,groups8778361861064173332t_real(H,S2)),groups8778361861064173332t_real(G,S2)) ) ) ) ) ).

% sum.related
tff(fact_7764_sum_Orelated,axiom,
    ! [R: fun_real_fun_real_o,S2: set_complex,H: fun_complex_real,G: fun_complex_real] :
      ( aa_real_o(aa_real_fun_real_o(R,zero_zero_real),zero_zero_real)
     => ( ! [X15: real,Y15: real,X24: real,Y23: real] :
            ( ( aa_real_o(aa_real_fun_real_o(R,X15),X24)
              & aa_real_o(aa_real_fun_real_o(R,Y15),Y23) )
           => aa_real_o(aa_real_fun_real_o(R,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X15),Y15)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X24),Y23)) )
       => ( aa_set_complex_o(finite3207457112153483333omplex,S2)
         => ( ! [X2: complex] :
                ( aa_set_complex_o(member_complex(X2),S2)
               => aa_real_o(aa_real_fun_real_o(R,aa_complex_real(H,X2)),aa_complex_real(G,X2)) )
           => aa_real_o(aa_real_fun_real_o(R,groups5808333547571424918x_real(H,S2)),groups5808333547571424918x_real(G,S2)) ) ) ) ) ).

% sum.related
tff(fact_7765_sum_Orelated,axiom,
    ! [R: fun_real_fun_real_o,S2: set_Code_integer,H: fun_Co3960896333048504454r_real,G: fun_Co3960896333048504454r_real] :
      ( aa_real_o(aa_real_fun_real_o(R,zero_zero_real),zero_zero_real)
     => ( ! [X15: real,Y15: real,X24: real,Y23: real] :
            ( ( aa_real_o(aa_real_fun_real_o(R,X15),X24)
              & aa_real_o(aa_real_fun_real_o(R,Y15),Y23) )
           => aa_real_o(aa_real_fun_real_o(R,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X15),Y15)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X24),Y23)) )
       => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,S2)
         => ( ! [X2: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(X2),S2)
               => aa_real_o(aa_real_fun_real_o(R,aa_Code_integer_real(H,X2)),aa_Code_integer_real(G,X2)) )
           => aa_real_o(aa_real_fun_real_o(R,groups1270011288395367621r_real(H,S2)),groups1270011288395367621r_real(G,S2)) ) ) ) ) ).

% sum.related
tff(fact_7766_sum_Orelated,axiom,
    ! [R: fun_rat_fun_rat_o,S2: set_nat,H: fun_nat_rat,G: fun_nat_rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(R,zero_zero_rat),zero_zero_rat)
     => ( ! [X15: rat,Y15: rat,X24: rat,Y23: rat] :
            ( ( aa_rat_o(aa_rat_fun_rat_o(R,X15),X24)
              & aa_rat_o(aa_rat_fun_rat_o(R,Y15),Y23) )
           => aa_rat_o(aa_rat_fun_rat_o(R,aa_rat_rat(plus_plus_rat(X15),Y15)),aa_rat_rat(plus_plus_rat(X24),Y23)) )
       => ( aa_set_nat_o(finite_finite_nat,S2)
         => ( ! [X2: nat] :
                ( aa_set_nat_o(member_nat(X2),S2)
               => aa_rat_o(aa_rat_fun_rat_o(R,aa_nat_rat(H,X2)),aa_nat_rat(G,X2)) )
           => aa_rat_o(aa_rat_fun_rat_o(R,groups2906978787729119204at_rat(H,S2)),groups2906978787729119204at_rat(G,S2)) ) ) ) ) ).

% sum.related
tff(fact_7767_sum_Orelated,axiom,
    ! [R: fun_rat_fun_rat_o,S2: set_int,H: fun_int_rat,G: fun_int_rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(R,zero_zero_rat),zero_zero_rat)
     => ( ! [X15: rat,Y15: rat,X24: rat,Y23: rat] :
            ( ( aa_rat_o(aa_rat_fun_rat_o(R,X15),X24)
              & aa_rat_o(aa_rat_fun_rat_o(R,Y15),Y23) )
           => aa_rat_o(aa_rat_fun_rat_o(R,aa_rat_rat(plus_plus_rat(X15),Y15)),aa_rat_rat(plus_plus_rat(X24),Y23)) )
       => ( aa_set_int_o(finite_finite_int,S2)
         => ( ! [X2: int] :
                ( aa_set_int_o(member_int(X2),S2)
               => aa_rat_o(aa_rat_fun_rat_o(R,aa_int_rat(H,X2)),aa_int_rat(G,X2)) )
           => aa_rat_o(aa_rat_fun_rat_o(R,groups3906332499630173760nt_rat(H,S2)),groups3906332499630173760nt_rat(G,S2)) ) ) ) ) ).

% sum.related
tff(fact_7768_sum_Orelated,axiom,
    ! [R: fun_rat_fun_rat_o,S2: set_complex,H: fun_complex_rat,G: fun_complex_rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(R,zero_zero_rat),zero_zero_rat)
     => ( ! [X15: rat,Y15: rat,X24: rat,Y23: rat] :
            ( ( aa_rat_o(aa_rat_fun_rat_o(R,X15),X24)
              & aa_rat_o(aa_rat_fun_rat_o(R,Y15),Y23) )
           => aa_rat_o(aa_rat_fun_rat_o(R,aa_rat_rat(plus_plus_rat(X15),Y15)),aa_rat_rat(plus_plus_rat(X24),Y23)) )
       => ( aa_set_complex_o(finite3207457112153483333omplex,S2)
         => ( ! [X2: complex] :
                ( aa_set_complex_o(member_complex(X2),S2)
               => aa_rat_o(aa_rat_fun_rat_o(R,aa_complex_rat(H,X2)),aa_complex_rat(G,X2)) )
           => aa_rat_o(aa_rat_fun_rat_o(R,groups5058264527183730370ex_rat(H,S2)),groups5058264527183730370ex_rat(G,S2)) ) ) ) ) ).

% sum.related
tff(fact_7769_sum__strict__mono,axiom,
    ! [A2: set_VEBT_VEBT,F: fun_VEBT_VEBT_real,G: fun_VEBT_VEBT_real] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,A2)
     => ( ( A2 != bot_bo8194388402131092736T_VEBT )
       => ( ! [X2: vEBT_VEBT] :
              ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X2),A2)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_VEBT_VEBT_real(F,X2)),aa_VEBT_VEBT_real(G,X2)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,groups2240296850493347238T_real(F,A2)),groups2240296850493347238T_real(G,A2)) ) ) ) ).

% sum_strict_mono
tff(fact_7770_sum__strict__mono,axiom,
    ! [A2: set_complex,F: fun_complex_real,G: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( ( A2 != bot_bot_set_complex )
       => ( ! [X2: complex] :
              ( aa_set_complex_o(member_complex(X2),A2)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(F,X2)),aa_complex_real(G,X2)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,groups5808333547571424918x_real(F,A2)),groups5808333547571424918x_real(G,A2)) ) ) ) ).

% sum_strict_mono
tff(fact_7771_sum__strict__mono,axiom,
    ! [A2: set_Code_integer,F: fun_Co3960896333048504454r_real,G: fun_Co3960896333048504454r_real] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A2)
     => ( ( A2 != bot_bo3990330152332043303nteger )
       => ( ! [X2: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X2),A2)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_Code_integer_real(F,X2)),aa_Code_integer_real(G,X2)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,groups1270011288395367621r_real(F,A2)),groups1270011288395367621r_real(G,A2)) ) ) ) ).

% sum_strict_mono
tff(fact_7772_sum__strict__mono,axiom,
    ! [A2: set_int,F: fun_int_real,G: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( ( A2 != bot_bot_set_int )
       => ( ! [X2: int] :
              ( aa_set_int_o(member_int(X2),A2)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(F,X2)),aa_int_real(G,X2)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,groups8778361861064173332t_real(F,A2)),groups8778361861064173332t_real(G,A2)) ) ) ) ).

% sum_strict_mono
tff(fact_7773_sum__strict__mono,axiom,
    ! [A2: set_real,F: fun_real_real,G: fun_real_real] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( ( A2 != bot_bot_set_real )
       => ( ! [X2: real] :
              ( aa_set_real_o(member_real(X2),A2)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,X2)),aa_real_real(G,X2)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,groups8097168146408367636l_real(F,A2)),groups8097168146408367636l_real(G,A2)) ) ) ) ).

% sum_strict_mono
tff(fact_7774_sum__strict__mono,axiom,
    ! [A2: set_VEBT_VEBT,F: fun_VEBT_VEBT_rat,G: fun_VEBT_VEBT_rat] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,A2)
     => ( ( A2 != bot_bo8194388402131092736T_VEBT )
       => ( ! [X2: vEBT_VEBT] :
              ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X2),A2)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_VEBT_VEBT_rat(F,X2)),aa_VEBT_VEBT_rat(G,X2)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,groups136491112297645522BT_rat(F,A2)),groups136491112297645522BT_rat(G,A2)) ) ) ) ).

% sum_strict_mono
tff(fact_7775_sum__strict__mono,axiom,
    ! [A2: set_complex,F: fun_complex_rat,G: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( ( A2 != bot_bot_set_complex )
       => ( ! [X2: complex] :
              ( aa_set_complex_o(member_complex(X2),A2)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_complex_rat(F,X2)),aa_complex_rat(G,X2)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,groups5058264527183730370ex_rat(F,A2)),groups5058264527183730370ex_rat(G,A2)) ) ) ) ).

% sum_strict_mono
tff(fact_7776_sum__strict__mono,axiom,
    ! [A2: set_Code_integer,F: fun_Code_integer_rat,G: fun_Code_integer_rat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A2)
     => ( ( A2 != bot_bo3990330152332043303nteger )
       => ( ! [X2: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X2),A2)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_Code_integer_rat(F,X2)),aa_Code_integer_rat(G,X2)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,groups6602215022474089585er_rat(F,A2)),groups6602215022474089585er_rat(G,A2)) ) ) ) ).

% sum_strict_mono
tff(fact_7777_sum__strict__mono,axiom,
    ! [A2: set_nat,F: fun_nat_rat,G: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( ( A2 != bot_bot_set_nat )
       => ( ! [X2: nat] :
              ( aa_set_nat_o(member_nat(X2),A2)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(F,X2)),aa_nat_rat(G,X2)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,groups2906978787729119204at_rat(F,A2)),groups2906978787729119204at_rat(G,A2)) ) ) ) ).

% sum_strict_mono
tff(fact_7778_sum__strict__mono,axiom,
    ! [A2: set_int,F: fun_int_rat,G: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( ( A2 != bot_bot_set_int )
       => ( ! [X2: int] :
              ( aa_set_int_o(member_int(X2),A2)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(F,X2)),aa_int_rat(G,X2)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,groups3906332499630173760nt_rat(F,A2)),groups3906332499630173760nt_rat(G,A2)) ) ) ) ).

% sum_strict_mono
tff(fact_7779_sum_Oinsert__if,axiom,
    ! [A2: set_VEBT_VEBT,G: fun_VEBT_VEBT_real,Xb: vEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,A2)
     => ( groups2240296850493347238T_real(G,insert_VEBT_VEBT(Xb,A2)) = $ite(aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xb),A2),groups2240296850493347238T_real(G,A2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_VEBT_VEBT_real(G,Xb)),groups2240296850493347238T_real(G,A2))) ) ) ).

% sum.insert_if
tff(fact_7780_sum_Oinsert__if,axiom,
    ! [A2: set_real,G: fun_real_real,Xb: real] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( groups8097168146408367636l_real(G,insert_real(Xb,A2)) = $ite(aa_set_real_o(member_real(Xb),A2),groups8097168146408367636l_real(G,A2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(G,Xb)),groups8097168146408367636l_real(G,A2))) ) ) ).

% sum.insert_if
tff(fact_7781_sum_Oinsert__if,axiom,
    ! [A2: set_int,G: fun_int_real,Xb: int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( groups8778361861064173332t_real(G,insert_int(Xb,A2)) = $ite(aa_set_int_o(member_int(Xb),A2),groups8778361861064173332t_real(G,A2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(G,Xb)),groups8778361861064173332t_real(G,A2))) ) ) ).

% sum.insert_if
tff(fact_7782_sum_Oinsert__if,axiom,
    ! [A2: set_complex,G: fun_complex_real,Xb: complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( groups5808333547571424918x_real(G,insert_complex(Xb,A2)) = $ite(aa_set_complex_o(member_complex(Xb),A2),groups5808333547571424918x_real(G,A2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(G,Xb)),groups5808333547571424918x_real(G,A2))) ) ) ).

% sum.insert_if
tff(fact_7783_sum_Oinsert__if,axiom,
    ! [A2: set_Code_integer,G: fun_Co3960896333048504454r_real,Xb: code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A2)
     => ( groups1270011288395367621r_real(G,insert_Code_integer(Xb,A2)) = $ite(aa_set5643387332196575111eger_o(member_Code_integer(Xb),A2),groups1270011288395367621r_real(G,A2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_Code_integer_real(G,Xb)),groups1270011288395367621r_real(G,A2))) ) ) ).

% sum.insert_if
tff(fact_7784_sum_Oinsert__if,axiom,
    ! [A2: set_VEBT_VEBT,G: fun_VEBT_VEBT_rat,Xb: vEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,A2)
     => ( groups136491112297645522BT_rat(G,insert_VEBT_VEBT(Xb,A2)) = $ite(aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xb),A2),groups136491112297645522BT_rat(G,A2),aa_rat_rat(plus_plus_rat(aa_VEBT_VEBT_rat(G,Xb)),groups136491112297645522BT_rat(G,A2))) ) ) ).

% sum.insert_if
tff(fact_7785_sum_Oinsert__if,axiom,
    ! [A2: set_real,G: fun_real_rat,Xb: real] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( groups1300246762558778688al_rat(G,insert_real(Xb,A2)) = $ite(aa_set_real_o(member_real(Xb),A2),groups1300246762558778688al_rat(G,A2),aa_rat_rat(plus_plus_rat(aa_real_rat(G,Xb)),groups1300246762558778688al_rat(G,A2))) ) ) ).

% sum.insert_if
tff(fact_7786_sum_Oinsert__if,axiom,
    ! [A2: set_nat,G: fun_nat_rat,Xb: nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( groups2906978787729119204at_rat(G,insert_nat(Xb,A2)) = $ite(aa_set_nat_o(member_nat(Xb),A2),groups2906978787729119204at_rat(G,A2),aa_rat_rat(plus_plus_rat(aa_nat_rat(G,Xb)),groups2906978787729119204at_rat(G,A2))) ) ) ).

% sum.insert_if
tff(fact_7787_sum_Oinsert__if,axiom,
    ! [A2: set_int,G: fun_int_rat,Xb: int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( groups3906332499630173760nt_rat(G,insert_int(Xb,A2)) = $ite(aa_set_int_o(member_int(Xb),A2),groups3906332499630173760nt_rat(G,A2),aa_rat_rat(plus_plus_rat(aa_int_rat(G,Xb)),groups3906332499630173760nt_rat(G,A2))) ) ) ).

% sum.insert_if
tff(fact_7788_sum_Oinsert__if,axiom,
    ! [A2: set_complex,G: fun_complex_rat,Xb: complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( groups5058264527183730370ex_rat(G,insert_complex(Xb,A2)) = $ite(aa_set_complex_o(member_complex(Xb),A2),groups5058264527183730370ex_rat(G,A2),aa_rat_rat(plus_plus_rat(aa_complex_rat(G,Xb)),groups5058264527183730370ex_rat(G,A2))) ) ) ).

% sum.insert_if
tff(fact_7789_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S6: set_VEBT_VEBT,T6: set_VEBT_VEBT,S2: set_VEBT_VEBT,I: fun_VE3165209603783583043T_VEBT,J: fun_VE3165209603783583043T_VEBT,T4: set_VEBT_VEBT,G: fun_VEBT_VEBT_uint32,H: fun_VEBT_VEBT_uint32] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,S6)
     => ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,T6)
       => ( ! [A3: vEBT_VEBT] :
              ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A3),minus_5127226145743854075T_VEBT(S2,S6))
             => ( aa_VEB5832953974749077948T_VEBT(I,aa_VEB5832953974749077948T_VEBT(J,A3)) = A3 ) )
         => ( ! [A3: vEBT_VEBT] :
                ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A3),minus_5127226145743854075T_VEBT(S2,S6))
               => aa_set_VEBT_VEBT_o(member_VEBT_VEBT(aa_VEB5832953974749077948T_VEBT(J,A3)),minus_5127226145743854075T_VEBT(T4,T6)) )
           => ( ! [B2: vEBT_VEBT] :
                  ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(B2),minus_5127226145743854075T_VEBT(T4,T6))
                 => ( aa_VEB5832953974749077948T_VEBT(J,aa_VEB5832953974749077948T_VEBT(I,B2)) = B2 ) )
             => ( ! [B2: vEBT_VEBT] :
                    ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(B2),minus_5127226145743854075T_VEBT(T4,T6))
                   => aa_set_VEBT_VEBT_o(member_VEBT_VEBT(aa_VEB5832953974749077948T_VEBT(I,B2)),minus_5127226145743854075T_VEBT(S2,S6)) )
               => ( ! [A3: vEBT_VEBT] :
                      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A3),S6)
                     => ( aa_VEBT_VEBT_uint32(G,A3) = zero_zero_uint32 ) )
                 => ( ! [B2: vEBT_VEBT] :
                        ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(B2),T6)
                       => ( aa_VEBT_VEBT_uint32(H,B2) = zero_zero_uint32 ) )
                   => ( ! [A3: vEBT_VEBT] :
                          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A3),S2)
                         => ( aa_VEBT_VEBT_uint32(H,aa_VEB5832953974749077948T_VEBT(J,A3)) = aa_VEBT_VEBT_uint32(G,A3) ) )
                     => ( groups8325533452322294502uint32(G,S2) = groups8325533452322294502uint32(H,T4) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_7790_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S6: set_VEBT_VEBT,T6: set_real,S2: set_VEBT_VEBT,I: fun_real_VEBT_VEBT,J: fun_VEBT_VEBT_real,T4: set_real,G: fun_VEBT_VEBT_uint32,H: fun_real_uint32] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,S6)
     => ( aa_set_real_o(finite_finite_real,T6)
       => ( ! [A3: vEBT_VEBT] :
              ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A3),minus_5127226145743854075T_VEBT(S2,S6))
             => ( aa_real_VEBT_VEBT(I,aa_VEBT_VEBT_real(J,A3)) = A3 ) )
         => ( ! [A3: vEBT_VEBT] :
                ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A3),minus_5127226145743854075T_VEBT(S2,S6))
               => aa_set_real_o(member_real(aa_VEBT_VEBT_real(J,A3)),minus_minus_set_real(T4,T6)) )
           => ( ! [B2: real] :
                  ( aa_set_real_o(member_real(B2),minus_minus_set_real(T4,T6))
                 => ( aa_VEBT_VEBT_real(J,aa_real_VEBT_VEBT(I,B2)) = B2 ) )
             => ( ! [B2: real] :
                    ( aa_set_real_o(member_real(B2),minus_minus_set_real(T4,T6))
                   => aa_set_VEBT_VEBT_o(member_VEBT_VEBT(aa_real_VEBT_VEBT(I,B2)),minus_5127226145743854075T_VEBT(S2,S6)) )
               => ( ! [A3: vEBT_VEBT] :
                      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A3),S6)
                     => ( aa_VEBT_VEBT_uint32(G,A3) = zero_zero_uint32 ) )
                 => ( ! [B2: real] :
                        ( aa_set_real_o(member_real(B2),T6)
                       => ( aa_real_uint32(H,B2) = zero_zero_uint32 ) )
                   => ( ! [A3: vEBT_VEBT] :
                          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A3),S2)
                         => ( aa_real_uint32(H,aa_VEBT_VEBT_real(J,A3)) = aa_VEBT_VEBT_uint32(G,A3) ) )
                     => ( groups8325533452322294502uint32(G,S2) = groups5944083974425963860uint32(H,T4) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_7791_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S6: set_real,T6: set_VEBT_VEBT,S2: set_real,I: fun_VEBT_VEBT_real,J: fun_real_VEBT_VEBT,T4: set_VEBT_VEBT,G: fun_real_uint32,H: fun_VEBT_VEBT_uint32] :
      ( aa_set_real_o(finite_finite_real,S6)
     => ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,T6)
       => ( ! [A3: real] :
              ( aa_set_real_o(member_real(A3),minus_minus_set_real(S2,S6))
             => ( aa_VEBT_VEBT_real(I,aa_real_VEBT_VEBT(J,A3)) = A3 ) )
         => ( ! [A3: real] :
                ( aa_set_real_o(member_real(A3),minus_minus_set_real(S2,S6))
               => aa_set_VEBT_VEBT_o(member_VEBT_VEBT(aa_real_VEBT_VEBT(J,A3)),minus_5127226145743854075T_VEBT(T4,T6)) )
           => ( ! [B2: vEBT_VEBT] :
                  ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(B2),minus_5127226145743854075T_VEBT(T4,T6))
                 => ( aa_real_VEBT_VEBT(J,aa_VEBT_VEBT_real(I,B2)) = B2 ) )
             => ( ! [B2: vEBT_VEBT] :
                    ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(B2),minus_5127226145743854075T_VEBT(T4,T6))
                   => aa_set_real_o(member_real(aa_VEBT_VEBT_real(I,B2)),minus_minus_set_real(S2,S6)) )
               => ( ! [A3: real] :
                      ( aa_set_real_o(member_real(A3),S6)
                     => ( aa_real_uint32(G,A3) = zero_zero_uint32 ) )
                 => ( ! [B2: vEBT_VEBT] :
                        ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(B2),T6)
                       => ( aa_VEBT_VEBT_uint32(H,B2) = zero_zero_uint32 ) )
                   => ( ! [A3: real] :
                          ( aa_set_real_o(member_real(A3),S2)
                         => ( aa_VEBT_VEBT_uint32(H,aa_real_VEBT_VEBT(J,A3)) = aa_real_uint32(G,A3) ) )
                     => ( groups5944083974425963860uint32(G,S2) = groups8325533452322294502uint32(H,T4) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_7792_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S6: set_real,T6: set_real,S2: set_real,I: fun_real_real,J: fun_real_real,T4: set_real,G: fun_real_uint32,H: fun_real_uint32] :
      ( aa_set_real_o(finite_finite_real,S6)
     => ( aa_set_real_o(finite_finite_real,T6)
       => ( ! [A3: real] :
              ( aa_set_real_o(member_real(A3),minus_minus_set_real(S2,S6))
             => ( aa_real_real(I,aa_real_real(J,A3)) = A3 ) )
         => ( ! [A3: real] :
                ( aa_set_real_o(member_real(A3),minus_minus_set_real(S2,S6))
               => aa_set_real_o(member_real(aa_real_real(J,A3)),minus_minus_set_real(T4,T6)) )
           => ( ! [B2: real] :
                  ( aa_set_real_o(member_real(B2),minus_minus_set_real(T4,T6))
                 => ( aa_real_real(J,aa_real_real(I,B2)) = B2 ) )
             => ( ! [B2: real] :
                    ( aa_set_real_o(member_real(B2),minus_minus_set_real(T4,T6))
                   => aa_set_real_o(member_real(aa_real_real(I,B2)),minus_minus_set_real(S2,S6)) )
               => ( ! [A3: real] :
                      ( aa_set_real_o(member_real(A3),S6)
                     => ( aa_real_uint32(G,A3) = zero_zero_uint32 ) )
                 => ( ! [B2: real] :
                        ( aa_set_real_o(member_real(B2),T6)
                       => ( aa_real_uint32(H,B2) = zero_zero_uint32 ) )
                   => ( ! [A3: real] :
                          ( aa_set_real_o(member_real(A3),S2)
                         => ( aa_real_uint32(H,aa_real_real(J,A3)) = aa_real_uint32(G,A3) ) )
                     => ( groups5944083974425963860uint32(G,S2) = groups5944083974425963860uint32(H,T4) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_7793_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S6: set_VEBT_VEBT,T6: set_int,S2: set_VEBT_VEBT,I: fun_int_VEBT_VEBT,J: fun_VEBT_VEBT_int,T4: set_int,G: fun_VEBT_VEBT_uint32,H: fun_int_uint32] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,S6)
     => ( aa_set_int_o(finite_finite_int,T6)
       => ( ! [A3: vEBT_VEBT] :
              ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A3),minus_5127226145743854075T_VEBT(S2,S6))
             => ( aa_int_VEBT_VEBT(I,aa_VEBT_VEBT_int(J,A3)) = A3 ) )
         => ( ! [A3: vEBT_VEBT] :
                ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A3),minus_5127226145743854075T_VEBT(S2,S6))
               => aa_set_int_o(member_int(aa_VEBT_VEBT_int(J,A3)),minus_minus_set_int(T4,T6)) )
           => ( ! [B2: int] :
                  ( aa_set_int_o(member_int(B2),minus_minus_set_int(T4,T6))
                 => ( aa_VEBT_VEBT_int(J,aa_int_VEBT_VEBT(I,B2)) = B2 ) )
             => ( ! [B2: int] :
                    ( aa_set_int_o(member_int(B2),minus_minus_set_int(T4,T6))
                   => aa_set_VEBT_VEBT_o(member_VEBT_VEBT(aa_int_VEBT_VEBT(I,B2)),minus_5127226145743854075T_VEBT(S2,S6)) )
               => ( ! [A3: vEBT_VEBT] :
                      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A3),S6)
                     => ( aa_VEBT_VEBT_uint32(G,A3) = zero_zero_uint32 ) )
                 => ( ! [B2: int] :
                        ( aa_set_int_o(member_int(B2),T6)
                       => ( aa_int_uint32(H,B2) = zero_zero_uint32 ) )
                   => ( ! [A3: vEBT_VEBT] :
                          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A3),S2)
                         => ( aa_int_uint32(H,aa_VEBT_VEBT_int(J,A3)) = aa_VEBT_VEBT_uint32(G,A3) ) )
                     => ( groups8325533452322294502uint32(G,S2) = groups5712668689793887828uint32(H,T4) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_7794_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S6: set_real,T6: set_int,S2: set_real,I: fun_int_real,J: fun_real_int,T4: set_int,G: fun_real_uint32,H: fun_int_uint32] :
      ( aa_set_real_o(finite_finite_real,S6)
     => ( aa_set_int_o(finite_finite_int,T6)
       => ( ! [A3: real] :
              ( aa_set_real_o(member_real(A3),minus_minus_set_real(S2,S6))
             => ( aa_int_real(I,aa_real_int(J,A3)) = A3 ) )
         => ( ! [A3: real] :
                ( aa_set_real_o(member_real(A3),minus_minus_set_real(S2,S6))
               => aa_set_int_o(member_int(aa_real_int(J,A3)),minus_minus_set_int(T4,T6)) )
           => ( ! [B2: int] :
                  ( aa_set_int_o(member_int(B2),minus_minus_set_int(T4,T6))
                 => ( aa_real_int(J,aa_int_real(I,B2)) = B2 ) )
             => ( ! [B2: int] :
                    ( aa_set_int_o(member_int(B2),minus_minus_set_int(T4,T6))
                   => aa_set_real_o(member_real(aa_int_real(I,B2)),minus_minus_set_real(S2,S6)) )
               => ( ! [A3: real] :
                      ( aa_set_real_o(member_real(A3),S6)
                     => ( aa_real_uint32(G,A3) = zero_zero_uint32 ) )
                 => ( ! [B2: int] :
                        ( aa_set_int_o(member_int(B2),T6)
                       => ( aa_int_uint32(H,B2) = zero_zero_uint32 ) )
                   => ( ! [A3: real] :
                          ( aa_set_real_o(member_real(A3),S2)
                         => ( aa_int_uint32(H,aa_real_int(J,A3)) = aa_real_uint32(G,A3) ) )
                     => ( groups5944083974425963860uint32(G,S2) = groups5712668689793887828uint32(H,T4) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_7795_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S6: set_VEBT_VEBT,T6: set_complex,S2: set_VEBT_VEBT,I: fun_co4340422964966740447T_VEBT,J: fun_VE5569070535936029429omplex,T4: set_complex,G: fun_VEBT_VEBT_uint32,H: fun_complex_uint32] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,S6)
     => ( aa_set_complex_o(finite3207457112153483333omplex,T6)
       => ( ! [A3: vEBT_VEBT] :
              ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A3),minus_5127226145743854075T_VEBT(S2,S6))
             => ( aa_complex_VEBT_VEBT(I,aa_VEBT_VEBT_complex(J,A3)) = A3 ) )
         => ( ! [A3: vEBT_VEBT] :
                ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A3),minus_5127226145743854075T_VEBT(S2,S6))
               => aa_set_complex_o(member_complex(aa_VEBT_VEBT_complex(J,A3)),minus_811609699411566653omplex(T4,T6)) )
           => ( ! [B2: complex] :
                  ( aa_set_complex_o(member_complex(B2),minus_811609699411566653omplex(T4,T6))
                 => ( aa_VEBT_VEBT_complex(J,aa_complex_VEBT_VEBT(I,B2)) = B2 ) )
             => ( ! [B2: complex] :
                    ( aa_set_complex_o(member_complex(B2),minus_811609699411566653omplex(T4,T6))
                   => aa_set_VEBT_VEBT_o(member_VEBT_VEBT(aa_complex_VEBT_VEBT(I,B2)),minus_5127226145743854075T_VEBT(S2,S6)) )
               => ( ! [A3: vEBT_VEBT] :
                      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A3),S6)
                     => ( aa_VEBT_VEBT_uint32(G,A3) = zero_zero_uint32 ) )
                 => ( ! [B2: complex] :
                        ( aa_set_complex_o(member_complex(B2),T6)
                       => ( aa_complex_uint32(H,B2) = zero_zero_uint32 ) )
                   => ( ! [A3: vEBT_VEBT] :
                          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A3),S2)
                         => ( aa_complex_uint32(H,aa_VEBT_VEBT_complex(J,A3)) = aa_VEBT_VEBT_uint32(G,A3) ) )
                     => ( groups8325533452322294502uint32(G,S2) = groups8736914816313324502uint32(H,T4) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_7796_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S6: set_real,T6: set_complex,S2: set_real,I: fun_complex_real,J: fun_real_complex,T4: set_complex,G: fun_real_uint32,H: fun_complex_uint32] :
      ( aa_set_real_o(finite_finite_real,S6)
     => ( aa_set_complex_o(finite3207457112153483333omplex,T6)
       => ( ! [A3: real] :
              ( aa_set_real_o(member_real(A3),minus_minus_set_real(S2,S6))
             => ( aa_complex_real(I,aa_real_complex(J,A3)) = A3 ) )
         => ( ! [A3: real] :
                ( aa_set_real_o(member_real(A3),minus_minus_set_real(S2,S6))
               => aa_set_complex_o(member_complex(aa_real_complex(J,A3)),minus_811609699411566653omplex(T4,T6)) )
           => ( ! [B2: complex] :
                  ( aa_set_complex_o(member_complex(B2),minus_811609699411566653omplex(T4,T6))
                 => ( aa_real_complex(J,aa_complex_real(I,B2)) = B2 ) )
             => ( ! [B2: complex] :
                    ( aa_set_complex_o(member_complex(B2),minus_811609699411566653omplex(T4,T6))
                   => aa_set_real_o(member_real(aa_complex_real(I,B2)),minus_minus_set_real(S2,S6)) )
               => ( ! [A3: real] :
                      ( aa_set_real_o(member_real(A3),S6)
                     => ( aa_real_uint32(G,A3) = zero_zero_uint32 ) )
                 => ( ! [B2: complex] :
                        ( aa_set_complex_o(member_complex(B2),T6)
                       => ( aa_complex_uint32(H,B2) = zero_zero_uint32 ) )
                   => ( ! [A3: real] :
                          ( aa_set_real_o(member_real(A3),S2)
                         => ( aa_complex_uint32(H,aa_real_complex(J,A3)) = aa_real_uint32(G,A3) ) )
                     => ( groups5944083974425963860uint32(G,S2) = groups8736914816313324502uint32(H,T4) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_7797_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S6: set_VEBT_VEBT,T6: set_Code_integer,S2: set_VEBT_VEBT,I: fun_Co3766629010929971184T_VEBT,J: fun_VE9219535225822561060nteger,T4: set_Code_integer,G: fun_VEBT_VEBT_uint32,H: fun_Co1554742298434871494uint32] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,S6)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,T6)
       => ( ! [A3: vEBT_VEBT] :
              ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A3),minus_5127226145743854075T_VEBT(S2,S6))
             => ( aa_Cod6507154402108480601T_VEBT(I,aa_VEB5935388676523804373nteger(J,A3)) = A3 ) )
         => ( ! [A3: vEBT_VEBT] :
                ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A3),minus_5127226145743854075T_VEBT(S2,S6))
               => aa_set5643387332196575111eger_o(member_Code_integer(aa_VEB5935388676523804373nteger(J,A3)),minus_2355218937544613996nteger(T4,T6)) )
           => ( ! [B2: code_integer] :
                  ( aa_set5643387332196575111eger_o(member_Code_integer(B2),minus_2355218937544613996nteger(T4,T6))
                 => ( aa_VEB5935388676523804373nteger(J,aa_Cod6507154402108480601T_VEBT(I,B2)) = B2 ) )
             => ( ! [B2: code_integer] :
                    ( aa_set5643387332196575111eger_o(member_Code_integer(B2),minus_2355218937544613996nteger(T4,T6))
                   => aa_set_VEBT_VEBT_o(member_VEBT_VEBT(aa_Cod6507154402108480601T_VEBT(I,B2)),minus_5127226145743854075T_VEBT(S2,S6)) )
               => ( ! [A3: vEBT_VEBT] :
                      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A3),S6)
                     => ( aa_VEBT_VEBT_uint32(G,A3) = zero_zero_uint32 ) )
                 => ( ! [B2: code_integer] :
                        ( aa_set5643387332196575111eger_o(member_Code_integer(B2),T6)
                       => ( aa_Cod8401701657498852871uint32(H,B2) = zero_zero_uint32 ) )
                   => ( ! [A3: vEBT_VEBT] :
                          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A3),S2)
                         => ( aa_Cod8401701657498852871uint32(H,aa_VEB5935388676523804373nteger(J,A3)) = aa_VEBT_VEBT_uint32(G,A3) ) )
                     => ( groups8325533452322294502uint32(G,S2) = groups8847630953604152069uint32(H,T4) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_7798_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S6: set_real,T6: set_Code_integer,S2: set_real,I: fun_Co3960896333048504454r_real,J: fun_re6978948838062957830nteger,T4: set_Code_integer,G: fun_real_uint32,H: fun_Co1554742298434871494uint32] :
      ( aa_set_real_o(finite_finite_real,S6)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,T6)
       => ( ! [A3: real] :
              ( aa_set_real_o(member_real(A3),minus_minus_set_real(S2,S6))
             => ( aa_Code_integer_real(I,aa_real_Code_integer(J,A3)) = A3 ) )
         => ( ! [A3: real] :
                ( aa_set_real_o(member_real(A3),minus_minus_set_real(S2,S6))
               => aa_set5643387332196575111eger_o(member_Code_integer(aa_real_Code_integer(J,A3)),minus_2355218937544613996nteger(T4,T6)) )
           => ( ! [B2: code_integer] :
                  ( aa_set5643387332196575111eger_o(member_Code_integer(B2),minus_2355218937544613996nteger(T4,T6))
                 => ( aa_real_Code_integer(J,aa_Code_integer_real(I,B2)) = B2 ) )
             => ( ! [B2: code_integer] :
                    ( aa_set5643387332196575111eger_o(member_Code_integer(B2),minus_2355218937544613996nteger(T4,T6))
                   => aa_set_real_o(member_real(aa_Code_integer_real(I,B2)),minus_minus_set_real(S2,S6)) )
               => ( ! [A3: real] :
                      ( aa_set_real_o(member_real(A3),S6)
                     => ( aa_real_uint32(G,A3) = zero_zero_uint32 ) )
                 => ( ! [B2: code_integer] :
                        ( aa_set5643387332196575111eger_o(member_Code_integer(B2),T6)
                       => ( aa_Cod8401701657498852871uint32(H,B2) = zero_zero_uint32 ) )
                   => ( ! [A3: real] :
                          ( aa_set_real_o(member_real(A3),S2)
                         => ( aa_Cod8401701657498852871uint32(H,aa_real_Code_integer(J,A3)) = aa_real_uint32(G,A3) ) )
                     => ( groups5944083974425963860uint32(G,S2) = groups8847630953604152069uint32(H,T4) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_7799_sum__nonneg__0,axiom,
    ! [S3: set_VEBT_VEBT,F: fun_VEBT_VEBT_real,I: vEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,S3)
     => ( ! [I3: vEBT_VEBT] :
            ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(I3),S3)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_VEBT_VEBT_real(F,I3)) )
       => ( ( groups2240296850493347238T_real(F,S3) = zero_zero_real )
         => ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(I),S3)
           => ( aa_VEBT_VEBT_real(F,I) = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_7800_sum__nonneg__0,axiom,
    ! [S3: set_real,F: fun_real_real,I: real] :
      ( aa_set_real_o(finite_finite_real,S3)
     => ( ! [I3: real] :
            ( aa_set_real_o(member_real(I3),S3)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(F,I3)) )
       => ( ( groups8097168146408367636l_real(F,S3) = zero_zero_real )
         => ( aa_set_real_o(member_real(I),S3)
           => ( aa_real_real(F,I) = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_7801_sum__nonneg__0,axiom,
    ! [S3: set_int,F: fun_int_real,I: int] :
      ( aa_set_int_o(finite_finite_int,S3)
     => ( ! [I3: int] :
            ( aa_set_int_o(member_int(I3),S3)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_int_real(F,I3)) )
       => ( ( groups8778361861064173332t_real(F,S3) = zero_zero_real )
         => ( aa_set_int_o(member_int(I),S3)
           => ( aa_int_real(F,I) = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_7802_sum__nonneg__0,axiom,
    ! [S3: set_complex,F: fun_complex_real,I: complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S3)
     => ( ! [I3: complex] :
            ( aa_set_complex_o(member_complex(I3),S3)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(F,I3)) )
       => ( ( groups5808333547571424918x_real(F,S3) = zero_zero_real )
         => ( aa_set_complex_o(member_complex(I),S3)
           => ( aa_complex_real(F,I) = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_7803_sum__nonneg__0,axiom,
    ! [S3: set_Code_integer,F: fun_Co3960896333048504454r_real,I: code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,S3)
     => ( ! [I3: code_integer] :
            ( aa_set5643387332196575111eger_o(member_Code_integer(I3),S3)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_Code_integer_real(F,I3)) )
       => ( ( groups1270011288395367621r_real(F,S3) = zero_zero_real )
         => ( aa_set5643387332196575111eger_o(member_Code_integer(I),S3)
           => ( aa_Code_integer_real(F,I) = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_7804_sum__nonneg__0,axiom,
    ! [S3: set_VEBT_VEBT,F: fun_VEBT_VEBT_rat,I: vEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,S3)
     => ( ! [I3: vEBT_VEBT] :
            ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(I3),S3)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_VEBT_VEBT_rat(F,I3)) )
       => ( ( groups136491112297645522BT_rat(F,S3) = zero_zero_rat )
         => ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(I),S3)
           => ( aa_VEBT_VEBT_rat(F,I) = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_7805_sum__nonneg__0,axiom,
    ! [S3: set_real,F: fun_real_rat,I: real] :
      ( aa_set_real_o(finite_finite_real,S3)
     => ( ! [I3: real] :
            ( aa_set_real_o(member_real(I3),S3)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_real_rat(F,I3)) )
       => ( ( groups1300246762558778688al_rat(F,S3) = zero_zero_rat )
         => ( aa_set_real_o(member_real(I),S3)
           => ( aa_real_rat(F,I) = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_7806_sum__nonneg__0,axiom,
    ! [S3: set_nat,F: fun_nat_rat,I: nat] :
      ( aa_set_nat_o(finite_finite_nat,S3)
     => ( ! [I3: nat] :
            ( aa_set_nat_o(member_nat(I3),S3)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(F,I3)) )
       => ( ( groups2906978787729119204at_rat(F,S3) = zero_zero_rat )
         => ( aa_set_nat_o(member_nat(I),S3)
           => ( aa_nat_rat(F,I) = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_7807_sum__nonneg__0,axiom,
    ! [S3: set_int,F: fun_int_rat,I: int] :
      ( aa_set_int_o(finite_finite_int,S3)
     => ( ! [I3: int] :
            ( aa_set_int_o(member_int(I3),S3)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_int_rat(F,I3)) )
       => ( ( groups3906332499630173760nt_rat(F,S3) = zero_zero_rat )
         => ( aa_set_int_o(member_int(I),S3)
           => ( aa_int_rat(F,I) = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_7808_sum__nonneg__0,axiom,
    ! [S3: set_complex,F: fun_complex_rat,I: complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S3)
     => ( ! [I3: complex] :
            ( aa_set_complex_o(member_complex(I3),S3)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(F,I3)) )
       => ( ( groups5058264527183730370ex_rat(F,S3) = zero_zero_rat )
         => ( aa_set_complex_o(member_complex(I),S3)
           => ( aa_complex_rat(F,I) = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_7809_sum__nonneg__leq__bound,axiom,
    ! [S3: set_VEBT_VEBT,F: fun_VEBT_VEBT_real,B4: real,I: vEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,S3)
     => ( ! [I3: vEBT_VEBT] :
            ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(I3),S3)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_VEBT_VEBT_real(F,I3)) )
       => ( ( groups2240296850493347238T_real(F,S3) = B4 )
         => ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(I),S3)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_VEBT_VEBT_real(F,I)),B4) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_7810_sum__nonneg__leq__bound,axiom,
    ! [S3: set_real,F: fun_real_real,B4: real,I: real] :
      ( aa_set_real_o(finite_finite_real,S3)
     => ( ! [I3: real] :
            ( aa_set_real_o(member_real(I3),S3)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(F,I3)) )
       => ( ( groups8097168146408367636l_real(F,S3) = B4 )
         => ( aa_set_real_o(member_real(I),S3)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(F,I)),B4) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_7811_sum__nonneg__leq__bound,axiom,
    ! [S3: set_int,F: fun_int_real,B4: real,I: int] :
      ( aa_set_int_o(finite_finite_int,S3)
     => ( ! [I3: int] :
            ( aa_set_int_o(member_int(I3),S3)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_int_real(F,I3)) )
       => ( ( groups8778361861064173332t_real(F,S3) = B4 )
         => ( aa_set_int_o(member_int(I),S3)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(F,I)),B4) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_7812_sum__nonneg__leq__bound,axiom,
    ! [S3: set_complex,F: fun_complex_real,B4: real,I: complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S3)
     => ( ! [I3: complex] :
            ( aa_set_complex_o(member_complex(I3),S3)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(F,I3)) )
       => ( ( groups5808333547571424918x_real(F,S3) = B4 )
         => ( aa_set_complex_o(member_complex(I),S3)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(F,I)),B4) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_7813_sum__nonneg__leq__bound,axiom,
    ! [S3: set_Code_integer,F: fun_Co3960896333048504454r_real,B4: real,I: code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,S3)
     => ( ! [I3: code_integer] :
            ( aa_set5643387332196575111eger_o(member_Code_integer(I3),S3)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_Code_integer_real(F,I3)) )
       => ( ( groups1270011288395367621r_real(F,S3) = B4 )
         => ( aa_set5643387332196575111eger_o(member_Code_integer(I),S3)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_Code_integer_real(F,I)),B4) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_7814_sum__nonneg__leq__bound,axiom,
    ! [S3: set_VEBT_VEBT,F: fun_VEBT_VEBT_rat,B4: rat,I: vEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,S3)
     => ( ! [I3: vEBT_VEBT] :
            ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(I3),S3)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_VEBT_VEBT_rat(F,I3)) )
       => ( ( groups136491112297645522BT_rat(F,S3) = B4 )
         => ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(I),S3)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_VEBT_VEBT_rat(F,I)),B4) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_7815_sum__nonneg__leq__bound,axiom,
    ! [S3: set_real,F: fun_real_rat,B4: rat,I: real] :
      ( aa_set_real_o(finite_finite_real,S3)
     => ( ! [I3: real] :
            ( aa_set_real_o(member_real(I3),S3)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_real_rat(F,I3)) )
       => ( ( groups1300246762558778688al_rat(F,S3) = B4 )
         => ( aa_set_real_o(member_real(I),S3)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_real_rat(F,I)),B4) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_7816_sum__nonneg__leq__bound,axiom,
    ! [S3: set_nat,F: fun_nat_rat,B4: rat,I: nat] :
      ( aa_set_nat_o(finite_finite_nat,S3)
     => ( ! [I3: nat] :
            ( aa_set_nat_o(member_nat(I3),S3)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(F,I3)) )
       => ( ( groups2906978787729119204at_rat(F,S3) = B4 )
         => ( aa_set_nat_o(member_nat(I),S3)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,I)),B4) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_7817_sum__nonneg__leq__bound,axiom,
    ! [S3: set_int,F: fun_int_rat,B4: rat,I: int] :
      ( aa_set_int_o(finite_finite_int,S3)
     => ( ! [I3: int] :
            ( aa_set_int_o(member_int(I3),S3)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_int_rat(F,I3)) )
       => ( ( groups3906332499630173760nt_rat(F,S3) = B4 )
         => ( aa_set_int_o(member_int(I),S3)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(F,I)),B4) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_7818_sum__nonneg__leq__bound,axiom,
    ! [S3: set_complex,F: fun_complex_rat,B4: rat,I: complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S3)
     => ( ! [I3: complex] :
            ( aa_set_complex_o(member_complex(I3),S3)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(F,I3)) )
       => ( ( groups5058264527183730370ex_rat(F,S3) = B4 )
         => ( aa_set_complex_o(member_complex(I),S3)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_complex_rat(F,I)),B4) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_7819_sum_Osetdiff__irrelevant,axiom,
    ! [A2: set_int,G: fun_int_uint32] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( groups5712668689793887828uint32(G,minus_minus_set_int(A2,collect_int(aTP_Lamm_any(G)))) = groups5712668689793887828uint32(G,A2) ) ) ).

% sum.setdiff_irrelevant
tff(fact_7820_sum_Osetdiff__irrelevant,axiom,
    ! [A2: set_complex,G: fun_complex_uint32] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( groups8736914816313324502uint32(G,minus_811609699411566653omplex(A2,collect_complex(aTP_Lamm_anz(G)))) = groups8736914816313324502uint32(G,A2) ) ) ).

% sum.setdiff_irrelevant
tff(fact_7821_sum_Osetdiff__irrelevant,axiom,
    ! [A2: set_Code_integer,G: fun_Co1554742298434871494uint32] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A2)
     => ( groups8847630953604152069uint32(G,minus_2355218937544613996nteger(A2,collect_Code_integer(aTP_Lamm_aoa(G)))) = groups8847630953604152069uint32(G,A2) ) ) ).

% sum.setdiff_irrelevant
tff(fact_7822_sum_Osetdiff__irrelevant,axiom,
    ! [A2: set_int,G: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( groups8778361861064173332t_real(G,minus_minus_set_int(A2,collect_int(aTP_Lamm_aob(G)))) = groups8778361861064173332t_real(G,A2) ) ) ).

% sum.setdiff_irrelevant
tff(fact_7823_sum_Osetdiff__irrelevant,axiom,
    ! [A2: set_complex,G: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( groups5808333547571424918x_real(G,minus_811609699411566653omplex(A2,collect_complex(aTP_Lamm_aoc(G)))) = groups5808333547571424918x_real(G,A2) ) ) ).

% sum.setdiff_irrelevant
tff(fact_7824_sum_Osetdiff__irrelevant,axiom,
    ! [A2: set_Code_integer,G: fun_Co3960896333048504454r_real] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A2)
     => ( groups1270011288395367621r_real(G,minus_2355218937544613996nteger(A2,collect_Code_integer(aTP_Lamm_aod(G)))) = groups1270011288395367621r_real(G,A2) ) ) ).

% sum.setdiff_irrelevant
tff(fact_7825_sum_Osetdiff__irrelevant,axiom,
    ! [A2: set_int,G: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( groups3906332499630173760nt_rat(G,minus_minus_set_int(A2,collect_int(aTP_Lamm_aoe(G)))) = groups3906332499630173760nt_rat(G,A2) ) ) ).

% sum.setdiff_irrelevant
tff(fact_7826_sum_Osetdiff__irrelevant,axiom,
    ! [A2: set_complex,G: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( groups5058264527183730370ex_rat(G,minus_811609699411566653omplex(A2,collect_complex(aTP_Lamm_aof(G)))) = groups5058264527183730370ex_rat(G,A2) ) ) ).

% sum.setdiff_irrelevant
tff(fact_7827_sum_Osetdiff__irrelevant,axiom,
    ! [A2: set_Code_integer,G: fun_Code_integer_rat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A2)
     => ( groups6602215022474089585er_rat(G,minus_2355218937544613996nteger(A2,collect_Code_integer(aTP_Lamm_aog(G)))) = groups6602215022474089585er_rat(G,A2) ) ) ).

% sum.setdiff_irrelevant
tff(fact_7828_sum_Osetdiff__irrelevant,axiom,
    ! [A2: set_int,G: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( groups4541462559716669496nt_nat(G,minus_minus_set_int(A2,collect_int(aTP_Lamm_aoh(G)))) = groups4541462559716669496nt_nat(G,A2) ) ) ).

% sum.setdiff_irrelevant
tff(fact_7829_sum__power__add,axiom,
    ! [Xb: uint32,M: nat,I5: set_nat] : groups833757482993574392uint32(aa_nat6935575117487981255uint32(aTP_Lamm_aoi(Xb),M),I5) = aa_uint32_uint32(times_times_uint32(aa_nat_uint32(power_power_uint32(Xb),M)),groups833757482993574392uint32(power_power_uint32(Xb),I5)) ).

% sum_power_add
tff(fact_7830_sum__power__add,axiom,
    ! [Xb: word_N3645301735248828278l_num1,M: nat,I5: set_nat] : groups2996710295995929986l_num1(aa_nat6251203789436639057l_num1(aTP_Lamm_aoj(Xb),M),I5) = aa_wor7274019993978516266l_num1(times_7065122842183080059l_num1(aa_nat5599461682457136004l_num1(power_2184487114949457152l_num1(Xb),M)),groups2996710295995929986l_num1(power_2184487114949457152l_num1(Xb),I5)) ).

% sum_power_add
tff(fact_7831_sum__power__add,axiom,
    ! [Xb: rat,M: nat,I5: set_nat] : groups2906978787729119204at_rat(aa_nat_fun_nat_rat(aTP_Lamm_aok(Xb),M),I5) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(Xb),M)),groups2906978787729119204at_rat(power_power_rat(Xb),I5)) ).

% sum_power_add
tff(fact_7832_sum__power__add,axiom,
    ! [Xb: int,M: nat,I5: set_nat] : groups3539618377306564664at_int(aa_nat_fun_nat_int(aTP_Lamm_aol(Xb),M),I5) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,Xb),M)),groups3539618377306564664at_int(aa_int_fun_nat_int(power_power_int,Xb),I5)) ).

% sum_power_add
tff(fact_7833_sum__power__add,axiom,
    ! [Xb: real,M: nat,I5: set_nat] : groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_aom(Xb),M),I5) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),M)),groups6591440286371151544t_real(aa_real_fun_nat_real(power_power_real,Xb),I5)) ).

% sum_power_add
tff(fact_7834_sum_OatLeastAtMost__rev,axiom,
    ! [G: fun_nat_nat,Nb: nat,M: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),set_or1269000886237332187st_nat(Nb,M)) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_aon(G),Nb),M)),set_or1269000886237332187st_nat(Nb,M)) ).

% sum.atLeastAtMost_rev
tff(fact_7835_sum_OatLeastAtMost__rev,axiom,
    ! [G: fun_nat_real,Nb: nat,M: nat] : groups6591440286371151544t_real(G,set_or1269000886237332187st_nat(Nb,M)) = groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_nat4085805313368488148t_real(aTP_Lamm_aoo(G),Nb),M),set_or1269000886237332187st_nat(Nb,M)) ).

% sum.atLeastAtMost_rev
tff(fact_7836_sum__pos2,axiom,
    ! [I5: set_VEBT_VEBT,I: vEBT_VEBT,F: fun_VEBT_VEBT_real] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,I5)
     => ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(I),I5)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_VEBT_VEBT_real(F,I))
         => ( ! [I3: vEBT_VEBT] :
                ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(I3),I5)
               => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_VEBT_VEBT_real(F,I3)) )
           => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),groups2240296850493347238T_real(F,I5)) ) ) ) ) ).

% sum_pos2
tff(fact_7837_sum__pos2,axiom,
    ! [I5: set_real,I: real,F: fun_real_real] :
      ( aa_set_real_o(finite_finite_real,I5)
     => ( aa_set_real_o(member_real(I),I5)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(F,I))
         => ( ! [I3: real] :
                ( aa_set_real_o(member_real(I3),I5)
               => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(F,I3)) )
           => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),groups8097168146408367636l_real(F,I5)) ) ) ) ) ).

% sum_pos2
tff(fact_7838_sum__pos2,axiom,
    ! [I5: set_int,I: int,F: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,I5)
     => ( aa_set_int_o(member_int(I),I5)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_int_real(F,I))
         => ( ! [I3: int] :
                ( aa_set_int_o(member_int(I3),I5)
               => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_int_real(F,I3)) )
           => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),groups8778361861064173332t_real(F,I5)) ) ) ) ) ).

% sum_pos2
tff(fact_7839_sum__pos2,axiom,
    ! [I5: set_complex,I: complex,F: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,I5)
     => ( aa_set_complex_o(member_complex(I),I5)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_complex_real(F,I))
         => ( ! [I3: complex] :
                ( aa_set_complex_o(member_complex(I3),I5)
               => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(F,I3)) )
           => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),groups5808333547571424918x_real(F,I5)) ) ) ) ) ).

% sum_pos2
tff(fact_7840_sum__pos2,axiom,
    ! [I5: set_Code_integer,I: code_integer,F: fun_Co3960896333048504454r_real] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,I5)
     => ( aa_set5643387332196575111eger_o(member_Code_integer(I),I5)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_Code_integer_real(F,I))
         => ( ! [I3: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(I3),I5)
               => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_Code_integer_real(F,I3)) )
           => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),groups1270011288395367621r_real(F,I5)) ) ) ) ) ).

% sum_pos2
tff(fact_7841_sum__pos2,axiom,
    ! [I5: set_VEBT_VEBT,I: vEBT_VEBT,F: fun_VEBT_VEBT_rat] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,I5)
     => ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(I),I5)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_VEBT_VEBT_rat(F,I))
         => ( ! [I3: vEBT_VEBT] :
                ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(I3),I5)
               => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_VEBT_VEBT_rat(F,I3)) )
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),groups136491112297645522BT_rat(F,I5)) ) ) ) ) ).

% sum_pos2
tff(fact_7842_sum__pos2,axiom,
    ! [I5: set_real,I: real,F: fun_real_rat] :
      ( aa_set_real_o(finite_finite_real,I5)
     => ( aa_set_real_o(member_real(I),I5)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_real_rat(F,I))
         => ( ! [I3: real] :
                ( aa_set_real_o(member_real(I3),I5)
               => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_real_rat(F,I3)) )
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),groups1300246762558778688al_rat(F,I5)) ) ) ) ) ).

% sum_pos2
tff(fact_7843_sum__pos2,axiom,
    ! [I5: set_nat,I: nat,F: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,I5)
     => ( aa_set_nat_o(member_nat(I),I5)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_nat_rat(F,I))
         => ( ! [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),groups2906978787729119204at_rat(F,I5)) ) ) ) ) ).

% sum_pos2
tff(fact_7844_sum__pos2,axiom,
    ! [I5: set_int,I: int,F: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,I5)
     => ( aa_set_int_o(member_int(I),I5)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_int_rat(F,I))
         => ( ! [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_7845_sum__pos2,axiom,
    ! [I5: set_complex,I: complex,F: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,I5)
     => ( aa_set_complex_o(member_complex(I),I5)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_complex_rat(F,I))
         => ( ! [I3: complex] :
                ( aa_set_complex_o(member_complex(I3),I5)
               => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(F,I3)) )
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),groups5058264527183730370ex_rat(F,I5)) ) ) ) ) ).

% sum_pos2
tff(fact_7846_sum__pos,axiom,
    ! [I5: set_VEBT_VEBT,F: fun_VEBT_VEBT_real] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,I5)
     => ( ( I5 != bot_bo8194388402131092736T_VEBT )
       => ( ! [I3: vEBT_VEBT] :
              ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(I3),I5)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_VEBT_VEBT_real(F,I3)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),groups2240296850493347238T_real(F,I5)) ) ) ) ).

% sum_pos
tff(fact_7847_sum__pos,axiom,
    ! [I5: set_complex,F: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,I5)
     => ( ( I5 != bot_bot_set_complex )
       => ( ! [I3: complex] :
              ( aa_set_complex_o(member_complex(I3),I5)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_complex_real(F,I3)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),groups5808333547571424918x_real(F,I5)) ) ) ) ).

% sum_pos
tff(fact_7848_sum__pos,axiom,
    ! [I5: set_Code_integer,F: fun_Co3960896333048504454r_real] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,I5)
     => ( ( I5 != bot_bo3990330152332043303nteger )
       => ( ! [I3: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(I3),I5)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_Code_integer_real(F,I3)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),groups1270011288395367621r_real(F,I5)) ) ) ) ).

% sum_pos
tff(fact_7849_sum__pos,axiom,
    ! [I5: set_int,F: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,I5)
     => ( ( I5 != bot_bot_set_int )
       => ( ! [I3: int] :
              ( aa_set_int_o(member_int(I3),I5)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_int_real(F,I3)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),groups8778361861064173332t_real(F,I5)) ) ) ) ).

% sum_pos
tff(fact_7850_sum__pos,axiom,
    ! [I5: set_real,F: fun_real_real] :
      ( aa_set_real_o(finite_finite_real,I5)
     => ( ( I5 != bot_bot_set_real )
       => ( ! [I3: real] :
              ( aa_set_real_o(member_real(I3),I5)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(F,I3)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),groups8097168146408367636l_real(F,I5)) ) ) ) ).

% sum_pos
tff(fact_7851_sum__pos,axiom,
    ! [I5: set_VEBT_VEBT,F: fun_VEBT_VEBT_rat] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,I5)
     => ( ( I5 != bot_bo8194388402131092736T_VEBT )
       => ( ! [I3: vEBT_VEBT] :
              ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(I3),I5)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_VEBT_VEBT_rat(F,I3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),groups136491112297645522BT_rat(F,I5)) ) ) ) ).

% sum_pos
tff(fact_7852_sum__pos,axiom,
    ! [I5: set_complex,F: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,I5)
     => ( ( I5 != bot_bot_set_complex )
       => ( ! [I3: complex] :
              ( aa_set_complex_o(member_complex(I3),I5)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_complex_rat(F,I3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),groups5058264527183730370ex_rat(F,I5)) ) ) ) ).

% sum_pos
tff(fact_7853_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_7854_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),groups2906978787729119204at_rat(F,I5)) ) ) ) ).

% sum_pos
tff(fact_7855_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_7856_sum_Osame__carrier,axiom,
    ! [C4: set_VEBT_VEBT,A2: set_VEBT_VEBT,B4: set_VEBT_VEBT,G: fun_VEBT_VEBT_uint32,H: fun_VEBT_VEBT_uint32] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,C4)
     => ( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(A2),C4)
       => ( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(B4),C4)
         => ( ! [A3: vEBT_VEBT] :
                ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A3),minus_5127226145743854075T_VEBT(C4,A2))
               => ( aa_VEBT_VEBT_uint32(G,A3) = zero_zero_uint32 ) )
           => ( ! [B2: vEBT_VEBT] :
                  ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(B2),minus_5127226145743854075T_VEBT(C4,B4))
                 => ( aa_VEBT_VEBT_uint32(H,B2) = zero_zero_uint32 ) )
             => ( ( groups8325533452322294502uint32(G,A2) = groups8325533452322294502uint32(H,B4) )
              <=> ( groups8325533452322294502uint32(G,C4) = groups8325533452322294502uint32(H,C4) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_7857_sum_Osame__carrier,axiom,
    ! [C4: set_real,A2: set_real,B4: set_real,G: fun_real_uint32,H: fun_real_uint32] :
      ( aa_set_real_o(finite_finite_real,C4)
     => ( aa_set_real_o(ord_less_eq_set_real(A2),C4)
       => ( aa_set_real_o(ord_less_eq_set_real(B4),C4)
         => ( ! [A3: real] :
                ( aa_set_real_o(member_real(A3),minus_minus_set_real(C4,A2))
               => ( aa_real_uint32(G,A3) = zero_zero_uint32 ) )
           => ( ! [B2: real] :
                  ( aa_set_real_o(member_real(B2),minus_minus_set_real(C4,B4))
                 => ( aa_real_uint32(H,B2) = zero_zero_uint32 ) )
             => ( ( groups5944083974425963860uint32(G,A2) = groups5944083974425963860uint32(H,B4) )
              <=> ( groups5944083974425963860uint32(G,C4) = groups5944083974425963860uint32(H,C4) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_7858_sum_Osame__carrier,axiom,
    ! [C4: set_complex,A2: set_complex,B4: set_complex,G: fun_complex_uint32,H: fun_complex_uint32] :
      ( aa_set_complex_o(finite3207457112153483333omplex,C4)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(A2),C4)
       => ( aa_set_complex_o(ord_le211207098394363844omplex(B4),C4)
         => ( ! [A3: complex] :
                ( aa_set_complex_o(member_complex(A3),minus_811609699411566653omplex(C4,A2))
               => ( aa_complex_uint32(G,A3) = zero_zero_uint32 ) )
           => ( ! [B2: complex] :
                  ( aa_set_complex_o(member_complex(B2),minus_811609699411566653omplex(C4,B4))
                 => ( aa_complex_uint32(H,B2) = zero_zero_uint32 ) )
             => ( ( groups8736914816313324502uint32(G,A2) = groups8736914816313324502uint32(H,B4) )
              <=> ( groups8736914816313324502uint32(G,C4) = groups8736914816313324502uint32(H,C4) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_7859_sum_Osame__carrier,axiom,
    ! [C4: set_Code_integer,A2: set_Code_integer,B4: set_Code_integer,G: fun_Co1554742298434871494uint32,H: fun_Co1554742298434871494uint32] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,C4)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(A2),C4)
       => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(B4),C4)
         => ( ! [A3: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(A3),minus_2355218937544613996nteger(C4,A2))
               => ( aa_Cod8401701657498852871uint32(G,A3) = zero_zero_uint32 ) )
           => ( ! [B2: code_integer] :
                  ( aa_set5643387332196575111eger_o(member_Code_integer(B2),minus_2355218937544613996nteger(C4,B4))
                 => ( aa_Cod8401701657498852871uint32(H,B2) = zero_zero_uint32 ) )
             => ( ( groups8847630953604152069uint32(G,A2) = groups8847630953604152069uint32(H,B4) )
              <=> ( groups8847630953604152069uint32(G,C4) = groups8847630953604152069uint32(H,C4) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_7860_sum_Osame__carrier,axiom,
    ! [C4: set_VEBT_VEBT,A2: set_VEBT_VEBT,B4: set_VEBT_VEBT,G: fun_VEBT_VEBT_real,H: fun_VEBT_VEBT_real] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,C4)
     => ( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(A2),C4)
       => ( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(B4),C4)
         => ( ! [A3: vEBT_VEBT] :
                ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A3),minus_5127226145743854075T_VEBT(C4,A2))
               => ( aa_VEBT_VEBT_real(G,A3) = zero_zero_real ) )
           => ( ! [B2: vEBT_VEBT] :
                  ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(B2),minus_5127226145743854075T_VEBT(C4,B4))
                 => ( aa_VEBT_VEBT_real(H,B2) = zero_zero_real ) )
             => ( ( groups2240296850493347238T_real(G,A2) = groups2240296850493347238T_real(H,B4) )
              <=> ( groups2240296850493347238T_real(G,C4) = groups2240296850493347238T_real(H,C4) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_7861_sum_Osame__carrier,axiom,
    ! [C4: set_real,A2: set_real,B4: set_real,G: fun_real_real,H: fun_real_real] :
      ( aa_set_real_o(finite_finite_real,C4)
     => ( aa_set_real_o(ord_less_eq_set_real(A2),C4)
       => ( aa_set_real_o(ord_less_eq_set_real(B4),C4)
         => ( ! [A3: real] :
                ( aa_set_real_o(member_real(A3),minus_minus_set_real(C4,A2))
               => ( aa_real_real(G,A3) = zero_zero_real ) )
           => ( ! [B2: real] :
                  ( aa_set_real_o(member_real(B2),minus_minus_set_real(C4,B4))
                 => ( aa_real_real(H,B2) = zero_zero_real ) )
             => ( ( groups8097168146408367636l_real(G,A2) = groups8097168146408367636l_real(H,B4) )
              <=> ( groups8097168146408367636l_real(G,C4) = groups8097168146408367636l_real(H,C4) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_7862_sum_Osame__carrier,axiom,
    ! [C4: set_complex,A2: set_complex,B4: set_complex,G: fun_complex_real,H: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,C4)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(A2),C4)
       => ( aa_set_complex_o(ord_le211207098394363844omplex(B4),C4)
         => ( ! [A3: complex] :
                ( aa_set_complex_o(member_complex(A3),minus_811609699411566653omplex(C4,A2))
               => ( aa_complex_real(G,A3) = zero_zero_real ) )
           => ( ! [B2: complex] :
                  ( aa_set_complex_o(member_complex(B2),minus_811609699411566653omplex(C4,B4))
                 => ( aa_complex_real(H,B2) = zero_zero_real ) )
             => ( ( groups5808333547571424918x_real(G,A2) = groups5808333547571424918x_real(H,B4) )
              <=> ( groups5808333547571424918x_real(G,C4) = groups5808333547571424918x_real(H,C4) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_7863_sum_Osame__carrier,axiom,
    ! [C4: set_Code_integer,A2: set_Code_integer,B4: set_Code_integer,G: fun_Co3960896333048504454r_real,H: fun_Co3960896333048504454r_real] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,C4)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(A2),C4)
       => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(B4),C4)
         => ( ! [A3: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(A3),minus_2355218937544613996nteger(C4,A2))
               => ( aa_Code_integer_real(G,A3) = zero_zero_real ) )
           => ( ! [B2: code_integer] :
                  ( aa_set5643387332196575111eger_o(member_Code_integer(B2),minus_2355218937544613996nteger(C4,B4))
                 => ( aa_Code_integer_real(H,B2) = zero_zero_real ) )
             => ( ( groups1270011288395367621r_real(G,A2) = groups1270011288395367621r_real(H,B4) )
              <=> ( groups1270011288395367621r_real(G,C4) = groups1270011288395367621r_real(H,C4) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_7864_sum_Osame__carrier,axiom,
    ! [C4: set_VEBT_VEBT,A2: set_VEBT_VEBT,B4: set_VEBT_VEBT,G: fun_VEBT_VEBT_rat,H: fun_VEBT_VEBT_rat] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,C4)
     => ( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(A2),C4)
       => ( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(B4),C4)
         => ( ! [A3: vEBT_VEBT] :
                ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A3),minus_5127226145743854075T_VEBT(C4,A2))
               => ( aa_VEBT_VEBT_rat(G,A3) = zero_zero_rat ) )
           => ( ! [B2: vEBT_VEBT] :
                  ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(B2),minus_5127226145743854075T_VEBT(C4,B4))
                 => ( aa_VEBT_VEBT_rat(H,B2) = zero_zero_rat ) )
             => ( ( groups136491112297645522BT_rat(G,A2) = groups136491112297645522BT_rat(H,B4) )
              <=> ( groups136491112297645522BT_rat(G,C4) = groups136491112297645522BT_rat(H,C4) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_7865_sum_Osame__carrier,axiom,
    ! [C4: set_real,A2: set_real,B4: set_real,G: fun_real_rat,H: fun_real_rat] :
      ( aa_set_real_o(finite_finite_real,C4)
     => ( aa_set_real_o(ord_less_eq_set_real(A2),C4)
       => ( aa_set_real_o(ord_less_eq_set_real(B4),C4)
         => ( ! [A3: real] :
                ( aa_set_real_o(member_real(A3),minus_minus_set_real(C4,A2))
               => ( aa_real_rat(G,A3) = zero_zero_rat ) )
           => ( ! [B2: real] :
                  ( aa_set_real_o(member_real(B2),minus_minus_set_real(C4,B4))
                 => ( aa_real_rat(H,B2) = zero_zero_rat ) )
             => ( ( groups1300246762558778688al_rat(G,A2) = groups1300246762558778688al_rat(H,B4) )
              <=> ( groups1300246762558778688al_rat(G,C4) = groups1300246762558778688al_rat(H,C4) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_7866_sum_Osame__carrierI,axiom,
    ! [C4: set_VEBT_VEBT,A2: set_VEBT_VEBT,B4: set_VEBT_VEBT,G: fun_VEBT_VEBT_uint32,H: fun_VEBT_VEBT_uint32] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,C4)
     => ( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(A2),C4)
       => ( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(B4),C4)
         => ( ! [A3: vEBT_VEBT] :
                ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A3),minus_5127226145743854075T_VEBT(C4,A2))
               => ( aa_VEBT_VEBT_uint32(G,A3) = zero_zero_uint32 ) )
           => ( ! [B2: vEBT_VEBT] :
                  ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(B2),minus_5127226145743854075T_VEBT(C4,B4))
                 => ( aa_VEBT_VEBT_uint32(H,B2) = zero_zero_uint32 ) )
             => ( ( groups8325533452322294502uint32(G,C4) = groups8325533452322294502uint32(H,C4) )
               => ( groups8325533452322294502uint32(G,A2) = groups8325533452322294502uint32(H,B4) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_7867_sum_Osame__carrierI,axiom,
    ! [C4: set_real,A2: set_real,B4: set_real,G: fun_real_uint32,H: fun_real_uint32] :
      ( aa_set_real_o(finite_finite_real,C4)
     => ( aa_set_real_o(ord_less_eq_set_real(A2),C4)
       => ( aa_set_real_o(ord_less_eq_set_real(B4),C4)
         => ( ! [A3: real] :
                ( aa_set_real_o(member_real(A3),minus_minus_set_real(C4,A2))
               => ( aa_real_uint32(G,A3) = zero_zero_uint32 ) )
           => ( ! [B2: real] :
                  ( aa_set_real_o(member_real(B2),minus_minus_set_real(C4,B4))
                 => ( aa_real_uint32(H,B2) = zero_zero_uint32 ) )
             => ( ( groups5944083974425963860uint32(G,C4) = groups5944083974425963860uint32(H,C4) )
               => ( groups5944083974425963860uint32(G,A2) = groups5944083974425963860uint32(H,B4) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_7868_sum_Osame__carrierI,axiom,
    ! [C4: set_complex,A2: set_complex,B4: set_complex,G: fun_complex_uint32,H: fun_complex_uint32] :
      ( aa_set_complex_o(finite3207457112153483333omplex,C4)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(A2),C4)
       => ( aa_set_complex_o(ord_le211207098394363844omplex(B4),C4)
         => ( ! [A3: complex] :
                ( aa_set_complex_o(member_complex(A3),minus_811609699411566653omplex(C4,A2))
               => ( aa_complex_uint32(G,A3) = zero_zero_uint32 ) )
           => ( ! [B2: complex] :
                  ( aa_set_complex_o(member_complex(B2),minus_811609699411566653omplex(C4,B4))
                 => ( aa_complex_uint32(H,B2) = zero_zero_uint32 ) )
             => ( ( groups8736914816313324502uint32(G,C4) = groups8736914816313324502uint32(H,C4) )
               => ( groups8736914816313324502uint32(G,A2) = groups8736914816313324502uint32(H,B4) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_7869_sum_Osame__carrierI,axiom,
    ! [C4: set_Code_integer,A2: set_Code_integer,B4: set_Code_integer,G: fun_Co1554742298434871494uint32,H: fun_Co1554742298434871494uint32] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,C4)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(A2),C4)
       => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(B4),C4)
         => ( ! [A3: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(A3),minus_2355218937544613996nteger(C4,A2))
               => ( aa_Cod8401701657498852871uint32(G,A3) = zero_zero_uint32 ) )
           => ( ! [B2: code_integer] :
                  ( aa_set5643387332196575111eger_o(member_Code_integer(B2),minus_2355218937544613996nteger(C4,B4))
                 => ( aa_Cod8401701657498852871uint32(H,B2) = zero_zero_uint32 ) )
             => ( ( groups8847630953604152069uint32(G,C4) = groups8847630953604152069uint32(H,C4) )
               => ( groups8847630953604152069uint32(G,A2) = groups8847630953604152069uint32(H,B4) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_7870_sum_Osame__carrierI,axiom,
    ! [C4: set_VEBT_VEBT,A2: set_VEBT_VEBT,B4: set_VEBT_VEBT,G: fun_VEBT_VEBT_real,H: fun_VEBT_VEBT_real] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,C4)
     => ( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(A2),C4)
       => ( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(B4),C4)
         => ( ! [A3: vEBT_VEBT] :
                ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A3),minus_5127226145743854075T_VEBT(C4,A2))
               => ( aa_VEBT_VEBT_real(G,A3) = zero_zero_real ) )
           => ( ! [B2: vEBT_VEBT] :
                  ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(B2),minus_5127226145743854075T_VEBT(C4,B4))
                 => ( aa_VEBT_VEBT_real(H,B2) = zero_zero_real ) )
             => ( ( groups2240296850493347238T_real(G,C4) = groups2240296850493347238T_real(H,C4) )
               => ( groups2240296850493347238T_real(G,A2) = groups2240296850493347238T_real(H,B4) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_7871_sum_Osame__carrierI,axiom,
    ! [C4: set_real,A2: set_real,B4: set_real,G: fun_real_real,H: fun_real_real] :
      ( aa_set_real_o(finite_finite_real,C4)
     => ( aa_set_real_o(ord_less_eq_set_real(A2),C4)
       => ( aa_set_real_o(ord_less_eq_set_real(B4),C4)
         => ( ! [A3: real] :
                ( aa_set_real_o(member_real(A3),minus_minus_set_real(C4,A2))
               => ( aa_real_real(G,A3) = zero_zero_real ) )
           => ( ! [B2: real] :
                  ( aa_set_real_o(member_real(B2),minus_minus_set_real(C4,B4))
                 => ( aa_real_real(H,B2) = zero_zero_real ) )
             => ( ( groups8097168146408367636l_real(G,C4) = groups8097168146408367636l_real(H,C4) )
               => ( groups8097168146408367636l_real(G,A2) = groups8097168146408367636l_real(H,B4) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_7872_sum_Osame__carrierI,axiom,
    ! [C4: set_complex,A2: set_complex,B4: set_complex,G: fun_complex_real,H: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,C4)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(A2),C4)
       => ( aa_set_complex_o(ord_le211207098394363844omplex(B4),C4)
         => ( ! [A3: complex] :
                ( aa_set_complex_o(member_complex(A3),minus_811609699411566653omplex(C4,A2))
               => ( aa_complex_real(G,A3) = zero_zero_real ) )
           => ( ! [B2: complex] :
                  ( aa_set_complex_o(member_complex(B2),minus_811609699411566653omplex(C4,B4))
                 => ( aa_complex_real(H,B2) = zero_zero_real ) )
             => ( ( groups5808333547571424918x_real(G,C4) = groups5808333547571424918x_real(H,C4) )
               => ( groups5808333547571424918x_real(G,A2) = groups5808333547571424918x_real(H,B4) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_7873_sum_Osame__carrierI,axiom,
    ! [C4: set_Code_integer,A2: set_Code_integer,B4: set_Code_integer,G: fun_Co3960896333048504454r_real,H: fun_Co3960896333048504454r_real] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,C4)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(A2),C4)
       => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(B4),C4)
         => ( ! [A3: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(A3),minus_2355218937544613996nteger(C4,A2))
               => ( aa_Code_integer_real(G,A3) = zero_zero_real ) )
           => ( ! [B2: code_integer] :
                  ( aa_set5643387332196575111eger_o(member_Code_integer(B2),minus_2355218937544613996nteger(C4,B4))
                 => ( aa_Code_integer_real(H,B2) = zero_zero_real ) )
             => ( ( groups1270011288395367621r_real(G,C4) = groups1270011288395367621r_real(H,C4) )
               => ( groups1270011288395367621r_real(G,A2) = groups1270011288395367621r_real(H,B4) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_7874_sum_Osame__carrierI,axiom,
    ! [C4: set_VEBT_VEBT,A2: set_VEBT_VEBT,B4: set_VEBT_VEBT,G: fun_VEBT_VEBT_rat,H: fun_VEBT_VEBT_rat] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,C4)
     => ( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(A2),C4)
       => ( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(B4),C4)
         => ( ! [A3: vEBT_VEBT] :
                ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A3),minus_5127226145743854075T_VEBT(C4,A2))
               => ( aa_VEBT_VEBT_rat(G,A3) = zero_zero_rat ) )
           => ( ! [B2: vEBT_VEBT] :
                  ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(B2),minus_5127226145743854075T_VEBT(C4,B4))
                 => ( aa_VEBT_VEBT_rat(H,B2) = zero_zero_rat ) )
             => ( ( groups136491112297645522BT_rat(G,C4) = groups136491112297645522BT_rat(H,C4) )
               => ( groups136491112297645522BT_rat(G,A2) = groups136491112297645522BT_rat(H,B4) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_7875_sum_Osame__carrierI,axiom,
    ! [C4: set_real,A2: set_real,B4: set_real,G: fun_real_rat,H: fun_real_rat] :
      ( aa_set_real_o(finite_finite_real,C4)
     => ( aa_set_real_o(ord_less_eq_set_real(A2),C4)
       => ( aa_set_real_o(ord_less_eq_set_real(B4),C4)
         => ( ! [A3: real] :
                ( aa_set_real_o(member_real(A3),minus_minus_set_real(C4,A2))
               => ( aa_real_rat(G,A3) = zero_zero_rat ) )
           => ( ! [B2: real] :
                  ( aa_set_real_o(member_real(B2),minus_minus_set_real(C4,B4))
                 => ( aa_real_rat(H,B2) = zero_zero_rat ) )
             => ( ( groups1300246762558778688al_rat(G,C4) = groups1300246762558778688al_rat(H,C4) )
               => ( groups1300246762558778688al_rat(G,A2) = groups1300246762558778688al_rat(H,B4) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_7876_sum_Omono__neutral__left,axiom,
    ! [T4: set_complex,S2: set_complex,G: fun_complex_uint32] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T4)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S2),T4)
       => ( ! [X2: complex] :
              ( aa_set_complex_o(member_complex(X2),minus_811609699411566653omplex(T4,S2))
             => ( aa_complex_uint32(G,X2) = zero_zero_uint32 ) )
         => ( groups8736914816313324502uint32(G,S2) = groups8736914816313324502uint32(G,T4) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_7877_sum_Omono__neutral__left,axiom,
    ! [T4: set_Code_integer,S2: set_Code_integer,G: fun_Co1554742298434871494uint32] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,T4)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(S2),T4)
       => ( ! [X2: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X2),minus_2355218937544613996nteger(T4,S2))
             => ( aa_Cod8401701657498852871uint32(G,X2) = zero_zero_uint32 ) )
         => ( groups8847630953604152069uint32(G,S2) = groups8847630953604152069uint32(G,T4) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_7878_sum_Omono__neutral__left,axiom,
    ! [T4: set_complex,S2: set_complex,G: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T4)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S2),T4)
       => ( ! [X2: complex] :
              ( aa_set_complex_o(member_complex(X2),minus_811609699411566653omplex(T4,S2))
             => ( aa_complex_real(G,X2) = zero_zero_real ) )
         => ( groups5808333547571424918x_real(G,S2) = groups5808333547571424918x_real(G,T4) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_7879_sum_Omono__neutral__left,axiom,
    ! [T4: set_Code_integer,S2: set_Code_integer,G: fun_Co3960896333048504454r_real] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,T4)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(S2),T4)
       => ( ! [X2: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X2),minus_2355218937544613996nteger(T4,S2))
             => ( aa_Code_integer_real(G,X2) = zero_zero_real ) )
         => ( groups1270011288395367621r_real(G,S2) = groups1270011288395367621r_real(G,T4) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_7880_sum_Omono__neutral__left,axiom,
    ! [T4: set_complex,S2: set_complex,G: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T4)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S2),T4)
       => ( ! [X2: complex] :
              ( aa_set_complex_o(member_complex(X2),minus_811609699411566653omplex(T4,S2))
             => ( aa_complex_rat(G,X2) = zero_zero_rat ) )
         => ( groups5058264527183730370ex_rat(G,S2) = groups5058264527183730370ex_rat(G,T4) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_7881_sum_Omono__neutral__left,axiom,
    ! [T4: set_Code_integer,S2: set_Code_integer,G: fun_Code_integer_rat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,T4)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(S2),T4)
       => ( ! [X2: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X2),minus_2355218937544613996nteger(T4,S2))
             => ( aa_Code_integer_rat(G,X2) = zero_zero_rat ) )
         => ( groups6602215022474089585er_rat(G,S2) = groups6602215022474089585er_rat(G,T4) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_7882_sum_Omono__neutral__left,axiom,
    ! [T4: set_complex,S2: set_complex,G: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T4)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S2),T4)
       => ( ! [X2: complex] :
              ( aa_set_complex_o(member_complex(X2),minus_811609699411566653omplex(T4,S2))
             => ( aa_complex_nat(G,X2) = zero_zero_nat ) )
         => ( groups5693394587270226106ex_nat(G,S2) = groups5693394587270226106ex_nat(G,T4) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_7883_sum_Omono__neutral__left,axiom,
    ! [T4: set_Code_integer,S2: set_Code_integer,G: fun_Code_integer_nat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,T4)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(S2),T4)
       => ( ! [X2: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X2),minus_2355218937544613996nteger(T4,S2))
             => ( aa_Code_integer_nat(G,X2) = zero_zero_nat ) )
         => ( groups7237345082560585321er_nat(G,S2) = groups7237345082560585321er_nat(G,T4) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_7884_sum_Omono__neutral__left,axiom,
    ! [T4: set_complex,S2: set_complex,G: fun_complex_int] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T4)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S2),T4)
       => ( ! [X2: complex] :
              ( aa_set_complex_o(member_complex(X2),minus_811609699411566653omplex(T4,S2))
             => ( aa_complex_int(G,X2) = zero_zero_int ) )
         => ( groups5690904116761175830ex_int(G,S2) = groups5690904116761175830ex_int(G,T4) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_7885_sum_Omono__neutral__left,axiom,
    ! [T4: set_Code_integer,S2: set_Code_integer,G: fun_Code_integer_int] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,T4)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(S2),T4)
       => ( ! [X2: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X2),minus_2355218937544613996nteger(T4,S2))
             => ( aa_Code_integer_int(G,X2) = zero_zero_int ) )
         => ( groups7234854612051535045er_int(G,S2) = groups7234854612051535045er_int(G,T4) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_7886_sum_Omono__neutral__right,axiom,
    ! [T4: set_complex,S2: set_complex,G: fun_complex_uint32] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T4)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S2),T4)
       => ( ! [X2: complex] :
              ( aa_set_complex_o(member_complex(X2),minus_811609699411566653omplex(T4,S2))
             => ( aa_complex_uint32(G,X2) = zero_zero_uint32 ) )
         => ( groups8736914816313324502uint32(G,T4) = groups8736914816313324502uint32(G,S2) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_7887_sum_Omono__neutral__right,axiom,
    ! [T4: set_Code_integer,S2: set_Code_integer,G: fun_Co1554742298434871494uint32] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,T4)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(S2),T4)
       => ( ! [X2: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X2),minus_2355218937544613996nteger(T4,S2))
             => ( aa_Cod8401701657498852871uint32(G,X2) = zero_zero_uint32 ) )
         => ( groups8847630953604152069uint32(G,T4) = groups8847630953604152069uint32(G,S2) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_7888_sum_Omono__neutral__right,axiom,
    ! [T4: set_complex,S2: set_complex,G: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T4)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S2),T4)
       => ( ! [X2: complex] :
              ( aa_set_complex_o(member_complex(X2),minus_811609699411566653omplex(T4,S2))
             => ( aa_complex_real(G,X2) = zero_zero_real ) )
         => ( groups5808333547571424918x_real(G,T4) = groups5808333547571424918x_real(G,S2) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_7889_sum_Omono__neutral__right,axiom,
    ! [T4: set_Code_integer,S2: set_Code_integer,G: fun_Co3960896333048504454r_real] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,T4)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(S2),T4)
       => ( ! [X2: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X2),minus_2355218937544613996nteger(T4,S2))
             => ( aa_Code_integer_real(G,X2) = zero_zero_real ) )
         => ( groups1270011288395367621r_real(G,T4) = groups1270011288395367621r_real(G,S2) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_7890_sum_Omono__neutral__right,axiom,
    ! [T4: set_complex,S2: set_complex,G: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T4)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S2),T4)
       => ( ! [X2: complex] :
              ( aa_set_complex_o(member_complex(X2),minus_811609699411566653omplex(T4,S2))
             => ( aa_complex_rat(G,X2) = zero_zero_rat ) )
         => ( groups5058264527183730370ex_rat(G,T4) = groups5058264527183730370ex_rat(G,S2) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_7891_sum_Omono__neutral__right,axiom,
    ! [T4: set_Code_integer,S2: set_Code_integer,G: fun_Code_integer_rat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,T4)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(S2),T4)
       => ( ! [X2: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X2),minus_2355218937544613996nteger(T4,S2))
             => ( aa_Code_integer_rat(G,X2) = zero_zero_rat ) )
         => ( groups6602215022474089585er_rat(G,T4) = groups6602215022474089585er_rat(G,S2) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_7892_sum_Omono__neutral__right,axiom,
    ! [T4: set_complex,S2: set_complex,G: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T4)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S2),T4)
       => ( ! [X2: complex] :
              ( aa_set_complex_o(member_complex(X2),minus_811609699411566653omplex(T4,S2))
             => ( aa_complex_nat(G,X2) = zero_zero_nat ) )
         => ( groups5693394587270226106ex_nat(G,T4) = groups5693394587270226106ex_nat(G,S2) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_7893_sum_Omono__neutral__right,axiom,
    ! [T4: set_Code_integer,S2: set_Code_integer,G: fun_Code_integer_nat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,T4)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(S2),T4)
       => ( ! [X2: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X2),minus_2355218937544613996nteger(T4,S2))
             => ( aa_Code_integer_nat(G,X2) = zero_zero_nat ) )
         => ( groups7237345082560585321er_nat(G,T4) = groups7237345082560585321er_nat(G,S2) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_7894_sum_Omono__neutral__right,axiom,
    ! [T4: set_complex,S2: set_complex,G: fun_complex_int] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T4)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S2),T4)
       => ( ! [X2: complex] :
              ( aa_set_complex_o(member_complex(X2),minus_811609699411566653omplex(T4,S2))
             => ( aa_complex_int(G,X2) = zero_zero_int ) )
         => ( groups5690904116761175830ex_int(G,T4) = groups5690904116761175830ex_int(G,S2) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_7895_sum_Omono__neutral__right,axiom,
    ! [T4: set_Code_integer,S2: set_Code_integer,G: fun_Code_integer_int] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,T4)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(S2),T4)
       => ( ! [X2: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X2),minus_2355218937544613996nteger(T4,S2))
             => ( aa_Code_integer_int(G,X2) = zero_zero_int ) )
         => ( groups7234854612051535045er_int(G,T4) = groups7234854612051535045er_int(G,S2) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_7896_sum_Omono__neutral__cong__left,axiom,
    ! [T4: set_VEBT_VEBT,S2: set_VEBT_VEBT,H: fun_VEBT_VEBT_uint32,G: fun_VEBT_VEBT_uint32] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,T4)
     => ( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(S2),T4)
       => ( ! [X2: vEBT_VEBT] :
              ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X2),minus_5127226145743854075T_VEBT(T4,S2))
             => ( aa_VEBT_VEBT_uint32(H,X2) = zero_zero_uint32 ) )
         => ( ! [X2: vEBT_VEBT] :
                ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X2),S2)
               => ( aa_VEBT_VEBT_uint32(G,X2) = aa_VEBT_VEBT_uint32(H,X2) ) )
           => ( groups8325533452322294502uint32(G,S2) = groups8325533452322294502uint32(H,T4) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_7897_sum_Omono__neutral__cong__left,axiom,
    ! [T4: set_real,S2: set_real,H: fun_real_uint32,G: fun_real_uint32] :
      ( aa_set_real_o(finite_finite_real,T4)
     => ( aa_set_real_o(ord_less_eq_set_real(S2),T4)
       => ( ! [X2: real] :
              ( aa_set_real_o(member_real(X2),minus_minus_set_real(T4,S2))
             => ( aa_real_uint32(H,X2) = zero_zero_uint32 ) )
         => ( ! [X2: real] :
                ( aa_set_real_o(member_real(X2),S2)
               => ( aa_real_uint32(G,X2) = aa_real_uint32(H,X2) ) )
           => ( groups5944083974425963860uint32(G,S2) = groups5944083974425963860uint32(H,T4) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_7898_sum_Omono__neutral__cong__left,axiom,
    ! [T4: set_complex,S2: set_complex,H: fun_complex_uint32,G: fun_complex_uint32] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T4)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S2),T4)
       => ( ! [X2: complex] :
              ( aa_set_complex_o(member_complex(X2),minus_811609699411566653omplex(T4,S2))
             => ( aa_complex_uint32(H,X2) = zero_zero_uint32 ) )
         => ( ! [X2: complex] :
                ( aa_set_complex_o(member_complex(X2),S2)
               => ( aa_complex_uint32(G,X2) = aa_complex_uint32(H,X2) ) )
           => ( groups8736914816313324502uint32(G,S2) = groups8736914816313324502uint32(H,T4) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_7899_sum_Omono__neutral__cong__left,axiom,
    ! [T4: set_Code_integer,S2: set_Code_integer,H: fun_Co1554742298434871494uint32,G: fun_Co1554742298434871494uint32] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,T4)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(S2),T4)
       => ( ! [X2: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X2),minus_2355218937544613996nteger(T4,S2))
             => ( aa_Cod8401701657498852871uint32(H,X2) = zero_zero_uint32 ) )
         => ( ! [X2: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(X2),S2)
               => ( aa_Cod8401701657498852871uint32(G,X2) = aa_Cod8401701657498852871uint32(H,X2) ) )
           => ( groups8847630953604152069uint32(G,S2) = groups8847630953604152069uint32(H,T4) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_7900_sum_Omono__neutral__cong__left,axiom,
    ! [T4: set_VEBT_VEBT,S2: set_VEBT_VEBT,H: fun_VEBT_VEBT_real,G: fun_VEBT_VEBT_real] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,T4)
     => ( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(S2),T4)
       => ( ! [X2: vEBT_VEBT] :
              ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X2),minus_5127226145743854075T_VEBT(T4,S2))
             => ( aa_VEBT_VEBT_real(H,X2) = zero_zero_real ) )
         => ( ! [X2: vEBT_VEBT] :
                ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X2),S2)
               => ( aa_VEBT_VEBT_real(G,X2) = aa_VEBT_VEBT_real(H,X2) ) )
           => ( groups2240296850493347238T_real(G,S2) = groups2240296850493347238T_real(H,T4) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_7901_sum_Omono__neutral__cong__left,axiom,
    ! [T4: set_real,S2: set_real,H: fun_real_real,G: fun_real_real] :
      ( aa_set_real_o(finite_finite_real,T4)
     => ( aa_set_real_o(ord_less_eq_set_real(S2),T4)
       => ( ! [X2: real] :
              ( aa_set_real_o(member_real(X2),minus_minus_set_real(T4,S2))
             => ( aa_real_real(H,X2) = zero_zero_real ) )
         => ( ! [X2: real] :
                ( aa_set_real_o(member_real(X2),S2)
               => ( aa_real_real(G,X2) = aa_real_real(H,X2) ) )
           => ( groups8097168146408367636l_real(G,S2) = groups8097168146408367636l_real(H,T4) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_7902_sum_Omono__neutral__cong__left,axiom,
    ! [T4: set_complex,S2: set_complex,H: fun_complex_real,G: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T4)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S2),T4)
       => ( ! [X2: complex] :
              ( aa_set_complex_o(member_complex(X2),minus_811609699411566653omplex(T4,S2))
             => ( aa_complex_real(H,X2) = zero_zero_real ) )
         => ( ! [X2: complex] :
                ( aa_set_complex_o(member_complex(X2),S2)
               => ( aa_complex_real(G,X2) = aa_complex_real(H,X2) ) )
           => ( groups5808333547571424918x_real(G,S2) = groups5808333547571424918x_real(H,T4) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_7903_sum_Omono__neutral__cong__left,axiom,
    ! [T4: set_Code_integer,S2: set_Code_integer,H: fun_Co3960896333048504454r_real,G: fun_Co3960896333048504454r_real] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,T4)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(S2),T4)
       => ( ! [X2: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X2),minus_2355218937544613996nteger(T4,S2))
             => ( aa_Code_integer_real(H,X2) = zero_zero_real ) )
         => ( ! [X2: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(X2),S2)
               => ( aa_Code_integer_real(G,X2) = aa_Code_integer_real(H,X2) ) )
           => ( groups1270011288395367621r_real(G,S2) = groups1270011288395367621r_real(H,T4) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_7904_sum_Omono__neutral__cong__left,axiom,
    ! [T4: set_VEBT_VEBT,S2: set_VEBT_VEBT,H: fun_VEBT_VEBT_rat,G: fun_VEBT_VEBT_rat] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,T4)
     => ( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(S2),T4)
       => ( ! [X2: vEBT_VEBT] :
              ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X2),minus_5127226145743854075T_VEBT(T4,S2))
             => ( aa_VEBT_VEBT_rat(H,X2) = zero_zero_rat ) )
         => ( ! [X2: vEBT_VEBT] :
                ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X2),S2)
               => ( aa_VEBT_VEBT_rat(G,X2) = aa_VEBT_VEBT_rat(H,X2) ) )
           => ( groups136491112297645522BT_rat(G,S2) = groups136491112297645522BT_rat(H,T4) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_7905_sum_Omono__neutral__cong__left,axiom,
    ! [T4: set_real,S2: set_real,H: fun_real_rat,G: fun_real_rat] :
      ( aa_set_real_o(finite_finite_real,T4)
     => ( aa_set_real_o(ord_less_eq_set_real(S2),T4)
       => ( ! [X2: real] :
              ( aa_set_real_o(member_real(X2),minus_minus_set_real(T4,S2))
             => ( aa_real_rat(H,X2) = zero_zero_rat ) )
         => ( ! [X2: real] :
                ( aa_set_real_o(member_real(X2),S2)
               => ( aa_real_rat(G,X2) = aa_real_rat(H,X2) ) )
           => ( groups1300246762558778688al_rat(G,S2) = groups1300246762558778688al_rat(H,T4) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_7906_sum_Omono__neutral__cong__right,axiom,
    ! [T4: set_VEBT_VEBT,S2: set_VEBT_VEBT,G: fun_VEBT_VEBT_uint32,H: fun_VEBT_VEBT_uint32] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,T4)
     => ( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(S2),T4)
       => ( ! [X2: vEBT_VEBT] :
              ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X2),minus_5127226145743854075T_VEBT(T4,S2))
             => ( aa_VEBT_VEBT_uint32(G,X2) = zero_zero_uint32 ) )
         => ( ! [X2: vEBT_VEBT] :
                ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X2),S2)
               => ( aa_VEBT_VEBT_uint32(G,X2) = aa_VEBT_VEBT_uint32(H,X2) ) )
           => ( groups8325533452322294502uint32(G,T4) = groups8325533452322294502uint32(H,S2) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_7907_sum_Omono__neutral__cong__right,axiom,
    ! [T4: set_real,S2: set_real,G: fun_real_uint32,H: fun_real_uint32] :
      ( aa_set_real_o(finite_finite_real,T4)
     => ( aa_set_real_o(ord_less_eq_set_real(S2),T4)
       => ( ! [X2: real] :
              ( aa_set_real_o(member_real(X2),minus_minus_set_real(T4,S2))
             => ( aa_real_uint32(G,X2) = zero_zero_uint32 ) )
         => ( ! [X2: real] :
                ( aa_set_real_o(member_real(X2),S2)
               => ( aa_real_uint32(G,X2) = aa_real_uint32(H,X2) ) )
           => ( groups5944083974425963860uint32(G,T4) = groups5944083974425963860uint32(H,S2) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_7908_sum_Omono__neutral__cong__right,axiom,
    ! [T4: set_complex,S2: set_complex,G: fun_complex_uint32,H: fun_complex_uint32] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T4)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S2),T4)
       => ( ! [X2: complex] :
              ( aa_set_complex_o(member_complex(X2),minus_811609699411566653omplex(T4,S2))
             => ( aa_complex_uint32(G,X2) = zero_zero_uint32 ) )
         => ( ! [X2: complex] :
                ( aa_set_complex_o(member_complex(X2),S2)
               => ( aa_complex_uint32(G,X2) = aa_complex_uint32(H,X2) ) )
           => ( groups8736914816313324502uint32(G,T4) = groups8736914816313324502uint32(H,S2) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_7909_sum_Omono__neutral__cong__right,axiom,
    ! [T4: set_Code_integer,S2: set_Code_integer,G: fun_Co1554742298434871494uint32,H: fun_Co1554742298434871494uint32] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,T4)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(S2),T4)
       => ( ! [X2: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X2),minus_2355218937544613996nteger(T4,S2))
             => ( aa_Cod8401701657498852871uint32(G,X2) = zero_zero_uint32 ) )
         => ( ! [X2: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(X2),S2)
               => ( aa_Cod8401701657498852871uint32(G,X2) = aa_Cod8401701657498852871uint32(H,X2) ) )
           => ( groups8847630953604152069uint32(G,T4) = groups8847630953604152069uint32(H,S2) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_7910_sum_Omono__neutral__cong__right,axiom,
    ! [T4: set_VEBT_VEBT,S2: set_VEBT_VEBT,G: fun_VEBT_VEBT_real,H: fun_VEBT_VEBT_real] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,T4)
     => ( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(S2),T4)
       => ( ! [X2: vEBT_VEBT] :
              ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X2),minus_5127226145743854075T_VEBT(T4,S2))
             => ( aa_VEBT_VEBT_real(G,X2) = zero_zero_real ) )
         => ( ! [X2: vEBT_VEBT] :
                ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X2),S2)
               => ( aa_VEBT_VEBT_real(G,X2) = aa_VEBT_VEBT_real(H,X2) ) )
           => ( groups2240296850493347238T_real(G,T4) = groups2240296850493347238T_real(H,S2) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_7911_sum_Omono__neutral__cong__right,axiom,
    ! [T4: set_real,S2: set_real,G: fun_real_real,H: fun_real_real] :
      ( aa_set_real_o(finite_finite_real,T4)
     => ( aa_set_real_o(ord_less_eq_set_real(S2),T4)
       => ( ! [X2: real] :
              ( aa_set_real_o(member_real(X2),minus_minus_set_real(T4,S2))
             => ( aa_real_real(G,X2) = zero_zero_real ) )
         => ( ! [X2: real] :
                ( aa_set_real_o(member_real(X2),S2)
               => ( aa_real_real(G,X2) = aa_real_real(H,X2) ) )
           => ( groups8097168146408367636l_real(G,T4) = groups8097168146408367636l_real(H,S2) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_7912_sum_Omono__neutral__cong__right,axiom,
    ! [T4: set_complex,S2: set_complex,G: fun_complex_real,H: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T4)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S2),T4)
       => ( ! [X2: complex] :
              ( aa_set_complex_o(member_complex(X2),minus_811609699411566653omplex(T4,S2))
             => ( aa_complex_real(G,X2) = zero_zero_real ) )
         => ( ! [X2: complex] :
                ( aa_set_complex_o(member_complex(X2),S2)
               => ( aa_complex_real(G,X2) = aa_complex_real(H,X2) ) )
           => ( groups5808333547571424918x_real(G,T4) = groups5808333547571424918x_real(H,S2) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_7913_sum_Omono__neutral__cong__right,axiom,
    ! [T4: set_Code_integer,S2: set_Code_integer,G: fun_Co3960896333048504454r_real,H: fun_Co3960896333048504454r_real] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,T4)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(S2),T4)
       => ( ! [X2: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X2),minus_2355218937544613996nteger(T4,S2))
             => ( aa_Code_integer_real(G,X2) = zero_zero_real ) )
         => ( ! [X2: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(X2),S2)
               => ( aa_Code_integer_real(G,X2) = aa_Code_integer_real(H,X2) ) )
           => ( groups1270011288395367621r_real(G,T4) = groups1270011288395367621r_real(H,S2) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_7914_sum_Omono__neutral__cong__right,axiom,
    ! [T4: set_VEBT_VEBT,S2: set_VEBT_VEBT,G: fun_VEBT_VEBT_rat,H: fun_VEBT_VEBT_rat] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,T4)
     => ( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(S2),T4)
       => ( ! [X2: vEBT_VEBT] :
              ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X2),minus_5127226145743854075T_VEBT(T4,S2))
             => ( aa_VEBT_VEBT_rat(G,X2) = zero_zero_rat ) )
         => ( ! [X2: vEBT_VEBT] :
                ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X2),S2)
               => ( aa_VEBT_VEBT_rat(G,X2) = aa_VEBT_VEBT_rat(H,X2) ) )
           => ( groups136491112297645522BT_rat(G,T4) = groups136491112297645522BT_rat(H,S2) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_7915_sum_Omono__neutral__cong__right,axiom,
    ! [T4: set_real,S2: set_real,G: fun_real_rat,H: fun_real_rat] :
      ( aa_set_real_o(finite_finite_real,T4)
     => ( aa_set_real_o(ord_less_eq_set_real(S2),T4)
       => ( ! [X2: real] :
              ( aa_set_real_o(member_real(X2),minus_minus_set_real(T4,S2))
             => ( aa_real_rat(G,X2) = zero_zero_rat ) )
         => ( ! [X2: real] :
                ( aa_set_real_o(member_real(X2),S2)
               => ( aa_real_rat(G,X2) = aa_real_rat(H,X2) ) )
           => ( groups1300246762558778688al_rat(G,T4) = groups1300246762558778688al_rat(H,S2) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_7916_sum_Osubset__diff,axiom,
    ! [B4: set_complex,A2: set_complex,G: fun_complex_real] :
      ( aa_set_complex_o(ord_le211207098394363844omplex(B4),A2)
     => ( aa_set_complex_o(finite3207457112153483333omplex,A2)
       => ( groups5808333547571424918x_real(G,A2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups5808333547571424918x_real(G,minus_811609699411566653omplex(A2,B4))),groups5808333547571424918x_real(G,B4)) ) ) ) ).

% sum.subset_diff
tff(fact_7917_sum_Osubset__diff,axiom,
    ! [B4: set_Code_integer,A2: set_Code_integer,G: fun_Co3960896333048504454r_real] :
      ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(B4),A2)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A2)
       => ( groups1270011288395367621r_real(G,A2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups1270011288395367621r_real(G,minus_2355218937544613996nteger(A2,B4))),groups1270011288395367621r_real(G,B4)) ) ) ) ).

% sum.subset_diff
tff(fact_7918_sum_Osubset__diff,axiom,
    ! [B4: set_complex,A2: set_complex,G: fun_complex_rat] :
      ( aa_set_complex_o(ord_le211207098394363844omplex(B4),A2)
     => ( aa_set_complex_o(finite3207457112153483333omplex,A2)
       => ( groups5058264527183730370ex_rat(G,A2) = aa_rat_rat(plus_plus_rat(groups5058264527183730370ex_rat(G,minus_811609699411566653omplex(A2,B4))),groups5058264527183730370ex_rat(G,B4)) ) ) ) ).

% sum.subset_diff
tff(fact_7919_sum_Osubset__diff,axiom,
    ! [B4: set_Code_integer,A2: set_Code_integer,G: fun_Code_integer_rat] :
      ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(B4),A2)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A2)
       => ( groups6602215022474089585er_rat(G,A2) = aa_rat_rat(plus_plus_rat(groups6602215022474089585er_rat(G,minus_2355218937544613996nteger(A2,B4))),groups6602215022474089585er_rat(G,B4)) ) ) ) ).

% sum.subset_diff
tff(fact_7920_sum_Osubset__diff,axiom,
    ! [B4: set_complex,A2: set_complex,G: fun_complex_nat] :
      ( aa_set_complex_o(ord_le211207098394363844omplex(B4),A2)
     => ( aa_set_complex_o(finite3207457112153483333omplex,A2)
       => ( groups5693394587270226106ex_nat(G,A2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups5693394587270226106ex_nat(G,minus_811609699411566653omplex(A2,B4))),groups5693394587270226106ex_nat(G,B4)) ) ) ) ).

% sum.subset_diff
tff(fact_7921_sum_Osubset__diff,axiom,
    ! [B4: set_Code_integer,A2: set_Code_integer,G: fun_Code_integer_nat] :
      ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(B4),A2)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A2)
       => ( groups7237345082560585321er_nat(G,A2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups7237345082560585321er_nat(G,minus_2355218937544613996nteger(A2,B4))),groups7237345082560585321er_nat(G,B4)) ) ) ) ).

% sum.subset_diff
tff(fact_7922_sum_Osubset__diff,axiom,
    ! [B4: set_complex,A2: set_complex,G: fun_complex_int] :
      ( aa_set_complex_o(ord_le211207098394363844omplex(B4),A2)
     => ( aa_set_complex_o(finite3207457112153483333omplex,A2)
       => ( groups5690904116761175830ex_int(G,A2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups5690904116761175830ex_int(G,minus_811609699411566653omplex(A2,B4))),groups5690904116761175830ex_int(G,B4)) ) ) ) ).

% sum.subset_diff
tff(fact_7923_sum_Osubset__diff,axiom,
    ! [B4: set_Code_integer,A2: set_Code_integer,G: fun_Code_integer_int] :
      ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(B4),A2)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A2)
       => ( groups7234854612051535045er_int(G,A2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups7234854612051535045er_int(G,minus_2355218937544613996nteger(A2,B4))),groups7234854612051535045er_int(G,B4)) ) ) ) ).

% sum.subset_diff
tff(fact_7924_sum_Osubset__diff,axiom,
    ! [B4: set_nat,A2: set_nat,G: fun_nat_rat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(B4),A2)
     => ( aa_set_nat_o(finite_finite_nat,A2)
       => ( groups2906978787729119204at_rat(G,A2) = aa_rat_rat(plus_plus_rat(groups2906978787729119204at_rat(G,minus_minus_set_nat(A2,B4))),groups2906978787729119204at_rat(G,B4)) ) ) ) ).

% sum.subset_diff
tff(fact_7925_sum_Osubset__diff,axiom,
    ! [B4: set_nat,A2: set_nat,G: fun_nat_int] :
      ( aa_set_nat_o(ord_less_eq_set_nat(B4),A2)
     => ( aa_set_nat_o(finite_finite_nat,A2)
       => ( groups3539618377306564664at_int(G,A2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups3539618377306564664at_int(G,minus_minus_set_nat(A2,B4))),groups3539618377306564664at_int(G,B4)) ) ) ) ).

% sum.subset_diff
tff(fact_7926_sum__diff,axiom,
    ! [A2: set_complex,B4: set_complex,F: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(B4),A2)
       => ( groups5808333547571424918x_real(F,minus_811609699411566653omplex(A2,B4)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,groups5808333547571424918x_real(F,A2)),groups5808333547571424918x_real(F,B4)) ) ) ) ).

% sum_diff
tff(fact_7927_sum__diff,axiom,
    ! [A2: set_Code_integer,B4: set_Code_integer,F: fun_Co3960896333048504454r_real] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A2)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(B4),A2)
       => ( groups1270011288395367621r_real(F,minus_2355218937544613996nteger(A2,B4)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,groups1270011288395367621r_real(F,A2)),groups1270011288395367621r_real(F,B4)) ) ) ) ).

% sum_diff
tff(fact_7928_sum__diff,axiom,
    ! [A2: set_complex,B4: set_complex,F: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(B4),A2)
       => ( groups5058264527183730370ex_rat(F,minus_811609699411566653omplex(A2,B4)) = minus_minus_rat(groups5058264527183730370ex_rat(F,A2),groups5058264527183730370ex_rat(F,B4)) ) ) ) ).

% sum_diff
tff(fact_7929_sum__diff,axiom,
    ! [A2: set_Code_integer,B4: set_Code_integer,F: fun_Code_integer_rat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A2)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(B4),A2)
       => ( groups6602215022474089585er_rat(F,minus_2355218937544613996nteger(A2,B4)) = minus_minus_rat(groups6602215022474089585er_rat(F,A2),groups6602215022474089585er_rat(F,B4)) ) ) ) ).

% sum_diff
tff(fact_7930_sum__diff,axiom,
    ! [A2: set_complex,B4: set_complex,F: fun_complex_int] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(B4),A2)
       => ( groups5690904116761175830ex_int(F,minus_811609699411566653omplex(A2,B4)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,groups5690904116761175830ex_int(F,A2)),groups5690904116761175830ex_int(F,B4)) ) ) ) ).

% sum_diff
tff(fact_7931_sum__diff,axiom,
    ! [A2: set_Code_integer,B4: set_Code_integer,F: fun_Code_integer_int] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A2)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(B4),A2)
       => ( groups7234854612051535045er_int(F,minus_2355218937544613996nteger(A2,B4)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,groups7234854612051535045er_int(F,A2)),groups7234854612051535045er_int(F,B4)) ) ) ) ).

% sum_diff
tff(fact_7932_sum__diff,axiom,
    ! [A2: set_nat,B4: set_nat,F: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_nat_o(ord_less_eq_set_nat(B4),A2)
       => ( groups2906978787729119204at_rat(F,minus_minus_set_nat(A2,B4)) = minus_minus_rat(groups2906978787729119204at_rat(F,A2),groups2906978787729119204at_rat(F,B4)) ) ) ) ).

% sum_diff
tff(fact_7933_sum__diff,axiom,
    ! [A2: set_nat,B4: set_nat,F: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_nat_o(ord_less_eq_set_nat(B4),A2)
       => ( groups3539618377306564664at_int(F,minus_minus_set_nat(A2,B4)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,groups3539618377306564664at_int(F,A2)),groups3539618377306564664at_int(F,B4)) ) ) ) ).

% sum_diff
tff(fact_7934_sum__diff,axiom,
    ! [A2: set_int,B4: set_int,F: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( aa_set_int_o(ord_less_eq_set_int(B4),A2)
       => ( groups8778361861064173332t_real(F,minus_minus_set_int(A2,B4)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,groups8778361861064173332t_real(F,A2)),groups8778361861064173332t_real(F,B4)) ) ) ) ).

% sum_diff
tff(fact_7935_sum__diff,axiom,
    ! [A2: set_int,B4: set_int,F: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( aa_set_int_o(ord_less_eq_set_int(B4),A2)
       => ( groups3906332499630173760nt_rat(F,minus_minus_set_int(A2,B4)) = minus_minus_rat(groups3906332499630173760nt_rat(F,A2),groups3906332499630173760nt_rat(F,B4)) ) ) ) ).

% sum_diff
tff(fact_7936_powser__sums__zero,axiom,
    ! [A: fun_nat_real] : aa_real_o(sums_real(aTP_Lamm_ajv(A)),aa_nat_real(A,zero_zero_nat)) ).

% powser_sums_zero
tff(fact_7937_sum__shift__lb__Suc0__0,axiom,
    ! [F: fun_nat_uint32,K: nat] :
      ( ( aa_nat_uint32(F,zero_zero_nat) = zero_zero_uint32 )
     => ( groups833757482993574392uint32(F,set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat),K)) = groups833757482993574392uint32(F,set_or1269000886237332187st_nat(zero_zero_nat,K)) ) ) ).

% sum_shift_lb_Suc0_0
tff(fact_7938_sum__shift__lb__Suc0__0,axiom,
    ! [F: fun_nat_rat,K: nat] :
      ( ( aa_nat_rat(F,zero_zero_nat) = zero_zero_rat )
     => ( groups2906978787729119204at_rat(F,set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat),K)) = groups2906978787729119204at_rat(F,set_or1269000886237332187st_nat(zero_zero_nat,K)) ) ) ).

% sum_shift_lb_Suc0_0
tff(fact_7939_sum__shift__lb__Suc0__0,axiom,
    ! [F: fun_nat_int,K: nat] :
      ( ( aa_nat_int(F,zero_zero_nat) = zero_zero_int )
     => ( groups3539618377306564664at_int(F,set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat),K)) = groups3539618377306564664at_int(F,set_or1269000886237332187st_nat(zero_zero_nat,K)) ) ) ).

% sum_shift_lb_Suc0_0
tff(fact_7940_sum__shift__lb__Suc0__0,axiom,
    ! [F: fun_nat_nat,K: nat] :
      ( ( aa_nat_nat(F,zero_zero_nat) = zero_zero_nat )
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(F),set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat),K)) = aa_set_nat_nat(groups3542108847815614940at_nat(F),set_or1269000886237332187st_nat(zero_zero_nat,K)) ) ) ).

% sum_shift_lb_Suc0_0
tff(fact_7941_sum__shift__lb__Suc0__0,axiom,
    ! [F: fun_nat_real,K: nat] :
      ( ( aa_nat_real(F,zero_zero_nat) = zero_zero_real )
     => ( groups6591440286371151544t_real(F,set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat),K)) = groups6591440286371151544t_real(F,set_or1269000886237332187st_nat(zero_zero_nat,K)) ) ) ).

% sum_shift_lb_Suc0_0
tff(fact_7942_sum_OatLeast0__atMost__Suc,axiom,
    ! [G: fun_nat_rat,Nb: nat] : groups2906978787729119204at_rat(G,set_or1269000886237332187st_nat(zero_zero_nat,aa_nat_nat(suc,Nb))) = aa_rat_rat(plus_plus_rat(groups2906978787729119204at_rat(G,set_or1269000886237332187st_nat(zero_zero_nat,Nb))),aa_nat_rat(G,aa_nat_nat(suc,Nb))) ).

% sum.atLeast0_atMost_Suc
tff(fact_7943_sum_OatLeast0__atMost__Suc,axiom,
    ! [G: fun_nat_int,Nb: nat] : groups3539618377306564664at_int(G,set_or1269000886237332187st_nat(zero_zero_nat,aa_nat_nat(suc,Nb))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups3539618377306564664at_int(G,set_or1269000886237332187st_nat(zero_zero_nat,Nb))),aa_nat_int(G,aa_nat_nat(suc,Nb))) ).

% sum.atLeast0_atMost_Suc
tff(fact_7944_sum_OatLeast0__atMost__Suc,axiom,
    ! [G: fun_nat_nat,Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),set_or1269000886237332187st_nat(zero_zero_nat,aa_nat_nat(suc,Nb))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(G),set_or1269000886237332187st_nat(zero_zero_nat,Nb))),aa_nat_nat(G,aa_nat_nat(suc,Nb))) ).

% sum.atLeast0_atMost_Suc
tff(fact_7945_sum_OatLeast0__atMost__Suc,axiom,
    ! [G: fun_nat_real,Nb: nat] : groups6591440286371151544t_real(G,set_or1269000886237332187st_nat(zero_zero_nat,aa_nat_nat(suc,Nb))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(G,set_or1269000886237332187st_nat(zero_zero_nat,Nb))),aa_nat_real(G,aa_nat_nat(suc,Nb))) ).

% sum.atLeast0_atMost_Suc
tff(fact_7946_sum_Onat__ivl__Suc_H,axiom,
    ! [M: nat,Nb: nat,G: fun_nat_rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(suc,Nb))
     => ( groups2906978787729119204at_rat(G,set_or1269000886237332187st_nat(M,aa_nat_nat(suc,Nb))) = aa_rat_rat(plus_plus_rat(aa_nat_rat(G,aa_nat_nat(suc,Nb))),groups2906978787729119204at_rat(G,set_or1269000886237332187st_nat(M,Nb))) ) ) ).

% sum.nat_ivl_Suc'
tff(fact_7947_sum_Onat__ivl__Suc_H,axiom,
    ! [M: nat,Nb: nat,G: fun_nat_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(suc,Nb))
     => ( groups3539618377306564664at_int(G,set_or1269000886237332187st_nat(M,aa_nat_nat(suc,Nb))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(G,aa_nat_nat(suc,Nb))),groups3539618377306564664at_int(G,set_or1269000886237332187st_nat(M,Nb))) ) ) ).

% sum.nat_ivl_Suc'
tff(fact_7948_sum_Onat__ivl__Suc_H,axiom,
    ! [M: nat,Nb: nat,G: fun_nat_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(suc,Nb))
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(G),set_or1269000886237332187st_nat(M,aa_nat_nat(suc,Nb))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(G,aa_nat_nat(suc,Nb))),aa_set_nat_nat(groups3542108847815614940at_nat(G),set_or1269000886237332187st_nat(M,Nb))) ) ) ).

% sum.nat_ivl_Suc'
tff(fact_7949_sum_Onat__ivl__Suc_H,axiom,
    ! [M: nat,Nb: nat,G: fun_nat_real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(suc,Nb))
     => ( groups6591440286371151544t_real(G,set_or1269000886237332187st_nat(M,aa_nat_nat(suc,Nb))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(G,aa_nat_nat(suc,Nb))),groups6591440286371151544t_real(G,set_or1269000886237332187st_nat(M,Nb))) ) ) ).

% sum.nat_ivl_Suc'
tff(fact_7950_sum_OatLeast__Suc__atMost,axiom,
    ! [M: nat,Nb: nat,G: fun_nat_rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
     => ( groups2906978787729119204at_rat(G,set_or1269000886237332187st_nat(M,Nb)) = aa_rat_rat(plus_plus_rat(aa_nat_rat(G,M)),groups2906978787729119204at_rat(G,set_or1269000886237332187st_nat(aa_nat_nat(suc,M),Nb))) ) ) ).

% sum.atLeast_Suc_atMost
tff(fact_7951_sum_OatLeast__Suc__atMost,axiom,
    ! [M: nat,Nb: nat,G: fun_nat_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
     => ( groups3539618377306564664at_int(G,set_or1269000886237332187st_nat(M,Nb)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(G,M)),groups3539618377306564664at_int(G,set_or1269000886237332187st_nat(aa_nat_nat(suc,M),Nb))) ) ) ).

% sum.atLeast_Suc_atMost
tff(fact_7952_sum_OatLeast__Suc__atMost,axiom,
    ! [M: nat,Nb: nat,G: fun_nat_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(G),set_or1269000886237332187st_nat(M,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(G,M)),aa_set_nat_nat(groups3542108847815614940at_nat(G),set_or1269000886237332187st_nat(aa_nat_nat(suc,M),Nb))) ) ) ).

% sum.atLeast_Suc_atMost
tff(fact_7953_sum_OatLeast__Suc__atMost,axiom,
    ! [M: nat,Nb: nat,G: fun_nat_real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
     => ( groups6591440286371151544t_real(G,set_or1269000886237332187st_nat(M,Nb)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(G,M)),groups6591440286371151544t_real(G,set_or1269000886237332187st_nat(aa_nat_nat(suc,M),Nb))) ) ) ).

% sum.atLeast_Suc_atMost
tff(fact_7954_dbl__inc__def,axiom,
    ! [Xb: word_N3645301735248828278l_num1] : neg_nu8115118780965096967l_num1(Xb) = plus_p361126936061061375l_num1(plus_p361126936061061375l_num1(Xb,Xb),one_on7727431528512463931l_num1) ).

% dbl_inc_def
tff(fact_7955_dbl__inc__def,axiom,
    ! [Xb: real] : aa_real_real(neg_nu8295874005876285629c_real,Xb) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Xb)),one_one_real) ).

% dbl_inc_def
tff(fact_7956_dbl__inc__def,axiom,
    ! [Xb: rat] : neg_nu5219082963157363817nc_rat(Xb) = aa_rat_rat(plus_plus_rat(aa_rat_rat(plus_plus_rat(Xb),Xb)),one_one_rat) ).

% dbl_inc_def
tff(fact_7957_dbl__inc__def,axiom,
    ! [Xb: int] : aa_int_int(neg_nu5851722552734809277nc_int,Xb) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb),Xb)),one_one_int) ).

% dbl_inc_def
tff(fact_7958_sum_OSuc__reindex__ivl,axiom,
    ! [M: nat,Nb: nat,G: fun_nat_rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
     => ( aa_rat_rat(plus_plus_rat(groups2906978787729119204at_rat(G,set_or1269000886237332187st_nat(M,Nb))),aa_nat_rat(G,aa_nat_nat(suc,Nb))) = aa_rat_rat(plus_plus_rat(aa_nat_rat(G,M)),groups2906978787729119204at_rat(aTP_Lamm_aop(G),set_or1269000886237332187st_nat(M,Nb))) ) ) ).

% sum.Suc_reindex_ivl
tff(fact_7959_sum_OSuc__reindex__ivl,axiom,
    ! [M: nat,Nb: nat,G: fun_nat_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,groups3539618377306564664at_int(G,set_or1269000886237332187st_nat(M,Nb))),aa_nat_int(G,aa_nat_nat(suc,Nb))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(G,M)),groups3539618377306564664at_int(aTP_Lamm_aoq(G),set_or1269000886237332187st_nat(M,Nb))) ) ) ).

% sum.Suc_reindex_ivl
tff(fact_7960_sum_OSuc__reindex__ivl,axiom,
    ! [M: nat,Nb: nat,G: fun_nat_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(G),set_or1269000886237332187st_nat(M,Nb))),aa_nat_nat(G,aa_nat_nat(suc,Nb))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(G,M)),aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_anu(G)),set_or1269000886237332187st_nat(M,Nb))) ) ) ).

% sum.Suc_reindex_ivl
tff(fact_7961_sum_OSuc__reindex__ivl,axiom,
    ! [M: nat,Nb: nat,G: fun_nat_real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(G,set_or1269000886237332187st_nat(M,Nb))),aa_nat_real(G,aa_nat_nat(suc,Nb))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(G,M)),groups6591440286371151544t_real(aTP_Lamm_anv(G),set_or1269000886237332187st_nat(M,Nb))) ) ) ).

% sum.Suc_reindex_ivl
tff(fact_7962_sum__Suc__diff,axiom,
    ! [M: nat,Nb: nat,F: fun_nat_rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(suc,Nb))
     => ( groups2906978787729119204at_rat(aTP_Lamm_aor(F),set_or1269000886237332187st_nat(M,Nb)) = minus_minus_rat(aa_nat_rat(F,aa_nat_nat(suc,Nb)),aa_nat_rat(F,M)) ) ) ).

% sum_Suc_diff
tff(fact_7963_sum__Suc__diff,axiom,
    ! [M: nat,Nb: nat,F: fun_nat_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(suc,Nb))
     => ( groups3539618377306564664at_int(aTP_Lamm_aos(F),set_or1269000886237332187st_nat(M,Nb)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(F,aa_nat_nat(suc,Nb))),aa_nat_int(F,M)) ) ) ).

% sum_Suc_diff
tff(fact_7964_sum__Suc__diff,axiom,
    ! [M: nat,Nb: nat,F: fun_nat_real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(suc,Nb))
     => ( groups6591440286371151544t_real(aTP_Lamm_aot(F),set_or1269000886237332187st_nat(M,Nb)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(F,aa_nat_nat(suc,Nb))),aa_nat_real(F,M)) ) ) ).

% sum_Suc_diff
tff(fact_7965_sum__atLeastAtMost__code,axiom,
    ! [F: fun_nat_uint32,A: nat,B: nat] : groups833757482993574392uint32(F,set_or1269000886237332187st_nat(A,B)) = set_fo8366116489143299838uint32(aTP_Lamm_aou(F),A,B,zero_zero_uint32) ).

% sum_atLeastAtMost_code
tff(fact_7966_sum__atLeastAtMost__code,axiom,
    ! [F: fun_nat_rat,A: nat,B: nat] : groups2906978787729119204at_rat(F,set_or1269000886237332187st_nat(A,B)) = set_fo1949268297981939178at_rat(aTP_Lamm_aov(F),A,B,zero_zero_rat) ).

% sum_atLeastAtMost_code
tff(fact_7967_sum__atLeastAtMost__code,axiom,
    ! [F: fun_nat_int,A: nat,B: nat] : groups3539618377306564664at_int(F,set_or1269000886237332187st_nat(A,B)) = aa_int_int(aa_nat_fun_int_int(set_fo2581907887559384638at_int(aTP_Lamm_aow(F),A),B),zero_zero_int) ).

% sum_atLeastAtMost_code
tff(fact_7968_sum__atLeastAtMost__code,axiom,
    ! [F: fun_nat_nat,A: nat,B: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(F),set_or1269000886237332187st_nat(A,B)) = aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(aTP_Lamm_aox(F),A),B),zero_zero_nat) ).

% sum_atLeastAtMost_code
tff(fact_7969_sum__atLeastAtMost__code,axiom,
    ! [F: fun_nat_real,A: nat,B: nat] : groups6591440286371151544t_real(F,set_or1269000886237332187st_nat(A,B)) = aa_real_real(set_fo3111899725591712190t_real(aTP_Lamm_aoy(F),A,B),zero_zero_real) ).

% sum_atLeastAtMost_code
tff(fact_7970_sum__mono2,axiom,
    ! [B4: set_VEBT_VEBT,A2: set_VEBT_VEBT,F: fun_VEBT_VEBT_real] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,B4)
     => ( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(A2),B4)
       => ( ! [B2: vEBT_VEBT] :
              ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(B2),minus_5127226145743854075T_VEBT(B4,A2))
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_VEBT_VEBT_real(F,B2)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups2240296850493347238T_real(F,A2)),groups2240296850493347238T_real(F,B4)) ) ) ) ).

% sum_mono2
tff(fact_7971_sum__mono2,axiom,
    ! [B4: set_real,A2: set_real,F: fun_real_real] :
      ( aa_set_real_o(finite_finite_real,B4)
     => ( aa_set_real_o(ord_less_eq_set_real(A2),B4)
       => ( ! [B2: real] :
              ( aa_set_real_o(member_real(B2),minus_minus_set_real(B4,A2))
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(F,B2)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups8097168146408367636l_real(F,A2)),groups8097168146408367636l_real(F,B4)) ) ) ) ).

% sum_mono2
tff(fact_7972_sum__mono2,axiom,
    ! [B4: set_complex,A2: set_complex,F: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,B4)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(A2),B4)
       => ( ! [B2: complex] :
              ( aa_set_complex_o(member_complex(B2),minus_811609699411566653omplex(B4,A2))
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(F,B2)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups5808333547571424918x_real(F,A2)),groups5808333547571424918x_real(F,B4)) ) ) ) ).

% sum_mono2
tff(fact_7973_sum__mono2,axiom,
    ! [B4: set_Code_integer,A2: set_Code_integer,F: fun_Co3960896333048504454r_real] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B4)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(A2),B4)
       => ( ! [B2: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(B2),minus_2355218937544613996nteger(B4,A2))
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_Code_integer_real(F,B2)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups1270011288395367621r_real(F,A2)),groups1270011288395367621r_real(F,B4)) ) ) ) ).

% sum_mono2
tff(fact_7974_sum__mono2,axiom,
    ! [B4: set_VEBT_VEBT,A2: set_VEBT_VEBT,F: fun_VEBT_VEBT_rat] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,B4)
     => ( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(A2),B4)
       => ( ! [B2: vEBT_VEBT] :
              ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(B2),minus_5127226145743854075T_VEBT(B4,A2))
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_VEBT_VEBT_rat(F,B2)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups136491112297645522BT_rat(F,A2)),groups136491112297645522BT_rat(F,B4)) ) ) ) ).

% sum_mono2
tff(fact_7975_sum__mono2,axiom,
    ! [B4: set_real,A2: set_real,F: fun_real_rat] :
      ( aa_set_real_o(finite_finite_real,B4)
     => ( aa_set_real_o(ord_less_eq_set_real(A2),B4)
       => ( ! [B2: real] :
              ( aa_set_real_o(member_real(B2),minus_minus_set_real(B4,A2))
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_real_rat(F,B2)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups1300246762558778688al_rat(F,A2)),groups1300246762558778688al_rat(F,B4)) ) ) ) ).

% sum_mono2
tff(fact_7976_sum__mono2,axiom,
    ! [B4: set_complex,A2: set_complex,F: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,B4)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(A2),B4)
       => ( ! [B2: complex] :
              ( aa_set_complex_o(member_complex(B2),minus_811609699411566653omplex(B4,A2))
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(F,B2)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups5058264527183730370ex_rat(F,A2)),groups5058264527183730370ex_rat(F,B4)) ) ) ) ).

% sum_mono2
tff(fact_7977_sum__mono2,axiom,
    ! [B4: set_Code_integer,A2: set_Code_integer,F: fun_Code_integer_rat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B4)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(A2),B4)
       => ( ! [B2: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(B2),minus_2355218937544613996nteger(B4,A2))
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_Code_integer_rat(F,B2)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups6602215022474089585er_rat(F,A2)),groups6602215022474089585er_rat(F,B4)) ) ) ) ).

% sum_mono2
tff(fact_7978_sum__mono2,axiom,
    ! [B4: set_nat,A2: set_nat,F: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,B4)
     => ( aa_set_nat_o(ord_less_eq_set_nat(A2),B4)
       => ( ! [B2: nat] :
              ( aa_set_nat_o(member_nat(B2),minus_minus_set_nat(B4,A2))
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(F,B2)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups2906978787729119204at_rat(F,A2)),groups2906978787729119204at_rat(F,B4)) ) ) ) ).

% sum_mono2
tff(fact_7979_sum__mono2,axiom,
    ! [B4: set_VEBT_VEBT,A2: set_VEBT_VEBT,F: fun_VEBT_VEBT_nat] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,B4)
     => ( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(A2),B4)
       => ( ! [B2: vEBT_VEBT] :
              ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(B2),minus_5127226145743854075T_VEBT(B4,A2))
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_VEBT_VEBT_nat(F,B2)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,groups771621172384141258BT_nat(F,A2)),groups771621172384141258BT_nat(F,B4)) ) ) ) ).

% sum_mono2
tff(fact_7980_sum_Oinsert__remove,axiom,
    ! [A2: set_VEBT_VEBT,G: fun_VEBT_VEBT_real,Xb: vEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,A2)
     => ( groups2240296850493347238T_real(G,insert_VEBT_VEBT(Xb,A2)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_VEBT_VEBT_real(G,Xb)),groups2240296850493347238T_real(G,minus_5127226145743854075T_VEBT(A2,insert_VEBT_VEBT(Xb,bot_bo8194388402131092736T_VEBT)))) ) ) ).

% sum.insert_remove
tff(fact_7981_sum_Oinsert__remove,axiom,
    ! [A2: set_complex,G: fun_complex_real,Xb: complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( groups5808333547571424918x_real(G,insert_complex(Xb,A2)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(G,Xb)),groups5808333547571424918x_real(G,minus_811609699411566653omplex(A2,insert_complex(Xb,bot_bot_set_complex)))) ) ) ).

% sum.insert_remove
tff(fact_7982_sum_Oinsert__remove,axiom,
    ! [A2: set_Code_integer,G: fun_Co3960896333048504454r_real,Xb: code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A2)
     => ( groups1270011288395367621r_real(G,insert_Code_integer(Xb,A2)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_Code_integer_real(G,Xb)),groups1270011288395367621r_real(G,minus_2355218937544613996nteger(A2,insert_Code_integer(Xb,bot_bo3990330152332043303nteger)))) ) ) ).

% sum.insert_remove
tff(fact_7983_sum_Oinsert__remove,axiom,
    ! [A2: set_VEBT_VEBT,G: fun_VEBT_VEBT_rat,Xb: vEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,A2)
     => ( groups136491112297645522BT_rat(G,insert_VEBT_VEBT(Xb,A2)) = aa_rat_rat(plus_plus_rat(aa_VEBT_VEBT_rat(G,Xb)),groups136491112297645522BT_rat(G,minus_5127226145743854075T_VEBT(A2,insert_VEBT_VEBT(Xb,bot_bo8194388402131092736T_VEBT)))) ) ) ).

% sum.insert_remove
tff(fact_7984_sum_Oinsert__remove,axiom,
    ! [A2: set_complex,G: fun_complex_rat,Xb: complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( groups5058264527183730370ex_rat(G,insert_complex(Xb,A2)) = aa_rat_rat(plus_plus_rat(aa_complex_rat(G,Xb)),groups5058264527183730370ex_rat(G,minus_811609699411566653omplex(A2,insert_complex(Xb,bot_bot_set_complex)))) ) ) ).

% sum.insert_remove
tff(fact_7985_sum_Oinsert__remove,axiom,
    ! [A2: set_Code_integer,G: fun_Code_integer_rat,Xb: code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A2)
     => ( groups6602215022474089585er_rat(G,insert_Code_integer(Xb,A2)) = aa_rat_rat(plus_plus_rat(aa_Code_integer_rat(G,Xb)),groups6602215022474089585er_rat(G,minus_2355218937544613996nteger(A2,insert_Code_integer(Xb,bot_bo3990330152332043303nteger)))) ) ) ).

% sum.insert_remove
tff(fact_7986_sum_Oinsert__remove,axiom,
    ! [A2: set_VEBT_VEBT,G: fun_VEBT_VEBT_nat,Xb: vEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,A2)
     => ( groups771621172384141258BT_nat(G,insert_VEBT_VEBT(Xb,A2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_VEBT_VEBT_nat(G,Xb)),groups771621172384141258BT_nat(G,minus_5127226145743854075T_VEBT(A2,insert_VEBT_VEBT(Xb,bot_bo8194388402131092736T_VEBT)))) ) ) ).

% sum.insert_remove
tff(fact_7987_sum_Oinsert__remove,axiom,
    ! [A2: set_complex,G: fun_complex_nat,Xb: complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( groups5693394587270226106ex_nat(G,insert_complex(Xb,A2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_complex_nat(G,Xb)),groups5693394587270226106ex_nat(G,minus_811609699411566653omplex(A2,insert_complex(Xb,bot_bot_set_complex)))) ) ) ).

% sum.insert_remove
tff(fact_7988_sum_Oinsert__remove,axiom,
    ! [A2: set_Code_integer,G: fun_Code_integer_nat,Xb: code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A2)
     => ( groups7237345082560585321er_nat(G,insert_Code_integer(Xb,A2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_Code_integer_nat(G,Xb)),groups7237345082560585321er_nat(G,minus_2355218937544613996nteger(A2,insert_Code_integer(Xb,bot_bo3990330152332043303nteger)))) ) ) ).

% sum.insert_remove
tff(fact_7989_sum_Oinsert__remove,axiom,
    ! [A2: set_VEBT_VEBT,G: fun_VEBT_VEBT_int,Xb: vEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,A2)
     => ( groups769130701875090982BT_int(G,insert_VEBT_VEBT(Xb,A2)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_VEBT_VEBT_int(G,Xb)),groups769130701875090982BT_int(G,minus_5127226145743854075T_VEBT(A2,insert_VEBT_VEBT(Xb,bot_bo8194388402131092736T_VEBT)))) ) ) ).

% sum.insert_remove
tff(fact_7990_sum_Oremove,axiom,
    ! [A2: set_VEBT_VEBT,Xb: vEBT_VEBT,G: fun_VEBT_VEBT_real] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,A2)
     => ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xb),A2)
       => ( groups2240296850493347238T_real(G,A2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_VEBT_VEBT_real(G,Xb)),groups2240296850493347238T_real(G,minus_5127226145743854075T_VEBT(A2,insert_VEBT_VEBT(Xb,bot_bo8194388402131092736T_VEBT)))) ) ) ) ).

% sum.remove
tff(fact_7991_sum_Oremove,axiom,
    ! [A2: set_complex,Xb: complex,G: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( aa_set_complex_o(member_complex(Xb),A2)
       => ( groups5808333547571424918x_real(G,A2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(G,Xb)),groups5808333547571424918x_real(G,minus_811609699411566653omplex(A2,insert_complex(Xb,bot_bot_set_complex)))) ) ) ) ).

% sum.remove
tff(fact_7992_sum_Oremove,axiom,
    ! [A2: set_Code_integer,Xb: code_integer,G: fun_Co3960896333048504454r_real] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A2)
     => ( aa_set5643387332196575111eger_o(member_Code_integer(Xb),A2)
       => ( groups1270011288395367621r_real(G,A2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_Code_integer_real(G,Xb)),groups1270011288395367621r_real(G,minus_2355218937544613996nteger(A2,insert_Code_integer(Xb,bot_bo3990330152332043303nteger)))) ) ) ) ).

% sum.remove
tff(fact_7993_sum_Oremove,axiom,
    ! [A2: set_VEBT_VEBT,Xb: vEBT_VEBT,G: fun_VEBT_VEBT_rat] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,A2)
     => ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xb),A2)
       => ( groups136491112297645522BT_rat(G,A2) = aa_rat_rat(plus_plus_rat(aa_VEBT_VEBT_rat(G,Xb)),groups136491112297645522BT_rat(G,minus_5127226145743854075T_VEBT(A2,insert_VEBT_VEBT(Xb,bot_bo8194388402131092736T_VEBT)))) ) ) ) ).

% sum.remove
tff(fact_7994_sum_Oremove,axiom,
    ! [A2: set_complex,Xb: complex,G: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( aa_set_complex_o(member_complex(Xb),A2)
       => ( groups5058264527183730370ex_rat(G,A2) = aa_rat_rat(plus_plus_rat(aa_complex_rat(G,Xb)),groups5058264527183730370ex_rat(G,minus_811609699411566653omplex(A2,insert_complex(Xb,bot_bot_set_complex)))) ) ) ) ).

% sum.remove
tff(fact_7995_sum_Oremove,axiom,
    ! [A2: set_Code_integer,Xb: code_integer,G: fun_Code_integer_rat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A2)
     => ( aa_set5643387332196575111eger_o(member_Code_integer(Xb),A2)
       => ( groups6602215022474089585er_rat(G,A2) = aa_rat_rat(plus_plus_rat(aa_Code_integer_rat(G,Xb)),groups6602215022474089585er_rat(G,minus_2355218937544613996nteger(A2,insert_Code_integer(Xb,bot_bo3990330152332043303nteger)))) ) ) ) ).

% sum.remove
tff(fact_7996_sum_Oremove,axiom,
    ! [A2: set_VEBT_VEBT,Xb: vEBT_VEBT,G: fun_VEBT_VEBT_nat] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,A2)
     => ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xb),A2)
       => ( groups771621172384141258BT_nat(G,A2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_VEBT_VEBT_nat(G,Xb)),groups771621172384141258BT_nat(G,minus_5127226145743854075T_VEBT(A2,insert_VEBT_VEBT(Xb,bot_bo8194388402131092736T_VEBT)))) ) ) ) ).

% sum.remove
tff(fact_7997_sum_Oremove,axiom,
    ! [A2: set_complex,Xb: complex,G: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( aa_set_complex_o(member_complex(Xb),A2)
       => ( groups5693394587270226106ex_nat(G,A2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_complex_nat(G,Xb)),groups5693394587270226106ex_nat(G,minus_811609699411566653omplex(A2,insert_complex(Xb,bot_bot_set_complex)))) ) ) ) ).

% sum.remove
tff(fact_7998_sum_Oremove,axiom,
    ! [A2: set_Code_integer,Xb: code_integer,G: fun_Code_integer_nat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A2)
     => ( aa_set5643387332196575111eger_o(member_Code_integer(Xb),A2)
       => ( groups7237345082560585321er_nat(G,A2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_Code_integer_nat(G,Xb)),groups7237345082560585321er_nat(G,minus_2355218937544613996nteger(A2,insert_Code_integer(Xb,bot_bo3990330152332043303nteger)))) ) ) ) ).

% sum.remove
tff(fact_7999_sum_Oremove,axiom,
    ! [A2: set_VEBT_VEBT,Xb: vEBT_VEBT,G: fun_VEBT_VEBT_int] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,A2)
     => ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xb),A2)
       => ( groups769130701875090982BT_int(G,A2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_VEBT_VEBT_int(G,Xb)),groups769130701875090982BT_int(G,minus_5127226145743854075T_VEBT(A2,insert_VEBT_VEBT(Xb,bot_bo8194388402131092736T_VEBT)))) ) ) ) ).

% sum.remove
tff(fact_8000_sum__diff1,axiom,
    ! [A2: set_VEBT_VEBT,F: fun_VEBT_VEBT_real,A: vEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,A2)
     => ( groups2240296850493347238T_real(F,minus_5127226145743854075T_VEBT(A2,insert_VEBT_VEBT(A,bot_bo8194388402131092736T_VEBT))) = $ite(aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A),A2),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,groups2240296850493347238T_real(F,A2)),aa_VEBT_VEBT_real(F,A)),groups2240296850493347238T_real(F,A2)) ) ) ).

% sum_diff1
tff(fact_8001_sum__diff1,axiom,
    ! [A2: set_complex,F: fun_complex_real,A: complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( groups5808333547571424918x_real(F,minus_811609699411566653omplex(A2,insert_complex(A,bot_bot_set_complex))) = $ite(aa_set_complex_o(member_complex(A),A2),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,groups5808333547571424918x_real(F,A2)),aa_complex_real(F,A)),groups5808333547571424918x_real(F,A2)) ) ) ).

% sum_diff1
tff(fact_8002_sum__diff1,axiom,
    ! [A2: set_Code_integer,F: fun_Co3960896333048504454r_real,A: code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A2)
     => ( groups1270011288395367621r_real(F,minus_2355218937544613996nteger(A2,insert_Code_integer(A,bot_bo3990330152332043303nteger))) = $ite(aa_set5643387332196575111eger_o(member_Code_integer(A),A2),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,groups1270011288395367621r_real(F,A2)),aa_Code_integer_real(F,A)),groups1270011288395367621r_real(F,A2)) ) ) ).

% sum_diff1
tff(fact_8003_sum__diff1,axiom,
    ! [A2: set_int,F: fun_int_real,A: int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( groups8778361861064173332t_real(F,minus_minus_set_int(A2,insert_int(A,bot_bot_set_int))) = $ite(aa_set_int_o(member_int(A),A2),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,groups8778361861064173332t_real(F,A2)),aa_int_real(F,A)),groups8778361861064173332t_real(F,A2)) ) ) ).

% sum_diff1
tff(fact_8004_sum__diff1,axiom,
    ! [A2: set_real,F: fun_real_real,A: real] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( groups8097168146408367636l_real(F,minus_minus_set_real(A2,insert_real(A,bot_bot_set_real))) = $ite(aa_set_real_o(member_real(A),A2),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,groups8097168146408367636l_real(F,A2)),aa_real_real(F,A)),groups8097168146408367636l_real(F,A2)) ) ) ).

% sum_diff1
tff(fact_8005_sum__diff1,axiom,
    ! [A2: set_VEBT_VEBT,F: fun_VEBT_VEBT_rat,A: vEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,A2)
     => ( groups136491112297645522BT_rat(F,minus_5127226145743854075T_VEBT(A2,insert_VEBT_VEBT(A,bot_bo8194388402131092736T_VEBT))) = $ite(aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A),A2),minus_minus_rat(groups136491112297645522BT_rat(F,A2),aa_VEBT_VEBT_rat(F,A)),groups136491112297645522BT_rat(F,A2)) ) ) ).

% sum_diff1
tff(fact_8006_sum__diff1,axiom,
    ! [A2: set_complex,F: fun_complex_rat,A: complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( groups5058264527183730370ex_rat(F,minus_811609699411566653omplex(A2,insert_complex(A,bot_bot_set_complex))) = $ite(aa_set_complex_o(member_complex(A),A2),minus_minus_rat(groups5058264527183730370ex_rat(F,A2),aa_complex_rat(F,A)),groups5058264527183730370ex_rat(F,A2)) ) ) ).

% sum_diff1
tff(fact_8007_sum__diff1,axiom,
    ! [A2: set_Code_integer,F: fun_Code_integer_rat,A: code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A2)
     => ( groups6602215022474089585er_rat(F,minus_2355218937544613996nteger(A2,insert_Code_integer(A,bot_bo3990330152332043303nteger))) = $ite(aa_set5643387332196575111eger_o(member_Code_integer(A),A2),minus_minus_rat(groups6602215022474089585er_rat(F,A2),aa_Code_integer_rat(F,A)),groups6602215022474089585er_rat(F,A2)) ) ) ).

% sum_diff1
tff(fact_8008_sum__diff1,axiom,
    ! [A2: set_int,F: fun_int_rat,A: int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( groups3906332499630173760nt_rat(F,minus_minus_set_int(A2,insert_int(A,bot_bot_set_int))) = $ite(aa_set_int_o(member_int(A),A2),minus_minus_rat(groups3906332499630173760nt_rat(F,A2),aa_int_rat(F,A)),groups3906332499630173760nt_rat(F,A2)) ) ) ).

% sum_diff1
tff(fact_8009_sum__diff1,axiom,
    ! [A2: set_real,F: fun_real_rat,A: real] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( groups1300246762558778688al_rat(F,minus_minus_set_real(A2,insert_real(A,bot_bot_set_real))) = $ite(aa_set_real_o(member_real(A),A2),minus_minus_rat(groups1300246762558778688al_rat(F,A2),aa_real_rat(F,A)),groups1300246762558778688al_rat(F,A2)) ) ) ).

% sum_diff1
tff(fact_8010_sum_Oub__add__nat,axiom,
    ! [M: nat,Nb: 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,Nb),one_one_nat))
     => ( groups2906978787729119204at_rat(G,set_or1269000886237332187st_nat(M,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),P2))) = aa_rat_rat(plus_plus_rat(groups2906978787729119204at_rat(G,set_or1269000886237332187st_nat(M,Nb))),groups2906978787729119204at_rat(G,set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),P2)))) ) ) ).

% sum.ub_add_nat
tff(fact_8011_sum_Oub__add__nat,axiom,
    ! [M: nat,Nb: 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,Nb),one_one_nat))
     => ( groups3539618377306564664at_int(G,set_or1269000886237332187st_nat(M,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),P2))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups3539618377306564664at_int(G,set_or1269000886237332187st_nat(M,Nb))),groups3539618377306564664at_int(G,set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),P2)))) ) ) ).

% sum.ub_add_nat
tff(fact_8012_sum_Oub__add__nat,axiom,
    ! [M: nat,Nb: 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,Nb),one_one_nat))
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(G),set_or1269000886237332187st_nat(M,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),P2))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(G),set_or1269000886237332187st_nat(M,Nb))),aa_set_nat_nat(groups3542108847815614940at_nat(G),set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),P2)))) ) ) ).

% sum.ub_add_nat
tff(fact_8013_sum_Oub__add__nat,axiom,
    ! [M: nat,Nb: nat,G: fun_nat_real,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,Nb),one_one_nat))
     => ( groups6591440286371151544t_real(G,set_or1269000886237332187st_nat(M,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),P2))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(G,set_or1269000886237332187st_nat(M,Nb))),groups6591440286371151544t_real(G,set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),P2)))) ) ) ).

% sum.ub_add_nat
tff(fact_8014_sum_Odelta__remove,axiom,
    ! [S2: set_VEBT_VEBT,A: vEBT_VEBT,B: fun_VEBT_VEBT_real,C: fun_VEBT_VEBT_real] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,S2)
     => ( groups2240296850493347238T_real(aa_fun1825138017709814396T_real(aa_fun910392650362481850T_real(aTP_Lamm_aoz(A),B),C),S2) = $ite(aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A),S2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_VEBT_VEBT_real(B,A)),groups2240296850493347238T_real(C,minus_5127226145743854075T_VEBT(S2,insert_VEBT_VEBT(A,bot_bo8194388402131092736T_VEBT)))),groups2240296850493347238T_real(C,minus_5127226145743854075T_VEBT(S2,insert_VEBT_VEBT(A,bot_bo8194388402131092736T_VEBT)))) ) ) ).

% sum.delta_remove
tff(fact_8015_sum_Odelta__remove,axiom,
    ! [S2: set_complex,A: complex,B: fun_complex_real,C: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S2)
     => ( groups5808333547571424918x_real(aa_fun8933909628229591020x_real(aa_fun8382785274855863872x_real(aTP_Lamm_apa(A),B),C),S2) = $ite(aa_set_complex_o(member_complex(A),S2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(B,A)),groups5808333547571424918x_real(C,minus_811609699411566653omplex(S2,insert_complex(A,bot_bot_set_complex)))),groups5808333547571424918x_real(C,minus_811609699411566653omplex(S2,insert_complex(A,bot_bot_set_complex)))) ) ) ).

% sum.delta_remove
tff(fact_8016_sum_Odelta__remove,axiom,
    ! [S2: set_Code_integer,A: code_integer,B: fun_Co3960896333048504454r_real,C: fun_Co3960896333048504454r_real] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,S2)
     => ( groups1270011288395367621r_real(aa_fun6341089502340155082r_real(aa_fun7644355603919855821r_real(aTP_Lamm_apb(A),B),C),S2) = $ite(aa_set5643387332196575111eger_o(member_Code_integer(A),S2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_Code_integer_real(B,A)),groups1270011288395367621r_real(C,minus_2355218937544613996nteger(S2,insert_Code_integer(A,bot_bo3990330152332043303nteger)))),groups1270011288395367621r_real(C,minus_2355218937544613996nteger(S2,insert_Code_integer(A,bot_bo3990330152332043303nteger)))) ) ) ).

% sum.delta_remove
tff(fact_8017_sum_Odelta__remove,axiom,
    ! [S2: set_VEBT_VEBT,A: vEBT_VEBT,B: fun_VEBT_VEBT_rat,C: fun_VEBT_VEBT_rat] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,S2)
     => ( groups136491112297645522BT_rat(aa_fun8105832842879916156BT_rat(aa_fun2693410532875676774BT_rat(aTP_Lamm_apc(A),B),C),S2) = $ite(aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A),S2),aa_rat_rat(plus_plus_rat(aa_VEBT_VEBT_rat(B,A)),groups136491112297645522BT_rat(C,minus_5127226145743854075T_VEBT(S2,insert_VEBT_VEBT(A,bot_bo8194388402131092736T_VEBT)))),groups136491112297645522BT_rat(C,minus_5127226145743854075T_VEBT(S2,insert_VEBT_VEBT(A,bot_bo8194388402131092736T_VEBT)))) ) ) ).

% sum.delta_remove
tff(fact_8018_sum_Odelta__remove,axiom,
    ! [S2: set_complex,A: complex,B: fun_complex_rat,C: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S2)
     => ( groups5058264527183730370ex_rat(aa_fun7622890401989837636ex_rat(aa_fun530470884365435460ex_rat(aTP_Lamm_apd(A),B),C),S2) = $ite(aa_set_complex_o(member_complex(A),S2),aa_rat_rat(plus_plus_rat(aa_complex_rat(B,A)),groups5058264527183730370ex_rat(C,minus_811609699411566653omplex(S2,insert_complex(A,bot_bot_set_complex)))),groups5058264527183730370ex_rat(C,minus_811609699411566653omplex(S2,insert_complex(A,bot_bot_set_complex)))) ) ) ).

% sum.delta_remove
tff(fact_8019_sum_Odelta__remove,axiom,
    ! [S2: set_Code_integer,A: code_integer,B: fun_Code_integer_rat,C: fun_Code_integer_rat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,S2)
     => ( groups6602215022474089585er_rat(aa_fun8169474305930283170er_rat(aa_fun1894662059565830609er_rat(aTP_Lamm_ape(A),B),C),S2) = $ite(aa_set5643387332196575111eger_o(member_Code_integer(A),S2),aa_rat_rat(plus_plus_rat(aa_Code_integer_rat(B,A)),groups6602215022474089585er_rat(C,minus_2355218937544613996nteger(S2,insert_Code_integer(A,bot_bo3990330152332043303nteger)))),groups6602215022474089585er_rat(C,minus_2355218937544613996nteger(S2,insert_Code_integer(A,bot_bo3990330152332043303nteger)))) ) ) ).

% sum.delta_remove
tff(fact_8020_sum_Odelta__remove,axiom,
    ! [S2: set_VEBT_VEBT,A: vEBT_VEBT,B: fun_VEBT_VEBT_nat,C: fun_VEBT_VEBT_nat] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,S2)
     => ( groups771621172384141258BT_nat(aa_fun8284231930576791164BT_nat(aa_fun8778301808485728350BT_nat(aTP_Lamm_apf(A),B),C),S2) = $ite(aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A),S2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_VEBT_VEBT_nat(B,A)),groups771621172384141258BT_nat(C,minus_5127226145743854075T_VEBT(S2,insert_VEBT_VEBT(A,bot_bo8194388402131092736T_VEBT)))),groups771621172384141258BT_nat(C,minus_5127226145743854075T_VEBT(S2,insert_VEBT_VEBT(A,bot_bo8194388402131092736T_VEBT)))) ) ) ).

% sum.delta_remove
tff(fact_8021_sum_Odelta__remove,axiom,
    ! [S2: set_complex,A: complex,B: fun_complex_nat,C: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S2)
     => ( groups5693394587270226106ex_nat(aa_fun2465681967280144180ex_nat(aa_fun2484745150758625324ex_nat(aTP_Lamm_apg(A),B),C),S2) = $ite(aa_set_complex_o(member_complex(A),S2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_complex_nat(B,A)),groups5693394587270226106ex_nat(C,minus_811609699411566653omplex(S2,insert_complex(A,bot_bot_set_complex)))),groups5693394587270226106ex_nat(C,minus_811609699411566653omplex(S2,insert_complex(A,bot_bot_set_complex)))) ) ) ).

% sum.delta_remove
tff(fact_8022_sum_Odelta__remove,axiom,
    ! [S2: set_Code_integer,A: code_integer,B: fun_Code_integer_nat,C: fun_Code_integer_nat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,S2)
     => ( groups7237345082560585321er_nat(aa_fun3076867170244285586er_nat(aa_fun7635586485349294009er_nat(aTP_Lamm_aph(A),B),C),S2) = $ite(aa_set5643387332196575111eger_o(member_Code_integer(A),S2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_Code_integer_nat(B,A)),groups7237345082560585321er_nat(C,minus_2355218937544613996nteger(S2,insert_Code_integer(A,bot_bo3990330152332043303nteger)))),groups7237345082560585321er_nat(C,minus_2355218937544613996nteger(S2,insert_Code_integer(A,bot_bo3990330152332043303nteger)))) ) ) ).

% sum.delta_remove
tff(fact_8023_sum_Odelta__remove,axiom,
    ! [S2: set_VEBT_VEBT,A: vEBT_VEBT,B: fun_VEBT_VEBT_int,C: fun_VEBT_VEBT_int] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,S2)
     => ( groups769130701875090982BT_int(aa_fun7144222995531728252BT_int(aa_fun7485955581636121530BT_int(aTP_Lamm_api(A),B),C),S2) = $ite(aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A),S2),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_VEBT_VEBT_int(B,A)),groups769130701875090982BT_int(C,minus_5127226145743854075T_VEBT(S2,insert_VEBT_VEBT(A,bot_bo8194388402131092736T_VEBT)))),groups769130701875090982BT_int(C,minus_5127226145743854075T_VEBT(S2,insert_VEBT_VEBT(A,bot_bo8194388402131092736T_VEBT)))) ) ) ).

% sum.delta_remove
tff(fact_8024_set__encode__def,axiom,
    nat_set_encode = groups3542108847815614940at_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% set_encode_def
tff(fact_8025_sum__strict__mono2,axiom,
    ! [B4: set_VEBT_VEBT,A2: set_VEBT_VEBT,B: vEBT_VEBT,F: fun_VEBT_VEBT_real] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,B4)
     => ( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(A2),B4)
       => ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(B),minus_5127226145743854075T_VEBT(B4,A2))
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_VEBT_VEBT_real(F,B))
           => ( ! [X2: vEBT_VEBT] :
                  ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X2),B4)
                 => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_VEBT_VEBT_real(F,X2)) )
             => aa_real_o(aa_real_fun_real_o(ord_less_real,groups2240296850493347238T_real(F,A2)),groups2240296850493347238T_real(F,B4)) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_8026_sum__strict__mono2,axiom,
    ! [B4: set_real,A2: set_real,B: real,F: fun_real_real] :
      ( aa_set_real_o(finite_finite_real,B4)
     => ( aa_set_real_o(ord_less_eq_set_real(A2),B4)
       => ( aa_set_real_o(member_real(B),minus_minus_set_real(B4,A2))
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(F,B))
           => ( ! [X2: real] :
                  ( aa_set_real_o(member_real(X2),B4)
                 => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(F,X2)) )
             => aa_real_o(aa_real_fun_real_o(ord_less_real,groups8097168146408367636l_real(F,A2)),groups8097168146408367636l_real(F,B4)) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_8027_sum__strict__mono2,axiom,
    ! [B4: set_complex,A2: set_complex,B: complex,F: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,B4)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(A2),B4)
       => ( aa_set_complex_o(member_complex(B),minus_811609699411566653omplex(B4,A2))
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_complex_real(F,B))
           => ( ! [X2: complex] :
                  ( aa_set_complex_o(member_complex(X2),B4)
                 => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(F,X2)) )
             => aa_real_o(aa_real_fun_real_o(ord_less_real,groups5808333547571424918x_real(F,A2)),groups5808333547571424918x_real(F,B4)) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_8028_sum__strict__mono2,axiom,
    ! [B4: set_Code_integer,A2: set_Code_integer,B: code_integer,F: fun_Co3960896333048504454r_real] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B4)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(A2),B4)
       => ( aa_set5643387332196575111eger_o(member_Code_integer(B),minus_2355218937544613996nteger(B4,A2))
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_Code_integer_real(F,B))
           => ( ! [X2: code_integer] :
                  ( aa_set5643387332196575111eger_o(member_Code_integer(X2),B4)
                 => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_Code_integer_real(F,X2)) )
             => aa_real_o(aa_real_fun_real_o(ord_less_real,groups1270011288395367621r_real(F,A2)),groups1270011288395367621r_real(F,B4)) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_8029_sum__strict__mono2,axiom,
    ! [B4: set_VEBT_VEBT,A2: set_VEBT_VEBT,B: vEBT_VEBT,F: fun_VEBT_VEBT_rat] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,B4)
     => ( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(A2),B4)
       => ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(B),minus_5127226145743854075T_VEBT(B4,A2))
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_VEBT_VEBT_rat(F,B))
           => ( ! [X2: vEBT_VEBT] :
                  ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X2),B4)
                 => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_VEBT_VEBT_rat(F,X2)) )
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,groups136491112297645522BT_rat(F,A2)),groups136491112297645522BT_rat(F,B4)) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_8030_sum__strict__mono2,axiom,
    ! [B4: set_real,A2: set_real,B: real,F: fun_real_rat] :
      ( aa_set_real_o(finite_finite_real,B4)
     => ( aa_set_real_o(ord_less_eq_set_real(A2),B4)
       => ( aa_set_real_o(member_real(B),minus_minus_set_real(B4,A2))
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_real_rat(F,B))
           => ( ! [X2: real] :
                  ( aa_set_real_o(member_real(X2),B4)
                 => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_real_rat(F,X2)) )
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,groups1300246762558778688al_rat(F,A2)),groups1300246762558778688al_rat(F,B4)) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_8031_sum__strict__mono2,axiom,
    ! [B4: set_complex,A2: set_complex,B: complex,F: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,B4)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(A2),B4)
       => ( aa_set_complex_o(member_complex(B),minus_811609699411566653omplex(B4,A2))
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_complex_rat(F,B))
           => ( ! [X2: complex] :
                  ( aa_set_complex_o(member_complex(X2),B4)
                 => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(F,X2)) )
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,groups5058264527183730370ex_rat(F,A2)),groups5058264527183730370ex_rat(F,B4)) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_8032_sum__strict__mono2,axiom,
    ! [B4: set_Code_integer,A2: set_Code_integer,B: code_integer,F: fun_Code_integer_rat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B4)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(A2),B4)
       => ( aa_set5643387332196575111eger_o(member_Code_integer(B),minus_2355218937544613996nteger(B4,A2))
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_Code_integer_rat(F,B))
           => ( ! [X2: code_integer] :
                  ( aa_set5643387332196575111eger_o(member_Code_integer(X2),B4)
                 => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_Code_integer_rat(F,X2)) )
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,groups6602215022474089585er_rat(F,A2)),groups6602215022474089585er_rat(F,B4)) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_8033_sum__strict__mono2,axiom,
    ! [B4: set_nat,A2: set_nat,B: nat,F: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,B4)
     => ( aa_set_nat_o(ord_less_eq_set_nat(A2),B4)
       => ( aa_set_nat_o(member_nat(B),minus_minus_set_nat(B4,A2))
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_nat_rat(F,B))
           => ( ! [X2: nat] :
                  ( aa_set_nat_o(member_nat(X2),B4)
                 => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(F,X2)) )
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,groups2906978787729119204at_rat(F,A2)),groups2906978787729119204at_rat(F,B4)) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_8034_sum__strict__mono2,axiom,
    ! [B4: set_VEBT_VEBT,A2: set_VEBT_VEBT,B: vEBT_VEBT,F: fun_VEBT_VEBT_nat] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,B4)
     => ( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(A2),B4)
       => ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(B),minus_5127226145743854075T_VEBT(B4,A2))
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_VEBT_VEBT_nat(F,B))
           => ( ! [X2: vEBT_VEBT] :
                  ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X2),B4)
                 => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_VEBT_VEBT_nat(F,X2)) )
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,groups771621172384141258BT_nat(F,A2)),groups771621172384141258BT_nat(F,B4)) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_8035_member__le__sum,axiom,
    ! [I: vEBT_VEBT,A2: set_VEBT_VEBT,F: fun_VEBT_VEBT_real] :
      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(I),A2)
     => ( ! [X2: vEBT_VEBT] :
            ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X2),minus_5127226145743854075T_VEBT(A2,insert_VEBT_VEBT(I,bot_bo8194388402131092736T_VEBT)))
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_VEBT_VEBT_real(F,X2)) )
       => ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,A2)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_VEBT_VEBT_real(F,I)),groups2240296850493347238T_real(F,A2)) ) ) ) ).

% member_le_sum
tff(fact_8036_member__le__sum,axiom,
    ! [I: complex,A2: set_complex,F: fun_complex_real] :
      ( aa_set_complex_o(member_complex(I),A2)
     => ( ! [X2: complex] :
            ( aa_set_complex_o(member_complex(X2),minus_811609699411566653omplex(A2,insert_complex(I,bot_bot_set_complex)))
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(F,X2)) )
       => ( aa_set_complex_o(finite3207457112153483333omplex,A2)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(F,I)),groups5808333547571424918x_real(F,A2)) ) ) ) ).

% member_le_sum
tff(fact_8037_member__le__sum,axiom,
    ! [I: code_integer,A2: set_Code_integer,F: fun_Co3960896333048504454r_real] :
      ( aa_set5643387332196575111eger_o(member_Code_integer(I),A2)
     => ( ! [X2: code_integer] :
            ( aa_set5643387332196575111eger_o(member_Code_integer(X2),minus_2355218937544613996nteger(A2,insert_Code_integer(I,bot_bo3990330152332043303nteger)))
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_Code_integer_real(F,X2)) )
       => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A2)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_Code_integer_real(F,I)),groups1270011288395367621r_real(F,A2)) ) ) ) ).

% member_le_sum
tff(fact_8038_member__le__sum,axiom,
    ! [I: int,A2: set_int,F: fun_int_real] :
      ( aa_set_int_o(member_int(I),A2)
     => ( ! [X2: int] :
            ( aa_set_int_o(member_int(X2),minus_minus_set_int(A2,insert_int(I,bot_bot_set_int)))
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_int_real(F,X2)) )
       => ( aa_set_int_o(finite_finite_int,A2)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(F,I)),groups8778361861064173332t_real(F,A2)) ) ) ) ).

% member_le_sum
tff(fact_8039_member__le__sum,axiom,
    ! [I: real,A2: set_real,F: fun_real_real] :
      ( aa_set_real_o(member_real(I),A2)
     => ( ! [X2: real] :
            ( aa_set_real_o(member_real(X2),minus_minus_set_real(A2,insert_real(I,bot_bot_set_real)))
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(F,X2)) )
       => ( aa_set_real_o(finite_finite_real,A2)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(F,I)),groups8097168146408367636l_real(F,A2)) ) ) ) ).

% member_le_sum
tff(fact_8040_member__le__sum,axiom,
    ! [I: vEBT_VEBT,A2: set_VEBT_VEBT,F: fun_VEBT_VEBT_rat] :
      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(I),A2)
     => ( ! [X2: vEBT_VEBT] :
            ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X2),minus_5127226145743854075T_VEBT(A2,insert_VEBT_VEBT(I,bot_bo8194388402131092736T_VEBT)))
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_VEBT_VEBT_rat(F,X2)) )
       => ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,A2)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_VEBT_VEBT_rat(F,I)),groups136491112297645522BT_rat(F,A2)) ) ) ) ).

% member_le_sum
tff(fact_8041_member__le__sum,axiom,
    ! [I: complex,A2: set_complex,F: fun_complex_rat] :
      ( aa_set_complex_o(member_complex(I),A2)
     => ( ! [X2: complex] :
            ( aa_set_complex_o(member_complex(X2),minus_811609699411566653omplex(A2,insert_complex(I,bot_bot_set_complex)))
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(F,X2)) )
       => ( aa_set_complex_o(finite3207457112153483333omplex,A2)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_complex_rat(F,I)),groups5058264527183730370ex_rat(F,A2)) ) ) ) ).

% member_le_sum
tff(fact_8042_member__le__sum,axiom,
    ! [I: code_integer,A2: set_Code_integer,F: fun_Code_integer_rat] :
      ( aa_set5643387332196575111eger_o(member_Code_integer(I),A2)
     => ( ! [X2: code_integer] :
            ( aa_set5643387332196575111eger_o(member_Code_integer(X2),minus_2355218937544613996nteger(A2,insert_Code_integer(I,bot_bo3990330152332043303nteger)))
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_Code_integer_rat(F,X2)) )
       => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A2)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_Code_integer_rat(F,I)),groups6602215022474089585er_rat(F,A2)) ) ) ) ).

% member_le_sum
tff(fact_8043_member__le__sum,axiom,
    ! [I: int,A2: set_int,F: fun_int_rat] :
      ( aa_set_int_o(member_int(I),A2)
     => ( ! [X2: int] :
            ( aa_set_int_o(member_int(X2),minus_minus_set_int(A2,insert_int(I,bot_bot_set_int)))
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_int_rat(F,X2)) )
       => ( aa_set_int_o(finite_finite_int,A2)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(F,I)),groups3906332499630173760nt_rat(F,A2)) ) ) ) ).

% member_le_sum
tff(fact_8044_member__le__sum,axiom,
    ! [I: real,A2: set_real,F: fun_real_rat] :
      ( aa_set_real_o(member_real(I),A2)
     => ( ! [X2: real] :
            ( aa_set_real_o(member_real(X2),minus_minus_set_real(A2,insert_real(I,bot_bot_set_real)))
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_real_rat(F,X2)) )
       => ( aa_set_real_o(finite_finite_real,A2)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_real_rat(F,I)),groups1300246762558778688al_rat(F,A2)) ) ) ) ).

% member_le_sum
tff(fact_8045_sum__natinterval__diff,axiom,
    ! [F: fun_nat_uint32,M: nat,Nb: nat] :
      groups833757482993574392uint32(aTP_Lamm_apj(F),set_or1269000886237332187st_nat(M,Nb)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb),minus_minus_uint32(aa_nat_uint32(F,M),aa_nat_uint32(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat))),zero_zero_uint32) ).

% sum_natinterval_diff
tff(fact_8046_sum__natinterval__diff,axiom,
    ! [F: fun_nat_rat,M: nat,Nb: nat] :
      groups2906978787729119204at_rat(aTP_Lamm_apk(F),set_or1269000886237332187st_nat(M,Nb)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb),minus_minus_rat(aa_nat_rat(F,M),aa_nat_rat(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat))),zero_zero_rat) ).

% sum_natinterval_diff
tff(fact_8047_sum__natinterval__diff,axiom,
    ! [F: fun_nat_int,M: nat,Nb: nat] :
      groups3539618377306564664at_int(aTP_Lamm_apl(F),set_or1269000886237332187st_nat(M,Nb)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb),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,Nb),one_one_nat))),zero_zero_int) ).

% sum_natinterval_diff
tff(fact_8048_sum__natinterval__diff,axiom,
    ! [F: fun_nat_real,M: nat,Nb: nat] :
      groups6591440286371151544t_real(aTP_Lamm_apm(F),set_or1269000886237332187st_nat(M,Nb)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(F,M)),aa_nat_real(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat))),zero_zero_real) ).

% sum_natinterval_diff
tff(fact_8049_sum__telescope_H_H,axiom,
    ! [M: nat,Nb: nat,F: fun_nat_rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
     => ( groups2906978787729119204at_rat(aTP_Lamm_apn(F),set_or1269000886237332187st_nat(aa_nat_nat(suc,M),Nb)) = minus_minus_rat(aa_nat_rat(F,Nb),aa_nat_rat(F,M)) ) ) ).

% sum_telescope''
tff(fact_8050_sum__telescope_H_H,axiom,
    ! [M: nat,Nb: nat,F: fun_nat_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
     => ( groups3539618377306564664at_int(aTP_Lamm_apo(F),set_or1269000886237332187st_nat(aa_nat_nat(suc,M),Nb)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(F,Nb)),aa_nat_int(F,M)) ) ) ).

% sum_telescope''
tff(fact_8051_sum__telescope_H_H,axiom,
    ! [M: nat,Nb: nat,F: fun_nat_real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
     => ( groups6591440286371151544t_real(aTP_Lamm_app(F),set_or1269000886237332187st_nat(aa_nat_nat(suc,M),Nb)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(F,Nb)),aa_nat_real(F,M)) ) ) ).

% sum_telescope''
tff(fact_8052_mask__eq__sum__exp,axiom,
    ! [Nb: nat] : minus_minus_uint32(aa_nat_uint32(power_power_uint32(numera9087168376688890119uint32(aa_num_num(bit0,one))),Nb),one_one_uint32) = groups833757482993574392uint32(power_power_uint32(numera9087168376688890119uint32(aa_num_num(bit0,one))),collect_nat(aa_nat_fun_nat_o(aTP_Lamm_dv,Nb))) ).

% mask_eq_sum_exp
tff(fact_8053_mask__eq__sum__exp,axiom,
    ! [Nb: nat] : minus_4019991460397169231l_num1(aa_nat5599461682457136004l_num1(power_2184487114949457152l_num1(numera7442385471795722001l_num1(aa_num_num(bit0,one))),Nb),one_on7727431528512463931l_num1) = groups2996710295995929986l_num1(power_2184487114949457152l_num1(numera7442385471795722001l_num1(aa_num_num(bit0,one))),collect_nat(aa_nat_fun_nat_o(aTP_Lamm_dv,Nb))) ).

% mask_eq_sum_exp
tff(fact_8054_mask__eq__sum__exp,axiom,
    ! [Nb: 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))),Nb)),one_one_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_dv,Nb))) ).

% mask_eq_sum_exp
tff(fact_8055_mask__eq__sum__exp,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)),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_dv,Nb))) ).

% mask_eq_sum_exp
tff(fact_8056_sum__gp__multiplied,axiom,
    ! [M: nat,Nb: nat,Xb: uint32] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
     => ( aa_uint32_uint32(times_times_uint32(minus_minus_uint32(one_one_uint32,Xb)),groups833757482993574392uint32(power_power_uint32(Xb),set_or1269000886237332187st_nat(M,Nb))) = minus_minus_uint32(aa_nat_uint32(power_power_uint32(Xb),M),aa_nat_uint32(power_power_uint32(Xb),aa_nat_nat(suc,Nb))) ) ) ).

% sum_gp_multiplied
tff(fact_8057_sum__gp__multiplied,axiom,
    ! [M: nat,Nb: nat,Xb: word_N3645301735248828278l_num1] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
     => ( aa_wor7274019993978516266l_num1(times_7065122842183080059l_num1(minus_4019991460397169231l_num1(one_on7727431528512463931l_num1,Xb)),groups2996710295995929986l_num1(power_2184487114949457152l_num1(Xb),set_or1269000886237332187st_nat(M,Nb))) = minus_4019991460397169231l_num1(aa_nat5599461682457136004l_num1(power_2184487114949457152l_num1(Xb),M),aa_nat5599461682457136004l_num1(power_2184487114949457152l_num1(Xb),aa_nat_nat(suc,Nb))) ) ) ).

% sum_gp_multiplied
tff(fact_8058_sum__gp__multiplied,axiom,
    ! [M: nat,Nb: nat,Xb: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
     => ( aa_rat_rat(times_times_rat(minus_minus_rat(one_one_rat,Xb)),groups2906978787729119204at_rat(power_power_rat(Xb),set_or1269000886237332187st_nat(M,Nb))) = minus_minus_rat(aa_nat_rat(power_power_rat(Xb),M),aa_nat_rat(power_power_rat(Xb),aa_nat_nat(suc,Nb))) ) ) ).

% sum_gp_multiplied
tff(fact_8059_sum__gp__multiplied,axiom,
    ! [M: nat,Nb: nat,Xb: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,one_one_int),Xb)),groups3539618377306564664at_int(aa_int_fun_nat_int(power_power_int,Xb),set_or1269000886237332187st_nat(M,Nb))) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,Xb),M)),aa_nat_int(aa_int_fun_nat_int(power_power_int,Xb),aa_nat_nat(suc,Nb))) ) ) ).

% sum_gp_multiplied
tff(fact_8060_sum__gp__multiplied,axiom,
    ! [M: nat,Nb: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),Xb)),groups6591440286371151544t_real(aa_real_fun_nat_real(power_power_real,Xb),set_or1269000886237332187st_nat(M,Nb))) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),M)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_nat_nat(suc,Nb))) ) ) ).

% sum_gp_multiplied
tff(fact_8061_sum_Oin__pairs,axiom,
    ! [G: fun_nat_rat,M: nat,Nb: nat] : groups2906978787729119204at_rat(G,set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),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))),Nb)))) = groups2906978787729119204at_rat(aTP_Lamm_apq(G),set_or1269000886237332187st_nat(M,Nb)) ).

% sum.in_pairs
tff(fact_8062_sum_Oin__pairs,axiom,
    ! [G: fun_nat_int,M: nat,Nb: nat] : groups3539618377306564664at_int(G,set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),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))),Nb)))) = groups3539618377306564664at_int(aTP_Lamm_apr(G),set_or1269000886237332187st_nat(M,Nb)) ).

% sum.in_pairs
tff(fact_8063_sum_Oin__pairs,axiom,
    ! [G: fun_nat_nat,M: nat,Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),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))),Nb)))) = aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_aps(G)),set_or1269000886237332187st_nat(M,Nb)) ).

% sum.in_pairs
tff(fact_8064_sum_Oin__pairs,axiom,
    ! [G: fun_nat_real,M: nat,Nb: nat] : groups6591440286371151544t_real(G,set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),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))),Nb)))) = groups6591440286371151544t_real(aTP_Lamm_apt(G),set_or1269000886237332187st_nat(M,Nb)) ).

% sum.in_pairs
tff(fact_8065_VEBT__internal_Oheight_Ocases,axiom,
    ! [Xb: vEBT_VEBT] :
      ( ! [A3: $o,B2: $o] : Xb != vEBT_Leaf((A3),(B2))
     => ~ ! [Uu2: option4927543243414619207at_nat,Deg2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] : Xb != aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uu2,Deg2,TreeList),Summary) ) ).

% VEBT_internal.height.cases
tff(fact_8066_sums__if_H,axiom,
    ! [G: fun_nat_real,Xb: real] :
      ( aa_real_o(sums_real(G),Xb)
     => aa_real_o(sums_real(aTP_Lamm_apu(G)),Xb) ) ).

% sums_if'
tff(fact_8067_sums__if,axiom,
    ! [G: fun_nat_real,Xb: real,F: fun_nat_real,Y: real] :
      ( aa_real_o(sums_real(G),Xb)
     => ( aa_real_o(sums_real(F),Y)
       => aa_real_o(sums_real(aa_fun1839206346889086768t_real(aTP_Lamm_apv(G),F)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Y)) ) ) ).

% sums_if
tff(fact_8068_mask__eq__sum__exp__nat,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)),aa_nat_nat(suc,zero_zero_nat)) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),collect_nat(aa_nat_fun_nat_o(aTP_Lamm_dv,Nb))) ).

% mask_eq_sum_exp_nat
tff(fact_8069_gauss__sum__nat,axiom,
    ! [Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_di),set_or1269000886237332187st_nat(zero_zero_nat,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),aa_nat_nat(suc,Nb))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% gauss_sum_nat
tff(fact_8070_double__gauss__sum,axiom,
    ! [Nb: nat] : aa_wor7274019993978516266l_num1(times_7065122842183080059l_num1(numera7442385471795722001l_num1(aa_num_num(bit0,one))),groups2996710295995929986l_num1(semiri8819519690708144855l_num1,set_or1269000886237332187st_nat(zero_zero_nat,Nb))) = aa_wor7274019993978516266l_num1(times_7065122842183080059l_num1(aa_nat5599461682457136004l_num1(semiri8819519690708144855l_num1,Nb)),plus_p361126936061061375l_num1(aa_nat5599461682457136004l_num1(semiri8819519690708144855l_num1,Nb),one_on7727431528512463931l_num1)) ).

% double_gauss_sum
tff(fact_8071_double__gauss__sum,axiom,
    ! [Nb: nat] : aa_rat_rat(times_times_rat(numeral_numeral_rat(aa_num_num(bit0,one))),groups2906978787729119204at_rat(semiri681578069525770553at_rat,set_or1269000886237332187st_nat(zero_zero_nat,Nb))) = aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,Nb)),aa_rat_rat(plus_plus_rat(aa_nat_rat(semiri681578069525770553at_rat,Nb)),one_one_rat)) ).

% double_gauss_sum
tff(fact_8072_double__gauss__sum,axiom,
    ! [Nb: nat] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),groups3539618377306564664at_int(semiri1314217659103216013at_int,set_or1269000886237332187st_nat(zero_zero_nat,Nb))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),one_one_int)) ).

% double_gauss_sum
tff(fact_8073_double__gauss__sum,axiom,
    ! [Nb: nat] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),groups7501900531339628137nteger(semiri4939895301339042750nteger,set_or1269000886237332187st_nat(zero_zero_nat,Nb))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,Nb)),aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,Nb)),one_one_Code_integer)) ).

% double_gauss_sum
tff(fact_8074_double__gauss__sum,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_set_nat_nat(groups3542108847815614940at_nat(semiri1316708129612266289at_nat),set_or1269000886237332187st_nat(zero_zero_nat,Nb))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,Nb)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(semiri1316708129612266289at_nat,Nb)),one_one_nat)) ).

% double_gauss_sum
tff(fact_8075_double__gauss__sum,axiom,
    ! [Nb: nat] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),groups6591440286371151544t_real(semiri5074537144036343181t_real,set_or1269000886237332187st_nat(zero_zero_nat,Nb))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),one_one_real)) ).

% double_gauss_sum
tff(fact_8076_double__arith__series,axiom,
    ! [A: word_N3645301735248828278l_num1,D: word_N3645301735248828278l_num1,Nb: nat] : aa_wor7274019993978516266l_num1(times_7065122842183080059l_num1(numera7442385471795722001l_num1(aa_num_num(bit0,one))),groups2996710295995929986l_num1(aa_wor1031616255390178039l_num1(aTP_Lamm_apw(A),D),set_or1269000886237332187st_nat(zero_zero_nat,Nb))) = aa_wor7274019993978516266l_num1(times_7065122842183080059l_num1(plus_p361126936061061375l_num1(aa_nat5599461682457136004l_num1(semiri8819519690708144855l_num1,Nb),one_on7727431528512463931l_num1)),plus_p361126936061061375l_num1(aa_wor7274019993978516266l_num1(times_7065122842183080059l_num1(numera7442385471795722001l_num1(aa_num_num(bit0,one))),A),aa_wor7274019993978516266l_num1(times_7065122842183080059l_num1(aa_nat5599461682457136004l_num1(semiri8819519690708144855l_num1,Nb)),D))) ).

% double_arith_series
tff(fact_8077_double__arith__series,axiom,
    ! [A: rat,D: rat,Nb: nat] : aa_rat_rat(times_times_rat(numeral_numeral_rat(aa_num_num(bit0,one))),groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aTP_Lamm_apx(A),D),set_or1269000886237332187st_nat(zero_zero_nat,Nb))) = aa_rat_rat(times_times_rat(aa_rat_rat(plus_plus_rat(aa_nat_rat(semiri681578069525770553at_rat,Nb)),one_one_rat)),aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(numeral_numeral_rat(aa_num_num(bit0,one))),A)),aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,Nb)),D))) ).

% double_arith_series
tff(fact_8078_double__arith__series,axiom,
    ! [A: int,D: int,Nb: nat] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_apy(A),D),set_or1269000886237332187st_nat(zero_zero_nat,Nb))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),one_one_int)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),D))) ).

% double_arith_series
tff(fact_8079_double__arith__series,axiom,
    ! [A: code_integer,D: code_integer,Nb: nat] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),groups7501900531339628137nteger(aa_Cod1578151519121547077nteger(aTP_Lamm_apz(A),D),set_or1269000886237332187st_nat(zero_zero_nat,Nb))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,Nb)),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,Nb)),D))) ).

% double_arith_series
tff(fact_8080_double__arith__series,axiom,
    ! [A: nat,D: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_aqa(A),D)),set_or1269000886237332187st_nat(zero_zero_nat,Nb))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(semiri1316708129612266289at_nat,Nb)),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,Nb)),D))) ).

% double_arith_series
tff(fact_8081_double__arith__series,axiom,
    ! [A: real,D: real,Nb: nat] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_aqb(A),D),set_or1269000886237332187st_nat(zero_zero_nat,Nb))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),one_one_real)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),D))) ).

% double_arith_series
tff(fact_8082_arith__series__nat,axiom,
    ! [A: nat,D: nat,Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_aqc(A),D)),set_or1269000886237332187st_nat(zero_zero_nat,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,Nb)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),D)))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% arith_series_nat
tff(fact_8083_Sum__Icc__nat,axiom,
    ! [M: nat,Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_di),set_or1269000886237332187st_nat(M,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,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_8084_double__gauss__sum__from__Suc__0,axiom,
    ! [Nb: nat] : aa_wor7274019993978516266l_num1(times_7065122842183080059l_num1(numera7442385471795722001l_num1(aa_num_num(bit0,one))),groups2996710295995929986l_num1(semiri8819519690708144855l_num1,set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat),Nb))) = aa_wor7274019993978516266l_num1(times_7065122842183080059l_num1(aa_nat5599461682457136004l_num1(semiri8819519690708144855l_num1,Nb)),plus_p361126936061061375l_num1(aa_nat5599461682457136004l_num1(semiri8819519690708144855l_num1,Nb),one_on7727431528512463931l_num1)) ).

% double_gauss_sum_from_Suc_0
tff(fact_8085_double__gauss__sum__from__Suc__0,axiom,
    ! [Nb: nat] : aa_rat_rat(times_times_rat(numeral_numeral_rat(aa_num_num(bit0,one))),groups2906978787729119204at_rat(semiri681578069525770553at_rat,set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat),Nb))) = aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,Nb)),aa_rat_rat(plus_plus_rat(aa_nat_rat(semiri681578069525770553at_rat,Nb)),one_one_rat)) ).

% double_gauss_sum_from_Suc_0
tff(fact_8086_double__gauss__sum__from__Suc__0,axiom,
    ! [Nb: nat] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),groups3539618377306564664at_int(semiri1314217659103216013at_int,set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat),Nb))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),one_one_int)) ).

% double_gauss_sum_from_Suc_0
tff(fact_8087_double__gauss__sum__from__Suc__0,axiom,
    ! [Nb: nat] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),groups7501900531339628137nteger(semiri4939895301339042750nteger,set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat),Nb))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,Nb)),aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,Nb)),one_one_Code_integer)) ).

% double_gauss_sum_from_Suc_0
tff(fact_8088_double__gauss__sum__from__Suc__0,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_set_nat_nat(groups3542108847815614940at_nat(semiri1316708129612266289at_nat),set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat),Nb))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,Nb)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(semiri1316708129612266289at_nat,Nb)),one_one_nat)) ).

% double_gauss_sum_from_Suc_0
tff(fact_8089_double__gauss__sum__from__Suc__0,axiom,
    ! [Nb: nat] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),groups6591440286371151544t_real(semiri5074537144036343181t_real,set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat),Nb))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),one_one_real)) ).

% double_gauss_sum_from_Suc_0
tff(fact_8090_gauss__sum,axiom,
    ! [Nb: nat] : groups3539618377306564664at_int(semiri1314217659103216013at_int,set_or1269000886237332187st_nat(zero_zero_nat,Nb)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),one_one_int))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% gauss_sum
tff(fact_8091_gauss__sum,axiom,
    ! [Nb: nat] : groups7501900531339628137nteger(semiri4939895301339042750nteger,set_or1269000886237332187st_nat(zero_zero_nat,Nb)) = aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,Nb)),aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,Nb)),one_one_Code_integer))),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) ).

% gauss_sum
tff(fact_8092_gauss__sum,axiom,
    ! [Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(semiri1316708129612266289at_nat),set_or1269000886237332187st_nat(zero_zero_nat,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,Nb)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(semiri1316708129612266289at_nat,Nb)),one_one_nat))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% gauss_sum
tff(fact_8093_arith__series,axiom,
    ! [A: int,D: int,Nb: nat] : groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_apy(A),D),set_or1269000886237332187st_nat(zero_zero_nat,Nb)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),one_one_int)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),D)))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% arith_series
tff(fact_8094_arith__series,axiom,
    ! [A: code_integer,D: code_integer,Nb: nat] : groups7501900531339628137nteger(aa_Cod1578151519121547077nteger(aTP_Lamm_apz(A),D),set_or1269000886237332187st_nat(zero_zero_nat,Nb)) = aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,Nb)),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,Nb)),D)))),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) ).

% arith_series
tff(fact_8095_arith__series,axiom,
    ! [A: nat,D: nat,Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_aqa(A),D)),set_or1269000886237332187st_nat(zero_zero_nat,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(semiri1316708129612266289at_nat,Nb)),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,Nb)),D)))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% arith_series
tff(fact_8096_sum__gp__offset,axiom,
    ! [Xb: rat,M: nat,Nb: nat] :
      groups2906978787729119204at_rat(power_power_rat(Xb),set_or1269000886237332187st_nat(M,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Nb))) = $ite(Xb = one_one_rat,aa_rat_rat(plus_plus_rat(aa_nat_rat(semiri681578069525770553at_rat,Nb)),one_one_rat),divide_divide_rat(aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(Xb),M)),minus_minus_rat(one_one_rat,aa_nat_rat(power_power_rat(Xb),aa_nat_nat(suc,Nb)))),minus_minus_rat(one_one_rat,Xb))) ).

% sum_gp_offset
tff(fact_8097_sum__gp__offset,axiom,
    ! [Xb: real,M: nat,Nb: nat] :
      groups6591440286371151544t_real(aa_real_fun_nat_real(power_power_real,Xb),set_or1269000886237332187st_nat(M,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Nb))) = $ite(Xb = one_one_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),one_one_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),M)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_nat_nat(suc,Nb))))),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),Xb))) ).

% sum_gp_offset
tff(fact_8098_gauss__sum__from__Suc__0,axiom,
    ! [Nb: nat] : groups3539618377306564664at_int(semiri1314217659103216013at_int,set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat),Nb)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),one_one_int))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% gauss_sum_from_Suc_0
tff(fact_8099_gauss__sum__from__Suc__0,axiom,
    ! [Nb: nat] : groups7501900531339628137nteger(semiri4939895301339042750nteger,set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat),Nb)) = aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,Nb)),aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,Nb)),one_one_Code_integer))),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) ).

% gauss_sum_from_Suc_0
tff(fact_8100_gauss__sum__from__Suc__0,axiom,
    ! [Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(semiri1316708129612266289at_nat),set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,Nb)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(semiri1316708129612266289at_nat,Nb)),one_one_nat))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% gauss_sum_from_Suc_0
tff(fact_8101_geometric__sums,axiom,
    ! [C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,C)),one_one_real)
     => aa_real_o(sums_real(aa_real_fun_nat_real(power_power_real,C)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),C))) ) ).

% geometric_sums
tff(fact_8102_geometric__sums,axiom,
    ! [C: complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,C)),one_one_real)
     => aa_complex_o(sums_complex(aa_com8181001469867462503omplex(power_power_complex,C)),aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,one_one_complex),aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,one_one_complex),C))) ) ).

% geometric_sums
tff(fact_8103_power__half__series,axiom,
    aa_real_o(sums_real(aTP_Lamm_aqd),one_one_real) ).

% power_half_series
tff(fact_8104_sums__zero,axiom,
    aa_real_o(sums_real(aTP_Lamm_aiq),zero_zero_real) ).

% sums_zero
tff(fact_8105_sums__zero,axiom,
    aa_nat_o(sums_nat(aTP_Lamm_en),zero_zero_nat) ).

% sums_zero
tff(fact_8106_sums__zero,axiom,
    aa_int_o(sums_int(aTP_Lamm_aqe),zero_zero_int) ).

% sums_zero
tff(fact_8107_sums__If__finite__set_H,axiom,
    ! [G: fun_nat_real,S2: real,A2: set_nat,S6: real,F: fun_nat_real] :
      ( aa_real_o(sums_real(G),S2)
     => ( aa_set_nat_o(finite_finite_nat,A2)
       => ( ( S6 = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,S2),groups6591440286371151544t_real(aa_fun1839206346889086768t_real(aTP_Lamm_aqf(G),F),A2)) )
         => aa_real_o(sums_real(aa_fun1839206346889086768t_real(aa_set1316517696749040435t_real(aTP_Lamm_aqg(G),A2),F)),S6) ) ) ) ).

% sums_If_finite_set'
tff(fact_8108_powser__sums__if,axiom,
    ! [M: nat,Z: real] : aa_real_o(sums_real(aa_real_fun_nat_real(aTP_Lamm_aqh(M),Z)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Z),M)) ).

% powser_sums_if
tff(fact_8109_powser__sums__if,axiom,
    ! [M: nat,Z: int] : aa_int_o(sums_int(aa_int_fun_nat_int(aTP_Lamm_aqi(M),Z)),aa_nat_int(aa_int_fun_nat_int(power_power_int,Z),M)) ).

% powser_sums_if
tff(fact_8110_int__sum,axiom,
    ! [F: fun_int_nat,A2: set_int] : aa_nat_int(semiri1314217659103216013at_int,groups4541462559716669496nt_nat(F,A2)) = groups4538972089207619220nt_int(aTP_Lamm_ais(F),A2) ).

% int_sum
tff(fact_8111_int__sum,axiom,
    ! [F: fun_nat_nat,A2: set_nat] : aa_nat_int(semiri1314217659103216013at_int,aa_set_nat_nat(groups3542108847815614940at_nat(F),A2)) = groups3539618377306564664at_int(aTP_Lamm_ait(F),A2) ).

% int_sum
tff(fact_8112_sum__subtractf__nat,axiom,
    ! [A2: set_VEBT_VEBT,G: fun_VEBT_VEBT_nat,F: fun_VEBT_VEBT_nat] :
      ( ! [X2: vEBT_VEBT] :
          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X2),A2)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_VEBT_VEBT_nat(G,X2)),aa_VEBT_VEBT_nat(F,X2)) )
     => ( groups771621172384141258BT_nat(aa_fun8284231930576791164BT_nat(aTP_Lamm_aqj(G),F),A2) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups771621172384141258BT_nat(F,A2)),groups771621172384141258BT_nat(G,A2)) ) ) ).

% sum_subtractf_nat
tff(fact_8113_sum__subtractf__nat,axiom,
    ! [A2: set_real,G: fun_real_nat,F: fun_real_nat] :
      ( ! [X2: real] :
          ( aa_set_real_o(member_real(X2),A2)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_real_nat(G,X2)),aa_real_nat(F,X2)) )
     => ( groups1935376822645274424al_nat(aa_fun211602392931233840al_nat(aTP_Lamm_aqk(G),F),A2) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups1935376822645274424al_nat(F,A2)),groups1935376822645274424al_nat(G,A2)) ) ) ).

% sum_subtractf_nat
tff(fact_8114_sum__subtractf__nat,axiom,
    ! [A2: set_int,G: fun_int_nat,F: fun_int_nat] :
      ( ! [X2: int] :
          ( aa_set_int_o(member_int(X2),A2)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(G,X2)),aa_int_nat(F,X2)) )
     => ( groups4541462559716669496nt_nat(aa_fun6506752150744536624nt_nat(aTP_Lamm_aql(G),F),A2) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups4541462559716669496nt_nat(F,A2)),groups4541462559716669496nt_nat(G,A2)) ) ) ).

% sum_subtractf_nat
tff(fact_8115_sum__subtractf__nat,axiom,
    ! [A2: set_nat,G: fun_nat_nat,F: fun_nat_nat] :
      ( ! [X2: nat] :
          ( aa_set_nat_o(member_nat(X2),A2)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(G,X2)),aa_nat_nat(F,X2)) )
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_aqm(G),F)),A2) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),A2)),aa_set_nat_nat(groups3542108847815614940at_nat(G),A2)) ) ) ).

% sum_subtractf_nat
tff(fact_8116_sum__SucD,axiom,
    ! [F: fun_nat_nat,A2: set_nat,Nb: nat] :
      ( ( aa_set_nat_nat(groups3542108847815614940at_nat(F),A2) = aa_nat_nat(suc,Nb) )
     => ? [X2: nat] :
          ( aa_set_nat_o(member_nat(X2),A2)
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(F,X2)) ) ) ).

% sum_SucD
tff(fact_8117_sum__eq__Suc0__iff,axiom,
    ! [A2: set_int,F: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( ( groups4541462559716669496nt_nat(F,A2) = aa_nat_nat(suc,zero_zero_nat) )
      <=> ? [X: int] :
            ( aa_set_int_o(member_int(X),A2)
            & ( aa_int_nat(F,X) = aa_nat_nat(suc,zero_zero_nat) )
            & ! [Xa3: int] :
                ( aa_set_int_o(member_int(Xa3),A2)
               => ( ( X != Xa3 )
                 => ( aa_int_nat(F,Xa3) = zero_zero_nat ) ) ) ) ) ) ).

% sum_eq_Suc0_iff
tff(fact_8118_sum__eq__Suc0__iff,axiom,
    ! [A2: set_complex,F: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( ( groups5693394587270226106ex_nat(F,A2) = aa_nat_nat(suc,zero_zero_nat) )
      <=> ? [X: complex] :
            ( aa_set_complex_o(member_complex(X),A2)
            & ( aa_complex_nat(F,X) = aa_nat_nat(suc,zero_zero_nat) )
            & ! [Xa3: complex] :
                ( aa_set_complex_o(member_complex(Xa3),A2)
               => ( ( X != Xa3 )
                 => ( aa_complex_nat(F,Xa3) = zero_zero_nat ) ) ) ) ) ) ).

% sum_eq_Suc0_iff
tff(fact_8119_sum__eq__Suc0__iff,axiom,
    ! [A2: set_Code_integer,F: fun_Code_integer_nat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A2)
     => ( ( groups7237345082560585321er_nat(F,A2) = aa_nat_nat(suc,zero_zero_nat) )
      <=> ? [X: code_integer] :
            ( aa_set5643387332196575111eger_o(member_Code_integer(X),A2)
            & ( aa_Code_integer_nat(F,X) = aa_nat_nat(suc,zero_zero_nat) )
            & ! [Xa3: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(Xa3),A2)
               => ( ( X != Xa3 )
                 => ( aa_Code_integer_nat(F,Xa3) = zero_zero_nat ) ) ) ) ) ) ).

% sum_eq_Suc0_iff
tff(fact_8120_sum__eq__Suc0__iff,axiom,
    ! [A2: set_nat,F: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( ( aa_set_nat_nat(groups3542108847815614940at_nat(F),A2) = aa_nat_nat(suc,zero_zero_nat) )
      <=> ? [X: nat] :
            ( aa_set_nat_o(member_nat(X),A2)
            & ( aa_nat_nat(F,X) = aa_nat_nat(suc,zero_zero_nat) )
            & ! [Xa3: nat] :
                ( aa_set_nat_o(member_nat(Xa3),A2)
               => ( ( X != Xa3 )
                 => ( aa_nat_nat(F,Xa3) = zero_zero_nat ) ) ) ) ) ) ).

% sum_eq_Suc0_iff
tff(fact_8121_sum__eq__1__iff,axiom,
    ! [A2: set_int,F: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( ( groups4541462559716669496nt_nat(F,A2) = one_one_nat )
      <=> ? [X: int] :
            ( aa_set_int_o(member_int(X),A2)
            & ( aa_int_nat(F,X) = one_one_nat )
            & ! [Xa3: int] :
                ( aa_set_int_o(member_int(Xa3),A2)
               => ( ( X != Xa3 )
                 => ( aa_int_nat(F,Xa3) = zero_zero_nat ) ) ) ) ) ) ).

% sum_eq_1_iff
tff(fact_8122_sum__eq__1__iff,axiom,
    ! [A2: set_complex,F: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( ( groups5693394587270226106ex_nat(F,A2) = one_one_nat )
      <=> ? [X: complex] :
            ( aa_set_complex_o(member_complex(X),A2)
            & ( aa_complex_nat(F,X) = one_one_nat )
            & ! [Xa3: complex] :
                ( aa_set_complex_o(member_complex(Xa3),A2)
               => ( ( X != Xa3 )
                 => ( aa_complex_nat(F,Xa3) = zero_zero_nat ) ) ) ) ) ) ).

% sum_eq_1_iff
tff(fact_8123_sum__eq__1__iff,axiom,
    ! [A2: set_Code_integer,F: fun_Code_integer_nat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A2)
     => ( ( groups7237345082560585321er_nat(F,A2) = one_one_nat )
      <=> ? [X: code_integer] :
            ( aa_set5643387332196575111eger_o(member_Code_integer(X),A2)
            & ( aa_Code_integer_nat(F,X) = one_one_nat )
            & ! [Xa3: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(Xa3),A2)
               => ( ( X != Xa3 )
                 => ( aa_Code_integer_nat(F,Xa3) = zero_zero_nat ) ) ) ) ) ) ).

% sum_eq_1_iff
tff(fact_8124_sum__eq__1__iff,axiom,
    ! [A2: set_nat,F: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( ( aa_set_nat_nat(groups3542108847815614940at_nat(F),A2) = one_one_nat )
      <=> ? [X: nat] :
            ( aa_set_nat_o(member_nat(X),A2)
            & ( aa_nat_nat(F,X) = one_one_nat )
            & ! [Xa3: nat] :
                ( aa_set_nat_o(member_nat(Xa3),A2)
               => ( ( X != Xa3 )
                 => ( aa_nat_nat(F,Xa3) = zero_zero_nat ) ) ) ) ) ) ).

% sum_eq_1_iff
tff(fact_8125_sum__roots__unity,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),Nb)
     => ( groups7754918857620584856omplex(aTP_Lamm_aqn,collect_complex(aa_nat_fun_complex_o(aTP_Lamm_adr,Nb))) = zero_zero_complex ) ) ).

% sum_roots_unity
tff(fact_8126_sum__nth__roots,axiom,
    ! [Nb: nat,C: complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),Nb)
     => ( groups7754918857620584856omplex(aTP_Lamm_aqn,collect_complex(aa_com5808801942499186519plex_o(aTP_Lamm_afd(Nb),C))) = zero_zero_complex ) ) ).

% sum_nth_roots
tff(fact_8127_sum__diff__nat,axiom,
    ! [B4: set_complex,A2: set_complex,F: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,B4)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(B4),A2)
       => ( groups5693394587270226106ex_nat(F,minus_811609699411566653omplex(A2,B4)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups5693394587270226106ex_nat(F,A2)),groups5693394587270226106ex_nat(F,B4)) ) ) ) ).

% sum_diff_nat
tff(fact_8128_sum__diff__nat,axiom,
    ! [B4: set_Code_integer,A2: set_Code_integer,F: fun_Code_integer_nat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B4)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(B4),A2)
       => ( groups7237345082560585321er_nat(F,minus_2355218937544613996nteger(A2,B4)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups7237345082560585321er_nat(F,A2)),groups7237345082560585321er_nat(F,B4)) ) ) ) ).

% sum_diff_nat
tff(fact_8129_sum__diff__nat,axiom,
    ! [B4: set_int,A2: set_int,F: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,B4)
     => ( aa_set_int_o(ord_less_eq_set_int(B4),A2)
       => ( groups4541462559716669496nt_nat(F,minus_minus_set_int(A2,B4)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups4541462559716669496nt_nat(F,A2)),groups4541462559716669496nt_nat(F,B4)) ) ) ) ).

% sum_diff_nat
tff(fact_8130_sum__diff__nat,axiom,
    ! [B4: set_nat,A2: set_nat,F: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,B4)
     => ( aa_set_nat_o(ord_less_eq_set_nat(B4),A2)
       => ( aa_set_nat_nat(groups3542108847815614940at_nat(F),minus_minus_set_nat(A2,B4)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),A2)),aa_set_nat_nat(groups3542108847815614940at_nat(F),B4)) ) ) ) ).

% sum_diff_nat
tff(fact_8131_sum__diff1__nat,axiom,
    ! [F: fun_VEBT_VEBT_nat,A2: set_VEBT_VEBT,A: vEBT_VEBT] :
      groups771621172384141258BT_nat(F,minus_5127226145743854075T_VEBT(A2,insert_VEBT_VEBT(A,bot_bo8194388402131092736T_VEBT))) = $ite(aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A),A2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups771621172384141258BT_nat(F,A2)),aa_VEBT_VEBT_nat(F,A)),groups771621172384141258BT_nat(F,A2)) ).

% sum_diff1_nat
tff(fact_8132_sum__diff1__nat,axiom,
    ! [F: fun_int_nat,A2: set_int,A: int] :
      groups4541462559716669496nt_nat(F,minus_minus_set_int(A2,insert_int(A,bot_bot_set_int))) = $ite(aa_set_int_o(member_int(A),A2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups4541462559716669496nt_nat(F,A2)),aa_int_nat(F,A)),groups4541462559716669496nt_nat(F,A2)) ).

% sum_diff1_nat
tff(fact_8133_sum__diff1__nat,axiom,
    ! [F: fun_real_nat,A2: set_real,A: real] :
      groups1935376822645274424al_nat(F,minus_minus_set_real(A2,insert_real(A,bot_bot_set_real))) = $ite(aa_set_real_o(member_real(A),A2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups1935376822645274424al_nat(F,A2)),aa_real_nat(F,A)),groups1935376822645274424al_nat(F,A2)) ).

% sum_diff1_nat
tff(fact_8134_sum__diff1__nat,axiom,
    ! [F: fun_nat_nat,A2: set_nat,A: nat] :
      aa_set_nat_nat(groups3542108847815614940at_nat(F),minus_minus_set_nat(A2,insert_nat(A,bot_bot_set_nat))) = $ite(aa_set_nat_o(member_nat(A),A2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),A2)),aa_nat_nat(F,A)),aa_set_nat_nat(groups3542108847815614940at_nat(F),A2)) ).

% sum_diff1_nat
tff(fact_8135_sums__le,axiom,
    ! [F: fun_nat_real,G: fun_nat_real,S3: real,Ta: real] :
      ( ! [N2: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(F,N2)),aa_nat_real(G,N2))
     => ( aa_real_o(sums_real(F),S3)
       => ( aa_real_o(sums_real(G),Ta)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,S3),Ta) ) ) ) ).

% sums_le
tff(fact_8136_sums__le,axiom,
    ! [F: fun_nat_nat,G: fun_nat_nat,S3: nat,Ta: nat] :
      ( ! [N2: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F,N2)),aa_nat_nat(G,N2))
     => ( aa_nat_o(sums_nat(F),S3)
       => ( aa_nat_o(sums_nat(G),Ta)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,S3),Ta) ) ) ) ).

% sums_le
tff(fact_8137_sums__le,axiom,
    ! [F: fun_nat_int,G: fun_nat_int,S3: int,Ta: int] :
      ( ! [N2: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,N2)),aa_nat_int(G,N2))
     => ( aa_int_o(sums_int(F),S3)
       => ( aa_int_o(sums_int(G),Ta)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,S3),Ta) ) ) ) ).

% sums_le
tff(fact_8138_sums__single,axiom,
    ! [I: nat,F: fun_nat_real] : aa_real_o(sums_real(aa_fun1839206346889086768t_real(aTP_Lamm_aqo(I),F)),aa_nat_real(F,I)) ).

% sums_single
tff(fact_8139_sums__single,axiom,
    ! [I: nat,F: fun_nat_nat] : aa_nat_o(sums_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_aqp(I),F)),aa_nat_nat(F,I)) ).

% sums_single
tff(fact_8140_sums__single,axiom,
    ! [I: nat,F: fun_nat_int] : aa_int_o(sums_int(aa_fun5565043122120104496at_int(aTP_Lamm_aqq(I),F)),aa_nat_int(F,I)) ).

% sums_single
tff(fact_8141_sums__add,axiom,
    ! [F: fun_nat_real,A: real,G: fun_nat_real,B: real] :
      ( aa_real_o(sums_real(F),A)
     => ( aa_real_o(sums_real(G),B)
       => aa_real_o(sums_real(aa_fun1839206346889086768t_real(aTP_Lamm_akr(F),G)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)) ) ) ).

% sums_add
tff(fact_8142_sums__add,axiom,
    ! [F: fun_nat_nat,A: nat,G: fun_nat_nat,B: nat] :
      ( aa_nat_o(sums_nat(F),A)
     => ( aa_nat_o(sums_nat(G),B)
       => aa_nat_o(sums_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_ako(F),G)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)) ) ) ).

% sums_add
tff(fact_8143_sums__add,axiom,
    ! [F: fun_nat_int,A: int,G: fun_nat_int,B: int] :
      ( aa_int_o(sums_int(F),A)
     => ( aa_int_o(sums_int(G),B)
       => aa_int_o(sums_int(aa_fun5565043122120104496at_int(aTP_Lamm_aqr(F),G)),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)) ) ) ).

% sums_add
tff(fact_8144_sums__mult2,axiom,
    ! [F: fun_nat_real,A: real,C: real] :
      ( aa_real_o(sums_real(F),A)
     => aa_real_o(sums_real(aa_real_fun_nat_real(aTP_Lamm_akz(F),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)) ) ).

% sums_mult2
tff(fact_8145_sums__mult,axiom,
    ! [F: fun_nat_real,A: real,C: real] :
      ( aa_real_o(sums_real(F),A)
     => aa_real_o(sums_real(aa_real_fun_nat_real(aTP_Lamm_aqs(F),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)) ) ).

% sums_mult
tff(fact_8146_sums__diff,axiom,
    ! [F: fun_nat_real,A: real,G: fun_nat_real,B: real] :
      ( aa_real_o(sums_real(F),A)
     => ( aa_real_o(sums_real(G),B)
       => aa_real_o(sums_real(aa_fun1839206346889086768t_real(aTP_Lamm_alk(F),G)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)) ) ) ).

% sums_diff
tff(fact_8147_sums__divide,axiom,
    ! [F: fun_nat_real,A: real,C: real] :
      ( aa_real_o(sums_real(F),A)
     => aa_real_o(sums_real(aa_real_fun_nat_real(aTP_Lamm_alm(F),C)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),C)) ) ).

% sums_divide
tff(fact_8148_sums__sum,axiom,
    ! [I5: set_VEBT_VEBT,F: fun_VE2989161857680785728t_real,Xb: fun_VEBT_VEBT_real] :
      ( ! [I3: vEBT_VEBT] :
          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(I3),I5)
         => aa_real_o(sums_real(aa_VEB3736018289492012273t_real(F,I3)),aa_VEBT_VEBT_real(Xb,I3)) )
     => aa_real_o(sums_real(aa_fun6147343916763992189t_real(aTP_Lamm_aqu(I5),F)),groups2240296850493347238T_real(Xb,I5)) ) ).

% sums_sum
tff(fact_8149_sums__sum,axiom,
    ! [I5: set_real,F: fun_re4238438383274954274t_real,Xb: fun_real_real] :
      ( ! [I3: real] :
          ( aa_set_real_o(member_real(I3),I5)
         => aa_real_o(sums_real(aa_real_fun_nat_real(F,I3)),aa_real_real(Xb,I3)) )
     => aa_real_o(sums_real(aa_fun628238740774635481t_real(aTP_Lamm_aqw(I5),F)),groups8097168146408367636l_real(Xb,I5)) ) ).

% sums_sum
tff(fact_8150_sums__sum,axiom,
    ! [I5: set_int,F: fun_int_fun_nat_real,Xb: fun_int_real] :
      ( ! [I3: int] :
          ( aa_set_int_o(member_int(I3),I5)
         => aa_real_o(sums_real(aa_int_fun_nat_real(F,I3)),aa_int_real(Xb,I3)) )
     => aa_real_o(sums_real(aa_fun2842512154077127769t_real(aTP_Lamm_aqy(I5),F)),groups8778361861064173332t_real(Xb,I5)) ) ).

% sums_sum
tff(fact_8151_sums__sum,axiom,
    ! [I5: set_nat,F: fun_nat_fun_nat_nat,Xb: fun_nat_nat] :
      ( ! [I3: nat] :
          ( aa_set_nat_o(member_nat(I3),I5)
         => aa_nat_o(sums_nat(aa_nat_fun_nat_nat(F,I3)),aa_nat_nat(Xb,I3)) )
     => aa_nat_o(sums_nat(aa_fun8466562714962205125at_nat(aTP_Lamm_aqz(I5),F)),aa_set_nat_nat(groups3542108847815614940at_nat(Xb),I5)) ) ).

% sums_sum
tff(fact_8152_sums__sum,axiom,
    ! [I5: set_complex,F: fun_co3258106519541737638omplex,Xb: fun_complex_complex] :
      ( ! [I3: complex] :
          ( aa_set_complex_o(member_complex(I3),I5)
         => aa_complex_o(sums_complex(aa_com8181001469867462503omplex(F,I3)),aa_complex_complex(Xb,I3)) )
     => aa_complex_o(sums_complex(aa_fun5003373910569599327omplex(aTP_Lamm_arb(I5),F)),groups7754918857620584856omplex(Xb,I5)) ) ).

% sums_sum
tff(fact_8153_sums__sum,axiom,
    ! [I5: set_int,F: fun_int_fun_nat_int,Xb: fun_int_int] :
      ( ! [I3: int] :
          ( aa_set_int_o(member_int(I3),I5)
         => aa_int_o(sums_int(aa_int_fun_nat_int(F,I3)),aa_int_int(Xb,I3)) )
     => aa_int_o(sums_int(aa_fun8710705519803144921at_int(aTP_Lamm_ard(I5),F)),groups4538972089207619220nt_int(Xb,I5)) ) ).

% sums_sum
tff(fact_8154_sums__sum,axiom,
    ! [I5: set_nat,F: fun_nat_fun_nat_real,Xb: fun_nat_real] :
      ( ! [I3: nat] :
          ( aa_set_nat_o(member_nat(I3),I5)
         => aa_real_o(sums_real(aa_nat_fun_nat_real(F,I3)),aa_nat_real(Xb,I3)) )
     => aa_real_o(sums_real(aa_fun7039663905532832253t_real(aTP_Lamm_are(I5),F)),groups6591440286371151544t_real(Xb,I5)) ) ).

% sums_sum
tff(fact_8155_Sum__Icc__int,axiom,
    ! [M: int,Nb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,M),Nb)
     => ( groups4538972089207619220nt_int(aTP_Lamm_dj,set_or1266510415728281911st_int(M,Nb)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Nb),aa_int_int(aa_int_fun_int_int(plus_plus_int,Nb),one_one_int))),aa_int_int(aa_int_fun_int_int(times_times_int,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_8156_sums__mult2__iff,axiom,
    ! [C: real,F: fun_nat_real,D: real] :
      ( ( C != zero_zero_real )
     => ( aa_real_o(sums_real(aa_fun1839206346889086768t_real(aTP_Lamm_arf(C),F)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,D),C))
      <=> aa_real_o(sums_real(F),D) ) ) ).

% sums_mult2_iff
tff(fact_8157_sums__mult__iff,axiom,
    ! [C: real,F: fun_nat_real,D: real] :
      ( ( C != zero_zero_real )
     => ( aa_real_o(sums_real(aa_fun1839206346889086768t_real(aTP_Lamm_akv(C),F)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),D))
      <=> aa_real_o(sums_real(F),D) ) ) ).

% sums_mult_iff
tff(fact_8158_sums__mult__D,axiom,
    ! [C: real,F: fun_nat_real,A: real] :
      ( aa_real_o(sums_real(aa_fun1839206346889086768t_real(aTP_Lamm_akv(C),F)),A)
     => ( ( C != zero_zero_real )
       => aa_real_o(sums_real(F),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),C)) ) ) ).

% sums_mult_D
tff(fact_8159_sums__Suc__imp,axiom,
    ! [F: fun_nat_real,S3: real] :
      ( ( aa_nat_real(F,zero_zero_nat) = zero_zero_real )
     => ( aa_real_o(sums_real(aTP_Lamm_anv(F)),S3)
       => aa_real_o(sums_real(F),S3) ) ) ).

% sums_Suc_imp
tff(fact_8160_sums__Suc,axiom,
    ! [F: fun_nat_real,L: real] :
      ( aa_real_o(sums_real(aTP_Lamm_anv(F)),L)
     => aa_real_o(sums_real(F),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,L),aa_nat_real(F,zero_zero_nat))) ) ).

% sums_Suc
tff(fact_8161_sums__Suc,axiom,
    ! [F: fun_nat_nat,L: nat] :
      ( aa_nat_o(sums_nat(aTP_Lamm_anu(F)),L)
     => aa_nat_o(sums_nat(F),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,L),aa_nat_nat(F,zero_zero_nat))) ) ).

% sums_Suc
tff(fact_8162_sums__Suc,axiom,
    ! [F: fun_nat_int,L: int] :
      ( aa_int_o(sums_int(aTP_Lamm_aoq(F)),L)
     => aa_int_o(sums_int(F),aa_int_int(aa_int_fun_int_int(plus_plus_int,L),aa_nat_int(F,zero_zero_nat))) ) ).

% sums_Suc
tff(fact_8163_sums__Suc__iff,axiom,
    ! [F: fun_nat_real,S3: real] :
      ( aa_real_o(sums_real(aTP_Lamm_anv(F)),S3)
    <=> aa_real_o(sums_real(F),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,S3),aa_nat_real(F,zero_zero_nat))) ) ).

% sums_Suc_iff
tff(fact_8164_sums__zero__iff__shift,axiom,
    ! [Nb: nat,F: fun_nat_real,S3: real] :
      ( ! [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),Nb)
         => ( aa_nat_real(F,I3) = zero_zero_real ) )
     => ( aa_real_o(sums_real(aa_fun1839206346889086768t_real(aTP_Lamm_arg(Nb),F)),S3)
      <=> aa_real_o(sums_real(F),S3) ) ) ).

% sums_zero_iff_shift
tff(fact_8165_sums__finite,axiom,
    ! [N: set_nat,F: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,N)
     => ( ! [N2: nat] :
            ( ~ aa_set_nat_o(member_nat(N2),N)
           => ( aa_nat_int(F,N2) = zero_zero_int ) )
       => aa_int_o(sums_int(F),groups3539618377306564664at_int(F,N)) ) ) ).

% sums_finite
tff(fact_8166_sums__finite,axiom,
    ! [N: set_nat,F: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,N)
     => ( ! [N2: nat] :
            ( ~ aa_set_nat_o(member_nat(N2),N)
           => ( aa_nat_nat(F,N2) = zero_zero_nat ) )
       => aa_nat_o(sums_nat(F),aa_set_nat_nat(groups3542108847815614940at_nat(F),N)) ) ) ).

% sums_finite
tff(fact_8167_sums__finite,axiom,
    ! [N: set_nat,F: fun_nat_real] :
      ( aa_set_nat_o(finite_finite_nat,N)
     => ( ! [N2: nat] :
            ( ~ aa_set_nat_o(member_nat(N2),N)
           => ( aa_nat_real(F,N2) = zero_zero_real ) )
       => aa_real_o(sums_real(F),groups6591440286371151544t_real(F,N)) ) ) ).

% sums_finite
tff(fact_8168_sums__If__finite,axiom,
    ! [P: fun_nat_o,F: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,collect_nat(P))
     => aa_int_o(sums_int(aa_fun5565043122120104496at_int(aTP_Lamm_arh(P),F)),groups3539618377306564664at_int(F,collect_nat(P))) ) ).

% sums_If_finite
tff(fact_8169_sums__If__finite,axiom,
    ! [P: fun_nat_o,F: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,collect_nat(P))
     => aa_nat_o(sums_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_ari(P),F)),aa_set_nat_nat(groups3542108847815614940at_nat(F),collect_nat(P))) ) ).

% sums_If_finite
tff(fact_8170_sums__If__finite,axiom,
    ! [P: fun_nat_o,F: fun_nat_real] :
      ( aa_set_nat_o(finite_finite_nat,collect_nat(P))
     => aa_real_o(sums_real(aa_fun1839206346889086768t_real(aTP_Lamm_arj(P),F)),groups6591440286371151544t_real(F,collect_nat(P))) ) ).

% sums_If_finite
tff(fact_8171_sums__If__finite__set,axiom,
    ! [A2: set_nat,F: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => aa_int_o(sums_int(aa_fun5565043122120104496at_int(aTP_Lamm_ark(A2),F)),groups3539618377306564664at_int(F,A2)) ) ).

% sums_If_finite_set
tff(fact_8172_sums__If__finite__set,axiom,
    ! [A2: set_nat,F: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => aa_nat_o(sums_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_arl(A2),F)),aa_set_nat_nat(groups3542108847815614940at_nat(F),A2)) ) ).

% sums_If_finite_set
tff(fact_8173_sums__If__finite__set,axiom,
    ! [A2: set_nat,F: fun_nat_real] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => aa_real_o(sums_real(aa_fun1839206346889086768t_real(aTP_Lamm_arm(A2),F)),groups6591440286371151544t_real(F,A2)) ) ).

% sums_If_finite_set
tff(fact_8174_lemma__termdiff2,axiom,
    ! [H: rat,Z: rat,Nb: nat] :
      ( ( H != zero_zero_rat )
     => ( minus_minus_rat(divide_divide_rat(minus_minus_rat(aa_nat_rat(power_power_rat(aa_rat_rat(plus_plus_rat(Z),H)),Nb),aa_nat_rat(power_power_rat(Z),Nb)),H),aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,Nb)),aa_nat_rat(power_power_rat(Z),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat))))) = aa_rat_rat(times_times_rat(H),groups2906978787729119204at_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_aro(H),Z),Nb),set_ord_lessThan_nat(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat))))) ) ) ).

% lemma_termdiff2
tff(fact_8175_lemma__termdiff2,axiom,
    ! [H: real,Z: real,Nb: nat] :
      ( ( H != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Z),H)),Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Z),Nb))),H)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Z),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat))))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,H),groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_arq(H),Z),Nb),set_ord_lessThan_nat(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat))))) ) ) ).

% lemma_termdiff2
tff(fact_8176_TBOUND__fi_H__adm,axiom,
    ! [Foo: fun_VE7674639578305544598at_nat] : comple8556566605269262190_VEBTi(partia7782936097874681665Ti_nat(heap_T3112222404744780921_VEBTi),partia1868168049876374393Ti_nat(heap_T7173139186834293313_VEBTi),aTP_Lamm_arr(Foo)) ).

% TBOUND_fi'_adm
tff(fact_8177_TBOUND__fi_H__adm,axiom,
    ! [Foo: fun_VE7674639578305544598at_nat] : comple5664458124825824345on_nat(partia6039416512482706817Ti_nat(heap_T7048022066654196708on_nat),partia7778075537949639097Ti_nat(heap_T7875578835903804844on_nat),aTP_Lamm_ars(Foo)) ).

% TBOUND_fi'_adm
tff(fact_8178_TBOUND__fi_H__adm,axiom,
    ! [Foo: fun_VE7674639578305544598at_nat] : comple5871978649864577227Heap_o(partia6726927458685305659Ti_nat(heap_Time_Heap_lub_o),partia3290229181235258227Ti_nat(heap_Time_Heap_ord_o),aTP_Lamm_art(Foo)) ).

% TBOUND_fi'_adm
tff(fact_8179_pochhammer__double,axiom,
    ! [Z: rat,Nb: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(times_times_rat(numeral_numeral_rat(aa_num_num(bit0,one))),Z)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)))),aa_nat_rat(comm_s4028243227959126397er_rat(Z),Nb))),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(plus_plus_rat(Z),divide_divide_rat(one_one_rat,numeral_numeral_rat(aa_num_num(bit0,one))))),Nb)) ).

% pochhammer_double
tff(fact_8180_pochhammer__double,axiom,
    ! [Z: real,Nb: nat] : aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),Z)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)))),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,Z),Nb))),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Z),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(aa_num_num(bit0,one))))),Nb)) ).

% pochhammer_double
tff(fact_8181_of__nat__code,axiom,
    ! [Nb: nat] : aa_nat5599461682457136004l_num1(semiri8819519690708144855l_num1,Nb) = semiri2846968517960172219l_num1(aTP_Lamm_aru,Nb,zero_z3563351764282998399l_num1) ).

% of_nat_code
tff(fact_8182_of__nat__code,axiom,
    ! [Nb: nat] : aa_nat_uint32(semiri2565882477558803405uint32,Nb) = semiri2064589214733661617uint32(aTP_Lamm_arv,Nb,zero_zero_uint32) ).

% of_nat_code
tff(fact_8183_of__nat__code,axiom,
    ! [Nb: nat] : aa_nat_rat(semiri681578069525770553at_rat,Nb) = semiri7787848453975740701ux_rat(aTP_Lamm_arw,Nb,zero_zero_rat) ).

% of_nat_code
tff(fact_8184_of__nat__code,axiom,
    ! [Nb: nat] : aa_nat_real(semiri5074537144036343181t_real,Nb) = aa_real_real(semiri7260567687927622513x_real(aTP_Lamm_arx,Nb),zero_zero_real) ).

% of_nat_code
tff(fact_8185_of__nat__code,axiom,
    ! [Nb: nat] : aa_nat_int(semiri1314217659103216013at_int,Nb) = aa_int_int(aa_nat_fun_int_int(semiri8420488043553186161ux_int(aTP_Lamm_ary),Nb),zero_zero_int) ).

% of_nat_code
tff(fact_8186_of__nat__code,axiom,
    ! [Nb: nat] : aa_nat_nat(semiri1316708129612266289at_nat,Nb) = aa_nat_nat(aa_nat_fun_nat_nat(semiri8422978514062236437ux_nat(aTP_Lamm_arz),Nb),zero_zero_nat) ).

% of_nat_code
tff(fact_8187_of__nat__code,axiom,
    ! [Nb: nat] : aa_nat_Code_integer(semiri4939895301339042750nteger,Nb) = aa_Cod1783974652533396728nteger(semiri4055485073559036834nteger(aTP_Lamm_asa,Nb),zero_z3403309356797280102nteger) ).

% of_nat_code
tff(fact_8188_floor__log__nat__eq__powr__iff,axiom,
    ! [B: nat,K: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
       => ( ( aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(log,aa_nat_real(semiri5074537144036343181t_real,B)),aa_nat_real(semiri5074537144036343181t_real,K))) = aa_nat_int(semiri1314217659103216013at_int,Nb) )
        <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Nb)),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,Nb),one_one_nat))) ) ) ) ) ).

% floor_log_nat_eq_powr_iff
tff(fact_8189_lessThan__iff,axiom,
    ! [I: real,K: real] :
      ( aa_set_real_o(member_real(I),set_or5984915006950818249n_real(K))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,I),K) ) ).

% lessThan_iff
tff(fact_8190_lessThan__iff,axiom,
    ! [I: rat,K: rat] :
      ( aa_set_rat_o(member_rat(I),set_ord_lessThan_rat(K))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,I),K) ) ).

% lessThan_iff
tff(fact_8191_lessThan__iff,axiom,
    ! [I: num,K: num] :
      ( aa_set_num_o(member_num(I),set_ord_lessThan_num(K))
    <=> aa_num_o(ord_less_num(I),K) ) ).

% lessThan_iff
tff(fact_8192_lessThan__iff,axiom,
    ! [I: nat,K: nat] :
      ( aa_set_nat_o(member_nat(I),set_ord_lessThan_nat(K))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),K) ) ).

% lessThan_iff
tff(fact_8193_lessThan__iff,axiom,
    ! [I: int,K: int] :
      ( aa_set_int_o(member_int(I),set_ord_lessThan_int(K))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,I),K) ) ).

% lessThan_iff
tff(fact_8194_lessThan__subset__iff,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa_set_rat_o(ord_less_eq_set_rat(set_ord_lessThan_rat(Xb)),set_ord_lessThan_rat(Y))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Y) ) ).

% lessThan_subset_iff
tff(fact_8195_lessThan__subset__iff,axiom,
    ! [Xb: num,Y: num] :
      ( aa_set_num_o(ord_less_eq_set_num(set_ord_lessThan_num(Xb)),set_ord_lessThan_num(Y))
    <=> aa_num_o(ord_less_eq_num(Xb),Y) ) ).

% lessThan_subset_iff
tff(fact_8196_lessThan__subset__iff,axiom,
    ! [Xb: nat,Y: nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(set_ord_lessThan_nat(Xb)),set_ord_lessThan_nat(Y))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Y) ) ).

% lessThan_subset_iff
tff(fact_8197_lessThan__subset__iff,axiom,
    ! [Xb: int,Y: int] :
      ( aa_set_int_o(ord_less_eq_set_int(set_ord_lessThan_int(Xb)),set_ord_lessThan_int(Y))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Y) ) ).

% lessThan_subset_iff
tff(fact_8198_floor__numeral,axiom,
    ! [V: num] : aa_real_int(archim6058952711729229775r_real,numeral_numeral_real(V)) = aa_num_int(numeral_numeral_int,V) ).

% floor_numeral
tff(fact_8199_floor__numeral,axiom,
    ! [V: num] : archim3151403230148437115or_rat(numeral_numeral_rat(V)) = aa_num_int(numeral_numeral_int,V) ).

% floor_numeral
tff(fact_8200_floor__one,axiom,
    aa_real_int(archim6058952711729229775r_real,one_one_real) = one_one_int ).

% floor_one
tff(fact_8201_floor__one,axiom,
    archim3151403230148437115or_rat(one_one_rat) = one_one_int ).

% floor_one
tff(fact_8202_pochhammer__0,axiom,
    ! [A: word_N3645301735248828278l_num1] : aa_nat5599461682457136004l_num1(comm_s6431939913906641691l_num1(A),zero_zero_nat) = one_on7727431528512463931l_num1 ).

% pochhammer_0
tff(fact_8203_pochhammer__0,axiom,
    ! [A: real] : aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,A),zero_zero_nat) = one_one_real ).

% pochhammer_0
tff(fact_8204_pochhammer__0,axiom,
    ! [A: rat] : aa_nat_rat(comm_s4028243227959126397er_rat(A),zero_zero_nat) = one_one_rat ).

% pochhammer_0
tff(fact_8205_pochhammer__0,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,A),zero_zero_nat) = one_one_nat ).

% pochhammer_0
tff(fact_8206_pochhammer__0,axiom,
    ! [A: int] : aa_nat_int(aa_int_fun_nat_int(comm_s4660882817536571857er_int,A),zero_zero_nat) = one_one_int ).

% pochhammer_0
tff(fact_8207_sum_OlessThan__Suc,axiom,
    ! [G: fun_nat_rat,Nb: nat] : groups2906978787729119204at_rat(G,set_ord_lessThan_nat(aa_nat_nat(suc,Nb))) = aa_rat_rat(plus_plus_rat(groups2906978787729119204at_rat(G,set_ord_lessThan_nat(Nb))),aa_nat_rat(G,Nb)) ).

% sum.lessThan_Suc
tff(fact_8208_sum_OlessThan__Suc,axiom,
    ! [G: fun_nat_int,Nb: nat] : groups3539618377306564664at_int(G,set_ord_lessThan_nat(aa_nat_nat(suc,Nb))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups3539618377306564664at_int(G,set_ord_lessThan_nat(Nb))),aa_nat_int(G,Nb)) ).

% sum.lessThan_Suc
tff(fact_8209_sum_OlessThan__Suc,axiom,
    ! [G: fun_nat_nat,Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),set_ord_lessThan_nat(aa_nat_nat(suc,Nb))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(G),set_ord_lessThan_nat(Nb))),aa_nat_nat(G,Nb)) ).

% sum.lessThan_Suc
tff(fact_8210_sum_OlessThan__Suc,axiom,
    ! [G: fun_nat_real,Nb: nat] : groups6591440286371151544t_real(G,set_ord_lessThan_nat(aa_nat_nat(suc,Nb))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(G,set_ord_lessThan_nat(Nb))),aa_nat_real(G,Nb)) ).

% sum.lessThan_Suc
tff(fact_8211_single__Diff__lessThan,axiom,
    ! [K: real] : minus_minus_set_real(insert_real(K,bot_bot_set_real),set_or5984915006950818249n_real(K)) = insert_real(K,bot_bot_set_real) ).

% single_Diff_lessThan
tff(fact_8212_single__Diff__lessThan,axiom,
    ! [K: nat] : minus_minus_set_nat(insert_nat(K,bot_bot_set_nat),set_ord_lessThan_nat(K)) = insert_nat(K,bot_bot_set_nat) ).

% single_Diff_lessThan
tff(fact_8213_single__Diff__lessThan,axiom,
    ! [K: int] : minus_minus_set_int(insert_int(K,bot_bot_set_int),set_ord_lessThan_int(K)) = insert_int(K,bot_bot_set_int) ).

% single_Diff_lessThan
tff(fact_8214_floor__diff__of__int,axiom,
    ! [Xb: real,Z: int] : aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),aa_int_real(ring_1_of_int_real,Z))) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_real_int(archim6058952711729229775r_real,Xb)),Z) ).

% floor_diff_of_int
tff(fact_8215_floor__diff__of__int,axiom,
    ! [Xb: rat,Z: int] : archim3151403230148437115or_rat(minus_minus_rat(Xb,aa_int_rat(ring_1_of_int_rat,Z))) = aa_int_int(aa_int_fun_int_int(minus_minus_int,archim3151403230148437115or_rat(Xb)),Z) ).

% floor_diff_of_int
tff(fact_8216_zero__le__floor,axiom,
    ! [Xb: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_real_int(archim6058952711729229775r_real,Xb))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb) ) ).

% zero_le_floor
tff(fact_8217_zero__le__floor,axiom,
    ! [Xb: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),archim3151403230148437115or_rat(Xb))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Xb) ) ).

% zero_le_floor
tff(fact_8218_numeral__le__floor,axiom,
    ! [V: num,Xb: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,V)),aa_real_int(archim6058952711729229775r_real,Xb))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,numeral_numeral_real(V)),Xb) ) ).

% numeral_le_floor
tff(fact_8219_numeral__le__floor,axiom,
    ! [V: num,Xb: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,V)),archim3151403230148437115or_rat(Xb))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,numeral_numeral_rat(V)),Xb) ) ).

% numeral_le_floor
tff(fact_8220_floor__less__zero,axiom,
    ! [Xb: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(archim6058952711729229775r_real,Xb)),zero_zero_int)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),zero_zero_real) ) ).

% floor_less_zero
tff(fact_8221_floor__less__zero,axiom,
    ! [Xb: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,archim3151403230148437115or_rat(Xb)),zero_zero_int)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),zero_zero_rat) ) ).

% floor_less_zero
tff(fact_8222_floor__less__numeral,axiom,
    ! [Xb: real,V: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(archim6058952711729229775r_real,Xb)),aa_num_int(numeral_numeral_int,V))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),numeral_numeral_real(V)) ) ).

% floor_less_numeral
tff(fact_8223_floor__less__numeral,axiom,
    ! [Xb: rat,V: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,archim3151403230148437115or_rat(Xb)),aa_num_int(numeral_numeral_int,V))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),numeral_numeral_rat(V)) ) ).

% floor_less_numeral
tff(fact_8224_zero__less__floor,axiom,
    ! [Xb: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_real_int(archim6058952711729229775r_real,Xb))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Xb) ) ).

% zero_less_floor
tff(fact_8225_zero__less__floor,axiom,
    ! [Xb: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),archim3151403230148437115or_rat(Xb))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),Xb) ) ).

% zero_less_floor
tff(fact_8226_floor__le__zero,axiom,
    ! [Xb: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(archim6058952711729229775r_real,Xb)),zero_zero_int)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),one_one_real) ) ).

% floor_le_zero
tff(fact_8227_floor__le__zero,axiom,
    ! [Xb: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,archim3151403230148437115or_rat(Xb)),zero_zero_int)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),one_one_rat) ) ).

% floor_le_zero
tff(fact_8228_one__le__floor,axiom,
    ! [Xb: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_real_int(archim6058952711729229775r_real,Xb))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Xb) ) ).

% one_le_floor
tff(fact_8229_one__le__floor,axiom,
    ! [Xb: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),archim3151403230148437115or_rat(Xb))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),Xb) ) ).

% one_le_floor
tff(fact_8230_floor__less__one,axiom,
    ! [Xb: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(archim6058952711729229775r_real,Xb)),one_one_int)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),one_one_real) ) ).

% floor_less_one
tff(fact_8231_floor__less__one,axiom,
    ! [Xb: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,archim3151403230148437115or_rat(Xb)),one_one_int)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),one_one_rat) ) ).

% floor_less_one
tff(fact_8232_floor__diff__numeral,axiom,
    ! [Xb: real,V: num] : aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),numeral_numeral_real(V))) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_real_int(archim6058952711729229775r_real,Xb)),aa_num_int(numeral_numeral_int,V)) ).

% floor_diff_numeral
tff(fact_8233_floor__diff__numeral,axiom,
    ! [Xb: rat,V: num] : archim3151403230148437115or_rat(minus_minus_rat(Xb,numeral_numeral_rat(V))) = aa_int_int(aa_int_fun_int_int(minus_minus_int,archim3151403230148437115or_rat(Xb)),aa_num_int(numeral_numeral_int,V)) ).

% floor_diff_numeral
tff(fact_8234_floor__numeral__power,axiom,
    ! [Xb: num,Nb: nat] : aa_real_int(archim6058952711729229775r_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,numeral_numeral_real(Xb)),Nb)) = aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,Xb)),Nb) ).

% floor_numeral_power
tff(fact_8235_floor__numeral__power,axiom,
    ! [Xb: num,Nb: nat] : archim3151403230148437115or_rat(aa_nat_rat(power_power_rat(numeral_numeral_rat(Xb)),Nb)) = aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,Xb)),Nb) ).

% floor_numeral_power
tff(fact_8236_floor__diff__one,axiom,
    ! [Xb: real] : aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),one_one_real)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_real_int(archim6058952711729229775r_real,Xb)),one_one_int) ).

% floor_diff_one
tff(fact_8237_floor__diff__one,axiom,
    ! [Xb: rat] : archim3151403230148437115or_rat(minus_minus_rat(Xb,one_one_rat)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,archim3151403230148437115or_rat(Xb)),one_one_int) ).

% floor_diff_one
tff(fact_8238_floor__divide__eq__div__numeral,axiom,
    ! [A: num,B: num] : aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,numeral_numeral_real(A)),numeral_numeral_real(B))) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_num_int(numeral_numeral_int,A)),aa_num_int(numeral_numeral_int,B)) ).

% floor_divide_eq_div_numeral
tff(fact_8239_numeral__less__floor,axiom,
    ! [V: num,Xb: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,V)),aa_real_int(archim6058952711729229775r_real,Xb))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,numeral_numeral_real(V)),one_one_real)),Xb) ) ).

% numeral_less_floor
tff(fact_8240_numeral__less__floor,axiom,
    ! [V: num,Xb: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,V)),archim3151403230148437115or_rat(Xb))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(numeral_numeral_rat(V)),one_one_rat)),Xb) ) ).

% numeral_less_floor
tff(fact_8241_floor__le__numeral,axiom,
    ! [Xb: real,V: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(archim6058952711729229775r_real,Xb)),aa_num_int(numeral_numeral_int,V))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,numeral_numeral_real(V)),one_one_real)) ) ).

% floor_le_numeral
tff(fact_8242_floor__le__numeral,axiom,
    ! [Xb: rat,V: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,archim3151403230148437115or_rat(Xb)),aa_num_int(numeral_numeral_int,V))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),aa_rat_rat(plus_plus_rat(numeral_numeral_rat(V)),one_one_rat)) ) ).

% floor_le_numeral
tff(fact_8243_one__less__floor,axiom,
    ! [Xb: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_real_int(archim6058952711729229775r_real,Xb))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,numeral_numeral_real(aa_num_num(bit0,one))),Xb) ) ).

% one_less_floor
tff(fact_8244_one__less__floor,axiom,
    ! [Xb: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),archim3151403230148437115or_rat(Xb))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,numeral_numeral_rat(aa_num_num(bit0,one))),Xb) ) ).

% one_less_floor
tff(fact_8245_floor__le__one,axiom,
    ! [Xb: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(archim6058952711729229775r_real,Xb)),one_one_int)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),numeral_numeral_real(aa_num_num(bit0,one))) ) ).

% floor_le_one
tff(fact_8246_floor__le__one,axiom,
    ! [Xb: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,archim3151403230148437115or_rat(Xb)),one_one_int)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),numeral_numeral_rat(aa_num_num(bit0,one))) ) ).

% floor_le_one
tff(fact_8247_floor__one__divide__eq__div__numeral,axiom,
    ! [B: num] : aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(B))) = aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),aa_num_int(numeral_numeral_int,B)) ).

% floor_one_divide_eq_div_numeral
tff(fact_8248_lessThan__def,axiom,
    ! [U: real] : set_or5984915006950818249n_real(U) = collect_real(aa_real_fun_real_o(aTP_Lamm_dp,U)) ).

% lessThan_def
tff(fact_8249_lessThan__def,axiom,
    ! [U: rat] : set_ord_lessThan_rat(U) = collect_rat(aa_rat_fun_rat_o(aTP_Lamm_dr,U)) ).

% lessThan_def
tff(fact_8250_lessThan__def,axiom,
    ! [U: num] : set_ord_lessThan_num(U) = collect_num(aTP_Lamm_dt(U)) ).

% lessThan_def
tff(fact_8251_lessThan__def,axiom,
    ! [U: nat] : set_ord_lessThan_nat(U) = collect_nat(aa_nat_fun_nat_o(aTP_Lamm_dv,U)) ).

% lessThan_def
tff(fact_8252_lessThan__def,axiom,
    ! [U: int] : set_ord_lessThan_int(U) = collect_int(aa_int_fun_int_o(aTP_Lamm_dx,U)) ).

% lessThan_def
tff(fact_8253_finite__set__of__finite__funs,axiom,
    ! [A2: set_VEBT_VEBT,B4: set_VEBT_VEBT,D: vEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,A2)
     => ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,B4)
       => finite6732971732205641050T_VEBT(collec3347073232890953880T_VEBT(aa_VEB992159746102714230VEBT_o(aa_set4017001947700194517VEBT_o(aTP_Lamm_asb(A2),B4),D))) ) ) ).

% finite_set_of_finite_funs
tff(fact_8254_finite__set__of__finite__funs,axiom,
    ! [A2: set_VEBT_VEBT,B4: set_real,D: real] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,A2)
     => ( aa_set_real_o(finite_finite_real,B4)
       => finite2897376015320999324T_real(collec112519529067239902T_real(aa_rea1931189138858785298real_o(aa_set3249680857378704241real_o(aTP_Lamm_asc(A2),B4),D))) ) ) ).

% finite_set_of_finite_funs
tff(fact_8255_finite__set__of__finite__funs,axiom,
    ! [A2: set_real,B4: set_VEBT_VEBT,D: vEBT_VEBT] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,B4)
       => finite1484216373293608330T_VEBT(collec7922731923894624716T_VEBT(aa_VEB1275781630754614706VEBT_o(aa_set1734275489953896617VEBT_o(aTP_Lamm_asd(A2),B4),D))) ) ) ).

% finite_set_of_finite_funs
tff(fact_8256_finite__set__of__finite__funs,axiom,
    ! [A2: set_real,B4: set_real,D: real] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( aa_set_real_o(finite_finite_real,B4)
       => finite4831739209692635500l_real(collec174519434428708266l_real(aa_rea3177242051037803890real_o(aa_set5769202425262026345real_o(aTP_Lamm_ase(A2),B4),D))) ) ) ).

% finite_set_of_finite_funs
tff(fact_8257_finite__set__of__finite__funs,axiom,
    ! [A2: set_VEBT_VEBT,B4: set_nat,D: nat] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,A2)
     => ( aa_set_nat_o(finite_finite_nat,B4)
       => finite9199480722951521088BT_nat(collec2629060299267094658BT_nat(aa_nat8961800256705610514_nat_o(aa_set6628163682670611349_nat_o(aTP_Lamm_asf(A2),B4),D))) ) ) ).

% finite_set_of_finite_funs
tff(fact_8258_finite__set__of__finite__funs,axiom,
    ! [A2: set_real,B4: set_nat,D: nat] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( aa_set_nat_o(finite_finite_nat,B4)
       => finite8414670351157798672al_nat(collect_fun_real_nat(aa_nat7275243809302487082_nat_o(aa_set2636817504407142141_nat_o(aTP_Lamm_asg(A2),B4),D))) ) ) ).

% finite_set_of_finite_funs
tff(fact_8259_finite__set__of__finite__funs,axiom,
    ! [A2: set_VEBT_VEBT,B4: set_int,D: int] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,A2)
     => ( aa_set_int_o(finite_finite_int,B4)
       => finite5021629703442324380BT_int(collec7674581316612673758BT_int(aa_int5874382713699637778_int_o(aa_set7680706651547224305_int_o(aTP_Lamm_ash(A2),B4),D))) ) ) ).

% finite_set_of_finite_funs
tff(fact_8260_finite__set__of__finite__funs,axiom,
    ! [A2: set_real,B4: set_int,D: int] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( aa_set_int_o(finite_finite_int,B4)
       => finite4236819331648601964al_int(collect_fun_real_int(aa_int2775185683442467186_int_o(aa_set704203600554081513_int_o(aTP_Lamm_asi(A2),B4),D))) ) ) ).

% finite_set_of_finite_funs
tff(fact_8261_finite__set__of__finite__funs,axiom,
    ! [A2: set_VEBT_VEBT,B4: set_complex,D: complex] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,A2)
     => ( aa_set_complex_o(finite3207457112153483333omplex,B4)
       => finite7606193347572897566omplex(collec4450279080940537952omplex(aa_com8209720474803639954plex_o(aa_set6707307212627643379plex_o(aTP_Lamm_asj(A2),B4),D))) ) ) ).

% finite_set_of_finite_funs
tff(fact_8262_finite__set__of__finite__funs,axiom,
    ! [A2: set_real,B4: set_complex,D: complex] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( aa_set_complex_o(finite3207457112153483333omplex,B4)
       => finite275139722573732590omplex(collec2475887038589252140omplex(aa_com7884460700185185646plex_o(aa_set4700154924931669219plex_o(aTP_Lamm_ask(A2),B4),D))) ) ) ).

% finite_set_of_finite_funs
tff(fact_8263_of__int__floor__le,axiom,
    ! [Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,aa_real_int(archim6058952711729229775r_real,Xb))),Xb) ).

% of_int_floor_le
tff(fact_8264_of__int__floor__le,axiom,
    ! [Xb: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,archim3151403230148437115or_rat(Xb))),Xb) ).

% of_int_floor_le
tff(fact_8265_floor__mono,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(archim6058952711729229775r_real,Xb)),aa_real_int(archim6058952711729229775r_real,Y)) ) ).

% floor_mono
tff(fact_8266_floor__mono,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Y)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,archim3151403230148437115or_rat(Xb)),archim3151403230148437115or_rat(Y)) ) ).

% floor_mono
tff(fact_8267_floor__less__cancel,axiom,
    ! [Xb: real,Y: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(archim6058952711729229775r_real,Xb)),aa_real_int(archim6058952711729229775r_real,Y))
     => aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y) ) ).

% floor_less_cancel
tff(fact_8268_floor__less__cancel,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,archim3151403230148437115or_rat(Xb)),archim3151403230148437115or_rat(Y))
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Y) ) ).

% floor_less_cancel
tff(fact_8269_floor__le__ceiling,axiom,
    ! [Xb: real] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(archim6058952711729229775r_real,Xb)),aa_real_int(archim7802044766580827645g_real,Xb)) ).

% floor_le_ceiling
tff(fact_8270_floor__le__ceiling,axiom,
    ! [Xb: rat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,archim3151403230148437115or_rat(Xb)),archim2889992004027027881ng_rat(Xb)) ).

% floor_le_ceiling
tff(fact_8271_lessThan__strict__subset__iff,axiom,
    ! [M: real,Nb: real] :
      ( aa_set_real_o(ord_less_set_real(set_or5984915006950818249n_real(M)),set_or5984915006950818249n_real(Nb))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,M),Nb) ) ).

% lessThan_strict_subset_iff
tff(fact_8272_lessThan__strict__subset__iff,axiom,
    ! [M: rat,Nb: rat] :
      ( aa_set_rat_o(ord_less_set_rat(set_ord_lessThan_rat(M)),set_ord_lessThan_rat(Nb))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,M),Nb) ) ).

% lessThan_strict_subset_iff
tff(fact_8273_lessThan__strict__subset__iff,axiom,
    ! [M: num,Nb: num] :
      ( aa_set_num_o(ord_less_set_num(set_ord_lessThan_num(M)),set_ord_lessThan_num(Nb))
    <=> aa_num_o(ord_less_num(M),Nb) ) ).

% lessThan_strict_subset_iff
tff(fact_8274_lessThan__strict__subset__iff,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_set_nat_o(ord_less_set_nat(set_ord_lessThan_nat(M)),set_ord_lessThan_nat(Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb) ) ).

% lessThan_strict_subset_iff
tff(fact_8275_lessThan__strict__subset__iff,axiom,
    ! [M: int,Nb: int] :
      ( aa_set_int_o(ord_less_set_int(set_ord_lessThan_int(M)),set_ord_lessThan_int(Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,M),Nb) ) ).

% lessThan_strict_subset_iff
tff(fact_8276_pochhammer__pos,axiom,
    ! [Xb: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,Xb),Nb)) ) ).

% pochhammer_pos
tff(fact_8277_pochhammer__pos,axiom,
    ! [Xb: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Xb)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_nat_rat(comm_s4028243227959126397er_rat(Xb),Nb)) ) ).

% pochhammer_pos
tff(fact_8278_pochhammer__pos,axiom,
    ! [Xb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Xb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,Xb),Nb)) ) ).

% pochhammer_pos
tff(fact_8279_pochhammer__pos,axiom,
    ! [Xb: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Xb)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(aa_int_fun_nat_int(comm_s4660882817536571857er_int,Xb),Nb)) ) ).

% pochhammer_pos
tff(fact_8280_lessThan__Suc,axiom,
    ! [K: nat] : set_ord_lessThan_nat(aa_nat_nat(suc,K)) = insert_nat(K,set_ord_lessThan_nat(K)) ).

% lessThan_Suc
tff(fact_8281_pochhammer__eq__0__mono,axiom,
    ! [A: real,Nb: nat,M: nat] :
      ( ( aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,A),Nb) = zero_zero_real )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),M)
       => ( aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,A),M) = zero_zero_real ) ) ) ).

% pochhammer_eq_0_mono
tff(fact_8282_pochhammer__eq__0__mono,axiom,
    ! [A: rat,Nb: nat,M: nat] :
      ( ( aa_nat_rat(comm_s4028243227959126397er_rat(A),Nb) = zero_zero_rat )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),M)
       => ( aa_nat_rat(comm_s4028243227959126397er_rat(A),M) = zero_zero_rat ) ) ) ).

% pochhammer_eq_0_mono
tff(fact_8283_pochhammer__neq__0__mono,axiom,
    ! [A: real,M: nat,Nb: nat] :
      ( ( aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,A),M) != zero_zero_real )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),M)
       => ( aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,A),Nb) != zero_zero_real ) ) ) ).

% pochhammer_neq_0_mono
tff(fact_8284_pochhammer__neq__0__mono,axiom,
    ! [A: rat,M: nat,Nb: nat] :
      ( ( aa_nat_rat(comm_s4028243227959126397er_rat(A),M) != zero_zero_rat )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),M)
       => ( aa_nat_rat(comm_s4028243227959126397er_rat(A),Nb) != zero_zero_rat ) ) ) ).

% pochhammer_neq_0_mono
tff(fact_8285_floor__le__round,axiom,
    ! [Xb: real] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(archim6058952711729229775r_real,Xb)),aa_real_int(archim8280529875227126926d_real,Xb)) ).

% floor_le_round
tff(fact_8286_floor__le__round,axiom,
    ! [Xb: rat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,archim3151403230148437115or_rat(Xb)),archim7778729529865785530nd_rat(Xb)) ).

% floor_le_round
tff(fact_8287_admissible__heap,axiom,
    ! [P: fun_Pr6004456690014451917_nat_o] : comple7368800650122905596on_nat(partia2080987842261039902Ti_nat(heap_T7048022066654196708on_nat),partia8535946980722491222Ti_nat(heap_T7875578835903804844on_nat),aTP_Lamm_asl(P)) ).

% admissible_heap
tff(fact_8288_admissible__heap,axiom,
    ! [P: fun_Pr7132087832597382491_nat_o] : comple8003709893544916456Heap_o(partia5551857090987368152Ti_nat(heap_Time_Heap_lub_o),partia2925774515620677392Ti_nat(heap_Time_Heap_ord_o),aTP_Lamm_asm(P)) ).

% admissible_heap
tff(fact_8289_admissible__heap,axiom,
    ! [P: fun_Pr4018130278567771477_nat_o] : comple8556566605269262190_VEBTi(partia7782936097874681665Ti_nat(heap_T3112222404744780921_VEBTi),partia1868168049876374393Ti_nat(heap_T7173139186834293313_VEBTi),aTP_Lamm_asn(P)) ).

% admissible_heap
tff(fact_8290_admissible__heap,axiom,
    ! [P: fun_Pr4917118001609057450_nat_o] : comple5664458124825824345on_nat(partia6039416512482706817Ti_nat(heap_T7048022066654196708on_nat),partia7778075537949639097Ti_nat(heap_T7875578835903804844on_nat),aTP_Lamm_aso(P)) ).

% admissible_heap
tff(fact_8291_admissible__heap,axiom,
    ! [P: fun_Pr3008164681728764094_nat_o] : comple5871978649864577227Heap_o(partia6726927458685305659Ti_nat(heap_Time_Heap_lub_o),partia3290229181235258227Ti_nat(heap_Time_Heap_ord_o),aTP_Lamm_asp(P)) ).

% admissible_heap
tff(fact_8292_le__floor__iff,axiom,
    ! [Z: int,Xb: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),aa_real_int(archim6058952711729229775r_real,Xb))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Z)),Xb) ) ).

% le_floor_iff
tff(fact_8293_le__floor__iff,axiom,
    ! [Z: int,Xb: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),archim3151403230148437115or_rat(Xb))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Z)),Xb) ) ).

% le_floor_iff
tff(fact_8294_floor__less__iff,axiom,
    ! [Xb: real,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(archim6058952711729229775r_real,Xb)),Z)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_int_real(ring_1_of_int_real,Z)) ) ).

% floor_less_iff
tff(fact_8295_floor__less__iff,axiom,
    ! [Xb: rat,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,archim3151403230148437115or_rat(Xb)),Z)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),aa_int_rat(ring_1_of_int_rat,Z)) ) ).

% floor_less_iff
tff(fact_8296_floor__add__int,axiom,
    ! [Xb: real,Z: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(archim6058952711729229775r_real,Xb)),Z) = aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),aa_int_real(ring_1_of_int_real,Z))) ).

% floor_add_int
tff(fact_8297_floor__add__int,axiom,
    ! [Xb: rat,Z: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,archim3151403230148437115or_rat(Xb)),Z) = archim3151403230148437115or_rat(aa_rat_rat(plus_plus_rat(Xb),aa_int_rat(ring_1_of_int_rat,Z))) ).

% floor_add_int
tff(fact_8298_int__add__floor,axiom,
    ! [Z: int,Xb: real] : aa_int_int(aa_int_fun_int_int(plus_plus_int,Z),aa_real_int(archim6058952711729229775r_real,Xb)) = aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,Z)),Xb)) ).

% int_add_floor
tff(fact_8299_int__add__floor,axiom,
    ! [Z: int,Xb: rat] : aa_int_int(aa_int_fun_int_int(plus_plus_int,Z),archim3151403230148437115or_rat(Xb)) = archim3151403230148437115or_rat(aa_rat_rat(plus_plus_rat(aa_int_rat(ring_1_of_int_rat,Z)),Xb)) ).

% int_add_floor
tff(fact_8300_le__floor__add,axiom,
    ! [Xb: real,Y: real] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(archim6058952711729229775r_real,Xb)),aa_real_int(archim6058952711729229775r_real,Y))),aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Y))) ).

% le_floor_add
tff(fact_8301_le__floor__add,axiom,
    ! [Xb: 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,archim3151403230148437115or_rat(Xb)),archim3151403230148437115or_rat(Y))),archim3151403230148437115or_rat(aa_rat_rat(plus_plus_rat(Xb),Y))) ).

% le_floor_add
tff(fact_8302_floor__power,axiom,
    ! [Xb: real,Nb: nat] :
      ( ( Xb = aa_int_real(ring_1_of_int_real,aa_real_int(archim6058952711729229775r_real,Xb)) )
     => ( aa_real_int(archim6058952711729229775r_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb)) = aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_real_int(archim6058952711729229775r_real,Xb)),Nb) ) ) ).

% floor_power
tff(fact_8303_floor__power,axiom,
    ! [Xb: rat,Nb: nat] :
      ( ( Xb = aa_int_rat(ring_1_of_int_rat,archim3151403230148437115or_rat(Xb)) )
     => ( archim3151403230148437115or_rat(aa_nat_rat(power_power_rat(Xb),Nb)) = aa_nat_int(aa_int_fun_nat_int(power_power_int,archim3151403230148437115or_rat(Xb)),Nb) ) ) ).

% floor_power
tff(fact_8304_floor__divide__of__int__eq,axiom,
    ! [K: int,L: int] : aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_int_real(ring_1_of_int_real,K)),aa_int_real(ring_1_of_int_real,L))) = aa_int_int(aa_int_fun_int_int(divide_divide_int,K),L) ).

% floor_divide_of_int_eq
tff(fact_8305_floor__divide__of__int__eq,axiom,
    ! [K: int,L: int] : archim3151403230148437115or_rat(divide_divide_rat(aa_int_rat(ring_1_of_int_rat,K),aa_int_rat(ring_1_of_int_rat,L))) = aa_int_int(aa_int_fun_int_int(divide_divide_int,K),L) ).

% floor_divide_of_int_eq
tff(fact_8306_pochhammer__nonneg,axiom,
    ! [Xb: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,Xb),Nb)) ) ).

% pochhammer_nonneg
tff(fact_8307_pochhammer__nonneg,axiom,
    ! [Xb: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Xb)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(comm_s4028243227959126397er_rat(Xb),Nb)) ) ).

% pochhammer_nonneg
tff(fact_8308_pochhammer__nonneg,axiom,
    ! [Xb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Xb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,Xb),Nb)) ) ).

% pochhammer_nonneg
tff(fact_8309_pochhammer__nonneg,axiom,
    ! [Xb: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Xb)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(aa_int_fun_nat_int(comm_s4660882817536571857er_int,Xb),Nb)) ) ).

% pochhammer_nonneg
tff(fact_8310_pochhammer__0__left,axiom,
    ! [Nb: nat] :
      aa_nat5599461682457136004l_num1(comm_s6431939913906641691l_num1(zero_z3563351764282998399l_num1),Nb) = $ite(Nb = zero_zero_nat,one_on7727431528512463931l_num1,zero_z3563351764282998399l_num1) ).

% pochhammer_0_left
tff(fact_8311_pochhammer__0__left,axiom,
    ! [Nb: nat] :
      aa_nat_uint32(comm_s6516030829397196305uint32(zero_zero_uint32),Nb) = $ite(Nb = zero_zero_nat,one_one_uint32,zero_zero_uint32) ).

% pochhammer_0_left
tff(fact_8312_pochhammer__0__left,axiom,
    ! [Nb: nat] :
      aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,zero_zero_real),Nb) = $ite(Nb = zero_zero_nat,one_one_real,zero_zero_real) ).

% pochhammer_0_left
tff(fact_8313_pochhammer__0__left,axiom,
    ! [Nb: nat] :
      aa_nat_rat(comm_s4028243227959126397er_rat(zero_zero_rat),Nb) = $ite(Nb = zero_zero_nat,one_one_rat,zero_zero_rat) ).

% pochhammer_0_left
tff(fact_8314_pochhammer__0__left,axiom,
    ! [Nb: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,zero_zero_nat),Nb) = $ite(Nb = zero_zero_nat,one_one_nat,zero_zero_nat) ).

% pochhammer_0_left
tff(fact_8315_pochhammer__0__left,axiom,
    ! [Nb: nat] :
      aa_nat_int(aa_int_fun_nat_int(comm_s4660882817536571857er_int,zero_zero_int),Nb) = $ite(Nb = zero_zero_nat,one_one_int,zero_zero_int) ).

% pochhammer_0_left
tff(fact_8316_sum_Onat__diff__reindex,axiom,
    ! [G: fun_nat_nat,Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_asq(G),Nb)),set_ord_lessThan_nat(Nb)) = aa_set_nat_nat(groups3542108847815614940at_nat(G),set_ord_lessThan_nat(Nb)) ).

% sum.nat_diff_reindex
tff(fact_8317_sum_Onat__diff__reindex,axiom,
    ! [G: fun_nat_real,Nb: nat] : groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_asr(G),Nb),set_ord_lessThan_nat(Nb)) = groups6591440286371151544t_real(G,set_ord_lessThan_nat(Nb)) ).

% sum.nat_diff_reindex
tff(fact_8318_sum__diff__distrib,axiom,
    ! [Q: fun_int_nat,P: fun_int_nat,Nb: int] :
      ( ! [X2: int] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(Q,X2)),aa_int_nat(P,X2))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups4541462559716669496nt_nat(P,set_ord_lessThan_int(Nb))),groups4541462559716669496nt_nat(Q,set_ord_lessThan_int(Nb))) = groups4541462559716669496nt_nat(aa_fun6506752150744536624nt_nat(aTP_Lamm_aql(Q),P),set_ord_lessThan_int(Nb)) ) ) ).

% sum_diff_distrib
tff(fact_8319_sum__diff__distrib,axiom,
    ! [Q: fun_nat_nat,P: fun_nat_nat,Nb: nat] :
      ( ! [X2: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(Q,X2)),aa_nat_nat(P,X2))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(P),set_ord_lessThan_nat(Nb))),aa_set_nat_nat(groups3542108847815614940at_nat(Q),set_ord_lessThan_nat(Nb))) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_aqm(Q),P)),set_ord_lessThan_nat(Nb)) ) ) ).

% sum_diff_distrib
tff(fact_8320_of__nat__floor,axiom,
    ! [R2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),R2)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,aa_int_nat(nat2,aa_real_int(archim6058952711729229775r_real,R2)))),R2) ) ).

% of_nat_floor
tff(fact_8321_of__nat__floor,axiom,
    ! [R2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),R2)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(semiri681578069525770553at_rat,aa_int_nat(nat2,archim3151403230148437115or_rat(R2)))),R2) ) ).

% of_nat_floor
tff(fact_8322_one__add__floor,axiom,
    ! [Xb: real] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(archim6058952711729229775r_real,Xb)),one_one_int) = aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),one_one_real)) ).

% one_add_floor
tff(fact_8323_one__add__floor,axiom,
    ! [Xb: rat] : aa_int_int(aa_int_fun_int_int(plus_plus_int,archim3151403230148437115or_rat(Xb)),one_one_int) = archim3151403230148437115or_rat(aa_rat_rat(plus_plus_rat(Xb),one_one_rat)) ).

% one_add_floor
tff(fact_8324_floor__divide__of__nat__eq,axiom,
    ! [M: nat,Nb: nat] : aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(semiri5074537144036343181t_real,M)),aa_nat_real(semiri5074537144036343181t_real,Nb))) = aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Nb)) ).

% floor_divide_of_nat_eq
tff(fact_8325_floor__divide__of__nat__eq,axiom,
    ! [M: nat,Nb: nat] : archim3151403230148437115or_rat(divide_divide_rat(aa_nat_rat(semiri681578069525770553at_rat,M),aa_nat_rat(semiri681578069525770553at_rat,Nb))) = aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Nb)) ).

% floor_divide_of_nat_eq
tff(fact_8326_le__mult__nat__floor,axiom,
    ! [A: real,B: real] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_int_nat(nat2,aa_real_int(archim6058952711729229775r_real,A))),aa_int_nat(nat2,aa_real_int(archim6058952711729229775r_real,B)))),aa_int_nat(nat2,aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)))) ).

% le_mult_nat_floor
tff(fact_8327_le__mult__nat__floor,axiom,
    ! [A: rat,B: rat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_int_nat(nat2,archim3151403230148437115or_rat(A))),aa_int_nat(nat2,archim3151403230148437115or_rat(B)))),aa_int_nat(nat2,archim3151403230148437115or_rat(aa_rat_rat(times_times_rat(A),B)))) ).

% le_mult_nat_floor
tff(fact_8328_nat__floor__neg,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),zero_zero_real)
     => ( aa_int_nat(nat2,aa_real_int(archim6058952711729229775r_real,Xb)) = zero_zero_nat ) ) ).

% nat_floor_neg
tff(fact_8329_floor__eq3,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Nb)))
       => ( aa_int_nat(nat2,aa_real_int(archim6058952711729229775r_real,Xb)) = Nb ) ) ) ).

% floor_eq3
tff(fact_8330_ceiling__altdef,axiom,
    ! [Xb: real] :
      aa_real_int(archim7802044766580827645g_real,Xb) = $ite(Xb = aa_int_real(ring_1_of_int_real,aa_real_int(archim6058952711729229775r_real,Xb)),aa_real_int(archim6058952711729229775r_real,Xb),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(archim6058952711729229775r_real,Xb)),one_one_int)) ).

% ceiling_altdef
tff(fact_8331_ceiling__altdef,axiom,
    ! [Xb: rat] :
      archim2889992004027027881ng_rat(Xb) = $ite(Xb = aa_int_rat(ring_1_of_int_rat,archim3151403230148437115or_rat(Xb)),archim3151403230148437115or_rat(Xb),aa_int_int(aa_int_fun_int_int(plus_plus_int,archim3151403230148437115or_rat(Xb)),one_one_int)) ).

% ceiling_altdef
tff(fact_8332_le__nat__floor,axiom,
    ! [Xb: nat,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,Xb)),A)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),aa_int_nat(nat2,aa_real_int(archim6058952711729229775r_real,A))) ) ).

% le_nat_floor
tff(fact_8333_ceiling__diff__floor__le__1,axiom,
    ! [Xb: real] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_real_int(archim7802044766580827645g_real,Xb)),aa_real_int(archim6058952711729229775r_real,Xb))),one_one_int) ).

% ceiling_diff_floor_le_1
tff(fact_8334_ceiling__diff__floor__le__1,axiom,
    ! [Xb: rat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,archim2889992004027027881ng_rat(Xb)),archim3151403230148437115or_rat(Xb))),one_one_int) ).

% ceiling_diff_floor_le_1
tff(fact_8335_floor__eq,axiom,
    ! [Nb: int,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(ring_1_of_int_real,Nb)),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,Nb)),one_one_real))
       => ( aa_real_int(archim6058952711729229775r_real,Xb) = Nb ) ) ) ).

% floor_eq
tff(fact_8336_real__of__int__floor__add__one__gt,axiom,
    ! [R2: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,R2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,aa_real_int(archim6058952711729229775r_real,R2))),one_one_real)) ).

% real_of_int_floor_add_one_gt
tff(fact_8337_real__of__int__floor__add__one__ge,axiom,
    ! [R2: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,R2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,aa_real_int(archim6058952711729229775r_real,R2))),one_one_real)) ).

% real_of_int_floor_add_one_ge
tff(fact_8338_real__of__int__floor__gt__diff__one,axiom,
    ! [R2: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,R2),one_one_real)),aa_int_real(ring_1_of_int_real,aa_real_int(archim6058952711729229775r_real,R2))) ).

% real_of_int_floor_gt_diff_one
tff(fact_8339_real__of__int__floor__ge__diff__one,axiom,
    ! [R2: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,R2),one_one_real)),aa_int_real(ring_1_of_int_real,aa_real_int(archim6058952711729229775r_real,R2))) ).

% real_of_int_floor_ge_diff_one
tff(fact_8340_pochhammer__rec,axiom,
    ! [A: word_N3645301735248828278l_num1,Nb: nat] : aa_nat5599461682457136004l_num1(comm_s6431939913906641691l_num1(A),aa_nat_nat(suc,Nb)) = aa_wor7274019993978516266l_num1(times_7065122842183080059l_num1(A),aa_nat5599461682457136004l_num1(comm_s6431939913906641691l_num1(plus_p361126936061061375l_num1(A,one_on7727431528512463931l_num1)),Nb)) ).

% pochhammer_rec
tff(fact_8341_pochhammer__rec,axiom,
    ! [A: real,Nb: nat] : aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,A),aa_nat_nat(suc,Nb)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),one_one_real)),Nb)) ).

% pochhammer_rec
tff(fact_8342_pochhammer__rec,axiom,
    ! [A: rat,Nb: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(A),aa_nat_nat(suc,Nb)) = aa_rat_rat(times_times_rat(A),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(plus_plus_rat(A),one_one_rat)),Nb)) ).

% pochhammer_rec
tff(fact_8343_pochhammer__rec,axiom,
    ! [A: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,A),aa_nat_nat(suc,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),one_one_nat)),Nb)) ).

% pochhammer_rec
tff(fact_8344_pochhammer__rec,axiom,
    ! [A: int,Nb: nat] : aa_nat_int(aa_int_fun_nat_int(comm_s4660882817536571857er_int,A),aa_nat_nat(suc,Nb)) = aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_nat_int(aa_int_fun_nat_int(comm_s4660882817536571857er_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),one_one_int)),Nb)) ).

% pochhammer_rec
tff(fact_8345_pochhammer__Suc,axiom,
    ! [A: rat,Nb: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(A),aa_nat_nat(suc,Nb)) = aa_rat_rat(times_times_rat(aa_nat_rat(comm_s4028243227959126397er_rat(A),Nb)),aa_rat_rat(plus_plus_rat(A),aa_nat_rat(semiri681578069525770553at_rat,Nb))) ).

% pochhammer_Suc
tff(fact_8346_pochhammer__Suc,axiom,
    ! [A: real,Nb: nat] : aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,A),aa_nat_nat(suc,Nb)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,A),Nb)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_nat_real(semiri5074537144036343181t_real,Nb))) ).

% pochhammer_Suc
tff(fact_8347_pochhammer__Suc,axiom,
    ! [A: int,Nb: nat] : aa_nat_int(aa_int_fun_nat_int(comm_s4660882817536571857er_int,A),aa_nat_nat(suc,Nb)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(aa_int_fun_nat_int(comm_s4660882817536571857er_int,A),Nb)),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_nat_int(semiri1314217659103216013at_int,Nb))) ).

% pochhammer_Suc
tff(fact_8348_pochhammer__Suc,axiom,
    ! [A: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,A),aa_nat_nat(suc,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,A),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(semiri1316708129612266289at_nat,Nb))) ).

% pochhammer_Suc
tff(fact_8349_pochhammer__Suc,axiom,
    ! [A: code_integer,Nb: nat] : aa_nat_Code_integer(comm_s8582702949713902594nteger(A),aa_nat_nat(suc,Nb)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(comm_s8582702949713902594nteger(A),Nb)),aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(A),aa_nat_Code_integer(semiri4939895301339042750nteger,Nb))) ).

% pochhammer_Suc
tff(fact_8350_pochhammer__rec_H,axiom,
    ! [Z: rat,Nb: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(Z),aa_nat_nat(suc,Nb)) = aa_rat_rat(times_times_rat(aa_rat_rat(plus_plus_rat(Z),aa_nat_rat(semiri681578069525770553at_rat,Nb))),aa_nat_rat(comm_s4028243227959126397er_rat(Z),Nb)) ).

% pochhammer_rec'
tff(fact_8351_pochhammer__rec_H,axiom,
    ! [Z: real,Nb: nat] : aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,Z),aa_nat_nat(suc,Nb)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Z),aa_nat_real(semiri5074537144036343181t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,Z),Nb)) ).

% pochhammer_rec'
tff(fact_8352_pochhammer__rec_H,axiom,
    ! [Z: int,Nb: nat] : aa_nat_int(aa_int_fun_nat_int(comm_s4660882817536571857er_int,Z),aa_nat_nat(suc,Nb)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Z),aa_nat_int(semiri1314217659103216013at_int,Nb))),aa_nat_int(aa_int_fun_nat_int(comm_s4660882817536571857er_int,Z),Nb)) ).

% pochhammer_rec'
tff(fact_8353_pochhammer__rec_H,axiom,
    ! [Z: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,Z),aa_nat_nat(suc,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Z),aa_nat_nat(semiri1316708129612266289at_nat,Nb))),aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,Z),Nb)) ).

% pochhammer_rec'
tff(fact_8354_pochhammer__rec_H,axiom,
    ! [Z: code_integer,Nb: nat] : aa_nat_Code_integer(comm_s8582702949713902594nteger(Z),aa_nat_nat(suc,Nb)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(Z),aa_nat_Code_integer(semiri4939895301339042750nteger,Nb))),aa_nat_Code_integer(comm_s8582702949713902594nteger(Z),Nb)) ).

% pochhammer_rec'
tff(fact_8355_pochhammer__product_H,axiom,
    ! [Z: rat,Nb: nat,M: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(Z),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),M)) = aa_rat_rat(times_times_rat(aa_nat_rat(comm_s4028243227959126397er_rat(Z),Nb)),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(plus_plus_rat(Z),aa_nat_rat(semiri681578069525770553at_rat,Nb))),M)) ).

% pochhammer_product'
tff(fact_8356_pochhammer__product_H,axiom,
    ! [Z: real,Nb: nat,M: nat] : aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,Z),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),M)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,Z),Nb)),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Z),aa_nat_real(semiri5074537144036343181t_real,Nb))),M)) ).

% pochhammer_product'
tff(fact_8357_pochhammer__product_H,axiom,
    ! [Z: int,Nb: 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,Nb),M)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(aa_int_fun_nat_int(comm_s4660882817536571857er_int,Z),Nb)),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,Nb))),M)) ).

% pochhammer_product'
tff(fact_8358_pochhammer__product_H,axiom,
    ! [Z: nat,Nb: 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,Nb),M)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,Z),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Z),aa_nat_nat(semiri1316708129612266289at_nat,Nb))),M)) ).

% pochhammer_product'
tff(fact_8359_pochhammer__product_H,axiom,
    ! [Z: code_integer,Nb: nat,M: nat] : aa_nat_Code_integer(comm_s8582702949713902594nteger(Z),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),M)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(comm_s8582702949713902594nteger(Z),Nb)),aa_nat_Code_integer(comm_s8582702949713902594nteger(aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(Z),aa_nat_Code_integer(semiri4939895301339042750nteger,Nb))),M)) ).

% pochhammer_product'
tff(fact_8360_sum_OlessThan__Suc__shift,axiom,
    ! [G: fun_nat_rat,Nb: nat] : groups2906978787729119204at_rat(G,set_ord_lessThan_nat(aa_nat_nat(suc,Nb))) = aa_rat_rat(plus_plus_rat(aa_nat_rat(G,zero_zero_nat)),groups2906978787729119204at_rat(aTP_Lamm_aop(G),set_ord_lessThan_nat(Nb))) ).

% sum.lessThan_Suc_shift
tff(fact_8361_sum_OlessThan__Suc__shift,axiom,
    ! [G: fun_nat_int,Nb: nat] : groups3539618377306564664at_int(G,set_ord_lessThan_nat(aa_nat_nat(suc,Nb))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(G,zero_zero_nat)),groups3539618377306564664at_int(aTP_Lamm_aoq(G),set_ord_lessThan_nat(Nb))) ).

% sum.lessThan_Suc_shift
tff(fact_8362_sum_OlessThan__Suc__shift,axiom,
    ! [G: fun_nat_nat,Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),set_ord_lessThan_nat(aa_nat_nat(suc,Nb))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(G,zero_zero_nat)),aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_anu(G)),set_ord_lessThan_nat(Nb))) ).

% sum.lessThan_Suc_shift
tff(fact_8363_sum_OlessThan__Suc__shift,axiom,
    ! [G: fun_nat_real,Nb: nat] : groups6591440286371151544t_real(G,set_ord_lessThan_nat(aa_nat_nat(suc,Nb))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(G,zero_zero_nat)),groups6591440286371151544t_real(aTP_Lamm_anv(G),set_ord_lessThan_nat(Nb))) ).

% sum.lessThan_Suc_shift
tff(fact_8364_sum__lessThan__telescope,axiom,
    ! [F: fun_nat_rat,M: nat] : groups2906978787729119204at_rat(aTP_Lamm_aor(F),set_ord_lessThan_nat(M)) = minus_minus_rat(aa_nat_rat(F,M),aa_nat_rat(F,zero_zero_nat)) ).

% sum_lessThan_telescope
tff(fact_8365_sum__lessThan__telescope,axiom,
    ! [F: fun_nat_int,M: nat] : groups3539618377306564664at_int(aTP_Lamm_aos(F),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_8366_sum__lessThan__telescope,axiom,
    ! [F: fun_nat_real,M: nat] : groups6591440286371151544t_real(aTP_Lamm_aot(F),set_ord_lessThan_nat(M)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(F,M)),aa_nat_real(F,zero_zero_nat)) ).

% sum_lessThan_telescope
tff(fact_8367_sum__lessThan__telescope_H,axiom,
    ! [F: fun_nat_rat,M: nat] : groups2906978787729119204at_rat(aTP_Lamm_ass(F),set_ord_lessThan_nat(M)) = minus_minus_rat(aa_nat_rat(F,zero_zero_nat),aa_nat_rat(F,M)) ).

% sum_lessThan_telescope'
tff(fact_8368_sum__lessThan__telescope_H,axiom,
    ! [F: fun_nat_int,M: nat] : groups3539618377306564664at_int(aTP_Lamm_ast(F),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_8369_sum__lessThan__telescope_H,axiom,
    ! [F: fun_nat_real,M: nat] : groups6591440286371151544t_real(aTP_Lamm_asu(F),set_ord_lessThan_nat(M)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(F,zero_zero_nat)),aa_nat_real(F,M)) ).

% sum_lessThan_telescope'
tff(fact_8370_sumr__diff__mult__const2,axiom,
    ! [F: fun_nat_rat,Nb: nat,R2: rat] : minus_minus_rat(groups2906978787729119204at_rat(F,set_ord_lessThan_nat(Nb)),aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,Nb)),R2)) = groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aTP_Lamm_asv(F),R2),set_ord_lessThan_nat(Nb)) ).

% sumr_diff_mult_const2
tff(fact_8371_sumr__diff__mult__const2,axiom,
    ! [F: fun_nat_int,Nb: nat,R2: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,groups3539618377306564664at_int(F,set_ord_lessThan_nat(Nb))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),R2)) = groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_asw(F),R2),set_ord_lessThan_nat(Nb)) ).

% sumr_diff_mult_const2
tff(fact_8372_sumr__diff__mult__const2,axiom,
    ! [F: fun_nat_Code_integer,Nb: nat,R2: code_integer] : aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(groups7501900531339628137nteger(F,set_ord_lessThan_nat(Nb))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,Nb)),R2)) = groups7501900531339628137nteger(aa_Cod1578151519121547077nteger(aTP_Lamm_asx(F),R2),set_ord_lessThan_nat(Nb)) ).

% sumr_diff_mult_const2
tff(fact_8373_sumr__diff__mult__const2,axiom,
    ! [F: fun_nat_real,Nb: nat,R2: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,groups6591440286371151544t_real(F,set_ord_lessThan_nat(Nb))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),R2)) = groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_asy(F),R2),set_ord_lessThan_nat(Nb)) ).

% sumr_diff_mult_const2
tff(fact_8374_sum_OatLeast1__atMost__eq,axiom,
    ! [G: fun_nat_nat,Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat),Nb)) = aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_anu(G)),set_ord_lessThan_nat(Nb)) ).

% sum.atLeast1_atMost_eq
tff(fact_8375_sum_OatLeast1__atMost__eq,axiom,
    ! [G: fun_nat_real,Nb: nat] : groups6591440286371151544t_real(G,set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat),Nb)) = groups6591440286371151544t_real(aTP_Lamm_anv(G),set_ord_lessThan_nat(Nb)) ).

% sum.atLeast1_atMost_eq
tff(fact_8376_sums__iff__shift,axiom,
    ! [F: fun_nat_real,Nb: nat,S3: real] :
      ( aa_real_o(sums_real(aa_nat_fun_nat_real(aTP_Lamm_anx(F),Nb)),S3)
    <=> aa_real_o(sums_real(F),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,S3),groups6591440286371151544t_real(F,set_ord_lessThan_nat(Nb)))) ) ).

% sums_iff_shift
tff(fact_8377_sums__split__initial__segment,axiom,
    ! [F: fun_nat_real,S3: real,Nb: nat] :
      ( aa_real_o(sums_real(F),S3)
     => aa_real_o(sums_real(aa_nat_fun_nat_real(aTP_Lamm_anx(F),Nb)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,S3),groups6591440286371151544t_real(F,set_ord_lessThan_nat(Nb)))) ) ).

% sums_split_initial_segment
tff(fact_8378_sums__iff__shift_H,axiom,
    ! [F: fun_nat_real,Nb: nat,S3: real] :
      ( aa_real_o(sums_real(aa_nat_fun_nat_real(aTP_Lamm_anx(F),Nb)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,S3),groups6591440286371151544t_real(F,set_ord_lessThan_nat(Nb))))
    <=> aa_real_o(sums_real(F),S3) ) ).

% sums_iff_shift'
tff(fact_8379_floor__split,axiom,
    ! [P: fun_int_o,Ta: real] :
      ( aa_int_o(P,aa_real_int(archim6058952711729229775r_real,Ta))
    <=> ! [I2: int] :
          ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,I2)),Ta)
            & aa_real_o(aa_real_fun_real_o(ord_less_real,Ta),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,I2)),one_one_real)) )
         => aa_int_o(P,I2) ) ) ).

% floor_split
tff(fact_8380_floor__split,axiom,
    ! [P: fun_int_o,Ta: rat] :
      ( aa_int_o(P,archim3151403230148437115or_rat(Ta))
    <=> ! [I2: int] :
          ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,I2)),Ta)
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Ta),aa_rat_rat(plus_plus_rat(aa_int_rat(ring_1_of_int_rat,I2)),one_one_rat)) )
         => aa_int_o(P,I2) ) ) ).

% floor_split
tff(fact_8381_floor__eq__iff,axiom,
    ! [Xb: real,A: int] :
      ( ( aa_real_int(archim6058952711729229775r_real,Xb) = A )
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,A)),Xb)
        & aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,A)),one_one_real)) ) ) ).

% floor_eq_iff
tff(fact_8382_floor__eq__iff,axiom,
    ! [Xb: rat,A: int] :
      ( ( archim3151403230148437115or_rat(Xb) = A )
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,A)),Xb)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),aa_rat_rat(plus_plus_rat(aa_int_rat(ring_1_of_int_rat,A)),one_one_rat)) ) ) ).

% floor_eq_iff
tff(fact_8383_floor__unique,axiom,
    ! [Z: int,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Z)),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,Z)),one_one_real))
       => ( aa_real_int(archim6058952711729229775r_real,Xb) = Z ) ) ) ).

% floor_unique
tff(fact_8384_floor__unique,axiom,
    ! [Z: int,Xb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Z)),Xb)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),aa_rat_rat(plus_plus_rat(aa_int_rat(ring_1_of_int_rat,Z)),one_one_rat))
       => ( archim3151403230148437115or_rat(Xb) = Z ) ) ) ).

% floor_unique
tff(fact_8385_less__floor__iff,axiom,
    ! [Z: int,Xb: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z),aa_real_int(archim6058952711729229775r_real,Xb))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,Z)),one_one_real)),Xb) ) ).

% less_floor_iff
tff(fact_8386_less__floor__iff,axiom,
    ! [Z: int,Xb: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z),archim3151403230148437115or_rat(Xb))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(aa_int_rat(ring_1_of_int_rat,Z)),one_one_rat)),Xb) ) ).

% less_floor_iff
tff(fact_8387_floor__le__iff,axiom,
    ! [Xb: real,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(archim6058952711729229775r_real,Xb)),Z)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,Z)),one_one_real)) ) ).

% floor_le_iff
tff(fact_8388_floor__le__iff,axiom,
    ! [Xb: rat,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,archim3151403230148437115or_rat(Xb)),Z)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),aa_rat_rat(plus_plus_rat(aa_int_rat(ring_1_of_int_rat,Z)),one_one_rat)) ) ).

% floor_le_iff
tff(fact_8389_le__mult__floor,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_real_int(archim6058952711729229775r_real,A)),aa_real_int(archim6058952711729229775r_real,B))),aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B))) ) ) ).

% le_mult_floor
tff(fact_8390_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,archim3151403230148437115or_rat(A)),archim3151403230148437115or_rat(B))),archim3151403230148437115or_rat(aa_rat_rat(times_times_rat(A),B))) ) ) ).

% le_mult_floor
tff(fact_8391_floor__correct,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,aa_real_int(archim6058952711729229775r_real,Xb))),Xb)
      & aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(archim6058952711729229775r_real,Xb)),one_one_int))) ) ).

% floor_correct
tff(fact_8392_floor__correct,axiom,
    ! [Xb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,archim3151403230148437115or_rat(Xb))),Xb)
      & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(plus_plus_int,archim3151403230148437115or_rat(Xb)),one_one_int))) ) ).

% floor_correct
tff(fact_8393_floor__eq4,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Nb)))
       => ( aa_int_nat(nat2,aa_real_int(archim6058952711729229775r_real,Xb)) = Nb ) ) ) ).

% floor_eq4
tff(fact_8394_floor__eq2,axiom,
    ! [Nb: int,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Nb)),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,Nb)),one_one_real))
       => ( aa_real_int(archim6058952711729229775r_real,Xb) = Nb ) ) ) ).

% floor_eq2
tff(fact_8395_floor__divide__real__eq__div,axiom,
    ! [B: int,A: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
     => ( aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),aa_int_real(ring_1_of_int_real,B))) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_real_int(archim6058952711729229775r_real,A)),B) ) ) ).

% floor_divide_real_eq_div
tff(fact_8396_power__diff__1__eq,axiom,
    ! [Xb: uint32,Nb: nat] : minus_minus_uint32(aa_nat_uint32(power_power_uint32(Xb),Nb),one_one_uint32) = aa_uint32_uint32(times_times_uint32(minus_minus_uint32(Xb,one_one_uint32)),groups833757482993574392uint32(power_power_uint32(Xb),set_ord_lessThan_nat(Nb))) ).

% power_diff_1_eq
tff(fact_8397_power__diff__1__eq,axiom,
    ! [Xb: word_N3645301735248828278l_num1,Nb: nat] : minus_4019991460397169231l_num1(aa_nat5599461682457136004l_num1(power_2184487114949457152l_num1(Xb),Nb),one_on7727431528512463931l_num1) = aa_wor7274019993978516266l_num1(times_7065122842183080059l_num1(minus_4019991460397169231l_num1(Xb,one_on7727431528512463931l_num1)),groups2996710295995929986l_num1(power_2184487114949457152l_num1(Xb),set_ord_lessThan_nat(Nb))) ).

% power_diff_1_eq
tff(fact_8398_power__diff__1__eq,axiom,
    ! [Xb: rat,Nb: nat] : minus_minus_rat(aa_nat_rat(power_power_rat(Xb),Nb),one_one_rat) = aa_rat_rat(times_times_rat(minus_minus_rat(Xb,one_one_rat)),groups2906978787729119204at_rat(power_power_rat(Xb),set_ord_lessThan_nat(Nb))) ).

% power_diff_1_eq
tff(fact_8399_power__diff__1__eq,axiom,
    ! [Xb: int,Nb: nat] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,Xb),Nb)),one_one_int) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb),one_one_int)),groups3539618377306564664at_int(aa_int_fun_nat_int(power_power_int,Xb),set_ord_lessThan_nat(Nb))) ).

% power_diff_1_eq
tff(fact_8400_power__diff__1__eq,axiom,
    ! [Xb: real,Nb: nat] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb)),one_one_real) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),one_one_real)),groups6591440286371151544t_real(aa_real_fun_nat_real(power_power_real,Xb),set_ord_lessThan_nat(Nb))) ).

% power_diff_1_eq
tff(fact_8401_one__diff__power__eq,axiom,
    ! [Xb: uint32,Nb: nat] : minus_minus_uint32(one_one_uint32,aa_nat_uint32(power_power_uint32(Xb),Nb)) = aa_uint32_uint32(times_times_uint32(minus_minus_uint32(one_one_uint32,Xb)),groups833757482993574392uint32(power_power_uint32(Xb),set_ord_lessThan_nat(Nb))) ).

% one_diff_power_eq
tff(fact_8402_one__diff__power__eq,axiom,
    ! [Xb: word_N3645301735248828278l_num1,Nb: nat] : minus_4019991460397169231l_num1(one_on7727431528512463931l_num1,aa_nat5599461682457136004l_num1(power_2184487114949457152l_num1(Xb),Nb)) = aa_wor7274019993978516266l_num1(times_7065122842183080059l_num1(minus_4019991460397169231l_num1(one_on7727431528512463931l_num1,Xb)),groups2996710295995929986l_num1(power_2184487114949457152l_num1(Xb),set_ord_lessThan_nat(Nb))) ).

% one_diff_power_eq
tff(fact_8403_one__diff__power__eq,axiom,
    ! [Xb: rat,Nb: nat] : minus_minus_rat(one_one_rat,aa_nat_rat(power_power_rat(Xb),Nb)) = aa_rat_rat(times_times_rat(minus_minus_rat(one_one_rat,Xb)),groups2906978787729119204at_rat(power_power_rat(Xb),set_ord_lessThan_nat(Nb))) ).

% one_diff_power_eq
tff(fact_8404_one__diff__power__eq,axiom,
    ! [Xb: int,Nb: nat] : aa_int_int(aa_int_fun_int_int(minus_minus_int,one_one_int),aa_nat_int(aa_int_fun_nat_int(power_power_int,Xb),Nb)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,one_one_int),Xb)),groups3539618377306564664at_int(aa_int_fun_nat_int(power_power_int,Xb),set_ord_lessThan_nat(Nb))) ).

% one_diff_power_eq
tff(fact_8405_one__diff__power__eq,axiom,
    ! [Xb: real,Nb: nat] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),Xb)),groups6591440286371151544t_real(aa_real_fun_nat_real(power_power_real,Xb),set_ord_lessThan_nat(Nb))) ).

% one_diff_power_eq
tff(fact_8406_geometric__sum,axiom,
    ! [Xb: rat,Nb: nat] :
      ( ( Xb != one_one_rat )
     => ( groups2906978787729119204at_rat(power_power_rat(Xb),set_ord_lessThan_nat(Nb)) = divide_divide_rat(minus_minus_rat(aa_nat_rat(power_power_rat(Xb),Nb),one_one_rat),minus_minus_rat(Xb,one_one_rat)) ) ) ).

% geometric_sum
tff(fact_8407_geometric__sum,axiom,
    ! [Xb: real,Nb: nat] :
      ( ( Xb != one_one_real )
     => ( groups6591440286371151544t_real(aa_real_fun_nat_real(power_power_real,Xb),set_ord_lessThan_nat(Nb)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb)),one_one_real)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),one_one_real)) ) ) ).

% geometric_sum
tff(fact_8408_pochhammer__product,axiom,
    ! [M: nat,Nb: nat,Z: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
     => ( aa_nat_rat(comm_s4028243227959126397er_rat(Z),Nb) = aa_rat_rat(times_times_rat(aa_nat_rat(comm_s4028243227959126397er_rat(Z),M)),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(plus_plus_rat(Z),aa_nat_rat(semiri681578069525770553at_rat,M))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),M))) ) ) ).

% pochhammer_product
tff(fact_8409_pochhammer__product,axiom,
    ! [M: nat,Nb: nat,Z: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
     => ( aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,Z),Nb) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,Z),M)),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Z),aa_nat_real(semiri5074537144036343181t_real,M))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),M))) ) ) ).

% pochhammer_product
tff(fact_8410_pochhammer__product,axiom,
    ! [M: nat,Nb: nat,Z: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
     => ( aa_nat_int(aa_int_fun_nat_int(comm_s4660882817536571857er_int,Z),Nb) = 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,Nb),M))) ) ) ).

% pochhammer_product
tff(fact_8411_pochhammer__product,axiom,
    ! [M: nat,Nb: nat,Z: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,Z),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,Z),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,Nb),M))) ) ) ).

% pochhammer_product
tff(fact_8412_pochhammer__product,axiom,
    ! [M: nat,Nb: nat,Z: code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
     => ( aa_nat_Code_integer(comm_s8582702949713902594nteger(Z),Nb) = 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,Nb),M))) ) ) ).

% pochhammer_product
tff(fact_8413_floor__divide__lower,axiom,
    ! [Q2: real,P2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Q2)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(ring_1_of_int_real,aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,P2),Q2)))),Q2)),P2) ) ).

% floor_divide_lower
tff(fact_8414_floor__divide__lower,axiom,
    ! [Q2: rat,P2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Q2)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(aa_int_rat(ring_1_of_int_rat,archim3151403230148437115or_rat(divide_divide_rat(P2,Q2)))),Q2)),P2) ) ).

% floor_divide_lower
tff(fact_8415_lemma__termdiff1,axiom,
    ! [Z: uint32,H: uint32,M: nat] : groups833757482993574392uint32(aa_nat6935575117487981255uint32(aa_uin9008539043801383600uint32(aTP_Lamm_asz(Z),H),M),set_ord_lessThan_nat(M)) = groups833757482993574392uint32(aa_nat6935575117487981255uint32(aa_uin9008539043801383600uint32(aTP_Lamm_ata(Z),H),M),set_ord_lessThan_nat(M)) ).

% lemma_termdiff1
tff(fact_8416_lemma__termdiff1,axiom,
    ! [Z: word_N3645301735248828278l_num1,H: word_N3645301735248828278l_num1,M: nat] : groups2996710295995929986l_num1(aa_nat6251203789436639057l_num1(aa_wor3973378241480532036l_num1(aTP_Lamm_atb(Z),H),M),set_ord_lessThan_nat(M)) = groups2996710295995929986l_num1(aa_nat6251203789436639057l_num1(aa_wor3973378241480532036l_num1(aTP_Lamm_atc(Z),H),M),set_ord_lessThan_nat(M)) ).

% lemma_termdiff1
tff(fact_8417_lemma__termdiff1,axiom,
    ! [Z: rat,H: rat,M: nat] : groups2906978787729119204at_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_atd(Z),H),M),set_ord_lessThan_nat(M)) = groups2906978787729119204at_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_ate(Z),H),M),set_ord_lessThan_nat(M)) ).

% lemma_termdiff1
tff(fact_8418_lemma__termdiff1,axiom,
    ! [Z: int,H: int,M: nat] : groups3539618377306564664at_int(aa_nat_fun_nat_int(aa_int431785320988601264at_int(aTP_Lamm_atf(Z),H),M),set_ord_lessThan_nat(M)) = groups3539618377306564664at_int(aa_nat_fun_nat_int(aa_int431785320988601264at_int(aTP_Lamm_atg(Z),H),M),set_ord_lessThan_nat(M)) ).

% lemma_termdiff1
tff(fact_8419_lemma__termdiff1,axiom,
    ! [Z: real,H: real,M: nat] : groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_ath(Z),H),M),set_ord_lessThan_nat(M)) = groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_ati(Z),H),M),set_ord_lessThan_nat(M)) ).

% lemma_termdiff1
tff(fact_8420_sum__gp__strict,axiom,
    ! [Xb: rat,Nb: nat] :
      groups2906978787729119204at_rat(power_power_rat(Xb),set_ord_lessThan_nat(Nb)) = $ite(Xb = one_one_rat,aa_nat_rat(semiri681578069525770553at_rat,Nb),divide_divide_rat(minus_minus_rat(one_one_rat,aa_nat_rat(power_power_rat(Xb),Nb)),minus_minus_rat(one_one_rat,Xb))) ).

% sum_gp_strict
tff(fact_8421_sum__gp__strict,axiom,
    ! [Xb: real,Nb: nat] :
      groups6591440286371151544t_real(aa_real_fun_nat_real(power_power_real,Xb),set_ord_lessThan_nat(Nb)) = $ite(Xb = one_one_real,aa_nat_real(semiri5074537144036343181t_real,Nb),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb))),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),Xb))) ).

% sum_gp_strict
tff(fact_8422_power__diff__sumr2,axiom,
    ! [Xb: uint32,Nb: nat,Y: uint32] : minus_minus_uint32(aa_nat_uint32(power_power_uint32(Xb),Nb),aa_nat_uint32(power_power_uint32(Y),Nb)) = aa_uint32_uint32(times_times_uint32(minus_minus_uint32(Xb,Y)),groups833757482993574392uint32(aa_uin238090125092722531uint32(aa_nat7697544214703702448uint32(aTP_Lamm_atj(Xb),Nb),Y),set_ord_lessThan_nat(Nb))) ).

% power_diff_sumr2
tff(fact_8423_power__diff__sumr2,axiom,
    ! [Xb: word_N3645301735248828278l_num1,Nb: nat,Y: word_N3645301735248828278l_num1] : minus_4019991460397169231l_num1(aa_nat5599461682457136004l_num1(power_2184487114949457152l_num1(Xb),Nb),aa_nat5599461682457136004l_num1(power_2184487114949457152l_num1(Y),Nb)) = aa_wor7274019993978516266l_num1(times_7065122842183080059l_num1(minus_4019991460397169231l_num1(Xb,Y)),groups2996710295995929986l_num1(aa_wor1031616255390178039l_num1(aa_nat6155939771436694596l_num1(aTP_Lamm_atk(Xb),Nb),Y),set_ord_lessThan_nat(Nb))) ).

% power_diff_sumr2
tff(fact_8424_power__diff__sumr2,axiom,
    ! [Xb: rat,Nb: nat,Y: rat] : minus_minus_rat(aa_nat_rat(power_power_rat(Xb),Nb),aa_nat_rat(power_power_rat(Y),Nb)) = aa_rat_rat(times_times_rat(minus_minus_rat(Xb,Y)),groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aa_nat298952458717073160at_rat(aTP_Lamm_atl(Xb),Nb),Y),set_ord_lessThan_nat(Nb))) ).

% power_diff_sumr2
tff(fact_8425_power__diff__sumr2,axiom,
    ! [Xb: int,Nb: 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,Xb),Nb)),aa_nat_int(aa_int_fun_nat_int(power_power_int,Y),Nb)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb),Y)),groups3539618377306564664at_int(aa_int_fun_nat_int(aa_nat6704147566042551216at_int(aTP_Lamm_atm(Xb),Nb),Y),set_ord_lessThan_nat(Nb))) ).

% power_diff_sumr2
tff(fact_8426_power__diff__sumr2,axiom,
    ! [Xb: real,Nb: nat,Y: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),Nb)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),Y)),groups6591440286371151544t_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_atn(Xb),Nb),Y),set_ord_lessThan_nat(Nb))) ).

% power_diff_sumr2
tff(fact_8427_diff__power__eq__sum,axiom,
    ! [Xb: uint32,Nb: nat,Y: uint32] : minus_minus_uint32(aa_nat_uint32(power_power_uint32(Xb),aa_nat_nat(suc,Nb)),aa_nat_uint32(power_power_uint32(Y),aa_nat_nat(suc,Nb))) = aa_uint32_uint32(times_times_uint32(minus_minus_uint32(Xb,Y)),groups833757482993574392uint32(aa_uin238090125092722531uint32(aa_nat7697544214703702448uint32(aTP_Lamm_ato(Xb),Nb),Y),set_ord_lessThan_nat(aa_nat_nat(suc,Nb)))) ).

% diff_power_eq_sum
tff(fact_8428_diff__power__eq__sum,axiom,
    ! [Xb: word_N3645301735248828278l_num1,Nb: nat,Y: word_N3645301735248828278l_num1] : minus_4019991460397169231l_num1(aa_nat5599461682457136004l_num1(power_2184487114949457152l_num1(Xb),aa_nat_nat(suc,Nb)),aa_nat5599461682457136004l_num1(power_2184487114949457152l_num1(Y),aa_nat_nat(suc,Nb))) = aa_wor7274019993978516266l_num1(times_7065122842183080059l_num1(minus_4019991460397169231l_num1(Xb,Y)),groups2996710295995929986l_num1(aa_wor1031616255390178039l_num1(aa_nat6155939771436694596l_num1(aTP_Lamm_atp(Xb),Nb),Y),set_ord_lessThan_nat(aa_nat_nat(suc,Nb)))) ).

% diff_power_eq_sum
tff(fact_8429_diff__power__eq__sum,axiom,
    ! [Xb: rat,Nb: nat,Y: rat] : minus_minus_rat(aa_nat_rat(power_power_rat(Xb),aa_nat_nat(suc,Nb)),aa_nat_rat(power_power_rat(Y),aa_nat_nat(suc,Nb))) = aa_rat_rat(times_times_rat(minus_minus_rat(Xb,Y)),groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aa_nat298952458717073160at_rat(aTP_Lamm_atq(Xb),Nb),Y),set_ord_lessThan_nat(aa_nat_nat(suc,Nb)))) ).

% diff_power_eq_sum
tff(fact_8430_diff__power__eq__sum,axiom,
    ! [Xb: int,Nb: 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,Xb),aa_nat_nat(suc,Nb))),aa_nat_int(aa_int_fun_nat_int(power_power_int,Y),aa_nat_nat(suc,Nb))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb),Y)),groups3539618377306564664at_int(aa_int_fun_nat_int(aa_nat6704147566042551216at_int(aTP_Lamm_atr(Xb),Nb),Y),set_ord_lessThan_nat(aa_nat_nat(suc,Nb)))) ).

% diff_power_eq_sum
tff(fact_8431_diff__power__eq__sum,axiom,
    ! [Xb: real,Nb: nat,Y: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_nat_nat(suc,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_nat_nat(suc,Nb))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),Y)),groups6591440286371151544t_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_ats(Xb),Nb),Y),set_ord_lessThan_nat(aa_nat_nat(suc,Nb)))) ).

% diff_power_eq_sum
tff(fact_8432_floor__divide__upper,axiom,
    ! [Q2: real,P2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Q2)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,P2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,P2),Q2)))),one_one_real)),Q2)) ) ).

% floor_divide_upper
tff(fact_8433_floor__divide__upper,axiom,
    ! [Q2: rat,P2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Q2)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,P2),aa_rat_rat(times_times_rat(aa_rat_rat(plus_plus_rat(aa_int_rat(ring_1_of_int_rat,archim3151403230148437115or_rat(divide_divide_rat(P2,Q2)))),one_one_rat)),Q2)) ) ).

% floor_divide_upper
tff(fact_8434_round__def,axiom,
    ! [Xb: real] : aa_real_int(archim8280529875227126926d_real,Xb) = aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(aa_num_num(bit0,one))))) ).

% round_def
tff(fact_8435_round__def,axiom,
    ! [Xb: rat] : archim7778729529865785530nd_rat(Xb) = archim3151403230148437115or_rat(aa_rat_rat(plus_plus_rat(Xb),divide_divide_rat(one_one_rat,numeral_numeral_rat(aa_num_num(bit0,one))))) ).

% round_def
tff(fact_8436_real__sum__nat__ivl__bounded2,axiom,
    ! [Nb: nat,F: fun_nat_Code_integer,K6: code_integer,K: nat] :
      ( ! [P5: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,P5),Nb)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(F,P5)),K6) )
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),K6)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,groups7501900531339628137nteger(F,set_ord_lessThan_nat(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K)))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,Nb)),K6)) ) ) ).

% real_sum_nat_ivl_bounded2
tff(fact_8437_real__sum__nat__ivl__bounded2,axiom,
    ! [Nb: nat,F: fun_nat_rat,K6: rat,K: nat] :
      ( ! [P5: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,P5),Nb)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,P5)),K6) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),K6)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups2906978787729119204at_rat(F,set_ord_lessThan_nat(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K)))),aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,Nb)),K6)) ) ) ).

% real_sum_nat_ivl_bounded2
tff(fact_8438_real__sum__nat__ivl__bounded2,axiom,
    ! [Nb: nat,F: fun_nat_int,K6: int,K: nat] :
      ( ! [P5: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,P5),Nb)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,P5)),K6) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K6)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,groups3539618377306564664at_int(F,set_ord_lessThan_nat(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K)))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),K6)) ) ) ).

% real_sum_nat_ivl_bounded2
tff(fact_8439_real__sum__nat__ivl__bounded2,axiom,
    ! [Nb: nat,F: fun_nat_nat,K6: nat,K: nat] :
      ( ! [P5: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,P5),Nb)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F,P5)),K6) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),K6)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),set_ord_lessThan_nat(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,Nb)),K6)) ) ) ).

% real_sum_nat_ivl_bounded2
tff(fact_8440_real__sum__nat__ivl__bounded2,axiom,
    ! [Nb: nat,F: fun_nat_real,K6: real,K: nat] :
      ( ! [P5: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,P5),Nb)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(F,P5)),K6) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),K6)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups6591440286371151544t_real(F,set_ord_lessThan_nat(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K)))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),K6)) ) ) ).

% real_sum_nat_ivl_bounded2
tff(fact_8441_one__diff__power__eq_H,axiom,
    ! [Xb: uint32,Nb: nat] : minus_minus_uint32(one_one_uint32,aa_nat_uint32(power_power_uint32(Xb),Nb)) = aa_uint32_uint32(times_times_uint32(minus_minus_uint32(one_one_uint32,Xb)),groups833757482993574392uint32(aa_nat6935575117487981255uint32(aTP_Lamm_att(Xb),Nb),set_ord_lessThan_nat(Nb))) ).

% one_diff_power_eq'
tff(fact_8442_one__diff__power__eq_H,axiom,
    ! [Xb: word_N3645301735248828278l_num1,Nb: nat] : minus_4019991460397169231l_num1(one_on7727431528512463931l_num1,aa_nat5599461682457136004l_num1(power_2184487114949457152l_num1(Xb),Nb)) = aa_wor7274019993978516266l_num1(times_7065122842183080059l_num1(minus_4019991460397169231l_num1(one_on7727431528512463931l_num1,Xb)),groups2996710295995929986l_num1(aa_nat6251203789436639057l_num1(aTP_Lamm_atu(Xb),Nb),set_ord_lessThan_nat(Nb))) ).

% one_diff_power_eq'
tff(fact_8443_one__diff__power__eq_H,axiom,
    ! [Xb: rat,Nb: nat] : minus_minus_rat(one_one_rat,aa_nat_rat(power_power_rat(Xb),Nb)) = aa_rat_rat(times_times_rat(minus_minus_rat(one_one_rat,Xb)),groups2906978787729119204at_rat(aa_nat_fun_nat_rat(aTP_Lamm_atv(Xb),Nb),set_ord_lessThan_nat(Nb))) ).

% one_diff_power_eq'
tff(fact_8444_one__diff__power__eq_H,axiom,
    ! [Xb: int,Nb: nat] : aa_int_int(aa_int_fun_int_int(minus_minus_int,one_one_int),aa_nat_int(aa_int_fun_nat_int(power_power_int,Xb),Nb)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,one_one_int),Xb)),groups3539618377306564664at_int(aa_nat_fun_nat_int(aTP_Lamm_atw(Xb),Nb),set_ord_lessThan_nat(Nb))) ).

% one_diff_power_eq'
tff(fact_8445_one__diff__power__eq_H,axiom,
    ! [Xb: real,Nb: nat] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),Xb)),groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_atx(Xb),Nb),set_ord_lessThan_nat(Nb))) ).

% one_diff_power_eq'
tff(fact_8446_TBOUND__adm,axiom,
    ! [Ta: fun_Pr8246910697231053897at_nat] : comple7368800650122905596on_nat(partia2080987842261039902Ti_nat(heap_T7048022066654196708on_nat),partia8535946980722491222Ti_nat(heap_T7875578835903804844on_nat),aTP_Lamm_aty(Ta)) ).

% TBOUND_adm
tff(fact_8447_TBOUND__adm,axiom,
    ! [Ta: fun_Pr8246910697231053897at_nat] : comple8003709893544916456Heap_o(partia5551857090987368152Ti_nat(heap_Time_Heap_lub_o),partia2925774515620677392Ti_nat(heap_Time_Heap_ord_o),aTP_Lamm_atz(Ta)) ).

% TBOUND_adm
tff(fact_8448_TBOUND__adm,axiom,
    ! [Ta: fun_Pr3615232783394198956at_nat] : comple8556566605269262190_VEBTi(partia7782936097874681665Ti_nat(heap_T3112222404744780921_VEBTi),partia1868168049876374393Ti_nat(heap_T7173139186834293313_VEBTi),aTP_Lamm_aua(Ta)) ).

% TBOUND_adm
tff(fact_8449_TBOUND__adm,axiom,
    ! [Ta: fun_Pr3615232783394198956at_nat] : comple5664458124825824345on_nat(partia6039416512482706817Ti_nat(heap_T7048022066654196708on_nat),partia7778075537949639097Ti_nat(heap_T7875578835903804844on_nat),aTP_Lamm_aub(Ta)) ).

% TBOUND_adm
tff(fact_8450_TBOUND__adm,axiom,
    ! [Ta: fun_Pr3615232783394198956at_nat] : comple5871978649864577227Heap_o(partia6726927458685305659Ti_nat(heap_Time_Heap_lub_o),partia3290229181235258227Ti_nat(heap_Time_Heap_ord_o),aTP_Lamm_auc(Ta)) ).

% TBOUND_adm
tff(fact_8451_refines__adm,axiom,
    ! [Ta: fun_Pr342922516890243977on_nat] : comple7368800650122905596on_nat(partia2080987842261039902Ti_nat(heap_T7048022066654196708on_nat),partia8535946980722491222Ti_nat(heap_T7875578835903804844on_nat),aa_fun6940291128091933217_nat_o(aTP_Lamm_aud,Ta)) ).

% refines_adm
tff(fact_8452_refines__adm,axiom,
    ! [Ta: fun_Pr238758252084143515Heap_o] : comple8003709893544916456Heap_o(partia5551857090987368152Ti_nat(heap_Time_Heap_lub_o),partia2925774515620677392Ti_nat(heap_Time_Heap_ord_o),aa_fun2387398806603316575ap_o_o(aTP_Lamm_aue,Ta)) ).

% refines_adm
tff(fact_8453_refines__adm,axiom,
    ! [Ta: fun_Pr4404027214751104635_VEBTi] : comple8556566605269262190_VEBTi(partia7782936097874681665Ti_nat(heap_T3112222404744780921_VEBTi),partia1868168049876374393Ti_nat(heap_T7173139186834293313_VEBTi),aa_fun3634491704933443261EBTi_o(aTP_Lamm_auf,Ta)) ).

% refines_adm
tff(fact_8454_refines__adm,axiom,
    ! [Ta: fun_Pr633315466666964198on_nat] : comple5664458124825824345on_nat(partia6039416512482706817Ti_nat(heap_T7048022066654196708on_nat),partia7778075537949639097Ti_nat(heap_T7875578835903804844on_nat),aa_fun2070105859800901863_nat_o(aTP_Lamm_aug,Ta)) ).

% refines_adm
tff(fact_8455_refines__adm,axiom,
    ! [Ta: fun_Pr940977554174145406Heap_o] : comple5871978649864577227Heap_o(partia6726927458685305659Ti_nat(heap_Time_Heap_lub_o),partia3290229181235258227Ti_nat(heap_Time_Heap_ord_o),aa_fun4060910825685108895ap_o_o(aTP_Lamm_auh,Ta)) ).

% refines_adm
tff(fact_8456_sum__split__even__odd,axiom,
    ! [F: fun_nat_real,G: fun_nat_real,Nb: nat] : groups6591440286371151544t_real(aa_fun1839206346889086768t_real(aTP_Lamm_aui(F),G),set_ord_lessThan_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aTP_Lamm_auj(F),set_ord_lessThan_nat(Nb))),groups6591440286371151544t_real(aTP_Lamm_auk(G),set_ord_lessThan_nat(Nb))) ).

% sum_split_even_odd
tff(fact_8457_floor__log__eq__powr__iff,axiom,
    ! [Xb: real,B: real,K: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B)
       => ( ( aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(log,B),Xb)) = K )
        <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),aa_int_real(ring_1_of_int_real,K))),Xb)
            & aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(plus_plus_int,K),one_one_int)))) ) ) ) ) ).

% floor_log_eq_powr_iff
tff(fact_8458_pochhammer__code,axiom,
    ! [A: word_N3645301735248828278l_num1,Nb: nat] :
      aa_nat5599461682457136004l_num1(comm_s6431939913906641691l_num1(A),Nb) = $ite(Nb = zero_zero_nat,one_on7727431528512463931l_num1,set_fo4709898541180519304l_num1(aTP_Lamm_aul(A),zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat),one_on7727431528512463931l_num1)) ).

% pochhammer_code
tff(fact_8459_pochhammer__code,axiom,
    ! [A: rat,Nb: nat] :
      aa_nat_rat(comm_s4028243227959126397er_rat(A),Nb) = $ite(Nb = zero_zero_nat,one_one_rat,set_fo1949268297981939178at_rat(aTP_Lamm_aum(A),zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat),one_one_rat)) ).

% pochhammer_code
tff(fact_8460_pochhammer__code,axiom,
    ! [A: real,Nb: nat] :
      aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,A),Nb) = $ite(Nb = zero_zero_nat,one_one_real,aa_real_real(set_fo3111899725591712190t_real(aTP_Lamm_aun(A),zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)),one_one_real)) ).

% pochhammer_code
tff(fact_8461_pochhammer__code,axiom,
    ! [A: int,Nb: nat] :
      aa_nat_int(aa_int_fun_nat_int(comm_s4660882817536571857er_int,A),Nb) = $ite(Nb = zero_zero_nat,one_one_int,aa_int_int(aa_nat_fun_int_int(set_fo2581907887559384638at_int(aTP_Lamm_auo(A),zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)),one_one_int)) ).

% pochhammer_code
tff(fact_8462_pochhammer__code,axiom,
    ! [A: code_integer,Nb: nat] :
      aa_nat_Code_integer(comm_s8582702949713902594nteger(A),Nb) = $ite(Nb = zero_zero_nat,one_one_Code_integer,aa_Cod1783974652533396728nteger(set_fo1084959871951514735nteger(aTP_Lamm_aup(A),zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)),one_one_Code_integer)) ).

% pochhammer_code
tff(fact_8463_pochhammer__code,axiom,
    ! [A: nat,Nb: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,A),Nb) = $ite(Nb = zero_zero_nat,one_one_nat,aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(aTP_Lamm_auq(A),zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)),one_one_nat)) ).

% pochhammer_code
tff(fact_8464_TBOUND__fi__adm,axiom,
    ! [Foo: fun_VE4436863797057549263at_nat] : comple7368800650122905596on_nat(partia2080987842261039902Ti_nat(heap_T7048022066654196708on_nat),partia8535946980722491222Ti_nat(heap_T7875578835903804844on_nat),aTP_Lamm_aur(Foo)) ).

% TBOUND_fi_adm
tff(fact_8465_TBOUND__fi__adm,axiom,
    ! [Foo: fun_VE4436863797057549263at_nat] : comple8003709893544916456Heap_o(partia5551857090987368152Ti_nat(heap_Time_Heap_lub_o),partia2925774515620677392Ti_nat(heap_Time_Heap_ord_o),aTP_Lamm_aus(Foo)) ).

% TBOUND_fi_adm
tff(fact_8466_TBOUND__fi__adm,axiom,
    ! [Foo: fun_Pr2564325226415181874at_nat] : comple8556566605269262190_VEBTi(partia7782936097874681665Ti_nat(heap_T3112222404744780921_VEBTi),partia1868168049876374393Ti_nat(heap_T7173139186834293313_VEBTi),aTP_Lamm_aut(Foo)) ).

% TBOUND_fi_adm
tff(fact_8467_TBOUND__fi__adm,axiom,
    ! [Foo: fun_Pr2564325226415181874at_nat] : comple5664458124825824345on_nat(partia6039416512482706817Ti_nat(heap_T7048022066654196708on_nat),partia7778075537949639097Ti_nat(heap_T7875578835903804844on_nat),aTP_Lamm_auu(Foo)) ).

% TBOUND_fi_adm
tff(fact_8468_TBOUND__fi__adm,axiom,
    ! [Foo: fun_Pr2564325226415181874at_nat] : comple5871978649864577227Heap_o(partia6726927458685305659Ti_nat(heap_Time_Heap_lub_o),partia3290229181235258227Ti_nat(heap_Time_Heap_ord_o),aTP_Lamm_auv(Foo)) ).

% TBOUND_fi_adm
tff(fact_8469_floor__log2__div2,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(log,numeral_numeral_real(aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,Nb))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(log,numeral_numeral_real(aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),one_one_int) ) ) ).

% floor_log2_div2
tff(fact_8470_floor__log__nat__eq__if,axiom,
    ! [B: nat,Nb: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Nb)),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,Nb),one_one_nat)))
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B)
         => ( aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(log,aa_nat_real(semiri5074537144036343181t_real,B)),aa_nat_real(semiri5074537144036343181t_real,K))) = aa_nat_int(semiri1314217659103216013at_int,Nb) ) ) ) ) ).

% floor_log_nat_eq_if
tff(fact_8471_sum__bounds__lt__plus1,axiom,
    ! [F: fun_nat_nat,Mm: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_anu(F)),set_ord_lessThan_nat(Mm)) = aa_set_nat_nat(groups3542108847815614940at_nat(F),set_or1269000886237332187st_nat(one_one_nat,Mm)) ).

% sum_bounds_lt_plus1
tff(fact_8472_sum__bounds__lt__plus1,axiom,
    ! [F: fun_nat_real,Mm: nat] : groups6591440286371151544t_real(aTP_Lamm_anv(F),set_ord_lessThan_nat(Mm)) = groups6591440286371151544t_real(F,set_or1269000886237332187st_nat(one_one_nat,Mm)) ).

% sum_bounds_lt_plus1
tff(fact_8473_pochhammer__times__pochhammer__half,axiom,
    ! [Z: rat,Nb: nat] : aa_rat_rat(times_times_rat(aa_nat_rat(comm_s4028243227959126397er_rat(Z),aa_nat_nat(suc,Nb))),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(plus_plus_rat(Z),divide_divide_rat(one_one_rat,numeral_numeral_rat(aa_num_num(bit0,one))))),aa_nat_nat(suc,Nb))) = groups73079841787564623at_rat(aTP_Lamm_auw(Z),set_or1269000886237332187st_nat(zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)),one_one_nat))) ).

% pochhammer_times_pochhammer_half
tff(fact_8474_pochhammer__times__pochhammer__half,axiom,
    ! [Z: real,Nb: nat] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,Z),aa_nat_nat(suc,Nb))),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Z),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(aa_num_num(bit0,one))))),aa_nat_nat(suc,Nb))) = groups129246275422532515t_real(aa_real_fun_nat_real(aTP_Lamm_aux,Z),set_or1269000886237332187st_nat(zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)),one_one_nat))) ).

% pochhammer_times_pochhammer_half
tff(fact_8475_sumr__cos__zero__one,axiom,
    ! [Nb: nat] : groups6591440286371151544t_real(aTP_Lamm_auy,set_ord_lessThan_nat(aa_nat_nat(suc,Nb))) = one_one_real ).

% sumr_cos_zero_one
tff(fact_8476_finite__nat__bounded,axiom,
    ! [S2: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,S2)
     => ? [K3: nat] : aa_set_nat_o(ord_less_eq_set_nat(S2),set_ord_lessThan_nat(K3)) ) ).

% finite_nat_bounded
tff(fact_8477_finite__nat__iff__bounded,axiom,
    ! [S2: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,S2)
    <=> ? [K4: nat] : aa_set_nat_o(ord_less_eq_set_nat(S2),set_ord_lessThan_nat(K4)) ) ).

% finite_nat_iff_bounded
tff(fact_8478_prod_Oneutral__const,axiom,
    ! [A2: set_nat] : groups705719431365010083at_int(aTP_Lamm_auz,A2) = one_one_int ).

% prod.neutral_const
tff(fact_8479_prod_Oneutral__const,axiom,
    ! [A2: set_int] : groups1705073143266064639nt_int(aTP_Lamm_ava,A2) = one_one_int ).

% prod.neutral_const
tff(fact_8480_prod_Oneutral__const,axiom,
    ! [A2: set_nat] : groups708209901874060359at_nat(aTP_Lamm_avb,A2) = one_one_nat ).

% prod.neutral_const
tff(fact_8481_of__nat__prod,axiom,
    ! [F: fun_int_nat,A2: set_int] : aa_nat_int(semiri1314217659103216013at_int,groups1707563613775114915nt_nat(F,A2)) = groups1705073143266064639nt_int(aTP_Lamm_ais(F),A2) ).

% of_nat_prod
tff(fact_8482_of__nat__prod,axiom,
    ! [F: fun_nat_nat,A2: set_nat] : aa_nat_real(semiri5074537144036343181t_real,groups708209901874060359at_nat(F,A2)) = groups129246275422532515t_real(aTP_Lamm_jo(F),A2) ).

% of_nat_prod
tff(fact_8483_of__nat__prod,axiom,
    ! [F: fun_nat_nat,A2: set_nat] : aa_nat_Code_integer(semiri4939895301339042750nteger,groups708209901874060359at_nat(F,A2)) = groups3455450783089532116nteger(aTP_Lamm_aiu(F),A2) ).

% of_nat_prod
tff(fact_8484_of__nat__prod,axiom,
    ! [F: fun_nat_nat,A2: set_nat] : aa_nat_int(semiri1314217659103216013at_int,groups708209901874060359at_nat(F,A2)) = groups705719431365010083at_int(aTP_Lamm_ait(F),A2) ).

% of_nat_prod
tff(fact_8485_of__nat__prod,axiom,
    ! [F: fun_nat_nat,A2: set_nat] : aa_nat_nat(semiri1316708129612266289at_nat,groups708209901874060359at_nat(F,A2)) = groups708209901874060359at_nat(aTP_Lamm_aiv(F),A2) ).

% of_nat_prod
tff(fact_8486_of__int__prod,axiom,
    ! [F: fun_nat_int,A2: set_nat] : aa_int_real(ring_1_of_int_real,groups705719431365010083at_int(F,A2)) = groups129246275422532515t_real(aTP_Lamm_aix(F),A2) ).

% of_int_prod
tff(fact_8487_of__int__prod,axiom,
    ! [F: fun_nat_int,A2: set_nat] : aa_int_rat(ring_1_of_int_rat,groups705719431365010083at_int(F,A2)) = groups73079841787564623at_rat(aTP_Lamm_avc(F),A2) ).

% of_int_prod
tff(fact_8488_of__int__prod,axiom,
    ! [F: fun_nat_int,A2: set_nat] : aa_int_int(ring_1_of_int_int,groups705719431365010083at_int(F,A2)) = groups705719431365010083at_int(aTP_Lamm_avd(F),A2) ).

% of_int_prod
tff(fact_8489_of__int__prod,axiom,
    ! [F: fun_int_int,A2: set_int] : aa_int_real(ring_1_of_int_real,groups1705073143266064639nt_int(F,A2)) = groups2316167850115554303t_real(aTP_Lamm_aiy(F),A2) ).

% of_int_prod
tff(fact_8490_of__int__prod,axiom,
    ! [F: fun_int_int,A2: set_int] : aa_int_rat(ring_1_of_int_rat,groups1705073143266064639nt_int(F,A2)) = groups1072433553688619179nt_rat(aTP_Lamm_aiz(F),A2) ).

% of_int_prod
tff(fact_8491_of__int__prod,axiom,
    ! [F: fun_int_int,A2: set_int] : aa_int_int(ring_1_of_int_int,groups1705073143266064639nt_int(F,A2)) = groups1705073143266064639nt_int(aTP_Lamm_aja(F),A2) ).

% of_int_prod
tff(fact_8492_prod_Oempty,axiom,
    ! [G: fun_nat_real] : groups129246275422532515t_real(G,bot_bot_set_nat) = one_one_real ).

% prod.empty
tff(fact_8493_prod_Oempty,axiom,
    ! [G: fun_nat_rat] : groups73079841787564623at_rat(G,bot_bot_set_nat) = one_one_rat ).

% prod.empty
tff(fact_8494_prod_Oempty,axiom,
    ! [G: fun_int_real] : groups2316167850115554303t_real(G,bot_bot_set_int) = one_one_real ).

% prod.empty
tff(fact_8495_prod_Oempty,axiom,
    ! [G: fun_int_rat] : groups1072433553688619179nt_rat(G,bot_bot_set_int) = one_one_rat ).

% prod.empty
tff(fact_8496_prod_Oempty,axiom,
    ! [G: fun_int_nat] : groups1707563613775114915nt_nat(G,bot_bot_set_int) = one_one_nat ).

% prod.empty
tff(fact_8497_prod_Oempty,axiom,
    ! [G: fun_real_real] : groups1681761925125756287l_real(G,bot_bot_set_real) = one_one_real ).

% prod.empty
tff(fact_8498_prod_Oempty,axiom,
    ! [G: fun_real_rat] : groups4061424788464935467al_rat(G,bot_bot_set_real) = one_one_rat ).

% prod.empty
tff(fact_8499_prod_Oempty,axiom,
    ! [G: fun_real_nat] : groups4696554848551431203al_nat(G,bot_bot_set_real) = one_one_nat ).

% prod.empty
tff(fact_8500_prod_Oempty,axiom,
    ! [G: fun_real_int] : groups4694064378042380927al_int(G,bot_bot_set_real) = one_one_int ).

% prod.empty
tff(fact_8501_prod_Oempty,axiom,
    ! [G: fun_nat_int] : groups705719431365010083at_int(G,bot_bot_set_nat) = one_one_int ).

% prod.empty
tff(fact_8502_prod_Oinfinite,axiom,
    ! [A2: set_nat,G: fun_nat_real] :
      ( ~ aa_set_nat_o(finite_finite_nat,A2)
     => ( groups129246275422532515t_real(G,A2) = one_one_real ) ) ).

% prod.infinite
tff(fact_8503_prod_Oinfinite,axiom,
    ! [A2: set_int,G: fun_int_real] :
      ( ~ aa_set_int_o(finite_finite_int,A2)
     => ( groups2316167850115554303t_real(G,A2) = one_one_real ) ) ).

% prod.infinite
tff(fact_8504_prod_Oinfinite,axiom,
    ! [A2: set_complex,G: fun_complex_real] :
      ( ~ aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( groups766887009212190081x_real(G,A2) = one_one_real ) ) ).

% prod.infinite
tff(fact_8505_prod_Oinfinite,axiom,
    ! [A2: set_Code_integer,G: fun_Co3960896333048504454r_real] :
      ( ~ aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A2)
     => ( groups9004974159866482096r_real(G,A2) = one_one_real ) ) ).

% prod.infinite
tff(fact_8506_prod_Oinfinite,axiom,
    ! [A2: set_nat,G: fun_nat_rat] :
      ( ~ aa_set_nat_o(finite_finite_nat,A2)
     => ( groups73079841787564623at_rat(G,A2) = one_one_rat ) ) ).

% prod.infinite
tff(fact_8507_prod_Oinfinite,axiom,
    ! [A2: set_int,G: fun_int_rat] :
      ( ~ aa_set_int_o(finite_finite_int,A2)
     => ( groups1072433553688619179nt_rat(G,A2) = one_one_rat ) ) ).

% prod.infinite
tff(fact_8508_prod_Oinfinite,axiom,
    ! [A2: set_complex,G: fun_complex_rat] :
      ( ~ aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( groups225925009352817453ex_rat(G,A2) = one_one_rat ) ) ).

% prod.infinite
tff(fact_8509_prod_Oinfinite,axiom,
    ! [A2: set_Code_integer,G: fun_Code_integer_rat] :
      ( ~ aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A2)
     => ( groups2555765274223993564er_rat(G,A2) = one_one_rat ) ) ).

% prod.infinite
tff(fact_8510_prod_Oinfinite,axiom,
    ! [A2: set_int,G: fun_int_nat] :
      ( ~ aa_set_int_o(finite_finite_int,A2)
     => ( groups1707563613775114915nt_nat(G,A2) = one_one_nat ) ) ).

% prod.infinite
tff(fact_8511_prod_Oinfinite,axiom,
    ! [A2: set_complex,G: fun_complex_nat] :
      ( ~ aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( groups861055069439313189ex_nat(G,A2) = one_one_nat ) ) ).

% prod.infinite
tff(fact_8512_cos__coeff__0,axiom,
    aa_nat_real(cos_coeff,zero_zero_nat) = one_one_real ).

% cos_coeff_0
tff(fact_8513_prod_Odelta_H,axiom,
    ! [S2: set_VEBT_VEBT,A: vEBT_VEBT,B: fun_VEBT_VEBT_real] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,S2)
     => ( groups2703838992350267259T_real(aa_fun1825138017709814396T_real(aTP_Lamm_ave(A),B),S2) = $ite(aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A),S2),aa_VEBT_VEBT_real(B,A),one_one_real) ) ) ).

% prod.delta'
tff(fact_8514_prod_Odelta_H,axiom,
    ! [S2: set_real,A: real,B: fun_real_real] :
      ( aa_set_real_o(finite_finite_real,S2)
     => ( groups1681761925125756287l_real(aa_fun6946558556880649704l_real(aTP_Lamm_avf(A),B),S2) = $ite(aa_set_real_o(member_real(A),S2),aa_real_real(B,A),one_one_real) ) ) ).

% prod.delta'
tff(fact_8515_prod_Odelta_H,axiom,
    ! [S2: set_nat,A: nat,B: fun_nat_real] :
      ( aa_set_nat_o(finite_finite_nat,S2)
     => ( groups129246275422532515t_real(aa_fun1839206346889086768t_real(aTP_Lamm_avg(A),B),S2) = $ite(aa_set_nat_o(member_nat(A),S2),aa_nat_real(B,A),one_one_real) ) ) ).

% prod.delta'
tff(fact_8516_prod_Odelta_H,axiom,
    ! [S2: set_int,A: int,B: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,S2)
     => ( groups2316167850115554303t_real(aa_fun3252798400008884712t_real(aTP_Lamm_avh(A),B),S2) = $ite(aa_set_int_o(member_int(A),S2),aa_int_real(B,A),one_one_real) ) ) ).

% prod.delta'
tff(fact_8517_prod_Odelta_H,axiom,
    ! [S2: set_complex,A: complex,B: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S2)
     => ( groups766887009212190081x_real(aa_fun8933909628229591020x_real(aTP_Lamm_avi(A),B),S2) = $ite(aa_set_complex_o(member_complex(A),S2),aa_complex_real(B,A),one_one_real) ) ) ).

% prod.delta'
tff(fact_8518_prod_Odelta_H,axiom,
    ! [S2: set_Code_integer,A: code_integer,B: fun_Co3960896333048504454r_real] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,S2)
     => ( groups9004974159866482096r_real(aa_fun6341089502340155082r_real(aTP_Lamm_avj(A),B),S2) = $ite(aa_set5643387332196575111eger_o(member_Code_integer(A),S2),aa_Code_integer_real(B,A),one_one_real) ) ) ).

% prod.delta'
tff(fact_8519_prod_Odelta_H,axiom,
    ! [S2: set_VEBT_VEBT,A: vEBT_VEBT,B: fun_VEBT_VEBT_rat] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,S2)
     => ( groups5726676334696518183BT_rat(aa_fun8105832842879916156BT_rat(aTP_Lamm_avk(A),B),S2) = $ite(aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A),S2),aa_VEBT_VEBT_rat(B,A),one_one_rat) ) ) ).

% prod.delta'
tff(fact_8520_prod_Odelta_H,axiom,
    ! [S2: set_real,A: real,B: fun_real_rat] :
      ( aa_set_real_o(finite_finite_real,S2)
     => ( groups4061424788464935467al_rat(aa_fun2468082234287226944al_rat(aTP_Lamm_avl(A),B),S2) = $ite(aa_set_real_o(member_real(A),S2),aa_real_rat(B,A),one_one_rat) ) ) ).

% prod.delta'
tff(fact_8521_prod_Odelta_H,axiom,
    ! [S2: set_nat,A: nat,B: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,S2)
     => ( groups73079841787564623at_rat(aa_fun765758790203053448at_rat(aTP_Lamm_avm(A),B),S2) = $ite(aa_set_nat_o(member_nat(A),S2),aa_nat_rat(B,A),one_one_rat) ) ) ).

% prod.delta'
tff(fact_8522_prod_Odelta_H,axiom,
    ! [S2: set_int,A: int,B: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,S2)
     => ( groups1072433553688619179nt_rat(aa_fun1481061427733998144nt_rat(aTP_Lamm_avn(A),B),S2) = $ite(aa_set_int_o(member_int(A),S2),aa_int_rat(B,A),one_one_rat) ) ) ).

% prod.delta'
tff(fact_8523_prod_Odelta,axiom,
    ! [S2: set_VEBT_VEBT,A: vEBT_VEBT,B: fun_VEBT_VEBT_real] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,S2)
     => ( groups2703838992350267259T_real(aa_fun1825138017709814396T_real(aTP_Lamm_avo(A),B),S2) = $ite(aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A),S2),aa_VEBT_VEBT_real(B,A),one_one_real) ) ) ).

% prod.delta
tff(fact_8524_prod_Odelta,axiom,
    ! [S2: set_real,A: real,B: fun_real_real] :
      ( aa_set_real_o(finite_finite_real,S2)
     => ( groups1681761925125756287l_real(aa_fun6946558556880649704l_real(aTP_Lamm_avp(A),B),S2) = $ite(aa_set_real_o(member_real(A),S2),aa_real_real(B,A),one_one_real) ) ) ).

% prod.delta
tff(fact_8525_prod_Odelta,axiom,
    ! [S2: set_nat,A: nat,B: fun_nat_real] :
      ( aa_set_nat_o(finite_finite_nat,S2)
     => ( groups129246275422532515t_real(aa_fun1839206346889086768t_real(aTP_Lamm_avq(A),B),S2) = $ite(aa_set_nat_o(member_nat(A),S2),aa_nat_real(B,A),one_one_real) ) ) ).

% prod.delta
tff(fact_8526_prod_Odelta,axiom,
    ! [S2: set_int,A: int,B: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,S2)
     => ( groups2316167850115554303t_real(aa_fun3252798400008884712t_real(aTP_Lamm_avr(A),B),S2) = $ite(aa_set_int_o(member_int(A),S2),aa_int_real(B,A),one_one_real) ) ) ).

% prod.delta
tff(fact_8527_prod_Odelta,axiom,
    ! [S2: set_complex,A: complex,B: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S2)
     => ( groups766887009212190081x_real(aa_fun8933909628229591020x_real(aTP_Lamm_avs(A),B),S2) = $ite(aa_set_complex_o(member_complex(A),S2),aa_complex_real(B,A),one_one_real) ) ) ).

% prod.delta
tff(fact_8528_prod_Odelta,axiom,
    ! [S2: set_Code_integer,A: code_integer,B: fun_Co3960896333048504454r_real] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,S2)
     => ( groups9004974159866482096r_real(aa_fun6341089502340155082r_real(aTP_Lamm_avt(A),B),S2) = $ite(aa_set5643387332196575111eger_o(member_Code_integer(A),S2),aa_Code_integer_real(B,A),one_one_real) ) ) ).

% prod.delta
tff(fact_8529_prod_Odelta,axiom,
    ! [S2: set_VEBT_VEBT,A: vEBT_VEBT,B: fun_VEBT_VEBT_rat] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,S2)
     => ( groups5726676334696518183BT_rat(aa_fun8105832842879916156BT_rat(aTP_Lamm_avu(A),B),S2) = $ite(aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A),S2),aa_VEBT_VEBT_rat(B,A),one_one_rat) ) ) ).

% prod.delta
tff(fact_8530_prod_Odelta,axiom,
    ! [S2: set_real,A: real,B: fun_real_rat] :
      ( aa_set_real_o(finite_finite_real,S2)
     => ( groups4061424788464935467al_rat(aa_fun2468082234287226944al_rat(aTP_Lamm_avv(A),B),S2) = $ite(aa_set_real_o(member_real(A),S2),aa_real_rat(B,A),one_one_rat) ) ) ).

% prod.delta
tff(fact_8531_prod_Odelta,axiom,
    ! [S2: set_nat,A: nat,B: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,S2)
     => ( groups73079841787564623at_rat(aa_fun765758790203053448at_rat(aTP_Lamm_avw(A),B),S2) = $ite(aa_set_nat_o(member_nat(A),S2),aa_nat_rat(B,A),one_one_rat) ) ) ).

% prod.delta
tff(fact_8532_prod_Odelta,axiom,
    ! [S2: set_int,A: int,B: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,S2)
     => ( groups1072433553688619179nt_rat(aa_fun1481061427733998144nt_rat(aTP_Lamm_avx(A),B),S2) = $ite(aa_set_int_o(member_int(A),S2),aa_int_rat(B,A),one_one_rat) ) ) ).

% prod.delta
tff(fact_8533_prod_Oinsert,axiom,
    ! [A2: set_VEBT_VEBT,Xb: vEBT_VEBT,G: fun_VEBT_VEBT_real] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,A2)
     => ( ~ aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xb),A2)
       => ( groups2703838992350267259T_real(G,insert_VEBT_VEBT(Xb,A2)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_VEBT_VEBT_real(G,Xb)),groups2703838992350267259T_real(G,A2)) ) ) ) ).

% prod.insert
tff(fact_8534_prod_Oinsert,axiom,
    ! [A2: set_real,Xb: real,G: fun_real_real] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( ~ aa_set_real_o(member_real(Xb),A2)
       => ( groups1681761925125756287l_real(G,insert_real(Xb,A2)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(G,Xb)),groups1681761925125756287l_real(G,A2)) ) ) ) ).

% prod.insert
tff(fact_8535_prod_Oinsert,axiom,
    ! [A2: set_nat,Xb: nat,G: fun_nat_real] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( ~ aa_set_nat_o(member_nat(Xb),A2)
       => ( groups129246275422532515t_real(G,insert_nat(Xb,A2)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(G,Xb)),groups129246275422532515t_real(G,A2)) ) ) ) ).

% prod.insert
tff(fact_8536_prod_Oinsert,axiom,
    ! [A2: set_int,Xb: int,G: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( ~ aa_set_int_o(member_int(Xb),A2)
       => ( groups2316167850115554303t_real(G,insert_int(Xb,A2)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(G,Xb)),groups2316167850115554303t_real(G,A2)) ) ) ) ).

% prod.insert
tff(fact_8537_prod_Oinsert,axiom,
    ! [A2: set_complex,Xb: complex,G: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( ~ aa_set_complex_o(member_complex(Xb),A2)
       => ( groups766887009212190081x_real(G,insert_complex(Xb,A2)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(G,Xb)),groups766887009212190081x_real(G,A2)) ) ) ) ).

% prod.insert
tff(fact_8538_prod_Oinsert,axiom,
    ! [A2: set_Code_integer,Xb: code_integer,G: fun_Co3960896333048504454r_real] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A2)
     => ( ~ aa_set5643387332196575111eger_o(member_Code_integer(Xb),A2)
       => ( groups9004974159866482096r_real(G,insert_Code_integer(Xb,A2)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_Code_integer_real(G,Xb)),groups9004974159866482096r_real(G,A2)) ) ) ) ).

% prod.insert
tff(fact_8539_prod_Oinsert,axiom,
    ! [A2: set_VEBT_VEBT,Xb: vEBT_VEBT,G: fun_VEBT_VEBT_rat] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,A2)
     => ( ~ aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xb),A2)
       => ( groups5726676334696518183BT_rat(G,insert_VEBT_VEBT(Xb,A2)) = aa_rat_rat(times_times_rat(aa_VEBT_VEBT_rat(G,Xb)),groups5726676334696518183BT_rat(G,A2)) ) ) ) ).

% prod.insert
tff(fact_8540_prod_Oinsert,axiom,
    ! [A2: set_real,Xb: real,G: fun_real_rat] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( ~ aa_set_real_o(member_real(Xb),A2)
       => ( groups4061424788464935467al_rat(G,insert_real(Xb,A2)) = aa_rat_rat(times_times_rat(aa_real_rat(G,Xb)),groups4061424788464935467al_rat(G,A2)) ) ) ) ).

% prod.insert
tff(fact_8541_prod_Oinsert,axiom,
    ! [A2: set_nat,Xb: nat,G: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( ~ aa_set_nat_o(member_nat(Xb),A2)
       => ( groups73079841787564623at_rat(G,insert_nat(Xb,A2)) = aa_rat_rat(times_times_rat(aa_nat_rat(G,Xb)),groups73079841787564623at_rat(G,A2)) ) ) ) ).

% prod.insert
tff(fact_8542_prod_Oinsert,axiom,
    ! [A2: set_int,Xb: int,G: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( ~ aa_set_int_o(member_int(Xb),A2)
       => ( groups1072433553688619179nt_rat(G,insert_int(Xb,A2)) = aa_rat_rat(times_times_rat(aa_int_rat(G,Xb)),groups1072433553688619179nt_rat(G,A2)) ) ) ) ).

% prod.insert
tff(fact_8543_prod_OlessThan__Suc,axiom,
    ! [G: fun_nat_real,Nb: nat] : groups129246275422532515t_real(G,set_ord_lessThan_nat(aa_nat_nat(suc,Nb))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,groups129246275422532515t_real(G,set_ord_lessThan_nat(Nb))),aa_nat_real(G,Nb)) ).

% prod.lessThan_Suc
tff(fact_8544_prod_OlessThan__Suc,axiom,
    ! [G: fun_nat_rat,Nb: nat] : groups73079841787564623at_rat(G,set_ord_lessThan_nat(aa_nat_nat(suc,Nb))) = aa_rat_rat(times_times_rat(groups73079841787564623at_rat(G,set_ord_lessThan_nat(Nb))),aa_nat_rat(G,Nb)) ).

% prod.lessThan_Suc
tff(fact_8545_prod_OlessThan__Suc,axiom,
    ! [G: fun_nat_int,Nb: nat] : groups705719431365010083at_int(G,set_ord_lessThan_nat(aa_nat_nat(suc,Nb))) = aa_int_int(aa_int_fun_int_int(times_times_int,groups705719431365010083at_int(G,set_ord_lessThan_nat(Nb))),aa_nat_int(G,Nb)) ).

% prod.lessThan_Suc
tff(fact_8546_prod_OlessThan__Suc,axiom,
    ! [G: fun_nat_nat,Nb: nat] : groups708209901874060359at_nat(G,set_ord_lessThan_nat(aa_nat_nat(suc,Nb))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,groups708209901874060359at_nat(G,set_ord_lessThan_nat(Nb))),aa_nat_nat(G,Nb)) ).

% prod.lessThan_Suc
tff(fact_8547_prod_Ocl__ivl__Suc,axiom,
    ! [G: fun_na2984781812408219459l_num1,M: nat,Nb: nat] :
      groups5040993209112964077l_num1(G,set_or1269000886237332187st_nat(M,aa_nat_nat(suc,Nb))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Nb)),M),one_on7727431528512463931l_num1,aa_wor7274019993978516266l_num1(times_7065122842183080059l_num1(groups5040993209112964077l_num1(G,set_or1269000886237332187st_nat(M,Nb))),aa_nat5599461682457136004l_num1(G,aa_nat_nat(suc,Nb)))) ).

% prod.cl_ivl_Suc
tff(fact_8548_prod_Ocl__ivl__Suc,axiom,
    ! [G: fun_nat_real,M: nat,Nb: nat] :
      groups129246275422532515t_real(G,set_or1269000886237332187st_nat(M,aa_nat_nat(suc,Nb))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Nb)),M),one_one_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,groups129246275422532515t_real(G,set_or1269000886237332187st_nat(M,Nb))),aa_nat_real(G,aa_nat_nat(suc,Nb)))) ).

% prod.cl_ivl_Suc
tff(fact_8549_prod_Ocl__ivl__Suc,axiom,
    ! [G: fun_nat_rat,M: nat,Nb: nat] :
      groups73079841787564623at_rat(G,set_or1269000886237332187st_nat(M,aa_nat_nat(suc,Nb))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Nb)),M),one_one_rat,aa_rat_rat(times_times_rat(groups73079841787564623at_rat(G,set_or1269000886237332187st_nat(M,Nb))),aa_nat_rat(G,aa_nat_nat(suc,Nb)))) ).

% prod.cl_ivl_Suc
tff(fact_8550_prod_Ocl__ivl__Suc,axiom,
    ! [G: fun_nat_int,M: nat,Nb: nat] :
      groups705719431365010083at_int(G,set_or1269000886237332187st_nat(M,aa_nat_nat(suc,Nb))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Nb)),M),one_one_int,aa_int_int(aa_int_fun_int_int(times_times_int,groups705719431365010083at_int(G,set_or1269000886237332187st_nat(M,Nb))),aa_nat_int(G,aa_nat_nat(suc,Nb)))) ).

% prod.cl_ivl_Suc
tff(fact_8551_prod_Ocl__ivl__Suc,axiom,
    ! [G: fun_nat_nat,M: nat,Nb: nat] :
      groups708209901874060359at_nat(G,set_or1269000886237332187st_nat(M,aa_nat_nat(suc,Nb))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Nb)),M),one_one_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,groups708209901874060359at_nat(G,set_or1269000886237332187st_nat(M,Nb))),aa_nat_nat(G,aa_nat_nat(suc,Nb)))) ).

% prod.cl_ivl_Suc
tff(fact_8552_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_nat_real,A2: set_nat] :
      ( ( groups129246275422532515t_real(G,A2) != one_one_real )
     => ~ ! [A3: nat] :
            ( aa_set_nat_o(member_nat(A3),A2)
           => ( aa_nat_real(G,A3) = one_one_real ) ) ) ).

% prod.not_neutral_contains_not_neutral
tff(fact_8553_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_VEBT_VEBT_real,A2: set_VEBT_VEBT] :
      ( ( groups2703838992350267259T_real(G,A2) != one_one_real )
     => ~ ! [A3: vEBT_VEBT] :
            ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A3),A2)
           => ( aa_VEBT_VEBT_real(G,A3) = one_one_real ) ) ) ).

% prod.not_neutral_contains_not_neutral
tff(fact_8554_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_real_real,A2: set_real] :
      ( ( groups1681761925125756287l_real(G,A2) != one_one_real )
     => ~ ! [A3: real] :
            ( aa_set_real_o(member_real(A3),A2)
           => ( aa_real_real(G,A3) = one_one_real ) ) ) ).

% prod.not_neutral_contains_not_neutral
tff(fact_8555_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_int_real,A2: set_int] :
      ( ( groups2316167850115554303t_real(G,A2) != one_one_real )
     => ~ ! [A3: int] :
            ( aa_set_int_o(member_int(A3),A2)
           => ( aa_int_real(G,A3) = one_one_real ) ) ) ).

% prod.not_neutral_contains_not_neutral
tff(fact_8556_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_nat_rat,A2: set_nat] :
      ( ( groups73079841787564623at_rat(G,A2) != one_one_rat )
     => ~ ! [A3: nat] :
            ( aa_set_nat_o(member_nat(A3),A2)
           => ( aa_nat_rat(G,A3) = one_one_rat ) ) ) ).

% prod.not_neutral_contains_not_neutral
tff(fact_8557_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_VEBT_VEBT_rat,A2: set_VEBT_VEBT] :
      ( ( groups5726676334696518183BT_rat(G,A2) != one_one_rat )
     => ~ ! [A3: vEBT_VEBT] :
            ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A3),A2)
           => ( aa_VEBT_VEBT_rat(G,A3) = one_one_rat ) ) ) ).

% prod.not_neutral_contains_not_neutral
tff(fact_8558_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_real_rat,A2: set_real] :
      ( ( groups4061424788464935467al_rat(G,A2) != one_one_rat )
     => ~ ! [A3: real] :
            ( aa_set_real_o(member_real(A3),A2)
           => ( aa_real_rat(G,A3) = one_one_rat ) ) ) ).

% prod.not_neutral_contains_not_neutral
tff(fact_8559_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_int_rat,A2: set_int] :
      ( ( groups1072433553688619179nt_rat(G,A2) != one_one_rat )
     => ~ ! [A3: int] :
            ( aa_set_int_o(member_int(A3),A2)
           => ( aa_int_rat(G,A3) = one_one_rat ) ) ) ).

% prod.not_neutral_contains_not_neutral
tff(fact_8560_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_VEBT_VEBT_nat,A2: set_VEBT_VEBT] :
      ( ( groups6361806394783013919BT_nat(G,A2) != one_one_nat )
     => ~ ! [A3: vEBT_VEBT] :
            ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A3),A2)
           => ( aa_VEBT_VEBT_nat(G,A3) = one_one_nat ) ) ) ).

% prod.not_neutral_contains_not_neutral
tff(fact_8561_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_real_nat,A2: set_real] :
      ( ( groups4696554848551431203al_nat(G,A2) != one_one_nat )
     => ~ ! [A3: real] :
            ( aa_set_real_o(member_real(A3),A2)
           => ( aa_real_nat(G,A3) = one_one_nat ) ) ) ).

% prod.not_neutral_contains_not_neutral
tff(fact_8562_prod_Oneutral,axiom,
    ! [A2: set_nat,G: fun_nat_int] :
      ( ! [X2: nat] :
          ( aa_set_nat_o(member_nat(X2),A2)
         => ( aa_nat_int(G,X2) = one_one_int ) )
     => ( groups705719431365010083at_int(G,A2) = one_one_int ) ) ).

% prod.neutral
tff(fact_8563_prod_Oneutral,axiom,
    ! [A2: set_int,G: fun_int_int] :
      ( ! [X2: int] :
          ( aa_set_int_o(member_int(X2),A2)
         => ( aa_int_int(G,X2) = one_one_int ) )
     => ( groups1705073143266064639nt_int(G,A2) = one_one_int ) ) ).

% prod.neutral
tff(fact_8564_prod_Oneutral,axiom,
    ! [A2: set_nat,G: fun_nat_nat] :
      ( ! [X2: nat] :
          ( aa_set_nat_o(member_nat(X2),A2)
         => ( aa_nat_nat(G,X2) = one_one_nat ) )
     => ( groups708209901874060359at_nat(G,A2) = one_one_nat ) ) ).

% prod.neutral
tff(fact_8565_prod_Oswap,axiom,
    ! [G: fun_nat_fun_nat_int,B4: set_nat,A2: set_nat] : groups705719431365010083at_int(aa_set2359242532610849469at_int(aTP_Lamm_avy(G),B4),A2) = groups705719431365010083at_int(aa_set2359242532610849469at_int(aTP_Lamm_awa(G),A2),B4) ).

% prod.swap
tff(fact_8566_prod_Oswap,axiom,
    ! [G: fun_nat_fun_int_int,B4: set_int,A2: set_nat] : groups705719431365010083at_int(aa_set6310687161026558745at_int(aTP_Lamm_awb(G),B4),A2) = groups1705073143266064639nt_int(aa_set3358596244511904025nt_int(aTP_Lamm_awd(G),A2),B4) ).

% prod.swap
tff(fact_8567_prod_Oswap,axiom,
    ! [G: fun_int_fun_nat_int,B4: set_nat,A2: set_int] : groups1705073143266064639nt_int(aa_set3358596244511904025nt_int(aTP_Lamm_awe(G),B4),A2) = groups705719431365010083at_int(aa_set6310687161026558745at_int(aTP_Lamm_awf(G),A2),B4) ).

% prod.swap
tff(fact_8568_prod_Oswap,axiom,
    ! [G: fun_int_fun_int_int,B4: set_int,A2: set_int] : groups1705073143266064639nt_int(aa_set7310040872927613301nt_int(aTP_Lamm_awg(G),B4),A2) = groups1705073143266064639nt_int(aa_set7310040872927613301nt_int(aTP_Lamm_awh(G),A2),B4) ).

% prod.swap
tff(fact_8569_prod_Oswap,axiom,
    ! [G: fun_nat_fun_nat_nat,B4: set_nat,A2: set_nat] : groups708209901874060359at_nat(aa_set6537093552120046177at_nat(aTP_Lamm_awi(G),B4),A2) = groups708209901874060359at_nat(aa_set6537093552120046177at_nat(aTP_Lamm_awj(G),A2),B4) ).

% prod.swap
tff(fact_8570_prod_Odistrib,axiom,
    ! [G: fun_nat_int,H: fun_nat_int,A2: set_nat] : groups705719431365010083at_int(aa_fun5565043122120104496at_int(aTP_Lamm_awk(G),H),A2) = aa_int_int(aa_int_fun_int_int(times_times_int,groups705719431365010083at_int(G,A2)),groups705719431365010083at_int(H,A2)) ).

% prod.distrib
tff(fact_8571_prod_Odistrib,axiom,
    ! [G: fun_int_int,H: fun_int_int,A2: set_int] : groups1705073143266064639nt_int(aa_fun6280345759651049192nt_int(aTP_Lamm_awl(G),H),A2) = aa_int_int(aa_int_fun_int_int(times_times_int,groups1705073143266064639nt_int(G,A2)),groups1705073143266064639nt_int(H,A2)) ).

% prod.distrib
tff(fact_8572_prod_Odistrib,axiom,
    ! [G: fun_nat_nat,H: fun_nat_nat,A2: set_nat] : groups708209901874060359at_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_awm(G),H),A2) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,groups708209901874060359at_nat(G,A2)),groups708209901874060359at_nat(H,A2)) ).

% prod.distrib
tff(fact_8573_prod__power__distrib,axiom,
    ! [F: fun_nat_int,A2: set_nat,Nb: nat] : aa_nat_int(aa_int_fun_nat_int(power_power_int,groups705719431365010083at_int(F,A2)),Nb) = groups705719431365010083at_int(aa_nat_fun_nat_int(aTP_Lamm_awn(F),Nb),A2) ).

% prod_power_distrib
tff(fact_8574_prod__power__distrib,axiom,
    ! [F: fun_int_int,A2: set_int,Nb: nat] : aa_nat_int(aa_int_fun_nat_int(power_power_int,groups1705073143266064639nt_int(F,A2)),Nb) = groups1705073143266064639nt_int(aa_nat_fun_int_int(aTP_Lamm_awo(F),Nb),A2) ).

% prod_power_distrib
tff(fact_8575_prod__power__distrib,axiom,
    ! [F: fun_nat_nat,A2: set_nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,groups708209901874060359at_nat(F,A2)),Nb) = groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_awp(F),Nb),A2) ).

% prod_power_distrib
tff(fact_8576_prod_Oswap__restrict,axiom,
    ! [A2: set_VEBT_VEBT,B4: set_nat,G: fun_VE7331017413731640256at_int,R: fun_VE6480751511521678896_nat_o] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,A2)
     => ( aa_set_nat_o(finite_finite_nat,B4)
       => ( groups6359315924273963643BT_int(aa_fun8666547004982707333BT_int(aa_fun127508041567870360BT_int(aTP_Lamm_awq(B4),G),R),A2) = groups705719431365010083at_int(aa_fun2071267779502525287at_int(aa_fun7598684367898235306at_int(aTP_Lamm_aws(A2),G),R),B4) ) ) ) ).

% prod.swap_restrict
tff(fact_8577_prod_Oswap__restrict,axiom,
    ! [A2: set_real,B4: set_nat,G: fun_real_fun_nat_int,R: fun_real_fun_nat_o] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( aa_set_nat_o(finite_finite_nat,B4)
       => ( groups4694064378042380927al_int(aa_fun7318160320247150695al_int(aa_fun4230461673125498710al_int(aTP_Lamm_awt(B4),G),R),A2) = groups705719431365010083at_int(aa_fun3188195345082886411at_int(aa_fun8006801000864960506at_int(aTP_Lamm_awv(A2),G),R),B4) ) ) ) ).

% prod.swap_restrict
tff(fact_8578_prod_Oswap__restrict,axiom,
    ! [A2: set_complex,B4: set_nat,G: fun_co7370231399432144932at_int,R: fun_co5676208753217392844_nat_o] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( aa_set_nat_o(finite_finite_nat,B4)
       => ( groups858564598930262913ex_int(aa_fun7310784931243472999ex_int(aa_fun7210183792663747796ex_int(aTP_Lamm_aww(B4),G),R),A2) = groups705719431365010083at_int(aa_fun1349975166126329225at_int(aa_fun7503395680861046006at_int(aTP_Lamm_awy(A2),G),R),B4) ) ) ) ).

% prod.swap_restrict
tff(fact_8579_prod_Oswap__restrict,axiom,
    ! [A2: set_Code_integer,B4: set_nat,G: fun_Co822469698833053651at_int,R: fun_Co1375048237525739357_nat_o] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A2)
     => ( aa_set_nat_o(finite_finite_nat,B4)
       => ( groups3188404863801439024er_int(aa_fun2505341505705002151er_int(aa_fun493137163814837797er_int(aTP_Lamm_awz(B4),G),R),A2) = groups705719431365010083at_int(aa_fun3278309868433077274at_int(aa_fun1276769135904165912at_int(aTP_Lamm_axb(A2),G),R),B4) ) ) ) ).

% prod.swap_restrict
tff(fact_8580_prod_Oswap__restrict,axiom,
    ! [A2: set_VEBT_VEBT,B4: set_int,G: fun_VE294566951569886236nt_int,R: fun_VE4235606543851232980_int_o] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,A2)
     => ( aa_set_int_o(finite_finite_int,B4)
       => ( groups6359315924273963643BT_int(aa_fun1410605457015381217BT_int(aa_fun1738282208931454104BT_int(aTP_Lamm_axd(B4),G),R),A2) = groups1705073143266064639nt_int(aa_fun2879915576171315815nt_int(aa_fun5708383307987270406nt_int(aTP_Lamm_axf(A2),G),R),B4) ) ) ) ).

% prod.swap_restrict
tff(fact_8581_prod_Oswap__restrict,axiom,
    ! [A2: set_real,B4: set_int,G: fun_real_fun_int_int,R: fun_real_fun_int_o] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( aa_set_int_o(finite_finite_int,B4)
       => ( groups4694064378042380927al_int(aa_fun7555737461897725707al_int(aa_fun1679075639499870878al_int(aTP_Lamm_axh(B4),G),R),A2) = groups1705073143266064639nt_int(aa_fun3996843141751676939nt_int(aa_fun129049606478405534nt_int(aTP_Lamm_axj(A2),G),R),B4) ) ) ) ).

% prod.swap_restrict
tff(fact_8582_prod_Oswap__restrict,axiom,
    ! [A2: set_complex,B4: set_int,G: fun_co333780937270390912nt_int,R: fun_co3431063785546946928_int_o] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( aa_set_int_o(finite_finite_int,B4)
       => ( groups858564598930262913ex_int(aa_fun5055871773053899019ex_int(aa_fun7361901416012365340ex_int(aTP_Lamm_axl(B4),G),R),A2) = groups1705073143266064639nt_int(aa_fun2158622962795119753nt_int(aa_fun607866698265288346nt_int(aTP_Lamm_axo(A2),G),R),B4) ) ) ) ).

% prod.swap_restrict
tff(fact_8583_prod_Oswap__restrict,axiom,
    ! [A2: set_Code_integer,B4: set_int,G: fun_Co3009391273526075439nt_int,R: fun_Co8353275306710069249_int_o] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A2)
     => ( aa_set_int_o(finite_finite_int,B4)
       => ( groups3188404863801439024er_int(aa_fun5771563549434713419er_int(aa_fun4265095070934345581er_int(aTP_Lamm_axq(B4),G),R),A2) = groups1705073143266064639nt_int(aa_fun4086957665101867802nt_int(aa_fun5679615576903115196nt_int(aTP_Lamm_axt(A2),G),R),B4) ) ) ) ).

% prod.swap_restrict
tff(fact_8584_prod_Oswap__restrict,axiom,
    ! [A2: set_VEBT_VEBT,B4: set_nat,G: fun_VE597728521808591204at_nat,R: fun_VE6480751511521678896_nat_o] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,A2)
     => ( aa_set_nat_o(finite_finite_nat,B4)
       => ( groups6361806394783013919BT_nat(aa_fun3621025987637128233BT_nat(aa_fun1603736613815373464BT_nat(aTP_Lamm_axu(B4),G),R),A2) = groups708209901874060359at_nat(aa_fun6249118799011721995at_nat(aa_fun2705500642357200114at_nat(aTP_Lamm_axv(A2),G),R),B4) ) ) ) ).

% prod.swap_restrict
tff(fact_8585_prod_Oswap__restrict,axiom,
    ! [A2: set_real,B4: set_nat,G: fun_real_fun_nat_nat,R: fun_real_fun_nat_o] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( aa_set_nat_o(finite_finite_nat,B4)
       => ( groups4696554848551431203al_nat(aa_fun2272639302901571595al_nat(aa_fun7039802393428226646al_nat(aTP_Lamm_axw(B4),G),R),A2) = groups708209901874060359at_nat(aa_fun7366046364592083119at_nat(aa_fun7718511627822915834at_nat(aTP_Lamm_axx(A2),G),R),B4) ) ) ) ).

% prod.swap_restrict
tff(fact_8586_mod__prod__eq,axiom,
    ! [F: fun_nat_int,A: int,A2: set_nat] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,groups705719431365010083at_int(aa_int_fun_nat_int(aTP_Lamm_axy(F),A),A2)),A) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,groups705719431365010083at_int(F,A2)),A) ).

% mod_prod_eq
tff(fact_8587_mod__prod__eq,axiom,
    ! [F: fun_int_int,A: int,A2: set_int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,groups1705073143266064639nt_int(aa_int_fun_int_int(aTP_Lamm_ani(F),A),A2)),A) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,groups1705073143266064639nt_int(F,A2)),A) ).

% mod_prod_eq
tff(fact_8588_mod__prod__eq,axiom,
    ! [F: fun_nat_nat,A: nat,A2: set_nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_anh(F),A),A2)),A) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,groups708209901874060359at_nat(F,A2)),A) ).

% mod_prod_eq
tff(fact_8589_prod__mono,axiom,
    ! [A2: set_nat,F: fun_nat_real,G: fun_nat_real] :
      ( ! [I3: nat] :
          ( aa_set_nat_o(member_nat(I3),A2)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(F,I3))
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(F,I3)),aa_nat_real(G,I3)) ) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups129246275422532515t_real(F,A2)),groups129246275422532515t_real(G,A2)) ) ).

% prod_mono
tff(fact_8590_prod__mono,axiom,
    ! [A2: set_VEBT_VEBT,F: fun_VEBT_VEBT_real,G: fun_VEBT_VEBT_real] :
      ( ! [I3: vEBT_VEBT] :
          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(I3),A2)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_VEBT_VEBT_real(F,I3))
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_VEBT_VEBT_real(F,I3)),aa_VEBT_VEBT_real(G,I3)) ) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups2703838992350267259T_real(F,A2)),groups2703838992350267259T_real(G,A2)) ) ).

% prod_mono
tff(fact_8591_prod__mono,axiom,
    ! [A2: set_real,F: fun_real_real,G: fun_real_real] :
      ( ! [I3: real] :
          ( aa_set_real_o(member_real(I3),A2)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(F,I3))
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(F,I3)),aa_real_real(G,I3)) ) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups1681761925125756287l_real(F,A2)),groups1681761925125756287l_real(G,A2)) ) ).

% prod_mono
tff(fact_8592_prod__mono,axiom,
    ! [A2: set_int,F: fun_int_real,G: fun_int_real] :
      ( ! [I3: int] :
          ( aa_set_int_o(member_int(I3),A2)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_int_real(F,I3))
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(F,I3)),aa_int_real(G,I3)) ) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups2316167850115554303t_real(F,A2)),groups2316167850115554303t_real(G,A2)) ) ).

% prod_mono
tff(fact_8593_prod__mono,axiom,
    ! [A2: set_nat,F: fun_nat_rat,G: fun_nat_rat] :
      ( ! [I3: nat] :
          ( aa_set_nat_o(member_nat(I3),A2)
         => ( 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,groups73079841787564623at_rat(F,A2)),groups73079841787564623at_rat(G,A2)) ) ).

% prod_mono
tff(fact_8594_prod__mono,axiom,
    ! [A2: set_VEBT_VEBT,F: fun_VEBT_VEBT_rat,G: fun_VEBT_VEBT_rat] :
      ( ! [I3: vEBT_VEBT] :
          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(I3),A2)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_VEBT_VEBT_rat(F,I3))
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_VEBT_VEBT_rat(F,I3)),aa_VEBT_VEBT_rat(G,I3)) ) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups5726676334696518183BT_rat(F,A2)),groups5726676334696518183BT_rat(G,A2)) ) ).

% prod_mono
tff(fact_8595_prod__mono,axiom,
    ! [A2: set_real,F: fun_real_rat,G: fun_real_rat] :
      ( ! [I3: real] :
          ( aa_set_real_o(member_real(I3),A2)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_real_rat(F,I3))
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_real_rat(F,I3)),aa_real_rat(G,I3)) ) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups4061424788464935467al_rat(F,A2)),groups4061424788464935467al_rat(G,A2)) ) ).

% prod_mono
tff(fact_8596_prod__mono,axiom,
    ! [A2: set_int,F: fun_int_rat,G: fun_int_rat] :
      ( ! [I3: int] :
          ( aa_set_int_o(member_int(I3),A2)
         => ( 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,A2)),groups1072433553688619179nt_rat(G,A2)) ) ).

% prod_mono
tff(fact_8597_prod__mono,axiom,
    ! [A2: set_VEBT_VEBT,F: fun_VEBT_VEBT_nat,G: fun_VEBT_VEBT_nat] :
      ( ! [I3: vEBT_VEBT] :
          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(I3),A2)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_VEBT_VEBT_nat(F,I3))
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_VEBT_VEBT_nat(F,I3)),aa_VEBT_VEBT_nat(G,I3)) ) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,groups6361806394783013919BT_nat(F,A2)),groups6361806394783013919BT_nat(G,A2)) ) ).

% prod_mono
tff(fact_8598_prod__mono,axiom,
    ! [A2: set_real,F: fun_real_nat,G: fun_real_nat] :
      ( ! [I3: real] :
          ( aa_set_real_o(member_real(I3),A2)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_real_nat(F,I3))
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_real_nat(F,I3)),aa_real_nat(G,I3)) ) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,groups4696554848551431203al_nat(F,A2)),groups4696554848551431203al_nat(G,A2)) ) ).

% prod_mono
tff(fact_8599_prod__nonneg,axiom,
    ! [A2: set_nat,F: fun_nat_int] :
      ( ! [X2: nat] :
          ( aa_set_nat_o(member_nat(X2),A2)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(F,X2)) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),groups705719431365010083at_int(F,A2)) ) ).

% prod_nonneg
tff(fact_8600_prod__nonneg,axiom,
    ! [A2: set_int,F: fun_int_int] :
      ( ! [X2: int] :
          ( aa_set_int_o(member_int(X2),A2)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(F,X2)) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),groups1705073143266064639nt_int(F,A2)) ) ).

% prod_nonneg
tff(fact_8601_prod__nonneg,axiom,
    ! [A2: set_nat,F: fun_nat_nat] :
      ( ! [X2: nat] :
          ( aa_set_nat_o(member_nat(X2),A2)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(F,X2)) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),groups708209901874060359at_nat(F,A2)) ) ).

% prod_nonneg
tff(fact_8602_prod__pos,axiom,
    ! [A2: set_nat,F: fun_nat_int] :
      ( ! [X2: nat] :
          ( aa_set_nat_o(member_nat(X2),A2)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(F,X2)) )
     => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),groups705719431365010083at_int(F,A2)) ) ).

% prod_pos
tff(fact_8603_prod__pos,axiom,
    ! [A2: set_int,F: fun_int_int] :
      ( ! [X2: int] :
          ( aa_set_int_o(member_int(X2),A2)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(F,X2)) )
     => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),groups1705073143266064639nt_int(F,A2)) ) ).

% prod_pos
tff(fact_8604_prod__pos,axiom,
    ! [A2: set_nat,F: fun_nat_nat] :
      ( ! [X2: nat] :
          ( aa_set_nat_o(member_nat(X2),A2)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(F,X2)) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),groups708209901874060359at_nat(F,A2)) ) ).

% prod_pos
tff(fact_8605_prod__ge__1,axiom,
    ! [A2: set_nat,F: fun_nat_real] :
      ( ! [X2: nat] :
          ( aa_set_nat_o(member_nat(X2),A2)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_nat_real(F,X2)) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),groups129246275422532515t_real(F,A2)) ) ).

% prod_ge_1
tff(fact_8606_prod__ge__1,axiom,
    ! [A2: set_VEBT_VEBT,F: fun_VEBT_VEBT_real] :
      ( ! [X2: vEBT_VEBT] :
          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X2),A2)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_VEBT_VEBT_real(F,X2)) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),groups2703838992350267259T_real(F,A2)) ) ).

% prod_ge_1
tff(fact_8607_prod__ge__1,axiom,
    ! [A2: set_real,F: fun_real_real] :
      ( ! [X2: real] :
          ( aa_set_real_o(member_real(X2),A2)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(F,X2)) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),groups1681761925125756287l_real(F,A2)) ) ).

% prod_ge_1
tff(fact_8608_prod__ge__1,axiom,
    ! [A2: set_int,F: fun_int_real] :
      ( ! [X2: int] :
          ( aa_set_int_o(member_int(X2),A2)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_int_real(F,X2)) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),groups2316167850115554303t_real(F,A2)) ) ).

% prod_ge_1
tff(fact_8609_prod__ge__1,axiom,
    ! [A2: set_nat,F: fun_nat_rat] :
      ( ! [X2: nat] :
          ( aa_set_nat_o(member_nat(X2),A2)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_nat_rat(F,X2)) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),groups73079841787564623at_rat(F,A2)) ) ).

% prod_ge_1
tff(fact_8610_prod__ge__1,axiom,
    ! [A2: set_VEBT_VEBT,F: fun_VEBT_VEBT_rat] :
      ( ! [X2: vEBT_VEBT] :
          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X2),A2)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_VEBT_VEBT_rat(F,X2)) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),groups5726676334696518183BT_rat(F,A2)) ) ).

% prod_ge_1
tff(fact_8611_prod__ge__1,axiom,
    ! [A2: set_real,F: fun_real_rat] :
      ( ! [X2: real] :
          ( aa_set_real_o(member_real(X2),A2)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_real_rat(F,X2)) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),groups4061424788464935467al_rat(F,A2)) ) ).

% prod_ge_1
tff(fact_8612_prod__ge__1,axiom,
    ! [A2: set_int,F: fun_int_rat] :
      ( ! [X2: int] :
          ( aa_set_int_o(member_int(X2),A2)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_int_rat(F,X2)) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),groups1072433553688619179nt_rat(F,A2)) ) ).

% prod_ge_1
tff(fact_8613_prod__ge__1,axiom,
    ! [A2: set_VEBT_VEBT,F: fun_VEBT_VEBT_nat] :
      ( ! [X2: vEBT_VEBT] :
          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X2),A2)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),aa_VEBT_VEBT_nat(F,X2)) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),groups6361806394783013919BT_nat(F,A2)) ) ).

% prod_ge_1
tff(fact_8614_prod__ge__1,axiom,
    ! [A2: set_real,F: fun_real_nat] :
      ( ! [X2: real] :
          ( aa_set_real_o(member_real(X2),A2)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),aa_real_nat(F,X2)) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),groups4696554848551431203al_nat(F,A2)) ) ).

% prod_ge_1
tff(fact_8615_prod_Ointer__filter,axiom,
    ! [A2: set_VEBT_VEBT,G: fun_VEBT_VEBT_real,P: fun_VEBT_VEBT_o] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,A2)
     => ( groups2703838992350267259T_real(G,collect_VEBT_VEBT(aa_fun8193769468198689164VEBT_o(aTP_Lamm_gh(A2),P))) = groups2703838992350267259T_real(aa_fun3597767889293385422T_real(aTP_Lamm_axz(G),P),A2) ) ) ).

% prod.inter_filter
tff(fact_8616_prod_Ointer__filter,axiom,
    ! [A2: set_real,G: fun_real_real,P: fun_real_o] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( groups1681761925125756287l_real(G,collect_real(aa_fun5868358341814070780real_o(aTP_Lamm_gi(A2),P))) = groups1681761925125756287l_real(aa_fun4448911596306054750l_real(aTP_Lamm_aya(G),P),A2) ) ) ).

% prod.inter_filter
tff(fact_8617_prod_Ointer__filter,axiom,
    ! [A2: set_nat,G: fun_nat_real,P: fun_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( groups129246275422532515t_real(G,collect_nat(aa_fun5937193402907943164_nat_o(aTP_Lamm_gj(A2),P))) = groups129246275422532515t_real(aa_fun5510454223943551838t_real(aTP_Lamm_ayb(G),P),A2) ) ) ).

% prod.inter_filter
tff(fact_8618_prod_Ointer__filter,axiom,
    ! [A2: set_int,G: fun_int_real,P: fun_int_o] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( groups2316167850115554303t_real(G,collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_gm(A2),P))) = groups2316167850115554303t_real(aa_fun7506669883404309598t_real(aTP_Lamm_ayc(G),P),A2) ) ) ).

% prod.inter_filter
tff(fact_8619_prod_Ointer__filter,axiom,
    ! [A2: set_complex,G: fun_complex_real,P: fun_complex_o] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( groups766887009212190081x_real(G,collect_complex(aa_fun2514317545224092796plex_o(aTP_Lamm_gk(A2),P))) = groups766887009212190081x_real(aa_fun8261057609414086622x_real(aTP_Lamm_ayd(G),P),A2) ) ) ).

% prod.inter_filter
tff(fact_8620_prod_Ointer__filter,axiom,
    ! [A2: set_Code_integer,G: fun_Co3960896333048504454r_real,P: fun_Code_integer_o] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A2)
     => ( groups9004974159866482096r_real(G,collect_Code_integer(aa_fun8214818745163655228eger_o(aTP_Lamm_ano(A2),P))) = groups9004974159866482096r_real(aa_fun7534015886319449118r_real(aTP_Lamm_aye(G),P),A2) ) ) ).

% prod.inter_filter
tff(fact_8621_prod_Ointer__filter,axiom,
    ! [A2: set_VEBT_VEBT,G: fun_VEBT_VEBT_rat,P: fun_VEBT_VEBT_o] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,A2)
     => ( groups5726676334696518183BT_rat(G,collect_VEBT_VEBT(aa_fun8193769468198689164VEBT_o(aTP_Lamm_gh(A2),P))) = groups5726676334696518183BT_rat(aa_fun7833968636320851834BT_rat(aTP_Lamm_ayf(G),P),A2) ) ) ).

% prod.inter_filter
tff(fact_8622_prod_Ointer__filter,axiom,
    ! [A2: set_real,G: fun_real_rat,P: fun_real_o] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( groups4061424788464935467al_rat(G,collect_real(aa_fun5868358341814070780real_o(aTP_Lamm_gi(A2),P))) = groups4061424788464935467al_rat(aa_fun2004904883527618314al_rat(aTP_Lamm_ayg(G),P),A2) ) ) ).

% prod.inter_filter
tff(fact_8623_prod_Ointer__filter,axiom,
    ! [A2: set_nat,G: fun_nat_rat,P: fun_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( groups73079841787564623at_rat(G,collect_nat(aa_fun5937193402907943164_nat_o(aTP_Lamm_gj(A2),P))) = groups73079841787564623at_rat(aa_fun696632391729880074at_rat(aTP_Lamm_ayh(G),P),A2) ) ) ).

% prod.inter_filter
tff(fact_8624_prod_Ointer__filter,axiom,
    ! [A2: set_int,G: fun_int_rat,P: fun_int_o] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( groups1072433553688619179nt_rat(G,collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_gm(A2),P))) = groups1072433553688619179nt_rat(aa_fun268837275264525578nt_rat(aTP_Lamm_ayi(G),P),A2) ) ) ).

% prod.inter_filter
tff(fact_8625_prod_Oshift__bounds__cl__Suc__ivl,axiom,
    ! [G: fun_nat_int,M: nat,Nb: nat] : groups705719431365010083at_int(G,set_or1269000886237332187st_nat(aa_nat_nat(suc,M),aa_nat_nat(suc,Nb))) = groups705719431365010083at_int(aTP_Lamm_aoq(G),set_or1269000886237332187st_nat(M,Nb)) ).

% prod.shift_bounds_cl_Suc_ivl
tff(fact_8626_prod_Oshift__bounds__cl__Suc__ivl,axiom,
    ! [G: fun_nat_nat,M: nat,Nb: nat] : groups708209901874060359at_nat(G,set_or1269000886237332187st_nat(aa_nat_nat(suc,M),aa_nat_nat(suc,Nb))) = groups708209901874060359at_nat(aTP_Lamm_anu(G),set_or1269000886237332187st_nat(M,Nb)) ).

% prod.shift_bounds_cl_Suc_ivl
tff(fact_8627_power__sum,axiom,
    ! [C: uint32,F: fun_nat_nat,A2: set_nat] : aa_nat_uint32(power_power_uint32(C),aa_set_nat_nat(groups3542108847815614940at_nat(F),A2)) = groups2278496514549435363uint32(aa_fun8367061318319979028uint32(aTP_Lamm_ayj(C),F),A2) ).

% power_sum
tff(fact_8628_power__sum,axiom,
    ! [C: word_N3645301735248828278l_num1,F: fun_nat_nat,A2: set_nat] : aa_nat5599461682457136004l_num1(power_2184487114949457152l_num1(C),aa_set_nat_nat(groups3542108847815614940at_nat(F),A2)) = groups5040993209112964077l_num1(aa_fun7778216322329253406l_num1(aTP_Lamm_ayk(C),F),A2) ).

% power_sum
tff(fact_8629_power__sum,axiom,
    ! [C: real,F: fun_nat_nat,A2: set_nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,C),aa_set_nat_nat(groups3542108847815614940at_nat(F),A2)) = groups129246275422532515t_real(aa_fun1538629612289296340t_real(aTP_Lamm_ayl(C),F),A2) ).

% power_sum
tff(fact_8630_power__sum,axiom,
    ! [C: int,F: fun_nat_nat,A2: set_nat] : aa_nat_int(aa_int_fun_nat_int(power_power_int,C),aa_set_nat_nat(groups3542108847815614940at_nat(F),A2)) = groups705719431365010083at_int(aa_fun1613598493704395220at_int(aTP_Lamm_aym(C),F),A2) ).

% power_sum
tff(fact_8631_power__sum,axiom,
    ! [C: int,F: fun_int_nat,A2: set_int] : aa_nat_int(aa_int_fun_nat_int(power_power_int,C),groups4541462559716669496nt_nat(F,A2)) = groups1705073143266064639nt_int(aa_fun2328901131235339916nt_int(aTP_Lamm_ayn(C),F),A2) ).

% power_sum
tff(fact_8632_power__sum,axiom,
    ! [C: nat,F: fun_nat_nat,A2: set_nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,C),aa_set_nat_nat(groups3542108847815614940at_nat(F),A2)) = groups708209901874060359at_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_ayo(C),F),A2) ).

% power_sum
tff(fact_8633_prod_Oshift__bounds__cl__nat__ivl,axiom,
    ! [G: fun_nat_int,M: nat,K: nat,Nb: nat] : groups705719431365010083at_int(G,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,Nb),K))) = groups705719431365010083at_int(aa_nat_fun_nat_int(aTP_Lamm_ayp(G),K),set_or1269000886237332187st_nat(M,Nb)) ).

% prod.shift_bounds_cl_nat_ivl
tff(fact_8634_prod_Oshift__bounds__cl__nat__ivl,axiom,
    ! [G: fun_nat_nat,M: nat,K: nat,Nb: nat] : groups708209901874060359at_nat(G,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,Nb),K))) = groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_anw(G),K),set_or1269000886237332187st_nat(M,Nb)) ).

% prod.shift_bounds_cl_nat_ivl
tff(fact_8635_prod__le__1,axiom,
    ! [A2: set_nat,F: fun_nat_real] :
      ( ! [X2: nat] :
          ( aa_set_nat_o(member_nat(X2),A2)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(F,X2))
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(F,X2)),one_one_real) ) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups129246275422532515t_real(F,A2)),one_one_real) ) ).

% prod_le_1
tff(fact_8636_prod__le__1,axiom,
    ! [A2: set_VEBT_VEBT,F: fun_VEBT_VEBT_real] :
      ( ! [X2: vEBT_VEBT] :
          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X2),A2)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_VEBT_VEBT_real(F,X2))
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_VEBT_VEBT_real(F,X2)),one_one_real) ) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups2703838992350267259T_real(F,A2)),one_one_real) ) ).

% prod_le_1
tff(fact_8637_prod__le__1,axiom,
    ! [A2: set_real,F: fun_real_real] :
      ( ! [X2: real] :
          ( aa_set_real_o(member_real(X2),A2)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(F,X2))
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(F,X2)),one_one_real) ) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups1681761925125756287l_real(F,A2)),one_one_real) ) ).

% prod_le_1
tff(fact_8638_prod__le__1,axiom,
    ! [A2: set_int,F: fun_int_real] :
      ( ! [X2: int] :
          ( aa_set_int_o(member_int(X2),A2)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_int_real(F,X2))
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(F,X2)),one_one_real) ) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups2316167850115554303t_real(F,A2)),one_one_real) ) ).

% prod_le_1
tff(fact_8639_prod__le__1,axiom,
    ! [A2: set_nat,F: fun_nat_rat] :
      ( ! [X2: nat] :
          ( aa_set_nat_o(member_nat(X2),A2)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(F,X2))
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,X2)),one_one_rat) ) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups73079841787564623at_rat(F,A2)),one_one_rat) ) ).

% prod_le_1
tff(fact_8640_prod__le__1,axiom,
    ! [A2: set_VEBT_VEBT,F: fun_VEBT_VEBT_rat] :
      ( ! [X2: vEBT_VEBT] :
          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X2),A2)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_VEBT_VEBT_rat(F,X2))
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_VEBT_VEBT_rat(F,X2)),one_one_rat) ) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups5726676334696518183BT_rat(F,A2)),one_one_rat) ) ).

% prod_le_1
tff(fact_8641_prod__le__1,axiom,
    ! [A2: set_real,F: fun_real_rat] :
      ( ! [X2: real] :
          ( aa_set_real_o(member_real(X2),A2)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_real_rat(F,X2))
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_real_rat(F,X2)),one_one_rat) ) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups4061424788464935467al_rat(F,A2)),one_one_rat) ) ).

% prod_le_1
tff(fact_8642_prod__le__1,axiom,
    ! [A2: set_int,F: fun_int_rat] :
      ( ! [X2: int] :
          ( aa_set_int_o(member_int(X2),A2)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_int_rat(F,X2))
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(F,X2)),one_one_rat) ) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups1072433553688619179nt_rat(F,A2)),one_one_rat) ) ).

% prod_le_1
tff(fact_8643_prod__le__1,axiom,
    ! [A2: set_VEBT_VEBT,F: fun_VEBT_VEBT_nat] :
      ( ! [X2: vEBT_VEBT] :
          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X2),A2)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_VEBT_VEBT_nat(F,X2))
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_VEBT_VEBT_nat(F,X2)),one_one_nat) ) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,groups6361806394783013919BT_nat(F,A2)),one_one_nat) ) ).

% prod_le_1
tff(fact_8644_prod__le__1,axiom,
    ! [A2: set_real,F: fun_real_nat] :
      ( ! [X2: real] :
          ( aa_set_real_o(member_real(X2),A2)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_real_nat(F,X2))
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_real_nat(F,X2)),one_one_nat) ) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,groups4696554848551431203al_nat(F,A2)),one_one_nat) ) ).

% prod_le_1
tff(fact_8645_prod_Orelated,axiom,
    ! [R: fun_real_fun_real_o,S2: set_nat,H: fun_nat_real,G: fun_nat_real] :
      ( aa_real_o(aa_real_fun_real_o(R,one_one_real),one_one_real)
     => ( ! [X15: real,Y15: real,X24: real,Y23: real] :
            ( ( aa_real_o(aa_real_fun_real_o(R,X15),X24)
              & aa_real_o(aa_real_fun_real_o(R,Y15),Y23) )
           => aa_real_o(aa_real_fun_real_o(R,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X15),Y15)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X24),Y23)) )
       => ( aa_set_nat_o(finite_finite_nat,S2)
         => ( ! [X2: nat] :
                ( aa_set_nat_o(member_nat(X2),S2)
               => aa_real_o(aa_real_fun_real_o(R,aa_nat_real(H,X2)),aa_nat_real(G,X2)) )
           => aa_real_o(aa_real_fun_real_o(R,groups129246275422532515t_real(H,S2)),groups129246275422532515t_real(G,S2)) ) ) ) ) ).

% prod.related
tff(fact_8646_prod_Orelated,axiom,
    ! [R: fun_real_fun_real_o,S2: set_int,H: fun_int_real,G: fun_int_real] :
      ( aa_real_o(aa_real_fun_real_o(R,one_one_real),one_one_real)
     => ( ! [X15: real,Y15: real,X24: real,Y23: real] :
            ( ( aa_real_o(aa_real_fun_real_o(R,X15),X24)
              & aa_real_o(aa_real_fun_real_o(R,Y15),Y23) )
           => aa_real_o(aa_real_fun_real_o(R,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X15),Y15)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X24),Y23)) )
       => ( aa_set_int_o(finite_finite_int,S2)
         => ( ! [X2: int] :
                ( aa_set_int_o(member_int(X2),S2)
               => aa_real_o(aa_real_fun_real_o(R,aa_int_real(H,X2)),aa_int_real(G,X2)) )
           => aa_real_o(aa_real_fun_real_o(R,groups2316167850115554303t_real(H,S2)),groups2316167850115554303t_real(G,S2)) ) ) ) ) ).

% prod.related
tff(fact_8647_prod_Orelated,axiom,
    ! [R: fun_real_fun_real_o,S2: set_complex,H: fun_complex_real,G: fun_complex_real] :
      ( aa_real_o(aa_real_fun_real_o(R,one_one_real),one_one_real)
     => ( ! [X15: real,Y15: real,X24: real,Y23: real] :
            ( ( aa_real_o(aa_real_fun_real_o(R,X15),X24)
              & aa_real_o(aa_real_fun_real_o(R,Y15),Y23) )
           => aa_real_o(aa_real_fun_real_o(R,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X15),Y15)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X24),Y23)) )
       => ( aa_set_complex_o(finite3207457112153483333omplex,S2)
         => ( ! [X2: complex] :
                ( aa_set_complex_o(member_complex(X2),S2)
               => aa_real_o(aa_real_fun_real_o(R,aa_complex_real(H,X2)),aa_complex_real(G,X2)) )
           => aa_real_o(aa_real_fun_real_o(R,groups766887009212190081x_real(H,S2)),groups766887009212190081x_real(G,S2)) ) ) ) ) ).

% prod.related
tff(fact_8648_prod_Orelated,axiom,
    ! [R: fun_real_fun_real_o,S2: set_Code_integer,H: fun_Co3960896333048504454r_real,G: fun_Co3960896333048504454r_real] :
      ( aa_real_o(aa_real_fun_real_o(R,one_one_real),one_one_real)
     => ( ! [X15: real,Y15: real,X24: real,Y23: real] :
            ( ( aa_real_o(aa_real_fun_real_o(R,X15),X24)
              & aa_real_o(aa_real_fun_real_o(R,Y15),Y23) )
           => aa_real_o(aa_real_fun_real_o(R,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X15),Y15)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X24),Y23)) )
       => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,S2)
         => ( ! [X2: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(X2),S2)
               => aa_real_o(aa_real_fun_real_o(R,aa_Code_integer_real(H,X2)),aa_Code_integer_real(G,X2)) )
           => aa_real_o(aa_real_fun_real_o(R,groups9004974159866482096r_real(H,S2)),groups9004974159866482096r_real(G,S2)) ) ) ) ) ).

% prod.related
tff(fact_8649_prod_Orelated,axiom,
    ! [R: fun_rat_fun_rat_o,S2: set_nat,H: fun_nat_rat,G: fun_nat_rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(R,one_one_rat),one_one_rat)
     => ( ! [X15: rat,Y15: rat,X24: rat,Y23: rat] :
            ( ( aa_rat_o(aa_rat_fun_rat_o(R,X15),X24)
              & aa_rat_o(aa_rat_fun_rat_o(R,Y15),Y23) )
           => aa_rat_o(aa_rat_fun_rat_o(R,aa_rat_rat(times_times_rat(X15),Y15)),aa_rat_rat(times_times_rat(X24),Y23)) )
       => ( aa_set_nat_o(finite_finite_nat,S2)
         => ( ! [X2: nat] :
                ( aa_set_nat_o(member_nat(X2),S2)
               => aa_rat_o(aa_rat_fun_rat_o(R,aa_nat_rat(H,X2)),aa_nat_rat(G,X2)) )
           => aa_rat_o(aa_rat_fun_rat_o(R,groups73079841787564623at_rat(H,S2)),groups73079841787564623at_rat(G,S2)) ) ) ) ) ).

% prod.related
tff(fact_8650_prod_Orelated,axiom,
    ! [R: fun_rat_fun_rat_o,S2: set_int,H: fun_int_rat,G: fun_int_rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(R,one_one_rat),one_one_rat)
     => ( ! [X15: rat,Y15: rat,X24: rat,Y23: rat] :
            ( ( aa_rat_o(aa_rat_fun_rat_o(R,X15),X24)
              & aa_rat_o(aa_rat_fun_rat_o(R,Y15),Y23) )
           => aa_rat_o(aa_rat_fun_rat_o(R,aa_rat_rat(times_times_rat(X15),Y15)),aa_rat_rat(times_times_rat(X24),Y23)) )
       => ( aa_set_int_o(finite_finite_int,S2)
         => ( ! [X2: int] :
                ( aa_set_int_o(member_int(X2),S2)
               => aa_rat_o(aa_rat_fun_rat_o(R,aa_int_rat(H,X2)),aa_int_rat(G,X2)) )
           => aa_rat_o(aa_rat_fun_rat_o(R,groups1072433553688619179nt_rat(H,S2)),groups1072433553688619179nt_rat(G,S2)) ) ) ) ) ).

% prod.related
tff(fact_8651_prod_Orelated,axiom,
    ! [R: fun_rat_fun_rat_o,S2: set_complex,H: fun_complex_rat,G: fun_complex_rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(R,one_one_rat),one_one_rat)
     => ( ! [X15: rat,Y15: rat,X24: rat,Y23: rat] :
            ( ( aa_rat_o(aa_rat_fun_rat_o(R,X15),X24)
              & aa_rat_o(aa_rat_fun_rat_o(R,Y15),Y23) )
           => aa_rat_o(aa_rat_fun_rat_o(R,aa_rat_rat(times_times_rat(X15),Y15)),aa_rat_rat(times_times_rat(X24),Y23)) )
       => ( aa_set_complex_o(finite3207457112153483333omplex,S2)
         => ( ! [X2: complex] :
                ( aa_set_complex_o(member_complex(X2),S2)
               => aa_rat_o(aa_rat_fun_rat_o(R,aa_complex_rat(H,X2)),aa_complex_rat(G,X2)) )
           => aa_rat_o(aa_rat_fun_rat_o(R,groups225925009352817453ex_rat(H,S2)),groups225925009352817453ex_rat(G,S2)) ) ) ) ) ).

% prod.related
tff(fact_8652_prod_Orelated,axiom,
    ! [R: fun_rat_fun_rat_o,S2: set_Code_integer,H: fun_Code_integer_rat,G: fun_Code_integer_rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(R,one_one_rat),one_one_rat)
     => ( ! [X15: rat,Y15: rat,X24: rat,Y23: rat] :
            ( ( aa_rat_o(aa_rat_fun_rat_o(R,X15),X24)
              & aa_rat_o(aa_rat_fun_rat_o(R,Y15),Y23) )
           => aa_rat_o(aa_rat_fun_rat_o(R,aa_rat_rat(times_times_rat(X15),Y15)),aa_rat_rat(times_times_rat(X24),Y23)) )
       => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,S2)
         => ( ! [X2: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(X2),S2)
               => aa_rat_o(aa_rat_fun_rat_o(R,aa_Code_integer_rat(H,X2)),aa_Code_integer_rat(G,X2)) )
           => aa_rat_o(aa_rat_fun_rat_o(R,groups2555765274223993564er_rat(H,S2)),groups2555765274223993564er_rat(G,S2)) ) ) ) ) ).

% prod.related
tff(fact_8653_prod_Orelated,axiom,
    ! [R: fun_nat_fun_nat_o,S2: set_int,H: fun_int_nat,G: fun_int_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(R,one_one_nat),one_one_nat)
     => ( ! [X15: nat,Y15: nat,X24: nat,Y23: nat] :
            ( ( aa_nat_o(aa_nat_fun_nat_o(R,X15),X24)
              & aa_nat_o(aa_nat_fun_nat_o(R,Y15),Y23) )
           => aa_nat_o(aa_nat_fun_nat_o(R,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,X15),Y15)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,X24),Y23)) )
       => ( aa_set_int_o(finite_finite_int,S2)
         => ( ! [X2: int] :
                ( aa_set_int_o(member_int(X2),S2)
               => aa_nat_o(aa_nat_fun_nat_o(R,aa_int_nat(H,X2)),aa_int_nat(G,X2)) )
           => aa_nat_o(aa_nat_fun_nat_o(R,groups1707563613775114915nt_nat(H,S2)),groups1707563613775114915nt_nat(G,S2)) ) ) ) ) ).

% prod.related
tff(fact_8654_prod_Orelated,axiom,
    ! [R: fun_nat_fun_nat_o,S2: set_complex,H: fun_complex_nat,G: fun_complex_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(R,one_one_nat),one_one_nat)
     => ( ! [X15: nat,Y15: nat,X24: nat,Y23: nat] :
            ( ( aa_nat_o(aa_nat_fun_nat_o(R,X15),X24)
              & aa_nat_o(aa_nat_fun_nat_o(R,Y15),Y23) )
           => aa_nat_o(aa_nat_fun_nat_o(R,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,X15),Y15)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,X24),Y23)) )
       => ( aa_set_complex_o(finite3207457112153483333omplex,S2)
         => ( ! [X2: complex] :
                ( aa_set_complex_o(member_complex(X2),S2)
               => aa_nat_o(aa_nat_fun_nat_o(R,aa_complex_nat(H,X2)),aa_complex_nat(G,X2)) )
           => aa_nat_o(aa_nat_fun_nat_o(R,groups861055069439313189ex_nat(H,S2)),groups861055069439313189ex_nat(G,S2)) ) ) ) ) ).

% prod.related
tff(fact_8655_prod_Oinsert__if,axiom,
    ! [A2: set_VEBT_VEBT,G: fun_VEBT_VEBT_real,Xb: vEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,A2)
     => ( groups2703838992350267259T_real(G,insert_VEBT_VEBT(Xb,A2)) = $ite(aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xb),A2),groups2703838992350267259T_real(G,A2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_VEBT_VEBT_real(G,Xb)),groups2703838992350267259T_real(G,A2))) ) ) ).

% prod.insert_if
tff(fact_8656_prod_Oinsert__if,axiom,
    ! [A2: set_real,G: fun_real_real,Xb: real] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( groups1681761925125756287l_real(G,insert_real(Xb,A2)) = $ite(aa_set_real_o(member_real(Xb),A2),groups1681761925125756287l_real(G,A2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(G,Xb)),groups1681761925125756287l_real(G,A2))) ) ) ).

% prod.insert_if
tff(fact_8657_prod_Oinsert__if,axiom,
    ! [A2: set_nat,G: fun_nat_real,Xb: nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( groups129246275422532515t_real(G,insert_nat(Xb,A2)) = $ite(aa_set_nat_o(member_nat(Xb),A2),groups129246275422532515t_real(G,A2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(G,Xb)),groups129246275422532515t_real(G,A2))) ) ) ).

% prod.insert_if
tff(fact_8658_prod_Oinsert__if,axiom,
    ! [A2: set_int,G: fun_int_real,Xb: int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( groups2316167850115554303t_real(G,insert_int(Xb,A2)) = $ite(aa_set_int_o(member_int(Xb),A2),groups2316167850115554303t_real(G,A2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(G,Xb)),groups2316167850115554303t_real(G,A2))) ) ) ).

% prod.insert_if
tff(fact_8659_prod_Oinsert__if,axiom,
    ! [A2: set_complex,G: fun_complex_real,Xb: complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( groups766887009212190081x_real(G,insert_complex(Xb,A2)) = $ite(aa_set_complex_o(member_complex(Xb),A2),groups766887009212190081x_real(G,A2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(G,Xb)),groups766887009212190081x_real(G,A2))) ) ) ).

% prod.insert_if
tff(fact_8660_prod_Oinsert__if,axiom,
    ! [A2: set_Code_integer,G: fun_Co3960896333048504454r_real,Xb: code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A2)
     => ( groups9004974159866482096r_real(G,insert_Code_integer(Xb,A2)) = $ite(aa_set5643387332196575111eger_o(member_Code_integer(Xb),A2),groups9004974159866482096r_real(G,A2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_Code_integer_real(G,Xb)),groups9004974159866482096r_real(G,A2))) ) ) ).

% prod.insert_if
tff(fact_8661_prod_Oinsert__if,axiom,
    ! [A2: set_VEBT_VEBT,G: fun_VEBT_VEBT_rat,Xb: vEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,A2)
     => ( groups5726676334696518183BT_rat(G,insert_VEBT_VEBT(Xb,A2)) = $ite(aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xb),A2),groups5726676334696518183BT_rat(G,A2),aa_rat_rat(times_times_rat(aa_VEBT_VEBT_rat(G,Xb)),groups5726676334696518183BT_rat(G,A2))) ) ) ).

% prod.insert_if
tff(fact_8662_prod_Oinsert__if,axiom,
    ! [A2: set_real,G: fun_real_rat,Xb: real] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( groups4061424788464935467al_rat(G,insert_real(Xb,A2)) = $ite(aa_set_real_o(member_real(Xb),A2),groups4061424788464935467al_rat(G,A2),aa_rat_rat(times_times_rat(aa_real_rat(G,Xb)),groups4061424788464935467al_rat(G,A2))) ) ) ).

% prod.insert_if
tff(fact_8663_prod_Oinsert__if,axiom,
    ! [A2: set_nat,G: fun_nat_rat,Xb: nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( groups73079841787564623at_rat(G,insert_nat(Xb,A2)) = $ite(aa_set_nat_o(member_nat(Xb),A2),groups73079841787564623at_rat(G,A2),aa_rat_rat(times_times_rat(aa_nat_rat(G,Xb)),groups73079841787564623at_rat(G,A2))) ) ) ).

% prod.insert_if
tff(fact_8664_prod_Oinsert__if,axiom,
    ! [A2: set_int,G: fun_int_rat,Xb: int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( groups1072433553688619179nt_rat(G,insert_int(Xb,A2)) = $ite(aa_set_int_o(member_int(Xb),A2),groups1072433553688619179nt_rat(G,A2),aa_rat_rat(times_times_rat(aa_int_rat(G,Xb)),groups1072433553688619179nt_rat(G,A2))) ) ) ).

% prod.insert_if
tff(fact_8665_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S6: set_VEBT_VEBT,T6: set_VEBT_VEBT,S2: set_VEBT_VEBT,I: fun_VE3165209603783583043T_VEBT,J: fun_VE3165209603783583043T_VEBT,T4: set_VEBT_VEBT,G: fun_VEBT_VEBT_real,H: fun_VEBT_VEBT_real] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,S6)
     => ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,T6)
       => ( ! [A3: vEBT_VEBT] :
              ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A3),minus_5127226145743854075T_VEBT(S2,S6))
             => ( aa_VEB5832953974749077948T_VEBT(I,aa_VEB5832953974749077948T_VEBT(J,A3)) = A3 ) )
         => ( ! [A3: vEBT_VEBT] :
                ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A3),minus_5127226145743854075T_VEBT(S2,S6))
               => aa_set_VEBT_VEBT_o(member_VEBT_VEBT(aa_VEB5832953974749077948T_VEBT(J,A3)),minus_5127226145743854075T_VEBT(T4,T6)) )
           => ( ! [B2: vEBT_VEBT] :
                  ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(B2),minus_5127226145743854075T_VEBT(T4,T6))
                 => ( aa_VEB5832953974749077948T_VEBT(J,aa_VEB5832953974749077948T_VEBT(I,B2)) = B2 ) )
             => ( ! [B2: vEBT_VEBT] :
                    ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(B2),minus_5127226145743854075T_VEBT(T4,T6))
                   => aa_set_VEBT_VEBT_o(member_VEBT_VEBT(aa_VEB5832953974749077948T_VEBT(I,B2)),minus_5127226145743854075T_VEBT(S2,S6)) )
               => ( ! [A3: vEBT_VEBT] :
                      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A3),S6)
                     => ( aa_VEBT_VEBT_real(G,A3) = one_one_real ) )
                 => ( ! [B2: vEBT_VEBT] :
                        ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(B2),T6)
                       => ( aa_VEBT_VEBT_real(H,B2) = one_one_real ) )
                   => ( ! [A3: vEBT_VEBT] :
                          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A3),S2)
                         => ( aa_VEBT_VEBT_real(H,aa_VEB5832953974749077948T_VEBT(J,A3)) = aa_VEBT_VEBT_real(G,A3) ) )
                     => ( groups2703838992350267259T_real(G,S2) = groups2703838992350267259T_real(H,T4) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
tff(fact_8666_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S6: set_VEBT_VEBT,T6: set_real,S2: set_VEBT_VEBT,I: fun_real_VEBT_VEBT,J: fun_VEBT_VEBT_real,T4: set_real,G: fun_VEBT_VEBT_real,H: fun_real_real] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,S6)
     => ( aa_set_real_o(finite_finite_real,T6)
       => ( ! [A3: vEBT_VEBT] :
              ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A3),minus_5127226145743854075T_VEBT(S2,S6))
             => ( aa_real_VEBT_VEBT(I,aa_VEBT_VEBT_real(J,A3)) = A3 ) )
         => ( ! [A3: vEBT_VEBT] :
                ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A3),minus_5127226145743854075T_VEBT(S2,S6))
               => aa_set_real_o(member_real(aa_VEBT_VEBT_real(J,A3)),minus_minus_set_real(T4,T6)) )
           => ( ! [B2: real] :
                  ( aa_set_real_o(member_real(B2),minus_minus_set_real(T4,T6))
                 => ( aa_VEBT_VEBT_real(J,aa_real_VEBT_VEBT(I,B2)) = B2 ) )
             => ( ! [B2: real] :
                    ( aa_set_real_o(member_real(B2),minus_minus_set_real(T4,T6))
                   => aa_set_VEBT_VEBT_o(member_VEBT_VEBT(aa_real_VEBT_VEBT(I,B2)),minus_5127226145743854075T_VEBT(S2,S6)) )
               => ( ! [A3: vEBT_VEBT] :
                      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A3),S6)
                     => ( aa_VEBT_VEBT_real(G,A3) = one_one_real ) )
                 => ( ! [B2: real] :
                        ( aa_set_real_o(member_real(B2),T6)
                       => ( aa_real_real(H,B2) = one_one_real ) )
                   => ( ! [A3: vEBT_VEBT] :
                          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A3),S2)
                         => ( aa_real_real(H,aa_VEBT_VEBT_real(J,A3)) = aa_VEBT_VEBT_real(G,A3) ) )
                     => ( groups2703838992350267259T_real(G,S2) = groups1681761925125756287l_real(H,T4) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
tff(fact_8667_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S6: set_real,T6: set_VEBT_VEBT,S2: set_real,I: fun_VEBT_VEBT_real,J: fun_real_VEBT_VEBT,T4: set_VEBT_VEBT,G: fun_real_real,H: fun_VEBT_VEBT_real] :
      ( aa_set_real_o(finite_finite_real,S6)
     => ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,T6)
       => ( ! [A3: real] :
              ( aa_set_real_o(member_real(A3),minus_minus_set_real(S2,S6))
             => ( aa_VEBT_VEBT_real(I,aa_real_VEBT_VEBT(J,A3)) = A3 ) )
         => ( ! [A3: real] :
                ( aa_set_real_o(member_real(A3),minus_minus_set_real(S2,S6))
               => aa_set_VEBT_VEBT_o(member_VEBT_VEBT(aa_real_VEBT_VEBT(J,A3)),minus_5127226145743854075T_VEBT(T4,T6)) )
           => ( ! [B2: vEBT_VEBT] :
                  ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(B2),minus_5127226145743854075T_VEBT(T4,T6))
                 => ( aa_real_VEBT_VEBT(J,aa_VEBT_VEBT_real(I,B2)) = B2 ) )
             => ( ! [B2: vEBT_VEBT] :
                    ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(B2),minus_5127226145743854075T_VEBT(T4,T6))
                   => aa_set_real_o(member_real(aa_VEBT_VEBT_real(I,B2)),minus_minus_set_real(S2,S6)) )
               => ( ! [A3: real] :
                      ( aa_set_real_o(member_real(A3),S6)
                     => ( aa_real_real(G,A3) = one_one_real ) )
                 => ( ! [B2: vEBT_VEBT] :
                        ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(B2),T6)
                       => ( aa_VEBT_VEBT_real(H,B2) = one_one_real ) )
                   => ( ! [A3: real] :
                          ( aa_set_real_o(member_real(A3),S2)
                         => ( aa_VEBT_VEBT_real(H,aa_real_VEBT_VEBT(J,A3)) = aa_real_real(G,A3) ) )
                     => ( groups1681761925125756287l_real(G,S2) = groups2703838992350267259T_real(H,T4) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
tff(fact_8668_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S6: set_real,T6: set_real,S2: set_real,I: fun_real_real,J: fun_real_real,T4: set_real,G: fun_real_real,H: fun_real_real] :
      ( aa_set_real_o(finite_finite_real,S6)
     => ( aa_set_real_o(finite_finite_real,T6)
       => ( ! [A3: real] :
              ( aa_set_real_o(member_real(A3),minus_minus_set_real(S2,S6))
             => ( aa_real_real(I,aa_real_real(J,A3)) = A3 ) )
         => ( ! [A3: real] :
                ( aa_set_real_o(member_real(A3),minus_minus_set_real(S2,S6))
               => aa_set_real_o(member_real(aa_real_real(J,A3)),minus_minus_set_real(T4,T6)) )
           => ( ! [B2: real] :
                  ( aa_set_real_o(member_real(B2),minus_minus_set_real(T4,T6))
                 => ( aa_real_real(J,aa_real_real(I,B2)) = B2 ) )
             => ( ! [B2: real] :
                    ( aa_set_real_o(member_real(B2),minus_minus_set_real(T4,T6))
                   => aa_set_real_o(member_real(aa_real_real(I,B2)),minus_minus_set_real(S2,S6)) )
               => ( ! [A3: real] :
                      ( aa_set_real_o(member_real(A3),S6)
                     => ( aa_real_real(G,A3) = one_one_real ) )
                 => ( ! [B2: real] :
                        ( aa_set_real_o(member_real(B2),T6)
                       => ( aa_real_real(H,B2) = one_one_real ) )
                   => ( ! [A3: real] :
                          ( aa_set_real_o(member_real(A3),S2)
                         => ( aa_real_real(H,aa_real_real(J,A3)) = aa_real_real(G,A3) ) )
                     => ( groups1681761925125756287l_real(G,S2) = groups1681761925125756287l_real(H,T4) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
tff(fact_8669_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S6: set_VEBT_VEBT,T6: set_int,S2: set_VEBT_VEBT,I: fun_int_VEBT_VEBT,J: fun_VEBT_VEBT_int,T4: set_int,G: fun_VEBT_VEBT_real,H: fun_int_real] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,S6)
     => ( aa_set_int_o(finite_finite_int,T6)
       => ( ! [A3: vEBT_VEBT] :
              ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A3),minus_5127226145743854075T_VEBT(S2,S6))
             => ( aa_int_VEBT_VEBT(I,aa_VEBT_VEBT_int(J,A3)) = A3 ) )
         => ( ! [A3: vEBT_VEBT] :
                ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A3),minus_5127226145743854075T_VEBT(S2,S6))
               => aa_set_int_o(member_int(aa_VEBT_VEBT_int(J,A3)),minus_minus_set_int(T4,T6)) )
           => ( ! [B2: int] :
                  ( aa_set_int_o(member_int(B2),minus_minus_set_int(T4,T6))
                 => ( aa_VEBT_VEBT_int(J,aa_int_VEBT_VEBT(I,B2)) = B2 ) )
             => ( ! [B2: int] :
                    ( aa_set_int_o(member_int(B2),minus_minus_set_int(T4,T6))
                   => aa_set_VEBT_VEBT_o(member_VEBT_VEBT(aa_int_VEBT_VEBT(I,B2)),minus_5127226145743854075T_VEBT(S2,S6)) )
               => ( ! [A3: vEBT_VEBT] :
                      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A3),S6)
                     => ( aa_VEBT_VEBT_real(G,A3) = one_one_real ) )
                 => ( ! [B2: int] :
                        ( aa_set_int_o(member_int(B2),T6)
                       => ( aa_int_real(H,B2) = one_one_real ) )
                   => ( ! [A3: vEBT_VEBT] :
                          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A3),S2)
                         => ( aa_int_real(H,aa_VEBT_VEBT_int(J,A3)) = aa_VEBT_VEBT_real(G,A3) ) )
                     => ( groups2703838992350267259T_real(G,S2) = groups2316167850115554303t_real(H,T4) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
tff(fact_8670_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S6: set_real,T6: set_int,S2: set_real,I: fun_int_real,J: fun_real_int,T4: set_int,G: fun_real_real,H: fun_int_real] :
      ( aa_set_real_o(finite_finite_real,S6)
     => ( aa_set_int_o(finite_finite_int,T6)
       => ( ! [A3: real] :
              ( aa_set_real_o(member_real(A3),minus_minus_set_real(S2,S6))
             => ( aa_int_real(I,aa_real_int(J,A3)) = A3 ) )
         => ( ! [A3: real] :
                ( aa_set_real_o(member_real(A3),minus_minus_set_real(S2,S6))
               => aa_set_int_o(member_int(aa_real_int(J,A3)),minus_minus_set_int(T4,T6)) )
           => ( ! [B2: int] :
                  ( aa_set_int_o(member_int(B2),minus_minus_set_int(T4,T6))
                 => ( aa_real_int(J,aa_int_real(I,B2)) = B2 ) )
             => ( ! [B2: int] :
                    ( aa_set_int_o(member_int(B2),minus_minus_set_int(T4,T6))
                   => aa_set_real_o(member_real(aa_int_real(I,B2)),minus_minus_set_real(S2,S6)) )
               => ( ! [A3: real] :
                      ( aa_set_real_o(member_real(A3),S6)
                     => ( aa_real_real(G,A3) = one_one_real ) )
                 => ( ! [B2: int] :
                        ( aa_set_int_o(member_int(B2),T6)
                       => ( aa_int_real(H,B2) = one_one_real ) )
                   => ( ! [A3: real] :
                          ( aa_set_real_o(member_real(A3),S2)
                         => ( aa_int_real(H,aa_real_int(J,A3)) = aa_real_real(G,A3) ) )
                     => ( groups1681761925125756287l_real(G,S2) = groups2316167850115554303t_real(H,T4) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
tff(fact_8671_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S6: set_VEBT_VEBT,T6: set_complex,S2: set_VEBT_VEBT,I: fun_co4340422964966740447T_VEBT,J: fun_VE5569070535936029429omplex,T4: set_complex,G: fun_VEBT_VEBT_real,H: fun_complex_real] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,S6)
     => ( aa_set_complex_o(finite3207457112153483333omplex,T6)
       => ( ! [A3: vEBT_VEBT] :
              ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A3),minus_5127226145743854075T_VEBT(S2,S6))
             => ( aa_complex_VEBT_VEBT(I,aa_VEBT_VEBT_complex(J,A3)) = A3 ) )
         => ( ! [A3: vEBT_VEBT] :
                ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A3),minus_5127226145743854075T_VEBT(S2,S6))
               => aa_set_complex_o(member_complex(aa_VEBT_VEBT_complex(J,A3)),minus_811609699411566653omplex(T4,T6)) )
           => ( ! [B2: complex] :
                  ( aa_set_complex_o(member_complex(B2),minus_811609699411566653omplex(T4,T6))
                 => ( aa_VEBT_VEBT_complex(J,aa_complex_VEBT_VEBT(I,B2)) = B2 ) )
             => ( ! [B2: complex] :
                    ( aa_set_complex_o(member_complex(B2),minus_811609699411566653omplex(T4,T6))
                   => aa_set_VEBT_VEBT_o(member_VEBT_VEBT(aa_complex_VEBT_VEBT(I,B2)),minus_5127226145743854075T_VEBT(S2,S6)) )
               => ( ! [A3: vEBT_VEBT] :
                      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A3),S6)
                     => ( aa_VEBT_VEBT_real(G,A3) = one_one_real ) )
                 => ( ! [B2: complex] :
                        ( aa_set_complex_o(member_complex(B2),T6)
                       => ( aa_complex_real(H,B2) = one_one_real ) )
                   => ( ! [A3: vEBT_VEBT] :
                          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A3),S2)
                         => ( aa_complex_real(H,aa_VEBT_VEBT_complex(J,A3)) = aa_VEBT_VEBT_real(G,A3) ) )
                     => ( groups2703838992350267259T_real(G,S2) = groups766887009212190081x_real(H,T4) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
tff(fact_8672_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S6: set_real,T6: set_complex,S2: set_real,I: fun_complex_real,J: fun_real_complex,T4: set_complex,G: fun_real_real,H: fun_complex_real] :
      ( aa_set_real_o(finite_finite_real,S6)
     => ( aa_set_complex_o(finite3207457112153483333omplex,T6)
       => ( ! [A3: real] :
              ( aa_set_real_o(member_real(A3),minus_minus_set_real(S2,S6))
             => ( aa_complex_real(I,aa_real_complex(J,A3)) = A3 ) )
         => ( ! [A3: real] :
                ( aa_set_real_o(member_real(A3),minus_minus_set_real(S2,S6))
               => aa_set_complex_o(member_complex(aa_real_complex(J,A3)),minus_811609699411566653omplex(T4,T6)) )
           => ( ! [B2: complex] :
                  ( aa_set_complex_o(member_complex(B2),minus_811609699411566653omplex(T4,T6))
                 => ( aa_real_complex(J,aa_complex_real(I,B2)) = B2 ) )
             => ( ! [B2: complex] :
                    ( aa_set_complex_o(member_complex(B2),minus_811609699411566653omplex(T4,T6))
                   => aa_set_real_o(member_real(aa_complex_real(I,B2)),minus_minus_set_real(S2,S6)) )
               => ( ! [A3: real] :
                      ( aa_set_real_o(member_real(A3),S6)
                     => ( aa_real_real(G,A3) = one_one_real ) )
                 => ( ! [B2: complex] :
                        ( aa_set_complex_o(member_complex(B2),T6)
                       => ( aa_complex_real(H,B2) = one_one_real ) )
                   => ( ! [A3: real] :
                          ( aa_set_real_o(member_real(A3),S2)
                         => ( aa_complex_real(H,aa_real_complex(J,A3)) = aa_real_real(G,A3) ) )
                     => ( groups1681761925125756287l_real(G,S2) = groups766887009212190081x_real(H,T4) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
tff(fact_8673_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S6: set_VEBT_VEBT,T6: set_Code_integer,S2: set_VEBT_VEBT,I: fun_Co3766629010929971184T_VEBT,J: fun_VE9219535225822561060nteger,T4: set_Code_integer,G: fun_VEBT_VEBT_real,H: fun_Co3960896333048504454r_real] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,S6)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,T6)
       => ( ! [A3: vEBT_VEBT] :
              ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A3),minus_5127226145743854075T_VEBT(S2,S6))
             => ( aa_Cod6507154402108480601T_VEBT(I,aa_VEB5935388676523804373nteger(J,A3)) = A3 ) )
         => ( ! [A3: vEBT_VEBT] :
                ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A3),minus_5127226145743854075T_VEBT(S2,S6))
               => aa_set5643387332196575111eger_o(member_Code_integer(aa_VEB5935388676523804373nteger(J,A3)),minus_2355218937544613996nteger(T4,T6)) )
           => ( ! [B2: code_integer] :
                  ( aa_set5643387332196575111eger_o(member_Code_integer(B2),minus_2355218937544613996nteger(T4,T6))
                 => ( aa_VEB5935388676523804373nteger(J,aa_Cod6507154402108480601T_VEBT(I,B2)) = B2 ) )
             => ( ! [B2: code_integer] :
                    ( aa_set5643387332196575111eger_o(member_Code_integer(B2),minus_2355218937544613996nteger(T4,T6))
                   => aa_set_VEBT_VEBT_o(member_VEBT_VEBT(aa_Cod6507154402108480601T_VEBT(I,B2)),minus_5127226145743854075T_VEBT(S2,S6)) )
               => ( ! [A3: vEBT_VEBT] :
                      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A3),S6)
                     => ( aa_VEBT_VEBT_real(G,A3) = one_one_real ) )
                 => ( ! [B2: code_integer] :
                        ( aa_set5643387332196575111eger_o(member_Code_integer(B2),T6)
                       => ( aa_Code_integer_real(H,B2) = one_one_real ) )
                   => ( ! [A3: vEBT_VEBT] :
                          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A3),S2)
                         => ( aa_Code_integer_real(H,aa_VEB5935388676523804373nteger(J,A3)) = aa_VEBT_VEBT_real(G,A3) ) )
                     => ( groups2703838992350267259T_real(G,S2) = groups9004974159866482096r_real(H,T4) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
tff(fact_8674_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S6: set_real,T6: set_Code_integer,S2: set_real,I: fun_Co3960896333048504454r_real,J: fun_re6978948838062957830nteger,T4: set_Code_integer,G: fun_real_real,H: fun_Co3960896333048504454r_real] :
      ( aa_set_real_o(finite_finite_real,S6)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,T6)
       => ( ! [A3: real] :
              ( aa_set_real_o(member_real(A3),minus_minus_set_real(S2,S6))
             => ( aa_Code_integer_real(I,aa_real_Code_integer(J,A3)) = A3 ) )
         => ( ! [A3: real] :
                ( aa_set_real_o(member_real(A3),minus_minus_set_real(S2,S6))
               => aa_set5643387332196575111eger_o(member_Code_integer(aa_real_Code_integer(J,A3)),minus_2355218937544613996nteger(T4,T6)) )
           => ( ! [B2: code_integer] :
                  ( aa_set5643387332196575111eger_o(member_Code_integer(B2),minus_2355218937544613996nteger(T4,T6))
                 => ( aa_real_Code_integer(J,aa_Code_integer_real(I,B2)) = B2 ) )
             => ( ! [B2: code_integer] :
                    ( aa_set5643387332196575111eger_o(member_Code_integer(B2),minus_2355218937544613996nteger(T4,T6))
                   => aa_set_real_o(member_real(aa_Code_integer_real(I,B2)),minus_minus_set_real(S2,S6)) )
               => ( ! [A3: real] :
                      ( aa_set_real_o(member_real(A3),S6)
                     => ( aa_real_real(G,A3) = one_one_real ) )
                 => ( ! [B2: code_integer] :
                        ( aa_set5643387332196575111eger_o(member_Code_integer(B2),T6)
                       => ( aa_Code_integer_real(H,B2) = one_one_real ) )
                   => ( ! [A3: real] :
                          ( aa_set_real_o(member_real(A3),S2)
                         => ( aa_Code_integer_real(H,aa_real_Code_integer(J,A3)) = aa_real_real(G,A3) ) )
                     => ( groups1681761925125756287l_real(G,S2) = groups9004974159866482096r_real(H,T4) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
tff(fact_8675_prod__dvd__prod__subset,axiom,
    ! [B4: set_complex,A2: set_complex,F: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,B4)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(A2),B4)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,groups861055069439313189ex_nat(F,A2)),groups861055069439313189ex_nat(F,B4)) ) ) ).

% prod_dvd_prod_subset
tff(fact_8676_prod__dvd__prod__subset,axiom,
    ! [B4: set_Code_integer,A2: set_Code_integer,F: fun_Code_integer_nat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B4)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(A2),B4)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,groups3190895334310489300er_nat(F,A2)),groups3190895334310489300er_nat(F,B4)) ) ) ).

% prod_dvd_prod_subset
tff(fact_8677_prod__dvd__prod__subset,axiom,
    ! [B4: set_complex,A2: set_complex,F: fun_complex_int] :
      ( aa_set_complex_o(finite3207457112153483333omplex,B4)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(A2),B4)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,groups858564598930262913ex_int(F,A2)),groups858564598930262913ex_int(F,B4)) ) ) ).

% prod_dvd_prod_subset
tff(fact_8678_prod__dvd__prod__subset,axiom,
    ! [B4: set_Code_integer,A2: set_Code_integer,F: fun_Code_integer_int] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B4)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(A2),B4)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,groups3188404863801439024er_int(F,A2)),groups3188404863801439024er_int(F,B4)) ) ) ).

% prod_dvd_prod_subset
tff(fact_8679_prod__dvd__prod__subset,axiom,
    ! [B4: set_nat,A2: set_nat,F: fun_nat_uint32] :
      ( aa_set_nat_o(finite_finite_nat,B4)
     => ( aa_set_nat_o(ord_less_eq_set_nat(A2),B4)
       => aa_uint32_o(aa_uin5924812838015166427nt32_o(dvd_dvd_uint32,groups2278496514549435363uint32(F,A2)),groups2278496514549435363uint32(F,B4)) ) ) ).

% prod_dvd_prod_subset
tff(fact_8680_prod__dvd__prod__subset,axiom,
    ! [B4: set_complex,A2: set_complex,F: fun_complex_uint32] :
      ( aa_set_complex_o(finite3207457112153483333omplex,B4)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(A2),B4)
       => aa_uint32_o(aa_uin5924812838015166427nt32_o(dvd_dvd_uint32,groups6230475983024736193uint32(F,A2)),groups6230475983024736193uint32(F,B4)) ) ) ).

% prod_dvd_prod_subset
tff(fact_8681_prod__dvd__prod__subset,axiom,
    ! [B4: set_Code_integer,A2: set_Code_integer,F: fun_Co1554742298434871494uint32] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B4)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(A2),B4)
       => aa_uint32_o(aa_uin5924812838015166427nt32_o(dvd_dvd_uint32,groups5586078468126652656uint32(F,A2)),groups5586078468126652656uint32(F,B4)) ) ) ).

% prod_dvd_prod_subset
tff(fact_8682_prod__dvd__prod__subset,axiom,
    ! [B4: set_nat,A2: set_nat,F: fun_nat_Code_integer] :
      ( aa_set_nat_o(finite_finite_nat,B4)
     => ( aa_set_nat_o(ord_less_eq_set_nat(A2),B4)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,groups3455450783089532116nteger(F,A2)),groups3455450783089532116nteger(F,B4)) ) ) ).

% prod_dvd_prod_subset
tff(fact_8683_prod__dvd__prod__subset,axiom,
    ! [B4: set_complex,A2: set_complex,F: fun_co8153928105989009288nteger] :
      ( aa_set_complex_o(finite3207457112153483333omplex,B4)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(A2),B4)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,groups8682486955453173170nteger(F,A2)),groups8682486955453173170nteger(F,B4)) ) ) ).

% prod_dvd_prod_subset
tff(fact_8684_prod__dvd__prod__subset,axiom,
    ! [B4: set_Code_integer,A2: set_Code_integer,F: fun_Co8499238673658658103nteger] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B4)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(A2),B4)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,groups3674199335183972705nteger(F,A2)),groups3674199335183972705nteger(F,B4)) ) ) ).

% prod_dvd_prod_subset
tff(fact_8685_prod__dvd__prod__subset2,axiom,
    ! [B4: set_VEBT_VEBT,A2: set_VEBT_VEBT,F: fun_VEBT_VEBT_nat,G: fun_VEBT_VEBT_nat] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,B4)
     => ( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(A2),B4)
       => ( ! [A3: vEBT_VEBT] :
              ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A3),A2)
             => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_VEBT_VEBT_nat(F,A3)),aa_VEBT_VEBT_nat(G,A3)) )
         => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,groups6361806394783013919BT_nat(F,A2)),groups6361806394783013919BT_nat(G,B4)) ) ) ) ).

% prod_dvd_prod_subset2
tff(fact_8686_prod__dvd__prod__subset2,axiom,
    ! [B4: set_real,A2: set_real,F: fun_real_nat,G: fun_real_nat] :
      ( aa_set_real_o(finite_finite_real,B4)
     => ( aa_set_real_o(ord_less_eq_set_real(A2),B4)
       => ( ! [A3: real] :
              ( aa_set_real_o(member_real(A3),A2)
             => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_real_nat(F,A3)),aa_real_nat(G,A3)) )
         => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,groups4696554848551431203al_nat(F,A2)),groups4696554848551431203al_nat(G,B4)) ) ) ) ).

% prod_dvd_prod_subset2
tff(fact_8687_prod__dvd__prod__subset2,axiom,
    ! [B4: set_complex,A2: set_complex,F: fun_complex_nat,G: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,B4)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(A2),B4)
       => ( ! [A3: complex] :
              ( aa_set_complex_o(member_complex(A3),A2)
             => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_complex_nat(F,A3)),aa_complex_nat(G,A3)) )
         => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,groups861055069439313189ex_nat(F,A2)),groups861055069439313189ex_nat(G,B4)) ) ) ) ).

% prod_dvd_prod_subset2
tff(fact_8688_prod__dvd__prod__subset2,axiom,
    ! [B4: set_Code_integer,A2: set_Code_integer,F: fun_Code_integer_nat,G: fun_Code_integer_nat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B4)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(A2),B4)
       => ( ! [A3: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(A3),A2)
             => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_Code_integer_nat(F,A3)),aa_Code_integer_nat(G,A3)) )
         => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,groups3190895334310489300er_nat(F,A2)),groups3190895334310489300er_nat(G,B4)) ) ) ) ).

% prod_dvd_prod_subset2
tff(fact_8689_prod__dvd__prod__subset2,axiom,
    ! [B4: set_VEBT_VEBT,A2: set_VEBT_VEBT,F: fun_VEBT_VEBT_int,G: fun_VEBT_VEBT_int] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,B4)
     => ( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(A2),B4)
       => ( ! [A3: vEBT_VEBT] :
              ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A3),A2)
             => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_VEBT_VEBT_int(F,A3)),aa_VEBT_VEBT_int(G,A3)) )
         => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,groups6359315924273963643BT_int(F,A2)),groups6359315924273963643BT_int(G,B4)) ) ) ) ).

% prod_dvd_prod_subset2
tff(fact_8690_prod__dvd__prod__subset2,axiom,
    ! [B4: set_real,A2: set_real,F: fun_real_int,G: fun_real_int] :
      ( aa_set_real_o(finite_finite_real,B4)
     => ( aa_set_real_o(ord_less_eq_set_real(A2),B4)
       => ( ! [A3: real] :
              ( aa_set_real_o(member_real(A3),A2)
             => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_real_int(F,A3)),aa_real_int(G,A3)) )
         => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,groups4694064378042380927al_int(F,A2)),groups4694064378042380927al_int(G,B4)) ) ) ) ).

% prod_dvd_prod_subset2
tff(fact_8691_prod__dvd__prod__subset2,axiom,
    ! [B4: set_complex,A2: set_complex,F: fun_complex_int,G: fun_complex_int] :
      ( aa_set_complex_o(finite3207457112153483333omplex,B4)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(A2),B4)
       => ( ! [A3: complex] :
              ( aa_set_complex_o(member_complex(A3),A2)
             => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_complex_int(F,A3)),aa_complex_int(G,A3)) )
         => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,groups858564598930262913ex_int(F,A2)),groups858564598930262913ex_int(G,B4)) ) ) ) ).

% prod_dvd_prod_subset2
tff(fact_8692_prod__dvd__prod__subset2,axiom,
    ! [B4: set_Code_integer,A2: set_Code_integer,F: fun_Code_integer_int,G: fun_Code_integer_int] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B4)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(A2),B4)
       => ( ! [A3: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(A3),A2)
             => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_Code_integer_int(F,A3)),aa_Code_integer_int(G,A3)) )
         => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,groups3188404863801439024er_int(F,A2)),groups3188404863801439024er_int(G,B4)) ) ) ) ).

% prod_dvd_prod_subset2
tff(fact_8693_prod__dvd__prod__subset2,axiom,
    ! [B4: set_VEBT_VEBT,A2: set_VEBT_VEBT,F: fun_VEBT_VEBT_uint32,G: fun_VEBT_VEBT_uint32] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,B4)
     => ( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(A2),B4)
       => ( ! [A3: vEBT_VEBT] :
              ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A3),A2)
             => aa_uint32_o(aa_uin5924812838015166427nt32_o(dvd_dvd_uint32,aa_VEBT_VEBT_uint32(F,A3)),aa_VEBT_VEBT_uint32(G,A3)) )
         => aa_uint32_o(aa_uin5924812838015166427nt32_o(dvd_dvd_uint32,groups8305177534072719291uint32(F,A2)),groups8305177534072719291uint32(G,B4)) ) ) ) ).

% prod_dvd_prod_subset2
tff(fact_8694_prod__dvd__prod__subset2,axiom,
    ! [B4: set_real,A2: set_real,F: fun_real_uint32,G: fun_real_uint32] :
      ( aa_set_real_o(finite_finite_real,B4)
     => ( aa_set_real_o(ord_less_eq_set_real(A2),B4)
       => ( ! [A3: real] :
              ( aa_set_real_o(member_real(A3),A2)
             => aa_uint32_o(aa_uin5924812838015166427nt32_o(dvd_dvd_uint32,aa_real_uint32(F,A3)),aa_real_uint32(G,A3)) )
         => aa_uint32_o(aa_uin5924812838015166427nt32_o(dvd_dvd_uint32,groups1111744456595050943uint32(F,A2)),groups1111744456595050943uint32(G,B4)) ) ) ) ).

% prod_dvd_prod_subset2
tff(fact_8695_prod_Osetdiff__irrelevant,axiom,
    ! [A2: set_nat,G: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( groups705719431365010083at_int(G,minus_minus_set_nat(A2,collect_nat(aTP_Lamm_ayq(G)))) = groups705719431365010083at_int(G,A2) ) ) ).

% prod.setdiff_irrelevant
tff(fact_8696_prod_Osetdiff__irrelevant,axiom,
    ! [A2: set_int,G: fun_int_int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( groups1705073143266064639nt_int(G,minus_minus_set_int(A2,collect_int(aTP_Lamm_ayr(G)))) = groups1705073143266064639nt_int(G,A2) ) ) ).

% prod.setdiff_irrelevant
tff(fact_8697_prod_Osetdiff__irrelevant,axiom,
    ! [A2: set_nat,G: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( groups708209901874060359at_nat(G,minus_minus_set_nat(A2,collect_nat(aTP_Lamm_ays(G)))) = groups708209901874060359at_nat(G,A2) ) ) ).

% prod.setdiff_irrelevant
tff(fact_8698_unbounded__k__infinite,axiom,
    ! [K: nat,S2: set_nat] :
      ( ! [M2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),M2)
         => ? [N7: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),N7)
              & aa_set_nat_o(member_nat(N7),S2) ) )
     => ~ aa_set_nat_o(finite_finite_nat,S2) ) ).

% unbounded_k_infinite
tff(fact_8699_infinite__nat__iff__unbounded,axiom,
    ! [S2: set_nat] :
      ( ~ aa_set_nat_o(finite_finite_nat,S2)
    <=> ! [M7: nat] :
        ? [N6: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M7),N6)
          & aa_set_nat_o(member_nat(N6),S2) ) ) ).

% infinite_nat_iff_unbounded
tff(fact_8700_infinite__nat__iff__unbounded__le,axiom,
    ! [S2: set_nat] :
      ( ~ aa_set_nat_o(finite_finite_nat,S2)
    <=> ! [M7: nat] :
        ? [N6: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M7),N6)
          & aa_set_nat_o(member_nat(N6),S2) ) ) ).

% infinite_nat_iff_unbounded_le
tff(fact_8701_central__binomial__lower__bound,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,numeral_numeral_real(aa_num_num(bit0,aa_num_num(bit0,one)))),Nb)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,Nb)))),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)),Nb))) ) ).

% central_binomial_lower_bound
tff(fact_8702_binomial__Suc__n,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,Nb)),Nb) = aa_nat_nat(suc,Nb) ).

% binomial_Suc_n
tff(fact_8703_binomial__n__n,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),Nb) = one_one_nat ).

% binomial_n_n
tff(fact_8704_binomial__0__Suc,axiom,
    ! [K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,zero_zero_nat),aa_nat_nat(suc,K)) = zero_zero_nat ).

% binomial_0_Suc
tff(fact_8705_binomial__1,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),aa_nat_nat(suc,zero_zero_nat)) = Nb ).

% binomial_1
tff(fact_8706_binomial__eq__0__iff,axiom,
    ! [Nb: nat,K: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K) = zero_zero_nat )
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),K) ) ).

% binomial_eq_0_iff
tff(fact_8707_binomial__Suc__Suc,axiom,
    ! [Nb: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,Nb)),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,Nb),K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),aa_nat_nat(suc,K))) ).

% binomial_Suc_Suc
tff(fact_8708_binomial__n__0,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),zero_zero_nat) = one_one_nat ).

% binomial_n_0
tff(fact_8709_zero__less__binomial__iff,axiom,
    ! [Nb: 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,Nb),K))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb) ) ).

% zero_less_binomial_iff
tff(fact_8710_choose__one,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),one_one_nat) = Nb ).

% choose_one
tff(fact_8711_binomial__eq__0,axiom,
    ! [Nb: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),K)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K) = zero_zero_nat ) ) ).

% binomial_eq_0
tff(fact_8712_Suc__times__binomial,axiom,
    ! [K: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,Nb)),aa_nat_nat(suc,K))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,Nb)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K)) ).

% Suc_times_binomial
tff(fact_8713_Suc__times__binomial__eq,axiom,
    ! [Nb: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,Nb)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,Nb)),aa_nat_nat(suc,K))),aa_nat_nat(suc,K)) ).

% Suc_times_binomial_eq
tff(fact_8714_binomial__symmetric,axiom,
    ! [K: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K)) ) ) ).

% binomial_symmetric
tff(fact_8715_choose__mult__lemma,axiom,
    ! [M: nat,R2: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),R2)),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K))),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K)),K)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),R2)),K)),K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),R2)),M)) ).

% choose_mult_lemma
tff(fact_8716_binomial__le__pow,axiom,
    ! [R2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,R2),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),R2)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Nb),R2)) ) ).

% binomial_le_pow
tff(fact_8717_zero__less__binomial,axiom,
    ! [K: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K)) ) ).

% zero_less_binomial
tff(fact_8718_Suc__times__binomial__add,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,A)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B))),aa_nat_nat(suc,A))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,B)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B))),A)) ).

% Suc_times_binomial_add
tff(fact_8719_binomial__Suc__Suc__eq__times,axiom,
    ! [Nb: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,Nb)),aa_nat_nat(suc,K)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,Nb)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K))),aa_nat_nat(suc,K)) ).

% binomial_Suc_Suc_eq_times
tff(fact_8720_choose__mult,axiom,
    ! [K: nat,M: nat,Nb: 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),Nb)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),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,Nb),K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),K))) ) ) ) ).

% choose_mult
tff(fact_8721_binomial__absorb__comp,axiom,
    ! [Nb: 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,Nb),K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)),K)) ).

% binomial_absorb_comp
tff(fact_8722_prod__int__eq,axiom,
    ! [I: nat,J: nat] : groups705719431365010083at_int(semiri1314217659103216013at_int,set_or1269000886237332187st_nat(I,J)) = groups1705073143266064639nt_int(aTP_Lamm_dj,set_or1266510415728281911st_int(aa_nat_int(semiri1314217659103216013at_int,I),aa_nat_int(semiri1314217659103216013at_int,J))) ).

% prod_int_eq
tff(fact_8723_small__lazy_H_Ocases,axiom,
    ! [Xb: product_prod_int_int] :
      ~ ! [D2: int,I3: int] : Xb != aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,D2),I3) ).

% small_lazy'.cases
tff(fact_8724_binomial__absorption,axiom,
    ! [K: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),aa_nat_nat(suc,K))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)),K)) ).

% binomial_absorption
tff(fact_8725_prod__int__plus__eq,axiom,
    ! [I: nat,J: nat] : groups705719431365010083at_int(semiri1314217659103216013at_int,set_or1269000886237332187st_nat(I,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),J))) = groups1705073143266064639nt_int(aTP_Lamm_dj,set_or1266510415728281911st_int(aa_nat_int(semiri1314217659103216013at_int,I),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),J)))) ).

% prod_int_plus_eq
tff(fact_8726_binomial__maximum,axiom,
    ! [Nb: nat,K: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% binomial_maximum
tff(fact_8727_binomial__antimono,axiom,
    ! [K: nat,K2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),K2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),K)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),Nb)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K2)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K)) ) ) ) ).

% binomial_antimono
tff(fact_8728_binomial__mono,axiom,
    ! [K: nat,K2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),K2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K2)),Nb)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K2)) ) ) ).

% binomial_mono
tff(fact_8729_binomial__maximum_H,axiom,
    ! [Nb: nat,K: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)),K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)),Nb)) ).

% binomial_maximum'
tff(fact_8730_binomial__le__pow2,axiom,
    ! [Nb: nat,K: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ).

% binomial_le_pow2
tff(fact_8731_choose__reduce__nat,axiom,
    ! [Nb: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),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,Nb),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,Nb),one_one_nat)),K)) ) ) ) ).

% choose_reduce_nat
tff(fact_8732_times__binomial__minus1__eq,axiom,
    ! [K: nat,Nb: 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,Nb),K)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),one_one_nat))) ) ) ).

% times_binomial_minus1_eq
tff(fact_8733_binomial__less__binomial__Suc,axiom,
    ! [K: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),aa_nat_nat(suc,K))) ) ).

% binomial_less_binomial_Suc
tff(fact_8734_binomial__strict__mono,axiom,
    ! [K: nat,K2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),K2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K2)),Nb)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K2)) ) ) ).

% binomial_strict_mono
tff(fact_8735_binomial__strict__antimono,axiom,
    ! [K: nat,K2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),K2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K))
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),Nb)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K2)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K)) ) ) ) ).

% binomial_strict_antimono
tff(fact_8736_central__binomial__odd,axiom,
    ! [Nb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ) ).

% central_binomial_odd
tff(fact_8737_binomial__addition__formula,axiom,
    ! [Nb: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),aa_nat_nat(suc,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,Nb),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,Nb),one_one_nat)),K)) ) ) ).

% binomial_addition_formula
tff(fact_8738_choose__two,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),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,Nb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% choose_two
tff(fact_8739_lessThan__Suc__atMost,axiom,
    ! [K: nat] : set_ord_lessThan_nat(aa_nat_nat(suc,K)) = set_ord_atMost_nat(K) ).

% lessThan_Suc_atMost
tff(fact_8740_atMost__Suc,axiom,
    ! [K: nat] : set_ord_atMost_nat(aa_nat_nat(suc,K)) = insert_nat(aa_nat_nat(suc,K),set_ord_atMost_nat(K)) ).

% atMost_Suc
tff(fact_8741_finite__nat__iff__bounded__le,axiom,
    ! [S2: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,S2)
    <=> ? [K4: nat] : aa_set_nat_o(ord_less_eq_set_nat(S2),set_ord_atMost_nat(K4)) ) ).

% finite_nat_iff_bounded_le
tff(fact_8742_sum__choose__upper,axiom,
    ! [M: nat,Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_ayt,M)),set_ord_atMost_nat(Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,Nb)),aa_nat_nat(suc,M)) ).

% sum_choose_upper
tff(fact_8743_sum__choose__lower,axiom,
    ! [R2: nat,Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_ayu,R2)),set_ord_atMost_nat(Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,R2),Nb))),Nb) ).

% sum_choose_lower
tff(fact_8744_choose__rising__sum_I2_J,axiom,
    ! [Nb: nat,M: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_ayv,Nb)),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,Nb),M)),one_one_nat)),M) ).

% choose_rising_sum(2)
tff(fact_8745_choose__rising__sum_I1_J,axiom,
    ! [Nb: nat,M: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_ayv,Nb)),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,Nb),M)),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat)) ).

% choose_rising_sum(1)
tff(fact_8746_sum__choose__diagonal,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_ayw(M),Nb)),set_ord_atMost_nat(M)) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,Nb)),M) ) ) ).

% sum_choose_diagonal
tff(fact_8747_vandermonde,axiom,
    ! [M: nat,Nb: nat,R2: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_ayx(M),Nb),R2)),set_ord_atMost_nat(R2)) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Nb)),R2) ).

% vandermonde
tff(fact_8748_atLeast1__atMost__eq__remove0,axiom,
    ! [Nb: nat] : set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat),Nb) = minus_minus_set_nat(set_ord_atMost_nat(Nb),insert_nat(zero_zero_nat,bot_bot_set_nat)) ).

% atLeast1_atMost_eq_remove0
tff(fact_8749_choose__row__sum,axiom,
    ! [Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(binomial,Nb)),set_ord_atMost_nat(Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb) ).

% choose_row_sum
tff(fact_8750_binomial,axiom,
    ! [A: nat,B: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),Nb) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_ayy(A),B),Nb)),set_ord_atMost_nat(Nb)) ).

% binomial
tff(fact_8751_polynomial__product__nat,axiom,
    ! [M: nat,A: fun_nat_nat,Nb: nat,B: fun_nat_nat,Xb: nat] :
      ( ! [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),I3)
         => ( aa_nat_nat(A,I3) = zero_zero_nat ) )
     => ( ! [J2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),J2)
           => ( aa_nat_nat(B,J2) = zero_zero_nat ) )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_ayz(A),Xb)),set_ord_atMost_nat(M))),aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_ayz(B),Xb)),set_ord_atMost_nat(Nb))) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aa_fun6650658898577385413at_nat(aTP_Lamm_azb(A),B),Xb)),set_ord_atMost_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Nb))) ) ) ) ).

% polynomial_product_nat
tff(fact_8752_choose__square__sum,axiom,
    ! [Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_azc,Nb)),set_ord_atMost_nat(Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)),Nb) ).

% choose_square_sum
tff(fact_8753_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))),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_8754_choose__linear__sum,axiom,
    ! [Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_azd,Nb)),set_ord_atMost_nat(Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat))) ).

% choose_linear_sum
tff(fact_8755_powr__int,axiom,
    ! [Xb: real,I: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),aa_int_real(ring_1_of_int_real,I)) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),I),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_int_nat(nat2,I)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_int_nat(nat2,aa_int_int(uminus_uminus_int,I))))) ) ) ).

% powr_int
tff(fact_8756_VEBT__internal_OminNull_Opelims_I1_J,axiom,
    ! [Xb: vEBT_VEBT,Y: $o] :
      ( ( aa_VEBT_VEBT_o(vEBT_VEBT_minNull,Xb)
      <=> (Y) )
     => ( aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),Xb)
       => ( ( ( Xb = vEBT_Leaf($false,$false) )
           => ( (Y)
             => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),vEBT_Leaf($false,$false)) ) )
         => ( ! [Uv2: $o] :
                ( ( Xb = vEBT_Leaf($true,(Uv2)) )
               => ( ~ (Y)
                 => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),vEBT_Leaf($true,(Uv2))) ) )
           => ( ! [Uu2: $o] :
                  ( ( Xb = vEBT_Leaf((Uu2),$true) )
                 => ( ~ (Y)
                   => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),vEBT_Leaf((Uu2),$true)) ) )
             => ( ! [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
                    ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uw2,Ux2),Uy2) )
                   => ( (Y)
                     => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uw2,Ux2),Uy2)) ) )
               => ~ ! [Uz2: product_prod_nat_nat,Va: nat,Vb2: list_VEBT_VEBT,Vc: vEBT_VEBT] :
                      ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(Uz2),Va,Vb2),Vc) )
                     => ( ~ (Y)
                       => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(Uz2),Va,Vb2),Vc)) ) ) ) ) ) ) ) ) ).

% VEBT_internal.minNull.pelims(1)
tff(fact_8757_of__nat__id,axiom,
    ! [Nb: nat] : aa_nat_nat(semiri1316708129612266289at_nat,Nb) = Nb ).

% of_nat_id
tff(fact_8758_negative__zle,axiom,
    ! [Nb: 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,Nb))),aa_nat_int(semiri1314217659103216013at_int,M)) ).

% negative_zle
tff(fact_8759_nat__neg__numeral,axiom,
    ! [K: num] : aa_int_nat(nat2,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K))) = zero_zero_nat ).

% nat_neg_numeral
tff(fact_8760_negative__zless,axiom,
    ! [Nb: 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,Nb)))),aa_nat_int(semiri1314217659103216013at_int,M)) ).

% negative_zless
tff(fact_8761_int__div__minus__is__minus1,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int)
     => ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B) = aa_int_int(uminus_uminus_int,A) )
      <=> ( B = aa_int_int(uminus_uminus_int,one_one_int) ) ) ) ).

% int_div_minus_is_minus1
tff(fact_8762_ceiling__divide__eq__div__numeral,axiom,
    ! [A: num,B: num] : aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,numeral_numeral_real(A)),numeral_numeral_real(B))) = aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,A))),aa_num_int(numeral_numeral_int,B))) ).

% ceiling_divide_eq_div_numeral
tff(fact_8763_fact__ge__self,axiom,
    ! [Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),aa_nat_nat(semiri1408675320244567234ct_nat,Nb)) ).

% fact_ge_self
tff(fact_8764_fact__mono__nat,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(semiri1408675320244567234ct_nat,M)),aa_nat_nat(semiri1408675320244567234ct_nat,Nb)) ) ).

% fact_mono_nat
tff(fact_8765_fact__less__mono__nat,axiom,
    ! [M: nat,Nb: 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),Nb)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(semiri1408675320244567234ct_nat,M)),aa_nat_nat(semiri1408675320244567234ct_nat,Nb)) ) ) ).

% fact_less_mono_nat
tff(fact_8766_int__cases,axiom,
    ! [Z: int] :
      ( ! [N2: nat] : Z != aa_nat_int(semiri1314217659103216013at_int,N2)
     => ~ ! [N2: nat] : Z != aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,N2))) ) ).

% int_cases
tff(fact_8767_int__of__nat__induct,axiom,
    ! [P: fun_int_o,Z: int] :
      ( ! [N2: nat] : aa_int_o(P,aa_nat_int(semiri1314217659103216013at_int,N2))
     => ( ! [N2: nat] : aa_int_o(P,aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,N2))))
       => aa_int_o(P,Z) ) ) ).

% int_of_nat_induct
tff(fact_8768_pos__zmult__eq__1__iff__lemma,axiom,
    ! [M: int,Nb: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,M),Nb) = one_one_int )
     => ( ( M = one_one_int )
        | ( M = aa_int_int(uminus_uminus_int,one_one_int) ) ) ) ).

% pos_zmult_eq_1_iff_lemma
tff(fact_8769_zmult__eq__1__iff,axiom,
    ! [M: int,Nb: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,M),Nb) = one_one_int )
    <=> ( ( ( M = one_one_int )
          & ( Nb = one_one_int ) )
        | ( ( M = aa_int_int(uminus_uminus_int,one_one_int) )
          & ( Nb = aa_int_int(uminus_uminus_int,one_one_int) ) ) ) ) ).

% zmult_eq_1_iff
tff(fact_8770_minus__int__code_I2_J,axiom,
    ! [L: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,zero_zero_int),L) = aa_int_int(uminus_uminus_int,L) ).

% minus_int_code(2)
tff(fact_8771_fact__ge__Suc__0__nat,axiom,
    ! [Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),aa_nat_nat(semiri1408675320244567234ct_nat,Nb)) ).

% fact_ge_Suc_0_nat
tff(fact_8772_dvd__fact,axiom,
    ! [M: nat,Nb: 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),Nb)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,M),aa_nat_nat(semiri1408675320244567234ct_nat,Nb)) ) ) ).

% dvd_fact
tff(fact_8773_int__cases4,axiom,
    ! [M: int] :
      ( ! [N2: nat] : M != aa_nat_int(semiri1314217659103216013at_int,N2)
     => ~ ! [N2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)
           => ( M != aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N2)) ) ) ) ).

% int_cases4
tff(fact_8774_int__zle__neg,axiom,
    ! [Nb: nat,M: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,M)))
    <=> ( ( Nb = zero_zero_nat )
        & ( M = zero_zero_nat ) ) ) ).

% int_zle_neg
tff(fact_8775_nonpos__int__cases,axiom,
    ! [K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),zero_zero_int)
     => ~ ! [N2: nat] : K != aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N2)) ) ).

% nonpos_int_cases
tff(fact_8776_negative__zle__0,axiom,
    ! [Nb: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,Nb))),zero_zero_int) ).

% negative_zle_0
tff(fact_8777_zmod__zminus2__eq__if,axiom,
    ! [A: int,B: int] :
      aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_int_int(uminus_uminus_int,B)) = $ite(aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) = zero_zero_int,zero_zero_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)),B)) ).

% zmod_zminus2_eq_if
tff(fact_8778_zmod__zminus1__eq__if,axiom,
    ! [A: int,B: int] :
      aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(uminus_uminus_int,A)),B) = $ite(aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) = zero_zero_int,zero_zero_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B))) ).

% zmod_zminus1_eq_if
tff(fact_8779_fact__diff__Suc,axiom,
    ! [Nb: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(suc,M))
     => ( aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,M)),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,M)),Nb)),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Nb))) ) ) ).

% fact_diff_Suc
tff(fact_8780_fact__div__fact__le__pow,axiom,
    ! [R2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,R2),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(semiri1408675320244567234ct_nat,Nb)),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),R2)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Nb),R2)) ) ).

% fact_div_fact_le_pow
tff(fact_8781_binomial__fact__lemma,axiom,
    ! [K: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1408675320244567234ct_nat,K)),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K)))),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K)) = aa_nat_nat(semiri1408675320244567234ct_nat,Nb) ) ) ).

% binomial_fact_lemma
tff(fact_8782_int__cases3,axiom,
    ! [K: int] :
      ( ( K != zero_zero_int )
     => ( ! [N2: nat] :
            ( ( K = aa_nat_int(semiri1314217659103216013at_int,N2) )
           => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2) )
       => ~ ! [N2: nat] :
              ( ( K = aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N2)) )
             => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2) ) ) ) ).

% int_cases3
tff(fact_8783_fact__eq__fact__times,axiom,
    ! [Nb: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),M)
     => ( aa_nat_nat(semiri1408675320244567234ct_nat,M) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1408675320244567234ct_nat,Nb)),groups708209901874060359at_nat(aTP_Lamm_di,set_or1269000886237332187st_nat(aa_nat_nat(suc,Nb),M))) ) ) ).

% fact_eq_fact_times
tff(fact_8784_not__zle__0__negative,axiom,
    ! [Nb: nat] : ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,Nb)))) ).

% not_zle_0_negative
tff(fact_8785_negative__zless__0,axiom,
    ! [Nb: nat] : aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,Nb)))),zero_zero_int) ).

% negative_zless_0
tff(fact_8786_negD,axiom,
    ! [Xb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),zero_zero_int)
     => ? [N2: nat] : Xb = aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,N2))) ) ).

% negD
tff(fact_8787_verit__less__mono__div__int2,axiom,
    ! [A2: int,B4: int,Nb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),B4)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(uminus_uminus_int,Nb))
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,B4),Nb)),aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),Nb)) ) ) ).

% verit_less_mono_div_int2
tff(fact_8788_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_8789_binomial__altdef__nat,axiom,
    ! [K: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(semiri1408675320244567234ct_nat,Nb)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1408675320244567234ct_nat,K)),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K)))) ) ) ).

% binomial_altdef_nat
tff(fact_8790_fact__div__fact,axiom,
    ! [Nb: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),M)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(semiri1408675320244567234ct_nat,M)),aa_nat_nat(semiri1408675320244567234ct_nat,Nb)) = groups708209901874060359at_nat(aTP_Lamm_di,set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat),M)) ) ) ).

% fact_div_fact
tff(fact_8791_neg__int__cases,axiom,
    ! [K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int)
     => ~ ! [N2: nat] :
            ( ( K = aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N2)) )
           => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2) ) ) ).

% neg_int_cases
tff(fact_8792_nat__mult__distrib__neg,axiom,
    ! [Z: int,Z5: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),zero_zero_int)
     => ( aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(times_times_int,Z),Z5)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_int_nat(nat2,aa_int_int(uminus_uminus_int,Z))),aa_int_nat(nat2,aa_int_int(uminus_uminus_int,Z5))) ) ) ).

% nat_mult_distrib_neg
tff(fact_8793_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_8794_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_8795_zdiv__zminus1__eq__if,axiom,
    ! [B: int,A: int] :
      ( ( B != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(uminus_uminus_int,A)),B) = $ite(aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) = zero_zero_int,aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B))),one_one_int)) ) ) ).

% zdiv_zminus1_eq_if
tff(fact_8796_zdiv__zminus2__eq__if,axiom,
    ! [B: int,A: int] :
      ( ( B != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_int_int(uminus_uminus_int,B)) = $ite(aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) = zero_zero_int,aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B))),one_one_int)) ) ) ).

% zdiv_zminus2_eq_if
tff(fact_8797_square__fact__le__2__fact,axiom,
    ! [Nb: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri2265585572941072030t_real,Nb)),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))) ).

% square_fact_le_2_fact
tff(fact_8798_zminus1__lemma,axiom,
    ! [A: int,B: int,Q2: int,R2: int] :
      ( aa_Pro4592068669816069595_int_o(eucl_rel_int(A,B),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q2),R2))
     => ( ( B != zero_zero_int )
       => aa_Pro4592068669816069595_int_o(eucl_rel_int(aa_int_int(uminus_uminus_int,A),B),
            aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,
                $ite(R2 = zero_zero_int,aa_int_int(uminus_uminus_int,Q2),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,Q2)),one_one_int))),
              $ite(R2 = zero_zero_int,zero_zero_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B),R2)))) ) ) ).

% zminus1_lemma
tff(fact_8799_minus__1__div__exp__eq__int,axiom,
    ! [Nb: nat] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) = aa_int_int(uminus_uminus_int,one_one_int) ).

% minus_1_div_exp_eq_int
tff(fact_8800_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_8801_int__bit__induct,axiom,
    ! [P: fun_int_o,K: int] :
      ( aa_int_o(P,zero_zero_int)
     => ( aa_int_o(P,aa_int_int(uminus_uminus_int,one_one_int))
       => ( ! [K3: int] :
              ( aa_int_o(P,K3)
             => ( ( K3 != zero_zero_int )
               => aa_int_o(P,aa_int_int(aa_int_fun_int_int(times_times_int,K3),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))) ) )
         => ( ! [K3: int] :
                ( aa_int_o(P,K3)
               => ( ( K3 != aa_int_int(uminus_uminus_int,one_one_int) )
                 => aa_int_o(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,K3),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))) ) )
           => aa_int_o(P,K) ) ) ) ) ).

% int_bit_induct
tff(fact_8802_m1mod2k,axiom,
    ! [Nb: nat] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) = 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))),Nb)),one_one_int) ).

% m1mod2k
tff(fact_8803_Maclaurin__lemma,axiom,
    ! [H: real,F: fun_real_real,J: fun_nat_real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),H)
     => ? [B8: real] : aa_real_real(F,H) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_fun1839206346889086768t_real(aTP_Lamm_aze(H),J),set_ord_lessThan_nat(Nb))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B8),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,H),Nb)),aa_nat_real(semiri2265585572941072030t_real,Nb)))) ) ).

% Maclaurin_lemma
tff(fact_8804_sb__dec__lem_H,axiom,
    ! [K: 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,aa_num_num(bit0,one))),K)),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K)))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K))),A)) ) ).

% sb_dec_lem'
tff(fact_8805_m1mod22k,axiom,
    ! [Nb: nat] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(uminus_uminus_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(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb))) = aa_int_int(aa_int_fun_int_int(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_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb))),one_one_int) ).

% m1mod22k
tff(fact_8806_VEBT__internal_OminNull_Opelims_I3_J,axiom,
    ! [Xb: vEBT_VEBT] :
      ( ~ aa_VEBT_VEBT_o(vEBT_VEBT_minNull,Xb)
     => ( aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),Xb)
       => ( ! [Uv2: $o] :
              ( ( Xb = vEBT_Leaf($true,(Uv2)) )
             => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),vEBT_Leaf($true,(Uv2))) )
         => ( ! [Uu2: $o] :
                ( ( Xb = vEBT_Leaf((Uu2),$true) )
               => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),vEBT_Leaf((Uu2),$true)) )
           => ~ ! [Uz2: product_prod_nat_nat,Va: nat,Vb2: list_VEBT_VEBT,Vc: vEBT_VEBT] :
                  ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(Uz2),Va,Vb2),Vc) )
                 => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(Uz2),Va,Vb2),Vc)) ) ) ) ) ) ).

% VEBT_internal.minNull.pelims(3)
tff(fact_8807_VEBT__internal_OminNull_Opelims_I2_J,axiom,
    ! [Xb: vEBT_VEBT] :
      ( aa_VEBT_VEBT_o(vEBT_VEBT_minNull,Xb)
     => ( aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),Xb)
       => ( ( ( Xb = vEBT_Leaf($false,$false) )
           => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),vEBT_Leaf($false,$false)) )
         => ~ ! [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
                ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uw2,Ux2),Uy2) )
               => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uw2,Ux2),Uy2)) ) ) ) ) ).

% VEBT_internal.minNull.pelims(2)
tff(fact_8808_sb__inc__lem_H,axiom,
    ! [A: int,K: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_int_int(uminus_uminus_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K)))
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),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,K)))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),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,K)))) ) ).

% sb_inc_lem'
tff(fact_8809_sb__dec__lem,axiom,
    ! [K: nat,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K))),A))
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K)))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K))),A)) ) ).

% sb_dec_lem
tff(fact_8810_binomial__code,axiom,
    ! [Nb: nat,K: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),K),
        zero_zero_nat,
        $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),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,Nb),K)),one_one_nat)),Nb),one_one_nat)),aa_nat_nat(semiri1408675320244567234ct_nat,K))) ) ).

% binomial_code
tff(fact_8811_one__mod__minus__numeral,axiom,
    ! [Nb: num] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))) = aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(adjust_mod,aa_num_int(numeral_numeral_int,Nb)),aa_Pro6799258434465214121nt_int(product_snd_int_int,unique5052692396658037445od_int(one,Nb)))) ).

% one_mod_minus_numeral
tff(fact_8812_minus__one__mod__numeral,axiom,
    ! [Nb: num] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_num_int(numeral_numeral_int,Nb)) = aa_int_int(aa_int_fun_int_int(adjust_mod,aa_num_int(numeral_numeral_int,Nb)),aa_Pro6799258434465214121nt_int(product_snd_int_int,unique5052692396658037445od_int(one,Nb))) ).

% minus_one_mod_numeral
tff(fact_8813_and__int_Opinduct,axiom,
    ! [A0: int,A1: int,P: fun_int_fun_int_o] :
      ( aa_Pro4592068669816069595_int_o(accp_P1096762738010456898nt_int(bit_and_int_rel),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A0),A1))
     => ( ! [K3: int,L2: int] :
            ( aa_Pro4592068669816069595_int_o(accp_P1096762738010456898nt_int(bit_and_int_rel),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,K3),L2))
           => ( ( ~ ( aa_set_int_o(member_int(K3),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int)))
                    & aa_set_int_o(member_int(L2),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int))) )
               => aa_int_o(aa_int_fun_int_o(P,aa_int_int(aa_int_fun_int_int(divide_divide_int,K3),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(divide_divide_int,L2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))) )
             => aa_int_o(aa_int_fun_int_o(P,K3),L2) ) )
       => aa_int_o(aa_int_fun_int_o(P,A0),A1) ) ) ).

% and_int.pinduct
tff(fact_8814_real__add__minus__iff,axiom,
    ! [Xb: real,A: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),aa_real_real(uminus_uminus_real,A)) = zero_zero_real )
    <=> ( Xb = A ) ) ).

% real_add_minus_iff
tff(fact_8815_floor__minus__divide__eq__div__numeral,axiom,
    ! [A: num,B: num] : aa_real_int(archim6058952711729229775r_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,numeral_numeral_real(A)),numeral_numeral_real(B)))) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,A))),aa_num_int(numeral_numeral_int,B)) ).

% floor_minus_divide_eq_div_numeral
tff(fact_8816_ceiling__minus__divide__eq__div__numeral,axiom,
    ! [A: num,B: num] : aa_real_int(archim7802044766580827645g_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,numeral_numeral_real(A)),numeral_numeral_real(B)))) = aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_num_int(numeral_numeral_int,A)),aa_num_int(numeral_numeral_int,B))) ).

% ceiling_minus_divide_eq_div_numeral
tff(fact_8817_numeral__mod__minus__numeral,axiom,
    ! [M: num,Nb: num] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))) = aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(adjust_mod,aa_num_int(numeral_numeral_int,Nb)),aa_Pro6799258434465214121nt_int(product_snd_int_int,unique5052692396658037445od_int(M,Nb)))) ).

% numeral_mod_minus_numeral
tff(fact_8818_minus__numeral__mod__numeral,axiom,
    ! [M: num,Nb: num] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_num_int(numeral_numeral_int,Nb)) = aa_int_int(aa_int_fun_int_int(adjust_mod,aa_num_int(numeral_numeral_int,Nb)),aa_Pro6799258434465214121nt_int(product_snd_int_int,unique5052692396658037445od_int(M,Nb))) ).

% minus_numeral_mod_numeral
tff(fact_8819_floor__minus__one__divide__eq__div__numeral,axiom,
    ! [B: num] : aa_real_int(archim6058952711729229775r_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(B)))) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_num_int(numeral_numeral_int,B)) ).

% floor_minus_one_divide_eq_div_numeral
tff(fact_8820_real__minus__mult__self__le,axiom,
    ! [U: real,Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,U),U))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Xb)) ).

% real_minus_mult_self_le
tff(fact_8821_minus__real__def,axiom,
    ! [Xb: real,Y: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),Y) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),aa_real_real(uminus_uminus_real,Y)) ).

% minus_real_def
tff(fact_8822_complex__mod__minus__le__complex__mod,axiom,
    ! [Xb: complex] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_complex_real(real_V1022390504157884413omplex,Xb))),aa_complex_real(real_V1022390504157884413omplex,Xb)) ).

% complex_mod_minus_le_complex_mod
tff(fact_8823_minus__integer__code_I2_J,axiom,
    ! [L: code_integer] : aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(zero_z3403309356797280102nteger),L) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,L) ).

% minus_integer_code(2)
tff(fact_8824_real__0__less__add__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Y))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,Xb)),Y) ) ).

% real_0_less_add_iff
tff(fact_8825_real__add__less__0__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Y)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Y),aa_real_real(uminus_uminus_real,Xb)) ) ).

% real_add_less_0_iff
tff(fact_8826_real__add__le__0__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Y)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),aa_real_real(uminus_uminus_real,Xb)) ) ).

% real_add_le_0_iff
tff(fact_8827_real__0__le__add__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Y))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,Xb)),Y) ) ).

% real_0_le_add_iff
tff(fact_8828_divide__powr__uminus,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),C)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),aa_real_real(uminus_uminus_real,C))) ).

% divide_powr_uminus
tff(fact_8829_realpow__square__minus__le,axiom,
    ! [U: real,Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,U),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% realpow_square_minus_le
tff(fact_8830_powr__neg__one,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),aa_real_real(uminus_uminus_real,one_one_real)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),Xb) ) ) ).

% powr_neg_one
tff(fact_8831_Divides_Oadjust__mod__def,axiom,
    ! [L: int,R2: int] :
      aa_int_int(aa_int_fun_int_int(adjust_mod,L),R2) = $ite(R2 = zero_zero_int,zero_zero_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,L),R2)) ).

% Divides.adjust_mod_def
tff(fact_8832_Bernoulli__inequality,axiom,
    ! [Xb: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Xb)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),Xb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),Xb)),Nb)) ) ).

% Bernoulli_inequality
tff(fact_8833_log__minus__eq__powr,axiom,
    ! [B: real,Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B)
     => ( ( B != one_one_real )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
         => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(log,B),Xb)),Y) = aa_real_real(aa_rea7340605660300375871l_real(log,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),aa_real_real(uminus_uminus_real,Y)))) ) ) ) ) ).

% log_minus_eq_powr
tff(fact_8834_powr__neg__numeral,axiom,
    ! [Xb: real,Nb: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),aa_real_real(uminus_uminus_real,numeral_numeral_real(Nb))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,Nb))) ) ) ).

% powr_neg_numeral
tff(fact_8835_cos__coeff__def,axiom,
    ! [X3: nat] :
      aa_nat_real(cos_coeff,X3) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),X3),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_real(semiri2265585572941072030t_real,X3)),zero_zero_real) ).

% cos_coeff_def
tff(fact_8836_minus__one__div__numeral,axiom,
    ! [Nb: num] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_num_int(numeral_numeral_int,Nb)) = aa_int_int(uminus_uminus_int,aa_Pro6799258434465214121nt_int(adjust_div,unique5052692396658037445od_int(one,Nb))) ).

% minus_one_div_numeral
tff(fact_8837_one__div__minus__numeral,axiom,
    ! [Nb: num] : aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))) = aa_int_int(uminus_uminus_int,aa_Pro6799258434465214121nt_int(adjust_div,unique5052692396658037445od_int(one,Nb))) ).

% one_div_minus_numeral
tff(fact_8838_sin__coeff__def,axiom,
    ! [X3: nat] :
      aa_nat_real(sin_coeff,X3) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),X3),zero_zero_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,X3),aa_nat_nat(suc,zero_zero_nat))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_real(semiri2265585572941072030t_real,X3))) ).

% sin_coeff_def
tff(fact_8839_minus__numeral__div__numeral,axiom,
    ! [M: num,Nb: num] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_num_int(numeral_numeral_int,Nb)) = aa_int_int(uminus_uminus_int,aa_Pro6799258434465214121nt_int(adjust_div,unique5052692396658037445od_int(M,Nb))) ).

% minus_numeral_div_numeral
tff(fact_8840_numeral__div__minus__numeral,axiom,
    ! [M: num,Nb: num] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))) = aa_int_int(uminus_uminus_int,aa_Pro6799258434465214121nt_int(adjust_div,unique5052692396658037445od_int(M,Nb))) ).

% numeral_div_minus_numeral
tff(fact_8841_sin__coeff__Suc,axiom,
    ! [Nb: nat] : aa_nat_real(sin_coeff,aa_nat_nat(suc,Nb)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(cos_coeff,Nb)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Nb))) ).

% sin_coeff_Suc
tff(fact_8842_cos__coeff__Suc,axiom,
    ! [Nb: nat] : aa_nat_real(cos_coeff,aa_nat_nat(suc,Nb)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(uminus_uminus_real,aa_nat_real(sin_coeff,Nb))),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Nb))) ).

% cos_coeff_Suc
tff(fact_8843_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_8844_upto_Opinduct,axiom,
    ! [A0: int,A1: int,P: fun_int_fun_int_o] :
      ( aa_Pro4592068669816069595_int_o(accp_P1096762738010456898nt_int(upto_rel),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A0),A1))
     => ( ! [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(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,I3),one_one_int)),J2) )
             => aa_int_o(aa_int_fun_int_o(P,I3),J2) ) )
       => aa_int_o(aa_int_fun_int_o(P,A0),A1) ) ) ).

% upto.pinduct
tff(fact_8845_signed__take__bit__Suc__bit0,axiom,
    ! [Nb: nat,K: num] : aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,aa_nat_nat(suc,Nb)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,K))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,Nb),aa_num_int(numeral_numeral_int,K))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% signed_take_bit_Suc_bit0
tff(fact_8846_signed__take__bit__Suc__minus__bit0,axiom,
    ! [Nb: nat,K: num] : aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,aa_nat_nat(suc,Nb)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,K)))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,Nb),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K)))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% signed_take_bit_Suc_minus_bit0
tff(fact_8847_signed__take__bit__Suc__bit1,axiom,
    ! [Nb: nat,K: num] : aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,aa_nat_nat(suc,Nb)),aa_num_int(numeral_numeral_int,aa_num_num(bit1,K))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,Nb),aa_num_int(numeral_numeral_int,K))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),one_one_int) ).

% signed_take_bit_Suc_bit1
tff(fact_8848_signed__take__bit__Suc__minus__bit1,axiom,
    ! [Nb: nat,K: num] : aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,aa_nat_nat(suc,Nb)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,K)))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,Nb),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K))),one_one_int))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),one_one_int) ).

% signed_take_bit_Suc_minus_bit1
tff(fact_8849_bot__nat__def,axiom,
    bot_bot_nat = zero_zero_nat ).

% bot_nat_def
tff(fact_8850_signed__take__bit__add,axiom,
    ! [Nb: nat,K: int,L: int] : aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,Nb),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,Nb),K)),aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,Nb),L))) = aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,Nb),aa_int_int(aa_int_fun_int_int(plus_plus_int,K),L)) ).

% signed_take_bit_add
tff(fact_8851_signed__take__bit__diff,axiom,
    ! [Nb: nat,K: int,L: int] : aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,Nb),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,Nb),K)),aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,Nb),L))) = aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,Nb),aa_int_int(aa_int_fun_int_int(minus_minus_int,K),L)) ).

% signed_take_bit_diff
tff(fact_8852_signed__take__bit__mult,axiom,
    ! [Nb: nat,K: int,L: int] : aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,Nb),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,Nb),K)),aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,Nb),L))) = aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,Nb),aa_int_int(aa_int_fun_int_int(times_times_int,K),L)) ).

% signed_take_bit_mult
tff(fact_8853_one__integer__def,axiom,
    one_one_Code_integer = aa_int_Code_integer(code_integer_of_int,one_one_int) ).

% one_integer_def
tff(fact_8854_less__eq__integer_Oabs__eq,axiom,
    ! [Xaa: int,Xb: int] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_int_Code_integer(code_integer_of_int,Xaa)),aa_int_Code_integer(code_integer_of_int,Xb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xaa),Xb) ) ).

% less_eq_integer.abs_eq
tff(fact_8855_plus__integer_Oabs__eq,axiom,
    ! [Xaa: int,Xb: int] : aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_int_Code_integer(code_integer_of_int,Xaa)),aa_int_Code_integer(code_integer_of_int,Xb)) = aa_int_Code_integer(code_integer_of_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Xaa),Xb)) ).

% plus_integer.abs_eq
tff(fact_8856_minus__integer_Oabs__eq,axiom,
    ! [Xaa: int,Xb: int] : aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_int_Code_integer(code_integer_of_int,Xaa)),aa_int_Code_integer(code_integer_of_int,Xb)) = aa_int_Code_integer(code_integer_of_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Xaa),Xb)) ).

% minus_integer.abs_eq
tff(fact_8857_times__integer_Oabs__eq,axiom,
    ! [Xaa: int,Xb: int] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_int_Code_integer(code_integer_of_int,Xaa)),aa_int_Code_integer(code_integer_of_int,Xb)) = aa_int_Code_integer(code_integer_of_int,aa_int_int(aa_int_fun_int_int(times_times_int,Xaa),Xb)) ).

% times_integer.abs_eq
tff(fact_8858_divide__integer_Oabs__eq,axiom,
    ! [Xaa: int,Xb: int] : aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_int_Code_integer(code_integer_of_int,Xaa)),aa_int_Code_integer(code_integer_of_int,Xb)) = aa_int_Code_integer(code_integer_of_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,Xaa),Xb)) ).

% divide_integer.abs_eq
tff(fact_8859_signed__take__bit__int__less__exp,axiom,
    ! [Nb: 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,Nb),K)),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) ).

% signed_take_bit_int_less_exp
tff(fact_8860_signed__take__bit__int__less__self__iff,axiom,
    ! [Nb: 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,Nb),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))),Nb)),K) ) ).

% signed_take_bit_int_less_self_iff
tff(fact_8861_signed__take__bit__int__greater__eq__self__iff,axiom,
    ! [K: int,Nb: 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,Nb),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))),Nb)) ) ).

% signed_take_bit_int_greater_eq_self_iff
tff(fact_8862_signed__take__bit__int__less__eq__self__iff,axiom,
    ! [Nb: 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,Nb),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))),Nb))),K) ) ).

% signed_take_bit_int_less_eq_self_iff
tff(fact_8863_signed__take__bit__int__greater__eq__minus__exp,axiom,
    ! [Nb: 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))),Nb))),aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,Nb),K)) ).

% signed_take_bit_int_greater_eq_minus_exp
tff(fact_8864_signed__take__bit__int__greater__self__iff,axiom,
    ! [K: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,Nb),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))),Nb))) ) ).

% signed_take_bit_int_greater_self_iff
tff(fact_8865_signed__take__bit__int__less__eq,axiom,
    ! [Nb: 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))),Nb)),K)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,Nb),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,Nb)))) ) ).

% signed_take_bit_int_less_eq
tff(fact_8866_signed__take__bit__int__eq__self,axiom,
    ! [Nb: 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))),Nb))),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))),Nb))
       => ( aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,Nb),K) = K ) ) ) ).

% signed_take_bit_int_eq_self
tff(fact_8867_signed__take__bit__int__eq__self__iff,axiom,
    ! [Nb: nat,K: int] :
      ( ( aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,Nb),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))),Nb))),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))),Nb)) ) ) ).

% signed_take_bit_int_eq_self_iff
tff(fact_8868_signed__take__bit__int__greater__eq,axiom,
    ! [K: int,Nb: 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))),Nb)))
     => 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,Nb)))),aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,Nb),K)) ) ).

% signed_take_bit_int_greater_eq
tff(fact_8869_sin__paired,axiom,
    ! [Xb: real] : aa_real_o(sums_real(aa_real_fun_nat_real(aTP_Lamm_azf,Xb)),aa_real_real(sin_real,Xb)) ).

% sin_paired
tff(fact_8870_signed__take__bit__numeral__minus__bit1,axiom,
    ! [L: num,K: num] : aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,aa_num_nat(numeral_numeral_nat,L)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,K)))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,aa_num_nat(pred_numeral,L)),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K))),one_one_int))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),one_one_int) ).

% signed_take_bit_numeral_minus_bit1
tff(fact_8871_cos__paired,axiom,
    ! [Xb: real] : aa_real_o(sums_real(aa_real_fun_nat_real(aTP_Lamm_azg,Xb)),aa_real_real(cos_real,Xb)) ).

% cos_paired
tff(fact_8872_ln__one__minus__pos__lower__bound,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(aa_num_num(bit0,one))))
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,Xb)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),Xb))) ) ) ).

% ln_one_minus_pos_lower_bound
tff(fact_8873_pred__numeral__simps_I1_J,axiom,
    aa_num_nat(pred_numeral,one) = zero_zero_nat ).

% pred_numeral_simps(1)
tff(fact_8874_Suc__eq__numeral,axiom,
    ! [Nb: nat,K: num] :
      ( ( aa_nat_nat(suc,Nb) = aa_num_nat(numeral_numeral_nat,K) )
    <=> ( Nb = aa_num_nat(pred_numeral,K) ) ) ).

% Suc_eq_numeral
tff(fact_8875_eq__numeral__Suc,axiom,
    ! [K: num,Nb: nat] :
      ( ( aa_num_nat(numeral_numeral_nat,K) = aa_nat_nat(suc,Nb) )
    <=> ( aa_num_nat(pred_numeral,K) = Nb ) ) ).

% eq_numeral_Suc
tff(fact_8876_pred__numeral__simps_I3_J,axiom,
    ! [K: num] : aa_num_nat(pred_numeral,aa_num_num(bit1,K)) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,K)) ).

% pred_numeral_simps(3)
tff(fact_8877_less__Suc__numeral,axiom,
    ! [Nb: nat,K: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Nb)),aa_num_nat(numeral_numeral_nat,K))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_num_nat(pred_numeral,K)) ) ).

% less_Suc_numeral
tff(fact_8878_less__numeral__Suc,axiom,
    ! [K: num,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_num_nat(numeral_numeral_nat,K)),aa_nat_nat(suc,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_num_nat(pred_numeral,K)),Nb) ) ).

% less_numeral_Suc
tff(fact_8879_le__Suc__numeral,axiom,
    ! [Nb: nat,K: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Nb)),aa_num_nat(numeral_numeral_nat,K))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),aa_num_nat(pred_numeral,K)) ) ).

% le_Suc_numeral
tff(fact_8880_le__numeral__Suc,axiom,
    ! [K: num,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,K)),aa_nat_nat(suc,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(pred_numeral,K)),Nb) ) ).

% le_numeral_Suc
tff(fact_8881_diff__Suc__numeral,axiom,
    ! [Nb: nat,K: num] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,Nb)),aa_num_nat(numeral_numeral_nat,K)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_num_nat(pred_numeral,K)) ).

% diff_Suc_numeral
tff(fact_8882_diff__numeral__Suc,axiom,
    ! [K: num,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_num_nat(numeral_numeral_nat,K)),aa_nat_nat(suc,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_num_nat(pred_numeral,K)),Nb) ).

% diff_numeral_Suc
tff(fact_8883_ln__le__cancel__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(ln_ln_real,Xb)),aa_real_real(ln_ln_real,Y))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y) ) ) ) ).

% ln_le_cancel_iff
tff(fact_8884_ln__less__zero__iff,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(ln_ln_real,Xb)),zero_zero_real)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),one_one_real) ) ) ).

% ln_less_zero_iff
tff(fact_8885_ln__gt__zero__iff,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(ln_ln_real,Xb))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xb) ) ) ).

% ln_gt_zero_iff
tff(fact_8886_ln__eq__zero__iff,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( ( aa_real_real(ln_ln_real,Xb) = zero_zero_real )
      <=> ( Xb = one_one_real ) ) ) ).

% ln_eq_zero_iff
tff(fact_8887_max__Suc__numeral,axiom,
    ! [Nb: nat,K: num] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(suc,Nb)),aa_num_nat(numeral_numeral_nat,K)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Nb),aa_num_nat(pred_numeral,K))) ).

% max_Suc_numeral
tff(fact_8888_max__numeral__Suc,axiom,
    ! [K: num,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_num_nat(numeral_numeral_nat,K)),aa_nat_nat(suc,Nb)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_num_nat(pred_numeral,K)),Nb)) ).

% max_numeral_Suc
tff(fact_8889_ln__le__zero__iff,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(ln_ln_real,Xb)),zero_zero_real)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),one_one_real) ) ) ).

% ln_le_zero_iff
tff(fact_8890_ln__ge__zero__iff,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(ln_ln_real,Xb))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Xb) ) ) ).

% ln_ge_zero_iff
tff(fact_8891_signed__take__bit__numeral__bit0,axiom,
    ! [L: num,K: num] : aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,aa_num_nat(numeral_numeral_nat,L)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,K))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,aa_num_nat(pred_numeral,L)),aa_num_int(numeral_numeral_int,K))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% signed_take_bit_numeral_bit0
tff(fact_8892_signed__take__bit__numeral__minus__bit0,axiom,
    ! [L: num,K: num] : aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,aa_num_nat(numeral_numeral_nat,L)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,K)))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,aa_num_nat(pred_numeral,L)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K)))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% signed_take_bit_numeral_minus_bit0
tff(fact_8893_signed__take__bit__numeral__bit1,axiom,
    ! [L: num,K: num] : aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,aa_num_nat(numeral_numeral_nat,L)),aa_num_int(numeral_numeral_int,aa_num_num(bit1,K))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,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) ).

% signed_take_bit_numeral_bit1
tff(fact_8894_polar__Ex,axiom,
    ! [Xb: real,Y: real] :
    ? [R4: real,A3: real] :
      ( ( Xb = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R4),aa_real_real(cos_real,A3)) )
      & ( Y = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R4),aa_real_real(sin_real,A3)) ) ) ).

% polar_Ex
tff(fact_8895_sin__x__le__x,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sin_real,Xb)),Xb) ) ).

% sin_x_le_x
tff(fact_8896_sin__le__one,axiom,
    ! [Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sin_real,Xb)),one_one_real) ).

% sin_le_one
tff(fact_8897_cos__le__one,axiom,
    ! [Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(cos_real,Xb)),one_one_real) ).

% cos_le_one
tff(fact_8898_log__def,axiom,
    ! [A: real,Xb: real] : aa_real_real(aa_rea7340605660300375871l_real(log,A),Xb) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(ln_ln_real,Xb)),aa_real_real(ln_ln_real,A)) ).

% log_def
tff(fact_8899_numeral__eq__Suc,axiom,
    ! [K: num] : aa_num_nat(numeral_numeral_nat,K) = aa_nat_nat(suc,aa_num_nat(pred_numeral,K)) ).

% numeral_eq_Suc
tff(fact_8900_ln__bound,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(ln_ln_real,Xb)),Xb) ) ).

% ln_bound
tff(fact_8901_ln__gt__zero__imp__gt__one,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(ln_ln_real,Xb))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xb) ) ) ).

% ln_gt_zero_imp_gt_one
tff(fact_8902_ln__less__zero,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(ln_ln_real,Xb)),zero_zero_real) ) ) ).

% ln_less_zero
tff(fact_8903_ln__gt__zero,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xb)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(ln_ln_real,Xb)) ) ).

% ln_gt_zero
tff(fact_8904_ln__ge__zero,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Xb)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(ln_ln_real,Xb)) ) ).

% ln_ge_zero
tff(fact_8905_ln__powr,axiom,
    ! [Xb: real,Y: real] :
      ( ( Xb != zero_zero_real )
     => ( aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),Y)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),aa_real_real(ln_ln_real,Xb)) ) ) ).

% ln_powr
tff(fact_8906_sin__x__ge__neg__x,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,Xb)),aa_real_real(sin_real,Xb)) ) ).

% sin_x_ge_neg_x
tff(fact_8907_sin__ge__minus__one,axiom,
    ! [Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_real_real(sin_real,Xb)) ).

% sin_ge_minus_one
tff(fact_8908_cos__ge__minus__one,axiom,
    ! [Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_real_real(cos_real,Xb)) ).

% cos_ge_minus_one
tff(fact_8909_pred__numeral__def,axiom,
    ! [K: num] : aa_num_nat(pred_numeral,K) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_num_nat(numeral_numeral_nat,K)),one_one_nat) ).

% pred_numeral_def
tff(fact_8910_lessThan__nat__numeral,axiom,
    ! [K: num] : set_ord_lessThan_nat(aa_num_nat(numeral_numeral_nat,K)) = insert_nat(aa_num_nat(pred_numeral,K),set_ord_lessThan_nat(aa_num_nat(pred_numeral,K))) ).

% lessThan_nat_numeral
tff(fact_8911_atMost__nat__numeral,axiom,
    ! [K: num] : set_ord_atMost_nat(aa_num_nat(numeral_numeral_nat,K)) = insert_nat(aa_num_nat(numeral_numeral_nat,K),set_ord_atMost_nat(aa_num_nat(pred_numeral,K))) ).

% atMost_nat_numeral
tff(fact_8912_ln__ge__zero__imp__ge__one,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(ln_ln_real,Xb))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Xb) ) ) ).

% ln_ge_zero_imp_ge_one
tff(fact_8913_ln__add__one__self__le__self,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),Xb))),Xb) ) ).

% ln_add_one_self_le_self
tff(fact_8914_ln__mult,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
       => ( aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Y)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(ln_ln_real,Xb)),aa_real_real(ln_ln_real,Y)) ) ) ) ).

% ln_mult
tff(fact_8915_ln__eq__minus__one,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( ( aa_real_real(ln_ln_real,Xb) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),one_one_real) )
       => ( Xb = one_one_real ) ) ) ).

% ln_eq_minus_one
tff(fact_8916_ln__div,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
       => ( aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Y)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(ln_ln_real,Xb)),aa_real_real(ln_ln_real,Y)) ) ) ) ).

% ln_div
tff(fact_8917_cos__two__neq__zero,axiom,
    aa_real_real(cos_real,numeral_numeral_real(aa_num_num(bit0,one))) != zero_zero_real ).

% cos_two_neq_zero
tff(fact_8918_ln__2__less__1,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(ln_ln_real,numeral_numeral_real(aa_num_num(bit0,one)))),one_one_real) ).

% ln_2_less_1
tff(fact_8919_ln__le__minus__one,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(ln_ln_real,Xb)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),one_one_real)) ) ).

% ln_le_minus_one
tff(fact_8920_ln__diff__le,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(ln_ln_real,Xb)),aa_real_real(ln_ln_real,Y))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),Y)),Y)) ) ) ).

% ln_diff_le
tff(fact_8921_ln__add__one__self__le__self2,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),Xb)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),Xb))),Xb) ) ).

% ln_add_one_self_le_self2
tff(fact_8922_ln__realpow,axiom,
    ! [Xb: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_real(ln_ln_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_real_real(ln_ln_real,Xb)) ) ) ).

% ln_realpow
tff(fact_8923_sin__gt__zero__02,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),numeral_numeral_real(aa_num_num(bit0,one)))
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(sin_real,Xb)) ) ) ).

% sin_gt_zero_02
tff(fact_8924_cos__two__less__zero,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(cos_real,numeral_numeral_real(aa_num_num(bit0,one)))),zero_zero_real) ).

% cos_two_less_zero
tff(fact_8925_cos__two__le__zero,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(cos_real,numeral_numeral_real(aa_num_num(bit0,one)))),zero_zero_real) ).

% cos_two_le_zero
tff(fact_8926_cos__is__zero,axiom,
    ? [X2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X2)
      & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X2),numeral_numeral_real(aa_num_num(bit0,one)))
      & ( aa_real_real(cos_real,X2) = zero_zero_real )
      & ! [Y5: real] :
          ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y5)
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y5),numeral_numeral_real(aa_num_num(bit0,one)))
            & ( aa_real_real(cos_real,Y5) = zero_zero_real ) )
         => ( Y5 = X2 ) ) ) ).

% cos_is_zero
tff(fact_8927_ln__one__minus__pos__upper__bound,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),Xb))),aa_real_real(uminus_uminus_real,Xb)) ) ) ).

% ln_one_minus_pos_upper_bound
tff(fact_8928_ln__powr__bound,axiom,
    ! [Xb: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(ln_ln_real,Xb)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),A)),A)) ) ) ).

% ln_powr_bound
tff(fact_8929_ln__powr__bound2,axiom,
    ! [Xb: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,aa_real_real(ln_ln_real,Xb)),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,A),A)),Xb)) ) ) ).

% ln_powr_bound2
tff(fact_8930_log__eq__div__ln__mult__log,axiom,
    ! [A: real,B: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( ( A != one_one_real )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B)
         => ( ( B != one_one_real )
           => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
             => ( aa_real_real(aa_rea7340605660300375871l_real(log,A),Xb) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(ln_ln_real,B)),aa_real_real(ln_ln_real,A))),aa_real_real(aa_rea7340605660300375871l_real(log,B),Xb)) ) ) ) ) ) ) ).

% log_eq_div_ln_mult_log
tff(fact_8931_cos__double__less__one,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),numeral_numeral_real(aa_num_num(bit0,one)))
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),Xb))),one_one_real) ) ) ).

% cos_double_less_one
tff(fact_8932_ln__one__plus__pos__lower__bound,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),Xb))) ) ) ).

% ln_one_plus_pos_lower_bound
tff(fact_8933_real__scaleR__def,axiom,
    ! [A: real,Xb: real] : aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,A),Xb) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),Xb) ).

% real_scaleR_def
tff(fact_8934_Maclaurin__minus__cos__expansion,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),zero_zero_real)
       => ? [T3: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),T3)
            & aa_real_o(aa_real_fun_real_o(ord_less_real,T3),zero_zero_real)
            & ( aa_real_real(cos_real,Xb) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_azh,Xb),set_ord_lessThan_nat(Nb))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,T3),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(aa_num_num(bit0,one)))),aa_nat_real(semiri5074537144036343181t_real,Nb))),pi)))),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb))) ) ) ) ) ).

% Maclaurin_minus_cos_expansion
tff(fact_8935_Maclaurin__cos__expansion2,axiom,
    ! [Xb: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => ? [T3: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),T3)
            & aa_real_o(aa_real_fun_real_o(ord_less_real,T3),Xb)
            & ( aa_real_real(cos_real,Xb) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_azh,Xb),set_ord_lessThan_nat(Nb))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,T3),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(aa_num_num(bit0,one)))),aa_nat_real(semiri5074537144036343181t_real,Nb))),pi)))),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb))) ) ) ) ) ).

% Maclaurin_cos_expansion2
tff(fact_8936_Maclaurin__sin__expansion3,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
       => ? [T3: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),T3)
            & aa_real_o(aa_real_fun_real_o(ord_less_real,T3),Xb)
            & ( aa_real_real(sin_real,Xb) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_azi,Xb),set_ord_lessThan_nat(Nb))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,T3),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(aa_num_num(bit0,one)))),aa_nat_real(semiri5074537144036343181t_real,Nb))),pi)))),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb))) ) ) ) ) ).

% Maclaurin_sin_expansion3
tff(fact_8937_Maclaurin__sin__expansion4,axiom,
    ! [Xb: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ? [T3: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),T3)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T3),Xb)
          & ( aa_real_real(sin_real,Xb) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_azi,Xb),set_ord_lessThan_nat(Nb))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,T3),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(aa_num_num(bit0,one)))),aa_nat_real(semiri5074537144036343181t_real,Nb))),pi)))),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb))) ) ) ) ).

% Maclaurin_sin_expansion4
tff(fact_8938_sin__pi__minus,axiom,
    ! [Xb: real] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,pi),Xb)) = aa_real_real(sin_real,Xb) ).

% sin_pi_minus
tff(fact_8939_cos__pi,axiom,
    aa_real_real(cos_real,pi) = aa_real_real(uminus_uminus_real,one_one_real) ).

% cos_pi
tff(fact_8940_cos__periodic__pi,axiom,
    ! [Xb: real] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),pi)) = aa_real_real(uminus_uminus_real,aa_real_real(cos_real,Xb)) ).

% cos_periodic_pi
tff(fact_8941_cos__periodic__pi2,axiom,
    ! [Xb: real] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,pi),Xb)) = aa_real_real(uminus_uminus_real,aa_real_real(cos_real,Xb)) ).

% cos_periodic_pi2
tff(fact_8942_sin__periodic__pi,axiom,
    ! [Xb: real] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),pi)) = aa_real_real(uminus_uminus_real,aa_real_real(sin_real,Xb)) ).

% sin_periodic_pi
tff(fact_8943_sin__periodic__pi2,axiom,
    ! [Xb: real] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,pi),Xb)) = aa_real_real(uminus_uminus_real,aa_real_real(sin_real,Xb)) ).

% sin_periodic_pi2
tff(fact_8944_cos__pi__minus,axiom,
    ! [Xb: real] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,pi),Xb)) = aa_real_real(uminus_uminus_real,aa_real_real(cos_real,Xb)) ).

% cos_pi_minus
tff(fact_8945_cos__minus__pi,axiom,
    ! [Xb: real] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),pi)) = aa_real_real(uminus_uminus_real,aa_real_real(cos_real,Xb)) ).

% cos_minus_pi
tff(fact_8946_sin__minus__pi,axiom,
    ! [Xb: real] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),pi)) = aa_real_real(uminus_uminus_real,aa_real_real(sin_real,Xb)) ).

% sin_minus_pi
tff(fact_8947_sin__npi2,axiom,
    ! [Nb: nat] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,pi),aa_nat_real(semiri5074537144036343181t_real,Nb))) = zero_zero_real ).

% sin_npi2
tff(fact_8948_sin__npi,axiom,
    ! [Nb: nat] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),pi)) = zero_zero_real ).

% sin_npi
tff(fact_8949_sin__npi__int,axiom,
    ! [Nb: int] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,pi),aa_int_real(ring_1_of_int_real,Nb))) = zero_zero_real ).

% sin_npi_int
tff(fact_8950_cos__pi__half,axiom,
    aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one)))) = zero_zero_real ).

% cos_pi_half
tff(fact_8951_sin__two__pi,axiom,
    aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),pi)) = zero_zero_real ).

% sin_two_pi
tff(fact_8952_cos__two__pi,axiom,
    aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),pi)) = one_one_real ).

% cos_two_pi
tff(fact_8953_sin__pi__half,axiom,
    aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one)))) = one_one_real ).

% sin_pi_half
tff(fact_8954_cos__periodic,axiom,
    ! [Xb: real] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),pi))) = aa_real_real(cos_real,Xb) ).

% cos_periodic
tff(fact_8955_sin__periodic,axiom,
    ! [Xb: real] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),pi))) = aa_real_real(sin_real,Xb) ).

% sin_periodic
tff(fact_8956_cos__2pi__minus,axiom,
    ! [Xb: real] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),pi)),Xb)) = aa_real_real(cos_real,Xb) ).

% cos_2pi_minus
tff(fact_8957_cos__npi2,axiom,
    ! [Nb: nat] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,pi),aa_nat_real(semiri5074537144036343181t_real,Nb))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Nb) ).

% cos_npi2
tff(fact_8958_cos__npi,axiom,
    ! [Nb: nat] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),pi)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Nb) ).

% cos_npi
tff(fact_8959_sin__2npi,axiom,
    ! [Nb: nat] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,Nb))),pi)) = zero_zero_real ).

% sin_2npi
tff(fact_8960_cos__2npi,axiom,
    ! [Nb: nat] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,Nb))),pi)) = one_one_real ).

% cos_2npi
tff(fact_8961_sin__2pi__minus,axiom,
    ! [Xb: real] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),pi)),Xb)) = aa_real_real(uminus_uminus_real,aa_real_real(sin_real,Xb)) ).

% sin_2pi_minus
tff(fact_8962_sin__int__2pin,axiom,
    ! [Nb: int] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),pi)),aa_int_real(ring_1_of_int_real,Nb))) = zero_zero_real ).

% sin_int_2pin
tff(fact_8963_cos__int__2pin,axiom,
    ! [Nb: int] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),pi)),aa_int_real(ring_1_of_int_real,Nb))) = one_one_real ).

% cos_int_2pin
tff(fact_8964_cos__3over2__pi,axiom,
    aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,numeral_numeral_real(aa_num_num(bit1,one))),numeral_numeral_real(aa_num_num(bit0,one)))),pi)) = zero_zero_real ).

% cos_3over2_pi
tff(fact_8965_sin__3over2__pi,axiom,
    aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,numeral_numeral_real(aa_num_num(bit1,one))),numeral_numeral_real(aa_num_num(bit0,one)))),pi)) = aa_real_real(uminus_uminus_real,one_one_real) ).

% sin_3over2_pi
tff(fact_8966_cos__npi__int,axiom,
    ! [Nb: int] :
      aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,pi),aa_int_real(ring_1_of_int_real,Nb))) = $ite(aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb),one_one_real,aa_real_real(uminus_uminus_real,one_one_real)) ).

% cos_npi_int
tff(fact_8967_cos__pi__eq__zero,axiom,
    ! [M: nat] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,pi),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M))))),numeral_numeral_real(aa_num_num(bit0,one)))) = zero_zero_real ).

% cos_pi_eq_zero
tff(fact_8968_sin__cos__npi,axiom,
    ! [Nb: nat] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)))),pi)),numeral_numeral_real(aa_num_num(bit0,one)))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Nb) ).

% sin_cos_npi
tff(fact_8969_pi__ge__zero,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),pi) ).

% pi_ge_zero
tff(fact_8970_cos__monotone__0__pi__le,axiom,
    ! [Y: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),Xb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),pi)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(cos_real,Xb)),aa_real_real(cos_real,Y)) ) ) ) ).

% cos_monotone_0_pi_le
tff(fact_8971_cos__mono__le__eq,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),pi)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),pi)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(cos_real,Xb)),aa_real_real(cos_real,Y))
            <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),Xb) ) ) ) ) ) ).

% cos_mono_le_eq
tff(fact_8972_cos__inj__pi,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),pi)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),pi)
           => ( ( aa_real_real(cos_real,Xb) = aa_real_real(cos_real,Y) )
             => ( Xb = Y ) ) ) ) ) ) ).

% cos_inj_pi
tff(fact_8973_sin__ge__zero,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),pi)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(sin_real,Xb)) ) ) ).

% sin_ge_zero
tff(fact_8974_pi__less__4,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_real,pi),numeral_numeral_real(aa_num_num(bit0,aa_num_num(bit0,one)))) ).

% pi_less_4
tff(fact_8975_pi__ge__two,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_eq_real,numeral_numeral_real(aa_num_num(bit0,one))),pi) ).

% pi_ge_two
tff(fact_8976_pi__half__neq__two,axiom,
    aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))) != numeral_numeral_real(aa_num_num(bit0,one)) ).

% pi_half_neq_two
tff(fact_8977_cos__mono__less__eq,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),pi)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),pi)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(cos_real,Xb)),aa_real_real(cos_real,Y))
            <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Xb) ) ) ) ) ) ).

% cos_mono_less_eq
tff(fact_8978_cos__monotone__0__pi,axiom,
    ! [Y: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Xb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),pi)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(cos_real,Xb)),aa_real_real(cos_real,Y)) ) ) ) ).

% cos_monotone_0_pi
tff(fact_8979_cos__monotone__minus__pi__0_H,axiom,
    ! [Y: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,pi)),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),Xb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(cos_real,Y)),aa_real_real(cos_real,Xb)) ) ) ) ).

% cos_monotone_minus_pi_0'
tff(fact_8980_sin__zero__iff__int2,axiom,
    ! [Xb: real] :
      ( ( aa_real_real(sin_real,Xb) = zero_zero_real )
    <=> ? [I2: int] : Xb = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(ring_1_of_int_real,I2)),pi) ) ).

% sin_zero_iff_int2
tff(fact_8981_pi__half__neq__zero,axiom,
    aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))) != zero_zero_real ).

% pi_half_neq_zero
tff(fact_8982_pi__half__less__two,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one)))),numeral_numeral_real(aa_num_num(bit0,one))) ).

% pi_half_less_two
tff(fact_8983_pi__half__le__two,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one)))),numeral_numeral_real(aa_num_num(bit0,one))) ).

% pi_half_le_two
tff(fact_8984_cos__monotone__minus__pi__0,axiom,
    ! [Y: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,pi)),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Xb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(cos_real,Y)),aa_real_real(cos_real,Xb)) ) ) ) ).

% cos_monotone_minus_pi_0
tff(fact_8985_cos__total,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),one_one_real)
       => ? [X2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X2)
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X2),pi)
            & ( aa_real_real(cos_real,X2) = Y )
            & ! [Y5: real] :
                ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y5)
                  & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y5),pi)
                  & ( aa_real_real(cos_real,Y5) = Y ) )
               => ( Y5 = X2 ) ) ) ) ) ).

% cos_total
tff(fact_8986_sincos__principal__value,axiom,
    ! [Xb: real] :
    ? [Y4: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,pi)),Y4)
      & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y4),pi)
      & ( aa_real_real(sin_real,Y4) = aa_real_real(sin_real,Xb) )
      & ( aa_real_real(cos_real,Y4) = aa_real_real(cos_real,Xb) ) ) ).

% sincos_principal_value
tff(fact_8987_pi__half__gt__zero,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one)))) ).

% pi_half_gt_zero
tff(fact_8988_pi__half__ge__zero,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one)))) ).

% pi_half_ge_zero
tff(fact_8989_m2pi__less__pi,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),pi))),pi) ).

% m2pi_less_pi
tff(fact_8990_sin__pi__divide__n__ge__0,axiom,
    ! [Nb: nat] :
      ( ( Nb != zero_zero_nat )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_nat_real(semiri5074537144036343181t_real,Nb)))) ) ).

% sin_pi_divide_n_ge_0
tff(fact_8991_minus__pi__half__less__zero,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))),zero_zero_real) ).

% minus_pi_half_less_zero
tff(fact_8992_cos__gt__zero,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(cos_real,Xb)) ) ) ).

% cos_gt_zero
tff(fact_8993_sin__gt__zero2,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(sin_real,Xb)) ) ) ).

% sin_gt_zero2
tff(fact_8994_sin__lt__zero,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,pi),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),pi))
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sin_real,Xb)),zero_zero_real) ) ) ).

% sin_lt_zero
tff(fact_8995_cos__60,axiom,
    aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit1,one)))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(aa_num_num(bit0,one))) ).

% cos_60
tff(fact_8996_sin__30,axiom,
    aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,aa_num_num(bit1,one))))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(aa_num_num(bit0,one))) ).

% sin_30
tff(fact_8997_sin__monotone__2pi__le,axiom,
    ! [Y: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),Xb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sin_real,Y)),aa_real_real(sin_real,Xb)) ) ) ) ).

% sin_monotone_2pi_le
tff(fact_8998_sin__mono__le__eq,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))),Y)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sin_real,Xb)),aa_real_real(sin_real,Y))
            <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y) ) ) ) ) ) ).

% sin_mono_le_eq
tff(fact_8999_sin__inj__pi,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))),Y)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))
           => ( ( aa_real_real(sin_real,Xb) = aa_real_real(sin_real,Y) )
             => ( Xb = Y ) ) ) ) ) ) ).

% sin_inj_pi
tff(fact_9000_cos__one__2pi__int,axiom,
    ! [Xb: real] :
      ( ( aa_real_real(cos_real,Xb) = one_one_real )
    <=> ? [X: int] : Xb = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(ring_1_of_int_real,X)),numeral_numeral_real(aa_num_num(bit0,one)))),pi) ) ).

% cos_one_2pi_int
tff(fact_9001_sin__le__zero,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,pi),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),pi))
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sin_real,Xb)),zero_zero_real) ) ) ).

% sin_le_zero
tff(fact_9002_cos__gt__zero__pi,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(cos_real,Xb)) ) ) ).

% cos_gt_zero_pi
tff(fact_9003_sin__less__zero,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(uminus_uminus_real,pi)),numeral_numeral_real(aa_num_num(bit0,one)))),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sin_real,Xb)),zero_zero_real) ) ) ).

% sin_less_zero
tff(fact_9004_cos__ge__zero,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(cos_real,Xb)) ) ) ).

% cos_ge_zero
tff(fact_9005_sin__monotone__2pi,axiom,
    ! [Y: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Xb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sin_real,Y)),aa_real_real(sin_real,Xb)) ) ) ) ).

% sin_monotone_2pi
tff(fact_9006_sin__mono__less__eq,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))),Y)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))
           => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sin_real,Xb)),aa_real_real(sin_real,Y))
            <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y) ) ) ) ) ) ).

% sin_mono_less_eq
tff(fact_9007_sin__total,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),one_one_real)
       => ? [X2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))),X2)
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X2),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))
            & ( aa_real_real(sin_real,X2) = Y )
            & ! [Y5: real] :
                ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))),Y5)
                  & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y5),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))
                  & ( aa_real_real(sin_real,Y5) = Y ) )
               => ( Y5 = X2 ) ) ) ) ) ).

% sin_total
tff(fact_9008_cos__one__2pi,axiom,
    ! [Xb: real] :
      ( ( aa_real_real(cos_real,Xb) = one_one_real )
    <=> ( ? [X: nat] : Xb = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,X)),numeral_numeral_real(aa_num_num(bit0,one)))),pi)
        | ? [X: nat] : Xb = aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,X)),numeral_numeral_real(aa_num_num(bit0,one)))),pi)) ) ) ).

% cos_one_2pi
tff(fact_9009_sin__pi__divide__n__gt__0,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_nat_real(semiri5074537144036343181t_real,Nb)))) ) ).

% sin_pi_divide_n_gt_0
tff(fact_9010_sincos__total__pi,axiom,
    ! [Y: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
     => ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = one_one_real )
       => ? [T3: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),T3)
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T3),pi)
            & ( Xb = aa_real_real(cos_real,T3) )
            & ( Y = aa_real_real(sin_real,T3) ) ) ) ) ).

% sincos_total_pi
tff(fact_9011_sin__expansion__lemma,axiom,
    ! [Xb: real,M: nat] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,M))),pi)),numeral_numeral_real(aa_num_num(bit0,one))))) = aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,M)),pi)),numeral_numeral_real(aa_num_num(bit0,one))))) ).

% sin_expansion_lemma
tff(fact_9012_cos__zero__iff__int,axiom,
    ! [Xb: real] :
      ( ( aa_real_real(cos_real,Xb) = zero_zero_real )
    <=> ? [I2: int] :
          ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),I2)
          & ( Xb = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(ring_1_of_int_real,I2)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one)))) ) ) ) ).

% cos_zero_iff_int
tff(fact_9013_sin__zero__iff__int,axiom,
    ! [Xb: real] :
      ( ( aa_real_real(sin_real,Xb) = zero_zero_real )
    <=> ? [I2: int] :
          ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),I2)
          & ( Xb = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(ring_1_of_int_real,I2)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one)))) ) ) ) ).

% sin_zero_iff_int
tff(fact_9014_cos__zero__lemma,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( ( aa_real_real(cos_real,Xb) = zero_zero_real )
       => ? [N2: nat] :
            ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2)
            & ( Xb = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N2)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one)))) ) ) ) ) ).

% cos_zero_lemma
tff(fact_9015_sin__zero__lemma,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( ( aa_real_real(sin_real,Xb) = zero_zero_real )
       => ? [N2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2)
            & ( Xb = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N2)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one)))) ) ) ) ) ).

% sin_zero_lemma
tff(fact_9016_cos__zero__iff,axiom,
    ! [Xb: real] :
      ( ( aa_real_real(cos_real,Xb) = zero_zero_real )
    <=> ( ? [N6: nat] :
            ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N6)
            & ( Xb = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N6)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one)))) ) )
        | ? [N6: nat] :
            ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N6)
            & ( Xb = aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N6)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))) ) ) ) ) ).

% cos_zero_iff
tff(fact_9017_sin__zero__iff,axiom,
    ! [Xb: real] :
      ( ( aa_real_real(sin_real,Xb) = zero_zero_real )
    <=> ( ? [N6: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N6)
            & ( Xb = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N6)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one)))) ) )
        | ? [N6: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N6)
            & ( Xb = aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N6)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))) ) ) ) ) ).

% sin_zero_iff
tff(fact_9018_cos__expansion__lemma,axiom,
    ! [Xb: real,M: nat] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,M))),pi)),numeral_numeral_real(aa_num_num(bit0,one))))) = aa_real_real(uminus_uminus_real,aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,M)),pi)),numeral_numeral_real(aa_num_num(bit0,one)))))) ).

% cos_expansion_lemma
tff(fact_9019_sincos__total__pi__half,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
       => ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = one_one_real )
         => ? [T3: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),T3)
              & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T3),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))
              & ( Xb = aa_real_real(cos_real,T3) )
              & ( Y = aa_real_real(sin_real,T3) ) ) ) ) ) ).

% sincos_total_pi_half
tff(fact_9020_sincos__total__2pi__le,axiom,
    ! [Xb: real,Y: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = one_one_real )
     => ? [T3: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),T3)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T3),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),pi))
          & ( Xb = aa_real_real(cos_real,T3) )
          & ( Y = aa_real_real(sin_real,T3) ) ) ) ).

% sincos_total_2pi_le
tff(fact_9021_sincos__total__2pi,axiom,
    ! [Xb: real,Y: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = one_one_real )
     => ~ ! [T3: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),T3)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_real,T3),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),pi))
             => ( ( Xb = aa_real_real(cos_real,T3) )
               => ( Y != aa_real_real(sin_real,T3) ) ) ) ) ) ).

% sincos_total_2pi
tff(fact_9022_Maclaurin__sin__expansion,axiom,
    ! [Xb: real,Nb: nat] :
    ? [T3: real] : aa_real_real(sin_real,Xb) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_azi,Xb),set_ord_lessThan_nat(Nb))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,T3),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(aa_num_num(bit0,one)))),aa_nat_real(semiri5074537144036343181t_real,Nb))),pi)))),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb))) ).

% Maclaurin_sin_expansion
tff(fact_9023_Maclaurin__cos__expansion,axiom,
    ! [Xb: real,Nb: nat] :
    ? [T3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,T3)),aa_real_real(abs_abs_real,Xb))
      & ( aa_real_real(cos_real,Xb) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_azh,Xb),set_ord_lessThan_nat(Nb))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,T3),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(aa_num_num(bit0,one)))),aa_nat_real(semiri5074537144036343181t_real,Nb))),pi)))),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb))) ) ) ).

% Maclaurin_cos_expansion
tff(fact_9024_Maclaurin__sin__expansion2,axiom,
    ! [Xb: real,Nb: nat] :
    ? [T3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,T3)),aa_real_real(abs_abs_real,Xb))
      & ( aa_real_real(sin_real,Xb) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_azi,Xb),set_ord_lessThan_nat(Nb))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,T3),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(aa_num_num(bit0,one)))),aa_nat_real(semiri5074537144036343181t_real,Nb))),pi)))),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb))) ) ) ).

% Maclaurin_sin_expansion2
tff(fact_9025_ln__series,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),numeral_numeral_real(aa_num_num(bit0,one)))
       => ( aa_real_real(ln_ln_real,Xb) = suminf_real(aa_real_fun_nat_real(aTP_Lamm_azj,Xb)) ) ) ) ).

% ln_series
tff(fact_9026_abs__ln__one__plus__x__minus__x__bound__nonpos,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(aa_num_num(bit0,one))))),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),Xb))),Xb))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ) ).

% abs_ln_one_plus_x_minus_x_bound_nonpos
tff(fact_9027_artanh__minus__real,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,Xb)),one_one_real)
     => ( aa_real_real(artanh_real,aa_real_real(uminus_uminus_real,Xb)) = aa_real_real(uminus_uminus_real,aa_real_real(artanh_real,Xb)) ) ) ).

% artanh_minus_real
tff(fact_9028_square__powr__half,axiom,
    ! [Xb: real] : aa_real_real(aa_rea7340605660300375871l_real(powr_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(aa_num_num(bit0,one)))) = aa_real_real(abs_abs_real,Xb) ).

% square_powr_half
tff(fact_9029_abs__sin__x__le__abs__x,axiom,
    ! [Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(sin_real,Xb))),aa_real_real(abs_abs_real,Xb)) ).

% abs_sin_x_le_abs_x
tff(fact_9030_lemma__interval__lt,axiom,
    ! [A: real,Xb: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),B)
       => ? [D2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D2)
            & ! [Y5: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),Y5))),D2)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),Y5)
                  & aa_real_o(aa_real_fun_real_o(ord_less_real,Y5),B) ) ) ) ) ) ).

% lemma_interval_lt
tff(fact_9031_abs__cos__le__one,axiom,
    ! [Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(cos_real,Xb))),one_one_real) ).

% abs_cos_le_one
tff(fact_9032_abs__sin__le__one,axiom,
    ! [Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(sin_real,Xb))),one_one_real) ).

% abs_sin_le_one
tff(fact_9033_sin__bound__lemma,axiom,
    ! [Xb: real,Y: real,U: real,V: real] :
      ( ( Xb = Y )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,U)),V)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),U)),Y))),V) ) ) ).

% sin_bound_lemma
tff(fact_9034_lemma__interval,axiom,
    ! [A: real,Xb: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),B)
       => ? [D2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D2)
            & ! [Y5: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),Y5))),D2)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),Y5)
                  & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y5),B) ) ) ) ) ) ).

% lemma_interval
tff(fact_9035_sin__zero__abs__cos__one,axiom,
    ! [Xb: real] :
      ( ( aa_real_real(sin_real,Xb) = zero_zero_real )
     => ( aa_real_real(abs_abs_real,aa_real_real(cos_real,Xb)) = one_one_real ) ) ).

% sin_zero_abs_cos_one
tff(fact_9036_sin__cos__le1,axiom,
    ! [Xb: real,Y: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sin_real,Xb)),aa_real_real(sin_real,Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(cos_real,Xb)),aa_real_real(cos_real,Y))))),one_one_real) ).

% sin_cos_le1
tff(fact_9037_abs__ln__one__plus__x__minus__x__bound__nonneg,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),Xb))),Xb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ) ).

% abs_ln_one_plus_x_minus_x_bound_nonneg
tff(fact_9038_abs__ln__one__plus__x__minus__x__bound,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xb)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(aa_num_num(bit0,one))))
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),Xb))),Xb))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ).

% abs_ln_one_plus_x_minus_x_bound
tff(fact_9039_pi__series,axiom,
    aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,aa_num_num(bit0,one)))) = suminf_real(aTP_Lamm_azk) ).

% pi_series
tff(fact_9040_monoseq__arctan__series,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xb)),one_one_real)
     => topolo6980174941875973593q_real(aa_real_fun_nat_real(aTP_Lamm_azl,Xb)) ) ).

% monoseq_arctan_series
tff(fact_9041_arctan__series,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xb)),one_one_real)
     => ( aa_real_real(arctan,Xb) = suminf_real(aa_real_fun_nat_real(aTP_Lamm_azm,Xb)) ) ) ).

% arctan_series
tff(fact_9042_summable__arctan__series,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xb)),one_one_real)
     => summable_real(aa_real_fun_nat_real(aTP_Lamm_azm,Xb)) ) ).

% summable_arctan_series
tff(fact_9043_tanh__ln__real,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_real(tanh_real,aa_real_real(ln_ln_real,Xb)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real)) ) ) ).

% tanh_ln_real
tff(fact_9044_zdvd1__eq,axiom,
    ! [Xb: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Xb),one_one_int)
    <=> ( aa_int_int(abs_abs_int,Xb) = one_one_int ) ) ).

% zdvd1_eq
tff(fact_9045_tanh__real__le__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(tanh_real,Xb)),aa_real_real(tanh_real,Y))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y) ) ).

% tanh_real_le_iff
tff(fact_9046_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_9047_zero__le__arctan__iff,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(arctan,Xb))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb) ) ).

% zero_le_arctan_iff
tff(fact_9048_arctan__le__zero__iff,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arctan,Xb)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),zero_zero_real) ) ).

% arctan_le_zero_iff
tff(fact_9049_tanh__real__nonpos__iff,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(tanh_real,Xb)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),zero_zero_real) ) ).

% tanh_real_nonpos_iff
tff(fact_9050_tanh__real__nonneg__iff,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(tanh_real,Xb))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb) ) ).

% tanh_real_nonneg_iff
tff(fact_9051_summable__rabs__cancel,axiom,
    ! [F: fun_nat_real] :
      ( summable_real(aTP_Lamm_azn(F))
     => summable_real(F) ) ).

% summable_rabs_cancel
tff(fact_9052_arctan__le__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arctan,Xb)),aa_real_real(arctan,Y))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y) ) ).

% arctan_le_iff
tff(fact_9053_arctan__monotone_H,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arctan,Xb)),aa_real_real(arctan,Y)) ) ).

% arctan_monotone'
tff(fact_9054_abs__zmult__eq__1,axiom,
    ! [M: int,Nb: int] :
      ( ( aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(times_times_int,M),Nb)) = one_one_int )
     => ( aa_int_int(abs_abs_int,M) = one_one_int ) ) ).

% abs_zmult_eq_1
tff(fact_9055_abs__div,axiom,
    ! [Y: int,Xb: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Y),Xb)
     => ( aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,Xb),Y)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(abs_abs_int,Xb)),aa_int_int(abs_abs_int,Y)) ) ) ).

% abs_div
tff(fact_9056_infinite__int__iff__unbounded__le,axiom,
    ! [S2: set_int] :
      ( ~ aa_set_int_o(finite_finite_int,S2)
    <=> ! [M7: int] :
        ? [N6: int] :
          ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,M7),aa_int_int(abs_abs_int,N6))
          & aa_set_int_o(member_int(N6),S2) ) ) ).

% infinite_int_iff_unbounded_le
tff(fact_9057_tanh__real__lt__1,axiom,
    ! [Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(tanh_real,Xb)),one_one_real) ).

% tanh_real_lt_1
tff(fact_9058_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_9059_nat__abs__mult__distrib,axiom,
    ! [W: int,Z: int] : aa_int_nat(nat2,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(times_times_int,W),Z))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,W))),aa_int_nat(nat2,aa_int_int(abs_abs_int,Z))) ).

% nat_abs_mult_distrib
tff(fact_9060_dvd__imp__le__int,axiom,
    ! [I: int,D: int] :
      ( ( I != zero_zero_int )
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),I)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,D)),aa_int_int(abs_abs_int,I)) ) ) ).

% dvd_imp_le_int
tff(fact_9061_summable__rabs__comparison__test,axiom,
    ! [F: fun_nat_real,G: fun_nat_real] :
      ( ? [N8: nat] :
        ! [N2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N8),N2)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_nat_real(F,N2))),aa_nat_real(G,N2)) )
     => ( summable_real(G)
       => summable_real(aTP_Lamm_azn(F)) ) ) ).

% summable_rabs_comparison_test
tff(fact_9062_summable__rabs,axiom,
    ! [F: fun_nat_real] :
      ( summable_real(aTP_Lamm_azn(F))
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,suminf_real(F))),suminf_real(aTP_Lamm_azn(F))) ) ).

% summable_rabs
tff(fact_9063_tanh__real__gt__neg1,axiom,
    ! [Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_real_real(tanh_real,Xb)) ).

% tanh_real_gt_neg1
tff(fact_9064_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_9065_zdvd__mult__cancel1,axiom,
    ! [M: int,Nb: int] :
      ( ( M != zero_zero_int )
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,M),Nb)),M)
      <=> ( aa_int_int(abs_abs_int,Nb) = one_one_int ) ) ) ).

% zdvd_mult_cancel1
tff(fact_9066_div__abs__eq__div__nat,axiom,
    ! [K: int,L: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(abs_abs_int,K)),aa_int_int(abs_abs_int,L)) = aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,K))),aa_int_nat(nat2,aa_int_int(abs_abs_int,L)))) ).

% div_abs_eq_div_nat
tff(fact_9067_even__abs__add__iff,axiom,
    ! [K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(abs_abs_int,K)),L))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(plus_plus_int,K),L)) ) ).

% even_abs_add_iff
tff(fact_9068_even__add__abs__iff,axiom,
    ! [K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(plus_plus_int,K),aa_int_int(abs_abs_int,L)))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(plus_plus_int,K),L)) ) ).

% even_add_abs_iff
tff(fact_9069_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_9070_monoseq__realpow,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),one_one_real)
       => topolo6980174941875973593q_real(aa_real_fun_nat_real(power_power_real,Xb)) ) ) ).

% monoseq_realpow
tff(fact_9071_summable__power__series,axiom,
    ! [F: fun_nat_real,Z: real] :
      ( ! [I3: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(F,I3)),one_one_real)
     => ( ! [I3: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(F,I3))
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Z)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z),one_one_real)
           => summable_real(aa_real_fun_nat_real(aTP_Lamm_azo(F),Z)) ) ) ) ) ).

% summable_power_series
tff(fact_9072_nat__intermed__int__val,axiom,
    ! [M: nat,Nb: 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),Nb) )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(F,aa_nat_nat(suc,I3))),aa_nat_int(F,I3)))),one_one_int) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
       => ( 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,Nb))
           => ? [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),Nb)
                & ( aa_nat_int(F,I3) = K ) ) ) ) ) ) ).

% nat_intermed_int_val
tff(fact_9073_incr__lemma,axiom,
    ! [D: int,Z: int,Xb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,Z),aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb),Z))),one_one_int)),D))) ) ).

% incr_lemma
tff(fact_9074_decr__lemma,axiom,
    ! [D: int,Xb: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb),Z))),one_one_int)),D))),Z) ) ).

% decr_lemma
tff(fact_9075_arctan__ubound,axiom,
    ! [Y: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(arctan,Y)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one)))) ).

% arctan_ubound
tff(fact_9076_arctan__one,axiom,
    aa_real_real(arctan,one_one_real) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,aa_num_num(bit0,one)))) ).

% arctan_one
tff(fact_9077_nat__ivt__aux,axiom,
    ! [Nb: nat,F: fun_nat_int,K: int] :
      ( ! [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),Nb)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(F,aa_nat_nat(suc,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,Nb))
         => ? [I3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I3),Nb)
              & ( aa_nat_int(F,I3) = K ) ) ) ) ) ).

% nat_ivt_aux
tff(fact_9078_arctan__lbound,axiom,
    ! [Y: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))),aa_real_real(arctan,Y)) ).

% arctan_lbound
tff(fact_9079_arctan__bounded,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))),aa_real_real(arctan,Y))
      & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(arctan,Y)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one)))) ) ).

% arctan_bounded
tff(fact_9080_nat0__intermed__int__val,axiom,
    ! [Nb: nat,F: fun_nat_int,K: int] :
      ( ! [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),Nb)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,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,Nb))
         => ? [I3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I3),Nb)
              & ( aa_nat_int(F,I3) = K ) ) ) ) ) ).

% nat0_intermed_int_val
tff(fact_9081_arctan__add,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xb)),one_one_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,Y)),one_one_real)
       => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(arctan,Xb)),aa_real_real(arctan,Y)) = aa_real_real(arctan,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Y)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Y)))) ) ) ) ).

% arctan_add
tff(fact_9082_machin__Euler,axiom,
    aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit1,aa_num_num(bit0,one)))),aa_real_real(arctan,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(aa_num_num(bit1,aa_num_num(bit1,one))))))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),aa_real_real(arctan,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,numeral_numeral_real(aa_num_num(bit1,one))),numeral_numeral_real(aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,one))))))))))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,aa_num_num(bit0,one)))) ).

% machin_Euler
tff(fact_9083_machin,axiom,
    aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,aa_num_num(bit0,one)))) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,aa_num_num(bit0,one)))),aa_real_real(arctan,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(aa_num_num(bit1,aa_num_num(bit0,one))))))),aa_real_real(arctan,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,one))))))))))) ).

% machin
tff(fact_9084_sum__pos__lt__pair,axiom,
    ! [F: fun_nat_real,K: nat] :
      ( summable_real(F)
     => ( ! [D2: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,aa_nat_nat(suc,zero_zero_nat))),D2)))),aa_nat_real(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,aa_nat_nat(suc,zero_zero_nat))),D2)),one_one_nat)))))
       => aa_real_o(aa_real_fun_real_o(ord_less_real,groups6591440286371151544t_real(F,set_ord_lessThan_nat(K))),suminf_real(F)) ) ) ).

% sum_pos_lt_pair
tff(fact_9085_arctan__double,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,Xb)),one_one_real)
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),aa_real_real(arctan,Xb)) = aa_real_real(arctan,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),Xb)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ) ).

% arctan_double
tff(fact_9086_summable__complex__of__real,axiom,
    ! [F: fun_nat_real] :
      ( summable_complex(aTP_Lamm_azp(F))
    <=> summable_real(F) ) ).

% summable_complex_of_real
tff(fact_9087_tan__periodic__pi,axiom,
    ! [Xb: real] : aa_real_real(tan_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),pi)) = aa_real_real(tan_real,Xb) ).

% tan_periodic_pi
tff(fact_9088_tan__npi,axiom,
    ! [Nb: nat] : aa_real_real(tan_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),pi)) = zero_zero_real ).

% tan_npi
tff(fact_9089_tan__periodic__n,axiom,
    ! [Xb: real,Nb: num] : aa_real_real(tan_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(Nb)),pi))) = aa_real_real(tan_real,Xb) ).

% tan_periodic_n
tff(fact_9090_tan__periodic__nat,axiom,
    ! [Xb: real,Nb: nat] : aa_real_real(tan_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),pi))) = aa_real_real(tan_real,Xb) ).

% tan_periodic_nat
tff(fact_9091_tan__periodic__int,axiom,
    ! [Xb: real,I: int] : aa_real_real(tan_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(ring_1_of_int_real,I)),pi))) = aa_real_real(tan_real,Xb) ).

% tan_periodic_int
tff(fact_9092_tan__periodic,axiom,
    ! [Xb: real] : aa_real_real(tan_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),pi))) = aa_real_real(tan_real,Xb) ).

% tan_periodic
tff(fact_9093_tan__45,axiom,
    aa_real_real(tan_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,aa_num_num(bit0,one))))) = one_one_real ).

% tan_45
tff(fact_9094_lemma__tan__total,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
     => ? [X2: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X2)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,X2),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))
          & aa_real_o(aa_real_fun_real_o(ord_less_real,Y),aa_real_real(tan_real,X2)) ) ) ).

% lemma_tan_total
tff(fact_9095_tan__gt__zero,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(tan_real,Xb)) ) ) ).

% tan_gt_zero
tff(fact_9096_tan__total,axiom,
    ! [Y: real] :
    ? [X2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))),X2)
      & aa_real_o(aa_real_fun_real_o(ord_less_real,X2),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))
      & ( aa_real_real(tan_real,X2) = Y )
      & ! [Y5: real] :
          ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))),Y5)
            & aa_real_o(aa_real_fun_real_o(ord_less_real,Y5),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))
            & ( aa_real_real(tan_real,Y5) = Y ) )
         => ( Y5 = X2 ) ) ) ).

% tan_total
tff(fact_9097_tan__monotone,axiom,
    ! [Y: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Xb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(tan_real,Y)),aa_real_real(tan_real,Xb)) ) ) ) ).

% tan_monotone
tff(fact_9098_tan__monotone_H,axiom,
    ! [Y: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))),Xb)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))
           => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Xb)
            <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(tan_real,Y)),aa_real_real(tan_real,Xb)) ) ) ) ) ) ).

% tan_monotone'
tff(fact_9099_tan__mono__lt__eq,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))),Y)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))
           => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(tan_real,Xb)),aa_real_real(tan_real,Y))
            <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y) ) ) ) ) ) ).

% tan_mono_lt_eq
tff(fact_9100_lemma__tan__total1,axiom,
    ! [Y: real] :
    ? [X2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))),X2)
      & aa_real_o(aa_real_fun_real_o(ord_less_real,X2),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))
      & ( aa_real_real(tan_real,X2) = Y ) ) ).

% lemma_tan_total1
tff(fact_9101_tan__minus__45,axiom,
    aa_real_real(tan_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,aa_num_num(bit0,one)))))) = aa_real_real(uminus_uminus_real,one_one_real) ).

% tan_minus_45
tff(fact_9102_tan__inverse,axiom,
    ! [Y: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_real_real(tan_real,Y)) = aa_real_real(tan_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one)))),Y)) ).

% tan_inverse
tff(fact_9103_tan__pos__pi2__le,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(tan_real,Xb)) ) ) ).

% tan_pos_pi2_le
tff(fact_9104_tan__total__pos,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
     => ? [X2: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X2)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,X2),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))
          & ( aa_real_real(tan_real,X2) = Y ) ) ) ).

% tan_total_pos
tff(fact_9105_tan__less__zero,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(uminus_uminus_real,pi)),numeral_numeral_real(aa_num_num(bit0,one)))),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(tan_real,Xb)),zero_zero_real) ) ) ).

% tan_less_zero
tff(fact_9106_tan__mono__le,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(tan_real,Xb)),aa_real_real(tan_real,Y)) ) ) ) ).

% tan_mono_le
tff(fact_9107_tan__mono__le__eq,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))),Y)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(tan_real,Xb)),aa_real_real(tan_real,Y))
            <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y) ) ) ) ) ) ).

% tan_mono_le_eq
tff(fact_9108_tan__bound__pi2,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,Xb)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,aa_num_num(bit0,one)))))
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(tan_real,Xb))),one_one_real) ) ).

% tan_bound_pi2
tff(fact_9109_arctan__unique,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))
       => ( ( aa_real_real(tan_real,Xb) = Y )
         => ( aa_real_real(arctan,Y) = Xb ) ) ) ) ).

% arctan_unique
tff(fact_9110_arctan__tan,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))
       => ( aa_real_real(arctan,aa_real_real(tan_real,Xb)) = Xb ) ) ) ).

% arctan_tan
tff(fact_9111_arctan,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))),aa_real_real(arctan,Y))
      & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(arctan,Y)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))
      & ( aa_real_real(tan_real,aa_real_real(arctan,Y)) = Y ) ) ).

% arctan
tff(fact_9112_tan__total__pi4,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,Xb)),one_one_real)
     => ? [Z3: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,aa_num_num(bit0,one)))))),Z3)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,Z3),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,aa_num_num(bit0,one)))))
          & ( aa_real_real(tan_real,Z3) = Xb ) ) ) ).

% tan_total_pi4
tff(fact_9113_complex__unimodular__polar,axiom,
    ! [Z: complex] :
      ( ( aa_complex_real(real_V1022390504157884413omplex,Z) = one_one_real )
     => ~ ! [T3: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),T3)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_real,T3),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),pi))
             => ( Z != aa_real_complex(complex2(aa_real_real(cos_real,T3)),aa_real_real(sin_real,T3)) ) ) ) ) ).

% complex_unimodular_polar
tff(fact_9114_Maclaurin__exp__lt,axiom,
    ! [Xb: real,Nb: nat] :
      ( ( Xb != zero_zero_real )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => ? [T3: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(abs_abs_real,T3))
            & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,T3)),aa_real_real(abs_abs_real,Xb))
            & ( aa_real_real(exp_real,Xb) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_azq,Xb),set_ord_lessThan_nat(Nb))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(exp_real,T3)),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb))) ) ) ) ) ).

% Maclaurin_exp_lt
tff(fact_9115_exp__le__cancel__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(exp_real,Xb)),aa_real_real(exp_real,Y))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y) ) ).

% exp_le_cancel_iff
tff(fact_9116_exp__eq__one__iff,axiom,
    ! [Xb: real] :
      ( ( aa_real_real(exp_real,Xb) = one_one_real )
    <=> ( Xb = zero_zero_real ) ) ).

% exp_eq_one_iff
tff(fact_9117_one__less__exp__iff,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(exp_real,Xb))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb) ) ).

% one_less_exp_iff
tff(fact_9118_exp__less__one__iff,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(exp_real,Xb)),one_one_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),zero_zero_real) ) ).

% exp_less_one_iff
tff(fact_9119_one__le__exp__iff,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(exp_real,Xb))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb) ) ).

% one_le_exp_iff
tff(fact_9120_exp__le__one__iff,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(exp_real,Xb)),one_one_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),zero_zero_real) ) ).

% exp_le_one_iff
tff(fact_9121_norm__cos__sin,axiom,
    ! [Ta: real] : aa_complex_real(real_V1022390504157884413omplex,aa_real_complex(complex2(aa_real_real(cos_real,Ta)),aa_real_real(sin_real,Ta))) = one_one_real ).

% norm_cos_sin
tff(fact_9122_Complex__mult__complex__of__real,axiom,
    ! [Xb: real,Y: real,R2: real] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_real_complex(complex2(Xb),Y)),aa_real_complex(real_V4546457046886955230omplex,R2)) = aa_real_complex(complex2(aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),R2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),R2)) ).

% Complex_mult_complex_of_real
tff(fact_9123_complex__of__real__mult__Complex,axiom,
    ! [R2: real,Xb: real,Y: real] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_real_complex(real_V4546457046886955230omplex,R2)),aa_real_complex(complex2(Xb),Y)) = aa_real_complex(complex2(aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R2),Xb)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R2),Y)) ).

% complex_of_real_mult_Complex
tff(fact_9124_complex__diff,axiom,
    ! [A: real,B: real,C: real,D: real] : aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,aa_real_complex(complex2(A),B)),aa_real_complex(complex2(C),D)) = aa_real_complex(complex2(aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),D)) ).

% complex_diff
tff(fact_9125_complex__scaleR,axiom,
    ! [R2: real,A: real,B: real] : aa_complex_complex(aa_rea2618301233319580611omplex(real_V2046097035970521341omplex,R2),aa_real_complex(complex2(A),B)) = aa_real_complex(complex2(aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R2),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R2),B)) ).

% complex_scaleR
tff(fact_9126_exp__ge__zero,axiom,
    ! [Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(exp_real,Xb)) ).

% exp_ge_zero
tff(fact_9127_not__exp__le__zero,axiom,
    ! [Xb: real] : ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(exp_real,Xb)),zero_zero_real) ).

% not_exp_le_zero
tff(fact_9128_Complex__eq__numeral,axiom,
    ! [A: real,B: real,W: num] :
      ( ( aa_real_complex(complex2(A),B) = numera6690914467698888265omplex(W) )
    <=> ( ( A = numeral_numeral_real(W) )
        & ( B = zero_zero_real ) ) ) ).

% Complex_eq_numeral
tff(fact_9129_complex__add,axiom,
    ! [A: real,B: real,C: real,D: real] : aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_real_complex(complex2(A),B)),aa_real_complex(complex2(C),D)) = aa_real_complex(complex2(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),D)) ).

% complex_add
tff(fact_9130_Complex__add__complex__of__real,axiom,
    ! [Xb: real,Y: real,R2: real] : aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_real_complex(complex2(Xb),Y)),aa_real_complex(real_V4546457046886955230omplex,R2)) = aa_real_complex(complex2(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),R2)),Y) ).

% Complex_add_complex_of_real
tff(fact_9131_complex__of__real__add__Complex,axiom,
    ! [R2: real,Xb: real,Y: real] : aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_real_complex(real_V4546457046886955230omplex,R2)),aa_real_complex(complex2(Xb),Y)) = aa_real_complex(complex2(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,R2),Xb)),Y) ).

% complex_of_real_add_Complex
tff(fact_9132_exp__gt__one,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(exp_real,Xb)) ) ).

% exp_gt_one
tff(fact_9133_exp__ge__add__one__self,axiom,
    ! [Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),Xb)),aa_real_real(exp_real,Xb)) ).

% exp_ge_add_one_self
tff(fact_9134_Complex__eq__neg__numeral,axiom,
    ! [A: real,B: real,W: num] :
      ( ( aa_real_complex(complex2(A),B) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W)) )
    <=> ( ( A = aa_real_real(uminus_uminus_real,numeral_numeral_real(W)) )
        & ( B = zero_zero_real ) ) ) ).

% Complex_eq_neg_numeral
tff(fact_9135_log__ln,axiom,
    ! [Xb: real] : aa_real_real(ln_ln_real,Xb) = aa_real_real(aa_rea7340605660300375871l_real(log,aa_real_real(exp_real,one_one_real)),Xb) ).

% log_ln
tff(fact_9136_complex__mult,axiom,
    ! [A: real,B: real,C: real,D: real] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_real_complex(complex2(A),B)),aa_real_complex(complex2(C),D)) = aa_real_complex(complex2(aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),D))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),D)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C))) ).

% complex_mult
tff(fact_9137_one__complex_Ocode,axiom,
    one_one_complex = aa_real_complex(complex2(one_one_real),zero_zero_real) ).

% one_complex.code
tff(fact_9138_Complex__eq__1,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_complex(complex2(A),B) = one_one_complex )
    <=> ( ( A = one_one_real )
        & ( B = zero_zero_real ) ) ) ).

% Complex_eq_1
tff(fact_9139_exp__ge__add__one__self__aux,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),Xb)),aa_real_real(exp_real,Xb)) ) ).

% exp_ge_add_one_self_aux
tff(fact_9140_lemma__exp__total,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Y)
     => ? [X2: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X2)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X2),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Y),one_one_real))
          & ( aa_real_real(exp_real,X2) = Y ) ) ) ).

% lemma_exp_total
tff(fact_9141_ln__ge__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),aa_real_real(ln_ln_real,Xb))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(exp_real,Y)),Xb) ) ) ).

% ln_ge_iff
tff(fact_9142_ln__x__over__x__mono,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(exp_real,one_one_real)),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(ln_ln_real,Y)),Y)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(ln_ln_real,Xb)),Xb)) ) ) ).

% ln_x_over_x_mono
tff(fact_9143_Complex__eq__neg__1,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_complex(complex2(A),B) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) )
    <=> ( ( A = aa_real_real(uminus_uminus_real,one_one_real) )
        & ( B = zero_zero_real ) ) ) ).

% Complex_eq_neg_1
tff(fact_9144_exp__le,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(exp_real,one_one_real)),numeral_numeral_real(aa_num_num(bit1,one))) ).

% exp_le
tff(fact_9145_exp__half__le2,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(exp_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(aa_num_num(bit0,one))))),numeral_numeral_real(aa_num_num(bit0,one))) ).

% exp_half_le2
tff(fact_9146_exp__bound,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(exp_real,Xb)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),Xb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ) ).

% exp_bound
tff(fact_9147_real__exp__bound__lemma,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(aa_num_num(bit0,one))))
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(exp_real,Xb)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),Xb))) ) ) ).

% real_exp_bound_lemma
tff(fact_9148_exp__ge__one__plus__x__over__n__power__n,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_nat_real(semiri5074537144036343181t_real,Nb))),Xb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),aa_nat_real(semiri5074537144036343181t_real,Nb)))),Nb)),aa_real_real(exp_real,Xb)) ) ) ).

% exp_ge_one_plus_x_over_n_power_n
tff(fact_9149_exp__ge__one__minus__x__over__n__power__n,axiom,
    ! [Xb: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_nat_real(semiri5074537144036343181t_real,Nb))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),aa_nat_real(semiri5074537144036343181t_real,Nb)))),Nb)),aa_real_real(exp_real,aa_real_real(uminus_uminus_real,Xb))) ) ) ).

% exp_ge_one_minus_x_over_n_power_n
tff(fact_9150_Maclaurin__exp__le,axiom,
    ! [Xb: real,Nb: nat] :
    ? [T3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,T3)),aa_real_real(abs_abs_real,Xb))
      & ( aa_real_real(exp_real,Xb) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_azq,Xb),set_ord_lessThan_nat(Nb))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(exp_real,T3)),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb))) ) ) ).

% Maclaurin_exp_le
tff(fact_9151_exp__lower__Taylor__quadratic,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),Xb)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),numeral_numeral_real(aa_num_num(bit0,one))))),aa_real_real(exp_real,Xb)) ) ).

% exp_lower_Taylor_quadratic
tff(fact_9152_log__base__10__eq2,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_real(aa_rea7340605660300375871l_real(log,numeral_numeral_real(aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,one))))),Xb) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(log,numeral_numeral_real(aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,one))))),aa_real_real(exp_real,one_one_real))),aa_real_real(ln_ln_real,Xb)) ) ) ).

% log_base_10_eq2
tff(fact_9153_tanh__real__altdef,axiom,
    ! [Xb: real] : aa_real_real(tanh_real,Xb) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_real_real(exp_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(aa_num_num(bit0,one)))),Xb)))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(exp_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(aa_num_num(bit0,one)))),Xb)))) ).

% tanh_real_altdef
tff(fact_9154_log__base__10__eq1,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_real(aa_rea7340605660300375871l_real(log,numeral_numeral_real(aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,one))))),Xb) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(ln_ln_real,aa_real_real(exp_real,one_one_real))),aa_real_real(ln_ln_real,numeral_numeral_real(aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,one))))))),aa_real_real(ln_ln_real,Xb)) ) ) ).

% log_base_10_eq1
tff(fact_9155_sin__tan,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,Xb)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))
     => ( aa_real_real(sin_real,Xb) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(tan_real,Xb)),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(tan_real,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ) ).

% sin_tan
tff(fact_9156_cos__tan,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,Xb)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))
     => ( aa_real_real(cos_real,Xb) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(tan_real,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ) ).

% cos_tan
tff(fact_9157_real__sqrt__eq__iff,axiom,
    ! [Xb: real,Y: real] :
      ( ( aa_real_real(sqrt,Xb) = aa_real_real(sqrt,Y) )
    <=> ( Xb = Y ) ) ).

% real_sqrt_eq_iff
tff(fact_9158_real__sqrt__zero,axiom,
    aa_real_real(sqrt,zero_zero_real) = zero_zero_real ).

% real_sqrt_zero
tff(fact_9159_real__sqrt__eq__zero__cancel__iff,axiom,
    ! [Xb: real] :
      ( ( aa_real_real(sqrt,Xb) = zero_zero_real )
    <=> ( Xb = zero_zero_real ) ) ).

% real_sqrt_eq_zero_cancel_iff
tff(fact_9160_real__sqrt__less__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sqrt,Xb)),aa_real_real(sqrt,Y))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y) ) ).

% real_sqrt_less_iff
tff(fact_9161_real__sqrt__le__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sqrt,Xb)),aa_real_real(sqrt,Y))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y) ) ).

% real_sqrt_le_iff
tff(fact_9162_real__sqrt__one,axiom,
    aa_real_real(sqrt,one_one_real) = one_one_real ).

% real_sqrt_one
tff(fact_9163_real__sqrt__eq__1__iff,axiom,
    ! [Xb: real] :
      ( ( aa_real_real(sqrt,Xb) = one_one_real )
    <=> ( Xb = one_one_real ) ) ).

% real_sqrt_eq_1_iff
tff(fact_9164_real__sqrt__gt__0__iff,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(sqrt,Y))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y) ) ).

% real_sqrt_gt_0_iff
tff(fact_9165_real__sqrt__lt__0__iff,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sqrt,Xb)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),zero_zero_real) ) ).

% real_sqrt_lt_0_iff
tff(fact_9166_real__sqrt__le__0__iff,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sqrt,Xb)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),zero_zero_real) ) ).

% real_sqrt_le_0_iff
tff(fact_9167_real__sqrt__ge__0__iff,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(sqrt,Y))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y) ) ).

% real_sqrt_ge_0_iff
tff(fact_9168_real__sqrt__lt__1__iff,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sqrt,Xb)),one_one_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),one_one_real) ) ).

% real_sqrt_lt_1_iff
tff(fact_9169_real__sqrt__gt__1__iff,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(sqrt,Y))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Y) ) ).

% real_sqrt_gt_1_iff
tff(fact_9170_real__sqrt__le__1__iff,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sqrt,Xb)),one_one_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),one_one_real) ) ).

% real_sqrt_le_1_iff
tff(fact_9171_real__sqrt__ge__1__iff,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(sqrt,Y))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Y) ) ).

% real_sqrt_ge_1_iff
tff(fact_9172_real__sqrt__abs2,axiom,
    ! [Xb: real] : aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Xb)) = aa_real_real(abs_abs_real,Xb) ).

% real_sqrt_abs2
tff(fact_9173_real__sqrt__mult__self,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sqrt,A)),aa_real_real(sqrt,A)) = aa_real_real(abs_abs_real,A) ).

% real_sqrt_mult_self
tff(fact_9174_real__sqrt__four,axiom,
    aa_real_real(sqrt,numeral_numeral_real(aa_num_num(bit0,aa_num_num(bit0,one)))) = numeral_numeral_real(aa_num_num(bit0,one)) ).

% real_sqrt_four
tff(fact_9175_real__sqrt__abs,axiom,
    ! [Xb: real] : aa_real_real(sqrt,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = aa_real_real(abs_abs_real,Xb) ).

% real_sqrt_abs
tff(fact_9176_real__sqrt__pow2__iff,axiom,
    ! [Xb: real] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(sqrt,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = Xb )
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb) ) ).

% real_sqrt_pow2_iff
tff(fact_9177_real__sqrt__pow2,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(sqrt,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = Xb ) ) ).

% real_sqrt_pow2
tff(fact_9178_real__sqrt__sum__squares__mult__squared__eq,axiom,
    ! [Xb: real,Y: real,Xaa: real,Ya: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xaa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Ya),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xaa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Ya),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% real_sqrt_sum_squares_mult_squared_eq
tff(fact_9179_complex__exp__exists,axiom,
    ! [Z: complex] :
    ? [A3: complex,R4: real] : Z = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_real_complex(real_V4546457046886955230omplex,R4)),aa_complex_complex(exp_complex,A3)) ).

% complex_exp_exists
tff(fact_9180_real__sqrt__le__mono,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sqrt,Xb)),aa_real_real(sqrt,Y)) ) ).

% real_sqrt_le_mono
tff(fact_9181_real__sqrt__mult,axiom,
    ! [Xb: real,Y: real] : aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Y)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sqrt,Xb)),aa_real_real(sqrt,Y)) ).

% real_sqrt_mult
tff(fact_9182_real__sqrt__divide,axiom,
    ! [Xb: real,Y: real] : aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Y)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(sqrt,Xb)),aa_real_real(sqrt,Y)) ).

% real_sqrt_divide
tff(fact_9183_real__sqrt__power,axiom,
    ! [Xb: real,K: nat] : aa_real_real(sqrt,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),K)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(sqrt,Xb)),K) ).

% real_sqrt_power
tff(fact_9184_real__sqrt__less__mono,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sqrt,Xb)),aa_real_real(sqrt,Y)) ) ).

% real_sqrt_less_mono
tff(fact_9185_real__sqrt__minus,axiom,
    ! [Xb: real] : aa_real_real(sqrt,aa_real_real(uminus_uminus_real,Xb)) = aa_real_real(uminus_uminus_real,aa_real_real(sqrt,Xb)) ).

% real_sqrt_minus
tff(fact_9186_real__sqrt__gt__zero,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(sqrt,Xb)) ) ).

% real_sqrt_gt_zero
tff(fact_9187_real__sqrt__eq__zero__cancel,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( ( aa_real_real(sqrt,Xb) = zero_zero_real )
       => ( Xb = zero_zero_real ) ) ) ).

% real_sqrt_eq_zero_cancel
tff(fact_9188_real__sqrt__ge__zero,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(sqrt,Xb)) ) ).

% real_sqrt_ge_zero
tff(fact_9189_real__sqrt__ge__one,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Xb)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(sqrt,Xb)) ) ).

% real_sqrt_ge_one
tff(fact_9190_real__div__sqrt,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),aa_real_real(sqrt,Xb)) = aa_real_real(sqrt,Xb) ) ) ).

% real_div_sqrt
tff(fact_9191_sqrt__add__le__add__sqrt,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Y))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(sqrt,Xb)),aa_real_real(sqrt,Y))) ) ) ).

% sqrt_add_le_add_sqrt
tff(fact_9192_le__real__sqrt__sumsq,axiom,
    ! [Xb: real,Y: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Xb)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Y)))) ).

% le_real_sqrt_sumsq
tff(fact_9193_sqrt2__less__2,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sqrt,numeral_numeral_real(aa_num_num(bit0,one)))),numeral_numeral_real(aa_num_num(bit0,one))) ).

% sqrt2_less_2
tff(fact_9194_real__less__rsqrt,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),Y)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_real_real(sqrt,Y)) ) ).

% real_less_rsqrt
tff(fact_9195_real__le__rsqrt,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),Y)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_real_real(sqrt,Y)) ) ).

% real_le_rsqrt
tff(fact_9196_sqrt__le__D,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sqrt,Xb)),Y)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ).

% sqrt_le_D
tff(fact_9197_real__sqrt__unique,axiom,
    ! [Y: real,Xb: real] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = Xb )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
       => ( aa_real_real(sqrt,Xb) = Y ) ) ) ).

% real_sqrt_unique
tff(fact_9198_real__le__lsqrt,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sqrt,Xb)),Y) ) ) ) ).

% real_le_lsqrt
tff(fact_9199_lemma__real__divide__sqrt__less,axiom,
    ! [U: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),U)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,U),aa_real_real(sqrt,numeral_numeral_real(aa_num_num(bit0,one))))),U) ) ).

% lemma_real_divide_sqrt_less
tff(fact_9200_real__sqrt__sum__squares__eq__cancel2,axiom,
    ! [Xb: real,Y: real] :
      ( ( aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) = Y )
     => ( Xb = zero_zero_real ) ) ).

% real_sqrt_sum_squares_eq_cancel2
tff(fact_9201_real__sqrt__sum__squares__eq__cancel,axiom,
    ! [Xb: real,Y: real] :
      ( ( aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) = Xb )
     => ( Y = zero_zero_real ) ) ).

% real_sqrt_sum_squares_eq_cancel
tff(fact_9202_real__sqrt__sum__squares__triangle__ineq,axiom,
    ! [A: real,C: real,B: real,D: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),D)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,C),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,D),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))) ).

% real_sqrt_sum_squares_triangle_ineq
tff(fact_9203_real__sqrt__sum__squares__ge2,axiom,
    ! [Y: real,Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% real_sqrt_sum_squares_ge2
tff(fact_9204_real__sqrt__sum__squares__ge1,axiom,
    ! [Xb: real,Y: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% real_sqrt_sum_squares_ge1
tff(fact_9205_sqrt__ge__absD,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xb)),aa_real_real(sqrt,Y))
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),Y) ) ).

% sqrt_ge_absD
tff(fact_9206_cos__45,axiom,
    aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,aa_num_num(bit0,one))))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(sqrt,numeral_numeral_real(aa_num_num(bit0,one)))),numeral_numeral_real(aa_num_num(bit0,one))) ).

% cos_45
tff(fact_9207_sin__45,axiom,
    aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,aa_num_num(bit0,one))))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(sqrt,numeral_numeral_real(aa_num_num(bit0,one)))),numeral_numeral_real(aa_num_num(bit0,one))) ).

% sin_45
tff(fact_9208_tan__60,axiom,
    aa_real_real(tan_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit1,one)))) = aa_real_real(sqrt,numeral_numeral_real(aa_num_num(bit1,one))) ).

% tan_60
tff(fact_9209_real__less__lsqrt,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sqrt,Xb)),Y) ) ) ) ).

% real_less_lsqrt
tff(fact_9210_sqrt__sum__squares__le__sum,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Y)) ) ) ).

% sqrt_sum_squares_le_sum
tff(fact_9211_sqrt__even__pow2,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_real_real(sqrt,aa_nat_real(aa_real_fun_nat_real(power_power_real,numeral_numeral_real(aa_num_num(bit0,one))),Nb)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,numeral_numeral_real(aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ) ).

% sqrt_even_pow2
tff(fact_9212_real__sqrt__ge__abs1,axiom,
    ! [Xb: real,Y: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xb)),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% real_sqrt_ge_abs1
tff(fact_9213_real__sqrt__ge__abs2,axiom,
    ! [Y: real,Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Y)),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% real_sqrt_ge_abs2
tff(fact_9214_sqrt__sum__squares__le__sum__abs,axiom,
    ! [Xb: real,Y: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(abs_abs_real,Xb)),aa_real_real(abs_abs_real,Y))) ).

% sqrt_sum_squares_le_sum_abs
tff(fact_9215_ln__sqrt,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_real(ln_ln_real,aa_real_real(sqrt,Xb)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(ln_ln_real,Xb)),numeral_numeral_real(aa_num_num(bit0,one))) ) ) ).

% ln_sqrt
tff(fact_9216_cos__30,axiom,
    aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,aa_num_num(bit1,one))))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(sqrt,numeral_numeral_real(aa_num_num(bit1,one)))),numeral_numeral_real(aa_num_num(bit0,one))) ).

% cos_30
tff(fact_9217_sin__60,axiom,
    aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit1,one)))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(sqrt,numeral_numeral_real(aa_num_num(bit1,one)))),numeral_numeral_real(aa_num_num(bit0,one))) ).

% sin_60
tff(fact_9218_arsinh__real__def,axiom,
    ! [Xb: real] : aa_real_real(arsinh_real,Xb) = aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real)))) ).

% arsinh_real_def
tff(fact_9219_complex__norm,axiom,
    ! [Xb: real,Y: real] : aa_complex_real(real_V1022390504157884413omplex,aa_real_complex(complex2(Xb),Y)) = aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% complex_norm
tff(fact_9220_real__sqrt__power__even,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
       => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(sqrt,Xb)),Nb) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ) ) ).

% real_sqrt_power_even
tff(fact_9221_arsinh__real__aux,axiom,
    ! [Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real)))) ).

% arsinh_real_aux
tff(fact_9222_real__sqrt__sum__squares__mult__ge__zero,axiom,
    ! [Xb: real,Y: real,Xaa: real,Ya: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xaa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Ya),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))) ).

% real_sqrt_sum_squares_mult_ge_zero
tff(fact_9223_arith__geo__mean__sqrt,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Y))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Y)),numeral_numeral_real(aa_num_num(bit0,one)))) ) ) ).

% arith_geo_mean_sqrt
tff(fact_9224_powr__half__sqrt,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(aa_num_num(bit0,one)))) = aa_real_real(sqrt,Xb) ) ) ).

% powr_half_sqrt
tff(fact_9225_tan__30,axiom,
    aa_real_real(tan_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,aa_num_num(bit1,one))))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_real_real(sqrt,numeral_numeral_real(aa_num_num(bit1,one)))) ).

% tan_30
tff(fact_9226_cos__x__y__le__one,axiom,
    ! [Xb: real,Y: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))),one_one_real) ).

% cos_x_y_le_one
tff(fact_9227_real__sqrt__sum__squares__less,axiom,
    ! [Xb: real,U: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,Xb)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,U),aa_real_real(sqrt,numeral_numeral_real(aa_num_num(bit0,one)))))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,Y)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,U),aa_real_real(sqrt,numeral_numeral_real(aa_num_num(bit0,one)))))
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),U) ) ) ).

% real_sqrt_sum_squares_less
tff(fact_9228_arcosh__real__def,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Xb)
     => ( aa_real_real(arcosh_real,Xb) = aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real)))) ) ) ).

% arcosh_real_def
tff(fact_9229_cos__arctan,axiom,
    ! [Xb: real] : aa_real_real(cos_real,aa_real_real(arctan,Xb)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% cos_arctan
tff(fact_9230_sin__arctan,axiom,
    ! [Xb: real] : aa_real_real(sin_real,aa_real_real(arctan,Xb)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% sin_arctan
tff(fact_9231_sqrt__sum__squares__half__less,axiom,
    ! [Xb: real,U: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,U),numeral_numeral_real(aa_num_num(bit0,one))))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,U),numeral_numeral_real(aa_num_num(bit0,one))))
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
           => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),U) ) ) ) ) ).

% sqrt_sum_squares_half_less
tff(fact_9232_sin__cos__sqrt,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(sin_real,Xb))
     => ( aa_real_real(sin_real,Xb) = aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(cos_real,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ) ).

% sin_cos_sqrt
tff(fact_9233_arctan__half,axiom,
    ! [Xb: real] : aa_real_real(arctan,Xb) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),aa_real_real(arctan,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))))) ).

% arctan_half
tff(fact_9234_cos__arcsin,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),one_one_real)
       => ( aa_real_real(cos_real,aa_real_real(arcsin,Xb)) = aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ) ) ).

% cos_arcsin
tff(fact_9235_sin__arccos__abs,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Y)),one_one_real)
     => ( aa_real_real(sin_real,aa_real_real(arccos,Y)) = aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ) ).

% sin_arccos_abs
tff(fact_9236_sin__arccos,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),one_one_real)
       => ( aa_real_real(sin_real,aa_real_real(arccos,Xb)) = aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ) ) ).

% sin_arccos
tff(fact_9237_arccos__1,axiom,
    aa_real_real(arccos,one_one_real) = zero_zero_real ).

% arccos_1
tff(fact_9238_arccos__minus__1,axiom,
    aa_real_real(arccos,aa_real_real(uminus_uminus_real,one_one_real)) = pi ).

% arccos_minus_1
tff(fact_9239_cos__arccos,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),one_one_real)
       => ( aa_real_real(cos_real,aa_real_real(arccos,Y)) = Y ) ) ) ).

% cos_arccos
tff(fact_9240_sin__arcsin,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),one_one_real)
       => ( aa_real_real(sin_real,aa_real_real(arcsin,Y)) = Y ) ) ) ).

% sin_arcsin
tff(fact_9241_arccos__0,axiom,
    aa_real_real(arccos,zero_zero_real) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))) ).

% arccos_0
tff(fact_9242_arcsin__1,axiom,
    aa_real_real(arcsin,one_one_real) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))) ).

% arcsin_1
tff(fact_9243_arcsin__minus__1,axiom,
    aa_real_real(arcsin,aa_real_real(uminus_uminus_real,one_one_real)) = aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one)))) ).

% arcsin_minus_1
tff(fact_9244_arccos__le__arccos,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),one_one_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arccos,Y)),aa_real_real(arccos,Xb)) ) ) ) ).

% arccos_le_arccos
tff(fact_9245_arccos__eq__iff,axiom,
    ! [Xb: real,Y: real] :
      ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xb)),one_one_real)
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Y)),one_one_real) )
     => ( ( aa_real_real(arccos,Xb) = aa_real_real(arccos,Y) )
      <=> ( Xb = Y ) ) ) ).

% arccos_eq_iff
tff(fact_9246_arccos__le__mono,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xb)),one_one_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Y)),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arccos,Xb)),aa_real_real(arccos,Y))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),Xb) ) ) ) ).

% arccos_le_mono
tff(fact_9247_arcsin__le__arcsin,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),one_one_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arcsin,Xb)),aa_real_real(arcsin,Y)) ) ) ) ).

% arcsin_le_arcsin
tff(fact_9248_arcsin__minus,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),one_one_real)
       => ( aa_real_real(arcsin,aa_real_real(uminus_uminus_real,Xb)) = aa_real_real(uminus_uminus_real,aa_real_real(arcsin,Xb)) ) ) ) ).

% arcsin_minus
tff(fact_9249_arcsin__eq__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xb)),one_one_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Y)),one_one_real)
       => ( ( aa_real_real(arcsin,Xb) = aa_real_real(arcsin,Y) )
        <=> ( Xb = Y ) ) ) ) ).

% arcsin_eq_iff
tff(fact_9250_arcsin__le__mono,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xb)),one_one_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Y)),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arcsin,Xb)),aa_real_real(arcsin,Y))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y) ) ) ) ).

% arcsin_le_mono
tff(fact_9251_arccos__lbound,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(arccos,Y)) ) ) ).

% arccos_lbound
tff(fact_9252_arccos__less__arccos,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),one_one_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(arccos,Y)),aa_real_real(arccos,Xb)) ) ) ) ).

% arccos_less_arccos
tff(fact_9253_arccos__less__mono,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xb)),one_one_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Y)),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(arccos,Xb)),aa_real_real(arccos,Y))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Xb) ) ) ) ).

% arccos_less_mono
tff(fact_9254_arccos__ubound,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arccos,Y)),pi) ) ) ).

% arccos_ubound
tff(fact_9255_arccos__cos,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),pi)
       => ( aa_real_real(arccos,aa_real_real(cos_real,Xb)) = Xb ) ) ) ).

% arccos_cos
tff(fact_9256_arcsin__less__arcsin,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),one_one_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(arcsin,Xb)),aa_real_real(arcsin,Y)) ) ) ) ).

% arcsin_less_arcsin
tff(fact_9257_arcsin__less__mono,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xb)),one_one_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Y)),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(arcsin,Xb)),aa_real_real(arcsin,Y))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y) ) ) ) ).

% arcsin_less_mono
tff(fact_9258_cos__arccos__abs,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Y)),one_one_real)
     => ( aa_real_real(cos_real,aa_real_real(arccos,Y)) = Y ) ) ).

% cos_arccos_abs
tff(fact_9259_arccos__cos__eq__abs,axiom,
    ! [Theta: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Theta)),pi)
     => ( aa_real_real(arccos,aa_real_real(cos_real,Theta)) = aa_real_real(abs_abs_real,Theta) ) ) ).

% arccos_cos_eq_abs
tff(fact_9260_arccos__lt__bounded,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(arccos,Y))
          & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(arccos,Y)),pi) ) ) ) ).

% arccos_lt_bounded
tff(fact_9261_arccos__bounded,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(arccos,Y))
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arccos,Y)),pi) ) ) ) ).

% arccos_bounded
tff(fact_9262_sin__arccos__nonzero,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),one_one_real)
       => ( aa_real_real(sin_real,aa_real_real(arccos,Xb)) != zero_zero_real ) ) ) ).

% sin_arccos_nonzero
tff(fact_9263_arccos__cos2,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,pi)),Xb)
       => ( aa_real_real(arccos,aa_real_real(cos_real,Xb)) = aa_real_real(uminus_uminus_real,Xb) ) ) ) ).

% arccos_cos2
tff(fact_9264_arccos__minus,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),one_one_real)
       => ( aa_real_real(arccos,aa_real_real(uminus_uminus_real,Xb)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,pi),aa_real_real(arccos,Xb)) ) ) ) ).

% arccos_minus
tff(fact_9265_cos__arcsin__nonzero,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),one_one_real)
       => ( aa_real_real(cos_real,aa_real_real(arcsin,Xb)) != zero_zero_real ) ) ) ).

% cos_arcsin_nonzero
tff(fact_9266_arccos,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(arccos,Y))
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arccos,Y)),pi)
          & ( aa_real_real(cos_real,aa_real_real(arccos,Y)) = Y ) ) ) ) ).

% arccos
tff(fact_9267_arccos__minus__abs,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xb)),one_one_real)
     => ( aa_real_real(arccos,aa_real_real(uminus_uminus_real,Xb)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,pi),aa_real_real(arccos,Xb)) ) ) ).

% arccos_minus_abs
tff(fact_9268_arccos__le__pi2,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arccos,Y)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one)))) ) ) ).

% arccos_le_pi2
tff(fact_9269_arcsin__lt__bounded,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))),aa_real_real(arcsin,Y))
          & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(arcsin,Y)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one)))) ) ) ) ).

% arcsin_lt_bounded
tff(fact_9270_arcsin__lbound,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))),aa_real_real(arcsin,Y)) ) ) ).

% arcsin_lbound
tff(fact_9271_arcsin__ubound,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arcsin,Y)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one)))) ) ) ).

% arcsin_ubound
tff(fact_9272_arcsin__bounded,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))),aa_real_real(arcsin,Y))
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arcsin,Y)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one)))) ) ) ) ).

% arcsin_bounded
tff(fact_9273_arcsin__sin,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))
       => ( aa_real_real(arcsin,aa_real_real(sin_real,Xb)) = Xb ) ) ) ).

% arcsin_sin
tff(fact_9274_le__arcsin__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(uminus_uminus_real,pi)),numeral_numeral_real(aa_num_num(bit0,one)))),Y)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),aa_real_real(arcsin,Xb))
            <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sin_real,Y)),Xb) ) ) ) ) ) ).

% le_arcsin_iff
tff(fact_9275_arcsin__le__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(uminus_uminus_real,pi)),numeral_numeral_real(aa_num_num(bit0,one)))),Y)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arcsin,Xb)),Y)
            <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_real_real(sin_real,Y)) ) ) ) ) ) ).

% arcsin_le_iff
tff(fact_9276_arcsin__pi,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))),aa_real_real(arcsin,Y))
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arcsin,Y)),pi)
          & ( aa_real_real(sin_real,aa_real_real(arcsin,Y)) = Y ) ) ) ) ).

% arcsin_pi
tff(fact_9277_arcsin,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))),aa_real_real(arcsin,Y))
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arcsin,Y)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))
          & ( aa_real_real(sin_real,aa_real_real(arcsin,Y)) = Y ) ) ) ) ).

% arcsin
tff(fact_9278_arccos__cos__eq__abs__2pi,axiom,
    ! [Theta: real] :
      ~ ! [K3: int] : aa_real_real(arccos,aa_real_real(cos_real,Theta)) != aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Theta),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(ring_1_of_int_real,K3)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),pi)))) ).

% arccos_cos_eq_abs_2pi
tff(fact_9279_exp__two__pi__i,axiom,
    aa_complex_complex(exp_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_real_complex(real_V4546457046886955230omplex,pi))),imaginary_unit)) = one_one_complex ).

% exp_two_pi_i
tff(fact_9280_exp__two__pi__i_H,axiom,
    aa_complex_complex(exp_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,imaginary_unit),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_real_complex(real_V4546457046886955230omplex,pi)),numera6690914467698888265omplex(aa_num_num(bit0,one))))) = one_one_complex ).

% exp_two_pi_i'
tff(fact_9281_norm__ii,axiom,
    aa_complex_real(real_V1022390504157884413omplex,imaginary_unit) = one_one_real ).

% norm_ii
tff(fact_9282_complex__i__mult__minus,axiom,
    ! [Xb: complex] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,imaginary_unit),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,imaginary_unit),Xb)) = aa_complex_complex(uminus1482373934393186551omplex,Xb) ).

% complex_i_mult_minus
tff(fact_9283_divide__i,axiom,
    ! [Xb: complex] : aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,Xb),imaginary_unit) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(uminus1482373934393186551omplex,imaginary_unit)),Xb) ).

% divide_i
tff(fact_9284_i__squared,axiom,
    aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,imaginary_unit),imaginary_unit) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).

% i_squared
tff(fact_9285_divide__numeral__i,axiom,
    ! [Z: complex,Nb: num] : aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,Z),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,numera6690914467698888265omplex(Nb)),imaginary_unit)) = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,imaginary_unit),Z))),numera6690914467698888265omplex(Nb)) ).

% divide_numeral_i
tff(fact_9286_power2__i,axiom,
    aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,imaginary_unit),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).

% power2_i
tff(fact_9287_exp__pi__i,axiom,
    aa_complex_complex(exp_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_real_complex(real_V4546457046886955230omplex,pi)),imaginary_unit)) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).

% exp_pi_i
tff(fact_9288_exp__pi__i_H,axiom,
    aa_complex_complex(exp_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,imaginary_unit),aa_real_complex(real_V4546457046886955230omplex,pi))) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).

% exp_pi_i'
tff(fact_9289_i__even__power,axiom,
    ! [Nb: nat] : aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,imaginary_unit),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),Nb) ).

% i_even_power
tff(fact_9290_complex__i__not__one,axiom,
    imaginary_unit != one_one_complex ).

% complex_i_not_one
tff(fact_9291_complex__i__not__numeral,axiom,
    ! [W: num] : imaginary_unit != numera6690914467698888265omplex(W) ).

% complex_i_not_numeral
tff(fact_9292_real__sqrt__inverse,axiom,
    ! [Xb: real] : aa_real_real(sqrt,aa_real_real(inverse_inverse_real,Xb)) = aa_real_real(inverse_inverse_real,aa_real_real(sqrt,Xb)) ).

% real_sqrt_inverse
tff(fact_9293_divide__real__def,axiom,
    ! [Xb: real,Y: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Y) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),aa_real_real(inverse_inverse_real,Y)) ).

% divide_real_def
tff(fact_9294_i__times__eq__iff,axiom,
    ! [W: complex,Z: complex] :
      ( ( aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,imaginary_unit),W) = Z )
    <=> ( W = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,imaginary_unit),Z)) ) ) ).

% i_times_eq_iff
tff(fact_9295_complex__i__not__neg__numeral,axiom,
    ! [W: num] : imaginary_unit != aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W)) ).

% complex_i_not_neg_numeral
tff(fact_9296_inverse__powr,axiom,
    ! [Y: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
     => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,aa_real_real(inverse_inverse_real,Y)),A) = aa_real_real(inverse_inverse_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,Y),A)) ) ) ).

% inverse_powr
tff(fact_9297_Complex__eq__i,axiom,
    ! [Xb: real,Y: real] :
      ( ( aa_real_complex(complex2(Xb),Y) = imaginary_unit )
    <=> ( ( Xb = zero_zero_real )
        & ( Y = one_one_real ) ) ) ).

% Complex_eq_i
tff(fact_9298_imaginary__unit_Ocode,axiom,
    imaginary_unit = aa_real_complex(complex2(zero_zero_real),one_one_real) ).

% imaginary_unit.code
tff(fact_9299_i__mult__Complex,axiom,
    ! [A: real,B: real] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,imaginary_unit),aa_real_complex(complex2(A),B)) = aa_real_complex(complex2(aa_real_real(uminus_uminus_real,B)),A) ).

% i_mult_Complex
tff(fact_9300_Complex__mult__i,axiom,
    ! [A: real,B: real] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_real_complex(complex2(A),B)),imaginary_unit) = aa_real_complex(complex2(aa_real_real(uminus_uminus_real,B)),A) ).

% Complex_mult_i
tff(fact_9301_forall__pos__mono__1,axiom,
    ! [P: fun_real_o,E: real] :
      ( ! [D2: real,E2: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,D2),E2)
         => ( aa_real_o(P,D2)
           => aa_real_o(P,E2) ) )
     => ( ! [N2: nat] : aa_real_o(P,aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,N2))))
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),E)
         => aa_real_o(P,E) ) ) ) ).

% forall_pos_mono_1
tff(fact_9302_sqrt__divide__self__eq,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(sqrt,Xb)),Xb) = aa_real_real(inverse_inverse_real,aa_real_real(sqrt,Xb)) ) ) ).

% sqrt_divide_self_eq
tff(fact_9303_i__complex__of__real,axiom,
    ! [R2: real] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,imaginary_unit),aa_real_complex(real_V4546457046886955230omplex,R2)) = aa_real_complex(complex2(zero_zero_real),R2) ).

% i_complex_of_real
tff(fact_9304_complex__of__real__i,axiom,
    ! [R2: real] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_real_complex(real_V4546457046886955230omplex,R2)),imaginary_unit) = aa_real_complex(complex2(zero_zero_real),R2) ).

% complex_of_real_i
tff(fact_9305_log__inverse,axiom,
    ! [A: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( ( A != one_one_real )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
         => ( aa_real_real(aa_rea7340605660300375871l_real(log,A),aa_real_real(inverse_inverse_real,Xb)) = aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(log,A),Xb)) ) ) ) ) ).

% log_inverse
tff(fact_9306_Complex__eq,axiom,
    ! [A: real,B: real] : aa_real_complex(complex2(A),B) = aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_real_complex(real_V4546457046886955230omplex,A)),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,imaginary_unit),aa_real_complex(real_V4546457046886955230omplex,B))) ).

% Complex_eq
tff(fact_9307_exp__plus__inverse__exp,axiom,
    ! [Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,numeral_numeral_real(aa_num_num(bit0,one))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(exp_real,Xb)),aa_real_real(inverse_inverse_real,aa_real_real(exp_real,Xb)))) ).

% exp_plus_inverse_exp
tff(fact_9308_complex__split__polar,axiom,
    ! [Z: complex] :
    ? [R4: real,A3: real] : Z = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_real_complex(real_V4546457046886955230omplex,R4)),aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_real_complex(real_V4546457046886955230omplex,aa_real_real(cos_real,A3))),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,imaginary_unit),aa_real_complex(real_V4546457046886955230omplex,aa_real_real(sin_real,A3))))) ).

% complex_split_polar
tff(fact_9309_plus__inverse__ge__2,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,numeral_numeral_real(aa_num_num(bit0,one))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),aa_real_real(inverse_inverse_real,Xb))) ) ).

% plus_inverse_ge_2
tff(fact_9310_real__inv__sqrt__pow2,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(inverse_inverse_real,aa_real_real(sqrt,Xb))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_real_real(inverse_inverse_real,Xb) ) ) ).

% real_inv_sqrt_pow2
tff(fact_9311_tan__cot,axiom,
    ! [Xb: real] : aa_real_real(tan_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one)))),Xb)) = aa_real_real(inverse_inverse_real,aa_real_real(tan_real,Xb)) ).

% tan_cot
tff(fact_9312_real__le__x__sinh,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(exp_real,Xb)),aa_real_real(inverse_inverse_real,aa_real_real(exp_real,Xb)))),numeral_numeral_real(aa_num_num(bit0,one)))) ) ).

% real_le_x_sinh
tff(fact_9313_real__le__abs__sinh,axiom,
    ! [Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xb)),aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(exp_real,Xb)),aa_real_real(inverse_inverse_real,aa_real_real(exp_real,Xb)))),numeral_numeral_real(aa_num_num(bit0,one))))) ).

% real_le_abs_sinh
tff(fact_9314_cmod__unit__one,axiom,
    ! [A: real] : aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_real_complex(real_V4546457046886955230omplex,aa_real_real(cos_real,A))),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,imaginary_unit),aa_real_complex(real_V4546457046886955230omplex,aa_real_real(sin_real,A))))) = one_one_real ).

% cmod_unit_one
tff(fact_9315_cmod__complex__polar,axiom,
    ! [R2: real,A: real] : aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_real_complex(real_V4546457046886955230omplex,R2)),aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_real_complex(real_V4546457046886955230omplex,aa_real_real(cos_real,A))),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,imaginary_unit),aa_real_complex(real_V4546457046886955230omplex,aa_real_real(sin_real,A)))))) = aa_real_real(abs_abs_real,R2) ).

% cmod_complex_polar
tff(fact_9316_powr__real__of__int,axiom,
    ! [Xb: real,Nb: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),aa_int_real(ring_1_of_int_real,Nb)) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Nb),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_int_nat(nat2,Nb)),aa_real_real(inverse_inverse_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_int_nat(nat2,aa_int_int(uminus_uminus_int,Nb))))) ) ) ).

% powr_real_of_int
tff(fact_9317_Maclaurin__sin__bound,axiom,
    ! [Xb: real,Nb: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(sin_real,Xb)),groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_azi,Xb),set_ord_lessThan_nat(Nb))))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(inverse_inverse_real,aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,Xb)),Nb))) ).

% Maclaurin_sin_bound
tff(fact_9318_Arg__minus__ii,axiom,
    aa_complex_real(arg,aa_complex_complex(uminus1482373934393186551omplex,imaginary_unit)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(uminus_uminus_real,pi)),numeral_numeral_real(aa_num_num(bit0,one))) ).

% Arg_minus_ii
tff(fact_9319_csqrt__ii,axiom,
    aa_complex_complex(csqrt,imaginary_unit) = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,one_one_complex),imaginary_unit)),aa_real_complex(real_V4546457046886955230omplex,aa_real_real(sqrt,numeral_numeral_real(aa_num_num(bit0,one))))) ).

% csqrt_ii
tff(fact_9320_Arg__ii,axiom,
    aa_complex_real(arg,imaginary_unit) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))) ).

% Arg_ii
tff(fact_9321_csqrt__1,axiom,
    aa_complex_complex(csqrt,one_one_complex) = one_one_complex ).

% csqrt_1
tff(fact_9322_csqrt__eq__1,axiom,
    ! [Z: complex] :
      ( ( aa_complex_complex(csqrt,Z) = one_one_complex )
    <=> ( Z = one_one_complex ) ) ).

% csqrt_eq_1
tff(fact_9323_power2__csqrt,axiom,
    ! [Z: complex] : aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(csqrt,Z)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = Z ).

% power2_csqrt
tff(fact_9324_divide__complex__def,axiom,
    ! [Xb: complex,Y: complex] : aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,Xb),Y) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,Xb),aa_complex_complex(invers8013647133539491842omplex,Y)) ).

% divide_complex_def
tff(fact_9325_of__real__sqrt,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_complex(real_V4546457046886955230omplex,aa_real_real(sqrt,Xb)) = aa_complex_complex(csqrt,aa_real_complex(real_V4546457046886955230omplex,Xb)) ) ) ).

% of_real_sqrt
tff(fact_9326_Arg__bounded,axiom,
    ! [Z: complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,pi)),aa_complex_real(arg,Z))
      & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(arg,Z)),pi) ) ).

% Arg_bounded
tff(fact_9327_complex__inverse,axiom,
    ! [A: real,B: real] : aa_complex_complex(invers8013647133539491842omplex,aa_real_complex(complex2(A),B)) = aa_real_complex(complex2(aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% complex_inverse
tff(fact_9328_cis__minus__pi__half,axiom,
    aa_real_complex(cis,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))) = aa_complex_complex(uminus1482373934393186551omplex,imaginary_unit) ).

% cis_minus_pi_half
tff(fact_9329_sinh__real__le__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sinh_real,Xb)),aa_real_real(sinh_real,Y))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y) ) ).

% sinh_real_le_iff
tff(fact_9330_sinh__real__nonneg__iff,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(sinh_real,Xb))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb) ) ).

% sinh_real_nonneg_iff
tff(fact_9331_sinh__real__nonpos__iff,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sinh_real,Xb)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),zero_zero_real) ) ).

% sinh_real_nonpos_iff
tff(fact_9332_norm__cis,axiom,
    ! [A: real] : aa_complex_real(real_V1022390504157884413omplex,aa_real_complex(cis,A)) = one_one_real ).

% norm_cis
tff(fact_9333_cis__zero,axiom,
    aa_real_complex(cis,zero_zero_real) = one_one_complex ).

% cis_zero
tff(fact_9334_cis__pi,axiom,
    aa_real_complex(cis,pi) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).

% cis_pi
tff(fact_9335_cis__pi__half,axiom,
    aa_real_complex(cis,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one)))) = imaginary_unit ).

% cis_pi_half
tff(fact_9336_cis__2pi,axiom,
    aa_real_complex(cis,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),pi)) = one_one_complex ).

% cis_2pi
tff(fact_9337_sinh__le__cosh__real,axiom,
    ! [Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sinh_real,Xb)),aa_real_real(cosh_real,Xb)) ).

% sinh_le_cosh_real
tff(fact_9338_cosh__real__nonpos__le__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),zero_zero_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(cosh_real,Xb)),aa_real_real(cosh_real,Y))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),Xb) ) ) ) ).

% cosh_real_nonpos_le_iff
tff(fact_9339_cosh__real__nonneg__le__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(cosh_real,Xb)),aa_real_real(cosh_real,Y))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y) ) ) ) ).

% cosh_real_nonneg_le_iff
tff(fact_9340_cosh__real__nonneg,axiom,
    ! [Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(cosh_real,Xb)) ).

% cosh_real_nonneg
tff(fact_9341_cosh__real__ge__1,axiom,
    ! [Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(cosh_real,Xb)) ).

% cosh_real_ge_1
tff(fact_9342_cis__mult,axiom,
    ! [A: real,B: real] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_real_complex(cis,A)),aa_real_complex(cis,B)) = aa_real_complex(cis,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)) ).

% cis_mult
tff(fact_9343_cis__divide,axiom,
    ! [A: real,B: real] : aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_real_complex(cis,A)),aa_real_complex(cis,B)) = aa_real_complex(cis,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)) ).

% cis_divide
tff(fact_9344_cosh__real__strict__mono,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(cosh_real,Xb)),aa_real_real(cosh_real,Y)) ) ) ).

% cosh_real_strict_mono
tff(fact_9345_cosh__real__nonneg__less__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(cosh_real,Xb)),aa_real_real(cosh_real,Y))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y) ) ) ) ).

% cosh_real_nonneg_less_iff
tff(fact_9346_cosh__real__nonpos__less__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),zero_zero_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(cosh_real,Xb)),aa_real_real(cosh_real,Y))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Xb) ) ) ) ).

% cosh_real_nonpos_less_iff
tff(fact_9347_arcosh__cosh__real,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_real(arcosh_real,aa_real_real(cosh_real,Xb)) = Xb ) ) ).

% arcosh_cosh_real
tff(fact_9348_DeMoivre,axiom,
    ! [A: real,Nb: nat] : aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_real_complex(cis,A)),Nb) = aa_real_complex(cis,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),A)) ).

% DeMoivre
tff(fact_9349_cis__conv__exp,axiom,
    ! [B: real] : aa_real_complex(cis,B) = aa_complex_complex(exp_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,imaginary_unit),aa_real_complex(real_V4546457046886955230omplex,B))) ).

% cis_conv_exp
tff(fact_9350_cosh__ln__real,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_real(cosh_real,aa_real_real(ln_ln_real,Xb)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),aa_real_real(inverse_inverse_real,Xb))),numeral_numeral_real(aa_num_num(bit0,one))) ) ) ).

% cosh_ln_real
tff(fact_9351_sinh__ln__real,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_real(sinh_real,aa_real_real(ln_ln_real,Xb)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),aa_real_real(inverse_inverse_real,Xb))),numeral_numeral_real(aa_num_num(bit0,one))) ) ) ).

% sinh_ln_real
tff(fact_9352_bij__betw__roots__unity,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => aa_set_complex_o(bij_betw_nat_complex(aTP_Lamm_azr(Nb),set_ord_lessThan_nat(Nb)),collect_complex(aa_nat_fun_complex_o(aTP_Lamm_adr,Nb))) ) ).

% bij_betw_roots_unity
tff(fact_9353_cot__less__zero,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(uminus_uminus_real,pi)),numeral_numeral_real(aa_num_num(bit0,one)))),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(cot_real,Xb)),zero_zero_real) ) ) ).

% cot_less_zero
tff(fact_9354_cot__npi,axiom,
    ! [Nb: nat] : aa_real_real(cot_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),pi)) = zero_zero_real ).

% cot_npi
tff(fact_9355_cot__periodic,axiom,
    ! [Xb: real] : aa_real_real(cot_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),pi))) = aa_real_real(cot_real,Xb) ).

% cot_periodic
tff(fact_9356_cot__gt__zero,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(cot_real,Xb)) ) ) ).

% cot_gt_zero
tff(fact_9357_tan__cot_H,axiom,
    ! [Xb: real] : aa_real_real(tan_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one)))),Xb)) = aa_real_real(cot_real,Xb) ).

% tan_cot'
tff(fact_9358_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_9359_pred__numeral__simps_I2_J,axiom,
    ! [K: num] : aa_num_nat(pred_numeral,aa_num_num(bit0,K)) = aa_num_nat(numeral_numeral_nat,aa_num_num(bitM,K)) ).

% pred_numeral_simps(2)
tff(fact_9360_semiring__norm_I26_J,axiom,
    aa_num_num(bitM,one) = one ).

% semiring_norm(26)
tff(fact_9361_semiring__norm_I28_J,axiom,
    ! [Nb: num] : aa_num_num(bitM,aa_num_num(bit1,Nb)) = aa_num_num(bit1,aa_num_num(bit0,Nb)) ).

% semiring_norm(28)
tff(fact_9362_semiring__norm_I27_J,axiom,
    ! [Nb: num] : aa_num_num(bitM,aa_num_num(bit0,Nb)) = aa_num_num(bit1,aa_num_num(bitM,Nb)) ).

% semiring_norm(27)
tff(fact_9363_eval__nat__numeral_I2_J,axiom,
    ! [Nb: num] : aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Nb)) = aa_nat_nat(suc,aa_num_nat(numeral_numeral_nat,aa_num_num(bitM,Nb))) ).

% eval_nat_numeral(2)
tff(fact_9364_one__plus__BitM,axiom,
    ! [Nb: num] : aa_num_num(aa_num_fun_num_num(plus_plus_num,one),aa_num_num(bitM,Nb)) = aa_num_num(bit0,Nb) ).

% one_plus_BitM
tff(fact_9365_BitM__plus__one,axiom,
    ! [Nb: num] : aa_num_num(aa_num_fun_num_num(plus_plus_num,aa_num_num(bitM,Nb)),one) = aa_num_num(bit0,Nb) ).

% BitM_plus_one
tff(fact_9366_bij__betw__nth__root__unity,axiom,
    ! [C: complex,Nb: nat] :
      ( ( C != zero_zero_complex )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => aa_set_complex_o(bij_be1856998921033663316omplex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_real_complex(real_V4546457046886955230omplex,aa_real_real(root(Nb),aa_complex_real(real_V1022390504157884413omplex,C)))),aa_real_complex(cis,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_complex_real(arg,C)),aa_nat_real(semiri5074537144036343181t_real,Nb))))),collect_complex(aa_nat_fun_complex_o(aTP_Lamm_adr,Nb))),collect_complex(aa_nat_fun_complex_o(aTP_Lamm_azs(C),Nb))) ) ) ).

% bij_betw_nth_root_unity
tff(fact_9367_assnle,axiom,
    ! [TreeLista: list_VEBT_VEBT,Tree_is: list_VEBT_VEBTi,X13a: array_VEBT_VEBTi,Summarya: vEBT_VEBT,X14a: vEBT_VEBTi] : aa_assn_o(aa_assn_fun_assn_o(entails,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_lis2508312872377092907i_assn(vEBT_L6296928887356842470_VEBTi(vEBT_vebt_assn_raw,TreeLista),Tree_is)),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_lis2508312872377092907i_assn(snga_assn_VEBT_VEBTi(X13a),Tree_is)),aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Summarya),X14a)))),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Summarya),X14a)),aa_lis2508312872377092907i_assn(snga_assn_VEBT_VEBTi(X13a),Tree_is))),aa_lis2508312872377092907i_assn(vEBT_L6296928887356842470_VEBTi(vEBT_vebt_assn_raw,TreeLista),Tree_is))) ).

% assnle
tff(fact_9368_int__ge__less__than2__def,axiom,
    ! [D: int] : int_ge_less_than2(D) = collec213857154873943460nt_int(produc4947309494688390418_int_o(aTP_Lamm_azt(D))) ).

% int_ge_less_than2_def
tff(fact_9369_real__root__zero,axiom,
    ! [Nb: nat] : aa_real_real(root(Nb),zero_zero_real) = zero_zero_real ).

% real_root_zero
tff(fact_9370_real__root__Suc__0,axiom,
    ! [Xb: real] : aa_real_real(root(aa_nat_nat(suc,zero_zero_nat)),Xb) = Xb ).

% real_root_Suc_0
tff(fact_9371_real__root__eq__iff,axiom,
    ! [Nb: nat,Xb: real,Y: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( ( aa_real_real(root(Nb),Xb) = aa_real_real(root(Nb),Y) )
      <=> ( Xb = Y ) ) ) ).

% real_root_eq_iff
tff(fact_9372_root__0,axiom,
    ! [Xb: real] : aa_real_real(root(zero_zero_nat),Xb) = zero_zero_real ).

% root_0
tff(fact_9373_ent__pure__pre__iff,axiom,
    ! [P: assn,B: $o,Q: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(entails,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,P),aa_o_assn(pure_assn,(B)))),Q)
    <=> ( (B)
       => aa_assn_o(aa_assn_fun_assn_o(entails,P),Q) ) ) ).

% ent_pure_pre_iff
tff(fact_9374_ent__pure__pre__iff__sng,axiom,
    ! [B: $o,Q: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(entails,aa_o_assn(pure_assn,(B))),Q)
    <=> ( (B)
       => aa_assn_o(aa_assn_fun_assn_o(entails,one_one_assn),Q) ) ) ).

% ent_pure_pre_iff_sng
tff(fact_9375_real__root__eq__0__iff,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( ( aa_real_real(root(Nb),Xb) = zero_zero_real )
      <=> ( Xb = zero_zero_real ) ) ) ).

% real_root_eq_0_iff
tff(fact_9376_real__root__less__iff,axiom,
    ! [Nb: nat,Xb: real,Y: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(root(Nb),Xb)),aa_real_real(root(Nb),Y))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y) ) ) ).

% real_root_less_iff
tff(fact_9377_real__root__le__iff,axiom,
    ! [Nb: nat,Xb: real,Y: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(root(Nb),Xb)),aa_real_real(root(Nb),Y))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y) ) ) ).

% real_root_le_iff
tff(fact_9378_real__root__eq__1__iff,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( ( aa_real_real(root(Nb),Xb) = one_one_real )
      <=> ( Xb = one_one_real ) ) ) ).

% real_root_eq_1_iff
tff(fact_9379_real__root__one,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_real(root(Nb),one_one_real) = one_one_real ) ) ).

% real_root_one
tff(fact_9380_ent__pure__post__iff,axiom,
    ! [P: assn,Q: assn,B: $o] :
      ( aa_assn_o(aa_assn_fun_assn_o(entails,P),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Q),aa_o_assn(pure_assn,(B))))
    <=> ( ! [H4: produc3658429121746597890et_nat] :
            ( aa_Pro653082453696054616_nat_o(rep_assn(P),H4)
           => (B) )
        & aa_assn_o(aa_assn_fun_assn_o(entails,P),Q) ) ) ).

% ent_pure_post_iff
tff(fact_9381_ent__pure__post__iff__sng,axiom,
    ! [P: assn,B: $o] :
      ( aa_assn_o(aa_assn_fun_assn_o(entails,P),aa_o_assn(pure_assn,(B)))
    <=> ( ! [H4: produc3658429121746597890et_nat] :
            ( aa_Pro653082453696054616_nat_o(rep_assn(P),H4)
           => (B) )
        & aa_assn_o(aa_assn_fun_assn_o(entails,P),one_one_assn) ) ) ).

% ent_pure_post_iff_sng
tff(fact_9382_real__root__lt__0__iff,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(root(Nb),Xb)),zero_zero_real)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),zero_zero_real) ) ) ).

% real_root_lt_0_iff
tff(fact_9383_real__root__gt__0__iff,axiom,
    ! [Nb: nat,Y: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(root(Nb),Y))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y) ) ) ).

% real_root_gt_0_iff
tff(fact_9384_real__root__ge__0__iff,axiom,
    ! [Nb: nat,Y: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(root(Nb),Y))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y) ) ) ).

% real_root_ge_0_iff
tff(fact_9385_real__root__le__0__iff,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(root(Nb),Xb)),zero_zero_real)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),zero_zero_real) ) ) ).

% real_root_le_0_iff
tff(fact_9386_real__root__lt__1__iff,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(root(Nb),Xb)),one_one_real)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),one_one_real) ) ) ).

% real_root_lt_1_iff
tff(fact_9387_real__root__gt__1__iff,axiom,
    ! [Nb: nat,Y: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(root(Nb),Y))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Y) ) ) ).

% real_root_gt_1_iff
tff(fact_9388_real__root__le__1__iff,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(root(Nb),Xb)),one_one_real)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),one_one_real) ) ) ).

% real_root_le_1_iff
tff(fact_9389_real__root__ge__1__iff,axiom,
    ! [Nb: nat,Y: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(root(Nb),Y))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Y) ) ) ).

% real_root_ge_1_iff
tff(fact_9390_real__root__pow__pos2,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
       => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(root(Nb),Xb)),Nb) = Xb ) ) ) ).

% real_root_pow_pos2
tff(fact_9391_real__root__inverse,axiom,
    ! [Nb: nat,Xb: real] : aa_real_real(root(Nb),aa_real_real(inverse_inverse_real,Xb)) = aa_real_real(inverse_inverse_real,aa_real_real(root(Nb),Xb)) ).

% real_root_inverse
tff(fact_9392_fr__rot,axiom,
    ! [A2: assn,B4: assn,C4: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(entails,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A2),B4)),C4)
     => aa_assn_o(aa_assn_fun_assn_o(entails,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,B4),A2)),C4) ) ).

% fr_rot
tff(fact_9393_fr__refl,axiom,
    ! [A2: assn,B4: assn,C4: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(entails,A2),B4)
     => aa_assn_o(aa_assn_fun_assn_o(entails,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A2),C4)),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,B4),C4)) ) ).

% fr_refl
tff(fact_9394_fr__rot__rhs,axiom,
    ! [A2: assn,B4: assn,C4: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(entails,A2),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,B4),C4))
     => aa_assn_o(aa_assn_fun_assn_o(entails,A2),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,C4),B4)) ) ).

% fr_rot_rhs
tff(fact_9395_ent__frame__fwd,axiom,
    ! [P: assn,R: assn,Ps: assn,F4: assn,Q: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(entails,P),R)
     => ( aa_assn_o(aa_assn_fun_assn_o(entails,Ps),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,P),F4))
       => ( aa_assn_o(aa_assn_fun_assn_o(entails,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,R),F4)),Q)
         => aa_assn_o(aa_assn_fun_assn_o(entails,Ps),Q) ) ) ) ).

% ent_frame_fwd
tff(fact_9396_ent__star__mono,axiom,
    ! [P: assn,P3: assn,Q: assn,Q6: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(entails,P),P3)
     => ( aa_assn_o(aa_assn_fun_assn_o(entails,Q),Q6)
       => aa_assn_o(aa_assn_fun_assn_o(entails,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,P),Q)),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,P3),Q6)) ) ) ).

% ent_star_mono
tff(fact_9397_real__root__mult,axiom,
    ! [Nb: nat,Xb: real,Y: real] : aa_real_real(root(Nb),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Y)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(root(Nb),Xb)),aa_real_real(root(Nb),Y)) ).

% real_root_mult
tff(fact_9398_real__root__mult__exp,axiom,
    ! [M: nat,Nb: nat,Xb: real] : aa_real_real(root(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Nb)),Xb) = aa_real_real(root(M),aa_real_real(root(Nb),Xb)) ).

% real_root_mult_exp
tff(fact_9399_real__root__minus,axiom,
    ! [Nb: nat,Xb: real] : aa_real_real(root(Nb),aa_real_real(uminus_uminus_real,Xb)) = aa_real_real(uminus_uminus_real,aa_real_real(root(Nb),Xb)) ).

% real_root_minus
tff(fact_9400_is__entails,axiom,
    ! [P: assn,Q: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(entails,P),Q)
     => aa_assn_o(aa_assn_fun_assn_o(entails,P),Q) ) ).

% is_entails
tff(fact_9401_real__root__commute,axiom,
    ! [M: nat,Nb: nat,Xb: real] : aa_real_real(root(M),aa_real_real(root(Nb),Xb)) = aa_real_real(root(Nb),aa_real_real(root(M),Xb)) ).

% real_root_commute
tff(fact_9402_real__root__divide,axiom,
    ! [Nb: nat,Xb: real,Y: real] : aa_real_real(root(Nb),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Y)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(root(Nb),Xb)),aa_real_real(root(Nb),Y)) ).

% real_root_divide
tff(fact_9403_real__root__pos__pos__le,axiom,
    ! [Xb: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(root(Nb),Xb)) ) ).

% real_root_pos_pos_le
tff(fact_9404_mod__frame__fwd,axiom,
    ! [Ps: assn,H: produc3658429121746597890et_nat,P: assn,R: assn,F4: assn] :
      ( aa_Pro653082453696054616_nat_o(rep_assn(Ps),H)
     => ( aa_assn_o(aa_assn_fun_assn_o(entails,P),R)
       => ( aa_assn_o(aa_assn_fun_assn_o(entails,Ps),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,P),F4))
         => aa_Pro653082453696054616_nat_o(rep_assn(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,R),F4)),H) ) ) ) ).

% mod_frame_fwd
tff(fact_9405_ent__star__mono__true,axiom,
    ! [A2: assn,A8: assn,B4: assn,B9: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(entails,A2),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A8),top_top_assn))
     => ( aa_assn_o(aa_assn_fun_assn_o(entails,B4),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,B9),top_top_assn))
       => aa_assn_o(aa_assn_fun_assn_o(entails,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A2),B4)),top_top_assn)),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A8),B9)),top_top_assn)) ) ) ).

% ent_star_mono_true
tff(fact_9406_ent__refl__true,axiom,
    ! [A2: assn] : aa_assn_o(aa_assn_fun_assn_o(entails,A2),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A2),top_top_assn)) ).

% ent_refl_true
tff(fact_9407_ent__true__drop_I1_J,axiom,
    ! [P: assn,Q: assn,R: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(entails,P),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Q),top_top_assn))
     => aa_assn_o(aa_assn_fun_assn_o(entails,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,P),R)),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Q),top_top_assn)) ) ).

% ent_true_drop(1)
tff(fact_9408_ent__true__drop_I2_J,axiom,
    ! [P: assn,Q: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(entails,P),Q)
     => aa_assn_o(aa_assn_fun_assn_o(entails,P),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Q),top_top_assn)) ) ).

% ent_true_drop(2)
tff(fact_9409_real__root__less__mono,axiom,
    ! [Nb: nat,Xb: real,Y: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(root(Nb),Xb)),aa_real_real(root(Nb),Y)) ) ) ).

% real_root_less_mono
tff(fact_9410_real__root__le__mono,axiom,
    ! [Nb: nat,Xb: real,Y: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(root(Nb),Xb)),aa_real_real(root(Nb),Y)) ) ) ).

% real_root_le_mono
tff(fact_9411_real__root__power,axiom,
    ! [Nb: nat,Xb: real,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_real(root(Nb),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),K)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(root(Nb),Xb)),K) ) ) ).

% real_root_power
tff(fact_9412_real__root__abs,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_real(root(Nb),aa_real_real(abs_abs_real,Xb)) = aa_real_real(abs_abs_real,aa_real_real(root(Nb),Xb)) ) ) ).

% real_root_abs
tff(fact_9413_real__root__gt__zero,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(root(Nb),Xb)) ) ) ).

% real_root_gt_zero
tff(fact_9414_real__root__strict__decreasing,axiom,
    ! [Nb: nat,N: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),N)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xb)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(root(N),Xb)),aa_real_real(root(Nb),Xb)) ) ) ) ).

% real_root_strict_decreasing
tff(fact_9415_sqrt__def,axiom,
    sqrt = root(aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% sqrt_def
tff(fact_9416_root__abs__power,axiom,
    ! [Nb: nat,Y: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_real(abs_abs_real,aa_real_real(root(Nb),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),Nb))) = aa_real_real(abs_abs_real,Y) ) ) ).

% root_abs_power
tff(fact_9417_real__root__pos__pos,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(root(Nb),Xb)) ) ) ).

% real_root_pos_pos
tff(fact_9418_real__root__strict__increasing,axiom,
    ! [Nb: nat,N: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),N)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),one_one_real)
           => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(root(Nb),Xb)),aa_real_real(root(N),Xb)) ) ) ) ) ).

% real_root_strict_increasing
tff(fact_9419_real__root__decreasing,axiom,
    ! [Nb: nat,N: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Xb)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(root(N),Xb)),aa_real_real(root(Nb),Xb)) ) ) ) ).

% real_root_decreasing
tff(fact_9420_odd__real__root__pow,axiom,
    ! [Nb: nat,Xb: real] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(root(Nb),Xb)),Nb) = Xb ) ) ).

% odd_real_root_pow
tff(fact_9421_odd__real__root__unique,axiom,
    ! [Nb: nat,Y: real,Xb: real] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),Nb) = Xb )
       => ( aa_real_real(root(Nb),Xb) = Y ) ) ) ).

% odd_real_root_unique
tff(fact_9422_odd__real__root__power__cancel,axiom,
    ! [Nb: nat,Xb: real] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_real_real(root(Nb),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb)) = Xb ) ) ).

% odd_real_root_power_cancel
tff(fact_9423_real__root__pow__pos,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
       => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(root(Nb),Xb)),Nb) = Xb ) ) ) ).

% real_root_pow_pos
tff(fact_9424_real__root__pos__unique,axiom,
    ! [Nb: nat,Y: real,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
       => ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),Nb) = Xb )
         => ( aa_real_real(root(Nb),Xb) = Y ) ) ) ) ).

% real_root_pos_unique
tff(fact_9425_real__root__power__cancel,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
       => ( aa_real_real(root(Nb),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb)) = Xb ) ) ) ).

% real_root_power_cancel
tff(fact_9426_real__root__increasing,axiom,
    ! [Nb: nat,N: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),one_one_real)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(root(Nb),Xb)),aa_real_real(root(N),Xb)) ) ) ) ) ).

% real_root_increasing
tff(fact_9427_log__root,axiom,
    ! [Nb: nat,A: real,B: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
       => ( aa_real_real(aa_rea7340605660300375871l_real(log,B),aa_real_real(root(Nb),A)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(log,B),A)),aa_nat_real(semiri5074537144036343181t_real,Nb)) ) ) ) ).

% log_root
tff(fact_9428_log__base__root,axiom,
    ! [Nb: nat,B: real,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B)
       => ( aa_real_real(aa_rea7340605660300375871l_real(log,aa_real_real(root(Nb),B)),Xb) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_real_real(aa_rea7340605660300375871l_real(log,B),Xb)) ) ) ) ).

% log_base_root
tff(fact_9429_ln__root,axiom,
    ! [Nb: nat,B: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B)
       => ( aa_real_real(ln_ln_real,aa_real_real(root(Nb),B)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(ln_ln_real,B)),aa_nat_real(semiri5074537144036343181t_real,Nb)) ) ) ) ).

% ln_root
tff(fact_9430_root__powr__inverse,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
       => ( aa_real_real(root(Nb),Xb) = aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_nat_real(semiri5074537144036343181t_real,Nb))) ) ) ) ).

% root_powr_inverse
tff(fact_9431_int__ge__less__than__def,axiom,
    ! [D: int] : int_ge_less_than(D) = collec213857154873943460nt_int(produc4947309494688390418_int_o(aTP_Lamm_azu(D))) ).

% int_ge_less_than_def
tff(fact_9432_big__assn__simp,axiom,
    ! [H: nat,TreeLista: list_VEBT_VEBT,L: nat,Xb: vEBT_VEBTi,Xaa: option_nat,X13a: array_VEBT_VEBTi,Tree_is: list_VEBT_VEBTi,Summarya: vEBT_VEBT,X14a: vEBT_VEBTi] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,H),size_s6755466524823107622T_VEBT(TreeLista))
     => aa_assn_o(aa_assn_fun_assn_o(entails,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,vEBT_vebt_delete(nth_VEBT_VEBT(TreeLista,H),L)),Xb)),aa_o_assn(pure_assn,Xaa = vEBT_vebt_mint(vEBT_vebt_delete(nth_VEBT_VEBT(TreeLista,H),L))))),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_lis2508312872377092907i_assn(snga_assn_VEBT_VEBTi(X13a),list_u6098035379799741383_VEBTi(Tree_is,H,Xb))),aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Summarya),X14a))),aa_lis2508312872377092907i_assn(vEBT_L1528199826722428489_VEBTi(minus_minus_set_nat(set_or4665077453230672383an_nat(zero_zero_nat,size_s6755466524823107622T_VEBT(TreeLista)),insert_nat(H,bot_bot_set_nat)),vEBT_vebt_assn_raw,TreeLista),Tree_is)))),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_lis2508312872377092907i_assn(snga_assn_VEBT_VEBTi(X13a),list_u6098035379799741383_VEBTi(Tree_is,H,Xb))),aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Summarya),X14a))),aa_o_assn(pure_assn,Xaa = vEBT_vebt_mint(vEBT_vebt_delete(nth_VEBT_VEBT(TreeLista,H),L))))),aa_lis2508312872377092907i_assn(vEBT_L6296928887356842470_VEBTi(vEBT_vebt_assn_raw,list_u1324408373059187874T_VEBT(TreeLista,H,vEBT_vebt_delete(nth_VEBT_VEBT(TreeLista,H),L))),list_u6098035379799741383_VEBTi(Tree_is,H,Xb)))) ) ).

% big_assn_simp
tff(fact_9433_local_Oext,axiom,
    ! [Y: nat,TreeLista: list_VEBT_VEBT,X13a: array_VEBT_VEBTi,Tree_is: list_VEBT_VEBTi,Summarya: vEBT_VEBT,X14a: vEBT_VEBTi] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),size_s6755466524823107622T_VEBT(TreeLista))
     => aa_assn_o(aa_assn_fun_assn_o(entails,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_lis2508312872377092907i_assn(snga_assn_VEBT_VEBTi(X13a),Tree_is)),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Summarya),X14a)),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,nth_VEBT_VEBT(TreeLista,Y)),nth_VEBT_VEBTi(Tree_is,Y))),aa_lis2508312872377092907i_assn(vEBT_L1528199826722428489_VEBTi(minus_minus_set_nat(set_or4665077453230672383an_nat(zero_zero_nat,size_s6755466524823107622T_VEBT(TreeLista)),insert_nat(Y,bot_bot_set_nat)),vEBT_vebt_assn_raw,TreeLista),Tree_is))))),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_lis2508312872377092907i_assn(snga_assn_VEBT_VEBTi(X13a),Tree_is)),aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Summarya),X14a))),aa_lis2508312872377092907i_assn(vEBT_L1528199826722428489_VEBTi(minus_minus_set_nat(set_or4665077453230672383an_nat(zero_zero_nat,size_s6755466524823107622T_VEBT(TreeLista)),insert_nat(Y,bot_bot_set_nat)),vEBT_vebt_assn_raw,TreeLista),Tree_is))),aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,nth_VEBT_VEBT(TreeLista,Y)),nth_VEBT_VEBTi(Tree_is,Y)))) ) ).

% local.ext
tff(fact_9434_atLeastLessThan__singleton,axiom,
    ! [M: nat] : set_or4665077453230672383an_nat(M,aa_nat_nat(suc,M)) = insert_nat(M,bot_bot_set_nat) ).

% atLeastLessThan_singleton
tff(fact_9435_txe,axiom,
    ! [Y: nat,TreeLista: list_VEBT_VEBT,Tree_is: list_VEBT_VEBTi,X13a: array_VEBT_VEBTi,Summarya: vEBT_VEBT,X14a: vEBT_VEBTi] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),size_s6755466524823107622T_VEBT(TreeLista))
     => aa_assn_o(aa_assn_fun_assn_o(entails,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,nth_VEBT_VEBT(TreeLista,Y)),nth_VEBT_VEBTi(Tree_is,Y))),aa_lis2508312872377092907i_assn(snga_assn_VEBT_VEBTi(X13a),Tree_is))),aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Summarya),X14a))),aa_lis2508312872377092907i_assn(vEBT_L1528199826722428489_VEBTi(minus_minus_set_nat(set_or4665077453230672383an_nat(zero_zero_nat,size_s6755466524823107622T_VEBT(TreeLista)),insert_nat(Y,bot_bot_set_nat)),vEBT_vebt_assn_raw,TreeLista),Tree_is))),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Summarya),X14a)),aa_lis2508312872377092907i_assn(snga_assn_VEBT_VEBTi(X13a),Tree_is))),aa_lis2508312872377092907i_assn(vEBT_L6296928887356842470_VEBTi(vEBT_vebt_assn_raw,TreeLista),Tree_is))) ) ).

% txe
tff(fact_9436_repack,axiom,
    ! [I: nat,TreeLista: list_VEBT_VEBT,Tree_is: list_VEBT_VEBTi,Rest: assn,X13a: array_VEBT_VEBTi,Summarya: vEBT_VEBT,X14a: vEBT_VEBTi] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s6755466524823107622T_VEBT(TreeLista))
     => aa_assn_o(aa_assn_fun_assn_o(entails,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,nth_VEBT_VEBT(TreeLista,I)),nth_VEBT_VEBTi(Tree_is,I))),Rest)),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_lis2508312872377092907i_assn(snga_assn_VEBT_VEBTi(X13a),Tree_is)),aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Summarya),X14a))),aa_lis2508312872377092907i_assn(vEBT_L1528199826722428489_VEBTi(minus_minus_set_nat(set_or4665077453230672383an_nat(zero_zero_nat,size_s6755466524823107622T_VEBT(TreeLista)),insert_nat(I,bot_bot_set_nat)),vEBT_vebt_assn_raw,TreeLista),Tree_is)))),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Rest),aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Summarya),X14a))),aa_lis2508312872377092907i_assn(snga_assn_VEBT_VEBTi(X13a),Tree_is))),aa_lis2508312872377092907i_assn(vEBT_L6296928887356842470_VEBTi(vEBT_vebt_assn_raw,TreeLista),Tree_is))) ) ).

% repack
tff(fact_9437_recomp,axiom,
    ! [I: nat,TreeLista: list_VEBT_VEBT,Tree_is: list_VEBT_VEBTi,X13a: array_VEBT_VEBTi,Summarya: vEBT_VEBT,X14a: vEBT_VEBTi] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s6755466524823107622T_VEBT(TreeLista))
     => aa_assn_o(aa_assn_fun_assn_o(entails,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,nth_VEBT_VEBT(TreeLista,I)),nth_VEBT_VEBTi(Tree_is,I))),aa_lis2508312872377092907i_assn(vEBT_L1528199826722428489_VEBTi(minus_minus_set_nat(set_or4665077453230672383an_nat(zero_zero_nat,size_s6755466524823107622T_VEBT(TreeLista)),insert_nat(I,bot_bot_set_nat)),vEBT_vebt_assn_raw,TreeLista),Tree_is))),aa_lis2508312872377092907i_assn(snga_assn_VEBT_VEBTi(X13a),Tree_is))),aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Summarya),X14a))),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Summarya),X14a)),aa_lis2508312872377092907i_assn(snga_assn_VEBT_VEBTi(X13a),Tree_is))),aa_lis2508312872377092907i_assn(vEBT_L6296928887356842470_VEBTi(vEBT_vebt_assn_raw,TreeLista),Tree_is))) ) ).

% recomp
tff(fact_9438_big__assn__simp_H,axiom,
    ! [H: nat,TreeLista: list_VEBT_VEBT,Xaa: vEBT_VEBT,L: nat,Xb: vEBT_VEBTi,Xba: option_nat,X13a: array_VEBT_VEBTi,Tree_is: list_VEBT_VEBTi,Summarya: vEBT_VEBT,X14a: vEBT_VEBTi] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,H),size_s6755466524823107622T_VEBT(TreeLista))
     => ( ( Xaa = vEBT_vebt_delete(nth_VEBT_VEBT(TreeLista,H),L) )
       => aa_assn_o(aa_assn_fun_assn_o(entails,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Xaa),Xb)),aa_o_assn(pure_assn,Xba = vEBT_vebt_mint(Xaa)))),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_lis2508312872377092907i_assn(snga_assn_VEBT_VEBTi(X13a),list_u6098035379799741383_VEBTi(Tree_is,H,Xb))),aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Summarya),X14a))),aa_lis2508312872377092907i_assn(vEBT_L1528199826722428489_VEBTi(minus_minus_set_nat(set_or4665077453230672383an_nat(zero_zero_nat,size_s6755466524823107622T_VEBT(TreeLista)),insert_nat(H,bot_bot_set_nat)),vEBT_vebt_assn_raw,TreeLista),Tree_is)))),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_lis2508312872377092907i_assn(snga_assn_VEBT_VEBTi(X13a),list_u6098035379799741383_VEBTi(Tree_is,H,Xb))),aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Summarya),X14a))),aa_o_assn(pure_assn,Xba = vEBT_vebt_mint(Xaa)))),aa_lis2508312872377092907i_assn(vEBT_L6296928887356842470_VEBTi(vEBT_vebt_assn_raw,list_u1324408373059187874T_VEBT(TreeLista,H,Xaa)),list_u6098035379799741383_VEBTi(Tree_is,H,Xb)))) ) ) ).

% big_assn_simp'
tff(fact_9439_ex__nat__less__eq,axiom,
    ! [Nb: nat,P: fun_nat_o] :
      ( ? [M7: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M7),Nb)
          & aa_nat_o(P,M7) )
    <=> ? [X: nat] :
          ( aa_set_nat_o(member_nat(X),set_or4665077453230672383an_nat(zero_zero_nat,Nb))
          & aa_nat_o(P,X) ) ) ).

% ex_nat_less_eq
tff(fact_9440_all__nat__less__eq,axiom,
    ! [Nb: nat,P: fun_nat_o] :
      ( ! [M7: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M7),Nb)
         => aa_nat_o(P,M7) )
    <=> ! [X: nat] :
          ( aa_set_nat_o(member_nat(X),set_or4665077453230672383an_nat(zero_zero_nat,Nb))
         => aa_nat_o(P,X) ) ) ).

% all_nat_less_eq
tff(fact_9441_atLeastLessThanSuc__atLeastAtMost,axiom,
    ! [L: nat,U: nat] : set_or4665077453230672383an_nat(L,aa_nat_nat(suc,U)) = set_or1269000886237332187st_nat(L,U) ).

% atLeastLessThanSuc_atLeastAtMost
tff(fact_9442_atLeast0__lessThan__Suc,axiom,
    ! [Nb: nat] : set_or4665077453230672383an_nat(zero_zero_nat,aa_nat_nat(suc,Nb)) = insert_nat(Nb,set_or4665077453230672383an_nat(zero_zero_nat,Nb)) ).

% atLeast0_lessThan_Suc
tff(fact_9443_subset__eq__atLeast0__lessThan__finite,axiom,
    ! [N: set_nat,Nb: nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(N),set_or4665077453230672383an_nat(zero_zero_nat,Nb))
     => aa_set_nat_o(finite_finite_nat,N) ) ).

% subset_eq_atLeast0_lessThan_finite
tff(fact_9444_atLeastLessThanSuc,axiom,
    ! [M: nat,Nb: nat] :
      set_or4665077453230672383an_nat(M,aa_nat_nat(suc,Nb)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb),insert_nat(Nb,set_or4665077453230672383an_nat(M,Nb)),bot_bot_set_nat) ).

% atLeastLessThanSuc
tff(fact_9445_prod__Suc__Suc__fact,axiom,
    ! [Nb: nat] : groups708209901874060359at_nat(suc,set_or4665077453230672383an_nat(aa_nat_nat(suc,zero_zero_nat),Nb)) = aa_nat_nat(semiri1408675320244567234ct_nat,Nb) ).

% prod_Suc_Suc_fact
tff(fact_9446_prod__Suc__fact,axiom,
    ! [Nb: nat] : groups708209901874060359at_nat(suc,set_or4665077453230672383an_nat(zero_zero_nat,Nb)) = aa_nat_nat(semiri1408675320244567234ct_nat,Nb) ).

% prod_Suc_fact
tff(fact_9447_atLeastLessThan__nat__numeral,axiom,
    ! [M: nat,K: num] :
      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),set_or4665077453230672383an_nat(M,aa_num_nat(pred_numeral,K))),bot_bot_set_nat) ).

% atLeastLessThan_nat_numeral
tff(fact_9448_atLeast1__lessThan__eq__remove0,axiom,
    ! [Nb: nat] : set_or4665077453230672383an_nat(aa_nat_nat(suc,zero_zero_nat),Nb) = minus_minus_set_nat(set_ord_lessThan_nat(Nb),insert_nat(zero_zero_nat,bot_bot_set_nat)) ).

% atLeast1_lessThan_eq_remove0
tff(fact_9449_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)))),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_9450_Sum__Ico__nat,axiom,
    ! [M: nat,Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_di),set_or4665077453230672383an_nat(M,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,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_9451_Chebyshev__sum__upper__nat,axiom,
    ! [Nb: nat,A: fun_nat_nat,B: fun_nat_nat] :
      ( ! [I3: nat,J2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I3),J2)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J2),Nb)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(A,I3)),aa_nat_nat(A,J2)) ) )
     => ( ! [I3: nat,J2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I3),J2)
           => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J2),Nb)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(B,J2)),aa_nat_nat(B,I3)) ) )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_awm(A),B)),set_or4665077453230672383an_nat(zero_zero_nat,Nb)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(groups3542108847815614940at_nat(A),set_or4665077453230672383an_nat(zero_zero_nat,Nb))),aa_set_nat_nat(groups3542108847815614940at_nat(B),set_or4665077453230672383an_nat(zero_zero_nat,Nb)))) ) ) ).

% Chebyshev_sum_upper_nat
tff(fact_9452_finite__atLeastZeroLessThan__integer,axiom,
    ! [U: code_integer] : aa_set5643387332196575111eger_o(finite6017078050557962740nteger,set_or8404916559141939852nteger(zero_z3403309356797280102nteger,U)) ).

% finite_atLeastZeroLessThan_integer
tff(fact_9453_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_9454_atLeastLessThanPlusOne__atLeastAtMost__int,axiom,
    ! [L: int,U: int] : set_or4662586982721622107an_int(L,aa_int_int(aa_int_fun_int_int(plus_plus_int,U),one_one_int)) = set_or1266510415728281911st_int(L,U) ).

% atLeastLessThanPlusOne_atLeastAtMost_int
tff(fact_9455_set__decode__0,axiom,
    ! [Xb: nat] :
      ( aa_set_nat_o(member_nat(zero_zero_nat),nat_set_decode(Xb))
    <=> ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Xb) ) ).

% set_decode_0
tff(fact_9456_forall__finite_I3_J,axiom,
    ! [Xb: nat,P: fun_nat_o] :
      ( ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),aa_nat_nat(suc,aa_nat_nat(suc,Xb)))
         => aa_nat_o(P,I2) )
    <=> ( aa_nat_o(P,zero_zero_nat)
        & ! [I2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),aa_nat_nat(suc,Xb))
           => aa_nat_o(P,aa_nat_nat(suc,I2)) ) ) ) ).

% forall_finite(3)
tff(fact_9457_Suc__0__mod__eq,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,zero_zero_nat)),Nb) = aa_o_nat(zero_n2687167440665602831ol_nat,Nb != aa_nat_nat(suc,zero_zero_nat)) ).

% Suc_0_mod_eq
tff(fact_9458_finite__atLeastLessThan__integer,axiom,
    ! [L: code_integer,U: code_integer] : aa_set5643387332196575111eger_o(finite6017078050557962740nteger,set_or8404916559141939852nteger(L,U)) ).

% finite_atLeastLessThan_integer
tff(fact_9459_finite__atLeastAtMost__integer,axiom,
    ! [L: code_integer,U: code_integer] : aa_set5643387332196575111eger_o(finite6017078050557962740nteger,set_or189985376899183464nteger(L,U)) ).

% finite_atLeastAtMost_integer
tff(fact_9460_Divides_Oadjust__div__eq,axiom,
    ! [Q2: int,R2: int] : aa_Pro6799258434465214121nt_int(adjust_div,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q2),R2)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Q2),aa_o_int(zero_n2684676970156552555ol_int,R2 != zero_zero_int)) ).

% Divides.adjust_div_eq
tff(fact_9461_set__decode__Suc,axiom,
    ! [Nb: nat,Xb: nat] :
      ( aa_set_nat_o(member_nat(aa_nat_nat(suc,Nb)),nat_set_decode(Xb))
    <=> aa_set_nat_o(member_nat(Nb),nat_set_decode(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ).

% set_decode_Suc
tff(fact_9462_Divides_Oadjust__div__def,axiom,
    ! [Qr: product_prod_int_int] : aa_Pro6799258434465214121nt_int(adjust_div,Qr) = aa_Pro6799258434465214121nt_int(produc8211389475949308722nt_int(aTP_Lamm_azv),Qr) ).

% Divides.adjust_div_def
tff(fact_9463_subset__decode__imp__le,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(nat_set_decode(M)),nat_set_decode(Nb))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb) ) ).

% subset_decode_imp_le
tff(fact_9464_forall__finite_I1_J,axiom,
    ! [P: fun_nat_o,I4: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),zero_zero_nat)
     => aa_nat_o(P,I4) ) ).

% forall_finite(1)
tff(fact_9465_set__decode__plus__power__2,axiom,
    ! [Nb: nat,Z: nat] :
      ( ~ aa_set_nat_o(member_nat(Nb),nat_set_decode(Z))
     => ( nat_set_decode(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)),Z)) = insert_nat(Nb,nat_set_decode(Z)) ) ) ).

% set_decode_plus_power_2
tff(fact_9466_set__decode__def,axiom,
    ! [Xb: nat] : nat_set_decode(Xb) = collect_nat(aa_nat_fun_nat_o(aTP_Lamm_azw,Xb)) ).

% set_decode_def
tff(fact_9467_Comparator__Generator_OAll__less__Suc,axiom,
    ! [Xb: nat,P: fun_nat_o] :
      ( ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),aa_nat_nat(suc,Xb))
         => aa_nat_o(P,I2) )
    <=> ( aa_nat_o(P,zero_zero_nat)
        & ! [I2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),Xb)
           => aa_nat_o(P,aa_nat_nat(suc,I2)) ) ) ) ).

% Comparator_Generator.All_less_Suc
tff(fact_9468_forall__finite_I2_J,axiom,
    ! [P: fun_nat_o] :
      ( ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),aa_nat_nat(suc,zero_zero_nat))
         => aa_nat_o(P,I2) )
    <=> aa_nat_o(P,zero_zero_nat) ) ).

% forall_finite(2)
tff(fact_9469_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_9470_and__int_Opelims,axiom,
    ! [Xb: int,Xaa: int,Y: int] :
      ( ( aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,Xb),Xaa) = Y )
     => ( aa_Pro4592068669816069595_int_o(accp_P1096762738010456898nt_int(bit_and_int_rel),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Xb),Xaa))
       => ~ ( ( Y = $ite(
                  ( aa_set_int_o(member_int(Xb),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int)))
                  & aa_set_int_o(member_int(Xaa),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int))) ),
                  aa_int_int(uminus_uminus_int,
                    aa_o_int(zero_n2684676970156552555ol_int,
                      ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Xb)
                      & ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Xaa) ))),
                  aa_int_int(aa_int_fun_int_int(plus_plus_int,
                      aa_o_int(zero_n2684676970156552555ol_int,
                        ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Xb)
                        & ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Xaa) ))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,Xb),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(divide_divide_int,Xaa),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ) )
           => ~ aa_Pro4592068669816069595_int_o(accp_P1096762738010456898nt_int(bit_and_int_rel),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Xb),Xaa)) ) ) ) ).

% and_int.pelims
tff(fact_9471_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_9472_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_9473_and__minus__numerals_I6_J,axiom,
    ! [Nb: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb)))),one_one_int) = one_one_int ).

% and_minus_numerals(6)
tff(fact_9474_and__minus__numerals_I2_J,axiom,
    ! [Nb: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb)))) = one_one_int ).

% and_minus_numerals(2)
tff(fact_9475_and__minus__numerals_I1_J,axiom,
    ! [Nb: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb)))) = zero_zero_int ).

% and_minus_numerals(1)
tff(fact_9476_and__minus__numerals_I5_J,axiom,
    ! [Nb: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb)))),one_one_int) = zero_zero_int ).

% and_minus_numerals(5)
tff(fact_9477_AND__upper2_H,axiom,
    ! [Y: int,Z: int,Xb: 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,Xb),Y)),Z) ) ) ).

% AND_upper2'
tff(fact_9478_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_9479_AND__upper2,axiom,
    ! [Y: int,Xb: 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,Xb),Y)),Y) ) ).

% AND_upper2
tff(fact_9480_AND__upper1,axiom,
    ! [Xb: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xb)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,Xb),Y)),Xb) ) ).

% AND_upper1
tff(fact_9481_AND__lower,axiom,
    ! [Xb: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xb)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,Xb),Y)) ) ).

% AND_lower
tff(fact_9482_AND__upper2_H_H,axiom,
    ! [Y: int,Z: int,Xb: 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,Xb),Y)),Z) ) ) ).

% AND_upper2''
tff(fact_9483_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_9484_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_9485_even__and__iff__int,axiom,
    ! [K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,K),L))
    <=> ( 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) ) ) ).

% even_and_iff_int
tff(fact_9486_and__int__rec,axiom,
    ! [K: int,L: int] :
      aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,K),L) = aa_int_int(aa_int_fun_int_int(plus_plus_int,
          aa_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_rec
tff(fact_9487_and__int__unfold,axiom,
    ! [K: int,L: int] :
      aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,K),L) = $ite(
        ( ( K = zero_zero_int )
        | ( L = zero_zero_int ) ),
        zero_zero_int,
        $ite(
          K = aa_int_int(uminus_uminus_int,one_one_int),
          L,
          $ite(L = aa_int_int(uminus_uminus_int,one_one_int),K,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(divide_divide_int,L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))))) ) ) ).

% and_int_unfold
tff(fact_9488_and__int_Oelims,axiom,
    ! [Xb: int,Xaa: int,Y: int] :
      ( ( aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,Xb),Xaa) = Y )
     => ( Y = $ite(
            ( aa_set_int_o(member_int(Xb),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int)))
            & aa_set_int_o(member_int(Xaa),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int))) ),
            aa_int_int(uminus_uminus_int,
              aa_o_int(zero_n2684676970156552555ol_int,
                ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Xb)
                & ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Xaa) ))),
            aa_int_int(aa_int_fun_int_int(plus_plus_int,
                aa_o_int(zero_n2684676970156552555ol_int,
                  ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Xb)
                  & ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Xaa) ))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,Xb),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(divide_divide_int,Xaa),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ) ) ) ).

% and_int.elims
tff(fact_9489_uint32_Osize__eq,axiom,
    ! [P2: uint32] : size_size_uint32(P2) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,one)))))) ).

% uint32.size_eq
tff(fact_9490_and__nat__numerals_I1_J,axiom,
    ! [Y: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_nat_nat(suc,zero_zero_nat)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Y))) = zero_zero_nat ).

% and_nat_numerals(1)
tff(fact_9491_and__nat__numerals_I3_J,axiom,
    ! [Xb: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Xb))),aa_nat_nat(suc,zero_zero_nat)) = zero_zero_nat ).

% and_nat_numerals(3)
tff(fact_9492_and__nat__numerals_I2_J,axiom,
    ! [Y: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_nat_nat(suc,zero_zero_nat)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Y))) = one_one_nat ).

% and_nat_numerals(2)
tff(fact_9493_and__nat__numerals_I4_J,axiom,
    ! [Xb: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Xb))),aa_nat_nat(suc,zero_zero_nat)) = one_one_nat ).

% and_nat_numerals(4)
tff(fact_9494_and__Suc__0__eq,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,Nb),aa_nat_nat(suc,zero_zero_nat)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% and_Suc_0_eq
tff(fact_9495_Suc__0__and__eq,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_nat_nat(suc,zero_zero_nat)),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% Suc_0_and_eq
tff(fact_9496_and__nat__unfold,axiom,
    ! [M: nat,Nb: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,M),Nb) = $ite(
        ( ( M = zero_zero_nat )
        | ( Nb = zero_zero_nat ) ),
        zero_zero_nat,
        aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ).

% and_nat_unfold
tff(fact_9497_and__nat__rec,axiom,
    ! [M: nat,Nb: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,M),Nb) = 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))),M)
            & ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb) ))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% and_nat_rec
tff(fact_9498_take__bit__num__simps_I1_J,axiom,
    ! [M: num] : aa_num_option_num(bit_take_bit_num(zero_zero_nat),M) = none_num ).

% take_bit_num_simps(1)
tff(fact_9499_take__bit__num__simps_I2_J,axiom,
    ! [Nb: nat] : aa_num_option_num(bit_take_bit_num(aa_nat_nat(suc,Nb)),one) = aa_num_option_num(some_num,one) ).

% take_bit_num_simps(2)
tff(fact_9500_take__bit__num__simps_I5_J,axiom,
    ! [R2: num] : aa_num_option_num(bit_take_bit_num(aa_num_nat(numeral_numeral_nat,R2)),one) = aa_num_option_num(some_num,one) ).

% take_bit_num_simps(5)
tff(fact_9501_pred__numeral__inc,axiom,
    ! [K: num] : aa_num_nat(pred_numeral,aa_num_num(inc,K)) = aa_num_nat(numeral_numeral_nat,K) ).

% pred_numeral_inc
tff(fact_9502_take__bit__num__simps_I3_J,axiom,
    ! [Nb: nat,M: num] : aa_num_option_num(bit_take_bit_num(aa_nat_nat(suc,Nb)),aa_num_num(bit0,M)) = aa_opt5369159329878751634on_num(case_o6005452278849405969um_num(none_num,aTP_Lamm_azx),aa_num_option_num(bit_take_bit_num(Nb),M)) ).

% take_bit_num_simps(3)
tff(fact_9503_take__bit__of__Suc__0,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),aa_nat_nat(suc,zero_zero_nat)) = aa_o_nat(zero_n2687167440665602831ol_nat,aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)) ).

% take_bit_of_Suc_0
tff(fact_9504_take__bit__num__simps_I4_J,axiom,
    ! [Nb: nat,M: num] : aa_num_option_num(bit_take_bit_num(aa_nat_nat(suc,Nb)),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(Nb),M))) ).

% take_bit_num_simps(4)
tff(fact_9505_take__bit__num__simps_I6_J,axiom,
    ! [R2: num,M: num] : aa_num_option_num(bit_take_bit_num(aa_num_nat(numeral_numeral_nat,R2)),aa_num_num(bit0,M)) = aa_opt5369159329878751634on_num(case_o6005452278849405969um_num(none_num,aTP_Lamm_azx),aa_num_option_num(bit_take_bit_num(aa_num_nat(pred_numeral,R2)),M)) ).

% take_bit_num_simps(6)
tff(fact_9506_take__bit__num__simps_I7_J,axiom,
    ! [R2: num,M: num] : aa_num_option_num(bit_take_bit_num(aa_num_nat(numeral_numeral_nat,R2)),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,R2)),M))) ).

% take_bit_num_simps(7)
tff(fact_9507_take__bit__numeral__minus__numeral__int,axiom,
    ! [M: num,Nb: 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,Nb))) = aa_option_num_int(case_option_int_num(zero_zero_int,aTP_Lamm_azy(M)),aa_num_option_num(bit_take_bit_num(aa_num_nat(numeral_numeral_nat,M)),Nb)) ).

% take_bit_numeral_minus_numeral_int
tff(fact_9508_take__bit__nat__eq,axiom,
    ! [K: int,Nb: 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,Nb),aa_int_nat(nat2,K)) = aa_int_nat(nat2,aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,Nb),K)) ) ) ).

% take_bit_nat_eq
tff(fact_9509_nat__take__bit__eq,axiom,
    ! [K: int,Nb: 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,Nb),K)) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),aa_int_nat(nat2,K)) ) ) ).

% nat_take_bit_eq
tff(fact_9510_take__bit__nat__less__eq__self,axiom,
    ! [Nb: 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,Nb),M)),M) ).

% take_bit_nat_less_eq_self
tff(fact_9511_take__bit__tightened__less__eq__nat,axiom,
    ! [M: nat,Nb: nat,Q2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,M),Q2)),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),Q2)) ) ).

% take_bit_tightened_less_eq_nat
tff(fact_9512_take__bit__mult,axiom,
    ! [Nb: nat,K: int,L: int] : aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,Nb),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,Nb),K)),aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,Nb),L))) = aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,Nb),aa_int_int(aa_int_fun_int_int(times_times_int,K),L)) ).

% take_bit_mult
tff(fact_9513_take__bit__diff,axiom,
    ! [Nb: nat,K: int,L: int] : aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,Nb),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,Nb),K)),aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,Nb),L))) = aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,Nb),aa_int_int(aa_int_fun_int_int(minus_minus_int,K),L)) ).

% take_bit_diff
tff(fact_9514_add__inc,axiom,
    ! [Xb: num,Y: num] : aa_num_num(aa_num_fun_num_num(plus_plus_num,Xb),aa_num_num(inc,Y)) = aa_num_num(inc,aa_num_num(aa_num_fun_num_num(plus_plus_num,Xb),Y)) ).

% add_inc
tff(fact_9515_num__induct,axiom,
    ! [P: fun_num_o,Xb: num] :
      ( aa_num_o(P,one)
     => ( ! [X2: num] :
            ( aa_num_o(P,X2)
           => aa_num_o(P,aa_num_num(inc,X2)) )
       => aa_num_o(P,Xb) ) ) ).

% num_induct
tff(fact_9516_take__bit__tightened__less__eq__int,axiom,
    ! [M: nat,Nb: nat,K: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
     => 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,Nb),K)) ) ).

% take_bit_tightened_less_eq_int
tff(fact_9517_take__bit__nonnegative,axiom,
    ! [Nb: 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,Nb),K)) ).

% take_bit_nonnegative
tff(fact_9518_take__bit__int__less__eq__self__iff,axiom,
    ! [Nb: 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,Nb),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_9519_inc_Osimps_I1_J,axiom,
    aa_num_num(inc,one) = aa_num_num(bit0,one) ).

% inc.simps(1)
tff(fact_9520_inc_Osimps_I2_J,axiom,
    ! [Xb: num] : aa_num_num(inc,aa_num_num(bit0,Xb)) = aa_num_num(bit1,Xb) ).

% inc.simps(2)
tff(fact_9521_inc_Osimps_I3_J,axiom,
    ! [Xb: num] : aa_num_num(inc,aa_num_num(bit1,Xb)) = aa_num_num(bit0,aa_num_num(inc,Xb)) ).

% inc.simps(3)
tff(fact_9522_add__One,axiom,
    ! [Xb: num] : aa_num_num(aa_num_fun_num_num(plus_plus_num,Xb),one) = aa_num_num(inc,Xb) ).

% add_One
tff(fact_9523_inc__BitM__eq,axiom,
    ! [Nb: num] : aa_num_num(inc,aa_num_num(bitM,Nb)) = aa_num_num(bit0,Nb) ).

% inc_BitM_eq
tff(fact_9524_BitM__inc__eq,axiom,
    ! [Nb: num] : aa_num_num(bitM,aa_num_num(inc,Nb)) = aa_num_num(bit1,Nb) ).

% BitM_inc_eq
tff(fact_9525_mult__inc,axiom,
    ! [Xb: num,Y: num] : aa_num_num(aa_num_fun_num_num(times_times_num,Xb),aa_num_num(inc,Y)) = aa_num_num(aa_num_fun_num_num(plus_plus_num,aa_num_num(aa_num_fun_num_num(times_times_num,Xb),Y)),Xb) ).

% mult_inc
tff(fact_9526_take__bit__decr__eq,axiom,
    ! [Nb: nat,K: int] :
      ( ( aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,Nb),K) != zero_zero_int )
     => ( aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,Nb),aa_int_int(aa_int_fun_int_int(minus_minus_int,K),one_one_int)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,Nb),K)),one_one_int) ) ) ).

% take_bit_decr_eq
tff(fact_9527_take__bit__nat__eq__self,axiom,
    ! [M: nat,Nb: 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))),Nb))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),M) = M ) ) ).

% take_bit_nat_eq_self
tff(fact_9528_take__bit__nat__less__exp,axiom,
    ! [Nb: 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,Nb),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ).

% take_bit_nat_less_exp
tff(fact_9529_take__bit__nat__eq__self__iff,axiom,
    ! [Nb: nat,M: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),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))),Nb)) ) ).

% take_bit_nat_eq_self_iff
tff(fact_9530_take__bit__nat__def,axiom,
    ! [Nb: nat,M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),M) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ).

% take_bit_nat_def
tff(fact_9531_take__bit__Suc__minus__bit1,axiom,
    ! [Nb: nat,K: num] : aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,aa_nat_nat(suc,Nb)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,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,Nb),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(inc,K))))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),one_one_int) ).

% take_bit_Suc_minus_bit1
tff(fact_9532_take__bit__int__less__exp,axiom,
    ! [Nb: 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,Nb),K)),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) ).

% take_bit_int_less_exp
tff(fact_9533_take__bit__int__def,axiom,
    ! [Nb: nat,K: int] : aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,Nb),K) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) ).

% take_bit_int_def
tff(fact_9534_take__bit__numeral__minus__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_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,K)))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,aa_num_nat(pred_numeral,L)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(inc,K))))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),one_one_int) ).

% take_bit_numeral_minus_bit1
tff(fact_9535_take__bit__nat__less__self__iff,axiom,
    ! [Nb: 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,Nb),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))),Nb)),M) ) ).

% take_bit_nat_less_self_iff
tff(fact_9536_take__bit__Suc__minus__bit0,axiom,
    ! [Nb: nat,K: num] : aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,aa_nat_nat(suc,Nb)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,K)))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,Nb),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K)))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% take_bit_Suc_minus_bit0
tff(fact_9537_take__bit__int__greater__eq__self__iff,axiom,
    ! [K: int,Nb: 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,Nb),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))),Nb)) ) ).

% take_bit_int_greater_eq_self_iff
tff(fact_9538_take__bit__int__less__self__iff,axiom,
    ! [Nb: 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,Nb),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))),Nb)),K) ) ).

% take_bit_int_less_self_iff
tff(fact_9539_take__bit__int__eq__self__iff,axiom,
    ! [Nb: nat,K: int] :
      ( ( aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,Nb),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))),Nb)) ) ) ).

% take_bit_int_eq_self_iff
tff(fact_9540_take__bit__int__eq__self,axiom,
    ! [K: int,Nb: 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))),Nb))
       => ( aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,Nb),K) = K ) ) ) ).

% take_bit_int_eq_self
tff(fact_9541_take__bit__incr__eq,axiom,
    ! [Nb: nat,K: int] :
      ( ( aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,Nb),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))),Nb)),one_one_int) )
     => ( aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,Nb),aa_int_int(aa_int_fun_int_int(plus_plus_int,K),one_one_int)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,Nb),K)) ) ) ).

% take_bit_incr_eq
tff(fact_9542_take__bit__numeral__minus__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_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,K)))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,aa_num_nat(pred_numeral,L)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K)))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% take_bit_numeral_minus_bit0
tff(fact_9543_take__bit__int__less__eq,axiom,
    ! [Nb: 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))),Nb)),K)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,Nb),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))),Nb))) ) ) ).

% take_bit_int_less_eq
tff(fact_9544_take__bit__int__greater__eq,axiom,
    ! [K: int,Nb: 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))),Nb))),aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,Nb),K)) ) ).

% take_bit_int_greater_eq
tff(fact_9545_signed__take__bit__eq__take__bit__shift,axiom,
    ! [Nb: nat,K: int] : aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,Nb),K) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,aa_nat_nat(suc,Nb)),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))),Nb)))),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) ).

% signed_take_bit_eq_take_bit_shift
tff(fact_9546_take__bit__minus__small__eq,axiom,
    ! [K: int,Nb: 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))),Nb))
       => ( aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,Nb),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))),Nb)),K) ) ) ) ).

% take_bit_minus_small_eq
tff(fact_9547_and__minus__numerals_I7_J,axiom,
    ! [Nb: 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,Nb)))),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,Nb))) ).

% and_minus_numerals(7)
tff(fact_9548_and__minus__numerals_I3_J,axiom,
    ! [M: num,Nb: 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,Nb)))) = 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,Nb))) ).

% and_minus_numerals(3)
tff(fact_9549_and__minus__numerals_I8_J,axiom,
    ! [Nb: 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,Nb)))),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,Nb))) ).

% and_minus_numerals(8)
tff(fact_9550_mask__nat__positive__iff,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(bit_se2002935070580805687sk_nat,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ).

% mask_nat_positive_iff
tff(fact_9551_and__minus__numerals_I4_J,axiom,
    ! [M: num,Nb: 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,Nb)))) = 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,Nb))) ).

% and_minus_numerals(4)
tff(fact_9552_less__eq__mask,axiom,
    ! [Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),aa_nat_nat(bit_se2002935070580805687sk_nat,Nb)) ).

% less_eq_mask
tff(fact_9553_mask__nonnegative__int,axiom,
    ! [Nb: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(bit_se2000444600071755411sk_int,Nb)) ).

% mask_nonnegative_int
tff(fact_9554_and__not__num_Osimps_I1_J,axiom,
    aa_num_option_num(bit_and_not_num(one),one) = none_num ).

% and_not_num.simps(1)
tff(fact_9555_less__mask,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(bit_se2002935070580805687sk_nat,Nb)) ) ).

% less_mask
tff(fact_9556_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_9557_and__not__num_Osimps_I2_J,axiom,
    ! [Nb: num] : aa_num_option_num(bit_and_not_num(one),aa_num_num(bit0,Nb)) = aa_num_option_num(some_num,one) ).

% and_not_num.simps(2)
tff(fact_9558_and__not__num_Osimps_I3_J,axiom,
    ! [Nb: num] : aa_num_option_num(bit_and_not_num(one),aa_num_num(bit1,Nb)) = none_num ).

% and_not_num.simps(3)
tff(fact_9559_take__bit__eq__mask__iff,axiom,
    ! [Nb: nat,K: int] :
      ( ( aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,Nb),K) = aa_nat_int(bit_se2000444600071755411sk_int,Nb) )
    <=> ( aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,Nb),aa_int_int(aa_int_fun_int_int(plus_plus_int,K),one_one_int)) = zero_zero_int ) ) ).

% take_bit_eq_mask_iff
tff(fact_9560_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_9561_Suc__mask__eq__exp,axiom,
    ! [Nb: nat] : aa_nat_nat(suc,aa_nat_nat(bit_se2002935070580805687sk_nat,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb) ).

% Suc_mask_eq_exp
tff(fact_9562_mask__nat__less__exp,axiom,
    ! [Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(bit_se2002935070580805687sk_nat,Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ).

% mask_nat_less_exp
tff(fact_9563_mask__half__int,axiom,
    ! [Nb: nat] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_nat_int(bit_se2000444600071755411sk_int,Nb)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = aa_nat_int(bit_se2000444600071755411sk_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)) ).

% mask_half_int
tff(fact_9564_mask__int__def,axiom,
    ! [Nb: nat] : aa_nat_int(bit_se2000444600071755411sk_int,Nb) = 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))),Nb)),one_one_int) ).

% mask_int_def
tff(fact_9565_mask__nat__def,axiom,
    ! [Nb: nat] : aa_nat_nat(bit_se2002935070580805687sk_nat,Nb) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)),one_one_nat) ).

% mask_nat_def
tff(fact_9566_and__not__num_Osimps_I8_J,axiom,
    ! [M: num,Nb: num] : aa_num_option_num(bit_and_not_num(aa_num_num(bit1,M)),aa_num_num(bit0,Nb)) = aa_opt5369159329878751634on_num(case_o6005452278849405969um_num(aa_num_option_num(some_num,one),aTP_Lamm_azz),aa_num_option_num(bit_and_not_num(M),Nb)) ).

% and_not_num.simps(8)
tff(fact_9567_take__bit__eq__mask__iff__exp__dvd,axiom,
    ! [Nb: nat,K: int] :
      ( ( aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,Nb),K) = aa_nat_int(bit_se2000444600071755411sk_int,Nb) )
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)),aa_int_int(aa_int_fun_int_int(plus_plus_int,K),one_one_int)) ) ).

% take_bit_eq_mask_iff_exp_dvd
tff(fact_9568_Code__Abstract__Nat_Otake__bit__num__code_I3_J,axiom,
    ! [Nb: nat,M: num] : aa_num_option_num(bit_take_bit_num(Nb),aa_num_num(bit1,M)) = case_nat_option_num(none_num,aTP_Lamm_baa(M),Nb) ).

% Code_Abstract_Nat.take_bit_num_code(3)
tff(fact_9569_arctan__def,axiom,
    ! [Y: real] : aa_real_real(arctan,Y) = the_real(aa_real_fun_real_o(aTP_Lamm_bab,Y)) ).

% arctan_def
tff(fact_9570_modulo__int__def,axiom,
    ! [K: int,L: int] :
      aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),L) = $ite(
        L = zero_zero_int,
        K,
        $ite(aa_int_int(sgn_sgn_int,K) = aa_int_int(sgn_sgn_int,L),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,L)),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,K))),aa_int_nat(nat2,aa_int_int(abs_abs_int,L))))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,L)),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(abs_abs_int,L)),aa_o_int(zero_n2684676970156552555ol_int,~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,L),K)))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,K))),aa_int_nat(nat2,aa_int_int(abs_abs_int,L))))))) ) ).

% modulo_int_def
tff(fact_9571_sgn__mult__dvd__iff,axiom,
    ! [R2: int,L: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,R2)),L)),K)
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,L),K)
        & ( ( R2 = zero_zero_int )
         => ( K = zero_zero_int ) ) ) ) ).

% sgn_mult_dvd_iff
tff(fact_9572_mult__sgn__dvd__iff,axiom,
    ! [L: int,R2: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,L),aa_int_int(sgn_sgn_int,R2))),K)
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,L),K)
        & ( ( R2 = zero_zero_int )
         => ( K = zero_zero_int ) ) ) ) ).

% mult_sgn_dvd_iff
tff(fact_9573_dvd__sgn__mult__iff,axiom,
    ! [L: int,R2: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,L),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,R2)),K))
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,L),K)
        | ( R2 = zero_zero_int ) ) ) ).

% dvd_sgn_mult_iff
tff(fact_9574_dvd__mult__sgn__iff,axiom,
    ! [L: int,K: int,R2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,L),aa_int_int(aa_int_fun_int_int(times_times_int,K),aa_int_int(sgn_sgn_int,R2)))
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,L),K)
        | ( R2 = zero_zero_int ) ) ) ).

% dvd_mult_sgn_iff
tff(fact_9575_div__eq__sgn__abs,axiom,
    ! [K: int,L: int] :
      ( ( aa_int_int(sgn_sgn_int,K) = aa_int_int(sgn_sgn_int,L) )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,K),L) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(abs_abs_int,K)),aa_int_int(abs_abs_int,L)) ) ) ).

% div_eq_sgn_abs
tff(fact_9576_int__sgnE,axiom,
    ! [K: int] :
      ~ ! [N2: nat,L2: int] : K != aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,L2)),aa_nat_int(semiri1314217659103216013at_int,N2)) ).

% int_sgnE
tff(fact_9577_ln__real__def,axiom,
    ! [Xb: real] : aa_real_real(ln_ln_real,Xb) = the_real(aa_real_fun_real_o(aTP_Lamm_bac,Xb)) ).

% ln_real_def
tff(fact_9578_sgn__mod,axiom,
    ! [L: int,K: int] :
      ( ( L != zero_zero_int )
     => ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,L),K)
       => ( aa_int_int(sgn_sgn_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),L)) = aa_int_int(sgn_sgn_int,L) ) ) ) ).

% sgn_mod
tff(fact_9579_ln__neg__is__const,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),zero_zero_real)
     => ( aa_real_real(ln_ln_real,Xb) = the_real(aTP_Lamm_gg) ) ) ).

% ln_neg_is_const
tff(fact_9580_zsgn__def,axiom,
    ! [I: int] :
      aa_int_int(sgn_sgn_int,I) = $ite(
        I = zero_zero_int,
        zero_zero_int,
        $ite(aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),I),one_one_int,aa_int_int(uminus_uminus_int,one_one_int)) ) ).

% zsgn_def
tff(fact_9581_div__sgn__abs__cancel,axiom,
    ! [V: int,K: int,L: int] :
      ( ( V != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,V)),aa_int_int(abs_abs_int,K))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,V)),aa_int_int(abs_abs_int,L))) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(abs_abs_int,K)),aa_int_int(abs_abs_int,L)) ) ) ).

% div_sgn_abs_cancel
tff(fact_9582_div__dvd__sgn__abs,axiom,
    ! [L: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,L),K)
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,K),L) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,K)),aa_int_int(sgn_sgn_int,L))),aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(abs_abs_int,K)),aa_int_int(abs_abs_int,L))) ) ) ).

% div_dvd_sgn_abs
tff(fact_9583_arccos__def,axiom,
    ! [Y: real] : aa_real_real(arccos,Y) = the_real(aa_real_fun_real_o(aTP_Lamm_bad,Y)) ).

% arccos_def
tff(fact_9584_eucl__rel__int__remainderI,axiom,
    ! [R2: int,L: int,K: int,Q2: int] :
      ( ( aa_int_int(sgn_sgn_int,R2) = aa_int_int(sgn_sgn_int,L) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,R2)),aa_int_int(abs_abs_int,L))
       => ( ( K = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Q2),L)),R2) )
         => aa_Pro4592068669816069595_int_o(eucl_rel_int(K,L),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q2),R2)) ) ) ) ).

% eucl_rel_int_remainderI
tff(fact_9585_div__noneq__sgn__abs,axiom,
    ! [L: int,K: int] :
      ( ( L != zero_zero_int )
     => ( ( aa_int_int(sgn_sgn_int,K) != aa_int_int(sgn_sgn_int,L) )
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,K),L) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(abs_abs_int,K)),aa_int_int(abs_abs_int,L)))),aa_o_int(zero_n2684676970156552555ol_int,~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,L),K))) ) ) ) ).

% div_noneq_sgn_abs
tff(fact_9586_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) ) ) )
         => ~ ! [R4: int,Q3: int] :
                ( ( A32 = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q3),R4) )
               => ( ( aa_int_int(sgn_sgn_int,R4) = aa_int_int(sgn_sgn_int,A22) )
                 => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,R4)),aa_int_int(abs_abs_int,A22))
                   => ( A1 != aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Q3),A22)),R4) ) ) ) ) ) ) ) ).

% eucl_rel_int.cases
tff(fact_9587_eucl__rel__int_Osimps,axiom,
    ! [A1: int,A22: int,A32: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(eucl_rel_int(A1,A22),A32)
    <=> ( ? [K4: int] :
            ( ( A1 = K4 )
            & ( A22 = zero_zero_int )
            & ( A32 = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,zero_zero_int),K4) ) )
        | ? [L4: int,K4: int,Q4: int] :
            ( ( A1 = K4 )
            & ( A22 = L4 )
            & ( A32 = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q4),zero_zero_int) )
            & ( L4 != zero_zero_int )
            & ( K4 = aa_int_int(aa_int_fun_int_int(times_times_int,Q4),L4) ) )
        | ? [R6: int,L4: int,K4: int,Q4: int] :
            ( ( A1 = K4 )
            & ( A22 = L4 )
            & ( A32 = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q4),R6) )
            & ( aa_int_int(sgn_sgn_int,R6) = aa_int_int(sgn_sgn_int,L4) )
            & aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,R6)),aa_int_int(abs_abs_int,L4))
            & ( K4 = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Q4),L4)),R6) ) ) ) ) ).

% eucl_rel_int.simps
tff(fact_9588_Code__Abstract__Nat_Otake__bit__num__code_I1_J,axiom,
    ! [Nb: nat] : aa_num_option_num(bit_take_bit_num(Nb),one) = case_nat_option_num(none_num,aTP_Lamm_bae,Nb) ).

% Code_Abstract_Nat.take_bit_num_code(1)
tff(fact_9589_pi__half,axiom,
    aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))) = the_real(aTP_Lamm_baf) ).

% pi_half
tff(fact_9590_pi__def,axiom,
    pi = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),the_real(aTP_Lamm_baf)) ).

% pi_def
tff(fact_9591_Code__Abstract__Nat_Otake__bit__num__code_I2_J,axiom,
    ! [Nb: nat,M: num] : aa_num_option_num(bit_take_bit_num(Nb),aa_num_num(bit0,M)) = case_nat_option_num(none_num,aTP_Lamm_bag(M),Nb) ).

% Code_Abstract_Nat.take_bit_num_code(2)
tff(fact_9592_divide__int__unfold,axiom,
    ! [K: int,M: nat,L: int,Nb: nat] :
      aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,K)),aa_nat_int(semiri1314217659103216013at_int,M))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,L)),aa_nat_int(semiri1314217659103216013at_int,Nb))) = $ite(
        ( ( aa_int_int(sgn_sgn_int,L) = zero_zero_int )
        | ( aa_int_int(sgn_sgn_int,K) = zero_zero_int )
        | ( Nb = zero_zero_nat ) ),
        zero_zero_int,
        $ite(aa_int_int(sgn_sgn_int,K) = aa_int_int(sgn_sgn_int,L),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Nb)),aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Nb)),aa_o_nat(zero_n2687167440665602831ol_nat,~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Nb),M)))))) ) ).

% divide_int_unfold
tff(fact_9593_modulo__int__unfold,axiom,
    ! [K: int,M: nat,L: int,Nb: nat] :
      aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,K)),aa_nat_int(semiri1314217659103216013at_int,M))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,L)),aa_nat_int(semiri1314217659103216013at_int,Nb))) = $ite(
        ( ( aa_int_int(sgn_sgn_int,L) = zero_zero_int )
        | ( aa_int_int(sgn_sgn_int,K) = zero_zero_int )
        | ( Nb = zero_zero_nat ) ),
        aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,K)),aa_nat_int(semiri1314217659103216013at_int,M)),
        $ite(aa_int_int(sgn_sgn_int,K) = aa_int_int(sgn_sgn_int,L),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,L)),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Nb))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,L)),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),aa_o_nat(zero_n2687167440665602831ol_nat,~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Nb),M))))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Nb))))) ) ).

% modulo_int_unfold
tff(fact_9594_divide__int__def,axiom,
    ! [K: int,L: int] :
      aa_int_int(aa_int_fun_int_int(divide_divide_int,K),L) = $ite(
        L = zero_zero_int,
        zero_zero_int,
        $ite(aa_int_int(sgn_sgn_int,K) = aa_int_int(sgn_sgn_int,L),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,K))),aa_int_nat(nat2,aa_int_int(abs_abs_int,L)))),aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,K))),aa_int_nat(nat2,aa_int_int(abs_abs_int,L)))),aa_o_nat(zero_n2687167440665602831ol_nat,~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,L),K)))))) ) ).

% divide_int_def
tff(fact_9595_arcsin__def,axiom,
    ! [Y: real] : aa_real_real(arcsin,Y) = the_real(aa_real_fun_real_o(aTP_Lamm_bah,Y)) ).

% arcsin_def
tff(fact_9596_Bit__Operations_Otake__bit__num__code,axiom,
    ! [Nb: nat,M: num] : aa_num_option_num(bit_take_bit_num(Nb),M) = produc478579273971653890on_num(aTP_Lamm_bal,product_Pair_nat_num(Nb,M)) ).

% Bit_Operations.take_bit_num_code
tff(fact_9597_sgn__div__eq__sgn__mult,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B) != zero_zero_int )
     => ( aa_int_int(sgn_sgn_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)) = aa_int_int(sgn_sgn_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)) ) ) ).

% sgn_div_eq_sgn_mult
tff(fact_9598_zero__le__sgn__iff,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(sgn_sgn_real,Xb))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb) ) ).

% zero_le_sgn_iff
tff(fact_9599_sgn__le__0__iff,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sgn_sgn_real,Xb)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),zero_zero_real) ) ).

% sgn_le_0_iff
tff(fact_9600_nat_Odisc__eq__case_I2_J,axiom,
    ! [Nat: nat] :
      ( ( Nat != zero_zero_nat )
    <=> aa_nat_o(case_nat_o($false,aTP_Lamm_dd),Nat) ) ).

% nat.disc_eq_case(2)
tff(fact_9601_nat_Odisc__eq__case_I1_J,axiom,
    ! [Nat: nat] :
      ( ( Nat = zero_zero_nat )
    <=> aa_nat_o(case_nat_o($true,aTP_Lamm_cy),Nat) ) ).

% nat.disc_eq_case(1)
tff(fact_9602_real__sgn__eq,axiom,
    ! [Xb: real] : aa_real_real(sgn_sgn_real,Xb) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),aa_real_real(abs_abs_real,Xb)) ).

% real_sgn_eq
tff(fact_9603_sgn__root,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_real(sgn_sgn_real,aa_real_real(root(Nb),Xb)) = aa_real_real(sgn_sgn_real,Xb) ) ) ).

% sgn_root
tff(fact_9604_sgn__eq,axiom,
    ! [Z: complex] : aa_complex_complex(sgn_sgn_complex,Z) = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,Z),aa_real_complex(real_V4546457046886955230omplex,aa_complex_real(real_V1022390504157884413omplex,Z))) ).

% sgn_eq
tff(fact_9605_less__eq__nat_Osimps_I2_J,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,M)),Nb)
    <=> aa_nat_o(case_nat_o($false,aa_nat_fun_nat_o(ord_less_eq_nat,M)),Nb) ) ).

% less_eq_nat.simps(2)
tff(fact_9606_max__Suc2,axiom,
    ! [M: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,M),aa_nat_nat(suc,Nb)) = aa_nat_nat(case_nat_nat(aa_nat_nat(suc,Nb),aa_nat_fun_nat_nat(aTP_Lamm_bam,Nb)),M) ).

% max_Suc2
tff(fact_9607_max__Suc1,axiom,
    ! [Nb: nat,M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(suc,Nb)),M) = aa_nat_nat(case_nat_nat(aa_nat_nat(suc,Nb),aa_nat_fun_nat_nat(aTP_Lamm_ban,Nb)),M) ).

% max_Suc1
tff(fact_9608_sgn__real__def,axiom,
    ! [A: real] :
      aa_real_real(sgn_sgn_real,A) = $ite(
        A = zero_zero_real,
        zero_zero_real,
        $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A),one_one_real,aa_real_real(uminus_uminus_real,one_one_real)) ) ).

% sgn_real_def
tff(fact_9609_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_9610_diff__Suc,axiom,
    ! [M: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),aa_nat_nat(suc,Nb)) = aa_nat_nat(case_nat_nat(zero_zero_nat,aTP_Lamm_di),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Nb)) ).

% diff_Suc
tff(fact_9611_sgn__power__injE,axiom,
    ! [A: real,Nb: nat,Xb: real,B: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sgn_sgn_real,A)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,A)),Nb)) = Xb )
     => ( ( Xb = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sgn_sgn_real,B)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,B)),Nb)) )
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
         => ( A = B ) ) ) ) ).

% sgn_power_injE
tff(fact_9612_sgn__power__root,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sgn_sgn_real,aa_real_real(root(Nb),Xb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,aa_real_real(root(Nb),Xb))),Nb)) = Xb ) ) ).

% sgn_power_root
tff(fact_9613_root__sgn__power,axiom,
    ! [Nb: nat,Y: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_real(root(Nb),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sgn_sgn_real,Y)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,Y)),Nb))) = Y ) ) ).

% root_sgn_power
tff(fact_9614_cis__Arg__unique,axiom,
    ! [Z: complex,Xb: real] :
      ( ( aa_complex_complex(sgn_sgn_complex,Z) = aa_real_complex(cis,Xb) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,pi)),Xb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),pi)
         => ( aa_complex_real(arg,Z) = Xb ) ) ) ) ).

% cis_Arg_unique
tff(fact_9615_split__root,axiom,
    ! [P: fun_real_o,Nb: nat,Xb: real] :
      ( aa_real_o(P,aa_real_real(root(Nb),Xb))
    <=> ( ( ( Nb = zero_zero_nat )
         => aa_real_o(P,zero_zero_real) )
        & ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
         => ! [Y3: real] :
              ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sgn_sgn_real,Y3)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,Y3)),Nb)) = Xb )
             => aa_real_o(P,Y3) ) ) ) ) ).

% split_root
tff(fact_9616_floor__real__def,axiom,
    ! [Xb: real] : aa_real_int(archim6058952711729229775r_real,Xb) = the_int(aa_real_fun_int_o(aTP_Lamm_bao,Xb)) ).

% floor_real_def
tff(fact_9617_Arg__correct,axiom,
    ! [Z: complex] :
      ( ( Z != zero_zero_complex )
     => ( ( aa_complex_complex(sgn_sgn_complex,Z) = aa_real_complex(cis,aa_complex_real(arg,Z)) )
        & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,pi)),aa_complex_real(arg,Z))
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(arg,Z)),pi) ) ) ).

% Arg_correct
tff(fact_9618_arctan__inverse,axiom,
    ! [Xb: real] :
      ( ( Xb != zero_zero_real )
     => ( aa_real_real(arctan,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),Xb)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sgn_sgn_real,Xb)),pi)),numeral_numeral_real(aa_num_num(bit0,one)))),aa_real_real(arctan,Xb)) ) ) ).

% arctan_inverse
tff(fact_9619_floor__rat__def,axiom,
    ! [Xb: rat] : archim3151403230148437115or_rat(Xb) = the_int(aTP_Lamm_bap(Xb)) ).

% floor_rat_def
tff(fact_9620_root__def,axiom,
    ! [Nb: nat,Xb: real] :
      aa_real_real(root(Nb),Xb) = $ite(Nb = zero_zero_nat,zero_zero_real,aa_real_real(the_in5290026491893676941l_real(top_top_set_real,aTP_Lamm_baq(Nb)),Xb)) ).

% root_def
tff(fact_9621_sgn__rat__def,axiom,
    ! [A: 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,uminus_uminus_rat(one_one_rat)) ) ).

% sgn_rat_def
tff(fact_9622_obtain__pos__sum,axiom,
    ! [R2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),R2)
     => ~ ! [S: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),S)
           => ! [T3: rat] :
                ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),T3)
               => ( R2 != aa_rat_rat(plus_plus_rat(S),T3) ) ) ) ) ).

% obtain_pos_sum
tff(fact_9623_less__eq__rat__def,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Y)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Y)
        | ( Xb = Y ) ) ) ).

% less_eq_rat_def
tff(fact_9624_GreatestI__nat,axiom,
    ! [P: fun_nat_o,K: nat,B: nat] :
      ( aa_nat_o(P,K)
     => ( ! [Y4: nat] :
            ( aa_nat_o(P,Y4)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y4),B) )
       => aa_nat_o(P,order_Greatest_nat(P)) ) ) ).

% GreatestI_nat
tff(fact_9625_Greatest__le__nat,axiom,
    ! [P: fun_nat_o,K: nat,B: nat] :
      ( aa_nat_o(P,K)
     => ( ! [Y4: nat] :
            ( aa_nat_o(P,Y4)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y4),B) )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),order_Greatest_nat(P)) ) ) ).

% Greatest_le_nat
tff(fact_9626_GreatestI__ex__nat,axiom,
    ! [P: fun_nat_o,B: nat] :
      ( ? [X_12: nat] : aa_nat_o(P,X_12)
     => ( ! [Y4: nat] :
            ( aa_nat_o(P,Y4)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y4),B) )
       => aa_nat_o(P,order_Greatest_nat(P)) ) ) ).

% GreatestI_ex_nat
tff(fact_9627_Arg__def,axiom,
    ! [Z: complex] :
      aa_complex_real(arg,Z) = $ite(Z = zero_zero_complex,zero_zero_real,fChoice_real(aTP_Lamm_bar(Z))) ).

% Arg_def
tff(fact_9628_new__addr__refl,axiom,
    ! [H: heap_e7401611519738050253t_unit,As3: set_nat] : hoare_new_addrs(H,As3,H) = As3 ).

% new_addr_refl
tff(fact_9629_diff__rat__def,axiom,
    ! [Q2: rat,R2: rat] : minus_minus_rat(Q2,R2) = aa_rat_rat(plus_plus_rat(Q2),uminus_uminus_rat(R2)) ).

% diff_rat_def
tff(fact_9630_signed__take__bit__eq__take__bit__minus,axiom,
    ! [Nb: nat,K: int] : aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,Nb),K) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,aa_nat_nat(suc,Nb)),K)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(suc,Nb))),aa_o_int(zero_n2684676970156552555ol_int,aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,K),Nb)))) ).

% signed_take_bit_eq_take_bit_minus
tff(fact_9631_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_9632_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_9633_signed__take__bit__nonnegative__iff,axiom,
    ! [Nb: 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,Nb),K))
    <=> ~ aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,K),Nb) ) ).

% signed_take_bit_nonnegative_iff
tff(fact_9634_bit__minus__numeral__Bit0__Suc__iff,axiom,
    ! [W: num,Nb: nat] :
      ( aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,W)))),aa_nat_nat(suc,Nb))
    <=> aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,W))),Nb) ) ).

% bit_minus_numeral_Bit0_Suc_iff
tff(fact_9635_bit__minus__numeral__Bit1__Suc__iff,axiom,
    ! [W: num,Nb: nat] :
      ( aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,W)))),aa_nat_nat(suc,Nb))
    <=> ~ aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,aa_num_int(numeral_numeral_int,W)),Nb) ) ).

% bit_minus_numeral_Bit1_Suc_iff
tff(fact_9636_bit__minus__numeral__int_I1_J,axiom,
    ! [W: num,Nb: num] :
      ( aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,W)))),aa_num_nat(numeral_numeral_nat,Nb))
    <=> aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,W))),aa_num_nat(pred_numeral,Nb)) ) ).

% bit_minus_numeral_int(1)
tff(fact_9637_bit__minus__numeral__int_I2_J,axiom,
    ! [W: num,Nb: num] :
      ( aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,W)))),aa_num_nat(numeral_numeral_nat,Nb))
    <=> ~ aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,aa_num_int(numeral_numeral_int,W)),aa_num_nat(pred_numeral,Nb)) ) ).

% bit_minus_numeral_int(2)
tff(fact_9638_bin__nth__minus__Bit0,axiom,
    ! [Nb: nat,W: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,W))),Nb)
      <=> aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,aa_num_int(numeral_numeral_int,W)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)) ) ) ).

% bin_nth_minus_Bit0
tff(fact_9639_bin__nth__minus__Bit1,axiom,
    ! [Nb: nat,W: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,W))),Nb)
      <=> aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,aa_num_int(numeral_numeral_int,W)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)) ) ) ).

% bin_nth_minus_Bit1
tff(fact_9640_bit__minus__int__iff,axiom,
    ! [K: int,Nb: nat] :
      ( aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,aa_int_int(uminus_uminus_int,K)),Nb)
    <=> aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,aa_int_int(bit_ri7919022796975470100ot_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,K),one_one_int))),Nb) ) ).

% bit_minus_int_iff
tff(fact_9641_not__int__def,axiom,
    ! [K: int] : aa_int_int(bit_ri7919022796975470100ot_int,K) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,K)),one_one_int) ).

% not_int_def
tff(fact_9642_and__not__numerals_I1_J,axiom,
    aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,one_one_int),aa_int_int(bit_ri7919022796975470100ot_int,one_one_int)) = zero_zero_int ).

% and_not_numerals(1)
tff(fact_9643_bit__not__int__iff_H,axiom,
    ! [K: int,Nb: nat] :
      ( aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,K)),one_one_int)),Nb)
    <=> ~ aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,K),Nb) ) ).

% bit_not_int_iff'
tff(fact_9644_not__int__div__2,axiom,
    ! [K: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(bit_ri7919022796975470100ot_int,K)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = aa_int_int(bit_ri7919022796975470100ot_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))) ).

% not_int_div_2
tff(fact_9645_even__not__iff__int,axiom,
    ! [K: 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(bit_ri7919022796975470100ot_int,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_not_iff_int
tff(fact_9646_and__not__numerals_I4_J,axiom,
    ! [M: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,M))),aa_int_int(bit_ri7919022796975470100ot_int,one_one_int)) = aa_num_int(numeral_numeral_int,aa_num_num(bit0,M)) ).

% and_not_numerals(4)
tff(fact_9647_and__not__numerals_I2_J,axiom,
    ! [Nb: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,one_one_int),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb)))) = one_one_int ).

% and_not_numerals(2)
tff(fact_9648_bit__imp__take__bit__positive,axiom,
    ! [Nb: nat,M: nat,K: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),M)
     => ( aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,K),Nb)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,M),K)) ) ) ).

% bit_imp_take_bit_positive
tff(fact_9649_and__not__num__eq__Some__iff,axiom,
    ! [M: num,Nb: num,Q2: num] :
      ( ( aa_num_option_num(bit_and_not_num(M),Nb) = aa_num_option_num(some_num,Q2) )
    <=> ( 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,Nb))) = aa_num_int(numeral_numeral_int,Q2) ) ) ).

% and_not_num_eq_Some_iff
tff(fact_9650_int__bit__bound,axiom,
    ! [K: int] :
      ~ ! [N2: nat] :
          ( ! [M3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),M3)
             => ( aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,K),M3)
              <=> aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,K),N2) ) )
         => ~ ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)
             => ( aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,K),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),one_one_nat))
              <=> ~ aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,K),N2) ) ) ) ).

% int_bit_bound
tff(fact_9651_and__not__numerals_I5_J,axiom,
    ! [M: num,Nb: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,M))),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb)))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,Nb)))) ).

% and_not_numerals(5)
tff(fact_9652_and__not__numerals_I7_J,axiom,
    ! [M: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,M))),aa_int_int(bit_ri7919022796975470100ot_int,one_one_int)) = aa_num_int(numeral_numeral_int,aa_num_num(bit0,M)) ).

% and_not_numerals(7)
tff(fact_9653_and__not__numerals_I3_J,axiom,
    ! [Nb: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,one_one_int),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb)))) = zero_zero_int ).

% and_not_numerals(3)
tff(fact_9654_and__not__num__eq__None__iff,axiom,
    ! [M: num,Nb: num] :
      ( ( aa_num_option_num(bit_and_not_num(M),Nb) = none_num )
    <=> ( aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,Nb))) = zero_zero_int ) ) ).

% and_not_num_eq_None_iff
tff(fact_9655_and__not__numerals_I9_J,axiom,
    ! [M: num,Nb: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,M))),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb)))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,Nb)))) ).

% and_not_numerals(9)
tff(fact_9656_and__not__numerals_I6_J,axiom,
    ! [M: num,Nb: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,M))),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb)))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,Nb)))) ).

% and_not_numerals(6)
tff(fact_9657_int__numeral__not__and__num,axiom,
    ! [M: num,Nb: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,M))),aa_num_int(numeral_numeral_int,Nb)) = aa_option_num_int(case_option_int_num(zero_zero_int,numeral_numeral_int),aa_num_option_num(bit_and_not_num(Nb),M)) ).

% int_numeral_not_and_num
tff(fact_9658_int__numeral__and__not__num,axiom,
    ! [M: num,Nb: 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,Nb))) = aa_option_num_int(case_option_int_num(zero_zero_int,numeral_numeral_int),aa_num_option_num(bit_and_not_num(M),Nb)) ).

% int_numeral_and_not_num
tff(fact_9659_bit__int__def,axiom,
    ! [K: int,Nb: nat] :
      ( aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,K),Nb)
    <=> ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb))) ) ).

% bit_int_def
tff(fact_9660_and__not__numerals_I8_J,axiom,
    ! [M: num,Nb: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,M))),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb)))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,Nb))))) ).

% and_not_numerals(8)
tff(fact_9661_not__int__rec,axiom,
    ! [K: int] : aa_int_int(bit_ri7919022796975470100ot_int,K) = 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_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_ri7919022796975470100ot_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ).

% not_int_rec
tff(fact_9662_Bit__Operations_Oset__bit__eq,axiom,
    ! [Nb: nat,K: int] : aa_int_int(aa_nat_fun_int_int(bit_se7879613467334960850it_int,Nb),K) = aa_int_int(aa_int_fun_int_int(plus_plus_int,K),aa_int_int(aa_int_fun_int_int(times_times_int,aa_o_int(zero_n2684676970156552555ol_int,~ aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,K),Nb))),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb))) ).

% Bit_Operations.set_bit_eq
tff(fact_9663_unset__bit__eq,axiom,
    ! [Nb: nat,K: int] : aa_int_int(aa_nat_fun_int_int(bit_se4203085406695923979it_int,Nb),K) = aa_int_int(aa_int_fun_int_int(minus_minus_int,K),aa_int_int(aa_int_fun_int_int(times_times_int,aa_o_int(zero_n2684676970156552555ol_int,aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,K),Nb))),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb))) ).

% unset_bit_eq
tff(fact_9664_take__bit__Suc__from__most,axiom,
    ! [Nb: nat,K: int] : aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,aa_nat_nat(suc,Nb)),K) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)),aa_o_int(zero_n2684676970156552555ol_int,aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,K),Nb)))),aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,Nb),K)) ).

% take_bit_Suc_from_most
tff(fact_9665_int__not__code_I1_J,axiom,
    aa_int_int(bit_ri7919022796975470100ot_int,zero_zero_int) = aa_int_int(uminus_uminus_int,one_one_int) ).

% int_not_code(1)
tff(fact_9666_bitNOT__integer__code,axiom,
    ! [I: code_integer] : aa_Cod1783974652533396728nteger(bit_ri7632146776885996613nteger,I) = aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,I)),one_one_Code_integer) ).

% bitNOT_integer_code
tff(fact_9667_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_9668_or__nat__numerals_I2_J,axiom,
    ! [Y: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_nat_nat(suc,zero_zero_nat)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Y))) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Y)) ).

% or_nat_numerals(2)
tff(fact_9669_or__nat__numerals_I4_J,axiom,
    ! [Xb: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Xb))),aa_nat_nat(suc,zero_zero_nat)) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Xb)) ).

% or_nat_numerals(4)
tff(fact_9670_or__minus__numerals_I6_J,axiom,
    ! [Nb: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb)))),one_one_int) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb))) ).

% or_minus_numerals(6)
tff(fact_9671_or__minus__numerals_I2_J,axiom,
    ! [Nb: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb)))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb))) ).

% or_minus_numerals(2)
tff(fact_9672_or__nat__numerals_I3_J,axiom,
    ! [Xb: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Xb))),aa_nat_nat(suc,zero_zero_nat)) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Xb)) ).

% or_nat_numerals(3)
tff(fact_9673_or__nat__numerals_I1_J,axiom,
    ! [Y: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_nat_nat(suc,zero_zero_nat)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Y))) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Y)) ).

% or_nat_numerals(1)
tff(fact_9674_and__minus__minus__numerals,axiom,
    ! [M: num,Nb: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))) = aa_int_int(bit_ri7919022796975470100ot_int,aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_num_int(numeral_numeral_int,M)),one_one_int)),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_num_int(numeral_numeral_int,Nb)),one_one_int))) ).

% and_minus_minus_numerals
tff(fact_9675_or__minus__minus__numerals,axiom,
    ! [M: num,Nb: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))) = aa_int_int(bit_ri7919022796975470100ot_int,aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_num_int(numeral_numeral_int,M)),one_one_int)),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_num_int(numeral_numeral_int,Nb)),one_one_int))) ).

% or_minus_minus_numerals
tff(fact_9676_not__bit__Suc__0__Suc,axiom,
    ! [Nb: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_nat_nat(suc,zero_zero_nat)),aa_nat_nat(suc,Nb)) ).

% not_bit_Suc_0_Suc
tff(fact_9677_bit__Suc__0__iff,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_nat_nat(suc,zero_zero_nat)),Nb)
    <=> ( Nb = zero_zero_nat ) ) ).

% bit_Suc_0_iff
tff(fact_9678_plus__and__or,axiom,
    ! [Xb: int,Y: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,Xb),Y)),aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,Xb),Y)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb),Y) ).

% plus_and_or
tff(fact_9679_OR__lower,axiom,
    ! [Xb: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xb)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),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,Xb),Y)) ) ) ).

% OR_lower
tff(fact_9680_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_9681_not__bit__Suc__0__numeral,axiom,
    ! [Nb: num] : ~ aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_nat_nat(suc,zero_zero_nat)),aa_num_nat(numeral_numeral_nat,Nb)) ).

% not_bit_Suc_0_numeral
tff(fact_9682_or__not__numerals_I1_J,axiom,
    aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,one_one_int),aa_int_int(bit_ri7919022796975470100ot_int,one_one_int)) = aa_int_int(bit_ri7919022796975470100ot_int,zero_zero_int) ).

% or_not_numerals(1)
tff(fact_9683_or__not__numerals_I2_J,axiom,
    ! [Nb: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,one_one_int),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb)))) = aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb))) ).

% or_not_numerals(2)
tff(fact_9684_or__not__numerals_I4_J,axiom,
    ! [M: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,M))),aa_int_int(bit_ri7919022796975470100ot_int,one_one_int)) = aa_int_int(bit_ri7919022796975470100ot_int,one_one_int) ).

% or_not_numerals(4)
tff(fact_9685_bit__nat__iff,axiom,
    ! [K: int,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_int_nat(nat2,K)),Nb)
    <=> ( 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),Nb) ) ) ).

% bit_nat_iff
tff(fact_9686_or__not__numerals_I3_J,axiom,
    ! [Nb: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,one_one_int),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb)))) = aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb))) ).

% or_not_numerals(3)
tff(fact_9687_or__not__numerals_I7_J,axiom,
    ! [M: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,M))),aa_int_int(bit_ri7919022796975470100ot_int,one_one_int)) = aa_int_int(bit_ri7919022796975470100ot_int,zero_zero_int) ).

% or_not_numerals(7)
tff(fact_9688_bit__nat__def,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,M),Nb)
    <=> ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))) ) ).

% bit_nat_def
tff(fact_9689_or__not__numerals_I6_J,axiom,
    ! [M: num,Nb: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,M))),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb)))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,Nb)))) ).

% or_not_numerals(6)
tff(fact_9690_OR__upper,axiom,
    ! [Xb: int,Nb: nat,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xb)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb))
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Y),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb))
         => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,Xb),Y)),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) ) ) ) ).

% OR_upper
tff(fact_9691_or__not__numerals_I5_J,axiom,
    ! [M: num,Nb: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,M))),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb)))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,Nb))))) ).

% or_not_numerals(5)
tff(fact_9692_Suc__0__or__eq,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_nat_nat(suc,zero_zero_nat)),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),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))),Nb))) ).

% Suc_0_or_eq
tff(fact_9693_or__Suc__0__eq,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,Nb),aa_nat_nat(suc,zero_zero_nat)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),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))),Nb))) ).

% or_Suc_0_eq
tff(fact_9694_or__nat__rec,axiom,
    ! [M: nat,Nb: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,M),Nb) = 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))),M)
            | ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb) ))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% or_nat_rec
tff(fact_9695_or__not__numerals_I9_J,axiom,
    ! [M: num,Nb: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,M))),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb)))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,Nb))))) ).

% or_not_numerals(9)
tff(fact_9696_or__not__numerals_I8_J,axiom,
    ! [M: num,Nb: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,M))),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb)))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,Nb))))) ).

% or_not_numerals(8)
tff(fact_9697_or__int__rec,axiom,
    ! [K: int,L: int] :
      aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,K),L) = aa_int_int(aa_int_fun_int_int(plus_plus_int,
          aa_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_se1409905431419307370or_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(divide_divide_int,L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ).

% or_int_rec
tff(fact_9698_or__nat__unfold,axiom,
    ! [M: nat,Nb: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,M),Nb) = $ite(
        M = zero_zero_nat,
        Nb,
        $ite(Nb = zero_zero_nat,M,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))) ) ).

% or_nat_unfold
tff(fact_9699_or__int__unfold,axiom,
    ! [K: int,L: int] :
      aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,K),L) = $ite(
        ( ( K = aa_int_int(uminus_uminus_int,one_one_int) )
        | ( L = aa_int_int(uminus_uminus_int,one_one_int) ) ),
        aa_int_int(uminus_uminus_int,one_one_int),
        $ite(
          K = zero_zero_int,
          L,
          $ite(L = zero_zero_int,K,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(ord_max_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(divide_divide_int,L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))))) ) ) ).

% or_int_unfold
tff(fact_9700_Bit__integer_Oabs__eq,axiom,
    ! [Xaa: int,Xb: $o] : bits_Bit_integer(aa_int_Code_integer(code_integer_of_int,Xaa),(Xb)) = aa_int_Code_integer(code_integer_of_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_o_int(zero_n2684676970156552555ol_int,(Xb))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Xaa))) ).

% Bit_integer.abs_eq
tff(fact_9701_xor__int__unfold,axiom,
    ! [K: int,L: int] :
      aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,K),L) = $ite(
        K = aa_int_int(uminus_uminus_int,one_one_int),
        aa_int_int(bit_ri7919022796975470100ot_int,L),
        $ite(
          L = aa_int_int(uminus_uminus_int,one_one_int),
          aa_int_int(bit_ri7919022796975470100ot_int,K),
          $ite(
            K = zero_zero_int,
            L,
            $ite(L = zero_zero_int,K,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(divide_divide_int,L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))))) ) ) ) ).

% xor_int_unfold
tff(fact_9702_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_9703_XOR__lower,axiom,
    ! [Xb: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xb)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),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,Xb),Y)) ) ) ).

% XOR_lower
tff(fact_9704_XOR__upper,axiom,
    ! [Xb: int,Nb: nat,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xb)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb))
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Y),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb))
         => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,Xb),Y)),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) ) ) ) ).

% XOR_upper
tff(fact_9705_xor__int__rec,axiom,
    ! [K: int,L: int] :
      aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,K),L) = aa_int_int(aa_int_fun_int_int(plus_plus_int,
          aa_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_se6526347334894502574or_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(divide_divide_int,L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ).

% xor_int_rec
tff(fact_9706_or__minus__numerals_I1_J,axiom,
    ! [Nb: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb)))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,one),aa_num_num(bitM,Nb)))) ).

% or_minus_numerals(1)
tff(fact_9707_or__minus__numerals_I5_J,axiom,
    ! [Nb: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb)))),one_one_int) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,one),aa_num_num(bitM,Nb)))) ).

% or_minus_numerals(5)
tff(fact_9708_xor__nat__numerals_I4_J,axiom,
    ! [Xb: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Xb))),aa_nat_nat(suc,zero_zero_nat)) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Xb)) ).

% xor_nat_numerals(4)
tff(fact_9709_xor__nat__numerals_I3_J,axiom,
    ! [Xb: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Xb))),aa_nat_nat(suc,zero_zero_nat)) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Xb)) ).

% xor_nat_numerals(3)
tff(fact_9710_xor__nat__numerals_I2_J,axiom,
    ! [Y: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_nat_nat(suc,zero_zero_nat)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Y))) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Y)) ).

% xor_nat_numerals(2)
tff(fact_9711_xor__nat__numerals_I1_J,axiom,
    ! [Y: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_nat_nat(suc,zero_zero_nat)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Y))) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Y)) ).

% xor_nat_numerals(1)
tff(fact_9712_or__minus__numerals_I8_J,axiom,
    ! [Nb: num,M: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb)))),aa_num_int(numeral_numeral_int,M)) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,M),aa_num_num(bit0,Nb)))) ).

% or_minus_numerals(8)
tff(fact_9713_or__minus__numerals_I4_J,axiom,
    ! [M: num,Nb: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb)))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,M),aa_num_num(bit0,Nb)))) ).

% or_minus_numerals(4)
tff(fact_9714_or__minus__numerals_I7_J,axiom,
    ! [Nb: num,M: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb)))),aa_num_int(numeral_numeral_int,M)) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,M),aa_num_num(bitM,Nb)))) ).

% or_minus_numerals(7)
tff(fact_9715_or__minus__numerals_I3_J,axiom,
    ! [M: num,Nb: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb)))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,M),aa_num_num(bitM,Nb)))) ).

% or_minus_numerals(3)
tff(fact_9716_or__not__num__neg_Osimps_I1_J,axiom,
    aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,one),one) = one ).

% or_not_num_neg.simps(1)
tff(fact_9717_or__not__num__neg_Osimps_I4_J,axiom,
    ! [Nb: num] : aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,aa_num_num(bit0,Nb)),one) = aa_num_num(bit0,one) ).

% or_not_num_neg.simps(4)
tff(fact_9718_or__not__num__neg_Osimps_I6_J,axiom,
    ! [Nb: num,M: num] : aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,aa_num_num(bit0,Nb)),aa_num_num(bit1,M)) = aa_num_num(bit0,aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,Nb),M)) ).

% or_not_num_neg.simps(6)
tff(fact_9719_or__not__num__neg_Osimps_I3_J,axiom,
    ! [M: num] : aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,one),aa_num_num(bit1,M)) = aa_num_num(bit1,M) ).

% or_not_num_neg.simps(3)
tff(fact_9720_or__not__num__neg_Osimps_I7_J,axiom,
    ! [Nb: num] : aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,aa_num_num(bit1,Nb)),one) = one ).

% or_not_num_neg.simps(7)
tff(fact_9721_or__not__num__neg_Osimps_I5_J,axiom,
    ! [Nb: num,M: num] : aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,aa_num_num(bit0,Nb)),aa_num_num(bit0,M)) = aa_num_num(bitM,aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,Nb),M)) ).

% or_not_num_neg.simps(5)
tff(fact_9722_or__not__num__neg_Osimps_I2_J,axiom,
    ! [M: num] : aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,one),aa_num_num(bit0,M)) = aa_num_num(bit1,M) ).

% or_not_num_neg.simps(2)
tff(fact_9723_or__not__num__neg_Osimps_I8_J,axiom,
    ! [Nb: num,M: num] : aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,aa_num_num(bit1,Nb)),aa_num_num(bit0,M)) = aa_num_num(bitM,aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,Nb),M)) ).

% or_not_num_neg.simps(8)
tff(fact_9724_or__not__num__neg_Oelims,axiom,
    ! [Xb: num,Xaa: num,Y: num] :
      ( ( aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,Xb),Xaa) = Y )
     => ( ( ( Xb = one )
         => ( ( Xaa = one )
           => ( Y != one ) ) )
       => ( ( ( Xb = one )
           => ! [M2: num] :
                ( ( Xaa = aa_num_num(bit0,M2) )
               => ( Y != aa_num_num(bit1,M2) ) ) )
         => ( ( ( Xb = one )
             => ! [M2: num] :
                  ( ( Xaa = aa_num_num(bit1,M2) )
                 => ( Y != aa_num_num(bit1,M2) ) ) )
           => ( ( ? [N2: num] : Xb = aa_num_num(bit0,N2)
               => ( ( Xaa = one )
                 => ( Y != aa_num_num(bit0,one) ) ) )
             => ( ! [N2: num] :
                    ( ( Xb = aa_num_num(bit0,N2) )
                   => ! [M2: num] :
                        ( ( Xaa = aa_num_num(bit0,M2) )
                       => ( Y != aa_num_num(bitM,aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,N2),M2)) ) ) )
               => ( ! [N2: num] :
                      ( ( Xb = aa_num_num(bit0,N2) )
                     => ! [M2: num] :
                          ( ( Xaa = aa_num_num(bit1,M2) )
                         => ( Y != aa_num_num(bit0,aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,N2),M2)) ) ) )
                 => ( ( ? [N2: num] : Xb = aa_num_num(bit1,N2)
                     => ( ( Xaa = one )
                       => ( Y != one ) ) )
                   => ( ! [N2: num] :
                          ( ( Xb = aa_num_num(bit1,N2) )
                         => ! [M2: num] :
                              ( ( Xaa = aa_num_num(bit0,M2) )
                             => ( Y != aa_num_num(bitM,aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,N2),M2)) ) ) )
                     => ~ ! [N2: num] :
                            ( ( Xb = aa_num_num(bit1,N2) )
                           => ! [M2: num] :
                                ( ( Xaa = aa_num_num(bit1,M2) )
                               => ( Y != aa_num_num(bitM,aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,N2),M2)) ) ) ) ) ) ) ) ) ) ) ) ) ).

% or_not_num_neg.elims
tff(fact_9725_int__numeral__or__not__num__neg,axiom,
    ! [M: num,Nb: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,Nb))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,M),Nb))) ).

% int_numeral_or_not_num_neg
tff(fact_9726_int__numeral__not__or__num__neg,axiom,
    ! [M: num,Nb: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,M))),aa_num_int(numeral_numeral_int,Nb)) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,Nb),M))) ).

% int_numeral_not_or_num_neg
tff(fact_9727_numeral__or__not__num__eq,axiom,
    ! [M: num,Nb: num] : aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,M),Nb)) = aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,Nb)))) ).

% numeral_or_not_num_eq
tff(fact_9728_xor__nat__unfold,axiom,
    ! [M: nat,Nb: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,M),Nb) = $ite(
        M = zero_zero_nat,
        Nb,
        $ite(Nb = zero_zero_nat,M,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))) ) ).

% xor_nat_unfold
tff(fact_9729_xor__nat__rec,axiom,
    ! [M: nat,Nb: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,M),Nb) = 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))),M) != ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% xor_nat_rec
tff(fact_9730_Suc__0__xor__eq,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_nat_nat(suc,zero_zero_nat)),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),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))),Nb)))),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))),Nb))) ).

% Suc_0_xor_eq
tff(fact_9731_xor__Suc__0__eq,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,Nb),aa_nat_nat(suc,zero_zero_nat)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),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))),Nb)))),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))),Nb))) ).

% xor_Suc_0_eq
tff(fact_9732_or__not__num__neg_Opelims,axiom,
    ! [Xb: num,Xaa: num,Y: num] :
      ( ( aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,Xb),Xaa) = Y )
     => ( aa_Pro7685007914727650815_num_o(accp_P3113834385874906142um_num(bit_or3848514188828904588eg_rel),product_Pair_num_num(Xb,Xaa))
       => ( ( ( Xb = one )
           => ( ( Xaa = one )
             => ( ( Y = one )
               => ~ aa_Pro7685007914727650815_num_o(accp_P3113834385874906142um_num(bit_or3848514188828904588eg_rel),product_Pair_num_num(one,one)) ) ) )
         => ( ( ( Xb = one )
             => ! [M2: num] :
                  ( ( Xaa = aa_num_num(bit0,M2) )
                 => ( ( Y = aa_num_num(bit1,M2) )
                   => ~ aa_Pro7685007914727650815_num_o(accp_P3113834385874906142um_num(bit_or3848514188828904588eg_rel),product_Pair_num_num(one,aa_num_num(bit0,M2))) ) ) )
           => ( ( ( Xb = one )
               => ! [M2: num] :
                    ( ( Xaa = aa_num_num(bit1,M2) )
                   => ( ( Y = aa_num_num(bit1,M2) )
                     => ~ aa_Pro7685007914727650815_num_o(accp_P3113834385874906142um_num(bit_or3848514188828904588eg_rel),product_Pair_num_num(one,aa_num_num(bit1,M2))) ) ) )
             => ( ! [N2: num] :
                    ( ( Xb = aa_num_num(bit0,N2) )
                   => ( ( Xaa = one )
                     => ( ( Y = aa_num_num(bit0,one) )
                       => ~ aa_Pro7685007914727650815_num_o(accp_P3113834385874906142um_num(bit_or3848514188828904588eg_rel),product_Pair_num_num(aa_num_num(bit0,N2),one)) ) ) )
               => ( ! [N2: num] :
                      ( ( Xb = aa_num_num(bit0,N2) )
                     => ! [M2: num] :
                          ( ( Xaa = aa_num_num(bit0,M2) )
                         => ( ( Y = aa_num_num(bitM,aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,N2),M2)) )
                           => ~ aa_Pro7685007914727650815_num_o(accp_P3113834385874906142um_num(bit_or3848514188828904588eg_rel),product_Pair_num_num(aa_num_num(bit0,N2),aa_num_num(bit0,M2))) ) ) )
                 => ( ! [N2: num] :
                        ( ( Xb = aa_num_num(bit0,N2) )
                       => ! [M2: num] :
                            ( ( Xaa = aa_num_num(bit1,M2) )
                           => ( ( Y = aa_num_num(bit0,aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,N2),M2)) )
                             => ~ aa_Pro7685007914727650815_num_o(accp_P3113834385874906142um_num(bit_or3848514188828904588eg_rel),product_Pair_num_num(aa_num_num(bit0,N2),aa_num_num(bit1,M2))) ) ) )
                   => ( ! [N2: num] :
                          ( ( Xb = aa_num_num(bit1,N2) )
                         => ( ( Xaa = one )
                           => ( ( Y = one )
                             => ~ aa_Pro7685007914727650815_num_o(accp_P3113834385874906142um_num(bit_or3848514188828904588eg_rel),product_Pair_num_num(aa_num_num(bit1,N2),one)) ) ) )
                     => ( ! [N2: num] :
                            ( ( Xb = aa_num_num(bit1,N2) )
                           => ! [M2: num] :
                                ( ( Xaa = aa_num_num(bit0,M2) )
                               => ( ( Y = aa_num_num(bitM,aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,N2),M2)) )
                                 => ~ aa_Pro7685007914727650815_num_o(accp_P3113834385874906142um_num(bit_or3848514188828904588eg_rel),product_Pair_num_num(aa_num_num(bit1,N2),aa_num_num(bit0,M2))) ) ) )
                       => ~ ! [N2: num] :
                              ( ( Xb = aa_num_num(bit1,N2) )
                             => ! [M2: num] :
                                  ( ( Xaa = aa_num_num(bit1,M2) )
                                 => ( ( Y = aa_num_num(bitM,aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,N2),M2)) )
                                   => ~ aa_Pro7685007914727650815_num_o(accp_P3113834385874906142um_num(bit_or3848514188828904588eg_rel),product_Pair_num_num(aa_num_num(bit1,N2),aa_num_num(bit1,M2))) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% or_not_num_neg.pelims
tff(fact_9733_num_Osize__gen_I3_J,axiom,
    ! [X33: num] : aa_num_nat(size_num,aa_num_num(bit1,X33)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(size_num,X33)),aa_nat_nat(suc,zero_zero_nat)) ).

% num.size_gen(3)
tff(fact_9734_num_Osize__gen_I1_J,axiom,
    aa_num_nat(size_num,one) = zero_zero_nat ).

% num.size_gen(1)
tff(fact_9735_num_Osize__gen_I2_J,axiom,
    ! [X23: num] : aa_num_nat(size_num,aa_num_num(bit0,X23)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(size_num,X23)),aa_nat_nat(suc,zero_zero_nat)) ).

% num.size_gen(2)
tff(fact_9736_dup__1,axiom,
    aa_Cod1783974652533396728nteger(code_dup,one_one_Code_integer) = aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)) ).

% dup_1
tff(fact_9737_dup_Oabs__eq,axiom,
    ! [Xb: int] : aa_Cod1783974652533396728nteger(code_dup,aa_int_Code_integer(code_integer_of_int,Xb)) = aa_int_Code_integer(code_integer_of_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb),Xb)) ).

% dup.abs_eq
tff(fact_9738_pow_Osimps_I1_J,axiom,
    ! [Xb: num] : aa_num_num(aa_num_fun_num_num(pow,Xb),one) = Xb ).

% pow.simps(1)
tff(fact_9739_cis__multiple__2pi,axiom,
    ! [Nb: real] :
      ( aa_set_real_o(member_real(Nb),ring_1_Ints_real)
     => ( aa_real_complex(cis,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),pi)),Nb)) = one_one_complex ) ) ).

% cis_multiple_2pi
tff(fact_9740_sin__times__pi__eq__0,axiom,
    ! [Xb: real] :
      ( ( aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),pi)) = zero_zero_real )
    <=> aa_set_real_o(member_real(Xb),ring_1_Ints_real) ) ).

% sin_times_pi_eq_0
tff(fact_9741_sin__integer__2pi,axiom,
    ! [Nb: real] :
      ( aa_set_real_o(member_real(Nb),ring_1_Ints_real)
     => ( aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),pi)),Nb)) = zero_zero_real ) ) ).

% sin_integer_2pi
tff(fact_9742_cos__integer__2pi,axiom,
    ! [Nb: real] :
      ( aa_set_real_o(member_real(Nb),ring_1_Ints_real)
     => ( aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),pi)),Nb)) = one_one_real ) ) ).

% cos_integer_2pi
tff(fact_9743_rat__inverse__code,axiom,
    ! [P2: rat] : quotient_of(inverse_inverse_rat(P2)) = aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(aTP_Lamm_bas),quotient_of(P2)) ).

% rat_inverse_code
tff(fact_9744_quotient__of__number_I3_J,axiom,
    ! [K: num] : quotient_of(numeral_numeral_rat(K)) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_num_int(numeral_numeral_int,K)),one_one_int) ).

% quotient_of_number(3)
tff(fact_9745_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_9746_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_9747_quotient__of__number_I5_J,axiom,
    ! [K: num] : quotient_of(uminus_uminus_rat(numeral_numeral_rat(K))) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K))),one_one_int) ).

% quotient_of_number(5)
tff(fact_9748_quotient__of__number_I4_J,axiom,
    quotient_of(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_9749_divide__rat__def,axiom,
    ! [Q2: rat,R2: rat] : divide_divide_rat(Q2,R2) = aa_rat_rat(times_times_rat(Q2),inverse_inverse_rat(R2)) ).

% divide_rat_def
tff(fact_9750_quotient__of__div,axiom,
    ! [R2: rat,Nb: int,D: int] :
      ( ( quotient_of(R2) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Nb),D) )
     => ( R2 = divide_divide_rat(aa_int_rat(ring_1_of_int_rat,Nb),aa_int_rat(ring_1_of_int_rat,D)) ) ) ).

% quotient_of_div
tff(fact_9751_quotient__of__denom__pos,axiom,
    ! [R2: rat,P2: int,Q2: int] :
      ( ( quotient_of(R2) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,P2),Q2) )
     => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Q2) ) ).

% quotient_of_denom_pos
tff(fact_9752_quotient__of__denom__pos_H,axiom,
    ! [R2: rat] : aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_Pro6799258434465214121nt_int(product_snd_int_int,quotient_of(R2))) ).

% quotient_of_denom_pos'
tff(fact_9753_rat__abs__code,axiom,
    ! [P2: rat] : quotient_of(abs_abs_rat(P2)) = aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(aTP_Lamm_bat),quotient_of(P2)) ).

% rat_abs_code
tff(fact_9754_rat__uminus__code,axiom,
    ! [P2: rat] : quotient_of(uminus_uminus_rat(P2)) = aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(aTP_Lamm_bau),quotient_of(P2)) ).

% rat_uminus_code
tff(fact_9755_rat__less__code,axiom,
    ! [P2: rat,Q2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,P2),Q2)
    <=> aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(aTP_Lamm_baw(Q2)),quotient_of(P2)) ) ).

% rat_less_code
tff(fact_9756_rat__floor__code,axiom,
    ! [P2: rat] : archim3151403230148437115or_rat(P2) = aa_Pro6799258434465214121nt_int(produc8211389475949308722nt_int(divide_divide_int),quotient_of(P2)) ).

% rat_floor_code
tff(fact_9757_rat__less__eq__code,axiom,
    ! [P2: rat,Q2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,P2),Q2)
    <=> aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(aTP_Lamm_bay(Q2)),quotient_of(P2)) ) ).

% rat_less_eq_code
tff(fact_9758_rat__sgn__code,axiom,
    ! [P2: rat] : quotient_of(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_9759_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_9760_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))) = divide_divide_rat(one_one_rat,numeral_numeral_rat(K)) ).

% Frct_code_post(5)
tff(fact_9761_rat__minus__code,axiom,
    ! [P2: rat,Q2: rat] : quotient_of(minus_minus_rat(P2,Q2)) = aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(aTP_Lamm_bba(Q2)),quotient_of(P2)) ).

% rat_minus_code
tff(fact_9762_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_9763_normalize__negative,axiom,
    ! [Q2: int,P2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Q2),zero_zero_int)
     => ( aa_Pro6970504127699319548nt_int(normalize,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,P2),Q2)) = 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,Q2))) ) ) ).

% normalize_negative
tff(fact_9764_normalize__denom__pos,axiom,
    ! [R2: product_prod_int_int,P2: int,Q2: int] :
      ( ( aa_Pro6970504127699319548nt_int(normalize,R2) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,P2),Q2) )
     => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Q2) ) ).

% normalize_denom_pos
tff(fact_9765_normalize__crossproduct,axiom,
    ! [Q2: int,S3: int,P2: int,R2: int] :
      ( ( Q2 != zero_zero_int )
     => ( ( S3 != zero_zero_int )
       => ( ( aa_Pro6970504127699319548nt_int(normalize,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,P2),Q2)) = aa_Pro6970504127699319548nt_int(normalize,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,R2),S3)) )
         => ( aa_int_int(aa_int_fun_int_int(times_times_int,P2),S3) = aa_int_int(aa_int_fun_int_int(times_times_int,R2),Q2) ) ) ) ) ).

% normalize_crossproduct
tff(fact_9766_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_9767_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_9768_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)) = uminus_uminus_rat(frct(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),B))) ).

% Frct_code_post(7)
tff(fact_9769_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))) = uminus_uminus_rat(frct(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),B))) ).

% Frct_code_post(8)
tff(fact_9770_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_9771_Frct__code__post_I4_J,axiom,
    ! [K: num] : frct(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_num_int(numeral_numeral_int,K)),one_one_int)) = numeral_numeral_rat(K) ).

% Frct_code_post(4)
tff(fact_9772_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))) = divide_divide_rat(numeral_numeral_rat(K),numeral_numeral_rat(L)) ).

% Frct_code_post(6)
tff(fact_9773_rat__times__code,axiom,
    ! [P2: rat,Q2: rat] : quotient_of(aa_rat_rat(times_times_rat(P2),Q2)) = aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(aTP_Lamm_bbc(Q2)),quotient_of(P2)) ).

% rat_times_code
tff(fact_9774_rat__divide__code,axiom,
    ! [P2: rat,Q2: rat] : quotient_of(divide_divide_rat(P2,Q2)) = aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(aTP_Lamm_bbe(Q2)),quotient_of(P2)) ).

% rat_divide_code
tff(fact_9775_rat__plus__code,axiom,
    ! [P2: rat,Q2: rat] : quotient_of(aa_rat_rat(plus_plus_rat(P2),Q2)) = aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(aTP_Lamm_bbg(Q2)),quotient_of(P2)) ).

% rat_plus_code
tff(fact_9776_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(
          a: int,
          a:= 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)),a)),aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_Pro6799258434465214121nt_int(product_snd_int_int,P2)),a)) ),
        $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(
            a: int,
            a:= aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(gcd_gcd_int,aa_Pro6799258434465214121nt_int(product_fst_int_int,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)),a)),aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_Pro6799258434465214121nt_int(product_snd_int_int,P2)),a)) ) ) ) ).

% normalize_def
tff(fact_9777_setceilmax,axiom,
    ! [S3: vEBT_VEBT,M: nat,Listy: list_VEBT_VEBT,Nb: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,S3),M)
     => ( ! [X2: vEBT_VEBT] :
            ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X2),set_VEBT_VEBT2(Listy))
           => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,X2),Nb) )
       => ( ( M = aa_nat_nat(suc,Nb) )
         => ( ! [X2: vEBT_VEBT] :
                ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X2),set_VEBT_VEBT2(Listy))
               => ( aa_nat_int(semiri1314217659103216013at_int,aa_VEBT_VEBT_nat(vEBT_VEBT_height,X2)) = aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(log,numeral_numeral_real(aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,Nb))) ) )
           => ( ( aa_nat_int(semiri1314217659103216013at_int,aa_VEBT_VEBT_nat(vEBT_VEBT_height,S3)) = aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(log,numeral_numeral_real(aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,M))) )
             => ( aa_nat_int(semiri1314217659103216013at_int,lattic8265883725875713057ax_nat(image_VEBT_VEBT_nat(vEBT_VEBT_height,insert_VEBT_VEBT(S3,set_VEBT_VEBT2(Listy))))) = aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(log,numeral_numeral_real(aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,M))) ) ) ) ) ) ) ).

% setceilmax
tff(fact_9778_height__compose__list,axiom,
    ! [Ta: vEBT_VEBT,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Ta),set_VEBT_VEBT2(TreeLista))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_VEBT_VEBT_nat(vEBT_VEBT_height,Ta)),lattic8265883725875713057ax_nat(image_VEBT_VEBT_nat(vEBT_VEBT_height,insert_VEBT_VEBT(Summarya,set_VEBT_VEBT2(TreeLista))))) ) ).

% height_compose_list
tff(fact_9779_gcd__1__int,axiom,
    ! [M: int] : aa_int_int(aa_int_fun_int_int(gcd_gcd_int,M),one_one_int) = one_one_int ).

% gcd_1_int
tff(fact_9780_max__ins__scaled,axiom,
    ! [Nb: nat,X14a: vEBT_VEBT,M: nat,X13a: list_VEBT_VEBT] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),aa_VEBT_VEBT_nat(vEBT_VEBT_height,X14a))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),lattic8265883725875713057ax_nat(insert_nat(aa_VEBT_VEBT_nat(vEBT_VEBT_height,X14a),image_VEBT_VEBT_nat(vEBT_VEBT_height,set_VEBT_VEBT2(X13a))))))) ).

% max_ins_scaled
tff(fact_9781_height__i__max,axiom,
    ! [I: nat,X13a: list_VEBT_VEBT,Foo: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s6755466524823107622T_VEBT(X13a))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_VEBT_VEBT_nat(vEBT_VEBT_height,nth_VEBT_VEBT(X13a,I))),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Foo),lattic8265883725875713057ax_nat(image_VEBT_VEBT_nat(vEBT_VEBT_height,set_VEBT_VEBT2(X13a))))) ) ).

% height_i_max
tff(fact_9782_max__idx__list,axiom,
    ! [I: nat,X13a: list_VEBT_VEBT,Nb: nat,X14a: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s6755466524823107622T_VEBT(X13a))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),aa_VEBT_VEBT_nat(vEBT_VEBT_height,nth_VEBT_VEBT(X13a,I)))),aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_VEBT_VEBT_nat(vEBT_VEBT_height,X14a)),lattic8265883725875713057ax_nat(image_VEBT_VEBT_nat(vEBT_VEBT_height,set_VEBT_VEBT2(X13a)))))))) ) ).

% max_idx_list
tff(fact_9783_gcd__neg__numeral__2__int,axiom,
    ! [Xb: int,Nb: num] : aa_int_int(aa_int_fun_int_int(gcd_gcd_int,Xb),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))) = aa_int_int(aa_int_fun_int_int(gcd_gcd_int,Xb),aa_num_int(numeral_numeral_int,Nb)) ).

% gcd_neg_numeral_2_int
tff(fact_9784_gcd__neg__numeral__1__int,axiom,
    ! [Nb: num,Xb: int] : aa_int_int(aa_int_fun_int_int(gcd_gcd_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))),Xb) = aa_int_int(aa_int_fun_int_int(gcd_gcd_int,aa_num_int(numeral_numeral_int,Nb)),Xb) ).

% gcd_neg_numeral_1_int
tff(fact_9785_Max__divisors__self__nat,axiom,
    ! [Nb: nat] :
      ( ( Nb != zero_zero_nat )
     => ( lattic8265883725875713057ax_nat(collect_nat(aa_nat_fun_nat_o(aTP_Lamm_wq,Nb))) = Nb ) ) ).

% Max_divisors_self_nat
tff(fact_9786_gcd__ge__0__int,axiom,
    ! [Xb: 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,Xb),Y)) ).

% gcd_ge_0_int
tff(fact_9787_gcd__mult__distrib__int,axiom,
    ! [K: int,M: int,Nb: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(abs_abs_int,K)),aa_int_int(aa_int_fun_int_int(gcd_gcd_int,M),Nb)) = aa_int_int(aa_int_fun_int_int(gcd_gcd_int,aa_int_int(aa_int_fun_int_int(times_times_int,K),M)),aa_int_int(aa_int_fun_int_int(times_times_int,K),Nb)) ).

% gcd_mult_distrib_int
tff(fact_9788_bezout__int,axiom,
    ! [Xb: int,Y: int] :
    ? [U2: int,V2: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,U2),Xb)),aa_int_int(aa_int_fun_int_int(times_times_int,V2),Y)) = aa_int_int(aa_int_fun_int_int(gcd_gcd_int,Xb),Y) ).

% bezout_int
tff(fact_9789_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_9790_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_9791_gcd__cases__int,axiom,
    ! [Xb: int,Y: int,P: fun_int_o] :
      ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xb)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
         => aa_int_o(P,aa_int_int(aa_int_fun_int_int(gcd_gcd_int,Xb),Y)) ) )
     => ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xb)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),zero_zero_int)
           => aa_int_o(P,aa_int_int(aa_int_fun_int_int(gcd_gcd_int,Xb),aa_int_int(uminus_uminus_int,Y))) ) )
       => ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),zero_zero_int)
           => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
             => aa_int_o(P,aa_int_int(aa_int_fun_int_int(gcd_gcd_int,aa_int_int(uminus_uminus_int,Xb)),Y)) ) )
         => ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),zero_zero_int)
             => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),zero_zero_int)
               => aa_int_o(P,aa_int_int(aa_int_fun_int_int(gcd_gcd_int,aa_int_int(uminus_uminus_int,Xb)),aa_int_int(uminus_uminus_int,Y))) ) )
           => aa_int_o(P,aa_int_int(aa_int_fun_int_int(gcd_gcd_int,Xb),Y)) ) ) ) ) ).

% gcd_cases_int
tff(fact_9792_gcd__unique__int,axiom,
    ! [D: int,A: int,B: int] :
      ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),D)
        & aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),A)
        & aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),B)
        & ! [E3: int] :
            ( ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,E3),A)
              & aa_int_o(aa_int_fun_int_o(dvd_dvd_int,E3),B) )
           => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,E3),D) ) )
    <=> ( D = aa_int_int(aa_int_fun_int_int(gcd_gcd_int,A),B) ) ) ).

% gcd_unique_int
tff(fact_9793_VEBT__internal_Oheight_Osimps_I2_J,axiom,
    ! [Uu: option4927543243414619207at_nat,Deg: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT] : aa_VEBT_VEBT_nat(vEBT_VEBT_height,aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uu,Deg,TreeLista),Summarya)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),lattic8265883725875713057ax_nat(image_VEBT_VEBT_nat(vEBT_VEBT_height,insert_VEBT_VEBT(Summarya,set_VEBT_VEBT2(TreeLista))))) ).

% VEBT_internal.height.simps(2)
tff(fact_9794_VEBT__internal_Oheight_Oelims,axiom,
    ! [Xb: vEBT_VEBT,Y: nat] :
      ( ( aa_VEBT_VEBT_nat(vEBT_VEBT_height,Xb) = Y )
     => ( ( ? [A3: $o,B2: $o] : Xb = vEBT_Leaf((A3),(B2))
         => ( Y != zero_zero_nat ) )
       => ~ ! [Uu2: option4927543243414619207at_nat,Deg2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
              ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uu2,Deg2,TreeList),Summary) )
             => ( Y != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),lattic8265883725875713057ax_nat(image_VEBT_VEBT_nat(vEBT_VEBT_height,insert_VEBT_VEBT(Summary,set_VEBT_VEBT2(TreeList))))) ) ) ) ) ).

% VEBT_internal.height.elims
tff(fact_9795_divide__nat__def,axiom,
    ! [M: nat,Nb: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Nb) = $ite(Nb = zero_zero_nat,zero_zero_nat,lattic8265883725875713057ax_nat(collect_nat(aa_nat_fun_nat_o(aTP_Lamm_bbh(M),Nb)))) ).

% divide_nat_def
tff(fact_9796_VEBT__internal_Oheight_Opelims,axiom,
    ! [Xb: vEBT_VEBT,Y: nat] :
      ( ( aa_VEBT_VEBT_nat(vEBT_VEBT_height,Xb) = Y )
     => ( aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_VEBT_height_rel),Xb)
       => ( ! [A3: $o,B2: $o] :
              ( ( Xb = vEBT_Leaf((A3),(B2)) )
             => ( ( Y = zero_zero_nat )
               => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_VEBT_height_rel),vEBT_Leaf((A3),(B2))) ) )
         => ~ ! [Uu2: option4927543243414619207at_nat,Deg2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uu2,Deg2,TreeList),Summary) )
               => ( ( Y = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),lattic8265883725875713057ax_nat(image_VEBT_VEBT_nat(vEBT_VEBT_height,insert_VEBT_VEBT(Summary,set_VEBT_VEBT2(TreeList))))) )
                 => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_VEBT_height_rel),aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uu2,Deg2,TreeList),Summary)) ) ) ) ) ) ).

% VEBT_internal.height.pelims
tff(fact_9797_bin__last__integer__nbe,axiom,
    ! [I: code_integer] :
      ( aa_Code_integer_o(bits_b8758750999018896077nteger,I)
    <=> ( aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(I),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) != zero_z3403309356797280102nteger ) ) ).

% bin_last_integer_nbe
tff(fact_9798_bij__betw__Suc,axiom,
    ! [M8: set_nat,N: set_nat] :
      ( aa_set_nat_o(bij_betw_nat_nat(suc,M8),N)
    <=> ( image_nat_nat(suc,M8) = N ) ) ).

% bij_betw_Suc
tff(fact_9799_gcd__1__nat,axiom,
    ! [M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,M),one_one_nat) = one_one_nat ).

% gcd_1_nat
tff(fact_9800_gcd__Suc__0,axiom,
    ! [M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,M),aa_nat_nat(suc,zero_zero_nat)) = aa_nat_nat(suc,zero_zero_nat) ).

% gcd_Suc_0
tff(fact_9801_gcd__pos__nat,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,M),Nb))
    <=> ( ( M != zero_zero_nat )
        | ( Nb != zero_zero_nat ) ) ) ).

% gcd_pos_nat
tff(fact_9802_image__Suc__atLeastLessThan,axiom,
    ! [I: nat,J: nat] : image_nat_nat(suc,set_or4665077453230672383an_nat(I,J)) = set_or4665077453230672383an_nat(aa_nat_nat(suc,I),aa_nat_nat(suc,J)) ).

% image_Suc_atLeastLessThan
tff(fact_9803_image__Suc__atLeastAtMost,axiom,
    ! [I: nat,J: nat] : image_nat_nat(suc,set_or1269000886237332187st_nat(I,J)) = set_or1269000886237332187st_nat(aa_nat_nat(suc,I),aa_nat_nat(suc,J)) ).

% image_Suc_atLeastAtMost
tff(fact_9804_range__mult,axiom,
    ! [A: real] :
      image_real_real(aa_rea7340605660300375871l_real(times_times_real,A),top_top_set_real) = $ite(A = zero_zero_real,insert_real(zero_zero_real,bot_bot_set_real),top_top_set_real) ).

% range_mult
tff(fact_9805_Max__divisors__self__int,axiom,
    ! [Nb: int] :
      ( ( Nb != zero_zero_int )
     => ( lattic8263393255366662781ax_int(collect_int(aa_int_fun_int_o(aTP_Lamm_wu,Nb))) = aa_int_int(abs_abs_int,Nb) ) ) ).

% Max_divisors_self_int
tff(fact_9806_gcd__diff2__nat,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),M)),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,M),Nb) ) ) ).

% gcd_diff2_nat
tff(fact_9807_gcd__diff1__nat,axiom,
    ! [Nb: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),M)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Nb)),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,M),Nb) ) ) ).

% gcd_diff1_nat
tff(fact_9808_zero__notin__Suc__image,axiom,
    ! [A2: set_nat] : ~ aa_set_nat_o(member_nat(zero_zero_nat),image_nat_nat(suc,A2)) ).

% zero_notin_Suc_image
tff(fact_9809_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_9810_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_9811_finite__int__iff__bounded,axiom,
    ! [S2: set_int] :
      ( aa_set_int_o(finite_finite_int,S2)
    <=> ? [K4: int] : aa_set_int_o(ord_less_eq_set_int(image_int_int(abs_abs_int,S2)),set_ord_lessThan_int(K4)) ) ).

% finite_int_iff_bounded
tff(fact_9812_finite__int__iff__bounded__le,axiom,
    ! [S2: set_int] :
      ( aa_set_int_o(finite_finite_int,S2)
    <=> ? [K4: int] : aa_set_int_o(ord_less_eq_set_int(image_int_int(abs_abs_int,S2)),set_ord_atMost_int(K4)) ) ).

% finite_int_iff_bounded_le
tff(fact_9813_gcd__mult__distrib__nat,axiom,
    ! [K: nat,M: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,M),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_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),Nb)) ).

% gcd_mult_distrib_nat
tff(fact_9814_bitval__bin__last__integer,axiom,
    ! [I: code_integer] : zero_n356916108424825756nteger(aa_Code_integer_o(bits_b8758750999018896077nteger,I)) = aa_Cod1783974652533396728nteger(bit_se3949692690581998587nteger(I),one_one_Code_integer) ).

% bitval_bin_last_integer
tff(fact_9815_bezout__nat,axiom,
    ! [A: nat,B: nat] :
      ( ( A != zero_zero_nat )
     => ? [X2: nat,Y4: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),X2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),Y4)),aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B)) ) ).

% bezout_nat
tff(fact_9816_bezout__gcd__nat_H,axiom,
    ! [B: nat,A: nat] :
    ? [X2: nat,Y4: 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),Y4)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),X2))
        & ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),X2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),Y4)) = 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),Y4)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),X2))
        & ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),X2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),Y4)) = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B) ) ) ) ).

% bezout_gcd_nat'
tff(fact_9817_gcd__is__Max__divisors__int,axiom,
    ! [Nb: int,M: int] :
      ( ( Nb != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(gcd_gcd_int,M),Nb) = lattic8263393255366662781ax_int(collect_int(aa_int_fun_int_o(aTP_Lamm_bbi(Nb),M))) ) ) ).

% gcd_is_Max_divisors_int
tff(fact_9818_bin__last__integer__code,axiom,
    ! [I: code_integer] :
      ( aa_Code_integer_o(bits_b8758750999018896077nteger,I)
    <=> ( aa_Cod1783974652533396728nteger(bit_se3949692690581998587nteger(I),one_one_Code_integer) != zero_z3403309356797280102nteger ) ) ).

% bin_last_integer_code
tff(fact_9819_image__Suc__lessThan,axiom,
    ! [Nb: nat] : image_nat_nat(suc,set_ord_lessThan_nat(Nb)) = set_or1269000886237332187st_nat(one_one_nat,Nb) ).

% image_Suc_lessThan
tff(fact_9820_image__Suc__atMost,axiom,
    ! [Nb: nat] : image_nat_nat(suc,set_ord_atMost_nat(Nb)) = set_or1269000886237332187st_nat(one_one_nat,aa_nat_nat(suc,Nb)) ).

% image_Suc_atMost
tff(fact_9821_atLeast0__lessThan__Suc__eq__insert__0,axiom,
    ! [Nb: nat] : set_or4665077453230672383an_nat(zero_zero_nat,aa_nat_nat(suc,Nb)) = insert_nat(zero_zero_nat,image_nat_nat(suc,set_or4665077453230672383an_nat(zero_zero_nat,Nb))) ).

% atLeast0_lessThan_Suc_eq_insert_0
tff(fact_9822_atLeast0__atMost__Suc__eq__insert__0,axiom,
    ! [Nb: nat] : set_or1269000886237332187st_nat(zero_zero_nat,aa_nat_nat(suc,Nb)) = insert_nat(zero_zero_nat,image_nat_nat(suc,set_or1269000886237332187st_nat(zero_zero_nat,Nb))) ).

% atLeast0_atMost_Suc_eq_insert_0
tff(fact_9823_lessThan__Suc__eq__insert__0,axiom,
    ! [Nb: nat] : set_ord_lessThan_nat(aa_nat_nat(suc,Nb)) = insert_nat(zero_zero_nat,image_nat_nat(suc,set_ord_lessThan_nat(Nb))) ).

% lessThan_Suc_eq_insert_0
tff(fact_9824_atMost__Suc__eq__insert__0,axiom,
    ! [Nb: nat] : set_ord_atMost_nat(aa_nat_nat(suc,Nb)) = insert_nat(zero_zero_nat,image_nat_nat(suc,set_ord_atMost_nat(Nb))) ).

% atMost_Suc_eq_insert_0
tff(fact_9825_range__mod,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( image_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_bbj,Nb),top_top_set_nat) = set_or4665077453230672383an_nat(zero_zero_nat,Nb) ) ) ).

% range_mod
tff(fact_9826_gcd__is__Max__divisors__nat,axiom,
    ! [Nb: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,M),Nb) = lattic8265883725875713057ax_nat(collect_nat(aa_nat_fun_nat_o(aTP_Lamm_bbk(Nb),M))) ) ) ).

% gcd_is_Max_divisors_nat
tff(fact_9827_image__add__int__atLeastLessThan,axiom,
    ! [L: int,U: int] : image_int_int(aa_int_fun_int_int(aTP_Lamm_bbl,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_9828_image__add__integer__atLeastLessThan,axiom,
    ! [L: code_integer,U: code_integer] : image_4470545334726330049nteger(aTP_Lamm_bbm(L),set_or8404916559141939852nteger(zero_z3403309356797280102nteger,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(U),L))) = set_or8404916559141939852nteger(L,U) ).

% image_add_integer_atLeastLessThan
tff(fact_9829_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,set_ord_lessThan_nat(aa_int_nat(nat2,U))) ) ) ).

% image_atLeastZeroLessThan_int
tff(fact_9830_bezw__aux,axiom,
    ! [Xb: nat,Y: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Xb),Y)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_Pro6799258434465214121nt_int(product_fst_int_int,bezw(Xb,Y))),aa_nat_int(semiri1314217659103216013at_int,Xb))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_Pro6799258434465214121nt_int(product_snd_int_int,bezw(Xb,Y))),aa_nat_int(semiri1314217659103216013at_int,Y))) ).

% bezw_aux
tff(fact_9831_image__minus__const__atLeastLessThan__nat,axiom,
    ! [C: nat,Xb: nat,Y: nat] :
      image_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_bbn,C),set_or4665077453230672383an_nat(Xb,Y)) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),Y),
        set_or4665077453230672383an_nat(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Xb),C),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Y),C)),
        $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y),insert_nat(zero_zero_nat,bot_bot_set_nat),bot_bot_set_nat) ) ).

% image_minus_const_atLeastLessThan_nat
tff(fact_9832_bin__last__integer_Oabs__eq,axiom,
    ! [Xb: int] :
      ( aa_Code_integer_o(bits_b8758750999018896077nteger,aa_int_Code_integer(code_integer_of_int,Xb))
    <=> ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Xb) ) ).

% bin_last_integer.abs_eq
tff(fact_9833_gcd__nat_Opelims,axiom,
    ! [Xb: nat,Xaa: nat,Y: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Xb),Xaa) = Y )
     => ( aa_Pro6150637604445964947_nat_o(accp_P4275260045618599050at_nat(gcd_nat_rel),aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Xb),Xaa))
       => ~ ( ( Y = $ite(Xaa = zero_zero_nat,Xb,aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xb),Xaa))) )
           => ~ aa_Pro6150637604445964947_nat_o(accp_P4275260045618599050at_nat(gcd_nat_rel),aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Xb),Xaa)) ) ) ) ).

% gcd_nat.pelims
tff(fact_9834_bitXOR__integer__unfold,axiom,
    ! [Xb: code_integer,Y: code_integer] :
      aa_Cod1783974652533396728nteger(bit_se3222712562003087583nteger(Xb),Y) = $ite(
        Xb = zero_z3403309356797280102nteger,
        Y,
        $ite(
          Xb = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer),
          aa_Cod1783974652533396728nteger(bit_ri7632146776885996613nteger,Y),
          bits_Bit_integer(aa_Cod1783974652533396728nteger(bit_se3222712562003087583nteger(aa_Cod1783974652533396728nteger(bits_b2549910563261871055nteger,Xb)),aa_Cod1783974652533396728nteger(bits_b2549910563261871055nteger,Y)),
            ~ aa_Code_integer_o(bits_b8758750999018896077nteger,Xb) = aa_Code_integer_o(bits_b8758750999018896077nteger,Y)) ) ) ).

% bitXOR_integer_unfold
tff(fact_9835_bin__rest__integer__code,axiom,
    ! [I: code_integer] : aa_Cod1783974652533396728nteger(bits_b2549910563261871055nteger,I) = aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(I),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) ).

% bin_rest_integer_code
tff(fact_9836_bin__rest__integer_Oabs__eq,axiom,
    ! [Xb: int] : aa_Cod1783974652533396728nteger(bits_b2549910563261871055nteger,aa_int_Code_integer(code_integer_of_int,Xb)) = aa_int_Code_integer(code_integer_of_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,Xb),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))) ).

% bin_rest_integer.abs_eq
tff(fact_9837_bitOR__integer__unfold,axiom,
    ! [Xb: code_integer,Y: code_integer] :
      aa_Cod1783974652533396728nteger(bit_se1080825931792720795nteger(Xb),Y) = $ite(
        Xb = zero_z3403309356797280102nteger,
        Y,
        $ite(
          Xb = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer),
          aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer),
          bits_Bit_integer(aa_Cod1783974652533396728nteger(bit_se1080825931792720795nteger(aa_Cod1783974652533396728nteger(bits_b2549910563261871055nteger,Xb)),aa_Cod1783974652533396728nteger(bits_b2549910563261871055nteger,Y)),
            ( aa_Code_integer_o(bits_b8758750999018896077nteger,Xb)
            | aa_Code_integer_o(bits_b8758750999018896077nteger,Y) )) ) ) ).

% bitOR_integer_unfold
tff(fact_9838_bitAND__integer__unfold,axiom,
    ! [Xb: code_integer,Y: code_integer] :
      aa_Cod1783974652533396728nteger(bit_se3949692690581998587nteger(Xb),Y) = $ite(
        Xb = zero_z3403309356797280102nteger,
        zero_z3403309356797280102nteger,
        $ite(
          Xb = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer),
          Y,
          bits_Bit_integer(aa_Cod1783974652533396728nteger(bit_se3949692690581998587nteger(aa_Cod1783974652533396728nteger(bits_b2549910563261871055nteger,Xb)),aa_Cod1783974652533396728nteger(bits_b2549910563261871055nteger,Y)),
            ( aa_Code_integer_o(bits_b8758750999018896077nteger,Xb)
            & aa_Code_integer_o(bits_b8758750999018896077nteger,Y) )) ) ) ).

% bitAND_integer_unfold
tff(fact_9839_UNIV__nat__eq,axiom,
    top_top_set_nat = insert_nat(zero_zero_nat,image_nat_nat(suc,top_top_set_nat)) ).

% UNIV_nat_eq
tff(fact_9840_wait__rule,axiom,
    ! [Nb: nat] : hoare_8945653483474564448t_unit(one_one_assn,aa_nat5861795706794010277t_unit(heap_Time_wait,Nb),aTP_Lamm_bbo) ).

% wait_rule
tff(fact_9841_take__bit__num__def,axiom,
    ! [Nb: nat,M: num] :
      aa_num_option_num(bit_take_bit_num(Nb),M) = $ite(aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),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,Nb),aa_num_nat(numeral_numeral_nat,M))))) ).

% take_bit_num_def
tff(fact_9842_drop__bit__nonnegative__int__iff,axiom,
    ! [Nb: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_nat_fun_int_int(bit_se8568078237143864401it_int,Nb),K))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K) ) ).

% drop_bit_nonnegative_int_iff
tff(fact_9843_drop__bit__minus__one,axiom,
    ! [Nb: nat] : aa_int_int(aa_nat_fun_int_int(bit_se8568078237143864401it_int,Nb),aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(uminus_uminus_int,one_one_int) ).

% drop_bit_minus_one
tff(fact_9844_num__of__nat__numeral__eq,axiom,
    ! [Q2: num] : aa_nat_num(num_of_nat,aa_num_nat(numeral_numeral_nat,Q2)) = Q2 ).

% num_of_nat_numeral_eq
tff(fact_9845_drop__bit__of__Suc__0,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se8570568707652914677it_nat,Nb),aa_nat_nat(suc,zero_zero_nat)) = aa_o_nat(zero_n2687167440665602831ol_nat,Nb = zero_zero_nat) ).

% drop_bit_of_Suc_0
tff(fact_9846_drop__bit__Suc__minus__bit0,axiom,
    ! [Nb: nat,K: num] : aa_int_int(aa_nat_fun_int_int(bit_se8568078237143864401it_int,aa_nat_nat(suc,Nb)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,K)))) = aa_int_int(aa_nat_fun_int_int(bit_se8568078237143864401it_int,Nb),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K))) ).

% drop_bit_Suc_minus_bit0
tff(fact_9847_drop__bit__numeral__minus__bit0,axiom,
    ! [L: num,K: num] : aa_int_int(aa_nat_fun_int_int(bit_se8568078237143864401it_int,aa_num_nat(numeral_numeral_nat,L)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,K)))) = aa_int_int(aa_nat_fun_int_int(bit_se8568078237143864401it_int,aa_num_nat(pred_numeral,L)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K))) ).

% drop_bit_numeral_minus_bit0
tff(fact_9848_drop__bit__Suc__minus__bit1,axiom,
    ! [Nb: nat,K: num] : aa_int_int(aa_nat_fun_int_int(bit_se8568078237143864401it_int,aa_nat_nat(suc,Nb)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,K)))) = aa_int_int(aa_nat_fun_int_int(bit_se8568078237143864401it_int,Nb),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(inc,K)))) ).

% drop_bit_Suc_minus_bit1
tff(fact_9849_drop__bit__numeral__minus__bit1,axiom,
    ! [L: num,K: num] : aa_int_int(aa_nat_fun_int_int(bit_se8568078237143864401it_int,aa_num_nat(numeral_numeral_nat,L)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,K)))) = aa_int_int(aa_nat_fun_int_int(bit_se8568078237143864401it_int,aa_num_nat(pred_numeral,L)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(inc,K)))) ).

% drop_bit_numeral_minus_bit1
tff(fact_9850_drop__bit__int__code_I2_J,axiom,
    ! [Nb: nat] : aa_int_int(aa_nat_fun_int_int(bit_se8568078237143864401it_int,aa_nat_nat(suc,Nb)),zero_zero_int) = zero_zero_int ).

% drop_bit_int_code(2)
tff(fact_9851_num__of__nat_Osimps_I1_J,axiom,
    aa_nat_num(num_of_nat,zero_zero_nat) = one ).

% num_of_nat.simps(1)
tff(fact_9852_numeral__num__of__nat,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_num_nat(numeral_numeral_nat,aa_nat_num(num_of_nat,Nb)) = Nb ) ) ).

% numeral_num_of_nat
tff(fact_9853_num__of__nat__One,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),one_one_nat)
     => ( aa_nat_num(num_of_nat,Nb) = one ) ) ).

% num_of_nat_One
tff(fact_9854_shiftr__integer__conv__div__pow2,axiom,
    ! [Nb: nat,Xb: code_integer] : aa_Cod1783974652533396728nteger(bit_se3928097537394005634nteger(Nb),Xb) = aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(Xb),aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Nb)) ).

% shiftr_integer_conv_div_pow2
tff(fact_9855_bin__rest__code,axiom,
    ! [I: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,I),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = aa_int_int(aa_nat_fun_int_int(bit_se8568078237143864401it_int,one_one_nat),I) ).

% bin_rest_code
tff(fact_9856_drop__bit__int__def,axiom,
    ! [Nb: nat,K: int] : aa_int_int(aa_nat_fun_int_int(bit_se8568078237143864401it_int,Nb),K) = aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) ).

% drop_bit_int_def
tff(fact_9857_drop__bit__nat__def,axiom,
    ! [Nb: nat,M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se8570568707652914677it_nat,Nb),M) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ).

% drop_bit_nat_def
tff(fact_9858_num__of__nat__double,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_num(num_of_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Nb)) = aa_num_num(bit0,aa_nat_num(num_of_nat,Nb)) ) ) ).

% num_of_nat_double
tff(fact_9859_num__of__nat__plus__distrib,axiom,
    ! [M: nat,Nb: 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),Nb)
       => ( aa_nat_num(num_of_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Nb)) = aa_num_num(aa_num_fun_num_num(plus_plus_num,aa_nat_num(num_of_nat,M)),aa_nat_num(num_of_nat,Nb)) ) ) ) ).

% num_of_nat_plus_distrib
tff(fact_9860_num__of__nat_Osimps_I2_J,axiom,
    ! [Nb: nat] :
      aa_nat_num(num_of_nat,aa_nat_nat(suc,Nb)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb),aa_num_num(inc,aa_nat_num(num_of_nat,Nb)),one) ).

% num_of_nat.simps(2)
tff(fact_9861_push__bit__nonnegative__int__iff,axiom,
    ! [Nb: 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,Nb),K))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K) ) ).

% push_bit_nonnegative_int_iff
tff(fact_9862_push__bit__of__Suc__0,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,Nb),aa_nat_nat(suc,zero_zero_nat)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb) ).

% push_bit_of_Suc_0
tff(fact_9863_drop__bit__push__bit__int,axiom,
    ! [M: nat,Nb: nat,K: int] : aa_int_int(aa_nat_fun_int_int(bit_se8568078237143864401it_int,M),aa_int_int(aa_nat_fun_int_int(bit_se545348938243370406it_int,Nb),K)) = aa_int_int(aa_nat_fun_int_int(bit_se8568078237143864401it_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Nb)),aa_int_int(aa_nat_fun_int_int(bit_se545348938243370406it_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),M)),K)) ).

% drop_bit_push_bit_int
tff(fact_9864_set__bit__nat__def,axiom,
    ! [M: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se7882103937844011126it_nat,M),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,M),one_one_nat)) ).

% set_bit_nat_def
tff(fact_9865_Bit__integer__code_I1_J,axiom,
    ! [I: code_integer] : bits_Bit_integer(I,$false) = aa_Cod1783974652533396728nteger(bit_se7788150548672797655nteger(one_one_nat),I) ).

% Bit_integer_code(1)
tff(fact_9866_flip__bit__nat__def,axiom,
    ! [M: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2161824704523386999it_nat,M),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,M),one_one_nat)) ).

% flip_bit_nat_def
tff(fact_9867_bit__push__bit__iff__int,axiom,
    ! [M: nat,K: int,Nb: nat] :
      ( aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,aa_int_int(aa_nat_fun_int_int(bit_se545348938243370406it_int,M),K)),Nb)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
        & aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,K),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),M)) ) ) ).

% bit_push_bit_iff_int
tff(fact_9868_Bit__Operations_Oset__bit__int__def,axiom,
    ! [Nb: nat,K: int] : aa_int_int(aa_nat_fun_int_int(bit_se7879613467334960850it_int,Nb),K) = aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,K),aa_int_int(aa_nat_fun_int_int(bit_se545348938243370406it_int,Nb),one_one_int)) ).

% Bit_Operations.set_bit_int_def
tff(fact_9869_bit__push__bit__iff__nat,axiom,
    ! [M: nat,Q2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,M),Q2)),Nb)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
        & aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,Q2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),M)) ) ) ).

% bit_push_bit_iff_nat
tff(fact_9870_flip__bit__int__def,axiom,
    ! [Nb: nat,K: int] : aa_int_int(aa_nat_fun_int_int(bit_se2159334234014336723it_int,Nb),K) = aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,K),aa_int_int(aa_nat_fun_int_int(bit_se545348938243370406it_int,Nb),one_one_int)) ).

% flip_bit_int_def
tff(fact_9871_shiftl__integer__conv__mult__pow2,axiom,
    ! [Nb: nat,Xb: code_integer] : aa_Cod1783974652533396728nteger(bit_se7788150548672797655nteger(Nb),Xb) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Xb),aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Nb)) ).

% shiftl_integer_conv_mult_pow2
tff(fact_9872_unset__bit__int__def,axiom,
    ! [Nb: nat,K: int] : aa_int_int(aa_nat_fun_int_int(bit_se4203085406695923979it_int,Nb),K) = aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,K),aa_int_int(bit_ri7919022796975470100ot_int,aa_int_int(aa_nat_fun_int_int(bit_se545348938243370406it_int,Nb),one_one_int))) ).

% unset_bit_int_def
tff(fact_9873_push__bit__int__def,axiom,
    ! [Nb: nat,K: int] : aa_int_int(aa_nat_fun_int_int(bit_se545348938243370406it_int,Nb),K) = aa_int_int(aa_int_fun_int_int(times_times_int,K),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) ).

% push_bit_int_def
tff(fact_9874_Bit__integer__code_I2_J,axiom,
    ! [I: code_integer] : bits_Bit_integer(I,$true) = aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(bit_se7788150548672797655nteger(one_one_nat),I)),one_one_Code_integer) ).

% Bit_integer_code(2)
tff(fact_9875_push__bit__nat__def,axiom,
    ! [Nb: nat,M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,Nb),M) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ).

% push_bit_nat_def
tff(fact_9876_push__bit__minus__one,axiom,
    ! [Nb: nat] : aa_int_int(aa_nat_fun_int_int(bit_se545348938243370406it_int,Nb),aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(uminus_uminus_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) ).

% push_bit_minus_one
tff(fact_9877_set__bit__integer__conv__masks,axiom,
    ! [Xb: code_integer,I: nat,B: $o] :
      generi2397576812484419408nteger(Xb,I,(B)) = $ite((B),aa_Cod1783974652533396728nteger(bit_se1080825931792720795nteger(Xb),aa_Cod1783974652533396728nteger(bit_se7788150548672797655nteger(I),one_one_Code_integer)),aa_Cod1783974652533396728nteger(bit_se3949692690581998587nteger(Xb),aa_Cod1783974652533396728nteger(bit_ri7632146776885996613nteger,aa_Cod1783974652533396728nteger(bit_se7788150548672797655nteger(I),one_one_Code_integer)))) ).

% set_bit_integer_conv_masks
tff(fact_9878_Code__Numeral_Opositive__def,axiom,
    code_positive = numera6620942414471956472nteger ).

% Code_Numeral.positive_def
tff(fact_9879_int__set__bit__True__conv__OR,axiom,
    ! [I: int,Nb: nat] : aa_o_int(generi8991105624351003935it_int(I,Nb),$true) = aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,I),aa_int_int(aa_nat_fun_int_int(bit_se545348938243370406it_int,Nb),one_one_int)) ).

% int_set_bit_True_conv_OR
tff(fact_9880_int__set__bit__False__conv__NAND,axiom,
    ! [I: int,Nb: nat] : aa_o_int(generi8991105624351003935it_int(I,Nb),$false) = aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,I),aa_int_int(bit_ri7919022796975470100ot_int,aa_int_int(aa_nat_fun_int_int(bit_se545348938243370406it_int,Nb),one_one_int))) ).

% int_set_bit_False_conv_NAND
tff(fact_9881_int__set__bit__conv__ops,axiom,
    ! [I: int,Nb: nat,B: $o] :
      aa_o_int(generi8991105624351003935it_int(I,Nb),(B)) = $ite((B),aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,I),aa_int_int(aa_nat_fun_int_int(bit_se545348938243370406it_int,Nb),one_one_int)),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,I),aa_int_int(bit_ri7919022796975470100ot_int,aa_int_int(aa_nat_fun_int_int(bit_se545348938243370406it_int,Nb),one_one_int)))) ).

% int_set_bit_conv_ops
tff(fact_9882_upto__aux__rec,axiom,
    ! [I: int,J: int,Js: list_int] :
      upto_aux(I,J,Js) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_int,J),I),Js,upto_aux(I,aa_int_int(aa_int_fun_int_int(minus_minus_int,J),one_one_int),cons_int(J,Js))) ).

% upto_aux_rec
tff(fact_9883_upto_Opsimps,axiom,
    ! [I: int,J: int] :
      ( aa_Pro4592068669816069595_int_o(accp_P1096762738010456898nt_int(upto_rel),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,I),J))
     => ( upto(I,J) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),J),cons_int(I,upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,I),one_one_int),J)),nil_int) ) ) ).

% upto.psimps
tff(fact_9884_upto_Opelims,axiom,
    ! [Xb: int,Xaa: int,Y: list_int] :
      ( ( upto(Xb,Xaa) = Y )
     => ( aa_Pro4592068669816069595_int_o(accp_P1096762738010456898nt_int(upto_rel),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Xb),Xaa))
       => ~ ( ( Y = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Xaa),cons_int(Xb,upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb),one_one_int),Xaa)),nil_int) )
           => ~ aa_Pro4592068669816069595_int_o(accp_P1096762738010456898nt_int(upto_rel),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Xb),Xaa)) ) ) ) ).

% upto.pelims
tff(fact_9885_int__lsb__numeral_I2_J,axiom,
    aa_int_o(least_4859182151741483524sb_int,one_one_int) ).

% int_lsb_numeral(2)
tff(fact_9886_int__lsb__numeral_I6_J,axiom,
    ! [W: num] : ~ aa_int_o(least_4859182151741483524sb_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,W))) ).

% int_lsb_numeral(6)
tff(fact_9887_int__lsb__numeral_I3_J,axiom,
    aa_int_o(least_4859182151741483524sb_int,aa_num_int(numeral_numeral_int,one)) ).

% int_lsb_numeral(3)
tff(fact_9888_int__lsb__numeral_I7_J,axiom,
    ! [W: num] : aa_int_o(least_4859182151741483524sb_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,W))) ).

% int_lsb_numeral(7)
tff(fact_9889_int__lsb__numeral_I4_J,axiom,
    aa_int_o(least_4859182151741483524sb_int,aa_int_int(uminus_uminus_int,one_one_int)) ).

% int_lsb_numeral(4)
tff(fact_9890_int__lsb__numeral_I8_J,axiom,
    ! [W: num] : ~ aa_int_o(least_4859182151741483524sb_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,W)))) ).

% int_lsb_numeral(8)
tff(fact_9891_int__lsb__numeral_I5_J,axiom,
    aa_int_o(least_4859182151741483524sb_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,one))) ).

% int_lsb_numeral(5)
tff(fact_9892_int__lsb__numeral_I9_J,axiom,
    ! [W: num] : aa_int_o(least_4859182151741483524sb_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,W)))) ).

% int_lsb_numeral(9)
tff(fact_9893_nth__upto,axiom,
    ! [I: int,K: nat,J: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),aa_nat_int(semiri1314217659103216013at_int,K))),J)
     => ( aa_nat_int(nth_int(upto(I,J)),K) = aa_int_int(aa_int_fun_int_int(plus_plus_int,I),aa_nat_int(semiri1314217659103216013at_int,K)) ) ) ).

% nth_upto
tff(fact_9894_length__upto,axiom,
    ! [I: int,J: int] : size_size_list_int(upto(I,J)) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,J),I)),one_one_int)) ).

% length_upto
tff(fact_9895_upto__rec__numeral_I1_J,axiom,
    ! [M: num,Nb: num] :
      upto(aa_num_int(numeral_numeral_int,M),aa_num_int(numeral_numeral_int,Nb)) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,Nb)),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,Nb))),nil_int) ).

% upto_rec_numeral(1)
tff(fact_9896_upto__rec__numeral_I4_J,axiom,
    ! [M: num,Nb: 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,Nb))) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))),cons_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,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,Nb)))),nil_int) ).

% upto_rec_numeral(4)
tff(fact_9897_upto__rec__numeral_I3_J,axiom,
    ! [M: num,Nb: num] :
      upto(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,Nb)) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_num_int(numeral_numeral_int,Nb)),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,Nb))),nil_int) ).

% upto_rec_numeral(3)
tff(fact_9898_upto__rec__numeral_I2_J,axiom,
    ! [M: num,Nb: num] :
      upto(aa_num_int(numeral_numeral_int,M),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))),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,Nb)))),nil_int) ).

% upto_rec_numeral(2)
tff(fact_9899_atLeastAtMost__upto,axiom,
    ! [I: int,J: int] : set_or1266510415728281911st_int(I,J) = set_int2(upto(I,J)) ).

% atLeastAtMost_upto
tff(fact_9900_upto__split2,axiom,
    ! [I: int,J: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),J)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,J),K)
       => ( upto(I,K) = append_int(upto(I,J),upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,J),one_one_int),K)) ) ) ) ).

% upto_split2
tff(fact_9901_upto__split1,axiom,
    ! [I: int,J: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),J)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,J),K)
       => ( upto(I,K) = append_int(upto(I,aa_int_int(aa_int_fun_int_int(minus_minus_int,J),one_one_int)),upto(J,K)) ) ) ) ).

% upto_split1
tff(fact_9902_atLeastLessThan__upto,axiom,
    ! [I: int,J: int] : set_or4662586982721622107an_int(I,J) = set_int2(upto(I,aa_int_int(aa_int_fun_int_int(minus_minus_int,J),one_one_int))) ).

% atLeastLessThan_upto
tff(fact_9903_upto_Oelims,axiom,
    ! [Xb: int,Xaa: int,Y: list_int] :
      ( ( upto(Xb,Xaa) = Y )
     => ( Y = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Xaa),cons_int(Xb,upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb),one_one_int),Xaa)),nil_int) ) ) ).

% upto.elims
tff(fact_9904_upto_Osimps,axiom,
    ! [I: int,J: int] :
      upto(I,J) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),J),cons_int(I,upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,I),one_one_int),J)),nil_int) ).

% upto.simps
tff(fact_9905_upto__rec1,axiom,
    ! [I: int,J: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),J)
     => ( upto(I,J) = cons_int(I,upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,I),one_one_int),J)) ) ) ).

% upto_rec1
tff(fact_9906_upto__rec2,axiom,
    ! [I: int,J: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),J)
     => ( upto(I,J) = append_int(upto(I,aa_int_int(aa_int_fun_int_int(minus_minus_int,J),one_one_int)),cons_int(J,nil_int)) ) ) ).

% upto_rec2
tff(fact_9907_bin__last__conv__lsb,axiom,
    aTP_Lamm_bbp = least_4859182151741483524sb_int ).

% bin_last_conv_lsb
tff(fact_9908_upto__split3,axiom,
    ! [I: int,J: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),J)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,J),K)
       => ( upto(I,K) = append_int(upto(I,aa_int_int(aa_int_fun_int_int(minus_minus_int,J),one_one_int)),cons_int(J,upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,J),one_one_int),K))) ) ) ) ).

% upto_split3
tff(fact_9909_concat__bit__Suc,axiom,
    ! [Nb: nat,K: int,L: int] : aa_int_int(aa_int_fun_int_int(bit_concat_bit(aa_nat_nat(suc,Nb)),K),L) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_concat_bit(Nb),aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),L))) ).

% concat_bit_Suc
tff(fact_9910_concat__bit__nonnegative__iff,axiom,
    ! [Nb: 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(Nb),K),L))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),L) ) ).

% concat_bit_nonnegative_iff
tff(fact_9911_concat__bit__assoc,axiom,
    ! [Nb: nat,K: int,M: nat,L: int,R2: int] : aa_int_int(aa_int_fun_int_int(bit_concat_bit(Nb),K),aa_int_int(aa_int_fun_int_int(bit_concat_bit(M),L),R2)) = aa_int_int(aa_int_fun_int_int(bit_concat_bit(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Nb)),aa_int_int(aa_int_fun_int_int(bit_concat_bit(Nb),K),L)),R2) ).

% concat_bit_assoc
tff(fact_9912_concat__bit__eq,axiom,
    ! [Nb: nat,K: int,L: int] : aa_int_int(aa_int_fun_int_int(bit_concat_bit(Nb),K),L) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,Nb),K)),aa_int_int(aa_nat_fun_int_int(bit_se545348938243370406it_int,Nb),L)) ).

% concat_bit_eq
tff(fact_9913_bit__concat__bit__iff,axiom,
    ! [M: nat,K: int,L: int,Nb: nat] :
      ( aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,aa_int_int(aa_int_fun_int_int(bit_concat_bit(M),K),L)),Nb)
    <=> ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),M)
          & aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,K),Nb) )
        | ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
          & aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,L),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),M)) ) ) ) ).

% bit_concat_bit_iff
tff(fact_9914_assn__aci_I11_J,axiom,
    ! [Xb: assn,Y: assn,A: assn,B: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(syntax7398250324933576852n_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Xb),Y)),A)
     => ( aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A),B) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,B),A) ) ) ).

% assn_aci(11)
tff(fact_9915_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_9916_upt__0__eq__Nil__conv,axiom,
    ! [J: nat] :
      ( ( upt(zero_zero_nat,J) = nil_nat )
    <=> ( J = zero_zero_nat ) ) ).

% upt_0_eq_Nil_conv
tff(fact_9917_upt__conv__Nil,axiom,
    ! [J: nat,I: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,J),I)
     => ( upt(I,J) = nil_nat ) ) ).

% upt_conv_Nil
tff(fact_9918_upt__merge,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,J),K) )
     => ( append_nat(upt(I,J),upt(J,K)) = upt(I,K) ) ) ).

% upt_merge
tff(fact_9919_int__sdiv__simps_I1_J,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(signed6714573509424544716de_int,A),one_one_int) = A ).

% int_sdiv_simps(1)
tff(fact_9920_upt__eq__Nil__conv,axiom,
    ! [I: nat,J: nat] :
      ( ( upt(I,J) = nil_nat )
    <=> ( ( J = zero_zero_nat )
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,J),I) ) ) ).

% upt_eq_Nil_conv
tff(fact_9921_int__sdiv__same__is__1,axiom,
    ! [A: int,B: int] :
      ( ( A != zero_zero_int )
     => ( ( aa_int_int(aa_int_fun_int_int(signed6714573509424544716de_int,A),B) = A )
      <=> ( B = one_one_int ) ) ) ).

% int_sdiv_same_is_1
tff(fact_9922_int__sdiv__simps_I3_J,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(signed6714573509424544716de_int,A),aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(uminus_uminus_int,A) ).

% int_sdiv_simps(3)
tff(fact_9923_sdiv__int__numeral__numeral,axiom,
    ! [M: num,Nb: num] : aa_int_int(aa_int_fun_int_int(signed6714573509424544716de_int,aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,Nb)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,Nb)) ).

% sdiv_int_numeral_numeral
tff(fact_9924_length__upt,axiom,
    ! [I: nat,J: nat] : size_size_list_nat(upt(I,J)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),I) ).

% length_upt
tff(fact_9925_upt__rec__numeral,axiom,
    ! [M: num,Nb: num] :
      upt(aa_num_nat(numeral_numeral_nat,M),aa_num_nat(numeral_numeral_nat,Nb)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,Nb)),cons_nat(aa_num_nat(numeral_numeral_nat,M),upt(aa_nat_nat(suc,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,Nb))),nil_nat) ).

% upt_rec_numeral
tff(fact_9926_int__sdiv__negated__is__minus1,axiom,
    ! [A: int,B: int] :
      ( ( A != zero_zero_int )
     => ( ( aa_int_int(aa_int_fun_int_int(signed6714573509424544716de_int,A),B) = aa_int_int(uminus_uminus_int,A) )
      <=> ( B = aa_int_int(uminus_uminus_int,one_one_int) ) ) ) ).

% int_sdiv_negated_is_minus1
tff(fact_9927_nth__upt,axiom,
    ! [I: nat,K: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),J)
     => ( aa_nat_nat(nth_nat(upt(I,J)),K) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K) ) ) ).

% nth_upt
tff(fact_9928_upt__eq__lel__conv,axiom,
    ! [L: nat,H: nat,Is1: list_nat,I: nat,Is2: list_nat] :
      ( ( upt(L,H) = append_nat(Is1,cons_nat(I,Is2)) )
    <=> ( ( Is1 = upt(L,I) )
        & ( Is2 = upt(aa_nat_nat(suc,I),H) )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,L),I)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),H) ) ) ).

% upt_eq_lel_conv
tff(fact_9929_upt__rec,axiom,
    ! [I: nat,J: nat] :
      upt(I,J) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J),cons_nat(I,upt(aa_nat_nat(suc,I),J)),nil_nat) ).

% upt_rec
tff(fact_9930_map__add__upt,axiom,
    ! [Nb: nat,M: nat] : map_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_bbq,Nb),upt(zero_zero_nat,M)) = upt(Nb,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Nb)) ).

% map_add_upt
tff(fact_9931_map__add__upt_H,axiom,
    ! [Ofs: nat,A: nat,B: nat] : map_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_bbq,Ofs),upt(A,B)) = upt(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),Ofs),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),Ofs)) ).

% map_add_upt'
tff(fact_9932_map__Suc__upt,axiom,
    ! [M: nat,Nb: nat] : map_nat_nat(suc,upt(M,Nb)) = upt(aa_nat_nat(suc,M),aa_nat_nat(suc,Nb)) ).

% map_Suc_upt
tff(fact_9933_upt__conv__Cons__Cons,axiom,
    ! [M: nat,Nb: nat,Ns: list_nat,Q2: nat] :
      ( ( cons_nat(M,cons_nat(Nb,Ns)) = upt(M,Q2) )
    <=> ( cons_nat(Nb,Ns) = upt(aa_nat_nat(suc,M),Q2) ) ) ).

% upt_conv_Cons_Cons
tff(fact_9934_upt__conv__Cons,axiom,
    ! [I: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
     => ( upt(I,J) = cons_nat(I,upt(aa_nat_nat(suc,I),J)) ) ) ).

% upt_conv_Cons
tff(fact_9935_upt__add__eq__append,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => ( upt(I,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K)) = append_nat(upt(I,J),upt(J,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K))) ) ) ).

% upt_add_eq_append
tff(fact_9936_upt__eq__append__conv,axiom,
    ! [I: nat,J: nat,Xs: list_nat,Ys: list_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => ( ( upt(I,J) = append_nat(Xs,Ys) )
      <=> ? [K4: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),K4)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K4),J)
            & ( upt(I,K4) = Xs )
            & ( upt(K4,J) = Ys ) ) ) ) ).

% upt_eq_append_conv
tff(fact_9937_atLeastLessThan__upt,axiom,
    ! [I: nat,J: nat] : set_or4665077453230672383an_nat(I,J) = set_nat2(upt(I,J)) ).

% atLeastLessThan_upt
tff(fact_9938_upt__append,axiom,
    ! [I: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
     => ( append_nat(upt(zero_zero_nat,I),upt(I,J)) = upt(zero_zero_nat,J) ) ) ).

% upt_append
tff(fact_9939_upt__Suc,axiom,
    ! [I: nat,J: nat] :
      upt(I,aa_nat_nat(suc,J)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J),append_nat(upt(I,J),cons_nat(J,nil_nat)),nil_nat) ).

% upt_Suc
tff(fact_9940_upt__Suc__append,axiom,
    ! [I: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => ( upt(I,aa_nat_nat(suc,J)) = append_nat(upt(I,J),cons_nat(J,nil_nat)) ) ) ).

% upt_Suc_append
tff(fact_9941_upt__eq__Cons__conv,axiom,
    ! [I: nat,J: nat,Xb: nat,Xs: list_nat] :
      ( ( upt(I,J) = cons_nat(Xb,Xs) )
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
        & ( I = Xb )
        & ( upt(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),one_one_nat),J) = Xs ) ) ) ).

% upt_eq_Cons_conv
tff(fact_9942_map__decr__upt,axiom,
    ! [M: nat,Nb: nat] : map_nat_nat(aTP_Lamm_bbr,upt(aa_nat_nat(suc,M),aa_nat_nat(suc,Nb))) = upt(M,Nb) ).

% map_decr_upt
tff(fact_9943_atLeastAtMost__upt,axiom,
    ! [Nb: nat,M: nat] : set_or1269000886237332187st_nat(Nb,M) = set_nat2(upt(Nb,aa_nat_nat(suc,M))) ).

% atLeastAtMost_upt
tff(fact_9944_atLeast__upt,axiom,
    ! [Nb: nat] : set_ord_lessThan_nat(Nb) = set_nat2(upt(zero_zero_nat,Nb)) ).

% atLeast_upt
tff(fact_9945_sgn__sdiv__eq__sgn__mult,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(aa_int_fun_int_int(signed6714573509424544716de_int,A),B) != zero_zero_int )
     => ( aa_int_int(sgn_sgn_int,aa_int_int(aa_int_fun_int_int(signed6714573509424544716de_int,A),B)) = aa_int_int(sgn_sgn_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)) ) ) ).

% sgn_sdiv_eq_sgn_mult
tff(fact_9946_atMost__upto,axiom,
    ! [Nb: nat] : set_ord_atMost_nat(Nb) = set_nat2(upt(zero_zero_nat,aa_nat_nat(suc,Nb))) ).

% atMost_upto
tff(fact_9947_signed__divide__int__def,axiom,
    ! [K: int,L: int] : aa_int_int(aa_int_fun_int_int(signed6714573509424544716de_int,K),L) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,K)),aa_int_int(sgn_sgn_int,L))),aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(abs_abs_int,K)),aa_int_int(abs_abs_int,L))) ).

% signed_divide_int_def
tff(fact_9948_Cauchy__iff2,axiom,
    ! [X5: fun_nat_real] :
      ( topolo4055970368930404560y_real(X5)
    <=> ! [J3: nat] :
        ? [M9: nat] :
        ! [M7: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M9),M7)
         => ! [N6: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M9),N6)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(X5,M7)),aa_nat_real(X5,N6)))),aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,J3)))) ) ) ) ).

% Cauchy_iff2
tff(fact_9949_sorted__list__of__set__atMost__Suc,axiom,
    ! [K: nat] : linord2614967742042102400et_nat(set_ord_atMost_nat(aa_nat_nat(suc,K))) = append_nat(linord2614967742042102400et_nat(set_ord_atMost_nat(K)),cons_nat(aa_nat_nat(suc,K),nil_nat)) ).

% sorted_list_of_set_atMost_Suc
tff(fact_9950_sorted__list__of__set__lessThan__Suc,axiom,
    ! [K: nat] : linord2614967742042102400et_nat(set_ord_lessThan_nat(aa_nat_nat(suc,K))) = append_nat(linord2614967742042102400et_nat(set_ord_lessThan_nat(K)),cons_nat(K,nil_nat)) ).

% sorted_list_of_set_lessThan_Suc
tff(fact_9951_fails__assert_H,axiom,
    ! [P: $o,H: heap_e7401611519738050253t_unit] :
      ( aa_hea7994211679605476167unit_o(time_f8834461667527620124t_unit(aa_o_H2459438917461573107t_unit(refine_Imp_assert,(P))),H)
    <=> ~ (P) ) ).

% fails_assert'
tff(fact_9952_upt__filter__extend,axiom,
    ! [U: nat,U3: nat,P: fun_nat_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,U),U3)
     => ( ! [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),U3) )
           => ~ aa_nat_o(P,I3) )
       => ( filter_nat(P,upt(zero_zero_nat,U)) = filter_nat(P,upt(zero_zero_nat,U3)) ) ) ) ).

% upt_filter_extend
tff(fact_9953_sort__upt,axiom,
    ! [M: nat,Nb: nat] : linord738340561235409698at_nat(aTP_Lamm_di,upt(M,Nb)) = upt(M,Nb) ).

% sort_upt
tff(fact_9954_sort__upto,axiom,
    ! [I: int,J: int] : linord1735203802627413978nt_int(aTP_Lamm_dj,upto(I,J)) = upto(I,J) ).

% sort_upto
tff(fact_9955_tl__upt,axiom,
    ! [M: nat,Nb: nat] : tl_nat(upt(M,Nb)) = upt(aa_nat_nat(suc,M),Nb) ).

% tl_upt
tff(fact_9956_hd__upt,axiom,
    ! [I: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
     => ( hd_nat(upt(I,J)) = I ) ) ).

% hd_upt
tff(fact_9957_VEBT_Osize_I3_J,axiom,
    ! [X11a: option4927543243414619207at_nat,X12a: nat,X13a: list_VEBT_VEBT,X14a: vEBT_VEBT] : aa_VEBT_VEBT_nat(size_size_VEBT_VEBT,aa_VEB5832953974749077948T_VEBT(vEBT_Node(X11a,X12a,X13a),X14a)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,size_list_VEBT_VEBT(size_size_VEBT_VEBT,X13a)),aa_VEBT_VEBT_nat(size_size_VEBT_VEBT,X14a))),aa_nat_nat(suc,zero_zero_nat)) ).

% VEBT.size(3)
tff(fact_9958_VEBT_Osize__gen_I1_J,axiom,
    ! [X11a: option4927543243414619207at_nat,X12a: nat,X13a: list_VEBT_VEBT,X14a: vEBT_VEBT] : aa_VEBT_VEBT_nat(vEBT_size_VEBT,aa_VEB5832953974749077948T_VEBT(vEBT_Node(X11a,X12a,X13a),X14a)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,size_list_VEBT_VEBT(vEBT_size_VEBT,X13a)),aa_VEBT_VEBT_nat(vEBT_size_VEBT,X14a))),aa_nat_nat(suc,zero_zero_nat)) ).

% VEBT.size_gen(1)
tff(fact_9959_entails__solve__init_I1_J,axiom,
    ! [P: assn,Q: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(fI_QUERY(P),Q),top_top_assn)
     => aa_assn_o(aa_assn_fun_assn_o(entails,P),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Q),top_top_assn)) ) ).

% entails_solve_init(1)
tff(fact_9960_FI__QUERY__def,axiom,
    ! [P: assn,Q: assn,F4: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(fI_QUERY(P),Q),F4)
    <=> aa_assn_o(aa_assn_fun_assn_o(entails,P),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Q),F4)) ) ).

% FI_QUERY_def
tff(fact_9961_frame__inference__init,axiom,
    ! [P: assn,Q: assn,F4: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(fI_QUERY(P),Q),F4)
     => aa_assn_o(aa_assn_fun_assn_o(entails,P),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Q),F4)) ) ).

% frame_inference_init
tff(fact_9962_entails__solve__init_I2_J,axiom,
    ! [P: assn,Q: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(fI_QUERY(P),Q),one_one_assn)
     => aa_assn_o(aa_assn_fun_assn_o(entails,P),Q) ) ).

% entails_solve_init(2)
tff(fact_9963_smod__int__range,axiom,
    ! [B: int,A: int] :
      ( ( B != zero_zero_int )
     => aa_set_int_o(member_int(aa_int_int(aa_int_fun_int_int(signed6292675348222524329lo_int,A),B)),set_or1266510415728281911st_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,aa_int_int(abs_abs_int,B))),one_one_int),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(abs_abs_int,B)),one_one_int))) ) ).

% smod_int_range
tff(fact_9964_smod__int__numeral__numeral,axiom,
    ! [M: num,Nb: num] : aa_int_int(aa_int_fun_int_int(signed6292675348222524329lo_int,aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,Nb)) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,Nb)) ).

% smod_int_numeral_numeral
tff(fact_9965_uint32_Osize__eq__length,axiom,
    aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,one)))))) = type_l796852477590012082l_num1(type_N8448461349408098053l_num1) ).

% uint32.size_eq_length
tff(fact_9966_smod__int__compares_I1_J,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(signed6292675348222524329lo_int,A),B)),B) ) ) ).

% smod_int_compares(1)
tff(fact_9967_smod__int__compares_I2_J,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(signed6292675348222524329lo_int,A),B)) ) ) ).

% smod_int_compares(2)
tff(fact_9968_smod__int__compares_I4_J,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(signed6292675348222524329lo_int,A),B)),zero_zero_int) ) ) ).

% smod_int_compares(4)
tff(fact_9969_smod__int__compares_I6_J,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(signed6292675348222524329lo_int,A),B)) ) ) ).

% smod_int_compares(6)
tff(fact_9970_smod__int__compares_I7_J,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(signed6292675348222524329lo_int,A),B)),zero_zero_int) ) ) ).

% smod_int_compares(7)
tff(fact_9971_smod__int__compares_I8_J,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),aa_int_int(aa_int_fun_int_int(signed6292675348222524329lo_int,A),B)) ) ) ).

% smod_int_compares(8)
tff(fact_9972_smod__mod__positive,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
       => ( aa_int_int(aa_int_fun_int_int(signed6292675348222524329lo_int,A),B) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) ) ) ) ).

% smod_mod_positive
tff(fact_9973_signed__modulo__int__def,axiom,
    ! [K: int,L: int] : aa_int_int(aa_int_fun_int_int(signed6292675348222524329lo_int,K),L) = aa_int_int(aa_int_fun_int_int(minus_minus_int,K),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(signed6714573509424544716de_int,K),L)),L)) ).

% signed_modulo_int_def
tff(fact_9974_smod__int__compares_I3_J,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,B)),aa_int_int(aa_int_fun_int_int(signed6292675348222524329lo_int,A),B)) ) ) ).

% smod_int_compares(3)
tff(fact_9975_smod__int__compares_I5_J,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(signed6292675348222524329lo_int,A),B)),aa_int_int(uminus_uminus_int,B)) ) ) ).

% smod_int_compares(5)
tff(fact_9976_smod__int__alt__def,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(signed6292675348222524329lo_int,A),B) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,A)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B))) ).

% smod_int_alt_def
tff(fact_9977_len__of__finite__1__def,axiom,
    ! [Xb: itself_finite_1] : type_l31302759751748491nite_1(Xb) = one_one_nat ).

% len_of_finite_1_def
tff(fact_9978_len__num1,axiom,
    ! [Uu: itself_Numeral_num1] : type_l4264026598287037465l_num1(Uu) = one_one_nat ).

% len_num1
tff(fact_9979_len__of__finite__2__def,axiom,
    ! [Xb: itself_finite_2] : type_l31302759751748492nite_2(Xb) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) ).

% len_of_finite_2_def
tff(fact_9980_len__of__finite__3__def,axiom,
    ! [Xb: itself_finite_3] : type_l31302759751748493nite_3(Xb) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one))) ).

% len_of_finite_3_def
tff(fact_9981_min__Suc__Suc,axiom,
    ! [M: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(suc,M)),aa_nat_nat(suc,Nb)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,M),Nb)) ).

% min_Suc_Suc
tff(fact_9982_min__0R,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,Nb),zero_zero_nat) = zero_zero_nat ).

% min_0R
tff(fact_9983_min__0L,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,zero_zero_nat),Nb) = zero_zero_nat ).

% min_0L
tff(fact_9984_min__minus_H,axiom,
    ! [M: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),K)),M) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),K) ).

% min_minus'
tff(fact_9985_min__minus,axiom,
    ! [M: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),K)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),K) ).

% min_minus
tff(fact_9986_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_9987_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_9988_min__pm,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,A),B)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B)) = A ).

% min_pm
tff(fact_9989_min__pm1,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B)),aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,A),B)) = A ).

% min_pm1
tff(fact_9990_rev__min__pm,axiom,
    ! [B: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,B),A)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B)) = A ).

% rev_min_pm
tff(fact_9991_rev__min__pm1,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B)),aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,B),A)) = A ).

% rev_min_pm1
tff(fact_9992_min__Suc__numeral,axiom,
    ! [Nb: nat,K: num] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(suc,Nb)),aa_num_nat(numeral_numeral_nat,K)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,Nb),aa_num_nat(pred_numeral,K))) ).

% min_Suc_numeral
tff(fact_9993_min__numeral__Suc,axiom,
    ! [K: num,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_num_nat(numeral_numeral_nat,K)),aa_nat_nat(suc,Nb)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_num_nat(pred_numeral,K)),Nb)) ).

% min_numeral_Suc
tff(fact_9994_min__diff,axiom,
    ! [M: nat,I: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),I)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),I)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,M),Nb)),I) ).

% min_diff
tff(fact_9995_nat__mult__min__right,axiom,
    ! [M: nat,Nb: nat,Q2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,Nb),Q2)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Q2)) ).

% nat_mult_min_right
tff(fact_9996_nat__mult__min__left,axiom,
    ! [M: nat,Nb: nat,Q2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,M),Nb)),Q2) = aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Q2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),Q2)) ).

% nat_mult_min_left
tff(fact_9997_concat__bit__assoc__sym,axiom,
    ! [M: nat,Nb: nat,K: int,L: int,R2: int] : aa_int_int(aa_int_fun_int_int(bit_concat_bit(M),aa_int_int(aa_int_fun_int_int(bit_concat_bit(Nb),K),L)),R2) = aa_int_int(aa_int_fun_int_int(bit_concat_bit(aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,M),Nb)),K),aa_int_int(aa_int_fun_int_int(bit_concat_bit(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Nb)),L),R2)) ).

% concat_bit_assoc_sym
tff(fact_9998_take__bit__concat__bit__eq,axiom,
    ! [M: nat,Nb: nat,K: int,L: int] : aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,M),aa_int_int(aa_int_fun_int_int(bit_concat_bit(Nb),K),L)) = aa_int_int(aa_int_fun_int_int(bit_concat_bit(aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,M),Nb)),K),aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Nb)),L)) ).

% take_bit_concat_bit_eq
tff(fact_9999_min__Suc2,axiom,
    ! [M: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,M),aa_nat_nat(suc,Nb)) = aa_nat_nat(case_nat_nat(zero_zero_nat,aa_nat_fun_nat_nat(aTP_Lamm_bbs,Nb)),M) ).

% min_Suc2
tff(fact_10000_min__Suc1,axiom,
    ! [Nb: nat,M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(suc,Nb)),M) = aa_nat_nat(case_nat_nat(zero_zero_nat,aa_nat_fun_nat_nat(aTP_Lamm_bbt,Nb)),M) ).

% min_Suc1
tff(fact_10001_mod__mod__power,axiom,
    ! [K: nat,M: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,K),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))),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,K),aa_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(ord_min_nat,M),Nb))) ).

% mod_mod_power
tff(fact_10002_int__set__bits__K__False,axiom,
    bit_bi6516823479961619367ts_int(aTP_Lamm_cy) = zero_zero_int ).

% int_set_bits_K_False
tff(fact_10003_int__set__bits__K__True,axiom,
    bit_bi6516823479961619367ts_int(aTP_Lamm_dd) = aa_int_int(uminus_uminus_int,one_one_int) ).

% int_set_bits_K_True
tff(fact_10004_bin__last__set__bits,axiom,
    ! [F: fun_nat_o] :
      ( bit_wf_set_bits_int(F)
     => ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),bit_bi6516823479961619367ts_int(F))
      <=> aa_nat_o(F,zero_zero_nat) ) ) ).

% bin_last_set_bits
tff(fact_10005_wf__set__bits__int__Suc,axiom,
    ! [F: fun_nat_o] :
      ( bit_wf_set_bits_int(aTP_Lamm_bbu(F))
    <=> bit_wf_set_bits_int(F) ) ).

% wf_set_bits_int_Suc
tff(fact_10006_wf__set__bits__int__const,axiom,
    ! [B: $o] : bit_wf_set_bits_int(aa_o_fun_nat_o(aTP_Lamm_mk,(B))) ).

% wf_set_bits_int_const
tff(fact_10007_ones,axiom,
    ! [Nb: nat,F: fun_nat_o] :
      ( ! [N5: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N5)
         => aa_nat_o(F,N5) )
     => bit_wf_set_bits_int(F) ) ).

% ones
tff(fact_10008_wf__set__bits__int_Ocases,axiom,
    ! [F: fun_nat_o] :
      ( bit_wf_set_bits_int(F)
     => ( ! [N2: nat] :
            ~ ! [N4: nat] :
                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),N4)
               => ~ aa_nat_o(F,N4) )
       => ~ ! [N2: nat] :
              ~ ! [N4: nat] :
                  ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),N4)
                 => aa_nat_o(F,N4) ) ) ) ).

% wf_set_bits_int.cases
tff(fact_10009_wf__set__bits__int_Osimps,axiom,
    ! [F: fun_nat_o] :
      ( bit_wf_set_bits_int(F)
    <=> ( ? [N6: nat] :
          ! [N9: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N6),N9)
           => ~ aa_nat_o(F,N9) )
        | ? [N6: nat] :
          ! [N9: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N6),N9)
           => aa_nat_o(F,N9) ) ) ) ).

% wf_set_bits_int.simps
tff(fact_10010_zeros,axiom,
    ! [Nb: nat,F: fun_nat_o] :
      ( ! [N5: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N5)
         => ~ aa_nat_o(F,N5) )
     => bit_wf_set_bits_int(F) ) ).

% zeros
tff(fact_10011_wf__set__bits__int__simps,axiom,
    ! [F: fun_nat_o] :
      ( bit_wf_set_bits_int(F)
    <=> ? [N6: nat] :
          ( ! [N9: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N6),N9)
             => ~ aa_nat_o(F,N9) )
          | ! [N9: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N6),N9)
             => aa_nat_o(F,N9) ) ) ) ).

% wf_set_bits_int_simps
tff(fact_10012_int__set__bits__unfold__BIT,axiom,
    ! [F: fun_nat_o] :
      ( bit_wf_set_bits_int(F)
     => ( bit_bi6516823479961619367ts_int(F) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_o_int(zero_n2684676970156552555ol_int,aa_nat_o(F,zero_zero_nat))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),bit_bi6516823479961619367ts_int(comp_nat_o_nat(F,suc)))) ) ) ).

% int_set_bits_unfold_BIT
tff(fact_10013_bin__rest__set__bits,axiom,
    ! [F: fun_nat_o] :
      ( bit_wf_set_bits_int(F)
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,bit_bi6516823479961619367ts_int(F)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = bit_bi6516823479961619367ts_int(comp_nat_o_nat(F,suc)) ) ) ).

% bin_rest_set_bits
tff(fact_10014_card_Ocomp__fun__commute__on,axiom,
    comp_nat_nat_nat(suc,suc) = comp_nat_nat_nat(suc,suc) ).

% card.comp_fun_commute_on
tff(fact_10015_Code__Numeral_Onegative__def,axiom,
    code_negative = comp_C3531382070062128313er_num(uminus1351360451143612070nteger,numera6620942414471956472nteger) ).

% Code_Numeral.negative_def
tff(fact_10016_Code__Target__Int_Onegative__def,axiom,
    code_Target_negative = comp_int_int_num(uminus_uminus_int,numeral_numeral_int) ).

% Code_Target_Int.negative_def
tff(fact_10017_shiftl__Suc__0,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_Sh3965577149348748681tl_nat,aa_nat_nat(suc,zero_zero_nat)),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb) ).

% shiftl_Suc_0
tff(fact_10018_shiftr__Suc__0,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_Sh2154871086232339855tr_nat,aa_nat_nat(suc,zero_zero_nat)),Nb) = aa_o_nat(zero_n2687167440665602831ol_nat,Nb = zero_zero_nat) ).

% shiftr_Suc_0
tff(fact_10019_msb__numeral_I1_J,axiom,
    ! [Nb: num] : ~ aa_int_o(most_s5051101344085556sb_int,aa_num_int(numeral_numeral_int,Nb)) ).

% msb_numeral(1)
tff(fact_10020_msb__1,axiom,
    ~ aa_int_o(most_s5051101344085556sb_int,one_one_int) ).

% msb_1
tff(fact_10021_msb__numeral_I2_J,axiom,
    ! [Nb: num] : aa_int_o(most_s5051101344085556sb_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))) ).

% msb_numeral(2)
tff(fact_10022_msb__bin__rest,axiom,
    ! [Xb: int] :
      ( aa_int_o(most_s5051101344085556sb_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,Xb),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))
    <=> aa_int_o(most_s5051101344085556sb_int,Xb) ) ).

% msb_bin_rest
tff(fact_10023_inj__Suc,axiom,
    ! [N: set_nat] : aa_set_nat_o(inj_on_nat_nat(suc),N) ).

% inj_Suc
tff(fact_10024_inj__on__diff__nat,axiom,
    ! [N: set_nat,K: nat] :
      ( ! [N2: nat] :
          ( aa_set_nat_o(member_nat(N2),N)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),N2) )
     => aa_set_nat_o(inj_on_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_bbn,K)),N) ) ).

% inj_on_diff_nat
tff(fact_10025_summable__reindex,axiom,
    ! [F: fun_nat_real,G: fun_nat_nat] :
      ( summable_real(F)
     => ( aa_set_nat_o(inj_on_nat_nat(G),top_top_set_nat)
       => ( ! [X2: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(F,X2))
         => summable_real(comp_nat_real_nat(F,G)) ) ) ) ).

% summable_reindex
tff(fact_10026_suminf__reindex__mono,axiom,
    ! [F: fun_nat_real,G: fun_nat_nat] :
      ( summable_real(F)
     => ( aa_set_nat_o(inj_on_nat_nat(G),top_top_set_nat)
       => ( ! [X2: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(F,X2))
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,suminf_real(comp_nat_real_nat(F,G))),suminf_real(F)) ) ) ) ).

% suminf_reindex_mono
tff(fact_10027_uint32__msb__test__bit,axiom,
    ! [Xb: uint32] :
      ( aa_uint32_o(most_s9063628576841037300uint32,Xb)
    <=> aa_nat_o(aa_uint32_fun_nat_o(bit_se5367290876889521763uint32,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,one)))))) ) ).

% uint32_msb_test_bit
tff(fact_10028_suminf__reindex,axiom,
    ! [F: fun_nat_real,G: fun_nat_nat] :
      ( summable_real(F)
     => ( aa_set_nat_o(inj_on_nat_nat(G),top_top_set_nat)
       => ( ! [X2: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(F,X2))
         => ( ! [X2: nat] :
                ( ~ aa_set_nat_o(member_nat(X2),image_nat_nat(G,top_top_set_nat))
               => ( aa_nat_real(F,X2) = zero_zero_real ) )
           => ( suminf_real(comp_nat_real_nat(F,G)) = suminf_real(F) ) ) ) ) ) ).

% suminf_reindex
tff(fact_10029_inj__sgn__power,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => aa_set_real_o(inj_on_real_real(aTP_Lamm_baq(Nb)),top_top_set_real) ) ).

% inj_sgn_power
tff(fact_10030_msb__uint32__code,axiom,
    ! [Xb: uint32] :
      ( aa_uint32_o(most_s9063628576841037300uint32,Xb)
    <=> aa_Code_integer_o(uint32_test_bit(Xb),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,one)))))) ) ).

% msb_uint32_code
tff(fact_10031_uint32__test__bit__def,axiom,
    ! [Xb: uint32,Nb: code_integer] :
      ( aa_Code_integer_o(uint32_test_bit(Xb),Nb)
    <=> $ite(
          ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Nb),zero_z3403309356797280102nteger)
          | aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,one)))))),Nb) ),
          aa_Code_integer_o(aa_uin5206268876086501546eger_o(aa_fun5262225984791465967eger_o(undefi9030187962712549518eger_o,bit_se5367290876889521763uint32),Xb),Nb),
          aa_nat_o(aa_uint32_fun_nat_o(bit_se5367290876889521763uint32,Xb),aa_Code_integer_nat(code_nat_of_integer,Nb)) ) ) ).

% uint32_test_bit_def
tff(fact_10032_nat__of__integer__numeral,axiom,
    ! [Nb: num] : aa_Code_integer_nat(code_nat_of_integer,aa_num_Code_integer(numera6620942414471956472nteger,Nb)) = aa_num_nat(numeral_numeral_nat,Nb) ).

% nat_of_integer_numeral
tff(fact_10033_nat__of__integer__code__post_I3_J,axiom,
    ! [K: num] : aa_Code_integer_nat(code_nat_of_integer,aa_num_Code_integer(numera6620942414471956472nteger,K)) = aa_num_nat(numeral_numeral_nat,K) ).

% nat_of_integer_code_post(3)
tff(fact_10034_nat__of__integer__1,axiom,
    aa_Code_integer_nat(code_nat_of_integer,one_one_Code_integer) = one_one_nat ).

% nat_of_integer_1
tff(fact_10035_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_10036_nat__of__integer__less__iff,axiom,
    ! [Xb: code_integer,Y: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),Xb)
     => ( 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,Xb)),aa_Code_integer_nat(code_nat_of_integer,Y))
        <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Xb),Y) ) ) ) ).

% nat_of_integer_less_iff
tff(fact_10037_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,set_ord_lessThan_nat(aa_Code_integer_nat(code_nat_of_integer,U))) ) ) ).

% image_atLeastZeroLessThan_integer
tff(fact_10038_integer__set__bit__code,axiom,
    ! [Xb: code_integer,Nb: code_integer,B: $o] :
      bits_integer_set_bit(Xb,Nb,(B)) = $ite(
        aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Nb),zero_z3403309356797280102nteger),
        aa_o_Code_integer(aa_Cod4043041029439962285nteger(aa_Cod3196565106923072233nteger(undefi7543901990407023336nteger,Xb),Nb),(B)),
        $ite((B),aa_Cod1783974652533396728nteger(bit_se1080825931792720795nteger(Xb),aa_Cod1783974652533396728nteger(bit_se7788150548672797655nteger(aa_Code_integer_nat(code_nat_of_integer,Nb)),one_one_Code_integer)),aa_Cod1783974652533396728nteger(bit_se3949692690581998587nteger(Xb),aa_Cod1783974652533396728nteger(bit_ri7632146776885996613nteger,aa_Cod1783974652533396728nteger(bit_se7788150548672797655nteger(aa_Code_integer_nat(code_nat_of_integer,Nb)),one_one_Code_integer)))) ) ).

% integer_set_bit_code
tff(fact_10039_uint32__shiftr__def,axiom,
    ! [Xb: uint32,Nb: code_integer] :
      aa_Cod8401701657498852871uint32(uint32_shiftr(Xb),Nb) = $ite(
        ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Nb),zero_z3403309356797280102nteger)
        | aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,one))))))),Nb) ),
        aa_Cod8401701657498852871uint32(aa_uin6552852722274082928uint32(aa_fun8838986389337097743uint32(undefi4150998591671624934uint32,bit_se3964402333458159761uint32),Xb),Nb),
        aa_uint32_uint32(aa_nat4578243230388868707uint32(bit_se3964402333458159761uint32,aa_Code_integer_nat(code_nat_of_integer,Nb)),Xb) ) ).

% uint32_shiftr_def
tff(fact_10040_uint32__shiftl__def,axiom,
    ! [Xb: uint32,Nb: code_integer] :
      aa_Cod8401701657498852871uint32(uint32_shiftl(Xb),Nb) = $ite(
        ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Nb),zero_z3403309356797280102nteger)
        | aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,one))))))),Nb) ),
        aa_Cod8401701657498852871uint32(aa_uin6552852722274082928uint32(aa_fun8838986389337097743uint32(undefi4150998591671624934uint32,bit_se5742574853984576102uint32),Xb),Nb),
        aa_uint32_uint32(aa_nat4578243230388868707uint32(bit_se5742574853984576102uint32,aa_Code_integer_nat(code_nat_of_integer,Nb)),Xb) ) ).

% uint32_shiftl_def
tff(fact_10041_uint32__set__bit__def,axiom,
    ! [Xb: uint32,Nb: code_integer,B: $o] :
      uint32_set_bit(Xb,Nb,(B)) = $ite(
        ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Nb),zero_z3403309356797280102nteger)
        | aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,one)))))),Nb) ),
        aa_o_uint32(aa_Cod7056071456392079164uint32(aa_uin937759467983514409uint32(aa_fun8001877339868165167uint32(undefi8935238026407452494uint32,generi1993664874377053279uint32),Xb),Nb),(B)),
        aa_o_uint32(aa_nat_fun_o_uint32(aa_uin7041821913289200502uint32(generi1993664874377053279uint32,Xb),aa_Code_integer_nat(code_nat_of_integer,Nb)),(B)) ) ).

% uint32_set_bit_def
tff(fact_10042_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_bbv,aa_Cod6977894654539884864nteger(code_divmod_integer(K),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))))) ).

% nat_of_integer_code
tff(fact_10043_fst__divmod__integer,axiom,
    ! [K: code_integer,L: code_integer] : aa_Pro5245656335456409788nteger(produc8508995932063986495nteger,aa_Cod6977894654539884864nteger(code_divmod_integer(K),L)) = aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(K),L) ).

% fst_divmod_integer
tff(fact_10044_snd__divmod__integer,axiom,
    ! [K: code_integer,L: code_integer] : aa_Pro5245656335456409788nteger(produc6174133586879617921nteger,aa_Cod6977894654539884864nteger(code_divmod_integer(K),L)) = aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(K),L) ).

% snd_divmod_integer
tff(fact_10045_divmod__integer__def,axiom,
    ! [K: code_integer,L: code_integer] : aa_Cod6977894654539884864nteger(code_divmod_integer(K),L) = aa_Cod6977894654539884864nteger(produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(K),L)),aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(K),L)) ).

% divmod_integer_def
tff(fact_10046_num__of__integer__code,axiom,
    ! [K: code_integer] :
      aa_Code_integer_num(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_bbw,aa_Cod6977894654539884864nteger(code_divmod_integer(K),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))))) ).

% num_of_integer_code
tff(fact_10047_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_bbx,aa_Cod6977894654539884864nteger(code_divmod_integer(K),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))))) ) ).

% int_of_integer_code
tff(fact_10048_int__of__integer__numeral,axiom,
    ! [K: num] : aa_Code_integer_int(code_int_of_integer,aa_num_Code_integer(numera6620942414471956472nteger,K)) = aa_num_int(numeral_numeral_int,K) ).

% int_of_integer_numeral
tff(fact_10049_one__integer_Orep__eq,axiom,
    aa_Code_integer_int(code_int_of_integer,one_one_Code_integer) = one_one_int ).

% one_integer.rep_eq
tff(fact_10050_plus__integer_Orep__eq,axiom,
    ! [Xb: code_integer,Xaa: code_integer] : aa_Code_integer_int(code_int_of_integer,aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(Xb),Xaa)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_Code_integer_int(code_int_of_integer,Xb)),aa_Code_integer_int(code_int_of_integer,Xaa)) ).

% plus_integer.rep_eq
tff(fact_10051_minus__integer_Orep__eq,axiom,
    ! [Xb: code_integer,Xaa: code_integer] : aa_Code_integer_int(code_int_of_integer,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(Xb),Xaa)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_Code_integer_int(code_int_of_integer,Xb)),aa_Code_integer_int(code_int_of_integer,Xaa)) ).

% minus_integer.rep_eq
tff(fact_10052_times__integer_Orep__eq,axiom,
    ! [Xb: code_integer,Xaa: code_integer] : aa_Code_integer_int(code_int_of_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Xb),Xaa)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_Code_integer_int(code_int_of_integer,Xb)),aa_Code_integer_int(code_int_of_integer,Xaa)) ).

% times_integer.rep_eq
tff(fact_10053_divide__integer_Orep__eq,axiom,
    ! [Xb: code_integer,Xaa: code_integer] : aa_Code_integer_int(code_int_of_integer,aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(Xb),Xaa)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_Code_integer_int(code_int_of_integer,Xb)),aa_Code_integer_int(code_int_of_integer,Xaa)) ).

% divide_integer.rep_eq
tff(fact_10054_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_10055_less__eq__integer_Orep__eq,axiom,
    ! [Xb: code_integer,Xaa: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Xb),Xaa)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_Code_integer_int(code_int_of_integer,Xb)),aa_Code_integer_int(code_int_of_integer,Xaa)) ) ).

% less_eq_integer.rep_eq
tff(fact_10056_int__of__integer__pow,axiom,
    ! [Xb: code_integer,Nb: nat] : aa_Code_integer_int(code_int_of_integer,aa_nat_Code_integer(power_8256067586552552935nteger(Xb),Nb)) = aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_Code_integer_int(code_int_of_integer,Xb)),Nb) ).

% int_of_integer_pow
tff(fact_10057_dup_Orep__eq,axiom,
    ! [Xb: code_integer] : aa_Code_integer_int(code_int_of_integer,aa_Cod1783974652533396728nteger(code_dup,Xb)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_Code_integer_int(code_int_of_integer,Xb)),aa_Code_integer_int(code_int_of_integer,Xb)) ).

% dup.rep_eq
tff(fact_10058_int__of__integer__less__iff,axiom,
    ! [Xb: code_integer,Y: code_integer] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_Code_integer_int(code_int_of_integer,Xb)),aa_Code_integer_int(code_int_of_integer,Y))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Xb),Y) ) ).

% int_of_integer_less_iff
tff(fact_10059_bin__last__integer_Orep__eq,axiom,
    ! [Xb: code_integer] :
      ( aa_Code_integer_o(bits_b8758750999018896077nteger,Xb)
    <=> ~ 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(code_int_of_integer,Xb)) ) ).

% bin_last_integer.rep_eq
tff(fact_10060_bin__rest__integer_Orep__eq,axiom,
    ! [Xb: code_integer] : aa_Code_integer_int(code_int_of_integer,aa_Cod1783974652533396728nteger(bits_b2549910563261871055nteger,Xb)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_Code_integer_int(code_int_of_integer,Xb)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% bin_rest_integer.rep_eq
tff(fact_10061_Bit__integer_Orep__eq,axiom,
    ! [Xb: code_integer,Xaa: $o] : aa_Code_integer_int(code_int_of_integer,bits_Bit_integer(Xb,(Xaa))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_o_int(zero_n2684676970156552555ol_int,(Xaa))),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,Xb))) ).

% Bit_integer.rep_eq
tff(fact_10062_uint32__shiftr__code,axiom,
    ! [W: uint32,Nb: code_integer] :
      rep_uint322(aa_Cod8401701657498852871uint32(uint32_shiftr(W),Nb)) = $ite(
        ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Nb),zero_z3403309356797280102nteger)
        | aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,one))))))),Nb) ),
        rep_uint322(aa_Cod8401701657498852871uint32(aa_uin6552852722274082928uint32(aa_fun8838986389337097743uint32(undefi4150998591671624934uint32,bit_se3964402333458159761uint32),W),Nb)),
        bit_se5176125413884933531l_num1(aa_Code_integer_nat(code_nat_of_integer,Nb),rep_uint322(W)) ) ).

% uint32_shiftr_code
tff(fact_10063_uint32__shiftl__code,axiom,
    ! [W: uint32,Nb: code_integer] :
      rep_uint322(aa_Cod8401701657498852871uint32(uint32_shiftl(W),Nb)) = $ite(
        ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Nb),zero_z3403309356797280102nteger)
        | aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,one))))))),Nb) ),
        rep_uint322(aa_Cod8401701657498852871uint32(aa_uin6552852722274082928uint32(aa_fun8838986389337097743uint32(undefi4150998591671624934uint32,bit_se5742574853984576102uint32),W),Nb)),
        bit_se837345729053750000l_num1(aa_Code_integer_nat(code_nat_of_integer,Nb),rep_uint322(W)) ) ).

% uint32_shiftl_code
tff(fact_10064_uint32_Oword__of__numeral,axiom,
    ! [Nb: num] : rep_uint322(numera9087168376688890119uint32(Nb)) = numera7442385471795722001l_num1(Nb) ).

% uint32.word_of_numeral
tff(fact_10065_size__uint32_Orep__eq,axiom,
    ! [Xb: uint32] : size_size_uint32(Xb) = size_s8261804613246490634l_num1(rep_uint322(Xb)) ).

% size_uint32.rep_eq
tff(fact_10066_uint32_Osize__eq__word__of,axiom,
    ! [P2: uint32] : size_size_uint32(P2) = size_s8261804613246490634l_num1(rep_uint322(P2)) ).

% uint32.size_eq_word_of
tff(fact_10067_uint32_Oeven__iff__word__of,axiom,
    ! [P2: uint32] :
      ( aa_uint32_o(aa_uin5924812838015166427nt32_o(dvd_dvd_uint32,numera9087168376688890119uint32(aa_num_num(bit0,one))),P2)
    <=> aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(numera7442385471795722001l_num1(aa_num_num(bit0,one))),rep_uint322(P2)) ) ).

% uint32.even_iff_word_of
tff(fact_10068_times__uint32_Orep__eq,axiom,
    ! [Xb: uint32,Xaa: uint32] : rep_uint322(aa_uint32_uint32(times_times_uint32(Xb),Xaa)) = aa_wor7274019993978516266l_num1(times_7065122842183080059l_num1(rep_uint322(Xb)),rep_uint322(Xaa)) ).

% times_uint32.rep_eq
tff(fact_10069_uint32_Oword__of__mult,axiom,
    ! [P2: uint32,Q2: uint32] : rep_uint322(aa_uint32_uint32(times_times_uint32(P2),Q2)) = aa_wor7274019993978516266l_num1(times_7065122842183080059l_num1(rep_uint322(P2)),rep_uint322(Q2)) ).

% uint32.word_of_mult
tff(fact_10070_divide__uint32_Orep__eq,axiom,
    ! [Xb: uint32,Xaa: uint32] : rep_uint322(aa_uint32_uint32(aa_uin3778660748068847871uint32(divide_divide_uint32,Xb),Xaa)) = divide1791077408188789448l_num1(rep_uint322(Xb),rep_uint322(Xaa)) ).

% divide_uint32.rep_eq
tff(fact_10071_uint32_Oword__of__div,axiom,
    ! [P2: uint32,Q2: uint32] : rep_uint322(aa_uint32_uint32(aa_uin3778660748068847871uint32(divide_divide_uint32,P2),Q2)) = divide1791077408188789448l_num1(rep_uint322(P2),rep_uint322(Q2)) ).

% uint32.word_of_div
tff(fact_10072_one__uint32_Orep__eq,axiom,
    rep_uint322(one_one_uint32) = one_on7727431528512463931l_num1 ).

% one_uint32.rep_eq
tff(fact_10073_uint32_Oword__of__diff,axiom,
    ! [P2: uint32,Q2: uint32] : rep_uint322(minus_minus_uint32(P2,Q2)) = minus_4019991460397169231l_num1(rep_uint322(P2),rep_uint322(Q2)) ).

% uint32.word_of_diff
tff(fact_10074_minus__uint32_Orep__eq,axiom,
    ! [Xb: uint32,Xaa: uint32] : rep_uint322(minus_minus_uint32(Xb,Xaa)) = minus_4019991460397169231l_num1(rep_uint322(Xb),rep_uint322(Xaa)) ).

% minus_uint32.rep_eq
tff(fact_10075_plus__uint32_Orep__eq,axiom,
    ! [Xb: uint32,Xaa: uint32] : rep_uint322(aa_uint32_uint32(plus_plus_uint32(Xb),Xaa)) = plus_p361126936061061375l_num1(rep_uint322(Xb),rep_uint322(Xaa)) ).

% plus_uint32.rep_eq
tff(fact_10076_uint32_Oword__of__add,axiom,
    ! [P2: uint32,Q2: uint32] : rep_uint322(aa_uint32_uint32(plus_plus_uint32(P2),Q2)) = plus_p361126936061061375l_num1(rep_uint322(P2),rep_uint322(Q2)) ).

% uint32.word_of_add
tff(fact_10077_uint32_Oword__of__power,axiom,
    ! [P2: uint32,Nb: nat] : rep_uint322(aa_nat_uint32(power_power_uint32(P2),Nb)) = aa_nat5599461682457136004l_num1(power_2184487114949457152l_num1(rep_uint322(P2)),Nb) ).

% uint32.word_of_power
tff(fact_10078_less__eq__uint32_Orep__eq,axiom,
    ! [Xb: uint32,Xaa: uint32] :
      ( aa_uint32_o(aa_uin5924812838015166427nt32_o(ord_less_eq_uint32,Xb),Xaa)
    <=> aa_wor4452366487969921764num1_o(ord_le3335648743751981014l_num1(rep_uint322(Xb)),rep_uint322(Xaa)) ) ).

% less_eq_uint32.rep_eq
tff(fact_10079_uint32_Oless__eq__iff__word__of,axiom,
    ! [P2: uint32,Q2: uint32] :
      ( aa_uint32_o(aa_uin5924812838015166427nt32_o(ord_less_eq_uint32,P2),Q2)
    <=> aa_wor4452366487969921764num1_o(ord_le3335648743751981014l_num1(rep_uint322(P2)),rep_uint322(Q2)) ) ).

% uint32.less_eq_iff_word_of
tff(fact_10080_uint32__test__bit__code,axiom,
    ! [W: uint32,Nb: code_integer] :
      ( aa_Code_integer_o(uint32_test_bit(W),Nb)
    <=> $ite(
          ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Nb),zero_z3403309356797280102nteger)
          | aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,one)))))),Nb) ),
          aa_Code_integer_o(aa_uin5206268876086501546eger_o(aa_fun5262225984791465967eger_o(undefi9030187962712549518eger_o,bit_se5367290876889521763uint32),W),Nb),
          aa_nat_o(bit_se6859397288646540909l_num1(rep_uint322(W)),aa_Code_integer_nat(code_nat_of_integer,Nb)) ) ) ).

% uint32_test_bit_code
tff(fact_10081_uint32__set__bit__code,axiom,
    ! [W: uint32,Nb: code_integer,B: $o] :
      rep_uint322(uint32_set_bit(W,Nb,(B))) = $ite(
        ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Nb),zero_z3403309356797280102nteger)
        | aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,one)))))),Nb) ),
        rep_uint322(aa_o_uint32(aa_Cod7056071456392079164uint32(aa_uin937759467983514409uint32(aa_fun8001877339868165167uint32(undefi8935238026407452494uint32,generi1993664874377053279uint32),W),Nb),(B))),
        generi5268133209446125161l_num1(rep_uint322(W),aa_Code_integer_nat(code_nat_of_integer,Nb),(B)) ) ).

% uint32_set_bit_code
tff(fact_10082_set__bit__uint32__code,axiom,
    ! [Xb: uint32,Nb: nat,B: $o] :
      aa_o_uint32(aa_nat_fun_o_uint32(aa_uin7041821913289200502uint32(generi1993664874377053279uint32,Xb),Nb),(B)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,one))))))),uint32_set_bit(Xb,aa_nat_Code_integer(code_integer_of_nat,Nb),(B)),Xb) ).

% set_bit_uint32_code
tff(fact_10083_integer__of__nat__1,axiom,
    aa_nat_Code_integer(code_integer_of_nat,one_one_nat) = one_one_Code_integer ).

% integer_of_nat_1
tff(fact_10084_one__uint32_Orsp,axiom,
    one_on7727431528512463931l_num1 = one_on7727431528512463931l_num1 ).

% one_uint32.rsp
tff(fact_10085_integer__of__nat__numeral,axiom,
    ! [Nb: num] : aa_nat_Code_integer(code_integer_of_nat,aa_num_nat(numeral_numeral_nat,Nb)) = aa_num_Code_integer(numera6620942414471956472nteger,Nb) ).

% integer_of_nat_numeral
tff(fact_10086_test__bit__uint32__code,axiom,
    ! [Xb: uint32,Nb: nat] :
      ( aa_nat_o(aa_uint32_fun_nat_o(bit_se5367290876889521763uint32,Xb),Nb)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,one)))))))
        & aa_Code_integer_o(uint32_test_bit(Xb),aa_nat_Code_integer(code_integer_of_nat,Nb)) ) ) ).

% test_bit_uint32_code
tff(fact_10087_shiftl__uint32__code,axiom,
    ! [Nb: nat,Xb: uint32] :
      aa_uint32_uint32(aa_nat4578243230388868707uint32(bit_se5742574853984576102uint32,Nb),Xb) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,one))))))),aa_Cod8401701657498852871uint32(uint32_shiftl(Xb),aa_nat_Code_integer(code_integer_of_nat,Nb)),zero_zero_uint32) ).

% shiftl_uint32_code
tff(fact_10088_shiftr__uint32__code,axiom,
    ! [Nb: nat,Xb: uint32] :
      aa_uint32_uint32(aa_nat4578243230388868707uint32(bit_se3964402333458159761uint32,Nb),Xb) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,one))))))),aa_Cod8401701657498852871uint32(uint32_shiftr(Xb),aa_nat_Code_integer(code_integer_of_nat,Nb)),zero_zero_uint32) ).

% shiftr_uint32_code
tff(fact_10089_uint32__divmod__code,axiom,
    ! [Xb: uint32,Y: uint32] :
      uint32_divmod(Xb,Y) = $ite(
        aa_uint32_o(aa_uin5924812838015166427nt32_o(ord_less_eq_uint32,numera9087168376688890119uint32(aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,one))))))))))))))))))))))))))))))))),Y),
        $ite(aa_uint32_o(aa_uin5924812838015166427nt32_o(ord_less_uint32,Xb),Y),aa_uin6954824371948101007uint32(produc1400373151660368625uint32(zero_zero_uint32),Xb),aa_uin6954824371948101007uint32(produc1400373151660368625uint32(one_one_uint32),minus_minus_uint32(Xb,Y))),
        $ite(
          Y = zero_zero_uint32,
          aa_uin6954824371948101007uint32(produc1400373151660368625uint32(div0_uint32(Xb)),mod0_uint32(Xb)),
          $let(
            q: uint32,
            q:= aa_uint32_uint32(aa_nat4578243230388868707uint32(bit_se5742574853984576102uint32,one_one_nat),uint32_sdiv(aa_uint32_uint32(aa_nat4578243230388868707uint32(bit_se3964402333458159761uint32,one_one_nat),Xb),Y)),
            $let(
              r: uint32,
              r:= minus_minus_uint32(Xb,aa_uint32_uint32(times_times_uint32(q),Y)),
              $ite(aa_uint32_o(aa_uin5924812838015166427nt32_o(ord_less_eq_uint32,Y),r),aa_uin6954824371948101007uint32(produc1400373151660368625uint32(aa_uint32_uint32(plus_plus_uint32(q),one_one_uint32)),minus_minus_uint32(r,Y)),aa_uin6954824371948101007uint32(produc1400373151660368625uint32(q),r)) ) ) ) ) ).

% uint32_divmod_code
tff(fact_10090_uint32_Oset__bits__aux__code,axiom,
    ! [F: fun_nat_o,Nb: nat,W: uint32] :
      set_bits_aux_uint32(F,Nb,W) = $ite(
        Nb = zero_zero_nat,
        W,
        $let(
          n: nat,
          n:= aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat),
          set_bits_aux_uint32(F,n,
            bit_se2966626333419230250uint32(aa_uint32_uint32(aa_nat4578243230388868707uint32(bit_se5742574853984576102uint32,one_one_nat),W),
              $ite(aa_nat_o(F,n),one_one_uint32,zero_zero_uint32))) ) ) ).

% uint32.set_bits_aux_code
tff(fact_10091_uint32__divmod__def,axiom,
    ! [Xb: uint32,Y: uint32] :
      uint32_divmod(Xb,Y) = $ite(Y = zero_zero_uint32,aa_uin6954824371948101007uint32(produc1400373151660368625uint32(aa_uint32_uint32(aa_uin3778660748068847871uint32(aa_fun8139942314387009082uint32(undefi5883245521155754897uint32,divide_divide_uint32),Xb),zero_zero_uint32)),aa_uint32_uint32(aa_uin3778660748068847871uint32(aa_fun8139942314387009082uint32(undefi5883245521155754897uint32,modulo_modulo_uint32),Xb),zero_zero_uint32)),aa_uin6954824371948101007uint32(produc1400373151660368625uint32(aa_uint32_uint32(aa_uin3778660748068847871uint32(divide_divide_uint32,Xb),Y)),aa_uint32_uint32(aa_uin3778660748068847871uint32(modulo_modulo_uint32,Xb),Y))) ).

% uint32_divmod_def
tff(fact_10092_uint32_Oset__bits__code,axiom,
    ! [P: fun_nat_o] : bit_bi705532357378895591uint32(P) = set_bits_aux_uint32(P,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,one)))))),zero_zero_uint32) ).

% uint32.set_bits_code
tff(fact_10093_div0__uint32__def,axiom,
    ! [Xb: uint32] : div0_uint32(Xb) = aa_uint32_uint32(aa_uin3778660748068847871uint32(aa_fun8139942314387009082uint32(undefi5883245521155754897uint32,divide_divide_uint32),Xb),zero_zero_uint32) ).

% div0_uint32_def
tff(fact_10094_uint32__sdiv__code,axiom,
    ! [Xb: uint32,Y: uint32] :
      rep_uint322(uint32_sdiv(Xb,Y)) = $ite(Y = zero_zero_uint32,rep_uint322(aa_uint32_uint32(aa_uin3778660748068847871uint32(aa_fun8139942314387009082uint32(undefi5883245521155754897uint32,divide_divide_uint32),Xb),zero_zero_uint32)),signed6753297604338940182l_num1(rep_uint322(Xb),rep_uint322(Y))) ).

% uint32_sdiv_code
tff(fact_10095_div__uint32__code,axiom,
    ! [Xb: uint32,Y: uint32] :
      aa_uint32_uint32(aa_uin3778660748068847871uint32(divide_divide_uint32,Xb),Y) = $ite(Y = zero_zero_uint32,zero_zero_uint32,uint32_div(Xb,Y)) ).

% div_uint32_code
tff(fact_10096_sshiftr__uint32__code,axiom,
    ! [Nb: nat,Xb: uint32] :
      aa_uint32_uint32(aa_nat4578243230388868707uint32(signed489701013188660438uint32,Nb),Xb) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,one))))))),
        aa_Cod8401701657498852871uint32(uint32_sshiftr(Xb),aa_nat_Code_integer(code_integer_of_nat,Nb)),
        $ite(aa_nat_o(aa_uint32_fun_nat_o(bit_se5367290876889521763uint32,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,one)))))),uminus_uminus_uint32(one_one_uint32),zero_zero_uint32) ) ).

% sshiftr_uint32_code
tff(fact_10097_uint32__div__def,axiom,
    ! [Xb: uint32,Y: uint32] : uint32_div(Xb,Y) = aa_Pro3737690986720249065uint32(produc9004433772639906525uint32,uint32_divmod(Xb,Y)) ).

% uint32_div_def
tff(fact_10098_uint32__sshiftr__def,axiom,
    ! [Xb: uint32,Nb: code_integer] :
      aa_Cod8401701657498852871uint32(uint32_sshiftr(Xb),Nb) = $ite(
        ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Nb),zero_z3403309356797280102nteger)
        | aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,one))))))),Nb) ),
        aa_uint32_uint32(aa_Cod3565511682404152816uint32(aa_fun7349710212561521167uint32(undefi424500106705823974uint32,signed489701013188660438uint32),Nb),Xb),
        aa_uint32_uint32(aa_nat4578243230388868707uint32(signed489701013188660438uint32,aa_Code_integer_nat(code_nat_of_integer,Nb)),Xb) ) ).

% uint32_sshiftr_def
tff(fact_10099_uint32__sshiftr__code,axiom,
    ! [W: uint32,Nb: code_integer] :
      rep_uint322(aa_Cod8401701657498852871uint32(uint32_sshiftr(W),Nb)) = $ite(
        ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Nb),zero_z3403309356797280102nteger)
        | aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,one))))))),Nb) ),
        rep_uint322(aa_uint32_uint32(aa_Cod3565511682404152816uint32(aa_fun7349710212561521167uint32(undefi424500106705823974uint32,signed489701013188660438uint32),Nb),W)),
        signed5000768011106662067l_num1(aa_Code_integer_nat(code_nat_of_integer,Nb),rep_uint322(W)) ) ).

% uint32_sshiftr_code
tff(fact_10100_integer__of__uint32__code,axiom,
    ! [Nb: uint32] :
      aa_uin2106300854182906631nteger(integer_of_uint32,Nb) = $ite(aa_nat_o(aa_uint32_fun_nat_o(bit_se5367290876889521763uint32,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,one)))))),aa_Cod1783974652533396728nteger(bit_se1080825931792720795nteger(intege5370686899274169573signed(bit_se6294004230839889034uint32(Nb,numera9087168376688890119uint32(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,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)))))))))))))))))))))))))))))))))),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,one))))))))))))))))))))))))))))))))),intege5370686899274169573signed(Nb)) ).

% integer_of_uint32_code
tff(fact_10101_uint32__sdiv__def,axiom,
    ! [Xb: uint32,Y: uint32] :
      uint32_sdiv(Xb,Y) = $ite(Y = zero_zero_uint32,aa_uint32_uint32(aa_uin3778660748068847871uint32(aa_fun8139942314387009082uint32(undefi5883245521155754897uint32,divide_divide_uint32),Xb),zero_zero_uint32),abs_uint32(signed6753297604338940182l_num1(rep_uint322(Xb),rep_uint322(Y)))) ).

% uint32_sdiv_def
tff(fact_10102_less__eq__uint32_Oabs__eq,axiom,
    ! [Xaa: word_N3645301735248828278l_num1,Xb: word_N3645301735248828278l_num1] :
      ( aa_uint32_o(aa_uin5924812838015166427nt32_o(ord_less_eq_uint32,abs_uint32(Xaa)),abs_uint32(Xb))
    <=> aa_wor4452366487969921764num1_o(ord_le3335648743751981014l_num1(Xaa),Xb) ) ).

% less_eq_uint32.abs_eq
tff(fact_10103_plus__uint32_Oabs__eq,axiom,
    ! [Xaa: word_N3645301735248828278l_num1,Xb: word_N3645301735248828278l_num1] : aa_uint32_uint32(plus_plus_uint32(abs_uint32(Xaa)),abs_uint32(Xb)) = abs_uint32(plus_p361126936061061375l_num1(Xaa,Xb)) ).

% plus_uint32.abs_eq
tff(fact_10104_minus__uint32_Oabs__eq,axiom,
    ! [Xaa: word_N3645301735248828278l_num1,Xb: word_N3645301735248828278l_num1] : minus_minus_uint32(abs_uint32(Xaa),abs_uint32(Xb)) = abs_uint32(minus_4019991460397169231l_num1(Xaa,Xb)) ).

% minus_uint32.abs_eq
tff(fact_10105_one__uint32__def,axiom,
    one_one_uint32 = abs_uint32(one_on7727431528512463931l_num1) ).

% one_uint32_def
tff(fact_10106_divide__uint32_Oabs__eq,axiom,
    ! [Xaa: word_N3645301735248828278l_num1,Xb: word_N3645301735248828278l_num1] : aa_uint32_uint32(aa_uin3778660748068847871uint32(divide_divide_uint32,abs_uint32(Xaa)),abs_uint32(Xb)) = abs_uint32(divide1791077408188789448l_num1(Xaa,Xb)) ).

% divide_uint32.abs_eq
tff(fact_10107_times__uint32_Oabs__eq,axiom,
    ! [Xaa: word_N3645301735248828278l_num1,Xb: word_N3645301735248828278l_num1] : aa_uint32_uint32(times_times_uint32(abs_uint32(Xaa)),abs_uint32(Xb)) = abs_uint32(aa_wor7274019993978516266l_num1(times_7065122842183080059l_num1(Xaa),Xb)) ).

% times_uint32.abs_eq
tff(fact_10108_uint32_Oof__word__numeral,axiom,
    ! [Nb: num] : abs_uint32(numera7442385471795722001l_num1(Nb)) = numera9087168376688890119uint32(Nb) ).

% uint32.of_word_numeral
tff(fact_10109_size__uint32_Oabs__eq,axiom,
    ! [Xb: word_N3645301735248828278l_num1] : size_size_uint32(abs_uint32(Xb)) = size_s8261804613246490634l_num1(Xb) ).

% size_uint32.abs_eq
tff(fact_10110_integer__of__uint32__signed__def,axiom,
    ! [Nb: uint32] :
      intege5370686899274169573signed(Nb) = $ite(aa_nat_o(aa_uint32_fun_nat_o(bit_se5367290876889521763uint32,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,one)))))),aa_uin2106300854182906631nteger(aa_fun3533256999457416010nteger(undefi4551866269130008737nteger,integer_of_uint32),Nb),aa_uin2106300854182906631nteger(integer_of_uint32,Nb)) ).

% integer_of_uint32_signed_def
tff(fact_10111_integer__of__uint32__signed__code,axiom,
    ! [Nb: uint32] :
      intege5370686899274169573signed(Nb) = $ite(aa_nat_o(aa_uint32_fun_nat_o(bit_se5367290876889521763uint32,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,one)))))),aa_uin2106300854182906631nteger(aa_fun3533256999457416010nteger(undefi4551866269130008737nteger,integer_of_uint32),Nb),aa_int_Code_integer(code_integer_of_int,semiri7338730514057886004m1_int(rep_uint32(Nb)))) ).

% integer_of_uint32_signed_code
tff(fact_10112_divmod__integer__eq__cases,axiom,
    ! [K: code_integer,L: code_integer] :
      aa_Cod6977894654539884864nteger(code_divmod_integer(K),L) = $ite(
        K = zero_z3403309356797280102nteger,
        aa_Cod6977894654539884864nteger(produc1086072967326762835nteger(zero_z3403309356797280102nteger),zero_z3403309356797280102nteger),
        $ite(
          L = zero_z3403309356797280102nteger,
          aa_Cod6977894654539884864nteger(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),aa_Cod6977894654539884864nteger(code_divmod_abs(K),L),produc6916734918728496179nteger(aTP_Lamm_bby(L),aa_Cod6977894654539884864nteger(code_divmod_abs(K),L)))) ) ) ).

% divmod_integer_eq_cases
tff(fact_10113_fst__divmod__abs,axiom,
    ! [K: code_integer,L: code_integer] : aa_Pro5245656335456409788nteger(produc8508995932063986495nteger,aa_Cod6977894654539884864nteger(code_divmod_abs(K),L)) = aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(abs_abs_Code_integer,K)),aa_Cod1783974652533396728nteger(abs_abs_Code_integer,L)) ).

% fst_divmod_abs
tff(fact_10114_snd__divmod__abs,axiom,
    ! [K: code_integer,L: code_integer] : aa_Pro5245656335456409788nteger(produc6174133586879617921nteger,aa_Cod6977894654539884864nteger(code_divmod_abs(K),L)) = aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(abs_abs_Code_integer,K)),aa_Cod1783974652533396728nteger(abs_abs_Code_integer,L)) ).

% snd_divmod_abs
tff(fact_10115_divmod__abs__code_I6_J,axiom,
    ! [J: code_integer] : aa_Cod6977894654539884864nteger(code_divmod_abs(zero_z3403309356797280102nteger),J) = aa_Cod6977894654539884864nteger(produc1086072967326762835nteger(zero_z3403309356797280102nteger),zero_z3403309356797280102nteger) ).

% divmod_abs_code(6)
tff(fact_10116_divmod__abs__code_I5_J,axiom,
    ! [J: code_integer] : aa_Cod6977894654539884864nteger(code_divmod_abs(J),zero_z3403309356797280102nteger) = aa_Cod6977894654539884864nteger(produc1086072967326762835nteger(zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(abs_abs_Code_integer,J)) ).

% divmod_abs_code(5)
tff(fact_10117_Rep__uint32_H__code,axiom,
    ! [Xb: uint32] : rep_uint32(Xb) = bit_bi5746210779246519537l_num1(aa_uint32_fun_nat_o(bit_se5367290876889521763uint32,Xb)) ).

% Rep_uint32'_code
tff(fact_10118_divmod__abs__def,axiom,
    ! [K: code_integer,L: code_integer] : aa_Cod6977894654539884864nteger(code_divmod_abs(K),L) = aa_Cod6977894654539884864nteger(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_10119_divmod__integer__code,axiom,
    ! [K: code_integer,L: code_integer] :
      aa_Cod6977894654539884864nteger(code_divmod_integer(K),L) = $ite(
        K = zero_z3403309356797280102nteger,
        aa_Cod6977894654539884864nteger(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),aa_Cod6977894654539884864nteger(code_divmod_abs(K),L),produc6916734918728496179nteger(aTP_Lamm_bbz(L),aa_Cod6977894654539884864nteger(code_divmod_abs(K),L))),
          $ite(
            L = zero_z3403309356797280102nteger,
            aa_Cod6977894654539884864nteger(produc1086072967326762835nteger(zero_z3403309356797280102nteger),K),
            aa_Pro4693618923521437820nteger(aa_fun2509385170507305080nteger(produc6499014454317279255nteger,uminus1351360451143612070nteger),
              $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,K),zero_z3403309356797280102nteger),aa_Cod6977894654539884864nteger(code_divmod_abs(K),L),produc6916734918728496179nteger(aTP_Lamm_bca(L),aa_Cod6977894654539884864nteger(code_divmod_abs(K),L)))) ) ) ) ).

% divmod_integer_code
tff(fact_10120_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_bcb(K),aa_Cod6977894654539884864nteger(code_divmod_abs(K),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))))) ).

% bit_cut_integer_code
tff(fact_10121_set__bits__int__def,axiom,
    ! [F: fun_nat_o] :
      bit_bi6516823479961619367ts_int(F) = $ite(
        ? [N6: nat] :
        ! [M7: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N6),M7)
         => ( aa_nat_o(F,M7) = aa_nat_o(F,N6) ) ),
        $let(
          n2: nat,
          n2:= ord_Least_nat(aTP_Lamm_bcc(F)),
          aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,n2),groups9116527308978886569_o_int(zero_n2684676970156552555ol_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)),map_nat_o(F,upt(zero_zero_nat,aa_nat_nat(suc,n2))))) ),
        zero_zero_int ) ).

% set_bits_int_def
tff(fact_10122_Least__eq__0,axiom,
    ! [P: fun_nat_o] :
      ( aa_nat_o(P,zero_zero_nat)
     => ( ord_Least_nat(P) = zero_zero_nat ) ) ).

% Least_eq_0
tff(fact_10123_Least__Suc,axiom,
    ! [P: fun_nat_o,Nb: nat] :
      ( aa_nat_o(P,Nb)
     => ( ~ aa_nat_o(P,zero_zero_nat)
       => ( ord_Least_nat(P) = aa_nat_nat(suc,ord_Least_nat(aTP_Lamm_bbu(P))) ) ) ) ).

% Least_Suc
tff(fact_10124_Least__Suc2,axiom,
    ! [P: fun_nat_o,Nb: nat,Q: fun_nat_o,M: nat] :
      ( aa_nat_o(P,Nb)
     => ( aa_nat_o(Q,M)
       => ( ~ aa_nat_o(P,zero_zero_nat)
         => ( ! [K3: nat] :
                ( aa_nat_o(P,aa_nat_nat(suc,K3))
              <=> aa_nat_o(Q,K3) )
           => ( ord_Least_nat(P) = aa_nat_nat(suc,ord_Least_nat(Q)) ) ) ) ) ) ).

% Least_Suc2
tff(fact_10125_set__bits__int__unfold_H,axiom,
    ! [F: fun_nat_o] :
      bit_bi6516823479961619367ts_int(F) = $ite(
        ? [N6: nat] :
        ! [N9: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N6),N9)
         => ~ aa_nat_o(F,N9) ),
        groups9116527308978886569_o_int(zero_n2684676970156552555ol_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)),map_nat_o(F,upt(zero_zero_nat,ord_Least_nat(aTP_Lamm_bcd(F))))),
        $ite(
          ? [N6: nat] :
          ! [N9: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N6),N9)
           => aa_nat_o(F,N9) ),
          $let(
            n2: nat,
            n2:= ord_Least_nat(aTP_Lamm_bce(F)),
            aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,n2),groups9116527308978886569_o_int(zero_n2684676970156552555ol_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)),append_o(map_nat_o(F,upt(zero_zero_nat,n2)),cons_o($true,nil_o)))) ),
          zero_zero_int ) ) ).

% set_bits_int_unfold'
tff(fact_10126_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_10127_VEBT__internal_Ovalid_H_Oelims_I3_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( ~ aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_valid,Xb),Xaa)
     => ( ( ? [Uu2: $o,Uv2: $o] : Xb = vEBT_Leaf((Uu2),(Uv2))
         => ( Xaa = one_one_nat ) )
       => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
              ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Mima,Deg2,TreeList),Summary) )
             => ( ( Deg2 = Xaa )
                & $let(
                    n2: nat,
                    n2:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                    $let(
                      m: nat,
                      m:= aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg2),n2),
                      ( ! [X: vEBT_VEBT] :
                          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),set_VEBT_VEBT2(TreeList))
                         => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_valid,X),n2) )
                      & aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_valid,Summary),m)
                      & ( size_s6755466524823107622T_VEBT(TreeList) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),m) )
                      & aa_opt7112393969400857165_nat_o(case_o184042715313410164at_nat(( ~ ? [X6: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,Summary),X6)
                            & ! [X: vEBT_VEBT] :
                                ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),set_VEBT_VEBT2(TreeList))
                               => ~ ? [X6: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,X),X6) ) ),produc6081775807080527818_nat_o(aa_nat8314798964848609180_nat_o(aa_nat7293937527460703909_nat_o(aa_VEB4248176028289774660_nat_o(aa_lis144135085560928841_nat_o(aTP_Lamm_bcf(Deg2),TreeList),Summary),n2),m))),Mima) ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.elims(3)
tff(fact_10128_valid__eq,axiom,
    ! [Ta: vEBT_VEBT,D: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_valid,Ta),D)
    <=> aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Ta),D) ) ).

% valid_eq
tff(fact_10129_valid__eq1,axiom,
    ! [Ta: vEBT_VEBT,D: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Ta),D)
     => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_valid,Ta),D) ) ).

% valid_eq1
tff(fact_10130_valid__eq2,axiom,
    ! [Ta: vEBT_VEBT,D: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_valid,Ta),D)
     => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Ta),D) ) ).

% valid_eq2
tff(fact_10131_VEBT__internal_Ovalid_H_Osimps_I1_J,axiom,
    ! [Uu: $o,Uv: $o,D: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_valid,vEBT_Leaf((Uu),(Uv))),D)
    <=> ( D = one_one_nat ) ) ).

% VEBT_internal.valid'.simps(1)
tff(fact_10132_VEBT__internal_Ovalid_H_Osimps_I2_J,axiom,
    ! [Mima2: option4927543243414619207at_nat,Deg: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Deg4: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_valid,aa_VEB5832953974749077948T_VEBT(vEBT_Node(Mima2,Deg,TreeLista),Summarya)),Deg4)
    <=> ( ( Deg = Deg4 )
        & $let(
            n2: nat,
            n2:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
            $let(
              m: nat,
              m:= aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg),n2),
              ( ! [X: vEBT_VEBT] :
                  ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),set_VEBT_VEBT2(TreeLista))
                 => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_valid,X),n2) )
              & aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_valid,Summarya),m)
              & ( size_s6755466524823107622T_VEBT(TreeLista) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),m) )
              & aa_opt7112393969400857165_nat_o(case_o184042715313410164at_nat(( ~ ? [X6: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,Summarya),X6)
                    & ! [X: vEBT_VEBT] :
                        ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),set_VEBT_VEBT2(TreeLista))
                       => ~ ? [X6: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,X),X6) ) ),produc6081775807080527818_nat_o(aa_nat8314798964848609180_nat_o(aa_nat7293937527460703909_nat_o(aa_VEB4248176028289774660_nat_o(aa_lis144135085560928841_nat_o(aTP_Lamm_bcf(Deg),TreeLista),Summarya),n2),m))),Mima2) ) ) ) ) ) ).

% VEBT_internal.valid'.simps(2)
tff(fact_10133_VEBT__internal_Ovalid_H_Oelims_I1_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: $o] :
      ( ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_valid,Xb),Xaa)
      <=> (Y) )
     => ( ( ? [Uu2: $o,Uv2: $o] : Xb = vEBT_Leaf((Uu2),(Uv2))
         => ( (Y)
          <=> ( Xaa != one_one_nat ) ) )
       => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
              ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Mima,Deg2,TreeList),Summary) )
             => ( (Y)
              <=> ~ ( ( Deg2 = Xaa )
                    & $let(
                        n2: nat,
                        n2:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                        $let(
                          m: nat,
                          m:= aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg2),n2),
                          ( ! [X: vEBT_VEBT] :
                              ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),set_VEBT_VEBT2(TreeList))
                             => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_valid,X),n2) )
                          & aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_valid,Summary),m)
                          & ( size_s6755466524823107622T_VEBT(TreeList) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),m) )
                          & aa_opt7112393969400857165_nat_o(case_o184042715313410164at_nat(( ~ ? [X6: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,Summary),X6)
                                & ! [X: vEBT_VEBT] :
                                    ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),set_VEBT_VEBT2(TreeList))
                                   => ~ ? [X6: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,X),X6) ) ),produc6081775807080527818_nat_o(aa_nat8314798964848609180_nat_o(aa_nat7293937527460703909_nat_o(aa_VEB4248176028289774660_nat_o(aa_lis144135085560928841_nat_o(aTP_Lamm_bcf(Deg2),TreeList),Summary),n2),m))),Mima) ) ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.elims(1)
tff(fact_10134_VEBT__internal_Ovalid_H_Oelims_I2_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_valid,Xb),Xaa)
     => ( ( ? [Uu2: $o,Uv2: $o] : Xb = vEBT_Leaf((Uu2),(Uv2))
         => ( Xaa != one_one_nat ) )
       => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
              ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Mima,Deg2,TreeList),Summary) )
             => ~ ( ( Deg2 = Xaa )
                  & $let(
                      n2: nat,
                      n2:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                      $let(
                        m: nat,
                        m:= aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg2),n2),
                        ( ! [X: vEBT_VEBT] :
                            ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),set_VEBT_VEBT2(TreeList))
                           => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_valid,X),n2) )
                        & aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_valid,Summary),m)
                        & ( size_s6755466524823107622T_VEBT(TreeList) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),m) )
                        & aa_opt7112393969400857165_nat_o(case_o184042715313410164at_nat(( ~ ? [X6: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,Summary),X6)
                              & ! [X: vEBT_VEBT] :
                                  ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),set_VEBT_VEBT2(TreeList))
                                 => ~ ? [X6: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,X),X6) ) ),produc6081775807080527818_nat_o(aa_nat8314798964848609180_nat_o(aa_nat7293937527460703909_nat_o(aa_VEB4248176028289774660_nat_o(aa_lis144135085560928841_nat_o(aTP_Lamm_bcf(Deg2),TreeList),Summary),n2),m))),Mima) ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.elims(2)
tff(fact_10135_VEBT__internal_Ovalid_H_Opelims_I1_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: $o] :
      ( ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_valid,Xb),Xaa)
      <=> (Y) )
     => ( aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(Xb),Xaa))
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( Xb = vEBT_Leaf((Uu2),(Uv2)) )
             => ( ( (Y)
                <=> ( Xaa = one_one_nat ) )
               => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(vEBT_Leaf((Uu2),(Uv2))),Xaa)) ) )
         => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Mima,Deg2,TreeList),Summary) )
               => ( ( (Y)
                  <=> ( ( Deg2 = Xaa )
                      & $let(
                          n2: nat,
                          n2:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                          $let(
                            m: nat,
                            m:= aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg2),n2),
                            ( ! [X: vEBT_VEBT] :
                                ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),set_VEBT_VEBT2(TreeList))
                               => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_valid,X),n2) )
                            & aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_valid,Summary),m)
                            & ( size_s6755466524823107622T_VEBT(TreeList) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),m) )
                            & aa_opt7112393969400857165_nat_o(case_o184042715313410164at_nat(( ~ ? [X6: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,Summary),X6)
                                  & ! [X: vEBT_VEBT] :
                                      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),set_VEBT_VEBT2(TreeList))
                                     => ~ ? [X6: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,X),X6) ) ),produc6081775807080527818_nat_o(aa_nat8314798964848609180_nat_o(aa_nat7293937527460703909_nat_o(aa_VEB4248176028289774660_nat_o(aa_lis144135085560928841_nat_o(aTP_Lamm_bcf(Deg2),TreeList),Summary),n2),m))),Mima) ) ) ) ) )
                 => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Mima,Deg2,TreeList),Summary)),Xaa)) ) ) ) ) ) ).

% VEBT_internal.valid'.pelims(1)
tff(fact_10136_VEBT__internal_Ovalid_H_Opelims_I2_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_valid,Xb),Xaa)
     => ( aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(Xb),Xaa))
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( Xb = vEBT_Leaf((Uu2),(Uv2)) )
             => ( aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(vEBT_Leaf((Uu2),(Uv2))),Xaa))
               => ( Xaa != one_one_nat ) ) )
         => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Mima,Deg2,TreeList),Summary) )
               => ( aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Mima,Deg2,TreeList),Summary)),Xaa))
                 => ~ ( ( Deg2 = Xaa )
                      & $let(
                          n2: nat,
                          n2:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                          $let(
                            m: nat,
                            m:= aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg2),n2),
                            ( ! [X: vEBT_VEBT] :
                                ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),set_VEBT_VEBT2(TreeList))
                               => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_valid,X),n2) )
                            & aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_valid,Summary),m)
                            & ( size_s6755466524823107622T_VEBT(TreeList) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),m) )
                            & aa_opt7112393969400857165_nat_o(case_o184042715313410164at_nat(( ~ ? [X6: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,Summary),X6)
                                  & ! [X: vEBT_VEBT] :
                                      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),set_VEBT_VEBT2(TreeList))
                                     => ~ ? [X6: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,X),X6) ) ),produc6081775807080527818_nat_o(aa_nat8314798964848609180_nat_o(aa_nat7293937527460703909_nat_o(aa_VEB4248176028289774660_nat_o(aa_lis144135085560928841_nat_o(aTP_Lamm_bcf(Deg2),TreeList),Summary),n2),m))),Mima) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.pelims(2)
tff(fact_10137_fi__match__entails,axiom,
    ! [M: list_P8527749157015355191n_assn] :
      ( ! [X2: produc6575502325842934193n_assn] :
          ( member7957490590177025114n_assn(X2,set_Pr1139785259514867910n_assn(M))
         => aa_Pro2491721552408928739assn_o(produc7274209992780475162assn_o(entails),X2) )
     => aa_assn_o(aa_assn_fun_assn_o(entails,foldr_assn_assn(times_times_assn,map_Pr8991440229025900053n_assn(produc9167289414957590229n_assn,M),one_one_assn)),foldr_assn_assn(times_times_assn,map_Pr8991440229025900053n_assn(produc2051961928117032727n_assn,M),one_one_assn)) ) ).

% fi_match_entails
tff(fact_10138_VEBT__internal_Ovalid_H_Opelims_I3_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( ~ aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_valid,Xb),Xaa)
     => ( aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(Xb),Xaa))
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( Xb = vEBT_Leaf((Uu2),(Uv2)) )
             => ( aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(vEBT_Leaf((Uu2),(Uv2))),Xaa))
               => ( Xaa = one_one_nat ) ) )
         => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Mima,Deg2,TreeList),Summary) )
               => ( aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Mima,Deg2,TreeList),Summary)),Xaa))
                 => ( ( Deg2 = Xaa )
                    & $let(
                        n2: nat,
                        n2:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                        $let(
                          m: nat,
                          m:= aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg2),n2),
                          ( ! [X: vEBT_VEBT] :
                              ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),set_VEBT_VEBT2(TreeList))
                             => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_valid,X),n2) )
                          & aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_valid,Summary),m)
                          & ( size_s6755466524823107622T_VEBT(TreeList) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),m) )
                          & aa_opt7112393969400857165_nat_o(case_o184042715313410164at_nat(( ~ ? [X6: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,Summary),X6)
                                & ! [X: vEBT_VEBT] :
                                    ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),set_VEBT_VEBT2(TreeList))
                                   => ~ ? [X6: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,X),X6) ) ),produc6081775807080527818_nat_o(aa_nat8314798964848609180_nat_o(aa_nat7293937527460703909_nat_o(aa_VEB4248176028289774660_nat_o(aa_lis144135085560928841_nat_o(aTP_Lamm_bcf(Deg2),TreeList),Summary),n2),m))),Mima) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.pelims(3)
tff(fact_10139_FI__RESULT__def,axiom,
    ! [M8: list_P8527749157015355191n_assn,UP: assn,UQ: assn,F4: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(fI_RESULT(M8,UP),UQ),F4)
    <=> ( ! [X: produc6575502325842934193n_assn] :
            ( member7957490590177025114n_assn(X,set_Pr1139785259514867910n_assn(M8))
           => aa_Pro2491721552408928739assn_o(produc7274209992780475162assn_o(entails),X) )
       => aa_assn_o(aa_assn_fun_assn_o(entails,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,foldr_assn_assn(times_times_assn,map_Pr8991440229025900053n_assn(produc9167289414957590229n_assn,M8),one_one_assn)),UP)),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,foldr_assn_assn(times_times_assn,map_Pr8991440229025900053n_assn(produc2051961928117032727n_assn,M8),one_one_assn)),UQ)),F4)) ) ) ).

% FI_RESULT_def
tff(fact_10140_FI__def,axiom,
    ! [M: list_P8527749157015355191n_assn,P2: assn,Q2: assn,Up: assn,Uq: assn,F: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(fi(M,P2,Q2,Up),Uq),F)
    <=> ( ! [X: produc6575502325842934193n_assn] :
            ( member7957490590177025114n_assn(X,set_Pr1139785259514867910n_assn(M))
           => aa_Pro2491721552408928739assn_o(produc7274209992780475162assn_o(entails),X) )
       => aa_assn_o(aa_assn_fun_assn_o(entails,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,foldr_assn_assn(times_times_assn,map_Pr8991440229025900053n_assn(produc9167289414957590229n_assn,M),one_one_assn)),P2)),Up)),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,foldr_assn_assn(times_times_assn,map_Pr8991440229025900053n_assn(produc2051961928117032727n_assn,M),one_one_assn)),Q2)),Uq)),F)) ) ) ).

% FI_def
tff(fact_10141_entails__solve__finalize_I2_J,axiom,
    ! [M8: list_P8527749157015355191n_assn] : aa_assn_o(aa_assn_fun_assn_o(fI_RESULT(M8,one_one_assn),one_one_assn),one_one_assn) ).

% entails_solve_finalize(2)
tff(fact_10142_frame__inference__finalize,axiom,
    ! [M8: list_P8527749157015355191n_assn,F4: assn] : aa_assn_o(aa_assn_fun_assn_o(fI_RESULT(M8,F4),one_one_assn),F4) ).

% frame_inference_finalize
tff(fact_10143_FI__p__nomatch,axiom,
    ! [M: list_P8527749157015355191n_assn,Ps2: assn,Qs: assn,Q2: assn,P2: assn,Up: assn,Uq: assn,F: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(fi(M,Ps2,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Qs),Q2),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,P2),Up)),Uq),F)
     => aa_assn_o(aa_assn_fun_assn_o(fi(M,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Ps2),P2),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Qs),Q2),Up),Uq),F) ) ).

% FI_p_nomatch
tff(fact_10144_FI__finalize,axiom,
    ! [M: list_P8527749157015355191n_assn,P2: assn,Up: assn,Q2: assn,Uq: assn,F: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(fI_RESULT(M,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,P2),Up)),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Q2),Uq)),F)
     => aa_assn_o(aa_assn_fun_assn_o(fi(M,P2,Q2,Up),Uq),F) ) ).

% FI_finalize
tff(fact_10145_entails__solve__finalize_I1_J,axiom,
    ! [M8: list_P8527749157015355191n_assn,P: assn] : aa_assn_o(aa_assn_fun_assn_o(fI_RESULT(M8,P),one_one_assn),top_top_assn) ).

% entails_solve_finalize(1)
tff(fact_10146_less__eq__char__simp,axiom,
    ! [B0: $o,B1: $o,B22: $o,B32: $o,B42: $o,B52: $o,B62: $o,B72: $o,C0: $o,C1: $o,C22: $o,C32: $o,C42: $o,C52: $o,C62: $o,C7: $o] :
      ( aa_char_o(ord_less_eq_char(aa_o_char(char2((B0),(B1),(B22),(B32),(B42),(B52),(B62)),(B72))),aa_o_char(char2((C0),(C1),(C22),(C32),(C42),(C52),(C62)),(C7)))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(foldr_o_nat(aTP_Lamm_bcg,cons_o((B0),cons_o((B1),cons_o((B22),cons_o((B32),cons_o((B42),cons_o((B52),cons_o((B62),cons_o((B72),nil_o))))))))),zero_zero_nat)),aa_nat_nat(foldr_o_nat(aTP_Lamm_bcg,cons_o((C0),cons_o((C1),cons_o((C22),cons_o((C32),cons_o((C42),cons_o((C52),cons_o((C62),cons_o((C7),nil_o))))))))),zero_zero_nat)) ) ).

% less_eq_char_simp
tff(fact_10147_less__char__simp,axiom,
    ! [B0: $o,B1: $o,B22: $o,B32: $o,B42: $o,B52: $o,B62: $o,B72: $o,C0: $o,C1: $o,C22: $o,C32: $o,C42: $o,C52: $o,C62: $o,C7: $o] :
      ( aa_char_o(ord_less_char(aa_o_char(char2((B0),(B1),(B22),(B32),(B42),(B52),(B62)),(B72))),aa_o_char(char2((C0),(C1),(C22),(C32),(C42),(C52),(C62)),(C7)))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(foldr_o_nat(aTP_Lamm_bcg,cons_o((B0),cons_o((B1),cons_o((B22),cons_o((B32),cons_o((B42),cons_o((B52),cons_o((B62),cons_o((B72),nil_o))))))))),zero_zero_nat)),aa_nat_nat(foldr_o_nat(aTP_Lamm_bcg,cons_o((C0),cons_o((C1),cons_o((C22),cons_o((C32),cons_o((C42),cons_o((C52),cons_o((C62),cons_o((C7),nil_o))))))))),zero_zero_nat)) ) ).

% less_char_simp
tff(fact_10148_integer__of__char__code,axiom,
    ! [B0: $o,B1: $o,B22: $o,B32: $o,B42: $o,B52: $o,B62: $o,B72: $o] : integer_of_char(aa_o_char(char2((B0),(B1),(B22),(B32),(B42),(B52),(B62)),(B72))) = aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,zero_n356916108424825756nteger((B72))),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)))),zero_n356916108424825756nteger((B62)))),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)))),zero_n356916108424825756nteger((B52)))),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)))),zero_n356916108424825756nteger((B42)))),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)))),zero_n356916108424825756nteger((B32)))),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)))),zero_n356916108424825756nteger((B22)))),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)))),zero_n356916108424825756nteger((B1)))),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)))),zero_n356916108424825756nteger((B0))) ).

% integer_of_char_code
tff(fact_10149_less__char__def,axiom,
    ! [C1: char,C22: char] :
      ( aa_char_o(ord_less_char(C1),C22)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_char_nat(comm_s629917340098488124ar_nat,C1)),aa_char_nat(comm_s629917340098488124ar_nat,C22)) ) ).

% less_char_def
tff(fact_10150_nat__of__char__less__256,axiom,
    ! [C: char] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_char_nat(comm_s629917340098488124ar_nat,C)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,one)))))))))) ).

% nat_of_char_less_256
tff(fact_10151_less__eq__char__def,axiom,
    ! [C1: char,C22: char] :
      ( aa_char_o(ord_less_eq_char(C1),C22)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_char_nat(comm_s629917340098488124ar_nat,C1)),aa_char_nat(comm_s629917340098488124ar_nat,C22)) ) ).

% less_eq_char_def
tff(fact_10152_range__nat__of__char,axiom,
    image_char_nat(comm_s629917340098488124ar_nat,top_top_set_char) = set_or4665077453230672383an_nat(zero_zero_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,one)))))))))) ).

% range_nat_of_char
tff(fact_10153_char_Osize_I2_J,axiom,
    ! [X1: $o,X23: $o,X33: $o,X42: $o,X52: $o,X62: $o,X7: $o,X8: $o] : size_size_char(aa_o_char(char2((X1),(X23),(X33),(X42),(X52),(X62),(X7)),(X8))) = zero_zero_nat ).

% char.size(2)
tff(fact_10154_char__of__integer__code,axiom,
    ! [K: code_integer] : char_of_integer(K) = produc4188289175737317920o_char(aTP_Lamm_bco,code_bit_cut_integer(K)) ).

% char_of_integer_code
tff(fact_10155_UNIV__char__of__nat,axiom,
    top_top_set_char = image_nat_char(unique3096191561947761185of_nat,set_or4665077453230672383an_nat(zero_zero_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,one))))))))))) ).

% UNIV_char_of_nat
tff(fact_10156_inj__on__char__of__nat,axiom,
    aa_set_nat_o(inj_on_nat_char(unique3096191561947761185of_nat),set_or4665077453230672383an_nat(zero_zero_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,one))))))))))) ).

% inj_on_char_of_nat
tff(fact_10157_String_Ochar__of__ascii__of,axiom,
    ! [C: char] : aa_char_nat(comm_s629917340098488124ar_nat,ascii_of(C)) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit1,one)))),aa_char_nat(comm_s629917340098488124ar_nat,C)) ).

% String.char_of_ascii_of
tff(fact_10158_DERIV__even__real__root,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),zero_zero_real)
         => has_fi5821293074295781190e_real(root(Nb),aa_real_real(inverse_inverse_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_nat_real(semiri5074537144036343181t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(root(Nb),Xb)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat))))),topolo2177554685111907308n_real(Xb,top_top_set_real)) ) ) ) ).

% DERIV_even_real_root
tff(fact_10159_has__real__derivative__neg__dec__left,axiom,
    ! [F: fun_real_real,L: real,Xb: real,S2: set_real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(Xb,S2))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,L),zero_zero_real)
       => ? [D2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D2)
            & ! [H6: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),H6)
               => ( aa_set_real_o(member_real(aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),H6)),S2)
                 => ( aa_real_o(aa_real_fun_real_o(ord_less_real,H6),D2)
                   => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,Xb)),aa_real_real(F,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),H6))) ) ) ) ) ) ) ).

% has_real_derivative_neg_dec_left
tff(fact_10160_has__real__derivative__pos__inc__left,axiom,
    ! [F: fun_real_real,L: real,Xb: real,S2: set_real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(Xb,S2))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),L)
       => ? [D2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D2)
            & ! [H6: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),H6)
               => ( aa_set_real_o(member_real(aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),H6)),S2)
                 => ( aa_real_o(aa_real_fun_real_o(ord_less_real,H6),D2)
                   => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),H6))),aa_real_real(F,Xb)) ) ) ) ) ) ) ).

% has_real_derivative_pos_inc_left
tff(fact_10161_has__real__derivative__neg__dec__right,axiom,
    ! [F: fun_real_real,L: real,Xb: real,S2: set_real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(Xb,S2))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,L),zero_zero_real)
       => ? [D2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D2)
            & ! [H6: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),H6)
               => ( aa_set_real_o(member_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),H6)),S2)
                 => ( aa_real_o(aa_real_fun_real_o(ord_less_real,H6),D2)
                   => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),H6))),aa_real_real(F,Xb)) ) ) ) ) ) ) ).

% has_real_derivative_neg_dec_right
tff(fact_10162_has__real__derivative__pos__inc__right,axiom,
    ! [F: fun_real_real,L: real,Xb: real,S2: set_real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(Xb,S2))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),L)
       => ? [D2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D2)
            & ! [H6: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),H6)
               => ( aa_set_real_o(member_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),H6)),S2)
                 => ( aa_real_o(aa_real_fun_real_o(ord_less_real,H6),D2)
                   => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,Xb)),aa_real_real(F,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),H6))) ) ) ) ) ) ) ).

% has_real_derivative_pos_inc_right
tff(fact_10163_DERIV__pow,axiom,
    ! [Nb: nat,Xb: real,S3: set_real] : has_fi5821293074295781190e_real(aTP_Lamm_bcp(Nb),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat)))),topolo2177554685111907308n_real(Xb,S3)) ).

% DERIV_pow
tff(fact_10164_DERIV__nonpos__imp__nonincreasing,axiom,
    ! [A: real,B: real,F: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( ! [X2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),X2)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X2),B)
             => ? [Y5: real] :
                  ( has_fi5821293074295781190e_real(F,Y5,topolo2177554685111907308n_real(X2,top_top_set_real))
                  & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y5),zero_zero_real) ) ) )
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(F,B)),aa_real_real(F,A)) ) ) ).

% DERIV_nonpos_imp_nonincreasing
tff(fact_10165_DERIV__nonneg__imp__nondecreasing,axiom,
    ! [A: real,B: real,F: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( ! [X2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),X2)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X2),B)
             => ? [Y5: real] :
                  ( has_fi5821293074295781190e_real(F,Y5,topolo2177554685111907308n_real(X2,top_top_set_real))
                  & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y5) ) ) )
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(F,A)),aa_real_real(F,B)) ) ) ).

% DERIV_nonneg_imp_nondecreasing
tff(fact_10166_deriv__nonneg__imp__mono,axiom,
    ! [A: real,B: real,G: fun_real_real,G2: fun_real_real] :
      ( ! [X2: real] :
          ( aa_set_real_o(member_real(X2),set_or1222579329274155063t_real(A,B))
         => has_fi5821293074295781190e_real(G,aa_real_real(G2,X2),topolo2177554685111907308n_real(X2,top_top_set_real)) )
     => ( ! [X2: real] :
            ( aa_set_real_o(member_real(X2),set_or1222579329274155063t_real(A,B))
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(G2,X2)) )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(G,A)),aa_real_real(G,B)) ) ) ) ).

% deriv_nonneg_imp_mono
tff(fact_10167_DERIV__mirror,axiom,
    ! [F: fun_real_real,Y: real,Xb: real] :
      ( has_fi5821293074295781190e_real(F,Y,topolo2177554685111907308n_real(aa_real_real(uminus_uminus_real,Xb),top_top_set_real))
    <=> has_fi5821293074295781190e_real(aTP_Lamm_bcq(F),aa_real_real(uminus_uminus_real,Y),topolo2177554685111907308n_real(Xb,top_top_set_real)) ) ).

% DERIV_mirror
tff(fact_10168_DERIV__fun__pow,axiom,
    ! [G: fun_real_real,M: real,Xb: real,Nb: nat] :
      ( has_fi5821293074295781190e_real(G,M,topolo2177554685111907308n_real(Xb,top_top_set_real))
     => has_fi5821293074295781190e_real(aa_nat_fun_real_real(aTP_Lamm_bcr(G),Nb),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(G,Xb)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)))),M),topolo2177554685111907308n_real(Xb,top_top_set_real)) ) ).

% DERIV_fun_pow
tff(fact_10169_DERIV__const__average,axiom,
    ! [A: real,B: real,V: fun_real_real,K: real] :
      ( ( A != B )
     => ( ! [X2: real] : has_fi5821293074295781190e_real(V,K,topolo2177554685111907308n_real(X2,top_top_set_real))
       => ( aa_real_real(V,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),numeral_numeral_real(aa_num_num(bit0,one)))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(V,A)),aa_real_real(V,B))),numeral_numeral_real(aa_num_num(bit0,one))) ) ) ) ).

% DERIV_const_average
tff(fact_10170_DERIV__const__ratio__const,axiom,
    ! [A: real,B: real,F: fun_real_real,K: real] :
      ( ( A != B )
     => ( ! [X2: real] : has_fi5821293074295781190e_real(F,K,topolo2177554685111907308n_real(X2,top_top_set_real))
       => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(F,B)),aa_real_real(F,A)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),A)),K) ) ) ) ).

% DERIV_const_ratio_const
tff(fact_10171_DERIV__const__ratio__const2,axiom,
    ! [A: real,B: real,F: fun_real_real,K: real] :
      ( ( A != B )
     => ( ! [X2: real] : has_fi5821293074295781190e_real(F,K,topolo2177554685111907308n_real(X2,top_top_set_real))
       => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(F,B)),aa_real_real(F,A))),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),A)) = K ) ) ) ).

% DERIV_const_ratio_const2
tff(fact_10172_DERIV__ln__divide,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => has_fi5821293074295781190e_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),Xb),topolo2177554685111907308n_real(Xb,top_top_set_real)) ) ).

% DERIV_ln_divide
tff(fact_10173_DERIV__neg__dec__left,axiom,
    ! [F: fun_real_real,L: real,Xb: real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(Xb,top_top_set_real))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,L),zero_zero_real)
       => ? [D2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D2)
            & ! [H6: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),H6)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_real,H6),D2)
                 => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,Xb)),aa_real_real(F,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),H6))) ) ) ) ) ) ).

% DERIV_neg_dec_left
tff(fact_10174_DERIV__pos__inc__left,axiom,
    ! [F: fun_real_real,L: real,Xb: real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(Xb,top_top_set_real))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),L)
       => ? [D2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D2)
            & ! [H6: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),H6)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_real,H6),D2)
                 => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),H6))),aa_real_real(F,Xb)) ) ) ) ) ) ).

% DERIV_pos_inc_left
tff(fact_10175_DERIV__neg__dec__right,axiom,
    ! [F: fun_real_real,L: real,Xb: real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(Xb,top_top_set_real))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,L),zero_zero_real)
       => ? [D2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D2)
            & ! [H6: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),H6)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_real,H6),D2)
                 => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),H6))),aa_real_real(F,Xb)) ) ) ) ) ) ).

% DERIV_neg_dec_right
tff(fact_10176_DERIV__pos__inc__right,axiom,
    ! [F: fun_real_real,L: real,Xb: real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(Xb,top_top_set_real))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),L)
       => ? [D2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D2)
            & ! [H6: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),H6)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_real,H6),D2)
                 => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,Xb)),aa_real_real(F,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),H6))) ) ) ) ) ) ).

% DERIV_pos_inc_right
tff(fact_10177_DERIV__local__const,axiom,
    ! [F: fun_real_real,L: real,Xb: real,D: real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(Xb,top_top_set_real))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D)
       => ( ! [Y4: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),Y4))),D)
             => ( aa_real_real(F,Xb) = aa_real_real(F,Y4) ) )
         => ( L = zero_zero_real ) ) ) ) ).

% DERIV_local_const
tff(fact_10178_ATP_Olambda__1,axiom,
    ! [Uu: product_prod_nat_nat] :
      aa_Pro7675874045790911655on_nat(aTP_Lamm_am,Uu) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,x12),one_one_nat),
        heap_T3487192422709364219on_nat(none_nat),
        $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_Pro4243173770044397973at_nat(product_snd_nat_nat,Uu)),xa),heap_T3487192422709364219on_nat(aa_nat_option_nat(some_nat,aa_Pro4243173770044397973at_nat(product_snd_nat_nat,Uu))),heap_T8222160169144143993on_nat(aa_nat5821310369566504398ap_nat(vEBT_VEBT_lowi(xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,x12),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aTP_Lamm_al(Uu))) ) ).

% ATP.lambda_1
tff(fact_10179_ATP_Olambda__2,axiom,
    ! [Uu: product_unit] : aa_Pro451875010097498139on_nat(aTP_Lamm_an,Uu) = case_o8344607093967974880at_nat(heap_T3487192422709364219on_nat(none_nat),aTP_Lamm_am,x11) ).

% ATP.lambda_2
tff(fact_10180_ATP_Olambda__3,axiom,
    ! [Uu: option_nat] : aa_option_nat_assn(aTP_Lamm_bb,Uu) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,mi),ma)),aa_nat_nat(suc,aa_nat_nat(suc,va)),treeList),summary)),tia)),aa_o_assn(pure_assn,Uu = aa_nat_option_nat(vEBT_vebt_pred(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,mi),ma)),aa_nat_nat(suc,aa_nat_nat(suc,va)),treeList),summary)),xa))) ).

% ATP.lambda_3
tff(fact_10181_ATP_Olambda__4,axiom,
    ! [Uu: nat] : aa_nat_real(aTP_Lamm_azk,Uu) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Uu)),one_one_real)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uu),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_nat))) ).

% ATP.lambda_4
tff(fact_10182_ATP_Olambda__5,axiom,
    ! [Uu: nat] : aa_nat_real(aTP_Lamm_aqd,Uu) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(aa_num_num(bit0,one)))),aa_nat_nat(suc,Uu)) ).

% ATP.lambda_5
tff(fact_10183_ATP_Olambda__6,axiom,
    ! [Uu: real] :
      ( aa_real_o(aTP_Lamm_baf,Uu)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Uu)
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Uu),numeral_numeral_real(aa_num_num(bit0,one)))
        & ( aa_real_real(cos_real,Uu) = zero_zero_real ) ) ) ).

% ATP.lambda_6
tff(fact_10184_ATP_Olambda__7,axiom,
    ! [Uu: product_unit] :
      aa_Pro451875010097498139on_nat(aTP_Lamm_ao,Uu) = heap_T5999496708990702694on_nat(aa_o_H2459438917461573107t_unit(refine_Imp_assert,
          ( ( some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,mi),ma)) = x11 )
          & ( aa_nat_nat(suc,aa_nat_nat(suc,va)) = x12 ) )),aTP_Lamm_an) ).

% ATP.lambda_7
tff(fact_10185_ATP_Olambda__8,axiom,
    ! [Uu: nat] : aa_nat_real(aTP_Lamm_auy,Uu) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(cos_coeff,Uu)),aa_nat_real(aa_real_fun_nat_real(power_power_real,zero_zero_real),Uu)) ).

% ATP.lambda_8
tff(fact_10186_ATP_Olambda__9,axiom,
    ! [Uu: nat] : aa_nat_nat(aTP_Lamm_bbr,Uu) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),aa_nat_nat(suc,zero_zero_nat)) ).

% ATP.lambda_9
tff(fact_10187_ATP_Olambda__10,axiom,
    ! [Uu: code_integer] : aa_Cod1783974652533396728nteger(aTP_Lamm_asa,Uu) = aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(Uu),one_one_Code_integer) ).

% ATP.lambda_10
tff(fact_10188_ATP_Olambda__11,axiom,
    ! [Uu: uint32] : aa_uint32_uint32(aTP_Lamm_arv,Uu) = aa_uint32_uint32(plus_plus_uint32(Uu),one_one_uint32) ).

% ATP.lambda_11
tff(fact_10189_ATP_Olambda__12,axiom,
    ! [Uu: word_N3645301735248828278l_num1] : aa_wor7274019993978516266l_num1(aTP_Lamm_aru,Uu) = plus_p361126936061061375l_num1(Uu,one_on7727431528512463931l_num1) ).

% ATP.lambda_12
tff(fact_10190_ATP_Olambda__13,axiom,
    ! [Uu: real] : aa_real_real(aTP_Lamm_arx,Uu) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Uu),one_one_real) ).

% ATP.lambda_13
tff(fact_10191_ATP_Olambda__14,axiom,
    ! [Uu: rat] : aa_rat_rat(aTP_Lamm_arw,Uu) = aa_rat_rat(plus_plus_rat(Uu),one_one_rat) ).

% ATP.lambda_14
tff(fact_10192_ATP_Olambda__15,axiom,
    ! [Uu: nat] : aa_nat_nat(aTP_Lamm_arz,Uu) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),one_one_nat) ).

% ATP.lambda_15
tff(fact_10193_ATP_Olambda__16,axiom,
    ! [Uu: int] : aa_int_int(aTP_Lamm_ary,Uu) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Uu),one_one_int) ).

% ATP.lambda_16
tff(fact_10194_ATP_Olambda__17,axiom,
    ! [Uu: int] :
      ( aa_int_o(aTP_Lamm_bbp,Uu)
    <=> ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Uu) ) ).

% ATP.lambda_17
tff(fact_10195_ATP_Olambda__18,axiom,
    ! [Uu: option_nat] : aa_option_nat_assn(aTP_Lamm_aq,Uu) = ex_ass463751140784270563_VEBTi(aa_opt2780285371362742984i_assn(aTP_Lamm_ap,Uu)) ).

% ATP.lambda_18
tff(fact_10196_ATP_Olambda__19,axiom,
    ! [Uu: num] : aa_num_option_num(aTP_Lamm_azz,Uu) = aa_num_option_num(some_num,aa_num_num(bit1,Uu)) ).

% ATP.lambda_19
tff(fact_10197_ATP_Olambda__20,axiom,
    ! [Uu: num] : aa_num_option_num(aTP_Lamm_azx,Uu) = aa_num_option_num(some_num,aa_num_num(bit0,Uu)) ).

% ATP.lambda_20
tff(fact_10198_ATP_Olambda__21,axiom,
    ! [Uu: int] : aa_int2565445551602696444nt_int(aTP_Lamm_bau,Uu) = aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(uminus_uminus_int,Uu)) ).

% ATP.lambda_21
tff(fact_10199_ATP_Olambda__22,axiom,
    ! [Uu: int] : aa_int2565445551602696444nt_int(aTP_Lamm_bat,Uu) = aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(abs_abs_int,Uu)) ).

% ATP.lambda_22
tff(fact_10200_ATP_Olambda__23,axiom,
    ! [Uu: nat] : aa_nat8803788312118491388at_nat(aTP_Lamm_mo,Uu) = aa_nat8803788312118491388at_nat(product_Pair_nat_nat,aa_nat_nat(suc,Uu)) ).

% ATP.lambda_23
tff(fact_10201_ATP_Olambda__24,axiom,
    ! [Uu: nat] : aa_nat_option_num(aTP_Lamm_bae,Uu) = aa_num_option_num(some_num,one) ).

% ATP.lambda_24
tff(fact_10202_ATP_Olambda__25,axiom,
    ! [Uu: num,Uua: nat] : aa_nat_option_num(aTP_Lamm_bak(Uu),Uua) = aa_num_option_num(case_num_option_num(aa_num_option_num(some_num,one),aTP_Lamm_bai(Uua),aTP_Lamm_baj(Uua)),Uu) ).

% ATP.lambda_25
tff(fact_10203_ATP_Olambda__26,axiom,
    ! [Uu: fun_nat_real,Uua: nat] :
      aa_nat_real(aTP_Lamm_apu(Uu),Uua) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua),zero_zero_real,aa_nat_real(Uu,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),one_one_nat)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% ATP.lambda_26
tff(fact_10204_ATP_Olambda__27,axiom,
    ! [Uu: int,Uua: int] :
      aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_bas,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_27
tff(fact_10205_ATP_Olambda__28,axiom,
    ! [Uu: option_int,Uua: int] :
      ( aa_int_o(aTP_Lamm_dy(Uu),Uua)
    <=> case_option_o_int($true,aa_int_fun_int_o(aTP_Lamm_dx,Uua),Uu) ) ).

% ATP.lambda_28
tff(fact_10206_ATP_Olambda__29,axiom,
    ! [Uu: option_nat,Uua: nat] :
      ( aa_nat_o(aTP_Lamm_dw(Uu),Uua)
    <=> aa_option_nat_o(case_option_o_nat($true,aa_nat_fun_nat_o(aTP_Lamm_dv,Uua)),Uu) ) ).

% ATP.lambda_29
tff(fact_10207_ATP_Olambda__30,axiom,
    ! [Uu: option_num,Uua: num] :
      ( aa_num_o(aTP_Lamm_du(Uu),Uua)
    <=> aa_option_num_o(case_option_o_num($true,aTP_Lamm_dt(Uua)),Uu) ) ).

% ATP.lambda_30
tff(fact_10208_ATP_Olambda__31,axiom,
    ! [Uu: option_rat,Uua: rat] :
      ( aa_rat_o(aTP_Lamm_ds(Uu),Uua)
    <=> case_option_o_rat($true,aa_rat_fun_rat_o(aTP_Lamm_dr,Uua),Uu) ) ).

% ATP.lambda_31
tff(fact_10209_ATP_Olambda__32,axiom,
    ! [Uu: option_real,Uua: real] :
      ( aa_real_o(aTP_Lamm_dq(Uu),Uua)
    <=> case_option_o_real($true,aa_real_fun_real_o(aTP_Lamm_dp,Uua),Uu) ) ).

% ATP.lambda_32
tff(fact_10210_ATP_Olambda__33,axiom,
    ! [Uu: nat,Uua: num] : aa_num_option_num(aa_nat3713591252005393295on_num(aTP_Lamm_bal,Uu),Uua) = case_nat_option_num(none_num,aTP_Lamm_bak(Uua),Uu) ).

% ATP.lambda_33
tff(fact_10211_ATP_Olambda__34,axiom,
    ! [Uu: option_set_int,Uua: set_int] :
      ( aa_set_int_o(aTP_Lamm_ed(Uu),Uua)
    <=> case_o223999843215110191et_int($false,ord_less_eq_set_int(Uua),Uu) ) ).

% ATP.lambda_34
tff(fact_10212_ATP_Olambda__35,axiom,
    ! [Uu: option_rat,Uua: rat] :
      ( aa_rat_o(aTP_Lamm_ee(Uu),Uua)
    <=> case_option_o_rat($false,aa_rat_fun_rat_o(ord_less_eq_rat,Uua),Uu) ) ).

% ATP.lambda_35
tff(fact_10213_ATP_Olambda__36,axiom,
    ! [Uu: option_num,Uua: num] :
      ( aa_num_o(aTP_Lamm_ef(Uu),Uua)
    <=> aa_option_num_o(case_option_o_num($false,ord_less_eq_num(Uua)),Uu) ) ).

% ATP.lambda_36
tff(fact_10214_ATP_Olambda__37,axiom,
    ! [Uu: option_nat,Uua: nat] :
      ( aa_nat_o(aTP_Lamm_eg(Uu),Uua)
    <=> aa_option_nat_o(case_option_o_nat($false,aa_nat_fun_nat_o(ord_less_eq_nat,Uua)),Uu) ) ).

% ATP.lambda_37
tff(fact_10215_ATP_Olambda__38,axiom,
    ! [Uu: option_int,Uua: int] :
      ( aa_int_o(aTP_Lamm_eh(Uu),Uua)
    <=> case_option_o_int($false,aa_int_fun_int_o(ord_less_eq_int,Uua),Uu) ) ).

% ATP.lambda_38
tff(fact_10216_ATP_Olambda__39,axiom,
    ! [Uu: nat,Uua: num] : aa_num_option_num(aTP_Lamm_bai(Uu),Uua) = aa_opt5369159329878751634on_num(case_o6005452278849405969um_num(none_num,aTP_Lamm_azx),aa_num_option_num(bit_take_bit_num(Uu),Uua)) ).

% ATP.lambda_39
tff(fact_10217_ATP_Olambda__40,axiom,
    ! [Uu: num,Uua: nat] : aa_nat_option_num(aTP_Lamm_bag(Uu),Uua) = aa_opt5369159329878751634on_num(case_o6005452278849405969um_num(none_num,aTP_Lamm_azx),aa_num_option_num(bit_take_bit_num(Uua),Uu)) ).

% ATP.lambda_40
tff(fact_10218_ATP_Olambda__41,axiom,
    ! [Uu: option_nat,Uua: list_VEBT_VEBTi] :
      aa_lis2508312872377092907i_assn(aa_opt2780285371362742984i_assn(aTP_Lamm_ap,Uu),Uua) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,summary),x14)),aa_lis2508312872377092907i_assn(snga_assn_VEBT_VEBTi(x13),Uua))),aa_lis2508312872377092907i_assn(vEBT_L6296928887356842470_VEBTi(vEBT_vebt_assn_raw,treeList),Uua))),
        aa_o_assn(pure_assn,
          ( ( x11 = some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,mi),ma)) )
          & ( x12 = aa_nat_nat(suc,aa_nat_nat(suc,va)) )
          & ( Uu = $ite(
                aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,ma),xa),
                aa_nat_option_nat(some_nat,ma),
                $let(
                  l: nat,
                  l:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                  $let(
                    h: nat,
                    h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                    $ite(
                      aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(treeList)),
                      $let(
                        minlow: option_nat,
                        minlow:= vEBT_vebt_mint(nth_VEBT_VEBT(treeList,h)),
                        $ite(
                          ( ( minlow != none_nat )
                          & aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_greater,aa_nat_option_nat(some_nat,l)),minlow) ),
                          aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,aa_nat_option_nat(some_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_option_nat(some_nat,h))),aa_nat_option_nat(vEBT_vebt_pred(nth_VEBT_VEBT(treeList,h)),l)),
                          $let(
                            pr: option_nat,
                            pr:= aa_nat_option_nat(vEBT_vebt_pred(summary),h),
                            $ite(
                              pr = none_nat,
                              $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,mi),xa),aa_nat_option_nat(some_nat,mi),none_nat),
                              aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,aa_nat_option_nat(some_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),pr)),vEBT_vebt_maxt(nth_VEBT_VEBT(treeList,the_nat(pr)))) ) ) ) ),
                      none_nat ) ) ) ) ) ))) ).

% ATP.lambda_41
tff(fact_10219_ATP_Olambda__42,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_azj,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Uua)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),one_one_nat))))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Uu),one_one_real)),aa_nat_nat(suc,Uua))) ).

% ATP.lambda_42
tff(fact_10220_ATP_Olambda__43,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_azf,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Uua)),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)),one_one_nat)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)),one_one_nat))) ).

% ATP.lambda_43
tff(fact_10221_ATP_Olambda__44,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_azg,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Uua)),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua))) ).

% ATP.lambda_44
tff(fact_10222_ATP_Olambda__45,axiom,
    ! [Uu: real,Uua: real] :
      ( aa_real_o(aa_real_fun_real_o(aTP_Lamm_bah,Uu),Uua)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))),Uua)
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Uua),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))
        & ( aa_real_real(sin_real,Uua) = Uu ) ) ) ).

% ATP.lambda_45
tff(fact_10223_ATP_Olambda__46,axiom,
    ! [Uu: real,Uua: real] :
      ( aa_real_o(aa_real_fun_real_o(aTP_Lamm_bab,Uu),Uua)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))),Uua)
        & aa_real_o(aa_real_fun_real_o(ord_less_real,Uua),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))
        & ( aa_real_real(tan_real,Uua) = Uu ) ) ) ).

% ATP.lambda_46
tff(fact_10224_ATP_Olambda__47,axiom,
    ! [Uu: code_integer,Uua: code_integer] :
      aa_Code_integer_int(aa_Cod1242632294114111009er_int(aTP_Lamm_bbx,Uu),Uua) = $let(
        l2: int,
        l2:= aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_Code_integer_int(code_int_of_integer,Uu)),
        $ite(Uua = zero_z3403309356797280102nteger,l2,aa_int_int(aa_int_fun_int_int(plus_plus_int,l2),one_one_int)) ) ).

% ATP.lambda_47
tff(fact_10225_ATP_Olambda__48,axiom,
    ! [Uu: nat,Uua: nat] :
      aa_nat_Code_integer(aa_nat2010205248571845816nteger(aTP_Lamm_mf,Uu),Uua) = $let(
        m2: code_integer,
        m2:= aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_nat_Code_integer(semiri4939895301339042750nteger,Uu)),
        $ite(Uua = zero_zero_nat,m2,aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(m2),one_one_Code_integer)) ) ).

% ATP.lambda_48
tff(fact_10226_ATP_Olambda__49,axiom,
    ! [Uu: nat,Uua: nat] :
      aa_nat_uint32(aa_nat6935575117487981255uint32(aTP_Lamm_lz,Uu),Uua) = $let(
        m2: uint32,
        m2:= aa_uint32_uint32(times_times_uint32(numera9087168376688890119uint32(aa_num_num(bit0,one))),aa_nat_uint32(semiri2565882477558803405uint32,Uu)),
        $ite(Uua = zero_zero_nat,m2,aa_uint32_uint32(plus_plus_uint32(m2),one_one_uint32)) ) ).

% ATP.lambda_49
tff(fact_10227_ATP_Olambda__50,axiom,
    ! [Uu: nat,Uua: nat] :
      aa_nat5599461682457136004l_num1(aa_nat6251203789436639057l_num1(aTP_Lamm_ma,Uu),Uua) = $let(
        m2: word_N3645301735248828278l_num1,
        m2:= aa_wor7274019993978516266l_num1(times_7065122842183080059l_num1(numera7442385471795722001l_num1(aa_num_num(bit0,one))),aa_nat5599461682457136004l_num1(semiri8819519690708144855l_num1,Uu)),
        $ite(Uua = zero_zero_nat,m2,plus_p361126936061061375l_num1(m2,one_on7727431528512463931l_num1)) ) ).

% ATP.lambda_50
tff(fact_10228_ATP_Olambda__51,axiom,
    ! [Uu: nat,Uua: nat] :
      aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_mc,Uu),Uua) = $let(
        m2: real,
        m2:= aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,Uu)),
        $ite(Uua = zero_zero_nat,m2,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,m2),one_one_real)) ) ).

% ATP.lambda_51
tff(fact_10229_ATP_Olambda__52,axiom,
    ! [Uu: nat,Uua: nat] :
      aa_nat_rat(aa_nat_fun_nat_rat(aTP_Lamm_mb,Uu),Uua) = $let(
        m2: rat,
        m2:= aa_rat_rat(times_times_rat(numeral_numeral_rat(aa_num_num(bit0,one))),aa_nat_rat(semiri681578069525770553at_rat,Uu)),
        $ite(Uua = zero_zero_nat,m2,aa_rat_rat(plus_plus_rat(m2),one_one_rat)) ) ).

% ATP.lambda_52
tff(fact_10230_ATP_Olambda__53,axiom,
    ! [Uu: nat,Uua: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_me,Uu),Uua) = $let(
        m2: nat,
        m2:= aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(semiri1316708129612266289at_nat,Uu)),
        $ite(Uua = zero_zero_nat,m2,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,m2),one_one_nat)) ) ).

% ATP.lambda_53
tff(fact_10231_ATP_Olambda__54,axiom,
    ! [Uu: nat,Uua: nat] :
      aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_md,Uu),Uua) = $let(
        m2: int,
        m2:= aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_int(semiri1314217659103216013at_int,Uu)),
        $ite(Uua = zero_zero_nat,m2,aa_int_int(aa_int_fun_int_int(plus_plus_int,m2),one_one_int)) ) ).

% ATP.lambda_54
tff(fact_10232_ATP_Olambda__55,axiom,
    ! [Uu: complex,Uua: real] :
      ( aa_real_o(aTP_Lamm_bar(Uu),Uua)
    <=> ( ( aa_complex_complex(sgn_sgn_complex,Uu) = aa_real_complex(cis,Uua) )
        & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,pi)),Uua)
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Uua),pi) ) ) ).

% ATP.lambda_55
tff(fact_10233_ATP_Olambda__56,axiom,
    ! [Uu: real,Uua: int] :
      ( aa_int_o(aa_real_fun_int_o(aTP_Lamm_bao,Uu),Uua)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Uua)),Uu)
        & aa_real_o(aa_real_fun_real_o(ord_less_real,Uu),aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(plus_plus_int,Uua),one_one_int))) ) ) ).

% ATP.lambda_56
tff(fact_10234_ATP_Olambda__57,axiom,
    ! [Uu: rat,Uua: int] :
      ( aa_int_o(aTP_Lamm_bap(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_10235_ATP_Olambda__58,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_azm,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Uua)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_nat)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_nat)))) ).

% ATP.lambda_58
tff(fact_10236_ATP_Olambda__59,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_azc,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_59
tff(fact_10237_ATP_Olambda__60,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_azq,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uua)),aa_nat_real(semiri2265585572941072030t_real,Uua)) ).

% ATP.lambda_60
tff(fact_10238_ATP_Olambda__61,axiom,
    ! [Uu: set_Code_integer,Uua: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(aa_set4044230130517860621eger_o(aTP_Lamm_aim,Uu),Uua)
    <=> ( aa_set5643387332196575111eger_o(ord_le1307284697595431911nteger(Uu),Uua)
        & aa_set5643387332196575111eger_o(finite6017078050557962740nteger,Uua) ) ) ).

% ATP.lambda_61
tff(fact_10239_ATP_Olambda__62,axiom,
    ! [Uu: set_complex,Uua: set_complex] :
      ( aa_set_complex_o(aa_set8643480048548153067plex_o(aTP_Lamm_ail,Uu),Uua)
    <=> ( aa_set_complex_o(ord_less_set_complex(Uu),Uua)
        & aa_set_complex_o(finite3207457112153483333omplex,Uua) ) ) ).

% ATP.lambda_62
tff(fact_10240_ATP_Olambda__63,axiom,
    ! [Uu: set_nat,Uua: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(aTP_Lamm_aij,Uu),Uua)
    <=> ( aa_set_nat_o(ord_less_set_nat(Uu),Uua)
        & aa_set_nat_o(finite_finite_nat,Uua) ) ) ).

% ATP.lambda_63
tff(fact_10241_ATP_Olambda__64,axiom,
    ! [Uu: set_int,Uua: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(aTP_Lamm_aik,Uu),Uua)
    <=> ( aa_set_int_o(ord_less_set_int(Uu),Uua)
        & aa_set_int_o(finite_finite_int,Uua) ) ) ).

% ATP.lambda_64
tff(fact_10242_ATP_Olambda__65,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_ayv,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_65
tff(fact_10243_ATP_Olambda__66,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_ayu,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_66
tff(fact_10244_ATP_Olambda__67,axiom,
    ! [Uu: nat,Uua: complex] :
      ( aa_complex_o(aa_nat_fun_complex_o(aTP_Lamm_adr,Uu),Uua)
    <=> ( aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Uua),Uu) = one_one_complex ) ) ).

% ATP.lambda_67
tff(fact_10245_ATP_Olambda__68,axiom,
    ! [Uu: nat,Uua: real] :
      ( aa_real_o(aa_nat_fun_real_o(aTP_Lamm_ads,Uu),Uua)
    <=> ( aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),Uu) = one_one_real ) ) ).

% ATP.lambda_68
tff(fact_10246_ATP_Olambda__69,axiom,
    ! [Uu: product_prod_nat_nat,Uua: nat] : aa_nat947733127081326238on_nat(aTP_Lamm_al(Uu),Uua) = heap_T8222160169144143993on_nat(aa_nat5821310369566504398ap_nat(vEBT_VEBT_highi(xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,x12),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat6020981482381522343on_nat(aTP_Lamm_ak(Uu),Uua)) ).

% ATP.lambda_69
tff(fact_10247_ATP_Olambda__70,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_azl,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_nat)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_nat))) ).

% ATP.lambda_70
tff(fact_10248_ATP_Olambda__71,axiom,
    ! [Uu: real,Uua: real] :
      ( aa_real_o(aa_real_fun_real_o(aTP_Lamm_bad,Uu),Uua)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Uua)
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Uua),pi)
        & ( aa_real_real(cos_real,Uua) = Uu ) ) ) ).

% ATP.lambda_71
tff(fact_10249_ATP_Olambda__72,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_apt(Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(Uu,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua))),aa_nat_real(Uu,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)))) ).

% ATP.lambda_72
tff(fact_10250_ATP_Olambda__73,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_apq(Uu),Uua) = aa_rat_rat(plus_plus_rat(aa_nat_rat(Uu,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua))),aa_nat_rat(Uu,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)))) ).

% ATP.lambda_73
tff(fact_10251_ATP_Olambda__74,axiom,
    ! [Uu: fun_nat_nat,Uua: nat] : aa_nat_nat(aTP_Lamm_aps(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_74
tff(fact_10252_ATP_Olambda__75,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_apr(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_75
tff(fact_10253_ATP_Olambda__76,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_aot(Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(Uu,aa_nat_nat(suc,Uua))),aa_nat_real(Uu,Uua)) ).

% ATP.lambda_76
tff(fact_10254_ATP_Olambda__77,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_aor(Uu),Uua) = minus_minus_rat(aa_nat_rat(Uu,aa_nat_nat(suc,Uua)),aa_nat_rat(Uu,Uua)) ).

% ATP.lambda_77
tff(fact_10255_ATP_Olambda__78,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_aos(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_78
tff(fact_10256_ATP_Olambda__79,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_ajv(Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uu,Uua)),aa_nat_real(aa_real_fun_nat_real(power_power_real,zero_zero_real),Uua)) ).

% ATP.lambda_79
tff(fact_10257_ATP_Olambda__80,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_ajw(Uu),Uua) = aa_rat_rat(times_times_rat(aa_nat_rat(Uu,Uua)),aa_nat_rat(power_power_rat(zero_zero_rat),Uua)) ).

% ATP.lambda_80
tff(fact_10258_ATP_Olambda__81,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_app(Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(Uu,Uua)),aa_nat_real(Uu,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),one_one_nat))) ).

% ATP.lambda_81
tff(fact_10259_ATP_Olambda__82,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_apn(Uu),Uua) = 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_82
tff(fact_10260_ATP_Olambda__83,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_apo(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_83
tff(fact_10261_ATP_Olambda__84,axiom,
    ! [Uu: fun_nat_uint32,Uua: nat] : aa_nat_uint32(aTP_Lamm_apj(Uu),Uua) = minus_minus_uint32(aa_nat_uint32(Uu,Uua),aa_nat_uint32(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),one_one_nat))) ).

% ATP.lambda_84
tff(fact_10262_ATP_Olambda__85,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_apm(Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(Uu,Uua)),aa_nat_real(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),one_one_nat))) ).

% ATP.lambda_85
tff(fact_10263_ATP_Olambda__86,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_apk(Uu),Uua) = 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_86
tff(fact_10264_ATP_Olambda__87,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_apl(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_87
tff(fact_10265_ATP_Olambda__88,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_asu(Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(Uu,Uua)),aa_nat_real(Uu,aa_nat_nat(suc,Uua))) ).

% ATP.lambda_88
tff(fact_10266_ATP_Olambda__89,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_ass(Uu),Uua) = minus_minus_rat(aa_nat_rat(Uu,Uua),aa_nat_rat(Uu,aa_nat_nat(suc,Uua))) ).

% ATP.lambda_89
tff(fact_10267_ATP_Olambda__90,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_ast(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_90
tff(fact_10268_ATP_Olambda__91,axiom,
    ! [Uu: fun_Co1554742298434871494uint32,Uua: code_integer] :
      ( aa_Code_integer_o(aTP_Lamm_aoa(Uu),Uua)
    <=> ( aa_Cod8401701657498852871uint32(Uu,Uua) = zero_zero_uint32 ) ) ).

% ATP.lambda_91
tff(fact_10269_ATP_Olambda__92,axiom,
    ! [Uu: fun_Co3960896333048504454r_real,Uua: code_integer] :
      ( aa_Code_integer_o(aTP_Lamm_aod(Uu),Uua)
    <=> ( aa_Code_integer_real(Uu,Uua) = zero_zero_real ) ) ).

% ATP.lambda_92
tff(fact_10270_ATP_Olambda__93,axiom,
    ! [Uu: fun_Code_integer_rat,Uua: code_integer] :
      ( aa_Code_integer_o(aTP_Lamm_aog(Uu),Uua)
    <=> ( aa_Code_integer_rat(Uu,Uua) = zero_zero_rat ) ) ).

% ATP.lambda_93
tff(fact_10271_ATP_Olambda__94,axiom,
    ! [Uu: fun_complex_uint32,Uua: complex] :
      ( aa_complex_o(aTP_Lamm_anz(Uu),Uua)
    <=> ( aa_complex_uint32(Uu,Uua) = zero_zero_uint32 ) ) ).

% ATP.lambda_94
tff(fact_10272_ATP_Olambda__95,axiom,
    ! [Uu: fun_complex_real,Uua: complex] :
      ( aa_complex_o(aTP_Lamm_aoc(Uu),Uua)
    <=> ( aa_complex_real(Uu,Uua) = zero_zero_real ) ) ).

% ATP.lambda_95
tff(fact_10273_ATP_Olambda__96,axiom,
    ! [Uu: fun_complex_rat,Uua: complex] :
      ( aa_complex_o(aTP_Lamm_aof(Uu),Uua)
    <=> ( aa_complex_rat(Uu,Uua) = zero_zero_rat ) ) ).

% ATP.lambda_96
tff(fact_10274_ATP_Olambda__97,axiom,
    ! [Uu: fun_int_uint32,Uua: int] :
      ( aa_int_o(aTP_Lamm_any(Uu),Uua)
    <=> ( aa_int_uint32(Uu,Uua) = zero_zero_uint32 ) ) ).

% ATP.lambda_97
tff(fact_10275_ATP_Olambda__98,axiom,
    ! [Uu: fun_int_real,Uua: int] :
      ( aa_int_o(aTP_Lamm_aob(Uu),Uua)
    <=> ( aa_int_real(Uu,Uua) = zero_zero_real ) ) ).

% ATP.lambda_98
tff(fact_10276_ATP_Olambda__99,axiom,
    ! [Uu: fun_int_rat,Uua: int] :
      ( aa_int_o(aTP_Lamm_aoe(Uu),Uua)
    <=> ( aa_int_rat(Uu,Uua) = zero_zero_rat ) ) ).

% ATP.lambda_99
tff(fact_10277_ATP_Olambda__100,axiom,
    ! [Uu: fun_int_nat,Uua: int] :
      ( aa_int_o(aTP_Lamm_aoh(Uu),Uua)
    <=> ( aa_int_nat(Uu,Uua) = zero_zero_nat ) ) ).

% ATP.lambda_100
tff(fact_10278_ATP_Olambda__101,axiom,
    ! [Uu: fun_nat_nat,Uua: nat] :
      ( aa_nat_o(aTP_Lamm_ays(Uu),Uua)
    <=> ( aa_nat_nat(Uu,Uua) = one_one_nat ) ) ).

% ATP.lambda_101
tff(fact_10279_ATP_Olambda__102,axiom,
    ! [Uu: fun_nat_int,Uua: nat] :
      ( aa_nat_o(aTP_Lamm_ayq(Uu),Uua)
    <=> ( aa_nat_int(Uu,Uua) = one_one_int ) ) ).

% ATP.lambda_102
tff(fact_10280_ATP_Olambda__103,axiom,
    ! [Uu: fun_int_int,Uua: int] :
      ( aa_int_o(aTP_Lamm_ayr(Uu),Uua)
    <=> ( aa_int_int(Uu,Uua) = one_one_int ) ) ).

% ATP.lambda_103
tff(fact_10281_ATP_Olambda__104,axiom,
    ! [Uu: code_integer,Uua: $o] : aa_o_char(aa_Cod7842134615204088097o_char(aTP_Lamm_bco,Uu),(Uua)) = produc4188289175737317920o_char(aTP_Lamm_bcn((Uua)),code_bit_cut_integer(Uu)) ).

% ATP.lambda_104
tff(fact_10282_ATP_Olambda__105,axiom,
    ! [Uu: produc3881548065746020326Ti_nat,Uua: fun_Pr4171499318505651870_VEBTi] : aa_fun943973709508947813_VEBTi(aTP_Lamm_abq(Uu),Uua) = aa_Pro1482084341531967383_VEBTi(produc3255295512018472142_VEBTi(aTP_Lamm_abp(Uua)),Uu) ).

% ATP.lambda_105
tff(fact_10283_ATP_Olambda__106,axiom,
    ! [Uu: produc3881548065746020326Ti_nat,Uua: fun_Pr342922516890243977on_nat] : aa_fun7698174670542568229on_nat(aTP_Lamm_abb(Uu),Uua) = aa_Pro2606730266749126658on_nat(produc8911080112929139129on_nat(aTP_Lamm_aba(Uua)),Uu) ).

% ATP.lambda_106
tff(fact_10284_ATP_Olambda__107,axiom,
    ! [Uu: produc3881548065746020326Ti_nat,Uua: fun_Pr342922516890243977on_nat] : aa_fun7698174670542568229on_nat(aTP_Lamm_aat(Uu),Uua) = aa_Pro2606730266749126658on_nat(produc8911080112929139129on_nat(aTP_Lamm_aas(Uua)),Uu) ).

% ATP.lambda_107
tff(fact_10285_ATP_Olambda__108,axiom,
    ! [Uu: produc3881548065746020326Ti_nat,Uua: fun_Pr238758252084143515Heap_o] : aa_fun6848502500920544641Heap_o(aTP_Lamm_zp(Uu),Uua) = aa_Pro3916655985231737292Heap_o(produc770043135277712853Heap_o(aTP_Lamm_zo(Uua)),Uu) ).

% ATP.lambda_108
tff(fact_10286_ATP_Olambda__109,axiom,
    ! [Uu: produc3960855945107176009Ti_nat,Uua: fun_Pr940977554174145406Heap_o] : aa_fun3870618101526761188Heap_o(aTP_Lamm_aal(Uu),Uua) = aa_Pro6173831900156065711Heap_o(produc5872130906356439992Heap_o(produc844971584604743726Heap_o(aTP_Lamm_aak(Uua))),Uu) ).

% ATP.lambda_109
tff(fact_10287_ATP_Olambda__110,axiom,
    ! [Uu: produc3960855945107176009Ti_nat,Uua: fun_Pr633315466666964198on_nat] : aa_fun1100322450986716680on_nat(aTP_Lamm_xl(Uu),Uua) = aa_Pro2727408364141080927on_nat(produc183673358652719894on_nat(produc3521483158534984758on_nat(aTP_Lamm_xk(Uua))),Uu) ).

% ATP.lambda_110
tff(fact_10288_ATP_Olambda__111,axiom,
    ! [Uu: produc3960855945107176009Ti_nat,Uua: fun_Pr633315466666964198on_nat] : aa_fun1100322450986716680on_nat(aTP_Lamm_wj(Uu),Uua) = aa_Pro2727408364141080927on_nat(produc183673358652719894on_nat(produc3521483158534984758on_nat(aTP_Lamm_wi(Uua))),Uu) ).

% ATP.lambda_111
tff(fact_10289_ATP_Olambda__112,axiom,
    ! [Uu: produc3960855945107176009Ti_nat,Uua: fun_Pr4404027214751104635_VEBTi] : aa_fun7494984083220284744_VEBTi(aTP_Lamm_vs(Uu),Uua) = aa_Pro721367001817565044_VEBTi(produc2943724498215716011_VEBTi(produc3258509807451162059_VEBTi(aTP_Lamm_vr(Uua))),Uu) ).

% ATP.lambda_112
tff(fact_10290_ATP_Olambda__113,axiom,
    ! [Uu: complex,Uua: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_aio,Uu),Uua) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(suc,Uua))),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Uu),Uua)) ).

% ATP.lambda_113
tff(fact_10291_ATP_Olambda__114,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_ain,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Uua))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uua)) ).

% ATP.lambda_114
tff(fact_10292_ATP_Olambda__115,axiom,
    ! [Uu: num,Uua: num] : aa_num_int(aTP_Lamm_azy(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_115
tff(fact_10293_ATP_Olambda__116,axiom,
    ! [Uu: nat,Uua: real] : aa_real_real(aTP_Lamm_baq(Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sgn_sgn_real,Uua)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,Uua)),Uu)) ).

% ATP.lambda_116
tff(fact_10294_ATP_Olambda__117,axiom,
    ! [Uu: $o,Uua: nat] : aa_nat_nat(aa_o_fun_nat_nat(aTP_Lamm_bcg,(Uu)),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_o_nat(zero_n2687167440665602831ol_nat,(Uu))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% ATP.lambda_117
tff(fact_10295_ATP_Olambda__118,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_azi,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(sin_coeff,Uua)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uua)) ).

% ATP.lambda_118
tff(fact_10296_ATP_Olambda__119,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_azh,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(cos_coeff,Uua)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uua)) ).

% ATP.lambda_119
tff(fact_10297_ATP_Olambda__120,axiom,
    ! [Uu: option_nat,Uua: vEBT_VEBTi] : aa_VEB2711320435582112505on_nat(aTP_Lamm_ab(Uu),Uua) = heap_T3669509953089699273on_nat(aa_VEB2711320435582112505on_nat(vEBT_vebt_maxti,Uua),aTP_Lamm_aa(Uu)) ).

% ATP.lambda_120
tff(fact_10298_ATP_Olambda__121,axiom,
    ! [Uu: code_integer,Uua: code_integer] :
      aa_Code_integer_num(aa_Cod6128727250572145423er_num(aTP_Lamm_bbw,Uu),Uua) = $let(
        l2: num,
        l2:= aa_Code_integer_num(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_121
tff(fact_10299_ATP_Olambda__122,axiom,
    ! [Uu: code_integer,Uua: code_integer] :
      aa_Code_integer_nat(aa_Cod5420483313623307717er_nat(aTP_Lamm_bbv,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_122
tff(fact_10300_ATP_Olambda__123,axiom,
    ! [Uu: real,Uua: real] :
      ( aa_real_o(aa_real_fun_real_o(aTP_Lamm_bac,Uu),Uua)
    <=> ( aa_real_real(exp_real,Uua) = Uu ) ) ).

% ATP.lambda_123
tff(fact_10301_ATP_Olambda__124,axiom,
    ! [Uu: code_integer,Uua: code_integer] : aa_Cod6977894654539884864nteger(aa_Cod8380929501330563836nteger(aTP_Lamm_un,Uu),Uua) = aa_Cod6977894654539884864nteger(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_124
tff(fact_10302_ATP_Olambda__125,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aTP_Lamm_um,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_125
tff(fact_10303_ATP_Olambda__126,axiom,
    ! [Uu: int,Uua: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_uo,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_126
tff(fact_10304_ATP_Olambda__127,axiom,
    ! [Uu: code_integer,Uua: code_integer] : aa_Cod6977894654539884864nteger(aa_Cod8380929501330563836nteger(aTP_Lamm_uq,Uu),Uua) = aa_Cod6977894654539884864nteger(produc1086072967326762835nteger(Uu),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Uua)) ).

% ATP.lambda_127
tff(fact_10305_ATP_Olambda__128,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aTP_Lamm_up,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_128
tff(fact_10306_ATP_Olambda__129,axiom,
    ! [Uu: int,Uua: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_ur,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_129
tff(fact_10307_ATP_Olambda__130,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_aux,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Uu),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(semiri5074537144036343181t_real,Uua)),numeral_numeral_real(aa_num_num(bit0,one)))) ).

% ATP.lambda_130
tff(fact_10308_ATP_Olambda__131,axiom,
    ! [Uu: rat,Uua: nat] : aa_nat_rat(aTP_Lamm_auw(Uu),Uua) = aa_rat_rat(plus_plus_rat(Uu),divide_divide_rat(aa_nat_rat(semiri681578069525770553at_rat,Uua),numeral_numeral_rat(aa_num_num(bit0,one)))) ).

% ATP.lambda_131
tff(fact_10309_ATP_Olambda__132,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_azd,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_132
tff(fact_10310_ATP_Olambda__133,axiom,
    ! [Uu: int,Uua: int] : aa_int_int(aa_int_fun_int_int(aTP_Lamm_azv,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_133
tff(fact_10311_ATP_Olambda__134,axiom,
    ! [Uu: nat,Uua: vEBT_VEBTi] : aa_VEB8194921341634505877Heap_o(aTP_Lamm_jp(Uu),Uua) = aa_nat2910048432523430448Heap_o(aa_VEB1455453220190609250Heap_o(vEBT_vebt_memberi,Uua),Uu) ).

% ATP.lambda_134
tff(fact_10312_ATP_Olambda__135,axiom,
    ! [Uu: set_Code_integer,Uua: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(aTP_Lamm_adw(Uu),Uua)
    <=> aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(Uua),Uu) ) ).

% ATP.lambda_135
tff(fact_10313_ATP_Olambda__136,axiom,
    ! [Uu: set_complex,Uua: set_complex] :
      ( aa_set_complex_o(aTP_Lamm_adv(Uu),Uua)
    <=> aa_set_complex_o(ord_le211207098394363844omplex(Uua),Uu) ) ).

% ATP.lambda_136
tff(fact_10314_ATP_Olambda__137,axiom,
    ! [Uu: set_nat,Uua: set_nat] :
      ( aa_set_nat_o(aTP_Lamm_adu(Uu),Uua)
    <=> aa_set_nat_o(ord_less_eq_set_nat(Uua),Uu) ) ).

% ATP.lambda_137
tff(fact_10315_ATP_Olambda__138,axiom,
    ! [Uu: set_int,Uua: set_int] :
      ( aa_set_int_o(aTP_Lamm_adx(Uu),Uua)
    <=> aa_set_int_o(ord_less_eq_set_int(Uua),Uu) ) ).

% ATP.lambda_138
tff(fact_10316_ATP_Olambda__139,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(ord_less_eq_nat,Uua),Uu) ) ).

% ATP.lambda_139
tff(fact_10317_ATP_Olambda__140,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_bbj,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Uua),Uu) ).

% ATP.lambda_140
tff(fact_10318_ATP_Olambda__141,axiom,
    ! [Uu: real,Uua: real] :
      ( aa_real_o(aa_real_fun_real_o(aTP_Lamm_dp,Uu),Uua)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Uua),Uu) ) ).

% ATP.lambda_141
tff(fact_10319_ATP_Olambda__142,axiom,
    ! [Uu: rat,Uua: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(aTP_Lamm_dr,Uu),Uua)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Uua),Uu) ) ).

% ATP.lambda_142
tff(fact_10320_ATP_Olambda__143,axiom,
    ! [Uu: num,Uua: num] :
      ( aa_num_o(aTP_Lamm_dt(Uu),Uua)
    <=> aa_num_o(ord_less_num(Uua),Uu) ) ).

% ATP.lambda_143
tff(fact_10321_ATP_Olambda__144,axiom,
    ! [Uu: nat,Uua: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_dv,Uu),Uua)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uua),Uu) ) ).

% ATP.lambda_144
tff(fact_10322_ATP_Olambda__145,axiom,
    ! [Uu: int,Uua: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_dx,Uu),Uua)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Uua),Uu) ) ).

% ATP.lambda_145
tff(fact_10323_ATP_Olambda__146,axiom,
    ! [Uu: real,Uua: real] : aa_real_real(aa_rea7340605660300375871l_real(aTP_Lamm_dz,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Uua),Uu) ).

% ATP.lambda_146
tff(fact_10324_ATP_Olambda__147,axiom,
    ! [Uu: rat,Uua: rat] : aa_rat_rat(aTP_Lamm_ea(Uu),Uua) = aa_rat_rat(times_times_rat(Uua),Uu) ).

% ATP.lambda_147
tff(fact_10325_ATP_Olambda__148,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_eb,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),Uu) ).

% ATP.lambda_148
tff(fact_10326_ATP_Olambda__149,axiom,
    ! [Uu: int,Uua: int] : aa_int_int(aa_int_fun_int_int(aTP_Lamm_ec,Uu),Uua) = aa_int_int(aa_int_fun_int_int(times_times_int,Uua),Uu) ).

% ATP.lambda_149
tff(fact_10327_ATP_Olambda__150,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_bbn,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uu) ).

% ATP.lambda_150
tff(fact_10328_ATP_Olambda__151,axiom,
    ! [Uu: nat,Uua: real] : aa_real_real(aTP_Lamm_bcp(Uu),Uua) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),Uu) ).

% ATP.lambda_151
tff(fact_10329_ATP_Olambda__152,axiom,
    ! [Uu: code_integer,Uua: code_integer] : aa_Cod1783974652533396728nteger(aTP_Lamm_bbm(Uu),Uua) = aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(Uua),Uu) ).

% ATP.lambda_152
tff(fact_10330_ATP_Olambda__153,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_bbq,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),Uu) ).

% ATP.lambda_153
tff(fact_10331_ATP_Olambda__154,axiom,
    ! [Uu: int,Uua: int] : aa_int_int(aa_int_fun_int_int(aTP_Lamm_bbl,Uu),Uua) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Uua),Uu) ).

% ATP.lambda_154
tff(fact_10332_ATP_Olambda__155,axiom,
    ! [Uu: list_VEBT_VEBTi,Uua: array_VEBT_VEBTi] : aa_arr8637723718858317150i_assn(aTP_Lamm_zq(Uu),Uua) = aa_lis2508312872377092907i_assn(snga_assn_VEBT_VEBTi(Uua),Uu) ).

% ATP.lambda_155
tff(fact_10333_ATP_Olambda__156,axiom,
    ! [Uu: code_integer,Uua: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(aTP_Lamm_wt,Uu),Uua)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Uua),Uu) ) ).

% ATP.lambda_156
tff(fact_10334_ATP_Olambda__157,axiom,
    ! [Uu: complex,Uua: complex] :
      ( aa_complex_o(aTP_Lamm_wp(Uu),Uua)
    <=> aa_complex_o(dvd_dvd_complex(Uua),Uu) ) ).

% ATP.lambda_157
tff(fact_10335_ATP_Olambda__158,axiom,
    ! [Uu: uint32,Uua: uint32] :
      ( aa_uint32_o(aa_uin5924812838015166427nt32_o(aTP_Lamm_wr,Uu),Uua)
    <=> aa_uint32_o(aa_uin5924812838015166427nt32_o(dvd_dvd_uint32,Uua),Uu) ) ).

% ATP.lambda_158
tff(fact_10336_ATP_Olambda__159,axiom,
    ! [Uu: word_N3645301735248828278l_num1,Uua: word_N3645301735248828278l_num1] :
      ( aa_wor4452366487969921764num1_o(aTP_Lamm_ws(Uu),Uua)
    <=> aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(Uua),Uu) ) ).

% ATP.lambda_159
tff(fact_10337_ATP_Olambda__160,axiom,
    ! [Uu: nat,Uua: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_wq,Uu),Uua)
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Uua),Uu) ) ).

% ATP.lambda_160
tff(fact_10338_ATP_Olambda__161,axiom,
    ! [Uu: int,Uua: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_wu,Uu),Uua)
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Uua),Uu) ) ).

% ATP.lambda_161
tff(fact_10339_ATP_Olambda__162,axiom,
    ! [Uu: vEBT_VEBT,Uua: real] : aa_rea2776067176874379751T_VEBT(aTP_Lamm_agi(Uu),Uua) = produc6931449550656315951T_VEBT(Uua,Uu) ).

% ATP.lambda_162
tff(fact_10340_ATP_Olambda__163,axiom,
    ! [Uu: vEBT_VEBT,Uua: $o] : aa_o_P4714545900465707367T_VEBT(aTP_Lamm_agj(Uu),(Uua)) = produc2982872950893828659T_VEBT((Uua),Uu) ).

% ATP.lambda_163
tff(fact_10341_ATP_Olambda__164,axiom,
    ! [Uu: vEBT_VEBT,Uua: nat] : aa_nat3316634782163766759T_VEBT(aTP_Lamm_agk(Uu),Uua) = produc599794634098209291T_VEBT(Uua,Uu) ).

% ATP.lambda_164
tff(fact_10342_ATP_Olambda__165,axiom,
    ! [Uu: vEBT_VEBT,Uua: int] : aa_int4520187544020889831T_VEBT(aTP_Lamm_agl(Uu),Uua) = produc3329399203697025711T_VEBT(Uua,Uu) ).

% ATP.lambda_165
tff(fact_10343_ATP_Olambda__166,axiom,
    ! [Uu: code_integer,Uua: code_integer] : aa_Cod6977894654539884864nteger(aTP_Lamm_agm(Uu),Uua) = aa_Cod6977894654539884864nteger(produc1086072967326762835nteger(Uua),Uu) ).

% ATP.lambda_166
tff(fact_10344_ATP_Olambda__167,axiom,
    ! [Uu: produc8923325533196201883nteger,Uua: fun_Pr5098669352780829690e_term] : aa_fun8425650785172306510nteger(aTP_Lamm_jw(Uu),Uua) = aa_Pro4291442876242880245nteger(produc565448451787809096nteger(Uua),Uu) ).

% ATP.lambda_167
tff(fact_10345_ATP_Olambda__168,axiom,
    ! [Uu: produc3658429121746597890et_nat,Uua: fun_Pr6460944699146269935_nat_o] : aa_fun179180238660531795et_nat(aTP_Lamm_jx(Uu),Uua) = aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(Uua),Uu) ).

% ATP.lambda_168
tff(fact_10346_ATP_Olambda__169,axiom,
    ! [Uu: product_prod_int_int,Uua: fun_Pr7763680713840462443e_term] : aa_fun9040010020517048364nt_int(aTP_Lamm_jz(Uu),Uua) = aa_Pro5748361610964203648nt_int(produc7320969046608631221nt_int(Uua),Uu) ).

% ATP.lambda_169
tff(fact_10347_ATP_Olambda__170,axiom,
    ! [Uu: product_prod_int_int,Uua: fun_in5730379987311621695e_term] : aa_fun3520829229142255404nt_int(aTP_Lamm_ka(Uu),Uua) = aa_Pro8800328608558162860nt_int(produc4729913460834754785nt_int(Uua),Uu) ).

% ATP.lambda_170
tff(fact_10348_ATP_Olambda__171,axiom,
    ! [Uu: produc8799533983703829792et_nat,Uua: fun_Pr6460944699146269935_nat_o] : aa_fun8682008811809929585et_nat(aTP_Lamm_jy(Uu),Uua) = aa_Pro1979504083478021276et_nat(produc7291855603632349880et_nat(Uua),Uu) ).

% ATP.lambda_171
tff(fact_10349_ATP_Olambda__172,axiom,
    ! [Uu: assn,Uua: assn] : aa_ass5585057816575862475n_assn(aTP_Lamm_kc(Uu),Uua) = aa_ass5585057816575862475n_assn(produc118845697133431529n_assn(Uua),Uu) ).

% ATP.lambda_172
tff(fact_10350_ATP_Olambda__173,axiom,
    ! [Uu: uint32,Uua: uint32] : aa_uin6954824371948101007uint32(aTP_Lamm_kd(Uu),Uua) = aa_uin6954824371948101007uint32(produc1400373151660368625uint32(Uua),Uu) ).

% ATP.lambda_173
tff(fact_10351_ATP_Olambda__174,axiom,
    ! [Uu: set_nat,Uua: heap_e7401611519738050253t_unit] : aa_hea7968205769688963091et_nat(aTP_Lamm_kb(Uu),Uua) = aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(Uua),Uu) ).

% ATP.lambda_174
tff(fact_10352_ATP_Olambda__175,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aTP_Lamm_ke,Uu),Uua) = aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Uua),Uu) ).

% ATP.lambda_175
tff(fact_10353_ATP_Olambda__176,axiom,
    ! [Uu: int,Uua: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_kf,Uu),Uua) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Uua),Uu) ).

% ATP.lambda_176
tff(fact_10354_ATP_Olambda__177,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_ayt,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uua),Uu) ).

% ATP.lambda_177
tff(fact_10355_ATP_Olambda__178,axiom,
    ! [Uu: set_VEBT_VEBT,Uua: vEBT_VEBT] :
      ( aa_VEBT_VEBT_o(aTP_Lamm_av(Uu),Uua)
    <=> aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Uua),Uu) ) ).

% ATP.lambda_178
tff(fact_10356_ATP_Olambda__179,axiom,
    ! [Uu: set_Pr958786334691620121nt_int,Uua: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aTP_Lamm_ba(Uu),Uua)
    <=> member5262025264175285858nt_int(Uua,Uu) ) ).

% ATP.lambda_179
tff(fact_10357_ATP_Olambda__180,axiom,
    ! [Uu: set_complex,Uua: complex] :
      ( aa_complex_o(aTP_Lamm_az(Uu),Uua)
    <=> aa_set_complex_o(member_complex(Uua),Uu) ) ).

% ATP.lambda_180
tff(fact_10358_ATP_Olambda__181,axiom,
    ! [Uu: set_real,Uua: real] :
      ( aa_real_o(aTP_Lamm_aw(Uu),Uua)
    <=> aa_set_real_o(member_real(Uua),Uu) ) ).

% ATP.lambda_181
tff(fact_10359_ATP_Olambda__182,axiom,
    ! [Uu: set_nat,Uua: nat] :
      ( aa_nat_o(aTP_Lamm_ax(Uu),Uua)
    <=> aa_set_nat_o(member_nat(Uua),Uu) ) ).

% ATP.lambda_182
tff(fact_10360_ATP_Olambda__183,axiom,
    ! [Uu: set_int,Uua: int] :
      ( aa_int_o(aTP_Lamm_ay(Uu),Uua)
    <=> aa_set_int_o(member_int(Uua),Uu) ) ).

% ATP.lambda_183
tff(fact_10361_ATP_Olambda__184,axiom,
    ! [Uu: vEBT_VEBT,Uua: vEBT_VEBT] :
      ( aa_VEBT_VEBT_o(aTP_Lamm_hh(Uu),Uua)
    <=> ( Uua = Uu ) ) ).

% ATP.lambda_184
tff(fact_10362_ATP_Olambda__185,axiom,
    ! [Uu: product_prod_int_int,Uua: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aa_Pro3952998155787061599_int_o(aTP_Lamm_hj,Uu),Uua)
    <=> ( Uua = Uu ) ) ).

% ATP.lambda_185
tff(fact_10363_ATP_Olambda__186,axiom,
    ! [Uu: complex,Uua: complex] :
      ( aa_complex_o(aTP_Lamm_hi(Uu),Uua)
    <=> ( Uua = Uu ) ) ).

% ATP.lambda_186
tff(fact_10364_ATP_Olambda__187,axiom,
    ! [Uu: real,Uua: real] :
      ( aa_real_o(aa_real_fun_real_o(aTP_Lamm_hm,Uu),Uua)
    <=> ( Uua = Uu ) ) ).

% ATP.lambda_187
tff(fact_10365_ATP_Olambda__188,axiom,
    ! [Uu: nat,Uua: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_hk,Uu),Uua)
    <=> ( Uua = Uu ) ) ).

% ATP.lambda_188
tff(fact_10366_ATP_Olambda__189,axiom,
    ! [Uu: int,Uua: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_hl,Uu),Uua)
    <=> ( Uua = Uu ) ) ).

% ATP.lambda_189
tff(fact_10367_ATP_Olambda__190,axiom,
    ! [Uu: fun_Code_integer_nat,Uua: code_integer] :
      ( aa_Code_integer_o(aTP_Lamm_afm(Uu),Uua)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_Code_integer_nat(Uu,Uua)) ) ).

% ATP.lambda_190
tff(fact_10368_ATP_Olambda__191,axiom,
    ! [Uu: fun_Pr5160692531401422620nt_nat,Uua: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aTP_Lamm_afe(Uu),Uua)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_Pro6801748904974264397nt_nat(Uu,Uua)) ) ).

% ATP.lambda_191
tff(fact_10369_ATP_Olambda__192,axiom,
    ! [Uu: fun_complex_nat,Uua: complex] :
      ( aa_complex_o(aTP_Lamm_afk(Uu),Uua)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_complex_nat(Uu,Uua)) ) ).

% ATP.lambda_192
tff(fact_10370_ATP_Olambda__193,axiom,
    ! [Uu: fun_nat_nat,Uua: nat] :
      ( aa_nat_o(aTP_Lamm_afg(Uu),Uua)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(Uu,Uua)) ) ).

% ATP.lambda_193
tff(fact_10371_ATP_Olambda__194,axiom,
    ! [Uu: fun_int_nat,Uua: int] :
      ( aa_int_o(aTP_Lamm_afi(Uu),Uua)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_int_nat(Uu,Uua)) ) ).

% ATP.lambda_194
tff(fact_10372_ATP_Olambda__195,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_auk(Uu),Uua) = aa_nat_real(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)),one_one_nat)) ).

% ATP.lambda_195
tff(fact_10373_ATP_Olambda__196,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_auj(Uu),Uua) = aa_nat_real(Uu,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)) ).

% ATP.lambda_196
tff(fact_10374_ATP_Olambda__197,axiom,
    ! [Uu: fun_fu1421233379579251956EBTi_o,Uua: fun_Pr4404027214751104635_VEBTi] :
      ( aa_fun2613385842899212575EBTi_o(aTP_Lamm_yr(Uu),Uua)
    <=> aa_fun2485637723310249309EBTi_o(Uu,produc3311654011758026858_VEBTi(produc2164094337957399884_VEBTi(Uua))) ) ).

% ATP.lambda_197
tff(fact_10375_ATP_Olambda__198,axiom,
    ! [Uu: fun_fu6711510597495717193_nat_o,Uua: fun_Pr633315466666964198on_nat] :
      ( aa_fun8910613404238449780_nat_o(aTP_Lamm_abu(Uu),Uua)
    <=> aa_fun5369152942363543986_nat_o(Uu,produc9121605208400662357on_nat(produc1757988346207259447on_nat(Uua))) ) ).

% ATP.lambda_198
tff(fact_10376_ATP_Olambda__199,axiom,
    ! [Uu: fun_fu2707964056798801139ap_o_o,Uua: fun_Pr940977554174145406Heap_o] :
      ( aa_fun3656762788152972374ap_o_o(aTP_Lamm_aby(Uu),Uua)
    <=> aa_fun339294776471787884ap_o_o(Uu,produc8906325494159015119Heap_o(produc8381543706267210711Heap_o(Uua))) ) ).

% ATP.lambda_199
tff(fact_10377_ATP_Olambda__200,axiom,
    ! [Uu: fun_real_real,Uua: real] : aa_real_real(aTP_Lamm_bcq(Uu),Uua) = aa_real_real(Uu,aa_real_real(uminus_uminus_real,Uua)) ).

% ATP.lambda_200
tff(fact_10378_ATP_Olambda__201,axiom,
    ! [Uu: fun_na1476630412504255114at_nat,Uua: product_prod_nat_nat] : aa_Pro262675715343573244at_nat(aTP_Lamm_lk(Uu),Uua) = aa_nat7603221163087113075at_nat(Uu,aa_Pro4243173770044397973at_nat(product_snd_nat_nat,Uua)) ).

% ATP.lambda_201
tff(fact_10379_ATP_Olambda__202,axiom,
    ! [Uu: fun_nat_o,Uua: product_prod_nat_nat] :
      ( aa_Pro6150637604445964947_nat_o(aTP_Lamm_lm(Uu),Uua)
    <=> aa_nat_o(Uu,aa_Pro4243173770044397973at_nat(product_snd_nat_nat,Uua)) ) ).

% ATP.lambda_202
tff(fact_10380_ATP_Olambda__203,axiom,
    ! [Uu: fun_in4211310926729683942nt_int,Uua: product_prod_int_int] : aa_Pro6970504127699319548nt_int(aTP_Lamm_lo(Uu),Uua) = aa_int7462565939943104719nt_int(Uu,aa_Pro6799258434465214121nt_int(product_snd_int_int,Uua)) ).

% ATP.lambda_203
tff(fact_10381_ATP_Olambda__204,axiom,
    ! [Uu: fun_int_o,Uua: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aTP_Lamm_lq(Uu),Uua)
    <=> aa_int_o(Uu,aa_Pro6799258434465214121nt_int(product_snd_int_int,Uua)) ) ).

% ATP.lambda_204
tff(fact_10382_ATP_Olambda__205,axiom,
    ! [Uu: fun_int_int,Uua: product_prod_int_int] : aa_Pro6799258434465214121nt_int(aTP_Lamm_ls(Uu),Uua) = aa_int_int(Uu,aa_Pro6799258434465214121nt_int(product_snd_int_int,Uua)) ).

% ATP.lambda_205
tff(fact_10383_ATP_Olambda__206,axiom,
    ! [Uu: fun_na1476630412504255114at_nat,Uua: product_prod_nat_nat] : aa_Pro262675715343573244at_nat(aTP_Lamm_kv(Uu),Uua) = aa_nat7603221163087113075at_nat(Uu,aa_Pro4243173770044397973at_nat(product_fst_nat_nat,Uua)) ).

% ATP.lambda_206
tff(fact_10384_ATP_Olambda__207,axiom,
    ! [Uu: fun_nat_o,Uua: product_prod_nat_nat] :
      ( aa_Pro6150637604445964947_nat_o(aTP_Lamm_kx(Uu),Uua)
    <=> aa_nat_o(Uu,aa_Pro4243173770044397973at_nat(product_fst_nat_nat,Uua)) ) ).

% ATP.lambda_207
tff(fact_10385_ATP_Olambda__208,axiom,
    ! [Uu: fun_in4211310926729683942nt_int,Uua: product_prod_int_int] : aa_Pro6970504127699319548nt_int(aTP_Lamm_kz(Uu),Uua) = aa_int7462565939943104719nt_int(Uu,aa_Pro6799258434465214121nt_int(product_fst_int_int,Uua)) ).

% ATP.lambda_208
tff(fact_10386_ATP_Olambda__209,axiom,
    ! [Uu: fun_int_o,Uua: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aTP_Lamm_lb(Uu),Uua)
    <=> aa_int_o(Uu,aa_Pro6799258434465214121nt_int(product_fst_int_int,Uua)) ) ).

% ATP.lambda_209
tff(fact_10387_ATP_Olambda__210,axiom,
    ! [Uu: fun_int_int,Uua: product_prod_int_int] : aa_Pro6799258434465214121nt_int(aTP_Lamm_ld(Uu),Uua) = aa_int_int(Uu,aa_Pro6799258434465214121nt_int(product_fst_int_int,Uua)) ).

% ATP.lambda_210
tff(fact_10388_ATP_Olambda__211,axiom,
    ! [Uu: fun_fu1178290271985267629EBTi_o,Uua: fun_Pr4171499318505651870_VEBTi] :
      ( aa_fun4096973365178274620EBTi_o(aTP_Lamm_abr(Uu),Uua)
    <=> aa_fun6333388201673717798EBTi_o(Uu,produc5159149307777246319_VEBTi(Uua)) ) ).

% ATP.lambda_211
tff(fact_10389_ATP_Olambda__212,axiom,
    ! [Uu: fun_fu4750890421531948930_nat_o,Uua: fun_Pr342922516890243977on_nat] :
      ( aa_fun7247719965521315857_nat_o(aTP_Lamm_acb(Uu),Uua)
    <=> aa_fun7244059194882734075_nat_o(Uu,produc1489253303066280154on_nat(Uua)) ) ).

% ATP.lambda_212
tff(fact_10390_ATP_Olambda__213,axiom,
    ! [Uu: fun_fu8886734894937297424ap_o_o,Uua: fun_Pr238758252084143515Heap_o] :
      ( aa_fun1852402840713402361ap_o_o(aTP_Lamm_acf(Uu),Uua)
    <=> aa_fun4656092467790859897ap_o_o(Uu,produc5685940877448195828Heap_o(Uua)) ) ).

% ATP.lambda_213
tff(fact_10391_ATP_Olambda__214,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_anv(Uu),Uua) = aa_nat_real(Uu,aa_nat_nat(suc,Uua)) ).

% ATP.lambda_214
tff(fact_10392_ATP_Olambda__215,axiom,
    ! [Uu: fun_nat_o,Uua: nat] :
      ( aa_nat_o(aTP_Lamm_bbu(Uu),Uua)
    <=> aa_nat_o(Uu,aa_nat_nat(suc,Uua)) ) ).

% ATP.lambda_215
tff(fact_10393_ATP_Olambda__216,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_aop(Uu),Uua) = aa_nat_rat(Uu,aa_nat_nat(suc,Uua)) ).

% ATP.lambda_216
tff(fact_10394_ATP_Olambda__217,axiom,
    ! [Uu: fun_nat_nat,Uua: nat] : aa_nat_nat(aTP_Lamm_anu(Uu),Uua) = aa_nat_nat(Uu,aa_nat_nat(suc,Uua)) ).

% ATP.lambda_217
tff(fact_10395_ATP_Olambda__218,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_aoq(Uu),Uua) = aa_nat_int(Uu,aa_nat_nat(suc,Uua)) ).

% ATP.lambda_218
tff(fact_10396_ATP_Olambda__219,axiom,
    ! [Uu: nat,Uua: num] : aa_num_option_num(aTP_Lamm_baj(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_219
tff(fact_10397_ATP_Olambda__220,axiom,
    ! [Uu: num,Uua: nat] : aa_nat_option_num(aTP_Lamm_baa(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(Uua),Uu))) ).

% ATP.lambda_220
tff(fact_10398_ATP_Olambda__221,axiom,
    ! [Uu: fun_li5930421888187426471o_assn,Uua: $o] : aa_o_assn(aTP_Lamm_gu(Uu),(Uua)) = ex_ass463751140784270563_VEBTi(aa_o_f6375964871679992646i_assn(aTP_Lamm_gt(Uu),(Uua))) ).

% ATP.lambda_221
tff(fact_10399_ATP_Olambda__222,axiom,
    ! [Uu: fun_li8825114373765265308i_assn,Uua: vEBT_VEBTi] : aa_VEBT_VEBTi_assn(aTP_Lamm_gs(Uu),Uua) = ex_ass463751140784270563_VEBTi(aa_VEB3739350836417479773i_assn(aTP_Lamm_gr(Uu),Uua)) ).

% ATP.lambda_222
tff(fact_10400_ATP_Olambda__223,axiom,
    ! [Uu: fun_li7023630697154767927t_assn,Uua: nat] : aa_nat_assn(aTP_Lamm_gq(Uu),Uua) = ex_ass463751140784270563_VEBTi(aa_nat6700791848136694008i_assn(aTP_Lamm_gp(Uu),Uua)) ).

% ATP.lambda_223
tff(fact_10401_ATP_Olambda__224,axiom,
    ! [Uu: fun_li3309467822378783623t_assn,Uua: option_nat] : aa_option_nat_assn(aTP_Lamm_go(Uu),Uua) = ex_ass463751140784270563_VEBTi(aa_opt2780285371362742984i_assn(aTP_Lamm_gn(Uu),Uua)) ).

% ATP.lambda_224
tff(fact_10402_ATP_Olambda__225,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_complex(aTP_Lamm_azr(Uu),Uua) = aa_real_complex(cis,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),pi)),aa_nat_real(semiri5074537144036343181t_real,Uua))),aa_nat_real(semiri5074537144036343181t_real,Uu))) ).

% ATP.lambda_225
tff(fact_10403_ATP_Olambda__226,axiom,
    ! [Uu: nat,Uua: option_nat] : aa_opt9163293676379396302on_nat(aTP_Lamm_a(Uu),Uua) = heap_T3487192422709364219on_nat(aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,aa_nat_option_nat(some_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,x12),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_option_nat(some_nat,Uu))),Uua)) ).

% ATP.lambda_226
tff(fact_10404_ATP_Olambda__227,axiom,
    ! [Uu: option_nat,Uua: option_nat] : aa_opt9163293676379396302on_nat(aTP_Lamm_aa(Uu),Uua) = heap_T3487192422709364219on_nat(aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,aa_nat_option_nat(some_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,x12),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),Uu)),Uua)) ).

% ATP.lambda_227
tff(fact_10405_ATP_Olambda__228,axiom,
    ! [Uu: fun_he7283999071544186133_VEBTi,Uua: heap_e7401611519738050253t_unit] : aa_hea6347871425501556311it_nat(aTP_Lamm_aci(Uu),Uua) = some_P696572436114257607it_nat(produc8770982656060504474it_nat(aa_hea7585900348394676422_VEBTi(Uu,Uua),produc584006145561248582it_nat(Uua,one_one_nat))) ).

% ATP.lambda_228
tff(fact_10406_ATP_Olambda__229,axiom,
    ! [Uu: fun_he4876066172508077696on_nat,Uua: heap_e7401611519738050253t_unit] : aa_hea5089399211691266754it_nat(aTP_Lamm_acj(Uu),Uua) = some_P2407035485129114418it_nat(produc61566615109097733it_nat(aa_hea6610145743731876273on_nat(Uu,Uua),produc584006145561248582it_nat(Uua,one_one_nat))) ).

% ATP.lambda_229
tff(fact_10407_ATP_Olambda__230,axiom,
    ! [Uu: fun_he8413527710729349838unit_o,Uua: heap_e7401611519738050253t_unit] : aa_hea8154171661607227764it_nat(aTP_Lamm_ack(Uu),Uua) = some_P468703482102919278it_nat(produc6655106138504972685it_nat(aa_hea7994211679605476167unit_o(Uu,Uua),produc584006145561248582it_nat(Uua,one_one_nat))) ).

% ATP.lambda_230
tff(fact_10408_ATP_Olambda__231,axiom,
    ! [Uu: nat,Uua: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_azw,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_231
tff(fact_10409_ATP_Olambda__232,axiom,
    ! [Uu: code_integer,Uua: nat] : aa_nat6383147911832638789nteger(aTP_Lamm_aup(Uu),Uua) = aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(Uu),aa_nat_Code_integer(semiri4939895301339042750nteger,Uua))) ).

% ATP.lambda_232
tff(fact_10410_ATP_Olambda__233,axiom,
    ! [Uu: word_N3645301735248828278l_num1,Uua: nat] : aa_nat5018596641434580727l_num1(aTP_Lamm_aul(Uu),Uua) = times_7065122842183080059l_num1(plus_p361126936061061375l_num1(Uu,aa_nat5599461682457136004l_num1(semiri8819519690708144855l_num1,Uua))) ).

% ATP.lambda_233
tff(fact_10411_ATP_Olambda__234,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_fun_real_real(aTP_Lamm_aun(Uu),Uua) = aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Uu),aa_nat_real(semiri5074537144036343181t_real,Uua))) ).

% ATP.lambda_234
tff(fact_10412_ATP_Olambda__235,axiom,
    ! [Uu: rat,Uua: nat] : aa_nat_fun_rat_rat(aTP_Lamm_aum(Uu),Uua) = times_times_rat(aa_rat_rat(plus_plus_rat(Uu),aa_nat_rat(semiri681578069525770553at_rat,Uua))) ).

% ATP.lambda_235
tff(fact_10413_ATP_Olambda__236,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_fun_nat_nat(aTP_Lamm_auq(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_236
tff(fact_10414_ATP_Olambda__237,axiom,
    ! [Uu: int,Uua: nat] : aa_nat_fun_int_int(aTP_Lamm_auo(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_237
tff(fact_10415_ATP_Olambda__238,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_bbt,Uu),Uua) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,Uu),Uua)) ).

% ATP.lambda_238
tff(fact_10416_ATP_Olambda__239,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_bbs,Uu),Uua) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,Uua),Uu)) ).

% ATP.lambda_239
tff(fact_10417_ATP_Olambda__240,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_ban,Uu),Uua) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Uu),Uua)) ).

% ATP.lambda_240
tff(fact_10418_ATP_Olambda__241,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_bam,Uu),Uua) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Uua),Uu)) ).

% ATP.lambda_241
tff(fact_10419_ATP_Olambda__242,axiom,
    ! [Uu: fun_complex_complex,Uua: complex] : aa_complex_real(aTP_Lamm_akm(Uu),Uua) = aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(Uu,Uua)) ).

% ATP.lambda_242
tff(fact_10420_ATP_Olambda__243,axiom,
    ! [Uu: fun_nat_complex,Uua: nat] : aa_nat_real(aTP_Lamm_akl(Uu),Uua) = aa_complex_real(real_V1022390504157884413omplex,aa_nat_complex(Uu,Uua)) ).

% ATP.lambda_243
tff(fact_10421_ATP_Olambda__244,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_akn(Uu),Uua) = aa_real_real(real_V7735802525324610683m_real,aa_nat_real(Uu,Uua)) ).

% ATP.lambda_244
tff(fact_10422_ATP_Olambda__245,axiom,
    ! [Uu: fun_VEBT_VEBT_nat,Uua: vEBT_VEBT] : aa_VEBT_VEBT_real(aTP_Lamm_jn(Uu),Uua) = aa_nat_real(semiri5074537144036343181t_real,aa_VEBT_VEBT_nat(Uu,Uua)) ).

% ATP.lambda_245
tff(fact_10423_ATP_Olambda__246,axiom,
    ! [Uu: fun_complex_nat,Uua: complex] : aa_complex_complex(aTP_Lamm_air(Uu),Uua) = aa_nat_complex(semiri8010041392384452111omplex,aa_complex_nat(Uu,Uua)) ).

% ATP.lambda_246
tff(fact_10424_ATP_Olambda__247,axiom,
    ! [Uu: fun_nat_nat,Uua: nat] : aa_nat_Code_integer(aTP_Lamm_aiu(Uu),Uua) = aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(Uu,Uua)) ).

% ATP.lambda_247
tff(fact_10425_ATP_Olambda__248,axiom,
    ! [Uu: fun_nat_nat,Uua: nat] : aa_nat_real(aTP_Lamm_jo(Uu),Uua) = aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(Uu,Uua)) ).

% ATP.lambda_248
tff(fact_10426_ATP_Olambda__249,axiom,
    ! [Uu: fun_nat_nat,Uua: nat] : aa_nat_nat(aTP_Lamm_aiv(Uu),Uua) = aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(Uu,Uua)) ).

% ATP.lambda_249
tff(fact_10427_ATP_Olambda__250,axiom,
    ! [Uu: fun_nat_nat,Uua: nat] : aa_nat_int(aTP_Lamm_ait(Uu),Uua) = aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(Uu,Uua)) ).

% ATP.lambda_250
tff(fact_10428_ATP_Olambda__251,axiom,
    ! [Uu: fun_int_nat,Uua: int] : aa_int_int(aTP_Lamm_ais(Uu),Uua) = aa_nat_int(semiri1314217659103216013at_int,aa_int_nat(Uu,Uua)) ).

% ATP.lambda_251
tff(fact_10429_ATP_Olambda__252,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_complex(aTP_Lamm_azp(Uu),Uua) = aa_real_complex(real_V4546457046886955230omplex,aa_nat_real(Uu,Uua)) ).

% ATP.lambda_252
tff(fact_10430_ATP_Olambda__253,axiom,
    ! [Uu: fun_complex_int,Uua: complex] : aa_complex_complex(aTP_Lamm_aiw(Uu),Uua) = aa_int_complex(ring_17405671764205052669omplex,aa_complex_int(Uu,Uua)) ).

% ATP.lambda_253
tff(fact_10431_ATP_Olambda__254,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_real(aTP_Lamm_aix(Uu),Uua) = aa_int_real(ring_1_of_int_real,aa_nat_int(Uu,Uua)) ).

% ATP.lambda_254
tff(fact_10432_ATP_Olambda__255,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_rat(aTP_Lamm_avc(Uu),Uua) = aa_int_rat(ring_1_of_int_rat,aa_nat_int(Uu,Uua)) ).

% ATP.lambda_255
tff(fact_10433_ATP_Olambda__256,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_avd(Uu),Uua) = aa_int_int(ring_1_of_int_int,aa_nat_int(Uu,Uua)) ).

% ATP.lambda_256
tff(fact_10434_ATP_Olambda__257,axiom,
    ! [Uu: fun_int_int,Uua: int] : aa_int_real(aTP_Lamm_aiy(Uu),Uua) = aa_int_real(ring_1_of_int_real,aa_int_int(Uu,Uua)) ).

% ATP.lambda_257
tff(fact_10435_ATP_Olambda__258,axiom,
    ! [Uu: fun_int_int,Uua: int] : aa_int_rat(aTP_Lamm_aiz(Uu),Uua) = aa_int_rat(ring_1_of_int_rat,aa_int_int(Uu,Uua)) ).

% ATP.lambda_258
tff(fact_10436_ATP_Olambda__259,axiom,
    ! [Uu: fun_int_int,Uua: int] : aa_int_int(aTP_Lamm_aja(Uu),Uua) = aa_int_int(ring_1_of_int_int,aa_int_int(Uu,Uua)) ).

% ATP.lambda_259
tff(fact_10437_ATP_Olambda__260,axiom,
    ! [Uu: fun_nat_uint32,Uua: nat] : aa_nat4578243230388868707uint32(aTP_Lamm_aou(Uu),Uua) = plus_plus_uint32(aa_nat_uint32(Uu,Uua)) ).

% ATP.lambda_260
tff(fact_10438_ATP_Olambda__261,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_fun_real_real(aTP_Lamm_aoy(Uu),Uua) = aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(Uu,Uua)) ).

% ATP.lambda_261
tff(fact_10439_ATP_Olambda__262,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : aa_nat_fun_rat_rat(aTP_Lamm_aov(Uu),Uua) = plus_plus_rat(aa_nat_rat(Uu,Uua)) ).

% ATP.lambda_262
tff(fact_10440_ATP_Olambda__263,axiom,
    ! [Uu: fun_nat_nat,Uua: nat] : aa_nat_fun_nat_nat(aTP_Lamm_aox(Uu),Uua) = aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(Uu,Uua)) ).

% ATP.lambda_263
tff(fact_10441_ATP_Olambda__264,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_fun_int_int(aTP_Lamm_aow(Uu),Uua) = aa_int_fun_int_int(plus_plus_int,aa_nat_int(Uu,Uua)) ).

% ATP.lambda_264
tff(fact_10442_ATP_Olambda__265,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_azn(Uu),Uua) = aa_real_real(abs_abs_real,aa_nat_real(Uu,Uua)) ).

% ATP.lambda_265
tff(fact_10443_ATP_Olambda__266,axiom,
    ! [Uu: fun_VE3708348504534315036i_assn,Uua: vEBT_VEBTi] : aa_VEBT_VEBTi_assn(aTP_Lamm_cb(Uu),Uua) = ex_ass463751140784270563_VEBTi(aa_VEB3739350836417479773i_assn(Uu,Uua)) ).

% ATP.lambda_266
tff(fact_10444_ATP_Olambda__267,axiom,
    ! [Uu: fun_Pr6039563626565002634i_assn,Uua: product_unit] : aa_Product_unit_assn(aTP_Lamm_cd(Uu),Uua) = ex_ass463751140784270563_VEBTi(aa_Pro7700840739073064379i_assn(Uu,Uua)) ).

% ATP.lambda_267
tff(fact_10445_ATP_Olambda__268,axiom,
    ! [Uu: fun_op8552747401449489031i_assn,Uua: option_nat] : aa_option_nat_assn(aTP_Lamm_bv(Uu),Uua) = ex_ass463751140784270563_VEBTi(aa_opt2780285371362742984i_assn(Uu,Uua)) ).

% ATP.lambda_268
tff(fact_10446_ATP_Olambda__269,axiom,
    ! [Uu: fun_o_4874018796564477717i_assn,Uua: $o] : aa_o_assn(aTP_Lamm_bz(Uu),(Uua)) = ex_ass463751140784270563_VEBTi(aa_o_f6375964871679992646i_assn(Uu,(Uua))) ).

% ATP.lambda_269
tff(fact_10447_ATP_Olambda__270,axiom,
    ! [Uu: fun_na8005944528675429559i_assn,Uua: nat] : aa_nat_assn(aTP_Lamm_bx(Uu),Uua) = ex_ass463751140784270563_VEBTi(aa_nat6700791848136694008i_assn(Uu,Uua)) ).

% ATP.lambda_270
tff(fact_10448_ATP_Olambda__271,axiom,
    ! [Uu: fun_Code_integer_o,Uua: code_integer] :
      ( aa_Code_integer_o(aTP_Lamm_aeb(Uu),Uua)
    <=> ~ aa_Code_integer_o(Uu,Uua) ) ).

% ATP.lambda_271
tff(fact_10449_ATP_Olambda__272,axiom,
    ! [Uu: fun_Pr3238374608701176162_int_o,Uua: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aTP_Lamm_ady(Uu),Uua)
    <=> ~ aa_Pro4592068669816069595_int_o(Uu,Uua) ) ).

% ATP.lambda_272
tff(fact_10450_ATP_Olambda__273,axiom,
    ! [Uu: fun_complex_o,Uua: complex] :
      ( aa_complex_o(aTP_Lamm_aea(Uu),Uua)
    <=> ~ aa_complex_o(Uu,Uua) ) ).

% ATP.lambda_273
tff(fact_10451_ATP_Olambda__274,axiom,
    ! [Uu: fun_char_o,Uua: char] :
      ( aa_char_o(aTP_Lamm_aee(Uu),Uua)
    <=> ~ aa_char_o(Uu,Uua) ) ).

% ATP.lambda_274
tff(fact_10452_ATP_Olambda__275,axiom,
    ! [Uu: fun_real_o,Uua: real] :
      ( aa_real_o(aTP_Lamm_aec(Uu),Uua)
    <=> ~ aa_real_o(Uu,Uua) ) ).

% ATP.lambda_275
tff(fact_10453_ATP_Olambda__276,axiom,
    ! [Uu: fun_nat_o,Uua: nat] :
      ( aa_nat_o(aTP_Lamm_aed(Uu),Uua)
    <=> ~ aa_nat_o(Uu,Uua) ) ).

% ATP.lambda_276
tff(fact_10454_ATP_Olambda__277,axiom,
    ! [Uu: fun_int_o,Uua: int] :
      ( aa_int_o(aTP_Lamm_adz(Uu),Uua)
    <=> ~ aa_int_o(Uu,Uua) ) ).

% ATP.lambda_277
tff(fact_10455_ATP_Olambda__278,axiom,
    ! [Uu: fun_nat_o,Uua: nat] :
      ( aa_nat_o(aTP_Lamm_bcc(Uu),Uua)
    <=> ! [M7: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Uua),M7)
         => ( aa_nat_o(Uu,M7)
          <=> aa_nat_o(Uu,Uua) ) ) ) ).

% ATP.lambda_278
tff(fact_10456_ATP_Olambda__279,axiom,
    ! [Uu: fun_nat_o,Uua: nat] :
      ( aa_nat_o(aTP_Lamm_bce(Uu),Uua)
    <=> ! [N9: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Uua),N9)
         => aa_nat_o(Uu,N9) ) ) ).

% ATP.lambda_279
tff(fact_10457_ATP_Olambda__280,axiom,
    ! [Uu: fun_nat_o,Uua: nat] :
      ( aa_nat_o(aTP_Lamm_bcd(Uu),Uua)
    <=> ! [N9: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Uua),N9)
         => ~ aa_nat_o(Uu,N9) ) ) ).

% ATP.lambda_280
tff(fact_10458_ATP_Olambda__281,axiom,
    ! [Uu: fun_Pr342922516890243977on_nat,Uua: fun_Pr342922516890243977on_nat] :
      ( aa_fun7247719965521315857_nat_o(aa_fun6940291128091933217_nat_o(aTP_Lamm_aud,Uu),Uua)
    <=> ! [X: produc3881548065746020326Ti_nat] : refine7594492741263601813on_nat(aa_Pro2606730266749126658on_nat(Uu,X),aa_Pro2606730266749126658on_nat(Uua,X)) ) ).

% ATP.lambda_281
tff(fact_10459_ATP_Olambda__282,axiom,
    ! [Uu: fun_Pr238758252084143515Heap_o,Uua: fun_Pr238758252084143515Heap_o] :
      ( aa_fun1852402840713402361ap_o_o(aa_fun2387398806603316575ap_o_o(aTP_Lamm_aue,Uu),Uua)
    <=> ! [X: produc3881548065746020326Ti_nat] : aa_Hea4522166450600650424ap_o_o(refine_Imp_refines_o(aa_Pro3916655985231737292Heap_o(Uu,X)),aa_Pro3916655985231737292Heap_o(Uua,X)) ) ).

% ATP.lambda_282
tff(fact_10460_ATP_Olambda__283,axiom,
    ! [Uu: fun_Pr4404027214751104635_VEBTi,Uua: fun_Pr4404027214751104635_VEBTi] :
      ( aa_fun2613385842899212575EBTi_o(aa_fun3634491704933443261EBTi_o(aTP_Lamm_auf,Uu),Uua)
    <=> ! [X: produc3960855945107176009Ti_nat] : refine5565527176597971370_VEBTi(aa_Pro721367001817565044_VEBTi(Uu,X),aa_Pro721367001817565044_VEBTi(Uua,X)) ) ).

% ATP.lambda_283
tff(fact_10461_ATP_Olambda__284,axiom,
    ! [Uu: fun_Pr633315466666964198on_nat,Uua: fun_Pr633315466666964198on_nat] :
      ( aa_fun8910613404238449780_nat_o(aa_fun2070105859800901863_nat_o(aTP_Lamm_aug,Uu),Uua)
    <=> ! [X: produc3960855945107176009Ti_nat] : refine7594492741263601813on_nat(aa_Pro2727408364141080927on_nat(Uu,X),aa_Pro2727408364141080927on_nat(Uua,X)) ) ).

% ATP.lambda_284
tff(fact_10462_ATP_Olambda__285,axiom,
    ! [Uu: fun_Pr940977554174145406Heap_o,Uua: fun_Pr940977554174145406Heap_o] :
      ( aa_fun3656762788152972374ap_o_o(aa_fun4060910825685108895ap_o_o(aTP_Lamm_auh,Uu),Uua)
    <=> ! [X: produc3960855945107176009Ti_nat] : aa_Hea4522166450600650424ap_o_o(refine_Imp_refines_o(aa_Pro6173831900156065711Heap_o(Uu,X)),aa_Pro6173831900156065711Heap_o(Uua,X)) ) ).

% ATP.lambda_285
tff(fact_10463_ATP_Olambda__286,axiom,
    ! [Uu: fun_Pr8246910697231053897at_nat,Uua: fun_Pr342922516890243977on_nat] :
      ( aa_fun7247719965521315857_nat_o(aTP_Lamm_aty(Uu),Uua)
    <=> ! [X: produc3881548065746020326Ti_nat] : aa_nat_o(time_T8353473612707095248on_nat(aa_Pro2606730266749126658on_nat(Uua,X)),aa_Pro3982081906321706362at_nat(Uu,X)) ) ).

% ATP.lambda_286
tff(fact_10464_ATP_Olambda__287,axiom,
    ! [Uu: fun_Pr8246910697231053897at_nat,Uua: fun_Pr238758252084143515Heap_o] :
      ( aa_fun1852402840713402361ap_o_o(aTP_Lamm_atz(Uu),Uua)
    <=> ! [X: produc3881548065746020326Ti_nat] : aa_nat_o(time_TBOUND_o(aa_Pro3916655985231737292Heap_o(Uua,X)),aa_Pro3982081906321706362at_nat(Uu,X)) ) ).

% ATP.lambda_287
tff(fact_10465_ATP_Olambda__288,axiom,
    ! [Uu: fun_Pr3615232783394198956at_nat,Uua: fun_Pr4404027214751104635_VEBTi] :
      ( aa_fun2613385842899212575EBTi_o(aTP_Lamm_aua(Uu),Uua)
    <=> ! [X: produc3960855945107176009Ti_nat] : aa_nat_o(time_T5737551269749752165_VEBTi(aa_Pro721367001817565044_VEBTi(Uua,X)),aa_Pro3471882085667088093at_nat(Uu,X)) ) ).

% ATP.lambda_288
tff(fact_10466_ATP_Olambda__289,axiom,
    ! [Uu: fun_Pr3615232783394198956at_nat,Uua: fun_Pr633315466666964198on_nat] :
      ( aa_fun8910613404238449780_nat_o(aTP_Lamm_aub(Uu),Uua)
    <=> ! [X: produc3960855945107176009Ti_nat] : aa_nat_o(time_T8353473612707095248on_nat(aa_Pro2727408364141080927on_nat(Uua,X)),aa_Pro3471882085667088093at_nat(Uu,X)) ) ).

% ATP.lambda_289
tff(fact_10467_ATP_Olambda__290,axiom,
    ! [Uu: fun_Pr3615232783394198956at_nat,Uua: fun_Pr940977554174145406Heap_o] :
      ( aa_fun3656762788152972374ap_o_o(aTP_Lamm_auc(Uu),Uua)
    <=> ! [X: produc3960855945107176009Ti_nat] : aa_nat_o(time_TBOUND_o(aa_Pro6173831900156065711Heap_o(Uua,X)),aa_Pro3471882085667088093at_nat(Uu,X)) ) ).

% ATP.lambda_290
tff(fact_10468_ATP_Olambda__291,axiom,
    ! [Uu: fun_VE4436863797057549263at_nat,Uua: fun_Pr342922516890243977on_nat] :
      ( aa_fun7247719965521315857_nat_o(aTP_Lamm_aur(Uu),Uua)
    <=> ! [X: vEBT_VEBTi,Xa3: nat] : aa_nat_o(time_T8353473612707095248on_nat(aa_nat947733127081326238on_nat(aa_VEB3566417204837707778on_nat(produc1489253303066280154on_nat(Uua),X),Xa3)),aa_nat_nat(aa_VEB205140289345309584at_nat(Uu,X),Xa3)) ) ).

% ATP.lambda_291
tff(fact_10469_ATP_Olambda__292,axiom,
    ! [Uu: fun_VE4436863797057549263at_nat,Uua: fun_Pr238758252084143515Heap_o] :
      ( aa_fun1852402840713402361ap_o_o(aTP_Lamm_aus(Uu),Uua)
    <=> ! [X: vEBT_VEBTi,Xa3: nat] : aa_nat_o(time_TBOUND_o(aa_nat2910048432523430448Heap_o(aa_VEB1455453220190609250Heap_o(produc5685940877448195828Heap_o(Uua),X),Xa3)),aa_nat_nat(aa_VEB205140289345309584at_nat(Uu,X),Xa3)) ) ).

% ATP.lambda_292
tff(fact_10470_ATP_Olambda__293,axiom,
    ! [Uu: fun_Pr2564325226415181874at_nat,Uua: fun_Pr4404027214751104635_VEBTi] :
      ( aa_fun2613385842899212575EBTi_o(aTP_Lamm_aut(Uu),Uua)
    <=> ! [X: produc3625547720036274456_VEBTi,Xa3: nat] : aa_nat_o(time_T5737551269749752165_VEBTi(aa_nat5408703056993926195_VEBTi(aa_Pro284253941917238388_VEBTi(produc2164094337957399884_VEBTi(Uua),X),Xa3)),aa_nat_nat(aa_Pro3057893768037969907at_nat(Uu,X),Xa3)) ) ).

% ATP.lambda_293
tff(fact_10471_ATP_Olambda__294,axiom,
    ! [Uu: fun_Pr2564325226415181874at_nat,Uua: fun_Pr633315466666964198on_nat] :
      ( aa_fun8910613404238449780_nat_o(aTP_Lamm_auu(Uu),Uua)
    <=> ! [X: produc3625547720036274456_VEBTi,Xa3: nat] : aa_nat_o(time_T8353473612707095248on_nat(aa_nat947733127081326238on_nat(aa_Pro1290670013494283871on_nat(produc1757988346207259447on_nat(Uua),X),Xa3)),aa_nat_nat(aa_Pro3057893768037969907at_nat(Uu,X),Xa3)) ) ).

% ATP.lambda_294
tff(fact_10472_ATP_Olambda__295,axiom,
    ! [Uu: fun_Pr2564325226415181874at_nat,Uua: fun_Pr940977554174145406Heap_o] :
      ( aa_fun3656762788152972374ap_o_o(aTP_Lamm_auv(Uu),Uua)
    <=> ! [X: produc3625547720036274456_VEBTi,Xa3: nat] : aa_nat_o(time_TBOUND_o(aa_nat2910048432523430448Heap_o(aa_Pro259339148018954309Heap_o(produc8381543706267210711Heap_o(Uua),X),Xa3)),aa_nat_nat(aa_Pro3057893768037969907at_nat(Uu,X),Xa3)) ) ).

% ATP.lambda_295
tff(fact_10473_ATP_Olambda__296,axiom,
    ! [Uu: fun_VE7674639578305544598at_nat,Uua: fun_Pr4404027214751104635_VEBTi] :
      ( aa_fun2613385842899212575EBTi_o(aTP_Lamm_arr(Uu),Uua)
    <=> ! [X: vEBT_VEBT,Xa3: vEBT_VEBTi,Xb4: nat] : aa_nat_o(time_T5737551269749752165_VEBTi(aa_nat5408703056993926195_VEBTi(aa_VEB4833742232818206615_VEBTi(aa_VEB6292075424854053302_VEBTi(produc3311654011758026858_VEBTi(produc2164094337957399884_VEBTi(Uua)),X),Xa3),Xb4)),aa_nat_nat(aa_VEB205140289345309584at_nat(aa_VEB5308151909866484295at_nat(Uu,X),Xa3),Xb4)) ) ).

% ATP.lambda_296
tff(fact_10474_ATP_Olambda__297,axiom,
    ! [Uu: fun_VE7674639578305544598at_nat,Uua: fun_Pr633315466666964198on_nat] :
      ( aa_fun8910613404238449780_nat_o(aTP_Lamm_ars(Uu),Uua)
    <=> ! [X: vEBT_VEBT,Xa3: vEBT_VEBTi,Xb4: nat] : aa_nat_o(time_T8353473612707095248on_nat(aa_nat947733127081326238on_nat(aa_VEB3566417204837707778on_nat(aa_VEB1030146501360450209on_nat(produc9121605208400662357on_nat(produc1757988346207259447on_nat(Uua)),X),Xa3),Xb4)),aa_nat_nat(aa_VEB205140289345309584at_nat(aa_VEB5308151909866484295at_nat(Uu,X),Xa3),Xb4)) ) ).

% ATP.lambda_297
tff(fact_10475_ATP_Olambda__298,axiom,
    ! [Uu: fun_VE7674639578305544598at_nat,Uua: fun_Pr940977554174145406Heap_o] :
      ( aa_fun3656762788152972374ap_o_o(aTP_Lamm_art(Uu),Uua)
    <=> ! [X: vEBT_VEBT,Xa3: vEBT_VEBTi,Xb4: nat] : aa_nat_o(time_TBOUND_o(aa_nat2910048432523430448Heap_o(aa_VEB1455453220190609250Heap_o(aa_VEB8672304581938048793Heap_o(produc8906325494159015119Heap_o(produc8381543706267210711Heap_o(Uua)),X),Xa3),Xb4)),aa_nat_nat(aa_VEB205140289345309584at_nat(aa_VEB5308151909866484295at_nat(Uu,X),Xa3),Xb4)) ) ).

% ATP.lambda_298
tff(fact_10476_ATP_Olambda__299,axiom,
    ! [Uu: fun_Pr6004456690014451917_nat_o,Uua: fun_Pr342922516890243977on_nat] :
      ( aa_fun7247719965521315857_nat_o(aTP_Lamm_asl(Uu),Uua)
    <=> ! [X: produc3881548065746020326Ti_nat,H4: heap_e7401611519738050253t_unit,H9: heap_e7401611519738050253t_unit,R6: option_nat,N6: nat] :
          ( aa_nat_o(heap_T306965388786959644on_nat(aa_Pro2606730266749126658on_nat(Uua,X),H4,H9,R6),N6)
         => aa_nat_o(aa_opt7444464167971770947_nat_o(aa_hea2436540527373857417_nat_o(aa_hea5443102814884448417_nat_o(aa_Pro190224997532733766_nat_o(Uu,X),H4),H9),R6),N6) ) ) ).

% ATP.lambda_299
tff(fact_10477_ATP_Olambda__300,axiom,
    ! [Uu: fun_Pr7132087832597382491_nat_o,Uua: fun_Pr238758252084143515Heap_o] :
      ( aa_fun1852402840713402361ap_o_o(aTP_Lamm_asm(Uu),Uua)
    <=> ! [X: produc3881548065746020326Ti_nat,H4: heap_e7401611519738050253t_unit,H9: heap_e7401611519738050253t_unit,R6: $o,N6: nat] :
          ( aa_nat_o(aa_o_fun_nat_o(heap_Time_effect_o(aa_Pro3916655985231737292Heap_o(Uua,X),H4,H9),(R6)),N6)
         => aa_nat_o(aa_o_fun_nat_o(aa_hea6792647905190977245_nat_o(aa_hea8977266003532396635_nat_o(aa_Pro3522642011054043020_nat_o(Uu,X),H4),H9),(R6)),N6) ) ) ).

% ATP.lambda_300
tff(fact_10478_ATP_Olambda__301,axiom,
    ! [Uu: fun_Pr4018130278567771477_nat_o,Uua: fun_Pr4404027214751104635_VEBTi] :
      ( aa_fun2613385842899212575EBTi_o(aTP_Lamm_asn(Uu),Uua)
    <=> ! [X: produc3960855945107176009Ti_nat,H4: heap_e7401611519738050253t_unit,H9: heap_e7401611519738050253t_unit,R6: vEBT_VEBTi,N6: nat] :
          ( aa_nat_o(aa_VEB8883426185042993134_nat_o(heap_T2071195472996403633_VEBTi(aa_Pro721367001817565044_VEBTi(Uua,X),H4,H9),R6),N6)
         => aa_nat_o(aa_VEB8883426185042993134_nat_o(aa_hea5256493101677210548_nat_o(aa_hea3862133550498874060_nat_o(aa_Pro9055585097396655438_nat_o(Uu,X),H4),H9),R6),N6) ) ) ).

% ATP.lambda_301
tff(fact_10479_ATP_Olambda__302,axiom,
    ! [Uu: fun_Pr4917118001609057450_nat_o,Uua: fun_Pr633315466666964198on_nat] :
      ( aa_fun8910613404238449780_nat_o(aTP_Lamm_aso(Uu),Uua)
    <=> ! [X: produc3960855945107176009Ti_nat,H4: heap_e7401611519738050253t_unit,H9: heap_e7401611519738050253t_unit,R6: option_nat,N6: nat] :
          ( aa_nat_o(heap_T306965388786959644on_nat(aa_Pro2727408364141080927on_nat(Uua,X),H4,H9,R6),N6)
         => aa_nat_o(aa_opt7444464167971770947_nat_o(aa_hea2436540527373857417_nat_o(aa_hea5443102814884448417_nat_o(aa_Pro3370481601460356387_nat_o(Uu,X),H4),H9),R6),N6) ) ) ).

% ATP.lambda_302
tff(fact_10480_ATP_Olambda__303,axiom,
    ! [Uu: fun_Pr3008164681728764094_nat_o,Uua: fun_Pr940977554174145406Heap_o] :
      ( aa_fun3656762788152972374ap_o_o(aTP_Lamm_asp(Uu),Uua)
    <=> ! [X: produc3960855945107176009Ti_nat,H4: heap_e7401611519738050253t_unit,H9: heap_e7401611519738050253t_unit,R6: $o,N6: nat] :
          ( aa_nat_o(aa_o_fun_nat_o(heap_Time_effect_o(aa_Pro6173831900156065711Heap_o(Uua,X),H4,H9),(R6)),N6)
         => aa_nat_o(aa_o_fun_nat_o(aa_hea6792647905190977245_nat_o(aa_hea8977266003532396635_nat_o(aa_Pro5855601548611308271_nat_o(Uu,X),H4),H9),(R6)),N6) ) ) ).

% ATP.lambda_303
tff(fact_10481_ATP_Olambda__304,axiom,
    ! [Uu: vEBT_VEBTi,Uua: nat] : aa_nat2910048432523430448Heap_o(aa_VEB1455453220190609250Heap_o(aTP_Lamm_acg,Uu),Uua) = heap_Time_Heap_o2(aTP_Lamm_zg) ).

% ATP.lambda_304
tff(fact_10482_ATP_Olambda__305,axiom,
    ! [Uu: vEBT_VEBTi,Uua: nat] : aa_nat947733127081326238on_nat(aa_VEB3566417204837707778on_nat(aTP_Lamm_acc,Uu),Uua) = heap_T5286843759275942675on_nat(aTP_Lamm_zf) ).

% ATP.lambda_305
tff(fact_10483_ATP_Olambda__306,axiom,
    ! [Uu: vEBT_VEBTi,Uua: nat] : aa_nat5408703056993926195_VEBTi(aa_VEB4833742232818206615_VEBTi(aTP_Lamm_abs,Uu),Uua) = heap_T1489671670754571048_VEBTi(aTP_Lamm_ys) ).

% ATP.lambda_306
tff(fact_10484_ATP_Olambda__307,axiom,
    ! [Uu: fun_Pr4171499318505651870_VEBTi,Uua: vEBT_VEBTi,Uub: nat] : aa_nat5408703056993926195_VEBTi(aa_VEB4833742232818206615_VEBTi(aTP_Lamm_abp(Uu),Uua),Uub) = aa_VEB836174949890266638_VEBTi(vEBT_c6028912655521741485_VEBTi(aa_nat3009484633486537743_VEBTi(aTP_Lamm_abo(Uu),Uub),aTP_Lamm_je(Uub)),Uua) ).

% ATP.lambda_307
tff(fact_10485_ATP_Olambda__308,axiom,
    ! [Uu: fun_Pr342922516890243977on_nat,Uua: vEBT_VEBTi,Uub: nat] : aa_nat947733127081326238on_nat(aa_VEB3566417204837707778on_nat(aTP_Lamm_aba(Uu),Uua),Uub) = aa_VEB2711320435582112505on_nat(vEBT_c6250501799366334488on_nat(aa_nat3457676333800472698on_nat(aTP_Lamm_aaz(Uu),Uub),aTP_Lamm_fi(Uub)),Uua) ).

% ATP.lambda_308
tff(fact_10486_ATP_Olambda__309,axiom,
    ! [Uu: fun_Pr342922516890243977on_nat,Uua: vEBT_VEBTi,Uub: nat] : aa_nat947733127081326238on_nat(aa_VEB3566417204837707778on_nat(aTP_Lamm_aas(Uu),Uua),Uub) = aa_VEB2711320435582112505on_nat(vEBT_c6250501799366334488on_nat(aa_nat3457676333800472698on_nat(aTP_Lamm_aar(Uu),Uub),aTP_Lamm_ez(Uub)),Uua) ).

% ATP.lambda_309
tff(fact_10487_ATP_Olambda__310,axiom,
    ! [Uu: fun_Pr238758252084143515Heap_o,Uua: vEBT_VEBTi,Uub: nat] : aa_nat2910048432523430448Heap_o(aa_VEB1455453220190609250Heap_o(aTP_Lamm_zo(Uu),Uua),Uub) = aa_VEB8194921341634505877Heap_o(vEBT_c6104975476656191286Heap_o(aa_nat1227502447850154924Heap_o(aTP_Lamm_zn(Uu),Uub),aTP_Lamm_jv(Uub)),Uua) ).

% ATP.lambda_310
tff(fact_10488_ATP_Olambda__311,axiom,
    ! [Uu: fun_VE7134199479003659425Heap_o,Uua: vEBT_VEBTi,Uub: nat] : aa_nat2910048432523430448Heap_o(aa_VEB1455453220190609250Heap_o(aTP_Lamm_ach(Uu),Uua),Uub) = aa_VEB8194921341634505877Heap_o(vEBT_c6104975476656191286Heap_o(aa_nat1227502447850154924Heap_o(aTP_Lamm_od(Uu),Uub),aTP_Lamm_jv(Uub)),Uua) ).

% ATP.lambda_311
tff(fact_10489_ATP_Olambda__312,axiom,
    ! [Uu: fun_VE8516060933855021849on_nat,Uua: vEBT_VEBTi,Uub: nat] : aa_nat947733127081326238on_nat(aa_VEB3566417204837707778on_nat(aTP_Lamm_ace(Uu),Uua),Uub) = aa_VEB2711320435582112505on_nat(vEBT_c6250501799366334488on_nat(aa_nat3457676333800472698on_nat(aTP_Lamm_nx(Uu),Uub),aTP_Lamm_fi(Uub)),Uua) ).

% ATP.lambda_312
tff(fact_10490_ATP_Olambda__313,axiom,
    ! [Uu: fun_VE8516060933855021849on_nat,Uua: vEBT_VEBTi,Uub: nat] : aa_nat947733127081326238on_nat(aa_VEB3566417204837707778on_nat(aTP_Lamm_acd(Uu),Uua),Uub) = aa_VEB2711320435582112505on_nat(vEBT_c6250501799366334488on_nat(aa_nat3457676333800472698on_nat(aTP_Lamm_nr(Uu),Uub),aTP_Lamm_ez(Uub)),Uua) ).

% ATP.lambda_313
tff(fact_10491_ATP_Olambda__314,axiom,
    ! [Uu: fun_VE8118064890490510894_VEBTi,Uua: vEBT_VEBTi,Uub: nat] : aa_nat5408703056993926195_VEBTi(aa_VEB4833742232818206615_VEBTi(aTP_Lamm_abt(Uu),Uua),Uub) = aa_VEB836174949890266638_VEBTi(vEBT_c6028912655521741485_VEBTi(aa_nat3009484633486537743_VEBTi(aTP_Lamm_nl(Uu),Uub),aTP_Lamm_je(Uub)),Uua) ).

% ATP.lambda_314
tff(fact_10492_ATP_Olambda__315,axiom,
    ! [Uu: nat,Uua: $o,Uub: $o] :
      aa_o_H4215342640551218896on_nat(aa_o_f5606600903069576627on_nat(aTP_Lamm_ez(Uu),(Uua)),(Uub)) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uu),
        $ite(
          (Uub),
          heap_T3487192422709364219on_nat(aa_nat_option_nat(some_nat,one_one_nat)),
          $ite((Uua),heap_T3487192422709364219on_nat(aa_nat_option_nat(some_nat,zero_zero_nat)),heap_T3487192422709364219on_nat(none_nat)) ),
        $ite(
          Uu = one_one_nat,
          $ite((Uua),heap_T3487192422709364219on_nat(aa_nat_option_nat(some_nat,zero_zero_nat)),heap_T3487192422709364219on_nat(none_nat)),
          heap_T3487192422709364219on_nat(none_nat) ) ) ).

% ATP.lambda_315
tff(fact_10493_ATP_Olambda__316,axiom,
    ! [Uu: fun_nat_real,Uua: fun_nat_real,Uub: nat] :
      aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_apv(Uu),Uua),Uub) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uub),aa_nat_real(Uua,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uub),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(Uu,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),one_one_nat)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% ATP.lambda_316
tff(fact_10494_ATP_Olambda__317,axiom,
    ! [Uu: fun_nat_real,Uua: fun_nat_real,Uub: nat] :
      aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_aui(Uu),Uua),Uub) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uub),aa_nat_real(Uu,Uub),aa_nat_real(Uua,Uub)) ).

% ATP.lambda_317
tff(fact_10495_ATP_Olambda__318,axiom,
    ! [Uu: num,Uua: code_integer,Uub: code_integer] :
      aa_Cod6977894654539884864nteger(aa_Cod8380929501330563836nteger(aTP_Lamm_my(Uu),Uua),Uub) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_num_Code_integer(numera6620942414471956472nteger,Uu)),Uub),aa_Cod6977894654539884864nteger(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))),aa_Cod6977894654539884864nteger(produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Uua)),Uub)) ).

% ATP.lambda_318
tff(fact_10496_ATP_Olambda__319,axiom,
    ! [Uu: num,Uua: nat,Uub: nat] :
      aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aTP_Lamm_mx(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_319
tff(fact_10497_ATP_Olambda__320,axiom,
    ! [Uu: num,Uua: int,Uub: int] :
      aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_mw(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_320
tff(fact_10498_ATP_Olambda__321,axiom,
    ! [Uu: set_nat,Uua: fun_nat_real,Uub: nat] :
      aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_arm(Uu),Uua),Uub) = $ite(aa_set_nat_o(member_nat(Uub),Uu),aa_nat_real(Uua,Uub),zero_zero_real) ).

% ATP.lambda_321
tff(fact_10499_ATP_Olambda__322,axiom,
    ! [Uu: set_nat,Uua: fun_nat_nat,Uub: nat] :
      aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_arl(Uu),Uua),Uub) = $ite(aa_set_nat_o(member_nat(Uub),Uu),aa_nat_nat(Uua,Uub),zero_zero_nat) ).

% ATP.lambda_322
tff(fact_10500_ATP_Olambda__323,axiom,
    ! [Uu: set_nat,Uua: fun_nat_int,Uub: nat] :
      aa_nat_int(aa_fun5565043122120104496at_int(aTP_Lamm_ark(Uu),Uua),Uub) = $ite(aa_set_nat_o(member_nat(Uub),Uu),aa_nat_int(Uua,Uub),zero_zero_int) ).

% ATP.lambda_323
tff(fact_10501_ATP_Olambda__324,axiom,
    ! [Uu: vEBT_VEBT,Uua: fun_VEBT_VEBT_uint32,Uub: vEBT_VEBT] :
      aa_VEBT_VEBT_uint32(aa_fun7720134271958203516uint32(aTP_Lamm_ajb(Uu),Uua),Uub) = $ite(Uu = Uub,aa_VEBT_VEBT_uint32(Uua,Uub),zero_zero_uint32) ).

% ATP.lambda_324
tff(fact_10502_ATP_Olambda__325,axiom,
    ! [Uu: vEBT_VEBT,Uua: fun_VEBT_VEBT_real,Uub: vEBT_VEBT] :
      aa_VEBT_VEBT_real(aa_fun1825138017709814396T_real(aTP_Lamm_ajh(Uu),Uua),Uub) = $ite(Uu = Uub,aa_VEBT_VEBT_real(Uua,Uub),zero_zero_real) ).

% ATP.lambda_325
tff(fact_10503_ATP_Olambda__326,axiom,
    ! [Uu: code_integer,Uua: fun_Co1554742298434871494uint32,Uub: code_integer] :
      aa_Cod8401701657498852871uint32(aa_fun8111848626357839434uint32(aTP_Lamm_ajg(Uu),Uua),Uub) = $ite(Uu = Uub,aa_Cod8401701657498852871uint32(Uua,Uub),zero_zero_uint32) ).

% ATP.lambda_326
tff(fact_10504_ATP_Olambda__327,axiom,
    ! [Uu: complex,Uua: fun_complex_uint32,Uub: complex] :
      aa_complex_uint32(aa_fun947202747272686700uint32(aTP_Lamm_ajf(Uu),Uua),Uub) = $ite(Uu = Uub,aa_complex_uint32(Uua,Uub),zero_zero_uint32) ).

% ATP.lambda_327
tff(fact_10505_ATP_Olambda__328,axiom,
    ! [Uu: complex,Uua: fun_complex_real,Uub: complex] :
      aa_complex_real(aa_fun8933909628229591020x_real(aTP_Lamm_ajk(Uu),Uua),Uub) = $ite(Uu = Uub,aa_complex_real(Uua,Uub),zero_zero_real) ).

% ATP.lambda_328
tff(fact_10506_ATP_Olambda__329,axiom,
    ! [Uu: real,Uua: fun_real_uint32,Uub: real] :
      aa_real_uint32(aa_fun39000138016665704uint32(aTP_Lamm_ajc(Uu),Uua),Uub) = $ite(Uu = Uub,aa_real_uint32(Uua,Uub),zero_zero_uint32) ).

% ATP.lambda_329
tff(fact_10507_ATP_Olambda__330,axiom,
    ! [Uu: real,Uua: fun_real_real,Uub: real] :
      aa_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_aji(Uu),Uua),Uub) = $ite(Uu = Uub,aa_real_real(Uua,Uub),zero_zero_real) ).

% ATP.lambda_330
tff(fact_10508_ATP_Olambda__331,axiom,
    ! [Uu: nat,Uua: fun_nat_uint32,Uub: nat] :
      aa_nat_uint32(aa_fun7322500112117231024uint32(aTP_Lamm_ajd(Uu),Uua),Uub) = $ite(Uu = Uub,aa_nat_uint32(Uua,Uub),zero_zero_uint32) ).

% ATP.lambda_331
tff(fact_10509_ATP_Olambda__332,axiom,
    ! [Uu: int,Uua: fun_int_uint32,Uub: int] :
      aa_int_uint32(aa_fun7636875859098149992uint32(aTP_Lamm_aje(Uu),Uua),Uub) = $ite(Uu = Uub,aa_int_uint32(Uua,Uub),zero_zero_uint32) ).

% ATP.lambda_332
tff(fact_10510_ATP_Olambda__333,axiom,
    ! [Uu: int,Uua: fun_int_real,Uub: int] :
      aa_int_real(aa_fun3252798400008884712t_real(aTP_Lamm_ajj(Uu),Uua),Uub) = $ite(Uu = Uub,aa_int_real(Uua,Uub),zero_zero_real) ).

% ATP.lambda_333
tff(fact_10511_ATP_Olambda__334,axiom,
    ! [Uu: vEBT_VEBT,Uua: fun_VEBT_VEBT_real,Uub: vEBT_VEBT] :
      aa_VEBT_VEBT_real(aa_fun1825138017709814396T_real(aTP_Lamm_ave(Uu),Uua),Uub) = $ite(Uu = Uub,aa_VEBT_VEBT_real(Uua,Uub),one_one_real) ).

% ATP.lambda_334
tff(fact_10512_ATP_Olambda__335,axiom,
    ! [Uu: vEBT_VEBT,Uua: fun_VEBT_VEBT_rat,Uub: vEBT_VEBT] :
      aa_VEBT_VEBT_rat(aa_fun8105832842879916156BT_rat(aTP_Lamm_avk(Uu),Uua),Uub) = $ite(Uu = Uub,aa_VEBT_VEBT_rat(Uua,Uub),one_one_rat) ).

% ATP.lambda_335
tff(fact_10513_ATP_Olambda__336,axiom,
    ! [Uu: code_integer,Uua: fun_Co3960896333048504454r_real,Uub: code_integer] :
      aa_Code_integer_real(aa_fun6341089502340155082r_real(aTP_Lamm_avj(Uu),Uua),Uub) = $ite(Uu = Uub,aa_Code_integer_real(Uua,Uub),one_one_real) ).

% ATP.lambda_336
tff(fact_10514_ATP_Olambda__337,axiom,
    ! [Uu: complex,Uua: fun_complex_real,Uub: complex] :
      aa_complex_real(aa_fun8933909628229591020x_real(aTP_Lamm_avi(Uu),Uua),Uub) = $ite(Uu = Uub,aa_complex_real(Uua,Uub),one_one_real) ).

% ATP.lambda_337
tff(fact_10515_ATP_Olambda__338,axiom,
    ! [Uu: real,Uua: fun_real_real,Uub: real] :
      aa_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_avf(Uu),Uua),Uub) = $ite(Uu = Uub,aa_real_real(Uua,Uub),one_one_real) ).

% ATP.lambda_338
tff(fact_10516_ATP_Olambda__339,axiom,
    ! [Uu: real,Uua: fun_real_rat,Uub: real] :
      aa_real_rat(aa_fun2468082234287226944al_rat(aTP_Lamm_avl(Uu),Uua),Uub) = $ite(Uu = Uub,aa_real_rat(Uua,Uub),one_one_rat) ).

% ATP.lambda_339
tff(fact_10517_ATP_Olambda__340,axiom,
    ! [Uu: nat,Uua: fun_nat_real,Uub: nat] :
      aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_avg(Uu),Uua),Uub) = $ite(Uu = Uub,aa_nat_real(Uua,Uub),one_one_real) ).

% ATP.lambda_340
tff(fact_10518_ATP_Olambda__341,axiom,
    ! [Uu: nat,Uua: fun_nat_rat,Uub: nat] :
      aa_nat_rat(aa_fun765758790203053448at_rat(aTP_Lamm_avm(Uu),Uua),Uub) = $ite(Uu = Uub,aa_nat_rat(Uua,Uub),one_one_rat) ).

% ATP.lambda_341
tff(fact_10519_ATP_Olambda__342,axiom,
    ! [Uu: int,Uua: fun_int_real,Uub: int] :
      aa_int_real(aa_fun3252798400008884712t_real(aTP_Lamm_avh(Uu),Uua),Uub) = $ite(Uu = Uub,aa_int_real(Uua,Uub),one_one_real) ).

% ATP.lambda_342
tff(fact_10520_ATP_Olambda__343,axiom,
    ! [Uu: int,Uua: fun_int_rat,Uub: int] :
      aa_int_rat(aa_fun1481061427733998144nt_rat(aTP_Lamm_avn(Uu),Uua),Uub) = $ite(Uu = Uub,aa_int_rat(Uua,Uub),one_one_rat) ).

% ATP.lambda_343
tff(fact_10521_ATP_Olambda__344,axiom,
    ! [Uu: vEBT_VEBT,Uua: fun_VEBT_VEBT_uint32,Uub: vEBT_VEBT] :
      aa_VEBT_VEBT_uint32(aa_fun7720134271958203516uint32(aTP_Lamm_ajl(Uu),Uua),Uub) = $ite(Uub = Uu,aa_VEBT_VEBT_uint32(Uua,Uub),zero_zero_uint32) ).

% ATP.lambda_344
tff(fact_10522_ATP_Olambda__345,axiom,
    ! [Uu: vEBT_VEBT,Uua: fun_VEBT_VEBT_real,Uub: vEBT_VEBT] :
      aa_VEBT_VEBT_real(aa_fun1825138017709814396T_real(aTP_Lamm_ajr(Uu),Uua),Uub) = $ite(Uub = Uu,aa_VEBT_VEBT_real(Uua,Uub),zero_zero_real) ).

% ATP.lambda_345
tff(fact_10523_ATP_Olambda__346,axiom,
    ! [Uu: code_integer,Uua: fun_Co1554742298434871494uint32,Uub: code_integer] :
      aa_Cod8401701657498852871uint32(aa_fun8111848626357839434uint32(aTP_Lamm_ajq(Uu),Uua),Uub) = $ite(Uub = Uu,aa_Cod8401701657498852871uint32(Uua,Uub),zero_zero_uint32) ).

% ATP.lambda_346
tff(fact_10524_ATP_Olambda__347,axiom,
    ! [Uu: complex,Uua: fun_complex_uint32,Uub: complex] :
      aa_complex_uint32(aa_fun947202747272686700uint32(aTP_Lamm_ajp(Uu),Uua),Uub) = $ite(Uub = Uu,aa_complex_uint32(Uua,Uub),zero_zero_uint32) ).

% ATP.lambda_347
tff(fact_10525_ATP_Olambda__348,axiom,
    ! [Uu: complex,Uua: fun_complex_real,Uub: complex] :
      aa_complex_real(aa_fun8933909628229591020x_real(aTP_Lamm_aju(Uu),Uua),Uub) = $ite(Uub = Uu,aa_complex_real(Uua,Uub),zero_zero_real) ).

% ATP.lambda_348
tff(fact_10526_ATP_Olambda__349,axiom,
    ! [Uu: real,Uua: fun_real_uint32,Uub: real] :
      aa_real_uint32(aa_fun39000138016665704uint32(aTP_Lamm_ajm(Uu),Uua),Uub) = $ite(Uub = Uu,aa_real_uint32(Uua,Uub),zero_zero_uint32) ).

% ATP.lambda_349
tff(fact_10527_ATP_Olambda__350,axiom,
    ! [Uu: real,Uua: fun_real_real,Uub: real] :
      aa_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_ajs(Uu),Uua),Uub) = $ite(Uub = Uu,aa_real_real(Uua,Uub),zero_zero_real) ).

% ATP.lambda_350
tff(fact_10528_ATP_Olambda__351,axiom,
    ! [Uu: nat,Uua: fun_nat_uint32,Uub: nat] :
      aa_nat_uint32(aa_fun7322500112117231024uint32(aTP_Lamm_ajn(Uu),Uua),Uub) = $ite(Uub = Uu,aa_nat_uint32(Uua,Uub),zero_zero_uint32) ).

% ATP.lambda_351
tff(fact_10529_ATP_Olambda__352,axiom,
    ! [Uu: nat,Uua: fun_nat_real,Uub: nat] :
      aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_aqo(Uu),Uua),Uub) = $ite(Uub = Uu,aa_nat_real(Uua,Uub),zero_zero_real) ).

% ATP.lambda_352
tff(fact_10530_ATP_Olambda__353,axiom,
    ! [Uu: nat,Uua: fun_nat_nat,Uub: nat] :
      aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_aqp(Uu),Uua),Uub) = $ite(Uub = Uu,aa_nat_nat(Uua,Uub),zero_zero_nat) ).

% ATP.lambda_353
tff(fact_10531_ATP_Olambda__354,axiom,
    ! [Uu: nat,Uua: fun_nat_int,Uub: nat] :
      aa_nat_int(aa_fun5565043122120104496at_int(aTP_Lamm_aqq(Uu),Uua),Uub) = $ite(Uub = Uu,aa_nat_int(Uua,Uub),zero_zero_int) ).

% ATP.lambda_354
tff(fact_10532_ATP_Olambda__355,axiom,
    ! [Uu: int,Uua: fun_int_uint32,Uub: int] :
      aa_int_uint32(aa_fun7636875859098149992uint32(aTP_Lamm_ajo(Uu),Uua),Uub) = $ite(Uub = Uu,aa_int_uint32(Uua,Uub),zero_zero_uint32) ).

% ATP.lambda_355
tff(fact_10533_ATP_Olambda__356,axiom,
    ! [Uu: int,Uua: fun_int_real,Uub: int] :
      aa_int_real(aa_fun3252798400008884712t_real(aTP_Lamm_ajt(Uu),Uua),Uub) = $ite(Uub = Uu,aa_int_real(Uua,Uub),zero_zero_real) ).

% ATP.lambda_356
tff(fact_10534_ATP_Olambda__357,axiom,
    ! [Uu: vEBT_VEBT,Uua: fun_VEBT_VEBT_real,Uub: vEBT_VEBT] :
      aa_VEBT_VEBT_real(aa_fun1825138017709814396T_real(aTP_Lamm_avo(Uu),Uua),Uub) = $ite(Uub = Uu,aa_VEBT_VEBT_real(Uua,Uub),one_one_real) ).

% ATP.lambda_357
tff(fact_10535_ATP_Olambda__358,axiom,
    ! [Uu: vEBT_VEBT,Uua: fun_VEBT_VEBT_rat,Uub: vEBT_VEBT] :
      aa_VEBT_VEBT_rat(aa_fun8105832842879916156BT_rat(aTP_Lamm_avu(Uu),Uua),Uub) = $ite(Uub = Uu,aa_VEBT_VEBT_rat(Uua,Uub),one_one_rat) ).

% ATP.lambda_358
tff(fact_10536_ATP_Olambda__359,axiom,
    ! [Uu: code_integer,Uua: fun_Co3960896333048504454r_real,Uub: code_integer] :
      aa_Code_integer_real(aa_fun6341089502340155082r_real(aTP_Lamm_avt(Uu),Uua),Uub) = $ite(Uub = Uu,aa_Code_integer_real(Uua,Uub),one_one_real) ).

% ATP.lambda_359
tff(fact_10537_ATP_Olambda__360,axiom,
    ! [Uu: complex,Uua: fun_complex_real,Uub: complex] :
      aa_complex_real(aa_fun8933909628229591020x_real(aTP_Lamm_avs(Uu),Uua),Uub) = $ite(Uub = Uu,aa_complex_real(Uua,Uub),one_one_real) ).

% ATP.lambda_360
tff(fact_10538_ATP_Olambda__361,axiom,
    ! [Uu: real,Uua: fun_real_real,Uub: real] :
      aa_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_avp(Uu),Uua),Uub) = $ite(Uub = Uu,aa_real_real(Uua,Uub),one_one_real) ).

% ATP.lambda_361
tff(fact_10539_ATP_Olambda__362,axiom,
    ! [Uu: real,Uua: fun_real_rat,Uub: real] :
      aa_real_rat(aa_fun2468082234287226944al_rat(aTP_Lamm_avv(Uu),Uua),Uub) = $ite(Uub = Uu,aa_real_rat(Uua,Uub),one_one_rat) ).

% ATP.lambda_362
tff(fact_10540_ATP_Olambda__363,axiom,
    ! [Uu: nat,Uua: fun_nat_real,Uub: nat] :
      aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_avq(Uu),Uua),Uub) = $ite(Uub = Uu,aa_nat_real(Uua,Uub),one_one_real) ).

% ATP.lambda_363
tff(fact_10541_ATP_Olambda__364,axiom,
    ! [Uu: nat,Uua: fun_nat_rat,Uub: nat] :
      aa_nat_rat(aa_fun765758790203053448at_rat(aTP_Lamm_avw(Uu),Uua),Uub) = $ite(Uub = Uu,aa_nat_rat(Uua,Uub),one_one_rat) ).

% ATP.lambda_364
tff(fact_10542_ATP_Olambda__365,axiom,
    ! [Uu: int,Uua: fun_int_real,Uub: int] :
      aa_int_real(aa_fun3252798400008884712t_real(aTP_Lamm_avr(Uu),Uua),Uub) = $ite(Uub = Uu,aa_int_real(Uua,Uub),one_one_real) ).

% ATP.lambda_365
tff(fact_10543_ATP_Olambda__366,axiom,
    ! [Uu: int,Uua: fun_int_rat,Uub: int] :
      aa_int_rat(aa_fun1481061427733998144nt_rat(aTP_Lamm_avx(Uu),Uua),Uub) = $ite(Uub = Uu,aa_int_rat(Uua,Uub),one_one_rat) ).

% ATP.lambda_366
tff(fact_10544_ATP_Olambda__367,axiom,
    ! [Uu: product_prod_nat_nat,Uua: option_nat,Uub: product_unit] :
      aa_Pro451875010097498139on_nat(aa_opt5089197864053947520on_nat(aTP_Lamm_ac(Uu),Uua),Uub) = $ite(
        Uua = none_nat,
        $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_Pro4243173770044397973at_nat(product_fst_nat_nat,Uu)),xa),heap_T3487192422709364219on_nat(aa_nat_option_nat(some_nat,aa_Pro4243173770044397973at_nat(product_fst_nat_nat,Uu))),heap_T3487192422709364219on_nat(none_nat)),
        heap_T2868974464944644318on_nat(aa_nat5408703056993926195_VEBTi(array_nth_VEBT_VEBTi(x13),the_nat(Uua)),aTP_Lamm_ab(Uua)) ) ).

% ATP.lambda_367
tff(fact_10545_ATP_Olambda__368,axiom,
    ! [Uu: nat,Uua: $o,Uub: $o] :
      aa_o_H4215342640551218896on_nat(aa_o_f5606600903069576627on_nat(aTP_Lamm_fi(Uu),(Uua)),(Uub)) = $ite(
        Uu = zero_zero_nat,
        $ite((Uub),heap_T3487192422709364219on_nat(aa_nat_option_nat(some_nat,one_one_nat)),heap_T3487192422709364219on_nat(none_nat)),
        heap_T3487192422709364219on_nat(none_nat) ) ).

% ATP.lambda_368
tff(fact_10546_ATP_Olambda__369,axiom,
    ! [Uu: code_integer,Uua: code_integer,Uub: code_integer] :
      aa_Cod6977894654539884864nteger(aa_Cod8380929501330563836nteger(aTP_Lamm_bca(Uu),Uua),Uub) = $ite(Uub = zero_z3403309356797280102nteger,aa_Cod6977894654539884864nteger(produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uua)),zero_z3403309356797280102nteger),aa_Cod6977894654539884864nteger(produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uua)),one_one_Code_integer)),aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uu)),Uub))) ).

% ATP.lambda_369
tff(fact_10547_ATP_Olambda__370,axiom,
    ! [Uu: code_integer,Uua: code_integer,Uub: code_integer] :
      aa_Cod6977894654539884864nteger(aa_Cod8380929501330563836nteger(aTP_Lamm_bby(Uu),Uua),Uub) = $ite(Uub = zero_z3403309356797280102nteger,aa_Cod6977894654539884864nteger(produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uua)),zero_z3403309356797280102nteger),aa_Cod6977894654539884864nteger(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_370
tff(fact_10548_ATP_Olambda__371,axiom,
    ! [Uu: code_integer,Uua: code_integer,Uub: code_integer] :
      aa_Cod6977894654539884864nteger(aa_Cod8380929501330563836nteger(aTP_Lamm_bbz(Uu),Uua),Uub) = $ite(Uub = zero_z3403309356797280102nteger,aa_Cod6977894654539884864nteger(produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uua)),zero_z3403309356797280102nteger),aa_Cod6977894654539884864nteger(produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uua)),one_one_Code_integer)),aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(Uu),Uub))) ).

% ATP.lambda_371
tff(fact_10549_ATP_Olambda__372,axiom,
    ! [Uu: nat,Uua: $o,Uub: $o] :
      aa_o_H7577119287286566245_VEBTi(aa_o_f5262348187908705736_VEBTi(aTP_Lamm_je(Uu),(Uua)),(Uub)) = $ite(
        Uu = zero_zero_nat,
        aa_VEB836174949890266638_VEBTi(heap_T3630416162098727440_VEBTi,vEBT_Leafi($true,(Uub))),
        $ite(Uu = one_one_nat,aa_VEB836174949890266638_VEBTi(heap_T3630416162098727440_VEBTi,vEBT_Leafi((Uua),$true)),aa_VEB836174949890266638_VEBTi(heap_T3630416162098727440_VEBTi,vEBT_Leafi((Uua),(Uub)))) ) ).

% ATP.lambda_372
tff(fact_10550_ATP_Olambda__373,axiom,
    ! [Uu: nat,Uua: array_VEBT_VEBTi,Uub: option_nat] :
      aa_opt9163293676379396302on_nat(aa_arr3757929795460847279on_nat(aTP_Lamm_fb(Uu),Uua),Uub) = $ite(Uub = none_nat,heap_T3487192422709364219on_nat(none_nat),heap_T2868974464944644318on_nat(aa_nat5408703056993926195_VEBTi(array_nth_VEBT_VEBTi(Uua),the_nat(Uub)),aa_opt1214731524852023858on_nat(aTP_Lamm_fa(Uu),Uub))) ).

% ATP.lambda_373
tff(fact_10551_ATP_Olambda__374,axiom,
    ! [Uu: fun_nat_o,Uua: fun_nat_real,Uub: nat] :
      aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_arj(Uu),Uua),Uub) = $ite(aa_nat_o(Uu,Uub),aa_nat_real(Uua,Uub),zero_zero_real) ).

% ATP.lambda_374
tff(fact_10552_ATP_Olambda__375,axiom,
    ! [Uu: fun_nat_o,Uua: fun_nat_nat,Uub: nat] :
      aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_ari(Uu),Uua),Uub) = $ite(aa_nat_o(Uu,Uub),aa_nat_nat(Uua,Uub),zero_zero_nat) ).

% ATP.lambda_375
tff(fact_10553_ATP_Olambda__376,axiom,
    ! [Uu: fun_nat_o,Uua: fun_nat_int,Uub: nat] :
      aa_nat_int(aa_fun5565043122120104496at_int(aTP_Lamm_arh(Uu),Uua),Uub) = $ite(aa_nat_o(Uu,Uub),aa_nat_int(Uua,Uub),zero_zero_int) ).

% ATP.lambda_376
tff(fact_10554_ATP_Olambda__377,axiom,
    ! [Uu: fun_VEBT_VEBT_uint32,Uua: fun_VEBT_VEBT_o,Uub: vEBT_VEBT] :
      aa_VEBT_VEBT_uint32(aa_fun2089568865546376974uint32(aTP_Lamm_anj(Uu),Uua),Uub) = $ite(aa_VEBT_VEBT_o(Uua,Uub),aa_VEBT_VEBT_uint32(Uu,Uub),zero_zero_uint32) ).

% ATP.lambda_377
tff(fact_10555_ATP_Olambda__378,axiom,
    ! [Uu: fun_VEBT_VEBT_real,Uua: fun_VEBT_VEBT_o,Uub: vEBT_VEBT] :
      aa_VEBT_VEBT_real(aa_fun3597767889293385422T_real(aTP_Lamm_anq(Uu),Uua),Uub) = $ite(aa_VEBT_VEBT_o(Uua,Uub),aa_VEBT_VEBT_real(Uu,Uub),zero_zero_real) ).

% ATP.lambda_378
tff(fact_10556_ATP_Olambda__379,axiom,
    ! [Uu: fun_Co1554742298434871494uint32,Uua: fun_Code_integer_o,Uub: code_integer] :
      aa_Cod8401701657498852871uint32(aa_fun7787524580160642142uint32(aTP_Lamm_anp(Uu),Uua),Uub) = $ite(aa_Code_integer_o(Uua,Uub),aa_Cod8401701657498852871uint32(Uu,Uub),zero_zero_uint32) ).

% ATP.lambda_379
tff(fact_10557_ATP_Olambda__380,axiom,
    ! [Uu: fun_complex_uint32,Uua: fun_complex_o,Uub: complex] :
      aa_complex_uint32(aa_fun9209306604415416350uint32(aTP_Lamm_ann(Uu),Uua),Uub) = $ite(aa_complex_o(Uua,Uub),aa_complex_uint32(Uu,Uub),zero_zero_uint32) ).

% ATP.lambda_380
tff(fact_10558_ATP_Olambda__381,axiom,
    ! [Uu: fun_complex_real,Uua: fun_complex_o,Uub: complex] :
      aa_complex_real(aa_fun8261057609414086622x_real(aTP_Lamm_ant(Uu),Uua),Uub) = $ite(aa_complex_o(Uua,Uub),aa_complex_real(Uu,Uub),zero_zero_real) ).

% ATP.lambda_381
tff(fact_10559_ATP_Olambda__382,axiom,
    ! [Uu: fun_real_uint32,Uua: fun_real_o,Uub: real] :
      aa_real_uint32(aa_fun7747623534341301918uint32(aTP_Lamm_ank(Uu),Uua),Uub) = $ite(aa_real_o(Uua,Uub),aa_real_uint32(Uu,Uub),zero_zero_uint32) ).

% ATP.lambda_382
tff(fact_10560_ATP_Olambda__383,axiom,
    ! [Uu: fun_real_real,Uua: fun_real_o,Uub: real] :
      aa_real_real(aa_fun4448911596306054750l_real(aTP_Lamm_anr(Uu),Uua),Uub) = $ite(aa_real_o(Uua,Uub),aa_real_real(Uu,Uub),zero_zero_real) ).

% ATP.lambda_383
tff(fact_10561_ATP_Olambda__384,axiom,
    ! [Uu: fun_nat_uint32,Uua: fun_nat_o,Uub: nat] :
      aa_nat_uint32(aa_fun7328656920723993502uint32(aTP_Lamm_anl(Uu),Uua),Uub) = $ite(aa_nat_o(Uua,Uub),aa_nat_uint32(Uu,Uub),zero_zero_uint32) ).

% ATP.lambda_384
tff(fact_10562_ATP_Olambda__385,axiom,
    ! [Uu: fun_int_uint32,Uua: fun_int_o,Uub: int] :
      aa_int_uint32(aa_fun1392314963150257310uint32(aTP_Lamm_anm(Uu),Uua),Uub) = $ite(aa_int_o(Uua,Uub),aa_int_uint32(Uu,Uub),zero_zero_uint32) ).

% ATP.lambda_385
tff(fact_10563_ATP_Olambda__386,axiom,
    ! [Uu: fun_int_real,Uua: fun_int_o,Uub: int] :
      aa_int_real(aa_fun7506669883404309598t_real(aTP_Lamm_ans(Uu),Uua),Uub) = $ite(aa_int_o(Uua,Uub),aa_int_real(Uu,Uub),zero_zero_real) ).

% ATP.lambda_386
tff(fact_10564_ATP_Olambda__387,axiom,
    ! [Uu: fun_VEBT_VEBT_real,Uua: fun_VEBT_VEBT_o,Uub: vEBT_VEBT] :
      aa_VEBT_VEBT_real(aa_fun3597767889293385422T_real(aTP_Lamm_axz(Uu),Uua),Uub) = $ite(aa_VEBT_VEBT_o(Uua,Uub),aa_VEBT_VEBT_real(Uu,Uub),one_one_real) ).

% ATP.lambda_387
tff(fact_10565_ATP_Olambda__388,axiom,
    ! [Uu: fun_VEBT_VEBT_rat,Uua: fun_VEBT_VEBT_o,Uub: vEBT_VEBT] :
      aa_VEBT_VEBT_rat(aa_fun7833968636320851834BT_rat(aTP_Lamm_ayf(Uu),Uua),Uub) = $ite(aa_VEBT_VEBT_o(Uua,Uub),aa_VEBT_VEBT_rat(Uu,Uub),one_one_rat) ).

% ATP.lambda_388
tff(fact_10566_ATP_Olambda__389,axiom,
    ! [Uu: fun_Co3960896333048504454r_real,Uua: fun_Code_integer_o,Uub: code_integer] :
      aa_Code_integer_real(aa_fun7534015886319449118r_real(aTP_Lamm_aye(Uu),Uua),Uub) = $ite(aa_Code_integer_o(Uua,Uub),aa_Code_integer_real(Uu,Uub),one_one_real) ).

% ATP.lambda_389
tff(fact_10567_ATP_Olambda__390,axiom,
    ! [Uu: fun_complex_real,Uua: fun_complex_o,Uub: complex] :
      aa_complex_real(aa_fun8261057609414086622x_real(aTP_Lamm_ayd(Uu),Uua),Uub) = $ite(aa_complex_o(Uua,Uub),aa_complex_real(Uu,Uub),one_one_real) ).

% ATP.lambda_390
tff(fact_10568_ATP_Olambda__391,axiom,
    ! [Uu: fun_real_real,Uua: fun_real_o,Uub: real] :
      aa_real_real(aa_fun4448911596306054750l_real(aTP_Lamm_aya(Uu),Uua),Uub) = $ite(aa_real_o(Uua,Uub),aa_real_real(Uu,Uub),one_one_real) ).

% ATP.lambda_391
tff(fact_10569_ATP_Olambda__392,axiom,
    ! [Uu: fun_real_rat,Uua: fun_real_o,Uub: real] :
      aa_real_rat(aa_fun2004904883527618314al_rat(aTP_Lamm_ayg(Uu),Uua),Uub) = $ite(aa_real_o(Uua,Uub),aa_real_rat(Uu,Uub),one_one_rat) ).

% ATP.lambda_392
tff(fact_10570_ATP_Olambda__393,axiom,
    ! [Uu: fun_nat_real,Uua: fun_nat_o,Uub: nat] :
      aa_nat_real(aa_fun5510454223943551838t_real(aTP_Lamm_ayb(Uu),Uua),Uub) = $ite(aa_nat_o(Uua,Uub),aa_nat_real(Uu,Uub),one_one_real) ).

% ATP.lambda_393
tff(fact_10571_ATP_Olambda__394,axiom,
    ! [Uu: fun_nat_rat,Uua: fun_nat_o,Uub: nat] :
      aa_nat_rat(aa_fun696632391729880074at_rat(aTP_Lamm_ayh(Uu),Uua),Uub) = $ite(aa_nat_o(Uua,Uub),aa_nat_rat(Uu,Uub),one_one_rat) ).

% ATP.lambda_394
tff(fact_10572_ATP_Olambda__395,axiom,
    ! [Uu: fun_int_real,Uua: fun_int_o,Uub: int] :
      aa_int_real(aa_fun7506669883404309598t_real(aTP_Lamm_ayc(Uu),Uua),Uub) = $ite(aa_int_o(Uua,Uub),aa_int_real(Uu,Uub),one_one_real) ).

% ATP.lambda_395
tff(fact_10573_ATP_Olambda__396,axiom,
    ! [Uu: fun_int_rat,Uua: fun_int_o,Uub: int] :
      aa_int_rat(aa_fun268837275264525578nt_rat(aTP_Lamm_ayi(Uu),Uua),Uub) = $ite(aa_int_o(Uua,Uub),aa_int_rat(Uu,Uub),one_one_rat) ).

% ATP.lambda_396
tff(fact_10574_ATP_Olambda__397,axiom,
    ! [Uu: fun_Pr238758252084143515Heap_o,Uua: nat,Uub: vEBT_VEBTi] : aa_VEB8194921341634505877Heap_o(aa_nat1164146675577659234Heap_o(aTP_Lamm_zi(Uu),Uua),Uub) = aa_nat2910048432523430448Heap_o(aa_VEB1455453220190609250Heap_o(produc5685940877448195828Heap_o(Uu),Uub),Uua) ).

% ATP.lambda_397
tff(fact_10575_ATP_Olambda__398,axiom,
    ! [Uu: fun_VE3708348504534315036i_assn,Uua: list_VEBT_VEBTi,Uub: vEBT_VEBTi] : aa_VEBT_VEBTi_assn(aa_lis5319508155010942429i_assn(aTP_Lamm_ca(Uu),Uua),Uub) = aa_lis2508312872377092907i_assn(aa_VEB3739350836417479773i_assn(Uu,Uub),Uua) ).

% ATP.lambda_398
tff(fact_10576_ATP_Olambda__399,axiom,
    ! [Uu: fun_VE7134199479003659425Heap_o,Uua: nat,Uub: vEBT_VEBTi] : aa_VEB8194921341634505877Heap_o(aa_nat1164146675577659234Heap_o(aTP_Lamm_ny(Uu),Uua),Uub) = aa_nat2910048432523430448Heap_o(aa_VEB1455453220190609250Heap_o(Uu,Uub),Uua) ).

% ATP.lambda_399
tff(fact_10577_ATP_Olambda__400,axiom,
    ! [Uu: fun_VE4144918862296792757_VEBTi,Uua: fun_Pr342922516890243977on_nat,Uub: vEBT_VEBTi] : aa_VEB836174949890266638_VEBTi(aa_fun5040531007017366174_VEBTi(aTP_Lamm_yn(Uu),Uua),Uub) = aa_fun765629416642892858_VEBTi(aa_VEB3118098574329375646_VEBTi(Uu,Uub),Uua) ).

% ATP.lambda_400
tff(fact_10578_ATP_Olambda__401,axiom,
    ! [Uu: fun_VE6677632329911286106Heap_o,Uua: fun_Pr342922516890243977on_nat,Uub: vEBT_VEBTi] : aa_VEB8194921341634505877Heap_o(aa_fun4002976272494211483Heap_o(aTP_Lamm_yp(Uu),Uua),Uub) = aa_fun4188577930510380905Heap_o(aa_VEB5612184156023240859Heap_o(Uu,Uub),Uua) ).

% ATP.lambda_401
tff(fact_10579_ATP_Olambda__402,axiom,
    ! [Uu: fun_VE6491116402277797979_VEBTi,Uua: fun_Pr238758252084143515Heap_o,Uub: vEBT_VEBTi] : aa_VEB836174949890266638_VEBTi(aa_fun8469889001190476422_VEBTi(aTP_Lamm_xz(Uu),Uua),Uub) = aa_fun238413853627933730_VEBTi(aa_VEB5816640871196325404_VEBTi(Uu,Uub),Uua) ).

% ATP.lambda_402
tff(fact_10580_ATP_Olambda__403,axiom,
    ! [Uu: fun_VE6943415934277719092Heap_o,Uua: fun_Pr238758252084143515Heap_o,Uub: vEBT_VEBTi] : aa_VEB8194921341634505877Heap_o(aa_fun2163459272138508211Heap_o(aTP_Lamm_yd(Uu),Uua),Uub) = aa_fun6848502500920544641Heap_o(aa_VEB6642844428796722333Heap_o(Uu,Uub),Uua) ).

% ATP.lambda_403
tff(fact_10581_ATP_Olambda__404,axiom,
    ! [Uu: fun_VE2006399705683203104omplex,Uua: complex,Uub: vEBT_VEBT] : aa_VEBT_VEBT_complex(aa_com2862003112617944841omplex(aTP_Lamm_amm(Uu),Uua),Uub) = aa_complex_complex(aa_VEB5608662635949521omplex(Uu,Uub),Uua) ).

% ATP.lambda_404
tff(fact_10582_ATP_Olambda__405,axiom,
    ! [Uu: fun_VE2989161857680785728t_real,Uua: nat,Uub: vEBT_VEBT] : aa_VEBT_VEBT_real(aa_nat4323386229672267365T_real(aTP_Lamm_aqt(Uu),Uua),Uub) = aa_nat_real(aa_VEB3736018289492012273t_real(Uu,Uub),Uua) ).

% ATP.lambda_405
tff(fact_10583_ATP_Olambda__406,axiom,
    ! [Uu: fun_VE597728521808591204at_nat,Uua: nat,Uub: vEBT_VEBT] : aa_VEBT_VEBT_nat(aa_nat7776146579971162249BT_nat(aTP_Lamm_alp(Uu),Uua),Uub) = aa_nat_nat(aa_VEB1122092716774696213at_nat(Uu,Uub),Uua) ).

% ATP.lambda_406
tff(fact_10584_ATP_Olambda__407,axiom,
    ! [Uu: fun_VE7331017413731640256at_int,Uua: nat,Uub: vEBT_VEBT] : aa_VEBT_VEBT_int(aa_nat3598295560461965541BT_int(aTP_Lamm_awr(Uu),Uua),Uub) = aa_nat_int(aa_VEB6167613734120275313at_int(Uu,Uub),Uua) ).

% ATP.lambda_407
tff(fact_10585_ATP_Olambda__408,axiom,
    ! [Uu: fun_VE294566951569886236nt_int,Uua: int,Uub: vEBT_VEBT] : aa_VEBT_VEBT_int(aa_int1343382402272391561BT_int(aTP_Lamm_axe(Uu),Uua),Uub) = aa_int_int(aa_VEB7166967446021329869nt_int(Uu,Uub),Uua) ).

% ATP.lambda_408
tff(fact_10586_ATP_Olambda__409,axiom,
    ! [Uu: fun_Co8905303033605938227omplex,Uua: complex,Uub: code_integer] : aa_Cod5621350732091595593omplex(aa_com7552888802784664052omplex(aTP_Lamm_ane(Uu),Uua),Uub) = aa_complex_complex(aa_Cod3397930797123873268omplex(Uu,Uub),Uua) ).

% ATP.lambda_409
tff(fact_10587_ATP_Olambda__410,axiom,
    ! [Uu: fun_Co3312552843764780407at_nat,Uua: nat,Uub: code_integer] : aa_Code_integer_nat(aa_nat5852537043073606456er_nat(aTP_Lamm_amh(Uu),Uua),Uub) = aa_nat_nat(aa_Cod8670293052248272184at_nat(Uu,Uub),Uua) ).

% ATP.lambda_410
tff(fact_10588_ATP_Olambda__411,axiom,
    ! [Uu: fun_Co822469698833053651at_int,Uua: nat,Uub: code_integer] : aa_Code_integer_int(aa_nat1674686023564409748er_int(aTP_Lamm_axa(Uu),Uua),Uub) = aa_nat_int(aa_Cod4492442032739075476at_int(Uu,Uub),Uua) ).

% ATP.lambda_411
tff(fact_10589_ATP_Olambda__412,axiom,
    ! [Uu: fun_Co3009391273526075439nt_int,Uua: int,Uub: code_integer] : aa_Code_integer_int(aa_int1390634949194299888er_int(aTP_Lamm_axr(Uu),Uua),Uub) = aa_int_int(aa_Cod5491795744640130032nt_int(Uu,Uub),Uua) ).

% ATP.lambda_412
tff(fact_10590_ATP_Olambda__413,axiom,
    ! [Uu: fun_Pr6039563626565002634i_assn,Uua: list_VEBT_VEBTi,Uub: product_unit] : aa_Product_unit_assn(aa_lis1661100524205913349t_assn(aTP_Lamm_cc(Uu),Uua),Uub) = aa_lis2508312872377092907i_assn(aa_Pro7700840739073064379i_assn(Uu,Uub),Uua) ).

% ATP.lambda_413
tff(fact_10591_ATP_Olambda__414,axiom,
    ! [Uu: fun_co7673397065115134084omplex,Uua: complex,Uub: complex] : aa_complex_complex(aa_com3342990543769683525omplex(aTP_Lamm_akd(Uu),Uua),Uub) = aa_complex_complex(aa_com3342990543769683525omplex(Uu,Uub),Uua) ).

% ATP.lambda_414
tff(fact_10592_ATP_Olambda__415,axiom,
    ! [Uu: fun_co3258106519541737638omplex,Uua: nat,Uub: complex] : aa_complex_complex(aa_nat3829984526574172007omplex(aTP_Lamm_ara(Uu),Uua),Uub) = aa_nat_complex(aa_com8181001469867462503omplex(Uu,Uub),Uua) ).

% ATP.lambda_415
tff(fact_10593_ATP_Olambda__416,axiom,
    ! [Uu: fun_co636942507509095880at_nat,Uua: nat,Uub: complex] : aa_complex_nat(aa_nat3391284541783913865ex_nat(aTP_Lamm_amc(Uu),Uua),Uub) = aa_nat_nat(aa_com6383720667567721865at_nat(Uu,Uub),Uua) ).

% ATP.lambda_416
tff(fact_10594_ATP_Olambda__417,axiom,
    ! [Uu: fun_co7370231399432144932at_int,Uua: nat,Uub: complex] : aa_complex_int(aa_nat8436805559129492965ex_int(aTP_Lamm_awx(Uu),Uua),Uub) = aa_nat_int(aa_com2205869648058525157at_int(Uu,Uub),Uua) ).

% ATP.lambda_417
tff(fact_10595_ATP_Olambda__418,axiom,
    ! [Uu: fun_co333780937270390912nt_int,Uua: int,Uub: complex] : aa_complex_int(aa_int6126987627036542017ex_int(aTP_Lamm_axm(Uu),Uua),Uub) = aa_int_int(aa_com3205223359959579713nt_int(Uu,Uub),Uua) ).

% ATP.lambda_418
tff(fact_10596_ATP_Olambda__419,axiom,
    ! [Uu: fun_op8552747401449489031i_assn,Uua: list_VEBT_VEBTi,Uub: option_nat] : aa_option_nat_assn(aa_lis7387135221810377032t_assn(aTP_Lamm_bu(Uu),Uua),Uub) = aa_lis2508312872377092907i_assn(aa_opt2780285371362742984i_assn(Uu,Uub),Uua) ).

% ATP.lambda_419
tff(fact_10597_ATP_Olambda__420,axiom,
    ! [Uu: fun_re7208826350674946562omplex,Uua: complex,Uub: real] : aa_real_complex(aa_com4766338864777664963omplex(aTP_Lamm_amq(Uu),Uua),Uub) = aa_complex_complex(aa_rea2618301233319580611omplex(Uu,Uub),Uua) ).

% ATP.lambda_420
tff(fact_10598_ATP_Olambda__421,axiom,
    ! [Uu: fun_re4238438383274954274t_real,Uua: nat,Uub: real] : aa_real_real(aa_nat_fun_real_real(aTP_Lamm_aqv(Uu),Uua),Uub) = aa_nat_real(aa_real_fun_nat_real(Uu,Uub),Uua) ).

% ATP.lambda_421
tff(fact_10599_ATP_Olambda__422,axiom,
    ! [Uu: fun_real_fun_nat_nat,Uua: nat,Uub: real] : aa_real_nat(aa_nat_fun_real_nat(aTP_Lamm_alt(Uu),Uua),Uub) = aa_nat_nat(aa_real_fun_nat_nat(Uu,Uub),Uua) ).

% ATP.lambda_422
tff(fact_10600_ATP_Olambda__423,axiom,
    ! [Uu: fun_real_fun_nat_int,Uua: nat,Uub: real] : aa_real_int(aa_nat_fun_real_int(aTP_Lamm_awu(Uu),Uua),Uub) = aa_nat_int(aa_real_fun_nat_int(Uu,Uub),Uua) ).

% ATP.lambda_423
tff(fact_10601_ATP_Olambda__424,axiom,
    ! [Uu: fun_real_fun_int_int,Uua: int,Uub: real] : aa_real_int(aa_int_fun_real_int(aTP_Lamm_axi(Uu),Uua),Uub) = aa_int_int(aa_real_fun_int_int(Uu,Uub),Uua) ).

% ATP.lambda_424
tff(fact_10602_ATP_Olambda__425,axiom,
    ! [Uu: fun_li8825114373765265308i_assn,Uua: vEBT_VEBTi,Uub: list_VEBT_VEBTi] : aa_lis2508312872377092907i_assn(aa_VEB3739350836417479773i_assn(aTP_Lamm_gr(Uu),Uua),Uub) = aa_VEBT_VEBTi_assn(aa_lis5319508155010942429i_assn(Uu,Uub),Uua) ).

% ATP.lambda_425
tff(fact_10603_ATP_Olambda__426,axiom,
    ! [Uu: fun_li3309467822378783623t_assn,Uua: option_nat,Uub: list_VEBT_VEBTi] : aa_lis2508312872377092907i_assn(aa_opt2780285371362742984i_assn(aTP_Lamm_gn(Uu),Uua),Uub) = aa_option_nat_assn(aa_lis7387135221810377032t_assn(Uu,Uub),Uua) ).

% ATP.lambda_426
tff(fact_10604_ATP_Olambda__427,axiom,
    ! [Uu: fun_li5930421888187426471o_assn,Uua: $o,Uub: list_VEBT_VEBTi] : aa_lis2508312872377092907i_assn(aa_o_f6375964871679992646i_assn(aTP_Lamm_gt(Uu),(Uua)),Uub) = aa_o_assn(aa_lis3121611389386064016o_assn(Uu,Uub),(Uua)) ).

% ATP.lambda_427
tff(fact_10605_ATP_Olambda__428,axiom,
    ! [Uu: fun_li7023630697154767927t_assn,Uua: nat,Uub: list_VEBT_VEBTi] : aa_lis2508312872377092907i_assn(aa_nat6700791848136694008i_assn(aTP_Lamm_gp(Uu),Uua),Uub) = aa_nat_assn(aa_lis7348694692495225848t_assn(Uu,Uub),Uua) ).

% ATP.lambda_428
tff(fact_10606_ATP_Olambda__429,axiom,
    ! [Uu: fun_o_4874018796564477717i_assn,Uua: list_VEBT_VEBTi,Uub: $o] : aa_o_assn(aa_lis3121611389386064016o_assn(aTP_Lamm_by(Uu),Uua),(Uub)) = aa_lis2508312872377092907i_assn(aa_o_f6375964871679992646i_assn(Uu,(Uub)),Uua) ).

% ATP.lambda_429
tff(fact_10607_ATP_Olambda__430,axiom,
    ! [Uu: fun_o_7171189317104958676_VEBTi,Uua: fun_Pr238758252084143515Heap_o,Uub: $o] : aa_o_H7577119287286566245_VEBTi(aa_fun6586487760926256075_VEBTi(aTP_Lamm_yf(Uu),Uua),(Uub)) = aa_fun238413853627933730_VEBTi(aa_o_f8811364424157995653_VEBTi(Uu,(Uub)),Uua) ).

% ATP.lambda_430
tff(fact_10608_ATP_Olambda__431,axiom,
    ! [Uu: fun_o_2223814739247024763Heap_o,Uua: fun_Pr238758252084143515Heap_o,Uub: $o] : aa_o_H7399033255965945470Heap_o(aa_fun7917812164634591790Heap_o(aTP_Lamm_yh(Uu),Uua),(Uub)) = aa_fun6848502500920544641Heap_o(aa_o_f6784619210364167924Heap_o(Uu,(Uub)),Uua) ).

% ATP.lambda_431
tff(fact_10609_ATP_Olambda__432,axiom,
    ! [Uu: fun_na1591161823130732966omplex,Uua: complex,Uub: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_amu(Uu),Uua),Uub) = aa_complex_complex(aa_nat3829984526574172007omplex(Uu,Uub),Uua) ).

% ATP.lambda_432
tff(fact_10610_ATP_Olambda__433,axiom,
    ! [Uu: fun_na8005944528675429559i_assn,Uua: list_VEBT_VEBTi,Uub: nat] : aa_nat_assn(aa_lis7348694692495225848t_assn(aTP_Lamm_bw(Uu),Uua),Uub) = aa_lis2508312872377092907i_assn(aa_nat6700791848136694008i_assn(Uu,Uub),Uua) ).

% ATP.lambda_433
tff(fact_10611_ATP_Olambda__434,axiom,
    ! [Uu: fun_nat_fun_nat_real,Uua: nat,Uub: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_akj(Uu),Uua),Uub) = aa_nat_real(aa_nat_fun_nat_real(Uu,Uub),Uua) ).

% ATP.lambda_434
tff(fact_10612_ATP_Olambda__435,axiom,
    ! [Uu: fun_nat_fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_aka(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(Uu,Uub),Uua) ).

% ATP.lambda_435
tff(fact_10613_ATP_Olambda__436,axiom,
    ! [Uu: fun_nat_fun_nat_int,Uua: nat,Uub: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_avz(Uu),Uua),Uub) = aa_nat_int(aa_nat_fun_nat_int(Uu,Uub),Uua) ).

% ATP.lambda_436
tff(fact_10614_ATP_Olambda__437,axiom,
    ! [Uu: fun_nat_fun_int_int,Uua: int,Uub: nat] : aa_nat_int(aa_int_fun_nat_int(aTP_Lamm_awc(Uu),Uua),Uub) = aa_int_int(aa_nat_fun_int_int(Uu,Uub),Uua) ).

% ATP.lambda_437
tff(fact_10615_ATP_Olambda__438,axiom,
    ! [Uu: fun_na7267038673063590618_VEBTi,Uua: fun_Pr342922516890243977on_nat,Uub: nat] : aa_nat5408703056993926195_VEBTi(aa_fun1166700169754672579_VEBTi(aTP_Lamm_yj(Uu),Uua),Uub) = aa_fun765629416642892858_VEBTi(aa_nat5169694059731795907_VEBTi(Uu,Uub),Uua) ).

% ATP.lambda_438
tff(fact_10616_ATP_Olambda__439,axiom,
    ! [Uu: fun_na979242554834862325Heap_o,Uua: fun_Pr342922516890243977on_nat,Uub: nat] : aa_nat2910048432523430448Heap_o(aa_fun1690681901636463158Heap_o(aTP_Lamm_yl(Uu),Uua),Uub) = aa_fun4188577930510380905Heap_o(aa_nat5421218410817480758Heap_o(Uu,Uub),Uua) ).

% ATP.lambda_439
tff(fact_10617_ATP_Olambda__440,axiom,
    ! [Uu: fun_na5845961453008041334_VEBTi,Uua: fun_Pr238758252084143515Heap_o,Uub: nat] : aa_nat5408703056993926195_VEBTi(aa_fun5587972357748661675_VEBTi(aTP_Lamm_xx(Uu),Uua),Uub) = aa_fun238413853627933730_VEBTi(aa_nat4209467000370889527_VEBTi(Uu,Uub),Uua) ).

% ATP.lambda_440
tff(fact_10618_ATP_Olambda__441,axiom,
    ! [Uu: fun_na8517577583218006873Heap_o,Uua: fun_Pr238758252084143515Heap_o,Uub: nat] : aa_nat2910048432523430448Heap_o(aa_fun7995322860690487374Heap_o(aTP_Lamm_yb(Uu),Uua),Uub) = aa_fun6848502500920544641Heap_o(aa_nat205861980368755650Heap_o(Uu,Uub),Uua) ).

% ATP.lambda_441
tff(fact_10619_ATP_Olambda__442,axiom,
    ! [Uu: fun_in1307110748760623106omplex,Uua: complex,Uub: int] : aa_int_complex(aa_com9156536935073639875omplex(aTP_Lamm_amz(Uu),Uua),Uub) = aa_complex_complex(aa_int3545933452204062147omplex(Uu,Uub),Uua) ).

% ATP.lambda_442
tff(fact_10620_ATP_Olambda__443,axiom,
    ! [Uu: fun_int_fun_nat_real,Uua: nat,Uub: int] : aa_int_real(aa_nat_fun_int_real(aTP_Lamm_aqx(Uu),Uua),Uub) = aa_nat_real(aa_int_fun_nat_real(Uu,Uub),Uua) ).

% ATP.lambda_443
tff(fact_10621_ATP_Olambda__444,axiom,
    ! [Uu: fun_int_fun_nat_nat,Uua: nat,Uub: int] : aa_int_nat(aa_nat_fun_int_nat(aTP_Lamm_alx(Uu),Uua),Uub) = aa_nat_nat(aa_int_fun_nat_nat(Uu,Uub),Uua) ).

% ATP.lambda_444
tff(fact_10622_ATP_Olambda__445,axiom,
    ! [Uu: fun_int_fun_nat_int,Uua: nat,Uub: int] : aa_int_int(aa_nat_fun_int_int(aTP_Lamm_arc(Uu),Uua),Uub) = aa_nat_int(aa_int_fun_nat_int(Uu,Uub),Uua) ).

% ATP.lambda_445
tff(fact_10623_ATP_Olambda__446,axiom,
    ! [Uu: fun_int_fun_int_int,Uua: int,Uub: int] : aa_int_int(aa_int_fun_int_int(aTP_Lamm_akg(Uu),Uua),Uub) = aa_int_int(aa_int_fun_int_int(Uu,Uub),Uua) ).

% ATP.lambda_446
tff(fact_10624_ATP_Olambda__447,axiom,
    ! [Uu: fun_fu411293875173739919on_nat,Uua: produc3881548065746020326Ti_nat,Uub: fun_Pr342922516890243977on_nat] : aa_fun7698174670542568229on_nat(aa_Pro8654539709784947602on_nat(aTP_Lamm_acl(Uu),Uua),Uub) = aa_Pro2606730266749126658on_nat(aa_fun943833532922116944on_nat(Uu,Uub),Uua) ).

% ATP.lambda_447
tff(fact_10625_ATP_Olambda__448,axiom,
    ! [Uu: fun_fu2426956696497423235Heap_o,Uua: produc3881548065746020326Ti_nat,Uub: fun_Pr238758252084143515Heap_o] : aa_fun6848502500920544641Heap_o(aa_Pro5165804456746783526Heap_o(aTP_Lamm_acm(Uu),Uua),Uub) = aa_Pro3916655985231737292Heap_o(aa_fun5049358401508434812Heap_o(Uu,Uub),Uua) ).

% ATP.lambda_448
tff(fact_10626_ATP_Olambda__449,axiom,
    ! [Uu: fun_fu1151686773700218227_VEBTi,Uua: produc3960855945107176009Ti_nat,Uub: fun_Pr4404027214751104635_VEBTi] : aa_fun7494984083220284744_VEBTi(aa_Pro7504462699960347282_VEBTi(aTP_Lamm_acn(Uu),Uua),Uub) = aa_Pro721367001817565044_VEBTi(aa_fun5250146550359294516_VEBTi(Uu,Uub),Uua) ).

% ATP.lambda_449
tff(fact_10627_ATP_Olambda__450,axiom,
    ! [Uu: fun_fu3620910142932681on_nat,Uua: produc3960855945107176009Ti_nat,Uub: fun_Pr633315466666964198on_nat] : aa_fun1100322450986716680on_nat(aa_Pro1535628157467025618on_nat(aTP_Lamm_aco(Uu),Uua),Uub) = aa_Pro2727408364141080927on_nat(aa_fun7861206497237518090on_nat(Uu,Uub),Uua) ).

% ATP.lambda_450
tff(fact_10628_ATP_Olambda__451,axiom,
    ! [Uu: fun_fu8308473583445096899Heap_o,Uua: produc3960855945107176009Ti_nat,Uub: fun_Pr940977554174145406Heap_o] : aa_fun3870618101526761188Heap_o(aa_Pro9147008720614807654Heap_o(aTP_Lamm_acp(Uu),Uua),Uub) = aa_Pro6173831900156065711Heap_o(aa_fun55487300827498300Heap_o(Uu,Uub),Uua) ).

% ATP.lambda_451
tff(fact_10629_ATP_Olambda__452,axiom,
    ! [Uu: nat,Uua: real,Uub: nat] :
      aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_aqh(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,
          $ite(Uub = Uu,one_one_real,zero_zero_real)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),Uub)) ).

% ATP.lambda_452
tff(fact_10630_ATP_Olambda__453,axiom,
    ! [Uu: nat,Uua: int,Uub: nat] :
      aa_nat_int(aa_int_fun_nat_int(aTP_Lamm_aqi(Uu),Uua),Uub) = aa_int_int(aa_int_fun_int_int(times_times_int,
          $ite(Uub = Uu,one_one_int,zero_zero_int)),aa_nat_int(aa_int_fun_nat_int(power_power_int,Uua),Uub)) ).

% ATP.lambda_453
tff(fact_10631_ATP_Olambda__454,axiom,
    ! [Uu: code_integer,Uua: code_integer,Uub: code_integer] :
      aa_Cod3168588076859124107eger_o(aa_Cod1096751537565085541eger_o(aTP_Lamm_bcb(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_454
tff(fact_10632_ATP_Olambda__455,axiom,
    ! [Uu: $o,Uua: code_integer,Uub: $o] : aa_o_char(aa_Cod7842134615204088097o_char(aTP_Lamm_bcn((Uu)),Uua),(Uub)) = produc4188289175737317920o_char(aa_o_f2742674633914528860o_char(aTP_Lamm_bcm((Uu)),(Uub)),code_bit_cut_integer(Uua)) ).

% ATP.lambda_455
tff(fact_10633_ATP_Olambda__456,axiom,
    ! [Uu: rat,Uua: int,Uub: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_bbg(Uu),Uua),Uub) = aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_bbf(Uua),Uub)),quotient_of(Uu)) ).

% ATP.lambda_456
tff(fact_10634_ATP_Olambda__457,axiom,
    ! [Uu: rat,Uua: int,Uub: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_bbe(Uu),Uua),Uub) = aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_bbd(Uua),Uub)),quotient_of(Uu)) ).

% ATP.lambda_457
tff(fact_10635_ATP_Olambda__458,axiom,
    ! [Uu: rat,Uua: int,Uub: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_bbc(Uu),Uua),Uub) = aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_bbb(Uua),Uub)),quotient_of(Uu)) ).

% ATP.lambda_458
tff(fact_10636_ATP_Olambda__459,axiom,
    ! [Uu: rat,Uua: int,Uub: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_bba(Uu),Uua),Uub) = aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_baz(Uua),Uub)),quotient_of(Uu)) ).

% ATP.lambda_459
tff(fact_10637_ATP_Olambda__460,axiom,
    ! [Uu: rat,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_bay(Uu),Uua),Uub)
    <=> aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(aa_int2464330468366073224_int_o(aTP_Lamm_bax(Uua),Uub)),quotient_of(Uu)) ) ).

% ATP.lambda_460
tff(fact_10638_ATP_Olambda__461,axiom,
    ! [Uu: rat,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_baw(Uu),Uua),Uub)
    <=> aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(aa_int2464330468366073224_int_o(aTP_Lamm_bav(Uua),Uub)),quotient_of(Uu)) ) ).

% ATP.lambda_461
tff(fact_10639_ATP_Olambda__462,axiom,
    ! [Uu: fun_nat_fun_int_int,Uua: set_nat,Uub: int] : aa_int_int(aa_set3358596244511904025nt_int(aTP_Lamm_awd(Uu),Uua),Uub) = groups705719431365010083at_int(aa_int_fun_nat_int(aTP_Lamm_awc(Uu),Uub),Uua) ).

% ATP.lambda_462
tff(fact_10640_ATP_Olambda__463,axiom,
    ! [Uu: fun_nat_fun_nat_int,Uua: set_nat,Uub: nat] : aa_nat_int(aa_set2359242532610849469at_int(aTP_Lamm_awa(Uu),Uua),Uub) = groups705719431365010083at_int(aa_nat_fun_nat_int(aTP_Lamm_avz(Uu),Uub),Uua) ).

% ATP.lambda_463
tff(fact_10641_ATP_Olambda__464,axiom,
    ! [Uu: fun_int_fun_nat_int,Uua: set_int,Uub: nat] : aa_nat_int(aa_set6310687161026558745at_int(aTP_Lamm_awf(Uu),Uua),Uub) = groups1705073143266064639nt_int(aa_nat_fun_int_int(aTP_Lamm_arc(Uu),Uub),Uua) ).

% ATP.lambda_464
tff(fact_10642_ATP_Olambda__465,axiom,
    ! [Uu: fun_int_fun_int_int,Uua: set_int,Uub: int] : aa_int_int(aa_set7310040872927613301nt_int(aTP_Lamm_awh(Uu),Uua),Uub) = groups1705073143266064639nt_int(aa_int_fun_int_int(aTP_Lamm_akg(Uu),Uub),Uua) ).

% ATP.lambda_465
tff(fact_10643_ATP_Olambda__466,axiom,
    ! [Uu: fun_nat_fun_nat_nat,Uua: set_nat,Uub: nat] : aa_nat_nat(aa_set6537093552120046177at_nat(aTP_Lamm_awj(Uu),Uua),Uub) = groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_aka(Uu),Uub),Uua) ).

% ATP.lambda_466
tff(fact_10644_ATP_Olambda__467,axiom,
    ! [Uu: set_int,Uua: fun_int_fun_nat_int,Uub: nat] : aa_nat_int(aa_fun8710705519803144921at_int(aTP_Lamm_ard(Uu),Uua),Uub) = groups4538972089207619220nt_int(aa_nat_fun_int_int(aTP_Lamm_arc(Uua),Uub),Uu) ).

% ATP.lambda_467
tff(fact_10645_ATP_Olambda__468,axiom,
    ! [Uu: set_complex,Uua: fun_co3258106519541737638omplex,Uub: nat] : aa_nat_complex(aa_fun5003373910569599327omplex(aTP_Lamm_arb(Uu),Uua),Uub) = groups7754918857620584856omplex(aa_nat3829984526574172007omplex(aTP_Lamm_ara(Uua),Uub),Uu) ).

% ATP.lambda_468
tff(fact_10646_ATP_Olambda__469,axiom,
    ! [Uu: set_int,Uua: fun_int_fun_nat_real,Uub: nat] : aa_nat_real(aa_fun2842512154077127769t_real(aTP_Lamm_aqy(Uu),Uua),Uub) = groups8778361861064173332t_real(aa_nat_fun_int_real(aTP_Lamm_aqx(Uua),Uub),Uu) ).

% ATP.lambda_469
tff(fact_10647_ATP_Olambda__470,axiom,
    ! [Uu: set_real,Uua: fun_re4238438383274954274t_real,Uub: nat] : aa_nat_real(aa_fun628238740774635481t_real(aTP_Lamm_aqw(Uu),Uua),Uub) = groups8097168146408367636l_real(aa_nat_fun_real_real(aTP_Lamm_aqv(Uua),Uub),Uu) ).

% ATP.lambda_470
tff(fact_10648_ATP_Olambda__471,axiom,
    ! [Uu: set_VEBT_VEBT,Uua: fun_VE2989161857680785728t_real,Uub: nat] : aa_nat_real(aa_fun6147343916763992189t_real(aTP_Lamm_aqu(Uu),Uua),Uub) = groups2240296850493347238T_real(aa_nat4323386229672267365T_real(aTP_Lamm_aqt(Uua),Uub),Uu) ).

% ATP.lambda_471
tff(fact_10649_ATP_Olambda__472,axiom,
    ! [Uu: fun_nat_fun_nat_real,Uua: set_nat,Uub: nat] : aa_nat_real(aa_set1166323925324055613t_real(aTP_Lamm_akk(Uu),Uua),Uub) = groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_akj(Uu),Uub),Uua) ).

% ATP.lambda_472
tff(fact_10650_ATP_Olambda__473,axiom,
    ! [Uu: set_nat,Uua: fun_nat_fun_nat_real,Uub: nat] : aa_nat_real(aa_fun7039663905532832253t_real(aTP_Lamm_are(Uu),Uua),Uub) = groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_akj(Uua),Uub),Uu) ).

% ATP.lambda_473
tff(fact_10651_ATP_Olambda__474,axiom,
    ! [Uu: fun_int_fun_int_int,Uua: set_int,Uub: int] : aa_int_int(aa_set7310040872927613301nt_int(aTP_Lamm_akh(Uu),Uua),Uub) = groups4538972089207619220nt_int(aa_int_fun_int_int(aTP_Lamm_akg(Uu),Uub),Uua) ).

% ATP.lambda_474
tff(fact_10652_ATP_Olambda__475,axiom,
    ! [Uu: fun_co7673397065115134084omplex,Uua: set_complex,Uub: complex] : aa_complex_complex(aa_set7743095458595710203omplex(aTP_Lamm_ake(Uu),Uua),Uub) = groups7754918857620584856omplex(aa_com3342990543769683525omplex(aTP_Lamm_akd(Uu),Uub),Uua) ).

% ATP.lambda_475
tff(fact_10653_ATP_Olambda__476,axiom,
    ! [Uu: fun_nat_fun_nat_nat,Uua: set_nat,Uub: nat] : aa_nat_nat(aa_set6537093552120046177at_nat(aTP_Lamm_akb(Uu),Uua),Uub) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_aka(Uu),Uub)),Uua) ).

% ATP.lambda_476
tff(fact_10654_ATP_Olambda__477,axiom,
    ! [Uu: set_nat,Uua: fun_nat_fun_nat_nat,Uub: nat] : aa_nat_nat(aa_fun8466562714962205125at_nat(aTP_Lamm_aqz(Uu),Uua),Uub) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_aka(Uua),Uub)),Uu) ).

% ATP.lambda_477
tff(fact_10655_ATP_Olambda__478,axiom,
    ! [Uu: fun_nat_real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_ajy(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uu,Uub)),aa_nat_real(aa_real_fun_nat_real(power_power_real,zero_zero_real),Uub))),aa_nat_real(Uua,Uub)) ).

% ATP.lambda_478
tff(fact_10656_ATP_Olambda__479,axiom,
    ! [Uu: fun_nat_rat,Uua: fun_nat_rat,Uub: nat] : aa_nat_rat(aa_fun765758790203053448at_rat(aTP_Lamm_ajx(Uu),Uua),Uub) = divide_divide_rat(aa_rat_rat(times_times_rat(aa_nat_rat(Uu,Uub)),aa_nat_rat(power_power_rat(zero_zero_rat),Uub)),aa_nat_rat(Uua,Uub)) ).

% ATP.lambda_479
tff(fact_10657_ATP_Olambda__480,axiom,
    ! [Uu: real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_aze(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(Uua,Uub)),aa_nat_real(semiri2265585572941072030t_real,Uub))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uub)) ).

% ATP.lambda_480
tff(fact_10658_ATP_Olambda__481,axiom,
    ! [Uu: set_VEBT_VEBT,Uua: nat,Uub: list_VEBT_VEBT] :
      ( aa_list_VEBT_VEBT_o(aa_nat1274389614848952367VEBT_o(aTP_Lamm_adj(Uu),Uua),Uub)
    <=> ( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(set_VEBT_VEBT2(Uub)),Uu)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_s6755466524823107622T_VEBT(Uub)),Uua) ) ) ).

% ATP.lambda_481
tff(fact_10659_ATP_Olambda__482,axiom,
    ! [Uu: set_Code_integer,Uua: nat,Uub: list_Code_integer] :
      ( aa_lis2813719629158071341eger_o(aa_nat2697030861539126eger_o(aTP_Lamm_adm(Uu),Uua),Uub)
    <=> ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(set_Code_integer2(Uub)),Uu)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_s3445333598471063425nteger(Uub)),Uua) ) ) ).

% ATP.lambda_482
tff(fact_10660_ATP_Olambda__483,axiom,
    ! [Uu: set_Pr5949110396991348497n_assn,Uua: nat,Uub: list_P8527749157015355191n_assn] :
      ( aa_lis1404944545740551261assn_o(aa_nat2893684320935325170assn_o(aTP_Lamm_adk(Uu),Uua),Uub)
    <=> ( ord_le171416862856029873n_assn(set_Pr1139785259514867910n_assn(Uub),Uu)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_s6829681357464350627n_assn(Uub)),Uua) ) ) ).

% ATP.lambda_483
tff(fact_10661_ATP_Olambda__484,axiom,
    ! [Uu: set_complex,Uua: nat,Uub: list_complex] :
      ( aa_list_complex_o(aa_nat5940423528545770789plex_o(aTP_Lamm_adl(Uu),Uua),Uub)
    <=> ( aa_set_complex_o(ord_le211207098394363844omplex(set_complex2(Uub)),Uu)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_s3451745648224563538omplex(Uub)),Uua) ) ) ).

% ATP.lambda_484
tff(fact_10662_ATP_Olambda__485,axiom,
    ! [Uu: set_real,Uua: nat,Uub: list_real] :
      ( aa_list_real_o(aa_nat3629845878562903719real_o(aTP_Lamm_adn(Uu),Uua),Uub)
    <=> ( aa_set_real_o(ord_less_eq_set_real(set_real2(Uub)),Uu)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_size_list_real(Uub)),Uua) ) ) ).

% ATP.lambda_485
tff(fact_10663_ATP_Olambda__486,axiom,
    ! [Uu: set_o,Uua: nat,Uub: list_o] :
      ( aa_list_o_o(aa_nat_fun_list_o_o(aTP_Lamm_ado(Uu),Uua),Uub)
    <=> ( ord_less_eq_set_o(set_o2(Uub),Uu)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_size_list_o(Uub)),Uua) ) ) ).

% ATP.lambda_486
tff(fact_10664_ATP_Olambda__487,axiom,
    ! [Uu: set_nat,Uua: nat,Uub: list_nat] :
      ( aa_list_nat_o(aa_nat5883842100386203267_nat_o(aTP_Lamm_adp(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(ord_less_eq_set_nat(set_nat2(Uub)),Uu)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_size_list_nat(Uub)),Uua) ) ) ).

% ATP.lambda_487
tff(fact_10665_ATP_Olambda__488,axiom,
    ! [Uu: set_int,Uua: nat,Uub: list_int] :
      ( aa_list_int_o(aa_nat3638697132715757351_int_o(aTP_Lamm_adq(Uu),Uua),Uub)
    <=> ( aa_set_int_o(ord_less_eq_set_int(set_int2(Uub)),Uu)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_size_list_int(Uub)),Uua) ) ) ).

% ATP.lambda_488
tff(fact_10666_ATP_Olambda__489,axiom,
    ! [Uu: set_VEBT_VEBT,Uua: nat,Uub: list_VEBT_VEBT] :
      ( aa_list_VEBT_VEBT_o(aa_nat1274389614848952367VEBT_o(aTP_Lamm_adb(Uu),Uua),Uub)
    <=> ( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(set_VEBT_VEBT2(Uub)),Uu)
        & ( size_s6755466524823107622T_VEBT(Uub) = Uua ) ) ) ).

% ATP.lambda_489
tff(fact_10667_ATP_Olambda__490,axiom,
    ! [Uu: set_Code_integer,Uua: nat,Uub: list_Code_integer] :
      ( aa_lis2813719629158071341eger_o(aa_nat2697030861539126eger_o(aTP_Lamm_ade(Uu),Uua),Uub)
    <=> ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(set_Code_integer2(Uub)),Uu)
        & ( size_s3445333598471063425nteger(Uub) = Uua ) ) ) ).

% ATP.lambda_490
tff(fact_10668_ATP_Olambda__491,axiom,
    ! [Uu: set_Pr5949110396991348497n_assn,Uua: nat,Uub: list_P8527749157015355191n_assn] :
      ( aa_lis1404944545740551261assn_o(aa_nat2893684320935325170assn_o(aTP_Lamm_adc(Uu),Uua),Uub)
    <=> ( ord_le171416862856029873n_assn(set_Pr1139785259514867910n_assn(Uub),Uu)
        & ( size_s6829681357464350627n_assn(Uub) = Uua ) ) ) ).

% ATP.lambda_491
tff(fact_10669_ATP_Olambda__492,axiom,
    ! [Uu: set_complex,Uua: nat,Uub: list_complex] :
      ( aa_list_complex_o(aa_nat5940423528545770789plex_o(aTP_Lamm_add(Uu),Uua),Uub)
    <=> ( aa_set_complex_o(ord_le211207098394363844omplex(set_complex2(Uub)),Uu)
        & ( size_s3451745648224563538omplex(Uub) = Uua ) ) ) ).

% ATP.lambda_492
tff(fact_10670_ATP_Olambda__493,axiom,
    ! [Uu: set_real,Uua: nat,Uub: list_real] :
      ( aa_list_real_o(aa_nat3629845878562903719real_o(aTP_Lamm_adf(Uu),Uua),Uub)
    <=> ( aa_set_real_o(ord_less_eq_set_real(set_real2(Uub)),Uu)
        & ( size_size_list_real(Uub) = Uua ) ) ) ).

% ATP.lambda_493
tff(fact_10671_ATP_Olambda__494,axiom,
    ! [Uu: set_o,Uua: nat,Uub: list_o] :
      ( aa_list_o_o(aa_nat_fun_list_o_o(aTP_Lamm_adg(Uu),Uua),Uub)
    <=> ( ord_less_eq_set_o(set_o2(Uub),Uu)
        & ( size_size_list_o(Uub) = Uua ) ) ) ).

% ATP.lambda_494
tff(fact_10672_ATP_Olambda__495,axiom,
    ! [Uu: set_nat,Uua: nat,Uub: list_nat] :
      ( aa_list_nat_o(aa_nat5883842100386203267_nat_o(aTP_Lamm_adh(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(ord_less_eq_set_nat(set_nat2(Uub)),Uu)
        & ( size_size_list_nat(Uub) = Uua ) ) ) ).

% ATP.lambda_495
tff(fact_10673_ATP_Olambda__496,axiom,
    ! [Uu: set_int,Uua: nat,Uub: list_int] :
      ( aa_list_int_o(aa_nat3638697132715757351_int_o(aTP_Lamm_adi(Uu),Uua),Uub)
    <=> ( aa_set_int_o(ord_less_eq_set_int(set_int2(Uub)),Uu)
        & ( size_size_list_int(Uub) = Uua ) ) ) ).

% ATP.lambda_496
tff(fact_10674_ATP_Olambda__497,axiom,
    ! [Uu: list_VEBT_VEBT,Uua: array_VEBT_VEBTi,Uub: list_VEBT_VEBTi] : aa_lis2508312872377092907i_assn(aa_arr7909941341466502432i_assn(aTP_Lamm_bf(Uu),Uua),Uub) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_lis2508312872377092907i_assn(snga_assn_VEBT_VEBTi(Uua),Uub)),aa_lis2508312872377092907i_assn(vEBT_L6296928887356842470_VEBTi(vEBT_vebt_assn_raw,Uu),Uub)) ).

% ATP.lambda_497
tff(fact_10675_ATP_Olambda__498,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_ayw(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_498
tff(fact_10676_ATP_Olambda__499,axiom,
    ! [Uu: vEBT_VEBT,Uua: set_VEBT_VEBT,Uub: vEBT_VEBT] :
      ( aa_VEBT_VEBT_o(aa_set2497645856053385599VEBT_o(aTP_Lamm_hx(Uu),Uua),Uub)
    <=> ( ( Uub = Uu )
        | aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Uub),Uua) ) ) ).

% ATP.lambda_499
tff(fact_10677_ATP_Olambda__500,axiom,
    ! [Uu: product_prod_int_int,Uua: set_Pr958786334691620121nt_int,Uub: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aa_set2224860389584585407_int_o(aTP_Lamm_ic(Uu),Uua),Uub)
    <=> ( ( Uub = Uu )
        | member5262025264175285858nt_int(Uub,Uua) ) ) ).

% ATP.lambda_500
tff(fact_10678_ATP_Olambda__501,axiom,
    ! [Uu: complex,Uua: set_complex,Uub: complex] :
      ( aa_complex_o(aa_set1742385355640705825plex_o(aTP_Lamm_ib(Uu),Uua),Uub)
    <=> ( ( Uub = Uu )
        | aa_set_complex_o(member_complex(Uub),Uua) ) ) ).

% ATP.lambda_501
tff(fact_10679_ATP_Olambda__502,axiom,
    ! [Uu: real,Uua: set_real,Uub: real] :
      ( aa_real_o(aa_set2323979645938365349real_o(aTP_Lamm_hy(Uu),Uua),Uub)
    <=> ( ( Uub = Uu )
        | aa_set_real_o(member_real(Uub),Uua) ) ) ).

% ATP.lambda_502
tff(fact_10680_ATP_Olambda__503,axiom,
    ! [Uu: nat,Uua: set_nat,Uub: nat] :
      ( aa_nat_o(aa_set_nat_fun_nat_o(aTP_Lamm_hz(Uu),Uua),Uub)
    <=> ( ( Uub = Uu )
        | aa_set_nat_o(member_nat(Uub),Uua) ) ) ).

% ATP.lambda_503
tff(fact_10681_ATP_Olambda__504,axiom,
    ! [Uu: int,Uua: set_int,Uub: int] :
      ( aa_int_o(aa_set_int_fun_int_o(aTP_Lamm_ia(Uu),Uua),Uub)
    <=> ( ( Uub = Uu )
        | aa_set_int_o(member_int(Uub),Uua) ) ) ).

% ATP.lambda_504
tff(fact_10682_ATP_Olambda__505,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_azu(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_505
tff(fact_10683_ATP_Olambda__506,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_aep(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_506
tff(fact_10684_ATP_Olambda__507,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_azt(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_507
tff(fact_10685_ATP_Olambda__508,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_aer(Uu),Uua),Uub)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Uu),Uub)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,Uub),Uua) ) ) ).

% ATP.lambda_508
tff(fact_10686_ATP_Olambda__509,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_aes(Uu),Uua),Uub)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,Uu),Uub)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Uub),Uua) ) ) ).

% ATP.lambda_509
tff(fact_10687_ATP_Olambda__510,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_aeq(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_510
tff(fact_10688_ATP_Olambda__511,axiom,
    ! [Uu: vEBT_VEBT,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_gb(Uu),Uua),Uub)
    <=> ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,Uu),Uub)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uua),Uub) ) ) ).

% ATP.lambda_511
tff(fact_10689_ATP_Olambda__512,axiom,
    ! [Uu: vEBT_VEBT,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_gc(Uu),Uua),Uub)
    <=> ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,Uu),Uub)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uub),Uua) ) ) ).

% ATP.lambda_512
tff(fact_10690_ATP_Olambda__513,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_bbk(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_513
tff(fact_10691_ATP_Olambda__514,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_bbi(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_514
tff(fact_10692_ATP_Olambda__515,axiom,
    ! [Uu: set_Code_integer,Uua: set_Code_integer,Uub: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(aa_set4044230130517860621eger_o(aTP_Lamm_acx(Uu),Uua),Uub)
    <=> ( minus_2355218937544613996nteger(Uub,Uu) = minus_2355218937544613996nteger(Uua,Uu) ) ) ).

% ATP.lambda_515
tff(fact_10693_ATP_Olambda__516,axiom,
    ! [Uu: set_complex,Uua: set_complex,Uub: set_complex] :
      ( aa_set_complex_o(aa_set8643480048548153067plex_o(aTP_Lamm_acw(Uu),Uua),Uub)
    <=> ( minus_811609699411566653omplex(Uub,Uu) = minus_811609699411566653omplex(Uua,Uu) ) ) ).

% ATP.lambda_516
tff(fact_10694_ATP_Olambda__517,axiom,
    ! [Uu: set_nat,Uua: set_nat,Uub: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(aTP_Lamm_acy(Uu),Uua),Uub)
    <=> ( minus_minus_set_nat(Uub,Uu) = minus_minus_set_nat(Uua,Uu) ) ) ).

% ATP.lambda_517
tff(fact_10695_ATP_Olambda__518,axiom,
    ! [Uu: set_int,Uua: set_int,Uub: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(aTP_Lamm_acv(Uu),Uua),Uub)
    <=> ( minus_minus_set_int(Uub,Uu) = minus_minus_set_int(Uua,Uu) ) ) ).

% ATP.lambda_518
tff(fact_10696_ATP_Olambda__519,axiom,
    ! [Uu: set_VEBT_VEBT,Uua: fun_VEBT_VEBT_o,Uub: vEBT_VEBT] :
      ( aa_VEBT_VEBT_o(aa_fun8193769468198689164VEBT_o(aTP_Lamm_gh(Uu),Uua),Uub)
    <=> ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Uub),Uu)
        & aa_VEBT_VEBT_o(Uua,Uub) ) ) ).

% ATP.lambda_519
tff(fact_10697_ATP_Olambda__520,axiom,
    ! [Uu: set_Code_integer,Uua: fun_Code_integer_o,Uub: code_integer] :
      ( aa_Code_integer_o(aa_fun8214818745163655228eger_o(aTP_Lamm_ano(Uu),Uua),Uub)
    <=> ( aa_set5643387332196575111eger_o(member_Code_integer(Uub),Uu)
        & aa_Code_integer_o(Uua,Uub) ) ) ).

% ATP.lambda_520
tff(fact_10698_ATP_Olambda__521,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_gl(Uu),Uua),Uub)
    <=> ( member5262025264175285858nt_int(Uub,Uu)
        & aa_Pro4592068669816069595_int_o(Uua,Uub) ) ) ).

% ATP.lambda_521
tff(fact_10699_ATP_Olambda__522,axiom,
    ! [Uu: set_complex,Uua: fun_complex_o,Uub: complex] :
      ( aa_complex_o(aa_fun2514317545224092796plex_o(aTP_Lamm_gk(Uu),Uua),Uub)
    <=> ( aa_set_complex_o(member_complex(Uub),Uu)
        & aa_complex_o(Uua,Uub) ) ) ).

% ATP.lambda_522
tff(fact_10700_ATP_Olambda__523,axiom,
    ! [Uu: set_real,Uua: fun_real_o,Uub: real] :
      ( aa_real_o(aa_fun5868358341814070780real_o(aTP_Lamm_gi(Uu),Uua),Uub)
    <=> ( aa_set_real_o(member_real(Uub),Uu)
        & aa_real_o(Uua,Uub) ) ) ).

% ATP.lambda_523
tff(fact_10701_ATP_Olambda__524,axiom,
    ! [Uu: set_nat,Uua: fun_nat_o,Uub: nat] :
      ( aa_nat_o(aa_fun5937193402907943164_nat_o(aTP_Lamm_gj(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(member_nat(Uub),Uu)
        & aa_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_524
tff(fact_10702_ATP_Olambda__525,axiom,
    ! [Uu: set_int,Uua: fun_int_o,Uub: int] :
      ( aa_int_o(aa_fun6243517787477317628_int_o(aTP_Lamm_gm(Uu),Uua),Uub)
    <=> ( aa_set_int_o(member_int(Uub),Uu)
        & aa_int_o(Uua,Uub) ) ) ).

% ATP.lambda_525
tff(fact_10703_ATP_Olambda__526,axiom,
    ! [Uu: vEBT_VEBT,Uua: fun_VEBT_VEBT_o,Uub: vEBT_VEBT] :
      ( aa_VEBT_VEBT_o(aa_fun8193769468198689164VEBT_o(aTP_Lamm_ij(Uu),Uua),Uub)
    <=> ( ( Uu = Uub )
        & aa_VEBT_VEBT_o(Uua,Uub) ) ) ).

% ATP.lambda_526
tff(fact_10704_ATP_Olambda__527,axiom,
    ! [Uu: product_prod_int_int,Uua: fun_Pr3238374608701176162_int_o,Uub: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aa_fun5349570537165229698_int_o(aTP_Lamm_il(Uu),Uua),Uub)
    <=> ( ( Uu = Uub )
        & aa_Pro4592068669816069595_int_o(Uua,Uub) ) ) ).

% ATP.lambda_527
tff(fact_10705_ATP_Olambda__528,axiom,
    ! [Uu: complex,Uua: fun_complex_o,Uub: complex] :
      ( aa_complex_o(aa_fun2514317545224092796plex_o(aTP_Lamm_ik(Uu),Uua),Uub)
    <=> ( ( Uu = Uub )
        & aa_complex_o(Uua,Uub) ) ) ).

% ATP.lambda_528
tff(fact_10706_ATP_Olambda__529,axiom,
    ! [Uu: real,Uua: fun_real_o,Uub: real] :
      ( aa_real_o(aa_fun5868358341814070780real_o(aTP_Lamm_io(Uu),Uua),Uub)
    <=> ( ( Uu = Uub )
        & aa_real_o(Uua,Uub) ) ) ).

% ATP.lambda_529
tff(fact_10707_ATP_Olambda__530,axiom,
    ! [Uu: nat,Uua: fun_nat_o,Uub: nat] :
      ( aa_nat_o(aa_fun5937193402907943164_nat_o(aTP_Lamm_im(Uu),Uua),Uub)
    <=> ( ( Uu = Uub )
        & aa_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_530
tff(fact_10708_ATP_Olambda__531,axiom,
    ! [Uu: int,Uua: fun_int_o,Uub: int] :
      ( aa_int_o(aa_fun6243517787477317628_int_o(aTP_Lamm_in(Uu),Uua),Uub)
    <=> ( ( Uu = Uub )
        & aa_int_o(Uua,Uub) ) ) ).

% ATP.lambda_531
tff(fact_10709_ATP_Olambda__532,axiom,
    ! [Uu: vEBT_VEBT,Uua: fun_VEBT_VEBT_o,Uub: vEBT_VEBT] :
      ( aa_VEBT_VEBT_o(aa_fun8193769468198689164VEBT_o(aTP_Lamm_id(Uu),Uua),Uub)
    <=> ( ( Uub = Uu )
        & aa_VEBT_VEBT_o(Uua,Uub) ) ) ).

% ATP.lambda_532
tff(fact_10710_ATP_Olambda__533,axiom,
    ! [Uu: product_prod_int_int,Uua: fun_Pr3238374608701176162_int_o,Uub: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aa_fun5349570537165229698_int_o(aTP_Lamm_if(Uu),Uua),Uub)
    <=> ( ( Uub = Uu )
        & aa_Pro4592068669816069595_int_o(Uua,Uub) ) ) ).

% ATP.lambda_533
tff(fact_10711_ATP_Olambda__534,axiom,
    ! [Uu: complex,Uua: fun_complex_o,Uub: complex] :
      ( aa_complex_o(aa_fun2514317545224092796plex_o(aTP_Lamm_ie(Uu),Uua),Uub)
    <=> ( ( Uub = Uu )
        & aa_complex_o(Uua,Uub) ) ) ).

% ATP.lambda_534
tff(fact_10712_ATP_Olambda__535,axiom,
    ! [Uu: real,Uua: fun_real_o,Uub: real] :
      ( aa_real_o(aa_fun5868358341814070780real_o(aTP_Lamm_ii(Uu),Uua),Uub)
    <=> ( ( Uub = Uu )
        & aa_real_o(Uua,Uub) ) ) ).

% ATP.lambda_535
tff(fact_10713_ATP_Olambda__536,axiom,
    ! [Uu: nat,Uua: fun_nat_o,Uub: nat] :
      ( aa_nat_o(aa_fun5937193402907943164_nat_o(aTP_Lamm_ig(Uu),Uua),Uub)
    <=> ( ( Uub = Uu )
        & aa_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_536
tff(fact_10714_ATP_Olambda__537,axiom,
    ! [Uu: int,Uua: fun_int_o,Uub: int] :
      ( aa_int_o(aa_fun6243517787477317628_int_o(aTP_Lamm_ih(Uu),Uua),Uub)
    <=> ( ( Uub = Uu )
        & aa_int_o(Uua,Uub) ) ) ).

% ATP.lambda_537
tff(fact_10715_ATP_Olambda__538,axiom,
    ! [Uu: set_VEBT_VEBT,Uua: fun_VEBT_VEBT_uint32,Uub: vEBT_VEBT] :
      ( aa_VEBT_VEBT_o(aa_fun6272599145770227614VEBT_o(aTP_Lamm_ahp(Uu),Uua),Uub)
    <=> ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Uub),Uu)
        & ( aa_VEBT_VEBT_uint32(Uua,Uub) != zero_zero_uint32 ) ) ) ).

% ATP.lambda_538
tff(fact_10716_ATP_Olambda__539,axiom,
    ! [Uu: set_VEBT_VEBT,Uua: fun_VEBT_VEBT_real,Uub: vEBT_VEBT] :
      ( aa_VEBT_VEBT_o(aa_fun1965331558590768990VEBT_o(aTP_Lamm_aib(Uu),Uua),Uub)
    <=> ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Uub),Uu)
        & ( aa_VEBT_VEBT_real(Uua,Uub) != zero_zero_real ) ) ) ).

% ATP.lambda_539
tff(fact_10717_ATP_Olambda__540,axiom,
    ! [Uu: set_Code_integer,Uua: fun_Co1554742298434871494uint32,Uub: code_integer] :
      ( aa_Code_integer_o(aa_fun8926496376162642256eger_o(aTP_Lamm_ahz(Uu),Uua),Uub)
    <=> ( aa_set5643387332196575111eger_o(member_Code_integer(Uub),Uu)
        & ( aa_Cod8401701657498852871uint32(Uua,Uub) != zero_zero_uint32 ) ) ) ).

% ATP.lambda_540
tff(fact_10718_ATP_Olambda__541,axiom,
    ! [Uu: set_complex,Uua: fun_complex_uint32,Uub: complex] :
      ( aa_complex_o(aa_fun436548534304416686plex_o(aTP_Lamm_ahx(Uu),Uua),Uub)
    <=> ( aa_set_complex_o(member_complex(Uub),Uu)
        & ( aa_complex_uint32(Uua,Uub) != zero_zero_uint32 ) ) ) ).

% ATP.lambda_541
tff(fact_10719_ATP_Olambda__542,axiom,
    ! [Uu: set_real,Uua: fun_real_uint32,Uub: real] :
      ( aa_real_o(aa_fun7556296611544133298real_o(aTP_Lamm_ahr(Uu),Uua),Uub)
    <=> ( aa_set_real_o(member_real(Uub),Uu)
        & ( aa_real_uint32(Uua,Uub) != zero_zero_uint32 ) ) ) ).

% ATP.lambda_542
tff(fact_10720_ATP_Olambda__543,axiom,
    ! [Uu: set_real,Uua: fun_real_real,Uub: real] :
      ( aa_real_o(aa_fun2895569901277098738real_o(aTP_Lamm_aid(Uu),Uua),Uub)
    <=> ( aa_set_real_o(member_real(Uub),Uu)
        & ( aa_real_real(Uua,Uub) != zero_zero_real ) ) ) ).

% ATP.lambda_543
tff(fact_10721_ATP_Olambda__544,axiom,
    ! [Uu: set_nat,Uua: fun_nat_uint32,Uub: nat] :
      ( aa_nat_o(aa_fun1182053767699555178_nat_o(aTP_Lamm_aht(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(member_nat(Uub),Uu)
        & ( aa_nat_uint32(Uua,Uub) != zero_zero_uint32 ) ) ) ).

% ATP.lambda_544
tff(fact_10722_ATP_Olambda__545,axiom,
    ! [Uu: set_nat,Uua: fun_nat_real,Uub: nat] :
      ( aa_nat_o(aa_fun6456149469358400426_nat_o(aTP_Lamm_aif(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(member_nat(Uub),Uu)
        & ( aa_nat_real(Uua,Uub) != zero_zero_real ) ) ) ).

% ATP.lambda_545
tff(fact_10723_ATP_Olambda__546,axiom,
    ! [Uu: set_int,Uua: fun_int_uint32,Uub: int] :
      ( aa_int_o(aa_fun6992502331602369202_int_o(aTP_Lamm_ahv(Uu),Uua),Uub)
    <=> ( aa_set_int_o(member_int(Uub),Uu)
        & ( aa_int_uint32(Uua,Uub) != zero_zero_uint32 ) ) ) ).

% ATP.lambda_546
tff(fact_10724_ATP_Olambda__547,axiom,
    ! [Uu: set_int,Uua: fun_int_real,Uub: int] :
      ( aa_int_o(aa_fun4444401208693448434_int_o(aTP_Lamm_aih(Uu),Uua),Uub)
    <=> ( aa_set_int_o(member_int(Uub),Uu)
        & ( aa_int_real(Uua,Uub) != zero_zero_real ) ) ) ).

% ATP.lambda_547
tff(fact_10725_ATP_Olambda__548,axiom,
    ! [Uu: set_VEBT_VEBT,Uua: fun_VEBT_VEBT_real,Uub: vEBT_VEBT] :
      ( aa_VEBT_VEBT_o(aa_fun1965331558590768990VEBT_o(aTP_Lamm_agv(Uu),Uua),Uub)
    <=> ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Uub),Uu)
        & ( aa_VEBT_VEBT_real(Uua,Uub) != one_one_real ) ) ) ).

% ATP.lambda_548
tff(fact_10726_ATP_Olambda__549,axiom,
    ! [Uu: set_VEBT_VEBT,Uua: fun_VEBT_VEBT_rat,Uub: vEBT_VEBT] :
      ( aa_VEBT_VEBT_o(aa_fun5889122065779655690VEBT_o(aTP_Lamm_ahh(Uu),Uua),Uub)
    <=> ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Uub),Uu)
        & ( aa_VEBT_VEBT_rat(Uua,Uub) != one_one_rat ) ) ) ).

% ATP.lambda_549
tff(fact_10727_ATP_Olambda__550,axiom,
    ! [Uu: set_Code_integer,Uua: fun_Co3960896333048504454r_real,Uub: code_integer] :
      ( aa_Code_integer_o(aa_fun3640226575523069072eger_o(aTP_Lamm_ahf(Uu),Uua),Uub)
    <=> ( aa_set5643387332196575111eger_o(member_Code_integer(Uub),Uu)
        & ( aa_Code_integer_real(Uua,Uub) != one_one_real ) ) ) ).

% ATP.lambda_550
tff(fact_10728_ATP_Olambda__551,axiom,
    ! [Uu: set_complex,Uua: fun_complex_real,Uub: complex] :
      ( aa_complex_o(aa_fun6195079535267300846plex_o(aTP_Lamm_ahd(Uu),Uua),Uub)
    <=> ( aa_set_complex_o(member_complex(Uub),Uu)
        & ( aa_complex_real(Uua,Uub) != one_one_real ) ) ) ).

% ATP.lambda_551
tff(fact_10729_ATP_Olambda__552,axiom,
    ! [Uu: set_real,Uua: fun_real_real,Uub: real] :
      ( aa_real_o(aa_fun2895569901277098738real_o(aTP_Lamm_agx(Uu),Uua),Uub)
    <=> ( aa_set_real_o(member_real(Uub),Uu)
        & ( aa_real_real(Uua,Uub) != one_one_real ) ) ) ).

% ATP.lambda_552
tff(fact_10730_ATP_Olambda__553,axiom,
    ! [Uu: set_real,Uua: fun_real_rat,Uub: real] :
      ( aa_real_o(aa_fun3209485225100452678real_o(aTP_Lamm_ahj(Uu),Uua),Uub)
    <=> ( aa_set_real_o(member_real(Uub),Uu)
        & ( aa_real_rat(Uua,Uub) != one_one_rat ) ) ) ).

% ATP.lambda_553
tff(fact_10731_ATP_Olambda__554,axiom,
    ! [Uu: set_nat,Uua: fun_nat_real,Uub: nat] :
      ( aa_nat_o(aa_fun6456149469358400426_nat_o(aTP_Lamm_agz(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(member_nat(Uub),Uu)
        & ( aa_nat_real(Uua,Uub) != one_one_real ) ) ) ).

% ATP.lambda_554
tff(fact_10732_ATP_Olambda__555,axiom,
    ! [Uu: set_nat,Uua: fun_nat_rat,Uub: nat] :
      ( aa_nat_o(aa_fun1118977351124560894_nat_o(aTP_Lamm_ahl(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(member_nat(Uub),Uu)
        & ( aa_nat_rat(Uua,Uub) != one_one_rat ) ) ) ).

% ATP.lambda_555
tff(fact_10733_ATP_Olambda__556,axiom,
    ! [Uu: set_int,Uua: fun_int_real,Uub: int] :
      ( aa_int_o(aa_fun4444401208693448434_int_o(aTP_Lamm_ahb(Uu),Uua),Uub)
    <=> ( aa_set_int_o(member_int(Uub),Uu)
        & ( aa_int_real(Uua,Uub) != one_one_real ) ) ) ).

% ATP.lambda_556
tff(fact_10734_ATP_Olambda__557,axiom,
    ! [Uu: set_int,Uua: fun_int_rat,Uub: int] :
      ( aa_int_o(aa_fun7094158214180239686_int_o(aTP_Lamm_ahn(Uu),Uua),Uub)
    <=> ( aa_set_int_o(member_int(Uub),Uu)
        & ( aa_int_rat(Uua,Uub) != one_one_rat ) ) ) ).

% ATP.lambda_557
tff(fact_10735_ATP_Olambda__558,axiom,
    ! [Uu: vEBT_VEBT,Uua: vEBT_VEBTi,Uub: $o] : aa_o_assn(aa_VEB6687300313434771872o_assn(aTP_Lamm_ej(Uu),Uua),(Uub)) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Uu),Uua)),aa_o_assn(pure_assn,(Uub) = aa_VEBT_VEBT_o(vEBT_VEBT_minNull,Uu))) ).

% ATP.lambda_558
tff(fact_10736_ATP_Olambda__559,axiom,
    ! [Uu: vEBT_VEBT,Uua: vEBT_VEBTi,Uub: option_nat] : aa_option_nat_assn(aa_VEB7611521244314475576t_assn(aTP_Lamm_ar(Uu),Uua),Uub) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Uu),Uua)),aa_o_assn(pure_assn,Uub = vEBT_vebt_mint(Uu))) ).

% ATP.lambda_559
tff(fact_10737_ATP_Olambda__560,axiom,
    ! [Uu: vEBT_VEBT,Uua: vEBT_VEBTi,Uub: option_nat] : aa_option_nat_assn(aa_VEB7611521244314475576t_assn(aTP_Lamm_as(Uu),Uua),Uub) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Uu),Uua)),aa_o_assn(pure_assn,Uub = vEBT_vebt_maxt(Uu))) ).

% ATP.lambda_560
tff(fact_10738_ATP_Olambda__561,axiom,
    ! [Uu: array_VEBT_VEBTi,Uua: list_VEBT_VEBTi,Uub: nat] : aa_nat_assn(aa_lis7348694692495225848t_assn(aTP_Lamm_jm(Uu),Uua),Uub) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_lis2508312872377092907i_assn(snga_assn_VEBT_VEBTi(Uu),Uua)),aa_o_assn(pure_assn,Uub = size_s7982070591426661849_VEBTi(Uua))) ).

% ATP.lambda_561
tff(fact_10739_ATP_Olambda__562,axiom,
    ! [Uu: array_o,Uua: list_o,Uub: nat] : aa_nat_assn(aa_lis7996121028773434071t_assn(aTP_Lamm_jj(Uu),Uua),Uub) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,snga_assn_o(Uu,Uua)),aa_o_assn(pure_assn,Uub = size_size_list_o(Uua))) ).

% ATP.lambda_562
tff(fact_10740_ATP_Olambda__563,axiom,
    ! [Uu: array_nat,Uua: list_nat,Uub: nat] : aa_nat_assn(aa_lis6041139530851742867t_assn(aTP_Lamm_jk(Uu),Uua),Uub) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,snga_assn_nat(Uu,Uua)),aa_o_assn(pure_assn,Uub = size_size_list_nat(Uua))) ).

% ATP.lambda_563
tff(fact_10741_ATP_Olambda__564,axiom,
    ! [Uu: array_int,Uua: list_int,Uub: nat] : aa_nat_assn(aa_lis2816288184639765743t_assn(aTP_Lamm_jl(Uu),Uua),Uub) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,snga_assn_int(Uu,Uua)),aa_o_assn(pure_assn,Uub = size_size_list_int(Uua))) ).

% ATP.lambda_564
tff(fact_10742_ATP_Olambda__565,axiom,
    ! [Uu: array_VEBT_VEBTi,Uua: list_VEBT_VEBTi,Uub: list_VEBT_VEBTi] : aa_lis2508312872377092907i_assn(aa_lis3169797078229555565i_assn(aTP_Lamm_agu(Uu),Uua),Uub) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_lis2508312872377092907i_assn(snga_assn_VEBT_VEBTi(Uu),Uua)),aa_o_assn(pure_assn,Uub = Uua)) ).

% ATP.lambda_565
tff(fact_10743_ATP_Olambda__566,axiom,
    ! [Uu: set_VEBT_VEBT,Uua: set_VEBT_VEBT,Uub: vEBT_VEBT] :
      ( aa_VEBT_VEBT_o(aa_set2497645856053385599VEBT_o(aTP_Lamm_gz(Uu),Uua),Uub)
    <=> ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Uub),Uu)
        & ~ aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Uub),Uua) ) ) ).

% ATP.lambda_566
tff(fact_10744_ATP_Olambda__567,axiom,
    ! [Uu: set_Pr958786334691620121nt_int,Uua: set_Pr958786334691620121nt_int,Uub: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aa_set2224860389584585407_int_o(aTP_Lamm_hd(Uu),Uua),Uub)
    <=> ( member5262025264175285858nt_int(Uub,Uu)
        & ~ member5262025264175285858nt_int(Uub,Uua) ) ) ).

% ATP.lambda_567
tff(fact_10745_ATP_Olambda__568,axiom,
    ! [Uu: set_complex,Uua: set_complex,Uub: complex] :
      ( aa_complex_o(aa_set1742385355640705825plex_o(aTP_Lamm_hc(Uu),Uua),Uub)
    <=> ( aa_set_complex_o(member_complex(Uub),Uu)
        & ~ aa_set_complex_o(member_complex(Uub),Uua) ) ) ).

% ATP.lambda_568
tff(fact_10746_ATP_Olambda__569,axiom,
    ! [Uu: set_real,Uua: set_real,Uub: real] :
      ( aa_real_o(aa_set2323979645938365349real_o(aTP_Lamm_ha(Uu),Uua),Uub)
    <=> ( aa_set_real_o(member_real(Uub),Uu)
        & ~ aa_set_real_o(member_real(Uub),Uua) ) ) ).

% ATP.lambda_569
tff(fact_10747_ATP_Olambda__570,axiom,
    ! [Uu: set_nat,Uua: set_nat,Uub: nat] :
      ( aa_nat_o(aa_set_nat_fun_nat_o(aTP_Lamm_he(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(member_nat(Uub),Uu)
        & ~ aa_set_nat_o(member_nat(Uub),Uua) ) ) ).

% ATP.lambda_570
tff(fact_10748_ATP_Olambda__571,axiom,
    ! [Uu: set_int,Uua: set_int,Uub: int] :
      ( aa_int_o(aa_set_int_fun_int_o(aTP_Lamm_hb(Uu),Uua),Uub)
    <=> ( aa_set_int_o(member_int(Uub),Uu)
        & ~ aa_set_int_o(member_int(Uub),Uua) ) ) ).

% ATP.lambda_571
tff(fact_10749_ATP_Olambda__572,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_bbh(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_572
tff(fact_10750_ATP_Olambda__573,axiom,
    ! [Uu: set_Pr3486312109254157132nteger,Uua: fun_Pr5098669352780829690e_term,Uub: produc8923325533196201883nteger] :
      ( aa_Pro4926521453222654329eger_o(aa_fun4505422719058378424eger_o(aTP_Lamm_acq(Uu),Uua),Uub)
    <=> member8978266922497061677nteger(aa_Pro4291442876242880245nteger(produc565448451787809096nteger(Uua),Uub),Uu) ) ).

% ATP.lambda_573
tff(fact_10751_ATP_Olambda__574,axiom,
    ! [Uu: set_Pr1312406793078187350et_nat,Uua: fun_Pr6460944699146269935_nat_o,Uub: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(aa_fun2638059832795814268_nat_o(aTP_Lamm_acr(Uu),Uua),Uub)
    <=> member4513763875858039863et_nat(aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(Uua),Uub),Uu) ) ).

% ATP.lambda_574
tff(fact_10752_ATP_Olambda__575,axiom,
    ! [Uu: set_Pr4040394263813967220et_nat,Uua: fun_Pr6460944699146269935_nat_o,Uub: produc8799533983703829792et_nat] :
      ( aa_Pro5643009357358181178_nat_o(aa_fun912680268133416158_nat_o(aTP_Lamm_acs(Uu),Uua),Uub)
    <=> member9174161442226468181et_nat(aa_Pro1979504083478021276et_nat(produc7291855603632349880et_nat(Uua),Uub),Uu) ) ).

% ATP.lambda_575
tff(fact_10753_ATP_Olambda__576,axiom,
    ! [Uu: set_Pr6700895561065273913nt_int,Uua: fun_Pr7763680713840462443e_term,Uub: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aa_fun3538162476766722827_int_o(aTP_Lamm_act(Uu),Uua),Uub)
    <=> member753276645541486746nt_int(aa_Pro5748361610964203648nt_int(produc7320969046608631221nt_int(Uua),Uub),Uu) ) ).

% ATP.lambda_576
tff(fact_10754_ATP_Olambda__577,axiom,
    ! [Uu: set_Pr4166680992137330533nt_int,Uua: fun_in5730379987311621695e_term,Uub: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aa_fun586359091965693151_int_o(aTP_Lamm_acu(Uu),Uua),Uub)
    <=> member5890090881338784710nt_int(aa_Pro8800328608558162860nt_int(produc4729913460834754785nt_int(Uua),Uub),Uu) ) ).

% ATP.lambda_577
tff(fact_10755_ATP_Olambda__578,axiom,
    ! [Uu: nat,Uua: complex,Uub: complex] :
      ( aa_complex_o(aa_com5808801942499186519plex_o(aTP_Lamm_afd(Uu),Uua),Uub)
    <=> ( aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Uub),Uu) = Uua ) ) ).

% ATP.lambda_578
tff(fact_10756_ATP_Olambda__579,axiom,
    ! [Uu: complex,Uua: nat,Uub: complex] :
      ( aa_complex_o(aa_nat_fun_complex_o(aTP_Lamm_azs(Uu),Uua),Uub)
    <=> ( aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Uub),Uua) = Uu ) ) ).

% ATP.lambda_579
tff(fact_10757_ATP_Olambda__580,axiom,
    ! [Uu: nat,Uua: vEBT_VEBTi,Uub: option_nat] : aa_option_nat_assn(aa_VEB7611521244314475576t_assn(aTP_Lamm_at(Uu),Uua),Uub) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,summary),Uua)),aa_o_assn(pure_assn,Uub = aa_nat_option_nat(vEBT_vebt_pred(summary),Uu))) ).

% ATP.lambda_580
tff(fact_10758_ATP_Olambda__581,axiom,
    ! [Uu: fun_fu9179568866395467601_VEBTi,Uua: fun_VE6677632329911286106Heap_o,Uub: fun_Pr342922516890243977on_nat] : aa_fun4188577930510380905Heap_o(aa_fun1993422656822580032Heap_o(aTP_Lamm_yq(Uu),Uua),Uub) = heap_T3040810144269856602EBTi_o(aa_fun765629416642892858_VEBTi(Uu,Uub),aa_fun4002976272494211483Heap_o(aTP_Lamm_yp(Uua),Uub)) ).

% ATP.lambda_581
tff(fact_10759_ATP_Olambda__582,axiom,
    ! [Uu: fun_fu9179568866395467601_VEBTi,Uua: fun_VE4144918862296792757_VEBTi,Uub: fun_Pr342922516890243977on_nat] : aa_fun765629416642892858_VEBTi(aa_fun1557956796225944984_VEBTi(aTP_Lamm_yo(Uu),Uua),Uub) = heap_T1006145433769338483_VEBTi(aa_fun765629416642892858_VEBTi(Uu,Uub),aa_fun5040531007017366174_VEBTi(aTP_Lamm_yn(Uua),Uub)) ).

% ATP.lambda_582
tff(fact_10760_ATP_Olambda__583,axiom,
    ! [Uu: fun_fu2606674006500944492ap_nat,Uua: fun_na979242554834862325Heap_o,Uub: fun_Pr342922516890243977on_nat] : aa_fun4188577930510380905Heap_o(aa_fun7768191950043460187Heap_o(aTP_Lamm_ym(Uu),Uua),Uub) = heap_Time_bind_nat_o(aa_fun4282859208648741077ap_nat(Uu,Uub),aa_fun1690681901636463158Heap_o(aTP_Lamm_yl(Uua),Uub)) ).

% ATP.lambda_583
tff(fact_10761_ATP_Olambda__584,axiom,
    ! [Uu: fun_fu2606674006500944492ap_nat,Uua: fun_na7267038673063590618_VEBTi,Uub: fun_Pr342922516890243977on_nat] : aa_fun765629416642892858_VEBTi(aa_fun3907257510474286387_VEBTi(aTP_Lamm_yk(Uu),Uua),Uub) = heap_T844888390831797134_VEBTi(aa_fun4282859208648741077ap_nat(Uu,Uub),aa_fun1166700169754672579_VEBTi(aTP_Lamm_yj(Uua),Uub)) ).

% ATP.lambda_584
tff(fact_10762_ATP_Olambda__585,axiom,
    ! [Uu: fun_fu335202653675969872Heap_o,Uua: fun_o_2223814739247024763Heap_o,Uub: fun_Pr238758252084143515Heap_o] : aa_fun6848502500920544641Heap_o(aa_fun4354595424708498391Heap_o(aTP_Lamm_yi(Uu),Uua),Uub) = heap_Time_bind_o_o(aa_fun6848502500920544641Heap_o(Uu,Uub),aa_fun7917812164634591790Heap_o(aTP_Lamm_yh(Uua),Uub)) ).

% ATP.lambda_585
tff(fact_10763_ATP_Olambda__586,axiom,
    ! [Uu: fun_fu335202653675969872Heap_o,Uua: fun_o_7171189317104958676_VEBTi,Uub: fun_Pr238758252084143515Heap_o] : aa_fun238413853627933730_VEBTi(aa_fun2042045670669332929_VEBTi(aTP_Lamm_yg(Uu),Uua),Uub) = heap_T5998771940306268294_VEBTi(aa_fun6848502500920544641Heap_o(Uu,Uub),aa_fun6586487760926256075_VEBTi(aTP_Lamm_yf(Uua),Uub)) ).

% ATP.lambda_586
tff(fact_10764_ATP_Olambda__587,axiom,
    ! [Uu: fun_fu4424605311360796457_VEBTi,Uua: fun_VE6943415934277719092Heap_o,Uub: fun_Pr238758252084143515Heap_o] : aa_fun6848502500920544641Heap_o(aa_fun5382224708498380120Heap_o(aTP_Lamm_ye(Uu),Uua),Uub) = heap_T3040810144269856602EBTi_o(aa_fun238413853627933730_VEBTi(Uu,Uub),aa_fun2163459272138508211Heap_o(aTP_Lamm_yd(Uua),Uub)) ).

% ATP.lambda_587
tff(fact_10765_ATP_Olambda__588,axiom,
    ! [Uu: fun_fu5786974620624140868ap_nat,Uua: fun_na8517577583218006873Heap_o,Uub: fun_Pr238758252084143515Heap_o] : aa_fun6848502500920544641Heap_o(aa_fun3258944081180477171Heap_o(aTP_Lamm_yc(Uu),Uua),Uub) = heap_Time_bind_nat_o(aa_fun6684961993577261245ap_nat(Uu,Uub),aa_fun7995322860690487374Heap_o(aTP_Lamm_yb(Uua),Uub)) ).

% ATP.lambda_588
tff(fact_10766_ATP_Olambda__589,axiom,
    ! [Uu: fun_fu4424605311360796457_VEBTi,Uua: fun_VE6491116402277797979_VEBTi,Uub: fun_Pr238758252084143515Heap_o] : aa_fun238413853627933730_VEBTi(aa_fun4276395798348166338_VEBTi(aTP_Lamm_ya(Uu),Uua),Uub) = heap_T1006145433769338483_VEBTi(aa_fun238413853627933730_VEBTi(Uu,Uub),aa_fun8469889001190476422_VEBTi(aTP_Lamm_xz(Uua),Uub)) ).

% ATP.lambda_589
tff(fact_10767_ATP_Olambda__590,axiom,
    ! [Uu: fun_fu5786974620624140868ap_nat,Uua: fun_na5845961453008041334_VEBTi,Uub: fun_Pr238758252084143515Heap_o] : aa_fun238413853627933730_VEBTi(aa_fun90388479857626845_VEBTi(aTP_Lamm_xy(Uu),Uua),Uub) = heap_T844888390831797134_VEBTi(aa_fun6684961993577261245ap_nat(Uu,Uub),aa_fun5587972357748661675_VEBTi(aTP_Lamm_xx(Uua),Uub)) ).

% ATP.lambda_590
tff(fact_10768_ATP_Olambda__591,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_azo(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uu,Uub)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),Uub)) ).

% ATP.lambda_591
tff(fact_10769_ATP_Olambda__592,axiom,
    ! [Uu: fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_ayz(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(Uu,Uub)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Uua),Uub)) ).

% ATP.lambda_592
tff(fact_10770_ATP_Olambda__593,axiom,
    ! [Uu: fun_nat_o,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_acz(Uu),Uua),Uub)
    <=> ( aa_nat_o(Uu,Uub)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uub),Uua) ) ) ).

% ATP.lambda_593
tff(fact_10771_ATP_Olambda__594,axiom,
    ! [Uu: fun_nat_nat,Uua: fun_nat_nat,Uub: nat] : aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_awm(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_594
tff(fact_10772_ATP_Olambda__595,axiom,
    ! [Uu: fun_nat_int,Uua: fun_nat_int,Uub: nat] : aa_nat_int(aa_fun5565043122120104496at_int(aTP_Lamm_awk(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_595
tff(fact_10773_ATP_Olambda__596,axiom,
    ! [Uu: fun_int_int,Uua: fun_int_int,Uub: int] : aa_int_int(aa_fun6280345759651049192nt_int(aTP_Lamm_awl(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_596
tff(fact_10774_ATP_Olambda__597,axiom,
    ! [Uu: fun_complex_complex,Uua: fun_complex_complex,Uub: complex] : aa_complex_complex(aa_fun3187328805393345264omplex(aTP_Lamm_ali(Uu),Uua),Uub) = aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,aa_complex_complex(Uu,Uub)),aa_complex_complex(Uua,Uub)) ).

% ATP.lambda_597
tff(fact_10775_ATP_Olambda__598,axiom,
    ! [Uu: fun_nat_real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_alk(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(Uu,Uub)),aa_nat_real(Uua,Uub)) ).

% ATP.lambda_598
tff(fact_10776_ATP_Olambda__599,axiom,
    ! [Uu: fun_int_int,Uua: fun_int_int,Uub: int] : aa_int_int(aa_fun6280345759651049192nt_int(aTP_Lamm_alj(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_599
tff(fact_10777_ATP_Olambda__600,axiom,
    ! [Uu: fun_VEBT_VEBT_nat,Uua: fun_VEBT_VEBT_nat,Uub: vEBT_VEBT] : aa_VEBT_VEBT_nat(aa_fun8284231930576791164BT_nat(aTP_Lamm_aqj(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_VEBT_VEBT_nat(Uua,Uub)),aa_VEBT_VEBT_nat(Uu,Uub)) ).

% ATP.lambda_600
tff(fact_10778_ATP_Olambda__601,axiom,
    ! [Uu: fun_real_nat,Uua: fun_real_nat,Uub: real] : aa_real_nat(aa_fun211602392931233840al_nat(aTP_Lamm_aqk(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_real_nat(Uua,Uub)),aa_real_nat(Uu,Uub)) ).

% ATP.lambda_601
tff(fact_10779_ATP_Olambda__602,axiom,
    ! [Uu: fun_nat_real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_aqf(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(Uua,Uub)),aa_nat_real(Uu,Uub)) ).

% ATP.lambda_602
tff(fact_10780_ATP_Olambda__603,axiom,
    ! [Uu: fun_nat_nat,Uua: fun_nat_nat,Uub: nat] : aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_aqm(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_603
tff(fact_10781_ATP_Olambda__604,axiom,
    ! [Uu: fun_int_nat,Uua: fun_int_nat,Uub: int] : aa_int_nat(aa_fun6506752150744536624nt_nat(aTP_Lamm_aql(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_604
tff(fact_10782_ATP_Olambda__605,axiom,
    ! [Uu: fun_complex_complex,Uua: fun_complex_complex,Uub: complex] : aa_complex_complex(aa_fun3187328805393345264omplex(aTP_Lamm_akp(Uu),Uua),Uub) = aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_complex_complex(Uu,Uub)),aa_complex_complex(Uua,Uub)) ).

% ATP.lambda_605
tff(fact_10783_ATP_Olambda__606,axiom,
    ! [Uu: fun_nat_real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_akr(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(Uu,Uub)),aa_nat_real(Uua,Uub)) ).

% ATP.lambda_606
tff(fact_10784_ATP_Olambda__607,axiom,
    ! [Uu: fun_nat_nat,Uua: fun_nat_nat,Uub: nat] : aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_ako(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_607
tff(fact_10785_ATP_Olambda__608,axiom,
    ! [Uu: fun_nat_int,Uua: fun_nat_int,Uub: nat] : aa_nat_int(aa_fun5565043122120104496at_int(aTP_Lamm_aqr(Uu),Uua),Uub) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(Uu,Uub)),aa_nat_int(Uua,Uub)) ).

% ATP.lambda_608
tff(fact_10786_ATP_Olambda__609,axiom,
    ! [Uu: fun_int_int,Uua: fun_int_int,Uub: int] : aa_int_int(aa_fun6280345759651049192nt_int(aTP_Lamm_akq(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_609
tff(fact_10787_ATP_Olambda__610,axiom,
    ! [Uu: fun_Code_integer_o,Uua: fun_Code_integer_o,Uub: code_integer] :
      ( aa_Code_integer_o(aa_fun8214818745163655228eger_o(aTP_Lamm_aeo(Uu),Uua),Uub)
    <=> ( aa_Code_integer_o(Uu,Uub)
        | aa_Code_integer_o(Uua,Uub) ) ) ).

% ATP.lambda_610
tff(fact_10788_ATP_Olambda__611,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_aek(Uu),Uua),Uub)
    <=> ( aa_Pro4592068669816069595_int_o(Uu,Uub)
        | aa_Pro4592068669816069595_int_o(Uua,Uub) ) ) ).

% ATP.lambda_611
tff(fact_10789_ATP_Olambda__612,axiom,
    ! [Uu: fun_complex_o,Uua: fun_complex_o,Uub: complex] :
      ( aa_complex_o(aa_fun2514317545224092796plex_o(aTP_Lamm_aen(Uu),Uua),Uub)
    <=> ( aa_complex_o(Uu,Uub)
        | aa_complex_o(Uua,Uub) ) ) ).

% ATP.lambda_612
tff(fact_10790_ATP_Olambda__613,axiom,
    ! [Uu: fun_nat_o,Uua: fun_nat_o,Uub: nat] :
      ( aa_nat_o(aa_fun5937193402907943164_nat_o(aTP_Lamm_ael(Uu),Uua),Uub)
    <=> ( aa_nat_o(Uu,Uub)
        | aa_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_613
tff(fact_10791_ATP_Olambda__614,axiom,
    ! [Uu: fun_int_o,Uua: fun_int_o,Uub: int] :
      ( aa_int_o(aa_fun6243517787477317628_int_o(aTP_Lamm_aem(Uu),Uua),Uub)
    <=> ( aa_int_o(Uu,Uub)
        | aa_int_o(Uua,Uub) ) ) ).

% ATP.lambda_614
tff(fact_10792_ATP_Olambda__615,axiom,
    ! [Uu: fun_Code_integer_o,Uua: fun_Code_integer_o,Uub: code_integer] :
      ( aa_Code_integer_o(aa_fun8214818745163655228eger_o(aTP_Lamm_aej(Uu),Uua),Uub)
    <=> ( aa_Code_integer_o(Uu,Uub)
        & aa_Code_integer_o(Uua,Uub) ) ) ).

% ATP.lambda_615
tff(fact_10793_ATP_Olambda__616,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_aef(Uu),Uua),Uub)
    <=> ( aa_Pro4592068669816069595_int_o(Uu,Uub)
        & aa_Pro4592068669816069595_int_o(Uua,Uub) ) ) ).

% ATP.lambda_616
tff(fact_10794_ATP_Olambda__617,axiom,
    ! [Uu: fun_complex_o,Uua: fun_complex_o,Uub: complex] :
      ( aa_complex_o(aa_fun2514317545224092796plex_o(aTP_Lamm_aei(Uu),Uua),Uub)
    <=> ( aa_complex_o(Uu,Uub)
        & aa_complex_o(Uua,Uub) ) ) ).

% ATP.lambda_617
tff(fact_10795_ATP_Olambda__618,axiom,
    ! [Uu: fun_nat_o,Uua: fun_nat_o,Uub: nat] :
      ( aa_nat_o(aa_fun5937193402907943164_nat_o(aTP_Lamm_aeg(Uu),Uua),Uub)
    <=> ( aa_nat_o(Uu,Uub)
        & aa_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_618
tff(fact_10796_ATP_Olambda__619,axiom,
    ! [Uu: fun_int_o,Uua: fun_int_o,Uub: int] :
      ( aa_int_o(aa_fun6243517787477317628_int_o(aTP_Lamm_aeh(Uu),Uua),Uub)
    <=> ( aa_int_o(Uu,Uub)
        & aa_int_o(Uua,Uub) ) ) ).

% ATP.lambda_619
tff(fact_10797_ATP_Olambda__620,axiom,
    ! [Uu: fun_nat_fun_nat_nat,Uua: set_nat,Uub: nat] : aa_nat_nat(aa_set6537093552120046177at_nat(aTP_Lamm_awi(Uu),Uua),Uub) = groups708209901874060359at_nat(aa_nat_fun_nat_nat(Uu,Uub),Uua) ).

% ATP.lambda_620
tff(fact_10798_ATP_Olambda__621,axiom,
    ! [Uu: fun_nat_fun_nat_int,Uua: set_nat,Uub: nat] : aa_nat_int(aa_set2359242532610849469at_int(aTP_Lamm_avy(Uu),Uua),Uub) = groups705719431365010083at_int(aa_nat_fun_nat_int(Uu,Uub),Uua) ).

% ATP.lambda_621
tff(fact_10799_ATP_Olambda__622,axiom,
    ! [Uu: fun_nat_fun_int_int,Uua: set_int,Uub: nat] : aa_nat_int(aa_set6310687161026558745at_int(aTP_Lamm_awb(Uu),Uua),Uub) = groups1705073143266064639nt_int(aa_nat_fun_int_int(Uu,Uub),Uua) ).

% ATP.lambda_622
tff(fact_10800_ATP_Olambda__623,axiom,
    ! [Uu: fun_int_fun_nat_int,Uua: set_nat,Uub: int] : aa_int_int(aa_set3358596244511904025nt_int(aTP_Lamm_awe(Uu),Uua),Uub) = groups705719431365010083at_int(aa_int_fun_nat_int(Uu,Uub),Uua) ).

% ATP.lambda_623
tff(fact_10801_ATP_Olambda__624,axiom,
    ! [Uu: fun_int_fun_int_int,Uua: set_int,Uub: int] : aa_int_int(aa_set7310040872927613301nt_int(aTP_Lamm_awg(Uu),Uua),Uub) = groups1705073143266064639nt_int(aa_int_fun_int_int(Uu,Uub),Uua) ).

% ATP.lambda_624
tff(fact_10802_ATP_Olambda__625,axiom,
    ! [Uu: fun_co7673397065115134084omplex,Uua: set_complex,Uub: complex] : aa_complex_complex(aa_set7743095458595710203omplex(aTP_Lamm_akc(Uu),Uua),Uub) = groups7754918857620584856omplex(aa_com3342990543769683525omplex(Uu,Uub),Uua) ).

% ATP.lambda_625
tff(fact_10803_ATP_Olambda__626,axiom,
    ! [Uu: fun_nat_fun_nat_real,Uua: set_nat,Uub: nat] : aa_nat_real(aa_set1166323925324055613t_real(aTP_Lamm_aki(Uu),Uua),Uub) = groups6591440286371151544t_real(aa_nat_fun_nat_real(Uu,Uub),Uua) ).

% ATP.lambda_626
tff(fact_10804_ATP_Olambda__627,axiom,
    ! [Uu: fun_nat_fun_nat_nat,Uua: set_nat,Uub: nat] : aa_nat_nat(aa_set6537093552120046177at_nat(aTP_Lamm_ajz(Uu),Uua),Uub) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(Uu,Uub)),Uua) ).

% ATP.lambda_627
tff(fact_10805_ATP_Olambda__628,axiom,
    ! [Uu: fun_int_fun_int_int,Uua: set_int,Uub: int] : aa_int_int(aa_set7310040872927613301nt_int(aTP_Lamm_akf(Uu),Uua),Uub) = groups4538972089207619220nt_int(aa_int_fun_int_int(Uu,Uub),Uua) ).

% ATP.lambda_628
tff(fact_10806_ATP_Olambda__629,axiom,
    ! [Uu: fun_nat_nat,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_ada(Uu),Uua),Uub)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(Uu,Uub)),Uua) ) ).

% ATP.lambda_629
tff(fact_10807_ATP_Olambda__630,axiom,
    ! [Uu: fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_anh(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(Uu,Uub)),Uua) ).

% ATP.lambda_630
tff(fact_10808_ATP_Olambda__631,axiom,
    ! [Uu: fun_nat_int,Uua: int,Uub: nat] : aa_nat_int(aa_int_fun_nat_int(aTP_Lamm_axy(Uu),Uua),Uub) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_nat_int(Uu,Uub)),Uua) ).

% ATP.lambda_631
tff(fact_10809_ATP_Olambda__632,axiom,
    ! [Uu: fun_int_int,Uua: int,Uub: int] : aa_int_int(aa_int_fun_int_int(aTP_Lamm_ani(Uu),Uua),Uub) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(Uu,Uub)),Uua) ).

% ATP.lambda_632
tff(fact_10810_ATP_Olambda__633,axiom,
    ! [Uu: fun_complex_complex,Uua: complex,Uub: complex] : aa_complex_complex(aa_com3342990543769683525omplex(aTP_Lamm_all(Uu),Uua),Uub) = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_complex_complex(Uu,Uub)),Uua) ).

% ATP.lambda_633
tff(fact_10811_ATP_Olambda__634,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_alm(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(Uu,Uub)),Uua) ).

% ATP.lambda_634
tff(fact_10812_ATP_Olambda__635,axiom,
    ! [Uu: fun_VEBT_VEBTi_assn,Uua: assn,Uub: vEBT_VEBTi] : aa_VEBT_VEBTi_assn(aa_ass4389913640333755968i_assn(aTP_Lamm_bs(Uu),Uua),Uub) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_VEBT_VEBTi_assn(Uu,Uub)),Uua) ).

% ATP.lambda_635
tff(fact_10813_ATP_Olambda__636,axiom,
    ! [Uu: fun_Pr6626665806141890376t_assn,Uua: assn,Uub: product_unit] : aa_Product_unit_assn(aa_ass3177796706412234594t_assn(aTP_Lamm_bt(Uu),Uua),Uub) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_Product_unit_assn(Uu,Uub)),Uua) ).

% ATP.lambda_636
tff(fact_10814_ATP_Olambda__637,axiom,
    ! [Uu: fun_complex_complex,Uua: complex,Uub: complex] : aa_complex_complex(aa_com3342990543769683525omplex(aTP_Lamm_akx(Uu),Uua),Uub) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(Uu,Uub)),Uua) ).

% ATP.lambda_637
tff(fact_10815_ATP_Olambda__638,axiom,
    ! [Uu: fun_option_nat_assn,Uua: assn,Uub: option_nat] : aa_option_nat_assn(aa_ass4112278883933122603t_assn(aTP_Lamm_bp(Uu),Uua),Uub) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_option_nat_assn(Uu,Uub)),Uua) ).

% ATP.lambda_638
tff(fact_10816_ATP_Olambda__639,axiom,
    ! [Uu: fun_li3352006551604695530i_assn,Uua: assn,Uub: list_VEBT_VEBTi] : aa_lis2508312872377092907i_assn(aa_ass1897699608592665680i_assn(aTP_Lamm_cu(Uu),Uua),Uub) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_lis2508312872377092907i_assn(Uu,Uub)),Uua) ).

% ATP.lambda_639
tff(fact_10817_ATP_Olambda__640,axiom,
    ! [Uu: fun_o_assn,Uua: assn,Uub: $o] : aa_o_assn(aa_assn_fun_o_assn(aTP_Lamm_br(Uu),Uua),(Uub)) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_o_assn(Uu,(Uub))),Uua) ).

% ATP.lambda_640
tff(fact_10818_ATP_Olambda__641,axiom,
    ! [Uu: fun_nat_assn,Uua: assn,Uub: nat] : aa_nat_assn(aa_assn_fun_nat_assn(aTP_Lamm_bq(Uu),Uua),Uub) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_nat_assn(Uu,Uub)),Uua) ).

% ATP.lambda_641
tff(fact_10819_ATP_Olambda__642,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_akz(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uu,Uub)),Uua) ).

% ATP.lambda_642
tff(fact_10820_ATP_Olambda__643,axiom,
    ! [Uu: fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_akw(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(Uu,Uub)),Uua) ).

% ATP.lambda_643
tff(fact_10821_ATP_Olambda__644,axiom,
    ! [Uu: fun_int_int,Uua: int,Uub: int] : aa_int_int(aa_int_fun_int_int(aTP_Lamm_aky(Uu),Uua),Uub) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(Uu,Uub)),Uua) ).

% ATP.lambda_644
tff(fact_10822_ATP_Olambda__645,axiom,
    ! [Uu: real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_arf(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uua,Uub)),Uu) ).

% ATP.lambda_645
tff(fact_10823_ATP_Olambda__646,axiom,
    ! [Uu: fun_nat_Code_integer,Uua: code_integer,Uub: nat] : aa_nat_Code_integer(aa_Cod1578151519121547077nteger(aTP_Lamm_asx(Uu),Uua),Uub) = aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_nat_Code_integer(Uu,Uub)),Uua) ).

% ATP.lambda_646
tff(fact_10824_ATP_Olambda__647,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_asy(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(Uu,Uub)),Uua) ).

% ATP.lambda_647
tff(fact_10825_ATP_Olambda__648,axiom,
    ! [Uu: fun_nat_rat,Uua: rat,Uub: nat] : aa_nat_rat(aa_rat_fun_nat_rat(aTP_Lamm_asv(Uu),Uua),Uub) = minus_minus_rat(aa_nat_rat(Uu,Uub),Uua) ).

% ATP.lambda_648
tff(fact_10826_ATP_Olambda__649,axiom,
    ! [Uu: fun_nat_int,Uua: int,Uub: nat] : aa_nat_int(aa_int_fun_nat_int(aTP_Lamm_asw(Uu),Uua),Uub) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(Uu,Uub)),Uua) ).

% ATP.lambda_649
tff(fact_10827_ATP_Olambda__650,axiom,
    ! [Uu: fun_real_real,Uua: nat,Uub: real] : aa_real_real(aa_nat_fun_real_real(aTP_Lamm_bcr(Uu),Uua),Uub) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(Uu,Uub)),Uua) ).

% ATP.lambda_650
tff(fact_10828_ATP_Olambda__651,axiom,
    ! [Uu: fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_awp(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(Uu,Uub)),Uua) ).

% ATP.lambda_651
tff(fact_10829_ATP_Olambda__652,axiom,
    ! [Uu: fun_nat_int,Uua: nat,Uub: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_awn(Uu),Uua),Uub) = aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_nat_int(Uu,Uub)),Uua) ).

% ATP.lambda_652
tff(fact_10830_ATP_Olambda__653,axiom,
    ! [Uu: fun_int_int,Uua: nat,Uub: int] : aa_int_int(aa_nat_fun_int_int(aTP_Lamm_awo(Uu),Uua),Uub) = aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(Uu,Uub)),Uua) ).

% ATP.lambda_653
tff(fact_10831_ATP_Olambda__654,axiom,
    ! [Uu: fun_li7499208429157053461_VEBTi,Uua: fun_VE4765431424076933413_VEBTi,Uub: list_VEBT_VEBTi] : aa_lis1839255523752068350_VEBTi(aa_fun2525796888956276588_VEBTi(aTP_Lamm_ze(Uu),Uua),Uub) = heap_T1006145433769338483_VEBTi(aa_lis1839255523752068350_VEBTi(Uu,Uub),Uua) ).

% ATP.lambda_654
tff(fact_10832_ATP_Olambda__655,axiom,
    ! [Uu: fun_o_2238389867833408620_VEBTi,Uua: fun_VE4765431424076933413_VEBTi,Uub: $o] : aa_o_H7577119287286566245_VEBTi(aa_fun432179723158315733_VEBTi(aTP_Lamm_yv(Uu),Uua),(Uub)) = heap_T1006145433769338483_VEBTi(aa_o_H7577119287286566245_VEBTi(Uu,(Uub)),Uua) ).

% ATP.lambda_655
tff(fact_10833_ATP_Olambda__656,axiom,
    ! [Uu: fun_o_8325599348076409421Heap_o,Uua: fun_o_2238389867833408620_VEBTi,Uub: $o] : aa_o_H7577119287286566245_VEBTi(aa_fun5660287117450118806_VEBTi(aTP_Lamm_yz(Uu),Uua),(Uub)) = heap_T5998771940306268294_VEBTi(aa_o_H7399033255965945470Heap_o(Uu,(Uub)),Uua) ).

% ATP.lambda_656
tff(fact_10834_ATP_Olambda__657,axiom,
    ! [Uu: fun_o_4158591058828729991ap_nat,Uua: fun_na7848093160509028170_VEBTi,Uub: $o] : aa_o_H7577119287286566245_VEBTi(aa_fun1800422248244663418_VEBTi(aTP_Lamm_zb(Uu),Uua),(Uub)) = heap_T844888390831797134_VEBTi(aa_o_H8739796202006397184ap_nat(Uu,(Uub)),Uua) ).

% ATP.lambda_657
tff(fact_10835_ATP_Olambda__658,axiom,
    ! [Uu: fun_na7848093160509028170_VEBTi,Uua: fun_VE4765431424076933413_VEBTi,Uub: nat] : aa_nat5408703056993926195_VEBTi(aa_fun6311797650338102561_VEBTi(aTP_Lamm_yw(Uu),Uua),Uub) = heap_T1006145433769338483_VEBTi(aa_nat5408703056993926195_VEBTi(Uu,Uub),Uua) ).

% ATP.lambda_658
tff(fact_10836_ATP_Olambda__659,axiom,
    ! [Uu: fun_na7848093160509028170_VEBTi,Uua: fun_VE7038733415900436564Heap_o,Uub: nat] : aa_nat2910048432523430448Heap_o(aa_fun3725866701989745153Heap_o(aTP_Lamm_yx(Uu),Uua),Uub) = heap_T3040810144269856602EBTi_o(aa_nat5408703056993926195_VEBTi(Uu,Uub),Uua) ).

% ATP.lambda_659
tff(fact_10837_ATP_Olambda__660,axiom,
    ! [Uu: fun_na2343073368905879023Heap_o,Uua: fun_o_2238389867833408620_VEBTi,Uub: nat] : aa_nat5408703056993926195_VEBTi(aa_fun5487288040762657696_VEBTi(aTP_Lamm_za(Uu),Uua),Uub) = heap_T5998771940306268294_VEBTi(aa_nat2910048432523430448Heap_o(Uu,Uub),Uua) ).

% ATP.lambda_660
tff(fact_10838_ATP_Olambda__661,axiom,
    ! [Uu: fun_na2343073368905879023Heap_o,Uua: fun_o_8325599348076409421Heap_o,Uub: nat] : aa_nat2910048432523430448Heap_o(aa_fun1108420279618462208Heap_o(aTP_Lamm_yy(Uu),Uua),Uub) = heap_Time_bind_o_o(aa_nat2910048432523430448Heap_o(Uu,Uub),Uua) ).

% ATP.lambda_661
tff(fact_10839_ATP_Olambda__662,axiom,
    ! [Uu: fun_na3533371890617757541ap_nat,Uua: fun_na7848093160509028170_VEBTi,Uub: nat] : aa_nat5408703056993926195_VEBTi(aa_fun49764924992962236_VEBTi(aTP_Lamm_zc(Uu),Uua),Uub) = heap_T844888390831797134_VEBTi(aa_nat5821310369566504398ap_nat(Uu,Uub),Uua) ).

% ATP.lambda_662
tff(fact_10840_ATP_Olambda__663,axiom,
    ! [Uu: fun_na3533371890617757541ap_nat,Uua: fun_na2343073368905879023Heap_o,Uub: nat] : aa_nat2910048432523430448Heap_o(aa_fun3424352875297450780Heap_o(aTP_Lamm_zd(Uu),Uua),Uub) = heap_Time_bind_nat_o(aa_nat5821310369566504398ap_nat(Uu,Uub),Uua) ).

% ATP.lambda_663
tff(fact_10841_ATP_Olambda__664,axiom,
    ! [Uu: product_prod_nat_nat,Uua: nat,Uub: option_nat] :
      aa_opt9163293676379396302on_nat(aa_nat9127621225095157975on_nat(aTP_Lamm_ad(Uu),Uua),Uub) = heap_T5999496708990702694on_nat(aa_o_H2459438917461573107t_unit(refine_Imp_assert,
          ( ( Uub = none_nat )
          = ( aa_nat_option_nat(vEBT_vebt_pred(summary),Uua) = none_nat ) )),aa_opt5089197864053947520on_nat(aTP_Lamm_ac(Uu),Uub)) ).

% ATP.lambda_664
tff(fact_10842_ATP_Olambda__665,axiom,
    ! [Uu: nat,Uua: nat,Uub: list_VEBT_VEBTi] : aa_lis1839255523752068350_VEBTi(aa_nat4125268785154830727_VEBTi(aTP_Lamm_xs(Uu),Uua),Uub) = heap_T7982501707604696571_VEBTi(aa_o_H2459438917461573107t_unit(refine_Imp_assert,size_s7982070591426661849_VEBTi(Uub) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,Uua))),aa_lis4026867407995182650_VEBTi(aa_nat1561540599483812487_VEBTi(aTP_Lamm_xr(Uu),Uua),Uub)) ).

% ATP.lambda_665
tff(fact_10843_ATP_Olambda__666,axiom,
    ! [Uu: nat,Uua: nat,Uub: list_VEBT_VEBTi] : aa_lis1839255523752068350_VEBTi(aa_nat4125268785154830727_VEBTi(aTP_Lamm_xp(Uu),Uua),Uub) = heap_T7982501707604696571_VEBTi(aa_o_H2459438917461573107t_unit(refine_Imp_assert,size_s7982070591426661849_VEBTi(Uub) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)),aa_lis4026867407995182650_VEBTi(aa_nat1561540599483812487_VEBTi(aTP_Lamm_xo(Uu),Uua),Uub)) ).

% ATP.lambda_666
tff(fact_10844_ATP_Olambda__667,axiom,
    ! [Uu: product_prod_nat_nat,Uua: nat,Uub: nat] : aa_nat947733127081326238on_nat(aa_nat6020981482381522343on_nat(aTP_Lamm_ak(Uu),Uua),Uub) = heap_T5999496708990702694on_nat(aa_o_H2459438917461573107t_unit(refine_Imp_assert,Uua = aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,x12),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat3999351741935936176on_nat(aa_nat3263736196148507005on_nat(aTP_Lamm_aj(Uu),Uua),Uub)) ).

% ATP.lambda_667
tff(fact_10845_ATP_Olambda__668,axiom,
    ! [Uu: vEBT_VEBT,Uua: fun_VEBT_VEBT_o,Uub: vEBT_VEBT] :
      ( aa_VEBT_VEBT_o(aa_fun8193769468198689164VEBT_o(aTP_Lamm_hr(Uu),Uua),Uub)
    <=> ( ( Uub != Uu )
       => aa_VEBT_VEBT_o(Uua,Uub) ) ) ).

% ATP.lambda_668
tff(fact_10846_ATP_Olambda__669,axiom,
    ! [Uu: product_prod_int_int,Uua: fun_Pr3238374608701176162_int_o,Uub: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aa_fun5349570537165229698_int_o(aTP_Lamm_hw(Uu),Uua),Uub)
    <=> ( ( Uub != Uu )
       => aa_Pro4592068669816069595_int_o(Uua,Uub) ) ) ).

% ATP.lambda_669
tff(fact_10847_ATP_Olambda__670,axiom,
    ! [Uu: complex,Uua: fun_complex_o,Uub: complex] :
      ( aa_complex_o(aa_fun2514317545224092796plex_o(aTP_Lamm_hv(Uu),Uua),Uub)
    <=> ( ( Uub != Uu )
       => aa_complex_o(Uua,Uub) ) ) ).

% ATP.lambda_670
tff(fact_10848_ATP_Olambda__671,axiom,
    ! [Uu: real,Uua: fun_real_o,Uub: real] :
      ( aa_real_o(aa_fun5868358341814070780real_o(aTP_Lamm_hs(Uu),Uua),Uub)
    <=> ( ( Uub != Uu )
       => aa_real_o(Uua,Uub) ) ) ).

% ATP.lambda_671
tff(fact_10849_ATP_Olambda__672,axiom,
    ! [Uu: nat,Uua: fun_nat_o,Uub: nat] :
      ( aa_nat_o(aa_fun5937193402907943164_nat_o(aTP_Lamm_ht(Uu),Uua),Uub)
    <=> ( ( Uub != Uu )
       => aa_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_672
tff(fact_10850_ATP_Olambda__673,axiom,
    ! [Uu: int,Uua: fun_int_o,Uub: int] :
      ( aa_int_o(aa_fun6243517787477317628_int_o(aTP_Lamm_hu(Uu),Uua),Uub)
    <=> ( ( Uub != Uu )
       => aa_int_o(Uua,Uub) ) ) ).

% ATP.lambda_673
tff(fact_10851_ATP_Olambda__674,axiom,
    ! [Uu: nat,Uua: nat,Uub: array_VEBT_VEBTi] : aa_arr4839786837114051083_VEBTi(aa_nat106358831840921632_VEBTi(aTP_Lamm_xq(Uu),Uua),Uub) = heap_T1006145433769338483_VEBTi(aa_nat5408703056993926195_VEBTi(vEBT_V739175172307565963ildupi,aa_nat_nat(suc,Uua)),aa_arr2912372873376718191_VEBTi(aTP_Lamm_xm(Uu),Uub)) ).

% ATP.lambda_674
tff(fact_10852_ATP_Olambda__675,axiom,
    ! [Uu: nat,Uua: nat,Uub: array_VEBT_VEBTi] : aa_arr4839786837114051083_VEBTi(aa_nat106358831840921632_VEBTi(aTP_Lamm_xv(Uu),Uua),Uub) = heap_T1006145433769338483_VEBTi(aa_nat5408703056993926195_VEBTi(vEBT_vebt_buildupi,aa_nat_nat(suc,Uua)),aa_arr2912372873376718191_VEBTi(aTP_Lamm_xm(Uu),Uub)) ).

% ATP.lambda_675
tff(fact_10853_ATP_Olambda__676,axiom,
    ! [Uu: array_VEBT_VEBTi,Uua: nat,Uub: nat] : aa_nat2910048432523430448Heap_o(aa_nat5592791454563588861Heap_o(aTP_Lamm_jr(Uu),Uua),Uub) = heap_Time_bind_nat_o(array_len_VEBT_VEBTi(Uu),aa_nat5592791454563588861Heap_o(aa_nat7429013343984486730Heap_o(aTP_Lamm_jq(Uu),Uua),Uub)) ).

% ATP.lambda_676
tff(fact_10854_ATP_Olambda__677,axiom,
    ! [Uu: nat,Uua: nat,Uub: array_VEBT_VEBTi] : aa_arr4839786837114051083_VEBTi(aa_nat106358831840921632_VEBTi(aTP_Lamm_xn(Uu),Uua),Uub) = heap_T1006145433769338483_VEBTi(aa_nat5408703056993926195_VEBTi(vEBT_V739175172307565963ildupi,Uua),aa_arr2912372873376718191_VEBTi(aTP_Lamm_xm(Uu),Uub)) ).

% ATP.lambda_677
tff(fact_10855_ATP_Olambda__678,axiom,
    ! [Uu: nat,Uua: nat,Uub: array_VEBT_VEBTi] : aa_arr4839786837114051083_VEBTi(aa_nat106358831840921632_VEBTi(aTP_Lamm_xt(Uu),Uua),Uub) = heap_T1006145433769338483_VEBTi(aa_nat5408703056993926195_VEBTi(vEBT_vebt_buildupi,Uua),aa_arr2912372873376718191_VEBTi(aTP_Lamm_xm(Uu),Uub)) ).

% ATP.lambda_678
tff(fact_10856_ATP_Olambda__679,axiom,
    ! [Uu: nat,Uua: option_nat,Uub: vEBT_VEBTi] : aa_VEB2711320435582112505on_nat(aa_opt1214731524852023858on_nat(aTP_Lamm_fa(Uu),Uua),Uub) = heap_T3669509953089699273on_nat(aa_VEB2711320435582112505on_nat(vEBT_vebt_minti,Uub),aa_opt7201247987542299399on_nat(aTP_Lamm_eq(Uu),Uua)) ).

% ATP.lambda_679
tff(fact_10857_ATP_Olambda__680,axiom,
    ! [Uu: nat,Uua: option_nat,Uub: vEBT_VEBTi] : aa_VEB2711320435582112505on_nat(aa_opt1214731524852023858on_nat(aTP_Lamm_er(Uu),Uua),Uub) = heap_T3669509953089699273on_nat(aa_VEB2711320435582112505on_nat(vEBT_vebt_maxti,Uub),aa_opt7201247987542299399on_nat(aTP_Lamm_eq(Uu),Uua)) ).

% ATP.lambda_680
tff(fact_10858_ATP_Olambda__681,axiom,
    ! [Uu: nat,Uua: nat,Uub: list_VEBT_VEBTi] : aa_lis1839255523752068350_VEBTi(aa_nat4125268785154830727_VEBTi(aTP_Lamm_xw(Uu),Uua),Uub) = heap_T5099337393651448672_VEBTi(array_615059503499738533_VEBTi(Uub),aa_nat106358831840921632_VEBTi(aTP_Lamm_xv(Uu),Uua)) ).

% ATP.lambda_681
tff(fact_10859_ATP_Olambda__682,axiom,
    ! [Uu: nat,Uua: nat,Uub: list_VEBT_VEBTi] : aa_lis1839255523752068350_VEBTi(aa_nat4125268785154830727_VEBTi(aTP_Lamm_xu(Uu),Uua),Uub) = heap_T5099337393651448672_VEBTi(array_615059503499738533_VEBTi(Uub),aa_nat106358831840921632_VEBTi(aTP_Lamm_xt(Uu),Uua)) ).

% ATP.lambda_682
tff(fact_10860_ATP_Olambda__683,axiom,
    ! [Uu: code_integer,Uua: code_integer,Uub: nat] : aa_nat_Code_integer(aa_Cod1578151519121547077nteger(aTP_Lamm_apz(Uu),Uua),Uub) = aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(Uu),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,Uub)),Uua)) ).

% ATP.lambda_683
tff(fact_10861_ATP_Olambda__684,axiom,
    ! [Uu: word_N3645301735248828278l_num1,Uua: word_N3645301735248828278l_num1,Uub: nat] : aa_nat5599461682457136004l_num1(aa_wor1031616255390178039l_num1(aTP_Lamm_apw(Uu),Uua),Uub) = plus_p361126936061061375l_num1(Uu,aa_wor7274019993978516266l_num1(times_7065122842183080059l_num1(aa_nat5599461682457136004l_num1(semiri8819519690708144855l_num1,Uub)),Uua)) ).

% ATP.lambda_684
tff(fact_10862_ATP_Olambda__685,axiom,
    ! [Uu: real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_aqb(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Uu),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Uub)),Uua)) ).

% ATP.lambda_685
tff(fact_10863_ATP_Olambda__686,axiom,
    ! [Uu: rat,Uua: rat,Uub: nat] : aa_nat_rat(aa_rat_fun_nat_rat(aTP_Lamm_apx(Uu),Uua),Uub) = aa_rat_rat(plus_plus_rat(Uu),aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,Uub)),Uua)) ).

% ATP.lambda_686
tff(fact_10864_ATP_Olambda__687,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_aqa(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_687
tff(fact_10865_ATP_Olambda__688,axiom,
    ! [Uu: int,Uua: int,Uub: nat] : aa_nat_int(aa_int_fun_nat_int(aTP_Lamm_apy(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_688
tff(fact_10866_ATP_Olambda__689,axiom,
    ! [Uu: uint32,Uua: nat,Uub: nat] : aa_nat_uint32(aa_nat6935575117487981255uint32(aTP_Lamm_att(Uu),Uua),Uub) = aa_nat_uint32(power_power_uint32(Uu),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),aa_nat_nat(suc,Uub))) ).

% ATP.lambda_689
tff(fact_10867_ATP_Olambda__690,axiom,
    ! [Uu: word_N3645301735248828278l_num1,Uua: nat,Uub: nat] : aa_nat5599461682457136004l_num1(aa_nat6251203789436639057l_num1(aTP_Lamm_atu(Uu),Uua),Uub) = aa_nat5599461682457136004l_num1(power_2184487114949457152l_num1(Uu),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),aa_nat_nat(suc,Uub))) ).

% ATP.lambda_690
tff(fact_10868_ATP_Olambda__691,axiom,
    ! [Uu: real,Uua: nat,Uub: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_atx(Uu),Uua),Uub) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),aa_nat_nat(suc,Uub))) ).

% ATP.lambda_691
tff(fact_10869_ATP_Olambda__692,axiom,
    ! [Uu: rat,Uua: nat,Uub: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aTP_Lamm_atv(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_692
tff(fact_10870_ATP_Olambda__693,axiom,
    ! [Uu: int,Uua: nat,Uub: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_atw(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_693
tff(fact_10871_ATP_Olambda__694,axiom,
    ! [Uu: uint32,Uua: nat,Uub: nat] : aa_nat_uint32(aa_nat6935575117487981255uint32(aTP_Lamm_aoi(Uu),Uua),Uub) = aa_nat_uint32(power_power_uint32(Uu),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),Uub)) ).

% ATP.lambda_694
tff(fact_10872_ATP_Olambda__695,axiom,
    ! [Uu: word_N3645301735248828278l_num1,Uua: nat,Uub: nat] : aa_nat5599461682457136004l_num1(aa_nat6251203789436639057l_num1(aTP_Lamm_aoj(Uu),Uua),Uub) = aa_nat5599461682457136004l_num1(power_2184487114949457152l_num1(Uu),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),Uub)) ).

% ATP.lambda_695
tff(fact_10873_ATP_Olambda__696,axiom,
    ! [Uu: real,Uua: nat,Uub: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_aom(Uu),Uua),Uub) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),Uub)) ).

% ATP.lambda_696
tff(fact_10874_ATP_Olambda__697,axiom,
    ! [Uu: rat,Uua: nat,Uub: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aTP_Lamm_aok(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_697
tff(fact_10875_ATP_Olambda__698,axiom,
    ! [Uu: int,Uua: nat,Uub: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_aol(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_698
tff(fact_10876_ATP_Olambda__699,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_aqc(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_699
tff(fact_10877_ATP_Olambda__700,axiom,
    ! [Uu: complex,Uua: fun_complex_complex,Uub: complex] : aa_complex_complex(aa_fun3187328805393345264omplex(aTP_Lamm_akt(Uu),Uua),Uub) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,Uu),aa_complex_complex(Uua,Uub)) ).

% ATP.lambda_700
tff(fact_10878_ATP_Olambda__701,axiom,
    ! [Uu: assn,Uua: fun_li3352006551604695530i_assn,Uub: list_VEBT_VEBTi] : aa_lis2508312872377092907i_assn(aa_fun6795544665460849810i_assn(aTP_Lamm_ct(Uu),Uua),Uub) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Uu),aa_lis2508312872377092907i_assn(Uua,Uub)) ).

% ATP.lambda_701
tff(fact_10879_ATP_Olambda__702,axiom,
    ! [Uu: real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_akv(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Uu),aa_nat_real(Uua,Uub)) ).

% ATP.lambda_702
tff(fact_10880_ATP_Olambda__703,axiom,
    ! [Uu: nat,Uua: fun_nat_nat,Uub: nat] : aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_aks(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uu),aa_nat_nat(Uua,Uub)) ).

% ATP.lambda_703
tff(fact_10881_ATP_Olambda__704,axiom,
    ! [Uu: int,Uua: fun_int_int,Uub: int] : aa_int_int(aa_fun6280345759651049192nt_int(aTP_Lamm_aku(Uu),Uua),Uub) = aa_int_int(aa_int_fun_int_int(times_times_int,Uu),aa_int_int(Uua,Uub)) ).

% ATP.lambda_704
tff(fact_10882_ATP_Olambda__705,axiom,
    ! [Uu: fun_VEBT_VEBTi_assn,Uua: assn,Uub: vEBT_VEBTi] : aa_VEBT_VEBTi_assn(aa_ass4389913640333755968i_assn(aTP_Lamm_dn(Uu),Uua),Uub) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Uua),aa_VEBT_VEBTi_assn(Uu,Uub)) ).

% ATP.lambda_705
tff(fact_10883_ATP_Olambda__706,axiom,
    ! [Uu: fun_Pr6626665806141890376t_assn,Uua: assn,Uub: product_unit] : aa_Product_unit_assn(aa_ass3177796706412234594t_assn(aTP_Lamm_do(Uu),Uua),Uub) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Uua),aa_Product_unit_assn(Uu,Uub)) ).

% ATP.lambda_706
tff(fact_10884_ATP_Olambda__707,axiom,
    ! [Uu: fun_option_nat_assn,Uua: assn,Uub: option_nat] : aa_option_nat_assn(aa_ass4112278883933122603t_assn(aTP_Lamm_dk(Uu),Uua),Uub) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Uua),aa_option_nat_assn(Uu,Uub)) ).

% ATP.lambda_707
tff(fact_10885_ATP_Olambda__708,axiom,
    ! [Uu: fun_o_assn,Uua: assn,Uub: $o] : aa_o_assn(aa_assn_fun_o_assn(aTP_Lamm_dm(Uu),Uua),(Uub)) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Uua),aa_o_assn(Uu,(Uub))) ).

% ATP.lambda_708
tff(fact_10886_ATP_Olambda__709,axiom,
    ! [Uu: fun_nat_assn,Uua: assn,Uub: nat] : aa_nat_assn(aa_assn_fun_nat_assn(aTP_Lamm_dl(Uu),Uua),Uub) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Uua),aa_nat_assn(Uu,Uub)) ).

% ATP.lambda_709
tff(fact_10887_ATP_Olambda__710,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_aqs(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Uua),aa_nat_real(Uu,Uub)) ).

% ATP.lambda_710
tff(fact_10888_ATP_Olambda__711,axiom,
    ! [Uu: uint32,Uua: fun_nat_nat,Uub: nat] : aa_nat_uint32(aa_fun8367061318319979028uint32(aTP_Lamm_ayj(Uu),Uua),Uub) = aa_nat_uint32(power_power_uint32(Uu),aa_nat_nat(Uua,Uub)) ).

% ATP.lambda_711
tff(fact_10889_ATP_Olambda__712,axiom,
    ! [Uu: word_N3645301735248828278l_num1,Uua: fun_nat_nat,Uub: nat] : aa_nat5599461682457136004l_num1(aa_fun7778216322329253406l_num1(aTP_Lamm_ayk(Uu),Uua),Uub) = aa_nat5599461682457136004l_num1(power_2184487114949457152l_num1(Uu),aa_nat_nat(Uua,Uub)) ).

% ATP.lambda_712
tff(fact_10890_ATP_Olambda__713,axiom,
    ! [Uu: real,Uua: fun_nat_nat,Uub: nat] : aa_nat_real(aa_fun1538629612289296340t_real(aTP_Lamm_ayl(Uu),Uua),Uub) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),aa_nat_nat(Uua,Uub)) ).

% ATP.lambda_713
tff(fact_10891_ATP_Olambda__714,axiom,
    ! [Uu: nat,Uua: fun_nat_nat,Uub: nat] : aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_ayo(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Uu),aa_nat_nat(Uua,Uub)) ).

% ATP.lambda_714
tff(fact_10892_ATP_Olambda__715,axiom,
    ! [Uu: int,Uua: fun_nat_nat,Uub: nat] : aa_nat_int(aa_fun1613598493704395220at_int(aTP_Lamm_aym(Uu),Uua),Uub) = aa_nat_int(aa_int_fun_nat_int(power_power_int,Uu),aa_nat_nat(Uua,Uub)) ).

% ATP.lambda_715
tff(fact_10893_ATP_Olambda__716,axiom,
    ! [Uu: int,Uua: fun_int_nat,Uub: int] : aa_int_int(aa_fun2328901131235339916nt_int(aTP_Lamm_ayn(Uu),Uua),Uub) = aa_nat_int(aa_int_fun_nat_int(power_power_int,Uu),aa_int_nat(Uua,Uub)) ).

% ATP.lambda_716
tff(fact_10894_ATP_Olambda__717,axiom,
    ! [Uu: assn,Uua: vEBT_VEBTi,Uub: vEBT_VEBTi] : aa_VEBT_VEBTi_assn(aa_VEB1545114916007340237i_assn(aTP_Lamm_ch(Uu),Uua),Uub) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Uu),aa_o_assn(pure_assn,Uub = Uua)) ).

% ATP.lambda_717
tff(fact_10895_ATP_Olambda__718,axiom,
    ! [Uu: assn,Uua: product_unit,Uub: product_unit] : aa_Product_unit_assn(aa_Pro6018021088111102263t_assn(aTP_Lamm_ci(Uu),Uua),Uub) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Uu),aa_o_assn(pure_assn,Uub = Uua)) ).

% ATP.lambda_718
tff(fact_10896_ATP_Olambda__719,axiom,
    ! [Uu: assn,Uua: option_nat,Uub: option_nat] : aa_option_nat_assn(aa_opt3861602675778688675t_assn(aTP_Lamm_ce(Uu),Uua),Uub) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Uu),aa_o_assn(pure_assn,Uub = Uua)) ).

% ATP.lambda_719
tff(fact_10897_ATP_Olambda__720,axiom,
    ! [Uu: assn,Uua: $o,Uub: $o] : aa_o_assn(aa_o_fun_o_assn(aTP_Lamm_cg(Uu),(Uua)),(Uub)) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Uu),aa_o_assn(pure_assn,(Uub) = (Uua))) ).

% ATP.lambda_720
tff(fact_10898_ATP_Olambda__721,axiom,
    ! [Uu: assn,Uua: nat,Uub: nat] : aa_nat_assn(aa_nat_fun_nat_assn(aTP_Lamm_cf(Uu),Uua),Uub) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Uu),aa_o_assn(pure_assn,Uub = Uua)) ).

% ATP.lambda_721
tff(fact_10899_ATP_Olambda__722,axiom,
    ! [Uu: fun_Code_integer_nat,Uua: code_integer,Uub: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(aTP_Lamm_afs(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_722
tff(fact_10900_ATP_Olambda__723,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_afo(Uu),Uua),Uub)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),
          $ite(Uub = Uua,aa_nat_nat(suc,aa_Pro6801748904974264397nt_nat(Uu,Uub)),aa_Pro6801748904974264397nt_nat(Uu,Uub))) ) ).

% ATP.lambda_723
tff(fact_10901_ATP_Olambda__724,axiom,
    ! [Uu: fun_complex_nat,Uua: complex,Uub: complex] :
      ( aa_complex_o(aa_com5808801942499186519plex_o(aTP_Lamm_afr(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_complex_nat(Uu,Uub)),aa_complex_nat(Uu,Uub))) ) ).

% ATP.lambda_724
tff(fact_10902_ATP_Olambda__725,axiom,
    ! [Uu: fun_nat_nat,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_afp(Uu),Uua),Uub)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),
          $ite(Uub = Uua,aa_nat_nat(suc,aa_nat_nat(Uu,Uub)),aa_nat_nat(Uu,Uub))) ) ).

% ATP.lambda_725
tff(fact_10903_ATP_Olambda__726,axiom,
    ! [Uu: fun_int_nat,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_afq(Uu),Uua),Uub)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),
          $ite(Uub = Uua,aa_nat_nat(suc,aa_int_nat(Uu,Uub)),aa_int_nat(Uu,Uub))) ) ).

% ATP.lambda_726
tff(fact_10904_ATP_Olambda__727,axiom,
    ! [Uu: fun_Code_integer_nat,Uua: fun_Code_integer_o,Uub: code_integer] :
      ( aa_Code_integer_o(aa_fun8214818745163655228eger_o(aTP_Lamm_afx(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_727
tff(fact_10905_ATP_Olambda__728,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_aft(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_728
tff(fact_10906_ATP_Olambda__729,axiom,
    ! [Uu: fun_complex_nat,Uua: fun_complex_o,Uub: complex] :
      ( aa_complex_o(aa_fun2514317545224092796plex_o(aTP_Lamm_afw(Uu),Uua),Uub)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),
          $ite(aa_complex_o(Uua,Uub),aa_complex_nat(Uu,Uub),zero_zero_nat)) ) ).

% ATP.lambda_729
tff(fact_10907_ATP_Olambda__730,axiom,
    ! [Uu: fun_nat_nat,Uua: fun_nat_o,Uub: nat] :
      ( aa_nat_o(aa_fun5937193402907943164_nat_o(aTP_Lamm_afu(Uu),Uua),Uub)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),
          $ite(aa_nat_o(Uua,Uub),aa_nat_nat(Uu,Uub),zero_zero_nat)) ) ).

% ATP.lambda_730
tff(fact_10908_ATP_Olambda__731,axiom,
    ! [Uu: fun_int_nat,Uua: fun_int_o,Uub: int] :
      ( aa_int_o(aa_fun6243517787477317628_int_o(aTP_Lamm_afv(Uu),Uua),Uub)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),
          $ite(aa_int_o(Uua,Uub),aa_int_nat(Uu,Uub),zero_zero_nat)) ) ).

% ATP.lambda_731
tff(fact_10909_ATP_Olambda__732,axiom,
    ! [Uu: fun_Code_integer_nat,Uua: fun_Code_integer_nat,Uub: code_integer] :
      ( aa_Code_integer_o(aa_fun4054834855839773292eger_o(aTP_Lamm_afn(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_732
tff(fact_10910_ATP_Olambda__733,axiom,
    ! [Uu: fun_Pr5160692531401422620nt_nat,Uua: fun_Pr5160692531401422620nt_nat,Uub: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aa_fun6302685600795359042_int_o(aTP_Lamm_aff(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_733
tff(fact_10911_ATP_Olambda__734,axiom,
    ! [Uu: fun_complex_nat,Uua: fun_complex_nat,Uub: complex] :
      ( aa_complex_o(aa_fun8723867720292580170plex_o(aTP_Lamm_afl(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_complex_nat(Uu,Uub)),aa_complex_nat(Uua,Uub))) ) ).

% ATP.lambda_734
tff(fact_10912_ATP_Olambda__735,axiom,
    ! [Uu: fun_nat_nat,Uua: fun_nat_nat,Uub: nat] :
      ( aa_nat_o(aa_fun8673969331427483654_nat_o(aTP_Lamm_afh(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_735
tff(fact_10913_ATP_Olambda__736,axiom,
    ! [Uu: fun_int_nat,Uua: fun_int_nat,Uub: int] :
      ( aa_int_o(aa_fun5425778157628386638_int_o(aTP_Lamm_afj(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_736
tff(fact_10914_ATP_Olambda__737,axiom,
    ! [Uu: fun_nat_real,Uua: nat,Uub: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_asr(Uu),Uua),Uub) = aa_nat_real(Uu,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),aa_nat_nat(suc,Uub))) ).

% ATP.lambda_737
tff(fact_10915_ATP_Olambda__738,axiom,
    ! [Uu: fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_asq(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_738
tff(fact_10916_ATP_Olambda__739,axiom,
    ! [Uu: fun_nat_real,Uua: nat,Uub: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_anx(Uu),Uua),Uub) = aa_nat_real(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)) ).

% ATP.lambda_739
tff(fact_10917_ATP_Olambda__740,axiom,
    ! [Uu: fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_anw(Uu),Uua),Uub) = aa_nat_nat(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)) ).

% ATP.lambda_740
tff(fact_10918_ATP_Olambda__741,axiom,
    ! [Uu: fun_nat_int,Uua: nat,Uub: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_ayp(Uu),Uua),Uub) = aa_nat_int(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)) ).

% ATP.lambda_741
tff(fact_10919_ATP_Olambda__742,axiom,
    ! [Uu: fun_Pr2118491707810114563at_nat,Uua: nat,Uub: nat] : aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aTP_Lamm_kq(Uu),Uua),Uub) = aa_Pro262675715343573244at_nat(Uu,aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Uua),Uub)) ).

% ATP.lambda_742
tff(fact_10920_ATP_Olambda__743,axiom,
    ! [Uu: fun_Pr4796943543331071514_nat_o,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_kr(Uu),Uua),Uub)
    <=> aa_Pro6150637604445964947_nat_o(Uu,aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Uua),Uub)) ) ).

% ATP.lambda_743
tff(fact_10921_ATP_Olambda__744,axiom,
    ! [Uu: fun_Pr2478283820680522243nt_int,Uua: int,Uub: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_ks(Uu),Uua),Uub) = aa_Pro6970504127699319548nt_int(Uu,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Uua),Uub)) ).

% ATP.lambda_744
tff(fact_10922_ATP_Olambda__745,axiom,
    ! [Uu: fun_Pr3238374608701176162_int_o,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_kt(Uu),Uua),Uub)
    <=> aa_Pro4592068669816069595_int_o(Uu,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Uua),Uub)) ) ).

% ATP.lambda_745
tff(fact_10923_ATP_Olambda__746,axiom,
    ! [Uu: fun_Pr982841511892225912nt_int,Uua: int,Uub: int] : aa_int_int(aa_int_fun_int_int(aTP_Lamm_ku(Uu),Uua),Uub) = aa_Pro6799258434465214121nt_int(Uu,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Uua),Uub)) ).

% ATP.lambda_746
tff(fact_10924_ATP_Olambda__747,axiom,
    ! [Uu: nat,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_arg(Uu),Uua),Uub) = aa_nat_real(Uua,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uu)) ).

% ATP.lambda_747
tff(fact_10925_ATP_Olambda__748,axiom,
    ! [Uu: fun_op8989173174992673105on_num,Uua: fun_num_option_num,Uub: num] : aa_num_option_num(aa_fun3636977833914307392on_num(aTP_Lamm_cr(Uu),Uua),Uub) = aa_opt5369159329878751634on_num(Uu,aa_num_option_num(Uua,Uub)) ).

% ATP.lambda_748
tff(fact_10926_ATP_Olambda__749,axiom,
    ! [Uu: fun_option_num_num,Uua: fun_num_option_num,Uub: num] : aa_num_num(aa_fun7142274290070961008um_num(aTP_Lamm_cn(Uu),Uua),Uub) = aa_option_num_num(Uu,aa_num_option_num(Uua,Uub)) ).

% ATP.lambda_749
tff(fact_10927_ATP_Olambda__750,axiom,
    ! [Uu: fun_option_num_int,Uua: fun_num_option_num,Uub: num] : aa_num_int(aa_fun2256179333612926594um_int(aTP_Lamm_co(Uu),Uua),Uub) = aa_option_num_int(Uu,aa_num_option_num(Uua,Uub)) ).

% ATP.lambda_750
tff(fact_10928_ATP_Olambda__751,axiom,
    ! [Uu: fun_o_o,Uua: fun_Pr4796943543331071514_nat_o,Uub: product_prod_nat_nat] :
      ( aa_Pro6150637604445964947_nat_o(aa_fun359341089595923442_nat_o(aTP_Lamm_cs(Uu),Uua),Uub)
    <=> aa_o_o(Uu,aa_Pro6150637604445964947_nat_o(Uua,Uub)) ) ).

% ATP.lambda_751
tff(fact_10929_ATP_Olambda__752,axiom,
    ! [Uu: fun_num_option_num,Uua: fun_num_num,Uub: num] : aa_num_option_num(aa_fun612943910911948912on_num(aTP_Lamm_cp(Uu),Uua),Uub) = aa_num_option_num(Uu,aa_num_num(Uua,Uub)) ).

% ATP.lambda_752
tff(fact_10930_ATP_Olambda__753,axiom,
    ! [Uu: fun_num_num,Uua: fun_num_num,Uub: num] : aa_num_num(aa_fun8048995283964520608um_num(aTP_Lamm_cj(Uu),Uua),Uub) = aa_num_num(Uu,aa_num_num(Uua,Uub)) ).

% ATP.lambda_753
tff(fact_10931_ATP_Olambda__754,axiom,
    ! [Uu: fun_num_int,Uua: fun_num_num,Uub: num] : aa_num_int(aa_fun3162900327506486194um_int(aTP_Lamm_ck(Uu),Uua),Uub) = aa_num_int(Uu,aa_num_num(Uua,Uub)) ).

% ATP.lambda_754
tff(fact_10932_ATP_Olambda__755,axiom,
    ! [Uu: fun_int_option_num,Uua: fun_num_int,Uub: num] : aa_num_option_num(aa_fun3858197967482829954on_num(aTP_Lamm_cq(Uu),Uua),Uub) = aa_int_option_num(Uu,aa_num_int(Uua,Uub)) ).

% ATP.lambda_755
tff(fact_10933_ATP_Olambda__756,axiom,
    ! [Uu: fun_int_num,Uua: fun_num_int,Uub: num] : aa_num_num(aa_fun5215869348392759218um_num(aTP_Lamm_cl(Uu),Uua),Uub) = aa_int_num(Uu,aa_num_int(Uua,Uub)) ).

% ATP.lambda_756
tff(fact_10934_ATP_Olambda__757,axiom,
    ! [Uu: fun_int_int,Uua: fun_num_int,Uub: num] : aa_num_int(aa_fun329774391934724804um_int(aTP_Lamm_cm(Uu),Uua),Uub) = aa_int_int(Uu,aa_num_int(Uua,Uub)) ).

% ATP.lambda_757
tff(fact_10935_ATP_Olambda__758,axiom,
    ! [Uu: fun_na1476630412504255114at_nat,Uua: nat,Uub: nat] : aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aTP_Lamm_kw(Uu),Uua),Uub) = aa_nat7603221163087113075at_nat(Uu,Uua) ).

% ATP.lambda_758
tff(fact_10936_ATP_Olambda__759,axiom,
    ! [Uu: fun_nat_o,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_ky(Uu),Uua),Uub)
    <=> aa_nat_o(Uu,Uua) ) ).

% ATP.lambda_759
tff(fact_10937_ATP_Olambda__760,axiom,
    ! [Uu: fun_in4211310926729683942nt_int,Uua: int,Uub: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_la(Uu),Uua),Uub) = aa_int7462565939943104719nt_int(Uu,Uua) ).

% ATP.lambda_760
tff(fact_10938_ATP_Olambda__761,axiom,
    ! [Uu: fun_int_o,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_lc(Uu),Uua),Uub)
    <=> aa_int_o(Uu,Uua) ) ).

% ATP.lambda_761
tff(fact_10939_ATP_Olambda__762,axiom,
    ! [Uu: fun_int_int,Uua: int,Uub: int] : aa_int_int(aa_int_fun_int_int(aTP_Lamm_le(Uu),Uua),Uub) = aa_int_int(Uu,Uua) ).

% ATP.lambda_762
tff(fact_10940_ATP_Olambda__763,axiom,
    ! [Uu: nat,Uua: array_VEBT_VEBTi,Uub: vEBT_VEBTi] : aa_VEB836174949890266638_VEBTi(aa_arr2912372873376718191_VEBTi(aTP_Lamm_xm(Uu),Uua),Uub) = aa_VEB836174949890266638_VEBTi(heap_T3630416162098727440_VEBTi,aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,Uu)),Uua),Uub)) ).

% ATP.lambda_763
tff(fact_10941_ATP_Olambda__764,axiom,
    ! [Uu: nat,Uua: $o,Uub: $o] :
      aa_o_H7399033255965945470Heap_o(aa_o_f869145629572826609Heap_o(aTP_Lamm_jv(Uu),(Uua)),(Uub)) = aa_o_H7399033255965945470Heap_o(heap_Time_return_o,
        $ite(
          Uu = zero_zero_nat,
          (Uua),
          $ite(Uu = one_one_nat,(Uub),$false) )) ).

% ATP.lambda_764
tff(fact_10942_ATP_Olambda__765,axiom,
    ! [Uu: nat,Uua: nat,Uub: option_nat] : aa_opt9163293676379396302on_nat(aa_nat9127621225095157975on_nat(aTP_Lamm_ep(Uu),Uua),Uub) = heap_T3487192422709364219on_nat(aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,aa_nat_option_nat(some_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uu),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_option_nat(some_nat,Uua))),Uub)) ).

% ATP.lambda_765
tff(fact_10943_ATP_Olambda__766,axiom,
    ! [Uu: nat,Uua: option_nat,Uub: option_nat] : aa_opt9163293676379396302on_nat(aa_opt7201247987542299399on_nat(aTP_Lamm_eq(Uu),Uua),Uub) = heap_T3487192422709364219on_nat(aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,aa_nat_option_nat(some_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uu),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),Uua)),Uub)) ).

% ATP.lambda_766
tff(fact_10944_ATP_Olambda__767,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] : aa_nat_assn(aa_nat_fun_nat_assn(aTP_Lamm_bd(Uu),Uua),Uub) = aa_o_assn(pure_assn,Uub = aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Uu),Uua)) ).

% ATP.lambda_767
tff(fact_10945_ATP_Olambda__768,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] : aa_nat_assn(aa_nat_fun_nat_assn(aTP_Lamm_bc(Uu),Uua),Uub) = aa_o_assn(pure_assn,Uub = aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Uu),Uua)) ).

% ATP.lambda_768
tff(fact_10946_ATP_Olambda__769,axiom,
    ! [Uu: vEBT_VEBT,Uua: vEBT_VEBTi,Uub: nat] : aa_nat2910048432523430448Heap_o(aa_VEB1455453220190609250Heap_o(aa_VEB8672304581938048793Heap_o(aTP_Lamm_abz,Uu),Uua),Uub) = heap_Time_Heap_o2(aTP_Lamm_zg) ).

% ATP.lambda_769
tff(fact_10947_ATP_Olambda__770,axiom,
    ! [Uu: vEBT_VEBT,Uua: vEBT_VEBTi,Uub: nat] : aa_nat947733127081326238on_nat(aa_VEB3566417204837707778on_nat(aa_VEB1030146501360450209on_nat(aTP_Lamm_abv,Uu),Uua),Uub) = heap_T5286843759275942675on_nat(aTP_Lamm_zf) ).

% ATP.lambda_770
tff(fact_10948_ATP_Olambda__771,axiom,
    ! [Uu: vEBT_VEBT,Uua: vEBT_VEBTi,Uub: nat] : aa_nat5408703056993926195_VEBTi(aa_VEB4833742232818206615_VEBTi(aa_VEB6292075424854053302_VEBTi(aTP_Lamm_yt,Uu),Uua),Uub) = heap_T1489671670754571048_VEBTi(aTP_Lamm_ys) ).

% ATP.lambda_771
tff(fact_10949_ATP_Olambda__772,axiom,
    ! [Uu: fun_Pr940977554174145406Heap_o,Uua: vEBT_VEBT,Uub: vEBT_VEBTi,Uuc: nat] : aa_nat2910048432523430448Heap_o(aa_VEB1455453220190609250Heap_o(aa_VEB8672304581938048793Heap_o(aTP_Lamm_aak(Uu),Uua),Uub),Uuc) = aa_VEB8194921341634505877Heap_o(vEBT_c6104975476656191286Heap_o(aa_nat1227502447850154924Heap_o(aa_VEB6636772557416003555Heap_o(aTP_Lamm_aaj(Uu),Uua),Uuc),aTP_Lamm_jv(Uuc)),Uub) ).

% ATP.lambda_772
tff(fact_10950_ATP_Olambda__773,axiom,
    ! [Uu: fun_Pr633315466666964198on_nat,Uua: vEBT_VEBT,Uub: vEBT_VEBTi,Uuc: nat] : aa_nat947733127081326238on_nat(aa_VEB3566417204837707778on_nat(aa_VEB1030146501360450209on_nat(aTP_Lamm_xk(Uu),Uua),Uub),Uuc) = aa_VEB2711320435582112505on_nat(vEBT_c6250501799366334488on_nat(aa_nat3457676333800472698on_nat(aa_VEB6681700655871274393on_nat(aTP_Lamm_xj(Uu),Uua),Uuc),aTP_Lamm_ez(Uuc)),Uub) ).

% ATP.lambda_773
tff(fact_10951_ATP_Olambda__774,axiom,
    ! [Uu: fun_Pr633315466666964198on_nat,Uua: vEBT_VEBT,Uub: vEBT_VEBTi,Uuc: nat] : aa_nat947733127081326238on_nat(aa_VEB3566417204837707778on_nat(aa_VEB1030146501360450209on_nat(aTP_Lamm_wi(Uu),Uua),Uub),Uuc) = aa_VEB2711320435582112505on_nat(vEBT_c6250501799366334488on_nat(aa_nat3457676333800472698on_nat(aa_VEB6681700655871274393on_nat(aTP_Lamm_wh(Uu),Uua),Uuc),aTP_Lamm_fi(Uuc)),Uub) ).

% ATP.lambda_774
tff(fact_10952_ATP_Olambda__775,axiom,
    ! [Uu: fun_Pr4404027214751104635_VEBTi,Uua: vEBT_VEBT,Uub: vEBT_VEBTi,Uuc: nat] : aa_nat5408703056993926195_VEBTi(aa_VEB4833742232818206615_VEBTi(aa_VEB6292075424854053302_VEBTi(aTP_Lamm_vr(Uu),Uua),Uub),Uuc) = aa_VEB836174949890266638_VEBTi(vEBT_c6028912655521741485_VEBTi(aa_nat3009484633486537743_VEBTi(aa_VEB600817258366935214_VEBTi(aTP_Lamm_vq(Uu),Uua),Uuc),aTP_Lamm_je(Uuc)),Uub) ).

% ATP.lambda_775
tff(fact_10953_ATP_Olambda__776,axiom,
    ! [Uu: fun_VE3775134097089128360on_nat,Uua: vEBT_VEBT,Uub: vEBT_VEBTi,Uuc: nat] : aa_nat947733127081326238on_nat(aa_VEB3566417204837707778on_nat(aa_VEB1030146501360450209on_nat(aTP_Lamm_abw(Uu),Uua),Uub),Uuc) = aa_VEB2711320435582112505on_nat(vEBT_c6250501799366334488on_nat(aa_nat3457676333800472698on_nat(aa_VEB6681700655871274393on_nat(aTP_Lamm_sz(Uu),Uua),Uuc),aTP_Lamm_fi(Uuc)),Uub) ).

% ATP.lambda_776
tff(fact_10954_ATP_Olambda__777,axiom,
    ! [Uu: fun_VE3775134097089128360on_nat,Uua: vEBT_VEBT,Uub: vEBT_VEBTi,Uuc: nat] : aa_nat947733127081326238on_nat(aa_VEB3566417204837707778on_nat(aa_VEB1030146501360450209on_nat(aTP_Lamm_abx(Uu),Uua),Uub),Uuc) = aa_VEB2711320435582112505on_nat(vEBT_c6250501799366334488on_nat(aa_nat3457676333800472698on_nat(aa_VEB6681700655871274393on_nat(aTP_Lamm_sk(Uu),Uua),Uuc),aTP_Lamm_ez(Uuc)),Uub) ).

% ATP.lambda_777
tff(fact_10955_ATP_Olambda__778,axiom,
    ! [Uu: fun_VE2018681615261037117_VEBTi,Uua: vEBT_VEBT,Uub: vEBT_VEBTi,Uuc: nat] : aa_nat5408703056993926195_VEBTi(aa_VEB4833742232818206615_VEBTi(aa_VEB6292075424854053302_VEBTi(aTP_Lamm_yu(Uu),Uua),Uub),Uuc) = aa_VEB836174949890266638_VEBTi(vEBT_c6028912655521741485_VEBTi(aa_nat3009484633486537743_VEBTi(aa_VEB600817258366935214_VEBTi(aTP_Lamm_rt(Uu),Uua),Uuc),aTP_Lamm_je(Uuc)),Uub) ).

% ATP.lambda_778
tff(fact_10956_ATP_Olambda__779,axiom,
    ! [Uu: fun_VE8181444127144381288Heap_o,Uua: vEBT_VEBT,Uub: vEBT_VEBTi,Uuc: nat] : aa_nat2910048432523430448Heap_o(aa_VEB1455453220190609250Heap_o(aa_VEB8672304581938048793Heap_o(aTP_Lamm_aca(Uu),Uua),Uub),Uuc) = aa_VEB8194921341634505877Heap_o(vEBT_c6104975476656191286Heap_o(aa_nat1227502447850154924Heap_o(aa_VEB6636772557416003555Heap_o(aTP_Lamm_qi(Uu),Uua),Uuc),aTP_Lamm_jv(Uuc)),Uub) ).

% ATP.lambda_779
tff(fact_10957_ATP_Olambda__780,axiom,
    ! [Uu: array_VEBT_VEBTi,Uua: nat,Uub: nat,Uuc: nat] :
      aa_nat2910048432523430448Heap_o(aa_nat5592791454563588861Heap_o(aa_nat7429013343984486730Heap_o(aTP_Lamm_jq(Uu),Uua),Uub),Uuc) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uua),Uuc),heap_T3040810144269856602EBTi_o(aa_nat5408703056993926195_VEBTi(array_nth_VEBT_VEBTi(Uu),Uua),aTP_Lamm_jp(Uub)),aa_o_H7399033255965945470Heap_o(heap_Time_return_o,$false)) ).

% ATP.lambda_780
tff(fact_10958_ATP_Olambda__781,axiom,
    ! [Uu: fun_nat_real,Uua: set_nat,Uub: fun_nat_real,Uuc: nat] :
      aa_nat_real(aa_fun1839206346889086768t_real(aa_set1316517696749040435t_real(aTP_Lamm_aqg(Uu),Uua),Uub),Uuc) = $ite(aa_set_nat_o(member_nat(Uuc),Uua),aa_nat_real(Uub,Uuc),aa_nat_real(Uu,Uuc)) ).

% ATP.lambda_781
tff(fact_10959_ATP_Olambda__782,axiom,
    ! [Uu: vEBT_VEBT,Uua: fun_VEBT_VEBT_real,Uub: fun_VEBT_VEBT_real,Uuc: vEBT_VEBT] :
      aa_VEBT_VEBT_real(aa_fun1825138017709814396T_real(aa_fun910392650362481850T_real(aTP_Lamm_aoz(Uu),Uua),Uub),Uuc) = $ite(Uuc = Uu,aa_VEBT_VEBT_real(Uua,Uuc),aa_VEBT_VEBT_real(Uub,Uuc)) ).

% ATP.lambda_782
tff(fact_10960_ATP_Olambda__783,axiom,
    ! [Uu: vEBT_VEBT,Uua: fun_VEBT_VEBT_rat,Uub: fun_VEBT_VEBT_rat,Uuc: vEBT_VEBT] :
      aa_VEBT_VEBT_rat(aa_fun8105832842879916156BT_rat(aa_fun2693410532875676774BT_rat(aTP_Lamm_apc(Uu),Uua),Uub),Uuc) = $ite(Uuc = Uu,aa_VEBT_VEBT_rat(Uua,Uuc),aa_VEBT_VEBT_rat(Uub,Uuc)) ).

% ATP.lambda_783
tff(fact_10961_ATP_Olambda__784,axiom,
    ! [Uu: vEBT_VEBT,Uua: fun_VEBT_VEBT_nat,Uub: fun_VEBT_VEBT_nat,Uuc: vEBT_VEBT] :
      aa_VEBT_VEBT_nat(aa_fun8284231930576791164BT_nat(aa_fun8778301808485728350BT_nat(aTP_Lamm_apf(Uu),Uua),Uub),Uuc) = $ite(Uuc = Uu,aa_VEBT_VEBT_nat(Uua,Uuc),aa_VEBT_VEBT_nat(Uub,Uuc)) ).

% ATP.lambda_784
tff(fact_10962_ATP_Olambda__785,axiom,
    ! [Uu: vEBT_VEBT,Uua: fun_VEBT_VEBT_int,Uub: fun_VEBT_VEBT_int,Uuc: vEBT_VEBT] :
      aa_VEBT_VEBT_int(aa_fun7144222995531728252BT_int(aa_fun7485955581636121530BT_int(aTP_Lamm_api(Uu),Uua),Uub),Uuc) = $ite(Uuc = Uu,aa_VEBT_VEBT_int(Uua,Uuc),aa_VEBT_VEBT_int(Uub,Uuc)) ).

% ATP.lambda_785
tff(fact_10963_ATP_Olambda__786,axiom,
    ! [Uu: code_integer,Uua: fun_Co3960896333048504454r_real,Uub: fun_Co3960896333048504454r_real,Uuc: code_integer] :
      aa_Code_integer_real(aa_fun6341089502340155082r_real(aa_fun7644355603919855821r_real(aTP_Lamm_apb(Uu),Uua),Uub),Uuc) = $ite(Uuc = Uu,aa_Code_integer_real(Uua,Uuc),aa_Code_integer_real(Uub,Uuc)) ).

% ATP.lambda_786
tff(fact_10964_ATP_Olambda__787,axiom,
    ! [Uu: code_integer,Uua: fun_Code_integer_rat,Uub: fun_Code_integer_rat,Uuc: code_integer] :
      aa_Code_integer_rat(aa_fun8169474305930283170er_rat(aa_fun1894662059565830609er_rat(aTP_Lamm_ape(Uu),Uua),Uub),Uuc) = $ite(Uuc = Uu,aa_Code_integer_rat(Uua,Uuc),aa_Code_integer_rat(Uub,Uuc)) ).

% ATP.lambda_787
tff(fact_10965_ATP_Olambda__788,axiom,
    ! [Uu: code_integer,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_aph(Uu),Uua),Uub),Uuc) = $ite(Uuc = Uu,aa_Code_integer_nat(Uua,Uuc),aa_Code_integer_nat(Uub,Uuc)) ).

% ATP.lambda_788
tff(fact_10966_ATP_Olambda__789,axiom,
    ! [Uu: complex,Uua: fun_complex_real,Uub: fun_complex_real,Uuc: complex] :
      aa_complex_real(aa_fun8933909628229591020x_real(aa_fun8382785274855863872x_real(aTP_Lamm_apa(Uu),Uua),Uub),Uuc) = $ite(Uuc = Uu,aa_complex_real(Uua,Uuc),aa_complex_real(Uub,Uuc)) ).

% ATP.lambda_789
tff(fact_10967_ATP_Olambda__790,axiom,
    ! [Uu: complex,Uua: fun_complex_rat,Uub: fun_complex_rat,Uuc: complex] :
      aa_complex_rat(aa_fun7622890401989837636ex_rat(aa_fun530470884365435460ex_rat(aTP_Lamm_apd(Uu),Uua),Uub),Uuc) = $ite(Uuc = Uu,aa_complex_rat(Uua,Uuc),aa_complex_rat(Uub,Uuc)) ).

% ATP.lambda_790
tff(fact_10968_ATP_Olambda__791,axiom,
    ! [Uu: complex,Uua: fun_complex_nat,Uub: fun_complex_nat,Uuc: complex] :
      aa_complex_nat(aa_fun2465681967280144180ex_nat(aa_fun2484745150758625324ex_nat(aTP_Lamm_apg(Uu),Uua),Uub),Uuc) = $ite(Uuc = Uu,aa_complex_nat(Uua,Uuc),aa_complex_nat(Uub,Uuc)) ).

% ATP.lambda_791
tff(fact_10969_ATP_Olambda__792,axiom,
    ! [Uu: list_VEBT_VEBT,Uua: nat,Uub: nat,Uuc: vEBT_VEBTi] : aa_VEB8194921341634505877Heap_o(aa_nat1164146675577659234Heap_o(aa_nat3058146694327238959Heap_o(aTP_Lamm_qj(Uu),Uua),Uub),Uuc) = aa_nat2910048432523430448Heap_o(aa_VEB1455453220190609250Heap_o(aa_VEB8672304581938048793Heap_o(vEBT_V854960066525838166emberi,nth_VEBT_VEBT(Uu,Uua)),Uuc),Uub) ).

% ATP.lambda_792
tff(fact_10970_ATP_Olambda__793,axiom,
    ! [Uu: nat,Uua: array_VEBT_VEBTi,Uub: option_nat,Uuc: product_unit] :
      aa_Pro451875010097498139on_nat(aa_opt5089197864053947520on_nat(aa_arr5509772438618118773on_nat(aTP_Lamm_oe(Uu),Uua),Uub),Uuc) = $ite(Uub = none_nat,heap_T3487192422709364219on_nat(none_nat),heap_T2868974464944644318on_nat(aa_nat5408703056993926195_VEBTi(array_nth_VEBT_VEBTi(Uua),the_nat(Uub)),aa_opt1214731524852023858on_nat(aTP_Lamm_fa(Uu),Uub))) ).

% ATP.lambda_793
tff(fact_10971_ATP_Olambda__794,axiom,
    ! [Uu: real,Uua: real,Uub: nat,Uuc: nat] : aa_nat_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_arq(Uu),Uua),Uub),Uuc) = groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_arp(Uu),Uua),Uub),set_ord_lessThan_nat(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),aa_nat_nat(suc,zero_zero_nat))),Uuc))) ).

% ATP.lambda_794
tff(fact_10972_ATP_Olambda__795,axiom,
    ! [Uu: rat,Uua: rat,Uub: nat,Uuc: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_aro(Uu),Uua),Uub),Uuc) = groups2906978787729119204at_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_arn(Uu),Uua),Uub),set_ord_lessThan_nat(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),aa_nat_nat(suc,zero_zero_nat))),Uuc))) ).

% ATP.lambda_795
tff(fact_10973_ATP_Olambda__796,axiom,
    ! [Uu: $o,Uua: $o,Uub: code_integer,Uuc: $o] : aa_o_char(aa_Cod7842134615204088097o_char(aa_o_f2742674633914528860o_char(aTP_Lamm_bcm((Uu)),(Uua)),Uub),(Uuc)) = produc4188289175737317920o_char(aa_o_f2742674633914528860o_char(aa_o_f9003794812476387647o_char(aTP_Lamm_bcl((Uu)),(Uua)),(Uuc)),code_bit_cut_integer(Uub)) ).

% ATP.lambda_796
tff(fact_10974_ATP_Olambda__797,axiom,
    ! [Uu: fun_nat_real,Uua: fun_nat_real,Uub: set_nat,Uuc: nat] : aa_nat_real(aa_set1166323925324055613t_real(aa_fun1400677770452585267t_real(aTP_Lamm_alh(Uu),Uua),Uub),Uuc) = groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_fun3023522663955195517t_real(aTP_Lamm_alg(Uu),Uua),Uuc),Uub) ).

% ATP.lambda_797
tff(fact_10975_ATP_Olambda__798,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_alf(Uu),Uua),Uub),Uuc) = groups4538972089207619220nt_int(aa_int_fun_int_int(aa_fun5911324870756489105nt_int(aTP_Lamm_ale(Uu),Uua),Uuc),Uub) ).

% ATP.lambda_798
tff(fact_10976_ATP_Olambda__799,axiom,
    ! [Uu: fun_complex_complex,Uua: fun_complex_complex,Uub: set_complex,Uuc: complex] : aa_complex_complex(aa_set7743095458595710203omplex(aa_fun1583801116089702097omplex(aTP_Lamm_ald(Uu),Uua),Uub),Uuc) = groups7754918857620584856omplex(aa_com3342990543769683525omplex(aa_fun6995112137416684571omplex(aTP_Lamm_alc(Uu),Uua),Uuc),Uub) ).

% ATP.lambda_799
tff(fact_10977_ATP_Olambda__800,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_alb(Uu),Uua),Uub),Uuc) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aa_fun6650658898577385413at_nat(aTP_Lamm_ala(Uu),Uua),Uuc)),Uub) ).

% ATP.lambda_800
tff(fact_10978_ATP_Olambda__801,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_axt(Uu),Uua),Uub),Uuc) = groups3188404863801439024er_int(aa_int1390634949194299888er_int(aTP_Lamm_axr(Uua),Uuc),collect_Code_integer(aa_int5993532643906289898eger_o(aa_fun7355488674016760700eger_o(aTP_Lamm_axs(Uu),Uub),Uuc))) ).

% ATP.lambda_801
tff(fact_10979_ATP_Olambda__802,axiom,
    ! [Uu: set_complex,Uua: fun_co333780937270390912nt_int,Uub: fun_co3431063785546946928_int_o,Uuc: int] : aa_int_int(aa_fun2158622962795119753nt_int(aa_fun607866698265288346nt_int(aTP_Lamm_axo(Uu),Uua),Uub),Uuc) = groups858564598930262913ex_int(aa_int6126987627036542017ex_int(aTP_Lamm_axm(Uua),Uuc),collect_complex(aa_int_fun_complex_o(aa_fun7279530946696969788plex_o(aTP_Lamm_axn(Uu),Uub),Uuc))) ).

% ATP.lambda_802
tff(fact_10980_ATP_Olambda__803,axiom,
    ! [Uu: set_real,Uua: fun_real_fun_int_int,Uub: fun_real_fun_int_o,Uuc: int] : aa_int_int(aa_fun3996843141751676939nt_int(aa_fun129049606478405534nt_int(aTP_Lamm_axj(Uu),Uua),Uub),Uuc) = groups4694064378042380927al_int(aa_int_fun_real_int(aTP_Lamm_axi(Uua),Uuc),collect_real(aa_int_fun_real_o(aa_fun2391495421969208892real_o(aTP_Lamm_aew(Uu),Uub),Uuc))) ).

% ATP.lambda_803
tff(fact_10981_ATP_Olambda__804,axiom,
    ! [Uu: set_VEBT_VEBT,Uua: fun_VE294566951569886236nt_int,Uub: fun_VE4235606543851232980_int_o,Uuc: int] : aa_int_int(aa_fun2879915576171315815nt_int(aa_fun5708383307987270406nt_int(aTP_Lamm_axf(Uu),Uua),Uub),Uuc) = groups6359315924273963643BT_int(aa_int1343382402272391561BT_int(aTP_Lamm_axe(Uua),Uuc),collect_VEBT_VEBT(aa_int8839263053405020433VEBT_o(aa_fun8807180201667926370VEBT_o(aTP_Lamm_aev(Uu),Uub),Uuc))) ).

% ATP.lambda_804
tff(fact_10982_ATP_Olambda__805,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_axb(Uu),Uua),Uub),Uuc) = groups3188404863801439024er_int(aa_nat1674686023564409748er_int(aTP_Lamm_axa(Uua),Uuc),collect_Code_integer(aa_nat7420681472272698950eger_o(aa_fun7373950593701497468eger_o(aTP_Lamm_ami(Uu),Uub),Uuc))) ).

% ATP.lambda_805
tff(fact_10983_ATP_Olambda__806,axiom,
    ! [Uu: set_complex,Uua: fun_co7370231399432144932at_int,Uub: fun_co5676208753217392844_nat_o,Uuc: nat] : aa_nat_int(aa_fun1349975166126329225at_int(aa_fun7503395680861046006at_int(aTP_Lamm_awy(Uu),Uua),Uub),Uuc) = groups858564598930262913ex_int(aa_nat8436805559129492965ex_int(aTP_Lamm_awx(Uua),Uuc),collect_complex(aa_nat_fun_complex_o(aa_fun4610599729497257276plex_o(aTP_Lamm_amd(Uu),Uub),Uuc))) ).

% ATP.lambda_806
tff(fact_10984_ATP_Olambda__807,axiom,
    ! [Uu: set_real,Uua: fun_real_fun_nat_int,Uub: fun_real_fun_nat_o,Uuc: nat] : aa_nat_int(aa_fun3188195345082886411at_int(aa_fun8006801000864960506at_int(aTP_Lamm_awv(Uu),Uua),Uub),Uuc) = groups4694064378042380927al_int(aa_nat_fun_real_int(aTP_Lamm_awu(Uua),Uuc),collect_real(aa_nat_fun_real_o(aa_fun6917832400337718588real_o(aTP_Lamm_aeu(Uu),Uub),Uuc))) ).

% ATP.lambda_807
tff(fact_10985_ATP_Olambda__808,axiom,
    ! [Uu: set_VEBT_VEBT,Uua: fun_VE7331017413731640256at_int,Uub: fun_VE6480751511521678896_nat_o,Uuc: nat] : aa_nat_int(aa_fun2071267779502525287at_int(aa_fun7598684367898235306at_int(aTP_Lamm_aws(Uu),Uua),Uub),Uuc) = groups6359315924273963643BT_int(aa_nat3598295560461965541BT_int(aTP_Lamm_awr(Uua),Uuc),collect_VEBT_VEBT(aa_nat1925708948643195573VEBT_o(aa_fun2405540735884763746VEBT_o(aTP_Lamm_aet(Uu),Uub),Uuc))) ).

% ATP.lambda_808
tff(fact_10986_ATP_Olambda__809,axiom,
    ! [Uu: set_real,Uua: fun_real_fun_nat_nat,Uub: fun_real_fun_nat_o,Uuc: nat] : aa_nat_nat(aa_fun7366046364592083119at_nat(aa_fun7718511627822915834at_nat(aTP_Lamm_axx(Uu),Uua),Uub),Uuc) = groups4696554848551431203al_nat(aa_nat_fun_real_nat(aTP_Lamm_alt(Uua),Uuc),collect_real(aa_nat_fun_real_o(aa_fun6917832400337718588real_o(aTP_Lamm_aeu(Uu),Uub),Uuc))) ).

% ATP.lambda_809
tff(fact_10987_ATP_Olambda__810,axiom,
    ! [Uu: set_VEBT_VEBT,Uua: fun_VE597728521808591204at_nat,Uub: fun_VE6480751511521678896_nat_o,Uuc: nat] : aa_nat_nat(aa_fun6249118799011721995at_nat(aa_fun2705500642357200114at_nat(aTP_Lamm_axv(Uu),Uua),Uub),Uuc) = groups6361806394783013919BT_nat(aa_nat7776146579971162249BT_nat(aTP_Lamm_alp(Uua),Uuc),collect_VEBT_VEBT(aa_nat1925708948643195573VEBT_o(aa_fun2405540735884763746VEBT_o(aTP_Lamm_aet(Uu),Uub),Uuc))) ).

% ATP.lambda_810
tff(fact_10988_ATP_Olambda__811,axiom,
    ! [Uu: set_Code_integer,Uua: fun_Co8905303033605938227omplex,Uub: fun_Co8975588572093281791plex_o,Uuc: complex] : aa_complex_complex(aa_fun7787660480616508700omplex(aa_fun963654016877434554omplex(aTP_Lamm_ang(Uu),Uua),Uub),Uuc) = groups8024822376189712711omplex(aa_com7552888802784664052omplex(aTP_Lamm_ane(Uua),Uuc),collect_Code_integer(aa_com4877492900066995176eger_o(aa_fun3025727905316630780eger_o(aTP_Lamm_anf(Uu),Uub),Uuc))) ).

% ATP.lambda_811
tff(fact_10989_ATP_Olambda__812,axiom,
    ! [Uu: set_int,Uua: fun_in1307110748760623106omplex,Uub: fun_in5065666251602127472plex_o,Uuc: complex] : aa_complex_complex(aa_fun7450476157521870733omplex(aa_fun6156949752960786844omplex(aTP_Lamm_anb(Uu),Uua),Uub),Uuc) = groups3049146728041665814omplex(aa_com9156536935073639875omplex(aTP_Lamm_amz(Uua),Uuc),collect_int(aa_complex_fun_int_o(aa_fun7455058932602313788_int_o(aTP_Lamm_ana(Uu),Uub),Uuc))) ).

% ATP.lambda_812
tff(fact_10990_ATP_Olambda__813,axiom,
    ! [Uu: set_nat,Uua: fun_na1591161823130732966omplex,Uub: fun_na1554462459066888140plex_o,Uuc: complex] : aa_complex_complex(aa_fun1423577888295648489omplex(aa_fun601428455815558228omplex(aTP_Lamm_amw(Uu),Uua),Uub),Uuc) = groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_amu(Uua),Uuc),collect_nat(aa_complex_fun_nat_o(aa_fun1708641150486296380_nat_o(aTP_Lamm_amv(Uu),Uub),Uuc))) ).

% ATP.lambda_813
tff(fact_10991_ATP_Olambda__814,axiom,
    ! [Uu: set_real,Uua: fun_re7208826350674946562omplex,Uub: fun_re7319081385389557872plex_o,Uuc: complex] : aa_complex_complex(aa_fun9084337531551611917omplex(aa_fun6739043078600750748omplex(aTP_Lamm_amr(Uu),Uua),Uub),Uuc) = groups5754745047067104278omplex(aa_com4766338864777664963omplex(aTP_Lamm_amq(Uua),Uuc),collect_real(aa_com1951324360330529881real_o(aa_fun7692540709613205436real_o(aTP_Lamm_aey(Uu),Uub),Uuc))) ).

% ATP.lambda_814
tff(fact_10992_ATP_Olambda__815,axiom,
    ! [Uu: set_VEBT_VEBT,Uua: fun_VE2006399705683203104omplex,Uub: fun_VE7631951351884037842plex_o,Uuc: complex] : aa_complex_complex(aa_fun2592672125884331625omplex(aa_fun8556514703313836300omplex(aTP_Lamm_amn(Uu),Uua),Uub),Uuc) = groups1794756597179926696omplex(aa_com2862003112617944841omplex(aTP_Lamm_amm(Uua),Uuc),collect_VEBT_VEBT(aa_com3651963455866140051VEBT_o(aa_fun7268111043052431330VEBT_o(aTP_Lamm_aex(Uu),Uub),Uuc))) ).

% ATP.lambda_815
tff(fact_10993_ATP_Olambda__816,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_amj(Uu),Uua),Uub),Uuc) = groups7237345082560585321er_nat(aa_nat5852537043073606456er_nat(aTP_Lamm_amh(Uua),Uuc),collect_Code_integer(aa_nat7420681472272698950eger_o(aa_fun7373950593701497468eger_o(aTP_Lamm_ami(Uu),Uub),Uuc))) ).

% ATP.lambda_816
tff(fact_10994_ATP_Olambda__817,axiom,
    ! [Uu: set_complex,Uua: fun_co636942507509095880at_nat,Uub: fun_co5676208753217392844_nat_o,Uuc: nat] : aa_nat_nat(aa_fun5527826185635525933at_nat(aa_fun4894236392271416310at_nat(aTP_Lamm_ame(Uu),Uua),Uub),Uuc) = groups5693394587270226106ex_nat(aa_nat3391284541783913865ex_nat(aTP_Lamm_amc(Uua),Uuc),collect_complex(aa_nat_fun_complex_o(aa_fun4610599729497257276plex_o(aTP_Lamm_amd(Uu),Uub),Uuc))) ).

% ATP.lambda_817
tff(fact_10995_ATP_Olambda__818,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_alz(Uu),Uua),Uub),Uuc) = groups4541462559716669496nt_nat(aa_nat_fun_int_nat(aTP_Lamm_alx(Uua),Uuc),collect_int(aa_nat_fun_int_o(aa_fun5439836533389078972_int_o(aTP_Lamm_aly(Uu),Uub),Uuc))) ).

% ATP.lambda_818
tff(fact_10996_ATP_Olambda__819,axiom,
    ! [Uu: set_real,Uua: fun_real_fun_nat_nat,Uub: fun_real_fun_nat_o,Uuc: nat] : aa_nat_nat(aa_fun7366046364592083119at_nat(aa_fun7718511627822915834at_nat(aTP_Lamm_alu(Uu),Uua),Uub),Uuc) = groups1935376822645274424al_nat(aa_nat_fun_real_nat(aTP_Lamm_alt(Uua),Uuc),collect_real(aa_nat_fun_real_o(aa_fun6917832400337718588real_o(aTP_Lamm_aeu(Uu),Uub),Uuc))) ).

% ATP.lambda_819
tff(fact_10997_ATP_Olambda__820,axiom,
    ! [Uu: set_VEBT_VEBT,Uua: fun_VE597728521808591204at_nat,Uub: fun_VE6480751511521678896_nat_o,Uuc: nat] : aa_nat_nat(aa_fun6249118799011721995at_nat(aa_fun2705500642357200114at_nat(aTP_Lamm_alq(Uu),Uua),Uub),Uuc) = groups771621172384141258BT_nat(aa_nat7776146579971162249BT_nat(aTP_Lamm_alp(Uua),Uuc),collect_VEBT_VEBT(aa_nat1925708948643195573VEBT_o(aa_fun2405540735884763746VEBT_o(aTP_Lamm_aet(Uu),Uub),Uuc))) ).

% ATP.lambda_820
tff(fact_10998_ATP_Olambda__821,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_azb(Uu),Uua),Uub),Uuc) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aa_fun6650658898577385413at_nat(aTP_Lamm_aza(Uu),Uua),Uuc)),set_ord_atMost_nat(Uuc))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Uub),Uuc)) ).

% ATP.lambda_821
tff(fact_10999_ATP_Olambda__822,axiom,
    ! [Uu: uint32,Uua: uint32,Uub: nat,Uuc: nat] : aa_nat_uint32(aa_nat6935575117487981255uint32(aa_uin9008539043801383600uint32(aTP_Lamm_asz(Uu),Uua),Uub),Uuc) = minus_minus_uint32(aa_uint32_uint32(times_times_uint32(aa_nat_uint32(power_power_uint32(aa_uint32_uint32(plus_plus_uint32(Uu),Uua)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))),aa_nat_uint32(power_power_uint32(Uu),Uuc)),aa_nat_uint32(power_power_uint32(Uu),Uub)) ).

% ATP.lambda_822
tff(fact_11000_ATP_Olambda__823,axiom,
    ! [Uu: word_N3645301735248828278l_num1,Uua: word_N3645301735248828278l_num1,Uub: nat,Uuc: nat] : aa_nat5599461682457136004l_num1(aa_nat6251203789436639057l_num1(aa_wor3973378241480532036l_num1(aTP_Lamm_atb(Uu),Uua),Uub),Uuc) = minus_4019991460397169231l_num1(aa_wor7274019993978516266l_num1(times_7065122842183080059l_num1(aa_nat5599461682457136004l_num1(power_2184487114949457152l_num1(plus_p361126936061061375l_num1(Uu,Uua)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))),aa_nat5599461682457136004l_num1(power_2184487114949457152l_num1(Uu),Uuc)),aa_nat5599461682457136004l_num1(power_2184487114949457152l_num1(Uu),Uub)) ).

% ATP.lambda_823
tff(fact_11001_ATP_Olambda__824,axiom,
    ! [Uu: real,Uua: real,Uub: nat,Uuc: nat] : aa_nat_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_ath(Uu),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Uu),Uua)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uuc))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uub)) ).

% ATP.lambda_824
tff(fact_11002_ATP_Olambda__825,axiom,
    ! [Uu: rat,Uua: rat,Uub: nat,Uuc: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_atd(Uu),Uua),Uub),Uuc) = minus_minus_rat(aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(aa_rat_rat(plus_plus_rat(Uu),Uua)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))),aa_nat_rat(power_power_rat(Uu),Uuc)),aa_nat_rat(power_power_rat(Uu),Uub)) ).

% ATP.lambda_825
tff(fact_11003_ATP_Olambda__826,axiom,
    ! [Uu: int,Uua: int,Uub: nat,Uuc: nat] : aa_nat_int(aa_nat_fun_nat_int(aa_int431785320988601264at_int(aTP_Lamm_atf(Uu),Uua),Uub),Uuc) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Uu),Uua)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))),aa_nat_int(aa_int_fun_nat_int(power_power_int,Uu),Uuc))),aa_nat_int(aa_int_fun_nat_int(power_power_int,Uu),Uub)) ).

% ATP.lambda_826
tff(fact_11004_ATP_Olambda__827,axiom,
    ! [Uu: real,Uua: real,Uub: nat,Uuc: nat] : aa_nat_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_arp(Uu),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Uua),Uu)),Uuc)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),Uuc))) ).

% ATP.lambda_827
tff(fact_11005_ATP_Olambda__828,axiom,
    ! [Uu: rat,Uua: rat,Uub: nat,Uuc: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_arn(Uu),Uua),Uub),Uuc) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(aa_rat_rat(plus_plus_rat(Uua),Uu)),Uuc)),aa_nat_rat(power_power_rat(Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),Uuc))) ).

% ATP.lambda_828
tff(fact_11006_ATP_Olambda__829,axiom,
    ! [Uu: nat,Uua: nat,Uub: vEBT_VEBTi,Uuc: option_nat] : aa_option_nat_assn(aa_VEB7611521244314475576t_assn(aa_nat2954700136012447621t_assn(aTP_Lamm_au(Uu),Uua),Uub),Uuc) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,nth_VEBT_VEBT(treeList,Uua)),Uub)),aa_o_assn(pure_assn,Uuc = aa_nat_option_nat(vEBT_vebt_pred(nth_VEBT_VEBT(treeList,Uua)),Uu))) ).

% ATP.lambda_829
tff(fact_11007_ATP_Olambda__830,axiom,
    ! [Uu: fun_nat_option_nat,Uua: fun_Pr4796943543331071514_nat_o,Uub: nat,Uuc: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aa_fun2218514334412473170_nat_o(aTP_Lamm_mu(Uu),Uua),Uub),Uuc)
    <=> ( ( aa_nat_option_nat(Uu,Uub) = aa_nat_option_nat(some_nat,Uuc) )
        & aa_Pro6150637604445964947_nat_o(Uua,aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Uub),Uuc)) ) ) ).

% ATP.lambda_830
tff(fact_11008_ATP_Olambda__831,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_mv(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_831
tff(fact_11009_ATP_Olambda__832,axiom,
    ! [Uu: fun_fu4810197545401214784nteger,Uua: fun_Pr7015759342434529115eger_o,Uub: fun_Pr5098669352780829690e_term,Uuc: produc8923325533196201883nteger] :
      ( aa_Pro4926521453222654329eger_o(aa_fun4505422719058378424eger_o(aa_fun5185621348090534806eger_o(aTP_Lamm_mp(Uu),Uua),Uub),Uuc)
    <=> ( ( aa_fun1271726682903785001nteger(Uu,Uub) = some_P6772290148444788224nteger(Uuc) )
        & aa_Pro4148064003102580676eger_o(Uua,aa_Pro4291442876242880245nteger(produc565448451787809096nteger(Uub),Uuc)) ) ) ).

% ATP.lambda_832
tff(fact_11010_ATP_Olambda__833,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_mq(Uu),Uua),Uub),Uuc)
    <=> ( ( aa_fun5824947343344787589et_nat(Uu,Uub) = some_P624177172695371229et_nat(Uuc) )
        & aa_Pro5643009357358181178_nat_o(Uua,aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(Uub),Uuc)) ) ) ).

% ATP.lambda_833
tff(fact_11011_ATP_Olambda__834,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_mr(Uu),Uua),Uub),Uuc)
    <=> ( ( aa_fun1716447855799474851et_nat(Uu,Uub) = some_P994283844982041979et_nat(Uuc) )
        & aa_Pro2671238023309943196_nat_o(Uua,aa_Pro1979504083478021276et_nat(produc7291855603632349880et_nat(Uub),Uuc)) ) ) ).

% ATP.lambda_834
tff(fact_11012_ATP_Olambda__835,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_ms(Uu),Uua),Uub),Uuc)
    <=> ( ( aa_fun251074914522852630nt_int(Uu,Uub) = some_P4184893108420464158nt_int(Uuc) )
        & aa_Pro2601676399929874711_int_o(Uua,aa_Pro5748361610964203648nt_int(produc7320969046608631221nt_int(Uub),Uuc)) ) ) ).

% ATP.lambda_835
tff(fact_11013_ATP_Olambda__836,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_mt(Uu),Uua),Uub),Uuc)
    <=> ( ( aa_fun3970441718214872642nt_int(Uu,Uub) = some_P4184893108420464158nt_int(Uuc) )
        & aa_Pro5451197003417746155_int_o(Uua,aa_Pro8800328608558162860nt_int(produc4729913460834754785nt_int(Uub),Uuc)) ) ) ).

% ATP.lambda_836
tff(fact_11014_ATP_Olambda__837,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_ayy(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_837
tff(fact_11015_ATP_Olambda__838,axiom,
    ! [Uu: nat,Uua: nat,Uub: array_VEBT_VEBTi,Uuc: nat] : aa_nat2910048432523430448Heap_o(aa_arr494317875797454629Heap_o(aa_nat857754347168699366Heap_o(aTP_Lamm_js(Uu),Uua),Uub),Uuc) = heap_Time_bind_nat_o(aa_nat5821310369566504398ap_nat(vEBT_VEBT_lowi(Uu),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uua),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat5592791454563588861Heap_o(aTP_Lamm_jr(Uub),Uuc)) ).

% ATP.lambda_838
tff(fact_11016_ATP_Olambda__839,axiom,
    ! [Uu: uint32,Uua: nat,Uub: uint32,Uuc: nat] : aa_nat_uint32(aa_uin238090125092722531uint32(aa_nat7697544214703702448uint32(aTP_Lamm_atj(Uu),Uua),Uub),Uuc) = aa_uint32_uint32(times_times_uint32(aa_nat_uint32(power_power_uint32(Uub),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),aa_nat_nat(suc,Uuc)))),aa_nat_uint32(power_power_uint32(Uu),Uuc)) ).

% ATP.lambda_839
tff(fact_11017_ATP_Olambda__840,axiom,
    ! [Uu: word_N3645301735248828278l_num1,Uua: nat,Uub: word_N3645301735248828278l_num1,Uuc: nat] : aa_nat5599461682457136004l_num1(aa_wor1031616255390178039l_num1(aa_nat6155939771436694596l_num1(aTP_Lamm_atk(Uu),Uua),Uub),Uuc) = aa_wor7274019993978516266l_num1(times_7065122842183080059l_num1(aa_nat5599461682457136004l_num1(power_2184487114949457152l_num1(Uub),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),aa_nat_nat(suc,Uuc)))),aa_nat5599461682457136004l_num1(power_2184487114949457152l_num1(Uu),Uuc)) ).

% ATP.lambda_840
tff(fact_11018_ATP_Olambda__841,axiom,
    ! [Uu: real,Uua: nat,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_atn(Uu),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Uub),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),aa_nat_nat(suc,Uuc)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uuc)) ).

% ATP.lambda_841
tff(fact_11019_ATP_Olambda__842,axiom,
    ! [Uu: rat,Uua: nat,Uub: rat,Uuc: nat] : aa_nat_rat(aa_rat_fun_nat_rat(aa_nat298952458717073160at_rat(aTP_Lamm_atl(Uu),Uua),Uub),Uuc) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(Uub),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),aa_nat_nat(suc,Uuc)))),aa_nat_rat(power_power_rat(Uu),Uuc)) ).

% ATP.lambda_842
tff(fact_11020_ATP_Olambda__843,axiom,
    ! [Uu: int,Uua: nat,Uub: int,Uuc: nat] : aa_nat_int(aa_int_fun_nat_int(aa_nat6704147566042551216at_int(aTP_Lamm_atm(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_843
tff(fact_11021_ATP_Olambda__844,axiom,
    ! [Uu: set_complex,Uua: fun_VE7631951351884037842plex_o,Uub: vEBT_VEBT,Uuc: complex] :
      ( aa_complex_o(aa_VEB8036846806599048395plex_o(aa_fun5896829868240219618plex_o(aTP_Lamm_amk(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_complex_o(member_complex(Uuc),Uu)
        & aa_complex_o(aa_VEB8036846806599048395plex_o(Uua,Uub),Uuc) ) ) ).

% ATP.lambda_844
tff(fact_11022_ATP_Olambda__845,axiom,
    ! [Uu: set_complex,Uua: fun_Co8975588572093281791plex_o,Uub: code_integer,Uuc: complex] :
      ( aa_complex_o(aa_Cod7908732550325689320plex_o(aa_fun2498529521300939132plex_o(aTP_Lamm_anc(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_complex_o(member_complex(Uuc),Uu)
        & aa_complex_o(aa_Cod7908732550325689320plex_o(Uua,Uub),Uuc) ) ) ).

% ATP.lambda_845
tff(fact_11023_ATP_Olambda__846,axiom,
    ! [Uu: set_complex,Uua: fun_re7319081385389557872plex_o,Uub: real,Uuc: complex] :
      ( aa_complex_o(aa_rea2382570592068293209plex_o(aa_fun789424914597169340plex_o(aTP_Lamm_amo(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_complex_o(member_complex(Uuc),Uu)
        & aa_complex_o(aa_rea2382570592068293209plex_o(Uua,Uub),Uuc) ) ) ).

% ATP.lambda_846
tff(fact_11024_ATP_Olambda__847,axiom,
    ! [Uu: set_complex,Uua: fun_na1554462459066888140plex_o,Uub: nat,Uuc: complex] :
      ( aa_complex_o(aa_nat_fun_complex_o(aa_fun6810266893190567484plex_o(aTP_Lamm_ams(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_complex_o(member_complex(Uuc),Uu)
        & aa_complex_o(aa_nat_fun_complex_o(Uua,Uub),Uuc) ) ) ).

% ATP.lambda_847
tff(fact_11025_ATP_Olambda__848,axiom,
    ! [Uu: set_complex,Uua: fun_in5065666251602127472plex_o,Uub: int,Uuc: complex] :
      ( aa_complex_o(aa_int_fun_complex_o(aa_fun9089661398657494332plex_o(aTP_Lamm_amx(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_complex_o(member_complex(Uuc),Uu)
        & aa_complex_o(aa_int_fun_complex_o(Uua,Uub),Uuc) ) ) ).

% ATP.lambda_848
tff(fact_11026_ATP_Olambda__849,axiom,
    ! [Uu: set_nat,Uua: fun_VE6480751511521678896_nat_o,Uub: vEBT_VEBT,Uuc: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(aa_fun1998352179508271774_nat_o(aTP_Lamm_aln(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_nat_o(member_nat(Uuc),Uu)
        & aa_nat_o(aa_VEB2314620471664397097_nat_o(Uua,Uub),Uuc) ) ) ).

% ATP.lambda_849
tff(fact_11027_ATP_Olambda__850,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_amf(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_nat_o(member_nat(Uuc),Uu)
        & aa_nat_o(aa_Cod1075288278386115654_nat_o(Uua,Uub),Uuc) ) ) ).

% ATP.lambda_850
tff(fact_11028_ATP_Olambda__851,axiom,
    ! [Uu: set_nat,Uua: fun_co5676208753217392844_nat_o,Uub: complex,Uuc: nat] :
      ( aa_nat_o(aa_complex_fun_nat_o(aa_fun8732346023647761980_nat_o(aTP_Lamm_ama(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_nat_o(member_nat(Uuc),Uu)
        & aa_nat_o(aa_complex_fun_nat_o(Uua,Uub),Uuc) ) ) ).

% ATP.lambda_851
tff(fact_11029_ATP_Olambda__852,axiom,
    ! [Uu: set_nat,Uua: fun_real_fun_nat_o,Uub: real,Uuc: nat] :
      ( aa_nat_o(aa_real_fun_nat_o(aa_fun3244926128379188796_nat_o(aTP_Lamm_alr(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_nat_o(member_nat(Uuc),Uu)
        & aa_nat_o(aa_real_fun_nat_o(Uua,Uub),Uuc) ) ) ).

% ATP.lambda_852
tff(fact_11030_ATP_Olambda__853,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_alv(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_853
tff(fact_11031_ATP_Olambda__854,axiom,
    ! [Uu: set_int,Uua: fun_VE4235606543851232980_int_o,Uub: vEBT_VEBT,Uuc: int] :
      ( aa_int_o(aa_VEB493578126231709133_int_o(aa_fun8464664609713939430_int_o(aTP_Lamm_axc(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_int_o(member_int(Uuc),Uu)
        & aa_int_o(aa_VEB493578126231709133_int_o(Uua,Uub),Uuc) ) ) ).

% ATP.lambda_854
tff(fact_11032_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_axp(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_11033_ATP_Olambda__856,axiom,
    ! [Uu: set_int,Uua: fun_co3431063785546946928_int_o,Uub: complex,Uuc: int] :
      ( aa_int_o(aa_complex_fun_int_o(aa_fun5644928480641789244_int_o(aTP_Lamm_axk(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_int_o(member_int(Uuc),Uu)
        & aa_int_o(aa_complex_fun_int_o(Uua,Uub),Uuc) ) ) ).

% ATP.lambda_856
tff(fact_11034_ATP_Olambda__857,axiom,
    ! [Uu: set_int,Uua: fun_real_fun_int_o,Uub: real,Uuc: int] :
      ( aa_int_o(aa_real_fun_int_o(aa_fun2967211649596192572_int_o(aTP_Lamm_axg(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_int_o(member_int(Uuc),Uu)
        & aa_int_o(aa_real_fun_int_o(Uua,Uub),Uuc) ) ) ).

% ATP.lambda_857
tff(fact_11035_ATP_Olambda__858,axiom,
    ! [Uu: set_VEBT_VEBT,Uua: fun_VE1909044406494083811eger_o,Uub: code_integer,Uuc: vEBT_VEBT] :
      ( aa_VEBT_VEBT_o(aa_Cod5040069908416755522VEBT_o(aa_fun1637734041313256866VEBT_o(aTP_Lamm_aez(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Uuc),Uu)
        & aa_Code_integer_o(aa_VEB1409644639000594780eger_o(Uua,Uuc),Uub) ) ) ).

% ATP.lambda_858
tff(fact_11036_ATP_Olambda__859,axiom,
    ! [Uu: set_VEBT_VEBT,Uua: fun_VE7631951351884037842plex_o,Uub: complex,Uuc: vEBT_VEBT] :
      ( aa_VEBT_VEBT_o(aa_com3651963455866140051VEBT_o(aa_fun7268111043052431330VEBT_o(aTP_Lamm_aex(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Uuc),Uu)
        & aa_complex_o(aa_VEB8036846806599048395plex_o(Uua,Uuc),Uub) ) ) ).

% ATP.lambda_859
tff(fact_11037_ATP_Olambda__860,axiom,
    ! [Uu: set_VEBT_VEBT,Uua: fun_VE6480751511521678896_nat_o,Uub: nat,Uuc: vEBT_VEBT] :
      ( aa_VEBT_VEBT_o(aa_nat1925708948643195573VEBT_o(aa_fun2405540735884763746VEBT_o(aTP_Lamm_aet(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Uuc),Uu)
        & aa_nat_o(aa_VEB2314620471664397097_nat_o(Uua,Uuc),Uub) ) ) ).

% ATP.lambda_860
tff(fact_11038_ATP_Olambda__861,axiom,
    ! [Uu: set_VEBT_VEBT,Uua: fun_VE4235606543851232980_int_o,Uub: int,Uuc: vEBT_VEBT] :
      ( aa_VEBT_VEBT_o(aa_int8839263053405020433VEBT_o(aa_fun8807180201667926370VEBT_o(aTP_Lamm_aev(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Uuc),Uu)
        & aa_int_o(aa_VEB493578126231709133_int_o(Uua,Uuc),Uub) ) ) ).

% ATP.lambda_861
tff(fact_11039_ATP_Olambda__862,axiom,
    ! [Uu: set_Code_integer,Uua: fun_Co8975588572093281791plex_o,Uub: complex,Uuc: code_integer] :
      ( aa_Code_integer_o(aa_com4877492900066995176eger_o(aa_fun3025727905316630780eger_o(aTP_Lamm_anf(Uu),Uua),Uub),Uuc)
    <=> ( aa_set5643387332196575111eger_o(member_Code_integer(Uuc),Uu)
        & aa_complex_o(aa_Cod7908732550325689320plex_o(Uua,Uuc),Uub) ) ) ).

% ATP.lambda_862
tff(fact_11040_ATP_Olambda__863,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_ami(Uu),Uua),Uub),Uuc)
    <=> ( aa_set5643387332196575111eger_o(member_Code_integer(Uuc),Uu)
        & aa_nat_o(aa_Cod1075288278386115654_nat_o(Uua,Uuc),Uub) ) ) ).

% ATP.lambda_863
tff(fact_11041_ATP_Olambda__864,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_axs(Uu),Uua),Uub),Uuc)
    <=> ( aa_set5643387332196575111eger_o(member_Code_integer(Uuc),Uu)
        & aa_int_o(aa_Cod8477617969808203498_int_o(Uua,Uuc),Uub) ) ) ).

% ATP.lambda_864
tff(fact_11042_ATP_Olambda__865,axiom,
    ! [Uu: set_complex,Uua: fun_co5676208753217392844_nat_o,Uub: nat,Uuc: complex] :
      ( aa_complex_o(aa_nat_fun_complex_o(aa_fun4610599729497257276plex_o(aTP_Lamm_amd(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_complex_o(member_complex(Uuc),Uu)
        & aa_nat_o(aa_complex_fun_nat_o(Uua,Uuc),Uub) ) ) ).

% ATP.lambda_865
tff(fact_11043_ATP_Olambda__866,axiom,
    ! [Uu: set_complex,Uua: fun_co3431063785546946928_int_o,Uub: int,Uuc: complex] :
      ( aa_complex_o(aa_int_fun_complex_o(aa_fun7279530946696969788plex_o(aTP_Lamm_axn(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_complex_o(member_complex(Uuc),Uu)
        & aa_int_o(aa_complex_fun_int_o(Uua,Uuc),Uub) ) ) ).

% ATP.lambda_866
tff(fact_11044_ATP_Olambda__867,axiom,
    ! [Uu: set_real,Uua: fun_re3424108668441675905eger_o,Uub: code_integer,Uuc: real] :
      ( aa_real_o(aa_Cod1315720040876898922real_o(aa_fun4247632250071955964real_o(aTP_Lamm_afa(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_real_o(member_real(Uuc),Uu)
        & aa_Code_integer_o(aa_rea4090721512283939562eger_o(Uua,Uuc),Uub) ) ) ).

% ATP.lambda_867
tff(fact_11045_ATP_Olambda__868,axiom,
    ! [Uu: set_real,Uua: fun_re7319081385389557872plex_o,Uub: complex,Uuc: real] :
      ( aa_real_o(aa_com1951324360330529881real_o(aa_fun7692540709613205436real_o(aTP_Lamm_aey(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_real_o(member_real(Uuc),Uu)
        & aa_complex_o(aa_rea2382570592068293209plex_o(Uua,Uuc),Uub) ) ) ).

% ATP.lambda_868
tff(fact_11046_ATP_Olambda__869,axiom,
    ! [Uu: set_real,Uua: fun_real_fun_nat_o,Uub: nat,Uuc: real] :
      ( aa_real_o(aa_nat_fun_real_o(aa_fun6917832400337718588real_o(aTP_Lamm_aeu(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_real_o(member_real(Uuc),Uu)
        & aa_nat_o(aa_real_fun_nat_o(Uua,Uuc),Uub) ) ) ).

% ATP.lambda_869
tff(fact_11047_ATP_Olambda__870,axiom,
    ! [Uu: set_real,Uua: fun_real_fun_int_o,Uub: int,Uuc: real] :
      ( aa_real_o(aa_int_fun_real_o(aa_fun2391495421969208892real_o(aTP_Lamm_aew(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_real_o(member_real(Uuc),Uu)
        & aa_int_o(aa_real_fun_int_o(Uua,Uuc),Uub) ) ) ).

% ATP.lambda_870
tff(fact_11048_ATP_Olambda__871,axiom,
    ! [Uu: set_nat,Uua: fun_na1554462459066888140plex_o,Uub: complex,Uuc: nat] :
      ( aa_nat_o(aa_complex_fun_nat_o(aa_fun1708641150486296380_nat_o(aTP_Lamm_amv(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_nat_o(member_nat(Uuc),Uu)
        & aa_complex_o(aa_nat_fun_complex_o(Uua,Uuc),Uub) ) ) ).

% ATP.lambda_871
tff(fact_11049_ATP_Olambda__872,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_afb(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_872
tff(fact_11050_ATP_Olambda__873,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_afc(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_873
tff(fact_11051_ATP_Olambda__874,axiom,
    ! [Uu: set_int,Uua: fun_in5065666251602127472plex_o,Uub: complex,Uuc: int] :
      ( aa_int_o(aa_complex_fun_int_o(aa_fun7455058932602313788_int_o(aTP_Lamm_ana(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_int_o(member_int(Uuc),Uu)
        & aa_complex_o(aa_int_fun_complex_o(Uua,Uuc),Uub) ) ) ).

% ATP.lambda_874
tff(fact_11052_ATP_Olambda__875,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_aly(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_875
tff(fact_11053_ATP_Olambda__876,axiom,
    ! [Uu: uint32,Uua: uint32,Uub: nat,Uuc: nat] : aa_nat_uint32(aa_nat6935575117487981255uint32(aa_uin9008539043801383600uint32(aTP_Lamm_ata(Uu),Uua),Uub),Uuc) = aa_uint32_uint32(times_times_uint32(aa_nat_uint32(power_power_uint32(Uu),Uuc)),minus_minus_uint32(aa_nat_uint32(power_power_uint32(aa_uint32_uint32(plus_plus_uint32(Uu),Uua)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc)),aa_nat_uint32(power_power_uint32(Uu),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc)))) ).

% ATP.lambda_876
tff(fact_11054_ATP_Olambda__877,axiom,
    ! [Uu: word_N3645301735248828278l_num1,Uua: word_N3645301735248828278l_num1,Uub: nat,Uuc: nat] : aa_nat5599461682457136004l_num1(aa_nat6251203789436639057l_num1(aa_wor3973378241480532036l_num1(aTP_Lamm_atc(Uu),Uua),Uub),Uuc) = aa_wor7274019993978516266l_num1(times_7065122842183080059l_num1(aa_nat5599461682457136004l_num1(power_2184487114949457152l_num1(Uu),Uuc)),minus_4019991460397169231l_num1(aa_nat5599461682457136004l_num1(power_2184487114949457152l_num1(plus_p361126936061061375l_num1(Uu,Uua)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc)),aa_nat5599461682457136004l_num1(power_2184487114949457152l_num1(Uu),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc)))) ).

% ATP.lambda_877
tff(fact_11055_ATP_Olambda__878,axiom,
    ! [Uu: real,Uua: real,Uub: nat,Uuc: nat] : aa_nat_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_ati(Uu),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uuc)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Uu),Uua)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc)))) ).

% ATP.lambda_878
tff(fact_11056_ATP_Olambda__879,axiom,
    ! [Uu: rat,Uua: rat,Uub: nat,Uuc: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_ate(Uu),Uua),Uub),Uuc) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(Uu),Uuc)),minus_minus_rat(aa_nat_rat(power_power_rat(aa_rat_rat(plus_plus_rat(Uu),Uua)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc)),aa_nat_rat(power_power_rat(Uu),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc)))) ).

% ATP.lambda_879
tff(fact_11057_ATP_Olambda__880,axiom,
    ! [Uu: int,Uua: int,Uub: nat,Uuc: nat] : aa_nat_int(aa_nat_fun_nat_int(aa_int431785320988601264at_int(aTP_Lamm_atg(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_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Uu),Uua)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))),aa_nat_int(aa_int_fun_nat_int(power_power_int,Uu),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc)))) ).

% ATP.lambda_880
tff(fact_11058_ATP_Olambda__881,axiom,
    ! [Uu: uint32,Uua: nat,Uub: uint32,Uuc: nat] : aa_nat_uint32(aa_uin238090125092722531uint32(aa_nat7697544214703702448uint32(aTP_Lamm_ato(Uu),Uua),Uub),Uuc) = aa_uint32_uint32(times_times_uint32(aa_nat_uint32(power_power_uint32(Uu),Uuc)),aa_nat_uint32(power_power_uint32(Uub),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uuc))) ).

% ATP.lambda_881
tff(fact_11059_ATP_Olambda__882,axiom,
    ! [Uu: word_N3645301735248828278l_num1,Uua: nat,Uub: word_N3645301735248828278l_num1,Uuc: nat] : aa_nat5599461682457136004l_num1(aa_wor1031616255390178039l_num1(aa_nat6155939771436694596l_num1(aTP_Lamm_atp(Uu),Uua),Uub),Uuc) = aa_wor7274019993978516266l_num1(times_7065122842183080059l_num1(aa_nat5599461682457136004l_num1(power_2184487114949457152l_num1(Uu),Uuc)),aa_nat5599461682457136004l_num1(power_2184487114949457152l_num1(Uub),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uuc))) ).

% ATP.lambda_882
tff(fact_11060_ATP_Olambda__883,axiom,
    ! [Uu: real,Uua: nat,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_ats(Uu),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uuc)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uub),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uuc))) ).

% ATP.lambda_883
tff(fact_11061_ATP_Olambda__884,axiom,
    ! [Uu: rat,Uua: nat,Uub: rat,Uuc: nat] : aa_nat_rat(aa_rat_fun_nat_rat(aa_nat298952458717073160at_rat(aTP_Lamm_atq(Uu),Uua),Uub),Uuc) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(Uu),Uuc)),aa_nat_rat(power_power_rat(Uub),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uuc))) ).

% ATP.lambda_884
tff(fact_11062_ATP_Olambda__885,axiom,
    ! [Uu: int,Uua: nat,Uub: int,Uuc: nat] : aa_nat_int(aa_int_fun_nat_int(aa_nat6704147566042551216at_int(aTP_Lamm_atr(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_885
tff(fact_11063_ATP_Olambda__886,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_ayx(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_886
tff(fact_11064_ATP_Olambda__887,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] :
      ( aa_int_o(aa_int_fun_int_o(aa_int2464330468366073224_int_o(aTP_Lamm_bax(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_887
tff(fact_11065_ATP_Olambda__888,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] :
      ( aa_int_o(aa_int_fun_int_o(aa_int2464330468366073224_int_o(aTP_Lamm_bav(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_888
tff(fact_11066_ATP_Olambda__889,axiom,
    ! [Uu: set_VEBT_VEBT,Uua: fun_VEBT_VEBT_real,Uub: fun_VEBT_VEBT_real,Uuc: vEBT_VEBT] :
      ( aa_VEBT_VEBT_o(aa_fun1965331558590768990VEBT_o(aa_fun5637434397837810486VEBT_o(aTP_Lamm_agw(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Uuc),Uu)
        & ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_VEBT_VEBT_real(Uua,Uuc)),aa_VEBT_VEBT_real(Uub,Uuc)) != one_one_real ) ) ) ).

% ATP.lambda_889
tff(fact_11067_ATP_Olambda__890,axiom,
    ! [Uu: set_VEBT_VEBT,Uua: fun_VEBT_VEBT_rat,Uub: fun_VEBT_VEBT_rat,Uuc: vEBT_VEBT] :
      ( aa_VEBT_VEBT_o(aa_fun5889122065779655690VEBT_o(aa_fun1978406156034293430VEBT_o(aTP_Lamm_ahi(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Uuc),Uu)
        & ( aa_rat_rat(times_times_rat(aa_VEBT_VEBT_rat(Uua,Uuc)),aa_VEBT_VEBT_rat(Uub,Uuc)) != one_one_rat ) ) ) ).

% ATP.lambda_890
tff(fact_11068_ATP_Olambda__891,axiom,
    ! [Uu: set_Code_integer,Uua: fun_Co3960896333048504454r_real,Uub: fun_Co3960896333048504454r_real,Uuc: code_integer] :
      ( aa_Code_integer_o(aa_fun3640226575523069072eger_o(aa_fun5110011637356352227eger_o(aTP_Lamm_ahg(Uu),Uua),Uub),Uuc)
    <=> ( aa_set5643387332196575111eger_o(member_Code_integer(Uuc),Uu)
        & ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_Code_integer_real(Uua,Uuc)),aa_Code_integer_real(Uub,Uuc)) != one_one_real ) ) ) ).

% ATP.lambda_891
tff(fact_11069_ATP_Olambda__892,axiom,
    ! [Uu: set_complex,Uua: fun_complex_real,Uub: fun_complex_real,Uuc: complex] :
      ( aa_complex_o(aa_fun6195079535267300846plex_o(aa_fun805843229300662576plex_o(aTP_Lamm_ahe(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_complex_o(member_complex(Uuc),Uu)
        & ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(Uua,Uuc)),aa_complex_real(Uub,Uuc)) != one_one_real ) ) ) ).

% ATP.lambda_892
tff(fact_11070_ATP_Olambda__893,axiom,
    ! [Uu: set_real,Uua: fun_real_real,Uub: fun_real_real,Uuc: real] :
      ( aa_real_o(aa_fun2895569901277098738real_o(aa_fun3693225987673163702real_o(aTP_Lamm_agy(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_real_o(member_real(Uuc),Uu)
        & ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(Uua,Uuc)),aa_real_real(Uub,Uuc)) != one_one_real ) ) ) ).

% ATP.lambda_893
tff(fact_11071_ATP_Olambda__894,axiom,
    ! [Uu: set_real,Uua: fun_real_rat,Uub: fun_real_rat,Uuc: real] :
      ( aa_real_o(aa_fun3209485225100452678real_o(aa_fun8459995508950063326real_o(aTP_Lamm_ahk(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_real_o(member_real(Uuc),Uu)
        & ( aa_rat_rat(times_times_rat(aa_real_rat(Uua,Uuc)),aa_real_rat(Uub,Uuc)) != one_one_rat ) ) ) ).

% ATP.lambda_894
tff(fact_11072_ATP_Olambda__895,axiom,
    ! [Uu: set_nat,Uua: fun_nat_real,Uub: fun_nat_real,Uuc: nat] :
      ( aa_nat_o(aa_fun6456149469358400426_nat_o(aa_fun5464864432877286986_nat_o(aTP_Lamm_aha(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_nat_o(member_nat(Uuc),Uu)
        & ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uua,Uuc)),aa_nat_real(Uub,Uuc)) != one_one_real ) ) ) ).

% ATP.lambda_895
tff(fact_11073_ATP_Olambda__896,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_ahm(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_nat_o(member_nat(Uuc),Uu)
        & ( aa_rat_rat(times_times_rat(aa_nat_rat(Uua,Uuc)),aa_nat_rat(Uub,Uuc)) != one_one_rat ) ) ) ).

% ATP.lambda_896
tff(fact_11074_ATP_Olambda__897,axiom,
    ! [Uu: set_int,Uua: fun_int_real,Uub: fun_int_real,Uuc: int] :
      ( aa_int_o(aa_fun4444401208693448434_int_o(aa_fun135101165924023862_int_o(aTP_Lamm_ahc(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_int_o(member_int(Uuc),Uu)
        & ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(Uua,Uuc)),aa_int_real(Uub,Uuc)) != one_one_real ) ) ) ).

% ATP.lambda_897
tff(fact_11075_ATP_Olambda__898,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_aho(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_int_o(member_int(Uuc),Uu)
        & ( aa_rat_rat(times_times_rat(aa_int_rat(Uua,Uuc)),aa_int_rat(Uub,Uuc)) != one_one_rat ) ) ) ).

% ATP.lambda_898
tff(fact_11076_ATP_Olambda__899,axiom,
    ! [Uu: set_VEBT_VEBT,Uua: fun_VEBT_VEBT_uint32,Uub: fun_VEBT_VEBT_uint32,Uuc: vEBT_VEBT] :
      ( aa_VEBT_VEBT_o(aa_fun6272599145770227614VEBT_o(aa_fun1156574460560677430VEBT_o(aTP_Lamm_ahq(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Uuc),Uu)
        & ( aa_uint32_uint32(plus_plus_uint32(aa_VEBT_VEBT_uint32(Uua,Uuc)),aa_VEBT_VEBT_uint32(Uub,Uuc)) != zero_zero_uint32 ) ) ) ).

% ATP.lambda_899
tff(fact_11077_ATP_Olambda__900,axiom,
    ! [Uu: set_VEBT_VEBT,Uua: fun_VEBT_VEBT_real,Uub: fun_VEBT_VEBT_real,Uuc: vEBT_VEBT] :
      ( aa_VEBT_VEBT_o(aa_fun1965331558590768990VEBT_o(aa_fun5637434397837810486VEBT_o(aTP_Lamm_aic(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Uuc),Uu)
        & ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_VEBT_VEBT_real(Uua,Uuc)),aa_VEBT_VEBT_real(Uub,Uuc)) != zero_zero_real ) ) ) ).

% ATP.lambda_900
tff(fact_11078_ATP_Olambda__901,axiom,
    ! [Uu: set_Code_integer,Uua: fun_Co1554742298434871494uint32,Uub: fun_Co1554742298434871494uint32,Uuc: code_integer] :
      ( aa_Code_integer_o(aa_fun8926496376162642256eger_o(aa_fun1119956648780089443eger_o(aTP_Lamm_aia(Uu),Uua),Uub),Uuc)
    <=> ( aa_set5643387332196575111eger_o(member_Code_integer(Uuc),Uu)
        & ( aa_uint32_uint32(plus_plus_uint32(aa_Cod8401701657498852871uint32(Uua,Uuc)),aa_Cod8401701657498852871uint32(Uub,Uuc)) != zero_zero_uint32 ) ) ) ).

% ATP.lambda_901
tff(fact_11079_ATP_Olambda__902,axiom,
    ! [Uu: set_complex,Uua: fun_complex_uint32,Uub: fun_complex_uint32,Uuc: complex] :
      ( aa_complex_o(aa_fun436548534304416686plex_o(aa_fun2751405167760845232plex_o(aTP_Lamm_ahy(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_complex_o(member_complex(Uuc),Uu)
        & ( aa_uint32_uint32(plus_plus_uint32(aa_complex_uint32(Uua,Uuc)),aa_complex_uint32(Uub,Uuc)) != zero_zero_uint32 ) ) ) ).

% ATP.lambda_902
tff(fact_11080_ATP_Olambda__903,axiom,
    ! [Uu: set_real,Uua: fun_real_uint32,Uub: fun_real_uint32,Uuc: real] :
      ( aa_real_o(aa_fun7556296611544133298real_o(aa_fun5572330763077102134real_o(aTP_Lamm_ahs(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_real_o(member_real(Uuc),Uu)
        & ( aa_uint32_uint32(plus_plus_uint32(aa_real_uint32(Uua,Uuc)),aa_real_uint32(Uub,Uuc)) != zero_zero_uint32 ) ) ) ).

% ATP.lambda_903
tff(fact_11081_ATP_Olambda__904,axiom,
    ! [Uu: set_real,Uua: fun_real_real,Uub: fun_real_real,Uuc: real] :
      ( aa_real_o(aa_fun2895569901277098738real_o(aa_fun3693225987673163702real_o(aTP_Lamm_aie(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_real_o(member_real(Uuc),Uu)
        & ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(Uua,Uuc)),aa_real_real(Uub,Uuc)) != zero_zero_real ) ) ) ).

% ATP.lambda_904
tff(fact_11082_ATP_Olambda__905,axiom,
    ! [Uu: set_nat,Uua: fun_nat_uint32,Uub: fun_nat_uint32,Uuc: nat] :
      ( aa_nat_o(aa_fun1182053767699555178_nat_o(aa_fun6957237400125433034_nat_o(aTP_Lamm_ahu(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_nat_o(member_nat(Uuc),Uu)
        & ( aa_uint32_uint32(plus_plus_uint32(aa_nat_uint32(Uua,Uuc)),aa_nat_uint32(Uub,Uuc)) != zero_zero_uint32 ) ) ) ).

% ATP.lambda_905
tff(fact_11083_ATP_Olambda__906,axiom,
    ! [Uu: set_nat,Uua: fun_nat_real,Uub: fun_nat_real,Uuc: nat] :
      ( aa_nat_o(aa_fun6456149469358400426_nat_o(aa_fun5464864432877286986_nat_o(aTP_Lamm_aig(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_nat_o(member_nat(Uuc),Uu)
        & ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(Uua,Uuc)),aa_nat_real(Uub,Uuc)) != zero_zero_real ) ) ) ).

% ATP.lambda_906
tff(fact_11084_ATP_Olambda__907,axiom,
    ! [Uu: set_int,Uua: fun_int_uint32,Uub: fun_int_uint32,Uuc: int] :
      ( aa_int_o(aa_fun6992502331602369202_int_o(aa_fun3369613587901038774_int_o(aTP_Lamm_ahw(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_int_o(member_int(Uuc),Uu)
        & ( aa_uint32_uint32(plus_plus_uint32(aa_int_uint32(Uua,Uuc)),aa_int_uint32(Uub,Uuc)) != zero_zero_uint32 ) ) ) ).

% ATP.lambda_907
tff(fact_11085_ATP_Olambda__908,axiom,
    ! [Uu: set_int,Uua: fun_int_real,Uub: fun_int_real,Uuc: int] :
      ( aa_int_o(aa_fun4444401208693448434_int_o(aa_fun135101165924023862_int_o(aTP_Lamm_aii(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_int_o(member_int(Uuc),Uu)
        & ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(Uua,Uuc)),aa_int_real(Uub,Uuc)) != zero_zero_real ) ) ) ).

% ATP.lambda_908
tff(fact_11086_ATP_Olambda__909,axiom,
    ! [Uu: vEBT_VEBT,Uua: vEBT_VEBTi,Uub: nat,Uuc: $o] : aa_o_assn(aa_nat_fun_o_assn(aa_VEB8176250900481067561o_assn(aTP_Lamm_bo(Uu),Uua),Uub),(Uuc)) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Uu),Uua)),aa_o_assn(pure_assn,(Uuc) = aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,Uu),Uub))) ).

% ATP.lambda_909
tff(fact_11087_ATP_Olambda__910,axiom,
    ! [Uu: vEBT_VEBT,Uua: vEBT_VEBTi,Uub: nat,Uuc: option_nat] : aa_option_nat_assn(aa_nat435388790147128531t_assn(aa_VEB1697702717351837573t_assn(aTP_Lamm_be(Uu),Uua),Uub),Uuc) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Uu),Uua)),aa_o_assn(pure_assn,Uuc = aa_nat_option_nat(vEBT_vebt_succ(Uu),Uub))) ).

% ATP.lambda_910
tff(fact_11088_ATP_Olambda__911,axiom,
    ! [Uu: nat,Uua: list_VEBT_VEBTi,Uub: array_VEBT_VEBTi,Uuc: vEBT_VEBTi] : aa_VEBT_VEBTi_assn(aa_arr8081119733880211216i_assn(aa_lis5880058344902901532i_assn(aTP_Lamm_bk(Uu),Uua),Uub),Uuc) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_lis2508312872377092907i_assn(snga_assn_VEBT_VEBTi(Uub),Uua)),aa_o_assn(pure_assn,Uuc = nth_VEBT_VEBTi(Uua,Uu))) ).

% ATP.lambda_911
tff(fact_11089_ATP_Olambda__912,axiom,
    ! [Uu: nat,Uua: list_Product_unit,Uub: array_Product_unit,Uuc: product_unit] : aa_Product_unit_assn(aa_arr8761957006411741156t_assn(aa_lis5628126305658872932t_assn(aTP_Lamm_bl(Uu),Uua),Uub),Uuc) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,snga_a4522542871529764173t_unit(Uub,Uua)),aa_o_assn(pure_assn,Uuc = nth_Product_unit(Uua,Uu))) ).

% ATP.lambda_912
tff(fact_11090_ATP_Olambda__913,axiom,
    ! [Uu: nat,Uua: list_option_nat,Uub: array_option_nat,Uuc: option_nat] : aa_option_nat_assn(aa_arr7483714166967448038t_assn(aa_lis7582702936096254407t_assn(aTP_Lamm_bh(Uu),Uua),Uub),Uuc) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,snga_assn_option_nat(Uub,Uua)),aa_o_assn(pure_assn,Uuc = aa_nat_option_nat(nth_option_nat(Uua),Uu))) ).

% ATP.lambda_913
tff(fact_11091_ATP_Olambda__914,axiom,
    ! [Uu: nat,Uua: list_o,Uub: array_o,Uuc: $o] : aa_o_assn(aa_arr2544151649598861476o_assn(aa_lis966790859614740569o_assn(aTP_Lamm_bj(Uu),Uua),Uub),(Uuc)) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,snga_assn_o(Uub,Uua)),aa_o_assn(pure_assn,(Uuc) = aa_nat_o(nth_o(Uua),Uu))) ).

% ATP.lambda_914
tff(fact_11092_ATP_Olambda__915,axiom,
    ! [Uu: nat,Uua: list_nat,Uub: array_nat,Uuc: nat] : aa_nat_assn(aa_arr3838850077245469638t_assn(aa_lis668879630561721847t_assn(aTP_Lamm_bi(Uu),Uua),Uub),Uuc) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,snga_assn_nat(Uub,Uua)),aa_o_assn(pure_assn,Uuc = aa_nat_nat(nth_nat(Uua),Uu))) ).

% ATP.lambda_915
tff(fact_11093_ATP_Olambda__916,axiom,
    ! [Uu: nat,Uua: list_int,Uub: array_int,Uuc: int] : aa_int_assn(aa_arr1589534196239669886t_assn(aa_lis459371148865260371t_assn(aTP_Lamm_bg(Uu),Uua),Uub),Uuc) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,snga_assn_int(Uub,Uua)),aa_o_assn(pure_assn,Uuc = aa_nat_int(nth_int(Uua),Uu))) ).

% ATP.lambda_916
tff(fact_11094_ATP_Olambda__917,axiom,
    ! [Uu: product_prod_nat_nat,Uua: nat,Uub: nat,Uuc: product_unit] : aa_Pro451875010097498139on_nat(aa_nat3999351741935936176on_nat(aa_nat3263736196148507005on_nat(aTP_Lamm_ah(Uu),Uua),Uub),Uuc) = heap_T2868974464944644318on_nat(aa_nat5408703056993926195_VEBTi(array_nth_VEBT_VEBTi(x13),Uub),aa_nat4669826966810537986on_nat(aa_nat107244183536939787on_nat(aTP_Lamm_ag(Uu),Uua),Uub)) ).

% ATP.lambda_917
tff(fact_11095_ATP_Olambda__918,axiom,
    ! [Uu: product_prod_nat_nat,Uua: nat,Uub: nat,Uuc: vEBT_VEBTi] :
      aa_VEB2711320435582112505on_nat(aa_nat4669826966810537986on_nat(aa_nat107244183536939787on_nat(aTP_Lamm_ag(Uu),Uua),Uub),Uuc) = $let(
        aktnode: vEBT_VEBT,
        aktnode:= nth_VEBT_VEBT(treeList,Uub),
        heap_T3669509953089699273on_nat(aa_VEB2711320435582112505on_nat(vEBT_vebt_minti,Uuc),aa_VEB3755160987679246317on_nat(aa_VEB5212316371209427943on_nat(aa_nat7793430520228346548on_nat(aa_nat8410684691732023425on_nat(aTP_Lamm_af(Uu),Uua),Uub),Uuc),aktnode)) ) ).

% ATP.lambda_918
tff(fact_11096_ATP_Olambda__919,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_aza(Uu),Uua),Uub),Uuc) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(Uu,Uuc)),aa_nat_nat(Uua,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))) ).

% ATP.lambda_919
tff(fact_11097_ATP_Olambda__920,axiom,
    ! [Uu: fun_complex_complex,Uua: fun_complex_complex,Uub: complex,Uuc: complex] : aa_complex_complex(aa_com3342990543769683525omplex(aa_fun6995112137416684571omplex(aTP_Lamm_alc(Uu),Uua),Uub),Uuc) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(Uu,Uub)),aa_complex_complex(Uua,Uuc)) ).

% ATP.lambda_920
tff(fact_11098_ATP_Olambda__921,axiom,
    ! [Uu: fun_nat_real,Uua: fun_nat_real,Uub: nat,Uuc: nat] : aa_nat_real(aa_nat_fun_nat_real(aa_fun3023522663955195517t_real(aTP_Lamm_alg(Uu),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uu,Uub)),aa_nat_real(Uua,Uuc)) ).

% ATP.lambda_921
tff(fact_11099_ATP_Olambda__922,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_ala(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_922
tff(fact_11100_ATP_Olambda__923,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_ale(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_923
tff(fact_11101_ATP_Olambda__924,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_axq(Uu),Uua),Uub),Uuc) = groups1705073143266064639nt_int(aa_Cod5491795744640130032nt_int(Uua,Uuc),collect_int(aa_Cod8477617969808203498_int_o(aa_fun1593725752699181308_int_o(aTP_Lamm_axp(Uu),Uub),Uuc))) ).

% ATP.lambda_924
tff(fact_11102_ATP_Olambda__925,axiom,
    ! [Uu: set_int,Uua: fun_co333780937270390912nt_int,Uub: fun_co3431063785546946928_int_o,Uuc: complex] : aa_complex_int(aa_fun5055871773053899019ex_int(aa_fun7361901416012365340ex_int(aTP_Lamm_axl(Uu),Uua),Uub),Uuc) = groups1705073143266064639nt_int(aa_com3205223359959579713nt_int(Uua,Uuc),collect_int(aa_complex_fun_int_o(aa_fun5644928480641789244_int_o(aTP_Lamm_axk(Uu),Uub),Uuc))) ).

% ATP.lambda_925
tff(fact_11103_ATP_Olambda__926,axiom,
    ! [Uu: set_int,Uua: fun_real_fun_int_int,Uub: fun_real_fun_int_o,Uuc: real] : aa_real_int(aa_fun7555737461897725707al_int(aa_fun1679075639499870878al_int(aTP_Lamm_axh(Uu),Uua),Uub),Uuc) = groups1705073143266064639nt_int(aa_real_fun_int_int(Uua,Uuc),collect_int(aa_real_fun_int_o(aa_fun2967211649596192572_int_o(aTP_Lamm_axg(Uu),Uub),Uuc))) ).

% ATP.lambda_926
tff(fact_11104_ATP_Olambda__927,axiom,
    ! [Uu: set_int,Uua: fun_VE294566951569886236nt_int,Uub: fun_VE4235606543851232980_int_o,Uuc: vEBT_VEBT] : aa_VEBT_VEBT_int(aa_fun1410605457015381217BT_int(aa_fun1738282208931454104BT_int(aTP_Lamm_axd(Uu),Uua),Uub),Uuc) = groups1705073143266064639nt_int(aa_VEB7166967446021329869nt_int(Uua,Uuc),collect_int(aa_VEB493578126231709133_int_o(aa_fun8464664609713939430_int_o(aTP_Lamm_axc(Uu),Uub),Uuc))) ).

% ATP.lambda_927
tff(fact_11105_ATP_Olambda__928,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_awz(Uu),Uua),Uub),Uuc) = groups705719431365010083at_int(aa_Cod4492442032739075476at_int(Uua,Uuc),collect_nat(aa_Cod1075288278386115654_nat_o(aa_fun2430183811687954940_nat_o(aTP_Lamm_amf(Uu),Uub),Uuc))) ).

% ATP.lambda_928
tff(fact_11106_ATP_Olambda__929,axiom,
    ! [Uu: set_nat,Uua: fun_co7370231399432144932at_int,Uub: fun_co5676208753217392844_nat_o,Uuc: complex] : aa_complex_int(aa_fun7310784931243472999ex_int(aa_fun7210183792663747796ex_int(aTP_Lamm_aww(Uu),Uua),Uub),Uuc) = groups705719431365010083at_int(aa_com2205869648058525157at_int(Uua,Uuc),collect_nat(aa_complex_fun_nat_o(aa_fun8732346023647761980_nat_o(aTP_Lamm_ama(Uu),Uub),Uuc))) ).

% ATP.lambda_929
tff(fact_11107_ATP_Olambda__930,axiom,
    ! [Uu: set_nat,Uua: fun_real_fun_nat_nat,Uub: fun_real_fun_nat_o,Uuc: real] : aa_real_nat(aa_fun2272639302901571595al_nat(aa_fun7039802393428226646al_nat(aTP_Lamm_axw(Uu),Uua),Uub),Uuc) = groups708209901874060359at_nat(aa_real_fun_nat_nat(Uua,Uuc),collect_nat(aa_real_fun_nat_o(aa_fun3244926128379188796_nat_o(aTP_Lamm_alr(Uu),Uub),Uuc))) ).

% ATP.lambda_930
tff(fact_11108_ATP_Olambda__931,axiom,
    ! [Uu: set_nat,Uua: fun_real_fun_nat_int,Uub: fun_real_fun_nat_o,Uuc: real] : aa_real_int(aa_fun7318160320247150695al_int(aa_fun4230461673125498710al_int(aTP_Lamm_awt(Uu),Uua),Uub),Uuc) = groups705719431365010083at_int(aa_real_fun_nat_int(Uua,Uuc),collect_nat(aa_real_fun_nat_o(aa_fun3244926128379188796_nat_o(aTP_Lamm_alr(Uu),Uub),Uuc))) ).

% ATP.lambda_931
tff(fact_11109_ATP_Olambda__932,axiom,
    ! [Uu: set_nat,Uua: fun_VE597728521808591204at_nat,Uub: fun_VE6480751511521678896_nat_o,Uuc: vEBT_VEBT] : aa_VEBT_VEBT_nat(aa_fun3621025987637128233BT_nat(aa_fun1603736613815373464BT_nat(aTP_Lamm_axu(Uu),Uua),Uub),Uuc) = groups708209901874060359at_nat(aa_VEB1122092716774696213at_nat(Uua,Uuc),collect_nat(aa_VEB2314620471664397097_nat_o(aa_fun1998352179508271774_nat_o(aTP_Lamm_aln(Uu),Uub),Uuc))) ).

% ATP.lambda_932
tff(fact_11110_ATP_Olambda__933,axiom,
    ! [Uu: set_nat,Uua: fun_VE7331017413731640256at_int,Uub: fun_VE6480751511521678896_nat_o,Uuc: vEBT_VEBT] : aa_VEBT_VEBT_int(aa_fun8666547004982707333BT_int(aa_fun127508041567870360BT_int(aTP_Lamm_awq(Uu),Uua),Uub),Uuc) = groups705719431365010083at_int(aa_VEB6167613734120275313at_int(Uua,Uuc),collect_nat(aa_VEB2314620471664397097_nat_o(aa_fun1998352179508271774_nat_o(aTP_Lamm_aln(Uu),Uub),Uuc))) ).

% ATP.lambda_933
tff(fact_11111_ATP_Olambda__934,axiom,
    ! [Uu: set_complex,Uua: fun_Co8905303033605938227omplex,Uub: fun_Co8975588572093281791plex_o,Uuc: code_integer] : aa_Cod5621350732091595593omplex(aa_fun9008897831946212043omplex(aa_fun8319717443755388649omplex(aTP_Lamm_and(Uu),Uua),Uub),Uuc) = groups7754918857620584856omplex(aa_Cod3397930797123873268omplex(Uua,Uuc),collect_complex(aa_Cod7908732550325689320plex_o(aa_fun2498529521300939132plex_o(aTP_Lamm_anc(Uu),Uub),Uuc))) ).

% ATP.lambda_934
tff(fact_11112_ATP_Olambda__935,axiom,
    ! [Uu: set_complex,Uua: fun_in1307110748760623106omplex,Uub: fun_in5065666251602127472plex_o,Uuc: int] : aa_int_complex(aa_fun3141780957025892107omplex(aa_fun323909820831547930omplex(aTP_Lamm_amy(Uu),Uua),Uub),Uuc) = groups7754918857620584856omplex(aa_int3545933452204062147omplex(Uua,Uuc),collect_complex(aa_int_fun_complex_o(aa_fun9089661398657494332plex_o(aTP_Lamm_amx(Uu),Uub),Uuc))) ).

% ATP.lambda_935
tff(fact_11113_ATP_Olambda__936,axiom,
    ! [Uu: set_complex,Uua: fun_na1591161823130732966omplex,Uub: fun_na1554462459066888140plex_o,Uuc: nat] : aa_nat_complex(aa_fun4713049822795992587omplex(aa_fun3479084698764006518omplex(aTP_Lamm_amt(Uu),Uua),Uub),Uuc) = groups7754918857620584856omplex(aa_nat3829984526574172007omplex(Uua,Uuc),collect_complex(aa_nat_fun_complex_o(aa_fun6810266893190567484plex_o(aTP_Lamm_ams(Uu),Uub),Uuc))) ).

% ATP.lambda_936
tff(fact_11114_ATP_Olambda__937,axiom,
    ! [Uu: set_complex,Uua: fun_re7208826350674946562omplex,Uub: fun_re7319081385389557872plex_o,Uuc: real] : aa_real_complex(aa_fun401690246322682763omplex(aa_fun7501666206714588442omplex(aTP_Lamm_amp(Uu),Uua),Uub),Uuc) = groups7754918857620584856omplex(aa_rea2618301233319580611omplex(Uua,Uuc),collect_complex(aa_rea2382570592068293209plex_o(aa_fun789424914597169340plex_o(aTP_Lamm_amo(Uu),Uub),Uuc))) ).

% ATP.lambda_937
tff(fact_11115_ATP_Olambda__938,axiom,
    ! [Uu: set_complex,Uua: fun_VE2006399705683203104omplex,Uub: fun_VE7631951351884037842plex_o,Uuc: vEBT_VEBT] : aa_VEBT_VEBT_complex(aa_fun1953711888481404901omplex(aa_fun5829985485761010584omplex(aTP_Lamm_aml(Uu),Uua),Uub),Uuc) = groups7754918857620584856omplex(aa_VEB5608662635949521omplex(Uua,Uuc),collect_complex(aa_VEB8036846806599048395plex_o(aa_fun5896829868240219618plex_o(aTP_Lamm_amk(Uu),Uub),Uuc))) ).

% ATP.lambda_938
tff(fact_11116_ATP_Olambda__939,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_amg(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_amf(Uu),Uub),Uuc))) ).

% ATP.lambda_939
tff(fact_11117_ATP_Olambda__940,axiom,
    ! [Uu: set_nat,Uua: fun_co636942507509095880at_nat,Uub: fun_co5676208753217392844_nat_o,Uuc: complex] : aa_complex_nat(aa_fun2265263913897893899ex_nat(aa_fun1127489917733235668ex_nat(aTP_Lamm_amb(Uu),Uua),Uub),Uuc) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_com6383720667567721865at_nat(Uua,Uuc)),collect_nat(aa_complex_fun_nat_o(aa_fun8732346023647761980_nat_o(aTP_Lamm_ama(Uu),Uub),Uuc))) ).

% ATP.lambda_940
tff(fact_11118_ATP_Olambda__941,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_alw(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_alv(Uu),Uub),Uuc))) ).

% ATP.lambda_941
tff(fact_11119_ATP_Olambda__942,axiom,
    ! [Uu: set_nat,Uua: fun_real_fun_nat_nat,Uub: fun_real_fun_nat_o,Uuc: real] : aa_real_nat(aa_fun2272639302901571595al_nat(aa_fun7039802393428226646al_nat(aTP_Lamm_als(Uu),Uua),Uub),Uuc) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_real_fun_nat_nat(Uua,Uuc)),collect_nat(aa_real_fun_nat_o(aa_fun3244926128379188796_nat_o(aTP_Lamm_alr(Uu),Uub),Uuc))) ).

% ATP.lambda_942
tff(fact_11120_ATP_Olambda__943,axiom,
    ! [Uu: set_nat,Uua: fun_VE597728521808591204at_nat,Uub: fun_VE6480751511521678896_nat_o,Uuc: vEBT_VEBT] : aa_VEBT_VEBT_nat(aa_fun3621025987637128233BT_nat(aa_fun1603736613815373464BT_nat(aTP_Lamm_alo(Uu),Uua),Uub),Uuc) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_VEB1122092716774696213at_nat(Uua,Uuc)),collect_nat(aa_VEB2314620471664397097_nat_o(aa_fun1998352179508271774_nat_o(aTP_Lamm_aln(Uu),Uub),Uuc))) ).

% ATP.lambda_943
tff(fact_11121_ATP_Olambda__944,axiom,
    ! [Uu: product_prod_nat_nat,Uua: nat,Uub: nat,Uuc: product_unit] : aa_Pro451875010097498139on_nat(aa_nat3999351741935936176on_nat(aa_nat3263736196148507005on_nat(aTP_Lamm_aj(Uu),Uua),Uub),Uuc) = heap_T5999496708990702694on_nat(aa_o_H2459438917461573107t_unit(refine_Imp_assert,Uub = aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,x12),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat3999351741935936176on_nat(aa_nat3263736196148507005on_nat(aTP_Lamm_ai(Uu),Uua),Uub)) ).

% ATP.lambda_944
tff(fact_11122_ATP_Olambda__945,axiom,
    ! [Uu: nat,Uua: list_VEBT_VEBT,Uub: list_VEBT_VEBTi,Uuc: option_nat] : aa_option_nat_assn(aa_lis7387135221810377032t_assn(aa_lis7458637400188059141t_assn(aTP_Lamm_bm(Uu),Uua),Uub),Uuc) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_o_assn(pure_assn,Uuc = vEBT_vebt_mint(nth_VEBT_VEBT(Uua,Uu)))),aa_lis2508312872377092907i_assn(vEBT_L6296928887356842470_VEBTi(vEBT_vebt_assn_raw,Uua),Uub)) ).

% ATP.lambda_945
tff(fact_11123_ATP_Olambda__946,axiom,
    ! [Uu: nat,Uua: list_VEBT_VEBT,Uub: list_VEBT_VEBTi,Uuc: option_nat] : aa_option_nat_assn(aa_lis7387135221810377032t_assn(aa_lis7458637400188059141t_assn(aTP_Lamm_bn(Uu),Uua),Uub),Uuc) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_o_assn(pure_assn,Uuc = vEBT_vebt_maxt(nth_VEBT_VEBT(Uua,Uu)))),aa_lis2508312872377092907i_assn(vEBT_L6296928887356842470_VEBTi(vEBT_vebt_assn_raw,Uua),Uub)) ).

% ATP.lambda_946
tff(fact_11124_ATP_Olambda__947,axiom,
    ! [Uu: product_prod_nat_nat,Uua: nat,Uub: nat,Uuc: product_unit] : aa_Pro451875010097498139on_nat(aa_nat3999351741935936176on_nat(aa_nat3263736196148507005on_nat(aTP_Lamm_ai(Uu),Uua),Uub),Uuc) = heap_T5999496708990702694on_nat(aa_o_H2459438917461573107t_unit(refine_Imp_assert,aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uub),size_s6755466524823107622T_VEBT(treeList))),aa_nat3999351741935936176on_nat(aa_nat3263736196148507005on_nat(aTP_Lamm_ah(Uu),Uua),Uub)) ).

% ATP.lambda_947
tff(fact_11125_ATP_Olambda__948,axiom,
    ! [Uu: fun_Pr238758252084143515Heap_o,Uua: array_VEBT_VEBTi,Uub: nat,Uuc: nat] : aa_nat2910048432523430448Heap_o(aa_nat5592791454563588861Heap_o(aa_arr9046104431811898738Heap_o(aTP_Lamm_zk(Uu),Uua),Uub),Uuc) = heap_Time_bind_nat_o(array_len_VEBT_VEBTi(Uua),aa_nat5592791454563588861Heap_o(aa_nat7429013343984486730Heap_o(aa_arr5803378942169532735Heap_o(aTP_Lamm_zj(Uu),Uua),Uub),Uuc)) ).

% ATP.lambda_948
tff(fact_11126_ATP_Olambda__949,axiom,
    ! [Uu: fun_VE7134199479003659425Heap_o,Uua: array_VEBT_VEBTi,Uub: nat,Uuc: nat] : aa_nat2910048432523430448Heap_o(aa_nat5592791454563588861Heap_o(aa_arr9046104431811898738Heap_o(aTP_Lamm_oa(Uu),Uua),Uub),Uuc) = heap_Time_bind_nat_o(array_len_VEBT_VEBTi(Uua),aa_nat5592791454563588861Heap_o(aa_nat7429013343984486730Heap_o(aa_arr5803378942169532735Heap_o(aTP_Lamm_nz(Uu),Uua),Uub),Uuc)) ).

% ATP.lambda_949
tff(fact_11127_ATP_Olambda__950,axiom,
    ! [Uu: nat,Uua: nat,Uub: list_VEBT_VEBTi,Uuc: product_unit] : aa_Pro3581650104493980720_VEBTi(aa_lis4026867407995182650_VEBTi(aa_nat1561540599483812487_VEBTi(aTP_Lamm_xr(Uu),Uua),Uub),Uuc) = heap_T5099337393651448672_VEBTi(array_615059503499738533_VEBTi(Uub),aa_nat106358831840921632_VEBTi(aTP_Lamm_xq(Uu),Uua)) ).

% ATP.lambda_950
tff(fact_11128_ATP_Olambda__951,axiom,
    ! [Uu: nat,Uua: nat,Uub: list_VEBT_VEBTi,Uuc: product_unit] : aa_Pro3581650104493980720_VEBTi(aa_lis4026867407995182650_VEBTi(aa_nat1561540599483812487_VEBTi(aTP_Lamm_xo(Uu),Uua),Uub),Uuc) = heap_T5099337393651448672_VEBTi(array_615059503499738533_VEBTi(Uub),aa_nat106358831840921632_VEBTi(aTP_Lamm_xn(Uu),Uua)) ).

% ATP.lambda_951
tff(fact_11129_ATP_Olambda__952,axiom,
    ! [Uu: assn,Uua: fun_VE905575822219544865i_assn,Uub: vEBT_VEBT,Uuc: vEBT_VEBTi] : aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(aa_fun4659777589554175228i_assn(aTP_Lamm_agg(Uu),Uua),Uub),Uuc) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Uu),aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(Uua,Uub),Uuc)) ).

% ATP.lambda_952
tff(fact_11130_ATP_Olambda__953,axiom,
    ! [Uu: assn,Uua: fun_VE8866928305347306583t_assn,Uub: vEBT_VEBT,Uuc: product_unit] : aa_Product_unit_assn(aa_VEB4153095751742533328t_assn(aa_fun1364258190791680492t_assn(aTP_Lamm_age(Uu),Uua),Uub),Uuc) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Uu),aa_Product_unit_assn(aa_VEB4153095751742533328t_assn(Uua,Uub),Uuc)) ).

% ATP.lambda_953
tff(fact_11131_ATP_Olambda__954,axiom,
    ! [Uu: assn,Uua: fun_VE6334927249922962060t_assn,Uub: vEBT_VEBT,Uuc: option_nat] : aa_option_nat_assn(aa_VEB931861714549014589t_assn(aa_fun1240582285364316732t_assn(aTP_Lamm_afy(Uu),Uua),Uub),Uuc) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Uu),aa_option_nat_assn(aa_VEB931861714549014589t_assn(Uua,Uub),Uuc)) ).

% ATP.lambda_954
tff(fact_11132_ATP_Olambda__955,axiom,
    ! [Uu: assn,Uua: fun_VE2104784059010244834o_assn,Uub: vEBT_VEBT,Uuc: $o] : aa_o_assn(aa_VEB5185540838851036891o_assn(aa_fun6967398642911097346o_assn(aTP_Lamm_agc(Uu),Uua),Uub),(Uuc)) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Uu),aa_o_assn(aa_VEB5185540838851036891o_assn(Uua,Uub),(Uuc))) ).

% ATP.lambda_955
tff(fact_11133_ATP_Olambda__956,axiom,
    ! [Uu: assn,Uua: fun_VE4931415397359990460t_assn,Uub: vEBT_VEBT,Uuc: nat] : aa_nat_assn(aa_VEB8640158415945897581t_assn(aa_fun7634423771582967228t_assn(aTP_Lamm_aga(Uu),Uua),Uub),Uuc) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Uu),aa_nat_assn(aa_VEB8640158415945897581t_assn(Uua,Uub),Uuc)) ).

% ATP.lambda_956
tff(fact_11134_ATP_Olambda__957,axiom,
    ! [Uu: $o,Uua: fun_nat_fun_nat_o,Uub: nat,Uuc: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aa_fun6699960423209655484_nat_o(aTP_Lamm_ox((Uu)),Uua),Uub),Uuc)
    <=> ( (Uu)
        & aa_nat_o(aa_nat_fun_nat_o(Uua,Uub),Uuc) ) ) ).

% ATP.lambda_957
tff(fact_11135_ATP_Olambda__958,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_oy((Uu)),Uua),Uub),Uuc)
    <=> ( (Uu)
        & aa_int_o(aa_int_fun_int_o(Uua,Uub),Uuc) ) ) ).

% ATP.lambda_958
tff(fact_11136_ATP_Olambda__959,axiom,
    ! [Uu: option4927543243414619207at_nat,Uua: nat,Uub: list_VEBT_VEBT,Uuc: vEBT_VEBT] : aa_VEB8692087746923902260T_VEBT(aa_lis2966823273985285125T_VEBT(aa_nat6705503460730637082T_VEBT(aa_opt2902932101092860132T_VEBT(aTP_Lamm_ot,Uu),Uua),Uub),Uuc) = produc6272728841373537334T_VEBT(Uu,produc1750349459881913976T_VEBT(Uua,produc6691630295680060561T_VEBT(Uub,Uuc))) ).

% ATP.lambda_959
tff(fact_11137_ATP_Olambda__960,axiom,
    ! [Uu: fun_He8247729123535230979_VEBTi,Uua: fun_o_6972675622308511895_VEBTi,Uub: $o,Uuc: $o] : aa_o_H7577119287286566245_VEBTi(aa_o_f5262348187908705736_VEBTi(aa_fun5654756598358634620_VEBTi(aTP_Lamm_fs(Uu),Uua),(Uub)),(Uuc)) = aa_Hea2545666323642607868_VEBTi(Uu,aa_o_H7577119287286566245_VEBTi(aa_o_f5262348187908705736_VEBTi(Uua,(Uub)),(Uuc))) ).

% ATP.lambda_960
tff(fact_11138_ATP_Olambda__961,axiom,
    ! [Uu: fun_He1783186893022644846on_nat,Uua: fun_o_6972675622308511895_VEBTi,Uub: $o,Uuc: $o] : aa_o_H4215342640551218896on_nat(aa_o_f5606600903069576627on_nat(aa_fun4045834922583578215on_nat(aTP_Lamm_fq(Uu),Uua),(Uub)),(Uuc)) = aa_Hea6259128614175166183on_nat(Uu,aa_o_H7577119287286566245_VEBTi(aa_o_f5262348187908705736_VEBTi(Uua,(Uub)),(Uuc))) ).

% ATP.lambda_961
tff(fact_11139_ATP_Olambda__962,axiom,
    ! [Uu: fun_He4624006784692404214Heap_o,Uua: fun_o_6972675622308511895_VEBTi,Uub: $o,Uuc: $o] : aa_o_H7399033255965945470Heap_o(aa_o_f869145629572826609Heap_o(aa_fun3457972572965721299Heap_o(aTP_Lamm_fu(Uu),Uua),(Uub)),(Uuc)) = aa_Hea7612170549154561063Heap_o(Uu,aa_o_H7577119287286566245_VEBTi(aa_o_f5262348187908705736_VEBTi(Uua,(Uub)),(Uuc))) ).

% ATP.lambda_962
tff(fact_11140_ATP_Olambda__963,axiom,
    ! [Uu: fun_He2623223239018661208_VEBTi,Uua: fun_o_4049874799390722306on_nat,Uub: $o,Uuc: $o] : aa_o_H7577119287286566245_VEBTi(aa_o_f5262348187908705736_VEBTi(aa_fun8772555301277947729_VEBTi(aTP_Lamm_fm(Uu),Uua),(Uub)),(Uuc)) = aa_Hea2832269215923769937_VEBTi(Uu,aa_o_H4215342640551218896on_nat(aa_o_f5606600903069576627on_nat(Uua,(Uub)),(Uuc))) ).

% ATP.lambda_963
tff(fact_11141_ATP_Olambda__964,axiom,
    ! [Uu: fun_He8640631688564546883on_nat,Uua: fun_o_4049874799390722306on_nat,Uub: $o,Uuc: $o] : aa_o_H4215342640551218896on_nat(aa_o_f5606600903069576627on_nat(aa_fun4373765338990682812on_nat(aTP_Lamm_fk(Uu),Uua),(Uub)),(Uuc)) = aa_Hea6597239020403563964on_nat(Uu,aa_o_H4215342640551218896on_nat(aa_o_f5606600903069576627on_nat(Uua,(Uub)),(Uuc))) ).

% ATP.lambda_964
tff(fact_11142_ATP_Olambda__965,axiom,
    ! [Uu: fun_He5988046180898962017Heap_o,Uua: fun_o_4049874799390722306on_nat,Uub: $o,Uuc: $o] : aa_o_H7399033255965945470Heap_o(aa_o_f869145629572826609Heap_o(aa_fun4676301296891262398Heap_o(aTP_Lamm_fo(Uu),Uua),(Uub)),(Uuc)) = aa_Hea662945835469240978Heap_o(Uu,aa_o_H4215342640551218896on_nat(aa_o_f5606600903069576627on_nat(Uua,(Uub)),(Uuc))) ).

% ATP.lambda_965
tff(fact_11143_ATP_Olambda__966,axiom,
    ! [Uu: fun_He6517945810935171576_VEBTi,Uua: fun_o_1207901889202581752Heap_o,Uub: $o,Uuc: $o] : aa_o_H7577119287286566245_VEBTi(aa_o_f5262348187908705736_VEBTi(aa_fun8320297987776248033_VEBTi(aTP_Lamm_fy(Uu),Uua),(Uub)),(Uuc)) = aa_Hea4465279032261640929_VEBTi(Uu,aa_o_H7399033255965945470Heap_o(aa_o_f869145629572826609Heap_o(Uua,(Uub)),(Uuc))) ).

% ATP.lambda_966
tff(fact_11144_ATP_Olambda__967,axiom,
    ! [Uu: fun_He6140748070426431971on_nat,Uua: fun_o_1207901889202581752Heap_o,Uub: $o,Uuc: $o] : aa_o_H4215342640551218896on_nat(aa_o_f5606600903069576627on_nat(aa_fun3614415378023021644on_nat(aTP_Lamm_fw(Uu),Uua),(Uub)),(Uuc)) = aa_Hea41712926108436044on_nat(Uu,aa_o_H7399033255965945470Heap_o(aa_o_f869145629572826609Heap_o(Uua,(Uub)),(Uuc))) ).

% ATP.lambda_967
tff(fact_11145_ATP_Olambda__968,axiom,
    ! [Uu: fun_He6597989805181038017Heap_o,Uua: fun_o_1207901889202581752Heap_o,Uub: $o,Uuc: $o] : aa_o_H7399033255965945470Heap_o(aa_o_f869145629572826609Heap_o(aa_fun3804294760359788590Heap_o(aTP_Lamm_ga(Uu),Uua),(Uub)),(Uuc)) = aa_Hea4824394052061459970Heap_o(Uu,aa_o_H7399033255965945470Heap_o(aa_o_f869145629572826609Heap_o(Uua,(Uub)),(Uuc))) ).

% ATP.lambda_968
tff(fact_11146_ATP_Olambda__969,axiom,
    ! [Uu: fun_Pr3650993782763339253T_VEBT,Uua: fun_o_8142510167905321106T_VEBT,Uub: $o,Uuc: $o] : aa_o_P8223745691486049304T_VEBT(aa_o_f7062522455407347339T_VEBT(aa_fun5380946117177639010T_VEBT(aTP_Lamm_ph(Uu),Uua),(Uub)),(Uuc)) = aa_Pro5421721056952420150T_VEBT(Uu,aa_o_P8223745691486049304T_VEBT(aa_o_f7062522455407347339T_VEBT(Uua,(Uub)),(Uuc))) ).

% ATP.lambda_969
tff(fact_11147_ATP_Olambda__970,axiom,
    ! [Uu: fun_Pr7471953115884902709VEBT_o,Uua: fun_o_8142510167905321106T_VEBT,Uub: $o,Uuc: $o] :
      ( aa_o_o(aa_o_fun_o_o(aa_fun449888776569364894un_o_o(aTP_Lamm_pj(Uu),Uua),(Uub)),(Uuc))
    <=> aa_Pro5475924895142249886VEBT_o(Uu,aa_o_P8223745691486049304T_VEBT(aa_o_f7062522455407347339T_VEBT(Uua,(Uub)),(Uuc))) ) ).

% ATP.lambda_970
tff(fact_11148_ATP_Olambda__971,axiom,
    ! [Uu: fun_Pr4796943543331071514_nat_o,Uua: fun_na4138050610385705875at_nat,Uub: nat,Uuc: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aa_fun7288258519458182931_nat_o(aTP_Lamm_kl(Uu),Uua),Uub),Uuc)
    <=> aa_Pro6150637604445964947_nat_o(Uu,aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(Uua,Uub),Uuc)) ) ).

% ATP.lambda_971
tff(fact_11149_ATP_Olambda__972,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_kn(Uu),Uua),Uub),Uuc)
    <=> aa_Pro4592068669816069595_int_o(Uu,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(Uua,Uub),Uuc)) ) ).

% ATP.lambda_972
tff(fact_11150_ATP_Olambda__973,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_ko(Uu),Uua),Uub),Uuc) = aa_Pro6799258434465214121nt_int(Uu,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(Uua,Uub),Uuc)) ).

% ATP.lambda_973
tff(fact_11151_ATP_Olambda__974,axiom,
    ! [Uu: fun_o_3001263932818058983T_VEBT,Uua: fun_o_fun_o_o,Uub: $o,Uuc: $o] : aa_o_P8223745691486049304T_VEBT(aa_o_f7062522455407347339T_VEBT(aa_fun4185041498930275652T_VEBT(aTP_Lamm_pl(Uu),Uua),(Uub)),(Uuc)) = aa_o_P8223745691486049304T_VEBT(Uu,aa_o_o(aa_o_fun_o_o(Uua,(Uub)),(Uuc))) ).

% ATP.lambda_974
tff(fact_11152_ATP_Olambda__975,axiom,
    ! [Uu: fun_o_67843798866261420at_nat,Uua: fun_nat_fun_nat_o,Uub: nat,Uuc: nat] : aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aa_fun9180491248050787109at_nat(aTP_Lamm_km(Uu),Uua),Uub),Uuc) = aa_o_P6312935399938478757at_nat(Uu,aa_nat_o(aa_nat_fun_nat_o(Uua,Uub),Uuc)) ).

% ATP.lambda_975
tff(fact_11153_ATP_Olambda__976,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_kp(Uu),Uua),Uub),Uuc) = aa_o_P3134438092330336605nt_int(Uu,aa_int_o(aa_int_fun_int_o(Uua,Uub),Uuc)) ).

% ATP.lambda_976
tff(fact_11154_ATP_Olambda__977,axiom,
    ! [Uu: fun_o_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_pn(Uu),Uua),(Uub)),(Uuc))
    <=> aa_o_o(Uu,aa_o_o(aa_o_fun_o_o(Uua,(Uub)),(Uuc))) ) ).

% ATP.lambda_977
tff(fact_11155_ATP_Olambda__978,axiom,
    ! [Uu: fun_o_o,Uua: fun_nat_fun_nat_o,Uub: nat,Uuc: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aa_fun6699960423209655484_nat_o(aTP_Lamm_kg(Uu),Uua),Uub),Uuc)
    <=> aa_o_o(Uu,aa_nat_o(aa_nat_fun_nat_o(Uua,Uub),Uuc)) ) ).

% ATP.lambda_978
tff(fact_11156_ATP_Olambda__979,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_kh(Uu),Uua),Uub),Uuc)
    <=> aa_o_o(Uu,aa_int_o(aa_int_fun_int_o(Uua,Uub),Uuc)) ) ).

% ATP.lambda_979
tff(fact_11157_ATP_Olambda__980,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_ki(Uu),Uua),Uub),Uuc) = aa_o_int(Uu,aa_int_o(aa_int_fun_int_o(Uua,Uub),Uuc)) ).

% ATP.lambda_980
tff(fact_11158_ATP_Olambda__981,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_kj(Uu),Uua),Uub),Uuc)
    <=> aa_int_o(Uu,aa_int_int(aa_int_fun_int_int(Uua,Uub),Uuc)) ) ).

% ATP.lambda_981
tff(fact_11159_ATP_Olambda__982,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_kk(Uu),Uua),Uub),Uuc) = aa_int_int(Uu,aa_int_int(aa_int_fun_int_int(Uua,Uub),Uuc)) ).

% ATP.lambda_982
tff(fact_11160_ATP_Olambda__983,axiom,
    ! [Uu: fun_nat_real,Uua: nat,Uub: nat,Uuc: nat] : aa_nat_real(aa_nat_fun_nat_real(aa_nat4085805313368488148t_real(aTP_Lamm_aoo(Uu),Uua),Uub),Uuc) = aa_nat_real(Uu,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)),Uuc)) ).

% ATP.lambda_983
tff(fact_11161_ATP_Olambda__984,axiom,
    ! [Uu: fun_nat_nat,Uua: nat,Uub: nat,Uuc: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_aon(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_984
tff(fact_11162_ATP_Olambda__985,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_baz(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_985
tff(fact_11163_ATP_Olambda__986,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_bbf(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_986
tff(fact_11164_ATP_Olambda__987,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_bbb(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_987
tff(fact_11165_ATP_Olambda__988,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_bbd(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_988
tff(fact_11166_ATP_Olambda__989,axiom,
    ! [Uu: set_VEBT_VEBT,Uua: set_VEBT_VEBT,Uub: vEBT_VEBT,Uuc: fun_VE3165209603783583043T_VEBT] :
      ( aa_fun8895443600707435607VEBT_o(aa_VEB992159746102714230VEBT_o(aa_set4017001947700194517VEBT_o(aTP_Lamm_asb(Uu),Uua),Uub),Uuc)
    <=> ! [X: vEBT_VEBT] :
          ( ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),Uu)
           => aa_set_VEBT_VEBT_o(member_VEBT_VEBT(aa_VEB5832953974749077948T_VEBT(Uuc,X)),Uua) )
          & ( ~ aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),Uu)
           => ( aa_VEB5832953974749077948T_VEBT(Uuc,X) = Uub ) ) ) ) ).

% ATP.lambda_989
tff(fact_11167_ATP_Olambda__990,axiom,
    ! [Uu: set_VEBT_VEBT,Uua: set_complex,Uub: complex,Uuc: fun_VE5569070535936029429omplex] :
      ( aa_fun896506121575208991plex_o(aa_com8209720474803639954plex_o(aa_set6707307212627643379plex_o(aTP_Lamm_asj(Uu),Uua),Uub),Uuc)
    <=> ! [X: vEBT_VEBT] :
          ( ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),Uu)
           => aa_set_complex_o(member_complex(aa_VEBT_VEBT_complex(Uuc,X)),Uua) )
          & ( ~ aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),Uu)
           => ( aa_VEBT_VEBT_complex(Uuc,X) = Uub ) ) ) ) ).

% ATP.lambda_990
tff(fact_11168_ATP_Olambda__991,axiom,
    ! [Uu: set_VEBT_VEBT,Uua: set_real,Uub: real,Uuc: fun_VEBT_VEBT_real] :
      ( aa_fun2793211803240935201real_o(aa_rea1931189138858785298real_o(aa_set3249680857378704241real_o(aTP_Lamm_asc(Uu),Uua),Uub),Uuc)
    <=> ! [X: vEBT_VEBT] :
          ( ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),Uu)
           => aa_set_real_o(member_real(aa_VEBT_VEBT_real(Uuc,X)),Uua) )
          & ( ~ aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),Uu)
           => ( aa_VEBT_VEBT_real(Uuc,X) = Uub ) ) ) ) ).

% ATP.lambda_991
tff(fact_11169_ATP_Olambda__992,axiom,
    ! [Uu: set_VEBT_VEBT,Uua: set_nat,Uub: nat,Uuc: fun_VEBT_VEBT_nat] :
      ( aa_fun9073248628033033085_nat_o(aa_nat8961800256705610514_nat_o(aa_set6628163682670611349_nat_o(aTP_Lamm_asf(Uu),Uua),Uub),Uuc)
    <=> ! [X: vEBT_VEBT] :
          ( ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),Uu)
           => aa_set_nat_o(member_nat(aa_VEBT_VEBT_nat(Uuc,X)),Uua) )
          & ( ~ aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),Uu)
           => ( aa_VEBT_VEBT_nat(Uuc,X) = Uub ) ) ) ) ).

% ATP.lambda_992
tff(fact_11170_ATP_Olambda__993,axiom,
    ! [Uu: set_VEBT_VEBT,Uua: set_int,Uub: int,Uuc: fun_VEBT_VEBT_int] :
      ( aa_fun6828103660362587169_int_o(aa_int5874382713699637778_int_o(aa_set7680706651547224305_int_o(aTP_Lamm_ash(Uu),Uua),Uub),Uuc)
    <=> ! [X: vEBT_VEBT] :
          ( ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),Uu)
           => aa_set_int_o(member_int(aa_VEBT_VEBT_int(Uuc,X)),Uua) )
          & ( ~ aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),Uu)
           => ( aa_VEBT_VEBT_int(Uuc,X) = Uub ) ) ) ) ).

% ATP.lambda_993
tff(fact_11171_ATP_Olambda__994,axiom,
    ! [Uu: set_real,Uua: set_VEBT_VEBT,Uub: vEBT_VEBT,Uuc: fun_real_VEBT_VEBT] :
      ( aa_fun2788233994425083827VEBT_o(aa_VEB1275781630754614706VEBT_o(aa_set1734275489953896617VEBT_o(aTP_Lamm_asd(Uu),Uua),Uub),Uuc)
    <=> ! [X: real] :
          ( ( aa_set_real_o(member_real(X),Uu)
           => aa_set_VEBT_VEBT_o(member_VEBT_VEBT(aa_real_VEBT_VEBT(Uuc,X)),Uua) )
          & ( ~ aa_set_real_o(member_real(X),Uu)
           => ( aa_real_VEBT_VEBT(Uuc,X) = Uub ) ) ) ) ).

% ATP.lambda_994
tff(fact_11172_ATP_Olambda__995,axiom,
    ! [Uu: set_real,Uua: set_complex,Uub: complex,Uuc: fun_real_complex] :
      ( aa_fun2856441533393264067plex_o(aa_com7884460700185185646plex_o(aa_set4700154924931669219plex_o(aTP_Lamm_ask(Uu),Uua),Uub),Uuc)
    <=> ! [X: real] :
          ( ( aa_set_real_o(member_real(X),Uu)
           => aa_set_complex_o(member_complex(aa_real_complex(Uuc,X)),Uua) )
          & ( ~ aa_set_real_o(member_real(X),Uu)
           => ( aa_real_complex(Uuc,X) = Uub ) ) ) ) ).

% ATP.lambda_995
tff(fact_11173_ATP_Olambda__996,axiom,
    ! [Uu: set_real,Uua: set_real,Uub: real,Uuc: fun_real_real] :
      ( aa_fun_real_real_o(aa_rea3177242051037803890real_o(aa_set5769202425262026345real_o(aTP_Lamm_ase(Uu),Uua),Uub),Uuc)
    <=> ! [X: real] :
          ( ( aa_set_real_o(member_real(X),Uu)
           => aa_set_real_o(member_real(aa_real_real(Uuc,X)),Uua) )
          & ( ~ aa_set_real_o(member_real(X),Uu)
           => ( aa_real_real(Uuc,X) = Uub ) ) ) ) ).

% ATP.lambda_996
tff(fact_11174_ATP_Olambda__997,axiom,
    ! [Uu: set_real,Uua: set_nat,Uub: nat,Uuc: fun_real_nat] :
      ( aa_fun_real_nat_o(aa_nat7275243809302487082_nat_o(aa_set2636817504407142141_nat_o(aTP_Lamm_asg(Uu),Uua),Uub),Uuc)
    <=> ! [X: real] :
          ( ( aa_set_real_o(member_real(X),Uu)
           => aa_set_nat_o(member_nat(aa_real_nat(Uuc,X)),Uua) )
          & ( ~ aa_set_real_o(member_real(X),Uu)
           => ( aa_real_nat(Uuc,X) = Uub ) ) ) ) ).

% ATP.lambda_997
tff(fact_11175_ATP_Olambda__998,axiom,
    ! [Uu: set_real,Uua: set_int,Uub: int,Uuc: fun_real_int] :
      ( aa_fun_real_int_o(aa_int2775185683442467186_int_o(aa_set704203600554081513_int_o(aTP_Lamm_asi(Uu),Uua),Uub),Uuc)
    <=> ! [X: real] :
          ( ( aa_set_real_o(member_real(X),Uu)
           => aa_set_int_o(member_int(aa_real_int(Uuc,X)),Uua) )
          & ( ~ aa_set_real_o(member_real(X),Uu)
           => ( aa_real_int(Uuc,X) = Uub ) ) ) ) ).

% ATP.lambda_998
tff(fact_11176_ATP_Olambda__999,axiom,
    ! [Uu: fun_Pr940977554174145406Heap_o,Uua: list_VEBT_VEBT,Uub: nat,Uuc: nat,Uud: vEBT_VEBTi] : aa_VEB8194921341634505877Heap_o(aa_nat1164146675577659234Heap_o(aa_nat3058146694327238959Heap_o(aa_lis6732064841504156908Heap_o(aTP_Lamm_zw(Uu),Uua),Uub),Uuc),Uud) = aa_nat2910048432523430448Heap_o(aa_VEB1455453220190609250Heap_o(aa_VEB8672304581938048793Heap_o(produc8906325494159015119Heap_o(produc8381543706267210711Heap_o(Uu)),nth_VEBT_VEBT(Uua,Uub)),Uud),Uuc) ).

% ATP.lambda_999
tff(fact_11177_ATP_Olambda__1000,axiom,
    ! [Uu: fun_VE8181444127144381288Heap_o,Uua: list_VEBT_VEBT,Uub: nat,Uuc: nat,Uud: vEBT_VEBTi] : aa_VEB8194921341634505877Heap_o(aa_nat1164146675577659234Heap_o(aa_nat3058146694327238959Heap_o(aa_lis6732064841504156908Heap_o(aTP_Lamm_pv(Uu),Uua),Uub),Uuc),Uud) = aa_nat2910048432523430448Heap_o(aa_VEB1455453220190609250Heap_o(aa_VEB8672304581938048793Heap_o(Uu,nth_VEBT_VEBT(Uua,Uub)),Uud),Uuc) ).

% ATP.lambda_1000
tff(fact_11178_ATP_Olambda__1001,axiom,
    ! [Uu: nat,Uua: option4927543243414619207at_nat,Uub: nat,Uuc: array_VEBT_VEBTi,Uud: vEBT_VEBTi] :
      aa_VEB836174949890266638_VEBTi(aa_arr2912372873376718191_VEBTi(aa_nat6445230840460233604_VEBTi(aa_opt5182041009420754510_VEBTi(aTP_Lamm_jd(Uu),Uua),Uub),Uuc),Uud) = case_o1356590567247012107at_nat($ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Uub),one_one_nat),aa_VEB836174949890266638_VEBTi(heap_T3630416162098727440_VEBTi,aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(Uua,Uub,Uuc),Uud)),aa_VEB836174949890266638_VEBTi(heap_T3630416162098727440_VEBTi,aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Uu),Uu)),Uub,Uuc),Uud))),aa_VEB6393524116121717046_VEBTi(aa_arr8059685486658508715_VEBTi(aa_nat7594302058568047468_VEBTi(aa_opt7720280829887018808_VEBTi(aTP_Lamm_jc(Uu),Uua),Uub),Uuc),Uud),Uua) ).

% ATP.lambda_1001
tff(fact_11179_ATP_Olambda__1002,axiom,
    ! [Uu: fun_Pr238758252084143515Heap_o,Uua: array_VEBT_VEBTi,Uub: nat,Uuc: nat,Uud: nat] :
      aa_nat2910048432523430448Heap_o(aa_nat5592791454563588861Heap_o(aa_nat7429013343984486730Heap_o(aa_arr5803378942169532735Heap_o(aTP_Lamm_zj(Uu),Uua),Uub),Uuc),Uud) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uub),Uud),heap_T3040810144269856602EBTi_o(aa_nat5408703056993926195_VEBTi(array_nth_VEBT_VEBTi(Uua),Uub),aa_nat1164146675577659234Heap_o(aTP_Lamm_zi(Uu),Uuc)),aa_o_H7399033255965945470Heap_o(heap_Time_return_o,$false)) ).

% ATP.lambda_1002
tff(fact_11180_ATP_Olambda__1003,axiom,
    ! [Uu: fun_VE7134199479003659425Heap_o,Uua: array_VEBT_VEBTi,Uub: nat,Uuc: nat,Uud: nat] :
      aa_nat2910048432523430448Heap_o(aa_nat5592791454563588861Heap_o(aa_nat7429013343984486730Heap_o(aa_arr5803378942169532735Heap_o(aTP_Lamm_nz(Uu),Uua),Uub),Uuc),Uud) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uub),Uud),heap_T3040810144269856602EBTi_o(aa_nat5408703056993926195_VEBTi(array_nth_VEBT_VEBTi(Uua),Uub),aa_nat1164146675577659234Heap_o(aTP_Lamm_ny(Uu),Uuc)),aa_o_H7399033255965945470Heap_o(heap_Time_return_o,$false)) ).

% ATP.lambda_1003
tff(fact_11181_ATP_Olambda__1004,axiom,
    ! [Uu: nat,Uua: nat,Uub: array_VEBT_VEBTi,Uuc: product_prod_nat_nat,Uud: option_nat] :
      aa_opt9163293676379396302on_nat(aa_Pro4885747571561342816on_nat(aa_arr7927785801258559901on_nat(aa_nat6148929329942773342on_nat(aTP_Lamm_es(Uu),Uua),Uub),Uuc),Uud) = $ite(
        Uud = none_nat,
        $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_Pro4243173770044397973at_nat(product_fst_nat_nat,Uuc)),Uu),heap_T3487192422709364219on_nat(aa_nat_option_nat(some_nat,aa_Pro4243173770044397973at_nat(product_fst_nat_nat,Uuc))),heap_T3487192422709364219on_nat(none_nat)),
        heap_T2868974464944644318on_nat(aa_nat5408703056993926195_VEBTi(array_nth_VEBT_VEBTi(Uub),the_nat(Uud)),aa_opt1214731524852023858on_nat(aTP_Lamm_er(Uua),Uud)) ) ).

% ATP.lambda_1004
tff(fact_11182_ATP_Olambda__1005,axiom,
    ! [Uu: nat,Uua: nat,Uub: array_VEBT_VEBTi,Uuc: vEBT_VEBTi,Uud: product_prod_nat_nat] :
      aa_Pro7675874045790911655on_nat(aa_VEB7519107058437837345on_nat(aa_arr2984472677043443222on_nat(aa_nat7372410979875760471on_nat(aTP_Lamm_ex(Uu),Uua),Uub),Uuc),Uud) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Uua),one_one_nat),
        heap_T3487192422709364219on_nat(none_nat),
        $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_Pro4243173770044397973at_nat(product_snd_nat_nat,Uud)),Uu),heap_T3487192422709364219on_nat(aa_nat_option_nat(some_nat,aa_Pro4243173770044397973at_nat(product_snd_nat_nat,Uud))),heap_T8222160169144143993on_nat(aa_nat5821310369566504398ap_nat(vEBT_VEBT_lowi(Uu),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uua),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_Pro4465521767867336112on_nat(aa_VEB1870382382330055850on_nat(aa_arr666314619328815135on_nat(aa_nat8253036615571765984on_nat(aTP_Lamm_ew(Uu),Uua),Uub),Uuc),Uud))) ) ).

% ATP.lambda_1005
tff(fact_11183_ATP_Olambda__1006,axiom,
    ! [Uu: nat,Uua: nat,Uub: array_VEBT_VEBTi,Uuc: vEBT_VEBTi,Uud: product_prod_nat_nat] :
      aa_Pro7675874045790911655on_nat(aa_VEB7519107058437837345on_nat(aa_arr2984472677043443222on_nat(aa_nat7372410979875760471on_nat(aTP_Lamm_fg(Uu),Uua),Uub),Uuc),Uud) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Uua),one_one_nat),
        heap_T3487192422709364219on_nat(none_nat),
        $ite(
          aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uu),aa_Pro4243173770044397973at_nat(product_fst_nat_nat,Uud)),
          heap_T3487192422709364219on_nat(aa_nat_option_nat(some_nat,aa_Pro4243173770044397973at_nat(product_fst_nat_nat,Uud))),
          $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_Pro4243173770044397973at_nat(product_snd_nat_nat,Uud)),Uu),heap_T3487192422709364219on_nat(none_nat),heap_T8222160169144143993on_nat(aa_nat5821310369566504398ap_nat(vEBT_VEBT_lowi(Uu),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uua),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_VEB3566417204837707778on_nat(aa_arr5230852142804057699on_nat(aa_nat6371187903253094776on_nat(aTP_Lamm_ff(Uu),Uua),Uub),Uuc))) ) ) ).

% ATP.lambda_1006
tff(fact_11184_ATP_Olambda__1007,axiom,
    ! [Uu: nat,Uua: nat,Uub: array_VEBT_VEBTi,Uuc: nat,Uud: nat] :
      aa_nat2910048432523430448Heap_o(aa_nat5592791454563588861Heap_o(aa_arr9046104431811898738Heap_o(aa_nat272996766647252915Heap_o(aTP_Lamm_jt(Uu),Uua),Uub),Uuc),Uud) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Uua),one_one_nat),
        aa_o_H7399033255965945470Heap_o(heap_Time_return_o,$false),
        $ite(
          Uu = Uuc,
          aa_o_H7399033255965945470Heap_o(heap_Time_return_o,$true),
          $ite(
            Uu = Uud,
            aa_o_H7399033255965945470Heap_o(heap_Time_return_o,$true),
            $ite(
              aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uu),Uuc),
              aa_o_H7399033255965945470Heap_o(heap_Time_return_o,$false),
              $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uud),Uu),aa_o_H7399033255965945470Heap_o(heap_Time_return_o,$false),heap_Time_bind_nat_o(aa_nat5821310369566504398ap_nat(vEBT_VEBT_highi(Uu),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uua),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_arr494317875797454629Heap_o(aa_nat857754347168699366Heap_o(aTP_Lamm_js(Uu),Uua),Uub))) ) ) ) ) ).

% ATP.lambda_1007
tff(fact_11185_ATP_Olambda__1008,axiom,
    ! [Uu: nat,Uua: option4927543243414619207at_nat,Uub: nat,Uuc: array_VEBT_VEBTi,Uud: vEBT_VEBTi] : aa_VEB2711320435582112505on_nat(aa_arr2955191963898321370on_nat(aa_nat5317369504323972463on_nat(aa_opt8624683241416710969on_nat(aTP_Lamm_fh(Uu),Uua),Uub),Uuc),Uud) = case_o8344607093967974880at_nat(heap_T3487192422709364219on_nat(none_nat),aa_VEB7519107058437837345on_nat(aa_arr2984472677043443222on_nat(aa_nat7372410979875760471on_nat(aTP_Lamm_fg(Uu),Uub),Uuc),Uud),Uua) ).

% ATP.lambda_1008
tff(fact_11186_ATP_Olambda__1009,axiom,
    ! [Uu: nat,Uua: option4927543243414619207at_nat,Uub: nat,Uuc: array_VEBT_VEBTi,Uud: vEBT_VEBTi] : aa_VEB2711320435582112505on_nat(aa_arr2955191963898321370on_nat(aa_nat5317369504323972463on_nat(aa_opt8624683241416710969on_nat(aTP_Lamm_ey(Uu),Uua),Uub),Uuc),Uud) = case_o8344607093967974880at_nat(heap_T3487192422709364219on_nat(none_nat),aa_VEB7519107058437837345on_nat(aa_arr2984472677043443222on_nat(aa_nat7372410979875760471on_nat(aTP_Lamm_ex(Uu),Uub),Uuc),Uud),Uua) ).

% ATP.lambda_1009
tff(fact_11187_ATP_Olambda__1010,axiom,
    ! [Uu: nat,Uua: option4927543243414619207at_nat,Uub: nat,Uuc: array_VEBT_VEBTi,Uud: vEBT_VEBTi] : aa_VEB8194921341634505877Heap_o(aa_arr2373296168907467658Heap_o(aa_nat8650107858160254923Heap_o(aa_opt6539295502271640599Heap_o(aTP_Lamm_ju(Uu),Uua),Uub),Uuc),Uud) = case_o1442776274061689234at_nat(aa_o_H7399033255965945470Heap_o(heap_Time_return_o,$false),produc3505292621261808240Heap_o(aa_arr9046104431811898738Heap_o(aa_nat272996766647252915Heap_o(aTP_Lamm_jt(Uu),Uub),Uuc)),Uua) ).

% ATP.lambda_1010
tff(fact_11188_ATP_Olambda__1011,axiom,
    ! [Uu: $o,Uua: $o,Uub: $o,Uuc: code_integer,Uud: $o] : aa_o_char(aa_Cod7842134615204088097o_char(aa_o_f2742674633914528860o_char(aa_o_f9003794812476387647o_char(aTP_Lamm_bcl((Uu)),(Uua)),(Uub)),Uuc),(Uud)) = produc4188289175737317920o_char(aa_o_f2742674633914528860o_char(aa_o_f9003794812476387647o_char(aa_o_f4183413333637126194o_char(aTP_Lamm_bck((Uu)),(Uua)),(Uub)),(Uud)),code_bit_cut_integer(Uuc)) ).

% ATP.lambda_1011
tff(fact_11189_ATP_Olambda__1012,axiom,
    ! [Uu: nat,Uua: list_VEBT_VEBTi,Uub: array_VEBT_VEBTi,Uuc: vEBT_VEBTi,Uud: array_VEBT_VEBTi] : aa_arr8637723718858317150i_assn(aa_VEB7841561841333234298i_assn(aa_arr6331738488200487899i_assn(aa_lis1967528460011584869i_assn(aTP_Lamm_ji(Uu),Uua),Uub),Uuc),Uud) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_lis2508312872377092907i_assn(snga_assn_VEBT_VEBTi(Uub),list_u6098035379799741383_VEBTi(Uua,Uu,Uuc))),aa_o_assn(pure_assn,Uud = Uub)) ).

% ATP.lambda_1012
tff(fact_11190_ATP_Olambda__1013,axiom,
    ! [Uu: nat,Uua: list_o,Uub: array_o,Uuc: $o,Uud: array_o] : aa_array_o_assn(aa_o_f8424462421971924018o_assn(aa_arr7807244294037787346o_assn(aa_lis9214373194942273159o_assn(aTP_Lamm_jf(Uu),Uua),Uub),(Uuc)),Uud) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,snga_assn_o(Uub,list_update_o(Uua,Uu,(Uuc)))),aa_o_assn(pure_assn,Uud = Uub)) ).

% ATP.lambda_1013
tff(fact_11191_ATP_Olambda__1014,axiom,
    ! [Uu: nat,Uua: list_nat,Uub: array_nat,Uuc: nat,Uud: array_nat] : aa_array_nat_assn(aa_nat8392338049721622586t_assn(aa_arr4802002299793635008t_assn(aa_lis5007279772662053221t_assn(aTP_Lamm_jg(Uu),Uua),Uub),Uuc),Uud) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,snga_assn_nat(Uub,list_update_nat(Uua,Uu,Uuc))),aa_o_assn(pure_assn,Uud = Uub)) ).

% ATP.lambda_1014
tff(fact_11192_ATP_Olambda__1015,axiom,
    ! [Uu: nat,Uua: list_int,Uub: array_int,Uuc: int,Uud: array_int] : aa_array_int_assn(aa_int2412489125148462906t_assn(aa_arr8066481932369824868t_assn(aa_lis2681267339321482597t_assn(aTP_Lamm_jh(Uu),Uua),Uub),Uuc),Uud) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,snga_assn_int(Uub,list_update_int(Uua,Uu,Uuc))),aa_o_assn(pure_assn,Uud = Uub)) ).

% ATP.lambda_1015
tff(fact_11193_ATP_Olambda__1016,axiom,
    ! [Uu: nat,Uua: nat,Uub: array_VEBT_VEBTi,Uuc: list_VEBT_VEBT,Uud: nat] : aa_nat2910048432523430448Heap_o(aa_lis5083133406047025005Heap_o(aa_arr6185401592279617542Heap_o(aa_nat1466035401240590747Heap_o(aTP_Lamm_qp(Uu),Uua),Uub),Uuc),Uud) = heap_Time_bind_nat_o(aa_nat5821310369566504398ap_nat(vEBT_VEBT_lowi(Uu),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uua),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat5592791454563588861Heap_o(aa_lis455817436297872826Heap_o(aa_arr1217223296832173011Heap_o(aa_nat6014765364524370664Heap_o(aTP_Lamm_qo(Uu),Uua),Uub),Uuc),Uud)) ).

% ATP.lambda_1016
tff(fact_11194_ATP_Olambda__1017,axiom,
    ! [Uu: nat,Uua: nat,Uub: array_VEBT_VEBTi,Uuc: list_VEBT_VEBT,Uud: product_unit] : aa_Pro3379321066291785715Heap_o(aa_lis4443831354813659076Heap_o(aa_arr8974760139355794305Heap_o(aa_nat808430574047188546Heap_o(aTP_Lamm_qq(Uu),Uua),Uub),Uuc),Uud) = heap_Time_bind_nat_o(aa_nat5821310369566504398ap_nat(vEBT_VEBT_highi(Uu),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uua),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_lis5083133406047025005Heap_o(aa_arr6185401592279617542Heap_o(aa_nat1466035401240590747Heap_o(aTP_Lamm_qp(Uu),Uua),Uub),Uuc)) ).

% ATP.lambda_1017
tff(fact_11195_ATP_Olambda__1018,axiom,
    ! [Uu: nat,Uua: nat,Uub: array_VEBT_VEBTi,Uuc: vEBT_VEBTi,Uud: nat] : aa_nat947733127081326238on_nat(aa_VEB3566417204837707778on_nat(aa_arr5230852142804057699on_nat(aa_nat6371187903253094776on_nat(aTP_Lamm_ff(Uu),Uua),Uub),Uuc),Uud) = heap_T8222160169144143993on_nat(aa_nat5821310369566504398ap_nat(vEBT_VEBT_highi(Uu),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uua),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat6020981482381522343on_nat(aa_VEB6424020099669959819on_nat(aa_arr7085280839797930604on_nat(aTP_Lamm_fe(Uua),Uub),Uuc),Uud)) ).

% ATP.lambda_1018
tff(fact_11196_ATP_Olambda__1019,axiom,
    ! [Uu: fun_Pr238758252084143515Heap_o,Uua: nat,Uub: nat,Uuc: array_VEBT_VEBTi,Uud: nat] : aa_nat2910048432523430448Heap_o(aa_arr494317875797454629Heap_o(aa_nat857754347168699366Heap_o(aa_nat8612037029889215215Heap_o(aTP_Lamm_zl(Uu),Uua),Uub),Uuc),Uud) = heap_Time_bind_nat_o(aa_nat5821310369566504398ap_nat(vEBT_VEBT_lowi(Uua),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uub),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat5592791454563588861Heap_o(aa_arr9046104431811898738Heap_o(aTP_Lamm_zk(Uu),Uuc),Uud)) ).

% ATP.lambda_1019
tff(fact_11197_ATP_Olambda__1020,axiom,
    ! [Uu: fun_VE7134199479003659425Heap_o,Uua: nat,Uub: nat,Uuc: array_VEBT_VEBTi,Uud: nat] : aa_nat2910048432523430448Heap_o(aa_arr494317875797454629Heap_o(aa_nat857754347168699366Heap_o(aa_nat8612037029889215215Heap_o(aTP_Lamm_ob(Uu),Uua),Uub),Uuc),Uud) = heap_Time_bind_nat_o(aa_nat5821310369566504398ap_nat(vEBT_VEBT_lowi(Uua),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uub),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat5592791454563588861Heap_o(aa_arr9046104431811898738Heap_o(aTP_Lamm_oa(Uu),Uuc),Uud)) ).

% ATP.lambda_1020
tff(fact_11198_ATP_Olambda__1021,axiom,
    ! [Uu: nat,Uua: array_VEBT_VEBTi,Uub: vEBT_VEBTi,Uuc: nat,Uud: nat] : aa_nat947733127081326238on_nat(aa_nat6020981482381522343on_nat(aa_VEB6424020099669959819on_nat(aa_arr7085280839797930604on_nat(aTP_Lamm_fe(Uu),Uua),Uub),Uuc),Uud) = heap_T2868974464944644318on_nat(aa_nat5408703056993926195_VEBTi(array_nth_VEBT_VEBTi(Uua),Uud),aa_nat4669826966810537986on_nat(aa_nat107244183536939787on_nat(aa_VEB216942459957533935on_nat(aa_arr2558136924908760784on_nat(aTP_Lamm_fd(Uu),Uua),Uub),Uuc),Uud)) ).

% ATP.lambda_1021
tff(fact_11199_ATP_Olambda__1022,axiom,
    ! [Uu: array_VEBT_VEBTi,Uua: list_VEBT_VEBT,Uub: nat,Uuc: nat,Uud: product_unit] : aa_Pro3379321066291785715Heap_o(aa_nat3811243159488243764Heap_o(aa_nat6178576981048102717Heap_o(aa_lis4519787659412806998Heap_o(aTP_Lamm_qk(Uu),Uua),Uub),Uuc),Uud) = heap_T3040810144269856602EBTi_o(aa_nat5408703056993926195_VEBTi(array_nth_VEBT_VEBTi(Uu),Uub),aa_nat1164146675577659234Heap_o(aa_nat3058146694327238959Heap_o(aTP_Lamm_qj(Uua),Uub),Uuc)) ).

% ATP.lambda_1022
tff(fact_11200_ATP_Olambda__1023,axiom,
    ! [Uu: nat,Uua: array_VEBT_VEBTi,Uub: vEBT_VEBT,Uuc: nat,Uud: option_nat] :
      aa_opt9163293676379396302on_nat(aa_nat9127621225095157975on_nat(aa_VEB8264456441901112694on_nat(aa_arr7060518384107407923on_nat(aTP_Lamm_of(Uu),Uua),Uub),Uuc),Uud) = heap_T5999496708990702694on_nat(aa_o_H2459438917461573107t_unit(refine_Imp_assert,
          ( ( Uud = none_nat )
          = ( aa_nat_option_nat(vEBT_vebt_succ(Uub),Uuc) = none_nat ) )),aa_opt5089197864053947520on_nat(aa_arr5509772438618118773on_nat(aTP_Lamm_oe(Uu),Uua),Uud)) ).

% ATP.lambda_1023
tff(fact_11201_ATP_Olambda__1024,axiom,
    ! [Uu: assn,Uua: fun_VE905575822219544865i_assn,Uub: vEBT_VEBT,Uuc: nat,Uud: list_VEBT_VEBTi] : aa_lis2508312872377092907i_assn(aa_nat6700791848136694008i_assn(aa_VEB320677594755043119i_assn(aa_fun8635824850342594777i_assn(aTP_Lamm_agh(Uu),Uua),Uub),Uuc),Uud) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Uu),aa_lis2508312872377092907i_assn(vEBT_L6296928887356842470_VEBTi(Uua,replicate_VEBT_VEBT(Uuc,Uub)),Uud)) ).

% ATP.lambda_1024
tff(fact_11202_ATP_Olambda__1025,axiom,
    ! [Uu: assn,Uua: fun_VE8866928305347306583t_assn,Uub: vEBT_VEBT,Uuc: nat,Uud: list_Product_unit] : aa_lis2705678021120627967t_assn(aa_nat5137481719278757952t_assn(aa_VEB7218668862568288351t_assn(aa_fun8456556664957245819t_assn(aTP_Lamm_agf(Uu),Uua),Uub),Uuc),Uud) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Uu),aa_lis2705678021120627967t_assn(vEBT_L8068554427805421084t_unit(Uua,replicate_VEBT_VEBT(Uuc,Uub)),Uud)) ).

% ATP.lambda_1025
tff(fact_11203_ATP_Olambda__1026,axiom,
    ! [Uu: assn,Uua: fun_VE6334927249922962060t_assn,Uub: vEBT_VEBT,Uuc: nat,Uud: list_option_nat] : aa_lis1398067190886521878t_assn(aa_nat5248546159757461219t_assn(aa_VEB2926451386496479642t_assn(aa_fun7129419364150962969t_assn(aTP_Lamm_afz(Uu),Uua),Uub),Uuc),Uud) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Uu),aa_lis1398067190886521878t_assn(vEBT_L8010285020845282001on_nat(Uua,replicate_VEBT_VEBT(Uuc,Uub)),Uud)) ).

% ATP.lambda_1026
tff(fact_11204_ATP_Olambda__1027,axiom,
    ! [Uu: assn,Uua: fun_VE2104784059010244834o_assn,Uub: vEBT_VEBT,Uuc: nat,Uud: list_o] : aa_list_o_assn(aa_nat5137362611963950667o_assn(aa_VEB7710612541095692010o_assn(aa_fun380857135891842193o_assn(aTP_Lamm_agd(Uu),Uua),Uub),Uuc),Uud) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Uu),aa_list_o_assn(vEBT_L7489408758114837031VEBT_o(Uua,replicate_VEBT_VEBT(Uuc,Uub)),Uud)) ).

% ATP.lambda_1027
tff(fact_11205_ATP_Olambda__1028,axiom,
    ! [Uu: assn,Uua: fun_VE4931415397359990460t_assn,Uub: vEBT_VEBT,Uuc: nat,Uud: list_nat] : aa_list_nat_assn(aa_nat7967224596217534867t_assn(aa_VEB8402767592425156170t_assn(aa_fun6018615163312975129t_assn(aTP_Lamm_agb(Uu),Uua),Uub),Uuc),Uud) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Uu),aa_list_nat_assn(vEBT_L8296926524756676353BT_nat(Uua,replicate_VEBT_VEBT(Uuc,Uub)),Uud)) ).

% ATP.lambda_1028
tff(fact_11206_ATP_Olambda__1029,axiom,
    ! [Uu: nat,Uua: option4927543243414619207at_nat,Uub: nat,Uuc: array_VEBT_VEBTi,Uud: option4927543243414619207at_nat] : aa_opt6266093870483638640Heap_o(aa_arr6198139058929667337Heap_o(aa_nat436397609992601886Heap_o(aa_opt4438414422311449448Heap_o(aTP_Lamm_qt(Uu),Uua),Uub),Uuc),Uud) = produc5946672270950774087Heap_o(aa_opt7844807959364121492Heap_o(aa_arr79663438428320465Heap_o(aa_nat2087714105643485586Heap_o(aa_opt9028207981337828190Heap_o(aTP_Lamm_qs(Uu),Uua),Uub),Uuc),Uud)) ).

% ATP.lambda_1029
tff(fact_11207_ATP_Olambda__1030,axiom,
    ! [Uu: nat,Uua: nat,Uub: array_VEBT_VEBTi,Uuc: vEBT_VEBTi,Uud: nat] : aa_nat5408703056993926195_VEBTi(aa_VEB4833742232818206615_VEBTi(aa_arr6831033733615195768_VEBTi(aa_nat7652049589537018893_VEBTi(aTP_Lamm_ip(Uu),Uua),Uub),Uuc),Uud) = aa_VEB836174949890266638_VEBTi(heap_T3630416162098727440_VEBTi,aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Uua),Uud)),Uu,Uub),Uuc)) ).

% ATP.lambda_1030
tff(fact_11208_ATP_Olambda__1031,axiom,
    ! [Uu: fun_Pr4171499318505651870_VEBTi,Uua: nat,Uub: option4927543243414619207at_nat,Uuc: nat,Uud: array_VEBT_VEBTi,Uue: vEBT_VEBTi] :
      aa_VEB836174949890266638_VEBTi(aa_arr2912372873376718191_VEBTi(aa_nat6445230840460233604_VEBTi(aa_opt5182041009420754510_VEBTi(aa_nat3009484633486537743_VEBTi(aTP_Lamm_abo(Uu),Uua),Uub),Uuc),Uud),Uue) = case_o1356590567247012107at_nat($ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Uuc),one_one_nat),aa_VEB836174949890266638_VEBTi(heap_T3630416162098727440_VEBTi,aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(Uub,Uuc,Uud),Uue)),aa_VEB836174949890266638_VEBTi(heap_T3630416162098727440_VEBTi,aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Uua),Uua)),Uuc,Uud),Uue))),aa_VEB6393524116121717046_VEBTi(aa_arr8059685486658508715_VEBTi(aa_nat7594302058568047468_VEBTi(aa_opt7720280829887018808_VEBTi(aa_nat6605875721633857101_VEBTi(aTP_Lamm_abn(Uu),Uua),Uub),Uuc),Uud),Uue),Uub) ).

% ATP.lambda_1031
tff(fact_11209_ATP_Olambda__1032,axiom,
    ! [Uu: vEBT_VEBT,Uua: nat,Uub: option4927543243414619207at_nat,Uuc: nat,Uud: array_VEBT_VEBTi,Uue: vEBT_VEBTi] :
      aa_VEB836174949890266638_VEBTi(aa_arr2912372873376718191_VEBTi(aa_nat6445230840460233604_VEBTi(aa_opt5182041009420754510_VEBTi(aa_nat3009484633486537743_VEBTi(aTP_Lamm_tw(Uu),Uua),Uub),Uuc),Uud),Uue) = case_o1356590567247012107at_nat($ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Uuc),one_one_nat),aa_VEB836174949890266638_VEBTi(heap_T3630416162098727440_VEBTi,aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(Uub,Uuc,Uud),Uue)),aa_VEB836174949890266638_VEBTi(heap_T3630416162098727440_VEBTi,aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Uua),Uua)),Uuc,Uud),Uue))),aa_VEB6393524116121717046_VEBTi(aa_arr8059685486658508715_VEBTi(aa_nat7594302058568047468_VEBTi(aa_opt7720280829887018808_VEBTi(aa_nat6605875721633857101_VEBTi(aTP_Lamm_tv(Uu),Uua),Uub),Uuc),Uud),Uue),Uub) ).

% ATP.lambda_1032
tff(fact_11210_ATP_Olambda__1033,axiom,
    ! [Uu: fun_VE8118064890490510894_VEBTi,Uua: nat,Uub: option4927543243414619207at_nat,Uuc: nat,Uud: array_VEBT_VEBTi,Uue: vEBT_VEBTi] :
      aa_VEB836174949890266638_VEBTi(aa_arr2912372873376718191_VEBTi(aa_nat6445230840460233604_VEBTi(aa_opt5182041009420754510_VEBTi(aa_nat3009484633486537743_VEBTi(aTP_Lamm_nl(Uu),Uua),Uub),Uuc),Uud),Uue) = case_o1356590567247012107at_nat($ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Uuc),one_one_nat),aa_VEB836174949890266638_VEBTi(heap_T3630416162098727440_VEBTi,aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(Uub,Uuc,Uud),Uue)),aa_VEB836174949890266638_VEBTi(heap_T3630416162098727440_VEBTi,aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Uua),Uua)),Uuc,Uud),Uue))),aa_VEB6393524116121717046_VEBTi(aa_arr8059685486658508715_VEBTi(aa_nat7594302058568047468_VEBTi(aa_opt7720280829887018808_VEBTi(aa_nat6605875721633857101_VEBTi(aTP_Lamm_nk(Uu),Uua),Uub),Uuc),Uud),Uue),Uub) ).

% ATP.lambda_1033
tff(fact_11211_ATP_Olambda__1034,axiom,
    ! [Uu: nat,Uua: nat,Uub: array_VEBT_VEBTi,Uuc: product_prod_nat_nat,Uud: option_nat,Uue: product_unit] :
      aa_Pro451875010097498139on_nat(aa_opt5089197864053947520on_nat(aa_Pro5009687527940591364on_nat(aa_arr722815569820881053on_nat(aa_nat5945554870259702194on_nat(aTP_Lamm_ru(Uu),Uua),Uub),Uuc),Uud),Uue) = $ite(
        Uud = none_nat,
        $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_Pro4243173770044397973at_nat(product_fst_nat_nat,Uuc)),Uu),heap_T3487192422709364219on_nat(aa_nat_option_nat(some_nat,aa_Pro4243173770044397973at_nat(product_fst_nat_nat,Uuc))),heap_T3487192422709364219on_nat(none_nat)),
        heap_T2868974464944644318on_nat(aa_nat5408703056993926195_VEBTi(array_nth_VEBT_VEBTi(Uub),the_nat(Uud)),aa_opt1214731524852023858on_nat(aTP_Lamm_er(Uua),Uud)) ) ).

% ATP.lambda_1034
tff(fact_11212_ATP_Olambda__1035,axiom,
    ! [Uu: nat,Uua: option4927543243414619207at_nat,Uub: nat,Uuc: array_VEBT_VEBTi,Uud: vEBT_VEBTi,Uue: product_prod_nat_nat] :
      aa_Pro1395359565518501564_VEBTi(aa_VEB6393524116121717046_VEBTi(aa_arr8059685486658508715_VEBTi(aa_nat7594302058568047468_VEBTi(aa_opt7720280829887018808_VEBTi(aTP_Lamm_jc(Uu),Uua),Uub),Uuc),Uud),Uue) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Uub),one_one_nat),aa_VEB836174949890266638_VEBTi(heap_T3630416162098727440_VEBTi,aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(Uua,Uub,Uuc),Uud)),heap_T844888390831797134_VEBTi(aa_nat5821310369566504398ap_nat(heap_Time_return_nat,aa_Pro4243173770044397973at_nat(product_fst_nat_nat,Uue)),aa_Pro3150526172571800645_VEBTi(aa_VEB4890588985164552255_VEBTi(aa_arr5592147402706875956_VEBTi(aa_nat5963232373128642933_VEBTi(aTP_Lamm_jb(Uu),Uub),Uuc),Uud),Uue))) ).

% ATP.lambda_1035
tff(fact_11213_ATP_Olambda__1036,axiom,
    ! [Uu: fun_Pr342922516890243977on_nat,Uua: nat,Uub: nat,Uuc: array_VEBT_VEBTi,Uud: vEBT_VEBTi,Uue: product_prod_nat_nat] :
      aa_Pro7675874045790911655on_nat(aa_VEB7519107058437837345on_nat(aa_arr2984472677043443222on_nat(aa_nat7372410979875760471on_nat(aa_nat8243565228001313504on_nat(aTP_Lamm_aaq(Uu),Uua),Uub),Uuc),Uud),Uue) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Uub),one_one_nat),
        heap_T3487192422709364219on_nat(none_nat),
        $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_Pro4243173770044397973at_nat(product_snd_nat_nat,Uue)),Uua),heap_T3487192422709364219on_nat(aa_nat_option_nat(some_nat,aa_Pro4243173770044397973at_nat(product_snd_nat_nat,Uue))),heap_T8222160169144143993on_nat(aa_nat5821310369566504398ap_nat(vEBT_VEBT_lowi(Uua),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uub),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_Pro4465521767867336112on_nat(aa_VEB1870382382330055850on_nat(aa_arr666314619328815135on_nat(aa_nat8253036615571765984on_nat(aa_nat2001721438062390249on_nat(aTP_Lamm_aap(Uu),Uua),Uub),Uuc),Uud),Uue))) ) ).

% ATP.lambda_1036
tff(fact_11214_ATP_Olambda__1037,axiom,
    ! [Uu: fun_VE8516060933855021849on_nat,Uua: nat,Uub: nat,Uuc: array_VEBT_VEBTi,Uud: vEBT_VEBTi,Uue: product_prod_nat_nat] :
      aa_Pro7675874045790911655on_nat(aa_VEB7519107058437837345on_nat(aa_arr2984472677043443222on_nat(aa_nat7372410979875760471on_nat(aa_nat8243565228001313504on_nat(aTP_Lamm_nq(Uu),Uua),Uub),Uuc),Uud),Uue) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Uub),one_one_nat),
        heap_T3487192422709364219on_nat(none_nat),
        $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_Pro4243173770044397973at_nat(product_snd_nat_nat,Uue)),Uua),heap_T3487192422709364219on_nat(aa_nat_option_nat(some_nat,aa_Pro4243173770044397973at_nat(product_snd_nat_nat,Uue))),heap_T8222160169144143993on_nat(aa_nat5821310369566504398ap_nat(vEBT_VEBT_lowi(Uua),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uub),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_Pro4465521767867336112on_nat(aa_VEB1870382382330055850on_nat(aa_arr666314619328815135on_nat(aa_nat8253036615571765984on_nat(aa_nat2001721438062390249on_nat(aTP_Lamm_np(Uu),Uua),Uub),Uuc),Uud),Uue))) ) ).

% ATP.lambda_1037
tff(fact_11215_ATP_Olambda__1038,axiom,
    ! [Uu: fun_Pr342922516890243977on_nat,Uua: nat,Uub: nat,Uuc: array_VEBT_VEBTi,Uud: vEBT_VEBTi,Uue: product_prod_nat_nat] :
      aa_Pro7675874045790911655on_nat(aa_VEB7519107058437837345on_nat(aa_arr2984472677043443222on_nat(aa_nat7372410979875760471on_nat(aa_nat8243565228001313504on_nat(aTP_Lamm_aay(Uu),Uua),Uub),Uuc),Uud),Uue) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Uub),one_one_nat),
        heap_T3487192422709364219on_nat(none_nat),
        $ite(
          aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uua),aa_Pro4243173770044397973at_nat(product_fst_nat_nat,Uue)),
          heap_T3487192422709364219on_nat(aa_nat_option_nat(some_nat,aa_Pro4243173770044397973at_nat(product_fst_nat_nat,Uue))),
          $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_Pro4243173770044397973at_nat(product_snd_nat_nat,Uue)),Uua),heap_T3487192422709364219on_nat(none_nat),heap_T8222160169144143993on_nat(aa_nat5821310369566504398ap_nat(vEBT_VEBT_lowi(Uua),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uub),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_VEB3566417204837707778on_nat(aa_arr5230852142804057699on_nat(aa_nat6371187903253094776on_nat(aa_nat1535423446758346053on_nat(aTP_Lamm_aax(Uu),Uua),Uub),Uuc),Uud))) ) ) ).

% ATP.lambda_1038
tff(fact_11216_ATP_Olambda__1039,axiom,
    ! [Uu: fun_VE8516060933855021849on_nat,Uua: nat,Uub: nat,Uuc: array_VEBT_VEBTi,Uud: vEBT_VEBTi,Uue: product_prod_nat_nat] :
      aa_Pro7675874045790911655on_nat(aa_VEB7519107058437837345on_nat(aa_arr2984472677043443222on_nat(aa_nat7372410979875760471on_nat(aa_nat8243565228001313504on_nat(aTP_Lamm_nw(Uu),Uua),Uub),Uuc),Uud),Uue) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Uub),one_one_nat),
        heap_T3487192422709364219on_nat(none_nat),
        $ite(
          aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uua),aa_Pro4243173770044397973at_nat(product_fst_nat_nat,Uue)),
          heap_T3487192422709364219on_nat(aa_nat_option_nat(some_nat,aa_Pro4243173770044397973at_nat(product_fst_nat_nat,Uue))),
          $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_Pro4243173770044397973at_nat(product_snd_nat_nat,Uue)),Uua),heap_T3487192422709364219on_nat(none_nat),heap_T8222160169144143993on_nat(aa_nat5821310369566504398ap_nat(vEBT_VEBT_lowi(Uua),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uub),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_VEB3566417204837707778on_nat(aa_arr5230852142804057699on_nat(aa_nat6371187903253094776on_nat(aa_nat1535423446758346053on_nat(aTP_Lamm_nv(Uu),Uua),Uub),Uuc),Uud))) ) ) ).

% ATP.lambda_1039
tff(fact_11217_ATP_Olambda__1040,axiom,
    ! [Uu: fun_Pr238758252084143515Heap_o,Uua: nat,Uub: nat,Uuc: array_VEBT_VEBTi,Uud: nat,Uue: nat] :
      aa_nat2910048432523430448Heap_o(aa_nat5592791454563588861Heap_o(aa_arr9046104431811898738Heap_o(aa_nat272996766647252915Heap_o(aa_nat7415925511494575164Heap_o(aTP_Lamm_zm(Uu),Uua),Uub),Uuc),Uud),Uue) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Uub),one_one_nat),
        aa_o_H7399033255965945470Heap_o(heap_Time_return_o,$false),
        $ite(
          Uua = Uud,
          aa_o_H7399033255965945470Heap_o(heap_Time_return_o,$true),
          $ite(
            Uua = Uue,
            aa_o_H7399033255965945470Heap_o(heap_Time_return_o,$true),
            $ite(
              aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uua),Uud),
              aa_o_H7399033255965945470Heap_o(heap_Time_return_o,$false),
              $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uue),Uua),aa_o_H7399033255965945470Heap_o(heap_Time_return_o,$false),heap_Time_bind_nat_o(aa_nat5821310369566504398ap_nat(vEBT_VEBT_highi(Uua),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uub),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_arr494317875797454629Heap_o(aa_nat857754347168699366Heap_o(aa_nat8612037029889215215Heap_o(aTP_Lamm_zl(Uu),Uua),Uub),Uuc))) ) ) ) ) ).

% ATP.lambda_1040
tff(fact_11218_ATP_Olambda__1041,axiom,
    ! [Uu: fun_VE7134199479003659425Heap_o,Uua: nat,Uub: nat,Uuc: array_VEBT_VEBTi,Uud: nat,Uue: nat] :
      aa_nat2910048432523430448Heap_o(aa_nat5592791454563588861Heap_o(aa_arr9046104431811898738Heap_o(aa_nat272996766647252915Heap_o(aa_nat7415925511494575164Heap_o(aTP_Lamm_oc(Uu),Uua),Uub),Uuc),Uud),Uue) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Uub),one_one_nat),
        aa_o_H7399033255965945470Heap_o(heap_Time_return_o,$false),
        $ite(
          Uua = Uud,
          aa_o_H7399033255965945470Heap_o(heap_Time_return_o,$true),
          $ite(
            Uua = Uue,
            aa_o_H7399033255965945470Heap_o(heap_Time_return_o,$true),
            $ite(
              aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uua),Uud),
              aa_o_H7399033255965945470Heap_o(heap_Time_return_o,$false),
              $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uue),Uua),aa_o_H7399033255965945470Heap_o(heap_Time_return_o,$false),heap_Time_bind_nat_o(aa_nat5821310369566504398ap_nat(vEBT_VEBT_highi(Uua),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uub),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_arr494317875797454629Heap_o(aa_nat857754347168699366Heap_o(aa_nat8612037029889215215Heap_o(aTP_Lamm_ob(Uu),Uua),Uub),Uuc))) ) ) ) ) ).

% ATP.lambda_1041
tff(fact_11219_ATP_Olambda__1042,axiom,
    ! [Uu: fun_Pr342922516890243977on_nat,Uua: nat,Uub: option4927543243414619207at_nat,Uuc: nat,Uud: array_VEBT_VEBTi,Uue: vEBT_VEBTi] : aa_VEB2711320435582112505on_nat(aa_arr2955191963898321370on_nat(aa_nat5317369504323972463on_nat(aa_opt8624683241416710969on_nat(aa_nat3457676333800472698on_nat(aTP_Lamm_aaz(Uu),Uua),Uub),Uuc),Uud),Uue) = case_o8344607093967974880at_nat(heap_T3487192422709364219on_nat(none_nat),aa_VEB7519107058437837345on_nat(aa_arr2984472677043443222on_nat(aa_nat7372410979875760471on_nat(aa_nat8243565228001313504on_nat(aTP_Lamm_aay(Uu),Uua),Uuc),Uud),Uue),Uub) ).

% ATP.lambda_1042
tff(fact_11220_ATP_Olambda__1043,axiom,
    ! [Uu: fun_Pr342922516890243977on_nat,Uua: nat,Uub: option4927543243414619207at_nat,Uuc: nat,Uud: array_VEBT_VEBTi,Uue: vEBT_VEBTi] : aa_VEB2711320435582112505on_nat(aa_arr2955191963898321370on_nat(aa_nat5317369504323972463on_nat(aa_opt8624683241416710969on_nat(aa_nat3457676333800472698on_nat(aTP_Lamm_aar(Uu),Uua),Uub),Uuc),Uud),Uue) = case_o8344607093967974880at_nat(heap_T3487192422709364219on_nat(none_nat),aa_VEB7519107058437837345on_nat(aa_arr2984472677043443222on_nat(aa_nat7372410979875760471on_nat(aa_nat8243565228001313504on_nat(aTP_Lamm_aaq(Uu),Uua),Uuc),Uud),Uue),Uub) ).

% ATP.lambda_1043
tff(fact_11221_ATP_Olambda__1044,axiom,
    ! [Uu: fun_VE8516060933855021849on_nat,Uua: nat,Uub: option4927543243414619207at_nat,Uuc: nat,Uud: array_VEBT_VEBTi,Uue: vEBT_VEBTi] : aa_VEB2711320435582112505on_nat(aa_arr2955191963898321370on_nat(aa_nat5317369504323972463on_nat(aa_opt8624683241416710969on_nat(aa_nat3457676333800472698on_nat(aTP_Lamm_nx(Uu),Uua),Uub),Uuc),Uud),Uue) = case_o8344607093967974880at_nat(heap_T3487192422709364219on_nat(none_nat),aa_VEB7519107058437837345on_nat(aa_arr2984472677043443222on_nat(aa_nat7372410979875760471on_nat(aa_nat8243565228001313504on_nat(aTP_Lamm_nw(Uu),Uua),Uuc),Uud),Uue),Uub) ).

% ATP.lambda_1044
tff(fact_11222_ATP_Olambda__1045,axiom,
    ! [Uu: fun_VE8516060933855021849on_nat,Uua: nat,Uub: option4927543243414619207at_nat,Uuc: nat,Uud: array_VEBT_VEBTi,Uue: vEBT_VEBTi] : aa_VEB2711320435582112505on_nat(aa_arr2955191963898321370on_nat(aa_nat5317369504323972463on_nat(aa_opt8624683241416710969on_nat(aa_nat3457676333800472698on_nat(aTP_Lamm_nr(Uu),Uua),Uub),Uuc),Uud),Uue) = case_o8344607093967974880at_nat(heap_T3487192422709364219on_nat(none_nat),aa_VEB7519107058437837345on_nat(aa_arr2984472677043443222on_nat(aa_nat7372410979875760471on_nat(aa_nat8243565228001313504on_nat(aTP_Lamm_nq(Uu),Uua),Uuc),Uud),Uue),Uub) ).

% ATP.lambda_1045
tff(fact_11223_ATP_Olambda__1046,axiom,
    ! [Uu: vEBT_VEBT,Uua: nat,Uub: option4927543243414619207at_nat,Uuc: nat,Uud: array_VEBT_VEBTi,Uue: product_unit] : aa_Pro3379321066291785715Heap_o(aa_arr4590259283924850188Heap_o(aa_nat2073207392357089185Heap_o(aa_opt2919655305572010731Heap_o(aa_nat8312540953585231916Heap_o(aTP_Lamm_qv(Uu),Uua),Uub),Uuc),Uud),Uue) = case_o1442776274061689234at_nat(aa_o_H7399033255965945470Heap_o(heap_Time_return_o,$false),produc3505292621261808240Heap_o(aa_arr9046104431811898738Heap_o(aa_nat272996766647252915Heap_o(aa_opt8863613227715573503Heap_o(aa_nat8734498203866763412Heap_o(aTP_Lamm_qu(Uu),Uua),Uub),Uuc),Uud)),Uub) ).

% ATP.lambda_1046
tff(fact_11224_ATP_Olambda__1047,axiom,
    ! [Uu: fun_Pr238758252084143515Heap_o,Uua: nat,Uub: option4927543243414619207at_nat,Uuc: nat,Uud: array_VEBT_VEBTi,Uue: vEBT_VEBTi] : aa_VEB8194921341634505877Heap_o(aa_arr2373296168907467658Heap_o(aa_nat8650107858160254923Heap_o(aa_opt6539295502271640599Heap_o(aa_nat1227502447850154924Heap_o(aTP_Lamm_zn(Uu),Uua),Uub),Uuc),Uud),Uue) = case_o1442776274061689234at_nat(aa_o_H7399033255965945470Heap_o(heap_Time_return_o,$false),produc3505292621261808240Heap_o(aa_arr9046104431811898738Heap_o(aa_nat272996766647252915Heap_o(aa_nat7415925511494575164Heap_o(aTP_Lamm_zm(Uu),Uua),Uuc),Uud)),Uub) ).

% ATP.lambda_1047
tff(fact_11225_ATP_Olambda__1048,axiom,
    ! [Uu: fun_VE7134199479003659425Heap_o,Uua: nat,Uub: option4927543243414619207at_nat,Uuc: nat,Uud: array_VEBT_VEBTi,Uue: vEBT_VEBTi] : aa_VEB8194921341634505877Heap_o(aa_arr2373296168907467658Heap_o(aa_nat8650107858160254923Heap_o(aa_opt6539295502271640599Heap_o(aa_nat1227502447850154924Heap_o(aTP_Lamm_od(Uu),Uua),Uub),Uuc),Uud),Uue) = case_o1442776274061689234at_nat(aa_o_H7399033255965945470Heap_o(heap_Time_return_o,$false),produc3505292621261808240Heap_o(aa_arr9046104431811898738Heap_o(aa_nat272996766647252915Heap_o(aa_nat7415925511494575164Heap_o(aTP_Lamm_oc(Uu),Uua),Uuc),Uud)),Uub) ).

% ATP.lambda_1048
tff(fact_11226_ATP_Olambda__1049,axiom,
    ! [Uu: $o,Uua: $o,Uub: $o,Uuc: $o,Uud: code_integer,Uue: $o] : aa_o_char(aa_Cod7842134615204088097o_char(aa_o_f2742674633914528860o_char(aa_o_f9003794812476387647o_char(aa_o_f4183413333637126194o_char(aTP_Lamm_bck((Uu)),(Uua)),(Uub)),(Uuc)),Uud),(Uue)) = produc4188289175737317920o_char(aa_o_f2742674633914528860o_char(aa_o_f9003794812476387647o_char(aa_o_f4183413333637126194o_char(aa_o_f2932186792858379797o_char(aTP_Lamm_bcj((Uu)),(Uua)),(Uub)),(Uuc)),(Uue)),code_bit_cut_integer(Uud)) ).

% ATP.lambda_1049
tff(fact_11227_ATP_Olambda__1050,axiom,
    ! [Uu: nat,Uua: nat,Uub: array_VEBT_VEBTi,Uuc: vEBT_VEBTi,Uud: nat,Uue: nat] :
      aa_nat5408703056993926195_VEBTi(aa_nat4282640288315822012_VEBTi(aa_VEB7687887442520674464_VEBTi(aa_arr2561095334397224705_VEBTi(aa_nat8715118376710468630_VEBTi(aTP_Lamm_ja(Uu),Uua),Uub),Uuc),Uud),Uue) = heap_T844888390831797134_VEBTi($ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uu),Uud),aa_nat5821310369566504398ap_nat(heap_Time_return_nat,Uud),aa_nat5821310369566504398ap_nat(heap_Time_return_nat,Uu)),aa_nat4282640288315822012_VEBTi(aa_nat7653869184832469701_VEBTi(aa_VEB4322370096674025769_VEBTi(aa_arr4526045539431534346_VEBTi(aa_nat9189989511310912415_VEBTi(aTP_Lamm_iz(Uu),Uua),Uub),Uuc),Uud),Uue)) ).

% ATP.lambda_1050
tff(fact_11228_ATP_Olambda__1051,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat,Uud: array_VEBT_VEBTi,Uue: vEBT_VEBTi] :
      aa_VEB836174949890266638_VEBTi(aa_arr2912372873376718191_VEBTi(aa_nat6445230840460233604_VEBTi(aa_nat3780430034849525329_VEBTi(aa_nat6565337644025228958_VEBTi(aTP_Lamm_iq(Uu),Uua),Uub),Uuc),Uud),Uue) = heap_T844888390831797134_VEBTi($ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uua),Uub),aa_nat5821310369566504398ap_nat(heap_Time_return_nat,Uub),aa_nat5821310369566504398ap_nat(heap_Time_return_nat,Uua)),aa_VEB4833742232818206615_VEBTi(aa_arr6831033733615195768_VEBTi(aa_nat7652049589537018893_VEBTi(aTP_Lamm_ip(Uu),Uuc),Uud),Uue)) ).

% ATP.lambda_1051
tff(fact_11229_ATP_Olambda__1052,axiom,
    ! [Uu: nat,Uua: nat,Uub: array_VEBT_VEBTi,Uuc: vEBT_VEBTi,Uud: product_prod_nat_nat,Uue: nat] : aa_nat947733127081326238on_nat(aa_Pro4465521767867336112on_nat(aa_VEB1870382382330055850on_nat(aa_arr666314619328815135on_nat(aa_nat8253036615571765984on_nat(aTP_Lamm_ew(Uu),Uua),Uub),Uuc),Uud),Uue) = heap_T8222160169144143993on_nat(aa_nat5821310369566504398ap_nat(vEBT_VEBT_highi(Uu),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uua),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat6020981482381522343on_nat(aa_Pro8919772687891686969on_nat(aa_VEB358289369652911795on_nat(aa_arr4041137963511513512on_nat(aa_nat1363181691939120105on_nat(aTP_Lamm_ev(Uu),Uua),Uub),Uuc),Uud),Uue)) ).

% ATP.lambda_1052
tff(fact_11230_ATP_Olambda__1053,axiom,
    ! [Uu: fun_Pr940977554174145406Heap_o,Uua: nat,Uub: nat,Uuc: array_VEBT_VEBTi,Uud: list_VEBT_VEBT,Uue: nat] : aa_nat2910048432523430448Heap_o(aa_lis5083133406047025005Heap_o(aa_arr6185401592279617542Heap_o(aa_nat1466035401240590747Heap_o(aa_nat7251126122756508648Heap_o(aTP_Lamm_aac(Uu),Uua),Uub),Uuc),Uud),Uue) = heap_Time_bind_nat_o(aa_nat5821310369566504398ap_nat(vEBT_VEBT_lowi(Uua),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uub),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat5592791454563588861Heap_o(aa_lis455817436297872826Heap_o(aa_arr1217223296832173011Heap_o(aa_nat6014765364524370664Heap_o(aa_nat4979062449914608309Heap_o(aTP_Lamm_aab(Uu),Uua),Uub),Uuc),Uud),Uue)) ).

% ATP.lambda_1053
tff(fact_11231_ATP_Olambda__1054,axiom,
    ! [Uu: fun_VE8181444127144381288Heap_o,Uua: nat,Uub: nat,Uuc: array_VEBT_VEBTi,Uud: list_VEBT_VEBT,Uue: nat] : aa_nat2910048432523430448Heap_o(aa_lis5083133406047025005Heap_o(aa_arr6185401592279617542Heap_o(aa_nat1466035401240590747Heap_o(aa_nat7251126122756508648Heap_o(aTP_Lamm_qb(Uu),Uua),Uub),Uuc),Uud),Uue) = heap_Time_bind_nat_o(aa_nat5821310369566504398ap_nat(vEBT_VEBT_lowi(Uua),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uub),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat5592791454563588861Heap_o(aa_lis455817436297872826Heap_o(aa_arr1217223296832173011Heap_o(aa_nat6014765364524370664Heap_o(aa_nat4979062449914608309Heap_o(aTP_Lamm_qa(Uu),Uua),Uub),Uuc),Uud),Uue)) ).

% ATP.lambda_1054
tff(fact_11232_ATP_Olambda__1055,axiom,
    ! [Uu: fun_Pr342922516890243977on_nat,Uua: nat,Uub: nat,Uuc: array_VEBT_VEBTi,Uud: vEBT_VEBTi,Uue: nat] : aa_nat947733127081326238on_nat(aa_VEB3566417204837707778on_nat(aa_arr5230852142804057699on_nat(aa_nat6371187903253094776on_nat(aa_nat1535423446758346053on_nat(aTP_Lamm_aax(Uu),Uua),Uub),Uuc),Uud),Uue) = heap_T8222160169144143993on_nat(aa_nat5821310369566504398ap_nat(vEBT_VEBT_highi(Uua),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uub),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat6020981482381522343on_nat(aa_VEB6424020099669959819on_nat(aa_arr7085280839797930604on_nat(aa_nat1161784880827100417on_nat(aTP_Lamm_aaw(Uu),Uub),Uuc),Uud),Uue)) ).

% ATP.lambda_1055
tff(fact_11233_ATP_Olambda__1056,axiom,
    ! [Uu: fun_Pr940977554174145406Heap_o,Uua: nat,Uub: nat,Uuc: array_VEBT_VEBTi,Uud: list_VEBT_VEBT,Uue: product_unit] : aa_Pro3379321066291785715Heap_o(aa_lis4443831354813659076Heap_o(aa_arr8974760139355794305Heap_o(aa_nat808430574047188546Heap_o(aa_nat9059937328556229451Heap_o(aTP_Lamm_aad(Uu),Uua),Uub),Uuc),Uud),Uue) = heap_Time_bind_nat_o(aa_nat5821310369566504398ap_nat(vEBT_VEBT_highi(Uua),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uub),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_lis5083133406047025005Heap_o(aa_arr6185401592279617542Heap_o(aa_nat1466035401240590747Heap_o(aa_nat7251126122756508648Heap_o(aTP_Lamm_aac(Uu),Uua),Uub),Uuc),Uud)) ).

% ATP.lambda_1056
tff(fact_11234_ATP_Olambda__1057,axiom,
    ! [Uu: fun_VE8181444127144381288Heap_o,Uua: nat,Uub: nat,Uuc: array_VEBT_VEBTi,Uud: list_VEBT_VEBT,Uue: product_unit] : aa_Pro3379321066291785715Heap_o(aa_lis4443831354813659076Heap_o(aa_arr8974760139355794305Heap_o(aa_nat808430574047188546Heap_o(aa_nat9059937328556229451Heap_o(aTP_Lamm_qc(Uu),Uua),Uub),Uuc),Uud),Uue) = heap_Time_bind_nat_o(aa_nat5821310369566504398ap_nat(vEBT_VEBT_highi(Uua),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uub),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_lis5083133406047025005Heap_o(aa_arr6185401592279617542Heap_o(aa_nat1466035401240590747Heap_o(aa_nat7251126122756508648Heap_o(aTP_Lamm_qb(Uu),Uua),Uub),Uuc),Uud)) ).

% ATP.lambda_1057
tff(fact_11235_ATP_Olambda__1058,axiom,
    ! [Uu: fun_VE8516060933855021849on_nat,Uua: nat,Uub: nat,Uuc: array_VEBT_VEBTi,Uud: vEBT_VEBTi,Uue: nat] : aa_nat947733127081326238on_nat(aa_VEB3566417204837707778on_nat(aa_arr5230852142804057699on_nat(aa_nat6371187903253094776on_nat(aa_nat1535423446758346053on_nat(aTP_Lamm_nv(Uu),Uua),Uub),Uuc),Uud),Uue) = heap_T8222160169144143993on_nat(aa_nat5821310369566504398ap_nat(vEBT_VEBT_highi(Uua),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uub),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat6020981482381522343on_nat(aa_VEB6424020099669959819on_nat(aa_arr7085280839797930604on_nat(aa_nat1161784880827100417on_nat(aTP_Lamm_nu(Uu),Uub),Uuc),Uud),Uue)) ).

% ATP.lambda_1058
tff(fact_11236_ATP_Olambda__1059,axiom,
    ! [Uu: fun_Pr342922516890243977on_nat,Uua: nat,Uub: array_VEBT_VEBTi,Uuc: vEBT_VEBTi,Uud: nat,Uue: nat] : aa_nat947733127081326238on_nat(aa_nat6020981482381522343on_nat(aa_VEB6424020099669959819on_nat(aa_arr7085280839797930604on_nat(aa_nat1161784880827100417on_nat(aTP_Lamm_aaw(Uu),Uua),Uub),Uuc),Uud),Uue) = heap_T2868974464944644318on_nat(aa_nat5408703056993926195_VEBTi(array_nth_VEBT_VEBTi(Uub),Uue),aa_nat4669826966810537986on_nat(aa_nat107244183536939787on_nat(aa_VEB216942459957533935on_nat(aa_arr2558136924908760784on_nat(aa_nat402565781209047397on_nat(aTP_Lamm_aav(Uu),Uua),Uub),Uuc),Uud),Uue)) ).

% ATP.lambda_1059
tff(fact_11237_ATP_Olambda__1060,axiom,
    ! [Uu: fun_VE8516060933855021849on_nat,Uua: nat,Uub: array_VEBT_VEBTi,Uuc: vEBT_VEBTi,Uud: nat,Uue: nat] : aa_nat947733127081326238on_nat(aa_nat6020981482381522343on_nat(aa_VEB6424020099669959819on_nat(aa_arr7085280839797930604on_nat(aa_nat1161784880827100417on_nat(aTP_Lamm_nu(Uu),Uua),Uub),Uuc),Uud),Uue) = heap_T2868974464944644318on_nat(aa_nat5408703056993926195_VEBTi(array_nth_VEBT_VEBTi(Uub),Uue),aa_nat4669826966810537986on_nat(aa_nat107244183536939787on_nat(aa_VEB216942459957533935on_nat(aa_arr2558136924908760784on_nat(aa_nat402565781209047397on_nat(aTP_Lamm_nt(Uu),Uua),Uub),Uuc),Uud),Uue)) ).

% ATP.lambda_1060
tff(fact_11238_ATP_Olambda__1061,axiom,
    ! [Uu: fun_Pr940977554174145406Heap_o,Uua: array_VEBT_VEBTi,Uub: list_VEBT_VEBT,Uuc: nat,Uud: nat,Uue: product_unit] : aa_Pro3379321066291785715Heap_o(aa_nat3811243159488243764Heap_o(aa_nat6178576981048102717Heap_o(aa_lis4519787659412806998Heap_o(aa_arr7619904339668570131Heap_o(aTP_Lamm_zx(Uu),Uua),Uub),Uuc),Uud),Uue) = heap_T3040810144269856602EBTi_o(aa_nat5408703056993926195_VEBTi(array_nth_VEBT_VEBTi(Uua),Uuc),aa_nat1164146675577659234Heap_o(aa_nat3058146694327238959Heap_o(aa_lis6732064841504156908Heap_o(aTP_Lamm_zw(Uu),Uub),Uuc),Uud)) ).

% ATP.lambda_1061
tff(fact_11239_ATP_Olambda__1062,axiom,
    ! [Uu: fun_VE8181444127144381288Heap_o,Uua: array_VEBT_VEBTi,Uub: list_VEBT_VEBT,Uuc: nat,Uud: nat,Uue: product_unit] : aa_Pro3379321066291785715Heap_o(aa_nat3811243159488243764Heap_o(aa_nat6178576981048102717Heap_o(aa_lis4519787659412806998Heap_o(aa_arr7619904339668570131Heap_o(aTP_Lamm_pw(Uu),Uua),Uub),Uuc),Uud),Uue) = heap_T3040810144269856602EBTi_o(aa_nat5408703056993926195_VEBTi(array_nth_VEBT_VEBTi(Uua),Uuc),aa_nat1164146675577659234Heap_o(aa_nat3058146694327238959Heap_o(aa_lis6732064841504156908Heap_o(aTP_Lamm_pv(Uu),Uub),Uuc),Uud)) ).

% ATP.lambda_1062
tff(fact_11240_ATP_Olambda__1063,axiom,
    ! [Uu: nat,Uua: nat,Uub: array_VEBT_VEBTi,Uuc: list_VEBT_VEBT,Uud: nat,Uue: nat] :
      aa_nat2910048432523430448Heap_o(aa_nat5592791454563588861Heap_o(aa_lis455817436297872826Heap_o(aa_arr1217223296832173011Heap_o(aa_nat6014765364524370664Heap_o(aTP_Lamm_qo(Uu),Uua),Uub),Uuc),Uud),Uue) = heap_T7577940988442287570unit_o(aa_o_H2459438917461573107t_unit(refine_Imp_assert,
          ( ( Uue = aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Uu),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uua),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) )
          & ( Uud = aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Uu),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uua),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) )),aa_nat3811243159488243764Heap_o(aa_nat6178576981048102717Heap_o(aa_lis4519787659412806998Heap_o(aa_arr7619904339668570131Heap_o(aa_nat1269258765096487380Heap_o(aTP_Lamm_qn(Uu),Uua),Uub),Uuc),Uud),Uue)) ).

% ATP.lambda_1063
tff(fact_11241_ATP_Olambda__1064,axiom,
    ! [Uu: product_prod_nat_nat,Uua: nat,Uub: nat,Uuc: vEBT_VEBTi,Uud: vEBT_VEBT,Uue: option_nat] : aa_opt9163293676379396302on_nat(aa_VEB3755160987679246317on_nat(aa_VEB5212316371209427943on_nat(aa_nat7793430520228346548on_nat(aa_nat8410684691732023425on_nat(aTP_Lamm_af(Uu),Uua),Uub),Uuc),Uud),Uue) = heap_T5999496708990702694on_nat(aa_o_H2459438917461573107t_unit(refine_Imp_assert,Uue = vEBT_vebt_mint(Uud)),aa_opt5089197864053947520on_nat(aa_VEB2490029800597546679on_nat(aa_VEB1055358423767010003on_nat(aa_nat4144897467549153500on_nat(aa_nat7557481500408122597on_nat(aTP_Lamm_ae(Uu),Uua),Uub),Uuc),Uud),Uue)) ).

% ATP.lambda_1064
tff(fact_11242_ATP_Olambda__1065,axiom,
    ! [Uu: vEBT_VEBT,Uua: nat,Uub: option4927543243414619207at_nat,Uuc: nat,Uud: array_VEBT_VEBTi,Uue: vEBT_VEBTi] : aa_VEB2711320435582112505on_nat(aa_arr2955191963898321370on_nat(aa_nat5317369504323972463on_nat(aa_opt8624683241416710969on_nat(aa_nat3457676333800472698on_nat(aTP_Lamm_ul(Uu),Uua),Uub),Uuc),Uud),Uue) = heap_T5999496708990702694on_nat(aa_o_H2459438917461573107t_unit(refine_Imp_assert,aa_VEBT_VEBT_o(vEBT_is_Node,Uu)),aa_VEB7937681756907646997on_nat(aa_arr5896957451705369866on_nat(aa_nat3269078897214415691on_nat(aa_opt5459845657693669015on_nat(aa_nat8168140587832489004on_nat(aTP_Lamm_uk(Uu),Uua),Uub),Uuc),Uud),Uue)) ).

% ATP.lambda_1065
tff(fact_11243_ATP_Olambda__1066,axiom,
    ! [Uu: vEBT_VEBT,Uua: nat,Uub: option4927543243414619207at_nat,Uuc: nat,Uud: array_VEBT_VEBTi,Uue: vEBT_VEBTi] : aa_VEB2711320435582112505on_nat(aa_arr2955191963898321370on_nat(aa_nat5317369504323972463on_nat(aa_opt8624683241416710969on_nat(aa_nat3457676333800472698on_nat(aTP_Lamm_ow(Uu),Uua),Uub),Uuc),Uud),Uue) = heap_T5999496708990702694on_nat(aa_o_H2459438917461573107t_unit(refine_Imp_assert,aa_VEBT_VEBT_o(vEBT_is_Node,Uu)),aa_VEB7937681756907646997on_nat(aa_arr5896957451705369866on_nat(aa_nat3269078897214415691on_nat(aa_opt5459845657693669015on_nat(aa_nat8168140587832489004on_nat(aTP_Lamm_ov(Uu),Uua),Uub),Uuc),Uud),Uue)) ).

% ATP.lambda_1066
tff(fact_11244_ATP_Olambda__1067,axiom,
    ! [Uu: vEBT_VEBT,Uua: nat,Uub: option4927543243414619207at_nat,Uuc: nat,Uud: array_VEBT_VEBTi,Uue: vEBT_VEBTi] : aa_VEB8194921341634505877Heap_o(aa_arr2373296168907467658Heap_o(aa_nat8650107858160254923Heap_o(aa_opt6539295502271640599Heap_o(aa_nat1227502447850154924Heap_o(aTP_Lamm_qw(Uu),Uua),Uub),Uuc),Uud),Uue) = heap_T7577940988442287570unit_o(aa_o_H2459438917461573107t_unit(refine_Imp_assert,aa_VEBT_VEBT_o(vEBT_is_Node,Uu)),aa_arr4590259283924850188Heap_o(aa_nat2073207392357089185Heap_o(aa_opt2919655305572010731Heap_o(aa_nat8312540953585231916Heap_o(aTP_Lamm_qv(Uu),Uua),Uub),Uuc),Uud)) ).

% ATP.lambda_1067
tff(fact_11245_ATP_Olambda__1068,axiom,
    ! [Uu: nat,Uua: nat,Uub: array_VEBT_VEBTi,Uuc: vEBT_VEBTi,Uud: product_prod_nat_nat,Uue: nat] : aa_nat5408703056993926195_VEBTi(aa_Pro3150526172571800645_VEBTi(aa_VEB4890588985164552255_VEBTi(aa_arr5592147402706875956_VEBTi(aa_nat5963232373128642933_VEBTi(aTP_Lamm_jb(Uu),Uua),Uub),Uuc),Uud),Uue) = heap_T844888390831797134_VEBTi(aa_nat5821310369566504398ap_nat(heap_Time_return_nat,aa_Pro4243173770044397973at_nat(product_snd_nat_nat,Uud)),aa_nat4282640288315822012_VEBTi(aa_VEB7687887442520674464_VEBTi(aa_arr2561095334397224705_VEBTi(aa_nat8715118376710468630_VEBTi(aTP_Lamm_ja(Uu),Uua),Uub),Uuc),Uue)) ).

% ATP.lambda_1068
tff(fact_11246_ATP_Olambda__1069,axiom,
    ! [Uu: nat,Uua: array_VEBT_VEBTi,Uub: vEBT_VEBTi,Uuc: nat,Uud: nat,Uue: vEBT_VEBTi] : aa_VEB2711320435582112505on_nat(aa_nat4669826966810537986on_nat(aa_nat107244183536939787on_nat(aa_VEB216942459957533935on_nat(aa_arr2558136924908760784on_nat(aTP_Lamm_fd(Uu),Uua),Uub),Uuc),Uud),Uue) = heap_T3669509953089699273on_nat(aa_VEB2711320435582112505on_nat(vEBT_vebt_maxti,Uue),aa_VEB9161639482783362738on_nat(aa_nat6300058732544452155on_nat(aa_nat7962384762897155012on_nat(aa_VEB4635808744058847656on_nat(aa_arr3352113509897533961on_nat(aTP_Lamm_fc(Uu),Uua),Uub),Uuc),Uud),Uue)) ).

% ATP.lambda_1069
tff(fact_11247_ATP_Olambda__1070,axiom,
    ! [Uu: fun_He8247729123535230979_VEBTi,Uua: fun_op5127434927081926045_VEBTi,Uub: option4927543243414619207at_nat,Uuc: nat,Uud: array_VEBT_VEBTi,Uue: vEBT_VEBTi] : aa_VEB836174949890266638_VEBTi(aa_arr2912372873376718191_VEBTi(aa_nat6445230840460233604_VEBTi(aa_opt5182041009420754510_VEBTi(aa_fun4118786628394255868_VEBTi(aTP_Lamm_fr(Uu),Uua),Uub),Uuc),Uud),Uue) = aa_Hea2545666323642607868_VEBTi(Uu,aa_VEB836174949890266638_VEBTi(aa_arr2912372873376718191_VEBTi(aa_nat6445230840460233604_VEBTi(aa_opt5182041009420754510_VEBTi(Uua,Uub),Uuc),Uud),Uue)) ).

% ATP.lambda_1070
tff(fact_11248_ATP_Olambda__1071,axiom,
    ! [Uu: fun_He1783186893022644846on_nat,Uua: fun_op5127434927081926045_VEBTi,Uub: option4927543243414619207at_nat,Uuc: nat,Uud: array_VEBT_VEBTi,Uue: vEBT_VEBTi] : aa_VEB2711320435582112505on_nat(aa_arr2955191963898321370on_nat(aa_nat5317369504323972463on_nat(aa_opt8624683241416710969on_nat(aa_fun7829374822385380327on_nat(aTP_Lamm_fp(Uu),Uua),Uub),Uuc),Uud),Uue) = aa_Hea6259128614175166183on_nat(Uu,aa_VEB836174949890266638_VEBTi(aa_arr2912372873376718191_VEBTi(aa_nat6445230840460233604_VEBTi(aa_opt5182041009420754510_VEBTi(Uua,Uub),Uuc),Uud),Uue)) ).

% ATP.lambda_1071
tff(fact_11249_ATP_Olambda__1072,axiom,
    ! [Uu: fun_He4624006784692404214Heap_o,Uua: fun_op5127434927081926045_VEBTi,Uub: option4927543243414619207at_nat,Uuc: nat,Uud: array_VEBT_VEBTi,Uue: vEBT_VEBTi] : aa_VEB8194921341634505877Heap_o(aa_arr2373296168907467658Heap_o(aa_nat8650107858160254923Heap_o(aa_opt6539295502271640599Heap_o(aa_fun7400618842136392575Heap_o(aTP_Lamm_ft(Uu),Uua),Uub),Uuc),Uud),Uue) = aa_Hea7612170549154561063Heap_o(Uu,aa_VEB836174949890266638_VEBTi(aa_arr2912372873376718191_VEBTi(aa_nat6445230840460233604_VEBTi(aa_opt5182041009420754510_VEBTi(Uua,Uub),Uuc),Uud),Uue)) ).

% ATP.lambda_1072
tff(fact_11250_ATP_Olambda__1073,axiom,
    ! [Uu: fun_He2623223239018661208_VEBTi,Uua: fun_op5783775701855621896on_nat,Uub: option4927543243414619207at_nat,Uuc: nat,Uud: array_VEBT_VEBTi,Uue: vEBT_VEBTi] : aa_VEB836174949890266638_VEBTi(aa_arr2912372873376718191_VEBTi(aa_nat6445230840460233604_VEBTi(aa_opt5182041009420754510_VEBTi(aa_fun8651712798330174033_VEBTi(aTP_Lamm_fl(Uu),Uua),Uub),Uuc),Uud),Uue) = aa_Hea2832269215923769937_VEBTi(Uu,aa_VEB2711320435582112505on_nat(aa_arr2955191963898321370on_nat(aa_nat5317369504323972463on_nat(aa_opt8624683241416710969on_nat(Uua,Uub),Uuc),Uud),Uue)) ).

% ATP.lambda_1073
tff(fact_11251_ATP_Olambda__1074,axiom,
    ! [Uu: fun_He8640631688564546883on_nat,Uua: fun_op5783775701855621896on_nat,Uub: option4927543243414619207at_nat,Uuc: nat,Uud: array_VEBT_VEBTi,Uue: vEBT_VEBTi] : aa_VEB2711320435582112505on_nat(aa_arr2955191963898321370on_nat(aa_nat5317369504323972463on_nat(aa_opt8624683241416710969on_nat(aa_fun5534811901068199356on_nat(aTP_Lamm_fj(Uu),Uua),Uub),Uuc),Uud),Uue) = aa_Hea6597239020403563964on_nat(Uu,aa_VEB2711320435582112505on_nat(aa_arr2955191963898321370on_nat(aa_nat5317369504323972463on_nat(aa_opt8624683241416710969on_nat(Uua,Uub),Uuc),Uud),Uue)) ).

% ATP.lambda_1074
tff(fact_11252_ATP_Olambda__1075,axiom,
    ! [Uu: fun_He5988046180898962017Heap_o,Uua: fun_op5783775701855621896on_nat,Uub: option4927543243414619207at_nat,Uuc: nat,Uud: array_VEBT_VEBTi,Uue: vEBT_VEBTi] : aa_VEB8194921341634505877Heap_o(aa_arr2373296168907467658Heap_o(aa_nat8650107858160254923Heap_o(aa_opt6539295502271640599Heap_o(aa_fun836855539625404138Heap_o(aTP_Lamm_fn(Uu),Uua),Uub),Uuc),Uud),Uue) = aa_Hea662945835469240978Heap_o(Uu,aa_VEB2711320435582112505on_nat(aa_arr2955191963898321370on_nat(aa_nat5317369504323972463on_nat(aa_opt8624683241416710969on_nat(Uua,Uub),Uuc),Uud),Uue)) ).

% ATP.lambda_1075
tff(fact_11253_ATP_Olambda__1076,axiom,
    ! [Uu: fun_He6517945810935171576_VEBTi,Uua: fun_op1763656958002597022Heap_o,Uub: option4927543243414619207at_nat,Uuc: nat,Uud: array_VEBT_VEBTi,Uue: vEBT_VEBTi] : aa_VEB836174949890266638_VEBTi(aa_arr2912372873376718191_VEBTi(aa_nat6445230840460233604_VEBTi(aa_opt5182041009420754510_VEBTi(aa_fun7082494938152799553_VEBTi(aTP_Lamm_fx(Uu),Uua),Uub),Uuc),Uud),Uue) = aa_Hea4465279032261640929_VEBTi(Uu,aa_VEB8194921341634505877Heap_o(aa_arr2373296168907467658Heap_o(aa_nat8650107858160254923Heap_o(aa_opt6539295502271640599Heap_o(Uua,Uub),Uuc),Uud),Uue)) ).

% ATP.lambda_1076
tff(fact_11254_ATP_Olambda__1077,axiom,
    ! [Uu: fun_He6140748070426431971on_nat,Uua: fun_op1763656958002597022Heap_o,Uub: option4927543243414619207at_nat,Uuc: nat,Uud: array_VEBT_VEBTi,Uue: vEBT_VEBTi] : aa_VEB2711320435582112505on_nat(aa_arr2955191963898321370on_nat(aa_nat5317369504323972463on_nat(aa_opt8624683241416710969on_nat(aa_fun878802566620225708on_nat(aTP_Lamm_fv(Uu),Uua),Uub),Uuc),Uud),Uue) = aa_Hea41712926108436044on_nat(Uu,aa_VEB8194921341634505877Heap_o(aa_arr2373296168907467658Heap_o(aa_nat8650107858160254923Heap_o(aa_opt6539295502271640599Heap_o(Uua,Uub),Uuc),Uud),Uue)) ).

% ATP.lambda_1077
tff(fact_11255_ATP_Olambda__1078,axiom,
    ! [Uu: fun_He6597989805181038017Heap_o,Uua: fun_op1763656958002597022Heap_o,Uub: option4927543243414619207at_nat,Uuc: nat,Uud: array_VEBT_VEBTi,Uue: vEBT_VEBTi] : aa_VEB8194921341634505877Heap_o(aa_arr2373296168907467658Heap_o(aa_nat8650107858160254923Heap_o(aa_opt6539295502271640599Heap_o(aa_fun2115832704338266106Heap_o(aTP_Lamm_fz(Uu),Uua),Uub),Uuc),Uud),Uue) = aa_Hea4824394052061459970Heap_o(Uu,aa_VEB8194921341634505877Heap_o(aa_arr2373296168907467658Heap_o(aa_nat8650107858160254923Heap_o(aa_opt6539295502271640599Heap_o(Uua,Uub),Uuc),Uud),Uue)) ).

% ATP.lambda_1078
tff(fact_11256_ATP_Olambda__1079,axiom,
    ! [Uu: fun_Pr3650993782763339253T_VEBT,Uua: fun_op8229906066235313459T_VEBT,Uub: option4927543243414619207at_nat,Uuc: nat,Uud: list_VEBT_VEBT,Uue: vEBT_VEBT] : aa_VEB8692087746923902260T_VEBT(aa_lis2966823273985285125T_VEBT(aa_nat6705503460730637082T_VEBT(aa_opt2902932101092860132T_VEBT(aa_fun7544404396881244796T_VEBT(aTP_Lamm_pg(Uu),Uua),Uub),Uuc),Uud),Uue) = aa_Pro5421721056952420150T_VEBT(Uu,aa_VEB8692087746923902260T_VEBT(aa_lis2966823273985285125T_VEBT(aa_nat6705503460730637082T_VEBT(aa_opt2902932101092860132T_VEBT(Uua,Uub),Uuc),Uud),Uue)) ).

% ATP.lambda_1079
tff(fact_11257_ATP_Olambda__1080,axiom,
    ! [Uu: fun_Pr7471953115884902709VEBT_o,Uua: fun_op8229906066235313459T_VEBT,Uub: option4927543243414619207at_nat,Uuc: nat,Uud: list_VEBT_VEBT,Uue: vEBT_VEBT] :
      ( aa_VEBT_VEBT_o(aa_lis4184305612057247525VEBT_o(aa_nat8798449810042143974VEBT_o(aa_opt7072554309604550194VEBT_o(aa_fun5124765266653172784VEBT_o(aTP_Lamm_pi(Uu),Uua),Uub),Uuc),Uud),Uue)
    <=> aa_Pro5475924895142249886VEBT_o(Uu,aa_VEB8692087746923902260T_VEBT(aa_lis2966823273985285125T_VEBT(aa_nat6705503460730637082T_VEBT(aa_opt2902932101092860132T_VEBT(Uua,Uub),Uuc),Uud),Uue)) ) ).

% ATP.lambda_1080
tff(fact_11258_ATP_Olambda__1081,axiom,
    ! [Uu: fun_o_3001263932818058983T_VEBT,Uua: fun_op541075054904862521VEBT_o,Uub: option4927543243414619207at_nat,Uuc: nat,Uud: list_VEBT_VEBT,Uue: vEBT_VEBT] : aa_VEB8692087746923902260T_VEBT(aa_lis2966823273985285125T_VEBT(aa_nat6705503460730637082T_VEBT(aa_opt2902932101092860132T_VEBT(aa_fun8362160726713724924T_VEBT(aTP_Lamm_pk(Uu),Uua),Uub),Uuc),Uud),Uue) = aa_o_P8223745691486049304T_VEBT(Uu,aa_VEBT_VEBT_o(aa_lis4184305612057247525VEBT_o(aa_nat8798449810042143974VEBT_o(aa_opt7072554309604550194VEBT_o(Uua,Uub),Uuc),Uud),Uue)) ).

% ATP.lambda_1081
tff(fact_11259_ATP_Olambda__1082,axiom,
    ! [Uu: fun_o_o,Uua: fun_op541075054904862521VEBT_o,Uub: option4927543243414619207at_nat,Uuc: nat,Uud: list_VEBT_VEBT,Uue: vEBT_VEBT] :
      ( aa_VEBT_VEBT_o(aa_lis4184305612057247525VEBT_o(aa_nat8798449810042143974VEBT_o(aa_opt7072554309604550194VEBT_o(aa_fun2455880738235528880VEBT_o(aTP_Lamm_pm(Uu),Uua),Uub),Uuc),Uud),Uue)
    <=> aa_o_o(Uu,aa_VEBT_VEBT_o(aa_lis4184305612057247525VEBT_o(aa_nat8798449810042143974VEBT_o(aa_opt7072554309604550194VEBT_o(Uua,Uub),Uuc),Uud),Uue)) ) ).

% ATP.lambda_1082
tff(fact_11260_ATP_Olambda__1083,axiom,
    ! [Uu: fun_Pr940977554174145406Heap_o,Uua: nat,Uub: option4927543243414619207at_nat,Uuc: nat,Uud: array_VEBT_VEBTi,Uue: option4927543243414619207at_nat] : aa_opt6266093870483638640Heap_o(aa_arr6198139058929667337Heap_o(aa_nat436397609992601886Heap_o(aa_opt4438414422311449448Heap_o(aa_nat1884043827438135593Heap_o(aTP_Lamm_aag(Uu),Uua),Uub),Uuc),Uud),Uue) = produc5946672270950774087Heap_o(aa_opt7844807959364121492Heap_o(aa_arr79663438428320465Heap_o(aa_nat2087714105643485586Heap_o(aa_opt9028207981337828190Heap_o(aa_nat2171064521259968883Heap_o(aTP_Lamm_aaf(Uu),Uua),Uub),Uuc),Uud),Uue)) ).

% ATP.lambda_1083
tff(fact_11261_ATP_Olambda__1084,axiom,
    ! [Uu: nat,Uua: option4927543243414619207at_nat,Uub: nat,Uuc: array_VEBT_VEBTi,Uud: option4927543243414619207at_nat,Uue: nat] : aa_nat6527088363260489928Heap_o(aa_opt7844807959364121492Heap_o(aa_arr79663438428320465Heap_o(aa_nat2087714105643485586Heap_o(aa_opt9028207981337828190Heap_o(aTP_Lamm_qs(Uu),Uua),Uub),Uuc),Uud),Uue) = produc5048428016959714504Heap_o(aa_nat3831830007592743808Heap_o(aa_opt3748239118003421002Heap_o(aa_arr581739785191926499Heap_o(aa_nat105892859602166008Heap_o(aa_opt8417070463701818562Heap_o(aTP_Lamm_qr(Uu),Uua),Uub),Uuc),Uud),Uue)) ).

% ATP.lambda_1084
tff(fact_11262_ATP_Olambda__1085,axiom,
    ! [Uu: fun_VE8181444127144381288Heap_o,Uua: nat,Uub: option4927543243414619207at_nat,Uuc: nat,Uud: array_VEBT_VEBTi,Uue: option4927543243414619207at_nat] : aa_opt6266093870483638640Heap_o(aa_arr6198139058929667337Heap_o(aa_nat436397609992601886Heap_o(aa_opt4438414422311449448Heap_o(aa_nat1884043827438135593Heap_o(aTP_Lamm_qf(Uu),Uua),Uub),Uuc),Uud),Uue) = produc5946672270950774087Heap_o(aa_opt7844807959364121492Heap_o(aa_arr79663438428320465Heap_o(aa_nat2087714105643485586Heap_o(aa_opt9028207981337828190Heap_o(aa_nat2171064521259968883Heap_o(aTP_Lamm_qe(Uu),Uua),Uub),Uuc),Uud),Uue)) ).

% ATP.lambda_1085
tff(fact_11263_ATP_Olambda__1086,axiom,
    ! [Uu: fun_Pr4404027214751104635_VEBTi,Uua: vEBT_VEBT,Uub: nat,Uuc: option4927543243414619207at_nat,Uud: nat,Uue: array_VEBT_VEBTi,Uuf: vEBT_VEBTi] :
      aa_VEB836174949890266638_VEBTi(aa_arr2912372873376718191_VEBTi(aa_nat6445230840460233604_VEBTi(aa_opt5182041009420754510_VEBTi(aa_nat3009484633486537743_VEBTi(aa_VEB600817258366935214_VEBTi(aTP_Lamm_vq(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf) = case_o1356590567247012107at_nat($ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Uud),one_one_nat),aa_VEB836174949890266638_VEBTi(heap_T3630416162098727440_VEBTi,aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(Uuc,Uud,Uue),Uuf)),aa_VEB836174949890266638_VEBTi(heap_T3630416162098727440_VEBTi,aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Uub),Uub)),Uud,Uue),Uuf))),aa_VEB6393524116121717046_VEBTi(aa_arr8059685486658508715_VEBTi(aa_nat7594302058568047468_VEBTi(aa_opt7720280829887018808_VEBTi(aa_nat6605875721633857101_VEBTi(aa_VEB7645014810616801796_VEBTi(aTP_Lamm_vp(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uuc) ).

% ATP.lambda_1086
tff(fact_11264_ATP_Olambda__1087,axiom,
    ! [Uu: fun_VE2018681615261037117_VEBTi,Uua: vEBT_VEBT,Uub: nat,Uuc: option4927543243414619207at_nat,Uud: nat,Uue: array_VEBT_VEBTi,Uuf: vEBT_VEBTi] :
      aa_VEB836174949890266638_VEBTi(aa_arr2912372873376718191_VEBTi(aa_nat6445230840460233604_VEBTi(aa_opt5182041009420754510_VEBTi(aa_nat3009484633486537743_VEBTi(aa_VEB600817258366935214_VEBTi(aTP_Lamm_rt(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf) = case_o1356590567247012107at_nat($ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Uud),one_one_nat),aa_VEB836174949890266638_VEBTi(heap_T3630416162098727440_VEBTi,aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(Uuc,Uud,Uue),Uuf)),aa_VEB836174949890266638_VEBTi(heap_T3630416162098727440_VEBTi,aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Uub),Uub)),Uud,Uue),Uuf))),aa_VEB6393524116121717046_VEBTi(aa_arr8059685486658508715_VEBTi(aa_nat7594302058568047468_VEBTi(aa_opt7720280829887018808_VEBTi(aa_nat6605875721633857101_VEBTi(aa_VEB7645014810616801796_VEBTi(aTP_Lamm_rs(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uuc) ).

% ATP.lambda_1087
tff(fact_11265_ATP_Olambda__1088,axiom,
    ! [Uu: product_prod_nat_nat,Uua: nat,Uub: nat,Uuc: vEBT_VEBTi,Uud: vEBT_VEBT,Uue: option_nat,Uuf: product_unit] :
      aa_Pro451875010097498139on_nat(aa_opt5089197864053947520on_nat(aa_VEB2490029800597546679on_nat(aa_VEB1055358423767010003on_nat(aa_nat4144897467549153500on_nat(aa_nat7557481500408122597on_nat(aTP_Lamm_ae(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf) = $ite(
        ( ( Uue != none_nat )
        & aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_greater,aa_nat_option_nat(some_nat,Uua)),Uue) ),
        heap_T3669509953089699273on_nat(aa_nat947733127081326238on_nat(aa_VEB3566417204837707778on_nat(aa_VEB1030146501360450209on_nat(vEBT_VEBT_vebt_predi,Uud),Uuc),Uua),aTP_Lamm_a(Uub)),
        heap_T3669509953089699273on_nat(aa_nat947733127081326238on_nat(aa_VEB3566417204837707778on_nat(aa_VEB1030146501360450209on_nat(vEBT_VEBT_vebt_predi,summary),x14),Uub),aa_nat9127621225095157975on_nat(aTP_Lamm_ad(Uu),Uub)) ) ).

% ATP.lambda_1088
tff(fact_11266_ATP_Olambda__1089,axiom,
    ! [Uu: nat,Uua: array_VEBT_VEBTi,Uub: vEBT_VEBTi,Uuc: nat,Uud: nat,Uue: vEBT_VEBTi,Uuf: option_nat] :
      aa_opt9163293676379396302on_nat(aa_VEB9161639482783362738on_nat(aa_nat6300058732544452155on_nat(aa_nat7962384762897155012on_nat(aa_VEB4635808744058847656on_nat(aa_arr3352113509897533961on_nat(aTP_Lamm_fc(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf) = $ite(
        ( ( Uuf != none_nat )
        & aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_less,aa_nat_option_nat(some_nat,Uuc)),Uuf) ),
        heap_T3669509953089699273on_nat(aa_nat947733127081326238on_nat(aa_VEB3566417204837707778on_nat(vEBT_vebt_succi,Uue),Uuc),aa_nat9127621225095157975on_nat(aTP_Lamm_ep(Uu),Uud)),
        heap_T3669509953089699273on_nat(aa_nat947733127081326238on_nat(aa_VEB3566417204837707778on_nat(vEBT_vebt_succi,Uub),Uud),aa_arr3757929795460847279on_nat(aTP_Lamm_fb(Uu),Uua)) ) ).

% ATP.lambda_1089
tff(fact_11267_ATP_Olambda__1090,axiom,
    ! [Uu: vEBT_VEBT,Uua: nat,Uub: option4927543243414619207at_nat,Uuc: nat,Uud: array_VEBT_VEBTi,Uue: vEBT_VEBTi,Uuf: product_prod_nat_nat] :
      aa_Pro1395359565518501564_VEBTi(aa_VEB6393524116121717046_VEBTi(aa_arr8059685486658508715_VEBTi(aa_nat7594302058568047468_VEBTi(aa_opt7720280829887018808_VEBTi(aa_nat6605875721633857101_VEBTi(aTP_Lamm_tv(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Uuc),one_one_nat),aa_VEB836174949890266638_VEBTi(heap_T3630416162098727440_VEBTi,aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(Uub,Uuc,Uud),Uue)),heap_T7982501707604696571_VEBTi(aa_o_H2459438917461573107t_unit(refine_Imp_assert,aa_VEBT_VEBT_o(vEBT_is_Node,Uu)),aa_Pro8034723660955798012_VEBTi(aa_VEB4262372667711301016_VEBTi(aa_arr9036764603281787897_VEBTi(aa_nat2771851207891374350_VEBTi(aa_opt7594342031245826264_VEBTi(aa_nat4475881176255417753_VEBTi(aTP_Lamm_tu(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf))) ).

% ATP.lambda_1090
tff(fact_11268_ATP_Olambda__1091,axiom,
    ! [Uu: fun_Pr4171499318505651870_VEBTi,Uua: nat,Uub: option4927543243414619207at_nat,Uuc: nat,Uud: array_VEBT_VEBTi,Uue: vEBT_VEBTi,Uuf: product_prod_nat_nat] :
      aa_Pro1395359565518501564_VEBTi(aa_VEB6393524116121717046_VEBTi(aa_arr8059685486658508715_VEBTi(aa_nat7594302058568047468_VEBTi(aa_opt7720280829887018808_VEBTi(aa_nat6605875721633857101_VEBTi(aTP_Lamm_abn(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Uuc),one_one_nat),aa_VEB836174949890266638_VEBTi(heap_T3630416162098727440_VEBTi,aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(Uub,Uuc,Uud),Uue)),heap_T844888390831797134_VEBTi(aa_nat5821310369566504398ap_nat(heap_Time_return_nat,aa_Pro4243173770044397973at_nat(product_fst_nat_nat,Uuf)),aa_Pro3150526172571800645_VEBTi(aa_VEB4890588985164552255_VEBTi(aa_arr5592147402706875956_VEBTi(aa_nat5963232373128642933_VEBTi(aa_nat7459085003528029950_VEBTi(aTP_Lamm_abm(Uu),Uua),Uuc),Uud),Uue),Uuf))) ).

% ATP.lambda_1091
tff(fact_11269_ATP_Olambda__1092,axiom,
    ! [Uu: fun_VE8118064890490510894_VEBTi,Uua: nat,Uub: option4927543243414619207at_nat,Uuc: nat,Uud: array_VEBT_VEBTi,Uue: vEBT_VEBTi,Uuf: product_prod_nat_nat] :
      aa_Pro1395359565518501564_VEBTi(aa_VEB6393524116121717046_VEBTi(aa_arr8059685486658508715_VEBTi(aa_nat7594302058568047468_VEBTi(aa_opt7720280829887018808_VEBTi(aa_nat6605875721633857101_VEBTi(aTP_Lamm_nk(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Uuc),one_one_nat),aa_VEB836174949890266638_VEBTi(heap_T3630416162098727440_VEBTi,aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(Uub,Uuc,Uud),Uue)),heap_T844888390831797134_VEBTi(aa_nat5821310369566504398ap_nat(heap_Time_return_nat,aa_Pro4243173770044397973at_nat(product_fst_nat_nat,Uuf)),aa_Pro3150526172571800645_VEBTi(aa_VEB4890588985164552255_VEBTi(aa_arr5592147402706875956_VEBTi(aa_nat5963232373128642933_VEBTi(aa_nat7459085003528029950_VEBTi(aTP_Lamm_nj(Uu),Uua),Uuc),Uud),Uue),Uuf))) ).

% ATP.lambda_1092
tff(fact_11270_ATP_Olambda__1093,axiom,
    ! [Uu: nat,Uua: nat,Uub: array_VEBT_VEBTi,Uuc: vEBT_VEBTi,Uud: list_VEBT_VEBT,Uue: vEBT_VEBT,Uuf: product_prod_nat_nat] :
      aa_Pro7675874045790911655on_nat(aa_VEB6310895951922269862on_nat(aa_lis7455679124549436279on_nat(aa_VEB7601412409763292529on_nat(aa_arr3206634112104793958on_nat(aa_nat2842375153920543911on_nat(aTP_Lamm_uf(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Uua),one_one_nat),
        heap_T3487192422709364219on_nat(none_nat),
        $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_Pro4243173770044397973at_nat(product_snd_nat_nat,Uuf)),Uu),heap_T3487192422709364219on_nat(aa_nat_option_nat(some_nat,aa_Pro4243173770044397973at_nat(product_snd_nat_nat,Uuf))),heap_T8222160169144143993on_nat(aa_nat5821310369566504398ap_nat(vEBT_VEBT_lowi(Uu),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uua),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_Pro4465521767867336112on_nat(aa_VEB6043830584257997615on_nat(aa_lis5985300925819340672on_nat(aa_VEB1019093884839074554on_nat(aa_arr1617095457630768239on_nat(aa_nat5782952277816596784on_nat(aTP_Lamm_ue(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf))) ) ).

% ATP.lambda_1093
tff(fact_11271_ATP_Olambda__1094,axiom,
    ! [Uu: nat,Uua: nat,Uub: array_VEBT_VEBTi,Uuc: vEBT_VEBTi,Uud: list_VEBT_VEBT,Uue: vEBT_VEBT,Uuf: product_prod_nat_nat] :
      aa_Pro7675874045790911655on_nat(aa_VEB6310895951922269862on_nat(aa_lis7455679124549436279on_nat(aa_VEB7601412409763292529on_nat(aa_arr3206634112104793958on_nat(aa_nat2842375153920543911on_nat(aTP_Lamm_oo(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Uua),one_one_nat),
        heap_T3487192422709364219on_nat(none_nat),
        $ite(
          aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uu),aa_Pro4243173770044397973at_nat(product_fst_nat_nat,Uuf)),
          heap_T3487192422709364219on_nat(aa_nat_option_nat(some_nat,aa_Pro4243173770044397973at_nat(product_fst_nat_nat,Uuf))),
          $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_Pro4243173770044397973at_nat(product_snd_nat_nat,Uuf)),Uu),heap_T3487192422709364219on_nat(none_nat),heap_T8222160169144143993on_nat(aa_nat5821310369566504398ap_nat(vEBT_VEBT_lowi(Uu),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uua),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_VEB1312864128696253885on_nat(aa_lis1761857624600698434on_nat(aa_VEB4172908723204034014on_nat(aa_arr2072991348567353407on_nat(aa_nat7185663501778404180on_nat(aTP_Lamm_on(Uu),Uua),Uub),Uuc),Uud),Uue))) ) ) ).

% ATP.lambda_1094
tff(fact_11272_ATP_Olambda__1095,axiom,
    ! [Uu: vEBT_VEBT,Uua: nat,Uub: option4927543243414619207at_nat,Uuc: nat,Uud: array_VEBT_VEBTi,Uue: nat,Uuf: nat] :
      aa_nat2910048432523430448Heap_o(aa_nat5592791454563588861Heap_o(aa_arr9046104431811898738Heap_o(aa_nat272996766647252915Heap_o(aa_opt8863613227715573503Heap_o(aa_nat8734498203866763412Heap_o(aTP_Lamm_qu(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Uuc),one_one_nat),
        aa_o_H7399033255965945470Heap_o(heap_Time_return_o,$false),
        $ite(
          Uua = Uue,
          aa_o_H7399033255965945470Heap_o(heap_Time_return_o,$true),
          $ite(
            Uua = Uuf,
            aa_o_H7399033255965945470Heap_o(heap_Time_return_o,$true),
            $ite(
              aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uua),Uue),
              aa_o_H7399033255965945470Heap_o(heap_Time_return_o,$false),
              $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uuf),Uua),aa_o_H7399033255965945470Heap_o(heap_Time_return_o,$false),produc1330493526443650053Heap_o(aa_arr6198139058929667337Heap_o(aa_nat436397609992601886Heap_o(aa_opt4438414422311449448Heap_o(aTP_Lamm_qt(Uua),Uub),Uuc),Uud),vEBT_c634343235235684882T_VEBT(aTP_Lamm_ot,aTP_Lamm_ou,Uu))) ) ) ) ) ).

% ATP.lambda_1095
tff(fact_11273_ATP_Olambda__1096,axiom,
    ! [Uu: fun_Pr940977554174145406Heap_o,Uua: vEBT_VEBT,Uub: nat,Uuc: option4927543243414619207at_nat,Uud: nat,Uue: array_VEBT_VEBTi,Uuf: product_unit] : aa_Pro3379321066291785715Heap_o(aa_arr4590259283924850188Heap_o(aa_nat2073207392357089185Heap_o(aa_opt2919655305572010731Heap_o(aa_nat8312540953585231916Heap_o(aa_VEB1825396708860521035Heap_o(aTP_Lamm_aai(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf) = case_o1442776274061689234at_nat(aa_o_H7399033255965945470Heap_o(heap_Time_return_o,$false),produc3505292621261808240Heap_o(aa_arr9046104431811898738Heap_o(aa_nat272996766647252915Heap_o(aa_opt8863613227715573503Heap_o(aa_nat8734498203866763412Heap_o(aa_VEB8906299275298315467Heap_o(aTP_Lamm_aah(Uu),Uua),Uub),Uuc),Uud),Uue)),Uuc) ).

% ATP.lambda_1096
tff(fact_11274_ATP_Olambda__1097,axiom,
    ! [Uu: fun_VE8181444127144381288Heap_o,Uua: vEBT_VEBT,Uub: nat,Uuc: option4927543243414619207at_nat,Uud: nat,Uue: array_VEBT_VEBTi,Uuf: product_unit] : aa_Pro3379321066291785715Heap_o(aa_arr4590259283924850188Heap_o(aa_nat2073207392357089185Heap_o(aa_opt2919655305572010731Heap_o(aa_nat8312540953585231916Heap_o(aa_VEB1825396708860521035Heap_o(aTP_Lamm_qh(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf) = case_o1442776274061689234at_nat(aa_o_H7399033255965945470Heap_o(heap_Time_return_o,$false),produc3505292621261808240Heap_o(aa_arr9046104431811898738Heap_o(aa_nat272996766647252915Heap_o(aa_opt8863613227715573503Heap_o(aa_nat8734498203866763412Heap_o(aa_VEB8906299275298315467Heap_o(aTP_Lamm_qg(Uu),Uua),Uub),Uuc),Uud),Uue)),Uuc) ).

% ATP.lambda_1097
tff(fact_11275_ATP_Olambda__1098,axiom,
    ! [Uu: vEBT_VEBT,Uua: nat,Uub: option4927543243414619207at_nat,Uuc: nat,Uud: array_VEBT_VEBTi,Uue: vEBT_VEBTi,Uuf: product_unit] : aa_Pro451875010097498139on_nat(aa_VEB7937681756907646997on_nat(aa_arr5896957451705369866on_nat(aa_nat3269078897214415691on_nat(aa_opt5459845657693669015on_nat(aa_nat8168140587832489004on_nat(aTP_Lamm_uk(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf) = produc2190226783763740553on_nat(aa_VEB5714833192853586542on_nat(aa_arr7759188791488537571on_nat(aa_nat7273124837800733860on_nat(aa_opt1247461068388908528on_nat(aa_nat309676689105111045on_nat(aTP_Lamm_uj(Uu),Uua),Uub),Uuc),Uud),Uue),vEBT_c634343235235684882T_VEBT(aTP_Lamm_ot,aTP_Lamm_ou,Uu)) ).

% ATP.lambda_1098
tff(fact_11276_ATP_Olambda__1099,axiom,
    ! [Uu: vEBT_VEBT,Uua: nat,Uub: option4927543243414619207at_nat,Uuc: nat,Uud: array_VEBT_VEBTi,Uue: vEBT_VEBTi,Uuf: product_unit] : aa_Pro451875010097498139on_nat(aa_VEB7937681756907646997on_nat(aa_arr5896957451705369866on_nat(aa_nat3269078897214415691on_nat(aa_opt5459845657693669015on_nat(aa_nat8168140587832489004on_nat(aTP_Lamm_ov(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf) = produc2190226783763740553on_nat(aa_VEB5714833192853586542on_nat(aa_arr7759188791488537571on_nat(aa_nat7273124837800733860on_nat(aa_opt1247461068388908528on_nat(aa_nat309676689105111045on_nat(aTP_Lamm_os(Uu),Uua),Uub),Uuc),Uud),Uue),vEBT_c634343235235684882T_VEBT(aTP_Lamm_ot,aTP_Lamm_ou,Uu)) ).

% ATP.lambda_1099
tff(fact_11277_ATP_Olambda__1100,axiom,
    ! [Uu: $o,Uua: $o,Uub: $o,Uuc: $o,Uud: $o,Uue: code_integer,Uuf: $o] : aa_o_char(aa_Cod7842134615204088097o_char(aa_o_f2742674633914528860o_char(aa_o_f9003794812476387647o_char(aa_o_f4183413333637126194o_char(aa_o_f2932186792858379797o_char(aTP_Lamm_bcj((Uu)),(Uua)),(Uub)),(Uuc)),(Uud)),Uue),(Uuf)) = produc4188289175737317920o_char(aa_o_f2742674633914528860o_char(aa_o_f9003794812476387647o_char(aa_o_f4183413333637126194o_char(aa_o_f2932186792858379797o_char(aa_o_f1143307120959153672o_char(aTP_Lamm_bci((Uu)),(Uua)),(Uub)),(Uuc)),(Uud)),(Uuf)),code_bit_cut_integer(Uue)) ).

% ATP.lambda_1100
tff(fact_11278_ATP_Olambda__1101,axiom,
    ! [Uu: nat,Uua: nat,Uub: array_VEBT_VEBTi,Uuc: vEBT_VEBTi,Uud: nat,Uue: nat,Uuf: nat] :
      aa_nat5408703056993926195_VEBTi(aa_nat4282640288315822012_VEBTi(aa_nat7653869184832469701_VEBTi(aa_VEB4322370096674025769_VEBTi(aa_arr4526045539431534346_VEBTi(aa_nat9189989511310912415_VEBTi(aTP_Lamm_iz(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf) = heap_T844888390831797134_VEBTi($ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uu),Uud),aa_nat5821310369566504398ap_nat(heap_Time_return_nat,Uu),aa_nat5821310369566504398ap_nat(heap_Time_return_nat,Uud)),aa_nat4282640288315822012_VEBTi(aa_nat7653869184832469701_VEBTi(aa_nat2287630409448422734_VEBTi(aa_VEB8340470449056147762_VEBTi(aa_arr7322700162901495443_VEBTi(aa_nat4321419670308649640_VEBTi(aTP_Lamm_iy(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf)) ).

% ATP.lambda_1101
tff(fact_11279_ATP_Olambda__1102,axiom,
    ! [Uu: fun_Pr4171499318505651870_VEBTi,Uua: nat,Uub: nat,Uuc: array_VEBT_VEBTi,Uud: vEBT_VEBTi,Uue: nat,Uuf: nat] :
      aa_nat5408703056993926195_VEBTi(aa_nat4282640288315822012_VEBTi(aa_VEB7687887442520674464_VEBTi(aa_arr2561095334397224705_VEBTi(aa_nat8715118376710468630_VEBTi(aa_nat5853910169739908579_VEBTi(aTP_Lamm_abl(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf) = heap_T844888390831797134_VEBTi($ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uua),Uue),aa_nat5821310369566504398ap_nat(heap_Time_return_nat,Uue),aa_nat5821310369566504398ap_nat(heap_Time_return_nat,Uua)),aa_nat4282640288315822012_VEBTi(aa_nat7653869184832469701_VEBTi(aa_VEB4322370096674025769_VEBTi(aa_arr4526045539431534346_VEBTi(aa_nat9189989511310912415_VEBTi(aa_nat4025588890810027244_VEBTi(aTP_Lamm_abk(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf)) ).

% ATP.lambda_1102
tff(fact_11280_ATP_Olambda__1103,axiom,
    ! [Uu: fun_VE8118064890490510894_VEBTi,Uua: nat,Uub: nat,Uuc: array_VEBT_VEBTi,Uud: vEBT_VEBTi,Uue: nat,Uuf: nat] :
      aa_nat5408703056993926195_VEBTi(aa_nat4282640288315822012_VEBTi(aa_VEB7687887442520674464_VEBTi(aa_arr2561095334397224705_VEBTi(aa_nat8715118376710468630_VEBTi(aa_nat5853910169739908579_VEBTi(aTP_Lamm_ni(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf) = heap_T844888390831797134_VEBTi($ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uua),Uue),aa_nat5821310369566504398ap_nat(heap_Time_return_nat,Uue),aa_nat5821310369566504398ap_nat(heap_Time_return_nat,Uua)),aa_nat4282640288315822012_VEBTi(aa_nat7653869184832469701_VEBTi(aa_VEB4322370096674025769_VEBTi(aa_arr4526045539431534346_VEBTi(aa_nat9189989511310912415_VEBTi(aa_nat4025588890810027244_VEBTi(aTP_Lamm_nh(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf)) ).

% ATP.lambda_1103
tff(fact_11281_ATP_Olambda__1104,axiom,
    ! [Uu: nat,Uua: nat,Uub: array_VEBT_VEBTi,Uuc: vEBT_VEBTi,Uud: list_VEBT_VEBT,Uue: vEBT_VEBT,Uuf: nat] : aa_nat947733127081326238on_nat(aa_VEB1312864128696253885on_nat(aa_lis1761857624600698434on_nat(aa_VEB4172908723204034014on_nat(aa_arr2072991348567353407on_nat(aa_nat7185663501778404180on_nat(aTP_Lamm_on(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf) = heap_T8222160169144143993on_nat(aa_nat5821310369566504398ap_nat(vEBT_VEBT_highi(Uu),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uua),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat6020981482381522343on_nat(aa_VEB4460395522175602758on_nat(aa_lis6278591930053894219on_nat(aa_VEB6094637369740064103on_nat(aa_arr5396080262017029960on_nat(aa_nat7337942549949891037on_nat(aTP_Lamm_om(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf)) ).

% ATP.lambda_1104
tff(fact_11282_ATP_Olambda__1105,axiom,
    ! [Uu: fun_Pr342922516890243977on_nat,Uua: nat,Uub: nat,Uuc: array_VEBT_VEBTi,Uud: vEBT_VEBTi,Uue: product_prod_nat_nat,Uuf: nat] : aa_nat947733127081326238on_nat(aa_Pro4465521767867336112on_nat(aa_VEB1870382382330055850on_nat(aa_arr666314619328815135on_nat(aa_nat8253036615571765984on_nat(aa_nat2001721438062390249on_nat(aTP_Lamm_aap(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf) = heap_T8222160169144143993on_nat(aa_nat5821310369566504398ap_nat(vEBT_VEBT_highi(Uua),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uub),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat6020981482381522343on_nat(aa_Pro8919772687891686969on_nat(aa_VEB358289369652911795on_nat(aa_arr4041137963511513512on_nat(aa_nat1363181691939120105on_nat(aa_nat4363128851979299954on_nat(aTP_Lamm_aao(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf)) ).

% ATP.lambda_1105
tff(fact_11283_ATP_Olambda__1106,axiom,
    ! [Uu: fun_VE8516060933855021849on_nat,Uua: nat,Uub: nat,Uuc: array_VEBT_VEBTi,Uud: vEBT_VEBTi,Uue: product_prod_nat_nat,Uuf: nat] : aa_nat947733127081326238on_nat(aa_Pro4465521767867336112on_nat(aa_VEB1870382382330055850on_nat(aa_arr666314619328815135on_nat(aa_nat8253036615571765984on_nat(aa_nat2001721438062390249on_nat(aTP_Lamm_np(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf) = heap_T8222160169144143993on_nat(aa_nat5821310369566504398ap_nat(vEBT_VEBT_highi(Uua),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uub),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat6020981482381522343on_nat(aa_Pro8919772687891686969on_nat(aa_VEB358289369652911795on_nat(aa_arr4041137963511513512on_nat(aa_nat1363181691939120105on_nat(aa_nat4363128851979299954on_nat(aTP_Lamm_no(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf)) ).

% ATP.lambda_1106
tff(fact_11284_ATP_Olambda__1107,axiom,
    ! [Uu: nat,Uua: nat,Uub: array_VEBT_VEBTi,Uuc: vEBT_VEBTi,Uud: product_prod_nat_nat,Uue: nat,Uuf: nat] : aa_nat947733127081326238on_nat(aa_nat6020981482381522343on_nat(aa_Pro8919772687891686969on_nat(aa_VEB358289369652911795on_nat(aa_arr4041137963511513512on_nat(aa_nat1363181691939120105on_nat(aTP_Lamm_ev(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf) = heap_T2868974464944644318on_nat(aa_nat5408703056993926195_VEBTi(array_nth_VEBT_VEBTi(Uub),Uuf),aa_nat4669826966810537986on_nat(aa_nat107244183536939787on_nat(aa_Pro1663420989643744413on_nat(aa_VEB6384125456408274455on_nat(aa_arr4162587047475849484on_nat(aa_nat5386638315231209293on_nat(aTP_Lamm_eu(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf)) ).

% ATP.lambda_1107
tff(fact_11285_ATP_Olambda__1108,axiom,
    ! [Uu: nat,Uua: list_VEBT_VEBT,Uub: vEBT_VEBT,Uuc: nat,Uud: nat,Uue: nat,Uuf: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aa_nat8314798964848609180_nat_o(aa_nat7293937527460703909_nat_o(aa_VEB4248176028289774660_nat_o(aa_lis144135085560928841_nat_o(aTP_Lamm_bcf(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Uue),Uuf)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uuf),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uu))
        & ! [I2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uud))
           => ( ? [X_1: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(Uua,I2)),X_1)
            <=> aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,Uub),I2) ) )
        & $ite(
            Uue = Uuf,
            ! [X: vEBT_VEBT] :
              ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),set_VEBT_VEBT2(Uua))
             => ~ ? [X6: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,X),X6) ),
            ( aa_nat_o(vEBT_V5917875025757280293ildren(Uuc,Uua),Uuf)
            & ! [X: nat] :
                ( 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,aa_num_num(bit0,one))),Uu))
               => ( aa_nat_o(vEBT_V5917875025757280293ildren(Uuc,Uua),X)
                 => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uue),X)
                    & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Uuf) ) ) ) ) ) ) ) ).

% ATP.lambda_1108
tff(fact_11286_ATP_Olambda__1109,axiom,
    ! [Uu: fun_Pr940977554174145406Heap_o,Uua: nat,Uub: nat,Uuc: array_VEBT_VEBTi,Uud: list_VEBT_VEBT,Uue: nat,Uuf: nat] :
      aa_nat2910048432523430448Heap_o(aa_nat5592791454563588861Heap_o(aa_lis455817436297872826Heap_o(aa_arr1217223296832173011Heap_o(aa_nat6014765364524370664Heap_o(aa_nat4979062449914608309Heap_o(aTP_Lamm_aab(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf) = heap_T7577940988442287570unit_o(aa_o_H2459438917461573107t_unit(refine_Imp_assert,
          ( ( Uuf = aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Uua),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uub),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) )
          & ( Uue = aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Uua),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uub),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) )),aa_nat3811243159488243764Heap_o(aa_nat6178576981048102717Heap_o(aa_lis4519787659412806998Heap_o(aa_arr7619904339668570131Heap_o(aa_nat1269258765096487380Heap_o(aa_nat3499107050721502685Heap_o(aTP_Lamm_aaa(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf)) ).

% ATP.lambda_1109
tff(fact_11287_ATP_Olambda__1110,axiom,
    ! [Uu: fun_VE8181444127144381288Heap_o,Uua: nat,Uub: nat,Uuc: array_VEBT_VEBTi,Uud: list_VEBT_VEBT,Uue: nat,Uuf: nat] :
      aa_nat2910048432523430448Heap_o(aa_nat5592791454563588861Heap_o(aa_lis455817436297872826Heap_o(aa_arr1217223296832173011Heap_o(aa_nat6014765364524370664Heap_o(aa_nat4979062449914608309Heap_o(aTP_Lamm_qa(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf) = heap_T7577940988442287570unit_o(aa_o_H2459438917461573107t_unit(refine_Imp_assert,
          ( ( Uuf = aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Uua),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uub),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) )
          & ( Uue = aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Uua),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uub),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) )),aa_nat3811243159488243764Heap_o(aa_nat6178576981048102717Heap_o(aa_lis4519787659412806998Heap_o(aa_arr7619904339668570131Heap_o(aa_nat1269258765096487380Heap_o(aa_nat3499107050721502685Heap_o(aTP_Lamm_pz(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf)) ).

% ATP.lambda_1110
tff(fact_11288_ATP_Olambda__1111,axiom,
    ! [Uu: nat,Uua: nat,Uub: array_VEBT_VEBTi,Uuc: vEBT_VEBT,Uud: product_prod_nat_nat,Uue: nat,Uuf: option_nat] :
      aa_opt9163293676379396302on_nat(aa_nat9127621225095157975on_nat(aa_Pro5040713308973091561on_nat(aa_VEB8830394031746293224on_nat(aa_arr3423563577619635969on_nat(aa_nat2635311902590537750on_nat(aTP_Lamm_rv(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf) = heap_T5999496708990702694on_nat(aa_o_H2459438917461573107t_unit(refine_Imp_assert,
          ( ( Uuf = none_nat )
          = ( aa_nat_option_nat(vEBT_vebt_pred(Uuc),Uue) = none_nat ) )),aa_opt5089197864053947520on_nat(aa_Pro5009687527940591364on_nat(aa_arr722815569820881053on_nat(aa_nat5945554870259702194on_nat(aTP_Lamm_ru(Uu),Uua),Uub),Uud),Uuf)) ).

% ATP.lambda_1111
tff(fact_11289_ATP_Olambda__1112,axiom,
    ! [Uu: nat,Uua: nat,Uub: array_VEBT_VEBTi,Uuc: list_VEBT_VEBT,Uud: nat,Uue: nat,Uuf: nat] : aa_nat2910048432523430448Heap_o(aa_nat5592791454563588861Heap_o(aa_nat7429013343984486730Heap_o(aa_lis5270594155775668615Heap_o(aa_arr7588197057154790688Heap_o(aa_nat5373961893125994933Heap_o(aTP_Lamm_qm(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf) = heap_T7577940988442287570unit_o(aa_o_H2459438917461573107t_unit(refine_Imp_assert,Uuf = size_s6755466524823107622T_VEBT(Uuc)),aa_nat3811243159488243764Heap_o(aa_nat6178576981048102717Heap_o(aa_nat5674868329695104454Heap_o(aa_lis4631090546836902751Heap_o(aa_arr3453073379018417564Heap_o(aa_nat4764294232906884829Heap_o(aTP_Lamm_ql(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf)) ).

% ATP.lambda_1112
tff(fact_11290_ATP_Olambda__1113,axiom,
    ! [Uu: fun_Pr633315466666964198on_nat,Uua: vEBT_VEBT,Uub: nat,Uuc: option4927543243414619207at_nat,Uud: nat,Uue: array_VEBT_VEBTi,Uuf: vEBT_VEBTi] : aa_VEB2711320435582112505on_nat(aa_arr2955191963898321370on_nat(aa_nat5317369504323972463on_nat(aa_opt8624683241416710969on_nat(aa_nat3457676333800472698on_nat(aa_VEB6681700655871274393on_nat(aTP_Lamm_xj(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf) = heap_T5999496708990702694on_nat(aa_o_H2459438917461573107t_unit(refine_Imp_assert,aa_VEBT_VEBT_o(vEBT_is_Node,Uua)),aa_VEB7937681756907646997on_nat(aa_arr5896957451705369866on_nat(aa_nat3269078897214415691on_nat(aa_opt5459845657693669015on_nat(aa_nat8168140587832489004on_nat(aa_VEB4878072573979604067on_nat(aTP_Lamm_xi(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf)) ).

% ATP.lambda_1113
tff(fact_11291_ATP_Olambda__1114,axiom,
    ! [Uu: fun_Pr633315466666964198on_nat,Uua: vEBT_VEBT,Uub: nat,Uuc: option4927543243414619207at_nat,Uud: nat,Uue: array_VEBT_VEBTi,Uuf: vEBT_VEBTi] : aa_VEB2711320435582112505on_nat(aa_arr2955191963898321370on_nat(aa_nat5317369504323972463on_nat(aa_opt8624683241416710969on_nat(aa_nat3457676333800472698on_nat(aa_VEB6681700655871274393on_nat(aTP_Lamm_wh(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf) = heap_T5999496708990702694on_nat(aa_o_H2459438917461573107t_unit(refine_Imp_assert,aa_VEBT_VEBT_o(vEBT_is_Node,Uua)),aa_VEB7937681756907646997on_nat(aa_arr5896957451705369866on_nat(aa_nat3269078897214415691on_nat(aa_opt5459845657693669015on_nat(aa_nat8168140587832489004on_nat(aa_VEB4878072573979604067on_nat(aTP_Lamm_wg(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf)) ).

% ATP.lambda_1114
tff(fact_11292_ATP_Olambda__1115,axiom,
    ! [Uu: fun_VE3775134097089128360on_nat,Uua: vEBT_VEBT,Uub: nat,Uuc: option4927543243414619207at_nat,Uud: nat,Uue: array_VEBT_VEBTi,Uuf: vEBT_VEBTi] : aa_VEB2711320435582112505on_nat(aa_arr2955191963898321370on_nat(aa_nat5317369504323972463on_nat(aa_opt8624683241416710969on_nat(aa_nat3457676333800472698on_nat(aa_VEB6681700655871274393on_nat(aTP_Lamm_sz(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf) = heap_T5999496708990702694on_nat(aa_o_H2459438917461573107t_unit(refine_Imp_assert,aa_VEBT_VEBT_o(vEBT_is_Node,Uua)),aa_VEB7937681756907646997on_nat(aa_arr5896957451705369866on_nat(aa_nat3269078897214415691on_nat(aa_opt5459845657693669015on_nat(aa_nat8168140587832489004on_nat(aa_VEB4878072573979604067on_nat(aTP_Lamm_sy(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf)) ).

% ATP.lambda_1115
tff(fact_11293_ATP_Olambda__1116,axiom,
    ! [Uu: fun_VE3775134097089128360on_nat,Uua: vEBT_VEBT,Uub: nat,Uuc: option4927543243414619207at_nat,Uud: nat,Uue: array_VEBT_VEBTi,Uuf: vEBT_VEBTi] : aa_VEB2711320435582112505on_nat(aa_arr2955191963898321370on_nat(aa_nat5317369504323972463on_nat(aa_opt8624683241416710969on_nat(aa_nat3457676333800472698on_nat(aa_VEB6681700655871274393on_nat(aTP_Lamm_sk(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf) = heap_T5999496708990702694on_nat(aa_o_H2459438917461573107t_unit(refine_Imp_assert,aa_VEBT_VEBT_o(vEBT_is_Node,Uua)),aa_VEB7937681756907646997on_nat(aa_arr5896957451705369866on_nat(aa_nat3269078897214415691on_nat(aa_opt5459845657693669015on_nat(aa_nat8168140587832489004on_nat(aa_VEB4878072573979604067on_nat(aTP_Lamm_sj(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf)) ).

% ATP.lambda_1116
tff(fact_11294_ATP_Olambda__1117,axiom,
    ! [Uu: fun_Pr940977554174145406Heap_o,Uua: vEBT_VEBT,Uub: nat,Uuc: option4927543243414619207at_nat,Uud: nat,Uue: array_VEBT_VEBTi,Uuf: vEBT_VEBTi] : aa_VEB8194921341634505877Heap_o(aa_arr2373296168907467658Heap_o(aa_nat8650107858160254923Heap_o(aa_opt6539295502271640599Heap_o(aa_nat1227502447850154924Heap_o(aa_VEB6636772557416003555Heap_o(aTP_Lamm_aaj(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf) = heap_T7577940988442287570unit_o(aa_o_H2459438917461573107t_unit(refine_Imp_assert,aa_VEBT_VEBT_o(vEBT_is_Node,Uua)),aa_arr4590259283924850188Heap_o(aa_nat2073207392357089185Heap_o(aa_opt2919655305572010731Heap_o(aa_nat8312540953585231916Heap_o(aa_VEB1825396708860521035Heap_o(aTP_Lamm_aai(Uu),Uua),Uub),Uuc),Uud),Uue)) ).

% ATP.lambda_1117
tff(fact_11295_ATP_Olambda__1118,axiom,
    ! [Uu: fun_VE8181444127144381288Heap_o,Uua: vEBT_VEBT,Uub: nat,Uuc: option4927543243414619207at_nat,Uud: nat,Uue: array_VEBT_VEBTi,Uuf: vEBT_VEBTi] : aa_VEB8194921341634505877Heap_o(aa_arr2373296168907467658Heap_o(aa_nat8650107858160254923Heap_o(aa_opt6539295502271640599Heap_o(aa_nat1227502447850154924Heap_o(aa_VEB6636772557416003555Heap_o(aTP_Lamm_qi(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf) = heap_T7577940988442287570unit_o(aa_o_H2459438917461573107t_unit(refine_Imp_assert,aa_VEBT_VEBT_o(vEBT_is_Node,Uua)),aa_arr4590259283924850188Heap_o(aa_nat2073207392357089185Heap_o(aa_opt2919655305572010731Heap_o(aa_nat8312540953585231916Heap_o(aa_VEB1825396708860521035Heap_o(aTP_Lamm_qh(Uu),Uua),Uub),Uuc),Uud),Uue)) ).

% ATP.lambda_1118
tff(fact_11296_ATP_Olambda__1119,axiom,
    ! [Uu: fun_Pr4171499318505651870_VEBTi,Uua: nat,Uub: nat,Uuc: array_VEBT_VEBTi,Uud: vEBT_VEBTi,Uue: product_prod_nat_nat,Uuf: nat] : aa_nat5408703056993926195_VEBTi(aa_Pro3150526172571800645_VEBTi(aa_VEB4890588985164552255_VEBTi(aa_arr5592147402706875956_VEBTi(aa_nat5963232373128642933_VEBTi(aa_nat7459085003528029950_VEBTi(aTP_Lamm_abm(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf) = heap_T844888390831797134_VEBTi(aa_nat5821310369566504398ap_nat(heap_Time_return_nat,aa_Pro4243173770044397973at_nat(product_snd_nat_nat,Uue)),aa_nat4282640288315822012_VEBTi(aa_VEB7687887442520674464_VEBTi(aa_arr2561095334397224705_VEBTi(aa_nat8715118376710468630_VEBTi(aa_nat5853910169739908579_VEBTi(aTP_Lamm_abl(Uu),Uua),Uub),Uuc),Uud),Uuf)) ).

% ATP.lambda_1119
tff(fact_11297_ATP_Olambda__1120,axiom,
    ! [Uu: fun_VE8118064890490510894_VEBTi,Uua: nat,Uub: nat,Uuc: array_VEBT_VEBTi,Uud: vEBT_VEBTi,Uue: product_prod_nat_nat,Uuf: nat] : aa_nat5408703056993926195_VEBTi(aa_Pro3150526172571800645_VEBTi(aa_VEB4890588985164552255_VEBTi(aa_arr5592147402706875956_VEBTi(aa_nat5963232373128642933_VEBTi(aa_nat7459085003528029950_VEBTi(aTP_Lamm_nj(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf) = heap_T844888390831797134_VEBTi(aa_nat5821310369566504398ap_nat(heap_Time_return_nat,aa_Pro4243173770044397973at_nat(product_snd_nat_nat,Uue)),aa_nat4282640288315822012_VEBTi(aa_VEB7687887442520674464_VEBTi(aa_arr2561095334397224705_VEBTi(aa_nat8715118376710468630_VEBTi(aa_nat5853910169739908579_VEBTi(aTP_Lamm_ni(Uu),Uua),Uub),Uuc),Uud),Uuf)) ).

% ATP.lambda_1120
tff(fact_11298_ATP_Olambda__1121,axiom,
    ! [Uu: fun_Pr342922516890243977on_nat,Uua: nat,Uub: array_VEBT_VEBTi,Uuc: vEBT_VEBTi,Uud: nat,Uue: nat,Uuf: vEBT_VEBTi] : aa_VEB2711320435582112505on_nat(aa_nat4669826966810537986on_nat(aa_nat107244183536939787on_nat(aa_VEB216942459957533935on_nat(aa_arr2558136924908760784on_nat(aa_nat402565781209047397on_nat(aTP_Lamm_aav(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf) = heap_T3669509953089699273on_nat(aa_VEB2711320435582112505on_nat(vEBT_vebt_maxti,Uuf),aa_VEB9161639482783362738on_nat(aa_nat6300058732544452155on_nat(aa_nat7962384762897155012on_nat(aa_VEB4635808744058847656on_nat(aa_arr3352113509897533961on_nat(aa_nat8985933306058800414on_nat(aTP_Lamm_aau(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf)) ).

% ATP.lambda_1121
tff(fact_11299_ATP_Olambda__1122,axiom,
    ! [Uu: fun_VE8516060933855021849on_nat,Uua: nat,Uub: array_VEBT_VEBTi,Uuc: vEBT_VEBTi,Uud: nat,Uue: nat,Uuf: vEBT_VEBTi] : aa_VEB2711320435582112505on_nat(aa_nat4669826966810537986on_nat(aa_nat107244183536939787on_nat(aa_VEB216942459957533935on_nat(aa_arr2558136924908760784on_nat(aa_nat402565781209047397on_nat(aTP_Lamm_nt(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf) = heap_T3669509953089699273on_nat(aa_VEB2711320435582112505on_nat(vEBT_vebt_maxti,Uuf),aa_VEB9161639482783362738on_nat(aa_nat6300058732544452155on_nat(aa_nat7962384762897155012on_nat(aa_VEB4635808744058847656on_nat(aa_arr3352113509897533961on_nat(aa_nat8985933306058800414on_nat(aTP_Lamm_ns(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf)) ).

% ATP.lambda_1122
tff(fact_11300_ATP_Olambda__1123,axiom,
    ! [Uu: nat,Uua: nat,Uub: array_VEBT_VEBTi,Uuc: list_VEBT_VEBT,Uud: nat,Uue: nat,Uuf: product_unit] : aa_Pro3379321066291785715Heap_o(aa_nat3811243159488243764Heap_o(aa_nat6178576981048102717Heap_o(aa_lis4519787659412806998Heap_o(aa_arr7619904339668570131Heap_o(aa_nat1269258765096487380Heap_o(aTP_Lamm_qn(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf) = heap_Time_bind_nat_o(array_len_VEBT_VEBTi(Uub),aa_nat5592791454563588861Heap_o(aa_nat7429013343984486730Heap_o(aa_lis5270594155775668615Heap_o(aa_arr7588197057154790688Heap_o(aa_nat5373961893125994933Heap_o(aTP_Lamm_qm(Uu),Uua),Uub),Uuc),Uud),Uue)) ).

% ATP.lambda_1123
tff(fact_11301_ATP_Olambda__1124,axiom,
    ! [Uu: fun_Pr940977554174145406Heap_o,Uua: nat,Uub: option4927543243414619207at_nat,Uuc: nat,Uud: array_VEBT_VEBTi,Uue: option4927543243414619207at_nat,Uuf: nat] : aa_nat6527088363260489928Heap_o(aa_opt7844807959364121492Heap_o(aa_arr79663438428320465Heap_o(aa_nat2087714105643485586Heap_o(aa_opt9028207981337828190Heap_o(aa_nat2171064521259968883Heap_o(aTP_Lamm_aaf(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf) = produc5048428016959714504Heap_o(aa_nat3831830007592743808Heap_o(aa_opt3748239118003421002Heap_o(aa_arr581739785191926499Heap_o(aa_nat105892859602166008Heap_o(aa_opt8417070463701818562Heap_o(aa_nat8831994779944892035Heap_o(aTP_Lamm_aae(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf)) ).

% ATP.lambda_1124
tff(fact_11302_ATP_Olambda__1125,axiom,
    ! [Uu: vEBT_VEBT,Uua: nat,Uub: option4927543243414619207at_nat,Uuc: nat,Uud: array_VEBT_VEBTi,Uue: vEBT_VEBTi,Uuf: option4927543243414619207at_nat] : aa_opt1657869255803490932on_nat(aa_VEB5714833192853586542on_nat(aa_arr7759188791488537571on_nat(aa_nat7273124837800733860on_nat(aa_opt1247461068388908528on_nat(aa_nat309676689105111045on_nat(aTP_Lamm_uj(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf) = produc2606485630176857543on_nat(aa_opt8950567732158611302on_nat(aa_VEB8223317440372349570on_nat(aa_arr6272259649854615779on_nat(aa_nat261593688111926264on_nat(aa_opt4414834345254747970on_nat(aa_nat2224702164617786371on_nat(aTP_Lamm_ui(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf)) ).

% ATP.lambda_1125
tff(fact_11303_ATP_Olambda__1126,axiom,
    ! [Uu: nat,Uua: option4927543243414619207at_nat,Uub: nat,Uuc: array_VEBT_VEBTi,Uud: vEBT_VEBTi,Uue: product_prod_nat_nat,Uuf: option4927543243414619207at_nat] : aa_opt8585651768545592073_VEBTi(aa_Pro4062752345418257877_VEBTi(aa_VEB8867863569893255025_VEBTi(aa_arr3828982135699130450_VEBTi(aa_nat5629247741253790183_VEBTi(aa_opt9205748023383517873_VEBTi(aTP_Lamm_tt(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf) = produc7864503674192730076_VEBTi(aa_opt7507675525305149691_VEBTi(aa_Pro1234189438217840325_VEBTi(aa_VEB660175962507042495_VEBTi(aa_arr8367260621226259636_VEBTi(aa_nat5784496157376858101_VEBTi(aa_opt81624670976341313_VEBTi(aTP_Lamm_ts(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf)) ).

% ATP.lambda_1126
tff(fact_11304_ATP_Olambda__1127,axiom,
    ! [Uu: fun_VE8181444127144381288Heap_o,Uua: nat,Uub: option4927543243414619207at_nat,Uuc: nat,Uud: array_VEBT_VEBTi,Uue: option4927543243414619207at_nat,Uuf: nat] : aa_nat6527088363260489928Heap_o(aa_opt7844807959364121492Heap_o(aa_arr79663438428320465Heap_o(aa_nat2087714105643485586Heap_o(aa_opt9028207981337828190Heap_o(aa_nat2171064521259968883Heap_o(aTP_Lamm_qe(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf) = produc5048428016959714504Heap_o(aa_nat3831830007592743808Heap_o(aa_opt3748239118003421002Heap_o(aa_arr581739785191926499Heap_o(aa_nat105892859602166008Heap_o(aa_opt8417070463701818562Heap_o(aa_nat8831994779944892035Heap_o(aTP_Lamm_qd(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf)) ).

% ATP.lambda_1127
tff(fact_11305_ATP_Olambda__1128,axiom,
    ! [Uu: vEBT_VEBT,Uua: nat,Uub: option4927543243414619207at_nat,Uuc: nat,Uud: array_VEBT_VEBTi,Uue: vEBT_VEBTi,Uuf: option4927543243414619207at_nat] : aa_opt1657869255803490932on_nat(aa_VEB5714833192853586542on_nat(aa_arr7759188791488537571on_nat(aa_nat7273124837800733860on_nat(aa_opt1247461068388908528on_nat(aa_nat309676689105111045on_nat(aTP_Lamm_os(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf) = produc2606485630176857543on_nat(aa_opt8950567732158611302on_nat(aa_VEB8223317440372349570on_nat(aa_arr6272259649854615779on_nat(aa_nat261593688111926264on_nat(aa_opt4414834345254747970on_nat(aa_nat2224702164617786371on_nat(aTP_Lamm_or(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf)) ).

% ATP.lambda_1128
tff(fact_11306_ATP_Olambda__1129,axiom,
    ! [Uu: $o,Uua: $o,Uub: $o,Uuc: $o,Uud: $o,Uue: $o,Uuf: $o,Uug: code_integer] : aa_Cod7842134615204088097o_char(aa_o_f2742674633914528860o_char(aa_o_f9003794812476387647o_char(aa_o_f4183413333637126194o_char(aa_o_f2932186792858379797o_char(aa_o_f1143307120959153672o_char(aa_o_f3038258172971468011o_char(aTP_Lamm_bch((Uu)),(Uua)),(Uub)),(Uuc)),(Uud)),(Uue)),(Uuf)),Uug) = char2((Uu),(Uua),(Uub),(Uuc),(Uud),(Uue),(Uuf)) ).

% ATP.lambda_1129
tff(fact_11307_ATP_Olambda__1130,axiom,
    ! [Uu: fun_Pr342922516890243977on_nat,Uua: nat,Uub: array_VEBT_VEBTi,Uuc: vEBT_VEBTi,Uud: nat,Uue: nat,Uuf: vEBT_VEBTi,Uug: option_nat] :
      aa_opt9163293676379396302on_nat(aa_VEB9161639482783362738on_nat(aa_nat6300058732544452155on_nat(aa_nat7962384762897155012on_nat(aa_VEB4635808744058847656on_nat(aa_arr3352113509897533961on_nat(aa_nat8985933306058800414on_nat(aTP_Lamm_aau(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug) = $ite(
        ( ( Uug != none_nat )
        & aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_less,aa_nat_option_nat(some_nat,Uud)),Uug) ),
        heap_T3669509953089699273on_nat(aa_nat947733127081326238on_nat(aa_VEB3566417204837707778on_nat(produc1489253303066280154on_nat(Uu),Uuf),Uud),aa_nat9127621225095157975on_nat(aTP_Lamm_ep(Uua),Uue)),
        heap_T3669509953089699273on_nat(aa_nat947733127081326238on_nat(aa_VEB3566417204837707778on_nat(produc1489253303066280154on_nat(Uu),Uuc),Uue),aa_arr3757929795460847279on_nat(aTP_Lamm_fb(Uua),Uub)) ) ).

% ATP.lambda_1130
tff(fact_11308_ATP_Olambda__1131,axiom,
    ! [Uu: fun_VE8516060933855021849on_nat,Uua: nat,Uub: array_VEBT_VEBTi,Uuc: vEBT_VEBTi,Uud: nat,Uue: nat,Uuf: vEBT_VEBTi,Uug: option_nat] :
      aa_opt9163293676379396302on_nat(aa_VEB9161639482783362738on_nat(aa_nat6300058732544452155on_nat(aa_nat7962384762897155012on_nat(aa_VEB4635808744058847656on_nat(aa_arr3352113509897533961on_nat(aa_nat8985933306058800414on_nat(aTP_Lamm_ns(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug) = $ite(
        ( ( Uug != none_nat )
        & aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_less,aa_nat_option_nat(some_nat,Uud)),Uug) ),
        heap_T3669509953089699273on_nat(aa_nat947733127081326238on_nat(aa_VEB3566417204837707778on_nat(Uu,Uuf),Uud),aa_nat9127621225095157975on_nat(aTP_Lamm_ep(Uua),Uue)),
        heap_T3669509953089699273on_nat(aa_nat947733127081326238on_nat(aa_VEB3566417204837707778on_nat(Uu,Uuc),Uue),aa_arr3757929795460847279on_nat(aTP_Lamm_fb(Uua),Uub)) ) ).

% ATP.lambda_1131
tff(fact_11309_ATP_Olambda__1132,axiom,
    ! [Uu: nat,Uua: nat,Uub: array_VEBT_VEBTi,Uuc: list_VEBT_VEBT,Uud: nat,Uue: nat,Uuf: nat,Uug: product_unit] :
      aa_Pro3379321066291785715Heap_o(aa_nat3811243159488243764Heap_o(aa_nat6178576981048102717Heap_o(aa_nat5674868329695104454Heap_o(aa_lis4631090546836902751Heap_o(aa_arr3453073379018417564Heap_o(aa_nat4764294232906884829Heap_o(aTP_Lamm_ql(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uud),Uuf),
        heap_T7577940988442287570unit_o(aa_o_H2459438917461573107t_unit(refine_Imp_assert,
            ( ( Uud = aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Uu),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uua),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) )
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uud),size_s6755466524823107622T_VEBT(Uuc)) )),aa_nat3811243159488243764Heap_o(aa_nat6178576981048102717Heap_o(aa_lis4519787659412806998Heap_o(aTP_Lamm_qk(Uub),Uuc),Uud),Uue)),
        aa_o_H7399033255965945470Heap_o(heap_Time_return_o,$false) ) ).

% ATP.lambda_1132
tff(fact_11310_ATP_Olambda__1133,axiom,
    ! [Uu: fun_Pr4404027214751104635_VEBTi,Uua: vEBT_VEBT,Uub: nat,Uuc: option4927543243414619207at_nat,Uud: nat,Uue: array_VEBT_VEBTi,Uuf: vEBT_VEBTi,Uug: product_prod_nat_nat] :
      aa_Pro1395359565518501564_VEBTi(aa_VEB6393524116121717046_VEBTi(aa_arr8059685486658508715_VEBTi(aa_nat7594302058568047468_VEBTi(aa_opt7720280829887018808_VEBTi(aa_nat6605875721633857101_VEBTi(aa_VEB7645014810616801796_VEBTi(aTP_Lamm_vp(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Uud),one_one_nat),aa_VEB836174949890266638_VEBTi(heap_T3630416162098727440_VEBTi,aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(Uuc,Uud,Uue),Uuf)),heap_T7982501707604696571_VEBTi(aa_o_H2459438917461573107t_unit(refine_Imp_assert,aa_VEBT_VEBT_o(vEBT_is_Node,Uua)),aa_Pro8034723660955798012_VEBTi(aa_VEB4262372667711301016_VEBTi(aa_arr9036764603281787897_VEBTi(aa_nat2771851207891374350_VEBTi(aa_opt7594342031245826264_VEBTi(aa_nat4475881176255417753_VEBTi(aa_VEB8590999954632166712_VEBTi(aTP_Lamm_vo(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug))) ).

% ATP.lambda_1133
tff(fact_11311_ATP_Olambda__1134,axiom,
    ! [Uu: fun_VE2018681615261037117_VEBTi,Uua: vEBT_VEBT,Uub: nat,Uuc: option4927543243414619207at_nat,Uud: nat,Uue: array_VEBT_VEBTi,Uuf: vEBT_VEBTi,Uug: product_prod_nat_nat] :
      aa_Pro1395359565518501564_VEBTi(aa_VEB6393524116121717046_VEBTi(aa_arr8059685486658508715_VEBTi(aa_nat7594302058568047468_VEBTi(aa_opt7720280829887018808_VEBTi(aa_nat6605875721633857101_VEBTi(aa_VEB7645014810616801796_VEBTi(aTP_Lamm_rs(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Uud),one_one_nat),aa_VEB836174949890266638_VEBTi(heap_T3630416162098727440_VEBTi,aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(Uuc,Uud,Uue),Uuf)),heap_T7982501707604696571_VEBTi(aa_o_H2459438917461573107t_unit(refine_Imp_assert,aa_VEBT_VEBT_o(vEBT_is_Node,Uua)),aa_Pro8034723660955798012_VEBTi(aa_VEB4262372667711301016_VEBTi(aa_arr9036764603281787897_VEBTi(aa_nat2771851207891374350_VEBTi(aa_opt7594342031245826264_VEBTi(aa_nat4475881176255417753_VEBTi(aa_VEB8590999954632166712_VEBTi(aTP_Lamm_rr(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug))) ).

% ATP.lambda_1134
tff(fact_11312_ATP_Olambda__1135,axiom,
    ! [Uu: fun_Pr633315466666964198on_nat,Uua: nat,Uub: nat,Uuc: array_VEBT_VEBTi,Uud: vEBT_VEBTi,Uue: list_VEBT_VEBT,Uuf: vEBT_VEBT,Uug: product_prod_nat_nat] :
      aa_Pro7675874045790911655on_nat(aa_VEB6310895951922269862on_nat(aa_lis7455679124549436279on_nat(aa_VEB7601412409763292529on_nat(aa_arr3206634112104793958on_nat(aa_nat2842375153920543911on_nat(aa_nat1315775135507255344on_nat(aTP_Lamm_xd(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Uub),one_one_nat),
        heap_T3487192422709364219on_nat(none_nat),
        $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_Pro4243173770044397973at_nat(product_snd_nat_nat,Uug)),Uua),heap_T3487192422709364219on_nat(aa_nat_option_nat(some_nat,aa_Pro4243173770044397973at_nat(product_snd_nat_nat,Uug))),heap_T8222160169144143993on_nat(aa_nat5821310369566504398ap_nat(vEBT_VEBT_lowi(Uua),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uub),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_Pro4465521767867336112on_nat(aa_VEB6043830584257997615on_nat(aa_lis5985300925819340672on_nat(aa_VEB1019093884839074554on_nat(aa_arr1617095457630768239on_nat(aa_nat5782952277816596784on_nat(aa_nat7314605301869835321on_nat(aTP_Lamm_xc(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug))) ) ).

% ATP.lambda_1135
tff(fact_11313_ATP_Olambda__1136,axiom,
    ! [Uu: fun_VE3775134097089128360on_nat,Uua: nat,Uub: nat,Uuc: array_VEBT_VEBTi,Uud: vEBT_VEBTi,Uue: list_VEBT_VEBT,Uuf: vEBT_VEBT,Uug: product_prod_nat_nat] :
      aa_Pro7675874045790911655on_nat(aa_VEB6310895951922269862on_nat(aa_lis7455679124549436279on_nat(aa_VEB7601412409763292529on_nat(aa_arr3206634112104793958on_nat(aa_nat2842375153920543911on_nat(aa_nat1315775135507255344on_nat(aTP_Lamm_se(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Uub),one_one_nat),
        heap_T3487192422709364219on_nat(none_nat),
        $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_Pro4243173770044397973at_nat(product_snd_nat_nat,Uug)),Uua),heap_T3487192422709364219on_nat(aa_nat_option_nat(some_nat,aa_Pro4243173770044397973at_nat(product_snd_nat_nat,Uug))),heap_T8222160169144143993on_nat(aa_nat5821310369566504398ap_nat(vEBT_VEBT_lowi(Uua),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uub),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_Pro4465521767867336112on_nat(aa_VEB6043830584257997615on_nat(aa_lis5985300925819340672on_nat(aa_VEB1019093884839074554on_nat(aa_arr1617095457630768239on_nat(aa_nat5782952277816596784on_nat(aa_nat7314605301869835321on_nat(aTP_Lamm_sd(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug))) ) ).

% ATP.lambda_1136
tff(fact_11314_ATP_Olambda__1137,axiom,
    ! [Uu: fun_Pr633315466666964198on_nat,Uua: nat,Uub: nat,Uuc: array_VEBT_VEBTi,Uud: vEBT_VEBTi,Uue: list_VEBT_VEBT,Uuf: vEBT_VEBT,Uug: product_prod_nat_nat] :
      aa_Pro7675874045790911655on_nat(aa_VEB6310895951922269862on_nat(aa_lis7455679124549436279on_nat(aa_VEB7601412409763292529on_nat(aa_arr3206634112104793958on_nat(aa_nat2842375153920543911on_nat(aa_nat1315775135507255344on_nat(aTP_Lamm_wb(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Uub),one_one_nat),
        heap_T3487192422709364219on_nat(none_nat),
        $ite(
          aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uua),aa_Pro4243173770044397973at_nat(product_fst_nat_nat,Uug)),
          heap_T3487192422709364219on_nat(aa_nat_option_nat(some_nat,aa_Pro4243173770044397973at_nat(product_fst_nat_nat,Uug))),
          $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_Pro4243173770044397973at_nat(product_snd_nat_nat,Uug)),Uua),heap_T3487192422709364219on_nat(none_nat),heap_T8222160169144143993on_nat(aa_nat5821310369566504398ap_nat(vEBT_VEBT_lowi(Uua),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uub),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_VEB1312864128696253885on_nat(aa_lis1761857624600698434on_nat(aa_VEB4172908723204034014on_nat(aa_arr2072991348567353407on_nat(aa_nat7185663501778404180on_nat(aa_nat4144773949132746529on_nat(aTP_Lamm_wa(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf))) ) ) ).

% ATP.lambda_1137
tff(fact_11315_ATP_Olambda__1138,axiom,
    ! [Uu: fun_VE3775134097089128360on_nat,Uua: nat,Uub: nat,Uuc: array_VEBT_VEBTi,Uud: vEBT_VEBTi,Uue: list_VEBT_VEBT,Uuf: vEBT_VEBT,Uug: product_prod_nat_nat] :
      aa_Pro7675874045790911655on_nat(aa_VEB6310895951922269862on_nat(aa_lis7455679124549436279on_nat(aa_VEB7601412409763292529on_nat(aa_arr3206634112104793958on_nat(aa_nat2842375153920543911on_nat(aa_nat1315775135507255344on_nat(aTP_Lamm_st(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Uub),one_one_nat),
        heap_T3487192422709364219on_nat(none_nat),
        $ite(
          aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uua),aa_Pro4243173770044397973at_nat(product_fst_nat_nat,Uug)),
          heap_T3487192422709364219on_nat(aa_nat_option_nat(some_nat,aa_Pro4243173770044397973at_nat(product_fst_nat_nat,Uug))),
          $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_Pro4243173770044397973at_nat(product_snd_nat_nat,Uug)),Uua),heap_T3487192422709364219on_nat(none_nat),heap_T8222160169144143993on_nat(aa_nat5821310369566504398ap_nat(vEBT_VEBT_lowi(Uua),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uub),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_VEB1312864128696253885on_nat(aa_lis1761857624600698434on_nat(aa_VEB4172908723204034014on_nat(aa_arr2072991348567353407on_nat(aa_nat7185663501778404180on_nat(aa_nat4144773949132746529on_nat(aTP_Lamm_ss(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf))) ) ) ).

% ATP.lambda_1138
tff(fact_11316_ATP_Olambda__1139,axiom,
    ! [Uu: fun_Pr940977554174145406Heap_o,Uua: vEBT_VEBT,Uub: nat,Uuc: option4927543243414619207at_nat,Uud: nat,Uue: array_VEBT_VEBTi,Uuf: nat,Uug: nat] :
      aa_nat2910048432523430448Heap_o(aa_nat5592791454563588861Heap_o(aa_arr9046104431811898738Heap_o(aa_nat272996766647252915Heap_o(aa_opt8863613227715573503Heap_o(aa_nat8734498203866763412Heap_o(aa_VEB8906299275298315467Heap_o(aTP_Lamm_aah(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Uud),one_one_nat),
        aa_o_H7399033255965945470Heap_o(heap_Time_return_o,$false),
        $ite(
          Uub = Uuf,
          aa_o_H7399033255965945470Heap_o(heap_Time_return_o,$true),
          $ite(
            Uub = Uug,
            aa_o_H7399033255965945470Heap_o(heap_Time_return_o,$true),
            $ite(
              aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uub),Uuf),
              aa_o_H7399033255965945470Heap_o(heap_Time_return_o,$false),
              $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uug),Uub),aa_o_H7399033255965945470Heap_o(heap_Time_return_o,$false),produc1330493526443650053Heap_o(aa_arr6198139058929667337Heap_o(aa_nat436397609992601886Heap_o(aa_opt4438414422311449448Heap_o(aa_nat1884043827438135593Heap_o(aTP_Lamm_aag(Uu),Uub),Uuc),Uud),Uue),vEBT_c634343235235684882T_VEBT(aTP_Lamm_ot,aTP_Lamm_ou,Uua))) ) ) ) ) ).

% ATP.lambda_1139
tff(fact_11317_ATP_Olambda__1140,axiom,
    ! [Uu: fun_VE8181444127144381288Heap_o,Uua: vEBT_VEBT,Uub: nat,Uuc: option4927543243414619207at_nat,Uud: nat,Uue: array_VEBT_VEBTi,Uuf: nat,Uug: nat] :
      aa_nat2910048432523430448Heap_o(aa_nat5592791454563588861Heap_o(aa_arr9046104431811898738Heap_o(aa_nat272996766647252915Heap_o(aa_opt8863613227715573503Heap_o(aa_nat8734498203866763412Heap_o(aa_VEB8906299275298315467Heap_o(aTP_Lamm_qg(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Uud),one_one_nat),
        aa_o_H7399033255965945470Heap_o(heap_Time_return_o,$false),
        $ite(
          Uub = Uuf,
          aa_o_H7399033255965945470Heap_o(heap_Time_return_o,$true),
          $ite(
            Uub = Uug,
            aa_o_H7399033255965945470Heap_o(heap_Time_return_o,$true),
            $ite(
              aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uub),Uuf),
              aa_o_H7399033255965945470Heap_o(heap_Time_return_o,$false),
              $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uug),Uub),aa_o_H7399033255965945470Heap_o(heap_Time_return_o,$false),produc1330493526443650053Heap_o(aa_arr6198139058929667337Heap_o(aa_nat436397609992601886Heap_o(aa_opt4438414422311449448Heap_o(aa_nat1884043827438135593Heap_o(aTP_Lamm_qf(Uu),Uub),Uuc),Uud),Uue),vEBT_c634343235235684882T_VEBT(aTP_Lamm_ot,aTP_Lamm_ou,Uua))) ) ) ) ) ).

% ATP.lambda_1140
tff(fact_11318_ATP_Olambda__1141,axiom,
    ! [Uu: nat,Uua: option4927543243414619207at_nat,Uub: nat,Uuc: array_VEBT_VEBTi,Uud: vEBT_VEBTi,Uue: list_VEBT_VEBT,Uuf: vEBT_VEBT,Uug: product_unit] : aa_Pro451875010097498139on_nat(aa_VEB1832073013541904922on_nat(aa_lis6328568131235813867on_nat(aa_VEB3884484516340040293on_nat(aa_arr5442694692229379418on_nat(aa_nat4425110079127750043on_nat(aa_opt7218664994477926375on_nat(aTP_Lamm_ug(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug) = case_o8344607093967974880at_nat(heap_T3487192422709364219on_nat(none_nat),aa_VEB6310895951922269862on_nat(aa_lis7455679124549436279on_nat(aa_VEB7601412409763292529on_nat(aa_arr3206634112104793958on_nat(aa_nat2842375153920543911on_nat(aTP_Lamm_uf(Uu),Uub),Uuc),Uud),Uue),Uuf),Uua) ).

% ATP.lambda_1141
tff(fact_11319_ATP_Olambda__1142,axiom,
    ! [Uu: nat,Uua: option4927543243414619207at_nat,Uub: nat,Uuc: array_VEBT_VEBTi,Uud: vEBT_VEBTi,Uue: list_VEBT_VEBT,Uuf: vEBT_VEBT,Uug: product_unit] : aa_Pro451875010097498139on_nat(aa_VEB1832073013541904922on_nat(aa_lis6328568131235813867on_nat(aa_VEB3884484516340040293on_nat(aa_arr5442694692229379418on_nat(aa_nat4425110079127750043on_nat(aa_opt7218664994477926375on_nat(aTP_Lamm_op(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug) = case_o8344607093967974880at_nat(heap_T3487192422709364219on_nat(none_nat),aa_VEB6310895951922269862on_nat(aa_lis7455679124549436279on_nat(aa_VEB7601412409763292529on_nat(aa_arr3206634112104793958on_nat(aa_nat2842375153920543911on_nat(aTP_Lamm_oo(Uu),Uub),Uuc),Uud),Uue),Uuf),Uua) ).

% ATP.lambda_1142
tff(fact_11320_ATP_Olambda__1143,axiom,
    ! [Uu: fun_Pr633315466666964198on_nat,Uua: vEBT_VEBT,Uub: nat,Uuc: option4927543243414619207at_nat,Uud: nat,Uue: array_VEBT_VEBTi,Uuf: vEBT_VEBTi,Uug: product_unit] : aa_Pro451875010097498139on_nat(aa_VEB7937681756907646997on_nat(aa_arr5896957451705369866on_nat(aa_nat3269078897214415691on_nat(aa_opt5459845657693669015on_nat(aa_nat8168140587832489004on_nat(aa_VEB4878072573979604067on_nat(aTP_Lamm_xi(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug) = produc2190226783763740553on_nat(aa_VEB5714833192853586542on_nat(aa_arr7759188791488537571on_nat(aa_nat7273124837800733860on_nat(aa_opt1247461068388908528on_nat(aa_nat309676689105111045on_nat(aa_VEB8598360222073186748on_nat(aTP_Lamm_xh(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),vEBT_c634343235235684882T_VEBT(aTP_Lamm_ot,aTP_Lamm_ou,Uua)) ).

% ATP.lambda_1143
tff(fact_11321_ATP_Olambda__1144,axiom,
    ! [Uu: fun_Pr633315466666964198on_nat,Uua: vEBT_VEBT,Uub: nat,Uuc: option4927543243414619207at_nat,Uud: nat,Uue: array_VEBT_VEBTi,Uuf: vEBT_VEBTi,Uug: product_unit] : aa_Pro451875010097498139on_nat(aa_VEB7937681756907646997on_nat(aa_arr5896957451705369866on_nat(aa_nat3269078897214415691on_nat(aa_opt5459845657693669015on_nat(aa_nat8168140587832489004on_nat(aa_VEB4878072573979604067on_nat(aTP_Lamm_wg(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug) = produc2190226783763740553on_nat(aa_VEB5714833192853586542on_nat(aa_arr7759188791488537571on_nat(aa_nat7273124837800733860on_nat(aa_opt1247461068388908528on_nat(aa_nat309676689105111045on_nat(aa_VEB8598360222073186748on_nat(aTP_Lamm_wf(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),vEBT_c634343235235684882T_VEBT(aTP_Lamm_ot,aTP_Lamm_ou,Uua)) ).

% ATP.lambda_1144
tff(fact_11322_ATP_Olambda__1145,axiom,
    ! [Uu: fun_VE3775134097089128360on_nat,Uua: vEBT_VEBT,Uub: nat,Uuc: option4927543243414619207at_nat,Uud: nat,Uue: array_VEBT_VEBTi,Uuf: vEBT_VEBTi,Uug: product_unit] : aa_Pro451875010097498139on_nat(aa_VEB7937681756907646997on_nat(aa_arr5896957451705369866on_nat(aa_nat3269078897214415691on_nat(aa_opt5459845657693669015on_nat(aa_nat8168140587832489004on_nat(aa_VEB4878072573979604067on_nat(aTP_Lamm_sy(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug) = produc2190226783763740553on_nat(aa_VEB5714833192853586542on_nat(aa_arr7759188791488537571on_nat(aa_nat7273124837800733860on_nat(aa_opt1247461068388908528on_nat(aa_nat309676689105111045on_nat(aa_VEB8598360222073186748on_nat(aTP_Lamm_sx(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),vEBT_c634343235235684882T_VEBT(aTP_Lamm_ot,aTP_Lamm_ou,Uua)) ).

% ATP.lambda_1145
tff(fact_11323_ATP_Olambda__1146,axiom,
    ! [Uu: fun_VE3775134097089128360on_nat,Uua: vEBT_VEBT,Uub: nat,Uuc: option4927543243414619207at_nat,Uud: nat,Uue: array_VEBT_VEBTi,Uuf: vEBT_VEBTi,Uug: product_unit] : aa_Pro451875010097498139on_nat(aa_VEB7937681756907646997on_nat(aa_arr5896957451705369866on_nat(aa_nat3269078897214415691on_nat(aa_opt5459845657693669015on_nat(aa_nat8168140587832489004on_nat(aa_VEB4878072573979604067on_nat(aTP_Lamm_sj(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug) = produc2190226783763740553on_nat(aa_VEB5714833192853586542on_nat(aa_arr7759188791488537571on_nat(aa_nat7273124837800733860on_nat(aa_opt1247461068388908528on_nat(aa_nat309676689105111045on_nat(aa_VEB8598360222073186748on_nat(aTP_Lamm_si(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),vEBT_c634343235235684882T_VEBT(aTP_Lamm_ot,aTP_Lamm_ou,Uua)) ).

% ATP.lambda_1146
tff(fact_11324_ATP_Olambda__1147,axiom,
    ! [Uu: $o,Uua: $o,Uub: $o,Uuc: $o,Uud: $o,Uue: $o,Uuf: code_integer,Uug: $o] : aa_o_char(aa_Cod7842134615204088097o_char(aa_o_f2742674633914528860o_char(aa_o_f9003794812476387647o_char(aa_o_f4183413333637126194o_char(aa_o_f2932186792858379797o_char(aa_o_f1143307120959153672o_char(aTP_Lamm_bci((Uu)),(Uua)),(Uub)),(Uuc)),(Uud)),(Uue)),Uuf),(Uug)) = produc4188289175737317920o_char(aa_o_f2742674633914528860o_char(aa_o_f9003794812476387647o_char(aa_o_f4183413333637126194o_char(aa_o_f2932186792858379797o_char(aa_o_f1143307120959153672o_char(aa_o_f3038258172971468011o_char(aTP_Lamm_bch((Uu)),(Uua)),(Uub)),(Uuc)),(Uud)),(Uue)),(Uug)),code_bit_cut_integer(Uuf)) ).

% ATP.lambda_1147
tff(fact_11325_ATP_Olambda__1148,axiom,
    ! [Uu: vEBT_VEBT,Uua: nat,Uub: option4927543243414619207at_nat,Uuc: nat,Uud: array_VEBT_VEBTi,Uue: vEBT_VEBTi,Uuf: product_prod_nat_nat,Uug: product_unit] : aa_Pro3581650104493980720_VEBTi(aa_Pro8034723660955798012_VEBTi(aa_VEB4262372667711301016_VEBTi(aa_arr9036764603281787897_VEBTi(aa_nat2771851207891374350_VEBTi(aa_opt7594342031245826264_VEBTi(aa_nat4475881176255417753_VEBTi(aTP_Lamm_tu(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug) = produc2624314226134418078_VEBTi(aa_Pro4062752345418257877_VEBTi(aa_VEB8867863569893255025_VEBTi(aa_arr3828982135699130450_VEBTi(aa_nat5629247741253790183_VEBTi(aa_opt9205748023383517873_VEBTi(aTP_Lamm_tt(Uua),Uub),Uuc),Uud),Uue),Uuf),vEBT_c634343235235684882T_VEBT(aTP_Lamm_ot,aTP_Lamm_ou,Uu)) ).

% ATP.lambda_1148
tff(fact_11326_ATP_Olambda__1149,axiom,
    ! [Uu: fun_Pr4171499318505651870_VEBTi,Uua: nat,Uub: nat,Uuc: array_VEBT_VEBTi,Uud: vEBT_VEBTi,Uue: nat,Uuf: nat,Uug: nat] :
      aa_nat5408703056993926195_VEBTi(aa_nat4282640288315822012_VEBTi(aa_nat7653869184832469701_VEBTi(aa_VEB4322370096674025769_VEBTi(aa_arr4526045539431534346_VEBTi(aa_nat9189989511310912415_VEBTi(aa_nat4025588890810027244_VEBTi(aTP_Lamm_abk(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug) = heap_T844888390831797134_VEBTi($ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uua),Uue),aa_nat5821310369566504398ap_nat(heap_Time_return_nat,Uua),aa_nat5821310369566504398ap_nat(heap_Time_return_nat,Uue)),aa_nat4282640288315822012_VEBTi(aa_nat7653869184832469701_VEBTi(aa_nat2287630409448422734_VEBTi(aa_VEB8340470449056147762_VEBTi(aa_arr7322700162901495443_VEBTi(aa_nat4321419670308649640_VEBTi(aa_nat695344571205917557_VEBTi(aTP_Lamm_abj(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug)) ).

% ATP.lambda_1149
tff(fact_11327_ATP_Olambda__1150,axiom,
    ! [Uu: fun_VE8118064890490510894_VEBTi,Uua: nat,Uub: nat,Uuc: array_VEBT_VEBTi,Uud: vEBT_VEBTi,Uue: nat,Uuf: nat,Uug: nat] :
      aa_nat5408703056993926195_VEBTi(aa_nat4282640288315822012_VEBTi(aa_nat7653869184832469701_VEBTi(aa_VEB4322370096674025769_VEBTi(aa_arr4526045539431534346_VEBTi(aa_nat9189989511310912415_VEBTi(aa_nat4025588890810027244_VEBTi(aTP_Lamm_nh(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug) = heap_T844888390831797134_VEBTi($ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uua),Uue),aa_nat5821310369566504398ap_nat(heap_Time_return_nat,Uua),aa_nat5821310369566504398ap_nat(heap_Time_return_nat,Uue)),aa_nat4282640288315822012_VEBTi(aa_nat7653869184832469701_VEBTi(aa_nat2287630409448422734_VEBTi(aa_VEB8340470449056147762_VEBTi(aa_arr7322700162901495443_VEBTi(aa_nat4321419670308649640_VEBTi(aa_nat695344571205917557_VEBTi(aTP_Lamm_ng(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug)) ).

% ATP.lambda_1150
tff(fact_11328_ATP_Olambda__1151,axiom,
    ! [Uu: nat,Uua: vEBT_VEBTi,Uub: nat,Uuc: nat,Uud: nat,Uue: nat,Uuf: $o,Uug: array_VEBT_VEBTi] :
      aa_arr4839786837114051083_VEBTi(aa_o_f5513965820244355694_VEBTi(aa_nat6231872576868200367_VEBTi(aa_nat3271849308718382136_VEBTi(aa_nat4180741733914107969_VEBTi(aa_nat6590260363808168906_VEBTi(aa_VEB8654527159822295598_VEBTi(aTP_Lamm_ir(Uu),Uua),Uub),Uuc),Uud),Uue),(Uuf)),Uug) = heap_T1006145433769338483_VEBTi($ite((Uuf),aa_nat5408703056993926195_VEBTi(aa_VEB4833742232818206615_VEBTi(vEBT_vebt_inserti,Uua),Uue),aa_VEB836174949890266638_VEBTi(heap_T3630416162098727440_VEBTi,Uua)),aa_arr2912372873376718191_VEBTi(aa_nat6445230840460233604_VEBTi(aa_nat3780430034849525329_VEBTi(aa_nat6565337644025228958_VEBTi(aTP_Lamm_iq(Uu),Uub),Uuc),Uud),Uug)) ).

% ATP.lambda_1151
tff(fact_11329_ATP_Olambda__1152,axiom,
    ! [Uu: nat,Uua: nat,Uub: array_VEBT_VEBTi,Uuc: vEBT_VEBTi,Uud: list_VEBT_VEBT,Uue: vEBT_VEBT,Uuf: product_prod_nat_nat,Uug: nat] : aa_nat947733127081326238on_nat(aa_Pro4465521767867336112on_nat(aa_VEB6043830584257997615on_nat(aa_lis5985300925819340672on_nat(aa_VEB1019093884839074554on_nat(aa_arr1617095457630768239on_nat(aa_nat5782952277816596784on_nat(aTP_Lamm_ue(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug) = heap_T8222160169144143993on_nat(aa_nat5821310369566504398ap_nat(vEBT_VEBT_highi(Uu),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uua),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat6020981482381522343on_nat(aa_Pro8919772687891686969on_nat(aa_VEB8035251902059208504on_nat(aa_lis5065134719111324425on_nat(aa_VEB1776269892753909763on_nat(aa_arr3452137061177157880on_nat(aa_nat8770382747262106937on_nat(aTP_Lamm_ud(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug)) ).

% ATP.lambda_1152
tff(fact_11330_ATP_Olambda__1153,axiom,
    ! [Uu: fun_Pr633315466666964198on_nat,Uua: nat,Uub: nat,Uuc: array_VEBT_VEBTi,Uud: vEBT_VEBTi,Uue: list_VEBT_VEBT,Uuf: vEBT_VEBT,Uug: nat] : aa_nat947733127081326238on_nat(aa_VEB1312864128696253885on_nat(aa_lis1761857624600698434on_nat(aa_VEB4172908723204034014on_nat(aa_arr2072991348567353407on_nat(aa_nat7185663501778404180on_nat(aa_nat4144773949132746529on_nat(aTP_Lamm_wa(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug) = heap_T8222160169144143993on_nat(aa_nat5821310369566504398ap_nat(vEBT_VEBT_highi(Uua),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uub),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat6020981482381522343on_nat(aa_VEB4460395522175602758on_nat(aa_lis6278591930053894219on_nat(aa_VEB6094637369740064103on_nat(aa_arr5396080262017029960on_nat(aa_nat7337942549949891037on_nat(aa_nat8478446459054042410on_nat(aTP_Lamm_vz(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug)) ).

% ATP.lambda_1153
tff(fact_11331_ATP_Olambda__1154,axiom,
    ! [Uu: fun_VE3775134097089128360on_nat,Uua: nat,Uub: nat,Uuc: array_VEBT_VEBTi,Uud: vEBT_VEBTi,Uue: list_VEBT_VEBT,Uuf: vEBT_VEBT,Uug: nat] : aa_nat947733127081326238on_nat(aa_VEB1312864128696253885on_nat(aa_lis1761857624600698434on_nat(aa_VEB4172908723204034014on_nat(aa_arr2072991348567353407on_nat(aa_nat7185663501778404180on_nat(aa_nat4144773949132746529on_nat(aTP_Lamm_ss(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug) = heap_T8222160169144143993on_nat(aa_nat5821310369566504398ap_nat(vEBT_VEBT_highi(Uua),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uub),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat6020981482381522343on_nat(aa_VEB4460395522175602758on_nat(aa_lis6278591930053894219on_nat(aa_VEB6094637369740064103on_nat(aa_arr5396080262017029960on_nat(aa_nat7337942549949891037on_nat(aa_nat8478446459054042410on_nat(aTP_Lamm_sr(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug)) ).

% ATP.lambda_1154
tff(fact_11332_ATP_Olambda__1155,axiom,
    ! [Uu: nat,Uua: nat,Uub: array_VEBT_VEBTi,Uuc: vEBT_VEBTi,Uud: nat,Uue: nat,Uuf: nat,Uug: nat] : aa_nat5408703056993926195_VEBTi(aa_nat4282640288315822012_VEBTi(aa_nat7653869184832469701_VEBTi(aa_nat2287630409448422734_VEBTi(aa_VEB8340470449056147762_VEBTi(aa_arr7322700162901495443_VEBTi(aa_nat4321419670308649640_VEBTi(aTP_Lamm_iy(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug) = heap_T844888390831797134_VEBTi(aa_nat5821310369566504398ap_nat(vEBT_VEBT_lowi(Uuf),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uua),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat4282640288315822012_VEBTi(aa_nat7653869184832469701_VEBTi(aa_nat2287630409448422734_VEBTi(aa_nat2878941900432676695_VEBTi(aa_VEB8966970847287376059_VEBTi(aa_arr4873757251330958748_VEBTi(aa_nat4224949603525894449_VEBTi(aTP_Lamm_ix(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug)) ).

% ATP.lambda_1155
tff(fact_11333_ATP_Olambda__1156,axiom,
    ! [Uu: fun_Pr342922516890243977on_nat,Uua: nat,Uub: nat,Uuc: array_VEBT_VEBTi,Uud: vEBT_VEBTi,Uue: product_prod_nat_nat,Uuf: nat,Uug: nat] : aa_nat947733127081326238on_nat(aa_nat6020981482381522343on_nat(aa_Pro8919772687891686969on_nat(aa_VEB358289369652911795on_nat(aa_arr4041137963511513512on_nat(aa_nat1363181691939120105on_nat(aa_nat4363128851979299954on_nat(aTP_Lamm_aao(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug) = heap_T2868974464944644318on_nat(aa_nat5408703056993926195_VEBTi(array_nth_VEBT_VEBTi(Uuc),Uug),aa_nat4669826966810537986on_nat(aa_nat107244183536939787on_nat(aa_Pro1663420989643744413on_nat(aa_VEB6384125456408274455on_nat(aa_arr4162587047475849484on_nat(aa_nat5386638315231209293on_nat(aa_nat7318487011899192278on_nat(aTP_Lamm_aan(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug)) ).

% ATP.lambda_1156
tff(fact_11334_ATP_Olambda__1157,axiom,
    ! [Uu: fun_VE8516060933855021849on_nat,Uua: nat,Uub: nat,Uuc: array_VEBT_VEBTi,Uud: vEBT_VEBTi,Uue: product_prod_nat_nat,Uuf: nat,Uug: nat] : aa_nat947733127081326238on_nat(aa_nat6020981482381522343on_nat(aa_Pro8919772687891686969on_nat(aa_VEB358289369652911795on_nat(aa_arr4041137963511513512on_nat(aa_nat1363181691939120105on_nat(aa_nat4363128851979299954on_nat(aTP_Lamm_no(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug) = heap_T2868974464944644318on_nat(aa_nat5408703056993926195_VEBTi(array_nth_VEBT_VEBTi(Uuc),Uug),aa_nat4669826966810537986on_nat(aa_nat107244183536939787on_nat(aa_Pro1663420989643744413on_nat(aa_VEB6384125456408274455on_nat(aa_arr4162587047475849484on_nat(aa_nat5386638315231209293on_nat(aa_nat7318487011899192278on_nat(aTP_Lamm_nn(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug)) ).

% ATP.lambda_1157
tff(fact_11335_ATP_Olambda__1158,axiom,
    ! [Uu: nat,Uua: array_VEBT_VEBTi,Uub: vEBT_VEBTi,Uuc: list_VEBT_VEBT,Uud: vEBT_VEBT,Uue: nat,Uuf: nat,Uug: product_unit] : aa_Pro451875010097498139on_nat(aa_nat3999351741935936176on_nat(aa_nat3263736196148507005on_nat(aa_VEB6005108705944495412on_nat(aa_lis8565807915274807301on_nat(aa_VEB8546404647211134847on_nat(aa_arr8208594633606790516on_nat(aTP_Lamm_oj(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug) = heap_T2868974464944644318on_nat(aa_nat5408703056993926195_VEBTi(array_nth_VEBT_VEBTi(Uua),Uuf),aa_nat4669826966810537986on_nat(aa_nat107244183536939787on_nat(aa_VEB6116742807553843114on_nat(aa_lis2576702180724645807on_nat(aa_VEB3484668625599252427on_nat(aa_arr4903404720305071532on_nat(aTP_Lamm_oi(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf)) ).

% ATP.lambda_1158
tff(fact_11336_ATP_Olambda__1159,axiom,
    ! [Uu: nat,Uua: array_VEBT_VEBTi,Uub: vEBT_VEBTi,Uuc: list_VEBT_VEBT,Uud: vEBT_VEBT,Uue: nat,Uuf: nat,Uug: vEBT_VEBTi] :
      aa_VEB2711320435582112505on_nat(aa_nat4669826966810537986on_nat(aa_nat107244183536939787on_nat(aa_VEB6116742807553843114on_nat(aa_lis2576702180724645807on_nat(aa_VEB3484668625599252427on_nat(aa_arr4903404720305071532on_nat(aTP_Lamm_oi(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug) = $let(
        aktnode: vEBT_VEBT,
        aktnode:= nth_VEBT_VEBT(Uuc,Uuf),
        heap_T3669509953089699273on_nat(aa_VEB2711320435582112505on_nat(vEBT_vebt_maxti,Uug),aa_VEB3755160987679246317on_nat(aa_VEB5212316371209427943on_nat(aa_nat7793430520228346548on_nat(aa_nat8410684691732023425on_nat(aa_VEB3020412023619788856on_nat(aa_VEB2867584020876442580on_nat(aa_arr7250883597209086773on_nat(aTP_Lamm_oh(Uu),Uua),Uub),Uud),Uue),Uuf),Uug),aktnode)) ) ).

% ATP.lambda_1159
tff(fact_11337_ATP_Olambda__1160,axiom,
    ! [Uu: nat,Uua: option4927543243414619207at_nat,Uub: nat,Uuc: array_VEBT_VEBTi,Uud: option4927543243414619207at_nat,Uue: nat,Uuf: list_VEBT_VEBT,Uug: vEBT_VEBT] :
      aa_VEB8530157447395038106Heap_o(aa_lis4690688361886208363Heap_o(aa_nat3831830007592743808Heap_o(aa_opt3748239118003421002Heap_o(aa_arr581739785191926499Heap_o(aa_nat105892859602166008Heap_o(aa_opt8417070463701818562Heap_o(aTP_Lamm_qr(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug) = heap_T7577940988442287570unit_o(aa_o_H2459438917461573107t_unit(refine_Imp_assert,
          ( ( Uua = Uud )
          & ( Uub = Uue ) )),aa_lis4443831354813659076Heap_o(aa_arr8974760139355794305Heap_o(aa_nat808430574047188546Heap_o(aTP_Lamm_qq(Uu),Uub),Uuc),Uuf)) ).

% ATP.lambda_1160
tff(fact_11338_ATP_Olambda__1161,axiom,
    ! [Uu: nat,Uua: nat,Uub: array_VEBT_VEBTi,Uuc: vEBT_VEBTi,Uud: list_VEBT_VEBT,Uue: vEBT_VEBT,Uuf: nat,Uug: nat] : aa_nat947733127081326238on_nat(aa_nat6020981482381522343on_nat(aa_VEB4460395522175602758on_nat(aa_lis6278591930053894219on_nat(aa_VEB6094637369740064103on_nat(aa_arr5396080262017029960on_nat(aa_nat7337942549949891037on_nat(aTP_Lamm_om(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug) = heap_T5999496708990702694on_nat(aa_o_H2459438917461573107t_unit(refine_Imp_assert,Uuf = aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Uu),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uua),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat3999351741935936176on_nat(aa_nat3263736196148507005on_nat(aa_VEB6005108705944495412on_nat(aa_lis8565807915274807301on_nat(aa_VEB8546404647211134847on_nat(aa_arr8208594633606790516on_nat(aa_nat1839186240859117749on_nat(aTP_Lamm_ol(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug)) ).

% ATP.lambda_1161
tff(fact_11339_ATP_Olambda__1162,axiom,
    ! [Uu: fun_Pr940977554174145406Heap_o,Uua: nat,Uub: nat,Uuc: array_VEBT_VEBTi,Uud: list_VEBT_VEBT,Uue: nat,Uuf: nat,Uug: nat] : aa_nat2910048432523430448Heap_o(aa_nat5592791454563588861Heap_o(aa_nat7429013343984486730Heap_o(aa_lis5270594155775668615Heap_o(aa_arr7588197057154790688Heap_o(aa_nat5373961893125994933Heap_o(aa_nat5062659879350109954Heap_o(aTP_Lamm_zz(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug) = heap_T7577940988442287570unit_o(aa_o_H2459438917461573107t_unit(refine_Imp_assert,Uug = size_s6755466524823107622T_VEBT(Uud)),aa_nat3811243159488243764Heap_o(aa_nat6178576981048102717Heap_o(aa_nat5674868329695104454Heap_o(aa_lis4631090546836902751Heap_o(aa_arr3453073379018417564Heap_o(aa_nat4764294232906884829Heap_o(aa_nat334188166746217062Heap_o(aTP_Lamm_zy(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug)) ).

% ATP.lambda_1162
tff(fact_11340_ATP_Olambda__1163,axiom,
    ! [Uu: fun_VE8181444127144381288Heap_o,Uua: nat,Uub: nat,Uuc: array_VEBT_VEBTi,Uud: list_VEBT_VEBT,Uue: nat,Uuf: nat,Uug: nat] : aa_nat2910048432523430448Heap_o(aa_nat5592791454563588861Heap_o(aa_nat7429013343984486730Heap_o(aa_lis5270594155775668615Heap_o(aa_arr7588197057154790688Heap_o(aa_nat5373961893125994933Heap_o(aa_nat5062659879350109954Heap_o(aTP_Lamm_py(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug) = heap_T7577940988442287570unit_o(aa_o_H2459438917461573107t_unit(refine_Imp_assert,Uug = size_s6755466524823107622T_VEBT(Uud)),aa_nat3811243159488243764Heap_o(aa_nat6178576981048102717Heap_o(aa_nat5674868329695104454Heap_o(aa_lis4631090546836902751Heap_o(aa_arr3453073379018417564Heap_o(aa_nat4764294232906884829Heap_o(aa_nat334188166746217062Heap_o(aTP_Lamm_px(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug)) ).

% ATP.lambda_1163
tff(fact_11341_ATP_Olambda__1164,axiom,
    ! [Uu: nat,Uua: array_VEBT_VEBTi,Uub: vEBT_VEBTi,Uuc: list_VEBT_VEBT,Uud: vEBT_VEBT,Uue: nat,Uuf: nat,Uug: product_unit] : aa_Pro451875010097498139on_nat(aa_nat3999351741935936176on_nat(aa_nat3263736196148507005on_nat(aa_VEB6005108705944495412on_nat(aa_lis8565807915274807301on_nat(aa_VEB8546404647211134847on_nat(aa_arr8208594633606790516on_nat(aTP_Lamm_ok(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug) = heap_T5999496708990702694on_nat(aa_o_H2459438917461573107t_unit(refine_Imp_assert,aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uuf),size_s6755466524823107622T_VEBT(Uuc))),aa_nat3999351741935936176on_nat(aa_nat3263736196148507005on_nat(aa_VEB6005108705944495412on_nat(aa_lis8565807915274807301on_nat(aa_VEB8546404647211134847on_nat(aa_arr8208594633606790516on_nat(aTP_Lamm_oj(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf)) ).

% ATP.lambda_1164
tff(fact_11342_ATP_Olambda__1165,axiom,
    ! [Uu: nat,Uua: nat,Uub: array_VEBT_VEBTi,Uuc: vEBT_VEBTi,Uud: product_prod_nat_nat,Uue: nat,Uuf: nat,Uug: vEBT_VEBTi] : aa_VEB2711320435582112505on_nat(aa_nat4669826966810537986on_nat(aa_nat107244183536939787on_nat(aa_Pro1663420989643744413on_nat(aa_VEB6384125456408274455on_nat(aa_arr4162587047475849484on_nat(aa_nat5386638315231209293on_nat(aTP_Lamm_eu(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug) = heap_T3669509953089699273on_nat(aa_VEB2711320435582112505on_nat(vEBT_vebt_minti,Uug),aa_VEB9161639482783362738on_nat(aa_nat6300058732544452155on_nat(aa_nat7962384762897155012on_nat(aa_Pro2771509044941117526on_nat(aa_VEB8816156001103073488on_nat(aa_arr6004237943432320069on_nat(aa_nat4340965201858767110on_nat(aTP_Lamm_et(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug)) ).

% ATP.lambda_1165
tff(fact_11343_ATP_Olambda__1166,axiom,
    ! [Uu: fun_Pr940977554174145406Heap_o,Uua: nat,Uub: nat,Uuc: array_VEBT_VEBTi,Uud: list_VEBT_VEBT,Uue: nat,Uuf: nat,Uug: product_unit] : aa_Pro3379321066291785715Heap_o(aa_nat3811243159488243764Heap_o(aa_nat6178576981048102717Heap_o(aa_lis4519787659412806998Heap_o(aa_arr7619904339668570131Heap_o(aa_nat1269258765096487380Heap_o(aa_nat3499107050721502685Heap_o(aTP_Lamm_aaa(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug) = heap_Time_bind_nat_o(array_len_VEBT_VEBTi(Uuc),aa_nat5592791454563588861Heap_o(aa_nat7429013343984486730Heap_o(aa_lis5270594155775668615Heap_o(aa_arr7588197057154790688Heap_o(aa_nat5373961893125994933Heap_o(aa_nat5062659879350109954Heap_o(aTP_Lamm_zz(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf)) ).

% ATP.lambda_1166
tff(fact_11344_ATP_Olambda__1167,axiom,
    ! [Uu: fun_VE8181444127144381288Heap_o,Uua: nat,Uub: nat,Uuc: array_VEBT_VEBTi,Uud: list_VEBT_VEBT,Uue: nat,Uuf: nat,Uug: product_unit] : aa_Pro3379321066291785715Heap_o(aa_nat3811243159488243764Heap_o(aa_nat6178576981048102717Heap_o(aa_lis4519787659412806998Heap_o(aa_arr7619904339668570131Heap_o(aa_nat1269258765096487380Heap_o(aa_nat3499107050721502685Heap_o(aTP_Lamm_pz(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug) = heap_Time_bind_nat_o(array_len_VEBT_VEBTi(Uuc),aa_nat5592791454563588861Heap_o(aa_nat7429013343984486730Heap_o(aa_lis5270594155775668615Heap_o(aa_arr7588197057154790688Heap_o(aa_nat5373961893125994933Heap_o(aa_nat5062659879350109954Heap_o(aTP_Lamm_py(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf)) ).

% ATP.lambda_1167
tff(fact_11345_ATP_Olambda__1168,axiom,
    ! [Uu: fun_Pr633315466666964198on_nat,Uua: vEBT_VEBT,Uub: nat,Uuc: option4927543243414619207at_nat,Uud: nat,Uue: array_VEBT_VEBTi,Uuf: vEBT_VEBTi,Uug: option4927543243414619207at_nat] : aa_opt1657869255803490932on_nat(aa_VEB5714833192853586542on_nat(aa_arr7759188791488537571on_nat(aa_nat7273124837800733860on_nat(aa_opt1247461068388908528on_nat(aa_nat309676689105111045on_nat(aa_VEB8598360222073186748on_nat(aTP_Lamm_xh(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug) = produc2606485630176857543on_nat(aa_opt8950567732158611302on_nat(aa_VEB8223317440372349570on_nat(aa_arr6272259649854615779on_nat(aa_nat261593688111926264on_nat(aa_opt4414834345254747970on_nat(aa_nat2224702164617786371on_nat(aa_VEB3185010184023139490on_nat(aTP_Lamm_xg(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug)) ).

% ATP.lambda_1168
tff(fact_11346_ATP_Olambda__1169,axiom,
    ! [Uu: fun_Pr633315466666964198on_nat,Uua: vEBT_VEBT,Uub: nat,Uuc: option4927543243414619207at_nat,Uud: nat,Uue: array_VEBT_VEBTi,Uuf: vEBT_VEBTi,Uug: option4927543243414619207at_nat] : aa_opt1657869255803490932on_nat(aa_VEB5714833192853586542on_nat(aa_arr7759188791488537571on_nat(aa_nat7273124837800733860on_nat(aa_opt1247461068388908528on_nat(aa_nat309676689105111045on_nat(aa_VEB8598360222073186748on_nat(aTP_Lamm_wf(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug) = produc2606485630176857543on_nat(aa_opt8950567732158611302on_nat(aa_VEB8223317440372349570on_nat(aa_arr6272259649854615779on_nat(aa_nat261593688111926264on_nat(aa_opt4414834345254747970on_nat(aa_nat2224702164617786371on_nat(aa_VEB3185010184023139490on_nat(aTP_Lamm_we(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug)) ).

% ATP.lambda_1169
tff(fact_11347_ATP_Olambda__1170,axiom,
    ! [Uu: fun_Pr4404027214751104635_VEBTi,Uua: nat,Uub: option4927543243414619207at_nat,Uuc: nat,Uud: array_VEBT_VEBTi,Uue: vEBT_VEBTi,Uuf: product_prod_nat_nat,Uug: option4927543243414619207at_nat] : aa_opt8585651768545592073_VEBTi(aa_Pro4062752345418257877_VEBTi(aa_VEB8867863569893255025_VEBTi(aa_arr3828982135699130450_VEBTi(aa_nat5629247741253790183_VEBTi(aa_opt9205748023383517873_VEBTi(aa_nat945706057799272946_VEBTi(aTP_Lamm_vn(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug) = produc7864503674192730076_VEBTi(aa_opt7507675525305149691_VEBTi(aa_Pro1234189438217840325_VEBTi(aa_VEB660175962507042495_VEBTi(aa_arr8367260621226259636_VEBTi(aa_nat5784496157376858101_VEBTi(aa_opt81624670976341313_VEBTi(aa_nat5602383413110735830_VEBTi(aTP_Lamm_vm(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug)) ).

% ATP.lambda_1170
tff(fact_11348_ATP_Olambda__1171,axiom,
    ! [Uu: vEBT_VEBT,Uua: nat,Uub: option4927543243414619207at_nat,Uuc: nat,Uud: array_VEBT_VEBTi,Uue: vEBT_VEBTi,Uuf: option4927543243414619207at_nat,Uug: nat] : aa_nat2075221075791484188on_nat(aa_opt8950567732158611302on_nat(aa_VEB8223317440372349570on_nat(aa_arr6272259649854615779on_nat(aa_nat261593688111926264on_nat(aa_opt4414834345254747970on_nat(aa_nat2224702164617786371on_nat(aTP_Lamm_ui(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug) = produc6124225815318652422on_nat(aa_nat8553908474689004666on_nat(aa_opt7284239819431652166on_nat(aa_VEB7959562887951478976on_nat(aa_arr4020752153017264181on_nat(aa_nat3060827894968391926on_nat(aa_opt346861859059968706on_nat(aa_nat3229958882251406039on_nat(aTP_Lamm_uh(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug)) ).

% ATP.lambda_1171
tff(fact_11349_ATP_Olambda__1172,axiom,
    ! [Uu: nat,Uua: option4927543243414619207at_nat,Uub: nat,Uuc: array_VEBT_VEBTi,Uud: vEBT_VEBTi,Uue: product_prod_nat_nat,Uuf: option4927543243414619207at_nat,Uug: nat] : aa_nat7217600295869438385_VEBTi(aa_opt7507675525305149691_VEBTi(aa_Pro1234189438217840325_VEBTi(aa_VEB660175962507042495_VEBTi(aa_arr8367260621226259636_VEBTi(aa_nat5784496157376858101_VEBTi(aa_opt81624670976341313_VEBTi(aTP_Lamm_ts(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug) = produc9050507437146595227_VEBTi(aa_nat1139649368474441743_VEBTi(aa_opt762848812907537627_VEBTi(aa_Pro4022776811541674023_VEBTi(aa_VEB3655672368276914755_VEBTi(aa_arr737849036559383332_VEBTi(aa_nat7609637249401053881_VEBTi(aa_opt3216071514793242883_VEBTi(aTP_Lamm_tr(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug)) ).

% ATP.lambda_1172
tff(fact_11350_ATP_Olambda__1173,axiom,
    ! [Uu: fun_VE3775134097089128360on_nat,Uua: vEBT_VEBT,Uub: nat,Uuc: option4927543243414619207at_nat,Uud: nat,Uue: array_VEBT_VEBTi,Uuf: vEBT_VEBTi,Uug: option4927543243414619207at_nat] : aa_opt1657869255803490932on_nat(aa_VEB5714833192853586542on_nat(aa_arr7759188791488537571on_nat(aa_nat7273124837800733860on_nat(aa_opt1247461068388908528on_nat(aa_nat309676689105111045on_nat(aa_VEB8598360222073186748on_nat(aTP_Lamm_sx(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug) = produc2606485630176857543on_nat(aa_opt8950567732158611302on_nat(aa_VEB8223317440372349570on_nat(aa_arr6272259649854615779on_nat(aa_nat261593688111926264on_nat(aa_opt4414834345254747970on_nat(aa_nat2224702164617786371on_nat(aa_VEB3185010184023139490on_nat(aTP_Lamm_sw(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug)) ).

% ATP.lambda_1173
tff(fact_11351_ATP_Olambda__1174,axiom,
    ! [Uu: fun_VE3775134097089128360on_nat,Uua: vEBT_VEBT,Uub: nat,Uuc: option4927543243414619207at_nat,Uud: nat,Uue: array_VEBT_VEBTi,Uuf: vEBT_VEBTi,Uug: option4927543243414619207at_nat] : aa_opt1657869255803490932on_nat(aa_VEB5714833192853586542on_nat(aa_arr7759188791488537571on_nat(aa_nat7273124837800733860on_nat(aa_opt1247461068388908528on_nat(aa_nat309676689105111045on_nat(aa_VEB8598360222073186748on_nat(aTP_Lamm_si(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug) = produc2606485630176857543on_nat(aa_opt8950567732158611302on_nat(aa_VEB8223317440372349570on_nat(aa_arr6272259649854615779on_nat(aa_nat261593688111926264on_nat(aa_opt4414834345254747970on_nat(aa_nat2224702164617786371on_nat(aa_VEB3185010184023139490on_nat(aTP_Lamm_sh(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug)) ).

% ATP.lambda_1174
tff(fact_11352_ATP_Olambda__1175,axiom,
    ! [Uu: fun_VE2018681615261037117_VEBTi,Uua: nat,Uub: option4927543243414619207at_nat,Uuc: nat,Uud: array_VEBT_VEBTi,Uue: vEBT_VEBTi,Uuf: product_prod_nat_nat,Uug: option4927543243414619207at_nat] : aa_opt8585651768545592073_VEBTi(aa_Pro4062752345418257877_VEBTi(aa_VEB8867863569893255025_VEBTi(aa_arr3828982135699130450_VEBTi(aa_nat5629247741253790183_VEBTi(aa_opt9205748023383517873_VEBTi(aa_nat945706057799272946_VEBTi(aTP_Lamm_rq(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug) = produc7864503674192730076_VEBTi(aa_opt7507675525305149691_VEBTi(aa_Pro1234189438217840325_VEBTi(aa_VEB660175962507042495_VEBTi(aa_arr8367260621226259636_VEBTi(aa_nat5784496157376858101_VEBTi(aa_opt81624670976341313_VEBTi(aa_nat5602383413110735830_VEBTi(aTP_Lamm_rp(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug)) ).

% ATP.lambda_1175
tff(fact_11353_ATP_Olambda__1176,axiom,
    ! [Uu: vEBT_VEBT,Uua: nat,Uub: option4927543243414619207at_nat,Uuc: nat,Uud: array_VEBT_VEBTi,Uue: vEBT_VEBTi,Uuf: option4927543243414619207at_nat,Uug: nat] : aa_nat2075221075791484188on_nat(aa_opt8950567732158611302on_nat(aa_VEB8223317440372349570on_nat(aa_arr6272259649854615779on_nat(aa_nat261593688111926264on_nat(aa_opt4414834345254747970on_nat(aa_nat2224702164617786371on_nat(aTP_Lamm_or(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug) = produc6124225815318652422on_nat(aa_nat8553908474689004666on_nat(aa_opt7284239819431652166on_nat(aa_VEB7959562887951478976on_nat(aa_arr4020752153017264181on_nat(aa_nat3060827894968391926on_nat(aa_opt346861859059968706on_nat(aa_nat3229958882251406039on_nat(aTP_Lamm_oq(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug)) ).

% ATP.lambda_1176
tff(fact_11354_ATP_Olambda__1177,axiom,
    ! [Uu: nat,Uua: nat,Uub: array_VEBT_VEBTi,Uuc: vEBT_VEBTi,Uud: product_prod_nat_nat,Uue: nat,Uuf: nat,Uug: vEBT_VEBTi,Uuh: option_nat] :
      aa_opt9163293676379396302on_nat(aa_VEB9161639482783362738on_nat(aa_nat6300058732544452155on_nat(aa_nat7962384762897155012on_nat(aa_Pro2771509044941117526on_nat(aa_VEB8816156001103073488on_nat(aa_arr6004237943432320069on_nat(aa_nat4340965201858767110on_nat(aTP_Lamm_et(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh) = $ite(
        ( ( Uuh != none_nat )
        & aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_greater,aa_nat_option_nat(some_nat,Uue)),Uuh) ),
        heap_T3669509953089699273on_nat(aa_nat947733127081326238on_nat(aa_VEB3566417204837707778on_nat(vEBT_vebt_predi,Uug),Uue),aa_nat9127621225095157975on_nat(aTP_Lamm_ep(Uua),Uuf)),
        heap_T3669509953089699273on_nat(aa_nat947733127081326238on_nat(aa_VEB3566417204837707778on_nat(vEBT_vebt_predi,Uuc),Uuf),aa_Pro4885747571561342816on_nat(aa_arr7927785801258559901on_nat(aa_nat6148929329942773342on_nat(aTP_Lamm_es(Uu),Uua),Uub),Uud)) ) ).

% ATP.lambda_1177
tff(fact_11355_ATP_Olambda__1178,axiom,
    ! [Uu: fun_Pr940977554174145406Heap_o,Uua: nat,Uub: nat,Uuc: array_VEBT_VEBTi,Uud: list_VEBT_VEBT,Uue: nat,Uuf: nat,Uug: nat,Uuh: product_unit] :
      aa_Pro3379321066291785715Heap_o(aa_nat3811243159488243764Heap_o(aa_nat6178576981048102717Heap_o(aa_nat5674868329695104454Heap_o(aa_lis4631090546836902751Heap_o(aa_arr3453073379018417564Heap_o(aa_nat4764294232906884829Heap_o(aa_nat334188166746217062Heap_o(aTP_Lamm_zy(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uue),Uug),
        heap_T7577940988442287570unit_o(aa_o_H2459438917461573107t_unit(refine_Imp_assert,
            ( ( Uue = aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Uua),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uub),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) )
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uue),size_s6755466524823107622T_VEBT(Uud)) )),aa_nat3811243159488243764Heap_o(aa_nat6178576981048102717Heap_o(aa_lis4519787659412806998Heap_o(aa_arr7619904339668570131Heap_o(aTP_Lamm_zx(Uu),Uuc),Uud),Uue),Uuf)),
        aa_o_H7399033255965945470Heap_o(heap_Time_return_o,$false) ) ).

% ATP.lambda_1178
tff(fact_11356_ATP_Olambda__1179,axiom,
    ! [Uu: fun_VE8181444127144381288Heap_o,Uua: nat,Uub: nat,Uuc: array_VEBT_VEBTi,Uud: list_VEBT_VEBT,Uue: nat,Uuf: nat,Uug: nat,Uuh: product_unit] :
      aa_Pro3379321066291785715Heap_o(aa_nat3811243159488243764Heap_o(aa_nat6178576981048102717Heap_o(aa_nat5674868329695104454Heap_o(aa_lis4631090546836902751Heap_o(aa_arr3453073379018417564Heap_o(aa_nat4764294232906884829Heap_o(aa_nat334188166746217062Heap_o(aTP_Lamm_px(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uue),Uug),
        heap_T7577940988442287570unit_o(aa_o_H2459438917461573107t_unit(refine_Imp_assert,
            ( ( Uue = aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Uua),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uub),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) )
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uue),size_s6755466524823107622T_VEBT(Uud)) )),aa_nat3811243159488243764Heap_o(aa_nat6178576981048102717Heap_o(aa_lis4519787659412806998Heap_o(aa_arr7619904339668570131Heap_o(aTP_Lamm_pw(Uu),Uuc),Uud),Uue),Uuf)),
        aa_o_H7399033255965945470Heap_o(heap_Time_return_o,$false) ) ).

% ATP.lambda_1179
tff(fact_11357_ATP_Olambda__1180,axiom,
    ! [Uu: fun_Pr633315466666964198on_nat,Uua: nat,Uub: option4927543243414619207at_nat,Uuc: nat,Uud: array_VEBT_VEBTi,Uue: vEBT_VEBTi,Uuf: list_VEBT_VEBT,Uug: vEBT_VEBT,Uuh: product_unit] : aa_Pro451875010097498139on_nat(aa_VEB1832073013541904922on_nat(aa_lis6328568131235813867on_nat(aa_VEB3884484516340040293on_nat(aa_arr5442694692229379418on_nat(aa_nat4425110079127750043on_nat(aa_opt7218664994477926375on_nat(aa_nat7200185421170774908on_nat(aTP_Lamm_xe(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh) = case_o8344607093967974880at_nat(heap_T3487192422709364219on_nat(none_nat),aa_VEB6310895951922269862on_nat(aa_lis7455679124549436279on_nat(aa_VEB7601412409763292529on_nat(aa_arr3206634112104793958on_nat(aa_nat2842375153920543911on_nat(aa_nat1315775135507255344on_nat(aTP_Lamm_xd(Uu),Uua),Uuc),Uud),Uue),Uuf),Uug),Uub) ).

% ATP.lambda_1180
tff(fact_11358_ATP_Olambda__1181,axiom,
    ! [Uu: fun_Pr633315466666964198on_nat,Uua: nat,Uub: option4927543243414619207at_nat,Uuc: nat,Uud: array_VEBT_VEBTi,Uue: vEBT_VEBTi,Uuf: list_VEBT_VEBT,Uug: vEBT_VEBT,Uuh: product_unit] : aa_Pro451875010097498139on_nat(aa_VEB1832073013541904922on_nat(aa_lis6328568131235813867on_nat(aa_VEB3884484516340040293on_nat(aa_arr5442694692229379418on_nat(aa_nat4425110079127750043on_nat(aa_opt7218664994477926375on_nat(aa_nat7200185421170774908on_nat(aTP_Lamm_wc(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh) = case_o8344607093967974880at_nat(heap_T3487192422709364219on_nat(none_nat),aa_VEB6310895951922269862on_nat(aa_lis7455679124549436279on_nat(aa_VEB7601412409763292529on_nat(aa_arr3206634112104793958on_nat(aa_nat2842375153920543911on_nat(aa_nat1315775135507255344on_nat(aTP_Lamm_wb(Uu),Uua),Uuc),Uud),Uue),Uuf),Uug),Uub) ).

% ATP.lambda_1181
tff(fact_11359_ATP_Olambda__1182,axiom,
    ! [Uu: fun_VE3775134097089128360on_nat,Uua: nat,Uub: option4927543243414619207at_nat,Uuc: nat,Uud: array_VEBT_VEBTi,Uue: vEBT_VEBTi,Uuf: list_VEBT_VEBT,Uug: vEBT_VEBT,Uuh: product_unit] : aa_Pro451875010097498139on_nat(aa_VEB1832073013541904922on_nat(aa_lis6328568131235813867on_nat(aa_VEB3884484516340040293on_nat(aa_arr5442694692229379418on_nat(aa_nat4425110079127750043on_nat(aa_opt7218664994477926375on_nat(aa_nat7200185421170774908on_nat(aTP_Lamm_su(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh) = case_o8344607093967974880at_nat(heap_T3487192422709364219on_nat(none_nat),aa_VEB6310895951922269862on_nat(aa_lis7455679124549436279on_nat(aa_VEB7601412409763292529on_nat(aa_arr3206634112104793958on_nat(aa_nat2842375153920543911on_nat(aa_nat1315775135507255344on_nat(aTP_Lamm_st(Uu),Uua),Uuc),Uud),Uue),Uuf),Uug),Uub) ).

% ATP.lambda_1182
tff(fact_11360_ATP_Olambda__1183,axiom,
    ! [Uu: fun_VE3775134097089128360on_nat,Uua: nat,Uub: option4927543243414619207at_nat,Uuc: nat,Uud: array_VEBT_VEBTi,Uue: vEBT_VEBTi,Uuf: list_VEBT_VEBT,Uug: vEBT_VEBT,Uuh: product_unit] : aa_Pro451875010097498139on_nat(aa_VEB1832073013541904922on_nat(aa_lis6328568131235813867on_nat(aa_VEB3884484516340040293on_nat(aa_arr5442694692229379418on_nat(aa_nat4425110079127750043on_nat(aa_opt7218664994477926375on_nat(aa_nat7200185421170774908on_nat(aTP_Lamm_sf(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh) = case_o8344607093967974880at_nat(heap_T3487192422709364219on_nat(none_nat),aa_VEB6310895951922269862on_nat(aa_lis7455679124549436279on_nat(aa_VEB7601412409763292529on_nat(aa_arr3206634112104793958on_nat(aa_nat2842375153920543911on_nat(aa_nat1315775135507255344on_nat(aTP_Lamm_se(Uu),Uua),Uuc),Uud),Uue),Uuf),Uug),Uub) ).

% ATP.lambda_1183
tff(fact_11361_ATP_Olambda__1184,axiom,
    ! [Uu: fun_Pr4404027214751104635_VEBTi,Uua: vEBT_VEBT,Uub: nat,Uuc: option4927543243414619207at_nat,Uud: nat,Uue: array_VEBT_VEBTi,Uuf: vEBT_VEBTi,Uug: product_prod_nat_nat,Uuh: product_unit] : aa_Pro3581650104493980720_VEBTi(aa_Pro8034723660955798012_VEBTi(aa_VEB4262372667711301016_VEBTi(aa_arr9036764603281787897_VEBTi(aa_nat2771851207891374350_VEBTi(aa_opt7594342031245826264_VEBTi(aa_nat4475881176255417753_VEBTi(aa_VEB8590999954632166712_VEBTi(aTP_Lamm_vo(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh) = produc2624314226134418078_VEBTi(aa_Pro4062752345418257877_VEBTi(aa_VEB8867863569893255025_VEBTi(aa_arr3828982135699130450_VEBTi(aa_nat5629247741253790183_VEBTi(aa_opt9205748023383517873_VEBTi(aa_nat945706057799272946_VEBTi(aTP_Lamm_vn(Uu),Uub),Uuc),Uud),Uue),Uuf),Uug),vEBT_c634343235235684882T_VEBT(aTP_Lamm_ot,aTP_Lamm_ou,Uua)) ).

% ATP.lambda_1184
tff(fact_11362_ATP_Olambda__1185,axiom,
    ! [Uu: fun_VE2018681615261037117_VEBTi,Uua: vEBT_VEBT,Uub: nat,Uuc: option4927543243414619207at_nat,Uud: nat,Uue: array_VEBT_VEBTi,Uuf: vEBT_VEBTi,Uug: product_prod_nat_nat,Uuh: product_unit] : aa_Pro3581650104493980720_VEBTi(aa_Pro8034723660955798012_VEBTi(aa_VEB4262372667711301016_VEBTi(aa_arr9036764603281787897_VEBTi(aa_nat2771851207891374350_VEBTi(aa_opt7594342031245826264_VEBTi(aa_nat4475881176255417753_VEBTi(aa_VEB8590999954632166712_VEBTi(aTP_Lamm_rr(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh) = produc2624314226134418078_VEBTi(aa_Pro4062752345418257877_VEBTi(aa_VEB8867863569893255025_VEBTi(aa_arr3828982135699130450_VEBTi(aa_nat5629247741253790183_VEBTi(aa_opt9205748023383517873_VEBTi(aa_nat945706057799272946_VEBTi(aTP_Lamm_rq(Uu),Uub),Uuc),Uud),Uue),Uuf),Uug),vEBT_c634343235235684882T_VEBT(aTP_Lamm_ot,aTP_Lamm_ou,Uua)) ).

% ATP.lambda_1185
tff(fact_11363_ATP_Olambda__1186,axiom,
    ! [Uu: nat,Uua: vEBT_VEBTi,Uub: vEBT_VEBT,Uuc: nat,Uud: nat,Uue: nat,Uuf: nat,Uug: $o,Uuh: array_VEBT_VEBTi] :
      aa_arr4839786837114051083_VEBTi(aa_o_f5513965820244355694_VEBTi(aa_nat6231872576868200367_VEBTi(aa_nat3271849308718382136_VEBTi(aa_nat4180741733914107969_VEBTi(aa_nat6590260363808168906_VEBTi(aa_VEB6485260183185377513_VEBTi(aa_VEB953658682601944675_VEBTi(aTP_Lamm_ta(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),(Uug)),Uuh) = heap_T1006145433769338483_VEBTi($ite((Uug),aa_nat5408703056993926195_VEBTi(aa_VEB4833742232818206615_VEBTi(aa_VEB6292075424854053302_VEBTi(vEBT_V3964819847710782039nserti,Uub),Uua),Uuf),aa_VEB836174949890266638_VEBTi(heap_T3630416162098727440_VEBTi,Uua)),aa_arr2912372873376718191_VEBTi(aa_nat6445230840460233604_VEBTi(aa_nat3780430034849525329_VEBTi(aa_nat6565337644025228958_VEBTi(aTP_Lamm_iq(Uu),Uuc),Uud),Uue),Uuh)) ).

% ATP.lambda_1186
tff(fact_11364_ATP_Olambda__1187,axiom,
    ! [Uu: fun_Pr4171499318505651870_VEBTi,Uua: nat,Uub: vEBT_VEBTi,Uuc: nat,Uud: nat,Uue: nat,Uuf: nat,Uug: $o,Uuh: array_VEBT_VEBTi] :
      aa_arr4839786837114051083_VEBTi(aa_o_f5513965820244355694_VEBTi(aa_nat6231872576868200367_VEBTi(aa_nat3271849308718382136_VEBTi(aa_nat4180741733914107969_VEBTi(aa_nat6590260363808168906_VEBTi(aa_VEB8654527159822295598_VEBTi(aa_nat7794021720236223159_VEBTi(aTP_Lamm_abc(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),(Uug)),Uuh) = heap_T1006145433769338483_VEBTi($ite((Uug),aa_nat5408703056993926195_VEBTi(aa_VEB4833742232818206615_VEBTi(produc5159149307777246319_VEBTi(Uu),Uub),Uuf),aa_VEB836174949890266638_VEBTi(heap_T3630416162098727440_VEBTi,Uub)),aa_arr2912372873376718191_VEBTi(aa_nat6445230840460233604_VEBTi(aa_nat3780430034849525329_VEBTi(aa_nat6565337644025228958_VEBTi(aTP_Lamm_iq(Uua),Uuc),Uud),Uue),Uuh)) ).

% ATP.lambda_1187
tff(fact_11365_ATP_Olambda__1188,axiom,
    ! [Uu: fun_VE8118064890490510894_VEBTi,Uua: nat,Uub: vEBT_VEBTi,Uuc: nat,Uud: nat,Uue: nat,Uuf: nat,Uug: $o,Uuh: array_VEBT_VEBTi] :
      aa_arr4839786837114051083_VEBTi(aa_o_f5513965820244355694_VEBTi(aa_nat6231872576868200367_VEBTi(aa_nat3271849308718382136_VEBTi(aa_nat4180741733914107969_VEBTi(aa_nat6590260363808168906_VEBTi(aa_VEB8654527159822295598_VEBTi(aa_nat7794021720236223159_VEBTi(aTP_Lamm_mz(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),(Uug)),Uuh) = heap_T1006145433769338483_VEBTi($ite((Uug),aa_nat5408703056993926195_VEBTi(aa_VEB4833742232818206615_VEBTi(Uu,Uub),Uuf),aa_VEB836174949890266638_VEBTi(heap_T3630416162098727440_VEBTi,Uub)),aa_arr2912372873376718191_VEBTi(aa_nat6445230840460233604_VEBTi(aa_nat3780430034849525329_VEBTi(aa_nat6565337644025228958_VEBTi(aTP_Lamm_iq(Uua),Uuc),Uud),Uue),Uuh)) ).

% ATP.lambda_1188
tff(fact_11366_ATP_Olambda__1189,axiom,
    ! [Uu: nat,Uua: array_VEBT_VEBTi,Uub: vEBT_VEBTi,Uuc: nat,Uud: nat,Uue: nat,Uuf: nat,Uug: $o,Uuh: vEBT_VEBTi] : aa_VEB836174949890266638_VEBTi(aa_o_f1343430628605260489_VEBTi(aa_nat4147387395820660958_VEBTi(aa_nat2236254811235047083_VEBTi(aa_nat5995673692660408824_VEBTi(aa_nat4988899645790341829_VEBTi(aa_VEB5075307909151313271_VEBTi(aa_arr5746817224660070252_VEBTi(aTP_Lamm_is(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),(Uug)),Uuh) = heap_T5099337393651448672_VEBTi(array_upd_VEBT_VEBTi(Uuf,Uuh,Uua),aa_o_f5513965820244355694_VEBTi(aa_nat6231872576868200367_VEBTi(aa_nat3271849308718382136_VEBTi(aa_nat4180741733914107969_VEBTi(aa_nat6590260363808168906_VEBTi(aa_VEB8654527159822295598_VEBTi(aTP_Lamm_ir(Uu),Uub),Uuc),Uud),Uue),Uuf),(Uug))) ).

% ATP.lambda_1189
tff(fact_11367_ATP_Olambda__1190,axiom,
    ! [Uu: fun_Pr633315466666964198on_nat,Uua: nat,Uub: nat,Uuc: array_VEBT_VEBTi,Uud: vEBT_VEBTi,Uue: list_VEBT_VEBT,Uuf: vEBT_VEBT,Uug: product_prod_nat_nat,Uuh: nat] : aa_nat947733127081326238on_nat(aa_Pro4465521767867336112on_nat(aa_VEB6043830584257997615on_nat(aa_lis5985300925819340672on_nat(aa_VEB1019093884839074554on_nat(aa_arr1617095457630768239on_nat(aa_nat5782952277816596784on_nat(aa_nat7314605301869835321on_nat(aTP_Lamm_xc(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh) = heap_T8222160169144143993on_nat(aa_nat5821310369566504398ap_nat(vEBT_VEBT_highi(Uua),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uub),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat6020981482381522343on_nat(aa_Pro8919772687891686969on_nat(aa_VEB8035251902059208504on_nat(aa_lis5065134719111324425on_nat(aa_VEB1776269892753909763on_nat(aa_arr3452137061177157880on_nat(aa_nat8770382747262106937on_nat(aa_nat388734249307221954on_nat(aTP_Lamm_xb(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh)) ).

% ATP.lambda_1190
tff(fact_11368_ATP_Olambda__1191,axiom,
    ! [Uu: fun_VE3775134097089128360on_nat,Uua: nat,Uub: nat,Uuc: array_VEBT_VEBTi,Uud: vEBT_VEBTi,Uue: list_VEBT_VEBT,Uuf: vEBT_VEBT,Uug: product_prod_nat_nat,Uuh: nat] : aa_nat947733127081326238on_nat(aa_Pro4465521767867336112on_nat(aa_VEB6043830584257997615on_nat(aa_lis5985300925819340672on_nat(aa_VEB1019093884839074554on_nat(aa_arr1617095457630768239on_nat(aa_nat5782952277816596784on_nat(aa_nat7314605301869835321on_nat(aTP_Lamm_sd(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh) = heap_T8222160169144143993on_nat(aa_nat5821310369566504398ap_nat(vEBT_VEBT_highi(Uua),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uub),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat6020981482381522343on_nat(aa_Pro8919772687891686969on_nat(aa_VEB8035251902059208504on_nat(aa_lis5065134719111324425on_nat(aa_VEB1776269892753909763on_nat(aa_arr3452137061177157880on_nat(aa_nat8770382747262106937on_nat(aa_nat388734249307221954on_nat(aTP_Lamm_sc(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh)) ).

% ATP.lambda_1191
tff(fact_11369_ATP_Olambda__1192,axiom,
    ! [Uu: nat,Uua: nat,Uub: array_VEBT_VEBTi,Uuc: vEBT_VEBTi,Uud: nat,Uue: nat,Uuf: nat,Uug: nat,Uuh: nat] : aa_nat5408703056993926195_VEBTi(aa_nat4282640288315822012_VEBTi(aa_nat7653869184832469701_VEBTi(aa_nat2287630409448422734_VEBTi(aa_nat2878941900432676695_VEBTi(aa_VEB8966970847287376059_VEBTi(aa_arr4873757251330958748_VEBTi(aa_nat4224949603525894449_VEBTi(aTP_Lamm_ix(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh) = heap_T844888390831797134_VEBTi(aa_nat5821310369566504398ap_nat(vEBT_VEBT_highi(Uuf),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uua),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat4282640288315822012_VEBTi(aa_nat7653869184832469701_VEBTi(aa_nat2287630409448422734_VEBTi(aa_nat2878941900432676695_VEBTi(aa_nat532351713702512864_VEBTi(aa_VEB7864109228220129220_VEBTi(aa_arr7707872583627363365_VEBTi(aa_nat3757413148909559610_VEBTi(aTP_Lamm_iw(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh)) ).

% ATP.lambda_1192
tff(fact_11370_ATP_Olambda__1193,axiom,
    ! [Uu: fun_Pr4171499318505651870_VEBTi,Uua: nat,Uub: nat,Uuc: array_VEBT_VEBTi,Uud: vEBT_VEBTi,Uue: nat,Uuf: nat,Uug: nat,Uuh: nat] : aa_nat5408703056993926195_VEBTi(aa_nat4282640288315822012_VEBTi(aa_nat7653869184832469701_VEBTi(aa_nat2287630409448422734_VEBTi(aa_VEB8340470449056147762_VEBTi(aa_arr7322700162901495443_VEBTi(aa_nat4321419670308649640_VEBTi(aa_nat695344571205917557_VEBTi(aTP_Lamm_abj(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh) = heap_T844888390831797134_VEBTi(aa_nat5821310369566504398ap_nat(vEBT_VEBT_lowi(Uug),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uub),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat4282640288315822012_VEBTi(aa_nat7653869184832469701_VEBTi(aa_nat2287630409448422734_VEBTi(aa_nat2878941900432676695_VEBTi(aa_VEB8966970847287376059_VEBTi(aa_arr4873757251330958748_VEBTi(aa_nat4224949603525894449_VEBTi(aa_nat7906690825329656190_VEBTi(aTP_Lamm_abi(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh)) ).

% ATP.lambda_1193
tff(fact_11371_ATP_Olambda__1194,axiom,
    ! [Uu: fun_VE8118064890490510894_VEBTi,Uua: nat,Uub: nat,Uuc: array_VEBT_VEBTi,Uud: vEBT_VEBTi,Uue: nat,Uuf: nat,Uug: nat,Uuh: nat] : aa_nat5408703056993926195_VEBTi(aa_nat4282640288315822012_VEBTi(aa_nat7653869184832469701_VEBTi(aa_nat2287630409448422734_VEBTi(aa_VEB8340470449056147762_VEBTi(aa_arr7322700162901495443_VEBTi(aa_nat4321419670308649640_VEBTi(aa_nat695344571205917557_VEBTi(aTP_Lamm_ng(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh) = heap_T844888390831797134_VEBTi(aa_nat5821310369566504398ap_nat(vEBT_VEBT_lowi(Uug),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uub),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat4282640288315822012_VEBTi(aa_nat7653869184832469701_VEBTi(aa_nat2287630409448422734_VEBTi(aa_nat2878941900432676695_VEBTi(aa_VEB8966970847287376059_VEBTi(aa_arr4873757251330958748_VEBTi(aa_nat4224949603525894449_VEBTi(aa_nat7906690825329656190_VEBTi(aTP_Lamm_nf(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh)) ).

% ATP.lambda_1194
tff(fact_11372_ATP_Olambda__1195,axiom,
    ! [Uu: fun_Pr633315466666964198on_nat,Uua: nat,Uub: array_VEBT_VEBTi,Uuc: vEBT_VEBTi,Uud: list_VEBT_VEBT,Uue: vEBT_VEBT,Uuf: nat,Uug: nat,Uuh: product_unit] : aa_Pro451875010097498139on_nat(aa_nat3999351741935936176on_nat(aa_nat3263736196148507005on_nat(aa_VEB6005108705944495412on_nat(aa_lis8565807915274807301on_nat(aa_VEB8546404647211134847on_nat(aa_arr8208594633606790516on_nat(aa_nat1839186240859117749on_nat(aTP_Lamm_vw(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh) = heap_T2868974464944644318on_nat(aa_nat5408703056993926195_VEBTi(array_nth_VEBT_VEBTi(Uub),Uug),aa_nat4669826966810537986on_nat(aa_nat107244183536939787on_nat(aa_VEB6116742807553843114on_nat(aa_lis2576702180724645807on_nat(aa_VEB3484668625599252427on_nat(aa_arr4903404720305071532on_nat(aa_nat907479165128101953on_nat(aTP_Lamm_vv(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug)) ).

% ATP.lambda_1195
tff(fact_11373_ATP_Olambda__1196,axiom,
    ! [Uu: fun_VE3775134097089128360on_nat,Uua: nat,Uub: array_VEBT_VEBTi,Uuc: vEBT_VEBTi,Uud: list_VEBT_VEBT,Uue: vEBT_VEBT,Uuf: nat,Uug: nat,Uuh: product_unit] : aa_Pro451875010097498139on_nat(aa_nat3999351741935936176on_nat(aa_nat3263736196148507005on_nat(aa_VEB6005108705944495412on_nat(aa_lis8565807915274807301on_nat(aa_VEB8546404647211134847on_nat(aa_arr8208594633606790516on_nat(aa_nat1839186240859117749on_nat(aTP_Lamm_so(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh) = heap_T2868974464944644318on_nat(aa_nat5408703056993926195_VEBTi(array_nth_VEBT_VEBTi(Uub),Uug),aa_nat4669826966810537986on_nat(aa_nat107244183536939787on_nat(aa_VEB6116742807553843114on_nat(aa_lis2576702180724645807on_nat(aa_VEB3484668625599252427on_nat(aa_arr4903404720305071532on_nat(aa_nat907479165128101953on_nat(aTP_Lamm_sn(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug)) ).

% ATP.lambda_1196
tff(fact_11374_ATP_Olambda__1197,axiom,
    ! [Uu: fun_Pr633315466666964198on_nat,Uua: nat,Uub: array_VEBT_VEBTi,Uuc: vEBT_VEBTi,Uud: list_VEBT_VEBT,Uue: vEBT_VEBT,Uuf: nat,Uug: nat,Uuh: vEBT_VEBTi] :
      aa_VEB2711320435582112505on_nat(aa_nat4669826966810537986on_nat(aa_nat107244183536939787on_nat(aa_VEB6116742807553843114on_nat(aa_lis2576702180724645807on_nat(aa_VEB3484668625599252427on_nat(aa_arr4903404720305071532on_nat(aa_nat907479165128101953on_nat(aTP_Lamm_vv(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh) = $let(
        aktnode: vEBT_VEBT,
        aktnode:= nth_VEBT_VEBT(Uud,Uug),
        heap_T3669509953089699273on_nat(aa_VEB2711320435582112505on_nat(vEBT_vebt_maxti,Uuh),aa_VEB3755160987679246317on_nat(aa_VEB5212316371209427943on_nat(aa_nat7793430520228346548on_nat(aa_nat8410684691732023425on_nat(aa_VEB3020412023619788856on_nat(aa_VEB2867584020876442580on_nat(aa_arr7250883597209086773on_nat(aa_nat6287673228857293642on_nat(aTP_Lamm_vu(Uu),Uua),Uub),Uuc),Uue),Uuf),Uug),Uuh),aktnode)) ) ).

% ATP.lambda_1197
tff(fact_11375_ATP_Olambda__1198,axiom,
    ! [Uu: fun_VE3775134097089128360on_nat,Uua: nat,Uub: array_VEBT_VEBTi,Uuc: vEBT_VEBTi,Uud: list_VEBT_VEBT,Uue: vEBT_VEBT,Uuf: nat,Uug: nat,Uuh: vEBT_VEBTi] :
      aa_VEB2711320435582112505on_nat(aa_nat4669826966810537986on_nat(aa_nat107244183536939787on_nat(aa_VEB6116742807553843114on_nat(aa_lis2576702180724645807on_nat(aa_VEB3484668625599252427on_nat(aa_arr4903404720305071532on_nat(aa_nat907479165128101953on_nat(aTP_Lamm_sn(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh) = $let(
        aktnode: vEBT_VEBT,
        aktnode:= nth_VEBT_VEBT(Uud,Uug),
        heap_T3669509953089699273on_nat(aa_VEB2711320435582112505on_nat(vEBT_vebt_maxti,Uuh),aa_VEB3755160987679246317on_nat(aa_VEB5212316371209427943on_nat(aa_nat7793430520228346548on_nat(aa_nat8410684691732023425on_nat(aa_VEB3020412023619788856on_nat(aa_VEB2867584020876442580on_nat(aa_arr7250883597209086773on_nat(aa_nat6287673228857293642on_nat(aTP_Lamm_sm(Uu),Uua),Uub),Uuc),Uue),Uuf),Uug),Uuh),aktnode)) ) ).

% ATP.lambda_1198
tff(fact_11376_ATP_Olambda__1199,axiom,
    ! [Uu: fun_Pr940977554174145406Heap_o,Uua: nat,Uub: option4927543243414619207at_nat,Uuc: nat,Uud: array_VEBT_VEBTi,Uue: option4927543243414619207at_nat,Uuf: nat,Uug: list_VEBT_VEBT,Uuh: vEBT_VEBT] :
      aa_VEB8530157447395038106Heap_o(aa_lis4690688361886208363Heap_o(aa_nat3831830007592743808Heap_o(aa_opt3748239118003421002Heap_o(aa_arr581739785191926499Heap_o(aa_nat105892859602166008Heap_o(aa_opt8417070463701818562Heap_o(aa_nat8831994779944892035Heap_o(aTP_Lamm_aae(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh) = heap_T7577940988442287570unit_o(aa_o_H2459438917461573107t_unit(refine_Imp_assert,
          ( ( Uub = Uue )
          & ( Uuc = Uuf ) )),aa_lis4443831354813659076Heap_o(aa_arr8974760139355794305Heap_o(aa_nat808430574047188546Heap_o(aa_nat9059937328556229451Heap_o(aTP_Lamm_aad(Uu),Uua),Uuc),Uud),Uug)) ).

% ATP.lambda_1199
tff(fact_11377_ATP_Olambda__1200,axiom,
    ! [Uu: fun_VE8181444127144381288Heap_o,Uua: nat,Uub: option4927543243414619207at_nat,Uuc: nat,Uud: array_VEBT_VEBTi,Uue: option4927543243414619207at_nat,Uuf: nat,Uug: list_VEBT_VEBT,Uuh: vEBT_VEBT] :
      aa_VEB8530157447395038106Heap_o(aa_lis4690688361886208363Heap_o(aa_nat3831830007592743808Heap_o(aa_opt3748239118003421002Heap_o(aa_arr581739785191926499Heap_o(aa_nat105892859602166008Heap_o(aa_opt8417070463701818562Heap_o(aa_nat8831994779944892035Heap_o(aTP_Lamm_qd(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh) = heap_T7577940988442287570unit_o(aa_o_H2459438917461573107t_unit(refine_Imp_assert,
          ( ( Uub = Uue )
          & ( Uuc = Uuf ) )),aa_lis4443831354813659076Heap_o(aa_arr8974760139355794305Heap_o(aa_nat808430574047188546Heap_o(aa_nat9059937328556229451Heap_o(aTP_Lamm_qc(Uu),Uua),Uuc),Uud),Uug)) ).

% ATP.lambda_1200
tff(fact_11378_ATP_Olambda__1201,axiom,
    ! [Uu: nat,Uua: nat,Uub: array_VEBT_VEBTi,Uuc: vEBT_VEBTi,Uud: list_VEBT_VEBT,Uue: vEBT_VEBT,Uuf: product_prod_nat_nat,Uug: nat,Uuh: nat] : aa_nat947733127081326238on_nat(aa_nat6020981482381522343on_nat(aa_Pro8919772687891686969on_nat(aa_VEB8035251902059208504on_nat(aa_lis5065134719111324425on_nat(aa_VEB1776269892753909763on_nat(aa_arr3452137061177157880on_nat(aa_nat8770382747262106937on_nat(aTP_Lamm_ud(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh) = heap_T5999496708990702694on_nat(aa_o_H2459438917461573107t_unit(refine_Imp_assert,Uug = aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Uu),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uua),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat3999351741935936176on_nat(aa_nat3263736196148507005on_nat(aa_Pro6111869828830443265on_nat(aa_VEB6942907905770183000on_nat(aa_lis7408950245444934749on_nat(aa_VEB6661098386149969529on_nat(aa_arr9010034949507926874on_nat(aa_nat5148509663038818543on_nat(aTP_Lamm_uc(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh)) ).

% ATP.lambda_1201
tff(fact_11379_ATP_Olambda__1202,axiom,
    ! [Uu: fun_Pr633315466666964198on_nat,Uua: nat,Uub: nat,Uuc: array_VEBT_VEBTi,Uud: vEBT_VEBTi,Uue: list_VEBT_VEBT,Uuf: vEBT_VEBT,Uug: nat,Uuh: nat] : aa_nat947733127081326238on_nat(aa_nat6020981482381522343on_nat(aa_VEB4460395522175602758on_nat(aa_lis6278591930053894219on_nat(aa_VEB6094637369740064103on_nat(aa_arr5396080262017029960on_nat(aa_nat7337942549949891037on_nat(aa_nat8478446459054042410on_nat(aTP_Lamm_vz(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh) = heap_T5999496708990702694on_nat(aa_o_H2459438917461573107t_unit(refine_Imp_assert,Uug = aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Uua),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uub),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat3999351741935936176on_nat(aa_nat3263736196148507005on_nat(aa_VEB6005108705944495412on_nat(aa_lis8565807915274807301on_nat(aa_VEB8546404647211134847on_nat(aa_arr8208594633606790516on_nat(aa_nat1839186240859117749on_nat(aa_nat5704251136259731006on_nat(aTP_Lamm_vy(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh)) ).

% ATP.lambda_1202
tff(fact_11380_ATP_Olambda__1203,axiom,
    ! [Uu: fun_VE3775134097089128360on_nat,Uua: nat,Uub: nat,Uuc: array_VEBT_VEBTi,Uud: vEBT_VEBTi,Uue: list_VEBT_VEBT,Uuf: vEBT_VEBT,Uug: nat,Uuh: nat] : aa_nat947733127081326238on_nat(aa_nat6020981482381522343on_nat(aa_VEB4460395522175602758on_nat(aa_lis6278591930053894219on_nat(aa_VEB6094637369740064103on_nat(aa_arr5396080262017029960on_nat(aa_nat7337942549949891037on_nat(aa_nat8478446459054042410on_nat(aTP_Lamm_sr(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh) = heap_T5999496708990702694on_nat(aa_o_H2459438917461573107t_unit(refine_Imp_assert,Uug = aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Uua),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uub),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat3999351741935936176on_nat(aa_nat3263736196148507005on_nat(aa_VEB6005108705944495412on_nat(aa_lis8565807915274807301on_nat(aa_VEB8546404647211134847on_nat(aa_arr8208594633606790516on_nat(aa_nat1839186240859117749on_nat(aa_nat5704251136259731006on_nat(aTP_Lamm_sq(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh)) ).

% ATP.lambda_1203
tff(fact_11381_ATP_Olambda__1204,axiom,
    ! [Uu: nat,Uua: nat,Uub: array_VEBT_VEBTi,Uuc: vEBT_VEBTi,Uud: list_VEBT_VEBT,Uue: vEBT_VEBT,Uuf: nat,Uug: nat,Uuh: product_unit] : aa_Pro451875010097498139on_nat(aa_nat3999351741935936176on_nat(aa_nat3263736196148507005on_nat(aa_VEB6005108705944495412on_nat(aa_lis8565807915274807301on_nat(aa_VEB8546404647211134847on_nat(aa_arr8208594633606790516on_nat(aa_nat1839186240859117749on_nat(aTP_Lamm_ol(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh) = heap_T5999496708990702694on_nat(aa_o_H2459438917461573107t_unit(refine_Imp_assert,Uug = aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Uu),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uua),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat3999351741935936176on_nat(aa_nat3263736196148507005on_nat(aa_VEB6005108705944495412on_nat(aa_lis8565807915274807301on_nat(aa_VEB8546404647211134847on_nat(aa_arr8208594633606790516on_nat(aTP_Lamm_ok(Uua),Uub),Uuc),Uud),Uue),Uuf),Uug)) ).

% ATP.lambda_1204
tff(fact_11382_ATP_Olambda__1205,axiom,
    ! [Uu: nat,Uua: array_VEBT_VEBTi,Uub: vEBT_VEBTi,Uuc: vEBT_VEBT,Uud: nat,Uue: nat,Uuf: vEBT_VEBTi,Uug: vEBT_VEBT,Uuh: option_nat] : aa_opt9163293676379396302on_nat(aa_VEB3755160987679246317on_nat(aa_VEB5212316371209427943on_nat(aa_nat7793430520228346548on_nat(aa_nat8410684691732023425on_nat(aa_VEB3020412023619788856on_nat(aa_VEB2867584020876442580on_nat(aa_arr7250883597209086773on_nat(aTP_Lamm_oh(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh) = heap_T5999496708990702694on_nat(aa_o_H2459438917461573107t_unit(refine_Imp_assert,Uuh = vEBT_vebt_maxt(Uug)),aa_opt5089197864053947520on_nat(aa_VEB2490029800597546679on_nat(aa_VEB1055358423767010003on_nat(aa_nat4144897467549153500on_nat(aa_nat7557481500408122597on_nat(aa_VEB3297963669270601860on_nat(aa_VEB3046651646968523646on_nat(aa_arr2696250042600076787on_nat(aTP_Lamm_og(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh)) ).

% ATP.lambda_1205
tff(fact_11383_ATP_Olambda__1206,axiom,
    ! [Uu: fun_Pr633315466666964198on_nat,Uua: nat,Uub: array_VEBT_VEBTi,Uuc: vEBT_VEBTi,Uud: list_VEBT_VEBT,Uue: vEBT_VEBT,Uuf: nat,Uug: nat,Uuh: product_unit] : aa_Pro451875010097498139on_nat(aa_nat3999351741935936176on_nat(aa_nat3263736196148507005on_nat(aa_VEB6005108705944495412on_nat(aa_lis8565807915274807301on_nat(aa_VEB8546404647211134847on_nat(aa_arr8208594633606790516on_nat(aa_nat1839186240859117749on_nat(aTP_Lamm_vx(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh) = heap_T5999496708990702694on_nat(aa_o_H2459438917461573107t_unit(refine_Imp_assert,aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uug),size_s6755466524823107622T_VEBT(Uud))),aa_nat3999351741935936176on_nat(aa_nat3263736196148507005on_nat(aa_VEB6005108705944495412on_nat(aa_lis8565807915274807301on_nat(aa_VEB8546404647211134847on_nat(aa_arr8208594633606790516on_nat(aa_nat1839186240859117749on_nat(aTP_Lamm_vw(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug)) ).

% ATP.lambda_1206
tff(fact_11384_ATP_Olambda__1207,axiom,
    ! [Uu: fun_VE3775134097089128360on_nat,Uua: nat,Uub: array_VEBT_VEBTi,Uuc: vEBT_VEBTi,Uud: list_VEBT_VEBT,Uue: vEBT_VEBT,Uuf: nat,Uug: nat,Uuh: product_unit] : aa_Pro451875010097498139on_nat(aa_nat3999351741935936176on_nat(aa_nat3263736196148507005on_nat(aa_VEB6005108705944495412on_nat(aa_lis8565807915274807301on_nat(aa_VEB8546404647211134847on_nat(aa_arr8208594633606790516on_nat(aa_nat1839186240859117749on_nat(aTP_Lamm_sp(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh) = heap_T5999496708990702694on_nat(aa_o_H2459438917461573107t_unit(refine_Imp_assert,aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uug),size_s6755466524823107622T_VEBT(Uud))),aa_nat3999351741935936176on_nat(aa_nat3263736196148507005on_nat(aa_VEB6005108705944495412on_nat(aa_lis8565807915274807301on_nat(aa_VEB8546404647211134847on_nat(aa_arr8208594633606790516on_nat(aa_nat1839186240859117749on_nat(aTP_Lamm_so(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug)) ).

% ATP.lambda_1207
tff(fact_11385_ATP_Olambda__1208,axiom,
    ! [Uu: nat,Uua: array_VEBT_VEBTi,Uub: vEBT_VEBTi,Uuc: nat,Uud: nat,Uue: nat,Uuf: nat,Uug: nat,Uuh: vEBT_VEBTi] : aa_VEB836174949890266638_VEBTi(aa_nat5661399468481021847_VEBTi(aa_nat5003802506003144480_VEBTi(aa_nat6388535185314696745_VEBTi(aa_nat3402329839695434930_VEBTi(aa_nat7160451295336493755_VEBTi(aa_VEB6809311486975643423_VEBTi(aa_arr4222437323582306304_VEBTi(aTP_Lamm_iu(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh) = heap_T5998771940306268294_VEBTi(aa_VEB8194921341634505877Heap_o(vEBT_VEBT_minNulli,Uuh),aa_VEB2625734152646192863_VEBTi(aa_nat3994193416293967532_VEBTi(aa_nat4708659386357353849_VEBTi(aa_nat6007137331526128070_VEBTi(aa_nat4759958082780230547_VEBTi(aa_nat5669347316383757024_VEBTi(aa_VEB8848578851935055378_VEBTi(aa_arr7350257379194205831_VEBTi(aTP_Lamm_it(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh)) ).

% ATP.lambda_1208
tff(fact_11386_ATP_Olambda__1209,axiom,
    ! [Uu: fun_Pr342922516890243977on_nat,Uua: nat,Uub: nat,Uuc: array_VEBT_VEBTi,Uud: vEBT_VEBTi,Uue: product_prod_nat_nat,Uuf: nat,Uug: nat,Uuh: vEBT_VEBTi] : aa_VEB2711320435582112505on_nat(aa_nat4669826966810537986on_nat(aa_nat107244183536939787on_nat(aa_Pro1663420989643744413on_nat(aa_VEB6384125456408274455on_nat(aa_arr4162587047475849484on_nat(aa_nat5386638315231209293on_nat(aa_nat7318487011899192278on_nat(aTP_Lamm_aan(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh) = heap_T3669509953089699273on_nat(aa_VEB2711320435582112505on_nat(vEBT_vebt_minti,Uuh),aa_VEB9161639482783362738on_nat(aa_nat6300058732544452155on_nat(aa_nat7962384762897155012on_nat(aa_Pro2771509044941117526on_nat(aa_VEB8816156001103073488on_nat(aa_arr6004237943432320069on_nat(aa_nat4340965201858767110on_nat(aa_nat1606558224950267919on_nat(aTP_Lamm_aam(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh)) ).

% ATP.lambda_1209
tff(fact_11387_ATP_Olambda__1210,axiom,
    ! [Uu: fun_VE8516060933855021849on_nat,Uua: nat,Uub: nat,Uuc: array_VEBT_VEBTi,Uud: vEBT_VEBTi,Uue: product_prod_nat_nat,Uuf: nat,Uug: nat,Uuh: vEBT_VEBTi] : aa_VEB2711320435582112505on_nat(aa_nat4669826966810537986on_nat(aa_nat107244183536939787on_nat(aa_Pro1663420989643744413on_nat(aa_VEB6384125456408274455on_nat(aa_arr4162587047475849484on_nat(aa_nat5386638315231209293on_nat(aa_nat7318487011899192278on_nat(aTP_Lamm_nn(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh) = heap_T3669509953089699273on_nat(aa_VEB2711320435582112505on_nat(vEBT_vebt_minti,Uuh),aa_VEB9161639482783362738on_nat(aa_nat6300058732544452155on_nat(aa_nat7962384762897155012on_nat(aa_Pro2771509044941117526on_nat(aa_VEB8816156001103073488on_nat(aa_arr6004237943432320069on_nat(aa_nat4340965201858767110on_nat(aa_nat1606558224950267919on_nat(aTP_Lamm_nm(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh)) ).

% ATP.lambda_1210
tff(fact_11388_ATP_Olambda__1211,axiom,
    ! [Uu: fun_Pr633315466666964198on_nat,Uua: vEBT_VEBT,Uub: nat,Uuc: option4927543243414619207at_nat,Uud: nat,Uue: array_VEBT_VEBTi,Uuf: vEBT_VEBTi,Uug: option4927543243414619207at_nat,Uuh: nat] : aa_nat2075221075791484188on_nat(aa_opt8950567732158611302on_nat(aa_VEB8223317440372349570on_nat(aa_arr6272259649854615779on_nat(aa_nat261593688111926264on_nat(aa_opt4414834345254747970on_nat(aa_nat2224702164617786371on_nat(aa_VEB3185010184023139490on_nat(aTP_Lamm_xg(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh) = produc6124225815318652422on_nat(aa_nat8553908474689004666on_nat(aa_opt7284239819431652166on_nat(aa_VEB7959562887951478976on_nat(aa_arr4020752153017264181on_nat(aa_nat3060827894968391926on_nat(aa_opt346861859059968706on_nat(aa_nat3229958882251406039on_nat(aa_VEB430102235636188558on_nat(aTP_Lamm_xf(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh)) ).

% ATP.lambda_1211
tff(fact_11389_ATP_Olambda__1212,axiom,
    ! [Uu: fun_Pr633315466666964198on_nat,Uua: vEBT_VEBT,Uub: nat,Uuc: option4927543243414619207at_nat,Uud: nat,Uue: array_VEBT_VEBTi,Uuf: vEBT_VEBTi,Uug: option4927543243414619207at_nat,Uuh: nat] : aa_nat2075221075791484188on_nat(aa_opt8950567732158611302on_nat(aa_VEB8223317440372349570on_nat(aa_arr6272259649854615779on_nat(aa_nat261593688111926264on_nat(aa_opt4414834345254747970on_nat(aa_nat2224702164617786371on_nat(aa_VEB3185010184023139490on_nat(aTP_Lamm_we(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh) = produc6124225815318652422on_nat(aa_nat8553908474689004666on_nat(aa_opt7284239819431652166on_nat(aa_VEB7959562887951478976on_nat(aa_arr4020752153017264181on_nat(aa_nat3060827894968391926on_nat(aa_opt346861859059968706on_nat(aa_nat3229958882251406039on_nat(aa_VEB430102235636188558on_nat(aTP_Lamm_wd(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh)) ).

% ATP.lambda_1212
tff(fact_11390_ATP_Olambda__1213,axiom,
    ! [Uu: fun_Pr4404027214751104635_VEBTi,Uua: nat,Uub: option4927543243414619207at_nat,Uuc: nat,Uud: array_VEBT_VEBTi,Uue: vEBT_VEBTi,Uuf: product_prod_nat_nat,Uug: option4927543243414619207at_nat,Uuh: nat] : aa_nat7217600295869438385_VEBTi(aa_opt7507675525305149691_VEBTi(aa_Pro1234189438217840325_VEBTi(aa_VEB660175962507042495_VEBTi(aa_arr8367260621226259636_VEBTi(aa_nat5784496157376858101_VEBTi(aa_opt81624670976341313_VEBTi(aa_nat5602383413110735830_VEBTi(aTP_Lamm_vm(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh) = produc9050507437146595227_VEBTi(aa_nat1139649368474441743_VEBTi(aa_opt762848812907537627_VEBTi(aa_Pro4022776811541674023_VEBTi(aa_VEB3655672368276914755_VEBTi(aa_arr737849036559383332_VEBTi(aa_nat7609637249401053881_VEBTi(aa_opt3216071514793242883_VEBTi(aa_nat8892849909372205124_VEBTi(aTP_Lamm_vl(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh)) ).

% ATP.lambda_1213
tff(fact_11391_ATP_Olambda__1214,axiom,
    ! [Uu: fun_VE3775134097089128360on_nat,Uua: vEBT_VEBT,Uub: nat,Uuc: option4927543243414619207at_nat,Uud: nat,Uue: array_VEBT_VEBTi,Uuf: vEBT_VEBTi,Uug: option4927543243414619207at_nat,Uuh: nat] : aa_nat2075221075791484188on_nat(aa_opt8950567732158611302on_nat(aa_VEB8223317440372349570on_nat(aa_arr6272259649854615779on_nat(aa_nat261593688111926264on_nat(aa_opt4414834345254747970on_nat(aa_nat2224702164617786371on_nat(aa_VEB3185010184023139490on_nat(aTP_Lamm_sw(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh) = produc6124225815318652422on_nat(aa_nat8553908474689004666on_nat(aa_opt7284239819431652166on_nat(aa_VEB7959562887951478976on_nat(aa_arr4020752153017264181on_nat(aa_nat3060827894968391926on_nat(aa_opt346861859059968706on_nat(aa_nat3229958882251406039on_nat(aa_VEB430102235636188558on_nat(aTP_Lamm_sv(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh)) ).

% ATP.lambda_1214
tff(fact_11392_ATP_Olambda__1215,axiom,
    ! [Uu: fun_VE3775134097089128360on_nat,Uua: vEBT_VEBT,Uub: nat,Uuc: option4927543243414619207at_nat,Uud: nat,Uue: array_VEBT_VEBTi,Uuf: vEBT_VEBTi,Uug: option4927543243414619207at_nat,Uuh: nat] : aa_nat2075221075791484188on_nat(aa_opt8950567732158611302on_nat(aa_VEB8223317440372349570on_nat(aa_arr6272259649854615779on_nat(aa_nat261593688111926264on_nat(aa_opt4414834345254747970on_nat(aa_nat2224702164617786371on_nat(aa_VEB3185010184023139490on_nat(aTP_Lamm_sh(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh) = produc6124225815318652422on_nat(aa_nat8553908474689004666on_nat(aa_opt7284239819431652166on_nat(aa_VEB7959562887951478976on_nat(aa_arr4020752153017264181on_nat(aa_nat3060827894968391926on_nat(aa_opt346861859059968706on_nat(aa_nat3229958882251406039on_nat(aa_VEB430102235636188558on_nat(aTP_Lamm_sg(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh)) ).

% ATP.lambda_1215
tff(fact_11393_ATP_Olambda__1216,axiom,
    ! [Uu: fun_VE2018681615261037117_VEBTi,Uua: nat,Uub: option4927543243414619207at_nat,Uuc: nat,Uud: array_VEBT_VEBTi,Uue: vEBT_VEBTi,Uuf: product_prod_nat_nat,Uug: option4927543243414619207at_nat,Uuh: nat] : aa_nat7217600295869438385_VEBTi(aa_opt7507675525305149691_VEBTi(aa_Pro1234189438217840325_VEBTi(aa_VEB660175962507042495_VEBTi(aa_arr8367260621226259636_VEBTi(aa_nat5784496157376858101_VEBTi(aa_opt81624670976341313_VEBTi(aa_nat5602383413110735830_VEBTi(aTP_Lamm_rp(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh) = produc9050507437146595227_VEBTi(aa_nat1139649368474441743_VEBTi(aa_opt762848812907537627_VEBTi(aa_Pro4022776811541674023_VEBTi(aa_VEB3655672368276914755_VEBTi(aa_arr737849036559383332_VEBTi(aa_nat7609637249401053881_VEBTi(aa_opt3216071514793242883_VEBTi(aa_nat8892849909372205124_VEBTi(aTP_Lamm_ro(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh)) ).

% ATP.lambda_1216
tff(fact_11394_ATP_Olambda__1217,axiom,
    ! [Uu: nat,Uua: array_VEBT_VEBTi,Uub: vEBT_VEBTi,Uuc: vEBT_VEBT,Uud: nat,Uue: nat,Uuf: vEBT_VEBTi,Uug: vEBT_VEBT,Uuh: option_nat,Uui: product_unit] :
      aa_Pro451875010097498139on_nat(aa_opt5089197864053947520on_nat(aa_VEB2490029800597546679on_nat(aa_VEB1055358423767010003on_nat(aa_nat4144897467549153500on_nat(aa_nat7557481500408122597on_nat(aa_VEB3297963669270601860on_nat(aa_VEB3046651646968523646on_nat(aa_arr2696250042600076787on_nat(aTP_Lamm_og(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui) = $ite(
        ( ( Uuh != none_nat )
        & aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_less,aa_nat_option_nat(some_nat,Uud)),Uuh) ),
        heap_T3669509953089699273on_nat(aa_nat947733127081326238on_nat(aa_VEB3566417204837707778on_nat(aa_VEB1030146501360450209on_nat(vEBT_VEBT_vebt_succi,Uug),Uuf),Uud),aa_nat9127621225095157975on_nat(aTP_Lamm_ep(Uu),Uue)),
        heap_T3669509953089699273on_nat(aa_nat947733127081326238on_nat(aa_VEB3566417204837707778on_nat(aa_VEB1030146501360450209on_nat(vEBT_VEBT_vebt_succi,Uuc),Uub),Uue),aa_nat9127621225095157975on_nat(aa_VEB8264456441901112694on_nat(aa_arr7060518384107407923on_nat(aTP_Lamm_of(Uu),Uua),Uuc),Uue)) ) ).

% ATP.lambda_1217
tff(fact_11395_ATP_Olambda__1218,axiom,
    ! [Uu: fun_Pr342922516890243977on_nat,Uua: nat,Uub: nat,Uuc: array_VEBT_VEBTi,Uud: vEBT_VEBTi,Uue: product_prod_nat_nat,Uuf: nat,Uug: nat,Uuh: vEBT_VEBTi,Uui: option_nat] :
      aa_opt9163293676379396302on_nat(aa_VEB9161639482783362738on_nat(aa_nat6300058732544452155on_nat(aa_nat7962384762897155012on_nat(aa_Pro2771509044941117526on_nat(aa_VEB8816156001103073488on_nat(aa_arr6004237943432320069on_nat(aa_nat4340965201858767110on_nat(aa_nat1606558224950267919on_nat(aTP_Lamm_aam(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui) = $ite(
        ( ( Uui != none_nat )
        & aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_greater,aa_nat_option_nat(some_nat,Uuf)),Uui) ),
        heap_T3669509953089699273on_nat(aa_nat947733127081326238on_nat(aa_VEB3566417204837707778on_nat(produc1489253303066280154on_nat(Uu),Uuh),Uuf),aa_nat9127621225095157975on_nat(aTP_Lamm_ep(Uub),Uug)),
        heap_T3669509953089699273on_nat(aa_nat947733127081326238on_nat(aa_VEB3566417204837707778on_nat(produc1489253303066280154on_nat(Uu),Uud),Uug),aa_Pro4885747571561342816on_nat(aa_arr7927785801258559901on_nat(aa_nat6148929329942773342on_nat(aTP_Lamm_es(Uua),Uub),Uuc),Uue)) ) ).

% ATP.lambda_1218
tff(fact_11396_ATP_Olambda__1219,axiom,
    ! [Uu: fun_VE8516060933855021849on_nat,Uua: nat,Uub: nat,Uuc: array_VEBT_VEBTi,Uud: vEBT_VEBTi,Uue: product_prod_nat_nat,Uuf: nat,Uug: nat,Uuh: vEBT_VEBTi,Uui: option_nat] :
      aa_opt9163293676379396302on_nat(aa_VEB9161639482783362738on_nat(aa_nat6300058732544452155on_nat(aa_nat7962384762897155012on_nat(aa_Pro2771509044941117526on_nat(aa_VEB8816156001103073488on_nat(aa_arr6004237943432320069on_nat(aa_nat4340965201858767110on_nat(aa_nat1606558224950267919on_nat(aTP_Lamm_nm(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui) = $ite(
        ( ( Uui != none_nat )
        & aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_greater,aa_nat_option_nat(some_nat,Uuf)),Uui) ),
        heap_T3669509953089699273on_nat(aa_nat947733127081326238on_nat(aa_VEB3566417204837707778on_nat(Uu,Uuh),Uuf),aa_nat9127621225095157975on_nat(aTP_Lamm_ep(Uub),Uug)),
        heap_T3669509953089699273on_nat(aa_nat947733127081326238on_nat(aa_VEB3566417204837707778on_nat(Uu,Uud),Uug),aa_Pro4885747571561342816on_nat(aa_arr7927785801258559901on_nat(aa_nat6148929329942773342on_nat(aTP_Lamm_es(Uua),Uub),Uuc),Uue)) ) ).

% ATP.lambda_1219
tff(fact_11397_ATP_Olambda__1220,axiom,
    ! [Uu: nat,Uua: nat,Uub: array_VEBT_VEBTi,Uuc: vEBT_VEBTi,Uud: product_prod_nat_nat,Uue: option4927543243414619207at_nat,Uuf: nat,Uug: list_VEBT_VEBT,Uuh: vEBT_VEBT,Uui: product_unit] : aa_Pro3581650104493980720_VEBTi(aa_VEB3467812963353200047_VEBTi(aa_lis3316967545134462464_VEBTi(aa_nat7280733210739239061_VEBTi(aa_opt2515449710316153439_VEBTi(aa_Pro2414373876686838441_VEBTi(aa_VEB2192875243258015011_VEBTi(aa_arr5702273427728458776_VEBTi(aa_nat1667688020712374873_VEBTi(aTP_Lamm_tq(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui) = aa_Pro1395359565518501564_VEBTi(produc737604151543542771_VEBTi(aa_VEB2281408947416558043_VEBTi(aa_lis148658726372936800_VEBTi(aa_nat3438343216000400545_VEBTi(aa_Pro9203685154717219507_VEBTi(aa_VEB4942273269570985133_VEBTi(aa_arr2286534195499807138_VEBTi(aa_nat2259444989923833827_VEBTi(aTP_Lamm_tp(Uu),Uua),Uub),Uuc),Uud),Uuf),Uug),Uuh)),the_Pr8591224930841456533at_nat(Uue)) ).

% ATP.lambda_1220
tff(fact_11398_ATP_Olambda__1221,axiom,
    ! [Uu: nat,Uua: nat,Uub: array_VEBT_VEBTi,Uuc: vEBT_VEBTi,Uud: nat,Uue: list_VEBT_VEBT,Uuf: vEBT_VEBT,Uug: nat,Uuh: nat,Uui: nat] :
      aa_nat5408703056993926195_VEBTi(aa_nat4282640288315822012_VEBTi(aa_nat7653869184832469701_VEBTi(aa_VEB681999334371813476_VEBTi(aa_lis3749794993760221801_VEBTi(aa_nat3264418630393705002_VEBTi(aa_VEB7195496841433988878_VEBTi(aa_arr953441552191549551_VEBTi(aa_nat1068536149936821380_VEBTi(aTP_Lamm_tn(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui) = heap_T844888390831797134_VEBTi($ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uu),Uuh),aa_nat5821310369566504398ap_nat(heap_Time_return_nat,Uuh),aa_nat5821310369566504398ap_nat(heap_Time_return_nat,Uu)),aa_nat4282640288315822012_VEBTi(aa_nat7653869184832469701_VEBTi(aa_nat2287630409448422734_VEBTi(aa_VEB3484995426995338861_VEBTi(aa_lis8107322492824698866_VEBTi(aa_nat5353086824354447155_VEBTi(aa_VEB4525472277563179415_VEBTi(aa_arr4621753813534489208_VEBTi(aa_nat369315079452798477_VEBTi(aTP_Lamm_tm(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui)) ).

% ATP.lambda_1221
tff(fact_11399_ATP_Olambda__1222,axiom,
    ! [Uu: fun_Pr4404027214751104635_VEBTi,Uua: nat,Uub: vEBT_VEBTi,Uuc: vEBT_VEBT,Uud: nat,Uue: nat,Uuf: nat,Uug: nat,Uuh: $o,Uui: array_VEBT_VEBTi] :
      aa_arr4839786837114051083_VEBTi(aa_o_f5513965820244355694_VEBTi(aa_nat6231872576868200367_VEBTi(aa_nat3271849308718382136_VEBTi(aa_nat4180741733914107969_VEBTi(aa_nat6590260363808168906_VEBTi(aa_VEB6485260183185377513_VEBTi(aa_VEB953658682601944675_VEBTi(aa_nat3348711059893815344_VEBTi(aTP_Lamm_uu(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),(Uuh)),Uui) = heap_T1006145433769338483_VEBTi($ite((Uuh),aa_nat5408703056993926195_VEBTi(aa_VEB4833742232818206615_VEBTi(aa_VEB6292075424854053302_VEBTi(produc3311654011758026858_VEBTi(produc2164094337957399884_VEBTi(Uu)),Uuc),Uub),Uug),aa_VEB836174949890266638_VEBTi(heap_T3630416162098727440_VEBTi,Uub)),aa_arr2912372873376718191_VEBTi(aa_nat6445230840460233604_VEBTi(aa_nat3780430034849525329_VEBTi(aa_nat6565337644025228958_VEBTi(aTP_Lamm_iq(Uua),Uud),Uue),Uuf),Uui)) ).

% ATP.lambda_1222
tff(fact_11400_ATP_Olambda__1223,axiom,
    ! [Uu: fun_VE2018681615261037117_VEBTi,Uua: nat,Uub: vEBT_VEBTi,Uuc: vEBT_VEBT,Uud: nat,Uue: nat,Uuf: nat,Uug: nat,Uuh: $o,Uui: array_VEBT_VEBTi] :
      aa_arr4839786837114051083_VEBTi(aa_o_f5513965820244355694_VEBTi(aa_nat6231872576868200367_VEBTi(aa_nat3271849308718382136_VEBTi(aa_nat4180741733914107969_VEBTi(aa_nat6590260363808168906_VEBTi(aa_VEB6485260183185377513_VEBTi(aa_VEB953658682601944675_VEBTi(aa_nat3348711059893815344_VEBTi(aTP_Lamm_qx(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),(Uuh)),Uui) = heap_T1006145433769338483_VEBTi($ite((Uuh),aa_nat5408703056993926195_VEBTi(aa_VEB4833742232818206615_VEBTi(aa_VEB6292075424854053302_VEBTi(Uu,Uuc),Uub),Uug),aa_VEB836174949890266638_VEBTi(heap_T3630416162098727440_VEBTi,Uub)),aa_arr2912372873376718191_VEBTi(aa_nat6445230840460233604_VEBTi(aa_nat3780430034849525329_VEBTi(aa_nat6565337644025228958_VEBTi(aTP_Lamm_iq(Uua),Uud),Uue),Uuf),Uui)) ).

% ATP.lambda_1223
tff(fact_11401_ATP_Olambda__1224,axiom,
    ! [Uu: nat,Uua: array_VEBT_VEBTi,Uub: vEBT_VEBTi,Uuc: vEBT_VEBT,Uud: nat,Uue: nat,Uuf: nat,Uug: nat,Uuh: $o,Uui: vEBT_VEBTi] : aa_VEB836174949890266638_VEBTi(aa_o_f1343430628605260489_VEBTi(aa_nat4147387395820660958_VEBTi(aa_nat2236254811235047083_VEBTi(aa_nat5995673692660408824_VEBTi(aa_nat4988899645790341829_VEBTi(aa_VEB257407171870985852_VEBTi(aa_VEB2935853492131583640_VEBTi(aa_arr6469760262597239545_VEBTi(aTP_Lamm_tb(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),(Uuh)),Uui) = heap_T5099337393651448672_VEBTi(array_upd_VEBT_VEBTi(Uug,Uui,Uua),aa_o_f5513965820244355694_VEBTi(aa_nat6231872576868200367_VEBTi(aa_nat3271849308718382136_VEBTi(aa_nat4180741733914107969_VEBTi(aa_nat6590260363808168906_VEBTi(aa_VEB6485260183185377513_VEBTi(aa_VEB953658682601944675_VEBTi(aTP_Lamm_ta(Uu),Uub),Uuc),Uud),Uue),Uuf),Uug),(Uuh))) ).

% ATP.lambda_1224
tff(fact_11402_ATP_Olambda__1225,axiom,
    ! [Uu: fun_Pr4171499318505651870_VEBTi,Uua: nat,Uub: array_VEBT_VEBTi,Uuc: vEBT_VEBTi,Uud: nat,Uue: nat,Uuf: nat,Uug: nat,Uuh: $o,Uui: vEBT_VEBTi] : aa_VEB836174949890266638_VEBTi(aa_o_f1343430628605260489_VEBTi(aa_nat4147387395820660958_VEBTi(aa_nat2236254811235047083_VEBTi(aa_nat5995673692660408824_VEBTi(aa_nat4988899645790341829_VEBTi(aa_VEB5075307909151313271_VEBTi(aa_arr5746817224660070252_VEBTi(aa_nat898395892396129965_VEBTi(aTP_Lamm_abd(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),(Uuh)),Uui) = heap_T5099337393651448672_VEBTi(array_upd_VEBT_VEBTi(Uug,Uui,Uub),aa_o_f5513965820244355694_VEBTi(aa_nat6231872576868200367_VEBTi(aa_nat3271849308718382136_VEBTi(aa_nat4180741733914107969_VEBTi(aa_nat6590260363808168906_VEBTi(aa_VEB8654527159822295598_VEBTi(aa_nat7794021720236223159_VEBTi(aTP_Lamm_abc(Uu),Uua),Uuc),Uud),Uue),Uuf),Uug),(Uuh))) ).

% ATP.lambda_1225
tff(fact_11403_ATP_Olambda__1226,axiom,
    ! [Uu: fun_VE8118064890490510894_VEBTi,Uua: nat,Uub: array_VEBT_VEBTi,Uuc: vEBT_VEBTi,Uud: nat,Uue: nat,Uuf: nat,Uug: nat,Uuh: $o,Uui: vEBT_VEBTi] : aa_VEB836174949890266638_VEBTi(aa_o_f1343430628605260489_VEBTi(aa_nat4147387395820660958_VEBTi(aa_nat2236254811235047083_VEBTi(aa_nat5995673692660408824_VEBTi(aa_nat4988899645790341829_VEBTi(aa_VEB5075307909151313271_VEBTi(aa_arr5746817224660070252_VEBTi(aa_nat898395892396129965_VEBTi(aTP_Lamm_na(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),(Uuh)),Uui) = heap_T5099337393651448672_VEBTi(array_upd_VEBT_VEBTi(Uug,Uui,Uub),aa_o_f5513965820244355694_VEBTi(aa_nat6231872576868200367_VEBTi(aa_nat3271849308718382136_VEBTi(aa_nat4180741733914107969_VEBTi(aa_nat6590260363808168906_VEBTi(aa_VEB8654527159822295598_VEBTi(aa_nat7794021720236223159_VEBTi(aTP_Lamm_mz(Uu),Uua),Uuc),Uud),Uue),Uuf),Uug),(Uuh))) ).

% ATP.lambda_1226
tff(fact_11404_ATP_Olambda__1227,axiom,
    ! [Uu: fun_Pr4171499318505651870_VEBTi,Uua: nat,Uub: nat,Uuc: array_VEBT_VEBTi,Uud: vEBT_VEBTi,Uue: nat,Uuf: nat,Uug: nat,Uuh: nat,Uui: nat] : aa_nat5408703056993926195_VEBTi(aa_nat4282640288315822012_VEBTi(aa_nat7653869184832469701_VEBTi(aa_nat2287630409448422734_VEBTi(aa_nat2878941900432676695_VEBTi(aa_VEB8966970847287376059_VEBTi(aa_arr4873757251330958748_VEBTi(aa_nat4224949603525894449_VEBTi(aa_nat7906690825329656190_VEBTi(aTP_Lamm_abi(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui) = heap_T844888390831797134_VEBTi(aa_nat5821310369566504398ap_nat(vEBT_VEBT_highi(Uug),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uub),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat4282640288315822012_VEBTi(aa_nat7653869184832469701_VEBTi(aa_nat2287630409448422734_VEBTi(aa_nat2878941900432676695_VEBTi(aa_nat532351713702512864_VEBTi(aa_VEB7864109228220129220_VEBTi(aa_arr7707872583627363365_VEBTi(aa_nat3757413148909559610_VEBTi(aa_nat7878045268284587783_VEBTi(aTP_Lamm_abh(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui)) ).

% ATP.lambda_1227
tff(fact_11405_ATP_Olambda__1228,axiom,
    ! [Uu: fun_VE8118064890490510894_VEBTi,Uua: nat,Uub: nat,Uuc: array_VEBT_VEBTi,Uud: vEBT_VEBTi,Uue: nat,Uuf: nat,Uug: nat,Uuh: nat,Uui: nat] : aa_nat5408703056993926195_VEBTi(aa_nat4282640288315822012_VEBTi(aa_nat7653869184832469701_VEBTi(aa_nat2287630409448422734_VEBTi(aa_nat2878941900432676695_VEBTi(aa_VEB8966970847287376059_VEBTi(aa_arr4873757251330958748_VEBTi(aa_nat4224949603525894449_VEBTi(aa_nat7906690825329656190_VEBTi(aTP_Lamm_nf(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui) = heap_T844888390831797134_VEBTi(aa_nat5821310369566504398ap_nat(vEBT_VEBT_highi(Uug),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uub),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat4282640288315822012_VEBTi(aa_nat7653869184832469701_VEBTi(aa_nat2287630409448422734_VEBTi(aa_nat2878941900432676695_VEBTi(aa_nat532351713702512864_VEBTi(aa_VEB7864109228220129220_VEBTi(aa_arr7707872583627363365_VEBTi(aa_nat3757413148909559610_VEBTi(aa_nat7878045268284587783_VEBTi(aTP_Lamm_ne(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui)) ).

% ATP.lambda_1228
tff(fact_11406_ATP_Olambda__1229,axiom,
    ! [Uu: nat,Uua: nat,Uub: array_VEBT_VEBTi,Uuc: vEBT_VEBTi,Uud: list_VEBT_VEBT,Uue: vEBT_VEBT,Uuf: product_prod_nat_nat,Uug: nat,Uuh: nat,Uui: product_unit] : aa_Pro451875010097498139on_nat(aa_nat3999351741935936176on_nat(aa_nat3263736196148507005on_nat(aa_Pro6111869828830443265on_nat(aa_VEB6942907905770183000on_nat(aa_lis7408950245444934749on_nat(aa_VEB6661098386149969529on_nat(aa_arr9010034949507926874on_nat(aa_nat5148509663038818543on_nat(aTP_Lamm_ua(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui) = heap_T2868974464944644318on_nat(aa_nat5408703056993926195_VEBTi(array_nth_VEBT_VEBTi(Uub),Uuh),aa_nat4669826966810537986on_nat(aa_nat107244183536939787on_nat(aa_Pro1663420989643744413on_nat(aa_VEB3438037858822377372on_nat(aa_lis3250066801158684525on_nat(aa_VEB8217410703897791847on_nat(aa_arr22699188990094940on_nat(aa_nat3630391635445309085on_nat(aTP_Lamm_tz(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh)) ).

% ATP.lambda_1229
tff(fact_11407_ATP_Olambda__1230,axiom,
    ! [Uu: nat,Uua: array_VEBT_VEBTi,Uub: vEBT_VEBTi,Uuc: nat,Uud: nat,Uue: nat,Uuf: nat,Uug: nat,Uuh: vEBT_VEBTi,Uui: $o] : aa_o_H7577119287286566245_VEBTi(aa_VEB2625734152646192863_VEBTi(aa_nat3994193416293967532_VEBTi(aa_nat4708659386357353849_VEBTi(aa_nat6007137331526128070_VEBTi(aa_nat4759958082780230547_VEBTi(aa_nat5669347316383757024_VEBTi(aa_VEB8848578851935055378_VEBTi(aa_arr7350257379194205831_VEBTi(aTP_Lamm_it(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),(Uui)) = heap_T1006145433769338483_VEBTi(aa_nat5408703056993926195_VEBTi(aa_VEB4833742232818206615_VEBTi(vEBT_vebt_inserti,Uuh),Uuf),aa_o_f1343430628605260489_VEBTi(aa_nat4147387395820660958_VEBTi(aa_nat2236254811235047083_VEBTi(aa_nat5995673692660408824_VEBTi(aa_nat4988899645790341829_VEBTi(aa_VEB5075307909151313271_VEBTi(aa_arr5746817224660070252_VEBTi(aTP_Lamm_is(Uu),Uua),Uub),Uuc),Uud),Uue),Uug),(Uui))) ).

% ATP.lambda_1230
tff(fact_11408_ATP_Olambda__1231,axiom,
    ! [Uu: nat,Uua: nat,Uub: array_VEBT_VEBTi,Uuc: vEBT_VEBTi,Uud: list_VEBT_VEBT,Uue: vEBT_VEBT,Uuf: product_prod_nat_nat,Uug: nat,Uuh: nat,Uui: vEBT_VEBTi] :
      aa_VEB2711320435582112505on_nat(aa_nat4669826966810537986on_nat(aa_nat107244183536939787on_nat(aa_Pro1663420989643744413on_nat(aa_VEB3438037858822377372on_nat(aa_lis3250066801158684525on_nat(aa_VEB8217410703897791847on_nat(aa_arr22699188990094940on_nat(aa_nat3630391635445309085on_nat(aTP_Lamm_tz(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui) = $let(
        aktnode: vEBT_VEBT,
        aktnode:= nth_VEBT_VEBT(Uud,Uuh),
        heap_T3669509953089699273on_nat(aa_VEB2711320435582112505on_nat(vEBT_vebt_minti,Uui),aa_VEB3755160987679246317on_nat(aa_VEB5212316371209427943on_nat(aa_nat7793430520228346548on_nat(aa_nat8410684691732023425on_nat(aa_Pro7328798252849467525on_nat(aa_VEB6402893080723908316on_nat(aa_VEB4808996286462526806on_nat(aa_arr4232376455131879627on_nat(aa_nat8394757134461757324on_nat(aTP_Lamm_ty(Uu),Uua),Uub),Uuc),Uue),Uuf),Uug),Uuh),Uui),aktnode)) ) ).

% ATP.lambda_1231
tff(fact_11409_ATP_Olambda__1232,axiom,
    ! [Uu: vEBT_VEBT,Uua: nat,Uub: option4927543243414619207at_nat,Uuc: nat,Uud: array_VEBT_VEBTi,Uue: vEBT_VEBTi,Uuf: option4927543243414619207at_nat,Uug: nat,Uuh: list_VEBT_VEBT,Uui: vEBT_VEBT] :
      aa_VEB3253283475358696116on_nat(aa_lis6893462050766468025on_nat(aa_nat8553908474689004666on_nat(aa_opt7284239819431652166on_nat(aa_VEB7959562887951478976on_nat(aa_arr4020752153017264181on_nat(aa_nat3060827894968391926on_nat(aa_opt346861859059968706on_nat(aa_nat3229958882251406039on_nat(aTP_Lamm_uh(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui) = heap_T5999496708990702694on_nat(aa_o_H2459438917461573107t_unit(refine_Imp_assert,
          ( ( Uuf = Uub )
          & ( Uug = Uuc )
          & aa_VEBT_VEBT_o(vEBT_is_Node,Uu) )),aa_VEB1832073013541904922on_nat(aa_lis6328568131235813867on_nat(aa_VEB3884484516340040293on_nat(aa_arr5442694692229379418on_nat(aa_nat4425110079127750043on_nat(aa_opt7218664994477926375on_nat(aTP_Lamm_ug(Uua),Uub),Uuc),Uud),Uue),Uuh),Uui)) ).

% ATP.lambda_1232
tff(fact_11410_ATP_Olambda__1233,axiom,
    ! [Uu: vEBT_VEBT,Uua: nat,Uub: option4927543243414619207at_nat,Uuc: nat,Uud: array_VEBT_VEBTi,Uue: vEBT_VEBTi,Uuf: option4927543243414619207at_nat,Uug: nat,Uuh: list_VEBT_VEBT,Uui: vEBT_VEBT] :
      aa_VEB3253283475358696116on_nat(aa_lis6893462050766468025on_nat(aa_nat8553908474689004666on_nat(aa_opt7284239819431652166on_nat(aa_VEB7959562887951478976on_nat(aa_arr4020752153017264181on_nat(aa_nat3060827894968391926on_nat(aa_opt346861859059968706on_nat(aa_nat3229958882251406039on_nat(aTP_Lamm_oq(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui) = heap_T5999496708990702694on_nat(aa_o_H2459438917461573107t_unit(refine_Imp_assert,
          ( ( Uuf = Uub )
          & ( Uug = Uuc )
          & aa_VEBT_VEBT_o(vEBT_is_Node,Uu) )),aa_VEB1832073013541904922on_nat(aa_lis6328568131235813867on_nat(aa_VEB3884484516340040293on_nat(aa_arr5442694692229379418on_nat(aa_nat4425110079127750043on_nat(aa_opt7218664994477926375on_nat(aTP_Lamm_op(Uua),Uub),Uuc),Uud),Uue),Uuh),Uui)) ).

% ATP.lambda_1233
tff(fact_11411_ATP_Olambda__1234,axiom,
    ! [Uu: nat,Uua: option4927543243414619207at_nat,Uub: nat,Uuc: array_VEBT_VEBTi,Uud: vEBT_VEBTi,Uue: product_prod_nat_nat,Uuf: option4927543243414619207at_nat,Uug: nat,Uuh: list_VEBT_VEBT,Uui: vEBT_VEBT] :
      aa_VEB5809408762104066889_VEBTi(aa_lis894883570680706766_VEBTi(aa_nat1139649368474441743_VEBTi(aa_opt762848812907537627_VEBTi(aa_Pro4022776811541674023_VEBTi(aa_VEB3655672368276914755_VEBTi(aa_arr737849036559383332_VEBTi(aa_nat7609637249401053881_VEBTi(aa_opt3216071514793242883_VEBTi(aTP_Lamm_tr(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui) = heap_T7982501707604696571_VEBTi(aa_o_H2459438917461573107t_unit(refine_Imp_assert,
          ( ( Uua = Uuf )
          & ( Uub = Uug ) )),aa_VEB3467812963353200047_VEBTi(aa_lis3316967545134462464_VEBTi(aa_nat7280733210739239061_VEBTi(aa_opt2515449710316153439_VEBTi(aa_Pro2414373876686838441_VEBTi(aa_VEB2192875243258015011_VEBTi(aa_arr5702273427728458776_VEBTi(aa_nat1667688020712374873_VEBTi(aTP_Lamm_tq(Uu),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui)) ).

% ATP.lambda_1234
tff(fact_11412_ATP_Olambda__1235,axiom,
    ! [Uu: fun_Pr633315466666964198on_nat,Uua: nat,Uub: nat,Uuc: array_VEBT_VEBTi,Uud: vEBT_VEBTi,Uue: list_VEBT_VEBT,Uuf: vEBT_VEBT,Uug: product_prod_nat_nat,Uuh: nat,Uui: nat] : aa_nat947733127081326238on_nat(aa_nat6020981482381522343on_nat(aa_Pro8919772687891686969on_nat(aa_VEB8035251902059208504on_nat(aa_lis5065134719111324425on_nat(aa_VEB1776269892753909763on_nat(aa_arr3452137061177157880on_nat(aa_nat8770382747262106937on_nat(aa_nat388734249307221954on_nat(aTP_Lamm_xb(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui) = heap_T5999496708990702694on_nat(aa_o_H2459438917461573107t_unit(refine_Imp_assert,Uuh = aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Uua),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uub),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat3999351741935936176on_nat(aa_nat3263736196148507005on_nat(aa_Pro6111869828830443265on_nat(aa_VEB6942907905770183000on_nat(aa_lis7408950245444934749on_nat(aa_VEB6661098386149969529on_nat(aa_arr9010034949507926874on_nat(aa_nat5148509663038818543on_nat(aa_nat8204191389585037116on_nat(aTP_Lamm_xa(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui)) ).

% ATP.lambda_1235
tff(fact_11413_ATP_Olambda__1236,axiom,
    ! [Uu: fun_VE3775134097089128360on_nat,Uua: nat,Uub: nat,Uuc: array_VEBT_VEBTi,Uud: vEBT_VEBTi,Uue: list_VEBT_VEBT,Uuf: vEBT_VEBT,Uug: product_prod_nat_nat,Uuh: nat,Uui: nat] : aa_nat947733127081326238on_nat(aa_nat6020981482381522343on_nat(aa_Pro8919772687891686969on_nat(aa_VEB8035251902059208504on_nat(aa_lis5065134719111324425on_nat(aa_VEB1776269892753909763on_nat(aa_arr3452137061177157880on_nat(aa_nat8770382747262106937on_nat(aa_nat388734249307221954on_nat(aTP_Lamm_sc(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui) = heap_T5999496708990702694on_nat(aa_o_H2459438917461573107t_unit(refine_Imp_assert,Uuh = aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Uua),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uub),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat3999351741935936176on_nat(aa_nat3263736196148507005on_nat(aa_Pro6111869828830443265on_nat(aa_VEB6942907905770183000on_nat(aa_lis7408950245444934749on_nat(aa_VEB6661098386149969529on_nat(aa_arr9010034949507926874on_nat(aa_nat5148509663038818543on_nat(aa_nat8204191389585037116on_nat(aTP_Lamm_sb(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui)) ).

% ATP.lambda_1236
tff(fact_11414_ATP_Olambda__1237,axiom,
    ! [Uu: nat,Uua: nat,Uub: array_VEBT_VEBTi,Uuc: vEBT_VEBTi,Uud: list_VEBT_VEBT,Uue: vEBT_VEBT,Uuf: product_prod_nat_nat,Uug: nat,Uuh: nat,Uui: product_unit] : aa_Pro451875010097498139on_nat(aa_nat3999351741935936176on_nat(aa_nat3263736196148507005on_nat(aa_Pro6111869828830443265on_nat(aa_VEB6942907905770183000on_nat(aa_lis7408950245444934749on_nat(aa_VEB6661098386149969529on_nat(aa_arr9010034949507926874on_nat(aa_nat5148509663038818543on_nat(aTP_Lamm_uc(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui) = heap_T5999496708990702694on_nat(aa_o_H2459438917461573107t_unit(refine_Imp_assert,Uuh = aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Uu),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uua),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat3999351741935936176on_nat(aa_nat3263736196148507005on_nat(aa_Pro6111869828830443265on_nat(aa_VEB6942907905770183000on_nat(aa_lis7408950245444934749on_nat(aa_VEB6661098386149969529on_nat(aa_arr9010034949507926874on_nat(aa_nat5148509663038818543on_nat(aTP_Lamm_ub(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh)) ).

% ATP.lambda_1237
tff(fact_11415_ATP_Olambda__1238,axiom,
    ! [Uu: fun_Pr633315466666964198on_nat,Uua: nat,Uub: nat,Uuc: array_VEBT_VEBTi,Uud: vEBT_VEBTi,Uue: list_VEBT_VEBT,Uuf: vEBT_VEBT,Uug: nat,Uuh: nat,Uui: product_unit] : aa_Pro451875010097498139on_nat(aa_nat3999351741935936176on_nat(aa_nat3263736196148507005on_nat(aa_VEB6005108705944495412on_nat(aa_lis8565807915274807301on_nat(aa_VEB8546404647211134847on_nat(aa_arr8208594633606790516on_nat(aa_nat1839186240859117749on_nat(aa_nat5704251136259731006on_nat(aTP_Lamm_vy(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui) = heap_T5999496708990702694on_nat(aa_o_H2459438917461573107t_unit(refine_Imp_assert,Uuh = aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Uua),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uub),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat3999351741935936176on_nat(aa_nat3263736196148507005on_nat(aa_VEB6005108705944495412on_nat(aa_lis8565807915274807301on_nat(aa_VEB8546404647211134847on_nat(aa_arr8208594633606790516on_nat(aa_nat1839186240859117749on_nat(aTP_Lamm_vx(Uu),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh)) ).

% ATP.lambda_1238
tff(fact_11416_ATP_Olambda__1239,axiom,
    ! [Uu: fun_VE3775134097089128360on_nat,Uua: nat,Uub: nat,Uuc: array_VEBT_VEBTi,Uud: vEBT_VEBTi,Uue: list_VEBT_VEBT,Uuf: vEBT_VEBT,Uug: nat,Uuh: nat,Uui: product_unit] : aa_Pro451875010097498139on_nat(aa_nat3999351741935936176on_nat(aa_nat3263736196148507005on_nat(aa_VEB6005108705944495412on_nat(aa_lis8565807915274807301on_nat(aa_VEB8546404647211134847on_nat(aa_arr8208594633606790516on_nat(aa_nat1839186240859117749on_nat(aa_nat5704251136259731006on_nat(aTP_Lamm_sq(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui) = heap_T5999496708990702694on_nat(aa_o_H2459438917461573107t_unit(refine_Imp_assert,Uuh = aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Uua),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uub),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat3999351741935936176on_nat(aa_nat3263736196148507005on_nat(aa_VEB6005108705944495412on_nat(aa_lis8565807915274807301on_nat(aa_VEB8546404647211134847on_nat(aa_arr8208594633606790516on_nat(aa_nat1839186240859117749on_nat(aTP_Lamm_sp(Uu),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh)) ).

% ATP.lambda_1239
tff(fact_11417_ATP_Olambda__1240,axiom,
    ! [Uu: fun_Pr633315466666964198on_nat,Uua: nat,Uub: array_VEBT_VEBTi,Uuc: vEBT_VEBTi,Uud: vEBT_VEBT,Uue: nat,Uuf: nat,Uug: vEBT_VEBTi,Uuh: vEBT_VEBT,Uui: option_nat] : aa_opt9163293676379396302on_nat(aa_VEB3755160987679246317on_nat(aa_VEB5212316371209427943on_nat(aa_nat7793430520228346548on_nat(aa_nat8410684691732023425on_nat(aa_VEB3020412023619788856on_nat(aa_VEB2867584020876442580on_nat(aa_arr7250883597209086773on_nat(aa_nat6287673228857293642on_nat(aTP_Lamm_vu(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui) = heap_T5999496708990702694on_nat(aa_o_H2459438917461573107t_unit(refine_Imp_assert,Uui = vEBT_vebt_maxt(Uuh)),aa_opt5089197864053947520on_nat(aa_VEB2490029800597546679on_nat(aa_VEB1055358423767010003on_nat(aa_nat4144897467549153500on_nat(aa_nat7557481500408122597on_nat(aa_VEB3297963669270601860on_nat(aa_VEB3046651646968523646on_nat(aa_arr2696250042600076787on_nat(aa_nat8399877474291611060on_nat(aTP_Lamm_vt(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui)) ).

% ATP.lambda_1240
tff(fact_11418_ATP_Olambda__1241,axiom,
    ! [Uu: fun_VE3775134097089128360on_nat,Uua: nat,Uub: array_VEBT_VEBTi,Uuc: vEBT_VEBTi,Uud: vEBT_VEBT,Uue: nat,Uuf: nat,Uug: vEBT_VEBTi,Uuh: vEBT_VEBT,Uui: option_nat] : aa_opt9163293676379396302on_nat(aa_VEB3755160987679246317on_nat(aa_VEB5212316371209427943on_nat(aa_nat7793430520228346548on_nat(aa_nat8410684691732023425on_nat(aa_VEB3020412023619788856on_nat(aa_VEB2867584020876442580on_nat(aa_arr7250883597209086773on_nat(aa_nat6287673228857293642on_nat(aTP_Lamm_sm(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui) = heap_T5999496708990702694on_nat(aa_o_H2459438917461573107t_unit(refine_Imp_assert,Uui = vEBT_vebt_maxt(Uuh)),aa_opt5089197864053947520on_nat(aa_VEB2490029800597546679on_nat(aa_VEB1055358423767010003on_nat(aa_nat4144897467549153500on_nat(aa_nat7557481500408122597on_nat(aa_VEB3297963669270601860on_nat(aa_VEB3046651646968523646on_nat(aa_arr2696250042600076787on_nat(aa_nat8399877474291611060on_nat(aTP_Lamm_sl(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui)) ).

% ATP.lambda_1241
tff(fact_11419_ATP_Olambda__1242,axiom,
    ! [Uu: nat,Uua: nat,Uub: array_VEBT_VEBTi,Uuc: vEBT_VEBTi,Uud: list_VEBT_VEBT,Uue: vEBT_VEBT,Uuf: product_prod_nat_nat,Uug: nat,Uuh: nat,Uui: product_unit] : aa_Pro451875010097498139on_nat(aa_nat3999351741935936176on_nat(aa_nat3263736196148507005on_nat(aa_Pro6111869828830443265on_nat(aa_VEB6942907905770183000on_nat(aa_lis7408950245444934749on_nat(aa_VEB6661098386149969529on_nat(aa_arr9010034949507926874on_nat(aa_nat5148509663038818543on_nat(aTP_Lamm_ub(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui) = heap_T5999496708990702694on_nat(aa_o_H2459438917461573107t_unit(refine_Imp_assert,aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uuh),size_s6755466524823107622T_VEBT(Uud))),aa_nat3999351741935936176on_nat(aa_nat3263736196148507005on_nat(aa_Pro6111869828830443265on_nat(aa_VEB6942907905770183000on_nat(aa_lis7408950245444934749on_nat(aa_VEB6661098386149969529on_nat(aa_arr9010034949507926874on_nat(aa_nat5148509663038818543on_nat(aTP_Lamm_ua(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh)) ).

% ATP.lambda_1242
tff(fact_11420_ATP_Olambda__1243,axiom,
    ! [Uu: nat,Uua: nat,Uub: array_VEBT_VEBTi,Uuc: vEBT_VEBTi,Uud: product_prod_nat_nat,Uue: nat,Uuf: list_VEBT_VEBT,Uug: vEBT_VEBT,Uuh: nat,Uui: nat] : aa_nat5408703056993926195_VEBTi(aa_nat4282640288315822012_VEBTi(aa_VEB2281408947416558043_VEBTi(aa_lis148658726372936800_VEBTi(aa_nat3438343216000400545_VEBTi(aa_Pro9203685154717219507_VEBTi(aa_VEB4942273269570985133_VEBTi(aa_arr2286534195499807138_VEBTi(aa_nat2259444989923833827_VEBTi(aTP_Lamm_to(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui) = heap_T844888390831797134_VEBTi(aa_nat5821310369566504398ap_nat(heap_Time_return_nat,aa_Pro4243173770044397973at_nat(product_snd_nat_nat,Uud)),aa_nat4282640288315822012_VEBTi(aa_nat7653869184832469701_VEBTi(aa_VEB681999334371813476_VEBTi(aa_lis3749794993760221801_VEBTi(aa_nat3264418630393705002_VEBTi(aa_VEB7195496841433988878_VEBTi(aa_arr953441552191549551_VEBTi(aa_nat1068536149936821380_VEBTi(aTP_Lamm_tn(Uu),Uua),Uub),Uuc),Uue),Uuf),Uug),Uuh),Uui)) ).

% ATP.lambda_1243
tff(fact_11421_ATP_Olambda__1244,axiom,
    ! [Uu: nat,Uua: nat,Uub: array_VEBT_VEBTi,Uuc: vEBT_VEBTi,Uud: product_prod_nat_nat,Uue: nat,Uuf: list_VEBT_VEBT,Uug: vEBT_VEBT,Uuh: nat,Uui: nat] : aa_nat5408703056993926195_VEBTi(aa_nat4282640288315822012_VEBTi(aa_VEB2281408947416558043_VEBTi(aa_lis148658726372936800_VEBTi(aa_nat3438343216000400545_VEBTi(aa_Pro9203685154717219507_VEBTi(aa_VEB4942273269570985133_VEBTi(aa_arr2286534195499807138_VEBTi(aa_nat2259444989923833827_VEBTi(aTP_Lamm_tp(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui) = heap_T844888390831797134_VEBTi(aa_nat5821310369566504398ap_nat(heap_Time_return_nat,aa_Pro4243173770044397973at_nat(product_fst_nat_nat,Uud)),aa_nat4282640288315822012_VEBTi(aa_VEB2281408947416558043_VEBTi(aa_lis148658726372936800_VEBTi(aa_nat3438343216000400545_VEBTi(aa_Pro9203685154717219507_VEBTi(aa_VEB4942273269570985133_VEBTi(aa_arr2286534195499807138_VEBTi(aa_nat2259444989923833827_VEBTi(aTP_Lamm_to(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh)) ).

% ATP.lambda_1244
tff(fact_11422_ATP_Olambda__1245,axiom,
    ! [Uu: fun_Pr4171499318505651870_VEBTi,Uua: nat,Uub: array_VEBT_VEBTi,Uuc: vEBT_VEBTi,Uud: nat,Uue: nat,Uuf: nat,Uug: nat,Uuh: nat,Uui: vEBT_VEBTi] : aa_VEB836174949890266638_VEBTi(aa_nat5661399468481021847_VEBTi(aa_nat5003802506003144480_VEBTi(aa_nat6388535185314696745_VEBTi(aa_nat3402329839695434930_VEBTi(aa_nat7160451295336493755_VEBTi(aa_VEB6809311486975643423_VEBTi(aa_arr4222437323582306304_VEBTi(aa_nat3616447687739732885_VEBTi(aTP_Lamm_abf(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui) = heap_T5998771940306268294_VEBTi(aa_VEB8194921341634505877Heap_o(vEBT_VEBT_minNulli,Uui),aa_VEB2625734152646192863_VEBTi(aa_nat3994193416293967532_VEBTi(aa_nat4708659386357353849_VEBTi(aa_nat6007137331526128070_VEBTi(aa_nat4759958082780230547_VEBTi(aa_nat5669347316383757024_VEBTi(aa_VEB8848578851935055378_VEBTi(aa_arr7350257379194205831_VEBTi(aa_nat728959005377815624_VEBTi(aTP_Lamm_abe(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui)) ).

% ATP.lambda_1245
tff(fact_11423_ATP_Olambda__1246,axiom,
    ! [Uu: fun_VE8118064890490510894_VEBTi,Uua: nat,Uub: array_VEBT_VEBTi,Uuc: vEBT_VEBTi,Uud: nat,Uue: nat,Uuf: nat,Uug: nat,Uuh: nat,Uui: vEBT_VEBTi] : aa_VEB836174949890266638_VEBTi(aa_nat5661399468481021847_VEBTi(aa_nat5003802506003144480_VEBTi(aa_nat6388535185314696745_VEBTi(aa_nat3402329839695434930_VEBTi(aa_nat7160451295336493755_VEBTi(aa_VEB6809311486975643423_VEBTi(aa_arr4222437323582306304_VEBTi(aa_nat3616447687739732885_VEBTi(aTP_Lamm_nc(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui) = heap_T5998771940306268294_VEBTi(aa_VEB8194921341634505877Heap_o(vEBT_VEBT_minNulli,Uui),aa_VEB2625734152646192863_VEBTi(aa_nat3994193416293967532_VEBTi(aa_nat4708659386357353849_VEBTi(aa_nat6007137331526128070_VEBTi(aa_nat4759958082780230547_VEBTi(aa_nat5669347316383757024_VEBTi(aa_VEB8848578851935055378_VEBTi(aa_arr7350257379194205831_VEBTi(aa_nat728959005377815624_VEBTi(aTP_Lamm_nb(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui)) ).

% ATP.lambda_1246
tff(fact_11424_ATP_Olambda__1247,axiom,
    ! [Uu: nat,Uua: nat,Uub: array_VEBT_VEBTi,Uuc: vEBT_VEBTi,Uud: nat,Uue: nat,Uuf: nat,Uug: nat,Uuh: nat,Uui: nat] : aa_nat5408703056993926195_VEBTi(aa_nat4282640288315822012_VEBTi(aa_nat7653869184832469701_VEBTi(aa_nat2287630409448422734_VEBTi(aa_nat2878941900432676695_VEBTi(aa_nat532351713702512864_VEBTi(aa_VEB7864109228220129220_VEBTi(aa_arr7707872583627363365_VEBTi(aa_nat3757413148909559610_VEBTi(aTP_Lamm_iw(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui) = heap_T844888390831797134_VEBTi(array_len_VEBT_VEBTi(Uub),aa_nat4282640288315822012_VEBTi(aa_nat7653869184832469701_VEBTi(aa_nat2287630409448422734_VEBTi(aa_nat2878941900432676695_VEBTi(aa_nat532351713702512864_VEBTi(aa_nat5473138614813200873_VEBTi(aa_VEB7525848083004366413_VEBTi(aa_arr912658909913700910_VEBTi(aa_nat5309850715826091203_VEBTi(aTP_Lamm_iv(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui)) ).

% ATP.lambda_1247
tff(fact_11425_ATP_Olambda__1248,axiom,
    ! [Uu: nat,Uua: nat,Uub: array_VEBT_VEBTi,Uuc: vEBT_VEBTi,Uud: nat,Uue: nat,Uuf: nat,Uug: nat,Uuh: nat,Uui: nat,Uuj: nat] :
      aa_nat5408703056993926195_VEBTi(aa_nat4282640288315822012_VEBTi(aa_nat7653869184832469701_VEBTi(aa_nat2287630409448422734_VEBTi(aa_nat2878941900432676695_VEBTi(aa_nat532351713702512864_VEBTi(aa_nat5473138614813200873_VEBTi(aa_VEB7525848083004366413_VEBTi(aa_arr912658909913700910_VEBTi(aa_nat5309850715826091203_VEBTi(aTP_Lamm_iv(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj) = $ite(
        ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uui),Uuj)
        & ~ ( ( Uu = Uud )
            | ( Uu = Uue ) ) ),
        heap_T1006145433769338483_VEBTi(aa_nat5408703056993926195_VEBTi(array_nth_VEBT_VEBTi(Uub),Uui),aa_nat5661399468481021847_VEBTi(aa_nat5003802506003144480_VEBTi(aa_nat6388535185314696745_VEBTi(aa_nat3402329839695434930_VEBTi(aa_nat7160451295336493755_VEBTi(aa_VEB6809311486975643423_VEBTi(aa_arr4222437323582306304_VEBTi(aTP_Lamm_iu(Uua),Uub),Uuc),Uue),Uuf),Uug),Uuh),Uui)),
        aa_VEB836174949890266638_VEBTi(heap_T3630416162098727440_VEBTi,aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Uud),Uue)),Uua,Uub),Uuc)) ) ).

% ATP.lambda_1248
tff(fact_11426_ATP_Olambda__1249,axiom,
    ! [Uu: fun_Pr633315466666964198on_nat,Uua: nat,Uub: array_VEBT_VEBTi,Uuc: vEBT_VEBTi,Uud: vEBT_VEBT,Uue: nat,Uuf: nat,Uug: vEBT_VEBTi,Uuh: vEBT_VEBT,Uui: option_nat,Uuj: product_unit] :
      aa_Pro451875010097498139on_nat(aa_opt5089197864053947520on_nat(aa_VEB2490029800597546679on_nat(aa_VEB1055358423767010003on_nat(aa_nat4144897467549153500on_nat(aa_nat7557481500408122597on_nat(aa_VEB3297963669270601860on_nat(aa_VEB3046651646968523646on_nat(aa_arr2696250042600076787on_nat(aa_nat8399877474291611060on_nat(aTP_Lamm_vt(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj) = $ite(
        ( ( Uui != none_nat )
        & aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_less,aa_nat_option_nat(some_nat,Uue)),Uui) ),
        heap_T3669509953089699273on_nat(aa_nat947733127081326238on_nat(aa_VEB3566417204837707778on_nat(aa_VEB1030146501360450209on_nat(produc9121605208400662357on_nat(produc1757988346207259447on_nat(Uu)),Uuh),Uug),Uue),aa_nat9127621225095157975on_nat(aTP_Lamm_ep(Uua),Uuf)),
        heap_T3669509953089699273on_nat(aa_nat947733127081326238on_nat(aa_VEB3566417204837707778on_nat(aa_VEB1030146501360450209on_nat(produc9121605208400662357on_nat(produc1757988346207259447on_nat(Uu)),Uud),Uuc),Uuf),aa_nat9127621225095157975on_nat(aa_VEB8264456441901112694on_nat(aa_arr7060518384107407923on_nat(aTP_Lamm_of(Uua),Uub),Uud),Uuf)) ) ).

% ATP.lambda_1249
tff(fact_11427_ATP_Olambda__1250,axiom,
    ! [Uu: fun_VE3775134097089128360on_nat,Uua: nat,Uub: array_VEBT_VEBTi,Uuc: vEBT_VEBTi,Uud: vEBT_VEBT,Uue: nat,Uuf: nat,Uug: vEBT_VEBTi,Uuh: vEBT_VEBT,Uui: option_nat,Uuj: product_unit] :
      aa_Pro451875010097498139on_nat(aa_opt5089197864053947520on_nat(aa_VEB2490029800597546679on_nat(aa_VEB1055358423767010003on_nat(aa_nat4144897467549153500on_nat(aa_nat7557481500408122597on_nat(aa_VEB3297963669270601860on_nat(aa_VEB3046651646968523646on_nat(aa_arr2696250042600076787on_nat(aa_nat8399877474291611060on_nat(aTP_Lamm_sl(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj) = $ite(
        ( ( Uui != none_nat )
        & aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_less,aa_nat_option_nat(some_nat,Uue)),Uui) ),
        heap_T3669509953089699273on_nat(aa_nat947733127081326238on_nat(aa_VEB3566417204837707778on_nat(aa_VEB1030146501360450209on_nat(Uu,Uuh),Uug),Uue),aa_nat9127621225095157975on_nat(aTP_Lamm_ep(Uua),Uuf)),
        heap_T3669509953089699273on_nat(aa_nat947733127081326238on_nat(aa_VEB3566417204837707778on_nat(aa_VEB1030146501360450209on_nat(Uu,Uud),Uuc),Uuf),aa_nat9127621225095157975on_nat(aa_VEB8264456441901112694on_nat(aa_arr7060518384107407923on_nat(aTP_Lamm_of(Uua),Uub),Uud),Uuf)) ) ).

% ATP.lambda_1250
tff(fact_11428_ATP_Olambda__1251,axiom,
    ! [Uu: fun_Pr4404027214751104635_VEBTi,Uua: nat,Uub: nat,Uuc: array_VEBT_VEBTi,Uud: vEBT_VEBTi,Uue: product_prod_nat_nat,Uuf: option4927543243414619207at_nat,Uug: nat,Uuh: list_VEBT_VEBT,Uui: vEBT_VEBT,Uuj: product_unit] : aa_Pro3581650104493980720_VEBTi(aa_VEB3467812963353200047_VEBTi(aa_lis3316967545134462464_VEBTi(aa_nat7280733210739239061_VEBTi(aa_opt2515449710316153439_VEBTi(aa_Pro2414373876686838441_VEBTi(aa_VEB2192875243258015011_VEBTi(aa_arr5702273427728458776_VEBTi(aa_nat1667688020712374873_VEBTi(aa_nat345951160182191842_VEBTi(aTP_Lamm_vk(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj) = aa_Pro1395359565518501564_VEBTi(produc737604151543542771_VEBTi(aa_VEB2281408947416558043_VEBTi(aa_lis148658726372936800_VEBTi(aa_nat3438343216000400545_VEBTi(aa_Pro9203685154717219507_VEBTi(aa_VEB4942273269570985133_VEBTi(aa_arr2286534195499807138_VEBTi(aa_nat2259444989923833827_VEBTi(aa_nat4303739465877327980_VEBTi(aTP_Lamm_vj(Uu),Uua),Uub),Uuc),Uud),Uue),Uug),Uuh),Uui)),the_Pr8591224930841456533at_nat(Uuf)) ).

% ATP.lambda_1251
tff(fact_11429_ATP_Olambda__1252,axiom,
    ! [Uu: fun_VE2018681615261037117_VEBTi,Uua: nat,Uub: nat,Uuc: array_VEBT_VEBTi,Uud: vEBT_VEBTi,Uue: product_prod_nat_nat,Uuf: option4927543243414619207at_nat,Uug: nat,Uuh: list_VEBT_VEBT,Uui: vEBT_VEBT,Uuj: product_unit] : aa_Pro3581650104493980720_VEBTi(aa_VEB3467812963353200047_VEBTi(aa_lis3316967545134462464_VEBTi(aa_nat7280733210739239061_VEBTi(aa_opt2515449710316153439_VEBTi(aa_Pro2414373876686838441_VEBTi(aa_VEB2192875243258015011_VEBTi(aa_arr5702273427728458776_VEBTi(aa_nat1667688020712374873_VEBTi(aa_nat345951160182191842_VEBTi(aTP_Lamm_rn(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj) = aa_Pro1395359565518501564_VEBTi(produc737604151543542771_VEBTi(aa_VEB2281408947416558043_VEBTi(aa_lis148658726372936800_VEBTi(aa_nat3438343216000400545_VEBTi(aa_Pro9203685154717219507_VEBTi(aa_VEB4942273269570985133_VEBTi(aa_arr2286534195499807138_VEBTi(aa_nat2259444989923833827_VEBTi(aa_nat4303739465877327980_VEBTi(aTP_Lamm_rm(Uu),Uua),Uub),Uuc),Uud),Uue),Uug),Uuh),Uui)),the_Pr8591224930841456533at_nat(Uuf)) ).

% ATP.lambda_1252
tff(fact_11430_ATP_Olambda__1253,axiom,
    ! [Uu: fun_Pr4404027214751104635_VEBTi,Uua: nat,Uub: nat,Uuc: array_VEBT_VEBTi,Uud: vEBT_VEBTi,Uue: nat,Uuf: list_VEBT_VEBT,Uug: vEBT_VEBT,Uuh: nat,Uui: nat,Uuj: nat] :
      aa_nat5408703056993926195_VEBTi(aa_nat4282640288315822012_VEBTi(aa_nat7653869184832469701_VEBTi(aa_VEB681999334371813476_VEBTi(aa_lis3749794993760221801_VEBTi(aa_nat3264418630393705002_VEBTi(aa_VEB7195496841433988878_VEBTi(aa_arr953441552191549551_VEBTi(aa_nat1068536149936821380_VEBTi(aa_nat1714504438291150161_VEBTi(aTP_Lamm_vh(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj) = heap_T844888390831797134_VEBTi($ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uua),Uui),aa_nat5821310369566504398ap_nat(heap_Time_return_nat,Uui),aa_nat5821310369566504398ap_nat(heap_Time_return_nat,Uua)),aa_nat4282640288315822012_VEBTi(aa_nat7653869184832469701_VEBTi(aa_nat2287630409448422734_VEBTi(aa_VEB3484995426995338861_VEBTi(aa_lis8107322492824698866_VEBTi(aa_nat5353086824354447155_VEBTi(aa_VEB4525472277563179415_VEBTi(aa_arr4621753813534489208_VEBTi(aa_nat369315079452798477_VEBTi(aa_nat3166855462611258970_VEBTi(aTP_Lamm_vg(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj)) ).

% ATP.lambda_1253
tff(fact_11431_ATP_Olambda__1254,axiom,
    ! [Uu: fun_VE2018681615261037117_VEBTi,Uua: nat,Uub: nat,Uuc: array_VEBT_VEBTi,Uud: vEBT_VEBTi,Uue: nat,Uuf: list_VEBT_VEBT,Uug: vEBT_VEBT,Uuh: nat,Uui: nat,Uuj: nat] :
      aa_nat5408703056993926195_VEBTi(aa_nat4282640288315822012_VEBTi(aa_nat7653869184832469701_VEBTi(aa_VEB681999334371813476_VEBTi(aa_lis3749794993760221801_VEBTi(aa_nat3264418630393705002_VEBTi(aa_VEB7195496841433988878_VEBTi(aa_arr953441552191549551_VEBTi(aa_nat1068536149936821380_VEBTi(aa_nat1714504438291150161_VEBTi(aTP_Lamm_rk(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj) = heap_T844888390831797134_VEBTi($ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uua),Uui),aa_nat5821310369566504398ap_nat(heap_Time_return_nat,Uui),aa_nat5821310369566504398ap_nat(heap_Time_return_nat,Uua)),aa_nat4282640288315822012_VEBTi(aa_nat7653869184832469701_VEBTi(aa_nat2287630409448422734_VEBTi(aa_VEB3484995426995338861_VEBTi(aa_lis8107322492824698866_VEBTi(aa_nat5353086824354447155_VEBTi(aa_VEB4525472277563179415_VEBTi(aa_arr4621753813534489208_VEBTi(aa_nat369315079452798477_VEBTi(aa_nat3166855462611258970_VEBTi(aTP_Lamm_rj(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj)) ).

% ATP.lambda_1254
tff(fact_11432_ATP_Olambda__1255,axiom,
    ! [Uu: nat,Uua: nat,Uub: array_VEBT_VEBTi,Uuc: vEBT_VEBTi,Uud: nat,Uue: list_VEBT_VEBT,Uuf: vEBT_VEBT,Uug: nat,Uuh: nat,Uui: nat,Uuj: nat] :
      aa_nat5408703056993926195_VEBTi(aa_nat4282640288315822012_VEBTi(aa_nat7653869184832469701_VEBTi(aa_nat2287630409448422734_VEBTi(aa_VEB3484995426995338861_VEBTi(aa_lis8107322492824698866_VEBTi(aa_nat5353086824354447155_VEBTi(aa_VEB4525472277563179415_VEBTi(aa_arr4621753813534489208_VEBTi(aa_nat369315079452798477_VEBTi(aTP_Lamm_tm(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj) = $let(
        xn2: nat,
        xn2:= 
          $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uu),Uug),Uug,Uu),
        heap_T844888390831797134_VEBTi($ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uu),Uuh),aa_nat5821310369566504398ap_nat(heap_Time_return_nat,Uu),aa_nat5821310369566504398ap_nat(heap_Time_return_nat,Uuh)),aa_nat4282640288315822012_VEBTi(aa_nat7653869184832469701_VEBTi(aa_nat2287630409448422734_VEBTi(aa_nat2878941900432676695_VEBTi(aa_VEB2417709626833497078_VEBTi(aa_lis5541900872984878331_VEBTi(aa_nat770714458426680252_VEBTi(aa_VEB4484153586529426336_VEBTi(aa_arr8317680606763408385_VEBTi(aa_nat4438223700708794134_VEBTi(aTP_Lamm_tl(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uuh),Uui),Uuj),xn2)) ) ).

% ATP.lambda_1255
tff(fact_11433_ATP_Olambda__1256,axiom,
    ! [Uu: fun_Pr4171499318505651870_VEBTi,Uua: nat,Uub: array_VEBT_VEBTi,Uuc: vEBT_VEBTi,Uud: nat,Uue: nat,Uuf: nat,Uug: nat,Uuh: nat,Uui: vEBT_VEBTi,Uuj: $o] : aa_o_H7577119287286566245_VEBTi(aa_VEB2625734152646192863_VEBTi(aa_nat3994193416293967532_VEBTi(aa_nat4708659386357353849_VEBTi(aa_nat6007137331526128070_VEBTi(aa_nat4759958082780230547_VEBTi(aa_nat5669347316383757024_VEBTi(aa_VEB8848578851935055378_VEBTi(aa_arr7350257379194205831_VEBTi(aa_nat728959005377815624_VEBTi(aTP_Lamm_abe(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),(Uuj)) = heap_T1006145433769338483_VEBTi(aa_nat5408703056993926195_VEBTi(aa_VEB4833742232818206615_VEBTi(produc5159149307777246319_VEBTi(Uu),Uui),Uug),aa_o_f1343430628605260489_VEBTi(aa_nat4147387395820660958_VEBTi(aa_nat2236254811235047083_VEBTi(aa_nat5995673692660408824_VEBTi(aa_nat4988899645790341829_VEBTi(aa_VEB5075307909151313271_VEBTi(aa_arr5746817224660070252_VEBTi(aa_nat898395892396129965_VEBTi(aTP_Lamm_abd(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uuh),(Uuj))) ).

% ATP.lambda_1256
tff(fact_11434_ATP_Olambda__1257,axiom,
    ! [Uu: fun_Pr4404027214751104635_VEBTi,Uua: nat,Uub: array_VEBT_VEBTi,Uuc: vEBT_VEBTi,Uud: vEBT_VEBT,Uue: nat,Uuf: nat,Uug: nat,Uuh: nat,Uui: $o,Uuj: vEBT_VEBTi] : aa_VEB836174949890266638_VEBTi(aa_o_f1343430628605260489_VEBTi(aa_nat4147387395820660958_VEBTi(aa_nat2236254811235047083_VEBTi(aa_nat5995673692660408824_VEBTi(aa_nat4988899645790341829_VEBTi(aa_VEB257407171870985852_VEBTi(aa_VEB2935853492131583640_VEBTi(aa_arr6469760262597239545_VEBTi(aa_nat7574034614035128846_VEBTi(aTP_Lamm_uv(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),(Uui)),Uuj) = heap_T5099337393651448672_VEBTi(array_upd_VEBT_VEBTi(Uuh,Uuj,Uub),aa_o_f5513965820244355694_VEBTi(aa_nat6231872576868200367_VEBTi(aa_nat3271849308718382136_VEBTi(aa_nat4180741733914107969_VEBTi(aa_nat6590260363808168906_VEBTi(aa_VEB6485260183185377513_VEBTi(aa_VEB953658682601944675_VEBTi(aa_nat3348711059893815344_VEBTi(aTP_Lamm_uu(Uu),Uua),Uuc),Uud),Uue),Uuf),Uug),Uuh),(Uui))) ).

% ATP.lambda_1257
tff(fact_11435_ATP_Olambda__1258,axiom,
    ! [Uu: fun_VE2018681615261037117_VEBTi,Uua: nat,Uub: array_VEBT_VEBTi,Uuc: vEBT_VEBTi,Uud: vEBT_VEBT,Uue: nat,Uuf: nat,Uug: nat,Uuh: nat,Uui: $o,Uuj: vEBT_VEBTi] : aa_VEB836174949890266638_VEBTi(aa_o_f1343430628605260489_VEBTi(aa_nat4147387395820660958_VEBTi(aa_nat2236254811235047083_VEBTi(aa_nat5995673692660408824_VEBTi(aa_nat4988899645790341829_VEBTi(aa_VEB257407171870985852_VEBTi(aa_VEB2935853492131583640_VEBTi(aa_arr6469760262597239545_VEBTi(aa_nat7574034614035128846_VEBTi(aTP_Lamm_qy(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),(Uui)),Uuj) = heap_T5099337393651448672_VEBTi(array_upd_VEBT_VEBTi(Uuh,Uuj,Uub),aa_o_f5513965820244355694_VEBTi(aa_nat6231872576868200367_VEBTi(aa_nat3271849308718382136_VEBTi(aa_nat4180741733914107969_VEBTi(aa_nat6590260363808168906_VEBTi(aa_VEB6485260183185377513_VEBTi(aa_VEB953658682601944675_VEBTi(aa_nat3348711059893815344_VEBTi(aTP_Lamm_qx(Uu),Uua),Uuc),Uud),Uue),Uuf),Uug),Uuh),(Uui))) ).

% ATP.lambda_1258
tff(fact_11436_ATP_Olambda__1259,axiom,
    ! [Uu: fun_VE8118064890490510894_VEBTi,Uua: nat,Uub: array_VEBT_VEBTi,Uuc: vEBT_VEBTi,Uud: nat,Uue: nat,Uuf: nat,Uug: nat,Uuh: nat,Uui: vEBT_VEBTi,Uuj: $o] : aa_o_H7577119287286566245_VEBTi(aa_VEB2625734152646192863_VEBTi(aa_nat3994193416293967532_VEBTi(aa_nat4708659386357353849_VEBTi(aa_nat6007137331526128070_VEBTi(aa_nat4759958082780230547_VEBTi(aa_nat5669347316383757024_VEBTi(aa_VEB8848578851935055378_VEBTi(aa_arr7350257379194205831_VEBTi(aa_nat728959005377815624_VEBTi(aTP_Lamm_nb(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),(Uuj)) = heap_T1006145433769338483_VEBTi(aa_nat5408703056993926195_VEBTi(aa_VEB4833742232818206615_VEBTi(Uu,Uui),Uug),aa_o_f1343430628605260489_VEBTi(aa_nat4147387395820660958_VEBTi(aa_nat2236254811235047083_VEBTi(aa_nat5995673692660408824_VEBTi(aa_nat4988899645790341829_VEBTi(aa_VEB5075307909151313271_VEBTi(aa_arr5746817224660070252_VEBTi(aa_nat898395892396129965_VEBTi(aTP_Lamm_na(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uuh),(Uuj))) ).

% ATP.lambda_1259
tff(fact_11437_ATP_Olambda__1260,axiom,
    ! [Uu: nat,Uua: array_VEBT_VEBTi,Uub: vEBT_VEBTi,Uuc: list_VEBT_VEBT,Uud: vEBT_VEBT,Uue: nat,Uuf: nat,Uug: nat,Uuh: nat,Uui: nat,Uuj: product_unit] : aa_Pro3581650104493980720_VEBTi(aa_nat7312404184930824517_VEBTi(aa_nat3221004963518969490_VEBTi(aa_nat1817389164722854239_VEBTi(aa_nat317164691840817580_VEBTi(aa_nat3080504576880736121_VEBTi(aa_VEB4815025940789304112_VEBTi(aa_lis3577763594685753089_VEBTi(aa_VEB8209137959338826235_VEBTi(aa_arr747291131267414768_VEBTi(aTP_Lamm_tf(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj) = heap_T1006145433769338483_VEBTi(aa_nat5408703056993926195_VEBTi(array_nth_VEBT_VEBTi(Uua),Uui),aa_nat5661399468481021847_VEBTi(aa_nat5003802506003144480_VEBTi(aa_nat6388535185314696745_VEBTi(aa_nat3402329839695434930_VEBTi(aa_nat7160451295336493755_VEBTi(aa_VEB779792351083059546_VEBTi(aa_lis578081645931084383_VEBTi(aa_VEB3237245999891841275_VEBTi(aa_arr5756643615615923676_VEBTi(aTP_Lamm_te(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui)) ).

% ATP.lambda_1260
tff(fact_11438_ATP_Olambda__1261,axiom,
    ! [Uu: fun_Pr633315466666964198on_nat,Uua: nat,Uub: nat,Uuc: array_VEBT_VEBTi,Uud: vEBT_VEBTi,Uue: list_VEBT_VEBT,Uuf: vEBT_VEBT,Uug: product_prod_nat_nat,Uuh: nat,Uui: nat,Uuj: product_unit] : aa_Pro451875010097498139on_nat(aa_nat3999351741935936176on_nat(aa_nat3263736196148507005on_nat(aa_Pro6111869828830443265on_nat(aa_VEB6942907905770183000on_nat(aa_lis7408950245444934749on_nat(aa_VEB6661098386149969529on_nat(aa_arr9010034949507926874on_nat(aa_nat5148509663038818543on_nat(aa_nat8204191389585037116on_nat(aTP_Lamm_wy(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj) = heap_T2868974464944644318on_nat(aa_nat5408703056993926195_VEBTi(array_nth_VEBT_VEBTi(Uuc),Uui),aa_nat4669826966810537986on_nat(aa_nat107244183536939787on_nat(aa_Pro1663420989643744413on_nat(aa_VEB3438037858822377372on_nat(aa_lis3250066801158684525on_nat(aa_VEB8217410703897791847on_nat(aa_arr22699188990094940on_nat(aa_nat3630391635445309085on_nat(aa_nat7077015166316254502on_nat(aTP_Lamm_wx(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui)) ).

% ATP.lambda_1261
tff(fact_11439_ATP_Olambda__1262,axiom,
    ! [Uu: fun_VE3775134097089128360on_nat,Uua: nat,Uub: nat,Uuc: array_VEBT_VEBTi,Uud: vEBT_VEBTi,Uue: list_VEBT_VEBT,Uuf: vEBT_VEBT,Uug: product_prod_nat_nat,Uuh: nat,Uui: nat,Uuj: product_unit] : aa_Pro451875010097498139on_nat(aa_nat3999351741935936176on_nat(aa_nat3263736196148507005on_nat(aa_Pro6111869828830443265on_nat(aa_VEB6942907905770183000on_nat(aa_lis7408950245444934749on_nat(aa_VEB6661098386149969529on_nat(aa_arr9010034949507926874on_nat(aa_nat5148509663038818543on_nat(aa_nat8204191389585037116on_nat(aTP_Lamm_rz(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj) = heap_T2868974464944644318on_nat(aa_nat5408703056993926195_VEBTi(array_nth_VEBT_VEBTi(Uuc),Uui),aa_nat4669826966810537986on_nat(aa_nat107244183536939787on_nat(aa_Pro1663420989643744413on_nat(aa_VEB3438037858822377372on_nat(aa_lis3250066801158684525on_nat(aa_VEB8217410703897791847on_nat(aa_arr22699188990094940on_nat(aa_nat3630391635445309085on_nat(aa_nat7077015166316254502on_nat(aTP_Lamm_ry(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui)) ).

% ATP.lambda_1262
tff(fact_11440_ATP_Olambda__1263,axiom,
    ! [Uu: fun_Pr633315466666964198on_nat,Uua: nat,Uub: nat,Uuc: array_VEBT_VEBTi,Uud: vEBT_VEBTi,Uue: list_VEBT_VEBT,Uuf: vEBT_VEBT,Uug: product_prod_nat_nat,Uuh: nat,Uui: nat,Uuj: vEBT_VEBTi] :
      aa_VEB2711320435582112505on_nat(aa_nat4669826966810537986on_nat(aa_nat107244183536939787on_nat(aa_Pro1663420989643744413on_nat(aa_VEB3438037858822377372on_nat(aa_lis3250066801158684525on_nat(aa_VEB8217410703897791847on_nat(aa_arr22699188990094940on_nat(aa_nat3630391635445309085on_nat(aa_nat7077015166316254502on_nat(aTP_Lamm_wx(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj) = $let(
        aktnode: vEBT_VEBT,
        aktnode:= nth_VEBT_VEBT(Uue,Uui),
        heap_T3669509953089699273on_nat(aa_VEB2711320435582112505on_nat(vEBT_vebt_minti,Uuj),aa_VEB3755160987679246317on_nat(aa_VEB5212316371209427943on_nat(aa_nat7793430520228346548on_nat(aa_nat8410684691732023425on_nat(aa_Pro7328798252849467525on_nat(aa_VEB6402893080723908316on_nat(aa_VEB4808996286462526806on_nat(aa_arr4232376455131879627on_nat(aa_nat8394757134461757324on_nat(aa_nat5552918743037330581on_nat(aTP_Lamm_ww(Uu),Uua),Uub),Uuc),Uud),Uuf),Uug),Uuh),Uui),Uuj),aktnode)) ) ).

% ATP.lambda_1263
tff(fact_11441_ATP_Olambda__1264,axiom,
    ! [Uu: fun_VE3775134097089128360on_nat,Uua: nat,Uub: nat,Uuc: array_VEBT_VEBTi,Uud: vEBT_VEBTi,Uue: list_VEBT_VEBT,Uuf: vEBT_VEBT,Uug: product_prod_nat_nat,Uuh: nat,Uui: nat,Uuj: vEBT_VEBTi] :
      aa_VEB2711320435582112505on_nat(aa_nat4669826966810537986on_nat(aa_nat107244183536939787on_nat(aa_Pro1663420989643744413on_nat(aa_VEB3438037858822377372on_nat(aa_lis3250066801158684525on_nat(aa_VEB8217410703897791847on_nat(aa_arr22699188990094940on_nat(aa_nat3630391635445309085on_nat(aa_nat7077015166316254502on_nat(aTP_Lamm_ry(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj) = $let(
        aktnode: vEBT_VEBT,
        aktnode:= nth_VEBT_VEBT(Uue,Uui),
        heap_T3669509953089699273on_nat(aa_VEB2711320435582112505on_nat(vEBT_vebt_minti,Uuj),aa_VEB3755160987679246317on_nat(aa_VEB5212316371209427943on_nat(aa_nat7793430520228346548on_nat(aa_nat8410684691732023425on_nat(aa_Pro7328798252849467525on_nat(aa_VEB6402893080723908316on_nat(aa_VEB4808996286462526806on_nat(aa_arr4232376455131879627on_nat(aa_nat8394757134461757324on_nat(aa_nat5552918743037330581on_nat(aTP_Lamm_rx(Uu),Uua),Uub),Uuc),Uud),Uuf),Uug),Uuh),Uui),Uuj),aktnode)) ) ).

% ATP.lambda_1264
tff(fact_11442_ATP_Olambda__1265,axiom,
    ! [Uu: fun_Pr633315466666964198on_nat,Uua: vEBT_VEBT,Uub: nat,Uuc: option4927543243414619207at_nat,Uud: nat,Uue: array_VEBT_VEBTi,Uuf: vEBT_VEBTi,Uug: option4927543243414619207at_nat,Uuh: nat,Uui: list_VEBT_VEBT,Uuj: vEBT_VEBT] :
      aa_VEB3253283475358696116on_nat(aa_lis6893462050766468025on_nat(aa_nat8553908474689004666on_nat(aa_opt7284239819431652166on_nat(aa_VEB7959562887951478976on_nat(aa_arr4020752153017264181on_nat(aa_nat3060827894968391926on_nat(aa_opt346861859059968706on_nat(aa_nat3229958882251406039on_nat(aa_VEB430102235636188558on_nat(aTP_Lamm_xf(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj) = heap_T5999496708990702694on_nat(aa_o_H2459438917461573107t_unit(refine_Imp_assert,
          ( ( Uug = Uuc )
          & ( Uuh = Uud )
          & aa_VEBT_VEBT_o(vEBT_is_Node,Uua) )),aa_VEB1832073013541904922on_nat(aa_lis6328568131235813867on_nat(aa_VEB3884484516340040293on_nat(aa_arr5442694692229379418on_nat(aa_nat4425110079127750043on_nat(aa_opt7218664994477926375on_nat(aa_nat7200185421170774908on_nat(aTP_Lamm_xe(Uu),Uub),Uuc),Uud),Uue),Uuf),Uui),Uuj)) ).

% ATP.lambda_1265
tff(fact_11443_ATP_Olambda__1266,axiom,
    ! [Uu: fun_Pr633315466666964198on_nat,Uua: vEBT_VEBT,Uub: nat,Uuc: option4927543243414619207at_nat,Uud: nat,Uue: array_VEBT_VEBTi,Uuf: vEBT_VEBTi,Uug: option4927543243414619207at_nat,Uuh: nat,Uui: list_VEBT_VEBT,Uuj: vEBT_VEBT] :
      aa_VEB3253283475358696116on_nat(aa_lis6893462050766468025on_nat(aa_nat8553908474689004666on_nat(aa_opt7284239819431652166on_nat(aa_VEB7959562887951478976on_nat(aa_arr4020752153017264181on_nat(aa_nat3060827894968391926on_nat(aa_opt346861859059968706on_nat(aa_nat3229958882251406039on_nat(aa_VEB430102235636188558on_nat(aTP_Lamm_wd(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj) = heap_T5999496708990702694on_nat(aa_o_H2459438917461573107t_unit(refine_Imp_assert,
          ( ( Uug = Uuc )
          & ( Uuh = Uud )
          & aa_VEBT_VEBT_o(vEBT_is_Node,Uua) )),aa_VEB1832073013541904922on_nat(aa_lis6328568131235813867on_nat(aa_VEB3884484516340040293on_nat(aa_arr5442694692229379418on_nat(aa_nat4425110079127750043on_nat(aa_opt7218664994477926375on_nat(aa_nat7200185421170774908on_nat(aTP_Lamm_wc(Uu),Uub),Uuc),Uud),Uue),Uuf),Uui),Uuj)) ).

% ATP.lambda_1266
tff(fact_11444_ATP_Olambda__1267,axiom,
    ! [Uu: fun_VE3775134097089128360on_nat,Uua: vEBT_VEBT,Uub: nat,Uuc: option4927543243414619207at_nat,Uud: nat,Uue: array_VEBT_VEBTi,Uuf: vEBT_VEBTi,Uug: option4927543243414619207at_nat,Uuh: nat,Uui: list_VEBT_VEBT,Uuj: vEBT_VEBT] :
      aa_VEB3253283475358696116on_nat(aa_lis6893462050766468025on_nat(aa_nat8553908474689004666on_nat(aa_opt7284239819431652166on_nat(aa_VEB7959562887951478976on_nat(aa_arr4020752153017264181on_nat(aa_nat3060827894968391926on_nat(aa_opt346861859059968706on_nat(aa_nat3229958882251406039on_nat(aa_VEB430102235636188558on_nat(aTP_Lamm_sv(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj) = heap_T5999496708990702694on_nat(aa_o_H2459438917461573107t_unit(refine_Imp_assert,
          ( ( Uug = Uuc )
          & ( Uuh = Uud )
          & aa_VEBT_VEBT_o(vEBT_is_Node,Uua) )),aa_VEB1832073013541904922on_nat(aa_lis6328568131235813867on_nat(aa_VEB3884484516340040293on_nat(aa_arr5442694692229379418on_nat(aa_nat4425110079127750043on_nat(aa_opt7218664994477926375on_nat(aa_nat7200185421170774908on_nat(aTP_Lamm_su(Uu),Uub),Uuc),Uud),Uue),Uuf),Uui),Uuj)) ).

% ATP.lambda_1267
tff(fact_11445_ATP_Olambda__1268,axiom,
    ! [Uu: fun_VE3775134097089128360on_nat,Uua: vEBT_VEBT,Uub: nat,Uuc: option4927543243414619207at_nat,Uud: nat,Uue: array_VEBT_VEBTi,Uuf: vEBT_VEBTi,Uug: option4927543243414619207at_nat,Uuh: nat,Uui: list_VEBT_VEBT,Uuj: vEBT_VEBT] :
      aa_VEB3253283475358696116on_nat(aa_lis6893462050766468025on_nat(aa_nat8553908474689004666on_nat(aa_opt7284239819431652166on_nat(aa_VEB7959562887951478976on_nat(aa_arr4020752153017264181on_nat(aa_nat3060827894968391926on_nat(aa_opt346861859059968706on_nat(aa_nat3229958882251406039on_nat(aa_VEB430102235636188558on_nat(aTP_Lamm_sg(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj) = heap_T5999496708990702694on_nat(aa_o_H2459438917461573107t_unit(refine_Imp_assert,
          ( ( Uug = Uuc )
          & ( Uuh = Uud )
          & aa_VEBT_VEBT_o(vEBT_is_Node,Uua) )),aa_VEB1832073013541904922on_nat(aa_lis6328568131235813867on_nat(aa_VEB3884484516340040293on_nat(aa_arr5442694692229379418on_nat(aa_nat4425110079127750043on_nat(aa_opt7218664994477926375on_nat(aa_nat7200185421170774908on_nat(aTP_Lamm_sf(Uu),Uub),Uuc),Uud),Uue),Uuf),Uui),Uuj)) ).

% ATP.lambda_1268
tff(fact_11446_ATP_Olambda__1269,axiom,
    ! [Uu: fun_Pr4404027214751104635_VEBTi,Uua: nat,Uub: option4927543243414619207at_nat,Uuc: nat,Uud: array_VEBT_VEBTi,Uue: vEBT_VEBTi,Uuf: product_prod_nat_nat,Uug: option4927543243414619207at_nat,Uuh: nat,Uui: list_VEBT_VEBT,Uuj: vEBT_VEBT] :
      aa_VEB5809408762104066889_VEBTi(aa_lis894883570680706766_VEBTi(aa_nat1139649368474441743_VEBTi(aa_opt762848812907537627_VEBTi(aa_Pro4022776811541674023_VEBTi(aa_VEB3655672368276914755_VEBTi(aa_arr737849036559383332_VEBTi(aa_nat7609637249401053881_VEBTi(aa_opt3216071514793242883_VEBTi(aa_nat8892849909372205124_VEBTi(aTP_Lamm_vl(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj) = heap_T7982501707604696571_VEBTi(aa_o_H2459438917461573107t_unit(refine_Imp_assert,
          ( ( Uub = Uug )
          & ( Uuc = Uuh ) )),aa_VEB3467812963353200047_VEBTi(aa_lis3316967545134462464_VEBTi(aa_nat7280733210739239061_VEBTi(aa_opt2515449710316153439_VEBTi(aa_Pro2414373876686838441_VEBTi(aa_VEB2192875243258015011_VEBTi(aa_arr5702273427728458776_VEBTi(aa_nat1667688020712374873_VEBTi(aa_nat345951160182191842_VEBTi(aTP_Lamm_vk(Uu),Uua),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj)) ).

% ATP.lambda_1269
tff(fact_11447_ATP_Olambda__1270,axiom,
    ! [Uu: fun_VE2018681615261037117_VEBTi,Uua: nat,Uub: option4927543243414619207at_nat,Uuc: nat,Uud: array_VEBT_VEBTi,Uue: vEBT_VEBTi,Uuf: product_prod_nat_nat,Uug: option4927543243414619207at_nat,Uuh: nat,Uui: list_VEBT_VEBT,Uuj: vEBT_VEBT] :
      aa_VEB5809408762104066889_VEBTi(aa_lis894883570680706766_VEBTi(aa_nat1139649368474441743_VEBTi(aa_opt762848812907537627_VEBTi(aa_Pro4022776811541674023_VEBTi(aa_VEB3655672368276914755_VEBTi(aa_arr737849036559383332_VEBTi(aa_nat7609637249401053881_VEBTi(aa_opt3216071514793242883_VEBTi(aa_nat8892849909372205124_VEBTi(aTP_Lamm_ro(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj) = heap_T7982501707604696571_VEBTi(aa_o_H2459438917461573107t_unit(refine_Imp_assert,
          ( ( Uub = Uug )
          & ( Uuc = Uuh ) )),aa_VEB3467812963353200047_VEBTi(aa_lis3316967545134462464_VEBTi(aa_nat7280733210739239061_VEBTi(aa_opt2515449710316153439_VEBTi(aa_Pro2414373876686838441_VEBTi(aa_VEB2192875243258015011_VEBTi(aa_arr5702273427728458776_VEBTi(aa_nat1667688020712374873_VEBTi(aa_nat345951160182191842_VEBTi(aTP_Lamm_rn(Uu),Uua),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj)) ).

% ATP.lambda_1270
tff(fact_11448_ATP_Olambda__1271,axiom,
    ! [Uu: fun_Pr633315466666964198on_nat,Uua: nat,Uub: nat,Uuc: array_VEBT_VEBTi,Uud: vEBT_VEBTi,Uue: list_VEBT_VEBT,Uuf: vEBT_VEBT,Uug: product_prod_nat_nat,Uuh: nat,Uui: nat,Uuj: product_unit] : aa_Pro451875010097498139on_nat(aa_nat3999351741935936176on_nat(aa_nat3263736196148507005on_nat(aa_Pro6111869828830443265on_nat(aa_VEB6942907905770183000on_nat(aa_lis7408950245444934749on_nat(aa_VEB6661098386149969529on_nat(aa_arr9010034949507926874on_nat(aa_nat5148509663038818543on_nat(aa_nat8204191389585037116on_nat(aTP_Lamm_xa(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj) = heap_T5999496708990702694on_nat(aa_o_H2459438917461573107t_unit(refine_Imp_assert,Uui = aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Uua),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uub),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat3999351741935936176on_nat(aa_nat3263736196148507005on_nat(aa_Pro6111869828830443265on_nat(aa_VEB6942907905770183000on_nat(aa_lis7408950245444934749on_nat(aa_VEB6661098386149969529on_nat(aa_arr9010034949507926874on_nat(aa_nat5148509663038818543on_nat(aa_nat8204191389585037116on_nat(aTP_Lamm_wz(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui)) ).

% ATP.lambda_1271
tff(fact_11449_ATP_Olambda__1272,axiom,
    ! [Uu: fun_VE3775134097089128360on_nat,Uua: nat,Uub: nat,Uuc: array_VEBT_VEBTi,Uud: vEBT_VEBTi,Uue: list_VEBT_VEBT,Uuf: vEBT_VEBT,Uug: product_prod_nat_nat,Uuh: nat,Uui: nat,Uuj: product_unit] : aa_Pro451875010097498139on_nat(aa_nat3999351741935936176on_nat(aa_nat3263736196148507005on_nat(aa_Pro6111869828830443265on_nat(aa_VEB6942907905770183000on_nat(aa_lis7408950245444934749on_nat(aa_VEB6661098386149969529on_nat(aa_arr9010034949507926874on_nat(aa_nat5148509663038818543on_nat(aa_nat8204191389585037116on_nat(aTP_Lamm_sb(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj) = heap_T5999496708990702694on_nat(aa_o_H2459438917461573107t_unit(refine_Imp_assert,Uui = aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Uua),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uub),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat3999351741935936176on_nat(aa_nat3263736196148507005on_nat(aa_Pro6111869828830443265on_nat(aa_VEB6942907905770183000on_nat(aa_lis7408950245444934749on_nat(aa_VEB6661098386149969529on_nat(aa_arr9010034949507926874on_nat(aa_nat5148509663038818543on_nat(aa_nat8204191389585037116on_nat(aTP_Lamm_sa(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui)) ).

% ATP.lambda_1272
tff(fact_11450_ATP_Olambda__1273,axiom,
    ! [Uu: nat,Uua: nat,Uub: array_VEBT_VEBTi,Uuc: vEBT_VEBTi,Uud: vEBT_VEBT,Uue: product_prod_nat_nat,Uuf: nat,Uug: nat,Uuh: vEBT_VEBTi,Uui: vEBT_VEBT,Uuj: option_nat] : aa_opt9163293676379396302on_nat(aa_VEB3755160987679246317on_nat(aa_VEB5212316371209427943on_nat(aa_nat7793430520228346548on_nat(aa_nat8410684691732023425on_nat(aa_Pro7328798252849467525on_nat(aa_VEB6402893080723908316on_nat(aa_VEB4808996286462526806on_nat(aa_arr4232376455131879627on_nat(aa_nat8394757134461757324on_nat(aTP_Lamm_ty(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj) = heap_T5999496708990702694on_nat(aa_o_H2459438917461573107t_unit(refine_Imp_assert,Uuj = vEBT_vebt_mint(Uui)),aa_opt5089197864053947520on_nat(aa_VEB2490029800597546679on_nat(aa_VEB1055358423767010003on_nat(aa_nat4144897467549153500on_nat(aa_nat7557481500408122597on_nat(aa_Pro3918843774365325431on_nat(aa_VEB7240715040874449526on_nat(aa_VEB3377150917682228498on_nat(aa_arr2747111264493904243on_nat(aa_nat610287491858673288on_nat(aTP_Lamm_tx(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj)) ).

% ATP.lambda_1273
tff(fact_11451_ATP_Olambda__1274,axiom,
    ! [Uu: nat,Uua: array_VEBT_VEBTi,Uub: vEBT_VEBTi,Uuc: list_VEBT_VEBT,Uud: vEBT_VEBT,Uue: nat,Uuf: nat,Uug: nat,Uuh: nat,Uui: nat,Uuj: product_unit] : aa_Pro3581650104493980720_VEBTi(aa_nat7312404184930824517_VEBTi(aa_nat3221004963518969490_VEBTi(aa_nat1817389164722854239_VEBTi(aa_nat317164691840817580_VEBTi(aa_nat3080504576880736121_VEBTi(aa_VEB4815025940789304112_VEBTi(aa_lis3577763594685753089_VEBTi(aa_VEB8209137959338826235_VEBTi(aa_arr747291131267414768_VEBTi(aTP_Lamm_tg(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj) = heap_T7982501707604696571_VEBTi(aa_o_H2459438917461573107t_unit(refine_Imp_assert,aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uui),size_s6755466524823107622T_VEBT(Uuc))),aa_nat7312404184930824517_VEBTi(aa_nat3221004963518969490_VEBTi(aa_nat1817389164722854239_VEBTi(aa_nat317164691840817580_VEBTi(aa_nat3080504576880736121_VEBTi(aa_VEB4815025940789304112_VEBTi(aa_lis3577763594685753089_VEBTi(aa_VEB8209137959338826235_VEBTi(aa_arr747291131267414768_VEBTi(aTP_Lamm_tf(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui)) ).

% ATP.lambda_1274
tff(fact_11452_ATP_Olambda__1275,axiom,
    ! [Uu: fun_Pr633315466666964198on_nat,Uua: nat,Uub: nat,Uuc: array_VEBT_VEBTi,Uud: vEBT_VEBTi,Uue: list_VEBT_VEBT,Uuf: vEBT_VEBT,Uug: product_prod_nat_nat,Uuh: nat,Uui: nat,Uuj: product_unit] : aa_Pro451875010097498139on_nat(aa_nat3999351741935936176on_nat(aa_nat3263736196148507005on_nat(aa_Pro6111869828830443265on_nat(aa_VEB6942907905770183000on_nat(aa_lis7408950245444934749on_nat(aa_VEB6661098386149969529on_nat(aa_arr9010034949507926874on_nat(aa_nat5148509663038818543on_nat(aa_nat8204191389585037116on_nat(aTP_Lamm_wz(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj) = heap_T5999496708990702694on_nat(aa_o_H2459438917461573107t_unit(refine_Imp_assert,aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uui),size_s6755466524823107622T_VEBT(Uue))),aa_nat3999351741935936176on_nat(aa_nat3263736196148507005on_nat(aa_Pro6111869828830443265on_nat(aa_VEB6942907905770183000on_nat(aa_lis7408950245444934749on_nat(aa_VEB6661098386149969529on_nat(aa_arr9010034949507926874on_nat(aa_nat5148509663038818543on_nat(aa_nat8204191389585037116on_nat(aTP_Lamm_wy(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui)) ).

% ATP.lambda_1275
tff(fact_11453_ATP_Olambda__1276,axiom,
    ! [Uu: fun_VE3775134097089128360on_nat,Uua: nat,Uub: nat,Uuc: array_VEBT_VEBTi,Uud: vEBT_VEBTi,Uue: list_VEBT_VEBT,Uuf: vEBT_VEBT,Uug: product_prod_nat_nat,Uuh: nat,Uui: nat,Uuj: product_unit] : aa_Pro451875010097498139on_nat(aa_nat3999351741935936176on_nat(aa_nat3263736196148507005on_nat(aa_Pro6111869828830443265on_nat(aa_VEB6942907905770183000on_nat(aa_lis7408950245444934749on_nat(aa_VEB6661098386149969529on_nat(aa_arr9010034949507926874on_nat(aa_nat5148509663038818543on_nat(aa_nat8204191389585037116on_nat(aTP_Lamm_sa(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj) = heap_T5999496708990702694on_nat(aa_o_H2459438917461573107t_unit(refine_Imp_assert,aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uui),size_s6755466524823107622T_VEBT(Uue))),aa_nat3999351741935936176on_nat(aa_nat3263736196148507005on_nat(aa_Pro6111869828830443265on_nat(aa_VEB6942907905770183000on_nat(aa_lis7408950245444934749on_nat(aa_VEB6661098386149969529on_nat(aa_arr9010034949507926874on_nat(aa_nat5148509663038818543on_nat(aa_nat8204191389585037116on_nat(aTP_Lamm_rz(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui)) ).

% ATP.lambda_1276
tff(fact_11454_ATP_Olambda__1277,axiom,
    ! [Uu: fun_Pr4404027214751104635_VEBTi,Uua: nat,Uub: nat,Uuc: array_VEBT_VEBTi,Uud: vEBT_VEBTi,Uue: product_prod_nat_nat,Uuf: nat,Uug: list_VEBT_VEBT,Uuh: vEBT_VEBT,Uui: nat,Uuj: nat] : aa_nat5408703056993926195_VEBTi(aa_nat4282640288315822012_VEBTi(aa_VEB2281408947416558043_VEBTi(aa_lis148658726372936800_VEBTi(aa_nat3438343216000400545_VEBTi(aa_Pro9203685154717219507_VEBTi(aa_VEB4942273269570985133_VEBTi(aa_arr2286534195499807138_VEBTi(aa_nat2259444989923833827_VEBTi(aa_nat4303739465877327980_VEBTi(aTP_Lamm_vi(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj) = heap_T844888390831797134_VEBTi(aa_nat5821310369566504398ap_nat(heap_Time_return_nat,aa_Pro4243173770044397973at_nat(product_snd_nat_nat,Uue)),aa_nat4282640288315822012_VEBTi(aa_nat7653869184832469701_VEBTi(aa_VEB681999334371813476_VEBTi(aa_lis3749794993760221801_VEBTi(aa_nat3264418630393705002_VEBTi(aa_VEB7195496841433988878_VEBTi(aa_arr953441552191549551_VEBTi(aa_nat1068536149936821380_VEBTi(aa_nat1714504438291150161_VEBTi(aTP_Lamm_vh(Uu),Uua),Uub),Uuc),Uud),Uuf),Uug),Uuh),Uui),Uuj)) ).

% ATP.lambda_1277
tff(fact_11455_ATP_Olambda__1278,axiom,
    ! [Uu: fun_VE2018681615261037117_VEBTi,Uua: nat,Uub: nat,Uuc: array_VEBT_VEBTi,Uud: vEBT_VEBTi,Uue: product_prod_nat_nat,Uuf: nat,Uug: list_VEBT_VEBT,Uuh: vEBT_VEBT,Uui: nat,Uuj: nat] : aa_nat5408703056993926195_VEBTi(aa_nat4282640288315822012_VEBTi(aa_VEB2281408947416558043_VEBTi(aa_lis148658726372936800_VEBTi(aa_nat3438343216000400545_VEBTi(aa_Pro9203685154717219507_VEBTi(aa_VEB4942273269570985133_VEBTi(aa_arr2286534195499807138_VEBTi(aa_nat2259444989923833827_VEBTi(aa_nat4303739465877327980_VEBTi(aTP_Lamm_rl(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj) = heap_T844888390831797134_VEBTi(aa_nat5821310369566504398ap_nat(heap_Time_return_nat,aa_Pro4243173770044397973at_nat(product_snd_nat_nat,Uue)),aa_nat4282640288315822012_VEBTi(aa_nat7653869184832469701_VEBTi(aa_VEB681999334371813476_VEBTi(aa_lis3749794993760221801_VEBTi(aa_nat3264418630393705002_VEBTi(aa_VEB7195496841433988878_VEBTi(aa_arr953441552191549551_VEBTi(aa_nat1068536149936821380_VEBTi(aa_nat1714504438291150161_VEBTi(aTP_Lamm_rk(Uu),Uua),Uub),Uuc),Uud),Uuf),Uug),Uuh),Uui),Uuj)) ).

% ATP.lambda_1278
tff(fact_11456_ATP_Olambda__1279,axiom,
    ! [Uu: fun_Pr4404027214751104635_VEBTi,Uua: nat,Uub: nat,Uuc: array_VEBT_VEBTi,Uud: vEBT_VEBTi,Uue: product_prod_nat_nat,Uuf: nat,Uug: list_VEBT_VEBT,Uuh: vEBT_VEBT,Uui: nat,Uuj: nat] : aa_nat5408703056993926195_VEBTi(aa_nat4282640288315822012_VEBTi(aa_VEB2281408947416558043_VEBTi(aa_lis148658726372936800_VEBTi(aa_nat3438343216000400545_VEBTi(aa_Pro9203685154717219507_VEBTi(aa_VEB4942273269570985133_VEBTi(aa_arr2286534195499807138_VEBTi(aa_nat2259444989923833827_VEBTi(aa_nat4303739465877327980_VEBTi(aTP_Lamm_vj(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj) = heap_T844888390831797134_VEBTi(aa_nat5821310369566504398ap_nat(heap_Time_return_nat,aa_Pro4243173770044397973at_nat(product_fst_nat_nat,Uue)),aa_nat4282640288315822012_VEBTi(aa_VEB2281408947416558043_VEBTi(aa_lis148658726372936800_VEBTi(aa_nat3438343216000400545_VEBTi(aa_Pro9203685154717219507_VEBTi(aa_VEB4942273269570985133_VEBTi(aa_arr2286534195499807138_VEBTi(aa_nat2259444989923833827_VEBTi(aa_nat4303739465877327980_VEBTi(aTP_Lamm_vi(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui)) ).

% ATP.lambda_1279
tff(fact_11457_ATP_Olambda__1280,axiom,
    ! [Uu: fun_VE2018681615261037117_VEBTi,Uua: nat,Uub: nat,Uuc: array_VEBT_VEBTi,Uud: vEBT_VEBTi,Uue: product_prod_nat_nat,Uuf: nat,Uug: list_VEBT_VEBT,Uuh: vEBT_VEBT,Uui: nat,Uuj: nat] : aa_nat5408703056993926195_VEBTi(aa_nat4282640288315822012_VEBTi(aa_VEB2281408947416558043_VEBTi(aa_lis148658726372936800_VEBTi(aa_nat3438343216000400545_VEBTi(aa_Pro9203685154717219507_VEBTi(aa_VEB4942273269570985133_VEBTi(aa_arr2286534195499807138_VEBTi(aa_nat2259444989923833827_VEBTi(aa_nat4303739465877327980_VEBTi(aTP_Lamm_rm(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj) = heap_T844888390831797134_VEBTi(aa_nat5821310369566504398ap_nat(heap_Time_return_nat,aa_Pro4243173770044397973at_nat(product_fst_nat_nat,Uue)),aa_nat4282640288315822012_VEBTi(aa_VEB2281408947416558043_VEBTi(aa_lis148658726372936800_VEBTi(aa_nat3438343216000400545_VEBTi(aa_Pro9203685154717219507_VEBTi(aa_VEB4942273269570985133_VEBTi(aa_arr2286534195499807138_VEBTi(aa_nat2259444989923833827_VEBTi(aa_nat4303739465877327980_VEBTi(aTP_Lamm_rl(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui)) ).

% ATP.lambda_1280
tff(fact_11458_ATP_Olambda__1281,axiom,
    ! [Uu: nat,Uua: array_VEBT_VEBTi,Uub: vEBT_VEBTi,Uuc: list_VEBT_VEBT,Uud: vEBT_VEBT,Uue: nat,Uuf: nat,Uug: nat,Uuh: nat,Uui: nat,Uuj: vEBT_VEBTi] : aa_VEB836174949890266638_VEBTi(aa_nat5661399468481021847_VEBTi(aa_nat5003802506003144480_VEBTi(aa_nat6388535185314696745_VEBTi(aa_nat3402329839695434930_VEBTi(aa_nat7160451295336493755_VEBTi(aa_VEB779792351083059546_VEBTi(aa_lis578081645931084383_VEBTi(aa_VEB3237245999891841275_VEBTi(aa_arr5756643615615923676_VEBTi(aTP_Lamm_te(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj) = heap_T5998771940306268294_VEBTi(aa_VEB8194921341634505877Heap_o(vEBT_VEBT_minNulli,Uuj),aa_VEB2625734152646192863_VEBTi(aa_nat3994193416293967532_VEBTi(aa_nat4708659386357353849_VEBTi(aa_nat6007137331526128070_VEBTi(aa_nat4759958082780230547_VEBTi(aa_nat5669347316383757024_VEBTi(aa_VEB6059947344347517207_VEBTi(aa_lis3097942893158148200_VEBTi(aa_VEB2760733029397411170_VEBTi(aa_arr2574505544232410071_VEBTi(aTP_Lamm_td(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj)) ).

% ATP.lambda_1281
tff(fact_11459_ATP_Olambda__1282,axiom,
    ! [Uu: fun_Pr4171499318505651870_VEBTi,Uua: nat,Uub: nat,Uuc: array_VEBT_VEBTi,Uud: vEBT_VEBTi,Uue: nat,Uuf: nat,Uug: nat,Uuh: nat,Uui: nat,Uuj: nat] : aa_nat5408703056993926195_VEBTi(aa_nat4282640288315822012_VEBTi(aa_nat7653869184832469701_VEBTi(aa_nat2287630409448422734_VEBTi(aa_nat2878941900432676695_VEBTi(aa_nat532351713702512864_VEBTi(aa_VEB7864109228220129220_VEBTi(aa_arr7707872583627363365_VEBTi(aa_nat3757413148909559610_VEBTi(aa_nat7878045268284587783_VEBTi(aTP_Lamm_abh(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj) = heap_T844888390831797134_VEBTi(array_len_VEBT_VEBTi(Uuc),aa_nat4282640288315822012_VEBTi(aa_nat7653869184832469701_VEBTi(aa_nat2287630409448422734_VEBTi(aa_nat2878941900432676695_VEBTi(aa_nat532351713702512864_VEBTi(aa_nat5473138614813200873_VEBTi(aa_VEB7525848083004366413_VEBTi(aa_arr912658909913700910_VEBTi(aa_nat5309850715826091203_VEBTi(aa_nat1165803401171981328_VEBTi(aTP_Lamm_abg(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj)) ).

% ATP.lambda_1282
tff(fact_11460_ATP_Olambda__1283,axiom,
    ! [Uu: fun_VE8118064890490510894_VEBTi,Uua: nat,Uub: nat,Uuc: array_VEBT_VEBTi,Uud: vEBT_VEBTi,Uue: nat,Uuf: nat,Uug: nat,Uuh: nat,Uui: nat,Uuj: nat] : aa_nat5408703056993926195_VEBTi(aa_nat4282640288315822012_VEBTi(aa_nat7653869184832469701_VEBTi(aa_nat2287630409448422734_VEBTi(aa_nat2878941900432676695_VEBTi(aa_nat532351713702512864_VEBTi(aa_VEB7864109228220129220_VEBTi(aa_arr7707872583627363365_VEBTi(aa_nat3757413148909559610_VEBTi(aa_nat7878045268284587783_VEBTi(aTP_Lamm_ne(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj) = heap_T844888390831797134_VEBTi(array_len_VEBT_VEBTi(Uuc),aa_nat4282640288315822012_VEBTi(aa_nat7653869184832469701_VEBTi(aa_nat2287630409448422734_VEBTi(aa_nat2878941900432676695_VEBTi(aa_nat532351713702512864_VEBTi(aa_nat5473138614813200873_VEBTi(aa_VEB7525848083004366413_VEBTi(aa_arr912658909913700910_VEBTi(aa_nat5309850715826091203_VEBTi(aa_nat1165803401171981328_VEBTi(aTP_Lamm_nd(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj)) ).

% ATP.lambda_1283
tff(fact_11461_ATP_Olambda__1284,axiom,
    ! [Uu: fun_Pr4171499318505651870_VEBTi,Uua: nat,Uub: nat,Uuc: array_VEBT_VEBTi,Uud: vEBT_VEBTi,Uue: nat,Uuf: nat,Uug: nat,Uuh: nat,Uui: nat,Uuj: nat,Uuk: nat] :
      aa_nat5408703056993926195_VEBTi(aa_nat4282640288315822012_VEBTi(aa_nat7653869184832469701_VEBTi(aa_nat2287630409448422734_VEBTi(aa_nat2878941900432676695_VEBTi(aa_nat532351713702512864_VEBTi(aa_nat5473138614813200873_VEBTi(aa_VEB7525848083004366413_VEBTi(aa_arr912658909913700910_VEBTi(aa_nat5309850715826091203_VEBTi(aa_nat1165803401171981328_VEBTi(aTP_Lamm_abg(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj),Uuk) = $ite(
        ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uuj),Uuk)
        & ~ ( ( Uua = Uue )
            | ( Uua = Uuf ) ) ),
        heap_T1006145433769338483_VEBTi(aa_nat5408703056993926195_VEBTi(array_nth_VEBT_VEBTi(Uuc),Uuj),aa_nat5661399468481021847_VEBTi(aa_nat5003802506003144480_VEBTi(aa_nat6388535185314696745_VEBTi(aa_nat3402329839695434930_VEBTi(aa_nat7160451295336493755_VEBTi(aa_VEB6809311486975643423_VEBTi(aa_arr4222437323582306304_VEBTi(aa_nat3616447687739732885_VEBTi(aTP_Lamm_abf(Uu),Uub),Uuc),Uud),Uuf),Uug),Uuh),Uui),Uuj)),
        aa_VEB836174949890266638_VEBTi(heap_T3630416162098727440_VEBTi,aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Uue),Uuf)),Uub,Uuc),Uud)) ) ).

% ATP.lambda_1284
tff(fact_11462_ATP_Olambda__1285,axiom,
    ! [Uu: fun_VE8118064890490510894_VEBTi,Uua: nat,Uub: nat,Uuc: array_VEBT_VEBTi,Uud: vEBT_VEBTi,Uue: nat,Uuf: nat,Uug: nat,Uuh: nat,Uui: nat,Uuj: nat,Uuk: nat] :
      aa_nat5408703056993926195_VEBTi(aa_nat4282640288315822012_VEBTi(aa_nat7653869184832469701_VEBTi(aa_nat2287630409448422734_VEBTi(aa_nat2878941900432676695_VEBTi(aa_nat532351713702512864_VEBTi(aa_nat5473138614813200873_VEBTi(aa_VEB7525848083004366413_VEBTi(aa_arr912658909913700910_VEBTi(aa_nat5309850715826091203_VEBTi(aa_nat1165803401171981328_VEBTi(aTP_Lamm_nd(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj),Uuk) = $ite(
        ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uuj),Uuk)
        & ~ ( ( Uua = Uue )
            | ( Uua = Uuf ) ) ),
        heap_T1006145433769338483_VEBTi(aa_nat5408703056993926195_VEBTi(array_nth_VEBT_VEBTi(Uuc),Uuj),aa_nat5661399468481021847_VEBTi(aa_nat5003802506003144480_VEBTi(aa_nat6388535185314696745_VEBTi(aa_nat3402329839695434930_VEBTi(aa_nat7160451295336493755_VEBTi(aa_VEB6809311486975643423_VEBTi(aa_arr4222437323582306304_VEBTi(aa_nat3616447687739732885_VEBTi(aTP_Lamm_nc(Uu),Uub),Uuc),Uud),Uuf),Uug),Uuh),Uui),Uuj)),
        aa_VEB836174949890266638_VEBTi(heap_T3630416162098727440_VEBTi,aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Uue),Uuf)),Uub,Uuc),Uud)) ) ).

% ATP.lambda_1285
tff(fact_11463_ATP_Olambda__1286,axiom,
    ! [Uu: nat,Uua: nat,Uub: array_VEBT_VEBTi,Uuc: vEBT_VEBTi,Uud: vEBT_VEBT,Uue: product_prod_nat_nat,Uuf: nat,Uug: nat,Uuh: vEBT_VEBTi,Uui: vEBT_VEBT,Uuj: option_nat,Uuk: product_unit] :
      aa_Pro451875010097498139on_nat(aa_opt5089197864053947520on_nat(aa_VEB2490029800597546679on_nat(aa_VEB1055358423767010003on_nat(aa_nat4144897467549153500on_nat(aa_nat7557481500408122597on_nat(aa_Pro3918843774365325431on_nat(aa_VEB7240715040874449526on_nat(aa_VEB3377150917682228498on_nat(aa_arr2747111264493904243on_nat(aa_nat610287491858673288on_nat(aTP_Lamm_tx(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj),Uuk) = $ite(
        ( ( Uuj != none_nat )
        & aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_greater,aa_nat_option_nat(some_nat,Uuf)),Uuj) ),
        heap_T3669509953089699273on_nat(aa_nat947733127081326238on_nat(aa_VEB3566417204837707778on_nat(aa_VEB1030146501360450209on_nat(vEBT_VEBT_vebt_predi,Uui),Uuh),Uuf),aa_nat9127621225095157975on_nat(aTP_Lamm_ep(Uua),Uug)),
        heap_T3669509953089699273on_nat(aa_nat947733127081326238on_nat(aa_VEB3566417204837707778on_nat(aa_VEB1030146501360450209on_nat(vEBT_VEBT_vebt_predi,Uud),Uuc),Uug),aa_nat9127621225095157975on_nat(aa_Pro5040713308973091561on_nat(aa_VEB8830394031746293224on_nat(aa_arr3423563577619635969on_nat(aa_nat2635311902590537750on_nat(aTP_Lamm_rv(Uu),Uua),Uub),Uud),Uue),Uug)) ) ).

% ATP.lambda_1286
tff(fact_11464_ATP_Olambda__1287,axiom,
    ! [Uu: fun_Pr4404027214751104635_VEBTi,Uua: nat,Uub: nat,Uuc: array_VEBT_VEBTi,Uud: vEBT_VEBTi,Uue: nat,Uuf: list_VEBT_VEBT,Uug: vEBT_VEBT,Uuh: nat,Uui: nat,Uuj: nat,Uuk: nat] :
      aa_nat5408703056993926195_VEBTi(aa_nat4282640288315822012_VEBTi(aa_nat7653869184832469701_VEBTi(aa_nat2287630409448422734_VEBTi(aa_VEB3484995426995338861_VEBTi(aa_lis8107322492824698866_VEBTi(aa_nat5353086824354447155_VEBTi(aa_VEB4525472277563179415_VEBTi(aa_arr4621753813534489208_VEBTi(aa_nat369315079452798477_VEBTi(aa_nat3166855462611258970_VEBTi(aTP_Lamm_vg(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj),Uuk) = $let(
        xn2: nat,
        xn2:= 
          $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uua),Uuh),Uuh,Uua),
        heap_T844888390831797134_VEBTi($ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uua),Uui),aa_nat5821310369566504398ap_nat(heap_Time_return_nat,Uua),aa_nat5821310369566504398ap_nat(heap_Time_return_nat,Uui)),aa_nat4282640288315822012_VEBTi(aa_nat7653869184832469701_VEBTi(aa_nat2287630409448422734_VEBTi(aa_nat2878941900432676695_VEBTi(aa_VEB2417709626833497078_VEBTi(aa_lis5541900872984878331_VEBTi(aa_nat770714458426680252_VEBTi(aa_VEB4484153586529426336_VEBTi(aa_arr8317680606763408385_VEBTi(aa_nat4438223700708794134_VEBTi(aa_nat6528503438221098723_VEBTi(aTP_Lamm_vf(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uui),Uuj),Uuk),xn2)) ) ).

% ATP.lambda_1287
tff(fact_11465_ATP_Olambda__1288,axiom,
    ! [Uu: fun_VE2018681615261037117_VEBTi,Uua: nat,Uub: nat,Uuc: array_VEBT_VEBTi,Uud: vEBT_VEBTi,Uue: nat,Uuf: list_VEBT_VEBT,Uug: vEBT_VEBT,Uuh: nat,Uui: nat,Uuj: nat,Uuk: nat] :
      aa_nat5408703056993926195_VEBTi(aa_nat4282640288315822012_VEBTi(aa_nat7653869184832469701_VEBTi(aa_nat2287630409448422734_VEBTi(aa_VEB3484995426995338861_VEBTi(aa_lis8107322492824698866_VEBTi(aa_nat5353086824354447155_VEBTi(aa_VEB4525472277563179415_VEBTi(aa_arr4621753813534489208_VEBTi(aa_nat369315079452798477_VEBTi(aa_nat3166855462611258970_VEBTi(aTP_Lamm_rj(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj),Uuk) = $let(
        xn2: nat,
        xn2:= 
          $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uua),Uuh),Uuh,Uua),
        heap_T844888390831797134_VEBTi($ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uua),Uui),aa_nat5821310369566504398ap_nat(heap_Time_return_nat,Uua),aa_nat5821310369566504398ap_nat(heap_Time_return_nat,Uui)),aa_nat4282640288315822012_VEBTi(aa_nat7653869184832469701_VEBTi(aa_nat2287630409448422734_VEBTi(aa_nat2878941900432676695_VEBTi(aa_VEB2417709626833497078_VEBTi(aa_lis5541900872984878331_VEBTi(aa_nat770714458426680252_VEBTi(aa_VEB4484153586529426336_VEBTi(aa_arr8317680606763408385_VEBTi(aa_nat4438223700708794134_VEBTi(aa_nat6528503438221098723_VEBTi(aTP_Lamm_ri(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uui),Uuj),Uuk),xn2)) ) ).

% ATP.lambda_1288
tff(fact_11466_ATP_Olambda__1289,axiom,
    ! [Uu: nat,Uua: nat,Uub: array_VEBT_VEBTi,Uuc: vEBT_VEBTi,Uud: nat,Uue: list_VEBT_VEBT,Uuf: vEBT_VEBT,Uug: nat,Uuh: nat,Uui: nat,Uuj: nat,Uuk: nat] : aa_nat5408703056993926195_VEBTi(aa_nat4282640288315822012_VEBTi(aa_nat7653869184832469701_VEBTi(aa_nat2287630409448422734_VEBTi(aa_nat2878941900432676695_VEBTi(aa_VEB2417709626833497078_VEBTi(aa_lis5541900872984878331_VEBTi(aa_nat770714458426680252_VEBTi(aa_VEB4484153586529426336_VEBTi(aa_arr8317680606763408385_VEBTi(aa_nat4438223700708794134_VEBTi(aTP_Lamm_tl(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj),Uuk) = heap_T844888390831797134_VEBTi(aa_nat5821310369566504398ap_nat(vEBT_VEBT_lowi(Uui),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uua),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat4282640288315822012_VEBTi(aa_nat7653869184832469701_VEBTi(aa_nat2287630409448422734_VEBTi(aa_nat2878941900432676695_VEBTi(aa_nat532351713702512864_VEBTi(aa_VEB5310930937212039423_VEBTi(aa_lis7835404322308837764_VEBTi(aa_nat8538784969185189829_VEBTi(aa_VEB5145764472262921513_VEBTi(aa_arr6568412587313337610_VEBTi(aa_nat1553062435445860255_VEBTi(aTP_Lamm_tk(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj),Uuk)) ).

% ATP.lambda_1289
tff(fact_11467_ATP_Olambda__1290,axiom,
    ! [Uu: fun_Pr4404027214751104635_VEBTi,Uua: nat,Uub: array_VEBT_VEBTi,Uuc: vEBT_VEBTi,Uud: list_VEBT_VEBT,Uue: vEBT_VEBT,Uuf: nat,Uug: nat,Uuh: nat,Uui: nat,Uuj: nat,Uuk: product_unit] : aa_Pro3581650104493980720_VEBTi(aa_nat7312404184930824517_VEBTi(aa_nat3221004963518969490_VEBTi(aa_nat1817389164722854239_VEBTi(aa_nat317164691840817580_VEBTi(aa_nat3080504576880736121_VEBTi(aa_VEB4815025940789304112_VEBTi(aa_lis3577763594685753089_VEBTi(aa_VEB8209137959338826235_VEBTi(aa_arr747291131267414768_VEBTi(aa_nat461013784042436401_VEBTi(aTP_Lamm_uz(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj),Uuk) = heap_T1006145433769338483_VEBTi(aa_nat5408703056993926195_VEBTi(array_nth_VEBT_VEBTi(Uub),Uuj),aa_nat5661399468481021847_VEBTi(aa_nat5003802506003144480_VEBTi(aa_nat6388535185314696745_VEBTi(aa_nat3402329839695434930_VEBTi(aa_nat7160451295336493755_VEBTi(aa_VEB779792351083059546_VEBTi(aa_lis578081645931084383_VEBTi(aa_VEB3237245999891841275_VEBTi(aa_arr5756643615615923676_VEBTi(aa_nat7386715292082006385_VEBTi(aTP_Lamm_uy(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj)) ).

% ATP.lambda_1290
tff(fact_11468_ATP_Olambda__1291,axiom,
    ! [Uu: fun_VE2018681615261037117_VEBTi,Uua: nat,Uub: array_VEBT_VEBTi,Uuc: vEBT_VEBTi,Uud: list_VEBT_VEBT,Uue: vEBT_VEBT,Uuf: nat,Uug: nat,Uuh: nat,Uui: nat,Uuj: nat,Uuk: product_unit] : aa_Pro3581650104493980720_VEBTi(aa_nat7312404184930824517_VEBTi(aa_nat3221004963518969490_VEBTi(aa_nat1817389164722854239_VEBTi(aa_nat317164691840817580_VEBTi(aa_nat3080504576880736121_VEBTi(aa_VEB4815025940789304112_VEBTi(aa_lis3577763594685753089_VEBTi(aa_VEB8209137959338826235_VEBTi(aa_arr747291131267414768_VEBTi(aa_nat461013784042436401_VEBTi(aTP_Lamm_rc(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj),Uuk) = heap_T1006145433769338483_VEBTi(aa_nat5408703056993926195_VEBTi(array_nth_VEBT_VEBTi(Uub),Uuj),aa_nat5661399468481021847_VEBTi(aa_nat5003802506003144480_VEBTi(aa_nat6388535185314696745_VEBTi(aa_nat3402329839695434930_VEBTi(aa_nat7160451295336493755_VEBTi(aa_VEB779792351083059546_VEBTi(aa_lis578081645931084383_VEBTi(aa_VEB3237245999891841275_VEBTi(aa_arr5756643615615923676_VEBTi(aa_nat7386715292082006385_VEBTi(aTP_Lamm_rb(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj)) ).

% ATP.lambda_1291
tff(fact_11469_ATP_Olambda__1292,axiom,
    ! [Uu: nat,Uua: array_VEBT_VEBTi,Uub: vEBT_VEBTi,Uuc: list_VEBT_VEBT,Uud: vEBT_VEBT,Uue: nat,Uuf: nat,Uug: nat,Uuh: nat,Uui: nat,Uuj: vEBT_VEBTi,Uuk: $o] : aa_o_H7577119287286566245_VEBTi(aa_VEB2625734152646192863_VEBTi(aa_nat3994193416293967532_VEBTi(aa_nat4708659386357353849_VEBTi(aa_nat6007137331526128070_VEBTi(aa_nat4759958082780230547_VEBTi(aa_nat5669347316383757024_VEBTi(aa_VEB6059947344347517207_VEBTi(aa_lis3097942893158148200_VEBTi(aa_VEB2760733029397411170_VEBTi(aa_arr2574505544232410071_VEBTi(aTP_Lamm_td(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj),(Uuk)) = heap_T7982501707604696571_VEBTi(aa_o_H2459438917461573107t_unit(refine_Imp_assert,(Uuk) = aa_VEBT_VEBT_o(vEBT_VEBT_minNull,nth_VEBT_VEBT(Uuc,Uui))),aa_o_f3680854670239116947_VEBTi(aa_VEB318638452350941999_VEBTi(aa_nat250827245977626168_VEBTi(aa_nat3169258303071516481_VEBTi(aa_nat904988376872050634_VEBTi(aa_nat6815201885744029651_VEBTi(aa_nat6697094627763402588_VEBTi(aa_VEB2406964849791727995_VEBTi(aa_lis5794688975234795520_VEBTi(aa_VEB4139799945145921820_VEBTi(aa_arr7513636907291371389_VEBTi(aTP_Lamm_tc(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj),(Uuk))) ).

% ATP.lambda_1292
tff(fact_11470_ATP_Olambda__1293,axiom,
    ! [Uu: fun_Pr633315466666964198on_nat,Uua: nat,Uub: nat,Uuc: array_VEBT_VEBTi,Uud: vEBT_VEBTi,Uue: vEBT_VEBT,Uuf: product_prod_nat_nat,Uug: nat,Uuh: nat,Uui: vEBT_VEBTi,Uuj: vEBT_VEBT,Uuk: option_nat] : aa_opt9163293676379396302on_nat(aa_VEB3755160987679246317on_nat(aa_VEB5212316371209427943on_nat(aa_nat7793430520228346548on_nat(aa_nat8410684691732023425on_nat(aa_Pro7328798252849467525on_nat(aa_VEB6402893080723908316on_nat(aa_VEB4808996286462526806on_nat(aa_arr4232376455131879627on_nat(aa_nat8394757134461757324on_nat(aa_nat5552918743037330581on_nat(aTP_Lamm_ww(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj),Uuk) = heap_T5999496708990702694on_nat(aa_o_H2459438917461573107t_unit(refine_Imp_assert,Uuk = vEBT_vebt_mint(Uuj)),aa_opt5089197864053947520on_nat(aa_VEB2490029800597546679on_nat(aa_VEB1055358423767010003on_nat(aa_nat4144897467549153500on_nat(aa_nat7557481500408122597on_nat(aa_Pro3918843774365325431on_nat(aa_VEB7240715040874449526on_nat(aa_VEB3377150917682228498on_nat(aa_arr2747111264493904243on_nat(aa_nat610287491858673288on_nat(aa_nat9025683123140566101on_nat(aTP_Lamm_wv(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj),Uuk)) ).

% ATP.lambda_1293
tff(fact_11471_ATP_Olambda__1294,axiom,
    ! [Uu: fun_VE3775134097089128360on_nat,Uua: nat,Uub: nat,Uuc: array_VEBT_VEBTi,Uud: vEBT_VEBTi,Uue: vEBT_VEBT,Uuf: product_prod_nat_nat,Uug: nat,Uuh: nat,Uui: vEBT_VEBTi,Uuj: vEBT_VEBT,Uuk: option_nat] : aa_opt9163293676379396302on_nat(aa_VEB3755160987679246317on_nat(aa_VEB5212316371209427943on_nat(aa_nat7793430520228346548on_nat(aa_nat8410684691732023425on_nat(aa_Pro7328798252849467525on_nat(aa_VEB6402893080723908316on_nat(aa_VEB4808996286462526806on_nat(aa_arr4232376455131879627on_nat(aa_nat8394757134461757324on_nat(aa_nat5552918743037330581on_nat(aTP_Lamm_rx(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj),Uuk) = heap_T5999496708990702694on_nat(aa_o_H2459438917461573107t_unit(refine_Imp_assert,Uuk = vEBT_vebt_mint(Uuj)),aa_opt5089197864053947520on_nat(aa_VEB2490029800597546679on_nat(aa_VEB1055358423767010003on_nat(aa_nat4144897467549153500on_nat(aa_nat7557481500408122597on_nat(aa_Pro3918843774365325431on_nat(aa_VEB7240715040874449526on_nat(aa_VEB3377150917682228498on_nat(aa_arr2747111264493904243on_nat(aa_nat610287491858673288on_nat(aa_nat9025683123140566101on_nat(aTP_Lamm_rw(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj),Uuk)) ).

% ATP.lambda_1294
tff(fact_11472_ATP_Olambda__1295,axiom,
    ! [Uu: fun_Pr4404027214751104635_VEBTi,Uua: nat,Uub: array_VEBT_VEBTi,Uuc: vEBT_VEBTi,Uud: list_VEBT_VEBT,Uue: vEBT_VEBT,Uuf: nat,Uug: nat,Uuh: nat,Uui: nat,Uuj: nat,Uuk: product_unit] : aa_Pro3581650104493980720_VEBTi(aa_nat7312404184930824517_VEBTi(aa_nat3221004963518969490_VEBTi(aa_nat1817389164722854239_VEBTi(aa_nat317164691840817580_VEBTi(aa_nat3080504576880736121_VEBTi(aa_VEB4815025940789304112_VEBTi(aa_lis3577763594685753089_VEBTi(aa_VEB8209137959338826235_VEBTi(aa_arr747291131267414768_VEBTi(aa_nat461013784042436401_VEBTi(aTP_Lamm_va(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj),Uuk) = heap_T7982501707604696571_VEBTi(aa_o_H2459438917461573107t_unit(refine_Imp_assert,aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uuj),size_s6755466524823107622T_VEBT(Uud))),aa_nat7312404184930824517_VEBTi(aa_nat3221004963518969490_VEBTi(aa_nat1817389164722854239_VEBTi(aa_nat317164691840817580_VEBTi(aa_nat3080504576880736121_VEBTi(aa_VEB4815025940789304112_VEBTi(aa_lis3577763594685753089_VEBTi(aa_VEB8209137959338826235_VEBTi(aa_arr747291131267414768_VEBTi(aa_nat461013784042436401_VEBTi(aTP_Lamm_uz(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj)) ).

% ATP.lambda_1295
tff(fact_11473_ATP_Olambda__1296,axiom,
    ! [Uu: fun_VE2018681615261037117_VEBTi,Uua: nat,Uub: array_VEBT_VEBTi,Uuc: vEBT_VEBTi,Uud: list_VEBT_VEBT,Uue: vEBT_VEBT,Uuf: nat,Uug: nat,Uuh: nat,Uui: nat,Uuj: nat,Uuk: product_unit] : aa_Pro3581650104493980720_VEBTi(aa_nat7312404184930824517_VEBTi(aa_nat3221004963518969490_VEBTi(aa_nat1817389164722854239_VEBTi(aa_nat317164691840817580_VEBTi(aa_nat3080504576880736121_VEBTi(aa_VEB4815025940789304112_VEBTi(aa_lis3577763594685753089_VEBTi(aa_VEB8209137959338826235_VEBTi(aa_arr747291131267414768_VEBTi(aa_nat461013784042436401_VEBTi(aTP_Lamm_rd(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj),Uuk) = heap_T7982501707604696571_VEBTi(aa_o_H2459438917461573107t_unit(refine_Imp_assert,aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uuj),size_s6755466524823107622T_VEBT(Uud))),aa_nat7312404184930824517_VEBTi(aa_nat3221004963518969490_VEBTi(aa_nat1817389164722854239_VEBTi(aa_nat317164691840817580_VEBTi(aa_nat3080504576880736121_VEBTi(aa_VEB4815025940789304112_VEBTi(aa_lis3577763594685753089_VEBTi(aa_VEB8209137959338826235_VEBTi(aa_arr747291131267414768_VEBTi(aa_nat461013784042436401_VEBTi(aTP_Lamm_rc(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj)) ).

% ATP.lambda_1296
tff(fact_11474_ATP_Olambda__1297,axiom,
    ! [Uu: fun_Pr4404027214751104635_VEBTi,Uua: nat,Uub: array_VEBT_VEBTi,Uuc: vEBT_VEBTi,Uud: list_VEBT_VEBT,Uue: vEBT_VEBT,Uuf: nat,Uug: nat,Uuh: nat,Uui: nat,Uuj: nat,Uuk: vEBT_VEBTi] : aa_VEB836174949890266638_VEBTi(aa_nat5661399468481021847_VEBTi(aa_nat5003802506003144480_VEBTi(aa_nat6388535185314696745_VEBTi(aa_nat3402329839695434930_VEBTi(aa_nat7160451295336493755_VEBTi(aa_VEB779792351083059546_VEBTi(aa_lis578081645931084383_VEBTi(aa_VEB3237245999891841275_VEBTi(aa_arr5756643615615923676_VEBTi(aa_nat7386715292082006385_VEBTi(aTP_Lamm_uy(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj),Uuk) = heap_T5998771940306268294_VEBTi(aa_VEB8194921341634505877Heap_o(vEBT_VEBT_minNulli,Uuk),aa_VEB2625734152646192863_VEBTi(aa_nat3994193416293967532_VEBTi(aa_nat4708659386357353849_VEBTi(aa_nat6007137331526128070_VEBTi(aa_nat4759958082780230547_VEBTi(aa_nat5669347316383757024_VEBTi(aa_VEB6059947344347517207_VEBTi(aa_lis3097942893158148200_VEBTi(aa_VEB2760733029397411170_VEBTi(aa_arr2574505544232410071_VEBTi(aa_nat2429737420813335448_VEBTi(aTP_Lamm_ux(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj),Uuk)) ).

% ATP.lambda_1297
tff(fact_11475_ATP_Olambda__1298,axiom,
    ! [Uu: fun_VE2018681615261037117_VEBTi,Uua: nat,Uub: array_VEBT_VEBTi,Uuc: vEBT_VEBTi,Uud: list_VEBT_VEBT,Uue: vEBT_VEBT,Uuf: nat,Uug: nat,Uuh: nat,Uui: nat,Uuj: nat,Uuk: vEBT_VEBTi] : aa_VEB836174949890266638_VEBTi(aa_nat5661399468481021847_VEBTi(aa_nat5003802506003144480_VEBTi(aa_nat6388535185314696745_VEBTi(aa_nat3402329839695434930_VEBTi(aa_nat7160451295336493755_VEBTi(aa_VEB779792351083059546_VEBTi(aa_lis578081645931084383_VEBTi(aa_VEB3237245999891841275_VEBTi(aa_arr5756643615615923676_VEBTi(aa_nat7386715292082006385_VEBTi(aTP_Lamm_rb(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj),Uuk) = heap_T5998771940306268294_VEBTi(aa_VEB8194921341634505877Heap_o(vEBT_VEBT_minNulli,Uuk),aa_VEB2625734152646192863_VEBTi(aa_nat3994193416293967532_VEBTi(aa_nat4708659386357353849_VEBTi(aa_nat6007137331526128070_VEBTi(aa_nat4759958082780230547_VEBTi(aa_nat5669347316383757024_VEBTi(aa_VEB6059947344347517207_VEBTi(aa_lis3097942893158148200_VEBTi(aa_VEB2760733029397411170_VEBTi(aa_arr2574505544232410071_VEBTi(aa_nat2429737420813335448_VEBTi(aTP_Lamm_ra(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj),Uuk)) ).

% ATP.lambda_1298
tff(fact_11476_ATP_Olambda__1299,axiom,
    ! [Uu: fun_Pr633315466666964198on_nat,Uua: nat,Uub: nat,Uuc: array_VEBT_VEBTi,Uud: vEBT_VEBTi,Uue: vEBT_VEBT,Uuf: product_prod_nat_nat,Uug: nat,Uuh: nat,Uui: vEBT_VEBTi,Uuj: vEBT_VEBT,Uuk: option_nat,Uul: product_unit] :
      aa_Pro451875010097498139on_nat(aa_opt5089197864053947520on_nat(aa_VEB2490029800597546679on_nat(aa_VEB1055358423767010003on_nat(aa_nat4144897467549153500on_nat(aa_nat7557481500408122597on_nat(aa_Pro3918843774365325431on_nat(aa_VEB7240715040874449526on_nat(aa_VEB3377150917682228498on_nat(aa_arr2747111264493904243on_nat(aa_nat610287491858673288on_nat(aa_nat9025683123140566101on_nat(aTP_Lamm_wv(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj),Uuk),Uul) = $ite(
        ( ( Uuk != none_nat )
        & aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_greater,aa_nat_option_nat(some_nat,Uug)),Uuk) ),
        heap_T3669509953089699273on_nat(aa_nat947733127081326238on_nat(aa_VEB3566417204837707778on_nat(aa_VEB1030146501360450209on_nat(produc9121605208400662357on_nat(produc1757988346207259447on_nat(Uu)),Uuj),Uui),Uug),aa_nat9127621225095157975on_nat(aTP_Lamm_ep(Uub),Uuh)),
        heap_T3669509953089699273on_nat(aa_nat947733127081326238on_nat(aa_VEB3566417204837707778on_nat(aa_VEB1030146501360450209on_nat(produc9121605208400662357on_nat(produc1757988346207259447on_nat(Uu)),Uue),Uud),Uuh),aa_nat9127621225095157975on_nat(aa_Pro5040713308973091561on_nat(aa_VEB8830394031746293224on_nat(aa_arr3423563577619635969on_nat(aa_nat2635311902590537750on_nat(aTP_Lamm_rv(Uua),Uub),Uuc),Uue),Uuf),Uuh)) ) ).

% ATP.lambda_1299
tff(fact_11477_ATP_Olambda__1300,axiom,
    ! [Uu: fun_VE3775134097089128360on_nat,Uua: nat,Uub: nat,Uuc: array_VEBT_VEBTi,Uud: vEBT_VEBTi,Uue: vEBT_VEBT,Uuf: product_prod_nat_nat,Uug: nat,Uuh: nat,Uui: vEBT_VEBTi,Uuj: vEBT_VEBT,Uuk: option_nat,Uul: product_unit] :
      aa_Pro451875010097498139on_nat(aa_opt5089197864053947520on_nat(aa_VEB2490029800597546679on_nat(aa_VEB1055358423767010003on_nat(aa_nat4144897467549153500on_nat(aa_nat7557481500408122597on_nat(aa_Pro3918843774365325431on_nat(aa_VEB7240715040874449526on_nat(aa_VEB3377150917682228498on_nat(aa_arr2747111264493904243on_nat(aa_nat610287491858673288on_nat(aa_nat9025683123140566101on_nat(aTP_Lamm_rw(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj),Uuk),Uul) = $ite(
        ( ( Uuk != none_nat )
        & aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_greater,aa_nat_option_nat(some_nat,Uug)),Uuk) ),
        heap_T3669509953089699273on_nat(aa_nat947733127081326238on_nat(aa_VEB3566417204837707778on_nat(aa_VEB1030146501360450209on_nat(Uu,Uuj),Uui),Uug),aa_nat9127621225095157975on_nat(aTP_Lamm_ep(Uub),Uuh)),
        heap_T3669509953089699273on_nat(aa_nat947733127081326238on_nat(aa_VEB3566417204837707778on_nat(aa_VEB1030146501360450209on_nat(Uu,Uue),Uud),Uuh),aa_nat9127621225095157975on_nat(aa_Pro5040713308973091561on_nat(aa_VEB8830394031746293224on_nat(aa_arr3423563577619635969on_nat(aa_nat2635311902590537750on_nat(aTP_Lamm_rv(Uua),Uub),Uuc),Uue),Uuf),Uuh)) ) ).

% ATP.lambda_1300
tff(fact_11478_ATP_Olambda__1301,axiom,
    ! [Uu: nat,Uua: array_VEBT_VEBTi,Uub: vEBT_VEBTi,Uuc: list_VEBT_VEBT,Uud: vEBT_VEBT,Uue: nat,Uuf: nat,Uug: nat,Uuh: nat,Uui: nat,Uuj: vEBT_VEBTi,Uuk: $o,Uul: product_unit] : aa_Pro3581650104493980720_VEBTi(aa_o_f3680854670239116947_VEBTi(aa_VEB318638452350941999_VEBTi(aa_nat250827245977626168_VEBTi(aa_nat3169258303071516481_VEBTi(aa_nat904988376872050634_VEBTi(aa_nat6815201885744029651_VEBTi(aa_nat6697094627763402588_VEBTi(aa_VEB2406964849791727995_VEBTi(aa_lis5794688975234795520_VEBTi(aa_VEB4139799945145921820_VEBTi(aa_arr7513636907291371389_VEBTi(aTP_Lamm_tc(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj),(Uuk)),Uul) = heap_T1006145433769338483_VEBTi(aa_nat5408703056993926195_VEBTi(aa_VEB4833742232818206615_VEBTi(aa_VEB6292075424854053302_VEBTi(vEBT_V3964819847710782039nserti,nth_VEBT_VEBT(Uuc,Uui)),Uuj),Uuh),aa_o_f1343430628605260489_VEBTi(aa_nat4147387395820660958_VEBTi(aa_nat2236254811235047083_VEBTi(aa_nat5995673692660408824_VEBTi(aa_nat4988899645790341829_VEBTi(aa_VEB257407171870985852_VEBTi(aa_VEB2935853492131583640_VEBTi(aa_arr6469760262597239545_VEBTi(aTP_Lamm_tb(Uu),Uua),Uub),Uud),Uue),Uuf),Uug),Uui),(Uuk))) ).

% ATP.lambda_1301
tff(fact_11479_ATP_Olambda__1302,axiom,
    ! [Uu: fun_Pr4404027214751104635_VEBTi,Uua: nat,Uub: nat,Uuc: array_VEBT_VEBTi,Uud: vEBT_VEBTi,Uue: nat,Uuf: list_VEBT_VEBT,Uug: vEBT_VEBT,Uuh: nat,Uui: nat,Uuj: nat,Uuk: nat,Uul: nat] : aa_nat5408703056993926195_VEBTi(aa_nat4282640288315822012_VEBTi(aa_nat7653869184832469701_VEBTi(aa_nat2287630409448422734_VEBTi(aa_nat2878941900432676695_VEBTi(aa_VEB2417709626833497078_VEBTi(aa_lis5541900872984878331_VEBTi(aa_nat770714458426680252_VEBTi(aa_VEB4484153586529426336_VEBTi(aa_arr8317680606763408385_VEBTi(aa_nat4438223700708794134_VEBTi(aa_nat6528503438221098723_VEBTi(aTP_Lamm_vf(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj),Uuk),Uul) = heap_T844888390831797134_VEBTi(aa_nat5821310369566504398ap_nat(vEBT_VEBT_lowi(Uuj),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uub),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat4282640288315822012_VEBTi(aa_nat7653869184832469701_VEBTi(aa_nat2287630409448422734_VEBTi(aa_nat2878941900432676695_VEBTi(aa_nat532351713702512864_VEBTi(aa_VEB5310930937212039423_VEBTi(aa_lis7835404322308837764_VEBTi(aa_nat8538784969185189829_VEBTi(aa_VEB5145764472262921513_VEBTi(aa_arr6568412587313337610_VEBTi(aa_nat1553062435445860255_VEBTi(aa_nat1501880811460822764_VEBTi(aTP_Lamm_ve(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj),Uuk),Uul)) ).

% ATP.lambda_1302
tff(fact_11480_ATP_Olambda__1303,axiom,
    ! [Uu: fun_VE2018681615261037117_VEBTi,Uua: nat,Uub: nat,Uuc: array_VEBT_VEBTi,Uud: vEBT_VEBTi,Uue: nat,Uuf: list_VEBT_VEBT,Uug: vEBT_VEBT,Uuh: nat,Uui: nat,Uuj: nat,Uuk: nat,Uul: nat] : aa_nat5408703056993926195_VEBTi(aa_nat4282640288315822012_VEBTi(aa_nat7653869184832469701_VEBTi(aa_nat2287630409448422734_VEBTi(aa_nat2878941900432676695_VEBTi(aa_VEB2417709626833497078_VEBTi(aa_lis5541900872984878331_VEBTi(aa_nat770714458426680252_VEBTi(aa_VEB4484153586529426336_VEBTi(aa_arr8317680606763408385_VEBTi(aa_nat4438223700708794134_VEBTi(aa_nat6528503438221098723_VEBTi(aTP_Lamm_ri(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj),Uuk),Uul) = heap_T844888390831797134_VEBTi(aa_nat5821310369566504398ap_nat(vEBT_VEBT_lowi(Uuj),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uub),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat4282640288315822012_VEBTi(aa_nat7653869184832469701_VEBTi(aa_nat2287630409448422734_VEBTi(aa_nat2878941900432676695_VEBTi(aa_nat532351713702512864_VEBTi(aa_VEB5310930937212039423_VEBTi(aa_lis7835404322308837764_VEBTi(aa_nat8538784969185189829_VEBTi(aa_VEB5145764472262921513_VEBTi(aa_arr6568412587313337610_VEBTi(aa_nat1553062435445860255_VEBTi(aa_nat1501880811460822764_VEBTi(aTP_Lamm_rh(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj),Uuk),Uul)) ).

% ATP.lambda_1303
tff(fact_11481_ATP_Olambda__1304,axiom,
    ! [Uu: nat,Uua: nat,Uub: array_VEBT_VEBTi,Uuc: vEBT_VEBTi,Uud: nat,Uue: list_VEBT_VEBT,Uuf: vEBT_VEBT,Uug: nat,Uuh: nat,Uui: nat,Uuj: nat,Uuk: nat,Uul: nat] : aa_nat5408703056993926195_VEBTi(aa_nat4282640288315822012_VEBTi(aa_nat7653869184832469701_VEBTi(aa_nat2287630409448422734_VEBTi(aa_nat2878941900432676695_VEBTi(aa_nat532351713702512864_VEBTi(aa_VEB5310930937212039423_VEBTi(aa_lis7835404322308837764_VEBTi(aa_nat8538784969185189829_VEBTi(aa_VEB5145764472262921513_VEBTi(aa_arr6568412587313337610_VEBTi(aa_nat1553062435445860255_VEBTi(aTP_Lamm_tk(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj),Uuk),Uul) = heap_T7982501707604696571_VEBTi(aa_o_H2459438917461573107t_unit(refine_Imp_assert,Uul = aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Uuj),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uud),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat7312404184930824517_VEBTi(aa_nat3221004963518969490_VEBTi(aa_nat1817389164722854239_VEBTi(aa_nat317164691840817580_VEBTi(aa_nat3080504576880736121_VEBTi(aa_nat5640937189053280966_VEBTi(aa_VEB5864305904528209917_VEBTi(aa_lis8083850882120436046_VEBTi(aa_nat6903794420489964771_VEBTi(aa_VEB7377128652718434837_VEBTi(aa_arr6679936031850374666_VEBTi(aa_nat7444525983491697995_VEBTi(aTP_Lamm_tj(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj),Uuk),Uul)) ).

% ATP.lambda_1304
tff(fact_11482_ATP_Olambda__1305,axiom,
    ! [Uu: fun_Pr4404027214751104635_VEBTi,Uua: nat,Uub: array_VEBT_VEBTi,Uuc: vEBT_VEBTi,Uud: list_VEBT_VEBT,Uue: vEBT_VEBT,Uuf: nat,Uug: nat,Uuh: nat,Uui: nat,Uuj: nat,Uuk: vEBT_VEBTi,Uul: $o] : aa_o_H7577119287286566245_VEBTi(aa_VEB2625734152646192863_VEBTi(aa_nat3994193416293967532_VEBTi(aa_nat4708659386357353849_VEBTi(aa_nat6007137331526128070_VEBTi(aa_nat4759958082780230547_VEBTi(aa_nat5669347316383757024_VEBTi(aa_VEB6059947344347517207_VEBTi(aa_lis3097942893158148200_VEBTi(aa_VEB2760733029397411170_VEBTi(aa_arr2574505544232410071_VEBTi(aa_nat2429737420813335448_VEBTi(aTP_Lamm_ux(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj),Uuk),(Uul)) = heap_T7982501707604696571_VEBTi(aa_o_H2459438917461573107t_unit(refine_Imp_assert,(Uul) = aa_VEBT_VEBT_o(vEBT_VEBT_minNull,nth_VEBT_VEBT(Uud,Uuj))),aa_o_f3680854670239116947_VEBTi(aa_VEB318638452350941999_VEBTi(aa_nat250827245977626168_VEBTi(aa_nat3169258303071516481_VEBTi(aa_nat904988376872050634_VEBTi(aa_nat6815201885744029651_VEBTi(aa_nat6697094627763402588_VEBTi(aa_VEB2406964849791727995_VEBTi(aa_lis5794688975234795520_VEBTi(aa_VEB4139799945145921820_VEBTi(aa_arr7513636907291371389_VEBTi(aa_nat1261261390489106578_VEBTi(aTP_Lamm_uw(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj),Uuk),(Uul))) ).

% ATP.lambda_1305
tff(fact_11483_ATP_Olambda__1306,axiom,
    ! [Uu: fun_VE2018681615261037117_VEBTi,Uua: nat,Uub: array_VEBT_VEBTi,Uuc: vEBT_VEBTi,Uud: list_VEBT_VEBT,Uue: vEBT_VEBT,Uuf: nat,Uug: nat,Uuh: nat,Uui: nat,Uuj: nat,Uuk: vEBT_VEBTi,Uul: $o] : aa_o_H7577119287286566245_VEBTi(aa_VEB2625734152646192863_VEBTi(aa_nat3994193416293967532_VEBTi(aa_nat4708659386357353849_VEBTi(aa_nat6007137331526128070_VEBTi(aa_nat4759958082780230547_VEBTi(aa_nat5669347316383757024_VEBTi(aa_VEB6059947344347517207_VEBTi(aa_lis3097942893158148200_VEBTi(aa_VEB2760733029397411170_VEBTi(aa_arr2574505544232410071_VEBTi(aa_nat2429737420813335448_VEBTi(aTP_Lamm_ra(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj),Uuk),(Uul)) = heap_T7982501707604696571_VEBTi(aa_o_H2459438917461573107t_unit(refine_Imp_assert,(Uul) = aa_VEBT_VEBT_o(vEBT_VEBT_minNull,nth_VEBT_VEBT(Uud,Uuj))),aa_o_f3680854670239116947_VEBTi(aa_VEB318638452350941999_VEBTi(aa_nat250827245977626168_VEBTi(aa_nat3169258303071516481_VEBTi(aa_nat904988376872050634_VEBTi(aa_nat6815201885744029651_VEBTi(aa_nat6697094627763402588_VEBTi(aa_VEB2406964849791727995_VEBTi(aa_lis5794688975234795520_VEBTi(aa_VEB4139799945145921820_VEBTi(aa_arr7513636907291371389_VEBTi(aa_nat1261261390489106578_VEBTi(aTP_Lamm_qz(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj),Uuk),(Uul))) ).

% ATP.lambda_1306
tff(fact_11484_ATP_Olambda__1307,axiom,
    ! [Uu: fun_Pr4404027214751104635_VEBTi,Uua: nat,Uub: array_VEBT_VEBTi,Uuc: vEBT_VEBTi,Uud: list_VEBT_VEBT,Uue: vEBT_VEBT,Uuf: nat,Uug: nat,Uuh: nat,Uui: nat,Uuj: nat,Uuk: vEBT_VEBTi,Uul: $o,Uum: product_unit] : aa_Pro3581650104493980720_VEBTi(aa_o_f3680854670239116947_VEBTi(aa_VEB318638452350941999_VEBTi(aa_nat250827245977626168_VEBTi(aa_nat3169258303071516481_VEBTi(aa_nat904988376872050634_VEBTi(aa_nat6815201885744029651_VEBTi(aa_nat6697094627763402588_VEBTi(aa_VEB2406964849791727995_VEBTi(aa_lis5794688975234795520_VEBTi(aa_VEB4139799945145921820_VEBTi(aa_arr7513636907291371389_VEBTi(aa_nat1261261390489106578_VEBTi(aTP_Lamm_uw(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj),Uuk),(Uul)),Uum) = heap_T1006145433769338483_VEBTi(aa_nat5408703056993926195_VEBTi(aa_VEB4833742232818206615_VEBTi(aa_VEB6292075424854053302_VEBTi(produc3311654011758026858_VEBTi(produc2164094337957399884_VEBTi(Uu)),nth_VEBT_VEBT(Uud,Uuj)),Uuk),Uui),aa_o_f1343430628605260489_VEBTi(aa_nat4147387395820660958_VEBTi(aa_nat2236254811235047083_VEBTi(aa_nat5995673692660408824_VEBTi(aa_nat4988899645790341829_VEBTi(aa_VEB257407171870985852_VEBTi(aa_VEB2935853492131583640_VEBTi(aa_arr6469760262597239545_VEBTi(aa_nat7574034614035128846_VEBTi(aTP_Lamm_uv(Uu),Uua),Uub),Uuc),Uue),Uuf),Uug),Uuh),Uuj),(Uul))) ).

% ATP.lambda_1307
tff(fact_11485_ATP_Olambda__1308,axiom,
    ! [Uu: fun_VE2018681615261037117_VEBTi,Uua: nat,Uub: array_VEBT_VEBTi,Uuc: vEBT_VEBTi,Uud: list_VEBT_VEBT,Uue: vEBT_VEBT,Uuf: nat,Uug: nat,Uuh: nat,Uui: nat,Uuj: nat,Uuk: vEBT_VEBTi,Uul: $o,Uum: product_unit] : aa_Pro3581650104493980720_VEBTi(aa_o_f3680854670239116947_VEBTi(aa_VEB318638452350941999_VEBTi(aa_nat250827245977626168_VEBTi(aa_nat3169258303071516481_VEBTi(aa_nat904988376872050634_VEBTi(aa_nat6815201885744029651_VEBTi(aa_nat6697094627763402588_VEBTi(aa_VEB2406964849791727995_VEBTi(aa_lis5794688975234795520_VEBTi(aa_VEB4139799945145921820_VEBTi(aa_arr7513636907291371389_VEBTi(aa_nat1261261390489106578_VEBTi(aTP_Lamm_qz(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj),Uuk),(Uul)),Uum) = heap_T1006145433769338483_VEBTi(aa_nat5408703056993926195_VEBTi(aa_VEB4833742232818206615_VEBTi(aa_VEB6292075424854053302_VEBTi(Uu,nth_VEBT_VEBT(Uud,Uuj)),Uuk),Uui),aa_o_f1343430628605260489_VEBTi(aa_nat4147387395820660958_VEBTi(aa_nat2236254811235047083_VEBTi(aa_nat5995673692660408824_VEBTi(aa_nat4988899645790341829_VEBTi(aa_VEB257407171870985852_VEBTi(aa_VEB2935853492131583640_VEBTi(aa_arr6469760262597239545_VEBTi(aa_nat7574034614035128846_VEBTi(aTP_Lamm_qy(Uu),Uua),Uub),Uuc),Uue),Uuf),Uug),Uuh),Uuj),(Uul))) ).

% ATP.lambda_1308
tff(fact_11486_ATP_Olambda__1309,axiom,
    ! [Uu: nat,Uua: nat,Uub: array_VEBT_VEBTi,Uuc: vEBT_VEBTi,Uud: nat,Uue: list_VEBT_VEBT,Uuf: vEBT_VEBT,Uug: nat,Uuh: nat,Uui: nat,Uuj: nat,Uuk: nat,Uul: nat,Uum: product_unit] : aa_Pro3581650104493980720_VEBTi(aa_nat7312404184930824517_VEBTi(aa_nat3221004963518969490_VEBTi(aa_nat1817389164722854239_VEBTi(aa_nat317164691840817580_VEBTi(aa_nat3080504576880736121_VEBTi(aa_nat5640937189053280966_VEBTi(aa_VEB5864305904528209917_VEBTi(aa_lis8083850882120436046_VEBTi(aa_nat6903794420489964771_VEBTi(aa_VEB7377128652718434837_VEBTi(aa_arr6679936031850374666_VEBTi(aa_nat7444525983491697995_VEBTi(aTP_Lamm_tj(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj),Uuk),Uul),Uum) = heap_T844888390831797134_VEBTi(aa_nat5821310369566504398ap_nat(vEBT_VEBT_highi(Uui),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uua),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat4282640288315822012_VEBTi(aa_nat7653869184832469701_VEBTi(aa_nat2287630409448422734_VEBTi(aa_nat2878941900432676695_VEBTi(aa_nat532351713702512864_VEBTi(aa_nat5473138614813200873_VEBTi(aa_VEB2726135844941024648_VEBTi(aa_lis3478650067825548685_VEBTi(aa_nat7061688277739565902_VEBTi(aa_VEB3820308522731820594_VEBTi(aa_arr165755620540720531_VEBTi(aa_nat5755912138998335400_VEBTi(aTP_Lamm_ti(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj),Uuk),Uul)) ).

% ATP.lambda_1309
tff(fact_11487_ATP_Olambda__1310,axiom,
    ! [Uu: fun_Pr4404027214751104635_VEBTi,Uua: nat,Uub: nat,Uuc: array_VEBT_VEBTi,Uud: vEBT_VEBTi,Uue: nat,Uuf: list_VEBT_VEBT,Uug: vEBT_VEBT,Uuh: nat,Uui: nat,Uuj: nat,Uuk: nat,Uul: nat,Uum: nat] : aa_nat5408703056993926195_VEBTi(aa_nat4282640288315822012_VEBTi(aa_nat7653869184832469701_VEBTi(aa_nat2287630409448422734_VEBTi(aa_nat2878941900432676695_VEBTi(aa_nat532351713702512864_VEBTi(aa_VEB5310930937212039423_VEBTi(aa_lis7835404322308837764_VEBTi(aa_nat8538784969185189829_VEBTi(aa_VEB5145764472262921513_VEBTi(aa_arr6568412587313337610_VEBTi(aa_nat1553062435445860255_VEBTi(aa_nat1501880811460822764_VEBTi(aTP_Lamm_ve(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj),Uuk),Uul),Uum) = heap_T7982501707604696571_VEBTi(aa_o_H2459438917461573107t_unit(refine_Imp_assert,Uum = aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Uuk),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uue),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat7312404184930824517_VEBTi(aa_nat3221004963518969490_VEBTi(aa_nat1817389164722854239_VEBTi(aa_nat317164691840817580_VEBTi(aa_nat3080504576880736121_VEBTi(aa_nat5640937189053280966_VEBTi(aa_VEB5864305904528209917_VEBTi(aa_lis8083850882120436046_VEBTi(aa_nat6903794420489964771_VEBTi(aa_VEB7377128652718434837_VEBTi(aa_arr6679936031850374666_VEBTi(aa_nat7444525983491697995_VEBTi(aa_nat8034297438400619220_VEBTi(aTP_Lamm_vd(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj),Uuk),Uul),Uum)) ).

% ATP.lambda_1310
tff(fact_11488_ATP_Olambda__1311,axiom,
    ! [Uu: fun_VE2018681615261037117_VEBTi,Uua: nat,Uub: nat,Uuc: array_VEBT_VEBTi,Uud: vEBT_VEBTi,Uue: nat,Uuf: list_VEBT_VEBT,Uug: vEBT_VEBT,Uuh: nat,Uui: nat,Uuj: nat,Uuk: nat,Uul: nat,Uum: nat] : aa_nat5408703056993926195_VEBTi(aa_nat4282640288315822012_VEBTi(aa_nat7653869184832469701_VEBTi(aa_nat2287630409448422734_VEBTi(aa_nat2878941900432676695_VEBTi(aa_nat532351713702512864_VEBTi(aa_VEB5310930937212039423_VEBTi(aa_lis7835404322308837764_VEBTi(aa_nat8538784969185189829_VEBTi(aa_VEB5145764472262921513_VEBTi(aa_arr6568412587313337610_VEBTi(aa_nat1553062435445860255_VEBTi(aa_nat1501880811460822764_VEBTi(aTP_Lamm_rh(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj),Uuk),Uul),Uum) = heap_T7982501707604696571_VEBTi(aa_o_H2459438917461573107t_unit(refine_Imp_assert,Uum = aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Uuk),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uue),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat7312404184930824517_VEBTi(aa_nat3221004963518969490_VEBTi(aa_nat1817389164722854239_VEBTi(aa_nat317164691840817580_VEBTi(aa_nat3080504576880736121_VEBTi(aa_nat5640937189053280966_VEBTi(aa_VEB5864305904528209917_VEBTi(aa_lis8083850882120436046_VEBTi(aa_nat6903794420489964771_VEBTi(aa_VEB7377128652718434837_VEBTi(aa_arr6679936031850374666_VEBTi(aa_nat7444525983491697995_VEBTi(aa_nat8034297438400619220_VEBTi(aTP_Lamm_rg(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj),Uuk),Uul),Uum)) ).

% ATP.lambda_1311
tff(fact_11489_ATP_Olambda__1312,axiom,
    ! [Uu: nat,Uua: nat,Uub: array_VEBT_VEBTi,Uuc: vEBT_VEBTi,Uud: nat,Uue: list_VEBT_VEBT,Uuf: vEBT_VEBT,Uug: nat,Uuh: nat,Uui: nat,Uuj: nat,Uuk: nat,Uul: nat,Uum: nat] : aa_nat5408703056993926195_VEBTi(aa_nat4282640288315822012_VEBTi(aa_nat7653869184832469701_VEBTi(aa_nat2287630409448422734_VEBTi(aa_nat2878941900432676695_VEBTi(aa_nat532351713702512864_VEBTi(aa_nat5473138614813200873_VEBTi(aa_VEB2726135844941024648_VEBTi(aa_lis3478650067825548685_VEBTi(aa_nat7061688277739565902_VEBTi(aa_VEB3820308522731820594_VEBTi(aa_arr165755620540720531_VEBTi(aa_nat5755912138998335400_VEBTi(aTP_Lamm_ti(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj),Uuk),Uul),Uum) = heap_T844888390831797134_VEBTi(array_len_VEBT_VEBTi(Uub),aa_nat4282640288315822012_VEBTi(aa_nat7653869184832469701_VEBTi(aa_nat2287630409448422734_VEBTi(aa_nat2878941900432676695_VEBTi(aa_nat532351713702512864_VEBTi(aa_nat5473138614813200873_VEBTi(aa_nat354071531552010866_VEBTi(aa_VEB545764574255792529_VEBTi(aa_lis7527251260247064854_VEBTi(aa_nat2109960713227968087_VEBTi(aa_VEB4528852985946928827_VEBTi(aa_arr8719208259821620636_VEBTi(aa_nat4030471164133998385_VEBTi(aTP_Lamm_th(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj),Uuk),Uul),Uum)) ).

% ATP.lambda_1312
tff(fact_11490_ATP_Olambda__1313,axiom,
    ! [Uu: nat,Uua: nat,Uub: array_VEBT_VEBTi,Uuc: vEBT_VEBTi,Uud: nat,Uue: list_VEBT_VEBT,Uuf: vEBT_VEBT,Uug: nat,Uuh: nat,Uui: nat,Uuj: nat,Uuk: nat,Uul: nat,Uum: nat,Uun: nat] :
      aa_nat5408703056993926195_VEBTi(aa_nat4282640288315822012_VEBTi(aa_nat7653869184832469701_VEBTi(aa_nat2287630409448422734_VEBTi(aa_nat2878941900432676695_VEBTi(aa_nat532351713702512864_VEBTi(aa_nat5473138614813200873_VEBTi(aa_nat354071531552010866_VEBTi(aa_VEB545764574255792529_VEBTi(aa_lis7527251260247064854_VEBTi(aa_nat2109960713227968087_VEBTi(aa_VEB4528852985946928827_VEBTi(aa_arr8719208259821620636_VEBTi(aa_nat4030471164133998385_VEBTi(aTP_Lamm_th(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj),Uuk),Uul),Uum),Uun) = $ite(
        ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uum),Uun)
        & ~ ( ( Uu = Uug )
            | ( Uu = Uuh ) ) ),
        heap_T7982501707604696571_VEBTi(aa_o_H2459438917461573107t_unit(refine_Imp_assert,Uum = aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Uuj),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uud),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat7312404184930824517_VEBTi(aa_nat3221004963518969490_VEBTi(aa_nat1817389164722854239_VEBTi(aa_nat317164691840817580_VEBTi(aa_nat3080504576880736121_VEBTi(aa_VEB4815025940789304112_VEBTi(aa_lis3577763594685753089_VEBTi(aa_VEB8209137959338826235_VEBTi(aa_arr747291131267414768_VEBTi(aTP_Lamm_tg(Uua),Uub),Uuc),Uue),Uuf),Uuh),Uui),Uuk),Uul),Uum)),
        aa_VEB836174949890266638_VEBTi(heap_T3630416162098727440_VEBTi,aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Uug),Uuh)),Uua,Uub),Uuc)) ) ).

% ATP.lambda_1313
tff(fact_11491_ATP_Olambda__1314,axiom,
    ! [Uu: fun_Pr4404027214751104635_VEBTi,Uua: nat,Uub: nat,Uuc: array_VEBT_VEBTi,Uud: vEBT_VEBTi,Uue: nat,Uuf: list_VEBT_VEBT,Uug: vEBT_VEBT,Uuh: nat,Uui: nat,Uuj: nat,Uuk: nat,Uul: nat,Uum: nat,Uun: product_unit] : aa_Pro3581650104493980720_VEBTi(aa_nat7312404184930824517_VEBTi(aa_nat3221004963518969490_VEBTi(aa_nat1817389164722854239_VEBTi(aa_nat317164691840817580_VEBTi(aa_nat3080504576880736121_VEBTi(aa_nat5640937189053280966_VEBTi(aa_VEB5864305904528209917_VEBTi(aa_lis8083850882120436046_VEBTi(aa_nat6903794420489964771_VEBTi(aa_VEB7377128652718434837_VEBTi(aa_arr6679936031850374666_VEBTi(aa_nat7444525983491697995_VEBTi(aa_nat8034297438400619220_VEBTi(aTP_Lamm_vd(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj),Uuk),Uul),Uum),Uun) = heap_T844888390831797134_VEBTi(aa_nat5821310369566504398ap_nat(vEBT_VEBT_highi(Uuj),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uub),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat4282640288315822012_VEBTi(aa_nat7653869184832469701_VEBTi(aa_nat2287630409448422734_VEBTi(aa_nat2878941900432676695_VEBTi(aa_nat532351713702512864_VEBTi(aa_nat5473138614813200873_VEBTi(aa_VEB2726135844941024648_VEBTi(aa_lis3478650067825548685_VEBTi(aa_nat7061688277739565902_VEBTi(aa_VEB3820308522731820594_VEBTi(aa_arr165755620540720531_VEBTi(aa_nat5755912138998335400_VEBTi(aa_nat7935733879562905205_VEBTi(aTP_Lamm_vc(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj),Uuk),Uul),Uum)) ).

% ATP.lambda_1314
tff(fact_11492_ATP_Olambda__1315,axiom,
    ! [Uu: fun_VE2018681615261037117_VEBTi,Uua: nat,Uub: nat,Uuc: array_VEBT_VEBTi,Uud: vEBT_VEBTi,Uue: nat,Uuf: list_VEBT_VEBT,Uug: vEBT_VEBT,Uuh: nat,Uui: nat,Uuj: nat,Uuk: nat,Uul: nat,Uum: nat,Uun: product_unit] : aa_Pro3581650104493980720_VEBTi(aa_nat7312404184930824517_VEBTi(aa_nat3221004963518969490_VEBTi(aa_nat1817389164722854239_VEBTi(aa_nat317164691840817580_VEBTi(aa_nat3080504576880736121_VEBTi(aa_nat5640937189053280966_VEBTi(aa_VEB5864305904528209917_VEBTi(aa_lis8083850882120436046_VEBTi(aa_nat6903794420489964771_VEBTi(aa_VEB7377128652718434837_VEBTi(aa_arr6679936031850374666_VEBTi(aa_nat7444525983491697995_VEBTi(aa_nat8034297438400619220_VEBTi(aTP_Lamm_rg(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj),Uuk),Uul),Uum),Uun) = heap_T844888390831797134_VEBTi(aa_nat5821310369566504398ap_nat(vEBT_VEBT_highi(Uuj),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uub),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat4282640288315822012_VEBTi(aa_nat7653869184832469701_VEBTi(aa_nat2287630409448422734_VEBTi(aa_nat2878941900432676695_VEBTi(aa_nat532351713702512864_VEBTi(aa_nat5473138614813200873_VEBTi(aa_VEB2726135844941024648_VEBTi(aa_lis3478650067825548685_VEBTi(aa_nat7061688277739565902_VEBTi(aa_VEB3820308522731820594_VEBTi(aa_arr165755620540720531_VEBTi(aa_nat5755912138998335400_VEBTi(aa_nat7935733879562905205_VEBTi(aTP_Lamm_rf(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj),Uuk),Uul),Uum)) ).

% ATP.lambda_1315
tff(fact_11493_ATP_Olambda__1316,axiom,
    ! [Uu: fun_Pr4404027214751104635_VEBTi,Uua: nat,Uub: nat,Uuc: array_VEBT_VEBTi,Uud: vEBT_VEBTi,Uue: nat,Uuf: list_VEBT_VEBT,Uug: vEBT_VEBT,Uuh: nat,Uui: nat,Uuj: nat,Uuk: nat,Uul: nat,Uum: nat,Uun: nat] : aa_nat5408703056993926195_VEBTi(aa_nat4282640288315822012_VEBTi(aa_nat7653869184832469701_VEBTi(aa_nat2287630409448422734_VEBTi(aa_nat2878941900432676695_VEBTi(aa_nat532351713702512864_VEBTi(aa_nat5473138614813200873_VEBTi(aa_VEB2726135844941024648_VEBTi(aa_lis3478650067825548685_VEBTi(aa_nat7061688277739565902_VEBTi(aa_VEB3820308522731820594_VEBTi(aa_arr165755620540720531_VEBTi(aa_nat5755912138998335400_VEBTi(aa_nat7935733879562905205_VEBTi(aTP_Lamm_vc(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj),Uuk),Uul),Uum),Uun) = heap_T844888390831797134_VEBTi(array_len_VEBT_VEBTi(Uuc),aa_nat4282640288315822012_VEBTi(aa_nat7653869184832469701_VEBTi(aa_nat2287630409448422734_VEBTi(aa_nat2878941900432676695_VEBTi(aa_nat532351713702512864_VEBTi(aa_nat5473138614813200873_VEBTi(aa_nat354071531552010866_VEBTi(aa_VEB545764574255792529_VEBTi(aa_lis7527251260247064854_VEBTi(aa_nat2109960713227968087_VEBTi(aa_VEB4528852985946928827_VEBTi(aa_arr8719208259821620636_VEBTi(aa_nat4030471164133998385_VEBTi(aa_nat552572580330102142_VEBTi(aTP_Lamm_vb(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj),Uuk),Uul),Uum),Uun)) ).

% ATP.lambda_1316
tff(fact_11494_ATP_Olambda__1317,axiom,
    ! [Uu: fun_VE2018681615261037117_VEBTi,Uua: nat,Uub: nat,Uuc: array_VEBT_VEBTi,Uud: vEBT_VEBTi,Uue: nat,Uuf: list_VEBT_VEBT,Uug: vEBT_VEBT,Uuh: nat,Uui: nat,Uuj: nat,Uuk: nat,Uul: nat,Uum: nat,Uun: nat] : aa_nat5408703056993926195_VEBTi(aa_nat4282640288315822012_VEBTi(aa_nat7653869184832469701_VEBTi(aa_nat2287630409448422734_VEBTi(aa_nat2878941900432676695_VEBTi(aa_nat532351713702512864_VEBTi(aa_nat5473138614813200873_VEBTi(aa_VEB2726135844941024648_VEBTi(aa_lis3478650067825548685_VEBTi(aa_nat7061688277739565902_VEBTi(aa_VEB3820308522731820594_VEBTi(aa_arr165755620540720531_VEBTi(aa_nat5755912138998335400_VEBTi(aa_nat7935733879562905205_VEBTi(aTP_Lamm_rf(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj),Uuk),Uul),Uum),Uun) = heap_T844888390831797134_VEBTi(array_len_VEBT_VEBTi(Uuc),aa_nat4282640288315822012_VEBTi(aa_nat7653869184832469701_VEBTi(aa_nat2287630409448422734_VEBTi(aa_nat2878941900432676695_VEBTi(aa_nat532351713702512864_VEBTi(aa_nat5473138614813200873_VEBTi(aa_nat354071531552010866_VEBTi(aa_VEB545764574255792529_VEBTi(aa_lis7527251260247064854_VEBTi(aa_nat2109960713227968087_VEBTi(aa_VEB4528852985946928827_VEBTi(aa_arr8719208259821620636_VEBTi(aa_nat4030471164133998385_VEBTi(aa_nat552572580330102142_VEBTi(aTP_Lamm_re(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj),Uuk),Uul),Uum),Uun)) ).

% ATP.lambda_1317
tff(fact_11495_ATP_Olambda__1318,axiom,
    ! [Uu: fun_Pr4404027214751104635_VEBTi,Uua: nat,Uub: nat,Uuc: array_VEBT_VEBTi,Uud: vEBT_VEBTi,Uue: nat,Uuf: list_VEBT_VEBT,Uug: vEBT_VEBT,Uuh: nat,Uui: nat,Uuj: nat,Uuk: nat,Uul: nat,Uum: nat,Uun: nat,Uuo: nat] :
      aa_nat5408703056993926195_VEBTi(aa_nat4282640288315822012_VEBTi(aa_nat7653869184832469701_VEBTi(aa_nat2287630409448422734_VEBTi(aa_nat2878941900432676695_VEBTi(aa_nat532351713702512864_VEBTi(aa_nat5473138614813200873_VEBTi(aa_nat354071531552010866_VEBTi(aa_VEB545764574255792529_VEBTi(aa_lis7527251260247064854_VEBTi(aa_nat2109960713227968087_VEBTi(aa_VEB4528852985946928827_VEBTi(aa_arr8719208259821620636_VEBTi(aa_nat4030471164133998385_VEBTi(aa_nat552572580330102142_VEBTi(aTP_Lamm_vb(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj),Uuk),Uul),Uum),Uun),Uuo) = $ite(
        ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uun),Uuo)
        & ~ ( ( Uua = Uuh )
            | ( Uua = Uui ) ) ),
        heap_T7982501707604696571_VEBTi(aa_o_H2459438917461573107t_unit(refine_Imp_assert,Uun = aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Uuk),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uue),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat7312404184930824517_VEBTi(aa_nat3221004963518969490_VEBTi(aa_nat1817389164722854239_VEBTi(aa_nat317164691840817580_VEBTi(aa_nat3080504576880736121_VEBTi(aa_VEB4815025940789304112_VEBTi(aa_lis3577763594685753089_VEBTi(aa_VEB8209137959338826235_VEBTi(aa_arr747291131267414768_VEBTi(aa_nat461013784042436401_VEBTi(aTP_Lamm_va(Uu),Uub),Uuc),Uud),Uuf),Uug),Uui),Uuj),Uul),Uum),Uun)),
        aa_VEB836174949890266638_VEBTi(heap_T3630416162098727440_VEBTi,aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Uuh),Uui)),Uub,Uuc),Uud)) ) ).

% ATP.lambda_1318
tff(fact_11496_ATP_Olambda__1319,axiom,
    ! [Uu: fun_VE2018681615261037117_VEBTi,Uua: nat,Uub: nat,Uuc: array_VEBT_VEBTi,Uud: vEBT_VEBTi,Uue: nat,Uuf: list_VEBT_VEBT,Uug: vEBT_VEBT,Uuh: nat,Uui: nat,Uuj: nat,Uuk: nat,Uul: nat,Uum: nat,Uun: nat,Uuo: nat] :
      aa_nat5408703056993926195_VEBTi(aa_nat4282640288315822012_VEBTi(aa_nat7653869184832469701_VEBTi(aa_nat2287630409448422734_VEBTi(aa_nat2878941900432676695_VEBTi(aa_nat532351713702512864_VEBTi(aa_nat5473138614813200873_VEBTi(aa_nat354071531552010866_VEBTi(aa_VEB545764574255792529_VEBTi(aa_lis7527251260247064854_VEBTi(aa_nat2109960713227968087_VEBTi(aa_VEB4528852985946928827_VEBTi(aa_arr8719208259821620636_VEBTi(aa_nat4030471164133998385_VEBTi(aa_nat552572580330102142_VEBTi(aTP_Lamm_re(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj),Uuk),Uul),Uum),Uun),Uuo) = $ite(
        ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uun),Uuo)
        & ~ ( ( Uua = Uuh )
            | ( Uua = Uui ) ) ),
        heap_T7982501707604696571_VEBTi(aa_o_H2459438917461573107t_unit(refine_Imp_assert,Uun = aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Uuk),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uue),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat7312404184930824517_VEBTi(aa_nat3221004963518969490_VEBTi(aa_nat1817389164722854239_VEBTi(aa_nat317164691840817580_VEBTi(aa_nat3080504576880736121_VEBTi(aa_VEB4815025940789304112_VEBTi(aa_lis3577763594685753089_VEBTi(aa_VEB8209137959338826235_VEBTi(aa_arr747291131267414768_VEBTi(aa_nat461013784042436401_VEBTi(aTP_Lamm_rd(Uu),Uub),Uuc),Uud),Uuf),Uug),Uui),Uuj),Uul),Uum),Uun)),
        aa_VEB836174949890266638_VEBTi(heap_T3630416162098727440_VEBTi,aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Uuh),Uui)),Uub,Uuc),Uud)) ) ).

% ATP.lambda_1319
tff(fact_11497_ATP_Olambda__1320,axiom,
    ! [Uu: option4927543243414619207at_nat,Uua: nat,Uub: list_VEBT_VEBT,Uuc: vEBT_VEBT] :
      ( aa_VEBT_VEBT_o(aa_lis4184305612057247525VEBT_o(aa_nat8798449810042143974VEBT_o(aa_opt7072554309604550194VEBT_o(aTP_Lamm_us,Uu),Uua),Uub),Uuc)
    <=> $true ) ).

% ATP.lambda_1320
tff(fact_11498_ATP_Olambda__1321,axiom,
    ! [Uu: $o,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_mm((Uu)),Uua),Uub)
    <=> (Uu) ) ).

% ATP.lambda_1321
tff(fact_11499_ATP_Olambda__1322,axiom,
    ! [Uu: $o,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_mn((Uu)),Uua),Uub)
    <=> (Uu) ) ).

% ATP.lambda_1322
tff(fact_11500_ATP_Olambda__1323,axiom,
    ! [Uu: vEBT_VEBT,Uua: real] : aa_real_VEBT_VEBT(aTP_Lamm_agq(Uu),Uua) = Uu ).

% ATP.lambda_1323
tff(fact_11501_ATP_Olambda__1324,axiom,
    ! [Uu: vEBT_VEBT,Uua: $o] : aa_o_VEBT_VEBT(aTP_Lamm_agr(Uu),(Uua)) = Uu ).

% ATP.lambda_1324
tff(fact_11502_ATP_Olambda__1325,axiom,
    ! [Uu: vEBT_VEBT,Uua: nat] : aa_nat_VEBT_VEBT(aTP_Lamm_ags(Uu),Uua) = Uu ).

% ATP.lambda_1325
tff(fact_11503_ATP_Olambda__1326,axiom,
    ! [Uu: vEBT_VEBT,Uua: int] : aa_int_VEBT_VEBT(aTP_Lamm_agt(Uu),Uua) = Uu ).

% ATP.lambda_1326
tff(fact_11504_ATP_Olambda__1327,axiom,
    ! [Uu: heap_T8145700208782473153_VEBTi,Uua: product_unit] : aa_Pro3581650104493980720_VEBTi(aTP_Lamm_gy(Uu),Uua) = Uu ).

% ATP.lambda_1327
tff(fact_11505_ATP_Olambda__1328,axiom,
    ! [Uu: heap_T8145700208782473153_VEBTi,Uua: produc3960855945107176009Ti_nat] : aa_Pro721367001817565044_VEBTi(aTP_Lamm_wm(Uu),Uua) = Uu ).

% ATP.lambda_1328
tff(fact_11506_ATP_Olambda__1329,axiom,
    ! [Uu: heap_T8145700208782473153_VEBTi,Uua: fun_Pr4404027214751104635_VEBTi] : aa_fun7494984083220284744_VEBTi(aTP_Lamm_zt(Uu),Uua) = Uu ).

% ATP.lambda_1329
tff(fact_11507_ATP_Olambda__1330,axiom,
    ! [Uu: heap_T5738788834812785303t_unit,Uua: product_unit] : aa_Pro1108561329362217960t_unit(aTP_Lamm_hf(Uu),Uua) = Uu ).

% ATP.lambda_1330
tff(fact_11508_ATP_Olambda__1331,axiom,
    ! [Uu: heap_T2636463487746394924on_nat,Uua: product_unit] : aa_Pro451875010097498139on_nat(aTP_Lamm_gx(Uu),Uua) = Uu ).

% ATP.lambda_1331
tff(fact_11509_ATP_Olambda__1332,axiom,
    ! [Uu: heap_T2636463487746394924on_nat,Uua: produc3960855945107176009Ti_nat] : aa_Pro2727408364141080927on_nat(aTP_Lamm_wn(Uu),Uua) = Uu ).

% ATP.lambda_1332
tff(fact_11510_ATP_Olambda__1333,axiom,
    ! [Uu: heap_T2636463487746394924on_nat,Uua: fun_Pr342922516890243977on_nat] : aa_fun7698174670542568229on_nat(aTP_Lamm_zr(Uu),Uua) = Uu ).

% ATP.lambda_1333
tff(fact_11511_ATP_Olambda__1334,axiom,
    ! [Uu: heap_T2636463487746394924on_nat,Uua: fun_Pr633315466666964198on_nat] : aa_fun1100322450986716680on_nat(aTP_Lamm_zu(Uu),Uua) = Uu ).

% ATP.lambda_1334
tff(fact_11512_ATP_Olambda__1335,axiom,
    ! [Uu: heap_Time_Heap_o,Uua: product_unit] : aa_Pro3379321066291785715Heap_o(aTP_Lamm_gv(Uu),Uua) = Uu ).

% ATP.lambda_1335
tff(fact_11513_ATP_Olambda__1336,axiom,
    ! [Uu: heap_Time_Heap_o,Uua: produc3960855945107176009Ti_nat] : aa_Pro6173831900156065711Heap_o(aTP_Lamm_wo(Uu),Uua) = Uu ).

% ATP.lambda_1336
tff(fact_11514_ATP_Olambda__1337,axiom,
    ! [Uu: heap_Time_Heap_o,Uua: fun_Pr238758252084143515Heap_o] : aa_fun6848502500920544641Heap_o(aTP_Lamm_zs(Uu),Uua) = Uu ).

% ATP.lambda_1337
tff(fact_11515_ATP_Olambda__1338,axiom,
    ! [Uu: heap_Time_Heap_o,Uua: fun_Pr940977554174145406Heap_o] : aa_fun3870618101526761188Heap_o(aTP_Lamm_zv(Uu),Uua) = Uu ).

% ATP.lambda_1338
tff(fact_11516_ATP_Olambda__1339,axiom,
    ! [Uu: heap_Time_Heap_nat,Uua: product_unit] : aa_Pro4051170119821641419ap_nat(aTP_Lamm_gw(Uu),Uua) = Uu ).

% ATP.lambda_1339
tff(fact_11517_ATP_Olambda__1340,axiom,
    ! [Uu: heap_e7401611519738050253t_unit,Uua: set_nat] : aa_set2130996941603788809t_unit(aa_hea6081028659853126223t_unit(aTP_Lamm_lf,Uu),Uua) = Uu ).

% ATP.lambda_1340
tff(fact_11518_ATP_Olambda__1341,axiom,
    ! [Uu: assn,Uua: product_unit] : aa_Product_unit_assn(aTP_Lamm_hg(Uu),Uua) = Uu ).

% ATP.lambda_1341
tff(fact_11519_ATP_Olambda__1342,axiom,
    ! [Uu: assn,Uua: produc6575502325842934193n_assn] : aa_Pro2979818503705714077n_assn(aTP_Lamm_agp(Uu),Uua) = Uu ).

% ATP.lambda_1342
tff(fact_11520_ATP_Olambda__1343,axiom,
    ! [Uu: assn,Uua: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(aTP_Lamm_lg,Uu),Uua) = Uu ).

% ATP.lambda_1343
tff(fact_11521_ATP_Olambda__1344,axiom,
    ! [Uu: assn,Uua: list_VEBT_VEBTi] : aa_lis2508312872377092907i_assn(aTP_Lamm_ei(Uu),Uua) = Uu ).

% ATP.lambda_1344
tff(fact_11522_ATP_Olambda__1345,axiom,
    ! [Uu: uint32,Uua: uint32] : aa_uint32_uint32(aa_uin3778660748068847871uint32(aTP_Lamm_lh,Uu),Uua) = Uu ).

% ATP.lambda_1345
tff(fact_11523_ATP_Olambda__1346,axiom,
    ! [Uu: real,Uua: vEBT_VEBT] : aa_VEBT_VEBT_real(aTP_Lamm_ago(Uu),Uua) = Uu ).

% ATP.lambda_1346
tff(fact_11524_ATP_Olambda__1347,axiom,
    ! [Uu: $o,Uua: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aTP_Lamm_mh((Uu)),Uua)
    <=> (Uu) ) ).

% ATP.lambda_1347
tff(fact_11525_ATP_Olambda__1348,axiom,
    ! [Uu: $o,Uua: complex] :
      ( aa_complex_o(aTP_Lamm_mg((Uu)),Uua)
    <=> (Uu) ) ).

% ATP.lambda_1348
tff(fact_11526_ATP_Olambda__1349,axiom,
    ! [Uu: $o,Uua: char] :
      ( aa_char_o(aTP_Lamm_ml((Uu)),Uua)
    <=> (Uu) ) ).

% ATP.lambda_1349
tff(fact_11527_ATP_Olambda__1350,axiom,
    ! [Uu: $o,Uua: real] :
      ( aa_real_o(aTP_Lamm_mj((Uu)),Uua)
    <=> (Uu) ) ).

% ATP.lambda_1350
tff(fact_11528_ATP_Olambda__1351,axiom,
    ! [Uu: $o,Uua: nat] :
      ( aa_nat_o(aa_o_fun_nat_o(aTP_Lamm_mk,(Uu)),Uua)
    <=> (Uu) ) ).

% ATP.lambda_1351
tff(fact_11529_ATP_Olambda__1352,axiom,
    ! [Uu: $o,Uua: int] :
      ( aa_int_o(aTP_Lamm_mi((Uu)),Uua)
    <=> (Uu) ) ).

% ATP.lambda_1352
tff(fact_11530_ATP_Olambda__1353,axiom,
    ! [Uu: nat,Uua: vEBT_VEBT] : aa_VEBT_VEBT_nat(aTP_Lamm_agn(Uu),Uua) = Uu ).

% ATP.lambda_1353
tff(fact_11531_ATP_Olambda__1354,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_li,Uu),Uua) = Uu ).

% ATP.lambda_1354
tff(fact_11532_ATP_Olambda__1355,axiom,
    ! [Uu: int,Uua: int] : aa_int_int(aa_int_fun_int_int(aTP_Lamm_lj,Uu),Uua) = Uu ).

% ATP.lambda_1355
tff(fact_11533_ATP_Olambda__1356,axiom,
    ! [Uu: fun_na5556145776640407093on_nat,Uua: produc3625547720036274456_VEBTi] : aa_Pro1290670013494283871on_nat(aTP_Lamm_wk(Uu),Uua) = Uu ).

% ATP.lambda_1356
tff(fact_11534_ATP_Olambda__1357,axiom,
    ! [Uu: fun_na2343073368905879023Heap_o,Uua: produc3625547720036274456_VEBTi] : aa_Pro259339148018954309Heap_o(aTP_Lamm_wl(Uu),Uua) = Uu ).

% ATP.lambda_1357
tff(fact_11535_ATP_Olambda__1358,axiom,
    ! [Uu: fun_na1476630412504255114at_nat,Uua: nat] : aa_nat8803788312118491388at_nat(aTP_Lamm_ll(Uu),Uua) = Uu ).

% ATP.lambda_1358
tff(fact_11536_ATP_Olambda__1359,axiom,
    ! [Uu: fun_nat_o,Uua: nat] : aa_nat_fun_nat_o(aTP_Lamm_ln(Uu),Uua) = Uu ).

% ATP.lambda_1359
tff(fact_11537_ATP_Olambda__1360,axiom,
    ! [Uu: fun_in4211310926729683942nt_int,Uua: int] : aa_int2565445551602696444nt_int(aTP_Lamm_lp(Uu),Uua) = Uu ).

% ATP.lambda_1360
tff(fact_11538_ATP_Olambda__1361,axiom,
    ! [Uu: fun_int_o,Uua: int] : aa_int_fun_int_o(aTP_Lamm_lr(Uu),Uua) = Uu ).

% ATP.lambda_1361
tff(fact_11539_ATP_Olambda__1362,axiom,
    ! [Uu: fun_int_int,Uua: int] : aa_int_fun_int_int(aTP_Lamm_lt(Uu),Uua) = Uu ).

% ATP.lambda_1362
tff(fact_11540_ATP_Olambda__1363,axiom,
    ! [Uu: code_integer,Uua: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(aTP_Lamm_lw,Uu),Uua) = Uua ).

% ATP.lambda_1363
tff(fact_11541_ATP_Olambda__1364,axiom,
    ! [Uu: heap_e7401611519738050253t_unit,Uua: set_nat] : aa_set_nat_set_nat(aa_hea108656230536133402et_nat(aTP_Lamm_lv,Uu),Uua) = Uua ).

% ATP.lambda_1364
tff(fact_11542_ATP_Olambda__1365,axiom,
    ! [Uu: assn,Uua: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(aTP_Lamm_lx,Uu),Uua) = Uua ).

% ATP.lambda_1365
tff(fact_11543_ATP_Olambda__1366,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_lu,Uu),Uua) = Uua ).

% ATP.lambda_1366
tff(fact_11544_ATP_Olambda__1367,axiom,
    ! [Uu: int,Uua: int] : aa_int_int(aa_int_fun_int_int(aTP_Lamm_ly,Uu),Uua) = Uua ).

% ATP.lambda_1367
tff(fact_11545_ATP_Olambda__1368,axiom,
    ! [Uu: $o,Uua: $o] : aa_o_P8223745691486049304T_VEBT(aa_o_f7062522455407347339T_VEBT(aTP_Lamm_ou,(Uu)),(Uua)) = undefi7074909574607331924T_VEBT ).

% ATP.lambda_1368
tff(fact_11546_ATP_Olambda__1369,axiom,
    ! [Uu: $o,Uua: $o] :
      ( aa_o_o(aa_o_fun_o_o(aTP_Lamm_ut,(Uu)),(Uua))
    <=> $false ) ).

% ATP.lambda_1369
tff(fact_11547_ATP_Olambda__1370,axiom,
    ! [Uu: nat,Uua: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_oz,Uu),Uua)
    <=> $true ) ).

% ATP.lambda_1370
tff(fact_11548_ATP_Olambda__1371,axiom,
    ! [Uu: int,Uua: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_pa,Uu),Uua)
    <=> $true ) ).

% ATP.lambda_1371
tff(fact_11549_ATP_Olambda__1372,axiom,
    ! [Uu: product_prod_nat_nat] : aa_Pro262675715343573244at_nat(aTP_Lamm_pt,Uu) = Uu ).

% ATP.lambda_1372
tff(fact_11550_ATP_Olambda__1373,axiom,
    ! [Uu: complex] : aa_complex_complex(aTP_Lamm_aqn,Uu) = Uu ).

% ATP.lambda_1373
tff(fact_11551_ATP_Olambda__1374,axiom,
    ! [Uu: word_N3645301735248828278l_num1] : aa_wor7274019993978516266l_num1(aTP_Lamm_df,Uu) = Uu ).

% ATP.lambda_1374
tff(fact_11552_ATP_Olambda__1375,axiom,
    ! [Uu: real] : aa_real_real(aTP_Lamm_dg,Uu) = Uu ).

% ATP.lambda_1375
tff(fact_11553_ATP_Olambda__1376,axiom,
    ! [Uu: rat] : aa_rat_rat(aTP_Lamm_dh,Uu) = Uu ).

% ATP.lambda_1376
tff(fact_11554_ATP_Olambda__1377,axiom,
    ! [Uu: num] : aa_num_num(aTP_Lamm_pu,Uu) = Uu ).

% ATP.lambda_1377
tff(fact_11555_ATP_Olambda__1378,axiom,
    ! [Uu: nat] : aa_nat_nat(aTP_Lamm_di,Uu) = Uu ).

% ATP.lambda_1378
tff(fact_11556_ATP_Olambda__1379,axiom,
    ! [Uu: int] : aa_int_int(aTP_Lamm_dj,Uu) = Uu ).

% ATP.lambda_1379
tff(fact_11557_ATP_Olambda__1380,axiom,
    ! [Uu: fun_fu6749172196426063865uint32] : aa_fun3431294132209178672uint32(aTP_Lamm_po,Uu) = Uu ).

% ATP.lambda_1380
tff(fact_11558_ATP_Olambda__1381,axiom,
    ! [Uu: fun_fu3466578569608634614eger_o] : aa_fun1804373999164127658eger_o(aTP_Lamm_pp,Uu) = Uu ).

% ATP.lambda_1381
tff(fact_11559_ATP_Olambda__1382,axiom,
    ! [Uu: fun_fu8139443716560175030uint32] : aa_fun3432716020469233706uint32(aTP_Lamm_pq,Uu) = Uu ).

% ATP.lambda_1382
tff(fact_11560_ATP_Olambda__1383,axiom,
    ! [Uu: fun_fu1303640425391265102uint32] : aa_fun3449071308854940378uint32(aTP_Lamm_pr,Uu) = Uu ).

% ATP.lambda_1383
tff(fact_11561_ATP_Olambda__1384,axiom,
    ! [Uu: fun_fu5030138910357066062uint32] : aa_fun953596214170897114uint32(aTP_Lamm_ps,Uu) = Uu ).

% ATP.lambda_1384
tff(fact_11562_ATP_Olambda__1385,axiom,
    ! [Uu: complex] : aa_complex_complex(aTP_Lamm_aip,Uu) = zero_zero_complex ).

% ATP.lambda_1385
tff(fact_11563_ATP_Olambda__1386,axiom,
    ! [Uu: uint32] : aa_uint32_uint32(aTP_Lamm_ek,Uu) = zero_zero_uint32 ).

% ATP.lambda_1386
tff(fact_11564_ATP_Olambda__1387,axiom,
    ! [Uu: real] : aa_real_real(aTP_Lamm_el,Uu) = zero_zero_real ).

% ATP.lambda_1387
tff(fact_11565_ATP_Olambda__1388,axiom,
    ! [Uu: rat] : aa_rat_rat(aTP_Lamm_em,Uu) = zero_zero_rat ).

% ATP.lambda_1388
tff(fact_11566_ATP_Olambda__1389,axiom,
    ! [Uu: nat] : aa_nat_real(aTP_Lamm_aiq,Uu) = zero_zero_real ).

% ATP.lambda_1389
tff(fact_11567_ATP_Olambda__1390,axiom,
    ! [Uu: nat] : aa_nat_nat(aTP_Lamm_en,Uu) = zero_zero_nat ).

% ATP.lambda_1390
tff(fact_11568_ATP_Olambda__1391,axiom,
    ! [Uu: nat] : aa_nat_int(aTP_Lamm_aqe,Uu) = zero_zero_int ).

% ATP.lambda_1391
tff(fact_11569_ATP_Olambda__1392,axiom,
    ! [Uu: int] : aa_int_int(aTP_Lamm_eo,Uu) = zero_zero_int ).

% ATP.lambda_1392
tff(fact_11570_ATP_Olambda__1393,axiom,
    ! [Uu: product_unit] : aa_Product_unit_assn(aTP_Lamm_bbo,Uu) = one_one_assn ).

% ATP.lambda_1393
tff(fact_11571_ATP_Olambda__1394,axiom,
    ! [Uu: nat] : aa_nat_nat(aTP_Lamm_avb,Uu) = one_one_nat ).

% ATP.lambda_1394
tff(fact_11572_ATP_Olambda__1395,axiom,
    ! [Uu: nat] : aa_nat_int(aTP_Lamm_auz,Uu) = one_one_int ).

% ATP.lambda_1395
tff(fact_11573_ATP_Olambda__1396,axiom,
    ! [Uu: int] : aa_int_int(aTP_Lamm_ava,Uu) = one_one_int ).

% ATP.lambda_1396
tff(fact_11574_ATP_Olambda__1397,axiom,
    ! [Uu: heap_e7401611519738050253t_unit] : aa_hea6347871425501556311it_nat(aTP_Lamm_ys,Uu) = none_P7832717587476222275it_nat ).

% ATP.lambda_1397
tff(fact_11575_ATP_Olambda__1398,axiom,
    ! [Uu: heap_e7401611519738050253t_unit] : aa_hea5089399211691266754it_nat(aTP_Lamm_zf,Uu) = none_P1551326421579882414it_nat ).

% ATP.lambda_1398
tff(fact_11576_ATP_Olambda__1399,axiom,
    ! [Uu: heap_e7401611519738050253t_unit] : aa_hea8154171661607227764it_nat(aTP_Lamm_zg,Uu) = none_P7668321371905463026it_nat ).

% ATP.lambda_1399
tff(fact_11577_ATP_Olambda__1400,axiom,
    ! [Uu: heap_e7401611519738050253t_unit] : aa_hea8112163199330556402it_nat(aTP_Lamm_zh,Uu) = none_P281974696781278558it_nat ).

% ATP.lambda_1400
tff(fact_11578_ATP_Olambda__1401,axiom,
    ! [Uu: produc120671012495760973it_nat] :
      ( aa_Pro2986953523225089223_nat_o(aTP_Lamm_cv,Uu)
    <=> $false ) ).

% ATP.lambda_1401
tff(fact_11579_ATP_Olambda__1402,axiom,
    ! [Uu: produc8047831477865546771it_nat] :
      ( aa_Pro6395689520723908999_nat_o(aTP_Lamm_cw,Uu)
    <=> $false ) ).

% ATP.lambda_1402
tff(fact_11580_ATP_Olambda__1403,axiom,
    ! [Uu: product_prod_nat_nat] :
      ( aa_Pro6150637604445964947_nat_o(aTP_Lamm_cz,Uu)
    <=> $false ) ).

% ATP.lambda_1403
tff(fact_11581_ATP_Olambda__1404,axiom,
    ! [Uu: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aTP_Lamm_ge,Uu)
    <=> $false ) ).

% ATP.lambda_1404
tff(fact_11582_ATP_Olambda__1405,axiom,
    ! [Uu: complex] :
      ( aa_complex_o(aTP_Lamm_gd,Uu)
    <=> $false ) ).

% ATP.lambda_1405
tff(fact_11583_ATP_Olambda__1406,axiom,
    ! [Uu: real] :
      ( aa_real_o(aTP_Lamm_gg,Uu)
    <=> $false ) ).

% ATP.lambda_1406
tff(fact_11584_ATP_Olambda__1407,axiom,
    ! [Uu: num] :
      ( aa_num_o(aTP_Lamm_cx,Uu)
    <=> $false ) ).

% ATP.lambda_1407
tff(fact_11585_ATP_Olambda__1408,axiom,
    ! [Uu: nat] :
      ( aa_nat_o(aTP_Lamm_cy,Uu)
    <=> $false ) ).

% ATP.lambda_1408
tff(fact_11586_ATP_Olambda__1409,axiom,
    ! [Uu: int] :
      ( aa_int_o(aTP_Lamm_gf,Uu)
    <=> $false ) ).

% ATP.lambda_1409
tff(fact_11587_ATP_Olambda__1410,axiom,
    ! [Uu: produc120671012495760973it_nat] :
      ( aa_Pro2986953523225089223_nat_o(aTP_Lamm_da,Uu)
    <=> $true ) ).

% ATP.lambda_1410
tff(fact_11588_ATP_Olambda__1411,axiom,
    ! [Uu: produc8047831477865546771it_nat] :
      ( aa_Pro6395689520723908999_nat_o(aTP_Lamm_db,Uu)
    <=> $true ) ).

% ATP.lambda_1411
tff(fact_11589_ATP_Olambda__1412,axiom,
    ! [Uu: product_prod_nat_nat] :
      ( aa_Pro6150637604445964947_nat_o(aTP_Lamm_de,Uu)
    <=> $true ) ).

% ATP.lambda_1412
tff(fact_11590_ATP_Olambda__1413,axiom,
    ! [Uu: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aTP_Lamm_pd,Uu)
    <=> $true ) ).

% ATP.lambda_1413
tff(fact_11591_ATP_Olambda__1414,axiom,
    ! [Uu: complex] :
      ( aa_complex_o(aTP_Lamm_pc,Uu)
    <=> $true ) ).

% ATP.lambda_1414
tff(fact_11592_ATP_Olambda__1415,axiom,
    ! [Uu: char] :
      ( aa_char_o(aTP_Lamm_pf,Uu)
    <=> $true ) ).

% ATP.lambda_1415
tff(fact_11593_ATP_Olambda__1416,axiom,
    ! [Uu: real] :
      ( aa_real_o(aTP_Lamm_pe,Uu)
    <=> $true ) ).

% ATP.lambda_1416
tff(fact_11594_ATP_Olambda__1417,axiom,
    ! [Uu: num] :
      ( aa_num_o(aTP_Lamm_dc,Uu)
    <=> $true ) ).

% ATP.lambda_1417
tff(fact_11595_ATP_Olambda__1418,axiom,
    ! [Uu: nat] :
      ( aa_nat_o(aTP_Lamm_dd,Uu)
    <=> $true ) ).

% ATP.lambda_1418
tff(fact_11596_ATP_Olambda__1419,axiom,
    ! [Uu: int] :
      ( aa_int_o(aTP_Lamm_pb,Uu)
    <=> $true ) ).

% ATP.lambda_1419
tff(fact_11597_ATP_Olambda__1420,axiom,
    ! [Uu: real] : aa_real_fun_nat_nat(aTP_Lamm_hn,Uu) = suc ).

% ATP.lambda_1420
tff(fact_11598_ATP_Olambda__1421,axiom,
    ! [Uu: $o] : aa_o_fun_nat_nat(aTP_Lamm_ho,(Uu)) = suc ).

% ATP.lambda_1421
tff(fact_11599_ATP_Olambda__1422,axiom,
    ! [Uu: nat] : aa_nat_fun_nat_nat(aTP_Lamm_hp,Uu) = suc ).

% ATP.lambda_1422
tff(fact_11600_ATP_Olambda__1423,axiom,
    ! [Uu: int] : aa_int_fun_nat_nat(aTP_Lamm_hq,Uu) = suc ).

% ATP.lambda_1423

% Helper facts (13)
tff(help_fequal_2_1_fequal_001t__Int__Oint_T,axiom,
    ! [X9: int,Y: int] :
      ( ( X9 != Y )
      | aa_int_o(aa_int_fun_int_o(fequal_int,X9),Y) ) ).

tff(help_fequal_1_1_fequal_001t__Int__Oint_T,axiom,
    ! [X9: int,Y: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(fequal_int,X9),Y)
      | ( X9 = Y ) ) ).

tff(help_fequal_2_1_fequal_001t__Nat__Onat_T,axiom,
    ! [X9: nat,Y: nat] :
      ( ( X9 != Y )
      | aa_nat_o(aa_nat_fun_nat_o(fequal_nat,X9),Y) ) ).

tff(help_fequal_1_1_fequal_001t__Nat__Onat_T,axiom,
    ! [X9: nat,Y: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(fequal_nat,X9),Y)
      | ( X9 = Y ) ) ).

tff(help_fequal_2_1_fequal_001t__Real__Oreal_T,axiom,
    ! [X9: real,Y: real] :
      ( ( X9 != Y )
      | aa_real_o(aa_real_fun_real_o(fequal_real,X9),Y) ) ).

tff(help_fequal_1_1_fequal_001t__Real__Oreal_T,axiom,
    ! [X9: real,Y: real] :
      ( ~ aa_real_o(aa_real_fun_real_o(fequal_real,X9),Y)
      | ( X9 = Y ) ) ).

tff(help_fChoice_1_1_fChoice_001t__Real__Oreal_T,axiom,
    ! [P: fun_real_o] :
      ( aa_real_o(P,fChoice_real(P))
      = ( ? [X6: real] : aa_real_o(P,X6) ) ) ).

tff(help_fequal_2_1_fequal_001t__Complex__Ocomplex_T,axiom,
    ! [X9: complex,Y: complex] :
      ( ( X9 != Y )
      | aa_complex_o(fequal_complex(X9),Y) ) ).

tff(help_fequal_1_1_fequal_001t__Complex__Ocomplex_T,axiom,
    ! [X9: complex,Y: complex] :
      ( ~ aa_complex_o(fequal_complex(X9),Y)
      | ( X9 = Y ) ) ).

tff(help_fequal_2_1_fequal_001t__VEBT____Definitions__OVEBT_T,axiom,
    ! [X9: vEBT_VEBT,Y: vEBT_VEBT] :
      ( ( X9 != Y )
      | aa_VEBT_VEBT_o(fequal_VEBT_VEBT(X9),Y) ) ).

tff(help_fequal_1_1_fequal_001t__VEBT____Definitions__OVEBT_T,axiom,
    ! [X9: vEBT_VEBT,Y: vEBT_VEBT] :
      ( ~ aa_VEBT_VEBT_o(fequal_VEBT_VEBT(X9),Y)
      | ( X9 = Y ) ) ).

tff(help_fequal_2_1_fequal_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
    ! [X9: product_prod_int_int,Y: product_prod_int_int] :
      ( ( X9 != Y )
      | aa_Pro4592068669816069595_int_o(aa_Pro3952998155787061599_int_o(fequal6010348034681699726nt_int,X9),Y) ) ).

tff(help_fequal_1_1_fequal_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
    ! [X9: product_prod_int_int,Y: product_prod_int_int] :
      ( ~ aa_Pro4592068669816069595_int_o(aa_Pro3952998155787061599_int_o(fequal6010348034681699726nt_int,X9),Y)
      | ( X9 = Y ) ) ).

% Conjectures (3)
tff(conj_0,hypothesis,
    tia = aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(x11,x12,x13),x14) ).

tff(conj_1,hypothesis,
    ( ( x11 = some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,mi),ma)) )
    & ( x12 = aa_nat_nat(suc,aa_nat_nat(suc,va)) ) ) ).

tff(conj_2,conjecture,
    hoare_7629718768684598413on_nat(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,summary),x14)),aa_lis2508312872377092907i_assn(snga_assn_VEBT_VEBTi(x13),tree_is))),aa_lis2508312872377092907i_assn(vEBT_L6296928887356842470_VEBTi(vEBT_vebt_assn_raw,treeList),tree_is)),heap_T5999496708990702694on_nat(aa_o_H2459438917461573107t_unit(refine_Imp_assert,$true),aTP_Lamm_ao),aTP_Lamm_aq) ).

%------------------------------------------------------------------------------